Abstract

Exact expressions are obtained in closed form for the phase and amplitude of the reflected light when a light beam is incident upon a multilayer of a number of homogeneous media. The assumption is made that Maxwell’s equations and Ohm’s law hold. These equations are applied to the special case of dielectric coated metals and the results are compared to ellipsometry measurements made in chromium slides covered with barium stearate films of known thickness. When the conductivity and the dielectric constant of the metal are allowed to assume complex values, and when precautions are taken to reduce the inaccuracy introduced by the residual oxide and gas layer on the metal, the agreement is good.

© 1962 Optical Society of America

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References

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  1. P. Drude, Ann. Physik 36, 865 (1889).
    [Crossref]
  2. L. Tronstad, Trans. Faraday Soc. 29, 502 (1933).
    [Crossref]
  3. A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
    [Crossref]
  4. J. B. Bateman and N. W. Harris, Ann. N. Y. Acad. Sci. 53, 1064 (1951).
    [Crossref]
  5. W. Weinstein, Vacuum 41, 3 (1954).
    [Crossref]
  6. J. A. Berning and P. H. Berning, J. Opt. Soc. Am. 50, 813 (1960).
    [Crossref]
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 490–516.
  8. J. R. Partington, An Advanced Treatise on Physical Chemistry (Longmans Green and Company, New York, 1953), Vol. IV, p. 156.
  9. J. A. Faucher, G. M. McManus, and H. T. Trurnit, J. Opt. Soc. Am. 48, 51 (1958).
    [Crossref] [PubMed]
  10. P. Drude, Ann. Physik 35, 508 (1888).
    [Crossref]
  11. H. J. Trurnit (to be published).
  12. A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
    [Crossref]
  13. K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
    [Crossref]

1960 (1)

1958 (1)

1954 (1)

W. Weinstein, Vacuum 41, 3 (1954).
[Crossref]

1951 (1)

J. B. Bateman and N. W. Harris, Ann. N. Y. Acad. Sci. 53, 1064 (1951).
[Crossref]

1949 (1)

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

1945 (1)

A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
[Crossref]

1935 (1)

K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
[Crossref]

1933 (1)

L. Tronstad, Trans. Faraday Soc. 29, 502 (1933).
[Crossref]

1889 (1)

P. Drude, Ann. Physik 36, 865 (1889).
[Crossref]

1888 (1)

P. Drude, Ann. Physik 35, 508 (1888).
[Crossref]

Bateman, J. B.

J. B. Bateman and N. W. Harris, Ann. N. Y. Acad. Sci. 53, 1064 (1951).
[Crossref]

Berning, J. A.

Berning, P. H.

Blodgett, K. B.

K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
[Crossref]

Drude, P.

P. Drude, Ann. Physik 36, 865 (1889).
[Crossref]

P. Drude, Ann. Physik 35, 508 (1888).
[Crossref]

Faucher, J. A.

Hanson, M.

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

Harris, N. W.

J. B. Bateman and N. W. Harris, Ann. N. Y. Acad. Sci. 53, 1064 (1951).
[Crossref]

McManus, G. M.

Partington, J. R.

J. R. Partington, An Advanced Treatise on Physical Chemistry (Longmans Green and Company, New York, 1953), Vol. IV, p. 156.

Rothen, A.

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
[Crossref]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 490–516.

Tronstad, L.

L. Tronstad, Trans. Faraday Soc. 29, 502 (1933).
[Crossref]

Trurnit, H. J.

H. J. Trurnit (to be published).

Trurnit, H. T.

Weinstein, W.

W. Weinstein, Vacuum 41, 3 (1954).
[Crossref]

Ann. N. Y. Acad. Sci. (1)

J. B. Bateman and N. W. Harris, Ann. N. Y. Acad. Sci. 53, 1064 (1951).
[Crossref]

Ann. Physik (2)

P. Drude, Ann. Physik 36, 865 (1889).
[Crossref]

P. Drude, Ann. Physik 35, 508 (1888).
[Crossref]

J. Am. Chem. Soc. (1)

K. B. Blodgett, J. Am. Chem. Soc. 57, 1007 (1935).
[Crossref]

J. Opt. Soc. Am. (2)

Rev. Sci. Instr. (2)

A. Rothen and M. Hanson, Rev. Sci. Instr. 20, 66 (1949).
[Crossref]

A. Rothen, Rev. Sci. Instr. 16, 26 (1945).
[Crossref]

Trans. Faraday Soc. (1)

L. Tronstad, Trans. Faraday Soc. 29, 502 (1933).
[Crossref]

Vacuum (1)

W. Weinstein, Vacuum 41, 3 (1954).
[Crossref]

Other (3)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill Book Company, Inc., New York, 1941), pp. 490–516.

J. R. Partington, An Advanced Treatise on Physical Chemistry (Longmans Green and Company, New York, 1953), Vol. IV, p. 156.

H. J. Trurnit (to be published).

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Figures (9)

Fig. 1
Fig. 1

A light wave incident upon a multilayered slab.

Fig. 2
Fig. 2

The general case of elliptically polarized light.

Fig. 3
Fig. 3

Phase difference versus conductivity for the n = 2 case, λ = 5460 Å, θ = 64°.

Fig. 4
Fig. 4

ρ versus conductivity for different permittivities in the n = 2 case. λ = 5460 Å, θ = 64°.

Fig. 5
Fig. 5

ρ versus the thickness of the center layer for different 2. λ = 5460 Å, θ = 64°.

Fig. 6
Fig. 6

Phase difference versus the thickness of the center layer for different permittivities of the center layer in the n = 3 case, λ = 5460 Å, θ = 64°.

Fig. 7
Fig. 7

Phase difference versus the thickness of the center layer for several incident angles. n = 3, λ = 5460 Å.

Fig. 8
Fig. 8

Amplitude reflectances coefficients for the s and p waves versus the thickness of the center layer for several incident angles. n = 3, λ = 5460 Å.

Fig. 9
Fig. 9

Amplitude reflectances coefficient for the s wave versus the thickness of the second layer for several incident angles. n = 3, λ = 5460 Å.

Tables (4)

Tables Icon

Table I Results of thickness calculation for sample A.

Tables Icon

Table II Results of thickness calculation for sample B.

Tables Icon

Table III Results of thickness calculation for sample A.

Tables Icon

Table IV Results of thickness calculation for sample B.

Equations (38)

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D m = m E m ,
B m = μ m m ,
J m = σ m E m ,
E m = E m exp [ i ( k m · r w t ) ] , B m = B m exp [ i ( k m · r w t ) ] ,
k m · E m = k m · B m = 0 ,
B m = ( c / w ) k m × E m ,
k m 2 k m · k m = ( w 2 / c 2 ) μ m ( m + i 4 π σ m / w ) ,
E n = B n = 0
θ m = θ m ,
k m sin θ m = k m + 1 sin θ m + 1 .
( k m ) x = k m sin θ m , ( k m ) x = k m sin θ m , ( k m ) y = 0 , ( k m ) y = 0 , ( k m ) z = k m cos θ m , ( k m ) z = k m cos θ m .
E m + E m
B m + B m ,
E m = E m { exp [ i ( k m x sin θ m w t ) ] } × { exp ( i k m z cos θ m ) } , E m = E m { exp [ i ( k m x sin θ m w t ) ] } × { exp ( i k m z cos θ m ) } , B m = B m { exp [ i ( k m x sin θ m w t ) ] } × { exp ( i k m z cos θ m ) } , B m = B m { exp [ i ( k m x sin θ m w t ) ] } × { exp ( i k m z cos θ m ) } .
[ E m E m ] or [ B m B m ] .
[ E s E s m + 1 ] = 1 1 q s [ 1 q s q s 1 ] [ E s E s m ] ,
[ B p m + 1 B p m + 1 ] = 1 1 q p [ 1 q p q p 1 ] [ B p m B p m ] .
q = ( a m a m + 1 ) / ( a m + a m + 1 ) ,
a s m = k m cos θ m / μ m
a p m = μ m cos θ m / k m .
[ E m , m + 1 E m , m + 1 ] = [ exp ( i k m L m cos θ m ) 0 0 exp ( i k m L m cos θ m ) ] [ E m 1 , m E m 1 , m ] ,
[ E s m 0 ] = [ I s n 1 , n ] [ M n 1 ] [ I s n 2 , n 1 ] × [ M 2 ] [ I s 1 , 2 ] [ E s 1 E s 1 ]
[ B p n 0 ] = [ I p ] [ M n 1 ] [ I p n 2 , n 1 ] × [ M 2 ] [ I p 1 , 2 ] [ B p 1 B p 1 ] ,
[ E s n 0 ] = [ P 11 s P 12 s P 21 s P 22 s ] [ E s 1 E s 1 ]
[ B p n 0 ] = [ P 11 p P 12 p P 21 p P 22 p ] [ B p 1 B p 1 ] ,
r s e i δ s E s 1 / E s 1 = P 21 s / P 22 s ,
r p e i δ p B p 1 / B p 1 = P 21 p / P 22 p ,
s = sin w t , p = ρ sin ( w t Δ ) ,
ρ e i Δ = r p exp ( i δ p ) / r s exp ( i δ s ) .
tan 2 Ω = tan 2 ψ cos Δ ,
tan ψ = ρ .
cos 2 ϕ = cos 2 ψ / cos 2 Ω .
cos 2 ψ = cos 2 ϕ cos 2 Ω , cos Δ = tan 2 Ω / tan 2 ψ .
r e i δ = q 12 + q 23 exp ( 2 i k 2 L cos θ 2 ) 1 + q 12 q 23 exp ( 2 i k 2 L cos θ 2 ) .
q 23 = r M e i δ M ,
r e i δ = q 12 + r M exp [ i ( 2 k 2 L cos θ 2 + δ M ) ] 1 + q 12 r M exp [ i ( 2 k 2 L cos θ 2 + δ M ) ] .
r 2 = q 12 2 + r M 2 + 2 q 12 r M cos ( 2 k 2 L cos θ 2 + δ M ) 1 + q 12 2 r M 2 + 2 q 12 r M cos ( 2 k 2 L cos θ 2 + δ M ) ,
δ = arc tan [ r M ( 1 q 12 2 ) sin ( 2 k 2 L cos θ 2 + δ M ) q 12 ( 1 + r M ) + r M ( 1 + q 12 2 ) cos ( 2 k 2 L cos θ 2 + δ M ) ] .