## Effects of gain and bandwidth on the multimode behavior of optoelectronic microwave oscillators

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

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]

## 2. The experimental system

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

_{3}Mach-Zehnder (MZ) modulator of DC and RF half-wave voltages

*V*, and subjected to a radio-frequency modulation voltage

_{B}*V*(

*t*). The modulated optical signal then travels through a 4-km long thermalized fiber delay-line, inducing a time-delay

*T*=20

*µ*s which corresponds to a free spectral range Ω

*/2*

_{T}*π*=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 Ω

_{0}/2

*π*=3 GHz, and the -3 dB bandwidth is ΔΩ/2

*π*=20 MHz (corresponding to a quality factor

*Q*=Ω

_{0}/ΔΩ=150). Therefore, there are ΔΩ/Ω

_{T}=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.

## 3. Multiple timescale analysis of the multimode behavior

*maximum gain mode*does not systematically win the modal competition.

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]

_{0}, the complex slowly-varying envelope 𝒜(

*t*)=|𝒜(

*t*)|

*e*

^{iψ(t)}of the quasi-sinusoidal microwave variable

*x*(

*t*) obeys

*μ*=ΔΩ/2 is the half-bandwidth of the RF filter,

*γ*=

*β*sin2

*ϕ*is the effective gain of the feedback loop, and Jc

_{1}is the

*Bessel-cardinal*function defined as Jc

_{1}(

*x*)=J

_{1}(

*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]

*≡𝒜(*

_{T}*t*-

*T*), and the phase matching condition

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]

*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

*µ*s. 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.

_{0},

*μ*and Ω

_{T}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*and |𝒜|

_{k}_{k}are associated to the timescales

*T*=

_{k}*ε*. The time derivative can also be expanded as

^{k}t*d*/

*dt*=∑

^{3}

_{k=0}

*ε*where

^{k}D_{k}*D*=

_{k}*∂*/

*∂T*. If we rescale the temporal parameters as

_{k}*μ*=

*ε*μ ^ and Ω

_{T}=

*ε*

^{2}

_{T}, the timescale components of the integral variable

*ε*:

*ε*

^{0}, we can deduce that

*u*

_{0}is sinusoidal, so that its integrand

*x*

_{0}may also be expressed as

*x*. The spectral splitting of Ω

_{0},

*μ*and Ω

_{T}suggests the following decomposition

*N*=

*μ*/Ω

_{T}is the number of cavity modes inside the bandwidth. The dynamics of the system is therefore ruled by the slow complex amplitude

*A*(

*T*

_{1}) associated to the carrier Ω

_{0}, and the even slower complex modal amplitudes

*a*(

_{n}*T*

_{3}) associated to the detuning frequencies

*n*Ω

_{T}. This modal decomposition enables the expansion of the amplitude according to

*A*(

*T*

_{1}) of the carrier is obtained by setting to zero the secular term in Eq. (6), thereby yielding

*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

*a*≡0 excepted for

_{n}*n*=0.

*a*(

_{n}*T*

_{3}) are obtained after projection onto

_{n}are defined in Eq. (11). The coupled system of Eqs. (12) and (13) constitutes the final result of the modal analysis. They may enable to understand the intrinsical mechanisms of the mode competition in OEOs.

*a*are subjected to a nonlinear global coupling, along with winding frequency terms which are stronger as the modal frequency detunings

_{n}*n*Ω

_{T}increase. It is worth noting that the modes have not been coupled phenomenologically as it is sometimes done in the literature: here, the coupling emerges naturally from the intrinsical nonlinearity of the system. This global nonlinear coupling is indeed very complex, if one remembers that terms as complicated as those of Eq. (11) are involved in the modal dynamics. This complexity is of course intrinsic to the original Eq. (2) ruling the total amplitude 𝒜 (

*t*), and which does not consider

*a priori*any form of modal structure. Here as in other multimode systems, the key advantage of modal expansion is therefore to track the dynamics of individual modes, and to give an explicit insight into the topological nature of their coupling.

## 4. Conclusion

## Acknowledgment

## References and links

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

2. | L. M. Narducci, |

3. | K. A. Winick, “Longitudinal mode competition in chirped grating distributed feedback lasers,” IEEE J. Quantum Electron. |

4. | A. M. Yacomotti, |

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. |

6. | T. Voigt, |

7. | C. Y. Wang, |

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. |

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

- X. S. Yao and L. Maleki, "Optoelectronic microwave oscillator," J. Opt. Soc. Am. B 13,1725-1735 (1996). [CrossRef]
- 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]
- K. A. Winick, "Longitudinal mode competition in chirped grating distributed feedback lasers," IEEE J. Quantum Electron. 35,1402-1411 (1999). [CrossRef]
- A. M. Yacomotti, et al, "Dynamics of multimode semiconductor lasers," Phys. Rev. A 69, 053816-1-9 (2004). [CrossRef]
- 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]
- T. Voigt, et al, "Experimental investigation of RiskenNummedalGrahamHaken laser instability in fiber ring lasers," Appl. Phys. B 79,175-183 (2004). [CrossRef]
- 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]
- 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]
- 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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