c 373 P-/W4 UC-NRLF 37*1 EXCHANGE THE CRYSTELLIPTOMETER An Instrument for the Polariscopic Analysis of Very Slender Beams of Light BY LE ROY D. WELD UNIVERST OF Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 1922 University of Iowa THE CRYSTELLIPTOMETER AN INSTRUMENT FOR THE POLARISCOPIC ANALYSIS OF VERY SLENDER BEAMS OF LIGHT BY LE ROY D. WELD I. INTRODUCTORY The determination of the elements of elliptic vibration of light, which has been transformed from a condition of plane to one of elliptical polarization, is found to reveal so much concerning the optical nature of the agencies effecting the transformation that it becomes a matter of considerable importance to be able to make such measurements with reasonable accuracy. It so happens that the ordinary methods, which are employed for the elliptical analy- sis, and most of which are too well known to need setting forth here, require a fair breadth of field, and would therefore not easily apply, for example, to a ray passing through a pin-hole. The research presented in this paper had its origin in a problem suggested to the writer some years ago by Dr. L. P. Sieg, and was begun in 1915. It has been carried out at Coe College by means of apparatus, much of which was kindly loaned for the purpose by the University of Iowa. Various investigators have attempted to obtain the optical constants of metals and other opaque substances by reflection methods, with indifferent success. It appears certain from the research of Tate 1 that the difficulty has been due, not to lack of precision in the analysis of the elliptically polarized light reflected from the opaque surfaces, but to the variation in surface condi- tions, brought about by the polishing process, which modified the nature of that light. Thus it was found impossible to prepare two mirrors of the same kind of metal, even by the same process, which would give consistent results, while different measure- ments, and even different kinds of reflection measurements, on 1 Tate, Phys. Rev., 34, p. 321, 1912. 67 68 ** **w/WEW> [J.O.S.A. & R.S.I., VI the same fresh surface, gave good agreement. Dr. Sieg's suggestion was that opaque crystals, with their natural, unpolished facets, might be prepared of such surface purity as to obviate this diffi- culty, and at the same time furnish material for an interesting experimental research in crystal optics. Comparatively few large opaque crystals can be obtained in perfect form. Drude 2 made some experiments long since with lead sulphide, and Muller 3 subsequently, with antimony sulphide, freshly exposed by cleavage, employing ordinary polariscopic methods. In 1916 the writer made a preliminary report 4 on the present research, which had already given unmistakable evidence of strong double refraction in minute hexagonal selenium crystals, and at the same meeting, 5 Dr. Sieg reported having found by direct photometric measurements that the selenium crystal has two different reflecting powers in the two principal directions. Mr. C. H. Skinner has given an account 6 of his interesting observations on selenium, using the ordinary Babinet compensator method, from which he concluded that the crystals, like artificially prepared surfaces, have in some respects their individual peculiarities, espe- cially in the longitudinal direction. The method adopted by the writer was presented to the Physical Society in 1917 but was published only in abstract 7 at that time, the research being then still in progress. It is applicable not only to the polariscopic analysis of light reflected from polished plane surfaces of any sort, however small (such as minute spikelets or flakes of selenium, tellurium, cadmium, etc.), but to light in slen- der beams from any source. Its application has recently been suggested, for example, to the comparison, from point to point, of the optical properties of metal film-deposits of non-uniform thickness. Furthermore, being a photographic method, it is adaptable to the ultra-violet, and the apparatus was designed with this in view. Its original application to crystal-reflected, Drude, Ann. d. Physik, 36, p. 532, 1889. 3 Muller, Neues Jahrbuch fur Mineralogie, 17, p. 187, 1903. 4 Weld, Proc. Iowa Acad. Sci., 1916, p. 233. 6 Sieg, Proc. Iowa Acad. Sci., 1916, p. 179. Skinner, Phys. Rev., N.S. 9, p. 148, 1917. 7 Weld, Phys. Rev., N.S. 11, p. 249, 1918. Jan., 1922] THE CRYSTELLIPTOMETER 69 elliptically polarized light is what suggested to the writer the name "crystelliptometer" for the distinctive apparatus employed. II. PRINCIPLE OF THE POLARISCOPIC METHOD As regards the polariscopic analysis itself, the method employed is a modification of that used by Voigt for the study of polarized ultra-violet. The principle is stated in Voigt's original article 8 and a more detailed account given by Mr. R. S. Minor 9 who applied it to artificially polished mirrors of steel, copper, silver, etc. A non-mathematical statement of the original method will first be given. Let us have a uniform parallel beam of elliptically polarized monochromatic light; to determine the two elements of the elliptic a vibration, which may be taken as the ratio r = -of the X to the Y o amplitude, and the phase advance V of the X ahead of the Y harmonic component (either positive or negative). 10 R C Y =1 \ \ Fig. 1 Cross hairs and quartz wedge systems. XV , reticule. C, compensator. R, rotator This beam first passes through a pair of quartz wedges similar to a Babinet compensator, except that they are fixed with refer- ence to each other (C, Fig. 1). The edges are, let us say, vertical, that is, parallel to the Y direction. As we now face the oncoming 8 Voigt, Physik. Zeitschr., 2, p. 203, 1901. 9 Minor, Ann. d. Physik, 10, p. 581, 1903. 10 The reason for here using the inverted V instead of the customary A will appear later (Sec. VII). 70 LE ROY D. WELD [J.O.S.A. & R.S.I., VI light and move across the emergent beam from left to right, we encounter a steady increase in the phase advance, so that, whereas the light through the center of the compensator has phase difference V (the same as the original light), that at distance x from the center has phase difference V+sx, where s is the com- pensator constant in degrees of phase per millimeter. Periodically, therefore, we shall come to regions of light plane polarized at some angle, viz., where V +sx = 0, TT, 2-Tr, .... v -^ H / \ Fig. 2 Showing action of rotator on a strip of plane-polarized light The next optical member is another pair of wedges, which will be called a rotator (R, Fig. 1). This consists of a wedge D of right- handed, and one L of left-handed, quartz, the optic axis of each being along the path of the light. It is easily seen that if the light entering the rotator be plane-polarized, it will suffer a net rotation of plane which will be the greater, the higher the point at which it traverses the two wedges. This rotation may be designated by qy, q being the rotator constant in degrees of azi- muth per millimeter. To an observer facing the light as it comes from the rotator, any one of the periodic vertical ribbons of plane-polarized light from the compensator will have been acted upon by the rotator as Jan., 1922] THE CRYSTELLIPTOMETER 71 shown in Fig. 2. Therefore as we traverse this emergent beam vertically along any one of these plane-polarized regions, we shall encounter, at intervals, light vibrating vertically, and half way between these, light vibrating horizontally (V and H, Fig. 2)." Finally, let the beam be passed through a Nicol set so as. to extinguish, say, the vertical component. Obviously, wherever we had in one of these plane-polarized strips a point of vertical vibration, we shall now have a black spot. The field will then be covered with black spots arranged in a regular pattern of vertical columns and horizontal rows, the exact design of which will depend upon the elements of the original elliptic vibration. Con- versely, these elements may be easily deduced from the observed arrangement of the spot-pattern. There is introduced into the Fig. 3 Spot-pattern for elliptically polarized light reflected from nickel. (Enlarged.) field a pair of cross hairs corresponding to X and Y axes and coinciding with the neutral lines of rotator and compensator, respectively. This reticule is photographed along with the spot- pattern and the coordinates of the spots determined by subsequent measurements on the plate. Such a spot-pattern, with the cross hairs, is shown, enlarged, in Fig. 3, in which the elliptic polariza- tion was produced by reflection from nickel. The equations relating to the processes just described, and the practical deduction of the elliptic elements therefrom, will be worked out in subsequent sections of the paper. 72 LE ROY D. WELD [J.O.S.A. & R.S.I., VI III. ADAPTATION or THE METHOD TO SLENDER BEAMS The above procedure, as followed by Voigt and others, ob- viously requires a beam of light whose .cross section is large enough to cover the entire spot-pattern, and hence would not be adapted at all, for example, to light reflected from one facet of a small crystal. The writer's modification of it consists simply in keeping the very slender parallel beam, so reflected, stationary, and mov- ing the whole analyzing system of quartz wedges, cross hairs, Nicol and camera back and forth perpendicularly across it, with AZIMUTK CIRCCEr? Fig. 4 E Diagrammatic layout of apparatus. 5, source. M, monochromator. L t , Ni, collimator and polarizer. K, reflecting surface mounted in front of drum D. X to E, crystelliptometer a sort of weaving motion, until the whole spot-pattern has-been covered. The effect is the same as if the beam had a cross section as large as the field thus traced out. Whenever any point of the analyzing system corresponding to one of the dark spots arrives at the beam, the latter is extinguished thereby, and the spot is left on the plate. In practice it is not necessary to cover the whole pattern, but only the regions of it in which the rows of spots are known approximately to lie; and this fact saves much time. Jan, 1922] THE CRYSTELLIPTOMETER 73 The assemblage of optical parts, which is given the lateral motion just described, together with its mountings, considered as a single instrument, is what has been referred to as the "crys tellip tome ter . " The arrangement of the writer's apparatus for the study of crystals is shown in Fig. 4. Light from the source S is focused by the quartz lens LI on the inlet slit of the monochromator M, from whose outlet slit the monochromatic light diverges and is collimated by the quartz lens L 2 . (In the earlier work, color filters were used instead of the monochromator, and the filtered light focused by LI directly upon the slit of the collimator.) The parallel beam from L 2 is plane-polarized in any desired azimuth by Nij a large polarizing prism with square ends and a glycerine film. Fig. 5 A, spot-pattern for light reflected for selenium in parallel position. B, in perpendicular position The crystal or other small reflector K is mounted on a spectrom- eter, by means of whose telescope T and verniers V any desired angle of incidence may be secured. The mounting can be rotated so that it is very easy to exhibit the double refraction of selenium, tellurium, etc, the spot-patterns obtained with axis horizontal and with axis vertical being quite visibly different, as seen in Fig. 5. The slender reflected beam, in general elliptically polarized, now enters the analyzing system X, C, R, N 2 of the crystelliptom- eter. X represents the cross hairs, which the writer has found it necessary to place in front of the compensator C. (See Sec. VI.) 74 LE ROY D. WELD [J.O.S.A. & R.S.I., VI The compensator wedges have an angle of 52', which gives a relative phase change of about 144 per horizontal millimeter with sodium light. R is the rotator, with wedge angle 24, which gives a rotation in azimuth of about 16 per vertical millimeter with sodium light. The dimensions of the field are about 15 x 30 mm. The wedge system and cross hairs are mounted adjustably in a brass tube which screws upon the tube containing the analyzing prism Ni. Nz is a duplicate of Ni, the two being interchangeable. They are rectangular prisms 15 x 30 x 30 mm consisting of Iceland spar wedges separated by glycerine, the angle of incidence on the inter- face being 65. The emergent (extraordinary) light vibrates parallel to the 15 mm dimension. The spot-pattern and cross hairs are photographed together, by means of the quartz lens 3, upon the plate P. Behind the plate is an eyepiece E, used in adjusting the focus and alignment before the plate is inserted. The plates are 1 x lj^ inch, with emulsion adapted to the wave-length, and are held in a diminutive plate holder. The whole crystelliptometer, from cross hairs X to eyepiece E, is contained in a tube about 85 cm long, so mounted on two pairs of guides at right angles that it can be given the weaving motion referred to in order to trace out the spot-pattern. This is accom- plished by means of two micrometer screws perpendicular to each other. One of these screws is driven by a worm-gear electric motor mechanism so as to move the instrument slowly across, along a line determined by the other screw, the speed varying with the exposure required. Furthermore, the whole tube, vith its micrometer mountings, can be rotated about its longitudinal axis through any desired angle, thus varying the relative inclina- tion of the elliptic vibration to be analyzed, with respect to the coordinate axes of the analyzing system. The extinction plane of the analyzer N 2 , which coincides with the Y axis, being first verti- cal, one spot-pattern is produced. On rotating the instrument through an angle 6 another is obtained, and so on; without, how- ever, modifying in any way the actual nature of the elliptic light under analysis. The equations used in the subsequent Jan., 1922] THE CRYSTELLIPTOMETER 75 theory make provision for this arbitrary angle 6, the advantage of which will then appear. The source of light at first employed was an open Nernst fila- ment, which has later been replaced by a special low-voltage tungsten ribbon lamp. In some preliminary work with ultra- violet, an iron arc was employed. A general view of the apparatus is shown in Fig. 6. Fig. 6 General view of crystelliptometer and accessory apparatus The spot-patterns are measured upon a micrometer comparator to one-hundredth of a millimeter. The lenses were removed from the microscope, and a pin-hole and hair-line substituted, a device suggested to the writer by Dr. Elmer Dershem. Not a little of the routine work consists of the p'ate measurements and their reduction. It has been found possible, with good, clear plates, to locate a spot by a single measurement with a probable error of less than one-hundredth of a millimeter. A selenium spot-pattern and the corresponding "comparison plate" (see Sec. VI) are shown 76 LE ROY D. WELD [J.O.S.A. & R.S.I., VI together as A, B in Fig. 7, and another similar pair, with greater wave-length, as C, D. The quartz lenses and wedges and the Iceland spar wedges for the Nicols, designed by the writer, were made by Hilger of London, as was also the monochromator; the comparator by Gaertner of Chicago. The remainder of the special apparatus, including the driving mechanism, was built at the University of Iowa by Messrs. M. H. Teeuwen and J. B. Dempster, assisted by the writer. Fig. 7 A, selenium spot-pattern. B, corresponding comparison plate. C, D, same, with greater wave-length. Note the greater spot-intervals IV. OUTLINE OF THE MATHEMATICAL THEORY OF THE ANALYZING SYSTEM The mathematical theory of the action of the analyzing system, as used in the writer's method, and of the determination of the elliptic elements, will now be briefly given. 11 In the notation here used, X and Y are components of the vibration displacement of the light, while x and y are coordinates of points of the field with reference to the axes. 11 This is necessary for the reason that the present method departs in certain essen- tial particulars from the original and gives rise to a different set of equations. Minor did not, for example, rotate the quartz wedge system through the arbitrary angle 6 which makes possible the least square adjustment of the observations, and which, as will be easily seen, also provides automatically for any difficulty due to the azimuth of the light under examination happening to be very small, without altering that light in any way. Jan., 1922] THE CRYSTELLIPTOMETER 77 Let the harmonic components of the elliptic vibration to be an- alyzed, as viewed by one facing the oncoming waves, be given by the equations X = a cos (cot-fV) (1) F = b cos cot (2) If the analyzing system by now rotated through an angle +6, these equations are transformed, with reference to the new axes, into X' = a cos B cos (cot +V) +b sin cos cot. (3) F' = b cos 6 cos cot- a sin 8 cos (cot+V) (4) Upon passage through the compensator, the F-component receives an advance of phase, varying with the abscissa x of the point where it passes through, and equal to sx. (3) and (4) then become X" = a cos cos (cot+V) +b sin 6 cos cot, . . . (5) F" = 6 f cos 0cos (tot+sx)-a sin 6 cos (cot+V +sx)(6) Upon passage through the rotator, there is a simple rotation of the axes of the ellipse, without further change, through an angle qy, which corresponds to a rotation qy of the coordinate axes, so that the components of the finally emergent light are X'" = a cos 6 cos qy cos (cot+V) +6 sin 6 cos 673; cos cot b cos 6 sin qy cos (cot +sx)+a sin 6 sin qy cos (cot+V +sx). _ (V) Y f " = (a corresponding long expression which we shall not need) (8) The light now passes through the Nicol N 2 , which shuts off the F-component all over the field; hence (8) is not needed. Only X'" gets through, and this will vanish, leaving the dark spots, at every point where x and y have such values as to render the expression (7) equal to zero all the time, that is, independently of the value of t. To fulfill this condition, dividing (7) by b cos 6, letting - = r, expanding the parentheses containing co/ and group- b ing the terms in sin cot and cos cot separately, we have [ r tan 6 sin qy sin (V +sx) +sin qy sin sx r cos qy sin V] sin cot 78 LE ROY D. WELD [J.O.S.A. & R.S.I., VI +[tan cos qy+ r cos qy cos V sin qy cos sx -f-r tan 9 sin qy cos (V+sx)] cos cot = 0. That this may be true independently of /, the coefficients must separately vanish, giving r tan 6 sin qy sin (V +sx) -f sin qy sin sx r cos qy sin V = . . (9) tan 6 [cos qy-\-r sin qy cos (V+s#)] -\-r cos g;y cos V sin qy cos sx = (10) Letting tan 6 = m, expanding the functions of V+stf, and collect- ing, these become [m sin qy cos sx + cos qy]r sin V -f- m sin qy sin s#. r cos V = sin qy sin s# (11) m sin y sin $#. r sin V -f [m sin g;y cos sx+ cos g;y]r cos V =smqycossxmcosqy (12) In these equations, x and y are the measured coordinates of any dark spot, 5 and q are the compensator and rotator constants, determined from the spot intervals, and m is the tangent of the known angle 6. Hence everything is known except r and V. Letting m sin qy cos s#+cos qy = H, m sin qy sin sx = K, sin qy sin sx =L, sin qy cos sx m cos qy = P, (11) and (12) become H-r smV+K-r cos V =L (14) K-r smV+H-r cos V =P (15) the solution of which, as simultaneous equations, gives the re- quired elliptic elements (13) LH-PK tan V = PH-LK It is thus theoretically possible to deduce the elements from the measured coordinates of any single spot. Jan., 1922] THE CRYSTELLIPTOMETER 79 V. APPLICATION TO SPOT-PATTERNS If we place under examination light which is plane-polarized at azimuth 45, so that r = 1 and V = 0, it will be seen from the manner of their formation, as described in Sec. II, that the spots will be symmetrically arranged with respect to the F-axis in equally spaced horizontal rows. This is the condition of things on the comparison plate, Fig. 7 B (see Sec. VI). Let the distance apart of the spots in the rows be d, and in the vertical columns, (Fig. 8). These are easily measured (see Sec. VI). The com- pensator constant s is equal to , and the rotator constant q d 1 80 equals , in degrees per millimeter (see Sec. II). o \ PH +* < k III! 1 1 +' 4 I t I I - I t 1 I I- 2 i t r * t t* 444 s 441- itt r t t t - 444 444* Fig. 8 Diagram of spots on a comparison plate, taken with plane-polarized light at 45 azimuth Fig. 9 Diagram of spots on a pattern taken with light elliptically polarized at phase difference 202 and azimuth 29 If now we introduce by any means a phase difference V between the X- and F-components, there will be a uniform lateral dis- placement a of the whole spot system, so that the coordinates of the spot 5, which in Fig. 8 are 0, , now become a, . Then, again, if a change is produced in the azimuth of the incident light, the alternate rows +1, 2, etc. will be displaced vertically one way, 80 LE ROY D. WELD [J.O.S.A. & R.S.I., VI and rows +2, 1, etc., the other way, through a certain amount a, so that now the coordinates of S are x a, y = -fa (a being neg- ative). These changes are shown in Fig. 9. 12 Introducing the above values of s, q, x, and y into (13), and resuming m = tan 0, we get [~\ r T r~ n 45+ -180 cos -360 +cos 45 -f -180 5 J La J L 8 _r [a "1 fa ~| 45 + - 180 sin -360 , 1 Fa T ' (18) L = sin 45 + - 180 sin ^360 , L 5 J Id J P = sin|45 H-^ 180 |cosfo60] -tan (9 cos|45H-- 180 1. L d J La J L S J While the attention is, indeed, fixed on one particular spot 5, 5 Fig. 9, whose coordinates are a and -fa, yet in actual practice it is expedient to make measurements on a number of spots, usually twenty or more, and deduce the position of S from them. Refer- ring to Fig. 9, it is clear that the abscissa of any spot in the nth vertical column (numbered along the bottom of the figure) is d (19) which gives n By measuring the abscissas of several spots in different rows, thus varying x and n, we obta'n as many independent observa- tions upon a. 17 If the azimuth is made 90, rows +1, +2, rows 1, 2, etc., will merge or dove-tail together in pairs forming continuous horizontal dark stripes. If it is made 0, the pairs of rows +1, 1, etc., will similarly coincide. This affords a good means of adjusting the quartz wedge system with respect to the previously adjusted analyzing Nicol (see Sec. VI). Jan., 1922] THE CRYSTELLIPTOMETER 81 Again, the ordinate of any spot in the *>th horizontal row (counted upward or downward from the X-axis) may be seen to be or .-(-1)'- ' + j.. ..(20) ( according as the row is above or below the X-axis), and we shall have, therefore, as many observations upon a as there are spots measured. The averages from these observations on a and a are easily obtained by means of formulas depending upon the particular selection of spots made. If the selection consists of an equal num- ber of vertical columns on each side of the F-axis and of horizontal rows above and below the X-axis (as should be the case for other reasons), these averaging formulas become quite simple, a, a, d and 5 being thus determined from the measurements on the plate, substitution of their values in (18) gives the necessary constants H, K, L, P, appearing in the expressions for r and tan V, Eqs. (16), (17), and the problem is solved so far as is possible from a sin- gle experiment. We may, however, expose other plates with different values of 0, obtained by rotating the crystelliptometer tube about its own axis. This does not alter the elliptic elements, but it gives new values of H, K, L, P. In such case, (14) and (15) may be used as observation equations of the first degree with r sin V and r cos V as unknowns, and we may obtain as many different pairs of them as there are measured plates, finally adjusting them by the method of least squares. It has been the writer's practice to assign a weight to each measured plate by means of the grading method, 13 each plate characteristic, such as clearness, symmetry of spots, etc., being graded separately. It should be stated that, except in cases where the number of spots on a plate available for measurement is too limited for precision, a 13 Weld, A Method of Assigning Weights to Original Observations, Science, 50, p. 461, 1919. 82 LE ROY D. WELD [J.O.S.A. & R.S.I., VI single plate, with 6 = 0, is sufficient, and a vast amount of labo- rious calculation is thus avoided. For with 6 = in (18), the elliptic elements, given by (16) and (17), reduce at once to (21) (22) It is only with light of very large wave-length that the number of spots in the field is likely to be so few as to require the repeated exposures. VI. MISCELLANEOUS DETAILS AND SOURCES OF ERROR The general arrangement of the apparatus as employed for the study of opaque crystals was explained in Sec. III. In order to adjust all the parts in proper relation to each other, use is made of a sensitive cathetometer set with reference to the pier on which the apparatus stands. By this means is secured the horizontality of the monochromator, the collimator, the polarized incident beam, the reflected beam from the crystal, and the cry stellip tome ter tube, taken in the order named. The plane of incidence and reflection is thus strictly horizontal, and the azi- muth of polarization, the arbitrary angle 6, and the vibration com- ponents X and Y are reckoned with reference to it. The crystal is mounted on the end of a small rod in front of the dark opening into a hollow drum, a black body, so to speak, which makes a perfectly dead back-ground. The mounting may be turned in a vertical plane, and is provided with a graduated circle, so as to give the crystal any desired angle from to 180 with the plane of reflection. The drum (D, Fig. 4) is placed on a spectrometer prism table for adjusting the angle of incidence, as previously explained. Considerable trouble has been experienced with the mono- chromator, inasmuch as no reliance can apparently be placed upon the wave-lengths indicated by it; and furthermore, the wave- length corresponding to any given setting is found to vary from day to day. In all final work it has therefore been customary to Jan., 1922] THE CRYSTELLIPTOMETER 83 divert the light emerging from the monochromator into a separate grating spectrometer (not shown in Fig. 4) and compare it with the sodium standard just before making each exposure. Another and more serious difficulty with the monochromator is the impur- ity of the light furnished, especially in the shorter wave-lengths. The accurate adjustment of the focus of the collimating and camera lenses is a matter of some importance, especially the latter. These quartz lenses are, of course, not achromatic, and the focus must be calibrated for wave-length. In the case of the collimator, it has sufficed to measure the focal length for one wave-length on an optical bench and calibrate the tube from the known dispersion of quartz. But any inaccuracy in the focus of the camera lens will result in displacements of the spot images and resultant errors in the elliptic elements, so that this requires greater precision. The method here employed is one devised by the writer and referred to as the "offset" or "broken prism" method. 14 The proper focus for a given wave-length may be thus obtained with a probable error of only one or two tenths of a millimeter, and it is easy to calibrate the focus tube accordingly. The need for a special precaution arises from the fact that the spot-pattern in the crystelliptometer appears to be a sort of virtual image lying in a definite plane. It is necessary to get the cross hairs accurately into that plane, otherwise there will be an apparent parallax between cross hairs and spots, and the results will be seriously affected if the light happens to be not strictly parallel to the crystelliptometer axis. Furthermore, the position of this virtual plane is found to vary systematically with the wave-length, so that the adjustment has to be made for each wave-length used. This is accomplished by mounting the cross hairs in a ring having a longitudinal micrometer movement in .the tube. The crystelliptometer is turned a little to right and left with respect to the beam of light and the reticule moved forward or backward until the parallax disappears. It has always been found necessary, in the visible spectrum, to place the reticule in front of the compensator, as in Fig. 4. 14 Weld, Some Precise Methods of Focusing Lenses, School Science and Mathe- matics, 18, p. 547, 1918. 84 LE ROY D. WELD [J.O.S.A. & R.S.I., VI The cross hairs in the writer's instrument are of very fine spun glass, their images representing on the plate the X- and F- axes of the spot-pattern. The ring in which they are mounted is provided with the usual lateral adjusting screws. It is inevitable that the reticule will get out of adjustment laterally, and for the purpose of determining this error (which would be serious if neg- lected), what are called comparison plates are taken at frequent intervals, using a full-sized beam of parallel, strictly plane- polarized light direct from the polarizing Nicol. The cross hairs are first given approximate adjustment visually, using this light. The comparison plate is then taken and the small errors of cross hair adjustment remaining, amounting to a few thousandths of a millimeter, are determined by measurements upon it and proper allowance made for them in the reduction of other plates. If a symmetrical pattern is selected, the adjustment of the horizontal cross hair is really immaterial, as the true Jf-axis can readily be found as the mean position of the horizontal rows of spots em- ployed in measuring any plate. Owing, no doubt, to slight inequality of the quartz wedge angles, the cross hair adjustment error is found, like the parallax, to vary systematically with the wave-length, and a new compari- son plate must therefore be taken for each wave-length used. It is very difficult, even with the spectrometer test, to keep the wave- length strictly constant through a series of experiments. Curiously enough, the most sensitive, and the final, check on wave-length has been found to be the spot-interval d (Figs. 8, 9), which can be measured with great precision (see below). The greater the wave-length, the farther apart are the spots, both horizontally and vertically, on account of the dependence of both the phase- relation change, and the rotation, in quartz, upon the wave- length. (See Fig. 7.) In practice, it is found advisable to determine, by means of auxiliary comparison plates, the relation of the cross hair error to d for light in each region of the spectrum used, and to deduce the required correction from the measured d on each elliptic plate. The accurate determination of the spot intervals d and 6 for each plate is an essentially vital part of the work. The most Jan., 1922] THE CRYSTELLIPTOMETER 85 probable value of d can be deduced fr m the plate measurements by forming observation equations from the abscissas of the spots taken by groups in each row, and applying the simple least square adjustment requisite to the case. This is all done very quickly by means of a formula which is the same for all spot-patterns similarly selected. With a symmetrical'spot-pattern 5 is simply twice the mean absolute ordinate of the spots (without sign). Thus no extra measurements are necessary for d and 8. But it will not do to rely on the comparison plate, or any other one plate, for the spot-intervals corresponding to a supposedly fixed wave-length, as variations of wave-length too slight to have noticeable effect on the ordinary properties of the light, will cause serious errors through this means. When, as usual, the arbitrary orientation (Sec. IV) is zero, it is seen by Eq. (21) that the ratio r of the two vibration components of the unknown light, which is the tangent of the azimuth, is given in terms of a and 5. These are determined from the y measure- ments alone. For some reason not certainly explained, it has been found that there is a persistent error in r as deduced from the measurements on the spot-pattern, whose value appears to be a linear function of r itself. The error is eliminated by first finding the uncorrected value of r from the plate measurements, and then taking a correction pattern with plane-polarized light having azimuth set for that value of r (which is thus accurately known). The measurement of this plate, and the comparison of the erro- neous value of r deduced therefrom with the true value given by the polarizer azimuth circle, afford the necessary correction, which sometimes amounts to as much as two or three degrees. This must be done for each wave-length used. These azimuth- correction plates, like the comparison plates, are taken with full- sized beam from the polarizer and require only a short exposure. One more detail of technique is worthy of note. It will be noticed from Fig. 7C, for example, that on some patterns the spots are not symmetrical vertically, but are decidedly triangular or cuneiform, and tend to run together in double rows. This makes it difficult to estimate the Y position of the spot nucleus with certainty in measuring these plates. It will not do to bisect the 86 LE ROY D. WELD [J.O.S.A. & R.S.I., VI spot with the micrometer in this direction. In order to ascertain the location of the nucleus within the spot, the means adopted has been to calculate, theoretically, the geometrical form of the concentric lines of equal intensity surrounding it, taking the con- stants from actual measurements on typical plates. The form of these lines, derived from an application of the mean value theorem, is cos(7-^) = 1 - J/ - (1 - r2)c0s2(?y .. ..(23) r sin 2qy in which If is a parameter depending on the intensity along the curve in question; for the nucleus, M=0. x and y are the coordinate Fig. 10 Lines of equal intensity about the spots of Fig. 5B, greatly enlarged variables, and the other quantities have the same meanings as in Sec. IV. Fig. 10 shows the nuclei and curves corresponding to M = 0, 0.25 and 0.50 on the plate shown in Fig. 5B. With the knowledge afforded by such a figure, it is easy to estimate with some accuracy what point within the spot is to be aimed at in making the measurement. The nucleus appears to represent approximately the center of gravity of the spot area, rather than to bisect its vertical dimension. Jan., 1922] THE CRYSTELLIPTOMETER 87 VII. TYPICAL APPLICATIONS. REFLECTION FROM SELENIUM CRYSTALS Among the preliminary tests of the cry stellip tome ter and the methods of using it set forth in the foregoing sections were the analysis of polarized light rendered elliptical by reflection from nickel and copper mirrors or from crystals of lead sulphide and tellurium, or by passage through sheets of mica; and a tryout of FresnePs equations for the rotation of plane-polarized light reflected from glass. The only problem, however, upon which serious attack has yet been made by the crystelliptometer method is the experimental part of an extensive research now in progress at the University of Iowa, viz., the optical laws of absorbing crystals. The general theory of this subject was handled with great thoroughness by Drude in his inaugural dissertation 15 many years ago, but until recently the only experimental data upon which tests of the theory might be based have been with reference to certain large crystals of the rhombic system. The unique elec- tro-optical properties of the hexagonal selenium crystal have sug- gested a revival of the subject, the outgrowth of which is the research in question. The few data given below are the first final results obtained by this method, and will serve to illustrate it. They are summarized from a long series of measurements on many selenium crystals, and involved the taking of nearly three hundred plates. The instrument- is now in the hands of other observers whose aim is to accumulate information regarding the optical properties of small crystals of various metallic sub- stances. Some of the finest crystals of hexagonal selenium ever prepared were kindly put at the writer's disposal by Dr. E. 0. Dieterich, who produced them by sublimation at the University of Iowa. They are 2 or 3 cm long and with facets often . 5 mm in width. From some hundreds of these, about a dozen superb specimens were selected and mounted for use with the crystelliptometer. Selenium has a tendency to twist and warp, and great care had to be exercised to select crystals with plane facets. Even with these it 16 Drude, Ann. d. Physik, 32, p. 584, 1887. 88 LE ROY D. WELD [J.O.S.A. & R.S.I., VI was usually found expedient to limit the illumination to only one or two millimeters of length, so that an area of one square milli- meter or less of reflecting surface was quite typical. Much of the work was carried on at wave-lengths near the mid- dle of the visible spectrum. The incident light was, in every case, plane-polarized at azimuth 45. Several crystals were tested in both horizontal and vertical positions at wave-length 0.5/z, and at incidence angles 45 and 60. At the other wave-lengths the incidence angle was maintained at 60. The wave-lengths prin- cipally used were 0.45/z, 0.50 /*, 0.55 /*, 0.65 ju, and 0.70;*, a range sufficient to give typical results. The wave-length 0.60/z was de- ferred to a separate research, for the reason that the data obtained by C. H. Skinner 16 and others with selenium contain certain anoma- lies near this point, while the writer's results are strongly sugges- tive of what, with a transparent substance, would correspond to an absorption band, in the neighborhood of this wave-length. The matter deserves, therefore, more minute investigation. ___A_=2_e__ . Fig. 11 A, incident vibrations, plane-polarized. B, C, elliptic vibrations viewed looking with and against the beam, respectively Upon experimenting with a number of crystals prepared at different times, it was found that, with one or two exceptions, they gave fairly consistent results, though all the crystals were several months old when the experiments were begun. Great care, however, 16 Loc. dt. Jan., 1922] THE CRYSTELLIPTOMETER 89 had been taken to keep them clean and away from contact with fumes or corrosive gases. It is quite possible that these exceptions were due to surfaces that had in some way become tarnished or contaminated, in spite of the precautions. After the first trials, three crystals were selected which gave the clearest spot-patterns, and subsequent work was confined to these. The data in the accompanying tables are the weighted means of the results obtained at the respective wave-lengths. It should be stated that those corresponding to 0.70ju have small relative weight. Elliptic Elements for Selenium Crystal at 60 Incidence Wave- length (Microns) Crystal Axis Parallel to Incidence Plane Crystal Axis Perpendicular to Incidence Plane A * A * 0.45 23 11' 35 22' 29 31' 24 25' .50 11 52 34 14 27 54 23 15 .55 4 26 32 27 20 41 24 26 .65 13 54 34 12 21 49 27 7 .70 11 24 31 13 14 38 26 7 The data refer to the light vibrations as they would appear to an observer stationed just behind, or within, the reflecting surface of the crystal. The plane-polarized incident light vibrating as represented in Fig. 11 A, the reflected light vibrates elliptically as in B. But as the latter is viewed through the crystelliptom- eter, the observer, facing the oncoming reflected beam, it would of course appear reversed, as in C. The phase difference V of Sec. IV, which is what the crystellipometer analysis gives, applies to Fig. 1 1 C ; while the A given in the tables below, corresponding to Fig. 11B, is simply V minus 180. >J> is the azimuth angle, whose a tangent is the amplitude ratio - or r of Sec. IV. 90 LE ROY D. WELD [J.O.S.A. & R.S.I., VI Two Incidences at Wave-length . 5 Micron Incidence Axis Parallel Axis Perpendicular A * A ^ 45 60 2 11' 11 52 44 52' 34 14 14 1' 27 54 37 46' 23 15 The values given in the second table are capable, according to Dr. R. P. Baker, who has recently investigated the application of Drude's theory to hexagonal crystals, of yielding the two sets of optical constants of selenium corresponding to this wave-length. It is expected that this calculation will appear in a subsequent paper along with data from the further experimental work now in progress. A. (De$'fes) A jar Tellurium A JOY 5 e/ en I'M m 30 40 SO TO SO Inclination (Dec ion (De^rets) {p for Tellurium for Selenium 20 30 SO 60 TO 80 > Inclination Jn el in*1io Fig. 12 Variations of A and S^ with inclination of crystal, to plane of incidence, for selenium and tellurium The above results refer only to the two principal positions of the crystal, namely, with the axis parallel to and perpendicular to the Jan., 1922] THE CRYSTELLIPTOMETER 91 incidence plane. Incidentally it was thought worth while to study the variations of the elliptic elements with the angle of inclination as the crystal is turned from one position to the other. Typical results are depicted in Fig. 12 with selenium and tellurium. No explanation is immediately apparent for the maxima and minima occurring in both cases in the value of A, but it is hoped that the mathematical theory may ultimately yield one. Further exper- iments of this kind would be desirable. In conclusion, the writer wishes to pay tribute to the "team work" among the various workers at the University of Iowa who have contributed toward the progress of the research, to one phase of which this paper is devoted. The generous cooperation of the staff of the Physics Department, and especially of Dr. Sieg, who suggested the problem, is very greatly appreciated. COE COLLEGE, CEDAR RAPIDS, IOWA, OCTOBER, 1921. *jj^ JOXiJjO YV 4 54;*>. '{,33 QC313 UNIVERSITY OF CALIFORNIA LIBRARY