Abstract

Chirped quasi-phase-matched (QPM) gratings offer essentially constant gain over wide bandwidths, making them promising candidates for short-pulse optical parametric amplifiers. However, we discovered that high-gain noncollinear processes can compete with the desired broadband gain of such amplifiers. Here, we investigate these noncollinear gain-guided modes both numerically and analytically, including longitudinal nonuniformity of the phase-matching profile, lateral localization of the pump beam, and the noncollinear propagation of the interacting waves.

© 2010 Optical Society of America

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References

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  1. I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
    [CrossRef]
  2. J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
    [CrossRef]
  3. I. Jovanovic, B. J. Comaskey, C. A. Ebbers, R. A. Bonner, D. M. Pennington, and E. C. Morse, “Optical parametric chirped-pulse amplifier as an alternative to Ti:sapphire regenerative amplifiers,” Appl. Opt. 41, 2923-2929 (2002).
    [CrossRef] [PubMed]
  4. S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
    [CrossRef]
  5. K. L. Baker, “Single-pass gain in a chirped quasi-phase-matched optical parametric oscillator,” Appl. Phys. Lett. 82, 3841-3843 (2003).
    [CrossRef]
  6. M. Charbonneau-Lefort, M. M. Fejer, and B. Afeyan, “Tandem chirped quasi-phase-matching grating optical parametric amplifier design for simultaneous group delay and gain control,” Opt. Lett. 30, 634-636 (2005).
    [CrossRef] [PubMed]
  7. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings I: practical design formulas,” J. Opt. Soc. Am. B 25, 463-480 (2008).
    [CrossRef]
  8. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Optical parametric amplifiers using chirped quasi-phase-matching gratings II: space-time evolution of light pulses,” J. Opt. Soc. Am. B 25, 683-700 (2008).
    [CrossRef]
  9. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
    [CrossRef]
  10. R. L. Byer, Nonlinear Optics, P.G.Harper and B.S.Wherrett, eds., (Academic, 1977).
  11. T. Kobayashi and A. Baltuska, “Sub-5-fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
    [CrossRef]
  12. M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
    [CrossRef]
  13. M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
    [CrossRef]
  14. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B 25, 1402-1413 (2008).
    [CrossRef]
  15. D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
    [CrossRef]
  16. M. M. Sushchik and G. I. Freidman, “The effect of nonuniformity of the amplitude and phase distribution of the pumping radiation on the spatial locking of parametrically amplified waves,” Radiophysics 13, 1043-1047 (1970).
    [CrossRef]
  17. S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
    [CrossRef]
  18. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  19. B. B. Afeyan and E. A. Williams, “A variational approach to parametric instabilities in inhomogeneous plasmas I: two model problems,” Phys. Plasmas 4, 3788-3802 (1997).
    [CrossRef]
  20. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).
  21. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Butterworth-Heinemann Ltd., 1977).
  22. B. B. Afeyan and E. A. Williams, “Stimulated Raman sidescattering with the effects of oblique incidence,” Phys. Fluids 28, 3397-3408 (1985).
    [CrossRef]
  23. D. Pesme, G. Laval, and R. Pellat, “Parametric instabilities in bounded plasmas,” Phys. Rev. Lett. 31, 203-206 (1973).
    [CrossRef]
  24. D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
    [CrossRef]
  25. F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
    [CrossRef]

2008 (3)

2005 (1)

2003 (1)

K. L. Baker, “Single-pass gain in a chirped quasi-phase-matched optical parametric oscillator,” Appl. Phys. Lett. 82, 3841-3843 (2003).
[CrossRef]

2002 (2)

1999 (1)

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

1998 (1)

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
[CrossRef]

1997 (2)

B. B. Afeyan and E. A. Williams, “A variational approach to parametric instabilities in inhomogeneous plasmas I: two model problems,” Phys. Plasmas 4, 3788-3802 (1997).
[CrossRef]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

1985 (1)

B. B. Afeyan and E. A. Williams, “Stimulated Raman sidescattering with the effects of oblique incidence,” Phys. Fluids 28, 3397-3408 (1985).
[CrossRef]

1977 (1)

F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
[CrossRef]

1974 (1)

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

1973 (2)

D. Pesme, G. Laval, and R. Pellat, “Parametric instabilities in bounded plasmas,” Phys. Rev. Lett. 31, 203-206 (1973).
[CrossRef]

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

1972 (1)

M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
[CrossRef]

1970 (1)

M. M. Sushchik and G. I. Freidman, “The effect of nonuniformity of the amplitude and phase distribution of the pumping radiation on the spatial locking of parametrically amplified waves,” Radiophysics 13, 1043-1047 (1970).
[CrossRef]

1967 (1)

D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
[CrossRef]

1966 (1)

S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

Afeyan, B.

Afeyan, B. B.

B. B. Afeyan and E. A. Williams, “A variational approach to parametric instabilities in inhomogeneous plasmas I: two model problems,” Phys. Plasmas 4, 3788-3802 (1997).
[CrossRef]

B. B. Afeyan and E. A. Williams, “Stimulated Raman sidescattering with the effects of oblique incidence,” Phys. Fluids 28, 3397-3408 (1985).
[CrossRef]

Backus, S.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
[CrossRef]

Baker, K. L.

K. L. Baker, “Single-pass gain in a chirped quasi-phase-matched optical parametric oscillator,” Appl. Phys. Lett. 82, 3841-3843 (2003).
[CrossRef]

Baltuska, A.

T. Kobayashi and A. Baltuska, “Sub-5-fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).

Bers, A.

F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
[CrossRef]

Bobroff, D. L.

D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
[CrossRef]

Bonner, R. A.

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

R. L. Byer, Nonlinear Optics, P.G.Harper and B.S.Wherrett, eds., (Academic, 1977).

Chambers, F. W.

F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
[CrossRef]

Charbonneau-Lefort, M.

Collier, J.

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

Collier, J. L.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Comaskey, B. J.

Danson, C. N.

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

DuBois, D. F.

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

Durfee, C. G.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
[CrossRef]

Ebbers, C. A.

Fejer, M. M.

Forslund, D. W.

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

Freidman, G. I.

M. M. Sushchik and G. I. Freidman, “The effect of nonuniformity of the amplitude and phase distribution of the pumping radiation on the spatial locking of parametrically amplified waves,” Radiophysics 13, 1043-1047 (1970).
[CrossRef]

Harris, S. E.

S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

Haus, H. A.

D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
[CrossRef]

Hernandez-Gomez, C.

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

Jovanovic, I.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Kapteyn, H. C.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
[CrossRef]

Kobayashi, T.

T. Kobayashi and A. Baltuska, “Sub-5-fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Butterworth-Heinemann Ltd., 1977).

Langley, A. J.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Laval, G.

D. Pesme, G. Laval, and R. Pellat, “Parametric instabilities in bounded plasmas,” Phys. Rev. Lett. 31, 203-206 (1973).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Butterworth-Heinemann Ltd., 1977).

Liu, C. S.

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Matousek, P.

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Morse, E. C.

Murnane, M. M.

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
[CrossRef]

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).

Pellat, R.

D. Pesme, G. Laval, and R. Pellat, “Parametric instabilities in bounded plasmas,” Phys. Rev. Lett. 31, 203-206 (1973).
[CrossRef]

Pennington, D. M.

Pesme, D.

D. Pesme, G. Laval, and R. Pellat, “Parametric instabilities in bounded plasmas,” Phys. Rev. Lett. 31, 203-206 (1973).
[CrossRef]

Rosenbluth, M. N.

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
[CrossRef]

Ross, I. N.

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

Sushchik, M. M.

M. M. Sushchik and G. I. Freidman, “The effect of nonuniformity of the amplitude and phase distribution of the pumping radiation on the spatial locking of parametrically amplified waves,” Radiophysics 13, 1043-1047 (1970).
[CrossRef]

Towrie, M.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Walczak, J.

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

White, R. B.

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

Williams, E. A.

B. B. Afeyan and E. A. Williams, “A variational approach to parametric instabilities in inhomogeneous plasmas I: two model problems,” Phys. Plasmas 4, 3788-3802 (1997).
[CrossRef]

B. B. Afeyan and E. A. Williams, “Stimulated Raman sidescattering with the effects of oblique incidence,” Phys. Fluids 28, 3397-3408 (1985).
[CrossRef]

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

Appl. Opt. (2)

J. Collier, C. Hernandez-Gomez, I. N. Ross, P. Matousek, C. N. Danson, and J. Walczak, “Evaluation of an ultrabroadband high-gain amplification technique for chirped pulse amplification facilities,” Appl. Opt. 36, 7486-7493 (1999).
[CrossRef]

I. Jovanovic, B. J. Comaskey, C. A. Ebbers, R. A. Bonner, D. M. Pennington, and E. C. Morse, “Optical parametric chirped-pulse amplifier as an alternative to Ti:sapphire regenerative amplifiers,” Appl. Opt. 41, 2923-2929 (2002).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

K. L. Baker, “Single-pass gain in a chirped quasi-phase-matched optical parametric oscillator,” Appl. Phys. Lett. 82, 3841-3843 (2003).
[CrossRef]

S. E. Harris, “Proposed backward wave oscillator in the infrared,” Appl. Phys. Lett. 9, 114-116 (1966).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

J. Appl. Phys. (1)

D. L. Bobroff and H. A. Haus, “Impulse response of active coupled wave systems,” J. Appl. Phys. 38, 390-403 (1967).
[CrossRef]

J. Opt. Soc. Am. B (3)

Meas. Sci. Technol. (1)

T. Kobayashi and A. Baltuska, “Sub-5-fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
[CrossRef]

Opt. Commun. (1)

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, “The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers,” Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (2)

B. B. Afeyan and E. A. Williams, “Stimulated Raman sidescattering with the effects of oblique incidence,” Phys. Fluids 28, 3397-3408 (1985).
[CrossRef]

F. W. Chambers and A. Bers, “Parametric interactions in an inhomogeneous medium of finite extent with abrupt boundaries,” Phys. Fluids 20, 466-468 (1977).
[CrossRef]

Phys. Plasmas (1)

B. B. Afeyan and E. A. Williams, “A variational approach to parametric instabilities in inhomogeneous plasmas I: two model problems,” Phys. Plasmas 4, 3788-3802 (1997).
[CrossRef]

Phys. Rev. Lett. (4)

D. Pesme, G. Laval, and R. Pellat, “Parametric instabilities in bounded plasmas,” Phys. Rev. Lett. 31, 203-206 (1973).
[CrossRef]

D. F. DuBois, D. W. Forslund, and E. A. Williams, “Parametric instabilities in finite inhomogeneous media,” Phys. Rev. Lett. 33, 1013-1016 (1974).
[CrossRef]

M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
[CrossRef]

M. N. Rosenbluth, R. B. White, and C. S. Liu, “Temporal evolution of a three-wave parametric instability,” Phys. Rev. Lett. 31, 1190-1193 (1973).
[CrossRef]

Radiophysics (1)

M. M. Sushchik and G. I. Freidman, “The effect of nonuniformity of the amplitude and phase distribution of the pumping radiation on the spatial locking of parametrically amplified waves,” Radiophysics 13, 1043-1047 (1970).
[CrossRef]

Rev. Sci. Instrum. (1)

S. Backus, C. G. Durfee, M. M. Murnane, and H. C. Kapteyn, “High power ultrafast lasers,” Rev. Sci. Instrum. 69, 1207-1223 (1998).
[CrossRef]

Other (4)

R. L. Byer, Nonlinear Optics, P.G.Harper and B.S.Wherrett, eds., (Academic, 1977).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Butterworth-Heinemann Ltd., 1977).

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Figures (21)

Fig. 1
Fig. 1

Numerical results for a gain parameter λ = 4 and the degenerate case α 1 = 1 , α 2 = 1 (i.e. k 1 = k 2 ): (a) 2-D plots of the amplitude of the signal and idler beams, A 1 and A 2 (logarithmic scale); (b) Peak amplitude; (c) Mode profile at position z ¯ = 15 .

Fig. 2
Fig. 2

Signal and idler mode shape at various positions along the direction of propagation.

Fig. 3
Fig. 3

Growth rate, K ¯ , as a function of (a) the gain parameter λ in the degenerate case α 1 = 1 , α 2 = 1 , and (b) the asymmetry parameter α 1 for λ = 2 and λ = 4 .

Fig. 4
Fig. 4

Graphical solution of the eigenvalue condition tan ( u π 2 ) = 2 λ ( u 2 ) 2 u , Eq. (59), for λ = 10 .

Fig. 5
Fig. 5

Normalized growth rate, K ¯ , of the fundamental mode ( n = 0 ) vs λ. This plot shows a comparison between the WKB quantization condition evaluated numerically in the case of a gaussian, top-hat (“flat”), and parabolic pump profiles; the analytical formula obtained assuming a parabolic pump in the infinite pump-strength limit, Eq. (69); and the numerical simulations using a gaussian pump.

Fig. 6
Fig. 6

Normalized growth rate, K ¯ , of the fundamental mode ( n = 0 ) vs α 1 , comparing the analytical formula, Eq. (69) with the numerical simulations. Note that the growth rate for α 1 is equal to that for 1 α 1 , corresponding to the exchange of the roles of signal and idler.

Fig. 7
Fig. 7

Example of a case where amplification is suppressed in a non-uniform medium: (a) 2-D plots of the beam amplitudes and (b) peak amplitudes for α 1 = 1 , λ = 3 and κ ¯ = 6 .

Fig. 8
Fig. 8

Example of a case where noncollinear gain-guided modes exist in a non-uniform medium even though phase mismatch is present: (a) 2-D plots of the beam amplitudes, (b) peak amplitudes and (c) peak phases for α 1 = 1 , λ = 4 and κ ¯ = 4 .

Fig. 9
Fig. 9

Transverse cuts at the end of the simulation ( z ¯ = 25 ) . The parameters used for this simulation are α 1 = 1 , α 2 = 1 , λ = 4 and κ ¯ = 4 .

Fig. 10
Fig. 10

2-D plots of the Fourier transforms in x ¯ . The parameters used for this simulation are α 1 = 1 , α 2 = 1 , λ = 4 and κ ¯ = 4 .

Fig. 11
Fig. 11

Growth rate K ¯ as a function of (a) the gain parameter λ, and (b) chirp rate κ ¯ , for the degenerate case α 1 = 1 .

Fig. 12
Fig. 12

Normalized growth rate, K ¯ λ = K γ 0 , as a function of normalized angle, 1 λ = tan θ 1 γ 0 w 0 , for the degenerate case α 1 = 1 , α 2 = 1 and various values of the Rosenbluth gain parameter κ ¯ λ = 1 λ R = κ γ 0 2 .

Fig. 13
Fig. 13

(a) Angle range for which gain-guided noncollinear modes exist, as a function of κ ¯ λ = 1 λ R , where λ R is the Rosenbluth gain parameter, and for the degenerate case α 1 = 1 , α 2 = 1 . The solid lines show the maximum and minimum boundaries of this region, and the dashed line shows the angle of maximum growth rate. (b) Maximum growth rate as a function of the chirp rate.

Fig. 14
Fig. 14

Angular dependence of the growth rate obtained analytically, Eq. (96). This plot shows K ¯ λ = K γ 0 as a function of tan θ 1 γ 0 w 0 , for fixed ratios of κ ¯ λ = κ γ 0 2 = 1 λ R (where λ R is the Rosenbluth gain parameter). This plot was obtained for the degenerate case k ¯ 12 = 1 , i.e. α 1 = 1 , α 2 = 1 .

Fig. 15
Fig. 15

Physical picture of gain-guided noncollinear modes in non-uniform media. (a) When L pwt > L deph , gain-guided modes do not exist because they are suppressed by the dephasing. (b) When L pwt < L deph , gain-guided modes exist because the waves escape the pump before experiencing dephasing. (c) However, when the angle is too large, gain-guided modes do not exist because the gain is too low.

Fig. 16
Fig. 16

Effect of diffraction on noncollinear gain-guided modes, for positive and negative chirp rates. The parameters are α 1 = 0.7 , α 2 = 1.4 , λ = 5 . The chirp rate is (a) κ ¯ = 3 and (b) κ ¯ = 3 .

Fig. 17
Fig. 17

(a) Beam amplitude and (b) their spectral content in the case of negative chirp rate. The parameters in this case are α 1 = 0.7 , α 2 = 1.4 , λ = 10 and κ ¯ = 2 .

Fig. 18
Fig. 18

Amplification of noncollinear modes in a non-uniform gain medium seeded collinearly, in the presence of diffraction. 2-D plots of the (a) amplitudes of the fields and (b) their Fourier transforms in x ¯ . The parameters are α 1 = α 2 = 0 , λ = 4 , κ = 2 and β 1 = β 2 = 0.01 .

Fig. 19
Fig. 19

Amplification of noncollinear modes in a non-uniform gain medium seeded collinearly, in the presence of diffraction. The amplification initially ceases when the nonuniformity is turned on, as predicted by the Rosenbluth model, but eventually resumes as noncollinear gain-guided modes are amplified. (a) Peak amplitude and (b) trajectory of the peak amplitude in k x space. The parameters are α 1 = α 2 = 0 , λ = 4 , κ = 2 and β 1 = β 2 = 0.01 .

Fig. 20
Fig. 20

(a) Product of threshold length and diffraction parameter, β = β 1 = β 2 , as a function of the dephasing rate, for λ = 4 . (b) Threshold length L th vs λ, for fixed β = β 1 = β 2 = 0.01 and κ = 2 .

Fig. 21
Fig. 21

Summary of noncollinear gain-guided modes in non-uniform phase-matched media. The parabolas describe the spatial dependence of the phase-matching angle θ 1 , pm ( z ) given by Eq. (19). A signal wave with incidence angle θ 1 is phase-matched at position z pm . If this angle lies in the proper range, a gain-guided mode with growth rate K ( θ 1 ) can be excited. If the chirp rate is positive, the gain length is limited to the range over which noncollinear phase-matching is possible (i.e. from z pm to z pm 0 , the collinear PPMP). If it is negative, the mode can grow indefinitely (i.e. until it reaches the limit of the medium).

Equations (104)

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2 E ̃ j 1 c 2 2 E ̃ j t 2 = μ 0 2 P ̃ j t 2 ,
P ̃ NL , 0 = 2 ε 0 d eff e i K g r E ̃ 1 E ̃ 2 ,
P ̃ NL , 1 = 2 ε 0 d eff e i K g r E ̃ 0 E ̃ 2 * ,
P ̃ NL , 2 = 2 ε 0 d eff e i K g r E ̃ 0 E ̃ 1 * .
E ̃ j = E j e i ( k j r ω j t ) ,
P ̃ NL , j = P NL , j e i ( k j r ω j t ) ,
2 E j + 2 i k j E j = μ 0 ω j 2 P NL , j .
E j z + k j x k j z E j x i 2 k j z 2 E j x 2 = i μ 0 ω j 2 2 k j z P NL , j .
E 0 = E ¯ 0 A 0 ( x ) ,
E 1 , 2 = ω 1 , 2 n 1 , 2 cos θ 1 , 2 A 1 , 2 ,
A 1 z + tan θ 1 A 1 x i 2 k 1 cos θ 1 2 A 1 x 2 = i γ 0 A 0 ( x ) A 2 * e i κ r ,
A 2 * z + tan θ 2 A 2 * x + i 2 k 2 cos θ 2 2 A 2 * x 2 = i γ 0 A 0 * ( x ) A 1 e i κ r ,
γ 0 = ω 1 ω 2 n 1 n 2 cos θ 1 cos θ 2 d eff c E ¯ 0
k 1 sin θ 1 + k 2 sin θ 2 = 0 ,
k 1 cos θ 1 + k 2 cos θ 2 = k p K g ( z pm ) .
sin θ 2 = k 1 k 2 sin θ 1 .
θ 1 ( z pm ) 2 [ k 0 k 1 k 2 K g ( z pm ) ] k 1 ( 1 + k 1 k 2 ) .
κ ( z ) = κ ( z z pm ) ,
θ 1 , pm ( z ) = θ 1 2 ( z pm ) 2 κ ( z z pm ) k 1 ( 1 + k 1 k 2 ) .
x ¯ x w 0 .
L pwt w 0 tan θ 1 tan θ 2 .
L pwt w 0 θ 1 k ¯ 12 ,
k ¯ 12 k 1 k 2 .
z ¯ z L pwt .
A 1 z ¯ + α 1 A 1 x ¯ i β 1 2 A 1 x ¯ 2 = i λ 1 2 A 0 ( x ¯ ) A 2 * e i ϕ ( z ¯ ) ,
A 2 * z ¯ + α 2 A 2 * x ¯ + i β 2 2 A 2 * x ¯ 2 = i λ 1 2 A 0 * ( x ¯ ) A 1 e i ϕ ( z ¯ ) ,
α 1 = | tan θ 1 tan θ 2 | 1 k ¯ 12 ,
α 2 = | tan θ 2 tan θ 1 | k ¯ 12 ,
β j = L pwt 2 k j cos θ j w 0 2 L pwt L diff , j ,
λ = ( γ 0 L pwt ) 2 .
κ ¯ ( z ¯ ) = κ ¯ ( z ¯ z ¯ pm ) ,
κ ¯ κ L pwt 2
ϕ ( z ¯ ) = κ ¯ 2 [ ( z ¯ z ¯ pm ) 2 ( z ¯ 0 z ¯ pm ) 2 ] ,
d A 1 d z = i γ 0 A 2 * e i ϕ ( z ) ,
d A 2 * d z = i γ 0 A 1 e i ϕ ( z ) ,
ϕ ( z ) = z 0 z κ ( z ) d z .
A 1 ( z ) = cosh ( γ 0 z ) ,
A 2 ( z ) = sinh ( γ 0 z ) .
L g = 1 γ 0 .
κ ( z ) = κ ( z z pm ) .
d 2 d z 2 a j + [ ( κ ( z ) 2 ) 2 + i κ 2 γ 0 2 ] a j = 0 .
L deph = 2 γ 0 | κ | .
G = exp ( π γ 0 2 | κ | ) .
λ R = γ 0 2 | κ | .
L deph = 2 λ R κ .
A 1 z ¯ + α 1 A 1 x ¯ = i λ 1 2 A 0 ( x ¯ ) A 2 * ,
A 2 * z ¯ + α 2 A 2 * x ¯ = i λ 1 2 A 0 * ( x ¯ ) A 1 .
A 0 ( x ¯ ) = exp ( x ¯ 2 ) .
A 1 , 2 ( z ¯ , x ¯ ) = e K ¯ z ¯ Ψ 1 , 2 ( x ¯ ) ,
L 1 L 2 A 1 λ | A 0 ( x ¯ ) | 2 A 1 α 2 d ln A 0 d x ¯ L 1 A 1 = 0 ,
d 2 A ̃ 1 d x ¯ 2 [ d ln A 0 d x ¯ i ( α 1 + α 2 ) k ¯ z ] d A ̃ 1 d x ¯ + [ k ¯ z 2 + λ | A 0 ( x ¯ ) | 2 i α 2 k ¯ z d ln A 0 d x ¯ ] A ̃ 1 = 0 ,
A ̃ 1 ( x ¯ ) = A 0 1 2 ( x ¯ ) e i 2 ( α 1 + α 2 ) k ¯ z x ¯ a 1 ( x ¯ ) .
d 2 a 1 d x ¯ 2 + Q ( x ¯ ) a 1 = 0
Q ( x ¯ ) = λ | A 0 ( x ¯ ) | 2 1 4 [ d ln A 0 d x ¯ i ( α 1 α 2 ) k ¯ z ] 2 + 1 2 d 2 ln A 0 d x ¯ 2 .
| A 0 | 2 = { 1 , | x ¯ | 1 0 , | x ¯ | > 1 . .. }
A ̃ 1 ( x ¯ ) { 0 , x ¯ < 1 e 1 2 ( α 1 + α 2 ) K ¯ x ¯ sin [ λ 1 4 ( α 1 α 2 ) 2 K ¯ 2 ( x ¯ + 1 ) ] , 1 < x ¯ < 1 e K ¯ x ¯ α 1 , x ¯ > 1 } .
cot 2 λ K ̃ 2 = K ̃ λ K ̃ 2 .
u = 2 λ K ̃ 2 .
tan ( u π 2 ) = 2 λ ( u 2 ) 2 u .
λ th , n = [ ( n + 1 2 ) π 2 ] 2 .
K ¯ n = 2 α 1 + 1 α 1 λ ( n + 1 ) 2 ( π 2 ) 2 ,
γ 0 | tan θ 1 tan θ 2 | × 2 w 0 = π 2 .
K n = 2 γ 0 α 1 + 1 α 1 1 ( n + 1 ) 2 π 2 4 λ .
a 1 exp ± i x ¯ Q d x ¯ .
x ¯ 1 x ¯ 2 Q d x ¯ = ( n + 1 2 ) π ,
K ¯ n = 2 α 1 + 1 α 1 λ ( n + 1 2 ) 2 ( π 2 ) 2 .
| A 0 | 2 1 x ¯ 2 .
k ¯ z = ± 2 i α 1 α 2 1 + 1 4 λ λ 1 2 λ + 1 4 ( 2 n + 1 ) .
K ¯ n 2 α 1 + 1 α 1 λ ( 2 n + 1 ) λ ,
λ th , n = ( 1 + 2 n ) 2 .
K n 2 γ 0 α 1 + 1 α 1 1 2 n + 1 λ .
a 1 ( x ¯ ) sin [ x ¯ 1 x ¯ Q ( x ¯ ) d x ¯ + φ ] ,
a 1 exp ± x ¯ Q d x ¯ A 0 ± 1 2 e i 2 ( α 1 α 2 ) k z x ¯ , | x ¯ | 1 .
A 1 ( z ¯ , x ¯ ) { A 0 ( x ¯ ) e K ¯ ( z ¯ x ¯ α 2 ) x ¯ 1 A 0 1 2 ( x ¯ ) e K ¯ [ z ¯ + ( α 1 + α 2 ) x ¯ ] sin [ x ¯ 1 x ¯ Q ( x ¯ ) d x ¯ + φ ] 1 x ¯ 1 e K ¯ ( z ¯ x ¯ α 1 ) x ¯ 1 , } ,
K ¯ 2 α 1 + 1 α 1 λ λ .
A 1 z ¯ + α 1 A 1 x ¯ i β 1 2 A 1 x ¯ 2 = i λ 1 2 A 0 ( x ¯ ) A 2 * ,
A 2 * z ¯ + α 2 A 2 * x ¯ + i β 2 2 A 2 * x ¯ = i λ 1 2 A 0 * ( x ¯ ) A 1 .
L 1 L 2 A 1 λ | A 0 ( x ¯ ) | 2 A 1 = 0 ,
d 2 A ̃ ̂ 1 d k ¯ x 2 + Q ̂ ( k ¯ x ) A ̃ ̂ 1 = 0 ,
Q ̂ ( k ¯ x ) = 1 + 1 λ ( k ¯ z + α 1 k ¯ x + β 1 k ¯ x 2 ) ( k ¯ z + α 2 k ¯ x β 2 k ¯ x 2 ) ,
k ¯ x * = { 0 ( α 1 α 2 β 1 + β 2 ) } .
k ¯ x * = { 0 2 w 0 k 1 sin θ 1 } .
A 1 z ¯ + α 1 A 1 x ¯ = i λ 1 2 A 0 ( x ¯ ) A 2 * e i ϕ ( z ¯ ) ,
A 2 * z ¯ + α 2 A 2 * x ¯ = i λ 1 2 A 0 * ( x ¯ ) A 1 e i ϕ ( z ¯ ) .
ϕ ( z ¯ ) = κ ¯ 2 [ ( z ¯ z ¯ pm ) 2 ( z ¯ 0 z ¯ pm ) 2 ] .
L 1 L 2 A 1 λ | A 0 ( x ¯ ) | 2 A 1 ( i κ ¯ ( z ¯ ) + α 2 d ln A 0 d x ¯ ) L 1 A 1 = 0 ,
A 1 ( z ¯ , x ¯ ) = B 1 ( z ¯ , x ¯ ) exp [ i α 1 α 1 α 2 ϕ ( z ¯ x ¯ α 1 ) ] ,
L 1 L 2 A 1 = L 1 L 2 B 1 + i κ ( z ¯ x ¯ α 1 ) L 1 B 1 .
L 1 L 2 B 1 λ | A 0 ( x ¯ ) | 2 B 1 + i α 2 κ ¯ x ¯ L 1 B 1 = 0 .
d 2 B ̃ 1 d x ¯ 2 + i [ ( α 1 + α 2 ) k ¯ z + κ ¯ x ¯ ] d B ̃ 1 d x ¯ + [ k ¯ z 2 + λ | A 0 ( x ¯ ) | 2 i α 2 κ ¯ x ¯ ] B ̃ 1 = 0.
B ̃ 1 ( x ¯ ) = b 1 ( x ¯ ) exp { i 2 [ ( α 1 + α 2 ) k ¯ z x ¯ + κ ¯ 2 x ¯ 2 ] } .
d 2 b 1 d x ¯ 2 + Q ( x ¯ ) b 1 ( x ¯ ) = 0 ,
Q ( x ¯ ) = λ | A 0 ( x ¯ ) | 2 + 1 4 [ ( α 1 α 2 ) k ¯ z + κ ¯ x ¯ ] 2 i 2 κ ¯ .
k ¯ z = 2 i α 1 + 1 α 1 1 κ ¯ 2 4 λ λ i κ ¯ 2 ( 2 n + 1 ) λ 1 κ ¯ 2 4 λ .
K ¯ = 2 α 1 + 1 α 1 1 κ ¯ 2 4 λ λ λ 1 κ ¯ 2 4 λ .
K = 2 γ 0 α 1 + 1 α 1 1 λ 4 λ R 2 1 1 λ 1 λ 4 λ R 2 ,
K γ 0 = 2 α 1 + 1 α 1 1 ( L pwt L deph ) 2 1 L g L pwt 1 ( L pwt L deph ) 2 .
L 1 L 2 A 1 λ | A 0 ( x ¯ ) | 2 A 1 i κ ¯ ( z ¯ ) L 1 A 1 = 0 ,
d 2 A ̃ ̂ 1 d k ¯ x 2 + Q ̂ ( k ¯ x ) A ̃ ̂ 1 = 0 ,
Q ̂ ( k ¯ x ) = 1 + 1 λ [ ( k ¯ z + α 1 k ¯ x + β 1 k ¯ x 2 ) ( k ¯ z + α 2 k ¯ x β 2 k ¯ x 2 ) κ ¯ ( z ¯ ) ( k ¯ z + α 1 k ¯ x + β 1 k ¯ x 2 ) ] .
k ¯ x * = 1 2 ( α 1 α 2 β 1 + β 2 ) ± 1 4 ( α 1 α 2 β 1 + β 2 ) 2 κ ¯ ( z ¯ ) β 1 + β 2 .
k 1 x * = ± k 1 θ 1 2 2 κ ( z ) 1 + k 1 k 2 .
L ¯ th 1 8 β ( κ ¯ λ ) 2 ,
L th , physical = k 1 w 0 2 4 λ R 2 = L diff , 1 8 λ R 2 ,

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