Abstract

A further extension of the iteration method for beam propagation calculation is presented that can be applied for volume Bragg gratings (VBGs) with extremely large grating strength. A reformulation of the beam propagation formulation is presented for analyzing the reflection of a laser beam by a deformed VBG. These methods will be shown to be very accurate and efficient. A VBG with generic z-dependent distortion has been analyzed using these methods.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. 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] [PubMed]
  9. P. Kaczmarski and P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 24, 675-676 (1988).
    [CrossRef]
  10. H. L. Rao, R. Scarmozzino, and R. M. Osgood Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830-832 (1999).
    [CrossRef]
  11. Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
    [CrossRef]
  12. H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389-391 (2000).
  13. P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
    [CrossRef]
  14. J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868 (1992).
    [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] [PubMed]
  16. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, pages 2909-2947 (1969).
  17. L. B. Glebov, J. Lumeau, S. Mokhov, V. Smirnov, and B. Y. Zeldovich, “Reflection of light by composite volume holograms: Fresnel corrections and Fabry-Perot spectral filtering,” J. Opt. Soc. Am. A 25, 751-764 (2008).
    [CrossRef]
  18. E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).
  19. H. Shu, “Analytic and numeric modeling of diode laser pumped Yb:YAG laser oscillators and amplifiers,” Ph.D. dissertation (University of Central Florida, 2003).
  20. H. Shu and M. Bass, “Three-dimensional computer model for simulating realistic solid-state lasers,” Appl. Opt. 46, 5687-5697 (2007).
    [CrossRef] [PubMed]
  21. V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. , 11, 1513-1517(1993).
    [CrossRef]

2008 (6)

2007 (2)

2002 (1)

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

2001 (1)

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

2000 (1)

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389-391 (2000).

1999 (1)

H. L. Rao, R. Scarmozzino, and R. M. Osgood Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

1996 (1)

1993 (1)

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. , 11, 1513-1517(1993).
[CrossRef]

1992 (1)

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868 (1992).
[CrossRef]

1988 (1)

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

1985 (1)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

1982 (1)

1978 (1)

1969 (1)

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

Ahmed, S.

Andrusyak, O.

Bass, M.

Betty, I.

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389-391 (2000).

Boggess, T. F.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Ciapurin, I.

Clarkson, W. A.

El-Refaei, H.

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389-391 (2000).

Glebov, L.

Glebov, L. B.

Glytsis, E. N.

Gourevitch, A.

Guha, S.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Ho, P. L.

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

Hong, J.

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868 (1992).
[CrossRef]

Hostutler, D. A.

Huang, W. P.

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868 (1992).
[CrossRef]

Jelger, P.

Kaczmarski, P.

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

Kim, J. W.

Kogelnik, H.

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

Lagasse, P. E.

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

Laurell, F.

Lu, Y. Y.

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

Lumeau, J.

Makino, T.

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868 (1992).
[CrossRef]

McComb, T.

Mizrahi, V.

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. , 11, 1513-1517(1993).
[CrossRef]

Moharam, M. G.

Mokhov, S.

Osgood, R. M.

H. L. Rao, R. Scarmozzino, and R. M. Osgood Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Rao, H. L.

H. L. Rao, R. Scarmozzino, and R. M. Osgood Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Richardson, M.

Sahu, J. K.

Scarmozzino, R.

H. L. Rao, R. Scarmozzino, and R. M. Osgood Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Sevian, A.

Shu, H.

Sipe, J. E.

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. , 11, 1513-1517(1993).
[CrossRef]

Smirl, A. L.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Smirnov, V.

Soileau, M. J.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Sudesh, V.

Thylen, L.

Van Stryland, E. W.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Vanherzeele, H.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Venus, G.

Wang, P.

Wei, S. H.

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Woodall, M. A.

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Yevick, D.

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389-391 (2000).

D. Yevick and L. Thylen, “Analysis of gratings by the beam-propagation method,” J. Opt. Soc. Am. 72, 1084-1089 (1982).
[CrossRef]

Young, L.

Zeldovich, B. Y.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

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

Electron. Lett. (1)

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

IEEE Photon. Technol. Lett. (3)

H. L. Rao, R. Scarmozzino, and R. M. Osgood Jr., “A bidirectional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Y. Y. Lu and S. H. Wei, “A new iterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

P. L. Ho and Y. Y. Lu, “A stable bidirectional propagation method based on scattering operators,” IEEE Photon. Technol. Lett. 13, 1316-1318 (2001).
[CrossRef]

J. Lightwave Technol. (2)

J. Hong, W. P. Huang, and T. Makino, “On the transfer matrix method for distributed-feedback waveguide devices,” J. Lightwave Technol. 10, 1860-1868 (1992).
[CrossRef]

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. , 11, 1513-1517(1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, “Two photon absorption, nonlinear refraction, and optical limiting in semiconductors,” Opt. Eng. 24, 613-623 (1985).

Opt. Express (1)

Opt. Lett. (4)

Other (2)

H. El-Refaei, D. Yevick, and I. Betty, “Stable and noniterative bidirectional beam propagation method,” IEEE Photon. Technol. Lett. 12, 389-391 (2000).

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 (5)

Fig. 1
Fig. 1

Schematic of the four-part division of a VBG.

Fig. 2
Fig. 2

Calculated intensity reflection (stars) versus deviation from the Bragg wavelength, together with the analytic calculation (open circles) using coupled-wave theory [16].

Fig. 3
Fig. 3

Calculated intensity reflection (stars) versus deviation from λ = 1.064 μm , together with the calculation using the matrix method described in [17] (solid curve).

Fig. 4
Fig. 4

Schematic of the considered VBG with asymmetric distortion.

Fig. 5
Fig. 5

Calculated power reflection as a function of f.

Equations (15)

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n = n 0 + Δ n · cos ( q 0 · r + φ ) ,
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 ,
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 1 ( z = 0 ) = A input , B 2 ( z = L 0 2 ) = B 3 ( z = L 0 2 ) , A 3 ( z = L 0 2 ) = A 2 ( z = L 0 2 ) , B 4 ( z = L 0 ) = 0 ,
n = n 0 + Δ n · cos [ ( q 0 + Δ q ) · r + φ ] + Δ n T ,
2 E + ( k 0 + Δ k ) 2 ε r E = 0 ,
E = A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r .
k 0 n 0 e A + q 0 = k 0 n 0 e B ,
2 [ A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r ] + ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) 2 · [ A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r ] + ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n · [ A · e i ( k 0 n 0 e A + q 0 + Δ q ) · r i φ + A · e i ( k 0 n 0 e A q 0 Δ q ) · r + i φ + B · e i ( k 0 n 0 e B + q 0 + Δ q ) · r i φ + B · e i ( k 0 n 0 e B q 0 Δ q ) · r + i φ ] = 0.
2 [ A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r ] + ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) 2 · [ A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r ] + ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n · [ A · e i ( k 0 n 0 e A + q 0 + Δ q ) · r i φ + B · e i ( k 0 n 0 e B q 0 Δ q ) · r + i φ ] = 0.
2 [ A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r ] + ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) 2 · [ A · e i k 0 n 0 e A · r + B · e i k 0 n 0 e B · r ] + ( k 0 + Δ k ) 2 ( n 0 + Δ n T ) Δ n · [ A · e i ( k 0 n 0 e B + Δ q ) · r i φ + B · e i ( k 0 n 0 e A Δ q ) · r + i φ ] = 0.
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 .
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 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 .
L ( x , y ) = L 0 + Δ L · e r 2 w 0 2 ,
Δ q ( x , y , z ) = f · q 0 · e r 2 w 0 2 · e 4 z L 0 ,

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