1. Introduction

Phase-shift laser range finders (PSLRFs) offer the advantages of high precision, large measurement range, and sufficient maturity, meeting the measurement requirements of aerospace equipment and large-component processing. PSLRFs employ electro–optical intensity modulators to generate intensity-modulated lasers [1]. The Mach-Zehnder modulator (MZM) is widely employed owing to its low driving voltage, large bandwidth, and low chirp signal [2]. Because the nonlinear transmission characteristic of MZMs results in a harmonic distortion in the transmitted signals, they generally operate at the quadrature point in PSLRFs.

Furthermore, MZMs suffer from drift, resulting in operating point fluctuations during long working periods [3, 4]. These fluctuations change the optical power and increase the harmonic components. These changes introduce considerable phase errors in photodetectors (PDs) as well as phase calculations, decreasing the measurement accuracy of PSLRFs [5–7]. Therefore, a high-stability operating point control method must be developed for MZMs to stabilize the optical power and reduce the harmonic components.

Numerous internal and external factors cause MZM drift including thermo-optic, pyroelectric, photorefractive, and dielectric relaxation effects [8]. However, MZM drifts cannot be completely suppressed by stabilizing the temperature, methods for enhancing the internal material properties and structure are not applicable to general situations [9,10]. The general method involves controlling the direct current (DC) bias voltage that can be used to determine the operating point of MZMs.

Various methods for controlling the DC bias voltage have been proposed in recent years. One group of these methods involves monitoring the output optical power of MZMs as feedback. The DC bias voltage is controlled to maintain a constant power level [11]. However, these methods have some limitations, such as susceptibility to fluctuations in the input laser intensity, optical path loss, and circuit drift, resulting in low operating point stability.

The other group of methods is based on an auxiliary dither signal mixed with the input DC voltage of the MZM. The DC voltage is regulated by monitoring the harmonics of the dither signal in the MZM output [12]. For example, because the amplitude of the second harmonic is zero at the quadrature point, canceling it produces the quadrature point of MZM. Wang and Kowalczyk proposed a control method at any operating point using the ratio of the fundamental to the second harmonics as feedback [13]. This method exhibited good anti-disturbance performance because it eliminated the influences of input laser power fluctuation and circuit drift [14]. However, this method loses the aforementioned advantages at the quadrature point where the second harmonic is easily masked by the spectrum noise floor (NF). Moreover, determining the drift direction at the quadrature point is challenging, that further reduces control stability. Therefore, the control stability at the quadrature point is generally lower than those at the other operating points.

Here, we studied the operating point control method for MZMs using the ratio as feedback to provide the quadrature point with the same excellent anti-disturbance performance as other operating points in the feedback control loop and enhance the control stability therein. We employed a cascaded MZMs structure to achieve the research objectives. In this structure, the controlled MZM still functioned at the quadrature point and was used as the PSLRF source. Concurrently, the output of the cascaded MZMs structure exhibited characteristics similar to those exhibited at the other operating points and was deployed as a feedback control loop. Moreover, the controlled MZM exhibited better control stability at the quadrature operating point than the previous method. Furthermore, we applied the proposed operating point control method to a PSLRF and obtained highly stable ranging results.

2. Principle of the Mach-Zehnder modulator operating point controller

2.1 Error analysis of the Mach-Zehnder modulator control method

When auxiliary dither signal ${V_{AC}} = a\sin ({\omega t} )$ and DC voltage, ${V_{DC}}$ are applied to the MZM, the output optical power is given as follows:

Ratio k is not affected by optical power loss or circuit-gain drift and is shown in Fig. 1.

However, the shot noise of the PD and the circuit thermal and quantization noises of analog-to-digital converters (ADCs), most of these noises are Gaussian white noise, cannot be avoided during the detection, processing, and sampling of MZM output signals. The NF and amplitude error are the effects of the aforementioned white noise in the spectrum, as calculated via fast Fourier transform (FFT). The spectrum NF level related to the Gaussian white noise is given as follows [15]:

 

Fig. 1. (a) Normalized amplitudes of fundamental and second harmonics as a function of the operating point $\Phi = {{{V_{DC}}\pi } / {{V_\pi }}} + {\varphi _0}$ and (b) The ratio k as a function of the operating point at ${a / {{V_\pi }}} = 0.1$.

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In addition, determining the drift direction by ratio k becomes challenging near the quadrature point as V₁ and V₂ are the extreme values. The general determination method involves shifting the current operating point to the left and right, with a minimum shift amount ${\sigma _\Phi }$. Thus, we assume that the operating point error expands two times owing to the determination of the drift direction. For example, $\beta {P_0}$=1 V, spectrum NF ${V_{nfloor}}$ is 1 mV, and Avg is 8. Figure 2 shows the operating point error.

 

Fig. 2. (a) Operating point error of the MZM as a function of the operating point $\Phi $, at ${a / {{V_\pi }}} = 0.1$. (b) Operating point error of the MZM at the quadrature point $({\Phi = {\pi / 2}} )$ with dither signals of different amplitudes.

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Compared with those at other operating points, the control stability at the quadrature operating points is lower and challenging to increase. Thus, we conclude that the irreducible spectrum NF and non-directivity are the major factors contributing to the large operation error at the quadrature point. However, for long-distance measurement applications, the quadrature operating point is essential for satisfactory photodetector conversion efficiency and phase-calculation accuracy.

Figure 2(b) shows the operating point error of the MZM at the quadrature point with dither signals of different amplitudes. The increase in the dither signal amplitude causes the decrease in ${{{J_1}} / {{J_2}}}$, further decreasing the operating point error. Therefore, increasing the amplitude of the dither signal can improve the control stability of the MZM operating point. However, using a larger dither signal increases the third-order intermodulation component of the dither signal and high-frequency modulated signal in the MZM output, which introduces a large phase error. In the PSLRFs, the dither on the MZM with the amplitude greater than $0.12{V_{\pi 1}}$, can introduce an additional phase error that is greater than ${0.01^ \circ }$. Therefore, a low-amplitude of dither signal is also necessary for the MZM operating point controller used in PSLRFs.

In this study, we propose a dual-MZM cascade structure for the operating point controller of the MZM. In this structure, the controller can operate at other operating points away from the quadrature point while maintaining the controlled MZM at the quadrature point with a low dither signal to improve the control stability. The controlled MZM still meets the requirements of distance measurement, and the higher stability of its operating point controller improves the stability of the distance results.

2.2 Controller principle based on the dual-MZM cascade structure

Figure 3 shows the cascade MZM structure, which includes a target MZM (MZM1) and an auxiliary MZM (MZM2). MZM2 exhibits the same performance as MZM1 (using the same type of modulator product). This structure can be called a dual-MZM cascade structure because the laser successively passes through MZM1 and MZM2.

 

Fig. 3. Dual-MZM cascade structure.

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The output optic power of this structure is given as follows:

The amplitude of the dither signal applied to the MZM2 is $0.8372{V_{\pi 2}}$; thus, ${J_1}({{m_2}} )\approx {J_2}({{m_2}} )$. The operating points of the MZM1 and MZM2 are defined as ${\Phi _1} = {m_{DC1}} + {\varphi _1}$ and ${\Phi _2} = {m_{DC2}} + {\varphi _2}$, respectively, and the amplitude functions of frequency components $4{\omega _1}$and$5{\omega _1}$, are defined as$Y({{\Phi _1},{\Phi _2}} )$ and $Z({{\Phi _1},{\Phi _2}} )$, respectively, in the PD.

When the operating points of the MZM2 are changed to${\Phi _2} + {\pi / 2}$, the following substitutions are made to functions Y and Z.

The following equations of the amplitudes are valid when ${\Phi _1}$and${\Phi _2}$ satisfy conditions $|{\tan {\Phi _1}} |\ge {{|{\tan {\Phi _2}} |\cdot {J_2}({{m_1}} )} / {{J_1}({{m_1}} )}}$and$|{\tan {\Phi _1}} |\ge {{{J_2}({{m_1}} )} / {[{{J_1}({{m_1}} )|{\tan {\Phi _2}} |} ]}}$, respectively.

The ratio function, $K({{\Phi _1},{\Phi _2}} )$, is defined as follows:

 

Fig. 4. (a)Calculation results of Eq. (13) are the functions of the operating point ${\Phi _1}$. (b)Ratio K as a function of operating point ${\Phi _1}$ at ${{{A_1}} / {{V_{\pi 1}}}} = 0.1$.

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Figure 4 shows several abnormal mutations in the entire operating area of the MZM1, because these operating points do not meet the aforementioned conditions. Therefore, the conditions of Eq. (13) determine the control range of the dual-MZM cascade structure and the range of the offset, that is the MZM2 operating point. According to the conditions of Eq. (13), the control range of the dual-MZM cascade structure is also related to the amplitude of the dither signal on the MZM1. Figure 5 shows the operating points of the MZM1 and MZM2 that satisfy conditions $|{\tan {\Phi _1}} |\ge {{|{\tan {\Phi _2}} |\cdot {J_2}({{m_1}} )} / {{J_1}({{m_1}} )}}$ and $|{\tan {\Phi _1}} |\ge {{{J_2}({{m_1}} )} / {({{J_1}({{m_1}} )|{\tan {\Phi _2}} |} )}}$ with different dither amplitudes in the colored region. When the operating point of the MZM1 is the quadrature point, the offset is avoided from points near ${\Phi _2} = n\pi /2,({n = 0,1,2 \cdots } )$ such that the control range of the MZM1 is extremely small. Concurrently, the drift direction of the MZM1 is clearly determined at the quadrature point. ${\Phi _2} = \pi /4 + n\pi /2,({n = 0,1,2 \cdots } )$ always correspond to the maximum control range; therefore, these are better offset points. Although the control range gradually shrinks with the increase in ${m_1}$, it is still sufficiently large to include the drift within one control period near the quadrature point of the MZM1.

 

Fig. 5. Control range of the dual-MZM cascade structure at different amplitudes ${A_1}$.

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Next, we analyze the operating point error of the MZM1, according to K in Eq. (14). In the FFT-calculated spectrum, we assume that NF is ${V_{nfloor}}$ and that the amplitude error of the spectral lines is ${V_{np}}$. When the MZM1 is at the quadrature point, amplitudes ${Y_1}$, ${Y_2}$, ${Z_1}$, and${Z_2}$ are not masked by spectrum NF and are measured using the same circuit. Therefore, their amplitude errors are ${V_{np}}$ and do not correlate with each other. Concurrently, the drift direction of the MZM1 can be clearly determined using ratio K. According to the combined uncertainty formula, the operating point error of the MZM1 is given as follows:

In this structure, ${\Phi _2}$ can be calculated using the ratio of the amplitudes of frequency components, $3{\omega _1}$ and $6{\omega _1}$ in the output of the cascaded MZM structure, as shown in Eq. (16), that is independent of the operating point of the MZM1. As previously stated in Fig. 2, the operating point error of the MZM2 is a considerably small value compared to that of the MZM1, due to ${A_2} = 0.8372{V_{\pi 2}}$ . Thus, the influence of ${\Phi _2}$ is not considered in Eq. (15).

Figure 6 shows the operating point error of the MZM1, $\beta {P_0}$=1 V, the spectrum NF ${V_{nfloor}}$=1 mV, and Avg = 8. The operating point of the MZM2 is ${\Phi _2} = {\pi / 4} + n\pi ,({n = 0,1,2 \cdots \cdots } )$, and these of the MZM1 without directivity are close to ${\Phi _1} = {{3\pi } / 4} + {{n\pi } / 2},({n = 0,1,2 \cdots \cdots } )$. The operating point error is not affected by the location of the operating points of the MZM1 within the appropriate range of the dual operating points, as shown in Fig. 6. Compared with the single-MZM control method, the control error in this structure is effectively reduced when the MZM1 is at the quadrature point with equal dither signals. The error can be further reduced by increasing Avg in this structure. However, this is not possible when using the single-MZM control method. However, the drift rate must be comprehensively considered when selecting an appropriate Avg. Therefore, the controller based on the dual-MZM cascade structure is expected to exhibit higher stability and still satisfy the requirements of PSLRFs.

 

Fig. 6. Operating point error of the MZM1 as a function of operating point ${\Phi _1}$, at ${{{A_1}} / {{V_{\pi 1}}}} = 0.1$.

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3. Implementation and setup of the controller

Figure 7 shows the PSLRF with the MZM operating point controller. The wavelength of the laser is 1550 nm. The output laser of the MZM1 is primarily used for the distance measurements. The remaining small part of the output laser of the MZM1 is transferred to the MZM2. This dual-MZM cascade structure and control circuit are combined to form the operating point controller. The output laser of MZM2 is converted into an electrical signal by the PD, and the electrical signal is filtered and amplified using the control circuit. To avoid FFT calculation errors owing to spectrum leakage, the sampling frequency of the ADC is used as an integer multiple of the frequency of the dither signal. Because the sampling frequency of the ADC is 150 kHz, and the FFT calculation uses 1024 points, the frequency of the dither signal applied to the MZM1 is ${{{\omega _1}} / {2\pi }} = {{10 \times 150({\textrm{kHz}} )} / {1024}} \approx 1464.84$ Hz. We use a 16-bit ADC with a sampling range between -10 V and 10 V. The FFT operation and real-time tracking cycle are also implemented using a digital signal processor (DSP). The DC voltages of the MZM1 and MZM2 are generated by a digital-to-analog converter (DAC), and the dither signals are generated by a direct digital synthesizer (DDS). The dithers of the MZM1 and MZM2 are synchronized. The clocks of the both dithers are homologous to reduce the influence of frequency fluctuations. Notably the phase synchronization between the two dithers is better than $0.1\pi $ in our controller, this slightly influences the controller performance.

 

Fig. 7. PSLRF using the operating point controller based on the dual-MZM cascade structure.

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The controller based on the dual-MZM cascade structure is expected to satisfy the following conditions: (1) two dither signals must be applied to the MZM1 and MZM2, and the frequencies must be fitted as ${\omega _2} = 3{\omega _1}$; (2) the amplitude of the dither signal applied to the MZM2 must be approximately $0.8372{V_{\pi 2}}$; and (3) the appropriate MZM2 operating point must be determined by the amplitude of the dither signal applied to the MZM1 and the controllable range.

In fact, ${A_2}$ is not accurate, and has a small error with $0.8372{V_{\pi 2}}$ in our experiment. According to the experimental observations, frequency components $4{\omega _1}$ and $5{\omega _1}$ are still not submerged by the NF of the spectrum, and K function is monotonous, near the quadrature point of the MZM1, that still agree with the conditions for the verification experiment. Therefore, the influence of this small amplitude error on the controller is not considered. Using a DDS with a variable amplitude, the ideal amplitude of the dither for the MZM2 $({{A_2} = 0.8372{V_{\pi 2}}} )$ can be determined accurately and automatically.

Figure 8 shows a simplified flowchart of the MZM operating point controller. The controller comprises two parts: searching and tracking. The objectives of the search include the quadrature point of the MZM1, the offset point, and the half-wave voltage of the MZM2. Furthermore, the dither signal of the MZM2 must be closed when searching for the quadrature point of the MZM1. Thereafter, the MZM1 is at the quadrature point, and the DC voltage of the MZM2 remains constant at ${V_{DC2}}$and ${V_{DC2}} + {{{V_{\pi 2}}} / 2}$ during the FFT period in turn. We sample the output of the MZM cascade structure and calculate amplitudes ${Y_1}$, ${Y_2}$, ${Z_1}$, ${Z_2}$,and ratio K; notably, K is the default target, ${K_{set}}$.

 

Fig. 8. Simplified flowchart of the operating point controller based on the dual-MZM cascade structure.

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After completing the search, we begin the tracking cycle. First, the operating point of the MZM2 is tracked at the early stage of each tracking cycle. The magnitude and direction of the operating point drift of the MZM2 are determined by ratio k₂ between the amplitudes of frequency components $3{\omega _1}$ and $6{\omega _1}$. We change the DC voltage of the MZM2 based on the drift conditions. After tracking the offset, we begin to track the operating point of the MZM1. We assume that the operating point of the MZM2 do not change during this process because of the low drift speed of the MZMs.

The DC voltage of the MZM2 remains at the present ${V_{DC2}}$ and ${V_{DC2}} + {{{V_{\pi 2}}} / 2}$ for the FFT period in turn, and K is calculated. The magnitude and direction of the operating point drift of the MZM1 are calculated using K. We change the DC voltage of the MZM1 based on the drift conditions, and complete one tracking cycle.

4. Experimental results

To confirm the performance of the MZM operating point controller, we test the stability of the average optical power of the MZM1 output using a Thorlabs S144C InGaAs integrating head and PM100D optical power meter. When dither signal ${V_1} = {A_1}\cos ({{\omega_1}t} )$ and DC voltage are applied to the MZM1, its average optical power can be obtained as follows:

We conduct an output average optical power test for the traditional control method (Section 2) and the proposed control method in 2 h. In these experiments, the two control methods employ the same laser source and dither signal amplitudes ${A_1} = 0.05{V_{\pi 1}}$(${J_0}({{m_1}} )\approx 0.995$). Both the MZM control methods are operated at the quadrature point. We perform the two control methods using the same hardware circuit. Figure 9 shows the two groups of average optical power data. The rated output power of the laser is 10 mW, however the MZM and transmission optical path experience power losses. The output optical power stability of traditional MZM operating point controller is 1.0%, and the corresponding operating point stability is ±0.59° according to Eq. (17). The output optical power stability of the proposed MZM operating point controller is 0.63%, and the corresponding operating point stability is ±0.36°. Thus, the proposed MZM operation control method improves the control stability by 39% at the quadrature point.

 

Fig. 9. Average optical power of the MZM1 with the (a) traditional and (b) proposed operating point controllers.

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We apply the two MZM operating control methods to the PSLRF (Fig. 7) and test its ranging stability. We modulate the intensity of the laser at a 200 MHz frequency. The output laser of the MZM1 is divided into the following two beams by the first depolarizing splitter (a nonpolarizing beam splitter prism; NPBS): measurement and reference. The reference beam directly penetrates the PD to generate a reference signal. The measurement beam is reflected onto the device by a mirror at the measured distance. Next, the measurement beam penetrates the PD to generate a measurement signal. The distance is calculated by phase difference between the measurement and reference signals. We employ down-conversion signal processing and an auto-digital phase-meter to calculate the phase difference, updating the phase data every 0.1 s.

Figure 10 shows bias voltage ${V_{DC}}$ applied to the controlled MZM1 and the ranging results obtained without the controller as well as those obtained using the traditional and proposed MZM operating point control methods. According to the ranging results obtained without the controller, the drift of the MZM introduces a large-ranging error on the order of millimeters. The PSLRF using the traditional MZM operating point controller exhibits a stability of $63\,{\mathrm{\mu} \mathrm{m}}({3\sigma } )$ in 1 h. Conversely, using the proposed MZM operating point controller, the stabilities of the ranging results are $17\,{\mathrm{\mu} \mathrm{m}}({3\sigma } )$ and $39\,{\mathrm{\mu} \mathrm{m}}({3\sigma } )$ in 1 min and 1 h, respectively. The experimental results confirms that the operating point controller based on the dual-MZM cascade structure exhibits higher operating point control stability at the quadrature point.

 

Fig. 10. DC voltage applied to the MZM1 and the ranging results without controller, as well as with the traditional and proposed operating point control methods. (a) DC voltage applied to the uncontrolled MZM. (b) Ranging results with the uncontrolled MZM. (c) DC voltage applied to the MZM employing the traditional operating point controller. (d) Ranging results with the traditional operating point controller. (e) DC voltage applied to the MZM using the proposed operating point controller. (f) Ranging results using the proposed operating point controller.

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In one control cycle, the operating point of the MZM1 maintains the control output of the previous control cycle, meanwhile the MZMs still drifts. The controller has a loop delay that affects the control performance. The delay time is the time of one control cycle, that is $\Delta t = N{{\sum Avg} / {{f_{sample}}}}$, sampling frequency is ${f_{sample}} = 150\textrm{kHz}$, the number of FFT points is $N = 1024$, and the sum of the average number performed in one control cycle is $\sum Avg = 20$ (we perform 16 average times to calculate${Y_1},{Y_2},{Z_1},{Z_2}$ and four average times to control the operating point of the MZM2.) The delay time $\Delta t$ is about 0.14 s. The maximum drift speed of the operating point measured in our laboratory environment is ${{{{0.08}^ \circ }} / \textrm{s}}$. This operating point error caused by the delay time is approximately ${0.011^ \circ }$ in theory; this is considerably less than the operating point stability of the controller in Fig. 9. Therefore, the effects of the delay time on the controller performance and PSLRF are not considered in our experiment. If we attempt to further improve the control stability by increasing the average times, the delay time and drift speed of the MZM will be the main limiting factors.

5. Conclusion

Operating-point drifts in MZMs result in large-range errors. Therefore, we investigate an operating point control method for an MZM in a PSLRF. Using a traditional controller based on the ratio of the fundamental to the second harmonics, we confirm that the stability of the MZM operating point control at the quadrature point is lower than those at other operating points. This is mainly attributed to the easy masking of the second harmonics by the spectrum NF and the nondeterministic drift direction. Laser applications require a quadrature point for the MZM, and a small dither signal for the controller. Accordingly, we propose an operating point control method for the MZM based on a dual-MZM cascade structure. The proposed control method solves the aforementioned problems and increases control stability at the quadrature point. Furthermore, we compare the operating point stabilities of the traditional and proposed methods using the same dither signal by measuring the output average optical power. The optical power stability of the proposed method is 0.63%, corresponding to the operating point stability of the MZM, that is ±0.36°. The proposed MZM operation control method increases the control stability by 39% at the quadrature point. Using the proposed MZM operating point controller, the stabilities of the ranging results are $17\,{\mathrm{\mu} \mathrm{m}}({3\sigma } )$ and $39\,{\mathrm{\mu} \mathrm{m}}({3\sigma } )$ in 1 min and 1 h, respectively. The experimental results confirm that the operating point controller based on the dual-MZM cascade structure exhibits higher control stability at the quadrature point than those at other points.

Although the implementation process of the proposed operating point control method is complicated and requires an additional MZM and additional set of control circuits, it provides important reference for applications requiring high stability. Specially, for the applications such as satellites baseline measurement of the synthetic aperture radar, the stability of ranging results is improved to $39\,{\mathrm{\mu} \mathrm{m}}({3\sigma } )$ meeting the requirements of long-term continuous work.