The study of nonlinear propagation in optical fibres dates back

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1 progress article Published online: 30 JANUARY 2009 doi: /nphoton Ten years of nonlinear optics in photonic crystal fibre John M. Dudley 1 and J. Roy Taylor 2 The year 2009 marks the tenth anniversary of the first report of white-light supercontinuum generation in photonic crystal fibre. This result had a tremendous impact on the field of nonlinear fibre optics and continues to open up new horizons in photonic science. Here we provide a concise and critical summary of the current state of nonlinear optics in photonic crystal fibre, identifying some of the most important and interesting recent developments in the field. We also discuss several emerging research directions and point out links with other areas of physics that are now becoming apparent. The study of nonlinear propagation in optical fibres dates back to the early 1970s and the first fabrication of low-loss silica fibre waveguides. Nonlinearity in fibres arises primarily from the intensity-dependent refractive index, but it is the way in which this nonlinearity and the fibre dispersion combine to influence light propagation that generates such a rich variety of dynamical effects. By the late 1980s, for example, it was known that nonlinearity and dispersion could balance to support stable soliton propagation, yet they could also interact in the neighbourhood of the fibre s zerodispersion wavelength to yield marked instabilities and the generation of broadband supercontinuum spectra. The study of photonic crystal fibres (PCF) began in the early 1990s, and was initially motivated by the desire to create fibres that guided by means of a photonic bandgap effect. The first successful PCF, however, guided because of an effective refractive index difference between a solid silica core and a surrounding cladding region containing a transverse microstructure of air holes. For a comprehensive review of this development, see ref. 1. Although index guiding in PCF is conceptually similar to guidance in conventional fibres, it was soon realized that the additional degrees of freedom offered by engineering the air-hole geometry yielded guidance properties unattainable in standard fibre. Specifically, the strong waveguide contribution to the dispersion meant that it was possible to obtain zero-dispersion wavelengths in the visible or near infrared region, very far from the intrinsic material value of fused silica (around 1.3 μm). The significance of such dispersion engineering for nonlinear optics was revealed in striking fashion at the 1999 Conference on Lasers and Electro-Optics, where Ranka et al. reported that they had obtained supercontinuum generation spanning 400 to 1,500 nm using only nanojoule-energy 100-fs pulses from a modelocked Ti:sapphire laser 2,3. The key feature of the PCF used in these experiments was that its zero dispersion was shifted to be close to the pump wavelength around 800 nm. A further contributing factor was the enhanced fibre nonlinearity owing to tight modal confinement in the core. Indeed, shortly afterwards, Broderick et al. discussed nonlinearity engineering in PCF in more detail, and reported careful experimental measurements of the enhanced nonlinear phase shifts that could be obtained 4. The scene was set for a decade of work that has profoundly altered the research landscape in nonlinear fibre optics. This research has led to numerous technological advances in high-brightness source development, and established important links with other areas of physics through the application of frequency combs to precision measurements of fundamental physical constants. Many aspects of this progress have been well documented in reviews of specific technical areas 5 8, and some applications have now moved beyond the research stage into commercial products. As we shall see, technological progress continues to be made on many fronts, and research into the fundamental aspects of nonlinear propagation in PCF remains as dynamic as ever. Connections with other areas of physics continue to be made, and recent work has linked propagation effects in PCF to analogous phenomena in hydrodynamics, thermodynamics and astrophysics. Not intended to be a comprehensive review, this progress article discusses a selection of these recent results, with the primary intention of illustrating how the rich diversity of PCF-based research is continuing to drive nonlinear photonics and physics in new and sometimes unexpected directions. The supercontinuum revolution The 1999 supercontinuum results attracted immediate attention because of their potential application in optical frequency metrology, allowing complex room-sized frequency chains to be replaced by compact benchtop systems. A historically oriented account of this research and its revolutionary impact has been provided by Hall and Hänsch 9. Interestingly, although the frequency metrology community quickly worked out how to generate stable supercontinua experimentally, developing a clear theoretical interpretation of the supercontinuum broadening took a little longer. In hindsight, this is surprising, as many previous studies of nonlinear fibre propagation had observed essentially the same spectral broadening processes, albeit using different source systems and generally in different wavelength regimes 10,11. Nonetheless, by around 2002, the supercontinuum generation process was well understood with the contributions of soliton fission, the Raman self-frequency shift and dispersive wave generation all identified (see ref. 5 and references therein). PCF-based frequency combs have now been widely adopted, and have led to many well-known successes such as precision measurements of fundamental constants of physics. With an eye on continued applications, current research is looking into ways of developing all-fibre-format comb systems with broad spectral coverage, technical robustness and high power. To this end, advances 1 Département d Optique P. M. Duffieux, Institut FEMTO-ST, CNRS UMR 6174, Université de Franche-Comté, Besançon, France. 2 Femtosecond Optics Group, Physics Department, Imperial College, Prince Consort Road, London SW7 2BW, UK. john.dudley@univ-fcomte.fr; jr.taylor@imperial.ac.uk nature photonics VOL 3 FEBRUARY

2 progress article Nature photonics doi: /nphoton a b OPO output nm Pump nm 20 Signal (db) Wavelength (nm) c Pump laser 1.3 m PCF Output Mirror Mirror Figure 1 Frequency conversion in non-supercontinuum experiments. Complementing the now-ubiquitous images of supercontinuum generation in PCF, this figure shows tunable narrowband spectral generation from a picosecond PCF optical parametric oscillator (OPO). a, The photograph and b, tuning curve show the tuning characteristics spanning the visible spectral range. c, Illustration showing the simplicity of the butt-coupled OPO cavity. Panel a image from John Harvey; data in b and c from ref. 15. in Yb-fibre-based sources with large mode area have been recently applied to develop a frequency comb system with more than 10 W average power. Significantly, this high power performance was obtained at the same time as submillihertz linewidth relative to a conventional Ti:sapphire comb 12. Of course, frequency metrology is not the only important application of broadband supercontinuum spectra, and there have been continued advances in fields such as spectroscopy, microscopy and optical coherence tomography. However, some of the most interesting progress in nonlinear applications of PCF has not involved broadband spectral generation at all, but rather the generation of narrowband frequency components through parametric frequency conversion. An area where this has been applied with particular success is in the generation of correlated photon pairs for quantum information applications. In contrast to experiments using nonlinear crystals, correlated photon pair generation in PCF at power levels suitable for multiphoton interference experiments has been demonstrated with only milliwatts of pump power. This represents an improvement of several orders of magnitude over bulk approaches 13. In other recent studies, narrowband parametric phasematching in PCF has been used to realize widely tunable low-threshold χ (3) picosecond optical parametric oscillators 14,15. Figure 1 shows results obtained using such a PCF-based oscillator pumped by a picosecond Ti:sapphire laser 15. By butt-coupling dichroic mirrors to either end of a highly nonlinear index-guiding PCF, tunable narrowband oscillation in the nm range has been demonstrated at peak power thresholds of only 15 W. The possibility of simultaneously obtaining broad tuning, narrowband oscillation and low gain threshold arises because of the characteristic dispersion and nonlinearity parameters of PCF, and may well lead to a new generation of compact visible light sources when combined with laser diode-based pump sources. Extending the physics of supercontinuum generation The basic features of what could be termed orthodox supercontinuum generation using femtosecond pulses are now well understood, and the detailed temporal and spectral structure seen in experiments is well reproduced using realistic numerical models. Interestingly, although the essential nonlinear propagation equations of fibre optics have been clearly expressed in the literature for many years, the recent studies of supercontinuum generation have highlighted the quantitative predictive power of careful numerical modelling. Simulations have played a central part in revealing subtle features of the nonlinear spectral broadening processes, such as coupling between the Raman and modulation-instability gain processes, and the effect of dispersion in the nonlinear response due to variation in the effective mode area in the fibre 5. In addition, the continued application of new diagnostic techniques, together with studies of supercontinuum generation using a wider range of pump sources, has led to improved physical insight and sometimes unexpected analogies with other areas of physics. For example, the use of time frequency spectrograms allows improved visualization of the femtosecond soliton propagation dynamics, and clarification of how spectral components on the long- and short-wavelength edges of the supercontinuum spectrum can interact despite frequency separations approaching an octave. Specifically, cross-phase modulation from long-wavelength solitons undergoing the soliton self-frequency shift has been shown to lead to energy localization or trapping in the normal dispersion regime, and the extension of the short-wavelength edge of the supercontinuum. Although this effect had been noticed in earlier studies in more conventional fibres 10,16, the results in PCF have motivated new theoretical work using a gravitational force analogy to provide analytical insight into the observed spectral structure 17. More recent research has also used a gravity-like analogy to speculate that features of supercontinuum generation dynamics can be interpreted in terms of an event horizon, an effect usually associated with astrophysical black holes 18. Recently, another form of trapping in supercontinuum generation has also been seen for the first time in PCF. Specifically, a series of experiments have used cut-back measurements (where the 86 nature photonics VOL 3 FEBRUARY

3 Nature photonics doi: /nphoton development of the supercontinuum is investigated along the length of the nonlinear fibre) to reveal the presence of previously unknown long-wavelength bound states, where two ejected solitons from the initial soliton fission mechanism become trapped by their mutual nonlinear interaction 19. Other current research into supercontinuum generation is studying the spectral properties observed using picosecond and nanosecond pulses from compact and relatively inexpensive pump sources. With such long pulses, however, the generated spectra typically suffer from shot-to-shot variations induced by modulation instability, and this can be a limiting factor in some potential applications. There is thus considerable interest in developing techniques that allow a degree of control into the shaping and stability of supercontinuum generation in this regime. Improved understanding into the mechanisms underlying these fluctuations has recently been provided through measurements of shot-to-shot statistics in a picosecond pumped supercontinuum 20. Of particular interest is the fact that fluctuations in the supercontinuum spectral structure were shown to be associated with the generation of optical rogue waves statistically rare events associated with the extreme redshift of long-wavelength Raman soliton pulses. Because these fluctuations have their origin in modulation instability, there are intriguing connections with the infamous and destructive freak waves observed on the surface of the ocean. Subsequent studies have examined these instabilities in more detail, focusing on the interaction dynamics of the initial instability growth and Raman soliton propagation 21. In fact, inspired by previous studies of induced modulation instability in the context of ultrashort pulse train generation, recent work has shown that induced supercontinuum generation using dual frequency fields can significantly modify the supercontinuum spectral characteristics 22,23. The first experiments on supercontinuum generation used femtosecond duration sources, but work was rapidly extended to picosecond duration pumps. More recently, the availability of highpower and compact nanosecond and quasi-continuous-wave (CW) sources has motivated much interest in experimental and theoretical studies of supercontinuum generation in this regime. With the choice of an appropriate fibre dispersion profile, experiments have reported supercontinua with tens of watts average power and output spectral densities of up to 100 mw nm 1 (refs 24,25). Numerical modelling has also shown that, even in this regime, trapping and scattering of dispersive waves by high-energy solitons play a dominant part in extending the spectral bandwidth to shorter wavelengths. From a fundamental viewpoint, the fact that broadening with a quasi-cw pump leads to the excitation of a very large number of solitons has meant that the effects of multiple soliton collisions and interactions are receiving renewed attention 26,27. In particular, recent theoretical studies suggest that new insights can be attained by adapting research from complex systems, turbulence and thermodynamics. Indeed, the first publications using these ideas are beginning to appear 28,29. Figure 2 uses results from numerical simulations to illustrate some of these new features of supercontinuum dynamics in the quasi-cw and long pulse regimes. Figure 2a uses the time wavelength spectrogram representation to show the complex evolution of a noise-seeded supercontinuum at various stages and the emergence of a large number of solitons on the long-wavelength edge of the pump. Understanding the collective dynamics of these solitons remains an important open area of research. In this context, Fig. 2b shows results from stochastic simulations of picosecond supercontinuum generation, superposing output spectra (grey curves) from 1,000 simulations as well as the calculated mean (red line). The expanded view of the long-wavelength edge shows how a small number of solitons undergo an increased redshift relative to the mean, representing the tail of a characteristic L-shaped probability distribution 20. It is the statistical analysis and distribution fitting of the amplitudes of these solitons in the tail that can reveal extreme a b Wavelength (μm) Wavelength (μm) Spectrum (10 db per div.) progress article m m Time (ps) Time (ps) m 25 m Time (ps) Time (ps) 900 1,000 1,100 1,200 Wavelength (nm) 1,300 1,220 1,240 1,260 Wavelength (nm) Figure 2 Unusual features of the quasi-cw or long pulse regime. a, Simulated spectrogram evolution for a high-power quasi-cw pumped supercontinuum using 170 W pump power and near-zero dispersion wavelength pumping. Following modulational instability, inspection shows soliton trapping of dispersive waves after 9 m and a clear one-to-one correspondence between soliton and dispersive wave components after 25 m (data from ref. 25). b, Results from 1,000 simulations of picosecond pumped supercontinuum generation seeded from noise showing a long tail of redshifted soliton pulses in the expanded subfigure of the longwavelength edge (data from ref. 21). value or rogue characteristics. Soliton interactions and collisions also seem to have an important role in the dynamics that generate these particular events. Hollow-core nonlinearities Another PCF milestone celebrating its tenth anniversary in 2009 is photonic bandgap guidance in a hollow-core (HC) structure 30. When compared with solid-core fibre, HC-PCF make use of a fundamentally different type of guidance, in which light is confined to the low-index core region by means of a two-dimensional photonic bandgap crystal. In the context of nonlinear optics, a unique feature of the HC-PCF is that it can be applied at both ends of the nonlinear spectrum when it is filled with air, low intrinsic nonlinearity allows high-power pulse delivery applications, whereas when filled with appropriate gas- and liquid-phase media, high intrinsic nonlinearity allows the observation of frequency conversion effects at very low power levels. More specifically, in the case of air-filled HC-PCF, the effective nonlinearity is reduced by as much as three orders of magnitude when compared with solid-core fibres. But because the overall dispersion is generally anomalous over most of the transmission profile, guidance of high-peak-power pulses by means of nonlinear soliton effects is nonetheless possible. Using a 3-m-long air-filled fibre, Ouzonov et al. 31 demonstrated guidance of 2.4-MW solitons of 110-fs duration around 1,500 nm, but the Raman self-frequency shift Wavelength (μm) Wavelength (μm) Spectrum (10 db per div.) nature photonics VOL 3 FEBRUARY

4 progress article a Probe 1,517 nm c Power (db a.u.) (J = 15) Δ p (J = 16) Δ c Wavelength (nm) Control 1,535 nm (J = 17) 1, Frequency (THz) b in air ultimately shifted the solitons towards the absorbing edge of the bandgap. However, using Raman-inactive xenon gas, they were able to avoid this effect and to report the propagation of 75-fs pulses at 5.5 MW (ref. 31). These results have motivated other exploratory studies in the field of ultrafast optics and source development. A natural follow-up has been higher-order soliton compression, again in xenon 32, and other studies have used HC-PCF in the construction of all-fibre-format chirped pulse amplifier systems operating with kilowatt peak powers 33. In this latter parameter regime, propagation makes use of the ultra-low nonlinearity of the HC-PCF to obtain purely dispersive recompression of high-power amplified pulses from compact fibre systems. Although subsequent development has allowed increased power scaling of these systems, it is unlikely that the all-fibre systems will be able to match the ultimate power performance of optimized bulk systems. Nonetheless, for applications where footprint and robustness are paramount, such systems remain commercially attractive. If the hollow core is filled with a nonlinear material, such as gas or liquid, bandgap guidance of light in the core allows greatly increased interaction lengths such that nonlinear effects can be observed at much reduced power levels. This potential for ultralow power gas-phase nonlinear optics has been beautifully demonstrated in a series of experiments by Benabid et al. 34, where stimulated Raman Transmission Transmission d Power (dbm) Absorption Detuning (GHz) EIT Detuning (GHz) kw 470 kw 90 kw 800 1,000 1,200 Wavelength (nm) Figure 3 HC-PCF allows the study of diverse nonlinear effects in gases and liquids. a, Resonant electromagnetically induced transparency (EIT) in acetylene using a 1.3-m PCF segment, where Δ p and Δ c are pump and control frequency offsets. b, In the absence of a control beam, tuning the probe yields Doppler-broadened absorption (top), whereas the presence of a 320-mW control beam yields induced transparency (bottom) (black: theory; grey: experiment). Panels a and b reproduced from ref. 36; 2005 APS. c, Generation of an octave-spanning frequency comb in kagome HC-PFC (inset; image from Fetah Benabid) through stimulated Raman scattering in molecular hydrogen 38 ( 2007 AAAS). d, Supercontinuum generation in a 5-cm water-filled HC-PCF (inset) at peak powers shown. Pump wavelength is 980 nm and the zero-dispersion wavelength is ~1 μm (data from ref. 40; inset 2008 OSA). Nature photonics doi: /nphoton scattering in H 2 -filled fibres was reported with thresholds up to 10 6 times lower than in the bulk configuration. A particularly elegant demonstration of the unique advantages of HC-PCF for nonlinear optics was the use of the photonic bandgap structure as a filter in such experiments, selectively allowing operation on the weaker rotational Raman bands by filtering and consequently suppressing vibrational stimulated Raman scattering 35. Another nonlinear process that has been studied at ultralow power levels in HC-PCF is electromagnetically induced transparency, with experiments in acetylene providing proof-of-principle results in the telecommunications wavelength band 36,37. These results may point to future application in quantum information technologies. Other recent applications of HC-PCF include the generation and guidance of multi-octave frequency combs 38 and the report of sub-watt continuous-wave pumped Raman laser action with bulk optical coupled end mirrors 39. Of course, the selective filling of HC-PCF for nonlinear optics is not restricted to gases; liquids can also provide nonlinear responses that can be used for frequency conversion. Indeed, some recent results have used an appropriate water-filled HC-PCF pumped by a high-power 980-nm source for supercontinuum generation 40. Figure 3 presents a selection of these results, illustrating the tremendous potential of HC-PCF for research in different areas of nonlinear optics. In the future, the high efficiency of stimulated Raman scattering in hydrogen and other gases may lead to compact sources for the generation of sub-cycle optical pulses for attoscience applications. Work is also underway to integrate HC-PCF nonlinear devices with more conventional solid-state technologies. Important practical developments include photonic microcells where HC-PCF are hermetically sealed to solid conventional optical fibres, and the use of femtosecond micromachining to yield additional functionalities 41,42. Draw-tower tapering An important recent advance in PCF fabrication has been the report of controlled tapering during the draw-tower phase. In particular, by varying the fibre drawing parameters, it is now possible to realize custom-tapered PCF with dispersion and nonlinearity parameters that can vary longitudinally on length scales of tens of metres. This technology has been demonstrated with both solid-core and hollow-core PCF, and tapered fibres of both types have found immediate application in pulse compression. This work has built on the well-established physics of soliton propagation in dispersiondecreasing fibre to develop tapered-pcf designs optimized for the compression of picojoule energy picosecond pulses around 1 μm to the ~50-fs regime 43. As with any soliton-based technique, there are tradeoffs between fibre dispersion and pulse energy. Nonetheless, compression of pulses with pulse energies greater than 50 nj at 800 nm has been possible using tapered hollow-core PCF (ref. 44). The possibility of custom tapering of HC-PCF in this way raises the possibility of simultaneous high power delivery and compression, whereas in the case of solid-core PCF, perhaps the most exciting prospect is the fabrication of precisely designed longitudinal dispersion and/or nonlinearity maps for controlled spectral broadening under general conditions 45. Figure 4 highlights the potential of this technology, showing the vast dispersion and nonlinearity parameter space possible with control of PCF dimensions (Fig. 4a and b) and results of soliton compression where a 15-fold reduction in pulse duration has been observed (Fig. 4c and d). Interestingly, when combined with a convex dispersion profile, propagation in a dispersion-decreasing fibre can lead to highly uniform and stable supercontinuum generation, and numerical studies extending ideas originally developed for telecommunications spectral slicing have recently explored this possibility with PCF (ref. 46). Experiments in this area would represent a logical next step. 88 nature photonics VOL 3 FEBRUARY

5 Nature photonics doi: /nphoton Emerging nonlinear waveguides Initial studies of nonlinearity in PCF focused on silica-based fibres, highlighting the unprecedented degree to which the nonlinear and dispersive properties could be engineered through waveguide design. In some of the most exciting recent work in this field, these ideas have now been applied to non-silica glass fibres and planar waveguides, promising the development of a new generation of nonlinear waveguide structure. The nonlinear response of a waveguide, γ = n 2 ω/ca eff, can be tailored in two ways: through material selection to modify the nonlinear refractive index, n 2 ; and through waveguide design such that the modal effective area, A eff, optimizes the nonlinear interaction (where ω is the angular frequency and c is the speed of light). One approach to developing PCF with higher nonlinearities has been to use non-silica compound glasses, because their higher linear refractive indices lead to increased modal confinement and their associated nonlinear refractive indices can be increased by orders of magnitude relative to silica 47. They are also attractive for applications at infrared wavelengths when losses in fused silica can become detrimental. A wide range of studies using PCF based on lead silicate, bismuth and tellurite glasses has been carried out 48, and a particularly impressive recent result has been the observation of supercontinuum generation in a tellurite PCF out to near 5 μm (ref. 49). Interestingly, parallel work using non-pcf fluoride fibre has achieved similarly impressive results, with reports of a highpower system with more than 1 W average power extending to 4 μm (ref. 50). In fact, these results provide a good example of the recent synergy that has developed between research using conventional fibres and that using PCF. Another approach has been to start from standard silica-based PCF, but to use post-tapering (for example using a flame-brush technique) to increase the effective nonlinear response by reducing the fibre dimensions to the submicrometre level. In such tapered fused silica nanowires, both supercontinuum generation and soliton compression have been reported 8. For applications in which the integration potential for telecommunications applications is critical, the ideas of dispersion and nonlinearity engineering from PCF research have been used to optimize the design of waveguides in materials such as silicon and chalcogenide glass. Experimental results studying supercontinuum generation and other nonlinear frequency conversion processes have been obtained and may point to a new class of nonlinear photonic device for future signal processing applications The study of nonlinear propagation in these new waveguides is also of interest from a theoretical perspective and is stimulating research into developing new and more general nonlinear propagation models 54. Another area of research into new materials is that of polymer PCF. Although not yet extensively studied for nonlinear applications, polymer fibres are nonetheless of great interest because they could be produced cheaply and yield transverse geometries that are difficult to realize in glass 47. Old physics, new directions Supercontinuum generation in photonic crystal fibres is a complex spectral broadening process that involves the interaction between a number of different nonlinear effects and the intrinsic linear dispersion of the fibre waveguide. Although many of these processes had been seen in experiments on nonlinear fibre optics before 1999, the new guidance properties of PCF made it particularly easy to generate octave-spanning spectra. PCF-supported supercontinuum generation found immediate application in frequency metrology, and also made it possible to study in detail previously unappreciated aspects of complex nonlinear pulse propagation in optical fibres. Experiments using HC PCF have taken gas- and liquidbased nonlinear optics to a new regime and opened important interactions with other fields of ultrafast optics. These studies over the past 10 years have seen a remarkable number of developments a c Λ d 2 µm 2 µm 17 m Dispersion-decreasing fibre progress article b 0 ps nm 1 km 1 d Output FWHM (fs) 25 ps nm 1 km Input 50 ps 30 µm 2 nm 1 km 1 15 µm Pulse energy (pj) by groups worldwide, and the quantitative agreement between intricate experiments and realistic numerical modelling is arguably as good as in any other domain of modern physics. The use of PCF has reduced the power demands for the study of nonlinear optics, and this has both allowed technological developments to advance and permitted fundamental studies to be undertaken with low-power laser systems. At the same time, the research has resulted in the development of unique and versatile instruments that have found commercial success and continue to contribute to fields ranging from precision frequency measurements to biomedical diagnostics. Many challenges remain. In the case of solid-core PCF, power scaling is a continual challenge, and will no doubt motivate continued work to design fibres with dispersion profiles tailored to available high-power pulsed sources. In the future, researchers will also need to develop techniques to generate shorter-wavelength radiation in the ultraviolet and to extend spectral broadening in the mid-infrared, but the physics groundwork has been done and essentially this work will be technology-driven. From a fundamental perspective, continuous-wave supercontinuum generation may well provide an intriguing platform for the study of collective fibre soliton dynamics. Known as the soliton gas, this is a regime of nonlinear optics that has not yet been amenable to widespread study. A challenge here will be to develop suitable measurement techniques for the study of intrinsically incoherent and noisy nonlinear optical processes. When one considers the past 10 years of research in context, it becomes clear that PCF can be well described as a unifying as well as an enabling technology. PCF appears as a common factor in groundbreaking experiments that have combined ideas and researchers from diverse domains, such as guided wave and gas-based nonlinear optics, ultrafast source development, nanophotonics, materials 4 75 ps nm 1 km 1 7 µm Air fill fraction d/ Λ τ p (sech 2 ) = 48 fs Figure 4 Draw-tower tapering allows fabrication of longitudinally varying dispersion and nonlinearity profiles, opening new possiblities for nonlinear optics in PCF. For the structure in a, the contours in b show variation in dispersion (red) and effective area (blue) as a function of pitch Λ and air-fill fraction d/λ (where d is the diameter of air holes) (data in b from ref. 45). Adiabatic soliton compression can be observed where the PCF structure varies longitudinally as in c. For dispersion varying from 33 ps nm 1 km 1 to 5 ps nm 1 km 1 at 1.06 μm, d shows how changing pulse energy optimizes compression (shown as full-width at half-maximum, FWHM). Minimum durations of 48 fs are obtained, with interferometric autocorrelation (inset) confirming the compressed pulse quality. Panels c and d are from ref. 43; 2007 OSA Pitch Λ (µm) nature photonics VOL 3 FEBRUARY

6 progress article science and clinical medicine. It is likely that great progress will continue in all of these fields, but perhaps the most genuine future breakthroughs will be made at the boundaries between disciplines. Another important lesson to take from the past decade is that careful scrutiny of the literature can reveal studies in earlier systems that provide physical and time-saving insight into current experiments using more advanced technologies. References 1. Russell, P. St. J. Photonic-crystal fibers. J. Lightwave Technol. 24, (2006). 2. Ranka, J. K., Windeler, R. S. & Stentz, A. J. Efficient visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm. Conference on Lasers and Electro-Optics (CLEO), Baltimore, postdeadline paper CPD8 (1999). 3. Ranka, J. K., Windeler, R. S. & Stentz, A. J. Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800 nm. Opt. Lett. 25, (2000). 4. 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Nonlinear optical phenomena in silicon waveguides: modeling and applications. Opt. Express 15, (2007). Acknowledgements J.M.D. thanks the Institut Universitaire de France for support. J.R.T. is a Royal Society Wolfson Research Merit Award holder. 90 nature photonics VOL 3 FEBRUARY

7 This article was downloaded by: [University of Bath] On: 30 January 2012, At: 04:16 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Modern Optics Publication details, including instructions for authors and subscription information: Linear and nonlinear optical properties of hollow core photonic crystal fiber F. Benabid a & P.J. Roberts a a Centre for Photonics & Photonic Materials, Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK Available online: 27 Jan 2011 To cite this article: F. Benabid & P.J. Roberts (2011): Linear and nonlinear optical properties of hollow core photonic crystal fiber, Journal of Modern Optics, 58:2, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

8 Journal of Modern Optics Vol. 58, No. 2, 20 January 2011, INVITED TOPICAL REVIEW Linear and nonlinear optical properties of hollow core photonic crystal fiber F. Benabid* and P.J. Roberts Centre for Photonics & Photonic Materials, Department of Physics, University of Bath, Claverton Down, Bath BA2 7AY, UK (Received 23 August 2010; final version received 18 November 2010) We review the optical guidance properties of hollow-core photonic crystal fibers. We follow a historical perspective to introduce the two major optical guidance mechanisms that were identified as operating in these fibers: photonic bandgap guidance and inhibited coupling guidance. We then review the modal properties of these fibers and assess the transmission loss mechanisms in photonic bandgap guiding hollow-core photonic crystal fiber. We dedicate a section to a review of the technical basics of hollow-core photonic crystal fiber fabrication and photonic microcell assembly. We review some of the early results on the use of hollow-core photonic crystal fiber for laser guiding micro-sized particles, as well as the generation of stimulated Raman scattering, electromagnetically induced transparency and laser frequency stabilization when the fiber core is filled with a gas-phase material. We conclude this review with a non-exhaustive list of prospects where hollow-core photonic crystal fiber could play a central role. Keywords: photonic bandgap fiber; photonic crystal fiber; nonlinear optics; coherent optics 1. Introduction Since the advent of quantum mechanics and the demonstration of the wave nature of matter, solidstate physics has embraced a great number of concepts and analytical tools from optics. However, we had to wait almost 100 years to witness a conceptual transfer from the field of solid-state and quantum mechanics to optics, a process which began in the late 1970s with the development of periodic photonic materials, also known as photonic crystals. The proposal of the photonic bandgap by Yablonovitch [1] and John [2], in the late 1980s, added a great deal of impetus to the study of these structures. Among the geometries that have emerged as a consequence of the photonic crystal concept, the photonic crystal fiber (PCF) is perhaps the most developed. The PCF is an optical fiber whose cladding comprises a 2D periodic array of air holes that run down the length of the fiber [3]. Numerous forms of PCF exist, and many schemes can be envisaged with which to categorize them. The classification chosen here is based on the material which forms the core region and through which the majority of the optical guided mode field travels. A PCF in which the core is hollow, in the sense of being filled with air or some other gas, or evacuated to effectively contain vacuum, will be referred to as hollow core photonic crystal fiber (HC-PCF) [4]. A fiber which contains solid material within the guiding core will be termed a solid core PCF (SC-PCF) and a fiber with a liquid-based core will be called a liquid core PCF (LC- PCF). Each form of fiber can be sub-categorized according to the mechanism responsible for guidance. The SC-PCF and LC-PCF forms can guide by a modified form of total internal reflection (TIR) if an appropriate average of the refractive index within the cladding is below the index within the core. This form of guidance does not require a bandgap and in fact is entirely analogous to the guidance in conventional optical fibers. The PCF geometry, even in the TIR guiding form, does however offer a greater deal of flexibility in guiding properties due to the dispersion shown by the effective index of the cladding [5]. One striking example of this is the endlessly single mode (ESM) characteristic that becomes possible if the low index regions within the cladding are sufficiently small [5]. Alternative forms of SC-PCF and LC-PCF exist that guide by means of a photonic bandgap created by the cladding. Such structures are generally characterized by a cladding which contains isolated higher index inclusions, which can either be formed by a liquid of appropriate refractive index [6], or by up-doped regions within the glass or polymer host [7]. The guidance of such PCF forms, to a large extent, can also be understood in terms of an antiresonant reflecting optical waveguide (ARROW) model [8]. *Corresponding author. f.benabid@bath.ac.uk ISSN print/issn online ß 2011 Taylor & Francis DOI: /

9 88 F. Benabid and P.J. Roberts The HC-PCF form can guide either by virtue of a photonic bandgap (the BG-HC-PCF), or in the absence of a bandgap, due to an inhibited coupling between the core guided modes and modes associated with the cladding (the IC-HC-PCF) [9,10]. The former form of fiber is characterized by low attenuation over a restricted range of wavelengths dictated by the bandgap, whereas the latter shows somewhat higher propagation loss but guides over much broader wavelength ranges. All forms of HC-PCF which offer robust guidance involve inner cladding structures which contain very little solid material, a filling fraction less than 10% by area being typical. In addition to the silica-made HC-PCF geometries mentioned above, other optical hollow-core waveguides have been studied. These are distinct either because they are made of materials other than silica, or the cladding structure shows only one-dimensional refractive index variations [11], or the guidance significantly differs from BG or IC forms [12]. Figure 1(a) shows a HC-PCF that guides by means of a PBG, whilst the one shown in Figure 1(b) guides via IC. The properties of these fibers will be discussed in detail in later sections. The figure also shows other hollow-core waveguide examples. For example, the structure in Figure 1(c) is a HC-PCF in the sense that the cladding surrounding the core forms a twodimensional photonic crystal. This fiber has the distinctive feature of guiding with a highly enhanced intensity within a central nano-scaled hollow core. Here, the light is confined in and close to the hollow region because of the discontinuity of the electric field at the interface between silica and the central nanohole [12]. Figure 1(d) shows a hollow-core fiber made of two chalcogenide glasses arranged in concentric layers to create a Bragg optical fiber [13,14]. With a judicious choice of the layers refractive indices and spacing, this fiber guides via a PBG over all the possible incident angles which correspond to propagation within the core medium ( omniguidance ) [11]. The hollow-core waveguide shown in Figure 1(e) guides by virtue of a resonant reflection optical wave-guidance (ARROW) mechanism. This structure has the attribute of being integrated within a silicon chip [15]. Historically, the BG-HC-PCF was in fact the first guiding form of PCF to be envisaged, having been proposed by Russell in 1991 [3]. The possibility of bandgap guidance in an air core at the relatively modest glass/air refractive index contrast was theoretically demonstrated by Birks et al. in 1995 [16], prior to its experimental demonstration in 1999 [17]. Initial interest in the BG-HC-PCF was fuelled by the hope that the propagation loss could be reduced below the level attainable with solid core fibers (approx db km 1 ). This possibility arises since the Rayleigh scattering coefficient of air within the guiding core, at relevant pressures, is many times below that of silica. So far the loss of these fibers has been reduced to 1.2 db km 1, and it is now understood that the limit to further loss reduction is set by an intrinsic roughness of the air glass interfaces in the structure [18]. This roughness, being of thermodynamic origin, can not be greatly reduced by any known modification of the fiber drawing procedure. Instead, the route to loss reduction has involved designing the fiber to exclude as much of the guided light as possible from the interfaces. Constraints on fiber geometry imposed by surface tension that acts during the fiber draw, and the need for resilience to micro- and macro-bend loss, has so far prevented further loss reduction. Despite the loss of the BG-HC-PCF remaining a little too high to be of direct interest for long-haul signal transmission, its value is sufficiently low for a large number of applications that require guidance in Figure 1. Hollow-core waveguides: (a) triangular bandgap HC-PCF, (b) Kagome HC-PCF, (c) nano-void fiber, (d) Bragg fiber, and (e) Chip integrated waveguide.

10 Journal of Modern Optics 89 a hollow core to become realizable. The mode chromatic dispersion of these fibers ranges from large negative to large positive values across the bandgap, which makes them appropriate for compression of laser pulses down to femtosecond (fs) duration [19]. The intrinsic nonlinearity of air at standard temperature and pressure (STP) is nearly 1000 times below that of silica, which enables the mode nonlinear coefficient for BG-HC-PCF to be of order 1000 times below the value for a solid fiber of comparable mode area. This enables far higher field intensities and peak pulse powers to be handled before the onset of nonlinear processes degrade the output pulse characteristics [20]. Many applications of HC-PCF exploit the possibility of introducing a gas into the guiding core. The variety of optical properties that can be found in different gas-phase materials, in conjunction with an appropriate choice of the gas pressure within the core, confers a great deal of flexibility to the nature of light gas interactions within a HC-PCF host. In addition to being a natural choice for some gas sensing applications, these fibers can be used in the study of nonlinear and quantum optical processes in a gas medium. The low intrinsic loss of the HC-PCF waveguide allows an unprecedented length over which a laser field can interact with a gas in a diffractionless fashion, with a mode area maintained typically at order 10 2 for the BG form, being the wavelength. This contrasts with the intrinsic diffractive nature of free-space laser beams that prevents a high field intensity being maintained beyond a few Rayleigh lengths, which generally amounts to a few centimeters at most. As a result of the extremely high interaction figure-of-merit afforded by HC-PCF [9], the power required for generating stimulated Raman scattering (SRS) in hydrogen has been lowered by a factor of more than one million whilst exhibiting a near quantum-limited conversion efficiency [21], and quantum effects such as electromagnetically induced transparency (EIT) are made possible in molecular gases [22,23]. As an example, using acetylene filled HC-PCF, EIT has been observed over a grid of absorption lines that spans the telecommunication band, with prospects for optical telecommunications and slow light applications [23]. Technological implementations of devices based on gas laser light interaction in HC-PCF will require a degree of compactness and integrability. Without these qualities, the devices will be confined to scientific exploration within the laboratory. Fortunately, the development of the robust HC-PCF based gas cell, called a photonic microcell (PMC), holds promise for the deployment of the devices [24]. Other applications involving HC-PCF exist, perhaps the most intriguing of which is the guidance of micron-sized particles by a laser beam propagating through the hollow core of the fiber [25]. Together with the demonstration of SRS in hydrogen in 2002 [9], this was in fact the first reported application involving HC-PCF, and enabled laser particle guidance over distances more than 100 times longer than had previously been reported [25]. With today s HC-PCF transmission performance, one could envisage the intrusion-free transportation of micron-sized objects such as biological cells over hundreds of meters. This paper reviews the progress made in the understanding of the guidance properties of HC- PCF, in the fabrication of these fibers, and finally in many of the applications that involve HC-PCF. In Section 2, the guidance mechanisms responsible for guidance in BG-HC-PCF and IC-HC-PCF are first reviewed. The guidance properties, including dispersion, power-in-glass fraction and mode effective area, are then discussed with typical values and trends given for the most important classes of fiber. In Section 3, the fabrication techniques employed for HC-PCF and the photonic microcell are reviewed. In Section 4, applications of the fibers are discussed. The list of applications covered is not exhaustive, but represents the most commonly reported uses of the fibers. After a description of laser-induced particle guidance within HC-PCF, low threshold SRS generation in two different HC-PCFs filled with hydrogen gas is treated. Experimental observations of EIT, both in acetylene filled and rubidium-vapor filled HC-PCF cores maintained at low pressure, are then considered. Finally, as a demonstration of the PMC potential, an all-fiber laser stabilization unit using acetylene filled HC-PCF is described. The final section of this chapter offers conclusions and discusses likely future developments in the rapidly advancing field of HC-PCF. 2. Photonic guidance properties of HC-PCF 2.1. Guidance mechanisms Out-of plane photonic bandgap guidance The mechanisms responsible for guidance in hollowcore photonic crystal fiber (HC-PCF) are very distinct from the total internal reflection based guidance in conventional fibers, since the core-index of a HC-PCF is lower than the effective index of the cladding. Low-loss bandgap guidance in HC-PCF becomes possible if no modes associated with cladding exist close to and just below the air lightline over one or more frequency ranges. The air lightline is here defined by the effective index value n eff ¼ 1, or equivalently a propagation constant ¼ kn eff, where k is the free-space wavenumber k ¼ 2p=. The E and H fields of the modes supported by the waveguide are understood as being

11 90 F. Benabid and P.J. Roberts given by the forms Eðr, z; tþ ¼eðrÞ exp½iðz!tþš and Hðr, z; tþ ¼hðrÞ exp½iðz!tþš, respectively, where Oz is the Cartesian axis direction which coincides with the fiber axis and r ¼ðx, yþ. The angular frequency is related to k by! ¼ c 0 k, c 0 being the speed of light in vacuum. Figure 2 shows schematically how and the local wavenumber are related within an air region, and also how is related to the wavenumber, k gl ¼ n gl k, within the glass host material. It must, however, be stressed that ray-tracing based descriptions of guidance are not appropriate for BG-HC-PCF fibers. Instead the modal properties of the photonic structure that forms the cladding of a HC-PCF is conventionally found by casting the full-vectorial Maxwell Helmholtz equation as an eigenvalue problem. This is done by considering the spectrum of the propagating modes with a propagation component out of the periodic plane (i.e. finite ) for an infinite photonic crystal structure. The transverse wave-vector equation governing the H-field in this out-of-plane configuration is expressed as r 2 þ k 2 n 2 h? þr ln n 2 ð rh? Þ ¼ 2 h?, ð1þ where h? ðþis r the component of hðrþ in the x y plane (i.e. normal to the fiber axis) and n(x, y) is the index profile of the photonic structure. Birks et al. [16] reported the possibility of an out-of-plane PBG based on a photonic structure comprising a triangular arrangement of air holes in a silica background. Their results showed that the step index (index ratio between the higher-index region and the lower-index one) between glass and air is sufficient to provide PBGs that straddle the air lightline. This refractive index contrast is much lower than that required to form a bandgap for the case of in-plane propagation in 2D periodic dielectric structures (i.e. ¼ 0) [3,4]. This is so because the relevant index step is the transverse one n gl,? =n air,? ¼ K gl,? =K air,?, which can reach very large values, instead of the fixed and low value of n gl =n air 1:46 (see Figure 2). Whilst the silica/air refractive index contrast is insufficient for the scattering from any geometrical arrangement of these constituents to be strong enough to open up a bandgap for in-plane propagation, i.e. ¼ kn eff ¼ 0, the scattering strength increases as increases. As increases towards n air k, the transverse index ratio reaches sufficiently high values to enable gaps to open up below the air lightline. Robust out-of-plane photonic bandgaps become possible when the cladding is substantially periodic, with a number of lattice types being possible. By far the most studied is the triangular lattice shown at the top of Figure 3(a) for circular holes of varying diameter compared to the pitch L. The bandgaps responsible for waveguiding in a fiber geometry where the majority of Figure 2. Schematic diagram showing how the propagation constant is related to effective index, as well as the local wavenumber in air and glass. the cladding is effectively periodic, with several crystal periods surrounding the core, can most conveniently be calculated by considering an infinite periodic reference lattice. The periodic part of the cladding constitutes a sub-region of this reference lattice. In the reference lattice, Bloch s theorem is applicable, which implies the modes can be labeled according to k BZ, a wavevector within the first Brillouin Zone associated with the lattice periodicity [26], and a band index n. Considering the wave-equation at a fixed frequency!, as in Equation (1), the mode field distributions can be constructed from their dependence within a single unit cell according to E!,kBZ,nðr þ R, zþ H!,kBZ,nðr þ R, zþ ¼ exp i½ n ð!,k BZ Þz þ k BZ R!tŠ e!,k BZ,nðÞ r, ð2þ h!,kbz,nðþ r where R is any real-space lattice vector. The propagation constant is a function of the continuous variables! and k BZ, as well as the discrete band index n. For each n, the mode propagation constant plotted as a function of! and k BZ defines the dispersion surface n ð!, k BZ Þ. The bandgaps are identified as regions in the! plane where no states are found for any n or k BZ. Instead of solving a non- Hermitian wave equation such as Equation (1) at a fixed frequency, the problem can be re-cast with frequency considered as the eigenvalue and a continuous parameter, so as to map out the dispersion surfaces according to! n ð, k BZ Þ. The resulting wave equation for the H-field can be cast in a Hermitian

12 Journal of Modern Optics 91 Figure 3. Evolution of the band structure with the air-filling fraction. (a) Band structure for a triangular lattice with different D/L (the air-filling fraction is deduced from pðd=lþ 2 =2ð3 1=2 Þ). (b) The evolution of the edges and the centre of the lowestfrequency bandgap as D/L increases. (c) The evolution of the bandwidth with D/L. (The color version of this figure is included in the online version of the journal.) form, which renders the calculation of the bands more efficient if material dispersion can be neglected. The modes of a periodic dielectric medium are direct analogues of those associated with electrons in periodic crystals, and schemes for calculating photonic bands are an illustration of a fruitful transfer between solid-state physics and guided-wave optics. Numerical methods for calculating the photonic band dispersion surfaces n ð!, k BZ Þ, or equivalently! n ð, k BZ Þ, together with the corresponding modal fields (e!,kbz, nðþ, r h!,kbz, nðþ)or(e r,kbz, nðþ,h r,kbz, nðþ), r can be based for example on a plane wave expansion [27,28] or a finite element based decomposition [29]. A useful means of encapsulating many important aspects of the spectrum associated with a periodic cladding lattice is provided by the photonic density of states (DOPS), which is considered as a function of both wavenumber, k, and effective infective index, n eff or equivalently the propagation constant. The number of electromagnetic modes per unit cell and per unit axial length (along 0z), within the infinitesimal wavenumber range k to k þ dk, and infinitesimal effective index range n eff to n eff þ dn eff, is given by ðk, n eff Þ dk dn eff, where the DOPS ðk, n eff Þ can be calculated according to: ðk, n eff Þ ¼ A cell ð2pþ 3 k X n ð BZ d 2 k BZ ðk k n ðkn eff, k BZ ÞÞ, 1 st in which A cell is the area of a unit cell, (u) is the Dirac delta-function and k n ð, k BZ Þ¼! n ð, k BZ Þ=c 0. Knowledge of the dispersion surfaces enables the DOPS to be constructed. The bandgaps are identified as regions in the k n eff plane where the DOPS is zero. The bottom of Figure 3(a) shows the DOPS centered on the dispersion curve of the vacuum-line (vertical red line) for such a triangular lattice with different air hole diameter-to-pitch ratios D/L. The DOPS has been expressed as a function of ð kþl and kl. The colors ð3þ

13 92 F. Benabid and P.J. Roberts Figure 4. DOPS of a triangular lattice with an air-filling fraction of 92%. (The color version of this figure is included in the online version of the journal.) indicate the density of states (DOPS) of the modes supported by the structure and the bandgap is depicted in white. The DOPS values are normalized to that of silica bulk material [30,31]. For D/L ¼ 90% and below, the band structure exhibits several narrow bandgaps in agreement with the results in [16]. As D/L increases, the bandgaps located in the higher frequency region tend to shrink and eventually disappear, while the bandgap which at D/L ¼ 90% is centered at kl 9 follows a clear trend: it becomes wider and its location shifts to the higher frequencies (see Figure 3(b)). Figure 3(c) shows the evolution of the bandwidth 1 of this bandgap with D/L, and indicates an apparent cutoff value for D/L above which the bandwidth widens at a very fast rate as D/L is increased. For example, an increase of D/L from 94% to 96% doubles the bandwidth. The wide bandgap for D/L ¼ 100%, shown for pure academic purposes as its fabrication is impossible, suggests that the bandgap formation is primarily due to the glass features located at the interstitials of the stacked air holes. In fact the shift in the frequency location and the widening of the bandgap trends continue for D/L4100% (i.e. for smaller but better isolated interstitial features). A later section will provide a physical explanation of this behavior. The sensitivity of the width of the lowest frequency bandgap to the air-filling fraction, which is related to D/L for such a triangular arrangement by pðd=lþ 2 =2ð3 1=2 Þ, partly explains the dramatically fast reduction in the transmission loss that has been reported for HC-PCF based on this structure [32,33]: early fibers were constrained to relatively low air-filling fraction, but improvements in gas pressure control during the fiber draw has now enabled higher air-filling fractions to become accessible [9,34]. The holes in the cladding of fibers fabricated with high air-filling fraction (480%), such as the ones reported in [32,33], deviate from a perfectly circular form. Under the different forces present during the fibre draw (see Section 3), the resulting cladding structure comprises air holes with a more hexagonal shape. Crucially for bandgap formation, these hexagonal shapes possess curved corners (see inset of Figure 4) so that glass nodes are formed at the junction of three cladding holes [35]. The evolution of the PBG with air-filling fraction keeps the same general trends as for the circular hole case. The air-filling fraction can now be higher than the upper limit of 91% for a perfect triangular lattice of circular holes, and hence the bandgap can be shifted to even higher frequency regions. This is exemplified by the DOPS diagram in Figure 4 which shows a PBG extending below the airline for a lattice structure that possesses 92% air-filling fraction. The central normalized frequency is around 15.5 which is much higher than the value 11 found for the case of an ideal triangular lattice of touching air holes with circular cross-section. The complicated singly-connected nature of the high index cladding component, exemplified in the insert to Figure 4, renders quantitative analytic approaches to the calculation of the dispersion surfaces and the bandgap regions unfeasible. Nevertheless, some characteristics of cladding modes and the process of bandgap formation can be qualitatively understood by employing simple toy-models. One can either approach the problem starting from a nearly free photon picture [2], or from a localized-basis tight binding picture, both of which are direct

14 Journal of Modern Optics 93 analogues of simple schemes employed for electronic band structure calculations in solid state physics. The latter approach provides the greater insight into the nature of the bandgap formation and resilience to the effects of disorder, and will be discussed below Intuitive models of photonic bandgap guidance The results shown above are based on numerically solving a wave equation such as the one shown in Equation (1). Though powerful, the numerical approach rarely provides strong insight on how a PBG forms and does not constitute a simple predictive design tool. In an endeavor to provide an intuitive tool, a model based on the photonic analogue of the tightbinding model in atomic physics has been employed in [36]. The analysis of the nature of cladding modes shows that the PBG is formed by three types of underlying cladding mode constituent: the light at the upper effective index band-edge is confined mainly to the glass nodes which exist where three glass struts meet (see the inset to Figure 4), whereas the lower effective index band-edge results from the interplay between a mode associated with the struts and another which is mainly localized within the holes. The details of the model are developed below. In order to show how a PBG forms in microstructured optical fibers we start by giving a general historical background, and then exemplify the formation of a PBG by using the photonic tight-binding model (PTBM) for a simple 1D array of optical fibers. Finally, we explain in the framework of PTBM the genesis of a PBG in a realistic triangular-lattice HC-PCF and identify the structural constituents of the photonic structure responsible for the photonic bands Historical background. Yariv and Yeh considered the nature of the cladding states which exist outside the bandgap spectral region. In particular, they studied how light propagates within different constituent features of a photonic crystal cladding [13,14]. In analyzing their proposal of a Bragg fiber (i.e. a fiber with a cladding made of concentric solid rings with alternating refractive indices) in the 1970s [13,14], the authors pointed out qualitatively that the allowed cladding bands are composed of several constituent modes mainly confined in the high-index layers. The number of modes is given by the number of these layers. As the frequency and the effective indices of the modes in a particular allowed band become higher, the band shrinks to a set of degenerate modes confined in the high-index layers. This observation clearly indicates the role of the resonance properties of the high-index layers in the formation of the bandgap [37]. Subsequently, Duguay et al. introduced the terminology of Anti-Resonant Reflective Optical Waveguides (ARROW) [38]. The ARROW model has since been successfully applied to some classes of photonic crystal fibers [8,39]. In these studies [8,39], the authors consider PCFs with a structure composed of a lowindex background material with isolated high-index cylindrical inclusions. They show, using the ARROW model, that the frequencies of transmission bands are primarily determined by the geometry and the modal properties of the individual high-index inclusions irrespective of the lattice constant value. Consequently, one can consider the high-index inclusions as optical resonators whose modal properties largely determine the properties of the formed bandgaps. More recently, Birks et al. [40] proposed a semianalytical approach to understanding the formation of bandgaps in all-solid bandgap PCFs with the same type of cladding structure as considered in [8,39]. The authors modeled the out-of-plane mode structure of the fiber cladding by employing an approximate technique related to the cellular method used in solid-state physics [26]. This method explicitly takes into account electromagnetic coupling between each high index cylinder within the cladding and, in contrast to the ARROW model, provides information on the width of the cladding pass-bands. The model also elucidates the hybridization which occurs between resonances associated with the localized high index regions and the low index regions which fit between them. The cellular method proposed in [40] provides important and useful insights into the guidance mechanism of all-solid PBG photonic crystal fibers which comprise arrangements of separated high index regions within a lower index background, and on the nature of bandgap formation within the claddings of these fibers. However, the complex topological structure of the silica air PCF cladding, such as possessed by the HC-PCF shown in Figure 1(a), makes it very challenging to identify the relevant optical resonators responsible for the PBG formation. Indeed, the singly-connected nature of the high index component of such fibers calls into question the applicability of the picture of guidance and bandgap formation based on the influence of a few constituent elementary resonator features. Consequently, in order to assess the applicability of a resonator picture as an aid to designing realistic HC-PCF, it is useful to develop experimental or theoretical tools aimed at identifying any relevant cladding features which may be present.

15 94 F. Benabid and P.J. Roberts Photonic tight-binding model: toy model for PBG formation. The tight binding model has been successfully used in atomic and solid-state physics to give an approximate description of the origin of electronic bands (allowed and forbidden) [26]. In this picture, electronic bands in a solid (e.g. a crystal) are considered to be the result of bringing together isolated and identical atomic sites to form the solid. This approach has the double advantage of holding a high degree of predictive power as well as being simple. Indeed, the model starts with the Hamiltonian of a single atom with its well defined energy levels and wavefunctions. As these sites move closer together, each single electronic state, whose wavefunctions (orbitals) were well localized within the potential well created by the single atom, interacts with corresponding electronic states at neighboring lattice sites to form a band of energy levels with delocalized wavefunctions (Bloch functions). The wavefunctions are all a linear superposition of the isolated atomic orbitals. For crystal symmetries pertinent to the current discussion, the upper energy level of the formed band is formed by an anti-symmetric wavefunction, in which the sign of the wavefunction changes upon moving from a chosen site to a nearest neighbor. The lower energy level is symmetric, i.e. it shows no sign oscillations, and is a more tightly-bonded state (see left-hand side panel of Figure 5). We expect that a simple transposition of the tight-binding model to a photonic structure will provide a wealth of information on the nature of the modes in the allowed bands and on the formation of bandgap (see right-hand side panel of Figure 5). Let us now consider a one-dimensional array of N identical optical fibers. These fibers consist of silica rods with a radius and are spaced by a pitch L. Figure 6(a) shows the dispersion curves for the fundamental mode and the degenerate second-order modes. For simplicity, the curves were deduced from an approximate analytical expression for the propagation constant of the fundamental and the second-order modes [41]: 2 ¼ðn co kþ 2 U2 1 k 2 2 ðk þ =Þ 2 : ð4þ Here, U 1 is a constant equal to for the fundamental mode and to for the second-order modes. is a function of the refractive index of the rod, n co, and that of the background, n bkg, and is given by ¼ð1 DÞ=n co ð2dþ 1=2, with D ¼ðn 2 co n2 bkg Þ=2n2 co. In our model n co is taken to 1.45 and n bkg ¼ 1, i.e. the index of air. The case of an array of N silica rods can be solved using an approach based on coupled mode theory, which is equivalent to a tight binding description. For the sake of simplicity we only account Figure 5. Schematic illustration on how allowed bands and band gaps form in the case of a crystal made of several and identical atomic sites and the corresponding situation for several identical waveguides. (The color version of this figure is included in the online version of the journal.) for the coupling due to adjacent rods. We consider a Gaussian profile for the rod index so that the coupling coefficient, C, between two identical rods is given by the following simple analytical expression which is valid for the fundamental and second order rod modes [42] CL ¼ ð2pþ 1=2 n co D ~ kl½ðnco ð2dþ 1=2 ÞkDŠ 1=2 E ð ~ 1Þð ~ 1 n co ð2dþ 1=2 kdþ : Here ~ ¼ =L is the normalized radius. The set of the coupled equations is reduced to an eigenvalue problem based on the following tri-diagonal N-dimensional matrix 0 1 C C C C C... 0 : ð6þ B A C Consequently the individual modes of the N isolated rods are found to form a band of N closely packed modes (a continuum for the case of N ¼1) whose propagation constants are given by: j L ¼ L 2 ðclþ 2 1=2cos jp, j ¼ 1, 2,..., N: N þ 1 ð7þ Figure 6(b) shows the dispersion of these modes for N ¼ 1000 and ¼ 0.45L. The plot clearly shows the formation of the allowed photonic bands whose spread ð5þ

16 Journal of Modern Optics 95 Figure 6. Photonic tight-binding model. (a) Dispersion curve of fundamental and second higher-order modes of a silica rod. Here the modes below the air-line are ignored. (b) Dispersion diagram of an array of silica rods. (c) Dispersion diagram of an array of silica rods after an appropriate transformation. (The color version of this figure is included in the online version of the journal.) in effective index increases as the normalized frequency becomes smaller. The plot also illustrates the existence of a PBG which extends to effective indices lower than 1, indicating the possibility of trapping light in a hollow defect. In order to draw a clear parallel with the tight-binding model used in atomic physics, Figure 6(c) shows the same dispersion information as Figure 6(b) but the index is replaced by ½ðn co kþ 2 2 Š 1=2. This figure shows the splitting of the fundamental mode and its transformation into an allowed band as kl decreases, which can be realized either by moving the rods closer to each other or reducing the frequency. Furthermore, by direct analogy with the tight-binding model, the mode of the lowest energy corresponds to the most strongly confined mode (tight-bonded mode). This mode is also named the fundamental space-filling mode in the community of PCF [5]. Finally, one notices that the PBG becomes weakly dependent on the pitch as kl increases and that asymptotically the band-edges are solely determined by dispersion curve for the mode of a single rod. This behavior is the central premise behind the ARROW model [23]. Consequently, the PTBM offers a simple and powerful framework for the study of the formation of PBG in photonic structures and provides a sound foundation for the ARROW model Photonic tight-binding model description of HC-PCF cladding. The previous section illustrated a toy model aimed at elucidating how a PBG forms using an approach analogous to the tight-binding model.

17 96 F. Benabid and P.J. Roberts Figure 7. (a) SEM of a HC-PCF with core guidance at 1064 nm. (b) Details of the cladding structure used for the numerical modeling. (c) Brillouin zone symmetry point nomenclature. (d ) Propagation diagram for the HC-PCF cladding lattice. Black represents zero DOPS and white maximum DOPS. The upper x-axis shows the corresponding wavelengths for a HC-PCF guiding at 800 nm (pitch L ¼ 2.15 mm). The solid and dotted lines represent the -point and J-point mode trajectories, respectively. The trajectory of the cladding modes on the edges of the PBG are represented in red for the interstitial apex mode with the near field (e), in blue for the silica strut mode ( f ) and in green for the air-hole mode (g). The first two modes are shown at an effective index of (dash-dotted white line), whereas the air-hole mode is shown at kl ¼ 15.5 and n eff y ¼ (The color version of this figure is included in the online version of the journal.) Here, we report on experimental and theoretical results which validates this approach for a realistic HC-PCF. Figure 7(d) to (g) show the numerically calculated density-of-states (DOPS) [31,43] and the modal properties of the HC-PCF cladding structure shown in Figure 7(a) and (b). Either side of the out-of-plane PBG region (shown in black in Figure 7(d)) are allowed bands where the cladding can support propagating modes (regions 1 and 2 in Figure 7(d)). These bands follow the PTBM in the sense that the bands shrink as kl increases. Of particular interest is the mode at the lower-index edge of region 1 since it determines the lower-frequency edge of the PBG of the cladding. The dispersion of this mode is represented in red in Figure 7(d) and its near-field (NF) is shown in Figure 7(e) at a representative normalized wavenumber kl. This confirms that the light is predominantly guided in the interstitial apexes. The apexes are thus identified as the most important optical resonators associated with the lower-frequency bandgap edge [44]. The nature of the modes at the upper-frequency edge of the PBG (i.e. the larger kl of the PBG at n eff 1) is more complicated than those at the lower-frequency edge. Indeed, the lower edge is formed by several and overlapping allowed photonic bands resulting in a band-edge being formed by a combination of the trajectories of two cladding modes of different symmetry. At frequencies (i.e. kl) below kl ¼ 16.9, this edge is due to a mode located at the J-point of the Brillouin zone (see Figure 7(c)), which guides predominantly in the air holes ( air-hole mode) shown by the dotted green curve in Figure 7(d)). Above kl ¼ 16.9, the edge is predominantly due to a mode located at the -point (Figure 7(f), blue traces in Figure 7(d)) guiding within and close to the silica struts ( strut mode) which join neighboring apexes, with little light penetrating into the apexes. Consequently, the silica struts directly limit the upper frequency of the fiber transmission band. These observations were corroborated experimentally by measuring spectrally-resolved near field (NF) images and spatially-resolved transmission spectra of 3 mm long HC-PCFs using a scanning near-field optical microscope (SNOM). Both the NF and spectra were taken by exciting the fiber with super-continuum light over a narrow angular range near the air lightline. The transmission spectra of the relevant modes show distinctive cut-offs relating to the fiber PBG location. Figure 8(a) shows the typical measured NF profiles when the fiber is excited by light near the lowerfrequency bandgap edge and the upper-frequency edge (Figure 8(a) (part 1) and 8(a) (parts 2 and 3), respectively. The NF is obtained by scanning the SNOM tip, with 0.15 mm step, over an area covering a few unit cells of the HC-PCF cladding structure. Figure 8(a) (part 1) clearly shows that the imaged mode corresponds to that of the apex resonator whilst Figure 8(a) (parts 2 and 3) show light confined predominantly in the struts and air, respectively. Moreover, when the tip is aligned with an interstitial apex, the transmission spectra of the two different fibers show a cut-off around the lower frequency edge of the PBG (see the solid black line on the lower graph of Figure 8(c) for the 1060 nm HC-PCF).

18 Journal of Modern Optics 97 Figure 8. (a) SNOM images of the (1) apex mode (2) strut mode and (3) air-hole mode of the fiber cladding. (b) Optical spectrum of the HC-PCF guiding around 800 nm taken with the SNOM tip aligned with the core (black line) and near an air-hole of the cladding (gray line). (c) Optical spectrum of the HC-PCF guiding around 1064 nm taken with the SNOM tip aligned (top) with the core, (bottom) with an interstitial apex (black solid line) and with an air hole of the cladding (gray doted line). The peaks around 1064 nm are due to the residual super-continuum pump. (The color version of this figure is included in the online version of the journal.) This corresponds to the apex mode frequency cut-off near the air-line in accordance with the numerical simulation (see Figure 7(d)). Similarly, for strut and air-hole modes, the transmission spectra show a clear cut-off at the short wavelength side of the HC-PCF transmission bandwidth (see the gray line of Figure 8(b) for the 800 nm HC-PCF and the gray dotted line of the lower graph of Figure 8(c) for the 1060 nm HC-PCF). However, due to the limited spatial resolution of the SNOM and to the hybridization between the two constituent resonators, the transmission spectra collected when the tip was aligned on top of a strut or in an air hole do not show a measurable difference in their frequency cut-off. Nevertheless, the above results using the SNOM do confirm that the PBG is formed by the interplay of three distinct resonators Broadband hollow core guidance Bandgap guidance enables low propagation loss to be attained, but the bandwidth for guidance is restricted by the achievable width of the bandgap close to the air lightline. Since the fiber must be mechanically sound, the glass nodes which can act as antiresonant reflecting optical waveguides are compromised by the necessary presence of connecting glass struts, which restricts the frequency range of the broadest bandgap in practical glass/air structures to about 40% of the bandgap central frequency. The bandgap fibers with high airfilling fraction claddings, that have recently become accessible to fabrication, enable higher order gaps to open up, but these are relatively narrow and separated by wide spectral regions where attenuation within the core will be high. Many applications either require or would benefit from an increased bandwidth of guidance. A class of fiber which provides broad frequency ranges of relatively low propagation loss is based on cladding structures which entail a connected network of nearly constant thickness glass struts. Examples of such fibers are shown in Figures 1(b) and 9(b). The guidance within the core shown by these fibers does not rely of the formation of cladding bandgaps. Instead, the cladding structure is designed in such a way that the cladding modes which are present close to the air lightline do not interact strongly with the core mode of interest. The strategy behind the design of broadband guiding fibers is, in many respects, opposite to that adopted for bandgap fibers. The latter rely on sizeable glass nodes where the struts cross to form the bandgap, and the glass struts which are necessary to connect them act as distinct resonators that tend to narrow the bandgaps. The broadband guiding fiber designs, on the other hand, require that the glass nodes are maintained small, since these possess resonances which can exist close to the air lightline at wavelengths distinct from the transverse resonances associated with the struts, thus narrowing the guiding bandwidth available. A common theme behind the design of hollow core fibers is to minimize, as far as possible, the number of

19 98 F. Benabid and P.J. Roberts Figure 9. (a) Top left: scanning electron micrograph of a triangular-lattice HC-PCF. Bottom left: near-field profile of the fundamental (HE 11 -like) air-guided core mode lying within a bandgap (center). Right: Band diagram showing the presence of the PBG. (b) Same as (a) but for Kagome-lattice HC-PCF. The fundamental mode lies within a continuum of cladding modes and the band diagram does not exhibit a PBG. (The color version of this figure is included in the online version of the journal.) distinct forms of resonator that operate close to the core mode effective index (which is very close to the air lightline, n eff ¼ 1). Aspects of the IC-HC-PCF guidance can be understood by considering a hypothetical glass/air Bragg fiber, which comprises concentric layers of silica in air, such that geometry possesses cylindrical symmetry, see Figure 10. Each annular shell of silica has the common thickness t. Due to the symmetry, the modes of this fiber geometry decouple according to an azimuthal field variation of the form expðim Þ in the z-components of their E and H fields; modes with different m do not interact. The confinement loss of the HE 11 core guided mode, which belongs to the m ¼ 1 mode class, can be reduced arbitrarily by incorporation of a sufficient number of glass shells, except close to transverse resonances given by the condition, kðn 2 gl 1Þ1=2 t ¼ p j, (j ¼ 1, 2, 3...), with n gl the refractive index of glass. The IC-HC-PCF also shows low transmission at wavelengths near where this resonance s Figure 10. A Bragg fiber comprising concentric silica shells of thickness t in air. is fulfilled. The Bragg fiber does possess cladding modes which can phase-match with the core mode, but symmetry precludes interaction since the former are from a different m mode class. The singly connected R t

20 Journal of Modern Optics 99 Figure 11. Dispersion properties of the fundamental-like mode in an example BG-HC-PCF. (a) The example structure, which incorporates features within the core surround which render this region antiresonant, a property which lowers loss. (b) Trajectories of HE 11 -like mode and those of nearby modes with the same symmetry. The HE 11 -like mode branches interact with steeper mode branches associated with core interface modes. (c) The GVD over the operational bandwidth of the fiber. For the example fiber, the bandwidth is constrained by anti-crossing events, but the bandgap edge can also curtail the useable wavelength range. (The color version of this figure is included in the online version of the journal.) nature of IC-HC-PCF dramatically lowers the symmetry from C 1, so symmetry-induced decoupling is not relied upon to give good core guidance. Instead, mode incompatibility primarily associated with very different phase variations of the modes inhibits the coupling Modal properties of HC-PCF At wavelengths of low-loss transmission loss, the core modes of both BG-HC-PCF and IC-HC-PCF are quite similar to those of a capillary fiber [45] with regard to the field distribution within the core and the mode effective index trajectory. The effective index trajectory does differ in detail, particularly for the BG-HC-PCF. This can be quantified in terms of the group velocity dispersion (GVD) variation which is the result of two contributions: waveguide dispersion due to confinement of most of the field to the core, and claddingrelated dispersion which is due to alterations in the field distribution within the cladding as the wavelength changes. The latter dispersion contribution increases sharply and dominates as the bandgap edges are approached, and is a necessary consequence of the loss experienced outside the bandgap; a Kramers Kronig relation relates loss to dispersion. The resulting dispersion for the BG-HC-PCF shows the characteristic S-shaped GVD form of bandgap guidance, and a red-shift of its zero-crossing relative to the central frequency of the transmission spectrum of the fiber due to the positive waveguide GVD contribution. The dispersion of an example BG-HC-PCF is shown in Figure 11. Anticrossing events occur between the HE 11 -like mode branch and modes associated with the core surround glass region. The impact of these events on the GVD mimics that of a band gap edge. The associated mode hybridization close to an anticrossing leads to an increase in the mode power in glass fraction,, for the HE 11 -like branch. Figure 12 shows for the example fiber, together with a quantity F which is a measure of the mode intensity at the glass interfaces. The latter provides information on the expected mode propagation loss, see Section Anticrossing events are discussed in more detail below. Light can be guided with low attenuation in an air core defect providing that the air core accommodates air-dominated modes within the (k,n eff ) region of the cladding PBG. In order for most of the field to be guided in the air core, the effective index n eff is necessarily below unity, with the value increasing towards unity as the core size increases and more mode power fraction resides in air. For typical cladding structures, the depth below n eff ¼ 1 that the

21 100 F. Benabid and P.J. Roberts (a) (b) η 0.01 FΛ kλ 0.1 kλ Figure 12. (a) The power in glass fraction of the HE 11 -like mode as a function of the normalized wavenumber kl. (b) The normalized measure of mode interface intensity, FL, defined in Equation (13) below, as a function of the normalized wavenumber kl. In both plots, the peaks around kl ¼ 15.4 and 18.1 are due to mode anti-crossing events. (The color version of this figure is included in the online version of the journal.) Figure 13. Experimental near-field profiles for a seven-cell defect core HC-PCF guiding at 1300 nm. (a) Fundamental mode HE 11 ;(b) and (c) high-order core modes TE 01 and TE 01, respectively; (d) anti-crossing between fundamental and surface modes; (e) high order core mode; ( f ) surface mode. bandgap extends is such that air-dominated guidance can be achieved with a circular core defect of diameter larger than the cladding pitch [14]. Within the PBG, the lowest-order fundamental core-guided mode (Figure 13(a)), which within the core is similar in field distribution to an HE 11 mode of a standard fiber, is the core-guided mode with the highest propagation constant and closest to the air lightline (HE 11 in Figure 13). For larger cores, higher-order air-guided modes (Figure 13(b), (c) and (e)) can propagate at lower n eff (HOM in Figure 13). Modes which are not primarily guided within the air core, but are localized within and close to the core-surround region, are typically also present within the bandgap. Despite much of the mode field residing in glass, the effective index of these modes can attain values below unity since the fields show quite rapid phase variations within the fiber cross-section. Within the PBG, the trajectory of the air-guided modes in the k n eff plane is flatter than those of the cladding modes and the coresurround guided modes. The core mode trajectories are almost horizontal in the DOPS diagram shown in Figure 14, which maps the DOPS as a function of and n eff. The core guided mode trajectory is limited

22 Journal of Modern Optics 101 Figure 14. The DOPS of a typical HC-PCF with a seven-cell core, designed to guide around 900 nm, plotted as a function of wavelength and effective index n eff. The colored lines represents the dispersion curves of different defect modes. (The color version of this figure is included in the online version of the journal.) on either side by the continuum of modes creating the PBG s edges (Figure 14). Close to the bandgap boundaries, the core mode couples to the continuum, leading to high loss and increased dispersion. This effectively limits the operating bandwidth of the hollow core PCF to a relatively narrow range. As a result, the pitch L of the fiber s cladding has to be chosen carefully for the fiber to guide at a given wavelength. In addition to this optical bandwidth limitation imposed by the photonic crystal cladding, a further constraint is caused by the interaction of the air-guided modes with surface modes. These modes, called surface mode by analogy to electronic surface states in solid state physics, originate at the interface between the photonic crystal cladding and the core defect (Figure 13(f)). Due to the large amount of light propagating in the silica core surround, the dispersion of these modes, represented as solid lines in Figure 11, has a steeper slope than the air-core modes and can occasionally intersect with them inside the PBG. If the surface and core mode have the same and some degree of symmetry and spatial overlap, an anticrossing occurs where the dispersion curves of the modes repel each other (see Figures 11 and 14), leading to a dramatic increase in the transfer of energy from the core mode to the high attenuation surface mode. These high attenuation regions at the frequency of these anti-crossings further reduce the operational bandwidth and affect the overall transmission of the HC-PCF. It is also interesting to note that these surface modes have two cut-off frequencies and as a consequence cannot be identified as a TIR guiding process.

23 102 F. Benabid and P.J. Roberts Figure 15. Bandgap cladding structures with (a) 85%, (b) 92% and (c) 96% air-filling fraction. All these cladding forms are amenable to fabrication Loss mechanisms Confinement The bandgap fibers can be rendered free of confinement loss over the bandgap range by ensuring a sufficient number of cladding periods surround the guiding core. In practice, for a cladding with a structure corresponding to Figure 15(a), 10 layers are sufficient to confine the HE 11 -like core mode over the majority of the bandgap such that confinement loss is below 1 db km 1. For a cladding of the form shown in Figure 15(b), just eight periods are required, and for the form shown in Figure 15(c), only six are needed. The requisite number of cladding periods for all these cladding types, which in air-filling fraction range from 85% to 96%, is readily incorporated into fabricated fibers, which leaves other loss mechanisms responsible for attenuation in BG-HC-PCF. Chief amongst these is scattering loss due to roughness at the glass air interfaces, which will be treated in the next subsection. The main loss mechanism in IC-HC-PCF is confinement; the absence of a photonic bandgap, and the presence of weak residual interaction between core and cladding mode constituents, implies that incorporating more and more cladding periods does not lead to progressive loss reduction. The number of cladding periods beyond which further loss reduction does not occur depends on the details of the cladding structure, but for the example structure shown in Figure 9(b), this threshold is found to be just two periods. The realization that most of the guided power resides outside the solid fiber constituent, and that confinement loss is constrained in practical designs to be of order 1 db m 1, has given impetus to the development of polymer-based HC-PCF [46]. If the fiber draw can be sufficiently well controlled when using such a material, the loss is expected to be similar to that of the silica-based IC-HC-PCF. In fact, it appears that the IC-HC-PCF are easier to fabricate when using polymers than the BG-HC-PCF forms, and the former have so far shown lower overall loss than the latter with use of this material Scattering loss Since most of the guided light in BG-HC-PCF resides in the core or the cladding holes, mode attenuation due to Rayleigh scattering loss within the silica is much reduced compared to all-solid fibers. The Rayleigh scattering coefficient of air at STP is many times below that of silica, so it was hoped that the loss of the BG- HC-PCF could be reduced below the figure of 0.16 db km 1 attainable with the solid fibers, which would open up exciting possibilities for deployment of the BG-HC-PCF as a long-haul communication fiber. Fabricated BG-HC-PCF showed considerably higher attenuation than this, yet the confinement loss was confirmed by calculation to be negligible. It was clear that the presence of the glass air interfaces was detrimental to the loss, as had been inferred from earlier studies of small-core TIR-PCF. The roughness at the interfaces is primarily due to surface capillary waves (SCWs) which become frozen-in as the glass (or polymer) moves through the glass transition during the fiber draw [18]. The time-scales associated with the glass-forming process are such that equilibrium thermodynamics are believed to be sufficient to describe the static SCWs that remain after the transition; the mean energy of each SCW component is k B T tr =2, where k B is the Boltzmann constant and T tr is the fictitious glass transition temperature [47]. The relative phase of each thermally excited SCW is random, resulting in a rough surface which must be described statistically. For an infinitely extended 2D interface, the energy per unit area of a SCW with 2D wavevector k SCW and amplitude A is given by E ¼ 1 4 jaj2 jk SCW j 2, ð8þ where the surface tension has been denoted by. This leads to a roughness power spectrum ~C ðk SCW Þ ¼ k BT g 1 j j 2 ð9þ k SCW

24 Journal of Modern Optics 103 for the extended interface. The roughness spectrum given in Equation (9) is scale-free, i.e. it describes a statistically fractal interface. The spectrum describing the rough interfaces within a PCF becomes quantized due to the closed nature of a hole perimeter. Ignoring any interaction between SCWs on neighboring holes and any effects related to the curvature of the (statistically averaged) hole interface, the roughness spectrum at a hole with perimeter length S can be approximated by ~C m ðþ ¼ k BT g 1 1 h i, S ks ðmþ2 þ 2 ð10þ where k ðmþ s ¼ 2pm=S is the azimuthal wavevector component, the index m takes all integer values and is the wavevector component along the hole axis direction. Atomic force microscope (AFM) measurements of the roughness spectrum on the interfaces within silica PCF are consistent with the form given in Equation (10) over the scale-range covered by the measurements (approximately 80 nm to 8 mm) [18]. The roughness measurement was only taken along the axial direction P of the hole, so only correspondence of 1 ~ m¼ 1 C m ðþ could be checked. If the SCWs are formed by an equilibrium thermodynamic process, they can not be altered greatly by modifying the fiber draw process. The level of roughness for a given geometry is dictated by the ratio T tr /. Adding dopants to the glass can alter the values of and T tr to some degree, but no additive is known that will lead to a significant decrease of the ratio T tr /. Given the seemingly immutable nature of the interface roughness, the strategy employed to reduce the loss has been to design the fibers such that they exclude light as much possible from the interfaces. This has either involved introducing features to the glass ring that surrounds the air core to render it antiresonant within the wavelength range of the bandgap, or thinning this core-surround ring which has the effect of dispelling unwanted core-surround related guided modes from the bandgap close to the air lightline. The former approach ultimately enables lower loss to be achieved, but suffers from a reduced useable bandwidth due to an increase in the number of core-surround related guided modes which reside in the bandgap near n eff ¼ 1. Conventional wisdom specifies that dispelling the unwanted core-surround related guided modes from the bandgap close to the air lightline, so their trajectories in n eff space do not come close to the trajectory of the wanted core mode, will lead to a reduction in the field intensity of the latter mode in the proximity of the core surround, including its interfaces. The idea is that interaction and consequent hybridization will decrease with increasing de-tuning. If the core-surround modes are rendered leaky by being pushed into the continuum of cladding modes, this will also tend to decrease the core-mode field intensity within the core-surround region due to any residual interaction. Such a core-surround mode expulsion can be achieved in a practical design by appropriately thinning the core surround ring [48]. Fabricated examples of fibers with thinned core surrounds and a high air-filling fraction within the cladding have indeed shown reduced loss, and also have the benefit of a broad low-loss operating wavelength range. The latter aspect also leads to a lower and more slowly varying dispersion, which is often beneficial in applications. It is noteworthy that, in the attainment of ultimate low loss, the favorable field exclusion properties of antiresonance can prevail over the detrimental effects associated with the encroachment of core-surround modes into the bandgap. The scattering loss incurred due to the hole interface roughness differs from the Rayleigh form familiar from scattering due to sub-wavelength inhomogeneities. Propagation within bulk glass is characterized by the 4 Rayleigh wavelength dependence, so if the mode power fraction within glass is (), the bulk-glass scattering loss within the bandgap of a BG-HC-PCF can be estimated from gl ðþ ¼ gl ð Þ 4, ð11þ where gl is the Rayleigh loss coefficient of the glass. This expression assumes the influence of the interfaces on inhomogeneities within the body of the glass which make up the fiber s microstructure is negligible. If the roughness at the air glass interfaces were sub-wavelength, the scattering loss it would cause could be estimated from ifaces ðþ ¼ ifaces F ðþ 4, ð12þ where ifaces is a coefficient which characterizes the level of interface roughness and the glass/air index contrast, and F() is defined by Ð FðÞ ¼ " 1=2 hole dsjej 2 0 Ð interfaces 0 cross-sectiondse^ ð HÞz, ð13þ where E and H are the electric and magnetic field distributions, respectively, of the core guided mode and z is the unit vector along the fiber axis. The interface roughness spectrum given in Equation (10), however, has significant weight at length scales of the order and longer than the optical wavelength. In fact, the roughness amplitude increases with increasing spatial

25 104 F. Benabid and P.J. Roberts scale, i.e. diminishing k-vector, so that the simple law given in Equation (12) can not be expected to hold. Perturbation theory can be invoked to calculate the scattering due to roughness that has a general spectral form. The resulting expression is complicated when roughness components exist on length scales of order the wavelength, but the increase in the spectral power with increasing length scale is reflected by a general increase in the scattering rate into modes with effective index values close to that of the guided mode, n 1. The scattering into modes which have a high proportion of modal power at interfaces close to the core is also favored. These observations suggest that coupling into bandgap guided core-surround modes will be particularly strong, and indeed an analysis of loss has been attempted based solely on mode-coupling to these CS modes [48]. Except for close to anti-crossing events, however, this latter analysis does not adequately describe the loss, and coupling to the full continuum of cladding modes needs to be considered. Such calculations, assuming a roughness spectrum of the form given in Equation (10), have not yet been attempted for bandgap fibers, but for TIR guiding PCF, the analysis predicts a weakened wavelength dependence compared to the Rayleigh scattering result given in Equation (11) [49]. A BG-HC-PCF only guides over the relatively narrow wavelength range associated with the bandgap, so it is pertinent to consider how the loss scales when the fiber structural dimensions are scaled in proportion to the central operating wavelength c. For a given fiber, c can conveniently be considered as the wavelength where the loss is minimized. This form of scaling enables a comparison of the expected loss of fibers which are fabricated according to the same design, but with different spatial scales and hence target wavelength range. Since the refractive index of glass does not change greatly over the wavelength range of interest, the scaling behaviour of the roughness scattering loss is dictated by the form of the roughness power spectrum. The roughness form given in Equation (10) gives rise to a loss dependence of the form ifaces ð c Þ ¼ L¼ð c = c0 ifaces ð c0 Þð c = c0 Þ 3, ð14þ ÞL 0 where c0 is some reference wavelength, and the stipulation L ¼ð c = c0 ÞL 0 under the equality sign reminds us that the structural dimensions scales with the wavelength. Alternative glasses or polymers can be considered in connection with reducing roughness scattering loss but, at least for glasses for which T tr and have been tabulated, these all appear to have a larger T tr / ratio than silica. Nevertheless, a possible route to loss reduction involves using glasses which transmit at longer wavelengths, since the mode propagation loss at a given roughness amplitude decreases rapidly with wavelength [49]. Glasses with low refractive index are preferred in connection with loss reduction, since the loss scales with ðn 2 g 1Þ2. Of the glasses with tabulated properties, ZEBLAN glass, which transmits with low attenuation in the mid-ir, has been identified as a candidate material from which to fabricate BG-HC- PCF that show lower loss than can be attained using silica. It remains to be seen whether the mechanical properties of ZEBLAN glass allow the intricate BG- HC-PCF geometry to be realized during a fiber draw. 3. Fabrication of hollow core photonic crystal fibers and photonic microcells 3.1. Fabrication procedure of HC-PCF The fabrication of all PCFs, independent of their cladding structure, follows a common core-procedure [50]. It is based on drawing a tube (usually made of silica) with a chosen wall thickness to a long length and subsequently dividing it into hundreds of 1 m long and 1 mm diameter capillaries (Figure 16(1)). These are then stacked by hand in the desired arrangement (Figure 16(2)) to form a stack. The stack is then fused and drawn to a number of 1 m long and a few millimeters diameter canes (Figure 16(3)). Finally, each cane is drawn into a fiber (Figure 16(4)). This procedure has proven to be sufficient to efficiently draw all kinds of solid-core PCF, along with the early HC-PCFs which have relatively low air-filling fraction (see e.g. Figure 17(a)). However, as the air-filling fraction becomes higher than 80%, which is required for most hollow-core band gap guiding fibers, the preservation of the integrity of the fiber structure becomes problematic. This is because the effects of surface tension and the spatial variation of viscosity during the fiber draw become important, especially when close to the jacket of the fiber: a glass tube which contains the fiber microstructure (see Figure 17(b) and (c)). The HC-PCF can be viewed as a periodic set of thin tubes of silica. Due to surface tension, these experience an inward directed force and consequently the tubes tend to collapse. In order to counter-balance this effect, an equal pressure directed outward is exerted by loading the tube with a gas flow [9]. Given the fact the surface tension depends on the diameter of the collapsing tube, the core, the cladding and the interface between the jacket and the stack are subjected to different gas pressures (see Figure 18). Using this technique, it is possible to draw jacketed HC-PCFs with extremely high air-filling fraction (487%) whilst keeping a high level of fiber integrity.

26 Journal of Modern Optics 105 Figure 16. Schematic diagram of the fabrication procedure of HC-PCF. (The color version of this figure is included in the online version of the journal.) Figure 17. SEM of different HC-PCF presented in chronological order. Part (a) is the first fabricated HC-PCF [17] and exhibits an air-filling fraction of 40% and transmission loss of a few 100 db m 1.(b) A HC-PCF exhibiting 60% of air-filling fraction and a transmission loss of 10 db m 1.(c) A HC-PCF with air-filling fraction 470% when drawn with a jacket using the conventional stack-draw technique (expansion of the core and distortion of the cladding structure). (d) Jacketed HC-PCF (airfilling fraction 487%) drawn using the modified stack-draw technique [9]. (e) A HC-PCF from BlazePhotonics. (The color version of this figure is included in the online version of the journal.) Figure 18. Schematic diagram of the pressurization procedure of HC-PCF. (The color version of this figure is included in the online version of the journal.) Application of the technique has lead to the state-ofthe-art HC-PCF, which were made for example by BlazePhotonics [50]. Historically, the drawing parameters have been chosen using trial and error, as well as using asymptotic expressions for the collapse and expansion ratio of the capillary diameter, induced by the surface tension and the applied pressure, respectively. The problem is simplified by assuming all the cladding holes of the cladding are collapsing or expanding

27 106 F. Benabid and P.J. Roberts (under pressurization) at the same rate. This implicitly assumes that the initial holes are identical. Consequently, one can limit the pressurization to three main sections: (1) the cane-jacket region, for which, by applying a negative pressure between the jacket and the cane, one can prevent the surface tension induced collapse, (2) the cladding, which is formed by identical holes and lastly (3) the core hole which goes through expansion and collapse at a different rate, and hence an independent pressure is applied. Based on this, one can not only keep the integrity of the cladding structure, but also tailor its shape Gas loading in HC-PCF and photonic microcell assembly Gas loading One of the most salient features of HC-PCF is its ability to confine any gas-phase material within its micrometer scale optical guiding core. The first demonstration of HC-PCF filled with a gas-phase material was reported in 2002 [9]. The gas loading of HC-PCF is based on applying a differential pressure between the two ends of the fiber. This is realized by placing each fiber end in a gas control chamber that can be evacuated (using a vacuum pump), or filled with high pressure gas (see Figure 19). In order to ensure a hermitic seal, the two ends of the HC-PCF are mounted onto tailored fiber holders. Two fiber holder designs are shown in Figure 20. The key advantage of the technique compared to simply placing the fiber inside a gas chamber is that the gas-loading into the fiber core is controllable and faster. Also, this technique can easily allow flushing of residual unwanted gases that have accumulated inside the core of the HC-PCF Photonic microcell assembly A photonic microcell (PMC) [24] comprises a length of gas-filled HC-PCF that is hermetically sealed at both ends to solid optical fibers with a minimum of splice loss. An example is shown in Figure 21. The formation of a PMC requires the gas-filled fiber to be sealed after being loaded with gas at the correct pressure. The sealing process is achieved by splicing the hollow-core PCF to standard optical fiber such as SMF. This process is carried out using a filament fusion splicer. Even though it is also possible to use arc fusion splicers [51], these, offer less control and repeatability than filament splicer models. Photographs of a splice between HC-PCF and SMF are shown in Figure 22; the structure of the HC-PCF is visible in the fiber on Figure 19. Photograph of the gas-control chambers used for evacuating and filling HC-PCF. Left: stainless steel chamber used for atomic vapors at high-vacuum levels. Right: brass chamber used for molecular gases. The fiber end is hermetically sealed inside the chamber, and an optical window allows for the coupling of light into the fiber. (The color version of this figure is included in the online version of the journal.) the right, and the core of the SMF is visible in the fiber on the left of the image. The typical splice loss achieved between a seven-cell core HC-PCF guiding at 1550 nm, and Corning SMF-28 is between 0.7 and 0.9 db. The greatest contribution to this figure is due to the mode mismatch between the fibers; the HC-PCF has a fundamental core guided mode with a shape that does not perfectly match the near-gaussian mode of a conventional single-mode fiber. Mismatch between core sizes of the two fibers also contributes. Together, the shape and size mismatch represents 0.6 db of the total loss in the example above. Further contributions arise from Fresnel reflection [52] at the air silica interface, typically giving a loss of 0.15 db. Figure 22(e) shows the end face of HC-PCF as seen when a splice between HC-PCF and SMF is broken. It can be seen that the cladding structure of the HC- PCF has recessed, giving a curved end face, which occurs due to surface tension effects which act during the splicing process as the fiber is heated. This effect can both lead to an increase or a decrease in the modefield mismatch, and therefore the total splice loss. Moreover, in addition to the splice loss, the Fresnel reflection leads to an etalon effect whereby an optical cavity is formed as a result of the reflections at either end of the HC-PCF in the microcell, with a free spectral range of c/(2l). For a typical microcell with a fiber length, L, of several meters, this frequency is MHz and can be very parasitic in applications such as coherent optics or laser metrology (see sections below). To reduce the reflection, it is possible to perform angled splices, with each fiber cleaved at a matching angle. By performing such a splice, with each

28 Journal of Modern Optics 107 Copper o -ring Holds rod in place in the chamber.. Brass rod Compresses o -ring against chamber. Copper o -ring Provides a vacuum seal between holder and chamber, together with knife edge on flange. Fiber HC-PCF to be filled with gas. Rubber o -ring Provides the seal between rod and gas chamber. End cap Compresses Viton o -rings. Viton o -ring Provides a vacuum seal around the fibre. End cap Compresses Viton o -ring to provide a vacuum seal around the fibre. Viton o -ring Provides a vacuum seal around the fibre. Holder body Attaches to gas control chamber using a standard conflat vacuum port fitting. Figure 20. (Left) Fiber holder design 1. This assembly allows the fiber end to be hermetically sealed within the gas control chamber, and easily removed without damage when required. The vacuum seal is provided by the compression of Viton o -rings around the fiber. This design is used for loading fibers with molecular gases. (Right) Fiber holder design 2. This assembly is used to hermetically seal a fiber within the stainless steel gas chambers used for UHV application such as atomic vapor loading. The standard knife-edge seal on the flange allows UHV connection to a vacuum port. (The color version of this figure is included in the online version of the journal.) Figure 21. A photonic microcell is a gas-filled HC-PCF hermetically spliced to a solid optical fiber. (The color version of this figure is included in the online version of the journal.) fiber cleaved at an angle of 8, it has been possible to reduce the reflection back into the guided mode by 44 db, resulting in a return loss of 5 60 db. This is much reduced compared to the value 16 db for a normal splice [53]. An example of an angled splice is shown in Figure 23. In its original version, the PMC assembly is formed by simply removing one end of the HC-PCF from a gas-control chamber, and splicing it to SMF. The same process is then repeated at the second end. Microcells formed using this technique have been used for stimulated Raman scattering in hydrogen gas and laser frequency stabilization [24] (sections below). This technique is only suitable for relatively high gas pressures. Indeed, the end of the fiber is open to the atmosphere until the splice has been completed, a process that with practice can be completed in approximately s. During this time, either gas will escape from a fiber filled at high pressure, or air will enter into a fiber filled at low pressure. This is modelled by considering HC-PCF which is sealed at one end, and open to the air at the other. Figures 24 and 25 show the actual pressure obtained in the microcell as a function of the time taken to perform the sealing splice; the initial acetylene pressures inside the two fibers are 10 bar (Figure 24) and 1 mbar (Figure 25), respectively. During the 30 s it takes to complete the splicing procedure, the pressure in the fiber initially at 10 bar will reduce to approximately 4.5 bar. When producing a high-pressure cell, the loss of gas can be approximately compensated for by initially filling the fiber at a higher pressure than required, and does not present a serious problem in the fabrication of such microcells. Considering the lowpressure 5 m long fiber, however, the pressure will rise to 300 mbar in the 30 s taken to perform the splice. This would be unacceptable for coherent optics applications, where the excess pressure would lead to an increase in collisional broadening of absorption lines, and decoherence in the system (sections below). In order to lift this limitation, a new technique has been developed where the fiber is prepared at the required vacuum pressure (Step 1, Figure 26) and loaded with He at slightly more than 1 bar before being spliced (Step 2). The pressure of He is chosen such that very little contamination enters during the splicing process. The high permeability of the He atoms through glass means that the initial vacuum pressure will be restored after a few hours, leaving a low insertion loss, compact, mbar pressure microcell. The

29 Pressure / bar 108 F. Benabid and P.J. Roberts Figure 22. (a) Photograph taken before the splice of triangular lattice HC-PCF (right) and conventional solid single-mode fiber (left), as seen by the camera on a Vytran 2000 filament fusion splicer; (b) completed splice between a Kagome-lattice HC-PCF and SMF; (c) and (d) electron micrographs of spliced fiber; (e) end face of HC-PCF when a splice is broken apart showing a curved surface. Figure 23. (a) Angled cleaves of HC-PCF and SMF; (b) the fibers are rotated and butt-coupled ready for splicing; (c) side view and (d) top view of the angle-spliced fibers. Position in fiber / m Average pressure / bar Time / s Time / s Figure 24. Left: pressure distribution within a 5 m long fiber with 12 mm core diameter as a function of time. The fiber is initially filled uniformly at 10 bar, and sealed at one end. The second end is open to a reservoir at 1 bar. Right: average pressure within the fiber as a function of time. This example represents a typical situation encountered when sealing a high-pressure photonic microcell. (The color version of this figure is included in the online version of the journal.)

30 Pressure / bar Journal of Modern Optics 109 Position in fiber / m Average pressure / bar Time / s 0.0 1E Time / s Figure 25. Left: pressure distribution within a 5 m long fiber with 12 mm core diameter as a function of time. The fiber is initially filled uniformly at 1 mbar, and sealed at one end. The second end is open to a reservoir at 1 bar. Right: average pressure within the fiber as a function of time, on a linear scale (black) and logarithmic scale (gray). The average pressure rises to 100 mbar within just a few seconds. (The color version of this figure is included in the online version of the journal.) Figure 26. Left: photonic microcell preparation process. The various steps are described in the text. Right: EIT trace when the HC-PCF is attached to the gas chamber (a) and while the photonic microcell is formed (b) and (c). Evolution of the EIT linewidth with the power of the coupling laser (expressed in terms of the Rabi frequency). (The color version of this figure is included in the online version of the journal.) right hand-side panel of Figure 26 shows electromagnetically induced transparency (EIT) traces when the fiber is attached to controlled gas chambers filled with acetylene under a pressure of 0.1 mbar (Figure 26(a)), and after the PMC has been formed (Figure 26(b) and (c)). The EIT linewidth evolution with the coupling power indicates that the initial pressure inside the HC- PCF is recovered [54]. 4. Application of hollow-core photonic crystal fibers The key factors in many of the applications involving laser matter interactions, whether they involve gas-phase material such as in nonlinear and quantum optics and laser frequency metrology, or micron-scaled particles for particle/atom guidance purposes, are: (i) diffraction-free propagation of light, (ii) low power loss, and (iii) light confinement to a very small area. This can be summarized by the maximizing of the following dimensionless figure-of-merit [9] f om ¼ ðl eff =A eff Þ. Here L eff is the effective constantintensity interaction length, A eff the effective crosssectional area and is the vacuum wavelength. A number of conventional approaches, such as tightfocusing a laser beam or using a fiber capillary, have been used to enhance this figure. The figure-of-merit obtained using these procedures, however, is orders

31 110 F. Benabid and P.J. Roberts of magnitude lower than can be achieved with HC- PCF, as illustrated in Figure 27. This shows that HC- PCF is an excellent host for all kinds of applications involving strong interaction of laser light with matter in general, and gases in particular. In the subsequent sections experimental results are shown which illustrate the huge improvements a HC-PCF can convey to the above-mentioned fields Particle guidance Ashkin [55] demonstrated that small particles could be propelled and suspended against gravity using only the force of radiation pressure. The use of radiation pressure provides a useful means for non-intrusive manipulation of microscopic objects, for example particles or biological entities. These applications are, however, intrinsically limited by the diffraction of the laser beam to sub-millimetre length scales, as strong lateral confinement requires tight beam focusing. Overcoming this limitation is of particular interest in many areas where transportation of micro-sized objects over longer distances is required. For stable guidance, including cornering, one requires constant beam intensity focused to a small spot size over many Rayleigh lengths, so that a hollow optical waveguide is a natural approach. Unlike glass capillaries, where the maximum guiding length one can hope for is typically a few millimeters for a laser power of 100 mw [56], in a HC-PCF, the guidance length can be many times longer. In [25], particle guidance in HC-PCF was demonstrated over a length of 150 mm with only 80 mw laser power. The transverse (gradient) force in this fiber would be sufficient to easily support the particle against the force of gravity (if the fiber were horizontal) or to steer the particle around sharp corners (if the fiber were bent). Such a strong gradient force with a comparable laser power would only be attainable over a distance of 0.6 mm using a focused beam in free space, or over 12 mm using a standard capillary fiber. The experimental set-up is described in [25]. A collimated CW argon ion laser, operating at nm, is focused and directed vertically upwards into the core of a HC-PCF. The particles are held on a vibrating glass plate located in the focused beam just below the fiber. The HC-PCF is held vertically above the glass plate. The length of the HC-PCF used in the experiments varied from 100 to 200 mm. The two steps of the experiment, namely the levitation of the particle from the glass plate and the guidance in the air-core of the fiber are monitored using two CCD cameras equipped with telescopes and connected to a monitor, a PC and a video recorder. Figure 28(b) shows the Figure 27. Figure-of-merit for a capillary fiber, a focused laser beam and four HC-PCFs with different loss figures. At a bore radius of 5 mm the HC-PCF with the lowest transmission loss (2dBkm 1 ) exhibits a figure-of-merit more than 10,000,000 higher than either the focused laser beam or the capillary. (The color version of this figure is included in the online version of the journal.) output face of the fiber when the input face is illuminated with white light. It has a core diameter of 20 mm, a pitch of 3 4 mm and an air filling factor of 70%. At the operational wavelength of nm, loss was measured to be 10 db m 1. Figure 29 shows a sequence of frames of polystyrene spheres of 5 mm diameter being levitated and guided over a 1 mm section of 150 mm long HC-PCF, localized at about 4 cm from the input end of the fiber at a speed of 1cms 1. Despite the high loss figure of 10 db m 1 for the HC-PCF used in the experiments, the guidance length achieved with the 80 mw laser power represents an improvement of more than one order of magnitude over previously reported results. With a loss below 10 db km 1, the possible guidance lengths would increase to a few 100 m Low threshold spectral line generation using stimulated Raman scattering The nature of light guidance in a HC-PCF implies this fiber is an excellent candidate for hosting nonlinear optical interactions where long interaction length, low linear transmission losses and small mode diameters are simultaneously required. This contrasts with conventional techniques which are all compromises. One field where HC-PCF has enabled dramatic progress is stimulated Raman scattering (SRS) in molecular gases, where conventional techniques require high power lasers (1 MW) to reach the

32 Journal of Modern Optics 111 Figure 28. (a) The transmission spectrum of the HC-PCF used here for particle levitation. (b) The micrograph of the exit end of a 5 cm long HC-PCF as seen from a microscope. (The color version of this figure is included in the online version of the journal.) Figure 29. Top row: a snap-shot sequence showing a polystyrene particle (pointed out by an arrow) being levitated in air and guided towards the core of a HC-PCF. Each frame corresponds to a captured scene size of mm 2. Bottom row: A sequence showing the guiding of the particle within the core of the HC-PCF. Each frame corresponds to a captured scene size of mm 2. Raman threshold. In the following sections, results are presented on the generation of vibrational SRS in hydrogen using a Kagome HC-PCF with a pump threshold 100 times lower than previously reported in any single-pass or multi-pass cell. Measurements on rotational SRS using a band-gap HC-PCF, which show photon conversion close to the quantum limit and pump power thresholds some one million lower than earlier reports, will also be discussed Vibrational SRS Generation of SRS using the vibrational resonance of hydrogen was the first demonstration of SRS in HC- PCF [9]. A Q-switched Nd:YAG laser, operating at 532 nm with a pulse duration of 6 ns, is coupled to the lowest-order air-guided mode of the HC-PCF filled with hydrogen. The initial length of the fiber was 1m. The HC-PCF had a cladding with the Kagome structure described in Section 2. Its transmission bandwidth covers the whole visible/ir range with a loss value of 1dBm 1 Figure 30(d) shows the nearfield patterns of the pump (532 nm), Stokes (683 nm) and anti-stokes (435 nm) components transmitted through the hydrogen-filled HC-PCF. The threshold level of the coupled energy was measured for different fiber lengths. For a fiber length of 1 m the threshold was found to be 0.8 mj for Stokes (i.e. 133 W of peak power) generation. As the Stokes power increases with increasing input power, an anti-stokes component appears. The measured threshold for this is 3.4 mj. Despite the absence of an optimization study, a photon conversion efficiency to the Stokes component of 30% is obtained for a 32 cm fiber length and a coupled energy of 4.5 mj. Moreover, for a fiber length and coupled power where the Stokes field has not yet suffered from significant loss, the ratio of anti-stokes to Stokes power reaches a maximum of 5%. This demonstration initiated a number of experiments using HC-PCF Quantum-limited conversion to the Stokes line using rotational SRS The generation of pure rotational SRS was undertaken using the same principle as was used to generate vibrational SRS in H 2. However, unlike in the preceding section, the fiber used here guides via a photonic bandgap, hence exhibiting a narrower bandwidth but much lower transmission loss (see Figure 31). This enables one to preferentially excite the rotational S 00 (1) transition of ortho-hydrogen whose Raman frequency shift of 18 THz is covered

33 112 F. Benabid and P.J. Roberts Figure 30. (a) The loss spectrum of the Kagome HC-PCF. (b) The SEM of the fiber. (c) The end face of the fiber when it is illuminated at the input face by a white light source. (d ) The near field pattern of the pump, the Stokes and the anti-stokes components, respectively. (The color version of this figure is included in the online version of the journal.) Figure 31. (a) SEM of the HC-PCF. (b) Transmission loss spectrum of the fiber. The vertical arrows show the location of the pump (P), the first Stokes (S) and the anti-stokes (AS). by the fiber transmission band. On the other hand, the limited bandwidth of the fiber inhibits the amplification of the vibrational transition Q 01 (1) Stokes component because the Raman frequency shift of 125 THz is much larger than the 50 THz of the HC-PCF bandwidth. These features make it ideal for applications (e.g. LIDAR, single frequency converters) where occurrence of additional Raman lines, due to either Raman cascade or wave mixing, can be highly undesirable. It also is an ideal host when extremely high conversion is needed within a restricted bandwidth. The experimental setup used here is basically identical to the one reported in [9], except that the pump signal was delivered by a passively Q-switched frequency-doubled Nd:YAG microchip laser with a maximum output energy of 2 mj, and is circularly polarized in order to have higher gain coefficient and to eliminate Stokes anti-stokes coupling [57]. The fiber has a core diameter of 7.2 mm and its cladding has a triangular lattice structure with a pitch of 3 mm. The initial fiber length was 35 m. Its transmission spectrum is 150 nm wide, centered around 1064 nm. Figure 32 shows the evolution of the ratio of the transmitted average power to that of the coupled power as the latter increases. Both the pump (1064 nm) and the first Stokes (1135 nm) components are tracked. The transmitted spectrum through 3 m long fiber is also shown for two different coupled energy values, showing the near depletion of the pump and almost full conversion to the first Stokes. The lowest threshold energy was 3nJ(3.9 W peak power), observed for a fiber of length 35 m (Figure 32(a)). This value is more than one million times lower than the lowest value reported in conventional experiments for rotational SRS generation [58 60].

34 Journal of Modern Optics 113 Figure 32. Evolution of the ratio of transmitted average power over that of the coupled average power for the pump (open circles) and the Stokes (solid circles) in the case of a fiber length of (a) 35 m and (b) 2.9 m. (c) The transmitted spectrum for two different values of coupled energy through a 3 m long fiber. The threshold level increased as the fiber length was reduced in agreement with the theoretical prediction. For fiber lengths shorter than 3 m, the ratio of the Stokes power to the coupled power reached a maximum of 86%. This corresponds to a photon conversion efficiency of 92%, which was the highest ever reported. Finally, with the advent of the PMC, rotational SRS in hydrogen was demonstrated in an all-fiber system PBG-limited comb generation using rotational SRS In this section, the experimental setup was similar to the one described in [9], except that the pump source was a Q-switched Nd:YVO 4 laser at 1047 nm. The laser pulse width could be varied from 6 to 45 ns with a linewidth of 260 GHz. A half-wave plate on a rotating stage and a polarizing beamsplitter were used to control the power of the pump beam, while a rotating quarter-wave plate was used to vary the pump polarization from linear to circular. The pump beam was coupled into 11 m of a HC-PCF filled with H 2 gas at 10 bar pressure in the manner described in [1]. The input power was monitored just before the coupling lens, as was the power of the back-reflected light. The output spectrum was recorded on an optical spectrum analyzer while the power and temporal profiles of the different spectral components were recorded using interference filters to select different wavelengths. The HC-PCF used in the experiment was the same as in [21], its transmission spectrum spanning 900 to 1200 nm (see Figure 33(a)). In the region nm, the loss spectrum exhibits an almost flat low-loss region, its lowest loss being about 67 db km 1 at 1060 nm. As pointed out in [21], all the vibrational bands (Raman shift 4155 cm 1 ) generated in the SRS process lie outside the low-loss transmission range of the fiber and are therefore suppressed. Consequently, the rotational SRS frequency components (Raman shifts 587 and 354 cm 1 ) experience greater overall gain. In the recorded output spectra (Figure 33(b)), up to eight different SRS lines were observed and identified. They are due both to the S 00 (1) rotational transition of ortho-hydrogen and to the S 00 (0) rotational transition of para-hydrogen. The S 00 (0) transition is usually hard to observe. However, due to the long interaction length and the suppression of vibrational SRS, this transition was routinely observed in the experiments. Generation of the first S 00 (1) Stokes band was achieved at extremely low thresholds (590 nj) for the shorter pulses used. The SRS spectrum shown in Figure 33 was generated by a pump with a pulse width of 13 ns and energy of less than 4 mj (peak power of 300 W). This is the first demonstration of such a comb-like spectrum arising from multiple-order SRS in H 2 using such a low pump energy [3,4]. The results could offer new prospects for generation of ultrashort pulses based on the Raman sideband technique [61,62].

35 114 F. Benabid and P.J. Roberts Figure 34. Evolution of the average transmitted power of the different Raman spectral components as the coupled pump energy is varied. The pulse width of the pump is 13 ns. Figure 33. (a) Transmission spectrum of a 20 m long HC-PCF (right-hand side axis) and the loss spectrum of the HC-PCF (left-hand axis). (b) The SRS spectrum transmitted through 11 m of HC-PCF filled with hydrogen pumped with laser light with 13 ns pulse width and a peak power of 300 W. (The color version of this figure is included in the online version of the journal.) Figure 34 shows the growth of the different spectral components as the input pump energy is increased. The S 00 (1) first-order Stokes wave at 1115 nm (ortho-s1) grew to exceed the power in the pump wave, and subsequently generated new Raman frequencies (cascaded SRS). These frequencies correspond to both types of rotational Raman transitions. Furthermore, the Stokes intensity of the S 00 (0) transition at 1161 nm (para-s1, pump: 1115 nm) is, surprisingly, stronger than both the corresponding para-s1, at 1087 nm, pumped by the input laser field and the ortho-s2 at 1193 nm. This is explained by the attenuation spectrum of the fiber. When the hydrogen is pumped by the laser field at 1047 nm, the ortho-s1 and ortho-s2 waves dominate the spectrum at the expense of the less efficient para-s2 transition. However, when the ortho- S1 wave (1115 nm) becomes strong enough to generate new Raman frequencies, the higher-order transitions of ortho-hydrogen lie in the high-loss region of the fiber spectrum and are, therefore, suppressed while the para-s1 transition at 1161 nm, which is located in a relatively low-loss wavelength, is enhanced CW-pumped rotational SRS The early results on the generation of stimulated Raman scattering (SRS) in H 2 -filled HC-PCF clearly indicate that Raman conversion in the CW regime could be possible with a pump power of only a few Watts. In such a single pass configuration, there are no restrictions on the converted wavelength and no requirement for a cumbersome cavity-locking system. Figure 35 shows Stokes generation using a CW pump laser. The figure also shows that the conversion is also extremely efficient, with 99.99% of the output power being at the converted Stokes wavelength. These unprecedented results could widen the wavelength range achievable with CW lasers, without compromising on the linewidth or laser power Coherent stimulated Raman scattering In addition to the dramatic drop in the pump power levels required for SRS generation in gas-phase materials such as hydrogen, the long interaction length and the tight light confinement offered by HC- PCF also altered the amplification regime of SRS Transient regime in HC-PCF The theory of Stokes amplification in stimulated Raman scattering (SRS) has been studied extensively within the classical framework by several authors, for example Wang [63], Carman et al. [64] and Akhmanov et al. [65], and quantum mechanically by Raymer and Mostowski [66]. These studies established the existence of two regimes of amplification for systems pumped by pulsed lasers. Firstly, a steady-state regime is reached after an interaction time t ss ¼ G ss T 2, where G ss is the steady-state net gain and is equal to g SS I P z, g SS being the Raman intensity gain coefficient at steady

36 Journal of Modern Optics 115 Figure 35. CW Raman gas fiber-laser. Left: Transmission spectrum (a) below and (b) above threshold. (c) Evolution of the output intensity profile with pump power, at pump and Stokes wavelengths. state, I P the pump intensity, z the interaction length and T 2 the dephasing time of the Raman medium. Secondly, there is a transient regime for interaction times shorter than G ss T 2. A commonly used rule-ofthumb distinguishes the transient from the steady-state regimes by comparing the dephasing time with the pulse duration, the transient regime being defined to be when the pulse duration is shorter than T 2 and the steady-state regime when it is longer than T 2 (see [67] and the references therein). This is justified by the fact that in most experiments the effective interaction length is limited to the Rayleigh range of the focused laser beam (for a single-pass configuration), or to around a meter in the case of multi-pass cells [68]. Consequently, the net gain remains relatively small, reaching extremely high values (4100) only for very high peak powers (1 MW). The need for powerful pulsed laser sources imposes not only technical limitations, but also fundamental ones due for example to the presence of self-focusing. As a result, there have been few previous experiments studying the SRS amplification regimes, and no report to date giving an explicit and comprehensive experimental account of Stokes amplification with pump pulsewidths spanning both regimes. Such a study could provide an experimental foundation for the various theoretical findings in the subject. Moreover, experimental observations of transient behavior in hydrogen, for example, have been reported only with pump pulses much shorter than a few hundred picoseconds (the typical value T 2 at pressure around 10 bar) [69]. With the advent of HC-PCF, effective interaction lengths of hundreds of meters have become possible. Consequently, the commonly established criterion for the separation of the transient regime from the steady-state is not necessary valid. Furthermore, SRS using HC-PCF presents major advantages (both technological and fundamental) which make studies of Stokes amplification in such structures useful. Due to the effectiveness of HC-PCFs as gas cells for SRS [9], these fibers are likely to be increasingly used as Raman converters or Raman lasers. In such a system the weakness of the offresonance Raman gain is overcome by enhancement factors greater than 10 6 thanks to an extremely long interaction length and tight optical confinement over the whole fiber length (see previous section). Secondly, HC-PCF offers an excellent tool for exploring fundamental aspects of SRS [21]. For example, one can reach the high-gain limit with low peak powers. For our typical configuration with an effective length of 10 m, a fiber core diameter of d ¼ 7 mm, a Raman gain coefficient g 3cmGW 1 and a pump power P ¼ 10 W, the gain can be calculated from the form G 1 gpl eff y ð15þ 2 d 2 to be G4100. In order to explore the effects of a long interaction length on the Stokes amplification process, measurements of the threshold energy required to generate the first Stokes of the S 00 (1) transition were carried out in [70] for different pulse widths. At threshold, all the other competing nonlinear processes can be ignored, the pump is effectively undepleted, and the net gain is nearly constant. Consequently, threshold measurements provide us with excellent conditions for quantitative experimental investigation and comparison with theoretical findings on Stokes amplification with different interaction times. In SRS, after a propagation

37 116 F. Benabid and P.J. Roberts distance z and an interaction time, the transmitted Stokes intensity I S ðz, Þ grows from noise given by Is sp ð0þ ¼ð1=2ÞGh S,2G being the Raman linewidth (i.e. G ¼ð2pT 2 Þ 1 ) and S the Stokes frequency. At threshold, the Stokes intensity has built up to a level such that ln I S ðz, Þ I sp S ð0þ ¼ G th ð16þ the net gain G th being a constant between 20 and 30. This represents the pump threshold condition, and in order to deduce it for different pulse widths, the approach of Raymer and Mostowski [66] was chosen. One can deduce analytical expressions for the energy threshold as a function of pulse width if we consider separately the three time regimes observed by Raymer and Mostowski [66]. The first regime includes interaction times shorter than 1 1=Ggz, when the scattering is spontaneous and the Stokes intensity is given by I sp S ðz, Þ ¼ð1=2ÞGgzðh SÞ, i.e. the power of the Stokes wave is independent of the pulse width. The second corresponds to timescales between 1 and 2 gz=g, for which the scattering is transient, and we obtain I tr S ðz, Þ h exp 2ð2gzGÞ 1=2 2G S : ð17þ 8p Finally, for interaction times larger than 2, the steadystate scattering is reached, in which case the Stokes expression in the high-gain limit reduces to I SS S ðz, Þ h G S expð gzþ: 1=2 2ðpgzÞ ð18þ Applying the threshold condition mentioned above for the steady-state regime and assuming square pulses, one finds the following threshold energy: E th P,SS ð PÞ¼G th ~P P þ E 0SS : ð19þ Here, ~P ¼ðg ss ðl eff y=a eff ÞÞ 1 which has the units of power, and the slope G th ~P is the steady-state threshold power. Because the steady-state expression (18) ceases to be valid for pulse widths shorter than 2, E 0SS is added empirically to fit the experimental data and corresponds to the offset between the Stokes intensities in the steady-state regime and the spontaneous regime [66] it was also used in previous experiments [60]. Expression (19) was evaluated for our experimental parameters (i.e. an effective area of 50 mm 2, an effective length of 10 m, a Raman gain of 1cmGW 1 [71], a dephasing time of 200 ps taken from [60], and Gth ¼ 20) and is plotted in Figure 36 (dotted-line). The slope of the theoretical curve Gth ~P agrees very well with the experimental data for pulse widths greater than 14 ns. The offset value of the energy threshold E 0SS is found to be 90 nj. Figure 36. Energy threshold of the pump for the generation of the first Stokes S 00 (1) as a function of the pump pulse width. The solid circles represent the experimental data. The dashed and dotted lines show the theoretical threshold for the transient and the steady-state regime, respectively. (The color version of this figure is included in the online version of the journal.) For the transient regime, the energy threshold can be written as: E th P,tr ð PÞ¼ ~P 2G G th 2: þ lnð4pþþlnðg P Þþ2G P ð20þ This expression exhibits a quadratic evolution of the threshold with the pump pulsewidth and does not depend on the dephasing time. This evolution is plotted in Figure 36 (dashed line) and shows good agreement with the experimental data for pulse widths shorter than 14 ns. The pulse width of 14 ns consequently represents the transition from transient scattering to the steady-state regime. This value is around 100 times longer than the typical pulse widths at which transient scattering is usually observed [69], because the parameters in our experiments are so different to those used previously. An estimate of the theoretical passage time 2 for the measured energy threshold 100 nj gives a value of 14.3 ns which is in excellent agreement with our experimental value. It is essential to note that the transience in our case is not related to the laser bandwidth. Indeed the laser bandwidth, which can affect the gain and the evolution in Raman amplification from the noise via the dispersion, is much smaller than the dispersion acceptance bandwidth [72] (i.e.! L ½gðpG= 2Dn g ÞŠ 1=2 [73], Dn g being the group index difference between the pump and the Stokes, and is measured to be 10 3 ) and hence can be ignored. The consequences of transient SRS with such long pulses are interesting

38 Journal of Modern Optics 117 and numerous, including possible phase pulling between the Stokes and the pump [69], and low threshold Raman solitons with quasi-cw pulses. The amplification dynamics in SRS also leaves fingerprints on the temporal profile of the transmitted pump and Stokes. Among the predictions in the treatment of Raymer and Mostowski [66] and also by Carman et al. [64] is that the transient behavior will usually lead to some sharpening of the Stokes pulse in the time domain and a delay with respect to the pump pulse. In view of the foregoing, the temporal profile of the output pulse was investigated for different pulsewidths and gains. The traces of the pump and the different Raman components were split using interference filters, calibrated and recorded. Figure 37 shows the temporal shapes at the output of the fiber. The transmitted pump (Figure 37, curve (b)), first-order Stokes (Figure 37, curve (c)) and the second-order Stokes (Figure 37, curve (d)) components are shown, together with the input pulse shape before entering the fiber (Figure 37, curve (a)) which is 7 ns wide and has an energy of 800 nj (i.e. steady state net gain of G 130). The shape of the total output pulse was the result of the co-propagating pulses at the original frequency and the ones generated in the SRS process and dominated by the two ortho-stokes. This was experimentally demonstrated by almost exactly reconstructing the total output pulse from the constituent pulses. The peak of the first-order Stokes pulse occurs at the tail of the pump pulse due to the gain accumulation as the Stokes pulse builds up [6] and accounts for the dip in the profile of the transmitted pump pulse. It builds up rapidly at the peak of the pump pulse and then falls off quickly at the trailing edge of the pump pulse [2]. The resulting delay of the first-order Stokes pulse peak relative to the peak of the pump pulse was measured to be around 2 ns. Some temporal narrowing of the Stokes pulse was observed; almost a factor of 2 with respect to the initial pump pulse. All these experimental observations are in agreement with transient SRS theory [64] Multi-octave spanning frequency comb generation The fact that the transient regime of Raman amplification, which was previously limited to femtosecond pump pulses, can be attained with nanosecond pulses using HC-PCF, has spawned interest from the ultrafast optics community. This is because the fiber can potentially be used to generate broad coherent frequency combs, by cascaded transient Raman scattering, in the nanosecond regime. These coherent combs can then be recombined to create attosecond pulse trains. Thanks to the broadband guidance of Figure 37. Temporal profiles of the different transmitted pulses. Curve (a): the temporal profile of 7 ns wide input pulse before entering the fiber; curve (b): the depleted pump; curve (c): the first-order Stokes; and curve (d): the secondorder Stokes of the S 00 (1) Raman transition. (The color version of this figure is included in the online version of the journal.) Kagome fibers, frequency combs of up to 45 spectral components, covering 1000 THz, have been generated and guided through the fiber with little loss (Figures 38 and 39) [10]. This solution presents a low-power alternative to other ultra-short pulse generation methods and does not pose any engineering challenges such as requiring cryogenic cooling Quantum optics and metrology using HC-PCF Electromagnetically induced transparency in acetylene filled HC-PCF Another field where the interaction length and field confinement play key roles is quantum optics. An example of the many phenomena in this field is electromagnetically induced transparency (EIT) [74]. This refers to an effect such that, in a medium driven by a control laser, a probe laser whose frequency is near an otherwise absorbing transition will experience a narrow transparency window at the center of the absorption profile. In addition, the transparency is accompanied with a very sharp change in dispersion. These features find compelling applications in topics as varied as ultraslow light [75] and light storage [76], laser cooling [77], nonlinear optics [78] and atomic clocks [79]. These applications are the driving motives behind the extensive study of EIT since its first experimental demonstration [80]. There the growing endeavors being undertaken aimed at finding ways to implement EIT in all-optical switching and for signal processing in optical communication, as well as in building blocks for quantum computing and teleportation.

39 118 F. Benabid and P.J. Roberts Figure 38. (a) SEM showing the cladding structure of the Kagome HC-PCF. (b) Lower magnification SEM showing the full structure of the fiber. (c) Optical spectrum showing the broadband guidance of the fiber. (d) Experimental setup for the generation of multi-octave frequency comb. Figure 39. Images and spectrum of the generated HSRS at the output of 1 m long hydrogen-filled Kagome fiber, for (a) a linearly polarized and (b) a circularly polarized laser input. (The color version of this figure is included in the online version of the journal.) Despite the consensus on the potential of EIT mentioned above, experiments have been restricted to atomic vapors (e.g. Rb), and few studies have addressed the occurrence of EIT in molecular systems [81]. Molecular systems have a number of distinctive features which could not only broaden our fundamental understanding of EIT-related phenomena by offering new test grounds but would also open new technological prospects. For instance, many molecular systems exhibit quantized and spectrally resolvable vibrational and regularly spaced rotational vibrational transitions which cover the whole VIS-IR spectrum. An appropriate combination of two transitions could form a three-level system in the L, V or cascade configurations where electromagnetically induced transparencies could in principle occur. Figure 40 schematically illustrates this in the case of rotational transitions between two vibrational states in a parallel band of a linear molecule [82]. The figure clearly shows that for a control laser which is on resonance with an absorption line (e.g. P(J þ 1), J being the rotational quantum number), one could observe transparencies with a probe laser tuned around either the line R(J 1), thus forming a L interaction configuration, or the R(J þ 1) line, forming a V interaction scheme. Consequently, the R-branch could be used as a comb of transparencies spanning several THz for future devices such as all-optical routers in telecommunications. In this case, molecules such as acetylene and hydrogen cyanide are the natural choice, since their 1 þ 3 and 2 1 bands, respectively, offer a comb of stable and regularly spaced ro-vibrational overtone