Abstract

Coupled mode theory for parallel waveguides is extended to include temporal variations of both the dielectric function of the photonic array and the input optical power. This formulation can be very useful for the design and comprehensive analysis of modern photonic devices, such as two-dimensional photonic crystals, represented by arrays of parallel waveguides. In the special case of a time-dependent input signal, but stationary dielectric constant, analytical solutions exist for the extended formulation. The accuracy and computer time of the formulation’s numerical solutions are examined against finite difference time domain and time-dependent beam propagation analyses of waveguide arrays.

© 2010 Optical Society of America

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References

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  1. A. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267-1277 (1972).
    [CrossRef]
  2. V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
    [CrossRef]
  3. V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
    [CrossRef]
  4. A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
    [CrossRef]
  5. A. A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90-99 (1986).
    [CrossRef]
  6. A. A. Hardy and E. Kapon, “Coupled mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966-971 (1996).
    [CrossRef]
  7. A. A. Hardy and W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528-534 (1986).
    [CrossRef]
  8. A. A. Hardy, “A unified approach to coupled mode phenomena,” IEEE J. Quantum Electron. 34, 1109-1116 (1998).
    [CrossRef]
  9. V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled mode theory,” J. Opt. Soc. Am. B 26, 1248-1255 (2009).
    [CrossRef]
  10. F. Luan, A. George, T. Hedley, G. Pearce, D. Bird, J. Knight, and P. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369-2371 (2004).
    [CrossRef] [PubMed]
  11. M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
    [CrossRef]
  12. C.-C. Shih and A. Yariv, “A theoretical model of linear electro-optical effect,” J. Phys. C 15, 825-846 (1982).
    [CrossRef]
  13. X. Sun, A. Zadok, M. Shearn, K. Diest, A. Ghaffari, H. Atwater, A. Scherer, and A. Yariv, “Electrically pumped hybrid evanescent Si/InGaAsP lasers,” Opt. Lett. 34, 1345-1347 (2009).
    [CrossRef] [PubMed]
  14. E. von Kamke, Differentialgleichungen-Lösungsmethoden und Lösungen (Verbesserte Auflage, 1959), Vol. II.
  15. S. Habraken and G. Nienhuis, “Modes of a rotating astigmatic optical cavity,” Phys. Rev. A 77, 053803 (2008).
    [CrossRef]
  16. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  17. A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
    [CrossRef]
  18. A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 2000).
  19. L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
  20. J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, The Wadsworth & Brooks/Cole Mathematics Series (Wadsworth & Brooks/Cole, 1989).
  21. A. A. Hardy and M. Ben-Artzi, “Expansion of an arbitrary field in terms of waveguide modes,” IEE Proc.: Optoelectron. 141, 16-20 (1994).
    [CrossRef]
  22. H. Kogelnik, “Theory of dielectric waveguides” in Integrated Optics, T.Tamir, ed., (Springer, 1975).
    [CrossRef]

2009 (2)

2008 (2)

S. Habraken and G. Nienhuis, “Modes of a rotating astigmatic optical cavity,” Phys. Rev. A 77, 053803 (2008).
[CrossRef]

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

2007 (1)

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

2005 (2)

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

2004 (1)

1998 (1)

A. A. Hardy, “A unified approach to coupled mode phenomena,” IEEE J. Quantum Electron. 34, 1109-1116 (1998).
[CrossRef]

1996 (1)

A. A. Hardy and E. Kapon, “Coupled mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966-971 (1996).
[CrossRef]

1994 (1)

A. A. Hardy and M. Ben-Artzi, “Expansion of an arbitrary field in terms of waveguide modes,” IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

1986 (2)

A. A. Hardy and W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528-534 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

1985 (1)

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

1982 (1)

C.-C. Shih and A. Yariv, “A theoretical model of linear electro-optical effect,” J. Phys. C 15, 825-846 (1982).
[CrossRef]

1972 (1)

Alkeskjold, T.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Angelis, C.

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

Atwater, H.

Ben-Artzi, M.

A. A. Hardy and M. Ben-Artzi, “Expansion of an arbitrary field in terms of waveguide modes,” IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

Bird, D.

Bjarklev, A.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Boiko, D.

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

Capobianco, A.

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

Coldren, L.

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Corzine, S.

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Diest, K.

Engan, H.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

George, A.

Ghaffari, A.

Haakestad, M.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Habraken, S.

S. Habraken and G. Nienhuis, “Modes of a rotating astigmatic optical cavity,” Phys. Rev. A 77, 053803 (2008).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Hardy, A. A.

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled mode theory,” J. Opt. Soc. Am. B 26, 1248-1255 (2009).
[CrossRef]

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

A. A. Hardy, “A unified approach to coupled mode phenomena,” IEEE J. Quantum Electron. 34, 1109-1116 (1998).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966-971 (1996).
[CrossRef]

A. A. Hardy and M. Ben-Artzi, “Expansion of an arbitrary field in terms of waveguide modes,” IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528-534 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

Hedley, T.

Kapon, E.

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled mode theory,” J. Opt. Soc. Am. B 26, 1248-1255 (2009).
[CrossRef]

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

A. A. Hardy and E. Kapon, “Coupled mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966-971 (1996).
[CrossRef]

Knight, J.

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides” in Integrated Optics, T.Tamir, ed., (Springer, 1975).
[CrossRef]

Locatelli, A.

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 2000).

Luan, F.

Modotto, D.

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

Nielsen, M.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Nienhuis, G.

S. Habraken and G. Nienhuis, “Modes of a rotating astigmatic optical cavity,” Phys. Rev. A 77, 053803 (2008).
[CrossRef]

Nusinsky, I.

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled mode theory,” J. Opt. Soc. Am. B 26, 1248-1255 (2009).
[CrossRef]

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

Pearce, G.

Pigozzo, F.

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

Riishede, J.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Russell, P.

Scherer, A.

Scolari, L.

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

Shearn, M.

Shih, C. -C.

C.-C. Shih and A. Yariv, “A theoretical model of linear electro-optical effect,” J. Phys. C 15, 825-846 (1982).
[CrossRef]

Shteeman, V.

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled mode theory,” J. Opt. Soc. Am. B 26, 1248-1255 (2009).
[CrossRef]

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

Snyder, A.

Streifer, W.

A. A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528-534 (1986).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

Strikwerda, J.

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, The Wadsworth & Brooks/Cole Mathematics Series (Wadsworth & Brooks/Cole, 1989).

Sun, X.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

von Kamke, E.

E. von Kamke, Differentialgleichungen-Lösungsmethoden und Lösungen (Verbesserte Auflage, 1959), Vol. II.

Yariv, A.

Zadok, A.

IEE Proc.: Optoelectron. (1)

A. A. Hardy and M. Ben-Artzi, “Expansion of an arbitrary field in terms of waveguide modes,” IEE Proc.: Optoelectron. 141, 16-20 (1994).
[CrossRef]

IEEE J. Lightwave Technol. (1)

A. A. Hardy and W. Streifer, “Coupled modes of multiwaveguide systems and phased arrays,” IEEE J. Lightwave Technol. LT-4, 90-99 (1986).
[CrossRef]

IEEE J. Quantum Electron. (5)

A. A. Hardy and E. Kapon, “Coupled mode formulations for parallel-laser resonators with application to vertical-cavity semiconductor-laser arrays,” IEEE J. Quantum Electron. 32, 966-971 (1996).
[CrossRef]

A. A. Hardy and W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. QE-22, 528-534 (1986).
[CrossRef]

A. A. Hardy, “A unified approach to coupled mode phenomena,” IEEE J. Quantum Electron. 34, 1109-1116 (1998).
[CrossRef]

V. Shteeman, D. Boiko, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to periodic large arrays of identical waveguides for photonic crystals applications,” IEEE J. Quantum Electron. 43, 215-224 (2007).
[CrossRef]

V. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Extension of coupled mode analysis to infinite photonic superlattices,” IEEE J. Quantum Electron. 44, 826-833 (2008).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. Haakestad, T. Alkeskjold, M. Nielsen, L. Scolari, J. Riishede, H. Engan, and A. Bjarklev, “Electrically tunable photonic bandgap guidance in a liquid-crystal-filled photonic crystal fiber,” IEEE Photon. Technol. Lett. 17, 819-821 (2005).
[CrossRef]

J. Lightwave Technol. (1)

A. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. LT-3, 1135-1146 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. C (1)

C.-C. Shih and A. Yariv, “A theoretical model of linear electro-optical effect,” J. Phys. C 15, 825-846 (1982).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Locatelli, D. Modotto, C. Angelis, F. Pigozzo, and A. Capobianco, “Time domain bidirectional beam propagation method for second harmonic generation in multilayers,” Opt. Quantum Electron. 37, 121-131 (2005).
[CrossRef]

Phys. Rev. A (1)

S. Habraken and G. Nienhuis, “Modes of a rotating astigmatic optical cavity,” Phys. Rev. A 77, 053803 (2008).
[CrossRef]

Other (6)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

A. Snyder and J. Love, Optical Waveguide Theory (Kluwer, 2000).

L. Coldren and S. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, The Wadsworth & Brooks/Cole Mathematics Series (Wadsworth & Brooks/Cole, 1989).

E. von Kamke, Differentialgleichungen-Lösungsmethoden und Lösungen (Verbesserte Auflage, 1959), Vol. II.

H. Kogelnik, “Theory of dielectric waveguides” in Integrated Optics, T.Tamir, ed., (Springer, 1975).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematics of a composite photonic structure containing two parallel waveguides.

Fig. 2
Fig. 2

(a) An array of two identical planar waveguides; (b) an array of three identical planar waveguides; (c) a truncated Gaussian—the temporal profile of the incoming electromagnetic field.

Fig. 3
Fig. 3

The input (at z = 0 , t = 0 ) and the output (at z = 300 μ m , t = 1.5   ps ) fields’ profile in the array of two coupled identical waveguides.

Fig. 4
Fig. 4

The input (at z = 0 , t = 0 ) and the output (at z = 300 μ m , t = 1.5   ps ) fields’ profile in the array of three coupled identical waveguides.

Fig. 5
Fig. 5

The relative error in the spatial distribution of the output fields at z = 300 μ m , t = 1.5   ps , computed by the TD CMT and TD BPM methods ( | E | method 2 | E | FDTD 2 ) / | E | FDTD 2 .

Tables (4)

Tables Icon

Table 1 Effective Propagation Constants of the Array Modes of the Structure, Depicted in Fig. 3

Tables Icon

Table 2 Effective Propagation Constants of the Array Modes of the Structure, Depicted in Fig. 4

Tables Icon

Table 3 Cumulative Error in the Electric Field Distribution at the Output Plane, z = 300 μ m , for the Structures, Depicted in Figs. 3, 4

Tables Icon

Table 4 Approximate Computer Time Required for the Analysis of the Pulse Propagation in the Structures, Depicted in Figs. 3, 4 (Four-Core Intel Processor Core 2 Quad Q6600, 2.40 GHz, RAM of 3.5 Gbytes)

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

E ̃ ( r , t ) = E ( r , t ) e i ω t ,
H ̃ ( r , t ) = H ( r , t ) e i ω t ,
× E ̃ ( r , t ) = B ̃ ( r , t ) t = [ i ω μ 0 H ( r , t ) μ 0 H ( r , t ) t ] e i ω t ,
× H ̃ ( r , t ) = D ̃ ( r , t ) t = [ i ω [ ε ( p ) ( x , y ) + Δ ε ( p ) ( x , y , z , t ) ] E ( r , t ) + Δ ε ( p ) ( x , y , z , t ) t E ( r , t ) + [ ε ( p ) ( x , y ) + Δ ε ( p ) ( x , y , z , t ) ] E ( r , t ) t ] e i ω t .
E t ( r , t ) = p = 1 N m = 1 n p u m ( p ) ( z , t ) E t m ( p ) ( x , y ) + Q t ( r , t ) ,
H t ( r , t ) = p = 1 N m = 1 n p u m ( p ) ( z , t ) H t m ( p ) ( x , y ) + R t ( r , t ) .
U ( z , t ) z + B ω U ( z , t ) t = i M ( z , t ) U ( z , t ) ,
M ( z , t ) = B + P 1 K ̃ ( z , t ) ,
U ( z , t ) = e i M z U 0 ( z , t ) = A 1 R ( z ) A U 0 ( z , t ) ,
σ I ̂ U s t = M s t U s t ,
M s t = B + P 1 K ̃ .
( 2 x 2 + 2 y 2 + 2 i k z + 2 i k 1 c t ) I ( x , y , z , t ) = 0.
{ E t ( r , t ) H t ( r , t ) } = ξ 1 ( z , t ) { E t 1 ( a ) ( x , y ) H t 1 ( a ) ( x , y ) } + ν = 2 ξ ν ( z , t ) { E t ν ( a ) ( x , y ) H t ν ( a ) ( x , y ) } = η 1 ( z , t ) { E t 1 ( b ) ( x , y ) H t 1 ( b ) ( x , y ) } + ν = 2 η ν ( z , t ) { E t ν ( b ) ( x , y ) H t ν ( b ) ( x , y ) } ,
{ E t ( r , t ) H t ( r , t ) } = { u a ( z , t ) E t 1 ( a ) ( x , y ) + u b ( z , t ) E t 1 ( b ) ( x , y ) + Q t ( r , t ) u a ( z , t ) H t 1 ( a ) ( x , y ) + u b ( z , t ) H t 1 ( b ) ( x , y ) + R t ( r , t ) } .
{ E t 1 ( a ) ( x , y ) H t 1 ( a ) ( x , y ) } = ν = 1 C ν ( a b ) { E t ν ( b ) ( x , y ) H t ν ( b ) ( x , y ) } ,
{ E t 1 ( b ) ( x , y ) H t 1 ( b ) ( x , y ) } = ν = 1 C ν ( b a ) { E t ν ( a ) ( x , y ) H t ν ( a ) ( x , y ) } ,
C ν ( a b ) = 2 z ̂ E t 1 ( a ) ( x , y ) × H t ν ( b ) ( x , y ) d x d y .
Q t ( r , t ) = [ ξ 1 ( z , t ) u a ( z , t ) u b ( z , t ) C 1 ( b a ) ] E t 1 ( a ) ( x , y ) + ν = 2 ( ξ ν ( z , t ) u b ( z , t ) C ν ( b a ) ) E t ν ( a ) ( x , y ) = [ η 1 ( z , t ) u b ( z , t ) u a ( z , t ) C 1 ( a b ) ] E t 1 ( b ) ( x , y ) + ν = 2 ( η ν ( z , t ) u a ( z , t ) C ν ( a b ) ) E t ν ( b ) ( x , y ) .
ξ 1 ( z , t ) = u a ( z , t ) + C 1 ( b a ) u b ( z , t ) ,
η 1 ( z , t ) = u b ( z , t ) + C 1 ( a b ) u a ( z , t ) .
D ̃ 1 ( r , t ) = [ ε ( x , y ) + Δ ε ( x , y , z , t ) ] E 1 ( r , t ) e i ω t ,
B ̃ 1 ( r , t ) = μ 0 H 1 ( r , t ) e i ω t ,
D ̃ 2 ( r , t ) = ε ( x , y ) E μ ( x , y ) e i β μ z e i ω t ,
B ̃ 2 ( r , t ) = μ 0 H μ ( x , y ) e i β μ z e i ω t .
E 1 t ( x , y , z , t ) = ν = 1 p ν ( z , t ) E t ν ( x , y ) ,
H 1 t ( x , y , z , t ) = ν = 1 p ν ( z , t ) H t ν ( x , y ) .
z ̂ d x d y z ( E t μ × H 1 t E 1 t × H t μ ) = T 1 + T 2 + T 3 + T 4 ,
T 1 = d x d y ( μ 0 H μ ( x , y ) t H 1 ( x , y ) ) ,
T 2 = i ω d x d y ( E μ ( x , y ) Δ ε ( x , y , z , t ) E 1 ( r , t ) ) ,
T 3 = d x d y ( E μ ( x , y ) E 1 ( r , t ) Δ ε ( x , y , z , t ) t ) ,
T 4 = d x d y ( E μ ( x , y ) [ ε ( x , y ) + Δ ε ( x , y , z , t ) ] t E 1 ( r , t ) ) ,
z ̂ d x d y z ( E t μ × H 1 t E 1 t × H t μ ) = p μ ( z , t ) z i β μ p μ ( z , t ) .
T 2 = i ω ν = 1 p ν ( z , t ) d x d y Δ ε ( x , y , z , t ) [ E t μ ( x , y ) E t ν ( x , y ) ε ( x , y ) ε ( x , y , z , t ) E z μ ( x , y ) E z ν ( x , y ) ] .
T 1 T 4 1 2 β μ ω p μ ( z , t ) t ,
p μ ( z , t ) z + β μ ω p μ ( z , t ) t = i β μ p μ ( z , t ) + i ω ν = 1 p ν ( z , t ) d x d y Δ ε ( x , y , z , t ) E t μ ( x , y ) E t ν ( x , y ) i ω ν = 1 p ν ( z , t ) d x d y ε ( x , y ) Δ ε ( x , y , z , t ) ε ( x , y , z , t ) E z μ ( x , y ) E z ν ( x , y ) .
p μ ( z ) z = i β μ p μ ( z ) + i ω ν = 1 p ν ( z ) d x d y Δ ε ( x , y , z ) E t μ ( x , y ) E t ν ( x , y ) i ω ν = 1 p ν ( z ) d x d y ε ( x , y ) Δ ε ( x , y , z ) ε ( x , y , z ) E z μ ( x , y ) E z ν ( x , y ) ,
β μ β μ + i β μ ω t ,
ε ( x , y , z ) ε ( x , y , z , t ) ,
Δ ε ( x , y , z ) Δ ε ( x , y , z , t ) .
B = [ B ( 1 , 1 ) 0 0 B ( N , N ) ] ,
P = [ P ( 1 , 1 ) P ( 1 , N ) P ( N , 1 ) P ( N , N ) ] ,
K ̃ = [ K ̃ ( 1 , 1 ) K ̃ ( 1 , N ) K ̃ ( N , 1 ) K ̃ ( N , N ) ] .
B l , m ( q , p ) = β m ( p ) δ q p δ l m ,
P l , m ( q , p ) = z ̂ E t m ( p ) ( x , y ) × H t l ( q ) ( x , y ) + E t m ( q ) ( x , y ) × H t l ( p ) ( x , y ) d x d y ,
K ̃ l , m ( q , p ) = ω Δ ε ( q ) ( x , y , z , t ) [ E t l ( q ) ( x , y ) E t m ( p ) ( x , y ) ε ( p ) ( x , y ) ε ( x , y , z , t ) E z l ( q ) ( x , y ) E z m ( p ) ( x , y ) ] d x d y ,
p , q = 1 , , N ,     l = 1 , , n q ,     m = 1 , , n p .
2 z ̂ E t m ( p ) ( x , y ) × H t l ( q ) ( x , y ) d x d y = δ m l δ p q .

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