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Injection locking and pulling phenomena in an optoelectronic oscillator

Zhiqiang Fan, Jun Su, Yue Lin, Di Jiang, Yuan Chen, Xiang Li, and Qi Qiu

Open Access Open Access

Abstract

Injection locking and pulling characteristics of a long-loop optoelectronic oscillator (OEO) that has a large number of closely-spaced longitudinal modes are theoretically analyzed and experimentally evaluated. A differential phase equation that relates the phase difference between the OEO and the injected microwave signal to its instantaneous beat angular frequency is derived in the time domain. Based on the differential phase equation, both the locking and pulling characteristics of an injection-locked OEO are studied, and the phase noise performance is analyzed. It is found that the locking and pulling performance depends upon three parameters, the initial frequency difference between the frequency of the signal generated by the free-running OEO and frequency of the injected microwave signal, the voltage ratio between the signal generated by the free-running OEO and the injected microwave signal, and the Q factor of the free-running OEO. The phase noise performance depends upon the locking range, the phase noise performance of the free-running OEO as well as that of the injected microwave signal. The analysis is validated experimentally. Excellent agreement is found between the theoretical analysis and the experimental demonstration.

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

1. Introduction

An optoelectronic oscillator (OEO) with ultra-low phase noise, benefiting from its powerful energy storage capacity due to the use of a long optical fiber [1,2], is an excellent candidate for high frequency and low phase noise microwave generation which can find applications in high-performance communications systems, radar, and electronic measurement systems [3–5]. However, an OEO with a long fiber feedback loop would generate multiple closely-spaced longitudinal modes [6], making single-mode oscillation hard to achieve. A few solutions have been proposed to achieve a single-mode oscillation with an increased sidemode suppression ratio (SMSR). For example, to achieve single-mode oscillation, we can use an OEO with dual or multiple loops. Due to the Vernier effect [7], the effective free spectral range (FSR) is increased, making the single-mode selection more effective. Another solution, parity-time symmetry [8], has recently been proposed to achieve single-mode oscillation. Since no ultra-narrow band optical filter is needed, the implementation is greatly simplified. Single-mode oscillation can also be realized based on injection locking [9]. When a mode with its frequency close to that of the injected microwave signal, the particular mode is selected. The key features of using injection locking include excellent side mode suppression and flexible frequency tunability.

Injection locking was firstly studied in electronic oscillators by Van der Pol when he solved a second-order differential equation on nonlinear oscillations [10]. It is a well-known scheme to realize a stable electrical oscillator, and it has been further investigated in greater depth since then [11–15]. These theoretical studies have advanced the development of high-performance electrical oscillators. Recently, injection locking has been employed in OEOs to reduce the side modes caused by a long fiber feedback loop [9,16–18]. In an injection-locked OEO, a reference signal with a frequency close to that of one selected mode of a free-running OEO is injected into the free-running OEO, and the selected mode will be more competitive than the other modes due to the higher initial energy provided by the injected microwave signal. Thus, the mode is selected, and the other modes are suppressed. Consequently, a single-mode oscillation with a frequency identical to that of the injected signal is achieved. An injection-locked OEO inherits the low phase noise property of a free-running OEO, which was validated by experiments [9,16]. In addition, the tuning ability of an injection-locked OEO with a fine-tuning step and a wide frequency tunable range was also experimentally demonstrated [9,17].

To better understand the injection-locking mechanism, theoretical analysis has been recently performed and reported [19,20]. It was found that the injected microwave signal will influent the phase noise and the frequency pulling characteristic of an OEO. However, in the analysis reported in [19,20], the fact that an OEO usually has a long loop with many closely-spaced modes was not considered. In this article, we will perform an analysis of an injection-locked OEO that has long loop length with a large number of closely-spaced longitudinal modes. A differential phase equation is developed in the time domain, which is employed to analyze the injection locking and pulling characteristics. The phase noise performance is also studied. The analysis is then validated by an experiment in which an OEO with different loop lengths is considered. The experimental results confirm the validity of the theoretical analysis.

2. Theoretical analyses

Figure 1 shows the schematic diagram of an injection-locked OEO. As can be seen, a light wave generated by a laser diode (LD) is coupled into a Mach-Zehnder modulator (MZM) via a polarization controller (PC). By tuning the PC, the polarization state of the incident light wave to the MZM can be controlled to be aligned with the principal axis of the MZM to minimize the polarization-dependent loss. An intensity-modulated signal is generated at the output of the MZM and is transmitted through a long single-mode fiber (SMF) and a variable optical delay line (VODL), and detected at a photodetector (PD). A microwave signal is generated, and it is then amplified by an electrical amplifier (EA), filtered by an electronic band-pass filter (BPF), and fed back to the MZM to close the OEO loop. To achieve injection locking, an external microwave signal from a microwave source is combined with the generated microwave signal via an electrical coupler (EC) and injected into the OEO. An electrical divider (ED) is used to tap part of the generated microwave signal for performance evaluation, including spectrum measurement by an electrical spectrum analyzer (ESA) and phase noise measurement by a signal analyzer (SA).

 figure: Fig. 1.

Fig. 1. The schematic diagram of an injection-locked OEO. LD: laser diode; MZM: Mach-Zehnder modulator; VODL: variable optical delay line; SMF: single-mode fiber; PD: photoelectric detector; EA: electrical amplifier; BPF: band-pass filter; ED: electrical divider; EC: electrical coupler; ESA: electrical spectrum analyzer.

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It should be noted that a free-running OEO has a loop structure with a series of closely-spaced oscillation modes separated by its FSR, given by fFSR = 1/T, where T is the time delay of the feedback loop. For example, if the time delay of the feedback loop is 100 ns or 100 µs, or the loop length is 20 m or 20 km, the FSR would be 10 kHz or 10 MHz. A longer loop length corresponds to a greater number of closely-spaced longitudinal modes. If the frequency of the injected microwave signal is equal or close to that of one mode, that particular mode is selected, and all other modes are suppressed. Then, the signal generated by the OEO is locked to the injected microwave signal, and the OEO undergoes single-mode oscillation. In other words, all the energy at different modes is concentrated on the selected oscillation mode when the OEO is in the locked state. If the frequency of the injected microwave signal is not equal to but not far from the frequency of the injected signal, the selected mode is pulled to make its frequency equal to the injected microwave signal. Although the selected mode is pulled to the injected mode, the large number of closely-spaced modes will still exist. Mode competition also exists among these oscillation modes. Note that the locking range should be less than the FSR. Otherwise, more than one mode would be selected, and thus the single-mode oscillation will not be implemented.

We assume that a reference signal, the initial state of the selected mode in the free-running OEO, can be expressed as

$${V_0}(t )= {E_0}\cos ({{\omega_0}t} )$$
where E0 and ω0 are the voltage and the angular frequency of the reference signal, respectively.

To analyze the pulling and locking characteristics of an injected OEO, we inject a microwave signal with a frequency close to that of one selected oscillation mode in a free-running OEO. Figure 2(b) shows the vector diagram of an injected OEO at a given instant, in which Eosc, Einj, and Egen are the voltages of the selected mode in the free-running OEO, the injected microwave signal, and the generated signal, respectively, and ωosc, ωinj, and ωgen are the angular frequencies of the selected mode in the free-running OEO, the injected microwave signal, and the generated signal, respectively. The instantaneous beat angular frequency in the injected OEO is given by Δωosc = ωosc - ωinj. If the phase difference between the selected mode in the free-running OEO and the injected microwave signal is ψ, the instantaneous beat angular frequency can be rewritten as Δωosc = dψ/dt according to the diagram of phase versus frequency for an oscillator, as shown in Fig. 2(b). The undisturbed beat angular frequency is given by Δω0 = ω0 - ωinj.

 figure: Fig. 2.

Fig. 2. (a) The vector diagram of an injected OEO at a given instant. (b) The diagram to show the phase versus frequency relationship for an oscillator.

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According to Fig. 2(a), it is clear that the lengths of the two red dashed lines are equal (L1 = L2), and the lengths of the two red solid lines are equal (L3 = L4). Thus, we have

$$\tan (\varphi )= \frac{{{E_{inj}}\sin (\psi )}}{{{E_{osc}} + {E_{inj}}\cos (\psi )}}$$
where φ is the phase difference between the selected mode in the free-running OEO and the generated signal of the injected OEO. Considering the OEO is a sinusoidal oscillator like a single-tuned oscillator, textbooks give the left-hand side of Eq. (2) as [11,12],
$$\tan (\varphi )= 2Q\frac{{{\omega _0} - {\omega _{osc}}}}{{{\omega _0}}}$$
where Q is the quality factor of the delay line loop in the OEO, which is given by Q = ω0T, where T is the time delay of the optoelectronic feedback loop.

In consequence, Eqs. (2) and (3) can be rewritten as an equation

$$\frac{{ - {E_{inj}}\sin (\psi )}}{{{E_{osc}} + {E_{inj}}\cos (\psi )}}\textrm{ = }2Q\frac{{{\omega _{osc}} - {\omega _0}}}{{{\omega _0}}}$$
where (ωosc - ω0) is also given by
$${\omega _{osc}} - {\omega _0}\textrm{ = }({{\omega_{osc}} - {\omega_{inj}}} )- ({{\omega_0} - {\omega_{inj}}} )= \Delta {\omega _{osc}} - \Delta {\omega _0} = {{d\psi } / {dt}} - \Delta {\omega _0}$$

Submitting (5) into (4), the differential phase equation in the time domain is given by [15],

$$\frac{{d\psi }}{{dt}}\textrm{ = }\Delta {\omega _0} - \frac{{{\omega _0}}}{{2Q}} \cdot \frac{{{E_{inj}}\sin (\psi )}}{{{E_{osc}} + {E_{inj}}\cos (\psi )}}$$

In the steady state, the left-hand side of (6) equals zero, and thus we have

$$\Delta {\omega _0}\textrm{ = }\frac{{{\omega _0}}}{{2Q}} \cdot \frac{{{E_{inj}}\sin (\psi )}}{{{E_{osc}} + {E_{inj}}\cos (\psi )}}$$

The locking range of the injection-locked OEO can be found by solving (7) for a maximum with a condition of cos(ψ) = -(Einj/Eosc). It is given by

$$\Delta {\omega _{0,\max }}\textrm{ = }\frac{{{\omega _0}}}{{2Q}} \cdot \frac{1}{{\sqrt {{{({{{{E_{osc}}} / {{E_{inj}}}}} )}^2}\textrm{ - }1} }}$$

By substituting Q = ω0T and fFSR = 1/T into (8), the locking range is rewritten as

$$\Delta {\omega _{0,\max }}\textrm{ = }\frac{{{f_{FSR}}}}{2} \cdot \frac{1}{{\sqrt {{{({{{{E_{osc}}} / {{E_{inj}}}}} )}^2}\textrm{ - }1} }}$$

Note that in the analysis here we denote the locking range by using the maximum beat frequency Δω0,max, and the overall locking range is, in fact, ±Δω0,max around ω0. In an injection-locked OEO, the locking range should be less than the FSR for the stable single-mode operation. This means Δω0,max < fFSR /2, and thus we have

$${E_{osc}} > \sqrt 2 {E_{inj}}$$

It means that the relationship between the voltages of the selected mode in a free-running OEO and the injected microwave signal should meet the condition given by (10) to realize a stable single-mode oscillation in an injection-locked OEO.

Then, we can assume Einj ≪ Eosc, the differential phase equation in the time domain given by (6) is rewritten as

$$\frac{{d\psi }}{{dt}}\textrm{ = }\Delta {\omega _0} - \frac{{{\omega _0}}}{{2Q}} \cdot \frac{{{E_{inj}}}}{{{E_{osc}}}}\sin (\psi )$$

In the steady state, the left-hand side of (11) equals zero, and thus we have

$$\Delta {\omega _0}\textrm{ = }\frac{{{\omega _0}}}{{2Q}} \cdot \frac{{{E_{inj}}}}{{{E_{osc}}}}\sin (\psi )$$

The locking range of the injection-locked OEO is found by solving (12) for a maximum, which occurs when sin(ψ) = 1, and thus the locking range is given by

$$\Delta {\omega _{0,\max }}\textrm{ = }\frac{{{\omega _0}}}{{2Q}} \cdot \frac{{{E_{inj}}}}{{{E_{osc}}}}$$

Again, by substituting Q = ω0T and fFSR = 1/T into (13), we have

$$\Delta {\omega _{0,\max }}\textrm{ = }\frac{{{f_{FSR}}}}{2} \cdot \frac{{{E_{inj}}}}{{{E_{osc}}}}$$

It means the locking range of an injected OEO depends on the voltage ratio between the injected microwave signal and the selected mode in a free-running OEO, and the FSR of the free-running OEO, which is determined by the time delay of the feedback loop.

By integrating (11), the phase difference ψ as a function of time is given by

$$\tan \frac{\psi }{2} = \frac{1}{K} + \frac{{\sqrt {{K^2} - 1} }}{K}\tan \left[ {\frac{{\Delta {\omega_0}({t - {t_0}} )}}{2} \cdot \frac{{\sqrt {{K^2} - 1} }}{K}} \right]$$
where t0 is an integration constant, and K is given by
$$K = 2\frac{{\Delta {\omega _0}}}{{{f_{FSR}}}}\frac{{{E_{osc}}}}{{{E_{inj}}}}$$

By comparing (14) and (16), an OEO is locked if |K| ≤ 1. On the other hand, the OEO is unlocked if |K| > 1.

According to (15), the phase difference ψ will grow at a periodically varying rate with the time increases uniformly if the OEO is in the unlocked state. Figure 3 shows the phase difference ψ as a function of time when the OEO is in the unlocked state. As shown in Fig. 3, the phase difference ψ varies periodically with a variation range from -π to π, and a variation period of (2πK)/[Δω0(K2−1)1/2. Thus, the instantaneous beat angular frequency of an injected OEO in the unlocked state is given by

$$\Delta {\omega _{osc}} = \Delta {\omega _0}\frac{{\sqrt {{K^2} - 1} }}{K}$$

By substituting (14) and (16), (17) can be rewritten as

$$\Delta {\omega _{osc}} = \Delta {\omega _{0,\max }}\sqrt {{K^2} - 1}$$
or
$$\Delta {\omega _{osc}} = \sqrt {\Delta {\omega _0}^2\textrm{ - }\Delta {\omega _{0,\max }}^2}$$

According to (16) to (19), the instantaneous beat angular frequency of an injected OEO is plotted as a function of the undisturbed beat angular frequency in Fig. 4(a) and (b). The instantaneous beat angular frequency of an injected OEO equals zero if the OEO is under the injection-locked condition, where |K| ≤ 1. It means the frequency of the injection-locked OEO equals the frequency of the injected microwave signal. On the other hand, if the OEO is unlocked, where |K| > 1, we have |Δωosc| < |Δω0|. It means a pulling phenomenon occurs in the injected OEO, here we, therefore, say the unlocked OEO is under the quasi-locked condition. Under the quasi-locked condition, the instantaneous beat angular frequency Δωosc increases from zero to close to the undisturbed beat angular frequency Δω0, when Δω0 increases from Δω0,max to πfFSR, or in other words, when K increases from 1 to πfFSR /Δω0,max. The phase difference ψ is also re-plotted as a function of time with different values of K in Fig. 4(c). When the undisturbed beat angular frequency Δω0 is close to the locking range Δω0,max, which means K is close to 1, the phase difference ψ has a jump for a short duration and a slow rise for a long duration. It means a strong pulling phenomenon occurs, which is consistent with the analysis of Fig. 4(a) and (b). If Δω0 is much larger than Δω0,max, or K is much larger than 1, the phase difference increases approximately linearly with time, and a weak pulling phenomenon occurs, which is also shown in Fig. 4(a) and (b).

 figure: Fig. 3.

Fig. 3. Phase difference ψ as a function of time when the OEO is in the unlocked state.

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 figure: Fig. 4.

Fig. 4. The relationship between the instantaneous beat angular frequency and the undisturbed beat angular frequency with (a) different voltage ratios of the injected microwave signal to the selected mode in the free-running OEO, and (b) different FSRs for the free-running OEO, respectively. (c) Phase difference ψ as a function of time under the quasi-locked condition with different K.

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Under the injection-locked condition, the selected mode of the free-running OEO is locked to the injected microwave signal, and all the other modes are suppressed because of the enhanced mode competitiveness for the selected mode. Under the quasi-locked condition, the selected mode of the free-running OEO is pulled to the injected microwave signal, and according to the output spectrum of an unlocked driven oscillator given by [14], the voltage of the selected mode in the quasi-locked OEO is given by

$${V_{ql - osc}} = {E_{osc}}{e^{j{\omega _{inj}}t}}{e^{j\psi (t )}}$$
which is deduced from (15), where t0 is assumed to zero and Vql-osc is the voltage of the selected mode, and the term of ejψ(t) is given by
$${e^{j\psi (t )}}\textrm{ = }j\tan \left( {\frac{\theta }{2}} \right) + \frac{{1 - {{\tan }^2}\left( {\frac{\theta }{2}} \right)}}{{j\tan \left( {\frac{\theta }{2}} \right)}}\sum\limits_{n = 1}^\infty {{{\left[ {j\tan \left( {\frac{\theta }{2}} \right)} \right]}^n}} {e^{jn({\Delta {\omega_{osc}}t + \theta } )}}$$
where θ is given by
$$\sin (\theta )\textrm{ = }\frac{1}{K},\begin{array}{{cc}} {}&{\left( { - \frac{\pi }{2} < \theta < \frac{\pi }{2}} \right)} \end{array}$$
or
$$\cos (\theta )= \frac{{\Delta {\omega _{osc}}}}{{\Delta {\omega _0}}}$$

Tables Icon

Table 1. Output Spectrum of a Quasi-locked OEO

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Considering a series of closely-spaced modes in an OEO due to a long feedback loop, the output spectrum of a quasi-locked OEO is given by

$${V_{ql - OEO}} = \sum\limits_{m = \lfloor{ - {{{\omega_{osc}}} / {{f_{FSR}}}}} \rfloor }^\infty {[{{E_{osc,m}}{e^{j({{\omega_{inj}} + m{f_{FSR}}} )t}}} ]} \;{e^{j\psi (t )}}$$
where m is an integer, which denotes m-th oscillation mode, Eosc,m is the voltage amplitude of the m-th mode. Note that m = 0 denotes the selected mode to be injection locked, and ⌊ ⌋ indicates rounding down to the nearest integer. Besides, (21) should be rewritten as
$${e^{j\psi (t )}}\textrm{ = }jk\tan \left( {\frac{\theta }{2}} \right) + \frac{{1 - {{\tan }^2}\left( {\frac{\theta }{2}} \right)}}{{j\tan \left( {\frac{\theta }{2}} \right)}}\sum\limits_{n = 1}^\infty {{{\left[ {j\tan \left( {\frac{\theta }{2}} \right)} \right]}^n}} {e^{jn({\Delta {\omega_{osc}}t + \theta } )}}$$
where k = 1 if m = 0, and k = 0 if m ≠ 0.

Figure 5(a) shows the output spectrum of a free-running OEO, where a series closely spaced oscillation modes exist. By contrast, according to (24) and (25), the output spectrum of a quasi-locked OEO is presented in Table 1 and Fig. 5(b). It is clear that there are numerous oscillation modes except the selected mode to be injected, and the FSR of the quasi-locked OEO is consistent with that of the free-running OEO. The output spectrum of the OEO under the injection-locked condition is also plotted in Fig. 5(c), where all the oscillation modes are suppressed except the selected mode.

 figure: Fig. 5.

Fig. 5. The output spectra of (a) a free-running OEO, (b) a quasi-locked OEO, and (c) an injection-locked OEO.

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Besides the output spectrum, phase noise is also a critical parameter to evaluate the performance of a generated microwave signal. According to [16–18], the phase noise of the injection-locked OEO depends on the phase noises of the injected microwave signal and the free-running OEO, which are not correlated [21,22]. Thus, we first analyze the phase noise transfer between the injected microwave signal and the generated microwave signal of an injection-locked OEO.

According to (1) and Fig. 2(a), we assume the initial state of the injected microwave signal is

$${V_{inj}}(t )= {E_{inj}}\cos ({{\omega_{inj}}t + \psi } )$$

After a short while, the instantaneous oscillations of the free-running OEO and the injected microwave signal can be expressed, respectively, by

$${V_{osc}}(t )= {E_{osc}}\cos ({{\omega_{osc}}t + {{\psi^{\prime}}_{osc}}} )= {E_{osc}}\cos [{{\omega_0}t + {\psi_{osc}}(t )} ]$$
$${V^{\prime}_{inj}}(t )= {E_{inj}}\cos ({{\omega_{inj}}t + {{\psi^{\prime}}_{inj}}} )= {E_{inj}}\cos [{{\omega_0}t + {\psi_{inj}}(t )} ]$$
where ψosc(t) and ψinj(t) are given, respectively, by
$${\psi _{osc}}(t )\textrm{ = }({{\omega_{osc}} - {\omega_0}} )t + {\psi ^{\prime}_{osc}}$$
$${\psi _{inj}}(t )= ({{\omega_{inj}} - {\omega_0}} )t + {\psi ^{\prime}_{inj}}$$

According to Fig. 2(a), by substituting (13), (29), and (30), Eq. (11) is rewritten as

$$\frac{{d[{{\psi_{inj}}(t )- {\psi_{osc}}(t )} ]}}{{dt}}\textrm{ = }\Delta {\omega _0} - \Delta {\omega _{0,\max }}\sin [{{\psi_{inj}}(t )- {\psi_{osc}}(t )} ]$$
where dψinj(t)/dt = Δω0, and the term of ψinj(t) - ψosc(t) has a small value in the locked condition. Thus, (31) is simplified to be
$$\frac{{d{\psi _{osc}}(t )}}{{dt}}\textrm{ = }\Delta {\omega _{0,\max }}[{{\psi_{inj}}(t )- {\psi_{osc}}(t )} ]$$
Taking the Laplace transforms to both sides in (32), we have
$$s{\psi _{osc}}(s )\textrm{ = }\Delta {\omega _{0,\max }}[{{\psi_{inj}}(s )- {\psi_{osc}}(s )} ]$$

Then, the phase transfer function between the injected microwave signal and the injection-locked OEO is given by

$${H_{inj}}(s )\textrm{ = }\frac{{{\psi _{osc}}(s )}}{{{\psi _{inj}}(s )}}\textrm{ = }\frac{1}{{1 + {s / {\Delta {\omega _{0,\max }}}}}}$$

Substituting s = j2πf and taking the square modulus, the phase noise transfer model between the injected microwave signal and the injection-locked OEO in the form of phase noise power spectrum is given by

$${|{{H_{inj}}({jf} )} |^2}\textrm{ = }\frac{{{S_{osc}}(f )}}{{{S_{inj}}(f )}}\textrm{ = }\frac{1}{{1 + {{({{f / {\Delta {f_{0,\max }}}}} )}^2}}}$$
where f is the frequency offset, Δf0,max = Δω0,max/(2π), Sosc(f) and Sinj(f) are the phase noise power spectra of the injection-locked OEO and the injected microwave signal, respectively. Figure 6 shows the phase noise transfer model between an injected microwave signal and an injection-locked OEO.

 figure: Fig. 6.

Fig. 6. The phase noise transfer model between an injected microwave signal and an injection-locked OEO.

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Keep in mind that the phase noise of the injection-locked OEO depends on the phase noises of the injected microwave signal and the free-running OEO, which are not correlated, the phase noise transfer model of an injection-locked OEO is shown in Fig. 7. As can be seen, the phase noise transfer model between the free-running OEO and the injection-locked OEO in the form of phase noise power spectrum is given by

$${|{{H_{free}}({jf} )} |^2}\textrm{ = }\frac{{{S_{osc}}(f )}}{{{S_{free}}(f )}}\textrm{ = }\frac{1}{{1 + {{({{{\Delta {f_{0,\max }}} / f}} )}^2}}}$$
where Sfree(f) is the phase noise power spectrum of the free-running OEO. Then, the phase noise transfer model of the injection-locked OEO in the form of the phase noise power spectrum is written as
$${S_{osc}}(f )\textrm{ = }{|{{H_{inj}}({jf} )} |^2}{S_{inj}}(f )\textrm{ + }{|{{H_{free}}({jf} )} |^2}{S_{free}}(f )$$

 figure: Fig. 7.

Fig. 7. The phase noise transfer model of an injection-locked OEO.

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It is clear that, according to (37), the phase noise of an injection-locked OEO depends on the phase noises of the injected microwave signal and the free-running OEO. There is a demarcation point at the offset frequency of Δf0,max, which depends on the voltage ratio between the injected microwave signal and the selected mode in the free-running OEO, as well as the FSR of the free-running OEO. If f≪Δf0,max, the phase noise of the injection-locked OEO is primarily determined by the phase noise of the injected microwave signal. If f≫Δf0,max, the phase noise of the injection-locked OEO is primarily determined by the phase noise of the free-running OEO. In other words, the near-carrier and far-carrier phase noises of the injection-locked OEO primarily depend on the phase noises of the injected microwave signal and the free-running OEO, respectively.

3. Experiment

An experiment is performed based on the setup in Fig. 1 to observe the locking and pulling phenomena in an injected OEO. A light wave with an optical power of 10 dBm at 1550 nm from the LD (Yenista optics TLS-AG) is sent to the MZM (OCLARO F-10) with a 3-dB bandwidth of 11 GHz via a PC. The modulated light wave is then passed through the SMF with a length of 2 km, 4 km, or 10 km, the VODL with a tuning range of 600 ps as well as a tuning resolution of 39.6 fs, and detected at the PD (Kang Guan KG-PD-10G) with a 3-dB bandwidth of 11 GHz. A microwave signal is generated, and it is amplified by the EA with a 3-dB bandwidth of 40 GHz, filtered by the BPF with a 3-dB bandwidth of 13 MHz at the center frequency of 10.664 GHz, and feedback to the MZM. A microwave signal with a frequency of 10.664 GHz generated by a microwave source (Anritsu MG3693C) is injected into the free-running OEO. The spectrum of the OEO is measured by the ESA (Anritsu MS2725C), and a 20-dB attenuator is connected to the input port of the ESA to protect the ESA.

Figure 8(a1) shows the measured electrical spectrum of the free-running OEO with a loop time delay of around 50 µs. A series of closely spaced modes are observed due to the large loop time delay. Injecting a microwave signal with a power of −20 dBm into the free-running OEO and tuning the injected microwave signal close to the primary oscillation mode of the free-running OEO, the primary mode is selected to oscillate, and all the other modes are suppressed when the free-running OEO is locked, the electrical spectrum is measured and shown in Fig. 8(a2). Then, tuning the injected microwave signal approach to the other modes, the electrical spectra are measured in Fig. 8(a3). As can be seen in Figs. 8(a2) and (a3), single-mode oscillations with SMSRs higher than 75 dB at different modes are obtained when they are selected to be locked, and the output powers are measured to be identical.

 figure: Fig. 8.

Fig. 8. The measured output spectra of different OEOs with the same loop length of 10 km. (a1) is the output spectrum of the free-running OEO, Center: 10.664 GHz, RBW: 30 Hz, Span: 100 kHz, (a2) and (a3) are the output spectra of injection-locked OEO, Center: 10.664 GHz, RBW: 30 Hz, Span: 100 kHz, (b1) to (b3) are the output spectra of unlocked OEO, (b1) and (b2) Center: 10.664 GHz, RBW: 30 Hz, Span: 10 kHz, (b3) Center: 10.664 GHz, RBW: 30 Hz, Span: 4 kHz.

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The output spectra in the quasi-locked conditions are also plotted in Fig. 8(b1) to (b3), and asymmetric sidebands are observed. As can be seen in Fig. 8(b1) and (b2), all of the sidebands except the injected microwave signal are on the opposite side of the oscillation mode of the free-running OEO from the injected microwave signal. When tuning the oscillation mode of the free-running OEO closer to the injected microwave signal, the output spectrum of the quasi-locked OEO is measured in Fig. 8(b3), and the sidebands are observed on both sides of the injected microwave signal. However, it should be noted that the sidebands are still asymmetric, and the left sideband, which is on the opposite side of the oscillation mode of the free-running OEO from the injected microwave signal, is larger than the right one. The small right sideband is due to the amplitude variation neglected in Eq. (11) [23]. In a word, the sidebands of quasi-locked OEOs are asymmetric. That is because of the different cycle numbers between the quasi-locked OEO and the injected microwave signal. To be precise, the quasi-locked OEO has more cycles than the injected microwave signal if the frequency of the oscillation mode of the free-running OEO is higher than the frequency of the injected microwave signal. In contrast, the quasi-locked OEO has fewer cycles than the injected microwave signal if the frequency of the oscillation mode of the free-running OEO is lower than the frequency of the injected microwave signal. Thus, the sidebands opposite the injected microwave signal side from the oscillation mode of the free-running OEO dominates at the output of the quasi-locked OEO.

The output spectra of the injection-locked OEO are shown in Fig. 9 when the injected microwave signal is tuned within the locking range. The frequency of the injection-locked OEO equals the frequency of the injected microwave signal, which means the instantaneous beat frequency is zero. It agrees well with the theoretical analysis given by (17) to (19), and Fig. 4. In addition, the output power of the injection-locked OEO is almost constant, and it is independent of the frequency and power of the injected microwave signal.

 figure: Fig. 9.

Fig. 9. The measured output spectra of injection-locked OEOs when the free-running OEO has a same loop length of around 10 km, a same oscillation frequency of 10.664 GHz, and the injected microwave signal has different frequencies. Center 10.664 GHz, RBW: 10 Hz, Span: 1 kHz.

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Keep in mind that the OEO has a series of closely spaced modes due to the long feedback-loop length. The output spectra of the OEO injected with a microwave signal having different powers and the same frequency of 10.664 GHz are measured and are shown in Fig. 10. Setting the injected power to be −5 dBm, the output spectra of the OEO injected by a microwave signal are measured and are shown in Fig. 10(a1 to 5) when one selected oscillation mode of the free-running OEO is tuned from finj + fFSR/2 to finj – fFSR/2, where finj is the frequency of the injected microwave signal. The output spectra of the OEO with different levels of injected powers of −10, −15, and −20 dBm are shown in Fig. 10(b1 to 5), 10(c1 to 5), and 10(d1 to 5), respectively.

 figure: Fig. 10.

Fig. 10. The measured output spectra of OEOs with different oscillation frequencies, a same loop length of around 10 km, a same injected frequency of 10.664 GHz, and different injected powers. (a1) – (a4) The power of the injected microwave signal is −5dBm. (b1) – (b4) The power of the injected microwave signal is−10dBm. (c1) – (c4) The power of the injected microwave signal is −15dBm. (d1) – (d4) The power of the injected microwave signal is−20dBm. All the figures have the same setup of Center: 10.664 GHz, RBW: 30 Hz, Span: 40 kHz.

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As can be seen in Fig. 10(a1) and (a5), the injected microwave signal is located in the middle of two oscillation modes of the OEO, and the same power at the frequency of the injected microwave signal is obtained. The same phenomenon is observed when the injected microwave signal has different powers, as shown in Fig. 10(b1) and (b5), Figs. 10(c1) and (c5), as well as Fig. 10(d1) and (d5). Comparing Fig. 10(a to d1) or Fig. 10(a to d5), the power at the frequency of the injected microwave signal linearly depends on the power of the injected microwave signal when the injected microwave signal is located in the middle of two oscillation modes of the OEO. Once the OEO is injection-locked, as shown in Fig. 10(a to d3), the power at the frequency of the injected microwave signal is fixed and equal. In other words, the output power of an injection-locked OEO is almost independent of the injected power when the injected energy is far less than the energy of the free-running OEO. That is because the energy of all modes in a free-running OEO is transmitted to the injected mode if the free-running OEO is injection locked, and the injected energy is far less than the energy of the free-running OEO. As can be seen in Fig. 10(a to d3), SMSRs higher than 75 dB are obtained when the injected microwave signal has different powers.

Figure 10(a to d2) and 10(a to d4) shows the output spectra in the quasi-locked condition. Asymmetric sidebands are observed at all the modes of the quasi-locked OEO, which agree well with the analysis given by (25), Table  1, and Fig. 5(b). The instantaneous beat frequency depends on the frequency difference between the injected microwave signal and the nearest oscillation mode in a free-running OEO to the injected microwave signal. It should be noted that the FSR of a quasi-locked OEO equals the FSR of a free-running OEO. The reason is that the FSR of an OEO depends on the loop length, and the injected microwave signal cannot affect the loop length of an OEO.

In an OEO injected by a microwave signal, the power at the frequency of the injected microwave signal is measured and shown in Fig. 11(a) and (b) when the injected microwave signal has different levels of power, and the OEO has different loop lengths, respectively. As can be seen in Fig. 11, the power at the frequency of the injected microwave signal is increased when the instantaneous beat frequency goes to zero, and the power is the highest when the OEO is injection-locked. The output power of an injection-locked OEO is independent of the power of the injected microwave signal and the loop length of the OEO. The power at the frequency of the injected microwave signal goes to the lowest when the instantaneous beat frequency is fFSR/2 or –fFSR/2, and the lowest power depends on the power of the injected microwave signal.

 figure: Fig. 11.

Fig. 11. The relationship between the instantaneous beat frequency and the power at the frequency of the injected microwave signal. (a) The loop length of the OEO is around 4 km and the powers of injected microwave signals are −10 dBm, −15 dBm and −20 dBm, respectively, Center: 10.664 GHz, span 50 kHz. (b) The power of injected microwave signal is −5 dBm and the loop lengths of OEOs are 2 km, 4 km and 10 km, respectively, Center: 10.664 GHz, span 100 kHz.

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Figure 12(a) and (b) show the relationship between the undisturbed beat frequency and the instantaneous beat frequency when the OEO has different loop lengths, and the injected microwave signal has different powers, respectively. When the injected microwave signal has a power of −5 dBm, as can be seen in Fig. 12(a), plots A and C show the relationship between the undisturbed beat frequency and the instantaneous beat frequency. A stronger pulling phenomenon is observed when the undisturbed beat frequency is close to the injection locking range than it is away from the locking range. The same phenomenon is also obtained in plots B and D when the loop length of the OEO is about 10 km. It is also observed that a longer loop length causes a smaller locking range, which agrees well with the simulation results in Fig. 4(b). In Fig. 12(b), the loop length of the OEO is chosen to be around 10 km, the injected microwave signal has different powers of −5 and −20 dBm, and then the relationships between the undisturbed and instantaneous beat frequencies are plotted as plots A or C, and plots B or D, respectively. It is found that a higher power of the injected microwave signal results in a larger injection locking range, which agrees well with the simulation results in Fig. 4(a).

 figure: Fig. 12.

Fig. 12. The relationship between the undisturbed beat frequency and the instantaneous beat frequency. (a) The OEOs have different loop lengths and a same injection power of −5 dBm, Plot A and plot C: the loop length is 2 km, Plot B and Plot D: the loop length is 10 km. (b) The OEOs have different injection powers and a same loop length of 10 km, Plot A and Plot C: the injection power is −5 dBm, Plot B and Plot D: the injection power is −20 dBm, (c) The relationship between the power at injection frequency and the undisturbed beat frequency, Plot A and Plot B have a same loop length of 2 km and a same power of −5 dBm.

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When the loop length of the OEO is about 2 km, and the power of the injected microwave signal is −5 dBm, the relationship between the undisturbed beat frequency and the instantaneous beat frequency is plotted as plot A in Fig. 12(a), the relationship between the undisturbed beat frequency and the power at the frequency of the injected microwave signal in the OEO is plotted as plot B in Fig. 12(c). It is observed that the power of the injection-locked OEO is a constant, and the frequency of the injection-locked OEO equals the frequency of the injected microwave signal. In the quasi-locked condition, the power at the frequency of the injected microwave signal is rapidly decreased, and the pulling phenomenon is also rapidly weakening as the instantaneous beat frequency is increased. The reason is that the effect of the injected microwave signal on the phase of the free-running OEO is reduced if the instantaneous beat frequency is increased. The reason is also described in Fig. 4(c).

Figure. 13(a) and (b) shows the effects of the loop length of the OEO and the power of the injected microwave signal on the locking range of an injection-locked OEO. It is obtained that the locking range of an injection-locked OEO linearly dependent on 1/L or (Pinj/Posc)−1/2, where L is the loop length of the OEO, Pinj and Posc are the power of the injected microwave signal and the power of the oscillation mode of the free-running OEO. In other words, the locking range of an injection-locked OEO depends linearly on the voltage ratio between the injected microwave signal and the selected mode in the free-running OEO, or the FSR of the free-running OEO. The measured results agree well with (14).

 figure: Fig. 13.

Fig. 13. The relationship between (a) the loop length of an OEO and locking range at different injection powers, in which L is the loop length, (b) the injection power and locking range at different loop lengths, in which PINJ/POSC is the ratio of the power of the injection signal to the power of free-running signal.

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The phase noise performance of an injection-locked OEO is also evaluated and shown in Fig. 14, where the free-running OEO is injected by a microwave signal generated by a microwave source (Agilent E8254A), and the phase noise is measured by a microwave signal analyzer (Agilent E5052B). The phase noises of the injection-locked OEOs with different loop lengths of around 70 m, 400 m, 4 km, and 10 km are measured and shown in Fig. 14(a), (b), (c), and (d), respectively. As can be seen in Fig. 14(a), the low-offset phase noise of the free-running OEO with a loop length of around 70 m is larger than that of E8254A. Once the OEO is injection-locked by E8254A, the phase noise the OEO at a lower frequency offset is reduced visibly, and the phase noise at a higher frequency offset stays the same with that of the free-running OEO. The same phenomenon is also observed in Fig. 14(b) and (c), and this phenomenon can be described by (37), the phase noise of the injection-locked OEO depends on the phase noises of the injected microwave signal and the free-running OEO, and there is a demarcation point at the offset frequency of Δf0,max. It should also be noted that the phase noise of an injection-locked OEO is almost following that of the free-running OEO in Fig. 14(d). The reason is that the location of the demarcation point depends on the locking range, which depends on the loop length of the OEO. A longer loop length of around 10 km in Fig. 14(d) induces a smaller locking range, which causes the demarcation point at a lower frequency offset, and thus the phase noise of the injection-locked OEO is almost the same as that of the free-running OEO.

 figure: Fig. 14.

Fig. 14. The measured phase noise of an OEO injected by a microwave signal with different powers when the OEO has different loop lengths of around (a) 70 m, (b) 400 m, (c) 4 km, and (d) 10 km, respectively.

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Besides, a longer loop length leads to a lower phase noise of an injection-locked OEO. As can be seen in Fig. 14, the phase noises of the injection-locked OEOs with different loop lengths of around 70 m, 400 m, 4 km, and 10 km are measured to be −109.6, −110.4, −126.7, and −130.1 dBc/Hz, respectively, at an offset frequency of 10 kHz, when the injected microwave signal has a power of −10 dBm and a phase noise of −115.1 dBc/Hz at 10-kHz frequency offset.

However, a long loop length in a free-running OEO causes a series of high peaks, as shown in Fig. 14. It is observed that a higher injected power in an injection-locked OEO induces lower peaks. For example, in Fig. 14(c), the peaks are reduced from −55.36 to −94.59, −99.79, −105.6, and −108.6 dBc/Hz when the OEO is not injected and the injected powers are −10, −5, 0, and 5 dBm, respectively. That is because a higher injected power would increase the mode competitiveness of an injected mode of the OEO, and thus the peaks are suppressed more strongly. It should be kept in mind that the injected power will affect the location of the demarcation point in an injection-locked OEO, according to (35) to (37), and thus the injected power should be carefully designed by considering the loop length of the OEO and the phase noise performance of the injected microwave signal to achieve the best phase noise performance for an injection-locked OEO.

4. Conclusion

In summary, the injection locking and pulling phenomena in an OEO with a large number of closely-spaced longitudinal modes resulting from the long loop, as well as the phase noise performance of the injection-locked OEO, were theoretically analyzed and experimentally evaluated. Excellent agreement was found between the theoretical analysis and the experimental demonstration. It was found that the locking and pulling characteristics of an injection-locked OEO depend upon three parameters, the initial frequency difference between the frequency of the signal generated by the free-running OEO and frequency of the injected microwave signal, the voltage ratio between the signal generated by the free-running OEO and the injected signal, as well as the Q factor of the free-running OEO. The phase noise performance of an injection-locked OEO depends upon two parameters, the phase noise performances of the free-running OEO and the injected signal, as well as the locking range of the injection-locked OEO. The analysis and the conclusion provide a guideline in designing an injection-locked OEO with the best phase noise performance.

Funding

Fundamental Research Funds for the Central Universities (ZYGX2015J050); National Natural Science Foundation of China (61971110).

Acknowledgements

We would like to thank Prof. Jianping Yao for his constructive suggestions and support on the phase noise measurement of the OEO. The work of Zhiqiang Fan was supported by the China Scholarship Council under Grant 201806070053.

Disclosures

The authors declare no conflicts of interest.

References

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Figures (14)
Fig. 1.
Fig. 1. The schematic diagram of an injection-locked OEO. LD: laser diode; MZM: Mach-Zehnder modulator; VODL: variable optical delay line; SMF: single-mode fiber; PD: photoelectric detector; EA: electrical amplifier; BPF: band-pass filter; ED: electrical divider; EC: electrical coupler; ESA: electrical spectrum analyzer.

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Fig. 2.
Fig. 2. (a) The vector diagram of an injected OEO at a given instant. (b) The diagram to show the phase versus frequency relationship for an oscillator.

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Fig. 3.
Fig. 3. Phase difference ψ as a function of time when the OEO is in the unlocked state.

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Fig. 4.
Fig. 4. The relationship between the instantaneous beat angular frequency and the undisturbed beat angular frequency with (a) different voltage ratios of the injected microwave signal to the selected mode in the free-running OEO, and (b) different FSRs for the free-running OEO, respectively. (c) Phase difference ψ as a function of time under the quasi-locked condition with different K.

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Fig. 5.
Fig. 5. The output spectra of (a) a free-running OEO, (b) a quasi-locked OEO, and (c) an injection-locked OEO.

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Fig. 6.
Fig. 6. The phase noise transfer model between an injected microwave signal and an injection-locked OEO.

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Fig. 7.
Fig. 7. The phase noise transfer model of an injection-locked OEO.

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Fig. 8.
Fig. 8. The measured output spectra of different OEOs with the same loop length of 10 km. (a1) is the output spectrum of the free-running OEO, Center: 10.664 GHz, RBW: 30 Hz, Span: 100 kHz, (a2) and (a3) are the output spectra of injection-locked OEO, Center: 10.664 GHz, RBW: 30 Hz, Span: 100 kHz, (b1) to (b3) are the output spectra of unlocked OEO, (b1) and (b2) Center: 10.664 GHz, RBW: 30 Hz, Span: 10 kHz, (b3) Center: 10.664 GHz, RBW: 30 Hz, Span: 4 kHz.

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Fig. 9.
Fig. 9. The measured output spectra of injection-locked OEOs when the free-running OEO has a same loop length of around 10 km, a same oscillation frequency of 10.664 GHz, and the injected microwave signal has different frequencies. Center 10.664 GHz, RBW: 10 Hz, Span: 1 kHz.

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Fig. 10.
Fig. 10. The measured output spectra of OEOs with different oscillation frequencies, a same loop length of around 10 km, a same injected frequency of 10.664 GHz, and different injected powers. (a1) – (a4) The power of the injected microwave signal is −5dBm. (b1) – (b4) The power of the injected microwave signal is−10dBm. (c1) – (c4) The power of the injected microwave signal is −15dBm. (d1) – (d4) The power of the injected microwave signal is−20dBm. All the figures have the same setup of Center: 10.664 GHz, RBW: 30 Hz, Span: 40 kHz.

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Fig. 11.
Fig. 11. The relationship between the instantaneous beat frequency and the power at the frequency of the injected microwave signal. (a) The loop length of the OEO is around 4 km and the powers of injected microwave signals are −10 dBm, −15 dBm and −20 dBm, respectively, Center: 10.664 GHz, span 50 kHz. (b) The power of injected microwave signal is −5 dBm and the loop lengths of OEOs are 2 km, 4 km and 10 km, respectively, Center: 10.664 GHz, span 100 kHz.

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Fig. 12.
Fig. 12. The relationship between the undisturbed beat frequency and the instantaneous beat frequency. (a) The OEOs have different loop lengths and a same injection power of −5 dBm, Plot A and plot C: the loop length is 2 km, Plot B and Plot D: the loop length is 10 km. (b) The OEOs have different injection powers and a same loop length of 10 km, Plot A and Plot C: the injection power is −5 dBm, Plot B and Plot D: the injection power is −20 dBm, (c) The relationship between the power at injection frequency and the undisturbed beat frequency, Plot A and Plot B have a same loop length of 2 km and a same power of −5 dBm.

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Fig. 13.
Fig. 13. The relationship between (a) the loop length of an OEO and locking range at different injection powers, in which L is the loop length, (b) the injection power and locking range at different loop lengths, in which PINJ/POSC is the ratio of the power of the injection signal to the power of free-running signal.

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Fig. 14.
Fig. 14. The measured phase noise of an OEO injected by a microwave signal with different powers when the OEO has different loop lengths of around (a) 70 m, (b) 400 m, (c) 4 km, and (d) 10 km, respectively.

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Tables (1)

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Table 1. Output Spectrum of a Quasi-locked OEO

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Equations (37)

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V 0 ( t ) = E 0 cos ⁡ ( ω 0 t )
tan ⁡ ( φ ) = E i n j sin ⁡ ( ψ ) E o s c + E i n j cos ⁡ ( ψ )
tan ⁡ ( φ ) = 2 Q ω 0 − ω o s c ω 0
− E i n j sin ⁡ ( ψ ) E o s c + E i n j cos ⁡ ( ψ )  =  2 Q ω o s c − ω 0 ω 0
ω o s c − ω 0  =  ( ω o s c − ω i n j ) − ( ω 0 − ω i n j ) = Δ ω o s c − Δ ω 0 = d ψ / d t − Δ ω 0
d ψ d t  =  Δ ω 0 − ω 0 2 Q ⋅ E i n j sin ⁡ ( ψ ) E o s c + E i n j cos ⁡ ( ψ )
Δ ω 0  =  ω 0 2 Q ⋅ E i n j sin ⁡ ( ψ ) E o s c + E i n j cos ⁡ ( ψ )
Δ ω 0 , max  =  ω 0 2 Q ⋅ 1 ( E o s c / E i n j ) 2  -  1
Δ ω 0 , max  =  f F S R 2 ⋅ 1 ( E o s c / E i n j ) 2  -  1
E o s c > 2 E i n j
d ψ d t  =  Δ ω 0 − ω 0 2 Q ⋅ E i n j E o s c sin ⁡ ( ψ )
Δ ω 0  =  ω 0 2 Q ⋅ E i n j E o s c sin ⁡ ( ψ )
Δ ω 0 , max  =  ω 0 2 Q ⋅ E i n j E o s c
Δ ω 0 , max  =  f F S R 2 ⋅ E i n j E o s c
tan ⁡ ψ 2 = 1 K + K 2 − 1 K tan ⁡ [ Δ ω 0 ( t − t 0 ) 2 ⋅ K 2 − 1 K ]
K = 2 Δ ω 0 f F S R E o s c E i n j
Δ ω o s c = Δ ω 0 K 2 − 1 K
Δ ω o s c = Δ ω 0 , max K 2 − 1
Δ ω o s c = Δ ω 0 2  -  Δ ω 0 , max 2
V q l − o s c = E o s c e j ω i n j t e j ψ ( t )
e j ψ ( t )  =  j tan ⁡ ( θ 2 ) + 1 − tan 2 ( θ 2 ) j tan ⁡ ( θ 2 ) ∑ n = 1 ∞ [ j tan ⁡ ( θ 2 ) ] n e j n ( Δ ω o s c t + θ )
sin ⁡ ( θ )  =  1 K , ( − π 2 < θ < π 2 )
cos ⁡ ( θ ) = Δ ω o s c Δ ω 0
V q l − O E O = ∑ m = ⌊ − ω o s c / f F S R ⌋ ∞ [ E o s c , m e j ( ω i n j + m f F S R ) t ] e j ψ ( t )
e j ψ ( t )  =  j k tan ⁡ ( θ 2 ) + 1 − tan 2 ( θ 2 ) j tan ⁡ ( θ 2 ) ∑ n = 1 ∞ [ j tan ⁡ ( θ 2 ) ] n e j n ( Δ ω o s c t + θ )
V i n j ( t ) = E i n j cos ⁡ ( ω i n j t + ψ )
V o s c ( t ) = E o s c cos ⁡ ( ω o s c t + ψ ′ o s c ) = E o s c cos ⁡ [ ω 0 t + ψ o s c ( t ) ]
V i n j ′ ( t ) = E i n j cos ⁡ ( ω i n j t + ψ ′ i n j ) = E i n j cos ⁡ [ ω 0 t + ψ i n j ( t ) ]
ψ o s c ( t )  =  ( ω o s c − ω 0 ) t + ψ o s c ′
ψ i n j ( t ) = ( ω i n j − ω 0 ) t + ψ i n j ′
d [ ψ i n j ( t ) − ψ o s c ( t ) ] d t  =  Δ ω 0 − Δ ω 0 , max sin ⁡ [ ψ i n j ( t ) − ψ o s c ( t ) ]
d ψ o s c ( t ) d t  =  Δ ω 0 , max [ ψ i n j ( t ) − ψ o s c ( t ) ]
s ψ o s c ( s )  =  Δ ω 0 , max [ ψ i n j ( s ) − ψ o s c ( s ) ]
H i n j ( s )  =  ψ o s c ( s ) ψ i n j ( s )  =  1 1 + s / Δ ω 0 , max
| H i n j ( j f ) | 2  =  S o s c ( f ) S i n j ( f )  =  1 1 + ( f / Δ f 0 , max ) 2
| H f r e e ( j f ) | 2  =  S o s c ( f ) S f r e e ( f )  =  1 1 + ( Δ f 0 , max / f ) 2
S o s c ( f )  =  | H i n j ( j f ) | 2 S i n j ( f )  +  | H f r e e ( j f ) | 2 S f r e e ( f )
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