Infrared refractive index of silicon David F. Edwards and Ellen Ochoa

University of California, Los Alamos Scientific Laborato­ry, P.O. Box 1663, Los Alamos, New Mexico 87545. Received 22 August 1980. 0003-6935/80/244130-02$00.50/0. © 1980 Optical Society of America. Accurate values of refractive indices are necessary for de­

signing sophisticated multielement optical components. Recent advances in lasers and laser applications have stressed the need for accurate index values farther in the IR. Re­fractive-index values for silicon, reported 23 years ago by Salzberg and Villa,1 have been reproduced in several widely used handbooks.2-4 They used an autocollimation method5

and reported values for the spectral region from 1.3570 to 11.04 μm.

Randall and Rawcliffe6 and Lowenstein et al.7 have used the channel spectrum technique to measure the refractive index of a number of materials in the far infrared {v < 350 cm-1). The present measurements bridge the spectral gap between the earlier results and these far-IR values.

We have measured the refractive index of optical quality silicon from 400 to 4000 cm -1 using the channel spectrum technique.6,7 Fringes of equal chromatic order8 are produced when radiation is multiply reflected between the parallel surfaces of the sample. The spacing between the fringes de­pends only on the sample thickness and the real part of the refractive index. Knowing the sample thickness and the wavelength, one can, in principle, determine the index. This technique has also been used by one of us9 to determine thicknesses for thin samples, assuming the index is known.

The silicon samples were from a single crystal boule,10 3-4 Ω-cm, n-type (phosphorous-doped), with a (111) orientation. The sample preparation technique is similar to that reported by Lazazzera11 for sapphire and ruby. The samples were 2.5 cm in diameter with a thickness of 0.09279 ± 0.00010 cm and parallel to a second of arc or better. The channel spectrum was measured from 400 to 4000 cm -1 with 0.06-cm-1 resolu­tion using a Nicolet model 7199 Fourier transform IR spec­trometer.12 This instrument has an effective aperture of ƒ/5, and the beam size is ~2 mm in diameter at the sample posi­tion. The sample was positioned to be normal to this beam, with the beam in the center of the sample. Since the samples are several times the beam size, no edge effects are ex­pected.

Table I. Index of Refraction of Silicon (26°C)

For the channel spectra, the order of the fringe m is given by

where n(σ) is the refractive index at wave number σ, and h is the metric thickness. As pointed out by Moss13 and others, a difficulty of the channel technique is determination of m. We avoided this problem by using a method suggested by Baumeister.14 Assigning an arbitrary order number to a fringe, the entire channel spectra are fit using the polyno­mial

4130 APPLIED OPTICS / Vol. 19, No. 2 4 / 1 5 December 1980

where the coefficients are determined by a least squares fit. The slope of this curve is

Assuming the index can be expressed by the polynomial

we can evaluate the ni coefficients by substituting Eq. (4) into Eq. (3) and comparing like terms. We have

We evaluated the coefficients in Eq. (5) from the ai coeffi­cients and generated a list of index values from Eq. (4). The coefficients in Eq. (5) were corrected for the beam angle fol­lowing the method of Randall and Rawcliffe.6 The index values evaluated by the above method are insensitive to the choice of the initial order number. A variation ±100 in m produced no changes in n in the fifth decimal place.

The Herzberger-type dispersion formula15'16 is one of the preferred forms for expressing indices,

where L = 1/(λ2 - 0.028) with the wavelength λ in μm. As explained by Herzberger,15 the 0.028 is the square of the mean asymptote for the short wavelength abrupt rise in index for fourteen materials (silicon included). We have fit the indices determined from Eq. (4) to the dispersion formula, Eq. (6). The coefficients are listed in Table I. The quality of the fit of the channel data to the dispersion formula, Eq. (6), is very good with differences in the fourth and fifth decimal places. Listed in Table I are selected index values calculated using Eq. (6) and the given coefficients. The uncertainty in the index values is determined to first order by the uncertainty in the sample thickness (1 part in 104). Also listed are the published index values of Salzberg and Villa.1 These older data are consistently lower than those reported here. These variations are not unexpected because we are considering silicon samples from two different crystals with different resistivities. Also, variations as large as 5 X 10~2 are known to exist between different melts of some materials.15

The extrapolation of Eq. (6) to 350 cm"1 gives an index of 3.4198, which agrees within the uncertainty with the value of Loewenstein et al.7 This agreement may be fortuitous, but our data join continuously with the far-IR data.7 Within the 400-4000-cm-1 spectral region, silicon has several absorption bands due to oxygen (8.3, 9.0, 19.4 μm) and an absorption at 625 c m - 1 due to a fundamental lattice vibration. Each of these absorptions are too weak (<10 cm - 1 ) to show up in the refractive index.

The authors wish to acknowledge the assistance of Vito Lazazzera for the sample preparation, Allen Gauler for mea­suring the sample thickness, Mariena Vasquez for measuring the spectra, and Lawrence Sherman for assisting with the computer fit of the data. We also wish to acknowledge the referee who called our attention to the necessary beam angle correction.

This work was sponsored by a grant from the U.S. Depart­ment of Energy.

References 1. C. D. Salzberg and J. J. Villa, J. Opt. Soc. Am. 47, 244 (1957). 2. D. E. Gray, Ed., American Institute of Physics Handbook

(McGraw-Hill, New York, 1972).

3. W. G. Driscoll, Ed., Handbook of Optics (McGraw-Hill, New York, 1978).

4. W. L. Wolfe, Ed., Handbook of Military Infrared Technology (Office of Naval Research, Washington, D.C., 1965).

5. E. D. McAlister, J. J. Villa, and C. D. Salzberg, J. Opt. Soc. Am. 46, 485 (1956).

6. C. M. Randall and R. D. Rawcliffe, Appl. Opt. 6, 1889 (1967). 7. E. V. Loewenstein and D. R. Smith, Appl. Opt. 10, 577 (1971); E.

V. Loewenstein, D. R. Smith, and Robert L. Morgan, Appl. Opt. 12, 398 (1973).

8. S. Tolansky, Philos. Mag. 36, 225 (1945). 9. D. F. Edwards and V. Lazazzera, J. Phys. E: 3, 156 (1970).

10. Grown by Glass Technology, Elmsford, N.Y. 11. V. Lazazzera, Laser Focus 14, 75 (1978). 12. Nicolet Instrument Corp., Madison, Wis. 13. T. S. Moss, Optical Properties of Semi-Conductors (Butterworths

Scientific Pub., London, 1959). 14. P. Baumeister, Optical Coating Laboratory, Inc.; private com­

munication. 15. M. Herzberger and C. D. Salzberg, J. Opt. Soc. Am. 52, 420

(1962). 16. M. Herzberger, Opt. Acta 6, 197 (1959).

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