Three-dimensional structures for enhanced transmission through a metallic film: Annular aperture arrays F. I. Baida* and D. Van Labeke Laboratoire d’Optique P.M. Duffieux, Centre National de La Recherche Scientifique Unite ´ Mixte de Recherche 6603, Institut de Microtechniques de Franche-Comte ´, Universite ´ de Franche-Comte ´, 25030 Besanc ¸on Cedex, France Received 14 March 2002; revised manuscript received 7 November 2002; published 25 April 2003 Light transmission through arranged nanostructures can be increased comparing to what was observed three years ago by Ebbesen et al. Nature, 391, 667 1998. The results obtained recently by Krishnan et al. Opt. Commun. 200,1 2001 with gold hole grating are confirmed, but a new design is suggested in order to enhance the light transmission. A three-dimensional finite difference time domain simulation was performed in order to obtain the spectral response of metallic periodic structures in the visible domain. The grating structure is modified by filling the central region of each hole with a concentric cylinder of smaller diameter. The grating now consists of a periodic array of annular apertures. Theoretical simulations were performed with such a subwavelength silver grating. It is shown that the transmission efficiency can reach 80% in the visible spectral range. DOI: 10.1103/PhysRevB.67.155314 PACS numbers: 78.20.Bh, 42.25.Fx I. INTRODUCTION In 1998, Ebbesen et al. 1 experimentally demonstrated that an ‘‘extraordinary’’ transmission of light could be obtained through subwavelength hole arrays in metallic films. Their results have stimulated new works and many papers have been dedicated to the subject of enhanced transmission. 2–12 Most of the theoretical studies were devoted to one- dimensional 1D grating. 2,4–8 To our knowledge, the first calculation of the transmission through a 2D grating was performed by Popov et al. 8 They used a Fourier modal method extended to crossed gratings. Generally the phenom- ena of enhanced transmission is interpreted in terms of sur- face plasmon resonance. 3–5,9,12 In Refs. 9 and 12, the trans- mission spectra of a 2D hole grating are calculated by using a modal expansion of the fields. In those papers, a very simple and efficient minimal model is developed which al- lows the authors to conclude that ‘‘the holes behave like subwavelength cavities for the evanescent waves coupling the surface plasmon on either sides of the films.’’ By using a differential method, Salomon et al. 10 numerically studied the transmission of a very thin 20 nm 2D hole grating. Refer- ences 2, 6, and 7 attribute the transmission enhancement to cavity resonances into the holes. In Ref. 11, Vigoureux analyses the Ebbesen experiment in terms of short range dif- fraction of evanescent waves. A very clear insight of the problem of enhanced transmis- sion is given by Popov et al. 8 and their discussion is at the origin of our paper. For a one-dimensional grating made of subwavelength slits engraved into a thick metal, there is al- ways a TEM mode without cutoff that can propagate through the slits without attenuation. The occurrence of this mode explains grating anomalies and very high transmission ob- tained for perfect metal. 13–15 Considering real metals with finite conductivity, the occurrence of losses in the metal gen- erally attenuates the transmission. But, if the field extension in the metal remains limited, a large transmission can be maintained. A nice example of enhanced transmission by a resonant cavity mode is presented in Ref. 7. A 80% diffrac- tion efficiency in transmission is obtained by illuminating a silver lamellar grating period 0.9 m, thickness 1.8 m with slits of 90 nm width. For a hole grating, the TEM mode without cutoff does not exist and the relative high transmission of Ebbesen grating is more difficult to be interpreted. By using a Fourier modal method, Popov et al. 8 have demonstrated the existence of a transmission channel which allows the resonant excitation of surface plasmon. The transmission is enhanced: for the circular aperture grating, the transmission efficiency can be larger than the ratio of hole surface to the total surface for some wave- lengths. But the effective transmission of the grating remains small. For example, the maximum of transmission experi- mentally observed in Ref. 12 is less than 8% zero-order transmission spectra through a hole grating in silver, hole diameter 200 nm, lattice constant 600 nm, silver thickness 250 nm. A theoretical calculation on a similar grating leads to a transmission around 8% Fig. 19 of Ref. 8 for a square hole grating in silver thickness 200 nm, square width 250 nm, period 900 nm. The purpose of this paper is to show that an effective larger transmission can be obtained by using another struc- ture. A TEM mode cannot exist in a circular waveguide with a simply connected cross section Ref. 16, Chap. 71. It ex- ists in an infinite slit. It does not exist in a circular hollow cavity. However, a coaxial structure made with a perfect metal has a TEM mode without cutoff. 8 In the following, by using the finite difference time do- main FDTD method, we have calculated the transmission of a biperiodic array of annular apertures made in gold or silver layer. We demonstrate that very large transmission ef- ficiencies can be obtained with such a structure. II. PRINCIPLE OF THE CALCULATIONS The principle of the FDTD method can be found in many textbooks. 17,18 The computing space area and time interval PHYSICAL REVIEW B 67, 155314 2003 0163-1829/2003/6715/1553147/$20.00 ©2003 The American Physical Society 67 155314-1