Solitons

Summer research project by Caleb Todd

Last summer, I underwent a research project in the department of physics at the University of Auckland. One of the department's most active fields of research is optics, and my supervisor, Associate Professor Miro Erkintalo, specialises in nonlinear photonics, which is the study of high-intensity light. A host of rich dynamics and structures can be observed in this regime; it balances theoretical interest and practical applications and has something which everyone can engage with. Since the mathematics that underpins nonlinear light maps closely to other physical systems, it also provides a convenient testing ground for the behaviour of phenomena that manifest in all fields of physics. One such phenomenon is the soliton.

Solitons are localised pulses that propagate without changing shape. A ubiquitous wave phenomenon, they were first identified by John Scott Russell when he noticed that water displaced by a boat in a canal continued to move through the channel at a constant speed and without flattening or otherwise dissipating. This is in contrast to ordinary waves, which broaden, narrow, or break, given time. For nearly a century, this discovery's importance was not fully appreciated; it was a curiosity with little to no use. However, solitons have become a hot topic in recent years and one of their uses won the 2005 Nobel prize in physics.

The solitons we will consider are not pulses of water, but light. If you take a stretch of optical fibre and join its ends to form a loop, you have what is known as a fibre ring resonator. Any light you send in will circulate for a long time before it is lost. If the light within the resonator is sufficiently intense, it can experience nonlinear effects that are not present with low-intensity light. For our purposes, we only need to know about one of these effects: the nonlinear refractive index.

The refractive index of a material determines the speed at which light propagates through it. A Nonlinear refractive index refers to a material's tendency to change its refractive index as the light's intensity changes. Pulses of light are more intense where their size is greatest (the peak of the pulse). So, across the profile (i.e. shape) of the pulse, the refractive index of the material will be changing. This intensity-dependent refractive index, also known as the Kerr effect, causes nonlinear self-focusing, where the pulse is contracted due to the variation in speed across its profile. Usually, the wavelength of a pulse is constant throughout, but the Kerr effect also shifts the pulse's leading edge towards higher (blue) frequencies and the trailing edge towards lower (red) frequencies. This is depicted in the graphic on the next page.

Image made by Emmanuel Boutet Licence: CC BY-SA 3.0

A double balance maintains a soliton's constant shape. On the one hand, driving must offset lost energy: a laser will continually send light into the resonator to compensate for the light which escapes. On the other hand, the width/shape of the soliton arises through a balance between nonlinear self-focusing and dispersive spread.

Dispersion refers to the way the speed of a wave changes as the frequency of the wave changes. If higher frequencies move slower, the material is called ‘normally dispersive’, and if lower frequencies move slower, it is called ‘anomalously dispersive’. In the anomalous dispersion regime, the leading (blue-shifted) edge of the pulse will move faster than the trailing (red-shifted) edge, causing the pulse to spread out. This dispersive spread compensates for nonlinear self-focusing, and the overall width of the soliton remains constant.

Solitons formed by this double balance in fibre ring resonators are known as Kerr cavity solitons. They have been studied extensively by research groups around the world, including the nonlinear photonics group here in the University of Auckland's physics department. One reason they have garnered so much interest is their use in generating optical frequency combs (light made up of a series of equally spaced frequencies). These frequency combs, whose development was awarded the Nobel prize in 2005, are used as high-precision tools in spectroscopy, optical clocks, metrology, and GPS technology.

One of the severe limitations, though, on solitons is the requirement to operate in the anomalously dispersive regime. Referring back to the double balance, the dispersive spread can only compensate the nonlinear self-focusing if the blue-shifted leading edge of the pulse moves faster than the red-shifted trailing edge. If we are in the normal dispersion regime, the situation is reversed, and the dispersion helps the self-focusing rather than hinders it. There is no balance. The issue is that optical fibre is only anomalously dispersive at specific wavelengths. For example, light in the visible spectrum is normally dispersive in optical fibre. So, if you want an optical frequency comb at wavelengths of light in the visible region, you won't be able to use a fibre ring resonator soliton. There is a way around this, though, which was the focus of my research project.

Kerr cavity solitons can be made to exist in the normal dispersion regime by exploiting higher-order dispersive effects. Whether we are in the normal or anomalous dispersion regime is determined by the second-order dispersion coefficient. That's the coefficient on the second-derivative term in the dispersion's Taylor series if that means anything to you*. In general, the coefficients at all orders will modify the soliton's behaviour, but the effects of coefficients beyond the second-order are usually inconsequentially small. However, it is possible to operate at wavelengths where the third-order coefficient is as significant or even more so than the second-order coefficient. This is useful because the effect of third-order dispersion on a soliton is to shift its centre frequency away from the frequency of the driving laser. In particular, with sufficiently strong third-order dispersion, the soliton can be pumped in the normal regime but have an anomalous centre frequency. This allows the double balance to be restored, even with normally dispersive pumping.

There are complications, though. The dispersion parameters are not the only factors in the behaviour of light within fibre ring resonators. Two other central parameters are the driving power (which we will call X) and the detuning of the driving frequency from the resonator's nearest resonant frequency (which we will call Δ). Solitons do not exist at every pair of X and Δ, and these parameters also determine their shape. In particular, larger values of Δ give rise to taller, narrower solitons, which is favourable in producing optical frequency combs because a narrower pulse is comprised of a broader frequency profile. Ideally, we would be able to increase Δ arbitrarily, but there is an upper limit in Δ of soliton existence at any given driving power.

When third-order dispersion can be neglected, there are well-understood bounds on soliton existence. An approximate upper limit in Δ can be determined for any given value of X. My project was to probe the existence range of solitons when the third-order dispersion cannot be neglected.

A plot depicting the upper and lower limits of the detunings at which solitons may exist. The purple and red curves are for when third-order dispersion is not accounted for. The blue and orange points are the upper and lower limits which I found for a given, non-negligible strength of third-order of dispersion. Linear fits for my data have been presented to guide the eye.

To do this requires us to turn to computers. The canonical model of light within fibre ring resonators is an extended form of the so-called Lugiato-Lefever equation. Solitons are pulsed solutions to this equation that maintain a constant shape in time. When third-order dispersion is included in the model, it is difficult to obtain analytical results which describe soliton characteristics as Δ and X change. However, we can use code on computers to simulate the Lugiato-Lefever equation, and by changing the parameters we can observe how the solitons change. In particular, you can pick values of X, Δ, and the third-order dispersion strength, generate a soliton, then increase Δ until the soliton no longer exists to find an existence upper limit in Δ.

I found that third-order dispersion drastically reduces that upper bound in Δ. The graph above shows how substantial the decrease in the upper limit is, even for a moderate third-order dispersion strength. This limits the usefulness of a third-order dispersion approach to introducing normally dispersive soliton frequency combs because large Δ's enable more efficient energy conversion from the driving laser into the frequency comb. Nonetheless, third-order dispersion solitons still comprise useful tools for producing frequency combs at new wavelengths.

One of solitons' most useful features is how their position and number can be precisely controlled by rapidly modulating the driving over time. Again, this is well-understood when third-order dispersion is negligible, but including third-order effects complicates things substantially. My summer research's natural progression is to investigate how third-order dispersion affects the manipulation of solitons when the driving is modulated. This is the focus of my BSc (Hons) research project.

If the properties of solitons with third-order dispersion can be quantified, we will be able to reliably access soliton regimes traditionally not accessible. Kerr cavity solitons underpin a substantial range of technologies, some of whose limitations can be reduced by the promising features which these new solitons possess. I look forward to seeing how this research develops, particularly the breadth of its impact on fields of science, from biology to chemistry to astronomy. The capacity for nonlinear optical systems which support solitons to be scaled down to micrometre sizes means that whatever advances are made could easily seep into everyday use. It is very possible that future generations of ubiquitous technologies like phones and computers will rely on Kerr cavity solitons.

*If it doesn’t, don’t worry. For interest's sake, though, a Taylor series is among the most important tools in a physicist’s kit. It describes the fact that pretty much any function f(x) you’re interested in (in this case, frequency as a function of wavenumber) can be represented by a polynomial. In general, the polynomial will have an infinite number of terms, with each term being of the form f(0) x , (the nth derivative of f at 0 multiplied by x to the power of n). However, close to x=0 the terms with larger exponents will quickly approach zero and can be safely ignored. If we are left with only a few terms, our job is greatly simplified because polynomials are much easier to analyse in general than arbitrary functions.