Vol. 57

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 57, NUMBER 5 MAY 1967

Refractive-Index Interpolation for Fused Silica*BERLYx BRIXNER

University of California, Los Alamos Scientific Laboratory, Los Alamos, New Mexico 87544

(Received 26 November 1966)

The advantages of a four-term Sellmeier-equation fit to refractive indexes of recently produced fused silicaare described. The four-term fit gives more accurate smoothing and interpolation of data than the three-term fit used previously and also indicates data of questionable accuracy. After the suspect data had beeneliminated, fits with an average deviation of 4.3-10-6 were obtained with both three- and four-term equa-tions over the range 0.21-2.32 p.

INDEx HEADINGS: Refractive index; Fused silica; Wavelengths; Absorption.

T O solve lens-design problems it is often necessaryTto smooth and interpolate refractive-index data,since the several techniques used to obtain these datatend to vary in precision. In general, the best averagevalues for a particular spectral region are desired. Theaim is to be able to estimate the accuracy of datapoints in order to detect faulty data, which can theneither be removed from consideration or, if included,can be given an estimated reliability.

This paper reports results obtained by fitting a four-term Sellmeier dispersion equation to refractive indicesof recently produced fused silica. The Sellmeier equa-tions fitted precision data accurately and smoothly.Suspected erroneous data are indicated prominently bylarge deviations. Because difficulty has been experi-enced in obtaining a solution with earlier computationmethods, there seems to have been no previous reportof the use of the four-term Sellmeier equation to fitrefractive-index data.1 At the Los Alamos ScientificLaboratory, a least-squares program has been developedthat has alwavs converged on a solution even for a ten-term equation. The calculations used for the study tobe described were made with W. Anderson's least-squares program (the FORTRAN code can be obtainedfrom the author or from William Anderson, Los AlamosScientific Laboratory) for fitting one- to ten-term Sell-meier equations [E2-1= I (X2 .a 2i- 1)/(X

2- a2 ?2)] to re-fractive-index data, where n is a refractive index, X thecorresponding wavelength in microns, a is a parameter,and i the index number of the equation term.2 Three

* Work done under the auspices of the U. S. Atomic EnergyCommission.

1 L. E. Sutton and 0. N. Stavroudis, J. Opt. Soc. Am. 51, 902(1961).

2 XV. Anderson (private communication).

improvements of the procedure have brought aboutthis more accurate calculation. First, the nearest mini-mum of the squared-error sum is determined at eachiteration in accordance with the Cauchy search method.3

Second, a constant is added to the principal-diagonalelements of the matrix4 and a search procedure isinstituted to find the optimum value for that constantat each iteration.2 Third, singular linear systems aretransformed into nonsingular systems by adding to thediagonal elements a constant of sufficient size.2

The errors of fit obtained with four-term equationsfitted to two sets of data are shown in Table I. Theparameters used are given in Table II. The refractiveindices used were from the extensive series of measure-ments at 60 wavelengths made at the National Bureauof Standards.5 The average curve-fit error is 9- 10-f.For the second set of data, some of the first set of datawere excluded because the initial curve-fitting trialsrevealed what seemed to be systematic errors. Althoughthe errors in parentheses are calculated from the ex-cluded data, they were not used in the fitting pro-cedure. The average curve-fit error given by the se-lected data is 4-10-6, as compared with a 7t 10-6average error obtained when the Malitson parameterswere used.5 This gain was obtained by eliminating allrefractive indices measured with absorption bands andall that showed large, systematic errors of fit. As aconsequence, all data for wavelengths greater than 2.3 Atwere lost. Points extrapolated with these parameters(X> 2.3 A) show the expected progressive divergencefrom the unused data (Table I, second series). The

I A. L. Cauchy, Compt. Rend. 25, 536 (1847).4 K. Levenberg, Quart. Appl. Math. 2, 164 (194).I. H. Malitson, J. Opt. Soc. Am. 55, 1205 (1965).

674

REFRACTIVE INDEX FOR FUSED SILICA

TABLE I. Fused silica: computed refractive indexes and fit errors X 106 (20'C).

Wavelength Spectral Four-term fit to 180 data points Four-term fit to 123 data pointsin microns source Computed Corning Dynasil G. E. Computed Corning Dynasil G. E.

0.2138560.2144380.2267470.2302090.2378330.2399380.2482720.2652040.2698850.2752780.2803470.2893600.2967280.3021500.3302590.3341480.3403650.3466200.3610510.3650150.4046560.4358350.4678160.4861330.5085820.5460740.5769590.5790650.5875610.5892620.6438470.6562720.6678150.7065190.8521110.8943501.013981.082971.128661.36221.395061.46951.529521.66061.6811.69321.709131.813071.970092.05812.15262.325422.43743.24393.26683.30263.4223.50703.55643.7067

ZnCdCdHgHgHgHgHgHgHgHgHgHgHgZnHgCdCdCdHgHgHgCdHCdHgHgHgHeNaCdHHeHeCsCsHgHeHgHgHgCsHgTCBaPolybHgHgHgHgHeTCBHgTCBPolyPolyPolyPolyPolyTCBTCB

1.5342891.5337061.5227551.5200881.5147391.5133771.5084081.5000361.4980521.4959171.4940421.4909901.4887321.4871921.4805341.4797581.4785781.4774621.4751231.4745331.4696131.4666901.4642921.4631271.4618651.4600821.4588511.4587741.4584691.4584101.4567111.4563741.4560751.4551541.4524751.4518451.4502501.4494121.4488751.4462131.4458371.4449741.4442661.4426661.4424091.4422551.4420521.4406921.4385121.4372161.4357621.4329221.4309511.4131381.4125251.4115541.4081941.4056831.4041751.399360

-11+5

+67-35+3-4-4

-39+2-2-6

+18-5-7+6-3

+15-11+9-5+5+8+5+5+2+0-0+1-2+0+2-0+0+1-2+2-2+1+1-7-2+4

+10-10+3

+12+5-0

+14+5

-15-4

-20+1-0

+13+26-33-28+7

-24-5

+64-38+13-8

-11-32-9+4

-11+22-10+0

+15+4+8-6-3-9+3+4+3+4-1-3-2+1-7-3-3-2-2+3-7-4-5-6+2

-15+3

+10+8

-15-5-1+8-0

+20-1

-18+3

-18+9

+10+11+28-27-30+15

-13-6

+69-30+9-1-3

-20+6+8

-13+20-2+6+8

+14-2-8-2

-15+6+6+6+6+3-9-1+1-4+1-0-0-5-2-5+1-2+2+3

-13-4

+11+2-7

+13+6+4

+13+20-3

-17-0

-11+5+5+9

+23-17-19+20

1.5342811.5336981.5227501.5200831.5147361.5133741.5084071.5000361.4980531.4959181.4940431.4909921.4887301.4871951.4805371.4797611.4785821.4774651.4751261.4745361.4696151.4666921.4642931.4631271.4618651.4600821.4588511.4587731.4584691.4584091.4567101.4563731.4560741.4551531.4524741.4518441.4502501.4494121.4488761.4462161.4458401.4449781.4442701.4426701.4424141.4422591.4420561.4406971.4385161.4372191.4357641.4329211.4309461.4130871.4124721.4114981.4081291.4056111.4041001.399278

-3+13

(+71)c(-30)

+6-1-3

(-319)

-3(-8)

(+16)-8

-10+3-6

+12-14+6(-8)+3+7+4+5+2+1+0+2-2+1+3+1+1+2-1+2-2+1-0

(-1.0)-5-0+6

(-14)(-2)+8+1-5

(+9)+2

(-17)-3

(-15)(+53)(+53)(+69)(+91)(+39)(+47)(+89)

-16+3

(+68)(-32)+16-5

-10(-32)-10+3

(-13)(+20)-13-2

+12+1+5-9-6

(-12)+1+3+2+4-1-2-2+2-7-2-2-1-1+4-6-4-5-6+1

(-18)+0+6+4

(-19)(-10)

- 5+4-5

(+ 15)-4

(-20)+4

(-13)(+61)(+63)(+67)(+93)(+45)(+45)(+97)

-5+2

(+73)(-25)+12+2-2

(-20)+5+7

(- 15)(+18)

-5+4+5

+11-5

-11-5

(-18)+4+5+5+6+3-8-1+2-4+2+1+1-4-1-4+1-2+2+2

(-16)-7+7-2

(-11)(+8)+2-0+8

(+15)-6

(-19)+1

(-6)(+57)(+58)(+65)(+88)(+55)(+56)

(+102)

I TCB = 1, 2, 4-Trichlorobenzene.b Poly =Polystyrene.-Errors in parentheses were not used in the fit procedure.

greatest smoothing of these less-accurately determineddata is obtained by calculating indices of refraction forX> 2.3 , with the parameters from the four-term fit toall the data (Table I, first series).

Refractive indices measured with absorption bandswere excluded after a few preliminary runs with a

four-term equation showed unusually large errors as-sociated with indices measured with absorption-bandspectra. The fact that between 1954 and 1965 apprecia-ble changes have been made in the standard wave-lengths (3.2432 , was changed to 3.2439, 3.2666 to3.2668, 3.3033 to 3.3026, 3.4188 to 3.422, and 3.5078

675May 1967

BERLYN BRIXNER

TABLE II. Sellmeier parameters and fit characteristics for fused silica.

Code NBS LASL LASL LASL LASLData points 180 180 180 123 1232 (Deviations) 2 (5.164-10-8) 4.906-10-8 3.731-10-8 3.752-10- 3.739-109Avg. deviation 11.9-10-6 11.3.-10-1 9.3 -10-6 4.3-10-6 4.3 -10-6Parameter Max. (1.2-10-3) 2.6.10-12 9.8-10-6 2.4-109 4.8-109slopes fmin. t1.3- 10-5) 1.8- 10-14 6.6-10-8 5.2- 10-l1 1.9.10-12

i=1 a=1 0.6961663 0.61145816 0.53186932 0.55105619 0.547884691 2 0.0684043 0.063264820 0.056857781 0.058686335 0.0584085792 3 0.4079426 0.49267859 0.57234217 0.55312773 0.556303872 4 0.1162414 0.11318188 0.11075886 0.11128041 0.111187953 5 0.8974794 0.90213984 0.066425330 0.92922006 0.475272673 6 9.896161 9.9180201 6.6776876 10.052067 8.59647074 7 1.1524322 6.70253344 8 12.206844 49.173434

to 3.5070),5 6 suggests that these wavelengths are ofuncertain accuracy. This idea is reinforced by studiesmade at LASL. A ten-year-old polystyrene absorption-wavelength standard underwent an irreversible wave-length change in the third significant figure after itwas heated to 60TC. In addition, the relatively weakband at 3.5070 y is likely to be "pulled" toward thestrong absorption band at 3.422 ,/i7 Finally, there is alarge systematic shift of the sign of the errors betweenthese wavelengths (-20 to +40) when the Malitsonparameters are used for calculation.8 For precise meas-urements, even the sharpest molecular bands shouldnot be preferred to precisely determined atomic lines.9

Argon has a number of intense lines in this same region.' 0

Exclusion of the remaining refractive-index valuesis not justified in detail because there is insufficientinformation about the measurements. To assist inanalyzing these deviations it would be helpful to haveplots of the spectra used, similar to those given inRefs. 7 and 10. Without these plots it is difficult tojustify the inclusion of data with fit errors muchgreater than the mean deviation of most of the otherdata and also much greater than the precision of meas-urement claimed for the spectrometers used. Eventhough nearly one third of the data points were ex-cluded from the fit, the fit errors calculated for theexcluded interpolatable points were on the averageonly 1- 10-1 greater than those calculated with the NBSparameters.

I W. S. Rodney and R. J. Spindler, J. Res. Natl. Bur. Std.(U. S.) 53, 187 (1954).

7E. K. Plyler, A. Danti, L. R. Blaine, and E. D. Tidwell,J. Res. Nat]. Bur. Std. (U. S.) 64, 31, 47 (1960).

8 Reference 5, p. 1206.9 Reference 7, p. 29.10 K. N. Rao, C. J. Humphreys, and D. H. Rank, Wavelength

Standards in tMe Infrared (Academic Press Inc., New York, 1966),p.36.

Table II gives four new sets of Sellmeier parameterstogether with the set of parameters from Ref. 5. Inaddition, it contains the following information aboutthe runs: the number of data points used, the averagefit of the data, the squared-error sum from the rungenerating the parameters reported, and the maximumand minimum slopes of the parameters as calculated inthe LASL program. The parameter slope refers to theleast-squares sum as a function of individual parameterchanges. Although the LASL three-term fit to the 180raw data points was only slightly superior to the NBSfit, and although the four-term fit to the 123 selecteddata points was hardly better than the three-term fit,the four-term fit proved superior when applied to the180 raw-data points. In this run, the average error offit was reduced from 11.9-10-6 to 9.3-10-6, and thesystematic errors were detected and eliminated.

The conclusion is that smaller deviations are ob-tained with a four-term Sellmeier equation fit to re-fractive-index data than with a three-term fit, if pointsof uncertain accuracy are included in the data and ifmaximum smoothing of the data is desired. However,with a series of precisely determined points, the four-term fit seems to be hardly more accurate than thethree-term fit.

Thanks go to William Anderson, whose versatilenew program is based on calculation procedures es-sential for precise work, to Thomas C. Doyle, whosedetailed analysis of the old procedures made it possibleto use them intelligently in this program, and to R. E.Von Holdt, R. Engleman, M. T. Menzel, R. K. Zeigler,and the late C. A. Lehman whose experimental codesaided understanding of the problems solved in thepresent code. Study of runs from the code Lehmanderived from the NBS three-term FAP code listingand notes by L. E. Sutton, was helpful.

676 Vol. 57