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Abstract
An optoelectronic oscillator (OEO) with improved sideband suppression by joint use of the spectral Vernier effect and parity-time (PT) symmetry is proposed and experimentally demonstrated. The spectral Vernier effect is implemented using two mutually coupled loops with different loop lengths, to increase the effective free spectral range (FSR). To further increase the mode selection capability to ensure stable single-frequency oscillation with an increased sidemode suppression ratio (SMSR), PT symmetry is implemented, in which the two mutually coupled loops are controlled with balanced gain and loss. Thanks to the combined effects, stable single-mode oscillation with a significantly increased SMSR is achieved. The proposed OEO is studied theoretically and evaluated experimentally. The results show that for a generated microwave signal at 10 GHz, the SMSR is 67.68 dB, which is increased by 11.20 dB or 26.05 dB, when using only the spectral Vernier effect or only the PT symmetry. Thanks to the long length of the longer loop, good phase noise performance is still maintained. The measurement shows that a phase noise as low as -124.5 dBc/Hz at an offset frequency of 10 kHz is achieved.
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1. Introduction
An optoelectronic oscillator (OEO) can be used to generate a microwave signal at a high frequency with an ultra-low phase noise [1,2], which can find applications in wireless communications, radar and electronic warfare systems [3–6]. The phase noise performance of an OEO is mainly determined by the Q factor of the feedback loop [7–9]. A longer loop length corresponds to a higher Q factor. For example, an OEO with a loop length of 16 km was implemented to generate a microwave signal at 10 GHz with a phase noise as low as −163 dBc/Hz at an offset frequency of 6 kHz [10]. However, an OEO with a long loop length will have a large number of densely spaced longitudinal modes, making mode election for single-frequency oscillation extremely challenging [11,12].
Recently, parity-time (PT) symmetry has been proven to be an effective solution for mode selection in various oscillators, including lasers [13–15], electronic microwave oscillators [16,17], and OEOs [18–24]. Thanks to the mode selection capability offered by PT symmetry, a high-Q factor optical or microwave filter is not needed, which would simplify greatly the system implementation. In general, an OEO with PT symmetry consists of two mutually coupled feedback loops having identical geometry, but with one having a gain and the other having a loss. When the gain and the loss coefficients are of the same magnitude and are greater than the coupling coefficient, PT symmetry is broken, which would strongly enhance the gain difference between the mainmode and the sidemodes [18–24], facilitating mode selection to enable single-mode oscillation without the need for a narrowband filter.
Although the employment of PT symmetry can help achieve stable single-mode oscillation in an OEO, the sidemode suppression ratio (SMSR) is still low. For applications such as high-sensitivity wireless systems and radar, low spurious microwave sources are highly needed [18–24]. For example, the PT-symmetric OEOs reported in [18,19] have SMSRs of only 30 and 45 dB, which are considered very low. The PT-symmetric OEOs with tunable frequency reported in [20,21] have SMSRs of 46 and 31 dB. A recent demonstration of an OEO based on two equivalent coupled loops implemented using a single physical loop, but two optical carriers corresponding to the gain and the loss loops, was reported [22]. Although the structure is simplified with improved stability, the SMSR was measured to be 46.75 dB, which is again considered low. An OEO based on the nonreciprocal modulation of a travelling-wave electro-optic modulator with, again, a single physical loop was demonstrated to show high operational stability, but the SMSR was measured to be 45 dB [23]. In [24], a PT-symmetric OEO employing an integrated mode-locked laser (MLL) to form a coupled OEO to facilitate frequency selection was demonstrated. The oscillation frequency of the OEO was determined by the pulse train repetition rate of the MLL. However, its SMSR is only 20 dB. A technique that can ensure stable single-frequency oscillation and a high SMSR is of critical importance for low spurious systems. A solution is to apply a joint use of the spectral Vernier effect and PT symmetry.
Generally, a PT symmetric system, including an OEO, requires perfect alignment between the localized eigenmodes in the two coupled loops, and the gain and loss coefficients should also be balanced. Recently, a more generalized PT symmetric system, known as the PT-reciprocal scaling (PTX) symmetry, was reported. In the system, the gain and loss coefficients are not required to be balanced, but can have a scaling factor between them [25,26]. The PTX provides a degree of freedom for the manipulation of a PT symmetric system. Similarly, a PT symmetric system with only part of the localized eigenmodes being aligned can also be employed for mode selection. For example, if the lengths of the two loops in an OEO are not equal, but with a rational scaling factor, part of the localized eigenmode can still be aligned, which would enhance the gain difference between the mainmode and the sidemodes, facilitating mode selection. This is the key feature that allows the implementation of the spectral Vernier effect and PT symmetry in one single system.
In this paper, we propose and experimentally demonstrate an OEO by joint use of the spectral Vernier effect and PT symmetry to generate a microwave signal with an ultra-low phase noise and a high SMSR. The spectral Vernier effect is implemented with two mutually coupled OEO loops with different lengths to increase the effective mode interval while maintaining a high Q factor [8,9]. The PT symmetry is achieved by controlling the powers of two linearly polarized optical carriers at two different wavelengths with the gain and loss being balanced by using a polarization controller (PC) and a polarization beam combiner (PBC). Thanks to the joint effects of the spectral Vernier effect and the PT symmetry, stable single-mode oscillation is achieved with a significantly increased SMSR. The proposed OEO is evaluated experimentally. For two mutually coupled OEO loops of 5 km and 100 m in length, a microwave signal at 10 GHz is generated with an SMSR of 67.68 dB, which is increased by 11.20 dB or 26.05 dB when using only the spectral Vernier effect or the PT symmetry. Thanks to the long length of the longer loop, the good phase noise performance is still maintained. The measurement shows that a phase noise as low as -124.5 dBc/Hz at an offset frequency of 10 kHz is achieved.
2. Principle
The schematic diagram of the proposed OEO is shown in Fig. 1. Two tunable laser sources (TLSs) are used to generate two optical carriers of different wavelengths. The two optical carriers are combined at a PBC through PC1 and PC2 and sent to a Mach-Zehnder modulator (MZM) through PC3. PC1 and PC2 are used to adjust the powers of the optical carriers at the output of the PBC by tuning the polarization alignment between the polarization directions of the optical carriers and the principal axes of the PBC. PC3 is used to reduce the polarization-dependent insertion loss at the MZM. Two wavelength division multiplexers (WDM1 and WDM2) connected by two single-mode fibers (SMFs) with different lengths of 5 km and 100 m are incorporated to realize two physical paths of different lengths and thus to achieve the spectral Vernier effect. The optical signals from WDM2 are detected at a photodetector (PD), which is amplified by an electrical amplifier (EA), filtered by an electric bandpass filter (EBF) and fed back to the MZM via a 1×2 microwave power splitter. The other port of the splitter is connected to an electrical spectrum analyzer (ESA) where the spectrum of the generated microwave signal is analyzed.
Fig. 1. Schematic diagram of the OEO with spectral Vernier effect and PT symmetry. TLS: tunable laser source; PC: polarization controller; PBC: polarizer beam combiner; MZM: Mach-Zehnder modulator; WDM: wavelength division multiplexer; SMF: single-mode fiber; PD: photodetector; EA: electric amplifier; EBF: electric bandpass filter; ESA: electrical spectrum analyzer.
The spectral Vernier effect requires that the oscillating mode in the dual-physical-loop OEO satisfies
where ${k_1}$ and ${k_2}$ are two integers, $\Delta {\nu _i} = {c / {{\tau _i}}}$ for i = 1, 2 are the mode spacing for the long and short OEO loops, ${\tau _1}$ and ${\tau _2}$ are the round-trip times of the long and short loops, respectively, which are given by ${\tau _i} = {\tau _{i,0}} + {D_i}({{\lambda_i} - {\lambda_0}} )$, ${\omega _n}$ is the angular frequency of the primary n-th order eigenmode, ${\tau _{i,0}}$ is the time delay for the reference wavelength ${\lambda _0}$ passing through SMF1 or SMF2, ${D_i}$ is the dispersion coefficient of SMF1 or SMF2, and ${\lambda _1}$ and ${\lambda _2}$ are the two carrier wavelengths from TLS1 and TLS2.The open-loop gain of the OEO loop is jointly determined by the frequency response ${g_e}(\Omega )$ of the electrical components, the frequency response ${g_o}(\Omega )$ of optical components including the modulation frequency response and the dispersion-induced power penalty (DIPP) function, which is given by [27]:
For simplicity, the modulation frequency response and the DIPP function are considered constant within the EBF bandwidth, thus ${g_o}(\Omega )= {g_o}$. Equation (2) can be expressed by its Taylor expansion,
On the other hand, PT symmetry can also be implemented to enhance the gain difference between the mainmode and sidemodes, to facilitate mode selection. The implementation of PT symmetry requires the eigenmodes of the two loops to be aligned and the gain and loss to be balanced. In our system, the alignment in eigenmodes is achieved by tuning the wavelengths of TLS1 and TLS2, to control the wavelength-dependent time delays in SMF1 and SMF2. The gain and loss balance is realized by varying the power of the optical carriers by tuning PC1 and PC2. With PT symmetry, the gain difference between the primary mode and the secondary mode of the OEO is given by [19]
Due to the small mode spacing of the OEO, we assume an small value for $\Delta {g_{DL}}$ such that $\Delta {g_{DL}} = g({{\Omega _n}} )- g({{\Omega _m}} )\ll {1 / {g({{\Omega _n}} )}} \ll g({{\Omega _n}} )$, Eq. (6) can be approximated as It can be seen the gain difference between the primary and secondary modes is enhanced by the dual-loop spectral Vernier effect due to the increased mode spacing, and it is further enhanced by the PT symmetry. By combining the spectral Vernier effect and the PT symmetry, single-mode oscillation with strongly suppressed sidemodes can be achieved.Figure 2 illustrates the principle of improving the SMSR by joint use of the spectral Vernier effect and PT symmetry. Figures 2(a) and (b) show the two sets of equally spaced eigenmodes of the long and short OEO loops, respectively. The dashed line shows the open-loop frequency response of the OEO, which has a slowly varying envelop as compared with the mode spacing of the two OEO loops. Figure 2(c) shows that by using the dual-loop structure, the eigenmode of the long loop is suppressed due to the spectral Vernier effect, and the gain difference between a primary mode and a secondary mode is increased as compared with a single-loop case, which is then further increased by introducing PT symmetry, as shown in Fig. 2(d). An increased SMSR can then be achieved.
Fig. 2. The eigenmodes of the OEO with (a) a long, (b) a short, (c) dual loops with spectral Vernier effect but without PT symmetry, and (d) dual loops with both spectral Vernier effect and PT symmetry.
3. Experimental results
The operation of the proposed OEO is experimentally evaluated based on the setup shown in Fig. 1. First, the open-loop frequency response is measured. To do so, the loop is opened at the gain probing point shown in Fig. 1 and a vector network analyzer (VNA) is connected. Figure 3 shows the open-loop frequency response of the OEO loop measured by the VNA. The light waves are generated by a four-port laser source (Keysight N7714A). The MZM (MX-LN-40, Photline Technologies) has a bandwidth of 30 GHz. The PD (XPDV2120RA, U2t) has a bandwidth of 50 GHz. The EA is implemented by cascading three EAs (SHF 806E) to provide a total electrical gain of over 75 dB. The EBF (GBPF100-28M-SMF, GWAVE) has a bandwidth of 28 MHz at a center frequency of 10 GHz. The open-loop frequency response of the OEO loop measured the VNA (Keysight N5224A) is shown in Fig. 3(a). Note that an EBF with a central frequency of 10 GHz and a bandwidth of 20 MHz is used to roughly select the oscillation frequency.
Fig. 3. (a) The open-loop frequency response of the OEO measured by a VNA connected between the MZM and the splitter (blue), and the calculated open-loop frequency response with enhanced spectral ripples with PT symmetry; (b) the zoom-in view, showing an enhanced gain ripple that can be used to suppress the sidemodes.
As shown in Fig. 3(a), the open-loop frequency response has a smooth profile, with noticeable ripples seen in the zoom-in view in Fig. 3(b). Figure 3(a) also shows the calculated open-loop frequency response when PT symmetry is introduced to enhance the ripples in the original frequency response spectrum, which results in strong ripples that enhances gain difference between mainmode and the sidemodes to achieve single-mode oscillation.
Figure 4 shows the spectra of the generated microwave signal with different measurement resolution bandwidths (RBWs) and different operation conditions. Specifically, when only TLS1 is turned on, only the long loop is closed, multimode oscillation is resulted due to the absence of effective mode selection mechanism, as can be seen in Fig. 4(a). The mode spacing is 40.80 kHz, corresponding to a loop length of 5 km.
Fig. 4. The spectra of the microwave signal generated by the OEO at a central frequency of 10 GHz with different measurement resolution bandwidths (RBWs) and different operation conditions. Multimode oscillation spectra of the OEO with (a) only the long loop, RBW: 3.9 kHz and (b) only the short loop, RBW: 200 kHz. (c) Single-mode oscillation spectrum when only the spectral Vernier effect is used, RBW: 1.5 kHz. (d) Single-mode oscillation spectrum when only PT symmetry is implemented in the long loop, RBW: 3.2 kHz. The spectra with improved SMSR by combing spectral Vernier effect and PT symmetry with two different RBWs of (e) 10 kHz and (f) 1.2 kHz. Inset: zoom-in view of the spectrum with 800-Hz frequency span.
When only TLS2 is turned on, only the short loop is closed, multimode oscillation is again resulted, but the mode spacing is 1.96 MHz, corresponding to a short loop length of 100 m, as shown in Fig. 4(b). When both TLS1 and TLS2 are turned on with each having an optical power of 12.50 dBm at 1557.4 nm and 1559.7 nm, both the long and the short loops are closed. Since the gain and loss of the two loops are not controlled to be balanced, PT symmetry is not enabled. In this case, only the Vernier effect is active, which leads to single-mode oscillation, but the SMSR is still high. As can be seen, the SMSR is 56.48 dB for the residual long-loop modes, as shown in Fig. 4(c).
Then, the wavelengths of TLS1 and TLS2 are tuned to 1557.4 nm and 1557.6 nm with optical powers of 10.9 and 9.5 dBm, respectively, and the polarization states of the two wavelengths are made orthogonal and aligned with the principal axis of the PBC by adjusting PC1 and PC2. Since the wavelengths of TLS1 and TLS2 are both in the passband of the WDM upper channel, both wavelengths will travel in the long loop. By tuning PC3 to control the polarization states of the two wavelengths, the powers of the two wavelengths after the MZM are changed, which enabled the gain and loss of the two polarization loops balanced and greater than the coupling coefficient, thus the PT symmetry is broken. As a result, the SMSR is increased [22]. As shown in Fig. 4(d), the SMSR is measured to be 41.63 dB.
Finally, both TLS1 and TLS2 are turned on and the powers are controlled to be 12.50 and 9.50 dBm, and the wavelengths are 1557.4 nm and 1559.7 nm. Since the wavelength of TLS2 is now in the passband of the WDM lower channel, the two wavelengths will travel in the long loop and short loop. PC1 and PC2 are adjusted again to make the gain coefficient higher than the coupling coefficient. By adjusting PC3 to balance the gain and loss in the loops, broken PT symmetry is achieved. Since both the spectral Vernier effect and PT symmetry are enabled, the SMSR is highly increased. As shown in Figs. 4(e) and (f), the SMSR is significantly increased to 67.68 dB. Thanks to the joint operation of spectral Vernier effect and PT symmetry, the SMSR is increased by 11.20 dB or 26.05 dB when using only the spectral Vernier effect or the PT symmetry.
The stability of the OEO is also evaluated. To do so, we let the OEO to operate in a lab environment for 10 minutes, no mode hoping and power fluctuations are observed.
The advantage of implementing an OEO with a long loop length is its high Q factor, which leads to the generation of a microwave signal with an ultra-low phase noise. Figure 5 shows the measured phase noise of a microwave signal generated by the OEO with both the spectral Vernier effect and PT symmetry. The phase noise of the microwave signal generated by the OEO are both analyzed by a signal analyzer (Keysight N9040B). The phase noise is -124.5 dBc/Hz for a carrier frequency of 10 GHz at an offset frequency of 10 kHz, and the side modes are lower than -90 dBc/Hz, which verifies the effectiveness of mode selection by joint use of the spectral Vernier effect and PT symmetry. The phase noise performance is mainly determined by the length of the long SMF in the loop, which is 5 km. If the length of the long optical fiber is increased, the phase noise of the generated microwave signal can be further reduced.
Fig. 5. The measured phase noise spectrum of a 10-GHz microwave signal generated by the OEO. The phase noise is -124.5 dBc/Hz at an offset frequency of 10 kHz.
4. Conclusion
In summary, we proposed and experimentally demonstrated an OEO to generate a microwave signal with a low phase noise and a high SMSR. Thanks to the joint use of the spectral Vernier effect and the PT symmetry, the mainmode to sidemode ratio was significantly increased, enabling the OEO to generate a single-mode microwave signal with a significantly increased SMSR. The proposed OEO was studied theoretically and evaluated experimentally. The experimental results showed that for a generated microwave signal at 10 GHz, the SMSR was measured 67.68 dB, which was increased by 11.20 dB or 26.05 dB, when only using the spectral Vernier effect or the PT symmetry. Thanks to the long length of the longer loop, the good phase noise performance was still maintained. The measurement shows that a phase noise as low as -124.5 dBc/Hz at an offset frequency of 10 kHz was achieved. The joint use of PT symmetry and the Vernier effect opens a new direction for the implementation of an OEO with significantly increased SMSR, which can find applications in microwave systems where a microwave source with ultra-low sidemodes is needed. Our results show the effectiveness of combining multiple mode-selection mechanisms. If an even-higher SMSR is needed, more mode-selection methods may be added to the proposed OEO, such as injection locking and narrowband filtering, to achieve an SMSR beyond 100 dB.
Funding
Guangdong Province Key Field R&D Program Project (2020B0101110002); National Key Research and Development Program of China (2021YFB2800804); National Natural Science Foundation of China (61905095, 62101214).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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