Abstract

We investigate theoretically the interaction of two or more beams intersecting at a small angle in a photorefractive crystal with a local type of holographic recording. All necessary diffraction orders are taken into account. It is shown that for photorefractive ferroelectrics the number of diffracted beams can be quite large. For negative nonlinearlity the high exponential gain previously predicted for three- or four-wave mixing does not exist if all diffraction orders are considered.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  2. L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Crystals (Calderon, Oxford, 1996).
  3. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [CrossRef]
  4. N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
    [CrossRef]
  5. N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
    [CrossRef]
  6. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).
  7. L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
    [CrossRef]
  8. A. Roy and K. Singh, “Effect of optical activity on higher-order self-diffraction in an absorptive photorefractive medium: transmission geometry for two-wave mixing,” J. Appl. Phys. 71, 5332–5337 (1992).
    [CrossRef]
  9. K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
    [CrossRef]
  10. K. H. Ringhofer and L. Solymar, “3-wave and 4-wave forward mixing in photorefractive materials,” J. Appl. Phys. (N.Y.) 48, 395–400 (1988).
    [CrossRef]
  11. D. C. Jones and L. Solymar, “Comparison of 2-wave and 3-wave forward mixing in bismuth silicon oxide—theory and experiment,” IEEE J. Quantum Electron. 27, 121–127 (1991).
    [CrossRef]
  12. D. R. Erbshloe and L. Solymar, “Unidirectional ring resonator in photorefractive bismuth silicon oxide with two pump beams,” Appl. Phys. Lett. 53, 1135–1137 (1988).
    [CrossRef]
  13. D. R. Erbshloe and L. Solymar, “Linear resonator in photorefractive BSO with two pump beams,” Electron. Lett. 24, 683–684 (1988).
    [CrossRef]
  14. I. C. Khoo and T. H. Liu, “Probe beam amplification via 2-wave and 4-wave mixing in a nematic liquid-crystal film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
    [CrossRef]
  15. H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
    [CrossRef] [PubMed]
  16. I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
    [CrossRef]
  17. B. L. Volodin, B. Kippeln, K. Meerholz, N. Peyghambarian, N. V. Kukhtarev, and H. J. Caulfield, “Study of non-Bragg orders in dynamic self-diffraction in a photorefractive polymer: experiment, theory and applications,” J. Opt. Soc. Am. B 13, 2261–2267 (1996).
    [CrossRef]
  18. D. C. Jones, S. F. Lyuksyutov, and L. Solymar, “Three-wave and four-wave forward phase-conjugate imaging in photorefractive bismuth silicon oxide,” Opt. Lett. 15, 935–937 (1990).
    [CrossRef] [PubMed]
  19. J. Takacs, H. C. Ellin, and L. Solymar, “Multiple forward phase conjugation in photorefractive bismuth silicate crystal,” Opt. Commun. 93, 223–226 (1992).
    [CrossRef]
  20. S. L. Sochava, R. C. Troth, and S. I. Stepanov, “Holographic interferometry using −1 diffraction order in photorefractive Bi12SiO20 and Bi12TiO20 crystals,” J. Opt. Soc. Am. B 9, 1521–1527 (1992).
    [CrossRef]
  21. L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1986).

1998

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

1996

1992

J. Takacs, H. C. Ellin, and L. Solymar, “Multiple forward phase conjugation in photorefractive bismuth silicate crystal,” Opt. Commun. 93, 223–226 (1992).
[CrossRef]

S. L. Sochava, R. C. Troth, and S. I. Stepanov, “Holographic interferometry using −1 diffraction order in photorefractive Bi12SiO20 and Bi12TiO20 crystals,” J. Opt. Soc. Am. B 9, 1521–1527 (1992).
[CrossRef]

A. Roy and K. Singh, “Effect of optical activity on higher-order self-diffraction in an absorptive photorefractive medium: transmission geometry for two-wave mixing,” J. Appl. Phys. 71, 5332–5337 (1992).
[CrossRef]

1991

D. C. Jones and L. Solymar, “Comparison of 2-wave and 3-wave forward mixing in bismuth silicon oxide—theory and experiment,” IEEE J. Quantum Electron. 27, 121–127 (1991).
[CrossRef]

1990

1989

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

1988

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

K. H. Ringhofer and L. Solymar, “3-wave and 4-wave forward mixing in photorefractive materials,” J. Appl. Phys. (N.Y.) 48, 395–400 (1988).
[CrossRef]

D. R. Erbshloe and L. Solymar, “Unidirectional ring resonator in photorefractive bismuth silicon oxide with two pump beams,” Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

D. R. Erbshloe and L. Solymar, “Linear resonator in photorefractive BSO with two pump beams,” Electron. Lett. 24, 683–684 (1988).
[CrossRef]

1987

I. C. Khoo and T. H. Liu, “Probe beam amplification via 2-wave and 4-wave mixing in a nematic liquid-crystal film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

1977

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

1969

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

1966

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Au, L. B.

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Buse, K.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Caulfield, H. J.

Eichler, H. J.

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Ellin, H. C.

J. Takacs, H. C. Ellin, and L. Solymar, “Multiple forward phase conjugation in photorefractive bismuth silicate crystal,” Opt. Commun. 93, 223–226 (1992).
[CrossRef]

Erbshloe, D. R.

D. R. Erbshloe and L. Solymar, “Unidirectional ring resonator in photorefractive bismuth silicon oxide with two pump beams,” Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

D. R. Erbshloe and L. Solymar, “Linear resonator in photorefractive BSO with two pump beams,” Electron. Lett. 24, 683–684 (1988).
[CrossRef]

Gaylord, T. K.

Gerwens, A.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Glotz, M.

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Jones, D. C.

D. C. Jones and L. Solymar, “Comparison of 2-wave and 3-wave forward mixing in bismuth silicon oxide—theory and experiment,” IEEE J. Quantum Electron. 27, 121–127 (1991).
[CrossRef]

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, “Three-wave and four-wave forward phase-conjugate imaging in photorefractive bismuth silicon oxide,” Opt. Lett. 15, 935–937 (1990).
[CrossRef] [PubMed]

Khoo, I. C.

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

I. C. Khoo and T. H. Liu, “Probe beam amplification via 2-wave and 4-wave mixing in a nematic liquid-crystal film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

Kippeln, B.

Kogelnik, H.

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

Korneev, N.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Krätzig, E.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Kukhtarev, N.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Kukhtarev, N. V.

Kummrov, A.

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Lindquist, R. G.

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Liu, T. H.

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

I. C. Khoo and T. H. Liu, “Probe beam amplification via 2-wave and 4-wave mixing in a nematic liquid-crystal film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

Lyuksyutov, S. F.

Magnusson, R.

Markov, V.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Mayorga, D.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Meerholz, K.

Michael, R. R.

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Normandin, R.

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Odulov, S.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

Peyghambarian, N.

Richter, K.

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Ringhofer, K. H.

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

K. H. Ringhofer and L. Solymar, “3-wave and 4-wave forward mixing in photorefractive materials,” J. Appl. Phys. (N.Y.) 48, 395–400 (1988).
[CrossRef]

Roy, A.

A. Roy and K. Singh, “Effect of optical activity on higher-order self-diffraction in an absorptive photorefractive medium: transmission geometry for two-wave mixing,” J. Appl. Phys. 71, 5332–5337 (1992).
[CrossRef]

Singh, K.

A. Roy and K. Singh, “Effect of optical activity on higher-order self-diffraction in an absorptive photorefractive medium: transmission geometry for two-wave mixing,” J. Appl. Phys. 71, 5332–5337 (1992).
[CrossRef]

Sochava, S. L.

Solymar, L.

J. Takacs, H. C. Ellin, and L. Solymar, “Multiple forward phase conjugation in photorefractive bismuth silicate crystal,” Opt. Commun. 93, 223–226 (1992).
[CrossRef]

D. C. Jones and L. Solymar, “Comparison of 2-wave and 3-wave forward mixing in bismuth silicon oxide—theory and experiment,” IEEE J. Quantum Electron. 27, 121–127 (1991).
[CrossRef]

D. C. Jones, S. F. Lyuksyutov, and L. Solymar, “Three-wave and four-wave forward phase-conjugate imaging in photorefractive bismuth silicon oxide,” Opt. Lett. 15, 935–937 (1990).
[CrossRef] [PubMed]

K. H. Ringhofer and L. Solymar, “3-wave and 4-wave forward mixing in photorefractive materials,” J. Appl. Phys. (N.Y.) 48, 395–400 (1988).
[CrossRef]

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

D. R. Erbshloe and L. Solymar, “Linear resonator in photorefractive BSO with two pump beams,” Electron. Lett. 24, 683–684 (1988).
[CrossRef]

D. R. Erbshloe and L. Solymar, “Unidirectional ring resonator in photorefractive bismuth silicon oxide with two pump beams,” Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

Stepanov, S.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Stepanov, S. I.

Takacs, J.

J. Takacs, H. C. Ellin, and L. Solymar, “Multiple forward phase conjugation in photorefractive bismuth silicate crystal,” Opt. Commun. 93, 223–226 (1992).
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Troth, R. C.

Volodin, B. L.

Yang, X.

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Appl. Phys. B: Lasers Opt.

N. Korneev, D. Mayorga, S. Stepanov, A. Gerwens, K. Buse, and E. Krätzig, “Enhancement of the photorefractive effect by homogeneous pyroelectric fields,” Appl. Phys. B: Lasers Opt. 66, 393–396 (1998).
[CrossRef]

Appl. Phys. Lett.

K. H. Ringhofer and L. Solymar, “New gain mechanism for wave amplification in photorefractive materials,” Appl. Phys. Lett. 53, 1039–1040 (1988).
[CrossRef]

D. R. Erbshloe and L. Solymar, “Unidirectional ring resonator in photorefractive bismuth silicon oxide with two pump beams,” Appl. Phys. Lett. 53, 1135–1137 (1988).
[CrossRef]

Bell Syst. Tech. J.

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

Electron. Lett.

D. R. Erbshloe and L. Solymar, “Linear resonator in photorefractive BSO with two pump beams,” Electron. Lett. 24, 683–684 (1988).
[CrossRef]

IEEE J. Quantum Electron.

I. C. Khoo and T. H. Liu, “Probe beam amplification via 2-wave and 4-wave mixing in a nematic liquid-crystal film,” IEEE J. Quantum Electron. 23, 171–173 (1987).
[CrossRef]

D. C. Jones and L. Solymar, “Comparison of 2-wave and 3-wave forward mixing in bismuth silicon oxide—theory and experiment,” IEEE J. Quantum Electron. 27, 121–127 (1991).
[CrossRef]

L. B. Au and L. Solymar, “Higher diffraction orders in photorefractive materials,” IEEE J. Quantum Electron. 24, 162–168 (1988).
[CrossRef]

J. Appl. Phys.

A. Roy and K. Singh, “Effect of optical activity on higher-order self-diffraction in an absorptive photorefractive medium: transmission geometry for two-wave mixing,” J. Appl. Phys. 71, 5332–5337 (1992).
[CrossRef]

J. Appl. Phys. (N.Y.)

K. H. Ringhofer and L. Solymar, “3-wave and 4-wave forward mixing in photorefractive materials,” J. Appl. Phys. (N.Y.) 48, 395–400 (1988).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

JETP Lett.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307–310 (1966).

Opt. Commun.

N. Kukhtarev, V. Markov, and S. Odulov, “Transient energy transfer during hologram formation in LiNbO3 in external electric field,” Opt. Commun. 23, 338–343 (1977).
[CrossRef]

J. Takacs, H. C. Ellin, and L. Solymar, “Multiple forward phase conjugation in photorefractive bismuth silicate crystal,” Opt. Commun. 93, 223–226 (1992).
[CrossRef]

Opt. Lett.

Phys. Rev. A

H. J. Eichler, M. Glotz, A. Kummrov, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent mixing in silicon,” Phys. Rev. A 35, 4673–4678 (1987).
[CrossRef] [PubMed]

Phys. Rev. B

I. C. Khoo, R. Normandin, T. H. Liu, R. R. Michael, and R. G. Lindquist, “Degenerate multiwave mixing and phase conjugation in silicon,” Phys. Rev. B 40, 7759–7766 (1989).
[CrossRef]

Other

L. Solymar and D. J. Cooke, Volume Holography and Volume Gratings (Academic, New York, 1986).

L. Solymar, D. J. Webb, and A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Crystals (Calderon, Oxford, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Beam diffraction on a thick grating can be represented as diffraction on a number of thin gratings stacked in the z direction. The evolution of the -1st- and the 1-st-order amplitudes as additional layers are added is shown by the thinner arrows.

Fig. 2
Fig. 2

Geometry of the multiple-beam interaction.

Fig. 3
Fig. 3

Amplification of the weak beam that results from its interaction with a pump as a function of the angle between beams for four values of medium gain parameter γ.

Fig. 4
Fig. 4

Number of diffraction orders that result from the interaction of two beams with equal amplitudes that have intensities higher than 1% of the initial pump beam intensity. Crystal length, 1 cm; γ=-3 cm-1 for filled squares, γ=+3 cm-1 for open squares.

Fig. 5
Fig. 5

Intensities of three orders of diffraction as a function of propagation length for an angle between beams of 0.01 rad and γ=0.3 cm-1. The principal period for this angle is 1.42 cm. The intensity of the -1st order is multiplied by 10, and that of the -2nd order by 1000.

Fig. 6
Fig. 6

Same as Fig. 5 but for γ=4 cm-1. Solid curve, 0th order; dashed curve, -1st order; dotted curve, -2nd order. Twenty-four diffraction orders are taken into account. No multiplication factors are used.

Fig. 7
Fig. 7

Weak-beam behavior (0th order) in a medium with negative nonlinearity for four numbers of beams, N. The initial amplitude of the pump (1st and -1st orders) is 1; that of the signal beam is 0.001. The angle between the 0th and the 1st orders is 0.01 rad; γ=-2 cm-1.

Fig. 8
Fig. 8

Interaction of two pumps and two signal beams. The initial amplitude of the pumps is 1; that of the signal beams is 0.001. N is the number of diffraction orders taken into account. The angle between the closest orders is 0.0057 rad; γ=-2 cm-1. For N=4, this is the optimal angle that corresponds to 1638 times amplification after 1-cm propagation length (not shown).

Equations (14)

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

σ0-σ1=±K.
E=i=-Si(z)exp(-jσir),
σi=σ0-iK,
n(r)=n0+½h=1[Δnh(z)exp(jhKr)+Δnh*(z)exp(-jhKr)],
cos ΘzSi+jϑiSi+jh=1(Si-hκh+Si+hκh*)=0,
ϑi=i(1-i)K2/(2β0),
κh(z)=k0Δnh(z)/2
κh=γ2Si*Si+hSi*Si,
zS-1+jϑ-1S-1+j2γ(S-1+S1*)=0,
zS1+j2γ[S-1*+S1]=0,
λ1,2=½[jϑ-1±(-ϑ-12-8γϑ-1)1/2]
S1(z)=S1(0)cosh(2 γ z)exp(-2 j γ z).
z S-1+jϑ-1S-1+jγ exp(-j γ z)=0,
S-1(z)=γγ-ϑ-1[exp(-jγz)-exp(-jϑ-1z)].

Metrics