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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 10 — May. 16, 2005
  • pp: 3719–3727
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Analysis of mode stability in a concave mirror vertical-cavity surface-emitting laser with an oxide aperture

Young-Gu Ju  »View Author Affiliations


Optics Express, Vol. 13, Issue 10, pp. 3719-3727 (2005)
http://dx.doi.org/10.1364/OPEX.13.003719


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Abstract

We applied the Fox-Li resonator theory to analyze the mode stability of concave mirror surface-emitting lasers. The numerical modeling incorporates the oxide aperture in the simple classical cavity by adding a non-uniform phase shifting layer to the flat mirror side. The calculation shows that there is a modal loss difference between the fundamental mode and the competing modes. The amount of loss difference depends upon cavity length and the thickness of the oxide aperture. In addition to loss difference, modal gain difference plays a key role in discriminating between the fundamental mode and the higher order transverse modes. The modal gain difference heavily depends upon the size of the oxide aperture and the field intensity distribution. To summarize, the geometry of the concave cavity affects the mode profile and the unique field profile of each transverse mode makes a difference in both modal loss and gain. Finally, this leads to a side-mode suppression.

© 2005 Optical Society of America

1. Introduction

Despite the importance of the issue and the success of the device fabrications, the progress of theoretical modeling in concave mirror vertical-cavity lasers is not as rapid as the experimental one. Even though modeling attempts have existed, they seem to rely on very complicated and unconventional approaches to analyze mode competition and mode stability [9

9. S. H. Park, Y. Park, and H. Jeon, “Theory of the mode stabilization mechanism in concave-micromirror-capped vertical-cavity surface-emitting lasers,” J. of Appl. Phys. 94, 1312–1317 (2003) [CrossRef]

]. In this paper, we analyze the concave mirror vertical-cavity with the classical resonator theory to demonstrate that the classical resonator theory is powerful enough to provide practical information about the mode profile of the dominant transverse modes and the competition between them. We also simulate various geometrical situations including variations of cavity length and oxide aperture.

2. Numerical modeling

The modeling of a concave mirror vertical-cavity is not as easy as it appears since it is an open-sided cavity. Most of the textbooks dealing with lasers include resonator theory, however, they also mention the difficulty of mode calculation of the open sided cavity [10

10. A. E. Siegman, Laser, (Oxford University Press, 1986), Chap. 14

]. Mathematically, resonator eigenmodes are described by the integral equation as shown in Eq. (1), however, the round-trip propagation kernel is generally found not to be a hermitian operator. This, in turn, means that the existence of a complete and orthogonal set of eigensolutions is not guaranteed. It is just accepted that transverse eigenmodes exist empirically. As a matter of fact, real lasers have never had any difficulty in finding such modes in which to oscillate. The Fox-Li theory [11

11. A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. of Quantum Electron. QE-4, 460–465 (1968) [CrossRef]

] relies on this physical process when searching for eigenmodes. The method is a computer simulation of the physical experiment of exciting a resonator externally and adjusting its length to resonate the various modes. The output at each resonance is purified by means of mode filters consisting of suitably adjusted resonators in tandem. For this reason, we adopted the Fox-Li theory to calculate the mode profile of concave mirror surface-emitting lasers.

The iterative integral equation for the field E(r, ϕ) is originally in a two-dimensional form. It is possible, however, to reduce it to a one-dimensional equation in Eq. (1) by assuming a sinusoidal azimuthal distribution of the field, as seen in Eq. (2).

Wl(n+1)(r)=0aKl(r,s)Wl(n)(s)sds
where{Wl(n):RadialfielddistributionfunctionKl(r,s):Kernal
(1)
E(r,φ)=Wl(r)ejlφ
(2)

This assumption is valid as long as the cavity has cylindrical symmetry. The reduction into a one-dimensional form saves a lot of computation time as compared with the two-dimensional form. It also means that a simulation, over a wide range of parametric space, is possible within a reasonable time.

Fig. 1. A schematic diagram of a concave mirror vertical-cavity with an oxide aperture.

The DBR structure is not included in the modeled cavity since it causes a uniform phase shift in a lateral direction, unlike the oxide aperture. It is assumed that the constant phase shift is absorbed into the flat mirror side. The tuning process in the Fox-Li theory adjusts the cavity length within the wavelength order so as to locate the resonance frequencies of the longitudinal modes. Therefore, ignoring the DBR structure in modeling may not incur serious problems, at least, in terms of phase shift. On the other hand, the omission of the DBR structure greatly simplifies numerical calculations.

Table 1. The parameter values used in the modeling

table-icon
View This Table

Fig. 2. Normalized field intensity of a mode obtained using the parameters in Table 1. Azimuthal mode number l=0. Phase shift=26°.

Fig. 3. Power loss per iteration of the mode from Fig. 2.

3. Results

In order to investigate mode stability, we scan several parameters over a range of values while the values of other parameters are held constant, as indicated in Table 1. One of the variable parameters is the cavity length and the results are presented in Fig. 4, Fig. 5 and Fig. 6. The three plots correspond to three different oxide apertures illustrating the effect of the oxide aperture on mode stability. The first one indicates the case where there is no oxide aperture, while the other two results refer to oxide apertures of different thicknesses. The thicknesses of the oxide in Fig. 5 and Fig. 6 are 0.03 µm and 0.06 µm, respectively.

Fig. 4. Power loss as a function of cavity length., No oxide aperture is present
Fig. 5. Power loss as a function of cavity length. The thickness of the oxide aperture is 0.03 µm.

As for the concave cavity with an oxide aperture, the cavity loss of the fundamental mode is larger than that of the cavity without it, as shown in Fig. 5. The competing higher order transverse modes, however, also experience a larger loss, which means a larger mode suppression ratio. By using the same criteria for mode stability and lasing conditions, the laser calculated in Fig. 5 can operate at a cavity length between 100 µm and 270 µm with mode stability. The loss difference between TEM00 and TEM01 changes depending upon the cavity length. The range between 170 µm and 230 µm is a window whereby the loss difference is small and thus, mode competition is intense. In this region, considerable optical power of the side mode can be observed along with that of the most dominant mode.

Fig. 6. Power loss as a function of cavity length. Thickness of the oxide aperture is 0.06 µm.

Fig. 7. Confinement factor v.s. cavity length.
Fig. 8. Transverse mode profiles used in Fig. 7. Radius of oxide aperture(rox)=6 µm, rox/mirror radius(a)=0.3

In this paper, we only demonstrate the simplicity and the feasibility of Fox-Li resonator theory in the analysis of a concave mirror vertical-cavity surface-emitting laser. More accurate modeling requires further research on thermal effect and cavity geometries conforming to real devices [17

17. A. Mooradian, “Coupled cavity high power semiconductor laser,” United States patent 6778582, (2004)

]. The sophistication of this theoretical work will be the topic of the future research.

4. Summary

References and links

1.

H. Martinsson, J. A. Vukusic, and A. Larsson, “Single-mode power dependence on surface relief size for mode-stabilized oxide-confined vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 12, 1129–1131 (2000). [CrossRef]

2.

E. W. Young, K. D. Choquette, S. L. Chuan, K. M. Geib, A. J. Fischer, and A. A. Allerman, “Singletransverse-mode vertical-cavity lasers under continuous and pulsed operation,” IEEE Photon. Technol. Lett. 13, 927–929 (2001) [CrossRef]

3.

Jong-Hwa Baek, Dae-Sung Song, In-Kag Hwang, Kum-Hee Lee, Y. H. Lee, Young-gu Ju, Takashi Kondo, Tomoyuki Miyamoto, and Fumio Koyama, “Transverse mode control by etch-depth tuning in 1120-nm GaInAs/GaAs photonic crystal vertical-cavity surface-emitting lasers,” Opt. Express 12, 859–867 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-859 [CrossRef] [PubMed]

4.

L.A. Coldren, S.P. DenBaars, O. Buchinsky, T. Margalith, D.A. Cohen, A.C. Abare, and M. Hansen, “Blue and Green InGaN VCSEL Technology,” Final Report 1997–98 for MICRO Project 97–033 (1997), http://www.ucop.edu/research/micro/97_98/97_033.pdf

5.

Si-Hyun Park, Jaehoon Kim, Heonsu Jeon, Tan Sakong, Sung-Nam Lee, Suhee Chae, Y. Park, Chang-Hyun Jeong, Geun-Young Yeom, and Yong-Hoon Cho, “Room-temperature GaN vertical-cavity surfaceemitting laser operation in an extended cavity scheme,” Appl. Phys. Lett. , 83, 2121–2121 (2003) [CrossRef]

6.

K. S. Kim, Y. H. Lee, B. Y. Jung, and C. K. Hwangbo, “Single mode operation of a curved-mirror vertical-emitting laser with an active distributed Bragg reflector,” Jpn. J. Appl. Phys. 41, L827–L829 (2002) [CrossRef]

7.

P. Brick, S. Lutgen, T. Albrecht, J. Luft, and W. Spath, “High-efficiency high-power semiconductor disc laser,” Proc. SPIE 4993, 50–56 (2003) [CrossRef]

8.

E. M. Strzelecka, J. G. McInerney, A. Mooradian, A. Lewis, A. V. Shchegrov, D. Lee, and Wats, “High-power high-brightness 980-nm lasers based on the extended cavity surface emitting lasers concept,” Proc. SPIE 4993, 57–67 (2003) [CrossRef]

9.

S. H. Park, Y. Park, and H. Jeon, “Theory of the mode stabilization mechanism in concave-micromirror-capped vertical-cavity surface-emitting lasers,” J. of Appl. Phys. 94, 1312–1317 (2003) [CrossRef]

10.

A. E. Siegman, Laser, (Oxford University Press, 1986), Chap. 14

11.

A. G. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. of Quantum Electron. QE-4, 460–465 (1968) [CrossRef]

12.

Young-Gu Ju, “Evaluation scheme for the design of power-optimized single mode vertical-cavity surface-emitting lasers,” Opt. Express 12, 2542–2547 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2542 [CrossRef] [PubMed]

13.

Y. G. Ju, J. H. Ser, and Y. H. Lee, “Analysis of metal-interlaced-grating vertical-cavity surface-emitting lasers using the modal method by modal expansion,” IEEE J. of Quantum Electron. 33, 589–595 (1997) [CrossRef]

14.

S. Corzine, Ph.D. dissertation, University of California at Santa Barbara, ch. 4. (1993)

15.

S. A. Riyopoulos, D. Dialetis, J. Liu, and B. Riely, “Generic Representation of Active Cavity VCSEL Eigenmodes by Optimized Waist Gauss-Laguerre Modes,” IEEE J. Sel. Top. Quantum Electron. 7, 312–27 (2001) [CrossRef]

16.

E. R. Hegblom, “Engineering oxide apertures in vertical cavity lasers,” Ph.D dissertation, University of California at Santa Barbara, ch. 4. (1999).

17.

A. Mooradian, “Coupled cavity high power semiconductor laser,” United States patent 6778582, (2004)

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(140.5960) Lasers and laser optics : Semiconductor lasers

ToC Category:
Research Papers

History
Original Manuscript: April 8, 2005
Revised Manuscript: May 2, 2005
Published: May 16, 2005

Citation
Young-Gu Ju, "Analysis of mode stability in a concave mirror vertical-cavity surface-emitting laser with an oxide aperture," Opt. Express 13, 3719-3727 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3719


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References

  1. H. Martinsson, J. A. Vukusic, and A. Larsson, �??Single-mode power dependence on surface relief size for mode-stabilized oxide-confined vertical-cavity surface-emitting lasers,�?? IEEE Photon. Technol. Lett. 12, 1129-1131 (2000). [CrossRef]
  2. E. W. Young, K. D. Choquette, S. L. Chuan, K. M. Geib, A. J. Fischer, and A. A. Allerman, �??Single-transverse-mode vertical-cavity lasers under continuous and pulsed operation,�?? IEEE Photon. Technol. Lett. 13, 927-929 (2001) [CrossRef]
  3. Jong-Hwa Baek, Dae-Sung Song, In-Kag Hwang, Kum-Hee Lee, and Y. H. Lee, Young-gu Ju, Takashi Kondo, Tomoyuki Miyamoto, and Fumio Koyama, "Transverse mode control by etch-depth tuning in 1120-nm GaInAs/GaAs photonic crystal vertical-cavity surface-emitting lasers," Opt. Express 12, 859-867 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-859">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-859</a> [CrossRef] [PubMed]
  4. L.A. Coldren, S.P. DenBaars, O. Buchinsky, T. Margalith, D.A. Cohen, A.C. Abare, and M. Hansen, �??Blue and Green InGaN VCSEL Technology,�?? Final Report 1997-98 for MICRO Project 97-033 (1997) , <a href="http://www.ucop.edu/research/micro/97_98/97_033.pdf">http://www.ucop.edu/research/micro/97_98/97_033.pdf</a>
  5. Si-Hyun Park, Jaehoon Kim, Heonsu Jeon, Tan Sakong, Sung-Nam Lee, Suhee Chae, Y. Park, Chang-Hyun Jeong and Geun-Young Yeom, and Yong-Hoon Cho, �??Room-temperature GaN vertical-cavity surface-emitting laser operation in an extended cavity scheme,�?? Appl. Phys. Lett., 83, 2121-2121 (2003) [CrossRef]
  6. K. S. Kim, Y. H. Lee, B. Y. Jung and C. K. Hwangbo, �??Single mode operation of a curved-mirror vertical-emitting laser with an active distributed Bragg reflector,�?? Jpn. J. Appl. Phys. 41, L827-L829 (2002) [CrossRef]
  7. P. Brick, S. Lutgen, T. Albrecht, J. Luft, W. Spath, �??High-efficiency high-power semiconductor disc laser,�?? Proc. SPIE 4993, 50-56 (2003) [CrossRef]
  8. E. M. Strzelecka, J. G. McInerney, A. Mooradian, A. Lewis, A. V. Shchegrov, D. Lee, Wats, �??High-power high-brightness 980-nm lasers based on the extended cavity surface emitting lasers concept,�?? Proc. SPIE 4993, 57-67 (2003) [CrossRef]
  9. S. H. Park, Y. Park, and H. Jeon, �??Theory of the mode stabilization mechanism in concave-micromirror-capped vertical-cavity surface-emitting lasers,�?? J. of Appl. Phys. 94, 1312-1317 (2003) [CrossRef]
  10. A. E. Siegman, Laser, (Oxford University Press, 1986), Chap. 14
  11. A. G. Fox and T. Li, �??Computation of optical resonator modes by the method of resonance excitation,�?? IEEE J. of Quantum Electron. QE-4, 460-465 (1968) [CrossRef]
  12. Young-Gu Ju, "Evaluation scheme for the design of power-optimized single mode vertical-cavity surface-emitting lasers," Opt. Express 12, 2542-2547 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2542">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2542</a> [CrossRef] [PubMed]
  13. Y. G. Ju, J. H. Ser, and Y. H. Lee, "Analysis of metal-interlaced-grating vertical-cavity surface-emitting lasers using the modal method by modal expansion," IEEE J. of Quantum Electron. 33, 589-595 (1997) [CrossRef]
  14. S. Corzine, Ph.D. dissertation, University of California at Santa Barbara, ch. 4. (1993)
  15. S. A. Riyopoulos, D. Dialetis, J. Liu, and B. Riely, "Generic Representation of Active Cavity VCSEL Eigenmodes by Optimized Waist Gauss-Laguerre Modes,�?? IEEE J. Sel. Top. Quantum Electron. 7, 312-27 (2001) [CrossRef]
  16. E. R. Hegblom, �??Engineering oxide apertures in vertical cavity lasers,�?? Ph.D dissertation, University of California at Santa Barbara, ch. 4. (1999).
  17. A. Mooradian, �??Coupled cavity high power semiconductor laser,�?? United States patent 6778582, (2004)

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