1. Introduction

The mid-IR wavelength range has become increasingly important in various areas such as spectroscopy [1, 2], LADAR [3] and astronomical instrumentation [4]. A supercontinuum in the mid-IR as a broadband and coherent source is of great importance [5–11], especially considering the limited availability of widely tunable lasers in the mid-IR: e.g., cascaded quantum lasers and inter-band lasers usually have discrete wavelengths and a small tuning range. There have been great advances in fiber-based supercontinuum generation (SCG), which can cover a wide wavelength range from 1.4 to 16 µm [5, 6, 9]. However, most of optical materials for CMOS-compatible photonic integration have their transparency windows below 6 µm [10, 11]. In this sense, Germanium has unique advantages [7, 8, 11–13]: it has a wide transparency window up to 14 µm, possesses a refractive index as high as 4.3 for tight mode confinement, and a high nonlinear Kerr index n₂, and more importantly, it is CMOS compatible. Ge-on-Si waveguide platforms have been studied for mid-IR SCG pumped at 3-5 µm [7, 8], where two-photon and three-photon absorptions are detrimental. On the other hand, due to the high refractive index, Ge waveguides typically have strong chromatic dispersion, which makes the newly generated frequencies in nonlinear processes quickly walk off relative to the pump pulse and stops nonlinear interactions between widely apart spectral components. Some waveguides have been proposed for dispersion tailoring in the near-IR, and among these structures, the slot waveguides can generate flattened dispersion with four zero-dispersion wavelengths (ZDWs) and are suitable for SCG [14, 15]. However, the low-index materials for a slot typically have a transparency window cut-off below 6 µm, and this would limit the use of the dispersion flattening technique [14, 15] in the mid-IR.

Here, we propose a dispersion-engineered Ge-on-Si waveguide with a low dispersion from 3 to 11 µm (~2 octaves), without using a slot. It is shown by simulation that an ultrawide supercontinuum can be generated with good coherence from 2 to 12 µm (~2.6 octaves), using a sub-ps pump pulse. We compare the optimum SCG with pulsewidths from 120 to 700 fs and show that using a long pulse one can still obtain similar spectral coverage. This means that the dispersion-flattened waveguides can be used for SCG under quite flexible conditions.

2. Device design and modeling

We design a dispersion-flattened Ge-on-Si waveguide for mid-IR SCG, as shown in Fig. 1. Silicon-on-Insulator (SOI) is a standard waveguide platform used for on-chip photonics in the near-IR. However, the buried SiO₂ layer has high absorption beyond 3.8 µm [16]. One solution is to form suspended structures by selectively removing the buried oxide [17]. The waveguide consists of a Ge strip and suspended Si membrane that is partially etched, and SiO₂ is removed to reduce propagation loss. Waveguide dimensions are: Ge height Hu = 2200 nm, Si ridge height Hl = 600 nm, Si slab thickness = 545 nm, SiO₂ substrate thickness = 3000 nm, Si width W = 920 nm, and sidewall angle = 87°. We use the quasi-TM mode.

 

Fig. 1 Dispersion-flattened Ge-on-Si waveguide, with SiO₂ substrate partially removed, for supercontinuum generation in the mid-IR.

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As shown in Fig. 2(a), the waveguide exhibits a saddle-shaped dispersion with low and flat anomalous dispersion between the two ZDWs. Dispersion, D, varies from 51 to 9.45 ps/(nm·km) between 3.37 and 10.43 µm. Different from the previous slot-based one [14, 15], this dispersion engineering technique does not require the formation of a slot, because most of low-index materials used for a slot become absorptive in the mid-IR. Here, Ge has a very high index contrast with air, which would cause strong waveguide dispersion. Thus, we add a material that has a refractive index relatively close to Ge so that the guided mode can extend as wavelength increases. This is why we have Si below Ge. The unique wavelength-dependent mode extension can be controlled by varying Si dimension, and then we obtain a desirable effective index as a function of wavelength and thus a flat and low dispersion [18]. Figure 3 shows the influence of waveguide dimensions (around the values given above) on dispersion properties. The parameter changes cause a dispersion variation of several ps/nm/km per nm, and the dispersion slope is slightly modified. We also randomly change the waveguide width, height and etch depth within a range of ± 1% and ± 2.5%. The dispersion’s sensitivity to the dimension variations is relatively small, and the slope is quite stable. Figure 2(a) also shows that the nonlinear coefficient γ decreases with wavelength significantly.

 

Fig. 2 (a) Flat and low dispersion is produced between 3.37 and 10.43 μm, a wavelength range around two octaves. The nonlinear coefficient γ decreases quickly with the wavelength. (b) The substrate leakage increases significantly with wavelength.

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Fig. 3 Dispersion profiles with structural parameters changed around the optimum values.

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We set the scattering loss of the waveguide to be 6 dB/cm. Although Si has increasingly higher absorption at a long wavelength beyond 8.5 µm [11], we use it as the substrate, so its influence is reduced greatly, because most of light is confined in Ge. Thus, the main source of propagation loss is the scattering loss and substrate leakage at long wavelengths, as seen in Fig. 2(b), which is 11 dB/cm at 12 µm.

To investigate SCG in the MIR range, we build the nonlinear pulse propagation model based on the generalized nonlinear Schrodinger equation [11], taking self-phase modulation (SPM), self-steepening, wavelength-dependent loss, and all-order dispersion into account:

We define A = A(z, t) as the complex amplitude of the pump pulse. In Eq. (1), α is the total linear propagation loss, including material absorption, substrate leakage and scattering loss. We consider all-order linear dispersion (each dispersion coefficient β_m is associated with the Taylor series expansion of the propagation constant β(ω) at the carrier frequency). The effective mode area A_eff is included in the nth-order coefficient ${\gamma }_n$of nonlinearity [11], which is defined as ${\gamma }_n={\omega }_0{\partial }^n\left[\gamma \left(\omega \right)/\omega \right]/\partial {\omega }^n$, where ${\omega }_0$ is the angular frequency of the carrier.

In our nonlinear simulations, we set the time window length to be 5 ps (i.e., frequency resolution Δf = 200 GHz), and the sub-ps pump pulse, centered at 6.57 µm, has a full width at half-maximum (FWHM) of 700 fs and a peak power of 400 W. Input pulse energy is 317 pJ. The pump wavelength is close to the middle of the low-dispersion band, where the γ value is 10.96 /W/m and D = 9.7 ps/(nm·km), which result in a soliton number of N = 56, and the higher-order modes are mostly cut-off around the pump wavelength, so the pump is coupled to the fundamental mode. Owing to the Ge bandgap wavelength at 1.6 μm and long-wavelength pumping in this work, one can ignore two-photon, three-photon and even four-photon absorption effects [7], so no nonlinear loss is included in the model.

3. Results and discussion

Figure 4(a) illustrates the supercontinuum evolution along the waveguide, and we note that the pulse spectrum experiences first slightly and then suddenly broadening at around 5 mm, which is explained as a positive feedback process in SPM and significant self-steepening effect. First, SPM generates red and blue frequency components on the leading and trailing edges of the pulse, respectively. Given the flat and low anomalous dispersion, the newly generated frequency components slightly walk toward each other, which results in pulse compression in the time domain, enabling the sharp edges of the pulse and the high peak power, which in turn enhances the SPM effect. More importantly, due to the wideband low dispersion, the pulse experiences an accumulated self-steepening effect [19], which becomes more obvious from 4 to 5 mm. The spectral coherence of the generated supercontinuum is examined over distance by calculating the complex degree of first-order coherence |g₁₂⁽¹⁾|(λ) [8] from 100 simulations with random quantum noise. As shown in Fig. 4(b), at 0.5 mm, the coherence is relatively low, because the input spectrum is not yet broadened much. As the pulse propagates, we obtain better coherence over a wider band. |g₁₂⁽¹⁾|(λ) is unity over the bandwidth of interest at 5.35 mm, which corresponds to good coherence.

 

Fig. 4 (a) Spectral evolution versus propagation distance. The spectrum of the pulse becomes significantly broadened, and the broadest and flattest spectrum are obtained with dispersive wave in the short wavelength at a propagation distance of about 5.35 mm. (b) Degree of coherence in SCG changes over distance, and good coherence is found at 5.35 mm.

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At a propagation distance of about 5.35 mm, the spectrum is greatly broadened, covering a wavelength range from 3.7 to 9.24 µm at the −30 dB level, which is about 1.35 octaves. This is a quite significant spectral broadening considering the pulse is sub-ps, owing to broadband low dispersion. Furthermore, the spectral power varies slightly from 2.5 to 4 µm and 8 to 12 µm, the flat spectrum can be used in the spectrum slice filtering technique and acquire Nyquist pulse trains [20]. The dispersive wave appears at about 2 µm, and the red frequency components can only extend to about 14 µm due to the high propagation loss as seen in Fig. 2(b). Figure 5 is the spectrogram evolution of the pump pulse. Self-steepening, associated with intensity-dependent group velocity, causes pulse peak delay and a sharp trailing edge of the optical pulse. Then, SPM induces very blue-shifted spectral components at the steepened trailing edge, so the main part of the spectrum extends more to the high-frequency end, as we can see in Fig. 5 at 5.35 mm. This causes the optical shock and the sudden spectral broadening. The optical shock has a temporal feature of 22 fs, close to a single cycle time. At a longer propagation distance, the dispersion is accumulated further, frequency components begin walk off from each other and the pulse breaks into multiple parts, and the spectral broadening is almost stopped.

 

Fig. 5 Spectrogram evolution of the pulse. Strong self-steepening and spectral broadening occur in the pulse propagation from 0 to 5.35 mm. We can see an optical shock on the trailing edge of the pulse as a result of the self-steepening effect and low dispersion, which has a temporal feature of 22 fs, almost a single cycle time at 6.57 µm.

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The peak power of input pulses plays an important role in SCG. Next, we change the peak power from 300 W to 600 W, when keeping the pulse width and wavelength to be 700 fs and 6.57 µm, respectively. The optimum spectra under these conditions are illustrated in Fig. 6(a). With a higher peak power, pulse compression and SCG occur earlier, while there is a small change in the spectral width and flatness of the optimum spectra. SPM is stronger with a higher peak power, so new frequency components are generated more quickly, which facilitate the dispersive wave generation. As the pulse peak power increases to 600 W, the required waveguide length can be reduced to 3.8 mm to obtain the optimum spectrum.

 

Fig. 6 (a) The best spectra generated with different peak powers of the input pulses, which illustrate that the broadest spectra can be obtained at a shorter distance with an increasing pump power. (b) The best spectra generated with different widths of the input pulses, which show that using a long pulse one can still obtain similar spectral coverage.

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The speed of the spectrum broadening and the flatness of the generated supercontinuum are associated with input pulse width. We change the FWHM of input pulse from 120 fs to 700 fs, when keeping the pulse wavelength at 6.57 µm, and find out according pulse peak power to enable similar SCG. As we can see from Fig. 6(b), an input pulse with a short pulse width can generate a slightly better supercontinuum in terms of spectrum flatness. However, we obtain quite similar spectral coverage, with two dispersive waves located at around 2 and 13 µm, respectively. It is noted that in the previous work [5, 9] an ultrashort pulse of 100 or 150 fs is employed for SCG in the mid-IR, which is only a few optical cycles at the pump wavelength. For on-chip applications, an input pulse from a femtosecond source will likely be spreading in time during the coupling from free space or fiber to a chip, and also there is often a tapered waveguide section before the nonlinear waveguide to increase coupling efficiency, which may cause pulse spreading as well. Thus, successful SCG using sub-ps pump pulses in Ge-on-Si waveguides shows a promising approach to on-chip SCG in the mid-IR. One can re-design the waveguide for SCG centered at 3-4 μm wavelengths, where fs sources typically have higher pulse energy, while three-photon absorption has to be considered there [7].

4. Conclusion

We have presented a novel Ge-on-Si waveguide that achieves dispersion flattening without using a slot structure, which enables a low-absorption wavelength range to be extended to 11 μm. Benefiting from the high nonlinearity and flattened dispersion, one can produce an ultrawide supercontinuum of 2.6 octaves, with even a sub-ps pump pulse. The spectrum covers a wavelength range from 2 to 12 μm, highly desirable for absorption spectroscopy and many other mid-IR applications.