## Integral equation approach for the analysis of high-power semiconductor optical amplifiers

Optics Express, Vol. 14, Issue 6, pp. 2398-2403 (2006)

http://dx.doi.org/10.1364/OE.14.002398

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### Abstract

For the first time, an integral equation approach for the numerical assessment of Semiconductor Optical Amplifiers (SOAs) is proposed. Performance comparisons between the suggested formulation and the traditional transfer matrix method are carried out in terms of the computation costs in solving the multi-wave mixing process in bidirectional, high-power SOAs. Computation efficiency improvement by more than an order of magnitude was observed with the proposed formulation, achieving better accuracy at equivalent spatial resolution.

© 2006 Optical Society of America

## 1. Introduction

1. C. Y. J Chu and H. Ghafouri-Shiraz, “Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using transfer matrices,” J. Lightwave Technol. **12**, 1378–1386 (1994). [CrossRef]

3. H. Lee, H. Yoon, Y. Kim, and J. Jeong, “Theoretical study of frequency chirping and extinction ratio of wavelength-converted optical signals by XGM and XPM using SOA’s,” IEEE J. Quantum Electron. **35**, 1213–1219 (1999). [CrossRef]

4. M. J. Connelly, “Wideband semiconductor optical amplifier steady-state numerical model,” IEEE J. Quantum Electron. **37**, 439–447 (2001). [CrossRef]

5. B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photonics Technol. Lett. **12**, 1486–1488 (2000). [CrossRef]

6. N. Park, P. Kim, J. Park, and L. K. Choi, “Integral form expansion of fiber Raman amplifier problem,” Opt. Fiber Technol., **11**, 111–130 (2005). [CrossRef]

7. J. Park, P. Kim, J. Park, H. Lee, and N. Park, “Closed integral form expansion of Raman equation for efficient gain optimization process,” IEEE Photonics Technol. Lett. **16**, 1649–1651 (2004). [CrossRef]

## 2. Formulation

8. M. A. Summerfield and R. S. Tucker, “Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. **5**, 839–850 (1999). [CrossRef]

*a*(

_{l}*z*) the complex amplitudes of the signal fields at the frequency

*ω*,

_{j}*z*the propagation axis,

*N*the carrier density,

*g*(

_{l}*N*) the modal gain,

*α*the linewidth enhancement factor, and

*γ*is the scattering loss per unit length.

_{sc}*ε*’s are inverse saturation powers from the nonlinearity,

_{m}*β*’s are equivalent linewidth enhancement factors accounting for gain and index modulation at all frequencies, and

_{m}*τ*’s are relaxation times associated with various nonlinear processes, such as carrier population pulsation (

_{m}*m*=

*cpp*), spectral hole burning (

*m*=

*shb*), and carrier heating (

*m*=

*ch*) [8

8. M. A. Summerfield and R. S. Tucker, “Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. **5**, 839–850 (1999). [CrossRef]

*i*,

*j*,

*k*’ It were made to consider the wave component generated at the frequency

*ω*through the four-wave-mixing effect in the SOA (Δ

_{l}*ω*=

_{ij}*ω*-

_{i}*ω*=

_{j}*ω*-

_{l}*ω*).

_{k}*z*; and is given by

*n*iterations, Eq. (2) becomes

*x*units of elemental segments (of length Δ

*z*, Fig. 1), we now successfully convert Eq. (3) into a matrix form (for

*y*number signals for each direction),

*n*is the iteration number, and

*,*

**A***, and*

**F***are defined as,*

**T****A**

^{0}for all segments, assuming transparent SOA.

**F**

^{0}using

**A**

^{0}

**A**

^{1}=

**F**

^{0}

**T**

**A**^{n}can be obtained, converging to the exact value. It is important to note here that, for the conventional coupled differential equation based approach, solutions are obtained for each segment in a sequential manner, while their values are repeatedly updated as the iteration procedure continues [Fig. 1(a)]. In contrast, the proposed formulation uses the full set of wave information (at every segment and wavelength) from the previous iterations (with

**F**^{(n-1)}- enabling a simpler and much faster calculation (note that the information on the SOA segments are updated with simple multiplication of two-dimensional matrices-without the need of solving differential equations for each segment).

## 3. Results

8. M. A. Summerfield and R. S. Tucker, “Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. **5**, 839–850 (1999). [CrossRef]

**5**, 839–850 (1999). [CrossRef]

^{th}order), ~30 times faster assessment of the result was possible when using the proposed approach.

## 4. Conclusion

## References and links

1. | C. Y. J Chu and H. Ghafouri-Shiraz, “Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using transfer matrices,” J. Lightwave Technol. |

2. | S. L. Zhang and J. J. O’Reilly, “Modelling of four-wave-mixing wavelength conversion in a semiconductor laser amplifier,” Physical Modelling of Semiconductor Devices, IEE Colloquium on, 5/1-5/6 (1995). |

3. | H. Lee, H. Yoon, Y. Kim, and J. Jeong, “Theoretical study of frequency chirping and extinction ratio of wavelength-converted optical signals by XGM and XPM using SOA’s,” IEEE J. Quantum Electron. |

4. | M. J. Connelly, “Wideband semiconductor optical amplifier steady-state numerical model,” IEEE J. Quantum Electron. |

5. | B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photonics Technol. Lett. |

6. | N. Park, P. Kim, J. Park, and L. K. Choi, “Integral form expansion of fiber Raman amplifier problem,” Opt. Fiber Technol., |

7. | J. Park, P. Kim, J. Park, H. Lee, and N. Park, “Closed integral form expansion of Raman equation for efficient gain optimization process,” IEEE Photonics Technol. Lett. |

8. | M. A. Summerfield and R. S. Tucker, “Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers,” IEEE J. Sel. Top. Quantum Electron. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(060.4510) Fiber optics and optical communications : Optical communications

(140.4480) Lasers and laser optics : Optical amplifiers

(250.5980) Optoelectronics : Semiconductor optical amplifiers

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: January 11, 2006

Revised Manuscript: March 3, 2006

Manuscript Accepted: March 7, 2006

Published: March 20, 2006

**Citation**

Young Jin Jung, Pilhan Kim, Jaehyoung Park, and Namkyoo Park, "Integral equation approach for the analysis of high-power semiconductor optical amplifiers," Opt. Express **14**, 2398-2403 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2398

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### References

- C. Y. J Chu, H. Ghafouri-Shiraz, "Analysis of gain and saturation characteristics of a semiconductor laser optical amplifier using transfer matrices," J. Lightwave Technol. 12, 1378-1386 (1994). [CrossRef]
- S. L. Zhang and J. J. O’Reilly, "Modelling of four-wave-mixing wavelength conversion in a semiconductor laser amplifier," Physical Modelling of Semiconductor Devices, IEE Colloquium on, 5/1-5/6 (1995).
- H. Lee, H. Yoon, Y. Kim, J. Jeong, "Theoretical study of frequency chirping and extinction ratio of wavelength-converted optical signals by XGM and XPM using SOA’s," IEEE J. Quantum Electron. 35, 1213-1219 (1999). [CrossRef]
- M. J. Connelly, "Wideband semiconductor optical amplifier steady-state numerical model," IEEE J. Quantum Electron. 37, 439-447 (2001). [CrossRef]
- B. Min, W. J. Lee, and N. Park, "Efficient formulation of Raman amplifier propagation equations with average power analysis," IEEE Photonics Technol. Lett. 12, 1486-1488 (2000). [CrossRef]
- N. Park, P. Kim, J. Park, L. K. Choi, "Integral form expansion of fiber Raman amplifier problem," Opt. Fiber Technol., 11, 111-130 (2005). [CrossRef]
- J. Park, P. Kim, J. Park, H. Lee, and N. Park, "Closed integral form expansion of Raman equation for efficient gain optimization process," IEEE Photonics Technol. Lett. 16, 1649-1651 (2004). [CrossRef]
- M. A. Summerfield, and R. S. Tucker, "Frequency domain model of multiwave mixing in bulk semiconductor optical amplifiers," IEEE J. Sel. Top. Quantum Electron. 5, 839-850 (1999). [CrossRef]

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