## Rate Equations for modeling dispersive nonlinearity in Fabry-Perot semiconductor optical amplifiers

Optics Express, Vol. 11, Issue 21, pp. 2689-2696 (2003)

http://dx.doi.org/10.1364/OE.11.002689

Acrobat PDF (146 KB)

### Abstract

We model the non-linear gain characteristics of a Fabry-Perot semiconductor optical amplifier using a modified photon density rate equation. Good agreement is found with experimental results, with the simulation accurately reproducing all the major characteristics of the amplifier. To our knowledge, this is the first calculation using only the rate equations that accurately predicts the gain and nonlinear behavior of FPSOAs.

© 2003 Optical Society of America

## Introduction

1. D. Wiedenmann, B. Moeller, R. Michalzik, and K.J. Ebeling, “Performance characteristics of vertical-cavity semiconductor laser amplifiers,” Electron. Lett. **32**, 342–343 (1996) [CrossRef]

3. J. Piprek, S. Bjorlin, and E. Bowers, “Design and analysis of vertical-cavity semiconductor optical amplifiers,” IEEE J. Quantum Electron. **37**, 127–134 (2001) [CrossRef]

2. C. Tombing, T. Saitoh, and T. Mukai, “Performance prediction for vertical-cavity semiconductor laser amplifiers,” IEEE J. Quantum Electron. **30**, 2491–2499 (1994) [CrossRef]

4. Daniel T. Cassidy, “Comparison of rate equation and Fabry-Perot approaches to modeling a diode laser” Appl. Opt. **22**, 3321–3326 (1983) [CrossRef] [PubMed]

3. J. Piprek, S. Bjorlin, and E. Bowers, “Design and analysis of vertical-cavity semiconductor optical amplifiers,” IEEE J. Quantum Electron. **37**, 127–134 (2001) [CrossRef]

5. M.J. Adams, J.V. Collins, and I.D. Henning, “Analysis of semiconductor laser optical amplifiers”, IEE Proc. J Optoelectron. **132**, 58–63, (1985). [CrossRef]

8. P. Wen, M. Sánchez, M. Gross, O. Kibar, and S. Esener, “New photon density rate equation for Fabry-Perot semiconductor optical amplifiers (FP SOAs),” in Physics and Simulation of Optoelectronic Devices X, Proc. SPIE **4646**, 243–250, (2002). [CrossRef]

## Theory

_{e}) and photon (N

_{p}) number in the laser cavity, with several substitutions to simplify the equations. In the carrier rate equation, there are three terms. The first represents the electrical injection of carriers, via the bias current

*I*, differential efficiency

*η*, and charge of an electron,

_{e}*q*. The middle term contains a recombination rate term γ

_{e}, which is the inverse of the electron lifetime, τ

_{e}:

*A,B,C*represent non-radiative, radiative and Auger recombination, respectively, and

*n*is the carrier density. The last term of the carrier rate equation is the consumption of carriers by stimulated emission, which is directly proportional to the gain rate (G) and the photon number.

*γ*is the photon loss rate, and

_{p}*R*is the spontaneous emission rate.

_{sp}*N*is the number of photons externally injected into the cavity, and

_{inj}*C*is a coupling coefficient. The gain is approximated using a linear expression, since only a narrow range of bias currents is to be considered:

_{p}*Γ*is the lateral confinement factor,

*a*is the differential gain coefficient,

*n*is the carrier density,

*n*the transparency carrier density,

_{0}*µ*is the group refractive index, and

_{g}*c*is the speed of light in vacuum. Thus

*v*is the group velocity of light in the laser cavity. This value is a weak function of the carrier density, but in the first order approximation used here, it can be considered a constant. The photon loss rate is defined as:

_{g}*α*, and

_{m}*α*is the internal loss, with

_{i}*τ*representing the photon lifetime. The mirror loss is approximated as a distributed loss over the length of the cavity (

_{p}*L*), and is a function of the mirror reflectivites

*R*and

_{1}*R*, which are the front and rear mirrors, respectively. The next term of the photon equation in Eq. (1) represents the spontaneous emission photons:

_{2}*β*, and

_{sp}*V*′ is the volume of the gain medium. For edge emitting FPSOAs, the entire laser cavity is the gain medium, but in a quantum well VCSOA, the gain medium is only a fraction of the laser cavity. This fact is accounted for by the use of a longitudinal confinement factor, which is designated

*Γ*.

_{l}*et al.*in [8

8. P. Wen, M. Sánchez, M. Gross, O. Kibar, and S. Esener, “New photon density rate equation for Fabry-Perot semiconductor optical amplifiers (FP SOAs),” in Physics and Simulation of Optoelectronic Devices X, Proc. SPIE **4646**, 243–250, (2002). [CrossRef]

*k*, the photon roundtrip time is

*τ*, and

_{RT}*η*is a fitting parameter. The single pass amplitude gain is given by

_{in}*G*and

_{s}*ϕ*represents the single pass phase change. Equation. (6) is derived from the summation of the optical field and its multiple reflections inside the laser cavity. This term differs from the expression in [8

8. P. Wen, M. Sánchez, M. Gross, O. Kibar, and S. Esener, “New photon density rate equation for Fabry-Perot semiconductor optical amplifiers (FP SOAs),” in Physics and Simulation of Optoelectronic Devices X, Proc. SPIE **4646**, 243–250, (2002). [CrossRef]

*η*, and a factor of (

_{in}*1*-

*R*). This factor results from the fact that in [8

_{1}**4646**, 243–250, (2002). [CrossRef]

*N*represents the photons delivered to the front face of the amplifier, and so a proportionality factor is required. The fitting parameter is justified as follows: Note that the injection term is made into a rate by dividing by the roundtrip time (

_{inj}*τ*). Some correction is required to account for the reality that the actual rate at which photons are added to the cavity is somewhat less than the term above, since it takes more than one roundtrip to build up the interference. Such a correction is included in the fitting parameter

_{RT}*η*, which also accounts for physical coupling loss in the optical system.

_{in}*β*is the linewidth enhancement factor, and

_{c}*ϕ*is the detuning from the resonant frequency. The carrier density without optical injection is given in the parameter

_{0}*n*

_{1}. Note the longitudinal confinement is present in these equations because the gain and index change occur only within the active region. The second term of the phase equation has been included in the calculations of this paper in order to model the nonlinear amplifier behavior. That term couples the optical phase to the carrier density in the amplifier, and Eq. (6) is a sensitive function of the phase,

*ϕ*. Thus, Eqs. (6) and (7) implicitly form an additional coupling between the photon density and carrier rate equations that is not captured by the traditional rate equations.

*η*is included in this conversion, to account for losses in the optical system before the detector. Equations (1–7) define all the expressions necessary to this simulation, but do not have a general analytic solution, so predictions are obtained numerically, using Mathematica (Wolfram Research). These equations are applied to a VCSOA in the next section, and the predictions compared to measured results.

_{out}## Experiment results

*Γ*and

*Γ*are defined by the overlap of the optical mode with the active gain region (QW). The VCSOA used is a proton implanted VCSEL manufactured by Emcore. The aperture size is 8 µm. Further details of the device and the experimental setup can be found in [10

_{l}10. M. Sánchez, P. Wen, M. Gross, and S. Esener, “Nonlinear gain in vertical-cavity semiconductor optical amplifiers.” IEEE Phot. Tech. Lett. **15**, 507–509, (2003). [CrossRef]

*β*,

_{c}*α*,

_{i}*η*and,

_{in}*η*are fixed by fitting the L-I curve of the laser. The internal loss is determined by fitting to the width of the gain window of the device at low input power (100nW), as in [10

_{out}10. M. Sánchez, P. Wen, M. Gross, and S. Esener, “Nonlinear gain in vertical-cavity semiconductor optical amplifiers.” IEEE Phot. Tech. Lett. **15**, 507–509, (2003). [CrossRef]

*η*and

_{in}*η*is also allowed when fitting the I/O characteristic. The value of

_{out}*η*presented here compares favorably (factor of 2) to our estimates based on calculation of the photon buildup time.

_{in}10. M. Sánchez, P. Wen, M. Gross, and S. Esener, “Nonlinear gain in vertical-cavity semiconductor optical amplifiers.” IEEE Phot. Tech. Lett. **15**, 507–509, (2003). [CrossRef]

## Conclusion

## References and Links

1. | D. Wiedenmann, B. Moeller, R. Michalzik, and K.J. Ebeling, “Performance characteristics of vertical-cavity semiconductor laser amplifiers,” Electron. Lett. |

2. | C. Tombing, T. Saitoh, and T. Mukai, “Performance prediction for vertical-cavity semiconductor laser amplifiers,” IEEE J. Quantum Electron. |

3. | J. Piprek, S. Bjorlin, and E. Bowers, “Design and analysis of vertical-cavity semiconductor optical amplifiers,” IEEE J. Quantum Electron. |

4. | Daniel T. Cassidy, “Comparison of rate equation and Fabry-Perot approaches to modeling a diode laser” Appl. Opt. |

5. | M.J. Adams, J.V. Collins, and I.D. Henning, “Analysis of semiconductor laser optical amplifiers”, IEE Proc. J Optoelectron. |

6. | G. P. Agrawal and N. K. Dutta, |

7. | M.J. Adams, “Time Dependent Analysis of Active and Passive Optical Bistability in Semiconductors”, IEE Proceedings J Optoelectron. |

8. | P. Wen, M. Sánchez, M. Gross, O. Kibar, and S. Esener, “New photon density rate equation for Fabry-Perot semiconductor optical amplifiers (FP SOAs),” in Physics and Simulation of Optoelectronic Devices X, Proc. SPIE |

9. | P Royo, R Koda, and L.A. Coldren, “Vertical cavity semiconductor optical amplifiers: comparison of Fabry-Perot and rate equation approaches.”, IEEE J. Quantum Electron. |

10. | M. Sánchez, P. Wen, M. Gross, and S. Esener, “Nonlinear gain in vertical-cavity semiconductor optical amplifiers.” IEEE Phot. Tech. Lett. |

11. | P. Wen, M. Sánchez, M. Gross, and S. Esener, “Vertical-cavity optical AND gate”, Opt. Commun. |

12. | T.E. Sale, |

13. | L Coldren and S. Corzine, |

14. | J.H. Shin, J.K Hwang, H Ha, and Y.H. Lee, “Anamalous above-threshold spontaneous emission in gainguided vertical-cavity surface-emitting lasers”, Appl. Phys. Lett. |

15. | KH Ha and YH Lee, “Determiniation of Cavity Loss in Proton Implanted Vertical-Cavity Surface Emitting Lasers,” Jpn. J. Appl. Phys. |

**OCIS Codes**

(140.3280) Lasers and laser optics : Laser amplifiers

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

(230.4320) Optical devices : Nonlinear optical devices

(250.5980) Optoelectronics : Semiconductor optical amplifiers

(270.3430) Quantum optics : Laser theory

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 3, 2003

Revised Manuscript: October 8, 2003

Published: October 20, 2003

**Citation**

Michael Sánchez, Pengyue Wen, Matthias Gross, and Sadik Esener, "Rate Equations for modeling dispersive nonlinearity in Fabry-Perot semiconductor optical amplifiers," Opt. Express **11**, 2689-2696 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-21-2689

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### References

- D. Wiedenmann, B. Moeller, R. Michalzik, and K.J.Ebeling, �??Performance characteristics of vertical-cavity semiconductor laser amplifiers,�?? Electron. Lett. 32, 342-343 (1996) [CrossRef]
- C. Tombing, T. Saitoh and T. Mukai, �??Performance prediction for vertical-cavity semiconductor laser amplifiers,�?? IEEE J. Quantum Electron. 30, 2491-2499 (1994) [CrossRef]
- J. Piprek, S. Bjorlin and E. Bowers, �??Design and analysis of vertical-cavity semiconductor optical amplifiers,�?? IEEE J. Quantum Electron. 37, 127-134 (2001) [CrossRef]
- Daniel T. Cassidy, �?? Comparison of rate equation and Fabry-Perot approaches to modeling a diode laser�?? Appl. Opt. 22, 3321-3326 (1983) [CrossRef] [PubMed]
- Adams, M.J., Collins, J.V., and Henning, I.D., �??Analysis of semiconductor laser optical amplifiers�??, IEE Proc. J Optoelectron. 132, 58-63, (1985). [CrossRef]
- G. P. Agrawal and N. K. Dutta, Semiconductor lasers, (Kluwer Academic, 1993)
- M.J. Adams, �??Time Dependent Analysis of Active and Passive Optical Bistability in Semiconductors�??, IEE Proceedings J Optoelectron. 132, 343-348, (1985). [CrossRef]
- P. Wen, M. Sánchez, M. Gross, O. Kibar, S. Esener, �??New photon density rate equation for Fabry-Perot semiconductor optical amplifiers (FP SOAs),�?? in Physics and Simulation of Optoelectronic Devices X, Proc. SPIE 4646, 243-250, (2002) [CrossRef]
- Royo, P; Koda, R; Coldren, L.A., �??Vertical cavity semiconductor optical amplifiers: comparison of Fabry- Perot and rate equation approaches.�??, IEEE J. Quantum Electron. 38, 279-84, (2002). [CrossRef]
- M. Sánchez, P. Wen, M. Gross, S. Esener, �??Nonlinear gain in vertical-cavity semiconductor optical amplifiers.�?? IEEE Phot. Tech. Lett. 15, 507-9, (2003). [CrossRef]
- P. Wen, M. Sánchez, M. Gross, S. Esener, �??Vertical-cavity optical AND gate�??, Opt. Commun. 219, 383-387, (2003) [CrossRef]
- T.E. Sale, Vertical Cavity Surface Emitting Lasers, (Research Studies Press, Somerset, England, 1995)
- Coldren, L, Corzine, S., Diode Lasers and Photonic Integrated Circuits, (Wiley-Interscience, New York, NY, 1995
- Shin, J.H., Hwang, J.K, Ha, H, Lee, Y.H., �??Anamalous above-threshold spontaneous emission in gain-guided vertical-cavity surface-emitting lasers�??, Appl. Phys. Lett. 68, 2180-2182, (1996) [CrossRef]
- Ha, KH, Lee, YH, �??Determiniation of Cavity Loss in Proton Implanted Vertical-Cavity Surface Emitting Lasers,�?? Jpn. J. Appl. Phys. 37, L372-L374, (1998) [CrossRef]

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