## Comprehensive computational model of single- and dual-loop optoelectronic oscillators with experimental verification |

Optics Express, Vol. 18, Issue 20, pp. 21461-21476 (2010)

http://dx.doi.org/10.1364/OE.18.021461

Acrobat PDF (1350 KB)

### Abstract

We describe a comprehensive computational model for single-loop and dual-loop optoelectronic oscillators (OEOs). The model takes into account the dynamical effects and noise sources that are required to accurately model OEOs. By comparing the computational and experimental results in a single-loop OEO, we determined the amplitudes of the white noise and flicker noise sources. We found that the flicker noise source contains a strong component that linearly depends on the loop length. Therefore, the flicker noise limits the performance of long-cavity OEOs (⪆ 5 km) at low frequencies (*f* < 500 Hz). The model for a single-loop OEO was extended to model the dual-loop injection-locked OEO (DIL-OEO). The model gives the phase-noise, the spur level, and the locking range of each of the coupled loops in the OEO. An excellent agreement between theory and experiment is obtained for the DIL-OEO. Due to its generality and accuracy, the model is important for both designing OEOs and studying the physical effects that limit their performance. We demonstrate theoretically that it is possible to reduce the first spur in the DIL-OEO by more than 20 dB relative to its original performance by changing its parameters. This theoretical result has been experimentally verified.

© 2010 Optical Society of America

## 1. Introduction

1. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. A **13**, 1725–1735 (1996). [CrossRef]

*Q*RF cavity. Such a high-

*Q*cavity makes possible the generation of a high-frequency signal whose phase noise is nearly independent of the oscillation frequency. Furthermore, the use of a long optical fiber and a tunable narrowband RF filter makes it possible to tune the oscillating signal over a very broad frequency range. These important advantages make the OEO an attractive candidate to replace classical oscillators such as multiplied quartz crystals or phase-locked dielectric resonator oscillators.

2. 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]

3. D. Dahan, E. Shumakher, and G. Eisenstein, “Self-starting ultralow-jitter pulse source based on coupled opto-electronic oscillators with an intracavity fiber parametric amplifier,” Opt. Lett. **30**, 1623–1625 (2005). [CrossRef] [PubMed]

*et al.*[2

2. 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]

1. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. A **13**, 1725–1735 (1996). [CrossRef]

*et al.*[5

5. Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. **32**, 2571–2573 (2007). [CrossRef]

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

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

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

*f*) noise source, gain saturation in the RF amplifiers, and the spectrum of the RF filter. By extracting the magnitude of the phase flicker noise amplitude from phase measurements in single-loop OEOs with different cavity lengths, we have found that the phase flicker noise contains a strong component with an amplitude that depends on the square-root of cavity length. Therefore, in long-cavity OEOs (

*L*> 5 km) the phase noise at low frequencies (

*f*< 500 Hz) is dominated by the phase flicker noise. A good quantitative agreement between theory and experiments is obtained for the phase noise spectrum and the spur levels of both the slave and the master loops in the DIL-OEO. In contrast to the reduced model that we previously described in [4], this model takes into account the full OEO dynamics, including the growth of the oscillator signal from noise, rather than assuming steady-state operation. Thus, unlike the model in [4], it is capable of determining the locking bandwidth of the two loops in the DIL-OEO. Moreover, this model takes into account the locations of the lumped elements in each loop, rather than treating them as distributed elements. As a consequence, this model can accurately describe the effect of large lumped coupling between the loops of the DIL-OEO, as well as gain, loss, and gain saturation. Since this computational model is based on the full physics, it is inherently more trustworthy than the reduced models. As is almost the case when comparing simplified or reduced models to more complete or full models, there is a tradeoff between computational speed and accuracy. Our view is that for modeling OEOs, this tradeoff favors the use of the full model. It runs quickly — taking less than two minutes of CPU time on a standard desktop computer for one set of parameters — and has allowed us to examine a broad parameter range.

## 2. Model description

**26**, 148–159 (2009). [CrossRef]

*μ*s for a 40 m loop, corresponding to a frequency of 5 MHz, and may be as long as 30

*μ*s for a 6 km loop, corresponding to 33 kHz. Finally, we have the frequency scale of the phase noise. We are interested in the frequency range from about 1 Hz to 100 kHz. A point that should be emphasized is that the last two scales are not well-separated. In most previous models [1

1. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. A **13**, 1725–1735 (1996). [CrossRef]

5. Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. **32**, 2571–2573 (2007). [CrossRef]

### 2.1. Modeling the oscillating signal and its noise in a single-loop OEO

**26**, 148–159 (2009). [CrossRef]

*V*

_{in}(

*t*), is approximately a sinusoidal wave with an angular carrier frequency

*ω*= 2π

_{c}*f*, a time-dependent phase

_{c}*ϕ*(

*t*), and a time-dependent amplitude

*V*

_{in}(

*t*). Since the OEO signal is narrow-band we assume that |

*dϕ/dt*| ≪

*ω*and

_{c}*mω*with

_{c}*m*> 1, where

*m*is an integer, but these higher harmonics are filtered out, so that it is sufficient to model only the propagation of the phasor that represents the signal around the angular carrier frequency

*ω*. The evolution of the phasor

_{c}*α*is the insertion loss in the modulator and the detector,

*P*

_{0}is the optical power at the modulator input,

*η*is a parameter determined by the extinction ratio of the modulator (1 +

*η*)/(1 −

*η*),

*ρ*is the responsivity of the photodetector,

*R*is the impedance at the output of the detector,

*V*

_{π,DC}and

*V*

_{π,AC}are the modulator half-wave voltages for the DC and AC voltages, respectively, and

*V*is the DC bias voltage.

_{B}**26**, 148–159 (2009). [CrossRef]

_{F}= 8 MHz. In our model, we used the experimentally measured transmission function that is shown in Fig. 2(b). To ensure that the RF filter has a casual response, a linear phase chirp was added to the measured transmission function that corresponds to a delay of 0.1

*μ*s.

*N*points, which implies a time separation or resolution time

*δt*=

*τ/N*. The number of points was chosen so that the simulation bandwidth Δ

*f*= 1/

*δt*is broader than the RF filter FWHM bandwidth, Ω

_{F}. Furthermore, we checked that the results did not change when we increased the number of points

*N*.

*ρ*

_{SN}, and the thermal noise power density,

*ρ*

_{th}, respectively. The photodetector’s shot noise power density is evaluated using the formula:

*ρ*

_{SN}= 2

*eI*

_{PD}

*R*, where

_{τ}denotes averaging over the round-trip time. The spectral power density of the thermal noise, given by

*ρ*

_{th}= (NF)

*k*

_{B}

*T*, is determined by the noise factor NF of the RF amplifiers. The noise factor was determined empirically, and in order to obtain the best match between theory and experiment we typically used a noise factor of NF = 4. We added noise to the oscillating phasor in the same manner that is described in [6

**26**, 148–159 (2009). [CrossRef]

*N*mutually independent noise variables

*w*= 1, …,

_{i}, i*N*, to the array of the oscillating phasor, such that the variance of the noise variables is set by the relation 〈|

*w*|

_{i}^{2}〉

_{τ}/2

*R*=

*ρ/τ*, where

*ρ*is the noise power density. A complex Gaussian distribution is assumed, and each of the real and imaginary parts of the noise variables is normally distributed with a variance

*ρR/τ*. The noise is added in the simulation after the photodetector and before each of the RF amplifiers with a noise power density of

*ρ*

_{SN}and

*ρ*

_{th}, respectively. We note that the main contribution of the thermal noise to the phase noise is from the noise that is added at the input of the first RF amplifier. As a result, the phase noise in our simulation is practically determined by the total white noise that is added between the photodetector and the first RF amplifier, which has a noise power density of

*ρ*

_{total}=

*ρ*

_{th}+

*ρ*

_{SN}.

*N*

_{RT}round trips in order to calculate the phase noise power spectral density at low frequencies. The phase noise of the oscillating signal is calculated using the Fourier transform of the accumulated phasor [6

**26**, 148–159 (2009). [CrossRef]

*T*

_{tot}where

*T*

_{tot}=

*N*

_{Rt}

*τ*is the overall accumulation time of the phasor in the simulation run. The power spectral density of the phase noise is then calculated by averaging the power spectral density that we obtain from individual simulation runs over

*N*

_{avg}realizations. In our simulations, we used a range of accumulation times

*T*

_{tot}= 10 – 100 ms, a range of resolution times

*δt*= 50 – 60 ns, and a range of numbers of realizations

*N*

_{avg}= 20 – 100. We checked the convergence of our computational results by verifying that the power spectral density of the phase noise did not change when we increased the resolution time

*δt*or the number of realizations

*N*

_{avg}.

### 2.2. Modeling the flicker noise

11. N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/*f ^{α}* power law noise generation,” IEEE Proc.

**83**, 802–827 (1995). [CrossRef]

*θ*= 1, …,

_{k}, k*M*, with an averaged spectrum of 〈

*S*(

_{θ}*f*)〉 =

*b*

_{−1}/

*f*. The length of the time series is determined by the ratio between the total accumulation time and the resolution time of the simulation,

*M*=

*T*

_{tot}/

*δt*=

*N*

_{RT}

*N*. We started with discrete white Gaussian noise in the time domain,

*w*. The variance of the white noise was set so that 〈

_{k}*w*〉

_{k}^{2}= 2

*πb*

_{−1}. The filtering in the simulation was implemented in the frequency domain, so that

*n*= 1, …,

*M, ν*= −1/2 + (

_{n}*n*− 1)/

*M*is the normalized fourier frequency, Θ(

*ν*) is the filtered noise,

_{n}*H*(

*ν*) is the filter response in the frequency domain given by

_{n}*W*(

*ν*) is the discrete Fourier transform of

_{n}*w*defined by

_{k}*ν*)

_{n}*θ*has an averaged spectrum of 〈

_{k}*S*(

_{θ}*f*)〉 =

*b*

_{−1}/

*f*. The series

*θ*is generated at the beginning of each simulation run. During each round trip, we use

_{k}*N*subsequent terms of the series to multiply the array of the phasor by exp(

*iθ*), so that by the end of the run all the terms of the series are used only once.

_{k}### 2.3. Modeling dual-loop OEOs

*τ*

_{1}and

*τ*

_{2}, when

*τ*

_{2}≤

*τ*

_{1}. Following the terminology in [2

2. 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]

*τ*

_{1}, as the master loop, and to the OEO loop with the shorter loop delay,

*τ*

_{2}, as the slave loop. We assume an arbitrary carrier frequency

*f*that is approximately equal to the expected oscillating frequency of the DIL-OEO, and we denote the phasors in the master loop and the slave loop with respect to the carrier frequency as

_{c}*a*

_{1}(

*t*) and

*a*

_{2}(

*t*), respectively.

*γ*= 1, 2:

_{ij}, i, j*a*(

_{i}*t*) and

*i*= 1) and in the slave loop (

*i*= 2), respectively. The forward injection coefficient, Γ

_{12}= |

*γ*

_{12}|

^{2}, represents the relative injected power from the master to the slave loop and the backward injection coefficient, Γ

_{21}= |

*γ*

_{21}|

^{2}, represents the relative injected power from the slave to the master loop. Unidirectional injection corresponds to Γ

_{21}= 0.

*a*

_{1}(

*t*) and

*a*

_{2}(

*t*) in a round trip in each of the OEO loops. In our model implementation, the phasors

*a*

_{1}(

*t*) and

*a*

_{2}(

*t*) were sampled with the same resolution time

*δt*, which is typically 50 nanoseconds and was always chosen so that

*τ*

_{2}/

*δt*=

*N*

_{2}is an exact integer. In a 40 m loop we typically chose

*N*

_{2}= 4. In this case, we find

*τ*

_{1}=

*N*

_{1}

*δt*=

*δτ*

_{1}, where |

*δτ*

_{1}| <

*δt*/2. In our simulations, we retained

*N*

_{2}values of

*a*

_{2}(

*t*) and

*N*

_{1}values of

*a*

_{1}(

*t*) in two separate arrays. The arrays of the phasor in the master loop and in the slave loop before the injection bridge are denoted by

*a*

_{1}(

*i*

_{1}) and

*a*

_{2}(

*i*

_{2}), respectively, such that

*i*

_{1}= 1, …,

*N*

_{1}and

*i*

_{2}= 1, …,

*N*

_{2}. The arrays of the phasor in the master loop and in the slave loop after the injection bridge are denoted by

*k*= 1, …,

*M*be the iteration index and

*M*be the number of the total accumulated terms in the simulation run in each of the loops, such that

*M*=

*T*

_{tot}/

*δt*, and let us assume that

*δτ*

_{1}= 0. We let

*i*

_{1}= mod (

*i,N*

_{1}) + 1 and

*i*

_{2}= mod(

*i,N*

_{2}) + 1. Thus, the variables

*i*

_{1}and

*i*

_{2}pass cyclically through the values

*i*

_{1}= 1, …,

*N*

_{1}and

*i*

_{2}= 1, …,

*N*

_{2}. In each iteration, we used

*a*

_{1}(

*i*

_{1}) and

*a*

_{2}(

*i*

_{2}), for which we calculated the phasors after the injection bridge,

*i*=

_{j}*N*= 1,2, we used the array of the phasor after the bridge,

_{j}, j*a*(

_{j}*i*), for the following round-trip. The evolution of the phasor after the bridge in each loop was calculated by taking into account the response of all the lumped components on the phasor, as well as the additive white noise and the multiplicative flicker noise. The evolution of the phasor array in each loop was calculated in the same manner as in the single-loop OEO model that was described in sub-section 2.1 and in [6

_{j}**26**, 148–159 (2009). [CrossRef]

*δτ*

_{1}≠ 0 by adding a constant phase shift of

*δϕ*

_{1}= −2

*πf*

_{c}δτ_{1}to the phasor array of the master loop after it propagates for a time

*N*

_{1}

*δt*that is approximately equal to the round-trip duration of the master loop. Our model’s ability to treat the case

*δτ*

_{1}≠ 0 allows us to take into account the incommensurability of the two loops, which is always present in practice.

**26**, 148–159 (2009). [CrossRef]

## 3. Comparison with experimental results

*V*

_{π,AC}= 5 V,

*V*

_{π,DC}= 3.15 V,

*V*= 2.6 V, and

_{B}*η*= 0.7. We used the specified photodetector responsivity: 0.8 A/W at DC and 0.55 A/W at 10 GHz. The measured optical power

*P*

_{0}is 17 mW and the impedance at the output of the photodetector was

*R*= 50 Ω. The first step of the comparison was to compare the theoretical and the measured RF power at the output of the detector. We added an effective loss between −0.4 dB and −0.9 dB to the model in order to match the measured RF power. The photodetector’s shot noise power density,

*ρ*

_{SN}, was then determined from the round-trip averaged photodetector’s current

*I*

_{PD}. We empirically set the noise factor NF of the RF amplifiers — and hence the thermal noise power density

*ρ*

_{th}= (NF)

*k*

_{B}

*T*— and the flicker noise coefficient

*b*

_{−1}, so that we obtained the best match between theory and experiment, as described in sub-section 3.1.

### 3.1. Loop length dependance of the phase flicker noise in a single-loop OEO

*τ*= 31.7

*μ*s. The figure shows that in order to obtain good agreement between theory and experiment, a phase flicker noise source must be added. Good agreement between theory and experiment was also obtained when the loop delay was equal to 0.5,2.7,7,10,15, or 27

*μ*s, corresponding to loop lengthes between 100 and 5400 m, when an appropriate amount of flicker noise was added. For all the OEO lengths, the measured oscillation power at the photodiode was −22 ± 1 dBm and the oscillating power at the output of the RF amplifiers was equal to 23.3 ± 0.3 dBm. The total white noise power density that was used in the theoretical model for all the loop delays was equal to

*ρ*= 9 × 10

_{N}^{−20}± 0.5 × 10

^{−20}W/Hz, which can be obtained by assuming that the photodetector is limited by shot noise and that the RF amplifiers have a noise factor NF = 4. The only free parameters in the simulation that we changed when the cavity length was varied were the flicker coefficient

*b*

_{−1}, which determines the power density of the phase flicker noise source, and the effective loss. We varied the effective loss in the model between −0.4 dB and −0.9 dB in order to match the theoretical RF power of the photodetector output to the measured power. We chose the flicker noise coefficient,

*b*

_{−1}so that the theoretical phase noise matches the measured phase noise. Figure 5 demonstrates the dependence of the extracted flicker noise coefficient on the cavity length. Each dot in Fig. 5 was extracted from the measured phase noise. The accuracy of each dot is limited by the noise in the measured data, which approximately equals 3 dB. The figure shows that the flicker coefficient can be divided into a noise component that is independent of the cavity length

*b*

_{−1}= 2 × 10

^{−12}and a component that depends on the cavity length

*L*. The component that does not depend on the loop length is consistent with the typical flicker noise power that is observed in RF amplifiers [10

10. 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]

*f*< 500 Hz) of long-cavity OEOs (

*L*> 5 km). The dependance of the flicker noise power on the loop length that was obtained in the single-loop OEO was successfully used as an input to our model for the DIL-OEO.

### 3.2. Comparison between theory and experiment in the DIL-OEO

_{12}= Γ

_{21}= 0 and Γ

_{11}= −0.3 dB, Γ

_{22}= −2.5. The other parameters used for the two loops were the same as as those we used in the model for the single-loop OEO. We included white noise and flicker noise in the model. The power density of the white noise was equal to

*ρ*

_{N,1}= 2.4 × 10

^{−20}W/Hz for the master loop and to

*ρ*

_{N,2}= 9 × 10

^{−21}W/Hz for the slave loop. The flicker noise coefficient that we used was equal to

*b*

_{−1}= 10

^{−11}for the master loop and

*b*

_{−1}= 10

^{−12}for the slave loop. The flicker noise coefficient of the slave loop is consistent with the flicker noise coefficient that was measured for RF amplifiers [10

10. 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]

_{11}= −0.3 dB, Γ

_{22}= −2.5 dB, and Γ

_{12}= Γ

_{21}= −20 dB. The coupling between the two loops was experimentally implemented using a phase shifter before the coupling bridge in the same loop, so that the coupling coefficients

*γ*are real numbers. We obtained good quantitative agreement between theory and experiment for the phase noise spectrum as well as for the spur levels in both OEO loops in the injection-locked case as demonstrated in Fig. 6(b). Figure 7(a) shows that the model accurately describes the reduction of the phase noise in the slave loop within the frequency locking range of the two loops. Within the locking range, the phase noise in the slave loop is mainly determined by the phase noise of the master loop. Fig. 7(b) shows that the first spur level in the injection-locked master loop is reduced by approximately 20 dB — from –95 dBc/Hz to – 115 dBc/Hz — compared to the spur level when it is free-running.

_{ij}## 4. Theoretical study of approaches to decrease the first spur in the master loop

_{12}≃ Γ

_{21}and Γ

_{11}≃ Γ

_{22}. We define the power injection ratio

*R*

_{Γ}= Γ

_{12}/Γ

_{11}= Γ

_{21}/Γ

_{22}, and we will determine the optimum performance as we vary this quantity and the slave loop’s delay time. In the computational work that we will present in this section, we have set Γ

_{11}= Γ

_{22}= 0.25. In the experimental verifications that we present here, we set Γ

_{11}to within 0.5 dB of Γ

_{22}.

*R*

_{Γ}varies. In Fig. 8(a), we present results when the slave loop has a loop delay of 0.2

*μ*s, corresponding to a loop length of approximately 40 m, and in Fig. 8(b), we present results when the slave loop has a loop delay of 2.0

*μ*s, corresponding to a loop length of approximately 400 m. We set

*R*

_{Γ}= −40 dB, −20 dB, and 0 dB. The other parameters are the same as those that we used for modeling the experimental setup that we described in the previous section. For comparison, we also show the free-running case in both Fig. 8(a) and Fig. 8(b). We recall that when the master loop is free-running, it functions as a single-loop OEO. In Fig. 9, we summarize the key results from Fig. 8 by showing the maximum spur level as a function of

*R*

_{Γ}. We find that the optimal power injection ratio in the original experimental setup [7], in which the slave loop delay is 0.2

*μ*s, is equal to −20 dB. We also find that by increasing the slave loop delay from 0.2 to 2.0

*μ*s, the spur level in the master loop is reduced from −115 dBc/Hz to −125 dBc/Hz when

*R*

_{Γ}= −20 dB and is further reduced to a level of −135 dBc/Hz by increasing

*R*

_{Γ}to −6 dB. We note that the phase noise of the injection-locked master loop in all the cases that we show in Fig. 9 is approximately equal to the phase noise of the free-running master loop. Thus, we theoretically predict a reduction of the first spur level in the injection-locked master loop by approximately 20 dB relative to the spur level in the original setup, while maintaining approximately the same low phase noise. We note that although Fig. 9 demonstrates the reduction of only the first spur level, other spurs are reduced as well. The second spur level, for example, is reduced from −123 dBc/Hz in the original setup to less than −140 dBc/Hz by increasing the slave loop length from 0.2

*μ*s to 2

*μ*s and by increasing the power injection ratio

*R*

_{Γ}from −20 dB to −6 dB.

*μ*s. The level of the first spur in the master loop is mainly determined by the phase noise of the free-running slave loop and the injection parameters. Thus, it is possible to decrease the spur level in the master loop by increasing the length of the slave loop, which decreases the slave loop’s free-running phase noise. However, increasing the length of the slave loop reduces its cavity mode spacing and can increase the level of the high-order spurs in the master loop in cases where they nearly coincide with spurs in the slave loop. If these spurs lie within the bandwidth of the RF filter, they can exceed the lower spurs in magnitude and degrade the performance of the DIL-OEO. We have not carried out a detailed study of this issue, as it has not been present in our experiments to date, in which the maximum slave loop delay has been 2.5

*μ*s. However, we note that this issue will place a practical limit on how long it is possible to make the slave loop

_{12}≃ Γ

_{21}and Γ

_{11}≃ Γ

_{22}— a point that we experimentally verified. The best spur reduction was achieved by increasing the slave loop-delay from 0.2

*μ*s to 2.5

*μ*s and by varying

*R*

_{Γ}. The loop-delay of 2.5

*μ*s is 0.5

*μ*s longer than in the original theoretical studies. The maximum reduction in the spur level was achieved by increasing

*R*

_{Γ}from −20 dB to −6 dB. To achieve this large power injection ratio, we implemented a new bridge that has Γ

_{11}= −7.5 dB and Γ

_{22}= −7.0 dB, so that this bridge has approximately 6 dB more loss than in our earlier experiments [7]. We implemented this bridge by using four 3-dB couplers, with two placed in each loop. In each loop, the front coupler was connected to the rear coupler in each loop and to the rear coupler in the other loop. With this configuration, we could obtain power injection ratios,

*R*

_{Γ}= Γ

_{12}/Γ

_{11}≃ Γ

_{21}/Γ

_{22}as large as 0 dB. We could then reduce

*R*

_{Γ}by adding attenuators between the front coupler of one loop and the rear coupler of another loop.

*μ*s, was −75 dBc/Hz. By injection-locking the master loop to a slave loop with

*τ*

_{2}= 0.2

*μ*s and

*R*

_{Γ}= −20 dB, the spur level decreased to −110 dBc/Hz. When we increased the slave loop delay to

*τ*

_{2}= 2.5

*μ*s and increased

*R*

_{Γ}to −6 dB, the spur level further decreased to −129 dBc/Hz. These values are both within 2 dB of the calculated results with this set of parameters.

*τ*

_{2}= 2.5

*μ*s with a power injection ratio

*R*

_{Γ}= −20 dB in the experiments, we could not maintain a stable phase lock between the master and the slave loops. The experimentally-measured spur level in this case was −109 dBc/Hz, instead of −120 dBc/Hz, as theoretically predicted. Our model assumes that the master and slave OEO loops are phase-locked, and it will not provide reliable answers unless there is a good lock. These results underline the importance of achieving a good phase lock — not only to achieve good agreement between theory and experiment, but also to obtain good performance from the OEO. We will say more about the conditions to achieve a good phase lock in a future publication.

_{21}from the slave loop to the master loop also reduces the first spur, but at the expense of increasing the master phase noise within the locking range. When the

*Q*factor of the master loop, as defined in [1

**13**, 1725–1735 (1996). [CrossRef]

*Q*factor of the slave loop,

*Q*

_{1}≫

*Q*

_{2}, the master phase noise in the injection-locked case is approximately unaffected by the locking as long as Γ

_{21}= Γ

_{12}≪ 1 [4].

## 5. Conclusions

## Acknowledgement

## References and links

1. | X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. A |

2. | 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. |

3. | D. Dahan, E. Shumakher, and G. Eisenstein, “Self-starting ultralow-jitter pulse source based on coupled opto-electronic oscillators with an intracavity fiber parametric amplifier,” Opt. Lett. |

4. | C. R. Menyuk, E. C. Levy, O. Okusaga, G. M. Carter, M. Horowitz, and W. Zhou, “An analytical model of the dual-injection-locked opto-electronic oscillator,” IFCS (2009). |

5. | Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. |

6. | E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling opto-electronic oscillators,” J. Opt. Soc. Am. B |

7. | 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,” IFCS (2009). |

8. | O. Okusaga, E. J. Adles, E. C. Levy, M. Horowitz, G. M. Carter, C. R. Menyuk, and W. Zhou, “Spurious mode suppression in dual injection-locked optoelectronic oscillators,” IFCS (2010). |

9. | E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, “Study of dual-loop optolectronic oscillators,” IFCS (2009). |

10. | 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 |

11. | N. J. Kasdin, “Discrete simulation of colored noise and stochastic processes and 1/ 83, 802–827 (1995). [CrossRef] |

**OCIS Codes**

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(230.0250) Optical devices : Optoelectronics

(230.4910) Optical devices : Oscillators

**ToC Category:**

Optical Devices

**History**

Original Manuscript: July 14, 2010

Revised Manuscript: September 3, 2010

Manuscript Accepted: September 7, 2010

Published: September 24, 2010

**Citation**

Etgar C. Levy, Olukayode Okusaga , Moshe Horowitz, Curtis R. Menyuk, Weimin Zhou, and Gary M. Carter, "Comprehensive computational model of single- and dual-loop optoelectronic
oscillators with experimental verification," Opt. Express **18**, 21461-21476 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21461

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### References

- X. S. Yao, and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. A 13, 1725-1735 (1996). [CrossRef]
- W. Zhou, and G. Blasche, "Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultralow spurious level," IEEE Trans. Microw. Theory Tech. 53, 929-933 (2005). [CrossRef]
- D. Dahan, E. Shumakher, and G. Eisenstein, "Self-starting ultralow-jitter pulse source based on coupled optoelectronic oscillators with an intracavity fiber parametric amplifier," Opt. Lett. 30, 1623-1625 (2005). [CrossRef] [PubMed]
- C. R. Menyuk, E. C. Levy, O. Okusaga, G. M. Carter, M. Horowitz, and W. Zhou, "An analytical model of the dual-injection-locked opto-electronic oscillator," IFCS (2009).
- Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, "Dynamic instabilities of microwaves generated with optoelectronic oscillators," Opt. Lett. 32, 2571-2573 (2007). [CrossRef]
- E. C. Levy, M. Horowitz, and C. R. Menyuk, "Modeling opto-electronic oscillators," J. Opt. Soc. Am. B 26, 148-159 (2009). [CrossRef]
- 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," IFCS (2009).
- O. Okusaga, E. J. Adles, E. C. Levy, M. Horowitz, G. M. Carter, C. R. Menyuk, and W. Zhou, "Spurious mode suppression in dual injection-locked optoelectronic oscillators," IFCS (2010).
- E. C. Levy, O. Okusaga, M. Horowitz, C. R. Menyuk, W. Zhou, and G. M. Carter, "Study of dual-loop optoelectronic oscillators," IFCS (2009).
- 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]
- N. J. Kasdin, "Discrete simulation of colored noise and stochastic processes and 1/ f〈 power law noise generation," IEEE Proc. 83, 802-827 (1995). [CrossRef]

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