I Introduction
In recent years, optical circuit switches (OCSs) were gradually applied to metro area networks and backbone networks to provide huge capacity. With the advent of 5G era, ubiquitous broadband applications, such as video sharing, spring up extensively. A single optical fiber with wavelength division multiplexing (WDM) technology may not be enough to carry the traffic between two adjacent nodes. Usage of multiple fibers in a network link at the same time has been a development trend of the nextgeneration network [14]. According to the prediction of [14], one link may need to use up to 80 parallel fibers in 2027. As a result, largescale OCSs are highly desirable in the near future.
However, as the key component of OCSs, traditional optical crossconnects (OXCs) cannot scale to a large port count. The function of OXCs is to provision connections in a nonblocking manner. As Fig. 1 shows, each port carries multiple wavelengths. The OXC can always build up a connection from an input to an output, if they have the same idle wavelength. An example is the connection at wavelength from input 3 to output 2 in Fig. 1. The traditional OXC is not scalable due to the following two reasons. In a classical OXC, there are wavelength selective switches (WSSs) at the input stage and WSSs at the output stage. Currently, the port count of the WSS is only , which limits the port count of the OXC to . Also, the OXC connects an input WSS to an output WSS via exactly one fiber, and thus contains fibers. When is large, the number of fibers will be very large and thus lead to a high cabling complexity. As [8] pointed out, high cabling complexity has become the development obstruction of the OCS.
Ia Previous Work
To address this problem, several previous works have been focused on the construction of largescale OXCs [1, 2, 15, 4, 5, 10, 7, 18].
The first one is HIER proposed by [1, 2]. The port count of HIER is , where is the degree of the node, i.e., the number of links that connect to other nodes, and is the number of fibers on each link. Ref. [1] first decomposes a WSS to a WSS, each output of which is attached by a WSS, and then replaces each WSS by a wavelength nonsensitive optical spaceswitching module. The motivation of such replacement is to reduce the system cost at the expense of the flexibility of wavelength switching. Thus, HIER is internally blocking. As [1] shows, the optical network equipped with HIERs should take into consideration the internal structure and the switching state of the nodes when it sets up an endtoend lightpath for a request. This may impose a high computation complexity on the routing process. Also, at the output side of HIER, there are delivery and coupling switch (DCSW) modules, each of which includes fibers. Typically, is a small integer, and thus is on the same order of . This implies that each DCSW module is a largescale module if is large. Another issue of HIER is that, each output port of the DCSW is an optical coupler (OC), which will lead to a large insertion loss when is large.
To cut down the cost and the insertion loss of HIER, [4, 5] proposed a twostage OXC. Different from HIER, this twostage structure replaces each WSS by a waveband selective switch (WBSS), and each OC by a cascaded structure including OCs and an WSS. Though the twostage OXC is more flexible than HIER, it is also internally blocking.
Ref. [10, 7] proposed a modular OXC that is constructed by interconnecting a set of smallsize OXCs in a ring topology. In particular, most of the ports of each smallsize OXC are the ports of the modular OXC, while the remaining ports are connected with other smallsize OXCs. This design is modular since it only employs smallsize OXCs. Another advantage is that it has a low cabling complexity. However, it is clear that this structure is internally blocking. Also, the insertion loss between different inputoutput pair is quite different. A connection that has to pass through a number of smallsize OXCs will suffer a large optical loss, which limits the scalability of this design.
Ref. [18] presented an arrayed waveguide grating (AWG) based OXC for highperformance computers. Different from that of the traditional OXC, each port of this OXC only carries one wavelength at the same time. In an AWGbased OXC, there are optical switches (OSs) at the input stage and AWGs at the output stage, where . The signal of a connection is generated by a tunable laser. The OXC selects an AWG for a connection via a OS and determines the output of the AWG by tuning the wavelength of the laser. This AWGbased OXC can route the connections in a nonblocking manner. However, the cabling complexity is high. Note that a classical OXC can carry connections at the same time. To provide the same capacity, we need an AWGbased OXC, which contains fibers. If , the cabling complexity is .
In summary, the existing designs for largescale OXCs are either internally blocking or not scalable in terms of cabling complexity.
IB Our Contributions
This paper proposes a method to construct a largescale OXC, which is scalable and nonblocking at each wavelength. This method is essentially a modularization process of traditional OXCs. There are three phases in this modularization process. In phase 1, we factorize the interconnection network between the input stage and the output stage of the traditional OXC into a set of smallsize interconnection networks. In phase 2, we decompose each WSS into a twostage cascaded structure of smallsize WSSs. In phase 3, we combine the smallsize interconnection networks with the smallsize WSSs to form a set of smallsize OXCs. At last, we obtain a modular OXC, which is actually a network of smallsize OXCs.
The proposed OXC possesses the following advantages:

It is modular since it can be constructed from a set of WSSs and OXC modules, where ;

It is internally nonblocking at each wavelength, and possesses the same routing process as the traditional OXC;

Its cabling complexity is small because the number of fiber links is , only of that of the traditional OXC;

Its physicallayer transmission performance and insertion loss are acceptable, meaning that it is feasible for practical applications.
The rest paper is organized as follows. In Section II, we introduce the structure and property of the traditional OXCs. We propose a threephase process to modularize the traditional OXC and finally obtain the modular OXC in Section III. We evaluate the performance of the modular OXC in terms of scalability, routing property, and physicallayer transmission performance in Section IV. Section V concludes this paper.
Ii Preliminary
This section introduces the structure and the function of the classical OXC, since it is the start point of our modular design. As Fig. 1 shows, an OXC consists of WSSs and WSSs with fibers in between. Each input (output) of the OXC carries wavelengths . We denote an OXC associated with wavelengths as , where the input WSSs are labelled by and the output WSSs are numbered by from top to bottom. As an example, the OXC in Fig. 1 is denoted by .
Iia Wss
There are several kinds of WSSs, such as microelectro mechanical system (MEMS) based WSS [3], liquid crystal on silicon (LCOS) based WSS [9], and planar lightwave circuit (PLC) based WSS [11]. Herein, we take the PLCbased WSS as an example to elaborate the function of the WSS.
The function of a WSS is to switch wavelengths to their desired outputs. In a PLCbased WSS, there are a wavelength demultiplexers (DeMuxs) at the input stage, OSs in the middle stage, and wavelength multiplexers (Muxs) at the output stage [11]. The input WDM signal is demultiplexed by the DeMux. Each demultiplexed wavelength is then fed to a OS attached at each output of the DeMux. Since each output of OS connects with an input of a Mux, it can switch the demultiplexed wavelength to the desired Mux. In other words, the WSS switches each wavelength separately by a dedicated OS such that the switching of different wavelengths is mutually independent [11]. Fig. 2 plots a PLCbased WSS associated with 4 wavelengths. It can either switch and to outputs 0 and 1, respectively, or route both of them to output 1.
The WSS is the inverse of a WSS. As long as no two signals at the same wavelength are fed to the WSS, it can multiplex the wavelengths from different inputs to the output. At each input, the WDM signal is first demultiplexed and each demultiplexed wavelength is then separately switched by a dedicated OS to the Mux. Fig. 2 illustrates the cases with and without contention.
IiB Shuffle Interconnection
The fibers in OXC is actually an interconnection network. In the following, we show that this interconnection network is actually an shuffle network [6].
Consider an interconnection network, where total inputs can be divided into groups, each of which contains inputs, and total outputs can be divided into groups, each of which contains outputs.
Definition 1.
An interconnection network is an shuffle network, denoted as , if the th input of the th input group connects to the th output of the th output group, where , .
In , the outputs of each WSS and the inputs of each WSS naturally constitute an input group and an output group, respectively. In particular, the th output of the th WSS connects with the th input of the th WSS. It follows that these fibers is an shuffle network. For example, the 36 fibers in OXC in Fig. 1 is a shuffle network , as Fig. 3 shows.
We use two bits to label an input (output) of , where the first bit and the second bit are called group address and port address, respectively. The th input of the th input group is labelled as and the th output of the th output group is labelled as , where , . With such a numbering scheme, a shuffle network possesses the following properties:

These is exactly one fiber between an input group and an output group;

If an input connects with an output, the bit address of the output is the exchange of the two subaddresses of the input.
That is, input connects with output . We denote the fiber between input and output as . As Fig. 3 illustrates, fiber links input 32 to output 23.
We say an interconnection network is functionally equivalent to an shuffle network if it has the features P1 and P2.
The interconnection pattern of can be delineated by an connectivity table, denoted by . In the table, the th row and the th column are respectively corresponding to the th input group and the th output group of . We denote the intersection of row and column as , which is corresponding to the fiber connecting input to output . We thus fill entry with , . As an example, the connectivity table is given by Fig. 4, where entry is corresponding to fiber .
IiC Routing Property of OXC
The function of the OXC is to setup a connection at a wavelength from an input to an output. We denote a connection from wavelength at input to at output as , where . According to the structure of , has a selfrouting property, that is, the path of is uniquely determined by its input address and its output address as follows:
It is clear that the connections at different wavelengths will not conflict with each other, since the WSS or the WSS can switch different wavelengths independently. In the following, we show that the OXC can also route the connections at the same wavelength in a nonblocking manner.
Lemma 1.
OXC is nonblocking at each wavelength.
Proof.
Consider an extreme case, where wavelength is idle only at input and output . In the following, we show that a connection can still be established in this case.
Wavelength is free at input , and thus is idle at all the outputs of WSS . Similarly, is idle at all the inputs of WSS. It follows that is available at fiber . We thus can build up a connection from input to output via wavelength . ∎
Iii Construction Of Modular OXCs
As Section I states, the number of fibers in the classical OXC is the square of the port count of the OXC, and thus increases very fast with the size of OXC. We thus start the modularization of the classical OXC with the factorization of the interconnection network. Specifically, we devise a threephase modularization process: (1) factorize the shuffle network into a set of smallsize shuffle networks, (2) decompose each WSS to a cascaded structure of smallsize WSSs, and (3) merge the decomposed WSSs with the factorized shuffle networks to form smallsize OXCs. In this way, we obtain a modular OXC, which is actually an interconnection network of smallsize OXCs.
Iiia Shuffle Network Factorization
In this part, we construct an modular shuffle network, denoted by , using a set of shuffle networks , where and . In the target shuffle network , there are input groups, each containing inputs, and output groups, each including outputs. The port count of is times that of . If we use s to construct , the inputs and the outputs of each shuffle network should be the inputs and the outputs of , respectively. Also, every inputs (outputs) of shuffle network should be grouped together as an input (output) group. To achieve this goal, we use an approach similar to that in [17], which factorizes the AWGbased shuffle networks with the help of connectivity table.
We denote the connectivity table of as , where each row and each column are corresponding to an input group and an output group of , respectively. According to properties of shuffle networks and the above discussion, a legitimate table should satisfy the following conditions:

Each entry represents one and only one fiber connecting a pair of input group and output group;

Each entry contains two addresses, and one address is the inverse of the two subaddress of the other one;

can be divided to subtables, each of which is the connectivity table of an shuffle network .
In the following, we show that such a table can be derived from the connectivity table of an shuffle network . Recall that is an table. We first apply a modulo operation to each number in . The row index and the column index change to and , respectively, where mod , and . Accordingly, entry varies from , to , . Thus, in the table after the modulo operation, the row index periodically increases from 0 to , and there are periods in total. The same thing happens with the column index . Every rows and columns in a period form a subtable, which is exactly the connectivity table of an shuffle network . In other words, the new table contains identical subtables. To distinguish these subtables, we add one bit, denoted by , before the row index , and one bit, denoted by , before the column index , where . After these operations, Table has the following changes:

Row changes to row since it is the th row in the th period, where ;

Column changes to column since it is the th column in the th period, where ;

The content in entry changes to , .
This yields a new table, denoted by . Also, we denote the subtable identified by and as . For example, after applying the modulo operations to table in Fig. 4, we obtain the connectivity table in Fig. 5. It is very easy to check that table satisfies requirements R1, R2 and R3.
According to table , we can construct modular shuffle network as follows:

For row , create an input group , which includes inputs, and divide this group into subgroups, each of which contains inputs, where the th input in the th subgroup is labelled as ;

For column , create an output group , which includes outputs, and divide this group into subgroups, each of which contains outputs, where the th output in the th subgroup is labelled as ;

Layout shuffle subnetworks vertically, where the th subnetwork is labelled as if ;

Link the ports of input subgroup to the ports of input group of , since the th period of the th row in is the th row of ;

Link the ports of output subgroup to the ports of output group of , since the th period of the th column in is the th column of .
The network constructed from table in Fig. 5 is plotted in Fig. 6. We show that table is indeed the connectivity table of . Consider input group and output group of . Input group 10 connects with subnetworks and , and output group 02 connects with and . Thus, the fiber between input group 10 and output group 02 must pass through . That is, the trace of the fiber is as follows:
As Fig. 6 plots, port of input subgroup (i.e., input ) connects with input of , and port of output subgroup (i.e., output ) connects with output of . According to property P2 of shuffle networks, input of connects with output . Thus, input connects with output . Accordingly, the content in entry of table in Fig. 5 is 1002, 0210. This example clearly shows that there is only one fiber, denoted as , connecting input group and output group , and the input address is the inverse of the two subaddresses of the output address . This indicates that has the following property.
Lemma 2.
Interconnection network is equivalent to shuffle network in terms of connectivity, where .
Proof.
Consider fiber in . If and , we have the following onetoone correspondence:
(1) 
According to properties P1 and P2 of , there is also a onetoone correspondence:
(2) 
Therefore, there is a oneone and onto mapping between in and in :
(3) 
This clearly demonstrates the equivalence between and in terms of connectivity. ∎
We thus call modular shuffle network.
We replace with in OXC , and obtain a new switching network, called . For example, Fig. 7 is the switching network called , which is produced by replacing shuffle network in Fig. 1 with modular shuffle network in Fig. 6.
As Fig. 7 illustrates, the th WSS that connects with input group of in now connects with input group of in , where . We thus relabel the WSS as . Similarly, we relabel WSS as , where . For example, WSS 3 that connects with input group 3 in Fig. 1 now connects with input group 10 of in Fig. 7, and thus is relabeled as 10. Similarly, WSS 2 in Fig. 1 is relabeled as 02.
IiiB WSS Decomposition
As [16] mentions, a WSS is functionally equivalent to a WSS, each output of which is cascaded by a WSS, where . Fig. 8 illustrates that a WSS can be replaced by a WSS cascaded by 2 WSSs.
We replace each WSS in by a WSS cascaded by WSSs, and obtain a new switching network . After the replacement, in Fig. 7 changes to in Fig. 9.
As Fig. 9 plots, the WSSs and the WSSs become the inputs and the outputs of . We label the WSS decomposed from WSS in as , and the WSS attached at the th output of WSS as , where . We label the WSSs at the outputs in the same way. Clearly, WSS connects with input subgroup of and thus input group of . Similarly, WSS connects with output subgroup of and thus output group of . For example, WSS 10 in Fig. 9 is decomposed from WSS in Fig. 7. The WSS at the 0th output of WSS 10 is labelled as , and it connects with input subgroup 100 of and thus input group of .
IiiC Merge Operation
According to the description in Section IIIB, a shuffle subnetwork connects with WSSs and WSSs . It is interesting to see that these components actually constitute a OXC . For instance, , WSSs 100,110,120, and WSSs 001,011,021 in Fig. 9 composes a OXC .
We thus perform merge operations. In the th merge, we combine shuffle subnetwork , WSSs and WSSs to a OXC, labelled by , where . It is clear that WSSs is now the th input of and WSSs is the th output of . As a result, we obtain a modular OXC, which is actually an interconnection network of WSSs, OXCs, and WSSs. We denote this modular OXC as . For example, after merge operations, changes to in Fig. 10. Also, WSSs 100 and WSS 021 are now the 0th input and the 2nd output of , respectively.
Fig. 11 plots an modular OXC , where . In particular, the connectivity of is as follows:

The th WSS at the input stage is labelled by , and the th WSS at the output stage is labelled by , where , , and ;

The th output of WSS connects with the th input of ;

The th input of WSS connects with the th output of ;
Iv Performance Evaluation
In this section, we evaluate the performance of OXC , where . We analyze the routing property in Section IVA, simulate the physicallayer performance in Section IVB, and discuss the scalability in Section IVC.
Iva Nonblocking Routing Property
Let be a connection at wavelength from WSS to WSS in , where . According to the connectivity of in Section IIIC, it is easy to know that the path of is uniquely determined by the input address and the output address as follows:
This indicates that has a selfrouting property similar to that of the classical OXC.
Since the WSSs or WSSs can switch different wavelengths independently, the connections at different wavelengths do not conflict with each other. In the following, we show that the connections at the same wavelength will not suffer internal contentions in .
Theorem 1.
The modular OXC is nonblocking at each wavelength.
Proof.
Consider an extreme case, where wavelength is idle only at input and output . In the following, we show that can be built up in this case.
Wavelength is idle at input , which implies that is idle at all the outputs of WSS. Similarly, is idle at all the inputs of WSS . According to the connectivity of , is free at both the link between output of WSS and input of , and that between input of WSS and output of . Lemma 1 shows that can set up a subconnection from its input to its output , as long as is idle at its input and its output . Thus, can be established, which proves this theorem. ∎
IvB Physicallayer Performance
We use a commercial software OptiSystem to simulate the transmission performance of a modular OXC , which contains 64 WSSs, 64 , and 64 WSSs. As Fig. 11 plots, a connection will pass through two WSSs and two WSSs. Since there is no WSS module in OptiSystem, we construct the WSS using 1 DeMux, 8 OSs, and 8 Muxs, according to the structure of PLCbased WSSs in [11].
Fig. 12(a) shows the simulation setup. We generate a 10Gb/s optical signal at wavelength 1552.53 nm using a MachZehnder Modulator (MZM), and inject it into a path containing 4 DeMuxs, 4 OSs, and 4 Muxs. At the receiver side, the optical signal goes through a variable optical attenuator (VOA) for power adjustment, an erbiumdoped fiber amplifier (EDFA) for power amplification, and an optical filter (OF) for spontaneous emission noise suppression. Via an avalanche photodiode (APD), the optical signal is converted to an electrical signal, which is finally fed to a biterror rate tester (BERT) to measure biterrorratio (BER) performance.
In the simulation, we measure the BER of the connection under different received optical powers (ROPs) by tuning the VOA. We compare the BER performance of the connection with that of a backtoback (BtB) signal that only passes through a short fiber link in Fig. 12(b). Compared to the BtB signal, the connection only suffers a power penalty of dB at the BER of . Furthermore, as [13] shows, the insertion loss of a LCOSbased WSS is dB, which is independent of the port count . Thus, the insertion loss of is dB, and will not increase with the port count of the OXC. This indicates that the modular OXC is scalable in terms of the transmission performance and the insertion loss.
To be costeffective, it is possible to replace each WSS at the input with a OC, which however may increase the insertion loss of the modular OXC. For example, if we replace each WSS at the input of , the insertion loss will be increased by 4 dB.
IvC Cabling Complexity
The modularization of OXCs not only scales down the size of employed optical components, but also cuts down the cabling complexity. As Fig. 11 shows, the number of fibers between two adjacent stages is , which is of that of the classical OXC. For example, a classical OXC contains fibers. The number of fibers is reduced to 2560, if we use WSSs and OXCs to construct a modular OXC . In particular, the number of cables is when .
Also, all the fibers inside each OXC can be capsulated in a standard module, and thus will not increase the cabling complexity of the entire network. An example of the integrated OXC module is the WSS reported by [12]. We expect that the size of OXC module will be large in the near future.
V Conclusion
In this paper, we propose a method to construct a largescale WSSbased OXC. Our analysis shows that the proposed OXC has the following advantages: (1) it is modular since it can be constructed from a set of smallsize WSSs and smallsize OXC modules; (2) it is scalable because its cabling complexity is smaller than that of the classical OXC; (3) its power penalty is small, and its insertion loss is acceptable and does not increase with the port count.
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