SUPPLEMENTARY INFORMATION

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1 Microscopic theory of the extraordinary optical transmission (supplementary information) Haitao Liu 1,2 and Philippe Lalanne 1 1 Laboratoire Charles Fabry de l Institut d Optique, CNRS, Univ Paris-Sud, Campus Polytechnique, RD 128, Palaiseau cedex, France 2 Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, Institute of Modern Optics, Nankai University, Tianjin , P. R. China. 1/ SPP-MODEL COUPLED-MODE EQUATIONS In this Section, we derive the coupled-mode equations that provide a SPP-model for the extraordinary optical transmission (EOT). For that purpose, we consider the scattering problem shown in Fig. A, where a metal-air interface is perforated by an array of N hole chains. The interface that is illuminated by an incident plane wave is periodic in the y- direction (periodicity a y ). For the sake of generality, we consider that the chains are located at arbitrary positions in the x-direction and we denote by a n the separation distance between the (n-1) th chain and the n th chain, n=0, 1,, N-1. We further denote by A n and B n the unknown SPP modal coefficients scattered in the positive and negative x-directions by the n th hole chain, and by c n the excitation coefficient of the fundamental Bloch mode supported by the n th chain. For the n th hole chain, the coupled-mode equations readily lead to A = M β( k ) + u τa + u ρb, (1a) n n x n n 1 n+ 1 n+ 1 B = M β( k ) + u τb + u ρa, (1b) n n x n+ 1 n+ 1 n n 1 c = M t( k ) + u αa + u αb, (1c) n n x n n 1 n+ 1 n+ 1 where u n =exp(ik SP a n ) and M n =w 1 w 2 w n (n>0, M 0 =1) is the in-plane phase retardation of the incident plane wave at the nth hole chain, w n being equal to exp(ik x a n ). For an array composed of N chains, Eqs. (1a)-(1c) lead to a set of 3N linear equations with 3N unknowns, the A n, B n and c n s. In the absence of any SPP incident fields on the left and 1

2 right side of the array, A -1 =0 and B N =0 (boundary conditions), the set of equations is easily solved by a matrix inversion or by an iterative procedure for very large values of N. Fig. A. Modal coefficients for a 1D finite array of hole chains located at arbitrary positions in x-direction and drilled in a metal substrate. The array is illuminated by a plane wave impinging at oblique incidence. Only three chains are shown for the sake of clarity. ψ ( ) PW denotes the incident plane wave with a k x wave vector k=k x x-k z z and with a TM polarization (magnetic vector along y-axis). + ψ SP and SP ψ represent the two SPP-modes propagating in positive and negative x-directions, and ψ 0 is the fundamental Bloch mode of an individual hole chain. For fully periodic structures as those considered in the main text (N= and a n =a x for any n), the SPP coupled-mode equations have to obey pseudo-periodic conditions (Floquet theorem), and we additionally have A n = wa n-1, B n = wb n-1, c n = wc n-1, (2) where w=exp(ik x a x ) is the phase-delay accumulated by the incident plane wave over a grating period, a x being the period in x-direction. By rewriting Eqs. (2) as A n =w n A 0, B n =w n B 0 and c n =w n c 0 and by inserting them into the set of Eqs. (1a)-(1b), one obtains β( kx)(1 wuτ) + β( kx) wuρ A0 =, (3a) (1 w uτ)(1 wuτ) u ρ B 1 1 β( kx)(1 w uτ) + β( kx) w uρ 0 = (1 w uτ)(1 wuτ) u ρ, (3b) 2

3 where u=exp(ik SP a x ). The excitation coefficient c 0 in Eq. (1c) is derived in a similar way. For periodic arrays of chains, the individual chain modes are all phased (c n =wc n-1 ). Provided that the chain distance a x is larger than the skin depth, the set of phased chainmodes forms a supermode, which is nothing else than the fundamental Bloch mode of the 2D hole array. Thus c 0 represents the excitation coefficient of this supermode. It is denoted t A (k x ) in the main text (see Fig. 1d, main text). One obtains 1 β( kx)( w uτ+ uρ) + β( kx)( w uτ+ uρ) A ( x) = ( x) + α t k t k u (1 w uτ)(1 wuτ) u ρ. (4a) Similar equations can be written for the rear interface illuminated by the fundamental Bloch supermode of the 2D array. By applying the above procedure, one obtains 2 1 uα ( w + w 2uτ + 2uρ) A ( x ) r k = r+. (4b) (1 w uτ)(1 wuτ) u ρ for the reflection coefficient r A (k x ) of the fundamental Bloch mode of the 2D hole array (Fig. 1d, main text). The essence of Eqs. (4a)-(4b), and especially of the shared denominator that results from a geometric summation over all chain contributions, is a multiple scattering process that involves the excitation of SPP modes by the incident field and their further scattering onto the infinite set of periodically-spaced chains. Under the assumption that the energy flow in the metal film is mediated only via the fundamental Bloch mode, the (0,0)-order transmittance T= t F (k x ) 2 of the membrane perforated by the 2D hole array is given by the classical Fabry-Perot formula t ( k )exp( ik nd) =, (4c) 1 r ( k )exp( i2 k nd) 2 A x 0 F( kx ) 2 A x 0 t where n is the complex effective index of the fundamental Bloch mode of the 2D hole array, and d is the film thickness. The set of Eqs. (4a)-(4c), which solely involves SPPmodes as the mediator for the interaction, provides closed-form expressions for the EOT. 3

4 2/ THE PHASE-MATCHING CONDITION FOR THE RESONANT PEAKS The important phase-matching condition that explains the presence of resonance peaks in the EOT is driven by the zeros of the denominator of t A (k x ) and r A (k x ). Its physics is discussed in details in the main text for normal incidence. For oblique incidence, k x 0, the same discussion holds and by using the denominator in Eqs. (4a) or (4b), one easily obtains that the resonance is achieved for Re(k SP )a x +arg(τ) ±k x a x, (5) modulo 2π. Thus, in comparison with the normal incidence case, the general phasematching condition incorporates an additional term, namely the phase-shift k x a x over a grating period. It is meaningful and can be simply interpreted as a condition of constructive interference for the SPP-modes generated by adjacent chains under delayed excitations ±k x a x, the plus and minus signs referring to SPP-modes propagating towards positive or negative x-directions. As shown by the comparison between the RCWA and model data in Fig. 2a (main text), the general condition quantitatively predicts the two transmission branches in the (ω, k x ) plane. Figure B illustrates the phase-matching condition and its consequences on the EOT phenomenon in the near-infrared. The results hold for normal incidence (k x =0) at a fixed wavelength, λ=0.974 µm, and for several values of the periodicity a x. It turns out that for the specific periodicities a x that satisfy Re(k SP )a x +arg(ρ+τ) = 0 modulo 2π (obtained at the crossings between the red and blue lines in Fig. Ba), the 2D-holearray Bloch mode scattering coefficients, t A and r A, exhibit sharp resonance peaks. Figure Bb shows the resonances of t A 2 as a function of a x. More specifically, since we find analytically that the maxima of t A 2 scale as 1/a x for lossless metals, we have normalized t A 2 to remove any damping associated to geometrical scattering effects. The normalized transmission (a x /λ) t A 2 exhibits sharp resonance peaks whenever the phasematching condition is fulfilled. The peak maxima of the red curve that corresponds to the SPP-model predictions gently decrease with increasing a x s because of the SPPattenuation that increases as the separation distance between adjacent chains increases. Since they are related to evanescent modes, the 2D-hole-array Bloch mode scattering coefficients are not constrained by energy-conservation laws and reach values well 4

5 larger than unity. This in turn allows an efficient funneling of light through the perforated membrane, in agreement with the conclusions of the classical modeexpansion approach [1]. Fig. B. Physical interpretation of the resonance through the multiplescattering SPP-model. a SPP phase matching condition -Re(k SP )a x =arg(ρ+τ). b Resonance peaks of the 2D-hole-array scattering transmittance t A 2. Actually, (a x /λ) t A 2 is plotted as a function of a x. Blue: RCWA, red: SPP model. The inset shows an enlarged view of the first resonance for a x /λ 1. All the plots are obtained for a y =0.94 µm, D=0.266 µm, λ=0.974 µm and normal incidence. Similar resonances are obtained for r A 2. The SPP-model also predicts that t A is very small (see inset in Fig. Bb) for the a x s values corresponding to the resonant excitation of a SPP mode on a flat interface through the reciprocal grating momentum (Re(k SP )±k x =0 modulo 2π/a x ). Such a x s are very close to and slightly larger than the a x s corresponding to the resonance peaks. Consistently with the transmittance spectra shown in Fig. 2b in the main text, Fig. Bb shows that the SPP-model underestimates the first resonance by a factor 2, but accurately predicts higher-order resonances for larger a x s. This can be understood by considering the respective weights of the CW and SPP-mode fields to the scattered light. 5

6 For a x /λ 1, the two fields are almost equal, but for longer separation distances, the SPPmode is the preponderant vector of the interaction between the adjacent chains. 3/ RELATIVE CONTRIBUTIONS OF SPP AND CW FIELDS We have also compared the SPP-model predictions with RCWA data in various regions of the spectrum, from the visible to thermal infrared, by scaling all the geometric parameters of the 2D hole arrays. For that purpose, the transmittance of subwavelength hole arrays with different periods (a=a x =a y ) has been calculated with the RCWA and with the SPP-model, using the frequency-dependent permittivity of gold given in [2]. Figure C shows the data obtained as a function of the normalized wavelength for three periodicities (a=0.68, 0.94 and 2.92 µm), corresponding to visible, near-infrared and infrared light illuminations. The two main characteristics of the transmission spectra, namely the decrease of the peak transmission and the relative blue-shift of the enhanced transmission as a increases, are well reproduced by the SPP-model. The striking correspondence between the wavelengths of the transmission peaks and those of the reflectance dips are also well predicted by the SPP-model. However, it is evident that the SPP-model becomes less and less accurate as the wavelength increases. For instance, the SPP-model almost completely fails to predict the 10% peak transmittance in the infrared spectral range (blue curves in Fig. C). This is due to the fact that the CW contribution, which is neglected in the SPPmodel, becomes the dominant vector of the multiple-scattering interaction process in the infrared. This trade is contemplated in Fig. D, where the magnetic field Re(H y ) scattered on the interface (z=0) by a 1D hole chain illuminated at normal incidence is plotted as a function of the normalized in-plane x- and y-coordinates. The upper row corresponds to the total scattered field, and the SPP and CW contributing to the total field are respectively shown in the second and third rows. We note that the CW contribution is merely independent of the metal dielectric properties, whereas the SPP contribution rapidly drops as the metal conductivity increases. At visible wavelengths, λ=0.706 µm, the SPP contribution dominates even at small distances from the chain. As the metal conductivity increases, the CW becomes largely preponderant. All this evidences that 6

7 the multiple-interaction process between adjacent hole chains at infrared wavelengths is basically driven by CWs rather than by SPP-modes. The EOT in metal films perforated by 2D hole arrays therefore encompasses two different waves, each of them being equally important in the interaction process. Fig. C. The multiple-scattering SPP-model fails at predicting the EOT of gold membranes perforated by 2D hole arrays at infrared frequencies. (a) Zeroorder transmittance spectra. (b) Zero-order reflectance spectra. Dotted curves: RCWA data. Solid curves: SPP-model predictions. The red, green and blue curves are obtained for a=0.68, 0.94 and 2.92 µm, respectively. Other parameters are d/a=0.2128, D/a=0.2828, and k x =0 (at normal incidence). Although more studies are needed for full assessment, we believe that the dualwave picture could be supported by experimental data in the future. Near-field measurements may directly evidence the dominant CW contribution in the far infrared, and the dual-wave model may be validated at visible frequencies by far-field transmittance measurements of chain doublets, in a spirit similar to that used in recent experiments with slit doublets, see Refs and 21 in the main text and [3] hereafter. 7

8 Fig. D. Dominant magnetic fields, Re(H y ), scattered on a gold interface by a hole chain illuminated by a normally-incident TM-polarized plane wave. Different gold dielectric constants ε g, corresponding to λ=0.706, and µm, are considered. The perfectly-conducting (PC) metal case is additionally shown in the upper right column. (a) Total field. The incident and specular reflected plane waves have been removed for the sake of clarity. (b) SPP contribution only. (c) Cylindrical wave contribution only. The geometrical parameters of the chain are a y /λ =0.965 and D/λ= All plots are obtained for an incident plane wave with a unitary magnetic field at the gold surface. 4/ NUMERICAL ANALYSIS The theoretical results in the main text rely on a series of computational results obtained with three-dimensional fully-vectorial calculations. Technical details and relevant references are provided in this Section. Nanohole array transmission spectra. The computation of the transmission and reflectance spectra of nanohole arrays are obtained with a grating solver, known as the Rigorous-Coupled-Wave-Analysis (RCWA). The method is a frequency modal method that operates in Fourier space. It has been first developed in the 80 s [4] and has been further refined to enhance the stability, the convergence performance and the computational efficiency [5-7]. For the calculations, we have used an in-house software [8]. The calculation mainly consists in calculating all the Bloch modes of the metallic gratings in Fourier space by solving an eigenproblem [9]. The Maxwell s differential equations are further integrated analytically in the z-direction through an S-matrix 8

9 algorithm [7]. This is achieved by matching the tangential field components at the grating interfaces with a Raleigh plane-wave expansion. For subwavelength periodicities, this solver provides highly accurate data that can be considered as virtually exact, as evidenced by previous comparisons with experimental results, for details see Ref. 10 in the main text. Elementary scattering coefficients of the hole chain. The elementary scattering coefficients defined in Fig. 1 (main text) play a central role in our theoretical analysis. Their computations that deserve special attention have been obtained with a generalized version of the RCWA, which allows to handle non-periodic structure with a supercell approach [10, 11]. The generalized method shares the same main characteristics as the RCWA (analytical integration in one space direction and sampling in Fourier domain in the two others, use of S-matrix algorithms ), but additionally incorporates perfectlymatched layers (PMLs) in the transverse directions to carefully handle the outgoing wave conditions. The PMLs are implemented as a nonlinear complex coordinate transform [11]. Referring first to the SPP scattering problem of Fig. 1a (main text), the analytical integration is performed in the x-direction. Fully periodic boundary conditions are applied in the y-direction (chain periodicity direction) and PMLs are incorporated in the z-direction. In the computation, the forward- and backwardpropagating SPP modes of the flat interface, which are otherwise known analytically, are calculated as eigenstates of the supercell transfer matrix of the flat interface surrounded by PMLs. The ρ and τ modal coefficients are straightforwardly derived from the S-matrix. Related calculations for the SPP scattering by metallic plasmonic crystals composed of finite-size triangular lattices of gold bumps have been successfully performed with the same approach, see [12] for details. The β(k x ) coefficients are derived for all k x by using a plane-wave decomposition of the electromagnetic field above the interface in the air medium. Since the z-direction is handled analytically, the computational window size can be made arbitrarily large, and the plane-wave decomposition is obtained with a high accuracy, even for highly oblique plane waves close to grazing incidence (k x k 0 ). Although the scattering coefficient α could have been computed by using the orthogonality relation of Bloch modes of the 1D hole chain, it is not estimated in this calculation. 9

10 The solution of the scattering-event problem of Fig. 1b is performed with an analytical integration in the z-direction with periodic boundaries in the y-direction and PMLs in the x-direction. The modal reflectance of the hole chain is obtained from the S- matrix. The plane-wave expansion coefficients are calculated using a plane-wave decomposition of the field in the air clad. Note that, in addition to the PMLs, real and linear coordinate transforms are applied in the x-direction to artificially increase the computational window without sacrificing the resolution requirements in Fourier space [13]. The SPP scattering coefficient α is obtained from a rigorous formalism based on a normal-mode-decomposition approach. It is analytically calculated as an overlap integral between the computed scattered field and the analytically known SPP-mode field at the flat interface [14]. The same formalism has been applied to separate the SPP and CW contributions from the total field scattered by the hole chain in Fig. 3 (main text) and in Fig. D of this document. Reciprocity. The scattering event of Fig. 1c has not been modelized since the associated scattering coefficients can all be derived from the previous calculations by virtue of the Lorentz reciprocity. Usually, this theorem that plays an important role in the SPP-model is established in the sense of the Poynting vector, i.e. with E H* products that guaranty energy conservation. For dissipative materials, like gold at optical frequencies, the classical conjugate form of Lorentz theorem is not valid, and one has to rely on the unconjugate general form of Lorentz theorem with E H pseudo- Poynting products [15]. Within this framework, all modes (the plane waves indeed, but also the Bloch modes of the hole chain for instance) obey orthogonality relations with E H products. In addition, the symmetrical property of the S-matrix, which guaranties that the SPP scattering coefficients β(k x ) in Fig. 1a and 1c for instance are strictly equal, is only achieved for a specific normalization of the modes. Normalization plays a central role in the coupled-mode equations. It corresponds to a unitary pseudo-poynting transverse mode flux, 1/2 S (E (m) H (m) ) ds =1, with E (m) and H (m) being the transverse electric and magnetic mode fields and S being the transverse cross-section of the mode. The scattering coefficient, t A, between the incident plane wave and the 2D hole-array evanescent Bloch supermode is obtained for this specific normalization. Since this 10

11 coefficient is not constrained by any energy conservation law, its modulus can be larger than one. Accuracy. For the square hole geometry considered in this work, no sampling error occurs and the only source of numerical inaccuracies is the inevitable truncation of Fourier series for numerical purpose. Convergence tests have been provided to estimate the accuracy of the computational results by incrementally increasing the number of Fourier coefficients retained in the computation. Convergence has been achieved in our calculations, letting us expect a relative error in the range of a few percents for all computational results. Typical spectra obtained for all the elementary scattering coefficients in the nearinfrared are shown in Fig. 1e (main text) and in Fig. E below. Fig. E. Wavelength dependence of the scattering coefficients α, β(k x =0), r and t(k x =0) of a single hole chain in a gold substrate illuminated under normal incidence. The chain period is a y =0.94 µm and the hole side length is D=0.266 µm. The related coefficients ρ and τ are given in Fig. 1e in the main text. References [1] Martin-Moreno, L., Garcia-Vidal, F. J., Lezec, H. J., Pellerin, K. M., Thio, T., Pendry, J. B. & Ebbesen, T. W. Theory of extraordinary optical transmission through subwavelength hole arrays. Phys. Rev. Lett. 86, (2001). [2] Palik, E. D. Handbook of Optical Constants of Solids, Part II, Academic, New York (1985). [3] Schouten, H. F., Kuzmin, N., Dubois, G., Visser, T. D., Gbur, G., Alkemade, P. F. A., Blok, H., Hooft, G. W., Lenstra, D. & Eliel, E. R. Plasmon-assisted two-slit transmission: 11

12 Young s experiment revisited. Phys. Rev. Lett. 94, (2005). [4] Gaylord, T. K. & Moharam, M. G. Analysis and Application of optical diffraction by grating. Proceedings of the IEEE 73, (1985). [5] Lalanne, P. & Morris, G. M. Highly improved convergence of the coupled-wave method for TM polarization. J. Opt. Soc. Am. A 13, (1996). [6] Li, L. Mathematical reflections on the Fourier modal method in grating theory. Mathematical Modeling in Optical Science, Frontiers in Applied Mathematics, Eds. G. Bao, L. Cowsar and W. Masters, pp , Society for Industrial and Applied Mathematics, Philadelphia, [7] Moharam, M. G., Pommet, D. A., Grann, E. B. & Gaylord, T. K. Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach. J. Opt. Soc. Am. A 12, (1995). [8] Hugonin, J. P. & Lalanne, P. Reticolo software for grating analysis, Institut d Optique, Palaiseau, France (2005). [9] Li, L. New formulation of the Fourier modal method for crossed surface-relief gratings. J. Opt. Soc. Am. A 14, (1997). [10] Silberstein, E., Lalanne, P., Hugonin, J. P. & Cao, Q. Use of grating theories in integrated optics. J. Opt. Soc. Am. A 18, (2001). [11] Hugonin, J. P. & Lalanne, P. Perfectly-matched-layers as nonlinear coordinate transforms: a generalized formalization. J. Opt. Soc. Am. A 22, (2005). [12] Baudrion, Weeber, J., Dereux, A., Lecamp, G., Lalanne, P. & Bozhevolnyi, S. Influence of the filling factor on the spectral properties of plasmonic crystals. Phys. Rev. B 74, (2006). [13] See the MM3 method in the Benchmark article: Besbes, M., Hugonin, J. P., Lalanne, P., van Haver, S., Janssen, O. T. A., Nugrowati, A. M., Xu, M., Pereira, S. F., Urbach, H. P., van de Nes, A. S., Bienstman, P., Granet, G., Moreau, A., Helfert, S., Sukharev, M., Seideman, T., Baida, F. I., Guizal, B. & Van Labeke, D. Numerical analysis of a slitgroove diffraction problem. JEOS: RP 2, (2007). [14] Lalanne, P., Hugonin, J. P. & Rodier, J. C. Theory of surface plasmon generation at nanoslit aperture. Phys. Rev. Lett. 95, (2005). [15] Snyder A. W. & Love, J. D. Optical Waveguide theory, Chapman and Hall, London, New York (1983). 12