Albert Ernest Smith
candidate for the degree of

Doctor of PhilosoPhy

Dissertation: The Effect of the Source Aperture
on Diffraction Grating Images
Outline of Studies
Major subject: Physics
Minor subject: Mathematics
Biographical Items
Born, November 1, 1927, Windham, Vermont
Undergraduate Studies, Atlantic Union College,
1944-49
Graduate Studies, Michigan State College, 1949 -
1954
Experience: Graduate Assistant, Michigan State
College, 1949-1955; Instructor in
Physics, Michigan State College,
1955-1954; Assistant Professor of
Physics, Union College, 1954 -

THE EFFECT OF THE SOURCE APERTURE
ON DIFFRACTION GRATING IMAGES

BF

Albert Ernest Smith

AN ABSTRACT

Submitted to the School of Graduate Studies
of Michigan State College of Agriculture
in partial fulfillment of the require-

ments for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1954

Approved :2 ézi 2 ‘éfi‘aé‘

 

Albert E. Smith
1.

In a previous work Smith and Hause (1) derived
an expression for the intensity distribution in the
Fraunhofer pattern of a diffraction grating with N
apertures as a function of the breadth of the il-
luminating aperture where the source aperture was
illuminated in the noncoherent mode. Almost simul-
taneously Takeyama, et a1. (2) arrived at the corre-
sponding expression for a grating with a small number
of apertures illuminated in the coherent mode. Both
of these expressions were extremely cumbersome in
case of a diffraction grating with a large number of
apertures. No experimental verification existed for
the coherent mode, and the verification for the non-
coherent mode was limited to the behavior of the
subsidiary maxima as a function of source breadth in
the case of a grating with a small number of apertures.

In this work both expressions mentioned were sim-
plified so that they could readily be used to find
the intensity distribution in the image formed by the
diffraction grating of many apertures. Particular
attention was given to the intensity distribution in
the principal maxima.

The results obtained for the diffraction grating
were compared with the results which Van Cittert (5)

Albert E. Smith
2.

obtained in his calculations of the intensity distri-
bution in the images formed by_a single diffracting
aperture. Within the limits of the assumptions used
to simplify the grating theory no differences were
found. In the grating theory it was assumed that the
source aperture was located on the axis of the grat-
113 and that the angle subtended by the source aper-
ture was sufficiently small that the first order
approximation could be used for the sins of the angle
and that N was large.

These results were verified in the laboratory by
direct measurement of the intensity in the grating
image. TwO different gratings were used of different
N values and measurements taken over a wide range of
values of the source aperture breadth. The results
obtained were compared graphically with the theo-
retical results. Three different modes of illumina-
tion as suggested by Stockbarger and Burns (4) were
used and the results of the comparison were inter-
preted in terms of the degree of coherence or nonco-
herence of the source aperture. Measurements were
also made on the image formed by the single aperture

as a means of verifying Van Cittert's calculations

Albert E. Smith

3.

and of checking the conclusion of the present cal-

culation that the two theories give the same result.

No difference was evident.

(1)

(2)

(3)

(4)

Smith A. E., and C. D. Hause, Fraunhofer Mul-
tiple Slit Diffraction Patterns with Finite
Sources,JDpt. Soc. Am. 3g; 426-450, (1952)

Takeyama, H., T. Kitahara, and T. Matubayasi,
0n the Mathematical Treatment of the Effect of
the Width of the Slit on Fraunhoferrs Diffrac-

 

tion PhenomenonTPart II),iSci. Hirosima Univ.
IS.A.) lg, 159-146, (1951)

 

Van Cittert, P.H., Zum Einfluss der Spaltbreite
auf die Intensitatsverteilung in Spektrallinien.

 

Stockbarger, D. C. and L. Burns, Line Shape as

a Function of Spectrograph Slit Irradiation. J.
Upto Soc. Am. 232, 37 -5 4, 53)

 

 

THE EFFECT OF THE SOURCE APERTURE
ON DIFFRACTION GRATING IMAGES

By

Albert Ernest Smith

A THESIS

Submitted to the School of Graduate Studies
of Michigan State College of Agriculture
in partial fulfillment of the require-

ments for the degree of

DOCTOR OF PHILOSOPHY

'Department of Physics and Astronomy

1954

5/4137?" 7
g/Jé7

ACKNOWIEDGEMENT

The author wishes to express his appreciation to
Dr. C. D. Hause, at whose suggestion this work was
undertaken, for his constant encouragement and
guidance in its completion. He is also grateful to

Dr. R. D. Spence for his suggestions and for checking
the final report.

WW

I.

II.

III.

IV.
V.

VI.

TABLE OF C ON TEN TS

INTRODUCTION . . . . . . . .
THEORY . . . . . . . . . . .
A. Coherent Illumination . .
B. Noncoherent Illumination.
DESCRIPTION OF EXPERIMENT. .
A. Coherent. . . . . . . . .
B. Noncoherent . . . . . . .
EXPERIMENTAL RESULTS . . . .
SUMMARY. . . . . . . . . . .
LIST OF REFERENCES . . . . .
APPENDIX . . . . . . . . . .

. . 24

LE.
1.
2.
e.
4.
5.
e.
7.
s.
9.

10.

11.

12.

15.

14.

15.

16.

17.
18.
19.
20.

The General Optical Arrangement .

LIST OF FIGURES

Components of the Rectangular Image . .

The Rectangular Amplitude Function. . .

Calculated Image Coherent Mode

'1

Noncoherent

2J/Ns.
41/113.
GA/Ns.
BVNs.
ZA/Ns.
4x/Ns.
Gi/Ns.

Central intensity calculated. Coherent Mode

Optical Arrangement

Grating C. Coherent

Noncoherent

for coherent mode .

" Broad Source mode .

Lens mode

Mode.

Central Intensity

and Half-intensity Breadth. e o I o o o e

Grating C. Coherent Mode,

Lens Mode.
half-intensity Breadth. .

Central Intensity and

Zl/Ns . . . . .

4N/Ns . . . . .
GA/Ns . . . . .

Page

12
12
2O
21
22
23
31
32
33
35
36
39
39
39

49
50
51
52

53

21.
22.
23.
24.

25.
26.
27.
28.

29.
30.
31.
32.
33.

34.
35.
36.
37.

38.
39.
40.
41.

42.
43.
44.

Grating C. Lens Mode
w n n w
w n n n
Grating C.

21/Ns

41/Ns . . . . .

Bl/Ns

Broad-source Mode.

intensity and Half-intensity

Grating C. Broad Source Mode.
a n n n

Grating C. Broad-source Mode

Grating B. Coherent Mode.

and Half-intensity Breadth .

Grating B. Coherent Mode.

0! I! n I!

I! n I. I!

fl 3! I! n
Grating B. Broad-source Mode.

Central

Breadth .

2X/Ns .

4A/Na e
Gl/Ns . .

21/Ns.
4l/Ns.
GE/Ns.
Bl/Ns.

Central

intensity and Half-intensity Breadth .

Grating B.
I! II I I
II II I! O!
Grating B. Lens Mode.

Broad-source Mode.

half-intensity Breadth . . .

Grating B. Lens Mode.
w n n w
I! N N It

Single Aperture.

2.7K/Ns

5.4‘YNs
8.15/Ns

2.7yN. . . . . .

504VN3. o o e e

8011/N80 e o e e

Coherent Mode.

Central

Intensity and Half-intensity Breadth . .

Single Aperture.

7' fl 1!

Coherent Mode.

21/1)
4W

671)

Central Intensity

Central Intensity and

54
55
56

57
58
59
6O

61
62
63
64
65

66
67
68
69

7O
71
72
73

74
75
76
77

45.

46.

47.
48.
49.

50.

51.
52.
53.
54.
55.
56.
57.

58.

Single Aperture. Coherent Mode. BX/D. .

Single Aperture. Broad-source Mode. Intensity

at the Center and Half-intensity Breadth.

Single Aperture. Broad-source Mode. 2.81/D
w n u a n 5055flD
I! n n '7 " 8.41/1)

Single Aperture. Lens Mode. Intensity at the

Center and Half-intensity Breadth . . . .
Single Aperture Lens Mode. 2.81/D . . .
N I! II " 5.61/13 0 e a

Single Aperture. Lens Mode 8.4l/D . . .

Grating B. Coherent Mode 14 X/Ns . . .
' " Source Aperture Removed . . . .
Single Aperture. Coherent Mede 5.5hfl).
u w w n 9.4§/D.

" " . Source Aperture Removed . . . .

78

79
80
81
82

83
84
85
86
94
95
96
97
98

I. INTRODUCTION

The problem of predicting the characteristics of
the image formed by an aberrationless Optical system
is one which depends to a large extent on the dif-
fraction effects produced by the components of the
system. If the amplitude and phase distributions in
one cross section of an optical system are known to-
gether with the geometry of the components which
follow this cross section, it is possible, at least
in principle, to determine the intensity distribution
for any other cross section desired. In practice,
however, the problem may be extremely difficult except
for certain definite planes of the system.

With these ideas in mind there are three places
of interest, places at which computation is reason-
ably simple, in the system to be considered: the ob-
ject or source element, the diffracting or image-
forming element, and the image or.foca1 plane. In
this work the problem will be taken as one in Fraun-
hofer diffraction, where a plane wave, or set of plane
waves, from the object source is incident on a dif-
fracting element and the resulting disturbance is
focused on an image plane to be examined.

A.technique which has been commonly used is to

calculate the pattern produced by a diffracting ele-

2.

ment when illuminated by a point or line source.

This method results in an expression which is a func-
tion of the characteristics of the diffracting ele-
ment and of the angles of incidence and of observa-
tion. If the source has finite dimensions, it is then
treated as being made up of a large number of elemen-
tary sources of infinitesimal dimensions, each of
which contributes to the final image. The manner in
which the elementary sources contribute to the final
image depends on their relative phases. Two limiting
cases exist for which the pattern may readily be cal-
culated. The source may be radiating as a unit, all
elements having the same phase, in which case the am-
plitudes are additive, or there may be a random dis-
tribution of phases in the source, in which case the
intensities are additive. These usually are referred
to respectively as the coherent and noncoherent modes
of illumination.

The ideas expressed here have been used by Van
Cittert (l) and others (2-4) to determine the charac-
teristics of the image formed by a single diffracting
aperture in a Fraunhofer system as a function of the
mode of illumination and of the dimensions of the ob-
ject source. The results of this study have been ap-
plied to predict the line-shape characteristic of a

prism spectroscOpe (5) and to predict the image-

3.

forming preperties of the microscope.

The double slit diffracting system has been treat-
ed extensively (6, 7) and the characteristics of the
images formed have been applied to the problem of de-
termining the diameters of stellar (8) and microsc0pic
objects (9). ‘

It has often been tacitly assumed by spectro-
scepists that the single-aperture results apply equally
well to the far more complicated system of the diffrac-
tion grating. The problem considered here is the cal-
culation of the intensity distribution in the grating
image as a function of source dimensions for both co-
herent and noncoherent illumination and a comparison
of these calculations with experimental results. The
results, both theoretical and experimental, indicate
to a very close approximation that grating apertures
do behave as single apertures when the gratings have
large numbers of lines. The work has been done only
for small incidence angles.

In a former work (10) the problem of a grating
with a small number of lines and a noncoherent source
was treated with particular reference to the behavior
of the subsidiary maxima as the breadth of the source
was increased. A theoretical treatment of the corre-
sponding coherent case was carried out almost simul-

taneously (11) with interesting predictions as to the

4.

variation of the intensity in the center of the
principal maxima as a function of source width; but
apparently no experimental verification exists.

The formal techniques for extending the problem
to the diffraction grating with any number of slits
was essentially complete with the treatment of the
small grating. In fact, in the treatment of the grat-
ing with a small number of slits expressions were
found which, although extremely cumbersome, were
equally valid for a large number of apertures. It
therefore seemed of interest to carry out the com-
putations for a grating consisting of a large number
of apertures and to attempt to simplify to such an
extent that it would be suitable for rapid computation.

The computations were carried out as a two dimen-
sional problem assuming rectangular sources which are
either completely coherent or completely noncoherent.
Since these conditions are not realizable experimen-
tally the results of the theory and the experiment
cannot be expected to agree completely. This and
other sources of difficulty will be discussed later.
There is. however, a considerable range of agreement
and the experimental results found can be taken as

verification of the principal predictions of the theory.

5.

II . THEORY

The optical system under consideration consists of
a source of finite breadth w located in the focal plane
of the collimating lens. The collimator is followed by
the diffracting element, a grating of N slits in the
case studied, and the pattern produced is imaged in the
focal plane of the camera lens.

The source subtends an angle a =W/f at the colli-
mator and hence the plane waves from the source are in-
cident on the grating with angles ranging from +°‘/2 to
- 0‘/2. The source radiates monochromatic light of wave-
length A. The separation of the apertures in the grat-
ing, the grating constant, will be a.

To find the effect produced in the image plane by
a line element of the object a method is used which is
similar to that of Born (12).

From a single element of the source a plane wave
with an amplitude namenuis incident on the grating
at an angle 1. Where 3': (:1, k=211/‘). and r is the
distance parameter. The amplitude at an observation
point P in the image plane will be, if the angle of ob-
servation of the diffraction is 9, the result of smmning
the contributions from all the grating apertures:

”ex"

3*“ Z 63m)" (M6+A~£~L\

Q: Owe (1)

M$O

.Q as monogamou m.“ Gofipowammav mo onGa 059 an}. HB 9333
.N}+ op N\§| Scam monaeb no moxep onqm mafia .H mm popwfimamop m.“ 00.30.... 23
.Ho pGoEoHo mafia w pom 00:33“qu .Ho mamas one .9» new Heoauao Hwaocow one H .wam

NZSQ szxwdw NKDEwu<
3:: 02:25:18 uozaom

_

_
a _

_

. m m.

.

_

_ Ob ..

 

.lllillli )allli

7a

The manner of finding the intensity expected at
the point P depends upon the phase relations between the
object elements. In the coherent case it will be as-
sumed that the source radiates as a unit with all ele-
ments in the same phase and having the same amplitude.

When this is true the resultant intensity is found to be
oak

I =[N2, where A = //'a.di (2)

whereas in the noncoherent case there is a completely

random.distribution of phases and the intensity can be

”[13? di _ (3)
- /2

If the anglese Gand i are assumed to be small, it is

written

possible to substitute for the sines of the angles the
angular values themselves. Simultaneously the exponent
involving r may be omitted as a constant multiplying

factor. Thus equation (1) is simplified to read
nu“

“new.”
a 2 a5 Z 6 (1')

Mao

A. COHERENT ILLUMINATION

A different set of numbers may be used to denote
the grating apertures, and equation (1'), when rewritten
in terms of these new numbers, becomes without changing

its value

“‘H '.a¢eu)
M 6
0.: “Z 6‘ where M__ m (4)
' 2
neg—M

Since n is always an integer N must be an odd number.
This restriction would appear to limit the generality

of the theory. It will be seen later, however, that if

N is large the contribution from the correction involved
if N-l is used in place of N has an absolute value less
than l/N of the total.

In order to obtain the amplitude from the complete

source, equation (4) is substituted into equation (2)

with the result

a». “M k (6+)
in s ‘

H: a./ Z E a“
-072

”:0"

Since this is a finite series it is possible to inter-

change the order of the operations without changing

the value A... M ‘9‘ W2. 1: ,
Z' in GB 3“ “ .1
H = (1., 6 f e d
Ana—H -°‘/2

Expanding the exponential terms and integrating
M=M
wmfisekmh‘i’z +1 5 Mm‘cOAi-smki'é]

M

H: 2.9: M
ks A‘s-H

The imaginary terms may be seen to vanish if rearranged,

since

A»(‘“)M.*~(’M‘Q) + “(M10 “(1'69 2 o AIM/m: AM. 1M3] :: 0
M

(flu) (N) ’ M 5 o

9.

The remaining term can be expanded to read

m2" III=M (5)
g: Q. Alba/“8.8 (“1‘2") A Mm’ié (ck-26)
:3; ____2___ .+ ___;1__.
rn nn
”1:-" , Mz—M
This expression may be simplified by letting
0'”: “333‘ .9'= 435—9 - (6)

In which case

ms." nu.” . . .
n = 9.9 [ Mm(°"+29')+ W49?

£6 m m (7)
MIL-H ”‘3'"
or MsM ma"
R : 9.22 I: ӣ2964. AW Mr]
Its an no (8)
Ins-M m..-”
where

+= o<'+29’ x~=°¢'—29'
x I (9)

making use of the facts that

MON) 4, m9”) ._ 2 MM! MM’IJ :x

0"“ (+NO I'FCT ’ 757

 

M
it is seen that
2," Ms"
H = 2:; [CU + M” +2 M%*”] (10)
1T M
M=' m:|
where (1323?. The factorlr is introduced to sim-
plify a later result. ”
The series of the form Z“?'which appears
twice in the expression for the amplitude is too cumber-
some to evaluate exactly when M (or N) is large. The
remainder of this analysis will be devoted to an attempt
to arrive at a reasonably close approximation to its

value in a more tractable form. Although the first two

10.

steps that follow are not necessary to the logic of the
final result they are helpful.in understanding the ap-
proximation which is being made.

Consider a function defined as follows:

“’0 2 “it“ . 0<X<21I
(11)
((30:21!) :9“) I xao,21r,...,2m , gun’s".

The Fourier development for f(x) is the infinite series
24414:“ (12)
m=l ”\

A theorem.proved by B6cher (13) states that the Fourier

development of the function f(x) converges to the value

f(x) at all points where the function is continuous and

to the value l/2[f(x+o) + f(x-ofl at all points where

the function is discontinuous. It converges uniformly
throughout any interval which does not include or reach
up to a point of discontinuity.

The function f(x) consists of a series of straight
lines with negative slopes and with finite discontinui-
ties occurring at intervals of x that are multiples of
2w. If as a first approximation M is set equal to in-
finity in equation (10) the result should give an ap-
proximation to the amplitude. on

. . , . -
H t? [w +: AfiN * Zm'arx] . (13)

M“

11.

This becomes upon substitution of equation (12)

H= SJ [0" +-(’(X*) +¥(X‘)] .

TT

(ILL)

If (11), the definition of (12), is now inserted, the

result obtained is
04)!“ <er( ()5)

+ ..
a. 0.. ' --x _X here I
F) — " [at “r /2 4] , u) o<¥‘<a1t

11'

which becomes by use of (9)

I , I I ’ (d'+2e'<2“
a: 3: 0"” “(Liz-e) - (“-53”) 0 . 0"
1r 2 ’ 0(6)! -29'<zw ,

This expression, when plotted as a function ofG', con-
sists of two series of straight lines with opposite
slopes displaced from.the origin by an.amount @é_as in
Fig. 2. The result as seen either graphically or ana-
lytically will be a series of rectangular images at
values of G'equal to multiples of fl as in Fig. 3.
Making use now of the periodic nature of the function
f(x) to evaluate the amplitude outside of the region
between.+$&andxfig, for example with positive values to

the right of + W2
_. - o<x’<2il'
Ream-webrm] - I
1x 2 ) ~2I<x <0
which becomes by equation (9)

929:1 °<’..(°L'_t3°') -(«'—26') °‘°"+’°"“( <17)
1r ’2 a ) -2(<c:'-26'<o

12.

.

X: ><>

Fig. 2 The components of the rectangular amplitude function.

 

 

 

 

 

 

 

 

 

— 1— —
_ hag . I—
-w o 1' 6'e»

Fig. 3 The rectangular amplitude function with no account
taken of the Gibbs phenomenon.

13.

It is now possible to sum up the results obtained in
equations (16) and (17) in a form which describes the

rectangular images.
H: 00’. J ”9é<9’< 01/5

He. 0 J —7r+°}2’ <9'<«°‘/'2 «wt 92”<e’<1-°‘/é.
It can be seen from the periodic nature of the Fourier
series that this pattern will repeat itself in such a
fashion that the images appear at intervals of u along
the G'axis. This picture, rough though it is, corre-
sponds in some particulars to the anticipated grating
image, and it would appear that there might be some uses
for such an approximation.

In treating the whole as a rectangular image, the
behavior of the approximation at the points of discon-
tinuity has thus far been ignored. It is, however, well
known that in the limit, as M approaches infinity, the
value of the Fourier series approaches the value of the
function only at points where there are no discontinu-
ities, and at points where there are discontinuities it
exceeds the value of the function by an amount which de-
pends on the value of the discontinuity. This means
that instead of the simple rectangular images pictured
in Fig. 3 the images would have local maxima at points
close to the points of discontinuity. This behavior

comes under the general heading of the Gibbs phenomenon,

ILL.

which also describes the behavior of the finite series
2“?“ It is possible to progress further with the
approximation by a.method which follows.

For any partial series (finite number of terms)
the sum oscillates about the value of the function with
deviations the magnitudes of which increase as the point
of discontinuity is approached. If M is allowed to in-
crease, the number of oscillations increases, and as has
been noted the partial sum converges to the value of the
function except at the discontinuities. It is possible
by the method which follows to locate the places of max-
imum deviation and to evaluate by another approximation
the difference between the partial sum and the value of
the function. From points located in this manner a
curve can be drawn which represents the amplitude dis-
tribution more closely than does the rectangular image.
Or, alternatively, the intensity distribution can be
constructed directly from.the expression obtained.

From.equation (12)

MaM‘
5x“) :Z A?)
m=| (19)

msM

8““) =2 f‘coa m1 <1! . m“

In:

N
‘1
r-w
N;

S
E
L——J
o.
X

15.

But
12%: 2 Awmfi%i '

and from (4) M'+‘% a N/2, from which it follows

that

5"“) -_-_ f)! [“VZ + ,2»... (“/21 dx

2“ X/z
0

+ x ” Nx/a ax (20)
e 2 MW: .

 

The difference between the partial sum S.(x) and
the function f(x) will be given by

H.,“) = 5N“) --F(x\

 

(21)
N7: Tr-X
Rum): Jed—[fl “a a 4?)
.Auwlvfih
=- -‘«/.+ +f Ax
‘2.Au~y z, (22)

As previously stated, the partial sum oscillates

about the value of the function so that R~(x) is alter-

nately positive and negative. The maximum values of

IRJx)I measured in a direction perpendicular to the x—

axis will occur when ¢¥JM=O .
x

l6.

ARK“) :- M “/2 = 0 , when X: Y4 123,—; ,!=’1.3r--(23)
dx ZMx/z

The evaluation of the integral involved in equation

 

 

(21) follows that of Bocher. ‘Addition and subtraction
‘ - h.
of the term I w Ax = ( ”£1” to the right
0

 

‘ 0
side of equation (21) gives the result that
N's/z .
RJ") :2 ——“/2 t 4114:: dx
.‘ I»: ,ebw’“%z __..1..~ww;] Jr,
01-, a 2M”: x
Ky; .
RH!) = -Tt/2 + anx + Be“) (24)
0
where
x
.. I -
Bum .. /2[MW/2(-i—;/2-%—) six. (25)

An integration of equation (25) by parts gives the

result
54:) = V" { [wa/e(=~.;1§%")]:+ (2.)
_. ‘X
[*mm(xzwe 2:; lam} .

The two functions which occur here,

2 MX/a -x a“; xécmx/z —2_ M1x/2.
x My; x‘ A...“ ”2

 

both become infinite when xg2fi2 But in an intervalo<x<b
where b<2t'both functions are continuous except at the

point x:O where they are not defined and they both ap-

17.

preach finite limits as x approaches zero. It is
therefore possible to find a positive constant K such

that

[a‘yjfiga’w < K , mifj‘x‘wza fif’Ve <K
for o<x<b , where b<2TI .
It then follows that

[8,.(X)[<2K/N . when o<x<b (27)

This inequality shows that Bu(x) approaches zero as N
tends to infinity for all values of x which satisfy
the restrictions. In the case of the grating, however,
there is the difficulty that N remains finite, although
large, and therefore B.(x) does not necessarily go to
zero. But if a further condition is put on b, that b
may never come close to 2“, that is to say that x must
remain small, then it follows that K will never become
very large andIBu(x) will be negligible in comparison
with the other terms in equation (24). RN(x) can then

be rewritten omitting the last term.

"2. -
R~(A : --_“/a “I” 'flil‘ AX
or
RxCx) = ”/2 + 81(N"/2) (29)

It is now possible to do two things which give infor-
mation as to the behavior of the resulting pattern.

First the values of the extrema of Ru(x) can be eval-

uated at the pointst=%§F.

18.

RNCXQ) '5 nit/2. + $5 (1W) I ,0: [12,3) . ..

There will be two such differences, one for x*and the
other for x'. When these differences are added to
f(x‘) and to f(x‘) at the intervals of 2fl/N and a smooth
curve is constructed through the points thus located,
the result is two curves with a set of local maxima
and minima. When plotted on theEfiaxis the origins of
the two curves, x20 and 16.0, are displaced froma'go
by amounts of -°"/2 and +°}72 respectively. The result-
ing amplitude is then the sum of these two curves.
As the value of<X'is increased the two curves slide
over each other introducing variations in the ampli-
tude distribution. It is readily seen that the maximum
value of the amplitude and thus of the intensity oocus
when the two first maxima of the two curves coincide
at 6.0, that is when Ot'u2W/N. Iftx'is increased beyond
this value, the intensity at the center decreases and
passes through a minimum, after which it oscillates as
a function oftX'but never again attains its first
value. For the larger values of¢X'the maximum values
of the intensity in the image pattern occur at the
sides of the image. They also oscillate as a function
ofo‘ , but at a higher level than the central values.
Alternatively it is possible to substitute equa-
tion (28) directly into equation (21) and the result

19.

into equation (10).

_. x) H
5"“) _ R'l + )0 (21)

s m == -‘"/2 + 34% + (Hg)
N

2 -x/a + 3; (W2)
H = 0%[d' 4‘72. ..x72 +Sc(N"/2J +9 (Kr/2] (10)

By use of equation (9), this becomes
H: 0% [Saifi/a(d'+29"( + g, iN(d'~26')/2}]
or in terms of the original variables at andQ ,

Fl .3 «445,1: 19.: («+2e)] + S: [aim-26)]; (30)

The intensity then follows as

_ “ks Q; nits (CL-2.63:”?- (31)
_|_::: ’S;[.q(u+29)]+ [Ti

20.

w-

 

IO’

 

 

 

Ck: ‘ ‘ ‘—
“°’& ‘0 “M 9—)
Fig. h Theoretical distribution of intensity
in the ima e of the coherent source of angular
widthu =2} Ns.

 

21.

l2.b

 

 

 

c j": 3 “72. 6—?
Fig. 5 Theoretical intensity distribution for
a coherent source of angular widthOI=H/Ns.

22.

H

not

 

 

 

 

,, I. i .
”"1 O “I;
9 --"

Fig. 6 Theoretical intensity distribution for
the coherent source of angular widthU=67k/Ns.

14*

23.

 

 

 

o 1 ‘ 4
.92 o «,2
. 9 —-» ’
Fig. 7 Theoretical intensity distribution for
the coherent source of angular width“ =3 Ns.

 

24.

B. NONCOHERENT ILLUMINATION
The equation (1') for the amplitude contribution

from a single line element of the source,
01:”!

 

' Simon)
0. =-. a. em + (1.)
Ila-O ’
may be rewritten
I ewkued)
a 2: Q6 '&6(6+i)
l - e’ .

The intensity contribution from the line source will

then be .
mane“) ' _ e‘WM‘W)

l-E' , .
Ia). 3' C1:[ 7:?“(9')” ] ' 73-1350“)

 

__ o‘ l -- mes(e+L)

_ ° I — mks (e+i\
a M2 Nggfi (Bi-i»

:La° -

 

M“ 333 (9“)

m-Jf-l .
2 at [N +2: (N-M) Cm N315 (94)] .

’4 (32)

25.

The intensity at the angle 6 from the entire
source of angular width oc will then be given by
equation (5).

I : [alt d3 . (3)
L2
0V2 m:N-| .
I: a:[ N +2 :( N-M) mNhs(9+“)]aé.(53)
-d/z m=| .

The order of Operations may be interchanged, and upon

expanding the cosine of the sum of two angles,

I : Nd +
Mall-l “/2.
25(N'M)f [wmkse mmksé ~MMXeGMmksL] AL.
mzl -"1

When the indicated integration is carried out the re-

sult is
m: N”!

‘1

I 3 N“ “Ks (”4%) ““69 “mist

Using equation (6) and writing in terms of the sum

and difference of angles
MIN

.. [:3 All. ' e' ' new-291]
I-N«+?3E ( “)1: m(ot+2 )+M(
4n=n
The 2/ks may be removed as a multiplying factor and

the term under the summation expanded as a group of

sums o

26.

 

 

' “an“ (d"+290 flfdedumldLQel
' ML + Z __
I : Na + NIZI M Ms: m
AWN"! ng-l . ' .
.— Z MMW'J-N') — Z Motto! ~26)
M:l M5!
The sum of the last two terms is known.
Mar" ,«s’f', . I
a hM(¢’+26’) - MU‘"29)
~« +~z m + m
Ms! M:(

ME (“4299/21 Mp1)— (a! '419’)/2]

MUW-fl-ZB'VZ]

. . _. L. ')
- ”(Macaw/z] WWW 29/2)

M [(0" *2 e"/21

 

 

Rewriting in terms of equation (9),

flaN'l «a» ‘

I‘m" +NZMX+ 4-sz?“ (:55)
._Y(x*) .. Y(X')
where
. Ny/ . ' (N—DX/
‘[(x\ = ’4“ z A“:- Z (:56)

 

,0Q~x4z
The two finite series which occur as the second

and third terms are of the same form as those which

27.

occurred in the coherent case. The contribution from
these two plus the first term will be of the same
form as the coherent amplitude expression. The one
difference is the location of the extreme, which is
controlled by the number of terms in the series.

The difference between the values of the rec-
tangular image and that of the partial sum will be,

as in equation (22),
" ' - x
PM!) = 3V2 + f M"? K' 0" l2 - (37)
O

2.4daflfi

And the location of the extrema will be given by

53.“) _ 45w (2 N-I)X/2_

_. : (3
dx 2 aux/2.

 

01'

(2N-I)Xn/a :flw 1 x,=':_‘_;‘;l£| ’ 1:523... (38)

If, as is required by the approximations made, N is
large and l is small, it will be found convenient
later to make the location approximately

x, -“-" JV“

OP
9;: 12,42.» , buxom (59)

Thus it is seen that the extrema occur here twice as
often as in the coherent case. The value of the dif-

ference at these points of extrema will be the same

28.

as in the coherent case except for the factor of N

which multiplies all of the first three terms.

RN“) :4; .155. + S; (NXaS I 1:1,2,3, . .. (28')

The remaining part of the intensity distribution,
the fourth and fifth terms, are still rather complex.

A.workab1e approximation may be found as follows:
at: ”-I

V?“ _ “ii/2. ‘5’:(N")Vz : diam:
__ ~ My; (55)

Mal

 

If x is small and N is large this may be rewritten

 

Aim: ””2.
Y“) 1"— 2 x (40)

The extreme values of this expression are found when

 

 

de “ a [1.3% -x NM’ l‘Wz em “a __ o
a}. _- x?
w Emmy [MM/2 - Nx cm Nut/a] : o
2 . (41)
This can be seen to be true when
NZ :3“ ' 3‘: “293...!” O1. [{x-ztflu "”2 (b) (42)
A graphical analysis shows that equation (42 b) is
satisfied when
Nut/a r“ (2’9“)w/2. , 33'12'3’” . (42 b')

SUbstitution of these results into (40) give the result

29.

Yb“ : O ' whfin X223'W/K ' ,R: [’2’3J , , .

Y“) =9- 2/x :2 N/(zm—u) ' ”he" x =ék'M/N (H3)
or writing in terms of of and 6’,
Y(die') = 0 I (Johan 6'=h/N"°‘72 I 6': g—fiT/K

= 2 9’=(2—~——"""_ av '= «2415-1:
got, 2“ /2. ' 6 2 2” (44)

The construction of intensity distribution from
the results found follows the same procedure as was
used to find the amplitude distributions of the co-
herent case. A rectangular image is described by the
first three terms in equation (35). The subtraction
of the Y terms at the points and of amounts given by
equation (44) completes the picture.

It is seen that since the terms in Y are positive
and at a maximum and are to be subtracted when R, has
a positive maximum value and are zero when Ru has a
maximum negative value that the net result for the
noncoherent case is going to be a rather smooth curve.
A construction of the intensity distribution by draw-
ing the curves bears out this result.

And again, alternatively, as in the coherent case,
equation (10), the intensity distribution may be eval-

uated as

 

30.

I a Not' + N 54v) + NSNOV) —Y(x*) 4.1M

45)
which becomes
I = N S E”“”‘7a] +'N 5i[‘”"”‘72]
_ M2 NX”/2 _ M72 (46)

 

x72 x72

01‘

I :2 N S.- [W’WQPNROE + N St [0“) 33;? (W298

_ .2[N%3(gzea ”M1[N§§(d'2°fl (4'7)
1% (a +16) thUVZB)

 

 

The construction of the curves was made with the
results as indicated in the accompanying figures.
Figs. (4) - (7) and (ll) are for the coherent case.
Fig. (4) - (7) show the intensity distribution as a
function of the observation angleG . Each curve rep-
resents a different value of the source aperture and
is calculated for the source width which will give
the maximum visibility of the structure in the image.
In addition to the curves for the intensity distri-
bution in the image for a given source width it is
possible to find the intensity in the center of the
image, ate :0 , as a function of“ . Thus equation

(31) becomes for the intensity at the center

31.

 

 

 

 

‘fii 5’ 42 B-—9
Fig. 8 Theoretical intensity distribution in'

the image of a noncoherent source of angular
widths: =21/Ns .

I2

32.

 

 

 

A . A

 

«a o ”i 6-»

Fig. 9 Theoretical distribution for a non-
coherent Source with°I=" NS.

[2.

33.

 

i

 

 

 

‘ L L

 

‘5”; 0 ”i

6 -9

Fig. 10 Theoretical distribution for a non-
coherent source with ot: Gl/Ns .

54.
It: [a S; (NSukW/qr
(48a)
Fig. (11) illustrates this result.
The construction for the noncoherent mode of 11-
lumination is illustrated in Figs. (8) - (10) and (12).
The intensity at the center of the pattern is given by

v. - N'OAS _ - z[(N-l)akS“/q
I. _. 2 S.[( 6g] 2 ~10“) MOW] (48b)

and the result is plotted in Fig. (12).

C. COMPARISON WITH THE SINGLE APERTURE

It is of interest to compare the results derived
here for the diffraction grating with those given by
Van Cittert, €€§§££3n (1), for the single diffracting
aperture. For the single aperture illuminated in the

coherent mode the intensity distribution independent

of constant multipliers is
I: [8}.(U’o49)d + S'WWHJ

This when translated into the terms which are used in

this work becomes
2
I = {Si [021 (oi-29%] 4. 34:0): “(26)“? (49)

Where D is the breadth of the diffracting aperture,

the result for the noncoherent illumination becomes

1' = s. [vs/,1 mm] + a {Damnefl

' 1iflh -
T (at 29) ~ 1. D1? (ok-tze) (50)

 

_i __ _____,____.
9;} (a-ze) 13%“ +2.6)

35.
I4

7

I3

(I)

 

 

”/Ns ‘Wus 5%; 0""

Fig. 11 The intensity at the center of the image of
the coherent source as a function of the angular
source width as calculated from the theory.

36.

 

 

J _A

o - . x .
2Ms W... 6V”: d a,

Fig. 12 The intensity at the center of the image from
the noncoherent source,cs a function of the angular
source width as calculated from the theory.

57.

These results should compare with equations (31)
and (47). When cognizance is taken of the fact that
the product (N-l)s=’Ngwhich is equivalent to the D of
the single aperture, is the total expanse of the grat-
ing aperture it is seen that the two appear to be
identical. That is to say, the grating aperture acts
within the limits of the approximations used as a
single diffracting aperture. The results stated for
the single aperture are subject only to the condition
of small angles of incidence. For large N it would be
expected that there would be no detectable difference
between the single aperture and the grating in either

of the two modes of illumination treated.

38.
III. DESCRIPTION OF EXPERIMENT

The experimental techniques used to verify the
preceding theory were essentially quite simple. The
source of light was a high.pressure mercury arc, air-
cooled so that it could be Operated for long periods
in a light-proof housing. The slit used as a source
aperture was a Spindler and Hoyer bilateral precision
slit. The collimating and camera lenses were achromats
of approximately one meter focal length. A.951-A
photomultiplier tube in an American Instrument Company
microphotometer was used to read the image intensity.
The photomultiplier tube was located behind an exit
slit constructed of a pair of razor blades with a
separation of 0.015 mm. The readings of the inten-
sity were taken from left to right on the image at

intervals of 0.05 mm on a precision screw.

A. COHERENT

In order to achieve coherent illumination of the
source aperture the lamp was placed at a distance of
10 meters from the aperture and a slit of less than
1 mm breadth was then placed about 10 cm from the lamp.
With this arrangement a very small part of the light
source was contributing to the excitation at the
source aperture and hence it was reasonable to regard

it as predominantly coherent.

39.

 

 

83323: go sees :83: one ma .93

A . I’C’!’
I.
flvl.
ls 1""

nofipmawEdHHH M0 opoE zpcoaonooz ose ma .wfim

‘ .v
I I p II - I
., .I..y’1.
. l .
ll..l.ll,.sl..s!
I...
. ~\‘ -.(Jtl‘ ..I.A\. ..
l 1‘}. . \(t ‘ Ix: II

can» pnosoao
mafi>fioooa waapowamafip

— {b

 

ohdphoam
coauom

 

40.

Essentially this same arrangement was used by
Stockbarger and Burns (5) in their work which was an
outgrowth of van Cittert's (l) theory. They used a
distance of two meters between the source and source
aperture and the curves which resulted were not ex-
amined in detail as to their comparison with theory.
Particular notice was made of the resolving power of
the instrument under different types of illumination
and the general characteristics of the line shape
which influence resolving power were noted.

Evidence that the arrangement is essentially
correct exists in the fact that originally a distance
of 5 meters was used between the source aperture and
the source with the result of a systematic difference
between the predicted patterns and the observed pat-
terns. The structure visibility was much less than
predicted and the positions of maximum visibility
were displaced toward source apertures wider than
predicted. These differences were markedly reduced
with the increase of the separation to 10 meters. It
was believed that little improvement would be noticed
by increasing the separation beyond this point and tie
dimunition in light intensity would not be tolerable.

There are at least two ways in which the source
aperture differs from the ideal coherent radiator

assumed in the theoretical treatment: (1) It is not a

41.

source in itself but a slit and as such has its own
characteristic diffraction pattern. This was found
to give little or no trouble for small openings of
the source aperture, but as the breadth of the aper—
ture was increased an effect was noticed which was
the result of the edge of the aperture acting as a
source. (2) There is some mixture of phases in the
plane of the aperture, some incoherence. This effect
does not lend itself to‘neasurement, but from a study
of the diffraction patterns it is seen to increase as

the breadth of the source increases. The setup is

illustrated in Fig.(13.)

B. NONCOHERENT

Two different methods of illumination are de-
scribed in the literature (5) as approaching the
noncoherent mode, a broad source placed before the
source aperture (broad source mode) and an image of
the light source focused on the source aperture
(lens mode). The arrangement for the broad source
mode is shown in Fig. (14) and the lens mode in Fig.
(15). The same apparatus was used for the noncoherent
as was used for the coherent case.

It is readily seen that complete noncoherence

will not be achieved by either of these arrangements

42.

since complete noncoherence would require that every
point of the light source illuminate every point of
the source aperture. The comparison between the the-
oretical and experimental results will indicate which

mode more closely approximates noncoherence.

IV EXPERI MENTAL RESULTS

To obtain the results represented in Figs. (16)-
(27) a Cenco transmission grating having 5000 lines/
inch and covered with a mask which left an aperture of
breadth 5.42 mm was used. The three different modes
of illumination were employed as illustrated in Figs.
(13) - (15).

The results for the coherent mode in Figs. (16)-
(19) are to be compared with the calculated curves in
Figs. (4) - (8). The experimental curves are construct-
ed by interpolating between the experimental points.
Sample sets of experimental points are included in
Figs. (16a) - (20a). The other curves which follow
are constructed in the same fashion but the experi-
mental points are not included. There were very few
instances when it was necessary to deviate from the
experimental points in the interest of smoothing the
resulting experimental curve. In no case did the
deviation exceed 11%. The experimental images
follow the predicted image in many respects. The
changes in the pattern occur when predicted and the
maximum visibility of the structure close to the
source widths predicted. The biggest differences
occur in the structure visibility which is not as
great as would be predicted for the completely coher-
ent source. The asymmetry which appears in Figs.(18)
and (19) illustrates a characteristic of the

44.

images of the coherent mode to be extremely sensi-
tive to the alignment of the diffracting element in
the beam. It is possible by rather laborious adjust-
ments to obtain almost completely symmetric images.
Some of these are shown later. The principal reason
for the sensitivity is the narrow beam present in the
coherent mode. For large values of the source aper-
ture most of the radiation will be contained within a
beam but little broader than the source aperture it-
self. The lack of structural visibility is without
doubt a result of the mixture of noncoherent elements.
This would tend to flatten out the image as seen by
comparison with the noncoherent mode.

The curve for the intensity at the center of the
image as a function of source width in Fig. (20) is
to be compared with the curve in Fig. (8). .aa might
be expected from the observations already made con-
cerning the visibility of the structure in the image,
the second minimum which occurs at the center will
not be as great relative to the first maximum as pre-
dicted. In other respects the two appear to be the
same. The curve is not carried beyond the place where
the structure appears in the image.

The maximum value of the intensity can be pre-
dicted from equation (31) to occur at a source width

01 ='27\/u6 (5') , The subsequent source widths

45.

for maximum visibility of the structure occur at in-
tegral multiples of the above value.

In all of the graphs for the coherent case shown
the first maxima occurs at the place predicted or ex-
tremely close to it. This was the case when the dis-
tance between the light source and the source aper-
ture was ten meters. When a distance of five meters
was used there appeared to be a systematic error in
that the maximum always appeared at a value of source
width somewhat greater than was predicted. .A mixture
of noncoherent elements would tend to produce suchd
deviation since in the noncoherent mode there is no
definite maximum at the center; but a value which the
central intensity appears to approach asymptotically.

The curve for the half-intensity breadth which
occurs also in.fig. (16) will be discussed later.

Figs. (20)- (23) for the broad source mode and
Figs. (24)- (27) for the lens mode represent approaches
to the noncoherent illumination and are to be com-
pared with Figs. (9)- (12). In general it is seen
that the broad source images have rounded tops and
the central intensity quickly approaches an almost
constant value. The lens mode apparently more closely
approximates the noncoherent case with its square
tapped images and more slowly ascending central in-

tensity.

46.

Comparison of the half-intensity breadths in
Figs. (16), (20), and (24) show some facts of interest.
In the coherent case the half-intensity breadth re-
mains at an almost constant value for a considerable
range of source widths and then increases at a linear
rate. In the broad source mode and in the lens mode
the half-intensity breadth begins to increase much
sooner. These facts in conjunction with the behavior
of the central intensity become important in the
choice of an optimum source width in a given spectral
instrument and in some other Optical instruments in
which the diffraction properties of the image become
important. It is often desired to obtain a maximum
intensity and still retain a relatively narrow image.
It can be seen from these graphs that there does exist
an Optimum source width which satisfies these condi-
tions. Further the optimum width for the coherent
case will be almost twice the value for the coherent
mode that it will be for the lens mode. The broad
source mode lies between the two only slightly greater
than that for the lens mode. This would further sub-
stantiate the conclusion that the lens mode is more
nearly noncoherent than the broad source mode.

These conclusions verify the predictions which
van Cittert (1) made for the half-intensity breadth

of the image formed by a rectangular aperture.

47.

The results represented in Figs. (28) - (40) were
essentially duplicates of those shown in Figs. (16) -
(17) except they were made with a different grating.
The grating used was of the transmission type having
90 lines/mm. An area of 3.53 mm breadth was used to
obtain the images observed. The details of these
images are essentially the same as those obtained with
the Cenco grating. There is some improvement as the
asymmetry observed before is not as apparent in these
images and the intensity at the center of the coherent
image as shown in Fig. (28) follows the predicted
curve more closely.

Measurements were also made with this grating in
the first order images to find any difference which
might occur as a result of the increase in diffrac-
tion angle. No observable effects appeared. This is
as would be expected since the theory does not actu-
ally require any limitations on the anglee . SineG
could be used instead of 9 without changing the form
of the expressions.

The set of Figs. (41)- (53) were the result of
measurements taken to find any differences which might
appear between the single diffracting aperture and the
diffraction grating. For the results pictured a rec-

tangular aperture of 4.15 mm breadth was used as the

48.

diffracting element. As predicted from the com-
parison of the grating theory with the aperture
theory no significant difference is apparent when

these images are compared with either of the pre-

vious sets.

49.

Fig. 16(3) Experimental points for Fig. 16

 

 

 

 

 

) T!
I4'
l2 ’
MT +415
8 , “o."
t 59
5 r “03
)
Q ' d10.2
2 b “0-!
p
O 1 0
° INQs cx-?

Fig. 16 Intensity at center of image and
half-intensity breadth as functions of
angular source breadth. Grating C - 3000
lines/inch; grating aperture, 3.u2 mm;
coherent mode. AB is expressed in units

of 10"3 radians.

50.

Fig. 17(a) Experimental points for Fig. 17

 

\Op

 

 

 

 

 

O

-03 -01 o 0.1- 0.1 9.9
Fig. 17 Grating C - image formed by coherent
mode. Source aperture of angular breadth

cl =23/Ns. 6 is expressed in units of

 

10'3 radians.

51.

Fig. 18(a) Experimental points for Fig. 18

is:

H

 

 

 

 

l I_ A

 

.o:n| .0:2 0 0.1 OJ!
Fig. 18 Grating C. Coherent mode of
illumination a = 2VNs .

52.

Fig. 19(a) Experimental points for Fig. 19

 

 

H

M

 

 

 

 

 

A n . n i 4 . I I _ I .
-O.6 -o.'l -O.1 o 0-2 0-” a5
6 a

Fig. 19 Grating C. Coherent mode of
illumination G = ‘VNs.

H

55.

 

 

 

”Vs: ' a -v
Fig. 20 Grating C, illuminated in the
lens mode. Intensity at the center of

the image and half-intensity breadth as
functions of the source width.

I).

8

0

1D

 

 

L

 

l

A ‘

 

Fig. 21 Grating C.

~cs -a2 0 a2 a4 0-»

Lens mode of illumination
e1: ZVNs.

I“

55.

 

 

 

A L A A A L 1

-O-6 ~03! -o.z o 0.2 0.! 0.6
. 6 fl
Fig. 22 Grating C. Lens mode of illumination
u = quS e

l 56.
w

 

 

 

 

 

-93 - 0.5 - as —o.7. o 0.2 6.4 o. 6 0.8

 

 

9—9
Fig. 23 Grating C. Lens mode of illumination

57.

 

 

 

.uL
l2 ' "0.5
r
W) .eJ
p
3 r (*0!
P A9
5L 1&3
r HA2
' “OJ
. T O
2E4“ <x-e

Fig. 2h Grating C illuminated in the
broad source mode. Intensity in the
center of the image and half-intensity
breadth as functions of the source width.

58.

 

 

 

 

‘ A n A L

-M -o.z o 0.2 0.4 e —»

Fig. 25 Grating C. Broad source mode of
illumination u = al/Ns .

 

II.

59.

 

 

 

L A A L

 

 

 

-ae -e,q 41?— o 0.2. on 0.6

Fig. 26 Grating C. Broad source mead? of

illumination u = “/Ns .

I!

la

60.

 

 

A

- 0.3 -o.5 -o.» 43.2.

4‘“

Fig. 27 Grating c.
illumination 0!: GVNs .

 

L a 1 A

a a. 2 OH O. 6

Broad source mead? of

0-?

 

 

61.

 

 

 

 

 

C" ”O."

.. A9
5‘. 1) 0.3
1‘, v0.7.

2 . «#0.!

p

a ‘ O

2&«5 a'-9

Fig. 28 Intensity at center of image and
half-intensity breadth as functions of
angular source breadth. Grating B - 90
lines/mm; grating aperture 3.50 mm; coherent

mode.JGis expressed in units of 10-3 radians.

62.

Ill»

 

 

 

 

0

-0.‘ - 03. O Q'- OH 9—,

Fig. 29 Grating B. Image formed by coherent
mode of illumination. Source aperture of
angular breadth a: 2VNs. is expressed in

units of 10-3 radians.

 

 

 

A A A A AA

l4

 

63.

 

 

A A__ A A l

 

 

~66 —Qfl . 0.2, o #03. a." 0-6
9 a

Fig. 30 Grating B. Coherent mode of
illumination ct = '1st .

"I

64.

 

 

 

 

 

A
- 4‘ A ‘ L A A ‘ A

-M ~o.e -o.« «n o u M M M
O -‘»

Fig. 31 Grating R. Coherent 2203.0 9..
Illumination ct: 65-33.

1
.F

“It

 

64.

 

 

 

 

A A A . ‘-

 

~60

-o.6 ~03 ~02. o 0.1 0.4 as 0.9
9 a

Fig. 31 Grating B. Coherent mode of
illumination ct: 61/Ns .

 

65.

 

 

'O.‘

~06 -ON ~02. 0 0.1 all 0.6
e ->
Fig. 32 Grating B. Coherent mode of
illumination cl: QVNS.

L

0.3

 

 

 

 

 

 

 

66.

m D

P
.2. J 0.6
w- A «at
a b 1 0'”
6 D 1>O.3
* r ‘ 0.2
2 i at
O L O

zx/us $"fi

Fig. 33 Grating B illuminated in the broad

59

source mode. Intensity in the center of the (:f
image and half-intensity breadth as functions‘

of the angular source width.

 

l2

IO

.1

58.

 

 

 

 

. A ‘ A L

 

“0" -002 0 0.2 o 0* e a

Fig. 25 Grating C. Broad source mode of
illumination °‘ = 2k/Ns .

H

-
0
7

59.

 

 

 

A l A A

 

.. 0.5 .¢_q -0. 2. O 0.7. O .‘l 0.6

Fig. 26 Grating C. Broad source mo%<?of
illumination u = “/Ns .

"I

la

60.

 

 

 

A A i A n A A _.

- 0.8 -0.6 '0! 41.2. 0 0. 2 OH O. a 0.8

Fig. 27 Grating C. Broad source moed? of
illumination 0!: GVNS.

61.

led- )

ct wo-u

.. A9
6" (10.3
IN «v0.2
1' H-OJ

 

 

Fig. 28 Intensity at center of image and
half-intensity breadth as functions Of
angular source breadth. Grating B - 90
lines/mm; grating aperture 3.50 mm; coherent

mode.leis expressed in units of 10"3 radians.

62.

m »

Em

 

 

4

 

.‘3

 

-031 -0; c 0.2- 0.1 9...,

Fig. 29 Grating B. Image formed by coherent
mode of illumination. Source aperture of
angular breadth at: 2VNs. is expressed in

units of 10-3 radians.

(4*

12b

lOi'

 

55.

 

 

J A - A l

.ae -os -az 0 «ex 03 as
9-»

Fig. 30 Grating B. Coherent mode of
illumination 0! == “VNS .

l1?

 

64.

 

 

 

A A A A R _ .h

-00 -O-6 -03 ~02. O 0.1 0.4 . as 0.?
9 a

Fig. 31 Grating B. Coherent mode of
illumination 0!: 61/Ns.

w»

v

 

65.

 

 

A A j A A A L A L

'0-8 '06 -O.'( - 0.1 O 0.2 all 0.6 0-!
e ->
Fig. 32 Grating B. Coherent mode of
illumination at z: 95/113.

66.

 

 

 

 

 

,qL

P
”1' d 0.6
Io- «at

at 1°"

I A6

6. 10.3
a} 10.2
2* Jul
0 O

 

 

ZX/u‘ $ "‘9

Fig. 33 Grating B illuminated in the broad ,
source mode. Intensity in the center of the f x
image and half-intensity breadth as functions‘
of the angular source width.

V

 

67.

 

 

l A

A

 

- D.“ — 0.1 o 0.?- o.“ e 9

Fig. 3h Grating B.) Broad source mode of
illumination d =2-VVNs. -

68.

I2'

 

 

0L A A ._ L

4.6 —o.'i —o.'2

Fig. 35 Grating B.

illumination e: = 5319’

 

4A A A A

o 0.1 an 0.5
9-45
Broad source mode of
NS .

69.

IO'

 

 

a j _A_ L A a A 4 A a
as -O.6 -O.'l - 0.2 O 0.2 03 0.6 at

9—?

Fig. 36 Grating B. Broad source mode of
illumination 0! = I’IYNSo

70.

 

 

H r
I

I2. 4 as
.

lo! ‘ O.‘
r

9 _ no.4

A0

6p ~03
t

#a 0&2
)

2 p 4 0-!
b

0- O

 

Fig. 37 Grating B illuminated in the lens
mode. Intensity at the center of the image
and the half-intensity breadth as functions
of the angular source width.

H

'71.

V

 

 

 

L A A A

-o.'l -o.z a 0.2 0.4 9 a

Fig. 38 Grating B. Lens mode of illumination
0 =27A/Ns.

'72.

I2b

(01 P

 

 

 

 

6“; -in,.i

-o.s 43.6 .0)! -0.'L o 0.2 0.1! 0.0 0.3
6 —--3

Fig. 39 Grating B. Lens mode of illumination

'73.

 

I'ir

IID

IO!

 

 

 

 

l. J A A L a L n J

-03 -05 - on -e.a o 0.2. 0!! 0.6 0.}
6 -?
Fig. ho Grating B. Lens mode of illumination
0‘ = aIVNS e

 

 

 

74.
\W’
'2'
'°( ‘»°5
8 b e 0-”
A9

6 +0.3

b
H r +0.2
2 " 90.!
O 0

 

Fig. hl Single aperture of n.10 mm width
illuminated in the coherent mode. Intensity
at the center and half-intensity breadth

as functions of the angular source width

49 is expressed in units of 10"3 radians.

75.

and

 

 

 

O A A A A A

-0.2- O 0.2 6 q

Fig. #2 Single aperture illuminated in the
coherent mode. Source aperture of angular

breadth 0<=Z)‘/D. ,Qis expressed in 10"3

radians.

ll

'76.

 

 

 

 

A A A A j j

-03 -o.z o on. 0.4 9 a

Fig. MB Single aperture. Coherent mode of
illumination 0‘ = ‘M/D.

N

I2

10

77.

 

Q \
A I x‘

 

 

I . J . j _

 

' 0.5 -O.‘l '0'). 0

Fig. uh Single aperture.
illumination on = Gl/D.

0.7. of! 0.5
e —-;
Coherent mode of

78.

W

7

'2-

 

 

 

0 A J l A ‘

-03 -o.6 -o.q -0a. a

Fig. AS Single aperture.
illumination 0! =8A/D.

 

 

,__\

. 0.1 03 0-6 0‘

Coherent mode of

79.

 

 

 

|¢[
'1” <f0.‘
IO-b ‘ 0"
‘$ 40.“
40
60 ‘ 03
W
*«5 40.1
¢ OJ
0

 

 

 

2"Vii: <i-da

Fig. #6 Single aperture of u.lO mm width
illuminated in the broad so.mode. Intensity
at the center and half-intensity breadth

as functions of angular source width.

I0

80.

 

 

* K

 

A l I A

-aH ~02 o 9.2 M e -9

Fig. #7 Single aperture illuminated in the
broad source mode. Source of angular breadth

0K: 2.8 7‘/D. 9 is expressed in 10-3 radians.

I2”

 

 

 

4e 4 i

'05 ‘Ofi '03, O 0.2 0.4 0.6

L L A

Fig. LLB Single aperture. Broad sougc?mode
of illumination a: SSA/D.

N

N)

82.

 

 

Q J

b

 

 

i
~o-I «as 4).! - on. a

Fig. #9 Single aperture.
of illumination q = 8.41/D.

0.2. 0.4 0‘ 0.8
9 --?

Broad source mode

83.

 

 

 

 

H»

ll‘ '1
P

[0» I 0.5
8’ do."
r A9

GP ‘0-3
p

HL J a2
»

2L ‘0'
t

0L 0

 

2.1/0 “—9

Fig. 50 Single aperture illuminated in
the lens mode. Intensity at the center
and half-intensity breadth as functions
of angular source breadth.

84.

 

 

 

 

O J l 9 I 1

Fig. 51 Single aperture illuminated in the
lens mode. Source aperture of angular breadth

cl = 2.81/D. 9 is expressed in 10"3 radians.

l2

'0

85.

‘

 

 

 

‘ O u
I l L 4: _I J_

 

-aa .03 -02. o 0.2. as 0.6

6—9
Fig. 52 Single aperture illuminated in the
lens mode cx =5.67\/D.

H

86.

 

 

 

 

 

I)

A _- A I

-o.a 45 ~03 -az a m. on 0.6 0.8

6 —v
Fig. 53 Single aperture illuminated in the lens
mode 'cx: 8.4ND.

‘vv

V SUMMARY

Starting with the basic relations for a diffrac-
tion grating illuminated by a line source under
Fraunhofer conditions the theory was extended to in-
clude a source of finite breadth. Two different
modes of illumination of this finite source were as-
sumfipd, the radiation in the plane of the source be-
ing either completely coherent or completely non-
coherent. The results for a diffraction grating of
many apertures were extremely cumbersome. But with
the limiting assumptions of a large number of aper-
tures and small incidence angles the results were
simplified to give expressions for the intensity dis-
tributions in the images as functions of the angular
breadth of the source aperture} the dimensions of the
diffraction grating, and the wavelength of the inci-
dent radiation.

The results were compared with the relations
which Van Cittert (1) found for single diffracting
apertures illuminated in the modes prescribed for the
grating. Within the limitations of the theory no
difference was found to exist. It is believed, bew-
ever, on the basis of a previous work (10) that slight
differences exist for a grating with a small number of

apertures.

88.

Experimental verification of the theory.was ob-
tained by illuminating the source aperture in ways
which more or lass approximate the assumed conditions.
The experimental images found were compared with the
images expected from the calculation. It was found
that in most details the experimental and theoret-
ical results were in agreement. Differences were
ascribed to a lack of complete coherence or nonco-
herence in the source aperture, to the narrow beam
of radiation present in the coherent mode of illumi-
nation, and to the fact that the source aperture
contributes to the system in some ways as a diffract-
ing element.

The images formed with the diffraction grating
are compared experimentally with images formed with a
single diffracting aperture as a means of checking the
identity of the two theoretical results. No observ-
able difference was found. These latter observations
also serve as a check on the theory of the single
aperture.

Two different experimental arrangements have been
suggested (5) as approximating the noncoherent mode of
illumination, see Figs. (14) and (15). It was found
that the lens mode of illumination more nearly fits

the results of the noncoherent theory.

(1)

VI LIST OF REFERENCES

Van Cittert, P. H., Zum Einfluss der Spaltbreite
auf die Intensitatsverteilung in Spektrallifiien.

. Z.VPhysik, fig, 547-563, ’Tlssoff

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

Sehuster, A., Optics of the Spectroscope.
Astrophy8., 21’ -210, (1905)

Wadsworth, F.L.O. On the Resolving_Power of
Telesco es and SpectroscOpes for Lines of
Finite—Eidth. Phil. Mag. gs, 55, 3172323, (1897)

Godfrey, G. H. Diffraction of Light from Sources
of Finite_Pimensions. Austral. Jour. Sci. Res.
A, 9;, 1-17, (1948)

 

Stockbarger, D. C. and L. Burns, Line Shape as a

Function of Spectrograph Slit IrradiationgnOpt.
SOC. 1m. Zé’ 37g.384, 53)

Michelson, A. A. On the Application of Inter-
ference Methods to Astronomical Measurements.
Fiji. Mags fl,j-21, (1890f

Mhnster, C. Unahnliche Abbildung und Ausmessung
von Nichtselbstleuchtern Ein Eeitra zum K uiv-
Elenzprinzip Ann. Physik, lfi, 6I9-g44, (E935)

Michelson, A. A. and F. G. Pease, Measurement of
the Diameter 9f -Orionis with the Interferometer
KstrOphys. §§, 249-259, (1921)

Gerhardt, U. Application of the Mighelson Stellar
Interferometer to the Measurement of Small Par-
ticles. ZeitSe f0 Phys. E, 697-7117, (1926)

Smith, 3. E. and C. D. Hause Fraunhofer Multiplg
Slit Diffraction Patterns withiFinite Sources. J.
Upto SOC. Am. 2g, 426-430, (1g52)

Takeyama, H., T. Kitahara, and T. Matubayasi.
On the Mathematical Treatment of the Effect of
theFWidth of‘the Slit on Fraunhofer‘s Diffraction

PhenomenonFTPart IIT.J,Sci.Hirosima Univ. KS.A.)

90.

(12) Born, M. Optik Julius Springer; Berlin (1953)
(Lithoprint by Edwards Bros., Ann Arbor) p. 163

(15) Bocher, M. Introduction to the Theory of Fourier's
Series. Ann. of Math. §_(S.§) 81-145, (1905:706)

(14) Carslaw, H. S. Introduction to the Theory of
Fourier's Seriesjand"Integrals ed. 3, Dover Pub.,
New‘YOrk, 289:510, (1930)

(15) Hopkins, H. H. On the Diffraction Theory of
Optical Images Proc. Roy. Soc. K? 217, 408-432,
(laseffi

 

APPENDIX

The difficulties which were experienced in the
case of coherent illumination were mentioned briefly
under the description of experimental results. In
order to show the limitations of the theory it would
appear advisable to describe these difficulties in
more detail with graphical illustration. Possible
explanations for the causes of the characteristics
will be offered.

In the coherent case it was observed that when
the total expanse of the diffracting element used was
smaller than the width of the source aperture that
some anomalies were present in the resulting image.
In particular, it was found that the structure of the
image lags behind the predicted structure, that is,
the amount of structure in terms of numbers of ob—
served local maxima was not as great as was predicted
by the theory. In addition there appeared for’dif-
fracting elements of small expanse and relatively
wide source apertures a small peak on each side of the
image which moved out from the image and became more
distinct as the source aperture was Opened.

In the first situation it was found that the maxi-
mum visibility of the structure predicted in the pat-

tern always appeared at source breadths slightly

92.

greater than those expected. The difference between
predicted and observed source widths was completely
negligible for small source apertures but became

the dominant factor as the source aperture breadth
became broader than the total expanse of the dif-
fracting element. It was further found that if the
slit used as the source aperture was completely re-
moved, the slit immediately in front of the lamp now
becoming the source, that the pattern was only slightly
influenced. In no case could more structure be found
in the presence of the source aperture than was

found with the source aperture removed. This would
indicate that when the source aperture was wider than
the diffracting element the aperture had little in-
fluence upon the resulting pattern. The image ob-
served was a Fresnel pattern with the source at the
lamp acting as the source for the pattern. In in-
termediate situations it would appear reasonable to
expect the pattern to be a mixture of the two types.
This problem is illustrated in Figs. (54) and (55).
Fig. (54) is a diffraction grating of aperture 5.5 mm
illuminated by a source of angular breadth .14 /Ns,
which would predict a pattern of seven maxima on the
basis of the coherent theory. It is seen in Fig. (55)

that there are only six maxima in the pattern with

93.

source aperture removed. When the source aperture
was widened beyond its width in Fig. (54) the six
peaks appeared, but not clearly, and they are there-
fore not pictured.

In the second situation where the diffracting
element was quite narrow at the start the pattern
with the source slit removed as in Fig. (59) only
showed two definite maxima. The number of maxima
and the general outline of the image depend upon the
dimensions of the system and the diffracting element.

The peaks which appear at each side in Figs. (56)
and (57) but which are not present in Fig. (59) can
be explained in terms of the source element which can
be said to exist at each edge of the aperture. The
peaks appear at the positions predicted by the Fraun-
hofer theory, which assumes that they are images of
line sources located at the aperture edges. These
line elements are noncoherent with respect to the
other elements of the aperture plane. This structure

is absent when the source aperture is removed.

91+.

W‘.

up

l0”

 

 

 

 

 

P \
0 Z J L . . L . __9

-|_2, -03 -o.II 0 OJ! ox m.
9 ——9
Fig. 5L1. Grating B illuminated in the coherent
mode «=Hl/Ns.

9S.

 

 

 

 

 

 

 

 

 

 

 

 

 

U
. a U

 

 

 

 

L a ' a 1 a . T L a a
4.2 - 0.8 —M O a)! 0.8 c.2. {.0
9 —--3

 

Fig. 55 Grating B with source aperture removed. Note
similarities to Fig. Sh.

96.

 

 

 

 

 

 

. b
b
6 .
If h
2 b
F

o L l l I __1 n i

- 0.5 - 0.6 - 0.3 O 03 0.6 03
9 --?

' Fig. 56 Single aperture illuminated in the coherent
mode. Note the structure beginning to appear at
each side of the, image “=5-5VD.

97.

 

 

 

 

4.2 -o.9 -O.6 -O.'5 0 0.3 0. 6 (1g 1.2
e ——7

Fig. 57 Single aperture illuminated in the coherent
mode as in Fig. 56. cl =?.4VD.

98.

Mr

 

 

 

 

 

 

° J... .7... -5 ; .3 .2. .13? z. .:
e—v

Fig. 58 Single aperture as in Figs. (56)-and (57)

with source aperture removed. \,

\_—.

"‘illillllifllilli“