## Abstract

An approach to reducing the chromatic dispersion due to coupling between input waveguides before the input slab for a synchronized-router-based flat-passband filter using a multiple-input arrayed waveguide grating (AWG) is proposed. The proposed method uses phase compensation at the waveguide array of the AWG by correction of waveguide lengths. The characteristics of the flat-passband filter that consists of a multiple-input AWG combined with cascaded Mach-Zehnder interferometers (MZIs) are simulated using a theoretical model of the multiple-input AWG based on Fourier optics and the coupled-mode theory. The simulation result reveals that the chromatic dispersion within the passband can be significantly reduced by using phase compensation and additional dummy waveguides at the input just before the slab.

© 2009 OSA

## 1. Introduction

In metropolitan and access area wavelength division multiplexing (WDM) networks, multi/demultiplexers should have a flat and wide spectral response to allow the concatenation of many filters. Various techniques have been proposed to flatten the passband of multi/demultiplexers [1–13] in the last decade. The techniques can be basically divided in two types, i.e. obtaining a rectangular focusing field profile or combining two synchronized routers. The latter type using a combination of two synchronized routers [4,5,8–13] is a promising approach to obtain low-loss and wide-passband characteristics. Synchronized-router-based filters using a Mach-Zehnder interferometer (MZI) or a three-arm interferometer for the input of an arrayed waveguide grating (AWG) have been reported [8,12,13] to obtain low-loss and wide-passband in a small chip. To analyze the performance of synchronized routers efficiently and comprehensively, we have developed a theoretical model of a multiple-input AWG [14,15] by extending the model based on Fourier optics [16–18]. We have also demonstrated a flat-passband multi/demultiplexer that consists of a multiple-input AWG combined with a cascaded MZI structure as an input router with steep passband and small intrinsic loss [19,20].

Meanwhile, small chromatic dispersion as well as low-loss and wide-passband is desirable for typical filter applications, especially for high bit-rate transmission systems. We found that if the input waveguides just before the input slab have narrower core and gap widths, then this is better in terms of insertion loss [14,19]. However, narrower core and gap widths lead to larger coupling between the input waveguides, which can affect the amplitude and phase characteristics of the circuit. The coupling can cause the increase of chromatic dispersion due to the phase shift in coupled light.

In this paper, we propose an approach to reducing the chromatic dispersion of a synchronized-router-based flat-passband filter using a multiple-input AWG by phase compensation at the waveguide array of the AWG. Doerr et al. [21] have reported the improvement of the port-to-port passband shape for the dynamic gain equalizer by appropriately changing the length of waveguides in the array. We apply the similar method to reducing the chromatic dispersion due to coupling between input waveguides before the input slab for flat-passband filter using a multiple-input AWG. In this paper, the principle of the phase compensation for the optical circuit consisting of a multiple-input AWG combined with a cascaded MZI structure is described and its characteristics are simulated using a theoretical model of the multiple-input AWG based on Fourier optics and the coupled-mode theory.

## 2. Principle

#### 2.1 Structure

The optical circuit of a flat-passband filter that consists of a multiple-input AWG combined with a cascaded MZI structure is shown in Fig. 1
. It consists of a multiple-input AWG with a free spectral range (FSR) of Δ*f _{FSR}* and a cascaded MZI structure [22–24] connected to the AWG input waveguides. The multiple-input AWG has

*M*input waveguides and a waveguide array consisting of 2

*I*+ 1 waveguides. Here, the length of the waveguides in the waveguide array is slightly changed for phase compensation from the normal design.

Reflecting the results in the previous report [14,15], we chose two stages of cascaded MZIs (i.e., the number of input waveguides before the slab *M* = 4) to achieve a small chip size as well as sufficient flatness. The two slabs in the AWG have the same focal length *z*. The waveguides in the array are connected to the edges of the two slabs with a waveguide interval of *d*. The signals from a first-stage MZI are demultiplexed by second-stage MZIs by setting the FSR of the second-stage MZIs, Δ*f _{MZI}*, to twice that of the first-stage one. The signals with four equally spaced frequencies

*f*

_{1}, …,

*f*

_{4}within one FSR of the second-stage MZIs are first divided by the first-stage MZI between the two groups

*f*

_{1},

*f*

_{3}and

*f*

_{2},

*f*

_{4}, and next divided by the second-stage MZIs into individual signals. The lower port of the upper second-stage MZI and the upper port of the lower second-stage MZI should cross each other so that the signals

*f*

_{1}, …,

*f*

_{4}are spatially arranged in this order at the input side of the AWG. To obtain an appropriate demultiplexing function, the channel spacing of the AWG should be the same value as the FSR of the final-stage MZIs, i.e., to Δ

*f*.

_{MZI}#### 2.2 Phase compensation

We derive a theoretical model of the flat-passband filter with phase compensation for low chromatic dispersion based on the model of the multiple-input AWG in our previous work [14].

Suppose there is uniform coupling to only adjacent input waveguides and the relative amplitude of coupled light is *δ*. The field distribution at the edge of the input slab illuminated by the *m*-th output of the cascaded MZI structure is expressed as

*u*(

^{o}_{in}*x*) is the mode field without coupling,

*x*is the position of the

_{m}*m*-th input waveguide along the edge of the input slab, Δ

*x*is the constant interval between two adjacent input waveguides, and

*χ*is the relative amplitude of path-through light.

Here, the length of the *i*-th waveguide ($-I\le i\le I$) in the waveguide array is slightly changed from the normal design of the AWG by *l _{i}* for phase compensation as

*L*is the constant difference in length between adjacent waveguides in the array and

*L*

^{0}is the length of the 0th waveguide without phase compensation. The phase delay of the

*i*-th waveguide for phase compensation,

*θ*, is expressed aswhere

_{i}*n*is the effective refractive index of the waveguides in the array and

_{a}*λ*

_{0}is the central wavelength. Here, the contribution of the phase compensation Θ(

*y*) is defined as

*y*is defined as Δ

*y*=

*λ*

_{0}

*z*/(

*n*) and

_{s}d*n*is the effective refractive index of the slab waveguide.

_{s}Based on our previous work [14], the transfer function of the multiple-input AWG for the *n*-th output waveguide located at *y _{n}* along the edge of the output slab,

*t*(

*y*;

_{n}*f*), is derived as the convolution between the transfer function without phase compensation and the contribution of the phase compensation Θ(

*y*) as

*D*(

_{N}*x*) is the Dirichlet kernel

*D*(

_{N}*x*) = sin(

*Nπx*)/sin(

*πx*) [25],

*E*(

*x*;

_{m}*f*) is the amplitude as a function of the optical frequency

*f*for the

*m*-th input waveguide connected at the position

*x*along the edge of the input slab,

_{m}*u*(

_{out}*y*) is the mode field function of the output waveguide at the interface to the output slab, and

*f*(

_{a}*x*) and

*f*(

_{b}*y*) are the images of the input-side and output-side mode field functions of a single waveguide in an array produced on the input and output edges of the slabs. ${g}_{1}(y)\otimes {g}_{2}(y)$ represents the convolution of periodical functions

*g*

_{1}(

*y*) and

*g*

_{2}(

*y*) with a period of Δ

*y*, i.e.,

*u*(

_{in}*y*), is sufficiently narrow so that the energy is substantially limited over the interval of the convolution, -Δ

*y*/2 <

*y*< Δ

*y*/2, the first convolution of the right hand in Eq. (5) is reduced to

*U*(

_{in}*id*) is the Fourier transform of

*u*(

_{in}*x*), i.e.,

*k*is the wave number defined as

*k*= 2

*πn*

_{s}/λ_{0}. Using Eq. (1), when there is a coupling to adjacent input waveguides,

*U*(

_{in}*id*) is reduced to

*U*(

^{0}_{in}*id*) is the Fourier transform of

*u*(

^{o}_{in}*x*). From Eq. (10), the distortion in phase delay due to the coupling is regarded as tan

^{−1}[2

*δ*cos(2

*πi*Δ

*x*/Δ

*y*)/

*χ*]. Hence, to reduce the chromatic dispersion due to the coupling, the phase delay for phase compensation should be

*θ’*is the constant. Therefore, to improve chromatic dispersion due to coupling characterized with the parameters

*δ*and

*χ*, the waveguides in the array should be lengthened by

*l*so as to give the phase change shown as Eq. (11).

_{i}Equation (10) is based on the assumption that all the input waveguides to the input slab have two adjacent waveguides. It implies that dummy waveguides are needed on both sides of input waveguides for entire compensation. The characteristics of the structure with dummy waveguides, as well as those without dummy waveguides, will be discussed in Section 4.

## 3. Model for coupling before input slab

To simulate coupling at the input waveguides before the input slab, we derive a model that includes the effect of coupling using the coupled-mode theory [26] based on our previous work [19].

Let *E*(*x _{m}*;

*f*) be the amplitude of the

*m-*th input waveguide ($0\le m\le M-1$) from cascaded MZIs before coupling.

*E*(

*x*;

_{m}*f*) is given by [14,24]

*n’*is the arbitrary output port number and $\tilde{\phi}$ is the constant phase. The amplitude at the interface to the input slab after coupling is expressed as

*T*is the

*M*x

*M*transfer matrix of input waveguides between cascaded MZIs and the input slab including coupling. Here, the uniform coupling between

*M*parallel and identical waveguides with a length

*L*is considered as a simple model of the coupling before the input slab. Under this simplification,

*T*is derived from a superposition of supermodes of the

*M*waveguides [27]. Each individual waveguide is assumed to support only one mode and the nearest neighbor coupling is taken into account. Letting the propagation constant of each individual waveguide be

*β*, the coupling coefficient be

*κ*, and the transfer function

*T*=

*e*[

^{-jβL}*t*] ($0\le i,j\le M-1$), the element

_{ij}*t*for

_{ij}*M*= 4 is given as a function of

*κL*by [19]

*π*/2 radians as the light is coupled to the next adjacent waveguide when

*κL*is small. When

*κL*is assumed to be small enough to satisfy the approximations $\mathrm{sin}\left(2\kappa L\mathrm{cos}\left({\scriptscriptstyle \frac{2\pi}{5}}\right)\right)\cong 2\kappa L\mathrm{cos}\left({\scriptscriptstyle \frac{2\pi}{5}}\right)$ and $\mathrm{cos}\left(2\kappa L\mathrm{cos}\left({\scriptscriptstyle \frac{2\pi}{5}}\right)\right)\cong 1$, the transfer matrix

*T*can be approximated by

*t*(|

_{ij}*i*–

*j*| = 1) is approximately –

*jκL*. Hence, compared with Eq. (1), phase delay due to the coupling to nearest adjacent waveguides is compensated when $\delta \cong \kappa L$ and $\chi \cong 1$ in Eq. (11).

When dummy waveguides are used on both sides of the input waveguides, the amplitude at the interface to the input slab after coupling is expressed as

*x*

_{-1}and

*x*are the positions of the dummy waveguides along the edge of the input slab. In this case, the dimension of the transfer matrix

_{M}*T*becomes (

*M*+ 2) x

*M*. When

*κL*is assumed to be as small as that in the derivation of Eq. (16), the transfer matrix

*T*for

*M*= 4 with the dummy waveguides is given by

## 4. Simulation results and discussion

Design parameters for the simulation are summarized in Table 1
. We set the simulation parameters for good performance in terms of flatness, minimum transmittance and crosstalk from the numerical analysis [14]. Here, 2*I _{P}* + 1 is defined as the number of waveguides in an array perfectly occupies one Brillouin zone [28,29] as

*u*(

^{o}_{in}*y*) and

*u*(

_{out}*y*) as

*w*is the spot size and

_{u}*η*is the coupling efficiency between

*u*(

^{o}_{in}*y*) and

*u*(

_{out}*y*).

The calculated spectral response and chromatic dispersion are plotted in Fig. 2
for various *κL* without phase compensation. When coupling occurs, the chromatic dispersion is no longer zero and the maximum absolute value within the passband increases as *κL* increases. Three valleys in the chromatic dispersion around the frequencies of the transition of light from one input waveguide to the next (i.e., *f/*Δ*f _{MZI}* = 0, ± 0.25) are caused by the change in the phase of the light launched into the input slab due to the dependence of the amplitude of the coupled light on frequency. The transmittance around the passband also decreases as the coupling coefficient

*κL*increases.

We calculated the characteristics with phase compensation. The phase delay for the phase compensation, calculated from Eq. (11), is plotted in Fig. 3
. The peak-to-peak values of the phase delay of 0.79 and 1.52 rad for *κL* = 0.2 and 0.4 correspond to the changes in the optical path lengths of about 0.19 and 0.38 μm at the wavelength of 1.55 μm, respectively. It implies that the fabrication resolution in optical path length of the order of a few tens of nanometers would be required. The transmittance and chromatic dispersion of the flat-passband filters with and without phase compensation for *κL* = 0.2 and 0.4 are plotted in Figs. 4
and 5
. The chromatic dispersion within the passband can be significantly reduced by using phase compensation. The chromatic dispersion within the passband of ± 0.35 x Δ*f _{MZI}* becomes –9.4 to 0.5 ps/nm by using phase compensation, whereas that is –19.7 to 5.4 ps/nm without phase compensation when

*κL*= 0.2. The chromatic dispersion near the edge of the passband is still larger than that around the center of the passband. This is because the coupling is asymmetrical when the light mainly propagates either side of the four input waveguides (i.e., the 0th or 3rd waveguide) and it leads to the deviation of the phase distortion from the compensated values. The amount of chromatic dispersion below 10 ps/nm may be allowable for some applications such as point-to-point communications with a bit rate below 10 Gb/s. However, when the filter is used for applications in which many multi/demultiplexers are cascaded (e.g., optical cross-connect), this amount of chromatic dispersion may be not allowable, especially, for high bit-rate signals such as 40 Gb/s.

To further reduce the chromatic dispersion near the edge of the passband, we also investigated the structure in which additional dummy waveguides on both sides of four input waveguides were arranged just before the slab as shown in Fig. 6
. The transmittance and chromatic dispersion of the flat-passband filters with the dummy input waveguides with and without phase compensation for *κL* = 0.2 and 0.4 are plotted in Figs. 7
and 8
. Compared with the result in Figs. 4 and 5, the chromatic dispersion around the edge of the passband can be further reduced to –0.3 to 0.1 ps/nm by using the additional dummy waveguides when *κL* = 0.2.

We should note that the chromatic dispersion slightly increases with larger *κL* values even if the phase is compensated because the approximations $\mathrm{sin}\left(2\kappa L\mathrm{cos}\left({\scriptscriptstyle \frac{2\pi}{5}}\right)\right)\cong 2\kappa L\mathrm{cos}\left({\scriptscriptstyle \frac{2\pi}{5}}\right)$ and $\mathrm{cos}\left(2\kappa L\mathrm{cos}\left({\scriptscriptstyle \frac{2\pi}{5}}\right)\right)\cong 1$ don’t hold any longer and the coupling to the waveguide next to the adjacent waveguide does not become negligible. The coupling can be reduced by introducing curved waveguides with a tapered core [19].

In the simulation, we considered the coupling between parallel and identical input waveguides for simplicity. Actually, the use of the waveguides with varying distance as shown in Fig. 6 is inevitable. Nevertheless, the chromatic dispersion would be reduced also for the structure using input waveguides with varying distance by taking the varying distance into account in simulating the chromatic dispersion and optimizing the parameters *δ* and *χ* in Eq. (11) as far as the coupling at this portion is small.

In some cases, phase errors induced by fabrication imperfections also become an issue. The amount of phase errors arising from fabrication imperfections depends on the fabrication process to be used. The proposed method using correction of waveguide lengths would be effective for the fabrication process in which small phase errors arising from fabrication imperfection are expected, such as silica-based planar lightwave circuit (PLC) technology. The distortion in phase delay due to coupling would also be compensated by changing optical path lengths in the waveguide array in post-fabrication process using ultraviolet irradiation [30], instead of changing predetermined waveguide lengths. The optical-path-length compensation in the post-fabrication process would be applicable to compensating for both phase errors arising from fabrication imperfections and the phase distortion due to coupling if the phase errors due to fabrication imperfections are large.

## 5. Conclusion

An approach to reducing the chromatic dispersion due to coupling between input waveguides before the input slab for a synchronized-router-based flat-passband multi/demultiplexer using a multiple-input AWG has been proposed. The proposed method uses phase compensation at the array by correction of waveguide lengths. The principle of the phase compensation for the optical circuit consisting of a multiple-input AWG combined with a cascaded MZI structure is described and its characteristics are simulated. The chromatic dispersion within the passband of ± 0.35 x Δ*f _{MZI}* can be significantly reduced to –9.4 to 0.5 ps/nm by using phase compensation, whereas that is –19.7 to 5.4 ps/nm without phase compensation when

*κL*= 0.2. The chromatic dispersion can be further reduced to –0.3 to 0.1 ps/nm by using additional dummy waveguides.

## Acknowledgment

This work in part was supported by a research-aid fund of the Foundation for Technology Promotion of Electronic Circuit Board, a research-aid fund of the Suzuki Foundation and a research-aid fund of the Asahi Glass Foundation.

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