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Dynamics and timing-jitter of regenerative RF feedback assisted resonant electro-optic frequency comb

Huanfa Peng, Peng Lei, Xiaopeng Xie, and Zhangyuan Chen

Open Access Open Access

Abstract

Optical frequency combs have impacted a wide range of applications. The optical cavity assisted resonant electro-optic (EO) comb features broad spectral bandwidth and deterministic repetition-rate. However, the optical pulse timing-jitter is limited by the phase noise of the external RF drive source. To overcome this limitation at high repetition-rate, a regenerative RF feedback assisted resonant EO comb has been experimentally developed, which shows low phase noise of the regenerative RF drive signal. However, yet to date, the nonlinear dynamics and timing-jitter of the resonant EO comb coupled with a regenerative RF feedback loop remain unexplored. Here, for the first time, we quantitatively investigate the dynamics and timing-jitter of the resonant EO comb generation coupled with an optoelectronic oscillator (OEO). Our model indicates the detuning of the seed laser and EO comb cavity resonance determines the nonlinear feedback RF gain of OEO. Numerical simulations reveal the emergence of the bifurcation behaviors of the regenerative RF signal and the distinct EO comb formation dynamics with the increasing feedback RF gain of OEO, which provide the guidelines to achieve stable regenerative resonant EO comb. Moreover, in the presence of the noisy nonlinear feedback RF gain induced by the laser-cavity detuning fluctuations, our timing-jitter model predicts the seed laser frequency noise will be converted to the regenerative RF phase noise and optical pulse timing-jitter under the stable oscillation state of OEO. We highlight the mechanism of the transduction of laser frequency noise to the RF phase noise is differ from the Mach-Zehnder modulator based OEO. Experimentally, we attain a 10 GHz resonant EO comb generation with phase noise of −130 dBc/Hz at 10 kHz offset for the regenerative 10 GHz RF signal and optical pulse timing-jitter of 52.8 fs with an integral range from 100 Hz to 1 MHz. The experimental results agree well with the theoretical models. Our findings offer the solid guidelines for the design of the regenerative RF feedback assisted stable resonant EO comb with an optimized timing-jitter performance.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical frequency comb is an optical signal consisting of hundreds or even thousands of highly stable, narrow spectral, equidistantly spaced, and phase-locked optical tones, which enables the coherent phase link between the RF and optical domains [1]. The optical frequency comb has impacted various applications, including frequency metrology, time-keeping, ranging, optical communication, optical and microwave frequency synthesis, spectroscopy, and astronomical spectrograph calibration [2]. At present, the optical frequency comb generation relies on four kinds of mechanisms. The first one is based on the solid or fiber mode-locked laser (MLL) [3]. The second category utilizes the chip-scale semiconductor MLL [2]. The third is the electro-optic (EO) frequency comb [4]. The forth approach is based on the chip-scale Kerr soliton comb [5]. Among them, the EO frequency comb offers the prospects of agile and deterministic repetition-rate, which is attractive for the applications. Usually, the EO frequency comb generator can be formed by the cascading of a multitude of EO modulators. However, without external nonlinear spectral broadening, the comb bandwidth is usually limited [6]. To achieve high efficient EO frequency comb with broad spectral bandwidth, the resonant EO frequency comb has been proposed and developed [7]. The key idea is to place an EO modulator inside an optical resonator or an optical cavity. The EO modulator is driven by a RF signal with the frequency matches the integer multiple of the optical resonator mode spacing. The use of the optical resonator enhances the EO interaction strength, which leads to the broadening of the comb bandwidth. In time domain, the generated EO frequency comb features narrow optical pulse train [8]. The phase noise of the external RF drive signal is demonstrated as the limitation of the timing-jitter of the optical pulse train [8]. In particular, the phase noise of the RF oscillator degrades with the carrier frequency [9], which makes it challenging to achieve low timing-jitter at high repetition-rate. To overcome this limitation, a regenerative RF feedback assisted resonant EO frequency comb generation system has been experimentally developed [10,11]. A portion of the output optical signal from the resonant EO comb generator is photodetected and fed back to drive the resonant EO comb generator after the RF amplification and bandpass RF filtering. The feedback loop forms an optoelectronic oscillator (OEO), which can generate a regenerative RF drive signal with ultra-low phase noise and phase noise nearly independent of oscillation frequency [12]. This way eliminates the requirement of the external RF drive source and provides the prospect to achieve a resonant EO comb with ultra-low timing-jitter and high repetition-rate, simultaneously. Despite the preliminary work for the theoretical description of the dynamics of the Mach-Zehnder modulator (MZM) based OEO [13–15], the use of the resonant EO comb generator in an OEO loop features the coupling of the cavity dynamics of the resonant EO comb generator and the nonlinear dynamics of the regenerative optoelectronic oscillation loop, which shows more dynamical properties but still remain unexplored. Moreover, the timing-jitter of the regenerative RF feedback assisted resonant EO comb is of high significance for the applications but the theoretical description is still lacking.

In this paper, we quantitatively investigate the dynamics of the regenerative RF signal, the resonant EO comb formation, and the optical pulse timing-jitter of the resonant EO comb generation coupled with an OEO cavity, for the first time. Analogously to the framework of Ikeda map method used for the modelling of the time-delayed MZM based OEO [16], we derived a nonlinear time-delayed differential equation to describe the dynamics of the regenerative RF signal and the resonant EO comb formation. Unlike the model of the MZM based OEO [16], our model takes into account the coupling of the resonant EO comb cavity and the OEO cavity, which reveals that the detuning of the seed laser and EO comb cavity resonance determines the nonlinear feedback RF gain of the OEO loop. Our numerical predictions show the amplitude of the regenerative RF signal undergoes non-oscillation, stable oscillation, period-doubling, and chaos with the increasing of the linear feedback RF gain, and the corresponding distinct EO comb formation dynamics. These results provide the guidelines to achieve stable regenerative resonant EO comb. Furthermore, in the presence of laser-cavity fluctuations in the nonlinear time-delayed differential equation for the regenerative RF dynamics, we build a phase noise model, which predicts the laser frequency noise will be converted to the phase noise of the regenerative RF signal under the stable oscillation state of OEO. We highlight that the mechanism of the conversion of laser frequency noise to the RF phase noise is differ from the MZM based OEO [16] due to the coupling of the OEO cavity and the resonant EO comb cavity. Experimentally, we attain an EO comb with 10 GHz comb spacing, and a 10 GHz regenerative RF signal with phase noise of −130 dBc/Hz at 10 kHz offset. The experimental results show the conversion of the seed laser frequency noise dominates the phase noise of the regenerative RF signal. The theoretical transduction of the laser frequency noise to the RF phase noise is calculated to be coincided with the experimental results. Our findings provide the solid guidelines for the design and timing-jitter optimization of the resonant EO comb generation without the use of external RF drive source.

The remaining paper is organized as follows. In Section 2, the system architecture and principle of the regenerative RF feedback assisted EO comb are presented. Afterwards, the dynamics of the regenerative RF signal, the dynamics of the EO comb formation and the optical pulse timing-jitter will be presented in Section 3. An experiment is implemented in Section 4 to verify the theoretical predictions. Finally, a conclusion is drawn in Section 5.

2. System architecture and operation principle

Figures 1(a) and 1(b) show the schematic diagrams of the resonant EO comb generator when driven by an external RF drive source and a regenerative RF signal, respectively. A continuous-wave laser is injected to the resonant EO comb generator after passing through an optical isolator. Here, the resonant EO comb generator is formed by a high speed EO phase modulator inside a Fabry-Perot (FP) cavity. The phase modulator is driven by a RF signal. A Bias-T is used to combine the RF drive signal and the DC bias signal. By tuning the voltage of the DC bias applied on the phase modulator, the frequency of the cavity resonance can be changed. The cavity resonance and the seed laser wavelength can be matched by either tuning the laser wavelength or the DC bias of the phase modulator. When the frequency of the RF drive signal matches the integer number of the free-spectral-range (FSR) of the FP cavity, the optical sidebands generated by the phase modulator are resonant, which leads to a broad comb bandwidth. In time domain, the EO comb features optical pulse train. In contrast to the scheme driven by an external RF drive source, the regenerative RF feedback assisted EO comb generation eliminates the external RF drive source, as shown in Fig. 1(b). After passing through a long fiber, the EO comb is converted to RF signal by a photodetector. The converted RF signal is fed back to drive the resonant EO comb generator after the RF amplification and the bandpass RF filtering. The center frequency of the bandpass RF filter matches a harmonic of the FSR of the FP cavity. This hybrid optoelectronic feedback loop forms an OEO, which can generate a regenerative RF signal with ultra-low phase noise and phase noise nearly independent of frequency. The regenerative RF signal frequency is determined by the bandpass RF filter and the FSR of the FP cavity. Unlike the former case by using external RF source, the evolution of the regenerative RF signal is influenced by the resonant EO comb generator due to the coupling of the resonant EO comb generator and the optoelectronic oscillation loop, which shows rich dynamics. Besides, the phase noise of the regenerative RF signal and the optical pulse timing-jitter are highly relevant to the noise sources of the feedback loop.

 figure: Fig. 1.

Fig. 1. Two system architectures of the resonant EO comb generation. (a) The resonant EO comb generation when driven by an external RF drive source. (b) The resonant EO comb generation when driven by a regenerative RF signal. CW: continuous-wave. EO: electro-optic. ESA: electrical spectrum analyzer.

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3. Theoretical models and numerical simulations

In the following sub-sections, we will firstly derive the time domain model of the regenerative RF feedback assisted EO comb generation. Afterwards, the temporal dynamics of the amplitude of the regenerative RF signal will be theoretically investigated. After that, the EO comb dynamics will be presented. Finally, we will show the optical pulse timing-jitter model.

3.1 Time domain model of the regenerative RF feedback assisted resonant EO comb

To mathematically describe the evolution of the regenerative RF signal and comb formation in Fig. 1(b), the framework of Ikeda map can be used [16]. Figure 2 shows the block diagram of the time domain model of the Ikeda-like OEO loop model, which consists of four elements in the feedback loop. The Ikeda variable of this model is the voltage of the regenerative RF signal, which is denoted by Vosc(t). It circulates in the clockwise direction and is subjected to the four elements of the feedback loop. The first element is the nonlinear electrical-to-optical (E/O) and optical-to-electrical (O/E) unit, which quantitatively shows the nonlinear relation of the voltage of the RF drive signal for the resonant EO comb generator and the output RF voltage of the photodetector. The second is the fiber delay line, which indicates the time delay of the optical signal after propagating through the fiber. The third element is the RF amplifier, which describes the RF gain of the feedback RF signal. The last one is the linear bandpass RF filter. To mathematically link the voltage of the input and output signals of the bandpass RF filter, we can use the integrodifferential operator ĤBF{Vosc(t)} [16]. In the following part, the differential equation for the dynamics of Vosc(t) will be derived.

 figure: Fig. 2.

Fig. 2. Block diagram of the time domain model by using the framework of Ikeda-like model. The Vosc(t) denotes the voltage of the regenerative RF signal, which circulates in the clockwise direction and is subjected to the four main elements of the feedback loop, the nonlinear electrical-to-optical (E/O) and optical-to-electrical (O/E) unit, the fiber delay line, the RF amplifier, and the bandpass RF filter.

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The linear bandpass RF filter can be considered as an input-output system obeying:

$${\hat{H}_{\textrm{BF}}}\{ {V_{\textrm{osc}}}(t)\} = {V_{\textrm{in}}}(t), $$
where Vin(t) and Vosc(t) are the input and output voltages of the bandpass RF filter, respectively. As the Ikeda-like model shown in Fig. 2, the input of the bandpass RF filter is given as:
$${V_{\textrm{in}}}(t) = g \cdot {f_{\textrm{NL}}}[{V_{\textrm{osc}}}(t - T)], $$
where fNL is the nonlinear function of the E/O and O/E unit, g is the voltage gain of the regenerative RF signal, T is the time delay. By combining Eqs. (1) and (2), the dynamics of the regenerative RF signal is governed by the Ikeda-like equation, which is yielded as:
$${\hat{H}_{\textrm{BF}}}\{ {V_{\textrm{osc}}}(t)\} = g \cdot {f_{\textrm{NL}}}[{V_{\textrm{osc}}}(t - T)]. $$

To get the full description of the evolution of the regenerative RF signal, two relations should be derived, which are the nonlinear function fNL, and the linear operator ĤBF{Vosc(t)}. First, we will derive the nonlinear function fNL of the E/O and O/E unit. This unit is formed by the resonant EO comb generator followed by a photodetector, as shown in Fig. 3(a). The input and output of this unit are the RF drive voltage of the EO comb generator Vosc(t) and the output voltage of the photodetector fNL{Vosc(t)}. A traveling-wave type phase modulator is placed in the FP cavity, which results in only the light co-propagating with the RF signal will be modulated [17]. A monochromatic light wave seeds the resonant EO comb generator. Mathematically, the electric field of the input light can be described as:

$${E_{\textrm{in}}}(t) = {E_{\textrm{in}}}{e^{i2\pi {v_0}t}}, $$
where Ein is the amplitude, i is the imaginary unit, t denotes the time, v0 is the laser frequency. The voltage of the sinusoidal RF drive signal applied on the phase modulator can be given as:
$${V_{\textrm{osc}}}(t) = {V_{\textrm{osc}}}\sin (2\pi {f_{\textrm{osc}}}t), $$
where Vosc is the amplitude, and fosc is the frequency. The optical phase shifts of the phase modulator due to the RF signal modulation and the DC bias can be expressed as:
$${\varphi _\textrm{m}}(t) = \pi \frac{{{V_{\textrm{osc}}}(t)}}{{{V_\mathrm{\pi }}}},\,\,\textrm{and}\,\,{\varphi _\textrm{B}} = \pi \frac{{{V_\textrm{B}}}}{{{V_\mathrm{\pi }}}}, $$
where Vπ is the half-wave voltage of the phase modulator, and VB is the DC bias voltage. When the RF modulation frequency matches a harmonic of the FSR of the FP cavity, the injection light wave and the phase modulated sidebands are resonantly enhanced. In this case, the output electric field of the resonant EO comb generator can be expressed as [17]:
$${E_{\textrm{out}}}(t) = {E_{\textrm{in}}}(t) \cdot \sqrt {{\eta _{FP}}} (t^{\prime}t^{\prime})\sum\limits_{k = 0}^{ + \infty } {{{(rr)}^k}{{(\sqrt \eta )}^{2k + 1}}\exp \{ i[ - (2k + 1){k_0}nL + (k + 1)({\varphi _\textrm{m}} + {\varphi _\textrm{B}})]\} }, $$
where ηFP is the power insertion loss coefficient of the FP cavity, $t^{\prime}$ is the amplitude transmission coefficient of the FP cavity, r is the amplitude reflection coefficient of the FP cavity, η is the single pass power loss coefficient of the FP cavity, k is an integer number, k0 is the wavenumber of the light wave in vacuum, n is the refractive index of the phase modulator, L is the physical length of the FP cavity. The relation of the amplitude transmission coefficient and the power reflection coefficient can be expressed as:
$$R = {r^2} = 1 - {(t^{\prime})^2}, $$
where R is the power reflection coefficient of the FP cavity mirror. The FSR of the FP cavity is:
$${f_{\textrm{FSR}}} = \frac{c}{{2nL}}, $$
where c is the light velocity in vacuum. By combining Eqs. (4)–(9), the output electric field of the EO comb generator can be computed as:
$${E_{\textrm{out}}}(t) = {E_{\textrm{in}}}(t)\sqrt {{\eta _{\textrm{FP}}}} \sqrt \eta (1 - R)\frac{{\exp \{ i[ - \frac{{\pi {\nu _0}}}{{{f_{\textrm{FSR}}}}} + {\varphi _\textrm{m}} + {\varphi _\textrm{B}}]\} }}{{1 - R\eta \exp \{ i[ - \frac{{2\pi {\nu _0}}}{{{f_{\textrm{FSR}}}}} + {\varphi _\textrm{m}} + {\varphi _\textrm{B}}]\} }}. $$

 figure: Fig. 3.

Fig. 3. (a) Schematic diagram of the E/O and O/E unit formed by the resonant EO comb generator followed by a photodetector. A traveling-wave type phase modulator (PM) is placed in the FP cavity, only the light co-propagating with the RF signal is modulated. (b) Schematic diagram of the bandpass RF filter formed by a series RLC circuit.

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The frequency of the seed laser v0 can be written as the sum of the frequency of the closet cavity resonance to the seed laser frequency and the frequency detuning, which is given as:

$${\nu _0} = M \cdot {f_{\textrm{FSR}}} + \delta v, $$
where M is the mode number of the closet cavity resonance to the seed laser, δv = v0-vcavity is the frequency detuning of the seed laser and the cavity resonance, vcavity is the frequency of the cavity resonance. When the frequency of the seed laser is close to the cavity resonance, then δv is much smaller than fFSR. In the right-hand side term of Eq. (10), the phase factor of the exponential component in the denominator can be reformulated as:
$$- \frac{{2\pi {\nu _0}}}{{{f_{\textrm{FSR}}}}} ={-} \frac{{2\pi \delta \nu }}{{{f_{\textrm{FSR}}}}} - 2M\pi. $$

Correspondingly, the output power of the EO comb generator can be given as:

$${P_{\textrm{out}}}(t) = {E_{\textrm{out}}}(t) \cdot E_{\textrm{out}}^\ast (t). $$

By combining Eq. (10), and Eqs. (12)–(13), the output power is derived as:

$${P_{\textrm{out}}}(t) = \frac{{\alpha {P_{\textrm{in}}}}}{{1 + F \cdot {{\sin }^2}[\theta + \pi \frac{{{V_{\textrm{osc}}}(t)}}{{2{V_\mathrm{\pi }}}}]}}, $$
where Pin=|Ein(t)|2 is the input optical power of the EO comb generator. In Eq. (14), the variables θ, F, and α are defined as:
$$\theta ={-} \frac{{\pi \delta \nu }}{{{f_{\textrm{FSR}}}}} + \pi \frac{{{V_\textrm{B}}}}{{2{V_\mathrm{\pi }}}}, $$
$$F = \frac{{4R\eta }}{{{{(1 - R\eta )}^2}}}, $$
and
$$\alpha = \frac{{{\eta _{\textrm{FP}}}\eta {{(1 - R)}^2}}}{{{{(1 - R\eta )}^2}}}, $$
where θ is the generalized laser-cavity detuning. When the EO comb is sent to a fast photodetector, the output voltage of the photodetector can be expressed as:
$${f_{\textrm{NL}}}\{ {V_{\textrm{osc}}}(t)\} = \Re {P_{\textrm{out}}}(t){Z_0}, $$
where ℜ and Z0 are the responsivity and load impedance of the photodetector, respectively. In Eq. (18), it shows the nonlinear relation of the input and output of the E/O and O/E unit. By substituting Eq. (14) into Eq. (18), and considering the time delay T of the fiber delay line, the time-delayed output of the E/O and O/E unit can be expressed as:
$${f_{\textrm{NL}}}\{ {V_{\textrm{osc}}}(t - T)\} = \frac{{\alpha \Re {P_{\textrm{in}}}{Z_0}}}{{1 + F \cdot {{\sin }^2}[\theta + \pi \frac{{{V_{\textrm{osc}}}(t - T)}}{{2{V_\mathrm{\pi }}}}]}}. $$

In Eq. (19), the right-hand side term multiplied with the linear RF gain g serves as the input of the bandpass RF filter. To quantitatively model the filtering characteristics of the bandpass RF filter in the OEO loop, we can use the series Resistance-Inductance-Capacitance (RLC) circuit model [18]. In Laplace domain, the transfer function of the series RLC circuit shown in Fig. 3(b) can be described as:

$$\frac{{{V_{\textrm{osc}}}(s)}}{{{V_{\textrm{in}}}(s)}} = \frac{1}{{{Q_\textrm{r}}s/{\omega _\textrm{r}} + {Q_\textrm{r}}{\omega _\textrm{r}}/s + 1}}, $$
where Vosc(s) and Vin(s) are the Laplace transforms of the voltage of the output and input signals of the bandpass RF filter, ωr and Qr are the resonance frequency and quality factor of the bandpass RF filter, respectively. The parameters ωr and Qr are given as:
$${\omega _\textrm{r}} = \frac{1}{{\sqrt {LC} }}, \;\textrm{and}\; {Q_\textrm{r}} = \frac{1}{Z}\sqrt {\frac{L}{C}},$$
where Z, L, and C are the resistance of the resistor, the inductance of the inductor, and the capacitance of the capacitor, respectively. By transforming the Eq. (20) into time domain, the relation of the input and output voltage of the bandpass RF filter can be derived as:
$$\frac{{{Q_\textrm{r}}}}{{{\omega _\textrm{r}}}}\frac{{d{V_{\textrm{osc}}}(t)}}{{dt}} + {Q_\textrm{r}}{\omega _\textrm{r}}\int\limits_{{t_0}}^t {{V_{\textrm{osc}}}(s^{\prime})} ds^{\prime} + {V_{\textrm{osc}}}(t) = {V_{\textrm{in}}}(t), $$
where t0 denotes the initial time of the integral. By comparing Eq. (22) with Eq. (1), the integrodifferential operator ĤBF{Vosc(t)} can be derived as:
$${\hat{H}_{\textrm{BF}}}\{ {V_{\textrm{osc}}}(t)\} = \frac{{{Q_\textrm{r}}}}{{{\omega _\textrm{r}}}}\frac{{d{V_{\textrm{osc}}}(t)}}{{dt}} + {Q_\textrm{r}}{\omega _\textrm{r}}\int\limits_{{t_0}}^t {{V_{\textrm{osc}}}(s^{\prime})} ds^{\prime} + {V_{\textrm{osc}}}(t). $$

To simplify the derived equations, we use the dimensionless variable x(t) to describe the evolution of the amplitude of the regenerative RF signal. The x(t) is defined as:

$$x(t) = \pi \frac{{{V_{\textrm{osc}}}(t)}}{{2{V_\mathrm{\pi }}}}. $$

By substituting Eqs. (19) and (23) into Eq. (3) and using the x(t) defined in Eq. (24), the dimensionless nonlinear time-delayed differential equation is derived as:

$$\frac{{{Q_\textrm{r}}}}{{{\omega _\textrm{r}}}}\frac{{dx(t)}}{{dt}} + {Q_\textrm{r}}{\omega _\textrm{r}}\int\limits_{{t_0}}^t {x(s^{\prime})} ds^{\prime} + x(t) = g\frac{\gamma }{{1 + F \cdot {{\sin }^2}[\theta + x(t - T)]}}, $$
where γ is defined as:
$$\gamma = \frac{{\pi \alpha \Re {P_{\textrm{in}}}{Z_0}}}{{2{V_\mathrm{\pi }}}}. $$

The Eq. (25) is the basis for the nonlinear dynamics and phase noise study of the regenerative RF feedback assisted EO comb.

3.2 Temporal dynamics of the amplitude of the regenerative RF signal

Physically, due to the narrow-band filtering of the bandpass RF filter, we can use the ansatz of quasi-continuous-wave solution for the regenerative RF signal, which is written as:

$$x(t) = \frac{1}{2}A(t){e^{i{\omega _{\textrm{osc}}}t}} + \frac{1}{2}{A^\ast }(t){e^{ - i{\omega _{\textrm{osc}}}t}} = u(t)\cos [{\omega _{\textrm{osc}}}t + \varphi (t)], $$
where A(t)=u(t)eiφ(t) is the slowly varying complex envelope of x(t), ωosc is the angular frequency, A*(t) is the complex conjugate of A(t), u(t) and φ(t) are the amplitude and the phase of A(t). The right-hand side term of Eq. (25) can be reformulated as:
$$\frac{{g\gamma }}{{1 + F \cdot {{\sin }^2}[\theta + x(t - T)]}} = \frac{{g\rho }}{{1 - \delta \cos [2\theta + 2x(t - T)]}}, $$
where the variables ρ and δ are defined as ρ=2γ/(2+F), and δ=F/(F+2). By using the Taylor series expansion, the Eq. (28) can be derived as:
$$\frac{{g\rho }}{{1 - \delta \cos [2\theta + 2x(t - T)]}} = g\rho \sum\limits_{k = 0}^{ + \infty } {\{ \delta \cos {{[2\theta + 2x(t - T)\} }^k}}. $$

In the right-hand side term of Eq. (29), the cosine of a sinusoidal function of angular frequency ωosc can be Fourier-expanded in harmonics of ωosc by using the Jacobi-Anger expansion. The Jacobi-Anger expansion is given as:

$${e^{iz\cos (\xi )}} = \sum\limits_{k ={-} \infty }^{ + \infty } {{i^k}{J_k}(z)} {e^{ik\xi }}, $$
where Jk is the kth-order Bessel function of the first kind. By inserting the ansatz of x(t) shown in Eq. (27) into Eq. (29), and discarding the harmonics of ωosc and DC term due to the bandpass RF filter is narrowly resonant around ωosc, the left-hand side term of Eq. (29) can be derived as [see Appendix A]:
$$\frac{{g\rho }}{{1 - \delta \cos [2\theta + 2x(t - T)]}} ={-} g \cdot \varsigma \cdot x(t - T), $$
where ς is the nonlinear feedback RF gain, which is derived as:
$$\varsigma = \frac{\rho }{{u(t - T)}}\sum\limits_{k = 1}^{ + \infty } {\frac{{{\delta ^k}}}{{{2^{k - 2}}}}\sum\limits_{l = 0}^{l < \frac{k}{2}} {C_k^l{J_1}[2(k - 2l)u(t - T)]\sin [2(k - 2l)\theta ]} }, $$
where $C_k^l$ is the number of l-permutations of k. By substituting Eq. (31) into Eq. (25), the dynamics of x(t) with oscillation frequency of ωosc can be derived as:
$$\frac{{{Q_\textrm{r}}}}{{{\omega _\textrm{r}}}}\frac{{dx(t)}}{{dt}} + {Q_\textrm{r}}{\omega _\textrm{r}}\int\limits_{{t_0}}^t {x(s^{\prime})} ds^{\prime} + x(t) ={-} g \cdot \varsigma \cdot x(t - T). $$

To further simplify the dynamics equation of x(t), we can derive the dynamics of the slowly varying complex envelope A(t). Due to A(t) varies much slowly than the carrier frequency ωosc, the following differential equation for A(t) can be derived [see Appendix B], which is:

$$\frac{{dA(t)}}{{dt}} ={-} \mu {e^{i\vartheta }}A(t) + \mu {e^{i\vartheta }} \cdot g \cdot \varsigma \cdot A(t - T), $$
where $\mu = {\omega _\textrm{r}}/\sqrt {4Q_\textrm{r}^2 + 1}$ and $\vartheta \textrm{ = arctan(1/}2{Q_\textrm{r}}\textrm{)}$. In the stationary oscillation state, A(t) should be a constant, which indicates the first order time derivative of A(t) is zero. It leads to A(t)=A(t-T). In this case, the amplitude of the regenerative RF signal obeys the nonlinear algebraic equation of gς=1. In order to graphically show the solutions of the amplitude of the regenerative RF signal, we reformulate the nonlinear algebraic equation as:
$$u(t - T) \cdot \varsigma = \frac{{u(t - T)}}{g}. $$

To find the solutions of the u(t-T), we can plot the curves of the terms in the right-hand and left-hand sides of Eq. (35) with respect to u(t-T). The abscissa value of the intersection points of the two curves are the solutions of u(t-T).

To investigate the temporal dynamics of the amplitude of the regenerative RF signal, numerical simulations based on the parameters listed in Table 1 are performed. The time-delayed differential equation shown in Eq. (34) is numerically solved by MATLAB with the function of dde23 [19]. The dde23 is intended to solve the time-delayed differential equation with constant time delay. The simulation parameters of the resonant EO comb generator are based on the device specifications of a commercial optical frequency comb generator (OFCG) from Optocomb Inc. with the model of WTEC-01-25. The time delay of the feedback loop is set to 10 us, which indicates 2 km fiber is used in the OEO loop. Moreover, the simulation time is 1 ms. The initial value of the Vosc(t) is set to 1 mV, which is assumed to be the amplified thermal noise at the output of the RF amplifiers within the passband of the bandpass RF filter.

Tables Icon

Table 1. Simulation parameters for the temporal dynamics of the regenerative RF signal

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Figures 4(a) – 4(c) show the evolution of the amplitude |A(t)| with respect to nature time t with linear feedback gain g of 68, 116, and 350, respectively. Note that the generalized laser-cavity detuning θ is set to be 0.1 in this case. With linear feedback RF gain of 68, the oscillation amplitude converges to constant in a short time, which indicates the stable oscillation. By contrast, the emergence of periodic amplitude modulation and chaos in the presence of larger linear feedback gain can be observed, as shown in Figs. 4(b) and 4(c). In the period-doubling state, the regenerative RF amplitude is periodically switched between two numbers. In the chaotic state, the amplitude of the regenerative RF signal randomly varies over time. The nonlinear modulation of the amplitude is unfavorable for the stable EO comb generation. To investigate the nonlinear dynamics of the amplitude, the bifurcation diagram for the amplitude |A(t)| is simulated, as shown in Fig. 4(d). It shows that the amplitude undergoes no oscillation (1), stable oscillation (2), cascaded period-doubling (3) and chaos (4) with the increasing of the linear feedback gain g. To operate the OEO in the stationary oscillation state, the linear feedback gain g should be in the range of 38 to 81. To find the solutions of the stable amplitude of the regenerative RF signal, the graphical representation of the oscillatory solution by using Eq. (35) is shown in Fig. 4(e). The blue curve shows the nonlinear term of the left-hand side of Eq. (35). The magenta and red curves show the term in the right-hand side of Eq. (35) with g of 38, and 81, respectively, which are the lower and upper bounds of the stationary state. The solution of the oscillation amplitude is the abscissa value of the nontrivial intersection point of the nonlinear blue curve and the linear dashed lines. The intersection points in the light blue region shows the no oscillation state. In the green region, we can find the solution for the stable oscillation. In the orange region, the system shows unstable oscillation, which is the instability region. Besides, to investigate the effects of the generalized laser-cavity detuning θ on the nonlinear term u(t-T)ς, the curves of the u(t-T)ς versus u(t-T) under different values of θ are plotted in Fig. 4(f). When θ is 0, the u(t-T)ς is calculated to be 0, as illustrated by the red curve. In this case, it leads to no RF oscillation in the feedback loop. When the θ is increased from 0.1 to 0.3, the peak of the u(t-T)ς is shifted to the right side. It requires higher linear RF gain g to achieve a unique nontrivial solution of Eq. (35).

 figure: Fig. 4.

Fig. 4. (a), (b), and (c): The evolution of the amplitude of the regenerative RF signal with linear feedback gain g of 68, 116, and 350, respectively. (d) The numerical bifurcation diagram for the amplitude of the regenerative RF signal with respect to the linear gain g. The numbers (1)–(4) with different colors indicate distinct bifurcation behaviours. (e) The graphical representation of Eq. (35). The solid blue curve is u(t-T)ς, and the dashed lines are the u(t-T)/g with boundary values of the feedback gain g (38 for magenta and 81 for red) in the stationary state. (f) The nonlinear term u(t-T)ς versus the u(t-T) with different generalized laser-cavity detuning θ. The magenta dashed line is the u(t-T)/g when g is 68. The a.u. denotes arbitrary unit.

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3.3 Dynamics of the regenerative EO comb formation

In this section, the formation of the optical waveform and spectrum of the EO comb will be investigated. The evolution of the optical waveform is governed by Eq. (14). In the following part, we will derive the power of the comb lines. By using the Taylor expansion, the electric field of the EO comb shown in Eq. (10) can be reformulated as:

$${E_{\textrm{out}}}(t) = {E_{\textrm{in}}}(t)\beta \sum\limits_{k = 1}^{ + \infty } {{{(R\eta )}^{k - 1}}\exp (ik2\theta )\exp \{ ik2u(t)\cos [{\omega _{\textrm{osc}}}t + \varphi (t)]\} }, $$
where β= (ηFPη)−1/2(1-R)exp(iπv0/fFSR). By using the Jacobi-Anger expansion shown in Eq. (30), the output electric field can be expanded as the superposition of the harmonics of ωosc, which is expressed as:
$${E_{\textrm{out}}}(t) = {E_{\textrm{in}}}\beta \sum\limits_{q ={-} \infty }^{ + \infty } {{i^q}\{ \sum\limits_{k = 1}^{ + \infty } {{{(R\eta )}^{k - 1}}\exp (i2k\theta )} {J_q}[2ku(t)]\} {e^{i\{{2\pi {v_0}t + q[{{\omega_{\textrm{osc}}}t + \varphi (t)} ]} \}}}}. $$

In Eq. (37), the electric field consists of a multitude of frequency components with equidistant frequency spacing of ωosc. With respect to the seed laser frequency v0, the complex amplitude of the electric field of the qth comb line is extracted as:

$${E_q}(t) = {E_{\textrm{in}}} \cdot \beta \cdot {i^q}\sum\limits_{k = 1}^{ + \infty } {{{(R\eta )}^{k - 1}}\exp (i2k\theta )} {J_q}[2ku(t)]. $$

Correspondingly, the optical power of the qth comb line is derived as:

$${P_q}(t) = |{E_q}(t){|^2} = {P_{\textrm{in}}}|\beta {|^2} \cdot |\sum\limits_{k = 1}^{ + \infty } {{{(R\eta )}^{k - 1}}\exp (i2k\theta )} {J_q}[2ku(t)]{|^2}. $$

The power of the comb line is affected by the generalized laser-cavity detuning θ and the amplitude of the regenerative RF signal u(t). The EO comb formation depends on the dynamics of the regenerative RF signal.

To investigate the resonant EO comb formation, numerical simulations of the Eqs. (14), (34), and Eq. (39) are performed. First, based on the Eq. (34), the amplitude of the RF signal will be solved. Afterwards, the RF amplitude will be substituted into Eqs. (14) and (39) to get the optical waveform and comb spectrum of the EO comb. Figure 5 shows the simulation results of the dynamics of the EO comb formation. Two cases are considered. For the first case, the OEO is operated in the stable oscillation state with linear feedback gain g of 68, as shown in Figs. 5(a) and 5(b). In contrast, the second case shows the comb formation when the OEO is operated in the period-doubling state with linear feedback gain g of 116, which is illustrated by Figs. 5(c) and 5(d). Note that the intensity of the optical waveform and the comb spectrum are normalized by the peak power. The top of Fig. 5(a) shows the evolution of the optical waveform. The time window for the time domain representation of the optical pulse is set to be the roundtrip time of the FP cavity. The bottom of Fig. 5(a) is the time slice of the optical waveform at the time of 0.5 ms, which shows the temporal optical waveform of the comb. Correspondingly, the top of Fig. 5(b) is the evolution of the comb spectrum. The bottom of Fig. 5(b) is the time slice of the comb spectrum at the time of 0.5 ms. The center wavelength of the seed laser is 1550.12 nm and 513 optical modes are simulated. In the beginning of the comb formation, the optical power is concentrated to the center wavelength, which leads to the flat optical waveform and less comb lines in the EO comb cavity, as shown in the top of Figs. 5(a) and 5(b) when the time is below 0.05 ms. When the RF oscillation is stable after the time of 0.05 ms, an optical pulse train with broad comb spectrum is attained, as illustrated in the bottom of Figs. 5(a) and 5(b). The optical waveform features two sets of time-interleaved optical pulse train, as indicated by the purple and red dashed arrows. The period of each set of optical pulse train is equal to the period of the regenerative RF signal, which is denoted by Tosc. Note that the pulse formation arises from the periodic changing of the magnitude of the denominator in Eq. (14). The pulse emerges when the denominator reaches the minima. With a sinusoidal RF drive signal, it will have two distinct minima for the denominator in one period of the RF drive signal, which leads to the time-interleaved two sets of optical pulse train. Figures 5(c) and 5(d) show the optical waveform and comb spectrum when the OEO is operated in the period-doubling state. In this case, the optical waveform is periodically switched between two distinct states, as indicated by the white and red dashed boxes in the top of Fig. 5(c). The optical waveform of the two states are illustrated in the bottom of Fig. 5(c). In state 1, the two sets of the optical pulse train have large time spacing. In contrast, the two sets of the optical pulse will be merged nearly to one set of optical pulse train in state 2. Correspondingly, the comb spectrum evolution in the top of Fig. 5(d) shows the periodic changing of the comb bandwidth. The comb spectrum features triangle shape with symmetrical comb lines with respect to the seed laser wavelength. In state 1, the comb bandwidth is much broader than that of the state 2.

 figure: Fig. 5.

Fig. 5. (a) Top: The evolution of the optical waveform versus nature time. Bottom: optical waveform in the stationary oscillation state. (b) Top: The evolution of the comb spectrum. Bottom: comb spectrum in the stationary oscillation state. The linear feedback gain g is 68. Tosc is the period of the regenerative RF signal. (c) Top: The evolution of the optical waveform when the OEO is operated in the period-doubling state. Bottom: the optical waveform of the two different states shown by the white and red dashed boxes in the top of Fig. 5(c). (d) Top: The evolution of the comb spectrum when the OEO is operated in the period-doubling state. Bottom: the time sliced comb spectrum in two different states shown by the white and red dashed boxes in the top of Fig. 5(d). The linear feedback gain g is 116.

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3.4 Phase noise of the regenerative RF signal and optical pulse timing-jitter

The phase noise of the OEO can be modeled by a Langevin approach in time domain [15]. Two kinds of noise sources, which are the additive noise and the multiplicative noise, will be converted to the close-in phase noise of the oscillation RF carrier [15]. The additive noise can be treated as the Langevin forcing term, which is added to the right-hand side of Eq. (33). The additive noise around the oscillation carrier of the regenerative RF signal can be written as [15]:

$${\xi _\textrm{a}}(t) = \frac{1}{2}{\varsigma _\textrm{a}}(t){e^{i{\omega _{\textrm{osc}}}t}} + \frac{1}{2}\varsigma _\textrm{a}^\ast (t){e^{ - i{\omega _{\textrm{osc}}}t}}, $$
where ςa(t) is the complex amplitude of the white Gaussian noise. The second noise source is the multiplicative noise, which is induced by the fluctuations of the OEO feedback RF gain [15]. Theoretically, the multiplicative noise can be modeled by the relative gain fluctuation coefficient ηm(t). By adding the additive noise to the right-hand side of Eq. (33), and considering the feedback gain fluctuations, the temporal dynamics of A(t) can be derived as [15]:
$$\frac{{dA(t)}}{{dt}} ={-} \mu {e^{i\vartheta }}A(t) + \mu {e^{i\vartheta }}g \cdot \varsigma \cdot [1 + {\eta _\textrm{m}}(t)] \cdot A(t - T) + \mu {e^{i\vartheta }}{\varsigma _\textrm{a}}(t). $$

Note that the additive noise is added at the output of the RF gain unit shown in Fig. 2. Besides, the feedback RF gain fluctuations can be caused by the fluctuation of the linear feedback gain g of the RF amplifier, or the fluctuation of the nonlinear feedback gain ς. It should be noted that the phase evolution of the regenerative RF signal in the presence of noise for different oscillation states of the OEO can be numerically simulated via Eq. (41). However, in the period-doubling and chaotic states shown in Figs. 4(b) and 4(c), the RF amplitudes are nonlinearly modulated, which lead to unstable EO combs. Here, we aim to investigate the timing-jitter of the stable EO comb, which is the case of stable oscillation state of OEO. A feasible way is to numerically simulate the stochastic nonlinear time-delayed differential equation. However, it requires small time step to guarantee the computational convergence, which leads to the increase of the computational effort to predict the long-term RF phase fluctuations. An alternative way is to analytically derive the RF phase noise spectrum after the linearization of Eq. (41) near the steady-state of OEO [15]. In this case, the magnitude of the total feedback RF gain gς is equal to unity. Moreover, the phase dynamics in Eq. (41) can be derived by using the Ito rules of stochastic calculus [15], which is given as:

$$\frac{{d\varphi (t)}}{{dt}} ={-} \mu [\varphi (t) - \varphi (t - T)] + \frac{\mu }{{2{Q_\textrm{r}}}}{\eta _\textrm{m}}(t) + \frac{\mu }{{|{{A_0}} |}}{\xi _{\mathrm{a,\varphi }}}(t), $$
where ξa,φ(t) is a real Gaussian white noise, A0 is the dimensionless RF amplitude in stationary oscillation state. Equation (42) shows the evolution of the slowly varying phase of the RF signal, which is the close-in carrier phase noise. To achieve the phase noise spectrum, the Eq. (42) can be transformed into Laplace domain as [20]:
$$\varphi (s) = [\frac{1}{{2{Q_\textrm{r}}}}{\eta _\textrm{m}}(s) + \frac{1}{{|{{A_0}} |}}{\xi _{\mathrm{a,\varphi }}}(s)] \cdot H(s), $$
where H(s) is the transfer function of the OEO loop, which is:
$$H(s) = \frac{\mu }{{s + \mu - \mu {e^{ - sT}}}}. $$

To suppress the side-modes of the OEO loop, the dual-loop structure OEO by using the Vernier effect to guarantee single-mode oscillation is widely used [21]. With a dual-loop OEO configuration, the transfer function can be described as [22]:

$${H_{\textrm{dual - loop}}}(s) = \frac{\mu }{{s + \mu - \mu [\varepsilon {e^{ - s{T_1}}} + (1 - \varepsilon ){e^{ - s{T_2}}}]}}, $$
where ε is the ratio between the RF amplitude of the first loop and total RF amplitude of the two loops, T1 and T2 are the time-delays of the two loops. In order to get the power spectral density (PSD) of the phase fluctuation, we take square modulus and substitute s = i2πf into Eq. (43). Note that f is the Fourier offset frequency from the RF carrier. The PSD of the phase noise of the regenerative RF signal can be derived as:
$${S_\varphi }(f) = [\frac{1}{{4Q_\textrm{r}^2}}|{\eta _\textrm{m}}(f){|^2} + \frac{1}{{{{|{{A_0}} |}^2}}}|{\xi _{\mathrm{a,\varphi }}}(f){|^2}] \cdot |H(f){|^2}. $$

The PSD of ξa,φ(t) is |ξa,φ(f)|2=2Da [15]. Da is the white noise spectral density. In the OEO loop, the additive phase noise is contributed by the thermal noise, shot noise and the relative intensity noise (RIN) of the seed laser. The Da can be given as [15]:

$${D_\textrm{a}} = \frac{{\pi {R_{\textrm{out}}}{g^2}}}{{8V_\mathrm{\pi }^2}}[{F_0}{k_\textrm{B}}{T_0} + 2e{I_{\textrm{ph}}}{Z_0} + {N_{\textrm{rin}}}I_{\textrm{ph}}^2{Z_0}], $$
where F0 is the noise figure of the RF amplifier, kB is the Boltzmann constant, T0 is the room temperature, e is the electron charge, Iph is the photocurrent, Nrin is the RIN of the seed laser, and Rout is the output impedance of the photodetector. Unlike the MZM based OEO, in addition to the multiplicative noise induced by the gain fluctuations of the RF amplifier, a unique multiplicative noise source induced by the nonlinear feedback RF gain fluctuations via laser-cavity detuning drift should be taken into account in our system. Due to the flicker noise of the RF amplifier and the fluctuation of laser-cavity detuning are uncorrelated, the PSD of the relative gain fluctuation of the OEO can be given as:
$$|{\eta _\textrm{m}}(f){|^2} = |{\eta _{\textrm{m,g}}}(f){|^2} + |{\eta _{\mathrm{m,\varsigma }}}(f){|^2}, $$
where |ηm,g(f)|2 and |ηm,ς(f)|2 are the PSD of the linear and nonlinear relative gain fluctuations. Usually, the multiplicative noise of the RF amplifier is flicker type, which is given as [15]:
$$|{\eta _{\textrm{m,g}}}(f){|^2} = 2{D_\textrm{m}}(1 + \frac{{{f_\textrm{H}}}}{{f + {f_\textrm{L}}}}), $$
where fL and fH are the low corner and high corner frequencies of the flicker noise, Dm is the noise floor of the flicker noise. In the presence of the generalized laser-cavity detuning fluctuation, the nonlinear gain ς is not a constant, which can be replaced by ς+Δς(t). Δς(t) is the fluctuation of the nonlinear feedback RF gain induced by the laser-cavity detuning drift. Mathematically, it can be described as:
$$\Delta \varsigma (t) = \frac{{d\varsigma }}{{d\theta }} \cdot \Delta \theta (t), $$
where Δθ(t) is the fluctuation of the generalized laser-cavity detuning. The relative gain fluctuation of the nonlinear feedback RF gain ς is given as:
$${\eta _{\mathrm{m,\varsigma }}}(t) = \frac{{\Delta \varsigma (t)}}{\varsigma }. $$

The generalized laser-cavity detuning is related to the fluctuations of the seed laser frequency, and the resonance of the EO comb cavity, as shown in Eq. (15). With the thermal stabilization of the FP cavity of the EO comb generator, we can assume that the effect of the cavity resonance fluctuation is much smaller than the seed laser frequency drift. Thus, by neglecting the fluctuation of the cavity resonance, the Δθ(t) can be derived as:

$$\Delta \theta (t) ={-} \frac{{\pi \Delta {v_0}(t)}}{{{f_{\textrm{FSR}}}}}, $$
where Δv0(t) is the laser frequency fluctuation. By combining Eqs. (50)–(52), the relative nonlinear feedback RF gain fluctuation can be derived as:
$${\eta _{\mathrm{m,\varsigma }}}(t) ={-} \frac{\pi }{{\varsigma {f_{\textrm{FSR}}}}} \cdot \frac{{d\varsigma }}{{d\theta }} \cdot \Delta {v_0}(t). $$

The PSD of ηm,ς(t) in Eq. (53) is:

$$|{\eta _{\mathrm{m,\varsigma }}}(f){|^2} = |\frac{\pi }{{\varsigma {f_{\textrm{FSR}}}}} \cdot \frac{{d\varsigma }}{{d\theta }}{|^2} \cdot {S_{\Delta v}}(f), $$
where SΔv(f) is the PSD of the laser frequency noise. The optical pulse timing-jitter can be calculated by the integral of the phase noise of the regenerative RF signal, which is [8]:
$${\tau _{jitter}} = \frac{1}{{{\omega _{\textrm{osc}}}}}\sqrt {\int\limits_{f = {f_1}}^{f = {f_2}} {{S_\varphi }(f)} df}, $$
where f1 and f2 are the lower and upper bounds of the integral. It should be noted that the analytical phase noise model shown in Eq. (46) shows high computational efficiency but with the neglect of the impacts of second-order amplitude fluctuations on the RF phase noise [15]. Moreover, the accuracy of the model also depends on the accurate modeling or fitting of the physical noise sources.

4. Experimental results and discussions

To verify the theoretical predictions, an experiment is performed. The setup is shown in Fig. 6. An OFCG (Optocomb Inc., WTEC-01-25) is used as the resonant EO comb generator. According to the specifications of the OFCG, the cavity power reflectivity R is 0.97, the half-wave voltage at 10 GHz is around 22 V, the single-pass power loss coefficient is around 0.98, the FSR is 2.5 GHz and the effective finesse of the cavity is around 60. The temperature of the F-P cavity of the OFCG is stabilized by a temperature controller, which improves the stability of the cavity resonance. A continuous-wave laser (Teraxion Inc., NLL) with wavelength of 1550.12 nm, linewidth of 5 kHz, and power of 80 mW is used as the seed laser of the OFCG. To compensate the optical loss of the OFCG and the optical fiber links, an erbium-doped fiber amplifier (EDFA) is used after the OFCG. Two single-mode fibers with lengths of 2 km and 0.5 km are used to form a dual-loop OEO to suppress the side-modes. Two photodiodes (PD1 and PD2, APIC Inc.) with responsivity of 0.95 A/W, saturation power of 50 mW, bandwidth of 20 GHz are used to convert the optical signals to electrical signals. Due to the low optical return loss (ORL) of the photodiodes, two optical isolators are utilized to decrease the optical reflection induced phase noise in the two fiber links [23]. The outputs of the two photodiodes are combined by an electrical combiner. Three low phase noise amplifiers (LPA, Analog Devices Inc., HMC-C072) with operation frequency range from 6 GHz to 12 GHz, gain of 11 dB, noise figure of 4.5 dB, and saturation power of 22 dBm are cascaded to provide sufficient RF gain for the OEO. An EBPF with the center frequency of 10 GHz and 3-dB bandwidth of 1 GHz is used to select the transmission resonance of the OFCG. The regenerative RF signal is fed back to drive the OFCG. A Bias-T (SHF Inc., BT65R) is used to combine the RF drive signal and the DC bias signal. A portion of the oscillation RF signal is monitored by an electrical spectrum analyzer (ESA).

 figure: Fig. 6.

Fig. 6. Experimental setup. CW: continuous-wave. OFCG: optical frequency comb generator. PM: phase modulator. EDFA: erbium-doped fiber amplifier. PD1, PD2: photodiodes. LPA: low phase noise amplifier. ESA: electrical spectrum analyzer.

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4.1 Transmission of the E/O and O/E unit

As analyzed in Section 3.1, the OFCG followed by the photodetector serves as the E/O and O/E unit in the OEO loop. First, the transmission response of the OFCG followed by the photodetector is tested by using a vector network analyzer (VNA, Keysight Inc., N5247A), as shown in Fig. 7(a). The OFCG is driven by the output of the VNA, and the photodetected comb is sent to the input port of the VNA. A multitude of narrow transmission resonances with center frequencies of the integer multiple of 2.5 GHz can be observed. The resonance frequencies are determined by the FSR of the OFCG. The 3-dB bandwidth of the resonances are around 40 MHz. The transmission with multiple resonances is unfavorable for the OEO to achieve single-mode oscillation. Here, we use a bandpass RF filter with 3-dB bandwidth of 1 GHz to select the 10 GHz transmission resonance, as illustrated in the inset of Fig. 7(a). Moreover, as theoretically analyzed in Section 3.2, the nonlinear feedback RF gain is related to the generalized laser-cavity detuning, which can be changed by the DC bias voltage. To investigate the influence of the DC bias on the nonlinear feedback RF gain, we test the 10 GHz fundamental signal of the photodetected comb after using an external 10 GHz signal to drive the OFCG, as shown in Fig. 7(b). A high level 10 GHz signal can be obtained by tuning the DC bias of the OFCG. However, the fundamental mode can be strongly suppressed by changing the DC bias, as shown by the inset of Fig. 7(b). The second harmonic level is much higher than that of the fundamental mode. It corresponds to the zero nonlinear feedback RF gain when zero generalized laser-cavity detuning is applied, as shown by the red curve in Fig. 4(f). It should be noted that a low nonlinear feedback RF gain is unfavorable for the OEO oscillation.

 figure: Fig. 7.

Fig. 7. (a) Measured electrical-optical transmission response of the OFCG followed by a photodetector by using a vector network analyzer (VNA). Inset: The response of the 10 GHz electrical bandpass filter and the selected transmission resonance. (b) Electrical spectrum of the photodetected comb when the OFCG is driven by an external 10 GHz RF signal. Inset: the electrical spectrum of the photodetected comb by changing the DC bias of the OFCG.

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4.2 Spectra of the regenerative RF signal and the EO comb

Figure 8(a) shows the measured electrical spectrum of the regenerative 10 GHz RF signal by an ESA (Keysight Inc., N9030A). The span and resolution bandwidth (RBW) are 10 MHz and 9.1 kHz, respectively. It shows a single oscillation mode with high spectral-purity. Correspondingly, a broadband comb spectrum with mode spacing of 10 GHz is achieved, as shown in Fig. 8(b). The comb spectrum follows the envelope of the simulated comb spectrum shown in Fig. 5. To make a comparison, we use the Eq. (39) to calculate the comb spectrum envelope, which is shown by the magenta dashed curve. The experimental comb spectrum envelope is well fitted by the theoretical model. As demonstrated in the preliminary work [17], the comb spectrum will be asymmetric in the presence of the frequency detuning of the RF drive signal and the cavity resonance spacing. For the resonant EO comb generation by using external RF signal, it is challenging to keep zero frequency detuning of the RF drive signal and the cavity resonance spacing. However, in our system, the frequency of the regenerative RF signal is determined by the transmission resonance shown in the inset of Fig. 7(a), which is equal to the integer multiple of the cavity resonance spacing. A robust zero frequency detuning can be kept, which results in long-term stable symmetrical comb spectrum.

 figure: Fig. 8.

Fig. 8. (a) Electrical spectrum of the 10 GHz regenerative RF signal. (b) Optical spectrum of the resonant EO comb. The dashed magenta line shows the theoretical comb envelope fitting by using the model shown in Eq. (39).

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4.3 Phase noise of the regenerative RF signal

The SSB phase noise of the regenerative 10 GHz RF signal is measured by a signal source analyzer (SSA, Keysight Inc., E5052B + E5053A), as shown by the blue curve in Fig. 9. The measured SSB phase noise of the 10 GHz RF signal is −130 dBc/Hz at 10 kHz offset. Correspondingly, the timing-jitter of the optical pulse is calculated to be 52.8 fs with integral range from 100 Hz to 1 MHz by using Eq. (55). The spurs at offset of integer multiple of 100 kHz are originate from the 2 km OEO loop length. To analyze the noise sources in the OEO loop, the quantities of the experimental parameters are measured. The input optical power of the photodiode is around 4 mW. At the output of the photodiode, the power of the 10 GHz RF signal is around 20 µW. The seed laser RIN is measured to be −158 dB/Hz at 10 GHz offset by using an RIN measurement system (SYCATUS Inc., A0010A). With these parameters, the estimated shot noise, thermal noise, and the RIN noise are illustrated by the purple, green, and magenta dashed lines, respectively. Note that the amplified spontaneous emission (ASE) noise of the EDFA will also contribute to the phase noise floor of the OEO. Furthermore, the contribution of the additive phase noise of the RF amplifiers is shown by the orange dashed curve, which is the multiplication of the dual-loop OEO transfer function and the additive phase noise of the RF amplifier. The additive phase noise of the RF amplifier is measured by a phase noise analyser (Rohde & Schwarz Inc., FSWP26), which shows the offset frequency range is from 100 Hz to 1 MHz. It indicates the phase noise floor of the regenerative RF signal is limited by the RIN noise of the seed laser. Note that the close-in phase noise below 100 kHz offset is not dominated by the additive phase noise of the RF amplifier, which indicates the close-in phase noise is limited by other noise source, such as the laser frequency noise analysed in Section 3.4. To make a comparison with the OFCG driven by an external RF synthesizer, the phase noise of the photodetected comb of the OFCG driven by external RF signal is measured, as shown by the red curve. The external 10 GHz RF drive signal is generated by a benchtop RF synthesizer (Keysight Inc., E8257D). Due to our system benefits from the high quality factor of the OEO loop, it shows the phase noise of the regenerative RF signal is lower than that of the photodetected 10 GHz comb of the OFCG driven by external RF signal. Due to the OEO phase noise is nearly independent of frequency, our system is promising to attain high repetition-rate EO comb with low timing-jitter.

 figure: Fig. 9.

Fig. 9. SSB phase noise of the regenerative RF signal (Blue curve), the noise sources (Dashed curves) and the photodetected comb by using an external 10 GHz RF drive signal (Red curve).

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4.4 Contribution of seed laser frequency noise to the regenerative RF phase noise

To investigate the contribution of the laser frequency noise to the regenerative RF signal phase noise and the optical pulse timing-jitter, two distinct lasers are utilized to perform a comparison. Figure 10(a) shows the measured frequency noise of the two seed lasers by an optical noise analyzer (SYCATUS, A0040A). Correspondingly, the SSB phase noise of the 10 GHz regenerative RF signals by using the two seed lasers are shown by the blue and red curves in Fig. 10(b), which reveals that the system can achieve better phase noise performance by using a lower frequency noise seed laser. Furthermore, it shows high correlation between the laser frequency noise and the RF phase noise of the regenerative signal. To quantitatively investigate the contribution factor of the laser frequency noise to the phase noise of the regenerative RF signal, we calculate the phase noise curves by using the theoretical phase noise model shown in Eq. (46), and substituting the measured laser frequency noise shown in Fig. 10(a) into the phase noise model. Note that we use the transfer function of dual-loop structure OEO shown in Eq. (45). The theoretical predictions of the RF phase noise are shown by the olive curve (1) and the magenta curve (2) for the two seed lasers based EO combs. The transduction factors of the laser frequency noise to the RF phase noise of the two scenarios are estimated to be the same number of −170 dB. Note that the transfer function of the OEO loop is not included in the transduction factor. By combining Eqs. (46) and (54) in Section 3.4, the theoretical transduction factor of the laser frequency noise to the RF phase noise can be derived as:

$${F_\textrm{C}} = \frac{1}{{4Q_\textrm{r}^2}}|\frac{\pi }{{\varsigma {f_{\textrm{FSR}}}}} \cdot \frac{{d\varsigma }}{{d\theta }}{|^2}. $$

In our experiment, the Qr of the bandpass RF filter is 10, and fFSR is 2.5 GHz. By using the values of θ, ς, and the stationary oscillation amplitude |A(t)| in the simulation part of the Section 3.2, the theoretical conversion factor is calculated to be −164 dB, which is the same order of magnitude as the experimental transduction factor. However, the laser frequency noise can also be converted to the RF phase noise via the chromatic dispersion of the fiber [24]. The conversion factor is calculated to be −215 dB in our system by using the chromatic dispersion of 2 km single-mode fiber at 1550 nm, which indicates the transduction of laser frequency noise to the RF phase noise via laser-cavity detuning fluctuation dominates the RF phase noise. Since the contribution of the laser frequency noise to the regenerative RF phase noise arises from the laser-cavity detuning fluctuation, the stabilization of the laser-cavity detuning or by using lower frequency noise seed laser is beneficial for the optical pulse timing-jitter optimization.

 figure: Fig. 10.

Fig. 10. (a) The frequency noise spectrum of the two seed lasers used in the experiments. (b) The blue and red curves are the measured SSB phase noise of the regenerative RF signals with the two seed lasers, respectively. The olive curve (1) and magenta curve (2) denote the calculated phase noise by using the model shown in Eq. (46) and with the laser frequency noise as the noise source. Note that the dual-loop transfer function of OEO shown in Eq. (45) is used.

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5. Summary

In summary, we have developed a theoretical model to investigate the dynamics and the timing-jitter of the regenerative RF feedback assisted EO comb. The numerical simulation results show the amplitude of the regenerative RF signal undergoes a multitude of bifurcation behaviors, which lead to distinct EO comb formation dynamics. Our phase noise model predicts the laser frequency noise will be converted to the phase noise of the regenerative RF signal, which limits the optical pulse timing-jitter. We attained a 10 GHz regenerative resonant EO comb, which features low phase noise of −130 dBc/Hz at 10 kHz offset for the regenerative RF signal and 52.8 fs optical pulse timing-jitter. The phase noise of the regenerative RF signal is lower than that of the commercial benchtop RF synthesizer. The experimental results agree well with the theoretical predictions. Our findings provide solid guidelines to achieve a stable regenerative resonant EO comb with optimized timing-jitter. Besides, due to the phase noise of the regenerative RF signal is nearly independent of frequency, it is promising to achieve ultra-low timing-jitter resonant EO comb at high repetition-rate. By incorporating the integrated resonant EO comb generator in a thin film lithium-niobate nanophotonic platform [25], our system is amenable to chip-scale regenerative feedback assisted resonant EO comb generator.

Appendix A: Derivation of the nonlinear feedback RF gain ς

To simplify the derivation, we use term A to represent the right-hand side term of Eq. (29). By using Euler equation, the term A can be rewritten as:

$$A = g\rho \sum\limits_{k = 0}^{ + \infty } {{{(\frac{\delta }{2})}^k}{{\{ {e^{i2[\theta + x(t - T)]}} + {e^{ - i2[\theta + x(t - T)]}}\} }^k}}. $$

For the sake of simplicity, we use the abbreviations xT, uT, and φT to represent x(t-T), u(t-T) and φ(t-T), respectively. By using the binomial expansion theorem, the term A is derived as:

$$A = g\rho \sum\limits_{k = 0}^{ + \infty } {{{(\frac{\delta }{2})}^k}\sum\limits_{l = 0}^k {C_k^l{e^{i2l(\theta + {x_T})}}{e^{ - i2(k - l)(\theta + {x_T})}}} }, $$
where $C_k^l$ is the number of l-permutations of k. Since the x(t-T) is a cosine function, we can use the Jacobi-Anger expansion shown in Eq. (30) to rewrite Eq. (58) as the superposition of the components in harmonics of ωosc, which is derived as:
$$A = g\rho \sum\limits_{k = 0}^{ + \infty } {{{(\frac{\delta }{2})}^k}\sum\limits_{l = 0}^k {C_k^l{e^{i2(2l - k)\theta }}\sum\limits_{q ={-} \infty }^{ + \infty } {{i^q}{J_q}[2(2l - k){u_T}]{e^{iq[{\omega _{\textrm{osc}}}(t - T) + {\varphi _T}]}}} } }. $$

By discarding the harmonics of ωosc and the DC term, and using the Euler equation and the Bessel function condition of J1(x)=-J−1(x), the term with frequency of ωosc is extracted as:

$$A = g\rho \sum\limits_{k = 1}^{ + \infty } {\frac{{{\delta ^k}}}{{{2^{k - 1}}}}\sum\limits_{l = 0}^k {C_k^l{e^{i2(2l - k)\theta }}i{J_1}[2(2l - k){u_T}]\cos [{\omega _{\textrm{osc}}}(t - T) + {\varphi _T}]} }. $$

The Eq. (60) can be rewritten as:

$$A = B \cdot \cos [{\omega _{\textrm{osc}}}(t - T) + {\varphi _T}], $$
where the coefficient B is given as:
$$B = g\rho \sum\limits_{k = 1}^{ + \infty } {\frac{{{\delta ^k}}}{{{2^{k - 1}}}}} \sum\limits_{l = 0}^{l < k/2} {C_k^l{e^{i2(2l - k)\theta }}i{J_1}[2(2l - k){u_T}] + C_k^{k - l}{e^{i2[2(k - l) - k]\theta }}i{J_1}\{ 2[2(k - l) - k]{u_T}\} }. $$

With the conditions of $C_k^l = C_k^{k - l}$, J1(x)=-J1(-x), the Eq. (62) can be reformulated as:

$$B ={-} g\rho \sum\limits_{k = 1}^{ + \infty } {\frac{{{\delta ^k}}}{{{2^{k - 2}}}}} \sum\limits_{l = 0}^{l < k/2} {C_k^l{J_1}[2(k - 2l){u_T}]\sin [2(k - 2l)\theta ]}. $$

By combining Eqs. (61) and (63), we can get the final expression of A as:

$$A ={-} \frac{{g\rho }}{{{u_T}}}\sum\limits_{k = 1}^{ + \infty } {\frac{{{\delta ^k}}}{{{2^{k - 2}}}}\sum\limits_{l = 0}^{l < k/2} {C_k^l{J_1}[2(k - 2l){u_T}]\sin [2(k - 2l)\theta ]} } {u_T}\cos [{\omega _{\textrm{osc}}}(t - T) + {\varphi _T}]. $$

Since x(t-T)=uTcos[ωosc(t-T)+φT], the Eq. (64) can be rewritten as:

$$A ={-} g \cdot \varsigma \cdot x(t - T), $$
where ς is the nonlinear feedback RF gain, which is given as:
$$\varsigma = \frac{\rho }{{{u_T}}}\sum\limits_{k = 1}^{ + \infty } {\frac{{{\delta ^k}}}{{{2^{k - 2}}}}\sum\limits_{l = 0}^{l < k/2} {C_k^l{J_1}[2(k - 2l){u_T}]\sin [2(k - 2l)\theta ]} }. $$

Appendix B: Derivation of the nonlinear dynamics of A(t)

Note that the left-hand side term of Eq. (33) has the integral of x(t). Mathematically, the integral term will also oscillate with the frequency of ωosc, but with different slowly varying envelope [15]. We will use the following ansatz to represent the integral term, which is given as:

$$\int\limits_{{t_0}}^t {x(s^{\prime})} ds^{\prime} = \frac{1}{2}B(t){e^{i{\omega _{\textrm{osc}}}t}} + \frac{1}{2}{B^\ast }(t){e^{ - i{\omega _{\textrm{osc}}}t}}, $$
where B(t) is the slowly varying complex envelope of the integral of x(t), which is related to A(t). By taking the first order time derivative on both sides of Eq. (67) and comparing it with Eq. (27), the relation of A(t) and B(t) is derived as:
$$A(t) = \frac{{dB(t)}}{{dt}} + i{\omega _\textrm{r}}B(t). $$

By substituting Eq. (67) into Eq. (33), the differential equation for B(t) is derived as:

$$[\frac{{{d^2}B(t)}}{{d{t^2}}} + (\frac{{{\omega _\textrm{r}}}}{{{Q_\textrm{r}}}} + 2i{\omega _\textrm{r}})\frac{{dB(t)}}{{dt}} + i\frac{{\omega _\textrm{r}^2}}{{{Q_\textrm{r}}}}B(t)]\frac{{{e^{i{\omega _{\textrm{osc}}}t}}}}{2} + c.c ={-} \frac{{{\omega _\textrm{r}}}}{{2{Q_\textrm{r}}}}g\varsigma [{e^{i{\omega _{\textrm{osc}}}(t - T) + i{\varphi _T}}} + c.c], $$
where c.c is the complex conjugate of the proceeding term. Since B(t) varies much slowly than the carrier frequency ωosc, the following conditions will be fulfilled, which are [15]:
$$|\frac{{{d^2}B(t)}}{{d{t^2}}}|\ll \frac{{{\omega _\textrm{r}}}}{{{Q_\textrm{r}}}}|\frac{{dB(t)}}{{dt}}|,\textrm{ and }|\frac{{dB(t)}}{{dt}}|\ll {\omega _\textrm{r}}|B(t)|. $$

By using the approximations in Eq. (70), the relation of A(t) and B(t) shown in Eq. (68) can be approximated as:

$$A(t) \approx i{\omega _\textrm{r}}B(t). $$

By combining Eqs. (69) and (71), the evolution of A(t) can be derived as:

$$\frac{{dA(t)}}{{dt}} ={-} \mu {e^{i\vartheta }}A(t) - \mu {e^{i\vartheta }}g \cdot \varsigma \cdot {e^{ - i{\omega _{\textrm{osc}}}T}}A(t - T), $$
where the variables μ and ϑ are defined as:
$$\mu = {\omega _\textrm{r}}/\sqrt {4Q_\textrm{r}^2 + 1} ,\textrm{ and }\vartheta \textrm{ = arctan(}1/2{Q_\textrm{r}}\textrm{)}. $$

According to the Barkhausen criterion of the oscillator [18], the round-trip phase shift of the OEO loop should be a multiple of 2π. The total RF phase shift is induced by the four elements in the feedback loop shown in Fig. 2. We can set the phase condition of exp(-iωoscT)=-1. The final differential equation for the slowly varying complex envelope A(t) is derived as:

$$\frac{{dA(t)}}{{dt}} ={-} \mu {e^{i\vartheta }}A(t) + \mu {e^{i\vartheta }}g \cdot \varsigma A(t - T). $$

Note that, to satisfy the phase condition of Barkhausen criterion, it requires $g\zeta>0$.

Funding

National Natural Science Foundation of China (61805003, 61690194).

Acknowledgments

The authors would like to thank SYCATUS for offering the optical noise analyzer (A0040A), the RIN measurement system (A0010A) and Rohde & Schwarz for offering the phase noise analyzer (FSWP26).

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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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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Figures (10)
Fig. 1.
Fig. 1. Two system architectures of the resonant EO comb generation. (a) The resonant EO comb generation when driven by an external RF drive source. (b) The resonant EO comb generation when driven by a regenerative RF signal. CW: continuous-wave. EO: electro-optic. ESA: electrical spectrum analyzer.

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Fig. 2.
Fig. 2. Block diagram of the time domain model by using the framework of Ikeda-like model. The Vosc(t) denotes the voltage of the regenerative RF signal, which circulates in the clockwise direction and is subjected to the four main elements of the feedback loop, the nonlinear electrical-to-optical (E/O) and optical-to-electrical (O/E) unit, the fiber delay line, the RF amplifier, and the bandpass RF filter.

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Fig. 3.
Fig. 3. (a) Schematic diagram of the E/O and O/E unit formed by the resonant EO comb generator followed by a photodetector. A traveling-wave type phase modulator (PM) is placed in the FP cavity, only the light co-propagating with the RF signal is modulated. (b) Schematic diagram of the bandpass RF filter formed by a series RLC circuit.

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Fig. 4.
Fig. 4. (a), (b), and (c): The evolution of the amplitude of the regenerative RF signal with linear feedback gain g of 68, 116, and 350, respectively. (d) The numerical bifurcation diagram for the amplitude of the regenerative RF signal with respect to the linear gain g. The numbers (1)–(4) with different colors indicate distinct bifurcation behaviours. (e) The graphical representation of Eq. (35). The solid blue curve is u(t-T)ς, and the dashed lines are the u(t-T)/g with boundary values of the feedback gain g (38 for magenta and 81 for red) in the stationary state. (f) The nonlinear term u(t-T)ς versus the u(t-T) with different generalized laser-cavity detuning θ. The magenta dashed line is the u(t-T)/g when g is 68. The a.u. denotes arbitrary unit.

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Fig. 5.
Fig. 5. (a) Top: The evolution of the optical waveform versus nature time. Bottom: optical waveform in the stationary oscillation state. (b) Top: The evolution of the comb spectrum. Bottom: comb spectrum in the stationary oscillation state. The linear feedback gain g is 68. Tosc is the period of the regenerative RF signal. (c) Top: The evolution of the optical waveform when the OEO is operated in the period-doubling state. Bottom: the optical waveform of the two different states shown by the white and red dashed boxes in the top of Fig. 5(c). (d) Top: The evolution of the comb spectrum when the OEO is operated in the period-doubling state. Bottom: the time sliced comb spectrum in two different states shown by the white and red dashed boxes in the top of Fig. 5(d). The linear feedback gain g is 116.

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Fig. 6.
Fig. 6. Experimental setup. CW: continuous-wave. OFCG: optical frequency comb generator. PM: phase modulator. EDFA: erbium-doped fiber amplifier. PD1, PD2: photodiodes. LPA: low phase noise amplifier. ESA: electrical spectrum analyzer.

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Fig. 7.
Fig. 7. (a) Measured electrical-optical transmission response of the OFCG followed by a photodetector by using a vector network analyzer (VNA). Inset: The response of the 10 GHz electrical bandpass filter and the selected transmission resonance. (b) Electrical spectrum of the photodetected comb when the OFCG is driven by an external 10 GHz RF signal. Inset: the electrical spectrum of the photodetected comb by changing the DC bias of the OFCG.

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Fig. 8.
Fig. 8. (a) Electrical spectrum of the 10 GHz regenerative RF signal. (b) Optical spectrum of the resonant EO comb. The dashed magenta line shows the theoretical comb envelope fitting by using the model shown in Eq. (39).

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Fig. 9.
Fig. 9. SSB phase noise of the regenerative RF signal (Blue curve), the noise sources (Dashed curves) and the photodetected comb by using an external 10 GHz RF drive signal (Red curve).

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Fig. 10.
Fig. 10. (a) The frequency noise spectrum of the two seed lasers used in the experiments. (b) The blue and red curves are the measured SSB phase noise of the regenerative RF signals with the two seed lasers, respectively. The olive curve (1) and magenta curve (2) denote the calculated phase noise by using the model shown in Eq. (46) and with the laser frequency noise as the noise source. Note that the dual-loop transfer function of OEO shown in Eq. (45) is used.

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Tables (1)

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Table 1. Simulation parameters for the temporal dynamics of the regenerative RF signal

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Equations (74)

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H ^ BF { V osc ( t ) } = V in ( t ) ,
V in ( t ) = g ⋅ f NL [ V osc ( t − T ) ] ,
H ^ BF { V osc ( t ) } = g ⋅ f NL [ V osc ( t − T ) ] .
E in ( t ) = E in e i 2 π v 0 t ,
V osc ( t ) = V osc sin ⁡ ( 2 π f osc t ) ,
φ m ( t ) = π V osc ( t ) V π , and φ B = π V B V π ,
E out ( t ) = E in ( t ) ⋅ η F P ( t ′ t ′ ) ∑ k = 0 + ∞ ( r r ) k ( η ) 2 k + 1 exp ⁡ { i [ − ( 2 k + 1 ) k 0 n L + ( k + 1 ) ( φ m + φ B ) ] } ,
R = r 2 = 1 − ( t ′ ) 2 ,
f FSR = c 2 n L ,
E out ( t ) = E in ( t ) η FP η ( 1 − R ) exp ⁡ { i [ − π ν 0 f FSR + φ m + φ B ] } 1 − R η exp ⁡ { i [ − 2 π ν 0 f FSR + φ m + φ B ] } .
ν 0 = M ⋅ f FSR + δ v ,
− 2 π ν 0 f FSR = − 2 π δ ν f FSR − 2 M π .
P out ( t ) = E out ( t ) ⋅ E out ∗ ( t ) .
P out ( t ) = α P in 1 + F ⋅ sin 2 [ θ + π V osc ( t ) 2 V π ] ,
θ = − π δ ν f FSR + π V B 2 V π ,
F = 4 R η ( 1 − R η ) 2 ,
α = η FP η ( 1 − R ) 2 ( 1 − R η ) 2 ,
f NL { V osc ( t ) } = ℜ P out ( t ) Z 0 ,
f NL { V osc ( t − T ) } = α ℜ P in Z 0 1 + F ⋅ sin 2 [ θ + π V osc ( t − T ) 2 V π ] .
V osc ( s ) V in ( s ) = 1 Q r s / ω r + Q r ω r / s + 1 ,
ω r = 1 L C , and Q r = 1 Z L C ,
Q r ω r d V osc ( t ) d t + Q r ω r ∫ t 0 t V osc ( s ′ ) d s ′ + V osc ( t ) = V in ( t ) ,
H ^ BF { V osc ( t ) } = Q r ω r d V osc ( t ) d t + Q r ω r ∫ t 0 t V osc ( s ′ ) d s ′ + V osc ( t ) .
x ( t ) = π V osc ( t ) 2 V π .
Q r ω r d x ( t ) d t + Q r ω r ∫ t 0 t x ( s ′ ) d s ′ + x ( t ) = g γ 1 + F ⋅ sin 2 [ θ + x ( t − T ) ] ,
γ = π α ℜ P in Z 0 2 V π .
x ( t ) = 1 2 A ( t ) e i ω osc t + 1 2 A ∗ ( t ) e − i ω osc t = u ( t ) cos ⁡ [ ω osc t + φ ( t ) ] ,
g γ 1 + F ⋅ sin 2 [ θ + x ( t − T ) ] = g ρ 1 − δ cos ⁡ [ 2 θ + 2 x ( t − T ) ] ,
g ρ 1 − δ cos ⁡ [ 2 θ + 2 x ( t − T ) ] = g ρ ∑ k = 0 + ∞ { δ cos ⁡ [ 2 θ + 2 x ( t − T ) } k .
e i z cos ⁡ ( ξ ) = ∑ k = − ∞ + ∞ i k J k ( z ) e i k ξ ,
g ρ 1 − δ cos ⁡ [ 2 θ + 2 x ( t − T ) ] = − g ⋅ ς ⋅ x ( t − T ) ,
ς = ρ u ( t − T ) ∑ k = 1 + ∞ δ k 2 k − 2 ∑ l = 0 l < k 2 C k l J 1 [ 2 ( k − 2 l ) u ( t − T ) ] sin ⁡ [ 2 ( k − 2 l ) θ ] ,
Q r ω r d x ( t ) d t + Q r ω r ∫ t 0 t x ( s ′ ) d s ′ + x ( t ) = − g ⋅ ς ⋅ x ( t − T ) .
d A ( t ) d t = − μ e i ϑ A ( t ) + μ e i ϑ ⋅ g ⋅ ς ⋅ A ( t − T ) ,
u ( t − T ) ⋅ ς = u ( t − T ) g .
E out ( t ) = E in ( t ) β ∑ k = 1 + ∞ ( R η ) k − 1 exp ⁡ ( i k 2 θ ) exp ⁡ { i k 2 u ( t ) cos ⁡ [ ω osc t + φ ( t ) ] } ,
E out ( t ) = E in β ∑ q = − ∞ + ∞ i q { ∑ k = 1 + ∞ ( R η ) k − 1 exp ⁡ ( i 2 k θ ) J q [ 2 k u ( t ) ] } e i { 2 π v 0 t + q [ ω osc t + φ ( t ) ] } .
E q ( t ) = E in ⋅ β ⋅ i q ∑ k = 1 + ∞ ( R η ) k − 1 exp ⁡ ( i 2 k θ ) J q [ 2 k u ( t ) ] .
P q ( t ) = | E q ( t ) | 2 = P in | β | 2 ⋅ | ∑ k = 1 + ∞ ( R η ) k − 1 exp ⁡ ( i 2 k θ ) J q [ 2 k u ( t ) ] | 2 .
ξ a ( t ) = 1 2 ς a ( t ) e i ω osc t + 1 2 ς a ∗ ( t ) e − i ω osc t ,
d A ( t ) d t = − μ e i ϑ A ( t ) + μ e i ϑ g ⋅ ς ⋅ [ 1 + η m ( t ) ] ⋅ A ( t − T ) + μ e i ϑ ς a ( t ) .
d φ ( t ) d t = − μ [ φ ( t ) − φ ( t − T ) ] + μ 2 Q r η m ( t ) + μ | A 0 | ξ a , φ ( t ) ,
φ ( s ) = [ 1 2 Q r η m ( s ) + 1 | A 0 | ξ a , φ ( s ) ] ⋅ H ( s ) ,
H ( s ) = μ s + μ − μ e − s T .
H dual - loop ( s ) = μ s + μ − μ [ ε e − s T 1 + ( 1 − ε ) e − s T 2 ] ,
S φ ( f ) = [ 1 4 Q r 2 | η m ( f ) | 2 + 1 | A 0 | 2 | ξ a , φ ( f ) | 2 ] ⋅ | H ( f ) | 2 .
D a = π R out g 2 8 V π 2 [ F 0 k B T 0 + 2 e I ph Z 0 + N rin I ph 2 Z 0 ] ,
| η m ( f ) | 2 = | η m,g ( f ) | 2 + | η m , ς ( f ) | 2 ,
| η m,g ( f ) | 2 = 2 D m ( 1 + f H f + f L ) ,
Δ ς ( t ) = d ς d θ ⋅ Δ θ ( t ) ,
η m , ς ( t ) = Δ ς ( t ) ς .
Δ θ ( t ) = − π Δ v 0 ( t ) f FSR ,
η m , ς ( t ) = − π ς f FSR ⋅ d ς d θ ⋅ Δ v 0 ( t ) .
| η m , ς ( f ) | 2 = | π ς f FSR ⋅ d ς d θ | 2 ⋅ S Δ v ( f ) ,
τ j i t t e r = 1 ω osc ∫ f = f 1 f = f 2 S φ ( f ) d f ,
F C = 1 4 Q r 2 | π ς f FSR ⋅ d ς d θ | 2 .
A = g ρ ∑ k = 0 + ∞ ( δ 2 ) k { e i 2 [ θ + x ( t − T ) ] + e − i 2 [ θ + x ( t − T ) ] } k .
A = g ρ ∑ k = 0 + ∞ ( δ 2 ) k ∑ l = 0 k C k l e i 2 l ( θ + x T ) e − i 2 ( k − l ) ( θ + x T ) ,
A = g ρ ∑ k = 0 + ∞ ( δ 2 ) k ∑ l = 0 k C k l e i 2 ( 2 l − k ) θ ∑ q = − ∞ + ∞ i q J q [ 2 ( 2 l − k ) u T ] e i q [ ω osc ( t − T ) + φ T ] .
A = g ρ ∑ k = 1 + ∞ δ k 2 k − 1 ∑ l = 0 k C k l e i 2 ( 2 l − k ) θ i J 1 [ 2 ( 2 l − k ) u T ] cos ⁡ [ ω osc ( t − T ) + φ T ] .
A = B ⋅ cos ⁡ [ ω osc ( t − T ) + φ T ] ,
B = g ρ ∑ k = 1 + ∞ δ k 2 k − 1 ∑ l = 0 l < k / 2 C k l e i 2 ( 2 l − k ) θ i J 1 [ 2 ( 2 l − k ) u T ] + C k k − l e i 2 [ 2 ( k − l ) − k ] θ i J 1 { 2 [ 2 ( k − l ) − k ] u T } .
B = − g ρ ∑ k = 1 + ∞ δ k 2 k − 2 ∑ l = 0 l < k / 2 C k l J 1 [ 2 ( k − 2 l ) u T ] sin ⁡ [ 2 ( k − 2 l ) θ ] .
A = − g ρ u T ∑ k = 1 + ∞ δ k 2 k − 2 ∑ l = 0 l < k / 2 C k l J 1 [ 2 ( k − 2 l ) u T ] sin ⁡ [ 2 ( k − 2 l ) θ ] u T cos ⁡ [ ω osc ( t − T ) + φ T ] .
A = − g ⋅ ς ⋅ x ( t − T ) ,
ς = ρ u T ∑ k = 1 + ∞ δ k 2 k − 2 ∑ l = 0 l < k / 2 C k l J 1 [ 2 ( k − 2 l ) u T ] sin ⁡ [ 2 ( k − 2 l ) θ ] .
∫ t 0 t x ( s ′ ) d s ′ = 1 2 B ( t ) e i ω osc t + 1 2 B ∗ ( t ) e − i ω osc t ,
A ( t ) = d B ( t ) d t + i ω r B ( t ) .
[ d 2 B ( t ) d t 2 + ( ω r Q r + 2 i ω r ) d B ( t ) d t + i ω r 2 Q r B ( t ) ] e i ω osc t 2 + c . c = − ω r 2 Q r g ς [ e i ω osc ( t − T ) + i φ T + c . c ] ,
| d 2 B ( t ) d t 2 | ≪ ω r Q r | d B ( t ) d t | ,  and  | d B ( t ) d t | ≪ ω r | B ( t ) | .
A ( t ) ≈ i ω r B ( t ) .
d A ( t ) d t = − μ e i ϑ A ( t ) − μ e i ϑ g ⋅ ς ⋅ e − i ω osc T A ( t − T ) ,
μ = ω r / 4 Q r 2 + 1 ,  and  ϑ  = arctan( 1 / 2 Q r ) .
d A ( t ) d t = − μ e i ϑ A ( t ) + μ e i ϑ g ⋅ ς A ( t − T ) .
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