## Abstract

We propose an original diffraction grating demultiplexer device with a very small footprint, designed for the silicon-on-insulator waveguide platform. The wavelength dispersive properties are provided by a second-order diffraction grating designed to be lithographically defined and etched in the sidewall of a curved Si waveguide. The grating is blazed to maximize the -1^{st} order diffraction efficiency. The diffracted light is coupled into the silicon slab waveguide via an impedance matching subwavelength grating gradient index antireflective interface. The waveguide is curved in order to focus the light onto the Rowland circle, where different wavelengths are intercepted by different receiver waveguides. The phase errors were substantially reduced using an apodized design with a chirped grating, which assures a constant effective index along the grating length. The simulated crosstalk is -30 dB. The device has 15 channels with a spacing of 25 nm, thus a broadband operational bandwidth of 375 nm. Its performance approaches the diffraction limit. The layout size is 90 µm×140 µm, which is the smallest footprint yet reported for a mux/dmux device of a similar performance.

©2008 Optical Society of America

## 1. Introduction

Wavelength (de)multiplexers [1–3] based on arrayed waveguide gratings (AWGs) and echelle gratings have been developed for Wavelength Division Multiplexed (WDM) communication networks, with new applications emerging, particularly in spectroscopy and sensing [3,5]. Devices have progressively been made smaller, resulting in higher integration density, more functionalities on a single chip and greater yields. In particular, the high index contrast of the silicon-on-insulator (SOI) platform allows for waveguides of sub-micrometer dimensions and waveguide bend radii as small as a few micrometers, thereby markedly reducing device size. Recently, several very compact AWGs have been reported using SOI waveguides [6–13]. However, difficulties with very compact AWGs, often Si wire based, include scattering loss and phase errors caused by waveguide sidewall roughness, both of which limit the crosstalk performance. Multiplexers based on planar waveguide echelle gratings avoid some of these problems, since the phase delays occur in the slab waveguide rather than in a waveguide array. Several demultiplexer devices based on waveguide echelle gratings have also recently been fabricated, both in glass (silica-on-silicon [14]) and SOI waveguides [15,16]. The main challenges in fabricating echelle-grating demultiplexers are to fabricate smooth vertical grating facets, control the polarization dependent wavelength shift, and reduce the polarization dependent loss [1]. These issues have been successfully overcome in echelle grating devices based on glass waveguides [14,17], but are not completely resolved in the SOI based devices.

Curved waveguide demultiplexers have only recently been conceptually suggested [18]. Here we propose an original coarse wavelength division (de)multiplexer (CWDM) based on a single curved waveguide with an etched sidewall grating. Similar to the echelle grating demultiplexers, a benefit of our design is that the waveguide phased array is not required. This implies an obvious advantage of a smaller device size compared to an AWG. Furthermore, since the light propagates in a single waveguide, the effects of overall loss and phase error accumulation due to waveguide imperfections are reduced. This is in contrast to an AWG, where power is split in the phasar array and each arm experiences its own loss and phase error (ultimately leading to crosstalk) [19]. Since the efficiency of scattering from sidewall roughness is directly related to the interaction length as well as any roughness periodicity, minimizing interaction length will reduce the negative effects of sidewall roughness. Compared to typical waveguide echelle gratings demultiplexers, metallization is not needed in our device. For an improved echelle grating reflectivity in thin (single-mode) SOI slab waveguides, the use of Bragg reflectors instead of metallizing have been suggested at the grating facets [20]. However, this may limit the maximum demultiplexer wavelength range, unlike in our device where broadband operation range is readily achieved (375 nm for the device reported in this paper). Such broad wavelength operation range is relevant for CWDM applications. Our device can also be extended to DWDM applications by increasing the Rowland radius of the demultiplexer, thereby creating greater channel separation at the focal plane. It is also an interesting feature of our device that, unlike echelle gratings, it allows a modification of both phase (for example, by pitch modification) and intensity (by grating apodization) of the diffracted field with no loss penalty, which is relevant for various applications, including mux/dmux passband widening.

## 2. Device principle

Light comprising multiple wavelengths is coupled from the external port (typically a single mode optical fiber) into the input channel waveguide (Fig. 1). The light propagating in the input waveguide is diffracted in a direction normal to the light propagation direction by the sidewall grating. We use a second-order grating with the pitch *Λ=λ _{0}/n_{eff}*, where

*λ*is demultiplexer central wavelength and

_{0}*n*is the effective index of the input waveguide fundamental mode. The grating is blazed to maximize diffraction efficiency for the -1

_{eff}^{st}diffraction order propagating towards the demultiplexer focal region. Conceptually, each grating tooth acts as a small prism partially reflecting the waveguide mode, via total internal reflection (TIR) at the grating facets, thereby increasing grating diffraction efficiency into the -1

^{st}order compared to the 1

^{st}order.

We use a subwavelength grating (SWG) with triangular teeth [21] for Fresnel reflectivity reduction at the boundary between the trench lateral to the Si waveguide and the slab region. Such a triangular SWG acts as a graded-index (GRIN) medium, thereby suppressing Fresnel reflection at the silicon-silica boundary. Since diffraction effects are suppressed due to the subwavelength nature of the grating, the light passing from the trench to the slab waveguide is affected by the effective average index of the GRIN SWG structure. Compared to single-layer AR coatings, our SWG GRIN structures can be readily produced using standard lithographic and etching techniques at the wafer level. Since our triangular SWG structure is based on the principle of an effective medium [22] in general and the GRIN effect in particular, it provides a larger spectral bandwidth compared to AR structures based on light interference.

The device focusing properties are provided by curving the waveguide along an arc of radius *f*=2*R* (where *f* is the focal length) and centered at the demultiplexer centre wavelength focal point (*F* in Fig. 1). The receiver waveguides start at the Rowland circle [23] of radius *R*, tangential to the curved input waveguide.

## 3. Analytical model

The device is initially modeled using the 2D Kirchhoff-Huygens diffraction integral [24]. The far-field *Ψ(x ^{′}, y^{′})* with coordinates

*x*

^{′}and

*y*

^{′}along the focal curve, that is the Rowland circle of radius

*R*, is adapted from the general case for our specific device whereby the integration path runs over the curved grating

*C(x,y)*:

In Eq. (1), *ψ(x, y)* is the near-field profile with coordinates *x* and *y* along the grating, *φ _{w}* is the phase accumulated in the input waveguide,

*φ*is the phase accumulated in the slab region,

_{s}*λ*is the wavelength,

*d=[(x*

^{′}-x)^{2}+(y^{′}-y)^{2}]

^{1/2}is the distance between a grating facet and a specific position along the focal curve, and geometry factor is

*G*=(cosα+cos

*γ*)/2, where α and

*γ*are the angles between the normal of the grating facet and the incoming and outgoing light wavevector respectively. For our device, the discretized Kirchhoff-Huygens diffraction integral, i.e. sum of light contribution from all facets

*j*=1, 2, …,

*N*is used.

The device schematic used in the analytical model is shown in Fig. 2. It is observed that the phase contributions are: *φ _{w}=k_{w}2RΔθ_{j}* and

*φ*, where

_{s}=k_{s}d_{j}*k*is the wavenumber in the waveguide,

_{w}*k*is the wavenumber in the slab, and

_{s}*Δθ*is the angle between the

_{j}*j*and

^{th}*1*facet (

^{st}*θ*). Since

_{j}-θ_{1}*d*does not vary significantly along the grating and for different focal locations; it is considered

_{j}*d*when it is outside the phase definition. Since the facet geometry is not specified in this model, the term

_{j}≈2R*G*, which is a constant when the grating is blazed, is considered to be unity. Distance between adjacent facets is |

*mλ*|, where

_{o}/n_{eff}*m*is the grating order,

*n*is the effective index of the curved guide, and

_{eff}*λ*is the center wavelength. Thus, field intensity along the focal curve is:

_{0}## 4. Device simulation

#### 4.1 Design parameters

The input channel waveguide length is *L*=100 µm and width is *a*=0.6 µm. The outer waveguide sidewall comprises the second-order blazed diffraction grating with a maximum modulation depth of 0.3 µm. This grating design yields less then 1% of the residual transmitted light in the waveguide fundamental mode at the end of the grating. The blazing orientation (see Fig. 1) is chosen to enhance diffraction efficiency into the -1^{st} order using the TIR effect. In order to reduce the Fresnel reflection loss and FP cavity effects, the diffracted light is coupled into the slab waveguide through a triangular SWG GRIN antireflective interface. The SWG comprises 1 µm long triangular segments with a pitch of 0.3 µm [21].

The geometry of the Rowland configuration was determined as follows. The receiver waveguide width *w*=1.4 µm and the waveguide pitch at the Rowland circle of 2.4 µm were chosen to ensure compact size yet avoid mode delocalization and minimize receiver-limited crosstalk. For such a waveguide, the numerical aperture angular full width is ~0.7 rad from FDTD simulations, measured at 1/e^{2} irradiance asymptotes. For a 100 µm long curved grating, using a 0.7 rad angular width corresponds to focal length of *f*=140 µm, thus the Rowland circle radius of *R*=70 µm. This geometry ensures that the numerical apertures of the receiver waveguides and the grating are matched. The resulting layout size is ~90 µm×140 µm.

#### 4.2 Analytical results

First using Eq. (2), the focal field was calculated for the following device parameters: *n*=3.476, *R*=70 µm, *N*=230, *λ _{0}*=1540 nm,

*m*=-1, grating length

*L=|Nmλ*~100 µm, full-angle of the grating arc of 0.7 rad, and Gaussian near-field profile (along the grating). Bulk index of silicon is used intentionally (instead of a lower 2D effective index) to keep this study as general as possible and independent of a given waveguide geometry. Appropriate adjustments would have to be made if a specific 3D device structure is simulated. Figure 3 shows the demultiplexer spectra for the wavelength range of

_{0}/n_{eff}|*λ*=1390 nm-1665 nm. The channel spacing is 25 nm, and calculated crosstalk is <-40 dB near the center wavelength.

#### 4.3 FDTD simulation parameters

To investigate the device further, 2D finite-difference time domain (FDTD) simulations were performed for a layout of 100 µm×3 µm on an SOI platform with material refractive indices *n _{Si}*=3.476 and

*n*=1.444. The simulations were carried out for one polarization, with the electric field in the plane of the layout in Fig. 1. The obvious limitation of 2D simulation with respect to a real 3D device is that vertical dimension is not accounted for; however, when the light is well confined vertically this is a reasonable approximation for the behavior of the device in the transverse plane for a proof-of-concept study. The main aspect that is gained in 3D simulations is insight into coupling efficiency to the slab region. The continuous wave (CW) Si waveguide fundamental mode was used as the input field. The mesh size used in the simulations was Δ

_{SiO2}*x*×Δ

*y*=20 nm×20 nm, while the time step was set according to the Courant limit of Δ

*t*≤1/(

*c*(1/Δx

^{2}+1/Δy

^{2})

^{1/2}). In order to minimize the simulation layout size and render the computation more efficient, the simulation was performed for straight rather than curved waveguide geometry and the light distribution in the focal plane was calculated using the far-field integral of the field profile in the proximity of the waveguide grating, as shown in Fig. 4.

#### 4.4 Unapodized waveguide grating

Simulations were first performed on a grating with a constant modulation depth of 0.3 µm and wavelength *λ*=1540 nm. Figure 5(a) shows the calculated phase and near-exponential field amplitude in the proximity of the grating. The calculated far-field profile is shown in Fig. 5(b). Four peaks are observed in the far field, at -42°, -23°, -5° and 28°, respectively. The peak with the maximum amplitude (at -5°) corresponds to the fundamental mode of the input waveguide. The satellite peaks arise from higher order excitation due to the abrupt impedance mismatch between the waveguide sections with and without the sidewall grating. In the following section we show that these satellite peaks can be suppressed by grating apodization.

#### 4.5 Grating apodization

The grating apodization is introduced in order to assure an adiabatic diffraction onset and to control the near-field intensity profile. Our aim is to obtain a near-field Gaussian profile, since the latter yields a far-field Gaussian distribution similar to the receiver waveguide fundamental mode. The blazed grating was apodized from an initial modulation depth of 30 nm to 300 nm maximum depth over the first 70 µm of the grating. The maximum grating depth (300 nm) was used for the remaining 30 µm of the grating length. Apodization function *y=y _{0}*exp

*(-x*was used, where

^{2}/2σ^{2})*y*

_{0}is the maximum grating modulation,

*x*is the position along the grating and σ is the variance of the Gaussian function. We used

*σ*=60 µm in the simulations.

Figure 6(a) shows the near-field amplitude and the phase calculated for an apodized grating. It is observed that the field distribution is, to a good approximation, Gaussian. However, it is apparent that the wave vectors (*k _{1}, k_{2},* and

*k*) at different positions along the grating length have noticeably different directions. This varying wavevector direction results in a varying phase tilt for different positions along the grating, which causes angular broadening of the far-field profile. The latter is observed in the calculated far-field angular distribution (Fig. 6(b)). This phase tilt variation along the grating arises from the effective index variation caused by apodization of the grating modulation depth, since the effective index is a weighted index average that is different for each periodic grating segment.

_{3}#### 4.6 Phase correction

The angular broadening caused by phase tilt variation can be compensated by chirping the grating pitch to ensure a uniform effective index along the grating length. A similar effect may also be achieved by gradually changing the waveguide width along the grating length.

We determined the required grating chirp by calculating the grating diffraction angle for several (constant) modulation depths in the range from 50 nm to 300 nm and for several values of grating pitch in the range from 400 nm to 520 nm (Fig. 7(a)). The corresponding pitch and modulation is then chosen such that the diffraction angle is constant, i.e. far-field peak *θ*=0° at the centre wavelength *λ*=1540 nm, along the grating with varied (apodized) modulation depth. A 3^{rd} order polynomial fit representing this pitch-depth dependency (Fig. 7(b)) is used to generate a layout script for the FDTD simulation of a phase-corrected demultiplexer design.

Figure 8(a) shows the near-field at *λ*=1540 nm for an apodized phase corrected grating. A substantial reduction in the phase tilt variation along the grating is observed compared to the previous design with no phase error correction (Fig. 6(a)). This results in a well resolved narrow far-field distribution with a full-width-at-half-maximum (FWHM) angular width of ~0.5° as shown in Fig. 8(b).

Figure 9 shows the calculated spectral response of a 15 channel demultiplexer in the wavelength range of *λ*=1415 nm-1765 nm, with a channel spacing of 25 nm. The passband angular width is ~0.5° at the FWHM. The calculated crosstalk is -30 dB, for a 20 nm×20 nm simulation mesh size, but this value is still a numerical artifact of the chosen mesh size and the true theoretical limit will be smaller (<-30 dB). If fact, Eq. (2) predicts a theoretical crosstalk limit of <-40 dB near the central wavelength.

Loss associated with the focal field mismatch with the receiver waveguide fundamental mode was estimated as the overlap integral of the fundamental mode of the central receiver waveguide with the focal field for *λ*=1540 nm. Loss of -1 dB was found for receiver waveguide aperture width *w*=1.4 µm. This was the minimum loss for a range of widths tested and is in agreement with our initial design based on 1/*e*
^{2} irradiance consideration (Section 4.1). Loss due to light diffraction into the 1^{st} order is -1.4 dB. Thus the intrinsic device loss is -2.4 dB.

## 5. Conclusion

In this paper we proposed and analyzed an original CWDM demultiplexer design, with a compact footprint of 90 µm×140 µm. It uses a blazed sidewall grating formed in a curved silicon waveguide. Grating depth apodization and pitch chirping were used to obtain the optimized performance. The designed device has 15 channels with a wavelength spacing of 25 nm, thus a broadband operational bandwidth of 375 nm. This new type of waveguide demultiplexer is particularly promising for applications in optical interconnect and CWDM providing advantages such as compact size, broadband operation, and ability to tailor the passband. Since the angular dispersion depends only on the central wavelength, extension of this device to dense wavelength dispersion multiplexing is possible by up-scaling its dimensions by a factor (25 nm)/Δλ_{DWDM}, which is approximately 30 for 100 GHz channels.

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