Abstract

The complex coefficients of the reflection of light from an anisotropic conducting medium are derived assuming one of the optical axes to be perpendicular to the reflecting surface. The results represent generalizations of those obtained previously. The theory of reflection is developed with special reference to studies of strain-induced optical anisotropies in metals. How off-null ellipsometry can be used to determine complex photoelastic constants of metals is described. Measurements made on polycrystalline Al and Cu samples at different wavelengths in the visible region are reported.

© 1986 Optical Society of America

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References

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  1. M. Garfinkel, I. J. Tiemann, W. E. Engeler, “Piezoreflectivity of the Noble Metals,” Phys. Rev. 148, 695 (1966).
    [CrossRef]
  2. U. Gerhardt, “Effect of Uniaxial and Hydrostatic Strain on the Optical Constants and the Electronic Structure of Copper,” Phys. Rev. 172, 651 (1968).
    [CrossRef]
  3. L. G. Holcomb, N. M. Bashara, “Elasto-optic Effect in Absorbing Materials,” J. Opt. Soc. Am. 61, 608 (1971).
    [CrossRef]
  4. O. Keller, K. Pedersen, “Strain of Metal Surfaces Determined by Optical Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 401, 60 (1983).
  5. D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979).
  6. V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, New York, 1980).
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  8. T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
    [CrossRef]
  9. T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
    [CrossRef]
  10. H. Ehrenreich, H. R. Philipp, “Optical Properties of Ag and Cu,” Phys. Rev. 128, 1622 (1962).
    [CrossRef]
  11. G. A. Burdick, “Energy Band Structure of Copper,” Phys. Rev. 129, 138 (1963).
    [CrossRef]

1983 (1)

O. Keller, K. Pedersen, “Strain of Metal Surfaces Determined by Optical Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 401, 60 (1983).

1972 (1)

T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
[CrossRef]

1971 (1)

1968 (1)

U. Gerhardt, “Effect of Uniaxial and Hydrostatic Strain on the Optical Constants and the Electronic Structure of Copper,” Phys. Rev. 172, 651 (1968).
[CrossRef]

1966 (1)

M. Garfinkel, I. J. Tiemann, W. E. Engeler, “Piezoreflectivity of the Noble Metals,” Phys. Rev. 148, 695 (1966).
[CrossRef]

1963 (1)

G. A. Burdick, “Energy Band Structure of Copper,” Phys. Rev. 129, 138 (1963).
[CrossRef]

1962 (1)

H. Ehrenreich, H. R. Philipp, “Optical Properties of Ag and Cu,” Phys. Rev. 128, 1622 (1962).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, New York, 1980).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

L. G. Holcomb, N. M. Bashara, “Elasto-optic Effect in Absorbing Materials,” J. Opt. Soc. Am. 61, 608 (1971).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Burdick, G. A.

G. A. Burdick, “Energy Band Structure of Copper,” Phys. Rev. 129, 138 (1963).
[CrossRef]

Ehrenreich, H.

H. Ehrenreich, H. R. Philipp, “Optical Properties of Ag and Cu,” Phys. Rev. 128, 1622 (1962).
[CrossRef]

Engeler, W. E.

M. Garfinkel, I. J. Tiemann, W. E. Engeler, “Piezoreflectivity of the Noble Metals,” Phys. Rev. 148, 695 (1966).
[CrossRef]

Garfinkel, M.

M. Garfinkel, I. J. Tiemann, W. E. Engeler, “Piezoreflectivity of the Noble Metals,” Phys. Rev. 148, 695 (1966).
[CrossRef]

Gerhardt, U.

U. Gerhardt, “Effect of Uniaxial and Hydrostatic Strain on the Optical Constants and the Electronic Structure of Copper,” Phys. Rev. 172, 651 (1968).
[CrossRef]

Ginzburg, V. L.

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, New York, 1980).

Holcomb, L. G.

Keller, O.

O. Keller, K. Pedersen, “Strain of Metal Surfaces Determined by Optical Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 401, 60 (1983).

Narasimhamurty, T. S.

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
[CrossRef]

Nelson, D. F.

D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979).

Pedersen, K.

O. Keller, K. Pedersen, “Strain of Metal Surfaces Determined by Optical Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 401, 60 (1983).

Philipp, H. R.

H. Ehrenreich, H. R. Philipp, “Optical Properties of Ag and Cu,” Phys. Rev. 128, 1622 (1962).
[CrossRef]

Sosnowski, T. P.

T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
[CrossRef]

Tiemann, I. J.

M. Garfinkel, I. J. Tiemann, W. E. Engeler, “Piezoreflectivity of the Noble Metals,” Phys. Rev. 148, 695 (1966).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

T. P. Sosnowski, “Polarization Mode Filters for Integrated Optics,” Opt. Commun. 4, 408 (1972).
[CrossRef]

Phys. Rev. (4)

H. Ehrenreich, H. R. Philipp, “Optical Properties of Ag and Cu,” Phys. Rev. 128, 1622 (1962).
[CrossRef]

G. A. Burdick, “Energy Band Structure of Copper,” Phys. Rev. 129, 138 (1963).
[CrossRef]

M. Garfinkel, I. J. Tiemann, W. E. Engeler, “Piezoreflectivity of the Noble Metals,” Phys. Rev. 148, 695 (1966).
[CrossRef]

U. Gerhardt, “Effect of Uniaxial and Hydrostatic Strain on the Optical Constants and the Electronic Structure of Copper,” Phys. Rev. 172, 651 (1968).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

O. Keller, K. Pedersen, “Strain of Metal Surfaces Determined by Optical Ellipsometry,” Proc. Soc. Photo-Opt. Instrum. Eng. 401, 60 (1983).

Other (4)

D. F. Nelson, Electric, Optic, and Acoustic Interactions in Dielectrics (Wiley, New York, 1979).

V. M. Agranovich, V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, New York, 1980).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

T. S. Narasimhamurty, Photoelastic and Electro-Optic Properties of Crystals (Plenum, New York, 1981).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(a) Schematic diagram showing the geometry of the complex optical wave vectors at the metal surface. (b) Orientation of the principal optical axes with respect to the plane of incidence.

Fig. 2
Fig. 2

Schematic illustration showing the components in the ellipsometric setup, the local Cartesian coordinate systems, and the positive sense of rotations for the appropriate angles.

Fig. 3
Fig. 3

Measurements of the linear (scale to the left) and quadratic (scale to the right) terms in the detected intensity as a function of strain for Cu.

Fig. 4
Fig. 4

Measurements of the linear and quadratic terms for Al at different wavelengths of the incident light.

Fig. 5
Fig. 5

Measurements of the linear term for Cu at different wavelengths of the incident light.

Fig. 6
Fig. 6

Real (∊R) and imaginary (∊I) parts of the relative dielectric constant of the Al sample at different wavelengths measured by null ellipsometry.

Fig. 7
Fig. 7

Real (∊R) and imaginary (∊I) parts of the relative dielectric constant of the Cu sample at different wavelengths measured by null ellipsometry.

Fig 8
Fig 8

Difference between the complex photoelastic constants p11 = p11R + ip11I and p12R = P12R + ip12I for the Al sample. The results at the various wavelengths are calculated from the experimental data in Figs. 4 and 6.

Fig 9
Fig 9

Difference between the complex photoelastic constants p11 = p11R + ip11I and p12R = P12R + ip12I for the Cu sample. The results at the various wavelengths are calculated from the experimental data in Figs. 5 and 7.

Fig. 10
Fig. 10

Dependence of Γ1, Γ2, and Γ3 on the angle of incidence for TE and TM polarized light calculated from the experimental data at near-normal incidence on Cu at λ = 570 nm.

Equations (66)

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{ ( ω c 0 ) 2 ɛ 1 k 2 2 k 3 2 k 1 k 2 k 1 k 3 k 2 k 1 ( ω c 0 ) 2 2 k 1 2 k 3 2 k 2 k 3 k 3 k 1 k 3 k 2 ( ω c 0 ) 2 3 k 1 2 k 2 2 } × { E 1 E 2 E 3 } = { 0 0 0 } ,
i N i 2 = 1 + χ ii ( ω ) + i σ ii ( ω ) 0 ω , i = 1 , 2 , 3 ,
k i 1 = k 0 sin θ cos φ ,
k i 2 = k 0 sin θ sin φ ,
k i 3 = k 0 cos θ ,
k r 1 = k t 1 R = k i 1 ,
k r 2 = k t 2 R = k i 2 ,
k t 1 I = k t 2 I = 0 ,
k r 3 = k i 3 .
k t 3 = k t 3 R + i k t 3 I = [ A ± ( A 2 B ) 1 / 2 ] 1 / 2 , k t 3 R > 0 ,
A = ½ [ ( 1 + 1 3 ) k i 1 2 + ( 1 + 2 3 ) k i 2 2 ( 1 + 2 ) k 0 2 ] ,
B = ( 1 2 k 0 2 1 k i 1 2 2 k i 2 2 ) [ k 0 2 1 3 ( k i 1 2 + k i 2 2 ) ] .
{ M 11 + k i 1 k i 2 k i 1 k t 3 + 0 0 0 0 0 0 k i 1 k i 2 M 22 + k i 2 k t 3 + 0 0 0 0 0 0 0 0 0 M 11 k i 1 k i 2 k i 1 k t 3 0 0 0 0 0 0 k i 1 k i 2 M 22 k i 2 k t 3 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 k t 3 + k i 2 0 k t 3 k i 2 0 k i 3 k i 2 k t 3 + 0 k i 1 k t 3 0 k i 1 k i 3 0 k i 1 0 0 0 0 0 0 k i 1 k i 2 k i 3 } { E t 1 + E t 2 + E t 3 + E t 1 E t 2 E t 3 E r 1 E r 2 E r 3 } = { 0 0 0 0 E i 1 E i 2 k i 2 E i 3 k i 3 E i 2 k i 3 E i 1 k i 1 E i 3 0 } ,
E t 1 ± = α ± E t 3 ± ,
E t 2 ± = β ± E t 3 ± ,
α ± = k i 1 k t 3 ± k i 2 2 M 22 ± M 11 ± M 22 ± k i 1 2 k i 2 2 ,
β ± = k i 2 k t 3 ± k i 1 2 M 11 ± M 11 ± M 22 ± k i 1 2 k i 2 2 .
E r 1 = α + E t 3 + + α E t 3 E i 1 ,
E r 2 = β + E t 3 + + β E t 3 E i 2 ,
E r 3 = k i 1 k i 3 E r 1 + k i 2 k i 3 E r 2 .
E t 3 ± = B C A D A ± B A B ± ,
A ± = ( k i 3 + k t 3 ± ) ( k i 2 α ± k i 1 β ± ) ,
B ± = ( k i 1 + k i 2 ) + ( k i 3 + k t 3 ± ) ( α ± + β ± ) + k i 1 + k i 2 k i 3 ( k i 1 α ± + k i 2 β ± ) ,
C = 2 k i 3 ( k i 2 E i 1 k i 1 E i 2 ) ,
D = [ 2 k i 3 + k i 1 ( k i 1 + k i 2 ) k i 3 ] E i 1 + [ 2 k i 3 + k i 2 ( k i 1 + k i 2 ) k i 3 ] E i 2 ( k i 1 + k i 2 ) E i 3 .
( E i 1 , E i 2 , E i 3 ) = ( sin φ , cos φ , 0 ) E i TE ,
( E i 1 , E i 2 , E i 3 ) = ( cos θ cos φ , cos θ sin φ , sin θ ) E i TM .
E r TE = E r 2 cos φ E r 1 sin φ ,
E r TM = ( E r 1 cos φ + E r 2 sin φ ) cos θ E r 3 sin θ .
{ E r TM E r TE } = { T 11 T 12 T 21 T 22 } { E i TM E i TE } .
T 11 = 1 2 [ F 1 + n + + n ( Φ + W ) cos θ ] F 1 + F 2 ( n + + n ) ,
T 22 = 1 + 2 [ n + + n + ( Φ + W ) cos θ ] F 1 + F 2 ( n + + n ) ,
T 21 = T 12 = ( 1 2 ) sin 2 φ F 1 + F 2 ( n + + n ) ,
n ± = k t 3 ± k 0 ,
Φ = 1 cos 2 φ + 2 sin 2 φ ,
W = ( 1 2 Φ sin 2 θ 1 sin 2 θ 3 ) 1 / 2 ,
F 1 = 1 cos θ [ 1 + 2 ( 1 + Φ ) sin 2 θ + W ( 2 sin 2 θ sin 2 θ 3 ) ] ,
F 2 = 1 + W .
T 21 = ½ ( 1 2 ) sin 2 φ cos θ 2 + ( 1 + ) [ ( sin 2 θ ) 1 / 2 sin 2 θ ] .
T 11 = T 22 = 1 N 1 + N ,
T 21 = T 12 = N 1 N 2 ( 1 + N ) 2 sin 2 φ ,
= ( 0 ) ( 0 ) p S ( 0 ) ,
i = ( 0 ) [ ( 0 ) ] 2 [ p 11 S i + p 12 ( S j + S k ) ] ,
E AO t e = T A te R ( A ) T S xy R ( C ) T C fs R ( C P ) T P te R ( P ) E LO xy ,
E AO te = ( T 11 cos A + T 21 sin A ) ( cos 2 C + ρ C sin 2 C ) + ( T 12 cos A + T 22 sin A ) ( 1 ρ C ) × sin C cos C ,
| S | = ½ ( 0 μ 0 ) 1 / 2 | E AO | 2 ,
I = | T 11 | 2 ( Δ A ) 2 + 2 | T 22 | 2 ( Δ C ) 2 2 Δ A Δ C [ T 11 R ( T 22 R T 22 I ) + T 11 I ( T 22 R + T 22 I ) ] + | T 21 | 2 2 Δ A ( T 11 R T 21 R + T 11 I T 21 I ) + 2 Δ C [ T 21 R ( T 22 R T 22 I ) + T 21 I ( T 22 R + T 22 I ) ] .
I = Γ 0 + Γ 1 ( S 1 S 2 ) 2 sin 2 2 φ + ( Γ 2 Δ A + Γ 3 Δ C ) ( S 1 S 2 ) sin 2 φ ,
Γ 0 = | T 11 | 2 [ ( Δ A ) 2 + 2 ( Δ C ) 2 + 2 Δ A Δ C ] ,
Γ 1 = | α | 2 | p 11 p 12 | 2 ,
Γ 2 = 2 [ ( α R T 11 R + α I T 11 I ) ( p 12 R p 11 R ) + ( α R T 11 I α I T 11 R ) ( p 12 I p 11 I ) ] ,
Γ 3 = 2 { [ α R ( T 11 R T 11 I ) + α I ( T 11 R + T 11 I ) ] ( p 12 R p 11 R ) + [ α R ( T 11 R + T 11 I ) α I ( T 11 R T 11 I ) ] ( p 12 I p 11 I ) } ,
α = α R + i α I = N 3 2 ( 1 + N ) 2 .
T 11 = 1 2 ( cos φ + sin φ ) ( A + B A B + ) sin θ cos θ [ k i 1 ( α A + α + A ) + k i 2 ( β A + β + A ) ] ,
T 22 = 1 2 k 0 cos θ A + B A B + { [ B k 0 sin θ + A ( cos φ sin φ ) ] × ( β + cos φ α sin φ ) ( + ) } ,
T 21 = 2 k 0 ( cos φ + sin φ ) A + B A B + × [ A + ( β cos φ α sin φ ) ( + ) ] ,
T 12 = 2 ( A + B A B + ) sin θ { ( α k i 1 + β k i 2 ) × [ B + k 0 sin θ + A + ( cos φ sin φ ) ] ( + ) } ,
T 11 = 1 2 ( cos φ + sin φ ) ( A + B A B + ) sin θ cos θ × [ k i 1 ( α A + α + A ) ( 1 2 ) ] .
T 11 = 1 2 ( cos φ + sin φ ) ( k i 1 k i 2 k 0 2 ( 1 2 ) k t 3 + k t 3 ( k t 3 + k t 3 ) N ( A + B A B + ) sin θ cos θ × { ( k i 1 2 + k i 2 2 ) [ k i 1 2 + k i 2 2 k i 3 ( k t 3 + + k t 3 ) k t 3 + k t 3 ] k 0 2 ( k i 1 2 2 + k i 2 2 1 ) } ,
T 22 = 1 2 k 0 3 cos θ k i 1 k i 2 ( 2 1 ) ( k i 1 + k i 2 ) N ( A + B A B + ) { k t 3 + ( k t 3 ) 3 k 0 cos θ + k t 3 + k t 3 1 sin 2 θ ( k i 1 2 1 + k i 2 2 2 ) + k t 3 + k t 3 k 0 2 k i 3 [ k i 1 2 + k i 2 2 1 sin 2 θ ( k i 1 2 2 + k i 2 2 1 ) ] k t 3 + k 0 2 [ k 0 2 1 2 ( k i 1 2 1 + k i 2 2 2 ) ] ( + ) } ,
T 21 = 2 k 0 4 ( cos φ + sin φ ) k i 1 2 k i 2 2 ( 1 2 ) 2 k t 3 + k t 3 ( k t 3 + k t 3 ) N ( A + B A B + ) ,
T 12 = 2 k 0 ( k i 1 + k i 2 ) N ( A + B A B + ) { [ ( k i 1 2 + k i 2 2 ) 2 k 0 2 ( k i 1 2 2 + k i 2 2 1 ) + ( k i 1 2 + k i 2 ) 2 ( k t 3 ) 2 ] × { ( k t 3 + ) 3 k t 3 k 0 cos θ + ( k t 3 + ) 2 k t 3 1 sin 2 θ ( k i 1 2 1 + k i 2 2 2 ) + k t 3 + k t 3 + k 0 2 k i 3 [ k i 1 2 + k i 2 2 1 sin 2 θ ( k i 1 2 2 + k i 2 2 1 ) ] k t 3 k 0 2 ( k 0 2 1 2 k i 1 2 1 k i 2 2 2 ) } ( + ) } .
N = ( M 11 + M 22 + k i 1 2 k i 2 2 ) ( M 11 M 22 k i 1 2 k i 2 2 ) .
N ( A + B A B + ) = k 0 2 k i 1 k i 2 ( k i 1 k i 2 ) × ( 1 2 ) ( k t 3 k t 3 + ) k t 3 + k t 3 × { ( k t 3 + + k t 3 ) k 0 2 ( 1 + W ) + 1 k i 3 [ k 0 2 ( k i 1 2 2 + k i 2 2 1 ) ( k i 1 2 + k i 2 2 ) 2 + k 0 2 k t 3 + k t 3 ] + k i 3 [ k 0 2 ( 1 + 2 ) ( k i 1 2 + k i 2 2 ) + k 0 2 W ] } ,
k t 3 + k t 3 = k 0 2 W ( 1 sin 2 θ 3 )
N ( A + B A B + ) = k 0 11 sin 3 θ cos φ sin φ ( cos φ + sin φ ) × ( 1 2 ) ( n n + ) n + n × [ F 1 + F 2 ( n + + n ) ] ,

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