Interactions in Regime 1

The top row of simulations in Figures 68-71 all show solitons propagating at an angle of incidence of 2° without any surface wave present in order to highlight the soliton’s behaviour in the absence of more complicated interactions. The figures all show that, in spite of the changes in a variety of parameters, the soliton behaviour remains broadly the same – with beams reflected by the interface, which is to be expected since the angle of incidence is significantly lower than the critical angle.

For a quasi-paraxial interaction between relatively narrow beams, there is one key trend that is immediately apparent (see Figure 68): the soliton reflects from the interface and the energy initially localised in the surface wave is scattered into medium 2. The collision itself thus deflects the surface wave, transforming it into an obliquely-propagating soliton- like beam (this wave can no longer be classed as a surface excitation since it is no longer bound to the interface). Both spatial structures retain their distinctive identities (i.e., there is no splitting or coalescence) and the interaction generates radiation. The surface wave also remains unstable against nonparaxial collisions. As the nonlinearity exponent is increased from q = 1 to q = 2, the propagation angle of the deflected surface wave (relative to the interface) approaches that of the refracted soliton. For q = 3, these two entities coalesce into a single high-intensity narrow filament.

At this point, the role played by  in the surface wave solutions becomes more apparent. While decreasing  (to capture broader beams) leaves the normalised soliton profile essentially unchanged, there is a large effect on the surface wave (whose peak intensity can increase dramatically). One can reasonably anticipate that such a change may have a profound impact on the system evolution (see Figure 69). For q = 1 and a quasiparaxial incidence angle, the surface wave clearly dominates the system: the soliton refracts from the interface and the collision induces only small modulations in the surviving surface wave [see Figure 69d)]. This type of surface-wave modulation has previously been referred to as ‘s imming’ [1]. For q = 2, coalescence has been uncovered wherein the energy of the surface wave becomes coupled into the trajectory of the reflected soliton [see Figure 69e)]. For q = 3 [see Figure 69f)], one finds a similar type of coalescence but the radiation pattern (particularly in medium 1) is becoming more visible in the solution.

q = 1 q = 2 q = 3

Figure 68. Top row: Soliton interaction with an interface ( = 0.01 and  = 2.0) at the quasi-paraxial incidence angle inc = 2° and with  = 2.5×103. Middle row: same configuration as the top row, but with a regime 1 surface wave (with  = 1.8min) travelling along the interface. Bottom row: similar configuration to the middle row, except with a nonparaxial incidence angle of inc = 10°.

This pattern has distinct fringes that may be connected to the interference between radiation shed by the dominant post-collision filament and radiation that has been totally internally-reflected by the interface. Another interesting and rather subtle aspect of Figure 69f) is the degree of spatial asymmetry in the solution (introduced by the material interface itself). For a nonparaxial incidence angle, the skimming mode appears in the q = 1 system (as opposed to inducing surface-wave deflection). For q = 2 and q = 3, phenomena that are qualitatively similar to those in Figure 68 are uncovered.

By maintaining the broader-beams regime ( = 1.0×103) but lowering the linear refractive index step (from  = 0.01 to  = 0.005), the peak amplitude of the surface wave is decreased.

q = 1 q = 2 q = 3

Figure 69. Top row: Soliton interaction with an interface ( = 0.01 and  = 2.0) at the quasi-paraxial incidence angle inc = 2° and with  = 1.0×103. Middle row: same configuration as the top row, but with a regime 1 surface wave (with  = 1.8min) travelling along the interface. Bottom row: similar configuration to the middle row, except with a nonparaxial incidence angle of inc = 10°.

One then finds similar qualitative behaviour to that encountered for narrower beams with a larger linear-index step (compare Figures 70 and 68). Here, it is interesting that for q = 3, the reflecting soliton is annihilated by the collision and the surface wave is transformed to an off-axis soliton-like beam in medium 2.

By weakening the self-focusing properties of medium 2 (e.g., reducing  = 2.0 to  = 1.5), the quasi-paraxial collision can start to excite skimming modes. Inspection of Figure 71 reveals that the longitudinal oscillations of the skimming are more rapid for q = 2 than for

q = 1. Nonparaxial collisions for q = 1 and q = 2 trigger skimming and surface-wave deflection, respectively. However, in the case of q = 3, simulations reveal splitting rather than coalescence (i.e., after the collision, there are three distinct beams propagating in medium 2).

q = 1 q = 2 q = 3

Figure 70. Top row: Soliton interaction with an interface ( = 0.005 and  = 2.0) at the quasi-paraxial incidence angle inc = 2° and with  = 1.0×10

3

. Middle row: same configuration as the top row, but with a regime 1 surface wave (with  = 1.8min) travelling along the interface. Bottom row: similar configuration to the middle row, except with a nonparaxial incidence angle of inc = 10°.

q = 1 q = 2 q = 3

Figure 71. Top row: Soliton interaction with an interface ( = 0.005 and  = 1.5) at the quasi-paraxial incidence angle inc = 2° and with  = 1.0×103. Middle row: same configuration as the top row, but with a regime 1 surface wave (with  = 1.8min) travelling along the interface. Bottom row: similar configuration to the middle row, except with a nonparaxial incidence angle of inc = 10°.