Spurious mode reduction in dual injection-locked optoelectronic oscillators

O. Okusaga; E. J. Adles; E. C. Levy; W. Zhou; G. M. Carter; C. R. Menyuk; M. Horowitz

doi:10.1364/OE.19.005839

1. Introduction

1.1. The single-loop optoelectronic oscillator

An optoelectronic oscillator (OEO) is a radio frequency (RF) oscillator that utilizes an optical fiber delay line as a high-Q resonant cavity [1, 2]. The low loss per unit length of standard single-mode optical fiber of 0.2 dB/km indicates that it is should be possible in principle to construct OEOs with delay lines as long as 16 km and Q-factors as high as 4 × 10⁶ [3]. The best commercially available RF oscillators have Q-factors on the order of 1×10⁵ [4]. The high Q of an OEO implies that the phase noise that it produces should be very low.

The OEO’s Q-factor is approximately equal to 2πfτ, where f is the OEO’s oscillation frequency and τ is its round trip time [1]. However, the frequency spacing between the oscillation modes of the OEO is inversely proportional to τ. For example, an OEO with a 4 km fiber delay line has a Q-factor of 200,000 but a mode spacing of only 50 kHz. Modes spaced so narrowly cannot be filtered using conventional RF filters.

1.2. Dual-cavity optoelectronic oscillators

Mode selectors with higher finesse than conventional RF filters are necessary to suppress the spurs of a high-Q OEO. Dual-cavity OEOs use a second optoelectronic loop or delay line as a high-finesse filter. Proposed dual-cavity designs include: the dual optoelectronic oscillator (DOEO), the coupled optoelectronic oscillator (COEO), and the dual injection-locked opto-electronic oscillator (DIL-OEO) [5–7].

The DOEO consists of a single optoelectronic loop with two fiber delay lines of different lengths. The lengths are chosen so that the two delay-lines only share a single common mode within the RF filter bandwidth. All other modes experience less gain per round trip and so are suppressed via gain competition.

The COEO consists of a fiber laser with a medium length delay-line of approximately 500 m length coupled to an optoelectronic loop via a common electro-optic modulator. The electro-optic loop locks the fiber laser’s mode spacing to the desired RF frequency and suppresses all but one of the fiber laser’s modes.

The DIL-OEO consists of a high-Q master OEO loop that is injection-locked to a relatively low-Q slave OEO loop. The slave OEO loop signal preferentially amplifies the desired master loop mode while suppressing all other spurious modes. In this work, we focus on the DIL-OEO, presenting the first systematic study of injection-locking in optoelectronic oscillators.

1.3. The dual injection-locked optoelectronic oscillator

Figure 1 shows a schematic illustration of the DIL-OEO. The DIL-OEO consists of two coupled OEO loops. The mode spacing of the master loop is typically between 10 and 100 kHz. The shorter slave loop is designed to support only a single mode within the bandwidth of its in-loop RF filter. We phase lock the two loops by injecting a portion of the master-loop signal into the slave loop, while, at the same time, injecting a portion of the slave-loop signal into the master loop. As we have shown in previous work, this bi-directional injection is required to achieve both low phase noise and low spurs [8]. Injecting a portion of the master-loop signal into the slave loop ensures that the DIL-OEO maintains a high Q. Injecting a portion of the slave-loop signal into the master loop ensures spur suppression and maintains a stable phase-lock between both loops. We refer to injection from the master loop to the slave and injection from the slave to the master loop as forward and reverse injection respectively.

Fig. 1 A schematic illustration of the dual injection-locked OEO.

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2. Theory

Our ultimate goal is to design an optimized DIL-OEO with both low phase noise and low spur levels. However, due to the complexity of the injection-locking process, it is difficult to determine the OEO parameter values that produce the best results. To assist us in accomplishing our objective, we have created a reduced model that provides qualitative insights into the injection-locking process in the DIL-OEO [9]. From these insights and our quantitatively accurate full model [10] that has been described in detail elsewhere [11], we were able to construct our optimized DIL-OEO.

2.1. The Yao-Maleki single-loop model

Our reduced model is based on the Yao-Maleki single-loop OEO model. In their original work, Yao and Maleki presented a reduced model of the single-loop OEO that allowed them to calculate the phase noise when the OEO operates in steady-state [1, 2]. We used similar methods to model the two coupled loops of the DIL-OEO. In order to better describe our DIL-OEO model, we begin with a review of the Yao-Maleki model.

We first define A(t), the complex envelope of the voltage at the output of the photodetector, by writing $V (t) = A (t) \sqrt{R} exp (j 2 π f_{0} t) + complex conjugate$ , where V(t) is the real voltage signal, R is the output impedance of the photodetector, and f₀ is the carrier frequency, which is typically near 10 GHz. The envelope is normalized so that the oscillating power P(t) = |A(t)|². Following the signal through one round trip of the OEO that is shown in Fig. 1, we find that A(t) must satisfy the delay-difference equation

A (t) = G M [A (t - τ)] A (t - τ) + S (t),

where τ is the round-trip time, G is the small-signal gain, M [A(t – τ)] is the gain saturation factor, and S(t) is the noise input from which the oscillator power grows.

In general, the gain saturation factor M [A(t – τ)] is a nonlinear function of the amplitude A(t). However, when the OEO operates at steady state, the gain GM is determined by the Barkhausen condition [12] and may be treated as a constant. Yao and Maleki assume that the gain saturation is due to the electro-optic modulator and has the form

M (P) = 2 {(\frac{P}{P_{sat}})}^{1 / 2} J_{1} [{(\frac{P_{sat}}{P})}^{1 / 2}],

where P_sat is the saturation power. In our experiments, we found that the gain saturation was due to the RF amplifiers and took the form

M (P) = \frac{1}{1 + P / P_{sat}} .

This difference has no effect on our steady-state model. When the gain GM is constant, Eq. (1) becomes linear, and its Fourier transform takes the form

\tilde{A} (ω) = (1 - Δ) \tilde{A} (ω) exp (- j ω τ) + \tilde{S} (ω),

where Δ ≡ 1 – GM is a small positive quantity and we have used tildes to indicate the Fourier transform.

From Eq. (4), we obtain immediately

\tilde{A} (ω) = \frac{\tilde{S} (ω)}{1 - (1 - Δ) exp (- j ω τ)} .

Squaring Eq. (5) and taking an ensemble average over the noise we find

P (ω) = \frac{N (ω)}{{| 1 - (1 - Δ) exp (- j ω τ) |}^{2}},

where P(ω) ≡ 〈|Ã(ω)|²〉, N(ω) ≡ 〈|S̃(ω)|²〉, and the brackets 〈·〉 indicate an ensemble average.

The minima of the denominator in Eq. (6) occur when ωτ = 2nπ, where n is an integer. If we assume strictly white input noise sources so that N(ω) = N₀, then the power spectral density at each resonant frequency is

P_{0} = N_{0} / {| Δ |}^{2} .

For frequencies close to the central oscillating frequency, where n = 0, we find that ωτ ≪ 1 and Eq. (6) reduces to the standard Lorentzian form

P (ω) = \frac{N (ω)}{Δ^{2} + ω^{2} τ^{2}} .

2.2. The reduced DIL-OEO model

We now present an overview of our reduced model of the DIL-OEO [9]. Figure 2 is a schematic illustration of the DIL-OEO that shows the field injection ratios γ_ij. In a manner analogous to the Yao-Maleki model, we represent the DIL-OEO amplitudes as

\begin{array}{l} A_{1} (t) = γ_{11} G_{1} M_{1} A_{1} (t - τ_{1}) + γ_{12} A_{2} (t) + S_{1} (t), \\ A_{2} (t) = γ_{22} G_{2} M_{2} A_{2} (t - τ_{2}) + γ_{21} A_{1} (t) + S_{2} (t), \end{array}

where the subscripts 1 and 2 refer to the master and slave loops respectively and the coefficients γ₂₁ and γ₁₂ are the forward and reverse injection ratios.

Fig. 2 A schematic illustration of the DIL-OEO showing the field coupling ratios γ_ij.

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Taking the Fourier transform of Eq. (9) and solving for the steady-state amplitudes, we obtain

\begin{array}{l} {\tilde{A}}_{1} (ω) = \frac{1}{D (ω)} {[1 - (1 - Δ_{2}) exp (- j ω τ_{2})] {\tilde{S}}_{1} (ω) + γ_{12} {\tilde{S}}_{2} (ω)}, \\ {\tilde{A}}_{2} (ω) = \frac{1}{D (ω)} {[1 - (1 - Δ_{1}) exp (- j ω τ_{1})] {\tilde{S}}_{2} (ω) + γ_{21} {\tilde{S}}_{1} (ω)}, \end{array}

where we let

D (ω) = [1 - (1 - Δ_{1})] exp (- j ω τ_{1}) [1 - (1 - Δ_{2})] exp (- j ω τ_{2}) - γ_{12} γ_{21},

with Δ₁ ≡ 1 – γ₁₁G₁M₁ and Δ₂ ≡ 1 – γ₂₂G₂M₂. In Eq. (10) and Eq. (11), we have incorporated the coefficients γ₁₁ and γ₂₂ into the terms (1 – Δ₁) and (1 – Δ₂). We assume that the noise sources are uncorrelated, so that the power spectral densities are given by

\begin{array}{l} P_{1} (ω) = \frac{1}{{| D (ω) |}^{2}} [{| 1 - (1 - Δ_{2}) exp (- j ω τ_{2}) |}^{2} N_{1} (ω) + γ_{12}^{2} N_{2} (ω)], \\ P_{2} (ω) = \frac{1}{{| D (ω) |}^{2}} [{| 1 - (1 - Δ_{1}) exp (- j ω τ_{1}) |}^{2} N_{2} (ω) + γ_{21}^{2} N_{1} (ω)] . \end{array}

Equation (12) is our basic equation. To complete the model, we must estimate the quantities Δ₁ and Δ₂. Using Eq. (11) at resonance so that exp[−jωτ₁] = exp[−jωτ₂] = 1, we find that

D_{0} \equiv {D (ω) |}_{ω = 0} = Δ_{1} Δ_{2} - γ_{12} γ_{21} .

The power spectra for the master and slave loops are given by Eq. (12). If we substitute the condition ω = 0 into Eq. (12), we find that the resonant power levels are given by

\begin{array}{l} {P_{1} (ω) |}_{ω = 0} = \frac{1}{{| D_{0} |}^{2}} [{{| Δ_{2} |}^{2} N_{1} (ω) |}_{ω = 0} + {{| γ_{12} |}^{2} N_{2} (ω) |}_{ω = 0}], \\ {P_{2} (ω) |}_{ω = 0} = \frac{1}{{| D_{0} |}^{2}} [{{| Δ_{1} |}^{2} N_{2} (ω) |}_{ω = 0} + {{| γ_{21} |}^{2} N_{1} (ω) |}_{ω = 0}] . \end{array}

We define P₁₀ ≡ P₁(ω)|_ω₌₀ and N₂₀ ≡ N₂(ω)|_ω₌₀, so that

\begin{matrix} P_{10} \geq \frac{{| γ_{12} |}^{2} N_{20}}{{| D_{0} |}^{2}}, \\ \Rightarrow {| D_{0} |}^{2} \leq {| γ_{12} |}^{2} N_{20} / P_{10} . \end{matrix}

Similarly, |D₀|² ≤ |γ₂₁|² N₁₀/P₂₀. Because our OEOs are high-Q oscillators, the noise-to-signal ratios are very small in both the master and slave loops. The ratios N₂0/P₁0 and N₁0/P₂0 are both less than 10⁻¹⁴ in all our OEOs. Therefore, from Eq. (15), we determine that |D(ω)| ≪ |γ₁₂γ₂₁|.

Since the ratio of the noise power to the signal power is set by |D(ω)| ≪ |γ₁₂γ₂₁|, it is useful to write

Δ_{1} = α γ_{21} + ɛ_{1}, Δ_{2} = α^{- 1} γ_{12} + ɛ_{2},

where α is a constant we will determine shortly, and both ɛ₁ and ɛ₂ are small compared to |γ₁₂| and |γ₂₁|. To determine the precise values of α, ɛ₁, and ɛ₂, we focus on the region around the central oscillating mode where n, the mode number, is equal to 0. In this region, from Eq. (12), it follows that

P_{1} (ω) \propto P_{2} (ω) \propto 1 / (ɛ_{1}^{2} + ω^{2} τ_{1}^{2})

with just white noise drivers when ωτ₁ ≪ 1 and τ₂ ≪ τ₁. Therefore,

ɛ_{1}^{2}

is the noise-to-signal ratio at resonance of the injection-locked OEO.

Solving for the average oscillating powers in the master and slave loops, we find that

\begin{matrix} W_{1} = \frac{1}{4 τ_{1} W_{1}} \frac{Δ_{2}^{2} N_{1 W} + {| γ_{12} |}^{2} N_{2 W}}{Δ_{1} Δ_{2} - γ_{12} γ_{21}}, \\ W_{2} = \frac{1}{4 τ_{1} W_{1}} \frac{Δ_{1}^{2} N_{2 W} + {| γ_{21} |}^{2} N_{1 W}}{Δ_{1} Δ_{2} - γ_{12} γ_{21}}, \end{matrix}

where W₁ and W₂ are the average oscillating powers in the master and slave loops, and N_1W and N_2W are the average noise powers in both loops. Substituting Eq. (16) into Eq. (17), we find that

α = \sqrt{W_{2} / W_{1}}

,

ɛ_{1} = \frac{N_{1 W}}{4 τ_{1} W_{1}} {(1 + \frac{W_{2}}{W_{1}} \frac{{| γ_{12} |}^{2}}{{| γ_{21} |}^{2}} \frac{N_{2 W}}{N_{1 W}})}^{- 1},

and ɛ₂ = (γ₁₂/γ₂₁)(N_2W/N_1W)ɛ₁. Since W₁ and W₂ are the average oscillating powers in the master and slave loops, α² is the ratio between the average oscillating powers of the master and slave loops. As we state in section 3.1, the oscillating power levels in our master and slave loops are equal due to the fact that we use amplifiers with identical saturation powers in both loops. Therefore α² ≈ 1 in our experiments.

Due to injection locking, the quantities Δ₁ and Δ₂ are many orders of magnitude greater than Δ in the single-loop OEO [9]. The magnitudes of Δ₁ and Δ₂, as well as their dependence on both γ₁₂ and γ₂₁ have important implications for spur suppression in DIL-OEOs, which we will explore in the following sections.

2.3. Effects of injection-locking on the close-in phase noise

From Eq. (12), we can extract qualitative information on both the phase noise and spurious mode levels in both loops of the DIL-OEO. In this section, we focus on the effects of injection-locking on the phase noise around the central oscillating tones of both the master and slave loops, where the mode number n = 0. We refer to the noise in this range as the close-in phase noise.

In order to simplify the analysis, we assume that W₁ = W₂, and N₁ = N₂. We also assume that γ₂₁ ≫ γ₁₂. Equation (12) is capable of quantitatively describing the DIL-OEO’s behavior over a wider range of parameters. We limit ourselves to the above assumptions to clarify the behavior. For frequencies close to the central resonant frequency of both the master and slave loops, we may write

\begin{array}{l} {| 1 - (1 - Δ_{2}) exp (- j ω τ_{2}) |}^{2} \approx Δ_{2}^{2} = {| γ_{21} |}^{2}, \\ {| 1 - (1 - Δ_{1}) exp (- j ω τ_{1}) |}^{2} \approx Δ_{1}^{2} + ω^{2} τ_{1}^{2} = {| γ_{12} |}^{2} + ω^{2} τ_{1}^{2} . \end{array}

Substituting Eq. (19) into Eq. (12) we obtain the following asymptotic approximations of the power spectral densities in both loops:

\frac{P_{1} (ω)}{N_{1} (ω)} = \frac{1}{ɛ_{1}^{2} + ω^{2} τ_{1}^{2}}, \frac{P_{2} (ω)}{N_{2} (ω)} = \frac{1 + ω^{2} τ_{1}^{2} / {| γ_{21} |}^{2}}{ɛ_{1}^{2} + ω^{2} τ_{1}^{2}} .

Equation (20) shows that for radial frequencies less than ω_L = |γ₁₂|/τ₁, the master and slave loop signals converge. Beyond ω_L, the slave loop signal reaches a plateau, while the master loop’s signal continues to decrease with increasing frequency. Physically, the frequency ω_L corresponds to the Leeson frequency, beyond which the phase of the slave loop is no longer locked to the phase of the master loop [12]. Increasing the master-to-slave injection ratio increases the Leeson frequency thereby reducing the slave loop’s noise at high offset frequencies.

2.4. Effects of injection-locking on spurious modes

Finally, we investigate the power at the spurs. When we are close to a spur of the master loop, so that ω = ω_n = 2πn/τ₁, where n = 1, 2, 3, we find

\begin{array}{l} {| 1 - (1 - Δ_{2}) exp (- j ω τ_{2}) |}^{2} \approx Δ_{2}^{2} + {(2 π n τ_{2} / τ_{1})}^{2} = {| γ_{21} |}^{2} + {(2 π n τ_{2} / τ_{1})}^{2}, \\ {| 1 - (1 - Δ_{1}) exp (- j ω τ_{1}) |}^{2} \approx Δ_{1}^{2} = {| γ_{12} |}^{2}, \end{array}

so that the spectral power densities in the master and slave loops become

\begin{array}{l} \frac{P_{1} (ω_{n})}{N_{1} (ω_{n})} = \frac{1 + {(2 π n / | γ_{21} | τ_{1})}^{2}}{ɛ_{1}^{2} + {| γ_{12} / γ_{21} |}^{2} {(2 π n τ_{2} / τ_{1})}^{2}}, \\ \frac{P_{2} (ω_{n})}{N_{2} (ω_{n})} = \frac{1}{ɛ_{1}^{2} + {| γ_{12} / γ_{21} |}^{2} {(2 π n τ_{2} / τ_{1})}^{2}} . \end{array}

Spur suppression in the master loop occurs because the denominator in Eq. (12) only approaches its minimum value when both master and slave loops are resonant. The master loop spurs are attenuated by increasing the reverse injection ratio, reducing the forward injection ratio, and by increasing the slave loop length. The experimental data we will present in this work is consistent with these results. However in the optimal operating regime in which γ₂₁ ∼ γ₁₂ and both are large we found that it is necessary to use a full model to achieve quantitative accuracy [10].

3. Experimental setup

In this section we present a detailed description of the DIL-OEO. We describe the master and slave loops as well as the bi-directional injection bridge. We also outline steps taken to optimize the phase noise and stability of the DIL-OEO.

3.1. Master and slave loops

Figure 1 shows a schematic illustration of the DIL-OEO. The master and slave loops each contain a CW laser, a lithium niobate modulator, a length of optical fiber, a photodetector, a series of RF amplifiers, directional couplers leading to and from the bi-directional injection bridge, an RF filter, and output couplers to the measurement system. The CW lasers are distributed feedback (DFB) semiconductor diode lasers with wavelengths of 1550 nm. We used a 20 mW laser in the master loop, and an 80 mW laser in the slave loop.

We used standard Corning SMF28e single-mode optical fiber in all our experiments. High-Q OEOs typically have loop lengths between 2 km and 20 km. For our injection-locking study, we used a 4 km fiber spool in the master loop. We used two different slave loop lengths: 44 m and 547 m. We chose the slave loop lengths so that the slave loop supported no spurs within 500 kHz of the central RF oscillating tone. The 547 m slave supported approximately 16 spurs within the RF filter bandwidth, whereas the 44 m slave loop supported none.

We used semiconductor p-i-n photodetectors in both loops. Our photodetector responsivities were approximately 0.7 A/W at 1550 nm. The photodetectors we used had bandwidths of 12 GHz at their 3 dB rolloff frequencies. We used AML1010PNA2101 low-phase-noise RF amplifiers in both loops. These amplifiers have phase noise levels of −155 dBc/Hz at 1 kHz offset frequency, and noise figures of 6.5 dB. All amplifiers had 22 dB of small-signal gain, saturation output powers of 24 dBm at 10 GHz, and bandwidths of 2 GHz at their 3 dB rolloff frequencies. The RF filters ensured that the DIL-OEOs oscillated in an 8-MHz-wide frequency range around 10 GHz. The slave loop also included a tunable RF phase shifter. We used the phase shifter to tune the resonant frequency of the slave loop close enough to one of the resonant modes of the master loop to ensure phase-lock between both lops.

We placed output couplers in both loops so that we could measure their oscillating signals independently. We placed the output couplers immediately after the RF amplifiers so that we measured the RF signal at its maximum amplitude in each loop. The output powers were approximately 15 dBm. We required high output powers to drive the modulators and mixers in our phase noise measurement system with minimal amplification. In all of our experiments, the equilibrium oscillating power at the output couplers of the master and slave loops were within 1 dB of each other. We achieved nearly identical equilibrium powers in both master and slave loops by using RF amplifiers with similar saturation powers in both loops.

3.2. Bi-directional injection bridge

For the purposes of our study, the key feature of the DIL-OEO is the bi-directional injection bridge. We designed the injection bridge to allow us to vary the forward and reverse injection ratios independently. Figure 2 shows a schematic illustration of the bi-directional injection bridge. Both the master and slave loops contain output and input couplers. The bi-directional bridge consists of two arms. One arm leads from the master loop’s output coupler to an isolator, a variable attenuator and into the slave loop’s input coupler. The other arm of the bridge leads from the slave loop’s output coupler to a variable isolator, and ends at the input coupler of the master loop. The isolators in each arm provide 1 db of loss in their forward directions while providing 20 dB of loss in their reverse directions. These isolators ensure that 99% of the master-to-slave injection occurs in the forward injection arm, and that 99% of the slave-to-master injection occurs in the reverse injection arm. The effective forward and reverse injection ratios are determined by the input and output couplers as well as the variable attenuators in the injection bridge. The couplers used in this work are all bi-directional RF power dividers/combiners. For weakly coupled DIL-OEOs, we used −6 dB dividers were used as input and output couplers in both master and slave loops. We varied the forward and reverse injection ratios by varying the attenuation in each arm of the injection bridge. We used −6 dB couplers to construct our weakly-coupled bridge, and −3 dB couplers in our strongly-coupled bridge. In weakly-coupled DIL-OEOs, the attenuation in the injection bridge did not affect the signal levels in the DIL-OEO at steady state. However, when using −3 dB dividers for strong coupling, attenuation in the bridge did affect the power levels in both master and slave loops at steady state. Changes in the steady-state power levels in either loop altered both the phase noise and spur levels of the DIL-OEO. When building our strongly coupled DIL-OEO, we found it best to place the bridge after the final RF amplifier in both loops. Gain saturation in the final RF amplifier ensured that the input signal levels into the bridge at steady state remained constant even as we varied the forward and reverse injection ratios.

4. Experimental results

In this section, we present experimental data on the effects of forward injection, reverse injection, and slave loop length on the DIL-OEO’s phase noise and spurs. We collected all experimental data using a cross-correlation photonic delay line measurement system [13]. Our goal was determine the forward and reverse injection ratios that led to maximum spur suppression while maintaining low phase noise in the DIL-OEO.

In our reduced model, the bi-directional injection bridge is represented by a 2-by-2 matrix that couples the oscillating fields in the master and slave loops. The complex coupling coefficients are given by γ_ij, where subscripts 1 and 2 refer to the master and slave loops respectively. Experimentally, we measure the signal powers not the electric fields; so we will use the power coupling coefficients

Γ_{i j} = 10 {log}_{10} {| γ_{i j} |}^{2} .

We define the forward injection level as the ratio of the power of the signal injected from the master loop to the slave loop to the power of the master loop signal at the point of injection. Conversely, we define the reverse injection level as the ratio of the power injected from the slave loop to the master loop to the power of the slave loop signal at the point of injection. Given Eq. (23), the forward and reverse injection ratios are equal to Γ₂₁ and Γ₁₂ respectively.

4.1. Forward and reverse injection

As noted above, the forward injection ratio is defined as the ratio of the master-to-slave injected power to the oscillating power in the master loop at the point of injection. From Eq. (23), the forward injection level is

Γ_{12} = 10 {log}_{10} {| γ_{12} |}^{2} .

To test the effects of forward injection on phase noise and spurious mode levels, for fixed master and slave loops lengths, we vary the forward injection ratio while keeping the other injection ratios fixed. For this forward injection test, the injection ratios Γ₁₁ and Γ₂₂ were fixed at −3 dB. The reverse injection ratio Γ₁₂ was fixed at −15 dB. The forward injection ratio Γ₂₁ was set to the following values: −15 dB, −18 dB, −25 dB, and −35 dB. For each forward injection ratio, phase noise and spurious mode data was collected from both the master and slave loops of the DIL-OEO.

We performed the same procedure to test the effects of reverse injection on phase noise and spurious mode levels. In the case of reverse injection, the forward injection ratio Γ₂₁ was fixed at −15 dB, while the reverse injection ratio Γ₁₂ was set to the following values: −15 dB, −18 dB, −25 dB, and −35 dB. As with forward injection, the phase noise and spurious mode data was collected from both the master and slave loops of the DIL-OEO.

Figure 3 shows the dependence of the master loop signal parameters on both forward and reverse injection. Figure 3(a) shows that forward injection is 4 times more effective than reverse injection at altering master loop phase noise at 1 kHz. Similarly, Fig. 3(b) shows that the effect of forward injection on phase noise at 10 kHz is 8 times greater than that of reverse injection. Finally, Fig. 3(c) shows that reverse injection is 8 times more effective than forward injection at altering spur levels in the master loop. In Fig. 3 we have included theoretical data for comparison. This theoretical data was calculated using Eq. (12) which is valid for all values of γ₁₂ and γ₂₁ used.

Fig. 3 Effect of increasing the injection ratio on (a) the phase noise at 1 kHz, (b) the phase noise at 10 kHz, and (c) the first spur level of the master loop. Theoretical data derived from Eq. (12) are included for comparison.

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Figure 4 shows similar results for the slave loop signal. Forward injection has a greater effect than reverse injection on close-in phase noise and the Leeson bandwidth. The effect of forward injection on Leeson bandwidth can be determined from Fig. 4(b) which shows the phase noise at 10 kHz. At 10 kHz, the phase noise in the slave is primarily determined by the noise plateau which in turn is determined by the Leeson bandwidth of the DIL-OEO. Figure 4(c) shows that, as we observed in the master loop, the slave spur level is more strongly affected by reverse injection than forward injection. However, unlike in the master loop, there exists an optimal reverse injection ratio above which spur levels rise. We observe no such optimal point for forward injection over the range of values investigated. The above data suggests that increasing both forward and reverse injection in unison reduces both phase noise and spurious mode levels. Again, in Fig. 4 we have included theoretical data for comparison. This theoretical data was calculated using Eq. (12) which is valid for all values of γ₁₂ and γ₂₁ used.

Fig. 4 Effect of increasing the injection ratio on (a) the phase noise at 1 kHz, (b) the phase noise at 10 kHz, and (c) the first spur level of the slave loop. Theoretical data derived from Eq. (12) are included for comparison.

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4.2. Slave loop length

The preceding experimental data was obtained from the DIL-OEO using a 44 m slave loop. The slave loop length was chosen to ensure that the slave supported only a one mode within the bandwidth of the 8 MHz bandpass filter. However, because of the relatively short delay, the free-running slave loop noise is 30 dB higher than the free-running master loop noise at 1 kHz. Theoretical models predict that increasing slave loop length should increase suppression of the first and second spurs of the DIL-OEO [9,10,14]. In this section, we present experimental data on the effects of increasing slave loop length on the phase noise and spurs in a DIL-OEO.

We replaced the 44 m fiber spool in the slave loop of the DIL-OEO with a 547 m fiber spool. Figure 5 shows a plot of the free-running phase noise data for both slave loops. The 547 m slave loop’s phase noise is 15 dB lower than the that of the 44m from 1 kHz to 100 kHz. However, the 547 m slave has a spur level of −88 dBc/Hz at 375 kHz. From the free-running data, we expect the longer slave to reduce the spurious modes of the DIL-OEO at 50 and 100 kHz while introducing an additional mode at 375 kHz. We constructed DIL-OEOs using both slave loops while fixing the master loop length at 4 km. We fixed the forward and reverse injection ratios at −18 dB.

Fig. 5 Phase noise data for free-running 44 m and 547 m slave loops. The 547 m slave has a spur at 375 kHz while the 44 m slave has no spurs over the entire measurement range.

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Figure 6 shows phase noise data from both DIL-OEOs. Fig. 6(a) shows that increasing the slave loop’s length had no effect on the master loop phase noise. However, increasing the slave loop length reduced the first and second spurs in the master loop by 20 and 8 dB respectively. All of the master loop’s spurs were reduced by increasing slave loop length including the spur closest to the free-running slave loop spur at 375 kHz.

Fig. 6 Phase noise and spurious mode data from free-running and injection-locked OEOs.

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Figure 6(b) shows the phase noise data from the slave loop. Increasing slave loop length decreased the first and second spurs by 28 and 8 dB respectively. In the long-slave-loop DIL-OEO, the spur levels rose as their offset frequencies approached that of the free-running slave loop spur. At 354 kHz, the long-slave-loop spur level was 17 dB higher than that of the short slave loop. However, the maximum spur level in the long slave loop (−113 dBc/Hz at 354 kHz) was 8 dB lower than the maximum spur level in the short slave loop (−105 dBc/Hz at 50 kHz). Furthermore, the phase noise at offset frequencies greater than 10 kHz was also reduced by increasing the slave loop length.

We note that the Leeson bandwidth, remained unchanged. The master and slave loop signals diverged at 1.5 kHz regardless of the slave loop’s length. Nevertheless, the slave loop phase noise beyond the Leeson bandwidth decreased as we increased the slave loop length. At high offset frequencies, the injection-locked slave loop signal benefited from the reduced phase noise of the free-running slave loop. We conclude that, as in the master loop, increasing slave loop length improves the slave loop signal. In subsequent sections, we use a 547 m slave loop in the DIL-OEO because its benefits outweigh the disadvantage of the additional spurious mode at 354 kHz.

4.3. Coupling strength

In Sec. 4.1, we presented experimental data that suggests that the phase noise and spurious mode levels in both master and slave loops of the DIL-OEO may be reduced by increasing the forward and reverse injection ratios in unison. Both our full and reduced models also suggest that increased coupling should improve DIL-OEO performance [9, 10, 14]. In order to investigate the effects of increasing both forward and reverse injection, we constructed a strong-injection bridge. The weak-injection bridge used in the forward and reverse injection studies −6 dB couplers and could provide at most −15 dB forward and reverse injection. In the strong-injection bridge, we used −3 dB couplers in both the forward and reverse injection arms.

We achieved forward and reverse injection ratios of up to −7 dB with the strong-injection bridge. However, the insertion losses Γ₁₁ and Γ₂₂ were increased to −7 dB. For the purposes of this study, we define a coupling coefficient C given by

C \equiv \frac{Γ_{21}}{Γ_{11}} \approx \frac{Γ_{12}}{Γ_{22}} .

Increasing C corresponds to increasing both forward and reverse injection ratios at the same rate.

In our experimental study of coupling strength, we fixed Γ₁₁ and Γ₂₂ of the DIL-OEO at −7 dB. We varied the coupling coefficient by varying Γ₁₂ and Γ₂₁ using the variable attenuators in the bi-directional bridge. The coupling coefficients used in the study were: 0 dB, −3 dB, −6 dB, −10 dB,and −20 dB. For each coupling coefficient, we measured the phase noise in both the master and slave loops of the DIL-OEO.

Figure 7 shows the phase noise at 1 kHz versus coupling coefficient for both free-running and injection-locked master and slave loops. Increasing coupling from −20 dB to 0 dB reduced the slave loop noise by 3 dB while leaving the master loop noise unchanged. Over the set of coupling coefficients used, the best noise performance at 1 kHz was obtained by setting the coupling coefficient to 0 dB. Setting C to either 0 dB or −6 dB ensured that the phase noise power spectral densities at 1 kHz of both master and slave loops were within 3 dB of the phase noise power spectral density of the free-running master loop.

Fig. 7 A plot of the phase noise at 1 kHz versus the coupling coefficient. The data were collected from the injection-locked master and slave loops. Phase noise at 1 kHz from the free-running, master, and slave loops are all included for comparison.

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Figure 8 shows the effect of coupling strength on the noise at 10 kHz. Again the effect of coupling on the slave loop signal is greater than the effect on the master loop signal. Increasing the coupling from −20 dB to 0 dB reduced the slave loop noise by 13 dB but had no effect on the master loop noise. The injection-locked master loop phase noise at 10 kHz was lower than the phase noise of the free-running master loop because the white noise power in the free-running slave loop was lower than that of the free-running master loop.

Fig. 8 A plot of the phase noise at 10 kHz versus the coupling coefficient. The data were collected from the injection-locked master and slave loops. Phase noise at 10 kHz from the free-running, master, and slave loops are all included for comparison.

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In the slave loop, we used a 19 dBm laser in order to minimize the white noise in the slave loop. In the master loop, on the other hand we used a 13 dBm laser that provides lower flicker noise than does the slave loop’s laser because the noise in the DIL-OEO is dominated by the master loop’s contribution at offset frequencies within the Leeson bandwidth. The noise beyond the Leeson bandwidth is a combination of the noise from the master and slave loops. As one consequence, the noise in the injection-locked slave loop for −20 dB coupling is higher than the noise in the free-running slave loop.

Figure 9 shows the spur level versus coupling strength in both master and slave loops. Using −6 dB coupling generated the lowest spurs in both loops. The lowest master loop spur level was −128 dBc/Hz, which is 50 dB lower than that of the free-running master loop. The lowest spur level in the slave was −134 dBc/Hz, which is within 4 dB of the noise level of the free-running slave loop at the same offset frequency.

Fig. 9 A plot of spur level versus coupling coefficient. The data was collected from the injection-locked master and slave loops.

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Figure 10 shows the phase noise data of the optimized DIL-OEO. Using −6 dB coupling provided the best combination of low phase noise and low spur levels in both the master and slave loops. The slave loop signal, in particular, benefited from increased coupling. We succeeded in reducing the spur level to within 4 dB of the noise level of the free-running slave loop while reducing the phase noise at 1 and 10 kHz by 13 and 7 dB respectively. The noise penalty of the injection-locked slave relative to the free-running master loop was 8 dB at 10 kHz and less than 3 dB at 1 kHz, while the spur reduction was over 60 dB. Our experimental results are consistent with the predictions of both our full and reduced models [9, 10].

Fig. 10 A plot of the phase noise data from a DIL-OEO with −6 dB coupling between master and slave loops. The phase noise was measured from both master and slave loops. Phase noise data from the free-running master and slave loops are included for comparison.

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5. Conclusion

We investigated the effects of forward and reverse injection on the steady-state signal of the DIL-OEO. We found that increasing both forward and reverse injection in tandem reduced spur levels in the master loop while also reducing phase noise in the slave loop. Increasing slave loop length also reduced spur levels in the loop. The improved understanding of injection-locking obtained from our experimental data, coupled with insights from our theoretical models, enabled us to optimize the phase noise and spur levels of the DIL-OEO.

We have experimentally constructed a DIL-OEO with the low phase noise of a long-loop OEO and the spurious mode levels of a short-loop OEO. Our results demonstrate that the dual injection-locked OEO is capable of combining the high-Q of a long cavity OEO with the single-mode behavior of a short loop OEO. Having circumvented the tradeoff between Q and mode-spacing, we have demonstrated that the DIL-OEO is a practical source of low phase noise RF signals.

References and links

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2. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996). [CrossRef]

3. D. A. Howe and A. Hati, “Low-noise X-band oscillator and amplifier technologies: Comparison and status,” in “Proceedings of the Joint IEEE International Frequency Control Symposium and Precise Time and Time Interval (PTTI) Systems and Applications Meeting,” (Vancouver, Canada, 2005), pp. 481–487.

4. D. Green, C. McNeilage, and J. H. Searls, “A low phase noise microwave sapphire loop oscillator,” in “Proceedings of the IEEE International Frequency Control Symposium,” (Miami, FL, 2006), pp. 852–860.

5. X. S. Yao and L. Maleki, “Ultra-low phase noise dual-loop optoelectronic oscillator,” in “Technical Digest of the Optical Fiber Communication Conference and Exhibit (OFC ’98),” (San Jose, CA, 1998), pp. 353–354.

6. X. S. Yao, L. Davis, and L. Maleki, “Coupled optoelectronic oscillators for generating both RF signal and optical pulses,” J. Lightwave Technol. 18, 73–78 (2000). [CrossRef]

7. W. Zhou and G. Blasche, “Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level,” IEEE Trans. Microwave Theory Tech. 53, 929–933 (2005). [CrossRef]

8. O. Okusaga, W. Zhou, E. C. Levy, M. Horowitz, G. M. Carter, and C. R. Menyuk, “Experimental and simulation study of dual injection-locked OEOs,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 875–879.

9. C. R. Menyuk, E. C. Levy, O. Okusaga, M. Horowitz, G. M. Carter, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator (DIL-OEO),” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 870–874.

10. E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B 26, 148–159 (2009). [CrossRef]

11. E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification,” Opt. Express 18, 21461–21476 (2010). [CrossRef] [PubMed]

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13. E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurement of microwave oscillators,” J. Opt. Soc. Am. B 22, 987–997 (2005). [CrossRef]

14. E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optoelectronic oscillators,” in “Proceedings of Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium,” (Besancon, France, 2009), pp. 505–507.

Spurious mode reduction in dual injection-locked optoelectronic oscillators

Abstract

1. Introduction

1.1. The single-loop optoelectronic oscillator

1.2. Dual-cavity optoelectronic oscillators

1.3. The dual injection-locked optoelectronic oscillator

2. Theory

2.1. The Yao-Maleki single-loop model

2.2. The reduced DIL-OEO model

2.3. Effects of injection-locking on the close-in phase noise

2.4. Effects of injection-locking on spurious modes

3. Experimental setup

3.1. Master and slave loops

3.2. Bi-directional injection bridge

4. Experimental results

4.1. Forward and reverse injection

4.2. Slave loop length

4.3. Coupling strength

5. Conclusion

References and links

Cited By

Figures (10)

Equations (25)

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