Abstract

Some theoretical considerations and experimental techniques in application of an ADP 45° X-cut four-crystal electrooptic modulator (EOM) to fast retardation modulation ellipsometry are described, emphasis being placed on the thermal stabilization. First to solve the thermal instability problem and to analyze systematic errors resulting from use of the EOM, the Jones matrix for the EOM is theoretically constructed which includes the effect of axial misalignment of the ADP crystals, multiple reflection at crystal-matching-dielectric liquid interfaces, temperature differences among the four crystals, and modulator cell windows. As a result, temperature-dependent factors in EOM characteristics are made clear. Second, a practical matrix form of the Jones matrix is determined from experiments on the bias-voltage dependence of the transmitted light intensity. Last, a thermal stabilization technique introduced by the matrix representation is demonstrated in the application of the EOM to ellipsometry, which enables us to obtain ellipsometric parameters of a sample, (Ψ,Δ), independent of the thermal drift and imperfections that accompany the EOM.

© 1983 Optical Society of America

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References

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  1. H. Takasaki, Appl. Opt. 5, 759 (1966).
    [CrossRef] [PubMed]
  2. T. Yamaguchi, H. Hasunuma, Sci. Light 16, 64 (1967).
  3. T. Kasai, Rev. Sci. Instrum. 47, 1044 (1976).
    [CrossRef]
  4. R. H. Muller, Surf. Sci. 56, 19 (1976).
    [CrossRef]
  5. A. Moritani, J. Nakai, Appl. Opt. 21, 3231 (1982).
    [CrossRef] [PubMed]
  6. A. Moritani, Y. Okuda, H. Kubo, J. Nakai, unpublished.
  7. D. W. Berreman, J. Opt. Soc. Am. 62, 502 (1972).
    [CrossRef]
  8. Part of this section has been reported by A. Moritani, J. Nakai, Trans. Inst. Electr. Commun. Eng. Jpn. J66-C, 129 (1983), in Japanese.
  9. W. L. Wolfe, in Handbook of Optics, W. G. Driscoll, Ed. (McGraw-Hill, New York, 1978), p. 7–1.
  10. Operating manual for modulation system 3025 (Coherent Associates, Danbury, Conn.).
  11. I. P. Kaminow, E. H. Turner, Proc. IEEE 54, 1374 (1966).
    [CrossRef]
  12. P. Yeh, J. Opt. Soc. Am. 69, 742 (1979).
    [CrossRef]
  13. D. W. Berreman, J. Opt. Soc. Am. 63, 1374 (1973).
    [CrossRef]
  14. D. E. Aspnes, J. Opt. Soc. Am. 63, 1380 (1973).
    [CrossRef]
  15. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1976).
  16. V. M. Bermudez, V. H. Ritz, Appl. Opt. 17, 542 (1978).
    [CrossRef] [PubMed]
  17. J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).
  18. G. E. Francois, F. M. Librecht, Appl. Opt. 11, 472 (1972).
    [CrossRef] [PubMed]
  19. R. T. Denton, F. S. Chen, A. A. Ballman, J. Appl. Phys. 38, 1611 (1967).
    [CrossRef]
  20. On this point, a P(polarizer)–Q(EOM)–S(sample)–A(analyzer) optical system has an advantage over a PSQA system in the application to ellipsometry.
  21. P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
    [CrossRef]

1983 (1)

Part of this section has been reported by A. Moritani, J. Nakai, Trans. Inst. Electr. Commun. Eng. Jpn. J66-C, 129 (1983), in Japanese.

1982 (1)

1979 (1)

1978 (1)

1976 (2)

T. Kasai, Rev. Sci. Instrum. 47, 1044 (1976).
[CrossRef]

R. H. Muller, Surf. Sci. 56, 19 (1976).
[CrossRef]

1973 (3)

1972 (2)

1967 (2)

T. Yamaguchi, H. Hasunuma, Sci. Light 16, 64 (1967).

R. T. Denton, F. S. Chen, A. A. Ballman, J. Appl. Phys. 38, 1611 (1967).
[CrossRef]

1966 (2)

H. Takasaki, Appl. Opt. 5, 759 (1966).
[CrossRef] [PubMed]

I. P. Kaminow, E. H. Turner, Proc. IEEE 54, 1374 (1966).
[CrossRef]

Aspnes, D. E.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1976).

Ballman, A. A.

R. T. Denton, F. S. Chen, A. A. Ballman, J. Appl. Phys. 38, 1611 (1967).
[CrossRef]

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1976).

Bermudez, V. M.

Berreman, D. W.

Chen, F. S.

R. T. Denton, F. S. Chen, A. A. Ballman, J. Appl. Phys. 38, 1611 (1967).
[CrossRef]

Denton, R. T.

R. T. Denton, F. S. Chen, A. A. Ballman, J. Appl. Phys. 38, 1611 (1967).
[CrossRef]

Dill, F. H.

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

Francois, G. E.

Hasunuma, H.

T. Yamaguchi, H. Hasunuma, Sci. Light 16, 64 (1967).

Hauge, P. S.

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, E. H. Turner, Proc. IEEE 54, 1374 (1966).
[CrossRef]

Kasai, T.

T. Kasai, Rev. Sci. Instrum. 47, 1044 (1976).
[CrossRef]

Kubo, H.

A. Moritani, Y. Okuda, H. Kubo, J. Nakai, unpublished.

Librecht, F. M.

Moritani, A.

Part of this section has been reported by A. Moritani, J. Nakai, Trans. Inst. Electr. Commun. Eng. Jpn. J66-C, 129 (1983), in Japanese.

A. Moritani, J. Nakai, Appl. Opt. 21, 3231 (1982).
[CrossRef] [PubMed]

A. Moritani, Y. Okuda, H. Kubo, J. Nakai, unpublished.

Muller, R. H.

R. H. Muller, Surf. Sci. 56, 19 (1976).
[CrossRef]

Nakai, J.

Part of this section has been reported by A. Moritani, J. Nakai, Trans. Inst. Electr. Commun. Eng. Jpn. J66-C, 129 (1983), in Japanese.

A. Moritani, J. Nakai, Appl. Opt. 21, 3231 (1982).
[CrossRef] [PubMed]

A. Moritani, Y. Okuda, H. Kubo, J. Nakai, unpublished.

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).

Okuda, Y.

A. Moritani, Y. Okuda, H. Kubo, J. Nakai, unpublished.

Ritz, V. H.

Takasaki, H.

Turner, E. H.

I. P. Kaminow, E. H. Turner, Proc. IEEE 54, 1374 (1966).
[CrossRef]

Wolfe, W. L.

W. L. Wolfe, in Handbook of Optics, W. G. Driscoll, Ed. (McGraw-Hill, New York, 1978), p. 7–1.

Yamaguchi, T.

T. Yamaguchi, H. Hasunuma, Sci. Light 16, 64 (1967).

Yeh, P.

Appl. Opt. (4)

IBM J. Res. Dev. (1)

P. S. Hauge, F. H. Dill, IBM J. Res. Dev. 17, 472 (1973).
[CrossRef]

J. Appl. Phys. (1)

R. T. Denton, F. S. Chen, A. A. Ballman, J. Appl. Phys. 38, 1611 (1967).
[CrossRef]

J. Opt. Soc. Am. (4)

Proc. IEEE (1)

I. P. Kaminow, E. H. Turner, Proc. IEEE 54, 1374 (1966).
[CrossRef]

Rev. Sci. Instrum. (1)

T. Kasai, Rev. Sci. Instrum. 47, 1044 (1976).
[CrossRef]

Sci. Light (1)

T. Yamaguchi, H. Hasunuma, Sci. Light 16, 64 (1967).

Surf. Sci. (1)

R. H. Muller, Surf. Sci. 56, 19 (1976).
[CrossRef]

Trans. Inst. Electr. Commun. Eng. Jpn. (1)

Part of this section has been reported by A. Moritani, J. Nakai, Trans. Inst. Electr. Commun. Eng. Jpn. J66-C, 129 (1983), in Japanese.

Other (6)

W. L. Wolfe, in Handbook of Optics, W. G. Driscoll, Ed. (McGraw-Hill, New York, 1978), p. 7–1.

Operating manual for modulation system 3025 (Coherent Associates, Danbury, Conn.).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1976).

J. F. Nye, Physical Properties of Crystals (Oxford U.P., London, 1957).

A. Moritani, Y. Okuda, H. Kubo, J. Nakai, unpublished.

On this point, a P(polarizer)–Q(EOM)–S(sample)–A(analyzer) optical system has an advantage over a PSQA system in the application to ellipsometry.

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Figures (7)

Fig. 1
Fig. 1

Structure of an ADP (or KDP etc.) 45° X-cut four-crystal electrooptic modulator (EOM). The four crystals are (1)–(4). The extraordinary and ordinary rays in the crystals are indicated by the broken and solid lines, respectively. V is the applied modulation voltage or dc bias voltage. XYZ indicates the crystal axis coordinate in which the optic axis is represented by Z. xyz shows the light propagation coordinate system in which z is in the propagating direction.

Fig. 2
Fig. 2

Experimental setup for the measurement of bias voltage dependence of the transmitted light flux. P and A represent the polarizer and the analyzer, respectively. EOM is the four-crystal electrooptic modulator. PM is the photomultiplier.

Fig. 3
Fig. 3

Dependence of the transmitted light intensity proportional to |T21|2 on the bias voltage measured with the experimental setup shown in Fig. 2; (b) was measured ~15 min after (a). Thermal drift and imperfections in the modulation characteristics are apparent in these curves which, however, maintain the 2-wavelength voltage period.

Fig. 4
Fig. 4

Dependence of the transmitted light intensity proportional to |T11|2 on the bias voltage measured with the experimental setup shown in Fig. 2. The variation of the intensity is due to the multiple reflection effect and the amplitude of the curve is below 0.4% of the total magnitude of light intensity.

Fig. 5
Fig. 5

Block diagram of the ellipsometer system. The light intensity I and the modulation voltage V are simultaneously read by the digital memory. The sampling controller feeds external triggers to the digital memory to control the number of sampling points and intervals of sampling of I and V.5,6 The sawtooth or triangular voltage is applied by the function generator through the EOM power amplifier to modulate the retardation with the EOM.

Fig. 6
Fig. 6

Time dependence of the coefficients a 1 a 4. Here n is the number of repeated measurements. The measurement intervals are ~3 min, so that it takes ~1 h to get the twenty datum points of each coefficient.

Fig. 7
Fig. 7

Polarizer azimuth dependence of the coefficients a 1 (a) and a 2 (b). Here, P is equal to Ψe in Eq. (41) as the measurement was made in the PQA straight-through optical arrangement.

Equations (47)

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( 1 0 0 exp ( 4 i δ ) ) ,
X 2 + Y 2 n o 2 + Z 2 n e 2 + 2 r 41 E X Y Z = 1.
( X Y Z ) = ( 1 η 0 η 1 θ 0 θ 1 ) ( X Y Z ) = B ( X Y Z ) ,
η = ϕ + ξ ,
x y z ( n ) = C ( n ) B ( n ) A ( n ) x y z A ( n ) 1 B ( n ) 1 C ( n ) 1 ,
x y z ( 1 ) = ( x x x y x z y x y y y z z x z y z z ) ,
x x = ( a + c + 2 b θ ) / D , x y = η ( D n o 2 a b c ) / 2 D , x z = ( b 2 c θ ) / D , y x = x y , y y = n o 2 y z = η ( D n o 2 a b c ) / 2 D , z x = x z , z y = y z , z z = ( a c 2 b θ ) / D ,
a = n e 2 + n o 2 , b = n e 2 n o 2 , c = 2 r 41 E X , D = ( a 2 b 2 c 2 ) / 2 .
x y = y x = y z = z y = 0
Δ ( 1 ) = ( 0 1 0 0 s 0 u 0 0 0 0 1 u 0 t 0 ) ,
s = x x x z 2 / z z , t = y y y z 2 / z z , u = x y x z y z / z z .
z Ψ = i ω c o Δ Ψ = i α Δ Ψ ,
Ψ = ( E x H y E y H x ) .
Ψ ( z ) = Ψ ( o ) exp ( i α q z ) .
det ( Δ q J ) = 0 ,
q 1 = s = 2 / ( n e 2 + n o 2 2 r 41 E X ) , q 2 = q 1 , q 3 = t = n o , q 4 = q 3 ,
Ψ 1 = E x 1 exp ( i α q 1 z ) ( 1 q 1 γ γ q 1 ) , Ψ 2 = E x 2 exp ( i α q 1 z ) ( 1 q 1 γ γ q 1 ) , Ψ 3 = E y 1 exp ( i α q 3 z ) ( γ γ q 3 1 q 3 ) , Ψ 4 = E y 2 exp ( i α q 3 z ) ( γ γ q 3 1 q 3 ) ,
γ = ( q 1 2 s ) / u = u / ( t s ) = ( q 3 2 t ) / u = 1 2 η .
P 11 = P 22 = cos α q 1 h , P 12 = i q 1 1 sin α q 1 h , P 13 = γ ( cos α q 1 h cos α q 3 h ) , P 14 = P 32 = i γ ( q 1 1 sin α q 1 h q 3 1 sin α q 3 h ) P 21 = i q 1 sin α q 1 h , P 23 = P 41 = i γ ( q 1 sin α q 1 h q 3 sin α q 3 h ) , P 33 = P 44 = cos α q 3 h , P 34 = i q 3 1 sin α q 3 h , P 43 = i q 3 sin α q 3 h , P 24 = P 31 = P 42 = P 13 ,
( T x n a T x T y n a T y ) = P 4 ( d ) P a 3 ( l ) P 3 ( d ) P a 2 ( l ) P 2 ( d ) P a 1 ( l ) P 1 ( d ) × ( E x + R x n a ( E x R x ) E y + R y n a ( E y R y ) ) ,
( T x T y ) = ( T 11 T 12 T 21 T 22 ) ( E x E y ) .
( T x n a T x T y n a T y ) = P 1 ( d ) ( E x n a E x E y n a E y ) .
T ( 1 ) = exp ( i α q 1 d ) ( 1 γ { 1 exp [ i ( δ 0 δ ) ] } γ { 1 exp [ i ( δ 0 δ ) ] } exp [ i ( δ 0 δ ) ] ) .
δ 0 = α ( n o 2 n e 2 + n o 2 ) d , δ = 2 α r 41 d V ( n e 2 + n o 2 ) 3 / 2 d o .
T = T ( 4 ) T ( 3 ) T ( 2 ) T ( 1 ) ,
T 11 = 1 , T 12 = γ 1 ( γ 1 γ 2 ) exp [ i ( δ 0 δ ) ] ( γ 2 γ 3 ) exp [ 2 i ( δ 0 δ ) ] ( γ 3 γ 4 ) exp [ i ( δ 0 3 δ ) ] γ 4 exp ( 4 i δ ) , T 21 = γ 4 + ( γ 3 γ 4 ) exp [ i ( δ 0 + δ ) ] + ( γ 2 γ 3 ) exp [ 2 i ( δ 0 + δ ) ] + ( γ 1 γ 2 ) exp [ i ( δ 0 + 3 δ ) ] γ 1 exp ( 4 i δ ) ] , T 22 = exp ( 4 i δ ) ,
T 12 = γ 1 ( γ 1 γ 2 ) exp [ i ( δ 01 δ ) ] ( γ 2 γ 3 ) exp [ i ( δ 01 + δ 02 2 δ ) ] ( γ 3 γ 4 ) exp [ i ( δ 01 + δ 02 δ 03 3 δ ) ] γ 4 exp [ i ( δ 01 + δ 02 δ 03 δ 04 4 δ ) ] , T 22 = exp [ i ( δ 01 + δ 02 δ 03 δ 04 4 δ ) ] .
T 11 = 1 + y 0 + y 2 exp ( 2 i δ ) + y 4 exp ( 4 i δ ) , T 22 = exp ( 4 i δ ) [ 1 + y 0 + y 2 exp ( 2 i δ ) + y 4 exp ( 4 i δ ) ] ,
I = I 0 | T 21 | 2 .
I = I 0 | T 11 | 2 .
T p = R ( Ψ w 2 ) W ( δ w 2 ) R ( Ψ w 2 ) T R ( Ψ w 1 ) W ( δ w 1 ) R ( Ψ w 1 ) ,
W ( δ ) = ( 1 0 0 1 + i δ ) .
T p 11 = 1 , T p 12 = x 3 x 2 exp [ i ( δ 1 2 δ ) ] x 1 exp [ i ( δ 1 δ 2 4 δ ) ] , T p 21 = x * 1 + x 2 exp [ i ( δ 2 + 2 δ ) ] x * 3 exp [ i ( δ 1 δ 2 4 δ ) ] , T p 22 = ( 1 + i x 4 ) exp [ i ( δ 1 δ 2 4 δ ) ] .
δ 1 = δ 01 + δ 02 , δ 2 = δ 03 + δ 04 .
x 1 r = γ 3 = γ 4 , x 1 i = δ w 2 cos Ψ w 2 sin Ψ w 2 , x 3 r = γ 1 = γ 2 , x 3 i = δ w 1 cos Ψ w 1 sin Ψ w 1 .
x 2 = γ 1 γ 3 , x 4 = δ w 1 cos 2 Ψ w 1 + δ w 2 cos 2 Ψ w 2 .
tan Ψ e = tan P / tan Ψ , Δ e = Δ ,
I = I 0 ( 1 + a 1 cos 4 δ + a 2 sin 4 δ + a 3 cos 2 δ + a 4 sin 2 δ ) ,
a 1 = a 0 [ ( cos 2 Ψ e + 2 x 3 r sin 2 Ψ e cos Δ e + x 4 sin 2 Ψ e sin Δ e ) cos ( δ 1 δ 2 ) + ( sin 2 Ψ e sin Δ e + 2 x 3 i sin 2 Ψ e cos Δ e ) x 4 cos 2 Ψ e ) sin ( δ 1 δ 2 ) ] , a 2 = a 0 [ ( sin 2 Ψ e sin Δ e + 2 x 3 i sin 2 Ψ e cos Δ e x 4 cos 2 Ψ e ) cos ( δ 1 δ 2 ) + ( cos 2 Ψ e + 2 x 3 r sin 2 Ψ e cos Δ e + x 4 sin 2 Ψ e sin Δ e ) sin ( δ 1 δ 2 ) ] , a 3 = 2 x 2 sin Ψ e cos Δ e cos δ 2 , a 4 = 2 x 2 sin 2 Ψ e cos Δ e sin δ 2 , a 0 = 1 + 2 x 1 r sin 2 Ψ e cos Δ e .
4 δ = E V + A p ;
E = 4 2 α r 41 d ( n e 2 + n o 2 ) 3 / 2 d o ,
I = I 0 ( 1 + a 1 cos 4 δ + a 2 sin 4 δ + a 3 cos 2 δ + a 4 sin 2 δ ) ,
4 δ = E V + G , G = A p ( δ 1 δ 2 ) , }
a 1 = cos 2 Ψ e + 2 x 1 r cos 2 Ψ e sin 2 Ψ e cos Δ e + 2 x 3 r sin 2 Ψ e cos Δ e + x 4 sin 2 Ψ e sin Δ e , a 2 = sin 2 Ψ e sin Δ e 2 x 1 r sin 2 2 Ψ e sin Δ e cos Δ e 2 x 3 i sin 2 Ψ e cos Δ e + x 4 cos 2 Ψ e , a 3 = 2 x 2 sin 2 Ψ e cos Δ e cos 1 2 ( δ 1 + δ 2 ) , a 4 = 2 x 2 sin 2 Ψ e cos Δ e sin 1 2 ( δ 1 + δ 2 ) . }
a 1 = cos 2 Ψ e , a 2 = sin 2 Ψ e sin Δ e a 3 = 0 , a 4 = 0
Ψ e = P , Δ e = 0
a 1 = cos 2 Ψ e , a 2 = sin 2 Ψ e sin Δ e 2 x 3 i sin 2 Ψ e cos Δ e .

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