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Chapter 3
Nonlinear propagation in SOI waveguides

3.1 Temporal solitons

Temporal optical solitons were first predicted by Akira Hasegawa in 1973 [130], and first observed experimentally by Linn Mollenauer and Roger Stolen in 1980 [13]. Since then, solitons have become fundamental to modern optics. They have been applied in pulsed laser systems [131], optical logic gates [132], and optical data lines. It is possible to transmit solitons over thousands of km [133]; this has not only been achieved in loops of fibre in a laboratory, but in a 2872 km communication line between the Australian cities of Perth and Adelaide [134].

Optical solitons were originally observed in conventional fibre [13]. More recently, the need for dispersion tailoring has seen photonic crystal fibre (PCF) used as a medium for soliton propagation. [135]. These fibres have an intricate transverse structure of glass and air-gaps which run through their entire length. By selecting the geometry of this structure, the group velocity dispersion (GVD) can be greatly altered, thus allowing for anomalous GVD in materials with a normal bulk GVD.

Interest in solitons has naturally carried over into silicon on insulator, which is an ideal medium as it provides both strong ultrafast nonlinearity [55, 56, 58], and (like PCF) the capability for substantial dispersion tailoring [47, 48, 49]. Solitons in silicon on insulator have been observed by groups at Columbia University [57] and Rochester University [63]. In section 3.3, the results of a collaboration with experimentalists at the University of Bath are presented, further strengthening the base of evidence for the existence of solitons in silicon waveguides.

These solitons can be analysed using the model derived in the previous chapter. In the absence of damping and higher order dispersion, the NLS equation (equation 2.85) admits an extremely well known bright soliton solution of the form [23, 8, 97, 98]

( ) E (ζ,τ) = √2q-sech √2q-τ eiqζ
(3.1)

where q (which must be positive) is the wavenumber. This solution (plotted in figure 3.1) requires anomalous GVD, such that p2 = –1 2.


Figure 3.1: The sech-like soliton solution, displayed as function of power against time. The peak power is given by 2q, whilst the FWHM duration is given by 2ln( √-) 1 + 2/√ -- 2q

This is not an exact solution when damping or higher order dispersion is reintroduced. In fact, for the general case, no exact soliton solution can exist. Although solitons can never retain their shape perfectly in a real system, they can certainly exist, as is shown in section 3.1.1. The loss of exactitude also corresponds to the absence of the phase-shift on collision property described in section 1.1.2. Whilst solutions to the ideal NLS equation can pass through one another with only a phase shift, this effect is broken for the more realistic model, as is shown in section 3.3.3 [23].

The special case of q = 12, gives E(ζ,τ) = sech(τ)eiζ/2, namely a pulse of unit duration and unit amplitude in the dimensionless units defined in section 2.2.4. In equation 3.1, the duration and the amplitude are both linked to q, and so by specifying one, the other will also be specified. For a given pulse duration, the corresponding power is known as the soliton threshold. It follows from the choice of dimensionless units that a pulse with unit duration has a unit soliton threshold. In real units, the soliton threshold is P0, as defined in equation 2.64.

In the spectral domain, equation 3.1 has the power profile

( ) P (ω) ∝ sech2 √π--ω 8q
(3.2)

where the spreading about the carrier frequency (which is given by ω = 0 due to the moving frame of reference) is a consequence of the pulse having a finite duration. The solution is in fact transform limited in that the bandwidth is as small as it possibly can be, and is limited only by this fundamental effect. For a 100fs second pulse at 1.5μm, this gives a FWHM spectral bandwidth of 24nm.

Physically, the sech-like soliton corresponds to a balance between self phase modulation (SPM) and dispersive pulse broadening. SPM is a nonlinear effect that occurs in optical pulses, and is the temporal analogue of spatial self-focussing [8]. At the leading edge of the pulse, the refractive index increases with time, due to the rising intensity. This gives the phase of the optical wave an extra time dependence, which retards the fundamental temporal oscillation of the electromagnetic wave, thus redshifting it. Conversely, at the trailing edge of the pulse, the reverse happens, causing a blueshift. In an anomalously dispersive medium, blue light (by definition) has a higher group-velocity than red light, and so without nonlinearity, the pulse would lengthen. (The terms red and blue are used figuratively, to describe longer and shorter wavelengths.) This, however, is counteracted by the SPM. The strength of SPM is power dependent, and at the soliton threshold, the two effects precisely cancel to give a soliton.

With normal GVD, the SPM acts to reinforce dispersive pulse broadening, and so a soliton will not be formed. Dark solitons are possible however [136], as the leading edge of the soliton corresponds to decreasing (rather than increasing) amplitude, and so SPM acts in the opposite direction. Bright solitons can also be seen in the normal regime, by using materials with a defocussing nonlinearity such as aluminium gallium arsenide (AlGaAs) [137], again reversing the direction of SPM.

This soliton in equation 3.1 is not the only one that exists for the ideal system. (It is, however, the most common, and is known as the fundamental soliton.) There exist so-called higher order solitons, which can be excited with an initial condition of E(τ) = N sech(τ), where N is an integer. These solutions change shape as they propagate (but return to their original state at periodic intervals) and are known as breathers [23]. They are, however, unlikely to be seen in silicon on insulator, as they break up in the presense of linear absorption [138], nonlinear absorption [123] and other deviations [23]. A breather can be thought of as a superposition of solitons, and perturbations will cause these to separate. This pulse fission is considered in section 3.2.1.

It should be noted that these soliton solutions can be formally derived from the NLS equation by a procedure known as the inverse scattering transform [139]. This is an extremely powerful method of solving nonlinear partial differential equations, which can also be used to extract solutions from other soliton-yielding equations such as the Sine-Gordon equation and Korteweg-de Vries equation [44]. For the purpose of this report, however, such a formal approach is unnecessary, and will not be considered further.

3.1.1 Soliton formation

Pulse evolution in a lossless waveguide was modelled numerically for a range of input powers, as is shown in figure 3.2. (The temporal pulse profile at ζ = 0 was specified, and this was advanced in ζ to the desired output value. The computational algorithm is described in appendix A.1.) The input consisted of transform limited pulses with a 1.5μm carrier frequency and a FWHM duration of 100fs. A waveguide 220nm in height and 420nm in width was chosen. The waveguide was also assumed to be topped with a 100nm thick layer of a material with refractive index 1.35, in order to simulate an etching mask. The dispersion relation of this geometry is given in figure 2.3.

This waveguide provides a strong anomalous GVD (at the chosen 1.5μm pump wavelength) of 3934 psnm–1 km–1. The zero dispersion wavelengths (ZDWs) are well removed from the pump, having values of 1.243μm and 1.736μm. This provides low higher order dispersion (with β3 = 0.00615). The waveguide geometry is therefore well suited for observing straightforward soliton evolution, without the complication of other effects.

When the wire is pumped with pulses having a peak power below the soliton threshold, the SPM is unable to fully compensate for the dispersive pulse broadening, and so the pulse duration increases. At the soliton threshold, a soliton is formed. Above the soliton threshold power, the focussing effect of SPM exceeds the dispersive pulse broadening, and we see pulse compression.


Figure 3.2: Propagation of 100fs sech-like pulses through a 220nm × 420nm SOI waveguide in the absence of damping. Shown over a 0.7mm (1 dispersion length) propagation distance. a) At low power εP0 (where ε ≪ 1, giving quasi-linear evolution), the pulse disperses. b) At 0.5P0 dispersion is suppressed but not eliminated. c) At P0 pulse broadening is precisely counteracted by SPM, forming a soliton. d) At 1.5P0 SPM starts to overcompensate for dispersive broadening, causing pulse compression.
Damped solitons

In a real SOI wire, the effect of damping significantly affects soliton evolution. We therefore assume a 2PA coefficient of ε2pa = 0.1. This value has been reported in the literature [124], and is that which will be extracted from experimental data in section 3.3.2. We also assume a linear damping of εl = 0.01, giving a real-unit attenuation of 1.3 dB cm–1, which is low but perfectly realisable [117, 118]. On the other hand, the effect of free charge carriers is negligible at these relatively low energies, and will only become important when we consider more energetic pulses in the following sections.

When we introduce this damping (figure 3.3), we no longer see a soliton at the threshold energy. Instead, we must increase the energy slightly to compensate for the energy loss.


Figure 3.3: Propagation of 100fs sech-like pulses through a 220nm × 420nm SOI waveguide with both linear and nonlinear damping included. Shown over a 0.7mm (1 dispersion length) propagation distance propagation distance. a) At low power εP0 (where ε ≪ 1), the pulse broadens. b) At 0.5P0 dispersive pulse broadening is suppressed but not eliminated. c) At P0 there is (unlike with the undamped case) still some pulse broadening. d) At 1.5P0 the best approximation to a soliton is formed, as the increased power compensates for the energy loss. (Note that after a slight change in pulse shape, it settles down into an unchanging waveform.)

3.1.2 Soliton compression

Pulse compression happens at powers above the soliton threshold, due to nonlinearity overcompensating for the dispersive pulse broadening [8]. This has been observed in PCF [39] and glass nanowires [140]. The soliton pulse compression effect is demonstrated in figure 3.4 for a 220nm × 420nm wire, showing that a 100fs pulse (with power 3.5P0) is nonlinearly compressed to 34fs. This effect is put to use in section 5.2, where a spatiotemporal soliton requiring an upper pulse duration of 80fs is generated from a 100fs pulse.

The 3.5P0 pulse gives a roughly optimal compression ratio, as it occurs at a power slightly below that of the first higher order soliton (which having twice the amplitude of the fundamental soliton, occurs at 4P0). Therefore the pulse isn’t a superposition of multiple solitons, and so a relatively clean compression effect can be seen. At higher powers (at 7P0 for instance, as is shown in figure 3.4) the pulse is a higher order soliton, which under perturbation will break up. This soliton fission is considered in more detail in section 3.2.1. (It is in fact possible to compress pulses further, by using tapered waveguides in which the GVD gradually decreases along the propagation length. As a soliton moves, the soliton threshold gradually drops, providing a gentle but sustained compression effect, allowing for compression ratios of over ten to one [141].)


Figure 3.4: Compression of a 100fs pulse in a 220nm × 420nm SOI waveguide. Shown over a 0.7mm (1 dispersion length) propagation distance. The model includes both linear and nonlinear damping. a) At 1.5P0 a soliton exists. b) At 2P0 pulse compression occurs. c) At 3.5P0 maximal pulse compression occurs, with the pulse being reduced to 34fs. d) At 7P0 the pulse splits.

3.1.3 Čerenkov radiation

The sech-like soliton is extremely robust, in that it can retain its basic shape in the presence of a range of perturbations [130]. This stability primarily results from the wavenumber of each frequency component within the soliton being different from the wavenumber of linear radiation at the same frequency [142]. Therefore, light can’t easily escape from the soliton, as it has no available mode to leak into. In the presence of HOD, however, the soliton’s dispersion relation may overlap with the linear dispersion relation at a frequency away from the pump. In this case, resonant radiation can occur [142], as the matching of wavenumbers will cause a continuous matching of phase over the soliton’s length. (This phase matching can also allow a soliton to interact with an external signal of continuous-wave radiation [143].) It is possible for resonant radiation to be excited in a different waveguide mode to the soliton [144, 145], although this unlikely to occur in SOI, as the only other waveguide mode (as mentioned in section 2.1.2) has little projection onto the fundamental mode. Resonant radiation has been observed in both glass fibres [146, 147, 148, 149], and silicon wires [57].

When a charged particle travels faster than (the refractive phase velocity of) light, Čerenkov radiation is emitted. This is the optical equivalent of a sonic boom, and is most famously seen as the blue glow surrounding water-cooled nuclear reactors, where it is emitted by high-energy electrons. An optical soliton has a physical presence in the form of an index-shifted region of material, and this may also travel faster than (the refractive phase velocity of) light and thus emit Čerenkov radiation. It can be shown that this emission of Čerenkov radiation is in fact the same phenomenon as the above resonant radiation [142, 150]. They differ, however, in that soliton radiation requires phase matching, which is a consequence of the fact that whilst electrons are much smaller than the wavelength of the light they emit, solitons are much larger.

We can calculate the frequency of resonance by taking the ideal soliton solution (equation 3.1) and adding a small perturbation ε(ζ,τ) to it, such that

(√ -- (√ -- ) ) E (ζ,τ) = 2qsech 2q τ + ε(ζ,τ) eiqζ
(3.3)

Substituting this into the full equation of motion (equation 2.85) gives

[ 2] √-- (√ -- ) (√ -- ) iqε+ ∂ε= i Dˆ– 1-∂2- 2q sech 2qτ + iDˆε+ 2iq(2ε+ ε∗) sech2 2qτ ∂ζ 2∂τ
(3.4)

where terms containing ε2 and ε3 have been discounted (as we are treating ε as a small perturbation, rather than a general correction). We have also removed damping. The left hand side of this equation admits sinusoidal solutions, and so the whole equation can be thought of as a linear oscillator in ε (oscillating not with time, but with space) driven by an oscillatory force of magnitude √ -- 2qsech(√ -- ) 2qτ. We can search for resonances by determining the natural frequency of the oscillator system, and then matching this with the spatial wavenumber q. As ε is already modulated by q, these resonances will happen when ε shows no oscillation with respect to ζ.

We therefore remove the driving terms (i.e. those not containing ε). We also neglect the 2iq(2ε+ ε∗) sech2 (√2q-τ) term (which describes the refractive index change induced by the soliton field). Whilst this term is of importance when calculating the radiation amplitude [151], it can be neglected when we merely wish to determine the frequency of the resonance. This gives

iqε+ ∂ε= iˆD ε ∂ζ
(3.5)

We are looking for linear waves, and thus for solutions of the form ε = ε'eikζ–iωτ. However, for resonance, we need zero ζ dependence, and so we set the wavenumber k = 0, giving ε = ε'e–iωτ. This yields the resonant condition

q = D (ω)
(3.6)

The left hand side of the equation gives the dispersion relation of the soliton (which is a constant due to the moving frame of reference), whilst the right hand side is the dispersion relation of linear waves. Solving this equation for ω gives the frequency of resonance (relative to the pump frequency, which is constructed so that ω = 0), as is demonstrated in figure 3.5. At the pump frequency the function D(ω ) has a local maximum at D(0) = 0 (which corresponds to the requirement of anomalous GVD). As q is positive, it follows that D(ω) must have a point of inflection between the pump and Čerenkov frequencies. This point of inflection corresponds (by definition) to zero GVD, and so the Čerenkov radiation will be emitted at normally dispersive wavelengths.

A commonly used approximation of this result [152, 153, 154] is to assume that the Čerenkov radiation occurs at the ZDW, rather than on the other side of it. In the simplest possible case of β2 and β3 only, this gives a frequency (relative to the pump) of [155]

ω = 3|β2| β3
(3.7)

In addition to starting with knowledge of the higher order dispersion and using it to calculate the Čerenkov frequency, the reverse can be also done. This is useful from an experimental standpoint, as determining the β3 value of a waveguide requires many precise measurements of its group velocity over multiple wavelengths (as is described in section 3.3.1). Therefore, by measuring the position of the Čerenkov peak, a value of β3 can be inferred [57, 155] using equation 3.7.

Numerical analysis and spectral recoil

In order to observe Čerenkov radiation, the above 220nm × 420 waveguide geometry is unsuitable, as the ZDWs are too far from the pump. Therefore, a 220nm × 380nm geometry was chosen, which shifts the red-end ZDW to 1.627μm. (The dispersion relation of this geometry is plotted in figure 2.3.) Modelling using this waveguide configuration is shown in figure 3.5. A 33.3fs pulse duration was used, as longer pulses (being spectrally narrower) were found to produce very little radiation. The solution to equation 3.6 (and thus the spectral position of the Čerenkov radiation) is obtained graphically in figure 3.5.


Figure 3.5: The output (after 0.6mm of propagation) of a 33.3fs pulse at 1.5μm fired into a 220nm × 380nm SOI guide is shown top. A soliton is formed (the left hand peak), producing Čerenkov radiation (the right hand peak). Equation 3.6 is solved graphically by matching the linear wavenumber D(ω) to the soliton wavenumber q = 4.5. Notably the soliton is slightly blueshifted, whilst the radiation is to the red of its predicted position. The model includes linear damping and 2PA.

Notably, the position of the Čerenkov peak (1.816μm) is slightly different from the predicted value (1.814μm). Similarly, the soliton itself is blueshifted slightly. To investigate this further, the spectral position of both were measured as a function of distance along the waveguide, as is shown in figure 3.6. This shows that the radiation peak is incrementally redshifted, whilst the soliton is incrementally blueshifted. This wavelength shifting is known as spectral recoil. This effect results from conservation of momentum, and causes the frequency of the soliton to be pushed away from the frequency of the emitted radiation [142]. (The effect will not be predicted by the above analysis, as it results from the driving terms removed between equations 3.4 and 3.5 [156].) This in turn causes the frequency of the radiation to be pushed further from the pump frequency [157].


Figure 3.6: Spectral position of the soliton and the radiation peak (for the system described in figure 3.5) as a function of propagation distance. (The radiation peak is not shown before 0.07mm, as there was no local maximum to measure.) The soliton is incrementally blueshifted from its 1.5μm starting wavelength, whilst the radiation is incrementally redshifted. Notably the radiation peak starts off at shorter wavelengths to that predicted by equation 3.6, which is due to the initially weak peak being superposed with the soliton tail, thus moving its apparent maximum. As the amplitude grows, this effect rapidly vanishes.

Whilst Čerenkov radiation has been observed in SOI, direct comparison to these experiments is problematic, as the only papers reporting the phenomenon seem to be those using the above method of deriving the third order dispersion from the Čerenkov frequency [57, 62]. Therefore, agreement between theory and experiment would be by construction, rather than a meaningful physical result.

In section 5.3 we will return to the subject of Čerenkov radiation, and consider what happens in arrays of waveguides.

3.2 Pulse fission and spectral broadening

Nonlinear processes can cause the spectral width of a pulse to be hugely broadened [4, 5], in what is known as supercontinuum generation. This process involves the complex interplay between a wide range of physical effects. These can include pulse compression [147], soliton fission [158, 159], Raman scattering [160], four wave mixing [160, 161], Čerenkov radiation [148], modulational instability [162], and a novel effect whereby radiation is trapped within a gravity-like potential produced by accelerating solitons [163]. Supercontinuum generation has been observed in PCF [4, 164, 165] and tapered conventional fibres [166, 164, 165]. A similar effect has been observed in SOI waveguides [62], but with a far smaller spectral range.

3.2.1 Spectral broadening by soliton fission

At powers much higher than the soliton threshold, the self focussing caused by self phase modulation will overwhelm dispersive pulse broadening, causing pulse compression followed by fission, as is shown in figure 3.7. (As mentioned above, this can be thought of in terms of the input pulse being a superposition of many solitons, which separate under perturbation.) This fission is a starting point for spectral broadening, as is shown in figure 3.8, which gives the spectral output for a range of input powers. At the highest power of 640P0 (corresponding to a superposition of 25 solitons) the pulse broadens into a continuum with 800nm bandwidth. This was found to be roughly optimal, as further increases in power lead to no increase of the spectral width. This is notably greater than that previously observed in SOI waveguides [62]. It is, however, much less than the 4000nm bandwidth that can be achieved in PCF [167], and so it is more appropriate to call this process continuum generation, rather than supercontinuum generation.

Whilst the influence of free charge carriers was negligible for the analysis in section 3.1 it becomes significant at these much higher powers. The carrier cross section can be calculated as εfc = ε2paT0P0σfcc/2ℏωSeff, but this is problematic as it requires knowledge of the soliton threshold P0, which cannot be scaled away. Therefore, a physically reasonable order-of-magnitude estimate of εfc ≃ 10–3(1 + 7.5i) was used.


Figure 3.7: Evolution of a 100fs pulse with peak power 640P0 over the first 0.14mm of propagation through a 220nm × 380nm waveguide. The pulse is greatly compressed (and thus spectrally broadened) and shortly afterwards breaks up into a pulse chain.


Figure 3.8: Spectral output after 1.2mm (ζ = 2) of a 220nm × 380nm wire pumped with a 100fs 1.5μm pulse. Peaks powers of 10P0, 80P0 and 640P0 are used. Increasing the power increases the spectral range of the output.

Figure 3.9 shows the 640P0 output plotted as a FROG diagram. This technique (which is explained in more detail in appendix A.2) expands the signal into a two-dimensional image in which both the frequency and timing of optical features can be resolved. The features form an "S" shape across the diagram, which is due to the dependence of group velocity upon wavelength. The two ZDWs are (by definition) extremal points of the group velocity (with the 1.23μm wavelength being a local maximum and the 1.63μm wavelength being a local minimum), and so the curve changes direction at these points.

It should be noted that the chirp of a pulse (i.e. the difference in frequency between its leading and trailing edge) can be gauged by observing the angle at which its corresponding FROG feature lies on the diagram. A chirpless pulse will appear as an ellipse with axes parallel to those of time and frequency, but when chirp is present, these axes will be rotated. Dispersive pulses will steadily gain a chirp as they propagate along a waveguide, whilst solitons will not. There are several pulses in figure 3.9 which appear to have very little chirp (despite having travelled two dispersion lengths), and thus are probably solitons.


Figure 3.9: Output after 1.2mm (ζ = 2) of a 220nm × 380nm wire pumped with a 100fs 1.5μm pulse with a peak power of 640P0. The region between the dashed vertical lines is anomalously dispersive, and within this there are several solitons.
Role of Čerenkov radiation

If solitons are present, then Čerenkov radiation may also be present. To predict the resonances, equation 3.6 needs to be generalised for an arbitrary soliton frequency ωsol. It can be shown that this general form is [151]

|| q+ D (ωsol)+ (ω– ωsol) dD-|| = D(ω) dω ωsol
(3.8)

As before, the left hand side is the dispersion relation of the soliton. Matching this to the linear dispersion relation D(ω) gives the resonant frequency. This analysis is performed for a variety of spectral peaks in figure 3.10. It is probable that the peak at 1.45μm is a soliton which is emitting the radiation at 1.85μm.


Figure 3.10: Čerenkov analysis for the signal displayed in figure 3.9. A variety of spectral peaks in the anomalous regime (labelled from a to e) are taken, and equation 3.8 solved graphically by matching the linear dispersion relation (dashed line) to the nonlinear relations (solid lines). An estimated soliton wavenumber of q = 0.5 was used. It can be seen that peak b (at 1.45μm) matches up well to a peak in the normal regime (at 1.85μm), suggesting that Čerenkov radiation is present. Whilst the other peaks don’t match up, it is possible that more Čerenkov resonances are present, but that either the soliton or the radiation has been spectrally shifted.

3.2.2 Pumping at the zero dispersion wavelength

It is not necessary to start with a soliton in order to observe spectral broadening, as pumping at the zero dispersion wavelength will also give a continuum. (In fact, pumping at the ZDW is the classic way of realising continuum generation [4].) For this, a 220nm × 330nm waveguide was chosen, giving a relatively small (by SOI standards) normal GVD of –1416 psnm–1 km–1 at a 1.5μm pump wavelength. For 100fs pulses, this gives a dispersion length of 1.93mm. The dispersion relation of this geometry is given in figure 2.3. The initial condition was modulated by eiωτ, where the frequency difference ω = 0.6228 was chosen to shift the pump wavelength to the ZDW at 1.487μm. The damping coefficient was rescaled to εl = 0.05 to account for the longer dispersion length.

The output spectra for multiple input powers are shown in figure 3.11. At the highest (and again, roughly optimal) power of 640P0, a continuum with bandwidth 550nm was generated. (The meaning of P0 is slightly obscured here, as solitons are no longer possible. However the combination of variables is still mathematically valid.)


Figure 3.11: Spectral output after 3.86mm (ζ = 2) of pulses with input powers of 10P0, 80P0 and 640P0 fired at the zero dispersion wavelength of a 220nm × 330nm wire. The pump wavelength was tuned to 1.487μm to match the ZDW.

A FROG diagram for the 640P0 case is shown in figure 3.12. There are several pulses in the anomalous regime, where solitons may potentially be formed. However all of the corresponding FROG features are strongly rotated. Furthermore, the angle of rotation is roughly tangential to the "S"-shaped curve, which suggests that the features are merely dispersive pulses.


Figure 3.12: Result after 3.86mm (ζ = 2) of a pulse with input power 640P0 fired at the zero dispersion wavelength of 220nm × 330nm wire. The pump wavelength was tuned to 1.487μm to match the ZDW.

3.2.3 Energy saturation

As the energy of the input pulse is increased, the output energy is increased by successively smaller amounts (see figure 3.13). This is caused by two-photon absorption, which being a two-photon effect has a far greater impact upon the high energy pulses, thus reducing the output energies to a similar value. Such an effect is well documented [57, 168, 169, 170]. At higher energies still, the effect of the free charge carriers excited by the 2PA becomes significant. Being a three-photon-effect (requiring two photons to excite a carrier, and a third to participate in a scattering or absorption event) it only becomes important at very high intensities. The energy out versus energy in was calculated, as is shown in figure 3.13.


Figure 3.13: Energy saturation for 220nm × 330nm wire pumped at ZDW. (Scale assumes power unit is 1W.)

Notably, when free carriers are included, the output energy can actually decrease as the input energy is increased. This effect (which has been observed experimentally [57, 72, 169, 170]) can be explained through hysteresis, as the effect of the free charge carriers is not instantaneous. Consider two input pulses, one of low energy, and one of high energy: The two pulses will be reduced to a similar saturation energy by two photon absorption. However, the higher energy pulse will have induced many more charge carriers. As the charge carriers remain, they will continue to absorb light even after the saturation energy has been reached. When more carriers are present, this absorption will be greater, and may cause output energy to decrease with increasing input energy.

The effect of the free charge carriers can be compared to that of a resistor in which the resistance increases with temperature. Devices used as "resettable fuses" can work like this, and consist of carbon particles embedded in a polymer matrix. The temperature increase due to excessive current will cause the polymer to expand, increasing the distance between the particles and causing the resistance to greatly increase, thus allowing the fuse to "blow".

Consider the action of electrical pulses on the following electrical circuit:

If the heating and cooling were instantaneous, the resettable fuse would simply act as a nonlinear resistor. By analogy with the two photon absorption, a saturation effect would be observed, whereby increasing the peak voltage of the input pulses would give diminishing returns. However, the fuse could never blow, because once the voltage dropped, the resistance would instantaneously fall, and the remainder of the pulse could propagate as normal. It is only when hysteresis is present—by the fuse remaining hot—that the fuse can blow. This is analogous to the remaining free charge carriers persisting, and continuing to absorb light.

3.3 Comparison to experiment

In this section numerical modelling is compared to experimental data gained in colloboration with others. A microchip was fabricated at the University of Glasgow by Marco Gnan, Marc Sorel and Richard De-La-Rue. Optical experiments were then performed upon the chip at the University of Bath by Wei Ding, William Wadsworth and Jonathan Knight. The results of this collaboration have been published in Optics Express [54].


Figure 3.14: Schematic cross section of the 260nm × 480nm nanowire fabricated by Gnan, Sorel and De-La-Rue. The silica wire sits on top of a ledge of silica, due to overetching of the base and is covered with a layer of hydrogen silsesquioxane (HSQ) etching mask. Electron micrographs are shown in figure 1.2.

The chip consisted of SOI nanowires 260nm in height and 480nm wide on a base of silica. (An overview of the chip’s fabrication is given in appendix B.1.) Electron micrographs of the wires are shown in figure 1.2, whilst a schematic cross section is given in figure 3.14. The wires were topped with a hydrogen silsesquioxane (HSQ) etching mask with a thickness of 100nm and a refractive index of about 1.35. They also sat upon a slight pedestal of silica (about 20nm in height), due to overetching of the base. The chip was 15mm in length, which is substantially longer than that used by other groups. (A more typical value is 5mm [57, 63].)

3.3.1 Measuring the linear dispersion

The initial work was concerned with the linear dispersion of the waveguides. The experiments yielded values of the group index at multiple wavelengths (see appendix B.1 for details). From this, the GVD could be calculated by fitting the values to a polynomial in ω, and then differentiating the polynomial. Due to great expense of taking a high-accuracy group velocity measurement, only nine experimental datapoints were available. This small number can cause problems with polynomial fitting, due to the effect described in section 2.1.2. (This problem becomes greater still when calculating the third order dispersion, as the polynomial has to be differentiated twice, amplifying the fitting pathologies further. It is therefore not surprising that some groups have attempted to calculate the HOD indirectly, as was mentioned in section 3.1.3.)


Figure 3.15: a) Group velocities measured at various wavelengths (gained experimentally by Ding, Wadsworth and Knight, as described in appendix B.2) shown by circles. The dashed line shows the theoretical values, derived using the techniques described in section 2.1.2. b) Dispersion parameter (given in units of psnm–1 km–1) The solid lines show the empirical values (derived from the group velocities by curve fitting with third, fourth and fifth order polynomials). The dashed line gives the theoretical values.

The group velocities and the derived GVDs are plotted in figure 3.15. By taking all forms of error into account, Ding, Wadsworth and Knight were able to extract dipsersion coefficients (expanded about 1.5μm) of β2 = –2.31 ± 0.04ps2m–1 and β3 = 0.0119 ± 0.0009ps3m–1. The dispersion fitting was done with third, fourth and fifth order polynomials. These fits all yielded values of similar magnitude (with the differences contributing to the above error bars), and so the polynomial problem was not fatal.

This dispersion relation, however, differs substantially from that calculated using the techniques described in section 2.1.2. (This is also plotted in figure 3.15.) The coefficients obtained were β2 = –3.06ps2m–1 and β3 = 0.0133ps3m–1, which are well outside the error bars of the empirical values. A number of explanations for this discrepancy are possible:

This discrepancy is slightly disconcerting when it comes to the experimental realisation of the other waveguide geometries assumed throughout this report. However it should be noted that the ultimate aim is not to make precise predictions for future experiments, but to gain qualitative predictions for systems with physically realistic parameters. Even if the parameters are not precise, it is safe to assume that they are reasonable.

This experimentally derived dispersion relation was used for further modelling. The scaled dispersion coefficients (for a dispersion length of 1.05mm) are p2 = –0.5 and p3 = 0.0151.

3.3.2 Nonlinear propagation and parameter fitting

The output spectra for waveguides pumped with 100fs FWHM 1.5μm pulses were measured at a range of input powers, as is shown in figure 3.16. (These experiments, performed with 800nm pulses from a Titanium Sapphire mode-locked laser which were down-converted to 1500nm by a β-barium-borate optical parametric amplifier, are described in appendix B.3.) The total power output was also measured, as is shown in figure 3.17. As the power is increased, the spectral width of the output broadens. There is also a saturation effect, whereby increased input power gives diminishing returns with respect to the output power.


Figure 3.16: Experimental output spectra (obtained by Ding, Wadsworth and Knight) compared with best-fit numerical results. The dotted line gives modelling without the effect of 3PA or FCC interactions. The dashed line includes the effect of FCC interactions, whilst the solid line (which gives the best fit) includes the effect of 3PA.


Figure 3.17: The experimental output powers (obtained by Ding, Wadsworth and Knight) for various input powers are given by the black squares. The dotted line gives modelling without the effect of 3PA or FCC interactions. The dashed line includes the effect of FCC interactions, whilst the solid line (which gives the best fit) includes the effect of 3PA.

The powers shown in figures 3.16 and 3.17 are not the peak powers of the pulses entering the waveguide, but the mean power of the pulse train fired at the waveguide. It can be shown (working in real units) that a sech-like pulse with peak power Pmax and FWHM duration TFWHM has a total energy PmaxTFWHM/ln( √-) 1+ 2. It follows that the peak power in the waveguide is given in terms of the mean incident power Pinc by

( √-) ν-ln-1-+--2- Pmax = TFWHMfrep Pinc
(3.9)

where frep is the pulse repetition rate, and ν is the fraction of the incident light coupled into the waveguide. These had values of frep = 250KHz and ν = 0.5%, yielding an all-inclusive conversion relation of Pmax = 1.77 × 105 Pinc.

Numerical modelling of the system was performed, as is shown in figures 3.16 and 3.17. The comparison of theory to experiment was made by treating the soliton threshold P0 and the scaled 2PA coefficient ε2pa as being free parameters, and then using them as fitting parameters to gain the best match. The first of these parameters doesn’t actually affect the numerical results, as it is scaled out of the model. Therefore, fitting is a simple matter of matching the experimental and theoretical spectra, and comparing their power values. The ε2pa parameter is more important, as it directly enters the model, and different values will give qualitatively different results. The model also included linear absorption, which was measured to be 3.4 dB cm–1.

In addition to free charge carriers, the effect of 3 photon absorption was also considered (and the coefficient ε3pa treated as a third fitting parameter). As mentioned in section 2.3.1, this effect is known when the photon energy is less than twice silicon’s ∼ 1.1eV indirect bandgap (precluding 2PA) [117], but here we consider it alongside 2PA.

The best fitting was found to occur roughly when the soliton threshold corresponded to an incident power of Pinc ≃ 2μW, giving an estimate for the soliton threshold of P0 = 0.36W, and thus (from equation 2.64) a nonlinear coefficient of γ = 2000W–1m–1. The best fit to the 2PA coefficient was found to be ε2pa = 0.1.

For higher order absorption, the best match to experiment was actually found without the effect of FCC included, and a 3PA coefficient of ε3pa = 0.05. However, it should be noted that 3PA and FCC absorption are very similar phenomena, as both involve three photons. (The difference between them is that FCC absorption exhibits hysteresis, whilst 3PA is instantaneous.) Therefore, the model may still be accounting for the carriers indirectly.

A notable problem when matching theory to experiment was the lack of information on the chirp of the laser system. Whilst the pulse’s duration was known to be 100fs, its bandwidth was slightly wider than that of a transform limited pulse, suggesting that it was chirped. (This input spectrum is given in figure 3.16.) There was no data, however, to suggest whether the chirp was frequency increasing, frequency decreasing, or something more complicated. Therefore, for the purpose of modelling, the best approximation to this unknown pulse makeup was simply a transform limited pulse. The effect of this can be seen when comparing the numerical and experimental output spectra, as whilst the positions of the spectral peaks agree, they appear broader in the experimental case.

3.3.3 Determining if solitons are present

It is probable that soliton propagation is present, and (from the above parameter fitting) the soliton threshold corresponds to the 2μW mean input power. A useful test for determining if a pulse is or isn’t a soliton involves the conservation of the parameter

√ ----- S = TFWHM Pmax
(3.10)

which is known as the soliton area. It can be shown that for an ideal sech-like soliton, the parameter is always given by S0 = 2ln( √ -) 1+ 2√ ------ |β2|/γ, where β2 is the GVD, and γ is the Kerr coefficient. This is independent of the soliton duration, thus making it a property of the waveguide only. The quantity is not fixed for pulses in general, and so provides a useful means of identifying solitons. Whilst the presence of higher order dispersion and damping will affect S0 slightly (as the sech-like soliton will no longer be an exact solution of the equation of motion) we can still expect it to remain constant over the propagation distance. Therefore, if we see a pulse with a constant value of S (similar to S0) we have strong evidence that it is a soliton. (We will return to this technique for detecting solitons in multi-wire systems, in sections 4.3.3, 4.4.3 and 5.2.3.)

As the experiments provided no information as to the duration of the output pulses, this analysis must rely upon numerical data. However, now that the model has been fine-tuned with experimentally derived data, this can be done to a reasonable degree of accuracy. Figure 3.18 shows S plotted for a range of input powers. For comparison, the temporal and spectral widths are also plotted. This plot shows a plateau with S ≃ S0, over the input power range 1μW to 3μW. This region coincides with a strong supression of temporal pulse broadening, suggesting that soliton-like effects are at work. The soliton area remains roughly constant over propagation distance (figure 3.19), with an input power of 2μW providing a good fit, and 1.7μW providing the optimum match.


Figure 3.18: a) Temporal duration as a function of input power. b) Spectral width as a function of input power. c) Soliton area parameter S as function of input power. The dashed line gives the ideal value S0. (The line styles are as defined in figures 3.16 and 3.17.)


Figure 3.19: Soliton area parameter S plotted over distance, for a variety of input powers. The dashed line gives the ideal value S0. Powers of 1.7μW and 2μW show a roughly conserved value similar to S0, suggesting that solitons are present.

The pulse at 2μW is a promising candidate for a soliton. Temporal evolution over distance (shown in figure 3.20) supports this hypothesis. Whilst temporal broadening does happen, this is over 14 dispersion lengths, and is much less than that seen without nonlinearity.


Figure 3.20: Temporal profile plotted over distance for a 100fs pulse in a 260nm × 480nm SOI waveguide. With nonlinearity (left) the pulse broadening is greatly suppressed in comparison to that without nonlinearity (right).

As a final test for the existence of solitons, the evolution of pulse pairs was considered, as is shown in figure 3.21. If two solitons are in close proximity, they will start to attract each other [8], which indeed is seen. In the ideal NLS model, we would expect the solitons to remain independent. Due to damping, however, we lose the property of two solitons being able to pass through one another with only a phase shift, and so the two solitons fuse into one. Integration of the equations of motion beyond the physical length of the waveguide was also performed, which revealed that the fused pulse remained in one piece.

Conversely, if a half-cycle phase-shift is introduced between the solitons, the attraction will be replaced by repulsion [8]. Again, this is seen in figure 3.21.


Figure 3.21: Evolution of pair of 100fs pulses with 340fs peak-to-peak separation in a 260nm × 480nm SOI waveguide. When the solitons are in phase (left), attraction occurs, and they seem to merge. When they are half a cycle out of phase (right) repulsion occurs.

In summary, the existence of solitons is supported by a range of tests. Firstly, pulse broadening is greatly suppressed, and this occurs a power corresponding to the soliton threshold. Secondly, the soliton area parameter S is close to the predicted value, and is roughly conserved over the length of propagation. Thirdly, the pulses exhibit the attractive and repulsive behaviour seen by soliton pairs.

3.4 Continuous wave propagation and modulational instability

Modulational instability (MI) occurs when small deviations from the waveform are reinforced by nonlinearity, generating spectral sidebands and causing the eventual collapse of the waveform into a chain of pulses [8]. It has been both predicted [171] and observed [172] in SOI waveguides.

The distinctive spectral pattern formed by modulational instability can be predicted analytically. As a starting point, we take a continuous wave (CW) solution of equation 2.85

√-- iPζ E = P e
(3.11)

where P is a dimensionless power scaling coefficient, and we have ignored the effect of damping. (This solution is not a constant, as it would be for the envelope in a linear system, due to the intensity dependence of the refractive index. Therefore, the corresponding change in wavenumber causes the phase of the solution to be modulated in ζ. It should be noted, however, that P approaches zero faster than √P-, and so at low powers, equation 3.11 tends towards being a constant.) We then perturb the CW solution as

(√ -- ) E = P +ε eiPζ
(3.12)

where the perturbation term ε(ζ,t) shares the same phase modulation as P. Substituting this into the equation 2.85 (again without damping) gives

( ) ∂ε-– iDˆε = i P ε+ Pε∗ + √P-ε2 + 2√P-εε∗ + ε2ε∗ ∂ζ
(3.13)

As we are only considering the early stages of the modulational instability, the perturbation will be small, and so we can reject all the higher order terms in ε to give

∂ε– iˆD ε = iP (ε+ ε∗) ∂ζ
(3.14)

As we have terms in both ε, and its conjugate, it is productive to use a trial function with a similar structure, namely the superposition of a wave, and its conjugated equivalent. Therefore we choose

ikmζ–iωmτ –ikmζ+iωmτ ε = ε1e + ε2e
(3.15)

where the relative wavenumber km has yet to be determined. Substituting in (and rearranging) gives

[ε1D (ωm )+ ε1P + ε∗2P – ε1k]eikmζ–iωmτ (3.16) + [ε D (– ω ) +ε P + ε∗P + ε k]e–ikmζ+iωmτ = 0 2 m 2 1 2
This is a valid solution, so long as the coefficients in front of the complex exponential terms are zero. This condition leads to a pair of coupled equations, which can be solved in km to provide a dispersion relation (giving the phase-matching condition required for MI to occur) of the form
√----------------------- km = Dodd(ωm) ± D2even(ωm )+ 2PDeven(ωm)
(3.17)

where the dispersion operator has been split into even and odd components as D = Deven + Dodd. These are defined as

D (ω) ≡ 1[D (ω )+ D (– ω)] (3.18) even 2 D (ω) ≡ 1[D (ω )– D (– ω)] (3.19) odd 2
so that Deven will contain only even powers of ω, and Dodd will contain only odd powers. At frequencies where km has a non-zero imaginary component, exponential growth will occur. Defining a growth rate parameter g ≡ 2ℑ(km) (where the factor of 2 converts the growth rate in amplitude to the growth rate in power) gives
( √ ---2--------------------- 2 |{ 2 – D even(ωm )– 2PDeven(ωm) ; – D even(ωm)– 2P Deven (ωm ) ≥ 0 g(ωm) = | ( 0 ; – D2even(ωm)– 2P Deven (ωm ) < 0
(3.20)

It is apparent from the –Deven2(ωm) – 2PDeven(ωm ) > 0 condition that modulational instability will only occur when Deven is negative. This condition generally requires requires anomalous GVD, but it is sometimes possible for far-detuned spectral lines (as mentioned below) to be generated in a normally dispersive system. It is also apparent from the symmetry of equation 3.20 that the spectrum will be symmetric in frequency, so that each radiation peak will have a complement peak on the other side of the pump frequency.

With anomalous GVD, but without higher order dispersion, this reduces to

( |{ |ωm|√4P--–-ω2m- ; 4P ≥ ω2m g(ω ) = m |( 2 0 ; 4P < ω m
(3.21)

This describes two spectral side-lobes, with maxima at

ω = ±√2P-- m
(3.22)

With higher order dispersion, it is typical to see a extra pair of modulational instability peaks far away from the pump [173, 174, 175]. From equation 3.20, it can be seen that the maxima of these peaks coincide with the maxima of x(ω) ≡–Deven2(ωm) – 2PDeven(ωm ). These maxima occur when dx/dω = 0, and so we can derive a necessary (but insufficient) condition for the positions of the MI peaks

Deven(ωm)+ P = 0
(3.23)

This equation can be solved numerically, after which the pathological values in which dx/dω = 0 doesn’t correspond to a positively-valued maximum are removed.

Modelling a real device

A 220nm × 380nm SOI geometry was chosen, due its relatively large HOD, and thus the possibility of seeing the extra pair of higher-order sidebands. The growth rate spectrum at multiple wavelengths is plotted in figure 3.22 and the positions of the growth rate maxima are plotted in figure 3.23. These show that for a pump wavelength of 1.5μm, four sidebands should be seen.


Figure 3.22: Modulational instability sideband growth rate g, plotted as a function of both the pump wavelength and the sideband wavelength for a 220nm × 380nm SOI waveguide. The solid background colour corresponds to a growth rate of zero. Anomalous GVD (on the pump axis) is in between the dashed vertical lines. The dotted line corresponds to ωm = 0 (i.e. the pump and sideband wavelengths being the same).


Figure 3.23: As for figure 3.22, but with the growth rate maxima plotted. A pump wavelength of 1.5μm (represented by the grey bar) should give four sideband frequencies.

Modelling was performed using 10ps rectangular pulses with power 10P0. Random noise (with a relative amplitude of 0.5%) was also added to help initiate the instability. The results of the simulation (together with an overlay of the predicted gain) is given in figure 3.24. The model included 2PA (ε2pa = 0.1) and linear absorption (εl = 0.01). Free charge carriers were accounted for by assuming a carrier absorption cross section of εfc = 10–3(1 + 7.5i). A FROG diagram for the resulting signal can be seen in figure 3.25.

This scheme is notably different to that considered in the literature. Previous work with silicon [85, 171, 172] assumes a scheme whereby the instability is seeded with light at one of the predicted gain wavelengths, which is then amplified. Here, the effect of a pulse on its own is considered. Whilst random noise is added in order to initiate the instability, this lacks any preferred frequency component.


Figure 3.24: Spectral output of a 220nm × 380nm SOI wire pumped with a 10ps pulse with power 10P0. After 1.2mm, spectral sidebands due to modulational instability are present. a) Model includes linear absorption and 2PA, but not FCC effects. The predicted gain is overlaid, and was calculated for P = 4P0 to account for the effect of damping. b) Model also includes FCC effects. The predicted gain is calculated for P = 2.5P0 to account for further damping.


Figure 3.25: FROG diagram for the signal (inclusive of free carrier interactions) in figure 3.24. The colouration is logarithmic, over a 40dB range. The bright vertical feature is the pump, whilst the MI sidebands are visible as the four patches in the upper part of the image. It can be seen that these have a far shorter duration (of the order 1ps) than the pump. The peak at 1.89μm peak precedes its spectral counterpart at 1.24μm, due to group velocity difference causing them to separate.

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