INFRAR ED PHOT OELASTICIT Y;

PRINCIPLES, METHODS, AND APPLICATIONS
BY

Gary Lee Cloud

AN ABSTRAC T

Submitted to
Michigan State University .
in partial fulfillment of the requirements
for the degree of

DOC TOR OF PHILOSOPHY

Department of Metallurgy,
Mechanics and Materials Science

1966

 

 

 

ABSTRACT

INFRARED PHOTOELASTICIT Y;
PRINCIPLES, METHODS, AND APPLICATIONS

By Gary Lee Cloud

An investigation was undertaken with the primary objective of
extending photoelastic methods of stress analysis into the infrared
portion of the electromagnetic spectrum. A review of the basic
concepts and equations of photoelasticity serves to demonstrate that
the methods of measurement of birefringence are not limited to the
visible Spectrum. Apparatus and techniques for observing and
measuring birefringence in the infrared were developed. These
methods were then employed in the study of stress birefringence
in a broad range of wave lengths for various materials, including
semiconductors.

Monochromatic radiation in the range of wavelengths between
405 nm and 1900 nm was obtained by selective filtering of spectral
sources and isolation of a narrow band of wavelengths from white
radiation through the use of interference filters. Type HR Polaroid
sheet served to polarize the infrared radiation, and discs of half-
wave retardation materials were used singly and stacked as quarter -
wave plates for the spectral region between 0. 7p. and 1. 9H . Plane-
wise observation of birefringence . was accomplished with an infrared
converter, and results were recorded photographically in the visible
and infrared up to 1. 2p. using special infrared films. Pointwise sensing

techniques were developed for the measurement of birefringence in the

 

GAR Y LEE CLOUD

spectral range between 1. 2p , the practical limit ofimaging devices,
and 1. 9p. . A goniometric method of fractional is ochromatic order
determination was improved and employed in the quantitiative studies
of material behavior.

The stress birefringence of Columbia Resin CR ~39 and Palatal
P—6 was investigated in the visible and near infrared by planewise
measurements. The stress-Optical properties of polycarbonate resin
between 1.1 and 1. 9 microns wavelength were measured with the
point-sensing and compensation techniques. Preliminary observation
and measurement of stress-induced birefringence in silicon were
accomplished with the infrared converter and point-sensor systems.
The physical and technological implications of the stress birefringence
of this nontransparent semiconductor are very interesting. Additional
limited experimentation with amorphous selenium Showed that this
material may also prove useful for instrumentation.

It is known that for many materials the order of isochromatic
is not simply inversely proportional to wave length. This dependence
of the photoelastic coefficients on wavelength is called dispersion of
birefringence in analogy to a similar phenomenon occurring in optics-
Examination of past descriptions of dispersion, which are partially
arbitrary, shows them to be inadequate in View of progress in photo-
elasticity, spectroscopy and polymer physics. New measures of
dispersion were developed. Using these new ideas, the dispersion of
birefringence of CR -3 9, P-6, and polycarbonate were derived from

the calibration data. It was found that the stress-Optic coefficient for

 

 

 

GAR Y LEE CLOUD

CR -39 changes by about 5% in the visible spectrum. The coefficient
for P-6 changes by 11% in the same range, indicating that serious
errors could result if constant coefficients were used in tests involving
more than one wavelength. The dispersion of birefringence may be
closely related to structural changes at microscopic levels, and, thus,
might serve as an indicator of mode of deformation as well as of
material behavior. The birefringence and dispersion of P-6 were found
to depend on Sign as well as magnitude of stress. Continued investi-
gation of these problems and others suggested by the research will
contribute immediately to the technology and application of photo-
elasticity in addition to providing insight into material behavior and

the physical basis of induced birefringence.

y. n "in. VIII»)! m ...

 

 

INFRAR ED PHOTOELASTICIT Y;

PRINCIPLES, METHODS, AND APPLICATIONS

BY

Gary Lee Cloud

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy,
Mechanics and Materials Science

1966

 

 

AC KNOWLEDGEMENT

The author expresses his sincere appreciation to Dr. Jerzy
T. Pindera for his suggestion of the experimental problem and his
guidance and support of the research. The dedicated efforts of
Dr. George E. Mase, who advised the author throughout his course
of study and the preparation of the dissertation, are gratefully
acknowledged. Finally, the author wishes to thank the members
of his doctoral committee, Dr. Lawrence E. Malvern, Dr. William
A. Bradley, and Dr. Robert H. Wasserman, for their encouragement

and direction.

ii

 

TABLE OF CONTENTS

Article
1. Statement of the Problem . . . . . .............
1.1. Summary of Development of Photoelasticity . . .
1. 2. Purpose and Scope of the Investigation ......
1. 3. Review of RelatedLiterature . . . . . ......
2. Techniques of Investigation .................
2.1. The Photoelastic Method .............
2. 1 . 1. Nature of Light and the
Electromagnetic Spectrum .......
2.1.2. Refraction, Retardation,
Birefringence . . ............
2.1. 3. Birefringence in Deformable
Bodies ...... . ...........
2. 1.4. The Equations of Photoelasticity . . . .
2.1. 5. Optical Dispersion and
Dispersion of Birefringence .......
2. 2. Infrared Photoelasticity Apparatus

and Technique ...............

2. 2.1. Optical Systems Used ..........

2. 2. 2. Elements of the Optical Systems

2. 2. 3. Measurement of Fractional
Isochromatic Order .......

3. Determinations of Optical Behavior of Materials

3.

3.

1.

2.

Descriptions Of Materials and Testing
Procedures . . ............... . . .

3.1.1. Tensile Calibration of CR -39 in
Visible and Near Infrared. . . .

3.1. 2. Tests of Palatal P—6 ..... .

3. 1 . 3. Birefringence of Polycarbonate
between 1 and 2 microns . . . . . . . .

3. 1 . 4. Preliminary Investigations Of
Birefringence of Semiconductors . . . .

Treatment of Data and Presentation of

Results .................. . . . .
3. 2.1. Calibration of P-6 and CR—39 .....
3. 2. 2. Calibration of Polycarbonate ..... .
3. 2. 3. Dispersion Of Birefringence of

CR-39, P-6, and Polycarbonate . . . .
3.2.4. Birefringence of Semiconductors . . . .

iii

 

18
22

30

35

35
37

63

73

73

73
79

83
88

9O

90
100

102
115

 

Article

4.

Discussion and Conclusion . . . . . . . . . . . . . .

4. 1. Summary of Techniques and Apparatus . .
4.2. Material Behavior . . . . ...... . . .
4.3. Continuation of the Research . . . . . . .

Bibliography........ ........ .....

iv

 

Page

117

117
121

134
136

Number
2.1
2.2
2.3
2.4

2.5

2.6

2.7

3.1

3.2a

3.2b

3.3

3.4

3.5

3.6a

3.6b

3.6c

3.7

3.8

LIST OF FIGURES

Title
The Electromagnetic Spectrum . . . . . . . . . . .
Infrared Polariscope Arrangements . . . . . . . .
Photoelastic Apparatus for the 1-2 Micron Region.
Emission of Helium Lamp in Infrared . . . . . . .

Transmission of Representative Interference
Filters I O O O O O O O O O I C I O O O O I O O O O O 0

Infrared Sensing Circuit . . . . . . . . . . . . . .

Reduction of Error in Measurement of
Fractional Isochromatic Order . . . . . . . . . . .

CR -39 Model, Stress Distribution,
Load History . I C O O O O O O O O C C C C O D I I U

Isochromatic Pattern in CR ~39 Model for
A = 546nm with Linear Polarization . . . . . . . .

Isochromatic Pattern in CR -39 Model for
A = 1083nm with Linear Polarization . . . . . . .

Ring Model of Cured Palatal P-6, Load History . .

Isochromatic Pattern in P-6 Ring for
A = 1083nm with Circular Polarization. . . . . . .

Beam Model of Cured Palatal P-6, Load HistOry .

Isochromatic Pattern in P-6 Beam for
)1 = 546nm with Circular Polarization . . . . . . .

Isochromatic Pattern in P-6 Beam for
A = 405nm with Linear Polarization . . . . . . . .

Isochromatic Pattern in P-6 Beam for
A = 668nm with Linear Polarization . . . . . . . .

Disc Model of Polycarbonate, Load History . . . .
Frirge Order for Constant Wavelength and

Calculated Stress Along Axis of Portion Of _ . .
CR '39 Model. 0 o o o o o ccccccc o o o c o s o

 

Page
144
145
146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

 

Number
3. 9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20a

3.20b

LIST OF FIGURES (continued)

Title

Fringe Order Per Unit Thickness for Constant

Wavelengths in P-6 Beam . . . . . . .

Fringe Order Per Unit Thickness for Constant

Wavelength in P-6 Ring . . .

ccccc o o o u o o

Retardation Per Unit Thickness Along Axis of
CR -39 Model for Various Wavelengths . . . . . .

Retardation Per Unit Thickness for Constant
Wavelengths and Stress by Curved Beam Theory
forP-6Beam .....

Retardation Per Unit Thickness for Constant
Wavelengths in P-6 Ring. . . .

Fringe Order and Retardation Per Unit Thickness

o o I

versus Stress for Constant Wavelength in CR -39 .

Examples of Fringe Order Per Unit Thickness at

Constant Wavelength Versus Stress From Curved

Beam Theory for P-6 Beam . .

Fringe Order for Constant Wavelength Versus

Corrected Stress in Polycarbonate . .

Representative Retardation Per Unit Thickness

at Constant Wavelengths Versus Corrected Stress

in Polycarbonate. . . . . . . .

Spectral Transmittance of Polycarbonate. . . . .

Normalized Retardation versus Wavelength
and Frequency for CR -39 . . .

R epre sentative Normalized R etardati on

Versus Wavelength for P-6 .

Representative Normalized Retardation
Versus Frequency for P-6. . .

vi

0 o o C o o

Page

162

163

164

165

166

167

168

169

170

171

172

173

174

Number

3.21

3.22a
3.22b
3.23a
3.23b

3.24

 

LIST OF FIGURES (continued)

Tide Page
Normalized Retardation Versus Wavelength
for Polycarbonate for 1p. < A < 2“. . . . . . . . . 175
Dispersion of Birefringence versus Wavelength
for CR -39 o o o o o o o o D a c a c o o I o o o o O o 176
Dispersion of Birefringence versus Frequency
forCR-390000uoooooooooooooocoo 177
Dispersion of Birefringence versus Wavelength
for P—6 3 o o o o o o o o o o o o o o o o o o o o o o 178
Dispersion of Birefringence versus Frequency
for P-6 o o o o o o o o I o o o o a I o o n o I o o o 179
Stress Birefringence in Silicon . . . . . . . . . . 180

vfi

 

 

 

Number
2. 1

2.2

2.3

4.1

LIST OF TABLES

Title

Monochromatic Radiation Sources and Filters

Manufacturers Specifications,
Varo Inc., Model 5500C Detect-IR-scope . .

PhotographicFilms..............

Stress Optical Data in Infrared for

Polycarbonate Re sin

viii

 

Page
44

52

54

126

 

 

Article

A

B

LIST OF APPENDICES

Title

Representative Measurements and Corrected
DataforP-6 Beamat). =546nm. . . . . . .

Dimensions of Tapered CR -39 Model and
Computation of Stress . . . . . . . . . . . . .

Radius of Curvature and Stress in P-6 Beam .

Computation Of Stress Difference in
Polycarbonate Disc. . . . . . . . . . . . . . .

Illustrative Determination of Fractional
Isochromatic Order in Polycarbonate Disc . .

Illustrative Normalized Retardation Data and
Its Statistical Treatment for CR-39 . . . . . .

ix

 

Page

181

182

183

185

186

187

 

HI

$741351

0

 

LIST OF SYMBOLS

Electric Intensity

Magnetic Induction

Dielectric coefficient

Magnetic permeability

Velocity of electromagnetic wave
Velocity of light in a vacuum
Wavelength of radiation

6

Micron = 10- meter

9

Nanometer = 10' meter = millimi:cron
Radiation frequency, cps

Wave number

Absolute index of refraction
Thickness of parallel refractive plate

Distance along optical path

Absolute retardation of rays vibrating in principal
planes 1 and 2

Absolute index of refraction of medium surrounding
optical system (usually air)

Absolute indices of refraction corresponding to principal planes

Relative retardation

Isochromatic order

Principal stress

Principal strain

Maximum value of stress in linear range on basis of

accepted deviation from linearity (0. 5%, 1%, etc.) and
given time

Fifi-El

 

 

 

6“ Maximum value of strain in linear range on basis of
accepted deviation from linearity (0. 5%, 1%, etc.) and
given time

c1, c2 Absolute stress-optical coefficients

CU Stress-optical coefficient

C€ Strain-optical coefficient

Sc Material stress-optical coefficient for d = 1 cm

nx Absolute index of refraction at wavelength A

nF, nc, nD Absolute-indices of refraction referred to F, C, D
Fraunhofer Lines

e Quarter-wave plate error in wavelength

An = 111 — n2 Difference between indexes of refraction of rays .
vibrating in principal planes; a measure of birefringence

An 0 Unit birefringence (for 0' = 1 kg/mmz)

RO Unit relative retardation, unit retardation

r Normalized relative retardation, normalized retardation

D Dispersion of birefringence

D: Normalized dispersion of birefringence in terms of

wavelength
D: Normalized dispersion of birefringence in terms of

frequency

xi

 

 

1. Statement of the Problem

1.1. Summary of Development of Photoelasticity

The discovery by Sir David Brewster in 1816 of stress-
induced birefringence in glass established the beginnings Of an
optical method of experimental stress analysis. Brewster's
discovery went almost unnoticed for nearly a century until Coker
and Filon (10) and others began to develop this interesting physical
phenomenon into a useful and powerful tool for science and engi-
neering. Fundamentally, the method rests upon the measurement
of the artificial double refraction in a material through the use of
polarized light and the subsequent relating of this birefringence to
the state of stress or strain in the material.

Since the time of Coker and Filon and their contemporaries,
the photoelastic method has been greatly expanded in usefulness and
technique, and it has been applied to a multitude of academic and
practical problems. The mention of all significant works in the
field of photoelastic analysis is neither necessary nor appropriate
here. Comprehensive bibliographies are to be found in the classical
work already cited, and in the book by Wolf (78) as well as the
handbook edited by Hetenyi (29).

The disciplines of experimental mechanics and related areas
of materials science have undergone considerable transformation in
recent years. Photoelasticity has served and Should continue to

serve as a simple, inexpensive and straightforward method for

 

 

partially determining the state of stress or strain in Simple and

complicated structural shapes. The technique and application of the
method is continually being refined and extended. A few of the
writings which may be indicative Of the general trend of thought
include the works of Haberlund on the photoelastic analysis of
plates (26), Kuske on wave propagation and the physical modeling
of birefringent phenomena (41, 42), Schwieger on impact and photo-
elastic coatings (71), Pindera in rheology and material behavior
(58, 59, 60), and, of course, a multitude of others. Reference to
one of the bibliographies mentioned above or current review papers
and articles (25, 61) will further illustrate the range of endeavor in
the application and improvement Of photoelasticity.

A second aspect of the contemporary development of
experimental mechanics appears in the form of a closer affiliation
with what might be termed the basic sciences. Experimental
methods of field analysis are generally drawn from fundamental
physical phenomena. Examples include the disproportionately large
change in resistance of a wire upon stretching, piezoelectric effects,
sonic transmission and reflection, magnetostriction, and stress -
induced double refraction (piezobirefringence). Many timeslthe
unifying basis of the physical measurements becomes lost or ignored
in the bifurcation of packaged publishable research problems and the
complications surrounding the technique of obtaining data and translating
it into meaningful form. Doubtless, the closer alliance of the technology

and application of measurement with the physical basis of the measure-

 

ment will contribute significantly in both areas. The experimental

methods and the analysis Of results will be simplified, improved,
and extended, while the basic science will receive the benefit of
experience, additional data, and the resulting improvement in
understanding of physical phenomena.

One phase of the broadening of the base and the range of
application of photomechanics is the utilization of a band Of
radiation which is wider than the visible spectrum. Such endeavor
may be expected to contribute to the technique and range of application
of photoelastic measurement. It should also furnish an addition to
the catalogs Of physical data. Then, taken as a part of the whole,
there should result an increased understanding of the puzzling
phenomenon of birefringence. Finally, this inquiry should lead to
an improved conception of the relationships between structure,
environment, history, and observed behavior of materials. Apart
from the academic, the import of such understanding for the
analysis and synthesis of material and for structural design is
great. The extension of the photoelastic method outside the visible

light spectrum forms the subject of this work.

1. 2. Purpose and Scope of the Investigation

In view of the possible benefits derivable from such a study
and the limited past inquiry into the problem, an investigation was
undertaken with the primary purpose of extending photoelastic stress
analysis into the infrared portion of the electromagnetic spectrum

and establishing the physical and technological validity of the result.

 

 

The fundamental goal was to be approached through pursuit

of a number of secondary experimental objectives:

1. To develop reliable apparatus, methods, and techniques
for performing quantitative photoelastic investigations in the spectral
region from the short wavelength limit of visible light up to about
2. 0 microns wavelength.

2. To obtain preliminary stress-birefringence data for
certain photoelastic materials from limited calibration tests using
infrared radiation.

3. To improve the method of description of dispersion of
birefringence and to measure the dispersion of certain materials
over the visible and infrared ranges in accordance with the proposed
definitions; also to make preliminary studies of the effects of stress
and strain field upon the dispersion of birefringence.

4. To conduct limited studies of the stress-birefringence of
various materials which are not transparent to visible light, in
particular the semiconductors, and to inquire into their suitability

for use as photoelastic materials in the given spectral range.

1. 3. Review of Related Literature
The records Of past investigations which are directly related
to the measurement of birefringence in the long—wave region are
limited in quantity. However, there exists much information in
areas associated with the development Of apparatus and techniques
and the interpretation of results of infrared photoelastic investigations.

Some Of the significant and more comprehensive references will be

 

 

cited here. Other works of more retricted application are referred

to in the sections Of this report where they contribute most to the
subject under discussion.

In 1957, Drechsler and Schreiner (17) used infrared radiation
for the observation Of stress-birefringence in certain polymers. The
results were obtained through the use Of light whose wavelengths
extended continuously through the spectral range between the minimum
wavelength transmitted by their visible light blocking filter and the
upper limit of the light source and the sensing devices. Observation
and photography were accomplished by means Of a device which
converted an image formed by infrared radiation of up to about 850
nanometers wavelength into a visible replica. The analysis was
qualitative. in nature. E. Hausmann and G. Junghamel in 1962 (28)
published information on their more quantitative investigations
into infrared photoelasticity. In the experiments described in the
abstract Of the last mentioned presentation, an infrared converter
was employed to Observe photoelastic patterns in simple, thin models
made of polyvinyl chloride and ebonite. These materials are trans-
parent tO visible light only in thicknesses smaller than those used in
the experiments. Nonmonochromatic radiation at a central wavelength
of about 800 nanometers was used. A contribution of quite a different
sort is due to Dietzel, Deeg, and Amrhein (15), who demonstrated
the existence of double refraction of microwave radiation in a ceramic
material.

Infrared radiation has been used in the observation of

birefringence generated by internal structure, residual stresses and

 

Slip in crystals. Dash (13) describes his experiments in relating
the residual birefringence in silicon crystals, which are generally
not transparent to visible light, to the crystal structure and growth
process. Nonmonochromatic infrared light Of somewhat less than
one micron was used, and an infrared converter facilitated
observation. Appel and Pontarelli (3) described an infrared
crystallographic polariscope in an unpublished lecture. Besse and
Desvignes (4) also developed an infrared polariscope for studies

of crystal structure and behavior. Apparently, none of these latter
investigations were concerned with the detection of birefringence
which was artifically induced by stress. This problem has been
pursued by Prus sin and Stevenson (68), who measured the strain-
Optic coefficients of silicon crystals using an infrared converter and
nonmonochromatic radiation at about 1. 2 p. wavelength. Jacob and
Schmidt-Tiedermann (34) used a similar technique in determining
thermally—induced stresses in semiconductor junctions. The last—
mentioned authors cite three other papers which describe measure-
ments of the piezobirefringence of silicon (23, 30, 24). Apparently
the experimental methods were similar, but the results differ
widely.

Although there exists relatively little information directly

related to the specific area of longwave photoelasticity, considerable

data on the use of longwave radiation in other areas of sc1ence are

available. Infrared light has become very important because of its

suitability for use in the detection of hot bodies, alarm systems,

and penetration of a broad class of materials. At the more fundamental

 

level, infrared spectroscopy has developed into a most useful tool

for the analysis of material composition and structure. These and

other military, scientific, and industrial applications of infrared

radiation have led to the extraordinary development of the technology

and theory associated with the generation, processing, and detection

of longwave radiation. As a further result, an immense variety

of apparatus for infrared studies is available commercially. Thus,
although the application of infrared radiation to photOelastic stress analysis
may have been neglected, the tools and knowledge are immediately
available.

It is not practical to discuss here even a representative
number of the recent books and papers dealing with the various
aspects of infrared technology. Mention of only a few of these is
sufficient for understanding of the work at hand. The comprehensive
work of Jamieson and his associates (35) serves as a text of theory
and a catalog of technological data, as does the book by Kruse,
McGlauchlin and McQuistan (40). More elementary is the text by
Hackforth (27). Interesting and useful data in the area of infrared
technology, and other fields as well, is contained in the encyclopedic
collection of Landolt-B'Ornstein (43) and the standard handbooks such
as the one published by the American Institute of Physics (2) and that
edited by Condon and Odishaw (l l ).

The basic works and bibliographies in photoelasticity have
already been mentioned. Additionally, there is the concise work my
Jessop and Harris (37) and also Frocht's classic text (20). Shurcliff's

treatment (75) of the subject of polarized light places this matter upon

 

a firm mathematical and conceptual foundation within the limits of

electromagnetic theory.

The science of Optics is closely allied with the subject of
photoelasticity. The classical and profound treatment of the
subject by Born and Wolf (8) is most useful for its development
of optics from the electromagnetic field equations. Among the
many basic texts treating geometrical optics, the one by Sears (72)
is mentioned only because of its Simplicity, correctness, and

general availability.

 

2. Techniques of Investigation

2. 1. The Photoelastic Method

The measurement Of stress—or—strain—induced birefringence
using polarized visible light has been highly developed, and the
techniques and theory have been treated extensively by many
authors. Duplication of this information is neither appropriate
nor necessary here. But, for the sake Of completeness and to
clearly establish the basis for extending photoelasticity into the
longwave region, it is desirable to discuss briefly the nature of
light, the equations of photoelasticity, the techniques of measure-
ment, typical apparatus, and certain aspects Of material behavior
which are pertinent to the original investigation.
2. l. 1. Nature Of light and the electromagnetic spectrum

The exact nature of light is a question that has long puzzled
scholars and scientists. Quite apart from this question, the
problem Of developing a system for the adequate description Of the
nature and the behavior of light has also received much study.
For most purposes of photoelasticity, it is sufficient to consider
light, including visible, ultraviolet, and infrared, as well as other
forms of radiation such as heat and radio waves to be energy in the
form of transverse electromagnetic waves. At any time, the wave
train constituting a "ray" of radiation can be completely described
by two vectors which are perpendicular to the direction Of the ray

and to each other. The first vector, E , is usually called the

i):

 

 

10

electric intensity vector, while the second, B, is ordinarily
known as the magnetic induction. Starting with the fundamental
physical quantity known as "charge" and Coulomb's experimental
law, the laws governing the behavior of a field of electromagnetic
waves can be developed rationally. This electromagnetic field
theory, which serves to describe the behavior of light waves, is
based on a system Of equations worked out by several investigators,
then refined, extended, and systemized by James Clerk Maxwell in
1873. The treatment Of the propagation of light waves by Maxwell's
equations is simplified considerably because the wavelength of
“light'' is ordinarily negligible in relation to the other dimensions
in common optical systems. The behavior of light is discussed
from the viewpoint Of electromagnetic theory in the comprehensive
book by Born and Wolf (8). Coker and Filon (10) treat the matter
especially as it is related to photoelasticity.

Emphasis must be given to the fact that field theory is only
a workable method for describing without contradiction most
observable phenomena associated with the transmission, reflection
and propagation of light. The system is not complete, nor does it
contain an expression of the exact physical nature of this radiation.
Neither does the theory adequately explain the effects of light in
changing the electrical conductivity of certain materials (photo-
conductive or photoresistive), generating electron emission in
others (photoemissive), being generated by electron bombardment
in luminescent materials, exciting a voltage in some materials

(photovoltaic), or even changing the magnetic properties of another class

 

 

ll

of substances (photoelectromagnetic). These and other effects of
radiation are partially explained by quantum wave mechanics in which
light is considered to be composed Of bundles or "photons" Of
energy. This theory provides a way Of describing certain effects
of light. It fails to account for certain other aspects of behavior.
which are well described by field theory. In order words,, tWO
methods Of describing the behavior Of light exist. Neither method
is complete or sufficient, but they do not contradict one another

or experimental observation. Both fail to express the fundamental
nature Of the radiation. This troublesome lack of completeness
and consistency in the description of the nature Of radiant energy
and its interaction with bodies results from the restrictions on
human senses and understanding. The picture is complete; man
can see only a small part Of it and he must rationalize the rest.

As will be seen, the various phenomena associated with
birefringence are adequately described on the basis of electro-
magnetic field theory. The exception, for part of this work, is
that a quantum affect of light is used in measuring the intensity of
radiation. This exception must go unattended as something
observable and useful but presently beyond precise physical

description.

In a nonconducting medium which is free of electric charge,

Maxwell's equations require that the electric vector E, which is

characteristic of the field, satisfy the wave equation:

 

 

12

2...
ya? = Kp i—E— (2.1)
dt
For this equation, Kp. : ~12 , where v is the velocity Of
v

propagation of the wave. It is the speed Of light or any other
electromagnetic disturbance, and in a vacuum it iS found to be
very nearly 3 x 1010 cm/sec. The values of K, the dielectric
coefficient, and p. , the magnetic permeability, in the above
equations depend on the choice of units.

A solution to the wave equation (2. 1) may be taken in the

form:

E- : Fcos i—Tr(x-vt) (2.2)

This solution represents a harmonic plane wave of wavelength A
which is propagated in the x-direction with velocity v. F is a
vector independent of x and t which specifies the amplitude and
plane Of the wave. Such a wave may be thought of as the simplest
solution to the governing equation. Light rays which are not
planar and/ or which are composed of more than one wavelength
can be described by a vector sum of more than one solution to the
wave equation. If the medium is a conductor, the analysis is more
complex, but will yield results which, within the limit Of the
present work, are the same.

As mentioned, the travelling wave of eq. (2. 2) always lies

in a single plane. Such a wave iS said to be linearly polarized,

the term ”plane polarized" being ambiguous.

 

13

Associated with each train Of light waves is a wavelength A
and velocity v. These two quantities are related to the frequency U

Of the wave by the basic wave relation:
v : 11A (2. 3)

Since the frequency of a wave is constant, the wavelength must
change directly with any changes in velocity which may occur when
the wave passes from one medium into another.

The effect of electromagnetic waves upon human senses
depends upon the wavelength or the frequency of the wave. The
wavelength Of light from ultraviolet through infrared is most

-6

10 meter)

conveniently expressed in terms of the micron (In
or the millimicron(lm/.4= 10-9 meter : 1 nanometer). In these
units, ordinary visible light covers the range from about 0. 4 micron
through approximately 0. 7 micron. The infrared range begins at
about 0. 7 micron and extends three decades up to the neighborhood
of 1 millimeter, where short radio waves begin. The so-called
radiant energy and radio Spectral regions are known to overlap,

and the two types are distinguished only by their sources, whether
black body or electrical oscillator. The usual units for frequency
of a wave are cycles per second (cps) designated V , and wave
number 77 . The frequency, wave number and wavelength of
radiation are related to each other and to the velocity of light in

a vacuum as follows:

C(ugsec)

V(cps) 2 MP») = MM) (2.4a)

 

 

14

— -1 _ 1 _ V(C s)
”(cm >-m- aw—m/i‘a ”'4‘”

A chart showing the frequencies and wavelengths corresponding

to the various regions of the electromagnetic spectrum is given
in Fig. 2. 1 along with an expanded diagram of the visible and
infrared portions important to this investigation.

In this report, the term "light" will be applied to radiation
of any wavelength whose primary function is illumination.
2.1. 2. Refraction, Retardation, Birefringence

The rate at which a light wave travels through a transmitting
medium depends only on the nature of the medium. In fact, the
governing wave equation (2. 1) indicates that the velocity of propagation
of any electromagnetic wave through a material depends only on the
dielectric constant and magnetic permeability of the material. The
ratio of the Speed of light in a vacuum to the speed of light in a
particular material is known as the absolute index of refraction, n,

Of that material. The index ordinarily varies with the wavelength of

light:

(2.5)

<IO

“x
If a light ray passes through a parallel plate Of refracting
material, the ray will emerge from the plate in a position behind or
ahead of the position which would have been occupied if the ray had
not been disturbed. If d is the thickness of the plate whose index
of refraction is n1, and if n0 is the index Of refraction of the
material surrounding the plate, then the distance by which the refracted

ray lags behind its unrefracted position is:

 

”Uni (2.6)

This is the “absolute retardation" of the ray of light by the refracting
plate which is immersed in a refracting medium. To avoid ambiguity,
the “relative refractive index" will not be used here.

In certain materials, the refractive index varies with
orientation. An analysis of the transmission of an electromagnetic
wave through such an anisotropic medium is outside the scope Of
this writing. The problem has been treated by several authors
including, for example, Coker and Filon (10). For this work it is
sufficient to observe that the propagation of a light wave through an
optically anisotropic material can be described in terms of the
refractive indexes corresponding to a certain set of orthogonal axes
in the material. These are known as the principal axes and may be
thought of as the axes around which the refractive indexes vary in a
symmetrical manner. The indexes of refraction belonging to these
principal axes may be termed the principal values of the refractive
index. It can be shown that a material having unequal principal
refractive indexes can transmit only light which is linearly polarized
along the principal axes normal to the path of the light. The two
polarized rays will travel through the medium at velocities dependent
upon the refractive indexes corresponding to the principal axes in
the plane of the wave front. These principal axes in the plane of
the wave front and their respective refractive indexes are directly
related to, but are not generally the same as the principal axes and

principal indexes of the medium. The relationship depends upon the

 

 

 

 

16

direction of the light ray, and the two sets Of axes and indices are
the same only if the light path is along one of the principal axes of
the transmitting material.

Suppose now that a ray of light is passed normally through
a plane parallel plate of thickness d and made of material which
is optically anisotropic. Let the principal values Of the refractive
index in the plane of the plate be n and n

1 2’

Upon striking the plate, the ray will be polarized into components

the two being unequal.

along the principal optical axes of the plate. The two components of
the original ray will traverse the plate at different speeds, and upon
leaving the plate, one component will be retarded with respect to the
other. If the components could be separated from each other upon
emergence from the plate, then two rays of mutually perpendicular
linearly polarized light would be obtained. The Nicol Prism and
certain other polarizers are based on this principle. Dichroic type
polarizers, of which Polaroid sheet is an example, function in a
similar manner. But, instead of separating the two rays, the
dichroic polarizer simply absorbs one of them.

If the ray incident upon the plate is not polarized, meaning
that it consists of a random assortment of polarized components,
then the polarized components emerging from the plate will
recombine (unless artificially separated) into a ray without a
constant preferred orientation of the electric vector. The effects
of the anisotropic material upon the unpolarized incident ray cannot

be detected by ordinary means.

 

 

17

Suppose, however, that the ray incident upon the plate
having unequal refractive indexes is linearly polarized. This ray
will be resolved by the plate into two components bearing a known
relationship to the original ray. The two components will each
be retarded by the plate by amounts which may be computed from
formula (2. 6) for the absolute retardation due to a refracting

plate.

)r—‘f- R = (.n2 -n0),% (2.7)

R :(nl-n 2
o

O

The difference between these two values Of absolute
retardation represents the amount by which one component of the
ray is retarded with respect to the other. This "relative
retardation”, R , is ‘ dependent on no, the refractive index of

the surrounding medium. This fact is often ignored.
R-R R —(n -n)—d— (2.8)
‘ ' " 1 2 n

The eye cannot determine whether or not light is polarized nor
can it detect changes in phase of light waves. Thus, the light
emerging from the plate just described will appear to be no
different than the light entering the plate. In turn, the unaided
eye cannot distinguish this from common unpolarized radiation.
However, the state of polarization of the light leaving the plate
differs from the known state of polarization of the incident ray.

This difference can be detected and measured.

A plate of material whose principal refractive indexes differ

is said to be doubly refracting or “birefringert”. The detection and

 

 

 

 

 

18

measurement of the relative retardation produced between the
components of a polarized ray passing through such a material
forms the basis of the photoelastic method.

2.1. 3. Birefringence in Deformable Bodies

Certain materials are naturally birefringent, and slabs of
these materials retard light rays in the same manner as the
birefringent plate of the previous section. Natural crystalline
substances which are doubly refracting include Calcite (or Iceland
Spar; CaO ° C02) and Sodium Nitrate (NaNO3) as the most common
examples. Both of these crystals are used in the manufacture of
birefringence type polarizers such as the Nicol prism.

In another broad class of materials, the birefringence can
be induced by stress or deformation. Materials which exhibit this
property of stress or strain birefringence include glass, many
plastics, certain rubbers, semiconductors, and some fluid materials.
This type of birefringence is a property of a large number of trans-
parent materials, although the birefringence may be small enough
to be unnoticeable.

The birefringence—producing interactions between the elements
of a material's structure and stress are not yet clearly known. Models
of the material behavior have been developed on the basis of the diphase
theory (20) and a multiphase theory (41), but these are meant to serve
only as representations rather than explanations or descriptions.
Other investigators, for instance Persoz (56) and Rao (70), have
concluded that in certain molecular structures the birefringence may

be due to stretching of certain molecular groups and also to the

 

 

 

 

l9

reorientation of some anisotropic side chains. The stretching
birefringence and orientational birefringence may add or oppose one'ani
other, or one of them may be missing entirely, depending on the
material and the conditions of the test. It is on such a basis that the
presence of a second-order transition point and other diphase
phenomena may be justified for real materials. Another approach
is discussed by Blumstein (7),who analytically extends the Chemist's
concept of rubber photoelasticity to the epoxy resins. Further
discussion of the microscopic mechanism of birefringence is not
within the limitations of the present investigation. But the question
must be diligently studied, both analytically and experimentally, in
order to place photoelasticity on a sound scientific basis and also

to contribute to the knowledge of material structure.

Although artificial double refraction cannot yet be firmly
related to changes in material, it can be connected to the history of
stress and/or strain at a point through experimentally derived
relationships. For example, if crl and 0'2 are the principal
stresses at a point in a two-dimensional "momentarily linearly
elastic" plate of optically sensitive material (generalized plane
stress problem), then it can be Shown experimentally that the
principal axes of refraction correspond tO the principal stress
axes. A polarized ray will, when passed through the plate, be split
into two components along the principal stress axes while passing
through the plate, and the two components will be retarded absolutely
and with respect to one another. The absolute retardations can be

expressed in terms Of the absolute photoelastic coefficients, C and

1

 

 

20

c2, which are generally functions of stress, strain, time, and

wavelength of radiation. The equations are

R

1 (c0'+c0'

1122)

d (2. 9a)

R

2 (CZO'1 + c102) d (2.9b)

The relative retardation is the difference of R1 and R2. In terms

of the absolute coefficients, this relative retardation is:

R =(c1-c2)(0'1-0'2)d (2.10)

Comparing (2. 10) with (2. 8), it seems reasonable to write

n1 “z
a : c1(cr1 ~02) and n—0 = c2(0‘1-o‘z) (2.11)

The absolute coefficients have been only rarely used in
photoelasticity up to the present. The relative retardation is usually
related to the stress state by means Of an empirical stress-optical

coefficient C0_ , which is related to the absolute coefficients by

C=c—c

U 1 2 (2.12)

The expression for relative retardation then becomes:
R=Co_(0'l-o‘2)d (2.13)

Alternatively, the relative retardation may be written in terms of

strain at a point through use of the experimentally-derived strain—

optical c oefficient, C
6

R = C€(€1 -€2) d (2.14)

 

 

21

If the expressions for relative retardation in terms of stress
and strain are to be identical, then the principal axes of stress and
strain must be coincident. If methods based on birefringence
measurement are to be extended to problems involving large
deformation and flow where the stress and strain axes may be,
different, then the interpretation of data and the meaning Of the
coefficients becomes more complex. The photoelastic analysis of
such problems has been adequately formulated for only the Simplest
cases.

The interpretation of birefringence measurements in terms
of stress or strain is not a primary purpose of this investigation.
Wherever this problem must be considered, discussion will be
restricted to "momentarily linearly elastic" material behavior.

This means that the birefringence, stress, and strain are linearly
related at the time of measurement. The stress-Optical and strain-
optical coefficients are then functions only Of time and wavelength

of radiation. The restriction to a linear range suggests the problem
of detecting the occurrence Of nonlinear behavior. More will be

said on this particular problem later in this writing. The definition
of “momentarily linear elastic" behavior and calibration data are
presented in the works of Pincbra(58, 59, 60). A summary Of past
investigations, additional data, and the definition Of the limit of the
linear range of behavior are presented by Pindera and Kiesling (64).

The coefficients relating retardation and stress or strain are
generally complicated functions of material and its history, time,

stress, strain, temperature, and wavelength of the light used. In

ham

 

 

 

22

practice, the coefficients are usually developed experimentally by
measuring under controlled conditions the relative retardation
produced by a specimen in which the stress or strain field is
known. The results of such a calibration test can be used to
explore unknown stress states through measurements of retardation.
From what has been said of the relationships between the stress
field and the principal refractive axes and the magnitude of the
birefringence, one would expect that the photoelastic method will
yield information on principal stress orientation and difference in
magnitude of principal stresses in a two dimensional body. The
equations governing this measurement are presented in the following
section.
2.1. 4. The Equations of Photoelasticity

Double refraction is measured through determination of the
relative retardation between two components of a ray of polarized
light that is passed through the birefringent body. The phenomenon
of destructive interference of light is useful in this measurement Of
relative retardation, which is usually a very small quantity. The
equations describing the light transmitted through two polarizing
plates with a model between have been presented in different forms
by many authors. The equations must be briefly re-examined here
for three reasons. First, it should be demonstrated that the
equations are applicable throughout the electromagnetic spectrum.
Secondly, the equations for the case of parallel polarizing plates,
(without quarter-wave plates) are not ordinarily presented, since it

is not common to utilize this arrangement of the polariscope. Much

 

23

of the data presented here was obtained with linear polarization and
with polarizers both parallel and crossed. All the data must be
properly interpreted. Finally, one of the methods of measuring
small values of relative retardation required knowledge Of the
general relationship between light intensity and the relative
inclinations of polarizers and model.

The development of the governing equations may begin with
consideration of the ray emerging from the first polarizer. If this
ray has a wavelength A , travels in the z direction with velocity
v in a medium (usually air) having refractive index no, and is
polarized in the yz plane, then the electric vector describing this

radiation is:

E=Acosi—fl(z-vt)j (2.14)

When the ray falls normally upon a stressed birefringent plate whose
principal stresses, 0‘1 and 0'Z , are inclined at angles 4) and

(b + 900 to the x axis, the ray will be split into components along
the principal stress axes, and the two components will propagate
through the plate at different velocities v1 and v2. Writing

separately the scalar components:

E1 = E sin (b = A cos 2A1 (z - vlt) sin 4) (2.14a)
E2 = E cos <13 = A cos .11 (z - vzt) cos d) (2.14b)

Here, attenuation of the wave due to reflection and absorption and

also changes in phase which may occur at interfaces are neglected.

 

24

They are not assumed zero but rather are assumed to be equal for
the two components of the original ray.

After traversing the thickness, d , of the plate, each
component of the ray will have been retarded and the magnitudes
of the orthogonal electric vectors describing the rays emerging from

the plate are:

211' (1

E1 = A‘cos T [ z - vt - (n1 -no)§]sin 4) (2.15a)
E - Acosél[z- t-(n -n )-d—]cos¢ (215b)
2 ‘ ' A V 2 ono ‘

The 'vector sum of the two components represents the electric

vector of the emergent ray. The particular nature of this vector
will be discussed shortly. For the moment, consider the ray which
will emerge from a second polarizer which is placed with its
transmitting axis at an angle s with the original axis Of polarization
in the path of the ray whose components are now travelling at equal
velocities but out of phase. The analyzer (the second polarizer) will
transmit only those components which are parallel to its axis. The
electric vector of the linearly—polarized emerging ray will be inclined

at angle e to the axis of the first polarizer, and its magnitude will

be:
E=Acos-Zl[z-vt-(n -n)£]sin¢sin(e+¢)
A 1 ono
(2.16)
+Acos'z—Tr z-vt-(n -n)-g-]cos¢cos(e+¢)
A 2 on

O

which becomes after considerable manipulation,

 

25

E = Asin[% (n1 n2)d—](sine5in 24> - cos ecos 24>)
o
+ n - 2n
ZTI’ n1 2 o d
SlnT(Z - vt — 2 F— )+
o
1; d
A cos[)\ (n1 n2)a;]cos e-
+ n - 2n
21r n1 2 o d
COST (z -vt 2 :12) (2.17)

The emergent light wave is thus seen to be the scalar sum Of two
rays of equal wavelength out of phase by g and with amplitude
depending on the retardation in the birefringent plate and the
orientation of the polarizer, analyzer and stress axes. The general

expression is not amenable to straightforward physical interpretation.

It is useful and instructive to examine the result for two special

900, and

cases. First, with polarizer and analyzer crossed, e

the magnitude of the electric vector reduces to:

E

—n2)§] sin 24>

O

A Sin['T;:—(n1

n +n -2n
2n 1 On_d_) (2.18)

. 2
SlnT(Z -vt————-2-————-
o

This is clearly a travelling wave of wavelength A and amplitude
dependent upon the relative retardation and the orientation of the
stress axes. The amplitude, and thus the intensity, of the wave
is zero if:
(1) ¢ = 0, 1% ; that is, if the polarizers are aligned with
with the principal stresses in the birefringent plate.

This phenomenon serves as a device for determining

principal stre s 5 directions.

 

 

26

(n1 - n2) d
(2) T = m = 0 or an integer. This serves to
0
measure relative retardation, (n1 - n2) d, in terms Of

the wavelength of radiation used.
0

Now, for e = 0 , that is with polarizers parallel, the

magnitude of the electric vector of the emergent wave is:

. d.
E = - A Sln[1 (n - n )—]cos 2(1)
A l 2 no

n —+n -2n

. 2n 1 2 o A
s1nT(z-vt—--——2—-—n,o)+
Tr d
Acos[x(n1-n2);o‘l
+-n ~2n
211' I11 2 o d
cosT(z-vt- 2 3;) (2.19)

This indicates that the resulting electric vector consists of two
separate component waves out of phase by 1% . The amplitude of
one wave depends only on relative retardation and the other depends
on retardation and stress direction. The magnitude of the resultant
wave is zero only when both components are zero simultaneously.

. n (’11 ' n2) m
ThIS occurs only when <19 = 4 and _A—n— d = —2— , m an
integer. If the elements are properly aligned, such a system can
be used in addition to the crossed system for the determination of
relative retardation. Such a measurement with light field linear
polarization is not common in photoelasticity. The calculations also
Show that the retardations common to both rays as they pass through
refractive media do not affect the results. These common phase
shifts can be eliminated by a suitable coordinate transformation, but

in practice theyrare uwally ignored entirely. Only the relative phase

differences must be considered.

 

 

27

Suppose now that the stressed birefringent plate is not uniformly
birefringent. This would occur if the stress field varied. The crossed
linear polariscope would not transmit rays which passed through points
in the model where the stresses were properly inclined or where the
relative retardation was an integer multiple of the wavelength. An
image of the plate constructed with the rays transmitted through
the polariscope would Show two sets of dark lines. One set, called
the isoclinics, would be the locus of points at which the principal
stress directions coincide with the directions of polarizer and
analyzer. The second family is the locus of points where the
relative retardation equals integral multiples of the wavelength,

d-
R=(n1-n2)-Ig=mA, m=0, 1, 2,... (2.20)

These lines of constant relative retardation are termed whole —order
isochromatic fringes. They serve to indicate lines in the plate
along which the stress (or strain) has reached certain discrete

values. If CU is the so—called stress-optical coefficient, then

 

1 2 C d (2°21)

If polarizer and analyzer are placed parallel, dark lines will
show up in an image of the plate only in those areas where the
principal stresses are inclined 450 to the polarizer axis and where
the relative retardation is a half-integer multiple of the wavelength.

These lines are termed half-order isochromatic fringes. Along such

a fringe,

 

 

 

_ d _ m _ 2.22
R_(n1-n2)fi:_2>., m—l, 2,... . ( )
the corresponding stresses are:
_ m A
0'1-02 — 2C d (2.23)

These stress values are seen to be midway between the values given
by the whole-order fringes. The two sets of isochromatics taken
together thus give a plot of principal stress difference (or strain
difference) at a network of points in the plate. Values of stress
difference at points in between the is ochromatics can be found by
interpolation or compensation. When these are coupled with the
stress directions found from the isoclinic fringes by suitable
computation, the complete stress field can be mapped. That is,
0'1, 0'2 and their inclinations can be determined for any point in
the birefringent plate. Much has been written regarding the
interpretation of photoelastic fringe patterns. Further discussion
here will serve no purpose within the limited scope of the
investigation.

Returning now to equation (2. 15) for the components of
light emerging from the birefringent plate, consider the special
case when (n1 — n2)-i=-4A and 4) = 450.

n
0

unit vectors along the principal axes, the electric vector can be

A

If p and 61‘ are the

written as follows:

V2

-- _ 217 _ _ d A
E _ _2 Aicos—x Z V’t-(n1 no—nolP
21r d A A
+COST[Z vt-(nl-nohg+z]q}

 

 

29

Eliminating the retardation common to both components by a

coordinate change and introducing an identity:

f = if cosEiT-(z-vt)$-‘Sin'z)\l(Z-Vt)al (2~20)

The two components are thus identical but out of phase by 1;: . The
magnitude of the vector is found by the Pythagorean Theorem to be

. . . . A .. .
constant. The 1ncl1nat10n of the vector With the p 3X18: IS

0. 33%- (z — vt) (2.21)

At any instant, the orientation of the light vector varies uniformly

with position 2 on the optical axis, that is, it describes a circu1ar
helix. Otherwise, for fixed 2, the light vector rotates uniformly

with time and so describes a circle. Radiation which can be

described by (2. 20) is said to be circularly polarized. The birefringent
plate which produced the circularly polarized light from that Which was
plane polarized is termed a quarter wave plate.

Circularly polarized light serves two useful functions in photo-
elastic measurements. First, it can be used to remove the,
orientational dependence of extinction of radiation which exists in a
polariscope employing only linear polarizers. This use is well
described in the standard texts and handbooks. The linear polar-
iscope is converted to circular by insertion of quarter-wave plates
between polarizer and stressed plate and between analyzer and stressed
plate. The equations governing the use of the circular polariscope do
not have to be re-examined here, since the data used quantitatively in

this work were taken using linear polarization. In general, these

 

30

equations are slightly simpler in form than those presented for the
linear system.

The second important use of circular polarization is for
precise determination Of relative retardation at a prescribed point
in a birefringent plate. The isochromatic fringe patterns obtained
with the crossed or parallel polariscope indicate the locus of points
where the retardations are integral and half-integral multiples of
wavelength. The retardation at points between whole and half—order
isochromatics can be found by two methods of goniometric
compensation in which quarter wave plates are used. The two
methods are discussed thoroughly in the literature (36, 47). A
slight modification Of one of these methods was used in this
investigation, and the procedure and equations will be discussed
where appropriate in a later section. The errors which may result
from imperfect quarter-wave plates will also be mentioned. ‘

2.1. 5. Optical Dispersion and Dispersion of Birefringence

For a given state Of stress and time, the birefringent
properties of materials depend to a certain limited extent upon the
wavelength or frequency of the radiation used in its measurement.
That is to say, the order, m, of isochromatic fringe as expressed
by equations (2. 20 - 2. 23) Of the preceding section cannot be considered
to be inversely proportional to the wavelength. Unfortunately, this
simple proportionality is still Often assumed.

The degree to which the stress-birefringence of a material
depends on wavelength may be expressed in terms of the principal

refractive indices, n and n

1 Z; the absolute stress-optical

 

 

 

31

coefficients, c1 and c2; or the relative stress-optical coefficient.
Co; . The corresponding strain-birefringence coefficients could
also be used. In order to attain complete understanding Of the
phenomena of double -refraction and dispersion, the variation of
the absolute coefficients and of refractive index with wavelength
must be explored. These absolute quantities have been used only
rarely in photoelasticity up to now. Hence, this discussion will

be restricted to the relationship between the frequency or wave — °
length of radiation and the relative stress-optical coefficient.

The dependence of the relative retardation upon the
vibrational frequency of radiation is commonly called the
dispersion of birefringence, because it results from the same
basic physical phenomena as the dependence of the refractive
index, n, on the radiation frequency. This variation of n with
wavelength is known as optical dispersion or, Simply, dispersion.
The dispersive effect in glasses and other materials is useful in a
prism, for example, since it allows the prism to separate colors.
The same effect detracts from the usefulness Of a lens, and much
effort is expended in developing lenses which have a minimum of
chromatic dispersion. The term dispersion was first applied in
the area of photoelasticity by H. Ambronn (l), and it is now in
common use.

Part of this investigation was concerned with the measure-
ment and adequate description of dispersion of birefringence. As

yet, there has been little agreement among the various investigators

as to the best method Of presenting dispersion data and the proper

 

32

interpretation of the data. Neither is experimental information very
complete or conclusive. Hence, the results and ideas of various
researchers must be examined briefly here. Also, Since the
dispersion of birefringence is SO closely allied with the much better
known phenomenon of ordinary optical dispersion, much can be
gained from a short discussion of some accepted definitions of the
dispersive power Of glasses.

There are a variety Of arbitrary definitions of optical dispersive
power. Examples include the "relative dispersion, " which is expresset
in terms of the indices of refraction at the wavelengths of certain

Fraunhofer lines,

'—"—_'1— (2. 21a)

and the "partial dispersion":
C F

A more general measure of dispersion involves the derivative Of the

quantity under consideration with respect to the independent variable.

On this basis, the optical dispersion of a medium is defined as the

rate Of change Of the index of refraction with respect to radiation

frequency or, respectively, wavelength:

3—3 (2. 22a)
Ell—:1 (2. 22b)

In the general case, the refractive index changes with radiation

frequency, and normally the index increases with increasing

 

 

 

33

frequency. That is,

dA ’ dv

This type of dispersion is called normal dispersion.

It is known that under Special circumstances in a narrow
range of frequencies near absorption bands, that is to say in regions

of resonant frequencies for the material, the refractive index of

transparent materials decreases with increasing frequency:

dn dn
__ > —— <
dA 0 , dV 0 (2. 24)

This phenomenon is called anomalous dispersion. According to the
theory of molecular optics, it results from the frequency dependence
of the molecular polarizability (11, 8).

The dispersion of birefringence of certain photoelastic
materials, including mainly glasses but also celluloid and gelatin,
has been investigated by several authors. A comprehensive survey
of research work done in this field and a related bibliography is
presented in the classical work of Coker and Filon (10). These
authors found that for many glasses the variation of stress-optical

coefficient with color is, on the whole, closely fitted by the formula:

CO
C - (2.25)

o‘ ‘1-(>.7>.)

where CO and A0 are constants for any given glass. In fact, these

constants indirectly describe the dispersion Of birefringence. It may

be pOinted out that the above formula was used for non—creeping glasses.

 

 

34

E. Monch in his paper (49) proposed the following relation

as a measure of the dispersion of birefringence:

(mA)
_ Na Hg
D — (mx)Na (2.26)

where the indexes Na and Hg denote the yellow sodium line and

the blue mercury line, respectively. In a later paper (53), E.

Monch and R. Lorek used the same formula in relating the dispersion
Of birefringence to red light (A : 655 nm) and the blue mercury line
(A = 436 nm).

The same measure of dispersion of birefringence is used by
Monch, Hartmann, Jira, and Lorek (50, 51, 52, 38, 45), who attempted
to relate the degree Of plastic deformation to the dispersion of
birefringence for several plastics. Positive results were obtained
only for celluloid. For this material, the dispersion as expressed
by formula (2. 26) may be considered a measure of plastic distortion.
This result confirms those published in 1924 by A. Ramspeck (69).

He found that in the elastic range the dispersion of birefringence in
celluloid is normal, with the stress-Optical coefficient for red being
about 3% smaller than for violet. However, beyond the elastic limit,
Co for red is greater than that for violet.

Unfortunately, such a definite relationship does not exist for
several other model materials. This has been demonstrated by
R. Lorek (45) for several polyester resins of Leguval type and by
J. T. Pindera (62, 63) for allyl polyester resin CR -39 and unsaturated

P°1Yester resin P-6. These negative results should be interpreted

 

 

 

35

to mean only that the relation between inelastic deformation and the
dispersion of birefringence measured according to formula (2. 26)
has not yet been found. Typical measurements of the dispersion of
birefringence as reported by different authors are presented in the
papers cited above. Especially interesting is the presence of
anomalous dispersion first discovered by Filon (10).

The definitions developed for this investigation into the
dispersion of birefringence of representative materials are

presented along with the data in Chapter 3.

2. 2. Infrared Photoelasticity Apparatus and Technique
2. 2.1. Optical Systems Used

The visible radiation which is ordinarily used in photO-
mechanic investigations comprises only a very narrow portion of
the complete electromagnetic Spectrum. Certain optical effects
which are readily observable under visible illumination Should also
be apparent with nonvisible radiation provided the human senses
are extended by suitable apparatus and the optical interactions are
not limited to certain wavelengths. Examination Of the photoelastic
equations presented indicates no limitation to the visible portion
of the spectrum. Thus, with appropriate techniques, meaningful
measurements of birefringence could be taken in other spectral
regions. The actual methods used would depend on the wavelengths
Used. The important thing is that the functions served are the same

in all spectral regions.

 

 

 

36

The experimental investigations considered here were carried
out in the range of wavelengths from visible violet to two microns.
A basic difference in the techniques of recording data necessitated
the use of three different Optical systems. Sketches of the systems
are presented in Figure 2. 2, and photographs of the optical bench
are shown in Figure 2. 3. The first of these systems are used for
investigations through visible up to 0. 7+1 , and was basically no
different than the polariscopes commonly used for photoelastic
investigations in visible light. The second system, quite Similar
to the first (Figure 2. 2a), was used for the 0. 7 through 1. 2):. range.
Both these systems provided means of planewise Observation and
measurement of birefringence. The third system, shown in
Figure 2. 2b and Figure 2. 3a - 2. 3c served for the pointwise
measurement of birefringence. It was developed for the extension
Of photoelastic measurement up to the 2 micron limit of this
investigation, although, in principle, the approach is not limited
to any particular wavelength range. The reasons for the particular
configuration Of this system will be discussed shortly.

The following paragraphs describe the Optical elements and
techniques used for performing the basic functions in the various
polariscope systems. Suggestions for improvement and for the
extension of the method into other spectral regions are also
presented. The chapter closes with a discussion of the special

methods used to measure fractional fringe orders and record data.

 

 

37

2. 2. 2. Elements Of the Optical Systems
A. Light Sources and Filters

The first item Of apparatus required in any polariscope is a
source of radiation. This radiation should be nearly monochromatic.
Several different wavelengths should be available, and the radiation
should be as intense as possible.

The most common method of producing intense monochromatic
light is through the isolation of certain bands from the emission of
spectrally pure sources. Such sources generally consist of lamps
containing small amounts of certain gaseous or metallic chemical
elements. Proper electrical excitation will cause the elements to
emit energy at wavelengths characteristic of the element. This
phenomenon is that commonly used in the spectroscopic chemical
identification of materials. When the emission lines are separated
from one another by selective filtering, monochromatic radiation
results. For most lamps Of this type, the emission lines of useful
intensity occur‘between deep ultraviolet and 1. 5 microns.

For this investigation, a low-pressure Mercury-vapor lamp
(General Electric, H-4) and a low-pressure Helium lamp (Philips,
93098) were used as two of three sources of radiation in the visible
and near infrared spectral ranges. The emission characteristics
of this particular Mercury lamp, , as well as other Mercury lamps,
are well known (66). The characteristics of the Helium lamp in
the infrared region are not readily available. Usually the output of
spectral lamps differs to a limited extent from the spectroscopic
emission of the elements involved due to effects of pressure, impurities,

presence Of electrodes, etc. The emission of both spectral lamps in

   

 

38

the near infrared was determined by means of an infrared spectro-
meter (Perkin-Elmer, Model 137-G). This device is intended to be
used in measuring the infrared transmission characteristics of
materials for identification purposes. The instrument was used as
an emission spectroscope by blocking the light path in the sample
side of the optical system and using mirrors to inject light from

the lamps. The result is a plot which compares the lamp emission
to the emission of the standard source (glo-bar) inside the instrument
over the wavelength range from 0. 85 to 2. 5 microns. The results
of these measurements agreed well with published data. Of
particular interest is the output of the Helium lamp. According to
spectroscopic tables, the emission Spectrum of Helium in the
infrared consists of three very intense lines whose wavelengths

are very close together. For practical purposes, that is for low
resolution of wavelengths, the emission is one narrow band centered
at 1. 083 microns. The absence of other lines close by allows this
line to be easily isolated. Figure 2. 4 shows a representative plot

of the emission of the Helium lamp, copied from those obtained with
the Perkin-Elmer spectrometer.

Since the Helium and Mercury lamps were also used as sources
of monochromatic visible light in accordance with common practice,
their emission in this range was checked with an ordinary prism-type
emission spectroscope (Hilger, No. D—26. 303, 26222). The measure-
ments agreed well with the characteristics usually ascribed to these

lamps.

 

 

39

Sources of radiation which is to some degree continuous over
a broad portion of the electromagnetic spectrum are very common,
since every body at temperature greater than absolute zero is an
emitter of such radiation. The intensity of radiation and the span
of wavelengths are functions only of the temperature and surface
and shape characteristics. The basic emission law describes the
radiation from what is known as the ideal blackbody, and the
emission of real, imperfect bodies can be described in terms of
blackbody emission. Such : emitters of radiation which is
continuously distributed with respect to wavelength form useful
sources of infrared light. By proper choice of body and temperature,
the emission can be centered in any given span of wavelengths. A
ribbon of tungsten wire or a carbon resistor, for example, if heated
to about 10000 Kelvin, will emit radiation with wavelengths distributed
between 0. 66 microns and 60 microns, with the peak emission at
3. 2 microns. If mmochromatic light is required, it must be selectively
filtered from the "white", or continuously distributed light by filters
or by means of a monochromater. The most serious problem associated
with the use of spectrally-continuous sources of longwave radiation is
that of obtaining great enough intensity. Ideally, any wavelength can
be produced. Unfortunately, as the temperature of the emitter is
decreased in order to obtain longer wavelengths, the intensity of the
radiation also decreases. Waves as long as 1 cm have been produced
by simple blackbody radiation, but they were almost undetectable (35).
At wavelengths above: about 1 mm, reasonably intense radiation must
usually be obtained by electronic resonance such as is used in ordinary

radio circuits .

 

 

40

Two spectrally-continuous sources of radiation were used in
the present investigation. The first was a General Electric “Point-
O-Light" lamp. This is basically a 100 watt incandescent lamp whose
tungsten ribbon is closely wound in order to concentrate the source
of radiation into a small area. The result is quite intense white
light with wavelengths ranging from about 400 nm up to a maximum
of about 3p. . The other continuum source, which was particularly
well suited for the production of intense visible and infrared radiation,
was a Sylvania “Concentrated Arc" lamp, type Kl OOP. Although
rated at only 100 watts, the total emission of this lamp comes from
an arc whose diameter is only 0. 059 in (1. 5mm), giving a brillance
of 24, 500 candles per square inch. The spectral distribution of such
a lamp is quite uniform from near ultraviolet through several microns.
A direct current power supply which also provided the required
2000 volts for ignition was supplied with the lamp.

Various other sources which are similar in principle to those
described here can be employed in infrared optical systems.

Papoular (55) describes the use of a water-cooled Mercury-vapor
lamp as a source of radiation in the 15—100 cm"1 range. Here the
Ihot bulb rather than the arc acts as the emitter. The same author
also has developed an oven with an aperture which acts as a black-
body radiator. Devices based on the same principle are available
commercially.

Regardless of the apparatus used to produce radiation, some

sort of filtering is usually required in order to meet the requirement

that the light be monochromatic. Many methods of selective filtering

 

 

 

41

of radiation have been investigated and developed, and these are well
described in the extensive literature on the subject. Filters are
conveniently characterized on the basis of their transmission curves
as "wide bandpass“ or ”narrow bandpass”. The latter term may be
applied to filters which transmit a band of wavelengths which is narrow
enough to be taken as monochromatic for the pirpose at hand. Wide-
band filters transmit a wider span of wavelengths, and are intended
to be used in blocking off a portion of the spectrum. Special types of
wide 'bandpass filters are the low -pass type, which transmits
radiation below a certain wavelength, and the high-pass variety
which transmits above a certain wavelength. These terms are only
descriptive, and actual plots of transmission versus wavelength are
required for the precise description and evaluation of light filters.
Several filters of each type were used in this research effort.
Initially, for the isolation of emission lines from the spectrally pure
sources, wide-band filters of high-pass, low-pass, and controlled
band width types were used singly and in combination. These filters
were mostly of the "transmission", "color”, or "Wratten" type
which transmit certain colors and absorb all others. Their
construction, use, and specifications are described in detail in
the excellent pamphlet by Eastman Kodak (39). Many other types of
transmission-absorption filters, many of them simply aqueous
solutions of common chemicals, are described by Landolt-Borstein (43)
and North American Philips (57), in addition to the common physics
and optics textbooks and handbooks. Plyer and Ball (65) describe

the use of deposited films of tellurium, bismuth, antimony, and

 

 

 

42

magnesium oxide as transmission filters for the infrared. Other
authors (74, 6) present similar information on plastics and organic
dyes.

Transmission-absorption filters which pass only a very
narrow band of wavelengths are difficult to manufacture or obtain.
When monochromatic light must be obtained by isolating closely
spaced spectral emission lines or a narrow wavelength band from
continuum radiation, then either a monochromator or narrow band-
pass "interferenceH filter must be employed. The filters are
usually preferred when intensities must be high. Such filters
consist of several layers of dielectric materials having carefully
controlled thicknesses, rather than the one layer characteristic
of transmission filters. The thickness, number, and material of
the layers depends on the peak wavelength and bandwidth desired.
The simultaneous occurrence of selective reflection, absorption
and destructive and constructive interference within the layers
causes the combination to transmit only the chosen band of wave-
lengths. Often the primary wavelength is accompanied by sidebands
which are harmonics or subharmonics of the center wavelength.

Several interference type filters which were supplied by
three different manufacturers were used for experimentation in the
visible and infrared ranges in conjunction with the various light
sources. The characteristics of two filters which are representative
of those used are shown in Figure 2. 5. The data plotted in Figure
2. 5a for Optics Technology filter no. 160-1 -l45 is from spectrometer

plots supplied by the maker. The other plot (2. 5b) for Infrared

 

 

43

Industries filter no. NB-l56-0-(1) was obtained with the Perkin-
Elmer spectrometer previously described in the paragraphs on
lamps.

Table 2. 1 summarizes the apparatus used for producing the
radiation used in the investigation. (see page 44).

Of the sources listed, two require further comment. First,
the radiation centered at 853 nm is not truly monochromatic. Super-
position of the transmission of the filter, the output of the lamp, and
the photographic film used with this combination (Kodak 1R135)
indicates that the combination has a band pass centered at about
853 nm with a half-width of 36 nm, or 4%. This bandwidth is
sufficiently narrow to make possible a measurement in the region
between 707 nm and 1083 nm, although the accuracy of measurement
would certainly be less than that obtained with more nearly mono-
chromatic light. Secondly, the combination listed for a wavelength
of 707 nm actually includes a second weaker line at 728 nm. The

effects of this added radiation were small enough so that no difficulties

arose.

B. Lens

With most practical polariscopes, at least one lens is required
to cause the rays of light to converge to a projection device or else
to form an image of the birefringent plate upon a screen. A second
lens is usually used to collect and collimate the radiation from the
light source. Several additional lenses may be employed to improve

the overall quality of the optical path. The sketches in Figure 2. Z

 

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mhmufih was moonsom Gofidfipdm ofiumaoflmoonoz .~ .N “3an

 

 

 

45

show two such lens systems as they were developed for this
investigation.

As may be expected, lenses suitable for the visible portion
of the electromagnetic spectrum may not function well in the infrared
due to changes in transmission characteristics of the lens material
and variations in refractive index with wavelength (optical dispersion).
More precisely, the transmission of ordinary optical glass is down
to 10% at about 3|.L wavelength. Special glasses have been developed
which transmit longwave radiation well. Examples include fused
quartz and high-silicate glasses, calcium aluminate glass, arsenic
trisulfide glasses, and other commercially available modified
glasses. Extensive data on these glass materials and others are
given in references already cited. In general, this class of materials
can be made to function as windows, prisms and lenses up to somewhere
between 4 and 10 microns wavelength.

Due to the cost of special glasses and their reduced transmission,
other materials should be considered for lenses to be used beyond
about 3p. . The material should be transparent in the desired wave-
length range, it must have a suitable and preferably uniform
refractive index in this range, and it must be readily available, hard,
easily worked, non-toxic and stable. Among materials which meet
these requirements to varying degrees are calcite, sodium chloride,
silver chloride, magnesium oxide, barium titanate, magnesium
fluoride and zinc sulfide. The last two materials are marketed
commercially by Kodak under the trade names Irtran I and Irtran II,

respectively, and they are well suited to the manufacture of lens.

I-

 

 

 

46

More recently, a type of glass which is highly doped with selenium
has been successful for use in the extended wavelength range. This
material is known commercially as Servofrax and is a development
of Servo Corporation of America. This company also markets a
variety of lenses for use at up to 16 microns.

Another possible source of inexpensive infrared lenses is
described in detail by Gebbie and Cannon and others (21,16, 22) and
has been verified in the course of this research. The element
selenium occurs in an amorphous form which is transparent between
0. 6p, and 25p. wavelength. The refractive index at about 1 micron is
2. 5 Lenses of this material can be machined from blanks or, more
simply, cast from the melt in a pyrex mold. The mechanical
characteristics of this material are good, but they can be improved
by adding a trace of arsenic. The resulting material is more glass-
like and is useful to about 25 microns, with a deep absorption band

at 12. 7p. (35).

C. Polarizers

The most common methods of polarization of light utilize the
phenomenon of double refraction. Two such methods were described
in Section 2.1. 2 of this report. The first of these is to simply separate
two orthogonally polarized rays of light (Nicol . Prism, etc), while
the other approach is to use birefringent crystals which also absorb
one of the rays as they travel through the crystal. Polarizing filters
which are based on the latter effect, which is known as dichroism,

are presently the most common due to their extensive development,

k

 

 

 

47

low cost, large size and ready availability. Dichroic polarizers
which consist of a sheet of ' plastic carrying a multitude of properly
aligned microcrystals of dichroic material are marketed commer-
cially under the patents of the Polaroid Corporation. This company
supplies an excellent brochure (67) containing full descriptions and
samples of their polarizing materials.

Polaroid dichroic sheet polarizers were used exclusively in
the investigation reported upon here. In the visible region, type
HN32 Polaroid material in 13" diameter discs was used. These
discs were sandwiched between glass sheets and mounted in
aluminum rings which allowed calibrated rotational freedom.

Type HR Polaroid filters were employed in the infrared range
between 0. 7p. and the 2p. limit of these filters. HR polarizers are Of
the dichroic type, but they differ considerably in construction from
the types used for visible light. They are provided in glass
laminated squares (presently also unlaminated) in sizes up to 4"
maximum. Pairs in two sizes, 2" and 4", were mounted in
calibrated aluminum rotating rings and used in the infrared phases
of the research.

The axes of polarization of all polaroids used in this study
were aligned with respect to the protractor markings on the carrier
rings through use of the known zero—degree isoclinic existing in a
disc in diametral compression. This method of calibration of the
polariscope is described by Frocht (20) and others, and it is found
to be a very sensitive and accurate device. Observation of the

isoclinic in the infrared system was carried out with the aid of an

hm

 

 

48

infrared converter to be described later, and the pattern was also
photographed.

Other methods of polarization must be used at longer wave-
lengths. For the broad range between 0. 7 and 25 microns, suitable
polarizers are advertised by Hilger and Watts, Ltd. This device
is said to function on the basis of selective transmission and
reflection through films of crystalline selenium which are properly
oriented with respect to the optical path (Brewster's angle). Polarizers
which work in a similar way in a more limited range of wavelengths
can be constructed of one or more plates of dark glass or other
infrared dielectric materials such as silver chloride. The principle
upon which these polarizers act is well-known and has been often
used in the visible region when more sophisticated polarizers were
not available.

A polarizer which operates on the "picket fence” idea has
been developed by Bird and Parrish (5). This filter consists of
a. very fine wire grid, formed by vacuum deposition on a plastic
master, and it polarizes very efficiently at wavelengths from 2n
to well beyond 1511.. Mitsuishi and his associates (48) have made a
polarizer of a stack of polyethylene sheets of appropriate. thicknesses
which serves from 14p. through 200p. except for the material absorption
bands. Certain crystals of semiconductors are known to be optically
anisotropic. Possibly polarizers could be manufactured from these

materials if they are uniformly birefringent or dichroic.

 

 

 

49

D. Quarter-wave Plates

For the spectral range of wavelengths between visible red
and 2 microns, circularly polarized light was produced from linearly
polarized light by straightforward extension of the methods commonly
used in the visible range. This technique calls for the use of a sheet
of birefringent material which, when oriented at 45° to the polari-
zation axes, will split a linear polarized beam into two equal
components and retard one component with respect to the other by
1/4 of the wavelength of the light. For this purpose various materials
such as polyvinyl—alcohol, mica, or stressed glass can be used. If
the relative retardation is not exactly X/4 , then elliptically polarized
light results. Ordinarily the use of imperfect circular polarized
light does not create error in photoelastic measurements (47). If
imperfect quarter-wave plates are used in the goniometric
compensation methods of fringe order measurements, then certain
errors do arise. These will be discussed and calculated where
appropriate in this report.

Polaroid Corporation supplies sheets of retardation material
which produce relative retardations of any specified value up to
about 1/4 micron. Material producing the latter extreme of
retardation is called half-wave retardation material for the visible
region. If a sheet of the material behaved the same in near infrared
as in visible, it could be expected to serve as a k/4 plate for the
infrared range around 1 micron. This was found to be the case, and
the half-wave material was extensively used for circular polarization

in the infrared. As the investigation was extended up toward 2 microns

I»

 

 

 

50

wavelength, the error in such a plate became too large to be ignored.
For example, at 2p , the half-wave retardation material would act

as a K/8 retarder, resulting in a serious loss in system efficiency
and possible increase of second-order errors and dichroic effects.
Since the sheets seemed to transmit and retard properly at these
wavelengths, it became apparent that more than one sheet of various
retardation materials could be stacked with their optical axes aligned
in order to obtain plates which produced more nearly the required
retardation of X/4 . The error could be minimized by use of properly
chosen sheets producing different retardation. For this investigation
two sheets of the half-wave material were successfully used to
produce a X/4 retarder for the range of wavelengths between 1. 5
and 2p. .

In actual practice, 5—inch discs of the retardation material
were clamped in retaining rings with the optical axes of the material
aligned with a peg on the periphery of the ring. The plates could
then be inserted at will into the optical system by slipping them into
supporting holders which were provided with a slot to accommodate
the pegs. The supporting holders were similar to the polarizer rings
and provided complete facility for calibrated angular rotation of the
X/4 plates. The system was simple and performed efficiently up

to the 2 micron limit of transmission of the retardation material.

E. Sensors
The fundamental requirement in the measurement of

birefringence with a polariscope is to measure variations in light

 

 

 

51

intensity with changes in load, location on the birefringent plate,
time, or relative position of the optical axes of polarizer, principal
stresses, and quarter-wave retarders. This observation of
intensity can be carried out planewise, meaning the relative
retardation may be observed at every point in the plate
simultaneously, or the measurement may be made pointwise and
the doubly refracting plate scanned. The advantages of planewise
observation are obvious. Unfortunately, this technique cannot be
used except in a certain narrow range of wavelengths, and point-by-
point methods must eventually be employed. Variations of both
methods were used in the investigation under consideration. Since
they differed considerably, they will be discussed separately.

The planewise observation of data is most desirable for
photoelastic analysis. The approach is based upon observation of
the fringe pattern, that is the locus of points of zero intensity, in
an image of the stressed birefringent model. The observation may
be directly by eye, by viewing through an intermediate device such
as a screen or ground glass, or by photographing the image.

Direct visual observation of- thec fringe pattern
has the distinct advantage of immediacy, and the disadvantage of
producing no permanent record from which precise measurements
can be taken. In the visible spectrum, this observation is
accomplished by placing the field lens close to the birefringent
model, then placing the eye at the focal point of the lens, which
then acts as a magnifying glass. Another approach is to cast a real

image of the model upon a reflective screen or translucent plate.

i:-

 

 

 

52

Such simple procedures are, of course, not adequate in the infrared
range where special devices and techniques must be used for direct
observation.

One such device which makes immediate visual observation
of infrared phenomena possible is the infrared converter. This
apparatus is a refinement of the well known "snooperscope" or
"sniperscope" developed for military and law-enforcement activity.
At the heart of the device is an infrared converter tube which is
powered by a small high-voltage, low—current power supply. The
converter tube consists of a photo cathode which emits electrons
from the portions of its area which are subjected to infrared
radiation. These electrons are electrically accelerated and focused
to fall upon an electroluminescent anode screen in the same pattern
at which they left the cathode. The electroluminescent screen, of
course, converts the energy of the electrons to visible radiation,
producing a visible image of the infrared image. The image of the
infrared illuminated scene is cast upon the cathode by means of
suitable optics. The infrared converter used in this study was a
high resolution type manufactured by Varo, Inc. Its characteristics

are listed in the following table.

Table 2. 2.

Manufacturer's Performance Specifications

Varo Inc. , Model 5500C Detect-IR-scope

Spectral response 0. 4p to 1. 2p. (S—l)

Peak response 0. 8

Magnification 1:1

Resolution 5. 5 minutes minimum

Linear Distortion 8% maximum at 120 from field center
Focus range 1' to infinity

 

 

 

53

The performance of this instrument, which is of high quality, is
seriously limited for scientific work by its rather narrow spectral
response and the distortion and lack of resolution. It represents
the simplest of only a very few possibilities for infrared photo-
elastic observation, and the device proved invaluable in the present
investigation.

For accurate measurement and later reference, it is
desirable to have a permanent record of the photoelastic pattern.
This record is best obtained photographically in a certain limited
band of wavelengths. The method has long been used for photo-
elastic research in the visible range of the spectrum. For the
recording of fringe patterns in the infrared, two photographic
techniques may be proposed, and both were used to some extent
in the experimental research.

The first method used was direct photography of the image
of the birefringent model through the field lens of polariscope.
The present availability of films and camera lenses which may be
used in the infrared limits photography to about 1. 2 microns
wavelength. The broad wavelength range used here necessitated
the use of several different types of film during any one test.
These are listed in Table 2. 3 for each wavelength up to the limit
of l. 08p. . Processing of negatives was in accord with manufacturers'

specifications.

 

 

 

 

54

Table 2. 3 .

Photographic Films

x , nm film type

405 Kodak Plus -x or Panatomic -x
436 Kodak Plus -x or Panatomic -x
546 Kodak Plus -x or Panatomic -x
578 Kodak Plus -x or Panatomic -x
587 Kodak Plus -x or Panatomic -x
668 Kodak Plus -x or Panatomic -x
707 Kodak High Speed Infrared
853 Kodak IR 135

1083 Kodak Type I-Z Spectroscopic

Of particular interest is the use of Infrared Spectroscopic
Film Type I-Z which was donated by Eastman Kodak for trial. This
film is intended for use in detecting infrared radiation. It is not
commonly employed for imaging. This film proved very useful, as
it is the only film available for photography in this extended region.
The results were limited only slightly by the very slow film speed
and rather large grain size. The camera used was a Nikon F,
35 mm, with lenses chosen to match the optical system, which was
slightly different for each experiment.

The second technique for recording the fringe pattern was
to use the camera in conjunction with the infrared image converter
already described. The 35mm Nikon F camera was coupled with
the eyepiece of the converter by means of the telemicroscope adaptor
supplied for this camera. Pictures were then taken of the visible

image on the converter's luminescent screen using ordinary panchromatic

¥

 

 

 

 

 

55

films. The results were acceptable, but not as good as those obtained
by direct photography with infrared film. The lower quality was due
to the distortion and lack of resolution in the optics of the converter.
The system could be improved to a point by careful design of the
apparatus used to couple converter and camera. Such a conversion-
photography system might prove best for use above the 1. 2p. limit
of films should suitable infrared converter tubes become available.
One other planewise recording technique was tried in this
work. This involved the use of screens which luminesce visibly
when exposed to infrared radiation. The principles and preparation
of infrared photolurninescent materials are well-known (18, 19, 44)
since they received a great deal of attention in the 1940's before
converter tubes were developed to the point of usefulness. It
would seem possible to construct a screen which would convert
infrared radiation to visible by coating a ground glass, for example,
with a paint composed of powdered luminescent material in a binder.
The infrared photoelastic image could then be observed visually or
photographed with ordinary film. Another possibility which has been
successfully employed in infrared emission spectroscopy is to coat
an ordinary film plate with such a material. The material glows
when an infrared image falls on the plate, and the plate is thus
exposed. It should be mentioned that most of the luminescent
materials (as opposed to, say photoemissive materials) must first
be exited by exposure to gamma rays, x-rays, or ultraviolet up to
about 350 nm wavelength. The material may then glow, and the

infrared radiation extinguish the glow (photoquench), or the material

 

 

56

may emit visible light only when subjected to infrared radiation
after excitation.

Here, first efforts were with an infrared luminescent screen
obtained from Eastman Kodak. This device is simply a card which
seems to be impregnated with the luminescent material and then
coated with plastic. It is apparently intended for checking the
emission of near infrared lasers, which, of course, provide very
intense radiation. The intensities in the infrared polariscope were
not great enough to cause the screen to luminesce. But, the polar-
ized and unpolarized monochromatic infrared radiation would bring
about luminescence of a rather low intensity. With stronger light
sources, the simple card would perhaps serve well for visual
observation of infrared photoelastic fringe patterns. In addition to
the Kodak screen, an infrared phosphor was compounded according
to one of the recipes in one of the references mentioned (44). This
particular photoluminescent material was made by heating for one
hour a wet mixture of 20 g zinc sulfide and O. 9 g manganese chloride
and salt flux to the fusion temperature of lOOOOC in a nitrogen
atmosphere. The resulting powder, after excitation by ultraviolet,
did phosphoresce when exposed to infrared. The efficiency of the
material appeared to be of the same order as that of the Kodak
Screen. The theoretical limit of use of phosphors is about Zu
wavelength. If more efficient chemical converters could be developed,
they may prove to be the only methods of planewise observation

between the 1. 2p. photographic limit and 2p .

 

 

57

Closed circuit television provides another interesting approach
to the problem of planewise observation with longwave radiation.
Radio Corporation of America is at the time of this writing developing
a vidicon (television camera) tube which will respond to radiation of
up to 2. 8 microns wavelength. The projected cost of this .tube is
rather high in comparison to that of other data-recording apparatus,
and the associated television circuitry would have to be furnished.

If the cost of such a system could be justified in terms of the benefits
of the results obtained, this device would make possible the immediate
observation and permanent recording, on magnetic tape or photograph,
of static and dynamic photoelastic phenomena through visible and up

to the extended wavelength limit mentioned. The wavelength span
would be over eight times the visible bandwidth. The technique

should be seriously considered for future application in this and other
areas of research.

Since the practical limit of available imaging systems is at
about 1. 2 microns, investigations at longer wavelengths must employ
pointwise light sensing techniques. The sensing element around which
such a system is constructed may be a phototube, photoconductor, or
some other device which produces a signal, usually electrical, when
Subjected to a change of light flux. If such a sensor were placed at a
given point in an image of the birefringent plate in a polariscope, it
would generate a signal related to the intensity of light passing through
the point in the plate corresponding to the chosen point in the image.
This satisfies the basic requirements of photoelastic measurement.

Recent military and industrial ‘ interest has furnished impetus to the

¥

 

 

58

development of various sensing elements and systems adaptable for
use in the polariscope. Only the system developed for experimentation
plus a few suggestions for improvement can be considered here out of
the near infinity of possibilities.

As mentioned, the planar imaging techniques available for
this work were applicable up to 1. 2 microns maximum. The
investigation was to be extended up to the 2 micron limit of the
polarizers. Thus, a point sensing system for the range of wavelengths
between 1. 2|.L and 2p had to be developed. On the basis of spectral
response, availability, cost, sensitivity, and time constant, an
infrared photoconductive sensor manufactured by Infrared Industries
was chosen. This is a lead sulfide sensor with a square sensing
area measuring 0. 5 mm x 0. 5 mm. The maximum dimension of
the sensitive area of the device is «(’2 x 0. 5 mm = 0. 71 mm. If the
image of the photoelastic model being tested is n times larger
than the model itself, the sensor receives light from a point on the
model whose maximum dimension is 0.71/n mm. Even with
magnifications of only 1 or 2 times, the sensing area may be
considered to be a point relative to the other dimensions of ordinary
models.

The lead sulfide sensor is one whose resistance changeslwith
changes in intensity of incident light. A circuit which converted
this resistance change to an easily-processed voltage signal was
developed. The circuit diagram is shown in Figure 2.6. This

circuit was enclosed in a small aluminum box along with the sensor. .

I ,_ ,,
, ,

 

 

59

C onnie c ti"o,n‘s v; 1;er .. .. power and signal plus a small hole for
the admission of radiation were provided in the box, which may

be seen in the photographs of Figure 2. 3. The sensing circuit

was intended to make the output signal as independent as possible
from fluctuations in the supply voltage. To reduce electrical noise,
it was found desirable to use batteries as the bias voltage source.
The nominal bias for the sensor proper is specified by the maker to
be about 33 vdc. To reduce noise still further without decreasing
sensitivity unduly, that is, to improve the signal-to-noise ratio,
the sensor bias was reduced to values somewhat below nominal.
Various values were used, but most experimentation was carried
out with a power supply of 22. 5 v, which gave a sensor bias of about
10 v.

The output of the sensing circuit is a voltage whose variation with
practical changes in light intensity was on the order of 10-6 volts, so
extensive amplification of the signal was necessary. To allow the
required high amplification and to reduce the noise resulting from cell
and circuit components, the "lock—on” amplifying technique was used.
This method requires that the information of interest be used to
modulate a fixed frequency AC carrier signal. The amplifier can then
be tuned to the carrier frequency, and the variation in the amplitude of
that signal monitored. Noise at frequencies outside the bandpass of the
amplifier 'is, rejected. Use of a narrow bandpass filter or amplifier
results in maximum improvement in the signal/noise figure. Ordinary
amplitude modulation broadcast radio will be recognized as a special

application of this general approach to information processing.

In practice, the carrier signal in an optical sensing system is

¥

 

 

 

 

60

best created by chopping the light beam at a constant known
frequency. In a polariscope or other system-where polaroid filters
can be introduced, this chopping is best done by rotating one polarizer
with respect to the other. A sinusoidal carrier results. However,
since the measurements performed in this work involved manual
manipulation of the analyzer, a separate device for chopping the
light beam had to be introduced. This element of the optical system
consisted of an aluminum chopping blade placed normal to the optical
axis and rotated at nearly constant speed of 960 rpm by means of
a Hurst permanent-magnet hysteresis—type synchronous motor. The
blade carried a row of 20 holes near its periphery arranged so that
the width of each hole was about equal to the distance between holes.
The bhde was located at a focal point of the optical system. The
blade then functioned as a shutter, switching the light on and off
at a known frequency of about 340 cps. The time constant of the
sensor was short enough (< l u-sec) so that it responded quite
faithfully to the variations in light. Thus, if nothing further disturbed
the light beam, the output of the sensing circuit was a voltage varying
with time in a waveform closely approximating a square wave whose
frequency was 340 cps. The first and largest Fourier component
of this square wave would be a sinusoidal wave at the first harmonic
of 340 cps. This component could be isolated and amplified at will
with suitably tuned amplifiers and filters.

Now, if there was introduced into the optical path a material
Whose transparency varied at a rate considerably less than the
carrier frequency, its effect would be to change the maximum

amplitude of the chopped light beam. The output of the sensor would

g

 

 

61

then be an amplitude modulated square wave. The amplifier output
would indicate the changes in transparency of the material as
variations of amplitude of the first sinusoidal component of the
carrier. For the problem at hand, of course, the object whose
transparency changed was just the combination of polarizers,
birefringent plate and quarter-wave plates. The transparency
changes were due to birefringence in the plate. If the polariscope
system was arranged to give dark field, whole-order isochromatics
in the area of the image cov,ered by the sensor would ideally appear
at the system output as a 340 cps sine wave of zero amplitude.

The amplifier used in this study was a General Radio Co.
type no. 1232-A tuneable amplifier with gain variable up to 120 db,
and 3 db bandpass of 6% of the center frequency. The extremely
sharp peak of the gain-frequency characteristic caused some
difficulty because slight variations in chopping frequency appeared

as variations in output. This problem was minimized but not

eliminated by the motor used. Incorporated into the amplifier is

a voltmeter which indicates the rms output voltage. Essentially

then, this meter indicated light intensity for a given amplifier gain

setting. For each set of readings, the gain was set to give a

maximum output of about 0. 8 volt, the maximum output being 1 volt

before clipping of the waveform begins to occur. The output of the

SYStem was also monitored on a Hewlett-Packard Oscilloscope. This

was particularly useful in checking frequency variations and preliminary

adjustment of the system.

 

 

 

62

The pointwise technique indicates only the intensity of light
pas sed by a small area of the model. In order to observe the
ph otoelastic fringe pattern in the whole model, it would be necessary
to scan the image of the model, plotting voltage (light intensity)
ve r sus position. This could be accomplished by incorporating into
the amplifier output a demodulator which in turn could be coupled
to an oscilloscope or one axis of an x-y recorder. The position
indication could be obtained by a slide -wire potentiometer or
differential transformer in the scanning mechanism. A still greater
improvement would result from moving the model itself with respect
to the fixed optical system. Intensity variations in the optical field
then would not affect the readings. Also, a much narrower and more
intense beam and a more compact optical system could be used,
since there is no need to illuminate the whole model at once. The
loading system used provided complete facility for horizontal and
Vertical motion of the loaded model, so the optical system was
designed with this latter approach in mind.

Rather than use the sensing system to measure absolute light
intensity, results were improved by using it to indicate only relative
intensity. Relative retardations in the birefringent plate were then
measured through the use of compensation methods. In this way,
accurate determination of fractional fringe order was possible without
Concern for a known or fixed relationship between system voltage
Output, light intensity, and fringe order. The compensation method

Used will be pursued further in Section 2. 2. 3.

 

 

63

F. Optical Bench and Loading

The optical bench and loading frame used in this research were
designed and built at Michigan State University. The bench is of heavy
cast iron girder construction with facility for leveling and lateral
movement. The optical elements of the polariscope are mounted on
heavy plates which slide on the smooth rails of the bench. The
loading frame is readily adaptable to any type of loading system, and
it may be moved in its plane independently of the optical bench. The

photographs of Figure 2. 3 show clearly the construction of these

items of apparatus.
2. Z. 3. Measurement of Fractional Isochromatic Order

The equations developed in Section 2.1. 4 indicate that for
given relative retardation in a birefringent plate, the isochromatic

Order, m, becomes smaller as the wavelength of light used in its

measurement is increased. With very long wavelengths and/ or

with materials of low optical sensitivity, the exact relative

retardation at a poi

by Simple interpolation between whole- and half-order isochromatic

fringes. Under such conditions, improved measurements of relative

retardation can be obtaine

0f compensation. Accurate fractional fringe order measurement was

A number of methods of fractional fringe order measurement,

01‘ compensation, have been developed. The best known methods

1‘equire that an auxiliary device, usually a calibrated retarding plate,

 

 

 

64

be placed in the optical path adjacent to the point in the model where
the retardation is to be determined. Two other useful methods of
fractional fringe order measurement are based on rotation of one of
the polarizing elements in the optical system. These goniometric
methods have the advantages of simplicity and no requirement for
auxiliary equipment. Basically the same in principle and manipulation,
the methods are known by the names of the men who developed them.
Mindlin (47) and Jessop (36) have analyzed the errors in the two
methods, and Jessop shows that the errors inherent in the Senarmont
technique are always smaller than those of the Tardy method. (This
fact seems to be not widely recognized, and many investigators use
the least accurate method purposely with the mistaken notion that
the method involving Z-quarter wave plates is always most accurate.) ~,'
Both the Tardy and Senarmont techniques require rotation of
the analyzer of the polariscope to produce zero light intensity at the
point on the model where the partial fringe order is desired. The
methods, then, are basically l'null location'I methods, and they are
ideally suited to systems Where the presence or intensity of light
is detected electronically. In this investigation, a slight variation
0f the more accurate Senarmont method was used for calibration
measurements of longer wavelengths. Thus, it is necessary to
examine briefly the procedures and governing equations.
Let a polarized ray of monochromatic light fall upon a
birefringent plate of thickness d whose principal axes are inclined
at 450 to the polarization axis. Referring to equations (2.15), and

recalling that retardations common to both components were shown

 

 

 

65

to be irrelevant, the components of the original ray as they emerge

fr cm the plate may be written in terms of the relative retardation:

«f2 2

1T
E1 — TACOST[Z—Vt]

«f2 2 d
E2 = TAcosTTr[z—vt-(n1—n2);1:]

Now, as the two component rays enter a quarter wave plate whose
axes are at 450 to the principal axes, that is parallel and

pe rpendicular to the polarizer axis, each component will be
divided into subcomponents by the quarterwave plate, and the

re sultant components as they enter the plate will be:

V2 V2
EX — —2' E1 — 7 E2 (2. 27a)
_ V2 «[2
EV — ~2— El + —2—" E2 (2.271))

Upon traversing the plate, the rays will be retarded absolutely by a

common amount and relative to one another by Z;- . The components
leaving the plate may be described as
_ 1 Zn 2n i
EX —§A { cos T[Z - vt] - cos x z vt (n1 - n2)no]}
_ .1. 2i A 2_" d 51
Ey—2A{cos x [z vt —4] +cos A [z -vt—(n1-n2)E—-4J}
0
or,
, 2n n1 'n2
EX—-A51nTz vt 2110d]
. ZTr nl _ n2
Sln T 71-1: d] (2.283.)

 

 

n -n
E ~Asin—Z“[z—vt—l 2c1]
y A 2n
0
n 11
Zn 1 2
cos T an d] (2.2810)

The rays are then passed through a second polarizer inclined at

(g — e) from the first polarizer, that is at angle a from the

C onfiguration in which whole order isochromatics appear dark. The
polarized ray leaving the polarizer will be the sum of part of each

of the components leaving the quarter-wave plate:

E = E cos ‘9 + E sin 9- (2.29)
x V
This becomes
. 211’ nl -112
: ._ —— —— _ '9-
E A 5111': x ( 2n ) d ]
o
n — n
. Zn 1 2
sm 3:— [z —vt - Zno d] (2.30)

The retardation of the wave leaving the analyzer is thus seen to be in

the form of a single traveling wave whose intensity isrthe square of

the amplitude,

17(n -n2)d

kn
o

2 2 1
[

I:A sin — v] (2°31)

The intensity is zero when the argument of the expression vanishes
01‘ equals a multiple of 1800. Only the first value is of any

Consequence.

_ s: o (2.32)

Thus, if the various optical axes in a polariscope are properly oriented

With respect to the principal stress axes at some point in a birefringent

 

 

 

67

plate, the amount of rotation of the analyzer from the crossed

c onfiguration required to produce zero intensity at the same point
i s proportional to the fraction of a whole isochromatic order
existing at that point. The fraction, mf, of a whole order of
retardation is easily calculated.

-n

f Kno it 1800

 

(2.33)

If mW is the next highest or next lowest whole fringe, the actual

retardation at the point being considered is, in terms of fringe

order:

_e_O

180°

 

(2.34)

The direction of the principal stresses can be obtained from observation
Of the isoclinics at the point or points in the birefringent plate where
the accurate measurement of retardation is desired. The nearest
Whole order may be determined by inspection of the isochromatic
field or by watching the isochromatics develop, whichever is easier.
After insertion of the quarter~wave plate and proper alignment of
the axes of the optical elements, rotation of the analyzer until
extinction of the light occurs will indicate the fractional order. The
Sign of the correction in equation (2. 33) depends on the ”handedness”
0f the quarter-wave plates, the material of the birefringent plate,
and the direction of rotation. This is best established for each
POlariscope by preliminary tests on a known simple stress field such

as exists in a tension or compression bar.

I i

   

!

 

 

68

The pointwise light detection scheme used for the investigations
in the longwave spectral range was especially compatible with the
goniometric methods of fringe order measurement. With the polar-
i s c ope properly arranged and with the sensing device located at
the desired point in the image of the model, it would normally only
be necessary to rotate the analyzer until the output of the sensing
circuit, that is the amplitude of the sinusoidal voltage, was zero.
The difference between preliminary and final values of the analyzer

Orientation indicates the fractional fringe order. At a certain loss
01' simplicity, the procedure was changed to overcome a procedural
and system weakness and improve accuracy.

Expression (2. 31) for the light intensity as a function of the

relative retardation and inclination of the polarizer from the crossed
position may be rewritten in terms of the deviation as:

1 1T(n1-n2)d

EAZ{l-c032[ -e—]} (2.33)

I: Xno

A plot of intensity versus the angular difference of the argument is
thus a cosine curve whose amplitude varies between I = O and

I = 1/2 A2. The Senarmont null detection technique reduces to a
Search for the values of the argument at which the intensity becomes
zero. Otherwise stated, the problem is to determine the angle at
Which a sine or cosine curve reaches minimum amplitude by
Observation of the amplitude. Since the slope of such a curve is
Zero at the minimum, the problem is poorly conditioned to begin
With. The detection of minimum amplitude is further complicated

by noise in the intensity detection system. In most electronic systems,

h

 

 

including the intensity detection circuit used in this investigation, a

c e rtain small amount of random noise will appear at the output
regardless of the signal amplitude. As the signal, in this case the
light intensity, is reduced in magnitude, it will dip below the noise
level and become undetectable at values greater than zero. This
makes location of the angle at which zero intensity occurs a virtual
impossibility. The possible error in determination of the angle
may be rather large because of the shape of the curve at its
minimum. This difficulty is represented pictorially in Figure 2. 7a.
Better results can be obtained by measuring the angles on
either side of the minimum where the amplitude reaches some
arbitrarily assigned value and the slope of the curve is greater.
Since the intensity is an even function of the difference between
retardation and inclination of the polarizer, the average of the two
Values of angle thus obtained will represent the angle at which the
intensity reaches the desired minimum. The greater accuracy

results from both the averaging process and the reduction in the

 

possible error in both measurements due to lessened effect of the
noise. Reference to Figure 2. 7b will clarify this point. Additional
improvement will also result from rotating the analyzer such that
the arbitrary amplitude is approached from, say, a greater value,
80 that the indetenninancy in the measurement of the two angles due
to noise will oppose one another, and these errors will cancel in
the average.

The procedures just described were employed with .

Success in the experiments involving goniometric compensation with

x 4‘

 

 

70

the point-sensing techniques in the longwave region. In practice,
the analyzer was rotated to give maximum output of the detection
circuit, and the amplifier gain adjusted to give a meter reading
of 80. The angle at which the amplitude of the output was minimum
wa s measured as closely as noise in the system would allow. Then,
the angles on each side of the minimum value at which the amplitude
of the output reached an arbitrary value such as half of maximum,
or 40 on the meter, were measured carefully, and these two values
recorded. The average obtained was compared with the angle giving
minimum output as a check. The two measurements were usually in
good agreement, but scatter was appreciably less with the method of
aVeraging the symmetrical points.

A complete discussion of the possible error in the compensation

measurement is not feasible here. The intensity of available radiation,
the transmission of the optical elements, and the sensitivity of the
Sensor varied with wavelength. The gain of the amplifier was
adjusted to give the same output at each wavelength. The resulting
large variations in system gain meant that the noise, and therefore,
the signal-noise ratio varied with wavelengths used. Let it be
sufficient to say that the errors were reduced as much as possible
Within the limits of the system and operator judgment. The scatter
in measured data will give some indication of the failure to reduce
random errors to absolute null.

One source of systematic error that could affect results in an
unbtectable way deserves further consideration. The derivation of the

equations governing the Senarmont compensation technique rested on

 

 

71

the assumption that the quarter-wave plate used produced a relative
retardation of exactly % . It is well -known that errors in quarter-
wave plates do not ordinarily affect the positions of isochromatic
fringes in a stress or strain field (47). Still, to avoid possible
secondary effects, isochromatic fringe data for this investigation
were obtained using linear polarization. Fringe patterns obtained
with the circular polariscope were used only for auxiliary purposes.
The remaining question is how the quarter-wave plate errors
contribute to errors in compensation measurements of fractional
fringe order. The problem is especially important here, since, as
will be recalled from previous discussions, half-wave plates for the
visible spectrum were used singly or stacked to serve in the infrared
range.

Jessop, in the paper already referenced (36), presents an
analysis of errors in the Tardy and Senarmont methods of
compensation. If e is the quarter wave plate error in wavelengths,

the maximum system error (not operator error) in the latter method

will be, in terms of wavelengths:

,_..
N
N
:1

d
m n— (2.34:)

O

m rive sin—(n -n

actual ~ indicated _

The nominal retardation of the quarter wave plates used was 0. 277p. .
One of them thus served as a perfect quarter wave plate at

A = 4 x O. 277 :1.108p., and two plates would serve at X = 2. 216)!”
The maximum error would occur at a value midway between these
anelengths, where it was necessary to change from one to two

Plates. The maximum error in retardation would then occur at

I“

 

 

 

72

X = l. 662. An ideal quarter-wave plate for this wavelength would
produce a retardation of 0. 4155p” The actual plates produced
retardation of 0. 277p and 0. 554p. . The error then is 0. 4155 -

0. 277 = 0. 554 - 0. 4155 = :1: 0.1375u. The error in terms of

wavelength is, thus,

_ 0.1385 _
e—d: ———1.662 _ $0.0833

Using equation (2. 34), the maximum reading error would then equal
1/2 1r (0. 0833)2 = O. 0109 wavelengths. In other words, inaccuracies
in the quarter—wave plates used would induce into the measurement

of fractional fringe order a maximum error of 1% of one fringe. If
the total isochromatic order at the point where the fringe order is
being measured by the compensation technique had a value m, then
the maximum measurement error due to the quarter—wave plates
would be r—1n% . This source of error may be considered negligible
within the limits of the experiments involving radiation of wavelengths

between 1 and 2 microns.

 

 

3. Determinations of Optical Behavior of Materials

The experimental techniques which were developed for the
performing of photoelastic investigations in the longwave spectral
range were employed. in experiments having several distinct but
related purposes. These objectives included calibration at various
wavelengths of three resins, the definition and measurement of
dispersion of birefringence, and preliminary investigation into

the birefringence of certain semiconductor materials.

3.1. Description of Materials and Testing Procedures
3.1.1. Tensile calibration of CR -39 in visible and near infrared

Representative stress-optical properties, including
dispersion data, were obtained for the common birefringent
plastic CR-39 from planewise measurements in the spectral
range from the visible up to over one micron.

Columbia Resin CR-39, an allyl polyester resin, is probably
at present the most widely used photoelastic material. Its optical
sensitivity and transparency make it attractive for photoelastic use,
even though it suffers seriously from high mechanical and optical
creep and undesirable edge effects. The mechanical and birefringent
properties of this material have been extensively explored by several
investigators including Coolidge (12), Pindera (58, 62, 63), Clark (9)
and Pindera and Kiesling (64). These previous investigations have
consisted of calibration and creep tests using visible radiation. The
Present tests were intended to supplement existing data and provide

Preliminary indications of the optical properties of CR -39 in the

infrared range.

73

 

 

 

 

74

The particular material studied was obtained in 3/ 8" thick-
ness from Cast Optics Corporation. Tapered models were tested
in tension after the method discussed by Frocht (20), developed by
Pindera (62) and further improved by Pindera and Kiesling (64).
This calibration technique calls for the use of a wedge-shaped model
having a small wedge angle so that, under tensile loading, the stress
state at any point consists of a radial tensile stress which is easily
calculated. The gradient of stress over the length of the model
allows a tensile stress—birefringence plot to be constructed from
one set of observations on one model at any given time after loading.
A sketch of the model tested is presented in Figure 3.1 along with
a plot of the stress along the model axis and the load history.

The specimen was supported in the moveable loading frame
described previously and loaded through a deadweight loading
system. The loading lever of the system, designed and built for
the photoelasticity laboratory, was one employing hardened knife
edges at all load pivot points. The load was transmitted uniformly
to the specimen through specially designed grips which clamped
the enlarged ends of the model between serrated brass faces.

Local effects which tend to fracture brittle plastic models near
load application points were thus avoided. Each grip also provided
means of moving the load application point perpendicularly to the
load axis in order that the load could be precisely aligned with the
model axis. The dead weight of 40. 8 kg, which placed a load of
204.1 kg on the spcimen was applied gradually over a period of

about 10 minutes. This was sufficient to create a maximum tensile

x

 

 

 

75

stress of 2.126 kg/mm2 in the uniform neck of the speCirnen. The

period of recording results was expected to be more than one hour,
due to the necessity of changing apparatus for the various wave-
lengths of light used. In order to reduce the effects of creep during
the recording period, data was not taken until 21 hours and again

at 66 hours after loading. Only the results obtained at 66 hours
were evaluated. A complete calibration would, of course, measure
creep which took place throughout this time period. Data from
similar experiments by a previous investigator (63) indicate that

dm

during the recording period, the optical-creep rate, —dt_ , should
2 2

be less than 4 x 10- per hour at a stress level of 2. 2 kg/cm
for light at X = 436 nm. During the 3—hour data-recording period

of this test, then, the change of maximum isochromatic order

should be less than 0. 75%. The tests were conducted under ambient
laboratory conditions.

In order to measure relative retardation in the tapered model
for the given load and time conditions, photoelastic fringe patterns
were recorded photographically using monochromatic light at several
wavelengths spread over a sprectral range of 0. 678 microns. Chosen
on the basis of availability and convenience at the time of testing were
wavelengths of 405 nm, 436 nm, 546 nm, 578 nm, 587 nm, 668 nm,
707 nm, 853 nm, 1083 nm. The methods for obtaining monochromatic
radiation at these wavelengths have been discussed in Section 2. 2. 2
and are summarized in Table 2.1 of this report. The variety of wave-

lengths means, of course, that it was necessary to change light sources

and filters throughout the testing period. Within the limits imposed by

 

 

76

the necessity of changing light sources and other equipment the
various wavelengths of light were taken in nonsystematic order so
that the effects of creep during the recording period would at most
appear in the data as scatter, rather than as a trend over the
spectral range. Duplicate data obtained near the beginning of the
recording period and again near the end indicated that the creep
effects were well within the expected limits. The numerous filters
used were mounted in an indexed filter wheel to facilitate speedy
changes of color during the test.

The optical bench and lenses were arranged in accordance
with the discussion of Chapter 2. Figure 2. 2a shows the arrange-
ment of the system for planewise observations. The condensing
lens used in this portion of the study was a simple convex-convex
type. The small ratio of focal length to diameter and resulting high
distortion and chromatic aberration made it difficult to illuminate
the field uniformly. Some of this difficulty with chromatism was
overcome by moving the lens slightly for the different wavelengths.
The field lens utilized was of plano-convex type with focal-length to
diameter ratio of about 3. This simple lens, while of better quality
than the condensing lens, created enough distortion so that some
difficulties were encountered in the analysis of results.

Since both visible light and infrared radiation were employed,
two different sets of polaroid and quarter-wave plates were required,
and these also had to be switched during the data-recording period.
The polarizers and quarter-wave plates used have already been

described. Those for the visible radiation were of 13" diameter.

 
 

 

77

The 4" square size of the largest available infrared polarizers (HR
sheet, glass-mounted) made it necessary to cover the rather large
CR -39 model in segments, as will be described shortly. One layer
of half-wave retardation material for the visible region was used for
each of the two quarter-wave plates for the infrared range between
0. 7 and l. 08 micron. Permanent recording of the photoelastic
patterns in the loaded CR -39 model was accomplished photographically.
Auxiliary observation in the portions of the test involving nonvisible
light was possible through the use of the infrared converter. Both
camera and converter were located at or near the focal point of the
field lens. The location of this focus varied considerably over the
extreme range of wavelengths due to chromatic dispersion in the lens,
so the camera and converter had to be moved slightly with changes .in
wavelength to previously determined focal points along the optical
axis. This was required to establish uniform illumination of the
field and also to ensure that only rays passing normal to the plane of
the model were used in constructing the image of the model on the
film or converter cathode screen.

A 35 mm Nikon F camera with lens chosen to match the
other optical elements was used in the recording of results. With
a lens of 55 mm focal length, maximum image size was attained on
the film when the camera was placed at the focus of the field lens and
the field lens placed as little distance as possible, about 6", from
the model. This camera and its lens, while of excellent optical
quality, was also subject to the effects of chromatic dispersion. A

special focusing mark is provided on the lens barrel for focusing with

gm

 

 

 

 

78

near infrared radiation at about 0. 8 micron. For focusing at l. 08
micron, another index mark was established that provided a
correction in focus twice that prescribed for 0. 8 microns. The
various films required for photography over the extended spectral
range have been summarized in Table 2. 3 of Section 2. 2. 2. The
film had to be changed along with polarizers, light sources, and
filters in the course of the recording period. In order to change
film, the camera had to be dismounted, the film changed in a light-
tight bag, and the camera remounted, aligned, and focused at
considerable expense of time. The camera aperture setting and
shutter speeds depended on film, wavelength and optical system
configuration. Nothing will be added by tabulating these here, but
they varied between £22 at 1/250 sec for A = 435 nm and f8 at 30
sec for X = 1083 nm. Processing of the negatives was in accordance
with manufacturer‘s instructions, with preference given to the
methods giving highest contrast and smallest grain size.

The four-inch maximum field in the infrared system required
that the photoelastic pattern in the large tapered CR -39 model be
photographed in segments. To accomplish this, a pointerwas attached
to the loading frame which could be raised and lowered electrically
with respect to the optical bench. Index lines were placed on a station-
ary scale attached to the base of the loading frame. These lines were
placed so that when the pointer was aligned with them, the loading
frame and model would be properly located with about 4” of the model
in the polariscope field. In this way, slightly more than 1/3 of the

model could be photographed at one time. By raising and lowering

 

 

79

the frame and model, the segments could be photographed in order.
Enough overlap existed between the segments to allow the photo-
graphs to be properly matched. The matching process was facilitated
by scratching gage lines on the surface of the specimen. Light and
dark field pictures using both linearly and circularly polarized light
were taken at each position on the model with one wavelength before
changing wavelength and repeating the cyclic photography of the
segments of the model. A total of 108 exposures was required to
complete the test. Representative photographs of the isochromatic
pattern in the head end of the model with radiation of 546 nm and 1083
nm are presented in Figure 3. 2a and 3. 2b. These patterns were
obtained with linear polarization oriented at 450 to the axis of the
specimen. The 450 isoclinic is readily apparent in the fillet above
the uniform neck of the model.
3.1. 2. Tests of Palatal P-6

Certain aspects of the mechanical-optical properties of the
polyester resin Palatal P—6 were explored photoelastically in the
visible and near infrared spectral range. The birefringent properties
of this material in the visible range have been investigated by the
authors previously cited in connection with their work with CR -3 9.
These researchers found that P—6, which is not commonly used for
photoelastic studies in the United States, possessed certain desirable
characteristics including good light transmission, high optical
sensitivity, and relative freedom from time -edge effects.

The material used in this study was donated by Badische Anilin

and Soda-Fabrik (Germany) in the form of raw liquid resin. Sheets

 

80

of the polyester were formed by casting between glass plates a mixture
of 100 parts fluid resin, 2 parts catalyst, and 0. 05 parts accelerator.
After hardening, the sheets were removed from the forms, aged 3
months, and then cured at 900C for 40 hours. The heating and cooling
rate of the curing cycle was set at a nearly uniform rate of SOC/hr.
The models were then machined from these sheets using standard

methods.

Two different models were cut from different sheets of the
P-6. Sketches of the models tested and their load histories appear
in Figures 3. 3 and 3. 5. The circular ring having an outer diameter
of 80 mm, inner diameter of 44 mm and thickness of 6.10 mm was
tested in diameteral compression early in the course of the
investigation. This particular test was directed towards a prelim-
inary measurement of the dispersion characteristics of P-6, so the
results are not as comprehensive as those obtained from the other
experiments. The load of 320 pounds was applied to the speCimen
by means of a dead-weight lever system in a stationary frame. The
load on the specimen was maintained at ambient laboratory conditions
for 50 hours, at which time the birefringent pattern was recorded
photographically over a period of 2. 5 hours.

The optical system used in this first test of P-6 was essentially
the same as that used in the test of CR —39 already described. Rather
than the optical bench, the polariscope elements were supported on
threaded rods mounted in drilled aluminum plates. The plates were
themselves mounted on pedestals consisting of a telescoping aluminum

column on a heavy steel base.

 

81

The photoelastic pattern in the P~6 ring was photographed
with radiation of 6 different wavelengths. The methods described
earlier were used to produce light having wavelengths of 405 nm,
436 nm, 546 nm, 578 nm, 853 nm, and 1083 nm. Photographic
procedures were identical to those used in the tensile test of
CR -3 9. Representative photographs may be seen in Figure 3. 4.

The second model of Palatal P—6 resin tested was a beam
300 mm long by 38 mm deep and 6. 05 mm thick. The model was
loaded in pure bending with forces of 5. 44 kg applied at each of the
third points. The model was supported in the moveable loading
frame used in the test of CR -3 9, and loaded through cylindrical
pins which were held in place by vee-blocks. The lever arm
supported on hardened knife edges was used to transfer the dead-
weight load to the lower support of the model. The load was applied
over a period of about 10 minutes. Recording of the data took place
at 49 hours after loading. From the results obtained in the visible
spectral region by the investigators already mentioned, the rate of
optical creep of P-6 at the maximum stress level was —— < 2 x 10—
per hour for )x = 435 nm. Thus, during the time of recording data,
the change of maximum isochromatic order was less than 0. 3%. As
with the test of CR =3 9, duplicate pictures taken at one wavelength
early and late in the test indicated this estimate of the creep effects
to be conservatively large.

The optical system used in this test of the P—6 beam was the
same as that employed in the tensile test of CR -3 9, with the exception

of the collimating lens and the field lens. The high distortion evident

 

82

in the lenses used in the previous tests indicated a need for higher
quality optics. Matched convex-convex lenses of 33 cm diameter
and l m focal length became available and were used in this
experiment involving P-6. The large span of wavelengths used
and the accompanying chromatic dispersion in the lens still
necessitated the moving of the recording apparatus to the focal
points for each wavelength, but the distortion in the image of the
model and the uneven field illumination were effectively eliminated.
The birefringent patterns in the bent beam were recorded
photographically with illumination having wavelengths of 405 nm,
436 nm, 546 nm, 578 nm, 587 nm, 668 nm, 707 nm, 853 nm, and
1083 nm. Auxiliary observation at the longer wavelengths was
possible with the infrared converter. The techniques and experi-
mental procedures were the same as those already described.
Since a longer-focal‘length field lens was used, it was necessary to
use a camera lens having a focal length of 85 mm. This optical
system allowed photography of the full length of the beam with
visible radiation. In the infrared, only the central portion of the
beam could be photographed due to the smaller size of the infrared
polarizers. The negatives obtained at )x = 1083 with spectroscopic
film were somewhat underexposed, and it was necessary to use a
mercury intensifier to increase negative density to the point where
good measurements could be obtained. Otherwise, processing of
the films was as before. Pictures were obtained with both circular
and linear polarization at each wavelength. Figures 3. 6a - 3. 6c

show the isochromatic pattern in the beam at )x = 546 nm with

 

83

circular polarization, and at A = 450 nm and X = 668 nm with linear
polarization. The dimensions of the beam and the large load used
suggest the possibility that the beam may have buckled slightly during
the test,’ thus changing the moment and stress distribution. The
symmetry of the isochromatic patterns obtained with circularly -
polarized light and the fact that the central'isochromatics form
closed loops indicate that buckling definitely did not occur.
3.1. 3. Birefringence in Polycarbonate between 1 and 2 microns
Polycarbonate resin possesses a number of characteristics
which make it attractive for use as a photoelastic and photoplastic
model material. . Ito (33) summarizes the basic properties of this
material and proposes its use in studies involving elastic and
plastic deformation. The optical sensitivity of the resin is higher
than that of other photoelastic materials, and the same is true of its
modulus of elasticity. Toughness and impact strength are also
higher than that of linear polymers. Of particular interest is the
fact that polycarbonate can be severely plastically deformed, even
in the cold state, Without affecting either strength of transparency.
Polycarbonate seemed especially attractive for this

investigation, because, unlike several other common plastics, it

was highly transparent (above ~ 75%) in the spectral region in which

the study was to be carried out.

The model tested was a disc of 78. 5 mm diameter and 4. 26 mm

thickness which was cut from a cylindrical billet of ”Lexan", one of

the Commercial-names applied to polycarbonate of 4—4—dioxpheny1

Pl‘Opane and phosgen. The model and its load history are given in

 

x

 

 

 

84

Figure 3. 7. Both sides of the rough plate sawed from the billet
were faced in a lathe after turning the outer diameter. The disc
was then polished as flat as possible on metallographic polishing
plates using three grades of abrasive paper. Reasonably good
optical surfaces were then produced by lapping the plate on rotating
metallograph laps using linen and felt laps with 500 and 320 levigated
alumina as the lapping medium. Some large scratches remained in
the surfaces, but the small scratches were removed and the faces
of the model did not tend to diffuse light. Observation of the model
in a visible light polariscope indicated a residual stress pattern in
the outer 1 cm of the disc which was probably caused by rapid
cooling of the extruded billet. The residual stress pattern near
the center of the model was not great. Since it did not interfere
with the experimentation, no attempt was made to relieve the
residual stresses by annealing.

The polycarbonate disc was loaded in diametral compression
using an arrangement very similar to that used in loading the P-6
beam. Common procedure for calibrating a material by testing a
disc in compression is to load the disc, wait the desired time,
then record the fringe pattern or observe directly the birefringence
at various points on the model where the stress can be easily
calculated. Usually points on the horizontal and vertial diameters
are chosen. A stress—fringe plot can then be constructed from the
calculated stress spectrum and the observed birefringence. In
this quantitative application of the pointwise sensing technique, the

scanning procedure was not used, and the test was conducted in a

 

85

manner similar to the ordinary tensile test. That is, the relative
retardation at a given point on the model was observed as the load
was increased. Since the measuring and recording of fringe order
for several different wavelengths at each load required about 0. 5 hr.
creep effects could lead to serious error. The effects of creep or
the results can be minimized in a test of this sort by applying an
increment of load, then waiting a specified time before recording
results. If the time periods between recording birefringence at a
given wavelength are always identical and the stresses were within
the linear limit for the material, the deviations would be null. In
this case, the stress-birefringence curve for each wavelength
would correspond to a different testing time. If, on the other hand,
the time periods Were kept uniform and the several wavelengths
taken in nonsystematic order, the experimental points would deviate
from a smooth curve by no more than the amount of creep occurring
during the recording period. This deviation could be reduced to
near zero by extending the interval between increasing the load and
beginning to measure birefringence. In any case, the smooth
curves through the experimental points, which may be scattered
some amount by creep effects, should represent the stress~retardation
characteristic at each wavelength for the same time. If the material
is not ”momentarily-linear-elastic", the situation is more complex,
but still meaningful.

The last described approach was used in the test under discussion.
An increment of load was applied, the measurements of fringe order were

begun after a period of 1 hour. The total interval between successive

 

 

86

increases of load was thus about 1. 5 hours. There were some slight
deviations from this schedule in the early phases of the experiment,

as may be observed by reference to Figure 3. 7. Maximum load on the
specimen was limited to 108. 8 kg due to difficulty with the loading
apparatus. The resulting stress levels near the center of the disc
were apparently within the linear limit, but very severe local
deformation did take place near the pins through which the load was
applied to the model. Load increments varied between 6. 45 and

2.27 kg (1 to 5 lbs), and total testing time was 12. 5 hours.

The optical system used in this investigation of the birefringence
in polycarbonate with infrared radiation between hi and 2p. wavelengths
has been described in Chapter 2 and pictured in Figures 2. 2 and 2. 3.
The general techniques have also been discussed. After aligning the
center of the disc with the optic axis of the polariscope, the 55 mm
diameter achromatic field lens was adjusted to focus a magnified
image of the model in the plane of the infrared sensor. Reasonably
high light intensity was maintained by using an image which was 2. 5
times larger than the object. Since the sensor had a sensing area of
0. 5 mm square, the largest dimension of the area on the model over

which the intensity was averaged by the sensor was 3: x l. 4 =

 

0. 28 mm. The sensor was carefully aligned with the center of the
image of the model as indicated by diametral lines scratched on the
model surface. These lines were carried close to, but not through,
the center of the model so as not to interfere with observation. This
preliminary adjusting and focusing was done with visible radiation and

checked in near infrared with the aid of the infrared converter. The

 

87

correction in positioning between visible (650 nm) and near infrared
(lu) was almost undiscernible, due no doubt to the use of small
diameter achromatic lenses in the optical system. For assurance,
the small adjustment in focus necessary between visible and near
infrared was doubled to establish the approximate correction for the
spectral region beyond 1 micron where visual observation was not
feasible. Since imaging was not required, small errors in focus
were of not great consequence. Laboratory temperature during the
test was 200 C.

Relative retardation in the disc of polycarbonate was measured
using radiation of eight different wavelengths. Chosen at the time of
the experiment because of availability of filters, response of the
sensor, and transmission of the optical system were wavelengths of
1.145|J.,1.244p.,1.348H,1.446p.,1. 550u,1. 56p, 1.650u, 1.90u.
After applying each increment of load and waiting the specified time,
the fractional isochromatic order existing in the center of the model
with radiation of each of the eight wavelengths was measured using
the detection and goniometric compensation methods outlined in
Section 2. 2. 3. The nearest whole—order isochromatic was determined
by observing the changes in light intensity for one or another of the
wavelengths as indicated by the output of the detection circuit while
the load was being increased.

The spectral transmittance of polycarbonate in the 0. 85 to
7. 5p. range was obtained through the use of the Perkin-Elmer Model

l37-G Spectrophotometer in its normal mode.

 

88

3.1. 4. Preliminary Investigations of Birefringence of Semiconductors

Inquiry into the possibility of using materials which are not
transparent to visible light for photoelastic studies using infrared
radiation was begun by investigating the birefringent properties of
two semiconductor materials. These two materials, selenium and
silicon, were chosen because of their known transparency to infrared
light and their similarity to common, brittle metallic structural
materials. Also, these semiconductors have been shown to be
birefringent to some limited extent by authors already mentioned
(3,13, 23, 24, 30, 34, 68).

The element selenium exists in several forms including two
crystalline types and the amorphous form, which is of particular
interest for optical applications. The latter type is easily produced
by quenching the molten element to its freezing point. Its structure
is similar to that of amorphous sulfur , as might be expected because
-of their relation in the periodic chart. In the melt, selenium exists
as a fluid mixture of entangled chains of atoms. When quenched,
this entangled chain structure is maintained, although the chains may
take up some degree of preferred orientation with respect one one
another (31, 32). The result is a solid which seems to bear some
similarity to the chain structure of the polymers, but which also
possesses some of the aspects of metallic crystalline forms.
Amorphous selenium is known to transmit radiation between 0. 6p.
(deep visible red) to 25p. wavelength, with absorption bands at 13. 5
and 20. 5p. (35, 54). The material is quite stable, somewhat brittle,

and the frozen form does not soften or begin to transform to the

 

 

89

crystalline until 500 C. As mentioned, birefringence caused by
internal stresses created during crystallization has been observed.

For this experiment, discs of amorphous selenium having
50 mm diameter and 6. 3 mm thickness were produced by casting.

The mold consisted of a partial ring of aluminum which was
sandwiched between pyrex plates. The molten selenium was poured
into the mold through the opening in the ring and the whole assembly
immersed in cold water to freeze the selenium. After solidification,
the disc was separated from the mold and the sprue broken off. In
this way good optical surfaces were produced, but cavitation in the
disc due to shrinkage created considerable difficulty.

The disc model of selenium was supported and loaded in the
same way as was the polycarbonate disc. The optical system was
similar to that used for planewise test of CR -39. Observations were
visual by means of the infrared converter at 0. 850 microns radiation
wavelength.

Silicon is a hard and brittle material which belongs to the same
periodic group as selenium. It transmits radiation having wavelengths
between about 1.1 and 25p . A disc of this material having a diameter
of 25 mm and thickness of 0. 914 mm was loaded in diametral
compression and the stress birefringence observed and measured.
Since silicon transmits radiation whose wavelengths are within the
sensitivity range of the infrared converter, this instrument was used
to observe visually the photoelastic effects. Photography of the
isoclinic and isochromatic fringes could also be accomplished with

the c onverter.

 

 

90

The point sensing and compensation techniques were employed
in the quantitative measurement of the stress-birefringence of silicon.
The apparatus, stress calculations, and most experimental procedures
were identical to those of the test of polycarbonate. In this case, the
retardation was measured immediately following the application of
an increment of load. The possibility of buckling the thin specimen
limited the range of load which could be used. The small thickness
of the specimen also caused the fringe order for the maximum load
to be on the order of 0.1. Recalling the previous calculation of error
in the compensation method, the possible error in measurement of
the fringe order in the silicon disc might be as high as 10%.

In keeping with the organization of this report, the observations
and measurements of birefringence in selenium and silicon are
presented at the end of the following section. Various aspects of
this and the other phases of the investigation are discussed further

in Chapter 4.

3. 2. Treatment of Data and Presentation of Results
3. 2.1. Calibration of P—6 and CR .39

The stress-birefringence properties and dispersion of
birefringence characteristics of P—6 and CR -39 for the near infrared
and visible spectral range were obtained from the photographed
isochromatic pattern through a simple but refined graphical procedure.
First, the locations of the whole and half-order is ochromatic fringes
along the chosen cross section of each model were determined by

measurement directly on the 35 mm negatives. A two-dimensional

 

91

measuring microscope manufactured by W. G. Pye and Co. Ltd.
(England) was used for this measurement. The highest magnification,
20 x, was used and the centers of the isochromatic bands were generally
well defined when the negatives were viewed through the microscope.
Accuracy of the instrument is specified by the maker to be :1: . 01 mm
when moved over its full range of 20 cm laterally and 10 cm
longitudinally. The accuracy is said to be proportionately greater

in relation to the smaller amount of traverse. Duplication of measure;
ment on about one -fourth of the negatives indicated a minimum
repeatability of measurement of :l: . 01 mm. Assuming the accuracy

of the device is an order of magnitude greater for the small 35 mm
negative, the repeatability in the more subjective process of locating
the center of the isochromatics may be taken as the extreme measure-
ment error. The value given above was more than sufficient for this
work.

The negatives were illuminated during measurement by
supporting the microscope over a fluorescent lamp, and usually a
green Kodak filter was placed directly under the negative. Both
whole and half order fringes were not located on all negatives. Where
the light isochromatic on the negative (whole order for dark field, etc.)
was very sharply defined and the fringes were closely spaced, only
that isochromatic was located. Where the fringes were broad due to
underexposure or other causes, both half and whole order were
located by measurement, and often several observations were made

and the results checked for consistency and then averaged.

 

92

To reduce the effects of chromatic dispersion and distortion
in the polariscope lens system and unequal shrinkage in the variety
of films used, the measurements from the negatives were normalized
on the basis of the measured distance between gage lines which had
been scribed on the specimens. These gage lines usually showed on
the negatives, otherwise the extreme dimensions of the measured
cross section of the model were used as the basis of the correction
factor. The poorer lens used in the tests of the P-6 ring and the
tapered CR —39 model introduced certain distortions which could not
be completely nulled by linear magntification factors. These errors
became apparent as consistent opposing deviations in the plotted
results, and they were minimized by the graphical processing of
the data.

Measurements of the isochromatic order in the tapered CR —39
model were taken along the longitudinal center line. For the P—6 ring,
a horizontal radius was chosen as the cross section across which
results were to be evaluated. The central cross section of the P-6
beam was selected for measurement. The chosen sections are
indicated by heavy lines in the sketches of the various models
presented in Figures 3.1, 3. 3 and 3. 5. It is worth mentioning again
that measurements used for quantitative analysis were taken only
from the pictures obtained with linearly-polarized radiation. A
representative set of measurements and corrected data may be seen
in Appendix A.

From the corrected measurements of the location of the iso-

chromatics, plots of fringe order versus location on the cross section

 

93

were constructed for each model. The different curves corresponding
to the various wavelengths used in any one test were plotted on the
same sheet. Greatly reduced copies of some of these curves for
the CR -39 model appear in Figure 3. 8; those for the P-6 beam in
Figure 3. 9; and plots for the P-6 ring are shown in Figure 3.10.
The original working plots were made in the largest practicable
size using 24 in. x 36 in. graph paper for maximum resolution and
accuracy. A smooth curve was drawn through the experimental
points for each wavelength. In general, the deviations of the
experimental points from the smooth curves were so small as to be
insignificant with the scaling used. This reflects the consistent good
results which might be expected from the technique of carefully
measuring negatives with an instrument capable of high accuracy.
From the data and curves of fringe order versus location on
the cross section, plots of relative retardation per unit thickness
versus location were constructed by multiplying fringe order by the
appropriate wavelength. Due to the smoothness and consistency in
the measured and corrected data, this was used in developing the
retardation curves, rather than using points read from the fringe
order curves as would normally be the reason for employing the
smoothing graphical technique. Essentially, this conversion from
a plot of isochromatic order to one of retardation per unit thickness
consists of introducing scale factors into the plots of isochromatic
order at each wavelength in order to put them on a common basis.
Dispersion of birefringence becomes apparent at this point, since

the retardation plots for different radiation would be concurrent if

— ‘

 

 

94

double refraction was independent of wavelength. The actual measure-
ment of dispersion is discussed fully in later paragraphs.

The original working graphs of relative retardation per unit
thickness versus location on the cross section of the model for
constant wavelength of radiation were all made to large scale on
24" x 36" paper. Reductions of representative curves are given in
Figures 3.11, 3.12 and 3.13 for each of the three models. Experi-
mental points are not shown as they would only contribute a degree
of confusion. However, all observed data, if shown, would lie
within the width of the line used in drawing the smooth curves.

The stress along the longitudinal centerline of the tapered
CR—39 model was calculated from the well known elasticity solution
for a wedge with a concentrated force at the apex (76). The dimensions
of the model were determined with the measuring microscope. These
measured dimensions and the computation of stress are given in

Appendix B. The results are plotted along with fringe order per

unit thickness in Figure 3. 8. The stress gradiation in the region of

the fillet between the uniform neck of the model and the tapered portion
was estimated by draWing a smooth curve between the stress level in
the neck and the last point on the plot of the stress in the wedge where
the is ochromatic pattern indicated the theoretical solution justified.

This portion of the graph and portions on succeeding plots derived

from this smooth approximation are shown as dotted lines because of

the decreased confidence in the evaluation of the stress in the fillet.

From the master diagrams, plots of fringe order per unit

thickness and relative retardation per unit thickness versus stress

 

 

 

95

for constant wavelength were constructed. These are the calibration
curves for CR -39 for the conditions of the test. Again, the original
graphs were drawn to large scale on 24" x 36" paper. Diminished
reproductions of the curves may be seen in Figure 3.14.

The curvature of the P-6 beam, which is apparent in the
presented photographs, and the obvious shift of the zero order iso-
chromatic plus variations in spacing of the isochromatics indicate
that the stress in the beam cannot be calculated by ordinary beam
theory and/ or the strain-optic or stress-optic relationship is not
linear over the range of stress and strain existing in the beam.
Previous reports indicate that for the given stress range and other
conditions of the test, the relative retardation-stress relationship
for cured P-6 is linear (63). That is, the stress is lower than the
linear limit and the material is behaving optically in a "momentarily
linearly elastic" manner. If the stress-strain relationship could
also be assumed to be momentarily linear, then exact or approximate
elasticity solutions could be used for computing the stress in the
curved beam.

In order to determine the radius of curvature of the deformed
beam, the coordinates of three points on the curved centerline scratched
on the model were determined from the negatives of the isochromatic
photographs using the measuring microscope. Since the central
portion of the beam was in uniform bending, the elastic curve should
be a circular arc. The three known coordinates were sufficient to
establish the location of the center of curvature and the radius of

curvature. The value of the latter was 139 cm.

 

 

96

Knowing the radius of curvature, the bending moment, and
the dimensions of the beam, the elastic bending stress could be
calculated by a variety of methods. The Winkler—Bach Solution (73)
for a curved beam is poorly conditioned for use with beams of
shallow curvature, even though the basic assumption that cross
sections of the beam remain plane is completely justified by the
symmetry of the particular problem. The exact elasticity solution (1 4)
is rather cumbersome. In view of the probable accuracy of the measure-
ment of radius of curvature, it is doubtful that the increased labor is
justified by the additional accuracy. An approximate solution which
is developed by assuming a parabolic strain distribution and then
applying least work principles is shown by Den Hartog (14) to provide
accuracy superior to the Winkler-Bach solution for certain problems.
Computational effort is minimum and the stress in a beam of large
radius of curvature is easily calculated without the appearance of
large errors due to minimal errors in the value of the curvature.
The beam curvature measurements, sample calculations of stress
by the approximate elastic solution, and an abbreviated table of
values appears in Appendix C. As is apparent in the solution, the
curved-beam correction terms amount to only about 1. 5% of the
total stress which would result from ordinary straight-beam theory
for this problem. This is slightly more than the accepted maximum
correction from the Winkler—Bach analysis of this beam. The computed
stress is shown in Figure 3.12 along with retardation per unit thickness
as a function of distance from the centroid of the cross section. The

presented plot is a reduction of the 24H x 36" original. Even in the

 

 

97

reduced graph, the differences between the fringe-order plot and the
stress plot are obvious. The shape is different and the point of zero
stress does not correspond to the location of zero retardation. This
suggests immediately a possible lack of clear correlation between
retardation and stress and/or faulty calculation of the stress.
Proceeding further with the intent of exploring the stress —
retardation relationship, a plot of these two quantities was constructed.
This graph, presented in Figure 3.15, shows the values of fringe
order per unit thickness at various distances from the beam axis as
obtained from the graph of Figure 3.12 versus the stresses computed
for corresponding locations on the cross section. The original plot
was in the larger size, and the figure is a 1/3 reduction which includes
only 3 of the 9 wavelengths. Points from the compression side of the
beam were plotted separately from points on the tension side. Smooth
curves could be drawn through the two sets of points separately. Both
plots for each wavelength were curved over their whole length. The
curvatures of the two were opposite and the two curves consistently
crossed at about 2 kg/mmz. The divergence of the curves above and
below the crossing point was too great to be ignored. As suspected
from qualitative examination of the isochromatic patterns, the
experimental deviation could have been decreased only if the curvature
correction had been greater. That is, the curves of m versus stress
as calculated from an elasticity solution for the two halves of the beam

would have been in close agreement had the radius of curvature been

as small as about 15 cm, or one ~tenth the measured value.

 

 

 

98

The inclusion of the small radial stress which is required for
the exact solution would tend to bring the curves closer for part of
their length, but would cause the remaining portions to drift further
apart. This component of the stress involves only terms containing
the natural logarithm of the ratio of two distances from the center of
curvature to points on the beam. For the very shallow curvature of
the curved beam, the radial stress must be very small. Also, this
stress approaches zero in the outer fibers where the divergence of
the measured data is greatest. Similarly, the imposition of tensile
loading by the loading apparatus would have an effect opposite to that
observed. The uniformity of the isochromatics in the central region
of the beam indicated that this situation did not exist anyway. Nor
can the variations be due to thickness variations. The slight variation
in thickness (. 002) of the cross section of the beam was almost entirely
compensated by the lateral expansion during deformation. Measurement
of the deformed cross section showed that the effects of thickness
variation were less than 0.1%.

It must be concluded that the difference between the stress-
optic curves derived from opposite sides of the P—6 beam is due to
nonlinear relationships between stress, strain, and relative retardation.
The symmetry of the problem implies that the central cross section
remains plane, meaning that strain is proportional to distance from
the centroidal axis within the limits of the small correction due to
geometry change. Thus, the stress-strain-optic curves for the
material and the conditions of the test must be quite complex. In
particular, the properties of the material in tension must be different

from those for compression. The subsequent redistribution of stress

 

. ! (:9 Hi. Ova-Plainti-

Elna-

 

99

and, separately, of birefringence invalidates any computation of stress
based on elasticity theory, approximate or exact. This conclusion is
supported by the dispersion of birefringence exhibited by this material.
Verification of the result must await testing in simple tension and
simple compression where the computation of stress is not ill-founded
upon faulty assumptions. Such results are missing in the literature.

As previously mentioned, the test of the ring of P—6 was the
first in this series of experiments to extend planewise photoelastic
investigation into the near infrared. The conditions of the test were
less carefully controlled, especially as regards load, than were the
later tests which have already been discussed. The plots of fringe
order per unit thickness and relative retardation per unit thickness
as a function of position on a horizontal radius have been presented
in Figures 3. 10 and 3.13. These results indicate that, with proper
calibration, photoelastic analysis of complex stress fields can be
carried out with infrared radiation of up to lu wavelength. Since the
stress-optical coefficients for the longer wavelengths and for the time
of recording data in the test of the ring are lacking, it is not practicable
to develop an experimental analysis of the stress distribution using
infrared radiation for comparison with existing experimental solutions
obtained with visible light. The similarity of the is ochromatic
distributions obtained with both visible and infrared radiation to each
other and to those presented by Frocht (20) indicate qualitatively the
correctness of the procedure.

Neither is it feasible to develop from the test of the ring stress-

optic data which are within the limits of accuracy of the other tests of

 

 

100

this investigation. Solutions giving the tangential stresses in a ring
under diametral compression are well known (76), but there appears
to be no convenient analytical solution for the radial stress. Both
must be known in order to correlate their difference with the
birefringence in the ring. The problem is further complicated by
the probability that the stresses near the inner boundary of the ring
were greater than the linear limit for the material and time of the
test. The resulting redistribution of stress would seem to make
difficult, if not impossible, the accurate analytical determination of
stress.

In addition to providing graphical evidence of what may be
expected from infrared photoelastic investigation of complicated
stress fields, the test of the P-6 ring provided additional interesting
information on the dispersion of birefringence of this material.

3. 2. 2. Calibration of Polycarbonate

The data from the test of the disc of polycarbonate are
presented in Figure 3.16 as a plot of fringe order per centimeter
thickness versus corrected stress for constant wavelengths and the
particular load path used. The figure is a one -quarter reduction of
the larger original. As will be recalled, this stress—optic information
reported here is for the geometric center of the disc which was in
diametral compression. The principal stresses at this point were
calculated from the standard analytical solution (20, 76), and their
algebraic difference taken as the independent variable. This
calculation is given in Appendix D. The isochromatic order was the

sum of the observed whole-order fringe plus the fractional order as

 

 

101

determined by the special goniometric compensation method developed
for this investigation. The conversion from analyzer rotation and
nearest whole fringe order observation to fractional orders of iso—
chromatics is illustrated in the chart of Appendix E. Due to the
presence of residual stress, residual birefringence, and the very
small but undetermined initial load, the curves of fringe order versus
theoretical stress did not pass through the origin. Rather, they
converged at a point corresponding to zero fringe order at a stress
difference of - . 016 kg/mmz. The data was subsequently corrected
by adding this value to the calculated difference between principal
stresses, and later results were based on these corrected values.
Smooth curves which best fitted the experimental data proved to be
straight lines, as might be expected from the rather low level of
applied stress and the short waiting periods.

Plots of relative retardation per unit thickness versus stress
for the various wavelengths between 1p. and 2p. were constructed from
the plots of fringe order per unit thickness. Since the m-O' curves
were linear, they could easily be transformed to retardation plots.
The resulting graphs of retardation per unit thickness versus stress
for constant wavelength were linear, of course, and their separation
from one another indicates the dispersion of birefringence for poly-
carbonate in the given wavelength range. Reductions of typical of
the original 24“ x 36" plots of relative retardation versus stress may
be seen in Figure 3.17.

The plot of the transmittance of this particular specimen of

polycarbonate for the 0. 85 - 2. Su region is given in Figure 3.18.

 

 

102

Additional measurements which were taken for the 2. 5 - 7. 5H range of
wavelengths indicated the material is not transparent there. This
data indicated the range of wavelengths which could be used. Evidently
such observationiis limited to the visible through 1. 6H and l. 7 -
2. Lu regions, although photoelastic data from the narrow passband
around 2. 2/4 might prove especially interesting. In addition to the
practical function of indicating the range of usefulness of poly-
carbonate, the transmittance curve also gave preliminary indication
of possible relationships between material structure and such
phenomena as piezobirefringence and dispersion of birefringence.
3. 2. 3. Dispersion of Birefringence of P—6, CR —3 9, and Polycarbonate
The relationships between birefringence, wavelength or frequency
of radiation, and stress which were derived from the photoelastic tests
of CR -3 9, P-6-and polycarbonate resins made possible the investigation
of the dependence of the double refraction of these materials upon the
wavelength of light. This phenomenon, which is called dispersion of
birefringence in analogy to a similar occurrence in optics, was
described and defined previously in this report. Previous experimental
results and analytical descriptions have also been discussed. In view
of recent progress, these past methods of description seemed partially
inadequate. New quantitative measures of photoelastic dispersion
were developed, and these proposed methods of description are given
following a discussion of the requirements which must be met. The
results of the experiments are presented in accordance with the
developed methods following a discussion of the relationships between
the various photoelastic parameters and the quantities involved in

dispersion measurements.

 

 

 

103

For a certain period of time after loading, birefringence is a
linear function of the difference between the principal stresses or the
principal strains. It seems that the dispersion of birefringence in
this linear range of behavior should be independent of stress level.
This has been demonstrated for Palatal P-6 and for CR —39 by J. T.
Pindera (62, 63) on the basis of the formula (2. 26) proposed by
Manch for the quantitative description of dispersion. But even in
the linear range, dispersion of birefringence as expressed by
formula (2. 26) can be either dependent or independent of the time
of loading.

The discrimination between linear and nonlinear behavior of
model materials is one of the fundamental problems of mechanical
model analysis (59). Related to this is the problem of models and
materials which are suitable for investigations involving elasto-
plastic deformations. Results previously cited show that the
dispersion of CR ~39 is constant with time, meaning it is uniform
over the nonlinear and linear range of stress. The dispersion of
cured P-6 has been found to be independent of stress level in the
linear range, but it decreases with time. A rather particular
relationship between dispersion and stress level exists for uncured
P-6.

Considering the problem from the point of view of the structure
of material and related optical properties, it seems that the lack of
a clearly-defined relationship between the degree of plastic deformation
and the dispersion of birefringence, such as that discovered by Manch

for celluloid, may result from an inadequate formulation of the

 

 

104

dispersion of birefringence. According to the equation of Manch, the
dispersion of birefringence is expressed as the relative change in
birefringence in an arbitrary spectral range of 0.153 microns or

0. 219 microns. There are two basic limitations df this measure.
First, the spectral range is very narrow. Secondly, the spectral
range is chosen quite arbitrarily, and it bears no relationship to the
physical and optical phenomena occurring in the material in the
linear and nonlinear range.

The formulas used in geometrical optics for the description
of dispersion of optical glasses do not satisfy the special aspects of
the dispersion of birefringence. Also, these definitions are referred
to arbitrary spectral ranges which are bounded by certain Fraunhofer
lines.

Coker and Filon described the dispersion of birefringence by
a nonlinear function of wavelength which was similar to Hartmann's
dispersion formula. They proved that their formula adequately
described the optical properties of many glasses. chever, as
early as 1907, L. N. G. Filon found that in the visible spectral
range some glasses exhibit irregularities which are similar to
anomalous dispersion. Formula (2. 25) appears to be quite satisfactory
for glasses exhibiting only negligible creep and behaving linearly in
practically the whole range of stress and strain at room temperature.
The behavior of the plastics commonly used for general mechanical
models, photoelastic models, and photoelastic coatings is much more

complicated. For example, Pindera (62) shows that in the process of

creep the stress—optical and strain-optical coefficients may approach

 

 

 

105

zero, while the dispersion of birefringence measured according to
formula (2. 26) remains essentially constant. The extensive data
presented by T. D. Maksutova (46) on the relations between
retardation, stress-optical coefficient, stress, strain, temperature,
and material composition illustrate the very complicated nature of
the involved phenomena. The extension of the photoelastic method
into the longwave region makes it necessary to choose convenient,
uniform, and adequate methods for the description of properties of
photoelastic materials in linear and nonlinear stress ranges and
over spectral ranges of 0. 4 - l. 2 microns, 0. 4 - 2. 5 microns, and
finally of 0. 4 - 7 microns.

One important aspect of the present problem is to take into
account recent progress in the theory of dispersion and spectroscopy
of high polymers along with the previously established relations
between the structure of polymers and their optical properties. This
already has been expressed by J. T. Pindera (62) and by R. Van
Geen (77). The new ideas and results presented by the last author
go well beyond classical methods of photoelasticity and are outside
the scope of the present work.

The proven existence of a kind of anomalous dispersion of
birefringence, even in the visible spectral range, indicates strongly
that not only the extrapolation but also the interpolation of dispersion
curves may lead to erroneous results.

On the basis of the preceding short analysis, some suggestions
for the adequate and convenient description of the dispersion of

birefringence can be formulated.

 

 

 

 

106

The dispersion should be presented as a continuous function
of frequency or wavelength of radiation in a possible wide spectral
range. It should be referred to unit thickness and to unit stress or
unit strain for a chosen time point on a creep or relaxation curve.

It seems that the unit birefringence:

0

An

n1 — n2 , for unit stress,

AU 2

(71 ~02 =1kg/mm

may often be chosen as the basic quantity more conveniently than
the coefficients CG and C6 which would then be derived quantities.
The relations for the dispersion of birefringence or the derived
expressions should facilitate the distinguishing of linear and nonlinear
behavior. Here, the term "linear range" may be taken to mean the
region wherein the deviation from nonlinearity is less than 1% (64).
Of course, it is important to present the dispersion in such a way that
it would be easy to calculate the value of C0_ or C6 (respectively,
So or SE) in a given spectral band.

To satisfy the above requirements, the following graphical
descriptions were chosen:

1. Relative retardation versus wavelength and/ or relative

retardation versus radiation frequency for unit principal stress

difference and unit thickness:

R0 2 ROM) and/ or
R0 = Row)
for 0' = l kg/mm2 and d0 = 1 cm.

 

ll, It). i-

 

107

These plots may be called ”Unit Retardation Curves". In a linear
range, these quantities may be expected to be independent of stress
level but usually dependent on time.

2. Ratio of unit retardation for any wavelength to unit
retardation for a standard wavelength versus wavelength or,

respectively, frequency:

0
r : ——-RO()\) : r()\)
R (X 0)
o
r : ROW) : N")
R (v0)
where )t = 546.1 nm (green mercury line) and for a given time after

0

loading with a constant force (t = constant). These curves will be
termed ”Normalized Retardation Curves. " It is apparent that in the
linear range this ratio should be independent of stress level but may
depend on time.

3. Rate of change of normalized retardation versus radiation

wavelength or frequency:

0 _ d3 _ o
D)‘ — dk — D (X)
o _ g: _ 0
DV _ d‘l/ _ D (U)

The name applied to these quantities will be "Normalized Dispersion
of Birefringence. " Again, this quantity should not depend on stress
magnitude in a linear range but may depend on time. This definition

is analogous to a previously mentioned definition of optical dispersion.

 

 

 

108

4. Normalized dispersion of birefringence versus time during
creep and relaxation for given )\ or U and several given stress
levels. It is expected that these curves would indicate the effects of
nonlinear behavior on the dispersion.

The important relationships which could be derived from the
preceding list of basic curves would include:

1. Stress-optical coefficient and strain—optical coefficient

versus radiation frequency or wavelength at constant time:

C
0‘

C0_(V) or C0_ = C0_()\)

C
e

H

C€(U) or C6 = C€()\)

at t = constant.
2. Stress—optical coefficient and strain—optical coefficient

versus time in creep and relaxation for chosen wavelengths:

C0 = CU(t)

C6 = C6(t)

for X = constant.

According to the ideas expressed in papers (59) and (64), the curves
discussed above achieve their full meaning if they are supplemented
by curves which describe the range of linear relationship between

stress, strain and birefringence:

 

 

 

 

 

 

109

Several useful and interesting relationships between the

involved quantities may be derived. Some of these are listed as

follows:

Unit R etardation:

 

o 0'0 do
R = R —— ——
0‘1 — 0‘2 d
=i(n1_n2) 0' (:3 2'9
no 1 2
: c d a : R00.)
0‘ o o
for t : constant
2
0'0 = l kg/mm
(1 = 1 cm
0
Normalized Retardation:
, = Row 2 Rm
Ro()\ ) R(>\0)
o
: (n1 -n2)>\ : Anx — CUM) _ r0.)
(n1 v-n2))\O Anxo C00. 0)

for t : constant

In a linear range:

C (1t)
1‘0) L

C60)
C€().O)

for t = constant

Hence,

CU (X) = CO (to) r()\)

 

110

Normalized Dispersion of Birefringence:-

dr 1 dROx)

D EX ‘ R(>\O) d).

 

ll

 

 

 

for t 2 constant

In a linear range; for t = constant

 

l dCO‘
D“) 2 Co‘ (x 0) dx
_ __1__ d_C_e
‘ c (x ) d).
E O

 

Manch's Formula for dispersion of Birefringence:

R —R
D _ X1 1.2
1,2 RX

’ l

 

I |

,_.

l
i

 

H
H
l

(C,I )xz
(c, 311

| I
._..-
I

for t : constant.

Also:

 

 

 

lll

Plots of normalized retardation versus wavelength and
frequency of radiation were developed according to the above ideas
from the graphs of fringe order and relative retardation versus
either location on the cross section or stress for each of the three
materials whose properties were investigated quantitatively. For
CR -39 and P-6, the use of visible light in addition to infrared
radiation allowed the taking of the mercury green spectral line at
X = 546 nm as the basis for the computation of normalized
retardation. The relative retardation at each wavelength was

divided by the retardation for K : 546 nm for each of several

different stre s 5 levels .

For CR-39, normalized retardation was computed for each
of the 9 wavelengths and at 18 different stress magnitudes between

0 and 2.1 kg/mmz. The normalized retardation for this material and

for the conditions of the test seemed independent of stress level.

Since no ordering of the 18 values of normalized retardation at each

Wavelength was apparent, the data was amenable to statistical treat—

ment. The mean and RMS deviation (standard deviation) were

computed for each group of 18 values. A sample of the data and the

Statistical calculations are given in Appendix F. The standard

deviation, Which represents a confidence limit of about 68%, and the

extreme range 0f values obtained for the 18 stress levels up to

2-1 kg/mmZ are shown in the plots of normalized retardation for

CR-3 9, versus radiation wavelength, wave number and frequency.

These are given in Figure 3.19 as reductions of the originals.

 

 

 

112

The dependence of the normalized retardation of Palatal P—6
upon radiation frequency, wave number and wavelength is indicated
in the plots of Figure 3. 20. The data for this material evidenced a
peculiar relationship between the optical behavior, radiation wave—
length, and sign and magnitude of stress. Because of the resultant
ordering of the values of normalized retardation with stress, statistical
treatment of the data was not appropriate. The values reported here
are based on the average over the outer thirds of the beam tested.
The deviations in this portion were generally not random, but the
extremes of the values as indicated on the plots indicate that the
averages are meaningful measures of the dispersion of birefringence
over the corresponding stress ranges. The dependence of the
normalized retardation upon the stress is readily apparent in the
figure, and an average over the whole cross section of the beam
would not be valid. The values obtained for the normalized
retardation in the inner third of the beam were generally smaller
than the values shown for the outer thirds.

Further testing would be required in order to establish clearly
the relationship between stress and normalized retardation in P-6.
As an indicator of what may be expected from such an investigation,
values of normalized retardation extracted from the less complete
study of the ring in diametral compression were compared with the
data obtained from the beam. Shown on one of the graphs of Figure
3. 20 are values of normalized retardation at locations on the cross
section of the ring where the fringe order per unit thickness for

X = 546 nm was about equal to the fringe order per unit thickness

 

 

 

113

at the centers of the outer thirds of the beam and for the same wave-
length. Presumably, the differences are due to the different nature
of the stress field, the different conditions of the test, and also the
multiplication of error inherent in this difficult measurement of the
small differences between large values of relative retardation.

Because of the peculiar nature of the fluctuation of normalized
retardation for both P-6 and CR —39 in the wavelength region between
about 550 nm and 700 nm, these portions of the plots are shown as
dotted lines. As will be discussed more fully later, these undulations
may be due to anomalous dispersion, in which case the irregularity
between the plotted points may be more drastic than shown.

The normalized retardation of polycarbonate as derived from
the results of the calibration test in the spectral range between lu
and 2g is shown in Figure 3. 21 as a function of wavelength. The
values shown were derived from the plots of fringe order versus
stress. Since these plots were linear for the limited stress range
used, the normalized retardation was independent of stress. No
measurements were taken with visible light, so the relative
retardation at K o = 1.145 was used as the norm in constructing
these plots. The scatter in experimental data at this wavelength was
less than or equal to the deviations at the other wavelengths. The
size of the experimental points sh0wn in the figure is indicative of
the extreme deviation in normalized retardation as calculated from
the maximum separation of the experimental data points from the

linear plots of fringe order versus stress at each wavelength. This

places a bound on the extreme possible error in the determination of

 

 

 

114

the normalized retardation. A more realistic evaluation of the possible
error would be based on the root mean square of the deviation (standard
deviation) of the experimental data from the smoothed straight line
curve.

The irregularities in the graphs of normalized retardation
are similar to those of P-6 and CR —3 9, and also those presented by
Filon (10) for glass. However, because of the decreased accuracy
of this test and the lack of subsequent verification by further testing,
the curve is shown dotted. Further improvement in testing would
clearly define the exact nature of the behavior of polycarbonate. The
presented data is perhaps indicative of what may be expected, and it
should be considered only as a first approximation to the normalized
retardation of this material.

The normalized dispersion of birefringence of P-6 and CR -39
was derived from tracings of the large original plots of normalized
retardation of these two materials. This measure of dispersion is
essentially the slope of the normalized retardation versus wavelength
or frequency graphs. The slope was measured by using the well-known
reflection technique to construct normals to the curves at several points
along its length. More points were chosen in regions where the slope
varied rapidly, and several trials were made at points in particularly
troublesome portions of the curves, reversing the mirror for each
trial. Inversion of the slope of the normal at each point yielded the
slope of the normalized retardation curve at that point. These data
appear plotted in Figure 3.22 and 3. 23. The portions of the curves
where the values of the normalized dispersion may be questioned are

represented by dotted lines.

 

 

 

115

3. 2. 4. Birefringence of Semiconductors

The data resulting from the measurements of stress-birefringence
in silicon were treated in the same manner as were the data for poly-
carbonate. Plots of the fringe order versus stress difference for
wavelengths of 1.145, 1. 348, l. 560, and l. 90 microns are presented
in Figure 3. 24. The original data was corrected so that the straight
line plots might pass through the origin. The stress optical coefficients

were computed from the slopes of the stress—optic curves. At )1 = 1.145u,

CG was found to be 12.5 x10.4 kg-lmmz; for )x :1.348p., Co- =
14.8 x 10‘4kg'1mm2; for x = 1.56011, c0r = 17.1 x10“4 kg‘lmmz
4 l 2

and for k 21.90u, Ccr = 20.8x10— kg_ mm .
The small thickness of the silicon disc limited the range of
loads that could be applied without buckling and subsequent fracture
of this brittle material. Since isochromatic order is proportional to
load and to thickness, the maximum order of isochromatic observable
in this test was about 0.18 fringes. Recalling the estimate of error
in the goniometric compensation technique it is apparent that
experimental deviations on the order of 10% might be expected in this
particular study. The increased possibility of error shows clearly in
the scatter of the experimental points in the graph. For this and other
reasons, the curves of Figure 3. 24 should be thought of only as the
first approximation to the stresswoptic characteristic of silicon.
Transmission of near infrared radiation by the amorphous
selenium model was high, although partial crystallization in the slower-

cooled central portion caused the light to be partially scattered. With

no load on the model, photoelastic observation indicated two things of

 

 

 

 

116

interest. First, the model appeared to act as a retardation plate.
With crossed polaroids, the model appeared light, and it nearly
extinguished the infrared light with the light field arrangement of
the polariscope. This retardation was apparent with both linear
and circular polarization. Whether this was caused by the anisotropy
and stresses from pouring and cooling, or it was a rotation of the
light vector due to structure (optical activity), as is produced by
aqueous solutions of sugar was not immediately apparent. Secondly,
a birefringent pattern was noticed near the edges of the model. This
was presumably due to residual stresses created by the rapid cooling
of the portion of the model in contact with the aluminum mold. The
birefringence due to cooling stresses has been observed previously
by other investigators already mentioned.

No double refraction under externally applied loads was
apparent in the selenium model. This was unexpected in view of

ea rlie r finding 5 .

 

 

 

4. Discussion and Conclusion

The methods and results of this investigation, as presented
in the previous chapters, may be examined for their contribution
and deficiency in providing insight into the phenomenon of double
refraction and its measurement. The original objectives of the
experimental program were met, and some additional questions
were answered. But more questions were created. The new
information provides additional grounds for speculation about the
nature and behavior of materials. The new questions can be

answered only by continued experimental investigation.

4. 1. Summary of Techniques and Apparatus
The birefringence of various materials was observed and
measured using monochromatic radiation at several different wave-
lengths in the visible and near infrared spectral ranges. The
fundamental character of the measurement and the interpretation
of data in this extended range of wavelengths were identical to
those which form the basis of all ordinary photoelastic investigations.
Assuming that visible light, infrared radiation and radio waves,
regardless of their true nature, may be considered to consist of
electromagnetic waves whose behavior are described by Maxwell's
equations, then the phenomena of double refraction and its measure—
ment through the use of polarized light are not limited by wavelength.
While the governing photoelastic equations and the functioning

of the optical system required for the observation and measurement of

117

 

 

118

birefringence are independent of the spectral range in which the work
is being performed, the techniques and apparatus employed must
necessarily change. The greatest modification is to the methods of
sensing and recording the photoelastic pattern. Direct visual
observation is limited to the relatively narrow range of wavelengths
between 0. 4 and 0. 7 microns. This is also the spectral range in
which most photoelastic analyses are performed. The improvement
of photographic methods facilitated the direct planewise recording of
results up to about 1. 2 microns maximum wavelength. A device
which converted an infrared image to a visible one was used for
observations up to 1. 3p wavelength. The visible image in this
converter could also be photographed, thus extending planewise
recording up to about 1. 3 microns, but the quality of results obtained
was not as good as those derived from other methods. For the wave-
length region beyond about 1. 2 microns and up to l. 9 microns, point
sensing methods were employed. The apparatus consisted of a very
small infrared-sensitive photoconductive element which was used in
conjunction with a circuit which sensed and measured the voltage drop
across the light—variable resistor. The sensor, when placed in the
plane of an image of a photoelastic model, indicated the variations in
the light intensity transmitted by the area of the model corresponding
to the point in the image at which the sensor was located. Scanning
the image pointwise would furnish a complete map of the birefringent
pattern in the model. This general method of pointwise detection
Should not be limited to any particular portion of the electromagnetic

Spectrum. The output of the sensing circuit could be calibrated directly

 

 

 

119

in terms of intensity and eventually of birefringence through use of
the general photoelastic equations. The approach used here was
more appropriate to a non—ideal system in which noise was present
in that indications only of local minimum or maximum light intensity
were required. This was sufficient for accurate measurement of
fractional fringe orders by methods of goniometric compensation
(analyzer rotation). In fact, the technique of goniometric compensation
was improved so that the only requirement on the light sensing apparatus
was that it dependably indicated the attainment of arbitrary pre—
established levels of intensity. Examination of the equations which
govern the analyzer rotation method showed that the transmitted
intensity is an evm function of the angular position of the analyzer
when measured from the angle at which the intensity is minimum.
The average of two angles on either side of the minimum where
the transmitted intensities were equal provided a measurement of the
angle at which the light was minimum. This method of averaging the
symmetrical points was justifiably found to be more free from the
errors due to system noise and operator error than the ordinary
method of locating directly the angle of minimum intensity.

Three optical systems were developed. One was suited for
the visible region, another for planewise investigation in the infrared,
and another to be used with the pointwise method in the farther infrared.
Monochromatic radiation was obtained by means of spectral lamps and
interference filtering of white light. Ordinary polaroid sheet was used
in the visible region, and type HR infrared polarizing material served

byond O. 71.1 wavelength. Circular polarization in the visible was

 

 

 

 

120

accomplished by the usual methods. Half—wave retardation plates
served in the infrared up to about 1. 6p. . Two of these plates were
stacked in order to obtain more accurate circular polarization for
the range of wavelength between 1. 611 and 2p . The latter figure is
nearly the practical transmission limit of this material. The
errors in measurement of is ochromatic order which could result
from the errors in the quarter wave plates used for the given
spectral range were shown to amount to about $070, where m is
the fringe order.

The techniques and equipment developed served more than
adequately the requirements of this early limited investigation into
longwave photoelasticity. Refinement and sophistication are possible
and necessary for continued experimentation in the spectral range of
0. 4 to 2 microns wavelength and for the extension of the research up
to the radio wavelengths. The results of the investigation indicate
that this extension of photoelastic analysis further into the longwave
region is physically possible and may be useful and enlightening in
basic studies of rheology and material structure as well as for the
solution of special stress analysis problems. The methods used here
provide a basis from which to work, and several of the possibilities
for further development of the apparatus have been described. The
major problems to be met in the extension to longer wavelengths appear
to be the production of monochromatic radiation of sufficient intensity
and the associated problem of developing a sensing system of small
enough area and adequate signal-to-noise ratio. The extended-range

Closed circuit television system and the longwave scanning radiometer

 

 

 

 

 

121

provide additional possibilities for planewise measurement in the far

infrared spectral range.

4. 2. Material Behavior

The test of Columbia Resin CR =3 9, which was mainly directed
towards the measurement of dispersion of birefringence, provided
representative values of stress=optical coefficients for this material
under the particular conditions of the test. For stresses within the

2

linear limit stress of about 1 kg/mm , for a temperature of 240 C,

at 66 hours after loading and for .X o : 546 nm, the stress—optical

4 l 2

coefficient, Co- (X 0) was found to be 3. 7 x10.= kg“ mm . The

corresponding material stress—fringe coefficient for a thickness of

1 cm, So ()x O), was 0.15 kg/mmz. The unit retardation for this

2

material under the given conditions and for or : l kg/mm and

o
1 cm thickness was ROM 0) = 3.7 microns.
The stress-optical coefficients for the conditions of the test
but for wavelengths other than R o = 546 nm may be computed from
the presented dispersion data for CR_39. Figure 3.19 indicates that
serious error may result from the assumption that the stress—optical
coefficients are constant even within the limited visible spectrum.
For the specimen tested, C0" changed by about 5% between 405 nm
and 650 nm wavelength, independent of stress level. This change
would have to be taken into account in any test where light of more
than one wavelength was used. Since the dispersion did not depend

on stress, it is reasonable to expect that it would not depend on the

Other conditions of a test, at least within the linear range. If this

 

 

 

122

presumption can be justified experimentally, then the dispersion data
may be used in conjunction with stress=optical coefficients more
accurate than those presented here. Emphasis should be given the
fact that accuracy of this measurement of dispersion of birefringence
is not directly related to the accuracy of the determination of stress-
optic behavior. This was due primarily to the errors in locating points
on the specimen with respect to an absolute reference, which in turn
resulted from the distortions of the inferior lens system and the
necessity of observing the model in segments. The dispersion
measurement required no such absolute location of points as long as
the correspondence between points on different photographs of the
same segment was established. This was ea51ly and accurately
accomplished regardless of the distortion in the images.

The results of the test of cured Palatal P-6 were not nearly
as simple or well defined as were the results for CR -39. The data
for this material evidenaad a peculiar relationship between optical
coefficients, wavelength of radiation, and sign and magnitude of

stress. In the preceding chapter it was reasoned from the results

summarized in Figures 3.10, 3.12 and 3.15 that the stress-optical

coefficients and/or strain must depend on the sign of stress. This

prohibits computation of the stress distribution in the beam or the

ring by ordinary elasticity methods or correlation between calculated

Stress and birefringence. Hence, the optical coefficients could not

be derived from the results of the tests. The normalized retardation

and dispersion as functions of wavelength and frequency as reported

here were base

 

 

 

123

beam tested. The deviations were generally not random, but the
ranges of values shown on the curves of Figures 3. 20 and 3. 23
indicate that the average is at least a meaningful measure of the
dispersion of birefringence of P—6. The dependence of the
normalized retardation on stress is readily apparent, and it would
not be valid to take an average over the entire cross section of the
beam. This particular behavior was verified by the initial, more
limited test of the ring. Agreement was good considering the much
different nature of the stress fields in the two specimens and the
dependence of dispersion upon the stress field.

This apparent dependence of the optical properties of cured
P-6 upon the character of the stress field appears partially contrary
to the results reported by Pindera (62, 63) for this material. However,
this author has observed a similar relationship between optical
coefficients and stress for the uncured Palatal P—6. It seems most
probable that the curing cycle used did not provide the optimum level
of polymerization of this particular mixture of resin and catalyst.
Rather, the polymerization was something between that of the uncured
state and the nominal for cured resin. This opinion is strengthened
by the fact that the value of the dispersion calculated by Monch's
formula (2. 26) from the given data for both tension and compression
agree well with each other (within 0. 3%) and are both within 0. 2% of
the value given by Pindera for cured P—6 at the same stress and time.
Further investigation under carefully controlled conditions are required
in order to demonstrate more clearly the effects of resin mixture and

curing upon the photoelastic properties. Perhaps this material will be

 

 

 

 

124

all the more interesting and useful because of its peculiar properties.
The possibility of utilizing the variation in dispersion as a measure of
sign and magnitude of elastic or plastic stress or strain, or only as an
indicator of the presence of plastic deformation, is immediately
apparent.

The data presented in Figures 3. 20 and 3.23 show that the
stress—optical coefficient of P—6, regardless of its actual value,
changes by approximately 11% in the limited visible range. The use
of a coefficient which is constant over any part of the spectrum would
probably lead to quite significant error in experiments involving
radiation of more than one wavelength.

The test of the disc of polycarbonate yielded values of the
material optical coefficients at 8 different wavelengths in the spectral
range from 1.14511. to l. 90H . Since the stressuoptic curves were
linear, the material behaved as one momentarily linearly elastic, and
the time after the placing an increment of load may be taken as the
time of the test. The lack of data in the visible range prohibits the
use of mercury green wavelength as the basis for calculating unit
retardation and the reference wavelength for expression of the
coefficients. The shortest wavelength of 1.14511 was chosen as the
reference wavelength because of the minimal scatter in experimental
data at this wavelength. Thus, for X o = 1.145 micron, time = l. 25 hrs. ,
temperature : 200 C, the stress optical coefficient, CO, ()x o)’ attained

4 — 2 . . . .
kg lmm ; the material stress—fringe coeff1c1ent,

Sq (A o), was 0.153 kg/mm2 for 1 cm thickness; and the unit retardation

RO(>\ O) for 0'0 : l kg/mmz and d0 = 1 cm was 7. 49 microns. These

a value of 7.49 x10”

 

 

 

125

results may be instructively compared to those obtained by Ito (33),
although this author indicates neither the wavelength of light used

nor the other conditions of the calibration test. He reports a stress—
fringe coefficient of 1. 0 fringe kg_1mm. Assuming that the calibration
experiment was conducted with visible light with )x = 546 nm, then the

4kg ‘1mm2 . This value,

stress-optical coefficient would be 5. 46 x 10—
while based on supposition of the actual conditions of test, is of the
same order as the result of the present investigation. The difference
is probably due to the unknown testing procedure and conditions,
especially the wavelength of light used, and possibly to the presence

of dispersion of birefringence. In this case the variation of the stress —

optical coefficient with wavelength would be opposite to the decrease
normally found. The dispersion would be of the anomalous type.

The average values of stress-optic coefficient, material
stress-fringe coefficient, unit retardation, and normalized retardation
of polycarbonate for the eight wavelengths are summarized in Table 4.1.
The normalized retardation for polycarbonate has been presented in
Figure 3. 21. Since the stress-optic relationships for each wavelength
were linear within the stress range employed, the coefficients and
measures of the dispersion are independent of stress. The measured
extreme variation of the stress-optical coefficients with wavelength or
frequency in the spectral range is seen to be about 4%. This variation
is only about twice the limit of accuracy of the particular test. Hence,
the values of normalized retardation reported should be considered
only the first approximation to the actual value. The dispersion data

for P-6 and CR -39 suggest that this effect is decreased in the range

 

 

 

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127

beyond 0. 8p. . If polycarbonate behaves in a similar manner, its
dispersion may be much greater in the visible range than in the band
between 1 and 2 microns.

As was stated earlier, polycarbonate resin is very tough and
impact resistant and can withstand extensive plastic deformation.
If the complete stress-strain—optic —time law for this material can
be found, it should serve for the photoelastic analysis of problems
involving both elastic and inelastic deformations. The first and
simplest problem of this type is one directed towards the detection
of initiation of inelastic deformation and the indication of the elastic-
plastic boundary. More complex analyses dealing with plastic flow
would rest upon the material behavior and the extension of the
similarity principles. Investigation along these lines was not within
the limitations of this work, but one observation is worthy of mention.
In the region of high plastic deformation under the loading points,
the polycarbonate retained its optical transparency. This is different
from the behavior of many ductile polymers which whiten and become
opaque or translucent when severely deformed.

Of some interest is the consistent irregularity of the experi-
mental normalized retardation curves of P—6, CR -3 9, and polycarbonate-
The order of recording data during the various experiments indicates
that these irregularities cannot have resulted from time effects.
Although they may be partly caused by other experimental error,
particularly in the case of polycarbonate, their appearance must be at
least partially due to anomalous dispersion of birefringence. A

similar behavior has been observed during extensive tests of glass

 

 

 

 

128

3453 by L. N. G. Filon (10). It should be pointed out that the presented
plots of normalized retardation do not show other possible irregularities
which may be important. The very presence of the fluctuations:
emphasizes the fact that the interpolation as well as the extrapolation
of optical data can lead to serious errors unless the procedure is
justified experimentally.

The source of anomalous dispersion and the reasons for its
appearance are open to speculation. Two related possibilities present
themselves. First, birefringence seems to result from variations in
basic physical structure such as the changes in orientation or length
of certain molecular or atomic groupings such as chains, side chains,
or crystals (56). At different wavelengths or frequencies of radiation,
the double refraction probably depends upon different structural groups.
There is no reason why the transition from one group to another should
be smooth or even continuous. On the basis of this simplified model,

a plot of birefringence versus wavelength or frequency might be
expected to have a jagged sawtooth form with unequal spacings and
non-similar teeth. If the functioning of two or more groups overlap,
then the plot may not lack continuity. Such a continuous change in
dielectric constant within one group or between a series of structural
units could contribute the property of normal or anomalous dispersion,
either of the simple optical variety or as related to dispersion of
birefringence.

The second factor which may serve as a source of irregularity
of dispersion phenomena may be found in the effect upon light waves

of molecular and atomic vibrations. The science of transmission

 

 

 

 

 

129

spectroscopy is based on the fact that light will be absorbed rather
than transmitted by a medium when its frequency corresponds to a
characteristic frequency of a natural mode of vibration of a molecular
or atomic group contained in a transmitting medium. The association
of absorption frequencies with molecular constituents provides a tool
for the identification of compounds and the investigation of molecular
structure. It is conceivable that the vibration of molecular groups
contributes to the dispersive properties of materials. In particular,
a condition of resonance between some element of the material
structure and the radiation would logically disturb the double refraction
properties in much the same way as it disturbs the transmittance and
therefore the ordinary optical refraction. The part which molecular
vibration plays in the appearance of the property of normal and,
particularly, anomalous dispersion might be determined from
correlation of dispersion data with infrared transmission spectroscopic
information. Unfortunately, most transmission data for wavelengths
between blue visible and about 1 micron yield little information about
the bonds present in the photoelastic plastics such as P-6 and CR -39.
In fact, transmittance curves for the shorter wavelengths are not
generally available.

The normalized retardation of polycarbonate as shown in Figure
3. 21 may be instructively compared to the spectral transmittance
represented in Figure 3.18. While the results of the test of polycarbonate
are not considered sufficiently accurate to warrant the construction of
precise conclusions, a certain correlation between the plots is apparent.

The greatest irregularities in normalized retardation seem to occur

 

 

 

130

near the absorption band around l. 6611 wavelength. The measurement
at 1. 651.1 , very close to the absorption wavelength, deviated most from
the other measurements. The trend of the other values of normalized
retardation is generally to decrease with increased wavelength. That
is, the dispersion is normal. The large value of normalized retardation
near the absorption frequency creates an area of anomalous dispersion.
The deviation seems too great to result from experimental errors.

The results seem to indicate that improvement and extension of
dispersion measurements and their subsequent comparison with
spectroscopic data for various materials represents a promising and
logical step along this course of experimentation.

Regardless of the factors which cause the properties of single
and double refraction to depend upon wavelength and frequency of
radiation, the investigation of this dependence will contribute to
materials science and stress analysis in two ways. First, the
relating of birefringence and dispersive properties, especially
anomalous dispersion, to material composition and structure will
eventually indicate strongly the source of the phenomenon of
birefringence. What there is in a material that makes it birefringent
is an interesting question from an academic viewpoint. Such correlation
is also of practical interest, since it would facilitate the finding of
other good photoelastic materials without extensive experimentation.

It might also make possible the designing of new, more efficient
birefringent materials.

Another possibility of more immediate practical significance

is that extended investigation of dispersion should lead to knowledge of

M

 

 

 

131

possible relationships between dispersion and mode of deformation.
Even if the real source of birefringence and dispersion cannot be
pinpointed, the dispersion might depend on rearrangements of certain
groups of molecules. This could be interpreted from a simple
phenomenological point of view in terms of the diphase and multi-
phase models of the structure of plastics. The results would be
applicable to studies involving inelastic (including plastic and visco-
elastic) behavior, three dimensional photoelasticity, and the design
of rheological models which are more representative of real physical
behavior.

Contribution to the understanding of physical properties such
as stress birefringence and also to the practical application of photo-
elasticity would be gained by the discovery and investigation of new
materials which are doubly refracting in some spectral region. The
possibility of finding such materials is expanded by the extension of
the range of useable wavelengths. Conversely, the finding of
materials which are transparent and birefringent in infrared or longer
wave radiation adds to the probability of correlating photoelastic
behavior with material structure.

Crystalline silicon, which was known to transmit light of
wavelength longer than about 1.111, was found to be stress birefringent
up to the 21.1 limit of this work. Both isoclinics and isochromatics in
a disc of this material were observed visually using the infrared
converter, and the relationships between is ochromatic order and
stress were determined through use of the point sensing and

compensation procedure. The stress optic coefficient for silicon was

 

 

 

 

132

computed from the results present in Figure 3. 24. For example,

4 2

at). =1.14511, c = 12.5x10' kg/mm , and at). = 1.9011,

a
Co = 20. 8 x 10-4 kg/mm-Z. These values should be taken to be

first approximations only, because of possible errors in the data
resulting from the low is ochromatic orders in the thin specimen.
Mechanical and optical anisotropy of the crystal was also ignored.

Both silicon and amorphous selenium could be used to meet
certain requirements in infrared instrumentation. A stressed plate
of silicon, or a disc of naturally birefringent selenium could be made
to serve as a quarter wave plate through the very wide transmission
bands of wavelengths which are characteristic of these materials.
These doubly-refracting materials might also serve as the core of
beam-splitting or dichroic polarizers for the infrared range. Various
other technological applications of the stress birefringence in infrared
of silicon suggest themselves. These, along with extended discussion
of the physical nature of birefringence in semiconductors, are not
within the reasonable limits of this work.

Emphasis must be given the fact that the material tests carried
out as part of this investigation were intended primarily as initial
studies to demonstrate the validity of longwave photoelastic stress
analysis and to illustrate the various future possibilities. The
experiments were performed under laboratory conditions with less
than acceptable control of temperature and humidity. The properties
of plastic materials are known to depend to a large extent upon ambient

conditions. Proper calibration in conjunction with accurate model

analysis must be accomplished under carefully controlled conditions

 

 

 

133

so that the relationship between stress—strain—time -temperature-
humidity-resin can be properly developed. These tests are tedious

to perform at their present stage of development. Attempts to

shorten and refine the testing procedure are known to be in progress (64).
The results of this investigation, if they may be taken as at least
representative, show that the involved relationships are very complex.
The very common, greatly simplified approach to the study of the
properties of model materials is not necessary and cannot be justified
except on the insecure basis of expediency.

The relative retardation which takes place in a birefringent
plate, regardless of its nature, was shown to depend upon the absolute
index of refraction of the medium (usually air) in which the optical
system is immersed. This dependence is ordinarily of no consequence
when measurements are made with visible light. But the transmission
and refractive properties of air, as well as any other photoelastic
material, may vary a great deal over a broader wavelength spectrum
and under varying conditions of temperature and humidity. Serious
error could result from ignoring the effects of environment upon the
photoelastic coefficients in the extended spectral region. In particular,
the use of vacuum wavelengths with relative (to air) indices of refraction
is inconsistent, although common. Also, failure to account precisely
for the variation of the refractive index of air may be apparent in the
data as dispersion of birefringence of the refracting plate.

It is hoped that the techniques and apparatus developed for the
extension of photomechanics into the longwave spectral region and the
results of the investigations into the optical properties of materials in

this extended range provide the tools, guidelines, and motivation for

¥

 

 

 

 

134

Continued exploration. Whatever the results of the additional research,
they will be beneficial to the studies of materials structure, material

behavior, and the experimental stress and strain analysis.

4. 3. Continuation of the Research

Several suggestions for the improvement and extension of the
investigation may be formulated. Some of these are enumerated
below in the form of specific proposals which should not be considered
exclusive of other interesting and potentially profitable possibilities:

(l) The techniques and apparatus for the conducting of photo-
elastic investigations should be continuously improved, and the upper
wavelength limit raised. The infrared spectroscopy region between
0. 8M and 35“ should be considered the minimum limit of the range
of infrared photoelasticity. Later extension to the microwave region
should prove instructive.

(2) The relationships between stress, strain, birefringence,
and dispersion of birefringence for various photoelastic materials
in wide ranges of wavelengths of light should be determined. Immediate
attention should be given to polycarbonate which has been seriously
neglected as a photoelastic material, but which appears promising
for use in the analysis of inelastic behavior.

(3) The longwave photoelastic data, including dispersion
information, should be carefully compared with infrared transmission
Spectroscopic data. Both of these classes of information should be
vieWed in light of what is known about the physics and chemistry of
material structure. In this way, models explaining stress and

orientational birefringence phenomena may be constructed and improved,

 

 

 

 

135

and eventually the physical basis of photoelasticity may be firmly
established.

(4) Investigation of the natural and stress-induced birefringence
and dichroism of other materials which are not necessarily transparent
to visible radiation would contribute to several areas of knowledge and
practical endeavor. Of particular scientific and technological interest
is a major study of birefringence in semiconductors.

(5) The dependence of double refraction upon sign of stress
difference as well as magnitude must be explored as part of the
rheological study and also to eliminate the possibility of serious error
resulting from the use of simple calibration results in the analysis of
more complex stress fields. As a generalization, the empirical
relationships between birefringence and stress path must be critically
examined to determine whether the simplified relationships are always
justified.

(6) The way in which photoelastic measurements are affected
by the refractive properties of the medium surrounding the measuring
system must be clearly established for all wavelengths. The findings
of such a study may form the basis of a critical evaluation of other

results, particularly dispersion data.

 

 

 

 

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46. Maksutova, T. D. , "On Optical and Mechanical Propertiesof
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Leningradsky Universitet,Leningrad, Nr. 2, 1963.

47. Mindlin, R. D. , "Analysis of Doubly Refracting Materials with
Circularly and Elliptically Polarized Light, " Journal Optical
Society ofAmerica, Vol. 27, (1937), p. 285.

48. Mitsuishi, A. , et al. , “Polarizer for the Far Infrared Region, "

Jrnl. Opt. Soc. America, Vol. 50, No. 9, p. 433 (1960).

49. M8nch, E., "Die Dispersion der Doppelbrechung als Plastizitatsmass

in der Spannungsoptik, " Zeitschrift fur Angewandte Physik,
8 Heft, 1954.
50. Manch, E. , “Der heutige Stand der Photoplastizitat, " Schweizer

Archiv fur angenwandte Wissenschaft und Technik, Heft 5, 1959.

 

 

 

52.

53.

54.

55.

56.

57.

58.

59.

 

141

Monch, E. , "Die Dispersion der Doppelbrechung als Mass fur
die Plastizitat bei Spannungsoptischen Versuchen, " Forschung,
21 Band, 1 Heft, 1955.

Mbnch, E. , and Hartmann, Th. , ”Untersuchung einiger Kunststoffe
auf ihre Eignung als photoplastisches Modellmaterial, " Zeitschrift
fur Angewandte Physik, 11 Band, 1 Heft, 1959.

Monch, E. , and Lorek, R. , "A Study of the Accuracy'and Limits of
Application of Plane Photoelastic Experiments, " Proceedings of
the International Symposium on Photoelasticity, edited by M. M.
Frocht, Pergamon Press, New York, 1963.

Moss, T. 8., Optical Properties of Semiconductors, Butterworths
Scientific Publications, London, 1959.

Papoular, P. , "Far IR Reference Sources for the Evaluation of
High Temperatures,” Infrared Physics, Vol. 4, No. 2, April
1964, p.137.

Persoz, B. , "Birefringence du Polystyrene Sous Contraints
Constante a Differentes Temperatures, " Analyse des Contraintes,
Vol. VI, No. 4, October 1961.

Philips Spectral Lamps, North American Philips, Technical
Brochure No. 10. 078BE 3/60.

Pindera, J. T. , "Einige Rheologische Probleme bei Spannungs—
optischen Untersuchungen, " Internationales Spannungscoptisches
Symposium, Deutsche Akademie der Wissenschaften zu Berlin,
Berlin, 10-15 April, 1961, Akademie-Verlag, Berlin, 1962.

Pindera, J. T. , “Remarks on the Properties of Model Materials, "
Paper presented at 1963 Annual Meeting of Society for Experimental

Analysis (to be published), August 3, 1964.

m

 

 

 

60.

61.

62.

63.

64.

65.

66.

67.
68.

 

142

Pindera, J. T. , "On Some Criteria of Evaluation of Model Materials, "
Presented at Second International Conference on Experimental
Stress Analysis, Paris, April, 1962.

Pindera, J. T. , "Recent European Research Work on Photoelasticity, "
Presented SESA Spring Meeting, Salt Lake City, Utah, May 6-8,
1964.

Pindera, J. T. , "Investigation of Some Rheological Photoelastic
Properties of Some Polyester Resins, ” Part I, II, III (in Polish),
Rozprawy Inzynierskie, No. 3 and 4, 1959.

Pindera, J. T. , "Rheological Properties of Model Materials, "

(in Polish, to be published in English), Wydawnictwa Naukowo-
Techniczne, Warzawa, 1962.

Pindera,J. T. , and Kiesling, E. W. , "On the Linear Range of
Behavior of Photoelastic and Model Materials, " Proceedings,

3. Internationale Tagung fur Experimentelle Spannungsanalyse,
Berlin, 1966, VDI-Verlag, Dusseldorf (W. Germany), 1966.

Plyler, E. K. , and Ball, J. J. , "Filters for the Infrared Region, "
Jrnl. Opt. Soc. America, V. 42, No. 4, April 1952, p. 266.

Plyler, E. K., and Peters, C. W., "Wavelengths for Calibration
of Prism Spectrometers, " Paper No. 2159, Journal of Research
of National Bureau of Standards, Vol. 45, No. 6, December 1950.

Polarized Light, No. F3374, Published by Polaroid Corporation.

Prussin, S. , and Stevenson, A., "Strain—Optical Coefficients of
Silicon for Infrared Light, " Journal Applied Physics, Vol. 30,
(1959), pp. 452/3.

Ramspeck, A. , "Anomalien der accidentellen Doppelbrechung bei

Zelluloid, " Annalen der Physik, 1924, p. 1962.

 

 

 

143

70. Rao, T. V. K., "Relation Between Photoelastic Properties and
Molecular Behavior in Certain Plastics, " Ph. D. Thesis,
Michigan State University, 1960.

71. Schwieger, H. , "A Simple Calculation of the Transverse Impact
on Beams and Its Experimental Verification, " Proceedings,
Society for Experimental Stress Analysis, Vol. 5, No. 11,
November 1965, p. 378.

72. Sears, F. W., Optics, Addison-Wesley Press, Inc., Cambridge,
Massachusetts, 1949.

73. Seeley, F. B., and Smith, J. 0., Advanced Mechanics of Materials,
John Wiley and Sons, New York, 1 952.

74. Shenk, J. H., Hodge, E. 5., Morris, R. J., Pickett, E. E., and
Brode, W. R. , "Plastic Filters for Visible and Near Infrared
Regions," Jrnl. Opt. Soc. America, V. 36, (1946), pp. 569-575.

75. Shurcliff, W. A., Polarized Light Production and Use, Harvard
University Press, Cambridge, Massachusetts, 1962.

76. Timoshenko, S., and Goodier, J. N., Theory of Elasticity,
McGraw-Hill Company, New York, 1951.

77. Van Geen, R. , "Dispersion Cromatique de l'effet photoelastique, "
Second International Conference on Stress Analysis, Paris,
April, 1962.

78. Wolf, H., Spannusungsoptik, Springer-Verlag, Berlin, 1961.

 

 

 

144

wave number cm— . -

10 10 10 104 102 1 ufz 10'410’610'8

4 101210

frequency cps

10 101

infrared

O

visual

 

10'1210'10 10'8 10'6 10'4 10‘2 1 102 104 106 108

wavelength, cm

wavelength, nm
400 600 800 1000 1200 1400 1600 1800 2000_

NEAR.V
UV VEUAL INFRARED
IuKEOGRAPHKiANDIMAGnkiDEWCm INFRARED

THERMOCOUPLES, BOLOMETERS AND OTHER SENSING DEVICES

   

             

    

    

 
  
        

2 1.5 1 0.75 0.5

wave number, 1 04 cm"1

Figure 2.1. The Electromagnetic Spectrum.

. ,.

 

 

 

 

145

x /4 analyzer
plate
/ lens

 
   

Z camera or
light converter
source f'l

model 1 ter

polarizer
lens

Figure 2. 2a. Optical System for Visible and Near Infrared.

condenser

e). /4 plat achromat
lens system lens
achromat
lens
I ,1

‘1‘ 1 i I .-
‘llil ‘ _===_.

hopper
analyzer
radiation

model
source

 
 
   
 
  

     

 
  
   

 

sensor

polarizer

Figure 2. 2b. Optical System for Long Wavelengths.

Figure 2. 2. Infrared Polariscope Arrangements.

m _

 

 

146

   

(a) complete optical bench (b) source, chopper, filters

 

(c) infrared sensing head (cover removed)

Figure 2. 3. Photoelastic Apparatus for the One to Two
Micron Region.

M; _._J

 

 

 

147

m.m

N.N

oomw

H.N

o.~

ooom

 

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w; h; o4 m4 *4 m; NJ HQ 04

 

oomm 000p 0005 ooow ooooH

ON

0
‘1‘

O
\0

cm

9:
ooom~

% ‘uois sttue

 

 

(a)

W
U1

W
O

N
U'l

'—‘ N
U1 0

percent transmis sion
|—‘
O

0.8 0.9

20

percent transmis sion

1.2 1.4

Figure 2. 5.

ML

148

x =l.l45p.

Optics Technology
160 - 1.145

HW = .035p. : 3%

 

1.0 1.1 1.2 1.3 1.4

wavelength

  
  
  
 
  

X 21.55511

Infrared Industries
CN 7049
NB -1 56-0-(1)

HW=.045,1 =2.9%

  

1.5 1.6 1.7 1.8 1.9

wavelength

Transmission of Representative Interference
Filters.

 

 

 

220K

Figure 2. 6.

149

LEAD SULFIDE
\INFRAR ED SENSOR

  
 

AMPLIFIER METER
GENERAL RADIO
Type No. 1232-A

OSCILLOSC OPE
HEWLETT -PACKAR D
Model No. 13 0A

Infrared Sensing Circuit.

 

 

 

 

150

  
 
     
   
      
 

 

  
     

 

      
    

I:—12—A2[1-cos 21;]
17(n - n )
2 where:1;=——;-—2 d-e
A “0
T e- : rotation of analyzer
possible
2 error
_A_ due to
4 noise
noise level for
S/N = 10
O -‘uxmmmx
0 IL 3.31 1T5_TT3_"ZEZTTflfl
4 2 4 4 2 4 4 2
27;
(a) direct detection of minimum intensity angle.
2
L
2

possible error
due to noise

91 TI'
4 2

(b) detection of minimum intensity angle by averaging.

Figure 2. 7. Reduction of Error in Measurement of

Fractional Is ochromatic Order.

M

 

 

 

(a)

2
5"

H
o

radial stress kg/mm
H

O

(b)

(C)

M

 

vi—Ld= 9.50

tensile model, CR -39 dimensions millimeters

0

U1

 

stress distribution

recording
I l period

    

0 10 20 30 40 50 60 70 80 9O
~0. 2 hr
t, hr.

loading history

Figure 3.1. CR -39 Model, Stress Distribution, Load History.

 

 

 

 

 

 

Figure 3. 2a. Isochromatic Pattern in CR -39 Model for

X = 546 nm with Linear Polarization.

 

 

153

   

Figure 3. 2b. Isochromatic Pattern in CR ~39 Model for

)x = 1083 nm with Linear Polarization.

 

 

154

89

   

dimensions millimete r s

(a) ring model of cured Palatal P-6

recording period fl.

   

10 20 30 40 50 60

(b) load history

Figure 3. 3. Ring Model of Cured Palatal P-6,
Load History.

 

 

 

 

155

   

Figure 3. 4. Isochromatic Pattern in P-6 Ring for

X = 1083 nm with Circular Polarization.

 

 

 

156

 

(a) beam model of cured Palatal P-6

 

49 hr.

1:) recording period
2. 5 hr.

 

10
~0.2 hr

t, hr.

(b) load history

Figure 3. 5. Beam Model of Cured P-6,
Load History.

 

 

 

 

on.

ati

157
Isochromatic Pattern in P-6 Beam for

)r = 546 nm with Circular Polariz

 

Figure 3. 6a.

 

 

 

158

 

 

 

 

Figure 3. 6b. Isochromatic Pattern in P—6 Beam for

ati on.

Polariz

x : 405 nm with Linear

 

 

 

 

 

 

m
m
4.
1_O_

 

                 

 

gig/Ago

Egg/g.

10

2%

"gr/.8

  

gay/6

O 0 0 0
O 8 IO 4. Z
1

Figure 3. 7. Disc Model of Polycarbonate,

 

 

stress, kg/rnm‘Z

 

161

 

1.5

E

is”
1.2 e15

H

O

U

.5

(6

E

O
0.8 210

0

O

.E
0.4

 

me

Figure 3. 8. Fringe Order for Constant Wavelength and
Calculated Stress Along Axis of Portion of
CR -39 Model.

 

 

20

16 12

Figure 3. 9.

 

162

x = 405
)r = 436
12
X = 546
)r = 578
)r = 587
"r 8 x = 668
E x = 707
U
"o" )1 = 853
E
4 X 21083
8 4 1 4 8 12 16 20
distance from (L, mm
4
8
12
16

Fringe Order per Unit Thickness for Constant

Wavelength in P—6 Beam.

 

 

 

 

12

outer side

10 8 6

Figure 3.10.

 

163

8
'1‘
E
o
i
E 4
inner side
4 2 \ 4 6 8 10 12
distance from (L, mm
4
X = 1083
X = 853
8
)1 = 578
16
k = 436
A : 405
20

Fringe Order Per Unit Thickness for Constant
Wavelength in P-6 Ring.

 

 

 

164

10

 

Figure 3.11. Retardation Per Unit Thickness Along Axis
of CR-39 Modelfor Various Wavelengths.

 

 

 

 

20 16

Figure 3. 12.

165

0'
N = 546
._. E
l E X =1083
E \
° 2"
14 1.6 ..
"C? b
\
,<
E
n
“U 2 0.8
\
m

12 8 4 « 4 8 12 16 20

distance from Ci , mm

2 0.8
4 1.6
6 2.4
8 3.2

Retardation Per Unit Thickness for Constant
Wavelengths and Stress by Curved Beam
Theory for P—6 Beam.

 

 

 

 

 

12

10

Figure 3.13.

166

4
'1‘
E
o
:1.
.3 2
\
,<
E
4 2 p 2 4 6 8 10
distance from (1.. mm
2 1x = 1083
)r = 546
4 x : 405
6
8

Retardation Per Unit Thickness for Constant

Wavelengths in P-6 Ring.

 

12

 

 

25

20

15

10

 

167

m = 111(0), X = const.
CR ~39, t = 66 hr.
d = 9. 50 mm

X = 546 nm

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2
0',kg/mm

 

“1.0 1.1 1.2 1.3 1.4 1.5
0‘, kg/mm2

Figure 3. 14. Fringe Order and Retardation Per Unit Thickness
Versus Stress for Constant Wavelength in CR —3 9.

 

 

16

12

m/d, cm

 

168

__ _ _ compression

tension /

// // / /)\:1083nm
/ //
//
/
// /
/ / /
/
0.5 1.0 1.5 2.0 2.5 3.0

computed bending stress, kg/mm‘Z

Figure 3.15. Examples of Fringe Order Per Unit Thickness

at Constant Wavelength Vers'us Stress From

Curved Beam Theory for P—6 Beam.

 

 

 

is ochromatic order

 

169

m = m(0‘ ), X = const.
2, 4 polycarbonate resin

X=1.145
t=1.25hr. d=4.27mm ' 1”

 

O X = 1.244p.
2.0 0 o X = 1.34811.
X 21.44611.
X 21.55011.
X = 1.56011.
1.6 X 21.65011.
X = 1.914
1.2
0.8
0.4
0 0.2 0.4 0.6 0.8 1.0
2
01-02, kg/mm

Figure 3.16. Fringe Order for Constant Wavelength

Versus Corrected Stress in Polycarbonate.

 

 

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172

 

 

 

r = r(X) O 68% confidence limit
CR-39, t = 66 hr. I range
0 C (X)
r=M—) =0.— 0<0'1<2.1kg/mm2
RO(X ) C (X )
1.04
1.02
1"1.00
\\\ 1000 1100
0 98 §\\ X, nm
- Xo=546 \§\\
0'96 below 0' 11' R°(X C)) = 3.7“
Co‘ (1 0) = 3.7 x10_6kgu1cn’12
_ ~ -2
so (X0) — 15 kg cm
r = r(v)
CR -39, t = 66 hr.
r = Roll/1 : :00!) O 68% confidence limit
RO(UO) 6(1/0) I range
1.04
1.02
$41.00
0-98 CP5
0< 0'1<2.1kg/mm2 §\\E\_§
0.96 crz = 0
belowou, Ron/O) = 3.711 6 1 2
C (U ) : 3.7x10- kg.‘ cm
a o 2

So_(1/O) = 15 kg cm

Figure 3.19. Normalized Retardation Versus Wavelength and

Frequency for CR -3 9.

 

 

 

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m = m(0' ), X = const.
silicon

d= 0.914 mm

 

0.2 0.4 0.6 0.8 1.0 1.2

stress difference (0'1 - 0'2), kg/mm

Figure 3. 24. Stress Birefringence in Silicon.

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APPENDIX B. Dimensions of Tapered CR-39 Model and
Computation of Stress.

Model thickness = 0. 374 in = 9. 50 mm

Width of uniform neck = 0. 398 in = 10.5 mm

Half angle of tapered portion = a = 6. 870 = 0.1199 rad.
Load = 405 1b = 204. 08 kg

The equation for the radial normal stress in a wedge of half
angle a which is loaded axially at the apex is given in terms of r,
the distance from the apex as:
P cos 9
r l )

rd(a+z sin2<1

Along the centerline where e = 0 for the given model:‘

 

a = MM—
r 9.50 r ( 0.1199 +% sin 0.2398)
: 90.r15 kg/mmz
in the uniform neck:
P 204. 08 2

“r = X : (9.50)(10.15) 22'126 kg/mm

182

 

 

 

 

APPENDIX C. Radius of Curvature and Stress in P-6 Beam.

Since the center segment of the beam is in uniform bending,
its curvature is theoretically constant. The equation of the elastic

curve is that of a circle of radius r and centered at (a,b):

(x-a) +(y-b)2=r2

The location of three points on the elastic curve is sufficient for the
determination of a, b, and r. From the 35 mm photographic
negative, the measured coordinates of three points on the scratched

centerline of the given beam while it was deformed are:

Trial 1 Trial 2
Film xxxviiij Film xxxviiif
Point x y Point x y
AL 11134 7409 LS 11793 7347
C 11440 7418 C 12229 7364
EL 11749 7434 RS 12666 7395

Letting the first point be the origin, the coordinates are

Trial 1 Trial 2
Point x y Point x y
AL 0 0 LS 0 0
C 306 9 C 436 17
EL 615 25 RS 873 48

The parameters of the elastic curve are calculated to be

183

 

 

184

Triall Trial 2
a -.252 cm -.316 cm
b 13.784 cm 13.72 cm
c 13.78 cm 13.73 cm

The ratio between model size and image size on the negative of trials
1 and 2 respectively were 10.33 and 9. 89. The average radius of
curvature of the beam from the two measurements is R = 139 cm

= 54. 8 inches.

The stress may be calculated from the approximate energy
solution given by Den Hartog (14), since the Winkler—Bach Solution
(also approximate) serves poorly for beams of small curvature. The
energy solution is known to give superior results which are very near

the exact theoretical solution.

" I Y 12R R
here: h 21.5
_ bh3 _
1 _ 12 - 0.670
M: 3422? —1.7sr>
P: 120
R = 54.8"
i
so o‘=6269(y+.00335— 54.8)

Since -. 75 E y E + . 75, the maximum deviation from ordinary beam
theory is only slightly greater than 1.5%, A plot of stress versus

measured values of y appears in the main body of the report.

 

 

 

 

APPENDIX D. Computation of Stress Difference in Polycarbonate Disc.

For a disc of radius "a" and thickness "d“ compressed along a
vertical diameter by load "P”, the horizontal and vertical normal

stresses at the center of the disc are given by the equations (25a):

and also

,7.
XY

so,

For the experiment,

 

 

 

_ P
0—x _ Trad

_ _ 3P
o-y wad

—0' —0' -0' ~ 4P
0'1 2— x y — wad
d=0.l68in, a=1.540in
0'1 -0'2 = 4.921 P psi

0.3460 x10-2 P kg/mm

185

2

 

 

 

 

 

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w-.2 2 $4. .+ 24+ 8+ $+ m22+ Nw+ com .2
N44 .2 2 ~4N .+ m .m4+ m .ww+ 26+ A622+ 5+ omm .2
mom .2 2 mom .+ m .mm+ m 502+ 3+ 22+ 8 2+ 644 .2 6
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93.2 N N44: rims- m.mm- No- m - em- 44N.2
43.2 N com: mm- 02. am- 2+ m2- .422

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E __ u m H mm A

 

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APPENDIX F. Illustrative Normalized Retardation Data and Its
Statistical Treatment for CR -3 9.

For a population of N values of some quantity r, the average, r ,

is given by

.. 1N
rz-fiéri

The standard deviation, 0' , (root mean square of deviation from

average) is then:

1

N
0' :[fi 211(r.-r)2]1/2

For X = 707 nm, the normalized retardation for N = 18 different

stress levels is listed below as quantity r. Statistically, approximately
68% of the values which might be measured for stresses within the
given limits would lie within the range encompassed by the average

plus and minus the standard deviation. Sample calculations follow:

2 6

3 3

x10-

rxlo' Iri-rlxlo- (ri-r)

980 3 9
979 2 4
977 O 0
974 3 9
976 l 1
977 0 0
975 2 4
972 5 5
978 1 l
976 1 1
977 0 0
975 2 4

187

 

 

sum

r:<10-

973
980
977

977
978

17577

II
D-'
00

977

3

 

 

 

 

 

 

 

      

' MIC

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