Guided Modes in the Plane Array of Optical Waveguides
I.Ya. Polishchuk, M.I. Gozman, A.A. Anastasiev, Yu.I. Polishchuk, S.V. Solov'ov, E.A. Tsyvkunova
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Guided Modes in the Plane Array of Optical Waveguides
I. Ya. Polishchuk
National Research Center Kurchatov Institute, Moscow, 123182, RussiaMax Planck Institute for the Physics of Complex Systems, D-01187 Dresden, GermanyMoscow Institute of Physics and Technology, Dolgoprudnii, 141700 Russia
A.A. Anastasiev and E.A. Tsyvkunova
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute),115409, Kashirskoe shosse, 31, Moscow, Russia
M.I. Gozman and S.V. Solov’ov
Moscow Institute of Physics and Technology, Dolgoprudnii, 141700 Russia
Yu.I. Polishchuk
National Research Center Kurchatov Institute, Moscow, 123182, Russia
Abstract
It is known that, for the isolated dielectric cylinder waveguide, there exists the cutoff frequency ω ∗ below which there are no guided (radiationless) modes. It is shown in the paper that the infiniteplane periodic array of such waveguides possesses guided modes in the frequency domain whichis below the frequency ω ∗ . So far as the finite array is concerned, the modes in this frequencydomain are weakly radiating ones, but their quality factor Q increases as Q ( N ) ∼ N , N beingthe number of the waveguides in the array. This dependence is obtained both numerically, usingthe multiple scattering formalism, and is justified within the framework a simple analytical model. . INTRODUCTION The optical waveguides are the inherent component of the optical and optoelectronicdevices, indispensable for the optical signals transmission between different parts of thesystem. The interaction between the closely spaced waveguides usually results in the unde-sirable effects that distorts the transmitted signal. However, in some cases, the interactionbetween the waveguides can be exploited for practical purpose. In particular, this concernsthe plane periodic arrays of waveguides, which is a special kind of low-dimensional photoniccrystals. The main feature of such systems is a band structure of the optical spectrumwhich defines its peculiar properties [1–4]. Low-dimensional photonic crystal composed ofthe parallel rods are of a special interest. For the first time, the band structure of the planearray composed of the semiconductor cylinders was investigated in Ref. [5]. Then a similarmetallic structure which took into account dissipation was computed in Ref. [6]. Supercon-ducting photonic crystals of such kind were considered in [7]. Photonic crystals of such kindare useful for various applications [8]. In particular, they reveal negative-angle refractionand reflection [9]. On the other hand, the arrays of parallel interacting cylinders is an idealsystem to simulate various physical phenomena inherent in condensed matter physics suchas Anderson localization, Bloch oscillations, Bloch-Zener tunneling, etc. [10–14].The electromagnetic filed which describes a guided mode in a waveguide is finite insidethe waveguide, while it is vanishing at a large distance from it. It is well known that, forthe isolated cylinder waveguide, the guided modes can exist only above the so-called cutofffrequency ω ∗ [15]. This is due to the fact that the conversion of the modes with the frequencybelow ω ∗ into a free photon is possible. This frequency depends on the material refractiveindex and the waveguide diameter. However, in various applied tasks, it may be necessaryto have a guided mode below the frequency ω ∗ for the given waveguides. In particular,such problem may appear in the following case. Indeed, along with the radiation, thereexist losses connected with the absorption by the material itself. If the frequency window islocated below the cutoff frequency ω ∗ , there arise a problem to shift the frequency ω ∗ intothis window region.In this paper, we investigate a possibility of a formation of high quality guided modes inthe plane waveguide array which are below the cutoff frequency of the isolated waveguide ω ∗ . Thus our aim is to suppress the radiation loss using the array of the waveguides. First,2e consider the infinite plane periodical array of the waveguides an show that, taking intoaccount the interaction between them, results in the appearance of the guided modes below ω ∗ . These modes possess the infinite Q − factor. However, actually we deal with the arrayof a finite size and the modes become low-radiating ones. For this reason, we investigatehow the Q − factor of the low-radiating modes depends on the number of the waveguides inthe array N . Using the multiple scattering formalism (MSF), it is shown numerically that Q ( N ) ∼ N . This formalism is based on the exact description of the electromagnetic waveswhich are scattered by the infinite cylinder [21]. Besides, we propose a simple model whichqualitatively explains this cubic dependence.Note that the effect of increasing the Q − factor for the radiative modes in the array ofthe interacting spherical particles with increasing the array size, was discovered in [16–19].The paper is organized as follows. In Section II we describe the MSF giving a brief deriva-tion of the principal relations. Based on the relations obtained, the numerical simulationfor the infinite and the finite arrays of cylinder waveguides is given in Section III. A clearqualitative explanation of the results obtained numerically is given in Section IV. II. MULTIPLE SCATTERING FORMALISM
Let us consider the array of N parallel cylindrical dielectric waveguides (see figure 1).The axes of the waveguides are in the xz -plane and are parallel to the z -axis. The arrayis equidistant, a being the distance between the axes of the nearest waveguides. All thewaveguides are assumed to have the same radius R and the same refractive index n . Therefractive index of the environment is n . The system of units where the speed of light c = 1is used.Let a guided mode with a frequency ω is excited. Because of the translation invariancein the z direction, all the components of the electromagnetic field describing the guidedmode depend on the coordinate z as e iβz , β being a propagation constant. Thus, all thecomponents of the electromagnetic field describing the guided mode are proportional to thefactor e − iωt + iβz . First, let us describe the electromagnetic field inside the waveguides. The3eld inside the j -th waveguide, being of a finite value, may be represented as follows˜ E j ( t, r ) = e − iωt + iβz P m =0 , ± ... e imφ j (cid:16) c jm ˜ N ω ′ βm ( ρ j ) − d jm ˜ M ω ′ βm ( ρ j ) (cid:17) , ˜ H j ( t, r ) = e − iωt + iβz n P m =0 , ± ... e imφ j (cid:16) c jm ˜ M ω ′ βm ( ρ j ) + d jm ˜ N ω ′ βm ( ρ j ) (cid:17) , (1)where r = ( x, y, z ) = ( ρ , z ), ρ j = | ρ − a j | < R , φ j is the polar angle for the vector ρ − a j (see figure 1), ω ′ = nω . The vector cylinder harmonics ˜ M ω ′ βm ( ρ j ) and ˜ N ω ′ βm ( ρ j ) are definedas follows ˜ N ω ′ βm ( ρ j ) = e r iβκ J ′ m ( κρ j ) − e φ mβκ ρ j J m ( κρ j ) + e z J m ( κρ j ) , (2)˜ M ω ′ βm ( ρ j ) = e r mω ′ κ ρ j J m ( κρ j ) + e φ iω ′ κ J ′ m ( κρ j ) , (3)here κ = p n ω − β , J m ( κρ j ) is the Bessel function, and the prime means the derivativewith respect to the argument κρ j . Thus, the guided mode inside the j -th waveguide isdetermined by the frequency ω , by the propagation constant β and by the partial amplitudes c jm , d jm . FIG. 1: The optical waveguide array. The polar coordinates of radius-vector r relative to differentwaveguides. Now let us turn to the electromagnetic field for the same guided mode outside of thearray. This field is a sum of the contributions induced by all the waveguides: E ( t, r ) = N X j =1 E j ( t, r ) , H ( t, r ) = N X j =1 H j ( t, r ) . (4)4or the guided mode, the contribution of the j -th waveguide should vanish at ρ j → ∞ .Therefore, one can write E j ( t, r ) = e − iωt + iβz P m e imφ j (cid:16) a jm N ω βm ( ρ j ) − b jm M ω βm ( ρ j ) (cid:17) , H j ( t, r ) = e − iωt + iβz n P m e imφ j (cid:16) a jm M ω βm ( ρ j ) + b jm N ω βm ( ρ j ) (cid:17) . (5)where ρ j > R . In (5), another kind of the vector cylinder harmonics is introduced N ω βm ( ρ j ) = e r iβκ H ′ m ( κ ρ j ) − e φ mβκ ρ j H m ( κ ρ j ) + e z H m ( κ ρ j ) , (6) M ω βm ( ρ j ) = e r mω κ ρ j H m ( κ ρ j ) + e φ iω κ H ′ m ( κ ρ j ) , (7)where H m ( κ ρ j ) is the Hankel function of the first kind, ω = n ω , κ = p n ω − β . Thus,the contribution of the j -th waveguide into the guided mode field outside of the array isdetermined by the frequency ω , by the propagation constant β , by the partial amplitudes a jm , b jm . Note that, for β = 0, expansions (1) and (5) transform into the correspondingexpressions in [21], however different notations are used there. Below, the factor e − iωt + iβz isomitted.Below in this paper, for the purpose of illustration of the effect, we confine ourselves tothe zero-harmonic approximation. This means that only the terms with m = 0 are takeninto account in (1) and (5). It is easy to convince ourselves that in this case the guidedmodes are either transverse magnetic modes (TM) or transverse electric (TE) ones. For theTM mode b j = d j = 0, while for the TE mode a j = c j = 0. As an example, let usconsider the TM modes. Then, equations (1) and (5) take the form˜ E j ( r ) = c j ˜ N ω ′ β ( ρ j ) , ˜ H j ( r ) = c j n ˜ M ω ′ β ( ρ j ) , ρ j < R (8) E j ( r ) = a j N ω β ( ρ j ) , H j ( r ) = a j n M ω β ( ρ j ) , ρ j > R (9)Here the notations a j , c j are used instead of a j , c j .Let R j be the radius-vector of a point on the surface of the j -th waveguide. Then thefields ˜ E j ( R j ), ˜ H j ( R j ) in Eq.(1) and the fields E ( R j ) H ( R j ) in Eq.(4) obey the boundaryconditions on the surface of this waveguide. Generally, there are six boundary conditions.However, for the TM-modes only two of them are required.[ E ( R j )] z = [ ˜ E j ( R j )] z , [ H ( R j )] φ = [ ˜ H j ( R j )] φ . (10)5hese equations determine the partial amplitude a j and c j for given ω and β. To representEqs.(10) in a convenient form, one should express the fields E l ( R j ) for l = j entering inEq.(4) in terms of the functions ˜ N using the Graph formula N ω βn ( ρ l ) M ω βn ( ρ l ) e inφ l = + ∞ X m =0 U ljnm ( ω, β ) ˜ N ω βm ( ρ j )˜ M ω βm ( ρ j ) e imφ j , (11)here U ljnm ( ω, β ) = H n − m ( κ a | l − j | ) [sign( j − l )] n − m . (12)For the m = 0 approximation one has N ω β ( ρ l ) ≈ U l − j ( ω, β ) ˜ N ω β ( ρ j ) , M ω β ( ρ l ) ≈ U l − j ( ω, β ) ˜ M ω β ( ρ j ) , (13)where U l − j ( ω, β ) = U lj ( ω, β ). Then, it follows from Eq.(9) that E l ( r ) = a l U l − j ( ω, β ) ˜ N ω β ( ρ j ) , H l ( r ) = a l n U l − j ( ω, β ) ˜ M ω β ( ρ j ) . (14)Thus, E ( r ) = a j N ω β ( ρ j ) + P l = j a l U l − j ( ω, β ) ˜ N ω β ( ρ j ) , H ( r ) = a j n M ω β ( ρ j ) + P l = j a l n U l − j ( ω, β ) ˜ M ω β ( ρ j ) . (15)Substituting (15) and (8) into (10), one obtains a j H ( κ R ) + P l = j a l U l − j ( ω, β ) J ( κ R ) = c j J ( κR ) ,a j n iω κ H ′ ( κ R ) + P l = j a l n U l − j ( ω, β ) iω κ J ′ ( κ R ) = c j n iω ′ κ J ′ ( κR ) . (16)Then the system of equations (16) is reduced to the form a j ¯ a ( ω, β ) − X l = j U l − j ( ω, β ) a l = 0 , (17) c j = ¯ c ( ω, β ) a j , (18)where ¯ a ( ω, β ) = n κ J ′ ( κR ) J ( κ R ) − n κ J ( κR ) J ′ ( κ R ) n κ J ( κR ) H ′ ( κ R ) − n κ J ′ ( κR ) H ( κ R ) , (19)¯ c ( ω, β ) = n κ { H ( κ R ) J ′ ( κ R ) − H ′ ( κ R ) J ( κ R ) } n κJ ′ ( κ R ) J ( κR ) − n κJ ( κ R ) J ′ ( κR ) . (20)6he terms U l − j ( ω, β ) in Eq.(17) desribe the interaction between the waveguides. If theterms U l − j ( ω, β ) in Eq.(17) are neglected, the poles of ¯ a ( ω, β ) or ¯ c ( ω, β ) determines theguied modes for the isolated wavegide.System of equations (17) possesses nontrivial solutions ifdet (cid:13)(cid:13)(cid:13)(cid:13) δ jl ¯ a ( ω, β ) − U l − j ( ω, β ) (cid:13)(cid:13)(cid:13)(cid:13) = 0 . (21)This equation relates the frequency of the guided mode ω and its propagation constant β implicitly.For the infinite periodical array of identical waveguides, the solution of Eq.(17) reads a j = a e ikaj , − π/a < k ≤ π/a. (22)In this case, the nontrivial solution exists if1¯ a ( ω, β ) − U ( ω, β, k ) = 0 , (23)where U ( ω, β, k ) = X l =0 U l ( ω, β ) e ikal . (24)Equation (23) determines the dispersion law ω ( β, k ). If the propagation constant β is real,the corresponding mode frequencies may be either real or complex. If the frequency isreal, the mode possesses an infinite Q − factor. Otherwise the mode has a finite lifetimeand the imaginary part of the frequency determines the mode decay rate. However, if thecorresponding quality factor is large, the mode may be considered as a guided one. III. NUMERICAL SIMULATION FOR THE INFINITE AND THE FINITE AR-RAYS.
Let us consider the infinite the array of the waveguides with the geometric parameters andthe refractive indices which are chosen to be close to those in Refs. [14]. The specific valuesof the parameters are taken so that the illustration of the results looks quite representative.For this reason, one takes the waveguide radius R = 1 . µ m, the refractive index of thewaveguides n = 1 . n = 0 . n = 1 . ω ∗ = 5 . µ m − . This7alue corresponds to the cutoff propagation constant β ∗ = 8 . µ m − . (Let us remind thatthe speed of light c = 1 and, therefore, the frequency has a dimension of the inverse length).The modes we are interested in, appear due to the interaction between the waveguides.For this reason, we assume that the neighbor waveguides touch each other, since in this casethe interaction reveals itself the most strongly. As an example, let us choose the propagationconstant β = 8 µ m − < β ∗ . The dispersion curve ω ( β, k ) , which is a numerical solution toEq. (23), is presented in Fig. 2 by the thick solid line. FIG. 2: The dispersion curve for the infinite array.
One sees that the curve is completely located within the domain β < n ω ( k ) < p β + k . A physical explanation to this fact is given below. This numerical results completely supportsthe kinematic criterion for the mode to be a radiationless one. Thus the infinite periodicalarray of the waveguides may possess the guides modes with the frequencies below the cutofffrequency of the single waveguide.The radiationless guided modes inherent in the periodical array found above (see Fig.2)possess the infinite quality factor Q = 2Re ω/ | Im ω | . This is due to the fact that the arrayis infinite. However, actually one deals with the arrays composed of a finite number of thewaveguides N . On the other hand, it is evident that for N ≫ Q depends on the number of thewaveguides N. Using Eq. (21) one can obtain numerically that, for a finite N ≫ , the highest qualityfactor is reached for the modes whose frequency is close to the upper edge of the Brillouin8 = 8 m -1 = 7 m -1 Q u a lit y f ac t o r Q Number of waveguides N = 6 m -1 (a) a = 2R (b) a = 3R Q u a lit y f ac t o r Q Number of waveguides N = 8 m -1 = 7 m -1 = 6 m -1 FIG. 3: The dependence of the quality factor on the number of the waveguides N for the planearray of waveguides. zone k ≈ π/a (such a feature is inherent also for the array of spherical particles [16, 17, 19]).The dependence of the quality factor on the number of the waveguides N just for modes withthe highest Q − factor is illustrated by two example: the waveguides are touching, a = 2 R, and the waveguides are spatially separated, a = 3 R . The three values for the propagationconstant smaller than β ∗ are taken: β = 6 µ m − , β = 7 µ m − , β = 8 µ m − < β ∗ . Theresults of the numerical simulation are presented in Fig. 3. The analysis of the dependenciesin this figure reveals a remarkable feature: for N >
10 the dependence the quality factor Q ( N ) ∼ N .Let us retrieve, using Eq. (MSF Main), the relation between the partial amplitudes a j . A typical dependence, obtained numerically, is presented, as an example, in Fig. 4 for thecase N = 15 . P a r ti a l a m p lit ud e , a j Number of the waveguide, j
FIG. 4: The partial amplitude for the mode with the highest quality factor. V. THE INTERPRETATION OF THE NUMERICAL RESULTS
Knowing the dependence ω ( β, k ) allows us to determine the features of the guided modes.Let us pay attention, that the reason for the mode to possesses a finite lifetime is a conversionof it into a free photon. That is the mode is a radiative one. First, let us consider a singlewaveguide. Then the mode is described by a frequency ω and a propagation constant β . Forthe conversion into a free photon to takes place, the photon wave vector q should satisfythe two conditions: | q | = n ω and q z = β . Since q z < | q | , the photon can be emitted only if β < n ω . In the opposite case β > n ω , the mode is a radiationless one and it is aguided one with the infinite quality factor. Let us turn to the infinite plane periodic array.In this case, a mode is determined by a quasi-wave vector k , in addition to the frequency ω and the propagation constant β . Thus, the wave vector q of the emitted free photonsatisfies three conditions: | q | = n ω , q z = β and q x = k . It is obvious that p q x + q z < | q | .So, the mode can be converted into a free photon only if p k + β < n ω . Thus, theinfinite periodical array possesses guided modes within the frequency domain, which obeysthe kinematic criterion β < n ω < p β + k , (25)where the single waveguide allow only the radiating modes. Note that, since nω ( β, k ) < p β + k , a guided mode may not exist for small k at all.Then, let us explain qualitatively the cubic dependence for the Q -factor found in theprevious section. First, let us consider the infinite array of the waveguides. Let A j ( t ) be theeffective time-dependent partial amplitude for the j -th waveguide, which characterizes thewaveguide as a whole. The time evolution of A j ( t ) may be approximately described by theequation which similar to a Schr¨odinger one i dA j dt ( t ) + V (cid:16) A j − ( t ) + A j +1 ( t ) (cid:17) = 0 . (26)Here V is the effective coupling between the nearest waveguides. Let us find the solutionfor this equation in the form A j ( t ) = A e − iωt + ikj , − π < k ≤ π. (27)Substituting (27) into (26) one obtains the dispersion law: ω ( k ) = − V cos k. (28)10ow let us turn to the finite array. As found above, the infinite array possesses the infinite Q, while the finite array possesses a large but a finite Q. (see Fig. 3). For this reason, itis natural to assume that this is connected with the availability of the edge waveguides inthe array which are responsible for the radiation of the photon. Based on this fact, one canwrite for the finite array the equation similar to Eq. (26): i dA j dt ( t ) + V (cid:16) − δ j (cid:17) A j − ( t ) + V (cid:16) − δ jN (cid:17) A j +1 ( t ) + iγ (cid:16) δ j + δ jN (cid:17) A j ( t ) = 0 . (29)The parameter γ ≪ V is responsible for the free photon emission. Using (27) one obtainsfrom (29) ω A j + V (1 − δ j ) A j − + V (1 − δ jN ) A j +1 + iγ ( δ j + δ jN ) A j = 0 . (30)Note that ω = ω ′ + iγ may be complex. For the particular j = N, this equation takes theform ( ω ′ + iγ ) A N + V A N − = 0 . (31)So, ω ′ + iγ = − V A N − A N . (32)The dependence for a j in Fig. 4 approaches zero at the edges of the array and resemblesa cosine one. For this reason, let us seek the solution to Eq. (30) in the form A j ∼ cos k ( j − N/ , (33) k being close to π − π/N . Substituting (28) and (33) into (32), one gets − V cos k + iγ = − V cos k ( N/ − kN/ . (34)Let k = π − π/N + x, (35)where x is complex and | x | ≪ π/N . For the sake of simplicity, let us assume N to be even.Then, substituting (35) into (34), one gets2 V cos (cid:16) πN − x (cid:17) + iγ = V sin ( π/N + N x/
N x/ . (36)11aking into account a smallness of the arguments in the trigonometric functions in (36) oneobtains: 2 V + iγ ≈ V (cid:18) πN x + 1 (cid:19) . (37)Then, since γ ≪ V , one has x ≈ πN (cid:16) − i γV (cid:17) . (38)Substituting (38) and (35) into (28), one gets: ω ≈ V − i π γN . (39)Then, the quality factor Q = 2Re ω | Im ω | = V N π γ . (40)reveal the sought-for cubic dependence. V. CONCLUSION
In this paper we investigated the guided modes in the array of coupled waveguides belowthe cutoff frequency of a single waveguide, i.e. in the frequency domain where the singlewaveguide possesses only the radiating modes. It is shown that the infinite periodic ar-ray possesses a band of the guided modes with the infinite Q -factor. In the case of thefinite array, the modes below the cutoff frequency are weakly radiating ones. Their qualityfactor increases with the number of waveguides as Q ( N ) ∼ N . These results are obtainednumerically using the multiple scattering formalism. A clear physical interpretation of thenumerical results is given.
Acknowledgements
The study is partially supported by the Russian Fund for Basic Research (Grant 16-02-00660). 12 eferences [1] Lourtioz J-M, Benisty H, Berger V, Gerard J-M, Maystre D and Tchelnokov A 2008
PhotonicCrystals: Towards Nanoscale Photonic Devices (Springer)[2] Joannopoulos J, Villeneuve P R and Fan S. 1997
Nature
143 - 9[3] Busch K, L¨olkes S, Wehrspohn R B, F¨oll H (Eds.) 2004
Photonic Crystals. Advances in Design,Fabrication, and Characterization (Wiley-VCH Verlag GmbH & Co. KGaA)[4] Longhi S 2009
Laser & Photon. Rev.
243 - 61[5] A.R. McGurn and A.A. Maradudin, Phys. Rev. B , 17576 (1993).[6] V. Kuzmiak and A.A. Maradudin, Phys. Rev. B , 7427 (1997).[7] O.L. Berman, Yu.E. Lozovik, S.E. Eiderman, and R.D. Coalson, Phys. Rev. B , 092505(2006).[8] N. A. Giannakis, J. E. Inglesfield, A. K. Jastrzebski, and P. R. Young, J. Opt. Soc. Am. B (6), (2013).[9] S. Belan and S. Vergeles, Opt. Mater. Express , 130 (2015).[10] M. J. Zheng, J. J. Xiao, and K. W. Yu, Phys. Rev. A , 033829 (2010).[11] F. Lederer, G. I. Stegemanb, D. N. Christodoulides , G. Assanto, M. Segev, Y. Silberberg,Phys. Rep. , 817-823, 14 AUGUST(2003).[13] A. Szameit,T. Pertsch, S. Nolte, and A. T¨unnermann, F. Lederer, Phys. Rev. A , 043804(2008).[14] F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. T¨unnermann, S. Longhi,Phys. Rev. Lett. , 076802 (2009).[15] Marcuse D 1972 Light transmission optics (Van Nostrand Reinhold Company)[16] Blaustein G S, Gozman M I, Samoylova O M, Polishchuk I Ya and Burin A L 2007
OpticsExpress
19] I. Ya. Polishchuk, M. I. Gozman, G. S. Blaustein, and A. L. Burin, Phys. Rev. E , 0266012010.[20] M.I. Gozman, Yu.I.Polishchuk, I.Ya.Polishchuk, E.A.Tsivkunova, Solid State Comm., , 16 (2015)[21] Van de Hulst H C, Light scattering by small particles, Dover Publications (Inc., New York)1981(Inc., New York)1981