text: [11]. We use a non-cooperative game-theoretic framework to dispatch and generations, and physical constraints, e.g., power study the strategic behavior of microgrids in the power grid ﬂow equations [14], [22], are not part of their model. Our work from a cross-layer perspective. Thus, microgrids are players is focused on the interactions in the lower layer, and it can be in this renewable energy game. Different from the centralized naturally extended to the two-level framework by regarding control of power systems which is based on the optimal power the generators as active players. ﬂow, we adopt Nash equilibrium as our solution concept B. Organization of the Paper to characterize the control actions of microgrids. To design The rest of the paper is organized as follows. In Section , we an automated energy management system, we develop two formulate the problem and present a game-theoretic framework update schemes including the iterative update algorithm and to model the generation control in microgrids. We analyze the random update algorithm which enable the independent deci- game in Section and propose three update schemes to ﬁnd the sion making of microgrids. However, they need the support equilibrium solution in Section . Implementation framework of from the communication network as depicted in Fig. 1 for the PMU-enabled distributed algorithm is developed in Section the information exchange. To make the control system more . Case studies are given in Section , and Section concludes intelligent and efﬁcient, we also propose a PMU-enabled the paper. distributed algorithm that only depends on the smart device, C. Notations and Conventions phasor measurement unit (PMU) [12], to measure the voltage Notations and conventions adopted in this paper are sum- angles at the buses. By using this scheme, each microgrid does (cid:48) ∈ R marized as follows: P := [P , P , ..., P ] N, θ := npraolnatydneoermse,dlyntoourpkdtnhaoteewpitotshwesetrsraptseeucgpiyﬁpcloienadlmyobbuyansttehdoefoggneenntehereraatptoihorsan,soobrfuatontchgaelner [[pθPe11rg,s,θcP2ri,2pg.,t.s..,.o.θ,NaPn]Ng(cid:48)dd∈∗](cid:48)Ri∈nNdiR;cPaNtdde;t:h=qe:[p=Pa11r,a[qPm122,e,qte.2.r.,,.aP.c.,NNhqideN]v(cid:48)di]∈n(cid:48)g∈RthNRedN,oPpd;tgdimS:u=a-l at its bus, which preserves high privacy. and equilibrium, respectively; Use of (n) in the superscript The distributed iterative algorithm developed in this work indicates the nth iteration. In addition, microgrid and player enhances the resilience of the grid by enabling the real-time refer to the same entity, and they are used interchangeably. response to system changes. Power systems tend to fail, such II. PROBLEM FORMULATION as generator breakdown and trigger of the power transmission lines, due to natural disasters, man-made mistakes and cyber In this section, we ﬁrst review some power ﬂow basics, and attacks [13], [14], [15]. Therefore, by investigating the two- then formulate the problem. A. Power Flow Preliminaries layer smart grid framework under these faulty scenarios, N { } we can assess the resiliency of the proposed PMU-enabled In a power grid, let := r, 1, 2, ..., N be a set of algorithm. N + 1 buses, where r denotes the slack bus. Denote P , Q , i i The contributions of this paper are summarized as follow. V and θ as the amount of active power injection, reactive i i 1) We establish a non-cooperative game-theoretic frame- power injection, voltage magnitude and voltage angle at bus work to model the renewable energy generation planning i, respectively. Then, the power ﬂow equations of the system of microgrids in the smart grid, especially by considering with refer(cid:88)ence to the slack bus r are the physical and economic constraints. 2) We propose two iterative and random update algorithms (cid:88) − − P = V V [G cos(θ θ ) + B sin(θ θ )], (1) for the decision making of microgrids and derive sufﬁ- i i j ij i j ij i j ∈N cient conditions to ensure their convergence to a unique j − − − Nash equilibrium. Q = V V [G sin(θ θ ) B cos(θ θ )], (2) i i j ij i j ij i j 3) We develop a resilient and fully distributed PMU- ∈N j enabled update algorithm for microgrids, and design its implementation control framework. Moreover, two for i, j = 1, 2, ..., N, where Gij and Bij are the real and major fault models of the smart grid are introduced to ima∈giCnary× part of the element (i, j) in the admittance matrix validate the resiliency of the distributed algorithm. Y N N of the power grid. Note that Vr and θr are both known, and speciﬁcally, θ = 0. r A. Related Work Let Pg and Pl be the power generation and power load at i i bus i, respectively. Then, the active power injection at bus i Game-theoretic methods have been used for generation satisﬁes planning and control of the distributed energy resources in smart grid [11], [16], [17], [18]. Zhu et al. [9] have proposed a − ∀ ∈ N P = Pg Pl, i . (3) non-cooperative game framework for distributed generation in i i i (cid:80) (cid:80) power systems, and considered economic and AC power ﬂow constraints. Our work extends their model and algorithms to a Moreover, by considering the balance of the grid, we have distributed and resilient fashion. A two-level game-theoretic ∈N Pg = ∈N Pl. i i i i approach has been used to model the demand response in DC approximation is usually used for fast calculations of the the cross-layer smart grid framework shown in Fig. 1 which power ﬂow [14]. Assume that the reactance is much smaller includes both user and utility side [19], [20], [21]. They also than the resistance on transmission lines; the voltage angles ∈ N ∈ N developed distributed algorithms to ﬁnd the equilibrium solu- θ , i , are small, and the voltages V , i , are equal i ∈ N i − ≈ − tion. However, their main focuses have been on the economic to 1. Then, Q = 0, i , and sin(θ θ ) θ θ , i i j i j

text: We

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text: (cid:80) − ≈ {PG } cos(θ θ ) 1. Therefore, power ﬂow equations can be by constraint (4). For convenience, denote − , Θ− by the i j − i i represented by a set of linear equations as P = (cid:54) B (θ Cartesian product of all players’ action sets except i’th one. In ∀ ∈ N i j=i ij i PG θ ), i, j , which can be written in a matrix form as addition, denote by the feasible set of power generation of j (cid:0) (cid:1) P all buses in the grid, which can be obtained by using through − P = Bθ. (4) (3). Then, the feasible renewable generation set of game G can PG ⊗ PG ∩PG be deﬁned by := ∈N . In addition, based on PG F i d i the set , we can obtain the feasible voltage angle proﬁle of Remark 1: Matrix B includes the imaginary components F − all players Θ through (4). Thus, the feasible action set of all of Y except the slack bus’s row and column. Since B is a { F } F PG × players Pg, θ ∈N can be deﬁned as := Θ . Note symmetric reduced Laplacian matrix, then, B is invertible by i i i d F F that game G is coupled through the power ﬂow equations. the Kirchhoff’s matrix-tree theorem [23]. PG × → R The cost function U : [0, π] for player i is given B. Game-Theoretic Framework i i by Consider the smart grid is composed of load buses and N { } ⊆ N generator buses. We denote := 1, 2, ..., N as the stheet owfinNddabnudsessoltahratpocwanerg. eTnheerdareteforreen,ebwuasbesleinendtehregiseest sNuchaares Ui(Pig, θi) = ψiPig + ζ(Pil − Pig) + 12ηi2θi2, i ∈ Nd, (5) d able to generate power for self-efﬁciency. In addition, denote N { } N \N := N +1, N +2, ..., N = as a set of N buses where ψ is the unit cost of generated power for player i, ζ g d d d N g i that are either PQ-loads or generators, where set excludes is the unit price of renewable energy for sale deﬁned by the ∈ N the slack bus r for notation clarity. For load bus i , it power market, and η is a weighting parameter that indicates ∈ N i has speciﬁed load Pl, and for generator bus j , it has the importance of regulations of voltage angle at bus i. Note i predetermined generation Pg. that in (5), player i prefers a small voltage angle η which j N i Furthermore, for buses in the set of , they are connected satisﬁes a condition of using DC approximated power ﬂow. d with the microgrids that are able to generate renewable ener- Then, the optimization problem for player i is gies. Their loads to be serviced are speciﬁed ahead of time, and their generations of renewable energy need to be determined. OP : min U (Pg, θ ) Note that each microgrid has a maximum generation, i.e., i Pig,θi i −i i 0 ≤ Pg ≤ Pg , i ∈ N . Pg depends on the available s.t. P = Bθ, i i,max d i,max ≤ ≤ ∈ N stored power of the microgrid, and it is dynamically changing 0 Pg Pg , i . i i,max d due to the intermittent nature of renewable generations. For ctshtoainstevp-eoanfpi-eecnrh,caear,ngdethieotfcfrasatnomrbeaewgegoerdkneevirsiaclseiztsueddisietocdodfnoysrnidaaemrgeicidv.ePIning,mPaaigdx,mdiiaftxiothinne, N[PaTgs∗hh]ee∈qrNeunile,iwbθrai∗bulme=g(eN[θnE∗e)]ra∈stoiNolun,tigoiasnmapeapGior i(insPtagdw∗c,hoθenr∗dset)r,anwionhepedlraegyaPemrgde∗c.a=An N i i d d i i d for buses in , we set the power generation of PQ-load bus beneﬁt from deviating from it through changing his strategy. g to 0, and the power load of generator bus to 0 without loss of The formal deﬁnition of NE of game G is as follows. generality,. Deﬁnition 1 (Nash Equilibrium of Game G with Coupled ∗ ∗ Before formulating the game-theoretic framework for this Constraints): The solution pair (Pg , θ ) constitutes a Nash d d power generation game, we make several assumptions as equilibrium point for the microgrids in the non-cooperative ∀ ∈ N follows. First, the topology of the whole power system is game G if, for i , d known to all microgrids. This is justiﬁable since the parameters ∗ ∗ ≤ ∀ ∈ ∗ ∗ of power transmission lines, such as resistance and reactance, Ui(Pig , θi ) Ui(Pig, θi), (Pig, θi) Φi(P−gi, θ−i), are often known. Second, the constraints of each microgrid ∗ ∗ where Φ (P−g , θ− ) is a projected constraint set deﬁned by are common information, and each microgrid is aware of ∗ i∗ i {i ∗ ∗ ∈ F} Φ (P−g , θ− ) := (Pg, θ ) : (Pg, θ ; P−g , θ− ) . the physical constraints when making decisions [24], [25]. i i i i i i i i i C. Team Problem This indicates that every microgrid should take the power ﬂow constraints (4) into account. Third, power generations For comparison, we formulate a team problem (TP) that (cid:88) are given for all generators, and microgrids take actions by captures the global optimality of the smart grid as responding to the generator network. This is reasonable since microgrids can regulate itself more quickly than the generators, TP : min α U (Pg, θ ) and they can be viewed as followers who respond to the Pig,θi i∈Nd−i i i i generators [9], [19], [20]. We also assume that PMU can s.t. P = Bθ, bmeicermogprliodysewd(cid:8)htiochmiesaaslurreeadtyheavmolattaugree atencghlneoalot(cid:9)gtyheinbuthsefosmr aarllt (cid:80) 0 ≤ Pig ≤ Pig,max, i ∈ Nd, ∈ ∈ N grid [12]. where α (0, 1), i , are the weights on microgrids N {PG } { } P i d Let G := dN, i , Θi i∈Nd, U{iPiG∈Nd,} be a strategic such that Ni=d1 αi = 1. Note that the weighting constant αi game with a set of N players. , Θ is the action set indicates the importance of microgrid i among all players, and of player i, wheredPG :=d {Pg ∈ R i| 0 i≤ Pg ≤ Pg }, it is chosen or assigned by a system coordinator in the smart P i i + i i,max and is the feasible set of active power injection deﬁned grid. ##LTLine##

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text: ##LTLine##


text: Renewable energy generations vs. Number of iterations under TDA Renewable energy generations vs. Number of iterations under TDA Renewable energy generations vs. Number of iterations under TDA 80 70 80 Microgrid 1 Microgrid 1 Microgrid 1 70 MMiiccrrooggrriidd 23 60 MMiiccrrooggrriidd 23 70 MMiiccrrooggrriidd 23 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 Iteration Iteration Iteration (a) (b) (c) Fig. 6. All three faults occur at time step 19. (a), (b) and (c) show the renewable energy generations of three microgrids related to the generator G3 turn-off, microgrid 3 shut down, and open-circuit of the transmission line 8-14 in the smart grid, respectively. − while M = M indicates that all players generate power at (1) g¯−(n) s Pl < γ < g¯−(n) + s Pmax: 1 2 i ii i i i ii i the equilibrium. In addition, we delete the rows and columns (cid:0) (cid:1) (cid:0) (cid:1) − ∗ corresponding to the players with the boundary power gener- ∆P(n+1) = P(n+1) P i i i ation in (9) which yields 1 (cid:88)− (cid:0)− 1 −(cid:1) ∗  1 s12 s13 · · · s1M1   P∗  = sii γi g¯−(ni) sii γi g¯−i  ss22...12 s11...1 sss122...132 ·.·..· sss2M12...121   P...12∗  = q†M1 == −s1ii1j(cid:54)=i∈(cid:88)Nd sijs P∆j∗P−(nP).j(n) sM11 sM12 · · · sM1(M1−1) 1 P∗ sii j(cid:54)=i∈Nd ij j sM1M1 sM1M1 sM1M1 M1 ≤ − ⇔(cid:0) HM1P∗d,M1 = q†M1, (cid:1)(18) (2) γi g¯−(ni) sii−Pil:(cid:0) − ∗ (cid:1) (cid:0) (cid:1) where q† := [q†, q†, ..., q† ](cid:48), q† := q − ∆Pi(n+1) = Pil Pi (negative) iatsh1ni∈iadt s{tthi1h(uM,es21,bt+Mh.o.1e.u1),nPMsdoMmal1u1ra}y+xt.io1Isn+toldPusi∗dtiri,(eoMM1cnt11l+yi2es2xf)oiaPslntMlmso1weaMa+xqns1u2dit+lhiisba.tr.ui.uHni+miMqsu1iieMs.is2aTPsihnMmevfo2aerxlreltioaibswfoloesnr., >= −s1iis1iiγi(cid:54) −(cid:88)∈g¯N−(ni)sij−∆Ps1iji(n)γ.i − g¯−∗ i The payoff function (7) for player i is quadratic and convex. j=i d ≤ − ≥ ≥ Fploar∗yethrei’scaospetsimγai∗l respg¯−onise wsiiithPilreaspnedctγtio P−gg¯i−iis +achsiieiPveimdaaxt, (3) γi g¯−(ni) + siiPim(cid:0)ax:− ∗ (cid:1) (cid:0) (cid:1) Pg = 0 and Pg = Pmax, respectively. ∆P(n+1) = Pmax P (positive) i i i i i i poNsseibxlte, wcaesearigsutehatthaptlatyheer bio, uin∈da{r1y, s2o,l.u..t,iMon1i}s, uanchiqieuvee.sOthnee < s1 γi −(cid:88)g¯−(ni) − s1 γi − g¯−∗ i boundary power at the equilibrium. This case is ruled out when ii ii w{Me 1co+ns1tr,uMct1(1+8)2.,A..n.,oNthde}r,phoasssibthlee ciansneerisptohwateprlagyeenrerja,tijon∈. = −s1ii j(cid:54)=i∈Nd sij∆Pj(n). TachhiisevceananobtetbteerapNayaosfhf eaqt uthileibbriouumndeairtyh.erO, tshienrcepopslsaiybelre jcacsaens To sum up, (cid:80) can be a combination of these two cases and can be easily < sij |∆P(n)|, if γ ≤ g¯−(n) − s Pl, eliminated through similar analysis. Therefore, the boundar(cid:4)y | |  j(cid:54)=(cid:80)i∈Nd sii j ≥i i ii i Nash equilibrium solution is unique. ∆P(n+1) or γ g¯−(n) + s Pmax, i | | i i ii i = sij ∆P(n) , otherwise. j(cid:54)=i∈Nd sii j APPENDIX C When the generated renewable energy of an arbitrary player PROOF OF THEOREM 2 i at the Nash equilibrium point is 0, i.e., P∗ = −Pl, then i i through similar analysis, we have − ∗ ∈ N (cid:80) PPgerimnoaeoxrfa. te>FdirsrPte,in∗dewe>ﬁanbel−e∆PeiPnl,ei(rng∀)yi :e=∈xcPeNpi(tnd,)ithanpdPliagy,ievrie’ns aatlltdimp.leaWysehtreesnp’ |∆Pi(n+1)|  =< 0, sij |∆P(n)|,if γi ≤ g¯−oit−hesriwiPisile,. n, we have the following: j(cid:54)=i∈Nd sii j )WM( noitareneg rewoP )WM( noitareneg rewoP )WM( noitareneg rewoP ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 5 and the fact that sysmetab is a command-line tool makes TABLE 1 scripting and batch processing straightforward. sysmetab direct vs. adjoint average computation times and ratios for the reduced and full E. Coli network with stationary data Using the adjoint approach is not the only innovative feature of the software. Another novel aspect is the tech- Direct Adjoint Ratio nique chosen for generating the code: starting from the original FML ﬁle, the Scilab program is entirely generated Reduced network using XSL transformations (http://www.w3.org/TR/xslt). State computation (4) 1.5 1.5 1 Cascade (5) vs. (7) 6 0.35 17.5 These transformations are speciﬁed in XSL stylesheets, writ- Gradient assembly (6) vs. (8) 0.9 0.05 17.2 ten in another XML dialect. XSL is very different from other Total 8.4 1.9 4.47 programming languages in use today in that it is a declar- ative (as opposed to imperative) language. XSL stylesheets Full network State computation (4) 3.5 3.5 1 have the advantage of being explicit, readable by humans Cascade (5) vs. (7) 47.2 1.5 31 and easy to debug and maintain. The program is generated Gradient assembly (6) vs. (8) 4.2 0.2 19 in several steps, the ﬁnal step being the transformation of Total 55 5.2 10.6 the XML program description into the actual Scilab pro- − Time unit=10 3 s. gram. This ﬁnal step may easily be adapted for other script languages such as Matlab (http:/www.mathworks.com) or TABLE 2 Julia (http://julialang.org). sysmetab direct vs. adjoint average computation times and ratios for The generated program makes full use of Scilab’s poten- the reduced and full E. Coli network with nonstationary data tial for improving the speed of calculations: vectorization and sparse matrices are used, and UMFPACK multi-frontal Direct Adjoint Ratio LU factorization [14] is used for calculating the state and Reduced network adjoint state. The optimization phase is performed using State computation (15) 21 21 1 the Feasible Sequential Quadratic Programming (FSQP) al- Cascade (16) vs. (19) 948 15 60 gorithm [15], which is available as a Scilab module. Lin- Gradient assembly (17) vs. (20) 11 16 0.70 earized statistics or Monte-Carlo simulations can be used to Total 980 53 18.4 determine conﬁdence intervals on estimated ﬂuxes. Where Full network Monte-Carlo simulations are used, Scilab allows for parallel State computation (15) 87 87 1 execution of the optimization algorithm in a multi-core Cascade (16) vs. (19) 11214 178 63 Gradient assembly (17) vs. (20) 118 90 1.3 architecture. Total 11420 355 32 6 NUMERICAL RESULTS − As an example, we considered the central metabolism of Time unit=10 3 s. E. Coli, described in [16] and provided in the network exam- ples in the influx_s distribution (the ﬁle e_coli.ftbl). from the reduced network (using the backwards tracing The experimental stationary data consist of mass spectrom- method [17]) of 1069 cumomers are contrasted with those etry measurements of the intermediate metabolites Suc, obtained from the full original network of 5455 cumomers. ICit, PEP, PGA, FruBP, Glc6P, Fru6P, Rib5P, Gnt6P, and of Since there are almost ﬁve times as many cumomers in the extracellular ﬂux of Acetate. For the adjoint vs. direct the full network as in the reduced network, the time needed comparison in the stationary case, the original data were to compute the state derivative rises. In the direct approach, used. For the nonstationary case, synthetic noisy data were the contribution of this step to the overall time increases the generated by a forward nonstationary simulation using ratio between the two approaches to approximately 10. In the estimated values of ﬂux from the stationary data and the adjoint approach, the size of the state derivative com- some realistic pool sizes (the ﬁle e_coli_ns.fml in the putation is in relation relation to overall time is smaller. The sysmetab distribution). distribution of overall time reported in Table 1 clearly shows The results presented in this section were obtained on a the efﬁciency of the adjoint computation in comparison with dedicated Linux server hosting two ten-core Xeon E5-2660 the direct computation. v2 (2.20 GHz) processors with Scilab 5.5.2. Unless explicitly 6.1.2 Nonstationary data stated otherwise, computations used only one core of the Using the same network, we obtained estimated ﬂux processor, and every reported running time is the median values from stationary data (given in the next section) and over ten runs. Although the running times themselves may combined them with arbitrary pool size values comprising be very different if different hardware is used, we assume pool size measurements available in [16]. The resulting that the ratios between the different running times will values were used to simulate the time course of labeling remain similar. states. We obtained the data by running the implicit ODE 6.1 Adjoint versus direct gradient computation scheme (15) up to a ﬁxed terminal time T, and selecting the 6.1.1 Stationary data simulated measurements at time values τ , j = 1, . . . , M. j Table 1 compares the average computation times from Gaussian noise was then added by considering the standard steps (5)-(6) in the direct approach with steps (7)-(8) in the deviation values of each of the original stationary labeling adjoint approach. 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text: 16 waveguide modes. [43] This transition is complicated by the presence of an (anti)-crossing between the two guided modes at ≈ ka/2π 0.476, before they come back together for a degeneracy at ka/2π = 0.5. It is also made more complicated by the fact that at the bandedge itself any linear combination of modes is itself a mode. The modes found by the eigensolver were not in fact a time-reversal pair. However we found such a pair by taking linear complex combinations of the eignesolver’s solutions. ∗ † Contact me at: bl9453@bristol.ac.uk [1] M. Arcari, I. So¨llner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl. Near-unity coupling eﬃciency of a quantum emitter to a photonic crystal waveguide. Phys. Rev. Lett., 113:093603, Aug 2014. [2] Toshihiko Baba. Slow light in photonic crystals. Nature Photonics, 2:465 – 473, Aug 2008. [3] T F Krauss. Slow light in photonic crystal waveguides. Journal of Physics D: Applied Physics, 40(9):2666, 2007. [4] J. F. Nye. Lines of circular polarization in electromagnetic wave ﬁelds. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 389(1797):279–290, 1983. [5] M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers. Observation of polarization singularities at the nanoscale. Phys. Rev. Lett., 102:033902, Jan 2009. [6] Michael Berry. Making waves in physics. Nature, 403:21, Jan 2000. [7] J. F. Nye and M. V. Berry. Dislocations in wave trains. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 336(1605):165–190, 1974. [8] Konstantin Y. Bliokh and Franco Nori. Transverse and longitudinal angular momenta of light. Physics Reports, 592:1 – 38, 2015. [9] Jeﬀrey F. Wheeldon and Henry Schriemer. Wyckoﬀ positions and the expression of polarization sin- gularities in photonic crystals. Opt. Express, 17(4):2111–2121, Feb 2009. [10] Konstantin Y. Bliokh, Daria Smirnova, and Franco Nori. Quantum spin hall eﬀect of light. Science, 348(6242):1448–1451, 2015. [11] A. B. Young, A. C. T. Thijssen, D. M. Beggs, P. Androvitsaneas, L. Kuipers, J. G. Rarity, S. Hughes, and R. Oulton. Polarization engineering in photonic crystal waveguides for spin-photon entanglers. Phys. Rev. Lett., 115:153901, Oct 2015. [12] Carlos Gonzalez-Ballestero, Alejandro Gonzalez-Tudela, Francisco J. Garcia-Vidal, and Esteban Moreno. Chiral route to spontaneous entanglement generation. Phys. Rev. B, 92:155304, Oct 2015. [13] R. J. Coles, D. M. Price, B. Royall J. E. Dixon, E. Clarke, A. M. Fox, P. Kok, M. S. Skolnick, and M. N. Makhonin. Chirality of nanophotonic waveguide with embedded quantum emitter for unidirectional spin transfer. arXiv:1506.02266, June 2015. ##LTLine## ##LTLine## ##LTLine##

text: waveguide

label: reference

text: modes.

text: This

text: transition

text: is

text: complicated

text: by

text: the

text: presence

text: of

text: an

text: (anti)-crossing

text: between

text: the

text: two

text: guided

text: modes

text: at

text: before

text: they

text: come

text: back

text: together

text: for

text: a

text: degeneracy

text: at

text: It

text: is

text: also

text: made

text: more

text: complicated

text: by

text: the

text: fact

text: that

text: at

text: the

text: bandedge

text: itself

text: any

text: linear

text: combination

text: of

text: modes

text: is

text: itself

text: a

text: mode.

text: The

text: modes

text: found

text: by

text: the

text: eigensolver

text: were

text: not

text: in

text: fact

text: a

text: time-reversal

text: pair.

text: However

text: we

text: found

text: such

text: a

text: pair

text: by

text: taking

text: linear

text: complex

text: combinations

text: of

text: the

text: eignesolver’s

text: solutions.

text: Contact

text: me

text: at:

label: footer

text: M.

text: Arcari,

text: I.

text: So¨llner,

text: A.

text: Javadi,

text: S.

text: Lindskov

text: Hansen,

text: S.

text: Mahmoodian,

text: J.

text: Liu,

text: H.

text: Thyrrestrup,

text: E.

text: H.

text: Lee,

text: J.

text: D.

text: Song,

text: S.

text: Stobbe,

text: and

text: P.

text: Lodahl.

text: Near-unity

text: coupling

text: eﬃciency

text: of

text: a

text: quantum

text: emitter

text: to

text: a

text: photonic

text: crystal

text: waveguide.

text: Phys.

text: Rev.

text: Lett.,

text: Aug

text: Toshihiko

text: Baba.

text: Slow

text: light

text: in

text: photonic

text: crystals.

text: Nature

text: Photonics,

text: Aug

text: T

text: F

text: Krauss.

text: Slow

text: light

text: in

text: photonic

text: crystal

text: waveguides.

text: Journal

text: of

text: Physics

text: D:

text: Applied

text: Physics,

text: J.

text: F.

text: Nye.

text: Lines

text: of

text: circular

text: polarization

text: in

text: electromagnetic

text: wave

text: ﬁelds.

text: Proceedings

text: of

text: the

text: Royal

text: Society

text: of

text: London

text: A:

text: Mathematical,

text: Physical

text: and

text: Engineering

text: Sciences,

text: M.

text: Burresi,

text: R.

text: J.

text: P.

text: Engelen,

text: A.

text: Opheij,

text: D.

text: van

text: Oosten,

text: D.

text: Mori,

text: T.

text: Baba,

text: and

text: L.

text: Kuipers.

text: Observation

text: of

text: polarization

text: singularities

text: at

text: the

text: nanoscale.

text: Phys.

text: Rev.

text: Lett.,

text: Jan

text: Michael

text: Berry.

text: Making

text: waves

text: in

text: physics.

text: Nature,

text: Jan

text: J.

text: F.

text: Nye

text: and

text: M.

text: V.

text: Berry.

text: Dislocations

text: in

text: wave

text: trains.

text: Proceedings

text: of

text: the

text: Royal

text: Society

text: of

text: London

text: A:

text: Mathematical,

text: Physical

text: and

text: Engineering

text: Sciences,

text: Konstantin

text: Y.

text: Bliokh

text: and

text: Franco

text: Nori.

text: Transverse

text: and

text: longitudinal

text: angular

text: momenta

text: of

text: light.

text: Physics

text: Reports,

text: Jeﬀrey

text: F.

text: Wheeldon

text: and

text: Henry

text: Schriemer.

text: Wyckoﬀ

text: positions

text: and

text: the

text: expression

text: of

text: polarization

text: sin-

text: gularities

text: in

text: photonic

text: crystals.

text: Opt.

text: Express,

text: Feb

text: Konstantin

text: Y.

text: Bliokh,

text: Daria

text: Smirnova,

text: and

text: Franco

text: Nori.

text: Quantum

text: spin

text: hall

text: eﬀect

text: of

text: light.

text: Science,

text: A.

text: B.

text: Young,

text: A.

text: C.

text: T.

text: Thijssen,

text: D.

text: M.

text: Beggs,

text: P.

text: Androvitsaneas,

text: L.

text: Kuipers,

text: J.

text: G.

text: Rarity,

text: S.

text: Hughes,

text: and

text: R.

text: Oulton.

text: Polarization

text: engineering

text: in

text: photonic

text: crystal

text: waveguides

text: for

text: spin-photon

text: entanglers.

text: Phys.

text: Rev.

text: Lett.,

text: Oct

text: Carlos

text: Gonzalez-Ballestero,

text: Alejandro

text: Gonzalez-Tudela,

text: Francisco

text: J.

text: Garcia-Vidal,

text: and

text: Esteban

text: Moreno.

text: Chiral

text: route

text: to

text: spontaneous

text: entanglement

text: generation.

text: Phys.

text: Rev.

text: B,

text: Oct

text: R.

text: J.

text: Coles,

text: D.

text: M.

text: Price,

text: B.

text: Royall

text: J.

text: E.

text: Dixon,

text: E.

text: Clarke,

text: A.

text: M.

text: Fox,

text: P.

text: Kok,

text: M.

text: S.

text: Skolnick,

text: and

text: M.

text: N.

text: Makhonin.

text: Chirality

text: of

text: nanophotonic

text: waveguide

text: with

text: embedded

text: quantum

text: emitter

text: for

text: unidirectional

text: spin

text: transfer.

text: June

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 the z = 0 plane of symmetry. Away from this plane, E becomes non-zero. For details, see [27]). z As periodic structures, photonic crystal waveguide eigenmodes are Bloch modes, and these modes possess strong longitudinal (E ), as well as transverse (E ), components of the electric ﬁeld proﬁle. x y The spatial dependence of both these components, and the phase diﬀerence between them, ensures a complex polarisation landscape in the waveguide’s vicinity. The polarisation state at each point can be conveniently summarised by drawing a local polarisation ellipse. The eccentricity, angle and handedness of the ellipse uniquely describes the relative magnitudes of E and E , as well as the x y phase between two. Figure 1 (c) and (d) show the local polarisation ellipses for both the W1 and glide waveguides near the waveguide core. The points of circular polarisation (C-points) are marked by yellow circles. III. METHODS We calculated the photonic crystal waveguide dispersion ω(k) and group velocity (v = dω) and g dk corresponding eigenmodes E (r) using a fr√equency domain eigensolver [28]. In the calculations we k used a 2D supercell of dimensions a x 11a 3 and a grid size of a/96, ensuring convergence of the eigenfrequencies to much better than 0.1% [29]. Such a ﬁne calculation grid was chosen in order to locate and track the positions of the C-points with high ﬁdelity. 2D calculations were chosen to save computational resources. However, we used an eﬀective index approximation [30] that gives a very good agreement with the full 3D calculations; the eigenmodes display a very high (> 95%) agreement to those calculated using a full 3D method. Such a high agreement makes the 2D method suitable for investigating the polarisation properties of the waveguides. Once the eigenmodes of the waveguide are known, we can analyse their polarisation properties. Generically, in any 3D monochromatic ﬁeld, there will exist planes along which the light is linearly polarised (L-planes) and lines along which it is circularly polarised (C-Lines) [4, 31]. Here, we are concerned with the 2D cross-section through the centre of the waveguide (the z = 0 plane), meaning that we will ﬁnd L-lines and C-points where this plane intersects with the L-planes and C-lines. The E component of the electric ﬁeld is zero in this plane of symmetry [27], which considerably z simpliﬁes the polarisation analysis [32]. The polarisation properties are quantiﬁed by the three Stokes parameters [33]: ##LTLine## ##LTLine##

text: the

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text: of

text: both

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text: the

text: phase

text: diﬀerence

text: between

text: them,

text: ensures

text: a

text: complex

text: polarisation

text: landscape

text: in

text: the

text: waveguide’s

text: vicinity.

text: The

text: polarisation

text: state

text: at

text: each

text: point

text: can

text: be

text: conveniently

text: summarised

text: by

text: drawing

text: a

text: local

text: polarisation

text: ellipse.

text: The

text: eccentricity,

text: angle

text: and

text: handedness

text: of

text: the

text: ellipse

text: uniquely

text: describes

text: the

text: relative

text: magnitudes

text: of

text: E

text: and

text: E

text: as

text: well

text: as

text: the

text: x

text: y

text: phase

text: between

text: two.

text: Figure

text: (c)

text: and

text: (d)

text: show

text: the

text: local

text: polarisation

text: ellipses

text: for

text: both

text: the

text: and

text: glide

text: waveguides

text: near

text: the

text: waveguide

text: core.

text: The

text: points

text: of

text: circular

text: polarisation

text: (C-points)

text: are

text: marked

text: by

text: yellow

text: circles.

text: III.

text: METHODS

text: We

text: calculated

text: the

text: photonic

text: crystal

text: waveguide

text: dispersion

text: ω(k)

text: and

text: group

text: velocity

text: (v

text: dω)

text: and

text: g

text: dk

text: corresponding

text: eigenmodes

text: E

text: (r)

text: using

text: a

text: fr√equency

text: domain

text: eigensolver

text: In

text: the

text: calculations

text: we

text: k

text: used

text: a

text: supercell

text: of

text: dimensions

text: a

text: x

text: and

text: a

text: grid

text: size

text: of

text: ensuring

text: convergence

text: of

text: the

text: eigenfrequencies

text: to

text: much

text: better

text: than

text: Such

text: a

text: ﬁne

text: calculation

text: grid

text: was

text: chosen

text: in

text: order

text: to

text: locate

text: and

text: track

text: the

text: positions

text: of

text: the

text: C-points

text: with

text: high

text: ﬁdelity.

text: calculations

text: were

text: chosen

text: to

text: save

text: computational

text: resources.

text: However,

text: we

text: used

text: an

text: eﬀective

text: index

text: approximation

text: that

text: gives

text: a

text: very

text: good

text: agreement

text: with

text: the

text: full

text: calculations;

text: the

text: eigenmodes

text: display

text: a

text: very

text: high

text: agreement

text: to

text: those

text: calculated

text: using

text: a

text: full

text: method.

text: Such

text: a

text: high

text: agreement

text: makes

text: the

text: method

text: suitable

text: for

text: investigating

text: the

text: polarisation

text: properties

text: of

text: the

text: waveguides.

text: Once

text: the

text: eigenmodes

text: of

text: the

text: waveguide

text: are

text: known,

text: we

text: can

text: analyse

text: their

text: polarisation

text: properties.

text: Generically,

text: in

text: any

text: monochromatic

text: ﬁeld,

text: there

text: will

text: exist

text: planes

text: along

text: which

text: the

text: light

text: is

text: linearly

text: polarised

text: (L-planes)

text: and

text: lines

text: along

text: which

text: it

text: is

text: circularly

text: polarised

text: (C-Lines)

text: Here,

text: we

text: are

text: concerned

text: with

text: the

text: cross-section

text: through

text: the

text: centre

text: of

text: the

text: waveguide

text: (the

text: z

text: plane),

text: meaning

text: that

text: we

text: will

text: ﬁnd

text: L-lines

text: and

text: C-points

text: where

text: this

text: plane

text: intersects

text: with

text: the

text: L-planes

text: and

text: C-lines.

text: The

text: E

text: component

text: of

text: the

text: electric

text: ﬁeld

text: is

text: zero

text: in

text: this

text: plane

text: of

text: symmetry

text: which

text: considerably

text: z

text: simpliﬁes

text: the

text: polarisation

text: analysis

text: The

text: polarisation

text: properties

text: are

text: quantiﬁed

text: by

text: the

text: three

text: Stokes

text: parameters

text: ##LTLine##

text: ##LTLine##