Abstract

Optical parametric amplifiers using chirped quasi-phase-matching (QPM) gratings offer the possibility of engineering the gain and group delay spectra. We give practical formulas for the design of such amplifiers. We consider linearly chirped QPM gratings providing constant gain over a broad bandwidth, sinusoidally modulated profiles for selective frequency amplification and a pair of QPM gratings working in tandem to ensure constant gain and constant group delay at the same time across the spectrum. The analysis is carried out in the frequency domain using Wentzel–Kramers–Brillouin analysis.

© 2008 Optical Society of America

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References

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  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]
  2. C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Design and implementation of a TW-class high-average power laser system,” IEEE J. Sel. Top. Quantum Electron. 4, 395-406 (1988).
    [CrossRef]
  3. P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
    [CrossRef]
  4. W. Joosen, P. Agostini, G. Petite, J. P. Chambaret, and A. Antonetti, “Broadband femtosecond infrared parametric amplification in β-BaB2O4,” Opt. Lett. 17, 133-135 (1992).
    [CrossRef] [PubMed]
  5. G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562-1564 (1995).
    [CrossRef] [PubMed]
  6. G. M. Gale, F. Hache, and M. Cavallari, “Broad-bandwidth parametric amplification in the visible: femtosecond experiments and simulations,” IEEE J. Sel. Top. Quantum Electron. 4, 224-229 (1998).
    [CrossRef]
  7. A. Dubietis, G. Jonusauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437-440 (1992).
    [CrossRef]
  8. I. Jovanovic, C. Ebbers, and C. P. J. Barty, “Hybrid chirped-pulse amplification,” Opt. Lett. 27, 1622-1624 (2002).
    [CrossRef]
  9. 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]
  10. I. Jovanovic, C. G. Brown, C. A. Ebbers, C. P. J. Barty, N. Forget, and C. L. Blanc, “Generation of high-contrast millijoule pulses by optical parametric chirped-pulse amplification in periodically poled KTiOPO4,” Opt. Lett. 30, 1036-1038 (2005).
    [CrossRef] [PubMed]
  11. A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000cm−1 bandwidth,” Appl. Phys. Lett. 72, 147-149 (1998).
    [CrossRef]
  12. A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268-2270 (1999).
    [CrossRef]
  13. T. Kobayashi and A. Baltuska, “Sub-5 fs pulse generation from a noncollinear optical parametric amplifier,” Meas. Sci. Technol. 13, 1671-1682 (2002).
    [CrossRef]
  14. 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]
  15. A. Galvanauskas, M. A. Arbore, M. M. Fejer, M. E. Fermann, and D. Harter, “Fiber-laser-based femtosecond parametric generator in bulk periodically poled LiNbO3,” Opt. Lett. 22, 105-107 (1997).
    [CrossRef] [PubMed]
  16. A. Galvanauskas, A. Hariharan, D. Harter, M. A. Arbore, and M. M. Fejer, “High-energy femtosecond pulse amplification in a quasi-phase-matched parametric amplifier,” Opt. Lett. 23, 210-212 (1998).
    [CrossRef]
  17. A. Galvanauskas, D. Harter, M. A. Arbore, M. H. Chou, and M. M. Fejer, “Chirped-pulse-amplification circuits for fiber amplifiers, based on chirped-period quasi-phase-matching gratings,” Opt. Lett. 23, 1695-1697 (1998).
    [CrossRef]
  18. T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265-1276 (1990).
    [CrossRef]
  19. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341-1343 (1997).
    [CrossRef]
  20. M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22, 865-867 (1997).
    [CrossRef] [PubMed]
  21. G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864-866 (1998).
    [CrossRef]
  22. G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, “Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping,” J. Opt. Soc. Am. B 17, 304-318 (2000).
    [CrossRef]
  23. G. Imeshev, M. M. Fejer, A. Galvanauskas, and D. Harter, “Pulse shaping by difference-frequency mixing with quasi-phase-matching gratings,” J. Opt. Soc. Am. B 18, 534-539 (2001).
    [CrossRef]
  24. K. L. Baker, “Single-pass gain in a chirped quasi-phase-matched optical parametric oscillator,” Appl. Phys. Lett. 82, 3841-3843 (2003).
    [CrossRef]
  25. T. Beddard, M. Ebrahimzadeh, T. D. Reid, and W. Sibbett, “Five-optical-cycle pulse generation in the mid infrared from an optical parametric oscillator based on aperiodically poled lithium niobate,” Opt. Lett. 25, 1052-1054 (2000).
    [CrossRef]
  26. 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]
  27. M. N. Rosenbluth, “Parametric instabilities in inhomogeneous media,” Phys. Rev. Lett. 29, 565-567 (1972).
    [CrossRef]
  28. 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 (2008), to be published.
  29. R. L. Byer, “Parametric oscillators and nonlinear materials,” in Nonlinear Optics, P.G.Harper and B.S.Wherrett, eds. (Academic, 1977).
  30. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  31. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (Springer-Verlag, 1999).
  32. J. Heading, An Introduction to Phase-Integral Methods (Wiley, 1962).
  33. M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972).
    [CrossRef]
  34. K. G. Budden, The Propagation of Radio Waves (Cambridge U. Press, 1988).
  35. G. Cirmi, D. Brida, C. Manzoni, M. Marangoni, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses in the near-infrared from a noncollinear optical parametric amplifier,” Opt. Lett. 32, 2396-2398 (2007).
    [CrossRef] [PubMed]
  36. J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett. 31, 604-606 (2006).
    [CrossRef] [PubMed]
  37. R. B. White, Asymptotic Analysis of Differential Equations (Imperial College Press, 2005).

2008 (1)

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 (2008), to be published.

2007 (1)

2006 (1)

2005 (2)

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 (3)

2001 (1)

2000 (2)

1999 (1)

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268-2270 (1999).
[CrossRef]

1998 (6)

A. Galvanauskas, A. Hariharan, D. Harter, M. A. Arbore, and M. M. Fejer, “High-energy femtosecond pulse amplification in a quasi-phase-matched parametric amplifier,” Opt. Lett. 23, 210-212 (1998).
[CrossRef]

A. Galvanauskas, D. Harter, M. A. Arbore, M. H. Chou, and M. M. Fejer, “Chirped-pulse-amplification circuits for fiber amplifiers, based on chirped-period quasi-phase-matching gratings,” Opt. Lett. 23, 1695-1697 (1998).
[CrossRef]

A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000cm−1 bandwidth,” Appl. Phys. Lett. 72, 147-149 (1998).
[CrossRef]

G. M. Gale, F. Hache, and M. Cavallari, “Broad-bandwidth parametric amplification in the visible: femtosecond experiments and simulations,” IEEE J. Sel. Top. Quantum Electron. 4, 224-229 (1998).
[CrossRef]

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

G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864-866 (1998).
[CrossRef]

1997 (3)

1995 (1)

1992 (3)

W. Joosen, P. Agostini, G. Petite, J. P. Chambaret, and A. Antonetti, “Broadband femtosecond infrared parametric amplification in β-BaB2O4,” Opt. Lett. 17, 133-135 (1992).
[CrossRef] [PubMed]

A. Dubietis, G. Jonusauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437-440 (1992).
[CrossRef]

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]

1990 (1)

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265-1276 (1990).
[CrossRef]

1988 (2)

C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Design and implementation of a TW-class high-average power laser system,” IEEE J. Sel. Top. Quantum Electron. 4, 395-406 (1988).
[CrossRef]

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

1972 (2)

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

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (3)

A. Shirakawa and T. Kobayashi, “Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000cm−1 bandwidth,” Appl. Phys. Lett. 72, 147-149 (1998).
[CrossRef]

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, “Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification,” Appl. Phys. Lett. 74, 2268-2270 (1999).
[CrossRef]

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

IEEE J. Quantum Electron. (3)

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398-403 (1988).
[CrossRef]

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]

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265-1276 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

C. G. Durfee, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Design and implementation of a TW-class high-average power laser system,” IEEE J. Sel. Top. Quantum Electron. 4, 395-406 (1988).
[CrossRef]

G. M. Gale, F. Hache, and M. Cavallari, “Broad-bandwidth parametric amplification in the visible: femtosecond experiments and simulations,” IEEE J. Sel. Top. Quantum Electron. 4, 224-229 (1998).
[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)

A. Dubietis, G. Jonusauskas, and A. Piskarskas, “Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal,” Opt. Commun. 88, 437-440 (1992).
[CrossRef]

Opt. Lett. (14)

I. Jovanovic, C. Ebbers, and C. P. J. Barty, “Hybrid chirped-pulse amplification,” Opt. Lett. 27, 1622-1624 (2002).
[CrossRef]

I. Jovanovic, C. G. Brown, C. A. Ebbers, C. P. J. Barty, N. Forget, and C. L. Blanc, “Generation of high-contrast millijoule pulses by optical parametric chirped-pulse amplification in periodically poled KTiOPO4,” Opt. Lett. 30, 1036-1038 (2005).
[CrossRef] [PubMed]

W. Joosen, P. Agostini, G. Petite, J. P. Chambaret, and A. Antonetti, “Broadband femtosecond infrared parametric amplification in β-BaB2O4,” Opt. Lett. 17, 133-135 (1992).
[CrossRef] [PubMed]

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, “Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator,” Opt. Lett. 20, 1562-1564 (1995).
[CrossRef] [PubMed]

M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, “Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped-period-poled lithium niobate,” Opt. Lett. 22, 1341-1343 (1997).
[CrossRef]

M. A. Arbore, O. Marco, and M. M. Fejer, “Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings,” Opt. Lett. 22, 865-867 (1997).
[CrossRef] [PubMed]

G. Imeshev, A. Galvanauskas, D. Harter, M. A. Arbore, M. Proctor, and M. M. Fejer, “Engineerable femtosecond pulse shaping by second-harmonic generation with Fourier synthetic quasi-phase-matching gratings,” Opt. Lett. 23, 864-866 (1998).
[CrossRef]

A. Galvanauskas, M. A. Arbore, M. M. Fejer, M. E. Fermann, and D. Harter, “Fiber-laser-based femtosecond parametric generator in bulk periodically poled LiNbO3,” Opt. Lett. 22, 105-107 (1997).
[CrossRef] [PubMed]

A. Galvanauskas, A. Hariharan, D. Harter, M. A. Arbore, and M. M. Fejer, “High-energy femtosecond pulse amplification in a quasi-phase-matched parametric amplifier,” Opt. Lett. 23, 210-212 (1998).
[CrossRef]

A. Galvanauskas, D. Harter, M. A. Arbore, M. H. Chou, and M. M. Fejer, “Chirped-pulse-amplification circuits for fiber amplifiers, based on chirped-period quasi-phase-matching gratings,” Opt. Lett. 23, 1695-1697 (1998).
[CrossRef]

T. Beddard, M. Ebrahimzadeh, T. D. Reid, and W. Sibbett, “Five-optical-cycle pulse generation in the mid infrared from an optical parametric oscillator based on aperiodically poled lithium niobate,” Opt. Lett. 25, 1052-1054 (2000).
[CrossRef]

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]

G. Cirmi, D. Brida, C. Manzoni, M. Marangoni, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses in the near-infrared from a noncollinear optical parametric amplifier,” Opt. Lett. 32, 2396-2398 (2007).
[CrossRef] [PubMed]

J. Huang, X. P. Xie, C. Langrock, R. V. Roussev, D. S. Hum, and M. M. Fejer, “Amplitude modulation and apodization of quasi-phase-matched interactions,” Opt. Lett. 31, 604-606 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

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

Rep. Prog. Phys. (1)

M. V. Berry and K. E. Mount, “Semiclassical approximations in wave mechanics,” Rep. Prog. Phys. 35, 315-397 (1972).
[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 (6)

K. G. Budden, The Propagation of Radio Waves (Cambridge U. Press, 1988).

R. B. White, Asymptotic Analysis of Differential Equations (Imperial College Press, 2005).

R. L. Byer, “Parametric oscillators and nonlinear materials,” in 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).

J. Heading, An Introduction to Phase-Integral Methods (Wiley, 1962).

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Figures (12)

Fig. 1
Fig. 1

Illustration of OPA in a chirped QPM grating.

Fig. 2
Fig. 2

Nonuniform grating profile showing the PPMP, the turning points, and the nature of the solutions in each region.

Fig. 3
Fig. 3

Grating profile with multiple PPMPs.

Fig. 4
Fig. 4

Gain spectrum of a linear profile comparing the numerical solution with the WKB solution and the Rosenbluth gain factor for various grating lengths. The numerical values used are γ 2 κ = 2 and (a) κ 1 2 L = 20 , (b) κ 1 2 L = 30 , and (c) κ 1 2 L = 40 .

Fig. 5
Fig. 5

Phase spectrum of a linear profile comparing the numerical solution with the WKB solution and the simplified expressions, Eqs. (36, 37), for various grating lengths. Plots (a)–(c) correspond to the signal, plots, (d)–(f) correspond to the idler. The numerical values used are γ 2 κ = 2 and (a), (d) κ 1 2 L = 20 ; (b), (e) κ 1 2 L = 30 ; and (c), (f) κ 1 2 L = 40 .

Fig. 6
Fig. 6

Group delay spectrum of a linear profile normalized with respect to the delay between the waves τ 1 v s 1 v i L = τ δ v L . The delays are relative to reference waves traveling at the signal and idler velocities, respectively. These plots compare the numerical solution with the WKB solution and the simplified expressions, Eqs. (36, 37), for various grating lengths. Plots (a)–(c) correspond to the signal, plots (d)–(f) correspond to the idler. The numerical values used are γ 2 κ = 2 and (a), (d) κ 1 2 L = 20 ; (b), (e) κ 1 2 L = 30 ; and (c), (f) κ 1 2 L = 40 .

Fig. 7
Fig. 7

Ripple reduction using tapering of the coupling coefficient: (a) coupling coefficient profile, (b) gain spectrum, (c) signal group delay spectrum, and (d) idler group delay spectrum. The group delays are defined with respect to reference waves traveling at the signal and idler velocities, respectively. The tapering profile is given by Eq. (44) with l 1 = l 2 = w 1 = w 2 = 0.04 × L . The gain parameter is γ 2 κ = 2 and the length is κ 1 2 L = 20 . The gain and group delay spectra without apodization were shown in Figs. 4, 6, top case.

Fig. 8
Fig. 8

Ripple reduction using tapering of the QPM profile: (a) grating profile, (b) gain spectrum, (c) signal group-delay spectrums and (d) idler group delay spectrum. The group delays are defined with respect to reference waves traveling at the signal and idler velocities, respectively. The grating profile is given by Eq. (45) with μ κ 0 1 2 = 100 and ν = 21 . The gain parameter is γ 2 κ 0 = 2 and the length is κ 1 2 L = 20 .

Fig. 9
Fig. 9

Sinusoidal profile for selective frequency amplification: (a) QPM grating profile and (b) amplification spectrum, comparing the numerical values with the Rosenbluth amplification formula. The cubic approximation to the grating profile gives the peak amplification (marked by the dots on the plot), while the linear approximation (dashed curve) is valid away from the peaks but not in the vicinity of the maxima. The numerical parameters in this example are γ 2 κ 0 = 2 , κ 0 1 2 L = 40 , k μ = 2 π ( L 3 ) , and μ = κ 0 k μ .

Fig. 10
Fig. 10

Tandem configuration.

Fig. 11
Fig. 11

Stokes diagram of the function Q ¯ defined by Eq. (A3). Solid curve, anti-Stokes curves; dashed curves, Stokes curves; zig–zag, branch cut.

Fig. 12
Fig. 12

Comparison between the WKB solution and the numerical solution for λ = 1 , A s 0 = 1 , A i 0 = 0 , L ¯ = 20 , and ζ pm located at the center of the grating: (a) signal, (b) idler. The insets give details on the amplitudes near the input.

Tables (1)

Tables Icon

Table 1 Numerical Values for the OPA Design

Equations (105)

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κ ( z , δ ω ) = Δ k ( δ ω ) K g ( z ) .
κ ( z ) = κ ( z , δ ω ) z = d K g ( z ) d z .
d A s d z = i γ ( z , δ ω ) A i * e i ϕ ( z , δ ω ) ,
d A i * d z = i γ ( z , δ ω ) A s e i ϕ ( z , δ ω ) .
ϕ ( z , δ ω ) = z 0 z κ ( z , δ ω ) d z ,
d 2 a s , i d z 2 + Q ( z ) a s , i = 0 ,
Q ( z ) = ( κ 2 i 2 γ γ ) 2 γ 2 + i κ 2 + 1 2 ( γ γ ) ,
( z tp 1 , z ) Q 1 4 ( z ) exp ( i z tp 1 z Q 1 2 ( z ) d z ) ,
( z , z tp 1 ) Q 1 4 ( z ) exp ( i z z tp 1 Q 1 2 ( z ) d z ) .
A s ( z L ) [ γ ( z L ) γ ( z 0 ) ] 1 2 e i ϕ ( z L ) 2 ( C + i C ) i [ z tp 2 , z tp 1 ] [ ( z L , z tp 2 ) i ( z tp 2 , z L ) ] ,
C + = 1 ( z tp 1 , z 0 ) [ ( 1 + γ 2 ( z 0 ) κ 2 ( z 0 ) ) A s 0 γ ( z 0 ) κ ( z 0 ) A i 0 * ] ,
C = 1 ( z 0 , z tp 1 ) [ γ 2 ( z 0 ) κ 2 ( z 0 ) A s 0 + γ ( z 0 ) κ ( z 0 ) A i 0 * ] ,
[ z tp 2 , z tp 1 ] exp ( i z tp 2 z tp 1 Q 1 2 ( z ) d z ) .
A s R ( A s 0 + i A i 0 * e i φ ( z 0 , z pm ) ) e g ( z tp 1 , z tp 2 ) ,
φ ( z 0 , z pm ) z 0 z pm κ ( z ) d z ,
g ( z tp 1 , z tp 2 ) z tp 1 z tp 2 ( γ 2 κ 2 4 ) 1 2 d z ,
G linear = exp ( π γ 2 ( z pm ) κ ( z pm ) ) ,
G cubic = exp [ ( 2 γ 4 ( z pm ) κ ) 1 3 1 1 1 u 6 d u ] ,
A s ( j ) = R ( j ) [ A s 0 ( j ) + i A i 0 ( j ) * e i φ ( z 0 ( 1 ) , z pm ( j ) ) ] e g ( z tp 1 ( j ) , z tp 2 ( j ) ) ,
A i ( j ) = R ( j ) [ A i 0 ( j ) + i A s 0 ( j ) * e i φ ( z 0 ( 1 ) , z pm ( j ) ) ] e g ( z tp 1 ( j ) , z tp 2 ( j ) ) .
A s ( z L ) = A s 0 e i κ L 2 [ cosh Γ L i κ 2 Γ sinh Γ L ] + i γ Γ A i 0 * e i κ L 2 sinh Γ L ,
A i ( z L ) = A i 0 e i κ L 2 [ cosh Γ L i κ 2 Γ sinh Γ L ] + i γ Γ A s 0 * e i κ L 2 sinh Γ L ,
Γ = γ 2 ( κ 2 ) 2
A s , i ( z L ) A s 0 γ 2 Γ e Γ L .
G uniform = 1 2 e γ L .
κ FWMH = ± 2 L γ 2 L 2 ( ln 2 ) 2 .
Δ k ( δ ω ) = k p k s k i k p k s 0 k ω ω s 0 δ ω k i 0 + k ω ω i 0 δ ω = k p k s 0 k i 0 + ( 1 v s 1 v i ) δ ω ,
κ ( δ ω ) = ( 1 v s 1 v i ) δ ω .
1 δ v 1 v s 1 v i .
Δ ω uniform = 4 δ v L γ 2 L 2 ( ln 2 ) 2 4 δ v γ ,
κ ( z , δ ω ) = κ ( z z pm 0 ) δ ω δ v ,
z pm = z pm 0 + δ ω κ δ v .
κ = κ ( z z pm ) .
L g = 4 γ κ .
Δ ω chirped = κ δ v ( L L g ) .
A s = A s 0 e π γ 2 κ ,
A i = i A s 0 * e π γ 2 κ e i κ ( z pm z 0 ) 2 2 .
G Rosenbluth = e π γ 2 ( δ ω ) κ .
τ i = z pm z 0 δ v .
τ i s = z L z pm δ v .
ln G Rosenbluth × Δ ω chirped π δ v γ 2 L .
ln G uniform × Δ ω uniform 4 δ v γ 2 L .
G 3 D = exp ( π γ 2 ( x , y , t ) κ ) d x d y d t ,
γ ( z ) γ max = a + b × tanh ( z z 0 l 1 w 1 ) × tanh ( L z + z 0 l 2 w 2 ) .
κ = κ 0 ( z z pm 0 ) + μ ( z z pm 0 L 2 ) ν ( 1 v s 1 v i ) δ ω ,
κ = κ 0 ( z z pm 0 ) μ sin [ k μ ( z z pm 0 ) ] ( 1 v s 1 v i ) δ ω ,
G = exp [ 4.17 ( γ 4 κ 0 k μ 2 ) 1 3 ] .
A s ( 1 ) = R ( 1 ) A s 0 e g ( 1 ) ,
A i ( 1 ) = i R ( 1 ) A s 0 * e g ( 1 ) e i φ ( z 0 ( 1 ) , z pm ( 1 ) ) ,
A s ( 2 ) = A s 0 * R ( 1 ) R ( 2 ) e g ( 1 ) + g ( 2 ) e i φ ( z pm ( 1 ) , z pm ( 2 ) ) .
ln G ( δ ω ) = π γ 2 ( z pm ( 1 ) ) κ ( z pm ( 1 ) ) + π γ 2 ( z pm ( 2 ) ) κ ( z p m ( 2 ) ) .
τ ( δ ω ) = z pm ( 1 ) z 0 ( 1 ) v s + z L ( 1 ) z pm ( 1 ) v i + z pm ( 2 ) z 0 ( 2 ) v i + z L ( 2 ) z pm ( 2 ) v s .
A i ( z ) = i γ A s * z 0 z e i ϕ ( z ) d z .
A i ( z ) i e i π 4 γ 2 π κ A s * e ( i 2 ) κ ( z 0 z pm ) 2 .
ζ = κ 0 ( z z pm ) ,
d 2 a s d ζ 2 + Q ¯ ( ζ ) a s = 0 ,
Q ¯ ( ζ ) = ( κ ¯ 2 i 4 λ λ ) 2 λ + i κ ¯ 2 + 1 4 ( λ λ ) .
a s ( ζ 0 ) = γ 1 2 ( ζ 0 ) A s 0 ,
d a s ( ζ 0 ) d ζ = γ 1 2 ( ζ 0 ) [ i λ 1 2 ( ζ 0 ) A i 0 * i 2 κ ¯ ( ζ 0 ) A s 0 1 4 λ ( ζ 0 ) λ ( ζ 0 ) A s 0 ] .
Q ¯ 1 4 exp ( ± i ζ [ Q ¯ ( ζ ) ] 1 2 d ζ ) .
( ζ 1 , ζ ) Q ¯ 1 4 exp ( i ζ 1 ζ [ Q ¯ ( ζ ) ] 1 2 d ζ ) ,
( ζ , ζ 1 ) Q ¯ 1 4 exp ( i ζ ζ 1 [ Q ¯ ( ζ ) ] 1 2 d ζ ) .
region 8 : ( ζ 1 , ζ ) d .
region 7 : ( ζ 1 , ζ ) d + i ( ζ , ζ 1 ) s .
region 6 : ( ζ 1 , ζ ) s + i ( ζ , ζ 1 ) d .
region 5 : ( ζ 1 , ζ ) s + i { ( ζ , ζ 1 ) d + i ( ζ 1 , ζ ) s } = i ( ζ , ζ 1 ) d .
region 3 : i [ ζ 2 , ζ 1 ] ( ζ , ζ 2 ) s ,
[ ζ 2 , ζ 1 ] exp ( i ζ 2 ζ 1 [ Q ( ζ ) ] 1 2 d ζ ) .
region 2 : i [ ζ 2 , ζ 1 ] ( ζ , ζ 2 ) d .
region 1 : i [ ζ 2 , ζ 1 ] { ( ζ , ζ 2 ) d i ( ζ 2 , ζ ) s } .
region 8 : ( ζ , ζ 1 ) s ,
region 7 : ( ζ , ζ 1 ) s ,
region 6 : ( ζ , ζ 1 ) d ,
region 5 : ( ζ , ζ 1 ) d + i ( ζ 1 , ζ ) s ,
region 3 : [ ζ 2 , ζ 1 ] ( ζ , ζ 2 ) s + i [ ζ 1 , ζ 2 ] ( ζ 2 , ζ ) d [ ζ 2 , ζ 1 ] ( ζ , ζ 2 ) s ,
region 2 : [ ζ 2 , ζ 1 ] ( ζ , ζ 2 ) d ,
region 1 : [ ζ 2 , ζ 1 ] { ( ζ , ζ 2 ) d i ( ζ 2 , ζ ) s } .
a s I C s + ( ζ 1 , ζ ) + C s ( ζ , ζ 1 ) , ζ ζ tp ,
a s II ( i C s + + C s ) ( ζ , ζ 1 ) , ζ tp ζ ζ tp ,
a s III ( i C s + + C s ) [ ζ 2 , ζ 1 ] ( ζ , ζ 2 ) { 1 i ( ζ 2 , ζ ) ( ζ , ζ 2 ) } , ζ ζ tp .
C s + 1 ( ζ 1 , ζ 0 ) γ 1 2 ( ζ 0 ) [ ( 1 + λ ( ζ 0 ) κ ¯ ( ζ 0 ) 2 ) A s 0 λ 1 2 ( ζ 0 ) κ ¯ ( ζ 0 ) A i 0 * ] ,
C s 1 ( ζ 0 , ζ 1 ) γ 1 2 ( ζ 0 ) ( λ ( ζ 0 ) κ ¯ 2 ( ζ 0 ) A s 0 + λ 1 2 ( ζ 0 ) κ ¯ ( ζ 0 ) A i 0 * ) .
a ζ 0 Q 1 2 d ζ = ( λ i 2 ) { ζ 0 2 λ i 2 ζ 0 2 4 ( λ i 2 ) 1 + ln [ ζ 0 2 λ i 2 + ζ 0 2 4 ( λ i 2 ) 1 ] } ζ 0 2 4 + ( λ i 2 ) ln ( ζ 0 λ 1 2 ) + λ 2 .
a ζ Q 1 2 d ζ i λ ( 1 i 4 λ ) ζ + i π λ 2 + π 4 .
b ζ L Q 1 2 d ζ ζ L 2 4 ( λ i 2 ) ln ( ζ L λ 1 2 ) λ 2 .
a b Q 1 2 d ζ = i π ( λ i 2 ) .
A s I ( ζ ) [ ( 1 + ϵ 2 ( ζ 0 ) ) A s 0 + ϵ ( ζ 0 ) A i 0 * ] exp ( i λ ln ζ ζ 0 ) [ 1 ϵ ( ζ ) ( A i 0 * + ϵ ( ζ 0 ) A s 0 A s 0 + ϵ ( ζ 0 ) A i 0 * ) exp ( i 2 ( ζ 2 ζ 0 2 ) 2 i λ ln ζ ζ 0 ) ] ,
A s II ( ζ ) 1 2 exp [ π λ 2 + λ 1 2 ζ i ( ζ 4 λ 1 2 + ζ 2 4 + λ ln ϵ ( ζ 0 ) λ 2 ) ] × [ ( 1 + ϵ 2 ( ζ 0 ) ) A s 0 ϵ ( ζ 0 ) A i 0 * ( A i 0 * ϵ ( ζ 0 ) A s 0 ) exp ( i Φ ( ζ 0 ) ) ] ,
A s III ( ζ ) exp ( π λ + i λ ln ζ ζ 0 ) F ( ζ ) [ F * ( ζ 0 ) A s 0 F ( ζ 0 ) e i Φ ( ζ 0 ) A i 0 * ] ,
F ( ζ ) = 1 ϵ ( ζ ) exp ( i Φ ( ζ ) ) ,
Φ ( ζ ) = ζ 2 2 + 2 λ ln ϵ ( ζ ) λ + π 2 ,
ϵ ( ζ ) = λ 1 2 ζ .
A s I ( ζ ) A s 0 e i λ ln ζ ζ 0 ,
A s II ( ζ ) 1 2 [ A s 0 + i A i 0 * e i ( ζ 0 2 2 λ ) ] exp [ π λ 2 + λ 1 2 ζ i ( ζ 4 λ 1 2 + ζ 2 4 λ 2 ) ] ,
A s III ( ζ ) e π λ [ A s 0 + i A i 0 * e i ( ζ 0 2 2 λ ) ] e i λ ln ζ ζ 0 .
ζ 2 ζ ̃ Q ¯ 1 2 d ζ κ ¯ 2 4 + i 2 ln ( ζ ̃ λ 1 2 ) .
ζ ̃ ζ L Q ¯ 1 2 d ζ 1 2 ζ ̃ ζ L κ ¯ d ζ + i 2 ln ( κ ¯ ( ζ L ) ζ ̃ ) .
ζ 2 ζ L Q ¯ 1 2 d ζ 1 2 0 ζ L κ ¯ d ζ + i 2 ln ( κ ¯ ( ζ L ) λ 1 2 ) .
( ζ 1 , ζ 0 ) 2 1 2 λ 1 4 ( ζ 0 ) e i 2 ζ 0 0 κ ( ζ ) d ζ ,
( ζ 0 , ζ 1 ) 2 1 2 λ 1 4 ( ζ 0 ) κ ( ζ 0 ) e i 2 ζ 0 0 κ ( ζ ) d ζ ,
( ζ 2 , ζ L ) 2 1 2 λ 1 4 ( ζ 0 ) κ ( ζ L ) e i 2 0 ζ L κ ( ζ ) d ζ ,
( ζ L , ζ 2 ) 2 1 2 λ 1 4 ( ζ 0 ) e i 2 0 ζ L κ ( ζ ) d ζ .
[ ζ 2 , ζ 1 ] = e i ζ 2 ζ 1 Q ¯ 1 2 d ζ i e ζ 1 * ζ 2 * ( λ κ ¯ 2 4 ) 1 2 d ζ ,
A s III [ ( 1 i λ 1 2 ( ζ 0 ) κ ( ζ 0 ) e i ζ 0 0 κ ( ζ ) d ζ ) A s 0 + i e i ζ 0 0 κ ( ζ ) d ζ ( 1 + i λ 1 2 ( ζ 0 ) κ ( ζ 0 ) e i ζ 0 0 κ ( ζ ) d ζ ) A i 0 * ] e ζ 1 * ζ 2 * ( λ κ ¯ 2 4 ) 1 2 d ζ ( 1 i λ 1 2 ( ζ L ) κ ( ζ L ) e i 0 ζ L κ ( ζ ) d ζ ) .
A s [ A s 0 + i A i 0 * e i ζ 0 0 κ ¯ d ζ ] e ζ 1 * ζ 2 * ( λ κ ¯ 2 4 ) 1 2 d ζ ,

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