Embedded Solitons: Four-Frequency Radiation, Front Propagation and Radiation Inhibition

Abstract

Optical solitons which remain radiationless in spite of having wavenumbers immersed in the spectrum of linear waves are rather unusual. This article shows that moving solitons of this type are solutions of an extended NLS equation with third and fourth-order dispersion, and a quintic nonlinearity. The mechanism which prevents the emission of radiation in these solitons is presented. The radiation emitted when these solitons are perturbed is also studied. This radiation exhibits four propagating fronts, and the velocities of these fronts are calculated and explained.

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

“…As these values satisfy the condition (18), the gap of forbidden wavenumbers seen in the figure is explained. The dashed horizontal lines indicate the wavenumbers C 1 and C 2 of the soliton solutions of Equation (6) (with γ 1 = 5 and γ 2 = 1) that will be determined in Section 4.…”

“…In Figure 3 we can see the shape of the dispersion relation defined by Equation (15) (5) and (6), defined by Equation (15), with 1/30, 1/2 and 1/80, which satisfy the condition (18). The dashed lines , indicate the wavenumbers and of the solution of Equation (6) given by Equations (52)- (55) with 5 and…”

“…However, if desired, it is also possible to use the procedure shown in Refs. [18,20] to construct a rigorous proof (not just a qualitative one) which shows that the absence of resonances between the solitons and the radiation modes is the consequence of a delicate balance between the linear and the nonlinear terms of Equation (6). This procedure consists in taking the FT of Equation (6), using a function of the form (52) to calculate the FT of the nonlinear terms.…”

“…(a) −iu ttt and/or u 4t : these higher-order derivatives are necessary to describe sub-picosecond pulses [1][2][3][4][5][6][7][8][9][10][11][12]; in particular, conditions for including u 4t and discarding −iu ttt are discussed in [7,9], (b) |u| 4 u: this higher-order nonlinearity is used when we want to describe the propagation of pulses when the light intensity approaches the values which produce the "saturation" of the refractive index [13][14][15][16][17][18][19][20], (c) i(|u| 2 u) t : this term is necessary to describe the self-steepening of the optical pulses [21][22][23], (d) u(|u| 2 ) t : this term is associated with the effect of Raman scattering [24][25][26][27].…”

“…The consequences of the presence of these forbidden frequency gaps will be investigated in the following section. The dashed horizontal lines (5) and (6), defined by Equation (15), with c 0 = 1/30, c 2 = 1/2 and c 4 = 1/80, which satisfy the condition (18). The dashed lines k = C 1,2 indicate the wavenumbers C 1 and C 2 of the solution of Equation (6) given by Equations (52)- (55) with γ 1 = 5 and…”