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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 18 — Sep. 4, 2006
  • pp: 8290–8297
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Enhanced cross phase modulation instability in birefringent photonic crystal fibers in the anomalous dispersion regime

Anh Tuan Nguyen, Kien Phan Huy, Edouard Brainis, Pawel Mergo, Jan Wojcik, Tomasz Nasilowski, Jurgen Van Erps, Hugo Thienpont, and Serge Massar  »View Author Affiliations


Optics Express, Vol. 14, Issue 18, pp. 8290-8297 (2006)
http://dx.doi.org/10.1364/OE.14.008290


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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

1 . K. Tai , A. Hasegawa , and A. Tomita , “ Observation of modulational instability in optical fibers ,” Phys. Rev. Lett. 56 , 135 – 138 ( 1986 ). [CrossRef] [PubMed]

] 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.

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

3 . J. E. Rothenberg , “ Modulational instability for normal dispersion ,” Phys. Rev. A 42 , 682 – 685 ( 1990 ). [CrossRef] [PubMed]

, 4

4 . P. D. Drummond , T. A. B. Kennedy , J. M. Dudley , R. Leonhardt , and J. D. Harvey , “ Cross-phase modulational instability in high-birefringence fibers ,” Opt. Commun. 78 , 137 – 142 ( 1990 ). [CrossRef]

, 5

5 . S. G. Murdoch , R. Leonhardt , and J. D. Harvey , “ Polarization modulation instability in weakly birefringent fibers ,” Opt. Lett. 20 , 866 – 868 ( 1995 ). [CrossRef] [PubMed]

, 6

6 . D. Amans , E. Brainis , M. Haelterman , Ph. Emplit , and S. Massar “ Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime ,” Opt. Lett. 30 , 1051 – 1053 ( 2005 ). [CrossRef] [PubMed]

]. 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

8 . G. Millot , A. Sauter , J. M. Dudley , L. Provino , and R. S. Windeler , “ Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber ,” Opt. Lett. 27 , 695 – 697 ( 2002 ). [CrossRef]

] and further studied theoretically in [7

7 . F. Biancalana and D. V. Skryabin , “ Vector modulational instabilities in ultra-small core optical fibres ”, J. Opt. A: Pure Appl. Opt. 6 301 – 306 ( 2004 ). [CrossRef]

] and experimentaly in [9

9 . B. Kibler , C. Billet , J. M. Dudley , R. S. Windeler , and G. Millot , “ Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers ,” Opt. Lett. 29 , 1903 – 1905 ( 2004 ). [CrossRef] [PubMed]

, 10

10 . A. Tonello , S. Pitois , S. Wabnitz , G. Millot , T. Martynkien , W. Urbanczyk , J. Wojcik , A. Locatelli , M. Conforti , and C. De Angelis , “ Frequency tunable polarization and intermodal modulation instability in high birefringence holey fiber ,” Opt. Express 14 , 397 – 404 ( 2006 ), http://www.opticsexpress.org/abstract.cfm?id=86923 . [CrossRef] [PubMed]

, 11

11 . J. S. Y. Chen , G. K. L. Wong , S. G. Murdoch , R. J. Kruhlak , R. Leonhardt , J. D. Harvey , N. Y. Joly , and J. C. Knight , “ Cross-phase modulation instability in photonic crystal fibers ,” Opt. Lett. 31 , 873 – 875 ( 2006 ). [CrossRef] [PubMed]

, 12

12 . R. J. Kruhlak , G. K. Wong , J. S. Chen , S. G. Murdoch , R. Leonhardt , J. D. Harvey , N. Y. Joly , and J. C. Knight , “ Polarization modulation instability in photonic crystal fibers ,” Opt. Lett. 31 , 1379 – 1381 ( 2006 ) [CrossRef] [PubMed]

]. In [9

9 . B. Kibler , C. Billet , J. M. Dudley , R. S. Windeler , and G. Millot , “ Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers ,” Opt. Lett. 29 , 1903 – 1905 ( 2004 ). [CrossRef] [PubMed]

] 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

13 . M. Fiorentino , P. L. Voss , J. E. Sharping , and P. Kumar , “ All-fibre photon pair source for quantum communications ,” IEEE Photon. Technol. Lett. 14 , 983 – 985 ( 2002 ). [CrossRef]

,14

14 . J. Rarity , J. Fulconis , J. Duligall , W. Wadsworth , and P. Russell , “ Photonic crystal fiber source of correlated photon pairs ,” Opt. Express 13 , 534 – 544 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?id=82392 . [CrossRef] [PubMed]

]. 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 Ω0, 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:

Axz+Δβ12Axt+iβ222Axt2=(Ax2+BAy2)Ax
AyzΔβ12Ayt+iβ222Ayt2=(Ay2+BAx2)Ay
(1)

where Ax,y are the slowly varying amplitudes along the slow and fast axis of the fiber, Δβ 1 = 1/vgx - 1/vgy is the group-velocity mismatch parameter, β 2 the group-velocity dispersion parameter, γ = (n 2Ω0)/(cAeff ) the nonlinear parameter (with n 2 the nonlinear index coefficient, c the speed of light in vacuum, Aeff 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 = (γ P 0/2) z, the times as t˜=γP0β2t and the fields as ∣Ãx,y2 = 2∣Ax,y2/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

α=Δβ1γP0β2.
(2)

When α ≫ 1 and β 2 < 0 (anomalous dispersion) there are two distinct regimes of instability [6

6 . D. Amans , E. Brainis , M. Haelterman , Ph. Emplit , and S. Massar “ Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime ,” Opt. Lett. 30 , 1051 – 1053 ( 2005 ). [CrossRef] [PubMed]

]:

  • Scalar Modulation Instability with spectral peak approximately located at detuning ωSMIγP0β2,
  • Cross Phase Modulation Instability with spectral peak approximately located at detuning ωCPMI ≈∣Δβ 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 α>23[2

2 . G. P. Agrawal , Nonlinear Fiber Optics, third ed ., Academic Press (San Diego) , 2001 .

].

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

Ax=Ax0e(Px+BPy)z(1+0eiωteik(ω)zax(ω)+e+iωteik*(ω)zax(ω))
Ay=Ay0e(Py+BPx)z(1+0eiωteik(ω)zay(ω)+e+iωteik*(ω)zay(ω))
(3)

where Px = ∣Ax02 and Py = ∣Ay02. 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 Px = Py = P 0/2. Upon inserting Eqs. (3) into Eqs. (1) and linearizing, one obtains four coupled homogenous equations for ax,yω). These equations have solutions only for four choices of k(ω) (eigenvalues of the linear system) given by

k˜2=±2ω˜4(B2+α2(ω˜2+2sgn(β2)))+ω˜2(ω˜2+α2+2sgn(β2))
(4)

when written in rescaled units = (2/γ P 0) k and ω˜=ωγP0β2 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 ω˜ = 1.0 and the CPMI centered near ω˜ = 3.3. The CPMI is completely polarized along the optical axis of the fiber. Associated with the SMI there are two identical eigenvalues (ω˜) 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 Imk(ωCPMI)=13γP0 of the CPMI is smaller than the maximum gain Imk(ωSMI)=12γP0 of the SMI by a factor 2/3.

As we decrease α, ωCPMI 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 0.4625γ 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 Δβ 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 Δβ 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 ω˜=53 and the maximum gain is ∣Im(k) ∣ = 5γ 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).

Fig. 1. Gain spectra for MI in highly birefringent fibers in the anomalous dispersion regime for different values of the birefringence Δβ 1. The horizontal and vertical axes are the dimensionless units ω˜=ωγP0β2 and = (2/γ P 0)k. In the different panels we change the value of the dimensionless parameter α=Δβ1γP0β2. In Panel (a) to (f) α successively takes the values 3, 1.43, 1, 0.75, 0.6 and 0.
The black curves are the values of ∣Im[ (ω˜)]∣ (when Im[] is negative, ie. when there is positive gain) for the four eigenvalues given by Eq. (4).
The red and blue curves describe qualitatively the polarization of the solutions. Specifically they are given by ∣Im[(ω˜)]∣∣ax,y (ω˜)∣22 (∣ax (ω˜)∣2 + ∣ay (ω˜)∣2), ie. they are proportional to the intensities along the fast and slow axis of the fiber, and are normalized to sum to the black curves.

3. Experimental observations

The experimental setup we use is identical to that described in [6

6 . D. Amans , E. Brainis , M. Haelterman , Ph. Emplit , and S. Massar “ Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime ,” Opt. Lett. 30 , 1051 – 1053 ( 2005 ). [CrossRef] [PubMed]

]. 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.

Fig. 2. Scanning Electron Micrograph of fiber 030904p4

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 = 33.2W, 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 = 37.7W and 39.8W. 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

6 . D. Amans , E. Brainis , M. Haelterman , Ph. Emplit , and S. Massar “ Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime ,” Opt. Lett. 30 , 1051 – 1053 ( 2005 ). [CrossRef] [PubMed]

]). 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.

Fig. 3. Spectra measured at the output of 6.6 meters of fiber 030904p4 when the light is injected at 45° to the axes of the fiber. The peak powers P 0 are 33.2 W (blue), 37.7 W (red), 39.8 W (green). Note that the pump at 1536 nm is not observed in the spectra as it is removed by the FBG.(The noise level in the case P 0 = 39.8 W is higher than in the other cases due to a different setting of the OSA.)

Finally, we have independently estimated some of the properties of the fiber. We used the numerical method described in [16

16 . T.P. White , B.T. Kuhlmey , R.C. McPhedran , D. Maystre , G. Renversez , C. Martijn de Sterke , and L.C. Botten , “ Multipole method for microstructured optical fibers. I. Formulation ,” J. Opt. Soc. Am. B 19 , 2322 – 2330 ( 2002 ). [CrossRef]

] to compute the group-velocity dispersion parameter β 2 = - 139 ps2/km and the effective mode area Aeff = 2.85 μm2 of the fiber at 1536 nm. From the latter we deduced the nonlinear parameter γ = (n 2Ω0)/(cAeff ) = 35.9W-1km-1. We also used the method described in [17

17 . T. Nasilowski , T. Martynkien , G. Statkiewicz , M. Szpulak , J. Olszewski , G. Golojuch , W. Urbanczyk , J. Wojcik , P. Mergo , M. Makara , F. Berghmans , and H. Thienpont , “ Temperature and pressure sensitivities of the highly birefringent photonic crystal fiber with core asymmetry ,” Appl. Phys. B 81 , 325 – 331 ( 2005 ). [CrossRef]

] to measure the phase birefringence of the fiber B = λ(βx - βy )/(2π) and the group birefringence G=BλdB. From those measurements it follows that the beat length at 1536 nm is approximately 16 mm. We also deduced from those measurements that Δβ 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 ps2/km, while by injecting at 45° to the axis, we used CPMI spectra -similar to those presented in Figs. 3 and 4- to estimate Δβ 1 = 0.4 ps/m. Though not perfectly consistent, the different estimates for β 2 and Δβ 1 are in qualitative agreement. In the future, we intend to report in more detail on the properties of this fiber.

Fig. 4. Spectra measured at the output of 6.6 meters of fiber 030904p4 when the injected peak power at 1536 nm is 61W. The red and the blue curves represent the light polarized along the slow and fast axis of the fiber respectively as measured after a PBS, whereas the black curve represents the total spectrum. Note that the pump has been removed by a FBG, and does not appear in the spectrum. The noise level is higher than in Fig. 4 due to other experimental conditions. The SMI peaks are barely visible above the background.

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.

References and links

1 .

K. Tai , A. Hasegawa , and A. Tomita , “ Observation of modulational instability in optical fibers ,” Phys. Rev. Lett. 56 , 135 – 138 ( 1986 ). [CrossRef] [PubMed]

2 .

G. P. Agrawal , Nonlinear Fiber Optics, third ed ., Academic Press (San Diego) , 2001 .

3 .

J. E. Rothenberg , “ Modulational instability for normal dispersion ,” Phys. Rev. A 42 , 682 – 685 ( 1990 ). [CrossRef] [PubMed]

4 .

P. D. Drummond , T. A. B. Kennedy , J. M. Dudley , R. Leonhardt , and J. D. Harvey , “ Cross-phase modulational instability in high-birefringence fibers ,” Opt. Commun. 78 , 137 – 142 ( 1990 ). [CrossRef]

5 .

S. G. Murdoch , R. Leonhardt , and J. D. Harvey , “ Polarization modulation instability in weakly birefringent fibers ,” Opt. Lett. 20 , 866 – 868 ( 1995 ). [CrossRef] [PubMed]

6 .

D. Amans , E. Brainis , M. Haelterman , Ph. Emplit , and S. Massar “ Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime ,” Opt. Lett. 30 , 1051 – 1053 ( 2005 ). [CrossRef] [PubMed]

7 .

F. Biancalana and D. V. Skryabin , “ Vector modulational instabilities in ultra-small core optical fibres ”, J. Opt. A: Pure Appl. Opt. 6 301 – 306 ( 2004 ). [CrossRef]

8 .

G. Millot , A. Sauter , J. M. Dudley , L. Provino , and R. S. Windeler , “ Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber ,” Opt. Lett. 27 , 695 – 697 ( 2002 ). [CrossRef]

9 .

B. Kibler , C. Billet , J. M. Dudley , R. S. Windeler , and G. Millot , “ Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers ,” Opt. Lett. 29 , 1903 – 1905 ( 2004 ). [CrossRef] [PubMed]

10 .

A. Tonello , S. Pitois , S. Wabnitz , G. Millot , T. Martynkien , W. Urbanczyk , J. Wojcik , A. Locatelli , M. Conforti , and C. De Angelis , “ Frequency tunable polarization and intermodal modulation instability in high birefringence holey fiber ,” Opt. Express 14 , 397 – 404 ( 2006 ), http://www.opticsexpress.org/abstract.cfm?id=86923 . [CrossRef] [PubMed]

11 .

J. S. Y. Chen , G. K. L. Wong , S. G. Murdoch , R. J. Kruhlak , R. Leonhardt , J. D. Harvey , N. Y. Joly , and J. C. Knight , “ Cross-phase modulation instability in photonic crystal fibers ,” Opt. Lett. 31 , 873 – 875 ( 2006 ). [CrossRef] [PubMed]

12 .

R. J. Kruhlak , G. K. Wong , J. S. Chen , S. G. Murdoch , R. Leonhardt , J. D. Harvey , N. Y. Joly , and J. C. Knight , “ Polarization modulation instability in photonic crystal fibers ,” Opt. Lett. 31 , 1379 – 1381 ( 2006 ) [CrossRef] [PubMed]

13 .

M. Fiorentino , P. L. Voss , J. E. Sharping , and P. Kumar , “ All-fibre photon pair source for quantum communications ,” IEEE Photon. Technol. Lett. 14 , 983 – 985 ( 2002 ). [CrossRef]

14 .

J. Rarity , J. Fulconis , J. Duligall , W. Wadsworth , and P. Russell , “ Photonic crystal fiber source of correlated photon pairs ,” Opt. Express 13 , 534 – 544 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?id=82392 . [CrossRef] [PubMed]

15 .

E. Brainis , D. Amans , and S. Massar , “ Scalar and vector modulation instabilities induced by vacuum fluctuations in fibers: Numerical study ”, Phys. Rev. A 71 , 023808 ( 2005 ) [CrossRef]

16 .

T.P. White , B.T. Kuhlmey , R.C. McPhedran , D. Maystre , G. Renversez , C. Martijn de Sterke , and L.C. Botten , “ Multipole method for microstructured optical fibers. I. Formulation ,” J. Opt. Soc. Am. B 19 , 2322 – 2330 ( 2002 ). [CrossRef]

17 .

T. Nasilowski , T. Martynkien , G. Statkiewicz , M. Szpulak , J. Olszewski , G. Golojuch , W. Urbanczyk , J. Wojcik , P. Mergo , M. Makara , F. Berghmans , and H. Thienpont , “ Temperature and pressure sensitivities of the highly birefringent photonic crystal fiber with core asymmetry ,” Appl. Phys. B 81 , 325 – 331 ( 2005 ). [CrossRef]

OCIS Codes
(190.3100) Nonlinear optics : Instabilities and chaos
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 19, 2006
Revised Manuscript: August 16, 2006
Manuscript Accepted: August 16, 2006
Published: September 1, 2006

Citation
Anh Tuan Nguyen, Kien Phan Huy, Edouard Brainis, Pawel Mergo, Jan Wojcik, Tomasz Nasilowski, Jurgen Van Erps, Hugo Thienpont, and Serge Massar, "Enhanced cross phase modulation instability in birefringent photonic crystal fibers in the anomalous dispersion regime," Opt. Express 14, 8290-8297 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-18-8290


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References

  1. K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986). [CrossRef] [PubMed]
  2. G. P. Agrawal, Nonlinear Fiber Optics, third ed., (Academic Press, San Diego, 2001).
  3. J. E. Rothenberg, "Modulational instability for normal dispersion," Phys. Rev. A 42, 682-685 (1990). [CrossRef] [PubMed]
  4. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, "Cross-phase modulational instability in high-birefringence fibers," Opt. Commun. 78, 137-142 (1990). [CrossRef]
  5. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, "Polarization modulation instability in weakly birefringent fibers," Opt. Lett. 20, 866-868 (1995). [CrossRef] [PubMed]
  6. D. Amans, E. Brainis, M. Haelterman, Ph. Emplit, and S. Massar "Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime," Opt. Lett. 30, 1051-1053 (2005). [CrossRef] [PubMed]
  7. F. Biancalana, and D. V. Skryabin, "Vector modulational instabilities in ultra-small core optical fibres," J. Opt. A: Pure Appl. Opt. 6301-306 (2004). [CrossRef]
  8. G. Millot, A. Sauter, J. M. Dudley, L. Provino, and R. S. Windeler, "Polarization mode dispersion and vectorial modulational instability in air-silica microstructure fiber," Opt. Lett. 27, 695-697 (2002). [CrossRef]
  9. B. Kibler, C. Billet, J. M. Dudley, R. S. Windeler, and G. Millot, "Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers," Opt. Lett. 29, 1903-1905 (2004). [CrossRef] [PubMed]
  10. A. Tonello, S. Pitois, S. Wabnitz, G. Millot, T. Martynkien, W. Urbanczyk, J. Wojcik, A. Locatelli, M. Conforti, and C. De Angelis, "Frequency tunable polarization and intermodal modulation instability in high birefringence holey fiber," Opt. Express 14, 397-404 (2006). [CrossRef] [PubMed]
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