A Symmetric Parity–Time Coupled Optoelectronic Oscillator Using a Polarization–Dependent Spatial Structure

Abstract

We propose and experimentally investigate a symmetric parity-time (PT) coupled optoelectronic oscillator (COEO) based on a polarization-dependent spatial structure. In such a COEO system, the gain/loss and coupling coefficients of two orthogonal polarization optical waves can be controlled by adjusting the polarization controller (PC) and the bias voltage of a Mach-Zehnder modulator (MZM). The single-mode selection of a microwave signal can be implemented by the PT symmetry breaking of a special mode. The performance of the proposed COEO is experimentally examined, and a 10.0 GHz microwave signal with a phase noise of −109.1 dBc/Hz @ 10 kHz and a side mode suppression ratio of 51.4 dB is generated. Moreover, an optical frequency comb with a comb tooth spacing of 10.0 GHz and a bandwidth of 100 GHz within a 10 dB amplitude variation can be simultaneously generated.

1. Introduction

High-frequency microwave signals with high spectral purity are widely used in wireless communication, sensing, and radar [1]. Conventionally, high-frequency microwave signals can be generated based on frequency multiplication of low-frequency signals obtained from electronic oscillators (EOs). However, such a scheme encounters complex and expensive system architectures, as well as a limited output power and bandwidth due to all-electronic frequency multipliers. In particular, the phase noise of the generated high-frequency microwave signal is a multiple of the driving low-frequency signal, and therefore a high-frequency microwave signal with a high spectral purity is relatively difficult to acquire [2]. Using an optoelectronic oscillator (OEO) consisting of low-cost optical/electrical components, these restrictions mentioned above can be resolved [3,4]. A typical OEO is composed of a laser source, an electro-optic modulator, a long fiber, a photo detector (PD), an electrical amplifier, and an electrical bandpass filter (E-BPF). Since the long fiber (~km) is taken as the energy storage element in the OEO loop, the cavity possesses a high-quality (high Q) factor. As a result, the spectral purity of the generated high-frequency microwave can be guaranteed. Nevertheless, the adoption of a long-distance optical fiber makes it difficult to achieve system miniaturization. In order to shorten the length of the optical fiber and maintain a low phase noise, a scheme of coupled optoelectrical oscillators (COEOs) is proposed [5,6,7]. For COEOs, an optoelectronic oscillator is coupled with an optical oscillator, in which an erbium-doped fiber amplifier (EDFA) is employed to improve the Q value of the fiber ring cavity. Therefore, the fiber length in COEOs can be shortened to several hundreds of meters [8]. Although the fiber length is relatively short in the COEO system, there are still multiple oscillation modes with a frequency interval of a few MHz, requiring mode selection technology for generating high-frequency microwaves with a high spectral purity.

Recently, parity–time (PT) symmetry, a concept rooted in quantum field theory [9], has opened a new perspective for mode selection in all-optical or OEO systems [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. For a system consisting of two mutually coupled resonators with the same geometry, when the gain of one resonator equals the loss of the other, the system is considered to be a PT symmetric system. The coupling coefficient, which characterizes the strength of the interaction between the resonators of the system, is another key factor [18]. Through adjusting the gain/loss and coupling coefficients between two resonators, and once PT symmetry breaking can be acquired for a particular mode, the gain of the mode will be significantly higher than that of other modes. As a result, single-mode oscillation can be implemented. Considering that a PT symmetric system composed of two spatially separated resonators is easily affected by environmental perturbations, PT symmetric systems based on a single spatial structure have been proposed, in which two different optical waves obtain gain and loss, respectively [15,16,17].

Although the technique of PT symmetric configurations for mode selection has been demonstrated in all-optical or OEO systems, it has never been used in a COEO system. In this work, we propose and experimentally investigate a PT symmetric COEO scheme using a single polarization-dependent spatial structure for the first time. Since two orthogonal polarized optical waves share the same geometric structure, they naturally satisfy the demand that a PT symmetric system should possess identical geometry for such two optical waves with different polarization directions. Such a scheme is relatively easy to implement since the gain/loss and coupling coefficients can be flexibly adjusted through varying the ratio of two orthogonal polarization optical waves and the bias voltage of a Mach–Zehnder modulator (MZM). Under suitable operation conditions, two mutually coupled orthogonal polarization optical waves with a balanced gain and loss can be obtained, namely, the system can achieve PT symmetry. Coarse frequency selection is realized via an E-BPF with a 3 dB bandwidth of 21.5 MHz. When the coupling coefficient is less than the gain coefficient, the COEO can achieve single-mode oscillation due to PT symmetry breaking. The proposed PT symmetric COEO is experimentally investigated, and a high purity 10.0 GHz microwave signal with a phase noise of −109.1 dBc/Hz @ 10 kHz and a side mode suppression ratio of 51.4 dB is obtained. The measured frequency drift within one hour is less than 1 kHz. Furthermore, an optical frequency comb with a comb-tooth spacing of 10.0 GHz and a 100 GHz bandwidth within 10 dB power variation is obtained in the system.

2. Experimental Setup and Operation Principle

The schematic of the experimental setup of the PT-symmetric COEO system is shown in Figure 1, in which an optical loop is coupled with an optoelectronic loop. The optical loop is composed of an erbium-doped fiber amplifier (EDFA), a Mach-Zehnder modulator (MZM, Fujitsu, Kawasaki, Japan, 40 GHz), an optical bandpass filter (O-BPF, ExFOXTM-50, EXFO, Quebec, QC, Canada), a squeeze-type polarization controller (PC) and two optical couplers (OCs). In the fiber ring cavity, the length of the cavity is about 100 m, and a 10 dB bandwidth of the O-BPF is 100 GHz. The optical fibers used in the system are non-polarization-maintaining fibers. OC1 divides the optical signal into two channels, where one is for optical signal output and the other is for closing the optical loop. The EDFA is utilized to provide a gain for two polarization optical waves and realize a high Q value of the resonator. The MZM and the PC are utilized to adjust the gain/loss and coupling coefficients of the two polarization optical waves. The bias voltage of the MZM is provided by a DC power supply (TRADEX LPS 202A, TRADEX, Beijing, China) with 1 mV accuracy. The optical signal containing such two polarization optical waves is sent to the optoelectronic feedback loop after passing through the OC2. The optoelectronic feedback loop consists of a photodetector (PD, XPDV2120R, Finisar, Sunnyvale, CA, USA, 50 GHz bandwidth), an electrical bandpass filter (E-BPF), two cascaded electrical amplifiers (EAs, Agilent 83006, Keysight, Santa Clara, CA, USA), and an electrical coupler (EC). The PD converts the optical signal into an electrical signal, and the E-BPF is used for coarsely selecting the frequency of the generated microwave. The EA is used to provide the gain of photoelectric loop oscillation. The EC divides the electrical signal into two channels, where one is for microwave signal output and the other is for closing the photoelectric feedback loop. The optical spectrum of the output optical field is measured by an optical spectrum analyzer (OSA, Aragon Photonics BOSA lite+, Aragon Photonics, Zaragoza, Spain, 20 MHz resolution bandwidth), and the power spectrum and phase noise of the output microwave signal are measured by an electrical spectrum analyzer (ESA, R&S FSW, Keysight, Santa Clara, CA, USA, 67 GHz bandwidth).

The MZM used in the system is a polarization-dependent device. The output light intensity after the MZM can be optimized by adjusting the PC before the MZM. Under a fixed bias voltage, the light passing through the MZM in this experimental setup can be effectively equivalent to passing through a linear polarizer. If the insertion loss is neglected, the Jones matrix of a system comprising a PC and a linear polarizer in tandem is [25]:

$$\mathrm{J}=\left[\begin{array}{cc}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathsf{\theta }& \mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }\\ \mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }& {\mathrm{s}\mathrm{i}\mathrm{n}}^2\mathsf{\theta }\end{array}\right].$$

(1)

Therefore, the relations between the electric fields before and after passing through the MZM when the lights complete one round trip in the cavity can be expressed as [16,25]:

$$\left[\begin{array}{c}{\mathrm{E}}_{\mathrm{x}}^1\\ {\mathrm{E}}_{\mathrm{y}}^1\end{array}\right]=\mathrm{G}\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathsf{\pi }\mathrm{V}}{2{\mathrm{V}}_{\mathsf{\pi }}}+\mathsf{\phi }\right)\right|\left[\begin{array}{cc}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathsf{\theta }& \mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }\\ \mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }& {\mathrm{s}\mathrm{i}\mathrm{n}}^2\mathsf{\theta }\end{array}\right]\left[\begin{array}{c}{\mathrm{E}}_{\mathrm{x}}^0\\ {\mathrm{E}}_{\mathrm{y}}^0\end{array}\right],$$

(2)

where the subscripts x and y represent two polarization directions associated with the birefringence in the squeeze-type polarization controller. The superscripts 0 and 1 represent the disparities in the optical signal behavior within the cavity before and after one round trip. E denotes the electric field. θ represents the angle between the minimum loss polarization direction of the MZM and the x polarization direction, which can be adjusted by controlling the PC. V_π denotes the voltage of the quadrature bias point of the MZM, V represents the bias voltage of the MZM, and φ is the phase difference stemming from the unequal length of two arms of the MZM. G represents the optical gain when θ is 0 and V is at the voltage of the maximum bias point during the round trip in the cavity.

According to Equation (2), the round-trip gain coefficients for the two polarization directions of Ex and Ey are

$${\mathrm{G}}_{\mathrm{x}}=\mathrm{G}\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathsf{\pi }\mathrm{V}}{2{\mathrm{V}}_{\mathsf{\pi }}}+\mathsf{\phi }\right)\right|{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathsf{\theta },$$

(3)

$${\mathrm{G}}_{\mathrm{y}}=\mathrm{G}\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathsf{\pi }\mathrm{V}}{2{\mathrm{V}}_{\mathsf{\pi }}}+\mathsf{\phi }\right)\right|{\mathrm{s}\mathrm{i}\mathrm{n}}^2\mathsf{\theta }.$$

(4)

The round-trip coupling coefficient is

$$\mathsf{{\rm K}}=\mathrm{G}\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathsf{\pi }\mathrm{V}}{2{\mathrm{V}}_{\mathsf{\pi }}}+\mathsf{\phi }\right)\right|\mathrm{s}\mathrm{i}\mathrm{n}\mathsf{\theta }\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }.$$

(5)

According to the definitions of the gain and coupling per time unit coefficients, the gain and coupling coefficients can be written as [16]

$${\mathrm{g}}_{\mathrm{x}}=\mathrm{l}\mathrm{n}{\mathrm{G}}_{\mathrm{x}}/\mathrm{T},$$

(6)

$${\mathrm{g}}_{\mathrm{y}}=\mathrm{l}\mathrm{n}{\mathrm{G}}_{\mathrm{y}}/\mathrm{T},$$

(7)

$$\mathsf{\kappa }=\mathrm{l}\mathrm{n}\mathsf{{\rm K}}/\mathrm{T}.$$

(8)

where T is the round-trip time of light in the cavity.

When the gain and loss are balanced, namely, ${\mathrm{G}}_{\mathrm{x}}{\mathrm{G}}_{\mathrm{y}}=1$ or ${\mathrm{g}}_{\mathrm{x}}{+\mathrm{g}}_{\mathrm{y}}=0$, then

$$\mathsf{\theta }=\pm \frac{1}{2}\mathrm{a}\mathrm{c}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{2}{{\mathrm{G}}_0\left(\mathrm{V}\right)}\right),$$

(9)

where ${\mathrm{G}}_0\left(\mathrm{V}\right)=\mathrm{G}\left|\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{\mathsf{\pi }\mathrm{V}}{2{\mathrm{V}}_{\mathsf{\pi }}}+\mathsf{\phi }\right)\right|$. Only when ${\mathrm{G}}_0\left(\mathrm{V}\right)\ge 2$ is it possible to achieve the gain and loss balance for the two orthogonal polarization optical waves in the system. Based on the above analysis, it is evident that by controlling the PC and the bias voltage of the MZM, the gain and coupling coefficients of the two polarized light signals in the system can be adjusted. As a result, after the two polarized light signals enter the optoelectronic feedback loop through OC2, two photonic circuits with controllable gain, loss, and coupling can be formed. When adjusting the PC and the bias voltage of the MZM to satisfy the PT symmetry breaking condition, the COEO can achieve single-mode microwave signal oscillation without requiring an ultra-narrow electrical filter. It should be noted that the E-BPF in this system is only used for coarsely selecting the frequency of microwaves. In the absence of the PT symmetry condition, the microwave signal in the system still oscillates in a multi-mode manner.

According to the representation of the Hamiltonian of a non-Hermitian two-component coupled system, the coupling equations of the proposed COEO can be expressed as [18]

$$\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left(\begin{array}{c}{\mathrm{G}}_{\mathrm{n}}\\ {\mathrm{L}}_{\mathrm{n}}\end{array}\right)=\left(\begin{array}{cc}{\mathsf{\omega }}_{\mathrm{n}}-\mathrm{i}{\mathrm{g}}_{\mathrm{n}}& \mathsf{\kappa }\\ \mathsf{\kappa }& {\mathsf{\omega }}_{\mathrm{n}}-\mathrm{i}{\mathrm{l}}_{\mathrm{n}}\end{array}\right)\left(\begin{array}{c}{\mathrm{G}}_{\mathrm{n}}\\ {\mathrm{L}}_{\mathrm{n}}\end{array}\right),$$

(10)

where G_n and L_n indicate the amplitudes of the nth mode of the gain and loss cavities, ${\mathsf{\omega }}_{\mathrm{n}}-\mathrm{i}{\mathrm{g}}_{\mathrm{n}}$ and ${\mathsf{\omega }}_{\mathrm{n}}-\mathrm{i}{\mathrm{l}}_{\mathrm{n}}$ represent the complex frequencies of the nth mode for each component, respectively, g_n and l_n are the corresponding gain and loss coefficients, and κ labels the coupling coefficient between the two photonic circuits. According to (10), the eigenfrequencies of the system can be described as

$${\mathsf{\omega }}_{\mathrm{n}}^{\pm }={\mathsf{\omega }}_{\mathrm{n}}-\mathrm{i}\mathsf{\chi }\pm \sqrt{{\mathsf{\kappa }}^2+{\mathsf{\Gamma }}^2},$$

(11)

where $\mathsf{\chi }=\frac{1}{2}\left({\mathrm{g}}_{\mathrm{n}}+{\mathrm{l}}_{\mathrm{n}}\right)$, $\mathsf{\Gamma }=\frac{\mathrm{i}}{2}\left({\mathrm{g}}_{\mathrm{n}}-{\mathrm{l}}_{\mathrm{n}}\right)$.

Under the PT symmetry condition, namely g_n= −l_n, (11) can be simplified as

$${\mathsf{\omega }}_{\mathrm{n}}^{\pm }={\mathsf{\omega }}_{\mathrm{n}}\pm \sqrt{{\mathsf{\kappa }}^2-{{\mathrm{g}}_{\mathrm{n}}}^2}.$$

(12)

By comparing the magnitude between κ and g_n in (12), the characteristics of the eigenfrequencies of the system can be revealed. For κ > g_n, the eigenfrequencies will have two branches in the real part, and the system will exhibit PT symmetry. Under this case, there are two eigenmodes with different frequencies, which implies that the two eigenmodes are non-degenerate. For κ = g_n, the eigenfrequencies will not bifurcate, and the system will appear at what is called the ‘exceptional point’ (EP), which means that the eigenmodes are degenerate. For κ < g_n, the eigenfrequencies possess two branches in the imaginary part, and the PT symmetry is spontaneously broken. Under this condition, the two eigenmodes share the same frequency, but the loop provides the gain for one eigenmode and the same amplitude loss for the other.

The PT-symmetric system more easily achieves single-mode oscillation than a system without PT symmetry because of the difference in the maximum gain contrast. For a system without PT symmetry, the maximum gain contrast is

$$∆{\mathrm{g}}_{\mathrm{m}\mathrm{a}\mathrm{x}}={\mathrm{g}}_0-{\mathrm{g}}_1,$$

(13)

where g₀ is the largest longitudinal mode gain and g₁ is the next-largest longitudinal mode gain. For the PT symmetric system, the maximum gain differential is expressed as [10]

$${∆\mathrm{g}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{P}\mathrm{T}}=\sqrt{{\mathrm{g}}_0^2-{\mathrm{g}}_1^2}.$$

(14)

The gain contrast ratio (enhancement factor) is given by [16]

$$\mathrm{F}=\frac{{∆\mathrm{g}}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{\mathrm{P}\mathrm{T}}}{∆{\mathrm{g}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}=\sqrt{\frac{{\mathrm{g}}_0+{\mathrm{g}}_1}{{\mathrm{g}}_0-{\mathrm{g}}_1}}.$$

(15)

Clearly, the PT symmetry breaking induces stronger gain enhancement and better mode selection.

3. Results and Discussion

The amplitude frequency response of the forward transmission coefficient (S₂₁) of the E-BPF is measured by a vector network analyzer (VNA, Ceyear 3672E) as shown in Figure 2. From this diagram, one can see that the center frequency of the E-BPF is at 10 GHz and the full-width half maximum (FWHM) of the E-BPF is approximately 21.5 MHz. The length of the fiber ring cavity is about 100 m and the corresponding longitudinal mode interval is 2 MHz. Without PT symmetry, multi-mode oscillation occurs in the optoelectronic feedback loop. Therefore, the E-BPF can be used for coarsely selecting the frequency of the generated microwave.

Initially, the PC was configured in a certain state, and the bias voltage of the MZM was adjusted from 0 V to 2.4 V in increments of 0.2 V. The evolution of the optical spectrum of the output optical signal with the bias voltage of the MZM is illustrated in Figure 3. The optical spectrum was measured at a central frequency of 193.4187 THz with a span of 200 GHz and a resolution bandwidth (RBW) of 1.25 GHz. It is noteworthy that when the bias voltage of the MZM is set to 1.2 V, the optical spectrum of the output optical signal exhibits a characteristic comb-like spectrum, which differs significantly from the other optical spectral characteristics in the optical spectral evolution chart. The evolution characteristics of the optical spectrum of the output optical signal with the bias voltage of the MZM can be further analyzed by measuring the evolution characteristics of the electrical spectrum of the output microwave signal under the same experimental conditions.

Figure 4 illustrates the evolution of the electrical spectrum of the microwave signal with the bias voltage of the MZM. The electrical spectrum was measured at a central frequency of 10.0 GHz with a span of 100 MHz and an RBW of 1 kHz. In the voltage range of 0 V to 1.0 V, the system generates microwave signals through multi-mode oscillation, and the number of oscillation modes decreases with increasing voltage. When the bias voltage of the MZM is set to 1.2 V, the system generates single-mode oscillation of microwave signals. In the voltage range of 1.4 V to 2.4 V, the microwave signals exhibit multi-mode oscillation, and the number of oscillation modes increases with increasing voltage. The inset in Figure 4 shows the evolution of the electrical spectrum of the microwave signal as the bias voltage of the MZM is varied from 1.0 V to 1.4 V with a step of 0.1 V, which is determined by the DC power supply. It can be seen that by varying the bias voltage of the MZM within 1.1~1.3 V, single-mode microwave oscillation can be achieved. As a result, the maximum offset range of the bias voltage to guarantee single-mode oscillation is approximately 0.2 V. This phenomenon indicates that by properly setting the PC and the bias voltage of the MZM, the COEO system can achieve single-mode oscillation of microwave signals. The optical spectral characteristics of the output optical signal and the electrical spectral characteristics of the output microwave signal under identical experimental conditions indicate that modulating the optical signal with a single-mode oscillating microwave signal results in an optical signal with a comb-like spectrum.

By controlling the bias voltage of the MZM at 1.2 V and fine-tuning the PC, it is possible to achieve more precise adjustment of the gain/loss and coupling coefficients of the two orthogonal polarization optical waves in the optoelectronic feedback loop of the COEO system. This fine-tuning enables further optimization of the optical comb-like characteristics of the output optical signal and enhances the side mode suppression ratio of the output microwave signal.

Figure 5 illustrates the measured optical spectrum of the generated optical signals at a central frequency of 193.4187 THz. Figure 5a,b shows the broadband optical spectrum of the output optical signal with a 2.1 V bias voltage of the MZM. Figure 5c,d shows the comb-like optical spectrum of the output optical signal with a 1.2 V bias voltage of the MZM. In Figure 5a,c, the optical spectrum was acquired using an RBW of 1.25 GHz (10.00 pm), while in Figure 5b,d, the RBW was set at 20 MHz (0.16 pm). In Figure 5c,d, it can be observed that the optical frequency comb generated by the COEO shows a comb-line spacing of 10.0 GHz and a 100 GHz bandwidth within 10 dB power variation. Figure 5 reveals that the variations in the gain/loss and coupling coefficients of the two orthogonal polarization optical waves coexisting in the COEO system can alter the characteristics of the optical spectrum of the output optical signal. When there is an imbalance in the gain and loss of two orthogonal polarization optical waves, the system fails to satisfy the PT symmetry condition. The multi-longitudinal-mode optical signal modulated by multi-mode microwave signals induces the manifestation of broadband optical spectral characteristics in the output optical signal. On the other hand, when equilibrium between the gain and loss of the two orthogonal polarization optical signals is achieved and the gain coefficient exceeds the coupling coefficient, the system enters a state of PT symmetry breaking. In this case, single-longitudinal-mode operation can be achieved, which is multi-modulated in a loop by a 10 GHz microwave signal. Due to the multiple modulation, the higher-order sideband possesses a power equivalent to the central carrier, and therefore the output optical spectrum has optical frequency comb characteristics [26]. The power fluctuation of the optical frequency comb is due to the combination of cascade modulation and the dispersion characteristics of the utilized optical devices in the loop.

Figure 6a–d shows the characteristics of the electrical spectrum of the microwave signal generated in the system. When a bias voltage of 2.1 V is applied to the MZM, the system fails to satisfy the PT symmetry conditions, resulting in the manifestation of multi-mode oscillations in the microwave signal, as shown in Figure 6a. Considering that the frequency interval between the two longitudinal modes in the system is only about 2 MHz, the E-BPF with a 3 dB bandwidth of 21.5 MHz serves the purpose of coarsely selecting the frequency of the generated microwave. As a result, multiple modes can oscillate. Adjusting the bias voltage of the MZM to 1.2 V can achieve a balance between the gain and loss coefficients within the two photonic circuits. Furthermore, when the coupling coefficient is less than the gain coefficient, the COEO can achieve single-mode oscillation based on PT symmetry breaking. Figure 6b–d demonstrates the spectral characteristics of single-mode oscillation under different spans and RBWs, respectively. It can be seen that the side mode is greatly suppressed, and the side mode suppression ratio is 51.4 dB.

Figure 7 demonstrates the phase noise of the generated microwave signal with a central frequency of 10 GHz under PT symmetry breaking. From this diagram, it can be seen that the phase noise at the offset frequency of 10 kHz is −109.1 dBc/Hz, and the highest side band is suppressed below −102.9 dBc/Hz. It should be pointed out that the relatively strong phase noise originates from the low Q-value of the fiber ring cavity adopted in this system. Through appropriately increasing the length of the fiber ring cavity and widening the bandwidth of the O-BPF, the phase noise of the generated microwave can be further reduced [27].

In addition, the stability of the generated microwave signal under PT symmetry breaking was also measured. The frequency drift of the single-frequency microwave signal is measured by the ‘Max Hold’ function of the ESA, as illustrated in Figure 8. During the course of a one-hour monitoring period, the frequency drift range of the microwave signal is approximately 0.65 kHz within 10 dB amplitude variation, and no mode hopping phenomena are observed.

4. Conclusions

In summary, a PT-symmetric COEO based on a single polarization-dependent spatial structure is proposed and experimentally verified. By adjusting the PC and the bias voltage of the MZM, for two mutually coupled and orthogonally polarized modes, the loop can provide a gain for one polarized mode and a loss with an identical amplitude for the other. In this condition, PT symmetry is realized in the COEO. Furthermore, when the gain exceeds the coupling coefficient, single-mode oscillation is achieved based on PT symmetry breaking for a special mode. We experimentally demonstrate that a pure microwave signal with a central frequency of 10.0 GHz is generated in the case of PT symmetry breaking in the COEO system. The phase noise of the signal is −109.1 dBc/Hz at the frequency offset of 10 kHz and the side mode suppression ratio is 51.4 dB. The frequency drift within one hour is less than 1 kHz. Additionally, an optical frequency comb with a comb line spacing of 10.0 GHz and a 100 GHz bandwidth within 10 dB power variation is generated.