A new type of microstructured fiber for mid-infrared light is introduced. The chalcogenide glass-based microporous fiber allows extensive dispersion engineering that enables design of flattened waveguide dispersion windows and multiple zero-dispersion points – either blue-shifted or red-shifted from the bulk material zero-dispersion point – including the spectral region of CO2 laser lines ∼10.6 μm. Supercontinuum simulations for a specific chalcogenide microporous fiber are performed that demonstrate the potential of the proposed microstructured fiber design to generate a broad continuum in the middle-infrared region using pulsed CO2 laser as a pump. In addition, an analytical description of the Raman response function of chalcogenide As2Se3 is provided, and a Raman time constant of 5.4 fs at the 1.54 μm pump is computed. What distinguishes the microporous fiber from the microwire, nanowire and other small solid-core designs is the prospect of extensive chromatic dispersion engineering combined with the low loss guidance created by the porosity, thus offering long interaction lengths in nonlinear media.
© 2010 OSA
Sources of broadband middle-infrared light are currently the subject of active research due to the numerous potential practical incentives for optical frequency metrology, spectroscopy, astronomy, optical tomography and infrared imagery, among others. Supercontinuum (SC) generation in microstructured optical fibers (MOF) fabricated with highly-nonlinear compound glasses have recently demonstrated a bright prospect for this approach [1,2]. The previous studies have largely focused on fiber designs featuring a small core isolated from the cladding by an air gap (rod-in-air) in the so-called “wagon wheel” geometry, and with pump seeds in the near-infrared telecommunication window. A chalcogenide As2Se3 step-index fiber and photonic crystal fiber (PCF) seeded at 2.5 μm with a 0.1 nJ, 100 fs at 1 kHz pulsed laser have reported a SC output extending from 2.1 to 3.2 μm [3,4]. While small solid core fibers greatly enhance the achievable optical nonlinearities, they generally offer limited freedom for engineering the group velocity’s chromatic dispersion (GVD). Typically in small-core fibers, the strong waveguide dispersion enables one to blue-shift the first zero dispersion point (ZDP) – with respect to the material zero dispersion wavelength (ZDM) of compound glasses – in the near-infrared range (1.0 – 2.5 μm) where several laser sources are conveniently available for pumping. The latter scheme was adopted in  where a bismuth-glass small solid-core MOF theoretically demonstrated a 3000 nm continuum bandwidth with a long-wavelength limit near 5 μm.
In this theoretical contribution, we introduce a novel type of MOF for the middle-infrared (mid-IR): the chalcogenide microporous fiber with subwavelength holes which enables superior control of the GVD compared to suspended core fibers of a “wagon wheel” type MOF. We demonstrate that with appropriate engineering of the microporosity, one can create zero-dispersion points in the mid-IR above the ZDM of the chalcogenide glass, and in the wavelength range of CO2 lasers (9.3 – 10.6 μm). We then show with numerical simulations the potential of seeding at these large wavelengths for opening a SC window inside the 5-12 μm region that has mostly remained out of reach so far.
2. Geometry and linear properties of microporous fibers
The use of a porous core with subwavelength holes for reducing absorption losses in the fundamental mode was first proposed for terahertz (THz) waveguides using polymers as the dielectric materials . The wave guiding properties of 3-layers and 4-layers porous polymer fibers were theoretically investigated via fully-vectorial finite element calculations [5,6], and later fabricated and characterized . Similar structures were later reported in [7,9]. These studies showed that porous polymer fibers – whose guiding mechanism is total internal reflection between a higher average refractive index porous core and a lower index cladding – compare advantageously over the corresponding rod-in-air fiber designs in terms of modal absorption losses and macrobending losses. Specifically, for a given loss level the porous core fiber enables superior field confinement (i.e. smaller Aeff) compared to an equivalent rod-in-air fiber. This property of porous core fibers warrants an investigation of their nonlinear properties in highly-nonlinear chalcogenide glasses for mid-IR applications such as supercontinuum generation where tight field confinement and low losses are generally desirable for enhancing the nonlinear processes. Moreover, one expects that porosity introduces an additional design parameter that can be used to manipulate the fiber dispersion properties to the benefit of the fiber’s non-linear response.
Arsenic-selenide (As2Se3) chalcogenide glass is chosen here as the dielectric material because of its wide bulk transparency (2 – 14 μm), high refractive index and large nonlinearity in the mid-IR spectrum. Although As2Se3 exhibits relative large losses (∼5 dB/m) at 10.6 μm after fiberization , it remains one of the most transparent glasses currently available for mid-IR optics. Moreover as we show in what follows, the porous core design may further reduces by more than half the modal absorption losses to the level of state-of-the-art hollow core fibers used in the power delivery of CO2 lasers .
The basic geometry of a microporous fiber is schematically depicted in Fig. 1 . In this design, the solid core is pierced with N layers of subwavelength holes of diameter d in a triangular lattice of pitch Λ (also of subwavelength dimension: ). The total outer diameter of the fiber core is given by , and an infinite air cladding is assumed here for convenience. Due to computational constraints and in order to simplify analysis, the present investigation focuses on the case of N = 4 layers of subwavelength holes. Still, the results and discussion presented thereafter provide a key direction for scaling to larger (N>4) microporous fibers. We emphasize that strict periodicity is not required for this waveguide operation since it is the overall average refractive index of the porous core that guides light via total internal reflection; not the photonic bandgap effect.Fig. 2).
Keeping the number of layers fixed to N = 4, the fundamental mode effective refractive index (neff) and field distributions (Ex, Ey, Ez, Sz) were computed through 2nd-order accurate fully-vectorial finite-element calculations with terminating PML layers. Scanning of geometrical parameters (Λ = 0.20, 0.25, 0.30, ..., 1.0μm; d = 0.12, 0.14, 0.16, ..., 0.90μm) was performed for a broad range of input wavelengths (λ = 3.0, 3.5, 4.0, ..., 16.0 μm) so as to evaluate the GVD in the vicinity of the principal wavelength of interest: 10.5 μm. Post-processing routines include: hundredfold increase in data density for neff (λ) via cubic spline interpolation, then fitting the interpolated neff (λ) curve with a polynomial of degree 11, and evaluating the GVD (ps/km∙nm) with the equation .
The porosity p of a given fiber, or in other words the areal density of the cross-section occupied by holes, is:Eq. (2) that a substantial fraction, exactly 24.7% for N = 4, of solid material still remains in the core even when d = Λ due to the material-filled interstices between the holes, and the six core-cladding interspaces located at the , and angles [see Fig. 1(a)].
The fraction (fα) of modal absorption to bulk material absorption loss can be evaluated to first-order approximation via the expression :Fig. 3(b) ] alongside that of fm [Fig. 3(a)]. We note that n mat = 2.7678 and α mat = 1.4 m−1 (or 4.81 dB/m) for As2Se3 at λ = 10.5 μm. Figure 3(b) shows that modal absorption in a porous fiber can be lowered by half the initial bulk absorption value for a diameter-to-pitch ratio with corresponding porosity . As expected and revealed on Figs. 3(a), 3(b), the percentage of modal-to-bulk absorption (fα) has a strong correlation with the cross-section areal density of material (fm) in the core. In other words, lower confinement losses are achieved in designs with large porosity.
The real value of the effective refractive index neff is plotted in Fig. 4(a) and again illustrates the close relationship with the fraction fα of modal field in the solid material pictured in Fig. 3(b). The effective mode area Aeff defined as Fig. 4(b) which indicates a region in the Λ-d parameter space where minimum optimization of Aeff can be achieved. In the present case, minimization of Aeff at λ = 10.5 μm is obtained for Λ = 0.40 μm and d→0 (i.e. the limit of the air-suspended core).
The value of the chromatic dispersion parameter D and dispersion slope at the specific wavelength λ = 10.5 μm are plotted as a function of (Λ, d) in Fig. 5(a) and Fig. 5(b) respectively. In these plots we can identify several regions of low and flattened dispersion for λ = 10.5 μm. Two such examples of low dispersion engineering in microporous fibers occur at (Λ = 0.5, d = 0.38)μm and (Λ = 0.7, d = 0.62)μm, for which the detailed dispersion curves are respectively shown on Fig. 6(a) and Fig. 6(b). In both figures the As2Se3 material dispersion is plotted (blue curve) to appreciate the strong contribution of the waveguide dispersion to the total GVD. The waveguide dispersion grows stronger as the core diameter gets smaller which can result in very steep dispersion slopes as predicted from the inverse relation between Aeff and the absolute GVD value . The latter effect help explains the sharp transition in Fig. 5(a) from positive to negative dispersion regime in going from Λ = 0.40 μm to Λ = 0.35 μm. A more significant feature shown on Figs. 6(a), 6(b) is the formation of several zero-dispersion points (ZDP), and most notably the creation of a ZDP at λ = 10.5 μm red-shifted from the ZDM of As2Se3 at 7.225μm.
The ability of microporous fibers to tailor the dispersive properties of the waveguide through several degrees of structural freedom (N, Λ, d), opens up the possibility of seeding with large wavelengths (5 – 12 μm) in the mid-IR for optimum phase-matching of nonlinear optical processes such as FWM and SC generation, as discussed in more detail in Section 4.
3. Nonlinear properties of microporous fibers
3.1 Effective nonlinearity in microporous fibers
The effective nonlinearity of the waveguides was evaluated via the nonlinear parameter γ defined as :Fig. 7(a) ] and inside the gas-filled holes [Fig. 7(b)] where Argon serves as exemplar nonlinear gas. The implemented values for the nonlinear index of As2Se3  and Argon  are m2/W and m2/W respectively.
From the standard definition of the nonlinear parameter , γ = 2πn 2/(λAeff), one anticipates that for given values of input wavelength (λ = 10.5 μm) and nonlinear index (n 2), the nonlinearity γ mat in the solid material is maximal where Aeff is minimized, as accordingly shown in Fig. 7(a) and Fig. 4(b). While γ mat may be maximized in the suspended rod limit (d→0), the conclusion is different for the nonlinear gas interaction where optimization of γ gas is obtained for d>0 and within a distinct ellipsoidal region of the Λ-d parameter space [see Fig. 7(b)]. Of greater interest is the recognition that not only one can have relatively high values of the γ mat and γ gas parameters, but also one could operate in the region of low and flattened dispersion [see Figs. 5(a), 5(b)] for which phase-matching with a given nonlinear optical process is optimized. Therefore, what distinguishes the microporous fiber from the microwire, nanowire and other small solid-core designs is the prospect of extensive chromatic dispersion engineering. Combined with the low attenuation created by the porosity, one realizes that microporous fibers enable long interaction lengths in nonlinear media.
Pertaining to nonlinear optical interactions in gases, wavelength conversion in the visible spectrum through stimulated Raman scattering has been demonstrated in a hollow core PCF filled with hydrogen gas . The same type of setup, using rubidium vapor instead of hydrogen, showed high-efficiency FWM-frequency conversion at low microwatt pump powers . An attractive scheme is to use the linear and nonlinear optical interactions in the gas-filled holes of microporous fibers for remote sensing and spectroscopy of gaseous analytes.
3.2 Nonlinear Schrodinger equation and the Raman response of As2Se3
The following numerical simulations of the nonlinear optical interactions in chalcogenide microporous fibers is based on the scalar nonlinear Schrodinger equation (NLSE):19]:Fig. 8(a) ] with recently published results . The parameter relates to the phonon oscillation frequency while defines the characteristic damping time of the network of vibrating atoms.
The Raman gain spectrum shown on Fig. 8(b) was obtained with a λ = 1.54 μm CW pump and calculated from where is the imaginary part of the Fourier transform of hR (t), and fR is the fractional contribution of the Raman response to the total nonlinear response: , where the first term accounts for the instantaneous electronic response. The m/W peak gain coefficient is located at the frequency shift of 229.3 cm−1 in quantitative accord with experimental measurements . From the curve we calculated a fractional contribution fR = 0.115 in close agreement with the previously reported value fR = 0.1 in .
On substituting Eq. (7) with the relevant numerical parameters into the last expression, we find TR = 5.40 fs, which is to our best knowledge the first time a numerical value of TR for As2Se3 has been proposed in the literature.
To cross-check the validity of the latter result, the Raman time constant was also estimated from the slope of the Raman gain spectrum that is assumed to vary linearly in the vicinity of the pump frequency :
Considering Fourier-transform-limited Gaussian pulses of λ = 1.54 μm carrier wavelength and duration τ FWHM ≥ 100 fs (i.e. for FWHM pulse spectral bandwidths: Δν FWHM ≤ 4.4 THz), the slope value TR = 4.25 fs can be adopted; while for τ FWHM ≈91 fs (Δν FWHM ≈4.84 THz) and τ FWHM ≈83 fs (Δν FWHM ≈5.3 THz) the corresponding respective values TR = 5.36 fs and TR = 7.44 fs may be more appropriate. As evidenced on Fig. 9, the linear approximation of the Raman gain slope is a rough one and becomes questionable for pulses with FWHM spectral bandwidths > 4.4 THz in which the increase in Raman gain significantly deviates from the nearly linear rate in the vicinity of the carrier frequency ν 0.
In the next sub-section, a uniform value of TR = 5.40 fs [derived earlier from Eq. (8)] was implemented for all numerical simulations. To convert the latter value for a different wavelength of interest, we applied the 1/λ scaling rule of the Raman gain coefficient [23,24] which yielded TR = 0.792 fs for λ = 10.5 μm.
4. Supercontinuum bandwidth simulations
Taking a 10 cm long microporous fiber with (Λ = 0.40, d = 0.24) μm as a reference case, SC simulations were performed by solving Eq. (7) via the symmetrized split-step Fourier method with implementation of the Kerr and Raman responses of As2Se3 and the first m = 10 Taylor series coefficients βm of the propagation constant β(ω). At the λ = 10.5 μm wavelength, the fiber is pumped in the anomalous dispersion regime (D = 5.6 ps/(km∙nm)) with effective mode area Aeff = 11 μm2 and nonlinear coefficient γ = 571 W−1km−1. We here restrict ourselves to short pulse pumping 100 fs ≤ τ FWHM ≤ 10 ps for which we have soliton orders Nsol > 5 (where with and LNL = 1/(γP 0)). In this regime, SC generation is mainly driven by an initial nonlinear temporal compression of the pulse with creation of higher-order solitons and their successive fission into Nsol fundamental solitons via intrapulse Raman scattering. The Nsol fundamental solitons ejected by the fission process have peak powers where P 0 is the peak input pulse power and s = 1, 2,..., Nsol denotes the order in which they are ejected from the higher-order soliton . It follows that the most powerful of these fundamental solitons is the first one with an increased peak power of . The parameter P 1 is of prime importance since it is the most energetic fundamental soliton that mostly produces dispersive waves – also referred as Cherenkov radiation – which generates new wavelengths.
In all simulations, unchirped Gaussian pulses were adopted. Polarization coupling effects, which are assumed to be small here, were not included in the calculation since the purpose of this particular study is the adequate estimation of the output spectral bandwidth; not its exact fine spectral structure.
In the first set of simulations the input pulse energy (E 0) was kept fixed at 0.9 nJ with varying FWHM durations: 10, 1 and 0.1 ps [Fig. 10(a) ]. Conversely in the second simulation set [Fig. 10(b)], the pulse duration (TFWHM) was kept fixed at 1 ps while the seed energy was varied: 0.2, 2.0 and 5.0 nJ. The generated SC spectrum for peak solitonic powers P 1 ≥ 2 kW presents a complicated multi-peak structure along with a fine structuring related to the Raman scattering and soliton fission processes . We applied a moving average to gently smooth the output spectra that is akin to performing a temporal average from the input shot-to-shot random noise seeds.
For pulse energy E 0 = 0.9 nJ and duration TFWHM = 10 ps the calculated soliton fission length of the fiber is Lfiss = 50 cm (where Lfiss = LD/Nsol), which is significantly longer than the actual length L = 10 cm of the fiber. Hence for this case the temporal compression and soliton fission dynamics are minimal and the spectrum takes instead a self-phase modulation (SPM) structure with 4 peaks. The spectrum on Fig. 10(b) corresponding to E 0 = 0.2 nJ is also dominated by a SPM-induced broadening. But in the latter case, even though the fission length is apparently short enough Lfiss = 3.3 cm < L, the reason behind the relatively small broadening is related to the peak solitonic power level (P 1 = 0.6 kW) which is too low to activate the soliton fission process and subsequent dispersive waves generation.
At the 0.9 nJ seed energy, the shorter pulses TFWHM = 1 ps and TFWHM = 0.1 ps correspond to fission lengths Lfiss = 1.57 cm and Lfiss = 0.4 mm respectively, thus soliton fission dominates and significant spectral broadening is observed on Fig. 10(a). In the case of the shortest pulse (TFWHM = 0.1 ps), we notice the formation of a two-peaked structure which concords with the description of power transfer from the pump to both blue-shifted and red-shifted dispersive waves . The same spectral splitting can be seen on the SC spectrum yielded by the E 0 = 5 nJ and TFWHM = 1 ps pulse with the exception here that modulation instability (MI) plays a prominent role as witnessed by the strong fluctuations in the spectral regions where the GVD is anomalous. The latter MI effect can be expected from the characteristic length of modulation instabilities (LMI ∼4LNL), which is ten times shorter than the fission length (Lfiss = 6.7 mm) in this configuration.
The broadest and un-segmented SC calculated from −20 dB below the peak output power, has a bandwidth of 3100 nm (between 8.5 and 11.6 μm) and is obtained with the 0.9 nJ pulse of 1 ps duration. Although a more detailed investigation is needed to accurately optimize the main parameters of the system (seed energy, pulse duration, wavelength, fiber geometry and length) for maximal broadening, the above theoretical study demonstrates the potential of dispersion-tailored microporous fibers for SC generation in the mid-IR.
The pulse with 0.9 nJ energy and 1 ps duration yields the broadest SC with an input peak power of P 0 = 0.846 kW and peak solitonic power P 1 = 3.2 kW. Considering the 11 μm2 effective mode area, the latter powers translates to sizable intensity values of 15.4 GW/cm2 and 58 GW/cm2 respectively. Previously, supercontinuum generation in an As2Se3 glass small-core step-index fiber and a PCF fiber have sustained ∼3.5 GW/cm2 intensities at 1 kHz rate with no material damage reported . Still, it remains unclear whether the proposed As2Se3 microporous fibers can sustain peak intensities ≥ 10 GW/cm2 due to lack of available data in that regime. Nevertheless, we emphasize that contrary to standard small-core fibers where peak intensity is centered at the solid glass center; the peak power in microporous fibers is mostly concentrated in the air holes [as shown in Fig. 1(b)] thus considerably reducing the risk of permanent damage to the glass structure.
The motivation for selecting a 10.5 μm seed lies on the current relative accessibility and cost-effectiveness of CO2 laser sources in this wavelength range; in contrast to the more expensive and relatively complex OPO technology.
The location of the seed wavelength (10.5 μm) near the multiphonon absorption edge of As2Se3 (∼11.5 μm) would most likely lead to thermal dissipation issues in a practical implementation. Thus other highly-nonlinear compound glasses with larger long-wavelength transmission edges such as TAS (Te-As-Se glass) may be used alternatively. Moreover, the rapid progress in quantum cascade lasers (QCL) peak powers and beam quality  indicate that these cheaper and compact sources could eventually be considered in the near future for seeding a SC with a pump inside the 5-9 μm region.
A novel type of microstructured fiber for mid-IR is proposed: the chalcogenide microporous fiber with subwavelength holes provides considerable chromatic dispersion engineering capabilities through proper tuning of its constitutive geometrical parameters (N, Λ and d). While the present study concentrates on small-core fibers with N = 4 layers of holes, we stress that this type of waveguide could also be suited for the design of large mode area fibers (LMA) where typically N >>4. The proposed microporous geometry is a simple design ready to be implemented using current PCF technology with relatively few changes in the fabrication process.
The results presented in this work clearly demonstrate the potential of dispersion-tailored microporous fibers for nonlinear-phase matching applications and for supercontinuum generation. In particular, for geometrical parameters Λ = 0.40 μm and d = 0.24 μm, numerical simulations of the nonlinear Schrodinger equation theoretically shows that a broad SC bandwidth of 3100 nm extending from 8.5 μm to 11.6 μm can be generated in a 10 cm long chalcogenide As2Se3 microporous fiber pumped with a 0.9 nJ picosecond pulse at 10.5 μm wavelength.
In summary, the tunable dispersive and attenuating properties of microporous fibers promise to provide significant flexibility for the design of linear and nonlinear middle-infrared applications.
This project is supported in part by the Fonds Québecois de la Recherche sur la Nature et les Technologies (FQRNT), and by a Collaborative Research and Development Grant in association with CorActive High-Tech Inc.
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