## Numerical simulation of vertical cavity surface emitting lasers

Optics Express, Vol. 2, Issue 4, pp. 163-168 (1998)

http://dx.doi.org/10.1364/OE.2.000163

Acrobat PDF (181 KB)

### Abstract

The semiconductor laser simulator MINILASE is being extended to simulate vertical cavity surface emitting lasers (VCSELs). The electronic system analysis for VCSELs is identical to that for edge emitting lasers. A brief discussion of the capabilities of MINILASE in this domain will be presented. In order to simulate VCSELs, the optical mode solver in MINILASE must be extended to handle the reduced index guiding and significant gain guiding typical of many VCSEL structures. A new approach to solving the optical problem which employs active cavity modes rather than the standard passive cavity modes is developed. This new approach results in an integral eigenvalue equation in required gain amplitudes and corresponding modal fields. Sample results from an early implementation of a gain eigenvalue solver are shown to clarify the possibilities of this approach.

© Optical Society of America

## 1. Introduction

1. D. Burak and R. Binder, “Cold-Cavity Vectorial Eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. **33**, 1205–1215 (1997). [CrossRef]

2. C.C. Lin and D.G. Deppe, “Self-Consistent Calculation of Lasing Modes in a Planar Microcavity,” J. Lightwave Technol. **13**, 575–580 (1995). [CrossRef]

3. H. Bissessur and K. Iga, “FD-BPM Modeling of Vertical Cavity Surface Emitting Lasers,” Proc. SPIE **2994**, 150–158 (1997). [CrossRef]

4. G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, and S.W. Corzine, “Comprehensive Numerical Modeling of Vertical-Cavity Surface-Emitting Lasers,” IEEE J. Quantum Electron. **32**, 607–616 (1996). [CrossRef]

4. G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, and S.W. Corzine, “Comprehensive Numerical Modeling of Vertical-Cavity Surface-Emitting Lasers,” IEEE J. Quantum Electron. **32**, 607–616 (1996). [CrossRef]

2. C.C. Lin and D.G. Deppe, “Self-Consistent Calculation of Lasing Modes in a Planar Microcavity,” J. Lightwave Technol. **13**, 575–580 (1995). [CrossRef]

## 2. Electronic simulation

6. S. Selberherr, *Analysis and Simulation of Semiconductor Devices* (Springer-Verlag, Wien-New York, 1984). [CrossRef]

8. M. Grupen and K. Hess, “Severe gain suppression due to dynamic carrier heating in quantum well lasers,” Appl. Phys. Lett. **70**, 808–810 (1997). [CrossRef]

9. M. Grupen and K. Hess, “Simulation of carrier transport and nonlinearities in quantum well laser diodes,” IEEE J. Quantum Electron. **34**, 120–140 (1998). [CrossRef]

9. M. Grupen and K. Hess, “Simulation of carrier transport and nonlinearities in quantum well laser diodes,” IEEE J. Quantum Electron. **34**, 120–140 (1998). [CrossRef]

## 3. Optical simulation

### 3.1 Theory

*S*is the mode intensity,

*G*is the gain (due to stimulated emission) for the mode under consideration,

*τ*

_{phot}is the photon lifetime of the mode, and

*R*

_{spon}is the amount of spontaneous emission which couples into the mode. In order to find

*G*,

*τ*

_{phot}, and

*R*

_{spon}for the mode under consideration, the optical modes must be obtained from the Maxwell equations without field-independent sources (spontaneous emission). In the conventional semiconductor laser theory [10], a set of simplifying assumptions are made in order to find the optical modes. The cavity geometry is usually assumed to be separable to some degree, and the gain is often treated as a small perturbation to the permittivity

*∊*(

*r*⃗,

*ω*). In practice, the optical cavity is often closed in order to find the modes, and the effect of the gain and cavity loss on the spatial mode pattern is neglected altogether [9

9. M. Grupen and K. Hess, “Simulation of carrier transport and nonlinearities in quantum well laser diodes,” IEEE J. Quantum Electron. **34**, 120–140 (1998). [CrossRef]

*∊*(

*r*⃗,

*ω*) and find instead the

*active*cavity modes of the VCSEL structure.

*r*⃗,

*ω*) into two parts as

_{cav}(

*r*⃗,

*ω*) is the potentially complex relative permittivity representing the passive cavity, and

*χ*

_{gain}(

*r*⃗,

*ω*) is the necessarily complex susceptibility representing the gain region of the laser. A susceptibility rather than a permittivity is used to represent the gain in order to clearly separate it from

*∊*

_{cav}, for reasons that will become apparent shortly. We can write the vector Helmholtz equation for the electric field as

*J*⃗

_{stim},(

*r*⃗,

*ω*) = -

*iω*

*∊*

_{o}

*χ*

_{gain}(

*r*⃗,

*ω*)

*E*⃗(

*r*⃗,

*ω*) is the field-dependent current representing stimulated emission in the gain region. The solution of the inhomogeneous wave equation Eq. (3) can be represented as

*G*⃡

_{cav}(

*r*⃗,

*r*⃗′,

*ω*) is the tensor Green’s function for radiation from a current source within the open VCSEL cavity defined by

*∊*

_{cav}.

*χ*

_{gain}must satisfy in order that the corresponding mode

*E*⃗ lase (be self-supporting) in the absence of spontaneous emission. With spontaneous emission present, the gain (and the mode) must approach this condition asymptotically. In order to arrive at a more useful form of Eq. (4), we note that spatial distribution of the gain susceptibility

*χ*

_{gain}can be calculated from the electronic system using MINILASE. We therefore write

*χ*

^{gain}(

*r*⃗,

*ω*) =

*K*(

*ω*)

*r*⃗,

*ω*), where

*K*is a complex constant that allows

*χ*

_{gain}to meet the lasing condition given as Eq. (4). Now Eq. (4) becomes

*G*⃡ (

*ω*) is an integral operator defined by

*E*⃡ are the mode field patterns and the eigenvalues

*K*are the gain amplitudes which define the lasing condition for the corresponding modes. This eigenmode equation is unusual as the eigenvalues are complex gain eigenvalues rather than complex frequency eigenvalues.

*K*are a continuous function of frequency. In order to obtain the frequencies at which the VCSEL can actually lase, the complex gain susceptibility necessary for the optical system must be matched to the complex gain susceptibility available from the laser electronic system in both magnitude and phase. This match will only occur at discrete frequencies. This determines the lasing frequencies of the laser. In addition, the required photon rate equation parameters may be calculated for each mode obtained in this fashion through use of the lasing mode pattern

*E*⃗ and the gain susceptibility required to lase

*χ*

_{gain}. For example, we may obtain the photon lifetime by using the fact that gain equals loss at threshold; hence 1/

*τ*

_{phot}will equal the total gain of the active cavity mode at threshold, which may be calculated as the overlap integral of the mode’s normalized field intensity pattern times the gain susceptibility required to lase.

*∊*

_{cav}out of the Green’s function and put it into the eigenvalue problem instead. Suppose we have a localized passive cavity feature which we don’t want to include in the passive cavity Green’s function

*G*⃡

_{cav}We may then split the electric permittivity further as

### 3.2 Implementation

*∊*

_{cav}is planar, and therefore an analytic Green’s function is available [11]. The wavelength-tall optical cavity is bounded above and below by distributed Bragg reflectors (DBRs). There is an oxide layer surrounding the active region in the plane of the active region (similar to the model of Lin and Deppe [2

2. C.C. Lin and D.G. Deppe, “Self-Consistent Calculation of Lasing Modes in a Planar Microcavity,” J. Lightwave Technol. **13**, 575–580 (1995). [CrossRef]

*r*⃗) ≡ -

*i*within the 9

*μ*m ×10

*μ*m×3nm quantum well active region, and

*r*⃗) ≡ 0 elsewhere. The entire layer which contains the oxide and the active region is given the background index of the oxide, which is effectively raised to the index of

*GaAs*inside the active region by setting

*χ*

_{cav}inside the active region to be equal to the difference between the two permittivities.

*K*(

*ω*) for this structure (the first two are essentially degenerate) plotted at constant intervals in frequency. Approximating the gain susceptibility available from the electronic system of the laser under bias as purely imaginary (neglecting the Kramers-Kronig relation for now), lasing is only possible at frequencies at which one of the gain eigenvalue curves crosses the real axis, such that

*χ*

_{gain}(

*r*⃗,

*ω*) =

*K*(

*ω*)

*r*⃗) is also purely imaginary. The gain eigenvalue at the crossing frequency defines the gain susceptibility required to lase. The large area active region considered in this example produces a very tight spacing of lasing modes in frequency, but the gain amplitudes required for lasing are significantly different because of the different overlaps of the self-consistently calculated lasing modes with the active region. Figure 3 shows the field patterns of the first and third lasing modes inside the active region. These modes are full-vector modes of the three dimensional cavity.

## Acknowledgement

## References

1. | D. Burak and R. Binder, “Cold-Cavity Vectorial Eigenmodes of VCSEL’s,” IEEE J. Quantum Electron. |

2. | C.C. Lin and D.G. Deppe, “Self-Consistent Calculation of Lasing Modes in a Planar Microcavity,” J. Lightwave Technol. |

3. | H. Bissessur and K. Iga, “FD-BPM Modeling of Vertical Cavity Surface Emitting Lasers,” Proc. SPIE |

4. | G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, and S.W. Corzine, “Comprehensive Numerical Modeling of Vertical-Cavity Surface-Emitting Lasers,” IEEE J. Quantum Electron. |

5. | M. Grupen, G. Kosinovsky, and K. Hess, “The effect of carrier capture on the modulation bandwidth of quantum well lasers,” in Proceedings of the International Electron Devices Meeting, (IEEE Electron Devices Society, Washington, D.C., 1993) pp. 23.6.1-23.6.4. |

6. | S. Selberherr, |

7. | M. Grupen, K. Hess, and G.H. Song, “Simulation of transport over heterojunctions,” in Proc. 4th International Conf. Simul. Semicon. Dev. Process., Vol. 4 (IEEE Electron Devices Society, Zurich, 1991) p. 303–311. |

8. | M. Grupen and K. Hess, “Severe gain suppression due to dynamic carrier heating in quantum well lasers,” Appl. Phys. Lett. |

9. | M. Grupen and K. Hess, “Simulation of carrier transport and nonlinearities in quantum well laser diodes,” IEEE J. Quantum Electron. |

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

11. | W.C. Chew, |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.5960) Lasers and laser optics : Semiconductor lasers

**ToC Category:**

Focus Issue: Quantum well laser design

**History**

Original Manuscript: October 15, 1997

Published: February 16, 1998

**Citation**

Benjamin Klein, Leonard Register, Matthew Grupen, and Karl Hess, "Numerical simulation of vertical cavity surface emitting lasers," Opt. Express **2**, 163-168 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-4-163

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### References

- D. Burak and R. Binder, "Cold-Cavity Vectorial Eigenmodes of VCSEL's," IEEE J. Quantum Electron. 33, 1205-1215 (1997). [CrossRef]
- C.C. Lin and D.G. Deppe, "Self-Consistent Calculation of Lasing Modes in a Planar Microcavity," J. Lightwave Technol. 13, 575-580 (1995). [CrossRef]
- H. Bissessur and K. Iga, "FD-BPM Modeling of Vertical Cavity Surface Emitting Lasers," Proc. SPIE 2994, 150-158 (1997). [CrossRef]
- G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, and S.W. Corzine, "Comprehensive Numerical Modeling of Vertical-Cavity Surface-Emitting Lasers," IEEE J. Quantum Electron. 32, 607-616 (1996). [CrossRef]
- M. Grupen, G. Kosinovsky, and K. Hess, "The eect of carrier capture on the modulation bandwidth of quantum well lasers," in Proceedings of the International Electron Devices Meeting, (IEEE Electron Devices Society, Washington, D.C., 1993) pp. 23.6.1-23.6.4.
- S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer-Verlag, Wien-New York, 1984). [CrossRef]
- M. Grupen, K. Hess, and G.H. Song, "Simulation of transport over heterojunctions," in Proc. 4th International Conf. Simul. Semicon. Dev. Process., Vol. 4 (IEEE Electron Devices Society, Zurich, 1991) p. 303-311.
- M. Grupen and K. Hess, "Severe gain suppression due to dynamic carrier heating in quantum well lasers," Appl. Phys. Lett. 70, 808-810 (1997). [CrossRef]
- M. Grupen and K. Hess, "Simulation of carrier transport and nonlinearities in quantum well laser diodes," IEEE J. Quantum Electron. 34, 120-140 (1998). [CrossRef]
- G.P. Agrawal and N.K. Dutta, Semiconductor Lasers, Second Edition (Van Nostrand Reinhold, New York, 1993) pp. 39-55.
- W.C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990) pp. 57-79, 375-418.

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