Abstract

The split step method is applied to the iteration of the beam propagation method for analyzing the reflection of a laser beam by a volume Bragg grating. The application of the split step method is made possible by a way to properly treat the grating coupling terms in the paraxial wave equations. This method is demonstrated to be accurate in addition to efficient and robust. After this modification, the iteration of the beam propagation method is suitable for analyzing finite beams in volume Bragg gratings, for which the grating strength might be large. It is also suitable for analyzing Bragg gratings with nonuniform grating structures.

© 2009 Optical Society of America

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References

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2947 (1969).
  2. S. Ahmed and E. N. Glytsis, “Comparison of beam propagation method and rigorous coupled-wave analysis for single and multiplexed volume gratings,” Appl. Opt. 35, 4426-4435 (1996).
    [CrossRef]
  3. P. Yeh, Optical Waves in Layered Media (Wiley, 2005).
  4. 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]
  5. R. Kashyap, Fiber Bragg Gratings (Academic, 1999), Chap. 4.
  6. 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]
  7. S. Huang, M. LeBlanc, M. M. Ohn, and R. M. Measures, “Bragg intragrating structural sensing,” Appl. Opt. 34, 5003-5009 (1995).
    [CrossRef]
  8. M. Prabhugoud and K. Peters, “Modified transfer matrix formulation for Bragg grating strain sensors,” J. Lightwave Technol. 22, 2302-2309 (2004).
    [CrossRef]
  9. G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6, 995-997 (1994).
    [CrossRef]
  10. T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894-937(1985).
    [CrossRef]
  11. 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]
  12. R. S. Chu, J. A. Kong, and T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555-1561 (1977).
    [CrossRef]
  13. B. Benlarbi, P. St. J. Russell, and L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63-72 (1982).
    [CrossRef]
  14. H. T. Hsieh, W. H. Liu, F. Havermeyer, C. Moser, and D. Psaltis, “Beam-widthdependent filtering properties of strong volume holographic gratings,” Appl. Opt. 45, 3774-3780 (2006).
    [CrossRef]
  15. 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]
  16. 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]
  17. D. Yevick and B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461-462(1989).
    [CrossRef]
  18. D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109-112 (1990).
    [CrossRef]
  19. M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990-3998 (1978).
    [CrossRef]
  20. J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics 10, 129-160 (1976).
    [CrossRef]
  21. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000). Chap. 7.
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    [CrossRef]
  23. H. Shu and M. Bass, “Analysis and optimization of the numerical calculation in the slowly decaying imaginary distance beam propagation method,” J. Lightwave Technol. 26, 3199-3206 (2008).
    [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. H. Shu and M. Bass, “Three-dimensional computer model for simulating realistic solid-state lasers,” Appl. Opt. 46, 5687-5697 (2007).
    [CrossRef]
  26. S. T. Hendow and S. A. Shakir, “Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetric systems,” Appl. Opt. 25, 1759-1764 (1986).
    [CrossRef]
  27. A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).
    [CrossRef]
  28. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989).
  29. H. A. Van Der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BICG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631-644(1992).
    [CrossRef]
  30. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598-1600 (1995).
    [CrossRef]

2009 (1)

2008 (2)

H. Shu and M. Bass, “Analysis and optimization of the numerical calculation in the slowly decaying imaginary distance beam propagation method,” J. Lightwave Technol. 26, 3199-3206 (2008).
[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]

2007 (2)

2006 (1)

2004 (1)

1996 (1)

1995 (2)

1994 (2)

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]

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

1992 (2)

W. P. Huang and C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

H. A. Van Der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BICG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631-644(1992).
[CrossRef]

1990 (1)

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109-112 (1990).
[CrossRef]

1989 (1)

D. Yevick and B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461-462(1989).
[CrossRef]

1987 (1)

1986 (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]

1982 (1)

B. Benlarbi, P. St. J. Russell, and L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63-72 (1982).
[CrossRef]

1980 (1)

1978 (1)

1977 (1)

1976 (1)

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics 10, 129-160 (1976).
[CrossRef]

1969 (1)

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

1967 (1)

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

Ahmed, S.

Bass, M.

Benlarbi, B.

B. Benlarbi, P. St. J. Russell, and L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63-72 (1982).
[CrossRef]

Chaudhuri, S. K.

W. P. Huang and C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Chi, S.

Chu, R. S.

Chu, S. T.

W. P. Huang and C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

de Sterke, C. M.

Feit, M. D.

M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990-3998 (1978).
[CrossRef]

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics 10, 129-160 (1976).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989).

Fleck, J. A.

M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990-3998 (1978).
[CrossRef]

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics 10, 129-160 (1976).
[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]

Glytsis, E. N.

Goldberg, A.

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).
[CrossRef]

Guo, Q.

Havermeyer, F.

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]

Hendow, S. T.

Hermansson, B.

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109-112 (1990).
[CrossRef]

D. Yevick and B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461-462(1989).
[CrossRef]

Hsieh, H. T.

Huang, S.

Huang, W. P.

W. P. Huang and C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295-305 (1992).
[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]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, 1999), Chap. 4.

Kogelnik, H.

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

Kong, J. A.

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.

Liu, W. H.

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.

Morris, J. R.

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics 10, 129-160 (1976).
[CrossRef]

Moser, C.

Ohn, M. M.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000). Chap. 7.

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]

Peters, K.

Poladian, L.

Prabhugoud, M.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989).

Psaltis, D.

Radic, S.

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

Russell, P. St. J.

B. Benlarbi, P. St. J. Russell, and L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63-72 (1982).
[CrossRef]

Sakuda, K.

Schey, H. M.

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).
[CrossRef]

Schwartz, J. L.

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).
[CrossRef]

Shakir, S. A.

Shu, H.

Sipe, J. E.

Solymar, L.

B. Benlarbi, P. St. J. Russell, and L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63-72 (1982).
[CrossRef]

Tamir, T.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989).

Van Der Vorst, H. A.

H. A. Van Der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BICG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631-644(1992).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989).

Xu, C. L.

W. P. Huang and C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. 10, 295-305 (1992).
[CrossRef]

Yamada, M.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

Yevick, D.

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109-112 (1990).
[CrossRef]

D. Yevick and B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461-462(1989).
[CrossRef]

Zeldovich, B. Y.

Am. J. Phys. (1)

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).
[CrossRef]

Appl. Opt. (9)

M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990-3998 (1978).
[CrossRef]

S. T. Hendow and S. A. Shakir, “Recursive numerical solution for nonlinear wave propagation in fibers and cylindrically symmetric systems,” Appl. Opt. 25, 1759-1764 (1986).
[CrossRef]

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]

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

S. Ahmed and E. N. Glytsis, “Comparison of beam propagation method and rigorous coupled-wave analysis for single and multiplexed volume gratings,” Appl. Opt. 35, 4426-4435 (1996).
[CrossRef]

H. T. Hsieh, W. H. Liu, F. Havermeyer, C. Moser, and D. Psaltis, “Beam-widthdependent filtering properties of strong volume holographic gratings,” Appl. Opt. 45, 3774-3780 (2006).
[CrossRef]

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]

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

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]

Appl. Phys. B (1)

B. Benlarbi, P. St. J. Russell, and L. Solymar, “Bragg diffraction of finite beams by thick gratings: two rival theories,” Appl. Phys. B 28, 63-72 (1982).
[CrossRef]

Applied Physics (1)

J. A. Fleck, Jr., J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Applied Physics 10, 129-160 (1976).
[CrossRef]

Bell Syst. Tech. J. (1)

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

Electron. Lett. (1)

D. Yevick and B. Hermansson, “Split-step finite difference analysis of rib waveguides,” Electron. Lett. 25, 461-462(1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Yevick and B. Hermansson, “Efficient beam propagation techniques,” IEEE J. Quantum Electron. 26, 109-112 (1990).
[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]

IEEE Photonics Technol. Lett. (1)

G. P. Agrawal and S. Radic, “Phase-shifted fiber Bragg gratings and their application for wavelength demultiplexing,” IEEE Photonics Technol. Lett. 6, 995-997 (1994).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. (2)

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]

SIAM J. Sci. Statist. Comput. (1)

H. A. Van Der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BICG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 13, 631-644(1992).
[CrossRef]

Other (4)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge U. Press, 1989).

P. Yeh, Optical Waves in Layered Media (Wiley, 2005).

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2000). Chap. 7.

R. Kashyap, Fiber Bragg Gratings (Academic, 1999), Chap. 4.

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Figures (4)

Fig. 1
Fig. 1

Schematic of the considered VBG.

Fig. 2
Fig. 2

(a) Numerically calculated reflectance for the total laser power versus deviation from the Bragg wavelength at λ = 1.064 μm for the normally incident Gaussian beam with a beam waist radius of 300 μm (stars), together with the analytically calculated reflectance for the intensity for a normally incident plane wave using coupled wave theory [1] (open circles). (b) Plot of the numerically calculated reflectance for the total laser power versus the deviation from λ = 1.064 μm for the normally incident Gaussian beam with a beam waist radius of 300 μm (stars), together with the calculated reflectance for the intensity for the same nonuniform VBG using a plane wave input presented in [16] (open circles).

Fig. 3
Fig. 3

Numerically calculated reflectance for the total laser power versus deviation from the Bragg wavelength at λ = 1.064 μm for a normally incident Gaussian beam with a beam waist radius of 30 μm (stars), together with the analytically calculated reflectance for the intensity for a normally incident plane wave using coupled wave theory [1] (open circles).

Fig. 4
Fig. 4

Field amplitude patterns of the calculated output beams at z = 0 . (a)  | B | on the x axis for y = 0 for λ = 1064 nm + 0.1 nm ; (b)  | B | on the xy plane for λ = 1064 nm + 0.1 nm ; (c)  | B | on the x axis for y = 0 for λ = 1064 nm 0.1 nm ; (d)  | B | on the xy plane for λ = 1064 nm 0.1 nm ; (e)  | B | on the x axis for y = 0 for λ = 1064 nm ; (f)  | B | on the xy plane for λ = 1064 nm . The dotted lines in (a), (c), and (e) represent | A input | for the input Gaussian beam. All the plots are scaled for comparison. The wavelengths chosen for this figure are also indicated in Fig. 3 by arrows.

Equations (17)

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

n = n 0 + Δ n · cos [ ( q 0 + Δ q ) · r + φ ] + Δ n T ,
2 i k 0 n 0 A z = P 1 · B + P 3 · A + 2 A x 2 + 2 A y 2 , 2 i k 0 n 0 B z = P 2 · A + P 3 · B + 2 B x 2 + 2 B y 2 ,
P 1 = ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n · e i Δ q z + i φ , P 2 = ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n · e i Δ q z i φ , P 3 = ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) 2 k 0 2 n 0 2 .
2 i k 0 n 0 A z = P 1 · B A · A + P 3 · A 2 A x 2 + 2 A y 2 , 2 i k 0 n 0 B z = P 2 · A B · B + P 3 · B 2 B x 2 + 2 B y 2 .
A ( x , y , z + Δ z ) = D x y · exp [ i 2 k 0 n 0 · z z + Δ z ( P 1 · B A + P 3 ) · d z ] · D x y · A ( x , y , z ) + O [ ( Δ z ) 3 ] ,
D x y = exp [ i Δ z 4 k 0 n 0 · ( 2 x 2 + 2 y 2 ) ] 1 i Δ z 8 k 0 n 0 · ( 2 x 2 + 2 y 2 ) 1 + i Δ z 8 k 0 n 0 · ( 2 x 2 + 2 y 2 )
ϕ ( x , y , z ) = D x y · A ( x , y , z )
[ 1 + i Δ z 8 k 0 n 0 · ( 2 x 2 + 2 y 2 ) ] · ϕ ( x , y , z ) = [ 1 i Δ z 8 k 0 n 0 · ( 2 x 2 + 2 y 2 ) ] · A ( x , y , z )
ϕ ( x , y , z + Δ z ) = exp [ i 2 k 0 n 0 · z z + Δ z ( P 1 · B A + P 3 ) · d z ] · ϕ ( x , y , z ) ,
d ϕ d z = i 2 k 0 n 0 · ( P 1 · B + P 3 · ϕ )
A ( x , y , z + Δ z ) = D x y · ϕ ( x , y , z + Δ z )
[ 1 + i Δ z 8 k 0 n 0 · ( 2 x 2 + 2 y 2 ) ] · A ( x , y , z + Δ z ) = [ 1 i Δ z 8 k 0 n 0 · ( 2 x 2 + 2 y 2 ) ] · ϕ ( x , y , z + Δ z )
d ϕ d z = i 2 k 0 n 0 · ( P 1 · B A · ϕ + P 3 · ϕ )
2 A ( x , y , z ) x 2 A ( x , y , z ) w 2 , 2 A ( x , y , z ) y 2 A ( x , y , z ) w 2 , 2 ϕ ( x , y , z ) x 2 ϕ ( x , y , z ) w 2 , 2 ϕ ( x , y , z ) y 2 ϕ ( x , y , z ) w 2 , 2 A ( x , y , z + Δ z ) x 2 A ( x , y , z + Δ z ) w 2 , 2 A ( x , y , z + Δ z ) y 2 A ( x , y , z + Δ z ) w 2 , 2 ϕ ( x , y , z + Δ z ) x 2 ϕ ( x , y , z + Δ z ) w 2 , 2 ϕ ( x , y , z + Δ z ) y 2 ϕ ( x , y , z + Δ z ) w 2 .
ϕ ( x , y , z ) A ( x , y , z ) 1 2 i Δ z 8 k 0 n 0 w 2 1 + 2 i Δ z 8 k 0 n 0 w 2 , ϕ ( x , y , z + Δ z ) A ( x , y , z + Δ z ) 1 + 2 i Δ z 8 k 0 n 0 w 2 1 2 i Δ z 8 k 0 n 0 w 2 .
2 Δ z 8 k 0 n 0 w 2 0.002.
A input = A 0 · exp ( x 2 + y 2 w 0 2 ) ,

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