Abstract

It has previously been realized that angle-resolved spectra (or far-field spectra) reflect in detail the complex dynamics of femtosecond pulses propagating in nonlinear dispersive media [J. Opt. Soc. Am. B 22, 862 (2005)], and thus represent a potentially useful experimental diagnostic tool. In this paper we extend an effective three-wave mixing approach previously applied to nonlinear X-waves [Phys. Rev. Lett. 92, 253901 (2004)] to the analysis of far-field spectra. Theoretical justification for the approach is presented, and a practical method is proposed that makes it possible to interpret all features of far-field spectra, and to extract quantitative information about the underlying intra-pulse dynamics.

© 2005 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

Appl. Opt.

Appl. Phys.B

M. Kolesik, G. Katona, J. V. Moloney and E. M. Wright, "Theory and simulation of supercontinuum generation in transparent bulk media," Appl. Phys.B 77, 185-195 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Phys. Rev. E

J. Trull, O. Jedrkiewicz, P. Di Trapani, A. Matijošius, A. Varanavičius, G. Valiulis, R. Danielius, E. Kucinskas, A. Piskarskas, and S. Trillo, "Spatiotemporal three-dimensional mapping of nonlinear X-waves," Phys. Rev. E 69, 026607-4 (2004).
[CrossRef]

D. Faccio, A. Matijošius, A. Dubietis, R. Piskarskas, A. Varanavičius, E. Gaizauskas, A. Piskarskas, A. Couairon, and P. Di Trapani, "Near- and far-field evolution of laser pulse filaments in Kerr media," Phys. Rev. E 72, 037601-4 (2005).
[CrossRef]

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604-11 (2004).
[CrossRef]

M. A. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606-10 (2004).
[CrossRef]

Phys. Rev. Lett.

M. Kolesik, E.M. Wright, and J.V. Moloney, "Dynamic nonlinear X-waves for femtosecond pulse propagation in water," Phys. Rev. Lett. 92, 253901-4 (2004).
[CrossRef] [PubMed]

M. Kolesik, G. Katona, J. V. Moloney, and E. M. Wright, "Physical factors limiting the spectral extent and band gap dependence of supercontinuum generation," Phys. Rev. Lett. 91, 043905-4 (2003).
[CrossRef] [PubMed]

Supplementary Material (2)

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Figures (10)

Fig. 1.
Fig. 1.

Tabulated complex susceptibility of water is used in the definition of the linear spectral propagator of the UPPE simulator. The vertical lines mark the wavelengths at which simulations were performed.

Fig. 2.
Fig. 2.

On-axis intensity as a function of propagation distance and local time in the frame moving with the pulse. This picture shows a pulse splitting event that generates two split-off daughter pulses. The latter propagate with slightly different group velocities that manifest themselves in different angles with respect to the horizontal axis of the bright, high-intensity locations.

Fig. 3.
Fig. 3.

On-axis nonlinear response as a function of propagation distance and local time in the frame moving with the pulse. Bright areas represent two nonlinear response peaks generated by the two daughter pulses created in the pulse-splitting event. The angles with respect to horizontal of the bright areas correspond to the “propagation velocities” of the response peaks. The blue region represents the negative susceptibility modification caused by free electrons generated by the pulse. The right panel is an animation (1.5MB) illustrating the evolution of the two response peaks.

Fig. 4.
Fig. 4.

Logarithmic far-field spectral power density of a fs pulse after propagating for 2 cm in water.

Fig. 5.
Fig. 5.

Phase matched regions induced by the trailing (upper panel) and leading (lower panel) peaks of the nonlinear response. The central line of each region (shown in blue) marks the loci of best phase-matching and consists of two distinct components. Each response peak generates one half of the central X-shaped feature.

Fig. 6.
Fig. 6.

Animation (2.4MB) illustrating the evolution of the far-field spectrum filling-up the union of the phase-matched regions.

Fig. 7.
Fig. 7.

Maximum (left) and arrival time (right) in the local pulse frame of two nonlinear response peaks. Symbols represent maxima located in the data depicted in Fig. 3. Full lines show parabolic fits used to obtain the peak velocities and decelerations (see Eq. (9)).

Fig. 8.
Fig. 8.

Far field spectrum generated in water by a femtosecond pulse centered at 1100 nm, i.e., in the anomalous GVD region, and for a propagation distance of 1.2 cm. Elliptic structures in the long-wavelength portion of the spectrum around the central pulse frequency are characteristic of the anomalous chromatic dispersion. Note that the spectrum extends well into the normal GVD region.

Fig. 9.
Fig. 9.

On-axis nonlinear response as a function of propagation distance and local time in the frame moving with the pulse for the pulse with central frequency in the anomalous GVD region.

Fig. 10.
Fig. 10.

Phase matched region induced by the leading peak of the nonlinear response. The central line shown in blue has two components. The elliptic component passes through the point corresponding to the carrier wave of the fundamental frequency.

Equations (15)

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E ( x , y , z , t ) = u , v , ω A u , v , ω ( z ) e i k z ( ω , u , v ) z i ( ωt ux vy ) ,
k z ( ω , k x , k y ) ω 2 ε ( ω ) c 2 k x 2 k y 2 ,
z A u , v , ω ( z ) = i ω 2 2 c 2 k z ( ω , u , v ) e i k z ( ω , u , v ) × d x d y d t V e i ( ωt ux vy ) Δ χ ( x , y , z , t ) E ( x , y , z , t ) .
Δ χ ( x , y , z , t ) r = 1,2 Δ χ r ( x , y , z , t z ν r ) .
Δ χ r ( x , y , z , t z ν r ) = m , n , ω ˜ Δ χ m , n , ω ˜ ( r ) ( z ) e i ω ˜ ( t z ν r ) + i ( mx + ny ) .
A k x , k y , ω ( z ) = i ω 2 2 c 2 k z ( ω , k x , k y ) × d z r , u , v , Ω Δ χ k x u , k y v , ω Ω ( r ) ( z ) A u , v , Ω ( z ) ×
exp [ iz ( k z ( ω , k x , k y ) z + ( ω Ω ) z v r + k z ( Ω , u , v ) ) ]
k z ( ω , k x , k y ) + k z ( Ω , u , v ) + ω Ω v r 2 π l r , r = 1,2 ,
k z ( ω , k x , k y ) + k z ( ω 0 , 0 , 0 ) + ω ω 0 v r 2 π l r , r = 1,2 ,
τ ( z ) = τ 0 + ( 1 v g 1 v r ) ( z z 0 ) + α ( z z 0 ) 2 ,
1 l r 2 1 l r 2 + α 2 l r 2 ( ω ω 0 ) 2
E ( x , y , z , t ) e i ω o t + i k z ( ω o , 0,0 ) z
Δ χ ( x , y , z , t ) exp [ ( z z 0 ) 2 l 2 ( t τ ( z ) ) 2 w t 2 ]
exp [ 4 ( k z ( ω , k x , k y ) + k z ( ω 0 , 0,0 ) + ω ω 0 v r ) 2 1 l 2 + α 2 l 2 ( ω ω 0 ) 2 ]
( k z ( ω , k x , k y ) + k z ( ω 0 , 0,0 ) + ω ω 0 v r ) 2 < P 4 ( 1 l 2 + α 2 l 2 ( ω ω 0 ) 2 )

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