Abstract

We have developed a comprehensive simulation model for accurately studying the dynamics in optoelectronic oscillators (OEOs). Although the OEO is characterized by three widely separated time scales, our model requires neither long run times nor a large amount of memory storage. The model generalizes the Yao–Maleki model and includes all of the physical effects in the Yao–Maleki model as well as other physical effects that are needed to calculate important features of the OEO dynamics, such as the impact of the fast response time of the modulator on the phase noise power spectral density, the fluctuations of the OEO output due to the input noise, the cavity mode competition during the OEO start-up, and temporal amplitude oscillations in steady state. We show that the absolute value of the phase noise is 23dB lower than predicted by the Yao–Maleki model. The Yao–Maleki model does not take into account amplitude noise suppression due to the fast time response of the modulator, which accounts for this difference. We show that a single cavity mode oscillates in the OEO at steady state, and this mode is determined by the noise that is present when the OEO is turned on. When the small-signal open-loop gain is higher than 2.31, we show that the OEO amplitude oscillates in steady state. This temporal amplitude oscillation can be suppressed by using a narrow filter. Our simulation model, once extended to include flicker (1f) noise and different amplifier and modulator designs, will enable its users to accurately design OEOs.

© 2008 Optical Society of America

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References

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  1. X. S. Yao and L. Maleki, “Converting light into spectrally pure microwave oscillation,” Opt. Lett. 21, 483-485 (1996).
    [Crossref] [PubMed]
  2. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. A 13, 1725-1735 (1996).
    [Crossref]
  3. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141-1149 (1996).
    [Crossref]
  4. X. S. Yao and L. Maleki, “Dual microwave and optical oscillator,” Opt. Lett. 22, 1867-1869 (1997).
    [Crossref]
  5. X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36, 79-84 (2000).
    [Crossref]
  6. 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]
  7. N. Yu, E. Salik, and L. Maleki, “Ultralow-noise mode-locked laser with coupled opto-electronic oscillator conguration,” Opt. Lett. 30, 1231-1233 (2005).
    [Crossref] [PubMed]
  8. W. Zhou, S. Weiss, and C. Fazi, “Developing RF-photonics components for the army's future combat systems,” in Proceedings of the 25th Army Science Conference (2004), paper NO-02 (Report A231334, available at http://www.stormingmedia.us/23/2313/A231334.html).
  9. 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]
  10. M. C. Li, “A high precision Doppler radar based on optical fiber delay loops,” IEEE Trans. Antennas Propag. 52, 3319-3328 (2004).
    [Crossref]
  11. 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]
  12. D. Eliyahu, K. Sariri, M. Kamran, and M. Tokhmakhian, “Improving short and long term frequency stability of the opto-electronic oscillator,” in Proceedings of the IEEE International Frequency Control Symposium (IEEE, 2002), pp. 580-583.
  13. F. W. J. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegan, eds. (Dover, 1972), pp. 355-434.
  14. A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart and Winston, 1976).
  15. G. S. Fishman, “Generating samples,” in Monte Carlo: Concepts, Algorithms, and Applications (Springer, 1996), pp. 145-254.
  16. L. S. Cutler and C. L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proc. IEEE 54, 136-154 (1966).
    [Crossref]
  17. D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multiloop optoelectronic oscillator,” in Proceedings of the 2003 IEEE International Frequency Control Symposium (IEEE, 2003), pp. 405-410.

2007 (1)

2005 (3)

2004 (1)

M. C. Li, “A high precision Doppler radar based on optical fiber delay loops,” IEEE Trans. Antennas Propag. 52, 3319-3328 (2004).
[Crossref]

2000 (1)

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36, 79-84 (2000).
[Crossref]

1997 (1)

1996 (3)

X. S. Yao and L. Maleki, “Converting light into spectrally pure microwave oscillation,” Opt. Lett. 21, 483-485 (1996).
[Crossref] [PubMed]

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

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

1966 (1)

L. S. Cutler and C. L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proc. IEEE 54, 136-154 (1966).
[Crossref]

Bendoula, R.

Blasche, G.

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]

Chembo, Y. K.

Colet, P.

Cutler, L. S.

L. S. Cutler and C. L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proc. IEEE 54, 136-154 (1966).
[Crossref]

Dahan, D.

Eisenstein, G.

Eliyahu, D.

D. Eliyahu, K. Sariri, M. Kamran, and M. Tokhmakhian, “Improving short and long term frequency stability of the opto-electronic oscillator,” in Proceedings of the IEEE International Frequency Control Symposium (IEEE, 2002), pp. 580-583.

D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multiloop optoelectronic oscillator,” in Proceedings of the 2003 IEEE International Frequency Control Symposium (IEEE, 2003), pp. 405-410.

Fazi, C.

W. Zhou, S. Weiss, and C. Fazi, “Developing RF-photonics components for the army's future combat systems,” in Proceedings of the 25th Army Science Conference (2004), paper NO-02 (Report A231334, available at http://www.stormingmedia.us/23/2313/A231334.html).

Fishman, G. S.

G. S. Fishman, “Generating samples,” in Monte Carlo: Concepts, Algorithms, and Applications (Springer, 1996), pp. 145-254.

Kamran, M.

D. Eliyahu, K. Sariri, M. Kamran, and M. Tokhmakhian, “Improving short and long term frequency stability of the opto-electronic oscillator,” in Proceedings of the IEEE International Frequency Control Symposium (IEEE, 2002), pp. 580-583.

Larger, L.

Li, M. C.

M. C. Li, “A high precision Doppler radar based on optical fiber delay loops,” IEEE Trans. Antennas Propag. 52, 3319-3328 (2004).
[Crossref]

Maleki, L.

N. Yu, E. Salik, and L. Maleki, “Ultralow-noise mode-locked laser with coupled opto-electronic oscillator conguration,” Opt. Lett. 30, 1231-1233 (2005).
[Crossref] [PubMed]

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36, 79-84 (2000).
[Crossref]

X. S. Yao and L. Maleki, “Dual microwave and optical oscillator,” Opt. Lett. 22, 1867-1869 (1997).
[Crossref]

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

X. S. Yao and L. Maleki, “Converting light into spectrally pure microwave oscillation,” Opt. Lett. 21, 483-485 (1996).
[Crossref] [PubMed]

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

D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multiloop optoelectronic oscillator,” in Proceedings of the 2003 IEEE International Frequency Control Symposium (IEEE, 2003), pp. 405-410.

Olver, F. W.

F. W. J. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegan, eds. (Dover, 1972), pp. 355-434.

Rubiola, E.

Salik, E.

Sariri, K.

D. Eliyahu, K. Sariri, M. Kamran, and M. Tokhmakhian, “Improving short and long term frequency stability of the opto-electronic oscillator,” in Proceedings of the IEEE International Frequency Control Symposium (IEEE, 2002), pp. 580-583.

Searle, C. L.

L. S. Cutler and C. L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proc. IEEE 54, 136-154 (1966).
[Crossref]

Shumakher, E.

Tavernier, H.

Tokhmakhian, M.

D. Eliyahu, K. Sariri, M. Kamran, and M. Tokhmakhian, “Improving short and long term frequency stability of the opto-electronic oscillator,” in Proceedings of the IEEE International Frequency Control Symposium (IEEE, 2002), pp. 580-583.

Weiss, S.

W. Zhou, S. Weiss, and C. Fazi, “Developing RF-photonics components for the army's future combat systems,” in Proceedings of the 25th Army Science Conference (2004), paper NO-02 (Report A231334, available at http://www.stormingmedia.us/23/2313/A231334.html).

Yao, X. S.

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36, 79-84 (2000).
[Crossref]

X. S. Yao and L. Maleki, “Dual microwave and optical oscillator,” Opt. Lett. 22, 1867-1869 (1997).
[Crossref]

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

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

X. S. Yao and L. Maleki, “Converting light into spectrally pure microwave oscillation,” Opt. Lett. 21, 483-485 (1996).
[Crossref] [PubMed]

Yariv, A.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart and Winston, 1976).

Yu, N.

Zhou, W.

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]

W. Zhou, S. Weiss, and C. Fazi, “Developing RF-photonics components for the army's future combat systems,” in Proceedings of the 25th Army Science Conference (2004), paper NO-02 (Report A231334, available at http://www.stormingmedia.us/23/2313/A231334.html).

IEEE J. Quantum Electron. (2)

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

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36, 79-84 (2000).
[Crossref]

IEEE Trans. Antennas Propag. (1)

M. C. Li, “A high precision Doppler radar based on optical fiber delay loops,” IEEE Trans. Antennas Propag. 52, 3319-3328 (2004).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

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]

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]

Opt. Lett. (5)

Proc. IEEE (1)

L. S. Cutler and C. L. Searle, “Some aspects of the theory and measurement of frequency fluctuations in frequency standards,” Proc. IEEE 54, 136-154 (1966).
[Crossref]

Other (6)

D. Eliyahu and L. Maleki, “Low phase noise and spurious level in multiloop optoelectronic oscillator,” in Proceedings of the 2003 IEEE International Frequency Control Symposium (IEEE, 2003), pp. 405-410.

D. Eliyahu, K. Sariri, M. Kamran, and M. Tokhmakhian, “Improving short and long term frequency stability of the opto-electronic oscillator,” in Proceedings of the IEEE International Frequency Control Symposium (IEEE, 2002), pp. 580-583.

F. W. J. Olver, “Bessel functions of integer order,” in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegan, eds. (Dover, 1972), pp. 355-434.

A. Yariv, Introduction to Optical Electronics, 2nd ed. (Holt, Rinehart and Winston, 1976).

G. S. Fishman, “Generating samples,” in Monte Carlo: Concepts, Algorithms, and Applications (Springer, 1996), pp. 145-254.

W. Zhou, S. Weiss, and C. Fazi, “Developing RF-photonics components for the army's future combat systems,” in Proceedings of the 25th Army Science Conference (2004), paper NO-02 (Report A231334, available at http://www.stormingmedia.us/23/2313/A231334.html).

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Figures (15)

Fig. 1
Fig. 1

Schematic of the OEO.

Fig. 2
Fig. 2

Phase noise spectral density (solid curve) as a function of the frequency offset obtained for an OEO with a loop delay of τ = 0.28 μ s , oscillation power of 14.77 dBm , filter bandwidth of Γ = 20 MHz , noise power spectral density of ρ N = 10 17 mW Hz , small-signal open-loop gain of G S = 1.5 , and a voltage gain of G A = 7.5 . The noise was calculated over M = 10,000 round trips. A least-squares fit of the phase noise curve (dashed-dotted line) yields 55.5 20 log 10 ( f ) .

Fig. 3
Fig. 3

Phase noise spectral density S ϕ ( f ) calculated for three different integration times: (a) M τ = ( b ) 8.4, 2.8, and (c) 0.84 ms .

Fig. 4
Fig. 4

Comparison between the phase noise spectral density, S ϕ ( f ) , calculated by averaging the phase noise over 350 simulations (solid curve) and by performing a least-squares fit to a single simulation result that is shown in Fig. 2 (dashed-dotted line). The curves are compared to that obtained by using the Yao–Maleki model [2] (dashed curve). The approximately 2.5 dB difference is due to the neglect in the Yao–Maleki model of the modulator’s fast gain response.

Fig. 5
Fig. 5

Comparison between a phase noise spectral density S ϕ ( f ) (dark gray curve), amplitude noise (light gray curve), and the complete RF spectrum S RF ( f ) (black curve), which are calculated by averaging the noise over 350 simulations. The comparison justifies the approximation that S RF ( f ) S ϕ ( f ) for a wide frequency range in which the amplitude noise is negligible relative to the phase noise.

Fig. 6
Fig. 6

Phase noise spectral density of the OEO at a frequency offset of 30 kHz as a function of loop delay τ (circles) and the filter bandwidth in each case was Γ = 5.6 τ . The least-squares fitting of the data points (solid line) is given by: 155.4 20 log 10 [ τ ( μ s ) ] , which is in agreement with the results of the Yao–Maleki model (dashed line): 152.8 20 log 10 [ τ ( μ s ) ] . The simulation phase noise was extracted by fitting a curve at each delay using a least-squares fit, as shown in Fig. 2. A difference between the models of approximately 2.5 dB is visible at all loop delays.

Fig. 7
Fig. 7

Comparison between phase noise spectral density at a frequency offset 30 kHz offset as a function of the spectral noise density power calculated by using the Yao–Maleki model (dashed line) and by using our simulation (circles). The least-squares fit of the simulation data points (solid line), given by 25.62 + ρ N ( dBm Hz ) , is compared to the result of the Yao–Maleki model (dashed line): 28.25 + ρ N ( dBm Hz ) .

Fig. 8
Fig. 8

Oscillation in one of several oscillating cavity modes where the RF filter bandwidth Γ is equal to 20 MHz and the round-trip time τ is equal to 2 μ s . (a) The RF spectrum (left) and the real part of the amplitude (right) when the steady-state cavity mode has a frequency f c . (b) The RF spectrum (left) and the real part of the amplitude (right) when the cavity mode that oscillates in steady state has a frequency f c + 1 τ . (c) The RF spectrum (left) and the real part of the amplitude (right) when the steady-state cavity mode has a frequency f c + 2 τ .

Fig. 9
Fig. 9

(a) Probability density function of the normalized oscillating frequency when Γ τ = 40 . The normalized oscillating frequency is distributed with a standard deviation of σ mod = 0.96 . (b) The standard deviation of the normalized oscillating frequency distribution as a function of Γ τ .

Fig. 10
Fig. 10

(a) Normalized amplitude and (b) phase noise obtained for a small-signal open-loop gain G S = 1.5 . The steady-state amplitude does not depend on time and is equal to 1.75 V π π . The other OEO parameters are the same as in Fig. 2.

Fig. 11
Fig. 11

Normalized amplitude obtained for a small-signal open-loop gain G S = 2.4 (solid curve). The normalized amplitude changes between 2.035 and 2.755 (dashed lines) as predicted by our threshold condition. The other OEO parameters are the same as in Fig. 2.

Fig. 12
Fig. 12

Minimum and maximum amplitude obtained for a small-signal open-loop gain G S = 2.4 as a function of the filter bandwidth. The other OEO parameters and the pulse shape are the same as in Fig. 2. The dashed lines show our threshold condition and the minimum and maximum amplitude excursions.

Fig. 13
Fig. 13

Normalized amplitude obtained for a small-signal open-loop gain G S = 2.75 . Amplitude oscillations with a period of 4 τ are obtained as predicted by our threshold condition. The other OEO parameters are the same as in Fig. 2.

Fig. 14
Fig. 14

(a) Amplitude and (b) phase obtained for a small-signal open-loop gain G S = 3 . The dashed line in (a) gives the amplitude derived from the averaged power [ 2 R P avg ( T ) ] 1 2 . The power is averaged over a time duration of T P = 5 τ (dashed curve) and is compared to the result of the Yao–Maleki model 2.68 V π π (dashed-dotted line). The simulation parameters are the same as in Fig. 2.

Fig. 15
Fig. 15

Dependence of the time-averaged oscillation power as a function of the small-signal open-loop gain calculated in the simulations (diamonds) and compared to the power from the Yao–Maleki model [2] (dashed curve). The averaging time was 10 τ . Good agreement is obtained between the results although the assumption in the Yao–Maleki model that the amplitude does not change in time is not valid for G S > 2.31 .

Equations (27)

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V in ( t , T ) = a in mod ( T ) cos [ ω c t + ϕ ( T ) ] = 1 2 a in mod ( T ) exp ( i ω c t ) + c.c. ,
V out ( t , T ) = V ph ( 1 η sin { π [ V in ( t , T ) V π + V B V π ] } ) ,
V ph = α P 0 ρ R G A 2 ,
V out ( t , T ) = V ph { 1 η sin ( π V B V π ) J 0 ( π a in mod ( T ) V π ) 2 η sin ( π V B V π ) m = 1 ( 1 ) m J 2 m ( π a in mod ( T ) V π ) cos [ 2 m ω c t + 2 m ϕ ( T ) ] 2 η cos ( π V B V π ) m = 0 ( 1 ) m J 2 m + 1 ( π a in mod ( T ) V π ) cos [ ( 2 m + 1 ) ω c t + ( 2 m + 1 ) ϕ ( T ) ] } ,
V out ( t , T ) = D.C. η V ph cos ( π V B V π ) J 1 ( π a in mod ( T ) V π ) exp [ i ϕ ( T ) ] exp ( i ω c t ) + H.H. + c.c. ,
a out amp ( T ) = 2 η cos ( π V B V π ) V ph J 1 ( π a in mod ( T ) V π ) exp [ i ϕ ( T ) ] .
a out fil ( T ) exp ( 2 π i f c T ) = T a in fil ( T ) exp ( 2 π i f c T ) f ( T T ) d T ,
a ( T ) = k = a ̃ ( f k ) exp ( 2 π i f k T ) ,
a ̃ out fil ( f k ) = F ( f k + f c ) a ̃ in fil ( f k ) ,
F ( f ) = f ( T ) exp ( 2 π i f T ) d f .
F ( f k + f c ) = i Γ 2 f k + f c f 0 i Γ 2 ,
f ( T ) = π Γ exp [ 2 π i ( f 0 + i Γ 2 ) T ] u ( T ) ,
a in , 2 τ l ( T ) = { a l 1 ( T ) for ( l 2 ) τ T < ( l 1 ) τ a l ( T ) for ( l 1 ) τ T < l τ } .
a ̃ out l , amp ( f ) = G A [ a ̃ in l , amp ( f ) + w ̃ l ( f ) ] ,
a ̃ M τ ( f ) F M τ [ a ( T ) ] = 1 M τ 0 M τ a M τ ( T + T 0 ) exp ( 2 π i f T ) d T .
ϕ ̃ M τ ( f ) = F M τ [ ϕ ( T ) ] .
S ϕ M τ ( f ) = ϕ ̃ M τ ( f ) 2 δ f .
S ϕ M τ ( f ) S RF M τ ( f ) = F M τ [ a ( T ) ] 2 2 R P osc δ f ,
A NL = 2 ρ N δ f R .
G S = η π V ph V π cos ( π V B V π ) ,
P avg ( T ) = 1 T P T P 2 T P 2 a ( T T ) 2 2 R d T ,
V out = G ( a in fil ) a in fil cos ( ω c t + ϕ ) ,
G ( a in fil ) = 2 G S V π π a in fil J 1 ( π a in fil V π ) .
π a l + 1 ( T ) V π = 2 G S J 1 ( π a l ( T ) V π ) .
x l = x l + 1 = f G S ( x l ) ,
f G S ( x ) = 2 G S J 1 ( x ) .
x l = x l + 2 = f G S [ f G S ( x l ) ] = f G S 2 ( x l ) .

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