Abstract

The influence of cross-phase modulation on third-harmonic generation is theoretically studied. Generalized phase-matching conditions for third-harmonic generation including pump-intensity-dependent phase shifts related to self- and cross-phase modulation effects are discussed. The phase mismatch between the pump and third-harmonic pulses is shown to vary from the leading edge of the pump pulse to its trailing edge, resulting in an asymmetric spectral broadening of the third harmonic.

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References

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Appl. Phys. B (2)

N. I. Koroteev and A. M. Zheltikov, "Chirp control in third-harmonic generation due to cross-phase modulation," Appl. Phys. B 67, 53-57 (1998).
[CrossRef]

A. B. Fedotov, A. M. Zheltikov, A. P. Tarasevitch, D. von der Linde, ?Enhanced spectral broadening of short laser pulses in high-numerical-aperture holey fibers,? Appl. Phys. B 73, 181-184 (2001).
[CrossRef]

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

JETP (1)

A. M. Zheltikov, N. I. Koroteev, and A. N. Naumov, ?Self-and cross-phase modulation accompanying third harmonic generation in a hollow waveguide,? JETP 88, 857-867 (1999).
[CrossRef]

JETP Lett. (2)

A. B. Fedotov, A. M. Zheltikov, L. A. Mel'nikov, A. P. Tarasevitch, D. von der Linde, ?Spectral broadening of femtosecond laser pulses in fibers with a photonic-crystal cladding,? JETP Lett. 71, 281-285 (2000).
[CrossRef]

D. A. Akimov, A. B. Fedotov, A. A. Podshivalov, A. M. Zheltikov, A. A. Ivanov, M. V. Alfimov, S. N. Bagayev, V. S. Pivtsov, T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, ?Spectral Superbroadening of Subnanojoule Cr: Forsterite Femtosecond Laser Pulses in a Tapered Fiber,? JETP Lett. 74, 460 - 463 (2001).
[CrossRef]

Laser Phys. (2)

A. B. Fedotov, V. V. Yakovlev, and A. M. Zheltikov, ?Generation of Cross-Phase-Modulated Third Harmonic of Unamplified Femtosecond Cr: Forsterite Laser Pulses in a Holey Fiber,? Laser Phys. 12, no. 2 (2002).

A. N. Naumov and A. M. Zheltikov, ?Cross-Phase Modulation in Short Light Pulses as a Probe for Gas Ionization Dynamics: The Influence of Group-DelayWalk-off Effects,? Laser Phys. 10, 923-926 (2000).

Opt. Lett. (8)

Phys. Rev. A (1)

Y. Wang and R. Dragila, ?Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,? Phys. Rev. A 41, 5645-5649 (1990)
[CrossRef] [PubMed]

Phys. Usp. (1)

A. M. Zheltikov, ?Holey fibers,? Phys. Usp. 170, 1203-1224 (2000).
[CrossRef]

Science (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, D. C. Allan, ?Single-mode photonic guidance of light in air,? Science 285, 1537-1539 (1999).
[CrossRef] [PubMed]

Other (2)

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge Univ. Press, Cambridge, 1990).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd edition (Academic, San Diego, 2001).

Supplementary Material (1)

» Media 1: MOV (174 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

The effective wave-vector mismatch between the third-harmonic field and the nonlinear polarization induced in the medium at the frequency of the third harmonic (curves 1, 2; left-hand axis) and the dimensionless frequency deviation of the pump pulse ω¯p (curves 3, 4; right-hand axis) as functions of the propagation coordinate z for ηh = 1.6. The third harmonic is generated in a fiber with n 2 = 3.2-10-16 cm2/W, γ 1 = γ 2, Δk = 2 cm-1, ζ= -0.2 cm-1 by a 30-fs pump pulse (5) with φ 0(ηp ) = αηp4, aα = 0.13, and an energy of 0.01 nJ (2, 4) and 0.3 nJ (1, 3).

Fig. 2.
Fig. 2.

(175 KB) Animation of (a) time- and (b) frequency-domain evolution of the pump and third-harmonic pulses involved in THG, SPM, and XPM processes in a fiber with n 2 = 3.2-10-16 cm2/W; C NL = Pp3; Pp and Ph are the powers of the pump and third-harmonic pulses, respectively; Sp = |∫ A(ηh + ζz, z) exp [iΩηh ] h |2 and Sh =|∫ B(ηh +, z) exp [iΩηh ] h |2 are the spectra of the pump and third-harmonic pulses, respectively. The pump pulse has an initial duration of 30 fs, an energy of 0.3 nJ, and the initial phase φ 0(ηp ) = αηp4, α= 0.13.

Equations (13)

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A ( η p , z ) = A 0 ( η p ) exp [ i φ spm ( η p , z ) ] ,
B ( η h , z ) = exp [ i φ xpm ( η h , z ) ]
× 0 z d z A 0 3 ( η h + ς z ) exp [ i Δ k z + 3 i φ spm ( η h + ζ z , z ) i φ xpm ( η h , z ) ] ,
φ spm ( η p , z ) = γ 1 A 0 ( η p ) 2 z .
φ xpm ( η h , z ) = 2 γ 2 0 z A 0 ( η h + ς z ) 2 d z
A 0 ( η p ) = A ˜ exp [ i φ 0 ( η p ) ] cosh [ η p ] .
φ xpm ( η h , z ) = 2 γ 2 A ˜ 2 ζ [ tanh ( η h + ζ z ) tanh ( η h ) ]
Δ φ ( η h , z ) = Δ k z 3 φ spm ( η h + ζ z , z ) 3 φ 0 ( η h + ζ z ) + φ xpm ( η h , z ) ,
Δ k eff ( η h , z ) = z [ Δ φ ( η h , z ) ] .
Δ k eff = Δ k + δ k xpm ( η h , z ) + δ k w ( η h , z ) ,
δ k xpm ( η h , z ) = 2 γ 2 A 0 ( η h + ς z ) 2 3 γ 1 A 0 ( η h + ς z ) 2
δ k w ( η h , z ) = 3 ς ω ¯ p ( η h + ς z , z )
ω ¯ p ( η h , z ) = η arg ( A ( η , z ) )

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