Light passing through subwavelength apertures

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers

Rev. Mod. Phys. 82, 729 – Published 12 March 2010

Abstract

This review provides a perspective on the recent developments in the transmission of light through subwavelength apertures in metal films. The main focus is on the phenomenon of extraordinary optical transmission in periodic hole arrays, discovered over a decade ago. It is shown that surface electromagnetic modes play a key role in the emergence of the resonant transmission. These modes are also shown to be at the root of both the enhanced transmission and beaming of light found in single apertures surrounded by periodic corrugations. This review describes both the theoretical and experimental aspects of the subject. For clarity, the physical mechanisms operating in the different structures considered are analyzed within a common theoretical framework. Several applications based on the transmission properties of subwavelength apertures are also addressed.

51 More

DOI:https://doi.org/10.1103/RevModPhys.82.729

Authors & Affiliations

F. J. Garcia-Vidal

Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid, E-28049 Madrid, Spain

L. Martin-Moreno

Instituto de Ciencia de Materiales de Aragón (ICMA) and Departamento de Física de la Materia Condensada, CSIC–Universidad de Zaragoza, E-50009 Zaragoza, Spain

T. W. Ebbesen

ISIS, Université de Strasbourg and CNRS, 67000 Strasbourg, France

L. Kuipers

Center for Nanophotonics, FOM Institute for Atomic and Molecular Physics, Science Park 104, 1098 XG Amsterdam, The Netherlands

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 82, Iss. 1 — January - March 2010

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Various types of aperture structures that are considered in this review.Reuse & Permissions
Figure 2
(Color online) Schematic representation of a unit cell of lengths $L_{x}$ and $L_{y}$ containing a finite set of indentations perforated on a metallic film of thickness $h$ . The different terms appearing in the set of equations (1) are also schematically depicted.Reuse & Permissions
Figure 3
Schematic definition of the different scattering coefficients for the EM fields at isolated interfaces dividing two semi-infinite media. (a) A plane wave impinging from medium I into medium II. (b) Incidence from a waveguide mode $α$ in medium II into medium III. (c) As in (b) but now the waveguide mode $α$ is backscattered in medium II or transmitted to medium I.Reuse & Permissions
Figure 4
Schematic of a single slit of width $a$ perforated on a metal film of thickness $h$ illuminated by $p$ -polarized radiation. For the case of $s$ -polarized incident radiation, vectors $E$ and $H$ are interchanged.Reuse & Permissions
Figure 5
Transmission (in arbitrary units) as a function of frequency for a single slit of width $75 μ m$ perforated on a metal film of thickness $28.2 mm$ . Inset: Normalized-to-area transmittance calculated using the coupled-mode method. From 413.Reuse & Permissions
Figure 6
(Color online) Normalized-to-area transmittance vs wavelength (in units of the cutoff wavelength of the fundamental TE mode $λ_{C} = 2 a$ ) through a single slit of width $a$ perforated on a PEC film of different thicknesses $h$ , ranging from $h = 0$ to $3 a$ . The normally incident plane wave is $s$ polarized.Reuse & Permissions
Figure 7
(Color online) Normalized-to-area transmittance for a normally incident plane wave vs $k_{ω} r$ for a circular hole of radius $r$ perforated on PEC films of different thicknesses $h$ . Dashed and dotted lines show the predictions of Eqs. (9, 10), respectively. The cutoff condition of the hole waveguide, $k_{ω} r = 1.85$ , is also depicted.Reuse & Permissions
Figure 8
(Color) Transmission spectra for single circular holes. Inset: Scanning electron micrograph (SEM) image of a single circular hole of radius $r = 135 nm$ perforated on a suspended silver film. Main panel: Transmittance spectra for normal incident radiation for a range of film thicknesses $h$ . Each curve is an average of spectra obtained for several isolated holes of the same size. From 98.Reuse & Permissions
Figure 9
(Color) Normal incidence transmission spectra of an isolated rectangular aperture for various linear polarizations, defined as the angle that forms the $E$ field with the long side of the rectangle. The thickness of the free-standing silver film is $h = 700 nm$ and the sides of the rectangle are $a_{x} = 210 nm$ and $a_{y} = 310 nm$ . From 98.Reuse & Permissions
Figure 10
(Color) Transmission spectra of isolated rectangular apertures with fixed $h = 300 nm$ and $a_{y} = 270 nm$ with various $a_{x}$ $(105 < a_{x} < 260 nm)$ , perforated through silver films. All structures are illuminated by a normal incident plane wave whose $E$ field points along the short side of the rectangle. From 98.Reuse & Permissions
Figure 11
(Color online) Transmission through a single rectangular hole. Inset: Schematic of a single rectangular hole of sides $a_{x}$ and $a_{y}$ perforated through a metallic film of thickness $h$ . The structure is illuminated by a $p$ -polarized plane wave. Main panel: Normalized-to-area transmittance vs wavelength (in units of the cutoff wavelength $λ_{C} = 2 a_{y}$ ) for a normally incident $p$ -polarized plane wave impinging on single rectangular holes for different ratios $a_{y} ∕ a_{x}$ . The thickness of the metal film is $h = a_{y} ∕ 3$ . From 147.Reuse & Permissions
Figure 12
(Color online) Normalized transmission amplitudes, at normal incidence, of random arrays of holes with dimensions (a) $a_{x} = a_{y} = 200 μ m$ , (b) $a_{x} = 70 μ m$ and $a_{y} = 390 μ m$ , and (c) $a_{x} = 70 μ m$ and $a_{y} = 655 μ m$ . In all three cases the metal thickness is $h = 17 μ m$ . Insets are scanning electronic microscope (SEM) images of the 2D random arrays. From 222.Reuse & Permissions
Figure 13
(Color online) Schematic of a 1D periodic array of slits (period $d$ , width of the slits $a$ ) perforated on a metallic film of thickness $h$ . The structure is illuminated by $p$ -polarized radiation, with the $E$ field lying in the $x − z$ plane.Reuse & Permissions
Figure 14
(Color) Transmission resonances in 1D transmission gratings. (a) Transmission efficiency as a function of both $ω$ and $k_{x}$ . Period of the grating, $d = 500 nm$ ; width of the slits, $a = 50 nm$ ; and thickness $h = 400 nm$ . (b) Branch $B − C$ of the dispersion relation. (c) Surface-cavity (SC) ratio for the transmission branch displayed in (b). From 255.Reuse & Permissions
Figure 15
Total transmittance (normalized to the unit cell of the array) of a normal-incidence $p$ -polarized plane wave vs wavelength (in units of $d$ ) for periodic arrays of slits of fixed width, $a = 0.2 d$ , and several thicknesses $h = 0.2 d$ , $0.4 d$ , $0.6 d$ , and $0.8 d$ . Full lines show the calculations in which the ten lowest TM waveguide modes are considered whereas dotted lines render the results when only the first waveguide mode (TEM $α = 0$ ) is included.Reuse & Permissions
Figure 16
Dependence of $ρ$ vs wavelength for three different values of the ratio between the width of the slits and period of the array, $a ∕ d$ . (a) The modulus of $ρ (R)$ and (b) its phase $(θ)$ , in units of $π$ .Reuse & Permissions
Figure 17
(Color) Transmission spectra versus dispersion relations of the surface EM modes. Left panels: Transmittance spectra as a function of both wave number and wave vector for $p$ -polarized light impinging at two different arrays of slits. Right panels: Dispersion relation for the EM guided modes as obtained from Eq. (18). (a) $a = h = 0.2 d$ . (b) $a = 0.2 d$ and $h = 1.0 d$ .Reuse & Permissions
Figure 18
(Color online) Transmission through 1D periodic arrays of slits. (a) Inset: SEM image of one of the 1D arrays analyzed in these experiments. The period of the array is $d = 750 nm$ , the width $a = 100 nm$ , and two different thicknesses are analyzed, $h = 400$ and $450 nm$ . Main panel: Zero-order transmittance of a 1D slit array illuminated by a normal incidence $p$ -polarized plane wave. (b) Dispersion diagram [spectral locations of the transmission peaks vs energy (in eV) and parallel momentum (in units of $2 π ∕ d$ )] for the sample analyzed in panel (a) with $h = 400 nm$ . From 302.Reuse & Permissions
Figure 19
SEM image of a 2D hole array of circular holes (diameter of $150 nm$ ) milled in a $260 − nm$ -thick Au film that is deposited on a glass substrate. The hole arrays count $30 \times 30$ holes and the period of the square array is $460 nm$ . Courtesy of Eric Laux.Reuse & Permissions
Figure 20
Zero-order transmission of subwavelength hole array in a Ag film on a quartz substrate for normal incidence; the lattice period is $600 nm$ , the film thickness is $200 nm$ , and the diameter of the circular holes is $150 nm$ . The markers indicate positions for the maxima and minima expected for SPP modes on either the silver-quartz or the silver-air interface (solid markers) and the Wood-Rayleigh anomalies associated with the same interfaces (dashed markers). From 154.Reuse & Permissions
Figure 21
Measured and predicted dispersion of the transmittance through a subwavelength hole array. The transmitted intensity is given by a gray scale. The loci of the maxima (full lines) related to the excitation of SPPs on a flat surface and the minima ascribed to the Wood-Rayleigh anomaly (dashed lines) are superimposed on the measured data. From 154.Reuse & Permissions
Figure 22
Gray-scale images show the zero-order transmittance for two holey silver films as a function of the frequency and the in-plane wave vector $k_{x}$ . (a) Circular hole geometry for a silver film on glass. The scale is linear and the maximum transmittance is 32%. (b) Square hole geometry for a silver free-standing film. The maximum transmittance is 50% in this case. Superimposed black curves correspond to the SPP dispersion relations of (a) glass-silver interfaces perforated by circular holes and (b) air-silver interfaces drilled with square holes. The white curves correspond to the SPP dispersion relation for a flat interface: (a) glass-silver and (b) air-silver. From 216.Reuse & Permissions
Figure 23
(Color) Transmission through 2D periodic arrays of holes. (a) Experimental zero-order transmittance of a gold film on a quartz substrate $(ϵ_{s} = 2.31)$ as a function of the dielectric constant of the superstrate $ϵ_{L}$ . The film thickness is $250 nm$ , the hole diameter is $200 nm$ , and the period of the square array is $600 nm$ . (b)–(d) Calculated cross-sectional views of the electric field amplitudes with various states of resonance: (b) off resonance in the asymmetric situation (transmission 0.005%), (c) on resonance in the asymmetric situation (transmission 0.5%), and (d) on resonance in the symmetric situation (transmission 20%). In the three cases, light is impinging from the top and the $E$ -field amplitude is normalized to the amplitude of the incident plane wave using a logarithmic color scale. From 211.Reuse & Permissions
Figure 24
Schematic of a 2D hole array of square holes of side $a$ forming a square array of period $d$ perforated through a metallic film of thickness $h$ . The incident EM field is a $p$ -polarized plane wave.Reuse & Permissions
Figure 25
(Color online) Transmittance through a hole array drilled in a PEC film. The side of the hole is $a = 0.4 d$ and the film thickness is $h = 0.2 d$ . The solid curve is the result from the converged multimode expansion. The dashed curve is for the single-mode expansion discussed in this work. The dotted curve corresponds to the minimal model (see text).Reuse & Permissions
Figure 26
(Color online) Transmittance for a $p$ -polarized plane wave impinging into a hole array within the single-mode and PEC approximations. The side of the hole is $a = 0.4 d$ and the film thickness is $h = 0.2 d$ . The solid curve is for normal incidence, while the dashed curve is for incidence at $θ = 5 °$ .Reuse & Permissions
Figure 27
(Color online) Real (discontinuous lines) and imaginary (continuous lines) parts of $ρ$ , the reflection amplitude for a Bloch combination of ${TE}_{01}$ waveguide modes impinging into vacuum. The Bloch wave vector forms either 0° with the normal to the surface or 5°. The side of the hole is $a = 0.4 d$ .Reuse & Permissions
Figure 28
(Color online) Approximate dispersion relations for geometrically induced SPPs in a semi-infinite holey PEC film computed without diffraction effects. The side of the square holes is $a = 0.4 d$ . A dielectric medium with dielectric constant $ϵ = 1$ (solid curve) or $ϵ = 9$ (dotted curve) fills the holes. The discontinuous line marks the spectral region of what will become leaky modes when diffraction effects are considered.Reuse & Permissions
Figure 29
(Color) Evolution of the transmission with the thickness of the metal film. Top panel: Normal-incidence transmittance spectra through a hole array in a PEC film for different metal thicknesses $h$ . The side of the holes is fixed at $a = 0.4 d$ . Middle panel: Spectral dependence of $| ρ |$ (black curve) and $e^{| q_{z} | h}$ (color lines for the corresponding metal thicknesses represented in the upper panel). Bottom panel: Spectral dependence of $| G - Σ |$ (continuous line) and $| G^{V} |$ (dashed lines). The vertical discontinuous lines are guides for the eye marking the crossings of $| ρ |$ with the different $e^{| q_{z} | h}$ curves or, alternatively, the cuts between $| G - Σ |$ and $| G^{V} |$ .Reuse & Permissions
Figure 30
(Color) $E$ -field amplitude evaluated at the two resonant wavelengths in the $x − z$ plane that cuts the square holes by their centers for the case $h = 0.4 d$ depicted in Fig. 29. (a) The lower wavelength (antisymmetric mode). (b) The amplitude for the peak appearing at a longer wavelength (symmetric mode).Reuse & Permissions
Figure 31
Evolution as a function of metal thickness $(h)$ of the threshold value for the radii of circular holes $(r_{\min})$ forming a 2D array in order to have 100% transmission at, at least, one resonant wavelength $λ_{R}$ (full line). The dashed line represents the linewidth associated with the transmission peak in hole arrays with $r = r_{\min}$ for each value of $h$ . The inset shows the spectral separation between $λ_{R}$ and the period of the array $d$ , $Δ = λ_{R} - d$ , evaluated at $r = r_{\min}$ for each value of $h$ .Reuse & Permissions
Figure 32
(Color online) Transmittance spectra through an array of circular holes in a free-standing silver film of thickness $h = 275 nm$ . The radius of the hole is $r = 134 nm$ and the period of the array is $d = 600 nm$ . The dielectric function of silver is taken from 301. The solid curve is the FDTD result, obtained with a discretization mesh of $2 nm$ . The dashed curve is the result obtained within the coupled-mode method and the SIBC approximation. The dotted curve is for the last case, but with the absorption in the metal set zero, i.e., $Im (ϵ_{M}) = 0$ .Reuse & Permissions
Figure 33
(Color) Dependence of EOT with the material properties. (a) Experimental transmission intensities as a function of the (1,0) transmission peak position for different metals. The samples consist of perforated metal slabs (thickness $250 nm$ ) deposited on a glass substrate. The period of the square array is varied between 200 and $500 nm$ and the ratio between the period of the array and the diameter of the holes is fixed at 1.75. (b) Calculated transmission spectra for an array of circular holes in a metal film deposited on a glass substrate for several metals. Calculations were performed with the FDTD method. The geometrical parameters considered are period $d = 400 nm$ , metal thickness $h = 250 nm$ , and hole radius $r = 114 nm$ . From (a) 317 and (b) 330.Reuse & Permissions
Figure 34
Measured dependence of the transmission of the (1,0) peak through an array of square holes as a function of size of the holes (double logarithmic representation). The thickness of the gold films is $200 nm$ and they are deposited on a glass substrate. All arrays investigated have a period of $425 nm$ . The dotted line obeys $T \propto d^{4}$ . The dashed line corresponds to $T \propto d^{9}$ . For smaller holes the increase in normalized transmission is easily faster than $d^{4}$ , while for the larger holes the normalized transmission saturates. From 390.Reuse & Permissions
Figure 35
(Color online) Measured transmission spectra for three different hole shapes: circular (diameter $190 nm$ ), rectangular ( $150 \times 225 {nm}^{2}$ ; aspect ratio 1.5), and rectangular ( $75 \times 225 {nm}^{2}$ ; aspect ratio 3). As the aspect ratio of the holes is increased from 1 to 3, the normalized transmission of the primary peak increases by one order of magnitude. From 208.Reuse & Permissions
Figure 36
Measured normalized transmission of nanohole arrays consisting of rectangular holes with varying aspect ratio for two different incident polarizations. The legends show the aspect ratios of the holes. The insets show the relative direction of the polarization with respect to the hole geometry. (a) The normalized transmission exhibits an increase for all peaks. The (1,0) peak exhibits broadening and a pronounced redshift as the aspect ratio is increased. (b) For $x$ -polarized light the opposite trend is observed: an increase in aspect ratio results in a decrease in normalized transmission. For the largest aspect ratios a small amount of polarization cross-talk is observed that can be attributed to minute alignment imperfections. From 389.Reuse & Permissions
Figure 37
(Color) Transmission spectra of rectangular holes with the lattice constant as a varying parameter, plotted for the polarization (a) perpendicular to the short edge and (b) perpendicular to the long edge. All samples were fabricated with the same rectangular holes $(260 \times 200 {nm}^{2})$ perforated on suspended Ag films of thickness $400 nm$ . From 96.Reuse & Permissions
Figure 38
Measured dispersion of the (1,0) peak for different aspect ratios. The dispersion changes as the aspect ratio is increased: the fitting lines decrease in slope and the dispersion around $k_{∥} = 0$ becomes flatter for a larger range of $k_{∥}$ . This indicates a stronger degree of localization in the transmission process. From 389.Reuse & Permissions
Figure 39
(Color online) Double logarithmic plot of transmission of the fundamental beam at $830 nm$ (right axis, dots) and the collected second harmonic signal (left axis, triangles) as a function of aspect ratio. The transmission of the fundamental beam exhibits a monotonic increase before leveling off at an aspect ratio of 1.6. The second harmonic shows a steady increase of slightly less than two orders of magnitude up to an aspect ratio of 1.6. Between aspect ratios of 1.6 and 2.3 a peak is observed in the second harmonic with a magnitude of a factor 50. The lines are guides for the eye. From 393.Reuse & Permissions
Figure 40
(Color online) Double logarithmic plot of two components of the effective second-order nonlinear susceptibility vs aspect ratio. Between aspect ratios of 0.36 and 2.8 no clear trend is observable in the effective nonlinear susceptibility. Between aspect ratios of 1.6 and 2.8 a peak in the effective susceptibility of a factor more than one order of magnitude occurs. From 393.Reuse & Permissions
Figure 41
(Color online) Dependence of EOT with the number of holes. (a) Experimental transmission spectra for 2D square arrays containing $5 \times 5$ , $11 \times 11$ , $21 \times 21$ , and $31 \times 31$ circular holes. The arrays were milled in $275 − nm$ -thick free-standing silver films with a period $d = 600 nm$ and a hole radius $r = 134 nm$ . Transmissions are normalized to the hole area. (b) Normalized-to-area transmission spectra obtained from the coupled mode method in which SIBC have been applied. The geometrical parameters are the same as in the experiments. Inset: Comparison between the coupled mode and FDTD results for an infinite square array. From 316.Reuse & Permissions
Figure 42
(Color online) Dependence of the transmission enhancement with the diameter of the holes. (a) Experimental and (b) theoretical ratios of the transmission of $40 \times 40$ arrays to the transmission of the corresponding single hole with diameters of 150, 200, 250, and $300 nm$ . All data are presented in a logarithmic scale for better visualization. From 316.Reuse & Permissions
Figure 43
Optical transmission spectrum of a Penrose array with a rhombus edge of $P = 450 nm$ and a hole diameter of $150 nm$ . The transmitted intensity $I$ is normalized to the incident one $I_{0}$ . The inset is a SEM image of the array milled through a $300 − nm$ -thick silver film. From 318.Reuse & Permissions
Figure 44
(Color online) Transmission $(T)$ spectra for the ordered case (broken line), the Penrose lattice (continuous line), and the random configuration (dots). In all three cases, $r = 130 nm$ , $h = 170 nm$ , and $N = 636$ . $T$ is normalized to the value obtained for $N$ independent holes. The inset shows the structure factor of the Penrose lattice considered.Reuse & Permissions
Figure 45
(Color) Transmission per hole (normalized to the single-hole transmission) displayed in a color scale. (a), (b) Ordered arrays with circular and square external shapes, respectively. Both patterns were computed at the resonant wavelength of both structures: $λ = 575 nm$ . The results for the Penrose lattice at the corresponding resonant wavelengths are shown for (c) $λ = 500 nm$ and (d) $λ = 585 nm$ . The geometrical parameters are the same as in Fig. 44.Reuse & Permissions
Figure 46
SEM images of two samples showing a single aperture surrounded by periodic corrugations. (a) 1D case: a single slit symmetrically flanked by grooves. (b) 2D case: the so-called bull’s eye structure, a cylindrical hole surrounded by circular trenches.Reuse & Permissions
Figure 47
(Color) Normalized-to-area transmittance spectra for different structures formed by a central slit symmetrically flanked by ten grooves on the input surface. The width of the slits is $40 nm$ and the thickness of the free-standing silver film is $h = 350 nm$ . (a) In these spectra the depth of the grooves is fixed at $w = 40 nm$ and the period of the groove array $d$ is varied between 500 and $800 nm$ . The black curve corresponds to the single-slit case that is also reproduced in the inset for clarity. (b) Here the period is fixed, $d = 650 nm$ , and the depth of the grooves is changed from $w = 10 nm$ up to $275 nm$ . The inset shows a SEM image of one of the devices analyzed. From 142.Reuse & Permissions
Figure 48
(Color) Collimation of light emerging from a single aperture. (a) Transmission spectra at $p$ -polarized normal incidence for a single slit flanked at the output surface by two periodic arrays of five grooves (one placed to the left and the other to the right of the slit). The width of the slit and grooves is $40 nm$ , the groove periodicity is $d = 500 nm$ , and the metal thickness is $h = 300 nm$ . The nominal depth of the grooves is $60 nm$ . The spectra are recorded at various collection angles from 0° to 25°. The inset shows a dispersion curve of the transmission peak locations versus parallel momentum. (b) Transmission spectra for the bull’s eye geometry: a $250 − nm$ -diameter cylindrical hole is milled into a $300 − nm$ -thick silver film. It is surrounded by five circular trenches with depths of $60 nm$ and the groove periodicity is $600 nm$ . The tail above $800 nm$ is an experimental artifact. From 233.Reuse & Permissions
Figure 49
(Color online) Schematic of a single slit of width $a$ in a metallic film of thickness $h$ . This single slit is surrounded symmetrically by $2 N$ grooves (of width $a$ and depth $w$ ) in both input and output surfaces. The structure is illuminated by a normal incident $p$ -polarized plane wave.Reuse & Permissions
Figure 50
(Color online) Normalized-to-area transmittance as a function of the number of grooves when only the input surface is corrugated. The width of the slit and grooves is $a = 0.08 d$ , the depth of the grooves is $w = 0.2 d$ , and the metal thickness is $h = 0.7 d$ . The notation $[N, 0]$ means that $2 N$ grooves are perforated in the input surface and 0 in the exit surface. From 142.Reuse & Permissions
Figure 51
(Color online) Normalized-to-area transmittance as a function of the number of grooves when only the exit surface of the slit is corrugated. As in Fig. 50, the width of the slit and grooves is $a = 0.08 d$ , the depth of the grooves is $w = 0.2 d$ , and the metal thickness is $h = 0.7 d$ . The notation $[0, N]$ means that $2 N$ grooves are milled at the exit surface and 0 at the input surface. The inset shows the transmittance for the [10, 0] and [10, 10] configurations. From 142.Reuse & Permissions
Figure 52
(Color) Contour plot of the transmittance versus groove depth $(w)$ and wavelength (both in units of $d$ ) when a central slit is flanked symmetrically by a total of 20 grooves. Here $a = 0.08 d$ and the thickness of the metal $h = 0.7 d$ . White dots correspond to the spectral locations of the surface EM modes of an infinite array of rectangular grooves with varying $w$ as calculated from Eq. (A28) with $k_{x} = 2 π ∕ d$ . From 142.Reuse & Permissions
Figure 53
(Color) Electric field amplitude associated with the transmission peak $λ = 1.1 d$ emerging when a single slit is flanked by a total of 20 grooves, placed symmetrically at its entrance surface. The geometrical parameters of the structure are as in Fig. 50.Reuse & Permissions
Figure 54
(Color) Contour plots for $I (θ, λ)$ (see the text) for two different groove depths: (a) $w = 0.02 d$ and (b) $w = 0.2 d$ . The width of the central slit and grooves is $a = 0.2 d$ . The total number of grooves that are placed symmetrically with respect to the central slit is 20. From 257.Reuse & Permissions
Figure 55
(Color) Electric field amplitude profile (in arbitrary units) as a function of $x$ and $z$ for the case $N = 10$ with $a = 0.08 d$ and $w = 0.17 d$ that corresponds to the optimal parameters for beaming. In this case $d$ is chosen to be $500 nm$ , as in the experimental samples discussed in Sec. 4A. This $E$ -field profile is evaluated at the resonant wavelength, $λ_{R} = 1.06 d$ . From 144.Reuse & Permissions
Figure 56
(Color) Plasmonic collimator for a quantum cascade laser. (a) A SEM micrograph of the quantum cascade facet patterned with 20 rings. The rings are 0.6 and $1.0 μ m$ deep with a periodicity of $7.8 μ m$ . The inset is a zoom-in view of the facet. (b) Measured 2D far-field intensity distribution for the device. Courtesy of Nanfang Yu.Reuse & Permissions
Figure 57
(Color) Photon sorting based on EOT. Left: SEM image of a triangular lattice of slits fabricated by FIB in a Au film. Inside each triangle, gratings with different periods to couple light of different wavelengths are superimposed. Each grating selectively sends the resonating color to a slit where it is transmitted to the opposite side. Right: Far-field image of the light emerging from the structure on the left. Courtesy of Eric Laux.Reuse & Permissions
Figure 58
(Color online) 1D Green’s dyadic function (real and imaginary parts) for a Ag-vacuum interface and $p$ -polarized light of $λ = 600 nm$ . The dashed curves correspond to the SIBC approach [Eq. (A17)], whereas the black curves are for a PEC [Eq. (A18)]. The dotted curves display the SPP contribution to $G$ , as given by Eq. (A20).Reuse & Permissions

Reviews of Modern Physics