Abstract

The weak induced-phase-modulation effects on the shape and the phase of a generated second-harmonic signal, due to the primary signal, are investigated in the regime of weak conversion and with velocity dispersion and absorption present. In the time domain a two-peaks structure appears in the shape of the signal, and in the frequency domain the spectral distribution is broadened compared with that of the second-harmonic signal generated in the absence of induced phase modulation.

© 1987 Optical Society of America

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References

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  1. J. T. Manassah, J. Opt. Soc. Am. B 4, 1235 (1987);J. T. Manassah, O. Cockings, “The effects of velocity dispersion on the time-domain distribution of a second harmonic signal,” submitted to Opt. Lett.
    [CrossRef]
  2. J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).
  3. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  4. J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
    [CrossRef]
  5. R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
    [CrossRef] [PubMed]

1987 (2)

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

J. T. Manassah, J. Opt. Soc. Am. B 4, 1235 (1987);J. T. Manassah, O. Cockings, “The effects of velocity dispersion on the time-domain distribution of a second harmonic signal,” submitted to Opt. Lett.
[CrossRef]

1986 (1)

J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
[CrossRef]

1985 (1)

J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).

Alfano, R.

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

Alfano, R. R.

J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
[CrossRef]

J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).

Bhargava, R.

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Fitzpatrick, B.

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

Ho, P.

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

Ho, P. P.

J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
[CrossRef]

J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).

Jimbo, T.

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

Manassah, J. T.

J. T. Manassah, J. Opt. Soc. Am. B 4, 1235 (1987);J. T. Manassah, O. Cockings, “The effects of velocity dispersion on the time-domain distribution of a second harmonic signal,” submitted to Opt. Lett.
[CrossRef]

J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
[CrossRef]

J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).

Mustafa, M.

J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).

Mustafa, M. A.

J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
[CrossRef]

Wang, Q. Z.

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

J. T. Manassah, M. A. Mustafa, R. R. Alfano, P. P. Ho, IEEE J. Quantum Electron. QE-22, 197 (1986).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Lett. (1)

J. T. Manassah, M. Mustafa, R. R. Alfano, P. P. Ho, Phys. Lett. 113A, 242 (1985).

Phys. Rev. A (1)

R. Alfano, Q. Z. Wang, T. Jimbo, P. Ho, R. Bhargava, B. Fitzpatrick, Phys. Rev. A 35459 (1987);R. Alfano, P. Ho, in Proceedings of the International Laser '86 Conference (Society for Optical and Quantum Electronics, Alexandria, Va., 1986), p. 84.
[CrossRef] [PubMed]

Other (1)

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

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Figures (4)

Fig. 1
Fig. 1

The magnitude of the second-harmonic envelope as a function of the normalized time (in units of primary pulse width): η′z = 6, αz = 0.9, ξ/η′ = 20.–·–·–, γ = 0;———, 2γa02/η = 0.5π.

Fig. 2
Fig. 2

(a) The position shift, away from the pump, of the pump companion peak as a function of the product of the Kerr constant by the primary pulse intensity. (b) The magnitude of this peak normalized to Γ = 0 peak: ξ/η′ = 20, η′z = 6, αz = 0.9. Γ = 2γa02/η′.

Fig. 3
Fig. 3

The normalized spectral distribution of the primary pulse as a function of the normalized frequency difference: η′z = 6, αz = 0.9, ξ/η′ = 20. –·–·–, γ = 0;———, 2γa02/η′ = 0.5π.

Fig. 4
Fig. 4

The normalized spectral distribution of the second-harmonic signal as a function of the normalized frequency difference: η′z = 6, αz = 0.9, ξ/η′ = 20. –·–·–, γ = 0;———, 2γa02/η′ = 0.5π.

Equations (21)

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A z + 1 υ 1 A t = i A * B exp [ i ( k 2 2 k 1 ) z ] + i γ × ( | A | 2 + 2 | B | 2 ) A ,
B z + 1 υ 2 B t = i A 2 exp [ i ( k 2 2 k 1 ) z ] + i γ × ( | B | 2 + 2 | A | 2 ) B α B ,
= μ 0 ω 1 c d 2 ,
γ = 3 ω 1 2 μ 0 χ ( 3 ) / 8 k 1 ,
A z + 1 υ 1 A t = i γ | A | 2 A ,
B z + 1 υ 2 B t = i A 2 exp [ i ( k 2 2 k 1 ) z ] + 2 i γ | A | 2 B α B .
A = a 0 F [ ( t z / υ 1 ) / τ ] [ exp { i γ a 0 2 F 2 [ ( t z / υ 1 ) / τ ] z } ,
B = i a 0 2 exp [ α z + i ( 2 γ a 0 2 ) 0 z F 2 ( U + η z ) d z ] × 0 z d z F 2 ( U + η z ) exp [ i 2 γ a 0 2 F 2 ( U + η z ) z i ξ z ] × exp [ α z i 2 γ a 0 2 0 z F 2 ( U + η z ) d z ] ,
U = ( t z / υ 2 ) / τ ,
η = n 2 g n 1 g c τ ,
ξ = 2 ω c ( n 2 p n 1 p ) .
A = a 0 sech [ ( t z / υ 1 ) / τ ] exp { i γ a 0 2 sech 2 [ ( t z / υ 1 ) / τ ] } ,
B = i a 0 2 exp { α z + i 2 γ a 0 2 η tanh ( U + η z ) } × 0 z d z sech 2 ( U + η z ) exp { ( α i ξ ) z + i 2 γ a 0 2 × [ sech 2 ( U + η z ) z 1 η tanh ( U + η z ) ] } .
B = i a 0 2 η exp { α z + i 2 γ a 0 2 η tanh ( U + η z ) + i ξ η U α η U } U U + η z d y sech 2 ( y ) × exp { ( α η i ξ η ) y + i 2 γ a 0 2 η × [ ( y U ) sech 2 ( y ) tanh y ] } .
| B ( U = η z ) | a 0 2 1 ( α 2 + ξ 2 ) 1 / 2 ,
| B ( U = 0 ) | a 0 2 exp ( α z ) 1 ( α 2 + ξ 2 ) 1 / 2 .
Ĩ 1 ( Δ 1 , Γ ) = | exp ( i Δ 1 U ) A ( U , Γ ) d U | 2 | A ( U , 0 ) d U | 2 ,
Ĩ 2 ( Δ 2 , Γ ) = | exp ( i Δ 2 U ) B ( U , Γ ) d U | 2 | B ( U , 0 ) exp ( i Δ 2 U ) d U | max 2 ,
Γ = 2 γ a 0 2 η ,
Δ 1 = ( ω ω 1 ) τ ,
Δ 2 = ( ω ω 2 ) τ .

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