Colloquium: Light scattering by particle and hole arrays

This Colloquium analyzes the interaction of light with two-dimensional periodic arrays of particles and holes. The enhanced optical transmission observed in the latter and the presence of surface modes in patterned metal surfaces is thoroughly discussed. A review of the most significant discoveries in this area is presented first. A simple tutorial model is then formulated to capture the essential physics involved in these phenomena, while allowing analytical derivations that provide deeper insight. Comparison with more elaborated calculations is offered as well. Finally, hole arrays in plasmon-supporting metals are compared to perforated perfect conductors, thus assessing the role of plasmons in these types of structures through analytical considerations. The developments that have been made in nanophotonics areas related to plasmons in nanostructures, extraordinary optical transmission in hole arrays, complete resonant absorption and emission of light, and invisibility in structured metals are illustrated in this Colloquium in a comprehensive, tutorial fashion.

Figure 1

(Color online) Extraordinary optical transmission in hole arrays. The measured transmittance (solid curves) is shown for apertures drilled in (a) gold and (b) silver films, from 84 and 96, respectively. The silver film is self-standing in air, while the gold film is deposited in quartz and immersed in an index-matching liquid. The lattice constant is $a=600\mathrm{nm}$ in Au and $a=750\mathrm{nm}$ in Ag. The transmittance of the perforated gold goes well above that predicted for noninteracting apertures in a perfect-conductor film or by Bethe’s formula for a thin screen (dashed curves). Rayleigh’s condition for the (1,0) and (1,1) beams becoming grazing are indicated by vertical dashed lines. Analytical results are shown as arrows in (a) and as a dashed curve in (b) (see Sec. 4D). The transmittance is presented vs wavelength in the dielectric environment of the metal, normalized to the lattice constant.

Figure 2

(Color online) Two-dimensional array of small identical particles illuminated by a light plane wave. ${\mathbf{k}}_{\Vert }$ is the momentum component parallel to the array. The particle at position ${\mathbf{R}}_n$ displays a dipole ${\mathbf{p}}_n$.

Figure 3

(Color online) Reflectance spectra of square arrays of perfectly conducting thin circular disks. The wavelength $\lambda$ is normalized to the lattice constant $a$. The disks radius is (a) $b=a∕5$ and (b) $b=a∕9$. Light is impinging normal to the array and 100% reflection is observed in these two cases at the maximum. Solid curves, full numerical results. Dashed curves, analytical model for ${\mid r\mid }^2$ [Eq. (11)].

Figure 4

(Color online) Lattice sum $G_{zz}\left({\mathbf{k}}_{\Vert }\right)$ [Eq. (4)] for a square lattice of period $a$ as a function of parallel momentum $k_{\Vert }$ and wavelength $\lambda$. The direction of ${\mathbf{k}}_{\Vert }$ is along one of the axes of the lattice.

Figure 5

(Color online) Babinet’s principle applied to disk and hole arrays. The transmittance (reflectance) of the disk array for light of a given polarization $\sigma$ ($s$ or $p$) is identical to the reflectance (transmittance) of the complementary hole array for orthogonal polarization ${\sigma }^{\prime }$ ($p$ or $s$, respectively).

Figure 6

(Color online) Geometrical construction of the condition of full transmission in a hole array. (a) Wavelength dependence of the real part of the lattice sum $G_{xx}$ [Eq. (4)] for $k_{\Vert }=0$. (b) Normal-incidence transmittance of a hole array complementary of the disk array of Fig. 3a $(b=a∕5)$: exact calculation (solid curve), analytical model of Eq. (11) (dashed curve), and Fano profile of Eq. (15) (dotted curve). The transmission minimum at $\lambda =a$ results from the divergence of $G_{xx}$, while the transmission maximum (see vertical dashed line) is derived from the condition that $\mathrm{Re}\left\{G_{xx}\right\}$ equals the inverse of the hole polarizability, according to Eq. (11).

Figure 7

(Color online) Response of a small hole in a perfect-conductor thick film. (a) The field scattered by a subwavelength aperture in response to external electric $\left(E^{\mathrm{ext}}\right)$ and magnetic $\left(H^{\mathrm{ext}}\right)$ fields is equivalent (at large distance compared to the radius $b$) to that of effective electric $\left(p\right)$ and magnetic $\left(m\right)$ dipoles, which allow defining polarizabilities (${\alpha }_E$ and ${\alpha }_M$, respectively) both on the same side as the external fields $\left({\alpha }_{\nu }\right)$ and on the opposite side $\left({\alpha }_{\nu }^{\prime }\right)$. Only the perpendicular component of the electric field and the parallel component of the magnetic field can be nonzero at the surfaces of the perfect-conductor film. (b),(c) Thickness dependence of the real part of the hole response functions $g_{\nu }^{\pm }$ for $\lambda ⪢b$ [the imaginary part satisfies Eq. (16)].

Figure 8

(Color online) Thickness dependence of the normal-incidence transmittance spectra of square arrays of circular holes drilled in perfect-conductor films, according to Eq. (19). The hole radius $b=0.2a$, the wavelength $\lambda$, and the film thickness $h$ are given relative to the period $a$ (see insets).

Figure 9

(Color online) Lattice surface modes in a perforated perfect conductor. (a) The contour plot shows the modulus of the specular reflection coefficient of a semi-infinite metal for incident $p$-polarized light as a function of wavelength $\lambda$ and parallel momentum $k_{\Vert }$ (see insets for parameters). The upper-right inset shows a detail of the reflectivity as compared to the mode position predicted by Eqs. (23, 24) (see arrow). A reflection coefficient larger than 1 is only possible for evanescent waves outside the light cone. (b) Lattice modes in a perforated thin film, as measured by 157 (symbols).

Figure 10

(Color online) Polarization of holes in perfect conductors. (a) Electrostatic electric-field flow lines for a circular hole drilled in a semi-infinite perfect conductor subject to an external field ${\mathbf{E}}^{\mathrm{ext}}$ perpendicular to the surface, giving rise to an electric dipole $p={\alpha }_E$ $E^{\mathrm{ext}}$ as seen from afar. (b) Magnetostatic magnetic-field flow lines for the same hole subject to an external parallel field ${\mathbf{H}}^{\mathrm{ext}}$ and leading to a magnetic dipole $m={\alpha }_M$ $H^{\mathrm{ext}}$. (c) Summary of polarizabilities for square and circular holes in perfect-conductor surfaces, normalized using the aperture area $S$. The values for the circular hole are taken from the $h⪢b$ limit of Fig. 7. The circular opening in a thin screen is analytical (12; 71), but we must correct the right-hand side of Eq. (24) by a factor of 4 in this case because of cooperative interaction between both sides of the film.

Figure 11

(Color online) Enhanced transmission driven by a localized resonance. The normal-incidence transmission of a circular aperture drilled in a perfect-conductor film and filled with dielectric material is represented for different values of the permittivity $ϵ$ (see labels). The transmitted power is normalized to the incoming flux within the hole area. Evidence of Fabry-Perot resonances is clear in the $ϵ=50$ spectrum.

Figure 12

(Color online) Interplay between localized (site) and extended (lattice) resonances. The contour plots show the zero-order beam transmittance of a square array of circular holes drilled in a perfect-conductor film and filled with dielectric material of permittivity $ϵ=50$ as a function of parallel momentum ${\mathbf{k}}_{\Vert }$ and wavelength $\lambda$. The orientation of ${\mathbf{k}}_{\Vert }$ and the ratios between the hole radius $b$, the lattice constant $a$, and the film thickness $h$ are specified in the insets. The light is (a) $p$ polarized and (b) $s$ polarized. A transmission coefficient larger than 1 is only possible for evanescent waves below the light cone.

Figure 13

(Color online) Summary of surface plasmon properties. (a) Surface plasmon dispersion relation for a Drude metal of bulk plasmon frequency ${\omega }_p$. (b) Extension of the plasmon field into the metal (skin depth), into the vacuum, and along the surface (propagation distance) for several metals, as obtained from measured optical constants (75; 113).

Figure 14

(Color online) Effective polarization strength of a silver sphere near a silver planar surface. The sphere radius is a tenth of the wavelength. The polarization is normalized to the sphere volume. The dielectric function of silver is from 75.

Figure 15

(Color online) Instantaneous induced electric field set up by a perpendicular electric dipole (see vertical arrows) sitting at a distance $\lambda ∕20$ from the surface of a metal described by Eq. (12) with ${\omega }_p=15\mathrm{eV}$ and damping $\eta =0.6\mathrm{eV}$ (typical of Al) at frequency $\omega ={\omega }_p∕2$. The electric-field component parallel to the surface (this is radial with respect to the position of the dipole) and the component along the surface normal are represented separately. Poynting vector flow lines are superimposed on the plot of the normal component.

Figure 16

(Color online) Dipole-dipole interaction in metallic surfaces. (a) Schematic representation of the scaling of dipole-dipole interactions for electric and magnetic dipoles with respect to their separation $R$ near a metallic surface. The interaction decays as $\exp \left(ikR\right)∕R^n$ near a perfect conductor or as $\exp \left(i{k}_{\Vert }^{\mathrm{SP}}R\right)∕R^m$ near a metal with a dominant surface plasmon (see insets for values of the exponents $n$ and $m$). (b) Dipole-dipole interaction near a silver surface at a wavelength of $750\mathrm{nm}$ (three upper solid curves) as compared with the plasmon-pole approximation (three upper dashed curves, see text). We also show the interaction at a wavelength of $10\mathrm{mm}$ (lower curve, perfect-conductor limit). The dipole-dipole separation vector $\mathbf{R}$ is taken along $\widehat{\mathbf{x}}$.

Figure 17

(Color online) Relation between the power radiated after transmission through a deep subwavelength hole $\left(I_{\text{free}}\right)$ and the power emanating as surface plasmons $\left(I_{\mathrm{SP}}\right)$ for gold and silver, derived in the small-hole limit. [We obtain the emitted light intensity from integration of the radial Poynting vector corresponding to the field of Eq. (28) plus the dipole direct field over the upper hemisphere, far from the hole. The SP intensity comes from the Poynting vector normal to a cylindrical surface centered at the hole, with the field given by Eq. (30).] The metal dielectric function is from 75.

Figure 18

(Color online) Lattice sums and lattice resonances in a square array of holes drilled in gold vs a perfect conductor. The real part of the exact lattice sum for interaction of parallel magnetic $\left(M\right)$ dipoles is shown for gold (black curve) and for a perfect conductor (PC, gray curve), as compared to analytical approximate expressions (symbols). The Rayleigh condition for a period $a=600\mathrm{nm}$ is indicated by gray and black vertical dashed lines for light in the dielectric $(\lambda ∕n=a)$ and for surface plasmons $({\lambda }_{\mathrm{SP}}=a)$, respectively. Changes in the inverse magnetic polarizability of circular holes of different size [horizontal lines, as obtained from Fig. 7b (see footnote 9)], lead to different wavelengths of the lattice surface modes, as indicated by vertical arrows for the condition that the real part of the denominator of Eq. (3) be zero.

Figure 19

(Color online) Normal-incidence absorbance of (i) a silver particle array embedded in silica (refraction index $n=1.45$), (ii) the same array near a planar silver-silica interface, and (iii) an array of silica inclusions buried in silver below a silver-silica interface. All particles are spheres of $200\mathrm{nm}$ in diameter. The arrays have square symmetry with lattice constant $a=500\mathrm{nm}$. The distance from the sphere surfaces to the planar interface is $10\mathrm{nm}$ in the buried silica particles and $900\mathrm{nm}$ for the silver particles. The Rayleigh conditions for the reduced wavelength of light in the silica $(\lambda ∕n=a)$ and for the wavelength of the silver-silica interface plasmon $({\lambda }_{\mathrm{SP}}=a)$ are indicated by arrows $A$ and $B$, respectively.