1. Introduction

Arrayed waveguide gratings (AWGs) which spatially separate and combine wavelength channels have been proposed for the implementation of multiple applications that embrace the fields of devices, systems, and networks [1, 2]. As a cost-effective device with good stability and high wavelength accuracy, the AWG is fabricated as a single component regardless of its channel count through employing the planar lightwave circuit (PLC) technology. Recently the explosive growth in demand for the large transmission capacity has led to an urgent need for the AWG with large channel counts. At the same time, a more compact and less expensive multi/demultiplexer may be preferred. However, as the channel count increases the size of the AWG tends to dramatically increase, which requires a larger area of wafers to contain the whole device and a more precise fabrication process that otherwise results in big phase errors. A 25-GHz-spaced 400-channel AWG fabricated using 1.5% -Δ silica-based waveguides on a 6-inch Si wafer has been reported [3], and the maximum channel count to date for a 4-inch wafer is 512, where the whole configuration was contained by folding the 7-cm-long slab waveguides [4]. In order to realize an AWG with more than 1000 channels that cover the S-, C-, and L-bands of optical fiber amplifiers, a method cascading a primary AWG and some secondary AWGs in a tandem configuration has been developed [5], and herein many sophisticated technologies have been employed.

If a super high index contrast silica-based PLC (Δ ≥ 1.5%) is used, the length of slab waveguides is the main factor resulting in a large AWG compared with the bending radius of arrayed waveguides. This can be explained as follows. According to the theory well developed in the phased-array radar technology, the adjacent waveguide spacing should be smaller than half of the free-space wavelength to avoid the undesired grating lobes which limit the deflection angle [6, 7]. Though this requirement does not pose a problem for microwave phased arrays, it is a major difficulty for AWGs in the optics domain. Because the coupling between arrayed waveguides should be sufficiently low, large waveguide spacings of multiple wavelengths are unavoidable. The maximum deflection angle of the phased array in a general AWG is limited due to the large waveguide spacing, and can be described as

where m is the grating order, Δλ _max is half of the operating wavelength range, n_s is the effective refractive index of the second FPR, and d is the adjacent spacing of arrayed waveguides. For example, In the 400-channel AWG reported in [3], the maximum deflection angle of the phased array is about ± 2.6°. Consequently, when an AWG with large channel counts is taken into consideration, a long slab waveguide, which ensures that the deflection range can contain all the output waveguides, is necessary.

A recent work about an optical phased array with elements irregularly spaced has been reported [8]. This irregular phased array has dramatically suppressed sidelobes which therefore have no limits to the deflection angle, and has potential to alleviate the above-mentioned problem. A wider deflection angle will result in a shorter slab waveguide and accordingly a smaller size of the device. In this paper, we present a concept of irregularly arrayed waveguide gratings (IAWGs). Based on Refs. [9, 10], an IAWG Fourier optics model is derived to describe the transmission characteristics between any arbitrary pair of input/output ports. This model is not only an extensive version of its precursors that is suitable for both IAWGs and conventional AWGs, but also provides a new understanding to the operation of these devices. Section 2 is devoted to developing the model completely, first for a general situation and then obtaining simpler expressions under the Gaussian approximation of the waveguide mode profiles. Section 3 provides a design example to illuminate the features of IAWGs and some ideas for further work. Section 4 presents the conclusions.

2. Theoretical Model

The aim of the model proposed in this work is to obtain a closed expression for the transmission coefficient t(d_i,d_o,v) which relates the complex amplitudes of the modes of the input and output waveguides located at positions d_i and d_o, respectively,at frequency v. Our approach will follow the notation in Refs. [9, 10]. The input, output and arrayed waveguides are characterized by their power normalized modal field profiles b_i(x), b_o(x) and b_g(x) respectively, which are defined with respect to a local x-axis centered in the corresponding waveguides.

2.1. Theory of Irregular Phased Arrays

The theory of irregular phased arrays has been described in Ref. [8], and the main results are given in the following. The part of arrayed waveguides in Fig. 1 shows the configuration of an irregular phased array. According to the reference, if the interelement spacing d_r and the optical path difference n_cΔl_r (Δl_r = l _r+1 -l_r) between adjacent waveguides satisfy

the constructive interference occurs at the only direction which is the so-called main lobe

Here l_r is the length of the r-th arrayed waveguide, n_c is the effective refractive index of arrayed waveguides while that of FPR1 and FPR2 is n_s, m_r is the grating order, λ ₀ is the center wavelength, Δλ _max is half of the operating wavelength range, the optical wavelength λ varies from λ ₀ - Δλ _max to λ ₀ + Δλ _max, r = 1,2,… ,N - 1, N is the number of arrayed waveguides, Δλ = λ ₀ - λ (∣Δλ∣ ≤ Δλ _max), and D is a proportionality coefficient responsible for the proportion of Δl_r to d_r, determining the dispersion capability of the phased array and the maximum deflection angle. D ranges from 1 to ∞, corresponding to the dispersion capability from maximum to minimum. In other words, when the optical wavelength is λ = λ ₀ - Δλ, the irregular phased array will produce a main lobe in the direction θ _main, while in the other directions the in-phase conditions are not satisfied and consequently sidelobes are very low.

Instead of only one grating order m that should be determined in regular phased arrays, a set of m_r (including N - 1 grating orders) has to be selected in an appropriate range to construct an irregular phased array. Besides the number of arrayed waveguides N, the choice of different combinations of m_r influences the sidelobe level in the far-field radiation pattern. An optimization of the combination of m_r is necessary to design an irregular phased array with high performance.

2.2. Device layout

The IAWG layout is schematically shown in Fig. 1. Like a regular AWG, the IAWG consists of input waveguides (IWs), FPR1 (free propagation region), arrayed waveguides (AWs), FPR2, and output waveguides (OWs). The difference lies in the arrayed waveguides, where the adjacent waveguide spacing d_r differs from each other and the length of each waveguide is increased by a variable amount Δl_r with respect to that preceding it. As shown in Fig. 1, the single-mode input waveguide is situated at an arbitrary position d_i of the input plane x ₀, followed by FPR1 with the focal length L_f. At the output plane of FPR1 (x ₁), the N AWs separated by a variable distance d_r are placed to sample the outgoing light. The light samples propagate through the individual arrayed waveguides to the input plane (x ₂) of FPR2, which has the same geometrical characteristics as FPR1. For an assigned wavelength, the light diffracted in FPR2 will illuminate with equal phase (apart from an integer multiple of 2π) the output plane (x ₃) where it is collected by a single-mode output waveguide situated at a corresponding position d_o.

 

Fig. 1. Layout of IAWGs, where the arrayed waveguides is irregularly spaced.

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2.3. Field at the IW

If the field of the monomode input waveguide in the input plane x ₀ is described as f ₀(x ₀,v) at frequency v, then the input waveguide located at the arbitrary position d_i can be written as

where b^v_i (x) is the power normalized mode field profile of the input waveguide at frequency v assumed to be centered at the origin. For simplicity of the notation, hereafter f ₀(x ₀,d_i,v) will be abbreviated as f ₀(x ₀) and will be only explicitly used when necessary.

2.4. The First FPR

The input light is launched into FPR1 and then excites the AWs. Under the paraxial approximation, the field just before the plane x ₁ in the Fraunhofer region can be obtained by the spatial Fourier transform of the input field distribution

where TF{} is the operation of the Fourier transform, F ₀(f) is the Fourier Transform of the input field f ₀(x ₀), f is the spatial frequency domain variable of the Fourier transform, and α is the wavelength focal length product, given by

where L_f is the focal length and n_s is the effective refractive index of FPR1.

After the plane x ₁, the field is partially coupled into the AWs. The total field distribution for all the AWs can be expressed as

where b_g(x ₁) centered at the origin is the fundamental mode profile (power normalized) in any of the arrayed waveguides, d ₀ = 0 is defined for convenience, d_s is half the aperture of the AWs,

and a_n is the coupling coefficient to waveguide n, which can be solved by the overlap integral between the illuminating field and the corresponding waveguide mode

Next, we will follow the method in Ref. [10] in which no approximations were made in the coupling coefficient a_n. If define

where * denotes a convolution operation, we have

Therefore Eq. (8) can be rewritten as

where

Here δ_w is a sum of delta functions. In plane x ₁ the places which the arrayed waveguides are located at are irregularly arranged by the sampling function δ_w(x ₁), which can not be expressed as the periodic form of an infinite sum shown in Refs. [9, 10]. Therefore, the rectangular window function is not needed in Eq. (14).

2.5. Field at the End of the AWs

The function of the AWs is to provide an appropriate phase retardation system for the N light beams passing through them, since the attenuation is neglected. The length of the n-th waveguide is given by

where l ₀ is the length of the shortest waveguide, n = 0,1,2,…,N - 1. Each Δl_r (the length difference between adjacent waveguides) is determined by the theory of irregular phased arrays:

where Δl ₀ = 0 is defined and thus m ₀ = 0. A set of grating orders m_r has to be selected to set the values of each Δl_r and d_r for constructing the AWs, as will be given later in the example. Applying Eq. (16), the phase change resulting from the waveguide n can be obtained

Introducing the phase changes into Eq. (13), the field at the output of the AWs (the plane x ₂) can be written as

Here ϕ(x ₂,v) is obtained by transforming Eq. (18)

where

and c is the light velocity.

2.6. The Second FPR

After propagation in the AWs and FPR2 the light recombines at the end of FPR2 (the plane x ₃). The spatial field distribution at the focal plane of FPR2 can be obtained through the Fourier transform of Eq. (19), i.e.,

Here F ₂(f) denotes the Fourier transform of f ₂(x ₂), and can be expressed as

where

Introducing Eqs. (25-27) into Eq. (22), we get the field distribution at the plane x ₃ as

To clarify the meaning of Eq. (28), we need to further calculate G ₁(f) , for which we first obtain the closed expression for F ₁(f), i.e.,

where the following properties of the Fourier transform have been applied:

Hence, from Eqs. (25) and (29) we have

In order to make the results clearer, the following approximation is employed

This approximation holds true in most practical devices where the FPR focal length L_f is large enough and accordingly the radiation pattern of each of the individual waveguides of the array (i.e., B_g(x ₃/α)) is approximately constant along the width of the input mode profile.

Finally, at the plane x ₃ Eq. (28) can be rewritten as

where

Eqs. (34,35) shows that the light illuminating at the plane x ₃ can be expressed as a product of a summation of N terms (corresponding to N AWs) and a profile of the radiation pattern of the arrayed waveguide (i.e., B_g(x ₃/α)). The summation indicates the light is focused through the convolution of the array configuration with the mode field profile of the input waveguides b_i(-x ₃). The illumination level at the plane x ₃, governed by the product of two radiation patterns, relates to the position of the input waveguide through the term B_g(d_i/α). If the arrayed waveguides are unequally spaced according to the principle of irregular phased arrays, only one diffraction order will be produced at the output plane x ₃. Moreover, Eqs. (34,35) can be transformed into Eqs. (21,22) in Ref. [10] when the arrayed waveguides are equally spaced.

2.7. Field in the OWs

The field f ₃(x ₃) will be picked up by a corresponding output waveguide located at the position d_o. The coupling coefficient t(d_i,d_o,v) can be calculated by the overlap integral between the illuminating field and the output power normalized waveguide mode profile, i.e.,

Introducing Eq. (34) into (36) we get

Here we assumed that the radiation pattern of each of the individual waveguides of the array (B_g(x ₃/α)) is approximately constant along the width of the mode profile of the output waveguide b_o(x ₃ - d_o), similar to the approximation in Eq. (33). Employing the following property of integral

Eq. (37) can be rewritten as

where the new function q_n is defined as

To get a meaningful result, we apply Eq. (35) and rewrite q_n(x) as

Eqs. (39,41) can also be rewritten as

Eqs. (39,41) give the transmission coefficient between any arbitrary pair of input/output ports in either conventional or irregular AWGs, since the conventional AWG is a special case of IAWGs. The function q_n(x) defined in Eq. (41) (or q(x) in Eq. (43)) is the transfer function between the input port and the output port, showing a new understanding of the operation of these devices. From Eq. (39) (or Eq. (42)), we have t(d_i,d_o,v) = t(d_o,d_i,v), which shows the reciprocity property as described in reference [10]. This means the IAWG is still reciprocal, even though the arrayed waveguides are irregularly spaced.

2.8. Gaussian approximation

The fundamental mode profiles of all the input, output, and arrayed waveguides in the device can be approximately expressed as a power normalized Gaussian function

where ω_i, ω_o, and ω_g denote the mode field radii of input, output, and arrayed waveguides respectively. In this situation, Eqs. (39,41) can be further expressed as

3. Design and Simulation

The design procedure of IAWGs is different from that of conventional AWGs due to the properties of the irregular phased array (arrayed waveguides), where the optimal choice of a set of grating orders m_r to minimize the sidelobe level is a big challenge at present. Therefore, the design procedure is divided into two steps. In the first step, the design requirements and the material properties are used to obtain the physical parameters of all parts except the arrayed waveguides, which are temporarily considered as an ensemble with the function of deflecting input beams. The second step is devoted to determining and optimizing the placement of each arrayed waveguide (equivalent to the selection of a set of grating orders m_r) for minimization of the sidelobe level. In the following, a design example is given and the features of IAWGs are illuminated.

3.1. First Design Step

The IAWG is designed using the parameters that are taken from previous literature and can be realized in the conventional PLC technology. Table 1 shows the elementary parameters for the IAWG, where λ ₀ is the center wavelength, Δλ _max is the half operating wavelength range, Δλ _ch is the channel spacing, C_N is the number of channels, n_c and n_s are the effective refractive indices of the AWs and the FPRs respectively, d _min is the minimum interelement spacing with which the coupling between adjacent arrayed waveguides is generally below -80 dB, and w_i, w_o, and w_g are the mode field radii of input, output, and arrayed waveguides, respectively. Similar to conventional AWGs, IAWGs have many degrees of design freedom according to different tradeoffs of performances, and below is one design process.

Table 1. Elementary parameters for the designed IAWG

View Table

According to the above process, we fix d _out=12 μm, L_f=68582 μm, D=10.5994, and N=1400. Up to now the geometrical parameters are set except the placement of AWs, which will be determined and optimized in the next step.

3.2. Second Design Step

The placement of AWs, determined by the selection of N - 1 grating orders (m_r), plays an important role in crosstalk. Hence the optimization of placement is necessary for a high-performance IAWG. Some optimization algorithms, such as simulated annealing and genetic algorithms which recently have been shown to be fairly successful at the reduction of sidelobe levels in irregular microwave phased-array antennas [11, 12, 13], could be introduced to optimize the placement pattern of AWs for the reduction of crosstalk. These techniques have so far only been successfully applied to the design optimization of small to moderate size arrays (i.e., arrays containing a few hundred elements at most) due to the limitation of available computer power, while a practical IAWG needs thousands of arrayed waveguides.

The selection range of grating orders is another important parameter influencing the performance of IAWGs. The minimum grating order relating to d _min can be fixed by Eq. (3). Allowing a wider range of selection has a higher probability to obtain better crosstalk, however, it will cause a larger average spacing of arrayed waveguides and consequently result in greater losses. The actual selection range should balance the insertion loss with crosstalk. Before optimization algorithms could be successfully employed, the determination of the selection range is experiential. As an example for the following simulation, the values of m_r are selected from 16 to 39, from which the geometrical parameters d_j (ranging from 12.5 μm to 30.49 μm) and Δl_j (ranging from 16.52 μm to 40.27 μm) that determine the placement of AWs can be obtained by Eqs. (2) and (3), respectively.

Ten sets of m_r have been randomly selected to numerically calculate the transmission characteristics. In order to focus our attention on how much random placement of AWs influences crosstalk, here we only calculated the last term of Eq. (45) (i.e., ∑q_n), neglecting the effect of profile terms. The maximum crosstalk among all the 1080 channels for each set of values ranges from -18.107734 dB to -21.529974 dB. We have tried to optimize placement of AWs by a simple genetic algorithm, which based on the true-valued encoding, i.e., a set of m_r is treated as a chromosome. The crossover technique used is an one-point crossover, and the mutation operator here replaces a small portion of genes with different ones. The fitness is the maximum crosstalk among all the channels. A constant population model is used here, so that the chromosomes with low fitnesses enter the population and the members of the population that have high fitnesses are replaced. The process was time-consuming and finally the maximum crosstalk of -21.967267 dB was obtained. This optimized set of m_r is used to construct the IAWG. We have used the same method to optimize an IAWG with 300 arrayed waveguides, where the average maximum crosstalk of ten random placement of AWs is -15.2394 dB. And the optimized result is -18.6323 dB.

The dimension of the IAWG primarily depends not only on the focal length of FPRs (i.e., L_f) but also on the length difference between the two outmost arrayed waveguides (i.e., ∑Δl_j). In this design, L_f = 68582 μm and ∑Δl_j = 38661 μm. The compact 2.5%-Δ silica-based waveguides are used and consequently the minimum bending radius of the waveguides can be 2 mm. The input/output ports with a 25 μm spacing at the end facets are arranged so that the circuit would be as small as possible [3]. The layout is schematically shown in Fig. 2, which size is as large as 6 × 12 cm. Such a dimension of the device can be contained in a 6-inch wafer.

 

Fig. 2. Layout of the designed IAWGs.

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3.3. Simulation Results and Discussion

Using the above parameters, the characteristics of output power versus the scan parameter for the IAWG is numerically calculated under Gaussian approximation. Fig. 3 shows the transmission spectra of all 1080 channels with the wavelengths ranging from 1435 nm to 1650 nm. The input light source centered on 1542.5 nm is demultiplexed into 1080 different wavelengths with the channel spacing of about 0.199 nm. Owing to the profile of B_g(d_o/α) in Eq. (42), the loss and the crosstalk in far-end channels are higher than those of central channels. The high loss of each channel is primarily caused by the narrow waveguide width and the large spacings of the arrayed waveguides (ranging from 12.5 μm to 30.49 μm). In practice a low-loss IAWG can be achieved if tapered waveguides are employed. However, it is not easy to consider the effect of tapered waveguides in the Fourier optics theory of IAWGs (or AWGs). Fig. 4 shows the transmission spectrum of an arbitrary channel. The left inset indicates the crosstalk to the adjacent channels, and the right one shows the details of the spectrum. An apparent difference of the transmission characteristic from conventional AWGs is that the crosstalk to far-end channels is higher than that to adjacent ones, which corresponds to about -22dB and -32dB, respectively.

 

Fig. 3. Transmission spectra of all 1080 channels with the wavelengths ranging from 1435 nm to 1650 nm and the channel spacing of 0.199 nm.

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The crosstalk of IAWGs is relatively high when compared with that of conventional ones. This is attributed to two aspects which need further investigations. One is that the optimization of the arrays with a large number of elements (AWs) is difficult according to the present computer power. Even though the concept of cascade architecture has been introduced, the optimization is still time-consuming and complicated. However, due to the development of fast algorithms and computer technologies, it is anticipated that the optimization will eventually be extended to the design of much larger arrays that could contain as many as thousands of elements [11, 12, 13]. The other is the fabrication limitation of large numbers of arrayed waveguides on a wafer when implemented by planar lightwave circuits (PLCs). Under the same optimization technology, the IAWG with more arrayed waveguides will have a lower crosstalk. And compared with PLCs, more arrayed waveguides can be fabricated on multilevel lightwave circuits (MLCs) [14], which will be possible to enable the high-performance IAWGs to be realized.

Since the high-performance IAWGs need more investigations, a trade-off work pattern can be utilized to achieve a relatively low crosstalk. According to the property of IAWGs shown in Fig. 4, the crosstalk among adjacent channels, which are about 287 successive channels (from 1.5136 μm to 1.5716 μm in Fig. 4), are much lower than that to far-end channels. If we only use less than half of the region, e.g., 128 successive channels (less than 143 successive channels is needed) at any position among the 1080 channels, the crosstalk level at each channel from all the other channels will be less than -32 dB. Fig. 5 shows three examples. The three subfigures sequentially indicate 128 successive channels in left, middle, and right regions of the 1080 channels, respectively. This new work pattern with 128 successive channels that can be arbitrarily moveable in the whole operating wavelength range has potential to be applied to all optical network where waveband conversion is used.

 

Fig. 4. Transmission spectrum of an arbitrary channel.

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Fig. 5. The new work pattern with 128 successive channels that can be arbitrarily movable among the 1080 channels.

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4. Conclusion

The principle of IAWGs has been proposed, which can be considered as an extension of conventional AWGs. A Fourier optics model that is able to calculate the transmission characteristics between any arbitrary pair of input/output ports for either conventional AWGs or IAWGs has been derived. In addition, we have provided a detail design procedure and an design example to illuminate the properties of IAWGs. The results show the IAWG has an arbitrary free spectral range and is suitable to provide a large number of channels with a relatively smaller circuit region, which the conventional AWGs lack. In addition, the tradeoff work pattern with 128 successive channels that can be arbitrarily moveable in the whole operating wavelength range can provide an even lower crosstalk and have potential applications in optical networks.