Elsevier

Wave Motion

Volume 93, March 2020, 102485
Wave Motion

Perfect Brewster transmission through ultrathin perforated films

https://doi.org/10.1016/j.wavemoti.2019.102485Get rights and content

Highlights

  • Perfect transmission.

  • Interface homogenization.

  • Microstructured film, Ultrathin film.

Abstract

We address the perfect transmission of a plane acoustic wave at oblique incidence on a perforated, sound penetrable or rigid, film in two-dimensions. It is shown that the Brewster incidence θ realizing so-called extraordinary transmission due to matched impedances varies significantly when the thickness e of the film decreases. For thick films, i.e. ke1 with k the incident wavenumber, the classical effective medium model provides an accurate prediction of the Brewster angle independent of e (this Brewster angle is denoted θB). However, for thinner films with ke<1, θ becomes dependent of e and it deviates from θB. To properly describe this shift, an interface model is used (Marigo et al., 2017) which accurately reproduces the spectra of ultrathin to relatively thick perforated films. Depending on the contrasts in the material properties of the film and of the surrounding matrix, decreasing the film thickness can produce an increase or a decrease of θ; it can also produce the disappearance of a perfect transmission or to the contrary its appearance.

Introduction

Extraordinary optical/acoustical transmission (EOT/EAT) refers to high transmission through films with subwavelength apertures (Fig. 1), while a single aperture would transmit light/sound very poorly [1]. Such high transmission is made possible thanks to collective effects of the holes. Collective effects can render the perforated film able to support surface waves, as observed in plasmonic structures [2] or able to support resonances of the Fabry–Perot type [3], [4]. But they can also be non-resonant when they lend to the film an effective impedance which matches that of the surrounding matrix at the so-called Brewster incidence [5], [6] and, in contrast with resonance-based mechanisms, the resulting EOT has the advantage to be mildly affected by the losses [7]. It has been shown that the Brewster incidence θ can be tuned by playing with the geometrical parameters of perforated rigid screens [8], [9] and this has been used to realize beam shifter [10] and flat lenses [11]; similar tuning has been obtained by playing on the material properties of sound penetrable films  [12], [13], [14].

From a theoretical point of view, perfect transmissions have been analyzed owing to effective medium theories based on classical homogenization. With h the spacing and k the incident wavenumber, classical homogenization predicts that for kh1 the perforated film can be replaced by an equivalent homogeneous one for which the scattering properties are explicitly known. Such approaches have been applied firstly to sound-rigid films in acoustics [15], [16] and their electromagnetic counterparts [5], [17] and they have been then generalized to more involved geometric perforations [8], [9] and to sound-penetrable films [12], [13]. By construction, effective medium theories aim to describe the effective propagation within the film and they disregard the evanescent field excited at its end boundaries. This is intuitively justified if the film is thick enough to ensure that the effects of wave propagation are dominant compared to that of the boundary layers due to the evanescent field. When the boundary layer effects cannot be neglected, other strategies of homogenization have to be sought. This may happen when they are as important as the effects of wave propagation [18], [19] or when they become dominant [20], [21], [22]. In the present case, we expect that the boundary layer effects will become dominant when ke1. In this limit, we already know that perfect transmission based on Fabry–Perot resonances cannot take place (they require ke>π). The aim of the present study is to characterize the perfect transmissions based on matched impedances for thin ke1 and ultrathin ke1 films. To do so, we use an effective interface model [22] which provides the condition of existence of perfect transmissions and when perfect transmission is possible, a prediction for θ.

The paper is organized as follows. In Section 2, the problem is set and typical variations of the Brewster angles with the film thickness are given. In Section 3, the classical effective medium model is recalled, resulting in a prediction of the Brewster incidence θ=θB, Eq. (8) valid for thick films ke>1. We also provide the predictions given by the effective interface model detailed in  [22] and the resulting Brewster angle θ, (13), valid for thin films ke<1. The variations of θ when going from thin to thick films are inspected and validated with direct numerics in Section 4. We collect in the appendices additional calculations and results.

Section snippets

Perfect transmission through perforated films

We consider propagation of acoustic waves in the harmonic regime with time dependence eiωt. The propagation is described by the wave equation for the pressure p of the form div1ρ(x)p+ω2χ(x)p=0,with the mass density ρ(x) and the isentropic compressibility χ(x) varying in space x=(x,y), see Fig. 1. In the perforated film, with thickness e and filling fraction φ, they are denoted ρi and χi=1(ρici2), with ci the sound speed; in the surrounding matrix, they are denoted ρm and χm=1(ρmcm2). With k=

Effective medium model and effective interface model

Effective models aim at simplifying the problem in the scattering region y(0,e). In the classical effective medium model, this is done by replacing the perforated film by a homogeneous and anisotropic one; in the effective interface model, this is done by imposing non intuitive jump conditions between y=0 and y=e. Both models are based on asymptotic homogenization, hence they are valid in the limit kh1 (and we shall see the predictions hold up to kh1 with reasonable errors). However, while

The case of sound-penetrable films

To begin with, we report in Fig. 3 the spectrum of the reflection computed numerically and the spectra given by the effective medium theory and given by the effective interface theory. The film is that of Fig. 2(a) and we extended the range of thickness to ke(101103) (with kh=1). As expected, the actual spectrum is well reproduced by the effective interface model for thin films and it is well reproduced by the effective medium model for thick films; in between, the two models are roughly

Conclusion

We have studied the scattering properties of perforated films with a focus on thin films. The conditions under which perfect transmissions are possible are modified when ultrathin devices are considered; in particular an extraordinary transmission observed for a thick film can disappear when reducing its thickness, and the reverse situation is possible as illustrated at the beginning of this paper in Fig. 2. In all cases, a shift in the Brewster incidence θ has been exhibited: for film

Acknowledgments

MLC acknowledges support of FONDECYT grant No. 1170411 and Millenium Nucleus Physics of Active Matter of the Millenium Scientific Initiative of the Ministry of Economy, Development and Tourism (Chile). CH thanks Conicyt grant Doctorado Nacional 21161479. The authors gratefully acknowledge the Laboratoire International Associé “Matière: Structure et Dynamique” (LIA-MSD, France-Chile).

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