Nonparaxial one-dimensional spatial solitons

Blair, Steve

doi:10.1063/1.1286265

Cited by 48 publications

(21 citation statements)

References 25 publications

(25 reference statements)

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“…By deploying this perturbative technique [24,25], the 2 2    operator has been replaced by an O() combination of higher-order (linear and nonlinear) derivatives with respect to the time variable .…”

Section: A Governing Equation and Galilean Boostsmentioning

confidence: 99%

Wave envelopes with second-order spatiotemporal dispersion. I. Bright Kerr solitons and cnoidal waves

et al. 2012

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Section: A Governing Equation and Galilean Boostsmentioning

confidence: 99%

Wave envelopes with second-order spatiotemporal dispersion. I. Bright Kerr solitons and cnoidal waves

et al. 2012

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“…The envelopes can be numerically evaluated by integrating Eq. (8) and are shown in Fig. 2 for different values of u 0 2 .…”

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confidence: 99%

Nonparaxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media

2004

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We investigate nonlinear propagation in the presence of the optical Kerr effect by relying on a rigorous generalization of the standard parabolic equation that includes nonparaxial and vectorial terms. We show that, in the ͑1 1 1͒-D case, both soliton and propagation-invariant pattern solutions exist (while the standard hyperbolic-secant function is not a solution). © 2004 Optical Society of America OCIS codes: 190.0190, 190.3270, 190.5530. Monochromatic optical propagation in the presence of a refractive-index distribution n͑r͒ n 0 1 dn͑r͒ is usually described by the scalar parabolic equation. This is a by-product of the Helmholtz equationand the scalar parabolic equation is derived from Eq.(1) in the paraxial approximation. More precisely, after writing the electric f ield as E͑r, t͒ A͑r Ќ , z͒exp͑ikz 2 ivt͒, where v is the angular frequency, k ͑v͞c͒n 0 , and introducing d as a typical length scale of variations in n͑r͒, we assume that the parameter h ͑Dn͞n͒ ͑l͞d͒ , , 1, which expresses the smallness of the relative variation of the refractive index over a wavelength scale. Besides, if the f ield is initially (approximately) transversely polarized, we neglect its longitudinal component A z with respect to the transverse one A Ќ ͑r Ќ , z͒, and we can derive, using the slowly varying approximation, the standard scalar parabolic equation:We can then automatically satisfy the divergence equation = ? ͑eE͒ 0 (where e e 0 n 2 ) by using it to derive the (small) longitudinal component A z . If the smallness parameter h becomes comparable to 1, then the approach presented above, which represents only the lowest-order approximation in h, fails, and higher-order terms have to be added to account for nonparaxial contributions. The recent progress in nanotechnology and the possibility of fabricating optical structures with subwavelength features are compelling reasons for generalizing Eq. (2) to the nonparaxial regime. This can be done rigorously if, after splitting the f ield into a transverse and a longitudinal part, one is able to derive an equation for the transverse part alone to first order in ≠͞≠z that contains terms to all orders in h. This task was recently carried out in the general case (linear or nonlinear) of a tensorial refractive-index distribution, $ n ͑r͒, in a manner that fully preserved the vectorial nature of the problem.

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“…In any case, the comparison between paraxial and exact solitons done, for example, by inspecting the relative existence curves, shows that our solitons are a definite entity, independent of any approximation scheme; the transition between the paraxial and the highly diffractive regime is very smooth and does not exhibit any kind of dramatic catastrophic behavior, as implied by the standard paraxial theory. Finally, we note that many attempts have been made in the past few years to deal with fully nonparaxial spatial solitons, both in the unsaturated [19][20][21][22] and in the saturated, 18,23,24 nonlinear regime. In particular, TM solitons have been considered in Refs.…”

Section: Discussionmentioning

confidence: 98%

Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell's equations

et al. 2005

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We prove that spatial Kerr solitons, usually obtained in the frame of a nonlinear Schrödinger equation valid in the paraxial approximation, can be found in a generalized form as exact solutions of Maxwell's equations. In particular, they are shown to exist, both in the bright and dark version, as TM, linearly polarized, exactly integrable one-dimensional solitons and to reduce to the standard paraxial form in the limit of small intensities. In the two-dimensional case, they are shown to exist as azimuthally polarized, circularly symmetric dark solitons. Both one-and two-dimensional dark solitons exhibit a characteristic signature in that their asymptotic intensity cannot exceed a threshold value in correspondence of which their width reaches a minimum subwavelength value.

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Nonparaxial one-dimensional spatial solitons

Cited by 48 publications

References 25 publications

Wave envelopes with second-order spatiotemporal dispersion. I. Bright Kerr solitons and cnoidal waves

Wave envelopes with second-order spatiotemporal dispersion. I. Bright Kerr solitons and cnoidal waves

Nonparaxial spatial solitons and propagation-invariant pattern solutions in optical Kerr media

Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell's equations

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