A 2-dimensional optical architecture for solving Hamiltonianpath problem based on micro ring resonators

Nadim Shakeri a, Saeed Jalili a,n, Vahid Ahmadi a, Aref Rasoulzadeh Zali a, Sama Goliaei b

a Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iranb University of Tehran, Tehran, Iran

a r t i c l e i n f o

Article history:Received 3 February 2014Received in revised form5 June 2014Accepted 16 June 2014

Keywords:Optical computingOptoelectronic devicesNP-complete problems

a b s t r a c t

The problem of finding the Hamiltonian path in a graph, or deciding whether a graph has a Hamiltonianpath or not, is an NP-complete problem. No exact solution has been found yet, to solve this problemusing polynomial amount of time and space. In this paper, we propose a two dimensional (2-D) opticalarchitecture based on optical electronic devices such as micro ring resonators, optical circulators andMEMS based mirror (MEMS-M) to solve the Hamiltonian Path Problem, for undirected graphs in lineartime. It uses a heuristic algorithm and employs nþ1 different wavelengths of a light ray, to checkwhether a Hamiltonian path exists or not on a graph with n vertices. Then if a Hamiltonian path exists, itreports the path. The device complexity of the proposed architecture is O(n2).

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The Hamiltonian path in a graph is a path which visits eachvertex exactly once. The problem of finding a Hamiltonian path in agiven graph is an NP-complete problem, and has many real worldapplications [1–3]. NP-complete is a complexity class includingmany real-world problems, which have not been solved withpolynomial algorithms on conventional computers, yet. Heuristicand approximation methods, which do not necessarily find exactsolution, have been proposed to find an effective solution for theHamiltonian path problem (HPP) [4–8]. In addition to non-exactmethods, several exact methods have also been provided for HPP[9], but no solution with polynomial resource has been found yet.

Using new computational capabilities of optical computing, onecan solve NP-complete problems on optical computers by usingphysical property of light such as high speed, massive parallelismnature, and the ability of splitting a light ray into several rays.Recently some optical approaches have been provided to solveHPP [10–16]. Oltean [12] arranges a graph and the light rays aredelayed within the nodes of the graph using a special delayingsystem through optical fibers. At the destination node, the raywhich has visited each node exactly once is searched. In thisapproach, the length of the optical fibers, used for delaying thesignals, increases exponentially while the intensity of the signaldecreases exponentially with the number of nodes that are

traversed. Another approach is to construct optical masks in thepreprocessing phase, and applying the masks consisting of anexponential number of locations, to solve the problem in efficienttime [10]. Later, Cohen et al. [14] proposed an optical solver forcombinatorial problems such as Hamiltonian cycle based on Nano-technology and lithography methods to produce the masks inNano-scale size. They solved an instance with 15 vertices. Shakedet al. [16] proposed a method for solving a bounded instance of theTraveling Salesman Problem (TSP) and HPP with maximum size15. This method exploits fast matrix–vector multiplication basedon an optical processor. Recently, one method based on filters [13]is proposed to solve this problem by using light. In this approachfirst, space solution of the problem is generated and then invalidsolutions are eliminated by filters.

In this paper, we propose a novel 2-dimensional (2-D) opticalarchitecture based on optoelectronic devices such as micro ringresonators (MRRs), optical circulators (OCs), and MEMS basedMirrors (MEMS-Ms). Employing this architecture, one can solvethe Hamiltonian Path Problem on undirected graphs in linear time.In this architecture, different wavelengths of light are used todisplay whether a Hamiltonian path does exist or not, and declaresa Hamiltonian path, if such a path exists.

The rest of the paper is organized as follows: Section 2 presentsthe exact definition of the HPP. Sections 3 and 4 discuss and explainthe basic element that used in the architecture and the main idea ofit, respectively. Section 5 presents simulation results and Section 6discusses the complexity of the proposed architecture and comparesit with other related works. Section 7 explains the conclusion of thepaper and suggests future research directions.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/optlastec

Optics & Laser Technology

http://dx.doi.org/10.1016/j.optlastec.2014.06.0080030-3992/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: sjalili@modares.ac.ir (S. Jalili).

Optics & Laser Technology 65 (2015) 56–65

2. The Hamiltonian path problem

Given an undirected graph, G¼(V, E), with |V|¼n nodes and astart node (vstart), the problem is to compute whether there is asequence of vertices such that from each vertex there is an edge tothe next vertex in the sequence, beginning with node vstartcontaining all nodes exactly once. The output for this decisionproblem is either YES or NO depending on whether the Hamilto-nian path does exist or not. A graph may have zero, one or severalHamiltonian paths. For example, in Fig. 1(a), paths v1–v4–v2–v3,v1–v4–v3–v2 and v1–v3–v2–v4 are Hamiltonian paths. Fig. 1(b) shows a graph which has no Hamiltonian path. Finding aHamiltonian path is a decision problem in graph theory and theproblem of determining whether a Hamiltonian path exists in agiven directed/undirected graph is NP-complete [17].

3. The micro ring resonator (MRR)

In this paper, micro ring resonator (MRR) is used as a buildingblock for the proposed solution to the HPP. Fig. 2(a) and (b) showsthe schematic structure of MRR and the Z-transform model of anadd-drop ring resonator filter as the basic cell, respectively. Weinvestigate MRR as an optical add-drop filter, which consists of acircularly and two straight waveguides. The straight waveguideswhich serve as input and output pathway for the light ray arepatterned close to the micro ring. MRR gets a light ray (includingλ1⋯λi⋯λn wavelengths) as input from the input port, drops aspecific wavelength (i.e., λi) from the input ray according to theproperties of MRR to its drop port, and sends the rest of the inputray (λ1⋯λi�1� λiþ1⋯λn wavelengths) to the throughput port.

In order to evaluate throughput and drop port fields, we usethe Z-transform [19] to model the basic cell optical performance.The resultant transfer function representing the transmitted fieldin the throughput port and drop port is simplified to:

Et2Ei

¼ t�tγz�1

1�t2γz�1ð1Þ

ErEi

¼ κ2ffiffiffiffiffiffiffiffiffiffiffiγz�1

p

1�t2γz�1ð2Þ

so that:

γ ¼ e�Γα; α¼ �2πneffLRλ

ð3Þ

κ¼ffiffiffiffiffiffiffiffiffiffiffiffi1�t2

p; z¼ e� jβLR ð4Þ

where LR¼2π Reff, t, κ, neff, λ and β are resonator length, transmis-sion coefficient and coupling coefficient, effective refractive index,wavelength, and phase shift coefficient respectively. γ, Γ and α areimaginary part of refractive index of the lossy medium in the MRRwaveguide, optical mode confinement factor in lossy medium andabsorption coefficient, respectively. Fig. 3 shows the performanceof the basic cell in terms of Reff¼3 μm, neff¼3.45 and κ2¼0.05 μm(Reff is effective radius of MRR). This figure indicates drop portintensity versus throughput port intensity in range of 1.5–1.56 μmwith periodic response characteristics. The drop port responseshown by dashed line.

4. The proposed architecture for the Hamiltonian pathproblem

The main idea of our proposed solution is to consider nþ1wavelengths λ1⋯λnþ1, where wavelength λi is mapped to vertex vi(i¼1,…,n), and wavelength λnþ1 is used to representing thesolution, if any exists. We construct a lattice using waveguides,similar to the structure of the adjacent matrix of the given graph,and provide an optical structure, DMB (Decision Making Block), oneach point corresponding to 1 values in the adjacent matrix.Starting from the up right corner of the lattice where a light ray(containing all wavelengths λ1⋯λnþ1) arrives, DMBs specify thetravel direction of the light ray, and MRRs close to row and columnwaveguides drop the wavelengths in such a way that finally, a lightray reaching the bottom of matrix containing exactly one wave-length λnþ1, represents the solution to the HPP.

In Section 4.1, we design an optical structure DMB, which willbe used in the next sections to solve the HPP. In Section 4.2, weprovide details of the architecture to solve the yes/no version ofthe HPP, and in Section 4.3, we extend the architecture to find theorder of vertices in the Hamiltonian path, if any exists.

4.1. The DMB structure

We design the optical DMB for propagating and directing thelight ray in the proposed architecture. If there is a specificwavelength in the incoming light ray then, the DMB redirectsthe light ray according to one of directions shown in Fig. 4. In theother case, when a specific wavelength does not exist in the light

Fig. 1. (a) Graph G that includes some Hamiltonian paths such as v1–v4–v2–v3.(b) An example of a graph which has no Hamiltonian path.

Fig. 2. (a) The schematic structure of MRR [18]. (b) The Z-transform model of the MRR [19].

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–65 57

ray input, the DMB lets the light ray continues its path withoutredirection.

Figs. 5 and 6 show the schematic structures of a DMB in detail.A DMB consists of an optical circulator (OC), an MRR which isdescribed in Section 3, and a MEMS-M.

4.1.1. Optical circulator (OC)OC is an optical component, containing of three ports in

circular order, where the incoming light ray from each port issend to the next port. This means that if light ray enters port 1 it isemitted from port 2, but if the emitted light ray is reflected back tothe circulator it comes out of port 3 (see the top part of Figs. 5 and6). The OC just lets the light ray pass through waveguides.

4.1.2. MEMS-MMEMS-M is a microelectromechanical system mirror based

switch, which consists of a detector and a sub-microsecond MEMSmicromirror switch [20–22]. The MEMS-M switch is normally onso can be switched off with a voltage. When a specific wavelength(i.e., λi) reaches its detector, it transforms the wavelength to avoltage for excitation of the micromirror gate of MEMS switch.According to Figs. 5 and 6 MEMS-M is placed on waveguide toreflect the transmitted light ray from OC to it again or lets it tocontinue its path through that waveguide.

4.1.3. DMB functionEach DMB is placed on the cross point of waveguides repre-

senting rows and columns of adjacent matrix of a given graph. Asshown in Fig. 5 we label the related waveguides to row i andcolumn j of adjacent matrix by Ri, and CLj, respectively, and place aDMB on their cross point. In this figure the incoming light raypropagates to DMB from right side of Ri waveguide. For a specificwavelength λj, the DMB sends the incoming light ray from rightside to the left side of Ri waveguide if and only if it does notcontain λj (Fig.5(b)). The incoming light ray is sent from right sideof waveguide Ri to down side of waveguide CLj by DMB if and onlyif the light ray contains wavelength λj (Fig.5(a)). In the otherwords, DMB specifies the direction of the incoming light ray

according to existence or absence of a specific wavelength, λj, inthe incoming light ray.

According to Fig. 5(a) the MRR in the DMB structure drops apercentage of wavelength λj from the incoming light ray to thewaveguide L which is connected to drop port of the MRR andsimultaneously the light ray including the remaining wavelengthλj is send to the throughput port that is connected to port 1 of theOC. The light ray enters the OC from port 1 and exits from port2 which is connected to the waveguide CLj. At the same time, thewavelength λj in the waveguide L is transformed to a specificvoltage for driving MEMS micromirror by the detector. Therefore,the normally on mirror gate is excited by this voltage and switchedoff. In this case, the mirror is deviated from state (1) to state(2) with rotation and letting the wavelengths of the light ray,which is exited from port 2 of OC, pass through CLj waveguide.

In the other case, when λj wavelength does not exist in the lightray input, this means that the detector does not detect anywavelength so there is no excitation on the MEMS-M. In this case,the MEMS-M which is normally on prevents passing the light raythrough CLj waveguide by reflecting it to port 2 of OC (see Fig. 5(b)). Here, the reflected light ray enters to OC again from port2 and comes out from port 3 and then it continues to its paththrough Ri waveguide.

As shown in Fig. 6(a), when the light ray propagates to DMBfrom top of CLj waveguide, the normally on MEMS-M is placed onRi waveguide. In this case, the OC is placed at the correspondingcross point of Ri and CLj waveguides so that its ports 1 and 2 areconnected to CLj and Ri waveguides, respectively. In this figure theMRR is placed close to CLj waveguide. By dropping a percentage ofλi-wavelength to the MRR drop port we can check whether there isλi wavelength or not in the input light ray.

According to Fig. 6(a) if λi wavelength exists in the input lightray then, a percentage of λi propagates to L waveguide which isconnected to the drop port of the MRR so it is transformed to avoltage by the detector. Simultaneously, the remaining λi wave-length with other wavelengths of the input light ray exits from thethrough port of MRR and enters port 1 of the OC. At the same time,the light ray exits from port 2 of the OC that is connected to Riwaveguide and the mirror gate which is stimulated by drivingvoltage switched off. In this case, the mirror is deviated from state(1) to state (2) and lets the transmitted light ray from port 2 of theOC continues its path through Ri waveguide.

In the other case, Fig. 6(b), if there is no λi wavelength, thenthere is no excitation on mirror gate and it stays on and onlyreflects the light ray that comes from port 2 of the OC. Then thelight ray propagates through port 3 of the OC to CLj waveguide andcontinues its path.

4.2. The architecture, solving yes/no version of the HPP

First in a preprocessing phase we label the graph verticesaccording to Algorithm 1. Therefore in one step we find the vertexwith minimum degree in a given graph, and we number them from1 to n in counterclockwise (lines 1–4). Then in the next steps fornumbered graph G with new matrix M, we label the numberedvertices from v1 to vn (lines 5–18). Fig. 7 shows functionality ofalgorithm 1 as preprocessing phase on an undirected graph.

The optical architecture used in the proposed solution is con-structed from the adjacency matrix Mn�n of the given graph, wheremij is 1 if and only if two vertices vi and vj are connected in the graph,and is 0 otherwise. The proposed architecture consists of n wave-guides corresponding to n rows of the adjacent matrix, in addition ton waveguides corresponding to n columns of the adjacent matrix, inthe form of a lattice. If there is an edge between vi and vj vertices of agraph, then a DMB is placed on the intersection point of Ri and CLjwaveguides. In other words, the DMB structures on the lattice

1.5 1.51 1.52 1.53 1.54 1.55 1.56

0.2

0.4

0.6

0.8

1

λ (μm)

Dro

p/Tr

ough

put P

orts

Inte

nsity

Throughput PortDrop Port

Fig. 3. Drop and throughput ports intensity (Reff¼3 μm, neff¼3.45 andκ2¼0.05 μm).

Fig. 4. The redirection path of the light ray in the DMB.

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–6558

represent 1 values in the matrix M.

Algorithm 1. Numbering and labeling the graph nodes

Input: Matrix M representing the graph G1. Find_Min_Degree_Node;2. If there is several nodes with same degree as minimum

degree then3. Select one of them randomly and say it number 1;4. Else get the node with minimum degree number

1 therefore get other nodes number 2…n incounterclockwise;

// for numbered graph G with new Matrix M do the followingsteps:

5. Node with number 1 gets label v1;6. j¼1;7. For k¼1 to n�1 {8. Find_Min_Degree_Connected_Nodes (i); // finding

next nodes with minimum degree connected to numberednode i which are not labeled

9. If there are several nodes with same degree asminimum degree then

10. Select vertex with less i;// get the next label to graph nodes11. If degree of the node with minimum degree greater

than 1 then {12. If k is odd then13. Get the node with number i label v(n�[(k-1)/2]);14. Else //k is even1.5. Get the node with number i label vj;}16. Else get its next label and again find node with minimum

degree which is connected to numbered node i and repeatlines 9–16;

Fig. 5. The schematic structure of elements in the DMB. Propagation of light ray from right side of Ri waveguide. (a) The light ray including λj. (b) The light ray without λj.

Fig. 6. The schematic structure of elements in the DMB. Propagation of light ray from top of CLj waveguide. (a) The light ray including λi. (b) The light ray without λi.

Fig. 7. Example of an undirected graph. One Hamiltonian path is indicated bybold lines.

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–65 59

17. jþþ;18. kþþ;}

Fig. 8 shows the proposed architecture for the given graph inFig. 7. In this figure MRRs are shown by circles which are used tofilter the wavelengths. Waveguides are shown as dark lines. Theyare used to carry the wavelengths of a light ray. In the proposedarchitecture, the MRRs close to CLj waveguide filter λj wavelengthand the MRRs close to Ri waveguide filter λi wavelength.

As shown in Fig. 8 the light ray is launched to propagate throughthe waveguide from the right side of the architecture. According toAlgorithm 2, starting from an arbitrary vertex vi, as initial vertex inHamiltonian path on Ri waveguide, first λi wavelength is filtered byan MRR (line 1). For each next vertex vj, If vi and vj vertices aredirectly connected, then in the corresponding cross point of Ri and CLjwaveguides the light ray is redirected to waveguides by a DMB. Itchecks whether there is λj wavelength in the propagated light ray ornot. So according to the presence or absence of λj wavelength itdecides to redirect the light ray in one of the directions shown inFig. 4. After redirecting the light ray in proper direction, λj wavelengthis filtered by an MRR (lines 3–8). Finally, there is an answer when allof n wavelengths of the light ray, i.e., λ1⋯λn, are filtered except λnþ1

(lines 11–13).

Algorithm 2. Solving HPP

Inputs: λ1, …, λnþ1 // A range of wavelengths in a light rayMatrix M: after preprocessing phase which represents thelabeled HPP graph GStart node viDirection Di: The direction that a light ray is entered thearchitecture and can be D1, D2, D3 or D4 on the row(R) or

thecolumn(CL)

Output: a light ray including just λnþ1 wavelength1. filter(λi);//MRR function as a filter2. for each next vertex vj of graph G on Rj or CLj waveguide in

direction Di

// operation of a decision making block (DMB)

3 if (M(i,j)¼¼1) // vi and vj are directly connectedthen{

filter_1_percentage(λj);// filter 1% of λj wavelengthto decide about redirection of light

4. if λj is not filtered then5. Update_Direction(i,j,Direction);//the OC

operation6. filter(λj); // end of DMB operation7. vi¼vj;//update vi as next initial vertex }8. else Update_Direction(i,j,Direction);9. Endfor;10. if λ1,…, λn wavelengths are filtered and only λnþ1

wavelength is remained then11. output¼1;12. else no Hamiltonian path exists

Algorithm 3 shows the propagation process of the light ray byan OC in a DMB as Update_Direction function. For two directlyconnected vertices vi and vj, if the light ray exists in Ri waveguideand λj wavelength is not filtered then, the light ray redirects to CLjwaveguide (D1 and D4 directions (see Fig. 4)).

In other case, when vi and vj are not directly connected or λjwavelength is not filtered, the light ray continues to propagate inits path. If the light ray exists in CLj waveguide and λi wavelengthhas not already been filtered, then the light ray redirects to Riwaveguide (D2 and D3 directions, see Fig. 4). Otherwise, if vi and vjare not directly connected or λi wavelength is not filtered, the lightray continues to propagate in its path.

For example in Fig. 7, since the graph has four vertices so weneed five wavelengths, λ1⋯λ5. For more simplicity, we just showan instance path starting from vertex v1 (bold lines in Fig. 7). Thelight ray propagates into R1 waveguide related to the row of initialvertex v1. It is coupled to an MRR close to R1 waveguide and thenλ1 is dropped or filtered. The light ray (except λ1) after arriving atcross point of 1st row, R1, and 4th column, CL4, is redirected by aDMB.

Fig. 8. The proposed architecture for the given graph in Fig.7.

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–6560

Algorithm 3. Propagation of the light ray in the proposedarchitecture

Update_Direction (i, j, Direction) {1. switch (Direction)2. case ‘D1'3. if vi and vj are directly connected and λj

wavelength is notfiltered then

4. the rest of wavelengths propagate in directionD2

5. else the light continue to its path in direction D1

6. case ‘D2'7. if vi and vj are directly connected and λi

wavelength is not7. filtered then8. the rest of wavelengths propagate in

direction D1

9. else the light continues to its path in direction D2

10. case ‘D3'11. if vi and vj are directly connected and λi

wavelength is notfiltered then

12. the rest of wavelengths propagate indirection D1

13. else the light continues to its path in direction D3

14. case ‘D4'1.5. if vi and vj are directly connected and λj

wavelength is not16. Filtered then17. the rest of wavelengths propagate in

direction D2

18. else the light continues to its path in direction D4

19. endswitch; }

In DMB, first the light ray (except λ1 wavelength) is coupled toMRR and a percentage of λ4-lightwave is dropped. Then it istransformed to a voltage by detector and so the excited mirror gateswitches off and deviates from state (1) to state (2) by rotation andlets the light ray redirect to CL4 waveguide. Then, the light ray iscoupled into an MRR close to CL4 waveguide and λ4 wavelength isfiltered. In the corresponding cross point of CL4 and R2 waveguides,DMB redirects the light ray (except λ1 and λ4 wavelengths) to R2waveguide. Here, they are coupled into an MRR close to R2waveguide and λ2 wavelength is filtered. After arriving at vertexv3 in the corresponding cross point of R2 and CL3 waveguides, thelight ray is directed to CL3 waveguide by DMB. It is coupled to theMRR close to CL3 waveguide then λ3 wavelength is filtered. Since λ3and λ4 wavelengths of the light ray has already been filtered, whenthe ray of light arrives at the next cross point of R3, CL3 and CL3, R4waveguides respectively, the DMB lets the light ray continues to itspath. Finally, the light ray including just λ5 wavelength comes out ofthe waveguide related to vertex v3, CL3. Thus there is an answer forthe HPP YES/NO problem.

4.3. The architecture, finding the vertices order in the Hamiltonianpath

Due to the complexity of the system and using several identicalrings, we have to modify the architecture to identify the wave-lengths which have been filtered and determine the order of graphvertices in the Hamiltonian path. Functionality of this method isbased on principle of binary values of numbers. When λi wave-length is dropped or filtered to a waveguide, according to binary

value of i index, one can identify the wavelength. Therefore we usethe drop port of an MRR for this purpose.

Fig. 9 shows the truth table of this concept for binary values ofthe wavelengths that dropped into the row waveguides BitR2–BitR0. For example, if λ3 wavelength is filtered in the row wave-guide then the corresponding BitR0 and BitR1 waveguides will beequal to 1 and output of the BitR2 waveguide will be 0. In otherword, values of BitR2–BitR0 are binary value of number 3 (i.e.,BitR2BitR1BitR0¼011). Also BitCL2–BitCL0 are used to indicate whichwavelength dropped to column waveguides.

The developed architecture is shown in Fig. 10. For finding thepath illustrated in Fig. 7, v1–v4–v2–v3, the horizontal and verticalwaveguides indicated by BitR2–BitR0 and BitCL2–BitCL0, respectivelyare used. The HPP graph used in this article (see Fig. 7) requiresthree bits to determine the wavelengths. These waveguides areapplied to indicate which wavelengths are filtered. Employingthem, one can determine the order of filtering of the wavelengths,or the order of vertices of the graph in a Hamiltonian path.

5. Simulation results

To design our 2-D architecture based on Si-MRRs we have usedthe operation principle of MRR in the resonance by

mλ¼ nrLr ð5Þwhere m is an integer and, λ, nr, Lr are the input wavelength, thereal part of micro ring refractive index and the resonator length.The resonance frequency of the resonator is shifted by any smallchange in the optical length (nrLr). This change can be achieved byLr variation. Lr is defined by

Lr ¼ 2πR ð6ÞR is the radius of an MRR. Therefore, for filtering or droppingvarious wavelengths we require multiple MRRs with different radiior an MRR with different refractive index (nr).

Fig. 11 indicates the effect of the ring radius variation on theintensity of drop port. As shown in this figure, the resonancefrequency of the resonator is shifted by any small change (0.05 μm)in the radius of MRRs. This small change causes different wave-lengths to be dropped or filtered by MRRs.

To investigate the effect of MRR radius variation through thefabrication process which leads to changes in the resonancewavelength, it should be noticed that for solving HPP we mustconsider these constraints to our system response. Therefore theeffect of radius tolerance on the resonance wavelength has beenconsidered in Fig. 12(a) and (b).

As shown in Fig. 12(a) and (b), by changing the MRR radius withΔR¼þ1 nm and ΔR¼�1 nm the resonance wavelength changesfrom λ1¼1.513 μm to λ1¼1.512 μm and from λ4¼1.503 μm to

Fig. 9. Binary values of lines BitR2, BitR1, and BitR0 for representing wavelengths.

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–65 61

λ4¼1.502 μm, respectively. So by considering technological restric-tion we can find minimum tolerance of ΔR, so we can design theoptical system with a considered real condition.

In Fig. 13 the drop port intensity is depicted versus a selectedrange of the wavelengths of the light ray for different values ofcoupling coefficient. As shown in this figure, the higher couplingcoefficient (κ) causes the wider FWHM (Full Width at Half Max-imum) of the MRR response. FWHM of the MRR response can beadjusted by coupling coefficient and absorption coefficient [23].Moreover, increasing the absorption coefficient (α) of the MRRwaveguide leads to a broadened FWHM of response [24]. To avoidoverlapping of the wavelengths that dropped by different MRRs inthe architecture we have to use the drop port response withsmaller FWHM.

To validate the 2-D proposed optical architecture we used thedesigned Si-MRRs to solve the Hamiltonian path. Here MRRs withdifferent radii have been used to drop (filter) different wavelengths.

MRRs close to Ri waveguides have the same radius and alsoMRRs close to CLj waveguides have the same radius in the 2-D

architecture. The light ray can enter waveguides of the architectureshown in Fig. 9 from the right side (direction D1 in Fig. 8). For theHamiltonian path of v1–v4–v2–v3 described earlier, a light rayentered R1 waveguide from the right side of it. λ1 wavelength(1.512 μm) using an MRR (with Reff¼3 μm) close to R1 waveguide isdropped and exit from BitR0 waveguide. Fig. 14 shows the output ofBitR0 waveguide. The output of other row and column waveguides(i.e., BitR1, BitR2, BitCL0, and BitCL2) is zero.

The light ray arrives at the corresponding cross point of R1 and CL4waveguides. Since v1 and v4 vertices are directly connected, the lightray is redirected from R1 waveguide to CL4 waveguide by DMB. Hereby an MRR with 3.05 μm radius, λ4 wavelength (1.503 μm) is filteredand exits from BitCL2 waveguide as shown in Fig. 15.

The output wavelength of BitCL2 indicates that the next traversedvertex in the Hamiltonian path is v4. At the cross point of CL4 and R2waveguides one DMB redirects the light ray (except λ1, λ4 wave-lengths) to R2 waveguide. Then they are coupled to the MRR with3.1 μm radius close to R2 waveguide and then λ2 wavelength(1.527 μm) is dropped to the horizontal waveguide of BitR1. The outputof other horizontal and vertical waveguides is zero. Fig. 16 shows theoutput of this waveguide. So v2 is the next vertex in the path. The lightray (except λ1, λ2, λ4 wavelengths) after redirection from R2 waveguideto CL3 waveguide by using a DMB is coupled to an MRR with 3.15 μmradius and λ3 wavelength (1.517 μm) is filtered.

Fig. 17 indicates the output of BitCL0 and BitCL1 waveguides.Output of other waveguides is zero and the next vertex is v3.Therefore, the Hamiltonian path is: v1–v4–v2–v3.

6. Discussion

In the provided 2-D optical architecture, for an undirectedgraph G¼(V, E) with n vertices, nþ1 distinct wavelengths are usedin the light ray that entered the proposed architecture. First in thepreprocessing phase the vertices of the given graph are labeled byAlgorithm 1, and then, the labeled graph is used as input HPPgraph for 2-D architecture.

Fig. 10. The developed architecture to find the Hamiltonian path using light waveguides BitR2, BitR1, BitR0 and BitCL2, BitCL1, BitCL0.

1.5 1.51 1.52 1.530

0.2

0.4

0.6

0.8

11

λ (μm)

Dro

p P

ort I

nten

sity

R=3 μm R=3.05 μm R=3.1 μm R=3.15 μm

Fig. 11. The effect of the ring radius variation on the drop port intensity.

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–6562

According to Algorithm 2, solving the HPP requires O(n) time(where n is number of the graph vertices), and the light propaga-tion in the architecture can be done in O(1) time (according toAlgorithm 3). Therefore our optical architecture takes a linear timeof O(n), to solve each HPP instance in undirected graph. Note that

the provided optical architecture solves the HPP in a heuristicmanner.

In our proposed architecture, according to Fig. 8, we require O(n2)MRR components (to filter the wavelengths in a light ray) and DMB (to

Fig. 12. The effect of radius variation on the resonance wavelength (a) ΔR¼þ1 nm (b) ΔR¼�1 nm.

Fig. 13. Drop port intensity spectrum for different values of coupling coefficientswith Reff¼3 μm and neff¼3.45.

Fig. 14. The output of BitR0 waveguide.

Fig. 15. The output of BitCL2 waveguide.

Fig. 16. The output of BitR1 waveguide.

N. Shakeri et al. / Optics & Laser Technology 65 (2015) 56–65 63

traverse each vertex exactly once). On the other hand, the devicecomplexity for each HPP graph with n vertices is O(n2). The number ofdifferent wavelengths in a light ray is linear according to n, where n isthe number of vertices in each HPP instance.

The architecture is based on MRR, which is made of silicon;therefore we use wavelengths in range of 1550–1556 nm. We usethe operation principle of MRR in the resonance condition and apply asmall change in the optical length of the MRR to filter differentwavelengths. This change is achieved by small variation in MRR radius.Recall that a tolerance in the MRR radius can lead to change in theresonance wavelength so we should notice that for solving HPP theseconstraints should be considered in our system response. Also weapply the absorbing properties of the MRR to drop a percentage of awavelength to decide about redirecting the light ray in the DBMstructure. The wavelength drop is obtained by increasing absorptioncoefficient, α, in resonance condition.

In MRR structure, the drop field intensity bandwidth deter-mines how many wavelengths we can use and drop in ourarchitecture. Because, the larger the coupling coefficient andMRR waveguide loss will result in the larger FWHM (optical 3 dBbandwidth) of the drop field resonance. So the FWHM of the dropfield resonance, and consequently the coupling coefficient, MRRloss condition, and ring resonator radius, determine the channel'scross talk condition. If we design a structure that these coefficientsare selected properly, we can design such filters that have no crosstalk with each other. Therefore, a HPP graph with maximum vertexthat can be solved by this architecture depends on the designcondition such as absorption coefficient (α), coupling coefficient(κ) and the range of selected wavelengths.

As we mentioned earlier, the higher the coupling coefficient causesthewider FWHM of the MRR response. For a given radius of MRR (Reff)coupling coefficient can be adjusted by the gap distance betweenstraight and MRR waveguides. Moreover increasing the absorptioncoefficient (α) of the MRRwaveguide leads to the broadened FWHM ofthe response. So by choosing proper parameters (κ, α) for a given Reff,one can estimate the maximum vertexes that the problem can besolved. We can solve an instance of HPP with 15 vertices byparameters κ2¼0.05, α¼4�10�4 and radius variation of 0.05 μm.

In the DMB structure we use a MEMS-M switch which consistsof a detector beside a MEMS micromirror. As we mentioned earlierthe detector transforms a wavelength to a desired voltage forexciting micromirror gate in MEMS-M switch. This detectoraccording to recent technology [24,25] is sufficiently fast to detecta wavelength. Typically, the switching time of the MEMS micro-mirror switch is in sub-microsecond range [20,21,22] and the MRRresponse is in nano range. Since we know the MEMS micromirror

is a microelectromechanical device so the only restricted compo-nent in our proposed architecture is MEMS-M switch. It should benoted that we just use the MEMS-M switch as an optical switch inDMB structure while we can apply any optical switch for thispurpose. In this case, using a MEMS-M, the time synchronicity ofthe proposed architecture depends on its operation time. As wemention earlier, the switching time of this switch is in sub-microsecond so we should adjust the running process in thisrange. As a result the proposed optical architecture is able worktruly in terms of time synchronicity.

The advantage of the proposed solution in this paper in compar-ison to the optical approaches which use optical fibers to make delaysin the light motion is that in the delay-based approaches, exponentiallength of optical fiber is required, but we use polynomial length ofwaveguides. For example, the delay-based approach requires approxi-mately 300 km of fiber to solve the HPP over a given graph with 17vertices [12]. Also in the approach which is used by Dolev et al. [10], a3-dimensional model that used themasks consisting of an exponentialnumber of locations according to the input size, while the number ofoptical components in our proposed solution is polynomial. Recently,one approach based on light filters has been proposed to solve theHamiltonian path problem [13]. In this approach first, space solution ofthe problem is generated and then invalid solutions are eliminatedfrom it. The area of each filter is divided into exponential cells (2n� log(n)) that represent potential solutions to the HPP. The resourcecomplexity of these approaches [10,12,13] is yet exponential.

7. Conclusion

In this paper, we have proposed a new optical architecture tosolve the Hamiltonian path problem (HPP). HPP is an NP-completeproblem which no polynomial time solution for it on the conven-tional computers has been reported yet.

In the proposed structure, waveguides are used to construct alattice corresponding to the adjacent matrix of the given graph.We have used optoelectronic devices such as micro ring resonators(MRRs), optical circulators (OCs) and MEMS based mirrors (MEMS-Ms), which consist a DMB structure, in the cross point correspond-ing to edges of the given graph, to specify the output direction ofthe incoming light ray, in such a way that reaching a specificwavelength to the end of the lattice, represents the Hamiltonianpath of the given graph. Here, we have simulated the proposedoptical architecture for a given 4-vertix graph, and we have shownthat the proposed architecture requires O(n2) optical components.

The future work directions will be focused on design anoptimized MEMS-M switch and design a suitable optical switchin the DMB. The next work will be optimization of the proposedarchitecture so that we able to find all possible responses andsolve a Traveling Salesman Problem (TSP). Also we will try togeneralize the proposed optical architecture to solve other hardcombinatorial problems.

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Fig. 17 The output of BitCL0 and BitCL1 waveguides.

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