Optical pump assisted broadband terahertz frequency comb

Composed of numerous highly coherent and equally spaced laser longitudinal modes, optical frequency combs^1–6 have many unique features, such as ultrahigh frequency stability and ultralow phase noise, which make them potential candidates for large capacity radio-frequency communication,⁷ sensor imaging,⁸ and spectral metrology.^9–11 In particular, since many molecules’ characteristic absorption spectral lines lie in the THz spectral range,¹² highly accurate material detection has come into reality by detecting the absorption of comb lines through dual-comb spectroscopy technology.¹³ There are several actively and positively mode-locked methods to produce THz frequency combs in THz quantum cascade lasers (QCLs), including the microwave modulating the laser current^14,15 and a graphene-coupled saturable absorber.¹⁶ Compared with the optical frequency combs traditionally generated by a mode-locked laser,¹⁷ THz QCLs are capable of generating THz frequency combs through the four-wave-mixing (FWM) nonlinear effect,¹⁸ which shows constant instantaneous power instead of a short pulse in the time domain and similar frequency modulated signals, due to very fast gain recovery¹⁹ in QCLs. Unfortunately, it is very difficult to generate many high-power modes only by using an electrical pump in THz QCLs due to orders of magnitude higher dispersion. Although dispersion compensation methods^20,21 have been proposed, it is almost impossible to reduce this dispersion to the level of mid-infrared QCLs. Furthermore, only the correctly compensated THz QCLs could generate combs effectively.

The optically pumped nonlinear parametric process originates from the highest third-order nonlinear susceptibility in optical waveguides, such as high nonlinear optical fibers^22–26 and silicon nanophotonic waveguides.^27–31 The active region of QCLs intrinsically shows much higher nonlinearity. A very large value of the third-order nonlinearity susceptibility has been measured experimentally, and obvious degenerate nonlinear parametric process has also been observed.³² Based on the high nonlinearity in QCLs and dispersion engineering, this parametric process enables amplification, and at the same time, the comb lines can also be symmetrically duplicated on the other side of the pump when phase-matching condition is achieved. Therefore, it could provide a wide frequency comb bandwidth in mid-infrared QCLs through phase-matching controlling.³³ However, such a phase-matching method may be difficult to be applied to THz QCLs because of the complex dispersion profile of the laser active region. Optical parametric process with high conversion efficiency usually requires the pump frequency to be near the zero-dispersion frequency, to meet the momentum conservation condition in the nonlinear process of FWM. Nevertheless, with the assistance of waveguide engineering, for example, a chirped corrugated mirror is integrated into the QCL waveguide to make the longer-wavelength wave travel further,²⁰ the position of the zero-dispersion frequency in the frequency domain of a THz QCL can be controlled artificially,³⁴ and then the broadband frequency comb can be effectively excited by optical pumping.

To achieve broadband frequency combs in THz QCLs, the optically pumped parametric process may play an interesting and significant role. In this work, we fully utilize the time evolution simulations of the THz QCLs comb based on a Maxwell–Bloch model to represent the dynamics of the comb modes and an optically assisted pump whose frequency locates near the zero-dispersion frequency of the waveguide to investigate the possibility of broadening the bandwidth of the THz-QCLs comb.

The schematic diagram of an optical pump assisted broadband frequency comb is illustrated in Fig. 1. In order to model the optical pump assisted THz frequency comb, we employ the optical Bloch equations to describe the dynamics of a THz QCLs frequency comb in the model of a two-level system, including the population interaction as well as coherences between the upper and ground states. Then, the Maxwell equation is utilized to establish the electromagnetic interaction between the frequency comb and a high-power single optical pump in the active laser cavity.

To model the THz QCLs frequency comb, Khurgin et al.³⁵ and Villares and Faist³⁶ theoretically proposed and developed self-frequency-modulated Maxwell–Bloch formalism by taking the assumption that the ground state population is negligible and three order perturbative approximation. Following similar methods and symbols, the final coupled equations that describe the time evolution of the complex amplitude of the slowly varying envelope of the optical fields of all comb modes and the optical pump are

where the complex envelope A_n of mode n is normalized to the saturation field $Asat=ℏ/u21τcohτ2$⁠, τ_coh represents the coherent time of the laser transition (which is the relaxation time between the intersubband transition states), τ₂ is the lifetime of the upper state, and u₂₁ is the dipole matrix element between the upper and lower radiation states. The first term on the right-hand side of the equation describes the gain and dispersion, and the second term represents the FWM nonlinear process. Furthermore, the parametric process also generates idler modes through the summation term. Note that we do not consider the pump consumption here.

The subscripts in the second term of Eq. (1) satisfy the relationship n + l = m + k, which represents the relationship of four-wave mixing terms, and a dimensionless time T is introduced to normalize the real time to the photon lifetime by T = t/2t_c. The gain for the mode n is written by

where g₀ is the gain of the intersubband transition, which is normalized to the lasing threshold, $g0=ω0τcNu212Δρp/2ε0n02ℏ$⁠. The round-trip time of the l_c length cavity is τ_rt = 2l_cn_g/c, where n₀ is the refractive index, n_g is the group index, and c is the velocity of speed. The dispersion term can be written by³⁵ 

where ω_n is the frequency of the nth mode with ω_n = ω₂₁ + nω (where ω₂₁ is the central resonant frequency of the laser transition and the mode spacing ω = 2π/τ_rt) and the empty cavity frequency ω_nc = ω₂₁ + Δω (where the frequency shift is $Δω=nπc(2ng2lc−nπc2β2)/2ng3lc2$ and the group velocity dispersion parameter is β₂ = d²k/dω²). The FWM coefficients B_kl and C_kl are given by

Meanwhile, the intermodal overlap factor κ_klmn is as follows:

Since there is one high power optical pump, the same FWM mechanism can induce high degenerate parametric amplification and wavelength conversion simultaneously, where optical gain and idler modes occur. Except for the optical pump, the initial conditions of Eq. (1) are set by the same small field amplitude and random phases that represent typical THz QCLs. In the simulations, in order to reduce the simulation time and the computational load, several simplifications are adopted, such as the limited number of modes, and the frequencies of the optical pump and zero-dispersion are artificially set exactly on the comb modes. All the parameters and typical values of the THz QCLs are listed in Table I.

First, we show the dynamics of the THz frequency combs with and without the optical pump. Note that the gain center of the THz QCLs is always set at 3.5 THz and the zero-dispersion frequency is offset by 30 modes (about 3.977 THz), which is always set at mode 0 in all the following simulations. The simulation results of output spectral power density, phase, and instantaneous power evolution are depicted in Fig. 2. As shown in Fig. 2(a), the power density spectrum of the THz frequency comb without the optical pump injected is composed of only less than 8 mode lines with very low power. This is because the gain center is far from the zero-dispersion frequency; thus, the dispersion is very large and no appropriate dispersion can be obtained by dispersion compensation as well. When an optical pump with the power of 0.3A_sat and frequency at mode −3 is employed, the power density spectrum is shown in Fig. 2(b). The optical pump not only amplifies the original frequency comb but also replicates the frequency comb, and as a result, the number of the frequency comb modes is increased to more than 50. Figure 2(c) shows the phase evolution of the modes −29, −30, and −31 with and without the optical pump, respectively. The slope of the phase evolution is due to the dispersion effect, and the magnitude of the slope represents the degree that the real mode frequency deviates from the equally spaced frequency without dispersion.³⁴ This frequency offset can be calculated directly by the following equation: $dω=d(argAn)/dt$⁠. Compared with the self-referencing phase-locked process without the optical pump that the values of frequency offset of mode −29, −30, and −31 are 0.2815, 0.3741, and 0.5656 GHz, respectively, the slope of the phase is suppressed to 0.2533, 0.2824, and 0.3350 GHz, which indicates that the modes are locked better. Meanwhile, the time of phase stabilization is slightly increased by introducing the single optical pump, from less than 5 ns to more than 10 ns. In the time domain, the instantaneous power eventually tends to be constant when the steady state is reached, and with the optical pump injected, the time to reach constant power output is significantly reduced compared with the case where no optical pump is employed, as shown in Fig. 2(d).

In order to study and optimize the bandwidth of the THz frequency comb, we first studied the impact of the pump frequency. As seen from Figs. 3(a) and 3(c), within a certain frequency range, the pump frequency close to the zero-dispersion frequency can increase the bandwidth of the THz frequency comb where zero-dispersion frequency is set at mode 0, note that we have set the field of the optical pump to be 0.4A_sat. When the pump frequency is detuned from the zero-dispersion frequency by repetition frequency ω, the number of comb lines increases to more than 80, which is one order of magnitude more than that without pump injection [Fig. 2(a)]. In Fig. 3(d), we show the phase evolution of mode −30 when the frequency offset of the optical pump relative to the zero-dispersion frequency is ω, 3ω, and 5ω, respectively. The closer the pump frequency is to the zero-dispersion frequency, the smaller the slope of the phase, indicating that the phase mismatch is suppressed, so that more optical power is converted; thus, the bandwidth of the frequency comb is effectively increased. In addition, from the time evolution of the instantaneous power shown in the inset of Fig. 3(d), it can be seen that the pump frequency has no effect on the evolution time when the THz QCLs comb reaches steady state, but only has a slight effect on the power of the comb.

Then, we investigate the effect of pump power on the instantaneous power and optical power spectral density of the THz frequency comb, as shown in Fig. 4. It is obvious that the higher the optical pump power, the stronger the power output of the frequency comb, but the bandwidth of the frequency comb is not expanded very much. It should be noted, however, that the power of the comb lines within the bandwidth is enhanced by a higher optical pump. In addition, by raising the injected optical pump power, we can reduce the time for the frequency comb to enter the steady state, as depicted in Fig. 4(d). Eventually, the constant power output of the THz frequency combs is achieved.

In conclusion, based on the numerical solution of the Maxwell–Bloch model and the coupled wave theory in the slowly varying envelope and a three order perturbative approximation, we have presented the dynamics of broadband THz frequency combs by introducing an assisted optical pump, taking into account the dispersion. FWM enables both the self-reference phase locking and nonlinear parametric process. We have shown that, with the optical pump injected, the number of the comb lines is more than 80 by setting the frequency of the optical pump close to the zero-dispersion frequency, resulting in a much broader frequency comb. This optical pump nonlinear parametric process could be very promising. On the one hand, a single pump parametric scheme can be simply equipped. On the other hand, not only could the newly generated idler modes during the parametric amplification process double the number of the optical frequency comb, but also carry the phase information conjugate with the original comb modes. Moreover, very low phase noise in the nonlinear parametric process can keep the frequency comb in good coherence, which is our future research. All these can enhance the detection ability of the QCLs THz frequency comb and, therefore, lead to a growing number of spectroscopic applications.

This work was funded by the National Natural Science Foundation of China (NSAF) (Grant No. U1730246).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.