## Abstract

We computationally investigate supercontinuum generation in an As_{2}S_{3} solid core photonic crystal fiber (PCF) with a hexagonal cladding of air holes. We study the effect of varying the system (fiber and input pulse) parameters on the output bandwidth. We find that there is significant variation of the measured bandwidth with small changes in the system parameters due to the complex structure of the supercontinuum spectral output. This variation implies that one cannot accurately calculate the experimentally-expected bandwidth from a single numerical simulation. We propose the use of a smoothed and ensemble-averaged bandwidth that is expected to be a better predictor of the bandwidth of the supercontinuum spectra that would be produced in experimental systems. We show that the fluctuations are considerably reduced, allowing us to calculate the bandwidth more accurately. Using this smoothed and ensemble averaged bandwidth, we maximize the output bandwidth with a pump wavelength of 2.8 μm and obtain a supercontinuum spectrum that extends from 2.5 μm to 6.2 μm with an uncertainty of ± 0.5 μm. The optimized bandwidth is consistent with prior work, but with a significantly increased accuracy.

© 2010 Optical Society of America

## 1. Introduction

Supercontinuum generation in the visible spectrum has become an important application of photonic crystal fibers (PCF) [1, 2]. In this spectral range the use of silica optical fibers is well established; however, for other spectral ranges, different materials are better suited [3]. While currently not as fully developed, mid-infrared (mid-IR) supercontinuum sources are showing much promise for many applications. Mid-IR sources have application to biological spectroscopy [4], optical frequency metrology [5], and optical tomography [6].

Supercontinuum generation in the mid-IR requires the use of materials that have low material loss in the wavelength range of interest, as well as appropriate nonlinear and dispersion characteristics. The chalcogenide glasses, in particular As_{2}Se_{3} and As_{2}S_{3}, are transparent at mid-IR wavelengths as long as 10 μm. Mid-IR supercontinuum generation has been experimentally demonstrated using several chalcogenide glasses. In particular, mid-IR supercontinuum generation has been shown using chalcogenide nanowires [7], a small-core wagon wheel tellurite PCF with an output spectrum of 0.8–4.9 μm [8], and an As_{2}Se_{3} planar waveguide with an output spectrum of 1.2–2.0 μm [9]. Mid-IR supercontinuum generation has also been theoretically investigated. Price et. al. [10] investigated supercontinuum generation from 2–5 μm in a variety of heavy metal glasses with both small-core and large mode area fibers with a 2 μm pump. In addition, an As_{2}Se_{3} hexagonal PCF [11] and an As_{2}Se_{3} micro-porous fiber have recently been studied [12].

An important role for theoretical and numerical studies is to find fiber geometries and input pulses that will give the best supercontinuum output for a given application, and, to this end, computational studies to optimize supercontinuum sources have been performed using both straightforward parameter searches [11] and genetic optimization methods [13]. These studies typically maximize the supercontinuum bandwidth by changing the dispersion — in particular the location of the zero dispersion wavelengths — and the nonlinear properties of the fiber.

One of the principal figures of merit by which a supercontinuum source is judged is the output bandwidth, although there are other measures of the supercontinuum output that are important for specific applications, such as the coherence properties of the spectrum [14] and the smoothness of the output spectrum [15]. When optimizing the bandwidth computationally, it is important to use a measure of the bandwidth that agrees with what would be obtained in an experimental system. In this paper, we investigate a method to provide a better estimate of the bandwidth that is observed in experiments, and apply this method to maximize the supercontinuum bandwidth of a mid-IR supercontinuum source using an As_{2}S_{3} PCF. In a previous study of the same supercontinuum source, we found a relatively large uncertainty in the optimum due to large fluctuations in the calculated bandwidth across the range of fiber and input pulse parameters (system parameters) [16]. Here, we investigate the variation of the output spectral bandwidth due to changes in the system parameters of the supercontinuum source. We use small steps in the pulse parameters to reveal the large variations in the calculated output bandwidth. These large variations would have led to false maxima if the sampling had been done less finely, as is almost universally the case in studies up to the present time.

To find an estimate of the experimentally expected bandwidth, we smooth and then ensemble-average the supercontinuum spectral output and find the bandwidth of the output spectra corresponding to an average over the optical spectrum analyzer bandwidth and the range of system parameter variations that can be expected in an experimental system. This approach replicates the procedure in which the output bandwidths are experimentally measured better than has been the case in computational studies to date, while significantly reducing the fluctuations in the calculated bandwidth. As a consequence, we expect to obtain the optimal system parameters and the maximum achievable bandwidth with more accuracy than has been the case in the past.

## 2. Simulating Supercontinuum Generation

We use the same computational method to simulate supercontinuum generation that is described in detail by Hu et. al. [11]. This approach requires three independent steps: First, for a given set fiber parameters, we use Comsol Multiphysics, a commercial full-vectorial mode-solver based on the finite element method, to calculate the chromatic dispersion of the guided mode over the wavelength range of interest. Second, we solve the generalized nonlinear Schrödinger equation (GNLSE) using the dispersion that was calculated in the previous step. Finally, we calculate the expected output bandwidth from the output spectrum. This process is explained in more detail in the following sections.

In this study, we use a PCF with a hexagonal geometry as shown schematically in Fig. 1. The two fiber parameters that affect the dispersion are *d*, the air-hole diameter, and Λ, the air-hole pitch. In order to keep the fiber endlessly single-moded, we fix the air-hole diameter-to-pitch ratio in our simulations at *d/*Λ = 0.4. This value has been found to be the largest that is consistent with single-mode operation [11]. It is advantageous to maximize *d/*Λ to decrease the effective index of the cladding material, and hence increase nonlinearity and modal confinement. Thus, we optimize the fiber geometry over the single parameter of the air-hole pitch Λ.

Previous studies have shown that using longer wavelength input sources affords great advantages in output bandwidth; however, the current choice of available mid-IR lasers is limited [17]. In this study, we use an input source wavelength of 2.8 μm; this is the longest wavelength that has been used in experiments at the Naval Research Laboratory for supercontinuum generation in chalcogenide fibers.

For all simulations, we use a fiber length of 0.5 m. This value is currently used in experiments and gives a good balance between the extra spectral power generated at longer wavelengths by the soliton self-frequency shift as the length increases and the negative effect of increased fiber loss.

The supercontinuum process is modeled by the generalized nonlinear Schrödinger equation (GNLSE) [18]:

*A*(

*z,t*) is the electric field envelope and

*Ã*(

*z, ω*) is its Fourier transform,

*a*(

*ω*) is the frequency-dependent fiber loss, and

*γ*is the Kerr coefficient. The Kerr coefficient is given by

*γ*=

*n*

_{2}

*ω*

_{0}

*/*(

*cA*

_{eff}), where

*n*

_{2}is the nonlinear refractive index,

*ω*

_{0}is the angular frequency of the optical carrier,

*c*is the speed of light, and

*A*

_{eff}is the fiber’s effective area. The quantity

*R*(

*t*) is the nonlinear response function defined by

*R*(

*t*) = (1 –

*f*)

_{R}*δ*(

*t*) +

*f*(

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

*δ*(

*t*), and the delayed Raman contribution,

*h*(

_{R}*t*). We use

*f*to denote the fraction of the nonlinear response function that is due to the Raman effect, and we use IFT{} to denote the inverse Fourier transform. The material nonlinearity in As

_{R}_{2}S

_{3}is taken as

*n*

_{2}= 3.285 × 10

^{−18}m

^{2}/W, the Raman fraction as

*f*= 0.2, and the nonlinear coefficient

_{R}*γ*is calculated for each air-hole pitch based on the effective mode area of the numerically calculated mode. The Raman gain response of As

_{2}S

_{3}was measured experimentally, and we used the Kramers-Kronig relations to find the complete Raman response function; the resulting data has been published in Fig. 2 of [19]. The GNLSE of Eq. (1) is solved numerically using the split-step Fourier method implemented in MATLAB [20].

In these optimization studies, we consider four air-hole pitches over the range of interest. They are 2.5 μm, 3.0 μm, 3.5 μm, and 4.0 μm. The chromatic dispersion as a function of frequency, *β*(*ω*), and leakage loss of a PCF with each of these pitches are plotted in Figs. 2(a) and 2(b), respectively. As the pitch increases, the anomalous dispersion region broadens and begins at longer wavelengths. In general, larger pitches are advantageous for a number of reasons. First, given a sufficiently long wavelength pump and sufficient pump power, the phase matching condition for four-wave mixing (FWM) has a wider separation between Stokes and anti-Stokes wavelengths for larger pitches. This wider separation results in a wider bandwidth of the spectral broadening in the first few centimeters of the fiber, which is dominated by FWM. Second, since the rate of the soliton self-frequency shift (SSFS) is proportional to dispersion, PCFs with a larger pitch have more broadening from the SSFS. The SSFS is the dominant source of the large spectral broadening that occurs after the initial FWM broadening [21]. However, a larger pitch in a PCF also leads to an increase in the effective mode area, which decreases modal confinement and hence nonlinearity, which in turn is detrimental to the supercontinuum broadening process.

## 3. Bandwidth Calculation

A difficulty with the optimization of supercontinuum generation using an automated bandwidth calculation is that only calculating the bandwidth over a limited set of points in the optimization parameter space can lead to a false maximum being found. This difficulty becomes apparent when we calculate the bandwidth for many finely resolved values of the pulse width, as shown in the next section. Large variations of the bandwidth for small changes in the pulse width are apparent, and if the parameter space is less well-resolved and single-shot simulations are used, erroneous optima can be found.

Moreover, these variations are not representative of the experimental system. In any experimental system, the input pulse to the fiber will contain random quantum noise, and the pulse width and energy will also change from pulse to pulse. Experimentally, the repetition rate of the laser pulse is much greater than the sampling rate of the spectral measurement; so, the experimental output spectrum is a time average of the actual supercontinuum output. Therefore, the measured output spectrum is an ensemble average over quantum noise and random fluctuations in the system parameters which consist of the input pulse and fiber parameters. Furthermore, experimental measurements have a finite spectral resolution, which also serves to smooth the spectrum and reduce the large variations in the output [10].

A typical supercontinuum spectra, such as the spectrum given in Fig. 3(a), has a complex structure, which is a result of complicated soliton dynamics and depends on the number of interacting solitons, their interactions, pulse widths and individual powers — all of which are sensitive functions of the system parameters. This complex structure is changed by any noise on the input pulse, and it has been shown that ensemble averaging over incoherent pulse noise leads to a smoothing of the output spectra [14].

A measure of the bandwidth that better represents the bandwidth that is experimentally measured can be found by approximating numerically the action of an optical spectral analyzer (OSA) and the experimental variations that can be expected in real systems and performing an ensemble average. Experimentally, the effect of an OSA is to convolve the spectrum with a filter function of finite width; we can do the same computationally. We first smooth the spectrum by convolving it in the spectral domain with a super-Gaussian function of width *λ _{w}*,

*S*(

_{f}*λ*) is the filtered spectral density,

*S*(

*λ*) is the input spectral density and Λ is chosen depending on the width of the filter function.

The effect of spectral smoothing is shown in Fig. 3, in which an example supercontinuum output spectrum is filtered with different filter widths (*λ _{w}*). In practice, typical OSAs have a resolution of approximately

*λ*= 10 nm for supercontinuum generation experiments. We considered filter widths up to 100 nm in order to investigate whether spectral smoothing by itself would be sufficient to avoid the need for a separate ensemble average over the system parameters.

_{w}To reduce the impact of peaks in the supercontinuum output affecting the bandwidth measurement we define the average output power spectral density (PSD) of a spectrum as the PSD of an equivalent ideal rectangular spectrum with the same total power and the same bandwidth as the spectrum being measured. This rectangular output spectrum has a constant PSD, PSD_{av} = *P*_{tot}/BW, where *P*_{tot} is the total power in the output spectrum, and the bandwidth (BW) is the width of the spectrum between the points 20 dB down from PSD_{av}. This definition is recursive; however it converges in a few iterations. Fig. 3(b) shows an example spectrum, with the equivalent spectrum used in the bandwidth definition.

## 4. Bandwidth from Averaging over Pulse Parameters

We apply the method of the previous section to the PCF described in Section 2 with *λ _{w}* = 100 nm. The simulations used an initial hyperbolic-secant pulse with a pulse width and a peak power that we varied independently. The pulse width and peak power were varied over a range consistent with the experimentally achievable range of these parameters. Figure 4(a) shows a plot of the output bandwidth versus the input pulse full-width half-maximum (FWHM) for air-hole pitches of 2.5 μm, 3.0 μm, 3.5 μm, and 4.0 μm. In all cases, the short wavelength end of the bandwidth that we observe was at 2.5 μm and is essentially fixed by fiber loss [16]. The bandwidth for the PCF with a 2.5 μm pitch remains much the same as the pulse width changes. When the pitch is increased to 3.0 μm, however, the output bandwidth increases markedly as the pulse width increases. Also the fluctuations in the bandwidth become large. They are on the order of 300 nm, or about 10%, for very small changes in the input pulse duration. These fluctuations in the measured bandwidth are even larger for a pitch of 3.5 μm, where the bandwidth fluctuates by as much as 800 nm, about 23%.

Figure 4(b) shows a plot of the output bandwidth as a function input pulse peak power with steps of 1 W for air-hole pitches of 2.5 μm, 3.0 μm, 3.5 μm, and 4.0 μm. Again, the short wavelength end of the bandwidth is at 2.5 μm. This plot exhibits behavior similar to that in Fig. 4(a), in that the output bandwidth varies greatly for a small change in pulse power. Again, the fluctuations are not as large for a pitch of 2.5 μm as they are for larger pitches. We note that for each pitch the bandwidth increases relatively smoothly to a peak power of around 300 W, after which there are large variations in the bandwidth.

It is clear from Fig. 4 that even after spectral smoothing with *λ _{w}* = 100 nm, there are still considerable variations in the measured bandwidth that do not appear in an experimental system. As filtering the spectral output is not enough to remove these unphysical variations, we now calculate an ensemble average of the supercontinuum output for a range of pulse parameters that could be expected in an experimental system.

Figures 5(a) and 5(b) show the same data as Figs. 4(a) and 4(b), respectively, for a 3.5 μm pitch, and also the bandwidth calculated by the ensemble average of the pulse spectrum for all simulations over a 10% variation in the input parameter (either pulse width or peak power), which is a typical value in experiments, and then calculating the bandwidth by the method described in the previous section. This bandwidth is plotted as a dashed line Figs. 5(a) and 5(b) and corresponds to the the bandwidth that would be experimentally observed with a variation in system parameters of 10%. The bandwidth increases with increasing pulse width and pulse peak power, but still has significant fluctuations.

Figure 5 makes clear the necessity of ensemble averaging to find an expected bandwidth. Spectral filtering alone is not enough to reduce the large-scale fluctuations in output power. Trying to determine an optimum set of parameters from Fig. 4 is unreliable.

## 5. Application to Optimization

In Fig. 6, we show the averaged bandwidth that we calculated with the procedure described in Sections 3 and 4, applied to the data of Fig. 4. We find that by keeping the pulse width fixed at 1.5 ps we can obtain about 4 μm of output bandwidth for a pitch of 3.5 μm and peak powers greater than 1300 W.

In a previous optimization study of the same supercontinuum source [16], we found that the optimal parameters were a 4.0 μm pitch, a pulse width of 1150 fs, and a peak input power of 1280 W which corresponded to a calculated output bandwidth of about 4.0 μm, similar to the maximum bandwidth in this study. The present results are a more reliable estimate of the expected maximum bandwidth and optimal fiber and input pulse parameters, because the previous results were obtained by taking large steps in the pulse parameters and neglected the sensitivity of supercontinuum generation to small changes in the system parameters.

The uncertainty in choosing an optimum from Fig. 5 can be as high as 800 nm, or around 23%. We can see that there is still unphysical variation in the bandwidth by considering the values at at the right hand side of Fig. 6(a) and 6(b) where the system parameters are almost the same; however, the values of bandwidth are not at the same value. The overall estimate of the error in the ensemble-average method gives an uncertainty of approximately 500 nm, or slightly more than 10%. Therefore, we find a maximum bandwidth of 3.5 μm with a pitch of 3.5 μm that extends between 2.5 and 6.2 μm with an uncertainty of ± 0.5 μm.

## 6. Conclusion

We have investigated the large variations in bandwidth that are found with small changes in the pulse parameters for supercontinuum generation in an As_{2}S_{3} hexagonal photonic crystal fiber. Due to these variations, calculating the bandwidth using a single system realization does not give a reliable estimate of the bandwidth that is expected in an experimental supercontinuum source. Therefore, it is important when optimizing a supercontinuum source to ensemble average over the range of system parameters that is expected in the experiments. In doing so we have calculated a more accurate estimate of the bandwidth from numerically generated supercontinuum output as well as estimated the uncertainty in the bandwidth both before and after this procedure.

We have shown that spectral smoothing and ensemble averaging with a 10% variation of the input pulse width and peak power reduces the large variations in the bandwidth, increasing the accuracy of the results. However, fluctuations, although reduced, still remain. We only average over one pulse parameter at a time, and it is likely that simultaneously averaging over multiple parameters and ensemble-averaging over the input pulse noise simultaneously would further reduce the fluctuations and correspond better to experiments. Further work must be done to investigate this point.

## Acknowledgments

The authors are grateful to I. Aggarwal, J. Sanghera, L. B. Shaw for useful discussions and for arranging for this work’s support. This work was supported by the Naval Research Laboratory.

## References and links

**1. **J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. **25**, 25–27 (2000). [CrossRef]

**2. **J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fibre,” Nat. Photonics **3**, 85–90 (2009). [CrossRef]

**3. **X. Feng, A. Mairaj, D. Hewak, and T. Monro, “Nonsilica Glasses for Holey Fibers,” J. Lightwave Technol.23, 2046–2054 (2005). [CrossRef]

**4. **P. Rolfe, “In vivo near-infrared spectroscopy,” Annu. Rev. Biomed. Eng. **2**, 715–754 (2000). [CrossRef]

**5. **R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” *Phys. Rev. Lett.*85, 2264–2267 (2000). [CrossRef] [PubMed]

**6. **I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air–silica microstructure optical fiber,” Opt. Lett.26, 608–610 (2001). [CrossRef]

**7. **D. I. Yeom, E. C. Mgi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett.33, 660–662 (2008). [CrossRef] [PubMed]

**8. **P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. 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). [CrossRef] [PubMed]

**9. **M. R. Lamont, B. Luther-Davies, D. Choi, S. Madden, and B. J. Eggleton, “Supercontinuum generation in dispersion engineered highly nonlinear (*γ* = 10 W/m) As_{2}S_{3} chalcogenide planar waveguide,” Opt. Express16, 14938–14944 (2008). [CrossRef] [PubMed]

**10. **J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm–5 μm,” IEEE J. Sel. Top. Quantum Electron. **13**(3), 738–749 (2007). [CrossRef]

**11. **J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the bandwidth of supercontinuum generation in As_{2}Se_{3} chalcogenide fibers,” Opt. Express18, 6722–6739 (2010). [CrossRef] [PubMed]

**12. **B. Ung and M. Skorobogatiy, “Chalcogenide microporous fibers for linear and nonlinear applications in the mid-infrared,” Opt. Express **18**, 8647–8659 (2010). [CrossRef] [PubMed]

**13. **W. Q. Zhang, S. V. Afshar, and T. M. Monro, “A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation,” Opt. Express **17**, 19311–19327 (2009). [CrossRef]

**14. **J. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. **27**, 1180–1182 (2002). [CrossRef]

**15. **F. G. Omenetto, N. A. Wolchover, M. R. Wehner, M. Ross, A. Efimov, A. J. Taylor, V. V. Kumar, A. K. George, J. C. Knight, N. Y. Joly, and P. S. Russell, “Spectrally smooth supercontinuum from 350 nm to 3 μm in sub-centimeter lengths of soft-glass photonic crystal fibers,” Opt. Express **14**, 4928–4934 (2006). [CrossRef] [PubMed]

**16. **R. J. Weiblen, J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Maximizing the Supercontinuum Bandwidth in As_{2}S_{3} Chalcogenide Photonic Crystal Fibers,” in Proc. Conference on Lasers and Electro-Optics (CLEO), San Jose, CA, paper CTuX7, (2010).

**17. **J. S. Sanghera, L. Brandon Shaw, and I.D. Aggarwal, “Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications,” IEEE J. Sel. Top. Quantum Electron. **15**, 114–119 (2009). [CrossRef]

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

**19. **J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Computational study of 3–5 μm source created by using supercontinuum generation in As_{2}S_{3} chalcogenide fibers with a pump at 2 μm,” Opt. Lett.35, 2907–2909 (2010). [CrossRef] [PubMed]

**20. **O. V. Sinkin, R. Holzlhner, 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]

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