Passive mode-locking and terahertz frequency comb generation in resonant-tunneling-diode oscillator

Optical frequency combs in the terahertz frequency range are long-awaited frequency standards for spectroscopy of molecules and high-speed wireless communications. However, a terahertz frequency comb based on a low-cost, energy-efficient, and room-temperature-operating device remains unavailable especially in the frequency range of 0.1 to 3 THz. In this paper, we show that the resonant-tunneling-diode (RTD) oscillator can be passively mode-locked by optical feedback and generate a terahertz frequency comb. The standard deviation of the spacing between the comb lines, i.e., the repetition frequency, is reduced to less than 420 mHz by applying external bias modulation. A simulation model successfully reproduces the mode-locking behavior by including the nonlinear capacitance of RTD and multiple optical feedback. Since the mode-locked RTD oscillator is a simple semiconductor device that operates at room temperature and covers the frequency range of 0.1 to 2 THz (potentially up to 3 THz), it can be used as a frequency standard for future terahertz sensing and wireless communications.

In this note, we show an analytical model to explain the optical feedback effect on the oscillation frequency, i.e., the modification of the frequency-voltage curve, appearances of multiple optical modes, and dependence of the mode spacing to the feedback amplitude.
Here, we use the model derived in the reference [2]. They modeled the RTD oscillator under optical feedback with the following circuit equation: Here, %&' is the capacitance of the resonator, ) is the radiation conductance, and )'" is the absolute value of the negative differential conductance. is the reflectivity of the return light, including the coupling efficiency. 2 ! is the phase delay of the return light, and is a constant phase shift of the return light. The condition for steady oscillation in equation (1) yields the following equation: " ≅ ! " − ? ! " sin( " + ) = ! " − sin( " + ) .
Here, is the oscillation frequency under the influence of the feedback, ! is the free-running oscillation frequency, and is the quality factor of the LCR resonator. " is the delay time, and is a feedback parameter. Equation (2) represents the change of the oscillation frequency due to optical feedback. Interestingly, the same equation as (2) can be obtained for a semiconductor laser under weak optical feedback [1].
Equation (2) gives the relationship between the oscillation frequency = /2 and the free- Figure S2 is the numerical plot for various feedback parameters .  Figure. S2a, the mode spacing is different from the free-spectral range of a Fabry-Perot cavity, i.e., 1/ " (= 5 GHz in Figure. S2). As the feedback amplitude increases more ( = 10 and 100), the number of the modes for a certain ! increases, and the mode spacing approaches 1/ " (Figure. S2b). The strong-field limit corresponds to the Fabry-Perot cavity.  Figure S3 shows the measured temporal waveform plotted for a longer span than that of Figure 2a.

Note 4 Detail of the fitting (1) Fitting in short spans
The exact value of the time span (164 ns) was determined so that the number of the contained data points is an integer power of 2, for the convenience in the analysis based on Fast Fourier Transform (FFT) algorithm such as the Hilbert transform performed in Supplementary Note 5. The fitting in the divided spans can be explicitly expressed as follows: Here, !,* ,-, ./0,* , !,* , and &,* are the fitting parameter in each short span, corresponding to those in equation (3) in the main text. & is derived from the area of the comb lines in the magnified spectra.
A constant-coefficient = 0.95 was multiplied to all the amplitudes to correct the difference of the amplitude between the spectrum measurement and the waveform measurement. Since the modes have different frequencies, the relation between the initial phases &,* depends on the choice of !,* . To express the phase relationship uniquely, we choose the time origin !,* as the timing when 1,* = 2,* stands, as described in the next section. As a fitting condition, we put a constraint that 1,* = 2,* . We also show how we choose the !,* in which we can uniquely represent it.
Let us consider equation (4), and the temporal evolution of the modal phases described as In the analysis of the mode-locking, we are interested in the relative phase between the modes. If we take the mode with index = 4 as the reference, the relative phase can be defined as We note that there is an arbitrariness in the choice of the time origin !,* . For example, we can choose the time = !,* + 0.5 ./0 as a new time origin !,* * . In that case, the relative phase can be written as with the initial relative phases of (Δ (,* * , Δ 1,* * , Δ 2,* * , Δ 3,* * , Δ 4,* * ) = (0, , 0,0,0). Hence, the phase relationship has countless number of the equivalent representations.
To represent the phase relationship uniquely, we choose the time where 1,* = 2,* stands as the origin. This condition identifies the time origin in the period of ./0 uniquely because it is only one time that the phases of the adjacent modes are equal.
We note that !,* is necessary because noise induces unexpected timing shifts on the oscillator.
-π 0 π Relative phase ∆ψ n (rad)  Figure S5a shows a typical heterodyne waveform (blue trace) and its envelope (gray trace). Here, we do not discuss amplitude modulation in detail due to the possible inaccuracy of the amplitude measurement. Figure S5b shows the instantaneous frequency of the typical heterodyne waveform and the corresponding terahertz frequency derived from the Hilbert transform. It has as large frequency modulation as the bandwidth of the comb spectrum in Figure 1c.  Figure S6 shows the change of the spectrum when the feedback from the mirror was decreased from the passively mode-locked state. The passive mode-locked state is represented by the equidistant peaks observed when the feedback amplitude is close to its maximum value in our setup. It disappeared when the feedback amplitude was decreased to less than 93 %.   Here, we discuss the conditions of the optical feedback and the modulation power to obtain the hybrid mode-locked state. We show that the passive mode-locking mechanism is necessary to obtain a broadband comb spectrum. Figures S7-2a, b, and c show the spectra measured in different optical-feedback conditions. Figure   S7-2a shows the spectra measured without feedback from the mirror. Figure S7-2b shows the spectra measured when the feedback with the feedback from the mirror, but its position was not suitable for the passive mode-locking. In these cases, a comb spectrum was not obtained for any modulation power.
When the feedback condition is suitable for the passive mode-locking, a small bias-voltage modulation was efficient to stabilize the comb. Figure S7-2c shows the spectra under various modulation conditions. Figure S7-2c (i) shows the passively mode-locked spectrum observed without the modulation. Figure S7-2c (ii) and (iii) show the spectra under the bias-voltage modulation of -40 dBm and -30 dBm, respectively. The modulation frequency was 1.0932 GHz, which was the same as the harmonic-comb spacing of the passively mode-locked state.
When the modulation amplitude is further increased, the condition to obtain the mode-locked state becomes complicated. The amplitudes of the modes did not change largely, and the hybrid modelocking was achieved. Figure S7-2c (iv) shows the spectrum under a modulation with a power of -20 dBm and a frequency of 1.0932 GHz. In this case, a comb spectrum was not obtained. Figure S7-2c (v) shows the spectrum under a modulation with a power of -20 dBm and a frequency of 1.1023 GHz.
In this case, a comb spectrum was obtained. It is a remained task to reveal the range of modulation frequency and amplitude where we can obtain the harmonic mode-locking.
Finally, we note that we investigated only a limited part of the vast parameter space, such as modulation frequencies, amplitude, and feedback parameters. Investigating such parameter space would be an important future task to understand the hybrid mode-locking mechanism.
For ( ), we used the following function: Equation (9) is our original function based on the previous studies showing that the nonlinear capacitance of the RTD is proportional to ,67 ( ) [5] [6] [7]. Equation (9) is obtained by taking the derivative of equation (8) and picking the major terms that contribute to the nonlinearity around the inflection point of ,67 ( ).
The parameters of equations (8) and (9) are shown in Table S8-1. Figure S8 The passive elements of the circuit were = 21.2 pH and @A%" = 3.3 mS. = is a white noise source. Shot noise has the root-mean-square current fluctuations of 162 µA and the single-sided bandwidth of 10 THz. We did not include the shot-noise enhancement of the RTD [8].
The optical feedback was modeled as the feedback current Here, is the reflectivity which includes the coupling efficiency, and " is the time delay. When we include several return lights from several surfaces, the contributions from these return lights were included as a summation: The simulation was performed using LTspice.  Here, we show how we determined ,67 ( ) used for the simulation. ;-,/:0 ⁄ is the differential conductance of the high-frequency circuit. It should be zero at the edge of the bias-voltage range where the oscillation takes place. The obtained ;-,/:0 ( ) is shown in Figure S8-2a.

(3) Determination of feedback delays and reflectivity
In the simulation, we included three return lights shown in Table S8-2. In this section, we describe how we determined the feedback delay and reflectivity of the return lights included in our model. Table S8-2 Parameters of the return lights included in the simulation. Here, # is the index to identify the return light. The optical length is = " /2, where " is the delay time, and is the speed of the light in the vacuum. The feedback parameter is = ? ! " Q as introduced in equation (3).
In the calculation of , we assumed ! =300 GHz and =8. As explained with a simple model in Supplementary Note 5, the frequency jump is originated from the optical feedback from a surface nearby the oscillator. Figure S8-4a shows frequency-voltage curves of three oscillators of the same design but have some variation in actual properties. We found that the number of the frequency jump is different for three oscillators. These characteristics cannot be explained with a single return light because it causes equidistant frequency jumps as expressed by equation (2). Figure S8-4b shows that a simulation reproduced them by assuming two reflection surfaces with the parameter ( " , ) of (19.7 ps, 10 #(.! ) and (178 ps, 10 #1.! ). We expect that those reflections are caused at the edge of the horn antenna structure [9] and at an optical mount of the oscillator. The variation of the three oscillators was reproduced by assuming a variation of the inductance .
Here, we did not correct the detailed discrepancy in the voltage value because we intend to reproduce the behavior of the oscillator qualitatively. We also ignored the increase of the simulated frequency at the high-voltage limit. The mode spacing, shown as ∆ in Figure S8-5, was 270 MHz.
As shown in Figure S8-5b, the simulation reproduced the frequency change by the mirror position and the hysteresis. Here, the optical length was swept around 500 mm. The mode spacing changed depending on the reflectivity . When the reflectivity was = 10 #1.(3 , the simulation well reproduced the mode spacing of 270 MHz. The feedback parameter is = 19, which is large enough to cause the multiple longitudinal modes.

(3-3) Confirmation of parameters
Finally, we confirmed that the three return lights estimated above reproduce the observed behavior.
The simulated frequency-voltage curve shown in Figure 4a well reproduced the frequency-voltage curve of Figure 3a. Hence, we utilized the parameters of the return lights discussed above.
We note that slight return lights from the lens shown in Figure 1a and the detector were neglected in the simulation. In the experiment, we confirmed that these return lights had a small effect on the oscillator, for example, a change of the linewidth in the CW oscillation state and a slight shift of the frequency-jump voltage. However, the amplitude of these return lights was estimated to be so small that we did not take these return lights in our simulation.

Note 9
Terahertz waveform in simulation (9-1) Temporal waveform corresponding to Fig. 4b Figure S9-1 Simulated terahertz waveform that corresponds to the harmonic comb spectrum in Figure   4b. The simulation includes noise with a standard deviation of 1/10 of the shot noise. a, Waveform for 20 ns (filling the region inside the envelope). b, Instantaneous frequency obtained from the terahertz waveform by the Hilbert transform. c, Magnified view of the terahertz waveform. Figure S9-1a shows the simulated terahertz waveform that corresponds to the harmonic frequency comb in Figure 4b. The oscillation frequency is so fast that the waveform is filling the envelope.
Clearly, it does not show a significant amplitude modulation. A magnified view of the terahertz waveform is shown in Figure S9-1c. Figure S9-1b is the instantaneous frequency obtained by the Hilbert transform of the terahertz waveform in the following steps: (i) The simulated waveform has harmonic components such as second harmonics around 600 GHz, third harmonics around 900 GHz, and the other higher harmonics.
To obtain the modulation of the fundamental frequency, these harmonic components and lowfrequency noise were removed with a band-pass filter of 250 GHz to 350 GHz. (ii) We applied the modulated with a period of approximately 1.2 ns, which is the inverse of the comb spacing of 835 MHz in Figure 4b. The frequency modulation was as large as the spectral bandwidth.   Figure S9-2c. Figure S9-2b is the instantaneous frequency obtained in the same procedure as Figure S9-1b. The frequency modulation is larger than that of Figure   S9-1b, reflecting the broader spectrum.

Note 10 Broadband frequency combs
Fig. S10 Broadband comb spectra simulated for the conditions shown in Table S10.

Table S10
Feedback conditions for Figure S10a, b, and c. The reflectivities at the reflection surfaces and their delays are shown. The spacing of the comb lines are also shown. S10b 10 #$ 10 #( 0 4.61 S10c 10 #$ 0 10 #( 0.296 Figure S10 shows comb spectra simulated for various feedback conditions, and Table S10 shows the conditions. The distances of the reflection surfaces are the same as those in Table S7-2. Depending on the feedback conditions, we obtained various broadband comb spectra.