## Abstract

We describe in detail a procedure for maximizing the bandwidth of supercontinuum generation in As_{2}Se_{3} chalcogenide fibers and the physics behind this procedure. First, we determine the key parameters that govern the design. Second, we find the conditions for the fiber to be endlessly single-mode; the fiber should be endlessly single-mode to maintain high nonlinearity and low coupling loss. We find that supercontinuum generation in As_{2}Se_{3} fibers proceeds in two stages — an initial stage that is dominated by four-wave mixing and a later stage that is dominated by the Raman-induced soliton self-frequency shift. Third, we determine the conditions to maximize the Stokes wavelength that is generated by four-wave mixing in the initial stage. Finally, we put all these pieces together to maximize the bandwidth. We show that it is possible to generate an optical bandwidth of more than 4 *μ*m with an input pump wavelength of 2.5 *μ*m using an As_{2}Se_{3} fiber with an air-hole-diameter-to-pitch ratio of 0.4 and a pitch of 3 *μ*m. Obtaining this bandwidth requires a careful choice of the fiber’s waveguide parameters and the pulse’s peak power and duration, which determine respectively the fiber’s dispersion and nonlinearity.

©2010 Optical Society of America

## 1. Introduction

Supercontinuum generation uses the Kerr effect and the Raman effect, in combination with dispersion in optical fibers, to broaden the bandwidth of an optical signal, and it has numerous applications to spectroscopy, pulse compression, and the design of tunable ultrafast femtosecond laser sources [1]. Silica fibers have been the predominant source of supercontinuum generation to date [1]–[3]. However, the longest wavelength that can be generated in silica fibers is below 2.5 *μ*m due to material losses. Supercontinuum generation beyond this wavelength requires fibers with longer infrared (IR) transmission windows, along with an appropriate choice of dispersion and nonlinearity. Price, *et al*. [4] have shown theoretically that it is possible to generate a mid-IR supercontinuum from 2 to 5 *μ*m using a bismuth-glass photonic crystal fiber (PCF) with a wagon-wheel structure. Domachuk, *et al*. [5] have experimentally generated a mid-IR supercontinuum with a spectral range of 0.8 to 4.9 *μ*m using a tellurite PCF with the same structure. However, the light power generated above 3 *μ*m is less than 5% of the total power in those two cases. Work on mid-IR supercontinuum generation has also been done using fluoride and sapphire fibers [6]–[8]. Shaw, *et al*. have reported experimental work that demonstrates supercontinuum generation from 2.1 to 3.2 *μ*m in an As_{2}Se_{3} chalcogenide PCF with one ring of air holes in a hexagonal structure [9].

As just noted, supercontinuum generation has been demonstrated in tellurite PCFs [5], as well as in As_{2}Se_{3} chalcogenide PCFs [9]. Compared to tellurite glass, chalcogenide glass has a larger refractive index and a higher nonlinear index, leading to a greater modal confinement and a higher nonlinearity. Moreover, the long-wavelength side of the transmission window of tellurite glass cuts off at about 3 *μ*m [5], [10], which is too low to allow the generation of significant supercontinuum radiation in the range of 4–10 *μ*m. By contrast, the long-wavelength side of the transmission window of As_{2}Se_{3} glass cuts off at about 10 *μ*m [11], [12].

In prior brief meeting reports, we reported that we could use the measured Raman gain, along with the Kramers-Kronig relations, to calculate the full time-domain Raman response in As_{2}Se_{3} chalcogenide fibers [13]. We also reported that we could use the measured index of refraction as a function of wavelength, along with the fiber’s waveguide parameters (fiber pitch and air-hole-diameter-to-pitch ratio), to calculate the total chromatic dispersion. We then used this data to solve the generalized nonlinear Schrödinger equation, and we reported in one case that we could reproduce the experimentally-obtained spectrum of the supercontinuum generation [14]. We then demonstrated that it is possible to obtain a 4 *μ*m bandwidth with a 2.5 *μ*m input pump by carefully choosing the As_{2}Se_{3} PCF’s waveguide parameters, as well as the input pulse’s peak power and duration [15]. In this paper, we describe for the first time the optimization procedure that we used to maximize the bandwidth of the supercontinuum generation in As_{2}Se_{3} PCFs. While the basic physical processes — chromatic dispersion, the Kerr effect, and the Raman effect — are the same as in silica fibers [13], material differences have an important effect on the interaction of these processes and the process of optimization. Here we discuss in detail the impact of the physical processes on the optimization, and we describe the similarities and differences with supercontinuum generation in silica fibers.

In our optimization studies, we considered a hexagonal geometry in which one air hole is missing, shown schematically in Fig. 1. This geometry is the most common in PCFs [16]. We assume that the fiber has five air-hole rings, which is a large enough number, so that the change in the PCF’s dispersion and loss is negligible when additional air-hole rings are added. This number of air-hole rings is practical in silica fibers, but has yet to be demonstrated in chalcogenide fibers. Our results would have to be revisited in PCFs with a smaller number of air-hole rings, but the design procedure would be unchanged. The fiber waveguide parameters that are optimized in our study are the air-hole-diameter-to-pitch ratio *d*/Λ and the pitch Λ.

In order to maximize the bandwidth of the supercontinuum generation, we must carefully optimize the fiber’s waveguide parameters, as well as the input pulse’s duration and peak power. This optimization is done by solving the generalized nonlinear Schrödinger equation, while varying these parameters, and determining when the bandwidth is maximized. The first step in the procedure is to verify that we can solve the generalized nonlinear Schrödinger equation with sufficient accuracy to reproduce the experimentally-measured supercontinuum spectrum. The inputs to this equation are the total nonlinear response and the total chromatic dispersion, which in turn must be determined from the measured Raman gain, Kerr coefficient, and index of refraction as a function of wavelength, as well as the assumed or measured waveguide parameters. In Sec. 2, we describe this procedure, and we validate it by showing that there is good agreement between the supercontinuum spectrum that is generated in an experiment and
the corresponding simulation. The second step in this procedure is to determine the waveguide parameters for which the fiber is single-mode. If several modes are present, the fundamental mode will couple to higher-order modes, reducing the nonlinear interaction. Hence, the fiber should be designed to operate in a single-mode regime. This work is presented in Sec. 3. Super-continuum generation in As_{2}Se_{3} chalcogenide fibers is essentially a two stage process. In the first stage, four-wave mixing seeds the solitons, which in the second stage undergo a Raman self-frequency shift. It is advantageous for the Stokes wavelength that is generated in the initial stage to be as large as possible since our simulations show that doing so increases the final bandwidth that is generated, moving it closer to the ultimate limit, which is determined by the longer zero-dispersion wavelength [17]. The Stokes wavelength is determined by the phase-matching conditions, which in turn depend on the waveguide parameters. Thus, the third step in our procedure is to determine waveguide parameters that lead to the largest possible Stokes wavelength. This work is presented in Sec. 4. In Sec. 5, we present our final optimization of the waveguide and pulse parameters from the solution of the generalized nonlinear Schrödinger equation, along with animations that elucidate the design procedures. We present the conclusions in Sec. 6. The basic process of supercontinuum generation in As_{2}Se_{3} chalcogenide fibers is the same as in silica fibers, but the details differ substantially, due to the material differences in the two types of fibers. We discuss the similarities and differences throughout this paper.

## 2. Determining the nonlinear response and the dispersion

The first step in maximizing the bandwidth is to determine the nonlinear response and the chromatic dispersion, which are needed to solve the generalized nonlinear Schröinger equation (GNLS) [1], [4],

$$=\mathrm{i\gamma}\left(1+\frac{i}{{\omega}_{0}}\frac{\partial}{\partial t}\right)[A\left(z,t\right){\int}_{-\infty}^{t}R\left(t-t\prime \right)\mid A\left(z,t\prime \right){\mid}^{2}\mathrm{dt}\prime ],$$

where *A*(*z*,*t*) is the electric field envelope as a function of distance along the fiber *z* and retarded time *t*. IFT{} denotes the inverse Fourier transform, where Ω is the transform variable and the tilda indicates the Fourier transform. The parameter *ω*
_{0} is angular carrier frequency, *β*(*ω*
_{0}) is the corresponding propagation constant, and *a* is the fiber loss. We set the wavelength-independent fiber loss *a* equal to 4.8 dB/m in agreement with Ref. 9. The quantity *β*
_{1}(*ω*
_{0}) is the first derivative of *β*(*ω*
_{0}). While one often writes the GNLS with a Taylor series, this calculation is usually done in practice using the inverse Fourier transform, as shown in Eq. (1), unless the Taylor expansion only has a small number of terms like 2 or 3 [1]. The parameter *γ*= *n*
_{2}
*ω*
_{0}/(*cA*
_{eff}) is the Kerr coefficient, where *n*
_{2} is the nonlinear refractive index, *c* is the speed of light, and *A*
_{eff} is the fiber’s effective area. The nonlinear response function, *R*(*t*) = (1 - *f _{R}*)

*δ*(

*t*)+

*f*(

_{R}h_{R}*t*), includes both the Kerr (instantaneous)

*δ*(

*t*) contribution and the Raman (delayed)

*h*(

_{R}*t*) contribution, where ∫

^{∞}

_{0}

*h*(

_{R}*t*) = 1 [13].

To determine the nonlinear response, we must experimentally measure *n*
_{2} and the Raman gain *g*(Ω) = (2*ω _{p}*/

*c*)

*n*

_{2}

*f*Im[

_{R}*H*(Ω)] [13], [18], where

_{R}*ω*and Im[

_{p}*H*(Ω)] represent the pump frequency and the imaginary part of the Fourier transform of

_{R}*h*(

_{R}*t*), respectively. Once Im[

*H*(Ω)] is known, we may use the Kramers-Kronig relations (Hilbert transformation) to determine Re[

_{R}*H*(Ω)] [13]. The inverse Fourier transform of

_{R}*H*(Ω) then yields

_{R}*h*(

_{R}*t*). For the nonlinear index

*n*

_{2}, we used

*n*

_{2}= 1.5 × 10

^{-17}m/W at a wavelength of 2.5

*μ*m, which we obtained from Fig. 5 of Ref. 19. Using the parameters reported in Ref. 19, we also find

*f*= 0.1. Finally, in Fig. 2(a), we show

_{R}*N*

^{″}(Ω), which is proportional to Im[

*H*(Ω)], but has been normalized so that its peak value is equal to 1. The red dashed curve and the blue solid curve show respectively

_{R}*N*

^{″}(Ω) for a silica fiber [13] and an As

_{2}Se

_{3}chalcogenide fiber, whose Raman gain was measured at the Naval Research Laboratory. Figure 2(b) shows the corresponding Hilbert transforms

*N*

^{′}(Ω). The Raman gain for the silica fiber has a peak at around 440 cm

^{-1}, and the Raman gain for the chalcogenide fiber has a peak at around 230 cm

^{-1}. Hence, the response time for the chalcogenide fiber is longer than the response time for the silica fiber, as shown in Fig. 3, and the Raman response must be taken into account for pulses that are as long as a picosecond, in contrast to silica fibers, where the finite time delay may be ignored for pulses that are longer than 300 fs. We will use the nonlinear response function that we have determined here throughout the remainder of this paper.

Optimizing the waveguide parameters to maximize the bandwidth of the supercontinuum generation has no impact on the nonlinear response function. Hence, one can determine this response function once and for all at the beginning of the study, as we have just done. By contrast, the waveguide parameters do affect the dispersion, which has two components — a material component that is calculated from the experimental measurements of the refractive index at the beginning of the study and a waveguide component that must be recalculated for each choice of the waveguide parameters. Throughout our study, we calculated the waveguide component of *β*(*ω*
_{0}) = *n*
_{eff}
*k*
_{0}, where *n*
_{eff} and *k*
_{0} are the effective index and wavenumber, by solving Maxwell’s equations to find the fundamental mode and its corresponding wavenumber at the frequency of interest. We calculated these quantities using COMSOL Multiphysics, a commercial full-vector solver based on the finite-element method. The dashed curve in Fig. 4 shows the material dispersion that is calculated from the refractive index as a function of wavelength that was measured at the Naval Research Laboratory for chalcogenide glass using ellipsometry [20], while the solid curve shows the total dispersion that is calculated for a chalcogenide PCF with one air-hole ring, an air-hole-diameter-to-pitch ratio of 0.8, and a core diameter (2Λ – *d*) of 10 *μ*m. With these parameters, we have Λ = 8.33 *μ*m and *d* = 6.67 *μ*m. While this PCF is multi-mode at 2.5 *μ*m, which is the input wavelength, most of the input power is launched into the fundamental mode [1]. Note that the use of one air-hole ring in this validation study differs from the use of five air-hole rings in our optimization studies.

In order to validate this procedure prior to optimizing the fiber waveguide and pulse parameters, we compared our solution of Eq. (1) to the experimental results that were briefly reported in Ref. 9. This experiment used a Spectra-Physics OPA-800C as an input source. The output pulses had a full-width at half maximum (FWHM) of 100 fs, a pulse energy of 100 pJ, and an input wavelength of 2.5 *μ*m. The spectrum was measured using a Jarrell-Ash 1/4 meter monochromator with appropriate gratings and an InSb detector. The system was calibrated using known sources. In our simulation, we solved Eq. (1) using the nonlinear response for the chalcogenide fiber shown in Fig. 3 and the total dispersion profile shown in Fig. 4. The pulse parameters are the same as in the experiment. In all simulations in this paper, including this one, we varied the number of node points in the time and the frequency domain, the magnitude of the time and the frequency windows, and the longitudinal step size, and we verified that the impact on the output spectrum is negligible. We use a resolution bandwidth of 1 nm to plot the spectrum. We used the split-step Fourier method to solve Eq. (1) [21]. We validated our code by comparing our results to Fig. 3(a) of Ref. 1.

In Fig. 5, the blue solid curve and the red dashed curve show the simulation and the corresponding experimental spectra after one meter of propagation. The amplitude of the theoretical results was adjusted by a constant factor to match the experiment, but no other adjustments were made. The simulation results are in good agreement with the experiment. While there are quantitative differences that we attribute to a combination of the longitudinal variation of the
fiber cross-section and wavelength-dependent coupling losses into the spectrum analyzer [22], the simulation accurately reproduces the bandwidth of 2.1 to 3.3 *μ*m for the points 20 dB down from the peak, which agrees well with the experimental bandwidth of 2.1 to 3.2 *μ*m. Accurate reproduction of the bandwidth has been the principal validation criterion in most simulations of supercontinuum generation [5], [22]–[25]. This result shows that the measured nonlinear response of the chalcogenide fiber in combination with dispersion can account for the bandwidth of the supercontinuum generation. The good agreement between simulation and experiment also supports the assumption that the supercontinuum is generated mostly in the fundamental mode. We found an error in our previous code, so that the blue solid curve in this picture is slightly different from Fig. 2(b) in Ref. 14.

## 3. Condition for single-mode operation

In order to maximize the bandwidth of the supercontinuum generation, it is important for the fiber to be single-mode in the wavelength range of interest. If several modes exist in the PCF, the fundamental mode will couple to higher-order modes that have a larger effective area and a lower nonlinearity, reducing the nonlinear interactions that are needed for broad bandwidth supercontinuum generation.

It has been shown that silica PCFs with some geometries can be single-mode for any wavelength [26]. Here, we will find the single-mode condition for chalcogenide fiber. The cutoff wavelength of a higher-order mode is determined by using the effective cladding index *n*
_{FSM}. The quantity *n*
_{FSM} is the effective index of the fundamental space-filling mode (FSM), which is defined as the fundamental mode of the infinite photonic crystal cladding if the core is absent [26]. Figure 6 shows *n*
_{FSM} as a function of the ratio of wavelength to pitch, *λ*/Λ, with different ratios of hole diameter to pitch, *d*/Λ, calculated by using the full-vectorial plane-wave method [27]. The refractive index of the background glass is set to 2.8, corresponding to the refractive index of As_{2}Se_{3} at a wavelength of 4 *μ*m. Material dispersion is ignored here since it has no impact on whether a fiber is single-mode or multi-mode. Note that *n*
_{FSM} strongly depends on both the ratio of wavelength to the pitch and the ratio of the hole diameter to the pitch. In a standard step-index fiber, the number of guided modes is determined by the *V*-value [28],

where, *a*, *λ*, *n*
_{cl}, and *n*
_{cl} represent the core radius, wavelength, core index, and cladding index, respectively. The value *V* must be less than 2.405 for a fiber to have a single mode. The effective *V* for a PCF may be defined as [29]

v where *a*
_{eff} is the effective core radius. We use *a*
_{eff} = Λ/√3 with the cutoff condition of
*V*
_{eff} = 2.405 [29]–[31]. Figure 7 shows the curves corresponding to *V*
_{eff} = 2.405 for different refractive indices of glass. The red solid curve represents silica with a refractive index of 1.45, which reproduces the result in Ref. 29. The blue and green solid curves represent refractive indices of 2.4 and 2.8, which correspond to the refractive indices of As_{2}S_{3} and As_{2}Se_{3} at a wavelength of 4 *μ*m, respectively. At each index, the parameter values in the area above the curve corresponding to *V*
_{eff} = 2.405 and to the left of the dashed line correspond to single-mode operation. While the single-mode region and multi-mode region in Fig. 7 vary as the refractive index changes, the endlessly single-mode region is almost unchanged at every index, which agrees with the conclusion in Ref. 32. The physical reason is as follows: When *λ*/Λ is small, we may approximate Eq. (3) as

Using data shown in Fig. 6 with small λ/Λ, we find that *n*
_{co} - *n*
_{FSM} is proportional to (*λ*/Λ)^{2} and depends on *d*/Λ exponentially. Hence, *V*
_{eff} in Eq. (4) is independent of *λ*/Λ at small *λ*/Λ, which is why the curve corresponding to *V*
_{eff} = 2.405 falls steeply as *d*/Λ is reduced in Fig. 7. Considering that glass typically has a refractive index range of 1.45 to 2.8 [10], the variation of √*n*
_{co} is small compared to the variation in *d*/Λ. Hence, a PCF becomes endlessly single-mode at nearly the same *d*/Λ with different refractive indices.

## 4. Initial supercontinuum generation using four-wave mixing

Next we consider the initial phase of supercontinuum generation in an As_{2}Se_{3} PCF that is dominated by four-wave mixing. The goal is to find the fiber waveguide parameters that maximize the Stokes wavelength since that also maximizes the final bandwidth from the entire process of supercontinuum generation. We will optimize Λ, setting *d*/Λ = 0.4. We set *d*/Λ = 0.4 since it is advantageous for this parameter to be as large as possible in order to minimize the leakage loss, as long as the fiber remains endlessly single-mode.

In the four-wave mixing process, two pump photons at the same frequency generate a Stokes photon and an anti-Stokes photon. Hence, when considering the four-wave mixing process in isolation from the other nonlinear and dispersive processes that affect the evolution, the generalized nonlinear Schrödinger equation (GNLS) simplifies to [33]

The wavelengths of the Stokes and anti-Stokes components are determined by the phase-matching conditions, and our design goal is to achieve the largest possible wavelength for the Stokes component. A larger initial Stokes wavelength leads to a larger final bandwidth after the second stage of supercontinuum generation that is dominated by the Raman self-frequency shift.

We are interested in the situation where the Stokes and anti-Stokes frequencies are far away from the peak of the Raman gain at 6.7 THz shown in Fig. 2(a) [34]. In this case, it is only necessary to take the Kerr effect into account, so that we obtain

We then replace *A*(*z*,*t*) by *A _{s}* exp[

*i*(

*n*/

_{s}ω_{s}z*c*-

*ω*)] +

_{s}t*A*exp[

_{a}*i*(

*n*/

_{a}ω_{a}z*c*-

*ω*)] + 2

_{a}t*A*exp[

_{p}*i*(

*n*/

_{p}ω_{p}z*c*-

*ω*)] + c.c., where

_{p}t*A*,

_{s}*A*, and

_{a}*A*are the amplitudes at the angular frequencies

_{p}*ω*,

_{s}*ω*, and

_{a}*ω*for the Stokes, anti-Stokes, and pump wave, respectively. The values

_{p}*n*,

_{s}*n*, and

_{a}*n*are effective refractive indices for the Stokes, anti-Stokes, and pump wave, respectively. We obtain two linear coupled equations for the Stokes and anti-Stokes fields,

_{p}where *P _{p}* = ∣

*A*∣

_{p}^{2}is the peak power of the pump and

*θ*equals (

*n*+

_{s}ω_{s}*n*- 2

_{a}ω_{a}*n*)/

_{p}ω_{p}*c*-6

*γ*(

*ω*)(1 -

_{p}*f*)

_{R}*P*. Note that

_{p}*ω*+

_{s}*ω*= 2

_{a}*ω*. To solve these equations, we introduce

_{p}*B*=

_{s}*A*exp[-4

_{s}*iγ*(

*ω*)(1-

_{p}*f*)

_{R}*P*] and

_{p}z*B*=

_{a}*A*exp[-4

_{a}*iγ*(

*ω*)(1-

_{p}*f*)

_{R}*P*] and obtain

_{p}zwhere the phase-matching condition is given by [3], [33], [35]

Again, we use the full dispersion curve to calculate the phase-matching condition. Figure 8 shows the phase-matching diagram calculated for the As_{2}Se_{3} chalcogenide PCF. The blue solid, dashed, and dotted curves present the phase-matching conditions for peak powers of 1, 0.1, and 0 kW, respectively, with *d*/Λ = 0.4 and Λ = 3 *μ*m.

Figure 9 shows the phase-matching diagram calculated for the As_{2}Se_{3} chalcogenide PCF for a peak power of 0.1 kW with different pitches. The green dotted, blue dashed, and red solid curves present the phase-matching conditions for pitches of 2, 3, and 4 *μ*m, respectively. Note that around the wavelength of 2.5 *μ*m, the four-wave mixing wavelengths generated by a PCF with Λ = 3 *μ*m are further apart from each other than are the four-wave mixing wavelengths generated by a PCF with Λ = 2 *μ*m. There are no Stokes and anti-Stokes wavelengths corresponding to the pump wavelength around 2.5 *μ*m for a PCF with Λ = 4 *μ*m. Thus, it is advantageous to use a pitch of 3 *μ*m, rather than 2 *μ*m or 4 *μ*m.

## 5. Maximizing the bandwidth

We now solve Eq. (1) computationally to verify that the optimal pitch of 3 *μ*m that we found in Sec. 4 indeed maximizes the supercontinuum bandwidth and to optimize the pulse parameters. We again set *d*/Λ = 0.4. Most of the spectral broadening occurs in the first few centimeters. In the simulations reported here, we set the fiber length equal to 0.1 m, since we have found that a longer fiber length does not increase the bandwidth of the supercontinuum generation and increases the loss. The effective areas and nonlinear coefficients are calculated for different pitches [33], [36]. which has a low-loss region from 2 *μ*m to about 10 *μ*m. Figure 10 shows the material loss, leakage loss, and total loss for a PCF with *d*/Λ = 0.4 and Λ = 3 *μ*m. We set the minimum wavelength-independent material loss at 4.8 dB/m, which is consistent with Ref. 9. We add exponentially increased loss below a wavelength of 2.1 *μ*m and above a wavelength of 8.8 *μ*m in accordance with the loss measurement for chalcogenide fiber with low hydrogen impurities [12]. Between 8.8 and 9.4 *μ*m, the loss is increased exponentially from 4.8 to 100 dB/m. Between 2.1 and 1.8 *μ*m, the loss is also increased exponentially from 4.8 to 100 dB/m. Leakage loss is calculated using COMSOL. Figure 11 shows the output spectra with pitches of 2, 3, and 4 *μ*m. The input pulse has a FWHM of 500 fs and a peak power of 1 kW. We define a total generated bandwidth as the bandwidth inside frequency limits that are 20 dB down from the peak of the spectrum. Using a pitch of 3 *μ*m, the total generated bandwidth is more than 4 *μ*m, as shown in Fig. 11.

Figure 12 shows a movie of a simulation of the spectrogram as the wave propagates along a PCF when Λ = 3 *μ*m. The black solid curve shows the group delay with respect to the wave at 2.5 *μ*m, which is in the normal dispersion regime when Λ = 3 *μ*s shown in Fig. 13. In the beginning, the effect of self-phase modulation first broadens the spectrum so that it extends into the region around 2.6 *μ*m and 2.7 *μ*m. At a distance of about 0.03 m, the power that is generated at a wavelength of 4–5 *μ*m due to four-wave mixing rises above the noise and visibly grows in strength. The power grows from noise at precisely the Stokes and anti-Stokes wavelengths that are predicted by the phase-matching condition, Eq. (9), which indicates that solitons are generated by four-wave mixing, consistent with the phase-matching diagram in Fig. 8. Instead, the power grows from zero at precisely the Stokes and anti-Stokes wavelengths predicted by the phase-matching condition, Eq. (9), which indicates that solitons are generated by four-wave mixing, consistent with the phase-matching diagram in Fig. 8. At a later stage, the solitons generated using four-wave mixing around 4–5 *μ*m are shifted to longer wavelengths due to the soliton self-frequency shift [37], [38]. For an ideal soliton with a fixed duration, the rate at which this shift occurs is proportional to the dispersion [37]. While the solitons in supercontinuum generation are not ideal, we have found, consistent with earlier observations [3], that as the wavelength increases and the dispersion decreases, the soliton’s duration remains roughly constant, and the soliton’s energy decreases, leading to a decrease in the rate at which
the soliton self-frequency shift occurs. The center frequency of the largest soliton is shifted to around 5.5 *μ*m. There is little additional shifting because of lower dispersion above 5 *μ*m, as shown in the dash-dotted curve in Fig. 13, and decreased power due to the fiber attenuation. The zero dispersion wavelength including material and waveguide dispersion for the chalcogenide PCF at the short wavelength side is around 2.7 *μ*m according to our calculation. We also found that soliton-soliton collisions transfer energy to red-shifted solitons [39]. Figure 12 shows that the effects of the soliton trapping dispersive wave and shifting the light to shorter wavelengths are small in this case [40].

Figure 14 shows the total generated bandwidth as a function of Λ, which peaks at Λ ≃ 2.8 *μ*m. When the pitch is small, the bandwidth that can be generated using the four-wave mixing process, as shown in the red curve of Fig. 8, is narrower than the wavelength generated using a longer pitch, as shown in the blue curves of Fig. 8. We also note that the anomalous dispersion region is narrow in the case of small pitch, as shown in the blue dashed curved of Fig. 13. Since the phase-matching condition in Eq. (9) is calculated from the dispersion relation, the phase-matching diagram is also closely related to the dispersion relation. The small pitch leads to two zero-dispersion wavelengths where the longer zero-dispersion wavelength limits the supercontinuum bandwidth, due to cancellation of the self-frequency shift [17]. More importantly, the soliton self-frequency shift ceases when the dispersion becomes small, which is well before zero-dispersion wavelength of 6.8 *μ*m shown in Fig. 13. When the pitch is large, the pump wavelength of 2.5 *μ*m is in the normal dispersion region, which leads to a small amount of supercontinuum generation [1]. Meanwhile, the effective area becomes large and the nonlinearity is low.

Figure 15 shows the total generated bandwidth as a function of the FWHM of the input pulse. The input peak power is fixed at 1 kW. As the FWHM increases from 100 fs to 500 fs, the total generated bandwidth increases roughly linearly, while as the FWHM increases from 500 fs to 1000 fs, the total generated bandwidth remains roughly constant. In all cases, we have found that the initial pulse breaks up into about 10 solitons, each of which has a duration of about 500 fs. It is not possible to identify the lower-amplitude solitons unambiguously. The total generated bandwidth is determined by the largest-amplitude soliton, since this soliton undergoes the largest Raman-induced self-frequency shift. The amplitude of this soliton depends sensitively on the pulse duration and fluctuates as the pulse duration increases. Figures 16(a) and (b) show the output spectrograms with an input pulse duration of 600 fs and 650 fs, respectively, using the same color scale. While in principle, the sum of the energy in all the solitons has increased when the input pulse duration increases from 600 fs to 650 fs, the amplitude of the largest soli-ton has decreased; so, there is a slight decrease in the final bandwidth. That said, the amplitude of the largest-amplitude soliton roughly increases from 100 to 500 fs and roughly saturates between 500 fs to 1000 fs, and this behaviour of the largest soliton’s amplitude is the origin of the behaviour that we observed in the total generated bandwidth.

Figure 17 shows the total generated bandwidth as a function of the input peak power. The FWHM of the input pulse is fixed at 500 fs. The total generated bandwidth increases as the peak power increases up to 1.4 kW, after which it is nearly flat. As before, when we varied the input pulse duration, the total generated bandwidth is determined by the largest-amplitude soliton. For the same reason, the bandwidth increases roughly linearly as the peak power increases until the peak power reaches 1.4 kW.

In all cases, the maximum wavelength to which a soliton can shift is given by the zero-dispersion wavelength of 6.8 *μ*m, when Λ = 3 *μ*m, as shown in Fig. 13. In practice the maximum wavelength is somewhat shorter. However, as just noted, the limitation on the maximum bandwidth that we observed is due to the saturation of the amplitude of the largest soliton.

In order to find out how fiber loss affects supercontinuum generation, we have run simulations with a FWHM of 500 fs, a peak power of 1 kW, and no fiber loss. The bandwidth is 0.3 *μ*m wider than the bandwidth generated when we include the fiber loss.

We have also run a simulation with a wavelength-dependent effective area for the nonlinear coefficient. Other simulation parameters are the same as those we used for generating Fig. 17. We obtain a bandwidth of 4.2 *μ*m using an input peak power of 2.0 kW, instead of the bandwidth of 4.7 *μ*m that is shown in Fig. 17, which was generated using wavelength-independent effective area. At an input peak power of 2.5 kW, the bandwidths are 4.7 *μ*m and 4.8 *μ*m using a wavelength-dependent effective area and a wavelength-independent effective area, respectively. The two methods generate almost the same bandwidth at large peak power because the bandwidth is not limited by the nonlinearity; instead it is limited by the dispersion, as we previously discussed. The physics for generating a supercontinuum is the same in the two cases.

## 6. Conclusions

In this paper, we have described in detail the design procedure that we used to maximize the bandwidth of supercontinuum generation in As_{2}Se_{3} chalcogenide PCFs. We find that a bandwidth of more than 4 *μ*m can be obtained with an input pulse at 2.5 *μ*m by carefully optimizing the waveguide parameters and the input pulse parameters. We began by describing how we use the measured Raman gain, Kerr coefficient, and material index as a function of wavelength to find the total nonlinear response and the total dispersion, which are needed to solve the GNLS. We verified that our solution of the GNLS matches the experiment in one case. Next, we found the waveguide parameters that are required for the fiber to operate in a single mode and to maximize the Stokes wavelength in the initial stage of supercontinuum generation that is dominated by four-wave mixing. Finally, we put all these pieces together to maximize the bandwidth from supercontinuum generation. We explained the physics that leads to the maximal bandwidth by showing that supercontinuum generation in As_{2}Se_{3} chalcogenide fibers is initially generated by four-wave mixing, followed by the soliton self-frequency shift once a train of solitons has formed. These basic physical processes are the same as in silica fibers, but the details differ significantly. A careful choice of the waveguide parameters and hence the dispersion profile, as well as the pulse parameters and hence the nonlinearity, must be made to maximize the bandwidth.

## References and links

**1. **J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

**2. **S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers,” J. Opt. Soc. Am. B **19**, 753–764 (2002). [CrossRef]

**3. **G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt. Express **10**, 1083–1098 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-20-1083. [PubMed]

**4. **J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropou-los, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. **13**, 738–749 (2007). [CrossRef]

**5. **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**, 7161–7168 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7161. [CrossRef] [PubMed]

**6. **C. Xia, M. Kumar, O. P. Kulkarni, M. N. Islam, F. L. Terry Jr., M. J. Freeman, M. Poulain, and G. Mazé, “Mid-infrared supercontinuum generation to 4.5 *μ*m in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. **31**, 2553–2555 (2006). [CrossRef] [PubMed]

**7. **C. Xia, Z. Xu, M. N. Islam, F. L. Terry Jr., M. J. Freeman, A. Zakel, and J. Mauricio, “10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 *μ*m with direct pulse pattern modulation,” IEEE J. Sel. Top. Quantum Electron. **15**, 422–4342009. [CrossRef]

**8. **J. H. Kim, M.-K. Chen, C.-E. Yang, J. Lee, S. (S.) Yin, P. Ruffin, E. Edwards, C. Brantley, and C. Luo, “Broadband IR supercontinuum generation using single crystal sapphire fibers,” Opt. Express **16**, 4085–4093 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-6-4085. [CrossRef] [PubMed]

**9. **L. B. Shaw, V. Q. Nguyen, J. S. Sanghera, I. D. Aggarwal, P. A. Thielen, and F. H. Kung, “IR supercontinuum generation in As-Se photonic crystal fiber,” in Proc. Advanced Solid State Photonics, Vienna, Austria, paper TuC5 (2005).

**10. **X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, “Nonsilica glass for holey fibers,” J. Lightwave Technol. **23**, 2046–2054 (2005). [CrossRef]

**11. **P. A. Thielen, L. B. Shaw, P. C. Pureza, V. Q. Nguyen, J. S. Sanghera, and I. D. Aggarwal, “Small-core As-Se fiber for Raman amplification,” Opt. Lett. **28**, 1406–1408 (2003). [CrossRef] [PubMed]

**12. **V. Q. Nguyen, J. S. Sanghera, P. Pureza, F. H. Kung, and I. D. Aggarwal, “Fabrication of Arsenic Selenide Optical Fiber with Low Hydrogen Impurities,” J. Am. Ceram. Soc. **85**, 2849–2851 (2002). [CrossRef]

**13. **R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B **6**, 1159–1166 (1989). [CrossRef]

**14. **J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Raman response function and supercontinuum generation in chalcogenide fiber,” in Proc. Conference on Lasers and Electro-Optics (CLEO), San Jose, CA, paper CMDD2, (2008).

**15. **J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Supercontinuum generation in an As2Se3-based chalcogenide PCF using four-wave mixing and soliton self-frequency shift,” in Proc. Conference on Optical Fiber Communications (OFC), San Diego, CA, paper OWU6, (2009).

**16. **J. C. Knight, “Photonic crystal fibres,” Nature **424**, 847–851 (2003). [CrossRef] [PubMed]

**17. **D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, “Soliton Self-Frequency Shift Cancellation in Photonic Crystal Fibers,” Science **301**, 1705–1708 (2003). [CrossRef] [PubMed]

**18. **L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. **32**, 391–393 (2007). [CrossRef] [PubMed]

**19. **R. E. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. B. Shaw, and I. D. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As_{2}Se_{3} chalcogenide fibers,” J. Opt. Soc. Am. B **21**, 1146–1155 (2004). [CrossRef]

**20. **H. G. Tompkins and W. A. McGahan, *Spectroscopic Ellipsometry and Reflectometry*, John Wiley & Sons, Inc., New York, 1999.

**21. **O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. **21**, 61–68 (2003). [CrossRef]

**22. **W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibers,” Nature **424**, 511–515 (2003). [CrossRef] [PubMed]

**23. **M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: Experiment and simulation,” Appl. Phys. B **81**, 363–367 (2005). [CrossRef]

**24. **M. L. V. Tse, P. Horak, F. Poletti, N. G. Broderick, J. H. Price, J. R. Hayes, and D. J. Richardson, “Supercontinuum generation at 1.06 *μ*m in holey fibers with dispersion flattened profiles,” Opt. Express **14**, 4445–4451 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-10-4445. [CrossRef] [PubMed]

**25. **C. Xia, M. Kumar, M. Cheng, R. S. Hegde, M. N. Islam, A. Galvanauskas, H. G. Winful, F. L. Terry Jr., M. J. Freeman, M. Poulain, and G. Mazé “Power scalable mid-infrared supercontinuum generation in ZBLAN fluoride fibers with up to 1.3 watts time-averaged power,” Opt. Express **15**, 865–871 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-3-865. [CrossRef] [PubMed]

**26. **T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

**27. **S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173. [CrossRef] [PubMed]

**28. **A. W. Snyder and J. D. Love, *Optical Waveguide Theory* (Chapman & Hall, London, 1983).

**29. **K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. **23**, 3580–3590 (2005). [CrossRef]

**30. **M. Koshiba, “Full-vector analysis of photonic crystal fibers using the finite element method,” IEICE Trans. Electron. **85-C**, 881–888 (2002).

**31. **M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. **29**1739–1741, (2004). [CrossRef] [PubMed]

**32. **N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. **30**, 1455–1457 (2005). [CrossRef] [PubMed]

**33. **G. P. Agrawal, *Nonlinear Fiber Optics* (Academic Press, New York, 2001).

**34. **M. H. Frosz, T. Sorensen, and O. Bang, “Nanoengineering of photonic crystal fibers for supercontinuum spectral shaping,” J. Opt. Soc. Am. B **23**, 1692–1699 (2006). [CrossRef]

**35. **G. Genty, “Supercontinuum generation in microstructured fibers and novel optical measurement techniques,” Ph.D. Dissertation, Helsinki University of Technology, Espoo, Finland.

**36. **J. Hu, B. S. Marks, C. R. Menyuk, J. Kim, T. F. Carruthers, B. M. Wright, T. F. Taunay, and E. J. Friebele, “Pulse compression using a tapered microstructure optical fiber,” Opt. Express **14**, 4026–4036 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-9-4026. [CrossRef] [PubMed]

**37. **J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. **11**, 662–664 (1986). [CrossRef] [PubMed]

**38. **C. L. Hagen, J. W. Walewski, and S. T. Sanders, “Generation of a continuum extending to the midinfrared by pumping ZBLAN fiber with an ultrafast 1550-nm source,” IEEE Photon. Technol. Lett. **18**, 91–93 (2006). [CrossRef]

**39. **M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express **14**, 9391–9407 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9391. [CrossRef] [PubMed]

**40. **A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nature Photonics **1**, 653–657 (2007). [CrossRef]