## Spontaneous lifetime in a dielectrically-apertured Fabry-Perot microcavity

Optics Express, Vol. 2, Issue 4, pp. 157-162 (1998)

http://dx.doi.org/10.1364/OE.2.000157

Acrobat PDF (232 KB)

### Abstract

We present calculations of the modification of the spontaneous emission rate from a point source dipole in a Fabry-Perot microcavity containing an optically thin dielectric aperture. The dielectric aperture is described as a passive current source which is driven by the spontaneous point source. The spontaneous emission rate is shown to depend on the details of the aperture design, and shows a strong enhancement on resonance due to 3-dimensional optical confinement by the dielectric aperture.

© Optical Society of America

## 1. Introduction

1. D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. **70**, 3443–3448 (1991). [CrossRef]

7. G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. **30**, 2314–2318 (1994). [CrossRef]

6. C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. **30**, 2304–2313 (1994). [CrossRef]

10. D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. **64**, 97–99 (1994). [CrossRef]

11. J.M. Dallesasse, N. Holonyak Jr., A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. **57**, 2844–2846 (1990). [CrossRef]

12. D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. **67**, 2494–2596 (1995). [CrossRef]

10. D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. **64**, 97–99 (1994). [CrossRef]

13. D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. **71**, 160–162 (1997). [CrossRef]

14. Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. **33**, 2319–2326 (1997). [CrossRef]

15. D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. **9**, 713–715 (1997). [CrossRef]

## 2. Calculation

_{R}in the center of the cavity. The dielectric constant inside and outside the cavity is taken as that of free space, which simplifies the notation. However, we note that the same treatment can be applied to the semiconductor dielectric microcavity, although the complexity of the math is increased.

**r**,ω) is the dielectric constant that satisfies the planar cavity boundary conditions. For the case of Fig. 1,

**J**(

**r**,ω) has two contributions which are the spontaneous point source

**J**

_{sp}(

**r**,ω) and the induced polarization current in the aperture

**J**

_{d}(

**r**, ω). The radiation field from

**J**(

**r**, ω) can be calculated directly from Maxwell’s equations with the precaution that reflections from the cavity mirrors must be considered. Taking the spontaneous source as having vector amplitude lying in the x-y plane, for a symmetrical system as shown in Fig 1 the z component of the radiation field in the center of the cavity is zero while the x and y components take the Fourier transformed form [13

13. D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. **71**, 160–162 (1997). [CrossRef]

14. Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. **33**, 2319–2326 (1997). [CrossRef]

**J**

_{d}(

**r**, ω) from Eq. (2). The solution is then reinserted back into the right-hand side of (4) to calculate the next higher order approximation. This process is continued through enough iterations to obtain the convergent spontaneous field in the presence of the dielectric aperture.

## 3. Results

1. D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. **70**, 3443–3448 (1991). [CrossRef]

2. G. Bjork, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures” Phys. Rev. A **44**, 669–681 (1991). [CrossRef] [PubMed]

^{3}r

**E**

^{*}(

**r**, t) ·

**J**

_{sp}(

**r**, t) with the time dependence approximated as e

^{-iωt}. The two cavity mirrors have the same field reflectivity of 0.995. To obtain rapid convergence in Eq. (4), we assume that the thin dielectric disk has a Gaussian distribution for χ

_{R}(x,y,ω)Δz

_{R}. The amount of dielectric confinement is then characterized by the index step χ

_{R}(0,0,ω)Δz

_{R}at the center of the cavity and the radius of the disk is taken as w

_{χR}= 2 μm. The influence of the aperture on the spontaneous emission rate is studied by choosing different values of χ

_{R}(0,0,ω)Δz

_{R}(real values) and finding the self-consistent solutions from Eq. (4) for a range of

*ω*Figure 2 shows the calculated results for three values of susceptibility given as (a) χ

_{R}(0,0,ω)Δz

_{R}= 0, (b) χ

_{R}(0,0,ω)Δz

_{R}=18Å, and (c) χ

_{R}(0,0,ω)Δz

_{R}=36Å. The case of (a) χ

_{R}(0,0,ω)Δz

_{R}= 0 (planar half-wave cavity), in particular, has been studied previously with the results fairly well understood. [7

7. G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. **30**, 2314–2318 (1994). [CrossRef]

8. C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. **75**, 4668–4672 (1994). [CrossRef]

6. C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. **30**, 2304–2313 (1994). [CrossRef]

8. C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. **75**, 4668–4672 (1994). [CrossRef]

_{R}(0,0,ω)Δz

_{R}=18Å. The cut-off of the waveguide modes due to the aperture leads to 3-dimensional confinement, and we see a peak form in the spontaneous emission rate close to resonance. As expected for a 3-dimensionally confined mode, a reduction in the spontaneous emission rate is obtained for frequencies either too far above or below resonance due to detuning. For a 3-dimensionally confined mode, a loss rate dependence is expected for the resonant frequency and this is also observed. The 3-dimensionally confined mode suffers loss due to both mirror transmission and waveguide propagation. [9

9. Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A **53**, 1036–1047 (1996). [CrossRef] [PubMed]

_{R}(0,0,ω)Δz

_{R}=36Å.

## 4. Summary

14. Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. **33**, 2319–2326 (1997). [CrossRef]

## Acknowledgments

## References and links

1. | D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. |

2. | G. Bjork, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures” Phys. Rev. A |

3. | K. Ujihara, “Spontaneous emission and the concept of effective area in a very short cavity with plane parallel dielectric mirrors,” Jpn. J. Appl. Phys., Part 2 |

4. | N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. |

5. | D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. |

6. | C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. |

7. | G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. |

8. | C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. |

9. | Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A |

10. | D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. |

11. | J.M. Dallesasse, N. Holonyak Jr., A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. |

12. | D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. |

13. | D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. |

14. | Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. |

15. | D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. |

16. | R.F. Harrington, |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Focus Issue: Quantum well laser design

**History**

Original Manuscript: October 8, 1997

Published: February 16, 1998

**Citation**

Q. Deng and D. Deppe, "Spontaneous lifetime in a dielectrically-apertured Fabry-Perot microcavity," Opt. Express **2**, 157-162 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-4-157

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### References

- D.G. Deppe and C. Lei, "Spontaneous emission from a dipole in a semiconductor microcavity," J. Appl. Phys. 70, 3443-3448 (1991). [CrossRef]
- G. Bjork, S. Machida, Y. Yamamoto, and K. Igeta, "Modification of spontaneous emission rate in planar dielectric microcavity structures" Phys. Rev. A 44, 669-681 (1991). [CrossRef] [PubMed]
- K. Ujihara, "Spontaneous emission and the concept of effective area in a very short cavity with plane parallel dielectric mirrors," Jpn. J. Appl. Phys. , Part 2 30, L901-L903 (1991). [CrossRef]
- N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, "Controllable enhancement of excitonic spontaneous emission in quantum microcavities," Appl. Phys. Lett. 58, 2735-2737 (1991). [CrossRef]
- D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, "Controlled spontaneous emission in room temperature semiconductor microcavities," Appl. Phys. Lett. 60, 3202-3205 (1992). [CrossRef]
- C.C. Lin, D.G. Deppe, and C. Lei, "Role of waveguide light emission in planar microcavities," IEEE J. Quantum Electron. 30, 2304-2313 (1994). [CrossRef]
- G. Bjork, "On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes," IEEE J. Quantum Electron. 30, 2314-2318 (1994). [CrossRef]
- C.C. Lin and D.G. Deppe, "Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities," J. Appl. Phys. 75, 4668-4672 (1994). [CrossRef]
- Q. Deng and D.G. Deppe, "Spontaneous-emission coupling from multiemitters to the quasimode of a Fabry-Perot microcavity," Phys. Rev. A 53, 1036-1047 (1996). [CrossRef] [PubMed]
- D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, "Native-oxide defined ring contact for low threshold vertical-cavity lasers," Appl. Phys. Lett. 64, 97-99 (1994). [CrossRef]
- J.M. Dallesasse, N. Holonyak, Jr., A.R. Sugg, T.A. Richard, and N. El-Zein, "Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures," Appl. Phys. Lett. 57, 2844-2846 (1990). [CrossRef]
- D.L. Huffaker and D.G. Deppe, "Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities," Appl. Phys. Lett. 67, 2494-2596 (1995). [CrossRef]
- D.G. Deppe and Q. Deng, "Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser," Appl. Phys. Lett. 71, 160-162 (1997). [CrossRef]
- Q. Deng and D.G. Deppe, "Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors," IEEE J. Quantum Electron. 33, 2319-2326 (1997). [CrossRef]
- D.G. Deppe, T.-H. Oh, and D.L. Huffaker, "Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity," IEEE Photonics Technol. Lett. 9, 713-715 (1997). [CrossRef]
- R.F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961) pg. 188.

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