Abstract

A scheme of combined reflection and transmission ellipsometry on light-transmitting ambient–film–substrate systems is proposed and the required sample design and instrument operation are investigated. A comparative study of the sensitivity of external and internal reflection and transmission ellipsometry is carried out based on unified linear approximations of the exact equations. These approximations are general in that an arbitrary initial film thickness is assumed. They are simple, because a complex sensitivity function is introduced whose real and imaginary projections determine the psi (ψ) and delta (Δ) sensitivity factors. Among the conclusions of this paper are the following. (1) External reflection ellipsometry near the Brewster angle of a transparent ambient–substrate system is extremely sensitive to the presence of very thin interfacial films. For example, films as thin as 10−5 Å of gold are readily detectable on glass substrates at an angle of incidence 0.3° below the Brewster angle, assuming a measuring wavelength of 5461 Å with an ellipsometer of 0.05° precision. (2) The formation of thin nonabsorbing films at the interface between transparent ambient and substrate media is not detectable, to first order, as a change in the ellipsometric angle ψ by either internal or external reflection or transmission ellipsometry. (3) The film-detection sensitivity of transmission ellipsometry increases monotonically with angle of incidence. (4) For each angle of external incidence there is a corresponding angle of internal incidence that leads to the same values of the reflection and transmission sensitivity functions. These angles are interrelated by Snell's law. (5) The ranges of validity of the linear approximation in reflection and transmission ellipsometry are comparable. The case of total internal reflection ellipsometry may lead to strong nonlinear behavior of ψ and Δ as functions of the film thickness in the range below 0.05 of the wavelength of light.

© 1975 Optical Society of America

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References

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  1. E. Passaglia, R. R. Stromberg, J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Std. Misc. Pub. 256 (U.S. Govt. Printing Office, Washington, 1964);N. M. Bashara, A. B. Buckman, A. C. Hall, Eds., Proceedings of the Symposium on Recent Developments in Ellipsometry (North-Holland, Amsterdam, 1969) [Surface Sci. 16, 1969].
  2. J. D. E. McIntyre, W. Hansen, in Advances in Electrochemistry and Electrochemical Engineering,R. H. Muller, Ed. (Wiley-Interscience, New York, 1973), Vol. 9, Chaps. 1 and 2.
  3. J. D. E. McIntyre, D. E. Aspnes, Surface Sci. 24, 417 (1971).
    [CrossRef]
  4. It may be necessary to choose a measuring wavelength such that these requirements are met over the thickness range of interest.
  5. B. D. Cahan, R. F. Spanier, Surface Sci. 16, 166 (1969).
    [CrossRef]
  6. D. E. Aspnes, Opt. Commun. 8, 222 (1973).
    [CrossRef]
  7. P. S. Hauge, F. H. Dill, IBM J Res. Devel. 17, 472 (1973).
    [CrossRef]
  8. S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surface Sci. 37, 548 (1973).
    [CrossRef]
  9. J. L. Ord, Surface Sci. 16, 155 (1969).
    [CrossRef]
  10. D. E. Confer, R. M. A. Azzam, N. M. Bashara, to be published.
  11. In differential reflection spectroscopy, we measure the fractional change of reflectance δRν/Rν, where Rν = RνRν*. We can write δRν/Rν = (δRν/Rν) + (δRν*/Rν*) = 2Re(δRν/Rν) = [2Re(Qν)] (δd1/d1), where Qν is given by Eq. (14). This result is in agreement with one obtained by McIntyre and Aspnes (Ref. 3).
  12. The zero-thickness psi and delta sensitivity factors for reflection ellipsometry were introduced by R. C. Smith and M. Hacskaylo (the first of Refs. 1, p. 83), who used the restricted Drude linear approximation.
  13. L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
    [CrossRef]
  14. C. E. Lebernight, B. Lustman, J. Opt. Soc. Am. 29, 59 (1939).
    [CrossRef]
  15. R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
    [CrossRef]
  16. D. K. Burge, H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964).
    [CrossRef]
  17. R. J. Archer, Manual on Ellipsometry (Gaertner Sci. Corp., Chicago, 1968).
  18. D. W. Peterson, N. M. Bashara, J. Opt. Soc. Am. 55, 845 (1965).
    [CrossRef]
  19. A. N. Saxena, J. Opt. Soc. Am. 55, 1061 (1965).
    [CrossRef]
  20. E. C. Rowe, I. Shewchun, J. Opt. Soc. Am. 59, 1385 (1969).
    [CrossRef]
  21. Burge and Bennett, Ref. 16, have pointed out several errors in the linear approximations used by earlier workers.
  22. Transmission ellipsometry on glass slides that are coated on both sides by anisotropic Langmuir-Blodgett layers were studied by D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
    [CrossRef]
  23. Internal reflection ellipsometry was used by E. C. Chan, J. P. Marton, J. Appl. Phy. 43, 4027 (1972), to study metal deposits on glass slides.See also M. J. Dignam, B. Rao, M. Moskovits, R. B. Stobie, Can. J. Chem. 49, 1115 (1971).
    [CrossRef]
  24. Tennyson Smith, J. Opt. Soc. Am. 58, 1069 (1968).
    [CrossRef]
  25. G. A. Bootsma, F. Meyer, Surface Sci. 14, 52 (1969).
    [CrossRef]
  26. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 61, 1236 (1971).
    [CrossRef]

1973 (3)

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

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

S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surface Sci. 37, 548 (1973).
[CrossRef]

1972 (2)

Transmission ellipsometry on glass slides that are coated on both sides by anisotropic Langmuir-Blodgett layers were studied by D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[CrossRef]

Internal reflection ellipsometry was used by E. C. Chan, J. P. Marton, J. Appl. Phy. 43, 4027 (1972), to study metal deposits on glass slides.See also M. J. Dignam, B. Rao, M. Moskovits, R. B. Stobie, Can. J. Chem. 49, 1115 (1971).
[CrossRef]

1971 (2)

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 61, 1236 (1971).
[CrossRef]

J. D. E. McIntyre, D. E. Aspnes, Surface Sci. 24, 417 (1971).
[CrossRef]

1969 (4)

B. D. Cahan, R. F. Spanier, Surface Sci. 16, 166 (1969).
[CrossRef]

J. L. Ord, Surface Sci. 16, 155 (1969).
[CrossRef]

G. A. Bootsma, F. Meyer, Surface Sci. 14, 52 (1969).
[CrossRef]

E. C. Rowe, I. Shewchun, J. Opt. Soc. Am. 59, 1385 (1969).
[CrossRef]

1968 (1)

1965 (2)

1964 (1)

1957 (1)

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
[CrossRef]

1939 (1)

1935 (1)

L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
[CrossRef]

Archer, R. J.

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
[CrossRef]

R. J. Archer, Manual on Ellipsometry (Gaertner Sci. Corp., Chicago, 1968).

Aspnes, D. E.

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

J. D. E. McIntyre, D. E. Aspnes, Surface Sci. 24, 417 (1971).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 61, 1236 (1971).
[CrossRef]

D. E. Confer, R. M. A. Azzam, N. M. Bashara, to be published.

Bashara, N. M.

Bennett, H. E.

Bootsma, G. A.

G. A. Bootsma, F. Meyer, Surface Sci. 14, 52 (1969).
[CrossRef]

Burge, D. K.

S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surface Sci. 37, 548 (1973).
[CrossRef]

D. K. Burge, H. E. Bennett, J. Opt. Soc. Am. 54, 1428 (1964).
[CrossRef]

Cahan, B. D.

B. D. Cahan, R. F. Spanier, Surface Sci. 16, 166 (1969).
[CrossRef]

Chan, E. C.

Internal reflection ellipsometry was used by E. C. Chan, J. P. Marton, J. Appl. Phy. 43, 4027 (1972), to study metal deposits on glass slides.See also M. J. Dignam, B. Rao, M. Moskovits, R. B. Stobie, Can. J. Chem. 49, 1115 (1971).
[CrossRef]

Confer, D. E.

D. E. Confer, R. M. A. Azzam, N. M. Bashara, to be published.

den Engelsen, D.

Transmission ellipsometry on glass slides that are coated on both sides by anisotropic Langmuir-Blodgett layers were studied by D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[CrossRef]

Dill, F. H.

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

Hansen, W.

J. D. E. McIntyre, W. Hansen, in Advances in Electrochemistry and Electrochemical Engineering,R. H. Muller, Ed. (Wiley-Interscience, New York, 1973), Vol. 9, Chaps. 1 and 2.

Hauge, P. S.

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

Jasperson, S. N.

S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surface Sci. 37, 548 (1973).
[CrossRef]

Lebernight, C. E.

Lustman, B.

Marton, J. P.

Internal reflection ellipsometry was used by E. C. Chan, J. P. Marton, J. Appl. Phy. 43, 4027 (1972), to study metal deposits on glass slides.See also M. J. Dignam, B. Rao, M. Moskovits, R. B. Stobie, Can. J. Chem. 49, 1115 (1971).
[CrossRef]

McIntyre, J. D. E.

J. D. E. McIntyre, D. E. Aspnes, Surface Sci. 24, 417 (1971).
[CrossRef]

J. D. E. McIntyre, W. Hansen, in Advances in Electrochemistry and Electrochemical Engineering,R. H. Muller, Ed. (Wiley-Interscience, New York, 1973), Vol. 9, Chaps. 1 and 2.

Meyer, F.

G. A. Bootsma, F. Meyer, Surface Sci. 14, 52 (1969).
[CrossRef]

O'Handley, R. C.

S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surface Sci. 37, 548 (1973).
[CrossRef]

Ord, J. L.

J. L. Ord, Surface Sci. 16, 155 (1969).
[CrossRef]

Peterson, D. W.

Rowe, E. C.

Saxena, A. N.

Shewchun, I.

Smith, Tennyson

Spanier, R. F.

B. D. Cahan, R. F. Spanier, Surface Sci. 16, 166 (1969).
[CrossRef]

Tronstad, L.

L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
[CrossRef]

IBM J Res. Devel. (1)

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

J. Appl. Phy. (1)

Internal reflection ellipsometry was used by E. C. Chan, J. P. Marton, J. Appl. Phy. 43, 4027 (1972), to study metal deposits on glass slides.See also M. J. Dignam, B. Rao, M. Moskovits, R. B. Stobie, Can. J. Chem. 49, 1115 (1971).
[CrossRef]

J. Electrochem. Soc. (1)

R. J. Archer, J. Electrochem. Soc. 104, 619 (1957).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Phys. Chem. (1)

Transmission ellipsometry on glass slides that are coated on both sides by anisotropic Langmuir-Blodgett layers were studied by D. den Engelsen, J. Phys. Chem. 76, 3390 (1972).
[CrossRef]

Opt. Commun. (1)

D. E. Aspnes, Opt. Commun. 8, 222 (1973).
[CrossRef]

Surface Sci. (5)

B. D. Cahan, R. F. Spanier, Surface Sci. 16, 166 (1969).
[CrossRef]

S. N. Jasperson, D. K. Burge, R. C. O'Handley, Surface Sci. 37, 548 (1973).
[CrossRef]

J. L. Ord, Surface Sci. 16, 155 (1969).
[CrossRef]

J. D. E. McIntyre, D. E. Aspnes, Surface Sci. 24, 417 (1971).
[CrossRef]

G. A. Bootsma, F. Meyer, Surface Sci. 14, 52 (1969).
[CrossRef]

Trans. Faraday Soc. (1)

L. Tronstad, Trans. Faraday Soc. 31, 1151 (1935).
[CrossRef]

Other (8)

Burge and Bennett, Ref. 16, have pointed out several errors in the linear approximations used by earlier workers.

It may be necessary to choose a measuring wavelength such that these requirements are met over the thickness range of interest.

E. Passaglia, R. R. Stromberg, J. Kruger, Eds., Ellipsometry in the Measurement of Surfaces and Thin Films, Natl. Bur. Std. Misc. Pub. 256 (U.S. Govt. Printing Office, Washington, 1964);N. M. Bashara, A. B. Buckman, A. C. Hall, Eds., Proceedings of the Symposium on Recent Developments in Ellipsometry (North-Holland, Amsterdam, 1969) [Surface Sci. 16, 1969].

J. D. E. McIntyre, W. Hansen, in Advances in Electrochemistry and Electrochemical Engineering,R. H. Muller, Ed. (Wiley-Interscience, New York, 1973), Vol. 9, Chaps. 1 and 2.

D. E. Confer, R. M. A. Azzam, N. M. Bashara, to be published.

In differential reflection spectroscopy, we measure the fractional change of reflectance δRν/Rν, where Rν = RνRν*. We can write δRν/Rν = (δRν/Rν) + (δRν*/Rν*) = 2Re(δRν/Rν) = [2Re(Qν)] (δd1/d1), where Qν is given by Eq. (14). This result is in agreement with one obtained by McIntyre and Aspnes (Ref. 3).

The zero-thickness psi and delta sensitivity factors for reflection ellipsometry were introduced by R. C. Smith and M. Hacskaylo (the first of Refs. 1, p. 83), who used the restricted Drude linear approximation.

R. J. Archer, Manual on Ellipsometry (Gaertner Sci. Corp., Chicago, 1968).

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Figures (13)

Fig. 1
Fig. 1

Prism substrate for fixed-angle-of-incidence operation; (a) external incidence and (b) internal incidence.

Fig. 2
Fig. 2

Semicylindrical substrate for variable-angle-of-incidence operation; (a) external incidence and (b) internal incidence.

Fig. 3
Fig. 3

Three-telescope ellipsometer for combined reflection and transmission ellipsometry. L is the light source and D.P. is the data processor. The other symbols are explained in the text.

Fig. 4
Fig. 4

Dependence of the magnitude | K ¯ R | and angle θ ¯ R of the complex sensitivity function K ¯ R on the angle of incidence ϕ for external reflection ellipsometry. In this and the subsequent figures | K ¯ | is in reciprocal angstroms and θ ¯ and ϕ are in degrees. Left: air–MgF2–glass; middle: air–Si–glass; right: air–Au–glass.

Fig. 5
Fig. 5

Angle-of-incidence dependence of the psi and delta sensitivity functions S ¯ ψ R and S ¯ Δ R for external reflection ellipsometry. In this and the subsequent figures S ¯ ψ and S ¯ Δ are in units of degrees per angstrom. Left: air–MgF2–glass; middle: air–Si–glass; right: air–Au–glass.

Fig. 6
Fig. 6

Angle-of-incidence dependence of the magnitude | K ¯ T | and angle θ ¯ T of the complex sensitivity function K ¯ T for external-incidence transmission ellipsometry. Left: air–MgF2–glass; middle: air–Si–glass; right: air–Au–glass.

Fig. 7
Fig. 7

Angle-of-incidence dependence of the psi and delta sensitivity functions S ¯ ψ T and S ¯ Δ T for external-incidence transmission ellipsometry. Left: air–MgF2–glass; middle: air–Si–glass; right: air–Au–glass.

Fig. 8
Fig. 8

Angle-of-incidence dependence of the magnitude | K ¯ R | and angle θ ¯ R of the complex sensitivity function K ¯ R and of the psi and delta sensitivity functions S ¯ ψ R and S ¯ Δ R for internal reflection ellipsometry. Glass–Si–air.

Fig. 9
Fig. 9

Angle-of-incidence dependence of the magnitude | K ¯ T | and angle θ ¯ T of the complex sensitivity function K ¯ T and of the psi and delta sensitivity functions S ¯ ψ T and S ¯ Δ T for internal-incidence transmission ellipsometry. Glass–Au–air.

Fig. 10
Fig. 10

Dependence of the ellipsometric angles psi (ψ) and delta (Δ) on the film thickness D (= d1 for the air–MgF2–glass system. In this and the subsequent figures ψ and Δ are in degrees. D is in angstroms. Left: external-reflection ellipsometry; right: external-incidence transmission ellipsometry.

Fig. 11
Fig. 11

Thickness dependence of psi and delta for the air–Si–glass system. Left: external-reflection ellipsometry; right: external-incidence transmission ellipsometry.

Fig. 12
Fig. 12

Thickness dependence of psi and delta for the air–Au–glass system. Left: external-reflection ellipsometry; right: external-incidence transmission ellipsometry.

Fig. 13
Fig. 13

Thickness dependence of the psi and delta for the glass–Au–air system (left) and the glass–Si–air system (right) in total-internal-reflection ellipsometry.

Tables (2)

Tables Icon

Table I External-Incidence Sensitivity Factorsa

Tables Icon

Table II Transformation Relationships Between External and Internal Incidence

Equations (74)

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ρ = R p / R s
R ν = [ r 01 ν + r 12 ν exp ( j 2 β ) ] / [ 1 + r 01 ν r 12 ν exp ( j 2 β ) ] ,
β = 2 π ( d 1 / λ ) ( N 1 2 N 0 2 sin 2 ϕ 0 ) 1 / 2
δ ρ / ρ = ( δ R p / R p ) ( δ R s / R s ) ,
R ν = N ν / D ν ,
N ν = r 01 ν + r 12 ν ξ , D ν = 1 + r 01 ν r 12 ν ξ ,
ξ = exp ( j 2 β ) = exp ( γ d 1 ) ,
γ = j ( 4 π / λ ) ( N 1 2 N 0 2 sin 2 ϕ 0 ) 1 / 2 .
δ R ν / R ν = ( δ N ν / N ν ) ( δ D ν / D ν ) .
δ R ν / R ν = [ ( 1 / N ν ) ( d N ν / d ξ ) ( 1 / D ν ) ( d D ν / d ξ ) ] δ ξ .
δ ξ = γ ξ δ d 1 .
d N ν / d ξ = r 12 ν , d D ν / d ξ = r 01 ν r 12 ν ,
δ R ν / R ν = γ d 1 ξ { [ r 12 ν / ( r 01 ν + r 12 ν ξ ) ] [ r 01 ν r 12 ν / ( 1 + r 01 ν r 12 ν ξ ) ] } ( δ d 1 / d 1 ) = Q ν ( δ d 1 / d 1 ) ,
Q ν = γ d 1 ξ { [ r 12 ν / ( r 01 ν + r 12 ν ξ ) ] [ r 01 ν r 12 ν / ( 1 + r 01 ν r 12 ν ξ ) ] } .
δ ρ / ρ = K ( δ d 1 / d 1 ) ,
K = Q p Q s , = γ d 1 ξ { [ r 12 p / ( r 01 p + r 12 p ξ ) ] [ r 01 p r 12 p / ( 1 r 01 p r 12 p ξ ) ] [ r 12 s / ( r 01 s + r 12 s ξ ) ] + [ r 01 s r 12 s / ( 1 + r 01 s r 12 s ξ ) ] } .
δ R ν / R ν = γ { [ r 12 ν / ( r 01 ν + r 12 ν ) ] [ r 01 ν r 12 ν / ( 1 + r 01 ν r 12 ν ) ] } δ d 1 = Q ¯ ν δ d 1 ,
Q ¯ ν = γ { [ r 12 ν / ( r 01 ν + r 12 ν ) ] [ r 01 ν r 12 ν / ( 1 + r 01 ν r 12 ν ) ] } ,
δ ρ / ρ = K ¯ δ d 1 ,
K ¯ = Q ¯ p Q ¯ s , = γ { [ r 12 p / ( r 01 p + r 12 p ) ] [ r 01 p r 12 p / ( 1 + r 01 p r 12 p ) ] [ r 12 s / ( r 01 s + r 12 s ) ] + [ r 01 s r 12 s / ( 1 + r 01 s r 12 s ) ] } .
ρ = T p / T s ,
T ν = t 01 ν t 12 ν exp ( j β ) / [ 1 + r 01 ν r 12 ν exp ( j 2 β ) ] ,
ρ = ( t 01 p / t 01 s ) ( t 12 p / t 12 s ) { [ 1 + r 01 s r 12 s exp ( j 2 β ) ] / [ 1 + r 01 p r 12 p exp ( j 2 β ) ] } = M ( D s / D p ) ,
M = ( t 01 p / t 01 s ) ( t 12 p / t 12 s ) ,
δ ρ / ρ = ( δ M / M ) + ( δ D s / D s ) ( δ D p / D p ) .
δ ρ / ρ = ( δ D s / D s ) ( δ D p / D p ) = [ ( 1 / D s ) ( d D s / d ξ ) ( 1 / D p ) ( d D p / d ξ ) ] δ ξ .
δ ρ / ρ = γ d 1 ξ { [ r 01 p r 12 p / ( 1 + r 01 p r 12 p ξ ) ] [ r 01 s r 12 s / ( 1 + r 01 s r 12 s ξ ) ] } ( δ d 1 d 1 ) , = K ( δ d 1 / d 1 ) ,
K = γ d 1 ξ { [ r 01 p r 12 p / ( 1 + r 01 p r 12 p ξ ) ] [ r 01 s r 12 s / ( 1 + r 01 s r 12 s ξ ) ] } .
δ ρ / ρ = γ { [ r 01 p r 12 p / ( 1 + r 01 p r 12 p ) ] [ r 01 s r 12 s / ( 1 + r 01 s r 12 s ) ] } δ d 1 , = K ¯ δ d 1 ,
K ¯ = γ { [ r 01 p r 12 p / ( 1 + r 01 p r 12 p ) ] [ r 01 s r 12 s / ( 1 + r 01 s r 12 s ) ] } ,
ρ = tan ψ exp ( j Δ ) ,
δ ρ / ρ = ( 2 δ ψ sin 2 ψ ) + j δ Δ ,
δ ψ = ( ½ sin 2 ψ ) Re ( δ ρ / ρ ) ,
δ Δ = Im ( δ ρ / ρ ) .
δ ψ = [ ( ½ sin 2 ψ ) Re ( K ) ] ( δ d 1 / d 1 ) ,
δ Δ = [ Im ( K ) ] ( δ d 1 / d 1 ) ,
δ ψ = [ ( ½ sin 2 ψ ¯ ) Re ( K ¯ ) ] δ d 1 ,
δ Δ = [ Im ( K ¯ ) ] δ d 1 ,
ψ = ψ 0 + S ψ ( δ d 1 / d 1 ) ,
Δ = Δ 0 + S Δ ( δ d 1 / d 1 ) ,
ψ = ψ ¯ + S ¯ ψ δ d 1 ,
Δ = Δ ¯ + S ¯ Δ δ d 1 ,
S ψ = ( ½ sin 2 ψ 0 ) Re ( K ) ,
S Δ = Im ( K ) ,
S ¯ ψ = ( ½ sin 2 ψ ¯ ) Re ( K ¯ ) ,
S ¯ Δ = Im ( K ¯ ) .
K = | K | exp ( j θ ) , K ¯ = | K ¯ | exp ( j θ ¯ ) ,
S ψ = ( 180 / π ) ( ½ sin 2 ψ 0 ) | K | cos θ ,
S Δ = ( 180 / π ) | K | sin θ ,
S ¯ ψ = ( 180 / π ) ( ½ sin 2 ψ ¯ ) | K ¯ | cos θ ¯ ,
S ¯ Δ = ( 180 / π ) | K ¯ | sin θ ¯ ,
ϕ B 02 ( air glass ) = tan 1 ( N glass / N air ) = tan 1 1.5 = 56.31 ° .
R ν = r 02 ν = ( r 01 ν + r 12 ν ) / ( 1 + r 01 ν r 12 ν ) , ( ν = p , s ) ,
r 01 p + r 12 p = 0 at ϕ = ϕ B 02 .
S ¯ ψ R = F 1 F 2 F 3 | K ¯ R | .
S ¯ ψ R = 0 at 0 ϕ 90 ° ,
S ¯ ψ R = 0 at ϕ = ϕ B 02
S ¯ ψ T = 0 at 0 ϕ 90 ° ,
r = r , t = ( 1 r 2 ) / t .
N 0 sin ϕ 0 = N 2 sin ϕ 2 , N 2 > N 1 .
ϕ C 20 = sin 1 ( N 0 / N 2 ) , = sin 1 ( 0.6667 ) = 41.82 ° ,
ϕ B 20 = tan 1 ( N 0 / N 2 ) = 90 ° tan 1 ( N 2 / N 0 ) = 90 ° ϕ B 02 = 33.69 ° ,
| K ¯ R | ( rad / Å )
| K ¯ T | ( rad / Å )
θ ¯ R ( deg )
θ ¯ T ( deg )
S ¯ Δ R ( deg / Å )
S ¯ Δ T ( deg / Å )
S ¯ ψ R ( deg / Å )
S ¯ ψ T ( deg / Å )
K ¯ R
K ¯ R
K ¯ T
K ¯ T

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