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Abstract
A method to realize pulse laser phase locking and homodyne detection is proposed, which can be used in lidar and continuous variable quantum key distribution (CVQKD) systems. Theoretical analysis shows that homodyne detection of pulse laser has a sensitivity advantage of more than 4 dB over heterodyne detection. An experimental verification setup was constructed to realize phase-locking and homodyne detection of pulse lasers at repetition rates from 50 kHz to 2.4 MHz. For 320 ns signal laser pulses at 300 kHz with peak power of -65 dBm, the phase error is 8.9° (mainly limited by the chirp effect in the modulation of signal laser), and the detection signal-to-noise ratio reaches 20.2 dB. When the peak power is reduced to -75 dBm, phase locking and homodyne detection can still be achieved. Homodyne detection based on phase locking could serve as a novel weak-laser-pulse receiving method with high sensitivity and anti-interference performance.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In pulsed lidar systems, repetitive laser pulses are transmitted to the object, and the reflected or scattered signal is received and processed to get information about the object. High detection sensitivity and good anti-interference performance are important in such lidars. There are two fundamental detection methods in lidar, direct detection and coherent detection. Direct detection of single photon was realized by Geiger-mode avalanche photodiode (APD) and superconducting nanowire single-photon detectors (SNSPDs) [1,2]. However, it is difficult to distinguish weak echo laser signal from bright environment lights, as the response spectrum of these detectors is relatively broad [2]. Although optical filters can be used to reduce interference, it will introduce insertion loss, and it is difficult to compress the bandwidth of optical filters to the level of the bandwidth of transmitted laser signal [3].
Coherent detection can function as an extremely selective filter and amplifier for echo signal, so as to enhance the anti-interference performance and improve the sensitivity. There are two coherent detection schemes, heterodyne and homodyne detection.
Heterodyne detection is widely studied in wind monitoring [4,5]. It usually harnesses acousto-optic frequency shifting to generate the frequency difference between signal laser and local laser [6,7]. The frequency difference is generally 80 MHz to 150 MHz, which is usually much larger than the bandwidth of transmitted laser pulse, whose time duration is generally 100 ns to 300 ns [8,9]. So broadband detection is required to receive the high frequency beating signal, which limits the transimpedance gain in the detector and causes larger thermal noise. In addition, in high frequency region, the voltage noise of the transimpedance amplifier causes larger current noise in the junction capacitance of the photodiode. Larger thermal noise and current noise will degrade the signal-to-noise ratio (SNR) and sensitivity.
For homodyne detection, echo signal is recovered in base band directly by phase-locking. Therefore, much smaller detection bandwidth, which is close to the bandwidth of the laser pulse, is enough for signal recovering, so as to effectively reduce thermal noise and current noise [10,11]. Optical phase locking, which is normally essential in homodyne detection, had been employed to synchronize phase of continuous-wave laser beams in optical communication [12], microwave photonics [13], coherent beam combining [14] and twin-field quantum key distribution (TF-QKD) [15] systems. However, phase-locking of pulse laser with continuous local laser has not been reported, to the best of our knowledge, which is the key technique in homodyne detection of pulse laser echo.
In addition, in the continuous variable quantum key distribution (CV-QKD) system, in order to improve the security, a true local oscillator (LO) CV-QKD scheme has been proposed [16,17]. In this solution, it is necessary to achieve phase synchronization between reference laser pulse and local laser. Currently, dynamical phase-compensation is mainly used to track the phase of reference pulses [18,19]. By phase-locking of the reference pulses to local laser, the stability of the phase synchronization could be improved, so as to improve the reliability of the system.
In this work, a phase locking and homodyne detection scheme for pulse laser is proposed, and the sensitivity advantage of homodyne detection is analyzed, compared to heterodyne detection. An experimental setup is built to verify the feasibility of pulse laser phase-locking, and high-sensitivity homodyne detection of pulse laser is achieved.
2. Principle of phase locking and homodyne detection of a pulse laser
Figure 1 shows the principle of homodyne detection of pulse laser. The key issue is phase locking of the weak pulse signal laser to the continuous local laser by a phase-locking loop (PLL). The signal laser field is given by:
The local laser field is given by:
where ELO, ωs and φLO are the amplitude, angular frequency and initial phase of the local field, respectively.Pulse signal laser and continuous local laser are sent to 90° optical hybrid, which produces 4 interference optical signals with phase differences of 0°, 90°, 180° and 270°. The optical signals with phase differences of 0° and 180° are received by I branch balanced detector. The optical power received by the two photo diodes of this balanced detector is given by:
The output voltage of the I branch balanced detector is:
Here, we assume no following amplifier is included in the detector. For sensitivity analysis, the effect of following amplifier could be neglected. ABPD(t) is the amplitude of the beat frequency signal. Δω is the angular frequency difference. Δφ is the phase difference. RL is the transimpedance gain, [Ω], which is the ratio of output voltage of transimpedance amplifier (TIA) and photocurrent. Its value equals the feedback resister of the TIA. R is the responsivity of photodiode, [A/W].
Similarly, the output voltage of the Q branch balanced detector is:
I branch beat signal is used as the phase error signal, which is sent to the loop filter. The produced frequency control signal is sent to the frequency modulation port of local laser, so as to close the loop. In phase-locked state, the signal laser and local laser have the same frequency and a fixed phase difference: Δω = 0, Δφ = π/2. The amplitude of I branch signal tends to be zero. Then the echo signal is recovered directly in Q branch.
3. Analysis of pulse laser spectrum and loop transfer function
3.1 Spectrum of pulse laser
The pulse signal laser is produced by intensity modulation of continuous wave laser. The light field is given by Eq. (1) when the modulation waveform is Gaussian. The spectrum of periodic pulse laser consists of fundamental frequency and each order of symmetric side band, which has an angular frequency of ωs ± nωm, where, ωm = 2π/T, T is the modulation period, n is integer. The power of fundamental frequency and side bands are:
Figure 2 shows the normalized power of the fundamental frequency and side bands. In the figure, the modulation frequency fm is 300 kHz, and the full width at half maxima, FWHM, of laser pulse is 400 ns, corresponding to τ = 340 ns. For periodically modulated pulse laser, the side band power is comparable to that of the fundamental frequency.
Fig. 2. Normalized power of fundamental frequency and symmetric side bands of periodically modulated pulse laser
For the periodic pulse laser generated by intensity modulation, the fundamental frequency and each side frequency have the same phase. Therefore, when the fundamental frequency is phase-locked to the continuous local laser, locking of each side band is achieved simultaneously, thereby pulse laser phase-locking is achieved. By properly setting the loop bandwidth to be wider than the linewidth of fundamental frequency and local laser by more than an order, the fundamental frequency can be locked. On the other hand, in order to avoid the interference of side frequency to the phase-locked loop, the loop bandwidth should be much smaller than the modulation frequency. This is the principle rule in setting loop bandwidth in pulse laser PLL.
In true LO CV-QKD systems, the reference pulses would have random pulse period and varied pulse amplitude within a certain range [18]. In this case, the spectrum width of each side band would be broadened. The spectrum of first side band starts at the minimum pulse rate, or the inverse of maximum time interval between adjacent pulses. As long as the minimum pulse rate is much larger than the loop bandwidth, phase locking between reference pulses and local laser could be realized.
3.2 System transfer function of phase-locking loop
The loop bandwidth could be obtained by studying the open-loop and system (closed-loop) transfer functions. They are given by:
By performing Fourier analysis to the signal, we obtain ${K_d} = \sqrt \pi \tau {V_I}\textrm{/}T$. In Eq. (8), KG is the gain of loop amplifier. K0 is tuning coefficient of local laser, in unit of rad/s/V. F(s) is the transfer function of the loop filter. For an active proportional-plus-integral filter, the transfer function is $F(s )= ({1 + s{\tau_2}} )/({s{\tau_1}} )$. FL(s) is the transfer function of the frequency modulation response of the local laser. For piezoelectric frequency modulation, the transfer function can be approximated by a first-order low-pass filter function, ${F_L}(s)\textrm{ = }({2\pi {f_H}} )/({s\textrm{ + }2\pi {f_H}} )$, where fH is the modulation bandwidth. τd is time delay of the loop.
Loop gain K, natural frequency ωn, and damping factor ζ are important parameters that determine the loop performance. They are given by:
Figure 3 shows the system transfer function for different loop gains. The main loop parameters are listed as follows: τ1=580 μs, τ2=5 μs, τd=23 ns, fH=100 kHz. These values are consistent with our experimental values. When the loop gain K increases from 6.3×103 Hz to 3.1×105 Hz, the corresponding loop natural frequency increases from 2π×5.65 kHz to 2π×39.79 kHz, the loop damping factor increases from 0.089 to 0.622, and the corresponding loop bandwidth increases from 8.80 kHz to 77.91 kHz. The line width of the NPRO (Nonplanar Ring Oscillator) laser is less than 1 kHz. Therefore, by careful setting of loop parameters, the loop bandwidth could be an order of magnitude wider than the fundamental linewidth of signal laser, meanwhile, reasonable damping factor could be got. In addition, when the pulse repetition frequency is much greater than the loop bandwidth, the fundamental frequency of signal laser can be locked to the local laser, and each order of side band keeps out of the loop bandwidth and does not affect the loop performance. Thus, the phase locking of the pulse laser and the continuous local laser could be realized.
4. Sensitivity analysis
4.1 Noise analysis
The main factor that affects the sensitivity of homodyne detection is detector noise. Three major noise sources are analyzed below.
- (1) Intensity noise of local laser and shot noise
For coherent laser detection, the local laser power incident on the detector is much higher than the signal laser power, and the intensity noise of the local laser is converted into photocurrent noise. The laser intensity noise power [V2] of I/Q branch detector is given by:
$${P_{IN,I}} = {\left( {\frac{1}{2}{k_3}^2{P_{LO}}R} \right)^2}({RIN} )\cdot {R_L}^2 \cdot \Delta f = \frac{1}{4}{k_3}^4{P_{LO}}^2{R^2}({RIN} )\cdot {R_L}^2 \cdot \Delta f,$$$${P_{IN,Q}} = {\left( {\frac{1}{2}{k_4}^2{P_{LO}}R} \right)^2}({RIN} )\cdot {R_L}^2 \cdot \Delta f = \frac{1}{4}{k_4}^4{P_{LO}}^2{R^2}({RIN} )\cdot {R_L}^2 \cdot \Delta f,$$where Δf is the detection bandwidth. The noise power is proportional to the detection bandwidth. The minimum detection bandwidth should be the spectral width of the photoelectrical pulse signal. For a Gaussian-shaped electrical pulse as Eq. (9), the spectral bandwidth is ${B_E} = \sqrt {2\ln (2 )} /({2\pi \tau } )$. The relationship between FWHM pulse width τp and τ is ${\tau _p} = \textrm{2}\sqrt {\ln (2 )} \cdot \tau$. Therefore, the minimum detection bandwidth should be $\Delta {f_{\min }} = {B_E} = \sqrt 2 \ln (2 )/({\pi {\tau_p}} )$.RIN is relative intensity noise of local laser (dB/Hz). In this work, a solid-state laser is used as a local laser. Its intensity noise mainly includes relaxation oscillation noise and low-frequency noise caused by pumping. Balanced detection could reduce the impact of RIN by more than 30 dB. The quantum limit of RIN is 2q/ I, where q is the fundamental charge (1.6 × 10−19 C) and I is the DC photocurrent. Taking this limit into the Eq. (11), the quantum limit of intensity noise power of the I/Q branch can be obtained as:
This is the shot noise power. Therefore, the intensity noise of local laser and shot noise are the same type of noise. Shot noise is the quantum limit of intensity noise. Therefore, the noise power actually generated on the detector is the larger one of PIN or PS, which is written as MAX[PIN, PS].
- (2) Thermal noise
The thermal noise power [V2] is given by:
where k is Boltzmann constant, T is temperature [K]. When the I/Q branch detectors have the same bandwidth and transimpedance gain, the thermal noise power is equal. - (3) Noise introduced by transimpedance amplifier
Transimpedance amplifiers (TIA) are usually used as preamplifiers for balanced photodetectors. For an amplifier with FET-input stage, the input voltage noise should be considered. The noise power introduced is given by:
where EN is input voltage noise [V/Hz1/2], CD is the sum of parasitic input capacitance of TIA and photodiode capacitance. The input voltage noise introduces current noise in CD, thereby produces output noise power, which is represented by the second term.
4.2 Sensitivity of homodyne detection
The following two conditions should be met for high sensitivity homodyne detection. First, in I branch, phase locking is realized and sufficiently small phase error is achieved. Second, in Q branch, sufficiently high signal-to-noise ratio is achieved. The required signal optical power, which is detection sensitivity, will be calculated for I/Q branch based on above two conditions.
In I branch, the phase error variance ($\sigma _\theta ^2$) is given by:
For a reliable phase-locked loop, the phase error σθ should be smaller than 10°, or 0.17 rad.
Therefore, in I branch, the required peak power of signal laser is:
The system requirement of the false alarm rate (FAR) determines the required signal laser power in Q branch. When FAR is better than 1%, SNR should be higher than 13.5 dB. SNR of Q branch is given by:
Therefore, the required peak power of signal laser for the Q branch is:
The homodyne detection sensitivity PS0 is sum of the signal optical power required by I and Q branch, PS0=PS0,I+PS0,Q. The signal optical power mentioned in this article refers to the peak power of signal laser pulse.
From above analysis, it can be seen that the sensitivity of homodyne detection is related to the parameters of the pulse signal laser, as well as the performance of the local laser and the balanced detector. The key parameter that needs to be optimized is the local laser power PLO. Other main parameters that affect the detection sensitivity include the local laser relative intensity noise RIN and transimpedance gain RL. First, we will evaluate the typical parameters of pulse laser. Since the bandwidth of the phase-locking loop is on the order of tens kHz, in order to suppress interference of side band of signal laser, the pulse repetition frequency should be much higher than the loop bandwidth. In the following quantitative analysis of sensitivity, we take a typical repetition frequency f = 300kHz, and pulse width τp = 400 ns, the corresponding detection bandwidth Δf = BE = 0.78 MHz. Generally, in a 90° optical hybrid, the signal light is equally distributed to the I/Q branch. In this article, the SNR of the Q branch is close to the reciprocal of $\sigma _\theta ^{2}$ in I branch. So we can use the setting that ${k_1} = {k_2} = {k_3} = {k_4} = \sqrt {2} /2$. By photoelectric feedback control of the intensity noise of local laser, combined with the balanced detection technique, the RIN can be reduced to -140 to -150 dB/Hz. The transimpedance amplifier of the balanced detector can be an operational amplifier with low input noise, such as OPA657 of TI Inc. Based on the above analysis, following parameters are used to calculate the detection sensitivity:
Figure 4 shows the homodyne detection sensitivity as a function of local laser power, for different transimpedance gain and RIN. According to the sensitivity formulas (16) and (18) of the I/Q branch, the four terms on the right side of the formulas correspond to the laser intensity noise, shot noise, thermal noise, and TIA noise, respectively. Among them, the first term is proportional to PLO, the second term is irrelevant to PLO, and the third and fourth terms are inversely proportional to PLO. Therefore, there is an optimal value of PLO, which makes the detection sensitivity reach a minimum. As the PLO increases, the laser intensity noise exceeds the shot noise, making the detection sensitivity increase sharply. The turning power point mainly depends on RIN. When RIN is -150 dB/Hz, the turning power is 3.28 dBm; when RIN increases to -140dB/Hz, the turning power decreases to -6.71 dBm. -10 dBm to +5 dBm is the commonly used local laser power range. If the local laser power is too large, the detector will be saturated, while if it is too small, the detection gain will be insufficient. Therefore, by reducing the RIN of the local laser to within -140dB/Hz, we can obtain an optimum sensitivity. In addition, as RL increases, the detection sensitivity further decreases, and the thermal noise and TIA noise are reduced to much less than the shot noise. The shot noise power is irrelevant to PLO, so the best detection sensitivity can be obtained in a larger PLO range. RL is mainly limited by the detection bandwidth. For high-speed TIA, when the detection bandwidth is about 1MHz, RL can take the order of MΩ. It can be seen from the figure that the detection sensitivity can reach -85 dBm, and for a pulse width of 400ns, the average number of photons contained in each pulse is 4 to 5.
Fig. 4. Homodyne detection sensitivity as a function of local laser power, for different transimpedance gain and RIN
Costas phase-locked loop is a commonly used homodyne detection implementation. It is normally applied in optical communication for phase shift keying (PSK) transmission at high data rates. By properly setting the relationship between loop bandwidth, pulse repetition frequency and fundamental linewidth of laser carrier, as proposed in this work, homodyne detection could also be realized in Costas loop. In principle, similar sensitivity could be achieved. However, as the phase error signal is produced by mixing I and Q branch signal in Costas loop, more noise and error will be included. So we suppose the implementation in this work is more concise and the phase error should be smaller than Costas loop.
4.3 Sensitivity comparison with heterodyne coherent detection
In heterodyne coherent detection, splitter prisms or fiber couplers are used as optical mixing elements, and balanced detector is used to receive two interference lights with a phase difference of 180°. Pulse signal is recovered by subsequent signal processing [20].
Taking the local and signal light field as Eq. (1) and Eq. (2), the optical power received by the two detection ports of the balanced detector are:
The output voltage of the balanced detector is:
In the heterodyne detection system, the signal laser is generally acousto-optic frequency-shifted in the transmission module, and the echo signal laser is mixed with the local laser in the receiving module. Therefore, there is a fixed frequency difference in heterodyne detection, which is equal to the acousto-optic frequency shift fH. The commonly used acousto-optic frequency shift range is 80∼150 MHz. Typical value of 80 MHz is used in the following calculations. The noise and sensitivity analysis method is similar to that of homodyne detection. The detection sensitivity is given by:
Heterodyne detection sensitivity could be calculated with same parameters as shown in Table 1 and SNR of 13.5 dB. The main difference is that, for heterodyne detection, the detection bandwidth should be wider than 80 MHz, while the signal processing bandwidth Δf is still 0.78 MHz. Figure 5 shows the heterodyne detection sensitivity as a function of local laser power. It can be seen that there still exists optimum local laser power, which makes the detection sensitivity reach a minimum. On the other hand, since the detection bandwidth is increased by two orders, the transimpedance gain RL should be in the order of kΩ, less than 4 kΩ for OPA657. As RL decreases, the detection sensitivity increases. In addition, for smaller RL, thermal noise and TIA noise have larger impact, which greatly reduces the optimal value range of PLO. Generally, with same system parameters, the sensitivity of homodyne detection is better than heterodyne detection by around 4-5 dB. In addition, heterodyne detection requires additional signal processing to demodulate the pulse signal, so the detection sensitivity will be further degraded.
Fig. 5. Heterodyne detection sensitivity as a function of local laser power, for different transimpedance gain and RIN
Table 1. Numerical values of the pulse OPLL system
5. Experimental setup and results
5.1 Experimental setup
The key issue to realize pulse laser homodyne detection is optical phase locking. In order to verify the feasibility of pulse laser phase locking, an experimental setup, as shown in Fig. 6, was constructed.
The local and signal laser are both 1064 nm Nd:YAG NPROs, with temperature and piezoelectricity (PZT) frequency control ports. The sizes of local and signal laser crystals are 3×12×13 mm3 and 3×6×7.2 mm3, respectively. The dihedral angles are 45° for both crystals. The output facets are coated to be 96% reflecting in the S polarization and 87% reflecting in the P polarization at 1064 nm with incidence angle of 45°. The laser output power is 0.51 W at 1.1 W pump power with slope efficiency of 54% and laser linewidth of about 1 kHz. The temperature of the laser crystals is precisely controlled to keep laser frequency stable. The control precision is ±0.01 K for 1 hour and ±0.03 K for 8 hours. The typical frequency drift is less than ±30 MHz for 1 hour and ±90 MHz for 8 hours. The PZT tuning response bandwidth is around 100 kHz, and the tuning coefficient for local laser is 1.6 MHz/V. The output lights of local and signal lasers are coupled into polarization maintaining (PM) fibers.
The pulse laser is modulated by a PM fiber intensity modulator. The modulator bandwidth is 10 GHz, with half-wave voltage of 3.6 V and extinction ratio of 26 dB. The phase difference between I/Q branch of the home-made 90° optical hybrid is 90°±5°, and the overall insertion loss of signal laser is about 3.5 dB, which is mainly attributed to the fiber coupling loss. Fiber attenuators are inserted in the optical path to adjust local and signal laser power. 75-μm InGaAs PIN photodiodes are used in the balanced detector with transimpedance gain of 2 kΩ. The gain of the following amplifier circuit is adjustable from 30 dB to 70 dB. In our experiments, the pulse repetition rate changes normally from 100 kHz to 1 MHz, and the pulse width varies from 960 ns to 96 ns. The basic linewidth of pulse signal changes from 0.33 MHz to 3.3 MHz. Though better SNR will be achieved by digital signal processing (DSP) technique, as we are now in short of DSP circuits, we use hardware bandwidth-limited filter to recover the signal and set the detection bandwidth to be about 3 MHz.
In the experiment, by adjusting the gain of the amplifier circuit, the output signal amplitude of the I branch detector is 0.4 V to 1 V to ensure sufficient loop gain; the output amplitude of the Q branch is adjusted to be higher than the noise amplitude of the oscilloscope to ensure measurement accuracy of SNR. Through balanced detection and photoelectric negative feedback control of the laser noise, the RIN of the local laser can be suppressed to about -140 dB/Hz.
The loop filter is an active proportional-plus-integral filter, with τ1=580 μs and τ2=5 μs. Its main function is generating the frequency control signal and locking the loop. The control signal is amplified by the PZT driver with a gain of G=9, and then sent to the PZT frequency modulation port of local laser.
5.2 Experimental results
According to the calculation results in Fig. 4, the local laser power is attenuated to about -5 dBm. An arbitrary function generator AFG3252, made by TEK Inc., is used to drive the intensity modulator to generate Gaussian laser pulses. The ratio of the electrical FWHM pulse width to the period is 0.12, which is fixed for Gaussian waveform in AFG3252. The resulting ratio of the laser FWHM pulse width to the period is 0.096. The locking performance was tested for different peak powers of signal laser that sent to the 90° hybrid. When signal laser power is greater than -65 dBm, stable laser phase locking could be achieved. Figure 7 shows the output signals of I/Q branch of the balanced detector in open-loop and closed-loop states when the signal power is -65 dBm. In open-loop state, there is a frequency difference between local laser and signal laser, and the output of I/Q branch is the beat signal of pulse signal laser with CW local laser. In closed-loop state, as the initial frequency difference is smaller than 600 kHz, the loop will get locked under the effect of feedback control. As the phase difference Δφ approaches π/2, the amplitude of the I branch signal is greatly reduced. By preliminary analysis, we suppose that the residual signal amplitude is attributed to the chirp effect introduced by modulation, that is, the amplitude modulation simultaneously causes the transient change of the laser phase. This transient phase change causes fluctuations of the phase error signal in the pulse duration. In phase-locked state, the Gaussian pulse waveform is recovered in Q branch, and homodyne detection of the pulse laser is realized. From Fig. 7(b), the SNR is estimated to be 20.2 dB.
As the signal laser power increases, the phase-locking stability is significantly improved. When the signal laser peak power is reduced to -75 dBm ∼ -65 dBm, the phase locking phenomenon can still be observed, but the stability is greatly degraded. The main reason is small changes of polarization state in the PM fiber, which can easily cause tiny power unbalance at the output ports of 90° hybrid, and a small DC component in the output of balanced detector. When the DC component is greater than the amplitude of the phase error signal, the loop will lose lock. Such small and random change in the polarization state will cause the loop to lose lock frequently. The laser polarization extinction ratio (PER) in PM fiber is normally around 23 dB. As a contrast, the PER of crystal polarizers can reach 50 dB. If the local laser is transmitted in space and polarized by crystal polarizers, much purer linear polarization state could be produced. The power balance of each branch of 90° hybrid could be improved substantially, and the problem of DC component can be significantly alleviated.
Referred to the phase error measurement of continuous laser phase-locking loop, the phase error measurement formula of pulse laser phase-locking loop could be given by:
where, σθ,m is the measured phase error, Vp,rms is the RMS value of the I-branch phase error signal within the pulse duration in closed-loop state, and VI is the amplitude of the I-branch beat signal in the open-loop state. According to this formula and experimental results in Fig. 7, Vp,rms = 62 mV and VI = 0.4 V, so the phase error σθ,m is 8.9°.Figure 8 shows the measured closed-loop phase error when the signal laser peak power is -50 dBm, -60 dBm, and -65 dBm, respectively. In the measurements, by adjusting the signal amplitude of the I branch, or the time constants of the loop filter, the loop damping factor is set to around 0.3, and the loop bandwidth is 20 kHz∼30 kHz. The theoretically optimum value for the damping factor is 0.707. However, limited by the loop delay and response bandwidth of the PZT frequency turning, the loop tends to be unstable for higher damping factors. The loop delay time is around 23 ns. It can be seen from the figure that when the signal laser peak power is -50 dBm and -60 dBm, the phase error is around 7°∼8°. When the power is reduced to -65dBm, the phase error slightly increases to around 8°∼9°. This also supports the supposition that the phase error is mainly attributed to the chirp effect caused by intensity modulation, rather than detection noise, such as shot noise and thermal noise. In the future experiments, low-chirp modulator should be used. As the pulse frequency increases from 200 kHz to 1 MHz, the pulse width decreases from 480 ns to 96 ns and the spectral bandwidth of Q branch signal increases from 0.65 MHz to 3.25 MHz. As the detection bandwidth is about 3 MHz, the signal power loss increases slowly and the phase error increases slowly with pulse frequency. The RMS of phase error is about 0.2° for signal laser peak power of -60 dBm and -50 dBm, and it is about 0.3° for peak power of -65 dBm. In addition, with the increase of the signal laser power, SNR of Q branch also increases correspondingly. When the signal laser peak power is -60 dBm and -50 dBm, SNR reaches 25 dB and 35 dB, respectively.
For laser pulses from 1 MHz to 2.4 MHz, phase-locking and homodyne detection can still be realized. But due to the reduction of the pulse width, the required detection bandwidth is larger. For the same pulse energy, similar phase error and SNR can be achieved, regardless of pulse repetition rates. As long as the detection noise density is nearly flat, the detection sensitivity of pulse energy could remain nearly unchanged. For homodyne detection in this work, the noise density increases slowly with frequency mainly because of TIA noise. By further suppressing the laser intensity noise to approach quantum limit, optimum sensitivity could be got at higher local laser power PLO. Then the laser intensity noise could be much higher than TIA noise, and the total noise density could be nearly flat.
It can be expected that as long as the detection bandwidth is sufficient, the system can achieve homodyne detection for even higher repetition pulses. For nanosecond laser pulses in true LO CV-QKD systems [16,18–19], as the pulse rate is in MHz or 10s MHz order which is much higher than the loop bandwidth, phase locking could be realized with the proposed method.
If there is no DC drift in the output of balanced detector caused by the polarization change, the detection sensitivity should reach -71.7 dBm, when SNR of Q branch drops to 13.5dB. According to Fig. 4, when the transimpedance gain is 2 kΩ, this detection sensitivity is 10.0 dB away from the theoretical calculation. The sources of the 10.0 dB degradation are estimated as follows: 1) 3.5 dB due to insertion loss of 90° hybrid, 2) 5.8 dB due to wide detection bandwidth, which is 3.8 times of the theoretical value, 3) 0.7 dB due to insertion loss of optical connectors. Therefore, the experimental results are fundamentally consistent with the theoretical calculations.
6. Conclusion
A method of homodyne detection of pulse laser is proposed. The sensitivity of homodyne and heterodyne detection is analyzed and compared theoretically. It is indicated that homodyne detection has a sensitivity advantage of more than 4 dB. The key issue for pulse laser homodyne detection is phase locking between pulse signal laser and continuous local laser. An experimental setup was constructed to realize phase locking and homodyne detection of high-repetition-rate laser pulses. The measured sensitivity is fundamentally consistent with the theoretical calculation. The phase error of the pulse laser phase-locking loop is mainly caused by the chirp effect in intensity modulation. It would be greatly reduced by using a low-chirp modulator. In addition, slight changes in the polarization state of local laser will cause drifting of the DC component of the phase error signal, which will cause the loop to lose lock. Therefore, free-space-transmitted local laser should be employed to ensure the stability of the polarization state, so as to improve the reliability of the loop.
Funding
PetroChina Innovation Foundation (2017D-5007-0603); Fundamental Research Funds for the Central Universities.
Disclosures
The authors declare no conflicts of interest.
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