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Abstract
We report simulations of pulse evolution along a 5 cm long silicon germanium (SiGe) photonic waveguide. Femtosecond laser pulses of duration 210 fs and wavelength of 4.7 µm close to the first zero dispersion wavelength (ZDW) of SiGe situated at 4.6 µm were used in pumping the SiGe photonic waveguide. Simulations were made by solving the generalised nonlinear Schrödinger equation for the femtosecond pump pulse evolution along the length of SiGe photonic waveguide using the fourth order Runge-Kutta in the interaction picture method. The simulated supercontinum spans the 2.5–8.5 µm spectral region comprising of molecular signatures for most hazardous and greenhouse gases making it attractive for environmental monitoring applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Mid infrared (MIR) radiation is crucial in life-science [1], industry, environment and security [2]. Throughout this manuscript, it is considered that near-infrared (NIR) stretches from 0.7 to 2.5 µm, MIR from 2.5 to 15 µm and far infra-red (FIR) from 0.015 to 1 mm. Owing to the interaction with light, most gases are accountable for absorption at particular wavelengths in MIR spectral region, enabling detection and quantification of low concentration elements [3]. Most of the atmospheric gas molecules have their absorption signatures in the 3 to 8 µm MIR spectral window [4]. On-chip broad and coherent light sources in the MIR are thus important for compact gas sensing devices. Hereof, supercontinuum generation (SCG) can efficiently produce coherent broadband light in a single and compact device that can be used in gas absorption spectroscopy [5]. SCG in silicon (Si) based photonic waveguides is one of the most promising technique for generating coherent broadband light in the MIR region that can be used in absorption spectroscopy [6]. Besides their compactness and having strong nonlinear property, silicon based photonics are of great interest as they are based on a mature and large-scale fabrication technology offered by complementary metal-oxide semiconductor (CMOS) technology [7].
The first SCG experiment on a silicon-based chip covering the MIR spectral region up to $5.5\,\mathrm{\mu}\textrm{m}$ was attained in 2015 [8]. The silicon sapphire waveguide was pumped at 3.75 µm with $320\,\textrm{fs}$ pulses and supercontinuum (SC) spanning from 1.9 to 5.5 µm was obtained. This SC was restricted by absorption in sapphire [9]. Apart from having low nonlinear losses, crystalline Si also possesses a wide transparent window (see Table 1). However, when implemented as a photonic waveguide, transparency for crystalline Si on insulator is limited to 3.7 µm as a result of presence of absorption in silica substrate [10]. Recently, silicon germanium-on-silicon (SiGe) photonic waveguide has been regarded as a crucial photonic material as it possesses the capacity to function well ayont the silica and sapphire absorption limit of 3.7 µm and 5.5 µm, respectively [11]. The transparency window for different semiconducting materials is illustrated in Table 1 and is described as the band where the absorption attenuation is at most $2\,\textrm{dB}/\textrm{cm}$ [12].
Having intriguing properties like strong nonlinearity, large refractive index and wide MIR transparent window (see Table 1), germanium (Ge) have gained astounding interest for MIR nonlinear applications in recent years [13]. Although Ge platforms have been envisioned for MIR nonlinear photonics, they possess high lattice mismatch between Ge and Si leading to unwanted dislocations during operation [12,14]. As such, SiGe is commonly used to combine the strong nonlinear properties of Ge (with ${\sim} 40\,\mathrm{\%}$ Ge content in an alloy) for improved control of waveguide nonlinearity and reduces waveguide – substrate lattice mismatch [15]. Thus, SiGe on Si platform takes advantage of the low lattice mismatch between SiGe and Si to overcome treading dislocations that exist in Ge on Si platform [7]. In this manuscript we investigate numerically SCG in SiGe photonic waveguide exploring the effect of peak power, pulse duration, pump wavelength and pulse chirp on SC spectra.
2. Numerical model
Simulations of SCG in $\textrm{SiGe}$ photonic waveguide were carried out by solving the generalised nonlinear Schrödinger equation (GNLSE) using the fourth order Runge-Kutta in the interaction picture (RK4IP) method. The GNLSE used in our simulations is given in Eq. (1) [16–18]:
The first and the second terms on the right hand side are the linear and nonlinear effects, respectively. $A({z,T} )$ is the slowly varying envelope of the electric field, $T\; = \; t\; - {\beta _1}z$ is retarded time ($t$ is physical time), and $\alpha $ is the linear waveguide loss. ${\beta _m}$ and $\gamma $ are the ${m^{th}}$ order dispersion and nonlinear coefficients evaluated at the carrier (centre) frequency for pump laser ${\omega _0}$, respectively. ${\tau _{sh}} \approx 1/{\omega _0}$ is the envelope self-steepening timescale [17] and $R({T^{\prime}} )$ is the nonlinear response function which is responsible for the electronic and nuclear contribution [18] and takes the form:
where ${h_R}(T )$ is the Raman response function which accounts for the vibration of silicon molecules within the waveguide as light propagates [19] while ${f_R}$ is the fractional contribution of the delayed Raman response which is found to be 0.043 from the normalization condition of $\mathop \smallint \limits_0^\infty {h_R}(T )dT = 1$ [20,21]. Munaweera and Wijewardena analytically described the Raman response function as governed by Eq. (3) [22].Equation (1) describes the variation with distance in the z direction, of the electric field amplitude as a function of linear attenuation, dispersion and nonlinearity which is responsible for nonlinear effects in the photonic waveguide. SCG involves the interaction between nonlinear and linear effects that takes place during the propagation of laser pulses in a photonic waveguide [23]. Regardless of being a linear effect, dispersion ($D(\omega )$) is of paramount importance in influencing the behaviour of nonlinear interactions in an optical waveguide [17]. The relationship between $D(\omega )$ and the group velocity dispersion (GVD) parameter ${\beta _2}(\omega )$ is given by Eq. (4) [24]:
The dispersion profile of a strip silicon germanium on silicon (SiGe) photonic waveguide used in the simulations was calculated in Lumerical MODE as described by Sinobad et al. [21] is shown in Fig. 1. The Lumerical MODE uses finite different eigenmode (FDE) that determines the optical properties of materials to compute waveguide dispersion. In our design, we assumed wavelength dependent material refractive index ($n(\lambda )$) for Si and SiGe at room temperature using Sellmeier equations. For Si, the refractive index took the form [11]:
Fig. 1. Dispersion profile for silicon germanium waveguide (calculated in Lumerical MODE). The cross section of SiGe photonic waveguide is provided in the inset.
For $\textrm{S}{\textrm{i}_{1 - x}}\textrm{G}{\textrm{e}_x}$ (or simply SiGe), the refractive index was approximated as [25]:
Here x is the Ge composition in an alloy and ${n_{\textrm{Si}}}(\lambda )$ is the refractive index of Si defined in Eq. (6). We modelled our waveguide with 40% of Ge by composition i.e. $x = 0.4$ giving $\textrm{S}{\textrm{i}_{0.6}}\textrm{G}{\textrm{e}_{0.4}}$.
The dispersion coefficients at a pump wavelength (${\lambda _0}$) of 4.7 µm obtained from the dispersion curve in Fig. 1 are presented in Table 2. Inclusion of high order dispersion, up to 12th order (see Table 2), in the SCG model increases accuracy in reproducing the SiGe mode dispersion in the 3–10 µm spectral range [23,24].
Table 2. Dispersion parameters used in simulation.
$\textrm{SiGe}$ waveguide of nonlinearity $\gamma = 0.36\,\textrm{W}^{-1}\textrm{m}^{ - 1}$ is pumped at 4.7 µm wavelength near the first ZDW situated at 4.6 µm (refer to Fig. 1). The $\textrm{SiGe}$ waveguide has a loss $\alpha = 0.38\,\mathrm{dB}/\textrm{cm}$ and a fractional contribution of delayed Raman response, ${f_R} = 0.043$ (taken to be equal to that of crystalline silicon as ${h_R}(T )$ for Ge is extremely weak hence insignificant ${f_R}$) [21].
In our simulations, we considered the Gaussian pulse, $A({0,T} )$ of the form:
Here ${T_0}$, ${P_0}$, T and C are the pulse duration, pulse peak power, instantaneous time along the waveguide and chirp parameter, respectively. An up-chirp (i.e. a phenomenon when frequency increases from the leading to the trailing edge) takes place when $C > 0$ while down-chirp occurs when $C < 0$ [18]. Thus, the sign of the chirp depends on the sign of C. Femtosecond pulses exhibit chirp which can decrease or increase dispersion of a photonic waveguide during light propagation depending on the sign of the input chirp [26]. ${T_0}$ and ${P_0}$ are related to the dispersion length (${L_D}$) and nonlinear length (${L_{NL}}$), respectively according to Eq. (9) [27].
The pulse parameters used in the simulations are given in Table 3. We used the peak power and pump wavelength of 2 kW and 4.7 µm since there are commercial systems that are currently available on the market e.g. tunable optical parametric amplifier (MIR OPA – fs) [29,31].
Table 3. Pulse parameters used in simulation.
The characteristic lengths, ${L_D}$ and ${L_{NL}}$ for our SiGe photonic waveguide were computed from Eq. (9) and found to be 54 cm and 0.14 cm, respectively giving $N = 19$.
3. Numerical simulation results and discussions
A pulse of parameters presented in Table 3 was simulated for its spectral and temporal evolution along a 5 cm long, $6.0\,\times 4.2\,\mathrm{\mu}\textrm{m}^2$ cross-section $\textrm{SiGe}$ photonic waveguide whose GVD parameters are presented in Table 2. The spectral and temporal pulse evolution along a silicon germanium waveguide is shown in Fig. 2.
Fig. 2. Simulated (a) input and output pulse (line plot), (b) spectral pulse evolution and (c) temporal pulse evolution along the SiGe photonic waveguide. The dispersive waves ($\textrm{DW}$) radiation emanates at longer wavelength.
The spectral evolution profiles in Fig. 2(b) show that the initial broadening mechanisms is due to pulse compression (during the first 1 cm of the photonic waveguide) set off by the higher-order soliton formation. The time domain pulse compression in Fig. 2(c) is caused by self-phase modulation (SPM). Soliton fission (SF) and DW radiation followed the initial pulse compression mechanism (as illustrated by Fig. 2(b)). SF roughly takes place at 1.5 cm of the photonic waveguide and is succeeded by DW generation. Figures 2(a & b) shows that the DW radiation manifests at long wavelengths ($\mathrm{\lambda } \approx 10\,\mathrm{\mu}\textrm{m}$) while a series of SF spans the 2.5 to $8.5\,\mathrm{\mu}\textrm{m}$ spectral range. From Figs. 2(b & c), the stimulated Raman scattering (SRS) and four wave mixing (FWM) define the final spectrum shape. Upon propagating a distance of at least 3 cm, the pulse components spread and do not intersect temporally (see Fig. 2(c)) so SCG ceases and the SC spectrum is solely influenced by linear losses.
Motivated by the need to fully understand the broadening structure, the pulse evolution slices at different propagating lengths in temporal and spectral domain were simulated and results plotted in Fig. 3. From the temporal plots (Fig. 3(a)), it is discernible that a single pulse is conserved during the entire propagation. Temporal red shifted pulse broadening become more pronounced after 1.5 cm propagation. The blue shifted broadening edges are much steeper in slices within 2 cm due to self-steepening (SS) effect. In spectral domain (Fig. 3(b)), the pulse bandwidth (BW) increase linearly with propagation distance within 2 cm (as illustrated Fig. 3(c)). The bandwidth increase become negligible after propagating 3.75 cm. This infers that SCG is completed within a propagation length of 3.75 cm (indicated by the dotted red line in Fig. 3(c)) as can be seen by the flatness of the graph.
Fig. 3. Pulse evolution slices at different propagation distances in (a) temporal and (b) spectral domain. (c) spectral pulse bandwidth at different propagation distances. The bandwidth is computed at full width half maximum (FWHM) of the spectral intensity profile by the second moment of the spectral intensity profile. The dotted red line indicated the completion of pulse broadening along the SiGe photonic waveguide as can be seen by the flatness of the graph.
3.1 Peak power and pulse duration effects on SCG
The pump wavelength, optical waveguide length and pulse duration parameters were kept constant to investigate the effects of pulse peak power on generated SC. The spectral pulse evolution for different peak powers after 5 cm propagation along the SiGe photonic waveguide at 4.7 µm pump wavelength and pulse duration of 210 fs is illustrated in Fig. 4(a). It is clear that the spectral bandwidth of generated SC increases with peak power. For instance, the $0.7\,\mathrm{\mu}\textrm{m}$ and 7.2 µm bandwidths are generated with pulse of peak powers of 0.1 kW and 10 kW, respectively. The increase in pulse bandwidth originates from the Kerr nonlinearity effects influenced by SPM and SS [29]. The pulse bandwidths for peak powers ranging from 0.1 kW and 10 kW are plotted in Fig. 4(c). Although the bandwidths increases with peak powers, it can be noted that the change is very small for higher than 4.55 kW peak powers (Fig. 4(c), indicated by the dotted red line) indicating the pulse peak power saturation of the $\textrm{SiGe}$ photonic waveguide.
Fig. 4. Pulse evolution at different (a) peak powers (b) pulse durations. Pulse bandwidth at different (c) peak powers and (d) pulse duration of a 5 cm long SiGe photonic waveguide. The dotted red line indicates the power saturation threshold for the SiGe photonic waveguide (as the graph become flat).
The effect of pulse duration on SC generated after 5 cm propagation through a $\textrm{SiGe}$ photonic waveguide is illustrated in Fig. 4(b). The pulse peak power and pump wavelength are kept constant at 2 kW and 4.7 µm, respectively. The pulse bandwidth change with an increase in pulse duration is extremely small. The bandwidth remains almost constant at 6 µm with variation of pulse duration from 100 fs to 300 fs. The bandwidth changes by 1 nm for a 0.65% pulse duration fluctuation (see Fig. 4(d)). The relation between the peak power (${P_o}$) and pulse duration at full width half maximum (${T_{FWHM}}$) is $E = {P_o}.{T_{FWHM}}$ with $\textrm{E}$ being pulse energy [32]. Considering that ${P_o}$ is kept constant, significant spectral broadening like the one in Fig. 4(a) cannot be observed. The long pulse duration has spectrum structure that oscillates more. Further, the oscillation dips around the pump wavelength are more significant.
3.2 Peak power and pulse duration fluctuation effects on SC intensity
SC sources can be regarded as high intensity broadband light sources [33]. Most SC applications like spectroscopy, optical coherence tomography and frequency metrology require stable light intensity for improved accuracy [34]. Thus, besides having broader spectrum, SC sources must also possess stable output intensities for aforementioned applications. Coupled peak power and pulse duration fluctuations effects on output intensity were characterised. Maximum intensities, ${I_o}$ at different coupled peak powers (from Fig. 4(a)) and pulse durations (from Fig. 4(b)) were investigated and plotted in Fig. 5. As it can be seen in Fig. 5, fluctuations in coupled peak power and pulse duration leads to SC intensity instability. In both cases, intensity increases with increase in coupled peak power or pulse duration. On average, 0.15% peak power fluctuation causes the SC intensity to change by 1 dB (see Fig. 5(a)). Intensity increases linearly with an increase in pulse duration i.e. intensity changes by 1 dB for a 3.3% pulse duration fluctuation (see Fig. 5(b)). Although intensity instability is very small for changes in both peak power and pulse duration, it is significant enough to cause challenges in spectroscopy, optical coherence tomography and frequency metrology measurements. Thus, pulses from the pump laser must maintain operational peak power and duration for applications that demands stable SC intensity.
Fig. 5. Effects of (a) coupled peak power and (b) pulse duration fluctuations on SC spectra intensity of a 5 cm SiGe photonic waveguide pumped at 4.7 µm. The peak power and pulse duration plots are fitted with Boltzmann and linear functions, respectively.
3.3 Pump wavelength and waveguide nonlinear coefficient effects on SCG
The pump wavelength was varied from 3.5 to 5 µm with the other pulse and photonic waveguide parameters kept constant. The spectral pulse evolution for different pump wavelengths is presented in Fig. 6(a). Spectral broadening is wavelength dependent as less spectral broadening is observed for 3.5 and 4 µm pump wavelength compared to significant broadening for 4.7 and 5 µm. The bandwidth variation with pump wavelength is also plotted in Fig. 6(c). The bandwidth is significant for pump wavelengths greater than 4.6 µm. The 3.5 and 4 µm pump wavelengths are less than the first ZDW hence the $\textrm{SiGe}$ photonic waveguide experiences normal GVD which leads to frequency up-chirp of the pulses [24]. In contrast, the $\textrm{SiGe}$ photonic waveguide when pumped with 4.7 and 5 µm pulses experiences anomalous GVD as it is pumped with wavelength greater than the first ZDW [29]. Broad SC is produced in the anomalous GVD due to the balance between GVD and SPM which leads to frequency down-chirp [30]. Thus, anomalous GVD regime is necessary for the generation of significant SC.
Fig. 6. Pulse evolution at different (a) pump wavelengths (b) waveguide nonlinearity. Pulse bandwidth at different (c) pump wavelengths and (d) waveguide nonlinearity for a 5 cm SiGe photonic waveguide.
$\textrm{SiGe}$ photonic waveguide nonlinear coefficient ($\gamma $) was varied while keeping the pulse parameters constant at $210\,\mathrm{fs}$ pulse duration, 2 kW peak power and 4.7 µm pump wavelength. The spectral plots obtained are illustrated in Fig. 6(b). It can be observed from Fig. 6(b) that the spectrum broadens more when $\gamma $ is increased. The bandwidth increases linearly with waveguide nonlinearity (see Fig. 6(d)). The increased broadening is attributed to the fact that $\gamma $ varies inversely with ${L_{NL}}$. That is, increasing $\gamma $ reduces ${L_{NL}}$ hence increasing the soliton number N. Higher N is associated with broader SC but degrades spectra coherence [17,32]. Thus, nonlinear effects like SPM, FWM and SRS become significant when a photonic waveguide of large nonlinear coefficient is used.
3.4 Chirped pulse propagation effects on SCG
Pulses emitted from laser sources are usually chirped and the study of chirp effects is of paramount importance [35]. Chirped pulses (with different chirp parameters) of 2 kW peak power and 210 fs duration were pumped at 4.7 µm wavelength close the first ZDW in the anomalous regime of the SiGe photonic waveguide. The evolution of chirped pulses after propagating 5 cm along a SiGe photonic waveguide was simulated and its impact on pulse shape and spectrum investigated (see Fig. 7).
Fig. 7. Pulse evolution for pulse (a) shapes and (b) spectra for different chirp parameters for a 5 cm long SiGe photonic waveguide.
Asymmetric variation (from the un-chirped pulse shape) of the output pulse shapes (Fig. 7(a)) and pulse spectrum (Fig. 7(b)) is evident. An input positive chirp ($C\; = \; 2$ and $C = 5$) increases the SC bandwidth (see Fig. 7(b)). Enhanced SC bandwidth can be attributed to the fact that the input positive chirp adds to the chirp produced by SPM close the pulse centre, thus increasing the initial pulse compression required for soliton fission [36]. At 5 cm along the SiGe photonic waveguide, the anomalous GVD (i.e. ${\beta _2} ={-} 2.93 \times {10^{ - 26}}$) has an influence on the SC bandwidth such that the pump pulse portion is compressed due to the opposite sign of chirp induced by anomalous GVD and SPM. The variation of negative chirped pulses (with $C\; = \; - 2$ and $C ={-} 5$) in the anomalous dispersion regime is limited on the leading edge of the pulse shape (see Fig. 7(a)) due to the same sign with an anomalous GVD. In contrast, the oscillations appear on both leading and lagging edges though more pronounced on the lagging edges of the positive chirp.
In general, the spectra broaden with an increase in chirp (see Fig. 8). Thus, the sign of the chirp parameter is very crucial in pulse propagation characteristics along the photonic waveguide as positive chirped pulses in anomalous dispersion regime enhances the bandwidth of the generated SC while negative chirp reduces the bandwidth. Also, an optimal chirp exists that can maximize the bandwidth of the generated SC. In our case, maximum bandwidth is achieved at an input chirp $C = 4$ (indicated by the flatness of the Boltzmann fit at the red dotted line of Fig. 8).
Fig. 8. Pulse bandwidth for chirped pulses with different chirp parameters for a 5 cm long SiGe photonic waveguide.
4. Conclusion
$\textrm{SiGe}$ photonic waveguide must be pumped close to the first ZDW with an appropriate peak power and pulse duration to produce a significant SC (due to significant soliton related dynamics as higher order dispersion terms become dominant) which spans a spectral region of 2.5–8.5 µm and is suitable for gas absorption spectroscopy. Further, we managed to show that higher nonlinear coefficient and pulse peak power produce broader SC as nonlinear effects that promotes SCG (e.g. SPM, FWM, SRS, DW and SF) become more significant. Although higher peak power produce broader SC, it also produces strong tails close to the pulse edges and poor flatness of spectral intensity profile. Also, Pulses with shorter duration gives smooth spectral intensity profile for the same peak power. Thus, significant SC can be realized when a photonic waveguide with high nonlinear coefficient (like SiGe) is pumped with pulses of short duration and moderate peak power. Also, variation in coupled peak power and pulse duration leads to pulse intensity instability of the generated SC which is detrimental to intensity sensitive applications like frequency metrology, spectroscopy and optical coherence tomography. Further, SPM effect plays a vital role in SC broadening processes as an input positive chirp can increase the SC bandwidth via a modified pulse compression phase needed by soliton fission.
Funding
African Laser Centre, Council for Scientific and Industrial Research (HLHA23X task ALC –R019).
Acknowledgments
The Authors acknowledges the National University of Science and Technology Research Board and Laser Research Institute for providing research facilities.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Code 1, Ref. [37].
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