Modified Luneburg Lens: How Well Does It Focus Surface Water Waves?

Abstract

An optical lens focuses light and a similar device can be developed to focus surface water waves. A detailed description of such hydrodynamic lenses is given, for which the focusing is induced by shaping the bathymetry of the bottom. Classically, the Luneburg lens uses a specific radial variation of the refractive index. The modified Luneburg lens (MLL) introduces an extra degree of freedom, permitting the focal point to be tuned. It is shown how to design the MLL for water waves, and then its performance is evaluated. Compared with a simple parabolic-shaped mount, the MLL is shown to be free of spherical aberration, resulting in a focus with larger intensity and smaller size of the focal point. Moreover, the focusing properties can be tuned and enhanced thanks to the possibility of changing the position of the focal point. The focusing quality of the MLL is described in all water-depth regimes (covering dispersive and non-dispersive waves) and the focusing of linear and nonlinear waves is revealed experimentally. The option of moving the focal point outside the lens, where the water depth is constant, may be useful when locating devices for harvesting wave energy.

1. Introduction

It is well known that when surface water waves pass across an elevated bathymetry, refraction often results in the amplification of waves behind it [1,2,3,4,5]. This phenomenon can be enhanced by adjusting the bathymetry so as to create a hydrodynamic lens: the focusing of the waves can then be used to increase the harvest efficiency of devices designed to extract power from the waves. A hydrodynamic lens can concentrate waves on a particular focal point by transforming straight crests of incident waves, just like an optical lens. These hydrodynamic lenses have attracted the attention of ocean engineers; they are promising because they might enable the effective utilization of wave energy [6,7], the remaining challenge being to increase the harvest efficiency of the lens.

According to Griffiths and Porter [8], the idea of focusing with underwater lenses was first suggested by Mehlum and Stamnes in 1978. Since then, amplification and focusing of surface water waves by variable bathymetry have been studied in several situations, including gradual changes in depth and abrupt changes in depth (submerged plates); see, for example, [3,8,9,10,11,12,13,14,15,16,17,18]. Based on homogenization theory, the interactions of water waves with lens-like structures made of periodic bottom-mounted vertical cylinders have been investigated in the long-wavelength limit [19,20,21]. New mechanisms for controlling surface water waves, mainly inspired by photonic crystals and metamaterials, have been revealed, such as negative effective gravity [22,23], epsilon-near-zero focusing [24], zero refractive index [25] or reflectionless shifters [26]. Also, Maxwell’s fishpond lens, working for a source point [27] and gradient index media [28], have been studied in the shallow-water limit.

To fix ideas, we can start by considering two-dimensional acoustic waves governed by the Helmholtz equation, $({\nabla }^2+k_0^2{n}^2)u=0$, where the refractive index n is specified and $k_0$ is the constant wavenumber in the region where $n=1$. The lens corresponds to the bounded region where $n\ne 1$; for simplicity, take this region to be a disc $0\le r<r_0$ (r is a plane polar coordinate) and assume that n is a function of r only. The classical Luneburg lens [29,30] corresponds to the choice $n\left(r\right)={\{2-{(r/r_0)}^2\}}^{1/2}$; it collects plane waves and focuses them at a point on the circle $r=r_0$. This focal point can be moved by using [31]

$$n\left(r\right)={\alpha }_{\mathrm{f}}^{-1}\sqrt{1+{\alpha }_{\mathrm{f}}^2-{(r/r_0)}^2},$$

where ${\alpha }_{\mathrm{f}}$ is a dimensionless parameter. This defines the modified Luneburg lens (MLL). We are interested in the hydrodynamic analog of this lens. Lenses created using inhomogeneous media have been named ‘gradient index’ (GRIN) devices. They have recently shown interesting properties in many domains of physics [27,32,33,34,35,36,37,38,39].

In this paper, we give a detailed description of the efficiency of a hydrodynamic MLL. We consider surface waves on water of arbitrary (finite) depth: we do not restrict to shallow water waves. Specifically, if $h_0$ is the constant water depth outside the region occupied by the lens and $k_0$ is the wavenumber there, we say that we are in the shallow-water regime if $k_0{h}_0\ll 1$.

In Section 2, the bathymetry profile for the MLL is determined as a function of the water-wave regime. It is shown that the bathymetry profile does not depend on frequency in the shallow-water regime ($k_0{h}_0\ll 1$). For larger values of $k_0{h}_0$, the bathymetry profile does depend on the frequency. In Section 3, the propagation of surface water waves over the MLL is analyzed for non-dispersive shallow-water waves. The classical Luneburg lens, for which the focal point lies at the rim of the lens, is studied in Section 3.1 and results are compared with a parabolic-shaped lens in Section 3.2. Then, tunable and improved focusing is presented with the MLL in Section 3.3. In Section 4, the propagation of surface water waves over the MLL is studied in a more general case, without restriction to shallow-water waves. The evolution of the quality of the focusing through a fixed lens is presented as a function of the frequency of the incident wave in Section 4.1. Also optimal focusing of dispersive waves is demonstrated in Section 4.2. Finally, experiments of the focusing of linear and nonlinear waves are presented in Section 5.

2. Bathymetry Profile for the Modified Luneburg Index

2.1. Index of the Luneburg Lens

The classical Luneburg lens is a GRIN lens with a spatially varying refractive index

$$n\left(r\right)=\sqrt{2-{(r/r_0)}^2},r\le r_0,$$

(1)

where r is the radial distance from the lens center and $r_0$ is the radius of the lens [29,30]. Outside the lens ($r>r_0$), $n\left(r\right)=1$. The Luneburg lens focuses an incident plane wave on a focal point diametrically opposed to the incident direction at $r=r_0$. A modified version of the Luneburg lens enables us to tune the location of the focal point at a distance $d_{\mathrm{f}}$ from the lens center [31]. This modified Luneburg lens (MLL) has $n\left(r\right)$ given by

$$n\left(r\right)={\alpha }_{\mathrm{f}}^{-1}\sqrt{1+{\alpha }_{\mathrm{f}}^2-{(r/r_0)}^2},r\le r_0,$$

(2)

with

$${\alpha }_{\mathrm{f}}=d_{\mathrm{f}}/r_0.$$

(3)

When ${\alpha }_{\mathrm{f}}<1$ (${\alpha }_{\mathrm{f}}>1$), the focal point lies inside (outside) the lens and when ${\alpha }_{\mathrm{f}}=1$, the MLL reduces to the classical Luneburg lens.

2.2. Application to Water Waves

As illustrated in Figure 1, to apply the concept of the GRIN lens to water waves, we can use a varying bathymetry to obtain a varying index. Indeed, water waves are described by the dispersion relation

$${\omega }^2=gk\tanh kh,$$

(4)

with $\omega$ the frequency, k the wavenumber, h the local water depth and g the gravity constant. The phase speed is $c_{\mathrm{p}}=\omega /k$. Outside the lens, the wavenumber is $k_0$, the constant water depth is $h_0$ and the phase speed is $c_0=\omega /k_0$. By definition, the refractive index is $n=c_0/c_{\mathrm{p}}=k/k_0$,

$$n=\sqrt{{\displaystyle \frac{k\tanh k_0{h}_0}{k_0\tanh kh}}}.$$

(5)

Equation (5) shows the dependency of n on the bathymetry through h. It also shows that a correspondence between n and h will depend on the chosen frequency $\omega$. This contrasts with the case of non-dispersive waves, where the correspondence between n and the inhomogeneity of the medium is obtained independently of the frequency. This also contrasts with the case of dispersive waves with a dispersion relation of the form $c_{\mathrm{p}}\propto {\omega }^{\gamma }$, leading also to a correspondence being frequency independent; see [36] in the case of flexural waves in elastic plates ($\gamma ={\displaystyle \frac{1}{2}}$ in this context). Coming back to water waves, there are two regimes where $c_{\mathrm{p}}\propto {\omega }^{\gamma }$. The first regime corresponds to the dispersionless shallow-water limit ($kh$ and $k_0{h}_0\ll 1$), thus $\gamma =0$. The second regime is the deep water limit ($kh$ and $k_0{h}_0\gg 1$), with $\gamma =-1$, but this is of no interest here: there would be no bathymetry for the waves to feel!

The analysis presented in the following sections is made as a function of the water wave regime defined by $k_0{h}_0$. In fact, we shall consider the water depth $h_0$ to be fixed so that increasing $k_0{h}_0$ will be equivalent to increasing the frequency. In general, for a given frequency corresponding to a given $k_0{h}_0$, the water depth profile h, see Figure 1, can be determined by solving (5) numerically. Fortunately, a good approximate solution of the dispersion relation (see Appendix A) provides an analytical expression for the bathymetry,

$$h(r;k_0{h}_0)={\displaystyle \frac{1}{k_0\tanh k_0{h}_0}}\mathrm{arctanh}\left({\displaystyle \frac{1}{{\left[n\left(r\right)\right]}^2}}\tanh \left[k_0{h}_0\tanh k_0{h}_0\right]\right).$$

(6)

The shallow-water case ($k_0{h}_0$ and $kh\ll 1$) leads to significant simplifications, with

$$h_{\mathrm{sw}}\left(r\right)=h_0/{\left[n\left(r\right)\right]}^2,$$

(7)

where $n\left(r\right)$ is given by (2).

Thus, the analysis can be divided into two main parts, as follows.

(i) Non-dispersive waves (shallow-water regime) where the bathymetry of the lens does not depend on the frequency, following Equation (7). The propagation of water waves through the lens is studied with the Shallow Water Equation (SWE) in Section 3.

(ii) Dispersive waves, where the bathymetry profile depends on the frequency, following Equation (6). To account for dispersion, the propagation of water waves is studied with the Mild Slope Equation (MSE) in Section 4.

2.3. The Mild Slope Equation

The Luneburg index, and the resulting variation in the bathymetry, only guarantees focusing of rays in the high-frequency limit. At finite frequencies, it makes sense to inspect whether or not the bathymetry produces the expected focusing. To have a hope of a good lensing effect to occur, the lens has to contain many wavelengths, which means $k_0{r}_0\gg 1$. On the other hand, the wave will be sensitive to the bathymetry for moderate values of $k_0{h}_0$, say $k_0{h}_0\sim 1$. Combining these two, it follows that the ratio $h_0/r_0\ll 1$, which means that the bathymetry of the lens is slowly varying. This is good news since it allows us to solve the water wave propagation problem using the Mild Slope Equation (MSE), thus avoiding solving the full three-dimensional problem. The MSE [40] governing the surface wave elevation $\eta$ is

$$\nabla ·(\alpha \nabla \eta )+\alpha k^2\eta =0,$$

(8)

where $\alpha =c_{\mathrm{p}}/c_{\mathrm{g}}$ and $c_{\mathrm{g}}$ is the group velocity,

$$c_{\mathrm{g}}={\displaystyle \frac{c_{\mathrm{p}}}{2}}\left(1+{\displaystyle \frac{2kh}{\sinh 2kh}}\right).$$

The MSE can be employed efficiently to predict linear wave propagation over an arbitrary bathymetry and is valid when $|\nabla h|\ll 1$ [41]. Note that many modified and extended forms of the MSE have been proposed [42,43,44,45,46,47,48] to include the effects of wave nonlinearity, seabed slope, wave breaking or bed friction.

When the depth is small compared to the wavelength ($k_0{h}_0\ll 1$), the MSE tends to the linear Shallow Water Equation (SWE),

$$\nabla ·(h\nabla \eta )+({\omega }^2/g)\eta =0,$$

(9)

which clearly recovers the index scaling $n^2\propto 1/h$ of (7).

3. The Modified Luneburg Lens in the Shallow-Water Regime

In this part, the focusing properties of the modified Luneburg lens (MLL) are analyzed when the depth is small compared to the wavelength. This is conducted with the SWE (9), which gives non-dispersive wave propagation with $\omega =k\sqrt{gh}$. The focusing of surface water waves by the classical Luneburg lens is revealed and improved in comparison with a parabolic conical mount. In addition, the tunability of the MLL in the position of the focal point is demonstrated.

3.1. Classical Luneburg Lens

The propagation of surface water waves through the classical Luneburg lens is investigated numerically in the shallow-water regime ($k_0{h}_0\ll 1$). Thus, the bathymetry profile is frequency-independent, and is defined by (7) with ${\alpha }_{\mathrm{f}}=1$. Numerical computations are carried out, solving the SWE by a finite difference method, see Appendix B. The incident wave propagates in the forward x-direction and has unit amplitude. Figure 2a,b present the real part and the intensity (defined as ${\left|\eta \right|}^2$) of the calculated field, respectively. The radius of the lens is chosen to be equal to three wavelengths ($r_0=3{\lambda }_0$, $k_0{r}_0=6\pi$).

Clear focusing of surface water waves is observed with the focal point of large intensity lying at the rim of the lens (the maximum of normalized intensity at the focal point is 22). The sizes of the focal point, $W_x$ and $W_y$, defined as transverse full width at half maximum intensity, are found to be $W_y=0.41{\lambda }_0$ along y-axis, following the Rayleigh criterion of $\lambda /2$, and $W_x=1.1{\lambda }_0$ along x-axis.

This configuration (classical Luneburg lens with ${\alpha }_{\mathrm{f}}=1$, in the shallow-water regime and with $r_0=3{\lambda }_0$) will be used in several comparisons below.

3.2. Improved Focusing Compared to Parabolic Shaped Lens

To demonstrate the enhanced focusing of the Luneburg lens compared to lenses with other profiles, calculations were made with a lens having a parabolic profile. We chose this profile because it is simple and because, as for the Luneburg lens, the focal point is located at the rim of the lens. The bathymetry profile of the parabolic lens, chosen to have the same maximum height $h_0/2$ and radius $r_0$ as the classical Luneburg lens, is defined as

$$h_{\mathrm{parabolic}}={\displaystyle \frac{h_0}{2}}\left(1-{(r/r_0)}^2\right),r\le r_0.$$

(10)

Figure 3 presents the real part and intensity of the calculated fields through the classical Luneburg lens ((a) and (b)) and the parabolic lens ((c) and (d)). Note that, compared to Section 3.1, the frequency has been increased so that $r_0=6{\lambda }_0$.

Concerning the Luneburg lens, the focusing properties are improved compared to the first reference case depicted in Figure 2, with an increase in the maximum intensity of the focal point. This result is expected: when increasing the frequency, results should become closer to the high-frequency limit of rays focusing at one point. By comparing with the focusing through the parabolic lens, Figure 3c,d, a better focusing is observed for the Luneburg lens, with smaller size and larger intensity of the focal point. The focusing of the parabolic lens suffers from a spherical aberration that occurs when the incident rays end up focusing at different points after passing the lens. The Luneburg lens, which has a refractive index that is higher at the center of the lens and gradually decreases near the edge of the lens, can eliminate the spherical aberration. An illustration of this effect can be observed by looking at the streamlines of the energy flux, which are represented as blue lines in Figure 3a,c. The rays converge better to the focal point for the Luneburg lens than for the parabolic lens, for which the rays end in a larger area. This is due to a smoother slope of the bathymetry profile of the Luneburg lens, allowing for the elimination of spherical aberration (the evolution of the maximum intensity and normalized size of the focal spot with the frequency is presented in more detail in Appendix C for both lenses). A larger increase in intensity of the focal point with frequency is revealed for the Luneburg lens compared to the parabolic lens. Its size along the x-axis is nearly equal to one wavelength for all frequencies, while that for the parabolic lens increases with frequency, reaching a value three times larger than that of the Luneburg lens.

These results demonstrate focusing of surface water waves through the classical Luneburg lens, and this focusing is enhanced compared to a parabolic mount. In order to obtain further improvement of the focusing, the modified Luneburg lens is studied in the next part. The shape of the modified Luneburg lens depends on the parameter ${\alpha }_{\mathrm{f}}$, Equation (3), allowing the position of the focal point to be moved. The focusing properties are then analyzed as a function of this degree of freedom.

3.3. Enhanced and Tunable Focusing with the Modified Luneburg Lens

The effect of the position of the focal point is now investigated. The bathymetry profile of the modified Luneburg lens, (7), is imposed with different values of ${\alpha }_{\mathrm{f}}$ and, as for Figure 2, $r_0=3{\lambda }_0$. Figure 4 illustrates the real part and normalized intensity fields when the focal point lies outside ((a), (b) ${\alpha }_{\mathrm{f}}=1.4$) and inside ((c), (d) ${\alpha }_{\mathrm{f}}=0.7$) the lens. The line and dotted circles represent the end of the lens and the distance $d_{\mathrm{f}}={\alpha }_{\mathrm{f}}{r}_0$, respectively. For different values of ${\alpha }_{\mathrm{f}}$, the position of the focal point is on the dotted circle. Compared with the reference result of Figure 2 (${\alpha }_{\mathrm{f}}=1$), we can observe an increase (decrease) in focusing properties when ${\alpha }_{\mathrm{f}}<1$ (${\alpha }_{\mathrm{f}}>1$). In fact, when the focal point gets closer to the center of the lens, its maximum intensity increases and its size decreases.

Note also that the field is more disturbed when ${\alpha }_{\mathrm{f}}$ decreases, which is due to an increase in the slope of the bathymetry profile when ${\alpha }_{\mathrm{f}}$ decreases. The bathymetry profile of the lens is larger when ${\alpha }_{\mathrm{f}}$ decreases, allowing the focal point to be in a shallower region. The change in water depth causes a change in wavelength. When the wave goes into a shallower region, its wavelength is reduced and then the energy is concentrated in a smaller region. So for two different values of ${\alpha }_{\mathrm{f}}$, the focal point will be located in a shallower region for the smaller one, providing a better quality of focusing. This physical feature has also been analyzed in studies of focusing by submerged plates [15,17] or elliptical plateau [8], where it has been shown that maximum elevation is in the shallower region above the plateau or plate.

When the parameter ${\alpha }_{\mathrm{f}}$ is small, the water depth at the focal point becomes very small compared with the wavelength, so that nonlinear effects should be taken into account. A simple way to assess when nonlinearities are important is to calculate the Ursell number $U=H{\lambda }^2/h^3$, where H is the wave amplitude. The nonlinear regime corresponds to $U\gg 100$ [41,49,50]. For example, in the cases ${\alpha }_{\mathrm{f}}=1.4$, ${\alpha }_{\mathrm{f}}=0.7$ and ${\alpha }_{\mathrm{f}}=0.2$, the Ursell numbers at the focal point are $U=35$, $U=144$ and $U=30,000$, respectively. So, when ${\alpha }_{\mathrm{f}}$ becomes small, the Ursell number becomes very large, showing strong nonlinearities. Although the Ursell number value can be decreased by adjusting the wavelength and the wave amplitude, this paper does not consider this aspect further. However, it could be interesting to study hydrodynamic lenses using nonlinear water-wave theories [41,49,50], especially for small values of ${\alpha }_{\mathrm{f}}$.

The MLL offers improved focusing properties when the focal point gets closer to the center of the lens. Alternatively, moving the focal point outside the lens could be useful in some applications because, for example, energy converters could be located where the bathymetry is horizontal.

The propagation of surface water waves through a Luneburg lens has been described for non-dispersive waves. It has been shown that focusing of the waves is reached at the edge of the Luneburg lens with one bathymetry profile for all frequencies belonging to the shallow-water regime. The properties of the focusing through the Luneburg lens have been shown to be better than those of a parabolic mount. Also, enhanced and tunable focusing have been shown to be possible with the MLL.

4. The Modified Luneburg Lens in the Non-Shallow-Water Regime

The propagation of waves through the MLL can be analyzed in all water-wave regimes with the Mild Slope Equation (8). We solve this equation by a finite difference method, see Appendix B, and the aim of this section is to analyze the capability of focusing of the MLL in this dispersive regime.

4.1. Evolution of the Quality of the Focusing with Frequency

The evolution of the quality of the focusing through a fixed bathymetry is described here as a function of the frequency of the incident wave. Recall that the water depth outside the lens, $h_0$, is fixed so that increasing $k_0{h}_0$ corresponds to increasing the frequency. We start from the reference case of Figure 2 ($k_0{h}_0=0.2$) with the bathymetry defined in shallow water (7), and then increase the frequency, thus leaving the shallow-water regime. Figure 5 presents (a) the evolution of the quality of the focusing with $k_0{h}_0$, and the intensity fields for (b) $k_0{h}_0=0.2$ (reference case in shallow water), (c) $k_0{h}_0=0.8$ and (d) $k_0{h}_0=1.9$. The quality of the focusing is normalized by its maximum and defined as the ratio between the maximum intensity I and the normalized sizes along y and x axes of the focal spot,

$$Q={\displaystyle \frac{I{\lambda }_0^2}{W_x{W}_y}}.$$

(11)

The values for the maximum intensity, normalized sizes and position of the focal point are presented in Appendix E.

It is observed in Figure 5a that the quality of the focusing increases until reaching a maximum at $k_0{h}_0=0.8$, and then decreases. Before $k_0{h}_0=0.8$, the shallow-water bathymetry provides focusing at the edge of the lens, Figure 5b, and, as explained in Section 3, the increase in quality with $k_0{h}_0$ is due to an increase in frequency (the lens contains more wavelengths). Then, starting from $k_0{h}_0=0.8$, the quality starts decreasing. This deterioration is due to dispersive effects (non-shallow-water regime), which leads to a shift of the focal point position to the outside of the lens, Figure 5d, and which results in a focal point with smaller intensity and larger sizes, see Appendix E. In fact, the bathymetry profile suited for shallow-water waves is no longer efficient for non-shallow-water frequencies. However, although the quality of the focusing is decreasing compared to the maximum quality, its deterioration is not important. Results are presented for a large range of $k_0{h}_0$ values and the intensity values and sizes of the focal point are still non-negligible compared to the maximum ones. Thus, the focusing can still have good properties for a frequency range around the maximum quality.

4.2. Focusing of Dispersive Waves at a Given Frequency

It is important to notice that it is also possible to adapt the bathymetry profile to obtain optimal focusing at the edge of the lens at a desired frequency. In fact, the bathymetry profile suited at the chosen frequency can be obtained from (6). The focusing on different water-wave regimes $k_0{h}_0$ is presented in Figure 6.

For each regime, the bathymetry profile $h(r;k_0{h}_0)$ is given by (6). As for the reference case of Figure 2 in shallow water, results are presented for the classical Luneburg lens (${\alpha }_{\mathrm{f}}=1$) and with $r_0=3{\lambda }_0$. Figure 6a,b present the intensity and real part of the calculated fields for $k_0{h}_0=1.2$ (intermediate regime) and $k_0{h}_0=3.4$ (deep water regime). In all the cases, optimal focusing is obtained with the focal point lying at the rim of the lens. In the deeper water regime, the field is more disturbed, which is due to an increase in the slope of the bathymetry profile, but the lens focusing quality is still good. Note that the condition of the MSE ($|\nabla h|\ll 1$) is valid in these calculations, with $|\nabla h|\simeq {10}^{-2}\ll 1$ for the higher bathymetry slope (for $k_0{h}_0=3.4$). Good focusing quality, as demonstrated in Section 3.3 for the shallow-water regime, is also obtained for other values of ${\alpha }_{\mathrm{f}}$. It is thus possible to focus surface water waves with the modified Luneburg lens in all water-wave regimes.

5. Experimental Results

The experiments are made for a water depth at rest $h_0$ = 15 mm, a lens of radius $r_0$ = 200 mm and for a bathymetry profile suited for the frequency $f_c={\omega }_c/\left(2\pi \right)=4.5$ Hz. This situation corresponds to an intermediate regime with $k_0{h}_0$ = 1.4083 (the waves are dispersive, as in Section 4). The analysis of the focusing through the Luneburg lens is made for a frequency range f = [2.5–6.5] Hz, which corresponds to $k_0{h}_0$ = [0.66–2.53]. The lens is situated in a waveguide having the size of the lens, plane waves are generated in front of the lens and an absorbing beach is placed at the end of the waveguide to avoid undesired reflection. The bathymetry for the lens is made by use of a 3D printer with a resolution of 0.2 mm.

5.1. Linear Part of the Field

The full space-time measurements of the surface wave elevation are performed by Fourier Transform Profilometry (FTP), which has been adapted for water waves measurements ([51,52,53]). In the FTP method, a fringe pattern is projected onto the free surface and observed from a different position by the camera. The surface elevation $\eta (x,y,t)$ is encoded in the fringe deformation in comparison with the original (undeformed) grating image. It is, therefore, the phase shift between the reference and deformed images that contains all information about the deformed surface. Owing to the temporal resolution, the total measured displacement field $\eta (x,y,t)$ can be easily expanded in $\eta (x,y,t)={\sum }_{m}Re\left[{\widehat{\eta }}_m(x,y)e^{im\omega t}\right]$ to extract the complex field that is later analyzed. The complex field ${\widehat{\eta }}_m(x,y)$ corresponds to the m-th harmonic (for $m>0$) of the fundamental $\omega$.

Figure 7 presents the real part and the normalized intensity field obtained by (b) (c) numerical simulations and (e) (f) experiments at the frequency $f_c=4.5$ Hz. The numerical simulations are performed by solving the original Mild Slope Equation by the finite difference method and by accounting for the attenuation of the waves due to viscous damping, see Appendix F. The center of the lens is at $\{x,y\}=0$. The experimental results represent the linear component of the fields $\widehat{{\eta }_1}$. The real part of the field without the lens is represented in Figure 7a,d for simulation and experiment, respectively. Quite an important attenuation of the waves along the propagation is observed in experiments, which is not in accordance with the simulation. This is mainly due to the experimental setup, where it is difficult to clean the surface enough to avoid the attenuation of the waves, and also due to the surface of the lens, which is not perfectly smooth. However, despite these experimental difficulties, clear focusing at the end of the lens is observed in the experiment. The properties of the focal point are very close in both cases. The axial position of the focal point is $x_{simu}=18.96$ for simulation and $x_{exp}=18.56$ for experiments. The sizes of the focal point along x and y axes are, respectively, $W_{x_{simu}}=0.76{\lambda }_0$ and $W_{x_{exp}}=0.69{\lambda }_0$, and $W_{y_{simu}}=0.29{\lambda }_0$ and $W_{y_{exp}}=0.36{\lambda }_0$, providing focusing properties below Rayleigh criterion in $\lambda /2$. Appendix G presents the evolution of the focusing with frequency.

5.2. Nonlinear Focusing

The incident wave frequency is chosen to be $f=2.5$ Hz in order to have important nonlinear components. The resulting fields after the FTP procedure are illustrated in Figure 8. The fundamental component (linear part) ${\widehat{\eta }}_1$, the second ${\widehat{\eta }}_2$, third ${\widehat{\eta }}_3$ and fourth harmonics ${\widehat{\eta }}_4$ of the field are presented. The harmonics are significant, with amplitude for the second, third and fourth harmonics of $37\%$, $22\%$ and $10\%$ of the first harmonic, respectively. It can be seen that the amplitudes of the higher harmonics become more important inside the lens, when the water depth is smaller, as evaluated with the Ursell number. The wave deformation through the lens is clearly observed and focusing at the rim of the lens is revealed for all the components of the total field. Despite some experimental difficulties, these results have shown focusing by the Luneburg lens, and moreover, the focusing of nonlinear fields has been revealed.

6. Conclusions

The focusing of surface water waves by a modified Luneburg lens has been studied for non-dispersive (shallow water) and dispersive waves. It has been shown that it is possible to build a lens free of spherical aberration, with the possibility to tune the distance between the center of the lens and the position of the focal point. It has also been revealed that the efficiency of this lens is much better than for a parabolic-shaped lens. The experimental demonstration of the Luneburg lens has confirmed its focusing effect, with observations indicating that nonlinear wave components are likewise concentrated by the lens.