Solitons in Nonlocal Media

5.3 Constant Optical Amplification
5.3 Constant Optical Amplification
In the simplest case, γ can be approximated by a constant, i.e., independent from both
the spatial coordinates (the spatial distribution of the pump) and the intensity I ∝ |A| 2
(i.e., no gain saturation). Fig. 5.2 shows a comparison between soliton propagation
in undoped NLC [fig. 5.2(a)] and dye-doped NLC [fig. 5.2(a)]: in the latter case the
power increases with s; moreover, breathing features are affected by the incremented
power. In this example, for γ = 100m −1 there are four maxima in intensity along s,
whereas for zero γ the maxima are only three. To better address this issue (i.e., the
breathing behavior versus gain), fig. 5.3 displays the results at λ = 633nm for initial
beam powers of 0.5, 1.0, 1.5 and 2.0mW, respectively, plotting the beam waist across
t versus propagation s, wt(s) = 2
It(t,s)t 2 dt
It(t,s)dt , with It(t, s) = d
0 |A|2 dx the intensity
averaged along x. The quantity wt is well suited to describe beam (soliton) breathing
during amplification (or attenuation if γ < 0) along s. From the numerical simulations I
find that wt(s) ∼ = wx(s) when w x/t are much smaller than d, wx being the waist across x
(fig. 5.3). As predicted for highly nonlocal solitons in undoped media (γ = 0) (31; 51),
the breathing period decreases as the initial power Pin increases. Therefore, for a fixed
γ = 0, the propagation distance between two waist minima reduces (increases) owing
to amplification (attenuation). A finite γ affects the mean beam waist, which reduces
(grows) with s for γ > 0 (γ < 0). I repeated the calculations at λ = 1064nm; fig. 5.4
shows the computed waist wt(s) for Pin = 0.5 and 2.0mW: self-confinement is obtained
only when the power is large enough to induce self-focusing. Moreover, since diffraction
is stronger than in the red (for the same powers), waist oscillation periods are larger
than in fig. 5.3 (see chapter 2 for details).
Figure 5.5 shows the beam profile for λ = 633nm at s = 1.5mm with Pin = 0.25mW
and γ = 0 or γ = 1000m −1 , respectively. In the second case, the gain suffices to en-
hance the nonlinear confinement and overcome diffraction: indeed, there is a region
where the waist begins to decrease. Afterwards, beam width oscillates due to the in-
terplay of nonlinear self-focusing and diffractive spreading.
After the investigation on beam breathing, I discuss the amplification and its rela-
tionship with the propagation coordinate s. To this extent, I can define the beam
power amplification at a fixed s as G(s) = P(s)/Pin, i.e. the ratio between the power
in s and in s = 0. I find that G(s) = exp(2γs) if the self-induced waveguide has
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