Hollow-core photonic-bandgap fibers provide a new geometry for the realization and enhancement of many nonlinear optical effects. Such fibers offer novel guidance and dispersion properties that provide an advantage over conventional fibers for various applications. In this review we summarize the nonlinear optics experiments that have been performed using these hollow-core fibers.
©2008 Optical Society of America
Hollow-core photonic bandgap fibers (HC-PBGFs) [1, 2] are a recent addition to the optical fibers which were developed to exhibit unique waveguiding and nonlinear properties. These fibers display great versatility in terms of hosting nonlinear materials and have been instrumental in pushing the boundaries of nonlinear optical applications. As will be discussed here, such fibers provide the potential for applications that require either ultrahigh or ultralow nonlinear response.
Standard single-mode step-index optical fibers, which form the backbone of telecommunications today, work on the principle of total internal reflection . The core, which is a higher refractive index glass rod surrounded by a lower index glass cladding, can support light guidance in a single mode with losses as low as 0.15 dB/km . Nevetheless, solid-core fibers have certain shortcomings. Despite the relatively low losses, the signal sent into the fibers eventually attenuates and needs amplification at regular intervals. The guided light mode is confined within the glass with an effective mode area of the order of 80 µm2, and as the intensity of light is increased, the fiber exhibits nonlinear effects such as stimulated Raman scattering (SRS), stimulated Brillouin scattering (SBS), and self phase modulation (SPM) . These nonlinearities along with dispersion can lead to distortions in communication signals that result in increased bit-error rates and also effectively put an upper limit on the power of the input signal launched into the fiber, which is relevant for other fiber-based applications such as pulse delivery.
Alternatively, the nonlinearity and scattering losses in air are roughly 1000 times lower than in glass, which has led to efforts to confine light to a low-index air region. This requires a radical shift in the approach towards waveguide structure design, which would be beneficial in a number of ways. The scattering losses due to Rayleigh scattering and due to material defects could be significantly decreased. The threshold powers for unwanted SRS and SBS could be greatly increased, which could not only benefit telecommunications, but also aid in fiber delivery of intense ultrashort pulses over long distances. Also, if the air core is evacuated and then deliberately filled with appropriate gases or liquids, such a fiber represents a novel geometry for achieving very strong nonlinear interactions between light and matter due to the strong confinement of light in a small mode area and to the extremely long interaction lengths. Last, but not the least, if the dispersion of these waveguides can be suitably tailored, nonlinear processes such as four-wave mixing and soliton generation can be greatly enhanced.
It should be noted that glass capillaries have been used succesfully for various nonlinear optical applications. However, such gas-filled capillaries can only be used for high-power pulses since the loss of the fundamental leaky mode increases rapidly as the diameter is decreased, which limits the core diameter to sizes ≥100 µm . In addition, glass capillaries offer minimal dispersion control in contrast to HC-PBGFs that allow for significant tuning of the dispersive properties.
2. Photonic bandgap fibers : A brief introduction
Historically, the concept of a photonic bandgap gained significant interest due to it’s potential to curtail spontaneous emission in optical devices [7, 8]. The seminal work of Anderson  and Mott  on localisation of electrons at defect sites in regular solids suggested that the same could be observed for electromagnetic waves in strongly scattering dielectric structures. The first suggestion for guiding light in a low index waveguide by surrounding it with Bragg structures was made by Yeh and Yariv . Yablonovitch  and John  showed similar and complimentary theoretical results describing prohibited photonic transmission at certain wavelengths and corresponding guidance due to localization at the defect. Various two-dimensional photonic bandgap structures were studied and proposed as possible long distance light guidance structures [14, 15]. Four years after the first calculation of 2-D photonic bandgaps in air-silica structures , the first critical demonstration of single-mode photonic bandgap guidance in air was made possible using a very similar structure .
The standard design of these HC-PBGF structures (for details see Ref. ), as shown in Fig. 1(a), consists of a stack of hollow capillaries with the 7 center capillaries missing, which forms the defect surrounded by a triangular array of air holes in silica. Light coupled into the defect site enters the bandgap region and is repeatedly scattered such that the light experiences destructive interference everywhere except in the defect site and is effectively guided down the fiber at wavelengths throughout the bandgap region. The spacing of the hollow capillaries (called pitch) and the index contrast determine the center wavelength and width of the bandgap. Early versions of HC-PBGFs had relatively high losses due to a strong coupling between the hollow-core modes and surface modes which are lossy and the absence of a full photonic bandgap. The use of long lengths of near-single-mode HC-PBGFs became practical only when the design and manufacturing process was sufficiently refined enough to achieve losses as low as 13 dB/km at 1550 nm  [see Figure 1(b)]. Since then, a variety of low-loss hollow-core bandgap fibers have become available with bandgaps centered over a variety of different wavelength ranges. The lowest loss fiber (1.2 dB/km) was designed using a larger core formed by the removal of 19 cells .
3. Stimulated Raman scattering and four-wave mixing
Raman scattering is an inelastic scattering process in the presence of a pump beam in which a material makes a transition from one vibrational energy level to another through an intermediate state that is dipole coupled to both the initial and final states . If the final state has higher (lower) energy than the initial state, the emitted photon is called the Stokes (anti-Stokes) photon and has a longer (shorter) wavelength than the pump photon. In the presence of a Stokes signal, this process can be stimulated and the signal wave is amplified. In addition to wavelength conversion, Raman scattering is useful for spectroscopically probing energy levels not directly dipole-connected with the ground state. The number of Stokes photons, ms, generated in the material varies with length z as ,
where mp is the number of pump photons, and A is a constant including phase velocity and Raman gain coefficient. Maintaining a high confinement for pump and signal photons together over a long interaction length will significantly enhance the conversion rate. In fact, the first demonstration of the use of HC-PBGFs for nonlinear interactions was performed for Raman sideband generation  in which different lengths of Kagome-structured fibers were filled with hydrogen at a pressure of 17 bar. A frequency-doubled Nd:YAG laser at 532 nm was coupled from one end to serve as the pump and with increasing pump powers, a vibrational Raman Stokes signal was observed at 683 nm. As shown in Fig. 2, a much weaker anti-Stokes signal at 435.2 nm was also observed at higher pump powers. The measured Raman threshold was 800 nJ (6-ns pulses) for Stokes and 3.4 µJ for the anti-Stokes signal, which represented values that were 2 orders of magnitude lower than that reported previously for the single-pass bulk configuration.
An intriguing application of the bandgap nature of the HC-PBGFs is selectively enhancing the rotational Raman signal by supressing the vibrational transition, which by suitable choice of fiber, lies outside the bandgap. By launching the pump in circular polarization  in a fiber filled with H2 at a pressure of 7 bar, almost 92% conversion efficiency to the rotational Stokes signal was achieved. The threshold was observed to be a function of the fiber length (3 nJ for 39 m and 20 nJ for 2.9 m for 0.8-ns pulses at 1.064 µm), which is 106 times lower than previously reported in bulk-geometries. Additionally, by changing the polarization and power in the pump, the power in various Stokes and anti-Stokes lines could be selectively controlled.
The use of lower-loss fibers allows longer lengths to be used, which can be especially useful for cascaded SRS processes in which the generated frequencies act as the pump fields for generating further sidebands. With sufficient bandwidth, it is possible to use this comb for ultrashort pulse generation. Using a low-loss fiber whose bandgap does not support vibrational Raman lines, it has been shown that a purely rotational Raman sideband comb spanning 80 THz can be generated . In this experiment, an 11-m long fiber with bandgap in the 900–1200 nm region (fiber attenuation: 67 dB/km) was filled with H2 at 10 bar and pumped at 1047 nm (100–300 nJ pulses) to generate this comb. The dependence of Raman threshold on pulsewidth was also studied in detail. In a recent striking result using a Kagome-structured fiber, generation of a comb spanning 325 to 2300 nm  was demonstrated. Figure 3 shows the generated Raman spectrum. Note that linear polarization excites the vibrational Raman spectrum with a spacing of 125 THz and circular polarization excites rotational and vibrational Raman spectrum, which has a spacing of 18 THz. In principle, it should be possible to generate sub-femtosecond pulses by properly adjusting the phases of these sidebands.
Four-wave mixing (FWM) is a parametric nonlinear process in which 3 photons of different frequencies interact to create a 4th photon via the third-order susceptibility χ(3) of a material. FWM is an important nonlinear effect that finds application in generation of new wavelengths, parametric amplification , squeezed states and entangled photon-pair generation . Konorov et al.  demonstrated FWM in a hollow-core fiber by mixing a frequency ω and its second harmonic 2ω to generate the third harmonic 3ω i.e. 3ω=2ω+2ω-ω. The hollow-core fiber was designed to support transmission at all three frequencies. A two-color pump consisting of 30-ps (30 – 50 µJ) pump pulses at 1.06 µm and its second-harmonic at 0.53 µm were coupled into a 9-cm-long fiber with an air-filled core. The authors observed an enhancement by a factor of 800 over previous FWM experiments in a focused free-space geometry. Fedotov et al.  demonstrated enhanced coherent anti-Stokes Raman scattering (CARS) where ωCARS=2ω 1-ω 2 was generated through FWM of the fields at ω 1 and ω 2. Using a 3-cm nitrogen-filled HC-PBGF with bandgap that supports the relevant wavelengths, the authors showed an enhancement in the CARS signal of the Q-branch of molecular nitrogen by a factor of 15 over bulk focused geometries. Konorov et al.  also demonstrated phase-matched FWM using a large-core HC-PBGF for millijoule nanosecond pulses, and applied this process to the sensing of water molecules on the inner fiber walls using phase-matched CARS for the O-H bond vibrational modes.
4. Applications to high-power pulse delivery and modelocked femtosecond fiber lasers
In a HC-PBGF, a large fraction (>92%) of the total light power is confined to the air core, which reduces the optical nonlinearities by as much as a factor of 1000 as compared to solid-core silica fibers. A detailed study on nonlinearities in HC-PBGFs due to silica-glass contribution has been recently published . Since the material dispersion experienced by the light mode is that of air, the total dispersion is dominated by the waveguide dispersion. For all the hollow-core fibers fabricated till date, the group-velocity dispersion (GVD) across most of the bandgap is anomalous (See Fig. 4). The magnitude of the GVD can be substantially greater than that in conventional step-index silica fibers and the presence of anomalous GVD with nonlinearity can give rise to optical solitons [30, 31]. For a medium in which the nonlinear refractive index n 2>0, the soliton condition in which nonlinearity and anomalous dispersion are balanced, occurs when the dispersion length Lds=τ 2 p/|β 2| is equal to the nonlinear length Lnl=λAeff/2πn 2 P where τp is the pulse duration, β 2 is the GVD coefficient, λ is the wavelength, Aeff is the effective mode area, and P is the peak pulse power. This condition yields the following expression for the peak power Psol of the soliton :
We note that for HC-PBGFs, since the n 2 is roughly 1000 times smaller, soliton pulses can propagate with peak powers 1000 times higher than in conventional single-mode fibers assuming the dispersion β 2 is comparable in both fibers.
Ouzounov et al.  and Luan et al.  demonstrated solitonic transmission of high-power pulses in PBGFs. In Ref.  the peak power of the pulses was 2.4 MW (110 fs pulse centered at 1470 nm) in an air-filled PBGF. Intrapulse Raman scattering, as a result of the Raman response of air, eventually shifted the wavelengths of the pulse to the edge of the bandgap of the fiber which limited the distance over which it could be propagated. By filling the core with a non Raman-active noble gas (Xe), they were able to avoid the Raman self-frequency shift and demonstrated delivery of 5.5 MW soliton pulses (75 fs at 1510 nm, see Fig. 5) through the fiber - an improvement of 100 times over previous results. Luan et al.  demonstrated soliton pulse delivery over 5 m of fiber at 800 nm with a lower Raman-self-frequency shift.
Ouzounov et al.  further showed that by launching higher-order solitons in Xe gas, it is possible to compress 150-fs pulses by a factor of 2 (shown in Fig. 6) and the compression factor is limited by third-order dispersion in the fiber. They predict much higher compression ratios for longer pulses. Recently, Gérôme et al.  reported delivery of sub-100 fs pulses over 8 m of tapered hollow-core fiber using a similar compression scheme. Konorov et al.  demonstrated that sub-gigawatt pulses in a hollow-core fiber filled with various gases including air can undergo self-action due to nonlinearities of the gases that change the output beam profile as a function of power. Irrespective of the input beam profiles, the pulses coming out of the fiber had a clean circular mode suggesting the possible formation of a Townes profile . Fedotov et al.  have recently demonstrated a striking result whereby using ionization-induced blue-shift in Raman-active gases, they were able to control not only the overall rate of soliton selffrequency shift but also its sign for high-peak-power pulse delivery. Loss of pulse-energy to ionization and the limited anomalous GVD range limit the distance over which the blue-shift can be maintained and eventually the red-shift due to the Raman effect dominates at larger distances.
Femtosecond modelocked fiber lasers  that operate in the soliton regime require an anomalous GVD fiber to complement the nonlinearity inside the cavity. Since silica fibers do not exhibit anomalous dispersion at wavelengths below 1.3 µm, prism pairs or gratings are typically used, which makes the lasers susceptible to mechanical and environmental instabilities. The large anomalous dispersion of HC-PBGFs along with their low effective nonlinearity provide a distinct advantage since large nonlinearities can distort the pulse quality and limit the power in the pulse . Lim et al.  demonstrated a ytterbium fiber laser which used such a HC-PBGF for dispersion control to generate positively chirped pulses that yielded 160 fs pulses after post-compression. Lim et al.  also demonstrated polarization control using the inherent birefringence of a HC-PBGF to achieve higher environmental stability and generate 70-fs, 1-nJ pulses. Ortaç et al.  demonstrated an all-fiber laser that used a HC-PBGF for post-compression of positively-chirped 15.4-ps pulses to 213-fs pulses.
5. Resonant optical interactions with atoms and molecules
Nonlinear optics at ultra-low light levels is critical for quantum information applications such as processing and cryptography [44, 45], which require single-photon manipulation [46, 47], generation, and storage . Many applications utilize the phenomenon of electromagnetically-induced transparency (EIT) , in which a strong control beam modifies the energy level structure of a medium, to not only eliminate the absorption for a weak resonant probe, but also enhance the nonlinearities of the medium. The strength of the nonlinear interaction depends both on the optical depth α=nσL, where n is the number density of absorbers (e.g., atoms),σ is the absorption cross-section, and L is the length over which the interaction occurs, and on the ratio of the guided light mode area and the scattering cross-section of the medium. Hollow-core PBGFs, with their ability to host atoms and molecules and to strongly confine light over very long interaction lengths provide an ideal geometry for performing such low-light nonlinear optics. These fiber-based devices can also be easily integrated into existing telecommunications networks. By similar arguments, the same geometry can be used for evoking a strong nonlinear response from materials with weak nonlinearities.
Ghosh et al.  and Benabid et al.  demonstrated coherent interactions with molecules inside HC-PBGFs using acetylene. Acetylene is a linear symmetric molecule with clean rovibrational bands in the telecommunication wavelength regime, but it has a very weak transition dipole-moment (|µ|2~1×10-4 debye2) per absorption line . Nevertheless, significant electromagnetically induced transparency (EIT) was observed in L-type interactions  as shown in Fig. 7(b), with up to 40% depth (limited only by available pump powers). Pulse delays up to 800 ps with 20-ns probe pulses were also demonstrated at communication wavelengths using this system. Benabid et al.  observed EIT in both L and V schemes and were able to achieve transparencies as high as 70% by operating at higher pump powers. These observations of nonlinear effects from weakly nonlinear gases illustrate the ability of hollow-core fiber geometries to enhance the nonlinear response of the medium.
Henningsen et al.  reported saturated absorption spectroscopy with acetylene and 13C substituted hydrogen cyanide (HCN) with pump powers as low as 10 mW. They also studied gas-filling dynamics inside the core of the fiber and report fundamental differences in gasflow dynamics between acetylene and HCN due to the permanent dipole moment of HCN. Couny et al.  fabricated all-fiber devices by splicing HC-PBGFs filled with acetylene to standard SMF fibers and demonstrated EIT at cryogenic temperatures. Hensley et al.  used femtosecond laser pulses to micromachine channels into the sides of HC-PBGFs and demonstrated variable pressure all-fiber gas cells. An improvement in the signal-to-noise ratio for the measurement made in Ref.  was reported by Thapa et al.  using large-core fibers to isolate core modes from surface modes. As a result of using thermal distribution of molecules at room temperature (υrms≃400m/s) in the small cores (~10 µm), the coherence time of the interaction of molecules with photons is limited by collisions between the molecules and the inner core walls , which yields linewidths of the order of 30 - 40 MHz. Hald et al.  have shown that by properly choosing operational parameters like pressure and pump power, it is possible to achieve small linewidths and still maintain good spectral resolution.
Alkali atoms (e.g., Rubidium) interact strongly with resonant light due to their large transition dipole-moments (|µRb|2~1×102 debye2), which makes them good candidates for systems for performing low-light level nonlinear optics. Alkali atoms also have distinct hydrogen-like energy levels, and magnetic moment coupling between the electrons and nucleus gives rise to a distinct ground-state hyperfine doublet structure. As a result, the energy-level structure in which many energy levels are not directly dipole connected is ideal for setting up two-photon coherences through an intermediate level. This forms the basis of many schemes for storing quantum information that can be manipulated and retrieved. For these reasons alkali-atom systems are very attractive for use in quantum information systems.
The principal difficulty in working with alkali atoms is their chemical reactivity. Alkali atoms react readily with most substances and preventing interactions with glass cell walls is challenging. Furthermore, collisions with cell walls can cause transitions between the non-radiatively linked energy levels that can destroy the delicate coherences created in the atomic systems resulting in a loss of quantum coherence. In hollow-core PBGFs for which the core area is small, this problem is quite severe, and the atom-wall interactions make it difficult to load the fiber with a uniform vapor. Coating cell walls with paraffin has been shown to reduce the collisional decoherence as well as keep the atoms from reacting with bulk-cell walls [58, 59]. Ghosh et al.  chemically coated the inner core walls of a HC-PBGF with a monolayer of organosilane (ODMS) to keep Rb atoms from reacting with the core wall. The Rb vapor was allowed to diffuse down the hollow-core and adsorb on the coating. They used light-induced atomic desorption to generate the vapor by desorbing atoms off the wall coatings using a second desorbing beam. Figure 8(a) shows the first demonstration of the generation of a metastable Rb vapor inside HC-PBGFs. By injecting a control beam into the fiber, EIT in Rb in the V-scheme (probe on D1 transition, pump on D2 transition) was demonstrated using pump powers as low as 10 nW, which is 1000 times lower than that used previously in bulk cells, which clearly demonstrated the potential of HC-PBGF geometries for low-light level nonlinear optics. More recently, Light et al.  have demonstrated Λ-scheme EIT with EIT depths of 5% in a large-core kagome-structure fiber coated with a polymer - PDMS and claimed linewidths as narrow as 6 MHz.
6. Opto-mechanical effects in hollow-core photonic bandgap fibers
Light beams, through their electric field intensities, can exert physical forces on objects as demonstrated by Ashkin . Electric fields induce temporary dipole moments in dielectrics, which induces a force that causes the dipole to be attracted towards the region of higher electric field. This effect is used in applications such as optical tweezers for handling small particles and for trapping of atoms with light . HC-PBGFs with light coupled in the core can be used for transporting and delivery of miniscule particles, atoms and biological materials over large distances regardless of bends. This was first demonstrated  using a 80-µm-core hollow-core fiber that was positioned above a vibrating plate which held 5-µm polystyrene spheres. An Ar-ion beam (514 nm, 80 mW cw) is then coupled into the fiber, and the light coupled out from the other end forms a diverging light field as shown in Fig. 9(a) which accelerates the spheres towards the core in the direction of increasing light intensity. The force experienced by the particle depends on its distance from the fiber as shown in Fig. 9(b), and the authors reported particle transport over a 150-mm distance.
Mandal et al.  have extended the result to demonstrate transport of polystyrene particles through liquid media. The core of a 19-cell HC-PBGF was selectively filled by collapsing the surrounding capillaries at one end and was then dipped into and filled with deionised water. Polystyrene particles were introduced into the deionised water reservoir and a 488-nm laser (210 mW) was coupled into the fiber from the other end to demonstrate transport of particles over a distance of 2 cm. This technique also demonstrated optically induced separation of particles into distinct bands which has potential applications to separation of particles based on size and mass.
In a recent experiment , transport of Rb atoms through a HC-PBGF was demonstrated. A beam of Rb atoms is generated using an oven and collimator slits. This beam is incident on a fiber through which light is coupled from the other side. The light is suitably red-detuned from resonance generating an attractive dipole potential for the Rb atoms. The transported atoms were recorded on the other side using a channeltron, and atom fluxes as high as 300 atoms/s were demonstrated.
The unique HC-PBGF goemetry has opened new doors for pushing the limits of nonlinear optics. The required threshold power for processes such as SRS has been considerably lowered, and the demonstrated conversion eficiencies have been greatly improved. The SRS technique has applications in generation of sub-femtosecond pulses, and numerous groups are working towards using the gas-filled hollow-core fibers as high-gain media for making efficient Raman laser oscillators.
HC-PBGFs can facilitate numerous quantum optical applications. The scheme outlined in Ref.  can be potentially realised down to single-photon levels in HC-PBGF geometries where it is possible to have strong atom-photon coupling. One of the main obstacles is the sizable decoherence that arises due to atom-wall collisions in the core. A solution would be to generate cold-atoms either inside the fiber core or outside the core and then transport them into the core. By controlling the transverse motion of atoms optically to control the collisional decoherence, several groups are trying to realise sensitive quantum-optical effects. By confining atoms and the light mode to volumes with transverse areas approaching the atomic scattering cross-sections, effects seen with high-Q cavities such as photon-blockade  can be studied. This system can also be used to store and interact quantum information by storing multiple light pulses along the fiber length. Using the recently developed fiber micromachining techniques , it might be possible to write electromagnetic cavities into hollow-core fibers by altering the refractive index of the fiber walls, thus improving the atom-photon interaction by orders of magnitude.
These fibers can be used to greatly enhance the nonlinearities for weakly nonlinear materials and can be implemented in many sensitive gas and liquid phase spectroscopic applications. Hollow-core PBGFs may find applications in high harmonic generation (HHG) [68, 69]. By filling a hollow-core PBGF with suitable non-Raman-active gases like the noble gases, the conversion efficiency of HHG can be greatly enhanced due to the long interaction lengths . By preparing a resonant medium like Rb vapor using a weak cw beam , it might be possible to demonstrate HHG with very high efficiency inside hollow-core PBGFs.
Physical transport of particles along hollow-core fibers can also have uses in biological applications. Atom transport inside HC-PBGFs can be applied to perform Sagnac atom-interferometry along closed loops which can be used for applications such as highly sensitive gyroscopes and accelerometers.
The authors gratefully acknowledge support by the Center for Nanoscale Systems, supported by the NSF under grant No. EEC-0117770, the Air Force Office of Scientific Research under contract No. F49620-03-0223, and DARPA under the Slow-Light program.
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