Chapter 4
Nonlinear propagation in coupled waveguide arrays

If two waveguides are placed in close proximity, light will be transferred between them, due to the evanescent field surrounding one overlapping with the core of the other. Such a device is known as a directional coupler, and has been realised in both fibres [176] and in SOI [177]. This can be generalised to an array of waveguides, which again has been realised in both fibres [178] and SOI.

The use of silicon adds a novel feature to such devices, arising from the strong dependence of the coupling on frequency. The frequency dependence of couplers is well known [179], but in silicon it is so strong that the second derivative of the coupling with respect to frequency becomes substantial, giving it a group velocity dispersion (GVD).

The GVD of the coupling is almost always normal, but by choosing the relative phases within the wires, its effective sign can be changed. By exciting a coupler mode where the two wires are half a cycle out of phase, the effective coupling GVD can be made anomalous. This can be used to realise both solitons (section 4.3) and modulational instability (section 4.5) in coupled systems where the individual waveguides are normally dispersive.

In waveguide arrays, this effect generalises into an interplay between group velocity dispersion and diffraction. Anomalous GVD can be achieved by using anomalous diffraction, in which neighboring waveguides are out of phase. Again, this can be used to realise multi-wire solitons, as is shown in section 4.4.2.

Anomalous diffraction has been a widely studied topic in recent years [180]. A substantial fraction of this interest is due to its close association with negative index refraction [181, 182]. (Materials with negative refractive index have been hypothesised since the 1960s [183], but have only recently been realised [184], and even then in suboptimal highly-lossy form. They are of prime importance to modern optics, as they promise exotic applications including "superlenses" which can resolve images at less than the wavelength of light [185], and even invisibility devices [186].) The similarity between refraction and dispersion results from the fact that both relate the wavenumber of propagation to another wavenumber. In the case of refraction, this wavenumber is in time (i.e. the frequency), whilst in diffraction it is the transverse spatial wavenumber. Therefore, reversing the sign of diffraction is analogous to reversing the sign of the refractive index. This chapter generalises that result, by showing that reversing the sign of diffraction is also akin to reversing the sign of dispersion.

The key results of this chapter have been published in Optics Express [187].

4.1 Modelling coupled SOI waveguides

The light guided by a silicon nanowire isn’t totally confined to the silicon itself, and will in part travel in the surrounding medium. This can be seen in figure 4.1, which shows the waveguide modes to have evanescent tails which extend into the surrounding air and silica. If two waveguides are placed in close proximity, then the evanescent field of one will significantly overlap with the core of the other, allowing light to be transferred between them.

A coupled waveguide system can be thought of as being one big waveguide having two principal modes, known as supermodes. In the case of identical waveguides (which is the only case considered in this report), one of these modes will be perfectly symmetric (in that the field profiles in the two wires will be mirror images of one another), whilst the other will be perfectly antisymmetric (in that the mirror-imaging will be accompanied by a reversal of field direction). An example of such a mode pairing is given in figure 4.1. If a pulse of light is fired into just one wire, a superposition of the two supermodes will be excited. These will (in general) have different propagation constants and so beating between the two modes will occur. Each wire will alternately see constructive and destructive interference, and so the pulse will move back and forth between wires [179].

To extend the model derived in chapter 2 to a coupled-wire system, we start by making what is known as the tight-binding approximation [188]. This assumes that the supermodes are linear superpositions of the single-wire modes. (It is named after a similar approximation in solid state physics, in which the wavefunction of a crystal is assumed to be a linear superposition of the wavefunctions of the individual sites [189, 190].) This approach suffers from two main limitations: Firstly, it ignores the existence of higher order photonic bands in which the peaks of intensity no longer correspond to the waveguide cores [191]. Secondly, there is an upper limit to the coupling strength that can be considered, as when waveguides are in very close proximity, slot-guiding modes will form, in which light is trapped between the waveguides by total internal reflection [192].

To procede, we write the envelopes of the symmetric and antisymmetric supermodes (given by A_s and A_a respectively) as

For the study of solitons, the coupling coefficient is usually assumed to be a constant [179]. For silicon wires, however, it is necessary to consider the full dependence on frequency. This is represented using the Taylor expansion δ = ∑ _m^m. Rewriting equations 4.3 and 4.4 using these coefficients gives

(4.7)

where x is a positive constant. Taylor expanding this in ω gives

(4.8)

where we can identify the linear term –δ₀x with δ₁ and the quadratic term δ₀x² with δ₂. It therefore follows (assuming that we are not using an exotic waveguide configuration with negative c₀) that δ₁ is negative and δ₂ is positive.

In the absence of all damping and nonlinearity, the two supermodes will propagate independently, with Schrödinger-like equations given by

where (unlike with a single wire) the zeroth and first order terms are still present, as they differ between the two modes and thus can’t be scaled away. (This is highly significant, as it is these differences that cause the coupling.) If we substitute equations 4.1 and 4.2 into equations 4.9 and 4.10, we obtain

Adding or subtracting these gives

These are simply the wave equations for an isolated waveguide, with a coupling term of the form

(4.15)

added on, where A' is the envelope for the neighbouring waveguide.

Coupling length

If we consider the simplest possible case of coupled waveguides, with all nonlinearity, frequency dependence and damping removed (i.e. the case of monochromatic light through a lossless waveguide at low power), we obtain

(4.20)

This gives the distance taken for transfer of light from one wire into another. This (in the same manner as the dispersion length) provides a very useful metric with which to gauge the intrinsic length scales of coupling. It is also possible to calculate the coupling length in a more direct manner, by performing numerical integration of the full Maxwell equations in a coupled wire system [194].

Conversion to dimensionless units

When converted into the familiar dimensionless units, the coupling term becomes

(4.21)

where the mth coupling coefficient is given by

(4.22)

It follows from the definition of the δ_m coefficients (in equations 4.5 and 4.6) that these coefficients are given by

(4.23)

By analogy with the dispersion operator given by equation 2.71, we can define a coupling operator Ĉ as

(4.24)

This gives (for a single wire interacting with a second wire with envelope E')

(4.25)

which after reintroducing damping gives

(4.26)

Waveguide arrays

The procedure for calculating waveguide coupling is linear, and so obeys the principle of superposition. Therefore, when multiple waveguides are present, the coupling between each pair of neighbours can be considered independently, and the resulting coupling terms simply added together. This can be extended to an arbitrarily large set of waveguides, so that the undamped equation for the nth waveguide is given by

(4.27)

where the out-of-bounds values of E_n (i.e. those beyond the ends of the row of waveguides) are identically zero. Reintroducing damping gives

(4.28)

When N wires are present, it follows that there must be N separate equations to describe free carrier evolution. The equation for the nth waveguide is given by

(4.29)

where ν_n is the carrier density in that particular wire.

4.2 Device specifications

The aim of this chapter is to control dispersion with coupling. To demonstrate this point, a normally dispersive waveguide was chosen, which in isolation would not be capable of exhibiting soliton propagation or modulational instability. Therefore, transverse dimensions of 220nm × 330nm were chosen, giving a normal GVD of –1416 psnm^–1 km^–1 at a 1.5μm pump wavelength (Figure 4.2). For 100fs pulses, this gives a dispersion length of 1.93mm. Such waveguides were placed 330nm apart, providing very strong coupling (as is shown in figure 4.2), but remaining within the tight-binding approximation (as is shown in figure 4.3). The mode profiles were shown above (in figure 4.1). The leading coupling coefficients are c₀ = 132.7, c₁ = –24.22 and c₂ = 2.254, whilst the dispersion coefficient is p₂ = 0.5.

4.3 Temporal solitons in a directional coupler

Directional couplers in silicon can support solitons, even if the constituent waveguides are normally dispersive at the pump frequency. The coupled system is described by the equations

Three things about these solutions are notable:

• Firstly, the solitons no longer travel at the group velocity of the waveguide (which is zero in the moving frame in which the NLS equation is constructed). Instead it is increased in velocity by 1/c₁ for the symmetric case, and reduced by 1/c₁ for the antisymmetric case. This effect has been observed for pulses in two-core optical fibres [195].
• Secondly, the wavenumber condition for soliton existence is no longer q > 0, but q > c₀ for the symmetric mode and q > –c₀ for the antisymmetric mode. These cutoff points correspond to the wavenumbers of linear propagation (c₀ for the symmetric mode and –c₀ for the antisymmetric mode) at which point the pump frequency of the soliton would be phase matched to linear radiation modes.
• The third, and most important result is that the effective values of group velocity dispersion are p₂ ± c₂, rather than just p₂. Therefore, the threshold condition for soliton formation is now p₂ < –c₂ for the symmetric mode, and p₂ < c₂ for the antisymmetric mode. This allows for the extension of the spectral window within which solitons are permitted into the normal regime. Due to the positive sign of c₂, this will occur in the antisymmetric mode. Conversely, a symmetric mode restricts the range of the soliton regime. The GVD of the coupling, and the effective dispersions of the supermodes are given in figure 4.4.

4.3.1 Stability analysis of the antisymmetric mode

Just because equations 4.41 and 4.42 are stationary solutions, doesn’t mean that the corresponding solitons are stable. They could be mathematically analogous to a needle which is placed perfectly upright, balanced upon its point, such that the slightest perturbation will cause it to fall. Therefore, it is necessary to see how the solutions react to perturbation. Such analysis is widely established for models with frequency-independent coupling [196, 197], but here it is extended to systems where the coupling affects both the group velocity and GVD.

We will start by analysing the stability of solitons in the antisymmetric mode. We take a solution of the form X = 0, Y = ye^iqζ (where ye^iqζ is the solution given by equation 4.42 written in the form of a real amplitude y multiplied by a complex phase e^iqζ). We then add perturbation terms to give

(4.47)

Transforming equations 4.45 and 4.46 likewise gives

Equations 4.50 and 4.51 exhibit a continuum of eigenvalues along the imaginary axis, which correspond to the set of all delocalised eigenfunctions. (This is derived in appendix C.2.) However when c₁² + < 0, a bandgap exists in λ_y over the range

(4.52)

and when / < 0, a bandgap in l_y exists over the range

(4.53)

These delocalised modes are not particularly important in themselves, as having zero real part they will not contribute to exponential growth and instability. The band-gap is significant however, as within it discrete eigenvalues can exist, which correspond to localised eigenfunctions [23]. These can have non-zero real part, and if positive will lead to exponential growth and thus instability. A search for discrete eigenvalues was made numerically, as is shown in appendix C.3.

It is known that a bifurcation point should occur at q = c₀, corresponding to the existence of asymmetric nonlinear supermodes [196, 197]. (These modes are considered in more detail in section 4.5.2, in the context of modulation instability.) However no discrete eigenvalues corresponding to instabilities into these modes were found, despite scanning across both q and the pump frequency. This is not entirely unexpected, as the exotic solitons that form in these asymmetric modes have a higher energy than those in the ordinary antisymmetric mode [196, 197], and thus the latter are already in the more stable state.

Intramodal perturbation

For the intramodal perturbation, the discrete eigenvalues (of which there are four) can be found analytically. (The resulting spectrum is shown in figure 4.5.) There is a 2-fold degenerate solution with eigenvalue l_y = 0 corresponding to the eigenfunction

(4.54)

and a second solution, again with 2-fold degeneracy and l_y = 0, with eigenfunction

(4.55)

None of these will contribute to instability, as they have no real part. A numerical search was made for additional discrete eigenvalues (again, using the method described in appendix C.3), but none were found. This suggests that the antisymmetric mode is also stable with respect to intramodal instability.

4.3.2 Bifurcation of the symmetric supermode

We can analyse the stability of the symmetric mode in the same way. (The solutions will not exist at a 1.5μm pump. Therefore a 1.4μm pump was used.) As before, we have a symmetric solution of the form X = xe^iqζ, Y = 0 (where xe^iqζ is the solution given by equation 4.41 written in the form of a real amplitude x multiplied by a complex phase e^iqζ), which we perturb to give

(4.60)

Transforming equations 4.58 and 4.59 likewise gives

Again, we assume solutions of the form

As with the antisymmetric case, 4.63 and 4.64 exhibit a continuum of eigenvalues along the imaginary axis, corresponding to the set of all delocalised eigenfunctions. (This is derived in appendix C.2.) When c₁² + < 0 a bandgap exists in λ_x over the range

(4.65)

and when / < 0, a bandgap in l_x exists over the range

(4.66)

It is known that in the absence of coupling GVD, a bifurcation point occurs at q = c₀, which (as for the instability mentioned in section 4.3.1) corresponds to the formation of an asymmetric mode [196, 197]. If we wish to find similar behaviour, we need to search for a discrete eigenvalue which gains a positive real part when q exceeds a critical value. This search was made numerically (using the method described in appendix C.3), with the results shown in figure 4.6. This shows that as q is increased, a pair of discrete eigenvalues in λ_x (with values summing to zero) enter the band-gap and migrate along the imaginary axis towards the origin. When they reach it, they start heading in opposite directions along the real axis, and thus the soliton will become unstable.

We therefore see a bifurcation when λ_x = 0. In the special case where c₁ = 0, we can find an analytical solution

(4.67)

subject to the condition

(4.68)

Without coupling (when c₂ = 0, and p₂ must be –), this reduces to q = c₀, and thus the result is a generalisation of that seen in references [196] and [197]. We will return to the self-focussing effects that give rise to this instability in section 4.5.2.

Intramodal perturbation

For the intramodal perturbation, the discrete eigenvalues can (just as for the antisymmetric case) be found analytically. There is a 2-fold degenerate solution with eigenvalue l_y = 0 corresponding to the eigenfunction

(4.69)

and a second solution, again with 2-fold degeneracy and l_y = 0, with eigenfunction

(4.70)

Again, none of these will contribute to instability, as they have no real part. A numerical search was made for additional discrete eigenvalues, but none were found. This suggests that (as for the antisymmetric case) there is no intramodal instability.

4.3.3 Soliton generation

Soliton evolution was modelled numerically. (See appendix A.1.) The supermodes can be generated directly by firing coherent light into the array such that both wires are excited. The antisymmetric mode can be generated by pumping at a non-zero angle of incidence, so that the differing path lengths cause an incremental phase shift between neighboring waveguides [198, 199]. It is more interesting, however, to fire light into just one wire, thus creating a superposition of the two modes, and then to rely upon the group-velocity difference noted above to split the modes. Pulse propagation for the linear regime is given in figure 4.7, and it can be seen that the pulse splits cleanly in two. Introducing nonlinearity (but not damping) in figure 4.8 shows that the antisymmetric pulse forms a soliton. The introduction of damping in figure 4.9 shows that the soliton survives.

Figure 4.7: Evolution of a pulse with peak power 15P0 fired into the 1st wire of a 2 wire system. (Amplitude derived as , which corresponds to incoherent mixing between the outputs of each wire.) Without damping or nonlinearity, the pulse splits into a pair of supermodal pulses due to the effect of coupling on group velocity.

The soliton like pulse shown in figure 4.9 can be investigated further, by using the soliton area parameter defined in section 3.3.3. As before, we define a parameter S = T_FWHM. For the ideal soliton solution (equation 4.42) the peak power (summed over both of the channels) is given (in terms of the intrinsic unit P₀) by

(4.71)

Similarly, the FWHM duration is given (in terms of the intrinsic unit T₀) by

(4.72)

This gives an ideal soliton area parameter of

(4.73)

Figure 4.10 shows the soliton area parameter plotted as a function of distance. It remains roughly constant, having a value close to S₀, thus showing that the pulse is a soliton.

4.4 Temporal solitons in multiwire waveguide arrays

4.4.1 Interplay between diffraction and dispersion

The above can be generalised to a system with an arbitrary number of waveguides, N. We start by rewriting the undamped equations of motion (equations 4.27) in a vector form, where contains the values of E_n. This gives

(4.74)

where the function f ≡||² acts independently on each element of the vector, and the matrix operator is defined as

(4.75)

where δ is the Kronecker symbol. (It should be noted that the matrix will be encountered once again in section 5.3, when the radiation emitted by spatiotemporal solitons is considered.)

In order to proceed, we need to diagonalise this matrix equation. We write the eigenvalue equation for as

(4.76)

which yields eigenvalues of the form

(4.77)

with corresponding (normalised) eigenvectors

(4.78)

It should be noted that the index k is arbitrary, and thus may be replaced with other indices, as is done for χ_j and _j below. The eigenvalue are distinct, as the argument of the cosine function is always within the range 0 < 𝜃 < π, over which the cosine function shows one-to-one correspondence. As is both self-adjoint and distinct-eigenvalued, it follows that the normalised eigenvectors must form an orthonormal basis. This can be seen explicitly by defining a scalar product operation

(4.79)

Using the trigonometric identity

(4.81)

where Ẽ_k is the amplitude of each modal component. (The prefactor is a matter of convenience, chosen so that in a few steps further on, equation 4.92 will have unit nonlinearity.)

The basis conforms to the condition that E₀ = E_N+1 = 0, meaning that the (non-existent) wires outside of the waveguide boundaries contain no light. It should also be noted that this basis is a generalisation of that used for the N = 2 case in section 4.3, with Ẽ₁ = X and Ẽ₂ = Y .

Substituting this into equation 4.27 and then taking the scalar product with any basis vector _j (to project out the jth mode) gives

(4.82)

where the effective dispersion of each mode is given by

(4.83)

We have not fully diagonalised the equations, due to the nonlinear terms. However, there are no longer any linear terms linking the components of equation 4.82. Therefore, in the low power limit the system is diagonal and so each component of equation 4.82 will evolve independently.

Normal and anomalous diffraction

It is apparent from equation 4.83 that the dispersion and coupling are interlinked, with the eigenvalue χ_j controlling the extent of this interaction. The fact that these eigenvalues are distinct means that each mode has a unique dispersion relation. The values of χ_j occur in pairs having opposite sign (plus a zero-eigenvalue when N is odd) due to the property that cos ≡–cos. This is highly significant, as by choosing the appropriate mode, it is possible to effectively reverse the signs of the coupling coefficients.

It is instructive to rewrite equations 4.77 and 4.78 as

(4.86)

In the limit of an infinitely large array, the transverse wavenumber becomes a continuous varying quantity. If we were to excite such an array with a coherent beam of light fired along the direction of the waveguides, then a mode with all the wires in phase (and thus κ_x = 0) would be excited. If the initial excitation was restricted to a finite region of the waveguide, then a spatial wavepacket centred about κ_x = 0 would be excited. Alternatively, by tilting the beam with respect to the direction of the waveguides, a phase shift would be introduced between each wire (due to the varying path length) thus exciting a mode with non-zero κ_x [199]. In general, such a beam will spread into the surrounding waveguides, in what is essentially a discrete version of diffraction. It can be shown, for instance, that the width of a Gaussian beam focused onto the array boundary will vary with propagation as [199]

(4.87)

where W₀ is the initial width. The diffraction coefficient χ_j''≡ ∂²χ_j/∂κ_x² = –χ_j specifies the magnitude of the diffraction. It is the spatial equivalent of dispersion, and is almost always negative [198].

At κ_x = 0, the diffraction coefficient possesses its minimum value of χ_j'' = –2. As κ_x is increased from 0, the diffraction coefficient becomes less negative, until at κ_x = π/2 the diffraction vanishes, and the beam propagates with constant width. Increasing κ_x beyond π/2 causes the diffraction coefficient to become positive, in what is known as anomalous diffraction [198]. At κ_x = π the diffraction coefficient reaches its maximum value of χ_j'' = 2. Notably, equation 4.87 is not dependent on the sign of χ_j'', and so this state broadens in exactly the same manner as the κ_x = 0 state. More generally, the states with 0 ≤|κ_x| < π/2 give normal diffraction, whilst the states with π/2 < |κ_x|≤ π give anomalous diffraction. It can also be seen that normal diffraction corresponds to positive values of χ_j, whilst anomalous diffraction corresponds to negative values.

The mode profiles for the κ_x = 0, π/2 and π cases are plotted in figure 4.11. For κ_x = 0, the wires are all in phase, whilst for κ_x = π every mode is maximally out of phase with its neighbours. The diffractionless mode at κ_x = π/2 corresponds to a mode in which every other wire has zero amplitude, as is shown in figure 4.11.

In the more general case, the relationship between the normally diffractive and anomalously diffractive modes can be seen by defining a matrix Ŝ as

(4.88)

where δ is the Kronecker symbol. This matrix anticommutes with and thus multiplication of an eigenvector by Ŝ provides the eigenvector with oppositely signed eigenvalue. Therefore, multiplication by Ŝ (which is self-inverse) switches back-and-forth between states with normal and anomalous diffraction. The operation has the effect of shifting the phase of every other wire by half a cycle, which in turn has the effect of reversing the inter-wire phase relationship between each neighboring pair. Therefore, each normally diffractive mode is paired with an anomalously diffractive mode having the same amount of light in each waveguide, and the same beam widening properties, but opposing inter-wire phase relationships.

It should be noted that the scheme described above is not the only means of achieving anomalous diffraction. The most direct means is to invert the sign of coupling (as mentioned in section 4.1) by using systems consisting of (for example) AlGaAs waveguides separated by a metamaterial consisting of AlGaAs rods in a sapphire matrix [193, 181]. It can also be achieved by using higher order photonic bands [191, 200].

When a finite array of wires of wires is used, κ_x becomes quantised to the values given by equation 4.86. Whilst the extremal κ_x = 0 and κ_x = π modes can only exist in an infinite array, the κ_x = π/2 diffractionless modes can be seen, and exist for all arrays with odd N.

Controlling dispersion with diffraction

From equation 4.83 it is apparent that for normally diffractive modes (with positive χ_j), the coupling dispersion operator Ĉ will be added to the dispersion operator , whilst for anomalously diffractive modes (with negative χ_j), Ĉ will be subtracted from . In otherwords, normal diffraction will cause the coupling dispersion to reinforce the waveguide dispersion, whilst anomalous diffraction will cause the coupling dispersion to counteract the waveguide dispersion.

As was described in section 4.1, the GVD of coupling is usually normal. However, by using anomalous diffraction, we can use it to create anomalous GVD. Figure 4.12 shows the parameter space (in wavelength and transverse wavenumber) giving anomalous GVD for a set of 220nm × 330nm waveguides placed 330nm apart. Examples of actual dispersion curves that can be created like this, are shown in figure 4.13.

As mentioned previously, there is a correspondence between anomalous diffraction and negative index refraction [181, 182]. Switching back into the unscaled units (defined in section 4.1), the supermodal wavenumber is given by β_mode = β + χ_jδ, and so the supermodal effective index is given by

(4.89)

Whilst χ_j is not capable of changing the sign of the refractive index, it is capable of reversing the sign of one of the contributing terms.

4.4.2 Supermodal soliton solutions

When only one mode Ẽ_j is present, equation 4.82 reduces to

(4.90)

This can be simplified using the identity

(4.92)

where the coefficient Γ_Nj is given by

(4.93)

The exceptional cases at j = /2 correspond to the κ_x = π/2 non-diffractive modes, in which the wires are effectively uncoupled. (They also correspond to the trivial N = 1 case, where only a single wire is present.) Given the lack of any interaction between dispersion and diffraction, these modes are not particularly interesting, and will not be considered further.

Taking this single-mode equation, and truncating the operators to second order gives

(4.94)

This has solutions of the form

(4.95)

From equations 4.78 and 4.81, it follows that the transformation out of the modal basis is made as

(4.96)

Doing so, gives multi-wire soliton solutions of the form

(4.97)

It can be seen that in the single wire N = 1 case (where p₂ = –, Γ_Nj = and χ_j = 0) this reduces to the conventional solution of E = seche^iqζ . The solutions are a further generalisation of previously known solutions [179], to which they reduce in the case N = 2 and c₁ = c₂ = 0.

4.4.3 Soliton formation

Soliton evolution was modelled numerically. As with the two wire case, the anomalously diffractive supermodes can be excited directly by pumping at a non-zero angle of incidence [198]. However, as before, the method of pumping a single wire and then letting the resulting superposition of modes separate was used. Pulse propagation for the linear regime is given in figure 4.14, and it can be seen that the input pulse splits into a train of six pulses. Introducing nonlinearity (but not damping) in figure 4.15 shows that the final pulse in the train forms a soliton. The introduction of damping in figure 4.16 shows that the soliton survives.

Figure 4.14: Evolution of a pulse with peak power 150P0 fired into the 3rd wire of a 6 wire system. (Amplitude derived as , which corresponds to incoherent mixing between the outputs of each wire.) Without damping or nonlinearity, the pulse splits into a train of 6 supermodal pulses (with numbering corresponding to j in equation 4.83).

The soliton-like j = 6 pulse shown in figure 4.16 can be investigated (as in section 4.3.3) by using the soliton area parameter. For the ideal soliton solution (equation 4.97) the peak power summed over all of the channels is given by

(4.98)

and the FWHM duration is given by

(4.99)

This gives an ideal soliton area parameter of

(4.100)

Figure 4.17 shows the soliton area parameter plotted as a function of distance. It remains roughly constant, having a value close to S₀, thus showing that the pulse is a soliton.

4.5 Modulation instability in couplers

Modulation instability (which was considered for a single wire in section 3.4) can also happen in coupled waveguide systems [201, 202]. As for soliton propagation, the conventional models assume a frequency independent coupling term. Likewise, when coupling GVD is introduced, it can be seen that the dynamics are greatly changed, and that modulation instability can occur in normally dispersive waveguides.

Equations 4.35 and 4.36 possess continuous wave solutions of the form

(4.101)

for the symmetric mode, and

(4.102)

for the antisymmetric mode. As in section 3.4, P is related to the optical power. It follows from the transformation in equations 4.32 and 4.33 that P is the power per wire, and so the total power in the system in given by 2P.

As with the single wire case, perturbations from these solutions can be reinforced by the nonlinearity, creating spectral sidebands. However, two very different phenomena can result, depending on whether these sidebands are emitted into the same mode as the pump signal, or into the other mode. Both cases are considered below.

4.5.1 Intra-modal modulation instability

Taking the antisymmetric solution, we can perturb it as

(4.103)

where ε is a small perturbation into the same mode. Substituting this into equation 4.36 gives

(4.104)

where terms in ε² and ε³ have been removed, as we are only considering the first stage of perturbation. This is the same as equation 3.14, except we have replaced the dispersion operator with a modified operator –Ĉ + c₀. We can therefore use the same procedure to derive an expression for the modulation instability side-lobes. The gain spectrum is given by

(4.105)

where (as before) the even operators are defined as

(4.108)

where as described in section 3.4, this is a necessary but insufficient condition, requiring the explicit removal of pathological solutions. The predicted modulation instability peak positions are given in figure 4.18, whilst the numerical spectra are given in figure 4.19. As with a single wire case, four modulation instability peaks can be seen. However, these become obscured in the presence of free carrier interactions. (As in section 3.2.1, a physically reasonable estimate for the carrier cross section of ε_fc ≃ 10^–3 was used.) It should be noted, however, that techniques exist to sweep away free charge carriers [59], and so a more favourable experimental result may be possible.

4.5.2 Cross-modal modulation instability

In the symmetric supermode, the effect of coupling GVD is to restrict the wavelength range in which conventional modulational instability is possible. A different picture emerges, however, when the effect of cross-modal instability is considered. This can be seen by taking the symmetric supermodal solution, and modifying it as

(4.111)

Unlike equation 4.105, this does depend on the value of c₀, as the explicit and implicit values reinforce one another. This is a result of the wavenumber mismatch between the mode of the signal and the mode of the emitted radiation. The gain maxima occur at the solutions to

(4.112)

This gain spectrum is notably different from the inter-modal case, in that we can have non-zero gain at the pump frequency. We can see this by setting ω_m = 0, which reduces equation 4.111 to

(4.113)

We therefore have a critical power at P_c = c₀, above which the pump itself is unstable.

Modal instability and asymmetric nonlinear supermodes

This phenomenon of the pump itself becoming unstable is well documented, and results from the fact that the modes themselves are altered by the nonlinearity [179]. These modes can be investigated by rewriting equations 4.30 and 4.31 in a different basis as [203]

(4.120)

where (in keeping with the above notation) P = gives the mean power per wire. If S₁, S₂ and S₃, are regarded as the Cartesian coordinates of a three-dimensional vector , then equation 4.120 defines a sphere of radius 2P, on which must lie. (This is extremely similar to the Stokes formation of optical polarisation, with being a modified Stokes vector [204], which is not surprising, as instead of having light split between two principal polarisation states, we have light split between two wires.)

We can proceed by noting that

(4.121)

is a conserved quantity. This allows us to combine equations 4.114, 4.115 and 4.116 into a single second order equation of the form [203]

(4.122)

This is an anharmonic oscillator (which oscillates in ζ). We can analyse its dynamics by defining a Lagrangian

(4.123)

which upon substitution into the Euler Lagrange equation

(4.124)

gives equation 4.122 as required. We can separate the Lagrangian, into a quasi-kinetic energy (dS₁/dζ dependent) term T and a quasi potential energy (S₁ dependent) term V, using the standard energy construction of the Lagrangian (L = T –V). This gives

It should be noted that S₁ measures the asymmetry between the intensities in each wire, such that a value different from S₁ = 0 corresponds to a mode in which one wire contains more light than the other. Therefore, the potential wells that form when c₀Γ > 2c₀² correspond to the influence of asymmetric modes. These asymmetric modes are the result of self focussing, whereby the intensity dependence of the refractive index causes a waveguide with more light to become more confining, thus reinforcing any asymmetry. (Self focussing in coupled systems is a well known phenomenon for both continuous wave [205] and for solitons [206], which will be considered in more detail in chapter 5.) This self focussing can only happen when there is an existing asymmetry, and so (when c₀Γ > 2c₀²) the symmetric supermode is metastable.

This behaviour can be understood by returning to the analogy of a needle placed perfectly upright, balanced upon its point (as mentioned in section 4.3.1). If the needle is completely isolated from all external forces, then it will remain in position. However, any perturbation will be reinforced by gravity, and the needle will topple over. Notably, the initial state has rotational symmetry (about the axis defined by the shaft of the needle), as do the equations of motion describing the system. When the needle topples (and the system falls into a "vacuum state"), spontaneous symmetry breaking occurs, in which a new parameter (namely the azimuthal angle of the needle’s position) emerges. In the coupler we also have a symmetry, namely an invariance under swapping the wires. This is present both in the initial conditions, and in the equations of motion (which can be seen by swapping E₁ and E₂ in equations 4.30 and 4.31. Under perturbation, a new parameter (namely the sign of S₁) emerges, and thus we have symmetry breaking into one of two states.

It is also possible to "seed" the instability by introducing a small asymmetric signal, which causes the system to collapse into a particular state. This effect has been suggested as the basis of optical switches, whereby a weak signal can control the direction of a much stronger signal [207].

The symmetric supermodal solution has E₁ = E₂ = eⁱ, giving a value for Γ (from equation 4.121) of Γ = 2P. This sets the seperatrix between the stable and unstable regimes at P = c₀, which agrees with the result derived from equation 4.113. We therefore see that the cross-modal modulation instability is a frequency-dependent generalisation of ordinary coupler instability. Notably, this means that self focussing can occur below the critical power, but that it happens indirectly, via the generation of frequencies away from the pump.

Numerical modelling

For a system with 220nm × 330nm wires placed 330nm apart pumped at 1.5μm, equation 4.112 can be solved to give two gain peaks at 1.33μm and 1.74μm. This system was modelled numerically, as is shown in figure 4.21. This shows that sub-critical-power self-focussing does occur via the generation of spectral sidebands. The effect seems to vanish in the presence of free charge carriers. However, as noted previously, it is possible to sweep away free charge carriers [59], and so it may be possible to observe the phenomenon in a real device.