Breaking the limitation of mode building time in an optoelectronic oscillator.

An optoelectronic oscillator (OEO) is a microwave photonic system with a positive feedback loop used to create microwave oscillation with ultra-low phase noise thanks to the employment of a high-quality-factor energy storage element, such as a fiber delay line. For many applications, a frequency-tunable microwave signal or waveform, such as a linearly chirped microwave waveform (LCMW), is also needed. Due to the long characteristic time constant required for building up stable oscillation at an oscillation mode, it is impossible to generate an LCMW with a large chirp rate using a conventional frequency-tunable OEO. In this study, we propose and demonstrate a new scheme to generate a large chirp-rate LCMW based on Fourier domain mode locking technique to break the limitation of mode building time in an OEO. An LCMW with a high chirp rate of 0.34 GHz/μs and a large time-bandwidth product of 166,650 is demonstrated.

Here we analysis the MPF as follows. Since the notch has usually quite narrow bandwidth while the optical carrier and the other sideband are far away, we assume the notch filter contributes only time delay and loss on them. Detach such delay and loss, transfer function of the notch filter can be described as 1 − ( − 0 ) where 0 is the notch center. After notch filter, the optical field, which contains sideband remains, optical carrier, and the other sideband, can also be equivalent to the original three optical bands deconstructed by to-be-blocked component of one sideband, as shown in Fig. S1b. Mathematically, the latter is in proportion to where OC ( ) is the phase variation of continuous wave (CW) light source and ℎ( ) is inverse Fourier transform of ( ).
At PD the four optical bands are mixed, where the original ±1 sidebands cancel each other exactly due to the phase-modulation-induced opposite sign. The recovered RF signal is then the mixing product of the optical carrier and the to-be-blocked component.

Supplementary Note 2: Oscillation Process of the FDML-OEO
Oscillation process from noise to stable frequency scanning Page 4 of 9 Supplementary Figure 2 shows the temporal traces of the entire oscillation process from noise to stable frequency scanning measured by a real-time electronic oscilloscope (Tektronics DPO70000, 100 GS/s sampling rate) 3 . The entire process could be divided by initial phase, transition phase and stable frequency scanning phase. Supplementary Figure 2  shows the frequency distributions and temporal waveforms of the initial and transition phases, respectively. The amplitude of generated temporal waveform is amplified with the increasing of round trips, where the power of real-time frequency distribution is also increased obviously.
Finally, in the stable frequency-scanning phase, a linearly chirped microwave waveform with periodical frequency scanning is generated, as shown in Supplementary Fig. 2 (c).

Supplementary Figure 2. Temporal trace of the FDML-OEO oscillation process from noise to stable frequency
scanning. a Initial phase. b Transition phase. c Stable scanning phase.

Oscillation process from single-frequency to stable frequency scanning
The oscillation process from single frequency to stable scanning is also recorded, as shown in Supplementary Fig. 3. This is achieved by altering injection current of the tunable laser source (TLS) from constant to a saw-tooth driving. Supplementary Figure 3 (a) shows the frequency distributions when the transition process just started. The initial signal frequency is the predominant component during the whole scanning because of injection locking effect. The others scanning components of the waveform is gradually amplified with the increasing of round trips during the transition process, as can be seen in Supplementary Fig. 3 (b), the non-uniform amplification is mainly caused by mode competition. Finally, a stable scanning signal is achieved, as shown in Supplementary Fig. 3 (c).

Supplementary Note 3: Signal Consistency of the of the FDML-OEO
An overlay of 12 traces for frequency scanning microwave waveforms from 8 to 12 GHz recorded in the Fast Frame mode of the Tektronix oscilloscope are shown in Supplementary   Fig. 4 (a). The span was set to be 40 μs in order to trace at least one period of the generated waveform. The sampling rate was set to be 25 GS/s rather than 100 GS/s in order to have more frames overlay. The 100,000 times zoom-in view of the overlaid 12 traces is shown in Supplementary Fig. 4 (b). Only very small jitters can be seen from Supplementary Fig. 4, which Page 6 of 9 indicates a good consistency of the generated chirped microwave waveform.

Supplementary Note 5: Phase noise improvement of the FDML-OEO
The relatively large phase noise close to the carrier is mainly caused by the ambient fluctuation because the OEO is sensitive to the environmental changes. Thus, a lower phase noise close to the carrier can be expected by using vibration and thermal isolation.
In addition, the phase locking technique, which is widely used for frequency stabilization of an oscillator, can also be used in our scheme to reduce the phase noise close to the carrier 4 .
Basically, an optical self-phase locked loop (SPLL) can be used to stabilize the FDML-OEO.
To do so, a portion of the OEO optical output before the PD is coupled out of the OEO loop and delayed by TD, and the phase of the delayed signal is compared with the phase of the microwave signal generated from OEO. The delay time TD should satisfy TD =lTround-trip, where Tround-trip is the round-trip time of the OEO loop and l is an integer. In this way, an error signal is obtained without an external reference oscillator. The error signal is then fed back to the OEO loop to change the effective loop length. The phase noise performance at low offset frequencies as well as the long-term frequency stability can be improved 4 .
On the other hand, the phase noise at a frequency-offset far from the carrier is affected by the side-modes of the OEO loop. The side-mode spacing is 45 kHz in our demonstrated system.
The side-modes cannot be well suppressed due to the wide bandwidth of the MPF, which is normally at least tens of megahertz. A series of peaks observed from the SSB phase noise measurement, shown in Fig. 7, corresponds to the beating between two adjacent modes which is 45 kHz, and its multiples, leading to a worse phase noise performance at a frequency-offset far from the carrier, as compared with the one from the arbitrary waveform generator (AWG).
A multi-loop OEO is a good candidate 5 to obtain low phase noise at a frequency-offset far from the carrier. Supplementary Figure 6