Abstract

We present a theory of ultrashort-pulse second-harmonic generation (SHG) in materials with longitudinally nonuniform quasi-phase-matching (QPM) gratings. We derive an expression for the output second-harmonic field generated in an arbitrary QPM grating from an arbitrary fundamental field, valid for arbitrary material dispersion in the undepleted-pump approximation. In the case when group-velocity dispersion can be neglected, a simple transfer-function relationship describes the SHG process. This SHG transfer function depends only on material properties and on the QPM grating design. We use this SHG transfer function to show that nonuniform QPM gratings can be designed to generate nearly arbitrarily shaped second-harmonic output pulses. We analyze in detail a technologically important example of pulse shaping: the generation of compressed second-harmonic pulses from linearly chirped fundamental input pulses. The efficiency of these interactions as well as the limits imposed by higher-order material dispersion are discussed.

© 2000 Optical Society of America

Full Article  |  PDF Article

Errata

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: errata," J. Opt. Soc. Am. B 18, 121-121 (2001)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-18-1-121

References

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    [CrossRef]
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    [CrossRef] [PubMed]
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1999

1998

1997

1996

1995

1994

E. Sidick, A. Knoesen, and A. Dienes, “Ultrashort-pulse second harmonic generation in quasi-phase-matched dispersive media,” Opt. Lett. 19, 266–268 (1994).
[CrossRef] [PubMed]

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

1992

B. Zysset, I. Biaggio, and P. Gunter, “Refractive indices of orthorhombic KNbO3. I. Dispersion and temperature dependence,” J. Opt. Soc. Am. B 9, 380–386 (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]

1991

1990

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]

1983

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

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

S. L. Shapiro, “Second harmonic generation in LiNbO3 by picosecond pulses,” Appl. Phys. Lett. 13, 19–21 (1968).
[CrossRef]

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

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

1962

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]

Biaggio, I.

Bierlein, J. D.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KTiOPO4 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]

Bortz, M. L.

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Brener, I.

Byer, R. L.

R. L. Byer, “Quasi-phasematched nonlinear interactions and devices,” J. Nonlinear Opt. Phys. Mater. 6, 549–592 (1997), and references therein.
[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]

Cha, M.

Cheng, L. K.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KTiOPO4 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 KTiOPO4 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]

Dienes, A.

Ebrahimzadeh, M.

Faller, P.

Fejer, M. M.

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, “Multiple channel wavelength conversion using engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).
[CrossRef]

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

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, 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]

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30, 34–35 (1994).
[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]

Fenimore, D. L.

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 Photonics Technol. Lett. 11, 650–652 (1999).
[CrossRef]

Fujimura, M.

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

Galperin, J.

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KTiOPO4 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]

Gunter, P.

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 Photonics Technol. Lett. 11, 650–652 (1999).
[CrossRef]

Imeshev, G.

Ishigame, Y.

Ishihara, H.

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]

Kan’an, A. M.

Karlsson, H.

Kato, M.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

Knoesen, A.

Kuck, S.

Laurell, F.

Leaird, D. E.

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.

Mizuuchi, K.

K. Mizuuchi and K. Yamamoto, “Waveguide second-harmonic generation device with broadened flat quasi-phase-matched response by use of a grating structure with located phase shifts,” Opt. Lett. 23, 1880–1882 (1998).
[CrossRef]

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

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 KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

Nishihara, H.

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]

Parameswaran, K. R.

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]

Proctor, M.

Ramabadran, U. B.

Reid, D. T.

Sato, H.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Schepler, K. L.

Shapiro, S. L.

S. L. Shapiro, “Second harmonic generation in LiNbO3 by picosecond pulses,” Appl. Phys. Lett. 13, 19–21 (1968).
[CrossRef]

Sibbett, W.

Sidick, E.

Small, D.

Suhara, T.

Y. Ishigame, T. Suhara, and H. Ishihara, “LiNbO3 waveguide second-harmonic-generation device phase matched with a fan-out domain-inverted grating,” Opt. Lett. 16, 375–379 (1991).
[CrossRef] [PubMed]

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]

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).

Von Richter, P.

Weiner, A. M.

A. M. Weiner, A. M. Kan’an, and D. E. Leaird, “High-efficiency blue generation by frequency doubling of femtosecond pulses in a thick nonlinear crystal,” Opt. Lett. 23, 1441–1443 (1998).
[CrossRef]

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]

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 Photonics Technol. Lett. 11, 650–652 (1999).
[CrossRef]

Yamamoto, K.

K. Mizuuchi and K. Yamamoto, “Waveguide second-harmonic generation device with broadened flat quasi-phase-matched response by use of a grating structure with located phase shifts,” Opt. Lett. 23, 1880–1882 (1998).
[CrossRef]

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Zelmon, D.

Zysset, B.

Appl. Phys. Lett.

S. L. Shapiro, “Second harmonic generation in LiNbO3 by picosecond pulses,” Appl. Phys. Lett. 13, 19–21 (1968).
[CrossRef]

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

Electron. Lett.

M. L. Bortz, M. Fujimura, and M. M. Fejer, “Increased acceptance bandwidth for quasi-phasematched second harmonic generation in LiNbO3 waveguides,” Electron. Lett. 30, 34–35 (1994).
[CrossRef]

IEEE J. Quantum Electron.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

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

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]

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 Photonics Technol. Lett.

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

J. Appl. Phys.

G. D. Boyd and D. A. Kleinman, “Parametric interactions of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[CrossRef]

J. Cryst. Growth

L. K. Cheng, L. T. Cheng, J. Galperin, P. A. Morris Hotsenpiller, and J. D. Bierlein, “Crystal growth and characterization of KTiOPO4 isomorphs from the self-fluxes,” J. Cryst. Growth 137, 107–115 (1994).
[CrossRef]

J. Nonlinear Opt. Phys. Mater.

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Figures (7)

Fig. 1
Fig. 1

Time-domain picture of SHG with a chirped QPM grating. Two different spectral components of the FH pulse, ω (dashed curve) and ω (solid curve) propagate through the crystal with the same group velocity u1. At a certain position in the grating determined by the local QPM period the FH component ω generates the SH component 2ω (dashed curve), which then freely travels to the end of the crystal with group velocity u2u1. The other FH component, ω, gets converted to SH component 2ω (solid curve) further in the crystal, and then this SH component travels with group velocity u2. Because u2u1, the 2ω and 2ω components acquire a particular time delay relative to each other and to the FH pulse.

Fig. 2
Fig. 2

Square of absolute value of the transfer function for the uniform grating, Eq. (33), |Dˆ(Ω)|2sinc2(ΩδνL/2) (solid curve) and the square of the absolute value of A12ˆ(Ω) for a Gaussian FH pulse, Eq. (37), |A12ˆ(Ω)|2exp(-τ02Ω2/4) for τ0=δνL such that L/Lgv=1 (dash-dotted curve) and τ0=0.2δνL such that L/Lgv=5 (dashed curve). |Dˆ(Ω)|2 and |A12ˆ(Ω)|2 are plotted as functions of normalized frequency (δνL)Ω.

Fig. 3
Fig. 3

(a), (c), normalized 1/e power half-width of a SH pulse, τ2/τ0, generated with a uniform grating of length L, as obtained from Eq. (39) for a Gaussian FH pulse of the form of Eq. (37). (b), (d), the efficiency-reduction factor, g, Eq. (64) (solid curves), and the (L/Lgv)g(L/Lgv) factor (dashed curves). τ2/τ0, g, and L/Lgv g(L/Lgv) are plotted as functions of L/Lgv on linear [(a) and (b)] and logarithmic [(c) and (d)] scales.

Fig. 4
Fig. 4

Normalized FH pulse length, τ1/τ0 [solid curve, Eq. (49)], normalized SH pulse length, τ2/τ0 [dashed curve, Eq. (54)], and normalized plane-wave efficiency, ηPW/η0PW [dash-dotted line, Eq. (69)], plotted as functions of normalized FH chirp, C1/τ02. A chirped grating with a normalized chirp Dg(τ0/δν)2=-0.1 is assumed.

Fig. 5
Fig. 5

Absolute value (solid curves) and phase (dashed curves) of representative integrals F(Ω), defined by Eq. (59), as a function of x0 for different values of X.

Fig. 6
Fig. 6

Effect of bandwidth truncation by a chirped grating of finite length L=Nδν/(2Dgτ0) on the SH pulse 1/e power half-width (solid curve) and output energy (dash-dotted curve). Both are normalized to respective values obtained with a grating of infinite length and plotted as functions of N.

Fig. 7
Fig. 7

SHG conversion efficiency η0conf as a function of the FH wavelength for representative first-QPM-order materials.

Tables (1)

Tables Icon

Table 1 Comparison of Nonlinear Coefficient, GVM Parameter, Figure of Merit, and Confocal Efficiency for Several QPM Materials

Equations (76)

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F(t)-+Fˆ(ω)exp(iωt)dω,
Fˆ(ω)12π-+F(t)exp(-iωt)dt,
×Eˆ(r, ω)=-iωBˆ(r, ω),
×Hˆ(r, ω)=iωDˆ(r, ω),
2z2Eˆ1(z, ω)+k2(ω)Eˆ1(z, ω)=0,
2z2Eˆ2(z, ω)+k2(ω)Eˆ2(z, ω)=-μ0ω2PˆNL(z, ω),
PˆNL(z, ω)=120χ(2)(z)-+Eˆ1(z, ω)Eˆ1(z, ω-ω)dω,
Ei(z, t)=Bi(z, t)exp(iωit-ikiz).
Eˆi(z, ω)=Bˆi(z, Ωi)exp(-ikiz),
Eˆi(z, ω)=Aˆi(z, Ωi)exp[-ik(ωi+Ωi)z].
Bˆi(z, Ωi)=Aˆi(z, Ωi)exp{-i[k(ωi+Ωi)-ki]z}.
Aˆi(0, Ωi)=Bˆi(0, Ωi),
Aˆi(z, 0)=Bˆi(z, 0).
zAˆ1(z, Ω1)=0,
zAˆ2(z, Ω)=-i μ0ω222k2PˆNL(z, Ω)exp[ik(ω2+Ω)z],
PˆNL(z, Ω)
=0d(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)=Aˆ1(Ω)exp[-ik(ω1+Ω)z].
Aˆ2(L, Ω)=-+Aˆ1(Ω)Aˆ1(Ω-Ω)dˆ[Δk(Ω,Ω)]dΩ,
Δk(Ω, Ω)=k(ω1+Ω)+k(ω1+Ω-Ω)-k(ω2+Ω),
dˆ(Δk)=-iγ-+d(z)exp(-iΔkz)dz,
Δk(Ω, Ω)=Δk(Ω)+Δk(Ω, Ω),
Δk(Ω)=Δk0+δνΩ+12δβΩ2+δk(Ω),
Δk(Ω, Ω)=β1(Ω2-ΩΩ)+δk(Ω, Ω),
Dˆ(Ω)=-iγ-+d(z)exp[-i(Δk0+δνΩ)z]dz.
Aˆ2(L, Ω)=Dˆ(Ω)A12ˆ(Ω),
A12ˆ(Ω)=-+Aˆ1(Ω)Aˆ1(Ω-Ω)dΩ.
A2(L, t)=-iγ[d(t/δν)exp(-iΔk0t/δν)]A12(t),
Bˆ2(L, Ω)=Dˆ(Ω)B12ˆ(Ω)exp(-iΩL/u2),
B12ˆ(Ω)=-+Bˆ1(z=0, Ω)Bˆ1(z=0, Ω-Ω)dΩ.
d(z)=mdm(z)=m|dm|exp(iKmz),
|dm|=2πmdeff sin(πmG),
Dˆ(Ω)=γL|dm|sinc[(Δk0-Km+Ωδν)L/2],
E1(z=0, t)=E1 exp[i(ω1+Ω1)t].
A12ˆ(Ω)=E12δ(Ω-2Ω1).
Aˆ2(L, Ω2=2Ω1)
=E12γL|dm|sinc[(Δk0-Km+Ω2δν)L/2].
B1(0, t)=E0 exp-t22τ02.
Aˆ1(Ω)=12πE0τ0 exp-12τ02Ω2.
Aˆ2(Ω)=γ2πE02L|dm|τ0 exp-14τ02Ω2sinc(ΩδνL/2),
Lgv=τ0|δν|.
dm(z)=|dm(z)|exp[iK0mz+iΦm(z)],
Km(z)=K0m+dΦmdz.
Dˆ(Ω)=Aˆ2(Ω)A12ˆ(Ω).
d(z)=i δν2πγexp(iΔk0z)-+ Aˆ2(Ω)A12ˆ(Ω)exp(iΩδνz)dΩ.
dm(z)=|dm|exp[iK0m(z-L/2)+iDg(z-L/2)2]×rect(z/L-1/2),
Dˆ(Ω)=γ|dm|-L/2+L/2 exp[-i(Δk0-K0m+δνΩ)z+iDgz2]dz,
Dˆ(Ω)=γ|dm|πDg exp-i δν2Ω24Dg,
B1(0, t)=E0 τ0τ02+iC1exp-t22(τ02+iC1),
τ1=τ02+(C1/τ0)2.
Aˆ1(Ω)=12πE0τ0 exp-12(τ02+iC1)Ω2.
Aˆ2(L, Ω)=12γ|dm|E02 1Dgτ02τ02+iC1×exp-12(τ02/2+iC2)Ω2,
C2=12(C1+δν2/Dg).
B2(L, t)=γ|dm|E02πDg τ0τ1τ02-iC1τ02+i2C2×exp-t22(τ02/2+iC2),
τ2=(τ02)2+(2C2/τ0).
Dg=Dgopt-δν2C1,
B2(L, t)=E2 exp-t2τ02,
E2=γ|dm|τ0τ1 πC1δνE02.
Dˆ(Ω)=γ|dm|πDg exp-i δν2Ω24DgF(Ω),
F(Ω)=1πx0-Xx0+X exp(ix2)dx,
ΔΩg=2DgLδν.
L=N2δνDgτ0.
L=N2τ1δν,
ηPWU2U1=n2n1-+|D(Ω)|2|A12ˆ(Ω)|2dΩ-+|Aˆ1(Ω)|2dΩ.
ηPW=22π2 |dm|2n1n2λ12E02L2g(L/Lgv),
η0PW=21.3 |dm|2n1n2λ12E02Lgv2.
ηPW=n2n1|A12ˆ(Ω=0)|2-+|Dˆ(Ω)|2dΩ-+|Aˆ1(Ω)|2dΩ.
-+|Dˆ(Ω)|2dΩ=2πγ2δν-+|d(z)|2dz=2πγ2δνfL2deffπm2,
ηPW=2πg(L/Lgv=2)δνLτ1fη0PW,
ηPW=πg(L/Lgv=2)δν2Dgτ1τ0η0PW.
ηconf=8π2πc0|dm|2δνn1n2λ13U1 LLgvg(L/Lgv)h,
η0conf=76.70λ13FOMU1,
FOM=|dm|2cδνn1n2.
ηconf=2πg(L/Lgv=2)τ0τ1fη0conf.
ηconf=2πg(L/Lgv=2)τ0τ11Nη0conf,

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