Phase measurement by using a forced delay-line oscillator and its application for an acoustic fiber sensor

Michael Fleyer; Moshe Horowitz

doi:10.1364/OE.26.009107

1. Introduction

Injection-locking of limit-cycle oscillators to a forcing signal has been extensively studied theoretically and experimentally [1–6], and is summarized in [7,8]. A free-running limit-cycle oscillator can be forced to oscillate at a frequency that is different than its natural oscillation frequency by injection of an external signal into its cavity. If the frequency of the external signal is equal to the natural frequency of the free-running oscillator, then the oscillation signal becomes phase-locked to the external signal and the relative phase between the two signals equals zero. However, if the frequency of the external signal is different than the natural frequency of the oscillator, the oscillator can be locked to the external signal only when the detuning between the two frequencies is within a certain frequency range, called locking range, which is proportional to the injection ratio [2]. In this case, a phase lag between the oscillator and the injected signal is obtained, as explained in [6]. This phase lag depends on the frequency detuning between the external signal and the natural frequency of the oscillator and it is also inversely proportional to the injection ratio. When the injection ratio is small, the phase lag is highly sensitive to changes in the natural frequency of the oscillator. In this manuscript, we study this effect and apply it to measure small phase changes in the cavity of oscillators.

Optical acoustic interferometric fiber sensors are based on measuring the phase change in an optical wave that propagates through a fiber due to a strain induced by the acoustic signal. This phase change is measured by using an optical interferometry technique [9–11]. Such sensors allow the detection of periodic length changes that can be on the order of few femtometers. However, the linear dynamic range in optical interferometers is limited since a change in the fiber length, on the order of the optical wavelength (∼ μm), results in deterioration of the measured signal. Changes in the phase of a radio frequency (RF) signal that is transmitted over a fiber can be used to detect environmental perturbations. Since the wavelength of the RF signal is on the order of few centimeters, large changes in the fiber may be detected without a distortion. However, since the induced phase changes in the fiber are proportional to the signal frequency, the use of an RF signal instead of an optical signal significantly decreases the sensor sensitivity. Therefore, there is a need to enhance the response and to decrease the noise in fiber sensors that are based on RF signals.

Free-running oscillators can be used as sensors due to the dependence of their oscillation frequency on environmental conditions. In particular, optoelectronic oscillators (OEOs) can be used as fiber sensors, as described in [12–15] and in a review paper [16]. Such oscillators generate ultra-low phase noise RF signals due to the long fiber that is inserted into their cavity [17]. The propagation time in the fiber depends on the environmental conditions and hence sensors, which are based on measuring the frequency of a free-running OEO, have been used in order to measure temperature or acoustic signals. The phase response of free-running oscillators may strongly enhance low-frequency phase fluctuations in their cavity. The transfer function of delay-line oscillators, such as OEOs, to phase perturbations can be derived by using the Lesson model [18, 19] with an effective quality factor Q that is proportional to the cavity length L (Q ∼ L). This transfer function for the acoustic signal highly depends on the acoustic frequency. Moreover, the change in the oscillator frequency is equal to the frequency of acoustic signal. Since the acoustic signal frequency may be as low as few Hz, while the carrier frequency of OEOs is on the order of 10 GHz, the measurement of the frequency change due to the acoustic signal is a challenging task. The acoustic signal can be measured by using a phase discriminator [12,20]. However, the transfer function of discriminators significantly attenuates signals at low frequency offsets that are smaller than the inverse of their delay time. For example, for a delay length of 2 km, the response of the discriminator is attenuated by 3 dB at a frequency of about 20 kHz. Changes in the oscillator frequency may be also measured by mixing its output with a local oscillator (LO) [13, 14]. However, the noise of the LO is usually significantly higher than the ultra-low phase noise of an oscillators, such as OEOs, and therefore, the sensitivity of the sensor is degraded. Moreover, to perform the mixing, the oscillator frequency should be stabilized to compensate large thermal drifts in its carrier frequency.

In this manuscript, we propose a new method to measure small phase changes by using a forced delay-line oscillator. The method is based on the high sensitivity of the phase of a forced delay-line oscillator to changes in its cavity length. The magnitude and the bandwidth of the oscillator response to its cavity changes, can be straightforward controlled by adjusting the injection ratio and the injected frequency. The oscillator phase is directly measured by mixing its output with the external source that is used to injection-lock the oscillator. This mixing also highly suppresses the effect of the injected noise on the device sensitivity. To understand the sensor operation and its limitations, we study theoretically the dynamics of an injection-locked delay-line oscillator due to a small perturbation, caused by changes in the cavity delay and noise. In previous works, the nonlinear dynamics of injection-locked delay-line oscillators has been studied by using the Stuart-Landau equation [21], assuming a nonlinear element with a cubic nonlinearity. The dynamics of such oscillators has also been studied by using a delay differential equation that models the signal propagation in the cavity, assuming a nonlinear element with a sinusoidal transfer function [22, 23]. Such a transfer function is used to model optical Mach-Zehnder modulators (MZM). It was assumed in both models that the filter inside the oscillator is a bandpass filter with a Lorentzian transfer function.

To accurately study oscillators, which contain RF amplifiers, there is a need to model nonlinear effects in such amplifiers. RF amplifiers have an instantaneous complex response, which causes gain saturation and amplitude-to-phase conversion [24]. Amplitude-to-phase conversion is also added in OEOs by photo-detectors [25,26]. In this work, we model the small signal dynamic response of a delay-line oscillator with an amplifier that has an instantaneous complex nonlinear response. The filter inside the oscillator cavity may have a general transfer function with a bandwidth that can be larger than the mode spacing of the delay-line oscillator. We also allow the injection frequency to be different than the frequency in which the filter transmission is maximal. The analysis is based on calculating the dynamic equations for phase and amplitude perturbations, caused by changes in the oscillator delay due to an acoustic signal, noise added by internal noise sources in the cavity, and noise added by the injected signal. We give a general solution to the linearized equations that describes the amplitude and the phase of the oscillator in the frequency domain. We also give a sufficient condition for the stability of the solution as a function of the various oscillator parameters.

Since the dynamic equations couple between phase and amplitude perturbations, the change in the oscillator amplitude cannot be neglected in the general solution. However, to analyze the performance of the sensor, described in this manuscript, we can approximate the general solution in case when the frequency of the phase perturbations is significantly smaller than the bandwidth of the cavity filter and the oscillator amplifier is deeply saturated. In this case, we give a simple solution for the dynamics of the oscillator phase, while the obtained change in the oscillator amplitude becomes small with respect to the phase change. The solution indicates that the oscillator response to a phase change can be modeled by an effective enhancement of the perturbation followed by a low-pass filter (LPF) with a cutoff frequency that is inversely proportional to the signal enhancement. The enhancement of the phase perturbations can be orders of magnitude greater than one and it is inversely proportional to the injection ratio and to the cosine function of the average phase lag between the oscillator and the injected signal. Therefore, by decreasing the injection ratio, we can reduce the bandwidth of the noise, while strongly enhancing the phase perturbation. We also present the dynamic equation for the oscillator phase in the time domain and show that it is similar to the linearized Adler equation [2,27] with an additional term due to the cavity delay [28]. We give a solution to the equation that indicates that the temporal response of the delay-line oscillator is a weighted integration of the signal and noise with an effective integration duration that is equal to the inverse of the cutoff frequency of the LPF, which was obtained in the frequency domain analysis. Therefore, by decreasing the injection ratio, we can increase the integration duration and improve the temporal signal to noise ratio (SNR).

We show that the noise at frequencies that are lower than the cutoff frequency of the effective LPF, is approximately equal to the noise of the external source that is used for the injection locking. Therefore, after mixing the oscillator output with the external source, we can highly suppress the injected noise at the mixer output. Hence, at frequencies that are lower than the cutoff frequency of the sensor, the sensitivity is limited only by the internal noise that is added in a single roundtrip in the oscillator cavity. This noise is significantly lower than the noise that is accumulated in the oscillator cavity. The mixer also enables to directly detect low-frequency acoustic signals. The output of the mixer can be digitized by using a low-frequency analog-to-digital (A/D) converter and the minimum signal frequency that can be detected is not limited by the high carrier frequency of the oscillator.

We experimentally demonstrate the use of a forced delay-line oscillator for obtaining an acoustic fiber sensor in an OEO. We could use an OEO with a short fiber length of about 200-m, without degrading the sensor performance. We inserted a piezo-electric fiber stretcher (PFS) into the OEO cavity to accurately change the cavity length. The measurements of the signal enhancement, the cutoff frequency, and the noise spectrum are in a good agreement with the theory. For a weak injection power ratio of −40 dB, a power enhancement of about 40 dB was obtained for signals with a frequency below 400 Hz. A 3 dB SNR was obtained for a periodic stretching of the fiber with an amplitude of 930 pm at 30 Hz and 340 pm at 450 Hz, for a measurement duration of one second. We have measured the temporal response to a step wave that was supplied to the PFS. The minimum detectable signal became larger as the injection ratio was decreased due to enhancement of the signal and the reduction of the noise bandwidth. By decreasing the injection power ratio from −30 to −40 dB, we obtained an improvement in the temporal SNR by about 10 dB for a measurement duration of 0.1 sec.

We note that self-injection-locking in RF oscillators [29,30] can be used to detect the Doppler frequency shift in a reflection of the oscillator signal that is back-injected into the cavity. It was shown in [29] that this method can improve the SNR with respect to measurements, performed by using a free-running oscillator. However, in free running oscillators the phase noise rapidly grows at low frequency offsets [19]. Therefore, the SNR that is obtained in [29], is lower by a factor of the injection ratio than that obtained in the forced oscillator, described in this work. Moreover, in [29] a discriminator was used to detect phase changes and hence, the minimum detectable frequency was limited. In [30], the reflected wave that is back-injected into the oscillator cavity generates signals that are shifted by the Doppler frequency, with respect to the natural frequency of the oscillator. Therefore, an homodyne detection can be used. However, the response time of the oscillator and false signals that are generated by wave mixing in the oscillator cavity may limit the dynamic range and the sensitivity of such a sensor.

2. Theoretical model for a forced injection-locked oscillator

Figure 1 shows a schematic description of a delay-line oscillator with an external forcing signal. We represent the oscillation signal at the entrance to the amplifier G by a phasor x_in(t) = Re {a(t)e^jφ(t)}, where a(t) is the amplitude of the oscillation and φ(t) is its phase. The nonlinear element G is modeled by a complex instantaneous response function such that x_out(t) = f_G [a(t)]e^jζ[a(t)]. The real-valued function f_G [a(t)] represents the effective nonlinear amplitude saturation and the real-valued function ζ [a(t)] represents the effective amplitude-to-phase conversion. Such a model can be used to accurately model RF amplifiers and photo-detectors [24–26]. We denote the total time-varying delay in a single roundtrip by τ(t), and the effective frequency transfer function of the bandpass filter by H(ω). We assume that the oscillator is injection-locked to an external signal with a frequency ω_inj. We also assume that signals at higher order harmonics of ω_inj that are generated by the nonlinear element have a frequency outside the bandwidth of the bandpass filter.

Fig. 1 Schematic description of a delay-line injection-locked oscillator which is used for measurement of small variations in its cavity delay. G is a nonlinear element with instantaneous amplitude saturation and with amplitude-to-phase conversion, τ(t) is a time dependent delay, H(ω) is the frequency response of a bandpass filter that determines the operating bandwidth of the oscillator, ω_inj is the frequency of an external signal that is injected into the oscillator, and x(t) denotes a phasor. The signal causes a small variation in the delay τ(t) that causes a phase fluctuation Δφ(t) in the oscillating signal with respect to the forcing signal. The oscillator phase variations are detected by mixing the oscillator output with the forcing signal.

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The change in the amplitude and phase of the phasor x_in(t) in a roundtrip is modeled as:

a (t) e^{j φ (t)} = x_{inj} (t) + h (t) * {{f [a (t^{'}) + n_{a} (t^{'})] e^{j {φ (t^{'}) + n_{φ} (t^{'})}}} |}_{t^{'} = t - τ (t)},

where f[a(t)] = f_G [a(t)] e^jζ[a(t)] is the complex response function of the amplifier, * denotes a convolution operator, h(t) is the impulse response function of the filter that is obtained by performing an inverse Fourier transform on the function H(ω), n_a(t) and n_φ(t) are the effective internal amplitude and phase noise sources of the oscillator that are added in each roundtrip to the oscillating signal at the entrance to the amplifier, and x_inj(t) =[b + n_a,inj(t)]e^jω_injt e^jn_φ,inj(t) is the external injected signal with frequency ω_inj, amplitude b, amplitude noise n_a,inj(t), and phase noise n_φ,inj(t),

In Appendix A, we develop the dynamic equations for the perturbation of the amplitude and phase of the oscillator when the oscillator is injection-locked to an external signal. The perturbations are caused by the change in the cavity delay τ due to the acoustic signal, due to noise that is added by the injected signal, and due to the effective internal noise source that is added in each roundtrip to the oscillation phasor x_in(t) at the entrance of the amplifier. To simplify the equations, we assume that the change in the delay τ is smaller than the oscillation wavelength. We give a general solution in Appendix A and then we give in Appendix B a sufficient condition for the stability of the solution and its dependence on the injected frequency with respect to the frequency response function of the filter. To simplify the general solution, we assume that the amplifier is deeply saturated and the bandwidth of the filter is significantly broader than both the signal frequency and the mode spacing between the natural cavity modes of the oscillator. These assumptions enable us to accurately model the sensor, described in this manuscript. We show in this case that the oscillator dynamics can be modeled by an equation for the oscillator phase, while the amplitude perturbation is very small.

2.1. Zero-order solution

In case that the oscillator is injection-locked to an external source with a frequency ω_inj, the steady-state solution, without a perturbation, is found by substituting the steady-state amplitude a(t) = a₀ and phase φ(t) = ω_injt + Δφ₀ to Eq. (1). Splitting the equation into real and imaginary parts gives:

a_{0} = f_{G} (a_{0}) H_{0} cos (ζ_{0} + ϕ_{0} - ω_{inj} τ_{0}) + b cos Δ ϕ_{0},

0 = f_{G} (a_{0}) H_{0} sin (ζ_{0} + ϕ_{0} - ω_{inj} τ_{0}) - b sin Δ φ_{0},

where Δφ₀ is the phase lag between the oscillating signal and the external signal due to the detuning between the external signal frequency and natural frequency of the oscillator, H₀ = |H(ω_inj)| and ϕ₀ = ∡H(ω_inj) are the amplitude and the phase of the bandpass filter at ω = ω_inj, respectively, ζ₀ = ζ(a₀) is the amplitude-to-phase conversion of the nonlinear element, and τ₀ is the unperturbed delay of the cavity.

The solution of the nonlinear Eqs. (2)−(3) gives the amplitude a₀ and the phase lag Δφ₀. This solution depends on the injection frequency ω_inj and on the injection amplitude b. We assume in this manuscript that injection is weak such that b/a₀ ≪ 1 and therefore, we neglect the change in the oscillation amplitude a₀ as a function of injection. We define r_inj = b/a₀ and Γ_inj = (b/a₀)² as the injection ratio for amplitude and power respectively, and obtain from Eq. (2) and Eq. (3):

sin (ω_{inj} τ_{0} - ϕ_{0} - ζ_{0}) = r_{inj} sin (ω_{inj} τ_{0} - ϕ_{0} - ζ_{0} - Δ φ_{0}) .

We define the natural oscillation frequency of the free-running oscillator ω_m = (2πm + ϕ₀ + ζ₀) /τ₀, where m is an integer number. We denote by ω_k, the natural frequency that is the closest to the injection frequency ω_inj. Assuming that the bandwidth of the filter is significantly broader than the frequency spacing between adjacent natural modes, such that ∡H(ω_inj) ≃ ∡H(ω_k), a solution to Eq. (4) exists only if

r_{inj} \geq | sin (ω_{inj} τ_{0} - ω_{k} τ_{0}) | .

Equation (5) defines the locking range that gives the maximal frequency offset of the external signal ω_inj, with respect to the natural frequency ω_k, where injection-locking can be obtained. The locking range is given by ω_LR = (sin⁻¹ r_inj)/τ₀. For a weak injection, r_inj ≪ 1, ω_LR ≈ r_inj/τ₀. We note that this definition of the locking range does not consider the stability of the oscillation, but only the range of frequencies where a solution to Eqs. (2)−(3) may be obtained.

The corresponding phase lag between the external signal and the oscillation, Δφ₀, is extracted from Eq. (4):

Δ φ_{0} = - {sin}^{- 1} {sin (ω_{inj} τ_{0} - ω_{k} τ_{0}) / r_{inj}} + (ω_{inj} τ_{0} - ω_{k} τ_{0}),

where |Δφ₀| ≤ π. In fact, it can be shown that a stable operation of the injection-locked oscillation requires that |Δφ₀| ≤ π/2 as obtained in Eq. (58) in Appendix B and in [22]. We note that Eq. (58) also indicates that the maximum phase lag |Δφ₀| depends on the injection ratio r_inj and on the injection frequency ω_inj with respect to the transfer function of the bandpass filter. Therefore, |Δφ₀| might be smaller than π/2. We also show that the oscillation may become unstable in case that the transmission of the bandpass filter at the injected frequency ω_inj is significantly lower than its maximal value.

2.2. Small-signal analysis of a forced injection-locked oscillator

In this section, we analyze the oscillator response to a weak change in its cavity delay and to noise that is added by the internal noise source and by the injected signal. The acoustic signal causes a weak perturbation δτ(t) to the oscillator delay τ₀ such that τ(t) = τ₀ + δτ(t) and ω_inj |δτ(t)| ≪ 1. In case that the last condition is not met, the response of the oscillator to the signal will become nonlinear. The amplitude and the phase of the oscillator fluctuate in time due to the signal and due to the added noise:

a (t) = a_{0} + Δ a_{sig} (t) + Δ a_{N} (t),

φ (t) = ω_{inj} t + Δ φ_{0} + Δ φ_{sig} (t) + Δ φ_{N} (t),

where Δa_sig(t) and Δφ_sig(t) are the amplitude and phase perturbations caused by the signal, and Δa_N(t) and Δφ_N(t) are the perturbations caused by the noise. In Appendix A, we derive the dynamic equations for the amplitude and phase of the oscillator [Eq. (39)]. The amplifier nonlinearity adds two coefficients to the model: γ_am = a₀f′_G(a₀)/f_G(a₀) and γ_pm = a₀ζ′(a₀) that describe the amplitude-to-amplitude (AM-AM) and the amplitude-to-phase (AM-PM) conversion of the amplifier, respectively.

2.2.1. Response of a forced injection-locked oscillator to a small change in its cavity delay

The acoustic signal causes a small change δτ(t) in the cavity delay. For the case of a weak injection, r_inj ≪ 1, and a filter with bandwidth that is significantly broader than the signal frequency, the oscillator phase perturbation due to the acoustic signal is obtained by using Eq. (48) and Eq. (51) that are derived in Appendix A:

Δ φ_{sig} (ω) = - ω_{inj} δ τ (ω) T_{s} (ω),

where δτ (ω) is the Fourier transform of δτ(t), Δφ_sig(ω) is the Fourier transform of Δφ_sig(t), and

T_{s} (ω) = \frac{1 - r_{inj} cos Δ φ_{0}}{1 - (1 - r_{inj} cos Δ φ_{0}) exp (- j ω τ_{0})}

is the transfer function of the signal to the oscillator phase. Assuming that signal frequency is smaller than the oscillator mode spacing such that ωτ₀ ≪ 1,

T_{s} (ω) \approx \frac{G_{s}}{1 + j ω τ_{r}},

where

G_{s} = (1 - r_{inj} cos Δ φ_{0}) / (r_{inj} cos Δ φ_{0})

is the effective enhancement factor, τ_r = τ₀ (1 + G_s), and ω_r = 1/τ_r is defined as the effective response bandwidth of the oscillator. Equation (11) indicates that at frequencies that are lower than ω_r, the signal perturbation is enhanced by a factor of G_s. To obtain high signal enhancement, such that G_s ≫ 1, the injection ratio should be weak, r_inj ≪ 1. In this case, the bandwidth ω_r also linearly decreases with the decreasing of r_inj. At frequencies that are higher than ω_r, the amplitude of the transfer function decreases as a function of the frequency.

Figure 2 shows the transfer function |T_s(ω_s)|², given in Eq. (11), versus the signal frequency ω_s for different injection ratios Γ_inj. The frequency in the figure is presented by the phase ω_sτ₀. Each line in Fig. 2 corresponds to a constant power enhancement |T_s(ω_s)|² in dB. Dashed red line shows the condition for which ω_s = ω_r. The enhancement is inversely proportional to the injection ratio and is nearly constant inside the device bandwidth ω_r. Figure 3 shows the signal transfer function at frequency ω_s, for which ω_sτ₀ = 10⁻³ rad, versus the injection ratio and the initial phase lag Δφ₀. The results are shown for oscillation frequency ω_inj, for which the transmission of the bandpass filter is maximal. Hence, the maximum absolute value of the phase lag that is required for a stable oscillation, assuming that r_inj ≪ 1, approximately equals π/2 −r_inj, as discussed in section 2.1. The signal is significantly enhanced with increasing of the absolute value of the phase lag due to the increase of the enhancement factor as given in Eq. (12).

Fig. 2 Transfer function |T_s(ω_s)|² as a function of the signal frequency ω_s and the injection power ratio Γ_inj; τ₀ is the average cavity delay. Different lines correspond to constant power enhancements in dB. Dashed red line corresponds to the frequency ω_s = ω_r. The phase lag Δφ₀ equals zero.

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Fig. 3 Transfer function |T_s(ω_s)|² of the signal at frequency ω_s, for which ω_sτ₀ = 10⁻³ rad, versus the phase lag Δφ₀ and the injection ratio Γ_inj. Different lines correspond to a constant enhancement in dB. The signal is significantly enhanced when the phase lag approaches 90°.

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To obtain the temporal response of the oscillator to small changes in its length, Eqs. (9)−(10) can be also written in the time domain:

Δ φ_{sig} (t) = (1 - r_{inj} cos Δ φ_{0}) [Δ φ_{sig} (t - τ_{0}) - ω_{inj} δ τ (t)] .

Equation (13) can be obtained by linearizing the Adler equation [28] that is modified to include the delay τ₀. However, we derive this equation in Appendix A and show that it is accurate only in case that the bandwidth of the phase fluctuations is negligible with respect to the filter bandwidth and the injection is weak. We also show in Appendix A that the change in the oscillator amplitude does not affect the oscillator phase and its magnitude is very weak in comparison with the phase change.

We solve Eq. (9) for the signal response and write the solution in the time domain by expanding the function T_s(ω) in Eq. (10) by a geometric series assuming that |1− r_inj cos Δφ₀| < 1. The solution for the signal δτ(t) is given by:

Δ φ_{sig} (t) = - ω_{inj} \sum_{k = 0}^{\infty} {(1 - r_{inj} cos Δ φ_{0})}^{k + 1} δ τ (t - k τ_{0}) .

In our experiments, we have measured the temporal response of the oscillator to a step function, δτ(t) = δτ₀u(t). In this case, the output signal phase is piecewise constant function with a period of the unperturbed loop delay τ₀, and the solution to Eq. (14) is a finite geometric series from k = 0 to k = ⌊t/τ₀⌋, where k is an integer number and ⌊t/τ₀⌋ is the floor function:

Δ φ_{step} (t) = - ω_{inj} δ τ_{0} \sum_{k = 0}^{⌊ t / τ_{0} ⌋} {(1 - r_{inj} cos Δ φ_{0})}^{k + 1} .

Equation (15) can be written as

Δ φ_{step} (t) = - ω_{inj} δ τ_{0} G_{s} [1 - {(1 - r_{inj} cos Δ φ_{0})}^{⌊ t / τ_{0} ⌋}],

For a weak injection, r_inj cos Δφ₀ ≪ 1, we use the approximation ln(1 − x) ≈ −x + O(x²) for |x| ≪ 1 and obtain:

{(1 - r_{inj} cos Δ φ_{0})}^{⌊ t / τ_{0} ⌋} = exp {⌊ t / τ_{0} ⌋ ln (1 - r_{inj} cos Δ φ_{0})} \approx e^{- t / τ_{r}},

where τ_r = τ₀ (1 + G_s) ≫ τ₀ is the effective integration time as obtained in Eq. (11). Therefore, the output phase for a unit step delay perturbation equals

Δ φ_{step} (t) \approx - ω_{inj} δ τ_{0} G_{s} (1 - e^{- t / τ_{r}}) .

We note that even though the input step function is, strictly speaking, not a bandwidth-limited function, the result in Eq. (18) is equal to the result that can obtained by using Eq. (20) with δτ(t) = δτ₀u(t).

The impulse response function h_s(t) can be obtained by performing a time derivation of the step response, given in Eq. (18):

h_{s} (t) \approx \frac{G_{s}}{τ_{r}} e^{- t / τ_{r}} u (t),

where u(t) is a unit step function. We note that a similar exponentially decaying impulse response was obtained by analyzing the phase response of simple resonators [19]. The output phase due to the signal equals:

Δ φ_{sig} (t) \approx - ω_{inj} \frac{G_{s}}{τ_{r}} \int_{- \infty}^{t} d t^{'} e^{- (t - t^{'}) / τ_{r}} δ τ (t^{'}) .

Equation (20) qualitatively indicates that the output signal at time t is approximately equal to a weighted integration of the input signal over a time period τ_r. The time constant τ_r is linearly proportional to the delay of the loop and to the enhancement factor. Therefore, τ_r increases with decreasing the injection ratio r_inj or increasing the initial phase lag |Δφ₀|. We note that in a free-running oscillator, r_inj = 0, and then G_s in Eq. (12) and τ_r in Eq. (11) approach infinity. Therefore, the long integration time of the noise may limit the detection of time-dependent signals, as explained in section 2.2.2 below.

2.2.2. Noise in a forced injection-locked oscillator

In this subsection, we calculate the phase noise power spectral density (PSD) of the forced injection-locked oscillator, Δφ_N(t), as defined in Eq. (8). We assume weak noise sources such that |n_a(t)| ≪ a₀, |n_φ(t)| ≪ 1, |n_φ,inj(t)| ≪ 1 and |n_a,inj(t)| ≪ b|n_φ,inj(t)|. Equation (52), derived in Appendix A, gives the Fourier transform of the oscillator noise Δφ_N(ω), where ω is the frequency offset with respect to the injected frequency. We note that the injection frequency ω_inj in our calculations can be significantly different than the frequency where the filter transmission is maximal. We use the same assumptions as we used for calculating the oscillator response to the acoustic signal: a weak injection (r_inj ≪ 1) and a filter bandwidth that is significantly broader than the frequency offset ω. We add for the noise calculation another assumption that the amplifier is deeply saturated such that |γ_am| ≪ 1. After using these approximations in Eq. (48), derived in Appendix A, we obtain the Fourier transform of the oscillator phase:

\begin{array}{l} Δ φ_{N} (ω) = & n_{φ, inj} (ω) T_{s} (ω) / G_{s} + r_{inj} n_{φ, inj} (ω) γ_{pm} sin Δ φ_{0} e^{- j ω τ_{0}} T_{s} (ω) \\ + {\tilde{n}}_{φ} (ω) e^{- j ω τ_{0}} T_{s} (ω), \end{array}

where

{\tilde{n}}_{φ} (ω) = n_{φ} (ω) + γ_{pm} e^{- j ω τ_{0}} n_{a} (ω) / a_{0}

is the effective internal phase noise source of the oscillator that includes the effect of the amplitude-to-phase noise conversion in the amplifier, which is modeled by the coefficient γ_pm. Note that the dependence of the oscillator phase noise Δφ_N(ω) on the frequency is proportional to the same transfer function of the signal, T_s(ω). The first term in Eq. (21) is caused due to the addition of the injected signal phase noise to the oscillator phase. The second term in Eq. (21) is obtained due to the conversion of the injected phase noise into amplitude noise which is then converted back into phase noise by the AM-PM effect in the amplifier. When the injection frequency is equal to the natural mode frequency of the oscillator, such that Δφ₀ = 0°, or when |γ_pm| ≪ 1, this effect can be neglected with respect to the first term in Eq. (21). The last term is caused by the internal noise source of the oscillator.

Assuming that ωτ₀ ≪ 1, we can use the approximation of T_s(ω) which is given by Eq. (11) to estimate the contribution of the internal and the external noise sources. At frequencies below the device bandwidth ω_r, T_s(ω) ≈ G_s ≫ 1. In this case, the injected noise n_φ,inj(t) is added by the first term to Eq. (21) to the oscillator noise with approximately a unit factor. Therefore, the effect of this external noise on the output signal can be significantly reduced by mixing the oscillator signal with the external signal, as will be described in section 3 below. In case that the frequency of the external source is stabilized, the injection locking of an ultra-low phase noise oscillator can be used to accurately determine the oscillator frequency while obtaining ultra-low phase noise at sufficiently high frequency offsets, as was observed experimentally in [31]. For low injection ratio, the contribution of the internal noise source ñ_φ(ω) to the phase noise Δφ_N(ω) is enhanced at low frequencies by a factor G_s ≫ 1 [third term in Eq. (21)].

To calculate the PSD of the forced oscillator, we assume that the internal noise source and the phase noise of the external signal are zero-mean, wide-sense stationary, and statistically independent. We denote by S_inj(ω) and S_φ̃(ω) the PSD of the external signal phase noise n_φ,inj(t) and the internal phase noise source of the oscillator ñ_φ(t), respectively. The PSD of the oscillator phase noise that is derived from Eq. (21) equals:

S_{N} (ω) = {(1 / G_{s})}^{2} {| T_{s} (ω) |}^{2} {| 1 + e^{- j ω τ_{0}} q_{s} (ω) γ_{pm} tan Δ φ_{0} |}^{2} S_{inj} (ω) + {| T_{s} (ω) |}^{2} S_{\tilde{φ} (ω)} .

We can now compare the phase noise of the injection-locked oscillator to the phase noise of a free-running oscillator. The phase noise PSD of a free-running delay-line oscillator at frequencies that are much smaller than the inverse of the cavity delay τ₀, such that ωτ₀ ≪ 1, equals [17,19]:

S_{OEO - free} (ω) = S_{\tilde{φ}} (ω) / {(ω τ_{0})}^{2},

where S_φ̃ is the PSD of the effective internal phase noise source of the oscillator. Hence, the phase noise of a free-running oscillator rapidly increases as the frequency offset decreases. This effect is obtained since the oscillator phase is not stabilized and is marginally stable. On the other hand, Eq. (23) indicates that in an injection-locked oscillator with a weak injection ratio (r_inj ≪ 1),

{| T_{s} (ω) |}^{2} \approx G_{s}^{2}

for frequencies that are smaller than the device bandwidth ω_r. Hence, in injection-locked oscillators the enhancement of the noise does not increase at low frequency offsets, as occurs in free-running oscillators. This effect in the injection-locked oscillator can be also understood from Eq. (20) that indicates that the injection of the forcing signal limits the effective integration period of the noise.

3. Calculation of the beating signal at the mixer output

In order to extract the low-frequency acoustic signal from the high-frequency output of the oscillator and in order to suppress the effect of the noise of the external source on the phase measurement, we multiply the oscillator output with the external signal by using a mixer, as shown in Fig. 1. In this section, we calculate the low-frequency signal and noise at the output of the mixer. The mixer translates the oscillator output at frequency ω_inj + ω into a low frequency ω, where ω is a low frequency offset that is equal to the signal frequency and ω_inj is the high carrier frequency. The mixer output is filtered by a low-pass filter to remove the frequency component at ω_inj + ω and it then can be sampled directly by an analog-to-digital converter. Therefore, the same notation ω that was used in Section 2.2 to denote a frequency offset, will be used here for the corresponding frequency at the mixer output. The outputs of the I and Q channels of an I/Q mixer are given by [32,33]:

\begin{array}{l} I (t) = v (t) cos [Δ φ_{0} + Δ φ_{B} (t) + ϕ_{P}], \\ Q (t) = v (t) sin [Δ φ_{0} + Δ φ_{B} (t) + ϕ_{P}], \end{array}

where

Δ φ_{B} (t) = n_{φ, inj} (t) - Δ φ_{N} (t) - Δ φ_{sig} (t),

is the beating phase and ϕ_P is a constant phase shift that could be controlled in our experiments by using a mechanical phase shifter. In our experiments, we have set ϕ_P = 0. The output amplitude v(t) = l · b · a(t) is a slowly varying amplitude of the beating signal where the factor l represents the mixer loss and the transmission of a coupler, used to tap the oscillator output, and a (t) and b are the amplitudes of the oscillator and the injected source, respectively. Power saturation in practical mixers reduces the fluctuations of the output amplitude v(t) [32].

The phase is calculated from the I and Q channels by using the identity: tan⁻¹ {Q(t)/I(t)} = Δφ_B(t) + Δφ₀ [33]. The beating phase, given in Eq. (26), gives the signal contaminated by noise. The noise in the beating signal equals Δφ_B,noise(t) = n_φ,inj(t) − Δφ_N (t). For small frequencies with respect to the cavity mode spacing such that ωτ₀ ≪ 1, we approximate 1 − (1− r_inj cos Δφ₀) exp (−jωτ₀) ≈ r_inj cos Δφ₀ + jωτ₀ in Eq. (21), and obtain the Fourier transform of the phase noise at the mixer output:

Δ φ_{B, noise} (ω) = n_{φ, inj} (ω) \frac{j ω τ_{0} + r_{inj} γ_{pm} sin Δ φ_{0}}{r_{inj} cos Δ φ_{0} + j ω τ_{0}} - \frac{e^{- j ω τ_{0}} {\tilde{n}}_{φ} (ω)}{r_{inj} cos Δ φ_{0} + j ω τ_{0}},

with a PSD:

S_{B, noise} (ω) = {(G_{s} + 1)}^{2} \frac{S_{inj} (ω) [{(ω_{r} τ_{0})}^{2} {(γ_{pm} tan Δ φ_{0})}^{2} + {(ω τ_{0})}^{2}] + S_{\tilde{φ} (ω)}}{1 + {(ω / ω_{r})}^{2}} .

Equation (28) indicates that in case that the amplitude-to-phase noise in the amplifier can be neglected, such that |γ_pm| ≪ 1 or when the detuning is small such that |tan Δφ₀| ≪ 1, the effect of the injected signal noise on the beating signal is suppressed by a factor (ωτ₀)² with respect to the internal noise source for ωτ₀ ≪ 1. This suppression is obtained since for frequencies ω ≪ ω_r, the noise of the external signal is approximately added to the oscillator noise, as indicated in Eq. (21), since T_s(ω) ≈ G_s. Therefore, the beating between the oscillator and the external signal highly suppresses the effect of the external source noise. Equation (28) indicates that at frequency ω = ω_r, the response of the beating signal to both internal and external noise sources decreases to half of its maximal response at ω = 0. By decreasing the injection ratio r_inj, the cutoff frequency ω_r can be decreased such that less noise is accumulated in the mixer output.

In a free-running oscillator, the phase noise rapidly increases as the frequency offset with respect to the carrier frequency decreases, as given in Eq. (24). Since the measured signal frequency is unknown, noise at the lower frequencies of the sensor bandwidth will limit the system sensitivity. Moreover, the measurement of a real-time signal is performed over a finite observation time. As the observation time increases, phase noise with lower frequencies will affect the measurement. These noise components will significantly increase the total average noise power over the observation time. Moreover, low-frequency noise components will also increase noise components at higher frequencies due to the correlation between the measured spectrum at different frequencies [34]. In contrast to a free-running oscillators, we obtain in Eq. (28) that in a forced oscillator S_B,noise(ω) ≈ (G_s + 1)²S_φ̃(ω), for ω ≪ ω_r. Hence, the response of the injection-locked oscillator almost does not depend on the frequency at frequencies ω ≪ ω_r. The advantages of the phase measurement by using an injection-locked oscillator can be also shown in the time domain. We have shown in Eq. (20) that the time response of the injection-locked oscillator can be qualitatively described by a weighted integration of the signal over a time period τ_r. Hence, noise components with a frequency that is higher than 1/τ_r are suppressed by the effective integration of the oscillator. This integration time can be increased by decreasing the injection ratio.

We can quantitatively demonstrate the improvement of the signal to noise ratio (SNR) for a step waveform signal that was analyzed in section 2.2.1. Equation (18) indicates that for t ≫ τ_r, the steady-state response to a step signal equals Δφ_sig = −ω_injδτ₀G_s with a power that is proportional to $Δ φ_{sig}^{2}$ . Assuming a weak injection r_inj ≪ 1 such that G_s/(1 + G_s) ≈ 1 and an observation duration that is significantly longer than τ_r, the SNR, defined as the ratio between the signal power and noise power at t > 0, can be calculated by using Eq. (28) for the phase noise spectrum:

SNR = {(ω_{inj} δ τ_{0})}^{2} {[\frac{1}{2 π} \int_{0}^{\infty} d ω \frac{{(ω τ_{0})}^{2} S_{inj} (ω) + S_{φ} (ω)}{1 + {(ω / ω_{r})}^{2}}]}^{- 1},

Since ω_r = [τ₀ (1 + G_s)]⁻¹, Eq. (29) indicates that as G_s increases, the transfer function for the noise becomes spectrally narrower and therefore the power of the noise is reduced with respect to the power of the signal. We note that the maximum signal that can be linearly detected is also inversely proportional to G_s. However, since G_s can be easily adjusted by changing the injected power, it can be controlled in real-time according to the signal power.

Another approach to detect the existence of signals in the presence of noise is based on integrating the signal over a time window τ and comparing the result to that measured in the previous adjacent time slot [35]. In case of a white noise, it is possible to improve the SNR by increasing time window τ. However, it was shown in [35] that if the spectral density function of the noise rapidly grows as the frequency is decreased, the SNR is deteriorated when the integration time τ is increased. Therefore the maximum integration time τ that can be used to increase the sensor sensitivity is limited. In the case of a forced oscillator, the oscillator response to noise at low frequencies (ω < ω_r) almost does not depend on the frequency. However, in a free-running oscillator, low-frequency noise is enhanced, as given in Eq. (24), and therefore, when the time window is increased, the SNR will be deteriorated, as given in Eq. (13) in [35].

4. Experimental results and comparison to theory

We have used an optoelectronic oscillator (OEO) to experimentally verify the theoretical results, which are given is sections 2 and 3, for the signal and noise of a forced oscillator and for the mixing product. Then, we have demonstrated the use of the forced injection-locked oscillator as an acoustic fiber sensor. Equations (11) and (23) indicate that the enhancement of the noise due to the internal noise sources in a forced injection-locked oscillator almost does not depend on the loop delay τ₀ for frequencies ω < ω_r. After mixing the oscillator output with the external source, the noise in that frequency region is mainly determined by the internal noise source that is added in a single roundtrip in the oscillator. The frequency ω_r depends on the injection ratio and on the loop delay τ₀. Therefore, by using a short cavity OEO and by reducing the injection ratio, it is possible to obtain a low cutoff frequency ω_r without the need to use an OEO with a very long cavity length. Short cavity OEOs enable to reduce noise, added by changes in environmental conditions along the fiber locations, which are not used for the acoustic signal measurement.

The detailed experiment setup is shown in Fig. 4. A continuous wave (CW) laser with a power of 14 dBm was fed into a linearly biased Mach-Zehnder modulator (MZM) with V_π = 4 V. The modulator output was connected to a L = 200 m fiber. Isolators at fiber ends were used to prevent back reflections. The light was converted into an electrical signal by a photo-detector (PD) with conversion efficiency of 0.75 A/W and amplified by amplifier G₁ with a small-signal gain of 37.7 dB and a saturation power P_1dB of 13.8 dBm. Part of the amplified signal was fed to a signal source analyzer (Keysight E5052A + E5053B) to measure the phase noise and the other portion passed through a bandpass filter (BPF) with a full width at half maximum (FWHM) bandwidth of 10 MHz around a central frequency of 10 GHz. An external signal was injected through a voltage-controlled RF switch into coupler C₃. The combined signals were fed to coupler C₄. Part of the output of C₄ was fed to the RF input of an I/Q mixer and the remaining part was fed into the MZM.

Fig. 4 Schematic description of the experiment setup. Laser is a CW laser, MZM is a Mach-Zehnder modulator, PFS is a piezo-electric fiber stretcher, L is a fiber with a length L, PD is a photo-detector, G₁ – G₃ are amplifiers, SSA is a signal source analyzer, BPF is a bandpass filter, C₁ – C₄ are directional couplers with coupling ratios of −6 dB, −6 dB, −20 dB and −5 dB, respectively. PC is a personal computer, ϕ is electro-mechanical phase shifter, and LPF is a low-pass filter with a cutoff frequency of 20 kHz.

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A tunable signal generator (Agilent N5183A) was used to injection-lock the OEO. The signal was amplified by an amplifier G₃ with 1-dB compression of 13 dBm and a small-signal gain of 11 dB. Part of the signal was injected into the OEO through a variable attenuator and an RF switch. Another part of the signal was supplied to the LO input of the I/Q mixer after passing through electro-mechanical phase shifter. The power of the signal and the LO at the inputs of the I/Q mixer was 0 dBm and 13 dBm, respectively. The I and Q outputs of the mixer passed through low-pass filter with a cutoff frequency of 20 kHz and fed into a low-frequency amplifier with a gain of 60 dB and noise figure (NF) of 4 dB. The amplifier output was sampled by using a 24-bit A/D converter at a rate of 60 kHz. A piezo-electric fiber stretcher (PFS) was inserted into the oscillator cavity in order to apply a controllable change in the delay. The stretching coefficient was 1.6 μm/V.

The RF power saturation in our experiments was mainly determined by amplifier G₁. The input power to this amplifier was −7.8 dBm and its output power was 17 dBm. This corresponds to a gain of 25 dB, which is much lower than the amplifier small-signal gain of 37.7 dB. Hence, G₁ was deeply saturated in our experiments. We have measured the amplitude-to-amplitude conversion efficiency (γ_am) by opening the loop of the system and injecting a CW RF signal with a controllable power to the input of amplifier G₁. We have measured the output power from the photo-detector versus the input power to G₁, and extracted the amplitude transfer function f_G. Then, we calculated γ_am = a₀f′_G(a₀)/f_G(a₀) by performing a numerical derivation of the function f_G. Figure 5 shows γ_am versus the input power to G₁. For operating input power of −7.8 dBm at the entrance to G₁, the AM-AM conversion efficiency is γ_am = 0.06. Hence, the oscillator amplifier is deeply saturated as assumed in the solutions of the theoretical model, given in Eq. (52).

Fig. 5 Amplitude-to-amplitude conversion coefficient γ_am, derived from the open-loop power transmission, measured between the input to amplifier G₁ and the output of the photo-detector. At operating condition, the power at the entrance of G₁ is −7.8 dBm and the system is deeply saturated with γ_am = 0.06.

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To accurately injection-lock the OEO to the external signal, the frequency of the external signal should be within the locking range of one of the OEO modes. The modes spacing in our experiment is determined by the 200-m length fiber, and it is equal to about 1 MHz. Therefore, there are about 10 modes that can potentially oscillate within the 10-MHz bandwidth of the BPF in the OEO cavity. In [31], we have demonstrated the injection-lock of an OEO into an external source. To maintain the OEO frequency within the locking range, an electrical phase shifter was inserted to the OEO cavity. In the experiments that are described in this manuscript, we have chosen to continuously lock the oscillator to the external source by adjusting the frequency of the external signal instead of tuning the OEO oscillation frequency. Before the external signal was injected to the OEO, we measured the frequency detuning between the external signal and a free-running OEO by sampling the outputs of the I/Q mixer [31,36]. After adjusting the injected signal frequency to obtain a frequency detuning of less than about 10 Hz, we have started to inject the external signal to the OEO by closing the RF switch, shown in Fig. 4. In this case, the relative phase between the OEO and the signal should be approximately zero. To compensate for the difference between the cable lengths that are connected to the mixer, we added an electro-mechanical phase shifter ϕ at the LO input of a mixer and adjusted it to obtain a zero phase shift such that the absolute amplitude of the Q channel becomes minimal and the amplitude of the I channel becomes maximal. In this case, the average measured phase 〈tan⁻¹ {Q(t)/I(t)}〉_t was approximately equal to zero. Then, we could adjust the phase lag Δφ₀, between about −π/2 to π/2, by adjusting the frequency of the tunable signal generator, as expected from Eq. (6). To compensate the drift in the natural frequency of the OEO due to environmental conditions, we have adjusted the synthesizer frequency every 1 second according to the measured average phase Δφ₀.

To compare theory to experiments, we need to measure the phase noise spectrum of the internal noise source of the OEO. Since this noise source is very weak, we have extracted its magnitude from the phase noise of the free-running OEO that was measured by using a signal source analyzer (SSA). Figure 6 shows the phase noise spectrum of the free-running oscillator S_OEO–free(ω) (green curve). The PSD of the internal noise source, S_φ̃(ω), which is added in one roundtrip, was extracted by using Eq. (24). The result is shown in the bottom blue curve of Fig. 6. The PSD of this noise can be modeled by an empirical fit, shown in the bottom red dashed curve in Fig. 6: S_φ̃(ω) = 2πb₋₁/ω + 2πb₀, where b₋₁ = −118 dBc/Hz is the flicker noise coefficient, and b₀ = −155 dBc/Hz is the white noise coefficient. The flicker noise coefficient b₋₁ is the same as measured in [37] for a fiber length of L = 200 m. The white noise in our experiments corresponds to the shot noise of the photo-detector, calculated for the measured input optical power of 7.3 dBm, a 50 Ω resistance, and a PD responsivity of 0.75 A/W. The measured phase noise of the external signal S_inj(ω) that was used in the experiments (yellow curve) and its empirical fit, calculated by using a spline fit (dashed purple curve), are also shown in Fig. 6.

Fig. 6 Phase noise of the free-running OEO (green curve) and the phase noise of the injected source (yellow curve), measured by using SSA. The internal phase noise of the OEO (bottom blue curve) was extracted from the noise of the free-running OEO by using Eq. (24). Empirical fits to the curves are shown by dashed curves.

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Figure 7 shows the spectrum of the amplitude noise (solid yellow curve) and the phase noise (solid blue curve) of the injection-locked OEO, which were measured by using SSA. The measured phase noise is compared to the calculated PSD (dashed red curve) that was obtained by using Eq. (23). The injection ratio was Γ_inj = −30 dB and the average phase lag was Δφ₀ = 0°. The calculated device bandwidth equals f_r = 4.8 kHz. The figure shows that a very good quantitative agreement is obtained between the theoretical and the measured phase noise spectra. The measurements indicate that the amplitude noise of the OEO is much smaller than its phase noise. This result is expected from Eq. (54) in Appendix A, since the amplifier is deeply saturated, the injection is weak, and the phase lag approximately equals zero.

Fig. 7 Phase noise (solid blue curve) and amplitude noise (solid yellow curve) spectra of the injected-locked OEO, measured by using SSA. The measured phase noise is compared with the PSD, calculated by using Eq. (23) (dashed red curve). The injection ratio was Γ_inj = −30 dB and the phase lag was Δφ₀ = 0°. The device bandwidth equals f_r = 4.8 kHz.

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Figure 8 shows the phase noise PSD at the output of the mixer S_B,noise(f) (solid blue curve), calculated by using Eq. (28). The result is also compared to the phase noise PSD of the injection-locked OEO, S_N(ω) (solid red curve), calculated by using Eq. (23), and to the measured phase noise of the external signal (dashed black curve), where f in these two curves is the frequency offset with respect to the carrier frequency f_inj. The injection power ratio is Γ_inj = −40 dB, and the initial phase lag Δφ₀ is set to zero. The vertical dashed blue curve marks the device bandwidth of f_r = ω_r / (2π). The figure helps to understand the effect of the internal and the external noise sources on the noise at the mixer output. The frequency f_r gives the effective bandwidth of the device. For f < f_r, the phase noise of the injection-locked oscillator is approximately equal to that of the external signal, as indicated in Eq. (23). This noise is therefore suppressed at the mixer output due to the beating with the external source, as indicated by Eq. (28). The dominant noise in this frequency region is caused by the internal noise source of the oscillator. At frequencies f > f_r, the phase noise at the mixer output is mainly determined by the high phase noise of the external source. This results are clearly demonstrated by setting the internal noise source to zero (ñ_φ = 0, as shown by the solid yellow curve in Fig. 8).

Fig. 8 Phase noise PSD at the output of the mixer S_B,noise(f) (solid blue curve), calculated for an injection ratio Γ_inj = −40 dB and an average phase lag Δφ₀ = 0. The result is compared to the calculated phase noise PSD of the injection-locked OEO, S_N(ω) (solid red curve) and to the measured phase noise of the external signal (dashed black curve), where f in these two curves represents the frequency offset with respect to the carrier frequency f_inj. The solid yellow curve in the figure gives the calculated phase noise PSD at the mixer output when the internal noise source is set to zero. The vertical dashed blue line marks the device bandwidth of f_r = 1.6 kHz. The results indicate that inside the device bandwidth, the effect of the injected noise is highly suppressed at the mixer output.

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Figure 9 shows the measured spectrum of the beating phase, obtained from the sampled I/Q outputs of the mixer. The peak at 30 Hz is due to a 0.05 V voltage that was supplied to the PFS and caused a periodic change in fiber length with an amplitude of about 80 nm. The injection power ratio was Γ_inj = −40 dB and the phase lag was Δφ₀ = 0°. The experimental results are compared to the theoretical phase noise PSD, calculated by using Eq. (28) (red curve), and to the calculated signal phase power (red dot). The comparison shows that a good agreement between the theoretical model and experiments is obtained. Around 30 Hz, the internal phase noise source of the OEO is caused by the flicker noise of the amplifier. The PSD of this noise source equals −130 dBc/Hz, as shown in Fig. 6. This noise is enhanced in the oscillator cavity by the strong enhancement factor of about G_s ≃40 dB.

Fig. 9 Spectrum of a beating signal and the noise, measured by using A/D converter (blue solid curve), for a sinusoidal signal at 30 Hz that was supplied to the PFS and caused a stretching of the fiber with an amplitude of about 80 nm. The result is compared to the calculated noise PSD (solid red curve) and to the signal power (red dot). The resolution bandwidth equals RBW = 1 Hz, the injection power ratio equals Γ_inj = −40 dB, and the average phase lag equals Δφ₀ = 0°.

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We define the minimum detectable fiber stretching as the amplitude of the fiber length change that gives a signal to noise ratio of 3 dB. The minimum detectable amplitude of the fiber stretching was 930 pm at 30 Hz and 340 pm at 450 Hz. The higher sensitivity at 450 Hz is obtained since the internal noise source of the oscillator becomes weaker around this frequency, as shown in Fig. 9. We note that the sensitivity of sensors that are based on optical interference, can be orders of magnitude better than obtained in the device described in this manuscript. However, the maximum linear dynamic range of optical interferometric systems is limited by the wavelength of the light (≈ 1 μm). In the device, described in this manuscript, the maximum signal should be lower than about r_inj cos Δφ₀λ_RF, where λ_RF = c/(nf_inj) = 2 cm is the wavelength of the RF wave that is transmitted in the fiber, f_inj = 10¹⁰ Hz is the oscillation frequency, c is a speed of light in a vacuum, and n is optical fiber refractive index. In case that the enhancement factor G_s equals 25 dB, the maximum signal should be lower than about 185 μm. The dynamic region can also be controlled in a straightforward manner by changing the injection ratio according to the desired signal.

We have also measured the magnitude of the transfer function of the signal, T_s(ω), for injection ratios of −20 dB, −30 dB and −40 dB (red dots in Fig. 10). The average measured phase lag was equal to Δφ₀ ≃ 0, and the amplitude of the fiber stretching was 150 nm. The result is compared with the theoretical results calculated by using Eq. (11). The difference between the theoretical and experimental results is less than 1 dB. The figure indicates that the signal enhancement inside the device bandwidth is proportional to the inverse of the injection ratio.

Fig. 10 Transfer function of the signal, |T_s(ω)|², calculated by using Eq. (11), for injection ratios of −20 dB, −30 dB and −40 dB and a phase lag Δφ₀ = 0 (blue curves), which is compared to the measured results (red dots).

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We have measured the temporal response function of the system to a step function signal and compared it to the theoretical result given in Eq. (18). We have supplied to the PFS a periodic square signal with an amplitude of 0.6 V, a width of 0.1 s, and a time period of 0.2 s. Such a step function results in fiber stretching of 0.96 μm, which corresponds to about 0.3 mrad phase shift at the RF oscillation signal. The I/Q mixer outputs were passed through a low-pass filter with a cutoff frequency of 20 kHz and sampled by a 24-bit A/D converter at a rate of 100 kHz. We have measured the time-dependent phase at the mixer output for different injection ratios and initial phase lags Δφ₀. Figure 11(a) shows the temporal phase for: (i) Γ_inj = −30 dB, Δφ₀ = 0°; (ii) Γ_inj = −30 dB, Δφ₀ = 60°; (iii) Γ_inj = −40 dB, Δφ₀ = 0° and (iv) Γ_inj = −40 dB, Δφ₀ = 60°. In our theoretical results, given in Eq. (12), the power enhancement factors $G_{s}^{2}$ that correspond to the results in curves (i)−(iv) are 30, 36, 40 and 46 dB, respectively, the characteristic times τ_r are equal to 36, 71, 113, and 225 μs, respectively and the corresponding bandwidths of the sensor f_r = ω_r/(2π) are equal to 9, 4.5, 2.8 and 1.4 kHz, respectively. In Fig. 11(b), we focus on the measured transient response of the system and compare it to the theoretical results, calculated by using Eq. (18) (dashed black curves). A good quantitative agreement is obtained between theory and experiments. The figure indicates that the output signal increases as the injection ratio is reduced or when the absolute value of the initial phase lag is increased.

Fig. 11 (a) Measured temporal response of the phase at the mixer output when the fiber was stretched by a square voltage with a period of 0.2 s that was supplied to the PFS and caused a change of 0.96 μm in the fiber length. The response was measured for (i) Γ_inj = −30 dB, Δφ₀ = 0°; (ii) Γ_inj = −30 dB, Δφ₀ = 60°; (iii) Γ_inj = −40 dB, Δφ₀ = 0°, and (iv) Γ_inj = −40 dB, Δφ₀ = 60°. The corresponding power enhancement factors of experiments (i) −(iv) are 30, 36, 40 and 46 dB, respectively. Figure 11(b) shows a close-up on the initial response that is compared to the theoretical response function (dashed black curves), calculated by using Eq. (18). The mixer outputs were filtered by a low-pass filter with a FWHM of 40 kHz and was sampled at a rate of 100 kSamples/sec.

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To demonstrate the improvement of the SNR as a function of the injection ratio and the phase lag, we have normalized the output phase by the mean amplitude of the steady-state output. Figure 12 shows the normalized measured output for curves (i) and (iv) in Fig. 11(a) with corresponding enhancement factors of 30 and 46 dB, respectively. The results clearly show that as the enhancement of the signal increases, the SNR is improved due to the suppression of high-frequency noise, which is caused by the narrowing of the effective transfer function. We note that the main noise in the results that correspond to the higher enhancement factor (purple curve) is caused by electromagnetic interference (EMI) at frequencies that are equal to a multiple factor of the power line frequency of 50 Hz.

Fig. 12 Output signals that correspond to graphs (i) (blue curve) and (iv) (purple curve) in Fig. 11(a) that are obtained after normalizing the measured phase by the mean steady-state phase. The enhancement factor equals 30 dB for curve (i) and 46 dB for curve (iv). The normalized noise decreases for the higher enhancement factor case.

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We have calculated the SNR in the experiments by:

\hat{SNR} = \frac{{〈 Δ φ_{B} (t) 〉}^{2}}{〈 Δ φ_{B}^{2} (t) 〉 - {〈 Δ φ_{B} (t) 〉}^{2}},

where Δφ_B(t) is the sampled phase,

〈 x (t) 〉 = (1 / T_{obs}) \int_{0}^{T_{obs}} dt

x(t) denotes a time-average over the time interval of T_obs = 0.1 s where the steady-state output phase was obtained. The SNR for measurements (i)−(iv) in Fig. 11(a) is equal to 16, 20, 22 and 25 dB, respectively. These results did not significantly change when the measurements were repeated. Therefore, the real-time SNR is improved with increasing the enhancement factor, as expected by the theory, given in Eq. (29).

5. Conclusion

We have demonstrated, theoretically and experimentally, a new method to measure small changes in the cavity length of oscillators. The method is based on the high sensitivity of a forced delay-line oscillator to changes in its cavity delay. We apply the method to demonstrate a new acoustic fiber sensor. We give a comprehensive theoretical analysis to accurately model a forced injection-locked delay-line oscillator in order to study its noise and its dynamic response to a small perturbation in its cavity delay. The analysis is based on calculating the response to perturbations, caused by a temporal change in the cavity length, internal noise sources, and external noise that is added by the injected signal. We take into account various effects such as instantaneous gain saturation and amplitude-to-phase conversion in the amplifier. We also consider a general filter in the cavity with a bandwidth that is larger than the mode spacing of the delay-line oscillator and we allow the injection frequency to be different than the central frequency of the filter. We give a general solution to the amplitude and phase changes in the oscillator and give a sufficient stability condition for the solution. A simple solution is given for the case when the amplifier is deeply saturated. The solution indicates that at frequencies that are located inside a certain bandwidth around the oscillation frequency, the oscillator noise is approximately equal to the noise of the injected signal. Therefore, the injected noise at these frequencies can be highly suppressed by mixing the oscillator output with the external source, used for the injection-locking. The noise of the mixing product inside the device bandwidth is only limited only by the internal noise source that is added in a single roundtrip in the oscillator. The mixing between the oscillator output and the external source also allows the detection of signals at very low frequencies, around a high carrier frequency, without a need for a frequency discriminator that strongly attenuates low-frequency signals.

Variations in the cavity length with a frequency that is smaller than the oscillator bandwidth are converted into large variations in the oscillator phase with respect to the phase of the injected signal. This signal enhancement inversely depends on the injection ratio and on the cosine of the average phase lag between the oscillator output and the external source. High signal enhancement of more than 40 dB was obtained theoretically and experimentally for a low power injection ratio of −40 dB. The bandwidth of the device is inversely proportional to the enhancement factor for low injection ratio. Therefore, the magnitude of the output signal and the bandwidth of its noise can be easily controlled in real-time. The use of a forced injected-locked oscillator for phase measurements instead of free-running oscillators that were used in previous works, offers significant advantages. The frequency response function of the forced oscillator, inside its bandwidth, does not depend on the measured frequency. This is in contrast to free-running oscillators, in which the frequency response function to the acoustic signal and noise rapidly grows as the signal frequency is decreased. We also present the dynamic equation for the oscillator phase in the time domain and show that this equation is the same as the equation obtained by linearizing the Adler equation that is modified to include the cavity delay. However, we do not neglect amplitude fluctuations in the derivation of this equation. The result indicates that the output signal is obtained by a weighted integration of the signal and noise over a duration that is equal to the inverse of the device bandwidth. The increase of the integration duration as the injection ratio is decreased, enables to improve the real-time signal to noise ratio of the measurement.

We verified the theoretical results in experiments, based on an optoelectronic oscillator (OEO), and applied the results to demonstrate an acoustic fiber sensor. We obtained a good agreement between the theoretical and experimental results for the noise and the signal at the output of the injection-locked OEO and at the output of the mixer. For a measurement duration of one second, a 3 dB signal to noise ratio (SNR) was obtained for a periodic stretching of the fiber with an amplitude of 930 pm at 30 Hz and 340 pm at 450 Hz. We also demonstrated the improvement of the SNR, measured in the time domain. The SNR for a step-function signal, was improved by 10 dB over a time measurement of 0.1 sec, by decreasing the power injection ratio from −30 to −40 dB. This result is obtained due to the reduction of the noise bandwidth when the injection ratio is decreased.

We believe that theoretical model, developed in this work, will be important to accurately analyze various types of forced injection-locked delay-line oscillators with instantaneous amplitude saturation, amplitude-to-phase conversion, and a general filter. The method for phase measurement, described in this manuscript, may also be important to develop various types of new sensors that are based on forced oscillators. We intend to study in the future the dynamic response of a forced delay-line oscillator to large changes in its cavity.

Appendix A Perturbation analysis of a forced delay-line oscillator

In this Appendix, we derive the dynamic equations for a perturbation in the amplitude and phase of a forced injection-locked limit-cycle oscillator with a fast saturable amplifier. The derivation is based on calculating the response to a weak perturbation, caused by temporal changes in the oscillator delay and by noise that is added by internal noise sources of the oscillator and by the noise of the injected signal. We obtain the dynamic equations for the amplitude and the phase of the oscillation. We give a general solution to the equations and in Appendix B, we give a sufficient condition for the oscillation. We also provide a simple solution for frequency offsets with respect to the carrier frequency, which are smaller than the filter bandwidth, in case that the injection ratio is weak and the amplifier is deeply saturated. The solution indicates that the signal can be accurately modeled by an equation for the oscillator phase while the obtained amplitude variation is very small with respect to the phase variation.

To calculate the dynamics of the injection-locked oscillator to perturbations, we assume solution of the form: a(t) = a₀ + Δa(t) and φ(t) = Δφ₀ + Δφ(t). The perturbations are caused by a small temporal change in the delay variation, δτ(t), such that τ(t) = τ₀ + δτ(t) and by the external and internal noise sources that are defined at section 2. We assume a weak delay perturbation such that ω_inj |δτ(t)| ≪ 1, an external noise source with a weak amplitude and phase noise, such that |n_φ,inj(t)| ≪ 1 and |n_a,inj(t)| ≪ b, and a weak internal noise source, such that |n_φ(t)| ≪ 1, |n_a(t)| ≪ a₀.

We substitute a(t), φ(t), and τ(t) into Eq. (1) and perform a first order approximation to the nonlinear amplifier response:

f [a_{0} + Δ a (t)] \approx f_{G} (a_{0}) e^{i ζ (a_{0})} + [f_{G}^{'} (a_{0}) + j ζ^{'} (a_{0}) f_{G} (a_{0})] e^{j ζ (a_{0})} Δ a (t),

and to the phase noise: exp [jn_φ,inj(t)] ≈ 1+ jn_φ,inj(t) and the phase lag: exp [jΔφ(t)] ≈ 1+ jΔφ(t), and obtain the following equations for the amplitude and phase perturbations:

\begin{array}{l} Δ a (t) = & {γ_{am} q_{s} (t) - γ_{pm} q_{c} (t)} * [Δ a_{0} (t - τ_{0}) + n_{a} (t - τ_{0})] \\ - a_{0} q_{c} (t) * [Δ φ (t - τ_{0}) - ω_{inj} δ τ (t) + n_{φ} (t - τ_{0})] \\ + b n_{φ, inj} (t) sin Δ φ_{0} + n_{a, inj} (t) cos Δ φ_{0}, \end{array}

\begin{array}{l} a_{0} Δ φ (t) = & {γ_{am} q_{c} (t) + γ_{pm} q_{s} (t)} * [Δ a (t - τ_{0}) + n_{a} (t - τ_{0})] \\ + a_{0} q_{s} (t) * [Δ φ (t - τ_{0}) - ω_{inj} δ τ (t) + n_{φ} (t - τ_{0})] \\ + b n_{φ, inj} (t) cos Δ φ_{0} - n_{a, inj} (t) sin Δ φ_{0}, \end{array}

where γ_am = a₀f′_G(a₀)/ f_G(a₀), γ_pm = a₀ζ′(a₀) are amplitude-to-amplitude and amplitude-to-phase conversion coefficients of the amplifier, respectively and the functions q_c(t) and q_s(t) are equal to

\begin{array}{l} q_{c} (t) = (1 - r_{inj} cos Δ φ_{0}) h_{im} (t, ω_{inj}) - r_{inj} sin Δ φ_{0} h_{r} (t, ω_{inj}), \\ q_{s} (t) = (1 - r_{inj} cos Δ φ_{0}) h_{r} (t, ω_{inj}) + r_{inj} sin Δ φ_{0} h_{im} (t, ω_{inj}), \end{array}

where h_r(t, ω_inj) = Re {h(t)e^−jω_injt e^−jϕ₀}/H₀, h_im(t, ω_inj) = Im {h(t)e^−jω_injt e^−jϕ₀}/H₀ are the real and the imaginary parts of the filter impulse response function for perturbations around a carrier frequency of ω_inj, H₀ = |H(ω_inj)|, ϕ₀ = ∡H(ω_inj), and r_inj = b/a₀ is the injection ratio.

It is convenient to write Eqs. (32)−(34) in Fourier domain:

\begin{array}{l} Δ a (ω) = & {γ_{am} q_{s} (ω) - γ_{pm} q_{c} (ω)} [Δ a (ω) + n_{a} (ω)] e^{- j ω τ_{0}} \\ - a_{0} q_{c} (ω) e^{- j ω τ_{0}} [Δ φ (ω) + n_{φ} (ω) - e^{j ω τ_{0}} ω_{inj} δ τ (ω)] \\ + b n_{φ, inj} (ω) sin Δ φ_{0} + n_{a, inj} (ω) cos Δ φ_{0}, \end{array}

\begin{array}{l} a_{0} Δ φ (ω) = & {γ_{am} q_{c} (ω) + γ_{pm} q_{s} (ω)} e^{- j ω τ_{0}} [Δ a (ω) + n_{a} (ω)] \\ + a_{0} q_{s} (ω) e^{- j ω τ_{0}} [Δ φ (ω) + n_{φ} (ω) - e^{j ω τ_{0}} ω_{inj} δ τ (ω)] \\ + b n_{φ, inj} (ω) cos Δ φ_{0} - n_{a, inj} (ω) sin Δ φ_{0}, \end{array}

where

\begin{array}{l} q_{c} (ω) = (1 - r_{inj} cos Δ φ_{0}) h_{im} (ω, ω_{inj}) - r_{inj} sin Δ φ_{0} h_{r} (ω, ω_{inj}), \\ q_{s} (ω) = (1 - r_{inj} cos Δ φ_{0}) h_{r} (ω, ω_{inj}) + r_{inj} sin Δ φ_{0} h_{im} (ω, ω_{inj}) \end{array}

with

\begin{array}{l} h_{r} (ω, ω_{inj}) = \frac{1}{2 H_{0}} [H (ω_{inj} + ω) e^{- j ϕ_{0}} + H^{*} (ω_{inj} - ω) e^{j ϕ_{0}}], \\ h_{im} (ω, ω_{inj}) = \frac{- j}{2 H_{0}} [H (ω_{inj} + ω) e^{- j ϕ_{0}} - H^{*} (ω_{inj} - ω) e^{j ϕ_{0}}] . \end{array}

We note that the functions q_c(ω) and q_s(ω) depend on the injected signal frequency ω_inj due to the dependence of the filter response function on the frequency. To solve coupled Eqs. (35)−(36), we write them in a matrix form:

\begin{array}{l} M (ω) [\begin{matrix} Δ a (ω) \\ a_{0} Δ φ (ω) \end{matrix}] & = V (ω) {a_{0} ω_{inj} δ τ (ω) [\begin{matrix} 0 \\ 1 \end{matrix}] + e^{- j ω τ_{0}} [\begin{matrix} n_{a} (ω) \\ a_{0} n_{φ} (ω) \end{matrix}]} \\ + [\begin{matrix} cos Δ φ_{0} \\ - sin Δ φ_{0} \end{matrix}] n_{a, inj} (ω) + [\begin{matrix} sin Δ φ_{0} \\ cos Δ φ_{0} \end{matrix}] b n_{φ, inj} (ω), \end{array}

where

M (ω) = I - e^{- j ω τ_{0}} V (ω),

I is an identity matrix, and V

V (ω) = [\begin{matrix} γ_{am} q_{s} (ω) - γ_{pm} q_{c} (ω) & - q_{c} (ω) \\ γ_{am} q_{c} (ω) + γ_{pm} q_{s} (ω) & q_{s} (ω) \end{matrix}] .

The state vector [Δa(ω), a₀Δφ(ω)]^T gives the solution for the amplitude and the phase perturbation of the oscillator. The matrix M(ω) describes the response of the oscillator to a weak perturbation, which is caused by a delay perturbation and by the internal and external noise sources. A general stability analysis of the system is beyond the scope of the current manuscript and will be discussed elsewhere. However, we give in Appendix B a sufficient stability condition for the case of a deeply saturated amplifier.

If the system is stable, the matrix M(ω) must be invertible for each ω. The inverse of M(ω) is given by

M^{- 1} (ω) d (ω) = I - e^{- j ω τ_{0}} [\begin{matrix} q_{s} (ω) & q_{c} (ω) \\ - γ_{am} q_{c} (ω) - γ_{pm} q_{s} (ω) & γ_{am} q_{s} (ω) - γ_{pm} q_{c} (ω) \end{matrix}],

where d(ω) is the determinant of the matrix M(ω) which equals:

\begin{array}{l} d (ω) = & e^{- 2 j ω τ_{0}} γ_{am} [q_{s}^{2} (ω) + q_{c}^{2} (ω)] + 1 - (1 + γ_{am}) e^{- j ω τ_{0}} q_{s} (ω) \\ + e^{- j ω τ_{0}} γ_{pm} q_{c} (ω) . \end{array}

Equations (39), (42) and (43) give an explicit solution to the signal and noise for a given parameters γ_pm, γ_am and for a given injected frequency ω_inj. However, since the solution is cumbersome, we give below simple solutions, assuming that the filter bandwidth is significantly broader than the maximum signal frequency and the injection is weak (r_inj ≪ 1). To calculate the noise, we also assume that the amplifier is deeply saturated such that |γ_am| ≪ 1. These assumptions are fulfilled in our experimental setup and the approximated solution is in a good quantitative agreement with our experimental results given in section 4.

The system response to the delay perturbation δτ(ω) that is caused by the signal, is calculated by using Eq. (39) and Eq. (42):

Δ a_{sig} (ω) = \frac{a_{0} ω_{inj} δ τ (ω)}{d (ω)} q_{c} (ω),

a_{0} Δ φ_{sig} (ω) = \frac{a_{0} ω_{inj} δ τ (ω)}{d (ω)} [- q_{s} (ω) + e^{- j ω τ_{0}} γ_{am} (q_{c}^{2} (ω) + q_{s}^{2} (ω))],

where Δa_sig (ω) and Δφ_sig (ω) are the oscillator amplitude and phase perturbations due to the signal.

We now use the assumptions that the filter bandwidth is significantly broader than the maximum signal frequency and that the injection is weak (r_inj ≪ 1) and approximate the filter response around the oscillation frequency ω_inj:

\begin{array}{l} H (ω_{inj} + ω) \approx & [H_{0} + H_{0}^{'} ω + \frac{1}{2} H_{0}^{″} ω^{2} + O (ω^{3})] \\ \times exp [j ϕ_{0} + j ϕ_{0}^{'} ω + j \frac{1}{2} ϕ_{0}^{″} ω^{2} + O (ω^{3})], \end{array}

where H′₀ and H″₀ are the first and second order derivatives of the amplitude transfer function of the filter |H(ω)|, calculated at the frequency ω_inj, and ϕ′₀ and ϕ″₀ are the derivatives of the filter phase ∠H(ω), calculated at the frequency ω_inj. The first derivative of the phase, ϕ′₀, describes the group delay added by the filter, and it is added to the loop delay τ₀. Substitution of Eq. (46) without the term ϕ′₀ to Eq. (38) gives:

\begin{array}{l} h_{r} (ω, ω_{inj}) \approx 1 + O (ω^{2}), \\ h_{im} (ω, ω_{inj}) \approx O (| ω |) . \end{array}

Assuming that the frequency ω is very small in comparison with the filter bandwidth such that the change in the filter response can be approximated by using only first order terms in Eq. (47), Eq. (37) gives:

\begin{array}{l} q_{c} (ω) \approx - r_{inj} sin Δ φ_{0}, \\ q_{s} (ω) \approx 1 - r_{inj} cos Δ φ_{0} . \end{array}

In case that r_inj ≪ 1, also

| q_{c} (ω) | ≪ | q_{s} (ω) | .

In RF amplifiers, |γ_am| ≤ 1 and |γ_pm| ≲ 1 [26] and therefore, |γ_amq_c(ω)| ≪ |γ_pmq_s(ω)| and |γ_pmq_c(ω)| ≪ 1. Using these relations, Eqs. (44)−(45) give that

{| Δ a_{sig} (ω) |}^{2} / {| a_{0} Δ φ_{sig} (ω) |}^{2} \approx r_{inj}^{2} {sin}^{2} Δ φ_{0} ≪ 1

. Hence, for a weak injection, the amplitude variation, caused by changes in the delay, is much smaller than the phase variation.

To calculate the phase variation due to the signal, we also simplify Eq. (43) by using the above mentioned approximations:

\begin{array}{l} d (ω) & \approx e^{- 2 j ω τ_{0}} γ_{am} q_{s}^{2} (ω) + 1 - (1 + γ_{am}) e^{- j ω τ_{0}} q_{s} (ω) \\ = [1 - e^{- j ω τ_{0}} q_{s} (ω)] [1 - γ_{am} e^{- j ω τ_{0}} q_{s} (ω)] . \end{array}

By substituting Eq. (50) in Eq. (45) and using Eq. (49), we obtain that the change in the oscillator phase as a result of the delay perturbations equals:

Δ φ_{sig} (ω) = - ω_{inj} δ τ (ω) \frac{q_{s} (ω)}{1 - q_{s} (ω) exp (- j ω τ_{0})} .

To avoid instability, it is required that |γ_amq_s(ω)| < 1 for each ω in Eq. (50). Therefore, the amplifier should be sufficiently saturated such that |γ_am| < 1/(1 − r_inj cos Δφ₀).

To calculate the phase noise of the injection-locked oscillator, we assume that the injection ratio is small (r_inj ≪ 1) and that the bandwidth of the filter is significantly broader than the frequency offset of the noise we are interested in. These assumptions have also been used in our calculation of the signal. However, for the calculation of the noise, we add another assumption that the amplifier is deeply saturated and therefore |γ_am| ≪ 1, such that |q_s(ω)γ_am| ≪ 1. We also neglect the amplitude noise of the external source, [n_a,inj(t) ≪ bn_φ,inj(t)], since this noise is usually very small in oscillators due to the amplitude saturation of their amplifiers [19]. We use the above mentioned assumptions in the solution, given in Eq. (42) and Eq. (43), and obtain the phase fluctuations due to the internal noise source of the oscillator and due to the phase noise of the injected signal:

\begin{array}{l} Δ φ_{N} (ω) = & r_{inj} r_{φ, inj} (ω) \frac{cos Δ φ_{0} + γ_{pm} sin Δ φ_{0} e^{- j ω τ_{0}} q_{s} (ω)}{1 - q_{s} (ω) exp (- j ω τ_{0})} \\ + {\tilde{n}}_{φ} (ω) \frac{e^{- j ω τ_{0}} q_{s} (ω)}{1 - q_{s} (ω) exp (- j ω τ_{0})}, \end{array}

where

{\tilde{n}}_{φ} (ω) = n_{φ} (ω) + γ_{pm} e^{- j ω τ_{0}} n_{a} (ω) / a_{0}

is the effective internal phase noise source that accounts for amplitude-to-phase noise conversion in the amplifier.

The oscillator amplitude noise can be calculated by using Eq. (35):

Δ a_{N} (ω) / a_{0} \approx r_{inj} sin Δ φ_{0} [e^{- j ω τ_{0}} Δ φ_{N} (ω) + n_{φ} (ω) + n_{φ, inj} (ω)] + γ_{am} e^{- j ω τ_{0}} n_{a} (ω) .

Although the amplifier is deeply saturated, an amplitude noise is obtained at the entrance to the amplifier due to an effective phase-to-amplitude conversion that is obtained when the frequency of the external signal does not equal to the natural frequency of the oscillator. The third term in the equation is obtained since the amplifier is not completely saturated, such that 1 ≫ |γ_am| ≠ 0. However, for a week injection ratio r_inj ≪ 1 and deep saturation |γ_am| ≪ 1, this noise is significantly lower than the phase noise Δa_N(ω)/a₀ ≪ Δφ_N(ω).

Appendix B Sufficient condition for stability

In this section, we give a sufficient condition for the stability of the forced injection-locked oscillator. The small-gain theorem [38] gives a sufficient condition for a bounded output amplitude, assuming that the input perturbations are bounded in L₂-norm (BIBO stability). Hence, if the output signal that is given in Eq. (42) is sufficiently small and does not change the parameters of the linearized system (γ_am and γ_pm), the BIBO stability implies that the system is locally stable [39]. The transfer function matrix, which is given by Eq. (40), describes a feedback system with single pass transfer function matrix V and a delay τ₀. According to the small-gain theorem, it is sufficient to require that maximal singular value of V is less than one to obtain a BIBO stability for a perturbation in the L₂-norm [38]. This condition does not depend on the delay. The singular values of the matrix V can be calculated by finding the eigenvalues of the matrix V^†(ω)V(ω), where V^† is the conjugate transpose of the matrix V. This method can be straightforwardly used to obtain a sufficient stability condition for an arbitrary value of amplifier parameters γ_am and γ_pm and for an arbitrary injection frequency ω_inj with respect to transfer function of the oscillator filter.

To provide a simple stability condition, we assume that |γ_pm| ≪ 1 and |γ_am| ≪ 1 as obtained in our experimental setup described in section 4. In this case, the maximal singular value is equal to

σ^{2} (ω_{inj}) = M (ω_{inj}) (r_{inj}^{2} - 2 r_{inj} cos Δ φ_{0} + 1),

where

\begin{array}{l} M (ω_{inj}) = & max_{ω} ({| h_{r} (ω, ω_{inj}) |}^{2} + {| h_{im} (ω, ω_{inj}) |}^{2}) \\ = \frac{1}{2 H_{0}^{2}} max_{ω} [{| H (ω_{inj} + ω) |}^{2} + {| H (ω_{inj} - ω) |}^{2}] . \end{array}

Note that M(ω_inj) ≥ 1, since H₀ = |H(ω_inj)|. Therefore, to obtain σ²(ω_inj) < 1, the following conditions must be fulfilled:

r_{inj} \geq max {| sin (ω_{inj} τ_{0} - ω_{k} τ_{0}) |, 1 - \frac{1}{\sqrt{M (ω_{inj})}}},

and

cos Δ φ_{0} > \frac{1}{2 r_{inj}} [r_{inj}^{2} + 1 - \frac{1}{M (ω_{inj})}] .

In Eq. (57), we also added the requirement, given in Eq. (5), which gives the minimum injection ratio as a function of the frequency detuning. Since M(ω_inj) ≥ 1, it follows from Eq. (58) that cos Δφ₀ > 0. Therefore, the sufficient stability condition implies that |Δφ₀| < π/2, as we required in the solution of Eq. (6).

The threshold for the minimum injection ratio and the maximum phase lag given in Eqs. (57)−(58) also depend on the oscillation frequency ω_inj. Therefore, the maximum phase lag, |Δφ₀|, can be significantly lower than π/2 when the transfer function of the filter at ω_inj is significantly lower than its maximal value.

Funding

Israel Science Foundation (ISF) of the Israeli Academy of Sciences (1797/16).

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Phase measurement by using a forced delay-line oscillator and its application for an acoustic fiber sensor

Abstract

1. Introduction

2. Theoretical model for a forced injection-locked oscillator

2.1. Zero-order solution

2.2. Small-signal analysis of a forced injection-locked oscillator

2.2.1. Response of a forced injection-locked oscillator to a small change in its cavity delay

2.2.2. Noise in a forced injection-locked oscillator

3. Calculation of the beating signal at the mixer output

4. Experimental results and comparison to theory

5. Conclusion

Appendix A Perturbation analysis of a forced delay-line oscillator

Appendix B Sufficient condition for stability

Funding

References and links

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Figures (12)

Equations (58)

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