Abstract

We theoretically analyze a method for matching group velocities of fundamental and second harmonic femtosecond pulses during phase matched frequncy doubling by predispersing the fundamental pulse with a prism. The method permits improved conversion efficiency by allowing crystal lengths of several millimeters without sacrificing second harmonic pulse duration. Second harmonic pulse energy and duration are analyzed for β-BaB2O4, and limiting experimental factors are discussed. The results show that the method is most advantageous for incident pulses between 0.1- and 1.0-ps duration and microjoule and higher energies and that second harmonic pulse duration and conversion efficiency are not highly sensitive to optical misalignments of the order of 1°.

© 1990 Optical Society of America

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  1. J. Comly, E. Garmire, “Second Harmonic Generation from Short Pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
    [CrossRef]
  2. W. H. Glenn, “Second Harmonic Generation by Picosecond Optical Pulses,” IEEE J. Quantum Electron. QE-5, 284–290 (1969).
    [CrossRef]
  3. I. V. Tomov, R. Fedosejevs, A. A. Offenberger, “Up-conversion of Subpicosecond Light Pulses,” IEEE J. Quantum Electron. QE-18, 2048–2055 (1982).
    [CrossRef]
  4. D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
    [CrossRef]
  5. D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
    [CrossRef]
  6. W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Ultraviolet Optical Parametric Oscillation in β-BaB2O4,” Appl. Phys. Lett. 54, 13–15 (1989).
    [CrossRef]
  7. S. A. Akmanov, A. I. Kovrygin, A. P. Sukhorukov, “Optical Harmonic Generation and Optical Frequency Multipliers,” in Quantum Electronics: A TreatiseH. Rabin, C. L. Tang, Eds. (Academic, New York, 1975), Vol. 1, Pt. B.
  8. D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second Harmonic Generation of Light by Focused Laser Beams,” Phys. Rev. 145, 338–379 (1966).
    [CrossRef]
  9. O. E. Martinez, J. P. Gordon, R. L. Fork, “Negative Group-Velocity Dispersion Using Refraction,” J. Opt. Soc. Am. A 1, 1003–1006 (1984); R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative Dispersion Using Pairs of Prisms,” Opt. Lett. 9, 150–152 (1984).
    [CrossRef] [PubMed]
  10. Z. Bor, “Distortion of Femtosecond Laser Pulses in Lenses,” Opt. Lett. 14, 119–121 (1989).
    [CrossRef] [PubMed]
  11. O. E. Martinez, “Achromatic Phase Matching for Second Harmonic Generation of Femtosecond Pulses,” IEEE J. Quantum Electron. QE-25, 2464–2468 (1989).
    [CrossRef]
  12. G. Szabo, Z. Bor, “Broadband Frequency Doubler for Femtosecond Pulses,” Appl. Phys. B 50, 51–54 (1990).
    [CrossRef]
  13. Z. Bor, B. Racz, “Group Velocity Dispersion in Prisms and its Application to Pulse Compression and Travelling Wave Excitation,” Opt. Commun. 54, 165–170 (1985); O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. 59, 229–232 (1986).
    [CrossRef]
  14. J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  15. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  16. S. J. Brosnan, R. L. Byer, “Optical Parametric Oscillator Threshold and Linewidth Studies,” IEEE J Quantum Electron. QE-15, 415–431 (1979).
    [CrossRef]
  17. J. D. Kafka, T. Baer, “Prism-Pair Dispersive Delay Lines in Optical Pulse Compression,” Opt. Lett. 12, 401–403 (1987).
    [CrossRef] [PubMed]
  18. W. H. Knox, M. C. Downer, R. L. Fork, C. V. Shank, “Amplified Femtosecond Optical Pulses and Continuum Generation at 5-kHz Repetition Rate,” Opt. Lett. 9, 552–554 (1984).
    [CrossRef] [PubMed]

1990

G. Szabo, Z. Bor, “Broadband Frequency Doubler for Femtosecond Pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

1989

Z. Bor, “Distortion of Femtosecond Laser Pulses in Lenses,” Opt. Lett. 14, 119–121 (1989).
[CrossRef] [PubMed]

W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Ultraviolet Optical Parametric Oscillation in β-BaB2O4,” Appl. Phys. Lett. 54, 13–15 (1989).
[CrossRef]

O. E. Martinez, “Achromatic Phase Matching for Second Harmonic Generation of Femtosecond Pulses,” IEEE J. Quantum Electron. QE-25, 2464–2468 (1989).
[CrossRef]

1988

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

1987

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

J. D. Kafka, T. Baer, “Prism-Pair Dispersive Delay Lines in Optical Pulse Compression,” Opt. Lett. 12, 401–403 (1987).
[CrossRef] [PubMed]

1985

Z. Bor, B. Racz, “Group Velocity Dispersion in Prisms and its Application to Pulse Compression and Travelling Wave Excitation,” Opt. Commun. 54, 165–170 (1985); O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

1984

1982

I. V. Tomov, R. Fedosejevs, A. A. Offenberger, “Up-conversion of Subpicosecond Light Pulses,” IEEE J. Quantum Electron. QE-18, 2048–2055 (1982).
[CrossRef]

1979

S. J. Brosnan, R. L. Byer, “Optical Parametric Oscillator Threshold and Linewidth Studies,” IEEE J Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

1969

W. H. Glenn, “Second Harmonic Generation by Picosecond Optical Pulses,” IEEE J. Quantum Electron. QE-5, 284–290 (1969).
[CrossRef]

1968

J. Comly, E. Garmire, “Second Harmonic Generation from Short Pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

1966

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second Harmonic Generation of Light by Focused Laser Beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Akmanov, S. A.

S. A. Akmanov, A. I. Kovrygin, A. P. Sukhorukov, “Optical Harmonic Generation and Optical Frequency Multipliers,” in Quantum Electronics: A TreatiseH. Rabin, C. L. Tang, Eds. (Academic, New York, 1975), Vol. 1, Pt. B.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Ashkin, A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second Harmonic Generation of Light by Focused Laser Beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Baer, T.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bor, Z.

G. Szabo, Z. Bor, “Broadband Frequency Doubler for Femtosecond Pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

Z. Bor, “Distortion of Femtosecond Laser Pulses in Lenses,” Opt. Lett. 14, 119–121 (1989).
[CrossRef] [PubMed]

Z. Bor, B. Racz, “Group Velocity Dispersion in Prisms and its Application to Pulse Compression and Travelling Wave Excitation,” Opt. Commun. 54, 165–170 (1985); O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

Bosenberg, W. R.

W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Ultraviolet Optical Parametric Oscillation in β-BaB2O4,” Appl. Phys. Lett. 54, 13–15 (1989).
[CrossRef]

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

Boyd, G. D.

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second Harmonic Generation of Light by Focused Laser Beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Brosnan, S. J.

S. J. Brosnan, R. L. Byer, “Optical Parametric Oscillator Threshold and Linewidth Studies,” IEEE J Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

Byer, R. L.

S. J. Brosnan, R. L. Byer, “Optical Parametric Oscillator Threshold and Linewidth Studies,” IEEE J Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

Cheng, L. K.

W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Ultraviolet Optical Parametric Oscillation in β-BaB2O4,” Appl. Phys. Lett. 54, 13–15 (1989).
[CrossRef]

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

Comly, J.

J. Comly, E. Garmire, “Second Harmonic Generation from Short Pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Downer, M. C.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Edelstein, D. C.

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Fedosejevs, R.

I. V. Tomov, R. Fedosejevs, A. A. Offenberger, “Up-conversion of Subpicosecond Light Pulses,” IEEE J. Quantum Electron. QE-18, 2048–2055 (1982).
[CrossRef]

Fork, R. L.

Garmire, E.

J. Comly, 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]

Gordon, J. P.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Kafka, J. D.

Kleinman, D. A.

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second Harmonic Generation of Light by Focused Laser Beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Knox, W. H.

Kovrygin, A. I.

S. A. Akmanov, A. I. Kovrygin, A. P. Sukhorukov, “Optical Harmonic Generation and Optical Frequency Multipliers,” in Quantum Electronics: A TreatiseH. Rabin, C. L. Tang, Eds. (Academic, New York, 1975), Vol. 1, Pt. B.

Martinez, O. E.

Offenberger, A. A.

I. V. Tomov, R. Fedosejevs, A. A. Offenberger, “Up-conversion of Subpicosecond Light Pulses,” IEEE J. Quantum Electron. QE-18, 2048–2055 (1982).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Racz, B.

Z. Bor, B. Racz, “Group Velocity Dispersion in Prisms and its Application to Pulse Compression and Travelling Wave Excitation,” Opt. Commun. 54, 165–170 (1985); O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

Shank, C. V.

Sukhorukov, A. P.

S. A. Akmanov, A. I. Kovrygin, A. P. Sukhorukov, “Optical Harmonic Generation and Optical Frequency Multipliers,” in Quantum Electronics: A TreatiseH. Rabin, C. L. Tang, Eds. (Academic, New York, 1975), Vol. 1, Pt. B.

Szabo, G.

G. Szabo, Z. Bor, “Broadband Frequency Doubler for Femtosecond Pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

Tang, C. L.

W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Ultraviolet Optical Parametric Oscillation in β-BaB2O4,” Appl. Phys. Lett. 54, 13–15 (1989).
[CrossRef]

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

Tomov, I. V.

I. V. Tomov, R. Fedosejevs, A. A. Offenberger, “Up-conversion of Subpicosecond Light Pulses,” IEEE J. Quantum Electron. QE-18, 2048–2055 (1982).
[CrossRef]

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Wachman, E. S.

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

Yariv, A.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

Appl. Phys. B

G. Szabo, Z. Bor, “Broadband Frequency Doubler for Femtosecond Pulses,” Appl. Phys. B 50, 51–54 (1990).
[CrossRef]

Appl. Phys. Lett.

J. Comly, E. Garmire, “Second Harmonic Generation from Short Pulses,” Appl. Phys. Lett. 12, 7–9 (1968).
[CrossRef]

D. C. Edelstein, E. S. Wachman, L. K. Cheng, W. R. Bosenberg, C. L. Tang, “Femtosecond Ultraviolet Pulse Generation in β-BaB2O4,” Appl. Phys. Lett. 52, 2211–2213 (1988).
[CrossRef]

W. R. Bosenberg, L. K. Cheng, C. L. Tang, “Ultraviolet Optical Parametric Oscillation in β-BaB2O4,” Appl. Phys. Lett. 54, 13–15 (1989).
[CrossRef]

IEEE J Quantum Electron.

S. J. Brosnan, R. L. Byer, “Optical Parametric Oscillator Threshold and Linewidth Studies,” IEEE J Quantum Electron. QE-15, 415–431 (1979).
[CrossRef]

IEEE J. Quantum Electron.

O. E. Martinez, “Achromatic Phase Matching for Second Harmonic Generation of Femtosecond Pulses,” IEEE J. Quantum Electron. QE-25, 2464–2468 (1989).
[CrossRef]

W. H. Glenn, “Second Harmonic Generation by Picosecond Optical Pulses,” IEEE J. Quantum Electron. QE-5, 284–290 (1969).
[CrossRef]

I. V. Tomov, R. Fedosejevs, A. A. Offenberger, “Up-conversion of Subpicosecond Light Pulses,” IEEE J. Quantum Electron. QE-18, 2048–2055 (1982).
[CrossRef]

J. Appl. Phys.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, A. Zalkin, “Optical, Mechanical, and Thermal Properties of Barium Borate,” J. Appl. Phys. 62, 1968–1983 (1987).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

Z. Bor, B. Racz, “Group Velocity Dispersion in Prisms and its Application to Pulse Compression and Travelling Wave Excitation,” Opt. Commun. 54, 165–170 (1985); O. E. Martinez, “Pulse Distortions in Tilted Pulse Schemes for Ultrashort Pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

Opt. Lett.

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, P. S. Pershan, “Interactions Between Light Waves in a Nonlinear Dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

D. A. Kleinman, A. Ashkin, G. D. Boyd, “Second Harmonic Generation of Light by Focused Laser Beams,” Phys. Rev. 145, 338–379 (1966).
[CrossRef]

Other

S. A. Akmanov, A. I. Kovrygin, A. P. Sukhorukov, “Optical Harmonic Generation and Optical Frequency Multipliers,” in Quantum Electronics: A TreatiseH. Rabin, C. L. Tang, Eds. (Academic, New York, 1975), Vol. 1, Pt. B.

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

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Figures (5)

Fig. 1
Fig. 1

Schematic of a noncollinear phase matching geometry for frequency doubling of femtosecond pulses in which group velocities of the fundamental and second harmonic are matched. The angle α′ represents the prism-induced tilt angle of the pulse front outside the doubling crystal. For ease of presentation, the slightly smaller angle α inside the crystal caused by the refraction angle at the crystal interface has not been shown.

Fig. 2
Fig. 2

Waveguide variation of the scheme shown in Fig. 1, in which the incident pulses are internally reflected down the length of the doubling crystal to increase the effective interaction length.

Fig. 3
Fig. 3

Calculated temporal broadening of second harmonic pulses as a function of BBO crystal length for four durations of 620-nm fundamental pulses in a group velocity matched condition (τg = 0). Note the severe broadening of pulses of <0.1 ps and modest broadening for pulse durations τ ≥ 0.1 ps. The calculation, based on Eq. (12), includes crystalliine group velocity dispersion but neglects prism-induced dispersion.

Fig. 4
Fig. 4

Calculated temporal broadening of second harmonic pulses in BBO as a function of small detuning from the optimum group velocity matching angle α′. Incident pulses at 620 nm with durations indicated and 4.5-mm crystal length are assumed.

Fig. 5
Fig. 5

Calculated second harmonic conversion efficiencies in BBO in a simultaneously phase and group velocity matched configuration (solid curves) and with α detuned by 1° from exact group velocity matching (dashed curves). Incident pulses at 620 nm, τ = 100 fs and a single overlap cycle length Lc = 2.3 mm are assumed; m denotes the number of overlap cycles in a waveguide geometry such as that shown in Fig. 2.

Tables (1)

Tables Icon

Table I Phase Matching θ and Group Velocity Matching Angles Inside α and Outside α′ a BBO Doubling Crystal for Various Fundamental Wavelengths λ

Equations (20)

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u 1 , 2 ( x , y , z , t ) = E 1 , 2 exp { - 4 w 0 2 [ x 2 + ( y cos α ± z sin α ) 2 ] } × exp [ - 4 τ 2 ( t - z v g cos α ) 2 ] exp [ i ω 1 c ( y sin α + z cos α - c t ) ] ,
u 1 , 2 ( x , y , z , t ) = - + d w E 1 , 2 ( ω , z ) exp { i [ ± k y y + k z ( ω ) z - ω t ] }
E 1 , 2 ( ω , z ) = τ E 4 π exp { - 4 w 0 2 [ x 2 + ( y cos α ± z sin α ) 2 ] } × exp [ - τ 2 16 ( ω - ω 1 ) 2 ] . k y = - k ( ω 1 ) sin α , and k z ( ω ) = k ( ω 1 ) cos α + k ( ω ) - k ( ω 1 ) cos α
u 3 ( x , y , z , t ) = - + d ω - + d ω E 3 ( ω + ω , z ) × exp { i ( k ( ω + ω ) z - ( ω + ω ) t ] } .
d d z [ E 3 ( ω + ω , z ) ] = i ( ω + ω ) 2 × μ 0 ɛ 3 d E 1 ( ω , z ) E 2 ( ω , z ) exp ( i Δ k z ) ,
d d z [ E 2 * ( ω , z ) ] = i ω 2 × μ 0 ɛ 2 d E 1 ( ω , z ) E 3 * ( ω + ω , z ) exp ( i Δ k z ) ,
d d z [ E 1 ( ω , z ) ] = i ω 2 × μ 0 ɛ 1 d E 3 ( ω + ω , z ) E 2 * ( ω , z ) exp ( - i Δ k z ) .
0 L r exp ( i Δ k z ) d z = L c exp ( i Δ k L c 2 ) ( sin Δ k L c 2 Δ k L c 2 ) .
u 3 ( x , y , z , t ) = i d τ 2 E 2 L c ω 1 32 π μ 0 ɛ ( 2 ω 1 ) × exp [ - 8 w 0 2 ( x 2 + y 2 cos 2 α ) ] v 3 ( z , t ) ,
v 3 ( z , t ) = - d ω - d ω ( sin Δ k L c 2 Δ k L c 2 ) exp { - τ 2 16 [ ( ω - ω 1 ) 2 + ( ω - ω 1 ) 2 ] } exp [ i ( Σ k ) L c 2 ] exp [ i ω + ω c ( z - c t ) ]
sin x x x + 1 6 x 3 + x = 1 + 1 6 x 2 + exp ( 1 6 x 2 ) ,
v 3 ( z , t ) = V exp [ - 4 τ 3 2 ( t - z c ) 2 ] exp { i [ ϕ + 2 ω 1 c ( z - c t ) ] } ,
V = 8 π ( { τ 4 16 + L c 2 [ k ( ω 1 ) ] 2 cos 2 α } { ( τ g 2 3 + τ 2 4 ) 4 + L c 2 [ k ( ω 1 ) cos α + 2 k ( 2 ω 1 ) ] 2 } ) - 1 / 4
τ 3 = 2 { τ g 2 3 + τ 2 4 + L c 2 [ k ( ω 1 ) cos α + 2 k ( 2 ω 1 ) ] 2 ( τ g 2 3 + τ 2 4 ) } 1 / 2
τ g = L c [ 1 cos α ( d k d ω ) ω - ( d k d ω ) 2 ω ] ,
k ( 2 ω 1 ) - 2 k ( ω 1 ) cos α = 0 ;
1 cos α ( d k d ω ) ω - ( d k d ω ) 2 ω = 0.
u 3 ( x , y , z , t ) = i d τ 2 E 2 L c ω 1 32 π μ 0 ɛ ( 2 ω 1 ) × exp [ - 8 w 0 2 ( x 2 + y 2 cos 2 α ) ] exp [ - 4 τ 3 2 ( z c - t ) 2 + i ϕ ( t ) ] .
= - d x - d y - d t [ 1 2 ɛ μ 0 u ( x , y , z , t ) u * ( x , y , z , t ) ] .
η 3 in = μ 0 3 ɛ 0 3 d 2 ω 1 2 τ 2 τ 3 V 2 L 2 32 2 π π 3 n 1 2 n 2 w 0 2 cos α in ,

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