## Abstract

Configuration of a novel compact dispersive and focusing device based on a Curved Waveguide Grating (CWG) is presented, which is essentially an integrated optic wavelength demultiplexer consisting of a curved stripe waveguide with tilted grating, a slab waveguide adjacent to it, and a set of output waveguides locate on focal line of the curved waveguide. Underlying wavelength demultiplexing mechanism of CWG is theoretically illustrated by employing the Fourier optics approach. Analysis shows that device based on CWG possesses fine wavelength resolution, compact configuration, and potentially low cost as well, which make it a promising wavelength demultiplexer, or a network performance monitor, in DWDM optical networks.

©2006 Optical Society of America

## 1. Introduction

Wavelength division multiplexing (WDM) multiplexers and demultiplexers are indispensable devices for the rapidly overspreading dense wavelength division multiplexing (DWDM) optical telecommunication networks. In the last decades, several integrated wavelength demultiplexer, such as Arrayed Waveguide Grating [1] (AWG) and Etched Diffraction Grating [2] (EDG), have been contrived and successfully employed in multi-wavelength telecommunication systems.

Besides AWG and EDG, a grating based planar waveguide optical analyzer has been proposed by C. K. Madsen, et al. [3], in which the stripe waveguide with tilted grating provides both dispersive and focusing functionalities. This device possesses more compact configuration than AWG, and avoids the stringent etching control required for high-efficiency EDGs in planar waveguide. However, due to its large imaging aberration over the wavelength range utilized in DWDM network, the imaging performance of the grating could hardly satisfy the wavelength resolution requirement for DWDM applications.

For the first time to our knowledge, we proposed a novel dispersive and focusing device configuration based on a Curved Waveguide Grating (CWG), which tapes out lightwave propagates in the stripe waveguide and focus the diffracted light of different wavelength to separate points on focal line of the curved waveguide, where output waveguides locate. In this paper, we described the CWG configuration, after which, an illustration is given on its wavelength dispersion and focusing mechanism by Fourier optics approach.

## 2. Device descriptions

Figure 1 shows the schematic layout of the CWG proposed in this paper. Light from an input fiber is launched into a single-mode stripe waveguide, a tilted grating is induced, by UV exposure or other methods, into the curved part of the waveguide, to tap the guided light out. The light taped out is captured by a slab waveguide adjacent to the stripe waveguide, where it is guided to the output waveguides locate on focal line (circumference of the Rowland circle for an arc-shaped waveguide, for instance) of the curved waveguide. Similar to the spectrum analyzer devised by Madsen [3], the slab waveguide is offset from the stripe waveguide by a reasonable distance, which avoids evanescent coupling between the stripe waveguide and the slab waveguide, and minimizes the distance that the radiated light diffracts before being guided in the slab waveguide as well.

## 3. Theoretical model

#### 3.1. Bragg diffraction of waveguide with grating

Figure 2 shows a local portion of the curved waveguide with grating. According to the grating diffraction theory [4], the longitudinal phase matching condition can be given as

Where *n _{eff}* is effective index of the guided mode,

*n*is the refractive index of stripe waveguide cladding layer,

_{c}*λ*is wavelength in vacuum, Λ is the grating period, as shown in Fig. 1,

*m*is known as grating order.

If center wavelength is expected to propagate perpendicular to the stripe waveguide (assuming the center wavelength of the input light and the designed center wavelength of CWG are the same), i.e. *θ=π/2*, correspondingly, the grating period can be determined from Eq. (1).

Where c is the light velocity, *λ _{0}* and

*ν*are the design wavelength and frequency, respectively.

_{0}#### 3.2. Dispersive and focusing mechanism of CWG

The Fourier optics approach has been conveniently and successfully employed to model the AWG by H. Takenouchi et al. [5] and P. Munoz et al. [6, 7]. In this paper, the Fourier model is borrowed and adapted to demonstrate the wavelength demultiplexing mechanism of CWG, which is essentially of similar structure, and thus possesses similar functionality, to the second free propagation region (FPR) in AWG.

### 3.2.1. Field at the input waveguide

Consider the lightwave propagates in the curved waveguide with grating, represented by its slowly varying amplitude in the longitude direction, as well as spatial field profile in the transverse section.

Where *y* is the coordinate perpendicular to the stripe waveguide in the wafer plane, as shown in Fig. 1, *f _{0}(y)* is the power normalized spatial field profile,

*U(x*is amplitude factor,

_{0})*β*is propagation constant of the stripe waveguide.

For the fundamental mode propagation in the stripe waveguide, *f _{0}(y)* can be approximately expressed as a power normalized Gaussian function

Where *ω _{0}* is the mode field radius, which depends on particular waveguide parameters, including core layer dimensions, as well as refractive index profile.

### 3.2.2. Diffraction field of the grating in waveguide

For a grating stated above, it suffices to take only the lightwave coupled from the guided mode to the radiation mode into consideration. According to the coupled-mode theory for guided-wave optics, amplitude of the guided mode would decrease exponentially in the propagation direction along the stripe waveguide, given a uniform coupling coefficient *C*. Therefore, amplitude factor of the guided mode is expressed as

assuming a uniform grating consists of *2N+1* grating lines in the arc-shaped waveguide, with the *N+1*th one at *x _{0}=0*.

Correspondingly, power flow through the waveguide can be written as

Where *ε* and *µ _{0}* are permittivity and permeability, of the waveguide material, respectively.

Substituting Eq. (6) into Eq. (7), we get

Considering the law of energy conservation, power flow of the diffracted light is expressed in terms of differential of the guided light power flow.

For convenience, each grating line in the waveguide is equivalent to a partially reflective “mirror”, with reflecting power flow given as

Substituting (8) into (10), we obtain

Compared with (8), we can get amplitude factor of diffracted light of grating line at x0

as depicted in Fig. 3.

For grating line at *x _{0}*, phase factor of diffractive light can be written as

with

Where *κ* represents phase shift of diffraction light with respect to the guided light, due to “reflection” at each single grating line.

We can now write diffraction field of the grating as

Where ∗ denotes convolution, *δ _{Λ}(x_{0})* is a summation of delta functions.

*g(x _{0})* is the spatial field distribution of the central grating line (at

*x0=0*), which is determined by the specific grating structure on the waveguide, and will be discussed in more detail in section 4.

### 3.2.3. Spatial distribution on the focal plane

To obtain the spatial distribution at the focal plane, coordinate *x*
_{1}, the Fourier transformation of (15) is calculated, yielding

where the different terms are

with analytical expression given as

and depicted in terms of amplitude and phase factor respectively in Fig. 4.

In Eq. (18), Eq. (20) to (22), *u* is the spatial frequency domain variable of the Fourier transformation and *α _{ν}* is equivalent to the wavelength focal length product in Fourier optics propagation

where *n _{s}* is refractive index of the slab waveguide,

*L*is focal length (radius of curvature) of the arc-shaped waveguide grating.

_{f}If *ν-ν _{0}≪ν_{0}*, the approximation

*ν=ν*holds, and then

_{0}By substitute Eq. (20) and Eq. (21) in to Eq. (17), the spatial distribution at the focal plane can be rewritten as

As can be observed, the field at the focal plane is composed of a summation of terms, each one be univocally characterized by an integer value *r*, known as the diffraction order.

For each diffraction order, the CWG directs input light of different temporal frequency to a different spatial position on the focal line, according to the specific value of *r* and *γ*. The Frequency Spatial Dispersion Parameter (FSDP) given by

Each output waveguide on the focal line selects a temporal frequency bandpass, the shape of which is given by *E _{1}(x_{1})*, the term

*G(x*introduces the loss nonuniformity, and the term

_{1})*φ(ν)*incorporates the phase delay information.

For CWG, similar to AWG, Spatial Free Spectral Range (SFSR), referred as the distance on the focal line corresponding to the focusing points of a fixed input frequency *ν* by two consecutive diffractions (i.e., *k* and *k+1*), can be acquired from Eq. (25) and given as

## 4. Discussion

From the deduction above, we know that CWG possesses dispersive and focusing functionalities in a manner similar to AWG.

As mentioned above, loss nonuniformity, *G(x _{1})*, is related by Fourier transformation relationship to g(x

_{0}), which is in turn dependent on specific grating structure on the waveguide. A waveguide grating with structure given in Fig. 5,

*g(x*can be approximated as a truncated

_{0})*f(x*, given by

_{0})Where, *W _{T}* and

*W*are projection of grating line in the direction perpendicular to the propagation direction of the guided light and the diffracted light, respectively.

_{L}*f*is the power normalized spatial field profile, as given in Eq. (5).

_{0}(x_{0})*rect(x*is given by

_{0}/W_{T})For a special case with a tilted angle of *π/4, W _{T}=W_{L}*, and grating lines with sufficiently large length to partially “reflect” guided light over all its transverse section spatially,

*g(x*can be reasonably approximated as a power normalized Gaussian function

_{0})Substituting (30) into (22), we can get

with *ω _{g}* the field radius of diffracted light at

*x*.

_{0}Similar to the planar optical spectrum analyzer proposed in [3], CWG is a promising candidate configuration for wavelength demultiplexing in DWDM optical networks, especially for network performance monitoring. Compared with the device proposed by Madsen, CWG have the same advantages that the spectrum analyzer enjoys, including simple configuration, minimal fabrication complexity, and therefore potentially low cost. Besides these advantages, CWG possesses finer spectrum resolution than the device proposed in Ref. [3].

The planar optical spectrum analyzer proposed in Ref. [3] focus diffracted light by a chirping grating. Because the ideal chirp for focusing is wavelength dependent and only one chirp function, ideally for center wavelength, for instance, can be realized in the device, the bandwidth over which high resolution will be obtained is limited. As a result, the resolution is affected heavily by imaging aberration, and will be imaging limited, rather than diffraction limited, for long grating length. As an instance, for a grating length of 20mm, the focused spot size is limited by the imaging aberration at longer wavelength, as demonstrated in Ref. [3].

On the other hand, by adopting a curved waveguide configuration, rather than a chirped tilted waveguide phase grating, as in Ref. [3], to focus the defracted light onto focal line of the curved waveguide with grating, CWG have a magnificently improved imaging functionality over a much wider wavelength range, which spells much finer wavelength resolution. This advantage is demonstrated by a numerical simulation, performed on size of focused spot (main lobe) for the CWG with Rowland circle configuration, results being given in Fig. 6. For convenience of contrasting, main CWG configuration parameters are chosen to be of the same value as, or close to, corresponding parameters of the device proposed in Ref. [3], with: focal length *L _{f}=107mm*, slab waveguide refractive index

*n*, center wavelength

_{s}=1.45*λ*, grating length

_{0}=1540nm*L*. As shown in Fig. 6, for grating length of 10mm, resulted spots sizes are in good agreement with corresponding diffraction-limited values; for grating length of 20mm, resulted spot sizes deviate gradually from diffraction-limited values as wavelength depart from center wavelength, owning to imaging aberration of Rowland circle configuration. However, the spot sizes deviation from diffraction limited value over 1515~1565nm is sufficiently slight, a good reason for the CWG configuration to be regarded as diffraction-limited. In addition, for grating length of 10mm, the focused spot size is

_{g}=10mm, 20mm*22.9 µ m*, much finer in contrast with corresponding value of ~100

*µ m*in Ref. [3], which demonstrate a finer resolution capability of the CWG configuration.

It is worthy to note that *E _{1}(x_{1})* possesses big sidelobes, as shown in Fig. 4, which means reduced coupling efficiency from grating to output waveguides, and high cross-talk between adjacent channels. Therefore, apodization approaches, through chirping or other technologies, are indispensable for obtaining high performance devices. Obviously,

*U*could be tailored by designing a grating with a varying coupling coefficient in its longitude direction. One possible approach is to tailor

_{d}(x_{0})*U*, envelop of the grating diffraction field amplitude factor, to a Gaussian profile, or a truncated Gaussian profile, as that at the end of the arrayed waveguide in AWG. Correspondingly, according to Eq. (18), we can get a

_{d}(x_{0})*E*of the form of a Gaussian form, or M function, as shown in Ref. [7], which matches highly to the mode field of the fundamental mode of output waveguide, and thus gives high coupling efficiency to output waveguides, and low cross-talk between adjacent channels as well, as the instance for AWG.

_{1}(x_{1})## 5. Conclusions

A novel dispersive and focusing device configuration, CWG, has been proposed. A theoretical analysis on its wavelength demultiplexing mechanism is conducted by Fourier optics approach. Compared with the spectrum proposed by Madsen, imaging functionality of CWG is magnificently improved over a wide wavelength range, which make it a potentially competitive device configuration for wavelength demultiplexing and network monitoring in DWDM optical networks.

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