## Abstract

A photonic-based reconfigurable microwave frequency divider using two cascaded dual-parallel Mach-Zehnder modulators (DP-MZMs) is proposed. The first DP-MZM is driven by the input microwave signal, whereas the second DP-MZM is incorporated in an optoelectronic oscillator (OEO) loop and driven by the feedback signal. By properly setting the working conditions of the two DP-MZMs, the frequency of the input microwave signal is divided and the frequency-divided signal will oscillate in the OEO loop, with a tunable frequency-division factor determined by the bias conditions of the DP-MZMs. An experiment is performed. The reconfigurable microwave frequency divider is demonstrated with a frequency-division factor of 1.5, 2.5, 2, or 3, and the phase noise of the frequency-divided signals is also evaluated, which has an improvement of 3.22, 7.60, 5.80, or 9.49 dB at 10-kHz frequency offset, respectively, compared with that of the input microwave signals.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nowadays, millimeter-wave signals with unique advantages of large bandwidth and anti-interference have attracted a lot of attention. It can be widely used in radar, wireless communication, and many other systems [1,2]. The millimeter-wave frequency synthesizer is a key module in such millimeter-wave systems, which can be implemented in many ways such as direct frequency synthesis [3,4], direct digital frequency synthesis [5], and phase-locked loop (PLL) frequency synthesis [6]. In the PLL frequency synthesizer, the frequency divider plays a very important role, whose performance can greatly affect the quality of the generated signals and thus determine the performance of the whole millimeter-wave system.

Conventionally, the frequency divider is implemented by pure-electronic devices. However, the electrical-based frequency dividers have some drawbacks of limited bandwidth and poor frequency tunability caused by the well-known “electronic bottlenecks”. In the past few years, microwave photonics [7] has been proved to be a promising solution to overcome these limitations. Different frequency dividers based on microwave photonics have been reported [8–16], taking the advantages of large bandwidth, good tunability, immunity to electromagnetic interference offered by it.

The method in [8] employed the nonlinear dynamics of semiconductor lasers to achieve frequency division, in which the slave laser was driven into the dynamical period-two state by optical injection. Both the fundamental microwave frequency and its subharmonic frequency could be locked simultaneously when an external microwave near either frequency was applied. Therefore, the method could generate the 1/2 frequency-divided signal. Some other methods based on optical injection were proposed to obtain higher frequency-division factors [9,10]. The method in [9] could achieve both the 1/3 and the 1/4 frequency division, and the method in [10] could achieve multiple frequency-division factors from 2 to 4. Another kind of frequency divider based on the cross-gain modulation effect between the injected optical signal and the intra-cavity lasing fields was proposed in [11], which achieved a frequency-division factor from 2 to 6. However, the operating frequency range of these methods is limited by the low speed of the carrier migration in the semiconductor lasers and semiconductor devices.

To address this issue, some frequency dividers based on the optoelectronic oscillator (OEO) were proposed. To begin with, a 1/2 frequency divider has been realized in [12]. In the system, a Mach-Zehnder modulator (MZM) was incorporated in the OEO. The MZM was biased at the null point and driven by the combination of the input microwave signal and the feedback signal from the OEO loop. When the oscillation frequency was half the frequency of the input microwave signal, a stable oscillation could be established. Besides 1/2 frequency dividers, several methods that can realize larger frequency-division factors were proposed. One method based on an OEO loop in [13] could realize 1/3 frequency division. In the system, the microwave signal was first converted to the optical domain and then injected into the OEO loop, which was composed of an MZM, a photodetector (PD), a phase shifter (PS), an electrical amplifier (EA), and an electrical band-pass filter (EBPF). When the MZM was biased at the null point and both the PS and the EA were properly set, the desired signal whose frequency was 1/3 of the frequency spacing between the two optical wavelengths sent to the OEO loop could be generated. To realize a tunable frequency-division factor, a frequency divider that can realize both 1/2 and 1/3 frequency division was proposed in [14] based on a dual-parallel Mach-Zehnder modulator (DP-MZM) in an OEO loop. The two sub-MZMs of the DP-MZM were driven by the input microwave signal and the oscillation signal from the OEO loop, respectively. The frequency-division factors could be switched through changing the DC biases of the two sub-MZMs, and the phase gain of the loop could be adjusted by changing the DC bias of the main-MZM. The frequency-division factor was increased up to 6 in [15]. In the method, an optical frequency comb generator was incorporated in the OEO loop. The input signal could be locked to the *N-*th comb line with *N* representing the frequency division factor, by matching the center frequency of the EBPF and the frequency of the input signal.

It is noted that all the photonic-based frequency dividers mentioned above only focus on realizing the integer frequency division. A method based on microwave photonics to realize the fractional frequency division was reported in [16], which achieved a frequency-division factor of 1.5. In the scheme, two MZMs biased at the null point were cascaded. The first MZM was driven by the input signal and the second MZM was placed in an OEO loop and driven by the feedback signal. When the phase and amplitude gain of the OEO loop were properly adjusted, a signal whose frequency was 2/3 of that of the input signal could be obtained.

To achieve more flexible frequency division to meet the needs of the frequency synthesizer, it is highly desirable that a frequency divider that can realize both the integer frequency division and the fractional frequency division with simultaneous tunable frequency-division factors can be implemented. In this paper, a reconfigurable frequency divider based on two cascaded DP-MZMs is proposed. The first DP-MZM is driven by the input microwave signal, whereas the second DP-MZM is incorporated in an OEO loop and driven by the feedback signal. By properly setting the working conditions of the two DP-MZMs, the frequency of the input microwave signal is divided and the frequency-divided signal will oscillate in the OEO loop, with a tunable frequency-division factor determined by the bias conditions of the DP-MZMs. Compared with the method in [16], the proposed scheme can realize not only the fractional frequency division but also the integer frequency division. The division factor can be switched by changing the bias voltages of the two DP-MZMs. 2/3, 2/5, 1/2, and 1/3 frequency division is experimentally verified. The phase noise of the frequency-divided signals is 3.22, 7.60, 5.80, and 9.49 dB better than that of the corresponding input signals at 10-kHz frequency offset.

## 2. Principle

The schematic diagram of the proposed photonic-based reconfigurable frequency divider is shown in Fig. 1, which mainly consists of a laser diode (LD), two cascaded DP-MZMs, an erbium-doped fiber amplifier (EDFA), a PD, a PS, a low-noise amplifier (LNA), an electrical attenuator (ATT), and a power amplifier (PA). A continuous-wave (CW) light wave from the LD is applied to the first DP-MZM (DP-MZM1), which is driven by the input microwave signal. By properly biasing DP-MZM1, +*m*th- and -*n*th-order (*m* and *n* are non-negative integers) optical sidebands can be generated with other optical sidebands suppressed as shown in Fig. 1(a). The output of DP-MZM1 can thus be expressed as

*m*th- and -

*n*th-order optical sidebands, ${f_c}$ and ${f_s}$ represent the frequencies of the optical carrier and the input microwave signal, and ${\varphi _m}$ and ${\varphi _n}$ represent the phase of the two optical sidebands.

Then the optical signal from DP-MZM1 is injected into DP-MZM2 and modulated by the feedback oscillation signal ${V_\textrm{0}}\textrm{cos(2}\pi {f_{oeo}}\textrm{t + }{\varphi _{oeo}}\textrm{)}$ from the OEO loop, with ${V_\textrm{0}}$, ${f_{oeo}}$, and ${\varphi _{oeo}}$ representing the amplitude, the frequency, and the phase of the oscillation signal. As done in DP-MZM1, DP-MZM2 is also biased to generate the + *p*th- and -*q*th-order (*p* and *q* are non-negative integers) optical sidebands with other optical sidebands well suppressed as shown in Fig. 1(b). Therefore, the optical signal from DP-MZM2 can be expressed as

The optical signal from DP-MZM2 is amplified by the EDFA and then detected in the PD. The photocurrent from the PD is

*G*represent the total loss of the optical link and the gain of the OEO loop, respectively. As can be seen from Eq. (3), four microwave signals with different frequencies as well as a direct current component are generated. To maintain the oscillation in the OEO loop, the oscillation frequency should be equal to one of the four frequencies.

For the microwave signal centered at (*m *+ *n*)*f _{s}*, the condition to maintain the oscillation is

*f*=(

_{oeo}*m*+

*n*)

*f*, which can only be used to achieve frequency multiplication but not frequency division. For the microwave signal centered at (

_{s}*p*+

*q*)

*f*, the condition to maintain the oscillation is

_{oeo}*f*=(

_{oeo}*p*+

*q*)

*f*, i.e.,

_{oeo}*p*+

*q*=1, which means the optical signal is single-sideband (SSB) modulated in DP-MZM2. Under this condition, the oscillation frequency has no relationship with the input microwave signal. If enough gain is provided, it will be a free-running OEO and the oscillation frequency should be selected by a narrowband EBPF. For the other two microwave signals centered at (

*m*+

*n*)

*f*±(

_{s}*p*+

*q*)

*f*, if the gain introduced by the amplifiers is greater than the total loss of the link and the phase gain of the loop is a multiple of 2

_{oeo}*π*, the OEO can oscillate stably when the following relationship is satisfied

If *m *+ *n *< *p *+ *q*±1establishes, a frequency divider can be implemented. In comparison, if *m *+ *n *≥* p *+ *q*±1, a signal with a frequency equal to or higher than the input frequency can be generated. It can also be seen from Eq. (5) that when *m*, *n*, *p*, and *q* are fixed, two frequencies at (*m *+ *n*)/(*p *+ *q*+1) *f _{s}* and (

*m*+

*n*)/(

*p*+

*q*-1)

*f*will oscillate in the OEO loop theoretically due to the injection of the input microwave signal. In general, the frequency response of the devices in the system decreases with the increase of the operating frequency. Therefore, if no other frequency-selective devices are used in the OEO loop, the frequency at (

_{s}*m*+

*n*)/(

*p*+

*q*+1)

*f*is more likely to oscillate in the OEO loop than that at (

_{s}*m*+

*n*)/(

*p*+

*q*-1)

*f*. In this paper, no narrowband EBPF is employed, so the frequency divider is demonstrated by generating the frequency-divided signal at (

_{s}*m*+

*n*)/(

*p*+

*q*+1)

*f*. Different frequency-division factors are achieved by changing the specific orders of the two optical sidebands generated by the two DP-MZMs. By using the DP-MZM,

_{s}*m*and

*n*(

*p*and

*q*) can be 1 and 0 via SSB modulation, 1 and 1 via carrier-suppressed double-sideband (CS-DSB) modulation, 1 and 3 via carrier-suppressed single-sideband (CS-SSB) modulation with relative high modulation index, and 2 and 2 via optical modulation for frequency-quadrupled signal generation.

## 3. Experimental results and discussion

An experiment based on the setup shown in Fig. 1 is performed to verify the proposed system. Regarding the analysis in Section 2, the scheme can realize the (*m *+ *n*)/(*p *+ *q*±1) frequency division, where *m*, *n*, *p*, and *q* are non-negative integers. In the experiment, *m*, *n*, *p*, and *q* are limited to equal or less than 2, because optical sidebands with even higher orders cannot be generated with other optical sidebands well suppressed without using optical filters.

Firstly, a 13-dBm CW light wave centered at 1554.43 nm from an LD (HLT-ITLA-C-20-0-1-FA) is injected into DP-MZM1 (FTM7961EX/301), which is modulated by the input microwave signal generated from a microwave signal generator (MSG, Agilent 83630B). The optical signal from DP-MZM1 is sent to DP-MZM2 (Fujitsu FTM7961EX/301). Then the modulated optical signal from DP-MZM2 is amplified by an EDFA (Amonics, EDFA-PA-35-B) and split by a 1:99 optical coupler. An optical spectrum analyzer (OSA, Ando AQ6317B) is connected to the 1% port, whereas the output from the 99% port is sent to a PD (Discovery Semiconductors, DSC-40S) with 16-GHz bandwidth. The photocurrent from the PD is phase-shifted by a PS (Sage 6705), amplified by an LNA (ALM/145-5023-293, 23-dB gain, 5.85 to 14.5 GHz), attenuated by an ATT (NORSAL IND), and then amplified by a PA (JAC1011-450BC, 35-dB gain, 6.2-11.2 GHz). The phase and amplitude of the electrical signal in the OEO loop can be continuously adjusted by the PS and the ATT. It is worth noting that no EBPFs are incorporated in the scheme because the amplifiers used in the experiment already have certain frequency selectivity. Finally, the electrical signal from the PA is divided into two parts by an electrical coupler (Narda 4456-2, 2-18 GHz), with one part fed back to DP-MZM2 and the other part monitored by an electrical spectrum analyzer (ESA, Keysight N9020B).

#### 3.1 Fractional frequency division

### 3.1.1 2/3 frequency division

According to Eq. (5), 2/3 frequency division can be implemented by setting all *m*, *n*, *p*, and *q* to 1, which can be realized by simply using an intensity modulator (IM). If DP-MZMs are used, ±1st-order optical sidebands can also be obtained by generating the same CS-DSB signal from the two sub-MZMs of a single DP-MZM. Another kind of method is generating an optical signal with odd-order optical sidebands suppressed from one sub-MZM, and then canceling the optical carrier using the pure optical carrier from the other sub-MZM. To simplify the experiment, two IMs (Fujitsu FTM7920FBA/301, Sumitomo T. MXH1.5-20PD-ADC-049-001) are used instead of the two DP-MZMs in this case.

The frequency of the 20-dBm input microwave signal is set to 12 GHz and both the modulators are biased at the null point to implement CS-DSB modulation. Then the phase gain and the amplitude gain of the OEO loop are properly adjusted so that a stable oscillation can be established in the OEO loop. The measured optical spectrum at the 1% output port of the optical coupler is shown in Fig. 2(a). It can be seen that the ±1st-order optical sidebands that serve as the optical carriers in the second modulator are suppressed due to the CS-DSB modulation, which are more than 9.33 dB lower than the four generated optical sidebands in the second modulator. The beating product between the two optical sidebands in the middle of the spectrum is an 8-GHz signal, which is then fed back to the second modulator to modulate the ±1st-order optical sidebands from the first modulator and sustain a stable oscillation in the OEO loop. The electrical spectrum of the feedback signal with a resolution bandwidth of 3.2 MHz is shown in Fig. 2(b). It can be seen that a 2/3 frequency-divided signal at 8 GHz is generated. The left and right insets in Fig. 2(c) are the zoom-in views of the 12-GHz input signal and the 8-GHz oscillation signal, respectively. The phase noise of the input signal and the oscillation signal is measured by the ESA and shown in Fig. 2(c). The phase noise of the 8-GHz frequency-divided signal is 3.22 dB lower than that of the input signal at 10-kHz frequency offset, which is consistent with the theoretical value of 3.52 dB.

The frequency tunability of the proposed system is further studied by varying the frequency of the input microwave signal from 9.75 to 12.75 GHz with a step of 0.75 GHz. As shown in Fig. 3, the frequency of the generated 2/3 frequency-divided signal varies from 6.5 to 8.5 GHz with a step of 0.5 GHz, whereas the power of the generated signal varies within a range of 1.6 dB. In the experiment, although no electrical filter is used, the amplifiers and other electrical devices have their operating bandwidth, thus limiting the demonstrated frequency range of the frequency divider. Some other unwanted frequencies are generated and centered at the input frequency, the 1/3 of the input frequency, and the 4/3 of the input frequency, which is caused by the non-ideal optical spectrum generated in the experiment. However, the unwanted frequencies are more than 21.9 dB less than the desired frequency-divided signals. In practical applications, a purer frequency-divided signal can be obtained by using an EBPF in the OEO loop.

The generation of other unwanted frequencies is mainly due to the non-ideal bias control and extinction ratio of the modulators. In our experiment, no bias control circuit is employed, so the modulators are not precisely biased at the null point. Furthermore, the 20-dB extinction ratio makes it difficult to completely eliminate unwanted optical sidebands even if the bias point is ideally set. Therefore, when a stable oscillation is established, the output of the two modulators will contain some unwanted optical sidebands as shown in Fig. 4. In the output of the first modulator, a residual optical carrier (purple dashed lines) appears, which is injected into the second modulator and modulated by the feedback signal from the OEO loop. Two unwanted optical sidebands (blue dashed lines) are thus generated. The unwanted optical sidebands can beat with the desired optical sidebands (blue solid lines), generating a 1/3 frequency-divided signal and a signal with a frequency identical to the input frequency. Besides, the optical carriers (purple dashed lines) also exists in the second IM. The beating product between the residual optical carriers and the desired optical sidebands can generate both the desired 2/3 frequency-divided component and unwanted 1/3 frequency-divided component. However, because of the relatively lower power of the residual optical sidebands, the desired 2/3 frequency-divided component will be dominant, which indicates that a slight bias drift will not have a significant impact on the performance of the frequency divider. Even if there is a large bias drift, the desired 2/3 frequency-divided signal can still be generated dominantly by employing an EBPF. However, it should be noted that if the output power of the amplifier is fixed and the residual optical carrier power is large, the OEO may be unable to oscillate because too much of the amplifiers’ power is allocated to the unwanted signals.

### 3.1.2 2/5 frequency division

The case of the 2/5 frequency division is also verified. According to Eq. (5), *m*, *n*, *p*, and *q* are set to 1, 1, 2, and 2, respectively. Therefore, DP-MZM1 should be operated as a CS-DSB modulator, and DP-MZM2 should be properly biased to generate the ±2nd-order optical sidebands. In the experiment, an IM that is biased at the null point and driven by a 17.5-GHz signal with 20-dBm power is used to replace DP-MZM1 and generates the ±1st-order optical sidebands. DP-MZM2 is driven by the feedback signal from the OEO loop via a 90° electrical hybrid coupler (Narda 4065, 7.5-16 GHz). The two sub-MZMs of DP-MZM2 are biased at the peak points and the main-MZM is biased at the null point to generate the ±2nd-order optical sidebands. Then the phase gain and the amplitude gain of the OEO loop are well adjusted so that the 2/5 frequency-divided signal at 7 GHz can be generated.

The measured optical and electrical spectra are shown in Figs. 5(a) and (b), respectively. It can be seen from Fig. 5(a) that four dominant optical sidebands are generated at the output of DP-MZM2, and the frequency difference between the two optical sidebands in the middle is 7 GHz, which can beat with each other to obtain the desired 2/5 frequency-divided signal. Then the signal is fed back to DP-MZM2 to maintain a stable oscillation in the OEO loop. It can be seen from the electrical spectrum that the 1/5, 2/5, 3/5, 4/5 frequency-divided components are generated, of which the 2/5 frequency-divided signal is dominant with its power at least 25.45 dB higher than that of others. The phase noise of the 7-GHz oscillation signal and the 17.5-GHz input signal is shown in Fig. 5(c). The phase noise of the desired oscillation signal is about 7.60 dB lower than that of the input signal at 10-kHz frequency offset, which agrees with the theoretical value of 7.96 dB.

The unwanted frequencies shown in the electrical spectrum are also generated due to the generation of other optical sidebands in the optical signal, as discussed in Section 3.1.1. When the stable oscillation is established, some unwanted optical sidebands with low power appear as shown in Fig. 6. When these unwanted optical sidebands beat with the desired optical sidebands, some frequencies that are 1/5, 3/5, and 4/5 of the input frequency are generated. Among these frequencies, only the frequency that is 2/5 of the input frequency can maintain the oscillation of the OEO loop in conjunction with the input signal. Therefore, the 2/5 frequency-divided signal is dominant in the spectrum. An EBPF can be employed to remove these unwanted frequencies.

Note that in this case, i.e., *m*=1, *n*=1, *p*=2, and *q*=2, the 2/3 frequency-divided signal can also be obtained according to Eq. (5). However, the corresponding 2/3 frequency-divided component cannot oscillate if the frequency of the input signal is 17.5 GHz because the frequency of the 2/3 frequency-divided signal is not within the operating bandwidth of the amplifiers. To verify the generation of the 2/3 frequency-divided signal under this case, the frequency of the input signal is adjusted to 10.5 GHz. The corresponding optical and electrical spectra are shown in Fig. 7. From the electrical spectrum, it is obvious that there are other frequency components besides the desired 2/3 frequency-divided signal. However, the 2/3 frequency-divided component is dominant, which is 25.40 dB higher than other components. In addition, the 2/5 frequency-divided frequency is out of the operating bandwidth of the amplifiers, so no 4.2-GHz component is observed in Fig. 7(b).

Comparing the two cases above, we can find that when the modulators, as well as the PS, the ATT, and the amplifiers, are properly adjusted to obtain the desired optical sidebands, the free-running oscillation components can be well suppressed and two kinds of signals whose frequencies are (*m *+ *n*)/(*p *+ *q*±1) of the input frequency as indicated by Eq. (5) can be obtained and fed back to the second modulator to maintain the stable oscillation of the OEO loop. However, the phase gain and the amplitude gain of the OEO loop are different for these two signals. Only the one whose frequency is within the operating bandwidth of the amplifiers and the phase gain is a multiple of 2π can be generated.

#### 3.2 Integer frequency division

### 3.2.1 1/2 frequency division

In the experiment of generating a 2/5 frequency-divided signal, it is found that when the optical carriers in both modulators and the ±1st-order optical sidebands in the second modulator are not well suppressed, 1/2 frequency division can be achieved. The sketch of the 1/2 frequency division is shown in Fig. 8. When an 18-GHz microwave signal from the MSG is used as the input signal and the PS, the ATT, and the amplifiers are properly set, the measured optical spectrum is shown in Fig. 9(a). It can be seen that the optical carrier in the middle is dominant. The frequency difference of the adjacent optical sidebands is 9 GHz, which means that a desired 1/2 frequency-divided signal can be generated. The output signal is then fed back to DP-MZM2 so that the OEO loop can oscillate stably. The electrical spectrum of the 1/2 frequency-divided signal is shown in Fig. 9(b). It can be seen that the power of the 1/2 frequency-divided signal is 37.67 dB larger than that of the 18-GHz input signal. If a purer 1/2 frequency-divided signal is required, an EBPF should be applied to remove the unwanted 18-GHz signal. The phase noise of the input signal and the frequency-divided signal is shown in Fig. 9(c). The phase noise of the frequency-divided signal is -89.98 dBc/Hz at 10-kHz frequency offset and that of the input signal is -84.18 dBc/Hz. A difference of 5.80 dB in the phase noise can be observed, which agrees with the theoretical value of 6.02 dB.

The case of 1/2 frequency division can also be realized when *m*=0, *n*=1, *p*=0, *q*=1 according to Eq. (5). Correspondingly, both the DP-MZMs should be biased to generate the CS-SSB signal with the optical carrier and one of the 1st-order optical sideband equal in power. The sketch of the outputs of the two DP-MZMs is shown in Fig. 10(a). In addition, there are other working conditions for the two DP-MZMs to achieve 1/2 frequency division, and some of them are shown in Figs. 10(b)–(d). The obtained optical signals can also be detected in the PD to generate a desired 1/2 frequency-divided signal. It can be found that, as long as there is an optical carrier and at least one 1st-order sideband in the output of the two DP-MZMs, even if there are other orders of sidebands, the same function of 1/2 frequency division can be achieved. The 1/2 frequency-divided signal has a similar electrical spectrum and phase noise performance as shown in Fig. 9.

### 3.2.2 1/3 frequency division

As indicated by Eq. (5), the parameters should be set to *m*=0, *n*=1, *p*=1, and *q*=1 to realize 1/3 frequency division. To implement SSB modulation in DP-MZM1, the modulator is slightly biased away from the CS-SSB modulation condition, so the optical carrier is not suppressed. In the experiment, the power of the carrier is tuned to be equal to that of the +1st-order optical sideband. DP-MZM2 should be a CS-DSB modulator, so an IM biased at the null point is used to simplify the bias control in the experiment, which is driven by a 19.5-GHz signal with 20-dBm power. When the phase gain and the amplitude gain are well adjusted, a 1/3 frequency-divided signal oscillates in the OEO loop. The optical and electrical spectra are shown in Figs. 11(a) and (b). It can be seen that the optical carrier and the +1st-order optical sideband from the first modulator are suppressed due to the CS-DSB modulation in the second modulator, which are more than 12.82 dB lower than the four dominant optical sidebands. After photodetection, the four dominant sidebands will generate signals with frequencies that are 1/3, 2/3, 1, and 5/3 of the frequency of the input signal. According to Eq. (5), only the 1/3 frequency-divided signal, and the signal with exactly the same frequency as the input signal can oscillate in the OEO. However, because the input frequency is 19.5 GHz, only the 1/3 frequency-divided signal and its 2nd-order harmonic are generated due to the limited operating bandwidth of the amplifiers. The phase noise of the frequency-divided signal is also measured and shown in Fig. 11(c). It can be seen that the phase noise of the generated signal is 9.49 dB lower than that of the original input signal at 10-kHz frequency offset, which shows very good consistency with the theoretical value of 9.54 dB.

It is found that DP-MZM1 can also be replaced by an IM that functions as a DSB modulator to generate the optical carrier and the ±1st-order sidebands to realize 1/3 frequency division. The bias condition of the second IM is kept unchanged to implement the CS-DSB modulation. The sketch of the outputs of the two modulators is shown in Fig. 12. It can be seen that the desired 1/3 frequency-divided component can also be obtained by beating the adjacent 1st-order optical sidebands generated from the second modulator at the PD, which can make the structure of the frequency divider easier to control. Due to the limited extinction ratio and the DC bias drift of the two modulators, the optical carriers are not ideally suppressed in the second modulator. Luckily, as shown in Fig. 12, the beating products between the residual optical carriers and the desired optical sidebands can also generate the desired 1/3 frequency-divided signal, so the influence from the limited extinction ratio and the DC bias drift is relatively small.

It is noted that temperature is an important factor affecting the system stability. Firstly, the bias condition of the two DP-MZMs will drift with temperature. Therefore, in the absence of the bias control circuit, the two modulators cannot be stabilized in an ideal working state for a long time. Secondly, the OEO loop is also affected by temperature. However, compared with a free-running OEO, this effect will be much smaller due to avoiding the use of long optical fibers, which is very sensitive to the temperature change. Fortunately, according to the previous analysis, once a stable OEO oscillation has been established, a small bias drift or loop length change caused by the temperature change will not have a significant impact on the frequency divider, which is mainly due to the injection locking. For all these different cases discussed above, the frequency-divided signals can stably oscillate in the OEO loop for at least 60 minutes at room temperature in our experiment without any control and adjustment, which shows very good operating stability of the system. However, when the temperature change is very large, the balanced state of the OEO loop will be broken, which will affect the system performance.

## 4. Conclusion

In conclusion, a reconfigurable microwave frequency divider based on two cascaded DP-MZMs has been proposed and experimentally studied. In theory, any frequency-division factor of (*p *+ *q*±1)/(*m *+ *n*), with *m*, *n*, *p*, and *q* representing non-negative integers, can be realized if the signal power is large enough to generate high-order optical sidebands in conjunction with optical filters. In the experiment, only the optical carriers, ±1st- and ±2nd-order optical sidebands are used for frequency division. By changing the DC biases of the two DP-MZMs, the cases of 2/3, 2/5, 1/2, and 1/3 frequency divisions are experimentally verified. Limited by the operating bandwidth of the amplifiers used in the experiment, the frequency of the generated signal is between 6.5-9 GHz with their phase noise reduced by about -20lg[(*m *+ *n*)/(*p *+ *q*±1)] compared with that of the input signals.

## Funding

National Natural Science Foundation of China (61971193); Natural Science Foundation of Shanghai (20ZR1416100); State Key Laboratory of Advanced Optical Communication Systems and Networks (2020GZKF005); Fundamental Research Funds for the Central Universities.

## Disclosures

The authors declare no conflicts of interest.

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