...... . .o
\ o 4 . R.)- no.1
; - u
.\\ . ... NM‘ .4 .u find“ . . H
y .wxu \. r. . 2.. .m ‘.
i. V\ ‘ C, - r U
3.1.... t c\ . u o

v“ f.“ r
M . ‘Uo. -“ WV 1. ‘1‘
. u . . . . .3 r y .
I ‘ uh £
v.5.” .% ... A... 0"- Q‘lfi co. ~
. V . was a. . a... a ..
A \. f. o a)“ o-m ..u RH “95"
n . . Xx. v‘ ..w u a .
u I. ... ‘ID
3‘... \ .8. 5 4“ r\9 . , ¢ / I
‘.. o s. t. f o ... u 0.. o
u .. § a w. A o ‘ o .. Q
Ou< O“ V . ' . I N. lihll
. .,.. ., . ‘1‘ n.
0‘ Q .\ .A 1 w. 0 .&.Uo .
.. . ...... . 3.. .. 5
”o. o. u .o O <- ou‘!‘ I}. C O. ‘ O I".
-.\‘, .\ .o a u .Jwb _ u
\. ... . . k .. m .4
...“ o a ‘h c a 5..
.N 3-... ‘ .- r‘.0~ .\‘ a O
H a 5.0 O I ....
\. .OIA ... t .4
r .. ..u a.

.3 :52::E_:__E_E__:_:2

 

[‘HESIS

 

llHllllllllllllllHINIIHNIIUllllllllllIIHIIIIHIIIIIHWI ; LIBRAR7Y

31293 01774 3885 . '
—~ Michigan State

University

 

 

 

This is to certify that the
thesis entitled

INTENSITY DISTRIBUTIONS IN FRAUNHOFER
MULTIPLE SLI'I' DIFFRACTION PATTERNS

presented by
Albert. Ernest Smith

has been accepted towards fulfillment
of the requirements for

“08. degree in PhIBICI

(9.0V.

 

 

Major professor

Date Mm—

0-169

 

PLACE IN RETURN BOX to remove this chedtout from your record.
To AVOID FINE retum on or before date due.
MAY BE RECALLED with earlier due date if requested.

DATE DUE DATE DUE DATE DUE

 

INTENSITY DISTRIBUTIONS IN FRAUNHOEER
MULTIPLE SLIT DIFFRACTION PATTERNS

by

.Albert Ernest Smith

A.Theeie

submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics

1951

6/:1/6‘”
' ,8}—

up
K

ACKNOWLEDGMENT

I wish to express my sincere appreciation to
Professor 0. D. Hause for the suggestion of this problem

and for encouragement and helpful direction toward its

WW

completion.

255872

TABLE OF CONTENTS

I. INTRODUCTION . . . . . . . PAGE 1
II. smrsouscr.. . . . . . .mcmg

III. DISK SOURCE . . . . . . . . PAGE 16

I. INTRODUCTION

The applications of Fraunhofer diffraction patterns have
been many and varied.

The diffraction patterns from rectangular and circular
apertures give a means of determining the ultimate detail which may
be expected in the images produced by optical instruments with.aper-
tures of this type.

The two-slit system yields a method for determining the
angular width of a small source. In this method, which was devel-
oped by Michelson,1 use is made of the fact that the diffraction
pattern from a very small source is made up of equally spaced maxima
and minima. If two line sources are displaced slightly with respect
to each other, the intensity distribution in the resulting diffraction
pattern is the sum of the intensities in the individual patterns.

It was found by Michelson that if the light source consists
of two separate slits which subtend an angle d at the diffracting

apertures and if 0‘ :- fi- X the resultant pattern no longer shows

3:.
maxima and minima of intensity, but has a constant intensity for all
angles of diffraction. )s is the wavelength of light used and _d_ is
the separation of the diffracting slits.

In this method use is made of the fact that two separate
line sources are incoherent with respect to each other. For this
reason the resulting intensity distribution in the diffraction pat-
tern is the sum of the intensity distributions from the individual

801113980

If the source used is a solid rectangle, it may be regard!

ed.ae a large number of line sources, and the result is an incoherent
source. It has been found here that the maxima and minima in the
resulting pattern disappear when 0‘ 3 f , where 0‘ is the angle sub-
tended by the width of the source at the diffracting slits. In.thc
case of a solid disk source, the fringes are found to disappear when
the diameter subtends an angle 0‘ = 122%: .

Calculations are also carried out by Michelson for a disk
source in which the illumination decreases toward the perimeter and
for two separated disk sources where the separation is large in com-
parison with the diameter of either source.

These results have been applied to the determination of
the diameters of double stars,2 to the separation of double stars,3
and to the measurement of the diameters of microscopic particles.”

Since in all these cases the intensity distribution in
the two-slit diffraction pattern has been used as a characteristic
which would allow assignment of source dimensions, it is of interest
to determine whether by adding slits between these two the effective
resolving power of the area in which the two slits are located may
be increased.

The term resolvigg 2233;, as it is used in this paper, is
defined as the smallest angle of source. subtended.at the diffract-
ing.apertures. to which dimensions may be assigned. It is assumed
that the overall dimensions of the diffractor do not increase beyond
a certain limit.

The purpose, then. of this paper has been to determine
whether the inclusion of additional slits between the original two

increases the effective resolving power of the area. that is, whether

dimensions may be accurately assigned to smaller angles than those
to which dimensions are assignable by the use of the two slits.

It has been found that when three slits are used an effect-
ive increase in resolving power of 33% is achieved. When four slits
are used, an effective increase of the order of 25% is gained. The
four-slit case is found to be more precise experimentally than the
three-slit.

Calculations are carried out for the case of a rectangular

source and for the case of a disk sou-cs.

II. SLIT SOURCE

It has been found5 that when light of wavelength )‘ eman-
ating from a line source is diffracted by 1 rectangular apertures
of width 9‘ and separated by a distance d, the intensity distribution

in the diffraction pattern may be represented as follows:

1 : 32:41:11M‘V"
.6‘ I"V

 

where:

g. 1r 6; (9 H) ,

r . n ‘4. (9 .2)

if we assume both the angle of diffraction, 6 , and the angle of
incidence, _i_, to be small. This expression may be further simplified
by assuming a small value of g and dealing only with the central part

of the diffraction pattern. In this case,

I. M‘NY
I: R" M‘Y

 

If the angle of incidence is zero (1 : o), the intensity
distribution has a principal maximum at p 9 = O and other principal
maxima on each side of it separated by N;l minima and NLZ submaxima.

For example, if N'= 2, the equation for the intensity

distribution becomes simply:
i = 4 R: w‘ r

and we have an intensity pattern of equally spaced maxima and minima.
If N = 3, there is a single submaximum which is 11% as intense as the
principal maximum. If N = 4, there are two submaxima of equal inten-
sity, 7.14% as intense as the principal maximum. If N = 5, there are
three submaxima, the center one of which is 14.095 as intense, and the
outer ones 6.2% as intense, as the principal maximum.

From these examples it is observed that as N increases,
the intensity of the submarine decreases in relation to that of the
principal maxim. In the case of the diffraction grating the sub-
maxima are of essentially negligible intensity except adjacent to
the principal maxima. It is important to note, however, that when N
is smll the submaxima are definitely visible.

These expressions and statements apply only to the patterns
for a source of infinitely small dimensions, a line or point source,
which assumes a single wave train being emitted from the source and
diffracted. Since this is never attainable in practice, there is a
problem of determining the intensity distribution in the pattern
when the source has finite dimensions.

A source flit of width 1 is assumed, the width small in

comparison to the length. This source may be divided into many line

sources of width d1. Each of these elements of source then produces
its own diffraction pattern:

1 e R.‘ “9"” - R:
‘“’ at}? 4;: 1r ‘4 (9+3)

M1. Nn’k (9+3)

 

 

and each of these will contribute to the diffraction pattern which
is observed.

Let d be the angle which the source width 1 subtends at
the diffracting apertures. It is readily seen that each element of
the source has its own angle of incidence, and if the incidence
angle of the center element is zero, these angles vary continuously
from - g to «o g.

If complete incoherence is assumed for the resultant source,
the intensities in the patterns produced by the source elements add.

Since the intensities are additive and the angles of in-

cidence vary continuously from - g *0 '9 g" , the expression for the

ensuing intensity distribution as an integral over _i may be written:
d/
1. 7- .
'1 .. R A“! N Y .
_- D A; '
. 1 w
—J/‘. A
6

 

Making use of the identity:

M‘NV- 1[N+ ...cN-I)m3.v +(V'1l‘45‘lV‘ *“flw43J

 

and the substitution: = n,

I ME£VAHM=1R$ -|) u: Nude“) +(N-1)ao‘fna(9 “)+

+ m1(~-r)'fl’v\ (9+‘)]’L‘; .

VIC:

And since cos (ME) = cosA cosB - sinA sinB,

111 R./% tfy 4- (N-I) (MNnkBml‘er -M\Rk9‘uh\u)

4’ (N‘mhn 7M9 w‘luu - 4... ‘hr k 6‘... 47m) +-

..+ [mun-hue unl(N-ll7Tu£ —
44'. a (Nah a9 44.. “WW “if A; .

This result when integrated term by term becomes:

" n?[*(N—3~lm1uk9Aé-Ttkos+
(“Ink .
)w¥1(k9M- d‘rikd 4-
41k
+ _J_ calm-thine “(w-431w]
HIV-4W“
This may be written as a sum:
N-t
WE [ N“ W *‘ KL“ “I wlk'flkewhtflkfl
3:“ Nut .

In the case of the double slit, (N z 2), this reduces to:

I; ‘- R:[1‘m\e + as: True/seems]
1H.

For complete disappearance of the fringes in this case it
is necessary that 0‘ = %I «g , at which value the intensity distribu-

tion is constant for all values of 8 . It is also observed from

inspection of the general relation that if at = 2:: , the intensity
distribution is constant for all a for any number of diffracting
slits. This, however, gives no increase in resolving power. In
order to attain such an increase it is necessary to consider the
submaxima.

In general the intensity distribution is represented by a
number of harmonic terms whose amplitudes depend on N . For values
of d other than d =% the intensity distribution has been obtained
graphically, and a recognizable pattern has been sought to which a
unique value of i could be assigned. Curves have been constructed
showing the intensity distribution for various values of at for
three, four, five, and six diffracting apertures.

It has been found in general that the first change in pat-
tern takes place at d = if. At this value of d the pattern is
intermediate between the original of NAZ submaxima obtained with a
line source and a pattern of N—3 submaxima. There is thus only a
slight residual secondary structure.

The graphs which are shown (figs. 1-4) are plots of inten-
sity against the angle 1' %9 , the latter being taken between 0°
and 180°. In all cases the lowest line is that for a line source.
This is plotted to a different scale than the others, but the rela—
tive intensities remain the same. The other lines are curves of
constant d , for larger values of D( . ' For the case N : 3, rig. 1,
it is observed that all trace of the submaximum disappears at

N = 235%. It will be recognized that this particular value of
CV might be difficult to determine precisely experimentally, since

beyond this the minimum merely becomes more narrow.

4000*

3000‘

2000‘

IOOO‘

IOOJ

 

 

Fig. 1

|20°

|80°

wk—

o.|>~

Fig. 2

3000

20001

l00‘

 

 

 

10
Fig. 3

 

1
20001
' 1000‘
on
loo~
1%
J: 1%
0° 60° 120° |80°
1r%e——’

3000 - Fie- ‘*

2000~

[000-

l00‘

 

 

0° ' 6 0°

N=6

I20°

I80°

NI—
“P

3:--

ml—
at»

12

In Fig. 2, N : it, the intensity distribution changes from

one having two submaxima to a pattern having one submaximum at

= 33—13; :1- . This value is quite precise and can be successfully
recognized. experimentally. The deviation from the general relation
if? is accounted for by the lack of symmetry in the intensity distribu-
tion for the small source. This lack of symmetry exists in all patterns
for I 2 h, but becomes negligible for higher values. Fig. 3, H = 5,
follows in similar manner.

In Fig. 1+, the six-slit case is carried through a greater
range of fi‘ to illustrate what occurs as the source slit is widened.
Changes in character of the pattern are seen to take place at

N =%§ . d =%d>:' d : ing. Not shown here is the final dis-
appearance of all submaxima, which takes place at approximately

“I : —1—— ->-‘ . There is also a minor structure in the principal max-

1.5 d
ima which appears for U) -;- § .

The stated purpose here is to obtain an increase in resolv-
ing power with the overall area occupied by the diffracting slits
limited to a constant value. Call this overall width 5. In the two-
slit case, d = s, and the disappearance of fringes takes place at

Q 1': £5 . When three slits are used in the same area, 3 = 2d and the
s

submaxim disappears at a = g g7 . In general, s : (N-l)d and the
change in the character of the pattern takes place when a : iii-l 2: .
s

This gives for H = 3 a source to which dimensions may be assigned
that is two-thirds the size of source which is resolvable by the two

slit, representing an increase in resolving power of 33%. In the case

of N = 1+, the change takes place when cl = 3355’:- , giving an increase

13

in resolving power of'about 23%. In general, the increase in
resolving power associated with this change in pattern is l%9%_ This
percentage decreases as Nincreases. Therefore, the greatest increase
in resolving power is found when N': 3, followed by‘a lesser increase
in N‘: #, etc.

Experimentally the three slit case presents difficulty, as
is expected from the graphs of intensity distribution. In the other
cases the transition is limited to a very small range of values for d
and may'be determined quite accurately.

Experimental verification was obtained quite simply with
a laboratory spectrometer. Diffracting apertures were ruled on
aluminized glass plates and.mounted on the spectrometer table. The
bilateral slit of the collimator became the variable source slit, and
the patterns were either observed visually with the telescope or
photographed with an attached camera. The width of the source slit
for a particular type pattern was determined by direct measurement
with a microscope comparator. With apparatus available and consider-
ing the scale of the experiments, the source width could be determined
with an accuracy of only 5-10% at best. In all cases, the width of
source slit could be determined more precisely from the type of pattern
produced than from direct measurement.

The pictures in the following pages may be compared with the
preceding graphs. Figm.5-7 are a sequence of patterns for N’: h and
may be compared with Fig. 2 for graphical description. Figs. 8-11

inclusive are the patterns for N': 6 and may be compared with the

graph in Fig. 4.

14

Fig. 5. N = 1+. Line
source.

Fig. 6. N = 1+.
First disappearance of

submaxima: 1 X

: 3.92E '

By direct measurement,
W = 0.18 mm.
By calculation,
w = 0.17 mm.

 

Fig. 7. N = 1+. Single
submaximum visible.

 

15

Fig. 8. N’= 6. Line
source.

 

Fig. 9. N = 6.
First disappearance of

submaxima:
O{=.l.§~

By direct measurement,
w = 0.12 mm.
By'calculation,
w : 0.11 mm.

 

 

Fig. 10. N = 6.
Three submaxima visible.

 

Fig. 11. N = 6.
Second disappearance of
submaxima:

-1.\

(X---

3 d
By direct measurement,
w‘= 0.22 mm.
By calculation,

w = 0.22 mm.

 

III. DISK SOURCE

Calculations have been carried out for a circular or disk
source as follows, again assuming the following expression for an

infinitely small source:
- l

:i :: F‘°’ ,4AaJL VP .

 

In this case a disk source of finite dimensions is divided into many
line sources of width g; and of length 2 -x2 , where g is the
radius of the source and 5 is the distance of any element from the
center.

Fbr any element of width.d§, the contribution to the dif-
fraction pattern may be written as follows:

at]: = {(t) A; M‘NV,
M‘V

 

17

Using the same identity as for the slit source:6

0‘1 =1t(3)di[%+ (N-t) ca‘JY +
(”‘33 C—m‘tV' +-~-+ calm-Of]

where, as before, Y = 7r A/A ( 9 44')
and makin the subs i 11 ion ‘ d
g t t t n 211' 4
AT. :1&(z\olifg+(N-t)(w“9 Lana ~44. some)

+ (AI-1M“: l nemlw; ~Alx9w1u1)+

~--+[¢&(N-I)u0 Loam-l) k3 -
“(N-0K9 4.0..("0‘131‘ ”u.

If it is assumed that the contribution of a single source
element to the intensity at the center of the diffraction pattern
is proportional to the area of the element, namely 2m dx,and
2 d is the angle subtended by a diameter of the source at the dif-
fracting apertures, source and diffractor being separated by a dis-

tance E,

wzkd , szi
{Ulst- ‘2 lK‘VdI-i‘ all;

If these substitutions are made and the resulting expression is in-
tegrated over the entire source, the integration limits are -‘ and

4d . the resulting intensity distribution is:

18

d u
I = H K] {1; «usual-«WW? (causes a.
--.r

_.Muses.u)+(N-l3’d‘d‘(¢h3k9mlwi
- Mia‘kh'i) + - - - -+. V0041 .

[W (Md “9 ohm-0k: ~A£A(N4)u0/¢.{a(N-t) kl] gfl‘

Since the sine terms are all odd functions they drop out of the

expression and there remains

I . wt/‘Igmt dbl-WE} thwui

"d
+(N~}3V{‘-jt wlthim + . .

+ m cam-ow Lam-chi 2d:

which is equivalent to

1
II =-‘S.(1 j. l:"€¥’tfii:zfi A“. i10FJXCAD'K51E“”l“uqfig‘dl
Il
+(N°1)Ln1v\5/ thflga- talkie A}. 4 .. . .
0

t ww-tse/‘mt Lo(N-|)Ki A; {
0

Further simplification is introduced by making the substitutions:

LLNz‘A/gl 3 \~\ =_ k,€1 :.ldTI<%g~°‘

This changes the limits since

when i = 0,
when i :d,

These substitutions give:

19

1: grief /' Hm AM

(N-I\ Cake} V7.13 when obst-
(N‘uwaha Z'mtmamuciu.+
'+Ln(N-I)h9/'m us(N-\\Mkoiu(

Upon integration this becomes:7

I e 8l<1w‘T/z§ g! + Q’fi) as: we Lbs)
+(N-£)M1Kel(1m)+ .

+ (JGOJ‘OT 9 S [(Nulh] f

(N 0M

where 31(n) is the Bessel function of first order. This may be

written:

N—l
1. 45.3.? T“ +0in) mine MM].

If this is expanded for H = 2,

I -= Ifl‘wl: 1‘. + u: h9 3.00]
a.

3' M

It may be seen from this that, since Ji(m) : 0 when m = 3.83, the

disappearance of fringes for the double slit takes place when m.2 3.83
= 2 "" 01' 2* - - "’ 1.22 — o
i“ ’ ‘rr d a

Another property of the Bessel function is that the values

for which it goes to zero are not simple multiples of each other. For

20

this reason it is seen that in the general case N 2, there is no
value of d expected for which all terms beyond the constant are
simultaneously zero, and therefore, there will be no value of source
diameter for which the intensity is a constant for all 9 .

The intensity distribution is plotted graphically for the
four-slit case in the same fashion as for the slit sources. The
curves showing the intensity distribution are given in Fig. 12. From
these curves it is seen that the first change in character of this
pattern takes place at 2‘ = 1.472;} g . The point of this change, how-
ever, is not as precise as for the slit source. There is a region
of 3 5% surrounding this value in which there are no graphically
visible submaxima.

Photographs of these patterns have been made and are shown
in Figs. 13-18. There are two series, the first consisting of a
source of constant diameter with the separation of the diffracting
apertures varied, the second, a constant aperture separation with

the source diameter varied.

21
Fig.12

4000 f
3000‘,

2000‘

|000i

l00‘

 

 

 

 

—1 I 1
0° (1 6 0°
‘1 7(63'--'

T

I20° 180°

CIRCULAR SOURCE

N=4

22

Constant Source Diameter

Fig. 13. Disk Source.
N': h.

Source diameter = 0.23 mm.
Separation of apertures
= 0.16 mm.

was

 

Fig. 1h. Disk Source.
Source diameter : 0.23 mm.
Separation of apertures

: 0.18 mm.

were

 

Fig. 15. Disk Source.
Source diameter a 0.23 nub
Separation of apertures

= 0.25 mm.

 

2 «Mai

 

 

23

Constant Separation of Diffracting Slits

Fig. 16. Disk Source.
N : M
Source diameter - 0.23 mm.
Separation of apertures
: 0.10 mm.

last
20“ u a.

Fig. 1?. Disk Source.
Source diameter : 0.h2 mm.
Separation of apertures

: 0.10 mm.

Zii = l‘gg

3:
4 d

Fig. 18. Disk Source.
Source diameter = 0.66 mm.
Separation of apertures

: 0.10 mm.

1.223
2“) u a

 

 

21+

11317131731038

1Michelson, A. A., "On the Application of Interference Methods to
Astronomical Methods," Phil. Mag” 32 (1890), p. 1

2Anderson, J. A., "Application of Michelson‘s Interferometer Method
to the Measurement of Close Double Stars," Astrophys. Jour., fl.

(1920). p. 263

3Michelson, A. A., "On the Application of Interferometer Methods to
Astronomical Measurement,“ Astrophys. Jour., i]; (1920), p. 257

Michelson, A. A., and Pease, F. 0., "Measurement of the Diameter of
0( Orionis with the Interferometer,“ Astrophys. Jour., 53 (1921),
p. 249

”Gerhardt, U., I'Application of the Michelson Stellar Interferometer
to the Measurement of Small Particles," Zeits. f. Physik, 35 (1926),

p. 697.

Gerhardt, U., and Baeyer, 0. v. "Lower Limit of the Michelson Stellar
Interferometer Method of Measurement,’ Zeits. f. Physik, 3i (1926),
p. 718.

5Jenkins, F. A., and White, H. E. Fundamentals of Phy_sical Opticjs,
New York, McGraw Hill 00. Inc., 1937

6Jackson, D.. Fourier Series and Orthogonal Polynomials, The Math-
ematical Association of America, 1941, p. 33, 31+.

7Gray, A., Mathews, G. B., and MacRoberts, T. M., A Treatise on Bessel
Functions, London, Macmillan and Co. Ltd.. 1931, p. 1+6.

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