II. SYSTEM EQUATION OF OEO IN PRESENCE OF INTERFERENCE

Let the RF input to the modulating grid of the Mach–Zehnder modulator (MZM) is given by

$$v_{in} \lpar t\rpar = V\lpar t\rpar e^{j\lsqb \omega_1 t + \theta \lpar t\rpar \rsqb }\semicolon \;$$

the synchronizing signal to be S(t) = Ee ^{jω₁ t} and the interfering tone to be I(t) = yEe ^{jω _i t} , where “ω _i ” is the interfering frequency which is lying away from the OEO's free-running frequency, “y” is the fraction which indicates the strength of the interfering tone with respect to synchronizing signal, and “Δω = ω₁ − ω _i ” is the de-tuning frequency. If “G(s)” is the transfer function of the RF tuned circuit, then it can be expressed as [Reference Gonorovsky14] 1/G(s) = 1 + Q((s/ω₀) + (ω₀/s)), where “Q” is the quality factor of the tank circuit.

To analyze the behavior of the OEO in presence of the interfering signal shown in Fig. 2, we will assume that the OEO, instead of being exposed simultaneously to both the synchronizing signal and the interfering tone, is initially locked to the synchronizing signal and then it is influenced by the interfering tone which does not throw the system out of synchronism. In this condition, the instantaneous phase of the OEO will fluctuate around the static phase error “θ ₁” at a rate depending upon the difference of frequencies between the synchronizing signal and the interfering tone. Similarly, the amplitude of the forced oscillator will also vary around the steady state value at a rate depending upon the frequency difference between the two signals. The static phase error “θ ₁” gives a measure of the initial detuning of the OEO with respect to the synchronizing signal [Reference Ray, Pramanik, Banerjee and Biswas15].

Now, in order to find the system equation of the OEO in presence of interference we assume that despite the presence of strong non-linearity of MZM, there exists stationary amplitude of the microwave oscillation [Reference Biswas, Chatterjee and Pal12] of the form

$$v_{in} \lpar t\rpar =V\lpar t\rpar \exp \lpar j\omega t + \theta \lpar t\rpar \rpar.$$

The output power [Reference Zhou and Blasche7, Reference Chatterjee, Pal and Biswas17] of the MZM can be expressed as

$$P\lpar t\rpar = \displaystyle{1 \over 2} \alpha P_0 \left[1 - \eta \sin \left(\displaystyle{v_{in} \lpar t\rpar + V_B \over V_\pi} \right)\right]\semicolon \;$$

where “α” is the fraction of insertion loss of the modulator, “V _π ” is the half-wave voltage, “V _B ” is the bias voltage, “P ₀” is the input optical power, “v _in (t)” is the input RF voltage to the modulator and, “η” determines the extinction ratio of the modulator. Therefore, the output voltage of the photo detector when the output of the MZ modulator shines on it is

$$\eqalign{V_0 \left(t \right)&= \rho Z_{ph} P\left(t - \tau \right)\cr &= \rho \sqrt{R^2 + \displaystyle{1 \over \omega ^2 C^2}} P\left(t - \tau \right)e^{ - j\tan^{ - 1} \left(\displaystyle{1 \over \omega CR} \right)} \cr & \cong \rho RP\left(t - \tau \right)e^{\displaystyle{1 \over \,j\omega \tau_1}}\semicolon \; }$$

where “ρ” is the sensitivity, “Z _ph  = R − (j/ω C)” is the output impedance of the photo-detector and “τ₁ = RC” is the time-constant of photo-detector. Hence using the above arguments, it is not difficult to show that the output of the MZM [Reference Biswas, Chatterjee and Pal12] can be expressed as

$$\eqalign{V_0 \lpar t\rpar & =- 2\eta V_{ph} \cos \left(\displaystyle{\pi V_B \over V_\pi} \right)J_1 \left(\displaystyle{\pi V\lpar t - \tau \rpar \over V_\pi} \right)\cr &\quad \times \sin \left[\omega \lpar t - \tau \rpar \right]e^{\displaystyle{1 \over \,j\omega \tau_1}} \cr &= \displaystyle{N\lpar V\lpar t - \tau\rpar \rpar \over V} \exp \lpar \!-\! s\tau\rpar v_{in} \lpar t\rpar \exp \left(\displaystyle{1 \over s\tau_1} \right)}$$

where,

$$\eqalign{&N\lpar V\lpar t - \tau \rpar \rpar = - 2\eta V_{ph} \cos \left(\displaystyle{\pi V_B \over V_\pi} \right)J_1 \left(\displaystyle{\pi V\lpar t - \tau \rpar \over V_\pi} \right)\, {\rm and} \cr &V_{ph} = \displaystyle{\rho R\alpha P_0 \over 2}.}$$

Thus, the closed-loop equation of the OEO in presence of injection is given by

(1) $$\eqalign{&\left[\displaystyle{N\lpar V\lpar t - \tau \rpar \rpar \over V} e^{ - s\tau } e^{\displaystyle{1 \over s\tau _1}} v_{in} \lpar t\rpar + Ee^{j\omega_1 t} + yEe^{j\omega _i t} \right]= \displaystyle{v_{in} \lpar t\rpar \over G.G\lpar s\rpar }.}$$

(2) $$\eqalign{&\left[\displaystyle{N\lpar V\lpar t - \tau \rpar \rpar \over V}e^{ - s\tau } e^{\displaystyle{1 \over s\tau_1}} + \displaystyle{E \over V}e^{- j\theta \lpar t\rpar} + \; y \displaystyle{E \over V} e^{ - j\lsqb \Delta \omega t + \theta \lpar t\rpar \rsqb} \right] v_{in} \lpar t\rpar \cr &\quad = \displaystyle{v_{in} \lpar t\rpar \over G.G\lpar s\rpar }.}$$

Again, the complex frequency can be written as [Reference Biswas, Chatterjee and Pal12]

(3) $$\left. \matrix{s = j\omega = \displaystyle{1 \over v_{in} \lpar t\rpar } \displaystyle{dv_{in} \lpar t\rpar \over dt} = \displaystyle{1 \over V\lpar t\rpar } \displaystyle{dV\lpar t\rpar \over dt} + j\left(\omega_1 + \displaystyle{d\theta \over dt} \right)\hfill \cr \displaystyle{1 \over j\omega} \cong \displaystyle{1 \over j\omega_1} + \displaystyle{1 \over \omega_1^2} \left(\displaystyle{1 \over V\lpar t\rpar } \displaystyle{dV\lpar t\rpar \over dt} + j\displaystyle{d\theta \over dt} \right)\hfill} \right\}.$$

Rewriting (2) with the help of (3)

(4) $$\eqalign{& N\lpar V\lpar t - \tau \rpar \rpar e^{ - j\omega _1 \tau} e^{\displaystyle{1 \over \,j\omega_1 \tau_1}} + Ee^{ - j\theta \lpar t\rpar } + yEe^{ - j\lsqb \Delta \omega t + \theta \lpar t\rpar \rsqb } \cr & = \left(\displaystyle{V \over G} \right) \left[1 + \displaystyle{Q \over V} \left(\displaystyle{1 \over \omega_0} + \displaystyle{\omega_0 \over \omega_1^2} \right) \displaystyle{dV \over dt} + \;jQ\left\{\left(\displaystyle{\omega_1 \over \omega_0} - \displaystyle{\omega_0 \over \omega_1} \right) \right. \right. \cr &\left.\left. + \left(\displaystyle{1 \over \omega_0} + \displaystyle{\omega_0 \over \omega_1^2} \right) \displaystyle{d\theta \over dt} \right\}\right].}$$

Under the assumption that V _B  = V _π, η = 1, πV _ph  = V _π and V _π = π [Reference Biswas, Chatterjee and Pal12], we have N[V(t − τ)] = 2J ₁ [V(t − τ)] and when ω₁ ≈ ω₀, the real part of equation (4) gives the amplitude equation as

(5) $$\eqalign{\displaystyle{dV \over dt} &= \left(\displaystyle{\omega_0 \over 2Q} \right) \left\{2GJ_1 \lsqb V\lpar t - \tau \rpar \rsqb \left\{\cos \lpar \omega_0 \tau \rpar + \displaystyle{\sin \lpar \omega_0 \tau \rpar \over \omega_0 \tau_1} \right\} - V \right\} \cr &\qquad + \left(\displaystyle{\omega_0 \over 2Q} \right) GE \lcub \cos \lpar \theta \rpar + y \cos \lpar \Delta \omega t + \theta \rpar \rcub.}$$

Similarly equating the imaginary part, the phase equation can be obtained as

(6) $$\eqalign{&2J_1 \lsqb V\lpar t - \tau \rpar \rsqb \left\{\sin \lpar \omega_1 \tau \rpar - \displaystyle{\cos \lpar \omega_1 \tau \rpar \over \omega_1 \tau_1} \right\} + E\lcub \sin \lpar \theta \lpar t\rpar \rpar \cr &\quad +y \sin \lpar \Delta \omega t+\theta \lpar t\rpar \rpar \rcub =- \left\{\displaystyle{QV \over G} \left[\left(\displaystyle{\omega_1 \over \omega_0} - \displaystyle{\omega_0 \over \omega_1} \right) \right. \right. \cr &\left. \left. \quad+ \left(\displaystyle{1 \over \omega_0} + \displaystyle{\omega_0 \over \omega_1^2} \right) \displaystyle{d\theta \over dt} \right] \right\}.}$$

Again for ω₁ ≈ ω₀, the phase balance equation can be rewritten as

(7) $$\eqalign{& \displaystyle{d\theta \over dt} = \lpar \omega_0 - \omega_1\rpar - \left(\displaystyle{\omega_0 \over 2Q} \right)\left[\displaystyle{2GJ_1 \lsqb V\lpar t - \tau \rpar \rsqb \over V} \left\{\sin \lpar \omega _0 \tau \rpar \right. \right. \cr &\left. \left. \quad - \displaystyle{\cos \lpar \omega _0 \tau\rpar \over \omega_0 \tau_1} \right\} \right] - \left(\displaystyle{\omega_0 \over 2Q} \right)\left[\left(\displaystyle{GE \over V} \right)\lcub \sin \lpar \theta \lpar t\rpar \rpar \right. \cr &\left. \quad+ y\sin \lpar \Delta \omega t + \theta \lpar t\rpar \rpar \rcub \right].}$$

The photo-detector impedance “R” is replaced by “R − (j/ωC)” but it has no effect in our chosen frequency range. The effect of incorporating the photo-detector impedance in the amplitude and phase equations (5) and (7) has not been reported by other authors also.

III. WEAK INTERFERENCE LYING AWAY FROM THE OEO'S FREE-RUNNING FREQUENCY

Let us assume the solution for (6) as

(8) $$\theta = \theta_1 + m\sin \lpar \Delta \omega t + \alpha\rpar \comma \;$$

where “θ ₁” is the steady-state phase error, and the modulation index is small for low-level interference. Using (8) in (7), we get

(9) $$\eqalign{& m\Delta \omega \cos \lpar \Delta \omega t + \alpha \rpar \cr &\quad = \lpar \omega_0 - \omega_1 \rpar - \left(\displaystyle{\omega_0 \over 2Q} \right) \left[\displaystyle{2GJ_1 [V\lpar t - \tau \rpar] \over V} \right. \cr &\left. \quad \left\{\sin \lpar \omega_0 \tau\rpar - \displaystyle{\cos \lpar \omega_0 \tau \rpar \over \omega_0 \tau_1} \right\} \right] - \left(\displaystyle{\omega_0 \over 2Q} \right) \left(\displaystyle{GE \over V} \right)\cr &\quad \times \left[\matrix{J_0 \lpar m\rpar \sin \theta _1 + 2J_1 \lpar m\rpar \cos \theta _1 \sin \lpar \Delta \omega t + \alpha \rpar \hfill \cr + \;yJ_0 \lpar m\rpar [\sin \theta_1 \cos \lpar \Delta \omega t\rpar + \cos \theta_1 \sin \lpar \Delta \omega t\rpar ] \hfill \cr + \;yJ_1 \lpar m\rpar [\cos \theta_1 \sin \alpha - \cos \alpha \sin \theta_1 ]\hfill} \right].}$$

Using harmonic balance method [Reference Ray, Pramanik, Banerjee and Biswas15], we get

(10) $$\eqalign{&2Q\left(\displaystyle{\omega_1 - \omega _0 \over \omega _0} \right)= \Omega \cr &\quad = - \left[\displaystyle{2GJ_1 [V\lpar t - \tau\rpar] \over V} \left\{\sin \lpar \omega_0 \tau \rpar - \displaystyle{\cos \lpar \omega_0 \tau\rpar \over \omega_0 \tau_1} \right\}\right]\cr & \quad - \left(\displaystyle{GE \over V} \right)\left[J_0 \left(m \right)\sin \theta_1 + yJ_1 \left(m \right)\sin \lpar \alpha - \theta_1\rpar \right].}$$

Similarly, denoting $p = \left(\displaystyle{\omega_0 \over 2Q} \right)\left(\displaystyle{GE \over V} \right).$

$$\displaystyle{m\Delta \omega \over p} \cos \alpha = - \left[2J_1 \lpar m\rpar \cos \theta_1 \sin \alpha + yJ_0 \lpar m\rpar \sin \theta_1 \right]\comma \;$$

and

(11) $$\displaystyle{m\Delta \omega \over p}\sin \alpha = 2J_1 \lpar m\rpar \cos \theta_1 \cos \alpha + yJ_0 \lpar m\rpar \cos \theta_1.$$

Hence, it is not difficult to show using (11) that

(12) $$\sin \lpar \alpha - \theta _1 \rpar = \displaystyle{m\left(\Delta \omega \bigg{/}p \right) \over yJ_0 \lpar m\rpar}.$$

Substituting the value of (12) in (10)

(13) $$\eqalign{\Omega &=- \left[\displaystyle{2GJ_1 [V\lpar t - \tau \rpar] \over V} \left\{\sin \lpar \omega_0 \tau \rpar - \displaystyle{\cos \lpar \omega_0 \tau \rpar \over \omega_0 \tau_1} \right\} \right]\cr &\quad - \left(\displaystyle{GE \over V} \right) \left[J_0 \lpar m\rpar \sin \theta_1 + \left(\displaystyle{m\Delta \omega \over p} \right) \displaystyle{J_1 \lpar m\rpar \over J_0 \lpar m\rpar} \right].}$$

Again, using (11), it is not difficult to show that

(14) $$\eqalign{\Omega &= - \left[\displaystyle{2GJ_1 [V\lpar t - \tau \rpar] \over V} \left\{\sin \lpar \omega_0 \tau \rpar - \displaystyle{\cos \lpar \omega_0 \tau \rpar \over \omega_0 \tau_1} \right\}\right]- \left(\displaystyle{GE \over V} \right)\cr & \quad \times \left[J_0 \lpar m\rpar \sin \theta_1 + \displaystyle{m\left(\Delta \omega \bigg{/}p \right)y^2 J_0 \lpar m\rpar J_1 \lpar m\rpar \over m^2 \left(\Delta \omega \bigg{/}p \right)^2 + \;4J_1 ^2 \lpar m\rpar \cos ^2 \theta_1} \right].}$$

Now, for low-level interference, “m” is small and J ₀(m) ≅ 1; J ₁ (m) ≅ m/2. Maximum value of “θ ₁” is ±(π/2). Hence from (14)

(15) $$\eqalign{\Omega & =- \left[\displaystyle{2GJ_1 [V (t - \tau)] \over V} \left\{\sin \lpar \omega _0 \tau \rpar - \displaystyle{\cos \lpar \omega _0 \tau \rpar \over \omega _0 \tau_1} \right\}\right]\cr &\quad - \left(\displaystyle{GE \over V} \right)\left[\sin \theta_1 + \left(\displaystyle{y^2 \over 2} \right)\displaystyle{\left(\Delta \omega \bigg{/}p \right)\over \left(\Delta \omega \bigg{/} p \right)^2 + \cos ^2 \theta_1} \right]\cr & =- \left[\displaystyle{2GJ_1 [V (t - \tau)] \over V} \left\{\sin \lpar \omega_0 \tau\rpar - \displaystyle{\cos \lpar \omega _0 \tau\rpar \over \omega_0 \tau_1} \right\}\right]\cr &\quad - \left(\displaystyle{GE \over V} \right)\left[\pm 1 + \displaystyle{y^2\bigg{/}2 \over \left(\Delta \omega \bigg{/}p \right)} \right].}$$

Now, $\displaystyle{y^2 \over 2} \displaystyle{p \over \Delta \omega}$ can be written as $\displaystyle{y^2 \over 2} \displaystyle{p \over \Delta \omega} = \Delta \omega - \sqrt{\lpar \Delta \omega \rpar ^2 - y^2 p}$ . Thus, the lock-range of the OEO in presence of the interfering tone is given by

(16) $$\eqalign{\Omega &= - \left[\displaystyle{2GJ_1 [V (t - \tau)] \over V} \left\{\sin \lpar \omega _0 \tau \rpar - \displaystyle{\cos \lpar \omega _0 \tau\rpar \over \omega_0 \tau _1} \right\}\right]\cr &\quad - \left(\displaystyle{GE \over V} \right)\left[1 + \left(\Delta \omega - \sqrt{\lpar \Delta \omega \rpar ^2 - y^2 p} \right)\right].}$$

In deriving the steady-state value for the lock range given by equation (16), the positive sign of (15) has been considered. Moreover, since “V(t) and θ(t)” are slowly varying functions of time [Reference Gonorovsky14], i.e. $\displaystyle{1 \over \omega_0} \left(\displaystyle{d\theta \over dt} \right)\ll 1$ and $\displaystyle{1 \over V}\left(\displaystyle{dV \over dt} \right)\ll 1$ ,

$$V\lpar t - \tau \rpar \approx V\lpar t\rpar - \tau \displaystyle{dV \over dt} \approx V\lpar t\rpar \left[1 - \displaystyle{\tau \over V} \displaystyle{dV \over dt} \right]\approx V\lpar t\rpar \comma \;$$

and thus in the steady state $J_1 [V (t - \tau)] \approx J_1 [V (t)] \simeq \displaystyle{V\lpar t\rpar \over 2}$ .

From (15), it is clear that in absence of interference

(17) $$\left. \Omega \right\vert_{Lower} = - \left[\displaystyle{2GJ_1 [V (t - \tau)] \over V} \sin \lpar \omega _0 \tau \rpar \right] - \left. \left(\displaystyle{GE \over V} \right) \right\vert_{\theta _1 = + {\pi \over 2}}.$$

(18) $$\left. \Omega \right\vert _{Upper}=- \left[\displaystyle{2GJ_1 \lsqb V\lpar t - \tau \rpar \rsqb \over V} \sin \lpar \omega_0 \tau \rpar \right] + \left. \left(\displaystyle{GE \over V} \right) \right\vert_{\theta_1 = - {\pi \over 2}}.$$

Hence Ω _Lower  > Ω _Upper , which is also confirmed from the experimental findings given in Table 1 and Fig. 8. This asymmetric nature of the locking range of an OEO has not been reported so far as far as the knowledge of the authors goes.

Fig. 3. Locking range of injection-synchronized OEO with frequency detuning.

Table 1. Locking range with fiber delay.

Laser frequency = 500 Mrad/s; free-running frequency = 12.002 MHz; RF gain = 3; synchronizing signal strength = 0.4 V; interfering signal strength = 0 V.

IV. RESPONSE OF OEO TO NOISY SIGNAL

Thermal noise is the unavoidable form of interference that affects the behavior of the closed-loop system shown in Fig. 4. The power spectral density of the thermal noise is flat, frequency independent and hence the name white noise. It is expedient to write the signal contaminated with additive white Gaussian noise while passing through the narrow-band tuned circuit in absence of interfering tone as

$$\eqalign{&v_n \left(t \right)= \sqrt 2\, V_C \left(t \right)\cos \left[{\omega _1 t+\theta \left(t \right)} \right]+\sqrt 2\, V_S \left(t \right)\sin \left[{\omega _1 t+\theta \left(t \right)} \right]\cr &\quad\quad =\sqrt {2\left[{V_C ^2 \left(t \right)+V_S ^2 \left(t \right)} \right]} e^{\,j\left[{\omega _1 t+\theta \left(t \right)- \theta _n \left(t \right)} \right]}\comma \; }$$

where “ $V_C \lpar t\rpar $ ” and “ $V_S \lpar t\rpar $ ” are the narrowband independent Gaussian variables with one-sided power spectral density “N ₀”. The closed loop equation in this case will be

(19) $$\eqalign{&\left[\displaystyle{N\lpar V\lpar t - \tau \rpar \rpar \over V\left(t \right)} e^{ - s\tau} v_{in} \lpar t\rpar + Ee^{j\lsqb \omega _1 t + \psi \lpar t\rpar \rsqb } \right. \cr &\quad \left. +\sqrt{2\left[V_C ^2 \lpar t\rpar +V_S ^2 \lpar t\rpar \right]} e^{j\lsqb \omega_1 t + \theta \lpar t\rpar - \theta _n \lpar t\rpar \rsqb } \right]= \displaystyle{v_{in} \lpar t\rpar \over G.G\lpar s\rpar }. \cr &\displaystyle{N\lsqb V\lpar t - \tau \rpar \rsqb \over V\lpar t\rpar } e^{ - s\tau } v_{in} \lpar t\rpar + \displaystyle{E \over V\lpar t\rpar } e^{j\phi \lpar t\rpar } v_{in} \lpar t\rpar \cr &\quad + \sqrt{2} \displaystyle{V_n \lpar t\rpar \over V\lpar t\rpar } e^{ - j\theta _n \lpar t\rpar } v_{in} \lpar t\rpar = \displaystyle{v_{in} \lpar t\rpar \over G.G\lpar s\rpar }.}$$

Following similar analysis, the amplitude and phase equations are given by

(20) $$\eqalign{&\displaystyle{dV \over dt} = \left(\displaystyle{\omega_0 \over 2Q} \right) \left\{2GJ_1 \lsqb V\lpar t - \tau \rpar \rsqb \cos \lpar \omega_0 \tau \rpar - V\lpar t\rpar \right\} + \left(\displaystyle{\omega_0 \over 2Q} \right)\cr &\qquad \times \left\{\displaystyle{GE \over V\lpar t\rpar} \cos \lsqb \phi \lpar t\rpar \rsqb + \sqrt{2} G\displaystyle{V_n \lpar t\rpar \over V\lpar t\rpar} \cos \lsqb \theta _n \lpar t\rpar \rsqb \right\}\comma \;}$$

and

(21) $$\eqalign{& \displaystyle{d\phi \over dt} = \displaystyle{d\psi \over dt} - \displaystyle{d\theta \over dt} \cr &=\left[\Delta \omega + \omega _m K_p \cos \lpar \omega _m t\rpar \right]+ \left(\displaystyle{\omega_0 \over 2Q} \right)\cr &\quad \times \left\{2G\displaystyle{J_1 \lsqb V\lpar t - \tau \rpar \rsqb \over V\lpar t\rpar } \sin \lpar \omega _0 \tau \rpar \right\} - \left(\displaystyle{\omega_0 \over 2Q} \right)\displaystyle{GE \over V\lpar t\rpar } \sin \lsqb \phi \lpar t\rpar \rsqb \cr & \quad+ \left(\displaystyle{\omega_0 \over 2Q} \right)\sqrt{2} G\displaystyle{V_n \lpar t\rpar \over V\lpar t\rpar } \sin \lsqb \theta _n \lpar t\rpar \rsqb.}$$

where $V_n \lpar t\rpar =\sqrt {V_C ^2 \lpar t\rpar +V_S ^2 \lpar t\rpar } $ and $\theta _n \lpar t\rpar =\tan ^{ - 1} \left[{\displaystyle{{V_S \lpar t\rpar \over V_C \lpar t\rpar }}} \right]$ . Choosing $\beta=\left({\displaystyle{{\omega _0 / 2QV_0 }}} \right)$ , we start with the assumption that the oscillator is in the locked state under the influence of the signal. Assuming an initial detuning, it is not hard to conjecture that the instantaneous phase error will consist of three terms, viz., (1) a dc component due to initial detuning, (2) a component at the modulating frequency, and (3) random fluctuations due to noise. As a result, the solution of (21) may be assumed to be of the form

(22) $$\phi=\phi _0+M\sin \lpar \omega _m t+\delta \rpar +\phi _n \lpar t\rpar \comma \;$$

where “ϕ _n (t)” is a zero mean random variable due to the incoming noise and it is to be remembered that the random variable “ϕ _n ” has a variance “σ_ϕ Reference Yao and Maleki ² ”. It is to be noted also that “ϕ _n ” is not truly Gaussian but if the input carrier-to-noise ratio is not very small, the probability density function of “ϕ _n ” approximates closely to that of a Gaussian variable, i.e. $p\lpar \phi _n \rpar = \displaystyle{1 \over \sigma_\phi \sqrt{2\pi}} e^{ - \left({\phi_n^2 \over 2\sigma_\phi^2} \right)}$ . It is not difficult to show using (21), (22) and statistical linearization techniques

(23) $$\eqalign{& \displaystyle{d\phi _0 \over dt} + M\omega_m \cos \lpar \omega _m t + \delta \rpar +\displaystyle{d\phi _n \over dt} \cr & = \left[\Delta \omega + \omega _m K_p \cos \omega _m t \right]+ \left(\displaystyle{\omega_0 \over 2Q} \right)GC\sin \lpar \omega_0 \tau\rpar + \beta GN_2 \lpar t\rpar \cr & \quad - \beta GE\left[\left\{e^{ - ^{{\sigma_\phi^2 \over 2}}} \sin \phi _0+\phi _n e^{ - ^{{\sigma_\phi^2 \over 2}}} \cos \phi_0 \right\}J_0 \lpar M\rpar \right. \cr &\quad \left. +2J_1 \lpar M\rpar \sin \lpar \omega _m t+\delta \rpar \left\{e^{ - ^{{\sigma_\phi^2 \over 2}}} \cos \phi_0 + \phi _n e^{ - ^{{\sigma_\phi ^2 \over 2}}} \sin \phi_0 \right\}\right]\comma \; }$$

where, $N_2 \lpar t\rpar =\sqrt 2\, V_n \lpar t\rpar \sin \lsqb \theta _n \lpar t\rpar \rsqb $ and $\left. \phi_0 \right]_{\max } = \pm \left(\displaystyle{\pi \over 2} - M - \sigma _\phi \right)$ ; “±” sign refers to the upper or lower side locking ranges for the OEO. Hence, the variance of the phase error is given by

(24) $$\eqalign{\sigma_\phi ^2 &= \lpar \beta G\rpar ^2 \displaystyle{N_0 \over 2} \vint_{ - \infty}^{\infty} \left\vert \displaystyle{1 \over s + \beta GEJ_0 \lpar M\rpar e^{{ - \sigma_\phi^2 \over 2}} \cos \lpar \phi _0\rpar } \right\vert^2 ds \cr &= \displaystyle{\pi \beta GN_0 \over 2E} \times \displaystyle{1 \over J_0 \lpar M\rpar e^{{ - \sigma_\phi^2 \over 2}} \sin \lpar M - \sigma _\phi \rpar }\comma \;}$$

where, “N ₀ = kT”, “k” is the Boltzmann's constant and “T” is the noise temperature.

Fig. 4. Experimental set-up for the study of locking range.

V. RESULTS AND DISCUSSIONS

Theoretical variation of lock range with frequency detuning of the OEO has been shown in Fig. 6, using MATHCAD 14.0 software. It clearly indicates that with frequency detuning (Δω), the lock range increases. Experimental results which are given in Table 2 and Fig. 6 agree well with that obtained theoretically. However, for our experimental set-up, we have chosen the fiber delay to be 10 µs on the RF signal carried by the optical signal. This corresponds to a 100 kHz mode spacing such that the next optical mode is situated at 12.102 MHz, shown in Fig. 7. It is to be noted that the free-running frequency of the OEO is 12.002 MHz. Hence, as the interference is swept away from the center frequency of the OEO, frequency pulling affect by the neighboring side band is observed, which will eventually decrease the lock range. Thus it is expedient to keep the frequency detuning low, hence avoiding the frequency pulling by the neighboring side bands of the OEO. In Fig. 5, the variation of locking range with injection amplitude has been studied (Table 3). The graph shows the variation of locking range with and without interfering signal. Asymmetry in locking range is noticed from Table 1 and Fig. 8. This interesting observation is also evident from equations (17) and (18). Finally, we have shown the effect of Gaussian noise on the system performance and have given a method to calculate the variance of the phase error.

Fig. 5. Experimental variation of locking range with injection amplitude.

Fig. 6. Experimental variation of locking range with frequency detuning, “o – Experimental data”, “solid line – Data fit”.

Fig. 7. Locking range with fiber delay.

Fig. 8. Experimental variation of locking range with injection amplitude.

Table 2. Locking-range of injection synchronized OEO with interference.

Laser frequency: 500 Mrad/s, delay: 10.00 µs, interference amplitude = 0.09 V; free running frequency: 12.002 MHz; RF amplifier gain = 3.

Table 3. Locking-range variation with injection amplitude in OEO.

Laser frequency: 500 Mrad/s, delay: 10.00 µs, interference amplitude = 0.09 V; free running frequency: 12.002 MHz; RF amplifier gain = 3.