Optoelectronic oscillators (OEOs) are promising solutions for generating microwave signals with low phase noise and wideband tunability, and they can be applied to converging systems such as communications, radars, and electronic warfare systems. However, a significant challenge remains in ensuring a low phase noise, wideband tunability, and ultra-high side mode suppression ratio (SMSR) simultaneously. Parity-time (PT) symmetry breaking provides an excellent tool for single-mode oscillation by exploiting the interplay between the gain and loss. The oscillation mode was previously fixed because the breaking of the PT symmetry cannot be accurately manipulated. Herein, we propose an OEO with selective PT-symmetry breaking showing a wideband tunability and ultra-high SMSR. The tunability of the proposed OEO is attributed to the selection of different modes to break the PT symmetry using a widely tunable microwave photonic filter (MPF). The large roll-off of the MPF significantly enhances the gain difference between the selected and competing modes. Consequently, both the output power and SMSR of the OEO increase. During the experiment, the measured oscillation frequency is tuned from 2.6 to 40 GHz. The output power of the selected mode is enhanced by 12.9 dB, and the maximal SMSR reaches up to 71.4 dB. Further, the measured phase noise of the OEO at 17.74 GHz reaches at a 10-kHz offset frequency. Exploration of the selective PT-symmetry breaking provides the possibility of developing classes of widely tunable OEOs with an ultra-high SMSR and excellent low phase noise simultaneously.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
RetractionThis article has been retracted. Please see:
Haitao Tang, Yuan Yu, and Xinliang Zhang, "Widely tunable optoelectronic oscillator based on selective parity-time-symmetry breaking: retraction," Optica 6, 1506-1506 (2019)
The generation of a microwave signal with a low phase noise and wideband tunability is of significant importance for applications such as communications, radars, and electronic warfare systems [1,2]. A traditional electronic oscillator can generate a microwave signal with a low phase noise only at low frequencies . To obtain microwave signals with high frequencies, frequency multiplication must be used, and the phase noise will be significantly deteriorated [4–7].
Optoelectronic oscillators (OEOs) are promising solutions for generating microwave signals with a low phase noise and wideband tunability, and in theory the phase noise performance will not be deteriorated with the increase in frequency [8–12]. The low phase noise is ensured by the high factor of the OEO cavity with a long fiber. For example, a state-of-the-art OEO can generate a 10-GHz microwave signal with a phase noise as low as at a 6-kHz offset frequency using a 16-km-long fiber delay line in the OEO loop . However, the long fiber loop results in an ultra-narrow eigenmode frequency spacing. Therefore, an ultra-narrow electrical bandpass filter (EBF) is required to achieve single-mode oscillation [10,14]. However, it remains significantly challenging to realize an EBF with a sufficiently narrow bandwidth, and hence mode hopping is quite difficult to effectively suppress. To overcome the mode hopping, dual-loop OEOs that use the Vernier effect have been adopted [15,16]. In fact, a dual-loop structure is equivalent to a two-tap microwave photonic filter (MPF), which cannot effectively suppress the competing modes or ensure a stable oscillation. Moreover, the central frequency of the ultra-narrow EBF is usually fixed, as is the oscillation frequency of the EBF-based OEO. To obtain a tunable OEO and effectively suppress the mode hopping, a tunable microwave filter with ultra-narrow passband is in high demand. A tunable MPF with a 3-dB bandwidth of only 150 kHz has been incorporated in the OEO to suppress the competing modes . Owing to the tunability of the MPF, the OEO can be tuned from 0 to 40 GHz. However, the phase noise is deteriorated by the amplified spontaneous emission noise of the semiconductor optical amplifier.
As an important supplement to mode selection, parity-time (PT) symmetry was initially exploited in single-mode lasers [17–21] and electronic oscillators . Through a precise manipulation of the gain, loss, and coupling ratio between the cross-coupled cavities, PT symmetry can be broken for a specific eigenmode . Therefore, a stable single-mode oscillation can be achieved [17,18]. Recently, PT-symmetry breaking has been demonstrated in OEOs [24,25]. A 9.8-GHz microwave signal with a phase noise as low as  and a 4.07-GHz microwave signal with a phase noise as low as  at a 10-kHz offset frequency were generated. However, the oscillation frequency is fixed and cannot be tuned because the mode with PT-symmetry breaking cannot be manipulated.
In this paper, we propose and demonstrate a widely tunable OEO based on selective PT-symmetry breaking. Here, only the selected mode can break the PT symmetry and oscillate successfully, whereas other modes neither decay nor grow but rather remain neutral. It should be stressed that the mode used to break the PT symmetry can be manipulated. By selecting a different mode to break the PT symmetry, the oscillation mode of the OEO can be changed. Hence, the oscillation frequency of the proposed OEO can be tuned. In our proposed OEO, selective PT-symmetry breaking is achieved using a widely tunable MPF, which is realized through stimulated Brillouin scattering (SBS) . Two cross-coupled optoelectronic hybrid cavities with identical loop length are designed to provide a test bed for PT symmetry. By precisely manipulating the gain, loss, and coupling ratio between the cross-coupled cavities, the PT symmetry can be broken only for the selected mode with the peak gain. Thus, the selected mode then oscillates, and single-mode oscillation can be achieved . In addition, an MPF with a large roll-off can further enhance the gain difference between the selected mode and competing modes in the PT symmetric system, which makes the selected mode oscillate more easily as compared with the PT-symmetry OEO without an MPF. Simultaneously, the selective breaking of PT symmetry can effectively increase the maximum output power by 12.91 dB in the desired mode owing to the enhanced gain difference compared with the case without PT symmetry. Owing to the excellent mode selectivity of the PT-symmetry breaking, we achieve a purely stable microwave signal at 17.74 GHz with a 71.4-dB side mode suppression ratio (SMSR) in the case of a 1-km loop length. To the best of our knowledge, this is the highest SMSR demonstrated in widely tunable OEOs. The measured phase noise of the OEO with a 2-km loop length is at a 10-kHz offset frequency. By tuning the central frequency of the MPF for the selection of different modes, the measured oscillation frequency is tuned from 2.6 to 40 GHz. In theory, the lower limit of the oscillation frequency can approach 20 MHz, which is limited by the bandwidth of the MPF. The upper frequency limit is only subjected to the bandwidths of the phase modulator (PM), photodetector (PD), and electrical amplifier (EA). If these devices with larger bandwidths are used, the tunability of the OEO can be further improved.
2. THEORETICAL ANALYSIS
Figure 1 shows the block diagram of the widely tunable OEO based on selective PT-symmetry breaking. The continuous-wave (CW) light emitted from a laser diode (LD) is launched into a PM through a polarization controller (PC1). The phase-modulated signal is then fed into a 1-km-high nonlinear fiber (HNLF) through an isolator, which is used to block the counter-propagating optical wave. The pump light is emitted from a tunable laser diode (TLD) and launched into the HNLF through the optical circulator. Hence, a bandpass MPF based on SBS is achieved. To exploit the PT symmetry, two identical cross-coupled optoelectronic hybrid cavities are designed. One cavity has a gain and the other has a loss of the same magnitude. This is achieved using PC3 and a polarization beam splitter (PBS) to split the optical signal into two channels. By adjusting PC3, the power splitting ratio after the PBS can be continuously adjusted. Therefore, the coupling ratio between the dual loops can be adjusted from 0% to 100% by adjusting PC3 . The two optical signals with a tunable power splitting ratio are then sent to a balanced photodetector (BPD) to recover the microwave signal. An additional advantage of using the BPD is that the phase noise of the OEO can be partially suppressed . Considering that the identical loop length in the dual loops is essential to satisfy the PT symmetry, we insert two optical tunable delay lines (OTDLs) to ensure the same loop length in both cavities. The recovered microwave signal from the BPD is amplified using an electrical amplifier (EA) through a combiner and then launched into a splitter; one of the signals is then fed back to the PM for recirculation, and the other is fed into an electrical spectrum analyzer (ESA) for measurement.
Under PT symmetry, no modes will oscillate because the eigenmode of the PT-symmetry system remains neutral, despite the presence of the gain and loss [17,28–30]. However, by tuning PC3 to make the gain/loss reach above a certain threshold, the PT symmetry may be spontaneously broken for the mode with the peak gain. Thus, this mode experiences a gain to oscillate, whereas the remaining modes maintain neutral. The incorporated MPF with a large roll-off can enhance the gain difference between the selected mode and competing modes and increase the output power of the OEO. The corresponding eigenfrequency is determined by the central frequency of the MPF. Accordingly, the oscillation frequency of the proposed PT-symmetric OEO can be widely tuned.
Theoretically, the coupling equations of the eigenmodes in the two cross-coupled cavities are given by1), the eigenfrequencies of the system can be derived as 2) can be simplified as 3) that when the gain/loss is smaller than the coupling ratio , frequency splitting in the system emerges, as shown the gray dotted line in Fig. 2. These modes neither decay nor grow, but rather remain neutral . By contrast, once the gain of one specific mode exceeds , this symmetry is broken, and a conjugate pair of amplifying and decaying modes at the same frequency appears. The amplifying mode experiences a much higher gain compared with the remaining modes and stably oscillates. It should be stressed that the oscillation frequency is determined by the central frequency of the MPF (see the blue solid line in Fig. 2) . In a traditional OEO, the single-mode oscillation only occurs if the net loop gain exceeds 1 for a specific mode. Clearly, this imposes a stringent constraint on the operating condition, particularly in the case of a close mode spacing and large open-loop response bandwidth. As shown in Fig. 2, the oscillation-mode gain difference can be described as 17]. According to Eq. (2), the maximum achievable gain difference can be derived and expressed as
3. RESULTS AND DISCUSSION
As shown in Fig. 1, the experiment of the proposed OEO based on selective breaking of PT symmetry is performed using available commercial devices. The LD and TLD are an NKT E15 with a maximum output power of 18 dBm. The 3-dB bandwidths of the PM (Covega Mach-40) and BPD (u2t BPRV2125AM) are both 40 GHz. A 1-km HNLF [nonlinear coefficient, ; dispersion slope, is used as the SBS medium and delay unit. The delay range of the two OTDLs (Santec ODL-330) is 0–300 ps. The operation bandwidth of the EA (SHF S804A) is 90 kHz to 60 GHz, and its gain can be adjusted from 18 to 22 dB. The open-loop response of the proposed OEO is measured using a vector network analyzer (Anritsu MS4647B). The spectra and phase noise of the generated microwave signal are measured using an ESA (Keysight N9030A), whose lower limit of the phase noise measurement is at an offset frequency of 10 kHz.
In the experiment, we tested the OEO with a loop length of approximately 1 km. The wavelengths of the signal and pump waves are 1550.12 and 1549.9 nm, respectively. The optical pump power before the HNLF is 13 dBm. It should be noted that the 1-km HNLF acts as both the SBS media and delay line unit. After the SBS interaction between the pump wave and phase modulated signal waves, the SBS-based MPF is obtained as shown in the dotted red curve in Fig. 3(a). The 3-dB bandwidth of the MPF is 20 MHz, which contains hundreds of eigenmodes. The electrical spectrum of the OEO based on the SBS-based MPF without PT symmetry is measured by the ESA, and the result is shown as the black curve in Fig. 3(a). It can be clearly seen that a serious mode competition exists, and the maximum output power is . To achieve single-mode oscillation and suppress competing modes effectively, a microwave filter with 3-dB bandwidth less than 200 kHz, which is equal to the mode spacing of OEO, is demanded without incorporating PT-symmetry breaking.
To provide a suitable test bed for the PT symmetry, we designed a dual-loop OEO with identical loop lengths. By tuning PC3, we can precisely manipulate one loop to achieve a gain and the other to achieve a loss of the same amplitude. Once the peak gain of the mode is above the gain/loss threshold , the PT symmetry is broken for this mode, which experiences gain for a successful oscillation. This eigenfrequency is determined by the central frequency of the MPF. Owing to the selective PT-symmetry breaking, we successfully achieved single-mode oscillation at 17.74 GHz as shown in Fig. 3(b). The electrical spectrum measurement is conducted with a span of 100 MHz and a resolution bandwidth (RBW) of 50 kHz. To observe the electrical spectrum more clearly, a zoomed-in view of the spectrum with a 1-MHz span is shown in Fig. 3(c). The oscillation mode can be clearly identified, as can two sets of supermodes. Their frequency interval is 10 kHz, which is caused by a slight difference in length between the two optical branches after the PBS as shown in Fig. 1. We then gradually tune OTDL1 and increase the delay by approximately 31 ps to compensate for the length difference between the two fiber delay lines. It can be seen that the two adjacent supermodes become closer and that the supermodes are gradually suppressed during the tuning process. When the fiber lengths of the two optical branches are identical, a perfect single-mode oscillation with an SMSR as high as 71.4 dB is achieved as shown in Fig. 3(d). The maximum output power is 9.41 dBm, which is 12.9 dB higher than that without PT symmetry. In fact, when the relative length difference between the two optical branches is maintained within , the degradation of the SMSR is less than 1 dB. The resolution of the adopted OTDL is 0.2 ps, which fully meets the needs of the experiment. The temperature variation may also affect the relative length difference between the two branches. However, during the experiment, once a stable single-mode oscillation is achieved, no additional super-mode is found, and the SMSR remains above 70 dB. Therefore, under laboratory conditions, the effect of the temperature variation on the relative length difference between the two branches is almost negligible. When a certain difference in the optical fiber length is set between the two optical branches, a dual-loop OEO with different loop lengths can also be used to enhance the side-mode suppression through the Vernier effect . However, the mode suppression is not as good as that based on the PT-symmetry breaking. For example, the SMSR in an SBS-based dual-loop OEO using Vernier effect is 35 dB , which is about 36 dB lower than that of our proposed OEO. The ultra-high SMSR is attributed to the fact that the PT-symmetry breaking can greatly enhance the mode gain difference and suppress the side modes more strongly.
In addition, the enhanced gain difference can strongly suppress the side modes and thus increase the short-term stability of the proposed OEO. We also monitor the oscillation frequency stability of the OEO during a 10-min period. The measured electrical spectra of the generated microwave signal at 17.74 GHz within the 10-min timeframe are shown in Fig. 4(a). Given the instability of the relative wavelengths of the two lasers, mode hopping still occurs. The variations in the accurate oscillation frequency of the proposed OEO over time are shown in Fig. 4(b). We can see that the first mode hopping occurs during the third minute, and a total of three mode hoppings occur during the 10-min timeframe. In the absence of mode hopping, the frequency drift is approximately 200 Hz. If long-term frequency stability of the OEO is required, the wavelength interval of the two lasers should be locked, and an additional feedback loop will be necessary.
As indicated in the previous analysis, the selective PT-symmetry breaking can allow the desired mode to be selected for oscillation. Thus, the proposed selective PT-symmetry OEO exhibits excellent tunability owing to the widely tunable MPF. By changing the central frequency of the MPF, the measured microwave signal is tuned from 2.6 to 40 GHz as shown in Fig. 5. In theory, the lower limit of the oscillation frequency can reach close to 20 MHz, which is limited by the bandwidth of the MPF. The upper frequency limit is only subjected to the bandwidths of the PM, PD, and EA. If these devices with larger bandwidths are used, the tunable range of the OEO can be further enlarged. In Fig. 5, some microwave spurs can also be seen. These spurs are just high-order harmonics caused by the nonlinearity in the OEO loop.
The phase noises of microwave signals with different frequencies are also measured using the ESA. Figure 6 shows the phase noise when the oscillation frequency is 7.82 GHz (black curve) and 17.74 GHz (red curve), respectively. At a 10-kHz frequency offset, the corresponding phase noises are and , respectively, and the side mode noises are all below , indicating that the selective PT-symmetry breaking possesses strong mode selectivity. According to Eq. (5), the MPF with a large roll-off can further enhance the oscillation-mode gain difference with respect to the competing modes in the PT-symmetry OEO. Therefore, the competing modes are further suppressed.
The maximal obtained SMSR based on PT-symmetry breaking is 71.4 dB, which is the largest SMSR in widely tunable OEOs reported in the literature. To further reduce the phase noise, an extended investigation on the proposed OEO is conducted by increasing the loop length to 2 km. The corresponding mode spacing is reduced to approximately one half, and the gain difference between the selected mode and the next-highest competing mode is much smaller. In this case, the selective breaking of the PT symmetry for the mode with the peak gain is more difficult. Accordingly, the unwanted multi-modes are more difficult to be completely suppressed. Figure 7 shows a zoomed-in view of the spectrum with a 300-kHz span and a 1-kHz RBW. The competing modes can be clearly identified with a mode spacing of 87 kHz, which is in good agreement with the loop length of approximately 2 km. The SMSR is 53 dB, which is 16 dB lower than that of the OEO with a 1-km loop length. Conversely, when the loop length of the OEO is reduced, according to Eq. (5), the gain difference is further increased, and the side modes will be suppressed lower than those in the OEO with a 1-km loop length. Consequently, the SMSR must be higher than 71.4 dB. Considering that the HNLF length can also affect the performance of the generated MPF , we carry out an investigation into the effect of the OEO loop length on the SMSR through a numerical simulation, in which the performance of the MPF in the OEO is set as invariant. Figure 8 shows the simulated SMSR of the OEO with different loop lengths under the same open-loop response. It can be seen that the SMSR of the OEO is indeed increased when the loop length is reduced. When the loop length is 250 m, the amplitude of the side mode is extremely low, and the SMSR is as high as 83 dB. Reducing the loop length can increase the SMSR and increase the linewidth of the microwave signal as shown in the inset in Fig. 8, which directly indicates that its phase noise is deteriorated . Therefore, there is a trade-off between the phase noise and SMSR.
The measured phase noise of the OEO with a 2-km loop length at 17.74 GHz is shown as the red curve in Fig. 9. It can be seen that the phase noise is at a 10-kHz offset frequency, which is 10 dB lower than that of the OEO with a 1-km loop length. It can be concluded that, although the SMSR is slightly degraded, the phase noise at a 10-kHz offset frequency is reduced by 10 dB. In addition, the phase noise performance of a commercial microwave source (Agilent E8247C) at 17.74 GHz is also measured, as shown the blue curve in Fig. 7 for comparison. It can be seen that the phase noise of the proposed OEO with a 2-km loop length is 20 dB lower than that of the commercial microwave source at a 10-kHz offset frequency. This is because the Agilent source utilizes a frequency multiplication to obtain microwave signals at high frequencies, and the phase noise is deteriorated significantly at high frequencies. However, the phase noise of the OEO will not be deteriorated with the frequency increase in theory, which is one of its most superior advantages. At low frequencies, the phase noise of the Agilent source is lower than that of the proposed OEO, as shown the green and purple curves in Fig. 9. The phase noise of the proposed OEO with a 2-km loop length is at a 10-kHz offset frequency. The measured phase noises of 1- and 2-GHz microwave signals from the Agilent source are both at a 10-kHz offset frequency, which are actually and according to the data sheet, respectively . This is because the typical limit of the phase noise measurement of the ESA is at a 10-kHz offset frequency according to the data sheet . A lower phase noise may be observed using an ESA with a lower measurement limit. At low offset frequencies, the phase noise of the OEO is higher than that of the commercial microwave source because the long fiber in the OEO can be affected by environmental perturbations. If desired, a phase-locked loop can be incorporated into the OEO loop to reduce the phase noise at a low offset frequency and improve the long-term stability.
As is well known, commercial OEO products have been developed, such as the HI-Q COMPACT OEO , which has a significant advantage in terms of the phase noise performance, and its output frequency can be changed between 10 and 12 GHz. To achieve higher frequencies, a commercial solution is to use frequency multiplication, but the phase noise can be degraded at high frequencies . Therefore, the best choice is to directly generate high-frequency microwave signals. We consider the PT-symmetry breaking as a good supplement to mode selection in conventional filter-based OEOs. In addition to a microwave filter, when the PT-symmetry breaking is also exploited in the loop, a longer fiber can be used. Consequently, a lower phase noise and higher SMSR can be achieved. Further, by selectively breaking the PT symmetry, the generated high-quality microwave signals can be tuned over a large frequency range.
In conclusion, we proposed and experimentally demonstrated a widely tunable OEO based on selective PT-symmetry breaking. Our experiment results indicate that the selective breaking of PT symmetry can be used to enforce single-mode oscillation in the optoelectronic hybrid cavity of an OEO. Single-mode oscillation at 17.74 GHz with a 71.4-dB SMSR is achieved in the case of a 1-km loop length. In our proposed OEO, the incorporated MPF with a large roll-off can enhance the oscillation-mode gain difference between the selected mode and competing modes in the PT-symmetry system. This can help select the mode with peak gain more easily compared with the case in which only a microwave filtering technique or PT symmetry is used. Further, the selective breaking of PT symmetry also enhances the maximum output power of the selected mode by 12.9 dB. Simultaneously, the oscillation frequency is tuned from 2.6 to 40 GHz. When the optical fiber length is increased to 2 km, the phase noise of the OEO is reduced to at a 10-kHz offset frequency at 17.74 GHz. Exploitation of the selective PT-symmetry breaking provides the possibility of developing classes of widely tunable OEO with an ultra-high SMSR and excellent low phase noise simultaneously.
National Natural Science Foundation of China (NSFC) (61501194, 11664009); Natural Science Foundation of Hubei Province (2015CFB231, 2014CFA004, 2016CFB370); Fundamental Research Funds for the Central Universities (HUST: 2016YXMS025).
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