Abstract

Narrowband localized wave packets that are nondispersing and nondiffracting in one transverse dimension are characterized in anomalously dispersive media by means of a Fourier approach. Depending on the group velocity, waves with a dispersion relationship characterized by real wavenumbers can be O or X waves, while we also find waves with evanescent wavenumbers.

© 2008 Optical Society of America

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References

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2008

H.E.Hernandez-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves (Wiley, 2008).
[CrossRef]

2007

2004

S. Longhi, Opt. Lett. 29, 147 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, Opt. Lett. 29, 1446 (2004).
[CrossRef] [PubMed]

M. Porras and P. Di Trapani, Phys. Rev. E 69, 066606 (2004).
[CrossRef]

A. Ciattoni and P. Di Porto, Phys. Rev. E 69, 056611 (2004).
[CrossRef]

P. Saari and K. Reivelt, Phys. Rev. E 69, 036612 (2004).
[CrossRef]

2003

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

M. Porras, G. Valiulis, and P. Di Trapani, Phys. Rev. E 68, 016613 (2003).
[CrossRef]

M. A. Porras, C. Conti, S. Trillo, and P. Di Trapani, Opt. Lett. 28, 1092 (2003).
[CrossRef]

2002

2001

1997

1978

1965

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965).

Averchi, A.

Bragheri, F.

Christodoulides, D. N.

Ciattoni, A.

A. Ciattoni and P. Di Porto, Phys. Rev. E 69, 056611 (2004).
[CrossRef]

Conti, C.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

M. A. Porras, C. Conti, S. Trillo, and P. Di Trapani, Opt. Lett. 28, 1092 (2003).
[CrossRef]

Di Porto, P.

A. Ciattoni and P. Di Porto, Phys. Rev. E 69, 056611 (2004).
[CrossRef]

Di Trapani, P.

M. Porras, A. Dubietis, A. Matijosius, R. Piskarskas, F. Bragheri, A. Averchi, and P. Di Trapani, J. Opt. Soc. Am. B 24, 581 (2007).
[CrossRef]

M. Porras and P. Di Trapani, Phys. Rev. E 69, 066606 (2004).
[CrossRef]

D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, Opt. Lett. 29, 1446 (2004).
[CrossRef] [PubMed]

M. A. Porras, C. Conti, S. Trillo, and P. Di Trapani, Opt. Lett. 28, 1092 (2003).
[CrossRef]

M. Porras, G. Valiulis, and P. Di Trapani, Phys. Rev. E 68, 016613 (2003).
[CrossRef]

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Dubietis, A.

Efremidis, N. K.

Fibich, G.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965).

Jedrkiewicz, O.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Longhi, S.

Malomed, B. A.

Matijosius, A.

Orlov, S.

Papanicolau, G. C.

Piskarskas, A.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

S. Orlov, A. Piskarskas, and A. Stabinis, Opt. Lett. 27, 2167 (2002).
[CrossRef]

Piskarskas, R.

Porras, M.

M. Porras, A. Dubietis, A. Matijosius, R. Piskarskas, F. Bragheri, A. Averchi, and P. Di Trapani, J. Opt. Soc. Am. B 24, 581 (2007).
[CrossRef]

M. Porras and P. Di Trapani, Phys. Rev. E 69, 066606 (2004).
[CrossRef]

M. Porras, G. Valiulis, and P. Di Trapani, Phys. Rev. E 68, 016613 (2003).
[CrossRef]

Porras, M. A.

M. A. Porras, C. Conti, S. Trillo, and P. Di Trapani, Opt. Lett. 28, 1092 (2003).
[CrossRef]

M. A. Porras, Opt. Lett. 26, 1364 (2001).
[CrossRef]

Reivelt, K.

P. Saari and K. Reivelt, Phys. Rev. E 69, 036612 (2004).
[CrossRef]

Rtsep, M.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965).

Saari, P.

Sönajalg, H.

Stabinis, A.

Trillo, S.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

M. A. Porras, C. Conti, S. Trillo, and P. Di Trapani, Opt. Lett. 28, 1092 (2003).
[CrossRef]

Trull, J.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Valiulis, G.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

M. Porras, G. Valiulis, and P. Di Trapani, Phys. Rev. E 68, 016613 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. E

M. Porras and P. Di Trapani, Phys. Rev. E 69, 066606 (2004).
[CrossRef]

A. Ciattoni and P. Di Porto, Phys. Rev. E 69, 056611 (2004).
[CrossRef]

M. Porras, G. Valiulis, and P. Di Trapani, Phys. Rev. E 68, 016613 (2003).
[CrossRef]

P. Saari and K. Reivelt, Phys. Rev. E 69, 036612 (2004).
[CrossRef]

Phys. Rev. Lett.

C. Conti, S. Trillo, P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, and J. Trull, Phys. Rev. Lett. 90, 170406 (2003).
[CrossRef] [PubMed]

Other

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965).

H.E.Hernandez-Figueroa, M.Zamboni-Rached, and E.Recami, eds., Localized Waves (Wiley, 2008).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Typical dispersion curves (insets, k x normalized to have unit maximum) in the region of the parameter plane θ 0 , V, where LWs exist. The velocity V m (crossover from O to X waves) and V s = V g cos θ 0 (at which Ω s = 0 ) are calculated for fused silica at ω 0 = 0.5 fs 1 . Point A refers to the LWs shown in Fig. 4.

Fig. 2
Fig. 2

(a)–(c) Normalized LW envelope Ψ Ψ max of paraxial O waves versus normalized coordinates X x δ Ω α 2 , (a) T τ δ Ω , (b) and (c) T τ σ calculated for fused silica at ω 0 = 0.5 fs 1 , θ 0 = 3 × 10 4 rad , V = V s = 1.967 × 10 8 ms 1 , and δ Ω = 3.742 × 10 11 s 1 : (a) explicit solution [Eq. (9)], (b) and (c) numerical integration [Eq. (8)] for the Gaussian spectra exp ( Ω 2 σ 2 ) shown (dashed curves) in (d) superimposed to dispersion curves k x ( Ω ) [Eq. (5), solid curve; from full Sellmeier equations, crosses], (e) relative bandwidth δ Ω ω 0 versus angle θ 0 for V = V s .

Fig. 3
Fig. 3

(a) X wave profile from Eq. (8) versus X x σ α 1 , T τ σ , with V = 2.5 × 10 8 ms 1 ; (b) associated Gaussian spectrum f ( Ω ) (dotted curve) and right branch of hyperbolic dispersion relationship (solid curve) for silica at ω 0 = 0.5 fs 1 , θ 0 = 0 .

Fig. 4
Fig. 4

LWs in Eq. (11) versus normalized coordinates X = x k 0 k 0 τ 0 and T = τ τ 0 for different orders: (a) m = 0 , (b) m = 1 , and (c) m = 2 . (d) Spectra (dotted curves) corresponding to cases (a)–(c) superimposed to the dispersion relationship β ( Ω ) = k 0 k 0 Ω (solid curve). Here ω 0 = 0.5 fs 1 and V = V g = 1.967 × 10 8 ms 1 .

Equations (11)

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E ( x , z , t ) = Ψ ( x , τ ) e i b z e i ω 0 t ,
E ̂ ( x , z , ω ) = e i b z + Ψ ( x , t z V ) e i ( ω ω 0 ) t d t
x 2 Ψ ̂ ( x , Ω ) + k x 2 ( ω ) Ψ ̂ ( x , Ω ) = 0 ,
k x = ( n = 0 1 n ! ω n k ω 0 Ω n ) 2 ( b + Ω V ) 2 .
k x ( Ω ) = α 2 Ω 2 + 2 α 1 Ω + α 0 ,
cos θ ( Ω ) = b + Ω V n = 0 1 n ! ω n k ω 0 Ω n ,
Ψ ̂ ( x , Ω ) = f ( Ω ) { cos ( k x ( Ω ) x ) sin ( k x ( Ω ) x ) ,
Ψ ( x , τ ) = 1 2 π k x ( Ω ) Re f ( Ω ) cos ( k x ( Ω ) x ) e i Ω τ d Ω ,
Ψ ( x , τ ) = 1 2 α 2 J 0 ( δ Ω α 2 x 2 + τ 2 ) e i Ω s τ
Ψ n ( x , τ ) = 1 2 π k x ( Ω ) Im f ( Ω ) e β ( Ω ) x e i Ω τ d Ω ,
Ψ ( x , τ ) = m ! 2 π 1 ( τ 0 + i τ + x k 0 k 0 ) m + 1 ± m ! 2 π 1 ( τ 0 i τ + x k 0 k 0 ) m + 1 ,

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