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

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References

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  1. 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).
  2. 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).
  3. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
    [CrossRef]
  4. R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1521 (1972);ibid.64, 128 (1974).
    [CrossRef]
  5. Proceedings of the Third International Conference on Ellipsometry, edited by N. M. Bashara and R. M. A. Azzam (North-Holland, Amsterdam, 1976), in press.
  6. M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, in Ref. 5.
  7. P. S. Hauge, in Ref. 5.
  8. D. J. De Smet, in Ref. 5.
  9. A. B. Buckman and N. M. Bashara, Phys. Rev. 174, 719 (1968).
    [CrossRef]
  10. A. B. Buckman, in Ref. 2, p. 193.
  11. B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
    [CrossRef]
  12. J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
    [CrossRef]
  13. W. A. Shurcliff, Polarized Light (Harvard University, Cambridge, 1962).
  14. D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).
  15. If Jss= 0, the Jones matrix [Eq. (1)] can still be normalized by factoring out another nonzero element, e.g., Jpp. The subsequent development has to be modified accordingly.
  16. 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 (δσss, δψi, δΔi) 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).
  17. They include the azimuth angles and optical properties of the individual elements of P and A.
  18. 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.
  19. The determination of (δψ^i,δΔ^i), i= 1, 2, 3, (or δJ^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 δ m′ = I′δ S′. δ m′ is a modified 6 × 1 measurement vector with elements (δmk− δm7) (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¯ψ3k-α¯ψ37),(α¯Δ3k-α¯Δ37)] in the k th row (k= 1, 2, …, 6); and δ S′ is a 6 × 1 sample-perturbation vector with elements δψ1, δΔ1, δψ2, δΔ2, δψ3. δ S′, and hence δJ^n, can be obtained from δ S′ = [I′]−1δ m′, where [I′]−1 is the inverse of I′. Once δ S′ has been determined, δσ^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).
  20. See, for example, F. L. McCrackin, J. Opt. Soc. Am. 6057 (1970). Notice that we assume unit amplitude for the electric vibration of the light leaving the polarizer.
    [CrossRef]
  21. Extension to the general case of arbitrary values of ρC and C is straightforward but the results are quite complicated.
  22. M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).
  23. R. M. A. Azzam, Opt. Commun. 16, 153 (1976).
    [CrossRef]
  24. This assumes that the principal axes of the dielectric tensors of the substrate and film have arbitrary but known orientation.

1976 (1)

R. M. A. Azzam, Opt. Commun. 16, 153 (1976).
[CrossRef]

1974 (1)

J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
[CrossRef]

1973 (1)

B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
[CrossRef]

1972 (2)

1970 (1)

1968 (1)

A. B. Buckman and N. M. Bashara, Phys. Rev. 174, 719 (1968).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 336 (1972).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1521 (1972);ibid.64, 128 (1974).
[CrossRef]

A. B. Buckman and N. M. Bashara, Phys. Rev. 174, 719 (1968).
[CrossRef]

M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, in Ref. 5.

Buckman, A. B.

A. B. Buckman and N. M. Bashara, Phys. Rev. 174, 719 (1968).
[CrossRef]

A. B. Buckman, in Ref. 2, p. 193.

Cahan, B. D.

J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
[CrossRef]

B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
[CrossRef]

Cardona, M.

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).

Clarke, D.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

De Smet, D. J.

D. J. De Smet, in Ref. 5.

Elshazly-Zaghloul, M.

M. Elshazly-Zaghloul, R. M. A. Azzam, and N. M. Bashara, in Ref. 5.

Grainger, J. F.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, New York, 1971).

Hauge, P. S.

P. S. Hauge, in Ref. 5.

Horkans, J.

J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
[CrossRef]

B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
[CrossRef]

McCrackin, F. L.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard University, Cambridge, 1962).

Yeager, E.

J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
[CrossRef]

B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
[CrossRef]

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

R. M. A. Azzam, Opt. Commun. 16, 153 (1976).
[CrossRef]

Phys. Rev. (1)

A. B. Buckman and N. M. Bashara, Phys. Rev. 174, 719 (1968).
[CrossRef]

Surface Sci. (2)

B. D. Cahan, J. Horkans, and E. Yeager, Surface Sci. 37, 559 (1973).
[CrossRef]

J. Horkans, B. D. Cahan, and E. Yeager, Surface Sci. 46, 1 (1974).
[CrossRef]

Other (17)

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 Jss= 0, the Jones matrix [Eq. (1)] can still be normalized by factoring out another nonzero element, e.g., Jpp. 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 (δσss, δψi, δΔi) 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^i,δΔ^i), i= 1, 2, 3, (or δJ^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 δ m′ = I′δ S′. δ m′ is a modified 6 × 1 measurement vector with elements (δmk− δm7) (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¯ψ3k-α¯ψ37),(α¯Δ3k-α¯Δ37)] in the k th row (k= 1, 2, …, 6); and δ S′ is a 6 × 1 sample-perturbation vector with elements δψ1, δΔ1, δψ2, δΔ2, δψ3. δ S′, and hence δJ^n, can be obtained from δ S′ = [I′]−1δ m′, where [I′]−1 is the inverse of I′. Once δ S′ has been determined, δσ^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).

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).

Extension to the general case of arbitrary values of ρC and C is straightforward but the results are quite complicated.

M. Cardona, Modulation Spectroscopy (Academic, New York, 1969).

A. B. Buckman, in Ref. 2, p. 193.

This assumes that the principal axes of the dielectric tensors of the substrate and film have arbitrary but known orientation.

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.

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Figures (2)

FIG. 1
FIG. 1

A beam of polarized light interacts with a linear nondepolarizing optical sample S with Jones matrix J. ps is a reference Cartesian coordinate system defined in the same manner for both the input (i) and output (o) beams.

FIG. 2
FIG. 2

A general ellipsometer arrangement with arbitrary polarizing (P) and analyzing (A) optics. S is the sample under measurement by MGE, and M is the source of modulation applied to S. The signal output D of the photodetector D can be measured by a lock-in amplifier LIA, which receives its reference signal from M. L represents the light source.

Tables (2)

Tables Icon

TABLE I Partial derivatives required for the evaluation of the instrument matrix I for the PSA ellipsometer using Eqs. (35) and (21).

Tables Icon

TABLE II Partial derivatives required for the evaluation of the instrument matrix I for the PCSA ellipsometer using Eqs. (35) and (21).

Equations (37)

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J = [ J p p J p s J s p J s s ] .
J s s 0 ,
J = J s s J n ,
J n = [ ρ 1 ρ 2 ρ 3 1 ] ,
ρ 1 = J p p / J s s ,             ρ 2 = J p s / J s s ,             ρ 3 = J s p / J s s .
ρ i = tan ψ i e j Δ i ,             i = 1 , 2 , 3 ,
J s s 2 = σ s s .
J n = J ¯ n + δ J ˜ n , σ s s = σ ¯ s s + δ σ ˜ s s ,
δ J ˜ n = δ J ^ n sin ω t , δ σ ˜ s s = δ σ ^ s s sin ω t .
ψ i = ψ ¯ i + δ ψ ˜ i ,             Δ i = Δ ¯ i + δ Δ ˜ i , δ ψ ˜ i = δ ψ ^ i sin ω t ,             δ Δ ˜ i = δ Δ ^ i sin ω t , i = 1 , 2 , 3.
δ J ^ n = [ δ ρ ^ 1 δ ρ ^ 2 δ ρ ^ 3 0 ] ,
δ ρ i = ( sec 2 ψ ¯ i δ ψ ^ i + j tan ψ ¯ i δ Δ ^ i ) e j Δ ¯ i , i = 1 , 2 , 3.
D = K σ s s f ( ψ 1 , Δ 1 , ψ 2 , Δ 2 , ψ 3 , Δ 3 ; a μ , b ν ) ,
δ D / D = ( δ σ s s / σ s s ) + i = 1 3 ( α ψ i δ ψ i + α Δ i δ Δ i ) .
α ψ i = ( 1 / f ) f / ψ i , α Δ i = ( 1 / f ) f / Δ i , i = 1 , 2 , 3.
δ ^ D / ¯ D = ( δ σ ^ s s / σ ¯ s s ) + i = 1 3 ( α ¯ ψ i δ ψ ^ i + α ¯ Δ i δ Δ ^ i ) ,
α ¯ ψ i = ( 1 / f ) f / ψ i ψ ¯ i , Δ ¯ i , α ¯ Δ i = ( 1 / f ) f / Δ i ψ ¯ i , Δ ¯ i .
δ m = δ ^ D / ¯ D .
δ m = I δ S ,
δ m = [ δ m 1 δ m 2 δ m 7 ] ;
I = [ 1 α ¯ ψ 11 α ¯ Δ 11 α ¯ ψ 21 α ¯ Δ 21 α ¯ ψ 31 α ¯ Δ 31 1 α ¯ ψ 12 α ¯ Δ 12 α ¯ ψ 22 α ¯ Δ 22 α ¯ ψ 32 α ¯ Δ 32 1 α ¯ ψ 17 α ¯ Δ 17 α ¯ ψ 27 α ¯ Δ 27 α ¯ ψ 37 α ¯ Δ 37 ] ;
δ S = [ δ σ ^ s s / σ ¯ s s δ ψ ^ 1 δ Δ ^ 1 δ ψ ^ 2 δ Δ ^ 2 δ ψ ^ 3 δ Δ ^ 3 ] .
δ S = I - 1 δ m ,
E i = [ E i p E i s ] ,
E i p = cos P ,             E i s = sin P ,
E i p = cos C cos ( P - C ) - ρ C sin C sin ( P - C ) , E i s = sin C cos ( P - C ) + ρ C cos C sin ( P - C ) ,
E D = T A R ( A ) J E i ,
R ( A ) = [ cos A sin A - sin A cos A ] ,             T A = [ 1 0 0 0 ] ,
E D = [ L 0 ] ,
L = J s s [ ( ρ 1 cos A + ρ 3 sin A ) E i p + ( ρ 2 cos A + sin A ) E i s ] .
D = K L 2 ,
D = K σ s s f ( ψ 1 , Δ 1 , ψ 2 , Δ 2 , ψ 3 , Δ 3 ; A , P )
f = l 1 2 + l 2 2 ,
l 1 = cos A cos P ( tan ψ 1 cos Δ 1 + tan P tan ψ 2 cos Δ 2 + tan A tan ψ 3 cos Δ 3 + tan A tan P ) , l 2 = cos A cos P ( tan ψ 1 sin Δ 1 + tan P tan ψ 2 sin Δ 2 + tan A tan ψ 3 sin Δ 3 ) .
l 1 = cos A [ tan ψ 1 sin ( 2 P + Δ 1 ) + tan ψ 2 cos Δ 2 + tan A tan ψ 3 sin ( 2 P + Δ 3 ) + tan A ] , l 2 = cos A [ tan ψ 1 cos ( 2 P + Δ 1 ) - tan ψ 2 sin Δ 2 + tan A tan ψ 3 cos ( 2 P + Δ 3 ) ] .
α ψ i = ( 2 l 1 l 1 / ψ i + 2 l 2 l 2 / ψ i ) / ( l 1 2 + l 2 2 ) , α Δ i = ( 2 l 1 l 1 / Δ i + 2 l 2 l 2 / Δ i ) / ( l 1 2 + l 2 2 ) , i = 1 , 2 , 3.
¯ D n = K σ ¯ s s f ( ψ ¯ 1 , Δ ¯ 1 , ψ ¯ 2 , Δ ¯ 2 , ψ ¯ 3 , Δ ¯ 3 , a μ n , b ν n ) , n = 1 , 2 , , 7 ,