THE INTRINSIC INTENSITY OF LIGHT TRANSMITTEDTHROUGH A SLIT AS A FUNCTION OF THE WIDTH

AND DEPTH OF THE SLIT AND OF THE WAVELENGTH OF THE LIGHT

BY

L. P. SIEG AND A. T. FANT

INTRODUCTION

It appears that very little work has been done on the absorptionof light by a narrow, deep slit. This offers a very interesting prob-lem, for even a simple observation indicates that there is appar-ently a much stronger absorption than one might anticipate. Inaddition there is a strong polarization of the light. This latterphenomenon is very striking, but it has not been studied quanti-tatively in this first work. Rayleigh1 has treated theoretically thepolarization of light transmitted by a narrow slit in an infinitelythin, opaque screen. That work serves in this present instanceprincipally to enable us to define a "deep" slit. A "deep" slitshall here be considered as one which causes polarization of inci-dent natural light such that the electric vector parallel to thelength of the slit is greater than that perpendicular to the slit. Inthis sense it is well nigh impossible to attain to a thin slit. Evena thin scratch in a very thin silver film on glass constitutes a"deep" slit. Stewart2 noticed strong polarization and absorptionby a narrow slit in a thick screen in some experiments performedin another connection. It was thought a matter of interest, and ofsome importance to investigate the intensity of light transmittedby a deep slit as a function of the width and depth of the slit,and of the wave length of the light. Further work should be doneon the variation of the material constituting the jaws of the slit.

I Rayleigh. Roy. Soc. Lond. Proc. A, 89, p. 194, 1913-14.2 G. W. Stewart. Abstract. Phys. Rev. 5, p. 73, 1915.

218

May, 19211 PASSAGE OF LIGHT THROUGH DEEP SLITS

APPARATUS AND METHOD OF PROCEDURE

A sketch of the arrangement of the apparatus used is shown inFig. 1. Light from a tungsten strip lamp was focused on S of

FIG. 1

I ori a 9---

.3

L5Diagram of Apparatus

the monochromatic illuminator, A. By means of prism P1 anydesired wave-length could be obtained at S2. A lens L2, placedso that 2 was at its principal focus, secured parallel light at S3.A mirror at M1 divided the pencil into two parts, one reflectedat right angles, the other passing straight through. S wasthe slit under investigation. It was formed by two plane steelsurfaces of a pair of Hoke Precision Gauges. One of the gaugeswas clamped securely to the bed of the interferometer, K; theother was set in wax in a frame attached to the carriage of theinterferometer, brought into contact with the first gauge, andallowed to remain in that position until the wax had hardened.The slit was then placed parallel to the direction of the light.A template, D, prevented any light from passing below or aboveS3. A short focus lens was placed at L3, the center of the edge of

219

-M,�' 1.1.7". :-------1

H, .

U

:"M=ti --V- I : It

::i

L. P. SIEG AND A. T. FANT [J.O.S.A., V

S3 nearest it being at its principal focus. This pencil was thenbrought to a focus by L on a series of parallel silver bands betweentwo right-angled prisms, P2. The image of the near edge of S3 ,

thus formed, was viewed through a low power microscope at C.That portion of the light which was reflected by Ml was, by

means of the mirrors M2- and M 3 , made incident normally on thenicols N1N2 , the rotation of one of which enabled one to match theintensities of the two images in P2. By means of L5 the light com-ing through N1N2 was focused on the silver bands in P2. Thusthere was formed at P2 a continuous band of light; the portionreflected from the silver being the image of the near edge of S3,

and the portion seen between the silver bands being the imageof S2, formed by the light coming by way of M1M2M3. It wasfound by careful trial that the results were not appreciably alteredby placing the microscope, C, in line with S2 53 ; and this positionbeing more convenient, the final measurements were taken withthat arrangement. By means of an auxiliary mirror at M4 and thespectrometer, B, the mean wave-length of the light used was de-termined.

Denoting by Io the intensity transmitted through the nicolswhen the angular separation of their transmission planes is zero,the intensity, I, for any other angle of separation, 0, is

I=Iocos'O , (1)

For a wide opening of S3 the intensities of the images at P2 weremade equal by the use of a neutral tint glass in conjunction withthe nicols. Letting Io =1, since only comparative values arewanted, (1) becomes,

I =COS (2)

After S 3 is opened to a certain width, the image at P2 becomes con-stant in intensity, any further opening of S3 simply serving tomake the image wider. Hence intrinsic intensities are obtained,not total transmitted light.

Contact of the surfaces of S3 was determined, after thoroughlycleaning the surfaces, by the deflection of a galvanometer, G,placed in series with a dry cell, and the surfaces of S3. Then bymeans of the interferometer screw the separation of the surfaces

220

May, 1921] PASSAGE OF LIGHT THROUGH DEEP SLITS 221

was ascertained. For each width of S3, three independent settingsof N2 were made, and the mean relative intensities were thus ob-tained.

RESULTS

The results; rediuced to tabular form would-occupy more spacethan isdesixable. However, their essential character can readilybe represented bya -few- typical curves. These are drawn toillustrate three -separate relations; I as a function of w, and Lconstant (Fig. 2); I as a function of L X and w constant (Fig. 3);and-I as a function of , w and L constant (Fig. 4). In the aboveX represents the wave-length, w the width of slit, and L thedepth of slit, all expressed in cms. The full line curves representthe experimental results: the broken line curves represent certaincalculated results based on two assumptions to be discussed in thenext section,

FIG. 2

'', t10z2

! S4.8 -d ,,3)/ -2..6 / X -

If0 1} X / / 1 i)-L=.381 cm.tf~i / bLfX/21L.80c_

)-L= I.05 c n¢ .2 /7J / , W-L= 2.413 I. / .0000573C

O .004 w .008 012 cm.The intensity as a function of the width of slit, for various depths

THEORETICAL CONSIDERATIONS

Undoubted4f the passage of a wave through such a deep channelis a complicated matter, in which diffraction, selective reflection,and possibly interference are the chief factors. An accurate testof any theory would necessitate a knowledge of the reflection

L. P. SIEC AND A. T. FANT

FIG. 3

w :.00197cn,

I A R _, n 2 On 1 xn :, - -

*b .t/ iu £.1 m

LThe intensity as a function of the depth of slit, the

width and wave length constant

FIG. 4

The intensity as a function of the wave-length,the width and depth constant

constants of steel (in this case) for both electric vectors, for manyangles of incidence, and for all wave-lengths employed in theexperimental work. Only one set of such constants, that recordedby Drude3 for sodium light, has been found. Steel is such avariable substance, that one is in some doubt as to the validity ofapplying his results to the particular samples chosen for this work.

Let us assume a wave front, a incident normally on a slit ofdepth L, and width w (Fig. 5). Consider a point on a from which

FIG. 5

I' I"l- I

III .

Ia I

f -~~~~~~~ .IS

1- f - -- V

< ~ ~~ ISection of jaws of the slit

3 Drude. Winckelmann's Handb. der Phys., Vol. III, p. 823. 1st Ed.

A

[U.C.S.A., V222

I

May, 1921] PASSAGE OF LIGHT THROUGH DEEP SLITS

a pencil is diffracted through an angle 0. The intensity of thisdiffracted pencil is proportional to sin2 a / a, where a is an auxili-ary angle defined by the relation

VrWa -sin 0 ..................... (3)

From Fig. 5 we have0 = tan-' mw/ L ...................... (4)

where we assume m to be some integral number. It can readilybe shown that the number of reflections of a diffracted pencilbefore emergence from the slit is m. Substituting for 0 in (3)its value given by (4), we obtain

ir w2 mn a.) ........................... ( )X\VM2W2 + L 2

Since m2w2 is small as compared with L2 , we may writer m2nX L

Negative values of 0 can be neglected on account of symmetry.Plotting the relation I= sin a/a2 , we obtain the well-known

(full line) curve in Fig. 6.. Beyond a = r there is a series of

FIG. 6

1.0 7 7

.8A iO

O 1I0 80010 /GA~Diffracted intensity as a function of the auxiliary angle a

rapidly decreasing maxima, which we shall here neglect. Theentire area under this curve is x, and the area under the curvefrom a = o to a = r was calculated to be approximately 0.45 r.

223

224 L. P. SIEG AND A. T. FANT [J.O.S.A., V

If we assume the intensity of the light passing through a givenslit, which is infinitely thin to be unity, the resultant intensityof light passing through a given deep slit will be given by thefollowing expression,

1 7r= T sin2a iR + T R .(7)0.45ir ~in i al 5i= a

where a represents the value of a for i reflections, R, and Rp the

reflection constants for the electric vector parallel and perpendicu-lar respectively to the plane of incidence, and m the number ofreflections when a = r. Strictly the expression (7) sums up,not a smooth curve, but a series of rectangles, which more andmore closely approximate the curve as m increases. A morerigorous expression than (7) could be formulated as a definiteintegral, but when one realizes that the R's are not constants,but are complicated functions of the angles a, the wellnigh insuper-able difficulty of evaluating the series becomes apparent. Asubstitution of Drude's4 values in (7) for a typical case, yieldedresults that conformed in a general way with the curves of Fig. 2,but it was quite apparent that the reflection constants given byhim were larger than those in the present work.

In view of the difficulty of using even (7), the following, ad-mittedly somewhat crude method was employed. Its chiefmerit consists in the fact that the results obtained by it agreefairly well with the experimental results, and that the labor ofcomputation was much abbreviated.

Then for the sake of simplicity, as stated above, let the intensitycurve (Fig. 6) be replaced by a straight line running from I = 1at a = 0, to I = 0 at a = 1600. From (6) we see that a is directlyproportional to m. Hence an m can be found that will make a

any desired value. Thus with a given depth of slit, L, and agiven X, we can assume various w's, and for each w we can deter-mine a for one reflection. To obtain the number of reflectionsfor a = 1600 we have merely to divide 1600 by the value justdetermined for m = 1.

Drude, loc. cit.

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May, 1921] PASSAGE or LIGHT, THROUGH DEEP SLITS

Let a represent the ordinate (Fig. 6, dotted curve) at a= 0, andm the number of reflections. The second ordinate is then, a- a/m;the third, a - 2a/m, etc. The intensity is given by the areas ofthese strips, each multiplied by the reflecting power, R (mergingR, and Rp), raised to the power (m-1). Since we wish to call thelength of the base unity, the base of each strip is 1/m. Hencethe expression for the intensity is

l a(a-a/m) + (a-a/m) + (a-2a/)R .nU 2 R+

+ [a-(m-1) a/mi + [a-ma/inl 2R 1

which reduces toa Rm + R m - (2m + 1)R + (2m-1) )

2in' (1-R) 2

(8)

Applying this to the present problem we have160 X L 8XL180 = 2 -9 W2 a = 1, R = 0.6180w2 9w2

For m we must employ the nearest integer, and m should not besmaller than about 5. Substituting in (8) the values for a and R,we obtain

I 032m2 0.6m+1 + 0.6m + 0.8 l - 1. 6 (9)

For widths of slit for which m was smaller than 5, the full lineintensity curve (Fig. 6) was employed, and areas were directlycounted.

The broken-line curves of Figs. 2, 3, and 4 represent the resultsobtained as described above. In Fig. 2, (1) is the experimentalcurve, (1') the corresponding theoretical curve; etc.

DISCUSSION OF RESULTS

In attempting a possible explanation of the experimental curves(Figs. 2, 3, and 4), two simple assumptions have been made;first, that the light is diffracted at the entrance to the deep slit;and second, that this diffracted light is weakened through succes-sive reflections. These lead to results that fit the experimentalcurves approximately, with the exception of the case representedin Fig. 4. The differences undoubtedly arise from the rather

225

L. P. SIEG AND A. T. FANT

rough assumptions, made for the sake of simplicity. The essential

character of the curves is, however, with one exception, true to

experiment. Those results in which I is plotted as a function

of X cover a much smaller range of values than do those others

where I is a function of w, and of L. The change of I with the

wave-length is small, and the discrepancy may not be so serious

as it at first appears.In practice then, even with very thin slits, a certain width of

slit must be attained before the intrinsic transmitted intensity

is approximately constant. Indeed, the strong polarization

of light transmitted through narrow slits is of itself sufficient

evidence that at least all of one vector is completely suppressed.

This result in practice probably applies only to an instrument

wherein intensities are matched quantitatively by permitting

light to pass through a slit of variable width, particularly when

the widths are small. In such cases serious error would result.

STATE UNIVERSITY OF IOWA,

Jan. 1921.

[J.O.S.A., V226