## Abstract

We study Cross Phase Modulational Instability (CPMI) -a particular form of vector modulational instability- in the anomalous dispersion regime in highly birefringent, strongly dispersive, optical fibers. When the pump power is high, the detuning of the Scalar Modulational Instability (SMI) is comparable to the detuning of the CPMI. The gain of the CPMI -which is usually much smaller than the gain of the SMI-, is then strongly enhanced and becomes much larger than the gain of the SMI. This theoretical prediction is well verified experimentally using small core photonic crystal fibers.

©2006 Optical Society of America

## 1. Introduction

Modulation Instability (MI) is an ubiquitous phenomena which arises whenever dispersion and nonlinearities contrive to make a continuous wave unstable. Optical fibers are one of the preferred medium in which to study how MI affects the propagation of light, since the small mode-field diameters and long propagation distances make these effects readily accessible to experiment. The first experimental observation of MI in optical fibers [1] concerns Scalar Modulation Instability (SMI) in which polarization effects are absent. In birefringent fibers the two polarizations are coupled through the cross Kerr nonlinearity and new processes, called Vector Modulation Instabilities (VMI), can occur.

Two kinds of VMI can occur in birefringent fibers: Polarisation MI in which the pump beam is launched along one of the axes of the fiber whereas the Stokes and anti-Stokes waves are both polarised along the other axis; and Cross Phase Modulation Instability (CPMI) in which the pump is launched at 45 degrees to the axes of the fiber whereas the Stokes and anti-Stokes waves are polarised along the fast and slow axes. Related to this division, it is convenient to divide experiments on VMI in birefringent fibers into two classes: the low-birefringence situation where walk-off is negligible, and the high-birefringence case where the beat length is smaller than all other relevant length scales (fiber length, dispersion length, nonlinearity length). We refer to [2] for an overview of VMI.

In optical fibers VMI has been studied both in normal and anomalous dispersion, and in the high and low birefringence regimes, see for instance [3, 4, 5, 6]. Recently interest has focused on studying MI in microstructured fibers. The observation of VMI in highly birefringent photonic crystal fibers (PCF) was first reported in [8] and further studied theoretically in [7] and experimentaly in [9, 10, 11, 12]. In [9] it was also discussed how structural irregularities in highly birefringent fibers affect CPMI in the anomalous dispersion regime, and can make it unobservable: an effect which we confirm, having ourselves been unable to observe CPMI in the anomalous dispersion regime in similar fibers.

In the present work we report observations of CPMI in the anomalous dispersion regime in highly birefringent photonic crystal fibers. The fiber we used was not designed to be birefringent, but small imperfections induced during manufacture give rise to a residual birefringence. The birefringence is therefore weak, but still sufficient to be in the “strongly birefringent” case, according to the terminology discussed above.

Interestingly in our experimental conditions the detuning at which the CPMI and SMI spectral peaks appear are comparable. In this regime we observe a strong enhancement of the CPMI at the expense of the SMI, an effect which we show is in agreement with theory. The effect which we study, namely that when two instabilities occur at similar detunings they can no longer be viewed separately as two independent physical phenomena, but instead must be considered collectively, is very general. We expect it will find applications in various fields of nonlinear dynamics such as parametric amplifiers.

Finally we note that one of the main motivations for our work is that we believe that VMI constitutes an interesting alternative, in particular in view of its tunability, to non classical light source based on SMI as demonstrated for instance in [13,14]. But developing these applications requires that VMI be first understood in full detail.

## 2. Light propagation in highly birefringent fibers

The evolution of the envelope of a light pulse centered on frequency Ω_{0}, propagating in a highly birefringent fiber (ie. sufficiently birefringent that coherent terms that oscillate over the beat length of the fiber can be neglected), and neglecting absorption, is described by the equations:

$$\frac{\partial {A}_{y}}{\partial z}-\frac{\Delta {\beta}_{1}}{2}\frac{\partial {A}_{y}}{\partial t}+i\frac{{\beta}_{2}}{2}\frac{{\partial}^{2}{A}_{y}}{\partial {t}^{2}}=\mathit{i\gamma}\left({\mid {A}_{y}\mid}^{2}+B{\mid {A}_{x}\mid}^{2}\right){A}_{y}$$

where *A*_{x,y}
are the slowly varying amplitudes along the slow and fast axis of the fiber, Δ*β*
_{1} = 1/*v*_{gx}
- 1/*v*_{gy}
is the group-velocity mismatch parameter, *β*
_{2} the group-velocity dispersion parameter, *γ* = (*n*
_{2}Ω_{0})/(*cA*_{eff}
) the nonlinear parameter (with *n*
_{2} the nonlinear index coefficient, *c* the speed of light in vacuum, *A*_{eff}
the effective mode area), and *B* = 2/3 the cross-phase modulation weight parameter in silica fibers.

In order to simplify the theoretical description, we can rescale the lengths in Eqs. (1) as *z̃* = (*γ*
*P*
_{0}/2) *z*, the times as $\tilde{t}=\sqrt{\frac{\gamma {P}_{0}}{\mid {\beta}_{2}\mid}}t$ and the fields as ∣*Ã*_{x,y}
∣^{2} = 2∣*A*_{x,y}
∣^{2}/*P*
_{0}. Here *P*
_{0} is the total power of the continuous wave that is injected in the fiber. Eqs. (1) then depend on a single dimensionless parameter

When *α* ≫ 1 and *β*
_{2} < 0 (anomalous dispersion) there are two distinct regimes of instability [6]:

- Scalar Modulation Instability with spectral peak approximately located at detuning ${\omega}_{\mathit{SMI}}\approx \sqrt{\frac{\gamma {P}_{0}}{\mid {\beta}_{2}\mid}},$
- Cross Phase Modulation Instability with spectral peak approximately located at detuning
*ω*_{CPMI}≈∣Δ*β*_{1}/*β*_{2}∣.

Consequently the dimensionless parameter *α* measures the ratio between the detunings of the CPMI and SMI spectral peaks (strictly speaking, this interpretation only holds when *α* ≫ 1). In the present paper we investigate the behavior of SMI and CPMI in highly birefringent fibers in the anomalous dispersion regime as the dimensionless ratio *α* changes, and in particular when it is of order 1. Note that in the normal dispersion regime the parameter *α* also plays an important role, since MI only occurs when $\alpha >\sqrt{\frac{2}{3}}$[2].

The stability of a continuous solution of Eqs. (1) is studied by taking as ansatz

$${A}_{y}={A}_{y}^{0}{e}^{\mathit{i\gamma}\left({P}_{y}+B{P}_{x}\right)z}\left(1+{\int}_{0}^{\infty}\mathit{d\omega}{e}^{-\mathit{i\omega t}}{e}^{\mathit{ik}\left(\omega \right)z}{a}_{y}\left(\omega \right)+{e}^{+\mathit{i\omega t}}{e}^{-i{k}^{*}\left(\omega \right)z}{a}_{y}\left(-\omega \right)\right)$$

where *P*_{x}
= ∣${A}_{x}^{0}$∣^{2} and *P*_{y}
= ∣${A}_{y}^{0}$∣^{2}. For simplicity, and as this is the case where the CPMI is largest, we hereafter take the pump to be equally distributed along the two polarization axis *P*_{x}
= *P*_{y}
= **P**
_{0}/2. Upon inserting Eqs. (3) into Eqs. (1) and linearizing, one obtains four coupled homogenous equations for *a*_{x,y}
(±*ω*). These equations have solutions only for four choices of *k*(*ω*) (eigenvalues of the linear system) given by

when written in rescaled units *k̃* = (2/*γ*
*P*
_{0}) *k* and $\tilde{\omega}=\frac{\omega}{\sqrt{\frac{\gamma {P}_{0}}{\mid {\beta}_{2}\mid}}}$ A perturbation at detuning *ω* will grow exponentially if Im[*k*(*ω*)] < 0 for at least one of the four solutions of Eq. (4). If this happens for even a single detuning *ω*, the injected continuous wave will be unstable.

To investigate this we have plotted in Fig. 1 the gain spectra for different values of the dimensionless parameter *α* when *β*
_{2} < 0, as well as curves describing how much the growing modes are polarized along the optical axes of the fiber.

Panel (a) corresponds to the case when *α*≫1. For the sake of illustration we have chosen *α* = 3 which is sufficiently large to exhibit the main features of this case. One finds two spectral peaks: the SMI centered near *$\tilde{\omega}$* = 1.0 and the CPMI centered near *$\tilde{\omega}$* = 3.3. The CPMI is completely polarized along the optical axis of the fiber. Associated with the SMI there are two identical eigenvalues *k̃*(*$\tilde{\omega}$*) with negative imaginary part. The eigenvectors associated with these eigenvalues are polarized linearly along the slow and fast axis of the fiber respectively. In the regime *α*≫1 the maximum gain $\mid \mathrm{Im}\phantom{\rule{.2em}{0ex}}k\left({\omega}_{\mathit{CPMI}}\right)\mid =\frac{1}{3}\gamma {P}_{0}$ of the CPMI is smaller than the maximum gain $\mid \mathrm{Im}\phantom{\rule{.2em}{0ex}}k\left({\omega}_{\mathit{SMI}}\right)\mid =\frac{1}{2}\gamma {P}_{0}$ of the SMI by a factor 2/3.

As we decrease *α*, *ω*_{CPMI}
decreases, and the CPMI peaks cease to be completely polarized along the optical axis of the fiber, while the SMI peaks become slightly polarised. Interestingly the gain of the CPMI increases, while that of the SMI decreases. By numerical search, we found that at *α* ≃ 1.43 both gains are equal to 0.4625*γ*
*P*
_{0}. This case is illustrated in panel (b).

When we further decrease *α* below 1.43, the gain of the CPMI becomes larger than the gain of the SMI. This is illustrated in panels (c) (*α* = 1) and (d) (*α* = 0.75).

When *α* becomes significantly smaller than 1, as in panel (e) (*α* = 0.60), it is no longer legitimate to talk of SMI and CPMI separately, since only a single spectral peak appears.

In panel (f) we illustrate the unphysical limit *α* = 0. (Indeed in the limit Δ*β*
_{1}→ 0 no polarisation effects should survive and one should recover only SMI. But to see this one should reinstate the coherent terms that were neglected in Eqs. (1)). The limit *α* = 0 corresponds to the case where Δ*β*
_{1} is very small, but nevertheless sufficiently large that eq. (3) is valid. Taking *α* = 0 in eq. (4) one finds two gain curves, one of which is strictly smaller than the other, and hence unobservable. The larger of the two curves has its maximum at $\tilde{\omega}=\sqrt{\frac{5}{3}}$ and the maximum gain is ∣Im(*k*) ∣ = 5*γ*
*P*
_{0}/6. (This should be compared to the maximum gain of the SMI when light is injected along the optical axis of the fiber, which is equal to *γ*
*P*
_{0}).

## 3. Experimental observations

The experimental setup we use is identical to that described in [6]. It consists of a Q-switched laser (Cobolt Tango) which produces quasi-gaussian pulses at 1536 nm with a 3.55 ns FWHM and a 2.5 kHz repetition rate. The peak power of the pulses exceeds 1 kW. Linear polarization of the pulses is ensured by passing the light through a Polarizing Beam Splitter (PBS), and the orientation of the polarization before injection into the fiber is changed with a half wave plate. After passing through the fiber, the pump is removed using a Fiber Bragg Grating (FBG). The sidebands which arise due to MI can then be analyzed using another PBS and an Optical Spectrum Analyzer (OSA).

In the experiment reported here we used 6.6 meters of photonic crystal fiber 030904p4. This fiber was drawn in-house by the Laboratory of Optical fiber Technology, University of Maria Curie-Sklodowska (UMCS) in Lublin, Poland. A Scanning Electron Micrograph (SEM) of the fiber is shown in Fig. 2. As mentioned above the birefringence in this fiber is due to manufacturing imperfections. In a longer piece of fiber (20m) we observed that the birefringence along the fiber fluctuates. This makes precise measurements of some of the properties of the fibre and of the MI difficult.

We injected light at 45° to the axis of the fiber. In Fig. 3 we plot the spectrum measured at the output of the fiber for different values of the injected power. At the lowest peak power *P*
_{0} = 33.2*W*, the gains of the SMI and CPMI are approximately equal, corresponding to *α* ≃ 1.43, see Fig. 1 (b). The two other curves correspond to peak powers *P*
_{0} = 37.7*W* and 39.8*W*. In these cases the gain of the CPMI is larger than the gain of the SMI, as predicted by theory, see Fig. 1 (c) and (d). Due to experimental limitations we were not able to inject sufficient power to observe the merging of the CPMI and SMI peaks predicted theoretically in Fig. 1 (e) and (f).

In Fig. 4 we have plotted the polarisation of the output spectra, thereby showing that, even when the gain of the CPMI is larger than the gain of the SMI, the CPMI peaks remain polarised, as predicted in panels (c) and (d) of Fig. 1.

In other measurements (not shown), we checked that the SMI and the CPMI peaks could both be stimulated by a classical signal, thereby confirming that they are the result of an instability. (We expect that the signal in Figs. 3 and 4 result from spontaneous amplification of vacuum fluctuations, in analogy with what was observed in [6]). We also carried out similar measurements on a longer piece of fiber (20m). Although the amplitudes of the MI peaks were larger, the results were more difficult to interpret. We ascribe this to the fact that the optical axis of the fiber changes along the fiber, and to significant absorption along the fiber.

The agreement between the theoretical predictions of Fig. 1 and our experimental observations is expected since in the present case the beat length *L*_{B}
≃ 16 mm is much smaller than all other length scales: the non linear length is *L*_{NL}
= 1/*γ*
*P*
_{0} ≃ 46 cm, the fiber length is 6.6 m, the walk off length is *L*_{WO}
= *T*
_{0}/Δ*β*
_{1} ≃ 8.9 km (with *T*
_{0} = 3.55 ns the FWHM of the pulse), and the group velocity dispersion length is *L*_{GVD}
= ${T}_{0}^{2}$/*β*
_{2} ≃ 9 10^{4} km (see the next paragraph for a discussion of the fiber parameters). Hence it is perfectly legitimate to use Eq. (1) in which the coherent terms are neglected. We further confirmed this conclusion by numerically simulating the propagation of light in the optical fiber. To this end we used the program described in [15] which includes the effect of the coherent terms neglected in Eq. (1). We find that the inclusion of these terms does not modify the predictions of Fig. 1, at least for the range of parameters (fiber length, non linear coefficient, dispersion, birefringence) relevant to the present experimental investigation.

Finally, we have independently estimated some of the properties of the fiber. We used the numerical method described in [16] to compute the group-velocity dispersion parameter *β*
_{2} = - 139 ps^{2}/km and the effective mode area *A*_{eff}
= 2.85 μm^{2} of the fiber at 1536 nm. From the latter we deduced the nonlinear parameter *γ* = (*n*
_{2}Ω_{0})/(*cA*_{eff}
) = 35.9W^{-1}km^{-1}. We also used the method described in [17] to measure the phase birefringence of the fiber *B* = *λ*(*β*_{x}
- *β*_{y}
)/(2*π*) and the group birefringence $G=B-\lambda \frac{\mathit{dB}}{\mathit{d\lambda}}.$ From those measurements it follows that the beat length at 1536 nm is approximately 16 mm. We also deduced from those measurements that Δ*β*
_{1} at 1536 nm should lie between 0.23 ps/m and 0.58 ps/m. We ascribe this uncertainty to the fact that, as mentioned before, the birefringence arises from manufacturing imperfections and therefore varies along the fiber. Finally from SMI spectra obtained when light is injected along one axis of the fiber we estimated that *β*
_{2} = -157 ps^{2}/km, while by injecting at 45° to the axis, we used CPMI spectra -similar to those presented in Figs. 3 and 4- to estimate Δ*β*
_{1} = 0.4 ps/m. Though not perfectly consistent, the different estimates for *β*
_{2} and Δ*β*
_{1} are in qualitative agreement. In the future, we intend to report in more detail on the properties of this fiber.

In summary we have investigated theoretically and experimentally the regime, reached at high pump power, where CPMI and SMI occur at similar detunings in a highly birefringent photonic crystal fiber in the anomalous dispersion regime. This is an interesting regime because the two instabilities can no longer be considered as independent physical processes, but must be analysed collectively. We observed, as predicted theoretically, an enhancement of the CPMI at the expense of the SMI in this regime. We were not able to observe the merging of the instabilities which we predict should occur at even higher pump powers.

## Acknowledgments

The authors acknowledge the support of the Fonds pour la formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA, Belgium), of the Interuniversity Attraction Pole pro-gram of the Belgian government under Grant IAP5-18, of the EU project QAP, and of the European Network of Excellence on Micro-Optics NEMO.

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