Chapter 5
Spatiotemporal solitons in waveguide arrays

In the previous chapter, solitons were considered that occupied the entire width of a waveguide array. In this chapter, spatial self-confinement is introduced, leading to spatiotemporal solitons, or "light bullets". Whilst a bullet is usually self-localised into a small region of a larger continuous medium [18, 29], these bullets are slightly different, in that they are self-localised into a few waveguides of a larger array. Therefore, they may be better described as discrete bullets.

In the case of continuous wave radiation, discrete spatial solitons were first proposed by Demetrios Christodoulides and Richard Joseph in 1988 [26] and experimentally realised 10 years later [27]. Spatiotemporal solitons have proved more elusive, as they are subject to difficulties including the need for sufficiently strong nonlinearity [35] and the need to balance the intrinsic length scales of dispersion and diffraction [34]. If the former condition is not met, then the bullet will require impractically high input powers, whilst if the latter condition is violated, the powers required for temporal and spatial localisation will greatly differ. Despite these problems, bullets have been tentatively observed in silica waveguides [32]. They have also been proposed in multi-core fibres, [208, 209], which may soon become a reality due to recent advances in the fabrication of such fibres [178].

There are two principal types of bullet: A central bullet is self-localised in the middle of an array, whilst an edge bullet is self-trapped against the array boundary [17, 28]. A related concept is the X-wave [210], but this consists of a wave-shape which is naturally diffractionless, rather than one in which diffraction is present but suppressed by nonlinearity.

In this chapter, silicon on insulator is proposed as a medium for realising bullets. A distinctive pattern of radiation emitted by such bullets is also predicted. The key results were published in Physical Review A [211].

5.1 Finding bullet solutions

Bullet solutions were found by removing the higher order dispersion terms from equation 4.27 to give

(5.1)

The spatial wavenumber q was fixed by assuming bullet solutions of the form E_n = F_ne^iqζ. Substituting this in yields

(5.2)

Therefore, we have reduced the system to a coupled set of real ordinary differential equations. (Technically speaking, the system will accept complex solutions, but since the equations are symmetric under the transformation {F_n}→{F_ne^iϕ}, these are likely to be trivial. Treating the equations as being complex will simply introduce another degree of freedom, and greatly complicate matters.) It can be shown analytically that soliton solutions (stable or unstable) exist when q > 2c₀ cos(π/(N + 1)). The reason for this fundamental cutoff is explained in section 5.3.

The equations were solved numerically (using a method described in appendix A.3), yielding bullet solutions. An example of a central bullet solution for q = 4, c₀ = 1 and N = 21 is given in figure 5.1. Central bullets like this, with a single principal wire in the middle are in fact a special case. It is also possible to have a central bullet with two principal wires of equal amplitude, as described in appendix D.

As was shown in section 4.4.1, the frequency dependence of coupling leads to a change in the effective dispersion relation of the waveguides. Bullets, like temporal solitons, require anomalous GVD, and so it may seem that we can extend the spectral range in which they exist by using anomalous diffraction. This will not work, however, as the switch from normal to anomalous diffraction will also switch the self-focussing to a self-defocussing [200, 212, 213]. Therefore, the effect of coupling GVD will always be to shrink the spectral range within which bullets are permitted. This was not a great problem, however, as waveguides with relatively weak coupling GVD were used.

5.1.1 Stability criteria

The stability of a bullet can be analysed by considering the total amount of energy within it. It may seem that this energy is independently a function of both q and c₀, but if we transform equation 5.2 using F'_n ≡F _n we obtain

(5.3)

where the parameter V is given by V ≡ q/c₀. This is extremely useful, as the numerical solution of equation 5.2 needs only one free parameter (namely V ), and thus from a 1-dimensional set of solutions, we can obtain all of the solutions from simple scaling. The energy in the bullet can be obtained by integrating the intensity |E|² over all time, and summing over all the wires. This gives

(5.4)

This can be written more concisely as

(5.5)

where the scaled energy function f_N is derived from the solution to 5.3 as

(5.6)

The function f_N differs between central and edge bullets, with the values being smaller for the latter. Decreasing the value of N reduces the energy slightly, as light cannot diffract beyond the edge of the array, and so a lesser nonlinearity is required to surpress it. Figure 5.2 shows the energy curves for both central and odd bullets. Notably, the curves have a minimum, corresponding to the lowest possible bullet energies.

Figure 5.2: Normalised energy of edge and central bullets fN = U/, as a function of V = q/c0 shown with N = ∞ for both types of solitons, and with N = 3 and N = 5 for edge and central solitons respectively. The unstable (∂U/∂q < 0) solitons are denoted by dashed lines. The energy unit P0T0 is typically in the vicinity of 100 to 1000 femtojoules.

It can be shown that the quantity ∂U/∂q must be positive for soliton stability [214], a condition known as the Vakhitov-Kolokolov criterion [215]. The physical reason for this can be seen by considering what happens when particles (in this case, photons) are added to a soliton. This addition will normally cause the soliton to decrease in duration, as a result of the nonlinear self focussing. If, however, the soliton increases in duration, the nonlinearity is not acting to confine the extra particles, and the soliton will be unstable. The energy U increases with the number of particles, whilst the wavenumber q increases with a reduction of duration. Therefore positive values of ∂U/∂q correspond to the stable case of extra particles giving shorter durations. This stability criterion can be rewritten as V ≥ V _vk, where V _vk is defined to be the position of the energy minimum.

As the number of wires will inevitably be finite, a potential problem arises in that the confinement may result from boundary conditions, rather than being intrinsic to the equations of motion. Furthermore, when Čerenkov radiation is considered in section 5.3, systems with a small number of wires are found to be of particular interest, making this problem more pressing.

Figure 5.3 shows how edge effects affect the values of V _vk and the corresponding minimum energy. In order to meaningfully compare central and edge solitons, we define ΔN as the number of wires separating the maximum intensity wire from the edge. This is given by N – 1 for an edge soliton, and /2 for a central soliton (in a system with odd N). The edge effects rapidly diminish as ΔN is increased, with the quasi-infinite regime being effectively reached at ΔN = 5 or 6. In this regime V _vk has values of 3.097 and 2.841 for central and edge bullets respectively. The corresponding scaled energies f_∞ are given by 5.771 and 5.340 respectively.

Even before this limit is reached, it can be seen that the corrections are relatively minor, indicating that the structures are still essentially bullets. Numerical modelling in section 5.2.4 supports this view: Bullets are simulated for an N = 5 system, but no pathalogical behaviour is seen.

In the case of ΔN = 1 however (corresponding to N = 2 for the edge soliton and N = 3 for the central soliton) it is difficult to determine the locations of the V = V _vk points. This is because the computations have to be performed near to the fundamental cutoff points at V = 2cos(π/(N + 1)) (see equation 5.25 in section 5.3) leading to numerical instability. Therefore the ΔN = 1 cases in figure 5.3 are shown with crosses.

Figure 5.3: Effect of the array edges on the (scaled) minimum soliton energy fN and the (scaled) critical wavenumber V vk at which it occurs. The horizontal axes show ΔN, the number of wires separating the maximum intensity wires from the boundary. The quasi-infinite regime is effectively reached at ΔN = 5 or 6. The energy unit P0T0 is typically in the vicinity of 100 to 1000 femtojoules.

5.1.2 Maximum duration

The lower bound to q/c₀ also imposes an upper bound to the bullet duration. (We define the duration of the bullet to be the FHWM of its central peak. Although this construction is fairly arbitrary, it gives a reasonable metric with which to gauge the overall time scales of the bullet.) By taking the solutions to 5.3 and measuring their duration, we gain the quantity τ_bull', which (like F') has been scaled by a factor of . Transforming back gives

(5.7)

where a ≡ τ_bull'/.

The parameter a (in the quasi-infinite case) was calculated as a function of V , as is shown in figure 5.4. If we require a stable bullet (with respect to Vakhitov Kolokolov instability), we can read off an upper bound for the bullet width (which occurs when V = V _vk), yielding a value of a_max ≈ 0.839 for a central bullet and a ≈ 0.816 for an edge bullet. Therefore, we can see that the duration of a stable bullet is limited by the expression

(5.8)

In this expression, equality corresponds to the case of minimum bullet energy (for the given value of c₀).

As this criterion impinges on the physical limits of the system (in particular, the need to generate ultrashort pulses of light), it is important to consider it in terms of real units. Abandoning our scaled units gives

(5.9)

where β₂ is the unscaled GVD, and L_C is the coupling length. The numerically derived constant a_max has been rescaled as α_max ≡ a_max/. In the quasi-infinite case, this has values of α_max ≈ 0.473 for central bullets and α_max ≈ 0.460 for edge bullets. Having a finite number of wires imposes slight upwards corrections to α_max, as is shown in figure 5.5.

5.1.3 Bullet energy versus soliton energy

It is instructive to consider the energy of a bullet in relation to that of a soliton travelling through an equivalent waveguide. The energy of the familiar E = e^iqzsech soliton solution is given by

(5.10)

Considering this in terms of the FWHM duration τ_sol gives.

(5.11)

If we combine the relations 5.5 and 5.8 for the case of minimum bullet energy, we can eliminate c₀ to give

(5.12)

where the constant u_τ ≡ a_maxf_N has (in the quasi-infinite case) a value of approximately 4.84 for a central bullet and 4.36 for an edge bullet. This relationship is identical to that for the soliton (equation 5.11), except that the energy within a bullet or edge bullet is larger than a soliton with the same duration. In the following section, this result will become important, as if a bullet is created from a pulse being injected into a single channel, that pulse will have a power above the soliton threshold, and so will be compressed.

5.2 Modelling of bullets in a realisable SOI device

5.2.1 Device specifications

Waveguides of the type used in chapter 3 were chosen. The first of these was the 220nm × 420nm waveguide, with ZDWs placed well away from the 1.5μm pump. As for the single wires, this was chosen to give straightforward soliton evolution, without the complication of other physical effects. The second type was the 220nm × 380nm waveguide, having a ZDW relatively close to the pump at 1.627μm, thus allowing for observation of Čerenkov radiation.

The upper limit to pulse duration (equation 5.9) greatly complicates the process of seeing bullets experimentally. This is for two reasons: Firstly a short pulse corresponds to a required high peak intensity, which makes the bullet vulnerable to nonlinear absorption. Secondly, the generation of ultrashort pulses is extremely difficult, and so there is a practical lower limit to the duration that can be used. A 100fs second system (like that used in section 3.3) was considered.

From equation 5.9 it is apparent that both the GVD and coupling length both need to be as high as possible. The selected 380nm and 420nm wires are close to the point of maximum dispersion, and have GVDs of 5400 fs² mm^–1 and 4700 fs² mm^–1 respectively.

The coupling length is far more problematic, as we must find a compromise between the need to increase the pulse duration, and the need to have a respectable degree of coupling along the chip. (Nonlinear wire confinement cannot be observed if the wires are too far apart for any significant coupling to happen in the first place.) As a compromise, wire separations of 700nm were chosen, providing coupling lengths of 5.6mm and 6.4mm for the 380nm and 420nm wires respectively.

The chosen parameters impose an upper limit to the bullet duration of about 80fs. This appears to be problematic, as it is shorter than the 100fs duration of the input pulse. However, from equations 5.11 and 5.12 it can be seen that a pulse with sufficient energy to form a bullet will be substantially above the soliton threshold for a single wire. Such a pulse will be compressed, as was documented for a wire with the same geometry in section 3.1.2. (The coupling is irrelevant here, as the evolution happens of a distance scale which is much less than the coupling length.) It was seen that a 100fs pulse (with power 3.5P₀) is compressed to 34fs after only 0.7mm propagation, which is well below the upper duration limit.

5.2.2 Bullet propagation in an ideal system

Before performing realistic modelling, it is instructive to consider the undamped system being pumped with a precalculated bullet solution. The bullet solutions gained in section 5.1 were inexact, as they ignored the effect of coupling dispersion, and higher order dispersion. Despite this, they propagated robustly through the simulation for both the central bullet (figure 5.9) and edge bullet (figure 5.10) cases, indicating that the approximation was valid. Conversely, when the nonlinearity was removed, the solutions dispersed in both time and space.

5.2.3 Bullet formation in a realistic system

We will now consider bullet evolution in a more realistic manner, and so we will include both linear absorption and 2PA. We will also excite the system with a sech-like pulse into a single wire, rather than a pre-existing bullet solution, thus emulating a more realistic experimental set-up.

By scanning across multiple values, the optimal input power for bullet formation was found to be 3.5 times the soliton threshold for a single wire. Below this power, the wire confinement is insufficient. Above this power, the pulses split temporally, due to them being a higher order solitons which separate under perturbation. By using 100fs pulses of this power, both bullets, and edge bullets could be created.

The data displayed in figures 5.11 and 5.12 is highly encouraging. Other tests were applied to analyse the data in more detail. One useful metric is the fraction of energy that remains in the central wire (or the edge wire, in the case of an edge bullet). This is displayed (as a function of distance) in figure 5.13. In both cases, there is an initial decline, as the pulse settles into a bullet. Next, there is a roughly flat region, coresponding to stable bullet propagation. The start of this flat region is marked by a sudden departure from the linear profile, indicating that nonlinear processes have, at that point, assumed a major role in the pulse evolution. Eventually, the parameter starts to decline, indicating that the bullet is breaking apart.

It follows from the scaling of equation 5.3 that the quantity S ≡ t_bull (where P is the peak power) will be conserved for an ideal bullet. (This parameter is similar to the soliton area defined in section 3.3.3 and used to detect the presence of temporal solitons.) The parameter can be used to analyse soliton formation, as is shown in figure 5.13. It should be noted that an increase of S corresponds to pulse broadening, whilst a decrease corresponds to compression or attenuation. Both figures show an initial regime of pulse compression (when conversely, evolution in the linear regime shows broadening). This coincides with the region where the pulse is diffracting. ∂S/∂ζ then reaches zero (as is expected from a bullet) at the same distance as the wire confinement starts to level off and becomes quasi-constant. This is notable, as two important indicators of bullet formation have happened simultaneously. Finally, S starts to increase, which corresponds to pulse broadening, and eventual decay.

5.2.4 Bullet formation in a small system

In order to study the edge effects, numerical simulations were performed for a system with a restricted number of wires, as is shown in figure 5.14 for a central bullet, and figure 5.15 for an edge bullet. Despite the close proximity to the waveguide boundaries, these bullets still show strong localisation, which is in sharp contrast to the results seen in the linear regime.

An analysis of propagation over distance was performed, as is shown in figure 5.16. Both the evolution of the S parameter, and the fraction of light in the pump wire show very similar profiles to those for the larger system, again demonstrating that the structures can be considered bullets. This result is important from the point of view of the Čerenkov radiation considered in section 5.3.3, as the instance of a small number of wires is of particular interest.

5.3 Bullet radiation

The emission of Čerenkov radiation from temporal solitons has been widely studied, as was discussed in section 3.1.3. In this section, we extend the topic to include discrete spatiotemporal solitons. The most notable result is that for an N wire system, N separate resonant frequencies are emitted. Some of these, however, may be "forbidden" for symmetry reasons.

We start by taking the E_n = F_ne^iqζ solution (which is exact in the absence of higher order dispersion and coupling dispersion) and perturbing it as

(5.13)

where ε_n is the perturbation for the nth wire. Substituting this into equation 4.27 (thus reintroducing the higher order dispersion and coupling dispersion), gives

This equation is a generalisation of equation 3.4 in section 3.1.3. As before, the left hand side admits sinusoidal solutions, and so can be though of as an oscillator, but we now have a set of N oscillators rather than just one. The oscillators (which, as before, oscillate in space rather than time) have amplitude ε_n, and are driven by forces specified by F_n. As before, we need to find resonances with the spatial wavenumber q, and so (due to the fact that ε is already modulated by q) we look for solutions where ε shows no oscillation with respect to ζ. As for the single wire system, we remove the driving terms (i.e. those containing F_n but not ε_n), and the refractive index changing F_n² term. This gives

(5.15)

Again, we look for linear wave solutions with no ζ dependence, and so we take

(5.16)

Substituting this in gives

(5.17)

This can be reframed as an eigenvalue problem

(5.18)

where the column vector holds the values of ε'_n, the matrix is defined as

(5.19)

(where δ is the Kronecker symbol) and the eigenvalues have been written as

(5.20)

where λ_j is the jth eigenvalue of , and ω_j is the corresponding resonant frequency. The matrix is the same as that defined by equation 4.75 in section 4.4.2. This is not surprising, as we are once again considering the linear supermodes of a multiwire system. As before, the eigenvalues are

(5.21)

with corresponding normalised eigenvectors of the form

(5.22)

The eigenvectors for a selection of values of N are shown in figure 5.17.

As was shown in section 4.4.1, each linear mode will see a different dispersion relation. Each of these dispersion relations will (in general) yield a different frequency of resonant radiation. Therefore, for an N wire array, we expect there to be N resonant frequencies. (However, a particular resonance will only be excited if the bullet has a non-zero projection on it, and as will shortly be shown, this is not always the case.)

Combining equations 5.20 and 5.21 gives

(5.23)

Therefore erenkov radiation should be observed at the values of ω which satisfy equation 5.23 for j = 1,2…N.

Bullet cutoff

This resonance analysis also explains the fundamental cutoff mentioned in section 5.1.1. At the pump frequency, the right hand side of equation 5.23 reduces to

(5.24)

Therefore, resonant solutions can only exist within a limited range, due to the finite range of the cosine function. However, if a bullet is to exist, the equation must not have a solution, because if it did, it would describe resonant interactions (between the bullet and the linear modes of the system) at the pump frequency, which would destroy the bullet immediately. Therefore, for bullet existence we require that

(5.25)

where (as before) V ≡ q/c₀. The lower bound on V increases with N, and tends towards 2 as N goes to infinity.

5.3.1 Symmetry considerations and "Forbidden" resonances

The waveguide array is symmetric, in that reversing the order of its wires has no effect on its dynamics. We can investigate this symmetry using the exchange matrix Ĵ, which is defined as

(5.26)

This has the effect of has the effect of reversing the order of the elements in a vector it is multiplied by, and so when applied to corresponds to reversal of the wire ordering. The matrix Ĵ has two distinct families of eigenvalues: The set of all symmetric vectors is an eigenspace of Ĵ, which corresponds to an eigenvalue of 1. For an N × N matrix, this is ⌈N/2⌉-fold degenerate (where the ⌈x⌉ brackets denote the rounding up of x to the nearest integer). Similarly, the set of all antisymmetric vectors is an eigenspace corresponding to an eigenvalue of –1. This is ⌊N/2⌋-fold degenerate (where the ⌊x⌋ brackets denote the rounding down of x to the nearest integer).

The matrix commutes with Ĵ, and thus its eigenvectors must also be eigenvectors of Ĵ. (On the other hand, it doesn’t follow what the eigenvectors of Ĵ are eigenvectors of due to the degeneracy of the former’s eigenvalues.) Therefore, for an N wire system, ⌈N/2⌉ of the modes will be symmetric, and the remaining ⌊N/2⌋ will be antisymmetric.

This is highly significant, as a symmetric bullet will not radiate into the antisymmetric modes. We can see this by taking equation 5.14 and replacing the wire index n with a reverse-ordered wire index ñ ≡ N + 1 – n. For a symmetric bullet we have F_n = F_ñ and for an antisymmetric mode we have ε_n = –ε_ñ. This gives

Transforming equation 5.27 out of the ñ notation and adding it to equation (5.14) gives

(5.28)

This lacks any form of driving term, and so for an initial condition of ε_n = 0, the solution will remain at ε_n = 0. Therefore, a symmetrical soliton will not radiate into the antisymmetric modes. As ⌊N/2⌋ antisymmetric modes are present, we can therefore predict that ⌊N/2⌋ spectral peaks will be "forbidden". For an edge soliton, however, the above argument will no longer hold (due to the F_n = F_ñ predicate being no longer true). Therefore, all of the modes are permitted for edge bullets.

5.3.2 Idealised Čerenkov generation

Before considering Čerenkov generation in the most realistic case, it is instructive to consider the simplified case of ideal bullet solutions propagating in an undamped medium. Equation 5.23 is solved graphically in figure 5.18 for a variety of cases, thus showing that N resonant frequencies are indeed present for an N-wire system. In figures 5.19 and 5.20, the results of numerical simulations for an N = 5 system are shown for central and edge bullets respectively. Instead of being plotted in a wire-by-wire basis, these figures are plotted in a supermodal basis. (In other words, they are expanded in the eigenvectors of .) This shows the absence of the antisymmetric modal components for the central bullets, thus explaining why some of the resonances are forbidden. As expected, figure 5.19 shows 3 Čerenkov peaks, whilst figure 5.20 shows 5.

Figure 5.19: Čerenkov generation in a system of five 220nm × 380nm wires, after 5.8mm of propagation (at ζ = 10) for an ordinary bullet. Expanded in terms of the eigenvectors of , which are labelled by eigenvalue. Two of the modes are precisely zero, due to them being antisymmetric and thus not part of the symmetric bullet solution. Therefore, only three Čerenkov peaks are generated.

Figure 5.20: Čerenkov generation in a system of five 220nm × 380nm wires, after 5.8mm of propagation (at ζ = 10) for an edge bullet. Displayed in terms of the eigenvectors of , arranged by eigenvalue. The edge bullet contains both symmetric and antisymmetric modes, and thus all five Čerenkov peaks are generated.

5.3.3 Čerenkov spectra in a realistic system

We will now consider Čerenkov radiation in the realistic case. Of particular interest is the scenario in which the number of peaks is small, allowing a set number of distinct Čerenkov peaks (as opposed to an indistinct continuum of many overlapping peaks) to be seen. This is particularly important from the point of seeing if certain resonances are forbidden.

Both central bullets and edge bullets were modelled, as is shown in figures 5.21, 5.22 and 5.23. As predicted the edge bullets show N Čerenkov peaks, whereas the central bullets show ⌈N/2⌉Čerenkov peaks.

Figure 5.21: Power spectrum summed over all wires (after 2.4mm propagation, ζ = 4) for N = 3 array of 220nm×380nm wires with 700nm separation. Shown for centre-wire input (top) and edge-wire input (bottom). As predicted, all 3 Čerenkov peaks are present for the edge soliton, but 1 of these is suppressed for the central soliton. The input pulse is taken as sech.