Optical fibers and waveguides provide unique and distinct environments for nonlinear optics, because of the combination of high intensities, long interaction lengths, and control of the propagation constants. They are also becoming of technological importance. The topic has a long history but continues to generate rapid development, most recently through the invention of the new forms of optical fiber collectively known as photonic crystal fibers. Some of the discoveries and ideas from the new fibers look set to have lasting influence in the broader field of guided-wave nonlinear optics. In this paper we introduce some of these ideas.
©2007 Optical Society of America
Guided-wave nonlinear optics is a fast-moving and exciting research field, producing a series of interesting ‒ even fascinating ‒ results over the past few years. At the same time, it is rapidly moving out of the optical laboratory and into a range of devices in fields ranging from telecommunications to the bio-sciences. It is not a new research area, but one with a rich history stretching back almost to the first low-loss waveguides. However, the past five years have seen something of a renaissance in nonlinear guided-wave optics as a number of new fundamental effects have been identified, while advances in laser and waveguide technology have meant that effects which might have previously been considered exotic can now be readily observed ‒ even with the unaided eye ‒ using small and inexpensive laser systems. It is worth considering why the field is so active ‒ after all, low-loss optical fibers, one prime environment for nonlinear guided-wave optics, have been readily available for over 20 years. One reason alluded to above is the increased availability of optical sources - the range of pulse lengths and spectral ranges of laser systems available at relatively low cost not only enables new generations of laboratory experiment but also demands the development of a new spectrum of applications. As important, however, has been the development of new or improved technologies and concepts in forming nonlinear waveguides, and the greatly increased breadth of our understanding of nonlinear optics in them. Proper consideration of the spectacular advances in optical sources over the past decade lies outside the scope of this paper. Advances in waveguiding structures and our understanding of their nonlinear response do not. This is not intended as a comprehensive review of the field but more a selection of recent developments relating to photonic crystal fibers. With this selection, we hope to convince those new to the field of their significance for the broader field of guided-wave nonlinear optics.
2. Foundations of guided-wave nonlinear optics
We do not usually expect light to interact with itself, especially in a vacuum. Even when light passes through a sheet of glass (or another transparent medium) we usually expect it to remain unchanged, apart from the well-known linear effects like Fraunhöfer diffraction, scattering and accumulation of a phase shift. However, when the optical intensities are high, the material responds to the electromagnetic nature of light, changing its properties. Light passing through this changed material is then itself affected, so that the light interaction with itself is mediated by the material. The material response determines how the light is itself affected, and can take several forms. It can respond instantaneously, as in the Kerr effect which gives the intensity-dependent refractive index and well-known effects such as self-phase modulation and four-wave mixing. It can respond with some delay, as in Raman scattering. Electrostriction can excite acoustic waves in the material, which themselves scatter light (Brillouin scattering) or two-photon processes can give rise to unexpected absorption and to the generation of free charge carriers. Naturally, these different effects manifest differently under different circumstances. The most influential factors in determining the nonlinear response in optical waveguides are the bulk properties of the material itself, the optical properties of the waveguide in which the light is confined, and the nature of the light being used in the experiments.
The bulk properties of waveguide materials differ widely. In semiconductor materials, optical experiments are frequently performed relatively close to the electronic band edge, and so the probability of multi-photon absorption and consequent free-carrier generation is relatively high. The Kerr response in semiconductors is also often high relative to glasses, while Raman scattering can be strong but is likely to be narrow-band. Glasses vary widely, but are typically used further from the band edge than semiconductors, and multiphoton absorption is less frequently a problem. Their exceptional transparency can lead to remarkably low attenuation (and hence long interaction lengths) while their Kerr and Raman responses are relatively low. However, the Raman gain band is very broad, which has a number of interesting implications such as broadband Raman amplification and the soliton self-frequency shift. Liquids have a long history in nonlinear fiber optics , and along with gases they possess a wide range of nonlinear properties. They are again becoming of interest in nonlinear fiber optics, due to the development of photonic bandgap fibers. An excellent introduction to nonlinear optical processes in optical fibers is the book by Agrawal .
Nonlinear optical effects in waveguides can be spectacular. One reason is that the high intensities in a single-mode guide and the long interaction lengths (compared to the Rayleigh length of a focused beam) promote nonlinear interactions. Perhaps more subtly, in designing the waveguide one has freedom to engineer the optical response over a broad parameter space, so as to enhance or suppress certain interactions. More than anything else, the ability to engineer the waveguide response, and the insight into what this enables, has led to continuing rapid development of the field over the passed decade. Nonlinear effects inevitably involve more than a single frequency or wavelength of light. As a result, optimizing a waveguide to enhance a particular nonlinear effect implies considering the variation of the waveguide properties with wavelength, usually referred to as the dispersion of the guide. Different dispersion conditions are critical in different situations. For example, four-wave mixing processes require phase-matching between the modes propagating at different frequencies, soliton propagation requires that the group-velocity dispersion (GVD) be anomalous (at least, most commonly!), and blue- or ultraviolet supercontinuum generation is facilitated by group-index matching to solitons propagating at much longer wavelengths. Consequently, the ability to engineer the guided mode dispersion curves over a wide range of wavelengths is critical to controlling and using nonlinear effects. Both of these features ‒ tight confinement over long lengths and control of dispersion curves ‒ are manifested in the various forms of photonic crystal fiber.
3. New fiber structures lead to new concepts and possibilities
So what is most of the current research in this field actually about? Several trends have appeared over the past years, including the rise of nonlinear silicon-based waveguides [3–6], fiber- and waveguide-based supercontinuum generation [7–9], the development of new forms of photonic crystal fibers (PCF) [10,11], and continued progress in continuous [12,13] and discrete [14,15] soliton physics. Relatively recent and undeveloped fields are those of slow light in fibers and waveguides [16–19] and atomic and molecular interactions based on gases in hollow-core fibers and near waveguides [20,21]. Many recent developments have arisen because of the new opportunities offered by new and improved waveguides. Figure 1(a) shows several examples of recent nonlinear waveguides reported in the literature, selected to illustrate how some of the various forms of photonic crystal fiber shown in Fig. 2 compare to other nonlinear waveguides. The plot is of optical attenuation (which limits the effective length of the nonlinear interaction) against the Kerr nonlinear coefficient γ, which reflects how rapidly a nonlinear phase shift builds up under a given pump power. It is worth noting that the plot spans eight orders of magnitude in attenuation and eleven orders of magnitude in nonlinear coefficient, reflecting the extraordinary range of such effects. Because the length of the waveguide or fiber is under our control, the useable length is usually the effective length, and so if all devices performed equally they would fall on a straight line on this plot. The nonlinear coefficient of the two different forms of photonic crystal fiber shown differ by over seven orders of magnitude.
Figure 1(b) shows the group-velocity dispersion (GVD) of bulk silica and several forms of photonic crystal fiber, also illustrated in Fig. 2. All conventional (silica) optical fibers have dispersion curves very similar to those of bulk silica, and only slightly affected by the waveguide dispersion. The dispersion-flattened PCF (illustrated in Fig. 2(b)) can be engineered to have almost no dispersion over a wide range of wavelengths around 1500nm . The PCF with a small core (similar to that shown in Fig. 2(c)) has several interesting features, including a zero-point in the GVD at short wavelengths, anomalous (i.e. positive) dispersion over a range of wavelengths where standard fiber dispersion is always normal [7, 28], and can have a negative dispersion slope at longer wavelengths culminating in a second zero-crossing of the curve . Finally, the hollow-core fiber (as shown in Fig. 2(a)) has only a single low-loss transmission window (centered around 800nm in the case shown) and within that window has both positive and negative dispersion . Our intention in the remainder of this paper is to describe some of the developments in nonlinear guided-wave physics which have resulted from the new fibers.
Although nonlinear optics in waveguides includes a wide range of effects, surely the most widespread and pervasive concept is that of a soliton . The recent past has seen studies of solitons, continuous and discrete, in both time and space. In PCF, solitons have been invoked in applications like wavelength conversion , pulse delivery and compression [13,32–34] and supercontinuum generation [7,35–37], and have also been the subject of a number of more fundamental studies.
As well as being common on a global length scale in the context of long-distance telecommunications, solitons are now readily identified in optical fibers of lengths of less than a centimeter . Time-domain measurements of solitons are impressive and widespread, and simultaneous measurement of the intensity and phase variation offers conclusive proof of soliton propagation. However, solitons can often be most immediately identified in the spectral domain, especially in optical fibers, where the Raman interaction with the glass of which the fiber is formed leads to an obvious soliton self-frequency shift. An example is shown in Fig. 3(a). The spectra show the appearance of two distinct peaks in addition to the pump, corresponding to solitons. In the case shown in Fig. 3(a) the dispersion slope of the fiber was negative. With longer propagation distance, the peaks are broadened spectrally as the soliton length becomes compressed in time. This is because the self-shifting soliton moves into a wavelength band where the GVD of the mode is significantly lower. The negative dispersion slope is associated with the appearance of a second zero-point in the GVD curve for fibers in which the dispersion is dominated by waveguiding, and this has the effect of changing the soliton evolution in other ways as well (see  and below.) Another widespread representation of data is as a cross-correlation frequency-resolved plot, of the type shown in Fig. 3(b). Such plots are especially valuable in identifying the interaction between different spectral components inside the waveguide. For example, in Fig 3(b) the interaction of solitons in the anomalous dispersion regime with the dispersive radiation co-propagating with similar delay at shorter wavelengths becomes very obvious. These effects are not always as apparent in other material systems. The very different nature of the Raman response of semiconductors to that of glasses, and the different magnitude relative to the Kerr response, means that the self-frequency shift in short semiconductor waveguides is weak (if not negligible), making identification of solitons in these waveguides far more difficult. The spectral signature is thus relatively weak , while the limited sample length coupled with the fact that the solitons occur at very low optical powers make simultaneous time- and frequency-resolved measurements difficult.
Numerous theoretical studies of optical solitons have been focused on the problems of their existence and stability with respect to small perturbations. This stability determines whether noise around the ideal soliton profile is exponentially amplified during propagation. Numerical modeling [36,37,39] and analytical theories of the supercontinuum as a whole and of its constituent parts [40,41] have demonstrated that the radiative processes occurring during development of a PCF-based supercontinuum can not be understood using this approach. Instead, these processes are analogous to the non-exponential growth of oscillations of an externally driven oscillator, when driving frequency approaches the resonance. There were only a small number of theoretical descriptions of such processes with optical solitons in the pre-PCF era, see, e.g., [42,43]. The difference between the two pictures is important, because radiation arising directly from the solitons and from the mixing of solitons with dispersive wave packets results in different parts of the supercontinuum spectra being coherent and reproducible from one pump pulse to another, which is not the case when the radiation at the new frequencies builds up from noise .
Solitons are non-dispersive pulses, so the variation of the propagation constants of the Fourier components in the soliton spectrum with frequency forms a straight line, implying the same group velocity across the soliton spectrum. As a result, the phase matching conditions derived for the four-wave mixing of solitons with dispersive waves give signal frequencies very different from those predicted using standard phase matching for dispersive waves . In addition, the solitons generated close to the zero GVD point are strongly affected by this proximity. They are then not perfect, but tend to emit dispersive waves - resonant or Cherenkov radiation [12,37,42,43]. The emitted waves can again interact with solitons causing further frequency conversion, which in combination with the fact that the carrier soliton frequency is itself changed via intrapulse Raman scattering causes spectral broadening leading ultimately to supercontinuum generation .
Intrapulse Raman scattering plays several important roles in the interaction of solitons with dispersive radiation and strongly influences features of the supercontinuum well away from the soliton frequency. The nonlinear response due to the Raman effect occurs on the scale of tens of femtoseconds and slows femtosecond-scale solitons down. In the range of anomalous GVD smaller group velocities correspond to longer wavelengths. Hence the solitons are constantly decelerated and simultaneously red shifted as they propagate. Commonly, the zero GVD wavelength in PCFs is blue-detuned from the soliton frequencies . As a result, the Raman effect shifts the soliton away from the zero-GVD point, decreasing the overlap of the soliton spectrum with the normal GVD range. The amplitude of the blue shifted Cherenkov radiation emitted by the soliton then drops significantly with propagation along the fiber length. Cherenkov radiation alone can not therefore explain the continued extension of the short wavelength edge of the supercontinuum towards higher frequencies for long propagation distances [7,39,45]. Dispersive resonant radiation emitted by solitons is itself not affected by the Raman effect and therefore propagates with constant group velocity. This leads to collisions between the decelerating solitons and the trailing blue-shifted wave packets generated by the Cherenkov or SPM mechanisms. The waves emerging from this interaction again trail behind the soliton and have frequencies which are further shifted towards shorter wavelengths. The entire process of collision and radiation emission is repeated in a cascaded way resulting in a quasi-continuous shift of the short wavelength edge of the supercontinuum . The critical aspect of this scenario ensuring its efficiency over the long propagation distances is that the dispersive spreading of the wave packets and their nonlinear spectral transformations by cross-phase modulation are compensated by a wave trapping mechanism [41,46] (see Fig. 3(b)). Radiation in the normal GVD range gets trapped in the local minima of the refractive index formed on one side (front of the pulse) by the increase of the index created by the soliton and on the other side (tail of the pulse) by the effective linear increase of the refractive index existing due to acceleration of the soliton, see Fig. 3(c) and [41,47]. This effective potential is analogous to the inertial forces present in accelerating frames of reference. Any wave packet at the blue edge of the continuum can be expanded into modes of this potential. The frequencies of the individual modes or of the superpositions of the low lying modes of the potential well emerging in the course of supercontinuum generation shift to shorter wavelengths continuously. If, however, the experiment is organized in such a way that the short-wavelength pulse is formed of the highly excited modes of the potential, then the frequency increase occurs in a stepwise fashion . Obviously the radiation trapped by the soliton is delayed together with the latter. However, for normal GVD the group velocity decreases towards shorter wavelengths, which is consistent with the blue shift of the radiation.
If the zero GVD point is red detuned from the soliton frequency, then the resonant radiation is faster than the decelerating soliton. In this case, the radiation can not get trapped by the soliton. However, the radiation is amplified with propagation in this case, because the Raman effect pulls the soliton closer and closer to the zero GVD point . Eventually the radiation gets so strong that the soliton recoils against it. As a result the soliton acceleration and the frequency shift drop significantly, and nearly disappear [12,48]. Although the radiation cannot interact with its parent soliton, in many experimental situations an energetic incident pulse will break up into multiple solitons. In that case the powerful radiation wave generated by the first and most delayed soliton can meet and reflect from the second soliton (and maybe others), initiating frequency conversion towards shorter wavelengths exactly as described above. However, in this case the radiation initially has a longer wavelength than the soliton. The frequencies of the soliton and radiation then converge as they propagate along the fiber, and the generated spectrum fills the middle part of the continuum without extending its edges [40,41,49]
5. Pulse compression
With the rapid spread of ultrashort-pulse laser technology looking set to continue with the development of increasingly robust and high-power fibre-based sources, the generation, delivery and compression of these pulses becomes increasingly important. New forms of optical fiber have made several contributions. For example, impressive compression of transform-limited ultrashort pulses around 800nm wavelength has been demonstrated by spectral broadening using self-phase modulation in a very short piece of highly nonlinear photonic crystal fiber, followed by recompression of the broadened spectrum using bulk optics [50,51]. Spectral compression (i.e. transform-limited temporal stretching) has been performed using a hollow-core fiber to apply an anomalous chirp to an ultrashort pulse before nonlinear amplification causing spectral narrowing [52,53]. Such fibers have been used for recompression in chirped-pulse amplifier and parabolic pulse configurations [54,55]. They have also been used as nonlinear fibers, enabling high-power (MW peak power) pulses to propagate as optical solitons over lengths of a few meters of fiber. The unavoidable pulse lengthening due to the intrinsic fiber attenuation (and also due to Raman self-scattering which shifts the soliton to longer wavelengths where the GVD is higher) has been reversed by the use of a tapered fiber, causing compression of 200fs input pulses to 100fs output pulses after propagating single-mode through 8m of fiber . An example is shown in Fig. 4. Fig. 4(a) shows the dispersion of the tapered fiber at the input and output, and Fig 4(b) shows the autocorrelation width of the pulses after propagation through 8 meters of fiber as a function of the pulse energy. This type of soliton compression, first demonstrated in PCF (of the solid-core variety) in  has been significantly extended in solid-core fibers in  to compress input pulses around 1060nm by factors of 15 and more.
6. Wavelength conversion and supercontinuum generation
The field of fiber- and waveguide-based supercontinuum generation has been intensely active over the past few years, and this research is leading to a range of new commercial products. Developments have recently been comprehensively reviewed in , and the reader is referred there for a full account of developments. However, a few trends and opportunities can be identified which merit comment. The new generation of fiber-based supercontinuum light sources has come about because of the development of photonic crystal fibers, but is being widely adopted because of the simultaneous availability of a range of new laser pump sources, including both low-power solid-state (microchip) lasers and mode-locked fiber lasers. Such sources are also relatively inexpensive. Supercontinuum spectra obtained using amplified mode-locked fiber lasers span much or all of the wavelength range 450 nm – 1700 nm and beyond, and can have spectral densities of up to several mW/nm. These systems typically have desirable high repetition rates and low noise. Similar ‒looking spectra can be obtained through the use of lower-repetition-rate sub-nanosecond microchip lasers, which are extremely compact and affordable, but have lower average powers and are frequently passively Q-switched. In terms of applications, however, several problems remain to be addressed. First, although these sources are impressive, many applications require just a few milliWatts of optical power, but in a specific spectral region. Indeed, for many applications, multi-watt laser systems have to be ruled out for safety and damage considerations. This suggests two important avenues of research. First, obtaining similar supercontinuum spectra but using lower optical pump powers would make these sources suitable for a wider range of applications. Secondly, being able to convert power efficiently into a specific spectral range would be more desirable for many applications than a flat ultra-broadband spectrum. In particular, several emerging applications in biological and biomedical imaging require spectra in the blue and near-ultraviolet, with low noise and powers in the range of a few milliWatts. Current systems are useful, but could be more so if their spectra extended further into the ultraviolet. Other applications require coherent supercontinua spanning the visible but using low-energy femtosecond pump pulses, or supercontinua extending further into the mid infrared. The solutions to these problems lie in identifying a suitable pump source, and then combining it with a fiber (or series of fibers) which has the appropriate nonlinearity and dispersion curve.
Some progress towards these objectives has already been reported. Initiation of nonlinear processes requires pumping a fiber near to the zero-GVD wavelength, especially if the intention is to generate significant energy at shorter wavelengths than the pump. Alternatively, phase-matched four-wave mixing can be used to generate a shorter-wavelength signal by pumping appropriate fibers in the normal-dispersion regime [59,60]. However, in the first case, a fiber which is optimized to initiate supercontinuum generation towards shorter wavelengths is not also ideal to generate the shortest possible wavelength. When using FWM, attainable signal wavelengths are usually restricted to red or visible frequencies because the effective gain decreases rapidly for shorter wavelengths, due to non-uniformities along the fiber length and fiber attenuation. Consequently, in order to generate shorter wavelengths, a second stage of nonlinear conversion is most effective. This can take place in a second nonlinear fiber , or a tapered fiber in which the properties vary along the length [62–65] (see Fig. 5). The shortest wavelengths are frequently generated through continuing interaction between long-wavelength solitons and co-propagating blue radiation (see the soliton chapter above), and so such a process is most effective if the entire supercontinuum propagates through the whole length of fiber with low loss. Given the very broad wavelength range, fiber-integrated solutions are preferable to avoid chromatic losses in bulk-optical elements. Currently, short-wavelength generation is limited by the variation of refractive index (and group index) of silica with wavelength, which makes both group-index matching and phase-matching increasingly difficult as the wavelength decreases. Possible solutions to this are to excite higher-order modes in an overmoded fiber  or use modal phase-matching from the fundamental to a higher-order mode . However, the generated radiation is then typically in an undesirable (albeit frequently visually striking) higher-order fiber mode. An elegant solution would be to use a fiber-based broadband mode converter  to utilize higher-order mode properties while maintaining fundamental-mode input and output. Ultimately, generation of blue and UV radiation will be limited by the attenuation of the fibers at short wavelengths, as both the intrinsic material losses and the scattering loss due to surface roughness increases.
Instead of using a cascade of nonlinear processes to generate a broadband supercontinuum, it is possible to target more discrete spectral bands by facilitating wavelength conversion to just a specific band. One possible method is to generate Raman-shifting solitons and then extend the fiber length until the desired wavelength is reached . However, this is limited to the generation of longer wavelengths than the pump, and to the soliton energy. Modulation instability and four-wave mixing offer more promising avenues, and are best pursued using longer pulses where self-phase modulation is not dominant. In that case, four-wave mixing processes can be used to transfer energy to much shorter wavelengths, although in reality the efficiency of conversion to the shortest wavelengths is again limited by fiber attenuation and non-uniformities along the fiber length . Four-wave mixing processes based on the phase matching with the pump in the normal dispersion regime, which is possible due to the non-zero higher-order dispersion, have recently been used to generate correlated photon pairs for quantum communications applications .
Nonlinear waveguide and fiber optics is a research area brimming with excitement and progress. New technologies are enabling both new insights into nonlinear physics and the development of new application areas, as well as visually and intellectually appealing new research results. One of the most exciting areas has been the development of the new forms of optical fiber described in this paper. However, a striking feature of developments of the past few years has been the generality of many of the observed effects: on the one hand, this could be construed as being because the most basic effects are well understood: on the other, it means that effects to be found in new structures and material systems can be readily understood and predicted. Nonetheless, the next few years look likely to be as exciting as those just passed. Photonic crystal fibers have resulted in a range of compact and stable supercontinuum sources. These will become increasingly versatile as their spectral coverage is extended and as better methods to convert power efficiently from a pump laser to a specific spectral band are developed. Tapered or concatenated fibers will help. Tapered fibers also enable efficient pulse compression and delivery at previously impossible wavelengths. Use of the dispersion properties of higher-order guided modes  in different types of fibers using efficient mode converters  enables extension of a number of the current limitations on optical fiber performance. Using fibers with hollow cores enables delivery of high-power femtosecond and picosecond pulses from the new generation of mode-locked fiber lasers around 1μm wavelength over fiber lengths of tens of meters. Combined with the new generation of fiber-based optical sources we expect nonlinear processes in PCF to become an integral part of laser spectral and temporal engineering and beam delivery.
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