Deep subwavelength ultrasonic imaging using optimized holey structured metamaterials

This paper reports the experimental demonstration of deep subwavelength ultrasonic imaging of defects in metallic samples with a feature size of λ/25 using holey-structured metamaterial lenses. Optimal dimensions of the metamaterial’s geometric parameters are determined using numerical simulation and the physics of wave propagation through holey lenses. The paper also shows how the extraordinary transmission capacity of holey structured metamaterials comes about by the coupling of higher frequencies in the incident ultrasonic wave field to resonant modes of the lens.


Optimal geometrical parameters of the lens
We can summarize results of the studies on optimal parameters of the holey structured metamaterial lens as follows. For better resolution, diameter of the holes should be λ/n, where n is an integer > =10.
Hole length should be an integer multiple of half the wavelength m(λ/2), as expected based on the Fabry-Perot resonance principle [2,3].
The hole periodicity should be (2p)(λ/n), where p is an integer. This can again be understood from the underlying physical principles of holey structured lenses [4]. Consider the expression for the Transmission Ratio (T) of the metamaterial lens (see [2,4] for example): where, d is the hole size, Λ is periodicity, Y is the admittance of the waveguide mode within the holes, k is the wave number, and L is the length of the metalens.
Approximating the overall transmission as a sum of the all multiple scatterings in the holes, the above equation can also be rewritten as below: where, 12 = is reflection amplitude of fundamental mode inside the hole (see additional information in ref. [4]). In deep subwavelength region | | = 1, and | 23 | ∝ 2( ) or ∝ 2 . Hence the periodicity value is required to be proportional to two times the hole size for extraordinary transmission.
Further if w is the wall thickness between the two holes, the periodicity can be written as the sum of the hole size and the wall thickness, Thus the optimal wall thickness as per above notation, can be shown to be

Direct reception of wave field using Laser Doppler Vibrometer (LDV)
The Laser Doppler Vibrometer (LDV) used in our studies, can be used to capture local wave fields including evanescent waves from the surface of the sample. However, the evanescent waves arising from the wave-defect interaction, if captured directly in this manner, would be weak and not be sufficiently strong to resolve sub-wavelength defect features. The metamaterial lens serves to not only transmit the wave field to the image plane, but also to amplify the components including evanescent fields at target frequencies.
In order to demonstrate this, we performed a simple line scan experiment, where the LDV was The above result can also be understood based on the physics of the interaction of evanescent waves with material boundaries/interfaces. Firstly we distinguish the following types of evanescent waves: 1. The first type consists of evanescent waves generated when homogenous (propagating) plane-crested waves impinge at a material interface or free boundary. In the context of bulk ultrasonic waves, several authors have studied this problem (see for example, Ref. [5])Evanescent waves generated due to such homogenous plane bulk elastic waves incident at interfaces are generally very weak and can be detected only at specific angles of incidence, e.g., close to Rayleigh critical angle.
2. The second type consists of the evanescent waves generated when plane-crested waves impinge at a defect due to the wave-defect interaction; the high-frequency evanescent waves generated due to wave-defect interaction that we are seeking to capture in our experiments are of this type.
3. Several authors have studied the interaction of evanescent waves with material boundaries, especially solid-fluid interfaces occurring in our case. Dechamps [6] for example, shows that waves transmitted into fluid due to evanescent waves incident from solid side are generally weak, except at large incidence angles.
With this background, we observe that in the wave scattering problem studied in our paper, plane longitudinal ultrasonic waves interact with defects in a metallic sample with the following physical phenomena occurring:  The main body of the normally incident (zero angle to interface normal) homogenous propagative wave field does not interact with the defect, and directly impinges on the material interface/boundarybased on Ref. [5], we know that such waves do not generate evanescent wave fields unless incident at angles close to Rayleigh critical angle (not valid in our problem as we consider normally incident plane waves).
 A part of the normally incident homogenous wave field interacts with the subwavelength defect, generating evanescent waves in the forward-scattered 'shadow' region of the defect. These evanescent waves then travel from the defect location to the surface of the sampleand it is these that we are seeking to capture.
However as per [6], such evanescent waves transmitted across solid-fluid interfaces are in general weak unless we look at large angles of incidence. Since the defects considered in our problem are of subwavelength dimensions, they can be treated as point-scatterers, and hence the forward-scattered wave fields will mainly be incident at normal angles (zero angles) at the interface. Thus, such defect-generated evanescent wave fields are very difficult to pick-up by regular methods. It is in picking up these waves that the metamaterial lenses are useful.

Optimal geometrical parameters of the metalens
Geometrical parameters of the metalens optimal for high transmission ratio (TR) were studied using finite element (FE) simulations, as described in the main body of the paper (page 3).
However, in view of a large number of trial cases studied, only a subset of the results showing the variation of TR with certain illustrative or key values for the parameters were presented in Figure 1 in order to improve the readability. Supplementary Figure 3 crowding of data points, for all above graphs the staring value on the ordinate is considered as 0.05 instead of 0). A subset of these results was presented in Figure 1 of the main body of the paper.