## Abstract

In this paper a novel grating-like integrated optics device is proposed, the CrossWaveguide Grating (XWG). The device is based upon a modified configuration of a traditional ArrayedWaveguide Grating (AWG). The Arrayed Waveguides part is changed, as detailed along this document, giving the device both the ability of multi/demultiplexing and power splitting/coupling. Design examples and transfer function simulations show good agreement with the presented theory. Finally, some of the envisaged applications are outlined.

©2005 Optical Society of America

## 1. Introduction

In the last decade, the research field of integration of optical devices in planar chips has received great attention and efforts from research institutions, universities and companies. A considerable number of integrated optics devices are already commercially available, while researchers focus on miniaturisation and combination of different optical devices, both passive and active, in optical chips. One of the key devices enabling high bandwidth core networks has been the Arrayed Waveguide Grating (AWG), first proposed by [1], and that has been subject of extensive research and commercial interest. The AWG is a passive integrated optics device that performs multi/demultiplexing of wavelengths and that has other interesting properties as the Free Spectral Range, enabling the possibility of designing a cyclic router [2]. In this paper we propose a modification of the AWGthat adds extra functionality and some novel and potentially interesting properties.

In Section 2, the device layout is presented and a theoretical model to describe its transfer function is developed. Some design examples are given in Section 3. The envisaged applications are presented in Section 4 and finally the conclusions are drawn in Section 5.

## 2. Device layout and modeling

#### 2.1. Device layout

The modification mentioned above on the AWG to get this novel device, consists on changing the layout of the arrayed waveguides or grating arms. In a regular AWG, two Free Propagation Regions, (FPRs) are connected by a set of waveguides. The path length difference between adjacent waveguides is changed incrementally by a constant amount, typically named in the literature as Δ*L*, starting from a short waveguide at the bottom part of the array, and ending in a long waveguide at the opposite side. This is shown in Fig. 1(a). The novel device proposed is named Cross Waveguide Grating (XWG) for shortness, though a better name could be the Crossing Focal Points Waveguide Grating (XFP-WG) (the motivation for this name will be understood by the reader later on). The XWG has also this feature, i.e., the path length between adjacent waveguides is changed incrementally by a fixed amount Δ*L*, as in the AWG. However, the arms layout is such that there are two identical sub-arrays starting from the middle waveguide, which is the shortest, to the upper and lower waveguides, which are the longest. This is shown in Fig. 1(b).

This structural change in the array gives rise to novel properties and substantial different behaviour between an AWGand a XWG. To give insight on how the XWG works, we develop a mathematical model, based on Fourier Optics, following the formulation nature and convention of our previous work for the AWG [3] and recently validated in [4]

#### 2.2. The first FPR

The first FPR is modelled as a spatial Fourier transform, in such a way that under a Gaussian assumption for the input waveguide mode field (power normalised):

where *w*_{i}
is the input waveguide (IW) mode field radius, hence the far field, before the grating arms, coordinate *x*
_{1} in Fig. 1(b), will be:

with *u* is the Fourier domain variable, and *α* the wavelength, λ, to FPR length, *L*_{f}
, product from Fourier Optics [5]:

where *c* is the speed of light in vacuum, *n*_{s}
the FPR refractive index and *ν* the frequency. The path length difference is set to an integer number of times *m* (known as the grating order) the device design wavelength λ_{0} within the grating waveguides [3]:

with *n*_{c}
the refractive index of a waveguide in the array.

#### 2.3. The grating arms

The grating arms collect part of the light reaching the end of the first FPR. This can be modelled as sampling the field distribution from Eq. (2) at each waveguide [3]. The distance between adjacent waveguides is given by *d*_{w}
. Hence, the expression for the overall field over the *x*
_{1} coordinate is given by:

with *N* (odd) the total number of AWs. For convenience the two sets of waveguides are splitted in two summations. Hence, the overall field distribution over coordinate *x*
_{2} can be derived from the last equation by introducing the following phase change for each waveguide:

where *β*=2*πn*_{c}
/λ is the propagation constant for an arrayed waveguide, *L*
_{0} the length of the centre waveguide, and *r* is an integer number in $\left[-\frac{N-1}{2},\frac{N-1}{2}\right]$. The modulus operation over the waveguide number *r* reflects the fact that there are two symmetric sub-arrays, as mentioned previously. The field distribution for each sub-array can be expressed as follows:

Notice the different sign in the exponentials due to the modulus in the phase change from Eq. (6). Using *M*=(*N*-1)/2, it is possible to rewrite the latter expressions as follows:

with Π(*x*)=1 for |*x*|≤1/2 and 0 elsewhere.

#### 2.4. The second FPR

The field distribution over *x*
_{3} due to the contribution of each sub-array is given by the following equations:

where ${f}_{M}^{+}$
(*x*
_{3}) and ${f}_{M}^{-}$(*x*
_{3}) are modified versions of the M function from our previous work [3]:

that correspond to the spatial Fourier transforms of the right side and left side of a Gaussian function, ${B}_{1}\left(x\right)={e}^{-{b}^{2}{x}^{2}}$, respectively, and their analytical expressions are:

Note also that in Eq. (11)
*γ* is as in [3] the frequency spatial dispersion:

These functions, Eqs. (15), have the same amplitude, but conjugate phases, as plotted in Fig. 2 along with the ordinary M function. Though from the equations it is not clear, from Fig. 2 it is possible to conclude that:

#### 2.5. End to end field transfer function

Finally, as in [3] the end to end field transfer function is obtained by performing the overlap integral of the total diffraction pattern over *x*
_{3}, *f*
_{3}(*x*
_{3},*ν*), with the output waveguide (OW) mode field, *b*_{o}
(*x*
_{3}):

where *d*_{o}
is the OW spacing and *q* the OW number. The result is valid for an IW centred on the first FPR, and as in [3] it can be completed if a different IW is used (i.e. not centred).

## 3. Device properties derived from the model

The key equation to understand the device functioning is Eq. (11). The important detail is inside the brackets of the M functions. For the same frequency increment (the *ν* variable), the beam from each array sub-set will move in opposite directions. This is illustrated in Fig. 3(a) with a blue and red beam. Therefore, the same signal frequency components will be present at different output waveguides, so the device acts as a demultiplexer and 3 dB coupler at the same time. That means that waveguides +1 and -1 in the figure, will have the same frequency response, also +2 and -2, and so on.

Notice also that the beams may move the other direction, as in Fig. 3(b), for a contrary frequency increment. Hence, the frequency response for each input-output waveguide pair, will have two passing bands. The separation of these bands depends on the output waveguide position. For example, the separation of the passing bands for output waveguides ±2 will be bigger that for the output waveguides ±1.

The XWG is has also a periodic transfer function, i.e. a Free Spectral Range, as the AWG. This can be understood and derived from the argument in Eq. (11) following the same procedure than in [3]. The result is the same Δ*ν*_{FSR}
=*ν*
_{0}/*m*.

Regarding the cross-talk between adjacent channels, it is expected to be bigger than in a regular AWG with similar design parameters. This is due to the truncation effect of the input FPR far field because of the two sub-arrays, Eq. (9). A similar truncation occurs in the AWG, but for the whole far field (see [3]) collected by the arrayed waveguides. The truncation appears at the end of the second FPR as a convolution of the spatial Fourier Transform of the input waveguide far field with a sinc function, i.e. the Fourier Transform of a Π function. For the XWG, each half of the field will form a beam. As the truncation is shorter in the *x*
_{2} dimension, the main lobe of the corresponding sinc at the end of the second FPR will be wider. This convolved with the input waveguide far field Fourier transform will yield higher side lobes in the frequency response. In the next section, results from a design example will be shown and the cross talk commented.

## 4. Design example

A XWG device is designed following the design methodology from [3] and simulated using the tools verified in [4]. As top level parameters, the XWG is designed to be a 1×5-channel device with 400 GHz channel spacing and an FSR of 3.2 THz (non cyclic). With this and the design wavelength λ_{0}=1.55 µ*m*, the grating order is m=60. InP substrate values were chosen for the device, using shallowly etched 3 microns width waveguides. The gap between arrayed waveguides 0.6 µm, with N=100 waveguides, the FPR length *L*_{f}
=0.544462 mm and the OW spacing *d*_{o}
=8.92 µm, following a Dragone mounting (two confocal spheres).

The transfer functions from the IW to the five OWs are shown in Fig. 4. First, the green line shows the transfer function for the OW number ‘0’, i.e. the centre OW. Its location is where the two beams from the sub-arrays collide, and therefore there is a single peak with higher energy than the rest. The second green peak corresponds to the one located one FSR away due to the XWG periodicity, which is similar to the AWG.

Second, for the rest of the output ports, not centred, in Fig. 3 note how, as predicted by Eq. 11 and explained above, two peaks appear for a given output port. Each peak corresponds to one of the two beams created from the sub-arrays, that slide over the output focal plane. Each one of the two peaks has also its corresponding peak at the FSR distance, again due to the periodicity of the XWG.

Moreover, the transfer function has the same response for ports located symmetrically around OW number ‘0’ (the centred one). These responses appear superimposed in the figure (dashed vs solid and different colours, refer to the figure caption), i.e. port +1 has the same transfer function as port -1, port +2 has the same transfer function as port -2, and so on. This provides the same signal at two different output locations.

For the example, the effect of fabrication inaccuracies in the arrayed waveguides was not taken into account. It can be included as for the AWG [6], giving raise to a noise floor level in the response. The cross talk, from Fig. 4 is around 20 dB for all the channels. Another effect not considered in the simulations is the loss non uniformity, i.e. the different power distribution for the side channels with respect to the centre one. This is related to the far field of the arrayed waveguides fundamental mode [3]. It is expected as in the AWG, that the side channels, for instance ±2, to have higher insertion loss than the middle channels, i.e. ±1.

## 5. Potential applications

From the properties, it is clear that the main abilities of the device are frequency selectivity plus power splitting (or coupling if operated in the reverse direction). Hence, it has the potential applications illustrated in Fig. 5. First in the field of distribution networks, where data traffic is delivered from a central location, the Head End, to different zones. The XWG, not only provides wavelength selectivity, but also the 3 dB splitting, such a way that two distant zones can be provided with the same data. Moreover two wavelengths can be delivered at the same time without the need of using the pass-band available one FSR away as in the AWG-based distribution networks. This is advantageous since the wavelengths are closer and therefore fall within the bandwidth of the optical amplifiers. However, to separate the individualwavelengths, sharper filters are required.

Second, in optical backbone networks, where individual optical channels are groomed in groups of wavelengths (i.e. wavebands). Bands of 2 wavelengths, or more if the FSR is used, can be formed and splitted to be transported through two sub-networks, as can be seen in the figure, therefore enabling diversity routing in the waveband layer. The advantage of this configuration is that with only one period a band of wavelengths can be formed, and moreover no extra 3 dB splitters are needed in each port. If only one period is used, two wavelengths are available per port. This opens the window for more possibilities, as for example both wavelengths carrying the same information to prevent a laser failure, enabling also protection in the optical channel layer. As the wavelengths are close together, they can be easily modulated with the same electrooptical modulator. As in the previous application, the challenge is in the optical channel layer to separate wavelengths that are close together, and therefore sharp optical filters are needed.

Third, in the field of integrated optic devices, a multi-wavelength laser can be built with an XWG and SOAs as shown in the Figure. The SOAs are placed in the ‘+’ outputs of the XWG ending in mirrors, forming a laser cavity with the input waveguide, where another mirror is used. The ‘-’ outputs are coupled to the input, so they are the means of extracting the lasing wavelengths from the cavity. Compared to a multi-wavelength laser with AWGs, the wavelengths are available in pairs, and not as a whole set. However, challenges will appear in the design of the SOAs to have enough gain to lase at two wavelengths, the more for the outer most ports where the two peaks are more separated. Also as commented above, a pair of wavelengths can be modulated with the same information.

Finally, and though other applications can be envisaged, we propose the use of the XWG for distribution networks. Using the ability of two wavelengths per port, one can deliver downstream traffic, while the other one can be delivered unmodulated. In the remote optical node, the wavelengths are separated by means of a simple Optical Add Drop Multiplexer (OADM), and the unmodulated wavelength is used to transmit data in the upstream direction, using an Electro-Optical Modulator (EOM). The wavelengths are not separated the FSR, but the smaller distance shown in Fig. 4. Therefore if bidirectional amplifiers are used, the downstream and upstream wavelengths are closer, avoiding the use of high bandwidth optical amplifiers. This also allows to share the cost of expensive devices for occasional upstream transmissions, as for instance a tunable laser.

## 6. Conclusions

In this paper, a novel device, the Cross Waveguide grating has been proposed, the theory describing its response has been developed and shows good agreement with the simulation of a design example. Envisaged applications have been briefly described, though this part is left open for further research.

## Acknowledgments

The authors wish to acknowledge the ePIXnet (European Network of Excellence on Photonic Integrated Components and Circuits) European FP6 research project. P. Muñoz also wishes to acknowledge the Universidad Politécnica de Valencia for funding his stay at TU/e via the Programa Incentivo a la Investigación 2004. P. Muñoz wishes to acknowledge Prof. M.K. Smit, X.J.M. Leijtens and J.H. den Besten from the Opto-Electronic Devices group at the Technical University of Eindhoven (TU/e) for their helpful discussions.

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