Chalcogenide microporous fibers for linear and nonlinear applications in the mid-infrared

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 blueshifted 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 Optical Society of America OCIS codes: (060.2390) Fiber optics, infrared; (060.4005) Microstructured fibers; (190.4370) Nonlinear optics, fibers; (320.6629) Supercontinuum generation. References and links 1. P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. B. Cordeiro, J. C. Knight, and F. G. Omenetto, “Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs,” Opt. Express 16(10), 7161–7168 (2008). 2. J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. 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Introduction
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 As 2 Se 3 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 [2] where a bismuthglass 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 CO 2 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.

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 [5].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 [8].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 claddingcompare 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 A eff ) compared to an equivalent rod-inair 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 (As 2 Se 3 ) 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 As 2 Se 3 exhibits relative large losses (∼5 dB/m) at 10.6 µm after fiberization [10], 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-theart hollow core fibers used in the power delivery of CO 2 lasers [11].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: λ Λ ≪ ).The total outer diameter of the fiber core is given by ( ) + Λ , 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.The refractive index' chromatic dispersion of bulk As 2 Se 3 glass is modeled (in Fig. 2) via the following Sellmeier equation [12]: where A 0 = 2.234921, A 1 = 0.24164, A 2 = 0.347441, A 3 = 1.308575, and λ is expressed in microns.The corresponding material zero-dispersion wavelength (ZDM) of As 2 Se 3 is 7.225µm.The wavelength-dependant absorption coefficient k (cm −1 ) was provided at discrete wavelengths and cubic spline interpolation was used to estimate values at intermediate wavelengths of interest (dotted blue curve in Fig. 2).Keeping the number of layers fixed to N = 4, the fundamental mode effective refractive index (n eff ) and field distributions (E x , E y , E z , S z ) 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.Postprocessing routines include: hundredfold increase in data density for n eff (λ) via cubic spline interpolation, then fitting the interpolated n eff (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: where the full diameter ( ) + Λ and the total number of holes in the fiber ( )  The fraction (f α ) of modal absorption to bulk material absorption loss can be evaluated to first-order approximation via the expression [13]:  The real value of the effective refractive index n eff is plotted in Fig. 4(a) and again illustrates the close relationship with the fraction f α of modal field in the solid material pictured in Fig. 3(b).The effective mode area A eff defined as [14]    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.

Effective nonlinearity in microporous fibers
The effective nonlinearity of the waveguides was evaluated via the nonlinear parameter γ defined as [16]: where 2 i n denotes the value of the nonlinear index in the solid material or the gaseous medium.As a matter of fact, the hollow nature of the microporous fiber and its flexible dispersive properties not only make this type of waveguide a viable candidate for performing nonlinear interactions in the glass material; but also within the gas-filled holes.Therefore the value of the nonlinear parameter γ was evaluated both in As 2 Se 3 glass [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 As 2 Se 3 [17] and Argon [18] are  From the standard definition of the nonlinear parameter [19], γ = 2πn 2 /(λA eff ), 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 A eff 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 [20].The same type of setup, using rubidium vapor instead of hydrogen, showed high-efficiency FWM-frequency conversion at low microwatt pump powers [21].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.

Nonlinear Schrodinger equation and the Raman response of As 2 Se 3
The following numerical simulations of the nonlinear optical interactions in chalcogenide microporous fibers is based on the scalar nonlinear Schrodinger equation (NLSE): where ( ) is the slowly-varying pulse envelope, and first-order approximations are used to include self-steepening effects and the delayed ionic Raman response function h R (t).Assuming a Lorentzian profile for the Raman gain spectrum, the Raman response function may be expressed in a convenient form [19]: where the parameters is the imaginary part of the Fourier transform of h R (t), and f R is the fractional contribution of the Raman response to the total nonlinear response: ( ) ( ) ( ) ( ) , where the first term accounts for the instantaneous electronic response.The curve we calculated a fractional contribution f R = 0.115 in close agreement with the previously reported value f R = 0.1 in [22].
The characteristic Raman time constant T R in Eq. ( 6) is defined as the first moment of the nonlinear response function [19]: On substituting Eq. ( 7) with the relevant numerical parameters into the last expression, we find T R = 5.40 fs, which is to our best knowledge the first time a numerical value of T R for As 2 Se 3 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 [19]: Based on the procedure described in [23], several values of T R can be calculated via linear regressions (shown on Fig. 9) depending on the input pulse duration.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: ∆ν FWHM ≤ 4.4 THz), the slope value T R = 4.25 fs can be adopted; while for τ FWHM ≈91 fs (∆ν FWHM ≈4.84 THz) and τ FWHM ≈83 fs (∆ν FWHM ≈5.3 THz) the corresponding respective values T R = 5.36 fs and T R = 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 T R = 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 T R = 0.792 fs for λ = 10.5 µm.

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 As 2 Se 3 and the first m = 10 Taylor series coefficients β m 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 A eff = 11 µm 2 and nonlinear coefficient γ = 571 W −1 km −1 .We here restrict ourselves to short pulse pumping 100 fs ≤ τ FWHM ≤ 10 ps for which we have soliton orders N sol > 5 (where 2 where P 0 is the peak input pulse power and s = 1, 2,..., N sol denotes the order in which they are ejected from the higher-order soliton [25].It follows that the most powerful of these fundamental solitons is the first one with an increased characteristic length of modulation instabilities (L MI ∼4L NL ), which is ten times shorter than the fission length (L fiss = 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 µm 2 effective mode area, the latter powers translates to sizable intensity values of 15.4 GW/cm 2 and 58 GW/cm 2 respectively.Previously, supercontinuum generation in an As 2 Se 3 glass small-core step-index fiber and a PCF fiber have sustained ∼3.5 GW/cm 2 intensities at 1 kHz rate with no material damage reported [4].Still, it remains unclear whether the proposed As 2 Se 3 microporous fibers can sustain peak intensities ≥ 10 GW/cm 2 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 CO 2 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 As 2 Se 3 (∼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 [27] 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.

Conclusion
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 As 2 Se 3 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 middleinfrared applications.
on the number of layers N. It follows that the fraction of #123626 -$15.00USD Received 1 Feb 2010; accepted 16 Mar 2010; published 9 Apr 2010 (C) 2010 OSA cross-sectional area occupied by solid glass material is given by 1 m f p = − .We note from 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 30 ± °, 90 ± ° and 150 ± ° angles [see Fig. 1(a)].
modal power distribution in the direction of mode propagation: + z.The value of f α as a function of geometrical parameters (Λ, d) is shown in a density plot [Fig.3(b)] alongside that of f m [Fig.3(a)].We note that n mat = 2.7678 and α mat = 1.4 m −1 (or 4.81 dB/m) for As 2 Se 3 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 0.70 d Λ ≈ with corresponding porosity 37% p ≈.As expected and revealed on Figs.3(a), 3(b), the percentage of modal-to-bulk absorption (f α ) has a strong correlation with the cross-section areal density of material (f m ) in the core.In other words, lower confinement losses are achieved in designs with large porosity.

Fig. 4 .
Fig. 4. (a) Real effective refractive index and (b) effective area of fundamental mode in N = 4 layers As2Se3 microporous fiber at λ = 10.5 µm as a function of parameters (Λ, d) plotted in Fig.4(b) which indicates a region in the Λ-d parameter space where minimum optimization of A eff can be achieved.In the present case, minimization of A eff at λ = 10.5 µm is obtained for Λ = 0.40 µm and d→0 (i.e. the limit of the air-suspended core).

Fig. 5 .
Fig. 5. (a) Dispersion parameter (ps/km•nm) and (b) dispersion slope (ps/km•nm 2 ) in N = 4 layers As2Se3 microporous fiber at λ = 10.5 µm as a function of geometrical parameters (Λ, d) The value of the chromatic dispersion parameter D and dispersion slope dD dλ 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 As 2 Se 3 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

1 τ relates to the phonon oscillation frequency while 2 τ
found to yield the best fit [solid line in Fig.8(a)] with recently published results[22].The parameter defines the characteristic damping time of the network of vibrating atoms.

Fig. 8 .
Fig. 8. (a) Raman response function h(t) of As2Se3 glass, and (b) corresponding Raman gain gR spectrum for pump wavelength λ = 1.54 µm.The Raman gain spectrum peak gain coefficient is located at the frequency shift of 229.3 cm −1 in quantitative accord with experimental measurements[17].From the
= 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 N sol fundamental solitons via intrapulse Raman scattering.The N sol fundamental solitons ejected by the fission process have peak powers