J. 0. S. A.

Improved Methods of Illumination for the Measurement of AccidentalDouble Refraction

YNGVE BJbRNSTHLLaboratory of Physical Chemistry, University of Upsala, Upsala, Sweden

(Received February 24, 1939)

A GREAT number of measurements havebeen made of the so-called accidental

double refraction induced in otherwise isotropicmaterials by magnetic, electrical and mechanicalinfluences. The resulting effects are generallyvery small. Therefore half-shade methods areindispensable for the measurements. It might besupposed that the methods of illumination,which have been used in a considerable numberof determinations, ought to be perfect and thatthe methods of measurement could be improvedno more. In the following I shall show that theillumination systems in many cases could havebeen improved and at the same time I shallmention some more suitable arrangements andsome of the precautions, that must be observedin the measurements of the small effects associ-ated with accidental double refraction.

The optical arrangements, which have beenused for magnetic, electric and mechanical doublerefraction show a certain degree of resemblance.

A light source Zo (Fig. 1), usually the exit slitof a monochromatic illuminator, is imaged bymeans of an illumination lens L, for instance ininfinity. The light passes through the polarizingprism N and the cell C, containing the doublerefracting medium. Ri and R2 are two circulardiaphragms at the ends of the cell. The lightemerging from C is elliptically polarized. Wewant to determine the position of the axis ofthe artificial crystal, the ellipticity and theorientation of the major axis of the ellipticalvibration. For the determination of these con-stants compensators or other devices (Fig. 1, K)-and a half-shade (H) are used. The half-shadeis observed by means of some kind of telescopicsystem. The sensibility of the arrangementincreases with the illumination on the retina,since only small effects are considered and thelevel of illumination is low.

Everyone who has, studied methods of de-termining small double refraction, knows howdifficult it is to avoid double refraction in cover-

glasses even if these are thin or made of specialannealed glass. The use of lenses between thepolarizing prisms, for instance as in microscopes,is out of the question and should be avoided inthe measurement of small effects. I thereforerefrain from discussing illuminating devices ofsuch a type.

The primary light source must possess anintrinsic brightness as high as possible. It isnecessary to use a high intensity arc lamp ofthe carbon or the mercury type. It often happensat high current densities that the lateral gradientof the brightness is considerable. Furthermorethis gradient is not constant but changes withtime. The light source, e.g., the crater in the arcof the special type of carbon, which must beused consists of a line spectrum, superimposedupon a continuous spectrum. Besides, thecarbons never are homogeneous. Photometricmeasurements show that the variations ofbrightness may be rather considerable, evenwithin a small part of the light source. Attentionmust therefore be paid to the fact that the slitZo is not evenly illuminated.

It must be admitted that by using an extracondenser and a slit the illumination of the

A

+ I1~Z_ L N

l l.I __I

RC.RI R2

IA

-- IE3I NV

H KN -rA

FIG. 1. Optical arrangement for the measurement ofaccidental double refraction.

different parts of Z may be made constant.Such an arrangement always means a loss oflight amounting up to 50 percent. Because ofthis fact we do not deal with this arrangementhere, as there are other means at hand.

There are two features, which, according tomy opinion, must be strictly observed in allmeasurements of small effects. Every point ofthe half-shade must receive light from all partsof the light source Zo in order that it may be

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VOLUME 29MAY, 1939

To

YNGVE BJORNSTAHL

evenly illuminated. Further, reflections must notoccur. Not only the directly reflected light,which enters the system, is objectionable, butalso-especially with colloids and high molecularweight substances-the diffused light of theTyndall type, which may arise. In the followingI shall indicate the possibilities existing ofeliminating the two sources of disturbance,mentioned above.

A faulty arrangement is indicated by greatzero-point variations in the measurements. Infact, the constancy of the zero-point position ofthe compensator determines the possibility ofmeasuring small effects.

Further discussion of the matter requires ashort survey of the illumination devices andtheir relations to the observation system andthe eye. Cf. for instance, Drude.'

Suppose H to be the surface of the diaphragmof the half-shade; P to be the surface of thediaphragm which limits the light beam throughone point of the half-shade; H' to be the surfaceof the image H, produced by the observationsystem (telescope); F' to be the surface of theimage of P; P' is the exit pupil; H' to bethe surface of the image of the half-shade on theretina; P" to be the surface of the pupil ofthe eye; and suppose the distance between therespective diaphragms to be designated by thefollowing scheme (a" meaning a constantcharacteristic of the eye)

H Pa

H' I-'

a'

HI P"

at/

If the observation system is arranged so thatthe plane of the exit pupil coincides with thepupil of the eye; if p5't pi and if the brightnessof the light source is io, the flux of light L whichcan be passed through the arrangement approxi-mately is-ignoring reflection losses-

HP H'P' H"P'L =io- = io- = io-.(1

a2 a'2 a//2

If iB is the angle at which the image of the half-shade is observed by the eye and if is thecorresponding solid angle,

For the illumination of the image on the retinawe get

B = L/H" = L/tBa"2. (3)

The illumination is determined by the size ofthe image P'. We must attempt to make thelight flux as great and 3 as small as possible inorder to obtain a strong illumination on theimage on the retina. On account of physiologicalcauses j3 cannot possibly be made smaller than2 . The absolute diameter of the half-shademay be of any size; by choosing a suitable tele-scopic arrangement the angle of observation may always be brought to assume the optimalvalue.

We want to find an expression for the maximallight flux which with different experimentalarrangements may be forced through the cell C(Fig. 1) with the diaphragms R1 and R2 and thehalf-shade H. We thereby presume that nopart of the light beam is intercepted after itspassage through H.

ARRANGEMENTS FOR MECHANICAL AND ELECTRIC

DOUBLE REFRACTION

For electric and mechanical double refractionin liquids the parts (the surfaces) limiting therays are to some extent similar. In the firstcase they are represented by the condenser-plates of the cell, in the second case by a limitedpart of two concentric cylinder surfaces. Thetwo cases may, therefore, advantageously betreated at the same time.

Method I

If a light source (Fig. 2) is reproduced in theplane Z by means of the lens L and the extremerays are limited by the cell diaphragm RI andby the half-shade diaphragm H, the half-shadewill not be evenly illuminated. Suppose that R1is the diaphragm which limits the beam passingthrough any point on the half-shade. This impliesHR 2 =R1 where H, R and R2 represent thediameters. If we designate the distance betweenR1 and H with b, we get for the light flux througha surface element (df) at the center of thehalf shade, the expression

(2) dL = io(dH)C3 = io(dH) Rl/b2, (4)

where io represents the intrinsic brightness for a

202

H",~a .1t2

I P. Drude, Lehrbuch der Oplik, third edition, p. 80.

MEASUREMENT OF DOUBLE REFRACTION

zR H

L

FIG. 3. Method I, reflections.FIG. 2. Method I, inhomogeneously illuminated half-shade.

part of the light source and R the surfacecorresponding to the diameter R, and repre-sents the solid angle. We presume c, to be small.For a surface element at the edge of H we obtain

dL1= ii(dH)&- = i 1(dH) Rl/b2 , (5)

where i means the brightness of the respectivepart of the light source. The different lightbeams come from different parts of the lightsource with different brightness, correspondingto different parts of the image Z in the figure.The expressions are still valid whether the imageZ of the light source coincides with the plane ofthe half-shade or is situated at infinity. Botharrangements have been used in the measure-ment of electric double refraction. Leiser aswell as several other scientists, e.g. Otterbein,Borchert3 have used the first method, whileSzivessy4 used the latter with "parallel light."*

As regards mechanical double refraction noattention was earlier paid to the illuminationdevices. Vorlander and Walter' did not use anyillumination lens. Sadran 6 made use of "parallellight."

We shall consider the conditions determiningthe reflections in the general case, when theimage Z of the light source is located after thehalf-shade diaphragm. If we examine Fig. 3, wefind that the reflections from the walls of thecell are produced as soon as Z is great, i.e., thecondition for the nonappearance of reflections is

2 R. Leiser, Diss. (Halle, 1910).3 G. Otterbein, Physik. Zeits. 35, 256 (1934); L. H.

Borchert, Physik. Zeits. 39, 156 (1938).4 G. Szivessy, Zeits. f. physik. Chemie 26, 326 (1924).* The expression "parallel light" is used for convenience

and does not imply that all the rays passing through thecell are strictly parallel but only that the light source islocated at the focus of the illumination lens.

6 W. Vorldnder and R. Walter, Zeits. f. physik. Chemie118, 1 (1925).

6 Ch. Sadran, J. de phys. et rad. 7, 263 (1936).

Z<R 1. It is impossible to diminish the reflectionsby introducing a diaphragm, for instance, be-tween the lens and R. An increase in thedimensions of the cell corresponding to anincrease in the diaphragms R and R2 mightappear of advantage. These dimensions, however,are fixed, their size depending on the availablevoltage and the amount of substance for electricaldouble refraction, and on the velocity gradientand the amount of substance available formechanical double refraction. Since Z increaseswith the distance from H, the disturbances dueto reflections are especially marked when Z issituated in infinity, i.e., with parallel light.tSummarizing it may be said that this methodmust be looked upon as of doubtful value fromboth theoretical and experimental point of view.

Assuming the brightness to be the same in allparts of the light source

L = ioHR11Y. (6)

As to the size of the half-shade diaphragmwe have H-R,; thus in the most favorable case

LI =ioRl2/b2 . (7)

Method IIIf the principle of an even illumination of the

half-shade is to be maintained, the conditionsbecome more complicated, i.e., certain restric-tions as regards the diaphragms are necessary.If Fig. 4 is considered, the image Z of the lightsource is found to be located after the half-shadeH. Z limits the beam through one point of H.In this way each element of H receives lightfrom the whole light source. If the diametersH and Z are given, the cell diaphragms R andR 2 must not intercept any rays. Alternatively,

t It should not be necessary to point out that no realimage of Z is formed; the light beam is before that inter-cepted by the objective of the telescope.

L

203

YNGVE BJORNSTAHL

we may assume RI and Z to be given, whereuponthe size of H must not be greater than necessaryto allow every ray through H and Z to passthrough R. Let b be the distance between R1and H, and c the distance between H and Z,the condition for homogeneous illumination iseasily obtained.

(8)

The half-shade cannot be placed at any pointafter the diaphragm R2; in any case

appearance of reflections is Z <R1 ; according tothe expression (12) at maximal light flux we have

thus

(15)Z= cRi/2b,

cRi/2b<Rl or c2b.

The maximum light flux to be used is

L"Il = ioR,2/36b2. (16)

From expression (12) it is evident that in anycase H is smaller than R1. The emerging light

b<<cRI/Z. (9)

The light flux passing through the cell, thehalf-shade and Z is

L=io(7r 2/16) (H2 Z 2 /C2); (10)

or by substitution of (8)

72 Z2 (cRl-bZ) 2

L = io- *~ (11)16 (b+c) 2 c2

Supposing R1, b and c to be constants, L hasa maximum for

Z=cR1/2b; H=cR,/2(b+c) (12)

R 12 c 2

L11=io I I. (13)16b2 c+b1

If c is allowed to vary we find that Li,

L

FIG. 4. Method II.

reaches its maximum value L'1i when the lightsource is imaged at infinity, i.e., when c= o

L'ii=ioRI 2 /16b 2 . (14)

To use parallel light, however, causes con-siderable reflections. See Fig. 5.

It may be of interest to examine the conditionsfor the occurrence of reflections with this experi-mental arrangement. As in case I, reflections arefound to appear as soon as the image Z of thelight source is great. The conditions for the non-

A--I IH

FIG. 5. Method II with parallel light, homogenouslyilluminated half-shade, strong reflections.

flux is small compared to that of method I.We obtain rational illumination by sacrificing thelight flux or the brightness of the image on theretina. Expression (16) indicates that the half-shade should be placed as close to R2 as possible,that is to say, near the cell. The method has beenused by the author in the measurement ofelectric double refraction.7

Method IIILet us consider other possibilities. A survey of

the different alternatives of placing the image Zof the light source is given in Fig. 6. The imageZ may be placed at any point within the shadedareas. A natural location is in the vicinity of thediaphragm R. We then get an arrangementaccording to Fig. 7. The half-shade H may belocated at any point after the cell diaphragm R2.If the diameter H is smaller than R2 the half-shade will be evenly illuminated; each point ofit receives light from the whole image Z, whichis supposed to cover the whole diaphragm RI.

For the light flux through the arrangement

L1ri = ioRiH/b2 (17)

where b represents the distance between R1 andthe half-shade H. It is possible to make the half-shade diaphragm as large as R1 .

(18)Lirr=ioR 12/b2.

7 Y. Bj6rnstAhl, Phil. Mag. 2, 701 (1926).

- l - Be: l - -

204

H--(cR,-bZ)I(b+c).

I

MEASUREMENT OF DOUBLE REFRACTION

L

FIG. 6. Illustrating the location of the half-shade.

Comparing this expression with Lrr we findthe last described method of obtaining evenillumination superior by far as regards the lightflux. As regards reflections, Fig. 7 shows thatextremely marked reflections appear; the methodis therefore of no value. It is evident that adiaphragm somewhere between the illuminationlens and the half-shade is ineffective, it is truethat it may to some extent weaken the reflec-tions, but the light flux and the brightness of theimage on the retina are reduced at the same time.In the following way it is, however, possible toarrange an illumination that is free from ob-jectionable features. In Fig. 8, Z is the lightsource, L the illumination lens, R 1 the first celldiaphragm and H the half-shade diaphragm.Between Zo and the illumination lens there is aplane Ho corresponding to the plane in whichthe half-shade is located. In this plane a dia-phragm is fixed, the image of which exactlycovers the half-shade diaphragm H. In this wayall the rays which otherwise would strike thewalls of the cell and cause objectionable reflec-tions, are intercepted without impending therays which pass through R1 and H. For the lightflux the same expression as before is obtained.

Returning to Fig. 6, we find that it is alsopossible to locate Z within the cell, although inthat case Z must be made smaller if an evenillumination of the half-shade is to be produced.It is possible to deduce the expression for thelight flux (see Fig. 9)

2 H2{R 1(b+x)-HX12

L=iO--L1 (19)

x is the distance between Z and RI, x<b. Theexpression has a maximum for x=0, that is,when the image of the light source coincideswith the diaphragm R 1.

The method represented by Fig. 8 possessesall the qualities of a satisfactory illuminationdevice as specified above. The method has beenused by Dr. Snellmann and the author in meas-urements on electric and mechanical doublerefraction. The superiority of this method ismost obvious with substances where diffusion oflight occurs.

ARRANGEMENT FOR MAGNETIC DOUBLE

REFRACTION

In the case of magnetic double refraction theconditions are more complicated. Hitherto it hasnot been necessary to pay any attention to thedivergence of the rays; the length of the cell iscomparatively great, and the optical path foroblique rim rays is enlarged in the proportion1/cos (/ 2 ), where a is the angle between theoblique extreme rays. As a is small, the corre-sponding correction can be disregarded. In amagnetic field, however, an influence of theFaraday effect on rays forming an angle differentfrom 900 with the field is superimposed. Theplane of polarization of the extreme rays willbe rotated an angle proportional to cos (/ 2). Inorder to avoid this influence a must be kept verysmall. The results therefore become somewhatmodified. If the different methods are to becompared the angle a between the extreme raysmust be kept constant.

If for magnetic double refraction H is theinduction and I a coordinate at right angles to

H

FIG. 7. Method III, reflections.

the field, then the expression fH 2 dl as a ruledetermines the effect. The expression reaches itshighest value for wedge-shaped pole pieces ofthe greatest available length, that is, a lengthcorresponding to the core of the electromagnetand increases when the air gap is reduced. On theother hand, the light flux will decrease with theair gap, which determines the diameter of thediaphragms at the end of the cell. Thus an airgap corresponding to a maximum of sensitivity

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YNGVE BJORNSTAHL

H In the most favorable case to be imagined*

(22)LI= io(72/16) (R2oa2/4).

FIG. 8. Method III.

exists. Furthermore the most favorable arrange-ment of the pole shoes is a function of theamount of substance available and of the lightabsorption.

In order to give an idea of the dimensions of

the cell in some preliminary experiments, carriedout by means of the large electromagnet of thelaboratory the following data may be of interest.The length of the pole shoe of the electromagnetwas 58 cm, and that of the cell about 70 cm.The diameter of the diaphragms at the end ofthe cell was about 0.8 cm.

Of course the remarks on p. 204 must be appliedhere too: this method is not rational.

Method InIn order to eliminate the inefficiencies which

are connected with the above described method,it is possible to modify method II. In Fig. 4,R1 and R2 represent the cell diaphragms, H thehalf-shade and Z the image of the light source.If the respective symbols correspond to thediameters of the diaphragms, we get

a= (Z+H)/b, (23)

where a is the angle between extreme rays.The condition for homogenous illumination ofthe half-shade may be written

HR1-ba. (24)Method In

The method I with parallel light as outlinedabove has been used in the investigations madeby the Cotton school ;8 the half-shade is locatedat a distance of about two meters from thecenter of the pole shoes. Parallel light has alsobeen used by Szivessy9 and by Boarse.10 Diffi-culties appeared in the attempts to diminish theinfluence of reflections. Szivessy covered theinside of the cell tube with a layer of carbon, a

perhaps somewhat questionable method; Boarsemade the tube of copper, treated with hydrogensulphide, in order to produce a light absorbingsurface.

Assuming the brightness io of the light sourceto be the same at all points in order to obtainsimple expressions, according to the expression(6) we have

L=ioHRi/b2 ; H=H7r/4; Rj=Ri2 7r/4; (20)

L = io(7r2/ 16) (12R12/b2) . (21)

They can be used even in the case of parallellight. Further HR 1 . For extreme rays, theangle between which is a, we have

b= (Rl+H)/a.

8 M. Scherer, Theses (Paris, 1934), p. 17.9 G. Szivessy, Zeits. f. Physik 18, 97 (1923).10 H. A. Boarse, Phys. Rev. 46, 187 (1936).

The condition for absence of reflections is Zc R1 .The light flux through the arrangement is

7r2 IH2 Z 2 7r2 (R 1 - ba)2 Ri 2 a2

LIr=io- -16 c2 16 (2Rl-ba) 2

(25)

If we do not pay any attention to the reflections,that is, to the condition Z<R1 , we get

7r2 ra(b+c) -Rn 2IL'=io-(R1-ba)2I ]

16 Cand (26)

L reaches its highest value for c= oo

L/ I I = io ( 7r 2/16) (R1-bat)2o2. (27)

It is evident that b must be made as small aspossible. The half-shade should be located close

L R, R2

z ~~H

FIG. 9. Method III, location of the half-shade in the cell.

* It must be remembered that for various reasons itis not always possible to make an experimental arrange-ment corresponding to "the most favorable case."

Z Ht L R.

Z, I

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-_ I ,

MEASUREMENT OF DOUBLE REFRACTION

to the second cell diaphragm R2. This is incon-venient because of the circumstance that thehalf-shade ought to be located in the vicinity ofthe compensator, the analyzing prism etc. Theseshould be located at some distance from themagnetic field. The method has earlier beenused by the author., 12

If a comparison is made with method I, thesame relation, which has been mentioned onp. 205 exists: method IIm gives a smallerlight flux.

Method IIIm

If we consider method III with the image ofthe light source on the first cell diaphragm RI(Fig. 10), we have for the light flux the ex-pression

L=io(7r2 /16)(Z2 H2 /b2 ); a= (Z+H)/b. (28)

If Z covers the first cell diaphragm

Z=R 1 ; L = o(ir2 /16)R,2(a-R/b) 2. (29)

In order to obtain an even illumination of H,

H_-Ri; b 2Rl/ca. (30)

That is to say, in the most favorable case72 R12

Lrii=io-- a2. (31)16 4

The reflections may be eliminated in the mannerindicated earlier by introducing a diaphragm Ho.If the light fluxes obtained by means of thedifferent methods are compared we find

L 1 11=1;

LI

LIII 12R1-bal2

Lii 4 R-bajIn an actual case,

1 Lill 9a= ; b=8Ocm; R1=0.8cm; =-.

200 LII 4

The last method, which locates the image ofthe light source at the first cell diaphragm, is,therefore, superior to all others, producing alight flux equal to that of the defective methodIm, but giving at the same time a correct illumi-nation.

11 Y. BjbrnstAhl, Phil. Mag. 42, 352 (1921).12Y. Bjornsthhl, Experimental Studies on the Accidental

Double Refraction in Colloids (Upsala, 1924).

If we compare Ll with the light flux ofmethod Im using parallel light, we have

LIII/L'II= R12/4(R,-b a) 2. (33)

Also in this case L 1 II>L'II, if 2ba>Ri.A considerable advantage of this third method

is also that the half-shade may be located at asuitable distance from the second cell diaphragm,that is, outside the magnetic field and close tothe compensator, analyzing prism etc.

L

V . X _ _

FIG. 10. Method III for a magnetic field.

The influence on the measurements of ellip-ticity, which results from not using strictlyparallel light through the specimen cell is thesame at the different methods of illumination,for a given angle a (see Fig. 10 and Fig. 5)between oblique extreme rays. Even the use ofso-called parallel light (cf. footnote on page 203)does not eliminate this influence, which can beestimated with the aid of the theory for theisochromatic surfaces. Since the value of a issupposed to be small this influence can beneglected.

By using the methods of illumination describedabove and a compensator for instance of Brace'stype it is now possible to carry out measurementson substances which could hardly be studiedwithout these arrangements. The phenomenonis especially striking with substances with highmolecular weight and with colloidal solutions, inwhich diffusion of the light is appreciable.

SUMMARY

It is shown that nearly all illumination deviceshitherto used in the measurement of magnetic,electric and mechanical double refraction areinefficient. A description is given of methods forproducing a rational illumination by means ofwhich it is possible to carry out measurements onsubstances that otherwise could not be in-vestigated.

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