Automatic phase-matching technique for cascaded electro-optic frequency combs

Jin Wang; Feifei Yin; Zikai Yin; Zheng Wang; Haoyan Xu; Yitang Dai; Kun Xu

doi:10.3788/COL202523.013901

1. Introduction

The optical frequency comb (OFC) is a breakthrough in the field of laser technology, providing a high-precision frequency source in a simple and elegant way. It establishes a two-way coherent connection between the optical domain and the radio frequency (RF) domain, effectively building a bridge between optical frequencies and microwave frequencies^[1]. It has been widely used in microwave signal processing^[2,3], spectroscopy^[4,5], high-performance waveform synthesis^[6,7], high-precision measurement^[8,9], optical communications^[10,11], and ultra-wideband microwave signal reception^[12]. Among the methods of realizing optical frequency combs, electro-optic frequency combs (EOFCs) generated by electro-optic modulators (EOMs) have unique advantages, such as adjustable repetition frequencies, high-power sidebands, and reconfigurable spectra. Their high flexibility and reliability make them widely applicable in practical scenarios^[13–15].

When a continuous-wave (CW) pump laser is modulated by a sinusoidal microwave signal, sidebands are generated on both sides of the pump frequency. Hence, the generation of EOFCs essentially involves the creation of the sum and the difference frequencies derived from the combination of the optical pump signal and the microwave modulation signal^[16]. Since the number of optical comb lines generated by a single modulator is limited, multiple EOMs are typically cascaded to produce an EOFC with an excellent frequency tunability, a large number of comb lines, and a flat spectrum^[17]. One of the key factors that determines the performance of these cascaded EOFCs is the phase relationship of the input RF signals between the multiple EOMs in the system^[18]. Due to physical manufacturing differences and non-ideal performance of RF transmission components, the laser optical path delay and the RF circuit delay lengths are generally not equal within the system. These factors will inevitably lead to a certain delay difference in the transmission of multi-input RF signals. In a cascade EOFC generation system, the common method to achieve phase synchronization involves adding a phase shifter before each modulator’s RF signal input and performing phase synchronization through manual blind adjustment^[19]. This method is inefficient and time-consuming, especially in EOFC scenarios based on multiple cascaded modulators. Another solution is to use a neural network algorithm, such as reinforcement learning, to train an automatically optimized EOFC, thereby replacing the manual tuning method^[20]. However, the training time of this process is time-consuming, with a complex network structure and limited interpretability. Therefore, in the EOFC system, it is of great significance to construct a high-efficiency phase synchronization method in a simpler and more efficient way.

In this Letter, we propose an automatic phase-matching method for EOFC generation. To be specific, optical carriers and symmetrical sidebands at certain positions are selected to analyze the relationship between their corresponding optical power values and the driving RF phases of the EOMs. Based on this, a microcontroller feedback circuit is designed to monitor optical power changes in real time and adjust the EOM’s driving RF phase. Experiments show that the comb lines of the EOFC are quickly optimized at different frequencies. Moreover, we also have verified the long-term stability of the entire system.

2. Principle

The proposed scheme is shown in Fig. 1, illustrating the experimental setup for the phase automatic optimization in the cascaded EOM for the optical frequency combs. In this scheme, we firstly use a laser to generate a stable and continuous optical carrier wave. This optical carrier then passes through a Mach–Zehnder modulator (MZM) and three optical phase modulators (PMs) to achieve multi-level modulation effects. The RF signals from the microwave source are evenly distributed into four channels to drive the four EOMs. To adjust the RF signal phase dynamically, an electronically controlled phase shifter is placed between the RF input and the PM, allowing real-time phase changes with high flexibility and accuracy. Some special optical combs are selected through the comb line selector and connected to the micro-controller unit (MCU). The photodiode (PD) circuit module and the analog-to-digital converter (ADC) circuit module process the signals, which then control the electronically controlled phase shifter for phase matching. Cascading multiple-phase modulators produce a richer and broader spectrum. The function of the MZM is based on the principle of time-to-frequency (TTF) mapping, the time domain waveform shape can be mapped to the frequency domain envelope. To ensure accurate MZM control, RF attenuators and amplifiers are used to fine-tune the RF power input so that the MZM operates at the quadrature bias point. This optimization measure flattens the optical comb generated by the cascaded EOM system^[21]. The MZM works at a quadrature bias point with a direct current (DC) bias voltage of $V_{bias}$ , and the working mode is a push-pull mode. $E_{0}$ is the input light intensity and $ω_{0}$ is the input light angular frequency.

Figure 1.Structural diagram of the automatic phase-matching system for optical frequency combs generated by cascaded electro-optic modulation. LD, laser diode; MZM, Mach–Zehnder modulator; EVOA, electronically variable optical attenuator; PA, power amplifier; PM, phase modulator; PS, phase shifter; MCU, microcontroller unit; OSA, optical spectrum analyzer.

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The output light field formula after the MZM and the first-stage PM is $E_{out 1} (t) \propto E_{0} \cdot e^{j ω_{0} t} \cdot {e^{j m_{m z m} \cos w t} + e^{j [m_{m z m} \cos (w t + π) + V_{bias}]}} \cdot e^{j m_{p} \cos (w t + φ_{1})} .$ (1)

The phase of PM’s RF signal is $φ_{1}$ , $m_{m z m}$ is the modulation index of MZM, and $m_{p}$ is the modulation index of PM. w is the angular frequency of the input RF signal. Convert the above formula into a Bessel function expression as $E_{out 1} (t) \propto E_{0} e^{j w_{0} t} \cdot \sum_{n = - \infty}^{+ \infty} \sum_{k = - \infty}^{+ \infty} [(e^{j n π} + e^{j V_{bias}}) e^{j k φ_{1}} \cdot J_{n} (m_{m z m}) J_{k} (m_{p}) e^{j (n + k) w t}] .$ (2)

Among them, $n$ is the order of the modulation sidebands generated by MZM, $k$ represents the order of the modulation sidebands generated by the PM, and $J_{n}$ ( $*$ ) is the corresponding Bessel function of the first type. Let $M = n + k$ from the above formula and we obtain $E_{out 1} (t) \propto E_{0} \cdot e^{j (ω_{0} + M ω) t} \cdot \sum_{n = - \infty}^{+ \infty} [(e^{j n π} + e^{j V_{bias}}) e^{j (M - n) φ_{1}} \cdot J_{n} (m_{m z m}) J_{M - n} (m_{p})] .$ (3)

Then, the strength of the $M$ -th comb line can be expressed as $I_{M} \propto E_{0}^{2} {| \sum_{n = - \infty}^{+ \infty} [(e^{j n π} + e^{j V_{bias}}) e^{j (M - n) φ_{1}} \cdot J_{n} (m_{m z m}) J_{M - n} (m_{p})] |}^{2} .$ (4)

The above formula shows that the phase difference between the MZM and the first-stage PM RF drive signals affects the amplitude of the generated optical frequency comb (OFC) lines^[22].

The red dashed line and solid line in Fig. 2(a) represent the highest symmetric left and right sidebands, respectively (one of the symmetrical sidebands on the carrier’s left is the negative sideband, while the one on the right is the positive sideband), in which two single intersection points represent phase-matching points. In contrast, other symmetric sidebands are shown to have more than two crossover points without any discernible power relationship between them. In the figure, the intersection of the phase difference of 0° and 360° can be considered to be coincident. Therefore, as shown in Fig. 2(b), the phase matching of the MZM and the PM is achieved by calculating the power difference of specific sidebands. The algorithm flow chart is shown in Fig. 2(c). Set the minimum difference range to flag_min, a near-zero threshold. After starting, initialize two ADC values (ADC1_1 and ADC1_2) for power measurements from two channels. Initialize $Δ$ for real-time power difference and P for tracking the phase shifter’s IO port status. The duration required for the algorithm to collect data from the PD and process it in the MCU for computation constitutes one iteration. Since the electrical phase shifter is controlled by 6 TTL levels, with level numbers ranging from 0 to 63, a total of 64 iterations are required to complete a phase scan from 0° to 360°. During the phase-matching optimization process, the number of iterations required can vary between 0 and 64, with a time consumption of less than 1000 ms. In a loop, calculate the current power difference $Δ$ and compare it to flag_min. If $Δ$ exceeds flag_min, then increment P to adjust the phase shifter. Repeat until the minimum power difference is achieved. Finally, output P, the phase-matching point between the MZM and the first-level PM.

Figure 2.Principle analysis and algorithm flow chart. (a) Changes in the OFC symmetric sideband optical power with the phase difference between MZM and PM. (b) Changes in the power difference between the two-order sidebands corresponding to the red line in (a). (c) Phase-matching algorithm of the MZM and PM flow chart.

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Introduce multiple PMs based on the previous phase matching, then the system output light field expression is Eq. (5), $A = \sum_{i = 1}^{3} \cos φ_{i}$ , $B = \sum_{i = 1}^{3} \sin φ_{i}$ , $\cos φ_{p} = \frac{A}{\sqrt{A^{2} + B^{2}}}$ , $\sin φ_{p} = \frac{B}{\sqrt{A^{2} + B^{2}}}$ . Similarly, we can derive Eq. (6) for each optical comb line, $E_{out 1} (t) \propto E_{0} e^{j w_{0} t} \cdot {e^{j m_{m z m} \cos w t} + e^{j [m_{m z m} \cos (w t + π) + V_{bias}]}} \cdot \prod_{i = 1}^{3} e^{j m_{p} \cos (w t + φ_{i})} \propto E_{0} e^{j w_{0} t} \cdot {e^{j m_{m z m} \cos w t} + e^{j [m_{m z m} \cos (w t + π) + V_{bias}]}} \cdot e^{j m_{p} \sqrt{A^{2} + B^{2}} \cos (w t + φ_{p})},$ (5) $I_{M} \propto E_{0}^{2} {| \sum_{n = - \infty}^{+ \infty} [(e^{j n π} + e^{j V_{bias}}) e^{j (M - n) φ_{p}} \cdot J_{n} (m_{m z m}) J_{M - n} (m_{p}^{'})] |}^{2},$ (6)where the phase of RF signal input to the $i$ th PM is $φ_{i}$ . In the above formula, $m_{p}^{'} = m_{p} \sqrt{A^{2} + B^{2}}$ . Cascading increases the modulation depth of the PM, increasing the amplitude of the otherwise attenuated frequency components. The number of optical comb lines will increase, and the particularity of the optical power difference change at the symmetry point of the optical comb line is no longer satisfied. As the comb broadens, the carrier optical power trends downward. The wider the optical comb is, the more stable the carrier’s optical power becomes.

It is necessary and convenient to analyze changes in carrier optical power. After the new PM is introduced, the relationship between the carrier optical power difference and the phase difference (the unit phase difference is 1°) is shown in Fig. 3(a). When the carrier optical power difference changes minimally, it is close to the phase matching state. As shown in Fig. 3(b), it is still satisfied for cascading more PMs.

Figure 3.Principle analysis and algorithm flow chart. (a) Secondary PM optical carrier optical power difference (change in unit phase difference). (b) Three-level PM optical carrier optical power difference (change in unit phase difference). (c) Phase-matching algorithm flow chart of the MZM and multi-level PM.

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Therefore, using the sliding window to find the flattest point of the carrier optical power difference and using the center of the window as the corresponding phase-matching point, the core of the algorithm is to track and analyze the changes in the optical power value of the optical carrier in the optical frequency comb. The flow chart is shown in Fig. 3(c). Initialize an array ADC2 for these power changes and define flag_var to record the minimum variance in the sliding window. $S$ is the number of electrical phase shifter levels corresponding to a full cycle of the phase change, and $n$ is recorded as the current electrical phase shifter level. After starting, the variations in optical power during a complete phase scan from 0° to 360° are recorded. For the matching of the second and third phase matches, a total of 64 iterations are required, with a time consumption of less than 1000 ms following data acquisition for processing. Use a sliding window of five points to scan the data, calculating the variance as a measure of dispersion. Continuously slide the window and update flag_var with the minimum variance. Upon finding the minimum variance and its phase shifter level, output the value to determine the phase-matching point for subsequent second and third phase match cascades.

3. Experimental Results

To validate the ability of our proposed method to generate OFC quickly with adjustable comb line spacing and more comb lines, we use the comb line selector to choose the three specific optical comb lines and design a circuit for experimental testing. The detailed experimental setup is illustrated in Fig. 1. The slicing bandwidth of this comb line selector is 12.5 GHz. The circuit uses the STM32 MCU, known for its fast computing speed and low cost. The PD module in the single-chip circuit has an output bandwidth of 100 kHz, a response rate of 0.9 A/W, and a gain of $2.7 \times 10^{6} V / W$ . The ADC has a quantization accuracy of 12 bits.

In the experiment, the output power of the laser was set to 19 dBm, and the center frequency of the laser was stable at 193.39 THz. The RF source frequency is 13 GHz and the output power is set to 16 dBm. The PM (EOspace, PM-5SES-20-PFA-U/V) half-wave voltage ( $V_{π}$ ) in the experiment is 4.2 V. First, select the three filtered channel slices: the two symmetrical points with the highest first-stage PM sideband power and the carrier position. Then, precise phase control is achieved through the MCU after EOFC filtering. The MCU uses a search algorithm to find the minimum sideband power difference and phase-match the first PM and MZM. Next, it calculates the minimum variance of the carrier optical power difference using a sliding window for the subsequent cascaded PMs. Finally, phase matching of all PM levels is achieved. After completing the phase matching of each level of PM, the experimental data will be read by a spectrometer and plotted in detail in Fig. 4. Under the conditions of a given random initial phase, an optical comb with up to 53 comb lines and a flatness within 3 dB is finally obtained. The measurement results show that this automatic phase-matching method enables the EOFC to achieve good performance.

Figure 4.Results of multi-level PM phase-matching EOFC. (a) First-level PM output spectrum. (b) Second-level PM output spectrum. (c) Third-level PM output spectrum.

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The time required for phase matching at different operating frequencies is measured. As shown in Fig. 5, the actual measurement results demonstrate that when the frequency of the input RF signal changes within the range of 10–14 GHz clearly, the automatic control method of this study consistently achieves adaptive optimization of the phase relationship in a relatively short time. At the same time, due to the randomness of the initial phase, the time required to complete phase matching will vary. However, this method ensures that the phase matching time is always within 2000 ms, reflecting its efficiency.

Figure 5.Time to complete phase matching at different operating frequencies.

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In addition, this study conducts detailed measurements of the flatness performance after phase matching is completed at different operating frequencies. As shown in Fig. 6, the actual measurement results show that when the frequency of the input RF signal changes within the range of 10–14 GHz, the optimal performance of the EOFC and the number of comb lines within 3 dB flatness reach 40 and above.

Figure 6.Number of comb lines within 3, 5, and 10 dB flatness at different operating frequencies.

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To evaluate the stability of the EOFC power generation system, we performed sampling and analysis using a high-precision spectrometer (YOKOGAWA AQ6370C). The measurement accuracy of the spectrometer for optical comb power measurement is 0.01 dB. The optical frequency comb spectrum is collected every 1.645 s for a total of 99 min, with a total of 3600 spectral data. The stability of the system is accurately measured by the fluctuation changes of a large number of sampling points over a long period of time. We randomly select an optical frequency comb line as the starting reference point and the peak of the optical frequency comb as the sampling point. The error bar plot in Fig. 7 illustrates the long-term stability of the optical power of the comb lines. The data points, represented by red dots, denote the average optical power, while the error bars indicate the standard deviation of these measurements. For most comb lines, the power deviation is typically less than 2 dBm. The majority of the error bars are relatively short, with standard deviations below 0.5 dB, indicating high stability and low variability in the measurements. It shows that the system output fluctuation is very small and the long-term operation stability is high, which ensures the reliability of the system as a light source for various applications.

Figure 7.System stability analysis chart.

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4. Conclusion

In summary, we propose an automatic phase-matching EOFC generation method, which has practical significance for generating OFC light sources with excellent performance. In the method proposed in this Letter, by accurately capturing the optical power changes of the EOFC comb line at three specific positions, the phase synchronization problem in the optical comb system generated by the cascaded EOM is effectively solved. At the same time, the experiment has verified the strong stability in the EOFC generation system under long-term operation. The obtained high-performance stable OFC light source will bring new development possibilities for future optical communication systems.

Category: Microwave Photonics

Received: Jun. 17, 2024

Accepted: Jul. 29, 2024

Published Online: Feb. 10, 2025

The Author Email: Feifei Yin (yinfeifei@bupt.edu.cn)

DOI:10.3788/COL202523.013901

CSTR:32184.14.COL202523.013901