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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, 2]. 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] – often referred to, in the literature, as scalar and vectorial modulation instabilities [4] – 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
where ωp is the pump angular frequency, ω+ and ω−, those of the generated photons, n+ and n− their modal effective indices, and np1 and np2 those of the pump modes (which can be distinct, due to polarization). The generated photons appear in the spectrum as lines at symmetrical frequencies with respect to that of the pump, known as “daughters”.
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, 6] 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, expanded to include vectorial effects in Ref. 8 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], or simply modulation instability (MI) [9]. Limiting the Taylor expansion to the fourth order, one obtains the phase matching condition
The γ parameter is the nonlinear coefficient
with the 4WM effective area
which can be approximated by the self-phase modulation effective area [4]
In Eq. (2), P is the pump power, Ω = ω+ − ωp = ωp − ω-, and βn = dnβ/dωn∣ωp. In particular, β2 is the usual modal dispersion coefficient.
- 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
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, 6], 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 ss → ff) or the opposite (ff → ss in the anomalous dispersion regime) [10].
- 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
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, 12] (XPMI [13], CPMI [14]) 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). 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(n∥–n⊥). 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.
The birefringence is characterized by the phase birefringence parameter, defined by Bϕ = nx–ny, the difference between the effective indices nx and ny of two eigenmodes mainly polarized along the x and y axes respectively. This definition is valid for any mode. However, for the sake of clarity, we herein limit the study to the fudamental mode, so that the effective indices nx and ny are those of the two versions LPx01 and LPy01 of the fundamental mode. In particular, the phase birefringence is related to the beat length Lb = λ/∣Bϕ∣, which is often used to characterize the fundamental mode birefringence.
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 = n–λ dn/dλ, the group birefringence is then defined as Bg = Bϕ – λ dBϕ/dλ. 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].
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 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
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], 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
Three types of fibers with large form birefringence, already manufactured by the standard technique of stacking-and-drawing rods and tubes [16,17,20], were studied. In these structures, the strategy to induce the form birefringence consists in breaking the core 6-fold symmetry : in the case illustrated in Fig. 1(a), two large holes decrease the core dimension in the y (vertical) direction; in the cases of the Fig. 1(b) and 1(c), replacing one or two capillary tubes by one or two rods increases the x (horizontal) core dimension. Birefringences of the order of some 10-3 can thus be obtained. The effective indices and birefringence are computed by using a finite elements commercial software Comsol FemlabTM, verified by calculations made with a home-made boundary integral sofware [21]. The computed solutions were also checked experimentally on various geometries for some, but not all, values of the parameters. Figure 2(a) gives an example of such calculations for the phase birefringence. The corresponding dispersion curves shown in Fig. 2(b) are obtained by fitting polynomials with the calculated effective index data and deriving them with respect to wavelength. Because of the microstructured geometry of studied fibers, the modal dispersion curves are sometimes far from being monotonous like that of silica. As an example, for d/λ = 0.3, the dispersion coefficient D shown in Fig. 2(b) [corresponding to β2 shown in Fig. 3(a)] exhibits two zero dispersion wavelengths (ZDW) in the wavelength range under study (0.3 ≤ λ/Λ ≤ 1).
Fig. 3. Spectral dependence of the second order β2 and fourth order β4 dispersion coefficients (βn = dnβ/dωn∣ωp). 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.
Phase matching curves yield the daughters’wavelengths as a function of the pump wavelength. For the sake of generality, we herein normalized the wavelength λ with respect to the pitch Λ. The picth chosen for our calculations is Λ = 2 μm, typical of fibers designed for pump sources in the visible or near IR range such as the popular Ti:sapphire and Nd:YAG lasers.
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]. 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, 22], 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 (ff → ss type process occurring in the normal dispersion regime, as opposed to the case reported in Ref. 10). 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 (ss → ff 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]: 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]. 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]: 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]. 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]: 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]. In birefringent waveguides, vectorial processes are also allowed and have been theoretically studied in the framework of the quantum theory of light [10, 33, 34]. Daughter frequencies are usually close to that of the pump [29, 35], but photon pairs with widely separated wavelengths have also been reported [30–32, 36, 37]. 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, 32].
Although made in fibers having a nominal 6-fold symmetry, the uncontrolled birefringence of the fibers used in Refs. 31,32 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.
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