APRIL, 1941

Electro-Optical Field Mapping

HANS MUELLERGeorge Eastman Research Laboratories of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Received February 14, 1941)

As a result of the discovery of an unusually large electro-optical effect in certain colloidalsolutions of bentonite it has become possible to study and measure inhomogeneous electricfields by an optical method which is analogous to that of photoelasticity. Observations withcrossed Polaroids show two orthogonal systems of lines. The isoclinic lines determine the fielddirection, the isochromatic lines give the field intensity in any point, of a plane field. Themethod employs a.c. potentials of less than 200 volts.

IT is well known that most liquids becomedoubly refracting under the influence of an

electric field E. The electro-optical birefringenceis uniaxial with the optical axis in the fielddirection. The ordinary and extraordinary com-ponents of a light wave traveling normal to thefield suffer a relative phase shift

= 2irlBE2,(1)

where I is the length of the optical path in theliquid and B is the Kerr constant. From measure-ments of o and of the direction of the opticalaxis it is, therefore, possible to infer the magni-tude and direction of the field intensity in anypoint of a plane field distribution. In principlethis electro-optical method of field mapping issimpler than the analogous problem of elasticstrain analysis by photoelasticity. In the latterthe optical effects are related to a tensor quantity,in the former they serve to find a vector. Inpractice, however, the electro-optical investiga-tions encounter greater difficulties. This is dueto the fact that, while large phase shifts amount-ing to several multiples of 2r can readily beproduced by means of elastic stresses, it hasuntil recently been impossible to create electro-optical double refraction of a similar magnitude.Even in nitrobenzene, though its Kerr constant(B= 1O-5 e.s.u.) is a thousand times larger thanthat of most other liquids, the production of aphase shift of 2 r requires a field of an intensityapproaching the electrical breakdown strengthand an excessively long light path in the liquid.Since field mapping becomes very laborious whenthe phase shifts are small, the electro-opticalmethod can be made practical only if substanceswith larger Kerr constants can be employed.

From the work of Procopju,l BjbrnstAhl,2 andErrera, Overbeek and Sack3 it is known thatmany suspensions, colloids and proteins4 possessKerr constants which exceed that of nitro-benzene by factors ranging from 103 to 107.Unfortunately many of these substances are notsuitable for our purpose for one or both of thefollowing reasons: They are not sufficientlytransparent and the length of the active lightpath is limited. The Kerr effect in colloidsapproaches a saturation value and although theKerr constant is large for small fields the maxi-mum birefringence frequently is smaller thanthat attainable in nitrobenzene. Due to theirelectrical conductivity the colloids are heated inmoderately strong fields. The heating createsconvection currents which give rise to streamingbirefringence and disturb the image formation.

In the course of a recent investigation 5 of theelectro-optical properties of colloids, carried outin collaboration with Dr. B. W. Sakmann, wehave discovered certain solutions of bentonite inwhich the above limiting factors are minimizedto such an extent that the electro-optical methodof field mapping becomes practical. These aque-ous solutions are obtained by dilution of gelsof yellow bentonite. The gels were prepared byProfessor E. A. Hauser6 and Dr. D. S. le Beauin the laboratories of chemical engineering of

' S. Procopju, Ann. de physique 10, 213 (1924).2 Y. Bj6rnstahl, Thesis, Upsala (1924).3 J. Errera, W. Overbeek and H. Sack, J. Chim. Phys.

32, 681 (1935).4 M. A. Lauffer, J. Am. Chem. Soc. 74, 147 (1939).'H. Mueller, Phys. Rev. 55, 508, 792 (1939); F. J.

Norton, Phys. Rev. 55, 668, 1939; H. Mueller and B. W.Sakmann, Phys. Rev. 56, 615 (1939).

6 E. A. Hauser and D. S. le Beau, J. Phys. Chem. 42,1031 (1938); 43, 1037 (1939); 45, 54 (1941).

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ELECTRO-OPTICAL FIELD MAPPING

Massachusetts Institute of Technology. Onlysuch sols are suitable in which the averagediameter of the clay particles is less than 20millimicrons. The gels are gradually diluted untilthe spontaneous birefringence disappears and therelaxation time of the streaming birefringence isreduced to about 1/100 sec. This occurs at aconcentration of approximately 2 percent byweight. The resulting yellowish liquid is trans-parent in thicknesses up to 10 cm. The Kerrconstant for 60-cycle a.c. fields is of the order ofmagnitude B= 10 e.s.u. The saturation value ofthe birefringence is so large that for a light pathof 1 cm the maximum phase shift exceeds thevalue 2 r. The saturation occurs for fields ofabout 150 volts/cm. For fields of 100 volts/cmthe heating effects become noticeable after aperiod of about one minute. They are not dis-turbing since the electro-optical effects can bephotographed with an exposure time of onesecond. The values of the Kerr constant and ofthe saturation birefringence vary with the par-ticle size, the sol concentration, the temperatureand the frequency of the electric field. Aftersome experience one is able to adjust the Kerrconstant to any convenient value by diluting orconcentrating the sol, but the large values of pcan be obtained only with the finely dispersedcolloids.

In order to demonstrate the practicability andreliability of the electro-optical method of fieldmapping by means of these colloids we havestudied the well-known field distribution aroundtwo parallel cylindrical conductors. If the con-ductors are oppositely charged the equipotentiallines and the lines of force are circles, as shownin Fig. 1. These lines are not recorded by theKerr effect. The electro-optical method furnishesthe lines of constant field intensity and the linesof constant field direction. This follows from thefollowing elementary fact: If the Kerr cell isplaced between crossed Nicols and the fielddirection forms an angle (3 with the plane ofvibration of the polarizer, the intensity of thetransmitted light is proportional to

I=sin 2 20 sin2 2. (2)

Hence the intensity is zero when 0 = 0 or 0 = 900.With an inhomogeneous plane field distributionin the Kerr cell we observe, therefore, a set of

dark lines which correspond to the locus ofpoints where the field direction is in the planeof vibration of either the analyzer or the polar-izer. The position of these "isoclinic" lines isindependent of the wave-length of the light; theyalways are black. They are not influenced by achange of the voltage between the conductors andare independent of the length of the Kerr cell.

By rotation of the crossed Nicols the isocliniclines can be shifted so as to pass through any

FIG. 1. The electric field around two oppositely chargedparallel rods. The thin circles are the equipotential linesand the lines of force (dotted). The thick lines are the linesof constant field intensity and correspond to the iso-chromatic lines. The hyperbolas (dot-dashed) are theisoclinic lines for various field directions.

arbitrarily chosen point in the field. The factthat an isoclinic line passes through a point Pwhen the plane of vibration of the polarizer is atan angle q1 with the horizontal direction impliesthat the field in this point forms one of the fourangles V', 14'90', V,+1800 with the horizontal.It usually is easy to decide which one of the fourpossibilities is the correct one.

For the field around the parallel rods theisoclinic lines can be calculated and are shownin Fig. 1. They are equilateral hyperbolas. Forevery Nicol position 4' we obtain two hyperbolaswhich intersect in the centers C. The asymptotesof one hyperbola form with the horizontal theangles 2'V and 2(P-7r). The asymptotes of thesecond hyperbola bisect the angles between thoseof the first hyperbola. For the case = 0 or4 f=900 one hyperbola degenerates into the twoaxes of symmetry.

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According to Eq. (2) the intensity of thetransmitted light vanishes also when the phaseshift p is a whole multiple of 2r. For a fixedlength of the optical path this occurs when thefield intensity assumes one of a series of criticalvalues E1, E2, ***, Ei, * . For our colloids Eq.(1) is not valid but is to be replaced by a morecomplicated relation of the general form

w, = 2 l(E2), (1')

where the function f depends on the character-istics of the colloid. Hence the critical field in-tensities have to be determined empirically forevery solution. For an inhomogeneous field theobservation through crossed Nicols will thereforeshow a second set of dark lines which are the lociof points where the field intensity has one of thecritical values. These lines are called "isodynami-cal" or "isochromatic." The first adjective isborrowed from the nomenclature used in ter-restrial magnetism. The second is used in photo-elasticity and is justified by the fact that inobservation with white light these lines arecolored because the phase shift depends on thewave-length of the light.

The isochromatic lines are shifted and can bemade to pass through any arbitrary point bychanging the potentials of the electrodes. Todetermine the field intensity in a point one makesuse of the law which states that the multiplica-tion of the potentials of all electrodes by acommon factor alters the field intensity in everypoint by the same factor. By following thechanges of the isochromatic lines with potentialvariations and from the color sequence it is easyto assign the proper Ei values to the variousisochromatic lines.

For the field around two parallel cylindricalconductors the isochromatic lines are given bythe equation

r2 =cos 26th(Eo2/E, 2-sin2 2 6) ,

where r and are polar coordinates about theorigin 0. Eo is the field intensity in 0 and hasthe value Eo= V/a In [(r 1 +a)/(a-r 1 )], wherea = (r1 r2)' is the distance OC (see Fig. 1) and V isthe potential between the rods. For Ei>Eo theisochromatic line consists of two rings near theelectrodes; for Ei-Eo these rings merge to forma figure 8 with a normal intersect at 0. For

Ei<Eo the isochromatic lines form a single ringwhich approaches circular shape for small valuesof Ei.

The isochromatic lines are independent of theposition of the crossed Nicols. If the latter wererotated at a high speed the isoclinic lines couldnot be observed and only the isochromatic lineswould appear. The same result is accomplishedby using circular polarizers, i.e., by sending thelight successively through a polarizer, X plate,Kerr cell, -A plate and an analyzer. For thisarrangement the transmitted intensity is pro-portional to sin2

12 and only the isochromaticlines appear.

The isochromatic lines are orthogonal trajec-tories of the isoclinic lines. This is true if theelectric field has no space charges. To prove thislaw one takes into account that the isochromaticlines are given by the relation

c(x, y) = (04/ax) 2 + (/ay) 2 = const.,

where '1 is the potential of the field. The iso-clinic lines satisfy the equation

ab/ayi(x, y)= =const.

acb/ax

From Laplace's equation Act=0 it follows

ac i ac ai

axax ay ay

and this is the orthogonality condition for c and i.The relation between the isoclinic and isochro-matic lines is therefore analogous to the relationbetween the lines of force and the equipotentiallines. Although the electro-optical observationsdo not furnish the latter for the field underinvestigation, the observed lines may representthe lines of force and the equipotential lines of anentirely different electric field problem. In theexample of the oppositely charged parallel rods,for instance, the observed lines are identical withthe lines of force and the equipotential lines ofthe field around two rods which carry chargesof the same sign. In other cases, where the iso-clinic lines begin and end within the field andnot at a boundary, the isochromatic lines can beinterpreted as the equipotential lines of a fieldwith space charges.

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FIG. 2. Electro-optical fieldmaps for the electric fieldaround two parallel rods. Thefirst three maps are madewith crossed Polaroids ori-ented at 00, 450 and 22.5 tothe horizontal, respectively.The last figure gives the iso-chromatic lines as observedwith circular polarizers. Thea.c. potential across the rodsis 90 volts. (The broad darklines are shadows of the leadsto the electrodes.)

While the success of the experimental methoddepends primarily on the choice of the colloid,a series of other factors also are important. Thecolloids will slowly coagulate owing to electro-chemical reactions when they are in contact withcopper, brass and especially with solder. Elec-trodes of nickel, nichrome or monel give nodifficulties, provided they are clean and are notleft in the solution for several hours. Platinumand gold-plated electrodes do not affect thecolloids for periods of several weeks. Containersof glass, Bakelite, lucite and several otherplastics, held together by Duco cement, serve asKerr cells. Since the bentonite solutions areelectrolytic conductors, with a specific resistancevarying between 1000 to 5000 ohm-cm, we aredealing with electric field distributions in aconductor. These fields satisfy Laplace's equationbut the boundary conditions differ from theelectrostatic case. The electrodes remain equi-potential surfaces, but at the surfaces of thecolloid the lines of force must be parallel to thesurface. Near the top, bottom and side surfacesof the cell the observed lines will, therefore,differ somewhat from the theoretical curves, butif the electrodes are in contact with the windowsof the Kerr cell the field distortions at the endsof the electrodes are negligible and the theoryfor the infinitely long rods is applicable. The

leads to the electrodes do not distort the fieldif the wires are insulated.

The micelles of bentonite are negativelycharged. In a d.c. field they migrate to thecathode where they coagulate. This can beprevented only by using alternating potentialsbetween the electrodes, because it has beenfound impossible to obtain isoelectric solutionswith uncharged particles. In ordinary liquids analternating field Eo cos cot creates an alternatingphase shift so=2rlBEo2 cos2 wt= o'(1+cos 2cot)and the isochromatic lines become blurred andcan be observed only with stroboscopic illumina-tion. In our colloids, however, the particles andthe viscosity of the solutions are so large thatabove a certain frequency the micelles are toosluggish to follow the rapid variations of thefield. Nevertheless, they are oriented by thefield because the orienting torque is proportionalto E2 and the mean square value of the fieldintensity does not vanish. For frequencies above1 kilocycle/sec. these colloids show therefore aKerr effect, but the electro-optical birefringenceis steady. In his thesis Dr. Sakmann 7 has shownthat at lower frequencies the phase shift is givenby an equation of the form

so = 2rl{A (Eo2) +B(Eo 2) cos (2wi- a) },B. W. Sakmann, Ph.D. Thesis, Massachusetts Insti-

tutue of Technology, 1941.

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HANS MUELLER

where the value of B vanishes for high fre-quencies. For 60-cycle a.c. the ratio B/A be-tween the amplitude of the alternating and thesteady part of the phase shift depends on theconcentration and temperature of the solutionand on the size of the particles. For those solsfor which the saturation value of A (E0

2) is largeenough, and which are sufficiently transparentfor the purpose of field mapping, this ratio isbetween 1/10 and 1/100 at room temperature.The isochromatic lines are, therefore, reasonablysharp even for continuous illumination. They aresomewhat sharper if we use a light sourceoperated on 60-cycle a.c.

It is advisable to cool the solutions before andbetween the measurements and to apply theelectric fields for short periods only. If the solsget heated above 40'C the convection currentsbecome increasingly annoying, and, since therelaxation time of the particles is reduced in theless viscous medium, the isochromatic lines be-come diffuse.

The experimental equipment and procedurefor electro-optical field mapping are very simple.In our experiments the Kerr cell is a glasstrough with parallel windows 6 X 6 cm wide. Theelectrodes are rods of nickel, 93 inch in diameterand 5 cm long. They are held in parallel position,2 inch apart, by two plates of lucite. The a.c.potential is applied over two rubber insulatedwires and is adjusted by means of a Variactransformer. Polarizer and analyzer are twosheets of Polaroid. The light source is an a.c.-operated Hg arc. For observations with mono-chromatic light we use a green filter. The lightis made parallel by a first lens, and a second lens

41T Ltr

so E E, 100 E.

FIG. 3. Typical calibration curve for the electro-opticaleffect in bentonite solutions for 60-cycle a.c. fields (involts per cm and for a light path of 5 cm).

focuses the transmitted light into the cameraobjective. We use Agfa Superpan films with anexposure time of 1 sec. The photographs in Fig. 2demonstrate the practicability of the method. Ina qualitative way the agreement between theobserved and the calculated isoclinic and iso-chromatic lines is satisfactory. Slight deviationsare noticeable for the outermost isochromatic lineand for the isoclinic lines near the edges of thephotographs. These deviations are due to thefinite size of the Kerr cell.

A preliminary quantitative test was carried outas follows. Two plane and parallel nickel elec-trodes, of the same length as the rods and witha plate separation of 1 cm, were inserted in thecolloid and the critical values E were foundfrom the voltages required to produce extinctionbetween circular polarizers. For the particularcolloid used the following values were found

El=42, E2=66, E3 =89,E4 = 118 r.m.s. volt/cm.

The diagram in Fig. 3 illustrates the deviation ofthe Kerr effect in colloids from the classicallaw (1). It should be noted that this type offield dependence is advantageous for our pur-pose; the isochromatic lines are not crowdedtogether in the region of high field intensities,as they would be if the quadratic law werevalid.

This same colloid was used to measure thefield intensity in the midpoint between thecylinders. This is done by recording the voltagesbetween the two rods for which the first, second,third and fourth isochromatic lines assume theshape of the numeral 8. These voltages are

V1 =41, V2=64, V3=87,V4=115 volt (effective).

The close agreement between the V and Eivalues, which at first appeared to us quitemysterious, is due to the following coincidence.In our experimental arrangement the distancesr1 and r (see Fig. 1) are 0.40 and 0.87 cm hencea= (rir2)-= 0.59 cm and therefore

Eo = V/2.30a log (ri+a)/(a-ri) = 1.02 V.

This means that the voltages V which are re-quired to produce in the midpoint a field Eo

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ELECTRO-OPTICAL FIELD MAPPING2

which coincides with one of the critical valuesEs is for our arrangement 2 percent smaller thanthe numerical values of Ei. The data, therefore,show that the method is quantitative andfurnishes the values of the field intensities withan accuracy of about 1 percent.

The calibration of the colloid can also be per-formed by the photographic method. This is doneby studying the field in a cylindrical condenser.From the diameter of the rings shown in Fig. 4the values of Ei can readily be calculated.

The photographs in Fig. 4 suggest a variety ofproblems which might be investigated by theelectro-optical method, as for instance, the edgecorrections of parallel plate condensers and thedistortion of a homogeneous field by obstacles ofvarious shapes. Inasmuch as these photographsserve to find the solution of complicated boundaryvalue problems involving Laplace's equation, themethod is apt to furnish valuable information forproblems in other fields where Laplace's equationis of importance, e.g., in hydrodynamics.

The experimental method can be improved invarious ways. The conductivity of the colloidscan be reduced by dialysis. Stroboscopic illumi-nation or the use of fields of higher frequencieswill improve the sharpness of the isochromaticlines. By employing a longer optical path andcolor photography it will be possible to determinefrom a single picture the field intensity in anypoint within a large area of the field. For accuratemeasurements it would be advisable to mountthe Kerr cell in a thermostat and to record the in-tensity variations by microphotometric methods.

FIG. 4. Electro-optical field maps. At left: isochromaticlines and the isoclinic lines for 22.50 of the field at the edgeof a parallel plate condenser. At right: isochromatic linesfor the field in a cylindrical condenser and the distortionby a rectangular insulator of a uniform field between twoparallel vertical plates.

The development of this new method wouldhave been impossible without the collaborationof Professor E. A. Hauser and Dr. D. S. le Beauand their co-workers who prepared the bentonitegels, and of Dr. B. W. Sakmann, whose investiga-tions clarified the electro-optical phenomena inthe colloidal solutions. It is a pleasure to ac-knowledge the continued interest and assistanceof these co-workers and to thank Mr. G. J.Yevick for technical assistance.

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