We report the experimental observation of scalar and cross-phase modulation instabilities by pumping a highly birefringent photonic crystal fiber in the normal dispersion regime at 45° to its principal polarization axes. Five sideband pairs (two scalar and three vector ones) are observed simultaneously in the spontaneous regime, four of which have a large frequency shift from the pump, in the range 79-93 THz. These results are in excellent agreement with phase-matching arguments and numerical simulations.
©2013 Optical Society of America
Modulation instability (MI) is a physical process in which the steady state of a system is destabilized by a weak perturbation at a different frequency, as a result of a balance between linear and nonlinear (NL) effects . In optics, MI is often described as a four-photon mixing process in which two pump photons are transferred to symmetric spectral sidebands the location of which is governed by energy conservation and phase matching requirements . In optical fibers, the Kerr-induced NL phase mismatch 2γP (with γ the fiber NL coefficient and P the pump peak power) is positive (due to the self-focusing nature of the Kerr nonlinearity in glass) so that the MI process requires a negative linear phase mismatch to occur. Among the many ways of satisfying this condition in single-mode optical fibers, one can dissociate scalar processes (involving a single polarization state) from vector ones (involving more than one polarization state), as illustrated in Table 1. In scalar MI (SMI), the negative linear phase mismatch can be obtained through anomalous second-order dispersion (β2 < 0, with β2 the second-order dispersion coefficient)  or negative fourth-order dispersion (β4 < 0, with β4 the fourth-order dispersion coefficient) [4,5] in the case of normal second-order dispersion, to name a few. Vector MI occurs when both polarization components of a fiber are simultaneously excited . We focus our analysis in this paper on the case where they are both equally excited (see Table 1), which leads to the maximal gain. In this situation, vector MI is known under two different forms. The first one, termed polarization MI (PMI), involves a coherent interaction between orthogonal polarization states which have a small wave vector mismatch. Although it was first predicted to occur in isotropic media , in most configurations the coherent coupling is assisted by a weak phase birefringence  (on the order of 10−6) which significantly contributes to the linear phase mismatch and therefore plays a major role in the phase-matching condition. Typical features of PMI include the fact that both sidebands have the same polarization state, which is orthogonal to the polarization state of the pump (see Table 1), and that their generation from a pump located in the normal dispersion regime has a power threshold, on the contrary to scalar MI. The second form of vector MI, termed cross-phase modulation MI (XPMI), is observed in optical fibers with a large phase birefringence Bp  (in the order of 10−5 and more) so that orthogonal polarization states have a large wave vector mismatch. Therefore XPMI arises from incoherent interactions of two linearly polarized modes on each of the principal axes of the fiber. A typical feature of XPMI is that it generates pairs of orthogonally polarized sidebands from a pump linearly polarized at 45° to the principal axes of the fiber (see Table 1). Unlike PMI, it has no power threshold since linear phase-matching is allowed .
Experimentally, spontaneous PMI was first observed in a low birefringence fiber , followed by deeper studies in the stimulated regime . Its first demonstration in an isotropic fiber  has been reported almost 30 years after its theoretical predictions and was made possible thanks to the development of spun fibers with negligible phase birefringence (of about 10−9). Experimental demonstrations of XPMI were easier thanks to the availability of highly birefringent fibers and were reported in the early nineties for a normal dispersion pump [9,10], although its demonstration in the anomalous dispersion pumping regime  had to wait almost 20 years (due to the fact that scalar MI also occurs in this configuration). The invention of photonic crystal fibers (PCFs) in the mid-nineties  allowed the exploration of new regimes (in which multiple sets of XPMI sidebands can be observed [16,17]) thanks to the new degree of freedom they provide in controlling their dispersion and birefringence properties [18,19]. Spontaneous XPMI  and PMI  in PCFs were first reported by pumping in the normal dispersion region. They were followed by the demonstration of a new regime in which higher-order dispersion allowed for very large XPMI shifts (about 100 THz) [16,22] which had however to be seeded to be observed beyond 45 THz due to fluctuations in the PCF core diameter . Despite the fact that XPMI was shown to be even more strongly affected by PCF structural irregularities for an anomalous-dispersion pump , it was nonetheless observed in this regime , together with SMI, albeit with restrictions to small sidebands shifts of less than 1.5 THz.
In this work, we focus on SMI  and XPMI processes assisted by higher-order dispersion  in a highly birefringent PCF. We report the simultaneous observation of 5 sidebands pairs (2 from SMI and 3 from XPMI, all spontaneously) by pumping at a specific region in the normal dispersion regime of a PCF with tailored dispersion and birefringence properties.
2. Fiber design
Our work follows findings from [16,17] which show that using suitably designed PCFs allows XPMI phase-matching requirements to be satisfied simultaneously for multiple frequencies, providing that the pump is located at a specific spectral region in the phase-matching diagram. We thus designed a PCF, with tailored dispersion and birefringence suitable for pumping at a wavelength of 1064 nm. PCF requirements include:
- - low normal dispersion at 1064 nm, to allow large sideband shifts through negative higher-order dispersion (second-order dispersion being positive);
- - high phase birefringence to favor XPMI over PMI;
- - single mode operation over the wavelength range of interest to avoid intermodal MI;
- - weak longitudinal fluctuations to avoid suppression of parametric gain.
From these requirements, we have identified a fiber design by simulating the PCF properties with a commercial mode solver based on a finite-element method (FEM). For the fabricated PCF, the hole diameter and spacing are 1.9 µm and 4.12 μm, except for the two larger holes surrounding the core (see scanning electron image in the inset of Fig. 1), which have a diameter of 2.43 μm. Zero dispersion wavelengths (ZDWs) are calculated as 1093 nm and 1095.5 nm respectively for the slow (high group index) and fast (low group index) principal axes of the PCF. The corresponding second- and fourth-order dispersion coefficients at 1064 nm are respectively β2 = 3.5 × 10−27 s2/m and β4 = −9.8 × 10−56 s4/m for the slow axis and β2 = 3.9 × 10−27 s2/m and β4 = −9.6 × 10−56 s4/m for the fast axis. The simulated nonlinear coefficient is γ = 7.3 W−1.km−1 at 1064 nm. The calculated phase and group birefringence are respectively 0.3 × 10−4 and 0.5 × 10−4 at 1064 nm. The measured group birefringence is 0.6 × 10−4 at this wavelength, which indicates very slight discrepancies between geometrical parameters used in the FEM simulations and actual PCF parameters.
3. Numerical results
First, degenerate SMI and XPMI phase-matching curves were calculated using the following set of equationsFigure 1 shows SMI and XPMI phase-matching curves obtained for a pump peak power P = Px + Py = 430 W. The dashed lines correspond to the SMI process obtained by launching the pump respectively to the fast (red lines) and slow (blue lines) polarization axes. In these cases, the polarization state of the MI sidebands matches that of the pump. The solid lines correspond to the XPMI process obtained by launching the pump at 45° to the principal axes of the fiber. In this case, both MI sideband pairs are orthogonally polarized. These phase-matching diagrams show the possibility of simultaneously generating up to 5 sidebands pairs (2 SMI ones and 3 XPMI ones) for pump wavelengths less than 1077 nm, excited with appropriate input polarization state (the vertical line represents a 1064 nm pump). They also show that 4 of these sideband pairs are expected to be generated at very large frequency shifts from the pump of approximately 90 THz (around 1550 nm and 810 nm). Our aim in this work is to provide an experimental demonstration of simultaneous spontaneous SMI and XPMI processes in this configuration, which has never been done to our knowledge.
The phase-matching curves of Fig. 1 shows the expected sideband frequencies, but it does not provide any information about the parametric gain. Therefore, in order to test the feasibility of this experiment, we first performed a numerical study with a set of coupled-mode generalized nonlinear Schrödinger equations (GNLSE). In the case of high enough phase birefringence, the coherent coupling between the two polarization components can be neglected and the set of coupled-mode GNLSEs can be written [2,25]Figure 2 shows the output spectrum on the fast (red line) and slow (blue line) fiber axes obtained from a 2 m-long PCF with an input CW power of 430 W launched at 45° to the principal axes of the fiber. As expected from the phase-matching diagram of Fig. 1, it exhibits 4 sidebands pairs with a detuning of around 90 THz, and one sideband pair located much closer to the pump (they are depicted by arrows with the same style code as in Fig. 1). The highest intensity sideband pairs have the same polarization state and are thus identified as SMI sidebands. The three remaining sideband pairs (located on both sides of the pump and of SMI sidebands) have orthogonal polarization states and are identified as XPMI sidebands. These numerical simulations thus show the possibility of simultaneously generating up to 5 MI sidebands pairs spontaneously.
To demonstrate this experimentally, we used a quasi-CW Q-switched Nd:YAG laser delivering 0.6 ns pulses at 1064.5 nm with a linewidth of approximately 50 pm and a linear polarization state. A polarizer and a half-wave plate were used to adjust the peak power and polarization state. The beam coming out of the PCF was collimated; its polarization state was analyzed before it was launched into an optical spectrum analyzer with a multimode fiber. In order to simultaneously excite both SMI and XPMI processes, the pump beam was launched into the fiber with a polarization state at 45° to the principal axes of the PCF.
The output spectrum displayed in Fig. 3(a) was obtained with an injected peak power of 430 W in a PCF length of 3 m. It exhibits 5 sideband pairs in accordance to the phase-matching diagrams of Fig. 1 and simulation results of Fig. 2. The discrepancy between the measured and expected sidebands from Fig. 1 is less than 1.2 THz and is due to slight discrepancies between actual and simulated PCF parameters used to calculate their properties. The small peak located around 1117 nm is due to stimulated Raman scattering; the 1220 nm and 1310 nm peaks identified are artifacts of the spectrometer. Figures 3(b), 3(c) and 3(d) show spectra measured around anti-Stokes, pump and Stokes regions respectively. The analyzer was oriented along the fast and slow principal axes of the fiber (shown by red and blue lines respectively). As expected from theory and from Figs. 1 and 2, the sideband pairs with the same polarization state are generated at SMI wavelengths while the ones with orthogonal polarization state correspond to expected XPMI wavelengths. Furthermore, the sideband pairs with the highest power corresponds to phase-matched SMI wavelengths, while the ones observed at expected phase-matched XPMI wavelengths have a lower power. These specific features confirm the identification of SMI and XPMI processes represented by arrows in Fig. 3(a).
5. Discussion and conclusion
These results show that spontaneous SMI and XPMI processes can be simultaneously excited in optical fibers with sufficiently high phase birefringence [14,24] by launching the pump polarization state at 45° to each of the principal axes of the fiber. In this configuration, SMI arises from the independent propagation of each orthogonal polarization state along each principal PCF axis, while XPMI arises from the incoherent coupling between both polarization states of each principal PCF axes. The novelty of our study over previous reports [14,24] is that very large XPMI shifts were targeted facilitated by higher-order dispersion . This requires extra-care to reduce longitudinal irregularities of the PCF structure, which are known to mitigate and even suppress parametric gain [16,23,26]. Longitudinal fluctuations were thus reduced similarly to the work of Refs [26,27], which allowed us to observe largely detuned XPMI sidebands spontaneously for the first time, while previous observations needed the process to be seeded for larger XPMI shifts (> 45 THz)  due to small core diameter fluctuations. However, we suspect that spurious short scale fluctuations remained present in our PCF which could explain the fact that we had to use a longer PCF in experiments (3 m) than in the numerical study (2 m), in which short scale PCF irregularities were not accounted for. These spurious short scale fluctuations might also be at the origin of the relatively low experimental polarization extinction ratio of XPMI sidebands, as observed in Figs. 3(b)-3(d).
In summary, we have reported the simultaneous observation of spontaneous SMI and XPMI by pumping in the normal dispersion regime of a PCF. Up to 5 MI sidebands pairs (2 from SMI and 3 from XPMI) were observed at the same time, 4 of which had a large frequency shift from the pump in the range 79-93 THz. These results could be used to demonstrate the first fiber optical parametric oscillator (FOPO) based on vector MI, which could lead to enhanced wavelength tunability and polarization control over scalar FOPOs . They could also find applications in the development of fiber sources producing photon pairs [29,30] or two-color pulses for nonlinear microscopy [31,32].
The authors acknowledge Bertrand Kibler for fruitful discussions. This work was partly supported by the Agence Nationale de la Recherche through the ANR FOPAFE project, by the French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and Fonds Européen de Développement Régional (FEDER) through the “Contrat de Projets Etat Région (CPER) 2007-2013” and the “Campus Intelligence Ambiante (CIA)”. EJRK acknowledges funding support from the UK Engineering and Physical Sciences Research Council.
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