Definiciones conceptuales del marco lógico

These acoustic metamaterials achieve a negative refractive index by exploiting local resonances rather than scattering and interference effects. Furthermore, the lateral dimensions of each unit cell have to be much smaller than the incident wavelength. This fact makes the metamaterials more suitable than phononic crystals for producing a negative refractive index, involving a negative effective bulk modulus Keff and a negative effective density ρeff. Since the cell dimensions are smaller than the wavelength, the spatial periodic modulation of the impedance occurs at the same resolution; hence, all the metamaterials properties can be related to an effective homogenous description of the Lorentz form. An accepted method to measure their physical properties is to evaluate the transmission and reflection coefficient when the metamaterial is excited by an acoustic wave. The measured coefficients are then mathematically inverted to recover the bulk modulus and the density [33].

One of the first examples of such acoustic metamaterials can be found in [34]. An array of sub-wavelength Helmotz resonators drilled inside an aluminum samples showed a left- handed behavior in the frequency range of 30-35 kHz. The design exploited in the latter work is depicted in Figure 3.8:

Figure 3.8: (a) an example of acoustic metamaterial fundamental structure geometry and its respective electric model. (b) 1D-array arrangement of the said structure (Taken from ref [34]).

Moreover, it was shown that a negative refraction index introduces surface oscillations characterized by a very large wave vector k. As for their electromagnetic counterpart, these

surface oscillations allow the evanescent field scattering by an object to be recovered, i.e. superlensing is possible.

It should be noted that the first work related to the study of such metamaterials has been done by simply testing new designs without the aid of a well-developed mathematical model. A first attempt to provide a coherent mathematical analysis on this field was given in [35]. The authors related the changing on the radius of locally resonant cylindrical structures to the values of the sound velocity and density inside the metamaterial. A two- dimensional numerical simulation has also been provided by Ambati et al. [36]. They related the evanescent field recovering ability of a metamaterial with its thickness, as well as the resulting resolution limit, which is related to the sub-wavelength dimensions. Later work on the Helmotz-resonator metamaterial explained that the arrangement in series or parallel leads to very different intervals in which the negative index of refraction can be achieved. These frequency gaps can be easily calculated by describing the sound propagation inside the metamaterials with an analogous electric model. It is worth mentioning that multi band-fold gap acoustic metamaterials have been explored. In particular, the behavior of a multiple band-gap can be predicted if a multiresonator mass-in-mass lattice system is taken into account as a mathematical model [37].

Of relevance to this thesis are the metamaterials that achieve super-resolution via Fabry-Pèrot resonances inside drilled sub-wavelength holes. The first theoretical approach was described in [38]. By exploiting numerical simulation on a single hole embedded in water or air, the researchers found that the frequency of resonance is related to thickness of the hole. Furthermore, the resonances appear at higher harmonics, showing that the behavior is related to the Fabry-Pèrot theory. Moreover, the authors pointed out the importance of (i)

considering the material surrounding the holes as being rigid, (ii) having a high acoustic impedance mismatch between the two media and (iii) using a planar incident acoustic wave. Using such designs, a simple block of brass material with sub-wavelength drilled holes provided a super-resolution that can be easily tuned by changing the distance between each hole [27]. Furthermore, a brass acoustic magnifying lenses which operates in the audible range show a resolution up to a fifth of the wavelength [39].

Other studies demonstrate possibilities for reaching a wider bandwidth behaviour in metamaterials. By exploiting solid inclusions in air, it is possible to reach a broad-band left- handed metamaterials [40]. Moreover, the band-gap can be controlled by scaling the geometry to suit another frequency range [41]. Metamaterials made by piezoelectric inclusions can also provide novel application. By varying the applied voltage level, a membrane of piezoelectric material embedded in a solid matrix changes its corresponding acoustic impedance [42]. Alternative studies on how to focus the sound with a metamaterial also exist. By changing the dimension along and the filling ratio of solid inclusions in air it is possible to fabricate a gradient index lenses. By changing the latter parameters, the focusing focal length can be tailored to particular applications [43]. Moreover, the geometry of the inclusions can be engineered with a genetic stochastic algorithm [44].

Highly anisotropic metamaterials can also be exploited to hide an object from an acoustic radar. Perforated plastic plates, which are well-suited to air coupled applications, or circuits of acoustic capacitors and inductors elements are capable of hiding a metal object – an example of acoustic cloaking [45]–[47] :

Figure 3.9: An example of 3D broadband acoustic cloaking (taken from ref [47]).

Since the resolution of measurements made in either medical ultrasound or NDT is limited to the wavelength of the ultrasonic wave by the natural diffraction limit, it is easy to understand that the use of high frequency ultrasonic signals increase the resolution of the ultrasonic inspection. In standard ultrasonic testing, broad-band coded signals such as Golay or Chirps can be used to enhance the resolution [48], as was described in Chapters 1 and 2. However, due to attenuation phenomena that affect the SNR, the frequency of the input signal cannot be increased arbitrarily. A trade-off exists between resolution and signal attenuation, and this must be taken into account for each measurement. Thus, the use of acoustic metamaterials could provide a possible way to avoid the frequency/attenuation constraint by creating focusing effects that could normally only be produced using higher frequencies.

Note that most applications of ultrasound occur at frequencies higher than those examined to date using such metamaterials, and there is still a need to build a metamaterial device that can be exploited at frequencies of up to 1 MHz or higher. So far, the limitations on the achievable metamaterial cell dimensions have forced researchers to produce metamaterial devices only in the low frequency acoustic range (100Hz to 20 kHz), and most often in air. As an example, to produce a Fabry-Pèrot metamaterial useful at 2 kHz, one has to be able to drill squared holes of 0.79 mm x 0.79 mm [27]. By rescaling the dimension to

suit higher frequency uses, it is evident that alternative ways to fabricate metamaterials have to be explored. This is a subject of a later chapter in this thesis.