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MEASUREMENTS OF FRESNEL

EL‘IFFRACTION PATTERNS“

Thesis ior the. Degree of M. 3‘
MICHIGAN STATE. COLLEGE

J G u 31 W C: 0 d w :.1 rd
1938

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DATE DUE DATE DUE DATE DUE

 

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Thesis for degree of M.S.
J. Guy Eoodward

1958

TABLE OE CCHTEJJJ

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EXPERIMENTAL DATA

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DISCUSSION OF ERRORS
CONCLUSION
AC}: $0231.19 DG'ziulffll'I ’1‘

BIBLIOGRAPHY

115913

Page

13
16
18
57
51
52
53

54

111130 72213:“) ’3 1013:"

Early in the 19th century Eresnel developed and puelished
his theory of diffraction. She theory has been used to predict
the distribution of energy in all types of diffraction arising
from a cylindrical wave front. In comparing the theoretical
diffraction patterns with the observed patterns, Eresnel and
other early experimenters were forced to limit their work to
visual observations of the positions of the lightest and dark-
est parts of the patterns because of a lack of instruments to
measure light intensity with any degree of accuracy. In re-
cent years, with the accompanying development of photoelectric
cells and methods of measuring small electric currents, it has
become possible to make an accurate determination of the en-
ergy distribution in any diffraction pattern and hence to test
Fresnel's theory for the intensity relationships as well as
for the linear extent of the patterns.

The results of the first successful attempt at making
such a measurement were published by Lyman of Harvard University
in 19291. This work consisted in photographing a pattern and
determining its contour from a microphotometer trace of the
photographic plate. The plates used were carefully calibrated
by means of an exposure-density curve made at the same time.

By this method Lyman showed the intensity ratios of maxima and
ndnima in a measured straight-edge pattern to check within 4%
of the corresponding ratios taken from the theoretical pattern.

2

In the summer of 1937 McClellan at Michigan State College did

-2-

some preliminary work in which he found an agreement within

about 10%, except at very low values of intensity, between the

measured and calculated diffraction patterns due to a single

narrow slit.

ERL‘BIIEL'S THEORY
It is not within the purpose of this discussion to develop
Fresnel's theory of diffraction. Such a development may be

found in almost any textbook of physical optics:3

, and only the
final result of the derivation is given here. Fig. 1 represents
in cross section the Optical system necessary for the diffrac-
tion of light in passing through a single slit. 1:. is a line
source of monochromatic light which travels out in all direc-
tions giving a cylindrical wave front S. If at 0, at a distance
a from L, a slit is placed symmetrical with the optical axis of
the system, the cylindrical wave front striking it will be dif-
fracted forming a pattern which can be seen in the plane of ob-

servation. According to Fresnel the intensity at any point P'

I 9
in this plane is given by I”: + y where
I
x '/:03 Ii'! dv
a
V 9
.v '/Bin 121 dv
o
and v n 8 M
ab A ’

Absing the wave length of the monochromatic light, 3 the length
of wave front used, and a and p the distances indicated in Fig. 1.
Thus we see that the relative intensities of different parts of
a pattern can be predicted providing the dimensions used are
known. The linear spread of the pattern is determined by the
following equation in which _d_ is the distance of P' from the

axis:

b)(a + b)
29.

d - v
These are the equations which will be used later to cal-
culate the theoretical diffraction patterns. Before they can
be used, however, the Fresnel Integrals shown above must be
evaluated. This work has been done by a number of men using
several different methods. The most complete table of these
integrals, running from O to 8 in steps of .005 for values of
v, has been compiled by Sparrow at the University of Virginia“.
With a single slit the only part of the cylindrical wave
front which is effective in sending light to the screen is a
length As which, for ordinary slit widths, is essentially the

same as the width of the slit. Corresponding to A 3 there is

a Av given by

4,34,, M

abA

The first step in applying the theory to a given case is
to determine the value of av. This gives the limits for the
Fresnel Integrals. For this value of Av the two values of x
at the ends of the interval are read off and subtracted alge-
braically to give Ax. The same process is followed to obtain
4y. These are squared and added to obtain the value of the

intensity for the midpoint v.

METHOD OF HEAs'dEHEIP -TD EQUITnZET UQED

The energy distribution in the various diffraction pat-
terns was measured by means of a Visitron type 53-AV photoelec-
tric cell mounted on a carriage which was moved by a lead screw
of 3/8 inch diameter and 1 mm. pitch. The cell was moved across
a pattern and readings of the photoelectric current produced
were taken at frequent intervals. To obtain resolution a nar-
row slit aperture was always used directly in front of the photo-
cell in the path of the light. The light sensitive device was
mounted at one end and the diffracting slit at the other end of
a light-tight camera approximately 570 cm. long. The camera
consisted of a sheet iron tube 8 inches in diameter terminating
in a wooden box 2 meters long at the photocell end. In the end
of this box could.be placed either a translucent screen for vis-
ual observation of a pattern, or a plate holder for use in photo-
graphing the pattern, or the metal box containing the photoelec-
tric cell for a direct measurement of the pattern. To eliminate

.
troublesome scattered light in the camera, cardboard baffles
were placed at suitable positions inside the tube.

The line source of lightupostulated by the theory was ap-
proximated by using a narrow slit illuminated from behind by a
mercury vapor arc. This are was a General Electric lab are
No. H-S of the capillary type. The light from it passed through
three glass filters to be rendered monochromatic, the filters
being Corning G34Y and Cambridge fotanical supnxy Nos. 8 and 16.
More will be said about this source in a separate section. The

source slit was on the optical axis and about 50 cm. from the

diffracting slit.

Two thin, steel straight-edges about 8 cm. long, one mount-
ed firmly on the end of the camera and the other held against
the first by two rubber band springs, served as the diffracting
slit. Different slit widths were obtained by inserting between
the two straight-edges small but accurately ground roller bear-
ings whose diameters were known. for each slit width two such
separators of the same diameter were used, one at the top and
the other at the bottom of the slit. The rubber bands prevent—
ed both the separators and the movable straight-edge of the
slit from slipping. The source as well as the end of the cam-
era holding the diffracting slit was in one room while the pho-
tocell and controls were in another at the opposite end of the
camera. During a set of measurements it was necessary to take
several readings with the photocell dark. This was made easy
by placing a shutter device directly before the diffracting
slit, which shutter could be controlled by the operator at the
far end of the camera.

The greatest difficulty encountered in this work was the
procuring of diffraction patterns wide enough to permit resolv-
ing the fine structure and.yet of sufficient intensity_te pro-
duce a measureable photoelectric current. The length of cam-
era used was found by trial to be about the optimum value as a
compromise between intensity and resolution. hhile the mer-
cury vapor are was an extremely bright eource, the light from
it passed through three glass filters, three narrow slits, and

six meters of air before reaching the sensitive surface of the

~8-

photoelectric cell. The resulting currents were quite feeble,
the greatest current measured in any pattern, with one excep-

13 ampere with all measured cur-

tion, being less than 5 x 10-
rents lying between 10'15 and 5 x 10'13 ampere.

Such currents are, of course, far below the range of any
galvanometer. since a quadrant electrometer would have been
very inconvient for this work a vacuum.tube ampliflying circuit
was resorted to. A General Electric FP54 electrometer tube
was used in the DuBridge-Browns’6 balanced circuit. The schem-
atic diagram tOgether with the constants of the circuit is
shown in Fig. 2.

The EP54 tube, the grid input resistor, and two 22-1/2
volt "C" batteries were all mounted in the same metal box with
the photoelectric cell. All other controls except R; and R.
were mounted in a separate copper box which served as a shield.
The tube and the control box were connected by a flexible
shielded cable. R; was a Leeds-Northrup Students' Potentio-
meter. It was calibrated to read directly in volts by means
of R. and the milliammeter in series with it, the potentiometer
having been previously checked against a weston standard Call
to find the correct value of the current. All batteries, with
the exception of the two "C" batteries supplying the accelera-
ting potential for the photocell, were heavy duty lead.plate
storage batteries. The galvanometer, a Leeds-Northrup type
0-11

H-52290 operating at about 1 ampere per millimeter, was

mounted on a heawy iron bleak supported by sponge rubber pads

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-10-

on a rigid shelf at one end of the room. It was quite free
from disturbing vibrations. A narrow line of light, from a .
specially constructed lamp incorporating a lens and a 32 C.P.
automobile headlamp bulb, was reflected from the galvanometer
mirror onto a translucent screen and scale at the operator's
desk at the far end of the room, giving a light lever of about
4 meters. Fig. 3 shows the complete set-up of apparatus.

With the set-up and equipment described in the two preced-
ing paragraphs the sensitivity to Changes in grid potential of
the FP54 was 5.5 x 10"6 volts per millimeter as registered by
the galvanometer spot on the translucent screen. Eith the 1010
ohm input resistor this voltage sensitivity corresponded to a
current sensitivity of 5.5 x 10"16 ampere per mdllineter. The
circuit was used as a null instrument. The zero position of
the galvanometer was first found, then during a set of measure-
ments the galvanometer spot was kept on the same place on the
‘scale. This was accomplished by adjusting the Students' Poten-
tiometer to compensate for the potential drop across the input
resistor with a given photoelectric current. The millivolt
range of this potentiometer was used.

A great many troubles, large and small, were met in build-
ing up the FP 54 amplifying circuit, but this is not at all an
unusual occurrence in working with electrometer tubes. However,
if all connections are good, the resistors are dependable, the
batteries are fully charged, and the shielding of the circuit
is complete this method can be used to great advantage in mebsur-

ing currents too small for a galvanometer. The DuBridge-Brown

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circuit used is especially designed to eliminate difficulties
due to fluctuations in the voltage of the supply batteries.
The circuit as used was balanced with a filament current of
86.1 milliamperes, which is sufficiently close to the rated
filament current of 90 milliamperes for the EP54. Even with
the circuit balanced there was usually a steady drifting of
the galvanometer spot away from the zero position. However,
this drift could be made small enough to be quite easily cor-
rected for in the data taken. The method of making this cor-

rection will be shown later.

LIHEARITY OE PHOTO” ECDRIC CELL

Although theory tells us that the photoelectric current
should be proportional to the intensity of the light falling
on the sensitive surface, it was thought wise to test the Visi-
tron 53-AV cell to ascertain the kind of response it gave through-
out the range of intensities measured in diffraction patterns.
This was done by tne inverse square method using the familiar
fact that light intensity is inversely proportional to the square
of the distance from the source.

The box containing the photocell and its associated equip-
ment was mounted at one end of an Optical bench two meters long.
The photoelectric currents were measured with a light source
placed at different positions on the Optical bench. These meas-
urements were made by the same null method described in the last
section. The source used here consisted of a 50 watt Mazda lamp
in a light tight container which had a small slot (about 1/8 inch
by 1/2 inch) opened in one side. This slot was covered by lay-
ers of blank newsprint paper. The paper served the double pur-
pose of giving a diffuse source of light and of cutting down the
intensity to desired values. When several readings of intensi-
ties and distances had been taken over the two meter range of
the bench, another layer of paper would be added to give a less
intense light and the range covered again with this new source.

In this manner the photocell was checked for values of
grid voltage changes from 2.5 x 10'4 to 400 x 10'4 volts. to
do this it was necessary to use from one to ten layers of paper.

The cell response was found to be strictly linear throughout.

-14-

Some of the data taken in this way for the range from 2.5 x 10‘4
to 160 x 10'4 volts are tabulated in Table I and shown graphi-
cally in Fig. 4. All intensities measured in diffraction pat-

terns lay within this range.

Table I

Linearity of Photoelectric Cell

I - k/d' I - intensity of light
d - distance from source
k a constant
V - photocell response measured.by change of grid
potential in volts 1 105
Subscripts of V indicate the number of layers of

paper covering the source.

a k/d v5 v8 v10
45 cm. 472 2390 570 258
56 519 1500 379 172
66 230 1140 280 129
as 135 530 164 74

115 74.3 .552 107 53

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-15-

SOURCE

Fresnel's theory postulates a monochromatic source. The
mercury vapor arc and filters used in this experimental work
come very close to this ideal. The spectrum of mercury shows
that there are no strong lines within about 50 angstroms of
the very strong 5461 A. line. The system.of glass filters used
‘will transmit a narrow band in the neighborhood of this line.
Since it is the only strong line in that region, the light pas-
sing through the filters wdll be highly monochromatic with.the
wavelength 5461 2. There will, of course, be a little light of
other wavelengths coming through, but the amount is so small as
to be negligible in comparison with the accuraqy 0f the other
measurements.

The three filters work as follows: The Corning G54Y cuts
out everything on the blue side of the 5461 A. line; the Camp
bridge Botanical filter No. 8 cuts out a band about 200 A. wide
on the red side immediately adjacent to the 5461 A. line: the
N0. 16 filter cuts out all the red which is beyond the band
out out by the No. 8. All three will transmit the 5461 3. line.
Two spectrographs, one of the unfiltered mercury arc and the
other taken with the three filters in place, are shown in Fig. 5.
The exposure times are the same (1 sec.) for both pictures.

The are was checked with a foot-candle meter to attest
the constanqy of illumination. It was found that when the line
voltage was constant there was no perceptible variation in in-

tensity. However, during the daytime the line voltage varied

-17..

enough to cause variations of one and two per cent in the source.
One particularly troublesome circuit when closed would cause the
source to lose as much as five per cent of its intensity. Be-
cause of these findings most of the pattern measurements were
made late at night and on week-ends when there were no notice-

able fluctuations in the line.

Fig. 5
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Mercury Vapor Arc

Unfiltered and Filtered

EXPERIMENTAL DATA

With the equipment described above it was possible to
measure different types of diffraction patterns including the
straight-edge, double-slit, and single-slit patterns. Data
are given for diffraction at two single slits of different
widths. In measuring a diffraction pattern three variables
were recorded for each setting, viz, the distance moved across
the pattern by the photoelectric cell, the change in grid poten-
tial as read on the students' Potentiometer, and the time at
which the reading was taken with reference to the time at which
the first reading was taken. To facilitate making measurements
against time the readings were taken with the same time inter-
val (45 seconds) between every two successive readings.

then measurements were being made, every eighth reading
was taken with the shutter closed and the photocell darkened
and without moving the photocell from the position of the pre-
vious reading. These "dark readings" were thus taken only
against time and had no dependence on the diffraction pattern.
They were used in making the correction for drift. To make
the correction, the "dark readings" were plotted with.voltages
as ordinates and time as abscissas and the points thus fixed
were connected.by a smooth curve. Fromlthis curve the error
due to drift of a voltage reading taken with the photocell il-
luminated by the diffraction pattern could be determined by
reading off the ordinate on the drift curve correSjonding to
the time at which the uncorrected measurement was made. This

correction would be added or subtracted from the potentiometer

-19-

reading depending on whether the drift decreased or increased
the readings as time went on.

Since the corrected diffraction pattern readings were to
be plotted with voltages as ordinates and distances moved as
abscissas, the simplest way of applying the drift correction
was to plot the pattern readings and dark readings on the same
voltage scale and, using dividers, to add or subtract the cor-
rection from the uncorrected pattern reading at the time of
plotting the latter. for this reason on the curves represent-
ing the patterns measured there will be two curves-~the drift
curve with voltages as ordinates and times as abscissas, and
the corrected pattern curve with voltages as ordinates and
distances moved as abscissas.

To obtain a more accurate experimental curve of any pat-
tern an average was taken of three measured curves in each
case. The average was obtained by reading the ordinates of
corresponding points of the three curves under consideration
to obtain a mean value of the ordinate at that point. Since
the initial readings of the three measurements of a given pat-
tern were seldom taken at the same point in the pattern, the
corresponding points on the curves used in obtaining the aver-
age had to be determined from the curves themselves and not
from.the data.

There now follow tables containing the experimental data.
The curves are plotted from the data in these tables. Combined
in one tabulation will be the data for all three measurements of
a single pattern, but the curves for the measurements will be on

separate pages.

-20..

Table II

a - 53.5 :I: .1 cm. b - 557.9 1 .5 cm. A- 5451 A.

AS ‘ 10586 t 0001 me AV ' 4034 1 0015

Dist. Grid pot. change in volts x 105

Time Moved A B C
.75 min. 0.0 mm. 10 11 20
1.50 .5 15 15 17
2.25 1.0 18 20 25
3.00 1.5 24 55 40
5.75 2.0 50 55 40
4.50 2.5 59 42 45
5.25 5.0 49 48 51
6.75 5.5 55 65 75
7.50 4.0 70 79 85
8.25 4.5 94 82 88
9.00 5.0 105 100 107
9.75 5.5 120 127 150
10.50 6.0 148 149 151
11.25 6.5 198 152 165
12.75 7.0 222 189 195
15.50 7.0 241 240 245
14.25 8.0 256 284 287
15.00 8.5 500 505 507
15.75 9.0 529 520 525
16.50 9.5 506 556 555
17.25 10.0 270 400 594
18.75 10.5 259 411 405
19.50 11.0 218 572 577

Time
20.25
21.00
21.75
22.50
25.25
24.75
25.50
26.25
27.00
27.75
28.50
29.25
50.75
51.50
52.25
55.00
55.75
54.50
55.25
56.75
57.50
58.25
59.00
59.75

40.50

min.

Dist.
Loved

11.5 mm.

12.0
12.5
15.0
15.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
22.0
22.5
25.0

25.5

-21-

A

250

242

271

277

271

257

257

259

271

514

296
270

252

519
557
557
576
575
565
555

554

412
468
496

474

427

415

299

285

Grid pot. change in volts x 105

C

574

575

564

557

555

567

412

472

490

442

428

416

525

505

290

-22-

Dist. Grid pot. change in volts x 105
Time moved A B C

41.25 min. 24.0 mm. 184 285 286

Dark Readings

 

0.00 0 0 0

6.00 16 55 41
12.00 56 7O 86
18.00 47 110 122
24.00 64 151 165
50.00 105 186 198
56.00 158 225 254

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Dist.
0.00 cm.
.05

.10

.15

.50
.55
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00

1.05

From curves of figs. 6, 7, and 8.

Z3185
A
8
12
15
16
20
27
55
55
49
70
79
91
116
164
184
201
215
262
286
262
224

190

-25-

Table III

Dist.
.11 cm.
.16
.21
.26
.51
.56
.41
.46
.51
.56
.61
.66
.71
.76
.81
.86
.91
.95

1.01

1.06

1.11

1.16

6
15
11
16
18
29
54
55
50
80

90

volt
x 10

1012

118

171

2

C
15

18

14

19

55
55
49
67
79
90
124
159
192

205

248

285

240

191

Dist.

0.00 cm.

.20
.25
.50
.55
.40
.45
.50
.55
.60
.65

.70

1.00

1.05

volts
x 10D
Average
9
15

14

19
29
55
55
49
72
85
94
119
165

192

190

-27-

volts volts volts

x 100 x 105 x 105
Dist. A Dist. 3. G Dist. Average
1.10 cm. 167 1.21cm. 181 168 1.10 cm. 172
1.15 178 1.26 190 165 1.15 178
1.20 187 1.51 199 176 1.20 187
1.25 214 1.56 212 200 1.25 209
1.50 218 1.41 225 205 1.50 215
1.55 210 1.46 211 198 1.55 206
1.40 190 1.51 196 185 1.40 190
1.45 166 1.56 181 175 1.45 175
1.50 165 1.61 182 167 1.50 171
1.55 190 1.66 200 a 187 1.55 195
1.60 226 1.71 257 227 1.60 250
1.65 274 1.76 285 278 1.65 279
1.70 265 1.81 500 280 1.70 282
1.75 226 1.86 265 251 1.75 247
1.80 197 1.91 250 218 1.80 215
1.85 182 1.96 218 205 1.85 202
1.90 172 2.01 195 184 1.90 184
1.95 145 2.06 161 155 1.95 155
2.00 101 2.11 154 115 2.00 117
2.05 82 2.16 97 95 2.05 91
2.10 72 2.21 86 87 2.10 82
2.15 60 2.26 75 68 2.15 67
2.20 42 2.51 51 47 2.20 47
2.25 55 2.56 56 55 2.25 55
2.50 54 2.41 57 29 2.50 55

2.55 28 2.55 28

-33-

Table IV
a I 55.5 a .1 cm. b s 567.9 i .5 cm. 4K= 5461 A.

48 - 1.979 :I; .001 mm. Av . 5.42 3 .015

Time Dist. Grid pot. change in volts x 105
Moved A B C
.75 min. 0.00 mm. 58 25 25
1.50 .5 40 24 51
2.25 1.0 48 52 42
5.00 1.5 65 47 62
5.75 2.0 81 56 78
4.50 2.5 90 67 90
5.25 5.0 117 85 119
6.75 5.5 152 115 145
7.50 4.0 175 157 166
8.25 4.5 214 162 208
9.00 5.0 259 204 257
9.75 5.5 285 251 284
10.50 6.0 504 246 501
11.25 6.5 555 281 552
12.75 7.0 550 507 565
15.50 7.5 527 285 555
14.25 8.0 298 260 295
15.00 8.5 280 246 280
15.75 9.0 265 240 268
16.50 9.5 241 217 241
17.25 10.0 250 200 255

18.75 10.5 260 214 270

-29 ..

Dist. Grid not. change in volts x 105
Time moved A B C
19.50 min. 11.0 mm. 505 259 525
20.25 11.5 528 297 545
21.00 12.0 521 290 516
21.75 12.5 274 246 272
22.50 15.0 252 216 255
23.25 15.5 "264 220 285
24.75 14.0 520 264 554
25.50 14.5 555 505 571
26.25 15.0 555 505 546
27.00 15.5 294 262 296
27.75 16.0 260 221 268
28.50 16.5 266 224 280
29.25 17.0 295 244 512
50.75 17.5 518 270 558
51.50 18.0 559 290 557
52.25 18.5 567 511 588
55.00 19.0 407 548 425
55.75 19.5 598 558 408
54.50 20.0 579 527 585
55.25 20.5 560 511 572
56.75 21.0 560 507 555
57.50 21.5 517 276 516
58.25 22.0 279 241 287
59.00 22.5 265 222 274

-50-

Dist. Grid pot. change in volts 2 105

Time Moved A B C
59.75 min. 25.0 mm. 250 210 258
40.50 25.5 227 184 228
41.25 24.0 222 175 218
42.75 24.5 211 170 219
45.50 25.0 200 161 211
44.25 25.5 195 154 209
45.00 26.0 195 165 205
45.75 26.5 186 156 202
46.50 27.0 184 155 202
47.25 27.5 185 155 202

Dark Readings

0.00 0 0 0

6.00 55 10 22
12.00 56 26 70
18.00 77 44 84
24.00 100 62 100
50.00 118 70 125
56.00 147 94 145
42.00 165 124 164

48.00 178 146 192

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FPO/‘7 Z4545 I *C‘

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Dist.
.05 cm.
.10
.15
.20
.25
.50
.55
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.00
1.05
1.10

1.15

volts
x 105

A
52
59
55
46
57
59
85
114
154
170
215
254
252
279
291
265
255
212
195
168
155
180

220

22
21
28
41
50
58
76
102
122
147
186
212
224
256
278
255
226
210
201
177
157
168

210

-54-

Table V

Dist.
.04 cm.
.09
.14
.19
.24 ‘
.29
.54
.59
.44
.49
.54
.59
.64
.69
.74
.79
.84
.89
.94
.99
1.04
1.09

1.14

volts
1 105
0

16
21
55
45
51
7O
91
108
145
145
215
255

277

267
220
205
192
165
151
170

225

From curves of Figs. 9, 10, and 11

Dist 0
1.05 cm.

.10

.50
.55
.40
.45
.50

.55

.70
.75

.80

1.00
1.05
1.10

1.15

volts

1 105

Average
27
22
27
40
51
56
77
102
121
155

195

220

287
252
225
207
195
170
154
173

218

-35-

volts voltg voltg
x 105 x 10 x 10
Dist. A B Dist. C Dist. Average

1.20 cm. 242 245 1.19am. 255 1.20 cm. 247

1.25 252 256 1.24 250 1.25 255
1.50 182 189 1.29 187 1.50 186
1.55 157 157 1.54 160 1.55 158
1.40 156 159 1.59 175 1.40 165
1.45 218 201 1.44 240 1.45 216
1.50 252 240 1.49 266 1.50 255
1.55 227 241 1.54 248 1.55 259
1.60 185 187 1.59 198 1.60 190
1.65 150 155 1.64 155 1.65 155
1.70 155 156 1.69 157 1.70 155
1.75 179 175 1.74 180 1.75 178
1.80 196 138 1.79 207 1.80 200
1.85 215 216 1.84 222 1.85 217
1.90 259 255 1.89 245 1.90 259
1.95 276 270 1.94 279 1.95 275
2.00 262 275 1.99 291 2.00 275
2.05 240 241 2.04 247 2.05 245
2.10 218 222 2.09 255 2.10 224
2.15 210 210 2.14 215 2.15 211
2.20 165 175 2.19 175 2.20 171
2.25 124 155 2.24 140 2.25 155
2.50 109 115 2.29 125 2.50 115

2.55

95

2.54 107

2.55

99

Dist.

2040 cm.

2.45

2.50

2.55

2.60

2.65

2.70

2.75

2.80

volts
x 100
A

68

-55-

volt
x 10g
Dist. C

2.59 cm. 76

2.44 59
2.49 55
2.54 45
2.59 58
2.64 29
2.69 25
2.74 20
2.79 16

Dist.

204:0 cm.

2.45

2.50

volts

x 105

Average
71
.58
49
57
28
27
19

15

11

('2 7
-L) -

3’2111111'1 50:! III 1'": ’25". 4071. I'_‘IC.‘.L 13337 .:5

Using the formulae given above, the values of AV for the
two cases discussed were determined. From a table of Fresnel
Integrals the intensities at various points in the atterns
were found. These are tabulated in Tables VI and VII and plot-
ted in 3135. 12 and 13. Before these curves can be compared
with the measured ones they must be corrected for the width of
the photoelectric cell aperture. The effect of this aperture
would be to lower the high points, to raise the low points,
and to average out the very fine structure.

This aperture correction can be made on a theoretical curve
by using the fact that the total quantity of light falling on
the photo—sensitive surface in a given time would be proportion-
al to the area under the theoretical curve bounded by the limits
giving the width of the aperture. This statement is illustrated
in Eig. 12 where the line mn represents the aperture width and
the area under the curve within these limits is efg . This area
is taken to be proportional to the average intensity at the mid-
point of the aperture. a planimeter was used to measure these
areas at points .50 mm. apart on the scale of the curve and these
readings used to construct patterns which could be compared with
the experimental curves. These data are shown in Tables VIII and
IX. The width of the aperture Was measured with a travelling
microscope and found to be .49 mm.

To construct the theoretical and experimental curves on the
same graph both must be adapted to the same scale. This can
easily be done by comparing the corresponding readings for the

exact center of the pattern. For example in the (iv - 4.54

~38—

pattern d - 0.00 cm. and I - 87 at the center of the theoreti-
can pattern and d = 1.37 cm. and I - 2.5 at the center of the
observed pattern drawn from the average values of Table III.
Hence to put these two patterns on the same scale the 1.37 cm.
must be added to each value of.g on the theoretical pattern and
each value to I on the theoretical pattern must be multiplied
by 215/e7 . 2.47.

If the centers of the calculated and observed patterns
for «V I 5.42 (the latter drawn from average values given in
Table V) are compared, it is found that a quantity 1.29 cm.

must be added to.g to give the corrected distances, a and

c'
the areas must be multiplied by a factor of 2.78. However,

when the factor 2.78 is used and the resulting curve plotted

on the same graph as the experimental curve it is very evident
that the center point of the latter curve is in error and can-
not be used in finding the multiplying factor. For this rea-
son the first maximum to the right of the center has been used
to obtain the multiplying factor of 2.53.

Since the theoretical curves are symmetrical with the cen-
ter of the pattern, readings will repeat themselves on either
side of the center. The data for the theoretical curves are
given in Tables VIII and IX. In Figs. 14 and 15 the correspond-
ing experimental and calculated durves are drawn together, the

former from data in Tables III and V and the latter from Tables

VIII and IX.

-59-

Table VI

a - 53.5 1 .1 cm. b = 567.9 a .5 cm. .A=5461 A.

as - 1.586 :t .001 mm. av = 4.34 5; .015

___y ____s
v A}: (by I d
0.00 1.56 .731 2.29 0.00 mm.
.10 1.42 .781 2.20 .42
.20 1.08 .906 1.99 .85
.30 .736 1.04 1.78 1.27
.40 .530 1.12 1.80 1.70
.45 .497 1.12 1.62 1.90
.50 .510 1.12 1.63 2.12
.60 .682 1.12 1.80 2.54
.70 1.02 1.21 2.23 2.97
.80 1.37 1.39 2.76 3.39
.85 1.49 1.46 2.95 3.60
.90 1.61 1.51 3.12 3.82
.95 1.56 1.51 3.07 4.02
1.00 1.51 1.44 2.95 4.24
1.10 1.42 1.12 2.54 4.66
1.20 1.44 .740 2.18 5.09
1.25 1.51 .605 2.12 5.30
1.30 1.59 .518 2.11 5.51
1.35 1.64 .477 2.12 5.72
1.40 1.66 .473 2.13 5.94
1.45 1.59 .476 2.07 6.15
1.50 1.44 .502 1.94 6.36

1.70
1.75
1.80
1.85
1.90
1.95
2.00
2.10
2.20
2.25
2.30
2.35
2.40
2.45
2.50
2.60
2.70

2.80

.778
.719
.701
.701

.687

.537
.297
.158
.138
.135
.075
.114
.078
.037
.000

.002

.086
.086
.049
.060

.063

-40-'

.240
.200
.183

.194

.283
.298
.235
.133
.118
.131
.063
.006
.000
.000

.041

1.52
1.09
.97
.91
.90
.90
.89
.83
.62
.40
.34
.32
.27
.34
.36
.34
.24
.14
.12

.15

.09
.05

.06

8.90

9.54

9.75

9.96
10.18
10.39
10.60
11.02
11.45
11.87
12.30
12.72
13.14
13.57
13.99

14.42

a ' 53.5 I .1 cm.

.41-

Table VII

b a 56709 i .5 cm.

A8 a 10979 i .001 mm.

0.00
.10
.20
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.90

1.00

1.05

1.10

1.15

1.25

1.30

.632
.745
1.04
1.32
1.39
1.39
1.30
1.19
1.08
.988
.924
.893
.887
.891
.857
.731
.670
.640

.660

.891

1.08

13y
.789
.865

1.06

1.28

.891
1.25
1.37
1.42
1.39
1.35
1.30

1.30

A = 5461 A.

Av - 5.42 i .02

1.42
1.61
2.10
2.60
2.71
2.71
2.55
2.29
2.03
1.78
1.59
1.48
1.46
1.56
1.75
1.98
2.04
2.06
2.05
2.09
2.19

2.38

0.00 mm.
.42
.85
1.27
1.48

1.70

2.75
2.97
3.18
3.39
3.82
4.24
4.45
4.66
4.87

5.09

-42-

___s .___2

v 11x 13y I d
1.35 1.30 1.32 2.61 5.72 mm.
1.40 1.46 1.39 2.85 5.94
1.45 1.54 1.44 2.98 6.15
1.50 1.54 1.44 2.98 6.36
1.55 1.49 1.35 2.84 6.57
1.60 1.44 1.19 2.63 6.78
1.70 1.49 .815 2.30 7.21
1.75 1.59 .686 2.28 7.42
1.80 1.69 .621 2.31 7.63
1.90 1.66 .615 2.28 8.06
2.00 1.32 .554 1.87 8.48
2.10 1.04 .364 1.40 8.90
2.20 1.02 .255 1.28 3.33
2.25 1.02 .262 1.26 9.54
2.30 .941 .297 1.24 9.75
2.35 .803 .331 1.13 9.96
2.40 .638 .333 .97 10.18
2.50 .423 .242 .67 10.60
2.55 .371 .28 .66 10.81
2.60 .334 .196 .64 11.02
2.65 .377 .226 .60 11.23
2.70 .318 .275 .59 11.35
2.75 .228 .309 .54 11.66
2.80 .144 .301 .45 11.87
2.90 .072 .211 .28 12.30

3.10

3.20

3.30

3.40

3.50

.018

.001

.004

.011

.066

-45-

l‘y

.270
.199
.118

.130

.29

.20

.12

.14

.15

d
13.14 mm.
13.57
13.99
14.42

14.84

M71677 /’

 

 

 

 

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—J AV: 4.34
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D/SZZI/VCZ" FPO/‘7 0671/7254? 0F #477254/1/

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d

0.00 cm.

.05

.40
.45
.50
.55
.60
.65
.70
.75
.80

.85

1.00
1.05
1.10

1.15

Table VIII

A? =- 4.34

area
87
84
73

67

87

107

114

98

82

80

78

68

49

15

-45-

d
1.29 cm.
1.34 1.24
1.39 1.19
1.44 1.14
1.49 1.09
1.54 1.04
1.59 .99
1.64 .94
1.69 .89
1.74 .84
1.79 .79
1.84 .74
1.89 .69
1.94 .64
1.99 .59
2.04 .54
2.09 .49
2.14 .44
2.19 .39
2.24 .34
2.29 .29
2.34 .24
2.39 .19
2.44 .14

area x 2.47
215

207

282

242

203

198

193

168

86
77

74

37
32

37

 

 

 

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I----K 014100447!” fZ’W 774515 m

f767 /4

4v = 5.42

d area
0.00 cm. 54
.05 65
.10 81
.15 100
.20 91
.25 70
.30 55
.35 58
.40 68
.45 75
.50 76
.55 88
.60 110
.65 108
.70 91
.75 87
.80 85
.85 72
.90 55
.95 49
1.00 42
1.05 27

1.10

24

Table IX

dc
1.37 cm.
1.42 1.32
1.47 1.27
1.52 1.22
1.57 1.17
1.62 1.12
1.67 1.07
1.72 1.02
1.77 .97
1.82 .92
1.87 .87
1.92 .82
1.97 .77
2.02 .72
2.07 .67
2.12 .62
2.17 .57
2.22 .52
2.27 .47
2.32 .42
2.37 .37
2.42 .32
2.47 .27

area

x 2.53

137

164

230

177

139

147

173

190

192

223

278

273

-49-

d area -dc area x 2.53
1.15 cm, 22 2.52 cm. .22 cm. 56
1.20 16 2.57 .17 40
1.25 10 2.62 .12 25
1.30 11 2.67 .07 28
1.35 9 2.72 .02 23
1.40 6 2.77 15

1.45 6 2.82 15

 

 

 

 

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-51-

D188U8310N OF ERRORS

Most of the measurements given in the above experimental
data could be made with a fair degree of accuracy. Any error
in AV would be due to errors in _a_, _b_, and As. The values of
IcV'used could not be more than 0.4% from the correct value.
Potentiometer settings could be made to 2 x 10"5 volts, or about
1% on higher values and about 6 or 7% at lower values. However,
by taking the average of three measured patterns the error due
to the potentiometer readings would be cut down somewhat, pro-
bably to about 0.5; at higher values on the curve. Another
source of error was found in measuring the aperture width. Due
to irregularities of and diffraction effects at the edges of
the slit aperture, the cross hairs of the travelling micro-
sc0pe could not be set as exactly as might be desired and, al-
though an average of several readings was used, the value .49 mm.
for the aperture width is probably still in error by 1%.

An investigation of the error introduced by the error in
13v showed that at maxima and minima a change of 0.2% in the
value of sev could change the intensity of the theoretical curve
by as much as 4% while on the more linear parts of the curve the
the change caused by changing Asv was quite small. This, it
seems, might explain a considerable part of the observed dis-
agreement between theoretical and experimental intensities.
Agreement is seen to be quite good where the slope of the I vs.

g_curves is rather constant, but at the points of abrupt change

the disagreement is more marked.

Except far down on the curves the difference between

theoretical and observed values is less than 83 and at most
points is much better. Taking into account all sources of er-
ror, 0.5% for potentiometer readings, 1% for aperture width,
and 475 due to the error in Av, the maximum possible error is
estimated to be about 5.5;:5. The remaining 2.5;; is probably

due to the finite width of the source slit. It can easily be
shown that the effect of the source slit width is to lower the
high points, raise the low points, and leave unchanged the lin-
ear parts of the pattern curves. This effect would be quite
noticeable with a camera having the dimensions given above and

with the source slit at a width any greater than about .01 mm.

COECLUSION
Within the limits of experimental error in measuring a
diffraction pattern photoelectrically, the single slit dif-
fraction patterns as observed and as calculated by Frosnel's

theory are identical.

-53-

A CIGIO'.‘4LEDG?~II¥£Q T
To Dr. C. Duane Hausa, who suggested this research and
whose assistance and advice were most helpful throughout,
and to Professor C. I. Chapman whose interest and coopera-

tion made this work possible the writer wishes to express

his sincere appreciation.

1.

2.

3.

4.

5.

6.

-54-

BIBLIOGRAPHY
T. N. Lyman--
Proceedings 2; the National Lcadegy of Sciences

Vol. 16. pg. 71 (1930)

C. E. McClellan--
"A Direct measurement of Intensities in a Fresnel
Diffraction Pattern"

Thesis for M. S. at Michigan itate College (1938)

Francis A. Jenkins and Harvey 3. nhite--
"Fundamentals of Physical Optics” pgs. 185-201

mcGrawéHill Book Company, Inc., Haw York (1937)

C. M. Sparrow--
"Table of Eresnel Integrals”

Edwards Brothers, Inc., Ann Arbor, Michigan (1934)

Lee A. DuBridge and Hart Brown--
"An Improved d.c. Amplifiying Circuit"
Review of scientific Instruments

Vol. 4, No. 10, pgs. 532-536 (October 1933)

P. A. macdonald--
"The Thermionic AmplificaLion of Direct Currents"

ngsics
V01. 7, Ho. 8, pg. 265 ff. (August 1936)

 

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