Abstract

A theoretical treatment of group-velocity dispersion and higher-order dispersion effects on the second-harmonic-generation (SHG) process with longitudinally nonuniform quasi-phase-matching (QPM) gratings is presented. We show how these dispersion terms can be accounted for in the design of a QPM-SHG pulse shaper. Our numerical simulation results show that, if the proper dispersion correction is included in the QPM grating design, one can generate sub-10-fs transform-limited pulses at 400 nm by doubling the output of a Ti:sapphire oscillator.

© 2000 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  37. L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
    [CrossRef]

2000 (1)

1999 (5)

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, “Low-noise amplification of high-power pulses in multimode fibers,” IEEE Photon. Technol. Lett. 11, 650–652 (1999).
[CrossRef]

P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, W. Sibbett, H. Karlsson, and F. Laurell, “Simultaneous femtosecond-pulse compression and second-harmonic generation in aperiodically poled KTiOPO4,” Opt. Lett. 24, 1071–1073 (1999).
[CrossRef]

P. Loza-Alvarez, D. T. Reid, P. Faller, M. Ebrahimzadeh, and W. Sibbett, “Simultaneous second-harmonic generation and femtosecond-pulse compression in aperiodically poled KTiOPO4 with a RbTiOAsO4-based optical parametric oscillator,” J. Opt. Soc. Am. B 16, 1553–1560 (1999).
[CrossRef]

V. G. Dmitriev and S. G. Grechin, “Multi frequency laser radiation harmonics generation in nonlinear crystals with regular domain structure,” in ICONO ’98: Nonlinear Optical Phenomena and Coherent Optics in Information Technologies, S. S. Chesnokov, V. P. Kandidov, and N. I. Koroteev, eds., Proc. SPIE 3733, 228–236 (1999).

S. Saltiel and Y. Deyanova, “Polarization switching as a result of cascading of two simultaneously phasematched quadratic processes,” Opt. Lett. 24, 1296–1298 (1999).
[CrossRef]

1998 (3)

1997 (5)

1995 (3)

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second harmonic generation. II. Non-transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1713–1722 (1995).
[CrossRef]

E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second harmonic generation. I. Transform-limited fundamental pulses,” J. Opt. Soc. Am. B 12, 1704–1712 (1995).
[CrossRef]

1994 (2)

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, “Simultaneous generation of red, green and blue light in a segmented KTP waveguide using a single source,” Electron. Lett. 30, 975–976 (1994).
[CrossRef]

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

1993 (1)

S. Lin, B. Wu, F. Xie, and C. Chen, “Phase matching retracing behavior for second harmonic generation in LiB3O5 crystal,” J. Appl. Phys. 73, 1029–1034 (1993).
[CrossRef]

1992 (2)

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 (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]

1983 (1)

A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. QE-19, 1276–1283 (1983).
[CrossRef]

1969 (2)

W. H. Glenn, “Second harmonic generation by picosecond optical pulses,” IEEE J. Quantum Electron. QE-5, 284–290 (1969).
[CrossRef]

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, “Nonstationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–757 (1969).

1968 (1)

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, “Nonstationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–757 (1969).

Arbore, M. A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Baldi, P.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

Bierlein, J. D.

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, “Simultaneous generation of red, green and blue light in a segmented KTP waveguide using a single source,” Electron. Lett. 30, 975–976 (1994).
[CrossRef]

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bosenberg, W. R.

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]

Chen, C.

S. Lin, B. Wu, F. Xie, and C. Chen, “Phase matching retracing behavior for second harmonic generation in LiB3O5 crystal,” J. Appl. Phys. 73, 1029–1034 (1993).
[CrossRef]

Cheng, L. K.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Cheng, L. T.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Chirkin, A. S.

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, “Nonstationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–757 (1969).

Chou, M. H.

Comly, J.

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

De Micheli, M. P.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

Delacourt, D.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

Deyanova, Y.

Dienes, A.

Dmitriev, V. G.

V. G. Dmitriev and S. G. Grechin, “Multi frequency laser radiation harmonics generation in nonlinear crystals with regular domain structure,” in ICONO ’98: Nonlinear Optical Phenomena and Coherent Optics in Information Technologies, S. S. Chesnokov, V. P. Kandidov, and N. I. Koroteev, eds., Proc. SPIE 3733, 228–236 (1999).

Ebrahimzadeh, M.

Eimerl, D.

Faller, P.

Fejer, M. M.

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]

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, 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]

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]

J.-P. Meyn and M. M. Fejer, “Tunable ultraviolet radiation by second-harmonic generation in periodically poled lithium tantalate,” Opt. Lett. 22, 1214–1216 (1997).
[CrossRef] [PubMed]

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]

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]

Fermann, M.

Fermann, M. E.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, “Low-noise amplification of high-power pulses in multimode fibers,” IEEE Photon. Technol. Lett. 11, 650–652 (1999).
[CrossRef]

Frigo, M.

M. Frigo and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” IEEE Trans. Acoust. Speech Signal Process. 3, 1381–1384 (1998).

Galperin, J.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Galvanauskas, A.

Garmire, E.

J. Comly and E. Garmire, “Second harmonic generation from short pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

Glenn, W. H.

W. H. Glenn, “Second harmonic generation by picosecond optical pulses,” IEEE J. Quantum Electron. QE-5, 284–290 (1969).
[CrossRef]

Grechin, S. G.

V. G. Dmitriev and S. G. Grechin, “Multi frequency laser radiation harmonics generation in nonlinear crystals with regular domain structure,” in ICONO ’98: Nonlinear Optical Phenomena and Coherent Optics in Information Technologies, S. S. Chesnokov, V. P. Kandidov, and N. I. Koroteev, eds., Proc. SPIE 3733, 228–236 (1999).

Harter, D.

Hofer, M.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, “Low-noise amplification of high-power pulses in multimode fibers,” IEEE Photon. Technol. Lett. 11, 650–652 (1999).
[CrossRef]

Hollberg, L.

Imeshev, G.

Johnson, S. G.

M. Frigo and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” IEEE Trans. Acoust. Speech Signal Process. 3, 1381–1384 (1998).

Jundt, D. H.

D. H. Jundt, “Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,” Opt. Lett. 22, 1553–1555 (1997).
[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]

Karlsson, H.

Knoesen, A.

Laurell, F.

Levenson, M. D.

Lin, S.

S. Lin, B. Wu, F. Xie, and C. Chen, “Phase matching retracing behavior for second harmonic generation in LiB3O5 crystal,” J. Appl. Phys. 73, 1029–1034 (1993).
[CrossRef]

Loza-Alvarez, P.

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]

Marco, O.

Meyn, J.-P.

Morris Hotsenpiller, P. A.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KiTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Ostrowsky, D. B.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

Papuchon, M.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pfister, O.

Proctor, M.

Reid, D. T.

Saltiel, S.

Sibbett, W.

Sidick, E.

Stegeman, G. I.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, “Simultaneous generation of red, green and blue light in a segmented KTP waveguide using a single source,” Electron. Lett. 30, 975–976 (1994).
[CrossRef]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, “Nonstationary phenomena and space–time analogy in nonlinear optics,” Sov. Phys. JETP 28, 748–757 (1969).

Sundheimer, M. L.

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, “Simultaneous generation of red, green and blue light in a segmented KTP waveguide using a single source,” Electron. Lett. 30, 975–976 (1994).
[CrossRef]

Trevino-Palacios, C. G.

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, “Simultaneous generation of red, green and blue light in room temperature periodically poled lithium niobate waveguides using single source,” Electron. Lett. 31, 1350–1351 (1995).
[CrossRef]

Van Baak, D. A.

Velsko, S. P.

Villeneuve, A.

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, “Simultaneous generation of red, green and blue light in a segmented KTP waveguide using a single source,” Electron. Lett. 30, 975–976 (1994).
[CrossRef]

Webb, M. S.

Weiner, A. M.

A. M. Weiner, “Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation,” IEEE J. Quantum Electron. QE-19, 1276–1283 (1983).
[CrossRef]

Wells, J. S.

Windeler, R. S.

M. Hofer, M. E. Fermann, A. Galvanauskas, D. Harter, and R. S. Windeler, “Low-noise amplification of high-power pulses in multimode fibers,” IEEE Photon. Technol. Lett. 11, 650–652 (1999).
[CrossRef]

Wu, B.

S. Lin, B. Wu, F. Xie, and C. Chen, “Phase matching retracing behavior for second harmonic generation in LiB3O5 crystal,” J. Appl. Phys. 73, 1029–1034 (1993).
[CrossRef]

Xie, F.

S. Lin, B. Wu, F. Xie, and C. Chen, “Phase matching retracing behavior for second harmonic generation in LiB3O5 crystal,” J. Appl. Phys. 73, 1029–1034 (1993).
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Figures (12)

Fig. 1
Fig. 1

Normalized cw tuning curves, |dˆ|2, for a uniform grating in the presence of GVD for (a) small and (b) large values of α=(β1/2-β2)/(Lδν2).

Fig. 2
Fig. 2

Dependence of the normalized SH pulse length τ2/τ0 as a function of normalized FH chirp C1/τ02 for several values of normalized GVD coefficient at the SH, β2/δντ0. A linearly chirped grating with length as given by Eq. (6.8) and with a normalized chirp Dg(τ0/δν)2=-0.1 is assumed.

Fig. 3
Fig. 3

Effect of uncompensated cubic phase on the length of the SH pulse generated in a linearly chirped grating in the presence of GVD at the SH. The normalized SH pulse length τ2/τ0 obtained from relation (6.11) when the FH chirp is selected according to Eq. (6.14) is plotted as a function of (β2/δντ0)/[Dg(τ0/δν)2].

Fig. 4
Fig. 4

Required length of a chirped grating, as given by Eq. (8.16), normalized to the grating length for the case β1=β2=0, as a function of (β1+2β2)/τ0|δν|. The two branches correspond to the two signs in the denominator of Eq. (8.16).

Fig. 5
Fig. 5

(a) Normalized grating k-vector, [K(z)-Δk0](τ0/δν), as obtained with Eq. (8.12), and (b) normalized grating amplitude, |d(z)|, as obtained with Eq. (8.11). Both are plotted as functions of normalized position in the grating, z/L, for several representative pairs of the normalized GVD coefficients, (β1/δντ0, β2/δντ0): solid curve (0, 0); dashed curve, (0.02, 0.10); dotted curve, (0.05, 0.15); dotted–dashed curve, (0.10, 0.25).

Fig. 6
Fig. 6

(a) Distribution of the grating period Λ(z) for compression in lithium tantalate. The dashed curve represents a grating designed according to the prescription of Section 8 to account for GVD at both the SH and the FH; the dotted–dashed curve, a linearly chirped grating designed according to Section 6 [Eqs. (6.6) and (6.12)] to account for GVD at the SH; the dotted curve, a linearly chirped grating designed without accounting for the GVD effects [Eqs. (6.6) and (6.9)]. (b) Distribution of the grating amplitude for a grating designed to account for GVD at both the SH and the FH [Eq. (8.11) (dashed curve)]. Also shown is the unmodulated amplitude of the grating (solid curve).

Fig. 7
Fig. 7

(a) Intensities and (b) phases of the SH pulses obtained from a Gaussian FH pulse by use of different gratings. The dashed curve represents a grating designed to account for GVD at both the SH and the FH according to prescription of Section 8; the dotted–dashed curve, a grating designed to account for GVD at the SH [Eqs. (6.6) and (6.12)]; the dotted curve, a grating designed without accounting for the GVD effects [Eqs. (6.6) and (6.9)]. Also shown is the ideal SH pulse that would have been generated if the grating had not introduced spectral truncation and exactly corrected the FH phase (solid curve). Note that these results are obtained from the numerical simulations with dispersion terms beyond GVD set to zero.

Fig. 8
Fig. 8

SH pulses generated from a Gaussian FH pulse when the material dispersion beyond GVD is included in the simulations. The dashed curve represents the SH pulse obtained when the QPM grating is designed according to the prescription of Section 7 to completely account for the material dispersion; the dotted–dashed curve, the SH pulse generated in a grating designed to account for dispersion only up to the GVD terms (Section 8). In both cases grating amplitude is unmodulated. Also shown is the ideal SH pulse (solid curve).

Fig. 9
Fig. 9

QPM period as a function of FH wavelength in lithium niobate (solid curve), lithium tantalate (dashed curve), and KTP (dotted curve).

Fig. 10
Fig. 10

GVM coefficient δν as a function of FH wavelength in lithium niobate (solid curve), lithium tantalate (dashed curve), and KTP (dotted curve).

Fig. 11
Fig. 11

FH GVD coefficient β1 as a function of the FH wavelength, and the SH GVD coefficient β2 as a function of the FH wavelength that generates the corresponding SH, in lithium niobate (solid curve), lithium tantalate (dashed curve) and KTP (dotted curve).

Fig. 12
Fig. 12

Ratio βi/δν as a function of the FH wavelength in lithium niobate (solid curve), lithium tantalate (dashed curve), and KTP (dotted curve).

Equations (90)

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d(z)=m=-dm(z)=m=-|dm(z)|exp[iK0mz+iφm(z)],
Km(z)=dΦmdz=K0m+dφmdz
|dm(z)|=2πmdeff sin[πmG(z)],
Eˆ(z, ω)=Aˆ(z, Ω)exp[-ik(ω0+Ω)z],
E(z, t)=B(z, t)exp(iω0t-ik0z),
zAˆ1(z, Ω)=0,
zAˆ2(z, Ω)=-iμ0ω222k2PˆNL(z, Ω)exp[ik(ω2+Ω)z],
PˆNL(z, Ω)=ε0d(z)-+Aˆ1(z, Ω)Aˆ1(z, Ω-Ω)×exp{-i[k(ω1+Ω)+k(ω1+Ω-Ω)]z}dΩ.
Aˆ1(z, Ω)=Aˆ1(Ω),
Eˆ1(z, ω)=Eˆ1(z=0, ω)exp[-ik(ω)z],
Aˆ2(L, Ω)=-+Aˆ1(Ω)Aˆ1(Ω-Ω)dˆ[Δk(Ω, Ω)]dΩ.
dˆ(Δk)=-iγ-+d(z)exp(-iΔkz)d z,
Δk(Ω, Ω)=k(ω1+Ω)+k(ω1+Ω-Ω)-k(ω2+Ω).
Δk(Ω, Ω)=Δk(Ω)+Δk(Ω, Ω),
Δk(Ω)Δk0+δνΩ+12δβΩ2+δk(Ω),
Δk(Ω, Ω)β1(Ω2-ΩΩ)+δk(Ω, Ω),
Dˆ0(Ω)=-iγ-+d(z)exp[-i(Δk0+δνΩ)z]d z.
Aˆ2(L, Ω)=Dˆ0(Ω)A12ˆ(Ω),
A12ˆ(Ω)=-+Aˆ1(Ω)Aˆ1(Ω-Ω)dΩ.
Dˆ2(Ω)=-iγ-+d(z)exp[-iΔk(Ω)z]d z,
Aˆ2(L, Ω)=Dˆ2(Ω)A12ˆ(Ω).
E1(z=0, t)=E1 exp[i(ω1+Ω1)t].
Aˆ1(Ω)=E1δ(Ω-Ω1).
d(z)=|d|exp(iK0z)rectzL-12,
Aˆ2(L, Ω=2Ω1)=E12dˆ[Δk(Ω)],
dˆ[Δk(Ω)]=γL|d|sinc{[Δk(Ω)-K0]L/2},
Δk(Ω)=2k(ω1+Ω/2)-k(ω2+Ω)
Δk(Ω)=Δk0+δνΩ+12 12β1-β2Ω2.
ΔΩg5.57L|δν|(1+3.87α2).
ΔΩg4.72L|β1/2-β2| 1+0.090α.
B1(t)=E0 exp-t22τ02.
Aˆ1(Ω)=12πE0τ0 exp-12τ02Ω2.
Aˆ1(Ω)=12πE0τ0 exp-12(τ02+iC1)Ω2.
B1(t)=E0 τ0τ02+iC1 exp-t22(τ02+iC1).
d(z)=|d|expiK0z-L2+iDgz-L22rectzL-12,
K(z)=K0+2Dgz-L2,
Dˆ0(Ω)=γ|d|πDg exp-i δν2Ω24Dg,
L=3δνDgτ0.
Dg=-δν2C1,
Dˆ2=γ|d|πDg expi 14β2LΩ2-i (δνΩ-β2Ω2/2)24Dg.
Bˆ2(L, Ω)exp-14τ02Ω2-i4 C1+β2L+δν2DgΩ2+i4 δνβ2DgΩ3-i16 β22DgΩ4.
Dg=-δν2C1+β2L.
C1=-δν2Dg-β2L,
C1=-δν2Dg 1+3β2δντ0.
ΔC16δν2Dg β2δντ0,
Eˆ1(z, ω)=Eˆ1(z=0, ω)exp[-ik(ω)z].
ΦFH=Φ1(Ω1)-k(ω1+Ω1)z=Φ1-k1z-(z/u1)Ω1-k˜1(Ω1)z,
k˜i(Ωi)=n=2 1n! dnk(ω)dωnω=ωiΩin.
PˆNL(z, Ω)=ε0d(z)exp[-2ik1z-i(z/u1)Ω]pˆ(z, Ω),
pˆ(z, Ω)=-+Eˆ1(0, ω1+Ω)Eˆ1(0, ω1+Ω-Ω)×exp[-ik˜1(Ω)z-ik˜1(Ω-Ω)z]dΩ,
ΦP=Φ(z0)-2k1z0-(z0/u1)Ω0+[pˆ(z0, Ω0)],
ΦSH=-k(ω2+Ω0)(L-z0)=-k2(L-z0)-[(L-z0)/u2]Ω0-k˜2(Ω0)(L-z0).
Φtotal(z0, Ω0)
=Φ(z0)-Δk0z0-k2L-δνz0Ω0-(L/u2)Ω0+[pˆ(z0, Ω0)]-k˜2(Ω0)(L-z0).
Ω0Φ2(Ω0)+δνz0+Lu2-Ω0[pˆ(z0, Ω0)]
+(L-z0) Ω0k˜2(Ω0)=0.
Eˆ2(L, ω2+Ω)=-iγ exp[-ik(ω2+Ω)L]×-+|d(z)|exp[iΦ(z)]pˆ(z, Ω)×exp(-iΔk0z-iδνΩz)×exp[ik˜2(Ω)z]dz,
Eˆ2(L, ω2+Ω)=-iγ|d(z0)||pˆ(z0, Ω)|×2π|Φ(z0)+{[pˆ(z0, Ω)]}|1/2×exp[iΦtotal(z0, Ω)],
ΦP=Φ(z0)-12 arctanC1+β1z0τ02-2k1z0-z0u1Ω0-14(C1+β1z0)Ω02.
ΦSH=-k2(L-z0)-L-z0u2Ω0-12β2Ω02(L-z0).
Φ2=-Lu1-ΔTΩ0-12C2Ω02,
Ω0=2 δν(L-z0)-ΔTC(z0),
C(z)=C1-2C2+(β1-2β2)z+2β2L.
Φ(z0)=12 arctanC1+β1z0τ02+Δk0z0-[δν(L-z0)-ΔT]2C(z0),
|d(z)|1π|C(z)| τ1(z)τ01/2δν+(β1-2β2) δν(L-z)-ΔTC(z),
τ1(z)={τ02+[C1(z)/τ0]2}1/2.
Aˆ2(L, Ω)=-iγA12ˆ(Ω)-+dzd˜(z)exp[-iμ(Ω)z],
μ(Ω)=Δk(Ω)-14β1Ω2=Δk0+δνΩ+12 12β1-β2Ω2+δk(Ω),
d˜(z)=d(z)τ02+iC1τ02+iC1(z)1/2
C1(z)=C1+β1z,
d(z)=i 12πγ τ02+iC1(z)τ02+iC11/2×-+dμ Aˆ2(L, Ω)A12ˆ(Ω) exp[iμ(Ω)z],
Bˆ2(L, Ω)=12πE20τ2×exp-12(τ22+iC2)Ω2-iLu1-ΔTΩ.
Aˆ2(L, Ω)=12πE20τ2 exp-12(τ22+iC2-iβ2L)Ω2-i(δνL-ΔT)Ω.
τ2=τ0/2.
d(z)=|d(z)|exp[iΦ(z)],
Φ(z)=12 arctanC1(z)τ02+Δk0z-[δν(L-z)-ΔT]2C(z)
|d(z)|=1γ E20E02 1π|C(z)| τ1(z)τ01/2δν+(β1-2β2) δν(L-z)-ΔTC(z),
K(z)=12β1 1τ12(z)+Δk0+2δν δν(L-z)-ΔTC(z)+(β1-2β2) [δν(L-z)-ΔT]2C(z)2.
K(0)=Δk0±δν ΔΩg2+12 12β1-β2ΔΩg22,
K(L)=Δk0δν ΔΩg2+12 12β1-β2ΔΩg22.
ΔT=12δνL C1-2C2+β1LC1-2C2+(β1+2β2)L/2.
L=3|C1-2C2|/τ0|δν|±(3/2)[(β1+2β2)/τ0],
|d(z)|=2πm deffM τ0τ1(z)|C(z)|1/2 1|δν| δν+(β1-2β2) δν(L-z)-ΔTC(z),
M=maxτ0τ1(0)|C(0)|1/22C(L)C(0)+C(L),
τ0τ1(L)|C(L)|1/22C(0)C(0)+C(L).
E20=2πm deffMLgvγE02.
B2ideal(L, t)=IFT|FT{[B1(t)]2}|,
B2TL(L, t)=IFT|FT[B2(L, t)]|.
B1(t)=E0 sech(t/τs),
Bˆ1(Ω1)=E0 exp-(τsgΩ1)2m2,

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