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ABSTRACT

FOURIER MASK OPTICS

by John Frederick Kelsey

The Foucault knife edge test, the schlieren system,
the phase contrast method, and the concepts of spatial filter-
ing and theta modulation are all characterized by the same
mathematical expressions. These concepts are grouped here
under the title "Fourier Mask Optics" where the word mask
includes phase masks as well as the usual sense, i.e., ampli-
tude masks. A general mathematical procedure is derived from
basic principles and the manner in which the aforementioned
concepts fit into this mathematical framework is shown by way
of a simple example of each case. Several illustrations of
interesting subjects are also shown, with emphasis on the

schlieren system and the special case of isochromates.

FOURIER MASK OPTICS
BY

John Frederick Kelsey

A THESIS

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

MASTER OF SCIENCE

Department of Physics and Astronomy

1965

ACKNOWLEDGEMENTS

I would like to thank Professor E. A. Hiedemann for
his guidance and helpful suggestions. Thanks are also due
Dr. B. D. Cook, Dr. Y. F. Bow, Dr. W. R. Klein, Dr. W. G. Mayer,
and the entire Ultrasonics Group for their aid, encouragement

and helpful criticism.

ii

CHAPTER

II.
7111.

IV.

VI.

VII.

INTRODUCTION

Table of Contents

THE FOURIER TRANSFORM PROPERTY OF A LENS . .......

OBJECT FUNCTIONS AND THEIR FOURIER TRANSFORMS ...

ILLUMINATION OF THE OBJECT ..... ................ .

THE MASK FUNCTION AND FINAL IMAGE ..

A. Spatial Filtering

B. Phase Contrast .......... ...... .

C. The Schlieren System .............

CONCLUSION .......

BIBLIOGRAPHY

iii

12
17
19
21

2h

#1
he

FIGURE

10.

11.

12.

List of Figures

The Conceptual Steps in Assembling a Fourier

Mask System ........

A Spherical Wavefront Concentric About the Origin
of the Mask Plane ........

The Spatial Frequency Composition of a Square

waveAmplitUde Object 0.00....OIOOOOOOOOOOOOOOO0.0

The Selective Viewing of Crossed Gratings at

various Angles .OOOOOOOOOOOOOOOOOOOO
Examples of Phase Contrast Microphotographs ......

The Spatial Frequency Response of Various Source-
MaSk Pairs .0...0.00.0000....OOOOOOOOOOOOOOOIOOOI.

The Effect of Regular and Random Source-Mask

Pairs ......OOOOOOOOOOOOOOOOOOIOOO.

Sample Null-Schlieren Photographs ....

goo

Phasor Diagram of a Glass Wedge in a Schlieren

Interferometer ....

Examples of Schlieren Interferometer

Photographs ....

The Central Order of Light Diffracted by an

Ultrasonic Wave ..

Isochromate Photographs of Various Ultrasonic

Beam Patterns ........

iv

22

23

27

3O

32
33

35

37

39

MO

INTRODUCTION

There are several optical systems which are characterized by
the fact that somewhere between the object and its image a distribution
of light is formed similar to the Fourier transform of the light distri-
bution leaving the object. By masking this Fourier pattern almost any
information from the object can be selectively removed or passed to the
image.

Examples of Fourier mask optical systems are the following:
the Foucault knife edge test; the schlieren system; the phase contrast
method and the concepts of spatial filtering and theta modulation. It
should be noted that the word mask as used here includes phase masks as
well as the usual sense, i. e., amplitude masks.

For a qualitative understanding of Fourier mask optics
consider the idealized null schlieren system shown in Fig. 1d and the

conceptual steps (Fig. la, b, c) in assembling the system.

“'-"I- '1"-'lll‘-"'-‘-‘l 1' ..

. lxfi “JR

\

| ' ll lu'lllll|"“"l|'"'--"‘ "- --‘UIU'III

Q Q Q I

k
S
a
M
r
e
.1
r
u
0
F
a
g
n
.1
1
..D
m
e
8
8
8
n
.1
S
p
e
t
S
1
a
..u
t
p
e
C
n
O
C
e
h
T

Fig. 1.

 

Part (a) of Fig. 1 will be recognized as a simple camera. An arrow in the
object plane B is illuminated by the light at the top of the figure. An

image of the arrow is formed by the lens L in the image plane 3'. In

2

part (b) the object is illuminated by collimated light from a small

axial source. The arrow is now imaged in silhouette at B' and there is

an image of the point source in the plane A'. In the next step the arrow

is removed and a mask is placed in the mask plane A' so as to block the

image of the point source. Since no light passes A' the image plane 3'

will be in darkness. Now, if an object is placed in the B plane which

deflects the light sufficiently to miss the mask then it will be imaged

on the B' plane by lens L2.
It will be shown later that in the approximation of ideal

optical components the light distribution approaching the mask plane A'

is just the Fourier transform of the light leaving the object plane B,

and in turn the light distribution of the image at B' is just the Fourier

transform of the light distribution leaving the mask plane A'. A distinc-

tion must be made between the light distribution approaching and leaving

the mask plane A' since a difference is introduced due to the presence

of the mask. The mask will be characterized by a mask function M such

that M.operating on the light distribution E approaching the mask plane

A' yields the light distribution E' = ME leaving the mask plane A'.

The mathematical procedure used to determine the light distri-

bution in the final image is as follows:

l. The Fourier transform of the light distribution leaving the

object gives the light distribution approaching the mask plane A'.

 

2. The light distribution approaching the mask plane is
multiplied by the mask function M yielding the light distribution
leaving the mask plane A'.

3. The Fourier transform of the light leaving the mask
plane yields the light distribution of the final image at the plane 3'.

In order to understand the limitations of the procedure
outlined above the Fourier transform properties of a lens will be
derived from Huygen's principle in Chapter II. In Chapter III a general
treatment of objects is given along with the resulting light distribu-
tion approaching the mask plane. Before proceeding to the final two
steps outlined above, i.e., the effect of mask and the final image,
Chapter IV is introduced to discuss various source configurations in the
plane A. Chapter V begins with a general discussion of the effect of
mask and the final image, and concludes with a treatment and illustra-
tions of those Fourier mask Optical systems which have become most

useful.

II. THE FOURIER TRANSFORM PROPERTY OF A LENS

The following is a derivation of Optical image formation similar

to that first given by Porter(1). Let us begin by examining the situation

between the lens L and the mask plane A'. (see Fig. l).

2 Consider the

space description of a perfectly Spherical wavefront of light formed by
the lens L2 and converging toward the origin of the plane A'. The ampli-

tude of the electric vector may be written as

EB = ER exp [-ikR], (l)

where ER is the electric field at R, k is the optical wave constant, and

R is the radius of the spherical surface as shown in Fig. 2.

   

   

 

Fig. 2. A Spherical wavefront concentric about the origin of
the Mask Plane.

The coordinates u', w' describe points on the plane A' and the coordinates
x, y, 2 describe points on the Spherical surface B. The equivalence of
this spherical surface B and the object plane B mentioned in the intro-
duction depends on the intervening lens L2 and will be discussed later
in this chapter.

The optical disturbance at an arbitrary point P'(u',w') near
the origin can be calculated by applying Huygen's principle, i.e., by
considering each element of the spherical surface B as a secondary source,
and summing the contributions from all of these secondary sources at the
point P'(u’,w'). Let Q(x,z) be an arbitrary point on the spherical sur-

face B, the disturbance at P'(u',w') due to a point source at Q(x,z) is

then given by

dE(u',w') = C 1((01) ,_E exp[-ikR] 3% dx dz, (2)

where C is the normalization factor and K03) is the familiar inclination

(2)

factor given by

Km) = g5 (1 + cos a), (3)
n
where<1 is the angle of diffraction, i.e., the angle between the normal
at Q(x,z) and the direction QP'. In this case<x is always small and so

one has

Km) z - 11%. (A)

Now, by including the constant -ik/n in the normalization factor C,

Eq. (2) can be written

dE(u',w') = C ER exP[-ikR] 25% dx dz. (5)

The disturbance at P'(u',w') due to the entire surface B is just

E(u',w') = of]. ER eXp[-ikR] 213%?ng dx dz. (6)

Considering the vectors R and S'as shown in Fig. 2 the distance QP'

can be written as
__'2 ..
QP = R + s - 2Ros. (7)
Since S'is much smaller than R'the distance QP' may be approximated by
-I 1 I I
QP == R -'§ (u x + w z). (8)
With these approximations Eq. (6) becomes

E(u',w') = £17. ER exp[-%1£(u'x + w'z)] dx dz. (9)
B

If the light distribution is not uniform over the surface
B, then ER will be a function of x and z and must be left inside the

integral. In general E will be a complex function which will be written

R
as E(x,z). Using this notation Eq. (9) now becomes

E(u',w') =§ fE(x,z) exp [71} (u'x + w'z)] dx dz. (10)
'3

From Eq. (10) it can be seen that the light distribution approaching the
mask plane A' is just the Fourier transform of the light distribution on
a spherical surface at a distance R very large compared to u' and w'.
Equation (10) is the basic expression of the Fourier transform
property of a lens and will be used extensively later.
A similar derivation shows that the Fourier transform of the

light distribution E'(u'w') leaving the mask plane A' gives the light

distribution in the image plane 3' as
E(x',z') = gflE'(u',w') exp [- %1-(-(x'u' + z'w')] du' dw', (ll)
2 ' 2

where R2 is the distance from the mask plane A' to the image plane B'.

Equations (10) and (11) along with the expression,
E' (ugw') = M(u',w') E(uyw'). (12)

constitute a set of equations giving the final image E(x',z') for a given
mask function and object, provided we can prove the equivalence of the
object plane B and the Spherical surface B.

(3)

It can be shown from the Eikonal equation that the Spherical
surface and the object plane are equivalent under certain restrictions.
To investigate these restrictions consider the light distribu-
tion approaching the mask plane A' due to a perfect lens L2 with a
circular aperture. The light on the spherical surface B is limited to a

region x2 + 22 < hz. Making the transformation to polar coordinates

defined by

r cos 9 = x, r sin e = z,

8 cos 6' = u', and s sin 9' = w', (13)
gives

u'x + w'z = rs cos (9 - 9'). (1h)

Due to the circular symmetry of the problem, no generality is lost by
setting 6' = O. The light distribution at the mask plane, given by

Eq. (10), for this case is

E(s,e') = gffflEhfl) exp[-%1-<- rs cos 9] rder. (15)

The integration over r has been taken only from the origin to the margin
h of the spherical surface, because the value of E(r,e) outside the limits
of integration is zero. Assuming uniform illumination, E(r,e) can be
taken as a constant in the region r < h and can be taken outside the
integral.

(1*)

Using an integral representation for the Bessel functions ,

i-n Zfl '
JS=— [1 +5 d, 16
n( ) 2“ 0 exp (na cos a)] a ( )

Eq. (15) may be written

E(S,9') = Cf Jo (RE r) r dr. (17)

After integration and normalization, Eq. (17) becomes

 

E (5,9') = (18)
which is the well known expression for the amplitude of the Airy pattern.
Noting that the first zero of J1 occurs when the argument khS/R equals

3.83, we have the familiar equation for the diameter of the central disk

1.22%
28 = -E7E—- . (19)

It is important to note from Eq. (19) that for a given wavelength the
Airy pattern does not depend on the distance R alone but rather on the
angle tan -1 (h/R). This, in essence, States that any spherical surface
subtending the same angle produces an identical Airy pattern. For the
case in point we choose the surface with R equal to f, the focal length

of the lens L2.

10

The Eikonal equation goes even further than Eq. (19) to state
that the light distribution on any wavefront of an optical system can
be used for these calculations, provided there are no intervening obstacles.
With this in mind the restrictions governing the equivalence of the object
plane B and the spherical surface B can be immediately written down as
follows:

1. The lens L2 must be of sufficient aperature so as not to
constitute an obstacle.

2. The lens must be good enough to convert a plane wavefront
into a Spherical wavefront.

It should be noted that for an object that diffracts light at
large angles, the first restriction limits the distance from the object
to the lens and the proximity of the object to a marginal ray. This
condition is exactly the resolving condition, so restriction 1 may be
restated as:

1. The lens must be able to resolve the object. This
restriction is self evident and merits no further mention.'

The second restriction, on the other hand, is very difficult
to meet rigorously, even with quality optics. This should not be sur-
prising Since the Foucalt knife edge test and the first schlieren systems
were originally developed to examine minute deviations from geometrical
perfection in optical components. The extent to which this geometrical
restriction may be relaxed depends on the desired sensitivity of the

Fourier mask system. For many objects the sensitivity need not be too

11

high. Nonetheless, insufficient lens or mirror quality is still the horror
of precision Fourier mask methods. Since, however, the problem of lens
quality at most is one of finance, we shall proceed idealistically by
treating only the case of a geometrically perfect optics.

With these limitations in mind the light distribution approaching
the mask plane A' shall henceforth be assumed to be exactly the Fourier
transform of the light in the object plane B and no further mention of the

resolving power of the lens L or the geometrical quality of the optical

2

system will be made.

12

III. OBJECT FUNCTIONS AND THEIR FOURIER TRANSFORMS

'AS outlined earlier, the first step in determining the final
image from a Fourier mask system is to write an expression for the light
distribution leaving the object, and to take its Fourier transform deter-
mining the light distribution approaching the mask plane A'.

Objects in general will introduce both phase and amplitude
variations. The light distribution leaving the object can, in general,
be written

E(x,z) = G(x,z) exp [iH(x,z)] (20)
Following the usual convention this expression will be called the object
function. The amplitude factor G(x,z) and the phase factor H(x,z) are
independent and may be expanded separately. Amplitude and phase factors
of interest usually fall into two categories: those varying slowly which
can best be expanded in power series; and those varying rapidly and show-
ing periodicity which are best expanded in Fourier series. Other objects
which are neither periodic nor smooth can still be treated using the
fortunate characteristic, that the Fourier transform is an integral or
addition process. This allows the expression of different parts of the
object in whichever expansion is best fitted and then summing the integrals.
The integral limits take care of discontinuities.

With this in mind, the four cases which will be treated in some

detail for the one dimensional case, are the following:

13

A. G(X) = gan X a

B. H(x) = E bn xn ; (21)
C. G(x) = § an Sin(nk*x + fig) ;

D. H(k) = § Vn Sin(nk*x + ah) .

These are sufficient to handle any object function.

A. Pure Amplitude Object, Power Series.
In practice, this case is not of much interest, so only the
linear case will be treated. The light distribution approaching the mask

plane given by the Eq. (10) becomes

 

I
E(u') = 2—' x exp[- 25 (u'x)] dx. (22)
f _h f
The solution is

kh, kh ,

' 2 C' cos f—’u _ sin E” u
E(u ) = 2111 E;-[ _-Kh—_T—- kh ' 2 lo (23)

r“ (TE—u)

It is interesting to note that E(u') at u' = O is finite, even though each
term in the bracket is infinite. Furthermore, the intensity approaching
the mask plane is symetric about u' = 0 even for the case of the density

wedge.

B. Pure Phase Object, Power Series
First consider the case where the phase factor H(x,z) is a

linear function of x. The amplitude function is a constant so Eq. (10)

1h

can be written

E(u') = ffif: exp [4:45:3- - 131)] dx. (21.)
This is just the Airy pattern again with a displacement of the origin to
u' equal to be/k as would be expected from inserting a glass wedge. The
next term in the series x2 is exactly the problem encountered when examin-
ing thermal gradients(5). Also, by examining small amounts of wave front

curvature introduced by the x2 term the light distribution just in front

of and just behind the focal plane can be determined (6).

C. Pure Amplitude Object, Fourier Series

Consider the case of a pure amplitude object of the form
G(x) = R an cos (nk*x + gfi), (25)

where k* is the wave constant of the basic periodicity of the object.

In the more recent literature, the quantity k*/2n = 1/x* is called the

Spatial frequency. For this case Eq. (10) becomes

a' exp [1g ]jhexp[-ix(£l-1-L+nk*)] dx. (26)
n n 11 f

E(u') =%

co
2
n=-w

For large h this expression approaches the Dirac delta function at the
values nk* = ku'/f. This condition can be more easily recognized in
terms of x as

7’; = nifi- (27)

which will be recognized as the grating equation. From Eqs. (26) and

(27) it is apparent that there will exist a set of discrete lines whose

15

intensity are given by

[\D

a2 (28)

n

I =
n

H'nNIO

So it can be seen that a lens truly Fourier analyzes the spatial frequen-
cies Of a pure amplitude Object illuminated with coherent light. It is
the masking of this spatial frequency Spectrum that is the basis of the

concept Of Spatial filtering(7’8).

D. Pure Phase Object, Fourier Series.
The case Of an arbitrary phase Object gives rise to a light

distribution in the mask plane Of the form
E(u') = th exp[£ V sin (nk*x + g ) + ikx] dx. (29)
f -h n n n

This integral has been solved explicitly(9). The light amplitude in the
nth diffraction order for large h is given by
---(V1) Jk2<V2)

Em(u') = E E ... Em =-m Jm.2k2-3k

2 3 3

(3O)
Jk3 (v3)~~ exp [-i(k2¢2 + 1.3.53 + mm.
This expression is difficult tO evaluate except for cases where higher
harmonics are present only in small amounts. This case has received con-
siderable attention because to certain approximations it characterizes the
Optical effect produced by an ultrasonic wave showing finite amplitude

distortion.

16

In the section dealing with isochromates we shall have occasion
to treat the case Of a pure Sinusoidal ultrasonic wave. For this case
Eq. (30) reduces to

Em(u'> = Jm<v1) . (31)

This result was first given by Raman and Nath(lo). From Eq. (31) it can
be seen that phase Objects are 223 simply frequency analyzed in the mask
plane, rather a single Spatial frequency phase Object gives rise to many
diffraction orders. Great care must be taken tO eliminate phase irregular-
ities over the Object if one wishes tO use an Optical system as a frequency

analyzer.

17

IV. ILLUMINATION OF THE OBJECT

In Chapter III enough examples were worked SO that the light

distribution approaching the mask plane can now be determined for most

Objects Of interest. Before proceeding tO the effects Of masks (Chapter V),

mention should be made Of how to treat source configurations in the plane A

Other than the point source which has been used exclusively until now.

If a second point source were added in the source plane A,

there would be two Fburier patterns in the mask plane A'. These Fourier

patterns which we called E(u',w') in Chapter III were developed assuming

a single point source illumination. The function E(u',w') developed in

this way is usually referred to as the spread function. For the remainder

Of this section we will call the Spread function S(u',w') and reserve

E(u',w') for the

If the
electric vectors
lap. If the two

intensity Of the

‘Sum Of the spread functions from various point sources.

two point sources mentioned above are coherent, the
Of the Spread functions will add in any region of over-
point sources have no fixed phase relationship, then the

spread functions will add in the mask plane.

For simplicity we take unit magnification between the source

and mask plane, then we may write the preceding statements as

E(u',w') =17 E(-u,-w) S(u' + u, w' + w) dA, (32)
A .

for the coherent

case, and

I(u'w') Eda]. I(-u,-w) |S(u' + u, w, + W)|2 dA: (33)
A

18

for the incoherent case, where E(-u,-w) and I(-u,-w) represent the source
configuration in the plane A. These are the general expressions needed
tO determine the light distribution approaching the mask plane for any
source and Object.

Later some qualitative mention will be made Of how different
source-mask pairs affect the resolution and Speed Of various Fourier
mask systems, but all theoretical treatment will be for the simple case
Of a single point source. For this case E(-u,-w) is a delta function at

u = w = O and Eq. (32) becomes
E(u'w') = S(u'1w')1 (3,4)

which was our original convention.

19

V. THE MASK FUNCTION AND FINAL IMAGE

AS was outlined earlier the mask function M(u',w') multiplied
by the light distribution E(u',w') approaching the mask plane yields the
light distribution E'(u',w') leaving the mask plane. The final image in

the plane B', is just the Fourier transform of E'(u',w') as shown by

E(x',z') = gaff M(u',w') E(u'w') exp[-%§ (u'x'+w'z')]du'dw'. (35)
A.

Equation (35) can be written more compactly in terms of the
Fourier transforms Of M(u',w') and E(u',w'). Utilizing convolution

theory, Eq. (35) becomes

E(x',z') if] E(x,z) mil + % , £1 + g) dxdz, (36)
B 2 2
where N is the Fourier transform Of M(u',w'). This form has the advantage
that we need not take the Fourier transform of the Object function E(x,z),
but we now must take the Fourier transform Of the mask function M(u',w').
This in general is easier tO do since mask functions are usually constants
over various regions Of A'.

DeSpite the mathematical simplicity Of Eq. (36) it is difficult
tO follow the physical significance Of the effect of masks on various
information from the Objects. we, therefore, shall return to the somewhat
more complicated Eq. (35). In this form it is easy to understand the
effect of mask on information from an Object, because the object function
can be written in two parts: one part that is affected by the mask, and

one part that is not. Since this will be the general procedure followed

20

it is worthwhile to begin by considering the case Of nO mask at all before
proceeding to specific Fourier mask systems. With no mask present E'(u',w')
= E(u',w') so the final image, given by Eq. (11), becomes
I I C I I ik I I I I I I
E(x ,z ) =‘R E(u ,w ) exp [wi- (u x + w z )] du dw . (37)
2 A' 2

The inverse Fourier transform of E(u',w') is

E(x,z) = gal/E(u',w') exp[?15£(u'x + w'z)] du'dw'. (38)
AI

By redefining the coordinate system Of the image plane as

x' = -'% x and z' = -'% z (39)

Eqs. (37) and (38) yield
f-E(x':z') = R2 E(x,z) ° (’40)

This is the result expected for no mask, that is, neglecting
the brightness,the image is just the Object inverted and magnified. With
the redefined coordinates x', z' the Object and its image will be identical

' respectively.

functions Of x, z and x', 2
We now have the mathematical procedure to handle those special
types Of Fourier mask systems which have developed into systems Of prac-
tical importance. We will begin by briefly treating the case Of Spatial
filtering and simple phase contrast, and finish the chapter with a treat-
ment Of three special types Of schlieren systems. The first, for viewing

very feeble phase Objects, is a null schlieren system. The second, for

making quantitative measurements, is a schlieren interferometer. The third,

21

for viewing ultrasonic beam patterns, is the isochromate method.

A. Spatial Filtering
It was shown earlier that a pure amplitude grating is exactly
frequency analyzed in the mask plane. The expression for the spatial
frequency distribution in the mask plane, given by Eq. (27), can be

rewritten as a vector equation and is given by

13/21:: 1' kS/21rf (£11)

From Eq. (hl) it is Obvious that a circular Obstacle Of radius b in the
center Of the mask plane is a high pass filter, blocking all Spatial
frequencies k*/2n < kb/2nf. Also a circular aperature is a low pass
filter and an annular aperature is a band pass filter. By Observing the
filtered image one can see just which Spatial frequencies go into making
up the Object.

As an example Of Spatial filtering Fig. 3 shows the Optical
analogy Of the common electronic practice Of using a square wave input tO
test "time frequency" response. The photographs on the right are the Spec-
tra Of frequencies that are allowed to pass, i.e., frequencies that are
not blocked out by the mask, and the resulting image is on the left. It
is possible to see in Fig. 3 the spatial frequency composition Of an amp-
litude Object.

The presence Of small amounts Of the even harmonics is due to the
fact that the Ronchi ruling used introduced phase irregularities through the

transparent parts Of the ruling.

22

 

all frequencies

 

0th and lst orders

.r‘nll..a\l‘.' .' u

I. .....afiI’I

V .‘II”. {‘90

‘EI‘I’II‘SQI

Iii? .31.-..‘3 I I

0“-I’OI I’li

-‘l.".“"

‘0 I... u ‘I‘.”-

u I -. , II III

 

a

0th, lst, and 3rd orders

.0 I l.s0 I.. II

. JDQIODCI.

 

 

lst, 3rd, and 5th orders

 

Fig. 3. The spatial frequency composition Of a square wave

amplitude Object

all
so. 1 0°
NO 2 h5°
NO. 3 90°
NO. h 135

 

Fig. A. The selective viewing of crossed gratings at various

angles

2%

Since Eq. (#1) is a vector equation there is an angle 9
associated with the vector B'and the grating wave vector E*. The
selective passing Of crossed gratings at various angles by placing a slit
at the same angle in the mask plane was the subject Of a most famous
(11) (12)
study by Abbe and is the essence Of the concept Of theta modulation .
As an example Of theta modulation Fig. h shows four crossed gratings
making up the numerals l, 2, 3, and h. Again the light distributions

that are allowed to pass the mask plane are shown on the right and the

final images are shown on the left.

B. Phase Contrast

<13) 1..

The phase contrast method was developed by Zernike
1935. It has been a revolutionary tool in examining microscopic bio-
logical Specimens which are Often transparent but dO introduce small phase
shifts in the Object function. Since the phase error H(x,z) is small, the

Object function can be written as
E(x,z) = Eo exp[iH(x,z)]¢3Eo [l + iH(x,z)], (#2)

where E: is the intensity which is constant everywhere over the Object
plane. The light distribution approaching the mask plane is given by

Eq. (10) as

25

E(u',w') = gfon exp D? (u' x + w'z)] dx dz
B

(”3)
+.%%éyaE01H(x,z) exp [%E (u'x + w'z)] dxdz .

The first term on the right Of Eq. (#3) gives an Airy pattern in the
center Of the mask plane. Since microscopic specimens usually contain
high spatial frequencies, the second term on the right gives a distribu- _
tion Of light generally at some distance from the central area Of the
mask plane.

Placing a quarter wave plate over the central area introduces

a masking function given by
mm = b [- g 1 (1..)

which Operates on the first integral to produce a n/2 phase retardation
and a fractional reduction Of amplitude ., b2 being the transmission Of
the phase plate. The mask function M(u',w') has little effect on the
second integral if the spatial frequency Of H(x,z) is sufficiently high to
give negligible contribution in the central area. The light distribution

leaving the mask plane then is

E'(u',w') = -%flEoib exp[-%E(u'x+ w'z)] dxdz
' B (1+5)
+ gflEoimmz) exp [-%1£ (u'x + w'z)] dx dz .
B
With the redefined coordinates x'z', the second Fourier transform gives

the image as being just the Object but with the constant term multiplied

by -ib. The image then is given by

26

E(x',z') = Eoi [-b + H(x,z)] (R6)

Now it can be seen from Eq. (#3) that H(x,z) now adds in parallel in the
image, not perpendicularly as in the Object. This means that the inten-
sity, which is the absolute value Of Eq. (#6), varies with the phase factor
H(x',z') and is no longer a constant as in the Object function Eq. (#2).
It can also be seen that the contrast can be controlled somewhat by varying
b.

Examples of microphotographs are shown in Fig. 5 with and without

the phase plate over the central area.

C. The Schlieren System

All systems used for visualization Of phase Objects by the use
Of Opaque Fourier mask are termed schlieren systems(1u). There are many
types due to various source-mask pairs and various configurations Of lenses
or mirrors. Three special cases will be treated here, the null schlieren,

which has been used as a model Fourier mask system from the outset, the

schlieren interferometer, and the case Of isochromates.

l. The Null Schlieren System
AS would be expected from the name, the null schlieren system
is used for viewing very feeble phase Objects. The mask Of the null system
just covers the source image in the mask plane. Let us begin by calculating
the system's sensitivity for the case of a small circular source-mask pair

Of radius a. The insertion Of an Object will produce a displaced image in

27

   

a. T88t18 ZOOX Phase Contrast b. Testis 200x Bright Field

      

c. Testis 860x Phase Contrast d. Testis 860x Bright Field

 

e. Intestine 200x Phase Contrast f. Intestine 200x Bright Field

    

d .,

g. Kidney 860x Phase Contrast h. Kidney 860x Bright Field

   

Fig. 5. Examples Of phase contrast micrOphotographs

28

the mask plane. The displacement d can be found from Chapter III for the

various types Of Objects as

d = El f refractive wedge (R7)
+ k* .
d = - k- f sinusoidal amplitude grating (#8)
k-X-
dn = n‘E— f sinusoidal phase grating (#9)
th

where b1 is defined in Eq. (213) and dn is the displacement Of the n

diffraction order for a phase grating. In the third case account must be
taken of the amount Of light in each order which is a function Of the

maximum phase Shift V. In this case the average displacement of light

can be written as
‘/'u' J (V) du'
d = n . (50)
fJn(V) du'

 

For a given displacement d the light in a circular image that

will pass a circular Obstacle Of the same radius a can be found from

geometrical considerations(15) as
S(d) - S l - g-[arc cos 2— _.Q_ 1 _.EE )1/2] (51)
_ O n 2a 2a #a2 ’

where So is the brightness Of the displaced image or the average brightness
in the case Of a phase grating. If a vertical slit source and a line mask

both Of width 2a were used then S(d) would be given by

S 'g— for d < 2a
O 2a

S(d) = (52)

S for d > 2a.
O

29

These two cases are plotted in part a and b Of Fig. 6.

The amount Of light passing the mask can be increased by adding
more small circles or lines, in the source and mask planes. This will
multiply the right hand sides Of Eq. (51) and (52) by the number of identi-
cal sources added. This only increases the speed Of the system just as
though one had used a brighter light bulb. It should be pointed out that
the new sources and masks pass nO new information to the final image. In
addition we must now take into account the possibility that the displaced
image Of a particular source might be blocked by a nearby mask. This
gives rise tO curves like Fig. 6c which show the light passing through a
Ronchi ruling source-mask pair as a function of the separation d.

Still another problem is encountered when multiple source-mask
pairs are used. A very small Object in the Object plane B will diffract
light uniformly over the mask plane. This coherent light passing through
the various parts Of the mask, eSpecially if the mask shows any regularity,
will tend tO interfere beyond the mask and may cause multiple images.
Figure 7a is a null schlieren photograph Of a loaded plexiglas beam taken
with a Ronchi ruling source-mask pair. This photograph very clearly shows
the undesirable effect Of multiple images.

The multiple images may be eliminated by randomizing the source-
mask pair. This does not eliminate interference in the final image, but
it does.render the interference unrecognizable. The final image then appears
as if there were just some loss Of resolution. Figure 7b is a null Schlieren

photograph Of the same loaded Plexiglas beam taken with the source-mask pair

S(d)

 

 

i l

-2a 0 2a #a

0d-
-#a

a. A point source and a dot mask

S(d)

 

 

L 1 1 l
-#a -2a 0 2a #a

b. A slit source and a line mask

oh-

S(d)

 

 

l l
-#a -2a 0 2a #a

c. A Ronchi ruling source-mask pair

Ob

S(d)

 

 

 

OLWJ

d. A random source-mask pair

 

 

Fig. 6. The

 

 

 

 

 

 

 

 

 

 

 

 

 

2: is

 

spatial frequency responce of various source-mask pairs

.. ‘ 93;.- ‘3‘:

 

0}.
931-3

31

shown in Fig. 6d. This photograph clearly shows only a single image, and
some loss Of resolution just as anticipated.

Figure 8 shows several null schlieren photographs taken with the
source-mask pair shown in Fig. 6d. These photographs seem tO be an

improvement over those shown in earlier literature.

.‘Q’ch,’ 9.?,¢‘-,Ig.o’.v.
~ .. . ‘ .‘ ‘4 .v I a
' ° ’ h“ ‘1 ' (0"...

.'.0‘ -- b. “A.-- -.-

 

Regular Source-Mask Pair

...-.....‘0.

 

Random Source-Mask Pair

Fig. 7. The Effect Of Regular or Random Source-Mask Pairs

33

 

a. A Warm Hand b. A Solder Gun

c. A Light Bulb 6. Acetone Vapor

 

 

e. A Burning Match f. Warm Breath

Fig. 8. Sample Null-Schlieren Photographs

'3#

2. The Schlieren Interferometer

It has been found(l6) that by placing an Obstacle in the mask
plane that is somewhat smaller than the image Of the point source one can
construct an interferometer which will produce photographs exactly like a
Mach-Zehnder interferometer that is adjusted so that one fringe covers the
entire field.

By way Of an explanation the case of a thin glass wedge in the
center Of the Object field will be treated for the one dimensional case.
As Shown in Fig. 9a the Object function can be taken in two parts. The
first E1 is due to the area surrounding the glass wedge, and its Fourier
transform in Fig. 9b is very similar to the Airy pattern for a full
aperture. The Fourier transform Of the light due to the wedge E2 is
just a displaced Airy pattern in the mask plane. A small Obstacle
is placed at the center Of the axial Airy pattern in Fig. 9c. This has
the effect Of subtracting a delta function in Fig.9d . ‘The Fourier
transform Of each term is taken in Fig. 9e and these are added to Obtain
the final image as Shown in Fig. 9f.

It can be seen from Fig. 9f that there will be a series Of fringes

in the central area. A bright fringe will appear at each interval given by
H092) = (N +1/2) 2n (53)

just as with the Mach-Zehnder interferometer.
Several examples Of schlieren interferometer photographs are

shown in Fig. 10.

 

 

 

 

 

 

 

EENI(lJI)

Fig. 9. Phasor Diagram

Interferometer

Of a Glass Wedge in a

 

 

 

Schlieren

36

 

         

         

11101-11 "in.
1...: 3:9: I
«31090 voootsqoab .t b
so: I090 .vzlcxi
mu #9 05919.1“ 0-¢6909005

   

 
  

     

.. .J.-a.-.--¢- Eur-H.341. I‘vuy- .Idul

L’s-J ...: I I.II a 1.

 

.IIQO
iosgi¢0o 3-06:5.
§I§IO»IA>OII nu .5. .

   

r .OVOIOiizoOozn
{Iottxn

    

   
     

 

 

(x')

2

constant + E

 

 

 

E(x')

Fig. 9 con't.

37

 

b. Ordinary Glass

a. A Hot Light Bulb

 

 

 

d. A Hot Solder Gun

C.

Water Between Two Flats

Examples of Schlieren Interferometer Photographs

Figure 10.

38
3. The Isochromate Effect

The isochromate effect was discovered in 1938 by Hiedemann and

(17).

Osterhammel It is an interesting and beautiful way tO view ultrasonic
beam patterns. The method has been used to determine ultrasonic transducer
alignment and also some absorption measurements have been made by this
method.

AS was shown in Chapter III, for a purely Sinusoidal ultrasonic

wave that meets the restrictions Of behaving like a phase grating the

normalized diffraction pattern in the mask plane is given by

I = J:(V), (5#)

n

where the maximum phase shift V for an ultrasonic wave is given by

V = k (“max - [10) D. (55)

Here (“max - no) D is the difference in Optical path length through the
undisturbed medium and through the most compressed part Of the sound wave.
TO shorten the notation let us define (”max - no) D a [D].

Now consider a mask that allows only the central diffraction

order to pass. The light passing the mask is given by

I' = J3 (v) . I (56)

The intensity Of the unobstructed central order is plotted verses the
maximum Optical path difference [D] for several colors of light in

Fig. 11.

39

  

 

 

 

|.O
)—
I n _ .
I (u) -5 '-
: mm
-\ an
o 1 ,‘ _.‘
uoool zoool 3006A ,_

[D]

Figure 11. The Central Order Of Light Diffracted by an
Ultrasonic Wave.

It is seen that for a white light source and an Optical path
length difference Of 1500A, the zero order will be essentially red. Since
the zero order is the only one that is allowed tO pass, all parts Of a
sound field that have D = 1500: will appear red and all parts Of the sound
field that have D = 21003 will appear blue, hence the name isochromate.

Since all Of the Bessel function pass through zeroes, similar pictures can
0

be made by passing any order.

Some examples of isochromate photographs Of various ultrasonic

beam patterns are shown in Fig. 12.

 

 

Figure 12. Isochromate Photographs Of Various Ultrasonic Beam Patterns

#1

VI. CONCLUSION

Throughout this paper an attempt has been made tO choose term-
inology and mathematical expressions that are directly associated with the
physical situation. For instance, if the usual procedure Of Optical math-

ematics had been followed, the final image distribution given by

E(x'z') = 3% f/‘muuww E(u',w') exp [Ti-‘5- (u'xw'z'ndu'dw', (57)
2 2 '
Al

would have been expressed by way Of the convolution theorem in terms of

the Fourier transforms Of M(u',w') and E(u',w'). This would be a much

neater form but it fails to give any physical meaning tO how the mask affects
various information from the Object.' Hence, a mathematical procedure has
been advanced rather than a single expression. This procedure has given
considerable insight into the Operation Of Fourier mask systems, and has

in the case Of many Of the photographs shown, led to the Optimum configura-

tion of sources, lenses, and mask for the particular Object in question.

10.

ll.
12.

13.

l#.

15.

16.

17.

A.

#2

VII. BIBLIOGRAPHY

B. Porter, Phil. Mag., 11: 15h (1906).

. G. Stokes, Trans. Camb. Phil. Soc., 2, l (l8#9).

. N. Watson, A Treatise on the Theory Of Bessel Functions

 

(Cambridge University Press 1922), p.20.

Born and E. Wolf. Principles Of Optics (Pergomon Press 1959),

 

p. 111.

. B. Temple, J. Opt. Soc. Am., #1, 91 (1957).

. Lommel, Abh. Bayer. Aked., 15, 233 (1885).

Marechal and P. Croce, Compt. Rend., 237, 706 (1953).

. L. O'Neill, I.R.E. Trans. - P.G.I.T., 2, 56 (1956).
. E. Hargrove, Private Communication.

. V. Raman and N. S. N. Nath, Proc. Indian Acad. Sci. §.h06

(1935).

Abbe, Archiv. f. Mikruskopische Anat., 2, #13 (1873).

. D. Armitage and A. w. Lohman, Appl. Opt., 3, 399 (1965).

Zernike, 2. Tech. Phys., 16, hsh (1935).
Toepler, Pogg. Amn., l3#,19# (1868).
Francon, Modern Applications Of Physical Optics,

(John Wiley and Sons 1963), p. 6#.

. L. Gayhart and R. Prescott, J. Opt. Soc. Am. 39, 5#0

(19#9).

. Hiedemann and K. Osterhammel, Proc. Indian Acad. Sci.,

§. 275 (1938)-

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