1. Introduction

Microstructured optical fibers (MF) have become one of the most attractive platforms for the design of advanced photonic structures. This is essentially due to some peculiar characteristics of these structures not usually found in conventional fibers, such as single mode propagation from ultra-violet to infra-red spectral regions, ease of tailoring modal effective area, and optimized dispersion properties [1–4]. A striking application of these fibers is for the generation of broadband supercontinuum light, since they allow broader bandwidth than those obtained with bulk silica or standard optical fibers [5]. Supercontinuum generation (SCG) in MFs involves a quite intricate combination of dispersive and nonlinear mechanisms, which have been extensively investigated by several research groups. The main mechanisms behind SCG in the anomalous dispersion region and femtosecond pulses are self phase modulation (SPM) followed by soliton fission [6–7], which breaks the injected pulses into a series of fundamental solitons, and soliton self frequency shift (SSFS), which spans the spectrum into the red region [8]. The fundamental solitons also emit phase matched dispersive waves into the blue region [6,8,9,10,11], leading to a spectrum spanning of more than one octave [12]. A viable alternative to explain the formation of the short wavelength edge of the supercontinuum has been recently proposed by [13], which explains this effect in terms of four wave mixing between solitons and dispersive waves.A comprehensive review of the literature on supercontinuum SCG in MFs is available in [5]. Much of the success of MFs for SCG is also due to its extraordinary design flexibility, which allows to control field confinement (and consequently the nonlinear effect) just by changing the air hole diameter (d) to pitch (λ) ratio, that is, aff= d/λ. This characteristic provides an enormous degree of freedom for the design of new structures, and has proved crucial for the enhancement of nonlinear effects. Additionally, by adequately changing the aff parameter one is able to dislocate the zero dispersion wavelength (ZDW) to the visible region [3–4], allowing the generation of SC [12].

Polarization effects are another essential aspect of the dynamics of SCG in MFs. Many studies have been carried out about this subject, particularly in [14,15] where an in-depth investigation of supercontinuum and soliton effects in photonic crystal fibers with vector degrees of freedom taken into account are carried out. Another polarization effect with important consequences to SCG is vector modulational instability (VMI) [16]. This effect has been successfully characterized both theoretically [17] and experimentally [18, 19] for ultra-small core and photonic crystal fibers, respectively. The utilization of highly birefringent fibers (HiBi) for SCG has demonstrated to be quite useful, as it keeps the polarization state preserved along the propagation for all the spectral components. Not surprisingly, MFs have demonstrated to be the most appropriate structure for the enhancement of this useful effect, notably due to the endless geometry variation possibilities that can be obtained. In fact, several approaches have been attempted in the literature aiming at increasing this effect. Ortigosa-Blanch et al [20], for example, have observed that a required birefringence can be achieved either by making the shape of the refractive index profile of the waveguide noncircular (form or shape birefringence) or by making the material of the fiber itself birefringent. They have fabricated a single-mode highly birefringent (HiBi) MF by creating a regular array of air holes of two different sizes disposed about a pure-silica core, resulting in a mode birefringence of 3.7×10^-3. Hansen et. al. [21] have designed a HiBi MF whose main geometric characteristic was an asymmetric core consisting of two neighboring rods, i.e., the core was formed by the omission of two adjacent air holes. The cladding of the fiber consisted of a highly regular triangular lattice of air holes, which resulted in a birefringence of 9.3×10^-4 for the fundamental mode. Elliptical air holes have also been utilized in MFs to provide increased birefringence effect, as suggested by [22]. The birefringence in that particular structure was found to be intrinsically very strong (≥ 0.01 in some cases). The HiBi MF described in [20] was later investigated by [23] which demonstrated that the increase of birefringence through reducing the size of the small holes may not be practical, since the light confinement property of the fiber is affected when the hole size becomes small. It was also demonstrated that the birefringence in this fiber could be further enhanced by increasing the size to pitch ratio for the bigger holes. Another way of increasing birefringent effect consists in producing a MF with elliptical core. In fact, such fiber has been successfully used for SCG in [24], which experimentally obtained ultrabroad supercontinuum (SC) extending from 400 to 1750nm. Besides preserving the polarization state, as expected, it was also observed in [24] that the two eigenpolarizations modes exhibited different dispersion characteristics, meaning that the properties of the generated continuum could be conveniently tuned. In applications where a well defined state of polarization is required, the pump laser polarization must be aligned along one of the fiber’s principal axis. A perfect alignment, on the other hand, is difficult to obtain and the state of polarization of the SC may indeed be affected. It has been shown in [16] that an error of 2° in the alignment angle can be detrimental for the state of polarization of the SC. In this case, a spectral broadening due to XPM will occur, perturbing the state of polarization in a broad range of wavelengths. Moreover, when the pulse is aligned closer to the fast axis the polarization state becomes more affected by virtue of vector modulation instability (VMI) [16]. Another interesting effect relative to SCG in HiBi MF, as observed by Lehtonen et al. [24], is that the eigenpolarization modes were uncoupled due to high group velocity mismatch. Consequently, the resulting SC was considered a linear combination of two supercontinua generated separately in each principal axis. The same effect was later observed by Proulx et al. [25]. Therefore, it would be interesting to extend these studies to a different MF structure where birefringent and nonlinearity parameters are both very high, and investigate at what power levels the coupling between eigenpolarization modes would actually take place.

In this work we suggest an improved MF design for SCG. This improved design, defined here as D-shaped MF, or simply D-MF, significantly enhances birefringent and nonlinear effects simultaneously. Microstructured fibers have also been produced in D-shaped profiles via side polishing technique as wideband tunable filters [26–27], and as tunable directional couplers [28]. Unfortunately, besides not producing long fiber lengths, this technique may also introduce severe scattering losses as a byproduct of the polishing. The D-MF structure discussed in this work is assumed to be produced with a regular stacking and drawing technology, which may contribute to significantly reduce any scattering losses due to surface roughness as observed for the polished cases. The approach we have proposed, which represents a considerable improvement over side polished D-MFs, alleviates all drawbacks of the side polishing technique. It consists in stacking capillaries and rods inside the jacket so that a D-shape profile can be obtained as desired during the perform assembly. The upper half inside the jacket is not occupied by rods or capillaries, therefore a large air tunnel will be created in that region with the jacket preserved along the whole length. Preform fabrication techniques for MFs have evolved considerably in the past few years, and MFs geometries with increased complexity have been successfully obtained by several research groups as well described in [29,30]. Regarding a possible fabrication of the present D-MF preform, the capillaries and rods could be held together, if necessary, by way of thin wires and fused together during an intermediate drawing process, as described in [29]. This would allow one to optimize the drawing process of the D-MF to its final dimension. This structure would also require a careful control of the pressure inside the air holes and tunnel during the drawing process in order to prevent them from collapsing (particularly the tunnel). Thus, there is a need for a careful balance of the internal pressure, surface tension, and glass viscosity at the draw temperature of the furnace [30]. In any case, our previous experience concerning collaborative work on MF fabrication [31] suggests that any shaped preform tends to retain that shape upon drawing. So stacking a D-shaped structure should be possible although some optimization on drawing conditions is certainly necessary. We have found for this design a birefringence value as high as 4.87×10^-3 for the adopted pump wavelength and a mode field diameter approximately one half of a corresponding whole MF, which makes this fiber quite attractive for SC generation. The increase in the birefringence for the D-MF was not followed by a corresponding increase in the number of modes, as is the case when the aff parameter is increased [21,23]. A second ZDW (at 1326(1445)nm for the slow(fast) axis) was also found for this fiber. The SCG was then investigated for different alignment angles between pump and the principal axes so that the influence of the orthogonal eigenpolarization modes on each other could be evaluated. We have numerically investigated at what power levels the coupling between eigenpolarization modes begins to take place, and found that the group velocity mismatch plays a significant role in the dynamics of power exchange between these modes due the high birefringence observed in this fiber. When the peak power of the pulse was set to 2kW, no coupling between eigenpolarized modes was observed, just like in [24, 25], indicating that the power level was not high enough to provide coupling before the pulses separate temporally. A subtle coupling between the orthogonal modes was observed when the peak power was increased to 5kW, indicating that for this power level the nonlinear coupling could take place before the temporal separation of the pulses. Increasing the peak power even further (10kW), the coupling became more evident, as one would expect. We have also performed an ellipticity analysis in order to verify the sensitivity of the SC polarization state to small deviations in the alignment angle of the pump laser with one of the principal axis. The results have shown that the change of polarization state is not as dramatic as in (whole) birefringent MFs, with the ellipticity of the spectra showing a less dramatic variation with wavelength. Even though the present analysis has been focused on a particular MF design, its main conclusions are still valid for other highly birefringent MFs geometries as well.

2. Numerical model

The dispersion characteristics for the slow and fast principal axes of HiBi fibers may differ significantly from each other. Therefore, assuming that the dispersion profiles are the same, as it is usually done for low birefringence fibers [32], may not be adequate in this case. A more rigorous model of the wavelength dependence of the nonlinearity coefficient and the modal effective area, an accurate calculation of the dispersion characteristics for both principal axes, and a precise modeling of the Raman scattering become fundamental for the analysis of such fibers [16].

Therefore, the modeling equations adopted here are the coupled nonlinear Schrödinger equations (CNLSE), where all the parameters mentioned above are included. They are defined in terms of linear polarization components A ₁ and A ₂ as follows [32–34].

where A_n (z,T) is the pulse envelope for the polarization n, the asterisk denotes the complex conjugate, the time T is in a reference frame moving at the group velocity (T = t − z(β₁₁ + β₁₂)/2), β_mm = ∂^m β_n/∂ω^m∣_ω0 is the m-th term of the Taylor series expansion for the propagation constant β_n(ω), δβ₁ = (β₁₁ − β₁₂)/2, Δβ = β₀₁ −β₀₂, γ_n = n ₂ω₀/(cA _eff,n), n ₂ ≈ 2.6 × 10^-20 m ²/W, c is the speed of light in vacuum, A _eff,n is the effective area for the polarization n and ω₀ is the carrier frequency. The shock term τ _SHOCK,n = 1/ω₀ + ∂/∂ω{ln[1/n _eff(ω)A _eff,n(ω))]}∣_ω0, where n _eff (ω) is the effective index, includes to first order the frequency dependence of A _eff,n[33, 35]. f_R = 0.18 is the Raman response contribution to the Kerr effect, f ₁ is obtained via experimental Raman response [36], f ₃ = (r/τ₂)exp(−t/τ ₂), with τ₂ = 32 fs and r defined as in [34], and f ₂ = f ₁ − 2f ₃.

In the derivation of the above equation it is assumed that the spectral width of the pulse envelope is less than ≈ ω₀/3, and that backward traveling waves are neglected, as described in [33]. The last two terms of first line represents the linear part of the system (absorption has been neglected). The second, third and fourth lines account for the nonlinear effects. The instantaneous effect is modeled in the second line, and includes SPM, XPM and degenerated FWM. The third and fourth lines account for the Raman effect. f ₁ and f ₃ arise from the parallel and orthogonal contributions of the Raman effect. The imaginary parts of the Fourier transforms of f ₁ and f ₃ are related to the parallel and orthogonal Raman gain, respectively, and can be experimentally measured. The real parts can be calculated from the corresponding imaginary parts via Kramers-Kronig relation. f ₂ accounts for a frequency shift without gain [37]. The derivative term involving the nonlinear part of Eq. (1) describes the intensity dependence of the group velocity and is usually associated with self steepening [32].

According to [32], the term responsible for the degenerate four-wave-mixing between orthogonal components can be neglected when the fiber length is much larger than the beat length. This is exactly the case for the present D-MF, where the beat length is 0.14mm. Nonetheless, we decided to keep this term for completeness in order to verify its residual contribution to the coupling between the eigenpolarized modes for the case where the incident pulse is aligned 45° along the slow axis. In fact, this term does not perform any role for peak powers of 2kW and 5kW, but it does have a slight influence when the peak power is increased to10kW.

The CNLSE is then solved with the hybrid split-step Fourier method, with the linear term propagated in the frequency domain and the nonlinear term propagated with a 5^th order Runge-Kutta-Fehlberg (RKF) method [33]. The modal characteristics of this fiber was obtained with an H-field finite difference based semi-vectorial successive over relaxation method (FD-SOR) [38]. The modal analysis is carried out in terms of horizontal (H_x) and vertical (H_y) polarizations. This method is quite robust and allows one to obtain the propagation constants β₀₁ and β₀₁ for the respective polarized modes, as well as their field distributions. The step size utilized in the simulations was 1.25μm. The photon number was conserved within 2.5% for all simulations presented in this work.

3. Numerical results

The proposed D-MF design is shown in Fig. 1(a) together with a whole MF, Fig. 1(b), for comparison. Observe that the jacket is still preserved in the D-MF case. The hole diameter d=1.4μm and the pitch λ=1.6μm. The choice of the D-MF geometry is made on the basis of the modal effective area versus wavelength curves for the fast and slow principal axes, as shown in Fig. 2 (Left). All curves are referred to as cuts of the whole MF, that is: half a MF, which is depicted in Fig. 1(a), (diamonds), half a MF plus d/4 (squares), half a MF plus d/2 (circles), and a whole MF, which is Fig. 1(b), (triangles). The dispersion characteristics of these structures are shown in Fig. 2 (Right), where the symbols are defined as before. The effective area of the present D-MF is calculated according to [32]. Observe that as the fiber cut increases, the ZDW, originally at 726(746)nm for the slow(fast) axis, becomes more blueshifted as a consequence of the changes in the field confinement. Not surprisingly, a second ZDW also appears, especially for the D-MF representing half a MF.

 

Fig. 1. Microstructured optical fiber geometries. (a) D-MF, and (b) whole MF. In both cases the air holes are completely circular.

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Fibers with two ZDW are quite interesting for SCG since they allow one to obtain dispersive waves in the infrared [39,9] and to cancel the soliton self frequency shift (SSF) [39]. A remarkable finding regarding a MF with two closely lying ZDW, as originally investigated by [40], is that the SCG becomes dominated by self phase modulation (SPM) [41]. The authors in [40] have shown that by properly designing the dispersion characteristics of MFs the soliton fission mechanism and noise amplification through modulation instabilities can be suppressed, paving the way to obtaining a supercontinuum extremely unsusceptible to input pulse parameter variations. Therefore, we see the improved D-MF design suggested in this work as a promising alternative for SCG since it provides new degrees of freedom to control field confinement, birefringence, and dispersion characteristics of MFs.

The present improved D-MF design offers several technical advantages when compared to previously available HiBi MFs, such as: 1) it provides not only extremely high birefringence effect values, but also a significant increase in the nonlinear parameter owing to a reduced modal effective area. The modal effective areas for the present D-MF are 0.84μm² and 0.92μm² for the slow and fast axes, respectively. As a comparison, the same parameter was 2.3μm², 3μm² and 8.5μm² for the HiBi fibers described in [24], [25], and [42], respectively; 2) it does not excite higher order or radiation modes other than the two polarization states of the fundamental mode. Some HiBi fibers are found to be multimode. The elliptical core structure investigated in [24] has, theoretically, at least three guided modes. The asymmetric core structure investigated by [21] in fact presented a higher order mode as well; 3) it does offer an unprecedented possibility of controlling its modal characteristics by introducing other materials (such as liquids) inside its large tunnel (as well as inside the air holes which are still preserved on its lower side). Therefore, not only the modal dispersion characteristics can be changed but also the modal effective area. All other MFs available (either normal or HiBi) allow the introduction of other materials inside the air holes only. Depending on the liquid’s viscosity, this task can be seriously compromised by virtue of the hole diameter. Liquids can of course be used on top of D-MF obtained via side polishing technique. The disadvantages of this technique are that it does not produce long lengths of fiber and also it does excite higher order modes owing to surface roughness, as mentioned previously. In addition, the fact that the side polished MF does not preserve the jacket on the polished side may reduce its range of applications. All the above drawbacks are alleviated with the present improved D-MF design.

A side effect of the extreme asymmetry of the D-MF in Fig. 1(a) is that it becomes highly birefringent (HiBi), which is seen as a clear advantage for SCG as pointed out by Lehtonen et al. [24]. In fact, the birefringence B = Δβλ/(2π) obtained for the present case is at least one order of magnitude larger than that of regular MFs, as shown in Fig. 3 (Left). The residual birefringence observed for the whole MF is actually due to the discretization in the finite difference mode solver. In [24], the authors have experimentally characterized a HiBi MF and observed two significant advantages over conventional fibers: 1) the polarization of all spectral components is preserved along the fiber length, 2) both eigenpolarization modes exhibit distinct dispersion characteristics, which can be advantageously used to tune the properties of the SC.

Based on the results discussed above, we have chosen as our model problem the half MF shown in Fig. 1(a). This fiber presents the smallest modal effective area and, consequently, the largest nonlinear parameter γ_n of all MF discussed here. The dramatic reduction in the modal effective area for this case is also favored by the large index contrast between the DMF core (silica) and the upper clad (air tunnel) which prevents the field from spreading towards the tunnel. Larger index contrast results in larger field confinements and, therefore, smaller modal effective areas. Without the air tunnel, which would be the case of a whole MF, a simple reduction in the core diameter would cause the mode to approach cutoff and, consequently, increase its modal effective area. For the proposed D-MF design, the dispersion of the slow(fast) axis is anomalous above the ZDW of 605(675)nm and normal above the ZDW of 1326(1445)nm. The group delay (GD) and the dispersion (D) profile for this particular D-MF are shown in Fig. 3 (Right) for both fast and slow principal axes. The advantages of this fiber design for SCG are as follows: increased nonlinear parameter (approximately twice as much as that for the corresponding whole fiber, once the mode effective area was reduced to ≈50%(54%) of the corresponding whole MF for the slow(fast) principal axis at 680nm), improved field confinement, ultra high birefringence, and distinct dispersion characteristics for both fast and slow axes. More importantly, it allows one to more effectively tailor these parameters magnitude by simply adjusting the MF’s D shape.

 

Fig. 2. Left: Modal effective area versus wavelength for different MF cuts. The upper limit is the whole MF (triangles). Right: Dispersion characteristics for the fast and slow principal axes.

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Fig. 3. Left: Birefringence as a function of wavelength for different MF cuts. Right: Group delay and dispersion for the D-MF depicted in Fig. 1(a).

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Next, we proceed with the analysis of SCG in the proposed D-MF, assuming the tunnel comprising the upper half of the structure in Fig. 1(a) is filled with air. The choice of the pumping wavelength as well as the group delay profile play an important role in the SCG dynamics, as pointed out in [8,24], and therefore they need to be carefully chosen. With this in mind, we decided to explore the fiber sensitivity to variations in the excitation conditions, and to investigate how the coupling of the orthogonal eigenpolarization modes, if any, interferes with the SCG in a fiber exhibiting such an extreme combination of parameters. The pump wavelength is 680nm, located close to the zero dispersion wavelength and also in the anomalous region of both principal axes. The input pulse is a hyperbolic secant with duration (FWHM) T_FWHM=100fs. We have considered three different alignment angles (0°, 45°, and 90°), and three different peak powers (2kW, 5kW, and 10kW) for the input pulse so that the field interaction between both eigenpolarization modes can be properly addressed. We have used the global D-MF dispersion characteristics to solve the CNLSE [5]. The corresponding Taylor series expansion terms for both principal axes, truncated at β₉, are listed in Table 1.

Table 1. Taylor series expansion terms for both principal axes, truncated at β₉ (λ=680nm).

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The peak powers and alignment angles were selected so as to verify how significant the interaction really is for the present fiber design. As one might expect, the magnitude of this interaction depends on both the dispersion characteristics and nonlinear parameter of the fiber. Therefore, different MF designs may present different degrees of interaction. The simulation results are summarized in Fig. 4(a), whose rows are defined in terms of the input peak power as follows: Top 2kW, middle 5kW, and bottom 10kW. Each column represents a specific alignment condition, that is, from left to right: along the slow axis (0°), along the fast axis (90°), along the slow axis at 45°, along the fast axis at 45° (knowing that at 45° the power is equally divided between both principal axes). The length of the D-MF is 8cm. A good way of checking if the interaction is actually occurring consists on simulating the SCG in each axis separately (0° and 90°) assuming the peak power is only half of its original value. This situation ideally represents a 45° alignment where absolutely no interaction has occurred, and will be referred here as the uncoupled case. We have also included three additional cases with peak powers of 1kW, 2.5kW, and 5kW, respectively, each aligned at 0° and 90° with respect to the slow axis, as shown in Fig. 4(b). In a perfectly non interaction condition, each row in Fig. 4(b) should exactly match its corresponding axis (slow and fast) at 45° in Fig. 4(a).

First consider the pump alignment along the slow and fast axes, first and second columns in Fig. 4(a), respectively. Observe that for any given peak power, the SC in the slow axis is spectrally broader than that in the fast axis. Yet, it presents a more pronounced gap in the vicinity of its ZDW. This behavior is somewhat expected since the pump wavelength is located further away from the ZDW for this axis. As a result, frequencies are generated further into the blue region of the spectrum increasing the bandwidth of the SC, but also increasing the gap in the vicinity of the ZDW [8]. We have observed that the SC becomes more redshifted after 10cm propagation distance in the slow axis when the peak power was set to 5kW and 10kW, and after 14cm propagation distance when the peak power was set to 2kW (not shown here). It happens because the soliton propagating in the slow axis experiences a larger dispersion and, consequently, a higher red-shift rate [43].

Next, consider a 45° pump alignment so that both principal axes are allowed to interact with each other. It is well known that in HiBi fibers the pulses in each axis travel with very different group velocities and separate temporally much earlier than the onset of the SC. This causes a drastic reduction in the coupling between the corresponding eigenpolarization modes and, consequently, the resulting SC could be considered as a linear combination of the continua generated in each axis [24]. This is also the case for the present fiber design when the peak power is kept in 2kW. Observe that the spectra at 45° for both principal axes (first row of Fig. 4(a)) are virtually the same as the uncoupled case shown in the corresponding row of Fig. 4(b), characterizing a negligible interaction as described in [24, 25]. This indicates that the power level is not high enough to provide coupling before the pulses separate temporally. A subtle coupling between the orthogonal modes was observed when the peak power was increased to 5kW, indicating that for this power level the nonlinear coupling could take place before the temporal separation of the pulses as shown in the second row of Figs. 4(a) and 4(b). Increasing the peak power even further (10kW), the coupling became more evident (third row of these figures), as one would expect. We have observed that XPM and orthogonal Raman effects play a significant role in this interaction between both orthogonal eigenpolarized modes, which was verified by turning off these terms, one at a time, at the beginning of simulation. The overall results indicate that the interaction between the eigenpolarization modes indeed may not play a significant role in the SCG as long as the peak power is kept at moderate values.

This particular fiber design presents a walk-off distance of 2.27mm for the adopted pulse duration. Another important quantity for the description of SCG is the soliton fission length, which is associated with the point where the spectral broadening of the injected higher order soliton is maximum [5]. Once the initial spectral broadening is governed by the interplay between anomalous GVD and SPM, the soliton fission length will be dependent of both L_NL and L_D [44], where L_NL = 1/(γP _o) is the nonlinear length and L_D = T ₀ ²/(∣β₂∣) is the dispersion length. In fact, soliton fission can be estimated as $L_{\mathrm{fiss}}\approx \sqrt{L_{\mathrm{NL}}{L}_D}$ [5]. As long as the two principal axes exhibit different dispersion profile, and consequently different dispersion lengths, soliton fission will occur at different stages of propagation, even when the power is equally split between the two principal axes.

To highlight these differences, we show the temporal evolution for a 45° pump alignment and peak powers of 2kW, 5kW and 10kW in Fig. 5. The estimated fission lengths for the slow(fast) axes and this three different peak powers are 22(101)mm, 14(64)mm, and 10(45)mm, respectively. The orthogonally polarized pulses propagating at the slow and fast axes travel at different group velocities and are completely separated in time after few millimeters of propagation. The soliton fission occurs at an earlier stage in the slow principal axis because the dispersion length is shorter for this axis. Moreover, the higher the peak power the shorter the soliton fission length (compare the temporal evolution at different pump peak powers). This figure is plotted in logarithmic scale and is the sum of the intensities of the pulse propagating in the slow axis with the one propagating in the fast axis.

The information about the dynamics of SCG can be better conveyed by way of spectrogram representations, which provide relative temporal positions of the frequency components of the supercontinum. The spectrogram is defined as [5]:

where g(t − τ)is a variable-delay gate function, and A(t) is the vectorial sum of A ₁ and A ₂. Thus, Fig. 6 shows the spectrograms relative to a 45° alignment obtained after 8cm of propagation and three different peak powers. The arrows indicate some of the solitons and the dispersive waves in the fast and slow axis. Observe that the solitons in the slow axis are temporally more separated than in the fast axis, which is a clear indication that the onset of the soliton fission has occurred in an earlier stage.

When the peak power is adjusted to 2kW the onset of the SC occurs at a much longer distance from the total walk-off, resulting in a clear temporal separation between the supercontinua in both axes as shown in Fig. 6(a). Consequently, no interaction is expected to occur in this situation. The increase of the peak power to 5kW causes the onset of the SC to occur at a shorter distance. The dispersive waves generated in the fast axis become temporally closer to the frequency components around 700nm in the slow axis, as shown in Fig. 6(b). This is expected to occur since the dispersive waves in the fast axis are located in a region where the group velocity is smaller than that in the slow axis around the pump wavelength (see Fig. 3 (Right)). When the peak power is further increased to 10kW, Fig. 6(c), the dispersive waves generated in the fast axis temporally overlap the frequency region around the pump wavelength in the slow axis. This overlapping occurs after the onset of the SC without producing any significant variation in the SC spectra.

 

Fig. 4. Supercontinua generated in an 8cm long D-MF. Top row 2kW, middle row 5kW, and bottom row 10kW. Each column represents a different alignment condition, so from left to right: along the slow axis (0°), along the fast axis (90°), along the slow axis at 45°, along the fast axis at 45°. (b) Top row 1kW, middle row 2.5kW, and bottom row 5kW. First and second columns indicate pump aligned at 0° and 90° with respect to the slow axis, respectively.

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Finally, we investigate how the polarization state of the SC spectra evolves in such an extremely high birefringence condition. To do so, the ellipticity of the output pulse was calculated according to [16].

where à ₁(λ) and à ₂(λ) are the Fourier transforms of A ₁(T) and A ₂(T), respectively. As in the previous cases, three different peak powers are considered, namely, 2kW, 5kW, and 10kW. We simulate an alignment error by choosing the pump alignment as 2° and 88° (0° and 90° stands for alignment along the slow and fast axes, respectively) as shown in Fig. 7, for an 8cm long D-MF. The pump wavelength as well as the input pulse format and duration are the same as before. Differently from the results in [16], the pulse with less energy did not evolve into a SC. The mechanism that prevented the energy transfer between both eigenpolarization modes from occurring is the group velocity mismatch. The low energy pulse splits temporally from the high energy pulse in the early stages of propagation, inhibiting XPM from taking place. The low energy pulse itself does not carry enough power to initiate the SC and, as a result, the ellipticity is practically zero, characterizing a well defined polarization state either for a 0° or 90° alignment (except for the region around the pump wavelength where the ellipticity is somewhat chaotic due to the presence of the undesired orthogonal pulse).

 

Fig. 5. Temporal evolution plotted in logarithmic scale obtained with the pump aligned at 45° with respect to the slow axis. The peak powers for each case are, from left to right, 2kW, 5kW, and 10kW.

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Fig. 6. Spectrogram of the SC generated after 8cm of D-MF. The input pulse is a hyperbolic secant with T_FWHM=100fs, λ = 680nm, and alignment angle of 45°. The peak powers are: a) 2kW, b) 5kW, and c) 10kW.

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Fig. 7. SC and ellipticity for pump laser aligned at different angles. (a) 2°, (b) 88°. Top row 2kW, middle row 5kw, and bottom row 10kW. SC in the slow(fast) axis is shown in blue(red).

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4. Conclusion

In this work we carried out a rigorous analysis of SCG for a D-MF design exhibiting extremely high birefringent and nonlinearity parameters. This fiber design represented a considerable improvement over previously available side polished D-MFs, once it alleviates all drawbacks relative to side polishing technique, namely, severe surface roughness on the polished side, and higher order and radiation modes excitation. In addition, the mode effective area was reduced to ≈50%(54%) of the corresponding whole MF for the slow(fast) axis at 680nm, which makes this fiber quite attractive for SCG, particularly when polarization maintaining is required. It is worth mentioning that a reduction of the whole MF to half of its original size, as illustrated in Figure 1(a), was not followed by a corresponding increase in the number of modes as it is the case of side polished D-MFs. A second ZDW (at 1326(1445)nm for the slow(fast) axis) was also found for present D-MF fiber. Due to its unusually high birefringent and nonlinearity parameters, it became necessary to investigate how sensitive the SC generated with this fiber actually was to variations of the pump alignment angle, and at which peak power level the coupling between orthogonal eigenpolarization modes actually began to take place. Such a study is an important contribution to fully understand the power exchange dynamics under extreme birefringence and nonlinear conditions. The group velocity mismatch indeed played a significant role due the high birefringence observed for this fiber. No coupling between eigenpolarized modes was observed when the peak power of the input pulse was set to 2kW, indicating that the power level was not high enough to provide coupling before the temporal separation of the pulses due group velocity mismatch. A subtle coupling between the orthogonal modes was observed when the peak power level was increased to 5kW, indicating that for this power level the nonlinear coupling could take place before the temporal separation of the pulses. Increasing the peak power even further (10kW), the coupling became more evident, as one would expect. We have also performed an ellipticity analysis for the D-MF, and the results have shown that the change of the polarization state is not as significant as in birefringent MFs, with the ellipticity of the spectra showing a less dramatic variation with wavelength. Even though the present analysis has been focused on a particular fiber design, its main conclusions are still valid for other HiBi MFs as long as their group velocity mismatch is also high. Therefore, we see the improved D-MF design suggested in this work as a promising alternative for SCG since it provides new degrees of freedom to control field confinement, birefringence, and dispersion characteristics of MFs. Sensor applications are also envisioned for this fiber by virtue of the large tunnel created above the core, enabling one to more easily inject liquid or gases inside the fiber.