With the coming of big data era, high-speed switching networks become more and more important. Currently, massive data exchange is very common within the information systems at different scales. On a multi-processor system, different processors work together through a switching network to provide high-power data processing capability [1, 2, 3]. This is also true for mega data centers, where more than servers  are interconnected via a large-scale switching network to deliver high-quality cloud computing service or high-performance computation service. Also, in an even larger area, high-capacity switching network is now indispensable to big data transfers among different data centers . For example, it is reported that the traffic rate per link in a data center has come to tens of Tb/s [5, 6], and will reach Pb/s in the near future [4, 5, 7].
In the meanwhile, optical shuffle-exchange networks (SENs) have exhibited several advantages in high-speed interconnection . First, as a kind of optical switches, an optical SEN can provide large switching capacity [4, 9, 10, 11], especially when it employs wavelength division multiplexing (WDM) technology . Second, the structure of SENs is very regular and thus easy to deploy [4, 13]. Third, the SEN has a less network diameter than other networks, and thus less implementation cost . Fourth, the routing algorithm of the SEN is simple since it has a self-routing property [14, 15]: the path between an input-output pair is uniquely determined by the addresses of the input and the output. Hence, optical SEN has been considered as a promising candidate to provide high-speed interconnection for different network application scenarios [1, 2, 4, 8, 16].
The SEN is a kind of multi-stage switching networks, as Fig. 1 illustrates. An SEN consists of cascaded switching stages, each of which is a shuffle network followed by an exchange network. For example, the SEN in Fig. 1 is a SEN with and . Each shuffle network is an perfect shuffle. It splits its inputs and its outputs into input groups and output groups, respectively, and connects an input group and an output group via one and only one link. Each exchange network contains crossbars, and each crossbar is attached to an output group of the shuffle network. In other words, each crossbar performs switching function for an output group. For example, each shuffle network in Fig. 1 is a perfect shuffle, and it partitions 8 inputs and 8 outputs into input groups and output groups, each of which is encircled by a dotted circle. Each exchange network in Fig. 1 consists of crossbars, and the inputs of a crossbar are actually the outputs of an output group of a shuffle network.
Though electrical SEN is already a mature technology, the design of scalable optical SENs remains a big challenge. Up to now, several kinds of optical components, such as micro-electromechanical system (MEMS) , arrayed waveguide grating (AWG) , and tunable wavelength converter (TWC) , have been available to construct optical switches. The power consumption of optical components typically increases with their size, which, however, cannot be very large, due to cost, synthesis difficulty, or physical-layer performance. For example, an AWG with a large port count induces serious coherent crosstalk when the same wavelength is fed into a number of inputs . Such coherent crosstalk causes signal degradation and is very hard to get rid of. Another example is TWC with large wavelength conversion range, which is powerful but expensive . Also, the number of wavelengths required by the network, referred to as wavelength granularity in this paper, should not be very large, since the number of wavelengths available in optical transmission window is limited . Furthermore, the number of links in each shuffle network should be carefully considered. It is well-known that the cabling complexity has become one of the development bottle-neck of high-speed switching nodes . At last, the optical SEN should be compact and easy for maintenance, which is also very important for practical applications. Though there exist several optical designs, they cannot meet all these requirements at the same time.
I-a Previous work
In the early phase, the design of optical SENs was mainly based on free-space optics (FSO) technology. Ref. [23, 24, 25] proposed several designs, employing customized complex lens, polarizing beam splitters and micro-blazed grating arrays. In [26, 27, 13], different kinds of exchange networks were implemented with spatial light modulators, ferroelectric liquid crystals, calcite crystals, and arrayed optical switches, respectively. However, these FSO devices are typically costly, bulky, and difficult to adjust and maintain . Therefore, the FSO-based schemes are not suitable for most of practical applications.
In  and , two kinds of fiber-based SENs were proposed to avoid the disadvantages of the FSO-based SEN. In such SENs, optical fiber was used to construct the optical shuffle networks, while the MEMSs  or the electric-optical switches  were employed to implement the optical exchange networks. One feature of such kind of SENs is that each fiber only carries one optical signal. Thus, there are fibers in each optical shuffle network if the port count of the SEN is . Clearly, the cabling complexity of optical shuffle networks in the SEN will be high if is large.
Recently, combination of AWGs and TWCs provides a new way to construct SENs . An AWG is an passive optical component , each port of which carries the same set of wavelengths. The function of AWGs is to forward the signal from an input to an output without any contention. On the other hand, the TWC can convert an input wavelength to any of the output wavelengths in the conversion range. With the TWCs at each input, the AWG can perform high-speed switching function [17, 30, 31, 32, 33, 34, 35]. In particular, a TWC can send an optical packet from an input to an output of the AWG, if it converts the incoming wavelength to one of wavelengths. Based on such feature, Ref.  proposed an AWG-based SEN with cascaded switching stages, each of which was an AWG with each output attached by a TWC. However, the design in  has several drawbacks as follows. Firstly, in order to eliminate coherent crosstalk, the AWG-based SEN in  does not make full use of the WDM property of AWGs. Only wavelength channels of each AWG are employed, and thus the AWG utilization is only . Secondly, when is large, the scalability of this design is not good due to the following reasons: 1) it needs large-scale AWGs, which results in large wavelength granularity; 2) the conversion range of the employed TWCs is large; 3) a complex interconnection between the TWCs and the AWG at each stage is required to avoid the coherent crosstalk induced by the large-scale AWG.
I-B Our Work
The focus of this paper is on the construction of AWG-based WDM SENs. Different from that in , the design of this paper takes full advantage of the WDM property of AWGs, such that the above-mentioned drawbacks can be avoided.
To achieve this goal, the important step is to construct a modular AWG-based WDM shuffle network, in which the wavelength channels of each AWG are fully utilized. We demonstrate that an AWG is functionally equivalent to a shuffle network by nature if the wavelength channels at each port are considered as a channel group. Based on this result, we devise a systematic method to construct a large-scale WDM shuffle network using a collection of small-size AWGs associated with the same wavelength set. We show that the cabling complexity of such modular AWG-based shuffle networks is remarkably cut down and serious coherent crosstalk is suppressed.
We then propose an AWG-based WDM SEN by combining AWG-based WDM shuffle networks and TWC modules, each of which is constructed from a set of TWCs. We show that the conversion range of TWCs in the network is reduced since the size of each employed AWG is small. We also study the routing and wavelength assignment (RWA) problem of the AWG-based WDM SEN. We show that the self-routing property and the nonblocking routing conditions of the classical SEN are also preserved in such AWG-based WDM SEN.
In summary, our main contribution includes:
We find, for the first time, the functional equivalence between a single AWG and a shuffle network;
We develop a systematic method to construct modular AWG-based shuffle networks, of which the AWG size, the wavelength granularity, and the cabling complexity are scaled down;
We design AWG-based WDM SENs, of which the coherent crosstalk is small and the utilization of the optical components is 100% if the input channels are all busy.
The rest of this paper is organized as follows. We first demonstrate the functional equivalence between an AWG and a shuffle network in Section II, and then we develop a systematic approach to construct a modular AWG-based WDM shuffle network in Section III. Based on the result in Section III, we design AWG-based WDM SENs in Section IV. At last, Section V shows that the self-routing property and the nonblocking conditions of classical SENs are also preserved by such WDM SENs. Finally, Section VI concludes this paper.
Ii Generalized AWG-Based Shuffle Network
AWG is a kind of passive wavelength router, which is able to provide exact one connection between each input and each output. Because of such unique feature, the AWG has been used to provide broadband optical interconnections for different applications, such as shuffle network in . However, the efficient way for AWGs to fulfill the function of shuffle networks is still unknown.
In this section, we show for the first time that an AWG is functionally equivalent to a shuffle network. In particular, we first recall the definition of shuffle networks in Section II-A, and then demonstrate why and how the AWG can be equivalent to a shuffle network in Section II-B.
Ii-a Generalized Shuffle Network
Consider an interconnection network with inputs and outputs. Assume that inputs can be equally divided into groups, each with ports, and outputs can be partitioned into groups, each with ports, where . The generalized shuffle network can be defined as follows.
An interconnection network is an generalized shuffle network, denoted by , if the th port of the th input group connects to the th port of the th output group, where and .
Fig. 2 plots an shuffle network , where 18 inputs and 18 outputs are evenly divided into 3 groups and 6 groups, respectively. Also, the 5th input of input group 0 connects with the 0th output of output group 5.
We assign each port a two-field address, of which the first field is called group field while the second field is port field. In particular, we assign a two-field address to the th input of the th input group, and a two-field address to the th output of the th output group, where the underlined numbers in the field address are used to represent the corresponding port and/or group sub-addresses. Under such numbering scheme, input connects with output . We thus denote this connection by . For example, input connects with output via connection in Fig. 2.
From the above definition, it is easy to show that shuffle networks have the following two features:
An input group connects with an output group via exact one connection;
An input connects to an output, whose address is formed by exchanging the two sub-addresses of the input address.
An interconnection network is functionally equivalent to shuffle network if it can fulfill these two connection features.
Ii-B Single-AWG based Shuffle Network
An AWG, denoted as , has inputs and outputs, which are labelled from top to bottom. AWG is associated with a wavelength set in a free spectrum range (FSR), where . Without loss of generality, we assume that in this section, and thus . This paper only considers the wavelength channels in the main FSR, because AWGs suffer physical performance degradation at the wavelength channels outside the main FSR . In the main FSR, each input of AWG carries wavelength channels, and each output has wavelength channels. Fig. 3 gives an example of a AWG .
The AWG is a passive wavelength router. In an AWG, input is connected to output via wavelength , where:
where . The wavelength routing property (1) clearly shows that input and output are connected by exact one wavelength channel , of which the connection is denoted by . Such connection feature is consistent with the feature F1 of shuffle network .
Also, the wavelength routing property (1) of can be described by an routing table, denoted by , where the th row and the th column are corresponding to input and output , and the entry at the intersection, denoted by , records the wavelength channel of connection . For example, Table I is the routing table of AWG in Fig. 3, and entry is the wavelength channel of . Therefore, the entries at row (column ) are corresponding to the wavelength channels at input (output ).
According to the structure of , we assign addresses to the input channels and the output channels as follows:
Input channel: Assign addresses to the wavelength channels at row (i.e., input group ) from left to right. The th entry at row is assigned with field address and two-tuple address .
Output channel: Assign addresses to the wavelength channels at column (i.e., output group ) from top to bottom. The th entry at column is assigned with field address and two-tuple address .
In the field address, the first field is called port field, and the second field is referred to as channel field.
With such numbering scheme, entry in delineates both input channel and output channel , which means they connect to each other through . To describe such kind of connectivity explicitly, we replace entry in with input channel and output channel , and obtain a new table, called connectivity table and denoted by . For example, Table II gives the connectivity table of , in which entry shows that input channel connects with output channel . This property is quite similar to the connection feature F2 of shuffle network .
Therefore, if the channels at one port are regarded as a channel group, and have the following equivalence:
A single AWG is equivalent to a shuffle network in terms of the connectivity between input channels and output channels.
This property holds due to the fact that each input-output pair of is connected via only one wavelength channel, which is elaborated as follows:
In , input connects to output via only one connection . It follows that there exists an one-one and onto mapping
According to the property of shuffle network , input group is connected to output group via a unique connection . Thus, there is an one-one and onto mapping
Thus, if each port of is regarded as a channel group, there is an one-one and onto mapping between connection in and connection in , i.e.,
This establishes Property 1. ∎
The space representation of is plotted in Fig. 3, where each line in the left side represents an input channel, each line in the right side stands for an output channel, and that in between is a connection connecting a pair of input channel and output channel. Fig. 3 clearly shows the one-one and onto mapping between the connection in and that in .
According to Property 1, we can use an AWG to construct an single-AWG based shuffle network , where . The cabling complexity is only , since we need fibers at the inputs and outputs when we use it. Also, the AWG contains wavelength channels, and thus the utilization of the AWG is 100% if all the input wavelength channels are busy. However, as either the number of inputs or that of outputs becomes large, the single-AWG based shuffle network is not scalable and will suffer from difficult synthesis technique, serious crosstalk, and large wavelength granularity, as we mention in Section I.
Iii Modular AWG-based Shuffle Network
In this section, we study the method to construct a modular AWG-based shuffle network, which is the key step for the design of AWG-based SENs. In particular, we consider how to devise using a set of AWGs, where and . From Section II, we establish the equivalence between a single AWG and a shuffle network through investigating the routing table of AWGs. Thus, Section III-A starts the construction from the design of the routing table of the modular AWG-based shuffle network, based on which we come up with a systematic method to achieve the construction in Section III-B.
Iii-a Routing Table of
We denote the modular AWG-based shuffle network to be constructed as . In , there are input wavelength channels and output wavelength channels. These input channels are divided into input groups, each with input channels, and output channels are divided into output groups, each with output channels.
Since the building blocks of are AWGs, of which each port carries the same set of wavelengths , we need AWGs to construct a with input channels and output channels. Also, according to the definition of , every inputs of AWGs should be grouped together as an input group, and every output as an output group.
The wavelength routing property of can also be described by a wavelength routing table, denoted by , where each row and each column correspond to an input group and an output group respectively. According to the above description, in general, a legitimate routing table should satisfy the following conditions:
As the wavelength granularity of is , the routing table can only contain different wavelengths;
As a kind of shuffle networks, each entry of this table can contain one and only one wavelength;
Since every inputs of make up an input group and every output is an output group, the wavelengths in must appear times in each row, and once in each column.
A routing table that meets the above conditions can be constructed from the routing table of . There are rows and columns in , in which each entry has just one wavelength. In particular, entry at the intersection of row and column contains wavelength channel , where and . If we apply modulo operation to all the numbers in , the row index remains the same as , the column-index changes to , and becomes since
It is easy to see that the column index periodically increases from 0 to , and there are totally periods in the table after modulo operations. According to (7), every columns within a period form a subtable, each of which is a routing table of . Thus, there are such identical subtables in the table. To distinguish these subtables, one more field is added to the left side of the column index, say . In particular, if a column is the th column of the table, it will be labelled by since it is the th column within the th period, where . For example, after performing modulo operations on Table I and relabeling the columns, we obtain Table III, which contains 2 routing tables of . It is clear that this new table satisfies three conditions mentioned above, and thus is the desired routing table of .