Light wheel buildup using a backward surface mode
aa r X i v : . [ phy s i c s . op ti c s ] S e p Light wheel buildup using a backward surface mode
R´emi Poll`es, Antoine Moreau, and G´erard Granet
Clermont Universit´e, Universit´e Blaise Pascal, LASMEA, BP 10448, F-63000 Clermont-FerrandCNRS, UMR 6602, LASMEA, F-63177 Aubi`ere
When a guided mode is excited in a dielectric slab coupled to a backward surface wave at the interface betweena dielectric and a left-handed medium, light is confined in the structure : this is a light wheel. Complex planeanalysis of the dispersion relation and coupled-mode formalism give a deep insight into the physics of thisphenomenon (lateral confinement and the presence of a dark zone).
OCIS codes:
Recently, a lamellar structure consisting of a conven-tional dielectric layer coupled to a left-handed material(LHM) layer has been proposed to confine light. Becauseof contradirectional power flows in the two layers, anexotic localized mode called ”light wheel” can be ex-cited [1, 2]. In this letter, we present our study of a newtype of light wheel that uses a backward surface mode atthe interface between an LHM and a right-handed mate-rial (RHM). The dispersion relation of the structure andthe coupled-mode formalism allows us to describe andexplain the field distribution in the structure.Let us first consider a surface wave propagating alongan interface between two semi-infinite media: an LHMand a RHM. The LHM is caracterized by its dielectricpermittivity ε and its magnetic permeability µ , whichare both negative. The permittivity of the RHM is ε ,its permeability is µ . Because of unusual properties ofLHM, such an interface supports surface guided modeswhich can be backward (i.e., present opposite phase andgroup velocities) [3, 4]. In TM polarization, the disper-sion relation for surface waves propagating along thisinterface can be written as [4] α = k µ ε X ( X − Y )( X − , (1)where X = | ε | ε , Y = | µ | µ , α is the propagation constantand k is the wavenumber in the vacuum. A backwardsurface wave is supported provided the inequality1 < X < /Y (2)is verified.The light wheel develops from the contradirectionalcoupling between this backward mode and a forwardguided mode. This coupling appears when the disper-sion curves of the two waveguides cross each other (i.e.when they are under phase matching conditions). Evenwhen taking the dispersive character of the LHM intoaccount, such a coupling is thus not difficult to obtainas long as the condition (2) is fulfilled. At a given wave-length λ , let us consider, for instance, that ε = − . µ = − .
5. Because that satisfies the previous con-ditions, a backward surface wave between such an LHM and the air ( ε = 1, µ = 1) exists for α = α = 1 . k according to Eq. (1).Let us obtain the coupling with a symmetrical dielec-tric slab waveguide as depicted Fig. 1. According to itsdispersion relation, the thickness of the waveguide sup-porting a forward guided mode for the propagation con-stant α , in TM polarization, is given by h = 2 γ arctan (cid:18) ε κ ε γ (cid:19) , (3)where κ = p α − ε µ k and γ = p ε µ k − α .For ε = 1, µ = 1, ε = 3 and µ = 1, the value ofthickness h is 0 . λ . In the following, we retain theabove characteristics.The above dielectric slab and LHM interface arebrought close together, forming the structure presentedin Fig. 1. We have calculated the analytical expression (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) h2 − h1 ε µ ε µ ε µ ε µ dielectric LHM z Fig. 1. Dielectric slab waveguide and the LHM interfaceseparated by a distance h and surrounded by a mediumcharacterized by ε and µ .of the dispersion relation of the whole structure, in TMpolarization: x [exp(2 jγ h ) + X ] [exp(2 κ h ) + X ][exp(2 jγ h ) − X ][exp(2 κ h ) − X ] = − , (4)where x i = κ i ε /κ ε i and X i = (1 − x i ) / (1 + x i ). Figure2 shows, in the alpha complex plane, the solutions of thedispersion relation (Eq. (4)) for several values of distance h , the frequency being fixed. When distance h betweenthe slab and the interface is large enough, the waveguides1re independent, the dispersion relation is thus verifiedfor α = α . But when distance h decreases, they becomecoupled and two complex-conjugate solutions appear.For instance at h = 0 . λ , α = (1 . ± . i ) k .These twin solutions have the same real part so that thecorresponding modes cannot be excited separately by asource and they form a light wheel as shown in Fig. 3.The nonzero imaginary part of α means that the field E ( x, z, t ) = E ( z, t ) exp( iαx ) decays along the x axis.Because the imaginary parts of the solutions are oppo-site, the twin modes decay in opposite directions. Thesize of the light wheel is moreover controled by ℑ ( α ),a characteristic width of the light wheel being given by L = ℑ ( α ) . Figure 2(a) shows that the smaller the dis-tance h is, the higher the imaginary part of the propa-gation constant is and so the smaller the light wheel willbe. However, Fig. 2(b) shows that the imaginary part of α presents a maximum reached for strong coupling for h = 0 . λ . The light wheel can, thus, confine the lightin a region as small as a characteristic width L = 1 . λ .Figure 3(a) shows the electromagnetic field created in the (b)(a) -0.3-0.2-0.100.10.20.3 0.95 1 1.05 1.1 1.15 1.2 1.251.00.500.250.180.12 1.00.500.250.180.12 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-0.006-0.004-0.00200.0020.0040.006 1.24725 1.2473 1.24735 1.24741.5 1.2 1.1 1.01.5 1.2 1.1 1.0 α I m ag i na r y pa r t o f / k α Real part of / k Fig. 2. Solutions of the dispersion relation in the α com-plex plane, for different values of h in wavelength, thedistance between the dielectric waveguide and the LHMsurface. Inset, zoom in the region around α .structure by a punctual source inside the dielectric layer.The interference pattern is due to the two contrarotativelight wheels, which are excited and interfere. In Fig. 3(b),the light wheel is excited by a beam using evanescentcoupling: a prism is placed above the dielectric slab andthe structure is illuminated by a gaussian beam comingfrom above. Here, the field distribution is less intuitive.In particular, a dark zone appears just below the inci-dent beam, in the center of the dielectric waveguide butnot at the plasmonic interface. The presence of a similardark zone has already been noticed but not explained [1].We have applied the coupled-mode theory (CMT) [5–7]to this contradirectional coupling to get an analyticalmodel able to account for this phenomenon.Let us consider two independent guided modes whosecomplex amplitudes are A and B . Mode a corresponds LHMDielectricLHMDielectric (b)(a)
Prism
Fig. 3. Modulus of the field represented in a domain 100 λ large, 5 λ high. The distance between the dielectric slaband the LHM interface is h = 0 . λ . (a) The punctualsource, placed in the dielectric waveguide, excites twocontrapropagative light wheels. (b) The light wheel isexcited by a an incident gaussian beam (angle: 33 . ◦ ,waist: 10 λ ) in a prism ( ε = 5, µ = 1). White arrowsindicate the propagation direction of light. These imagesare obtained using the numerical method described in [9].to a right traveling wave in the dielectric slab, whereasmode b travels to the left on the LHM interface. If thetwo guides are brought close together, they become cou-pled: energy is exchanged between them. Hence in thiscontradirectional coupling case and under phase match-ing conditions, the complex amplitudes A and B dependon x , obeying relations of the type [6] dAdx = κ ∗ B, (5) dBdx = κA, (6)where κ is the coupling coefficient. Its value is given bythe imaginary part of the solution α of dispersion relationEq. (4). For weak coupling as well as for strong coupling,we indeed have κ = i ℑ ( α ).Here, the evanescent coupling used to excite mode a is assumed to be weak so that the guided modes areunmodified by the presence of the prism. An approachsuggested by [8] is to consider the guided mode excitedby a set of punctual sources situated inside the waveg-uide with an amplitude distribution along the x directiongiven by the incident beam. Each punctual source has anamplitude proportional to the amplitude of the incidentfield at the prism interface just above.Let us then determine A ( x ) and B ( x ), the modesamplitudes created by a punctual source localized at x =0 whose the amplitude is equal to one, i.e., a source givenby the expression S ( x ) = δ ( x ), where δ is the Diracdistribution. A and B can be seen as Green’s functions[8]. The solutions of Eqs. (5) and (6) are A ( x ) = H ( x ) exp( −| κ | x ) − H ( − x ) exp( | κ | x ) , (7) B ( x ) = exp (cid:16) − i π (cid:17) [ H ( x ) exp( −| κ | x ) + H ( − x ) exp( | κ | x )] , (8)2here H is the Heaviside function. Notice the π/ A and B in each domains x > x < S ( x ), the amplitudes of the modes are ob-tained by convolution of the source expression S with theGreen’s functions A and B . Thus, for a gaussian beamwhose the waist size is w , i.e., an amplitude distribution S ( x ) = S exp (cid:0) − x /w (cid:1) , we get A ( x ) = S √ π (cid:18) κ w (cid:19) × (cid:20) exp( −| κ | x ) erfc (cid:18) − xw + | κ | w (cid:19) − exp( | κ | x ) erfc (cid:18) xw + | κ | w (cid:19)(cid:21) , (9) B ( x ) = S √ π (cid:18) κ w − i π (cid:19) × (cid:20) − exp( −| κ | x ) erfc (cid:18) − xw + | κ | w (cid:19) + exp( κx ) erfc (cid:18) xw + | κ | w (cid:19)(cid:21) , (10)where erfc is the complementary error function.Figure 4 shows the modulus of the field distribution inthe middle of the dielectric slab of Fig. 3(b). It is com-pared to the theoretical mode profile | A ( x ) | computedfor κ = 0 . i , given by the dispersion relation. Be-cause the coupling with the prism is weak, the coupledmodes are relatively undisturbed and Eq. (9) accuratelydescribes the field in the slab. Both curves show a min-imum for x = 0 which corresponds to the dark zone inthe middle of the dielectric waveguide. The light wheel x (in wavelength) M odu l u s o f t he f i e l d ( i n a r b i t r a r y un i t s ) Fig. 4. Modulus of the field in the middle of the dielec-tric waveguide (blue curve) and | A ( x ) | the modulus ofthe theoretical amplitude of the mode a (red curve). Pa-rameters of the structure and the beam are as in Fig. 3. S is chosen arbitrarily.phenomenon can be summarized from Eqs. (9) and (10)as follows: the guided mode is excited in the dielectricwaveguide towards the right. It is then transferred by contradirectional coupling to the plasmonic backwardmode with a − π/ x = 0, and the energy is transferred to the dielectricwaveguide for x <
0, with another phase shift of − π/
2. Inthe dielectric slab, the right part and the left part of thelight wheel are thus in phase opposition. This cannot beseen in Fig. 3(a) because the source is ponctual. In Fig.3(b), because the evanescent coupling is equivalent to aspatially extended source, the parts of the light wheelwhich are in phase opposition ”overlap” in the dielectricslab and a dark zone appears.When we consider a lossy LHM, which is more likely,the light wheel and the dark zone still exist as shownin [1]. In this case, the model can be simply extended bychanging Eq. (6) into dB/dx = κA − κ l B , where κ l isthe extinction coefficient.Finally, the complex plane analysis and the CMT as-sociated with Ulrich’s approach of evanescent couplingallow a very accurate description and a deep understand-ing of the light wheel phenomenon and its universal fea-tures. Beyond the fact that a light wheel can be usedto confine light, this phenomenon can be used for beamreshaping [9], and an analytical model is in this contextparticularly useful. References
1. P. H. Tichit, A. Moreau and G. Granet, “Localizationof light in lamellar structure with left-handed medium:the Light Wheel,” Opt. Expr.10025(2010).