Broadband and broadangle extraordinary acoustic transmission through subwavelength apertures surrounded by fluids

We present a mechanism for ultra-broadband transmission of acoustic waves through subwavelength hole arrays. Different fluids surrounding (or filling) the holes are considered, which allows tuning of the band of maximum transmission to different angles of incidence. For certain configurations, this band of total transmission may appear at very small incident angles, making the system ‘invisible’ to sound at almost normal incidence. Analytical expressions for the specific incident angle, and for maximum transmission at that angle, are provided for any fluid-system configuration.


Introduction
Since the discovery of the resonant phenomenon of extraordinary optical transmission [1,2], the study of electromagnetic (EM) waves propagating through subwavelength hole arrays has attracted much attention. This discovery was subsequently transferred to acoustic waves [3][4][5][6], despite the differences between the vector EM and scalar acoustic cases. Recently, a non-resonant mechanism providing total transmission over ultrabroad bandwidths at specific incident angles (referred as 'plasmonic Brewster angles') has been reported for EM waves [7], and equivalently, for acoustic ones [8][9][10][11]. Both cases are impedance matching phenomena, and this condition is only provided for symmetric configurations where the same medium at the incident and transmission regions are considered.
In this paper we study theoretically the transmission of acoustic waves through subwavelength apertures in the general case when different fluids fill the reflection and transmission semi-infinite spaces, and also the apertures themselves. We provide analytical expressions for the angle and for the transmittance on this condition, which we show depend on the hole size, array period, L, and fluids properties (their density, ρ, and speed of sound, c), but not on the film thickness, h, as this is an impedance matching phenomenon. We obtain broadband transmission not only in frequency but also in incident angles close to normal incidence.

Resonant Brewster-like angle
2.1. Homogeneous system (c 1 ¼ c 2 ¼ c 3 and ρ 1 ¼ ρ 2 ¼ ρ 3 ) Consider the schematic in figure 1: a perfectly rigid (PR) screen of thickness h, perforated with two-dimensional (2D) holes periodically arranged in a square lattice (with period L). The shape of the holes does not strongly influence the impedance matching condition, being mainly dependent on hole area, A. In this paper we consider either squares of side a, or circles of radius r, and we investigate the effect in transmission when different fluids are considered. The PR approximation, in which the externally applied energy vanishes inside the rigid body and transmission through the plates themselves is neglected, is an excellent approach to treat stiff materials, like steel or brass [12], when they are in proximity to flexible materials or air. The theoretical formalism used throughout this paper is based on this PR approximation and it consists of a modal expansion of the pressure and velocity fields in the different regions. Waves are expanded into Bloch modes in the reflection (I) and transmission (III) regions (characterized by ρ c ( , ) 1 1 and ρ c ( , 3 3 ), respectively), and the field inside the holes (characterized by ρ c ( , ) 2 2 ) is written as a linear combination of the eigenmodes. Imposing the appropriate matching conditions at the interfaces (I-II and II-III), only the amplitudes of the z-component of the , and ρ c ( , ) 3 3 , respectively, are considered. The angle of incidence with respect to the normal is θ.
velocity field at the entrance (v) and exit ( ′ v ) sides of the cavities must be computed to obtain an entire field mapping comprising far field and near field distributions. Transmission is then defined as where I 0 measures the overlap between the incident plane wave and the fundamental mode inside the hole, Σ represents the bouncing back and forth of the acoustic waves inside the cavities, and the term ν G is linked to the coupling between the two sides of the screen through the holes. Finally, G I, III account for the acoustic coupling between the fundamental eigenmode and all the diffractive waves (in regions I and III, respectively). Analytical expressions for all these quantities are defined in the appendix. Figure 2 shows the angular transmission spectra for the simplest case of an homogeneous system (i.e., = = c c c 3 ). In this calculation, we consider an array with square holes of side = a L 0.53, and thickness = h L 1.0. This set of geometrical parameters clearly shows three types of transmission mechanisms associated with these systems: that associated with surface modes at λ θ ⩾ + L · (1 sin ( )), which depends on both the incident wavelength in region I (λ) and angle (θ); Fabry-Pérot (FP) resonances at wavelengths close to λ = h l 2 (with l = 1, 2, ...), which do not depend on the incident angle; and finally, an ultrabroadband resonance of total transmission, = T 1, at a given Brewster-like angle θ =°76.7 B , which does not depend on either the incident wavelength (for λ ⩽ h 2 ) or film thickness. As it has been previously demonstrated, these phenomena are achieved under some form of resonant condition [4], or when the system is anomalously impedance matched [8][9][10] at θ = F cos ( ) B , with = F A L 2 the 'filling factor' of a unit cell. Clearly, for homogeneous systems, the tunability of the ultrabroadband resonance to small incident angles is limited by the geometry of the system. However, in the following, we will show that the incorporation of different fluids filling the system allows for tuning the broadband transmission to any incident angle, even at normal incidence.
It is important to understand the role of G I, III in the mechanism of broadband transmission, which accounts for the acoustic coupling between the fundamental eigenmode inside the hole and all the diffractive waves in each semi-infinite (reflection or transmission) space. On the one hand, for subwavelength apertures, λ > ≫ L a r , , diffraction effects can be neglected as a first approximation, so the effective impedances in the reflection G I , and transmission regions G III (equations (A.8) and (A.9) in appendix), reduce to where we have defined ζ ρ = c j j j with = j 1, 2 or 3 labeling regions I, II, and III, respectively. On the other hand, the impedance of the hole (relative to that of medium I) is defined as ζ ζ = Z 2 1 . Since the broadband transmission results from an impedance matching phenomenon, the presence of three dissimilar fluids in the system (in regions I, II, and III) will provide two conditions for impedance matching (i.e., = G Z I and = G Z III ), and therefore, two Brewsterlike angles Let us first analyze the behavior of the transmittance in a symmetric system, i.e., a 'sandwich' configuration such that fluids in regions I and III are the same (with = = c c c 1 3 , and  (3) is shown with dashed lines. The set of geometrical parameters and fluids are considered in order to provide = T 1 at small incident angle values, reaching even maximum transmission at θ =°0 (panel (a)). Note also that the system in panel (a) is 'invisible' for sound, for both a wide range of incident angles and wavelengths. Similar results (not shown here) are also obtained when the holes are filled with methanol, ethylether, or isopentane, and also when the porosity (total hole area) is less than 0.25%. All previous results can be understood as an impedance matching phenomenon where ζ ζ < 2 will provide a matching condition occurring at large incident angles, while for ζ ζ > 2 , impedance matching brings small θ values. For experimental development, we expect that the inclusion of thin membranes to prevent fluid mixing, and the effects of fluid viscosity will not strongly affect these results. Any resonances associated with the thin membranes could be tuned out of the region of interest by appropriate tensioning.   . As for those parameters, the impedance matching conditions occur for θ =°71.7 B and θ =°39.5 B III , only the latter provides a broadband of maximum transmission. Note that square holes in panel (c) present maximum transmission at the corresponding Brewster angle, supporting our conclusions that this non-resonant phenomenon is valid irrespective of the hole shape.

Maximum transmission
One fundamental difference between symmetric and asymmetric configurations relies on the maximum transmission, T R . As it was previously demonstrated [4], maximum transmission in subwavelength apertures in homogeneous configurations is given by = T I G 4 R 0 2 i , where G i is the imaginary part of G (see definitions in the appendix). Finally we extend this previous work to the general case of asymmetric systems, and we provide analytical expressions for T R as a function of the filling factor and fluid properties. The maximum transmittance is obtained following a previous work for EM waves in subwavelength holes for any dielectric environment [13]. In both the EM and acoustic cases we are solving the wave equation, and boundary conditions imposed to the electric (E) and magnetic (H) fields are analogous to those imposed to the pressure (p) and normal component of the velocity (v z ), respectively. Therefore, by solving the sound wave equation and applying some approximations (see appendix for further details), we find for acoustic waves that maximum transmittance is given by

Conclusions
In conclusion, we have shown that arrays of 2D subwavelength apertures in the general case when any fluid filling the system is considered, present an ultrabroadband of total transmission of acoustic waves at specific incident angles. By considering different fluids in the system, we have shown that total transmission can be tuned to any incident angle, finding even maximum transmission in a wide range of incident angles and wavelengths at the same time. We have provided analytical expressions based on the perfect rigid (PR) approximation of general application for the specific incident angle, and also for the maximum intensity at this angle.  Now, the pressure and velocity fields are related through z3 1 3 0 2 2 1/2 . Within the PR approximation, the modes inside the apertures coincide with the waveguide modes of those apertures (which are known analytically for some geometries [4,6,14]). In the case of subwavelength holes, considering only the first eigenmode (0 ) inside the apertures provides a good approximation for the total transmittance. We express the pressure and velocity field in region II as . A and B are unknown expansion coefficients that can be calculated imposing appropriate boundary conditions to the acoustic fields at the interfaces of the system (z = 0 and z = h). The normal component of the velocity v z is continuous everywhere along the I-II and II-III interfaces, but the pressure p is continuous only at the openings.