In 1944, Bethe found the transmittance of electromagnetic (EM) waves through a tiny hole in a perfectly conducting screen varies as being proportional to (
However, in 1998, Ebbesen
It is well known that acoustic and EM waves share a lot of wave phenomena, but they have something in difference. In nature, acoustic wave is a scalar longitudinal wave in inviscid fluids, while EM wave is a vector transverse wave. Consequently, a subwavelength hole has no cutoff for acoustic wave, but does for EM wave, which underlies the distinct transmissions of acoustic/EM waves through a hole in an ideally rigid/conducting screen. The acoustic transmission of a single hole approaches a constant, 8/π2, dislike the EM case, with decreasing the ratio
Transmission/diffraction by an acoustical grating is an old problem, and the previous investigations addressed some cases: one-dimensional (1D) periodic slits in a rigid screen [31,32], a single hole in a thick wall [33,34], and a 1D grating composed of parallel steel rods with finite grating thickness [35,36]. Here we studied the acoustic transmissions through two structures: (1) a two-dimensional array (square lattice) of subwavelength hole and (2) a single hole surrounded by the surface periodic grooves. It is found that the acoustic transmission phenomenon for the structured thin plates is analogous completely to the case of EM wave, except for the transmission phase. For the hole array in thick plates, the transmission peaks are related to the Fabry-Perot-like (FP-like) resonances inside the holes and can occur to the frequencies well below Wood’s anomalies.
2. Ultrasonic measurements
In our experiments, the measurements of far field transmissions of acoustic waves in the ultrasonic frequency regime (0.2–2.0 MHz) were performed in a large water tank. Two immersion transducers were employed as ultrasonic generator and receiver, and the sample was placed at a rotation stage located between the two transducers at an appropriate distance. The sample could be rotated, so that the oblique incidences were measured. The ultrasonic pulse was incident upon the sample and the transmitted signal was collected by the receiver, collinear with the incident wave. Transmission magnitude,
In the context, the term “transmission” when referring to the spectrum means the amplitude ratio,
Apart from measuring the transmission spectrum, we also implemented point-by-point scanning to detect the pressure field distribution in the transmission process. A pinducer (1.5 mm in diameter) replaced the receiving transducer and was located at a distance,
3. Experimental results
3.1. Enhanced transmission of acoustic waves through hole array structure
First we measured the acoustic transmission of a hole array with the hole diameter
We also investigated the dependence of the transmission peak on the lattice constant. Figure 2 shows the normal transmissions of the hole arrays with identical lattice constant
Figure 3 shows the transmission spectra at oblique incidence measured with the incident angle
for the Wood’s anomaly frequency
In recent investigations, it is demonstrated that the SF resonance can be responsible for enhanced transmissions of EM waves through subwavelength hole arrays [26,27]. We have considered that the acoustic surface wave at the brass-water interface might play no role in the present transmission phenomenon, and shown that the SF resonance holds for acoustic waves by generalizing the proof of EM waves [38,39]. The SF resonance has some spectral features: the resonant wavelength is determined essentially by the lattice constant and is very sensitive to the incidence angle with accompanied by Wood’s anomalies. Here, the experimental results for the enhanced acoustic transmission through the hole array in the 0.5 mm thick plate manifests the features of SF resonance.
When the plate thickness becomes larger, the situations begin to divide for two types of waves. For EM wave, the transmission peak will diminish after the metallic film/plate becomes thick enough, because the holes have the cutoff. In sharp contrast, there is no cutoff for acoustic waves to propagate through the holes. When the thickness is large enough, for instance
3.2. Enhanced transmission of acoustic waves through bull’s eye structure
Soon after the discovery of extraordinary optical transmission through a metallic film with two-dimensional array of sub-wavelength holes, it was found that there can be enhanced and collimated transmission through a single sub-micron hole surrounded by finite periodic rings of indentations (denoted as bull’s eye) . We also examined the transmission of a bull’s eye structure for acoustic waves. The bull’s eye structure, shown in the inset of Figure 5, was fabricated by patterning both sides of a thin brass plate with concentric periodic grooves around a single cylindrical hole. The thickness of the brass plate is 1.6 mm, and the diameter of the central hole is 0.5 mm. The groove period is 2.0 mm, and there are a total of 15 grooves. The width and depth of each groove are 0.5 mm and 0.3 mm, respectively.
In Figure 5, we showed the measured transmittances as a function of frequency for both bull’s eye structure and the reference sample (a smooth brass plate of the same thickness). It can be seen that there is a transmission peak at 0.71 MHz for bull’s eye structure, while such peak is missing for the reference sample. In Figure 5, we also plotted the power transmittance calculated by using COMSOL MULTIPHYSICS, a commercial finite-element solver. It can be seen that the predicted peak position agrees well with the experimental data. However, the measured transmittance is much lower than that predicted and the precise reason for this disagreement is yet to be uncovered.
For ultrasonic waves in water, wavelength corresponding to 0.71 MHz is 2.1 mm, which is slightly larger than the groove period of bull’s eye, 2.0 mm. This close correspondence is a strong clue indicating that the enhanced transmittance is due to the diffraction effect. It has been shown that enhanced acoustic wave transmission through hole arrays in perfectly rigid thin plate, where there can be no surface wave, may be related (and understood via Babinet’s principle) to “resonant” reflection by its complementary structure, i.e., planar arrays of perfectly rigid disks [26, 39]. In fact, both were found to be associated with the
Besides the transmission enhancement, the collimation effect of the bull’s eye structure is very striking [4,40]. As shown in Figure 6(a), the far-field acoustic wave on the transmission side is also in the form of a tight beam with a lateral dimension not exceeding the groove periodicity. The full width at half maximum (FWHM) divergence is ±2o. As analyzed above, it is the coherent scattering which leads to the emergence of a strongly collimated beam in the far-field region. In Figures 6(b) and 6(c) we also plotted the scanned results at a distance of about 15 wavelengths from the transmission side of the surface, for both the bull’s eye structure and the reference sample. Compared with the reference sample, the collimation effect for bull’s eye structure is very evident. In addition, it is found by simulation that both the intensity of the acoustic wave field around the central hole region, as well as the collimation effect, would increase with the number of concentric grooves. This is reasonable, since the coherent scattering effect becomes stronger if more concentric grooves are involved.
4.1. Fabry-Perot resonances tuned via diffraction evanescent waves
For the hole arrays, we measured the acoustic samples with various plate thicknesses ranging from 0.5 mm to 3.1 mm, as plotted in the inset of Figure 7. In theory, we employed the mode expansion method to calculate analytically the transmission . We found that the observed transmission peaks are the manifestation of a type of resonance mode that has FP and SF resonances as the two limits. The diffraction evanescent modes play an important role in interpolating between the two limits. To make explicit the role of diffraction evanescent waves, we retained the lowest cylindrical mode inside the holes and 5 lowest plane wave modes outside the holes, and obtained the resonant mode equation as
in which , is the diffraction evanescent wavevector, and
In Figure 7 we show the measured and calculated FPEV resonance frequencies plotted as a function of inverse plate thickness. The FP condition is indicated by the black dashed straight lines, for
It is seen that as the ratio
It is interesting to note that the transmission of microwave through a metallic grating of 1D slits has the similar FPEV resonances for the incident polarization with E-field perpendicular to the silts, see open symbols in Figure 7 . This is because the slits have no cutoff to the perpendicular polarization of EM waves, in the same physics as the holes to acoustic waves.
4.2. The effective fluid model for thick plates
For a very small
The above argument is applicable under long wavelength limit (
where is the wavenumber of the incidence wave. The calculated results (solid lines) are shown in Figures 4(a) and 4(b) to compare with the experimental data (open circles) for two samples with identical thickness 2.3 mm, and identical lattice constant 2.0 mm, but different hole diameters 1.2 and 0.5 mm. Good agreement between the calculations and the experiments is seen at the frequencies below the Wood’s anomaly (0.75 MHz), which verifies the applicability of the effective fluid model at normal incidence.
In Figure 4(c), we see two flat bands appear below 0.75 MHz, and they are the first and second FP resonances, see Figure 4(a) where the phase values indicate the order of FP resonances. The open circles superimposed are the variations of the transmission peaks of the effective fluid layer which are obtained from the FP resonance condition at oblique incidence,. The agreement between the calculated variations of the transmission peaks and the measured results indicates the applicability of the effective fluid model persists to a range of incidence angle. The discrepancy at
The effective fluid model allows us to use the holey or slotted hard plate to realize an acoustic medium, and provides some freedom to design acoustic materials, because some material parameters, difficult to be tuned, are related simply to the structural factors of the hole array or the slits. In Figure 9, we illustrated conceptually an acoustic prism made of such slotted hard plate. A detailed discussion is seen in reference .
We investigated experimentally the acoustic transmission through subwavelength holes fabricated on brass plates at normal and oblique incidence within ultrasonic frequencies regime. The transmission phenomena for both hole array and bull’s eye structure in thin brass plates, analogous to the observed enhanced transmission of EM waves through subwavelength hole arrays in a metallic film, exhibit the transmission enhancement because of the SF resonance. At the peak frequency, the transmission phase is nearly −π, indicating the out-of-phase oscillations of the acoustic field at two surfaces of the plate. For the hole array in thick brass plates, the transmission peaks of acoustic waves are related to the FP-like resonances inside the holes and therefore occur well below Wood’s anomaly, since a hole has no cutoff frequency for acoustic propagation. By varying the plate thickness or channel length, one makes the transition from the FP resonance (thick plate limit) to the SF resonance (thin plate limit). Between the two limits there can be interesting deviation from FP resonance conditions, owing to the interaction of the diffraction evanescent waves. In the case of thick plates, the structure can be viewed as a new fluid with effective mass density and bulk modulus scaled, under long wavelength limit, by a factor of area fraction of the holes. The effective medium model describes well the transmission properties of the hole array within a range of incidence angle.
Our discussion assumed the approximation of hard plates and did not take acoustic surface waves into account. With acoustic surface waves being involved, transmissions of acoustic waves through structured plates have found far richer and more complicated physical phenomena in the past few years and will attract more attentions in the future [43-45]. Although this subject is an old problem, its new phenomena may appear from time to time and the underlying mechanism waits to be unlocked.
We would like to thank our collaborators in this project. B. Hou is supported by the National Natural Science Foundation of China (Grant No. 11104198) and a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). W. Wen is supported by RGC Grant HKUST2/CRF/11G.
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