## Abstract

The 3D finite-difference time-domain (FDTD) method is used to analyze the polarization effects in two kinds of linearly tapered optical waveguides: slab waveguides with only lateral tapers and rectangular cross section waveguides with both lateral and vertical tapers. For the slab waveguides, each guided mode of both the back reflected and output powers are determined and compared. For rectangular cross section waveguides, the output power of TE and TM modes with respect to taper length are computed and compared.

© 2003 Optical Society of America

## 1. Introduction

Tapered waveguides have been under study for a long time due to their ability to manipulate and transform optical mode sizes to connect optical devices of different cross-sectional dimensions in optical integrated circuits (OICs) [1]. To achieve a highly efficient power coupling, the structure must operate under radiation-loss-free and mode-conversion-free conditions. Although several studies have shown that an adiabatic taper can be used [2–6], these structures require precise control over taper shape and index profile. It is almost impossible to make such tapers using practical waveguide fabrication techniques. As a result, a number of methods have been developed to analyze the radiation loss of tapered waveguides, including intrinsic mode calculations [7], step transition method [8], coupled mode calculations [9], beam propagation method [10,11] and finite-difference third-order simplified wave equation method [12,13]. Those methods agree that, if a taper is long enough and has a small enough taper slope, the taper can achieve low loss. Early investigations have shown that 2D linear tapers [14] provide 90% coupling efficiency and more complicated taper shapes such as parabolic [10], exponential [15], Gaussian and hyperbolic [7] increase this percentage by modest amounts.

However, few papers have discussed the polarization effect in the tapered waveguides — how would the polarization change in the tapered waveguides? In this paper, we present a full wave analysis of the polarization effect in both linearly tapered slab waveguide and rectangular waveguides based on 3D-FDTD methods. For the laterally tapered slab waveguide, the back reflected power and transmitted power are computed. In addition, the reflected and transmitted power of each guided mode is calculated. For rectangular waveguide, we use TE and TM single mode sources to excite the waveguide, respectively. The back reflected power, radiation loss and transmitted power versus taper length for different polarization (TE and TM) are computed and compared. In Section 2, structures of these two waveguides are described. The numerical method used is presented in Section 3. In Section 4, simulation results and discussion are presented, followed by a brief summary in Section 5.

## 2. Tapered slab waveguide and rectangular waveguide

Figure 1 shows the 3D tapered slab waveguide under consideration. It has the following parameters: The height is H=0.228µm, the width of the input side is W1=1.824µm, the width of the output side is W2=0.72µm, the taper length is L=40µm, the length between the taper and the input side is L1=10µm, the length between the taper and the output side is L2=5µm; the slope of the taper is Φ=0.79°, the index inside the waveguide is n=3.5, the index outside is n=1.0. The wavelength considered here is 1µm. So, the taper is actually multimode-to-multimode. Figure 2 shows the top view and the side view of this waveguide.

Figure 3 shows the rectangular tapered waveguide. The dimension of the input side is 2.4×2.4 µm; dimension of the output side is 0.6×0.6 µm. The taper length L changes from 3µm to 15µm, under such condition, the lateral taper slope Φ varies from 3.4° to 16.7°; the vertical taper slope θ varies from 6.8° to 30.9°. The length between taper and the input side L1=2µm, the length between the taper and the output side is L2=12µm. Similar with the slab waveguide, the refractive index inside and outside the waveguide is 3.5 and 1.0, respectively.The wavelength is also 1.0µm. Considering that the top view of this waveguide is the same with Fig. 2(a), Fig. 4 only shows the side view of this waveguide.

According to Marcatili’s method [16], such a waveguide supports both *TE*^{z}
or ${E}_{\mathit{\text{pq}}}^{y}$
and *TM*^{z}
or ${E}_{\mathit{\text{pq}}}^{z}$
modes. Here, we use the lowest mode of both *TE*^{z}
and *TM*^{z}
to excite the waveguide, respectively, in order to find the relation between the polarization change and the taper length. Next, we will introduce the analysis method in detail.

## 3. Analysis method

Using Marcatili’s method, the magnetic field of *TE*^{z}
mode in the 3D optical waveguide can be expressed as:

Here, regions 1–3 are explained in Fig. 5. The important assumption of this method is that the electromagnetic field in the shaded area in Fig. 5 is small enough to be neglected since the electromagnetic field of the well-guided mode decays quite rapidly in the cladding region. Using the orthogonality of the guided modes, Eq. (1) can be interpreted as:

Here, *m* and *n* represent the mode number; A is the amplitude constant, B stands for the mode distribution in the cross section. In order to find the power carried by each guided mode, the amplitude has to be found. From orthogonality, we have:

Also, recalling that the field in the optical waveguide can be written as the sum of each different mode:

In Eq. (4), I stands for the highest mode number the waveguide can support. Therefore, combining Eqs. (3) and (4), we have

Note that *B*_{mn}
(*y, z*) and *C*_{mn}
are known from Eqs. (1) and (3), respectively, so the amplitude constant on each guided mode is then given from Eq. (5):

Once the amplitude is determined, the power carried by each mode can be calculated. To calculate the transmitted power distribution, substitute ${H}_{\mathit{\text{total}}}^{z}$ into Eq. (6) with the transmitted field ${H}_{\mathit{\text{out}}}^{z}$ . To calculate the back-reflected power distribution, substitute ${H}_{\mathit{\text{total}}}^{z}$ into Eq. (6) with the reflected field ${H}_{\mathit{\text{reflect}}}^{z}$ .

We use 3D-FDTD methods to conduct the numerical simulation. The source is introduced on the connecting boundary, as shown in Fig. 6. As a result, the field before the connecting boundary, in region 1, is the reflected and back-scattered field. The field after the connecting boundary, in region 2, is the total field. The shaded region is the PML absorbing boundary.

The computational cost is great due to the full 3D-FDTD simulation. So we wrote the computer program based on message passing interface (MPI) in order to realize the parallel computing on our cluster with 10 computers [17,18].

## 4. Results and discussion

#### 4.1. Laterally tapered slab waveguide

The waveguide is excited with the ${\mathit{\text{TE}}}_{\mathit{11}}^{z}$
mode. The wavelength is set to λ=1µm. The computational step size was set to 0.02µm. Figure 7 shows the steady state field components in the middle *xy* plane. Because it is multimode-to-multimode, the field distribution is not perfectly regular. This is more obvious in *E*_{z}
and *H*_{y}
field components in case that they are the dominant components of *TM*^{z}
mode. The amplitude of the dominant components of *TE*^{z}
mode is much bigger than those of the *TM*^{z}
mode.

Figure 8 shows the normalized amplitude and the power of each mode at the output port. The amplitude is normalized to the source ${\mathit{\text{TE}}}_{11}^{z}$
mode. The power is normalized to the total output power. Our simulation shows that although 100% power of ${\mathit{\text{TE}}}_{11}^{z}$
mode is excited, it remains 81% at the output, some power is coupled into higher order *TE* modes and *TM* modes, some is back reflected and scattered, the other is radiated and absorbed by the PML absorbing boundary. In the output power, ${\mathit{\text{TE}}}_{31}^{z}$
accounts for 15% while in the other *TE* modes, the power is relatively small. Compared with *TE* mode, *TM* mode power is small and the result shows that the power is more likely to couple into higher order *TM* mode. ${\mathit{\text{TM}}}_{42}^{z}$ mode accounts for 0.76%, ${\mathit{\text{TM}}}_{41}^{z}$ mode accounts for 0.48%, ${\mathit{\text{TM}}}_{22}^{z}$ mode accounts for 0.30%, ${\mathit{\text{TM}}}_{21}^{z}$ mode accounts for 0.07%.

Figure 9 shows the normalized amplitude and the power of each mode of the back-reflected field. We should point out that the back reflected power is around 5.4% of the source and our figure only included the relative dominant value so that some lower modes such as ${\mathit{\text{TE}}}_{42}^{z}$, ${\mathit{\text{TE}}}_{52}^{z}$, ${\mathit{\text{TM}}}_{11}^{z}$, ${\mathit{\text{TM}}}_{12}^{z}$ are not plotted. Our results show that the ${\mathit{\text{TE}}}_{11}^{z}$ mode is still the major contributor to the back-reflected field, it accounts for more than 85% of the total back-reflected power. For *TM* mode, the value of most of the modes is small except ${\mathit{\text{TM}}}_{21}^{z}$ and ${\mathit{\text{TM}}}_{42}^{z}$, which take 0.72% and 0.31% respectively.

#### 4.2. Tapered rectangular waveguide

In order to find the relation between polarization change and the taper length, we repeated the numerical experiment several times with different taper length, from 3µm to 15µm. We used ${\mathit{\text{TE}}}_{11}^{z}$ and ${\mathit{\text{TM}}}_{11}^{z}$ modes as sources, respectively and found that *TM* modes have less radiation loss and better polarization stability. Figure 10 shows the radiation loss comparison and back-reflected loss comparison. As you can see, the radiation loss of *TE* mode is almost twice the radiation loss of *TM* mode, for taper length longer than 6µm. They reach the same level only when the taper length is equal to 3µm. As for back-reflected loss, both of them remain at thevery low level except at 3µm, the *TM* mode outreaches *TE* mode by 7% with respect to the source power.

Figure 11 shows the percentage of output power with respect to source power versus taper length, Fig. 11(a) corresponds to the *TM* source and Fig. 11(b) corresponds to the *TE* source. Our simulation shows that for the same taper length, *TM* mode has less polarization change compared with *TE* mode. In other words, *TM* mode has better polarization stability. For example, when the taper length is 9µm, if the source is *TM*, then 81.6% power remains in *TM* mode, 11.8% power changes into *TE* mode, but if the source is *TE*, only 43% power remains in *TE* mode, 28.5% power changes into *TM* mode. What’s more, our simulation shows that the polarization remains well with long taper length and tends to change when the taper length becomes short. This is reasonable since in the short taper, the mode profile changes so fast that the EM field no longer satisfies the adiabatic boundary conditions. As a result the polarization will change in order to adapt to the new boundary condition. As is well know, an infinitely long taper is perfectly adiabatic in which case the polarization should remain unchanged, since it is in fact a straight waveguide. Figure 11 also shows that both the *TM* mode output power and *TE* mode output power decrease rapidly due to the increasing radiation loss and back-reflection loss, when the taper length becomes shorter than 9µm.

## 5. Conclusions

Polarization effects in linearly tapered waveguide are studied with full wave 3D-FDTD methods. Two kinds of tapered structures are presented and studied. For the tapered slab waveguide that we studied, more than 81% output power remains in the source mode: ${\mathit{\text{TE}}}_{11}^{z}$ mode, 15% power couples into ${\mathit{\text{TE}}}_{31}^{z}$. Although *TM* output mode power is less than 2%, most of them are coupled into higher order modes. ${\mathit{\text{TM}}}_{42}^{z}$ mode accounts for 0.76%, ${\mathit{\text{TM}}}_{41}^{z}$ mode accounts for 0.48%. For rectangular waveguide, our simulation showed that *TM* mode has better polarization stability and less radiation loss. Under the same condition, the radiation loss of *TE* mode is almost twice that of the *TM* mode. In addition, less power is changed into other polarization if the taper is excited with *TM* mode than *TE* mode.

Even though two examples of linearly tapered waveguides were discussed in details in the paper, the analysis can be generalized to include any taper shapes such as parabolic, exponential, Gaussian and hyperbolic tapers, which will be studied in the future.

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