## Beam propagation behavior in a quasi-stadium laser diode

Optics Express, Vol. 2, Issue 2, pp. 21-28 (1998)

http://dx.doi.org/10.1364/OE.2.000021

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### Abstract

The beam propagation behavior of a quasi-stadium laser diode is theoretically investigated. The resonator that we analyzed consists of one flat end-mirror, one convex curved end-mirror and two straight side wall mirrors. The cavity dimension is much larger than the oscillation wavelength. We derived one-dimensional Huygen’s integral equations for this laser cavity and carried out eigenmode calculations using the Fox and Li mode calculation method taking into account the effect of the side wall reflections and visualized the propagation beams. Unique beam propagation behaviors were obtained. These results well agree with our previous experimental results.

© Optical Society of America

## 1. Introduction

2. E. J. Heller, “Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits,” Phys. Rev. Lett. **53**, 1515 (1984) [CrossRef]

3. E. J. Heller and S. Tomsovic, “Postmodern Quantum Mechanics,” Phys. Today **46**, 38 (1993) [CrossRef]

2. E. J. Heller, “Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits,” Phys. Rev. Lett. **53**, 1515 (1984) [CrossRef]

9. T. Fukushima, S. A. Biellak, Y. Sun, C. G. Fanning, Y. Cheng, S. S. Wong, and A. E. Siegman, Lasing Characteristics of a Quasi-Stadium Laser Diode, in *Conference on Lasers and Electro-Optics*, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 227.

9. T. Fukushima, S. A. Biellak, Y. Sun, C. G. Fanning, Y. Cheng, S. S. Wong, and A. E. Siegman, Lasing Characteristics of a Quasi-Stadium Laser Diode, in *Conference on Lasers and Electro-Optics*, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 227.

## 2. Device structure and theoretical model

9. T. Fukushima, S. A. Biellak, Y. Sun, C. G. Fanning, Y. Cheng, S. S. Wong, and A. E. Siegman, Lasing Characteristics of a Quasi-Stadium Laser Diode, in *Conference on Lasers and Electro-Optics*, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 227.

*L*and width

*W*are 660 and 60μm, respectively. The radius of curved end-mirror

*R*is 60μm. The width of flat end-mirror

*W*is 20μm. The side walls are separated from both cavity ends by distance

_{r}*W*, which is 90μm. The purpose of the open unpumped corner regions is to suppress the higher order ray trajectories.

_{s}*E*(

_{f}*x*) with uniform intensity and phase as an initial condition. The field distribution on the curved end-mirror

_{f}, z_{f}*E*(

_{c}*x*) is calculated by using following one-dimensional Huygen′s integral equation from the images of the flat end-mirror including the virtual images,

_{c},z_{c}*λ*and

*k*= 2

*π*/

*λ*= 2

*πn*/

_{eff}*λ*

_{0}are oscillation wavelength and wavenumber inside the laser cavity.

*λ*

_{0}is wavelength in a vacuum and

*n*is effective index of the laser diode.

_{eff}*n*indicates the number of the flat end-mirror images as shown in Fig. 2a. cos

*θ*is the obliquity factor which depends on the angle

*θ*between the line element (

*x*)-(

_{f},z_{f}*x*) and the normal to the surface element

_{c},z_{c}*dx*.

_{f}*n*is the number of the curved end-mirror images.

*θ*is the angle between the line element (

*x*)-(

_{f},z_{f}*x*) and the normal to the surface element

_{c},z_{c}*dr*.

*x*and

*y*. During these Huygen’s integrals, we took into account the effect of the open unpumped corner regions by removing the field propagating from the surface element

*dx*or

_{f}*dr*whose line element (

*x*)-(

_{f},z_{f}*x*) crosses the open regions of the side walls. These round-trip calculations are repeated until the round-trip eigenvalue

_{c},z_{c}*γ*converges. We defined

*γ*as,

*m*is the number of the round-trip.

## 3. Calculation results

### 3.1 Spectrum of round-trip eigenvalue

*n*as 3.3, so that the calculation results agree with our experimental results.[9

_{eff}*Conference on Lasers and Electro-Optics*, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 227.

*n*is the number of the virtual images taken into account on each side of the laser cavity during the eigenmode calculation. When

_{v}*n*= 0, the resonator becomes simple unstable resonator without side wall reflections. In this case, the round-trip eigenvalue is relatively small. When we take one virtual image on each side of the laser cavity (

_{v}*n*= 1), the round-trip eigenvalue spectrum shows some ripples. It is speculated that these ripples are caused by the interference among the beam propagating directly and the beams reflected at the side walls. As the number of the virtual images increases, the round-trip eigenvalue increases and the spectrum becomes more complex shape, namely, some narrow and sharp peaks appear in the spectrum. However, there is no difference between the spectrums calculated for

_{v}*n*= 2 and

_{v}*n*= 4. The reason is that the open unpumped corner regions in the laser cavity restrict the higher order ray trajectories. It is found that

_{v}*n*= 2 is enough number to analyze this laser cavity.

_{v}### 3.2 Output beam patterns propagating from both end-mirrors

*Conference on Lasers and Electro-Optics*, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 227.

### 3.3 Beam propagation behavior inside the laser cavity

2. E. J. Heller, “Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits,” Phys. Rev. Lett. **53**, 1515 (1984) [CrossRef]

*R*/2 away from the top of the curved end-mirror. These spots again diverge and propagate toward the flat end-mirror with large beam divergence. Eventually the same fringe pattern is formed on the flat end-mirror after one round-trip.

## 4. Conclusions

## Acknowledgments

## Footnotes

* | Present address, KLA-Tencor Corporation, 1 Technology Drive, Milpitas, California 95035 |

** | Present address, Bell Laboratories Lucent Technologies, Holmdel, New Jersey 07733 |

## References and links

1. | M. C. Gutzwiller, |

2. | E. J. Heller, “Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits,” Phys. Rev. Lett. |

3. | E. J. Heller and S. Tomsovic, “Postmodern Quantum Mechanics,” Phys. Today |

4. | S. W. McDonald and A. N. Kaufman, “Wave Chaos in the Stadium: Statistical Properties of Short-Wave Solutions of the Helmholts Equation,” Phys. Rev. A |

5. | S. Tomsovic and E. J. Heller, “Semiclassical Dynamics of Chaotic Motion: Unexpected Long-Time Accuracy,” Phys. Rev. Lett. |

6. | S. A. Biellak, “Reactive Ion Etched Unstable and Stable Semiconductor Diode Lasers,” Ph.D. thesis, Stanford University (1995) |

7. | S. A. Biellak, C. G. Fanning, Y. Sun, S. S. Wong, and A. E. Siegman, High Power Diffraction Limited Reactive-Ion-Etched Unstable Resonator Diode Lasers, in |

8. | Y. Sun, “Lateral Mode Control of Semiconductor Lasers,” Ph.D. thesis, Stanford University (1995) |

9. | T. Fukushima, S. A. Biellak, Y. Sun, C. G. Fanning, Y. Cheng, S. S. Wong, and A. E. Siegman, Lasing Characteristics of a Quasi-Stadium Laser Diode, in |

10. | A. E. Siegman, |

**OCIS Codes**

(140.2020) Lasers and laser optics : Diode lasers

(140.3300) Lasers and laser optics : Laser beam shaping

(140.3410) Lasers and laser optics : Laser resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 21, 1997

Revised Manuscript: October 4, 1997

Published: January 5, 1998

**Citation**

Takehiro Fukushima, S. Biellak, Yan Sun, and Anthony Siegman, "Beam propagation behavior in a quasi-stadium laser diode," Opt. Express **2**, 21-28 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-2-21

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### References

- M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York Inc., 1990)
- E. J. Heller, "Bound-State Eigenfunctions of Classically Chaotic Hamiltonian Systems: Scars of Periodic Orbits," Phys. Rev. Lett. 53, 1515 (1984) [CrossRef]
- E. J. Heller and S. Tomsovic, "Postmodern Quantum Mechanics," Phys. Today 46, 38 (1993) [CrossRef]
- S. W. McDonald and A. N. Kaufman, "Wave Chaos in the Stadium: Statistical Properties of Short-Wave Solutions of the Helmholts Equation," Phys. Rev. A 37, 3067 (1988) [CrossRef] [PubMed]
- S. Tomsovic and E. J. Heller, "Semiclassical Dynamics of Chaotic Motion: Unexpected Long-Time Accuracy," Phys. Rev. Lett. 67, 664 (1991) [CrossRef] [PubMed]
- S. A. Biellak, "Reactive Ion Etched Unstable and Stable Semiconductor Diode Lasers," Ph.D. thesis, Stanford University (1995)
- S. A. Biellak, C. G. Fanning, Y. Sun, S. S. Wong, and A. E. Siegman, "High Power Diffraction Limited Reactive-Ion-Etched Unstable Resonator Diode Lasers," in Conference on Lasers and Electro-Optics, Vol. 8 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), p. 4.
- Y. Sun, "Lateral Mode Control of Semiconductor Lasers," Ph.D. thesis, Stanford University (1995)
- T. Fukushima, S. A. Biellak, Y. Sun, C. G. Fanning, Y. Cheng, S. S. Wong, and A. E. Siegman, "Lasing Characteristics of a Quasi-Stadium Laser Diode," in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), p. 227.
- A. E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986)

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