Injection Locking and Noise Reduction of Resonant Tunneling Diode Terahertz Oscillator

We studied the injection-locking properties of a resonant-tunneling-diode terahertz oscillator in the small-signal injection regime with a frequency-stabilized continuous THz wave. The linewidth of the emission spectrum dramatically decreased to less than 120 mHz (HWHM) from 4.4 MHz in the free running state as a result of the injection locking. We experimentally determined the amplitude of injection voltage at the antenna caused by the injected THz wave. The locking range was proportional to the injection amplitude and consistent with Adler's model. As increasing the injection amplitude, we observed decrease of the noise component in the power spectrum, which manifests the free-running state, and alternative increase of the injection-locked component. The noise component and the injection-locked component had the same power at the threshold injection amplitude as small as $5\times10^{-4}$ of the oscillation amplitude. This threshold behavior can be qualitatively explained by Maffezzoni's model of noise reduction in general limit-cycle oscillators.


I. INTRODUCTION
Compact and stable terahertz (THz) sources are highly desired for future THz imaging and THz wireless telecommunication systems. THz oscillators with resonant tunneling diodes (RTDs) are good candidates for compact THz sources that can operate at room temperature. [1][2][3][4] Many advances have led to the fundamental oscillation frequency now being up to 1.98 THz. 5 The output power has also been increased; it is up to 0.4 mW at 530 -590 GHz for a single oscillator 6 and 0.73 mW at 1 THz for a large-scale array. 7 One of the major concerns in regard to the RTD THz oscillator is its large linewidth in the free-running state. It is typically 10 MHz for an oscillator operating around several hundred GHz, 8,9 where the statistical property and the origin of the noise have yet to be determined.
Applications such as communications and RADAR require a narrow linewidth and frequency tunability.
They also require the oscillator to synchronize with a frequency reference in order to perform homodyne or heterodyne detection.
The most commonly used methods to stabilize the frequency of the oscillators are injection locking [10][11][12][13][14] and phase-locked loop (PLL), 15 which have complementary properties. 16 PLL has an advantage in controlling the long-time frequency drift, but it is difficult to suppress the high-frequency noise faster than the loop-propagation delay time. Injection locking can often achieve the suppression of the high-frequency noise in a reasonable injection condition. However, if the free-running frequency drifts far away from the injection frequency, it is difficult to keep the injection locking. For the RTD THz oscillators, an intensive study on spectral narrowing by PLL is already reported. 17 The injection locking of the RTD THz oscillator has been discussed in the context of sensitive THz-wave detection. 18,19 While these studies including subharmonic injection locking 20  In this study, we investigated the injectionlocking properties of the RTD THz oscillator in the small-signal injection regime without an optical feedback effect. We used isolators for THz waves 21 to reduce optical feedback from the detection system. In the visible frequency range, such devices are well established and used in injection-locking experiments. [22][23][24] In the THz frequency range, such devices are still under development. 21 To observe the noise reduction behavior by injection locking, we stabilized continuous THz sources to have a linewidth below 120 mHz and utilized them for precise spectroscopy and injection locking.
We performed measurements and analysis to obtain the spectra in which the free-running frequency of the RTD THz oscillator exactly equals to the injection frequency. This approach enabled us to obtain noise spectra which can be analyzed with a simple theory. 25 We found that the injection locking caused

II. HETERODYNE THZ DETECTION AND INJECTION LOCKING SYSTEMS
We have constructed a heterodyne spectralmeasurement system and an injection-locking system operating in the THz frequency region.    We prepared three semiconductor lasers whose frequencies are stabilized to the independent frequency-comb lines separated by integer multiples of 100 MHz. We set their frequencies ( " , # , $ ) so that their differences would each be the desired THz frequencies, i.e., the LO frequency %& = # − " , and the injection frequency *+, = $ − # . The linewidths of the THz waves were less than 120 mHz. In the locking range experiments (section IV), we had to sweep the injection frequency with an interval of less than 100 MHz. Only in that case did we stabilize the laser frequencies using a wavelength meter (Ångstrom WS7/30 IR, HighFinesse GmbH); this resulted in a linewidth of a few hundred kHz. We describe details of the narrow-band THz wave generation in Supplementary Section 1.

B. Specifications of the heterodyne detection system
In the heterodyne detection system, the THz LO signal was generated by UTC-PD1 in Fig.   1. The LO signal and the emission of the RTD THz oscillator were combined using a wiregrid polarizer (WG). The typical power of the LO signal reflected by the WG was 10 µW.
The mixed signal was detected by a Fermilevel managed barrier diode (FMBD), 27

C. Injection-locking system with isolators
Another continuous THz wave was generated with UTC-PD2 (see Figure 1) and used for injection locking. The injection power was changed using a pair of WGs. In the weaker region we also changed the laser intensity incident on UTC-PD2. This is because the extinction ratio of the WGs was not high enough to reduce the amplitude of injection field precisely down to 10 /$ . The

IV. LOCKING RANGE
We swept the injection frequency and measured the emission power spectra of the RTD THz oscillator (Fig. 3). The blue trace in the top shows the free-running spectra with a vertical dashed line indicating the position of To distinguish the injection-locked state from the injection-pulled state, we used the same criterion as a previous study 24 : if the height of the peak at the injection frequency is more than 20 dB larger than that of the sidebands, it is an injection-locked state. When the injection frequency becomes less than %,1*+ shown as a red dotted line, sidebands appear (group C); the oscillator is no more injection locked but injection pulled again. The spacing of the sidebands increases as the injection frequency depart from the free-running frequency. The frequency range from %,1*+ to %,123 is the locking range.
We measured the locking range for various amplitudes of injection field. (1) where 9 is the free-running frequency, is the Q-factor of the resonator, and <=> is the amplitude of oscillation voltage at the antenna. The factor = *+, <=> ⁄ is commonly called the injection ratio. Equation

A. Noise reduction threshold
It is important to explore the minimum injection strength to injection-lock the oscillator. We set the injection frequency at the center of the free-running spectrum and changed the amplitude of the injection field. We call the broad peak the noise component.

SUPPLEMENTARY MATERIAL
See the supplementary material for further details.

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

Section 1 THz wave generation with UTC-PD
Here, we describe the details of narrow-band continuous THz wave generation by differential frequency photomixing with uni-traveling-carrier photodiodes (UTC-PDs). In this method, we injected outputs from two CW lasers to a UTC-PD. The frequency difference of the lasers was set to the desired THz frequency. We stabilized the frequencies of the CW lasers using a wavelength meter (WLM) or an optical frequency comb. Figure S1 shows the feedback configuration using a WLM (Ångstrom WS7/30 IR, HighFinesse GmbH). Merit of this method was that the center THz frequency was continuously tunable. We used three frequency-tunable laser diodes (LDs) operating at 1.  (∆ hij ) was 10 /T Hz in 1 second. Hence, the expected linewidth of the 300 GHz signal is estimated to be ∆ × ∆ hij~3 00 mHz, which is comparable to the measured value (120 mHz). Figure S2 (b) shows the feedback configuration. In this setup, we controlled all the LDs.
We characterized the spectral resolution of the heterodyne measurement system by using the LO signal and the input signal stabilized with a frequency comb. Figure S3 shows the setup for the characterization. The heterodyne detection part is the same as the one shown in Fig.1 (a). Figure S4 shows the measured power spectrum. The half-width at half maximum (HWHM) of the heterodyne spectrum is 120 mHz, which is limited by the best RBW of 240 mHz of the spectrum analyzer operated in real-time spectrum analyzer (RTSA) mode. We can see some minor sidebands 40 dB less than the carrier signal at several MHz from the center, which results from the spectra of the feedback-controlled laser. Due to the very low signal level, those sidebands do not matter in this study.

Section 2 Structure and properties of THz isolators
In this section, we describe the structure and the properties of the THz isolators. We have constructed THz isolators, which has been originally proposed by Shalaby. 1 Figure S5 is the photograph of an isolator. The isolator consists of an anisotropic Sr-ferrite magnet of approximately 2 mm-thick (Himeji Denshi Co., Ltd.) and two wire-grid polarizers (WG). The thickness of the magnet is determined to set the Faraday rotation angle to approximately 45 degrees. It was not precisely 45 degrees, and we adjusted the angle of the WGs to minimize the backward transmission. The magnet and the WGs are tilted so that the THz wave reflected at the surfaces does not return to the signal source. The isolator was constructed on a breadboard to change its position without changing the configuration inside the isolator. Specifications of two isolators at 322 GHz are summarized in Table ST1.  The small forward transmission comes from an absorption in the Sr-ferrite magnets.

Investigation of magnets without absorption loss and improvement of the forward transmission
should be an important challenge in future. Figure S6 shows schematics of the evaluation setup of the forward transmission and the backward transmission. We generated a THz wave using a UTC-PD, and measured its power using an FMBD. The transmission of isolator 2 was measured. Isolator 1 was used to avoid the formation of a standing THz wave in Figure S6  where th2+=,r is the forward transmission power, th2+=,u is the backward transmission power, and *+ is the incident power.
In Fig. S7, we show the details of the experimental setup of Fig. 1, especially focusing the polarization of the THz waves. The angles of the WGs are set to obtain a large signal in the heterodyne detection and high injection power. The reflection of the RTD emission from UTC-PD2 and the FMBD was eliminated by the isolators. The round-trip attenuation to the reflected field amplitude was v r u = 6 × 10 /$ . Several factors, such as the reflectivity at UTC-PD2 and the FMBD, would also have reduced the amplitude of the feedback field by more than one order. From these results, we expect that the amplitude ratio of the voltage at the antenna caused by the optical feedback and the oscillation voltage is less than 5 × 10 /T , which is the threshold for the injection locking. This enabled us to measure the intrinsic properties of the injection locking in the small-signal injection regime.

Section 3 Optical feedback effect on the RTD THz oscillator
Here we describe the optical feedback effect on the RTD THz oscillator. When we remove the isolator for the detection path, a reflected THz wave of a small amplitude from the FMBD goes back to the RTD THz oscillator. We compared the emission power spectrum with and without the isolator for the detection path. We scanned the position of the RTD THz oscillator along the z-direction (relative distance: ) to change the time delay of the feedback as shown in Fig. S8. Figure S9 (a) shows the emission power spectrum measured with moving the RTD THz oscillator without the isolator (a). One can see that the oscillation spectrum is largely affected by the distance . When we use the isolator (Fig. S9 (b)), the oscillation frequency is almost independent of the position of the RTD oscillator. The small change in the oscillation frequency is due to the frequency fluctuation of the RTD oscillator, as shown in the control experiment ( Fig. S9 (c)). One possible explanation for the spectral change in Fig. S9

Section 4 Experimental setup for absolute output power measurement
We measured the emission power of the RTD THz oscillator using the setup shown in Fig.   S10. The emission was modulated in a square-wave on-off shape with an optical chopper. The modulation frequency was 11 Hz. The detector was a calibrated pyroelectric detector (THz 20, SLT Sensor & Lasertechnik GmbH). The detected signal was measured with a lock-in amplifier. In this measurement, we did not use an isolator because the standing THz wave that significantly affect the precision of the power measurement was not formed.

Section 5 Fluctuation of the free-running frequency and a post-selection method
The center frequency of the RTD THz oscillator in the free-running state fluctuated in time.
In Supplementary Section 5 (1), we show the basic properties of the fluctuation. In Supplementary Section 5 (2), we show how much the fluctuation affects the emission power spectra. In Supplementary Section 5 (3), we describe the post-selection analysis for characterize the spectra without the effect of the fluctuation. We used this method to take data shown in Fig.   2 (a), Fig. 5, and Fig. 6.

(1) Properties of free-running-frequency fluctuation
We measured a spectrogram, which is a series of emission power spectrum over time using the spectrum analyzer (MXA 9020B, Keysight Technologies Inc.) in RTSA mode. Figure S11 (a) shows a bare spectrogram in 1 second measured in the free-running case. The time between each trace is 115 µs. We can see fluctuation in the instantaneous center frequency { ( ). We derive the center frequency as the spectral centroid. Figure S11 Figure S11 (c) shows its power spectrum. We can see that the frequency noise has 1⁄ spectrum. This indicates that a parameter which affects the free-running frequency fluctuates with 1⁄ spectrum. A possible candidate for such a parameter is the capacitance of the RTD. The origin of the 1⁄ fluctuation cannot be determined from this measurement. (2) Impact of the fluctuation on the spectra Figure S12 shows the spectrogram measured with the injection signal of several amplitudes.
The horizontal axis is the frequency offset from the injection frequency. Figure S12 (a) shows the free-running case with fluctuation. Figure S12 (b) shows the spectrogram measured with very weak injection. We can see a narrow peak, i.e., injection-locked component at the injection frequency and a broad peak, i.e., noise component fluctuating in time. Figure S12 (c) shows the spectrogram taken with the maximum-amplitude injection in our setup. We can see only a narrow peak at the center with no fluctuating component. This means that the RTD THz oscillator is perfectly injection locked.
In Figure S13, we show spectra at several timings extracted from Fig. S12 to show the spectral shape clearly. As we described above, the noise component in Fig. S13 (a) and (b) fluctuates in time. We emphasize that the spectral shapes in Fig. S13 (b) are different. This is because the relation between the injection frequency and the free-running frequency is different for each trace. Hence, we need to choose spectra in which these two frequencies coincide in order to discuss the noise spectra with a simple model. If the injection signal is sufficiently strong, we observed perfectly injection-locked spectra as shown in Fig. S13 (c).

(3) Post-selection method
First, we explain the post-selection analysis in the case of Fig. 5 (a). Figure S14 (a) shows a bare spectrogram (10,000 traces) measured when a signal is injected close to the free-running frequency with a normalized injection amplitude of = 1.6 × 10 /# . The horizontal axis is the frequency offset to the injection frequency. To pick-up the spectra whose free-running frequency is the same as the injection frequency, we performed the following selection procedure: First, we calculated the spectral centroid at each moment, as shown in Fig. S14 (b).
Then, we sorted the spectra by the spectral centroid, as shown in Fig. S15 (a). Figure S15 (b) shows the corresponding spectral centroid. Next, we picked-up spectra with the same spectral centroid as the injection frequency (50 traces from 10,000 traces) and averaged them to obtain the spectra shown in Fig. 5 (a). We note that this method can be extended to pick-up a spectrum with an arbitrary offset-frequency between the injection-locked component and the noise component.
In the case of Fig. 2 (a), we measured a spectrogram of the free-running RTD oscillator in the same way and picked-up the spectra with a common center frequency.

Section 6 Measurement of injection amplitude and injection ratio
In this section, we describe how to determine the injection amplitude • *+, and injection ratio = • *+, • <=> ⁄ . • *+, is the injection amplitude at the antenna caused by the injection electric field. • <=> is the oscillation amplitude of the RTD oscillator itself measured at the antenna. In this section, tilde on • *+, and • <=> are used to clarify that these symbols represent the amplitude of the ac voltage at THz frequency.

(1) Measurement of the injection voltage ( • )
We performed a square-law detection of the injection THz wave using the RTD THz oscillator 2,3 with a setup shown in Fig. S16. The injection THz wave was generated with a UTC-PD. The laser incident on the UTC-PD was modulated square-wave on-off shape at 9.7 MHz.
The bias voltage of the RTD THz oscillator was set to 406 mV, where no oscillation took place. Figure S17 (a) shows the current-voltage curve † ‡ ( ) in the voltage range of 300-500 mV.
We can utilize its nonlinearity for THz-wave detection. When a signal of • *+, cos is injected, time-averaged current changes from that without the injection. The difference is where † ‡ (c) ( ) represents the n-th order derivative of † ‡ ( ).
To obtain the derivative coefficients, we fitted † ‡ ( ) around the bias voltage of 406 mV.
where ∆ = − 9 , and ‹d is the bias voltage of 406 mV. The fitting result is shown as the dashed curve in Fig. S17 (b). Table ST2 shows the fitting parameters and standard deviations as well as ∆ c (5 mV) in Eq. (S6-3) for = 2 and 4. We can see that ∆ # (5 mV) ≫ ∆ T (5 mV).
Hence, the higher order terms in Eq. (S6-3) is small in the case of • *+, ≤ 5 mV, which holds true in our experiment as described later.   Figure S18 shows the equivalent circuit for the measurement. The RTD THz oscillator was composed of an RTD, an LCR circuit, and a MIM capacitor. The bias voltage was applied to the RTD THz oscillator through the DC port of a bias tee. We used a bias tee with a frequency range of 0.1-6000 MHz. The modulated injection field caused a modulated square-law detection signal, which is coupled to a lock-in amplifier through the RF port of the bias tee. The squarelaw detection current at the feed point ∆ ‹d is represented as where • *+, is the injection amplitude at the antenna. The voltage measured at the lock-in amplifier is described as follows: We also confirmed that %ª« was proportional to • *+, by attenuating the injection field with a pair of WGs. We note that • *+, in the case of no attenuation slightly varied from experiment to experiment depending on the conditions such as optical alignment. In each experiment, we first measured • *+, in the case of no attenuation, and derived other • *+, values for various attenuations using the proportionality.
• *+, has two significant figures because the related quantities † ‡ ©© , %ª« and the transmission of the WG pair has two significant figures. (

2) Estimation of the oscillation voltage ( • )
We estimated the oscillation voltage from the radiation power. The assumed equivalent circuit of the RTD THz oscillator is shown in Fig. S19. The RTD THz oscillator is composed of a negative resistance − and an LCR resonator, which corresponds to the antenna. The oscillation amplitude • <=> can be represented as where <ºt is the emission power, and h is the radiative resistance. The emission power was typically 8 µW, as shown in Figure 1 (c). The antenna was a half-wavelength antenna. We assume its radiative resistance as 150 Ω, as in the previous study on the RTD THz oscillator of a similar structure. 2 Therefore, we can derive the oscillation voltage as • <=> = 50 mV. (S6 − 10) Here, • <=> has one significant figure in the above derivation because <ºt has only one significant figure due to noise in the measurement, and the assumption of h has uncertainty due to the unknown effective refractive index of the substrate. We verify this coupling efficiency by comparing the beam area and the effective antenna area.
We estimate the beam radius focused on the antenna ( 9$ ) with the formula for the Gausssian beam: Here, = 100 mm is the focal length of the parabolic mirror, = 6.3 /mm is the wavenumber, and 9" = 8 mm is the radius of the parallel beam. From this formula, we can derive that 9$ = 3 mm. It is known that the effective area of a dipole antenna can be represented as where λ is the effective THz wavelength at the antenna. The antenna length of the RTD THz oscillator was measured as 166 µm with an optical microscope. This corresponds to the half of the effective wavelength, so the antenna is designed for = 332 µm and the effective area of the dipole antenna is i = 0.014 mm # . Hence, the coupling efficiency derived from the beam shape is This value is comparable to the value of Eq. (S6-13). Figure S20 shows an equivalent circuit of an RTD oscillator for the circuit simulations. We  Figure S21 shows the free-running frequency, the free-running linewidth, and the locking range calculated with the constant RTD capacitance. Here, the locking range was independent of the bias voltage. Figure S22 represents the result for the case of the time-varying capacitance.

Section 7 Circuit simulation to verify the effect of voltage-dependent capacitance
Here, the locking range was bias-voltage dependent, and it was asymmetric at several bias points. Hence, these results support our statement that the voltage-dependent capacitance results in the bias-voltage dependent and asymmetric locking range.

Section 8 Peak area derivation and correction
In this section, we show how to derive the spectral area shown in Fig. 5 (b). We fitted the narrow and broad peaks in Fig.5 (a) with the Lorentzian functions. The HWHMs of both peaks were independent of the normalized injection amplitude . The HWHM of the narrow peak was about 50 kHz, which corresponds to the RBW of the spectrum analyzer. The HWHM of the broad peak was 4.4 MHz.
Here, the spectral height and the area of the broad peak were underestimated in the RTSA because of their noisiness and the frequency fluctuation. To correct this, we executed an independent total power measurement. In this measurement, we modulated the emission of the RTD THz oscillator using an optical chopper and performed a square-law detection with an FMBD. We compared the emission power in the free-running condition and the injectionlocked condition. Here, the injection amplitude was the maximum value in our setup, i.e., the normalized injection amplitude was about unity. We found that the total emission power was almost the same; the power difference in the two cases was less than 2%, which was comparable to the noise level of the measurement. We also confirmed that the total emission power was constant for various injection amplitudes.
From above experiments, we determined the correction factor for the spectral area of the broad peaks as 1.7 to keep the total peak area constant. We multiplied the spectral areas of the broad peaks by the factor, and normalized all the spectral areas with the total spectral area at = 1 to obtain Fig. 5 (b).

Section 9 Effect of flicker noise on the output noise spectra
In this section, we look into the detail of the output noise spectra ( Fig. 6 (a)). We show that the spectra can be explained as a result of a flicker noise source, especially in the high frequency part, where the output noise is small compared to the signal. The flicker noise is the noise with the power spectrum of /Ç (0<α<2). 5  where is the amplitude, and 9 is the oscillation frequency without a noise effect, and is a periodic function with a period of 2 . ∆ ( ) is the phase fluctuation, and ∆ ( ) is the amplitude fluctuation due to the noise source, such as the current fluctuation. In the output noise spectrum, the amplitude noise is usually much smaller than the phase noise. 7 We ignore the amplitude noise here. Fitting result of the measured noise spectra with Eq. (S9-4). Dashed lines are the fitting curve.
In Fig. S23 (a), we again show the noise spectra of Fig. 6 (a). In the free-running case ( 8 = 0), the output noise spectrum has an Δ /$ tail in the high frequency part and a flat region in the middle. The Δ /$ slope may result from the up conversion of the input Δ /" noise to the carrier frequency by the oscillator. This effect is well-known as the Leeson effect described as 6,7 Ì,rhii (Δ ) = ¾ 9 2 ∆ ¿ # Ì,*+jºt (Δ ) (S9 − 2) in the frequency range of Δ ≪ 9 2 ⁄ . Here, Ì,rhii (Δ ) is the output phase noise spectrum and is the Q-factor of the resonator. Ì,*+jºt ( ) is the power spectrum of the input phase fluctuation defined as where ℱ means the Fourier transform. We note that the variable is not the frequency offset but the frequency around DC. In Leeson's formula (S9-2), there is a problem that the output noise spectrum diverges at ∆ = 0. It is because this theory can be applied only in the high frequency region where the output noise is small. Therefore, we focus only on the high frequency part where the divergence does not matter.
In the injection-locked case of 8 = 13, 30, and 66 MHz, each output noise spectrum is composed of an approximately Δ /$ tail and an Δ /Ç (~0.8) part at the center. Here, the value of is determined by the fitting with the following function: