## Abstract

The performance of a polarization splitter based on vertically coupled microring resonator is rigorously investigated by a combination of a 3D full vectorial film mode matching method with a 3D full vectorial coupled mode theory. The spectral responses of the structure for TE and TM mode are calculated, together with eigenmodes of uncoupled waveguides and scattering matrix of coupling region. The result shows that the response of microring resonator is indeed strongly polarization dependent and the resonance wavelengths are different for TE and TM mode. Such property allows for the design of wavelength-sensitive integrated polarization splitter. The influence of geometrical parameters on splitting ratio is investigated and the results indicate that the structure can have a splitting ratio greater than 20dB at 1.55µm.

© 2006 Optical Society of America

## 1. Introduction

Polarization splitters, which can separate the two orthogonal polarization states, are essential components in photonics. Integrated polarization splitters offer the possibility of integration with other optical elements as well as the advantages of compactness. Various types of integrated polarization splitters have been reported in literatures [1–3].

Currently integrated photonic devices based on microring resonators have been widely studied because of their compact size and attractive spectral and polarization properties suitable for very large-scale integrated (VLSI) photonic circuits [4]. The idea of microring resonator type polarization splitter was first proposed by Klunder [5]. The working principles can be explained in terms of polarization dependent resonance wavelength (PDW): due to the difference in modal index and coupling coefficients for TE and TM mode, the responses of microring resonators show strong polarization dependence and the resonance wavelengths may be different for TE and TM mode. Thus, microring resonators can drop one polarized light to the drop port and transfer the other to the through port at specific wavelength.

In this work, we analyze a wavelength sensitive polarization splitter based on vertically coupled microring resonator as sketched in Fig. 1. A frequency domain model based on 3D full vectorial film mode matching method (FMM) [6–7] and coupled mode theory (CMT) [8–9] is employed to obtain the eigenmodes of uncoupled bent and straight waveguides, the scattering matrix of coupling region and finally the spectral responses of the entire structure. FMM is a rigorous and efficient way to calculate the eigenmodes of a real waveguide structure. If the geometries of waveguides are not too complicated, FMM is superior to other numerical methods such as finite-difference method, because the method tackles the problem semi-analytically and avoids unnecessary discretization. CMT has been applied extensively in integrated photonics as an intuitive tool for the analysis of interaction of modes in different waveguides which are brought into close proximity [9]. Using the eigenmodes of uncoupled waveguides as basis modes, we can obtain the scattering matrix which describes the behavior of coupling region by CMT. Then the spectral property of the entire structure can be evaluated in a straightforward way, and the simulation result indicates that pronounced polarization dependence allows for the design of wavelength sensitive polarization splitters. To the best of our knowledge, a polarization splitter based on microring resonator is modeled for the first time and the full vectorial nature of the model enables very accurate description of the device.

## 2. Theory and numerical results

Figure 2(a) sketches the basic geometry of the vertically coupled microring resonator under investigation. The ring resonator is symmetrically coupled to two identical straight waveguides. The straight waveguides with width v=1.0µm, thickness s=0.5µm, and refractive index *n*_{s}
=1.98 is embedded in a substrate with index *n*_{g}
=1.45. The bent waveguide of width w=1.0µm, thickness t=0.5µm and radius R=45.01µm with refractive index *n*_{b}
=2.0 is placed on the top of the substrate surface, covered by a material with refractive index *n*_{a}
=1.4017. The relative position of the straight core is defined by the horizontal gap d and the vertical position b between them, where d=R-H can be positive or negative values.

According to the standard resonator model [10], we assume that the interaction between straight and bent waveguide is confined in two “coupling regions” which are enclosed by the blue lines in Fig. 2(b) and individual modes propagate independently in their respective waveguides outside the regions. Thus, the structure can be divided into two couplers (coupler I and II) which are connected by two bends with length of L/2. In order to predict the spectral response of the resonator, we need a description of the wave propagation along the bends, the analysis of the behavior of the coupling region, and finally a framework to combine these individual parts.

Three steps are taken to model the structure. That is, the propagation constants and the eigenmodes of straight and bent waveguides are computed using FMM; the scattering matrix describing the behavior of coupling region is obtained by means of CMT and the spectral responses of microring resonator for TE and TM mode are calculated.

#### 2.1. Calculation of eigenmodes of straight and bent waveguides using FMM

In the FMM mode-solving arithmetic, waveguide cross section is divided into a number of vertical slices. In each slice, the mode of entire structure is expressed as a superposition of the slice eigenmodes. In order to achieve adequate accuracy, sufficient local guided and radiation modes, with corresponding propagation constants, have to be computed. Matching the fields of these slices at the vertical interfaces, we can determine the modes of the entire structure [7].

We calculated the eigenmodes of the individual waveguides described in Fig. 2. Both the two waveguides support only fundamental TE and TM mode. Figure 3 and Fig. 4 show the cross sections of the two waveguides with horizontal gap *d*=-0.5*µm* and vertical position *b*=0.5*µm* as well as the corresponding modal profiles. Table 1 lists the propagation constants of modes in those waveguides. Since an electromagnetic field propagating through a bend region would lose energy due to radiation, *γ*_{b}
=*β*-*iα* is complex value, where *β* and *α* is phase constant and attenuation constant.

FMM numerical parameters are vertical computational window [*x*
_{min}, *x*
_{max}]and numbers of slice mode *M*. For straight waveguide, [*x*
_{min}, *x*
_{max}]=[-6.3*µm*,3.7*µm*] and *M*=100. For bent waveguide, [*x*
_{min}, *x*
_{max}]=[-4.0*µm*,4.0*µm*] and *M*=200. Operation wavelength *λ*=1.55*µm*.

#### 2.2. Vertical coupling between straight waveguide and bent waveguide

In this section, the coupling region shown in Fig. 5 is analyzed. CMT is established on the assumption that the field of the coupling region can be represented as a linear superposition of the modes of uncoupled waveguides. In our cases, each waveguide in isolation supports only one guided mode per polarization direction, and different polarizations are not expected to interact significantly because the hybridization of the modes is negligible, hence the field in the coupling region can be written as:

(* E_{b}*,

*), (*

**H**_{b}*E*

_{s},

*) are the fields of the bent and straight waveguides.*

**H**_{s}*A*

_{b},

*A*

_{s}are unknown amplitudes. Using Lorentz reciprocity theorem, we can deduce the coupled mode equation [11]:

where *C*
^{-1} and *K* are two 2×2 matrices, and their elements are based on overlap integrals of different fields [8–10]. Solving the coupled mode Eq. (2), we can obtain the scattering matrix s which relates (*A*_{s}
(*z*_{o}
), *A*_{s}
(*z*_{o}
)) to (*A*_{s}
(*z*_{i}
), *A*_{s}
(*z*_{i}
)) (see Fig. 5) [8–9]:

In order to demonstrate the interaction between the two waveguides, we consider that the evolution * S*(z) was obtained by applying the calculation procedure to a series of computational windows with fixed lower boundary

*z*

_{i}and varying upper boundary at

*z*∈[

*z*

_{i},

*z*

_{0}].

In Fig. 6, the evolution of the squared elements of scattering matrix is presented as functions of *z* and these can be considered as power evolutions in the coupling region. Taking the third picture for an example, |*S*_{ss}
|^{2} and |*S*_{bs}
|^{2} can be interpreted as the local power in the straight and bent waveguide respectively, assuming a unit power of TM polarized light is launched into the straight waveguide. It can be easily seen that initially most of the power is confined in straight waveguide, then after a certain distance, the bend mode gets excited and the interaction of the two waveguides becomes significant, and finally modes in two waveguides become stationary at the end of the coupling region. In Fig. 7, some snapshots of the process described above were shown.

The CMT equations are integrated with numerical parameters Δ*x*=20*nm*, Δ*y*=20*nm*, Δ*z*=1.0*µm*; [*x*_{b}
,*x*_{t}
]=[-3.0*µm*,3.0*µm*], [*y*_{i}
,*y*_{o}
]=[-6.0*µm*,4.0*µm*], [*y*_{i}
,*y*_{o}
]=[-18.0*µm*,18.0*µm*]. The interaction of waves with vacuum wavelength *λ*=1.55*µm* is studied.

#### 2.3. Spectrum and splitting ratio of the entire structure

Let *S*_{bb}
=|*S*_{bb}
| exp(*iφ*) and *S*_{ss}
-*S*_{bs}*S*_{sb}
/*S*_{bb}
=*d* exp(*iφ*), then the dropped and through power is given by the equations below [10].

In order to obtain the spectrum responses, we need to repeat the entire procedure mentioned above for different wavelengths because the coupler scattering matrices and the propagation constants of the bends are wavelength dependent. That is a very tedious procedure and if only a narrow wavelength range is of interest, we can utilize the faster spectrum evaluation technique described in [10] and [11].

Three steps are taken to evaluate the spectrum [10–11]. The propagation constants of bend waveguide and the coupler scattering matrix are calculated at a few wavelengths, then a modified scattering matrix *S*_{m}
which is expected to be slow wavelength dependent is recalculated by extracting phase gains for propagation along the bends in coupling region, and finally the actual spectrum can be obtained by complex interpolations of relevant quantities. It should be noticed that *S*_{m}
is regarded as the scattering matrix of a coupler with zero length where the interaction takes place instantaneously at z=0 and this modification is compensated by redefining the lengths of the bends outside the coupling region as *L*=2*πR*. In our cases, the resonator spectrums are calculated by interpolating the bend mode propagation constants and modified scattering matrices for three different wavelengths, that is 1.545µm, 1.550µm and 1.555µm (quadratic interpolation).

Figure 9 summarizes the spectral responses for TE and TM modes for three different vertical positions at the same horizontal gap. It can be easily seen that 1.55µm is in the pass band of TM mode and out of the pass band of TE mode. If TE and TM polarized light at 1.55µm are launched into the input port simultaneously, the resonator will drop TM polarized light to the drop port and transmit TE polarized light to the though port. In this way, two orthogonal polarization states are split and transferred to different output ports.

The polarization splitting ratios of the device is defined as:

*SR*
_{1}=10log_{10}(${P}_{\mathit{\text{drop}}}^{\mathit{\text{TM}}}$
/${P}_{\mathit{\text{drop}}}^{\mathit{\text{TE}}}$
and *SR*
_{2}=10log_{10}(${P}_{\mathit{\text{through}}}^{\mathit{\text{TE}}}$
/P^{TM}
_{through}).

Table 2 summarized the splitting ratios at 1.550µm for splitter with geometrical parameters depicted in Fig. 9.

Figure 9 and Table 2 indicate that when the vertical position b is increased, the interaction between two waveguides becomes weaker and the resonator shows sharper spectral characteristics thus most of TE light will be transmitted to through port at 1.55µm. Therefore splitting ratio is higher as b is increased. In fact, microring resonator is a direct waveguide analogy of a Fabry-Perot resonator. The interaction of two waveguides plays the same role in microring resonator as facet reflectivity does in F-P resonator. The splitting ratios can be as high as 20dB, when the vertical position b is 0.9µm.

Figure 10 demonstrates the relevance of splitting ratios and the horizontal gap d for different vertical positions at 1.55µm. In the through port, as d deviates from the value where the splitting ratio gets its maximum, the spectrum can become so sharp that 1.55µm is out of the pass band of TM mode. In that case, the splitting ratio will decrease significantly and that can explain why the curve in the left picture drops rapidly at both sides of the maximum. In the drop port, however, sharp spectrum means very little TE light will appear in the drop port at 1.55µm thus the splitting ratio can be very high and that is the reason why the curve in the right picture shows an opposite curvature to the left one. According to Fig. 9 and Fig. 10, the horizontal gap and vertical position are key parameters for the design of polarization splitters and their values should be carefully chosen in order to get sufficient splitting ratios at the two output ports.

The wavelength dependence of splitting ratios is depicted in Fig. 11. It can be easily seen from the Fig. 11 that splitting ratios of the splitter are indeed strongly wavelength dependent. Consequently, microring resonator type polarization splitters can only work in a very narrow wavelength range; however, the spectral characteristics are expected to be improved by other methods, such as cascading several resonators with different resonance wavelengths.

It is very important to point out that the model presented here only takes into account radiation loss which is very small (see table 1). In reality, however, the spectrum can not be as nice and deep as shown in Fig. 9 because of scattering loss form the sidewalls.

## 3. Conclusion

In this work, a 3D frequency domain model of vertically coupled microring resonator based on full vectorial FMM and CMT has been employed to obtain the eigenmodes of uncoupled waveguides, the scattering matrix of coupling region, and finally the spectral responses for TE and TM mode. The result indicates that the resonance wavelengths of the two orthogonal polarization states are different because of the difference in modal index and scattering matrix for different polarizations, hence the different polarized light can be transfer to different output port at specific wavelength. The influence of geometrical parameters on splitting ratio is investigated and the results indicate that the structure can have a splitting ratio greater than 20dB at 1.55µm.

## Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant No. 60577007), the Science Fund for Distinguished Young Scholars of Hubei Province (Grant No. 2006ABB017) and the Program for New Century Excellent Talents in Ministry of Education of China (Grant No. NCET-04-0715). Xinlun Cai would like to thank Dr Prkna for his great help in modeling the 3-D waveguide.

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