## Numerical Methods for modeling Photonic-Crystal VCSELs |

Optics Express, Vol. 18, Issue 15, pp. 16042-16054 (2010)

http://dx.doi.org/10.1364/OE.18.016042

Acrobat PDF (1648 KB)

### Abstract

We show comparison of four different numerical methods for simulating Photonic-Crystal (PC) VCSELs. We present the theoretical basis behind each method and analyze the differences by studying a benchmark VCSEL structure, where the PC structure penetrates all VCSEL layers, the entire top-mirror DBR, a fraction of the top-mirror DBR or just the VCSEL cavity. The different models are evaluated by comparing the predicted resonance wavelengths and threshold gains for different hole diameters and pitches of the PC. The agreement between the models is relatively good, except for one model, which corresponds to the effective index method. The simulation results elucidate the strength and weaknesses of the analyzed methods; and outline the limits of applicability of the different models.

© 2010 Optical Society of America

## 1. Introduction

1. D. S. Song, S. H. Kim, H. G. Park, c. K. Kim, and Y. H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. **80**, 3901–3903 (2002). [CrossRef]

2. A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. **41**, 325–326 (2005). [CrossRef]

3. T. Czyszanowski, M. Dems, and K. Panajotov, “Single mode condition and modes discrimination in photonic-crystal 1.3 µm AlInGaAs/InP VCSEL,” Opt. Express **15**, 5604–5609 (2007). [CrossRef] [PubMed]

4. T. S. Kim, A. J. Danner, D. M. Grasso, E. W. Young, and K. D. Choquette, “Single fundamental mode photonic-crystal vertical cavity surface emitting laser with 9 GHz bandwidth,” Electron. Lett. **40**, 1340–1341 (2004). [CrossRef]

6. F. Romstad, S. Bischoff, M. Juhl, S. Jacobsen, and D. Birkedal, “Photonic crystals for long-wavelength single-mode VCSELs,” Proc. SPIE **6908**, 69080C (2008). [CrossRef]

7. D. S. Song, Y. J. Lee, H. W. Choi, and Y. H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers,” Appl. Phys. Lett. **82**, 3182–3184 (2003). [CrossRef]

8. M. Dems, T. Czyszanowski, and K. Panajotov, “Highly birefringent and dichroic photonic-crystal VCSEL design,” Opt. Commun **281**, 3149–3152 (2008). [CrossRef]

## 2. Computational Methods

9. G. R. Hadley, “Effective index model for vertical-cavity surface-emitting lasers,” Opt. Lett. **20**, 1483–1485 (1995). [CrossRef] [PubMed]

### 2.1. Coupled Mode Model (CMM)

10. G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys Rev. A **63**, 23816 (2001). [CrossRef]

### 2.2. Finite Element Method (FEM)

*E*(

**r**) and complex angular frequency (

*ω*). The imaginary part of the angular frequency gives the rate at which the intensity of the laser mode attenuates in the lack of pumping. The threshold gain can be estimated by dividing the cold-cavity modal loss with the total confinement factor for the quantum wells. However, if the built-in mode confinement is very low and significant gain-guiding occurs near the threshold condition, the cold-cavity model is not appropriate. An estimated threshold gain is then incorporated into the complex refractive index of the active region, and the obtained modal loss is used as a correction to the assumed threshold gain.

11. S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE T. Antenn. Propag. **44**, 1630–1639 (1996). [CrossRef]

*E*(

**r**) = ∑

*(*

_{i}c_{i}E^{i}**r**), an algebraic eigenproblem can be derived from the equivalent variational functional:

*c*denotes the column vector of the unknown coefficients,

**A**and

**B**matrices can be found elsewhere [12

12. P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Techn. **25**, 2427–2434 (2007). [CrossRef]

12. P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Techn. **25**, 2427–2434 (2007). [CrossRef]

13. P. Nyakas, G. Varga, Z. Puskás, N. Hashizume, T. Kárpáti, T. Veszprémi, and G. Zsombok, “Self-consistent real three-dimensional simulation of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B **23**, 1761–1769 (2006). [CrossRef]

### 2.3. Plane-Wave Admittance Transfer (PWAT)

14. M. Dems, R. Kotynski, and K. Panajotov, “Plane-wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express **13**, 3196–3207 (2005). [CrossRef] [PubMed]

^{2}is diagonal. The fields are matched using the admittance transfer technique and the eigenmode if found by searching for a complex frequency that yields a non-zero EM field distribution. In order to obtain a laser threshold gain, the imaginary part of the quantum-well refractive index is modified in each step of the root-finding algorithm, so that the imaginary part of the eigenfrequency is equal to zero.

### 2.4. Plane-Wave Effective Index (PWEI)

16. M. Dems and K. Panajotov, “Modeling of single- and multimode photonic-crystal planar waveguides with plane-wave admittance method,” Appl. Phys. B **89**, 19–23 (2007). [CrossRef]

16. M. Dems and K. Panajotov, “Modeling of single- and multimode photonic-crystal planar waveguides with plane-wave admittance method,” Appl. Phys. B **89**, 19–23 (2007). [CrossRef]

## 3. Results

3. T. Czyszanowski, M. Dems, and K. Panajotov, “Single mode condition and modes discrimination in photonic-crystal 1.3 µm AlInGaAs/InP VCSEL,” Opt. Express **15**, 5604–5609 (2007). [CrossRef] [PubMed]

*g*) as functions of the hole diameters for the photonic-crystal pitch Λ equal to 4.0 µm. The pitch of 4.0 µm is a typical value and has been kept fixed for the above mentioned cases (i) to (iv). In Figs. 6 and 7 we show the results determined for different sizes of the photonic-crystal lattice constant for cases (i) and (iii). From these graphs one can observe that all of the presented numerical methods show similar tendencies in the simplest cases, however, there are some quantitative discrepancies. Furthermore there exist some qualitative differences in some of the plots.

*λ*/32 and much finer

*λ*/10000 resolutions. The obtained value of 1.367 nm was subtracted from all the FEM wavelengths. After such operation, all the analyzed methods predict the same resonant wavelength with good agreement. The only exception is PWEI for the most shallow holes (Fig. 4d), in which case a visible drop of

*λ*for increasing diameter of holes is observed. This behavior can be explained by the fact that this drop is similar to the one predicted by all methods for deeper holes (Figs. 4a–c). The shallow holes do not differ qualitatively from deep ones for the effective index approach (i. e. the only difference is in the effective index in the hole-region projected into the

*x*-

*y*plane).

*d*/Λ > 0.4. This effect is not predicted by PWEI, which—as a variation of the effective index method—is incapable of properly considering scattering losses. Similarly, it even stronger underestimates the threshold gain for shallow holes (Fig. 4d), where scattering is much larger due to the weaker mode confinement. Furthermore, this weak confinement forces an à priori introduction of an estimated threshold gain in the quantum wells for FEM computations, as described in section 2.2. The same procedure is utilized by FEM also in other configurations, like for

*d*/Λ = 0.1.

*d*/Λ = 0.8 raises to 570 cm

^{−1}, which is much closer value to the two other methods. However, in such a case the requirement for computer resources and simulation times increases so strongly that we decided not to perform additional simulations with even higher numbers of plane waves. In order to investigate the origin of the predicted threshold gain by the different models, we analyzed the vertical (i. e. in a plane perpendicular to the VCSEL layers) cross-sections of the electric field profiles computed with CMM, FEM and PWAT. They are presented in Fig. 8. As one can see, CMM and FEM predict diagonally directed fields originating near the air holes in the cavity, which correspond to the diffraction scattering. However, in case of the PWAT with the default number of plane-waves, this effect is much weaker, especially when compared with the veritcally emitted flux.

*d*/Λ ratio set to 0.5; their mutual gain difference does not exceed 17%. It is noteworthy that for Λ larger than 6µm the CMM is unable to find accurate solutions. It is because the coupling coefficient matrix becomes close to singular or badly spaced when the total air hole volume is very large. This singular or badly-spaced matrix is difficult to be accurately inversed or solved for eigenvalues, using Matlab. However, we may say that for typical PC-VCSEL designs, the total air hole volume is much smaller. Here, the CMM can accurately anticipate the threshold gain. Fig. 7b shows the threshold gains for the holes etched only in the top DBR. Contrary to the wholly etched case in Fig. 7a, the PWEI shows considerable discrepancy from the other methods, for small pitches. This is caused by poor mode confinement and strong scattering for small Λ. Therefore, the correct threshold gain can not be predicted by effective index approaches in these cases, including PWEI.

## 4. Conclusions

## Acknowledgments

## References and links

1. | D. S. Song, S. H. Kim, H. G. Park, c. K. Kim, and Y. H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. |

2. | A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. |

3. | T. Czyszanowski, M. Dems, and K. Panajotov, “Single mode condition and modes discrimination in photonic-crystal 1.3 µm AlInGaAs/InP VCSEL,” Opt. Express |

4. | T. S. Kim, A. J. Danner, D. M. Grasso, E. W. Young, and K. D. Choquette, “Single fundamental mode photonic-crystal vertical cavity surface emitting laser with 9 GHz bandwidth,” Electron. Lett. |

5. | S. Bischoff, F. Romstad, M. Juhl, M. H. Madsen, J. Hanberg, and D Birkedal, “2.5 Gbit/s modulation of 1300 nm single-mode photonic crystal VCSELs,” in “Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD),” (2006), OFA6. |

6. | F. Romstad, S. Bischoff, M. Juhl, S. Jacobsen, and D. Birkedal, “Photonic crystals for long-wavelength single-mode VCSELs,” Proc. SPIE |

7. | D. S. Song, Y. J. Lee, H. W. Choi, and Y. H. Lee, “Polarization-controlled, single-transverse-mode, photonic-crystal, vertical-cavity, surface-emitting lasers,” Appl. Phys. Lett. |

8. | M. Dems, T. Czyszanowski, and K. Panajotov, “Highly birefringent and dichroic photonic-crystal VCSEL design,” Opt. Commun |

9. | G. R. Hadley, “Effective index model for vertical-cavity surface-emitting lasers,” Opt. Lett. |

10. | G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surface-emitting lasers,” Phys Rev. A |

11. | S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE T. Antenn. Propag. |

12. | P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Techn. |

13. | P. Nyakas, G. Varga, Z. Puskás, N. Hashizume, T. Kárpáti, T. Veszprémi, and G. Zsombok, “Self-consistent real three-dimensional simulation of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B |

14. | M. Dems, R. Kotynski, and K. Panajotov, “Plane-wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express |

15. | M. Dems, “Plane-wave admittance method and its applications to modeling semiconductor lasers and planar photonic-crystal structures,” Ph.D. thesis, Technical University of Lodz (2007). |

16. | M. Dems and K. Panajotov, “Modeling of single- and multimode photonic-crystal planar waveguides with plane-wave admittance method,” Appl. Phys. B |

17. | P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(140.5960) Lasers and laser optics : Semiconductor lasers

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 12, 2010

Revised Manuscript: June 10, 2010

Manuscript Accepted: June 10, 2010

Published: July 14, 2010

**Citation**

Maciej Dems, Il-Sug Chung, Peter Nyakas, Svend Bischoff, and Krassimir Panajotov, "Numerical Methods for modeling Photonic-Crystal VCSELs," Opt. Express **18**, 16042-16054 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-16042

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### References

- . D. S. Song, S. H. Kim, H. G. Park, c. K. Kim, and Y. H. Lee, “Single-fundamental-mode photonic-crystal verticalcavity surface-emitting lasers,” Appl. Phys. Lett. 80, 3901–3903 (2002). [CrossRef]
- . A. J. Danner, T. S. Kim, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity laser with improved output power,” Electron. Lett. 41, 325–326 (2005). [CrossRef]
- . T. Czyszanowski, M. Dems, and K. Panajotov, “Single mode condition and modes discrimination in photoniccrystal 1.3 μm AlInGaAs/InP VCSEL,” Opt. Express 15, 5604–5609 (2007). [CrossRef] [PubMed]
- . T. S. Kim, A. J. Danner, D. M. Grasso, E. W. Young, and K. D. Choquette, “Single fundamental mode photonic crystal vertical cavity surface emitting laser with 9 GHz bandwidth,” Electron. Lett. 40, 1340–1341 (2004). [CrossRef]
- . S. Bischoff, F. Romstad, M. Juhl, M. H. Madsen, J. Hanberg, and D. Birkedal, “2.5 Gbit/s modulation of 1300 nm single-mode photonic crystal VCSELs,” in “Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD),” (2006), OFA6.
- . F. Romstad, S. Bischoff, M. Juhl, S. Jacobsen, and D. Birkedal, “Photonic crystals for long-wavelength singlemode VCSELs,” Proc. SPIE 6908, 69080C (2008). [CrossRef]
- . D. S. Song, Y. J. Lee, H. W. Choi, and Y. H. Lee, “Polarization-controlled, single-transverse-mode, photoniccrystal, vertical-cavity, surface-emitting lasers,” Appl. Phys. Lett. 82, 3182–3184 (2003). [CrossRef]
- . M. Dems, T. Czyszanowski, and K. Panajotov, “Highly birefringent and dichroic photonic-crystal VCSEL design,” Opt. Commun 281, 3149–3152 (2008). [CrossRef]
- . G. R. Hadley, “Effective index model for vertical-cavity surface-emitting lasers,” Opt. Lett. 20, 1483–1485 (1995). [CrossRef] [PubMed]
- . G. P. Bava, P. Debernardi, and L. Fratta, “Three-dimensional model for vectorial fields in vertical-cavity surfaceemitting lasers,” Phys Rev. A 63, 23816 (2001). [CrossRef]
- . S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE T. Antenn. Propag. 44, 1630–1639 (1996). [CrossRef]
- . P. Nyakas, “Full-vectorial three-dimensional finite element optical simulation of vertical-cavity surface-emitting lasers,” IEEE J. Lightwave Techn. 25, 2427–2434 (2007). [CrossRef]
- . P. Nyakas, G. Varga, Z. Puskás, N. Hashizume, T. Kárpáti, T. Veszprémi, and G. Zsombok, “Self-consistent real three-dimensional simulation of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 23, 1761–1769 (2006). [CrossRef]
- . M. Dems, R. Kotynski, and K. Panajotov, “Plane-wave admittance method — a novel approach for determining the electromagnetic modes in photonic structures,” Opt. Express 13, 3196–3207 (2005). [CrossRef] [PubMed]
- . M. Dems, “Plane-wave admittance method and its applications to modeling semiconductor lasers and planar photonic-crystal structures,” Ph.D. thesis, Technical University of Lodz (2007).
- . M. Dems and K. Panajotov, “Modeling of single- and multimode photonic-crystal planar waveguides with planewave admittance method,” Appl. Phys. B 89, 19–23 (2007). [CrossRef]
- . P. Bienstman, R. Baets, J. Vukusic, A. Larsson, M. J. Noble, M. Brunner, K. Gulden, P. Debernardi, L. Fratta, G. P. Bava, W. Wenzel, K. Klein, C. Conradi, R. Pregla, S. A. Riyopoulos, J.-F. P. Seurin, and S. L. Chuang, “Comparison of optical VCSEL models on the simulation of oxide-confined devices,” IEEE J. Quantum Electron. 37, 1618–1631 (2001). [CrossRef]

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