## Abstract

Two-dimensional (2D) optical wavelength demultiplexing is demonstrated by employing multilevel arrayed waveguides as a 2D diffraction grating, named the 2D arrayed waveguide grating (2D-AWG). Since the monochromatic lightwave is diffracted by the 2D-AWG to a series of periodic spots with 2D diffraction orders in both x and y directions while the dispersion direction is never parallel to the x or y direction, we can obtain 2D wavelength demultiplexing exploiting diffraction orders of either the x or y direction. One of the two dispersion components is designed much larger than the other, and the correspondent spatial free spectral range component is set properly to ensure high diffraction efficiency. The input and output ports can also be designed based on the multilevel lightwave circuit (MLC), and their level planes can be tuned parallel to that of the MLC-based 2D-AWG, which makes it feasible to integrate the 2D-AWG with the input port and/or the output port. It provides a promising way to realize large-scale and high-density optical multiplexers/demultiplexers.

©2004 Optical Society of America

## 1. Introduction

The optical wavelength multiplexer/demultiplexer (MUX/DEMUX) is the crucial element for dense-wavelength-division-multiplexing (DWDM) optical communication systems. The diffraction grating technology is one of the best choices to achieve large-scale MUXs/DEMUXs [1–6]. Integration offers the advantages of compactness, reliability, and reduced packaging costs, and so two types of diffraction-grating-based integrated waveguide MUXs/DEMUXs, etched diffraction gratings (EDGs) [3,4] and arrayed-waveguide gratings (AWGs) [5–7], have been widely investigated and become the leading MUX/DEMUX technologies in recent years. However, since the wavelength dispersing distribution of the diffraction grating is along lines, diffraction gratings commonly used nowadays are employing the simple one-dimensional diffractive periodic configuration, and either the inputs for multiplexing or the demultiplexed outputs of the grating-based MUX/DEMUX are in one dimension, which limits the scale of the WDM channel number in a way. In practice, although the 400-channel MUX/DEMUX has been realized in a single wafer by applying the high-index-contrast silica planar waveguide technology [8], it is difficult to fabricate AWGs with more channels due to the size limitation of fabrication equipments [7].

Two-dimensional (2D) wavelength multiplexing/demultiplexing is one of the best ways to expand the scale of the grating-based MUX/DEMUX though the multichip-cascaded configuration has been applied successfully to over-1000-channel operation [9]. It is well-known that the multilevel lightwave circuit (MLC) [10] is a solution to high-density and large-scale optical integration. In this paper, we use multilevel arrayed waveguides to form a two-dimensional diffraction grating, named two-dimensional arrayed-waveguide grating (2D-AWG). Employing this 2D-AWG and taking advantage of its property of diffraction orders, we demonstrate two-dimensional wavelength demultiplexing.

## 2. Diffraction of 2D-AWG

Figure 1a shows the optical system for two-dimensional multiplexing/demultiplexing, in which the two end surfaces of the 2D-AWG are positioned at the focal planes of the lenses as well as one end of both the input and output ports. The schematic diagram of the proposed MLC-based 2D-AWG is illustrated in Fig. 1(b). Its waveguide levels are set to be in the x-z coordinate plane and its two end surfaces are in the x-y coordinate plane. In the end surfaces, the waveguide cores can be arranged in various periodic formats, while Fig. 1(b) presents the simple periodic rectangular grid, in which the periodic lengths in the x and y directions are *d _{x}* and

*d*, respectively.

_{y}*d*also indicates the level thickness of the 2D-AWG. The number of the waveguides in every level is assumed to be

_{y}*N*and the total number of the levels is

_{x}*N*. The length of the lth waveguide in the

_{y}*k*th level can be described as:

where *L*
_{0}=*L*
_{11} is the length of the first waveguide in the first level, and Δ*L*
_{x} and Δ*L*
_{y} are the length differences between every two adjacent waveguides in the same level and between two correspondent waveguides of two adjacent levels, respectively. Figure 1(c) shows the schematic waveguide layout of a waveguide level of the 2D-AWG. The input and output ports, as shown in Fig. 1(a), are used to couple light into or out of the optical system, respectively.

Based on the diffraction grating theory, we have the following approximate equation of the diffracted optical field in the focal plane if the monochromatic light is coupled into the optical system:

where *x _{l}*=

*x*

_{1}+

*ld*(

_{x}*l*=1,2,…,

*N*) and

_{x}*y*=

_{l}*y*

_{1}+

*kd*(

_{y}*k*=1,2,…,

*N*) indicate the position of the

_{y}*l*th waveguide in the

*k*th level in the output end surface of the 2D-AWG,

*υ*is the frequency of the input light,

*c*is the light speed in vacuum,

*n*is the effective index of the optical waveguides,

_{eff}*L*is the focal length of the two lenses,

_{f}*C*is the normalized optical power in the

_{lk}*l*th waveguide in the

*k*th level, and

*G*(

*x,y*) is the far field of the optical waveguide mode of the individual waveguides of the 2D-AWG. Here it is assumed that

*L*>>

_{f}*d*and

_{x}*d*. With this equation, we can obtain that the input light at a certain frequency

_{y}*υ*is diffracted to a series of spots distributed in two dimensions in the focal plane, corresponding to two-dimensional diffraction orders (m

_{x}, m

_{y}) as shown in Fig. 2. These diffracted spots form a two-dimensional periodic pattern with a rectangular grid.

The dispersion direction of the optical system employing the proposed 2D-AWG, like that of traditional diffraction-grating-based MUXs/DEMUXs, is in one dimension. We have the spatial dispersions for this optical system employing the 2D-AWG as follows:

where *r*
^{2}=*x*
^{2}+*y*
^{2}, ${D}_{\upsilon x}=\frac{\mathit{dx}}{d\upsilon}\approx \frac{\Delta {L}_{x}{n}_{\mathit{eff}}}{{d}_{x}{\upsilon}_{\mathit{mx}}}{L}_{f}$ and ${D}_{\upsilon y}=\frac{\mathit{dy}}{d\upsilon}\approx \frac{\Delta {L}_{y}{n}_{\mathit{eff}}}{{d}_{y}{\upsilon}_{\mathit{my}}}{L}_{f}$ are the x- and y-direction components of *D _{λ}*, respectively, and ${\upsilon}_{\mathit{mx}}=\frac{{m}_{x}c}{\Delta {L}_{x}{n}_{\mathit{eff}}}$ and ${\upsilon}_{\mathit{mx}}=\frac{{m}_{x}c}{\Delta {L}_{x}{n}_{\mathit{eff}}}$ are defined as the

*m*th- and

_{x}*m*th-order diffractive central frequencies, respectively. As shown in Fig. 2, the dispersion direction is at an angle of

_{y}*θ*with respect to the x direction:

In fact, the waveguides in every level, as shown in Fig. 1(c), compose a one-dimensional AWG (1D-AWG) in the x direction. All these x-direction 1D-AWGs have identical *m _{x}*th-order diffractive central frequency of

*υ*, spatial dispersion of

_{mx}*D*, and free spectral range (FSR) of

_{υx}*FSR*. Similarly, the y-direction correspondent waveguides in all levels form a series of y-direction 1D-AWGs, and have identical

_{υx}*m*th-order diffractive central frequency of

_{y}*υ*, spatial dispersion of

_{my}*D*, and FSR of

_{υy}*FSR*.

_{υy}## 3. Two-Dimensional Wavelength Demultiplexing

As mentioned in Section 2, the dispersion direction of the 2D-AWG-based optical MUX/DEMUX shown in Fig. 1(a) is in one dimension. But since the monochromatic lightwave is diffracted to a series of spots distributed in two dimensions, the wavelength dispersing distribution is along a series of dispersion lines, which are determined by the two-dimensional multiple diffraction orders. These dispersion lines are all in the dispersion direction, parallel to each other, as shown in Fig. 2. The simplest way to achieve wavelength multiplexing is to set the output receiving channels along any one of the diffraction lines. Here we do not consider the issue of the diffraction efficiency.

Since the dispersion direction is never parallel to the x or y direction, while the lines formed by the diffraction spots with the same x- or y-direction diffraction order (*m _{x}* or

*m*) are parallel to the y or x direction, respectively, we can always achieve two-dimensional wavelength demultiplexing employing a set of dispersion lines that have the same x-(or y-) direction diffraction order and continuous y-(or x-) direction diffraction orders. Figure 3 presents an output scheme for two-dimensional wavelength multiplexing, in which the diffraction lines have the same y-direction diffraction order and continuous x-direction diffraction orders.

_{y}In the output scheme shown in Fig. 3, the wavelength-demultiplexed two-dimensional output is in the periodic parallelogram format. One side of the parallelogram is in the dispersion direction, defined as the u direction, and the other side is assumed in the v direction, which is at an angle of *φ* with respect to the x direction:

The periodic lengths in the u and v directions are given by the following equations, respectively:

which also indicate the output separations of the adjacent wavelength-demultiplexed channels in the u and v directions, respectively. The output coordinate of the channel with the central frequency *υ _{i,j}* (

*i*=1, 2, …,

*N*=1, 2, …,

_{u, j}*N*), as shown in Fig. 3, can be written as:

_{v}In the above equations, *N _{u}* is the total number of the designed channels along every dispersion line,

*N*is the total number of the dispersion lines, and Δ

_{v}*υ*is the designed frequency-based channel spacing. Then it can be derived that the light at the frequency ${\upsilon}_{{N}_{u}+1,j}$ will find its ((

*m*)th,

_{x}-j*m*th)-order diffracted spot at the position (${x}_{1,j+1}$, ${y}_{1,j+1}$) as well as its ((

_{y}*m*

_{x-j}+1)th,

*m*th)-order diffracted spot at the position (${x}_{{N}_{u}+1,j}$, ${y}_{{N}_{u}+1,j}$). Therefore, the output scheme shown in Fig. 3 can be used for

_{y}*N*×

_{u}*N*two-dimensional wavelength demultiplexing with the fixed channel spacing of Δ

_{v}*υ*.

With the above analysis and Eq. (2), we designed and simulated a 2D-AWG-based optical DEMUX with 10×10 channels and 50 GHz channel spacing. Figure 4 presents the two-dimensional wavelength demultiplexing. It should be pointed that the far-field profile *G*(*x,y*) in Eq. (2) and the angle *φ* are important parameters to control the diffraction efficiency of the 2D-AWG-based optical DEMUX.

## 4. Demultiplexing Scheme for MLC-Based Integration

As analyzed in Section 3, the wavelength-demultiplexed two-dimensional output of the 2D-AWG-based optical DEMUX in the focal plane exhibits the periodic parallelogram format. Since the demultiplexed light propagates in the z direction, the output port with the wavelength-demultiplexing scheme shown in Fig. 3 can be fabricated by the MLC technology. The waveguide levels can be set in either the u-z or v-z plane. However, as mentioned in the above analysis, the dispersion direction is never in the x or y direction in the optical system for two-dimensional wavelength demultiplexing. It means that the level plane of the MLC-based output port will never be parallel to that of the 2D-AWG if the u-z plane is chosen as the basic plane of the output port, and so we can conclude that the integration of the output port with the 2D-AWG based on the MLC technology is not feasible in this case.

The v direction is randomly tunable if the diffraction efficiency issue is not taken into account. If *FSR _{υx}* is designed equal to

*N*Δ

_{u}*υ, φ*will be 90°, given by Eq. (10), and the v direction will be in the y direction. Similarly, to tune the v-direction to be in the x-direction, we can choose a set of the diffraction lines that have the same x-direction diffraction order and continuous y-direction diffraction orders and set

*FSR*equal to

_{υy}*N*Δ

_{u}*υ*. A 2D-AWG-based optical DEMUX for MLC-based integration was designed and simulated, and its wavelength-demultiplexed two-dimensional output is shown in Fig. 5, which covers the wavelength band from 1523 nm to 1563 nm and has a channel spacing of 50 GHz.

Integration is always expected for high-density operation. In this section, it is proved that the 2D-AWG and the input/output port can be integrated employing MLCs. If the laser direct writing technology for 3D waveguide fabrication [11] becomes mature and practical, we can even achieve full integration of the 2D-AWG-based MUX/DEMUX. Integrating with two-dimensional multimode-interference (MMI) couplers [12] is also a possible way to obtain fully integrated 2D-AWG-based optical MUXs/DEMUXs for two-dimensional wavelength demultiplexing.

## 5. Conclusion

Based on the above analysis, it can be concluded that the MLC-based 2D-AWG can be used to achieve two-dimensional wavelength demultiplexing by taking advantage of the property of diffraction orders. The input and output ports can also be made based on the MLC technology. Analysis shows that the level plane of the MLC-based input and output ports can be designed parallel to that of the 2D-AWG, which makes it feasible to monolithically integrate the 2D-AWG with the input port and/or the output port. It brings us a potential way to realize large-scale and high-density optical MUXs/DEMUXs.

## Acknowledgments

This work was supported by the Major State Basic Research Development Program under Grant G1999033104 and by the National Natural Science Foundation of China under Grant 60377030.

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