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

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References

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  1. X. S. Yao, and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. A 13, 1725-1735 (1996).
    [CrossRef]
  2. 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]
  3. 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]
  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. 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]
  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 optoelectronic 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 22, 987-997 (2005).
    [CrossRef]
  11. 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]

2009 (1)

2007 (1)

2005 (3)

1996 (1)

X. S. Yao, and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. A 13, 1725-1735 (1996).
[CrossRef]

1995 (1)

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]

Bendoula, R.

Blasche, G.

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]

Chembo, Y. K.

Colet, P.

Dahan, D.

Eisenstein, G.

Horowitz, M.

Huang, S.

Kasdin, N. J.

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]

Larger, L.

Levy, E. C.

Maleki, L.

Menyuk, C. R.

Rubiola, E.

Salik, E.

Shumakher, E.

Tavernier, H.

Yao, X. S.

X. S. Yao, and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. A 13, 1725-1735 (1996).
[CrossRef]

Yu, N.

Zhou, W.

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]

IEEE Proc. (1)

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]

IEEE Trans. Microw. Theory Tech. (1)

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]

J. Opt. Soc. Am. A (1)

X. S. Yao, and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. A 13, 1725-1735 (1996).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (2)

Other (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).

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).

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Figures (10)

Fig. 1
Fig. 1

Schematic illustration of the single-loop OEO configuration

Fig. 2
Fig. 2

(a) The measured electrical power at the output of the 3 RF amplifiers as a function of the input power. (b) The measured spectrum of the RF filter as measured experimentally and used in the model.

Fig. 3
Fig. 3

Schematic description of the DIL-OEO. The DIL-OEO operates in a master-slave configuration. The longer loop, referred to as the master loop, generates a harmonic signal with a very low phase noise. The shorter loop, referred to as the slave loop, is used to decrease the amplitude of the spurs. Part of the master signal, Γ12, is injected into the slave loop, as indicated schematically by the solid arrow. The dashed arrow indicates that part of the slave signal, Γ21, is coupled back into the master loop.

Fig. 4
Fig. 4

Comparison between the experimentally measured noise spectral density, S RF(f) (solid gray line) and the noise spectral density that is calculated by using a model that excludes flicker noise (dashed-dotted green line) and a generalized model that includes flicker noise (solid red line). All the theoretical curves were calculated assuming the same white noise, oscillation power, and small signal gain. The OEO loop-delay equals τ = 31.7 μs and the phase flicker coefficient equals b −1 = 10−11.

Fig. 5
Fig. 5

Dependence of the phase flicker coefficient on the loop length as extracted from single-loop OEO measurements (red dots). The dependence was found to be approximately linear (dashed line): b −1 = 2×10−12 + 2.5×10−15 L, where the fiber length L is measured in meters. The accuracy of the extracted dots is limited by the accuracy of the measured data, which is approximately 3 dB. Therefore, the error-bars of the extracted dots are equal to 3 dB.

Fig. 6
Fig. 6

(a) Phase noise of the master loop (blue) and the slave loop (red) when the loops are free-running and function as single-loop OEOs. (b) Phase noise of the master loop (magenta) and the slave loop (green) when the loops are injection-locked. The inset zooms in on the first spur in the master and slave loops. Good agreement is achieved between the experimental results (thin lines or light colors) and the theoretical results (thick lines or dark colors) when the loops are injection-locked. The injection power coefficients were Γ11 = −0.3 dB, Γ22 = −2.5 dB, and Γ12 = Γ21 = −20 dB. Theory shows that the first spur in the master loop is about 20 dB lower than in the unlocked case.

Fig. 7
Fig. 7

(a) Calculated phase noise for the free-running slave loop (red) and for the injection-locked slave loop (green) compared to experimental results (thin lines or light colors). The phase noise within the locking range is determined by the master loop. (b) The calculated first spur of the injection-locked master loop (magenta) compared to the spur in the free-running loop (blue). The spur is reduced by approximately 20 dB by injection-locking. The theoretical results (thick lines or dark colors) are also compared to the corresponding experimental results (thin lines or light colors).

Fig. 8
Fig. 8

Calculated spectrum of the phase noise in the neighborhood of the first spur of the master loop as R Γ varies for τ 1 = 20 μs and (a) τ 2 = 0.2 μs, (b) τ 2 = 2 μs. We show results for R Γ = −40 dB, −20 dB, and 0 dB. For comparison, we also show the spur level when the master loop is free-running and functions as a single-loop OEO.

Fig. 9
Fig. 9

Calculated dependence of the spur level on the power injection ratio when τ 2 = 0.2 μs (red triangles) and when τ 2 = 2 μs (blue circles).

Fig. 10
Fig. 10

Experimentally measured spectrum of the phase noise in the vicinity of the first spur. The spur level of the free-running master loop, which has a loop delay of τ 1 = 20 μs, was −75 dBc/Hz (red). By injection-locking the master loop to a slave loop with a loop delay of τ 2 = 0.2 μs and a power injection ratio of R Γ = −20 dB, we reduced the first spur to −110 dBc/Hz (green). Increasing the slave loop-delay to τ 2 = 2.5 μs, and keeping the same power injection ratio, we measured a spur of −109 dBc/Hz (cyan). The spur was reduced to −129 dBc/Hz by increasing the power injection ratio to R Γ = −6 dB (magenta).

Equations (7)

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V in ( t ) = | a in mod ( t ) | cos [ ω c t φ ( t ) ] = 1 2 a in mod ( t ) exp ( i ω c t ) + c . c . ,
a out PD ( T ) = α P 0 η ρ R cos ( π V B / V π , DC ) J 1 ( π | a in mod ( T ) | / V π , AC ) exp [ i φ ( T ) ] .
Θ ( ν n ) = H ( ν n ) W ( ν n ) ,
H ( ν n ) = [ 1 exp ( 2 π i ν n ) ] 1 / 2 , H ( ν M / 2 + 1 ) = 0 ,
W ( ν n ) = k = 1 M w k exp [ 2 π i ( k 1 ) ( n 1 ) / M ] .
θ k = ( 1 / M ) n = 1 M Θ ( ν n ) exp [ 2 π i ( k 1 ) ( n 1 ) / M ] .
( a 1 ( t ) a 2 ( t ) ) = ( γ 11 γ 21 γ 12 γ 22 ) ( a 1 ( t ) a 2 ( t ) ) .

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