Abstract

Ultrabroadband optical parametric generation in the near-IR (~135 THz, 1.15-2.4 µm) is demonstrated using bismuth triborate, BiB3O6 (BIBO), in a collinear geometry. The white light continuum energy obtained with a single stage reached 15 µJ (internal conversion efficiency of ≈7%). Integral pulse durations as short as 63 fs were derived from the recorded FROG traces, comparable to the 45 fs pulse duration of the 1 kHz Ti:sapphire regenerative amplifier used for pumping at 800 nm.

© 2008 Optical Society of America

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  1. R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
    [CrossRef]
  2. G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003).
    [CrossRef]
  3. A. Brodeur and S. L. Chin, "Ultrafast white-light continuum generation and self-focusing in transparent condensed media," J. Opt. Soc. Am. B 16, 637-650 (1999).
    [CrossRef]
  4. I. Nikolov, A. Gaydardzhiev, I. Buchvarov, P. Tzankov, F. Noack, and V. Petrov, "Ultrabroadband continuum amplification in the near infrared using BiB3O6 nonlinear crystals pumped at 800 nm," Opt. Lett. 32, 3342-3344 (2007).
    [CrossRef] [PubMed]
  5. H. Hellwig, J. Liebertz, and L. Bohaty, "Exceptional large nonlinear optical coefficients in the monoclinic bismuth borate BiB3O6 (BIBO)," Solid State Commun. 109, 249-251 (1999).
    [CrossRef]
  6. A. Birmontas, A. Piskarskas, and A. Stabinis, "Dispersion anomalies of tuning characteristics and spectrum of an optical parametric oscillator," Sov. J. Quantum Electron. 13, 1243-1246 (1983) [transl. from Kvantovaya Elektron. (Moscow) 10, 1881-1884 (1983)].
    [CrossRef]
  7. N. Umemura, K. Miyata, and K. Kato, "New data on the optical properties of BiB3O6," Opt. Mater. 30, 532-534 (2007).
    [CrossRef]
  8. X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
    [CrossRef]
  9. V. Petrov, F. Noack, P. Tzankov, M. Ghotbi, M. Ebrahim-Zadeh, I. Nikolov, and I. Buchvarov, "High-power femtosecond optical parametric amplification at 1 kHz in BiB3O6 pumped at 800 nm," Opt. Express 15, 556-563 (2007).
    [CrossRef] [PubMed]
  10. M. S. Webb, D. Eimerl, and S. P. Velsko, "Wavelength insensitive phase-matched second-harmonic generation in partially deuterated KDP," J. Opt. Soc. Am. B 9, 1118-1127 (2007).
    [CrossRef]
  11. D. N. Nikogosyan, Nonlinear Optical Crystals, (Springer, New York, 2005).
  12. B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
    [CrossRef]
  13. M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
    [CrossRef]
  14. S. N. Orlov, E. V. Pestryakov, and Yu. N. Polivanov, "Optical parametric amplification with bandwidth exceeding an octave," Quantum Electron. 34, 477-479 (2004) [transl. from Kvantovaya Elektron. (Moscow) 34, 477-481 (2004)].
    [CrossRef]

2007 (4)

2006 (1)

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

2004 (1)

S. N. Orlov, E. V. Pestryakov, and Yu. N. Polivanov, "Optical parametric amplification with bandwidth exceeding an octave," Quantum Electron. 34, 477-479 (2004) [transl. from Kvantovaya Elektron. (Moscow) 34, 477-481 (2004)].
[CrossRef]

2003 (1)

G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

1999 (2)

A. Brodeur and S. L. Chin, "Ultrafast white-light continuum generation and self-focusing in transparent condensed media," J. Opt. Soc. Am. B 16, 637-650 (1999).
[CrossRef]

H. Hellwig, J. Liebertz, and L. Bohaty, "Exceptional large nonlinear optical coefficients in the monoclinic bismuth borate BiB3O6 (BIBO)," Solid State Commun. 109, 249-251 (1999).
[CrossRef]

1993 (2)

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

1983 (1)

A. Birmontas, A. Piskarskas, and A. Stabinis, "Dispersion anomalies of tuning characteristics and spectrum of an optical parametric oscillator," Sov. J. Quantum Electron. 13, 1243-1246 (1983) [transl. from Kvantovaya Elektron. (Moscow) 10, 1881-1884 (1983)].
[CrossRef]

1982 (1)

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Banfi, G. P.

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

Bareika, B.

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Birmontas, A.

A. Birmontas, A. Piskarskas, and A. Stabinis, "Dispersion anomalies of tuning characteristics and spectrum of an optical parametric oscillator," Sov. J. Quantum Electron. 13, 1243-1246 (1983) [transl. from Kvantovaya Elektron. (Moscow) 10, 1881-1884 (1983)].
[CrossRef]

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Bohaty, L.

H. Hellwig, J. Liebertz, and L. Bohaty, "Exceptional large nonlinear optical coefficients in the monoclinic bismuth borate BiB3O6 (BIBO)," Solid State Commun. 109, 249-251 (1999).
[CrossRef]

Brodeur, A.

Buchvarov, I.

Canalias, C.

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

Cerullo, G.

G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

Chin, S. L.

Danielius, R.

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

De Silvestri, S.

G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

Deng, D.

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

Di Trapani, P.

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

Dikchyus, G.

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Ebrahim-Zadeh, M.

Eimerl, D.

Fragemann, A.

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

Gaydardzhiev, A.

Ghotbi, M.

Guo, D.

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

Hellwig, H.

H. Hellwig, J. Liebertz, and L. Bohaty, "Exceptional large nonlinear optical coefficients in the monoclinic bismuth borate BiB3O6 (BIBO)," Solid State Commun. 109, 249-251 (1999).
[CrossRef]

Kato, K.

N. Umemura, K. Miyata, and K. Kato, "New data on the optical properties of BiB3O6," Opt. Mater. 30, 532-534 (2007).
[CrossRef]

Laurell, F.

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

Li, M.

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

Liebertz, J.

H. Hellwig, J. Liebertz, and L. Bohaty, "Exceptional large nonlinear optical coefficients in the monoclinic bismuth borate BiB3O6 (BIBO)," Solid State Commun. 109, 249-251 (1999).
[CrossRef]

Liu, X.

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

Miyata, K.

N. Umemura, K. Miyata, and K. Kato, "New data on the optical properties of BiB3O6," Opt. Mater. 30, 532-534 (2007).
[CrossRef]

Nikolov, I.

Noack, F.

Orlov, S. N.

S. N. Orlov, E. V. Pestryakov, and Yu. N. Polivanov, "Optical parametric amplification with bandwidth exceeding an octave," Quantum Electron. 34, 477-479 (2004) [transl. from Kvantovaya Elektron. (Moscow) 34, 477-481 (2004)].
[CrossRef]

Pasiskevicius, V.

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

Pestryakov, E. V.

S. N. Orlov, E. V. Pestryakov, and Yu. N. Polivanov, "Optical parametric amplification with bandwidth exceeding an octave," Quantum Electron. 34, 477-479 (2004) [transl. from Kvantovaya Elektron. (Moscow) 34, 477-481 (2004)].
[CrossRef]

Petrov, V.

Piskarskas, A.

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

A. Birmontas, A. Piskarskas, and A. Stabinis, "Dispersion anomalies of tuning characteristics and spectrum of an optical parametric oscillator," Sov. J. Quantum Electron. 13, 1243-1246 (1983) [transl. from Kvantovaya Elektron. (Moscow) 10, 1881-1884 (1983)].
[CrossRef]

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Polivanov, Yu. N.

S. N. Orlov, E. V. Pestryakov, and Yu. N. Polivanov, "Optical parametric amplification with bandwidth exceeding an octave," Quantum Electron. 34, 477-479 (2004) [transl. from Kvantovaya Elektron. (Moscow) 34, 477-481 (2004)].
[CrossRef]

Righini, R.

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

Sirutkaitis, V.

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Stabinis, A.

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

A. Birmontas, A. Piskarskas, and A. Stabinis, "Dispersion anomalies of tuning characteristics and spectrum of an optical parametric oscillator," Sov. J. Quantum Electron. 13, 1243-1246 (1983) [transl. from Kvantovaya Elektron. (Moscow) 10, 1881-1884 (1983)].
[CrossRef]

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Tiihonen, M.

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

Tzankov, P.

Umemura, N.

N. Umemura, K. Miyata, and K. Kato, "New data on the optical properties of BiB3O6," Opt. Mater. 30, 532-534 (2007).
[CrossRef]

Velsko, S. P.

Webb, M. S.

Xu, Z.

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

Appl. Phys. B (1)

M. Tiihonen, V. Pasiskevicius, A. Fragemann, C. Canalias, and F. Laurell, "Ultrabroadband gain in optical parametric generator with periodically poled KTiOPO4," Appl. Phys. B 85, 73-77 (2006).
[CrossRef]

J. Appl. Phys. (1)

X. Liu, D. Deng, M. Li, D. Guo, and Z. Xu, "Retracing behaviour of the phase-matching angle of nonlinear crystals in optical parametric oscillators," J. Appl. Phys. 74, 2989-2991 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

R. Danielius, A. Piskarskas, A. Stabinis, G. P. Banfi, P. Di Trapani, and R. Righini, "Traveling-wave parametric generation of widely tunable, highly coherent femtosecond light pulses," J. Opt. Soc. Am. 10, 2222-2232 (1993).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Lett. (1)

Opt. Mater. (1)

N. Umemura, K. Miyata, and K. Kato, "New data on the optical properties of BiB3O6," Opt. Mater. 30, 532-534 (2007).
[CrossRef]

Quantum Electron. (1)

S. N. Orlov, E. V. Pestryakov, and Yu. N. Polivanov, "Optical parametric amplification with bandwidth exceeding an octave," Quantum Electron. 34, 477-479 (2004) [transl. from Kvantovaya Elektron. (Moscow) 34, 477-481 (2004)].
[CrossRef]

Rev. Sci. Instrum. (1)

G. Cerullo and S. De Silvestri, "Ultrafast optical parametric amplifiers," Rev. Sci. Instrum. 74, 1-18 (2003).
[CrossRef]

Solid State Commun. (1)

H. Hellwig, J. Liebertz, and L. Bohaty, "Exceptional large nonlinear optical coefficients in the monoclinic bismuth borate BiB3O6 (BIBO)," Solid State Commun. 109, 249-251 (1999).
[CrossRef]

Sov. J. Quantum Electron. (2)

A. Birmontas, A. Piskarskas, and A. Stabinis, "Dispersion anomalies of tuning characteristics and spectrum of an optical parametric oscillator," Sov. J. Quantum Electron. 13, 1243-1246 (1983) [transl. from Kvantovaya Elektron. (Moscow) 10, 1881-1884 (1983)].
[CrossRef]

B. Bareika, A. Birmontas, G. Dikchyus, A. Piskarskas, V. Sirutkaitis, and A. Stabinis, "Parametric generation of picosecond continuum in near-infrared and visible ranges on the basis of a quadratic nonlinearity," Sov. J. Quantum Electron. 12, 1654-1656 (1982) [transl. from Kvantovaya Elektron. (Moscow) 9, 2534-2536 (1982)].
[CrossRef]

Other (1)

D. N. Nikogosyan, Nonlinear Optical Crystals, (Springer, New York, 2005).

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Figures (10)

Fig. 1.
Fig. 1.

Gain bandwidth (FWHM) of BIBO and BBO, analytically calculated for collinear type e→oo interaction at λ P=800 nm, a crystal length of 5 mm and pump intensity of 50 GW/cm2 using only the first, second, third and fourth order Taylor series expansion terms of the wave-vector mismatch Δk, respectively.

Fig. 2.
Fig. 2.

(a). Normalized parametric gain in BIBO (negative type-I e→oo interaction in the x-z plane) for different pump wavelengths (indicated) at exact phase-matching angles, for a crystal length of 5 mm and pump intensity of 50 GW/cm2, (b) gain bandwidth (FWHM) as estimated from (a), (c) phase-matching curves at different pump wavelengths and crystal temperatures (indicated) for the same process and SHG type-I (oo→e) at room temperature. The absolute gain G for (a) decreases from 1.41×108 at λ P=720 nm to 2.91×107 at λ P=780 nm and to 5.56×106 at λ P=840 nm.

Fig. 3.
Fig. 3.

Refractive index of BIBO for type-I SHG in the x-z plane. The wavelength ranges are determined by the validity of the Sellmeier expansions used (474–3083 nm) [7]. Red curves correspond to oo→e and blue curves to ee→o phase-matching. The squares indicate the points where the first derivatives of the refractive index for the fundamental and second harmonic are equal (broadband SHG phase-matching corresponding to λ F in Table 1) and the circles indicate the points of vanishing second derivative (corresponding to twice the “magic” pump wavelength or 2λ P in Table 1).

Fig. 4.
Fig. 4.

OPG output energy obtained with the 3-mm (a) and 5-mm (b) thick BIBO crystals versus internal phase-matching angle relative to normal incidence. The average pump intensity (1/2 of the peak on-axis level) is 150 GW/cm2 (a) and 60 GW/cm2 (b).

Fig. 5.
Fig. 5.

OPG spectra obtained with the 3-mm (a) and 5-mm (b) thick BIBO crystals, recorded with an InGaAs spectrometer at five different internal angles of the pump beam. The pump intensity is 100 GW/cm2 (a) and 60 GW/cm2 (b).

Fig. 6.
Fig. 6.

Parametric gain of BIBO for collinear type e→oo interaction at λP =800 nm, calculated for several fixed phase-matching angles close to degeneracy (θ=11.05°). The crystal length, 3 mm (a) and 5 mm (b), and the pump intensity, 100 GW/cm2 (a) and 60 GW/cm2 (b), correspond to the experimental conditions in Fig. 5.

Fig. 7.
Fig. 7.

Spectrum (a), XFROG trace (b) and cross-correlation function with Gaussian fit (c) of the OPG output with the 3-mm thick BIBO crystal; pump intensity: 100 GW/cm2.

Fig. 8.
Fig. 8.

Spectrum (a), XFROG trace (b) and cross-correlation function with Gaussian fit (c) of the OPG output with the 5-mm thick BIBO crystal; pump intensity: 60 GW/cm2.

Fig. 9.
Fig. 9.

Spectrum (a), XFROG trace (b) and cross-correlation function with Gaussian fit (c) of the OPG output with the 3-mm thick BIBO crystal; pump intensity: 150 GW/cm2.

Fig. 10.
Fig. 10.

Group delay derived from the XFROG traces for the three cases depicted in Figs. 7–9.

Tables (1)

Tables Icon

Table 1. Parameters of several crystals that can be used in ultrabroadband OPA/OPG schemes pumped below 1 µm: λ P is the “magic” pump wavelength for which the signal/idler GVD vanishes near degeneracy; λ F is the fundamental wavelength for broadband SHG, Δλ/Δν correspond to the wavelength / frequency range for maximum gain bandwidth calculated at the 1/2 level (crystal length 5 mm, pump intensity 50 GW/cm2), θ/φand Λ are the phase-matching angle and the period for degenerate operation at the given λ P in birefringent and quasi phase-matching, respectively, and the GVM parameter 1/ν P-1/ν S,I is calculated for degenerate operation at λ P.*

Equations (8)

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G = 1 + Γ 2 g 2 sinh 2 ( gL )
G = 1 4 exp ( 2 g L )
Δ k = Δ k 0 + ( k S ω S k I ω I ) Δ ω + 1 2 ! ( 2 k S ω S 2 + 2 k I ω I 2 ) ( Δ ω ) 2 +
+ 1 3 ! ( 3 k S ω S 3 3 k I ω I 3 ) ( Δ ω ) 3 + 1 4 ! ( 4 k S ω S 4 + 4 k ω I 4 ) ( Δ ω ) 4 . . . . . .
Δ ν = 2 ( ln 2 ) 1 2 π ( Γ L ) 1 2 1 ν S 1 ν I 1
Δ ν = 2 ( ln 2 ) 1 4 π ( Γ L ) 1 4 2 k S ω S 2 + 2 k I ω I 2 1 2
Δ ν = ( 144 ln 2 ) 1 6 π ( Γ L ) 1 6 3 k S ω S 3 3 k I ω I 3 1 3
Δ ν = 2 ( 9 ln 2 ) 1 8 π ( Γ L ) 1 8 4 k S ω S 4 + 4 k I ω I 4 1 4

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