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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 12 — Jun. 9, 2008
  • pp: 9067–9072
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Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators

Y. Kouomou Chembo, Laurent Larger, Ryad Bendoula, and Pere Colet  »View Author Affiliations


Optics Express, Vol. 16, Issue 12, pp. 9067-9072 (2008)
http://dx.doi.org/10.1364/OE.16.009067


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Abstract

We show that when subjected to high gain, delay-line optoelectronic oscillators can display a strongly multimode behavior depending on the feedback bandwidth. We found that this dynamical regime may arise when the bandwidth of the feedback loop spans over several hundreds of ring cavity-modes, and also when the oscillator is switched on abruptly. Such a persistent multimode regime is detrimental to the performances of this system which is normally intended to provide ultra-pure and single-mode microwaves. We experimentally evidence this multimode dynamics and we propose a theory to explain this undesirable feature.

© 2008 Optical Society of America

1. Introduction

Narrow-bandwidth optoelectronic oscillators (OEOs) can generate ultra-pure microwave frequencies for aerospace and telecommunication applications [1

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

]. Single-mode OEOs are able to produce very sharp-peaked radio-frequencies, with extremely low phase noise, so that they are expected to be extremely useful in various areas of physics and technology where particularly high spectral purity is required. However, under certain conditions, we have observed that the OEO’s behavior is highly multimode, something incompatible with its intrinsic purpose which is to provide ultra-pure microwave frequencies through single-mode operation. In fact, if multi-longitudinal mode dynamics has been thoroughly studied in other oscillators like lasers (see for example refs. [2

2. L. M. Narducci, al et, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986). [CrossRef] [PubMed]

, 3

3. K. A. Winick, “Longitudinal mode competition in chirped grating distributed feedback lasers,” IEEE J. Quantum Electron. 35, 1402–1411 (1999). [CrossRef]

, 4

4. A. M. Yacomotti, et al, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69, 053816-1-9 (2004). [CrossRef]

, 5

5. J. L. Font, R. Vilaseca, F. Prati, and E. Roldán, “Coexistence of single-mode and multi-longitudinal mode emission in the ring laser model,” Opt. Commun. 261, 336–341 (2006). [CrossRef]

, 6

6. T. Voigt, et al, “Experimental investigation of RiskenNummedalGrahamHaken laser instability in fiber ring lasers,” Appl. Phys. B 79, 175–183 (2004). [CrossRef]

, 7

7. C. Y. Wang, et al, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802(R)-1-4 (2007). [CrossRef]

]), no study has focused on the case of OEOs. The aim of this article is therefore to evidence this multimode behavior in OEOs, and to give a theoretical insight into the mode coupling mechanisms, through time-domain modeling.

Fig. 1. Experimental setup for the single-loop OEO.

2. The experimental system

The system under study is a single-loop OEO [1

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

], corresponding to the experimental set-up presented in Fig. 1. In the optoelectronic feedback loop, a continuous-wave semiconductor laser of optical power P feeds a LiNbO3 Mach-Zehnder (MZ) modulator of DC and RF half-wave voltages VπDCVπRF=4.2V . This MZ modulator is biased with a voltage VB, and subjected to a radio-frequency modulation voltage V(t). The modulated optical signal then travels through a 4-km long thermalized fiber delay-line, inducing a time-delay T=20 µs which corresponds to a free spectral range ΩT/2π=1/T=50 kHz. A fast amplified photodiode with a conversion factor S converts the optical signal into an electrical one, which is amplified by a microwave amplifier (RF driver) with gain G. In the electrical path, a narrow-band microwave filter is inserted in order to select the frequency range for the amplified modes; its central frequency is Ω0/2π=3 GHz, and the -3 dB bandwidth is ΔΩ/2π=20 MHz (corresponding to a quality factor Q0/ΔΩ=150). Therefore, there are ΔΩ/ΩT=400 ring-cavity modes inside the bandwidth, and of course several thousands outside. All optical and electrical losses are gathered in a single attenuation factor κ.

Experimentally, when the system is switched on by continuous tuning of the OEO loop gain from below to above threshold, it oscillates in a single-mode fashion, and with an ultra-low phase noise as expected. But when the switching is abrupt above threshold, the OEO remains highly multimode even after waiting for very long transients, as it can be seen in Fig. 2. The multimode dynamics is sustained in a 4 MHz span around the central frequency of the RF filter, and the power difference between the oscillating modes and the damped modes within the filter bandwidth is approximately equal to 40 dB. This result clearly demonstrates that depending on the number of cavity modes within the RF filter bandwidth, OEOs may display a complex multimode dynamics, depending on the switching-on procedure.

Fig. 2. Experimental radio-frequency spectrum of the OEO after 30 seconds (dashed line) and after 2 hours (continuous line), in a 10 MHz window. The modal competition can be considered in this case as permanent.

3. Multiple timescale analysis of the multimode behavior

Our aim is to investigate the theoretical origin for this persistent multimode behavior in OEOs with the time-domain model introduced in ref. [8

8. Y. Kouomou 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]

]. The dynamics of the dimensionless microwave variable x(t)=πV(t)2VπRF is ruled by [9

9. Y. Kouomou Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic microwave oscillators,” IEEE J. Quantum Electron. (in press).

]

x+1ΔΩdxdt+Ω02ΔΩt0tx(s)ds=βcos2[x(tT)+ϕ],
(1)

where β=πκSGP2VπRF is the normalized feedback gain and ϕ=πVB2VπDC is the Mach-Zehnder offset phase. Then, around the carrier frequency Ω0, the complex slowly-varying envelope 𝒜(t)=|𝒜(t)|e (t) of the quasi-sinusoidal microwave variable x(t) obeys

𝒜.=μ𝒜+2μγJc1[2𝒜T]𝒜T,
(2)

where μ=ΔΩ/2 is the half-bandwidth of the RF filter, γ=β sin2ϕ is the effective gain of the feedback loop, and Jc1 is the Bessel-cardinal function defined as Jc1(x)=J1(x)/x [8

8. Y. Kouomou 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]

]. We have also adopted the conventional notation 𝒜T≡𝒜(t-T), and the phase matching condition eiΩ0T=1 has been considered.

We have simulated the deterministic Eq. (2) with two different initial conditions, corresponding to a smooth and abrupt start for the OEO: the smooth case corresponds the bifurcation diagrams as in [8

8. Y. Kouomou 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]

], where the below to above threshold condition is crossed with a continuous increase of the single-mode amplitude; the abrupt start corresponds to a situation where Gaussian random numbers are considered as initial condition in the interval [-T,0]. The result for this latter case is displayed in Fig. 3, and it can be seen that after one second, the dynamics of the OEO is highly multimode. The oscillating modes are within a 3 MHz bandwidth and 40 dB above the damped modes, in excellent agreement with the experimental result. It should be emphasized that 1 second is a macroscopic timescale for this system, for which the largest time-scale is T=20 µs. Hence, this simulation confirms that a persistent multimode behavior is observed, which may or not lead to a single-mode behavior after a transient process which is very long.

Fig. 3. Numerical simulation of the radio-frequency spectrum of the OEO after 1 s, in a 10 MHz window. The thin line represents the result of the simulation, while the thick line represents an averaging of the spectrum with a 125 kHz resolution, in order to facilitate comparison with the oscilloscope display of Fig. 2.

To provide an analytical insight into the multimode behavior of the OEO, we use a multiple time scale analysis to decompose the dynamics according to the three timescales Ω0, μ and ΩT which are separated by the same three orders of magnitude. Considering the smallness (or separation) parameter ε~10-3, the initial microwave variable oscillations can be expanded as

x(t)=k=03εkxk(T0,T1,T2,T3)+𝒪(ε4),
(3)

while its amplitude expands as

𝒜(t)=k=03εk𝒜k(T0,T1,T2,T3)+𝒪(ε4),
(4)

where xk and |𝒜|k are associated to the timescales Tk=εkt. The time derivative can also be expanded as d/dt=∑3 k=0 εkDk where Dk=/∂Tk. If we rescale the temporal parameters as μ=εμ^ and ΩT=ε 2 Ω^ T, the timescale components of the integral variable u(t)=t0tx(s)ds obey the following equations at the various orders of ε:

  • Order ε 0
    D02u0+Ω02u0=0
    (5)
  • Order ε 1
    D02u1+Ω02u1=2D0D1u02μ̂D0u0
    +2μ̂γJ1[2𝒜0T]×cos[Ω0T0+ψT]
    (6)
  • Order ε 2
    D02u2+Ω02u2=2D0D1u1(D12+2D0D2)u02μ̂(D0u1+D1u0)
    +2μ̂γ{2𝒜1TJ1[2𝒜0T]}×cos[Ω0T0+ψT]
    (7)
  • Order ε 3
    D02u3+Ω02u3=2D0D1u2(D12+2D0D2)u1
    2(D0D3+D1D2)u02μ̂(D0u2+D1u1+D2u0)
    +2μ̂γ{2𝒜2TJ1[2𝒜0T]+2𝒜1T2J1[2𝒜0T]}
    ×cos[Ω0T0+ψT]
    (8)

These equations correspond to the various dynamical behaviors at each timescale. From Eq. (5) corresponding to order ε 0, we can deduce that u 0 is sinusoidal, so that its integrand x 0 may also be expressed as

x0=12𝒜(T1,T2,T3)eiΩ0T0+12𝒜*(T1,T2,T3)eiΩ0T0,
(9)

which corresponds to our initial hypothesis of quasi-sinusoidal oscillations for x. The spectral splitting of Ω0, μ and ΩT suggests the following decomposition

𝒜(T1,T2,T3)=A(T1)n=NNan(T3)einΩ̂TT2,
(10)

𝒜0=Ω0ANNaneinΩ̂TT2
𝒜1=12Ai(A*T1AA*AT1)NNaneinΩ̂TT2
𝒜2=12Ω0AAT12NNaneinΩ̂TT2+12Ω̂TA
×[NNaneinΩ̂TT2NNnan*einΩ̂TT2+c.c.][NNaneinΩ̂TT2],
(11)

where “c. c.” stands for the complex conjugate of the preceding term.

The dynamical equation ruling the dynamics of the slow variation A(T 1) of the carrier is obtained by setting to zero the secular term in Eq. (6), thereby yielding

AT1=μ̂A+2μ̂γJ1[2𝒜0T][2𝒜0T]AT.
(12)

It is interesting to note that the amplitude A(T 1) which represents the fastest dynamics of the microwave envelope obeys an equation similar to the original Eq. (2). In fact, both equations are identical in the single-mode approximation, that is an≡0 excepted for n=0.

anT3=inΩ̂TΩ0{μ̂+1AAT1}an
+2μ̂γATAan×{2𝒜2TJ1[2𝒜0T]+2𝒜1T2J1[2𝒜0T]}[2AT·NNaneinΩ̂TT2],
(13)

4. Conclusion

Acknowledgment

Y. K. C. acknowledges a research grant from the Région de Franche-Comté, France. The authors also acknowledge financial support from the Ministerio de Educación y Ciencia (Spain) and from FEDER under grant TEC2006-10009 (PhoDeCC).

References and links

1.

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

2.

L. M. Narducci, al et, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842–1854 (1986). [CrossRef] [PubMed]

3.

K. A. Winick, “Longitudinal mode competition in chirped grating distributed feedback lasers,” IEEE J. Quantum Electron. 35, 1402–1411 (1999). [CrossRef]

4.

A. M. Yacomotti, et al, “Dynamics of multimode semiconductor lasers,” Phys. Rev. A 69, 053816-1-9 (2004). [CrossRef]

5.

J. L. Font, R. Vilaseca, F. Prati, and E. Roldán, “Coexistence of single-mode and multi-longitudinal mode emission in the ring laser model,” Opt. Commun. 261, 336–341 (2006). [CrossRef]

6.

T. Voigt, et al, “Experimental investigation of RiskenNummedalGrahamHaken laser instability in fiber ring lasers,” Appl. Phys. B 79, 175–183 (2004). [CrossRef]

7.

C. Y. Wang, et al, “Coherent instabilities in a semiconductor laser with fast gain recovery,” Phys. Rev. A 75, 031802(R)-1-4 (2007). [CrossRef]

8.

Y. Kouomou 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]

9.

Y. Kouomou Chembo, L. Larger, and P. Colet, “Nonlinear dynamics and spectral stability of optoelectronic microwave oscillators,” IEEE J. Quantum Electron. (in press).

OCIS Codes
(130.0250) Integrated optics : Optoelectronics
(190.3100) Nonlinear optics : Instabilities and chaos
(350.4010) Other areas of optics : Microwaves

ToC Category:
Optoelectronics

History
Original Manuscript: March 13, 2008
Revised Manuscript: April 23, 2008
Manuscript Accepted: April 27, 2008
Published: June 4, 2008

Citation
Y. Kouomou Chembo, Laurent Larger, Ryad Bendoula, and Pere Colet, "Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators," Opt. Express 16, 9067-9072 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-9067


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References

  1. X. S. Yao and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. B 13,1725-1735 (1996). [CrossRef]
  2. L. M. Narducci,  et al, "Mode-mode competition and unstable behavior in a homogeneously broadened ring laser," Phys. Rev. A 33,1842-1854 (1986). [CrossRef] [PubMed]
  3. K. A. Winick, "Longitudinal mode competition in chirped grating distributed feedback lasers," IEEE J. Quantum Electron. 35,1402-1411 (1999). [CrossRef]
  4. A. M. Yacomotti,  et al, "Dynamics of multimode semiconductor lasers," Phys. Rev. A 69, 053816-1-9 (2004). [CrossRef]
  5. J. L. Font, R. Vilaseca, F. Prati, and E. Roldán, "Coexistence of single-mode and multi-longitudinal mode emission in the ring laser model," Opt. Commun. 261,336-341 (2006). [CrossRef]
  6. T. Voigt,  et al, "Experimental investigation of RiskenNummedalGrahamHaken laser instability in fiber ring lasers," Appl. Phys. B 79,175-183 (2004). [CrossRef]
  7. C. Y. Wang,  et al, "Coherent instabilities in a semiconductor laser with fast gain recovery," Phys. Rev. A 75, 031802(R)-1-4 (2007). [CrossRef]
  8. Y. Kouomou 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]
  9. Y. Kouomou Chembo, L. Larger, and P. Colet, "Nonlinear dynamics and spectral stability of optoelectronic microwave oscillators," IEEE J. Quantum Electron. (in press).

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