## Simulation of resonant cavity enhanced (RCE) photodetectors using the finite difference time domain (FDTD) method

Optics Express, Vol. 12, Issue 20, pp. 4829-4834 (2004)

http://dx.doi.org/10.1364/OPEX.12.004829

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### Abstract

The resonant cavity enhanced (RCE) photodetectors is analyzed using the finite difference time domain (FDTD) method. Unlike the analytical models, FDTD includes all of the essential considerations such as the cavity build-up time, standing wave effect and the refractive index profiles across every layer. The fully numerical implementation allows it to be used as a verification of the analytical models. The simulation is demonstrated in terms of time and space enabling one to visualize how the field inside the cavity builds up. The results are compared with the analytical models to point out the subtle differences and assumptions made in the analytical models.

© 2004 Optical Society of America

## 1. Introduction

*et al*. [1

1. K. Kishino, M. S. Ûnlü, J. I. Chyi, J. Reed, L. Arsenault, and H. Morkoc, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electronics , **27**, 2025–2034 (1991). [CrossRef]

2. F. Y. Huang, A. Salvador, X. Gui, N. Teraguchi, and H. Morkoc, “Resonant-cavity GaAs/InGaAs/AlAs photodiodes with a periodic absorber structure,” Appl. Phys. Lett. **63**, 141–143 (1993). [CrossRef]

1. K. Kishino, M. S. Ûnlü, J. I. Chyi, J. Reed, L. Arsenault, and H. Morkoc, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electronics , **27**, 2025–2034 (1991). [CrossRef]

## 2. Formulation of FDTD

*E*⃗

_{m}and

*E*⃖

_{m}denote the field traveling to the right and left respectively, and the subscript refers to the

*m*th layer.

*and*

**E***in three dimensions. For simplicity, we will consider a one-dimensional problem with electric and magnetic field components,*

**H***E*and

_{x}*H*, propagating along the

_{y}*z*-direction through a RCE structure shown in Fig. 1. Since this structure contains an absorptive region, we need to write the Maxwell’s curl equations in a more general form, by including a loss term specified by the conductivity such as:

*J*=

_{x}*σE*. Here

_{z}*ε*and

_{r}*σ*denote the relative permittivity and the conductivity, respectively. If we take the finite differences for both the space and time derivatives, then the above equations can be expressed as [7

7. S. C. Hagness and R. M. Joseph, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations,” Radio Science , **31**, 931–941 (1996). [CrossRef]

*n*in the superscripts actually means a time

*t*=

*Δt*·

*n*. And

*k*in parentheses represent distance

*z*=

*Δz*·

*k*. In order to consider the conductivity, which contributes the absorption factor, we use the following relationship [8]:

*ℜ*and

*ℑ*are the reflected and transmitted power, respectively. Since

*ℜ*and

*ℑ*contain the entire structure of the RCE, including the active region and DBR layers, this expression can be used for predicting the performance of the detector.

## 3. Results and discussion

### 3.1 Optical field distribution

*μ*m. There is a great deal of field enhancement inside the resonant cavity in case (a) while the enhancement is weaker in case (b). In this simulation, we have used GaAs for the surrounding regions and InGaAs for the absorption region. The indices of these regions are 3.5 and 3.52, respectively. The top and bottom mirrors are AlGaAs/GaAs DBR stacks designed for 0.9

*μ*m. The index of refraction of AlGaAs and GaAs are 2.97 and 3.5, respectively. Figure 3 shows the energy distribution inside the cavity as a function of time. It can be seen that the steady-state condition is reached at around 540 fs. This build-up time will be an important factor in RCE photodetectors designed for high-speed operations. Although 540 fs corresponds to the THz region of operation, in practice, carrier lifetimes will limit the speed of operation to the GHz range. For instance, the transit time of the Schottky PD’s with a 0.3

*μ*m depletion layer was shown to be 3 ps [10

10. M. GÖkkavas, B. M. Onat, E. Özbay, E. P. Ata, J. Xu, E. Towe, and M. S. Ûnlü, “Design and optimization of high-speed resonant cavity enhanced Schottky photodiodes,” IEEE J. Quantum Electronics , **35**, 208–215 (1999). [CrossRef]

11. M. S. Ûnlü and S. Strite, “Resont cavity enhanced photonic devices,” J. Appl. Phys. **78**, 607–639 (1995). [CrossRef]

*τ*

_{RT}is the time required for photons to make one round trip in the optical cavity, and

*Loss*is the total decay during this trip. In this simulation, the cavity length of 2

*μ*m results in

*τ*

_{RT}≈ 48 fs and

*Loss*≈ 0.53 using typical parameters. From these we get a photon lifetime around 90 fs. This lifetime is significantly smaller than the 540 fs obtained from FDTD. This is mainly due to the fact that equation (8) does not consider the propagation time through the DBR stacks. The DBR stacks are treated as a single reflector with lumped phase shifts at either end. An analytical formulation for the field build-up inside a distributed structure is not trivial which the FDTD can handle quite easily and accurately.

### 3.2 Quantum efficiency

*P*and

_{f}*P*are forward and backward traveling wave respectively inside the structure. The absorption region is defined by an absorption coefficient (

_{b}*α*) and thickness (

*d*).

*L*and

_{1}*L*refer to the separations between the absorption region and the top and bottom mirrors respectively. The absorption coefficient of the material around the absorption region is expressed as

_{2}*α*. With this assumption, they were able to produce a simple equation at the resonant condition, i.e., 2

_{ex}*βL*+

*ψ*+

_{1}*ψ*= 2

_{2}*mπ*(

*m*= 1, 2, 3⋯):

*R*and

_{1}*R*. In practice, the index of refraction of the absorption region is different from the index of refraction of the surrounding regions. Even though this may be a small difference, it is still desirable to include this for self-consistency. In addition, the amplitude and phase of the reflection from the DBR stacks are a function of wavelength. One of the advantages of the FDTD method is that the device does not have to conform to an analytically describable structure. It is simple to develop but powerful for almost any configuration. It should be pointed out that a one-dimensional FDTD is not nearly as time-consuming or CPU-intensive as their three-dimensional counter-parts. The examples shown in this paper were performed using Matlab on a 2GHz Pentium IV computer.

_{2}## 4. Conclusion

## References and links

1. | K. Kishino, M. S. Ûnlü, J. I. Chyi, J. Reed, L. Arsenault, and H. Morkoc, “Resonant cavity-enhanced (RCE) photodetectors,” IEEE J. Quantum Electronics , |

2. | F. Y. Huang, A. Salvador, X. Gui, N. Teraguchi, and H. Morkoc, “Resonant-cavity GaAs/InGaAs/AlAs photodiodes with a periodic absorber structure,” Appl. Phys. Lett. |

3. | A. Srinivasan, S. Murtaza, J. C. Campbell, and B. G. Streetman, “High quantum efficiency dual wavelength resonant-cavity photodetector,” Appl. Phys. Lett. |

4. | B. Temelkuran, E. Ozbay, J. P. Kavanaugh, G. Tuttle, and K. M. Ho, “Resonant cavity enhanced detectors embedded in photonic crystals,” Appl. Phys. Lett. |

5. | Y. H. Zhang, H. T. Luo, and W. Z. Shen, “Study on the quantum efficiency of resonant cavity enhanced GaAs far-infrared detectors,” J. Appl. Phys. |

6. | C. Li, Q. Yang, H. Wang, J. Yu, Q. Wang, Y. Li, J. Zhou, H. Huang, and X. Ren, “Back-incident SiGe-Si multiple quantum-well resonant-cavity-enhanced photodetectors for 1.3-μm operation,” IEEE Photonics Tech. J. |

7. | S. C. Hagness and R. M. Joseph, “Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations,” Radio Science , |

8. | D. K. Cheng, |

9. | M. S. Ûnlü, G. Ulu, and M. GÖkkavas, “Resonant cavity enhanced photodetectors,” in Photodetectors and Fiber Optics, H. S. Nalwa, ed. (Academic Press, San Diego, Calif., 2001), pp. 97–201. |

10. | M. GÖkkavas, B. M. Onat, E. Özbay, E. P. Ata, J. Xu, E. Towe, and M. S. Ûnlü, “Design and optimization of high-speed resonant cavity enhanced Schottky photodiodes,” IEEE J. Quantum Electronics , |

11. | M. S. Ûnlü and S. Strite, “Resont cavity enhanced photonic devices,” J. Appl. Phys. |

12. | M. Born and E. Wolf, |

**OCIS Codes**

(040.0040) Detectors : Detectors

(040.5160) Detectors : Photodetectors

(230.0230) Optical devices : Optical devices

(230.5160) Optical devices : Photodetectors

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 4, 2004

Revised Manuscript: September 22, 2004

Published: October 4, 2004

**Citation**

Jang Pyo Kim and Andrew Sarangan, "Simulation of resonant cavity enhanced (RCE) photodetectors using the finite difference time domain (FDTD) method," Opt. Express **12**, 4829-4834 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4829

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### References

- K. Kishino, M. S. �?nlü, J. I. Chyi, J. Reed, L. Arsenault, and H. Morkoc, �??Resonant cavity-enhanced (RCE) photodetectors,�?? IEEE J. Quantum Electronics, 27, 2025-2034 (1991). [CrossRef]
- F. Y. Huang, A. Salvador, X. Gui, N. Teraguchi, and H. Morkoc, �??Resonant-cavity GaAs/InGaAs/AlAs photodiodes with a periodic absorber structure,�?? Appl. Phys. Lett. 63, 141-143 (1993). [CrossRef]
- A. Srinivasan, S. Murtaza, J. C. Campbell, and B. G. Streetman, �??High quantum efficiency dual wavelength resonant-cavity photodetector,�?? Appl. Phys. Lett. 66, 535-537 (1995). [CrossRef]
- B. Temelkuran, E. Ozbay, J. P. Kavanaugh, G. Tuttle, and K. M. Ho, �??Resonant cavity enhanced detectors embedded in photonic crystals,�?? Appl. Phys. Lett. 72, 2376-2378 (1998). [CrossRef]
- Y. H. Zhang, H. T. Luo, and W. Z. Shen, �??Study on the quantum efficiency of resonant cavity enhanced GaAs far-infrared detectors,�?? J. Appl. Phys. 91, 5538-5544 (2002). [CrossRef]
- C. Li, Q. Yang, H. Wang, J. Yu, Q. Wang, Y. Li, J. Zhou, H. Huang, X. Ren, �??Back-incident SiGe-Si multiple quantum-well resonant-cavity-enhanced photodetectors for 1.3-µm operation,�?? IEEE Photonics Tech. J. 12, 1373-1375 (2000). [CrossRef]
- S. C. Hagness, R. M. Joseph, �??Subpicosecond electrodynamics of distributed Bragg reflector microlasers: Results from finite difference time domain simulations,�?? Radio Science, 31, 931-941 (1996). [CrossRef]
- D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Menlo Park, 1992).
- M. S. �?nlü, G. Ulu, and M. Gökkavas, �??Resonant cavity enhanced photodetectors,�?? in Photodetectors and Fiber Optics, H. S. Nalwa, ed. (Academic Press, San Diego, Calif., 2001), pp. 97-201.
- M. Gökkavas B. M. Onat, E. �?zbay, E. P. Ata, J. Xu, E. Towe, M. S. �?nlü, �??Design and optimization of high-speed resonant cavity enhanced Schottky photodiodes,�?? IEEE J. Quantum Electronics, 35, 208-215 (1999). [CrossRef]
- M. S. �?nlü, S. Strite, �??Resont cavity enhanced photonic devices,�?? J. Appl. Phys. 78, 607-639 (1995). [CrossRef]
- M. Born, E. Wolf, Principles of Optics (Pergamon Press, Oxford, U. K., 1980).

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