Abstract

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

1. Introduction

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 [1]. 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 [2]. 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) [3] 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 [6]. 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 [7], in most configurations the coherent coupling is assisted by a weak phase birefringence [8] (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 [9] (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 [10].

Tables Icon

Table 1. Polarization state of the pump (top row) and sidebands (middle row) with respect to fiber principle axes (represented as black dotted lines) for various types of MI with a linearly polarized pump. The bottom row indicates required typical values of phase birefringence Bp.

Experimentally, spontaneous PMI was first observed in a low birefringence fiber [11], followed by deeper studies in the stimulated regime [12]. Its first demonstration in an isotropic fiber [13] 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 [14] 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 [15] 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 [20] and PMI [21] 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 [16]. Despite the fact that XPMI was shown to be even more strongly affected by PCF structural irregularities for an anomalous-dispersion pump [23], it was nonetheless observed in this regime [24], together with SMI, albeit with restrictions to small sidebands shifts of less than 1.5 THz.

In this work, we focus on SMI [5] and XPMI processes assisted by higher-order dispersion [16] 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.

 figure: Fig. 1

Fig. 1 SMI (dashed lines) and XPMI (solid lines) phase-matching curves as a function of pump wavelength calculated for the PCF described in the text. The pump power is set to 430 W. Red and blue lines represent sidebands generated respectively on the fast and slow axis of the PCF. The vertical line corresponds to the 1064 nm pump wavelength. Inset: scanning electron image of the fabricated PCF.

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3. Numerical results

First, degenerate SMI and XPMI phase-matching curves were calculated using the following set of equations

βP,x+βP,y=βS+βAS+γ(Px+Py)and2ωP=ωS+ωAS
where ωP is the angular pump frequency, βP,x and βP,y are the propagation constants of the pump modes, which can be distinct due to phase birefringence and Px and Py are their power. ωS and ωAS are the angular Stokes and anti-Stokes frequencies and βS and βAS are the modal effective indices at these frequencies. All effective indices were simulated using a commercial FEM mode solver for the geometrical PCF parameters described above. Figure 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]

Axzk1ik+1k!βk,xkAxtk=iγ(|Ax|2+23|Ay|2)Ax
Ayzk1ik+1k!βk,ykAytk=iγ(|Ay|2+23|Ax|2)Ay
here Ax and Ay are the field amplitude along orthogonal polarization axes x and y, βn,x and βn,y are the nth-order dispersion terms along these polarization axes and γ is the fiber nonlinear coefficient. Simulations were performed with the calculated PCF parameters described above, taking into account the whole dispersion curve. Stimulated Raman scattering and fiber attenuation were not taken into account as their impact was negligible in the whole study. The input field was a monochromatic continuous-wave (CW) with additional noise modeled as one half-photon per spectral mode with a random phase on each spectral bin. 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.

 figure: Fig. 2

Fig. 2 Numerical simulation of the output spectra on the fast (red line) and slow (blue line) PCF polarization axes. Arrows depict expected phase-matched wavelengths from Fig. 1, with the same style code.

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4. Experiments

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).

 figure: Fig. 3

Fig. 3 (a) Measured output spectrum for a pump peak power of 430 W and input polarization state at 45° of the PCF principal axes. (b), (c), (d) Close-up on the anti-Stokes (b), pump (c) and Stokes (d) spectral regions for the fast axis (red lines) and slow axis (blue lines).

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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 [16]. 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) [16] 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 [28]. 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].

Acknowledgments

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.

References and links

1. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009). [CrossRef]  

2. G. P. Agrawal, Nonlinear Fiber Optics, Fourth Edition (Academic Press, 2006).

3. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986). [CrossRef]   [PubMed]  

4. S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226(1-6), 415–422 (2003). [CrossRef]  

5. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225–2227 (2003). [CrossRef]   [PubMed]  

6. S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1048–1058 (1997). [CrossRef]  

7. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486 (1970).

8. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38(4), 2018–2021 (1988). [CrossRef]   [PubMed]  

9. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990). [CrossRef]   [PubMed]  

10. 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(2), 137–142 (1990). [CrossRef]  

11. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866–868 (1995). [CrossRef]   [PubMed]  

12. G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, “Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber,” J. Opt. Soc. Am. B 15(4), 1266–1277 (1998). [CrossRef]  

13. P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, “Isotropic polarization modulational instability and domain walls in spun fibers,” Appl. Phys. Lett. 75(19), 2873–2875 (1999). [CrossRef]  

14. D. Amans, E. Brainis, M. Haelterman, P. Emplit, and S. Massar, “Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime,” Opt. Lett. 30(9), 1051–1053 (2005). [CrossRef]   [PubMed]  

15. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef]   [PubMed]  

16. J. S. Chen, G. K. 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(7), 873–875 (2006). [CrossRef]   [PubMed]  

17. S. Virally, N. Godbout, S. Lacroix, and L. Labonté, “Two-fold symmetric geometries for tailored phase-matching in birefringent solid-core air-silica microstructured fibers,” Opt. Express 18(10), 10731–10741 (2010). [CrossRef]   [PubMed]  

18. F. Biancalana and D. V. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A, Pure Appl. Opt. 6(4), 301–306 (2004). [CrossRef]  

19. A. Tonello and S. Wabnitz, “Switching off polarization modulation instabilities in photonic crystal fibers,” IEEE Photon. Technol. Lett. 18(8), 953–955 (2006). [CrossRef]  

20. 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(9), 695–697 (2002). [CrossRef]   [PubMed]  

21. 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(10), 1379–1381 (2006). [CrossRef]   [PubMed]  

22. E. A. Zlobina, S. I. Kablukov, and S. A. Babin, “Phase matching for parametric generation in polarization maintaining photonic crystal fiber pumped by tunable Yb-doped fiber laser,” J. Opt. Soc. Am. B 29(8), 1959–1967 (2012). [CrossRef]  

23. 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(16), 1903–1905 (2004). [CrossRef]   [PubMed]  

24. A. T. Nguyen, K. Phan Huy, E. Brainis, P. Mergo, J. Wojcik, T. Nasilowski, J. Van Erps, H. Thienpont, and S. Massar, “Enhanced cross phase modulation instability in birefringent photonic crystal fibers in the anomalous dispersion regime,” Opt. Express 14(18), 8290–8297 (2006). [CrossRef]   [PubMed]  

25. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23(2), 174–176 (1987). [CrossRef]  

26. A. Mussot, A. Kudlinski, R. Habert, I. Dahman, G. Mélin, L. Galkovsky, A. Fleureau, S. Lempereur, L. Lago, D. Bigourd, T. Sylvestre, M. W. Lee, and E. Hugonnot, “20 THz-bandwidth continuous-wave fiber optical parametric amplifier operating at 1 µm using a dispersion-stabilized photonic crystal fiber,” Opt. Express 20(27), 28906–28911 (2012). [CrossRef]   [PubMed]  

27. B. Stiller, S. M. Foaleng, J.-C. Beugnot, M. W. Lee, M. Delqué, G. Bouwmans, A. Kudlinski, L. Thévenaz, H. Maillotte, and T. Sylvestre, “Photonic crystal fiber mapping using Brillouin echoes distributed sensing,” Opt. Express 18(19), 20136–20142 (2010). [CrossRef]   [PubMed]  

28. R. T. Murray, E. J. R. Kelleher, S. V. Popov, A. Mussot, A. Kudlinski, and J. R. Taylor, “Synchronously pumped photonic crystal fiber-based optical parametric oscillator,” Opt. Lett. 37(15), 3156–3158 (2012). [CrossRef]   [PubMed]  

29. J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534–544 (2005). [CrossRef]   [PubMed]  

30. J. A. Slater, J.-S. Corbeil, S. Virally, F. Bussières, A. Kudlinski, G. Bouwmans, S. Lacroix, N. Godbout, and W. Tittel, “Microstructured fiber source of photon pairs at widely separated wavelengths,” Opt. Lett. 35(4), 499–501 (2010). [CrossRef]   [PubMed]  

31. S. Lefrancois, D. Fu, G. R. Holtom, L. Kong, W. J. Wadsworth, P. Schneider, R. Herda, A. Zach, X. Sunney Xie, and F. W. Wise, “Fiber four-wave mixing source for coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 37(10), 1652–1654 (2012). [CrossRef]   [PubMed]  

32. T. Gottschall, M. Baumgartl, A. Sagnier, J. Rothhardt, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber-based source for multiplex-CARS microscopy based on degenerate four-wave mixing,” Opt. Express 20(11), 12004–12013 (2012). [CrossRef]   [PubMed]  

References

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  • |

  1. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
    [Crossref]
  2. G. P. Agrawal, Nonlinear Fiber Optics, Fourth Edition (Academic Press, 2006).
  3. K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986).
    [Crossref] [PubMed]
  4. S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226(1-6), 415–422 (2003).
    [Crossref]
  5. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225–2227 (2003).
    [Crossref] [PubMed]
  6. S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1048–1058 (1997).
    [Crossref]
  7. A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486 (1970).
  8. S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38(4), 2018–2021 (1988).
    [Crossref] [PubMed]
  9. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990).
    [Crossref] [PubMed]
  10. 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(2), 137–142 (1990).
    [Crossref]
  11. S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866–868 (1995).
    [Crossref] [PubMed]
  12. G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, “Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber,” J. Opt. Soc. Am. B 15(4), 1266–1277 (1998).
    [Crossref]
  13. P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, “Isotropic polarization modulational instability and domain walls in spun fibers,” Appl. Phys. Lett. 75(19), 2873–2875 (1999).
    [Crossref]
  14. D. Amans, E. Brainis, M. Haelterman, P. Emplit, and S. Massar, “Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime,” Opt. Lett. 30(9), 1051–1053 (2005).
    [Crossref] [PubMed]
  15. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996).
    [Crossref] [PubMed]
  16. J. S. Chen, G. K. 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(7), 873–875 (2006).
    [Crossref] [PubMed]
  17. S. Virally, N. Godbout, S. Lacroix, and L. Labonté, “Two-fold symmetric geometries for tailored phase-matching in birefringent solid-core air-silica microstructured fibers,” Opt. Express 18(10), 10731–10741 (2010).
    [Crossref] [PubMed]
  18. F. Biancalana and D. V. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A, Pure Appl. Opt. 6(4), 301–306 (2004).
    [Crossref]
  19. A. Tonello and S. Wabnitz, “Switching off polarization modulation instabilities in photonic crystal fibers,” IEEE Photon. Technol. Lett. 18(8), 953–955 (2006).
    [Crossref]
  20. 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(9), 695–697 (2002).
    [Crossref] [PubMed]
  21. 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(10), 1379–1381 (2006).
    [Crossref] [PubMed]
  22. E. A. Zlobina, S. I. Kablukov, and S. A. Babin, “Phase matching for parametric generation in polarization maintaining photonic crystal fiber pumped by tunable Yb-doped fiber laser,” J. Opt. Soc. Am. B 29(8), 1959–1967 (2012).
    [Crossref]
  23. 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(16), 1903–1905 (2004).
    [Crossref] [PubMed]
  24. A. T. Nguyen, K. Phan Huy, E. Brainis, P. Mergo, J. Wojcik, T. Nasilowski, J. Van Erps, H. Thienpont, and S. Massar, “Enhanced cross phase modulation instability in birefringent photonic crystal fibers in the anomalous dispersion regime,” Opt. Express 14(18), 8290–8297 (2006).
    [Crossref] [PubMed]
  25. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23(2), 174–176 (1987).
    [Crossref]
  26. A. Mussot, A. Kudlinski, R. Habert, I. Dahman, G. Mélin, L. Galkovsky, A. Fleureau, S. Lempereur, L. Lago, D. Bigourd, T. Sylvestre, M. W. Lee, and E. Hugonnot, “20 THz-bandwidth continuous-wave fiber optical parametric amplifier operating at 1 µm using a dispersion-stabilized photonic crystal fiber,” Opt. Express 20(27), 28906–28911 (2012).
    [Crossref] [PubMed]
  27. B. Stiller, S. M. Foaleng, J.-C. Beugnot, M. W. Lee, M. Delqué, G. Bouwmans, A. Kudlinski, L. Thévenaz, H. Maillotte, and T. Sylvestre, “Photonic crystal fiber mapping using Brillouin echoes distributed sensing,” Opt. Express 18(19), 20136–20142 (2010).
    [Crossref] [PubMed]
  28. R. T. Murray, E. J. R. Kelleher, S. V. Popov, A. Mussot, A. Kudlinski, and J. R. Taylor, “Synchronously pumped photonic crystal fiber-based optical parametric oscillator,” Opt. Lett. 37(15), 3156–3158 (2012).
    [Crossref] [PubMed]
  29. J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534–544 (2005).
    [Crossref] [PubMed]
  30. J. A. Slater, J.-S. Corbeil, S. Virally, F. Bussières, A. Kudlinski, G. Bouwmans, S. Lacroix, N. Godbout, and W. Tittel, “Microstructured fiber source of photon pairs at widely separated wavelengths,” Opt. Lett. 35(4), 499–501 (2010).
    [Crossref] [PubMed]
  31. S. Lefrancois, D. Fu, G. R. Holtom, L. Kong, W. J. Wadsworth, P. Schneider, R. Herda, A. Zach, X. Sunney Xie, and F. W. Wise, “Fiber four-wave mixing source for coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 37(10), 1652–1654 (2012).
    [Crossref] [PubMed]
  32. T. Gottschall, M. Baumgartl, A. Sagnier, J. Rothhardt, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber-based source for multiplex-CARS microscopy based on degenerate four-wave mixing,” Opt. Express 20(11), 12004–12013 (2012).
    [Crossref] [PubMed]

2012 (5)

2010 (3)

2009 (1)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[Crossref]

2006 (4)

2005 (2)

2004 (2)

2003 (2)

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226(1-6), 415–422 (2003).
[Crossref]

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225–2227 (2003).
[Crossref] [PubMed]

2002 (1)

1999 (1)

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, “Isotropic polarization modulational instability and domain walls in spun fibers,” Appl. Phys. Lett. 75(19), 2873–2875 (1999).
[Crossref]

1998 (1)

1997 (1)

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1048–1058 (1997).
[Crossref]

1996 (1)

1995 (1)

1990 (2)

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990).
[Crossref] [PubMed]

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(2), 137–142 (1990).
[Crossref]

1988 (1)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38(4), 2018–2021 (1988).
[Crossref] [PubMed]

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23(2), 174–176 (1987).
[Crossref]

1986 (1)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986).
[Crossref] [PubMed]

1970 (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486 (1970).

Amans, D.

Atkin, D. M.

Babin, S. A.

Baumgartl, M.

Berkhoer, A. L.

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486 (1970).

Beugnot, J.-C.

Biancalana, F.

F. Biancalana and D. V. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A, Pure Appl. Opt. 6(4), 301–306 (2004).
[Crossref]

Bigourd, D.

Billet, C.

Birks, T. A.

Bouwmans, G.

Brainis, E.

Bussières, F.

Chen, J. S.

Coen, S.

Corbeil, J.-S.

Dahman, I.

Delqué, M.

Drummond, P. D.

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(2), 137–142 (1990).
[Crossref]

Dudley, J. M.

Duligall, J.

Emplit, P.

Fleureau, A.

Foaleng, S. M.

Fu, D.

Fulconis, J.

Galkovsky, L.

Godbout, N.

Gottschall, T.

Habert, R.

Haelterman, M.

Harvey, J. D.

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986).
[Crossref] [PubMed]

Herda, R.

Holtom, G. R.

Hugonnot, E.

Jauregui, C.

Joly, N. Y.

Kablukov, S. I.

Kelleher, E. J. R.

Kennedy, T. A. B.

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(2), 137–142 (1990).
[Crossref]

Kibler, B.

Knight, J. C.

Kockaert, P.

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, “Isotropic polarization modulational instability and domain walls in spun fibers,” Appl. Phys. Lett. 75(19), 2873–2875 (1999).
[Crossref]

Kong, L.

Kruhlak, R. J.

Kudlinski, A.

Labonté, L.

Lacroix, S.

Lago, L.

Lee, M. W.

Lefrancois, S.

Lempereur, S.

Leonhardt, R.

Limpert, J.

Maillotte, H.

Massar, S.

Mélin, G.

Menyuk, C. R.

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23(2), 174–176 (1987).
[Crossref]

Mergo, P.

Millot, G.

Murdoch, S. G.

Murray, R. T.

Mussot, A.

Nasilowski, T.

Nguyen, A. T.

Ostrovsky, L. A.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[Crossref]

Phan Huy, K.

Pitois, S.

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226(1-6), 415–422 (2003).
[Crossref]

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, “Isotropic polarization modulational instability and domain walls in spun fibers,” Appl. Phys. Lett. 75(19), 2873–2875 (1999).
[Crossref]

Popov, S. V.

Provino, L.

Rarity, J.

Rothenberg, J. E.

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990).
[Crossref] [PubMed]

Rothhardt, J.

Russell, P.

Russell, P. St. J.

Sagnier, A.

Sauter, A.

Schneider, P.

Seve, E.

Skryabin, D. V.

F. Biancalana and D. V. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A, Pure Appl. Opt. 6(4), 301–306 (2004).
[Crossref]

Slater, J. A.

Stiller, B.

Sunney Xie, X.

Sylvestre, T.

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986).
[Crossref] [PubMed]

Taylor, J. R.

Thévenaz, L.

Thienpont, H.

Tittel, W.

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986).
[Crossref] [PubMed]

Tonello, A.

A. Tonello and S. Wabnitz, “Switching off polarization modulation instabilities in photonic crystal fibers,” IEEE Photon. Technol. Lett. 18(8), 953–955 (2006).
[Crossref]

Trillo, S.

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1048–1058 (1997).
[Crossref]

Tünnermann, A.

Van Erps, J.

Virally, S.

Wabnitz, S.

A. Tonello and S. Wabnitz, “Switching off polarization modulation instabilities in photonic crystal fibers,” IEEE Photon. Technol. Lett. 18(8), 953–955 (2006).
[Crossref]

G. Millot, E. Seve, S. Wabnitz, and M. Haelterman, “Observation of induced modulational polarization instabilities and pulse-train generation in the normal-dispersion regime of a birefringent optical fiber,” J. Opt. Soc. Am. B 15(4), 1266–1277 (1998).
[Crossref]

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1048–1058 (1997).
[Crossref]

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38(4), 2018–2021 (1988).
[Crossref] [PubMed]

Wadsworth, W.

Wadsworth, W. J.

Windeler, R. S.

Wise, F. W.

Wojcik, J.

Wong, G. K.

Wong, G. K. L.

Zach, A.

Zakharov, V. E.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[Crossref]

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486 (1970).

Zlobina, E. A.

Appl. Phys. Lett. (1)

P. Kockaert, M. Haelterman, S. Pitois, and G. Millot, “Isotropic polarization modulational instability and domain walls in spun fibers,” Appl. Phys. Lett. 75(19), 2873–2875 (1999).
[Crossref]

IEEE J. Quantum Electron. (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23(2), 174–176 (1987).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Tonello and S. Wabnitz, “Switching off polarization modulation instabilities in photonic crystal fibers,” IEEE Photon. Technol. Lett. 18(8), 953–955 (2006).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

F. Biancalana and D. V. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A, Pure Appl. Opt. 6(4), 301–306 (2004).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

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(2), 137–142 (1990).
[Crossref]

S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. 226(1-6), 415–422 (2003).
[Crossref]

Opt. Express (6)

J. Rarity, J. Fulconis, J. Duligall, W. Wadsworth, and P. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534–544 (2005).
[Crossref] [PubMed]

A. T. Nguyen, K. Phan Huy, E. Brainis, P. Mergo, J. Wojcik, T. Nasilowski, J. Van Erps, H. Thienpont, and S. Massar, “Enhanced cross phase modulation instability in birefringent photonic crystal fibers in the anomalous dispersion regime,” Opt. Express 14(18), 8290–8297 (2006).
[Crossref] [PubMed]

T. Gottschall, M. Baumgartl, A. Sagnier, J. Rothhardt, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber-based source for multiplex-CARS microscopy based on degenerate four-wave mixing,” Opt. Express 20(11), 12004–12013 (2012).
[Crossref] [PubMed]

S. Virally, N. Godbout, S. Lacroix, and L. Labonté, “Two-fold symmetric geometries for tailored phase-matching in birefringent solid-core air-silica microstructured fibers,” Opt. Express 18(10), 10731–10741 (2010).
[Crossref] [PubMed]

A. Mussot, A. Kudlinski, R. Habert, I. Dahman, G. Mélin, L. Galkovsky, A. Fleureau, S. Lempereur, L. Lago, D. Bigourd, T. Sylvestre, M. W. Lee, and E. Hugonnot, “20 THz-bandwidth continuous-wave fiber optical parametric amplifier operating at 1 µm using a dispersion-stabilized photonic crystal fiber,” Opt. Express 20(27), 28906–28911 (2012).
[Crossref] [PubMed]

B. Stiller, S. M. Foaleng, J.-C. Beugnot, M. W. Lee, M. Delqué, G. Bouwmans, A. Kudlinski, L. Thévenaz, H. Maillotte, and T. Sylvestre, “Photonic crystal fiber mapping using Brillouin echoes distributed sensing,” Opt. Express 18(19), 20136–20142 (2010).
[Crossref] [PubMed]

Opt. Lett. (11)

R. T. Murray, E. J. R. Kelleher, S. V. Popov, A. Mussot, A. Kudlinski, and J. R. Taylor, “Synchronously pumped photonic crystal fiber-based optical parametric oscillator,” Opt. Lett. 37(15), 3156–3158 (2012).
[Crossref] [PubMed]

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(9), 695–697 (2002).
[Crossref] [PubMed]

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(10), 1379–1381 (2006).
[Crossref] [PubMed]

J. A. Slater, J.-S. Corbeil, S. Virally, F. Bussières, A. Kudlinski, G. Bouwmans, S. Lacroix, N. Godbout, and W. Tittel, “Microstructured fiber source of photon pairs at widely separated wavelengths,” Opt. Lett. 35(4), 499–501 (2010).
[Crossref] [PubMed]

S. Lefrancois, D. Fu, G. R. Holtom, L. Kong, W. J. Wadsworth, P. Schneider, R. Herda, A. Zach, X. Sunney Xie, and F. W. Wise, “Fiber four-wave mixing source for coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 37(10), 1652–1654 (2012).
[Crossref] [PubMed]

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(16), 1903–1905 (2004).
[Crossref] [PubMed]

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225–2227 (2003).
[Crossref] [PubMed]

S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866–868 (1995).
[Crossref] [PubMed]

D. Amans, E. Brainis, M. Haelterman, P. Emplit, and S. Massar, “Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime,” Opt. Lett. 30(9), 1051–1053 (2005).
[Crossref] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996).
[Crossref] [PubMed]

J. S. Chen, G. K. 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(7), 873–875 (2006).
[Crossref] [PubMed]

Phys. Rev. A (2)

S. Wabnitz, “Modulational polarization instability of light in a nonlinear birefringent dispersive medium,” Phys. Rev. A 38(4), 2018–2021 (1988).
[Crossref] [PubMed]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

S. Trillo and S. Wabnitz, “Bloch wave theory of modulational polarization instabilities in birefringent optical fibers,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1048–1058 (1997).
[Crossref]

Phys. Rev. Lett. (1)

K. Tai, A. Hasegawa, and A. Tomita, “Observation of modulational instability in optical fibers,” Phys. Rev. Lett. 56(2), 135–138 (1986).
[Crossref] [PubMed]

Physica D (1)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[Crossref]

Sov. Phys. JETP (1)

A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP 31, 486 (1970).

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, Fourth Edition (Academic Press, 2006).

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Figures (3)

Fig. 1
Fig. 1 SMI (dashed lines) and XPMI (solid lines) phase-matching curves as a function of pump wavelength calculated for the PCF described in the text. The pump power is set to 430 W. Red and blue lines represent sidebands generated respectively on the fast and slow axis of the PCF. The vertical line corresponds to the 1064 nm pump wavelength. Inset: scanning electron image of the fabricated PCF.
Fig. 2
Fig. 2 Numerical simulation of the output spectra on the fast (red line) and slow (blue line) PCF polarization axes. Arrows depict expected phase-matched wavelengths from Fig. 1, with the same style code.
Fig. 3
Fig. 3 (a) Measured output spectrum for a pump peak power of 430 W and input polarization state at 45° of the PCF principal axes. (b), (c), (d) Close-up on the anti-Stokes (b), pump (c) and Stokes (d) spectral regions for the fast axis (red lines) and slow axis (blue lines).

Tables (1)

Tables Icon

Table 1 Polarization state of the pump (top row) and sidebands (middle row) with respect to fiber principle axes (represented as black dotted lines) for various types of MI with a linearly polarized pump. The bottom row indicates required typical values of phase birefringence Bp.

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

β P,x + β P,y = β S + β AS +γ( P x + P y ) and 2 ω P = ω S + ω AS
A x z k1 i k+1 k! β k,x k A x t k =iγ( | A x | 2 + 2 3 | A y | 2 ) A x
A y z k1 i k+1 k! β k,y k A y t k =iγ( | A y | 2 + 2 3 | A x | 2 ) A y

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