Abstract

The second harmonic generation in a thin β-barium borate crystal is used to measure χ (2) cascading phenomena in the spectral domain. The harmonic generation is induced by two pulses produced by spectrally filtering a femtosecond pulse and centered at the wavelength λ-Δλ and λ+Δλ. New spectral components appear in spectral density of both the fundamental and harmonic pulses. High order cascading phenomena are evidenced. In good agreement with theoretical predictions, for large phase mismatch the evolution of the spectra demonstrates the competition between cascaded χ(2) and χ(3) phenomena.

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References

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  1. L. A. Ostrovskii, "Self action of light in crystals," JETP Lett. 5, 272 (1967)
  2. R. Maleck Rassoul, A. Ivanov, E. Freysz, A. Ducasse and F. Hache, "Second harmonic generation under phase velocity and group velocity mismatch: influence of cascading, self-phase and cross-phase modulation," Opt. Lett. 22, 268 (1997)
    [CrossRef]
  3. S. Cussat Blanc, R. Maleck Rassoul, A. Ivanov, E. Freysz and A. Ducasse, "Influenec of cascading phenomena on a type I second-harmonic wave generated by an intense femtosecond pulse: application to the measurement of the effective second order coefficient," Opt. Lett. 22, 268 (1998)
  4. Y. Baek, R. Schiek and G.I. Stegeman, "All-optical switching in a hybrid Mach-Zehnder interferometer as a result of cascaded second-order nonlinearity," Opt. Lett. 20, 2168 (1995)
    [CrossRef]
  5. N.R. Belashenkov, S.V. Gagarskiiand M.V. Inochskin, "Nonlinear refraction of light on second-harmonic generation," Opt. Spectrosc. 66, 806 (1989)
  6. R. Desalvo, D.J. Hagan, M. Sheik-Bahae, G. Stegeman, and E.W. Van Stryland, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28 (1992)
    [CrossRef] [PubMed]
  7. R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas and G.P. Banfi, "Self diffraction through cascaded second order frequency-mixing effect in �-barium borate," Opt. Lett. 18, 574 (1993)
    [CrossRef]
  8. H. Tan, G.P. Banfi and A. Tomeselli, "Optical frequency mixing through cascaded second-order processes in -barium borate," Appl. Phys. Lett. 63, 2472 (1993)
    [CrossRef]
  9. A. Varanavicius, A. Bubietis, A. Berzanskis, R. Danielius, A. Piskarskas, " Near-degenerate cascaded four-wave mixing in a optical parametric apmplifier," Opt. Lett. 22, 1603 (1997)
    [CrossRef]
  10. K. Schneider, S. Schiller, "Multiple conversion and optical limiting in subharmonic-pumped parametric oscillator," Opt. Lett. 22, 363 (1997)
    [CrossRef] [PubMed]
  11. J.A. Armstrong, N. Bloembergen, J. Ducuing and P.S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev 127, 1918 (1962)
    [CrossRef]
  12. L. Collatz, "The Numerical Treatment of Differential Equations, (Springer-Verlag-Berlin-Heidelberg-New-York, 1966)
  13. A. Berzanski, R. Danielius, A. Piskarskas, A. Stabinis, "Parametrically induced light diffraction in crystal with second order susceptibility," Appl. Phys. B 60, 421 (1995)
    [CrossRef]

Other

L. A. Ostrovskii, "Self action of light in crystals," JETP Lett. 5, 272 (1967)

R. Maleck Rassoul, A. Ivanov, E. Freysz, A. Ducasse and F. Hache, "Second harmonic generation under phase velocity and group velocity mismatch: influence of cascading, self-phase and cross-phase modulation," Opt. Lett. 22, 268 (1997)
[CrossRef]

S. Cussat Blanc, R. Maleck Rassoul, A. Ivanov, E. Freysz and A. Ducasse, "Influenec of cascading phenomena on a type I second-harmonic wave generated by an intense femtosecond pulse: application to the measurement of the effective second order coefficient," Opt. Lett. 22, 268 (1998)

Y. Baek, R. Schiek and G.I. Stegeman, "All-optical switching in a hybrid Mach-Zehnder interferometer as a result of cascaded second-order nonlinearity," Opt. Lett. 20, 2168 (1995)
[CrossRef]

N.R. Belashenkov, S.V. Gagarskiiand M.V. Inochskin, "Nonlinear refraction of light on second-harmonic generation," Opt. Spectrosc. 66, 806 (1989)

R. Desalvo, D.J. Hagan, M. Sheik-Bahae, G. Stegeman, and E.W. Van Stryland, "Self-focusing and self-defocusing by cascaded second-order effects in KTP," Opt. Lett. 17, 28 (1992)
[CrossRef] [PubMed]

R. Danielius, P. Di Trapani, A. Dubietis, A. Piskarskas, D. Podenas and G.P. Banfi, "Self diffraction through cascaded second order frequency-mixing effect in �-barium borate," Opt. Lett. 18, 574 (1993)
[CrossRef]

H. Tan, G.P. Banfi and A. Tomeselli, "Optical frequency mixing through cascaded second-order processes in -barium borate," Appl. Phys. Lett. 63, 2472 (1993)
[CrossRef]

A. Varanavicius, A. Bubietis, A. Berzanskis, R. Danielius, A. Piskarskas, " Near-degenerate cascaded four-wave mixing in a optical parametric apmplifier," Opt. Lett. 22, 1603 (1997)
[CrossRef]

K. Schneider, S. Schiller, "Multiple conversion and optical limiting in subharmonic-pumped parametric oscillator," Opt. Lett. 22, 363 (1997)
[CrossRef] [PubMed]

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

L. Collatz, "The Numerical Treatment of Differential Equations, (Springer-Verlag-Berlin-Heidelberg-New-York, 1966)

A. Berzanski, R. Danielius, A. Piskarskas, A. Stabinis, "Parametrically induced light diffraction in crystal with second order susceptibility," Appl. Phys. B 60, 421 (1995)
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Experimental set-up

Fig. 2.
Fig. 2.

a-b: Spectrum of the SH and FF pulses at the exit of the crystal for Δk=0. c-d: FF spectrum before and after the filtering of the laser pulses.

Fig. 3.
Fig. 3.

a: Theoretical (—) and experimental (∙) evolutions of the peak at 2ω of the SH spectrum versus the total FF pulse energy. b: Theoretical (—) and experimental evolution of the peak at ω-3δω (∘) and ω+3δω (∙) of the FF spectrum versus the total input FF pulse energy. c: Theoretical (—), numerical (----) and experimental (∙) evolution of the peak at ω0±3δω for 50 µJ of input FF pulse energy versus the phase mismatch Δk.

Equations (7)

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ρ 1 ( z , t ) = ρ 10 ( t ) / ch ( ω χ eff ( 2 ) ρ 10 ( t ) z / ( 2 nc ) )
ρ 2 ( z , t ) = ρ 10 ( t ) th ( ω χ eff ( 2 ) ρ 10 ( t ) z / ( 2 nc ) )
A 1 z = 1 2 i γ χ eff ( 2 ) A 2 A 1 * exp ( i Δ kz ) + i γ χ ( 3 ) ( A 1 2 A 1 + 2 A 2 2 A 1 )
A 2 z = 1 2 χ eff ( 2 ) A 1 2 exp ( i Δ kz ) + i γ χ ( 3 ) ( A 2 2 A 2 + 2 A 1 2 A 2 )
A 1 ( z , t ) = a 10 ( t ) cos ( δ ω t ) ( 1 + i 3 4 γ a 10 ( t ) 2 χ eff ( 3 ) z ) + i 1 4 γ a 10 ( t ) 3 χ eff ( 3 ) z cos ( ± 3 δ ω t )
χ eff ( 3 ) = χ ( 3 ) + χ casc ( 3 )
χ casc ( 3 ) = γ χ eff ( 2 ) χ eff ( 2 ) 4 Δ k [ 1 ( sin c ( Δ k z ) + i sin c ( Δ k z 2 ) sin ( Δ k z 2 ) ) ]

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