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Ultralong Wave Focusing via Generalized Luneburg Lens

In this paper, a novel gradient index (GRIN) structural lens based on the concept of generalized Luneburg lens (GLL) is proposed. This lens allows for the realization of double foci and localization of energy flow between the two focal spots, thereby achieving ultralong focusing. The double-foci GRIN lens consists of two concentric circular regions with varying thickness defined in a thin plate structure. The two concentric circular regions are designed to realize continuous change of refractive indices with different profiles. Numerical simulations and experimental studies are performed to obtain the maximum displacement amplitude, full length at half maximum (FLHM), and full width at half maximum (FWHM) of the focal region of the lens. The results demonstrate that ultralong subwavelength focusing can be achieved for a broadband frequency range. In addition, our results show that the FLHM and FWHM can be tailored through the design of the focal length of the GLL. This offers a simple and flexible approach of engineering the GLL focusing characteristics and energy distributions for many applications.


Introduction
Gradient index (GRIN) lens has attracted much attention in recent years for operation and control of optical and acoustic wave propagation [1][2][3][4]. A GRIN lens entails a spatial variation of refractive index, which can be achieved by locally changing the geometric parameters of subwavelength unit cells. The effective refractive index of a GRIN lens depends on the filling ratio of unit cell scatterers [5,6]. The first GRIN lens for structural waves was proposed by Lin et al., which demonstrated the manipulation of the structural wave propagation along the desired trajectories by tailoring the filling ratio of unit cells [7]. For structural wave propagation in thin plates, the effective velocity of structural wave was shown to be directly related to the plate thickness, therefore, the GRIN lens can be achieved through varying the plate thickness [8][9][10][11].
One of the most interesting applications of structural wave propagation in plates is the vibration-based energy harvesting [12][13][14][15]. Focusing of plane structural waves with GRIN lens has been investigated for achieving enhanced energy harvesting. As environmental vibrations are usually in the far field, these excitations can be approximated as plane wave excitations and the energy can be collected at the focusing location of the lens. For example, Tol et al. [16] proposed a GRIN lens consisting of cylinder arrays with changing height and demonstrated that a piezoelectric transducer placed at the lens focus could produce a higher power of up to three times. Zhao et al. [17] designed a GRIN structural Luneburg lens based on continuous variation of plate thickness and demonstrated energy harvesting enhancement of nine times at the focal location of the lens. In another study, Zhao et al. [18] investigated a planar GRIN lens with a varying thickness, which achieved an amplitude of displacement at the focus about 30~40 times of that of incident wave. It should be noted that in all of the above-mentioned techniques the mechanical vibration energy was focused on a small single focus having a short full length at half maximum (FLHM).
Based on the principles of Luneburg lens [19][20][21][22], Sochacki [23] proposed a multi-foci optical GRIN lens. By changing refractive index profiles, the focal point can be tuned to locate either inside or outside of the lens and multiple focal points can be generated. Specifically, a double-foci generalized Luneburg lens (GLL) enables the positions of two focal points to be tailorable and allow energy localization between the two focal spots. Double-foci GLLs have been investigated in several recent studies. For example, Mao et al. [24] explored a tunable nanojet obtained with generalized optical Luneburg lens for achieving ultralong subwavelength focusing between the two foci. Chou et al. [25] proposed a double-foci GLL working in the microwave regime based on a metasurface consisting of circular metallic patches fabricated on a grounded dielectric substrate.
In this work, inspired by the concept of GLL in the electromagnetic regime, we explore a broadband double foci GLL for structural wave manipulation, which enables the realization of an ultralong focal region between the two foci. The refractive index variation in the structural GLL is realized by changing the lens thickness in order to tune the velocity of structural waves.

Generalized Luneburg Lens Design
The principle of the proposed structural GLL is illustrated in Figure 1(a), the lens is composed of two concentric circles with outer radius of R and inner radius of R'. As a plane wave interacts with the structural GLL, double foci can be produced. All the rays passing through the inner circle are focused with a focal length F1, and all the rays passing through the outer circle are focused with a focal length F2. The configuration of rays in such a medium can be described as follows, with the detailed derivations provided in [23]: where r is radial distance, = / , is the angle between the radius distance r and the tangential line, and = ( ) sin ( ). Note that Pa is parameter within the range of 0 to R.
The inner circle radius can be calculated based on the value of Pa; that is, is the refractive index of the inner circle. Based on Equation (1), the refractive index profile is obtained as: When flexural waves propagate through a thin plate structure with a variable thickness, the phase velocity cp is a function of the plate thickness h, which can be expressed as = = √ℎ 0 /ℎ. Therefore, the variable thickness profile of the double-foci GLL defined on a thin plate structure is provided as: In our numerical and experimental studies, we chose the following representative geometric parameters: h0 = 4 mm, R= 100 mm, and F1=1.2R. Different focal lengths F2 (F2= 1.6R and F2 = 2.5R) were considered. Based on Equations (2) and (3), the distribution of refractive indices along the radial distance is obtained, as shown in Figure 1

Numerical and Experimental Studies
The proposed GLL can be analysed by using different numerical methods, such as the spectral method [26]. In this study, we performed numerical simulations based on the finite

Conclusions
In conclusion, a novel design of gradient refraction index (GRIN) structural lens based on the generalized Luneburg lens (GLLs) was investigated, The lens was achieved by using a variable thickness concentric circular structure. This simple lens design renders a continuous gradient index, enabling broadband, ultralong subwavelength focusing of flexural wave.
Numerical and experimental studies were carried out to explore the ultralong focusing properties of the proposed lens. The results show that the GRIN lens can achieve a large FLHM up to 15 at the focusing region, while the FWHM is limited to a subwavelength scale for a broadband frequency range. This work is expected to benefit vibration-based energy harvesting techniques for many applications.

Conflict of Interest
The authors declare no conflict of interest.

Data Access Statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.