1. Introduction

Light scattering from subwavelength holes drilled in metals has been the subject of long-standing interest [1] motivated by phenomena such as extraordinary light transmission [2] that challenges the severe (a/λ)⁴ cut-off predicted by Bethe [1] for the transmission cross section of single holes of small radius a compared to the wavelength λ. In particular, nearly 100% transmission has been achieved for close-to-perfect conductors in the microwave [3, 4] and THz domains [5]. Quite different from perfect conductors, real metals are capable of sustaining surface plasmons that were readily recognized to mediate the interaction among arrayed holes at visible and near-infrared frequencies [2, 6]. Furthermore, the strong correlation of the transmission enhancement with the lattice periodicity in both of these metallic regimes has prompted rigorous descriptions of the effect in terms of dynamical diffraction [7, 8].

Transmission resonances in individual holes offer an additional handle to achieve extraordinary effects. These resonances can be triggered by filling the hole with high-index-of-refraction material [9]. The combination of lattice resonances [10, 11] in hole arrays and site resonances at specific hole positions can be anticipated to yield interesting properties in line with recent studies of light reflection on metal surfaces patterned with nanocavities that support localized modes [12].

In the present work, we offer a systematics to study the phenomenology associated to light transmission through subwavelength hole arrays in perfect-conductor films, which permits us to establish the existence of full transmission resonances for arbitrarily narrow holes in thick metallic screens. This extends previous work on holes in infinitely thin metallic screens [13]. Furthermore, individual-hole resonances are obtained by filling the holes with high-index-of-refraction materials. This gives rise to enhanced subwavelength transmission, which is demonstrated both theoretically and experimentally. Finally, the complex scenario that is presented when transmission resonances of individual holes are combined with resonances originating in the array periodicity is elucidated within our analysis.

Our discussion begins by considering single holes (Sec. 2), for which we discuss the following points:

The numerical results for single holes are based upon rigorous solution of Maxwell’s equations where two different methods have been used independently: (i) the boundary element method described in Ref. [9] and consisting in representing the hole boundaries by equivalent magnetic charges and currents that are calculated self-consistently by imposing the appropriate boundary conditions on the fields; and (ii) an expansion of the electromagnetic field in terms of modes of a cylindrical cavity inside the hole and cylindrical waves outside the hole, where the expansion coefficients are determined by imposing continuity of the parallel components of the electric and magnetic fields on both planes limiting the film [14]. The results obtained from these two methods for 300 discretization points in the latter and 20 modes in the hole cavity in the former cannot be distinguished on the scale of the figures.

Our main emphasis is placed on hole arrays (Sec. 3), for which we formulate a simple model that becomes exact in the narrow-hole limit. In this model, each hole is represented by equivalent induced dipoles on either side of the film, as mentioned above. These dipoles are actually proportional to the fields at the top and bottom ends of the hole. The fields acting on a given hole are the external fields plus the sum of the fields produced by the rest of the holes. In a periodic hole array illuminated by a plane wave, the parallel momentum k _∥ is fixed by the conditions of incidence, so that all holes have the same values of the dipoles, except for a phase factor related to k _∥, and the mentioned sums reduce to structure factors that permit writing a simple self-consistent equation for the problem, involving only the dipoles in the central hole. This simple model allows us to extract the following information:

The present model focusses on small-hole arrays in finite-thickness screens. For very thin screens a different model has been formulated that makes use of Babinet’s principle to relate the hole array problem to the complementary geometry of a disk array [15]. The latter can be carried out for disks (holes) or arbitrary dimensions relative to wavelength and spacing, but it reduces to the present model when the holes are small enough that they can be represented by equivalent induced dipoles. In the thin-screen limit, the dipoles on both sides of the film are equal in magnitude and opposite in direction, which reduces the problem even further [15].

2. Single holes

In his pioneering development, Bethe [1] showed that the scattered far-field from a hole drilled in an infinitely-thin perfect-conductor screen can be assimilated to that of a magnetic dipole parallel to the screen plus an electric dipole perpendicular to it. Narrow holes can be still represented by induced dipoles in thick screens (see Fig. 1). This allows defining electric (E) and magnetic (M) polarizabilities both on the same side as the applied field (α_v, with v =E,M) and on the opposite side (α´v). Flux conservation under arbitrary illumination leads to a new exact optical-theorem type of relationship between these polarizabilities.

Indeed, by considering two plane waves incident on either side of the film and by imposing that the incoming energy flux equals the outgoing one (the polarizabilities of the hole enter only in the latter), since perfect conductors are unable to absorb energy, one obtains the condition

where k = 2π/λ is the momentum of light in free space. The remaining real parts of g^±_v = (α_v ± α´_v)^-1 are obtained numerically from the field scattered by a single hole [14, 9] as compared to a dipolar pattern, and they are represented in Figs. 1(b)–(c) for empty holes.

In the thin-film limit, Figs. 1(b)–(c) show that ∣Re{g^±_v}∣ → ∞, so that only g ^- _v becomes relevant. This corresponds to the case where α´_v =-α_v, and Eq. (1) reduces to Im{1/α_v}= -4k ³/3. Invoking Babinet’s principle, one can claim that this same expression must apply to the polarizability of a disk, and indeed it is satisfied by any polarizable particle, except that the right hand side contains an extra factor of 2 that is accounted for by the fact that the hole responds to the relevant components of the field at the plane of the film, which have exactly twice the value of those of the external field.

When a subwavelength hole is filled with dielectric material of sufficiently high permittivity ε, hole-cavity resonances can exist thanks to the reduction of the wavelength by a factor of √ε. These resonances give rise to enhanced transmission [9], as shown in Fig. 2(a) by rigorous numerical solution of Maxwell’s equations for the isolated hole problem (curves) [14, 9]. Furthermore, experimental evidence of this effect is provided in Fig. 2(b), which compares the transmission of microwaves through subwavelength holes drilled in a steel film and filled with either ε = 10.2 dielectric material of the kind used in electronic-circuit boards or air. Two horn antennas were used to perform the measurements on a square steel screen of side equal to 50 cm. The distances between horns and film are shown in the inset. A 5-fold enhancement in the transmission is observed with the dielectric as compared to the empty hole.

The width of these resonances is dictated by coupling of the cavity modes to the continuum of light states outside the film. The resonances are of Fabry-Perot origin for larger thickness [15], but the transmission line shapes are actually determined from the noted coupling to the continua outside the film, as described by Fano [16]. In particular, the vanishing of transmission when g ⁺ _M = g ^- _M, i.e., α´_M = 0, is a signature of a Fano resonance (see inset in Fig. 2(a) for ε = 50). The coupling strength drops rapidly for large ε, due in part to small transmission through the dielectric-air interface as a result of the large impedance mismatch. The larger ε, the narrower the resonance, and the higher the transmission maxima.

Incidentally, the transmission cross-section of the hole can be obtained from our effective dipole model by noticing that the transmitted magnetic far field in response to a unit-electric-field plane wave incident normal to the film is H=f _Mexp(ikr)/r, with the magnetic transmission amplitude defined as f _M = 2k ²±´_M[x̂-(̂r ∙ x̂)r̂], where the incident magnetic field is taken along x and the factor of 2 results from full reflection on the perfect-conductor film. In the Gaussian units used throughout this paper, the cross section it then obtained by integrating ∣f _M∣² over the transmission hemisphere. The cross section normalized to the area of the hole is then given by 16k ⁴∣α´_M∣²/3a ² and represented by symbols in Fig. 2(a) for ε = 50. This compares remarkably well with the exact result (curve) for small a/λ.

3. Periodic hole arrays

Periodic arrays of sufficiently narrow and spaced holes can also be described by perpendicular electric dipoles p and p´ and parallel magnetic dipoles m and m´, where primed (unprimed) quantities are defined on the entry (exit) side of the film as determined by the incoming light [see Fig. 1(a)]. We consider first a unit-electric-field p-polarized plane wave incident on a hole array with parallel momentum k _∥ along the x axis, so that the external (incident plus reflected) field in the absence of the holes has parallel magnetic field H^ext_y = 2 along the y direction and perpendicular electric field E^ext_z = -2k _∥/k along z. One can write the following set of multiple-scattering equations for the self-consistent dipoles, that respond both to the external field and to the field scattered by the other holes [10, 11]:

 

Fig. 1. (a) The field scattered by a subwavelength hole drilled in a perfect-conductor film in response to external electric (E ⁰) and magnetic (H ⁰) fields is equivalent (at a large distance compared to the radius a) to that of effective electric (p) and magnetic (m) dipoles, which allow defining polarizabilities (α_E and α_M, respectively) both on the same side as the external fields (α_v) and on the opposite side (α´_v). Only the perpendicular component of the electric field and the parallel component of the magnetic field induce dipoles. (b)- (c) Real part of the hole response functions g^±_v for ε = μ = 1.

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Fig. 2. (a) Light transmittance through a circular hole drilled in a perfect-conductor film and filled with dielectric material for different values of the permittivity ε (see labels). The transmitted power is normalized to the incoming flux times the hole area. The ratio of the film thickness to the hole radius is 0.1. The left inset shows the transmission for ε = 50 in log scale. (b) Ratio of transmission for ε = 10.2 dielectric filling and ε = 1 (air) under the same conditions as in (a): theory (solid curve) vs experiment (symbols). The transmission through an infinite ε = 10.2 dielectric of the same thickness is shown as a dashed curve for comparison (dashed and solid curves approach each other in the high-frequency limit).

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where G_j and H describe the fields induced at a given hole by the other holes. Noticing that the dipoles depend on hole positions R = (x,y) only via phase factors exp(ik _∥ x), one finds

The solution of the above equations can be written

with

The zeroth-order transmission of the holey film is then obtained from the far field set up by the infinite 2D array of induced dipoles, T = ∣(2πk ²)/(Ak_z)(m´ - p´k _∥/k)∣². Here, A is the lattice unit-cell area and $k_z=\sqrt{k^2-k_{\parallel }^2}$.

The above expression for T can be derived by writing the electric field at the position r ₁ produced by a magnetic dipole m´ oriented along y and an electric dipole p´ oriented along z, both located the position r, as an integral over momentum components parallel to the film using the transformations

where we have used the notation k = (k _∥,k_z) and r = (R, z). Now, multiplying this expression by a phase factor exp(ik _∥ x) and summing for all sites r = (R,0) of the hole array, the exponentials give rise to a sum over Δ -functions in k´ _∥, differing from the incident parallel momentum k_∥ = k_∥ x̂ by reciprocal surface lattice vectors. Keeping only the zeroth-order transmitted beam (i.e., the term corresponding to the origin of the reciprocal lattice), the transmitted electric field reduces to

Finally, the transmission coefficient T must equal ∣E^trans∣², since the incident electric field has unit magnitude.

Similar considerations for s-polarized light show that its transmittance reduces to T = ∣₂πkm´/A∣², with magnetic dipoles parallel to k _∥ and no electric dipoles whatsoever (E ^ext _z = 0). More precisely, m±m´ = (2k_z/k)/(g ^± _M -G_x), from where one obtains

The last identity in Eq. (4) is derived from Eq. (1) and from the exact relation Im{G_x} = 2πk_z/A-2k ³/3 for propagating light (k _∣∣ < k).

The reflectivity of the hole array if just given by R = 1-T for these non-absorbing structures, so we will only study the transmission properties, keeping in mind that transmission maxima (minima) are accompanied by reflection minima (maxima), and vanishing (100%) transmission corresponds also to 100% (vanishing) reflection.

The performance of the hole array is dominated by divergences in the lattice sums when the diffraction orders (n, l) go grazing. More precisely, for a square lattice of spacing d and for k _∣∣ along x, the sums G_j in Eq. (2) go to +∞ as

In particular, G_y and G_z (p polarization) diverge on the lowest-frequency side of all grazing diffraction orders, as illustrated in Fig. 3, whereas G_x (s polarization) diverges only for l ≠ 0 (non-straight curves). This entails different peak structure patterns for s- and p-polarized light (see Fig. 4), with the former exhibiting transmission maxima near l ≠ 0 divergences only.

Interestingly, straightforward algebra shows that Eq. (4) predicts 100% transmission whenever the condition

is fulfilled for λ > d. Eq. (6) is a second-order algebraic equation in Re{G_x} that admits positive real solutions when

a condition that can be easily satisfied near l ≠ 0 grazing diffraction orders, where G_x can be chosen arbitrarily large within a narrow range of wavelengths [see Eq. (5)]. It should be noted that the difference g ⁺ _M - g ^- _M falls off rapidly to zero when the film thickness t is made much larger than the hole radius a for empty holes [see Fig. 1(b)]. However, for fixed t/a ratio and angle of incidence, the left hand side of (7) reduces to a positive real constant times λA/a ³, leading to the conclusion that 100% transmission is possible at a wavelength close to the noted divergences (e.g., λ ~ d for normal incidence on a square lattice of spacing d) regardless how narrow the holes are as compared to the film thickness, provided the separation between holes (or equivalently A) is made sufficiently large.

The interaction between site and lattice resonances is explored in Fig. 4 through the transmittance of square lattices of holes filled with materials of different permittivity for various values of the t/a ratio, for both p-polarized and s-polarized incident light. The dominant features of these plots can be classified as follows:

 

Fig. 3. Lattice sum G_z [Eq. (2)] for a square lattice of period d as a function of parallel momentum k _∣ and wavelength.

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Nearly-dispersionless site resonances occur both for p- and s-polarized light. In the examples of Fig. 4, no such resonances are observed for ε = 1, but clear indication of a site resonance is shown in Fig. 4(e) for s polarization and ε = 10, which corresponds roughly to the resonance of Fig. 2(b). Notice that this resonance is suppressed near the light line as evidence of the divergence of the lattice sums in that region (Wood’s anomaly). This site resonance is more strongly mixed with lattice resonances for p polarization, as shown in Fig. 4(b). Actually, mixing of site and lattice resonances occurs all throughout Fig. 4 [see avoided level crossings in Figs. 4(c),(g) near k _∥ = π/d], and methods recently introduced for slit arrays [18] could be readily applied to quantify and describe the degree of localized/de-localized character.

As ε increases the site resonances are pushed downwards in frequency [e.g., the resonance just discussed in Figs. 4(b),(e) nearly halves its frequency in Figs. 4(c),(f), where a higher-frequency resonance enters the plot]. Furthermore, higher-frequency site resonances are accompanied by vanishing of the transmission [see point (iii) above].

When comparing s and p polarization, the latter contains resonances that parallel those of the former and also additional resonances where G_z diverges but G_x does not. This is clear when comparing Figs. 4(a),(d). Furthermore as a rule of thumb Fig. 4 indicates that the degree of mixing of lattice and site resonances is stronger for p polarization. This seems plausible when one realizes that only p-polarized light involves normal electric fields at the surface, keeping in mind that a normal electric field suffers more strongly from the presence of an air-dielectric interface due to local conservation of the electric displacement as compared to a magnetic field.

 

Fig. 4. Zeroth order light transmittance through square arrays of circular holes drilled in perfect-conductor films as a function of parallel momentum k _∥ and wavelength λ. The ratio of the hole radius to the lattice spacing is taken as a/d = 0.2. Different values of the film thickness t and the dielectric constant inside the holes ε are considered, as shown by labels. Both p-polarized (H parallel to the film) and s-polarized (E parallel to the film) incident light are examined.

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Fig. 5. Normal-incidence transmittance (top), lattice sums and hole response functions (middle), and hole polarizability [bottom; see Fig. 1(a)] under the same conditions as in Fig. 4(j) (a/d = 0.2, t/a = 0.1, ε = 100).

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For large values of ε the transmission is dominated by site resonances, which absorb the effect of lattice resonances to a large extent [see Figs. 4(g)–(l)]. The latter become then very narrow and their presence is mainly observed via avoided level crossings.

Finally, it should be noted that for incident evanescent light with k < k _|| < 2π/d - k, the lattice sums satisfy Im{G_j}=-2k ³/3 and Im{H}=0. This implies that Δ_± [Eq. (3)] is real and can vanish for specific combinations of k and k _∣, leading to simultaneous infinite transmittance and reflectance in what constitute film-bound resonances, as recently predicted for related metal structures [19, 20].

4. Conclusions

A simple and powerful formalism has been used to analyze transmission through hole arrays leading to surprising results such as 100% transmission for thick perfect-conductor films perforated by arbitrarily narrow holes with the appropriate hole-lattice spacing. The fact that 100% transmission is possible in non-absorbing structures has been pointed out before [4, 21], but our theory shows rigorously that this is also possible for holes however small.

For isolated holes, both theoretical and experimental evidence of transmission resonances obtained by filling a hole with large-index-of-refraction material have been presented, resulting in considerable transmission enhancement. The existence of discrete states localized at the hole cavity gives rise to Fano-like transmission resonances by coupling to the continuum of states outside the film. The Fano-profile character of these resonances are evidenced by wavelengths of vanishing transmission close to transmission maxima.

Finally, filled-hole arrays have been shown to exhibit a colorful phenomenology, including new types of suppressed transmission and a complicated interplay between hole-site resonances and lattice resonances that is satisfactorily explained within the present approach. More detailed experimental and theoretical studies of these kinds of filled-hole structures are still needed in order to fully exploit their potential for technological applications.