## Abstract

We propose and demonstrate a frequency multiplying optoelectronic oscillator with nonlinearly-coupled double loops based on two cascaded Mach–Zehnder modulators, to generate high frequency microwave signals using only low-frequency devices. We find the final oscillation modes are only determined by the length of the master oscillation loop. Frequency multiplying signals are generated via nonlinearly-coupled double loops, the output of one loop being used to modulate the other. In the experiments, microwave signals at 10 GHz with −121 dBc/Hz phase noise at 10 kHz offset and 20 GHz with −112.8 dBc/Hz phase noise at 10 kHz offset are generated. Meanwhile, their side-mode suppression ratios are also evaluated and the maximum ratio of 70 dB is obtained.

©2013 Optical Society of America

## 1. Introduction

Optoelectronic oscillators (OEOs) with pure emitted microwave signals has attracted great attention recently due to its numerous potential applications and advantages in wireless communication system, radar systems, modern electronic warfare [1,2], optical signal processing [3–6], high sensitivity sensors [7], microwave signals detection [8], etc. Due to the electronic bottleneck of the electrical and electro-optical devices used, the generated frequency of conventional OEOs is limited to only few 10s of GHz [9]. To extend the operational frequency range, some frequency multiplying OEOs (FM-OEOs) concepts have been suggested [10–16]. For instance, a frequency-doubling OEO using a double-drive Mach–Zehnder modulator (MZM) and a frequency shifter was proposed in [10], where the MZM was biased at the minimum transmission point to realize double sideband suppressed carrier (DSB-SC) modulation. In order to generate higher frequency band of the microwave signals, a polarization modulator (PolM) aided by two polarizers was applied [11]. Moreover, a frequency-doubling OEO can also be realized using a phase modulator and a phase-shifted fiber Bragg grating (PS-FBG), albeit with the use of an optical notch filter [12]. However, a frequency-doubling OEO without an optical notch filter has been reported in [13], employing a dual parallel Mach Zehnder modulator (DP-MZM). Recently, a frequency-quadrupling OEO for multichannel up conversion was proposed and demonstrated based on two cascaded polarization modulators [14].

In this paper, a FM-OEO with nonlinearly-coupled double loops is established based on two cascaded MZMs. Two MZMs are applied to compose the nonlinearly-coupled double loops. The master loop, Loop_{1}, is designed to be relatively short in order to generate a single-mode microwave signal at the output, which the slave loop, Loop_{2}, is significantly longer in order to enhance the Q of the FM-OEO signal. A theoretical analysis is performed, which is validated with an experiment by adjusting the optical variable delay lines (OVDL) in Loop_{1} and Loop_{2}. The generation of frequency-multiplying microwave signals at 10GHz and 20GHz are demonstrated. To the best of our knowledge, this is the first FM-OEO scheme without optical filter, dual-parallel Mach-Zehnder modulator or polarization modulator.

## 2. Operation principle

The implementation of FM-OEO incorporates nonlinearly-coupled double loops using two MZMs as
shown in Fig. 1. The output light from first MZM (MZM_{1}) is equally split into two parts via an
optical coupler (OC). One part of the output signal is sent through to the second MZM
(MZM_{2}) while the other part is directed towards Loop_{2} through a longer
length single mode fibre (SMF) (several hundred meters) and an OVDL, after which it is converted
into an electrical signal using a photodetector (PD), PD_{2}, before being fed back into
the modulator part of MZM_{2}.

Suppose the electrical input signal ${V}_{1}(t)$ to MZM_{1} is a sinusoidal wave with an angular frequency of ${\omega}_{m}$, an amplitude of ${V}_{RF1}$, and a DC bias of ${V}_{b1}$, then

_{1}can be expressed by

_{1}. Then expand the left-hand side of Eq. (2) with Bessel functions:

The electrical signal *V*_{2} (*t*), proportional to${\left|{E}_{1}(t+{\tau}_{2})\right|}^{2}$, is delayed by a time τ_{2} and fed back into MZM_{2}. According to Eq. (3), *V*_{2}(*t*) can be expressed as follows:

_{2}, and ${G}_{A2}$ is the voltage gain of EA

_{2}.The output electric field of MZM

_{2}can be written as:

_{2}. Thus the output of PD

_{1}${E}_{3}\left(t\right)$can be simply described as follows:

_{1}.

It is clear from Eq. (6) that there are many harmonic components of ${\omega}_{m}$, such as frequency-doubling signal and -quadrupling signals. A particular signal can be isolated using a suitable RF filter with a sufficiently narrow bandwidth to block all other harmonic components. Using such filter, Eq. (6) can be re-written as:

_{1}at any instant time is the summation of all circulating filed in the system. When the FM-OEO oscillates stably, the total output of PD

_{1}can be expressed as:

_{1}, $n$ is the number of times the field has circulated around Loop

_{1}and ${G}_{e}\left({\omega}_{m}\right)$ is the effective open-loop gain, given by

*R*is the load impedance of PD

_{1}.

Once the FM-OEO oscillates stably, only the oscillation frequency components with the minimum loss and those total loop phase shift being a multiple of $2\pi $ can oscillate. Therefore, the oscillation frequency must meet the phase matching condition as:

where the oscillation frequency peaks will locate atTherefore, the relationship between frequency stability and loop delay can be described by:

where $\Delta \omega $ is the variation of angular frequency, $\Delta f$is the variation of frequency and $\Delta \tau $ is the variation of delay.Obviously, the final oscillation frequencies are determined by the delay of loop_{1 ${\tau}_{1}$}, but not the delay of loop_{2} ${\tau}_{2}$. From Eq. (6), it is apparent that ${\tau}_{2}$ only influences the initial phase of the oscillation signals, which is starkly different to the principle of the second loop in conventional dual-loop-OEO systems [17], whereby the longer loop presents low spurious peaks due to unwanted interferences. Only those modes that are closest to the filter center frequency and meet both the phase matching conditions of long loop and short loop will oscillate since they have constructive interferences. In other words, the double loops are linearly-coupled and determine the final oscillation modes altogether. However, the double loops are nonlinearly-coupled in our FM-OEO scheme, in which the output of one loop is used to modulate the other. This is quite an asymmetrical configuration because the two loops play different roles to guarantee the stable operation of the system. Loop_{1} acts as the master oscillation loop and determines the oscillation modes, while Loop_{2} plays a role as a slave oscillation loop to reduce the detrimental effect of the multiplicative phase noise and dump delay-induced spurious peaks [18].

## 3. Experiment result and discussion

An experiment based on the scheme shown in Fig. 1 is carried out. The DFB laser has a fixed wavelength of 1550.6 nm and the EDFA has a saturated output optical power of 23 dBm. The MZMs from JDSU have a 3 dB bandwidth of 20 GHz, whose half-wave voltages of 5.7 V. The PDs have a 3 dB bandwidth of 30 GHz with a responsivity of 0.9 A/W. The EAs with a bandwidth of 2~33 GHz and a maximum gain of 48 dB is applied to compensate the loop loss. The EBPF has a 3 dB bandwidth of 30 MHz with the center frequency of 10 GHz. The length of Loop_{1} is nearly 10 m with a time delay τ_{1} of 50ns, while the length of Loop_{2} is nearly 300 m and its delay τ_{2} is 1.5μs. An OVDL with the maximum delay of 330ps is used to adjust the feedback phase and tune the oscillation mode when needed. An ESA Agilent 8593E with input frequency ranging from 9 kHz to 23 GHz is used to observe the output microwave signals.

When the loop is closed, the single frequency oscillation of OEO is realized, as is shown in
Fig. 2. By appending OVDL_{1} in Loop_{1}, the initial oscillating signal is
10.00383 GHz with${\tau}_{OVDL1}\text{=}0$ps, as is shown in Fig.
2(a). By adjusting OVDL_{1}, the output spectrum of the generated oscillation
signal is changed to 10.00360 GHz with ${\tau}_{OVDL1}\text{=10}$ps, as is shown in Fig.
2(b). It is apparent that the relationship between frequency stability and the loop delay
is in agreement with Eq. (6). However, Fig. 2(c) and Fig. 2(d)
show that there is little change in the frequency of the oscillation signal when adjusting
OVDL_{2}. This strongly supports the case that the final oscillation modes are solely
determined by the delay of Loop_{1}, and not that of Loop_{2}.

To evaluate the side-mode suppression ratio (SMSR), the comparative analysis between FM-OEO and the conventional double-loop-OEO [17] with the same length of loops is made in Fig. 3(a), (b). The SMSR of FM-OEO is nearly 70 dB, which is more than 30 dB higher than that of the conventional double-loop OEO. The side-modes are sufficiently suppressed due to the function of the nonlinearly-coupled double loops.

Figure 4 shows the microwave signals at 10 GHz and 20 GHz. Due to the limit on the bandwidth of the employed ESA, only the frequency-doubled microwave signal at 20 GHz can be measured. As is seen in Fig. 4, the spectrum of the fundamental-frequency component is 21 dB higher than that of the frequency-doubled microwave signal. To investigate the spectral quality of the generated microwave signal, the single-sideband (SSB) phase noises of the signals are measured by the frequency discriminator method [19]. Figure 5 shows the results. As a comparison, the phase noise spectrums for different operating conditions are also shown in Fig. 5. The phase noises of four different signals at a 10-kHz offset frequency are −94.6 dBc/Hz,−106.9 dBc/Hz, −121 dBc/Hz and −112.8 dBc/Hz,respectively. Compared with the single-loop OEO, the SSB phase noise is reduced remarkably in the conventional double-loop OEO, as is shown in Fig. 5(a), (b). The FM-OEO is also superior to the conventional double-loop OEO in its less SSB phase noise due to the elimination of multiplicative phase noise, as is shown in Fig. 5(b), (c). The frequency-doubled microwave signal has 8.2 dB phase noise degradation, compared with that of the fundamental-frequency microwave signal. Theoretically, the phase noise of a frequency-doubled signal should have a phase noise degradation of about $10{\mathrm{log}}_{10}{2}^{2}\approx 6$dB.

## 4. Conclusion

A FM-OEO with nonlinearly-coupled double loops based on two cascaded MZMs is proposed and its mathematical model is demonstrated. In a conventional double-loop OEO, both loops are coupled linearly and play a similar role, while in our system the loops play different roles and the coupling is nonlinear to form the master-slave oscillation loops. The final oscillation modes are only determined by the length of the master loop, which accords with the theoretical analysis. The SMSR of FM-OEO is nearly 70 dB, which is more than 30 dB higher than that of the conventional double-loop OEO. The SSB phase noise of 10 GHz microwave signal is −121 dBc/Hz at a 10-kHz offset frequency. We believe that the proposed FM-OEO can find applications in wireless communication, radars devices, electronic warfare and optical signal processing.

## Acknowledgments

We thank the reviewers for their comments that substantially improved this work by the National Basic Research Program of China (973 Program 2012CB315703) and the National Natural Science Foundation of China (No. 61275027).

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