## Abstract

In this paper, the procedure to optimize flat-top Arrayed Waveguide Grating (AWG) devices in terms of transmission and dispersion properties is presented. The systematic procedure consists on the stigmatization and minimization of the Light Path Function (LPF) used in classic planar spectrograph theory. The resulting geometry arrangement for the Arrayed Waveguides (AW) and the Output Waveguides (OW) is not the classical Rowland mounting, but an arbitrary geometry arrangement. Simulation using previous published enhanced modeling show how this geometry reduces the passband ripple, asymmetry and dispersion, in a design example.

©2003 Optical Society of America

## 1. Introduction

The Arrayed Waveguide Grating, AWG, is a complex and versatile device used in several fields of optics communications [1]. The application range is very wide, comprising optical signal processing [2][3], WDM multi/demultiplexion [4], fiber ring lasers [5], wavelength interleavers [6] and wavelength routers [7], to name a few. The AWG splits the frequencies of an input light wave, which are focused to different output spatial positions. This is attained by an special arrangement, or mounting, of planar waveguides. It consists on two free space slab couplers, SCs, [8] joined by a set of waveguides, the arrayed waveguides (AWs). The light is fed to the first SC, and diffracted to the AWs. The length of consecutive waveguides in the array is increased by a constant amount, Δ*L*. This introduces a frequency dependent phase change to the light traveling through the waveguides. The latter, combined with the second SC, splits the input frequencies over the second SC output, where additional waveguides, the output waveguides (OWs), are laid to collect the desired frequencies.

With the increasing traffic demand due basically to the provission of high bandwidth consumming services, new challenges arise to the device engineers in general, and in particular to AWG designers, which have to provide large channel count, very flat top and low dispersion devices. This can be attained to some extent by using classic AWG design based on the Rowland mounting, since it provides one aberration-free (stigmatic) point, plus the cancellation of aberrations up to second order over the focal line [9].

However, several problems arise: first, to make a flat top device, i.e., an AWG with rectangular shape transmittance, multimode structures as input feeders are used. For instance, multimode interference, MMI, couplers [10][11] or parabolic horns [12] can be used. The output field amplitude profile of these structures is such that the overlap of its image on the output plane, with the OutputWaveguide, OW, or receiver field, yields a nearly flat top passband in the frequency domain. Nevertheless, the phase distribution of these feeders’ fields is not constant, so the prize that needs to be paid in order to attain flat top is an additional passband dispersion (inherent to the design). Secondly, in addition to this dispersion, another one due to fabrication errors appears. Consecuently, it is compulsory to reduce the design induced dispersion, while keeping the passband as flat and symmetric as possible.

So far, and to the best of our knowledge, the AWG designs are based on Rowland mountings, but further flexibility may be obtained by investigating other geometric arrangements for the mounting. We propose in this paper a methode to produce and analyze an arbitrary geometry arrangement, departing from a classical Rowland mounting design, to improve the flatness, asymmetry and passband dispersion properties inherent to the flat top Rowland AWG design.

The paper is structured as follows: Section 2 presents the Light Path Function theory from [9] and the systematic procedure to solve the equations numericaly with two stigmatic points. On Section 3, a design example and the results are presented in graphs and animations to show the evolution of the passband properties from the Rowland mounting design to the optimized geometry design. Finally, the conclusions are drawn in Section 4.

## 2. Light path function vectorial minimization

Grating-like and planar spectrograph devices are analyzed using the *Light Path Function*, LPF, [9], which tights together basic properties of the light as phase, wavelength and constructive interference, allowing for a simple, but yet powerfull, analysis of imaging systems as the AWG. Consider the layout of Fig. 1. The LPF is defined as:

where *n _{s}*(

*λ*) and

*n*(

_{c}*λ*) are the refractive index in the slabs or Free Propagation Regions, FPR, and the arrayed waveguides index, AW, respectiveley. The fist three terms in the equation take into account the propagation length difference through FPR

_{1}, AWs and FPR

_{2}. This difference must be an integer number of times the wavelength considered for it to interfere construtively in the ouput point, that is,

*D*. Hence,

*G*(

*x*) takes an integer value for the AW at (

*x, z*(

_{G}*x*)).

*G*(

*x*) is known as the

*grating function*of the grating-like device. Abreviations can be introduced for the terms in the expression, yielding:

Hence, *F _{i}*(

*x*) represents the path difference on the input side of the AWG. Δ

*L*(

*x*) is the length increment along the curved section between the center AW, laid from

*O*to

_{I}*O*, and an arbitrary waveguide, laid from

*P*and

_{I}*P*. Finally,

*F*(

_{d}*x*) is the path difference at the output side. In Fig. 1,

*z*(

_{G}*x*) represents the

*grating line*of the device, which for simplicity will be assumed the same for the input and output part of the device. The

*focal line*of the device is represented as

*z*(

_{f}*x*).

If not only a single point *x* is considered, but a set of points **x** (a vector), then Eq. (2) can be rewritten in a vectorial way:

where the bold symbols indicate that the functions and variables involved are vectors. The unknown parameters are then **x** and **z**
_{G}(*x*). The solution for a particular output point *D*, will yield a geometrical arrangement **x** and **z**
_{G}(*x*), for a given wavelength *λ _{D}* that cancels the LPF, i.e.,

**F**(

**x**)=0. The output points for which the LPF cancels, are known as

*stigmatic points*of the waveguide mounting [9]. Notice that on a first approach, there are three unkown geometrical variables or functions,

**x**,

**z**

_{G}(

*x*) and Δ

**L**(

**x**). Hence, and as proposed in [13], three stigmatic points could be used as initial conditions to solve equation 3, which would be turned into a system of three equations. However, to preserve the focusing properties of the AWG, we believe that the AW path length increment, Δ

**L**(

**x**) must be kept linear. Therefore, only two unknowns are left in the problem, which are

**x**and

**z**

_{G}(

*x*), and only two stigmatic points must be given as initial conditions for Eq. (3). Consequently, the procedure to find the mounting, both at the AW side and the focal line side, must be divided into two steps:

#### Step 1: AWs position

The system of equations to find the solution of **F**(**x**)=0, with two stigmatic points as initial conditions, is given by:

where the bold symbols indicate that the functions and variables involved are vectors. The unknown parameters are then **x** and **z**
_{G}(*x*). The input data needed for the problem is:

- Two stigmatic points
*D*=(_{k}*x*) and their associated wavelength_{d,k}, z_{d,k}*λ*for_{k}*k*=1, 2. - The refractive indexes of the slab and arrayed waveguides,
*n*(_{s}*λ*) and*n*(_{c}*λ*) respectively. - The position of the input waveguide, IW, which will be assumed to be centered and at a distance
*R*from*O*(see Fig. 1)._{I} - The groove function
**G**(**x**), which is a vector of integers ranging from -*N*/2 to*N*/2-1 (with N even). - The waveguide length increment function, Δ
**L**(**x**).

Hence, we are looking for the vectors **x** and **z**
_{G}(*x*), satisfying the stigmatic points, the groove function and the waveguide length increment function Δ**L**(**x**).

The system can be solved numericaly using MATLAB, with the fsolve function, which solves non linear equations by a least squares method. Initial conditions must be provided to fsolve, for the unknown parameters. To start from a known point, a Rowland mounting configuration is given as the initial step to fsolve. The Rowland mounting is described by:

where *d _{w}* is the AW spacing. The waveguide increment function is fixed to:

where *λ*
_{0}=*c/f*
_{0} is the AWG design wavelength. This preserves the focusing properties of the AWG, as mentioned before.

#### Step 2: OWs positions

Once the central section of the device is defined, by (**x**,**z**
_{G}(*x*)), the OWs positions can be calculated solving the following equation:

where the position of OW *k*, (*x _{k}, z_{k}*) for which the pass band center is

*f*=

_{k}*c/λ*is found. The initial condition is based, as mentioned before, on the Rowland mounting. The AWG physical lengths and parameters used as the starting point can be derived using any design methodology. For the example presented below, the methodology from [14] and [15] is used.

_{k}## 3. Results and discussion

The design example used in this section is based on a 160 channels with 25 GHz channel spacing AWG design, with a channel grid with lower granularity than the ITU G.692 50 GHz grid [16], the standard recommendation for optical WDM systems. The starting design is a Rowland mounting with the following parameters derived from the procedures in [14] and [15]: design frequency *f*
_{0}=194 THz (*f*
_{0}=*c/λ*
_{0}, see equation 7), slab length *R*=45804.120 *µ*m, AW spacing *d _{w}*=10.44

*µ*m, OW spacing

*d*=19.75

_{ow}*µ*m, number of AW

*N*=1000, diffraction order

*m*=32, path length increment at the design frequency Δ

*L*(

*f*

_{0})=34.0562 µm and slab refractive index

*n*(λ

_{s}_{0})=1.4305. A 14 microns mode from a parabolic horn is used to make the device flat top as in [12]. The rest of the waveguides are

*W*=6

*µ*m width, with

*n*=1.4551 and

_{core}*n*=1.4451. The waveguides modes are found using a mode solver [17]. The reader may notice this is a large device which in turn will be more sensitive to fabrication errors [18][19]. This example is used to demonstrate the need for an optimized design with low inherent passband deformation and dispersion.

_{cladding}The initial stigmatic points positions chosen in our methodology correspond to the first and last channel positions. If the aberrations are kept low (LPF cancels) for the latter, the aberrations are expected to be low too for the rest of channels, located in positions with lower angular departure from the OW mounting centre.

If the stigmatic point positions are over the Rowland circle, the result obtained for the transmission and dispersion will be very similar to the ones for the Rowland mounting configuration, and therefore no optimization will take place. The stigmatic point position can be calculated by geometrical means [1], using the grating equation, which is the first order derivative of the LPF, equaled to zero [9]. In the example described below, the stigmatic point x position is increased by 15 %. Animation graphs showing the starting (Rowland) and optimization steps results are accompanying this paper. The optimization steps correspond to increments in the x direction of 0, 3, 6, 9, 12 and 15 percent, from the mentioned original position of the stigmatic point over the Rowland mounting. The step size choice does not follow any optimization rule. It is choosen to show the results evolution properly. When the position of the stigmatic points is changed along the x-direction, the optimized geometry has approximately the same focusing distance, on the second slab, than the Rowland mounting arrangement. However, the device angular dispersion [1] is slightly different, and this have direct impact into the pass band shape and dispersion.

Using the Rowland design values as the starting geometry for the procedure described in the previous section, and with the shifted stigmatic point mentioned above, the obtained geometry is depicted in Figures 2 and 3. From the figures, the grating line side geometry is very close to the classical grating line used in AWGs [9], except for an slight displacement of the AWs in the z direction (see also Fig. 1). The classical grating line is a circle of radius *R*, centered at (0,R) on both slab couplers (see Fig. 1), while the obtained grating line is not symmetric. This is shown in Fig. 2, where the short AWs (smaller numbers) are further away from the (0,R) point, while the long AWs are nearer to the mentioned point. Conversely, on the focal line side, the focal line obtained is between 10 and 20 microns appart from the Rowland circle.

The arbitrary geometry is then tested using a Fresnel diffraction integral based simulator. As the grating and focal lines are no longer confocal circles, the Fresnel integral approximation by a Fourier Transform is not valid [20]. Thefore, the modeling from [14] has been properly modified to use Fresnel diffraction integrals. With this modeling, the normalized transmission obtained is shown in a blue plot in Fig. 4. Consequently, the flatness and band pass side steep is improved. However, the transmission floor is slightly increased. In Fig. 4 and enlarged view of the latter is shown too. From the upper plot, an improvement in the insertion loss and flatness can be observed. Moreover, the lower plot shows significant improvement in the passband dispersion.

Other relevant parameters in the pass band are: first, the bandwidth, at 0.5, 1, 3 and 20 dB points; second, the pass band ripple, defined as the difference between the maximum and minimum transmission within the pass band; and third, the passband asymmetry, defined as the difference between the maximum and second maximum in the transmission pass band. The bandwidth evolution along the optimization steps is shown in Fig. 5. It is observed that there is a trade off between the pass band dispersion and the bandwidth: the latter is reduced to lower the dispersion. However the reduction is between 10 and 16 percent for all the bandwidth points, which is not significant.

Finally, the ripple and pass band asymmetry evolution along the optimization steps are shown in Fig. 6. Whilst the ripple is signifcantly reduced, i.e., the flatness of the passing band is improved, the asymmetry is kept low, as it was in the original design^{1}.

## 4. Conclusions

In this paper, a method has been proposed to enhance the AWG performance, using the mounting geometry as an additional degree of freedom. This implies the departure from the traditional Rowland mounting geometry. The procedure to improve the transmission and dispersion properties of the AWG is based on two stigmatic point arbitrary mountings. The equations and procedure to find the arbitrary mounting satisfying two stigmatic points have been introduced. A design example has been presented. The results exhibit significant improvement of the AWG transmission and dispersion within the pass band.

## Acknowledgements

The authors wish to acknowledge the European Comission funding through IST-2001-37435-LABELS and IST-2001-32786-NEFERTITI programs, the Spanish CICYT funding through TIC 2002-04344-PROFECIA, and the Alcatel Optronics (UK) AWG design team, Daniel Ortega, Antoine Pujol and Jim Bonar for their helpful discussions and collaboration.

## Footnotes

^{1} | In fact asymmetry should not be too large in the ideal design, but only after fabrication when errors in the FPR come into place. However, this is out of the scope of this paper. |

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