GGuided modes in photonic structureswith left-handed components
P Markoˇs
Faculty of Mathematics, Physics and Informatics, Comenius University in BratislavaMlynsk´a dolina 2, 842 28 Bratislava, SlovakiaE-mail: [email protected]
Abstract.
The spectrum of guided modes of linear chain of dielectric and left-handedcylinders is analyzed. The structure of eigenfrequences is much more richer if cylindersare made from the left-handed material with both permittivity and permeabilitynegative. The number of guided modes is much larger, and their interaction withincident electromagnetic wave is much stronger. For some value of the wave vector, noguided modes were found. We discuss how these specific properties of guided modescorrespond to folded bands, observed recently in photonic structures with left-handedcomponents.PACS numbers: 42.70.Qs
Keywords : photonic structures, Fano resonances, left-handed materials
Submitted to:
J. Phys. B: At. Mol. Phys. a r X i v : . [ phy s i c s . op ti c s ] J a n uided modes in photonic structures with left-handed components
1. Introduction
Spatial periodicity of photonic structures composed from two dielectric materials isresponsible for a broad variety of interesting physical phenomena. Band structure offrequency spectrum, appearance of gaps of forbidden frequencies [1, 2, 3], bound states[4], surface states [5] and guided modes in periodic structures [6, 7] represent only afew of them. Artificial metal – composite with negative effective permittivity can beconstructed when one material is supplied by metal [8, 9, 10].Discovery of left-handed materials [11, 12] with simultaneously negativepermittivity and permeability, naturally addresses the question whether such materialscould be used in the construction of new photonic structures and how negativity’sof permittivity and permeability influence properties of resulting composite. Alreadyin one dimensional systems, a new gap associated with zero mean value of refractiveindex was observed [13]. In two dimensional photonic crystals, the use of left-handedcomponents led to the existence of folded bands with zero density of states for acertain interval of wave vectors [14, 15]. Numerical calculation of the transmissioncoefficients for photonic crystals with left-handed cylinders led to unexpected numericalinstabilities when standard numerical methods were used [16]. Typical example of suchinstability is shown below in figure 8 which shows numerical data for transmissioncoefficient obtained by the transfer matrix algorithm for different spatial discretizationgive inconsistent results, which do not converge even in the limit of the most accurateaccessible numerical approximation. Physically, such numerical instability indicates thatthe spatial distribution of electromagnetic field inside the photonic structure is stronglyinhomogeneous [17].In this paper we suggest that unusual properties of periodic photonic structureswith left-handed components are due to excitation of broad spectra of guided modeswhich interact with incident electromagnetic wave. We show that indeed photonicstructures with left-handed component support the excitation of a broad variety ofspatially inhomogeneous guided modes which strongly influence the transmission ofelectromagnetic waves.The paper is organized as follows: in Section 2 we describe the model and derivemathematical equations for guided modes in the most simple photonic structure: thelinear chain of cylinders. Obtained system of linear equations is used in Section 3 tocalculate of complete spectrum of guided modes both inside and outside the light cone.In Section 4 we use our numerical method for the calculation of the transmission throughof plane electromagnetic wave through our structures. For dielectric cylinders, we provedthat Fano resonance observed in the transmission spectra are due to the excitation ofleaky guided modes [1, 6, 7]. Much stronger interaction of transmitted wave with guidedmodes is found in structures containing left-handed cylinders. Conclusion is given inSection 5. uided modes in photonic structures with left-handed components xyz Figure 1.
Structure consists from periodic linear chain of homogeneous cylindersof infinite length in the z direction. Cylinders have relative permittivity ε andpermeability µ . We consider ε = 12 and µ = 1 for the dielectric cylinder, and ε = − µ = − y = 0. Central cylinder is located at x, y = 0 ,
0. The structureis periodic in the x direction with period a . The radius of cylinders is R = 0 . a .Electromagnetic wave impinges on the structure in y direction with polarization E (cid:107) z or H (cid:107) z . Since the structure is periodic in the x direction, it is sufficient to consideronly one unit cell.
2. The model
The structure of interest consists of an infinite periodic chain of cylinders embeddedin the vacuum. (figure 1) Distance between cylinders, a , defines spatial periodicity ofthe structure along the x direction. Cylinders are infinite along the z direction, haveradius R = 0 . a and are made from homogeneous material with relative permittivity ε and permeability µ . We chose ε = +12, µ = +1 for dielectric cylinders, and ε = − µ = − x direction with wave vector q and frequency ω . If ( q, ω ) lies inside the light cone (leaky modes with q < ω/c ), thesestates could couple to incident electromagnetic field. Consider first the E z polarized electromagnetic wave. For the central cylinder, weexpress the electric intensity e z and tangential component of the magnetic intensity h t in terms of Bessel functions [18]. Inside the cylinder ( r ≤ R ) we find e in z ( r, ϕ ) = J α +0 + 2 (cid:88) k> α + k J k cos( kϕ ) + 2 i (cid:88) k> α − k J k sin( kϕ ) − iZh in t ( r, ϕ ) = J (cid:48) α +0 + 2 (cid:88) k> α + k J (cid:48) k cos( kϕ ) + 2 i (cid:88) k> α − k J (cid:48) k sin( kϕ ) (1)In equation 1 we used J k = J k (2 πnr/λ ) , J (cid:48) k = J (cid:48) k (2 πnr/λ ) (2) uided modes in photonic structures with left-handed components J k , J (cid:48) k are the Bessel function ( k = 0 , , . . . ) and their derivatives, λ = 2 πc/ω isthe wavelength of electromagnetic field in vacuum, n = √ εµ is the index of refraction ofthe cylinder and Z = Z (cid:112) µ/ε ( c is the light velocity in vacuum and Z is the vacuumimpedance). Unknown parameters α + and α − determines amplitudes of even ( ∝ cos kϕ )and odd ( ∝ sin kϕ ) cylindrical waves, respectively.The electric and magnetic fields outside the central cylinder consist from twocontributions: the first one is the field scattered on the central cylinder, e z ( r, ϕ ) = H β +0 + 2 (cid:88) k> β + k H k cos( kϕ ) + 2 i (cid:88) k> β − k H k sin( kϕ ) − iZ h t ( r, ϕ ) = H (cid:48) β +0 + 2 (cid:88) k> β + k H (cid:48) k cos( kϕ ) + 2 i (cid:88) k> β − k H (cid:48) k sin( kϕ ) (3)( r ≥ R ) with free parameters β + and β − . Here, H k = H k (2 πr/λ ) , H (cid:48) k = H (cid:48) k (2 πr/λ ) . (4)and H k ( z ) = J k ( z ) + iY k ( z ) is the first Hankel function [18, 19].The second contribution to the external fields is represented by fields scatteredfrom all other cylinders. For the n th one ( n = ± , ± , . . . , N s ) we express electric andmagnetic intensity in coordinates associated with the center of the cylinder (figure 2) e nz ( ξ n , θ n ) = H ( w n ) β + n + 2 (cid:88) k> β + nk H k ( w n ) cos( kθ n )+ 2 i (cid:88) k> β − nk H k ( w n ) sin( kθ n ) − iZ h nt n ( ξ n , θ n ) = H (cid:48) ( w n ) β + n + 2 (cid:88) k> β + nk H (cid:48) k ( w n ) cos( kθ n )+ 2 i (cid:88) k> β − nk H (cid:48) k ( w n ) sin( kθ n ) Z h nr n ( ξ n , θ n ) = H ( w n ) β + n + 2 (cid:88) k> β + nk kw n H k ( w n ) cos( kθ n )+ 2 i (cid:88) k> β − nk kw n H k ( w n ) sin( kθ n ) (5)where we use w n = 2 πξ n /λ (6)Thanks to the periodicity of the model, coefficients β ± nk of the n th cylinder couldbe expressed in terms of coefficients β ± k as β ± nk = e iqna β ± k (7)Coefficients α and β could be calculated from the requirement of the continuity offields e z and h ϕ at the surface of the cylinder: e in z ( R − , ϕ ) = e out z ( R + , ϕ ) = e z ( R + ) + (cid:88) n (cid:54) =0 e nz ( ζ n , θ n ) (8) uided modes in photonic structures with left-handed components ϕ θθ n -n ξξ nana n-n Figure 2.
Parameters used in derivation of equation 5. and h in t ( R − ) = h out t = h t ( R + , ϕ )+ (cid:88) n (cid:54) =0 (cid:2) h nt n ( ξ n , θ n ) cos α n − h nr n ( w n , θ n ) sin α n (cid:3) (9)where α n = θ n − ϕ .Before solving this system of equations, we first transform all fields in equation 5as a functions of R and ϕ . This can be done with the use of the Gegenbauer formulafor cylindrical functions (figure 3) H m ( w ) e ± imχ = + ∞ (cid:88) k = −∞ H m + k ( u ) J k ( v ) e ± ikα (10)[19] and similar formula for the first derivative H (cid:48) m ( w ): H (cid:48) m ( w ) e ± imχ = + ∞ (cid:88) k = −∞ H (cid:48) m + k ( u ) J k ( v ) e ± ikα (11)Inserting (11) into equations (5) we express, after some algebra, the fields at the outerboundary of the cylinder in the form e out z ( R + , ϕ ) = N (cid:88) k,m =0 B km β + m cos kϕ + N (cid:88) k,m =1 C km β − m sin kϕh out t ( R + , ϕ ) = N (cid:88) k,m =0 B’ km β + m cos kϕ + N (cid:88) k,m =1 C’ km β − m sin kϕ (12)Where we introduced N as the highest order Bessel function used in the expressions ofthe fields. Explicit form of matrices B , B’ , C and C’ is given in the Appendix A. uwv χα Figure 3.
Parameters used in Gegenbauer’s relation, equations 10 and 11 uided modes in photonic structures with left-handed components e iz ( r, ϕ ) = J ( v ) + N (cid:88) k> e + k cos( kϕ ) + i N (cid:88) k> e − k sin( kϕ ) − iZ h iϕ ( r, ϕ ) = J (cid:48) ( v ) + N (cid:88) k> h + k cos( kϕ ) + i N (cid:88) k> h − k sin( kϕ ) (13)Inserting fields (1) and 12) into equations 8 and 9 we obtain the set of linearequations for even modes: A k α + k = (cid:80) m B km β + m + e + k ζ A’ k α + k = (cid:80) m B’ km β + m + h + k (14)( k = 0 , , . . . , N ) and similar equations for odd modes A k α − k = (cid:80) m C km β − m + e − k ζ A’ k α − k = (cid:80) m C’ km β − m + h − k (15)( k = 1 , , . . . , N ). Here, ζ = Z/Z , A k = J k (2 − δ k ) and A’ k = J (cid:48) k (2 − δ k ) (equation1). In the next step, we remove α + from equations 14 and obtain the system of linearequations for parameters β + : (cid:88) m (cid:20) B km − ζ J k J (cid:48) k B’ km (cid:21) β + m = e + k − ζ J k J (cid:48) k h + k , k = 0 , , . . . , N (16)and equivalent equation for parameters β − : (cid:88) m (cid:20) C km − ζ J k J (cid:48) k C’ km (cid:21) β − m = e − k − ζ J k J (cid:48) k h − k , k = 1 , , . . . , N (17)With known parameters β ± , we find α ± from equations 14 and 16.In numerical analysis, we consider the number of cylinders N s = 100 − N ≤
22. We found that consideration of N = 12 modes is sufficientin most of analyzed problems. Transmission coefficient can be calculated as the ratio of the y -component of thePoynting vector S x ( y p ), calculated for any y p > r to the incident Poynting vector, S iy T = S x ( y p ) S ix = (cid:90) + a/ − a/ d x e z ( x, y p ) h ∗ x ( x, y p ) / (cid:90) + a/ − a/ d x e iz ( x, y p )( h ix ( x, y p )) ∗ (18) uided modes in photonic structures with left-handed components a / λ - l n | A kk | even odd qa / 2 a / π λ Figure 4.
Guided modes of the chain of dielectric cylinders ( ε = 12, µ = 1). Leftfigure shows the frequency dependence of diagonal elements ln A kk for q = 0 (equations20 and 21. Four even modes and three odd modes can be identified. Right figurespresents the spectrum of E z polarized Open and closed symbols correspond to evenmodes ( n = 0 and 1) and odd mode n = 1. Owing to the periodicity of the problem,it is sufficient to consider wave vectors 0 < q < π/a [1]. The above analysis Can be repeated for the H z polarized modes. It turns out that finalequations (16) and 17) remain the same if we substitute ζ → ζ (19)and exchange of e x and h z in expression for the transmission coefficient (18).
3. Guided modes
Consider first equations 16 and 17 without incident electromagnetic wave. Then, thespectrum of guided modes can be found from the requirements of zero determinantdet (cid:20) B km − ζ J k J (cid:48) k B’ km (cid:21) = 0 (20)and det (cid:20) C km − ζ J k J (cid:48) k C’ km (cid:21) = 0 (21)for even and odd modes, respectively.With the use of Gauss-Jordan elimination method [20], we transform matrices inequations (20) and (21) to diagonal form and plot the frequency dependence of inverse ofobtained diagonal elements A kk . This enables us not only to find the eigenfrequency ofguided modes and their lifetime [21], but also identify their symmetry. As an example,we show in the left figure 4 resonances of leaky guided modes with zero wave vector q = 0. uided modes in photonic structures with left-handed components aq / 2 a / π λ Figure 5.
Spectrum of guided modes of the chain of cylinders made from theleft-handed material ( ε = − µ = −
1) cylinders. There are no modes around q c ≈ . × π/a . Open and closed symbols correspond to even and odd modes,respectively. Note that guided modes changes symmetry when q crosses q c . Right figure 4 presents eigenfrequences of guided modes as a function of wave vector q identified from the position of resonances in corresponding diagonal elements. Wefound two even modes with k = 0 and 1 and one odd mode k = 1, which agree with ourexpectation [1].The spectrum of guided modes for left-handed cylinders is more complicated andconsists of series of even and odd modes. In contrast to dielectric cylinders, theeigenfrequences of guided modes decreases when mode index k increases. The numberof modes depends on the model parameters and increases when absolute value of therefractive index increases.Another unexpected property of guided modes is the existence of ”critical value”of the wave vector q c ( q c ≈ . × π/a in our model) for which no guided mode exists(figure 5). Preliminary analysis of other left-handed structures with µ = − q c depends neither on (negative) permittivity nor on the radiusof cylinder.
4. Interaction of external field with guided modes
We solve equations 16 and 17 and calculate transmission coefficient for incident planewave with wavelength λ propagating along the y direction. First, we express thecoefficients E ± k and h ± k e + k = h + k = J k ( v )(1 + ( − k ) / (1 + δ k ) k = 0 , , . . .e − k = h − k = J k ( v )(1 − ( − k ) , k = 1 , , . . . (22)( v = 2 πR/λ ). From known coefficients β + and β − we find spatial distribution of fieldsand calculate transmission coefficient given by equation 18. uided modes in photonic structures with left-handed components T r a n s m i ss i on a / λ β β β Figure 6.
Transmission of the E z polarized plane wave propagating through aninfinite 1D chain of dielectric cylinders. Top panel displays the transmission coefficientas a function of a/λ , calculated by the transfer matrix method and by the formula(18). Three Fano resonances correspond to maxims of coefficients β shown in middlepanel. Real part of the intensity of electric field inside the cylinder for three resonancefrequencies, a/λ = 0 . , .
759 and 0 .
958 is shown in the bottom panel.
Figure 6 shows the transmission coefficient for the E z polarized plane electromagneticwave as a function of wavelength. Obtained results agree very well with numericaldata calculated by the transfer matrix method [23, 24]. Typical Fano resonances in thetransmission spectra [6, 7] are clearly visible and could be associated with excitationof corresponding guided modes [22] in cylinder chain. Owing to the symmetry of theincident wave which does not depend on x , only modes symmetric with respect to x → − x could be excited. Two even modes with k = 2 and k = 4 and one odd mode( k = 3) have been excited. Bottom panels show the real part of electric field inside thecylinder and confirm predicted symmetry of excited states.Similar results have been obtained also for the H z polarized incident wave shownin figure 7. In this case, two resonances of k = 0 mode are clearly visible. The lowerone, together with the first odd resonance, deform strongly the transmission spectrum.Three resonances observed at higher frequencies could be easily detected both from thetransmission spectra and from the frequency dependence of parameters β . uided modes in photonic structures with left-handed components T r a n s m i ss i on a / 0246 λ β β β β Figure 7.
Transmission of the H z polarized plane wave propagating through aninfinite chain of dielectric cylinders. Top panel displays the transmission coefficientas a function of a/λ , calculated by the transfer matrix method and by the formulaequation 18. Fano resonances correspond to maxims of coefficients β shown in middlepanel. Real part of the intensity of magnetic field inside the cylinder for three resonancefrequencies, a/λ = 0 . , .
865 and 0 .
931 is shown in the bottom panel.
The same analysis of the transmission spectra of cylinders made from the left-handedmedium ( ε = − µ = −
1) is more difficult, since standard numerical techniques(transfer matrix method, RCWA ) seem not to be suitable for the calculation of thetransmission coefficient. As shown in the left panel of figure 8, results obtained bythe transfer matrix method with three discretization of the unit cell provide us withcompletely different transmission coefficients. Such failure, obtained also when thetransmission was calculated by the RCWA method indicates that the eigenstates of theelectromagnetic field inside the structure are strongly inhomogeneous. Consequently,excited guided modes cannot be expanded in a finite series in the the basis ofeigenfrequences, used in the transfer matrix method.The assumption of strong spatial inhomogeneity is confirmed by frequencydependence of coefficients β shown in middle panel of figure 8. Incident electromagneticwave excites in the left-handed chain a series of guided modes. The resonant frequenciesof these modes lie very close to each other (in fact, since the resonances have a finitewidth, they overlap). Also, absolute values of coefficients β are in order of magnitude uided modes in photonic structures with left-handed components T r a n s m i sss i on a / λ a / even odd λ a / T r a n s m i ss i on λ Figure 8.
Left: transmission of the E z polarized electromagnetic wave through thelinear chain of left-handed cylinders calculated by the transfer-matrix method withthree different space discretization (120, 240 and 360 mesh points per unit cell). Middle:spectrum of coefficients β . Note the scale of the vertical axis. Right panel shows thetransmission coefficient calculated by the present method (equation 18 ). Bottompanel shows real part of the intensity of electric field inside the left-handed cylinder forfrequencies associated with resonances of even modes ( a/λ = 0 . , .
078 and 0 . . , .
047 and 0 . larger than in the case of dielectric cylinder.Right panel of figure 8 shows the transmission coefficient obtained by equation (18).Two minims in the transmission coefficient coincide with eigenfrequences of excitationof guided modes. The spatial distribution of electric field shown in the bottom panelconfirm that many resonances are excited simultaneously – in contrast to dielectricstructure, we cannot easily identify the order of excited modes.The above mentioned problem of numerical instability is not actual whentransmission of the H z polarized wave is calculated (figure 9). Now, numerical dataobtained by the transfer matrix method agree perfectly with those found by the presentmethod (left panel). As shown in the right panel, resonances of guided modes are veryweak and broad, so that they have negligible influence on the transmission coefficient. uided modes in photonic structures with left-handed components a / T r a n s m i ss i on λ a / even odd λ Figure 9.
Left: Transmission of H z -polarized plane wave through the chain of left-handed cylinders calculated by the transfer matrix method and from equation 18.Similarly to the case of dielectric cylinders, two numerical methods give identical result,because Fano resonances (shown in the right panel) are much weaker than in the caseof E z polarized wave (Fig. 8).
5. Conclusion
We presented physical and numerical analysis of the guided modes of linear chain ofcylinders, made either from the dielectric or from the left-handed material. Comparisonof these two structures shows new phenomena could be observed when permittivityand permeability possess negative values. We proved that the transmission ofelectromagnetic waves through the photonic structure with left-handed components isstrongly influenced by guided modes excited in the structure.The spectrum of guided modes contains much more branches. In contrast todielectric structure, the eigenfrequency of these modes decreases when mode index k increases. Also, there are no guided modes for the wave vector close to some criticalvalue q c . We suppose that this absence of guided modes corresponds to the folding offrequency bands observed recently [15, 14].Very rich spectrum of guided modes is responsible for numerical instabilities whenthe transmission coefficients is calculated by standard methods. Since eigenmodes ofthe left-handed structure strongly differs from plane waves, any numerical algorithm,based on the expansion of the fields into the plane waves (transfer matrix or RCWA)fails to recover true transmission coefficient. uided modes in photonic structures with left-handed components
6. Appendix A
For k, m = 1 , , . . . N B , J k ≡ J k (2 πR/λ ) and H k given by equation 4 we obtain the N + 1 × N + 1 matrix B B = H + N s (cid:88) n =1 qH ( un ) J B m = N s (cid:88) n =1 H m ( un ) J × [( − m e iqan + e − iqan ] B k = N s (cid:88) n =1 H k ( un ) J k × e iqan + ( − k e − iqan ] B km = 2 H k δ km + N s (cid:88) n =1 [ e iqan ( − m − k + e − iqan ][ H m − k ( un ) + ( − k H m + k ( un )] J k (23)and the N × N matrix C : C km = 2 H k δ km + N s (cid:88) n =1 [ e iqan ( − m − k + e − iqan ] × [ H m − k ( un ) − ( − k H m + k ( un )] J k (24)Matrices B’ and C’ could be obtained from B and C , respectively, by substitutions J k → J (cid:48) k , H k → H (cid:48) k (25) Acknowledgment
This work was supported by the Slovak Research and Development Agency underthe contract No. APVV-0108-11 and by the Agency VEGA under the contract No.1/0372/13.
References [1] Joannopoulos J D, Johnson S G Winn J N and Meade R G 2008
Photonic Crystals: Molding theFlow of Light
Optical Properties of Photonic Crystals (Berlin, Heidelberg: Springer, 2005)[3] Soukoulis C M (editor) 2001
Photonic Crystals and Light Localization in the 21st Century (Dordrecht: Kluver)[4] Robertson W M et al.
Phys. Rev. Lett. et al. Optics Lett. (1993)[6] Fan S and Joannopoulos J D 2002 Phys. Rev. B et al. Phys. Rev. Lett. Phys. Rev. Lett. Optics Lett. Science , 1780[12] Engheta N and Ziolkowski R W (Editors) 2006
Metamaterials: Physics and EngineeringExplorations. (New York: J. Wiley & Sons) uided modes in photonic structures with left-handed components [13] Li J et al Phys. Rev. Lett. et al Phys. Rev. B et al New J. Phys. private communication [17] Asatryan A A, Dossou K B and Botten L C 2008 Australian Institute of Physics, 18th nationalCongress [18] Stratton J A 1941
Electromagnetic Theory (New York: Mc Graw-Hill Comp.)[19] Abramowitz M and Stegun I A 1965
Handbook of Mathematical Functions (Dover Publ.)[20] Press W H et al
Numerical Recipes (Cambridge: Cambridge University Press)[21] Pfeiffer C A, Economou E N and Ngai K L 1974
Phys. Rev. B , 3038[22] Astratov V N et al J. Light. Technol. Phys. Rev. Lett. Phys. Rev. E65