## Abstract

The 400-channel 25-GHz-spacing SOI-based planar waveguide demultiplexer employing a concave grating across C- and L-bands is proposed in this paper. For the high polarization dependence, the waveguides are designed for supporting the TE mode only. To reduce the spherical aberration of the concave grating, the values of the maximum half divergent angle of the light source and minimum effective half width of the fundamental mode of the ridge waveguide are determined. We use a design example to show the spectral characteristics of the proposed design. Simulation results show that the proposed design provides better spectral characteristics and smaller die size.

© 2010 Optical Society of America

## 1. Introduction

With the explosive growth of Internet, the copper coaxial cable can not meet the increasing demand for the bandwidth while the optical fiber can provide greater bandwidth. In a wavelength division multiplexing (WDM) system, each wavelength is treated as a separate channel and it is possible to increase the bandwidth by increasing the channel number rather than increasing the bit rate. When the channel spacing is less than 100 GHz (0.8 nm), the technology is well know as dense wavelength division multiplexing (DWDM) [1]. International Telecommunication Union Telecommunication (ITU-T) Standardization Sector recommends several bands for lower transmission losses in a silica-based single-mode fiber, including S-, C-, and L-bands [1–3]. The maximum channel number in these bands depends greatly on what kinds of the demultiplexers are used. In recent years, there has been considerable interest in developing integrated planar waveguide demultiplexers, such as arrayed waveguide gratings (AWGs) [2–6] and planar waveguide concave gratings [7–11], due to the advantages of low insertion loss, low crosstalk, high possibilities of mass production, and high spectral resolution. However, AWGs has inherent limits due to larger die size, lower free spectral range (FSR), and greater sensitivity to the environment. The purpose of this research us to design a planar waveguide demultiplexer employing a concave grating. This research may provide an alternative to a planar waveguide demultiplexer with the high channel number and narrow channel spacing in an optical communication system.

## 2. Design and Simulation

In this paper, we do the research on a planar waveguide concave grating demultiplexer which is designed on a silicon-on-insulator (SOI) wafer. The concave grating is based on the recursive definition of the centers of the facet positions, which was first proposed by McGreer in 1996 [7]. There are three types of the concave gratings in the literature including Rowland circle, Taylor expansion, and recursive definition types, respectively [12]. Simulation results showed that the recursive definition type suffers from less spherical aberration than the other two types. The schematic figure of the planar waveguide demultiplexer employing a concave grating is shown in Fig. 1. The single-mode SOI wafer consists of a 200-to-300-nm-thick top silicon layer, and a 1-*μ* m-thick buried oxide layer, and a 500-*μ* m-thick silicon substrate while the light is transmitted in the top silicon layer. Therefore, the top silicon layer is the core layer and the buried oxide layer is the cladding layer. The refractive indices of the silicon and oxide materials, considered in this paper, are 3.50 and 1.45, respectively, and the cross-sectional view of the ridge waveguide (input and output waveguides) is shown in Fig. 2. The width and thickness of the core layer for the ridge waveguide are denoted as *w*_{si} and *t*_{si}, respectively. Since the effective indices of the TE and TM modes are highly polarization-dependent, the waveguide structures are designed for supporting the TE mode only without the design of the polarization compensator [5, 8]. So the thicknesses *t*_{si} of the core layer for the ridge and slab waveguides are specifically designed for the low propagation loss of the TE mode but high propagation loss of the TM mode.

Fig. 3 shows the light diffracted and focused by the concave grating, where A is the grating pole, B is the boundary of the concave grating on one side, C is the center of the grating curvature, P is the position of the light source, Q is the position of the focal point, *α* is the incident angle of the light at the grating pole, *β* is the diffraction angle of the light at the grating pole, *R* is the effective radius of the grating curvature, *r*_{1,0} is the distance between A and P, and *r*_{2,0} is the distance between A and Q. For the triangles ACX and BPX, we can obtain

and

Similarly, we can obtain

For the small arc angle *δγ*, the arc length AB can be approximated as the tangent length
$\overline{\mathrm{AB}}$. When the arc angles *δγ*, *δσ*, and *δρ* are small, they can be expressed as

The diffraction equation of the grating can be expressed as

where *n*_{eff} is the effective index in the slab waveguide, *d* is the grating period along the grating chord, *m* is the diffraction order, and *λ* is the wavelength of the light. After we differentiate Eq. (7), we can obtain

Taking Eqs. (2) to (6) into Eq. (8), we can obtain

which is the so-called focal equation of the concave grating [13]. To reduce the spherical aberration of the concave grating, the maximum arc angle *δγ*_{max} must be determined so we define the deviation function *f*(*δγ*) of the approximation as [14]

Simulation results show that for the value of *f*(*δγ*) lower than 0.1 % the arc angle *δγ* must be lower than 8.8° as shown in Fig. 4. For simplification, the maximum arc angle *δγ*_{max}, the maximum half central angle of the grating curvature, is chosen as 8.0° and then maximum
${\overline{\mathrm{AB}}}_{max}$ can be obtained from Eq. (4) as

Taking Eq. (11) into Eq. (5), we can obtain the maximum arc angle *δσ*_{max}, the maximum half divergent angle of the light source as [13]

We use a design example to show the spectral characteristics of the proposed design. Recommended by ITU-T, the C-band is defined in the range 1528 to 1561 nm and the L-band is defined in the range 1561 to 1610 nm [1–3]. For the design of the SOI-based planar waveguide concave grating demultiplexer across C- and L-bands, the center wavelength *λ*_{0} is chosen as 1570 nm. By using the effective-index method [15], Fig. 5 shows the propagation losses due to the leakages to the silicon substrate versus the thickness *t*_{si} of the core layer (top silicon layer) for both modes at a center wavelength of 1570 nm when the thickness of the cladding layer (buried oxide layer) is chosen as 1 *μ* m. It shows that when the thickness *t*_{si} of the core layer is lower than 220 nm, the propagation loss of the TM mode dramatically increases from about 9.0 dB/cm. So the thickness *t*_{si} is chosen as 220 nm for supporting the TE mode only as in [10, 16]. For the single-mode ridge waveguide, the width *w*_{si} of the core layer must be smaller than 500 nm [16]. So the width *w*_{si} is chosen as 500 nm as in [10, 16]. Then the effective half width *w*_{0} of the fundamental mode of the ridge waveguide along the *x*′-axis can be obtained from the Beam-PROP software (RSoft, Inc.) as 237 nm. The 500-nm-wide silicon photonic wire waveguides with a bending radius of few micrometers allow further reduce the die size [10, 16]. By using the effective-index method, the effective index *n*_{eff,TE0} of the fundamental TE mode in the slab waveguide can be obtained as 2.85 with the negligible propagation loss.

The scalar diffraction theory is valid when the grating period *d* is large as compared to the wavelength of the light, so *d* is chosen as 10 *μ* m. For the FSR larger than the bandwidth across the C- and L-bands, the diffraction order *m* is chosen as 18 and the FSR (= *λ*_{0}/*m*) can be obtained as 87 nm as in [3]. For no overlaps of the positions of the input waveguide and all the output waveguides, the incident angle *α* of the input waveguide at the grating pole is chosen as 32.0°. When *n*_{eff,TE0}, *d*, *α*, *m*, and *λ*_{0} are determined, the diffraction angle *β*_{0} of the design output waveguide at the grating pole can be obtained from Eq. (7) as 27.5°. For the small die size with the acceptable crosstalk between adjacent channels, the distance *r*_{1,0} and *r*_{2,0} are chosen as 45 mm (*r*_{1,0} = *r*_{2,0} = 45 mm). When the grating pole is chosen at the origin of the coordinates, the coordinate positions, (*a*_{1}, *b*_{1}) and (*a*_{2}, *b*_{2}), of the ends of the input and center output waveguides can be obtained as (*r*_{1,0} · sin*α*, *r*_{1,0} · cos*α*) and (*r*_{2,0} · sin*β*_{0}, *r*_{2,0} · cos*β*_{0}), respectively, as shown in Fig. 6. For the design of the concave grating, the grating period *d* is constant along the grating chord. When the *x*-axis coordinate position *x _{i}* of the center of the

*i*th grating facet is chosen as

*x*=

_{i}*i*·

*d*, the

*y*-axis coordinate position

*y*of the center of the

_{i}*i*th grating facet can be obtained from the root of the constraint function [7]. When

*α*,

*β*

_{0},

*r*

_{1,0}, and

*r*

_{2,0}are determined, the effective radius

*R*of the grating curvature can then be obtained from Eq. (9) as 51.87 mm. Then the maximum arc angle

*δσ*

_{max}can be obtained from Eq. (12) as 7.8°. According to the theory of the guided wave [15], the half angle

*δσ*of the Gaussian beam divergence at 1/

*e*amplitude on the

*x*′

*y*′-plane can be expressed as

From Eqs. (12) and (13), the minimum effective half width *w*_{0,min} can be obtained as 1.287 *μ* m. Therefore, we need a spot size converters to change the effective half width *w*_{0} from 237 nm to 1.287 *μ* m. For a 2-*μ* m-wide 40-nm-shallow-etched waveguide, the effective half width *w*′_{0} of the fundamental mode along the *x*′- or *x*″-axis can be obtained from the BeamPROP software (RSoft, Inc.) as 1.287 *μ* m. The spot size converters can be achieved by a two-step etch process at the ends of the input and output waveguides as shown in Fig. 7. It can also reduce the transition losses between the ridge waveguide and slab waveguide [10].

For a maximum arc angle *δσ*_{max} of 7.8°, the total illuminated grating periods *N* can be obtained as 1446. Figure 8 shows the simulated TE-mode spectral responses of 400 channels with a channel spacing of 25 GHz (0.2 nm), which are obtained from the overlap integral of the image field at the end of the output waveguide and the fundamental mode field of the output waveguide [9, 11]. The simulated insertion losses of 400 channels, which include the propagation loss, undesired-order loss, and the excess loss, range from 6.20 to 6.75 dB. In our case, the propagation loss of the TE mode is negligible. The undesired-order loss of the center channel, which comes from the diffraction of the light into the undesired adjacent four orders, is 5.92 dB. The excess loss of the center channel, which comes from the amplitude mismatch between the image field at the end of the output waveguide and the fundamental mode field of the output waveguide, is 0.28 dB. The crosstalk between adjacent channels is defined as the maximum signal received from adjacent channels within -1-dB passband bandwidth. For the same center wavelength, diffraction order, channel spacing, and channel number, the worst crosstalk in our case is -30 dB, while that in [3] is -20 dB. And the die size of the proposed demultiplexer employing a concave grating is 41 × 32 mm^{2}, while that of the demultiplexer employing an AWG in [3] is 124 × 64 mm^{2}. So the proposed design provides an alternative to a planar waveguide demultiplexer with higher spectral resolution, lower crosstalk, and smaller die size compared with those in [3].

## 3. Conclusion

In this paper, 400-channel 25-GHz spacing SOI-based planar waveguide demultiplexer employing a concave grating across C- and L-bands is proposed. For the high polarization dependence, the thickness and width of the ridge waveguide are specifically designed for supporting the TE mode only. To reduce the spherical aberration of the concave grating, we determine the values of the maximum half divergent angle of the light source and minimum effective half width of the fundamental mode of the ridge waveguide. The spot size converters are used at the end of the input and output waveguides to change the effective half width of the fundamental mode of the ridge waveguide. For a design example, simulation results show that the proposed design provides a worst crosstalk between adjacent channels of -30 dB and a remarkable die size of 41 × 32 mm^{2}. The proposed design provides an alternative to a planar waveguide demultiplexer with higher spectral resolution, lower crosstalk, and smaller die size.

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