Degenerate four wave mixing in solid core photonic bandgap fibers is studied theoretically. We demonstrate the possibility of generating parametric gain across bandgaps, and propose a specific design suited for degenerate four wave mixing when pumping at 532nm. The possibility of tuning the efficiency of the parametric gain by varying the temperature is also considered. The results are verified by numerical simulations of pulse propagation.
© 2008 Optical Society of America
The invention of the photonic crystal fiber (PCF) has had an immense impact on research in nonlinear fiber optics during the last 10 years. Mainly because of the possibilities of making very small cores, and thus large nonlinearities, and because the dispersion properties of PCFs can be tailored, which allow anomalous group velocity dispersion (GVD) at visible wavelengths . Together these two properties make it possible to study a large variety of nonlinear effects, such as soliton formation, four wave mixing (FWM), and supercontinuum generation . The first PCFs had a cladding, which consisted of air-holes placed in a periodic pattern around a defect site created by omitting an air-hole. The guiding mechanism of such a fiber can be understood intuitively by the concept of total internal reflection, since the solid silica core of the fiber has a higher index of refraction than the surrounding cladding, which is made up of high and low index regions consisting of silica and air, respectively. A fascinating feature of PCFs is that if the refractive index of the material in the holes is raised above the refractive index of silica, the structure can still guide light if certain conditions are satisfied. Such fibers have been constructed in various ways. One way is to infiltrate the air-holes of a standard PCF with a liquid that has a higher index of refraction than silica, this has been demonstrated using both high index oils  and liquid crystals . Another way of constructing such a fiber is by using the stack and draw technique known from PCFs with air-holes, but where the silica tubes have been replaced by doped silica rods with an index of refraction slightly higher than silica . The guiding mechanism of this type of fiber can clearly not be explained by the concept of modified total internal reflection. Instead various models have been developed to explain the guiding in these fibers. In one model the guiding is explained by considering the photonic crystal that constitutes the cladding of the fiber. It turns out that even for low index contrast such a photonic crystal possesses bandgaps for out of plane propagation . If light with a frequency lying in one of these bandgaps is injected into the core of the fiber, it will stay confined to the core due to the photonic bandgap existing in the cladding region. Another way of explaining the guiding of this type of fiber is by using a two dimensional analog of the planar anti-resonant reflecting optical waveguide (ARROW) . In this model the guidance is explained solely by the resonant properties of a cladding cylinder. Since a cladding cylinder is just an ordinary step index fiber, their properties can conveniently be described using the well known V-parameter. From the theory of step index fibers we know that a mode is in resonance with the cladding at cut off, so guided modes are expected when the V-parameter is far from a cut off. Therefore in the ARROW-model the guiding properties do not depend on the specific arrangement of the high index cylinders. In spite of its simplicity, the ARROW-model predicts the position of the transmission windows of these fibers with a high accuracy, and therefore fibers guiding light using this principle are also often referred to as ARROW-fibers instead of photonic bandgap fibers. The presence of the resonances in the transmission spectrum of ARROW-fibers gives rise to large normal and anomalous waveguide dispersion for wavelengths near resonance. This effect can be exploited to create fibers with zero dispersion in the blue end of the visible spectrum, and also fibers with multiple zero dispersion points are possible, since every transmission window possesses a zero dispersion frequency. In PCFs with air-hole claddings it requires sub-micron cores and high air fill fractions to achieve zero dispersion wavelengths below 600nm . PCFs with air-hole claddings that have several regions of anomalous dispersion separated by regions of normal dispersion can also be produced, but this requires modifications of the standard triangular design where all the cladding holes have the same radii, and the core is defined by omitting a single hole . Therefore ARROW-PCFs offer a unique testbed for investigation of dispersive properties in the visible spectrum. The price one has to pay for these properties is the presence of high loss regions in the transmission spectrum at frequencies where the core mode is nearly in resonance with a guided modes of the cladding cylinders.
In this work we study degenerate four wave mixing (DFWM) in ARROW-PCFs, and investigate the possibilities of pumping in one bandgap, and obtaining a parametric gain in another bandgap. In standard air-hole PCFs parametric gains at frequencies highly detuned from the pump have also been achieved [9, 10], but in general it is only possible when the pump is either in the anomalous dispersion region, or in the normal dispersion region close to the zero dispersion wavelength. Thus, it is difficult to obtain a parametric gain in air-hole PCFs when pumping below 600nm. Propagation of ultrashort pulses in ARROW-PCFs has been studied earlier both theoretically and experimentally in  and , where it was shown how femto-second pulses propagating close to a zero dispersion wavelength radiate energy in the form of a blueshifting dispersive wave, while the pump pulse is red-shifting due to the Raman effect, a phenomena also known from air-hole PCFs [13, 14]. In another recent work it was shown how Raman generation can be limited to the first Stokes order, by using a photonic bandgap fiber where the higher order Stokes frequencies lie in a low transmission region . But so far, to the best of our knowledge, neither FWM, or effects between two or more bandgaps in ARROW-PCFs have been studied before.
The rest of this paper is organized as follows, in Section 2 we discuss the basics of DFWM in waveguides. Then in the following section we present fiber designs capable of achieving DFWM across bandgaps, and demonstrate this by numerically simulating propagation of pico-second pulses. Section 4 summarizes the main conclusions.
2. Basic theory of degenerate four wave mixing
Four wave mixing is a third-order nonlinear parametric process. Qualitatively it can be described as a process where two photons with frequencies ω 1 and ω 2 are annihilated, and two new photons at frequencies ω 3 and ω 4 are generated. When the process is described in the time domain it is usually referred to as modulational instability. Energy conservation requires that ω 1+ω 2=ω 3+ω 4. In order to have the process occurring effectively phase matching is also required. If nonlinearities are neglected the phase mismatch is given by Δβ=β(ω 4)+β(ω 3)-β(ω 2)-β(ω 1), where β(ω) is the propagation constant at the frequency ω. In this work we are mainly interested in the process denoted as degenerate four wave mixing, where ω 1=ω 2=ωp. Conservation of energy then requires that the two generated photons have frequencies ω 3=ωp-Ω and ω 4=ωp+Ω, i.e. the generated frequencies are placed symmetrically around the pump frequency ωp. The low frequency component ωa=ωp-Ω, is denoted the Stokes frequency, while the high frequency component ωas=ωp+Ω, is denoted the anti-Stokes frequency. If the nonlinear response is assumed to consist of only an instantaneous Kerr-part, and the mode profile of the guided mode is constant, the parametric gain can be shown to be given by , where P 0 is the power of the pump, κ=2P 0 γ+Δβ is the nonlinear phase mismatch, and γ is the nonlinear parameter given by γ=2πn 2/(λ 0 Aeff), where n 2 is the nonlinear refractive index, λ 0 is the pump wavelength, and Aeff is the effective area of the guided mode, here assumed not to vary with wavelength. Therefore, in order to have a gain, we must have P 0 γ>|κ|/2, which can also be formulated as -4P 0 γ<Δβ<0, i.e. the linear phase-shift Δβ must be negative, but greater than -4P 0 γ. The highest possible gain is achieved if Δβ=-2P 0 γ, where the gain is gmax=2P 0 γ. When using pulses for FWM the walk-off length is also an important parameter, since the pump, Stokes, and anti-Stokes signals should be overlapping in time to get an efficient gain. The walk-off length is given by LW=T 0|ν -1 gp-ν -1 g1|-1, where T 0 is the pulse duration, νgp is the group velocity at the pump wavelength, and v g1 is the group velocity at the Stokes or anti-Stokes wavelength. The walk-off length is then the physical distance two simultaneously launched pulses with wavelengths λp and λ 1 should travel before their temporal separation is T 0.
3. Infiltrated ARROW-fibers
The fibers we study all have a triangular structure of high index rods around a defect where a rod has been omitted. To calculate the guided modes and eigenfrequencies we use a commercial finite element tool , where leakage losses are estimated by including a cylindrical perfectly matched layer (PML) around the boundary of the fiber. All the fibers considered in the following have 6 rings of holes around the core defect, which is also the case for many commercially available PCFs. We are mainly interested in calculating the phase-mismatch Δβ, which is strongly dependent on the chromatic dispersion of the fiber, therefore we include both the material dispersion of silica and the high index oils in our calculations. For the material dispersion of silica we use a Sellmeier equation, and for the material dispersion of the high index oils we use Cauchy polynomials that have been fitted to experimentally measured indices of refraction . The absorption of light propagating in the oil has been measured at 12 different wavelengths from 365nm to 1550nm by the manufacturer, and we interpolate these results to calculate the absorption loss in the fibers.
3.1. Transmission properties
An infiltrated fiber allows more freedom than an all-solid structure, since a large range of liquids exists with refractive indices ranging from below silica up to about 1.8. A bit of caution is needed when using liquids for optical waveguides, since many high index liquids have high absorbance in some regions of the visible and near infrared spectrum. When estimating the absorption loss in the liquid a crucial parameter is the modal overlap defined as 
where E λ (x,y) is the profile of the guided mode at the wavelength λ, and n(x,y) is the refractive index profile of the fiber. The integral in the numerator is taken only over the high index regions, while the integral in the denominator is taken over the whole transverse structure. When dealing with ARROW-fibers η is usually less than 5%, unless the wavelength is lying close to a resonance. The contribution to the linear loss of the fiber due to absorption in the liquid is calculated as αabs, fiber=ηαabs,oil, where αabs,oil is the linear absorption coefficient of the liquid. The total linear loss of the fiber is estimated by adding the leakage loss and absorption loss, since we neglect losses stemming from absorption in silica. At wavelengths far from resonance the absorption loss is dominating the total linear loss, while at wavelengths close to a resonance the leakage loss adds a significant contribution to the total linear loss.
In the following we propose a design suited for generating a DFWM-band in the 4th. bandgap when pumping at 532nm in the 3rd. bandgap. This design has Λ=3.1µm and d=1.23µm. The holes are infiltrated with a high index oil that has n=1.60 at λ=589.3nm . We have found this design by initially neglecting material dispersion, and calculating β (ω) for designs with different relative hole diameters but fixed values of the refractive indices. Now, within the scalar approximation one may derive a scaling law  describing the shift of the β (ω) curve as the refractive indices are changed. This allows us to approximately account for the frequency-dependent shifts in the refractive indices of silica and infiltration liquids caused by material dispersion, and also to predict results for different infiltration liquids based on a single calculation. In this way, we could efficiently check a range of fiber designs for the possibility of phase-matching between bandgaps, and found several designs where this could be achieved. For the most promising designs we have finally calculated β (ω) using a full-vectorial method with the material dispersion included, to verify that the phase-match was not just an artifact of the scalar approximation. Generally we found that for the process to occur at least one of the three wavelengths involved in the process will experience a loss of several dB/m, but further optimization in order to achieve lower losses still remains unexplored.
In Fig. 1(a) we have plotted the modal field overlap η as defined in Eq. (1) in the 3rd and 4th. bandgap of the considered ARROW-fiber (Λ=3.1µm and d=1.23µm, nLiquid≈1.60). We see that in the centers of the bandgaps less than 1% of the field is residing in the high index regions. On the upper x-axis of the plots in Fig. 1 the V-parameter of the individual high-index rods is shown, which makes it easy to identify the cut-offs between the bandgaps, since these coincide with the zero points of the Bessel functions J 0(V) and J 1(V). The spikes in the 4th. bandgap around 425nm and in the 3rd. bandgap around 510nm come from cut-offs of higher order modes (LP41 and LP31, respectively). In Fig. 1(b) and (c) the leakage loss and propagation loss is shown. The leakage loss arises because the finite number of holes around the core allows a small fraction of the field to leak out, and could therefore be reduced by increasing the number of holes. We see that the leakage loss is significantly higher in the 4th. bandgap than in the 3rd. This happens because the rod modes couple more strongly in the even-numbered bandgaps than in the odd-numbered bandgaps . The propagation loss also includes absorption losses in the liquid, and arises since η is nonzero. η will always increase near a bandgap edge, since the core mode is able to couple to a certain mode of the individual rods here. This is an intrinsic property of ARROW-PCFs, and the losses arising due to this cannot be minimized simply by adding more rings of holes to the structure. In Fig. 1(d) the effective area of the guided mode is plotted. In the same plot we have also shown the effective area of the guided mode for an airhole PCF with the same hole diameter and pitch. We see that around the centers of the bandgaps the area of the mode in the ARROW-fiber is smaller than in the air-hole PCF, meaning that the nonlinearity of the ARROW-fiber can actually be even larger than in the air-hole PCF at certain wavelengths.
As described in Section 2 the linear phase mismatch Δβ is an important parameter when calculating the efficiency of DFWM. In Fig. 2(a) (solid curve) we have plotted Δβ as a function of wavelength for a pump at λ=532nm. The fiber has the same structure as the one considered in Fig. 1. Since we have a parametric gain when the linear phase-mismatch satisfies -4P 0 γ<Δβ<0, we see that the fiber has two narrow regions with gain on each side of the pump. For the fiber considered here γ≈0.023(Wm)-1 at 532nm, so for a peak power of 10kW, we have a gain when -920m-1<Δβ<0. Also in Fig. 2(a) the linear phase-mismatch is shown for two other fibers, where the whole structure (pitch and hole diameter) has been up- or down-scaled by a half percent. We see that these variations changes the linear phase mismatch significantly. We therefore conclude that in order to achieve linear phase-match between two different bandgaps in an ARROW-PCF the tolerances on variations in the structural parameters of the fiber are very low. For state of the art air-hole PCFs it is often stated that the variations along the fiber are about one percent for several meters of fiber . For the relatively short pieces of fiber we are interested in here, it is therefore fair to assume that these variations are much less than a 1%.
One advantage of the liquid infiltrated fibers over their all solid counterparts is the possibility of changing their properties by varying the temperature slightly. This is possible since many liquids have a negative temperature coefficient, which is more than an order of magnitude higher than the positive temperature coefficient of silica . The oil we consider here has a temperature coefficient of dn/dT=-4.37·10-4 K -1. Therefore one can optimize the phase-mismatch by varying the temperature of the fiber. In Fig. 2(b) we show the phase-mismatch for a structure where the temperature is raised a few degrees above room temperature. We see that also small variations in the temperature have a high impact on the linear phase-mismatch. By adjusting the temperature only slightly we can therefore optimize the spectral width of the region with parametric gain. Besides the phase-match criteria, we must also consider the walk-off length, which can be calculated from the group velocity as described in Section 2. In Fig. 2(c) the group velocity is plotted as a function of wavelength for both the considered ARROW-fiber and the similar air-hole PCF. We see that in general there will be a walk-off if we consider effects between two different bandgaps, since group velocity match between two wavelengths lying in different bandgap is only possible if one of the wavelengths is lying close to the bandgap edge. When comparing the group velocity in the ARROW-fiber and the air-hole PCF we see that the group velocity mismatches are similar in magnitude in the considered wavelength spectrum. For a pump at 532nm and the corresponding anti-Stokes wavelength (465nm) we find that the walk-off length for a 10ps pulse is about 20cm. As already mentioned at the beginning of this section, linear losses stemming from absorption in the liquid and leakage of the field can have a detrimental effect on the efficiency of the FWM-process. For the fiber considered here, it is the anti-Stokes wavelength that experiences the greatest linear loss. As seen in Fig. 1(c) the linear loss at the pump, Stokes, and anti-Stokes wavelength are about 0.4dB/m, 0.9dB/m, and 7dB/m, respectively. Hence losses should not be neglected when simulating pulse propagation in these fibers.
The part of the linear loss that arises because of absorption in the oil will heat up the oil. To estimate how large this effect will be in the ARROW-PCF, we assume that the absorbed energy is distributed evenly between the 6 holes in the first ring around the core defect. If the absorption loss is 1dB/m, and we assume that the energy absorbed is distributed uniformly along the entire fiber there will be a temperature change of the order of 10-3K, for a single 10ps pulse with a peak power of 10kW propagating through 1.0m of fiber. The characteristic time τ for the heat diffusion can be estimated as τ=(ρ 0 C)R 2/κ , where ρ 0 is the density of the material, C is the heat capacity per unit mass, κ is the thermal conductivity, and R is a characteristic length scale of the structure. Using the parameter values for silica and a radius of 3.1µm (equal to the pitch in the considered fiber), we find a characteristic time for the heat diffusion of the order of 10-5s, so if the repetition rate of the pump laser is in the kHz range heating of the oil will only be moderate.
3.3. Pulse propagation
Various models for simulating propagation of short and ultrashort pulses in optical waveguides have been developed over the last 30 years. In most of these models the profile of the guided mode is assumed not to change with frequency. This is a valid approximation for many types of optical waveguides, but for ARROW-PCFs this approximation becomes invalid, since the field spreads a considerable amount of its energy into the high index rods near a resonance, while at the centers of the bandgaps the mode can be confined stronger to the defect than in a similar air-hole PCF, as shown in the previous section. Therefore we use a model that allows changes of the guided mode profile . The model resembles the modified nonlinear Schrödinger equation often used for simulating propagation of light pulses in nonlinear waveguides , but it allows a variation of the effective area of the guided mode, which is correct to first order . The unusual dispersion of ARROW-fibers does not allow the Taylor expansion of the propagation constant as a function of frequency which is often used in air-hole PCFs, instead we apply the dispersion operator directly in the frequency domain . In the region separating the 3rd. and 4th. bandgap it is not possible to calculate a guided mode, since the core mode is in resonance with the LP12 mode of a single high index rod. In this region we include an artificial loss of 100dB/m in the simulations shown here. We have carried out simulations with different losses in the regions between the two bandgaps, and observed that the output spectrum is only weakly dependent on the magnitude of this loss. In Fig. 3(a) we have plotted the group velocity dispersion of the fiber with hole diameter d=1.23µm and pitch Λ=3.1µm heated 3°C above room temperature. In the following we will use this fiber for our pulse simulations. In Fig. 3(b) we show the output spectrum for a 10ps (FWHM) Sech-shaped pulse with a peak power of 10kW, and a wavelength of λ 0=532nm. The length of the fiber is 1.5m. From the linear phase-mismatch plotted in Fig. 2(b) we see that Stokes and anti-Stokes peaks are expected around 620nm and 460nm, respectively. The output spectrum in Fig. 3(b) shows that the Stokes and anti-Stokes peaks appear at the predicted positions, which are indicated by the arrows in the top of the plot. We see that especially for the anti-Stokes wavelength the result of the approximate theory in Section 2 agrees well with the result of the full simulation, even though the simulation includes both self-steepening, Raman response, and a wavelength dependent effective area of the guided mode, while the approximate theory only includes an instantaneous non-linear response. For the Stokes wavelength there is a noticeable difference between the wavelength predicted by the approximate theory, and the peak in the output spectrum. A more detailed investigation reveals that the Raman response is responsible for this discrepancy. We demonstrate this by carrying out a simulation without the Raman response. The result of this simulation is also shown in Fig. 3(b) (dashed line). In this case the theoretically predicted Stokes wavelength is consistent with the intensity peak in the simulated spectrum. To understand why the Raman effect changes the output spectrum, we have calculated the energy pr. pulse in the 4th. band-gap as a function of propagation distance, as shown in Fig. 3(e). The total input energy pr. pulse is 113nJ, and the result with and without the Raman effect included is shown. In both cases the energy grows exponentially during the first 20cm of the propagation, which corresponds well with the walk-off distance calculated in Section 3.2. The ratio between the energy in the 4th. bandgap without and with the Raman effect included is expected to evolve approximately as exp(0.36P 0 γz), since the Raman effect reduces the instantaneous part of the nonlinear response (and hence the parametric gain) by 18% . With the parameters used here this ratio should be ~106 after 20cm propagation, which is in reasonable agreement with Fig. 3(e). After 20cm of propagation no further amplification of the anti-Stokes signal takes place, and the energy in the 4th. bandgap starts to decrease exponentially, because of the propagation loss of ~7dB/m at the anti-Stokes wavelength. Without the Raman response the energy decreases all the way out to 1.5m, but with the Raman effect included a cascade of Stokes and anti-Stokes Raman peaks eventually reach the regions with parametric gain, and start acting as a seed for DFWM. For the simulations shown here this happens around 90cm. The appearance of the anti-Stokes Raman peaks happens because of a self-induced phase-matching between the pump and the Stokes Raman peak . At this propagation length the pump has broadened in time and frequency because of GVD and self phase modulation, and red-shifted because of the Raman effect. It is therefore not surprising that our simplified theory does not predict the correct position of the Stokes peak around 620nm. After 1.1m of propagation energy is not transferred effectively into the 4th. bandgap, and therefore the energy in the 4th. bandgap again starts to decrease. In Fig. 3(c) and (d) we have shown spectrograms of the pulses without and with the Raman effect respectively. The spectrograms are shown for the propagation distance where the energy in the 4th bandgap is highest (24cm without the Raman effect, and 112cm with the Raman effect). In the spectrogram for the simulation without the Raman effect the generated Stokes and anti-Stokes signals can clearly be identified. For the simulation with the Raman response we can identify both Stokes and anti-Stokes DFWM signals, and the higher order Raman peaks (separated by 13.2THz) that seed the DFWM-process.
In summary, we have investigated the possibility of obtaining parametric gain across bandgaps in an infiltrated bandgap fiber. A necessary condition for parametric gain is a small linear phase mismatch, and our analysis shows that this can indeed be obtained for wavelengths lying in different bandgaps. We further show that because of the unique dispersion properties of ARROW-fibers, it is possible to achieve small linear phase mismatches, even at the blue edge of the visible spectrum, which is not possible in standard air-hole PCFs unless their cores are extremely small. To confirm the results, we have simulated pulse propagation in such a fiber using a model that includes mode profile dispersion and a wavelength dependent loss, which are two common features of ARROW-fibers not shared by their air-hole equivalents.
In this work we considered a design suited for pumping at 532nm. This was mainly done to illustrate how far down in the spectrum such effects are possible, and it should be emphasized that in principle the system could be optimized to work at any wavelength throughout the visible spectrum, by choosing the appropriate fiber parameters and liquid. We also show how the parametric gain can be changed by varying the temperature of the fiber by only a few degrees. Further, our analysis shows that for experimental realization of parametric gain across bandgaps, it is essential to use a fiber with only marginal structural variations along the fiber axis, since the phase-mismatch is highly dependent on the transverse structure of the fiber.
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