Abstract

A standard 2 × 2 matrix method-used in thin-film optics is applied to planar multilayer optical waveguides. All modes are required to satisfy substrate-to-cover field-transfer equations that reduce to the equation γcm11 + γcγsm12 + m21 + γsm22 = 0 for bound modes and leaky waves. Expressions are derived for the field profiles and the power in each medium. A first-order perturbation theory is developed and applied to absorbing multilayer guides and to the reflection of plane waves from the prism-loaded lossy multilayer guide. The latter leads to experimental arrangements for measuring losses in which the gap thickness and propagation constant are accessible parameters.

© 1984 Optical Society of America

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References

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  1. G. Al-Jumaily, S. D. Browning, A. F. Turner, “Polarization-insensitive refracting system for integrated optics,” J. Opt. Soc. Am. 72, 1822 (A) (1982).
  2. H. Ito, H. Inaba, “Efficient phase matched second-harmonic generation method in four-layered optical-waveguide structure,” Opt. Lett. 2, 139–141 (1978).
    [CrossRef] [PubMed]
  3. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971).
    [CrossRef] [PubMed]
  4. P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
    [CrossRef]
  5. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).
  6. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  7. P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in stratified media. 1. General theory,” J. Opt. Soc. Am. periodic 67, 423–438 (1977).
    [CrossRef]
  8. J. F. Revelli, D. Sarid, “Prism coupling into clad uniform optical waveguides,” J. Appl. Phys. 51, 3566–3575 (1980).
    [CrossRef]
  9. J. F. Revelli, “Enhancement of prism coupling efficiency in uniform optical waveguides: a correction,” J. Appl. Phys. 52, 3185–3189 (1981).
    [CrossRef]
  10. J. F. Revelli, “Mode analysis and prism coupling for multilayered optical waveguides,” Appl. Opt. 20, 3158–3167 (1981).
    [CrossRef] [PubMed]
  11. J. T. Chilwell, “Optical waveguiding in multilayer thin film stacks and the prism coupler: a matrix method,” Doctoral thesis (University of Otago, Dunedin, New Zealand, 1982).
  12. J. T. Chilwell, I. J. Hodgkinson, “Thin-film-matrix description of optical multilayer planar waveguides,” J. Opt. Soc. Am. 72, 1821 (A) (1982).
  13. I. J. Hodgkinson, J. T. Chilwell, “Reflection of plane waves from a prism-loaded lossy multilayer waveguide,” J. Opt. Soc. Am. 72, 1744–1745 (1982).
  14. S. A. Shakir, A. F. Turner, “Method of poles for multilayer thin-film waveguides,” Appl. Phys. A 29, 151–155 (1982).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  16. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5.
    [CrossRef]
  17. R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8.
  18. A. Thelen, “Design of multilayer interference filters,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1969), Vol. 5.
  19. In standard notation, we have k= 2πλ−1= ωc-1, where λ is the vacuum wavelength and c the vacuum speed of light. Also, μ is the magnetic permeability and ɛ is the electric permittivity of the medium. The refractive index is given by n= (μɛ)1/2(μ0ɛ0)−1/2.
  20. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  21. R. Ulrich, “Theory of the prism-film coupler by plane-wave analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
    [CrossRef]
  22. J. E. Sipe, J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. 72, 288–295 (1982).
    [CrossRef]
  23. Lotspeich has presented an approximate explicit solution of the propagation constant for the modal-dispersion function of a symmetric single-film waveguide [J. F. Lotspeich, “Explicit general eigenvalue solutions for dielectric slab waveguides,” Appl. Opt. 14, 327–335 (1975)].
    [CrossRef] [PubMed]
  24. To this end, Hardey et al. have derived expressions for the number of TE modes in periodic multilayer waveguides [A. Hardy, E. Kapon, A. Katzir, “Expression for the number of guided TE modes in periodic multilayer waveguides,” J. Opt. Soc. Am. 71, 1283–1285 (1981)].
    [CrossRef]
  25. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  26. The propagation constant is real or complex depending on the type of solution taken for Eq. (24) and whether or not the waveguide is perfect with real refractive indexes or dissipative. To summarize: (1) for bound modes (ns< β> nc), β is real in the nondissipative case and complex in the dissipative case; (2) for radiation modes (β< nc and/or ns), β is always real (or imaginary when evanescent radiation modes are considered; see Ref. 6, p. 27); (3) for leaky waves (β< nc and/or ns), β is always complex.
  27. Z. Knittl, Optics of Thin Films (Wiley, London, 1976), pp. 229–233.
  28. R. Ulrich, W. Prettl, “Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973).
    [CrossRef]
  29. R. Th. Kersten, “The prism–film coupler as a precision instrument. Part II. Measurements of refractive index and thickness of leaky wave guides,” Opt. Acta 22, 515–521 (1975).
    [CrossRef]
  30. R. Ulrich, W. Prettl [“Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973)] have classified leaky waves into two groups according to the properties of the stack. Lummer–Gehrcke waves occur when the waveguide can support bound modes. However, if the maximum refractive index of the waveguide is found to be the cover or substrate index, only leaky waves can be supported; these are referred to as low-index waves.
    [CrossRef]
  31. J. Kane, H. Osterberg, “Optical characteristics of planar guided modes,” J. Opt. Soc. Am. 54, 347–352 (1964).
    [CrossRef]
  32. The incident plane waves in the prism have real β, and β is a constant throughout the system. Note that this situation can be thought of as radiation modes of a particular class of waveguide.
  33. J. T. Chilwell, “Prism coupler jig: interference fringes enable observation of the coupling gap,” Appl. Opt. 21, 1310–1319 (1982).
    [CrossRef] [PubMed]
  34. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965).
  35. J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” in Physics of Thin Films, G. Hass, M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10.
  36. J. Ebert, H. Pannhorst, H. Küster, H. Welling, “Scatter losses of broadband interference coatings,” Appl. Opt. 18, 818–822 (1979).
    [CrossRef] [PubMed]

1982 (6)

G. Al-Jumaily, S. D. Browning, A. F. Turner, “Polarization-insensitive refracting system for integrated optics,” J. Opt. Soc. Am. 72, 1822 (A) (1982).

J. T. Chilwell, I. J. Hodgkinson, “Thin-film-matrix description of optical multilayer planar waveguides,” J. Opt. Soc. Am. 72, 1821 (A) (1982).

I. J. Hodgkinson, J. T. Chilwell, “Reflection of plane waves from a prism-loaded lossy multilayer waveguide,” J. Opt. Soc. Am. 72, 1744–1745 (1982).

S. A. Shakir, A. F. Turner, “Method of poles for multilayer thin-film waveguides,” Appl. Phys. A 29, 151–155 (1982).
[CrossRef]

J. E. Sipe, J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. 72, 288–295 (1982).
[CrossRef]

J. T. Chilwell, “Prism coupler jig: interference fringes enable observation of the coupling gap,” Appl. Opt. 21, 1310–1319 (1982).
[CrossRef] [PubMed]

1981 (3)

1980 (1)

J. F. Revelli, D. Sarid, “Prism coupling into clad uniform optical waveguides,” J. Appl. Phys. 51, 3566–3575 (1980).
[CrossRef]

1979 (1)

1978 (1)

1977 (1)

P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in stratified media. 1. General theory,” J. Opt. Soc. Am. periodic 67, 423–438 (1977).
[CrossRef]

1975 (2)

1973 (2)

R. Ulrich, W. Prettl [“Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973)] have classified leaky waves into two groups according to the properties of the stack. Lummer–Gehrcke waves occur when the waveguide can support bound modes. However, if the maximum refractive index of the waveguide is found to be the cover or substrate index, only leaky waves can be supported; these are referred to as low-index waves.
[CrossRef]

R. Ulrich, W. Prettl, “Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973).
[CrossRef]

1971 (1)

1970 (2)

1964 (1)

Al-Jumaily, G.

G. Al-Jumaily, S. D. Browning, A. F. Turner, “Polarization-insensitive refracting system for integrated optics,” J. Opt. Soc. Am. 72, 1822 (A) (1982).

Becher, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Browning, S. D.

G. Al-Jumaily, S. D. Browning, A. F. Turner, “Polarization-insensitive refracting system for integrated optics,” J. Opt. Soc. Am. 72, 1822 (A) (1982).

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Chilwell, J. T.

J. T. Chilwell, I. J. Hodgkinson, “Thin-film-matrix description of optical multilayer planar waveguides,” J. Opt. Soc. Am. 72, 1821 (A) (1982).

I. J. Hodgkinson, J. T. Chilwell, “Reflection of plane waves from a prism-loaded lossy multilayer waveguide,” J. Opt. Soc. Am. 72, 1744–1745 (1982).

J. T. Chilwell, “Prism coupler jig: interference fringes enable observation of the coupling gap,” Appl. Opt. 21, 1310–1319 (1982).
[CrossRef] [PubMed]

J. T. Chilwell, “Optical waveguiding in multilayer thin film stacks and the prism coupler: a matrix method,” Doctoral thesis (University of Otago, Dunedin, New Zealand, 1982).

Eastman, J. M.

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” in Physics of Thin Films, G. Hass, M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10.

Ebert, J.

Hardy, A.

Heavens, O. S.

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

Hodgkinson, I. J.

J. T. Chilwell, I. J. Hodgkinson, “Thin-film-matrix description of optical multilayer planar waveguides,” J. Opt. Soc. Am. 72, 1821 (A) (1982).

I. J. Hodgkinson, J. T. Chilwell, “Reflection of plane waves from a prism-loaded lossy multilayer waveguide,” J. Opt. Soc. Am. 72, 1744–1745 (1982).

Hong, C.

P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in stratified media. 1. General theory,” J. Opt. Soc. Am. periodic 67, 423–438 (1977).
[CrossRef]

Inaba, H.

Ito, H.

Jacobsson, R.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5.
[CrossRef]

R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8.

Kane, J.

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

Kapon, E.

Katzir, A.

Kersten, R. Th.

R. Th. Kersten, “The prism–film coupler as a precision instrument. Part II. Measurements of refractive index and thickness of leaky wave guides,” Opt. Acta 22, 515–521 (1975).
[CrossRef]

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), pp. 229–233.

Küster, H.

Lotspeich, J. F.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Osterberg, H.

Pannhorst, H.

Prettl, W.

R. Ulrich, W. Prettl, “Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973).
[CrossRef]

R. Ulrich, W. Prettl [“Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973)] have classified leaky waves into two groups according to the properties of the stack. Lummer–Gehrcke waves occur when the waveguide can support bound modes. However, if the maximum refractive index of the waveguide is found to be the cover or substrate index, only leaky waves can be supported; these are referred to as low-index waves.
[CrossRef]

Revelli, J. F.

J. F. Revelli, “Enhancement of prism coupling efficiency in uniform optical waveguides: a correction,” J. Appl. Phys. 52, 3185–3189 (1981).
[CrossRef]

J. F. Revelli, “Mode analysis and prism coupling for multilayered optical waveguides,” Appl. Opt. 20, 3158–3167 (1981).
[CrossRef] [PubMed]

J. F. Revelli, D. Sarid, “Prism coupling into clad uniform optical waveguides,” J. Appl. Phys. 51, 3566–3575 (1980).
[CrossRef]

Sarid, D.

J. F. Revelli, D. Sarid, “Prism coupling into clad uniform optical waveguides,” J. Appl. Phys. 51, 3566–3575 (1980).
[CrossRef]

Shakir, S. A.

S. A. Shakir, A. F. Turner, “Method of poles for multilayer thin-film waveguides,” Appl. Phys. A 29, 151–155 (1982).
[CrossRef]

Sipe, J. E.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Thelen, A.

A. Thelen, “Design of multilayer interference filters,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1969), Vol. 5.

Tien, P. K.

Turner, A. F.

S. A. Shakir, A. F. Turner, “Method of poles for multilayer thin-film waveguides,” Appl. Phys. A 29, 151–155 (1982).
[CrossRef]

G. Al-Jumaily, S. D. Browning, A. F. Turner, “Polarization-insensitive refracting system for integrated optics,” J. Opt. Soc. Am. 72, 1822 (A) (1982).

Ulrich, R.

R. Ulrich, W. Prettl, “Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973).
[CrossRef]

R. Ulrich, W. Prettl [“Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973)] have classified leaky waves into two groups according to the properties of the stack. Lummer–Gehrcke waves occur when the waveguide can support bound modes. However, if the maximum refractive index of the waveguide is found to be the cover or substrate index, only leaky waves can be supported; these are referred to as low-index waves.
[CrossRef]

P. K. Tien, R. Ulrich, “Theory of prism-film coupler and thin-film light guides,” J. Opt. Soc. Am. 60, 1325–1337 (1970).
[CrossRef]

R. Ulrich, “Theory of the prism-film coupler by plane-wave analysis,” J. Opt. Soc. Am. 60, 1337–1350 (1970).
[CrossRef]

Welling, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Yariv, A.

P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in stratified media. 1. General theory,” J. Opt. Soc. Am. periodic 67, 423–438 (1977).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in stratified media. 1. General theory,” J. Opt. Soc. Am. periodic 67, 423–438 (1977).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. (2)

R. Ulrich, W. Prettl [“Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973)] have classified leaky waves into two groups according to the properties of the stack. Lummer–Gehrcke waves occur when the waveguide can support bound modes. However, if the maximum refractive index of the waveguide is found to be the cover or substrate index, only leaky waves can be supported; these are referred to as low-index waves.
[CrossRef]

R. Ulrich, W. Prettl, “Planar leaky lightguides and couplers,” Appl. Phys. 1, 55–68 (1973).
[CrossRef]

Appl. Phys. A (1)

S. A. Shakir, A. F. Turner, “Method of poles for multilayer thin-film waveguides,” Appl. Phys. A 29, 151–155 (1982).
[CrossRef]

J. Appl. Phys. (2)

J. F. Revelli, D. Sarid, “Prism coupling into clad uniform optical waveguides,” J. Appl. Phys. 51, 3566–3575 (1980).
[CrossRef]

J. F. Revelli, “Enhancement of prism coupling efficiency in uniform optical waveguides: a correction,” J. Appl. Phys. 52, 3185–3189 (1981).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. periodic (1)

P. Yeh, A. Yariv, C. Hong, “Electromagnetic propagation in stratified media. 1. General theory,” J. Opt. Soc. Am. periodic 67, 423–438 (1977).
[CrossRef]

Opt. Acta (1)

R. Th. Kersten, “The prism–film coupler as a precision instrument. Part II. Measurements of refractive index and thickness of leaky wave guides,” Opt. Acta 22, 515–521 (1975).
[CrossRef]

Opt. Lett. (1)

Other (15)

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, New York, 1972).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

J. T. Chilwell, “Optical waveguiding in multilayer thin film stacks and the prism coupler: a matrix method,” Doctoral thesis (University of Otago, Dunedin, New Zealand, 1982).

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5.
[CrossRef]

R. Jacobsson, “Inhomogeneous and coevaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8.

A. Thelen, “Design of multilayer interference filters,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1969), Vol. 5.

In standard notation, we have k= 2πλ−1= ωc-1, where λ is the vacuum wavelength and c the vacuum speed of light. Also, μ is the magnetic permeability and ɛ is the electric permittivity of the medium. The refractive index is given by n= (μɛ)1/2(μ0ɛ0)−1/2.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

The propagation constant is real or complex depending on the type of solution taken for Eq. (24) and whether or not the waveguide is perfect with real refractive indexes or dissipative. To summarize: (1) for bound modes (ns< β> nc), β is real in the nondissipative case and complex in the dissipative case; (2) for radiation modes (β< nc and/or ns), β is always real (or imaginary when evanescent radiation modes are considered; see Ref. 6, p. 27); (3) for leaky waves (β< nc and/or ns), β is always complex.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976), pp. 229–233.

The incident plane waves in the prism have real β, and β is a constant throughout the system. Note that this situation can be thought of as radiation modes of a particular class of waveguide.

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

J. M. Eastman, “Scattering by all-dielectric multilayer bandpass filters and mirrors for lasers,” in Physics of Thin Films, G. Hass, M. H. Francombe, eds. (Academic, New York, 1978), Vol. 10.

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Figures (7)

Fig. 1
Fig. 1

The multilayer stack consisting of J planar films lying between the semi-infinite cover and substrate media. Also depicted is the relationship of the electric- and magnetic-field vectors to the polarization-dependent field amplitudes U, V, and W for the TE (left) and TM (right) polarizations.

Fig. 2
Fig. 2

The modal-dispersion function χ versus the propagation constant β for the TE polarization. The plot for the TM polarization is similar. The four-layer waveguide is characterized by (cover to substrate) nc = 1, n1 = 1.66, n2 = 1.53, n3 = 1.6, n4 = 1.66, ns = 1.5, and all film thickness are 500 nm. The wavelength taken in the calculation is 632.8 nm. The values of the effective indexes are given in Table 3.

Fig. 3
Fig. 3

The magnitudes of the amplitudes of the total field components (|U| and |V|) for each of the TE modes supported by the waveguide given in Fig. 2. The solid curves represent the magnitude of Ez; the broken curves represent the magnitude of Hy. Each field amplitude has been normalized to a maximum value of unity.

Fig. 4
Fig. 4

The form of the field amplitude U in the jth layer.

Fig. 5
Fig. 5

Maximum coupling of TE waves from the prism to the guide occurs when the face of the prism is placed at the turning point of the field U.

Fig. 6
Fig. 6

An ideal lossless multilayer guide (G) perturbed by small changes in refractive index (G′) and loaded with a prism (G″).

Fig. 7
Fig. 7

Photomultiplier traces illustrating the reflectance dip from a 580-nm lead fluoride film on a fused silica substrate (a) as a function of the thickness dc of the gap between the prism and the guide (β held constant at 1.5538) and (b) as a function of β with dc = 174 nm. The results indicate an extinction coefficient of 5 × 10−4 for the lead fluoride film (n1 = 1.754, ns = 1.457, nc = 1.000, np = 1.696, λ = 632.8 nm).

Tables (4)

Tables Icon

Table 1 Polarization-Dependent Parameters ρ and γ and Field Amplitudes U, V, and Wa

Tables Icon

Table 2 The TE Lummer–Gehrcke Leaky Waves Supported by the Four-Layer Guide of Fig. 2

Tables Icon

Table 3 Relative Percentage Power in Each Stratum of the Waveguide Given in Fig. 2

Tables Icon

Table 4 Exact and Approximate Effective Indexes for the Guide of Fig. 2 with Absorptiona

Equations (93)

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H = n E z 0 ,
ρ = { 0 TE 1 TM ,
α = n cos θ = ( n 2 - β 2 ) 1 / 2 ,
β = n sin θ .
γ = { n cos θ / z 0 TE z 0 cos θ / n TM .
V = γ i k α d U d x ,
W = β γ α U ,
U = 1 i k γ α d V d x .
( U + V + ) = ( 1 γ ) U + ,
( U - V - ) = ( 1 - γ ) U - .
( U j - 1 V j - 1 ) = M j ( U j V j ) .
M j = ( cos Φ j - i γ j sin Φ j - i γ j sin Φ j cos Φ j ) ,
M = j = 1 J M j .
( 1 + r c s γ c ( 1 - r c s ) = M ( 1 γ s ) t c s .
r c s = γ c m 11 + γ c γ s m 12 - m 21 - γ s m 22 γ c m 11 + γ c γ s m 12 + m 21 + γ s m 22 ,
t c s = 2 γ c γ c m 11 + γ c γ s m 12 + m 21 + γ s m 22 .
R = r 2 ,
T = Re ( γ s ) Re ( γ c ) t 2
p x = ½ Re ( U V * ) exp [ i k ( β - β * ) y ] ,
p y = ½ Re ( U W * ) exp [ i k ( β - β * ) y ] ,
r c s = exp ( - 2 i ϕ c s ) ,
ϕ c s = tan - 1 [ m 21 + γ s m 22 i γ c ( m 11 + γ s m 12 ) ] .
ϕ c s = tan - 1 [ ( n c n s ) 2 ρ ( β 2 - n s 2 ) 1 / 2 ( n c 2 - β 2 ) 1 / 2 ] ,             0 ϕ c s π / 2.
( U V ) [ cos ( k α c x + ϕ c s ) i γ c sin ( k α c x + ϕ c s ) ] ,             x 0.
( U V ) ( 1 γ s ) exp ( i k α s x ) ,             x x s .
( U c V c ) = M ( U s V s ) .
( 1 - γ c ) U c = M ( 1 γ s ) U s .
χ M ( β ) γ c m 11 + γ c γ s m 12 + m 21 + γ s m 22 = 0.
maximum film index > β > n c , n s .
tan Φ 1 = - i ( γ c + γ s ) ( γ 1 + γ c γ s / γ 1 ) - 1 ,
2 Φ 1 - 2 ϕ 1 c - 2 ϕ 1 s = 2 π
l = i k ( β - β * )
α s r α s i = η s 2 κ s - β β ,
U = U s exp [ ± k α s ( x - x s ) ] exp [ ± i k α s ( x - x s ) ] ,             x x s ,
p x = ± ½ γ s U s 2 exp [ ± 2 k α s ( x - x s ) ] exp ( - 2 k β y ) ,             x x s ,
( U V ) = [ U j cos [ k α j ( x - x j ) ] + i γ j V j sin [ k α j ( x - x j ) ] i γ j U j sin [ k α j ( x - x j ) ] + V j cos [ k α j ( x - x j ) ] ] .
U V = ( i / 2 ) ( γ j U j 2 + V j 2 γ j - 1 ) sin [ 2 k α j ( x - x j ) ] + U j V j cos [ 2 k α j ( x - x j ) ] ,
U 2 - V 2 γ j - 2 = Q j 2 ,
U = 0 ,             d V d x = 0             at x = x j + 1 k α j tan - 1 ( i γ j U j V j ) ,
V = 0 ,             d U d x = 0             at x = x j + 1 k α j tan - 1 ( i V j γ j U j ) .
γ j U j V j - 1 < 1 ,
γ j U j V j - 1 > 1.
( U V ) = Q j { cos [ k α j ( x - x j ) + ϕ j s ] i γ j sin [ k α j ( x - x j ) + ϕ j s ] } ,
P y j = ½ x j - 1 x j U W d x = β ρ γ j 2 α j x j - 1 x j U 2 d x .
P y j = ( β ρ / 4 k α j 2 ) [ ( γ j U j 2 - V j 2 / γ j ) Φ j + i ( U j - 1 V j - 1 - U j V j ) ] .
P y c = i ( β ρ γ c U c 2 / 4 k α c 2 )
P y s = i ( β ρ γ s U s 2 / 4 k α s 2 ) .
d U p / d x = 0             and             V p = 0
d V p / d x = 0             and             U p = 0
d d x U p + 2 = 0.
( U p V p ) = ( 1 + r p s γ p ( 1 - r p s ) ) U p +
U p + = γ p U p + V p 2 γ p .
( U p V p ) = [ cos ( k α c d c ) - i γ c sin ( k α c d c ) - i γ c sin ( k α c d c ) cos ( k α c d c ) ] M ( 1 γ s )
d U p d x = i k α c γ c V p ,             d V p d x = i k α c γ c U p .
U p V p * - U p * V p = 0 ,
Re ( U p ) Im ( V p ) = Im ( U p ) Re ( V p ) .
r p s = ( γ p U p - V p ) / ( γ p U p + V p ) ,
d r p s d x = 2 γ p ( d U p d x V p - U p d V p d x ) / ( γ p U p + V p ) 2
d c = i λ 4 π α c ln r c s
r min = ( γ p - γ c ) + ( γ p + γ c ) r c s / r c s ( γ p + γ c ) + ( γ p - γ c ) r c s / r c s ,
r c s = γ c m 11 + γ c γ s m 12 - m 21 - γ s m 22 γ c m 11 + γ c γ s m 12 + m 21 + γ s m 22 .
n j = n j ( 1 + κ j a ) ,
β ρ = β ρ ( 1 + b ) ,
α j = α j [ 1 + ( n j 2 / α j 2 ) κ j a - ( β ρ 2 / α j 2 ) b ]
γ j = γ j [ 1 + ( n j 2 / α j 2 - 2 ρ ) κ j a - ( β ρ 2 / α j 2 ) b ] ,
M j M j + A j a + B j b ,
A j = - κ j α j 2 { n j 2 Φ j sin Φ j i γ j [ n j 2 Φ j cos Φ j - ( n j 2 - 2 ρ α j 2 ) sin Φ j ] i γ j [ n j 2 Φ j cos Φ j + ( n j 2 - 2 ρ α j 2 ) sin Φ j ] n j 2 Φ j sin Φ j } ,
B j = β ρ 2 α j 2 × [ Φ j sin Φ j i γ j ( Φ j cos Φ j - sin Φ j ) i γ j ( Φ j cos Φ j + sin Φ j ) Φ j sin Φ j ] .
M = j = 1 J M j M + A a + B b ,
A = k = 1 J ( j = 1 k - 1 M j ) A k ( j = k + 1 J M j ) ,
χ M = γ c 11 + γ c γ s m 12 + m 21 + γ s m 22 = 0.
S A a + S B b = 0 ,
S A = χ A + γ c ( m 11 + γ s m 12 ) ( n c 2 α c - 2 - 2 ρ ) κ c + γ s ( γ c m 12 + m 22 ) ( n s 2 α s - 2 - 2 ρ ) κ s ,
S B = χ B - γ c ( m 11 + γ s m 12 ) β ρ 2 α c - 2 - γ s ( γ c m 12 + m 22 ) β m ρ 2 α s - 2 .
b = - S A S B a
β = β ρ ( 1 + b )
r p s = γ p m 11 + γ p γ s m 12 - m 21 - γ s m 22 γ p m 11 + γ p γ s m 12 + m 21 + γ s m 22 ,
( 1 1 / γ c γ c 1 ) + ( 2 ρ - n c 2 α c 2 ) κ c ( 0 1 / γ c - γ c 0 ) a + β ρ 2 α c 2 ( 0 1 / γ c - γ c 0 ) b + ( 1 - 1 / γ c - γ c 1 ) c ,
C = exp ( 2 i Φ c )
r p s = ( γ p - γ c ) S A a + ( γ p - γ c ) S B b + ( γ p + γ c ) S C C ( γ p + γ c ) S A a + ( γ p + γ c ) S B b + ( γ p - γ c ) S C C .
S C = - 2 γ c ( m 11 + γ s m 12 ) .
b = γ c 2 + γ p 2 2 γ p γ c S A S B a
C = γ c 2 - γ p 2 2 γ p γ c S A S C a .
d c = 1 i 2 k α c ln [ ( γ c 2 - γ p 2 ) S A a 2 γ p γ c S C ] .
δ a a = 2 ( ± 2 + 1 ) ,
δ b b = ± i 4 γ p γ c ( γ p 2 + γ c 2 ) ,
δ C C = 8 γ p 2 γ c 2 ( γ p 2 - γ c 2 ) 2 { - 1 ± [ 1 - ( γ p 2 - γ c 2 ) 2 4 γ p 2 γ c 2 ] 1 / 2 } .
δ d c = ± i λ 2 π α c sinh - 1 [ 2 γ p γ c i ( γ p 2 - γ c 2 ) ] ,
δ b = ± i 2 S A S B a .
cos ϕ 1 c cos ϕ J s ( 1 γ c / γ 1 2 γ c 1 ) M ( 1 γ s / γ J 2 γ s 1 )
M a ( U a V a ) = a ( U a V a ) ,
M I M a M II ( 1 γ s ) = M I M a ( U a V a ) = a M I ( U a V a ) = a U c ( 1 - γ c ) .
( cos Φ e - i γ e sin Φ e - i γ e sin Φ e cos Φ e ) .

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