Modulated generalized ellipsometry
JOSA, Vol. 66, Issue 6, pp. 520-524 (1976)
http://dx.doi.org/10.1364/JOSA.66.000520
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Abstract
We extend ellipsometry to the direct measurement of small perturbations of the Jones matrix of any linear nondepolarizing optical sample (system) subjected to a modulating stimulus such as temperature, stress, or electric or magnetic field. The methodology of this technique, to be called Modulated Generalized Ellipsometry (MGE), is presented. First an ellipsometer with arbitrary polarizing and analyzing optics is assumed, and subsequently the discussion is specialized to a conventional ellipsometer having either the polarizer-sample-analyzer (PSA) or the polarizer-compensator-sample-analyzer (PCSA) arrangement. MGE provides the tool for the systematic study of thermo-optical, piezo-optical, electro-optical, magneto-optical, and other allied effects for both isotropic and anisotropic materials that may be examined in either transmission or reflection. MGE is also applicable to (1) modulation spectroscopy of anisotropic media, (2) the study of electrochemical reactions on optically anisotropic electrodes, and (3) the extension of AIDER (angle-of-incidence-derivative ellipsometry and reflectometry) to the determination of the optical properties of anisotropic film-substrate systems.
© 1976 Optical Society of America
Citation
R. M. A. Azzam, "Modulated generalized ellipsometry," J. Opt. Soc. Am. 66, 520-524 (1976)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-66-6-520
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References
- Ellipsometry in the Measurement of Surfaces and Thin Films, edited by E. Passaglia, R. Stromberg, and J. Kruger, Nat. Bur. Stand. (U.S.) Misc. Publ. No. 256 (U. S. Government Printing Office, Washington, D. C., 1964).
- Proceedings of the Symposium on Recent Developments in Ellipsometry, edited by N. M. Bashara, A. B. Buckman, and A. C. Hall (North-Holland, Amsterdam, 1969).
- R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
- R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1521 (1972); ibid. 64, 128 (1974).
- Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976), in press.
- M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, in Ref. 5.
- P. S. Hauge, in Ref. 5.
- D. J. De Smet, in Ref. 5.
- A. B. Buckman and N. M. Bashara, Phys. Rev. 174, 719 (1968).
- A. B. Buckman, in Ref. 2, p. 193.
- B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
- J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
- W. A. Shurcliff, Polarized Light (Harvard University, Cambridge, 1962).
- D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).
- If J_{ss} = 0, the Jones matrix [Eq. (1)] can still be normalized by factoring out another nonzero element, e. g., J_{pp}. The subsequent development has to be modified accordingly.
- This choice of time dependence is adopted only for simplicity. The analysis is applicable to any waveform of periodic modulation M(t) impressed on the sample given that the resulting optical perturbations (δσ_{ss}, δψ_{i}, δΔ_{i}) also have the same waveform as, and are in phase with, the modulation. Thus, we may replace sinωt by M(t) in Eqs. (9) and (10). Furthermore, the step that leads from Eq. (14) to Eq. (16) remains valid when M(t) replaces sinωt. The caret over a quantity that varies as M(t) signifies the peak value of M(t).
- They include the azimuth angles and optical properties of the individual elements of P and A.
- A lock-in amplifier can be used for the precise measurement of the small ac signal received by the photodetector (see Fig. 2). The reference signal to the lock-in amplifier is derived from the same source of modulation M applied to the sample. The procedure is similar to that discussed in Refs. 9 and 10.
- The determination of (δψ⌃_{t}, δΔ⌃_{t}), i = 1, 2, 3, (or δĴ_{n}) can be separated from the determination of δσ⌃_{ss}/σ¯_{ss} by subtracting one (e. g., the seventh) of the seven equations represented by Eq. (16) from the remaining six. This gives a matrix equation of the form δm′ = I′ δS′. δm′ is a modified 6 × 1 measurement vector with elements (δm_{k} - δm_{7}) (k = 1, 2, …, 6); I′ is a modified 6 × 6 instrument matrix with elements [(α¯_{Ψ1k}-α_{Ψ17}),(α¯_{Δ1k}-α¯_{Δ17}),(α¯_{Ψ2k}-α_{Ψ27}),(α_{Δ2k}-α_{Δ27}),(α¯_{Ψ3k}-α_{Ψ27}),(α¯_{Δ3k}-α¯_{Δ37})] in the kth row (k = 1, 2, …, 6); and δS′ is a 6 × 1 sample-perturbation vector with elements δΨ_{1}, δΔ_{1}, δΨ_{2}, δΔ_{2}, δΨ_{3}, δΔ_{3}. δS′, and hence δĴ_{n}, can be obtained from δS′ = [I′]^{-1}δm′, where [I′]^{-1} is the inverse of I′. Once δS′ has been determined, [Equation]_{ss}/σ¯_{ss} can be calculated from any one of Eqs. (16). This procedure has the obvious advantage of requiring the inversion of a 6×6 matrix (I′) instead of a 7×7 matrix (I).
- See, for example, F. L. McCrackin, J. Opt. Soc. Am. 60 57 (1970). Notice that we assume unit amplitude for the electric vibration of the light leaving the polarizer.
- Extension to the general case of arbitrary values of ρ_{C} and C is straightforward but the results are quite complicated.
- M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).
- R. M. A. Azzam, Opt. Commun. 16, 153 (1976).
- This assumes that the principal axes of the dielectric tensors of the substrate and film have arbitrary but known orientation.
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