Optimizing bend loss in optical waveguide channel routing on photonic integrated circuits

Silicon photonics (Si-photonics) has been established as a potential technology that integrates both electronic and optical circuits on single integrated circuits (ICs) in order to satisfy the increasing demand for high speed and low power in the emerging market of ICs. It has opened up the research directions in the domain of design automation for photonic ICs. On the physical layout of the optical circuits, it is a challenging task to obtain the optimal routing of optical waveguides, while minimizing all the parameters like the number of tracks, total bend loss, worst signal loss, total propagation loss and total crossing loss. In this paper, we proposed two non-Manhattan grid-based methods for reducing the bend loss, worst signal loss and tracks in optical channel routing. First, a 0–1 integer linear programming-based algorithm called minimizing bend loss (MBL) is proposed, which minimizes the total bend loss (TBL) and the worst signal loss (WSL) while reducing the number of tracks (T) over the state-of-the-art technique. The execution time of MBL is very high for the large input. Hence next, a scalable heuristic called reducing bend loss (RBL) is presented that provides a better balance between the reduction of the TBL and T over the state-of-the-art and MBL algorithms. Simulation results show that MBL can reduce the TBL and the WSL by an average of 57.9% and 63.1%, respectively, with an average increase of 12% in T over state-of-the-art algorithms. The simulation results show that the RBL reduces the TBL and the WSL by an average of 39.7% and 41.3%, respectively, with an average increase of 23.7% in T over state-of-the-art algorithms.


Introduction
Due to the continuous increase in transistor count and increasing demand for high-speed and low-power consumption in the integrated circuits, Moore's law [1] is reaching its limit. 1 Moreover, the need for high-performance and lowpower circuits requires other non-transistor alternatives to the traditional very-large-scale integration (VLSI) industry. Hence, researchers are investigating several alternative technologies, such as reversible computing, quantum computing, silicon photonics and quantum photonics [3]. Among all, silicon photonics (Si-photonics), which is the integration of photonics and electronics on a single integrated circuit (IC) chip, is one of the potential technologies for next-generation computing [4]. This technology is rapidly growing as a platform for large-scale photonic integrated circuits (PICs) due to low fabrication cost, smaller footprint, low-power consumption and CMOS compatibility to basic CMOS electronics [1]. The advancement in PICs has opened a wide variety of applications in the field of telecommunications, photonic signaling, data center, high-performance computations, optical network-on-chip (ONoC), chip designing, medical diagnostics, and biosensing [3].
ONoC is one of the promising applications, where in place of metal wires optical interconnects/waveguides are used to satisfy the need for high bandwidth with lesser power dissipation. Current ONoC architectures can be classified into two categories: wavelength-routed and controlnetwork-based [5]. The layout of these optical networks is determined with the topology-mesh, torus, ring and mixed [6]. However, design of topologies should limit the number of laser sources and optical links, as laser source coupling is one of the main contributors to the chip packaging costs. Further, as the number of photonic elements and size of optical interconnects increase , the need for physical layout optimization (placement and routing) arises. However, for the complete adoption of large-scale photonic networks and interconnects, the design and fabrication of optical components first need to address the operational (optical power, on-chip temperature variations) and fabrication (optical waveguide properties, signal losses, area) challenges [6].
For the past few years, researchers have focused on the placement and global routing [7][8][9][10] of the on-chip optical components and interconnect. In many optical networks [11,12], the column-based placement of the photonic elements is performed to minimize signal loss and area. This columnbased arrangement of components makes the vertical routing region (channels) between the devices, leading to the channel routing problem. To date, several researchers have presented techniques for channel routing in PICs for minimizing the signal loss and area [13][14][15]. While obtaining the channel routing of optical waveguides, the bend geometry has a vital role in reducing the bend loss. The sharper bends having the smaller bend angles have more impact on the signal loss. Therefore, the minimization of bend loss is a primary objective in optical waveguide routing. This work focuses on the removal of bendings having higher losses. The contributions of this paper are summarized as follows.

A 0-1 integer linear programming (ILP) is formulated
for minimizing the bend loss (MBL) while reducing the number of tracks in optical channel routing. 2. A new grid-based restricted routing is proposed to eliminate the sharper bends having smaller bend angles. 3. A heuristic is proposed for reducing the bend loss (RBL) in the optical channel routing. 4. A comparative analysis of the simulated testcases shows that the proposed ILP formulation outperforms the stateof-the-art techniques in terms of the total bend loss, the worst signal loss, and the number of tracks.
The rest of the paper is organized as follows: Section 2 discusses the basic concepts and the prior works in optical channel routing. The motivation and problem formulation are discussed under Sect. 3, respectively. A 0-1 integer linear programming (ILP) formulation of optical waveguide channel routing for minimizing the bend loss, worst signal loss while reducing the number of tracks is presented in Sect. 4. Further, the proposed heuristic for optical channel routing is presented in Sect. 5. The results and comparative study are discussed under Sect. 6. Finally, Sect. 7 concludes this paper.

Basic preliminaries and prior works
This section presents the basic preliminaries and the related works in the domain of optical channel routing.

Basic preliminaries
This section provides a brief overview of the basics of optical channel routing. Table 1 represents all the symbols used in this work.

Channel grid
The channel routing of optical waveguides in the proposed work follows a grid-based non-Manhattan routing. Channel is represented as a space between the list of optical pins that are present on its two opposite sides. Figure 1 depicts the grid structure used in our work, where toplist and bottomlist are two lists of optical terminals along a rectangular channel whose width and height are represented as W and H, respectively. The terminals of both lists are assigned with a number p (1 ≤ p ≤ N ). The terminals with the same number are connected with an optical interconnect p during routing. To implement a channel routing, the channel is divided into a grid of uniform cells with the width and height of w and h, respectively. Each row in the grid contains different permutations of terminal numbers that decide the path of optical interconnects within a channel. These rows are called tracks. Hence, the number of tracks (T) decides the size/area of the channel.

Routing paths and bends
As mentioned above, our work is based on the non-Manhattan grid routing, where an optical waveguide always routes from bottom to top boundary terminals. Hence, from any point inside the channel grid, a waveguide can possibly move in five directions, i.e., left (L), right (R), left-diagonal (LD), right-diagonal (RD), and vertical up (U), as shown in Fig. 2a. On a track, when an optical waveguide changes its direction from the previous track direction, then it results in a bend in its path. The closed view of a grid cell in Fig. 1 shows that the cell diagonal results in two different angles of = tan −1 ( h w ) and = (90 − ) with its boundary. Hence, the movement of an optical waveguide in the channel grid results in five different types of bends with an angle of , , + , + , and 2 , where represents a 90 • angle. Figure 2b-f shows all the possible types of routing bends that can occur while routing from bottom to top directions. A Boolean variable for a channel grid point (i 1 , j 1 ) with an incoming optical waveguide o routing from a location of (i 2 , j 2 ) Eout o,i 1 ,j 1 ,i 2 ,j 2 A Boolean variable for a channel grid point (i 1 , j 1 ) with an outgoing optical waveguide o routing to a location of (i 2 , j 2 )

Types of losses
Based on the exploration of different routing geometries following three types of losses (with dB unit), bending loss, crossing loss, and length loss can be treated as the penalty in optical planar routing.
i) Bend Loss: For a given optical bend with two unit-length waveguides, bend radius (R), and a bend angle ( ). The bend loss (BL) can be computed with Eq. (1) [14].
which indicates the bend loss BL( ) is directly proportional to the bend angle and inversely proportional to the bend radius. ii) Propagation Loss: The propagation loss ( PL i ) of a given optical waveguide i is proportional to the length ( i ) and is defined as PL i = 3 ⋅ 10 −9 ⋅ i . From the viewpoint of optical signal loss, propagation/length loss on an optical waveguide is very small [14]. iii) Crossing Loss: When two optical waveguide cross each other, it results in some signal loss, which is named as crossing loss. According to the prior experimental work, it has been detected that the crossing loss of two optical nets is constant, i.e., in the range of (0.1-0.2 dB) per crossing.

Total signal loss (TSL)
TSL is the sum of the total bend loss (TBL), the total propagation loss (TPL) and the total crossing loss (TCL).

Worst signal loss (WSL)
The routing of each optical waveguide reduces the optical signal strength along its path due to the above-mentioned losses. For an N number of different routed paths, the worst signal loss (WSL) is defined as the maximum signal loss among these paths.
where oss j is the optical loss for an optical waveguide j along its routed path.

Prior works
In the routing of optical interconnects with the photonic integrated devices, researchers have explored many techniques while satisfying the optical properties and the design oss j constraint rules like the minimum distance between waveguides, waveguide width, and area [14,15], since the complete routing can be realized by performing the detailed/ channel routing after global routing. Hence, in the past channel routing has been performed on PICs for minimizing the total signal loss (TSL) and area [14,15], where the area is regarded as the function of the total length of waveguides and the number of tracks used in channel routing.
Condart et al. [16] presented the first work on optical channel routing, which was the adaptation of basic VLSI placement and routing in the optical domain. The optical routing is performed on Manhattan grid-based channels without considering optical waveguide geometries and crossing properties, resulting in a higher signal loss due to sharp bends. In [13] and [14], a 2SWAP channel routing based on the traditional SWAP [17] algorithm is proposed that shows the reduction in the total number of crossing and bends. 2SWAP reduces the 90 • bends by using the additional bends of 135 • . Hence, 2SWAP [14] reduces the signal loss than [16] with an overhead of increased number of tracks, i.e., area. Another non-Manhattan-based channel routing is introduced in [14], which used the basic left edge algorithm with the knock-knee method. This method shows the track reduction at the cost of higher bend loss compared to 2SWAP. Moreover, [14] follows a restricted constraint of 90 • crossing between two crossed waveguides, which is not mandatory for on-chip optical interconnects. A more flexible channel routing (TNR) has been presented in [15] that removed 90 • crossing constraints while maintaining a crossing angle in the range of 60 • -120 • . A new rectangular grid-based non-Manhattan routing model in TNR showed the reduction in bend loss by replacing the sharpen bends with smooth bends. In TNR, both the tracks and bends are reduced by using the existing modified two-pass optimality-oriented hierarchical bubble sorting algorithm [18]. However, still the presence of 90 • bends in TNR results in the higher bend loss. Moreover, this new routing within the rectangular grid model may violate the design rule check of minimum spacing distance among the waveguides.

Motivation and problem formulation
In this section, we discuss the motivation and problem statement of our work.

Motivation
The main objective of any routing problem is to route all the waveguides while reducing the TSL and area, i.e., tracks (T). In the routing of optical waveguides, bend loss and number of crossing are the two significant figures of merits in designing high-density PICs [19,20]. The higher radiation loss at the sharp bends limits the density of a PIC. Different approaches have been proposed in the literature to reduce TBL, where the curvilinear nature of optical waveguides plays an essential role [20]. Due to the curvilinear shapes, the performance of the optical circuit depends on the number of bends, bend radius, and width of the waveguides [21]. As discussed above, the routing grid in [15] considers the optical waveguide geometries and shows the reduction in bend loss compared to the prior work. In this work, we have used the channel grid structure similar to the [15]. However, to further reduce the bend loss, a restricted horizontal routing is used to eliminate the and bends. Further, the lack of multi-layer routing results in the crossing between waveguides, which can be reduced by controlled crossings. Moreover, each crossing has a constant loss. Therefore, bend loss minimization is of pivotal importance to design high-density PICs. Further, there is an overhead between TBL and T reduction. In this work, we are balancing these parameters with an ILP formulation to minimize the bend loss while reducing the area and WSL.

Problem formulation
This work aims to provide a routing path to the optical waveguides within a rectangular region/channel, where the two opposite sides, i.e., bottom and top boundaries of the channel, have fixed terminal pins. Each pin that corresponds to the same waveguide must be connected from bottom to top while satisfying the routing constraints. The channel is assumed to be a grid ( Fig. 1), where each net can route in five directions from the current grid point, as shown in Fig. 2.
Input: Given the bottom list and the top list of N number of optical waveguides.
Output: The routing path for each given optical waveguide from bottom list to the top list within the channel.
Objective: The main objectives of this channel routing are: (i) to minimize the total bend loss (TBL), (ii) to minimize the worst (maximum) signal loss (WSL) along the optical path, and (iii) to minimize the number of tracks (T), which has been defined as a multi-objective weighted function f as: where , , and are user-defined weighted parameters.

0-1 Integer linear programming (ILP) formulation for minimizing bend loss (MBL) in optical channel routing
The routing of each optical waveguide reduces the optical signal strength along its path due to various losses (as mentioned in Sect. 2.1). Hence in addition to optimize the TBL, WSL should also be reduced. Further, any routing aims to reduce the area. Therefore, the T should also be reduced to decrease the total channel area. In this section, a constraintbased 0-1 integer linear programming (ILP) formulation for optical channel routing has been introduced to minimize the bend loss (MBL). For the ILP formulation, we have considered the same N × T grid-based optical channel, as mentioned in Sect. 2.1, where N is the number of terminals and T is the number of tracks. We have formulated a 0-1 ILP model for minimizing the multi-objective function f as follows: where TBL, WSL are defined as Eqs.  Fig. 2) at every grid location (i, j) can be identified with Eqs. (5) to (6).
where Ein o,i 1 ,j 1 ,i 2 ,j 2 is a Boolean variable for a channel grid point (i 1 , j 1 ) with an incoming optical waveguide o routing from (i 2 , j 2 ) location, Eout o,i 1 ,j 1 ,i 2 ,j 2 is a Boolean variable for a channel grid point (i 1 , j 1 ) with an outgoing optical waveguide o routing to (i 2 , j 2 ) location. Equation (5) calculates the presence or absence of an bend at (i, j) grid location that would be better understood with Fig. 2b. Ein o,i,j,i−1,j , Ein o,i,j,i,j−1 , and Ein o,i,j,i,j+1 are the incoming optical waveguides (o) from the previous track ( i − 1 ) in vertical up (U) direction, from the same track (i) in left (L) and right (R) directions, respectively to the (i, j) location. Eout o,i,j,i,j−1 , Eout o,i,j,i,j+1 and Eout o,i,j,i+1,j are the outgoing optical waveguide (o) from the (i, j) grid location routing to the same track (i) in L, R directions, and to the next track ( i + 1 ) in U direction, respectively. Similarly, the remaining bends can be interpreted with Fig. 2b-f, respectively.
2) Routing Constraints: The routing of an optical waveguide from a grid location follows some restrictions to maintain the bottom-up paths for all the given waveguides. These restrictions are explained in Eqs. (8) to (10) and diagrammatically presented in Fig. 3.
The routing constraints for the bottom boundary ( r 0,j ) are illustrated in Fig. 3a-c and explained in Eq. (8). In Fig. 3a, b, the routing constraints for the bottom-left ( j = 1 ) and bottom-right ( j = N ) are shown where the waveguide (o) can move to the next track in two possible directions (U or R) and (L or U), respectively, while Fig. 3c represents the routing constraints for the remaining bottom points ( 2 ≤ j ≤ N ), where a waveguide can move to the next track by taking one of the possible path (L or U or R).
Similarly, the routing constraints for all remaining grid points along the channel tracks are shown in Fig. 3d-n, which are formally explained with Eq. (9). The routing constraints for a grid point at the left boundary (other than the bottom boundary) of the channel, i.e., i ≥ 1 and j = 1 , are represented in Fig. 3d-f. As shown in Fig. 3d Fig. 3f. Similarly, the routing constraints for the right boundary points (other than the bottom boundary), i.e., i ≥ 1 and j = N , are depicted in Fig. 3g-i. Otherwise, for the remaining internal grid points, i.e., i ≥ 1 and 2 < j < N , the five different routing constraints are shown in Fig. 3j-n. When an optical waveguide comes from the previous track at these internal points, it can route in any of the five different paths (Fig. 3j-l). However, as shown in Fig. 3m-n, the incoming waveguide from the same track can route to four possible paths. The routing constraints can be presented in Eq. (10) as follows: Equation (10) defines the routing constraints at each grid point. If there is any empty pin along the channel boundary, then there may be some grid points through which no waveguide is routed; hence, for such cases, r i,j = 0 . However, when there is no empty pin, then for the two-sided channel routing r i,j should be 1 for satisfying the design rule check (DRC) of photonic layout geometry.
3) Crossing Constraints: Furthermore, whenever a waveguide route in the horizontal L or R direction from a grid location of ( i, j 1 ) to ( i, j 2 ), then it may cross with another waveguide that was already routed at ( i, j 2 ) location. Hence, each grid point is assigned with a Boolean variable c i,j representing the crossing of waveguides at (i, j) grid location.
The crossing constraints are defined in Eqs. (11) to (13) and diagrammatically presented in Fig. 4.
The waveguides at the bottom-boundary grid points do not cross each other; hence, c 0,j = 0 , as mentioned in Eq. (11). However, for all the internal grid points with i > 0 , there may be the crossing of two waveguides with each other; hence, c i,j = 0 or 1, as mentioned in Eq. (12). Along the left-boundary, if a waveguide o 1 is already passed from an (i, j) location as shown in Fig. 4a or Fig. 4b, then there is only one possible routing path for o 2 at (i, j) with the incoming from L and outgoing to RD. Similarly, the crossing constraints along the right-boundary grid points are shown in Fig. 4c, 4d. The crossing constraints for the internal grid locations are presented in Fig. 4e-r. When the previously routed waveguide ( o 1 ) is present, as shown in Fig. 4e, f, then o 2 can cross (i, j) in two possible ways, i.e., from L to (U or R) or from R to (U or L). To satisfy the physical design rule, o 2 cannot route to RD. However, in Fig. 4g, o 2 can route (13) in three different ways as represented. Similarly, all the cases are defined as depicted in Fig. 4h- Further, the implementation of MBL is explained in Algorithm 1 that has toplist, bottomlist and N as inputs. As mentioned in lines 2-4, an ILP model is formulated for t different grids with the size of [N × 1 , N × 2 , ..., N × N] using Eqs. (5) to (13). Then, the minimized cost function is performed on all t routings to find the best routing results. Hence, the total time complexity for MBL is equal to O(tN 2 ) = O(N 3 ) . In the next section, a heuristic is proposed for optical channel routing.

Reducing bend loss (RBL) in optical channel routing
The existing TNR algorithm shows the horizontal movements of optical waveguides along tracks, which reduces the number of tracks as compared to 2SWAP. Hence in this work, sharing of two optical waveguides along one track has been considered to reduce the tracks. Further, the horizontal movement of optical waveguides in TNR routing has no restriction that results in a higher number of (90 • ) bends and a few bends. The presence of these bends results in a higher bend loss due to sharp angles. Since the main objective of this work is to reduce the total bend loss, hence we have introduced a restricted routing of optical waveguides, as shown in Fig. 5. Figure 5a shows the unrestricted horizontal routing, where an optical waveguide takes two bends, resulting a bend loss of 0.738 dB. However, restricting the routing with a diagonal before and after the horizontal path reduces the total bend loss to 0.482 dB, as shown in Fig. 5b. In this section, a heuristic for optical channel routing has been discussed that uses restricted movements. A heuristic approach for the on-chip channel routing for reducing the bend loss (RBL) is explained in Algorithm 2.
The RBL is a grid-based channel routing algorithm, where all the waveguides are routed from bottom to top terminal list. The aim of this algorithm is to route all the waveguides from the bottom terminal list to the top terminals while reducing the total bend loss (TBL) and the number of tracks (T) inside the channel. As aforementioned, the TBL can be reduced by using the restricted horizontal routing, which replaces the smaller angles, i.e., , angles with the larger angles, i.e., + , + , and 2 angles. Algorithm 2 describes the RBL algorithm that has two inputs toplist and bottomlist, where toplist and bottomlist are the terminal list of top and bottom boundaries of the optical channel. The four data variables start, end, minpos, and maxpos are used in RBL, where start and end variables point to the start and end locations of the unrouted terminal list in the current track; however, minpos and maxpos point to those locations of the unrouted waveguide that have minimum and maximum terminal numbers, respectively, in the current track. Initially, no waveguide is routed; hence, start and end are initialized with first and last indices of bottomlist (line 1). The maxpos and minpos are initiated to 0 (line 1), which are further updated with the FindMaxMin function (line 3) that returns the maxpos and minpos from the terminal list in the range of start and end. On each track, waveguides are routed by performing one or two horizontal left or right moves. The waveguide movement depends on the maxpos and minpos on the current track, which is described in Algorithm 2 with the following cases: case (i) routes maxpos waveguide at the end location by performing a RightMove (line 7-8), which has been explained in Algorithm 3, case (ii) routes minpos waveguide at the start location by performing a LeftMove (line 10), and case (iii) shows the sharing of tracks by two waveguides while performing both right and LeftMove (lines [14][15]. The horizontal routing in all three cases follows the restricted routing by replacing the U → L∕R → U moves with RD → R → RD and LD → L → LD for right and left routing, respectively (as depicted in Fig. 5). When minpos becomes higher than the maxpos + 1 , then the horizontal path for the waveguides at minpos and maxpos overlaps with each other (line 6-11). Hence, case (i) and (ii) route a single waveguide in horizontal right or left direction from maxpos or minpos, respectively (lines 7-10). However, when the minpos location is smaller as compared to maxpos + 1 , then the same track can be simultaneously used by two waveguides for both right and left moves (lines [13][14][15][16]. The selection of case (i) or case (ii) depends on R diff and L diff , where R diff and L diff are the difference of maxpos and minpos from the end and start, respectively. When R diff is more than L diff , then RightMove is performed on waveguide at maxpos (line 8). Otherwise, LeftMove is performed on waveguide at minpos (line 10). Further, the complete routing for these three cases has been outlined in detail in Algorithm 3.
1) RightMove: The RightMove of RBL has been explained in Algorithm 3, that takes the current track bottomlist, maxpos, start, and end as its input. In addition to the horizontal right routing, this algorithm also updates the locations of the remaining waveguides. All the waveguides on the left side of maxpos and on the right side of end are moved in one vertical up direction, i.e., U (1) (lines 3-4). The horizontal restricted routing has been detailed in lines 6-11, where r denotes the number of steps to be taken by waveguide at maxpos in the right direction. When r ≥ 0 , then the maxpos waveguide routes by taking one-step right-diagonal ( RD (1) ), r-step right ( R (r) ), and one-step right-diagonal ( RD (1) ), which has been presented as RD (1) ⟶ R (r) ⟶ RD (1) . However, when maxpos is immediately left to the end location, then waveguide does not require any horizontal move. Hence, the waveguide route by taking one-step right-diagonal ( RD (1) ) and then one-step vertical up ( U (1) ) moves, that is represented as RD (1) ⟶ U (1) . The remaining waveguides positioned in between ( max + 1, end ) move one-step leftdiagonal, i.e., LD (1) (lines 12-13). At the end, end, T, and bottomlist are updated (lines 15-17).
2) LeftMove: Similar to the RightMove, a restricted horizontal left routing at minpos has been implemented. In the LeftMove algorithm, waveguides replace the RD and R moves of Algorithm 3 with LD and L directional moves. In

Simulation results and discussion
This section presents the simulation results with the comparative analysis of the proposed algorithms.

Simulation results
In this section, the proposed MBL and RBL are compared with the existing 2SWAP [14] and TNR [15] algorithms in terms of signal loss and area used. All the algorithms are implemented in Python 3.7.4 and run on Intel Core i7-9750H CPU 2.60GHz machine with 8 GB memory. In order to perform the comparative analysis, different testcases are randomly generated with a varying number of terminals (N).
An illustration of optical channel routing implementation for the prior works (2SWAP [14] and TNR [15]) and the proposed algorithms (MBL, RBL) is presented with an example in Fig. 6, which provides a better understanding of bending effects while routing. As the smaller bend radius introduces the excess loss and crosstalk between different waveguides and the larger bending radii occupy a larger footprint. Hence, the minimum and maximum bend radius are satisfied by considering w and h as 1 and 1.732 units, respectively, resulting in = 90 • , = 60 • , + = 150 • , + = 120 • , 2 = 120 • angles [2,15]. The BL( ) , BL( ) , BL( + ) , BL( + ) , and BL(2 ) are 0.592, 0.369, 0.204, 0.204 and 0.074 (dB), respectively (from Eq. (1)). Figure 6a shows the optical channel routing for the 2SWAP, which takes a total of 20 bends with a bend angle of + and a total number of 6 tracks (T). Hence, TBL for 2SWAP is 1.5 dB. The routing result of TNR is represented in Fig. 6b, which uses a total of 6, 7 and 3 number of , + , and 2 bends. Hence, TBL for TNR is 3.4 dB, which is higher as compared to 2SWAP. This increase in TBL is due to the presence of smaller bends ( ) in TNR. However, T in TNR is reduced to 4, resulting in a smaller channel area than 2SWAP. Similarly, the calculated TBL for MBL, and RBL algorithms are 1.3, and 2.3 dB, respectively. The absence of and bends in RBL results (a) (b) (c) (d) Fig. 6 An example of channel routing using a 2SWAP [14], b TNR [15], c MBL, and d RBL  in the reduction of TBL with an overhead of an increased T compared to TNR, as shown in Fig. 6d. Further, this overhead is removed in MBL, which has both the least number of tracks ( T = 4 ) and TBL (1.3 dB), as depicted in Fig. 6c. As shown in Tables 2, 3, and 4, ten testcases (TC-1 to TC-10) with N = (10, 15, … , 100) are randomly generated that have two terminal lists named as toplist and bottomlist along the top and bottom boundaries of the optical channel, respectively. Both lists are initialized with a different integer number in the range of [1, n]. In Table 2, Nbends(x) denote the total number of bends with x-angle. The proposed MBL is simulated and compared for N ≤ 30 due to the high execution time. Hence, a '-' in Tables 2, 3, and 4 indicate that the testcase is not being run for the respective N value. In Table 3, TBL is the total bend loss in dB, TPL is the total propagation loss in 10 −6 units, TCL is the total crossing loss, and WSL is the worst signal loss in dB. In Table 4, T denotes the number of tracks, and TWL is the total waveguide length in the optical channel grid. A comparative study of the proposed MBL and RBL with the previous algorithms is shown in Tables 2, 3, and 4.
Since the objective of our work is to minimize the signal loss while reducing the area. Hence, the comparative results in Table 3 show that the proposed MBL outperforms in terms of signal loss as compared to the previous algorithms and the proposed RBL while maintaining a better balance for tracks compared to 2SWAP and RBL (as shown in Table 4). Tables 3 and 4 show that the MBL has an average reduction of 57.9% and 63.1% in TBL and WSL, respectively, with an increase of 12.7% and 9.8% in T and TWL, respectively, compared to TNR. However, compared to 2SWAP, MBL shows an average reduction of 44.8%, 38.1%, 25.4%, 5.8% in TBL, WSL, T, and TWL, respectively. Further, in comparison with RBL, MBL shows an average reduction of 48.1%, 43.7%, 12.9%, 26.9% in TBL, WSL, T, and TWL, respectively, for first five testcases (TC-1 to TC-5). Hence, these results show that the proposed MBL outperforms the prior works and the proposed RBL in terms of signal loss. However, MBL faces the scalability issue when the input size is increased. Therefore, we proposed a heuristic RBL which resolves this issue.
In Table 3, the simulation results show 2SWAP algorithm shows the least TBL as compared to TNR and the proposed RBL. However, it has a cost of higher T, as compared in Table 4. Hence, 2SWAP requires a larger chip area to implement. In Table 2, the simulation results show RBL does not have and bends, which is due to the restricted horizontal routing that eliminated the smaller bend angles. This elimination of smaller bends helps to reduce the TBL in RBL compared to TNR. In terms of T, TNR uses the least tracks at the cost of higher TBL and WSL, which results in the reduced signal strength at the output. The proposed RBL algorithm provides balanced results between 2SWAP and TNR while reducing TBL, WSL, TWL and T. In RBL, all the testcases except TC-1 (red text in Table 3) show the reduction of TBL, WSL and T in comparison with TNR. This is due to the higher number of bendings in RBL and fewer bends in TNR for the smaller input terminals. Hence, the proposed RBL algorithm shows a better reduction in all the routing parameters for the larger testcases having N > 10 . The comparative results show that the proposed RBL has an average reduction of 24.5%, 1.2% and 0.21% in T and TWL, and TPL, respectively, with an increase of 44.4% and 35.9% in TBL and WSL, respectively, compared to 2SWAP. This increase in Table 4 Comparative result of the proposed MBL and RBL algorithms with the previous algorithms in terms of area, i.e., tracks (T), and total wire length (TWL) ↓ and ↑ indicate the reduction and increment percentage, respectively TBL and WSL is due to the increasing number of bendings in RBL, as shown in Table 2. However, in comparison with TNR, the proposed RBL has an average reduction of 39.7%, 2.2%, and 41.3%, and in TBL, TPL, and WSL, respectively, with an increase of 23.7% and 1.2% in T and TWL, respectively. Therefore, these results conclude that the increase in T reduces the bends, which results in the reduction of signal loss with an overhead of increased area. Further, Figs. 7 and 8 show the average comparative results of 30 different testcases having N = 10 and 15, respectively. The total execution time for the average analysis is 78 and 248 hours for N = 10 and 15, respectively. Since the execution time for N > 15 is high, therefore we have considered an average comparative analysis for the 30 different testcases for N = 10 and 15 only. Figures 7a, c and 8a and c show that MBL has the least TBL and WSL compared to 2SWAP, TNR, and RBL, which is due to the minimization of smaller bends and total number of bends. This minimization of bend and worst signal losses shows the increment in T compared to the optimal method, i.e., TNR (Figs. 7d and 8d), which further increases the TWL (Figs. 7e and 8e) and TPL (Figs. 7b and 8b).
Hence, the simulation results suggest that there is a tradeoff between the total bend loss (i.e., signal loss) and the number of tracks (i.e., chip area). MBL can reduce both the signal loss and chip area compared to the proposed RBL. For some smaller testcases with N ≤ 10 , RBL does not perform well, whereas MBL is preferred over RBL. However, in the ILP formulation, MBL needs to consider a large number of Boolean variables for incoming ( N + 3 ) and outgoing ( N + 3 ) waveguides through each grid point ( N × T ), and it also has a large number of bending constraints ( 5 × N × T ), routing constraints ( N × T ), and crossing constraints ( N × T ). Hence, for some larger testcases, it becomes infeasible to get the routing solutions as the ILP program requires high computational power and time. In such cases, RBL provides good and balanced routing results even compared to the state-of-the-art algorithms. So, RBL is sound and complete, i.e., guaranteed to solve the routing problems for a given valid channel definition.

Discussion
The gridless optical routing in [7] presented a list of routing geometries where the optical waveguides are bent curvily. In the proposed grid-based channel routing, the optical waveguide curvilinear nature can be satisfied in the postprocessing step, as shown in Fig. 9.

Conclusions
In this paper at first, we propose an ILP formulation for minimizing the bend loss (MBL) in the optical channel routing for photonic integrated circuits. The proposed MBL results in the minimization of total bend loss and worst signal loss while reducing the number of tracks in comparison with the state-of-art techniques. However, the execution time of MBL is very high for larger testcases. Hence, a scalable heuristic for reducing the bend loss (RBL) is proposed. The proposed RBL algorithm eliminates the smaller bending, which shows a balanced result between the bend loss and track reduction, as compared to the state-of-the-art techniques. RBL reduces the bend loss with an overhead of increased number of tracks, which, in turn, increases the waveguide length and the channel area. Data availability Enquiries about data availability should be directed to the authors.