1. Introduction

Chalcogenide glasses exhibit low loss transmission in the short- and mid-wavelength infrared (short-, mid-IR) part of the spectrum, and thus lend themselves to waveguide fabrication for applications in the medical, military, and sensing arenas [1]. Furthermore, their strong, intrinsic nonlinear Kerr effect, which can be as much as ~ 900× that in silica [2], has led to demonstrations of high speed, all optical signal processing in a compact, planar waveguide geometry for next generation optical communications systems [3, 4]. The need for broadband sources in many of these applications at wavelengths beyond the ~ 2 µm transmission window limit associated with silica has prompted theoretical [5–7] and experimental [8–11] investigations into supercontinuum (SC) generation in chalcogenide photonic crystal fibers (PCFs), fiber tapers and planar waveguides. In these waveguides the ability to tailor the dispersion, which in the bulk material is large and normal, is central to their functionality. The basic mechanism behind SC generation in these geometries [12] is the fission of a high order optical soliton as a result of effects such as third order dispersion (TOD), dispersion of the nonlinear coefficient and the Raman effect, or noise (modulation instability regime). Additional broadening towards short wavelengths is effected by resonant dispersive wave emission and four–wave mixing involving fundamental solitons in the presence of TOD. The long wavelength limit of the SC is extended by the soliton self-frequency shift (SSFS) [13], which is facilitated by intrapulse Raman scattering. The enhancement of the Raman effect in chalcogenides over that in silica corresponds roughly with the increase in the Kerr nonlinearity.

In and of itself, the SSFS has relevance to the realization of tunable sources [14] and, in chalcogenide glasses, can be exploited for generating light in the mid-IR. This process of redshifting pulses also forms the basis of an all optical, analogue to digital conversion (ADC) method [15, 16] in which a train of soliton pulses is modulated by the analogue signal. Each pulse then experiences a redshift, the magnitude of which is determined by the individual pulse amplitude. In this way, the SSFS operates as a power-to-frequency conversion mechanism, and a set of interleaving spectral filters and binary detectors serve to encode digitally the resulting signal. In many of the SC studies in chalcogenide the rôle of the Raman response has either been ignored [5] or remained unclear beyond a net redshift of the spectral center of mass [9,11]. Specific discussion of the contribution of the SSFS to the spectral width of a SC generated in an As₂S₃ PCF has been presented recently [6]. The Raman spectrum used by these authors is notable in that it does not display any low frequency vibrational states.

In this paper we employ a Raman spectrum measured down to low wavenumbers to theoretically analyze the SSFS performance in an As₂S₃ planar waveguide. First we investigate the effect of differences in the Raman spectra of silica and As₂S₃ upon the frequency shift rate for an isolated, fundamental soliton, and then perform a detailed simulation of higher order soliton propagation in a chalcogenide ridge waveguide to demonstrate the generation of a Raman soliton exhibiting a redshift over hundreds of nanometers. The simulation parameters were chosen to reflect a realistic experimental situation which clearly demonstrates the dynamics of a Raman soliton.

2. Background and Theory

It is instructive to examine the Raman spectrum of As₂S₃ using that of silica as a standard. The normalized (see below), experimentally measured Raman spectra for silica [17] and As₂S₃ [18] are shown in Fig. 1(a). It should be noted, however, that the actual Raman gain coefficient in the latter is hundreds of times higher. The most salient difference between the two curves is that the main resonant peak in the chalcogenide at a detuning of ~ 10 THz is narrower and roughly three times as high as that in silica at ~ 13 THz. In addition, the spectrum for silica extends to higher frequencies than that for As₂S₃, although the data for the latter stops at 16.5 THz. Nonetheless, since the small peak in the As₂S₃ spectrum at ~ 15 THz corresponds to S-S bonds, the lightest pair of constituents in an As–deficient amorphous structure, no higher resonance is expected. In constrast, in the region of low frequency shifts below ~ 6 THz, the two curves are almost identical. The question arises as to whether the different shapes of these spectra result in a differing performance of the SSFS in the two materials, since this phenomenon is a consequence of the scattering process which they describe. Here we investigate this issue and find that, in spite of the different Raman spectra, the SSFS in silica and As₂S₃ has very similar properties.

The most general description of nonlinear phenomena in optical waveguides is produced by a generalized nonlinear Schröodinger equation (GNSE) [19, 20] which governs the evolution of the pulse envelope F(z,t) in the longitudinal (z) and temporal (t) coordinates. Such an equation can be written in Fourier space, where a tilde denotes a Fourier transform, as [20]

∂_zF̃(z,δ) = i[β(ω)−β ₁ δ−β ₀]F̃(z,δ)

 

Fig. 1. (a) Raman spectra and (b) spectral response functions R for bulk samples of As₂S₃ (solid blue lines) and silica (broken red lines).

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where

and ∣F̃(z,δ)∣² is the spectral energy density. Propagation in a single mode is assumed with propagation constant β. Its derivatives with respect to angular frequency ω = ω ₀+δ are β_m=∂^m_ωβ, with ω ₀ being a constant reference frequency. The quantities denoted by n and n ₂ are the linear and nonlinear indices of the material, respectively, with the latter including the electronic (Kerr) and nuclear (Raman) contributions. The vacuum speed of light is c, the effective index is n _eff = cβ/ω, and A _eff is the modal effective area. The nonlinear response function is defined as [19]

where the integral of the causal Raman response function h_R(t) is normalized to unity, and f_R is the fractional contribution of the Raman component to the total nonlinearity. We then associate the normalized Raman spectrum with the dissipative part, viz. α_R(ω) = 2Im[h̃_R(ω)]. In practice, the response function is obtained from the experimentally measured spontaneous scattering spectrum [21].

In the context of the SSFS for a single, isolated soliton, this general theory may be replaced by a simpler treatment [22–24]. This simplified theory furnishes an equation describing the evolution with distance z of the center frequency $\overline{\omega }$ of a soliton, viz.

where β ₂ is the group velocity dispersion (GVD) (assumed anomalous), T_s is the soliton pulse duration, and R is the dimensionless spectral response function defined as

The factor 1−f_R in the denominator of the RHS of Eq. (4) arises from our definition of the nonlinear parameter,

which contains only the Kerr part of the nonlinearity. The implication in this is the assumption of fundamental soliton pulse durations ≪100 fs, in which case the temporal Raman response may not be treated as instantaneous. When treating longer pulses the factor 1−f_R should be removed.

Since the GVD and the nonlinear parameter depend on frequency, the pulse duration evolves as the soliton frequency evolves according to

Incorporating the conservation of photon number into this expression yields

where E ₀ = E($\overline{\omega }$ ₀), and $\overline{\omega }$ ₀ is the initial soliton frequency.

In order to establish a meaningful basis for comparison of two waveguides with different nonlinear indices and dispersive properties, we can normalize the longitudinal spatial coordinate z to the nonlinear length which, for a fundamental soliton, is equal to the dispersive length, L_NL = L_D = T ² _s /∣β ₂∣. In doing so Eq. (4) is transformed to

where ζ = z/L_NL. Thus, for propagation of solitons of the same duration over an equal number of nonlinear lengths, variations in the frequency shifts in silica and As₂S₃ arise only through variations in f_R and R in the two materials.

Here it is instructive to examine the effect of the differences in the Raman spectra of As₂S₃ and silica upon the SSFS rate in the two materials. Through Eq. (5) this effect resides solely in the spectral response function R, which is plotted in Fig. 1(b) for the two cases. The two curves are qualitatively very similar, and the quantitative difference between them is never more than 25% over the pulse durations considered (0–500 fs). Thus, despite the disparities between the spectra shown in Fig. 1(a), the effect of the Raman spectrum upon the SSFS in As₂S₃ and silica is almost the same in both cases. The other material parameter pertaining to the Raman response which influences the SSFS rate is f_R. The magnitude of this quantity in silica is well established as f_R = 0.21 [25]. The corresponding value in As₂S₃ is less well determined, though an estimate may be obtained from pump–probe Raman gain measurements [26] using a rearrangement of an expression for the Raman gain to give,

where γ is the nonlinear coefficient (for long pulses) at the pump, ω _peak is the peak gain angular frequency, L _eff = (1−exp(−αL))/α is the effective length derived from the linear loss α and the actual length L, and dg_{R dB}/d P ₀ is the derivative of the Raman gain g_R in dB with respect to the peak power of the pump P ₀. The value obtained from published data [26] is f_R = 0.2 which, within the error of ~ 10% is equal to that in silica. Thus, since both f_R and R(T_s) are roughly the same in both materials, the Raman effect in the chalcogenide is of equal strength to that in silica, if normalized with respect to the total material nonlinearity.

A result of the assumption of equal duration fundamental soliton pulses is that the difference in the nonlinear lengths is determined solely by the GVD. Thus, the higher intrinsic nonlinearity in the chalcogenide facilitates only reduced power requirements through P ₀ = ∣β ₂∣/(T ² _sγ), where P ₀ is the soliton peak power. That is, the chalcogenide, purely by virtue of its higher nonlinearity, does not necessarily furnish any reduction in the propagation length required for a given frequency shift compared to that in silica. Such an advantage may only be obtained through dispersion engineering, but this may be performed in both silica and chalcogenide. Furthermore, TOD becomes important in practice as increasing GVD experienced by a redshifting soliton leads to pulse broadening in the temporal domain and an associated reduction in the peak power, which retards nonlinear effects such as the SSFS. Thus, the importance of considering the dispersion in comparing two specific architectures is paramount. In order to concentrate on the nonlinear properties of the chalcogenide, we consider only a single structure in Section 3, an As₂S₃ waveguide, and use it to demonstrate the qualitative behaviour of Raman solitons in this material.

3. Modelling

As₂S₃ ridge waveguides have proven to be a versatile geometry from the dispersion engineering perspective [27], and we employ such a structure for the purpose of performing detailed simulations of the SSFS in this material. We chose a waveguide with a cross-sectional area of 1.2 µm², a height : width ratio of 1 : 2, and an etch depth of half the height [27]. The light-guiding properties of this structure were calculated using the commercially available COMSOL software package, and are displayed in Fig. 2. The GVD for the TM (E_y) polarized mode is anomalous (β ₂ = −122 ps²/km) at 1550 nm and its effective mode area is 0.9 µm². The nonlinear index is set at 3×10⁻¹⁸ m²/W with a Raman fraction of f_R = 0.2 and, since it is the dispersive and nonlinear properties of the waveguide that are of interest here, loss is neglected.

The spectral trajectory of an individual soliton pulse may be obtained through integration of Eq. (4), which assumes that the TOD may be treated as a perturbation; i.e. ε = ∣β ₃∣/(∣β ₂∣T_s) ≪ 1. Of particular interest here are solitons with durations of 10’s of femtoseconds, which are readily obtained in supercontinuum generation from the fission of longer pulses which approximate a higher order soliton [12]. For a fundamental soliton injected into the ridge waveguide at 1550 nm with T_s = 20 fs, we have ε = 0.5, implying a substantial interaction with resonant, normally dispersive waves. Thus, to study a more realistic situation, we have simulated the propagation of an initially sech-shaped pulse with a pulse width at FWHM of 100 fs and peak power of 88 W (soliton order N = 3.9) along a 20 cm ridge waveguide by numerically solving Eq. (1) (we remind the reader that there is no assumption regarding the pulse width in this equation). The numerical integration, performed with the split–step Fourier method, involved a spectral domain of 300 THz discretized into 2¹⁵ bins and a local error threshold of 10⁻⁴ [28]. A spectrogram of the final pulse is shown in Fig. 3, along with the spectral evolution with distance along the waveguide. Fission of the initial pulse occurs within ~ 1 cm and a feature on the long wavelength edge of the spectrum is observed to undergo a continuous redshift throughout propagation. In the final pulse this feature is a temporally and spectrally localized pulse in the anomalous dispersion regime, redshifted ~ 450 nm with respect to the pump wavelength, with a short train of blue-shifted, normally dispersive waves [29]. Thus it bears all the hallmarks of a Raman soliton. This clearly shows how the SSFS acts in As₂S₃ as it does in silica to extend the red edge of a SC [12, 30].

 

Fig. 2. Light guiding properties of As₂S₃ rib waveguide TM mode with predominantly vertically (y) polarized electric field: (a) normalised vertical electric field (E_y) profile, (b) GVD, and (c) effective mode area. The base of the chalcogenide layer in (a) lies at y = 0.

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4. Discussion and Conclusion

We have studied the SSFS in an As₂S₃ ridge waveguide and considered the effects of the material properties upon this process, using silica as a benchmark. The quantitative change in the shift rate due to differences in the Raman spectra of the two materials was found to be less than 25 % for soliton pulse durations less than 500 fs, while the respective fractional contributions of the Raman response to the total nonlinearity in each material was estimated to be equivalent. Thus, taking into account the higher nonlinearity of the chalcogenide by operating at lower peak powers, the understanding and performance of the SSFS in silica should be directly transferrable to As₂S₃. However, the most uncertain aspect of this analysis is the value of f_R in As₂S₃ and, as the SSFS rate essentially scales linearly with this quantity, it should be the subject of a dedicated measurement.

 

Fig. 3. (a) Spectrogram of final pulse and (b) spectral evolution of initial sech pulse with N = 3.9 propagated along a 20 cm ridge waveguide. The colour dB scale is the same for both subfigures.

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Detailed simulation of pulse propagation in this architecture has unambiguously shown a Raman soliton emerging from the fission of an initial pulse and redshifting ~ 450 nm from the pump wavelength over a distance of 20 cm. This confirms that the SSFS can act to extend the red edge of supercontinua in this material, and supports the viability of SSFS–based functions, such as tunable mid–IR sources and all–optical ADC, in As₂S₃ waveguides.

In the intra-pulse scattering process which underlies the SSFS, fundamental solitons with a pulse duration ≳ 100 fs interact predominantly with the low frequency region of the Raman spectrum below ~ 6 THz. The data reproduced in Fig. 1(a) show the normalized scattering spectra for silica and As₂S₃ to be almost identical in this range, which leads to the near equivalence of the spectral response functions shown in Fig. 1(b) at these pulse durations. The root of this correspondence in the spectra is unlikely to be a coincidence, and may be due to similarities in the van der Waals forces which determine the motion of the SiO₄ tetrahedra and the AsS₃ pyramids in the two amorphous structures [31]. In contrast, the SSFS is expected to occur very differently in crystalline materials in which the Raman spectrum consists of discrete peaks. The lack of low–frequency Raman–active phonon modes in such materials implies that the SSFS does not operate on sufficiently long pulses.