Abstract

The mathematical foundation of the iteration of the beam propagation method is described in this paper. Detailed analytical analyses are presented. These analyses explain further how the iteration of the beam propagation method works, and provide the reason why it works. Some additional comments are also presented on the formulation and practical implementation in analyzing Bragg gratings in different situations.

© 2010 Optical Society of America

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  2. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
    [CrossRef]
  3. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
    [CrossRef]
  4. T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  5. A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
    [CrossRef]
  6. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  7. K. O. Hill, “Aperiodic distributed-parameter waveguides for integrated optics,” Appl. Opt. 13, 1853–1856 (1974).
    [CrossRef] [PubMed]
  8. H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).
  9. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  10. G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. 24, 2407–2414 (1988).
    [CrossRef]
  11. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
    [CrossRef] [PubMed]
  12. S. Huang, M. LeBlanc, M. M. Ohn, and R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003–5009 (1995).
    [CrossRef] [PubMed]
  13. J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
    [CrossRef]
  14. H. Shu and M. Bass, “Modeling the reflection of a laser beam by a deformed highly reflective volume Bragg grating,” Appl. Opt. 46, 2930–2938 (2007).
    [CrossRef] [PubMed]
  15. H. Shu, S. Mokhov, B. Y. Zeldovich, and M. Bass, “More on analyzing the reflection of a laser beam by a deformed highly reflective volume Bragg grating using iteration of the beam propagation method,” Appl. Opt. 48, 22–27 (2009).
    [CrossRef]
  16. H. Shu, “Split step solution in the iteration of the beam propagation method for analyzing Bragg gratings,” Appl. Opt. 48, 4794–4800 (2009).
    [CrossRef] [PubMed]
  17. H. Shu, “Analytic and numeric modeling of diode laser pumped Yb:YAG laser oscillators and amplifiers,” Ph.D. dissertation (University of Central Florida, 2003).
  18. H. Shu and M. Bass, “Three-dimensional computer model for simulating realistic solid-state lasers,” Appl. Opt. 46, 5687–5697 (2007).
    [CrossRef] [PubMed]
  19. P. Kaczmarski and P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
    [CrossRef]
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).
  21. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).
  22. G. R. Hadley, “Wide-angle beam propagation using Padé approximant operators,” Opt. Lett. 17, 1426–1428 (1992).
    [CrossRef] [PubMed]
  23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
    [CrossRef]
  24. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  25. D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
    [CrossRef]
  26. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  27. H. Shu, “Quartic form of the slowly decaying imaginary distance beam propagation method,” Appl. Opt. 48, 4056–4061 (2009).
    [CrossRef] [PubMed]
  28. S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098–2105 (1994).
    [CrossRef]

2009 (3)

2008 (1)

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
[CrossRef]

2007 (2)

2003 (1)

H. Shu, “Analytic and numeric modeling of diode laser pumped Yb:YAG laser oscillators and amplifiers,” Ph.D. dissertation (University of Central Florida, 2003).

2001 (1)

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

1999 (1)

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

1995 (1)

1994 (2)

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098–2105 (1994).
[CrossRef]

J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
[CrossRef]

1993 (1)

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

1992 (1)

1988 (2)

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. 24, 2407–2414 (1988).
[CrossRef]

P. Kaczmarski and P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

1987 (1)

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1981 (1)

1980 (1)

1977 (2)

J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
[CrossRef]

A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

1976 (1)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).

1974 (1)

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1965 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Agrawal, G. P.

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. 24, 2407–2414 (1988).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Bass, M.

Bobeck, A. H.

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. 24, 2407–2414 (1988).
[CrossRef]

Chen, J. C.

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098–2105 (1994).
[CrossRef]

de Sterke, C. M.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
[CrossRef]

Hadley, G. R.

Hagan, D. J.

Hellstrom, J. E.

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
[CrossRef]

Hill, K. O.

Huang, S.

Huang, W. P.

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Jacobsson, B.

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
[CrossRef]

Jungling, S.

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098–2105 (1994).
[CrossRef]

Kaczmarski, P.

P. Kaczmarski and P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kong, J. A.

Kovsh, D. I.

Lagasse, P. E.

P. Kaczmarski and P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

Laurell, F.

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
[CrossRef]

LeBlanc, M.

Magnusson, R.

Measures, R. M.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. 70, 300–304 (1980).
[CrossRef]

Mokhov, S.

Nakamura, M.

A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

Ohn, M. M.

Pasiskevicius, V.

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
[CrossRef]

Poladian, L.

Sakuda, K.

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Shu, H.

Sipe, J. E.

Southwell, W. H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

Van Stryland, E. W.

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Xu, C. L.

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Yamada, M.

Yang, S.

Yariv, A.

A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

Zeldovich, B. Y.

Appl. Opt. (9)

K. O. Hill, “Aperiodic distributed-parameter waveguides for integrated optics,” Appl. Opt. 13, 1853–1856 (1974).
[CrossRef] [PubMed]

M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474–3478 (1987).
[CrossRef] [PubMed]

D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, “Nonlinear optical beam propagation for optical limiting,” Appl. Opt. 38, 5168–5180 (1999).
[CrossRef]

S. Huang, M. LeBlanc, M. M. Ohn, and R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003–5009 (1995).
[CrossRef] [PubMed]

H. Shu and M. Bass, “Modeling the reflection of a laser beam by a deformed highly reflective volume Bragg grating,” Appl. Opt. 46, 2930–2938 (2007).
[CrossRef] [PubMed]

H. Shu and M. Bass, “Three-dimensional computer model for simulating realistic solid-state lasers,” Appl. Opt. 46, 5687–5697 (2007).
[CrossRef] [PubMed]

H. Shu, S. Mokhov, B. Y. Zeldovich, and M. Bass, “More on analyzing the reflection of a laser beam by a deformed highly reflective volume Bragg grating using iteration of the beam propagation method,” Appl. Opt. 48, 22–27 (2009).
[CrossRef]

H. Shu, “Quartic form of the slowly decaying imaginary distance beam propagation method,” Appl. Opt. 48, 4056–4061 (2009).
[CrossRef] [PubMed]

H. Shu, “Split step solution in the iteration of the beam propagation method for analyzing Bragg gratings,” Appl. Opt. 48, 4794–4800 (2009).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (2)

H. Kogelnik, “Filter response of nonuniform almost-periodic structures,” Bell Syst. Tech. J. 55, 109–126 (1976).

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Electron. Lett. (1)

P. Kaczmarski and P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675–676 (1988).
[CrossRef]

IEEE J. Quantum Electron. (5)

G. P. Agrawal and A. H. Bobeck, “Modeling of distributed feedback semiconductor lasers with axially-varying parameters,” IEEE J. Quantum Electron. 24, 2407–2414 (1988).
[CrossRef]

J. E. Hellstrom, B. Jacobsson, V. Pasiskevicius, and F. Laurell, “Finite beams in reflective volume Bragg gratings: theory and experiments,” IEEE J. Quantum Electron. 44, 81–89 (2008).
[CrossRef]

A. Yariv and M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098–2105 (1994).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol. (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (3)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

H. Shu, “Analytic and numeric modeling of diode laser pumped Yb:YAG laser oscillators and amplifiers,” Ph.D. dissertation (University of Central Florida, 2003).

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Figures (3)

Fig. 1
Fig. 1

Schematic of the counterpropagating laser beams in a Bragg grating.

Fig. 2
Fig. 2

Schematic of the counterpropagating laser beams in a Bragg grating divided into two parts.

Fig. 3
Fig. 3

Schematic of the counterpropagating laser beams in a Bragg grating divided into four parts.

Equations (48)

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2 i k 0 n 0 A z = k 0 2 n 0 Δ n B e i φ ,
2 i k 0 n 0 B z = k 0 2 n 0 Δ n A e i φ ,
A ( z = 0 ) = A input ,
B ( z = L 0 ) = 0 ,
B ( z = 0 ) = i e i φ A input   tanh ( S ) ,
A ( 1 ) ( z ) = A input .
B ( 1 ) ( z ) = i k 0 Δ n 2 A input e i φ ( z L 0 ) .
A ( 2 ) ( z ) = A input + ( k 0 Δ n 2 ) 2 A input ( 1 2 z 2 L 0 z ) .
B ( 2 ) ( z ) = i k 0 Δ n 2 A input e i φ ( z L 0 ) + i ( k 0 Δ n 2 ) 3 A input e i φ ( 1 3 L 0 3 + 1 6 z 3 1 2 L 0 z 2 ) .
A ( 3 ) ( z ) = A input + ( k 0 Δ n 2 ) 2 A input ( 1 2 z 2 L 0 z ) + ( k 0 Δ n 2 ) 4 A input ( 1 3 L 0 3 z + 1 24 z 4 1 6 L 0 z 3 ) ,
B ( 3 ) ( z ) = i k 0 Δ n 2 A input e i φ ( z L 0 ) + i ( k 0 Δ n 2 ) 3 A input e i φ ( 1 3 L 0 3 + 1 6 z 3 1 2 L 0 z 2 ) + i ( k 0 Δ n 2 ) 5 A input e i φ ( 2 15 L 0 5 + 1 6 L 0 3 z 2 1 24 L 0 z 4 + 1 120 z 5 ) .
B ( 1 ) ( z = 0 ) = i A input e i φ S ,
B ( 2 ) ( z = 0 ) = i A input e i φ ( S 1 3 S 3 ) ,
B ( 3 ) ( z = 0 ) = i A input e i φ ( S 1 3 S 3 + 2 15 S 5 ) .
B ( ) ( z = 0 ) = i A input e i φ ( S 1 3 S 3 + 2 15 S 5 17 315 S 7 + ) .
B ( ) ( z = 0 ) = i A input e i φ   tanh ( S ) .
for   Part   1 :     A 1 ( z = 0 ) = A input ,     B 1 ( z = z 12 ) = B 2 ( z = z 12 ) ,
for   Part   2 :     A 2 ( z = z 12 ) = A 1 ( z = z 12 ) ,     B 2 ( z = L 0 ) = 0.
B 1 ( 0 ) ( z ) = B 1 , previous ( n ) ( z ) ,
A 1 ( z = 0 ) = A input ,
B 1 ( z = z 12 ) = B 2 ( z = z 12 ) = ξ previous + Δ ξ ,
A 1 ( 1 ) ( z ) = A 1 , previous ( n + 1 ) ( z ) .
B 1 ( 1 ) ( z ) = B 1 , previous ( n + 1 ) ( z ) + Δ ξ .
A 1 ( 2 ) ( z ) = A 1 , previous ( n + 2 ) ( z ) i k 0 Δ n 2 Δ ξ e i φ z .
B 1 ( 2 ) ( z ) = B 1 , previous ( n + 2 ) ( z ) + Δ ξ + ( k 0 Δ n 2 ) 2 Δ ξ 1 2 ( z 2 z 12 2 ) .
A 1 ( 3 ) ( z ) = A 1 , previous ( n + 3 ) ( z ) i k 0 Δ n 2 Δ ξ e i φ z i ( k 0 Δ n 2 ) 3 Δ ξ e i φ ( 1 6 z 3 1 2 z 12 2 z ) ,
B 1 ( 3 ) ( z ) = B 1 , previous ( n + 3 ) ( z ) + Δ ξ + ( k 0 Δ n 2 ) 2 Δ ξ 1 2 ( z 2 z 12 2 ) + ( k 0 Δ n 2 ) 4 Δ ξ ( 5 24 z 12 4 + 1 24 z 4 1 4 z 12 2 z 2 ) .
B 1 ( ) ( z = 0 ) = B 1 , previous ( ) ( z = 0 ) + Δ ξ ( 1 1 2 S 1 2 + 5 24 S 1 4 61 720 S 1 6 + ) = i A input e i φ ( S 1 1 3 S 1 3 + 2 15 S 1 5 17 315 S 1 7 + ) + ξ previous ( 1 1 2 S 1 2 + 5 24 S 1 4 61 720 S 1 6 + ) + Δ ξ ( 1 1 2 S 1 2 + 5 24 S 1 4 61 720 S 1 6 + ) ,
B 1 ( ) ( z = 0 ) = i A input e i φ   tanh ( S 1 ) + ξ previous   sec   h ( S 1 ) + Δ ξ   sec   h ( S 1 ) .
B 2 ( 0 ) ( z ) = B 2 , previous ( n ) ( z ) ,
A 2 ( z = z 12 ) = A 1 ( z = z 12 ) = η previous + Δ η ,
B 2 ( z = L 0 ) = 0 ,
A 2 ( 1 ) ( z ) = A 2 , previous ( n + 1 ) ( z ) + Δ η .
B 2 ( 1 ) ( z ) = B 2 , previous ( n + 1 ) ( z ) + i k 0 Δ n 2 Δ η e i φ ( z L 0 ) .
A 2 ( 2 ) ( z ) = A 2 , previous ( n + 2 ) ( z ) + Δ η + ( k 0 Δ n 2 ) 2 Δ η ( 1 2 z 2 1 2 z 12 2 L 0 z + L 0 z 12 ) .
B 2 ( 2 ) ( z ) = B 2 , previous ( n + 2 ) ( z ) + i k 0 Δ n 2 Δ η e i φ ( z L 0 ) + i ( k 0 Δ n 2 ) 3 Δ η e i φ ( 1 6 z 3 + 1 3 L 0 3 1 2 z 12 2 z + 1 2 z 12 2 L 0 1 2 L 0 z 2 + L 0 z 12 z L 0 2 z 12 ) .
B 2 ( ) ( z = z 12 ) = B 2 , previous ( ) ( z = z 12 ) i Δ η e i φ ( S 2 1 3 S 2 3 + 2 15 S 2 5 17 315 S 2 7 + ) = i η previous e i φ ( S 2 1 3 S 2 3 + 2 15 S 2 5 17 315 S 2 7 + ) i Δ η e i φ ( S 2 1 3 S 2 3 + 2 15 S 2 5 17 315 S 2 7 + ) ,
B 2 ( ) ( z = z 12 ) = i η previous e i φ tanh ( S 2 ) i Δ η e i φ   tanh ( S 2 ) .
2 i k 0 n 0 A z = k 0 2 n 0 Δ n B e i φ + 2 A x 2 + 2 A y 2 ,
2 i k 0 n 0 B z = k 0 2 n 0 Δ n A e i φ + 2 B x 2 + 2 B y 2 ,
B ( ) ( z = 0 ) = i η e i φ ( S 1 3 S 3 + 2 15 S 5 17 315 S 7 + ) + ξ ( 1 1 2 S 2 + 5 24 S 4 61 720 S 6 + ) + W ,
2 i k 0 n 0 A z = ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n B e i Δ q z + i φ + [ ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) 2 k 0 2 n 0 2 ] A + 2 A x 2 + 2 A y 2 ,
2 i k 0 n 0 B z = ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n A e i Δ q z i φ + [ ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) 2 k 0 2 n 0 2 ] B + 2 B x 2 + 2 B y 2 ,
n = n 0 + Δ n   cos [ ( q 0 + Δ q ) r + φ ] + Δ n T ,
A = R ( z ) ϕ m ( x , y ) ,
B = S ( z ) ϕ m ( x , y ) ,
k 0 2 [ ( n 0 + Δ n w ) 2 n 0 2 ] A + 2 A x 2 + 2 A y 2 = k 0 2 [ ( n eff m ) 2 n 0 2 ] A ,
k 0 2 [ ( n 0 + Δ n w ) 2 n 0 2 ] B + 2 B x 2 + 2 B y 2 = k 0 2 [ ( n eff m ) 2 n 0 2 ] B ,

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