## 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

Resonant cavity enhanced (RCE) photodetectors can have quantum efficiencies close to unity. This is achieved by utilizing reflectors around the active region. The photons make multiple passes across the active region, improving the probability of absorption. Kishino *et al*. [1] presented an analytical formulation for the first time in 1991. A key assumption in their model was the inclusion of the standing wave effect (SWE) separate from the initial formulation. Many of the papers [2–6] published since 1991 cite the formula directly and treat the SWE separately in the same manner. While this formulation is correct, it makes the interpretation of several physical effects difficult.

Another aspect of this formulation is that the reflection at the interfaces between the absorption layer and surrounding layers are generally ignored. Although this may be a small effect due to the relatively low index differences between the layers, it is still desirable to have a self-consistent model that accounts for the complete refractive index profile inside the cavity, not just the Bragg mirrors. For this purpose, we developed a finite difference time domain (FDTD) method to carry out a rigorous analysis of RCE structures. In this paper, the FDTD method is discussed in terms of not only accuracy and scope of validity but also simplicity and insightfulness. Compared to the analytical model presented in [1] (hereafter referred to as the analytical model), this method leads to comparatively similar but more accurate and insightful results. This approach is an efficient way to visualize how the field inside the cavity propagates and builds in intensity as a function of time and space.

## 2. Formulation of FDTD

Figure 1 shows the schematic of the RCE structure. The structure consists of a top DBR stack, the absorption region and a bottom DBR stack. In the FDTD method, each stack is modeled with a propagation element and an interfacial element.

Since the field at every point is a superposition of forward and backward propagating waves, we can write:

where *E*⃗ _{m} and *E*⃖ _{m} denote the field traveling to the right and left respectively, and the subscript refers to the *m*th layer.

In free space, the time-dependent Maxwell’s curl equations can be written in term of * E* and

*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:

where *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]

In these equations, *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]:

The external quantum efficiency of the detector is the ratio of photon absorption that results in a photocurrent compared to the incident photons. If we assume only the active layer absorbs light, the quantum efficiency can be written as [9]

where *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

The FDTD method can be used to obtain the optical field distribution in a RCE photodetector without deriving any extra formulae. Figure 2(a), (b) show the position dependent cavity field distribution in the case with no absorption and in the case with absorption, respectively, at a wavelength of 0.9 *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], which is in the GHz range. Nevertheless, unlike the standard photodetectors, which have a photon lifetime of nearly zero, RCE detectors have an upper limit to their frequency of operation determined by the cavity buildup time. The time required to build the optical field inside the cavity has thus far been calculated using a simple cavity model as [11]

where *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

Kishino *et al*. [1] presented a detailed analysis showing a near unity quantum efficiency in a RCE photodetector. In their derivation, the standing wave effect was accounted for as an after-the-fact effect to include the interference of the forward and backward waves. The power absorbed in the absorption region (*P _{l}*) was shown to be related to the incident power

*P*as:

_{i}where *P _{f}* and

*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⋯):

An alternative way to this interpretation would be to calculate the total electric field first and then interpret the absorbed power from the field distribution. This approach will ensure that the standing wave effect is included self-consistently without introducing it afterwards.

Another important aspect is that the analytical formulae ignored the distributed nature of the refractive index changes across each layer. The amplitude and phase of the DBR reflector was lumped into a single parameter such as *R _{1}* and

*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}Our results were compared with the analytical model and the transfer matrix method (TMM) [12]. Most of the results from the FDTD were consistent with the analytical model except for the oscillations shown in Fig. 4. Notice that the FDTD and the TMM results agree almost exactly for the quantum efficiency. This difference between the analytical model and the FDTD arises from the standing wave effect. It should be noted that we have a different maximum quantum efficiency compared to the analytical model at the same normalized absorption coefficient in both cases. For instance, we have less than 20% quantum efficiency when the normalized absorption coefficient is 0.17 while the analytical model predicts more than 90% as in Fig. 4. This clearly demonstrates that one has to be careful in choosing the optimum thickness in order to get maximum quantum efficiency for a given structure.

In addition, the small difference in the envelope in Fig. 4 is attributed to the refractive index change between the absorption and the surrounding regions, which the FDTD and the TMM can capture quite accurately. Comparing the three different simulations, we conclude that the refractive index mainly determines the local maxima position of the envelope, that is, it shifts the curve slightly.

## 4. Conclusion

We have analyzed the RCE photodetectors using the finite difference time domain (FDTD) method in order to accurately predict the cavity build-up time, and to include the standing wave effect and the refractive index difference across each layer in a self-consistent manner.

The FDTD method is developed in terms of time and space in which we can visualize how the beam propagates and builds up in the RCE-type structures. The simulation shows that steady-state condition is reached at 540 fs, which is much longer than the photon lifetime predicted using a simple analytical model. This build-up time could be an important factor in RCE photodetectors designed for high-speed operation. In practice, however, carrier lifetime may limit the speed of detector down to the GHz region. The FDTD method demonstrates a comparatively similar pattern to the analytical model but represents more accurate analysis with fewer assumptions.

It is clear from the results that, when the normalized absorption coefficient is 0.17, the analytical model predicts that more than 90 % of the quantum efficiency can be achieved. However, the FDTD method predicts a quantum efficiency of less than 20 %. Therefore, this effectively demonstrates that the SWE plays a crucial role because the oscillatory nature of the light confinement determines the local maxima and minima as a function of normalized absorption coefficient. As a result, one has to carefully choose the optimum thickness in order to get maximum quantum efficiency for a given structure. In addition, the refractive index difference across each layer is contributed to the alternative shape of envelope of the quantum efficiency, so it should also be considered for the rigorous analysis of the RCE photodetector.

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