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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 10 — May. 10, 2010
  • pp: 10731–10741
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Two-fold symmetric geometries for tailored phase-matching in birefringent solid-core air-silica microstructured fibers

Stéphane Virally, Nicolas Godbout, Suzanne Lacroix, and Laurent Labonté  »View Author Affiliations


Optics Express, Vol. 18, Issue 10, pp. 10731-10741 (2010)
http://dx.doi.org/10.1364/OE.18.010731


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Abstract

The effect of birefringence in 2-fold-symmetric microstructured optical fibers on the phase matching conditions for four-wave mixing is analyzed. The three general types of four-wave mixing are considered. General features are obtained through analytic expansions of phase-matching formulas. Three commonly used designs of fibers are analyzed numerically. Particular designs allow the generation of specified wavelengths, supercontinuum or entangled photons.

© 2010 Optical Society of America

1. Introduction

Solid core air-silica microstructured fibers (MSF) allow for strong guided core mode confinement, so that several nonlinear effects can take place in fiber strands typically one meter long for easily attainable pump powers. Due to the glass central symmetry, only odd-order nonlinear processes may occur in silica cores. They are mainly third order processes, i.e. associated with the χ(3) nonlinear susceptibility coefficicient. Most MSFs present a 6-fold symmetry so that their theoretical birefringence is zero. However, in spite of the care taken to manufacture them, the symmetry is never perfect and the fibers exhibit an uncontrolled form birefringence, i.e., of pure geometrical origin. This can perturb the nonlinear phenomena, such as four wave mixing (4WM), making experimental results difficult to interpret. Moreover, this detrimental birefringence depends on wavelength and may vary along the fiber axis. A better strategy is then to induce the birefringence by reducing the nominally 6-fold to a 2-fold symmetry [1

1. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. Mangan, T. Birks, and P. S. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25(18), 1325 (2000), URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-18-1325. [CrossRef]

, 2

2. M. Steel and R. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19(4), 495–503 (2001), URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=920847. [CrossRef]

]. The flexibility of MSF designs makes it possible to tailor their birefringence properties while preserving a strong confinement in their core.

The purpose of this article is to examine four wave mixing (4WM) phase matching conditions in nominally birefringent silica core MSFs. As we extract relevant parameters to be taken into account in the design of tailored phase matching conditions, we show that various 2-fold symmetric geometries lead to different dispersion curves that can be exploited to manipulate those parameters.

The outline of this article is the following. In Section 2, the different types of 4WM in birefringent MSFs [3

3. R. Stolen, M. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6(5), 213 (1981), URLhttp://ol.osa.org/abstract.cfm?URI=ol-6-5-213. [CrossRef] [PubMed]

] – often referred to, in the literature, as scalar and vectorial modulation instabilities [4

4. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2006).

] – are examined. A nomenclature is proposed to clarify the different cases that may occur. Approximate equations for phase matching conditions are derived to highlight the influence of the different orders of the modal dispersion parameters. Section 3 discusses the essential features of the birefringence of three typical structures for solid core air-silica MSFs, with the emphasis on the birefringence spectral dependence. Section 4 presents results obtained from numerical calculations for different fiber structures and different phase matching types, as defined in Section 2. The results are discussed in Section 5 with regard to different applications, in different 4WM regimes, before the conclusion of Section 6.

2. Scalar and vectorial modulation instabilities

In the 4WM process, two pump photons are destroyed and two photons are generated at different wavelengths, provided total energy and momentum are preserved. We herein only consider degenerate 4WM for which the two pump photons are at the same wavelength. The following equations must then be satisfied

ω++ω=2ωpandn+ω++nω=(np1+np2)ωp,
(1)

Due to the χ(3) silica symmetry rules, only three types of 4WM phase matching processes may occur for a given fiber mode (e.g., the fundamental mode): the copolarized or C type, the orthogonal or O type, and the mixed or M type.

As detailed in the following, each type corresponds to a somewhat different phase matching momentum conservation equation, leading in turn to a different phase matching equation condition. In a first approximation, one can derive the phase matching condition from a Taylor expansion of β (ω) = kn(ω) around ωp, the pump frequency (k is here the wavenumber in vacuum). Although more accurate values may be obtained by means of numerical calculations, Taylor expansions to the second order [5

5. S. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866 (1995), URL http://ol.osa.org/abstract.cfm?URI=ol-20-8-866. [CrossRef] [PubMed]

, 6

6. R. Kruhlak, G. K. L. Wong, J. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. 31(10), 1379 (2006), URL http://ol.osa.org/abstract.cfm?URI=ol-31-10-1379. [CrossRef] [PubMed]

] provide good approximations for solutions close to the pump wavelengths. Additional solutions, further away from the pump, are obtained by a development to the fourth order, first proposed in Ref. 7

7. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225 (2003), URLhttp://ol.osa.org/abstract.cfm?URI=ol-28-22-2225. [CrossRef] [PubMed]

, expanded to include vectorial effects in Ref. 8

8. F. Biancalana and D. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A: Pure Appl. Opt. 6(4), 301–306 (2004), URLhttp://stacks.iop.org/1464-4258/6/i=4/a=002?key=crossref.048f95044d026e6efb96f637f8ad2229. [CrossRef]

and detailed herein.

  • In the C type (copolarized), the two pump photons and the two generated photons polarizations are parallel – thus the copolarized appellation. It is also known as the scalar modulation instability (SMI) [4

    4. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2006).

    ], or simply modulation instability (MI) [9

    9. G. K. L. Wong, A. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B 22(11), 2505 (2005), URLhttp://josab.osa.org/abstract.cfm?URI=josab-22-11-2505. [CrossRef]

    ]. Limiting the Taylor expansion to the fourth order, one obtains the phase matching condition

    Ω2=6β2β4(1±123β4β22γP).
    (2)

    The γ parameter is the nonlinear coefficient

    γ=kn2Aeff.
    (3)

    with the 4WM effective area

    Aeff=Aψ+2dAAψ2dAAψP12dAAψP22dAANLψ+ψψP1ψP2dA
    (4)

    which can be approximated by the self-phase modulation effective area [4

    4. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2006).

    ]

    Aeff(Aψ2dA)2ANLψ4dA.
    (5)

  • In the O type (orthogonal), the pump is polarized along a birefringence axis, the photons being generated on the other. One then finds the phase matching condition

    Ω2=6β2β4(1±123β4β22[k(nn)γP3])
    (6)

    where n and n are the effective indices at pump wavelength in the polarization direction respectively orthogonal and parallel to that of the pump photons. This process is sometimes referred to as polarization modulation instability (PMI) [5

    5. S. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866 (1995), URL http://ol.osa.org/abstract.cfm?URI=ol-20-8-866. [CrossRef] [PubMed]

    , 6

    6. R. Kruhlak, G. K. L. Wong, J. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. 31(10), 1379 (2006), URL http://ol.osa.org/abstract.cfm?URI=ol-31-10-1379. [CrossRef] [PubMed]

    ], or as type I-VMI, by analogy with the second order nonlinear photon generation in crystals. Depending on the dispersion regime, one can have the pump photons polarized along the slow (s) axis of the fiber and the daughter photons polarized along the fast (f) axis (the process usually occuring in the normal dispersion regime and being sometimes labeled ssff) or the opposite (ffss in the anomalous dispersion regime) [10

    10. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009), URLhttp://www.ncbi.nlm.nih.gov/pubmed/19293896. [CrossRef] [PubMed]

    ].

  • In the M type (mixed), the pump polarization direction is 45° tilted with respect to the x and y birefringence axes. The two generated photons are then polarized each on a different axis. The phase matching wavelengths are then obtained by solving the two equations

    β4Ω4±4Δβ3Ω3+12β2Ω2±12Δβ1Ω+12γP=0
    (7)

    with βn = (βnx+βny)/2 and Δβn = βnx-βny. In particular, Δβ1 = Bg is the group birefringence (see Section 3). This process is sometimes referred to as cross phase polarization modulation instability [11

    11. J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990), URLhttp://pra.aps.org/abstract/PRA/v42/i1/p682_1. [CrossRef] [PubMed]

    , 12

    12. P. Drummond, T. 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), URLhttp://linkinghub.elsevier.com/retrieve/pii/003040189090110F. [CrossRef]

    ] (XPMI [13

    13. J. Chen, G. K. L. Wong, S. Murdoch, R. Kruhlak, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Cross-phase modulation instability in photonic crystal fibers,” Opt. Lett. 31(7), 873 (2006), URLhttp://ol.osa.org/abstract.cfm?URI=ol-31-7-873. [CrossRef] [PubMed]

    ], CPMI [14

    14. A. 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 (2006), URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8290. [CrossRef] [PubMed]

    ]) or as type II-VMI, by analogy with the second order nonlinear photon generation in crystals.

Note that these phase matching conditions are derived for a given mode, assumed to be the fundamental mode, but that, in addition to similar phase matching conditions for higher order guided modes, intermodal processes are also possible, (e.g, involving LP01 and LP11 modes such as in Refs. 15–17

15. R. Stolen and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982), URL http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1071660. [CrossRef]

). For a given fiber, there is a large number of possible phase matching processes, so that a proper design of the fiber geometry makes it possible to obtain almost arbitrary daughter wavelengths with arbitrary polarizations for a given pump wavelength.

An important aspect, revealed by the above approximate equations, is the equivalence of pump power and positive birefringence in the phase matching processes: see, for example, the γP/3 term influence in Eq. (6) being equivalent to that of k(nn). In certain experiments, it can be difficult to distinguish the two phenomena. Note, in addition, that due to the uncontrolled birefringence of real MSFs, the degenerescence is lifted and these three types may coexist, leading in turn to possible multiple daughter lines or their spectral broadening.

3. Birefringence in selected 2-fold symmetric geometries

Examples of 2-fold symmetric geometries generating high birefringence in MSFs are presented in Fig. 1.

Fig. 1. Usual types of birefringent solid core air-silica MSFs under study: (A) two larger holes limit the core dimension; (B) the core consists of two missing holes; (C) the core consists of three missing holes. While in the former case, the birefringence is induced by a reduction in the core vertical y-dimension, in these two latter cases, the birefringence is induced by an increase in the core horizontal x-dimension. Symmetry x- and y-axes thus correspond to the largest and smallest dimension of the core, respectively. In the Fiber A case, the two different holes can be smaller than the others, thus making the y-dimension larger than the x one, as in Ref. 10.

Note that, in addition to this phase birefringence, one also defines a group birefringence, which is, most of the time, the actually measured parameter. The birefringence of fibers is indeed usually characterized through an interferometric measurement using a broadband source, which gives access to the difference of the group indices associated with the x and y fundamental mode components. By generalizing the group index concept ng = nλ dn/, the group birefringence is then defined as Bg = Bϕλ dBϕ/. It was theoretically predicted and experimentally observed that the phase and group birefringences of MSFs are of the same order of magnitude but have opposite signs [18

18. L. Labonté, D. Pagnoux, P. Roy, F. Bahloul, and M. Zghal, “Numerical and experimental analysis of the birefringence of large air fraction slightly unsymmetrical holey fibres,” Opt. Commun. 262(2), 180–187 (2006), URL http://linkinghub.elsevier.com/retrieve/pii/S0030401805014008. [CrossRef]

].

Note also that, due to additional space provided in the core (fibers B and C) or to a decrease of the equivalent cladding index (fiber A) in one direction, two spatial modes or more may be guided in the cores of these 2-fold symmetric structures. This is different from the behavior of MSFs presenting a 6-fold symmetric structure, which can be endlessly single-mode.

Using a simple analytical rectangular model, Ref. 19

19. L. Labonté, E. Pone, M. Skorobogatiy, N. Godbout, S. Lacroix, and D. Pagnoux, “Analysis of the birefringence of solid-core air-silica microstructured fibers,” Proc. SPIE pp. 73,570N–73,570N-11 (2009), URL http://link.aip.org/link/PSISDG/v7357/i1/p73570N/s1\&Agg=doi.

explains the origin of the birefringence of the 2-fold symmetric structures such as those represented in Fig. 1. The main conclusions, drawn from this model and confirmed by numerical results, are the following.

  • First, extremely large birefringences can be attained in the typical asymmetric MSFs shown in Fig. 1. As for any birefringence of geometrical origin, the fundamental LP01 core-mode phase birefringence is largely wavelength dependent. It is shown to follow roughly a power-law type dependence

    Bϕ=α·λm
    (8)

    where α is a constant and typical values of m are found to lie between 2.5 and 3, depending on the structure. As an example, the phase birefringence Bϕ, together with the corresponding D dispersion coefficient, as functions of λ, are shown in Fig. 2 for Fiber B of Fig. 1. The power-law of Eq. (8), which was first derived empirically for MSFs presenting uncontrolled small birefringence [18

    18. L. Labonté, D. Pagnoux, P. Roy, F. Bahloul, and M. Zghal, “Numerical and experimental analysis of the birefringence of large air fraction slightly unsymmetrical holey fibres,” Opt. Commun. 262(2), 180–187 (2006), URL http://linkinghub.elsevier.com/retrieve/pii/S0030401805014008. [CrossRef]

    ], is still valid for all three structures of Fig. 1 and for any dimension (i.e., any hole/pitch ratio) over a large wavelength range. For the particular example of Fig. 2, one has m ≈ 2.52, the law being valid all over the domain of variation of the parameters, except at large wavelengths for very small fractions of air (d/Λ < 0.4). Whenever the phase birefringence may be modelled by Eq. (8), the group birefringence may be written as Bg = α(1-m)λm which would be exactly equal to −Bϕ for m = 2 and −2Bϕ for m = 3. Thus, the group birefringence also exhibits a large wavelength dependence, but with opposite sign.

  • Second, for a given wavelength and a given structure, the birefringence increases with the equivalent index step, which is expected. In cases of Fibers B and C, the fundamental LP01 core-mode phase birefringence increases with the hole diameter d (keeping the pitch Λ constant). In case of Fiber A, the larger the difference between the large and the small hole diameters (D and d respectively), the larger the birefringence is. In a first approximation, the fundamental LP01 core-mode phase birefringence is proportional to (D-d)/Λ.
Fig. 2. Spectral dependence of (a) the phase birefringence Bϕ, and (b) the average modal dispersion D = (Dx +Dy)/2. The various curves are calculated for the fundamental mode of Fiber B (see Fig. 1), and correspond to various fractions of air d/Λ. For a step Λ=2μm, the values of this parameter, indicated for each curve, represent the holes radius expressed in μm. Note that the silica dispersion was taken into account for these calculations.

4. Calculation of the phase matching curves

Fig. 3. Spectral dependence of the second order β2 and fourth order β4 dispersion coefficients (βn = dnβ/dωnωp). The curves are calculated for the fundamental mode of Fiber B (see Fig. 1), and correspond to various fractions of air d/Λ. For a step Λ = 2μm, the values of this parameter, indicated for each curve, represent the holes radius expressed in μm. Note that the silica dispersion was taken into account for these calculations.

Figure 3 shows the variation of β2 and β4 with wavelength for Fiber B as shown in Fig. 1. Since the fiber is birefringent these values are actually averaged (βn = (βnx+βny)/2).

Figure 4 shows the phase matching curves of the three (C, O, and M) types obtained for Fiber B. They are not obtained by using the approximate formulations of Eqs. (2)–(7) but by using numerical resolution of the exact Eqs. (1). All the phase matching curves are calculated for γP = 0 to highlight the sole effect of birefringence. The only difference between Figs. 4(a) and 4(b) is the hole diameter: d1 = 0.8 and d2 = 1.6 μm respectively.

Fig. 4. C, O, and M phase matching curves for Fiber B as per Fig. 1. Those figures show daughter wavelengths (λout) plotted vs. pump wavelengths (λin). In both cases, the step is Λ=2 μm, but the diameter of the holes is different: d1 = 0.8 and d2 = 1.6 μm, respectively. All these phase matching curves are calculated for γP = 0.

Concentrating on Fig. 4(a), one first observes on the C type curves (solid lines) the typical features of the phase matching curves in the scalar approximation, with the bisector as a trivial Self Phase Modulation solution and non-trivial 4WM solutions when β2 > 0 (short wavelength side). Note that, due to birefringence, non trivial solutions are no more degenerate. The points where the non trivial solution curves intersects the bisector indicate the two zero dispersion wavelengths (ZDW) slightly separated in this case. (Note also that SMI solutions for β2 < 0 are not present here, as P = 0.) The second important feature of Fig. 4(a) is that, due to the birefringence, other O and M type phase matching curves (respectively dashed and dotted lines) are present, well separated from the C type curve. They look like SMI curves, wich is expected since birefringence plays a role analogous to the power in the phase matching equations, as mentioned above.

When comparing Figs. 4(a) and 4(b), one observes first that the main features of Fig. 4(a) are preserved in Fig. 4(b). Due to the larger holes in the latter case, the ZDW is moving to shorter wavelengths, which is expected [20

20. P. S. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006), URL http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-24-12-4729. [CrossRef]

]. The birefringence being also larger, the splitting effect, i.e., separation of the O and M curves from the C curve, is more visible in this case. In both figures, one observes the offset of the phase matching curves from the bisector increasing with the wavelength, due to the birefringence increasing with wavelength, as predicted by Eq. (8) and illustrated in Fig. 2. These figures show that, for a birefringent fiber, for the fundamental mode and a given pump wavelength, the number of daughter pairs depends on the dispersion regime: up to 8 pairs of daughters may be generated in the short wavelength side corresponding to the normal regime, while only 2 may be obtained in the anomalous regime (large wavelength side).

Figure 5 highlights the features associated with different fiber geometries. It shows the O-type phase matching curves, obtained for the three fiber geometries shown in Fig. 1, whose parameters are adjustable. As already mentioned in case of Figs. 4, the offset of the phase matching curves from the bisector increases with the wavelength, due to the corresponding birefringence increase. Fibers B and C having similar geometries show similar behaviors, as seen in Fig. 5(a), and are to be compared with the O type curves of Figs. 4 (the only difference being the value of d). The corresponding phase matching curves are somewhat more separated from the bisector in the Fiber C case, due to the more oblong core shape inducing larger birefringence than in the Fiber B case.

Fig. 5. O-type phase matching curves for the three form birefringent MSFs as per Fig. 1. For all fibers, the step is Λ = 2 μm and the diameter of the small holes is d = 1.4 μm. For fiber A, the diameter of the large holes is D = 2.2 μm. The geometry of the associated fiber structures is shown in inset. All these phase matching curves are calculated for γP = 0. Vertical lines represents ZDWs.

The curves for Fiber A shown in Fig. 5(b) exhibit somewhat different features. From a general viewpoint, the phase matching curves exhibit junctions in the neighborhoods of ZDW. Actually, the number of solutions for a given pump wavelength mainly depends on the sign of the ratio β2/β4 [7

7. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225 (2003), URLhttp://ol.osa.org/abstract.cfm?URI=ol-28-22-2225. [CrossRef] [PubMed]

, 22

22. M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-Based Highly Nonlinear Fibers and Their Application,” IEEE J. Sel. Top. Quantum Electron. 15(1), 103–113 (2009), URL http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4773307. [CrossRef]

], as shown by Eqs. (2) and (6). Whenever β2/β4 > 0, only one phase matching process exists in the O type configuration: the pump photons are polarized along the fast (f) and the daughter photons are generated along the slow (s) axis (ffss type process occurring in the normal dispersion regime, as opposed to the case reported in Ref. 10

10. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009), URLhttp://www.ncbi.nlm.nih.gov/pubmed/19293896. [CrossRef] [PubMed]

). This is the case for very short and very long wavelengths in Fig. 5(b) and long wavelengths in Fig. 5(a). The opposite is true (ssff process) when three matches coexist. This case occurs for example in the mid-wavelength range (in the anomalous dispersion regime) in Fig. 5(b) and short wavelength range (in the norrmal dispersion regime) in Fig. 5(a).

5. Applications

Two types of applications of the in-fiber 4WM processes are to be distinguished depending on the generation regime: stimulated as opposed to spontaneous regime. In the latter case, photons are ideally generated in single pairs. In the stimulated regime, generated photons stimulate the process resulting in exponential growth of the signal with the pump power. This case occurs for example in Optical Parametric Oscillators (OPOs) where the signal is recycled in a resonant cavity.

  • Optical parametric oscillator (OPO) sources [23

    23. Y. Deng, Q. Lin, F. Lu, G. P. Agrawal, and W. Knox, “Broadly tunable femtosecond parametric oscillator using a photonic crystal fiber,” Opt. Lett. 30(10), 1234 (2005), URL http://ol.osa.org/abstract.cfm?URI=ol-30-10-1234. [CrossRef] [PubMed]

    ]: Conventional OPOs, based on second-order nonlinear effect in crystals, require careful alignment in order to achieve optimal operation. Fiber OPOs eliminate the need for alignment and allow further integration with fiber components. Thanks to core light confinement and low losses, optical fiber thus appears as an ideal medium for OPO operation. Using a microstructured fiber permits in addition to customize the modal properties, particularly the dispersion. This has been exploited to build all-fiber OPOs by adujusting the fiber ZDW to existing high-power laser pump wavelengths [24

    24. J. Sharping, “Microstructure fiber based optical parametric oscillators,” J. Lightwave Technol. 26(14), 2184–2191 (2008), URL http://www.opticsinfobase.org/JLT/abstract.cfm?uri=JLT-26-14-2184. [CrossRef]

    ]. Using VMI instead of SMI as parametric gain source would permit to attain unconventional wavelengths in the mid-IR and UV-visible range for potential sensing and biomedicine applications.
  • Supercontinuum (SC) sources [25–27

    25. M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett. 82(14), 2197 (2003), URL http://apl.aip.org/applab/v82/i14/p2197_s1. [CrossRef]

    ]: For a given mode, up to three phase matches may coexist for given pump wavelength and polarization orientation with respect to the fiber axes, and a given phase matching configuration (C, O or M). See the mid-wavelength range of Fig. 5(b) for an O type example. In addition to this multiple phase matching conditions, different types may coexist for an off-axis pump polarization, if the pump power is sufficiently high. Due to the inherently few mode core structure, higher order mode as well as intermodal phase matching conditions in the C, O or M configuration can also occur [16

    16. C. Lesvigne, V. Couderc, A. Tonello, P. Leproux, A. Barthélémy, S. Lacroix, F. Druon, P. Blandin, M. Hanna, and P. Georges, “Visible supercontinuum generation controlled by intermodal four-wave mixing in microstructured fiber,” Opt. Lett. 32(15), 2173 (2007), URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-15-2173. [CrossRef] [PubMed]

    ]. All these phase matches lead to multiple pairs of daughters, sometimes difficult to interpret. The multiplicity of possible daughter lines is conversely beneficial for the SC sources, since SC generation builds on 4WM processes. The fact that daughters may be well separated from the pump source favors the generation in the visible short wavelength and near-UV ranges, which are more difficult to generate than the long wavelength side easily attainable through Stokes Raman gain.
  • Photon pair sources [28

    28. K. Garay-Palmett, H. McGuinness, O. Cohen, J. Lundeen, R. Rangel-Rojo, A. U’ren, M. Raymer, C. McKinstrie, S. Radic, and I. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870 (2007), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-22-14870. [CrossRef] [PubMed]

    ]: Spontaneous 4WM in solid-core MSFs has also been exploited to demonstrate non-classical photon sources to be used in quantum information experiments such as quantum cryptography, teleportation, remote coin tossing or computation. Most of the experiments to date used the scalar or C type 4WM process in which the pump photons and the scattered photons have the same polarization [29–32

    29. J. Sharping, J. Chen, X. Li, P. Kumar, and R. Windeler, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12(14), 3086 (2004), URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3086. [CrossRef] [PubMed]

    ]. In birefringent waveguides, vectorial processes are also allowed and have been theoretically studied in the framework of the quantum theory of light [10

    10. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009), URLhttp://www.ncbi.nlm.nih.gov/pubmed/19293896. [CrossRef] [PubMed]

    , 33

    33. E. Brainis, “Four-photon scattering in birefringent fibers,” Phys. Rev. A 79(2), 023840 (2009), URL http://link.aps.org/doi/10.1103/PhysRevA.79.023840. [CrossRef]

    , 34

    34. O. Cohen, J. Lundeen, B. Smith, G. Puentes, P. Mosley, and I. Walmsley, “Tailored Photon-Pair Generation in Optical Fibers,” Phys. Rev. Lett. 102(12), 123603 (2009), URL http://prl.aps.org/abstract/PRL/v102/i12/e123603. [CrossRef] [PubMed]

    ]. Daughter frequencies are usually close to that of the pump [29

    29. J. Sharping, J. Chen, X. Li, P. Kumar, and R. Windeler, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12(14), 3086 (2004), URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3086. [CrossRef] [PubMed]

    , 35

    35. J. Fan and A. Migdall, “Generation of cross-polarized photon pairs in a microstructure fiber with frequency-conjugate laser pump pulses,” Opt. Express 13(15), 5777 (2005), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-15-5777. [CrossRef] [PubMed]

    ], but photon pairs with widely separated wavelengths have also been reported [30–32

    30. J. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534 (2005), URL http://www.opticsexpress.org/abstractcfm?URI=OPEX-13-2-534. [CrossRef] [PubMed]

    , 36

    36. B. Smith, P. Mahou, O. Cohen, J. Lundeen, and I. Walmsley, “Photon pair generation in birefringent optical fibers,” Opt. Express 17(26), 23589 (2009), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-17-26-23589. [CrossRef]

    , 37

    37. A. McMillan, J. Fulconis, M. Halder, C. Xiong, J. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17(8), 6156 (2009), URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-8-6156. [CrossRef] [PubMed]

    ]. One of the benefits of such a scheme is that one of the daughter photon lays in the telecommunication range (around the wavelength of 1550 nm) where networks already exist and components are readily available, the other being in the visible range, where detectors quantum efficiency is large when compared to that of near IR range detectors. This type of source may then be used as a single photon source at the telecommunication wavelength, heralded by the visible photon emission. It may also be developed into a source of quantum entanglement, suitable among others for quantum repeaters required to break the distance barrier of quantum key distribution. Properties of phase matching curves may also be exploited to design fibers suitable for inherently monochromatic single photon sources [10

    10. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009), URLhttp://www.ncbi.nlm.nih.gov/pubmed/19293896. [CrossRef] [PubMed]

    , 32

    32. C. Söller, B. Brecht, P. Mosley, L. Zang, A. Podlipensky, N. Joly, P. S. Russell, and C. Silberhorn, “Bridging Visible and Telecom Wavelengths with a Single-Mode Broadband Photon Pair Source,” ArXiv (2009), URL http://arxiv.org/abs/0908.2932.

    ].

    Although made in fibers having a nominal 6-fold symmetry, the uncontrolled birefringence of the fibers used in Refs. 31

    31. J. 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), URL http://www.ncbi.nlm.nih.gov/pubmed/20160797. [CrossRef] [PubMed]

    ,32

    32. C. Söller, B. Brecht, P. Mosley, L. Zang, A. Podlipensky, N. Joly, P. S. Russell, and C. Silberhorn, “Bridging Visible and Telecom Wavelengths with a Single-Mode Broadband Photon Pair Source,” ArXiv (2009), URL http://arxiv.org/abs/0908.2932.

    is thought to play a role and designs using largely birefringent fibers such as those of Fig. 1 would probably lead to more performant sources in terms of polarization and monochromaticity.

    When compared to non-classical photon pair sources based on parametric down conversion in nonlinear crystals, these all-fiber sources have the advantage of relatively straight-forward coupling into optical fiber which make them particularly interesting for long distance quantum communication.

6. Conclusion

We studied some features of scalar and vectorial modulation instabilities phase matching curves in birefringent nonlinear mircrostructured fibers. Three types of fibers have been examined, corresponding to fibers already manufactured using the standard stack-and-draw technique. We first showed the general features of the fiber birefringence and dispersion when varying the fiber geometry and the pump wavelength. The induced birefringence can be large, possibly leading to phase matching daugthter wavelengths very far from the pump wavelength. This in turn makes it necessary to use brute force for solving the phase matching equations. Nevertheless, we also derived the approximate equations obtained from Taylor expansions in the pump wavelength neighborhood, thus delineating the role of the different parameters. In particular, we highlighted the equivalence between pump power and birefringence and underlined the key role played by the sign of the β2/β4 ratio in the phase matching processes.

We also clarified the different configurations of possible Vectorial Modulation Instabilities and proposed a new appelation (C, O, and M types) reminding the relative polarisation orientation of the pump and the daughters photons.

This study is beneficial for several applications involving the design of birefringent nonlinear fibers, such as all-fiber light sources whether classical or specifically designed for quantum information processing.

References and links

1.

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. Mangan, T. Birks, and P. S. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25(18), 1325 (2000), URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-18-1325. [CrossRef]

2.

M. Steel and R. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19(4), 495–503 (2001), URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=920847. [CrossRef]

3.

R. Stolen, M. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6(5), 213 (1981), URLhttp://ol.osa.org/abstract.cfm?URI=ol-6-5-213. [CrossRef] [PubMed]

4.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2006).

5.

S. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866 (1995), URL http://ol.osa.org/abstract.cfm?URI=ol-20-8-866. [CrossRef] [PubMed]

6.

R. Kruhlak, G. K. L. Wong, J. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. 31(10), 1379 (2006), URL http://ol.osa.org/abstract.cfm?URI=ol-31-10-1379. [CrossRef] [PubMed]

7.

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225 (2003), URLhttp://ol.osa.org/abstract.cfm?URI=ol-28-22-2225. [CrossRef] [PubMed]

8.

F. Biancalana and D. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A: Pure Appl. Opt. 6(4), 301–306 (2004), URLhttp://stacks.iop.org/1464-4258/6/i=4/a=002?key=crossref.048f95044d026e6efb96f637f8ad2229. [CrossRef]

9.

G. K. L. Wong, A. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B 22(11), 2505 (2005), URLhttp://josab.osa.org/abstract.cfm?URI=josab-22-11-2505. [CrossRef]

10.

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009), URLhttp://www.ncbi.nlm.nih.gov/pubmed/19293896. [CrossRef] [PubMed]

11.

J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990), URLhttp://pra.aps.org/abstract/PRA/v42/i1/p682_1. [CrossRef] [PubMed]

12.

P. Drummond, T. 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), URLhttp://linkinghub.elsevier.com/retrieve/pii/003040189090110F. [CrossRef]

13.

J. Chen, G. K. L. Wong, S. Murdoch, R. Kruhlak, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Cross-phase modulation instability in photonic crystal fibers,” Opt. Lett. 31(7), 873 (2006), URLhttp://ol.osa.org/abstract.cfm?URI=ol-31-7-873. [CrossRef] [PubMed]

14.

A. 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 (2006), URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8290. [CrossRef] [PubMed]

15.

R. Stolen and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982), URL http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1071660. [CrossRef]

16.

C. Lesvigne, V. Couderc, A. Tonello, P. Leproux, A. Barthélémy, S. Lacroix, F. Druon, P. Blandin, M. Hanna, and P. Georges, “Visible supercontinuum generation controlled by intermodal four-wave mixing in microstructured fiber,” Opt. Lett. 32(15), 2173 (2007), URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-15-2173. [CrossRef] [PubMed]

17.

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(1), 397 (2006), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-14-1-397. [CrossRef] [PubMed]

18.

L. Labonté, D. Pagnoux, P. Roy, F. Bahloul, and M. Zghal, “Numerical and experimental analysis of the birefringence of large air fraction slightly unsymmetrical holey fibres,” Opt. Commun. 262(2), 180–187 (2006), URL http://linkinghub.elsevier.com/retrieve/pii/S0030401805014008. [CrossRef]

19.

L. Labonté, E. Pone, M. Skorobogatiy, N. Godbout, S. Lacroix, and D. Pagnoux, “Analysis of the birefringence of solid-core air-silica microstructured fibers,” Proc. SPIE pp. 73,570N–73,570N-11 (2009), URL http://link.aip.org/link/PSISDG/v7357/i1/p73570N/s1\&Agg=doi.

20.

P. S. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006), URL http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-24-12-4729. [CrossRef]

21.

E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing,” Opt. Express 15(16), 10231 (2007), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-10231. [CrossRef] [PubMed]

22.

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, “Silica-Based Highly Nonlinear Fibers and Their Application,” IEEE J. Sel. Top. Quantum Electron. 15(1), 103–113 (2009), URL http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4773307. [CrossRef]

23.

Y. Deng, Q. Lin, F. Lu, G. P. Agrawal, and W. Knox, “Broadly tunable femtosecond parametric oscillator using a photonic crystal fiber,” Opt. Lett. 30(10), 1234 (2005), URL http://ol.osa.org/abstract.cfm?URI=ol-30-10-1234. [CrossRef] [PubMed]

24.

J. Sharping, “Microstructure fiber based optical parametric oscillators,” J. Lightwave Technol. 26(14), 2184–2191 (2008), URL http://www.opticsinfobase.org/JLT/abstract.cfm?uri=JLT-26-14-2184. [CrossRef]

25.

M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett. 82(14), 2197 (2003), URL http://apl.aip.org/applab/v82/i14/p2197_s1. [CrossRef]

26.

Z. Zhu and T. G. Brown, “Polarization properties of supercontinuum spectra generated in birefringent photonic crystal fibers,” J. Opt. Soc. Am. B 21(2), 249 (2004), URL http://josab.osa.org/abstract.cfm?URI=josab-21-2-249. [CrossRef]

27.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Opt. 78(4), 1135–1184 (2006), URL http://link.aps.org/doi/10.1103/RevModPhys.78.1135. [CrossRef]

28.

K. Garay-Palmett, H. McGuinness, O. Cohen, J. Lundeen, R. Rangel-Rojo, A. U’ren, M. Raymer, C. McKinstrie, S. Radic, and I. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15(22), 14870 (2007), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-22-14870. [CrossRef] [PubMed]

29.

J. Sharping, J. Chen, X. Li, P. Kumar, and R. Windeler, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12(14), 3086 (2004), URL http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-14-3086. [CrossRef] [PubMed]

30.

J. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13(2), 534 (2005), URL http://www.opticsexpress.org/abstractcfm?URI=OPEX-13-2-534. [CrossRef] [PubMed]

31.

J. 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), URL http://www.ncbi.nlm.nih.gov/pubmed/20160797. [CrossRef] [PubMed]

32.

C. Söller, B. Brecht, P. Mosley, L. Zang, A. Podlipensky, N. Joly, P. S. Russell, and C. Silberhorn, “Bridging Visible and Telecom Wavelengths with a Single-Mode Broadband Photon Pair Source,” ArXiv (2009), URL http://arxiv.org/abs/0908.2932.

33.

E. Brainis, “Four-photon scattering in birefringent fibers,” Phys. Rev. A 79(2), 023840 (2009), URL http://link.aps.org/doi/10.1103/PhysRevA.79.023840. [CrossRef]

34.

O. Cohen, J. Lundeen, B. Smith, G. Puentes, P. Mosley, and I. Walmsley, “Tailored Photon-Pair Generation in Optical Fibers,” Phys. Rev. Lett. 102(12), 123603 (2009), URL http://prl.aps.org/abstract/PRL/v102/i12/e123603. [CrossRef] [PubMed]

35.

J. Fan and A. Migdall, “Generation of cross-polarized photon pairs in a microstructure fiber with frequency-conjugate laser pump pulses,” Opt. Express 13(15), 5777 (2005), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-15-5777. [CrossRef] [PubMed]

36.

B. Smith, P. Mahou, O. Cohen, J. Lundeen, and I. Walmsley, “Photon pair generation in birefringent optical fibers,” Opt. Express 17(26), 23589 (2009), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-17-26-23589. [CrossRef]

37.

A. McMillan, J. Fulconis, M. Halder, C. Xiong, J. Rarity, and W. J. Wadsworth, “Narrowband high-fidelity all-fibre source of heralded single photons at 1570 nm,” Opt. Express 17(8), 6156 (2009), URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-8-6156. [CrossRef] [PubMed]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: February 11, 2010
Revised Manuscript: March 29, 2010
Manuscript Accepted: May 3, 2010
Published: May 7, 2010

Citation
Stéphane Virally, Nicolas Godbout, Suzanne Lacroix, and Laurent Labonté, "Two-fold symmetric geometries for tailored phase-matching in birefringent solid-core air-silica microstructured fibers," Opt. Express 18, 10731-10741 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-10-10731


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References

  1. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. Mangan, T. Birks, and P. S. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25(18), 1325 (2000), URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-25-18-1325. [CrossRef]
  2. M. Steel, and R. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19(4), 495–503 (2001), URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=920847. [CrossRef]
  3. R. Stolen, M. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6(5), 213 (1981), URL http://ol.osa.org/abstract.cfm?URI=ol-6-5-213. [CrossRef] [PubMed]
  4. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Boston, 2006).
  5. S. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulation instability in weakly birefringent fibers,” Opt. Lett. 20(8), 866 (1995), URL http://ol.osa.org/abstract.cfm?URI=ol-20-8-866. [CrossRef] [PubMed]
  6. R. Kruhlak, G. K. L. Wong, J. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Polarization modulation instability in photonic crystal fibers,” Opt. Lett. 31(10), 1379 (2006), URL http://ol.osa.org/abstract.cfm?URI=ol-31-10-1379. [CrossRef] [PubMed]
  7. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. 28(22), 2225 (2003), URL http://ol.osa.org/abstract.cfm?URI=ol-28-22-2225. [CrossRef] [PubMed]
  8. F. Biancalana, and D. Skryabin, “Vector modulational instabilities in ultra-small core optical fibres,” J. Opt. A: Pure Appl. Opt. 6(4), 301–306 (2004), URL http://stacks.iop.org/1464-4258/6/i=4/a=002?key=crossref.048f95044d026e6efb96f637f8ad2229. [CrossRef]
  9. G. K. L. Wong, A. Chen, S. Murdoch, R. Leonhardt, J. D. Harvey, N. Joly, J. C. Knight, W. J. Wadsworth, and P. S. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B 22(11), 2505 (2005), URL http://josab.osa.org/abstract.cfm?URI=josab-22-11-2505. [CrossRef]
  10. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17(6), 4670–4676 (2009), URL http://www.ncbi.nlm.nih.gov/pubmed/19293896. [CrossRef] [PubMed]
  11. J. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42(1), 682–685 (1990), URL http://pra.aps.org/abstract/PRA/v42/i1/p682_1. [CrossRef] [PubMed]
  12. P. Drummond, T. 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), URL http://linkinghub.elsevier.com/retrieve/pii/003040189090110F. [CrossRef]
  13. J. Chen, G. K. L. Wong, S. Murdoch, R. Kruhlak, R. Leonhardt, J. D. Harvey, N. Joly, and J. C. Knight, “Crossphase modulation instability in photonic crystal fibers,” Opt. Lett. 31(7), 873 (2006), URL http://ol.osa.org/abstract.cfm?URI=ol-31-7-873. [CrossRef] [PubMed]
  14. A. 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 (2006), URL http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8290. [CrossRef] [PubMed]
  15. R. Stolen, and J. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18(7), 1062–1072 (1982), URL http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1071660. [CrossRef]
  16. C. Lesvigne, V. Couderc, A. Tonello, P. Leproux, A. Barthélémy, S. Lacroix, F. Druon, P. Blandin, M. Hanna, and P. Georges, “Visible supercontinuum generation controlled by intermodal four-wave mixing in microstructured fiber,” Opt. Lett. 32(15), 2173 (2007), URL http://www.opticsinfobase.org/abstract.cfm?URI=ol-32-15-2173. [CrossRef] [PubMed]
  17. 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(1), 397 (2006), URL http://www.opticsexpress.org/abstract.cfm?URI=oe-14-1-397. [CrossRef] [PubMed]
  18. L. Labonté, D. Pagnoux, P. Roy, F. Bahloul, and M. Zghal, “Numerical and experimental analysis of the birefringence of large air fraction slightly unsymmetrical holey fibres,” Opt. Commun. 262(2), 180–187 (2006), URL http://linkinghub.elsevier.com/retrieve/pii/S0030401805014008. [CrossRef]
  19. L. Labonté, E. Pone, M. Skorobogatiy, N. Godbout, S. Lacroix, and D. Pagnoux, “Analysis of the birefringence of solid-core air-silica microstructured fibers,” Proc. SPIE pp. 73,570N–73,570N–11 (2009), URL http://link.aip.org/link/PSISDG/v7357/i1/p73570N/s1\&Agg=doi.
  20. P. S. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006), URL http://www.opticsinfobase.org/JLT/abstract.cfm?URI=JLT-24-12-4729. [CrossRef]
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