|
|
[next] [tail] [table of contents]
This report is fundamentally about Nonlinear Optics. This is the study of how light behaves at high intensities, and how to exploit this behaviour to gain extreme control over light. Nonlinearity is important because it allows light to interact with light. In a system where electric polarisation is simply proportional to the electric field, waveforms can be superposed without any change to the dynamics of the individual components; if two waveforms meet, they will simply pass through one another. Consequently, if you pass light with a given set of frequency components through a linear medium, you can only ever get those frequencies out. When optical nonlinearity is present, however, waveforms can mix or self-interact to produce an entirely new waveform.
Examples of nonlinear phenomena include second harmonic generation [1], in which photons are essentially combined to give photons with twice the energy; four wave mixing in which three frequency components interact to produce [2] or amplify [3] a fourth; supercontinuum generation [4, 5], in which light with a very broad frequency spectrum can be generated from a narrow-spectrum pulse; electromagnetically induced transparency [6] in which absorption at a particular frequency is eliminated by optically inducing destructive interference between the corresponding quantum states; stimulated Raman scattering [7] in which photons are swapped for photons with shifted energy, with attendant excitation or relaxation of the medium; and modulational instability [8] in which small deviations from a waveform are reinforced by nonlinearity, causing it to break up into a chain of pulses. Nonlinearity can also be used to create an environment for exotic linear phenomena. These include "slow light" in which light can be drastically slowed, or even halted and "fast light" in which the group velocity of light exceeds the speed of light in vacuum, or even becomes negative, causing a pulse to move towards the source [9].
An extremely important nonlinear phenomenon—and the subject of this report—is the soliton.
A soliton is a localised and self-sustaining wave motion that can arise in many non-linear systems. They contain a range of frequency components, and are bound together by the balance between the dispersion of these frequency components and the nonlinearity of the system.
The archetypal example of a soliton is the "Wave of Translation" observed by John Scott Russell in 1834. When a boat moving along a canal suddenly stopped, the surge of water at its bow continued to move, and "assum[ed] the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed" [10]. This description highlights an extremely important feature of solitons, namely their ability to retain their shape as they propagate, despite the inevitable presence of noise and damping. This feature is one of the principal reasons for interest in solitons. In fact, the very name "soliton" (as opposed to the more general term, "solitary wave") reflects the fact that solitons can be regarded as discrete particles, rather than waves.
Russell’s wave of translation is more specifically a temporal soliton. That is, at any point in time, the soliton only occupies a finite region of space along the direction of propagation. The name derives from the fact that an observer at a given point along the direction of propagation will only see the soliton for a limited time duration. Temporal solitons are also seen in magnetostatic waves [11], sound waves [12] and optical fibres [13]. They have also been proposed as a mechanism for many natural phenomena, including tsunamis [14] and signal propagation through neurons [15].
In contrast to the temporal soliton is the spatial soliton, in which the wave motion is confined in one or more directions transverse to the direction of propagation. (As with temporal solitons, the separation of frequency components is being nonlinearly suppressed, but this frequency is spatial rather than temporal.) A prime example of this is the optical self trapping described below, whereby diffractive spreading of a beam of light is cancelled by nonlinearity. Spatial solitons have also been observed in magnetic materials [16]. A relative of the spatial soliton is the edge soliton, in which the wave is self-confined against the edge of the propagation medium [17]. Solitons can fulfil the requirements for both spatial and temporal solitons [18], in which case they are called spatiotemporal solitons.
A concept related to the soliton is the topological soliton or topological defect. This has particle like properties which result from the medium being twisted or distorted in such a way as for it to be impossible to deform the defect into a non-defect through a continuous series of intermediate states. Instead, the solutions are split into topologically distinct families, which can only be switched between by the intervention of another physical effect. Examples include structural defects in liquid crystals [19], Falaco cells (vortex pairs that can occur on the surface of liquids) [20] and vortices in liquid helium [21]. Of particular interest is the possibility that spacetime itself may contain topological defects known as cosmic strings [22]. Topological solitons can be said to carry a topological charge which is a conserved number, describing the exact form of the defect. Solitons carrying such a charge are considered in chapter 6.
In some systems, it is possible for an absence of energy to behave as a soliton. These dark solitons or antisolitons consist of a region of reduced amplitude set against a continuous wave background. (In this context, a non-dark soliton is known as a bright soliton.) Like bright solitons, dark solitons can be both temporal and spatial in nature [23].
The resemblance of solitons to particles suggests that the converse may be true, and that elementary particles such as electrons are themselves solitons [24]. (There is as yet no experimental evidence to support this theory, and so it remains highly speculative.)
In the field of optics, temporal solitons were first observed in silica fibres by Linn Mollenauer and Roger Stolen in 1980 [13]. These solitons are sustained by the Kerr effect, whereby the refractive index increases in proportion to optical intensity. This causes self phase modulation in which the leading edge of the pulse is red-shifted, whilst the trailing edge is blueshifted. In the presence of anomalous group velocity dispersion (in which blue light has a higher group velocity than red light, as opposed to the normal group velocity dispersion which is more commonly encountered) the dispersion and the nonlinearity can balance to form a soliton. Temporal solitons in single waveguides are studied in chapter 3, whilst temporal solitons in coupled waveguide arrays are considered in chapter 4.
Spatial solitons are seen in the form of self trapping of optical beams [25]. This occurs when beam spreading due to diffraction is counteracted by self focussing, in which optically-induced refractive index increase causes the beam to converge. This effect can also be thought of as the light creating its own waveguide. Spatial solitons are not studied directly in this report, but attention is given to a related phenomenon known as the spatial discrete soliton. These occur in arrays of coupled waveguides, whereby the natural inter-waveguide coupling is suppressed by nonlinearity. Spatial discrete solitons were first proposed by Demetrios Christodoulides and Richard Joseph in 1988 [26] and were first realised experimentally in 1998 [27].
Optical spatiotemporal solitons are known as light bullets [18], where the light is both localised in the direction of propagation, and self trapped in the transverse direction. Edge bullets are also possible, whereby the light is trapped against the edge of the medium [17, 28]. In chapter 5 spatiotemporal discrete solitons, in which the properties of the temporal soliton and spatial discrete solitons are combined [17, 29, 30] are considered. Light bullets (and spatiotemporal solitons in general) have proved highly elusive, with only a few examples (e.g. [31, 32, 33]) seen to date. Difficulties include the need to balance the intrinsic length scales of dispersion and diffraction [34] (so that the powers needed to balance the two are equivalent), and the need for sufficiently strong nonlinearity [35] to reduce the power requirements to practical levels.
The Kerr effect is not the only form of optical nonlinearity capable of producing optical solitons. In materials lacking inversion symmetry (such as lithium niobate or β-barium-borate) electric polarisation as a function of electric field can have a quadratic component (as opposed to the cubic nonlinearity of the Kerr effect), and this can also be used to support solitons [36, 37]. Raman-effect solitons have been observed in a variety of forms; these include hybrid bright-dark temporal solitons in which a bright-soliton of Stokes-shifted light co-moves with a dark-soliton in the optical pump [38], and more conventional bright solitons [39] (which are considered in chapter 6). Solitons have also been observed in systems where stimulated Brillouin scattering (the result of photon-phonon interactions [40]) is the source of the nonlinearity [41]. It has even been suggested that spatial solitons supported by the fundamental nonlinearity of the vacuum may exist [42, 43]. (These, however, require intensities beyond currently available laser technology.)
Determining if a waveform is or isn’t a soliton is a non-trivial problem. It becomes more complicated still in light of the fact that the mathematical models yielding exact soliton solutions are only approximations to physical reality. As such, the solitons that occur in nature are sometimes referred to as quasi-solitons. These, however, must retain several important characteristics. Firstly, although deviations from an exact model will generally cause a quasi-soliton’s shape to change as it propagates, there must still be strong supression of this shape change. For example, a temporal quasi-soliton in a optical fibre may lengthen slightly as it propagates, but this increase should be far smaller than that seen without nonlinearity. Throughout this report, the difference between propagation with and without nonlinearity is used to demonstrate quasi-soliton formation.
Another crucial feature of any soliton (or quasi soliton) is that the localisation must be intrinsic to the bulk properties of the propagation medium, and not imposed by any artificial constraints. Mathematically speaking, the soliton should be a intrinsic property of the equations of motion of the propagation medium, rather than something that is forced by boundary conditions. For example, a pulse of monochromatic light being fired through a linear medium would not constitute a temporal soliton, as the localisation would be purely a result of the initial pulse being finite in time. Similarly, light travelling through an optical fibre would not – under ordinary circumstances – constitute a spatial soliton, as the localisation would be simply a result of the light undergoing total internal reflection at the fibre’s boundaries. (For a topological defect, this distinction is more complicated, as the separation between the topological families is generally the result of boundary conditions. However, these conditions do not push the solution into one particular family and so they don’t force a soliton to be present.) In chapter 5 this distinction between self-localisation and boundary condition localisation will become important, as spatial localisation in finite arrays of waveguides is considered.
Some sources (e.g. [44]) draw an explicit distinction between the terms "soliton" and "solitary wave", by stating that in addition to the requirement of self-sustaining localisation, a soliton must also be able to pass through another soliton with no ultimate effect other than a phase shift. However, in the context of nonlinear optics this requirement is often inappropriate. For example, optical phenomena which otherwise resemble solitons are widely observed to fuse, split or annihilate on collision [45]. (This is related to the above distinction between solitons and quasisolitons, as whilst the nonlinear Schrödinger equation derived in chapter 2 supports solitons with the phase-shift-only property, this is broken by the modifications that must be made in order to accurately describe a real system [23].) As the particle-like behaviour remains, it is unhelpful to exclude these objects as being solitons. Therefore, the phase-shift-only requirement is commonly disregarded by many sources. There does, however, remain an informal requirement that something interesting should happen on collision. In section 3.3.3 pulse fusion is used as a test to see if quasi-solitons are present.
Another test arises from the fact that many soliton solutions possess non-trivial conservation laws, in which a particular combination of variables should remain constant. In sections 3.3.3 and 5.2.3 such laws are used to gauge quasi-soliton formation.
For clarity, the term "quasi-soliton" will be dropped throughout the rest of this report.
Silicon on insulator (SOI) devices consist of nanoscale silicon waveguides sitting on top of an insulator base (which is almost always silica). These waveguides are commonly referred to as photonic nanowires (or just "wires"), where "photonics" is a general term used to describe the emission, manipulation and detection of photons. They typically operate at wavelengths in the vicinity of the 1.55μm infrared telecom band (which corresponds to the maximum transparency of silica fibre). A schematic of the type of SOI device considered in this report is given in figure 1.1, whilst an electron micrograph is given in figure 1.2.
Silicon has an extremely high refractive index (approximately 3.5 in the band of interest [46]), allowing for tight-confinement of light. This confinement can be spectacular, with light having a wavelength greater than a micron being confined to waveguides with transverse dimensions of only a few hundred nanometres. The strong dielectric boundary effects resulting from this confinement allows for dispersion tailoring, in which the dispersion relation can be modified by altering the geometry of the waveguide. In particular, the group velocity dispersion (GVD) can be made anomalous [47, 48]. This allows for solitons, despite silicon’s strongly normal GVD at these wavelengths. This dispersion can be enormous, with values three orders of magnitude greater than that in silica fibres attainable [49]. Silicon’s high refractive index can also be used to trap light in a gap between wires by total internal reflection, in an arrangement known as a slot waveguide [50, 51, 52, 53].
|
|
|
|
Silicon has a large Kerr nonlinearity [55, 56], which is greatly enhanced by the tight confinement that silicon nanowires give [57]. As with dispersion, the nonlinear coefficient can exceed that of silica fibres by three orders of magnitude [58]. Furthermore, this nonlinearity is ultrafast [58], so that in comparison to the frequency of the light, it responds effectively instantaneously to the changing electric field. These effects allow for temporal solitons [48, 59, 60, 61] with record-breakingly small peak powers (for sub-picosecond optical pulses) of only a few watts. [54, 62, 63, 64].
Silicon on insulator devices have a wide range of applications [59, 65, 66, 67]. One such example is the ring resonator [68, 69] which can be used as an optical delay line in optical signal processing [70, 71]. On-chip supercontinuum source are also possible [62, 64, 72]. Raman amplifiers are another application [73, 74, 75], with demonstration devices having been constructed [76, 77, 78, 79]. Raman lasers have been fabricated [80, 81], as have wavelength converters [82]. Devices utilising four wave mixing are also of interest [83, 84], with both amplifiers [85] and wavelength converters [86] having been demonstrated. Signal modulation is another area of importance [87, 88, 89, 90], as is signal switching [65, 91] and signal detection [92].
Ultimately, silicon on insulator promises to have a major impact upon future technology. A primary example is the realisation of optical data lines within microchips [93], which would greatly improve the performance of complex integrated circuits by allowing different regions to communicate at greater speed [94]. The ability to place optical components on a silicon chip would also greatly benefit the construction of optical signal transceivers, as all the functions could be performed by a single chip, rather than multiple components [95]. Another potential application is all-optical signal routing, in which data packets in telecommunication networks are directly switched in optical form (rather than being detected, processed in electrical form, and then reemitted).
In this report, silicon on insulator devices are considered as on-chip laboratories with which to investigate a wide range of nonlinear optical phenomena. A theoretical approach is used, but close links to experimental reality are maintained throughout.
In chapter 2 a model of optical propagation through SOI waveguides is derived, taking into account a range of physical effects, and (so far as possible) using physical parameters based upon experimental data. This includes the material dispersion of the silicon and silica, waveguide dispersion, linear absorption due to light escaping the waveguide, nonlinear absorption, and the effect of free charge carriers on optical transmission.
In chapter 3, light propagation through a single wire is modelled, revealing a variety of physical effects. A comparison to third-party experimental data is made, providing strong evidence for the existence of solitons in silicon wires.
In chapters 4 and 5 multiwire arrays are modelled. In chapter 4 it is shown that inter-wire diffraction is intimately linked to dispersion, and that by exploiting this it is possible to see temporal solitons in arrays of normally dispersive wires. In chapter 5, silicon on insulator is shown to be an excellent medium for realising spatiotemporal solitons. A distinctive pattern of radiation emitted by these "optical bullets" is predicted.
In chapter 6, solitons supported by the Raman effect (rather than Kerr nonlinearity) are considered. A new class of soliton solutions are derived, which have the novel property of existing even when the set of frequency components comprising the soliton are phase-mismatched.
A summary of original findings and published works is given in chapter 7.
[next] [front] [table of contents]
