Abstract

A subtractive Kramers-Kronig method is presented for obtaining the optical constants, n and k, of a uniform absorbing film on a substrate using the transmittance spectrum of a single film thickness. We give the results of tests on the reliability of our method, demonstrate the usefulness of the method when only transmittance data are available, and give examples of thin films whose optical constants depend on their thickness.

© 1985 Optical Society of America

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References

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  1. K. F. Palmer, J. A. Roux, B. E. Wood, “The Infrared Optical Properties of Mixtures of Molecular Species at 20 K,” AEDC-TR-80-30 (AD-A094214) (Jan.1981); also AIAA paper 83-1452.
  2. O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965).
  3. R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous Determination of Optical Constants of Absorbing Films by Reflectance and Transmittance Measurements,” Appl. Opt. 23, 1197 (1984).
    [CrossRef] [PubMed]
  4. S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute Infrared Intensity Measurements in Thin films: II. Solids Deposited on Halide Plates,” J. Chem. Phys. 38, 3474 (1963).
    [CrossRef]
  5. P.-O. Nilsson, “Determination of Optical Constants from Intensity Measurements at Normal Incidence,” Appl. Opt. 7, 435 (1968).
    [CrossRef] [PubMed]
  6. L. Harris, J. K. Beasley, A L. Loeb, “Reflection and Transmission of Radiation by Metal Films and the Influence of Non-absorbing Backings,” J. Opt. Soc. Am. 41, 604 (1951).
    [CrossRef]
  7. J. U. White, W. M. Ward, “Effects of Interference Fringes in Infrared Absorption Cells,” Anal. Chem. 37, (1965).
    [CrossRef] [PubMed]
  8. D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Indust. Appl. Math. 11, 431 (1963).
    [CrossRef]
  9. C. W. Peterson, B. W. Knight, “Causality Calculations in the Time Domain: An Efficient Alternative to the Kramers-Kronig Method,” J. Opt. Soc. Am. 63, 1238 (1973).
    [CrossRef]
  10. D. W. Johnson, “A Fourier Series Method for Numerical Kramers-Kronig Analysis,” J. Phys. A 8, 490 (1975).
    [CrossRef]
  11. F. W. King, “A Fourier Series Algorithm for the Analysis of Reflectance Data,” J. Phys. C 10, 3199 (1977).
    [CrossRef]
  12. R. K. Ahrenkiel, “Modified Kramers-Kronig Analysis of Optical Spectra,” J. Opt. Soc. Am. 61, 1651 (1971).
    [CrossRef]
  13. K. E. Peiponen, “On the Properties of the Complex Refractive Index of Lorentzian Type,” Phys. Scr. 21, 181 (1980).
    [CrossRef]
  14. R. W. Ditchburn, Light (Academic, New York, 1976), Vols. 1 and 2.
  15. J. A. Roux, B. F. Wood, A. M. Smith, “IR Optical Properties of Thin H2O, NH3, CO2 Cryofilms,” AEDC-TR-79-57 (AD-A074913) (Sept.1979).
  16. K. E. Tempelmeyer, D. W. Mills, “Refractive Index of Carbon Dioxide Cryodeposit,” J. Appl. Phys. 39, 2968 (1968).
    [CrossRef]
  17. In cases where some observed transmittance values are negative, we can establish a new base line for the transmittance spectrum by adding a very small constant to every original transmittance value so that the new transmittance spectrum will not have any negative (or zero) values. It is not clear, however, what the additive constant should be. The establishment of the criteria for choosing the constant is very important because relatively small adjustments of the additive constant can cause variances in the smallest transmittances of several hundred percent, or more, and can create very noticeable differences in the n spectra computed by skktrans, as we observed in the case of the thicker N2/CO2 films.
  18. B. E. Wood, Arvin/Calspan Field Services, Inc.; private communication.
  19. K. F. Palmer, M. Z. Williams, “Optical Constant Determination of Thin Films Condensed on Transmitting and Reflecting Surfaces,” AEDC-TR-83-64.
  20. E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relations in Determining the Phase Shift Occurring upon Reflection of Light from Thin Dielectric Layers,” Opt. Spektrosk. 24, 257 (1968)[Opt. Spectrosc. 24, 32 (1968)].
  21. E. A. Lupashko, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relationships to Calculate the Phase of the Wave Reflected from Thin Dielectric Layers,” Opt. Spektrosk. 29, 789 (1970)[Opt. Spectrosc. 29, 419 (1970)].
  22. F. Wooten, Optical Properties of Solids (Academic, New York, 1972).
  23. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon,New York, 1960).

1984 (1)

1981 (1)

K. F. Palmer, J. A. Roux, B. E. Wood, “The Infrared Optical Properties of Mixtures of Molecular Species at 20 K,” AEDC-TR-80-30 (AD-A094214) (Jan.1981); also AIAA paper 83-1452.

1980 (1)

K. E. Peiponen, “On the Properties of the Complex Refractive Index of Lorentzian Type,” Phys. Scr. 21, 181 (1980).
[CrossRef]

1977 (1)

F. W. King, “A Fourier Series Algorithm for the Analysis of Reflectance Data,” J. Phys. C 10, 3199 (1977).
[CrossRef]

1975 (1)

D. W. Johnson, “A Fourier Series Method for Numerical Kramers-Kronig Analysis,” J. Phys. A 8, 490 (1975).
[CrossRef]

1973 (1)

1971 (1)

1970 (1)

E. A. Lupashko, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relationships to Calculate the Phase of the Wave Reflected from Thin Dielectric Layers,” Opt. Spektrosk. 29, 789 (1970)[Opt. Spectrosc. 29, 419 (1970)].

1968 (3)

K. E. Tempelmeyer, D. W. Mills, “Refractive Index of Carbon Dioxide Cryodeposit,” J. Appl. Phys. 39, 2968 (1968).
[CrossRef]

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relations in Determining the Phase Shift Occurring upon Reflection of Light from Thin Dielectric Layers,” Opt. Spektrosk. 24, 257 (1968)[Opt. Spectrosc. 24, 32 (1968)].

P.-O. Nilsson, “Determination of Optical Constants from Intensity Measurements at Normal Incidence,” Appl. Opt. 7, 435 (1968).
[CrossRef] [PubMed]

1965 (1)

J. U. White, W. M. Ward, “Effects of Interference Fringes in Infrared Absorption Cells,” Anal. Chem. 37, (1965).
[CrossRef] [PubMed]

1963 (2)

D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Indust. Appl. Math. 11, 431 (1963).
[CrossRef]

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute Infrared Intensity Measurements in Thin films: II. Solids Deposited on Halide Plates,” J. Chem. Phys. 38, 3474 (1963).
[CrossRef]

1951 (1)

Ahrenkiel, R. K.

Beasley, J. K.

Botten, L. C.

Ditchburn, R. W.

R. W. Ditchburn, Light (Academic, New York, 1976), Vols. 1 and 2.

Harris, L.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965).

Johnson, D. W.

D. W. Johnson, “A Fourier Series Method for Numerical Kramers-Kronig Analysis,” J. Phys. A 8, 490 (1975).
[CrossRef]

King, F. W.

F. W. King, “A Fourier Series Algorithm for the Analysis of Reflectance Data,” J. Phys. C 10, 3199 (1977).
[CrossRef]

Knight, B. W.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon,New York, 1960).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon,New York, 1960).

Loeb, A L.

Lupashko, E. A.

E. A. Lupashko, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relationships to Calculate the Phase of the Wave Reflected from Thin Dielectric Layers,” Opt. Spektrosk. 29, 789 (1970)[Opt. Spectrosc. 29, 419 (1970)].

Lupaskho, E. A.

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relations in Determining the Phase Shift Occurring upon Reflection of Light from Thin Dielectric Layers,” Opt. Spektrosk. 24, 257 (1968)[Opt. Spectrosc. 24, 32 (1968)].

Maeda, S.

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute Infrared Intensity Measurements in Thin films: II. Solids Deposited on Halide Plates,” J. Chem. Phys. 38, 3474 (1963).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Indust. Appl. Math. 11, 431 (1963).
[CrossRef]

McKenzie, D. R.

McPhedran, R. C.

Mills, D. W.

K. E. Tempelmeyer, D. W. Mills, “Refractive Index of Carbon Dioxide Cryodeposit,” J. Appl. Phys. 39, 2968 (1968).
[CrossRef]

Miloslavskii, V. K.

E. A. Lupashko, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relationships to Calculate the Phase of the Wave Reflected from Thin Dielectric Layers,” Opt. Spektrosk. 29, 789 (1970)[Opt. Spectrosc. 29, 419 (1970)].

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relations in Determining the Phase Shift Occurring upon Reflection of Light from Thin Dielectric Layers,” Opt. Spektrosk. 24, 257 (1968)[Opt. Spectrosc. 24, 32 (1968)].

Netterfield, R. P.

Nilsson, P.-O.

Palmer, K. F.

K. F. Palmer, J. A. Roux, B. E. Wood, “The Infrared Optical Properties of Mixtures of Molecular Species at 20 K,” AEDC-TR-80-30 (AD-A094214) (Jan.1981); also AIAA paper 83-1452.

K. F. Palmer, M. Z. Williams, “Optical Constant Determination of Thin Films Condensed on Transmitting and Reflecting Surfaces,” AEDC-TR-83-64.

Peiponen, K. E.

K. E. Peiponen, “On the Properties of the Complex Refractive Index of Lorentzian Type,” Phys. Scr. 21, 181 (1980).
[CrossRef]

Peterson, C. W.

Roux, J. A.

K. F. Palmer, J. A. Roux, B. E. Wood, “The Infrared Optical Properties of Mixtures of Molecular Species at 20 K,” AEDC-TR-80-30 (AD-A094214) (Jan.1981); also AIAA paper 83-1452.

J. A. Roux, B. F. Wood, A. M. Smith, “IR Optical Properties of Thin H2O, NH3, CO2 Cryofilms,” AEDC-TR-79-57 (AD-A074913) (Sept.1979).

Schatz, P. N.

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute Infrared Intensity Measurements in Thin films: II. Solids Deposited on Halide Plates,” J. Chem. Phys. 38, 3474 (1963).
[CrossRef]

Shklyarevskii, I. N.

E. A. Lupashko, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relationships to Calculate the Phase of the Wave Reflected from Thin Dielectric Layers,” Opt. Spektrosk. 29, 789 (1970)[Opt. Spectrosc. 29, 419 (1970)].

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relations in Determining the Phase Shift Occurring upon Reflection of Light from Thin Dielectric Layers,” Opt. Spektrosk. 24, 257 (1968)[Opt. Spectrosc. 24, 32 (1968)].

Smith, A. M.

J. A. Roux, B. F. Wood, A. M. Smith, “IR Optical Properties of Thin H2O, NH3, CO2 Cryofilms,” AEDC-TR-79-57 (AD-A074913) (Sept.1979).

Tempelmeyer, K. E.

K. E. Tempelmeyer, D. W. Mills, “Refractive Index of Carbon Dioxide Cryodeposit,” J. Appl. Phys. 39, 2968 (1968).
[CrossRef]

Thyagarajan, G.

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute Infrared Intensity Measurements in Thin films: II. Solids Deposited on Halide Plates,” J. Chem. Phys. 38, 3474 (1963).
[CrossRef]

Ward, W. M.

J. U. White, W. M. Ward, “Effects of Interference Fringes in Infrared Absorption Cells,” Anal. Chem. 37, (1965).
[CrossRef] [PubMed]

White, J. U.

J. U. White, W. M. Ward, “Effects of Interference Fringes in Infrared Absorption Cells,” Anal. Chem. 37, (1965).
[CrossRef] [PubMed]

Williams, M. Z.

K. F. Palmer, M. Z. Williams, “Optical Constant Determination of Thin Films Condensed on Transmitting and Reflecting Surfaces,” AEDC-TR-83-64.

Wood, B. E.

K. F. Palmer, J. A. Roux, B. E. Wood, “The Infrared Optical Properties of Mixtures of Molecular Species at 20 K,” AEDC-TR-80-30 (AD-A094214) (Jan.1981); also AIAA paper 83-1452.

B. E. Wood, Arvin/Calspan Field Services, Inc.; private communication.

Wood, B. F.

J. A. Roux, B. F. Wood, A. M. Smith, “IR Optical Properties of Thin H2O, NH3, CO2 Cryofilms,” AEDC-TR-79-57 (AD-A074913) (Sept.1979).

Wooten, F.

F. Wooten, Optical Properties of Solids (Academic, New York, 1972).

AEDC-TR-80-30 (AD-A094214) (1)

K. F. Palmer, J. A. Roux, B. E. Wood, “The Infrared Optical Properties of Mixtures of Molecular Species at 20 K,” AEDC-TR-80-30 (AD-A094214) (Jan.1981); also AIAA paper 83-1452.

Anal. Chem. (1)

J. U. White, W. M. Ward, “Effects of Interference Fringes in Infrared Absorption Cells,” Anal. Chem. 37, (1965).
[CrossRef] [PubMed]

Appl. Opt. (2)

J. Appl. Phys. (1)

K. E. Tempelmeyer, D. W. Mills, “Refractive Index of Carbon Dioxide Cryodeposit,” J. Appl. Phys. 39, 2968 (1968).
[CrossRef]

J. Chem. Phys. (1)

S. Maeda, G. Thyagarajan, P. N. Schatz, “Absolute Infrared Intensity Measurements in Thin films: II. Solids Deposited on Halide Plates,” J. Chem. Phys. 38, 3474 (1963).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. A (1)

D. W. Johnson, “A Fourier Series Method for Numerical Kramers-Kronig Analysis,” J. Phys. A 8, 490 (1975).
[CrossRef]

J. Phys. C (1)

F. W. King, “A Fourier Series Algorithm for the Analysis of Reflectance Data,” J. Phys. C 10, 3199 (1977).
[CrossRef]

J. Soc. Indust. Appl. Math. (1)

D. W. Marquardt, “An Algorithm for Least-Squares Estimation of Nonlinear Parameters,” J. Soc. Indust. Appl. Math. 11, 431 (1963).
[CrossRef]

Opt. Spektrosk. (2)

E. A. Lupaskho, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relations in Determining the Phase Shift Occurring upon Reflection of Light from Thin Dielectric Layers,” Opt. Spektrosk. 24, 257 (1968)[Opt. Spectrosc. 24, 32 (1968)].

E. A. Lupashko, V. K. Miloslavskii, I. N. Shklyarevskii, “Use of the Kramers-Kronig Dispersion Relationships to Calculate the Phase of the Wave Reflected from Thin Dielectric Layers,” Opt. Spektrosk. 29, 789 (1970)[Opt. Spectrosc. 29, 419 (1970)].

Phys. Scr. (1)

K. E. Peiponen, “On the Properties of the Complex Refractive Index of Lorentzian Type,” Phys. Scr. 21, 181 (1980).
[CrossRef]

Other (8)

R. W. Ditchburn, Light (Academic, New York, 1976), Vols. 1 and 2.

J. A. Roux, B. F. Wood, A. M. Smith, “IR Optical Properties of Thin H2O, NH3, CO2 Cryofilms,” AEDC-TR-79-57 (AD-A074913) (Sept.1979).

In cases where some observed transmittance values are negative, we can establish a new base line for the transmittance spectrum by adding a very small constant to every original transmittance value so that the new transmittance spectrum will not have any negative (or zero) values. It is not clear, however, what the additive constant should be. The establishment of the criteria for choosing the constant is very important because relatively small adjustments of the additive constant can cause variances in the smallest transmittances of several hundred percent, or more, and can create very noticeable differences in the n spectra computed by skktrans, as we observed in the case of the thicker N2/CO2 films.

B. E. Wood, Arvin/Calspan Field Services, Inc.; private communication.

K. F. Palmer, M. Z. Williams, “Optical Constant Determination of Thin Films Condensed on Transmitting and Reflecting Surfaces,” AEDC-TR-83-64.

O. S. Heavens, Optical Properties of Thin Films (Dover, New York, 1965).

F. Wooten, Optical Properties of Solids (Academic, New York, 1972).

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon,New York, 1960).

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Figures (7)

Fig. 1
Fig. 1

Geometry depicting analytical model for a thin film formed on a thick substrate.

Fig. 2
Fig. 2

Refractive index of solid CO2 on 20 K germanium.

Fig. 3
Fig. 3

Absorption index of solid CO2 on 20 K germanium.

Fig. 4
Fig. 4

Refractive index of solid N2/CO2 mixture (74.7%/25.3%) on 20 K germanium computed at several thicknesses.

Fig. 5
Fig. 5

Absorption index of solid N2/CO2 mixture (74.7%/25.3%) on 20 K germanium computed at several thicknesses.

Fig. 6
Fig. 6

Absorption index of solid N2/NH3 mixture (85%/15%) on 20 K germanium computed at several thicknesses.

Fig. 7
Fig. 7

Plot of phase shift ϕ = ϕ0 + 2m at ν0 = 2144 cm−1 of N2/CO2 (74.7%/25.3%) films on 20 K germanium.

Equations (48)

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T ( ν ) = [ n 3 ( ν ) / n 0 ( ν ) ] t ̂ * ( ν ) t ̂ ( ν ) ,
t ̂ = ( n 2 / n 3 ) 1 / 2 T 23 1 / 2 t ̂ 012 exp ( β 2 d 2 ) / [ 1 R 210 R 23 exp ( 4 β 2 d 2 ) ] 1 / 2 .
T j m = ( n m / n j ) t ̂ j m * t ̂ j m ,
t ̂ j m = 2 n ̂ j / ( n ̂ j + n ̂ m ) ,
R j m = r ̂ j m * r ̂ j m ,
r ̂ j m = ( n ̂ j n ̂ m ) / ( n ̂ j + n ̂ m ) .
t ̂ 012 = t ̂ 01 t ̂ 12 exp ( i γ ̂ 1 d 1 ) / [ 1 + r ̂ 01 r ̂ 12 exp ( 2 i γ ̂ 1 d 1 ) ] ,
R 210 = r ̂ 210 * r ̂ 210 ,
r ̂ 210 = [ r ̂ 21 + r ̂ 10 exp ( 2 i γ ̂ 1 d 1 ) ] / [ 1 + r ̂ 21 r ̂ 10 exp ( 2 i γ ̂ 1 d 1 ) ] ,
β 2 = 2 π ν k 2 ,
γ ̂ j = 2 π ν n ̂ j ,
t ̂ ( ν ) = [ n 0 ( ν ) T ( ν ) / n 3 ( ν ) ] 1 / 2 exp [ i ϕ ( ν ) ] ,
ϕ ( ν ) + 2 m π = 2 ν π P 0 ln | t ̂ ( ν ) | d ν ν 2 ν 2 + 2 π ν d 1 = ν π P 0 ln T ( ν ) ν 2 ν 2 d ν + 2 π ν d 1 ,
ϕ ( ν ) 0 + 2 m π = ν 0 π P 0 ln T ( ν ) ν 2 ν 0 2 d ν + 2 π ν 0 d 1 .
ϕ ( ν ) + 2 m π = ν ν 0 [ ϕ ( ν 0 ) + 2 m π ] + ν ( ν 0 2 ν 2 ) π P 0 ln T ( ν ) ( ν 2 ν 2 ) ( ν 2 ν 0 2 ) d ν .
ϕ ( ν 0 ) = tan 1 { Im [ t ̂ ( ν 0 ) ] / Re [ t ̂ ( ν 0 ) ] } ,
k ln [ T s / ( I 3 / I 0 ) ] / ( 4 π ν d 1 ) ,
T s = T 02 T 23 exp ( 2 β 2 d 2 ) / [ 1 R 20 R 23 exp ( 4 β 2 d 2 ) ]
T s = ( n 3 / n 0 ) t ̂ s * t ̂ s ,
t ̂ s = t ̂ 23 t ̂ 02 exp ( i γ ̂ 2 d 2 ) / [ 1 r ̂ 23 r ̂ 20 exp ( a i γ ̂ 2 d 2 ) ] ,
k ( ν ) = k max / [ 1 + ( 4 / γ 2 ) ( ν ν 1 ) 2 ] .
n ( ν ) n 0 + j = 1 N k max j ν ν L j ( ν ν j ) ( ν L j ν j ) ν L j 2 ( ν ν j ) 2 + ν 2 ( ν L j ν j ) 2 .
ν L j = ν j γ j / 2
ν ( ν 0 2 ν 2 ) π P 0 ν l ln T ( ν ) ( ν 2 ν 0 2 ) ( ν 2 ν 2 ) d ν ν 2 π ( 1 ν ln | ν l + ν ν l ν | 1 ν 0 ln | ν l + ν 0 ν l ν 0 | ) ,
ν ( ν 0 2 ν 2 ) π P ν u ln T ( ν ) ( ν 2 ν 0 2 ) ( ν 2 ν 2 ) d ν ν ( ν 0 2 ν 2 ) ν u 4 ln T ( ν u ) π [ ν 0 2 + ν 2 ν u ν 0 4 ν 4 + 1 3 ν u 3 ν 0 2 ν 2 + 1 ν 0 4 ( ν 0 2 ν 2 ) ( 1 2 ν 0 ln | ν u + ν 0 ν u ν 0 | ) + 1 ν 4 ( ν 0 2 ν 2 ) ( 1 2 ν ln | ν u + ν ν u ν | ) ] .
n g ( ν ) = A + B L + C L 2 + D ν 2 + E ν 4 , A = 3.880 , B = 3.91707 × 10 9 cm 2 , C = 1.63492 × 10 17 cm 4 , D = 600 cm 2 , E = 5.3 × 10 8 cm 4 , L = ( ν 2 2.8 × 10 8 cm 2 ) 1 . }
ϕ ( ν ) + 2 m π [ 4 ν P 0 ν k ( ν ) ν 2 ν 2 d ν + 2 π ν ] d 1 ,
t ̂ = t ̂ 23 t ̂ 012 exp ( i γ ̂ 2 d 2 ) / [ 1 r ̂ 23 r ̂ 210 exp ( 2 i γ ̂ 2 d 2 ) ] .
R 0123 ( ν ) = r ̂ 0123 * ( ν ) r ̂ 0123 ( ν ) ,
r ̂ 0123 = r ̂ 012 + r ̂ 23 t ̂ 210 t ̂ 012 exp ( 2 i γ ̂ 2 d 2 ) / [ 1 r ̂ 23 r 210 exp ( 2 i γ ̂ 2 d 2 ) ]
R 0123 = R 012 + T 210 T 012 R 23 exp ( 4 β 2 d 2 ) / [ 1 R 210 R 23 exp ( 4 β 2 d 2 ) ]
r ̂ 012 = r ̂ 01 r 12 exp ( 2 i γ ̂ 1 d 1 ) / [ 1 + r ̂ 01 r ̂ 12 exp ( 2 i γ ̂ 1 d 1 ) ] ,
R 012 ( ν ) = r ̂ 012 * ( ν ) r ̂ 012 ( ν ) .
| r ̂ 012 | = R 012 1 / 2 ,
ϕ ( ν ) + 2 m π = 2 ν π P 0 ln | r ̂ 012 ( ν ) | d ν ν 2 ν 2 + 2 j = 1 Z [ tan 1 ν I j ν R j ν tan 1 ν I j ν R j + ν ] + 2 l = 1 Z 0 ( 1 ) l tan 1 ν 0 l ν .
T = T 012 T 23 exp ( 2 β 2 d 2 ) / [ 1 R 210 R 23 exp ( 4 β 2 d 2 ) ] ,
| t ̂ 012 | 2 = | t ̂ 01 | 2 | t ̂ 12 | 2 exp ( 4 π ν k d 1 ) 1 + 2 | r ̂ 01 | | r ̂ 12 | exp ( 4 π ν k d 1 ) cos ( θ 01 + θ 12 + 4 π ν n d 1 ) + | r ̂ 01 | 2 | r ̂ 12 | 2 exp ( 8 π ν k d 1 ) ,
| r ̂ 210 | 2 = | r ̂ 21 | 2 + 2 | r ̂ 21 | | r ̂ 10 | exp ( 4 π ν k d 1 ) cos ( θ 10 θ 21 + 4 π ν n d 1 ) + | r ̂ 10 | 2 exp ( 8 π ν k d 1 ) 1 + 2 | r ̂ 21 | | r ̂ 10 | exp ( 4 π ν k d 1 ) cos ( θ 10 + θ 21 + 4 π ν n d 1 ) + | r ̂ 21 | 2 | r ̂ 10 | 2 exp ( 8 π ν k d 1 ) ,
Δ ν ( 2 n d 1 ) 1
Δ d 1 ( 2 ν n ) 1 ,
ϕ ( ν ) + 2 m π [ 4 ν P 0 ν k ( ν ) d ν ν 2 ν 2 + 2 π ν ] d 1 2 ν π P 0 | r ̂ 01 | | r ̂ 12 | exp ( 4 π ν k d 1 ) cos ( θ 01 + θ 12 + 4 π ν n d 1 ) d ν ν 2 ν 2 ν π P 0 | r ̂ 01 | 2 | r ̂ 12 | 2 exp ( 8 π ν k 2 d 2 ) d ν ν 2 ν 2 + 4 ν d 2 P 0 ν k 2 ( ν ) d ν ν 2 ν 2 ν π P 0 | r ̂ 23 | 2 | r ̂ 210 | 2 exp ( 8 π ν k 2 d 2 ) d ν ν 2 ν 2 ν π P 0 ln | r ̂ 01 | 2 | t ̂ 12 | 2 | t ̂ 23 | 2 d ν ν 2 ν 2 .
m ν d 1 [ 1 π j = 1 N k max j ln | ν 2 ( ν j + γ j / 2 ) 2 ν 2 + ( ν j γ j / 2 ) 2 | + 1 ] ϕ ( ν ) 2 π .
̂ ( ν ) = n ̂ 2 ( ν ) 0 ,
̂ ( ν ) = n ̂ 2 ( ν ) 2 i σ c ν ,
ln T ( ν ) ln T 0 ,
t ̂ j m ( ν ) 1 + ( ν p m 2 ν p j 2 ) / ( 4 ν 2 ) , r ̂ j m ( ν ) ( ν p m 2 ν p j 2 ) / ( 4 ν 2 ) = t ̂ j m ( ν ) 1 ,
| t ̂ 012 | 2 = t ̂ 012 * t ̂ 012 [ 1 + ν p 2 2 / ( 4 ν 2 ) + ] 2 , | r ̂ 210 | 2 = r ̂ 210 * r ̂ 210 [ 2 ν p 1 2 ( ν p 1 2 ν p 2 2 ) + ν p 2 4 + ] / ( 16 ν 4 ) ,
T = t ̂ * t ̂ 1 ν p 2 / ( 8 ν 4 ) + or , ln T ν p 2 4 / ( 8 ν 4 ) .

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