## Abstract

We observe experimentally, for the first time to our knowledge, the simultaneous emission of two strong conjugate resonant dispersive waves by optical solitons. The effect is observed in a small waveguiding glass feature within the cladding of a Kagome hollow-core photonic crystal fiber. We demonstrate theoretically that the phenomenon is attributed to the unusually high fourth-order dispersion coefficient of the waveguiding feature.

© 2008 Optical Society of America

The generation of a broad supercontinuum (SC) in solid-core photonic crystal fibers (PCFs) is a well-studied and established effect (see [1] and references therein). In the femtosecond regime, that is when pumping the optical fiber with a femtosecond pulse in the anomalous dispersion spectral range, the SC generation process is mainly dominated by solitonic dynamics [1]. In addition to the formation of a high-order soliton in the anomalous dispersion spectral range and its subsequent fission into lower-order solitons, each soliton component is sometimes accompanied by the emission of resonant dispersive waves (RDWs) in a spectral range, where the dispersion can be normal [2]. The latter phenomenon entails the emission of narrowband quasi-linear pulses at specific frequencies and results from a special kind of phase matching (PM) between the optical soliton propagation constant and those of the emitted radiation components [3, 4]. The number of the emitted radiation components per soliton and their frequencies are determined by the zero roots of the underlying PM equation, which depends strongly on the high-order dispersion coefficients. For example, as it was theoretically pointed out by Wai *et al.* [4] and Akhemediev and Karlsson [3] that there can be only one RDW in the case of a medium that possesses a dominant third-order dispersion (3OD) coefficient ${\beta}_{3}$. In the case where the medium dispersion is dominated by a fourth-order dispersion (4OD) coefficient ${\beta}_{4}$, these authors pointed to the possibility of two RDWs. Prior to this Letter, only the 3OD induced RDW has been experimentally identified in the spectra generated within solid-core PCFs [1, 2].

In this Letter, we observe and explain, for the first time to our knowledge, the simultaneous emission of two conjugate RDWs. The two emissions comprise a strong radiation in the visible spectral range of $400\u2013500\text{\hspace{0.17em}}\mathrm{nm}$ and a weak radiation in the near-infrared range of $\sim 1600\text{\hspace{0.17em}}\mathrm{nm}$. These radiated components are part of a distinctively different SC generation mechanism from the one previously reported using a conventional solid-core PCF. The new SC is generated by coupling femtosecond pulses into one glass feature within the cladding of a Kagome-lattice hollow-core photonic crystal fiber (HC-PCF) [5, 6], [see Fig. 1B ].

Figure 1 shows the physical structure and summarizes the linear properties of the fiber and the employed waveguiding feature located within the fiber cladding. The fiber is a Kagome-lattice HC-PCF with a pitch of $12.5\text{\hspace{0.17em}}\mu \mathrm{m}$ [see Fig. 1A] and a strut-thickness of $\sim 550\text{\hspace{0.17em}}\mathrm{nm}$ [6]. The waveguiding feature is a glass node in the fiber cladding that is located at the intersection of the planes, which forms the tessellating star-of-David cladding geometry. The glass-feature is approximately a $1\text{\hspace{0.17em}}\mu \mathrm{m}\times 0.8\text{\hspace{0.17em}}\mu \mathrm{m}$ rectangular bridge [Fig. 1B]. Under the appropriate launching conditions, the light is readily guided and confined within the bridge, as is illustrated by the near-field of the light transmitted over an $\sim 50\text{\hspace{0.17em}}\mathrm{cm}$ length of fiber [Fig. 1C]. The rectangular shape confers a strong birefringence to this optical waveguide as illustrated in Fig. 1D. Figure 1D shows the calculated group velocity dispersion (GVD) spectrum for the two polarized modes associated with the fundamental spatial mode. The mode whose polarization is aligned with axis 1 [Fig. 1B] is labeled polarization 1, while polarization 2 corresponds to the mode whose polarization is aligned with axis 2.

Polarization 1 mode shows a stronger curvature in its GVD wavelength variation, and more importantly, the GVD is anomalous over a spectral band ranging from $630\text{\hspace{0.17em} to}1020\text{\hspace{0.17em}}\mathrm{nm}$. This gives rise to two zero-GVD wavelengths (ZDW), but the GVD of polarization 2 is maintained as normal. This characteristic differs notably from the GVD dependence of a typical PCF whose core is of similar dimensions [7]. A similar GVD was reported in [8] using a particular PCF, but radiation of dispersive waves mediated by 4OD was observed nor explored [9].

This difference in the dispersion properties of the two polarized modes in the bridge has different implications for each polarization’s nonlinear response. Figure 2 shows the generated spectrum when a waveguiding bridge of an $\sim 30\text{\hspace{0.17em}}\mathrm{cm}$ long fiber is excited by a $100\text{\hspace{0.17em}}\mathrm{fs}$ laser pulse from a frequency-doubled mode-locked erbium fiber laser emitting at $\sim 780\text{\hspace{0.17em}}\mathrm{nm}$ with a repetition frequency of $80\text{\hspace{0.17em}}\mathrm{MHz}$ and an average coupled power of $50\text{\hspace{0.17em}}\mathrm{mW}$. The spectrum is collected using two different spectrometers: a multichannel photospectrometer for the $300\u2013900\text{\hspace{0.17em}}\mathrm{nm}$ range and an optical spectrum analyzer for the range of $800\u20131800\text{\hspace{0.17em}}\mathrm{nm}$. When the laser pulse polarization is aligned with polarization 1, the generated spectrum that reaches the fiber output displays evidence of strong effects owing to soliton formation and fission. This is illustrated in Fig. 2A by the peaks at the red side of the pump soliton [1]. The spectrum exhibits two particularly distinctive peaks that rise simultaneously; one is a strong peak at $470\text{\hspace{0.17em}}\mathrm{nm}$ (labeled BRDW for blueshifted RDW) and the second is located at $1570\text{\hspace{0.17em}}\mathrm{nm}$ (labeled RRDW for redshifted RDW). The BRDW has the general features of RDW that have been previously observed and theoretically explained by the effect of 3OD [2]. However, there is no prior report on RRDW, which we will show is in fact the BRDW conjugate resonant dispersive emission from the pump soliton around $780\text{\hspace{0.17em}}\mathrm{nm}$. It is worth noting that these two peaks are not present for the case of polarization 2 owing to the absence of solitonic dynamics. This is expected from the normal dispersion of polarization 2 mode, and consequently the generated spectrum is likely to be due to the interplay of self-phase modulation, four-wave mixing, and stimulated Raman scattering [1]. Furthermore, the two experimental spectra indicate that the nonlinearity-induced polarization cross coupling is a relatively weak effect. In light of this we limit the rest of this Letter to the study of polarization 1. To verify the nature of BRDW and RRDW we first numerically simulated the pulse propagation in the bridge and second experimentally examined the spectral location of the conjugate RDW with the laser pump wavelength.

The numerical simulation was carried out by solving the standard generalized nonlinear Schrödinger equation (GNLSE) [7]: $i{\partial}_{z}\psi +\widehat{D}\left(i{\partial}_{t}\right)\psi +\gamma \cdot \widehat{N}\left({\mid \psi \mid}^{2}\right)\psi =0$. Here $\psi (z,t)$ is the complex amplitude of the pulse electric field envelope, *z* is the propagation distance, *t* is time, $\gamma \equiv {\omega}_{0}{n}_{2}\u2215c{A}_{\mathrm{eff}}\left(\lambda \right)$ is the fiber nonlinear coefficient, where ${n}_{2}$ is the Kerr coefficient of silica glass, and ${A}_{\mathrm{eff}}\left(\lambda \right)$ is the effective mode area of the glass bridge whose wavelength dependence is shown in Fig. 3A . In Fig. 3A the fraction of power in the air is also shown so as to distinguish the fraction of the coupled power that experiences the nonlinearity owing to silica. The nonlinear operator is defined as $\widehat{N}\left({\mid \psi \mid}^{2}\right)\equiv \alpha {\mid \psi \mid}^{2}+(1-\alpha )\int R(t-{t}^{\prime}){\mid \psi \left({t}^{\prime}\right)\mid}^{2}\mathrm{d}{t}^{\prime}$, where *α* is the Kerr/Raman nonlinearity ratio ($\alpha =0.82$ in silica glass) and $R\left(t\right)$ is the Raman response function [7]. The dispersion is considered using the dispersion operator $\widehat{D}\left(i{\partial}_{t}\right)\equiv {\sum}_{m=2}^{M}{\beta}_{m}{\left(i{\partial}_{t}\right)}^{m}\u2215m\text{!}$ within the GNLSE, where ${\beta}_{m}$ is the $m\text{th}$ order dispersion coefficient [7].

The dispersion of the present waveguiding bridge exhibits a characteristically high 4OD contribution relative to 3OD [1]. Figure 3B shows the spectra of ${\beta}_{3}$ and the contribution of 4OD relative to that of 3OD in the dispersion operator, ${\beta}_{4}\Delta \omega \u22154$. Here $\Delta \omega $ is the frequency shift from the pump-soliton frequency, ${\omega}_{0}$. The spectra show that in the spectral range of anomalous dispersion, 4OD has a comparable contribution in the dispersion operator to that of 3OD, and the ratio of ${\beta}_{4}\Delta \omega \u22154$ to ${\beta}_{3}$ is typically 1 order of magnitude larger than in a solid-core PCF with a similar core size at a wavelength near $800\text{\hspace{0.17em}}\mathrm{nm}$ [1]. As a consequence, the PM condition $D(\omega -{\omega}_{0})={\scriptstyle \frac{1}{2}}\gamma {\mid {\psi}_{0}\mid}^{2}$ required for a soliton with central amplitude ${\psi}_{0}$ to emit dispersive waves will be fulfilled at more than one optical frequency. This special regime is illustrated in Fig. 3C, where the generated spectrum calculated for a different propagation length is shown. Figure 3C reproduces the main features of the measured spectrum [see Fig. 2A]. This includes the well documented soliton-fission generated lower-order solitons and their subsequent Raman self-frequency shifts [1]. More importantly, the calculated spectrum emulates the experimental spectrum by showing the simultaneous formation, in the very early stage of propagation, of two spectral components around 500 and $1600\text{\hspace{0.17em}}\mathrm{nm}$. The evidence of the phenomenon underlying their generation is given in Fig. 3D, which shows that the zero crossings of the PM curve $D(\omega -{\omega}_{0})-{\scriptstyle \frac{1}{2}}\gamma {\mid {\psi}_{0}\mid}^{2}$ closely match the wavelengths of the two spectral components. This strongly indicates that the two spectral components are indeed the conjugate RDWs as predicted by Akhemediev and Karlsson [3]. To further corroborate the nature of the experimentally observed radiation components, we measured the dependence of their spectral location on the pump wavelength. This was accomplished by using a tunable mode-locked femtosecond Ti:sapphire oscillator. The pulse duration is $\sim 25\text{\hspace{0.17em}}\mathrm{fs}$, and the repetition rate is $85\text{\hspace{0.17em}}\mathrm{MHz}$. Figure 4 shows the spectral evolution of the pump soliton and the two RDWs as the center wavelength of the laser is changed from $770\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}840\text{\hspace{0.17em}}\mathrm{nm}$. The experimental data are shown by points, while the solid curves represent the theoretical results. The agreement between the two sets of data is excellent, thus providing further evidence on the RDWs nature of the two spectral peaks.

In conclusion, we have experimentally observed, for the first time, the simultaneous generation of two quasi-symmetric peaks of RDWs emitted by solitons in a glass waveguide. Their convenient spectral locations, their polarization, and their favorable temporal properties, which are inherited from the mode-locked input laser pulse, make them excellent candidates for ultrashort-pulsed laser sources.

The authors are grateful to C. Locke, J. McFerran, and M. Frosz for useful discussions. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) (UK) and Welch Foundation (grant A1547).

**1. **J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. **78**, 1135 (2006). [CrossRef]

**2. **A. V. Husakou and J. Herrmann, Phys. Rev. Lett. **87**, 203901 (2001). [CrossRef] [PubMed]

**3. **N. Akhmediev and M. Karlsson, Phys. Rev. A **51**, 2602 (1995). [CrossRef] [PubMed]

**4. **P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, Opt. Lett. **11**, 464 (1986). [CrossRef] [PubMed]

**5. **F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, Science **298**, 399 (2002). [CrossRef] [PubMed]

**6. **F. Couny, F. Benabid, and P. S. Light, Opt. Lett. **31**, 3574 (2006). [CrossRef] [PubMed]

**7. **F. Biancalana, D. V. Skryabin, and A. V. Yulin, Phys. Rev. E **70**, 016615 (2004). [CrossRef]

**8. **K. M. Hilligsøe, T. Andersen, H. Paulsen, C. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. Hansen, and J. Larsen, Opt. Express **12**, 1045 (2004). [CrossRef] [PubMed]

**9. **M. Frosz, P. Falk, and O. Bang, Opt. Express **13**, 6181 (2005). [CrossRef] [PubMed]