## Abstract

In this work, we demonstrate via computer simulation the single mode and zero birefringence conditions for photonic wires with height and width less than 600 nm. We report on the simulation conditions for both single mode and zero birefringence in silicon-on-insulator photonic wires and sub-micron rib waveguides using a 3-dimensional imaginary beam propagation method. The results show that operation in both single mode and zero birefringence is possible under certain circumstances and that the conditions are restricted by fabrication processes where birefringence is strongly dependent upon waveguide dimensions. A matrix of waveguide parameters has been identified at both operating wavelengths of 1310 nm and 1550 nm, which can satisfy single mode and zero birefringence conditions simultaneously. This is to provide a general design rule for waveguides in small dimensions on the order of hundreds of nanometres.

©2007 Optical Society of America

## 1. Introduction

Rib waveguides have been one of the primary building blocks for silicon photonics and Chan *et al*. have successfully addressed the issues of single mode and zero birefringence conditions at the micron scale [1]. With the advent of device miniaturization for improved performance and high density integration, a surge in research activities has focused on photonic wires (PW). Various components have been demonstrated, such as ring resonators [2], modulators [3], arrayed waveguide gratings [4], and one dimensional photonic bandgap microcavities [5]. To date, most research has been conducted on large cross sectional rib waveguides (~1 µm) and/or photonic wires with little emphasis placed on submicron rib waveguide structures. These sub-micron rib structures have several advantages, one of which is the implementation of high speed modulators based on the depletion mechanism for high optical confinement and low loss [3]. To the best of our knowledge, single mode and zero birefringence conditions have not been sufficiently addressed in photonic wires and submicron rib waveguides. Even though PW based devices have been widely demonstrated, they are usually designed only for single mode operation at 1550 nm [6–8].

In this paper, we study the stringent conditions of single mode and zero birefringence operation (ZBC) in two different waveguiding systems: (1) submicron rib waveguides and (2) photonic wires. Our modeling is based on a similar approach conducted at the micron scale on a cross-sectional rib structure as in our previous work [1]. This approach has been implemented on various silicon-on-insulator (SOI) devices [2],[4] and has shown good agreement with experimental data in [2]. The modeling in this work utilized the 3-D imaginary beam propagation method (IDBPM) to solve for the modal profile and effective indices [9] at the communication wavelengths (*λ _{0}*) of 1310 nm and 1550 nm. Also, our simulation results for PWs are compared with an MIT planewave calculation [10] and existing experimental results with good agreement [6]–[8]. This paper is organized as follows. In Section 2, we will discuss the device modeling, structure setup, and mesh grid parameters. Section 3 addresses aspects pertaining to rib waveguides, in particular SMC and ZBC and whether it is possible to satisfy both conditions simultaneously. In Section 4, we will extend the analysis of SMC and ZBC, using similar techniques to address the PWs and we conclude in Section 5.

## 2. Device modeling

The single mode simulation was set up with both strip and rib waveguides based on silicon (ng=3.477) on silicon dioxide (ns=1.444) with an upper cladding of silicon dioxide. SOI PWs with a height of 220 nm to 600 nm are analyzed, whereas rib waveguides are only analyzed at a 300 nm and a 400 nm Si-overlayer. This is to match the Si-overlayer with existing modulator dimensions for homogeneous integration [3]. Computational parameters such as domains along *x, y* and *z* axes should be carefully optimized to include the entire structure and allow the mode field to be sufficiently close to zero at the boundaries. In order to accurately determine the resultant modes in the waveguide under study and upon a mesh optimization exercise, we employed a grid mesh of 5 nm uniformly in all three directions. The waveguiding structures are shown in Fig. 1, which consist of 0.75 µm on both upper cladding *t* and lower cladding *t _{c}*. It is worth mentioning that loss of the waveguides is strongly dependent on

*t*due to substrate leakage and has been reported elsewhere [11]. Hence, in our simulation, we only consider

_{c}*t*≥0.7µm for its low leakage.

_{c}The approach uses an arbitrary field launched off-axis for each of the dominant polarizations. This excites all possible modes propagating in the* z* direction. To expedite the computation time, the mode number, *m*, is restricted to 2. Also, the notation *TE _{m}* and

*TM*is used throughout the following discussion.

_{m}## 3. Sub-micron Rib Waveguides

#### 3.1 Single mode condition

The single mode analysis for submicron rib waveguide is restricted to 300 nm and 400 nm Si overlayers. The simulation is carried out at both wavelengths of 1310 nm and 1550 nm. In the simulation, iterative mode-solving is used to obtain the field shape and propagation constant for the fundamental mode as well as higher order modes. The computations are performed using the IDBPM; it is, in general, an alternative, significantly faster approach than the correlation method used in [1]. The length of the waveguide is set at 10 mm, since the method is iterative and will automatically stop if it converges before the specified length. The launch field represents a Gaussian field launched off axis at one third of the waveguide height and width, thus exciting higher order modes in the structure. In the simulation, the waveguide height (*H*) is fixed, while the etch depth (*D*) is kept constant. The width (*W*) is increased in steps of 10nm until the next higher order mode is supported by the waveguide structure. Thus, the boundary of single mode/multimode operation can be computed by the minimum waveguide width in which the first order mode is allowed to exist in the waveguide. The single mode analysis for rib waveguides uses notation as in [1], where the data is presented in absolute waveguide dimensions as compared to a *W/H* ratio as presented by other authors. This is to give a clear indication of the single mode/multimode boundaries accompanied by the waveguides’ physical structure.

Our simulation results for rib waveguides at *H*=300 nm and *H*=400 nm at both *λ _{0}*=1310 nm and

*λ*=1550 nm are presented in Fig. 2. It can be noted that a gradual increase of the etch depth at 300 nm at

_{0}*λ*=1310 nm creates a wider region for the single mode condition. For

_{0}*H*=300 nm,

*D*=280 nm, the range of

*W*increases from 180 nm to 310 nm. If we compared with

*D*at 200nm, this is a significant increase of

*W*by 325%. On the contrary, for

*H*=400nm, the range of

*W*needed to satisfy the single mode condition decreases by 25%, implying that the single mode condition becomes more stringent with increasing

*D*at larger

*H*.

We also extended our investigation of the single mode condition to a wavelength of 1550 nm. The boundary conditions are shown in Fig. 3, which illustrates the two lowest *TE* and *TM* modes as a function of *W* and *D* for both silicon overlayers of 300 nm and 400 nm.

At *λ _{0}*=1550 nm, a similar trend is observed at both

*H*=300 nm and

*H*=400 nm as shown in Fig. 3. The results suggest that a broader region of single mode condition can be achieved by increasing

*D*. Also, the differences between SMC for different polarizations are more significant in sub-micron rib waveguides compared to their micron counterparts [1] due to the domination of the boundary conditions for both

*TE*and

*TM*polarizations at the sub-micron scale. However, the propagation loss increases for sub-micron waveguides which is caused by strong mode field intensity enhancement at the waveguide edges, resulting in light scattering due to sidewall roughness. Thus, there is a trade-off between waveguide size and propagation loss.

#### 3.2 Birefringence

Another important condition in sub-micron rib waveguides is the ZBC for maintaining phase consistency for *TE* and *TM* polarizations, and for better performance in most interferometric devices. The polarization dependence in PWs increases due to the differing shapes of *TE* and *TM* modes, hence it is believed that a stringent condition on polarization independence is imposed by tight geometrical dimensions and fabrication resolution. Previously, we have shown that ZBC is possible for rib waveguides by an appropriate choice of width and height geometry [1] and Ye *et al*. have also demonstrated this through engineering stress of the thickness *t* of SiO_{2} [12]. Hence, in this work we defined our cladding and substrate SiO_{2} layers for the rib waveguides to be symmetrical (*t=t _{c}*=0.75µm) and evaluated the effective indices of the fundamental quasi-TE and quasi-TM modes for different widths and heights. The ZBC is defined as the difference between the fundamental TE and TM modes

*(N*. Each of the modes has a corresponding

_{eff}(TE)-N_{eff}(TM)*N*. Therefore, in reality, only the single mode range as depicted in Fig. 2 and Fig. 3 needs to be considered. However, we looked into the region of deeper

_{eff}*D*ranging from 200nm to 280nm for

*H*=300nm, and 300nm to 380nm for

*H*= 400nm, for completeness.

The geometrical birefringence of the modes is due to a different alignment of the *E*-fields of the *TE* and *TM* modes. The index difference between the guiding and cladding regions in the vertical direction is much smaller than in the lateral direction. As a result, the *TE* and *TM* modes have different effective indices, which are a function of *W*. Figures 4(a) to 4(d) are graphical representations of the birefringent curves. It can be observed that for *H*=300 nm, the birefringent curves do not cross the *x*-axis. The discontinuities of the birefringent curves suggest that beyond the critical dimension, no mode is allowed to propagate; hence there will be no equivalent effective index. Therefore at *H*=300 nm, the graphs clearly indicate that no possible zero birefringence condition is present for this structure at both *λ _{0}*=1310 nm and

*λ*=1550 nm. Similarly, for

_{0}*H*=300 nm, ZBC is not possible at

*λ*=1310 nm as shown in Fig. 4(c).

_{0}Figure 4(d) shows the birefringent curves for *H*=400 nm at *λ _{0}*=1550 nm. Here, a critical width is defined where the effective indices of the

*TE*and

*TM*polarizations are equal. At

*D*=360 nm and

*D*=380 nm the birefringent curves cross the x-axis and continue to extend into the negative region. Above the critical width, the

*TE*mode has a higher effective index as most power is confined under the rib region which allows the higher-order modes to couple to the slab region. If the width is reduced, the effective index of the

*TE*and

*TM*modes becomes similar, and further reducing the width beyond the critical width increases the effective index of

*TM*modes, thus causing the effective index difference to become negative. The critical width corresponds to the point where the mode profile is nearly symmetrical for both

*TE*and

*TM*polarizations, thereby leading to a ZBC waveguide.

#### 3.3 Single mode and zero birefringence waveguides

Thus far, the requirements for achieving SMC and ZBC are presented separately. In order to identify waveguides which are able to fulfill SMC and ZBC simultaneously, the zero birefringent points in Fig. 4(d) have to be located first. If we now plot the locus of the points that cross the ZBC axis and combine the SMC mode-map in Fig. 3(b), we can demonstrate that both conditions can be met under certain circumstances. It is to be noted that this exercise is only applicable for rib waveguides with *H*=400nm at *λ _{0}*=1550nm. For the other rib structure analyses done in Section 3.1 and 3.2, even though SMC exists in most cases, ZBC is largely absent.

Figure 5 shows a plot of both SMC and the birefringence-free locus for *H*=400 nm at a wavelength of 1550 nm. Two dimensions are identified at *D*=360 nm, *W*=277 nm and *D*=380 nm, *W*=318 nm. These figures suggest that it is possible to achieve single mode condition for both polarizations while maintaining polarization independence at a fixed waveguide height. Theoretically, possible points in the multimode region may exhibit ZBC, hence a point on the birefringence-free locus that is below the single-mode boundary for both *TE* and *TM* modes should be chosen. However, for this particular case, all identified ZBC dimensions are single mode for both polarizations.

## 4. Photonic Wire

#### 4.1 Single mode condition

The single mode analysis for a photonic wire (PW) involves the same approach as described in Section 3.1. Figure 6 shows the simulated result for a PW single mode condition (SMC) at *λ _{0}*=1310 nm and

*λ*=1550 nm. The single mode region is governed by two distinct conditions; (1)

_{0}*TE*cutoff (upper limit) and (2)

_{1}*TM*fundamental mode (lower limit) due to the differing mode shapes of different polarizations. Both

*TE*and

*TM*polarizations are considered here because the analysis will be extended to solve for zero birefringence. We computed a matrix of waveguide dimensions by varying the width,

*W*, from 150 nm to 600 nm in steps of 20 nm until the next higher order mode is computed, thus formulating the SMC/MMC (multimode condition) transition boundary. If we examine a waveguide dimension of 300 nm×350 nm, the result indicates that the waveguide is single mode at 1550 nm, with a cutoff wavelength longer than 1310 nm; this implies that the SMC is more relaxed at longer wavelengths. To verify the accuracy of our simulation, we compared our result with existing experimental data, and plot this data as circles indicated in Fig. 5(b). We can see that the experimental results fall into their respective regions as proposed by various authors [6–8]. Also, if we compare the results in [10] at

*H*=300nm and

*λ*=1550nm (indicated as squares in Fig. 5(b)) with our simulated results, we observe a high degree of similarity. However, their results only specify SMC for the

_{0}*TE*polarization.

From Fig. 5, it can be noted that the *TE* polarization has a more stringent criterion. This is due to the *TE* polarization having two close boundaries where the tangential *E* field must be met. If the boundaries of the *TM _{0}* and

*TE*cutoffs are rigorously fitted, we can obtain an equation that describes the SMC at wavelength 1550 nm as follows:

_{1}Equation (1) describes the width and height for a PW to operate in a single mode. PWs operating at different wavelengths can be solved using the same approach.

#### 4.2. Birefringence

To compute the birefringence curves for PW, we used the same approach described in Section 3.2. Hence, we can produce a graphical representation of the ZBC for different waveguide heights and widths as shown in Fig. 6. These results show that the condition for *N _{eff}(TE)=N_{eff}(TM)* is possible by using an appropriate height and width combination. Curves crossing the birefringence point at zero on the x-axis indicate the ZBC. For each waveguide height

*H*, there is a possible width

*W*for the waveguide to have zero birefringence. However, small variations in PW dimensions, for example, a±10 nm uncertainly on waveguide width, will result in a change in birefringence of 0.02 for 300 nm height. When we compared the birefringence at 300 nm to a larger height of 600 nm, we noticed that the birefringence increased by an order of magnitude (±10 nm uncertainly, for height 300 nm the birefringence

is 0.02; this is compared to a height 600 nm where birefringence is 0.002). This clearly demonstrates the strong polarization dependence in PW due to decreasing the dimensionality. Also, it indicates that the polarization issues for PW are not only restricted by waveguide geometry but also rely on process technology. To date low loss SOI PW fabricated using deep UV lithography have shown an improvement in accuracy of ±5 nm [1] which suggests the possibility of achieving a near ZBC PW with better process controllability and improved accuracy.

#### 4.3. Single mode and zero birefringence photonic wires

SMC and ZBC can be achieved individually in PW. Can we achieve both conditions simultaneously? If we extract the ZBC points in Fig. 6 and form a locus of ZBC for different waveguide dimensions, we effectively obtain a matrix of PW waveguide dimensions that can satisfy this condition. This locus is superimposed on the SMC in Fig. 5 for *λ _{0}*=1330 nm and

*λ*=1550 nm; hence we acquired a set of device parameters that signify SMC and ZBC operation.

_{0}Figure 7 shows the SMC and ZBC locus at *λ _{0}*=1310 nm and

*λ*=1550 nm. In order to satisfy both requirements of SMC and ZBC, a point on the ZBC locus where it intersects within the singlemode region is chosen. These results give a clear indication of the device dimensions where SMC and ZBC are found. At longer wavelengths, a PW that is able to satisfy both conditions simultaneously can occur in larger waveguides. This relaxes the fabrication constraints. However, coupling of light at submicron scale into small waveguides may prove to be difficult.

_{0}It is interesting to note that the intersection between the birefringence locus and the *TE _{1}* cutoff boundary line determines the maximum PW height that allows both conditions. The results in Fig. 7 also reflect that if the PW is symmetrical (

*H=W*) and also satisfies the single mode condition for both

*TE*and

*TM*polarizations, SMC and ZBC are likely to be achieved simultaneously.

## 5. Summary

We have studied the possibility of achieving single mode and zero-birefringence conditions based on SOI photonic wires and submicron rib waveguides. It has been shown that three

distinctive regions, namely no modes, single mode and multimode regimes, existed for PWs and submicron rib waveguides. The condition that governed the SMC is determined by the stringent boundaries of *TM _{0}* and

*TE*. The polarized modes are strongly dependent upon waveguide dimensions which result in an increase in birefringence of an order of magnitude thief the waveguide height is reduced from 600 nm to 300 nm in a PW. In the rib structure, SMC and ZBC only co-exist for

_{1}*H*<400 nm. These results clearly establish the required dimensions of PW and submicron rib waveguides to satisfy both SMC and ZBC simultaneously.

## Acknowledgments

The authors would like to thank Branislav Timotijevic from University of Surrey for insightful discussions and A. Serban from Exploit Technologies, Agency Science and Technology, Singapore for his support in this work.

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