Improved analytical model of phase-locking dynamics in unlocked-driven optoelectronic oscillators under RF injection locking

Optics Communications 437 (2019) 184–192

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Improved analytical model of phase-locking dynamics in unlocked-driven optoelectronic oscillators under RF injection locking Abhijit Banerjee a ,∗, Jayjeet Sarkar b a b

Department of Electronics and Communication Engineering, Academy of Technology, Adisaptagram, Hooghly 712121, West Bengal, India Department of Applied Electronics and Instrumentation Engineering, Academy of Technology, Adisaptagram, Hooghly 712121, West Bengal, India

ARTICLE

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Keywords: Phase-locking dynamics Injection locking Beat frequency Lock-range and spectral components

ABSTRACT In this paper, we propose an improved closed form large injection perturbation analytical model for accurately studying the phase-locking dynamics and frequency pulling phenomenon in optoelectronic oscillator (OEO) for radio frequency (RF) signal injection. We provide formulas for the lock-range, and the beat frequency, which are general and accurate in estimating the degree of phase perturbation in unlocked-driven OEO under weak as well as strong injection signal level. We also give closed form expressions for the spectral components of the unlocked-driven oscillator. It is shown that our model is capable of predicting the phase-locking and estimating the frequency pulling of RF driven OEO under weak and strong injection signal level. The accuracy of the proposed model is verified by the simulations.

1. Introduction An OEO is a crucial component of modern microwave photonics systems to provide a spectrally pure and stable microwave reference signals [1–3]. OEOs are useful as well for optical communications [4– 7], photonic neuromorphic computing [8,9], sensing [10], measurement [11] and detection [12]. On the last two decades several configurations of OEO such as multiple loops [13–15] and coupled loops [16,17] have been developed to improve the phase noise performance of the oscillator. Also, OEOs with new topologies such as injection-locked (IL) OEO [18,19], dual ILOEO [20–22], phase-lock loop (PLL) locked OEO [23], IL-PLL locked OEO [24] and transposed frequency OEO [4] were proposed by several researchers to improve long term frequency stability and reduce phase noise at low-offset frequencies. Injectionpulling on the other hand, typically proves undesirable. The power amplifier output in a RF transceiver contains large spectral components in the vicinity of the oscillator free-running frequency, leaking through the package and substrate to the oscillator and causing frequency pulling. However, one should have a deep understanding of the injection-locking phenomena to exploit its merit in intelligent potential applications and reduce its effects whenever undesirable. Surprisingly, it is the unlockeddriven state rather than the locked state of the oscillator that will give us the clear picture of the locking dynamics within the oscillator. Several papers [25–27] have been written on the injection-locking dynamics of RF oscillators. It appears that few attempts have been made till now to analyze the injection-locking dynamics of an autonomous OEO to an injection RF signal. In [28], from the Arnold tongues theory, the author

has focused on resonance frequencies on the phase-locking process. The stability properties and the fixed points of the RF driven OEO are determined. This study does not describe the frequency-pulling effects on an OEO under RF signal injection. A. Banerjee gave an analytical model to explain the precise detail of the chain of events of injection locking procedure for independent sinusoidal RF injection signal [29, 30]. That model is incapable of describing the phase-locking dynamics under strong injection level. Based on [31], Banerjee presented a large injection perturbation analytical model to predict the influence of amplitude perturbation on the frequency pulling in RF driven single-loop microwave OEO. This model cannot accurately estimate the degree of phase perturbation and the spectral components of the unlocked driven OEO under strong injection level. Sarkar et al. [32] have investigated the frequency pulling phenomenon of an unlocked-driven OEO due to weak RF signal injection considering the phase perturbation only. These deficiencies have motivated us to develop a more improved analytical model of phase dynamics of unlocked driven OEO under RF injection level. This paper provides an improved closed form large injection analytical model for accurately predicting the phase locking dynamics and the frequency pulling phenomenon of unlocked driven optoelectronic microwave oscillator under weak as well as strong injection condition. This analytical modeling has potential application to SONET optical network [33] and mutual injection pulling between two OEOs in broadband trans-receiver. The rest of this paper is organized as follows. In Section 2 some prior works on small injection analytical model are introduced which are necessary for presenting the main results of this paper.

∗ Corresponding author. E-mail addresses: [email protected] (A. Banerjee), [email protected] (J. Sarkar).

https://doi.org/10.1016/j.optcom.2018.12.044 Received 25 August 2018; Received in revised form 7 December 2018; Accepted 10 December 2018 Available online 21 December 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.

A. Banerjee and J. Sarkar

Optics Communications 437 (2019) 184–192

𝑘 fixed. This means that the output carrier frequency of the oscillator is ‘pulled’ towards that of the injection signal. At some definite value of 𝛺, the beat frequency becomes zero, indicating that output carrier frequency of the oscillator is pulled up to the injection signal frequency. We can say that the phase synchronization has been established between the local oscillation and the forcing signal. The forced OEO output signal was derived as [29] ( ) 𝑣𝑂 (𝑡) = 𝑟𝑉𝑂𝜋 sin 𝜔1 𝑡 + 𝜙𝑐 [∝ ] √ (7) ∑ {( ) } + 2𝑉𝑂𝜋 𝑋 2 − 1 (−1)𝑛+1 𝑟𝑛 sin 𝜔1 − 𝑛𝜔𝑏 𝑡 + 𝜙𝑛

Section 3 focuses on improved large injection perturbation analytical model to analyze the phase-locking dynamics. Simulation results are given in Section 4, while Section 5 represents some conclusions. 2. Background (Small injection analytical model) In this section, we describe the small injection perturbation analytical model of unlocked-driven single-loop optoelectronic microwave oscillators proposed in [29], is repeated here to familiarize with the prior development of the phase-locking phenomenon in unlocked{ ( )} driven OEO. Assume the injection RF signal 𝑣𝑠 (𝑡) = 𝑉𝑆 exp 𝑗 𝜔1 𝑡 , { ( )} free-running OEO output signal 𝑣𝑂 (𝑡) = 𝑉𝑂 (𝑡) exp 𝑗 𝜔𝑂 𝑡 , and the { ( )} perturbed output of the OEO 𝑣𝑂 (𝑡) = 𝑉𝑂 (𝑡) exp 𝑗 𝜔1 𝑡 − 𝜙 (𝑡) , where 𝜔1 and 𝑉𝑆 are the frequency and fixed amplitude of the injection signal, respectively. 𝜔𝑂 is the free-running frequency of the OEO and 𝑉𝑂 is the amplitude of the output signal. 𝜙 (t) denotes the instantaneous phase difference between the OEO and the RF injection signal. The RF filter is a first-order linear band-pass filter (BPF) and its transfer function [34] 1 𝐻 (𝑠) = 1+𝑄 𝜔 ∕𝑠+𝑠∕𝜔 , where 𝑠 = 𝑗𝜔 and Q is the Quality factor of the ( 𝑂 𝑂) filter. Assuming the bandwidths of the BPF and RF amplifier are much smaller than the microwave frequency 𝜔𝑂 and neglecting higher-order harmonic terms, the amplitude and phase governing equations of the unlocked-driven OEO were derived as [29] [ ( ) ] 𝑑𝑉𝑂𝜋 (𝑡) 2 = 𝜇 𝛾 1 − 0.125𝑉𝑂𝜋 (𝑡) cos (𝛺𝜏) − 1 𝑉𝑂𝜋 (𝑡) 𝑑𝑡 (1) + 𝜇𝛾𝑉𝑆𝜋 cos (𝛺𝜏 + 𝜑 (𝑡)) ( ) 𝜇𝛾𝑉𝑆𝜋 𝑑𝜑 (𝑡) 2 = 𝛺 + 𝜇𝛾 1 − 0.125𝑉𝑂𝜋 sin (𝛺𝜏 + 𝜑 (𝑡)) (𝑡) sin (𝛺𝜏) + 𝑑𝑡 𝑉𝑂𝜋 (𝑡)

𝑛=1

√ ) ( where√𝜙𝑛 = 𝜙𝑐 + 𝑛 𝜔𝑏 𝑡𝑂 − 𝛼𝑂 , 𝜙𝑐 = 𝛺𝜏, 𝑟 = 𝑋 − 𝑋 2 − 1 and 𝛼𝑂 = tan−1 𝑋 2 − 1. In (7), the first term represents the RF injection signal with frequency 𝜔1 and the second term represents a large number of spectral components separated by beat frequency 𝜔𝑏 . 3. Improved large injection perturbation analytical model In this section, we will give the improved analytical model of a single-loop OEO under the influence of RF signal injection and analyze the phase locking phenomena based on the model. We first derive the locking equation for an OEO under injection. Fig. 2 shows a simplified OEO model, which consists of a non-linear element, a microwave filter, an optical fiber, a photo detector and a feedback path. The RF signal injection is modeled as an additive input 𝑉𝑆 which combines with the feedback signal 𝑉𝑂 to produce the signal 𝑉𝑋 . In this figure, 𝑉𝑆 represents the injection signal with constant normalized amplitude 𝑉𝑆𝜋 and frequency 𝜔1 . The output signal 𝑉𝑂 has a constant normalized amplitude 𝑉𝑂𝜋 and 𝜙 is the phase difference between the injection signal and the output signal. 𝜃 is the phase shift introduced by the filter circuit at any frequency 𝜔 and is given by [34] ( ( )) 2𝑄 𝜔𝑂 − 𝜔 𝜃 = tan−1 (8) 𝜔𝑂

(2)

where 𝜇 = 𝜔𝑂 ∕2𝑄 is the half-bandwidth of the BPF. 𝛾 is the effective ( ) loop-gain and 𝑉𝑝ℎ is the photo-detector voltage [29]. 𝑉𝑆𝜋 = 𝜋𝑉𝑆 ∕𝑉𝜋1 ( ) and 𝑉𝑂𝜋 = 𝜋𝑉𝑂 ∕𝑉𝜋1 express normalized amplitude of the injection ( ) signal and output signal, respectively. 𝛺 = 𝜔1 − 𝜔𝑂 denotes the frequency detuning between the free-running oscillation and the injection signal. 𝑉𝜋1 is modulator half-wave voltage of the driven oscillator. (See Fig. 1.) It had been assumed that 𝑉𝑆𝜋 is small compared to 𝑉𝑂𝜋 and 𝛺 is positive. In such a case, the injection signal will only perturb the phase of the oscillator √ which will oscillate with the unperturbed steady amplitude 𝑉𝑂𝑆𝜋 = 2 2 (𝛾 − 1) ∕𝛾, which can be obtained by putting 𝑉𝑆𝜋 = 0 and 𝑑𝑉𝑂𝜋 (𝑡) = 0 in (1). Thus, phase dynamics equation (2) will become 𝑑𝑡

and 𝛺𝜏 is the phase shift due to the time delay 𝜏 induced by the physical length and dispersion of the optical fiber. When the OEO oscillates with the free running frequency 𝜔𝑂 , the output signal 𝑉𝑂 must be in phase with the combined signal 𝑉𝑋 . When the OEO is driven by a RF injection signal of slightly different frequency 𝜔1 , the filter and the optical fiber introduces the necessary phase shift (𝛺𝜏 + 𝜃) to maintain oscillation at the frequency of the RF injection signal. From Fig. 2, the combined signal 𝑉𝑋 is given by ( ) ( ) 𝑉𝑋 = 𝑉𝑆𝜋 cos 𝜔1 𝑡 + 𝑉𝑂𝜋 cos 𝜔1 𝑡 − 𝜙 (9) ( ) ( ) ( ) = 𝑉𝑆𝜋 + 𝑉𝑂𝜋 cos 𝜙 cos 𝜔1 𝑡 + 𝑉𝑂𝜋 sin 𝜙 sin 𝜔1 𝑡 .

𝑑𝜙 (𝑡) = 𝛺 + 𝜇 sin (𝛺𝜏) − 𝑑 sin (𝛺𝜏 + 𝜙 (𝑡)) (3) 𝑑𝑡 ( ) where 𝑑 = 𝜇𝛾 𝑉𝑆𝜋 ∕𝑉𝑂𝑆𝜋 . In the unlocked-driven region (i.e., 𝛺 > 𝑘), the solution of (3) is known to have the form [29] { }] [ √ √ ) (𝑘 + 𝜇 sin (𝑘𝜏)) 𝑋 2 − 1 ( 𝑋2 − 1 1 −1 + tan 𝑡 − 𝑡𝑂 𝜙 = 2 tan 𝑋 𝑋

Converting the right hand side of (9) to a single sinusoid, we obtain ( ) ( ) 𝑉𝑆𝜋 + 𝑉𝑂𝜋 cos 𝜙 𝑉𝑋 = cos 𝜔1 𝑡 − 𝜓 (10) cos 𝜓

− 𝛺𝜏 (4) [ ] where 𝑋 = (𝛺 + 𝜇 sin (𝛺𝜏)) ∕ (𝑘 + 𝜇 sin (𝑘𝜏)) is interpreted as the normalized frequency detuning, 𝑡𝑂 is an integration constant and 𝑋 > 1.𝑘 denotes the lock-range in presence of only phase perturbation and can be obtained by numerically solving the equation 𝑘 + 𝜇 sin (𝑘𝜏) − 𝑑 = 0. When 𝑋 = 1, and since |sin (𝛺𝜏 + 𝜙)|max = 1, 𝜙 is constant. It indicates that the oscillator is phase synchronized with the injection RF signal. The beat-frequency during pull-in and the average value of the forced OEO frequency are known to be, respectively, given by [29] √ 𝑑𝜙 = (𝑘 + 𝜇 sin (𝑘𝜏)) 𝑋 2 − 1 𝑑𝑡 √ 𝜔𝑂 = 𝜔𝑂 + 𝛺 + (𝑘 + 𝜇 sin (𝑘𝜏)) 𝑋 2 − 1 𝜔𝑏 =

𝑉𝑂𝜋 sin 𝜙 . (11) 𝑉𝑆𝜋 + 𝑉𝑂𝜋 cos 𝜙 ( √ ) Using the relation cos 𝜓 = 1∕ 1 + tan 2𝜓 , (10) may be rewritten in the form ( ) 𝑉𝑋 = 𝑉𝑋𝑂𝜋 cos 𝜔1 𝑡 − 𝜓 (12) √ 2 + 𝑉 2 + 2𝑉 𝑉 where 𝑉𝑋𝑂𝜋 = 𝑉𝑆𝜋 𝑆𝜋 𝑂𝜋 cos 𝜙. The output optical power 𝑂𝜋 of the MZ modulator (which provides the nonlinearity in the oscillatory circuit) with RF half-wave voltage 𝑉𝜋1 and DC voltage 𝑉𝜋𝐷𝐶 , can be written as ( { [ ]}) 1 (13) 𝑃 (𝑡) = 𝛼𝑃𝑂 1 − 𝜂 sin 𝜋 𝑉𝑋∕𝑉𝜋1 + 𝑉𝐵∕𝑉𝜋𝐷𝐶 2 where 𝑉𝐵 is the dc bias voltage, 𝜂 is a parameter determined by the extinction ratio of the modulator (1 + 𝜂) ∕ (1 − 𝜂), 𝑃𝑂 is the input optical where

(5) (6)

From (5) it may be observed that the beat frequency gradually decreases when frequency detuning 𝛺 is decreased keeping lock-range 185

tan 𝜓 =

A. Banerjee and J. Sarkar

Optics Communications 437 (2019) 184–192

Fig. 1. Schematic diagram of a single-loop OEO under RF signal injection.

Fig. 2. Electrical equivalent block diagram of an OEO under injection and its vector diagram.

power, and 𝛼 is the insertion loss, respectively. Substituting (12) into (13) and then neglecting all the higher order terms, we obtain ) ( ( ) 𝜋𝑉𝑋𝑂𝜋 cos 𝜔1 𝑡 − 𝜓 (14) 𝑃 (𝑡) = 𝛼𝜂𝑃𝑂 𝐽1 𝑉𝜋

Combining (17) and (18), we obtain the differential equation describing the phase variation in the OEO expressed as

This 𝑃 (𝑡) )}will ] experience a phase shift [ { signal ( tan−1 𝜇1 𝜔1 − 𝜔𝑂 − 𝑑𝜓 + 𝛺𝜏 while passing through the RF filter 𝑑𝑡 and the optical fiber. Thus, the output signal is given by ( ) 𝜋𝑉𝑋𝑂𝜋 𝑃 (𝑡) = 𝛼𝜂𝜌𝐺𝐴 𝑃𝑂 𝐽1 𝑉𝜋1 ( { ( )} ) 𝑑𝜓 1 × cos 𝜔1 𝑡 − tan−1 𝜔1 − 𝜔𝑂 − − 𝛺𝜏 − 𝜓 (15) 𝜇 𝑑𝑡

(19)

𝜇𝐾1 sin (𝛺𝜏 + 𝜙) 𝜇 sin 𝛺𝜏 𝑑𝜙 =𝛺+ [ ]−[ ] 𝑑𝑡 cos (𝛺𝜏) + 𝐾1 cos (𝛺𝜏 + 𝜙) cos (𝛺𝜏) + 𝐾1 cos (𝛺𝜏 + 𝜙)

It is noted that (19) includes the effects of time delay of the optical fiber, strong injection signal and the non-linearity associated with the filter, giving a good insight into the OEO phase dynamics. In the absence of time delay 𝜏(i.e., 𝜏 = 0) and when the injection signal is much smaller than the output signal (i.e., 𝐾1 ≪ 1), (19) reduces to the familiar Adler’s equation [35], as per expectation. When the strength of the injection RF signal is large, both amplitude and phase of the OEO are perturbed by the injection signal [31]. In such a situation, the nonlinear coupled differential equations (19) and (1) can be used to characterize the phaselocking dynamics of RF driven OEO irrespective of the injection signal strength constraint. However, the complete closed form expressions of phase 𝜙 (𝑡) and amplitude 𝑉𝑂 (𝑡) does not be derived as simultaneous analytical solutions of (19) and (1) are almost impossible. Let us simplify the situation by assuming that the OEO oscillates with unperturbed steady amplitude 𝑉𝑂𝑆𝜋 i.e., the amplitude perturbation is negligible. We are ignoring the impact of amplitude perturbation in large injection perturbation model as it is impossible to solve (16) considering the effect of amplitude perturbation. Therefore, we are going to base our large perturbation analytical model on (16). By solving Eq. (19), the desired phase difference 𝜙 (𝑡) is obtained characterizing the overall phase response of the RF driven oscillator. When the oscillator is phase locked to the injection signal, phase difference 𝜙 must be constant,

where 𝐺𝐴 is the gain of the RF amplifier and 𝜌 is the conversion factor of the photo-detector. For stable oscillation of the OEO from (15), we obtain { ( )} 𝑑𝜓 1 𝜙 = 𝜓 + tan−1 𝜔1 − 𝜔𝑂 − + 𝛺𝜏 (16) 𝜇 𝑑𝑡 From (11), it can be shown that tan (𝜙 − 𝜓) =

𝑉𝑆𝜋 sin 𝜙 𝐾1 sin 𝜙 = 𝑉𝑂𝜋 + 𝑉𝑆𝜋 cos 𝜙 1 + 𝐾1 cos 𝜙

(17)

( ) and also 𝑑𝜓 ≈ 𝑑𝜙 , where 𝐾1 = 𝑉𝑆𝜋 ∕𝑉𝑂𝜋 is the injection ratio. From 𝑑𝑡 𝑑𝑡 (16), we obtain 𝑑𝜙 = 𝛺 − 𝜇 tan (𝜙 − 𝜓 − 𝛺𝜏) 𝑑𝑡

(18) 186

A. Banerjee and J. Sarkar

Optics Communications 437 (2019) 184–192

i.e.,

𝑑𝜙 𝑑𝑡

𝛺=

𝜇𝐾1 sin (𝛺𝜏 + 𝜙) − 𝜇 sin (𝛺𝜏) cos (𝛺𝜏) + 𝐾1 cos (𝛺𝜏 + 𝜙)

= 0 and from (19) it follows that (20)

To find the lock-range of the RF driven oscillator, we take the derivative of 𝛺 with respect to 𝜙 and equate that to 0 and finally we obtain (21)

𝜇 cos (𝛺𝜏 + 𝜙) + 𝛺 sin (𝛺𝜏 + 𝜙) = 0 From (21), it can be shown that sin (𝛺𝜏 + 𝜙) = √

1

(22)

1 + (𝛺∕𝜇)2 (𝛺∕𝜇) . cos (𝛺𝜏 + 𝜙) = √ 1 + (𝛺∕𝜇)2

Fig. 3. Large injection perturbation normalized frequency detuning factor 𝑋𝑆′ as a function of normalized frequency detuning 𝑋𝑆 for five different injection ratios 𝐾1 : 0.05-(red), 0.2-(green), 0.3-(blue), 0.4-(black) and 0.5-(cient). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

(23)

Then by inserting (22)–(23) into (20) and after few steps of mathematical simplification, we get lock-range of the OEO under strong injection signal level as 𝐾𝑆 = √

𝜇𝐾1

(24)

(1 + 𝜇𝜏)2 − 𝐾12

and

Next we are going to solve phase dynamics equation (19) ignoring the amplitude perturbation and assuming the unlocked operating condition of the RF driven OEO. Eq. (19) can also be written in the following format: ( ) ( ) 𝛽 − 𝐾1 sec2 𝛺𝜏+𝜙 + 2𝐾1 2 1 ( ) ( ) [ ( ) ] 𝜇𝐾1 𝑋 𝛽 − 𝐾 + 𝛼 tan 2 𝛺𝜏+𝜙 − 2 tan 𝛺𝜏+𝜙 + [𝑋 (𝛽 + 𝐾 ) + 𝛼 ] 1

2

2

1

(25) where 𝑋 = 𝛺∕𝜇𝐾1 , 𝛼 = sin (𝛺𝜏) ∕𝐾1 and 𝛽 = cos (𝛺𝜏). To extract the detailed picture of the injection-pulling dynamics, we have to find out the closed form expressions of unlocked-driven OEO spectral components. From the elaborate expression of Eq. (25), it is almost impossible to derive closed form expressions of phase perturbation or spectral components. Let us now simplify the situation by that the contribution of 2𝐾1 is negligible compared to ( assuming ) 𝛽 − 𝐾1 (sec2 ((𝛺𝜏 ) + 𝜙) ∕2). From Table 1, we can see that the lowest value of( 𝛽 − 𝐾1) sec2 ((𝛺𝜏 + 𝜙) ∕2) ∕2𝐾1 is at least 2.1115 at 𝐾1 = 0.30, making 𝛽 − 𝐾1 sec2 ((𝛺𝜏 + 𝜙) ∕2) always greater than 2𝐾1 within 30% of injection ratio. With this assumption, (25) can be easily solved to find the phase 𝜙 (𝑡) as function of normalized frequency detuning and injection ratio as √ √ ⎤ ⎡ 𝑝 𝑋𝑆2 − 1 𝑝𝐾𝑆 𝑋𝑆2 − 1 ( )⎥ ⎢ 𝑝 + tan 𝑡 − 𝑡𝑂 ⎥ − (𝛺𝜏) (26) 𝜙 (𝑡) = 2 tan−1 ⎢ 𝑋𝑆 2 ⎥ ⎢ 𝑋𝑆 ⎦ ⎣ √ ( ) 𝛽 − 𝐾1 , (27) where 𝑝 = 𝛽 + 𝐾1 𝛺 𝑋𝑆 = (28) 𝐾𝑆

Also, the large injection perturbation average frequency of the unlockeddriven OEO can be written as √ 𝜛𝑂𝑠 = 𝜔𝑂 + 𝛺 − 𝑝𝐾𝑆 𝑋𝑆2 − 1 (33) Under weak injection signal level, 𝐾1 ≪ 1, Eqs. (32) and (33) nearly transform into (5) and (6) respectively, i.e. 𝜔𝑏𝑠 ≈ 𝜔𝑏 and 𝜛𝑂𝑠 = 𝜛𝑂 . Thus the 𝜔𝑏𝑠 can estimate the beat frequency more accurately than 𝜔𝑏 under strong injection signal level, as 𝜔𝑏𝑠 is more sensitive to injection ratio 𝐾1 through large perturbation strength factor 𝑝. Large injection perturbation beat frequency 𝜔𝑏𝑠 is the more general and accurate closed form analytical expression of beat frequency of the unlocked-driven OEO under weak as well as in strong injection perturbation applications. Next we are going to study the effect of normalized frequency detuning 𝑋𝑆 on the large injection perturbation normalized frequency detuning factor 𝑋𝑆′ . Using Eq. (31), large injection perturbation normalized frequency detuning factor 𝑋𝑆′ is plotted as a function of normalized frequency detuning 𝑋𝑆 for the selected values of injection ratios 𝐾1 in Fig. 3. This figure shows that in the fast beating state (i.e., when 𝑋𝑆 is large), as 𝐾1 increases, 𝑋𝑆′ decreases rapidly at a fixed 𝑋𝑆 . However, as the oscillator approaches towards the quasi-locked state and very close to the locked state (i.e., 𝑋𝑆 is decreased), the dependence of 𝑋𝑆′ on 𝐾1 is decreased significantly. If 𝑋𝑆 is decreased further, 𝑋𝑆′ becomes unity, indicating that the OEO is phase-locked by the injection signal. Also, we can see that when 𝐾1 is low (i.e. weak injection signal), large injection perturbation normalized frequency detuning factor 𝑋𝑠′ becomes equal to 𝑋𝑆 , as per expectation. √Fig. 4 shows large injection perturbation normalized beat frequency

and 𝑡𝑂 is an integration constant. Differentiating (26) and using the results (27), (28), one can easily show 𝑑𝜙 (𝑡) 𝑑𝑡

[ { ∝ √ √ ∑ ( ) 2 = 𝑝𝐾𝑆 𝑋𝑆 − 1 × 1 + 2 (−1)𝑛 𝑟𝑛𝑆 cos 𝑛 𝑝𝐾𝑆 𝑋𝑆2 − 1 𝑡 − 𝑡𝑂 𝑛=1

]

𝑝 𝑋𝑆2 − 1 as a function of normalized frequency detuning 𝑋𝑆 for five different values of injection ratios 𝐾1 . It is observed that for a fixed 𝑋𝑆 , normalized beat frequency increases as 𝐾1 is increased. Also, as 𝑋𝑆 is decreased keeping 𝐾1 constant, normalized beat frequency decreases. It

(29) where

𝑟𝑆 = 𝑋𝑆′ −

√

𝑋𝑆′2 − 1,

(31)

where 𝑚 = (1 + 𝜇𝜏). The closed form expression (29) describes the phase perturbation of the unlocked-driven OEO under RF signal injection. For weak RF signal injection, a similar closed form phase perturbation equation was derived earlier in [29, equation (22)]. For small injection level, 𝐾1 ≪ 1, 𝑝 ≈ 1, 𝑟𝑆 ≈ 𝑟, 𝐾𝑆 ≈ 𝑘 and thus 𝑋𝑆′ ≈ 𝑋𝑆 ≈ 𝑋, as derived in [29]. Consequently, large injection perturbation equation (29) nearly transforms into Eq. (22) of [29], indicating that (26) measures the phase perturbation of the OEO under weak injection signal level also. The large injection perturbation beat frequency of the unlocked driven OEO can be obtained from (29) as √ (32) 𝜔𝑏𝑠 = 𝑝𝐾𝑆 𝑋𝑆2 − 1

× 𝑑𝜙 (𝑡) = 𝑑𝑡

} √ √ − tan−1 𝑋𝑆2 − 1 − tan−1 𝐾1 𝑋𝑆2 − 1

√ √ 2 √ √ ⎞ √⎛ √⎜ 𝛼 𝑚2 − 𝐾12 ⎟ 𝑋 √ 2 √⎜𝛽 + 𝑋𝑆′ = √ 𝑆 ⎟ − 𝐾1 √ 𝑋𝑆 ⎟ 𝑚2 − 𝐾12 ⎜ ⎠ ⎝

(30) 187

A. Banerjee and J. Sarkar

Optics Communications 437 (2019) 184–192 Table 1 ( ) Calculation of 𝛽 − 𝐾1 sec2 ((𝛺𝜏 + 𝜙) ∕2) at different phase angle for different injection ratios. ( ) 𝛽 − 𝐾1 sec2 ((𝛺𝜏 + 𝜙) ∕2) ∕2𝐾1 for different 𝐾1 values (𝛺𝜏 + 𝜙) (deg) 𝐾1 = 0.1 𝐾1 = 0.15 𝐾1 = 0.2 𝐾1 = 0.25 90 100 110 120 130 140 150 160 170

8.00 9.6811 12.1584 16.00 22.3956 34.1945 59.7128 132.6537 526.5844

5.00 6.0507 7.5990 10.00 13.9973 21.3716 37.3205 82.9086 329.00

3.50 4.2355 5.3193 7.00 9.7981 14.9601 26.1244 58.0360 230.00

2.60 3.1464 3.9515 5.20 7.2786 11.1132 19.4067 43.1125 171.00

𝐾1 = 0.30 2.115 2.4203 3.0396 4.00 5.5989 8.5486 14.9282 33.1634 131.00

√

𝑋𝑆′2 − 1 ( ) 𝑋𝑆′ + cos 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 √( √ ) ( ) 𝑋𝑆2 − 1 𝐾1 𝑋𝑆 1 − 𝐾12 sin 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 × − ( ). √( ) 𝑋𝑆′ + cos 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 𝑋𝑆2 − 1 1 + 𝐾12 ×

(39) Also, considering (26), (27), (31) and (29), we obtain √( )( ) 𝑋𝑆2 − 1 𝐾12 − 1 𝐾1 𝑋𝑆2 cos 𝜙 (𝑡) = − ( )+ ( ) 1 + 𝐾12 𝑋𝑆2 − 1 1 + 𝐾12 𝑋𝑆2 − 1 ( ) sin 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 × ′ ( ) 𝑋𝑆 + cos 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 √ √ 𝑋𝑆′2 − 1 + 𝐾1 𝑋𝑆′ 𝑋𝑆′2 − 1 × ′ ( ). 𝑋𝑆 + cos 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓

√ Fig. 4. Plot of large injection perturbation normalized beat frequency 𝑝 𝑋𝑆2 − 1 with the normalized frequency detuning 𝑋𝑆 for five different injection ratios 𝐾1 : 0.05-(red), 0.2-(green), 0.3-(blue), 0.4-(black) and 0.5-(cient). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

means that the frequency separation between the spectral components of the unlocked driven OEO output signal is decreased due to the phase perturbation imposed by the injection signal. Thus, the pulled oscillator is approaching towards the locked state from the fast-beat state through the quasi-locked state. Accurate closed form expressions are required for the spectral components of the unlocked-driven OEO in terms of the large injection perturbation beat frequency 𝜔𝑏𝑠 and injection ratio 𝐾1 to extract the clear picture of the phase-locking dynamics of the RF driven OEO. The unlocked-driven OEO output signal is written as ( ) 𝑣𝑂 (𝑡) = 𝑉𝑂𝜋 cos 𝜔1 𝑡 − 𝜙 (𝑡) = 𝑉𝑂𝜋 cos 𝜔1 𝑡 cos 𝜙 (𝑡) + 𝑉𝑂𝜋 sin 𝜔1 𝑡 sin 𝜙 (𝑡) .

Using (39) and (40) in (32), a closed form large injection perturbation expression for the spectral components of the unlocked-driven OEO is obtained as √ ( ) 𝑣𝑂 (𝑡) = 𝑉𝑂𝜋 𝑟𝑆 sin 𝜔1 𝑡 − 𝜀 + 2𝑉𝑂𝜋 𝑋𝑆′2 − 1 [∝ [( ) √( ∑ ) × 𝑋𝑆2 − 1 𝑡 (−1)𝑛+1 𝑟𝑛𝑆 sin 𝜔1 − 𝑛𝑝𝐾𝑆 𝑛=1

( +

(34) where

Using the results (26), (27) and (31), we obtain √ √( )( ) 𝑋𝑆 1 − 𝐾12 𝑋𝑆2 − 1 𝐾12 − 1 sin 𝜙 (𝑡) = ( )+ ( ) 1 + 𝐾12 𝑋𝑆2 − 1 1 + 𝐾12 𝑋𝑆2 − 1 √ 𝑋𝑆′2 − 1 × ′ ( ) 𝑋𝑆 + cos 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 √( √ ) ( ) 𝐾1 𝑋𝑆 1 − 𝐾12 𝑋𝑆2 − 1 sin 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 − × ( ) √( ) 𝑋𝑆′ + cos 2𝛽𝑆 − 𝛽𝑂𝑆 − 𝜓 1 + 𝐾12 𝑋𝑆2 − 1 √ ( ) 2𝛽𝑆 = 𝑝𝐾𝑆 𝑋𝑆2 − 1 𝑡 − 𝑡𝑂 , √ 𝛽𝑂𝑆 = tan−1 𝑋𝑆2 − 1, √ and 𝜓 = tan−1 𝐾1 𝑋𝑆2 − 1.

𝑛𝑝𝐾𝑆

)] ] √( ) 𝑋𝑆2 − 1 𝑡𝑂 + 𝑛𝜓 + 𝑛𝛽𝑂𝑆 − 𝜀

𝐾 𝑋 𝜀 = tan−1 √ 1 𝑆 . 1 − 𝐾12

(41)

(42)

The pull-in spectra of the unlocked-driven OEO can be evaluated by taking the Fourier transform of (41). Next we are going to investigate how the amplitudes of different harmonics present in the pulled OEO output signal is affected by decreasing the frequency detuning at different injection ratios 𝐾1 . Fig. 5 plots the variation of normalized amplitudes of different harmonics against the normalized frequency detuning 𝑋𝑆 for different values of 𝐾1 . Fig. 6(a) shows that as 𝑋𝑆 is decreased keeping normalized injection signal amplitude 𝐾1 fixed, normalized amplitude of the carrier decreases and consequently, strength of the sideband at the injection frequency increases due to the frequency pulling of the unlocked-driven OEO. It is also observed that as 𝐾1 increases, strength of the carrier signal decreases and normalized amplitude of the sideband at the injection frequency is increased, irrespective of 𝑋𝑆 . Finally, under phase-locked state when 𝑋𝑆 = 1, the OEO output signal contains a single component at the injection signal frequency. Fig. 6(b) shows that for a fixed 𝐾1 , there exists a value of 𝑋𝑆 for which the normalized amplitude of the sideband is maximized (i.e., peaking of the normalized amplitude) both for second and third sideband. The normalized amplitude peaking is dependent on injection ratio 𝐾1 , and it occurs in a relatively higher values of 𝑋𝑆 when 𝐾1 is increased for a particular sideband. It is also

(35) where

(40)

(36) (37) (38)

Using (29) in (33), provides √ √( )( ) 𝑋𝑆 1 − 𝐾12 𝑋𝑆2 − 1 𝐾12 − 1 sin 𝜙 (𝑡) = ( )+ ( ) 1 + 𝐾12 𝑋𝑆2 − 1 1 + 𝐾12 𝑋𝑆2 − 1 188

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Optics Communications 437 (2019) 184–192

Fig. 5. Output spectra of pulled OEO. (a), (b) Fast-beat state (spectrum distribution is asymmetric), (c) quasi-locked state (amplitude of the carrier is equal to that of injection signal), (d) unlocked-driven state (all sidebands except the injection signal is on the opposite side of the oscillator carrier from the injection signal), and (e) locked state of the OEO.

Fig. 7. Relative variation of normalized amplitudes of different harmonics present in the unlocked-driven OEO output signal with normalized frequency detuning 𝑋𝑆 for three different injection ratios 𝐾1 = 0.05, 0.2 and 0.4 for (a), (b) and (c), respectively. Injection signal (red), carrier signal (First harmonic n = 1) (black), second harmonic n = 2 (blue) and third harmonic n = 3 (green). The co-ordinate of the quasi-locked state for 𝐾1 = 0.05, 0.2 and 0.4 are (1.118, 0.6181), (1.124, 0.6180) and (1.146, 0.6181), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the quasi-locked state increases. Also, when 𝐾1 is varied, the equal normalized amplitude of the carrier and that of the sideband at injection frequency is not affected. Next, we study the effects of normalized frequency detuning 𝑋𝑆 and injection ratios 𝐾1 on the phase angle of the harmonic component present in the unlocked driven OEO output signal. Fig. 8, Figs. 9 and 10 plot the variations of phase angle parameters 𝜀, 𝜓 and 𝛽𝑂𝑆 against 𝑋𝑆 for different values of 𝐾1 . As 𝑋𝑠 decreases, the phase angle 𝜀, 𝜓 and 𝛽𝑂𝑆 decreases and finally under locked condition only 𝜓 and 𝛽𝑂𝑆 becomes zero and also non zero value of 𝜀 under locked state indicates the phase difference between the free running carrier signal and the injection signal. From Eq. (37), we can see that 𝛽𝑂𝑆 is independent of 𝐾1 . At a fixed 𝑋𝑆 , as 𝐾1 is increased, the phase angle 𝜀 and 𝜓 increases. When 𝑋𝑆 is decreased the dependence of both 𝜀 and 𝜓 on 𝐾1 is decreased significantly and the oscillator approaches towards the quasi-locked and locked state of the oscillator.

Fig. 6. Normalized amplitudes of different harmonics present in the unlocked-driven OEO output signal versus normalized frequency detuning 𝑋𝑆 for three different injection ratios 𝐾1 : 0.05 (cient), 0.3 (blue) and 0.5 (red). (a) Injection signal (- -) and carrier signal (First harmonic n = 1) (-) (b) second harmonic n = 2 (-) and third harmonic n = 3 (- -). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

noticed that as 𝑋𝑆 decreases keeping 𝐾1 constant, the second sideband achieves maxima first then third sideband and higher order sidebands sequentially achieves their maxima, very close to the quasi-locked state of the oscillator. We therefore need to investigate how the quasi-locked state is affected by increasing the normalized injection ratio 𝐾1 . We plot in Fig. 7 the normalized amplitudes of different harmonics against 𝑋𝑆 for different values of 𝐾1 . We can see that as 𝐾1 increases, the normalized frequency detuning 𝑋𝑆 at which the pulled OEO achieves 189

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Optics Communications 437 (2019) 184–192

Fig. 8. Phase angle 𝜀 of harmonic components versus normalized frequency detuning 𝑋𝑆 for four different injection ratios 𝐾1 : 0.05-(red), 0.2-(green), 0.3-(blue), and 0.4-(black). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Phase angle 𝜓 of harmonic components versus normalized frequency detuning 𝑋𝑆 for four different injection ratios 𝐾1 : 0.05-(red), 0.2-(green), 0.3-(blue), and 0.4-(black). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Variation of phase angle 𝛽𝑜𝑠 of harmonic components with normalized frequency detuning 𝑋𝑆 . Table 2 Summary of RF driven OEO system parameter values. Parameter symbol Description

Value

𝑉𝐵

DC bias voltage

3.14 V

𝑉𝜋1 𝐺𝐴 𝜌 𝜂 𝛼 𝑃𝑠𝑎𝑡 𝑃𝑂 𝜏

Modulator half-wave voltage of oscillator RF amplifier gain Conversion factor Parameter determined by extinction ratio of modulator Insertion loss Saturation power of the photo-detector Input optical power Time-delay

3.14 V 7.5 2.2 1.0202 1 dB 24 dBm 18 dBm 10 μsec.

Fig. 11. Spectra of the output RF signal of the unlocked driven OEO at different injection signal frequencies. The free-running frequency of the OEO is 11.9290 GHz. The injection ratio is kept constant at 0.3.

[ ] 0.9993616×109 𝑠 and bandwidth 20.00 MHz is 𝐻 (𝑠) = 2 . 𝑠 +0.9993616×109 𝑠+(2𝜋×11.929)2 ×1018 The RF filter was realized using the transfer function block available in the Simulink library. The parameter values used in the simulation are listed in Table 2. The corresponding effective loop-gain 𝛾 = 0.5 [28], which is required to obtain a perfectly pure microwave signal of frequency 11.9290 GHz. We have implemented our simulation model by discretizing the output RF signal using an array containing n = 50000 points, which induces a finite resolution time 𝛿𝑇 = 𝜏∕𝑁. The time delay was set to 𝜏 = 10 μs. As a result, the frequency resolution, 𝛿𝑓 = 1∕𝜏, was about 𝛿𝑓 = 0.1 MHz and the simulation bandwidth, 𝛥𝑓 = 1∕𝛿𝑇 was about 𝛥𝑓 = 5 GHz. It should be noted that the number of points

4. Simulation results To validate the accuracy of the large injection perturbation analytical model in predicting the phase-locking dynamics of the OEO, a single loop optoelectronic microwave oscillator was designed having freerunning frequency of 11.9290 GHz with steady state output voltage of 2.273 V and simulated in the MATLAB/Simulink environment. The transfer function of the RF filter having center frequency 11.9290 GHz 190

A. Banerjee and J. Sarkar

Optics Communications 437 (2019) 184–192 Table 3 Beat frequency of the unlocked-driven OEO. Parameters

𝐾1 = 0.05 𝐾𝑆 = 0.3004 kHz 𝑝 = 1.0513

𝐾1 = 0.30 𝐾𝑆 = 3.1449 kHz 𝑝 = 1.3628

Frequency detuning (kHz)

Beat frequency simulation (kHz)

Beat frequency (kHz) (% Error) Small injection perturbation model

Beat frequency (kHz) (% Error) Large injection perturbation model

45.2

(49.6/50.6)

32.7

(35.4/35.8)

22.6

(24.6/24.4)

12.9

(14.3/13.7)

49.3 (0.538/2.57) 34.7 (2.035/3.07) 23.8 (3.05/2.46) 14.2 (0.81/−3.65)

47.518 (4.21/6.15) 34.376 (2.82/3.98) 23.757 (3.25/2.8) 13.558 (5.39/0.80)

70.1

(103.5/104.0)

62.9

(94.0/94.3)

48.5

(69.8/71.6)

27.5

(40.9/40.0)

77.4 (25.20/25.02) 74.3 (20.95/21.21) 53.2 (23.81/25.70) 31.9 (21.89/20.25)

95.436 (7.81/8.20) 85.613 (8.95/9.20) 65.957 (5.46/7.85) 37.21 (8.95/6.85)

N, should be chosen sufficiently high to ensure that the simulation bandwidth will be sufficiently broader than the RF filter bandwidth. Table 3 gives a comparison of the beat frequency obtained from the simulation, the small injection perturbation analytical model [Eq. (5)] and the large injection perturbation analytical model [Eq. (32)], for different detuning frequencies at two different injection ratios 𝐾1 . The percentage errors in estimating the beat frequency with respect to the simulation have been calculated and are presented in Table 3. When 𝐾1 = 0.05 [i.e., low level injection], the small and large injection perturbation analytical results closely match with each other and also with the simulated results. For high level injection, (𝐾1 = 0.30), the large percentage error of the small injection perturbation model indicates the inaccuracy and inefficiency of the small injection perturbation analytical model in predicting the frequency pulling of oscillator. From Table 3 we can see that the large injection perturbation analytical model accurately predicts the phase-locking of the driven OEO for both weak and strong RF signal injection. Fig. 11 shows a number of simulated spectra for RF signal of injection ratio 0.3 and different injection signal frequency. The output signal spectra are all discrete and sideband distribution is non-symmetrical about the carrier, which was also experimentally reported by Stover [27] and Hakki [26]. They observed similar spectra for unlocked-driven tunnel-diode oscillators with free-running frequency of 11.3720 GHz. It is also observed that as frequency detuning deceases keeping 𝐾1 constant, the second sideband achieves maxima injection first (1.962, 11.9290022 GHz) (Fig. b) then third sideband (1.357, 11.9289999 GHz) (Fig. c).

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5. Conclusion In summary, we have developed a novel theoretical model for describing the phase-locking phenomenon of optoelectronic microwave oscillators under RF signal injection. We note that the large injection perturbation analytical model is capable of predicting the injectionlocking behavior of the RF driven OEO in weak as well as in large injection perturbation applications. Simulation results indicate that the proposed model can accurately estimate the frequency pulling of the OEO for microwave signal generation. Our results open new perspectives for optimization and engineering of the injection-locking and injection-pulling processes in OEO for microwave signal generation. References [1] X.S. Yao, L. Maleki, Optoelectronic microwave oscillator, J. Opt. Soc. Am. B 13 (8) (1996) 1725–1735. 191

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