Optical and physical parameters of Plexiglas 55 and Lexan
Roy M. Waxler, Deane Horowitz, and Albert Feldman
The following parameters have been obtained for Plexiglas 55 and Lexan: refractive index n at 486.1 nm,589.3 nm, and 656.3 nm, the thermooptic constant dn/dT at 632.8 nm, the linear thermal expansion coeffi-cient; the photoelastic constants q11 , ql 2, Pl, and P12; and the elastic moduli cl, c1 2, s1l, and s1 2. The ex-perimental value for the density derivative of refractive index p(dnldp)T deviates by only a small amountfrom the value calculated from the Lorentz-Lorenz equation. This is the expected result for molecular sol-ids. The density variation with temperature is the dominant contribution to dn/dT.
Introduction
In recent years transparent plastic materials havecome into wide use in the fabrication of optical com-ponents.1 In order to make optimal use of these ma-terials, component designers require basic optical andmaterial parameters such as refractive index n, ther-mooptic coefficient dn/dT, linear thermal expansioncoefficient a, elastic moduli ii and sj, and photoelasticconstants qj and Pij. In this paper we report mea-surements of the above properties on two commercialplastic materials, Plexiglas 55, a methyl methacrylatepolymer, and Lexan, a polycarbonate.2
Both plastics were obtained in the form of 6.4-mmthick plates with polished faces. The Plexiglas 55 wasobtained in the as-cast form, and examination of thematerial in a polariscope showed it to have essentiallyzero birefringence. We were unable to obtain Lexanthat was free from birefringence. The Lexan we usedhad undergone a stretching operation in'manufacture'consequently, a polariscopic examination of the materialshowed an optical path difference of six wavelengths oflight with X = 632.8 nm. The double refraction wasuniform over the whole plate with some irregularity atthe edges.
Refractive Index
A Hilger-Watts refractometer was used to obtainrefractive index data on the two plastics. The methodof measurement described by Fishter3 requires that thesample be in the form of a plate with one 90° angle that
The authors are with U.S. National Bureau of Standards, Institutefor Materials Research, Inorganic Materials Division, Washington,D.C. 20234.
fits into a glass V-block sample holder. A drop ofindex-matching fluid is placed between the contactingsurfaces. A sodium lamp was used to obtain data at589.3 nm, and a hydrogen Plucker tube was used toobtain data at 486.1 nm and 656.3 nm. The refractiveindex values at these wavelengths were plotted on agraph, and a curve was fitted empirically through thevalues. A value at 632.8 nm was taken from the curveand used as the index value for all the following work.The refractive indexes at the four wavelengths are givenin Table I for Plexiglas 55 and Lexan. The estimatedaccuracy is better than one in the fourth decimalplace.
Photoelastic and Elastic ConstantsAn isotropic material has two independent elastic
compliance coefficients s and S12 and two independentpiezooptic coefficients qll and ql2.4 In order to obtainthese four coefficients, a minimum of four independentmeasurements is required. We have obtained thesecoefficients in Plexiglas 55 and Lexan by interferometricmeasurements of optical path changes in specimenssubjected to uniaxial or hydrostatic stress. Polarimetricmeasurements of stress-induced birefringence were alsomade.
Specimens of both materials were cut in the form ofrectangular parallelepipeds with dimensions 50 mm X6.4 mm X 6.4 mm. Special care was taken to cut theLexan so that the principal axes of this doubly refractingspecimen coincided with the edges of the prism. Anexamination of the Lexan specimen between crossedpolarizers with collimated He-Ne radiation indicatedthat a principal axis was parallel to the long edge of theprism to within 0.1 deg of arc.
The unpolished faces of the specimens were precisionground on a surface grinder. The polished faces werethen repolished on a lap with ceric oxide slurry so thatFizeau type interference fringes in the shape of bull's-eye rings were formed in the center of each specimen.
Table I. Optical Characterization of Plexiglas 55 and Lexan
Plexiglas 55 Lexan
n(486.1 nm) 1.5014 1.5995n(589.3 nm) 1.4950 1.5854n(632.8 nm) 1.4934 1.5816n(656.3 nm) 1.4928 1.5800q11 26.7 X 10-12 Pa-1 -4.6 X 10-12 Pa1q12 25.5 X 10-12 Pa'1 34.6 X 10-12 Pa'P11 0.300 0.252P 12 0.297 0.321s1l 303 X 10-12 Pa'1 403 X 10-12 Pa'512 -108 X_10-12 Pa-1 -165 X 10-12 Pa 1
c11 5.46 X 109 Pa 5.79 X 109 Pacl2 3.03 X 109 Pa 4.02 X 109 Pap(nl/bp)exp 0.496 0.589AO 0.149 0.180a 67.9 X 10-6 C-1 65.5 x 10-6 OC-1dn/dT 105 X 10-6 C-1 -107 X 10-6 C- 1
(on/;3T)p -4.0 x 10-6 aC-1 8.7 x 10-6 OC-
The photoelastic and elastic constants were measuredas follows: a specimen was subjected to a uniaxialstress, which produced a change of optical pathlengthwithin the specimen. This optical path change resultedin fringe shifts that were measured by Twyman-Greenand Fizeau interferometry. A He-Ne laser operatingat 632.8 nm was the light source, and the interferencefringes were detected by a silicon matrix vidicon cameraand observed on a TV monitor. The apparatus forapplying the stress and the method for mounting thespecimen in the stressing apparatus have been describedin the literature.5
The shift of interference fringes per unit of appliedstress AN/IXP is given by
(AN/lXP) = (2t/X)[(n3 2)q - (n - 1)121 (1)
for a Twyman-Green interferometer, and by
(AXNIAP) = (2t/X)[(n 3/2)q - ns121 (2)
for a Fizeau interferometer, where t = specimen thick-ness, X = wavelength of radiation, q = q11 for radiationpolarized along the stress axis, and q = q 12 for radiationpolarized perpendicular to the stress axis.6 7 The shiftof Fizeau fringes per unit of applied hydrostatic pres-sure is
The apparatus for applying hydrostatic pressure hasalso been described in the literature.8
By the use of Eqs. (1)-(3) we obtain the coefficientsq11, q12, s11, and s1 2. As an added check, we also mea-sure the stress-induced birefringence. Using a polari-metric technique we obtain the fringe shift per unit ofapplied stress, which is given by
(ANB)/(AP) = (tn3 )/ (q11 - q12), (4)
where the radiation makes two passes through thespecimen.
During the course of the measurements, it was foundthat the fringes would drift gradually with time. We,
therefore, decided to wait for 5 min after changing thespecimen loading before recording the fringe shift.After this time interval, the fringe drift with time be-came small. We found that successive measurementsgave reproducible data with this procedure.
Each measurement was repeated five times, afterwhich the average value and the standard deviationwere computed. The results were inserted into Eqs.(1)-(4), and the values for q11, q12, sil, and S12 werecomputed by a least-squares procedure in which theappropriate weighting factors were computed from thestandard deviations. The results are shown in TableI. The estimated accuracy for the computed coeffi-cients is 2%.9 Also shown in Table I are values for theelastic stiffness constants c1 l and c12 and values for theelastooptic constants P11 and P12, which were computedfrom piezooptic constants and the elastic complian-ces.
Coefficient of Linear Thermal Expansion
The coefficient of linear thermal expansion of bothplastics was determined by an optical interferencemethod in which the specimen, approximately 6.4 mmthick, acts as a spacer between two optical flats. 7 12"13
Collimated laser radiation incident upon the flats isreflected so that Fizeau interference fringes are pro-duced, and these fringes are observed to shift as ,thetemperature of the specimen is varied. The change ofspecimen thickness is related to the fringe shift ANby
At = (X/2)AN. (5)
The expansion coefficient is then
a = (ilto)(dt/dT), (6)
where to = the specimen thickness at room tempera-ture.
The following experimental procedure was employedfor obtaining a. The specimen assembly was placed ina variable temperature apparatus.7 A helium atmo-sphere of approximately 1-2 Torr was maintained aboutthe specimen. Two type E thermocouples placed incontact with the specimen were used to monitor thetemperature. The temperature was increased byapplying power to the heater, resulting in a fringe shiftthat was detected by a PMT and recorded on a strip-chart recorder. In-each experimental run, the tem-perature was slowly increased by arr increment of ap-proximately 40° C.- The temperature was then held inquasi-equilibrium for about one-half hour until an ac-curate determination of an integral fringe shift had beenmade. The fringe count and the temperature were re-corded and the process was then repeated until the en-tire range of temperatures, from -160'C to 600C, wascovered. From these data, values of a were calculated.We then fitted a third-degree polynomial in T to thesevalues of a, where the standard deviation of the fit wasless than 2 X 10-6 0C-1. Values of a obtained from the
fitted equation are presented at 200C intervals in TableII for Plexiglas 55 and in Table III for Lexan. The ex-perimental data and the fitted curves for a are shownin Fig. 1. Data on Plexiglas 55 obtained by Rohm andHaas Company are represented by triangles in Fig. 1,and it can be seen that there is reasonably good agree-ment with the values reported in this study. The ir-regular form of the curve for Lexan is believed to be dueto the stretched state of the material.
Thermooptic Coefficient
The thermooptic coefficient dn/dT was found bymeasuring the shift of Fizeau fringes generated by aplane parallel plate (to 6.4 mm) as a function oftemperature. The fringe shift AN, which is a measureof the optical path change within the specimen, dependson both the change of refractive index with temperatureAn and the change of thickness with temperature At.Because At can be calculated from the values of a, wecan obtain An by using the equation
An = [(ANX)/(2to) - no(St)/(to)][1 + (At)/(to)]-, (7)
where no = room temperature refractive index. Weobtain dn/dT by taking the derivative of Eq. (7) withrespect to temperature.
The experimental procedure for measuring dn/dT isidentical with the procedure for determining a. Theresulting values of dn/dT were then fitted to a thirddegree polynomial in T. The standard deviation of thefit was less than 2 X 10-6 C-1 . Values for dn/dT ob-tained from the fitted equation are presented at 20'Cintervals in Table II for Plexiglas 55 and in Table III forLexan. The experimental data and the computedcurves are shown in Fig. 2.
0
-4
Td
-
-8
-12
-16
0 100
TEMPERATURE (C)
-200 -100 0 100
TEMPERATURE (C)
Fig. 1. Coefficient of linear thermal expansion as a function oftemperature for Plexiglas 55 and Lexan. (The values representedby triangles have been calculated from data reported by Rohm and Fig. 2. Thermooptic coefficient as a function of temperature for
A theoretical expression for p(on/p)T can be derivedfrom the Lorentz-Lorenz (L-L) equation1 4-17
(n2 - 1)/(n2 + 2) = (47r/3)MO3, (8)
where M = density of light scatterers, a = polarizabilityper scatterer, and p = density. If we assume that f isindependent of p, then p enters Eq. (8) only via M, whichis proportional to p. Using this assumption, we ob-tain
p(n/op)L-L = (n2 -1)(n 2 + 2) ()6n
However, in solids f does depend on density; hence, acorrection factor is usually introduced to reconcile theexperimental determination of p(onIOp)T with Eq. (9).Thus,
p(bnZ)p)exp = (1 - o)p(an/P)L-L, (10)
where the correction factor AO is called the strain po-larizability constant. 1 4 -17 We have calculated AO forPlexiglas 55 and Lexan (see Table I), which are molec-ular solids, and we found that for both materials AO issmall compared with unity indicating close agreementwith the L-L equation. In a similar study on lone pairsemiconductors, which are also molecular solids,Kastner18 has found only small deviations from the L-Lequation. Kastner noted that molecular solids are ex-pected to obey closely the L-L equation because of theweak binding between the molecular units; hence onewould expect AO to be small compared with unity forthese materials.
The change of refractive index with temperature 1 4 1 9
depends on a density- (or volume-) dependent term plusa change at constant density due solely to the temper-ature change itself:
(dn/dT) =-3ap(an/6p)T + (an/aT)p, (1 1)
where p(an/p)T = (n 3 /6)(pil + 2p12). 8 Each of thequantities in Eq. (11} is listed in Table I for both of theplastics. We note first that dn/dT for both plastics isdetermined predominantly by the change of density.This behavior is characteristic of most single crystalmaterials but is not characteristic of inorganic oxideglasses, where (on/oT)p makes the major contributionto dn/dT.19
This work was supported in part by the U.S. Air ForceMaterials Laboratory at Wright-Patterson Air ForceBase.
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the materials used in this study are necessarily the best available.
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