Abstract

We show that when the ratio rp/rs of the reflection amplitudes for the electromagnetic p and s waves is taken to be 1 at normal incidence, it will have the value −1 at grazing incidence. This result is valid for sharp or diffuse interfacial profiles, for internal as well as external reflections, and in the presence of absorption and anisotropy within the reflecting layer or its substrate. (The anisotropy of the dielectric function is limited to a difference in the response of the system to electric fields perpendicular or parallel to the interface, characterized by and |.) Under these conditions, there will always be at least one angle of incidence at which the real part of rp/rs, is zero. Under the same conditions, the reflected s and p electric fields at grazing incidence are out of phase with the incident electric fields, thus producing destructive interference at the mirror’s edge in Lloyd’s mirror experiment.

© 1985 Optical Society of America

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References

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  1. S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
    [CrossRef]
  2. D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980); “Adsorption at the liquid–vapor interface of a binary liquid mixture,”J. Chem. Phys. 73, 3366–3371 (1980); “Pretransition order on the surface of a nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 89, 319–325 (1982); “Ellipsometry of liquid surfaces,” J. Phys. (Paris) 44, 147–154 (1983).
  3. J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).
  4. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sec. 68.
  5. J. Lekner, “Exact reflection amplitudes for the Rayleigh profile,” Physica 116A, 235–247 (1982).
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Sec. 1.5.
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.
  8. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Sec. 13.8 and 28.10.
  9. J. Lekner, P. J. Castle, “Local fields near the surface of a crystalline dielectric,” Physica 101A, 89–98 (1980); “Variation of the local field through the liquid-vapour interface,” Physica 101A, 99–111 (1980).
  10. J. Lekner, “Anisotropy of the dielectric function within a liquid–vapour interface,” Mol. Phys. 49, 1385–1400 (1983).
    [CrossRef]
  11. G. Birkhoff, G.-C. Rota, Ordinary Differential Equations (Blaisdell, Waltham, Mass., 1969).

1983 (1)

J. Lekner, “Anisotropy of the dielectric function within a liquid–vapour interface,” Mol. Phys. 49, 1385–1400 (1983).
[CrossRef]

1982 (2)

J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).

J. Lekner, “Exact reflection amplitudes for the Rayleigh profile,” Physica 116A, 235–247 (1982).

1980 (2)

J. Lekner, P. J. Castle, “Local fields near the surface of a crystalline dielectric,” Physica 101A, 89–98 (1980); “Variation of the local field through the liquid-vapour interface,” Physica 101A, 99–111 (1980).

D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980); “Adsorption at the liquid–vapor interface of a binary liquid mixture,”J. Chem. Phys. 73, 3366–3371 (1980); “Pretransition order on the surface of a nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 89, 319–325 (1982); “Ellipsometry of liquid surfaces,” J. Phys. (Paris) 44, 147–154 (1983).

1969 (1)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.

Beaglehole, D.

D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980); “Adsorption at the liquid–vapor interface of a binary liquid mixture,”J. Chem. Phys. 73, 3366–3371 (1980); “Pretransition order on the surface of a nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 89, 319–325 (1982); “Ellipsometry of liquid surfaces,” J. Phys. (Paris) 44, 147–154 (1983).

Birkhoff, G.

G. Birkhoff, G.-C. Rota, Ordinary Differential Equations (Blaisdell, Waltham, Mass., 1969).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Sec. 1.5.

Castle, P. J.

J. Lekner, P. J. Castle, “Local fields near the surface of a crystalline dielectric,” Physica 101A, 89–98 (1980); “Variation of the local field through the liquid-vapour interface,” Physica 101A, 99–111 (1980).

Jasperson, S. N.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Sec. 13.8 and 28.10.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sec. 68.

Lekner, J.

J. Lekner, “Anisotropy of the dielectric function within a liquid–vapour interface,” Mol. Phys. 49, 1385–1400 (1983).
[CrossRef]

J. Lekner, “Exact reflection amplitudes for the Rayleigh profile,” Physica 116A, 235–247 (1982).

J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).

J. Lekner, P. J. Castle, “Local fields near the surface of a crystalline dielectric,” Physica 101A, 89–98 (1980); “Variation of the local field through the liquid-vapour interface,” Physica 101A, 99–111 (1980).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sec. 68.

Rota, G.-C.

G. Birkhoff, G.-C. Rota, Ordinary Differential Equations (Blaisdell, Waltham, Mass., 1969).

Schnatterly, S. E.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Sec. 13.8 and 28.10.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Sec. 1.5.

Mol. Phys. (1)

J. Lekner, “Anisotropy of the dielectric function within a liquid–vapour interface,” Mol. Phys. 49, 1385–1400 (1983).
[CrossRef]

Physica (4)

J. Lekner, “Exact reflection amplitudes for the Rayleigh profile,” Physica 116A, 235–247 (1982).

D. Beaglehole, “Ellipsometric study of the surface of simple liquids,” Physica 100B, 163–174 (1980); “Adsorption at the liquid–vapor interface of a binary liquid mixture,”J. Chem. Phys. 73, 3366–3371 (1980); “Pretransition order on the surface of a nematic liquid crystal,” Mol. Cryst. Liq. Cryst. 89, 319–325 (1982); “Ellipsometry of liquid surfaces,” J. Phys. (Paris) 44, 147–154 (1983).

J. Lekner, “Second-order ellipsometric coefficients,” Physica 113A, 506–520 (1982).

J. Lekner, P. J. Castle, “Local fields near the surface of a crystalline dielectric,” Physica 101A, 89–98 (1980); “Variation of the local field through the liquid-vapour interface,” Physica 101A, 99–111 (1980).

Rev. Sci. Instrum. (1)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Other (5)

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Sec. 68.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965), Sec. 1.5.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Sec. 13.8 and 28.10.

G. Birkhoff, G.-C. Rota, Ordinary Differential Equations (Blaisdell, Waltham, Mass., 1969).

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Equations (17)

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2 E y + ω 2 c 2 E y = 0 ,
d 2 E d z 2 + ( ω 2 c 2 - K 2 ) E = 0.
q 2 ( z ) = ( z ) ω 2 c 2 - K 2
exp ( i q 1 z ) + r s exp ( - i q 1 z ) E ( z ) t s exp ( i q 2 z ) .
( z ) = ½ ( 1 + 2 ) - ½ ( 1 - 2 ) tanh [ ( z - z 0 ) / 2 a ] .
E ( z ) = { exp ( i q 1 z ) + r s exp ( - i q 1 z ) , z < z 1 α A ( z ) + β B ( z ) , z 1 z z 2 t s exp ( i q 2 z ) , z > z 2 .
r s = exp ( 2 i q 1 z 1 ) q 1 q 2 ( A 1 B 2 - B 1 A 2 ) + i q 1 ( A 1 B 2 - B 1 A 2 ) + i q 2 ( A 1 B 2 - B 1 A 2 ) - ( A 1 B 2 - B 1 A 2 ) q 1 q 2 ( A 1 B 2 - B 1 A 2 ) + i q 1 ( A 1 B 2 - B 1 A 2 ) - i q 2 ( A 1 B 2 - B 1 A 2 ) + ( A 1 B 2 - B 1 A 2 ) .
r s exp ( 2 i q 1 z 2 ) q 1 - q 2 q 1 + q 2 as Δ z 0.
d d z ( 1 d B d z ) + ( ω 2 c 2 - K 2 ) B = 0.
exp ( i q 1 z ) - r p exp ( - i q 1 z ) B ( z ) ( 2 1 ) 1 / 2 t p exp ( i q 2 z ) .
cos θ 1 1 exp ( i K x ) [ exp ( i q 1 z ) + r p exp ( - i q 1 z ) ] E x cos θ 2 1 t p exp ( i K x + i q 2 z ) ,
- sin θ 1 1 exp ( i K x ) [ exp ( i q 1 z ) - r p exp ( - i q 1 z ) ] E z - sin θ 2 1 t p exp ( i K x + i q 2 z ) .
B ( z ) = { exp ( i q 1 z ) - r p exp ( - i q 1 z ) , z < z 1 γ C ( z ) + δ D ( z ) , z 1 z z 2 ( 2 1 ) 1 / 2 t p exp ( i q 2 z ) , z > z 2
r p = - exp ( 2 i q 1 z 1 ) Q 1 Q 2 ( C 1 D 2 - D 1 C 2 ) + i Q 1 ( C 1 D 2 - D 1 C 2 ) + i Q 2 ( C 1 D 2 - D 1 C 2 ) - ( C 1 D 2 - D 1 C 2 ) Q 1 Q 2 ( C 1 D 2 - D 1 C 2 ) + i Q 1 ( C 1 D 2 - D 1 C 2 ) - i Q 2 ( C 1 D 2 - D 1 C 2 ) + ( C 1 D 2 - D 1 C 2 ) ,
r p - exp ( 2 i q 1 z 1 ) Q 1 - Q 2 Q 1 + Q 2 as Δ z 0.
d 2 E d z 2 + ( ω 2 c 2 - K 2 ) E = 0
d d z ( 1 d B d z ) + ( ω 2 c 2 - K 2 ) B = 0.

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