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.

© 2001 Optical Society of America

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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. Letters 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. Letters 20, 2168 (1995)
    [CrossRef]
  5. N.R. Belashenkov, S.V. Gagarskiiand, and 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]
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    [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, and A. Piskarskas, “Near-degenerate cascaded four-wave mixing in a optical parametric apmplifier”, Opt. Lett. 22, 1603 (1997)
    [CrossRef]
  10. K. Schneider and S. Schiller, “Multiple conversion and optical limiting in subharmonic-pumped parametric oscillator”, Opt. Lett. 22, 363 (1997)
    [CrossRef] [PubMed]
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    [CrossRef]
  12. L. Collatz, “The Numerical Treatment of Differential Equations, Springer-Verlag-Berlin-Heidelberg-New-York, 1966”
  13. A. Berzanski, R. Danielius, A. Piskarskas, and A. Stabinis, “Parametrically induced light diffraction in crystal with second order susceptibility”, Appl. Phys. B 60, 421 (1995)
    [CrossRef]

1998 (1)

1997 (3)

1995 (2)

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. Letters 20, 2168 (1995)
[CrossRef]

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

1993 (2)

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 β-barium borate,” Opt. Letters 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]

1992 (1)

1989 (1)

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

1967 (1)

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

1962 (1)

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]

Armstrong, J.A.

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]

Baek, Y.

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. Letters 20, 2168 (1995)
[CrossRef]

Banfi, G.P.

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 β-barium borate,” Opt. Letters 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]

Belashenkov, N.R.

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

Berzanski, A.

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

Berzanskis, A.

Bloembergen, N.

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]

Bubietis, A.

Collatz, L.

L. Collatz, “The Numerical Treatment of Differential Equations, Springer-Verlag-Berlin-Heidelberg-New-York, 1966”

Cussat Blanc, S.

Danielius, R.

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

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

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 β-barium borate,” Opt. Letters 18, 574 (1993)
[CrossRef]

Desalvo, R.

Di Trapani, P.

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 β-barium borate,” Opt. Letters 18, 574 (1993)
[CrossRef]

Dubietis, A.

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 β-barium borate,” Opt. Letters 18, 574 (1993)
[CrossRef]

Ducasse, A.

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)

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. Letters 22, 268 (1997)
[CrossRef]

Ducuing, J.

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]

Freysz, E.

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)

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. Letters 22, 268 (1997)
[CrossRef]

Gagarskiiand, S.V.

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

Hache, F.

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. Letters 22, 268 (1997)
[CrossRef]

Hagan, D.J.

Inochskin, M.V.

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

Ivanov, A.

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)

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. Letters 22, 268 (1997)
[CrossRef]

Maleck Rassoul, R.

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)

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. Letters 22, 268 (1997)
[CrossRef]

Ostrovskii, L. A.

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

Pershan, P.S.

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]

Piskarskas, A.

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

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

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 β-barium borate,” Opt. Letters 18, 574 (1993)
[CrossRef]

Podenas, D.

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 β-barium borate,” Opt. Letters 18, 574 (1993)
[CrossRef]

Schiek, R.

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. Letters 20, 2168 (1995)
[CrossRef]

Schiller, S.

Schneider, K.

Sheik-Bahae, M.

Stabinis, A.

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

Stegeman, G.

Stegeman, G.I.

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. Letters 20, 2168 (1995)
[CrossRef]

Tan, H.

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]

Tomeselli, A.

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]

Van Stryland, E.W.

Varanavicius, A.

Appl. Phys. B (1)

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

Appl. Phys. Lett. (1)

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]

JETP Lett. (1)

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

Opt. Lett. (4)

Opt. Letters (3)

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 β-barium borate,” Opt. Letters 18, 574 (1993)
[CrossRef]

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. Letters 20, 2168 (1995)
[CrossRef]

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. Letters 22, 268 (1997)
[CrossRef]

Opt. Spectrosc. (1)

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

Phys. Rev (1)

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]

Other (1)

L. Collatz, “The Numerical Treatment of Differential Equations, Springer-Verlag-Berlin-Heidelberg-New-York, 1966”

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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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