## Abstract

We present an extension of the AWG model and design procedure described in [1] to incorporate multimode interference, MMI, couplers. For the first time to our knowledge, a closed formula for the passing bands bandwidth and crosstalk estimation plots are derived.

© Optical Society of America

## 1 Introduction

Arrayed waveguide gratings are called to be key devices in the implementation of optical layer functions, due to their spatial and spectral particular characterisitics [1, 2]. The potential can be increased if the AWG response is optimized in order to perform filtering as close as possible to the ideal filter. More ideal filtering can be attained through different ways proposed in the literature, such as parabolic waveguide horns [3], arrayed waveguide amplitude and phase apodisation [4], the use of Y-branches [5] and multimode interference, MMI, couplers [6]. Several AWG models and design procedures have been developed in the literature [1–8]. In particular we extend the one developed in [1] to include MMI couplers, still holding the analytical flavour of the model, and expressions for the bandwidth, crosstalk plots and design hints are presented.

This paper is organized as follows. In the next section, the AWG model from [1] is reviewed. Section III explains how this model is modified to incorporate MMI couplers to the input waveguides, IW’s. Section IV provides new design equations, plots and procedure hints. Finally, on Section V, an example comparing two AWG, with and without MMI couplers, is provided.

## 2 AWG Theoretical Model Review

This section reviews the AWG model previously developed in [1] for standard AWG’s. The AWG physical layout is shown in Fig. 1, where the abbreviations used along this paper are defined. It consists of a set of input and output waveguides, IW’s and OW’s respectively. The two sets of waveguides are joined by two free space couplers, or free propagation regions, FPR’s, [9, 10] connected by a set of waveguides named arrayed waveguides, AW’s. The AW’s act as a grating between the two FPR’s [11]. The lenght of each waveguide in the array is increased by a fixed amount, Δ*l*, with respect to that preceeding it. The inset on the upper left corner of Fig. 1 shows the waveguide layout with its corresponding parameters, the waveguide width, *W _{x}*, the gap between waveguides,

*G*, and the waveguide spacing,

_{x}*d*, where

_{x}*x*=

*i*,

*w*,

*o*, corresponding to IW’s, AW’s and OW’s respectively. The inset on the upper right side of the figure, shows the FPR’s layout. It consists of two sets of waveguides, the IW’s (OW’s) and the AW’s. The AW’s are positioned over a circumference of radius

*L*, which is called the focal length, whose center is located in the central IW (OW), CIW (COW). The rest of the IW’s are located over a circumference of diameter

_{f}*L*, called the

_{f}*Rowland circle*[5], as shown in the figure.

It is costumary on the literature to describe the field inside the waveguides by a Gaussian function [2, 5, 7]. Consider the spatial field distribution in the central IW, CIW, described by the following power normalized Gaussian function [2]:

where *ω _{i}* is the mode field radius, which is related to the waveguide width,

*W*[12]. The spatial field distribution in front of the AW’s can be expressed by the spatial Fourier transform of Eq. (1):

_{i}where *α* is:

with *c* the light speed, *L _{f}* the focal length of the FPR’s,

*n*its refractive index and

_{s}*ν*

_{0}the central design frequency of the AWG, i.e., the frequency corresponding to the center of the passing band from the CIW to the central output waveguide, COW. The total field distribution for the arrayed waveguides can be derived from the summation of the fundamental modes in the waveguides, each one weighted by a factor corresponding to the overlap integral between Eq. (2) and the following expression[13]:

where *ω _{g}* the modal field radius of the AW’s. For a set of

*N*the field distribution over

*x*

_{1}can be rewritten as follows:

with Π $\left(\frac{{x}_{1}}{N{d}_{w}}\right)$, being the pi function, whose expression is:

and *δ _{ω}* (

*x*

_{1}) is a summation of delta function

This description of the slab-array interface is an approximation in order to obtain suitable closed expressions for the device response. In order to obtain the proper transition loss in the interface under computer simulation, the corresponding overlap integrals have to be performed.

The length difference between two consecutive AW’s, Δ*l*, is set to an integer multiple, *m*, of the design wavelength in the waveguides:

*m* is known as the grating order, and *n*
_{c} is the refraction index in the waveguides. The value of Δ*l* ensures that the light wave from the CIW (*p*=0), focuses on to the central output waveguide, COW (*q*=0), at the design frequency *ν*
_{0}. The constant length increment between consecutive waveguides, is incorporated into Eq. (5) to yield the field distribution over *x*
_{2}:

where *ϕ*(*x*
_{2}, *ν*) is defined as:

To obtain the field distribution over *x*
_{3} in front of the OW’s, the Fourier transform of Eq. (9) is used to yield:

where *γ* is the *frequency-spatial dispersion parameter*, FSDP, relating the temporal frequencies of the input waveform to the spatial position at the output plane:

and *B _{g}* (

*x*

_{3}) is the Fourier transform of the modal field expression in the AW’s:

The function *f _{M}* (

*x*

_{3}) is the Fourier transform of the truncated Gaussian function, corresponding to the first two terms at the left hand side of Eq. (9):

The result of Eq. (12) encloses all the information of the AWG response. The loss non uniformity is introduced by the term *B _{g}* (

*x*

_{3}). The baseline temporal delay of the waveforms travelling through the AWG is incorporated by

*ψ*(

*ν*). The summation term on the right hand side of the equation, is responsible of the shape of the passing bands, and the spatial repetition of the response over the focal plane of the second FPR, depending both on the space coordinate

*x*

_{3}and the frequency. The spatial repetition period of the response is called

*Spatial Free Spectral Range*, SFSR:

which is a measure of the distance between the different diffraction orders of the AWG over the focal plane of the second FPR, for a given frequency. The *Frequency Free Spectral Range* is the frequency difference between two adjacent spatial diffraction orders that makes them focus to the same point in the *x*
_{3} plane [1]:

It is possible to rewrite Eq. (12) depending on these latter parameters (using Eqs. (13), (16) and (17)):

where Δ*ν _{FSR}*

_{,0}is the FSR for the AWG design frequency,

*ν*

_{0}. This expression illustrates how for a given diffraction order,

*r*, different frequencies focus to different points on the

*x*

_{3}plane. For

*ν*=

*ν*

_{0}, the order

*r*=

*m*is focused to the COW, as pointed previously.

Finally, it is possible to calculate the energy from Eq. (18) coupled to the fundamental mode in each OW, evaluating its overlap integral with the field distribution over *x*
_{3}, described by Eq. (18):

where *q* is the OW number, *d*
_{o} is the OW spacing and *β*
_{o} (*x*
_{3}) is the OW mode profile similar to Eq. (1). This expression corresponds to the field transmission coefficiente from the CIW, number 0, to an arbitrary OW, *q*. It is possible to derive an expression for an arbitrary pair of IW-OW, *t _{p,q}* (

*ν*) [1].

## 3 IW’s with MMI couplers

For the MMI to give a flat bandpass response, we consider a two-fold imaging configuration, where, as stated in [6] the output field from the MMI device is composed by the superposition of two spatially shifted versions of the input field to the MMI. Since this field is given by a Gaussian, the output field distribution can be expressed as the sumation of two Gaussian functions, as ilustrated in Fig. 2. The proposed MMI has a length, *L _{m}*:

where ζ_{0} and ζ_{1} are the effective indices of the fundamental and first order MMI coupler modes respectively. With this configuration, a center fed Gaussian field like the one in Eq. (1), is converted into a double Gaussian one, whose normalized power expression is [6]:

where Δ*x _{m}* is the peak separation between the two Gaussians. This separtion is closely related to the width and index contrast in the MMI. For high contrast MMI’s,

*W*=2Δ

_{m}*x*[6, 14]. Since this input field will be imaged to the OW’s over

_{m}*x*

_{3}, the overlap integral, i.e. the convolution, with the OW fundamental mode profile, will yield the desired rectangular shape for the passing band. The Fourier transform of Eq. (21) will yield the field distribution in front of the AW’s:

The expression for the field distribution over *x*
_{3}, Eq. (12), can be easily modified using the shifting properties of the Fourier transform:

$$+{f}_{M}\left({x}_{3}-r\frac{\alpha}{{d}_{w}}+\frac{\nu}{\gamma}-\frac{\Delta {x}_{m}}{2}\right)]$$

The field for any OW can be calculated inserting this expression into Eq. (19).

## 4 Modified Design Equations

#### 4.1 Channel Bandpass 3 dB Bandwidth

A new expression to evaluate the channel bandpass 3 dB bandwidth of an AWG with MMI couplers, can be derived under the simplifications detailed in [1], which reduce the field distribution over *x*
_{3}, Eq. (23), to the Gaussian functions inside *f _{M}* (

*x*

_{3}). Eq. (19) with

*q*=0, for the transmission between the CIW-COW pair, can be used to derive the expression for the channel bandpass 3 dB bandwidth:

The fall from the maximum to 3 dB can be calculated using a normalized to the maximum version of the later result:

For 1:1 imaging between the input plane, *x*
_{0}, and the ouput plane, *x*
_{3}, with identical input and output waveguides [6]:

This condition reduces Eq. (25) to:

with $x=\frac{\Delta \nu}{{\omega}_{o}\gamma}$. The zero of this last equation is located at *x*=1.6173, and hence, with Δ*ν*=Δ*ν′ _{bw}*/2, the expression for the bandwidth is:

which compared to the one obtained in [1] without MMI’s:

proves that an increase of nearly two times in the bandwidth is attained for the same AWG configuration.

#### 4.2 Cross Talk

The cross talk can be defined as the ratio between the amount of energy coupled from one IW to an OW, i.e. the “desired” OW, and the amount of energy coupled from the same IW to and adjacent OW to the desired OW, as described in [1]. Hence, it is possible to calculate the cross talk using the following expression:

where *x*
_{1n}=*x*
_{1}/*Nd _{w}* is the normalized length in front of the AW’s,

*x*

_{1n}∈[-0.5, 0.5],

*u*=

_{n}*Nd*is the normalized Fourier domain variable, σ is related to the AW’s illumination uniformity, and

_{w}u*σ*is the ratio between the OW spacing and the OW modal field radius:

_{o}In Eq. (30), *β _{i}* (

*u*)=

_{n}*F*{

*B*(

_{i}*x*

_{1n})}, where

*B*(

_{i}*x*

_{1n}) and

*β*(

_{o}*u*) are the illumination and the OW mode profile respectively, for the normalized coordinates. For the case of MMI couplers, the modified illumination from Eq. (22) must be used instead of

_{n}*B*(

_{i}*x*

_{1n}). It is possible to write a similar normalized expression in this case, assuming that

*ω*=

_{i}*ω*and Δ

_{o}*x*=2

_{m}*ω*:

_{i}and the normalized OW modal field profile is:

In both expressions constants have been neglected since the cross talk is defined as the ratio $\left(\mid \frac{{t}_{\mathrm{0,1}}(\sigma ,{\sigma}_{o})}{{t}_{\mathrm{0,1}}(\sigma ,0)}\mid \right)$. The integral in Eq. (30) is solved numericaly yielding the following engineering plots. Fig. 3 shows a surface plot for the cross talk and Fig. 4 shows a parametrized plot of the cross talk versus σ for some values of the ratio *σ _{o}*. The interpretation of this result is analog to the one corresponding to the ordinary AWG. Low uniformity of the illumination, Eq. (33), using small values for σ, yields low cross talk values. In general, reducing σ lowers the side lobes of

*β′*(

_{i}*u*), but also widens its main lobe. For small values of

_{n}*σ*, the cross talk value shows no dependence on σ, since despite the side lobes are lower, significant energy is coupled the main lobe of

_{o}*β′*(

_{i}*u*).

_{n}For example, if *σ _{o}*=5 in Fig. 4, the lower

*σ*, the lower the cross talk. A reduction of

*σ*can be attained for instance increasing the number of AW’s,

*N*, in Eq. (31), but it is not the only way due to the complex dependencies among the AWG parameters, as described in the design procedure elaborated in [1].

The results in the latter figures are a lowerbound of the cross talk, since the field in the waveguides is assumed to be a Gaussian function, but in fact, “real” fields show exponential decaying tails outside the waveguides [13], so a higher cross talk penalty has to be expected.

## 5 MMI-AWG Simulation Results

In order to compare two AWG’s, one with ordinary waveguides, and one with MMI at the IW’s, the design procedure described in [1] is followed starting in both cases with the same High Level Requirements, HLR, summarized in Table 1. For the MMI-based AWG, the new bandwidth equation and cross-talk plot are used. This yields different Physical Parameters, PhP, than for the ordinary AWG. The common PhP used for both cases are the refractive indices of the FPR’s and AW’s, *n _{s}*=1.4529 and

*n*

_{c}=1.453 respectively, the normalized waveguide frequency,

*V*=3, the waveguide width,

*W*=14

_{x}*µm*, the number of AW’s,

*N*=128 and the shortest waveguide length,

*l*

_{0}=26mm. The module and delay CIW-COW responses obtained for the MMI-based AWG are ploted in Fig. 6. The desired HLR are well matched after simulation as Table 1 states.

Both module responses from the ordinary and MMI-based AWG’s, are compared in Fig. 7. For the same HLR, a cross-talk reduction and optimized filtering, flateness and bandpass shape, are achieved, though some extra insertion loss is reported.

## 6 Conclusions

An extension of a previous developed AWG model and design procedure has been presented. A new formula for the MMI-based AWG bandwidth and new cross talk estimation plots have been developed. Results show perfect matching with this new design rules.

## Acknowledgements

The authors wish to acknowledge financial support from the Spanish CICYT via projects TEL-99-0437 and TIC98-0346. P. Muñoz wishes to acknowledge an FPI grant funding from UPV.

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