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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 7 — Mar. 29, 2010
  • pp: 6722–6739
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Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers

Jonathan Hu, Curtis R. Menyuk, L. Brandon Shaw, Jasbinder S. Sanghera, and Ishwar D. Aggarwal  »View Author Affiliations


Optics Express, Vol. 18, Issue 7, pp. 6722-6739 (2010)
http://dx.doi.org/10.1364/OE.18.006722


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Abstract

We describe in detail a procedure for maximizing the bandwidth of supercontinuum generation in As2Se3 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 As2Se3 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 μm with an input pump wavelength of 2.5 μm using an As2Se3 fiber with an air-hole-diameter-to-pitch ratio of 0.4 and a pitch of 3 μm. 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

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

]. Silica fibers have been the predominant source of supercontinuum generation to date [1

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

]–[3

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]

]. However, the longest wavelength that can be generated in silica fibers is below 2.5 μm 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

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]

] have shown theoretically that it is possible to generate a mid-IR supercontinuum from 2 to 5 μm using a bismuth-glass photonic crystal fiber (PCF) with a wagon-wheel structure. Domachuk, et al. [5

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]

] have experimentally generated a mid-IR supercontinuum with a spectral range of 0.8 to 4.9 μm using a tellurite PCF with the same structure. However, the light power generated above 3 μm 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

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 μm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. 31, 2553–2555 (2006). [CrossRef] [PubMed]

]–[8

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]

]. Shaw, et al. have reported experimental work that demonstrates supercontinuum generation from 2.1 to 3.2 μm in an As2Se3 chalcogenide PCF with one ring of air holes in a hexagonal structure [9

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).

].

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 As2Se3 chalcogenide fibers [13

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]

]. 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

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).

]. We then demonstrated that it is possible to obtain a 4 μm bandwidth with a 2.5 μm input pump by carefully choosing the As2Se3 PCF’s waveguide parameters, as well as the input pulse’s peak power and duration [15

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).

]. In this paper, we describe for the first time the optimization procedure that we used to maximize the bandwidth of the supercontinuum generation in As2Se3 PCFs. While the basic physical processes — chromatic dispersion, the Kerr effect, and the Raman effect — are the same as in silica fibers [13

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]

], 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.

Fig. 1. Photonic crystal fiber geometry.

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

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

]. 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 As2Se3 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

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]

]. 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 As2Se3 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

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

], [4

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]

],

A(z,t)z+a2AiIFT{[β(ω0+Ω)β(ω0)Ωβ1(ω0)]A˜(z,ω)}
=(1+iω0t)[A(z,t)tR(tt)A(z,t)2dt],
(1)

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

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).

. 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

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

]. 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 - fR)δ(t)+fRhR(t), includes both the Kerr (instantaneous) δ(t) contribution and the Raman (delayed) hR(t) contribution, where ∫ 0 hR(t) = 1 [13

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]

].

Fig. 2. (a) The imaginary part of third-order susceptibility, which is proportional to the Raman gain. It has been normalized to unity at the peak gain. (b) The real part of the third-order susceptibility is shown in the same arbitrary units. The red dashed curve and the blue solid curve show respectively N (Ω) for a silica fiber [13] and an As2Se3 chalcogenide fiber, whose Raman gain was measured at the Naval Research Laboratory.

Fig. 3. Raman response function. In these figures, the blue solid curve and the red dashed curve represent chalcogenide fiber and silica fiber, respectively.

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

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).

. 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 μm. 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

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]

]. We validated our code by comparing our results to Fig. 3(a) of Ref. 1

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

.

Fig. 4. The dashed curve and solid curve represent material dispersion for chalcogenide glass and total dispersion for a chalcogenide PCF with one air-hole ring, respectively.
Fig. 5. The blue solid curve and the red dashed curve show the simulation result and corresponding experimental result [9]. The dispersion is calculated with d/Λ = 0.8.
Fig. 6. The effective index of the fundamental space-filling mode n FSM as a function of the ratio of wavelength to pitch, λ/Λ, with different ratios of hole diameter to pitch, d/Λ.

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

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]

]. 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

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]

]. 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

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]

]. The refractive index of the background glass is set to 2.8, corresponding to the refractive index of As2Se3 at a wavelength of 4 μm. 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

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

],

V=2πλa(nco2ncl2)1/2,
(2)

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

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

]

Veff=2πλaeff(nco2nFSM2)1/2,
(3)

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

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

]–[31

31. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 291739–1741, (2004). [CrossRef] [PubMed]

]. 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

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

. The blue and green solid curves represent refractive indices of 2.4 and 2.8, which correspond to the refractive indices of As2S3 and As2Se3 at a wavelength of 4 μm, 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

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]

. The physical reason is as follows: When λ/Λ is small, we may approximate Eq. (3) as

Fig. 7. The curves corresponding to V eff = 2.405 for different refractive indices of glass. The red, blue, and green solid solid curves represent refractive indices of 1.45, 2.4, and 2.8, respectively.
Veff22ncoπ3λ/Λ(nconFSM)1/2.
(4)

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

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]

], 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

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

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

]

A(z,t)z=(ωp)[A(z,t)tR(tt)A(z,t)2dt].
(5)

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

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]

]. In this case, it is only necessary to take the Kerr effect into account, so that we obtain

A(z,t)z=(ωp)(1fR)A(z,t)A(z,t)2.
(6)

We then replace A(z,t) by As exp[i(nsωsz/c - ωst)] + Aa exp[i(naωaz/c - ωat)] + 2Ap exp[i(npωpz/c- ωpt)] + c.c., where As, Aa, and Ap are the amplitudes at the angular frequencies ωs, ωa, and ωp for the Stokes, anti-Stokes, and pump wave, respectively. The values ns, na, and np 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,

dAsdz=2(ωp)(1fR)[2PpAs+Ppexp(iθz)Aa*],
(7.a)
dAa*dz=2(ωp)(1fR)[2PpAa*+Ppexp(+iθz)As],
(7.b)

where Pp = ∣Ap2 is the peak power of the pump and θ equals (nsωs + naωa - 2npωp)/c-6γ(ωp)(1 - fR)Pp. Note that ωs + ωa = 2ωp. To solve these equations, we introduce Bs = As exp[-4(ωp)(1- fR)Ppz] and Ba = Aa exp[-4(ωp)(1- fR)Ppz] and obtain

dBsdz=2(ωp)(1fR)Ppexp(iκz)Ba*,
(8.a)
dBa*dz=2(ωp)(1fR)Ppexp(+iκz)Bs,
(8.b)

where the phase-matching condition is given by [3

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]

], [33

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

], [35

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

]

κ=(nsωs+naωa2npωp)/c+2(1fR)γ(ωp)Pp=0.
(9)

Again, we use the full dispersion curve to calculate the phase-matching condition. Figure 8 shows the phase-matching diagram calculated for the As2Se3 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 μm.

Figure 9 shows the phase-matching diagram calculated for the As2Se3 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 μm, respectively. Note that around the wavelength of 2.5 μm, the four-wave mixing wavelengths generated by a PCF with Λ = 3 μm are further apart from each other than are the four-wave mixing wavelengths generated by a PCF with Λ = 2 μm. There are no Stokes and anti-Stokes wavelengths corresponding to the pump wavelength around 2.5 μm for a PCF with Λ = 4 μm. Thus, it is advantageous to use a pitch of 3 μm, rather than 2 μm or 4 μm.

Fig. 8. The phase-matching diagram is calculated for an As2Se3 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 μm.
Fig. 9. The phase-matching diagram is calculated for an As2Se3 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 μm, respectively.
Fig. 10. The blue solid, green solid, and red dashed curves show respectively the material loss, leakage loss, and total loss for a PCF with d/Λ = 0.4 and Λ = 3 μm.

5. Maximizing the bandwidth

We now solve Eq. (1) computationally to verify that the optimal pitch of 3 μm 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

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

], [36

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]

]. which has a low-loss region from 2 μm to about 10 μm. Figure 10 shows the material loss, leakage loss, and total loss for a PCF with d/Λ = 0.4 and Λ = 3 μm. 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 μm and above a wavelength of 8.8 μm in accordance with the loss measurement for chalcogenide fiber with low hydrogen impurities [12

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]

]. Between 8.8 and 9.4 μm, the loss is increased exponentially from 4.8 to 100 dB/m. Between 2.1 and 1.8 μm, 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 μm. 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 μm, the total generated bandwidth is more than 4 μm, 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 μm. The black solid curve shows the group delay with respect to the wave at 2.5 μm, which is in the normal dispersion regime when Λ = 3 μs 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 μm and 2.7 μm. At a distance of about 0.03 m, the power that is generated at a wavelength of 4–5 μm 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 μm are shifted to longer wavelengths due to the soliton self-frequency shift [37

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

], [38

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]

]. For an ideal soliton with a fixed duration, the rate at which this shift occurs is proportional to the dispersion [37

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

]. While the solitons in supercontinuum generation are not ideal, we have found, consistent with earlier observations [3

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]

], 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 μm. There is little additional shifting because of lower dispersion above 5 μm, 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 μm according to our calculation. We also found that soliton-soliton collisions transfer energy to red-shifted solitons [39

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]

]. 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

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]

].

Fig. 11. The green dashed, blue solid, and red dotted curves show the output spectra with pitches of 2, 3, and 4 μm, respectively. The input pulse has a FWHM of 500 fs. The input peak power is set at 1 kW.
Fig. 12. (Media 1) Movie of a simulation of the spectrogram as the wave propagates along the PCF. The black solid curve shows the group delay with respect to the wave at 2.5 μm.
Fig. 13. The blue dashed, blue dash-dotted, and blue dotted curves present the chromatic dispersion for a PCF with Λ = 2 μm, Λ = 3 μm, and Λ = 4 μm, respectively.
Fig. 14. The total generated bandwidth as a function of Λ. The input source has a peak power of 1 kW and FWHM of 500 fs.
Fig. 15. The total generated bandwidth as a function of the FWHM of the input pulse. The PCF has a pitch of 3 μm. The input source has a peak power of 1 kW.

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.

Fig. 16. Spectrograms with (a) an input FWHM of 600 fs and (b) an input FWHM of 650 fs using the same color scale.

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 μm, when Λ = 3 μm, 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 μm 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 μm using an input peak power of 2.0 kW, instead of the bandwidth of 4.7 μm 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 μm and 4.8 μm 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.

Fig. 17. The total generated bandwidth as a function of input peak power. The PCF has a pitch of 3 μm. The input source has a FWHM of 500 fs.

6. Conclusions

In this paper, we have described in detail the design procedure that we used to maximize the bandwidth of supercontinuum generation in As2Se3 chalcogenide PCFs. We find that a bandwidth of more than 4 μm can be obtained with an input pulse at 2.5 μm 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 As2Se3 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

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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]

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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]

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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 μm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. 31, 2553–2555 (2006). [CrossRef] [PubMed]

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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 μm 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 As2Se3 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]

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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]

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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]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2390) Fiber optics and optical communications : Fiber optics, infrared
(320.6629) Ultrafast optics : Supercontinuum generation

ToC Category:
Ultrafast Optics

History
Original Manuscript: November 20, 2009
Revised Manuscript: March 2, 2010
Manuscript Accepted: March 5, 2010
Published: March 17, 2010

Citation
Jonathan Hu, Curtis R. Menyuk, L. Brandon Shaw, Jasbinder S. Sanghera, and Ishwar D. Aggarwal, "Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers," Opt. Express 18, 6722-6739 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-6722


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References

  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. Petropoulos, 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, "Midinfrared supercontinuum generation to 4.5 μm 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, J. Mauricio, "10.5 W time-averaged power mid-IR supercontinuum generation extending beyond 4 μm 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 As2Se3 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.
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