1. Introduction

A Luneburg lens is a spherical device that can transform the spherical wave of a point source placed on its surface into plane wave on the opposite side of the lens [1]. This lens is composed of gradient refractive index (GRIN) material and is particularly attractive due to its aberration-free focusing ability. More importantly, its spherical symmetry property allows for a very wide field-of-view. A variety of Luneburg lens applications have been proposed since its theoretical design in 1944 [2–6]. Among those applications, the Luneburg lens was mainly used as the key component for high gain microwave antennas. The interest in this field has been heightened in the recent past due to speculated possibility to engineer the left-handed metamaterials, or to allow for continuously varying material properties to steer the incident wave at will such as for the transformation optics (TO) [7]. Currently, most of the proposed Luneburg lens in literature are based on metamaterial which consists of many sub-wavelength resonators [4–6]. These metamaterial lenses are usually cheap and easy to fabricate. However, they are limited in bandwidth due to the highly dispersive nature of unit cells [8]. On the other hand, Luneburg lens based on traditional dielectric materials are usually broadband but bulky and difficult to fabricate [9, 10], especially for three-dimensional (3D) applications.

In the microwave regime, the manufacture of a Luneburg lens is usually approximated by a series of concentric inhomogeneous dielectric shells, whose electric permittivity is varied discretely. However, it is difficult to produce such shells with desired permittivity and shape accuracy. Moreover, with a large number of shells, the air gap effects can alter the lens performances significantly [11]. Alternatively, the variation in permittivity can be achieved by drilling holes in finite-thick dielectric slices in such a manner that the hole size is small in the centre and radially increases till the edge [12]. Though the underlying principle to achieve the effective response of this material is acceptable, but the cost and time involved (drilling different sized holes in different slices and then etching them back so as to form the radial lens) is unsuitable for the application at hand. Further, the scheme itself is intricate to design spherically symmetric lens. Besides, the performance of the design would be limited by the size of the holes, that is, the operating wavelength must be at least one-tenth of the size of the largest hole in the structure for a satisfactory operation of the device. Recent advances in the 3D printing or additive manufacturing prompted photopolymer jetting to design the lens by considering a unit-cell and then radially varying the size of that cell to achieve a variable permittivity for the desired design [13]. However, this kind of unit-cell is usually direction dependent and, hence, results in spatial dispersion and limitations on the operating frequency.

Despite many successful Luneburg lens applications in the microwave frequencies, its realization at optical frequencies is more complicated. With the help of electron-beam lithography technology, two-dimensional (2D) Luneburg lenses have been demonstrated in both silicon photonics [14, 15] and plasmonic devices [16]. The basic idea is to control the thickness of the dielectric layer of the device for required refractive index. The varying permittivity profile can also be engineered using graded plasmonic crystals (GPCs) imbued in a background material. This method was presented in [17] to design tunable GRIN optics using GPCs with radii-varying semiconductor rods. However, despite many papers have been published demonstrating 2D Luneburg lens, a manufacturing scheme for a 3D Luneburg lens has never been demonstrated in the visible frequency range, in the authors’ knowledge. One realisation at 48 THz has been proposed using the same concept as the 3D printed Luneburg lens in the microwaves [18], that is, by linearly varying the dimensions of the unit cell from the centre till the edge; principally, the unit-cell used is inherently direction and polarisation dependent.

In this paper, we propose to design 3D Luneburg lens using composites: the desired electromagnetic properties can be engineered by judiciously varying the volume fraction of the inclusion-to-host materials. The Luneburg lens is discretized into layers of scatterers where volume fraction of each layer is determined by using effective medium theory (EMT). We adopted three packing approaches for geometric arrangement of inclusions and designed composite Luneburg lenses in both microwave and optical frequencies. The proposed scheme usually results in a large number of inclusion scatterers which renderers thenumerical modelling intractable for commercial softwares. Herein, we opt to use a full-wave dipole-moment-based method [19] for the precise analysis of the interaction between the incident wave and the inclusion scatterers in the composite. We demonstrate the effectiveness of the proposed scheme by showing the near-field response and far-field beam steering of the lens antennas.

2. Principle, design and simulation of composites

2.1. Luneburg lens and effective medium theory

Assuming the relative permeability ${\mu }_r=1$, the permittivity profile of a conventional Luneburg lens can then be expressed as:

The graded permittivity profile can be realized using the well-known EMT, where a complex electromagnetic medium consist of two or more materials can be approximated with a homogeneous effective medium. This homogenization process makes it much easier in designing new materials with desired electromagnetic properties than it would be solely experimentally. In the literature, there are already many analytical models can be found for calculating of effective dielectric permittivity of a mixture [20, 21]. The Maxwell Garnett (MG) mixing rule [21] based on the Lord Rayleigh formula [22] is one of the oldest but also the most popular approach. For the case of a 2-phase mixture using spherical inclusions, the MG formula describes the effective permittivity of the composite in terms of the permittivities of both materials and volume fraction of inclusions-to-host media as follows:

It is worth noting that the MG mixing rule has many restrictions, such as much smaller particle size compared to wavelength, low particle concentration, off-resonances of the particles [23]. For the case of particle concentration, it is commonly believed that the MG formula becomes inaccurate when the volume fraction of inclusions is greater than 0.3. This is usually explained by the fact that higher-order multipoles effects start to dominate when the scatterers are close to each other while the MG theory does not take into consideration of non-dipolar interactions [24]. However, as contrary to this belief, it has been shown that for certain configurations, for example, particles can be considered random and uncorrelated, the MG formula remain surprisingly accurate at high particle density, in the presence of strong mutual interactions [25]. Another work on optical properties of Au@SiO2 particles has also experimentally demonstrated that the MG formula works quite well with volume fractions as high as 0.5 [26]. Nevertheless, we must stress that the MG mixing rule was used here for convenience. Other more accurate effective medium methods can also be adopted if necessary. For example, in [27] the authors extended the MG formula to include the effects of higher order multipole effects. The new method was reported to work well with random particle arrays having volume fraction greater than 0.3. Similarly, in [28], the authors proposed a new EMT that can even describe highly concentrated, resonant plasmonic particles by using dressed polarizability to describe dipolar interactions between nearby particles. Although these methods come at the cost of higher complexity, they can serve as supplementary guidelines for high volume fraction cases. It should be noted that Luneburg lens requires relatively low permittivity values ranging from 1 to 2 which makes the highest required volume fractions below the 0.3 threshold in our microwave and optical lens designs. With the MG formula, we propose to design the Luneburg lens by a controlled variation of the volume fraction of the inclusion-to-host layer by layer. The appropriate volume fraction of the scatterers in a host medium for realizing the discretized permittivity values given in Eq. (2) can then be calculated in the following form:

 

Fig. 1 Selection rules for the effective permittivity modelling. A desired value of the effective permittivity can be achieved by using either dielectric inclusions (in microwaves) or plasmonic inclusions (in optics). Dielectric inclusions require higher volume fraction as compared to that of plasmonic inclusions. Both the dielectric and plasmonic inclusions are considered lossless.

Download Full Size | PDF

Assuming the background medium is air, e.g., ${\epsilon }_h=1$, the relationship among the effective permittivity, volume fraction, and the permittivity of the inclusion scatterers is summarized in Fig. 1. In principle, a desired value of the effective permittivity can be realized by tuning either the volume fraction or the permittivity values of the inclusions. More obviously, the diagram is divided into top left and bottom right regions by materials with positive and negative permittivities, respectively. For conventional positive permittivity materials, the higher the inclusion scatterer permittivity, the lower the volume fraction is required to achieve a particular value of the effective permittivity. On the other hand, for negative permittivity materials like plasmonic inclusions, the smaller the permittivity, the higher the volume fraction is required. In comparison, by employing plasmonic inclusions the desired value of the effective permittivity can be achieved at lower volume fractions than it is using positive permittivity materials. The underlying physics principle is that the plasmonic inclusions cover a wider scattering cross-section as compared to that of conventional dielectrics. The disadvantage of using plasmonic scatterers, however, is that the desired values of permittivity with low-loss can be obtained in optics regime only. Further, plasmonic inclusions involve higher dispersion rates [29]. Nonetheless, if the application at hand requires a narrowband operation in optics, plasmonic inclusions are still a suitable option. The permittivity of a plasmonic material can be calculated by free-electron Drude model as:

2.2. Lens design

Varying the volume fraction of the inclusion scatterers can be simply achieved by changing the number of scatterers in the mixture. For the ease of realization, we assume identical spheres are used as inclusion scatterers for all layers. When the volume fractions of inclusion scatterers for each layer have been calculated, the subsequent step is to pack each layer with spheres. Packing of identical hard spheres has a long history due to its fundamental importance to many scientific disciplines like chemistry, physics and biology. The classic sphere packing problem asks what is the largest packing density one can achieve in a given volume. There already exists an extensive study in literature that covers this fundamental question. The highest packing density can be achieved is about 0.74 and 0.64 for regular lattice packing and random packing, respectively [30]. Here we adopted three packing methods to realize our composite Luneburg lens: simple cubic packing, hexagonal close packing and random packing. The simple cubic packing is based on arranging the lattice point on each corner of the cube. Each sphere on a lattice point is equally spaced between adjacent spheres. As simple cubic packing is not close packing, the packing density is 0.52. In the hexagonal close packing, spheres are stacked together layer by layer in an A-B-A-B-A pattern. The top view of each layer is a simple honeycomb-like tessellation. The B layer can be considered as a slightly shifted version of the A layer since each sphere in the B layer has to rest on top of three spheres in the A layer. The random sphere packing, in contrast to regular lattice packing, is a collection of random spheres. A direct way to obtain a cluster of randomly distributed spheres is to use the random sequential addition (RSA) algorithm [31]. This method sequentially generates random spheres (drawn from uniform distribution) to fill in a volume. Each time a new generated sphere will only be accepted if it does not overlap with existing spheres. This acceptation and rejection process can be very time consuming when the number of spheres is huge. More importantly, the algorithm can easily get stuck in finding available space to position a new sphere when the volume fraction reaches the saturation limit. It has been shown that the saturation limit is about 0.55 for the 2D case (circular disks) and 0.38 for the 3D case [32]. However, according to our test, the RSA already becomes very slow when the volume fraction reaches 0.25. To make it worse, the required volume fraction in the center of the lens can easily reach 0.3 with inclusion material permittivity equals 10. To alleviate this issue, we adopt a new hard sphere packing algorithm based on event-driven molecular dynamics [33] for high volume fraction cases. This algorithm runs more efficiently than the RSA algorithm and can be also applied for non-spherical particles.

 

Fig. 2 Illustration of the subtraction step of the innermost layer. Scatterers are first generated in a cube and then discarded if located outside the layer. Building example based on (a) simple cubic packing and (b) hexagonal close packing. The insets demonstrate the fully generated scatterers in a cube.

Download Full Size | PDF

With the dimension and permittivity of the inclusion scatterer determined, we can calculate the required number of scatteres in each layer according to Eqs. (2) and (4). The lens construction is done layer by layer. When building a certain layer, the inclusion scatterers are first generated to fill a entire unit cube with approximately the required number of scatterers. Then the scatterers are scaled down to the desired radius. Next, scatterers that lie outside the current layer will be discarded. As the scatterers are not dimensionless, there will always be scatterers that touch the inner or outer surface of the current layer. This issue is illustrated in Fig. 2 for the subtraction step of the innermost layer, where the color difference indicates the sphere heights. As can be seen from the figure, the scatterers that touch the surface have been subtracted and resulted in empty spaces inside the layer. This subtraction process tends to lower down the volume fraction in the current layer and hence results in a reduced value of effective permittivity. While this is actually a discretization issue, the inaccuracy can be mitigated by using scatterers with smaller radius. Further calibration can be employed by designing each layer with higher volume fraction than theoretically required.

 

Fig. 3 Visualization of 10 GHz Luneburg lens cut in half and viewed from top. (a) Conventional Luneburg lens using 6 dielectric layers. Homogenized Luneburg lens with 6 composite layers based on (b) simple cubic packing, (c) hexagonal close packing and (d) random packing. The insets demonstrate the three designs from another angle.

Download Full Size | PDF

Based on aforementioned packing schemes, three 10 GHz microwave Luneburg lens were designed and demonstrated in Fig. 3, together with a conventional dielectric lens for comparison. The 3D lens were cut in half and viewed from top to give a clear demonstration of the scatterer distribution. Clearly, scatterers are more densely packed in layers that are closer to the center. This is expected as the permittivity profile of a Luneburg lens gradually increases from edge to center. Both the cubic and hexagonal schemes show symmetrical properties in scatterer distribution while the random scheme is totally random.

2.3. Numerical simulation

The proposed composite lens scheme can easily result in about 10 thousand inclusion scatterers, which rendered the numerical solution using a commercial software intractable; because a commercial software needs to discretize each inclusion scatterer using surface basis functions (usually Rao-Wilton-Glisson (RWG) basis), which dramatically increases the computational complexity. To prevent the computational breakdown in such problems, the method of equivalent dipole moments (MEDM) [19] was utilized for efficient numerical modelling of composites, where we treat each scatterer as a dipole to calculate the coupling between the individual scatterers and to calculate the self-coupling term analytically. More specifically, the self-coupling term (on-diagonal entries in the moment matrix) has been analytically computed using an electrostatic approximation as the field is constant in the scatterer while the coupling terms (off-diagonal entries in the moment matrix) are calculated by replacing each scatterer with a dipole having an equivalent dipole moment to that of the scatterer and sensing the field at the observation point. The basis functions used are one to two orders of magnitude smaller than that of the operating wavelength which makes this technique particularly suitable for the simulation of materials composed of subwavelength particles. The point-dipole model allows the precise solution to the full-wave mathematical formulation by matrix inversion method as in the method of moments (MoM).

3. Results and discussions

3.1. Microwave Luneburg lens

The proposed scheme provides a means to realize composite designs based on off-the-shelf materials. In microwaves, zirconium tin titanate (ZST, chemical formula: $Z{r}_{0.8}S{n}_{0.2}Ti{O}_4$) ceramic can be synthesised to achieve a relative dielectric constant of 37.1 for a quality factor of 5000 at 10 GHz [34]. Small-sized dielectric scatterers of the ZST can be dispersed into a low-dielectric matrix to design volume-fraction-based EM devices.

 

Fig. 4 Calculated achieved permittivity based on the MG formula. The lens radius is 2λ. The inclusion scatterer radius is varied from 0.062λ to 0.041λ and has a relative permittivity of 37.1. (a) Simple cubic packing, (b) hexagonal close packing and (c) random packing. (d) Calculated achieved permittivity comparison after calibration.

Download Full Size | PDF

Based on discussions in the previous section, we designed several 10 GHz Luneburg lens using ZST inclusion particles. The lens has a diameter of $4\lambda$ and the inclusion particle radius varying from $0.062\lambda$ to $0.041\lambda$. The realized effective permittivity based on the three design approaches are shown in Figs. 4(1), 4(b) and 4(c), respectively. It can be noticed that all three approaches have been affected by the subtraction process, especially when the particle radius is $0.062\lambda$. Moreover, the deviation increases as the layer number decreases. This is because the volume of each layer and the desired volume fraction of particles varies inversely as the layer number decreases. The innermost layer has the smallest volume but requires the highest volume fraction of inclusion scatterers, as compared to other layers. In other words, one has to pack with relatively larger particles in the center area and hence results in higher volume reduction in the subtraction process. This issue can be alleviated by using smaller sized inclusion particles as can be seen in all three figures using scatterer radius of $0.050\lambda$ and $0.041\lambda$. Nevertheless, all three approaches are still more or less deviated from the theoretical curve even using particles with radius of $0.041\lambda$. Continuing decreasing the size of the inclusion particles will inevitably result in high computation load and the improvement does not scale well accordingly. When compared with each other, the cubic approach achieves the lowest accuracy while the random approach is between the cubic approach and the hexagonal approach. This is expected since the maximum packing density of thee approaches varies in the same manner. To realize the effective permittivity in a more accurate way, one has to calibrate the scatterer packing by setting up higher effective permittivity/volume fraction for each layer instead of using Eq. (2). Figure 4(d) demonstrates the resulted permittivity curves of the three approaches when calibrated against the desired permittivity profile. The random approach curve was averaged over 10 sets of independent simulations. Clearly, all three approaches achieved much better results than their uncalibrated counterparts. In comparison, the random approach outperforms the other two as the permittivity curve almost overlaps with the desired one. The achieved permittivity values across 10 simulations are also very stable as the maximum standard deviation occurs at layer 2 with a value of only 0.01. The two uniform lattice approaches, however, still exist a small degree of deviation from layer 3 to layer 1. This is because the scatterer radius is not small enough to be neglected and also as both approaches assume symmetrical scatterer arrangement, increasing the number of scatterers in one dimension can result in dozens of scatterers in total. As a consequence, it will be very difficult to accurately calibrate the volume fraction of scatterers in the center region. On the other hand, the random approach poses no such restriction. The volume fractions of scatterers can be precisely calibrated by gradually adding random scatterers to the volume.

It is worth noting that although the subtraction process results in discarded scatterers near the surface of each layer, the volume density between the surface areas has not changed at all. This discontinuity will of course affect the overall effective permittivity of the current layer. The actual achieved permittivity value might lies between the theoretical value and that of the calculated one using the actual number of scatterers. Nevertheless, the focus of this paper is to present a composite design methodology rather than to achieve the exact true effective permittivity profile. Besides, the design parameters based on the MG formula are approximate, and the approximation relaxes accuracy.

 

Fig. 5 Component y of electric field values when excited by a y-polarised 10 GHz Hertzian dipole placed at the focal point (x = 0, y = 0, z = −0.06) in metre. Three calibrated approaches by (a) simple cubic packing, (b) hexagonal close packing and (c) random packing with inclusion scatterer radius equals 0.041λ. d) Electric field values of a conventional ideal Luneburg lens simulated using the CST studio. Two uncalibrated random schemes with inclusion scatterer radius at (e)0.062λ and (f) 0.050λ.

Download Full Size | PDF

The final lens structures based on the three calibrated schemes are shown in Fig. 3 where the total number of inclusion particles used are 14329, 14139 and 14343, respectively. The highest volume fractions for the three microwave lens designs are 0.24, 0.33 and 0.27, respectively. A Hertzian dipole was placed at a focal point ($x=0,y=0,z=-0.06$) at the lens surface. Figure 5 presents the corresponding electric-field intensity maps around the lens at thex-z plane. Additionally, a conventional discrete-layer-lens with the theoretical effective permittivity profile was simulated in a commercial EM software CST for comparison. It can be observed that the three approaches all successfully transformed the spherical wave into plane wave, in a similar way like the CST simulation example. This agreement indicates that all three packing methods are working well for composite homogenization purposes and have at least achieved effective permittivity values that are close to the desired ones. To understand the effects of the calibration procedure and using smaller scatterer radius, near-field plots of two uncalibrated random approaches were also presented for comparison as shown in Figs. 5(e) and 5(f). The spherical wavefront to plane wavefront transformation becomes more obvious as the scatterer radius decreases and the calibrated scheme with $0.041\lambda$ scatterer radius gives the best performance. This variation trend agrees well with their corresponding achieved permittivity values.

 

Fig. 6 Far-field gain plot of 10 GHz microwave Luneburg lens based on the (a) simple cubic packing, (b) hexagonal close packing, (c) random packing and (d) conventional dielectric layers. A y-polarized dipole is varied on the focal curve in 7 steps (from 180° to 360°) in the azimuthal plane. Labels on the outer circle indicate the ϕ variation (in degrees) in the azimuthal plane, gain labels on circles are in dBi, and the gain values below 0 have been suppressed.

Download Full Size | PDF

The far-field gain pattern of the three calibrated lenses are demonstrated in Figs. 6(a), 6(b) and 6(c), together with the result of the conventional dielectric lens in Fig 6(d). The location of the dipole was varied along the focal curve in the azimuthal plane from 180° to 360° in 7 equal steps. While the conventional ideal lens is rotational symmetry, we only showed one pattern for simplicity. The average achieved gains over the 7 directions are 18.07 dBi (max: 18.34, min: 17.71), 17.25 dBi (max: 17.74, min: 17.08) and 18.10 dBi (max: 18.73, min: 17.08) for the three approaches, respectively. This result is close to the conventional lens which achieved the gain of 18.2 dBi simulated with the CST solver. Further, as can be seen from the figure, all three approaches are relatively stable in scanning across 180 degrees with standard deviations of the gain as 0.34 dB, 0.22 dB and 0.38 dB, respectively. It can also be observed that the two regular lattice approaches show symmetrical sidelobes in the gain plot while the random approach does not due to the random nature of the device. When compared with the conventional ideal lens, the composite lenses exhibit larger sidelobes and backscattering. This is likely to be caused by the imperfect and non-smooth surface of the composite lenses since.

3.2. Optical Luneburg lens

In optical frequencies, plasmonic inclusions can be utilized to yield similar results as dielectric inclusions, though at a lower volume fraction of inclusions-to-host. Therefore, the proposed lens design can be translated into the optics regime, where we make use of low-loss plasmonic inclusions to demonstrate the functionality of the Luneburg lens. The inclusion-nanoparticle permittivity can be calculated via Eq. (5) and the parameter values used were the following: ${\omega }_p=2\pi \times 2.15e15$ Hz, $\hslash {\tau }^{-1}=0.5$ [24] is the phenomenological electron-motion damping rate adjusted to match the imaginary part of ε. We choose the permittivity value at 450 THz, because at this frequency the gold permittivity has low loss value. The relative value of the calculated permittivity using Eq. (5) is -13.81 and that of the imaginary part (extinction coefficient) is 0.61. Gold (Au) nanoparticles of diameter 47 nm ($0.035\lambda$) were used to obtain a six layer lens of diameter 2.67 μm ($4\lambda$). Gold nanoparticle-based composites, however, are much more dispersive (as evident from Eq. (5)) and there exist only certain values of frequency where low-loss permittivity can be achieved. The demonstrated optical lens can perform, therefore, in a narrow frequency band only.

 

Fig. 7 Calculated achieved permittivity of the three calibrated approaches at 450 THz. The lens radius is 1.4 µm and the inclusion scatterer radius is 46.5 nm. The inclusion relative permittivity is -13.81 and the imaginary part of permittivity is 0.61.

Download Full Size | PDF

The three packing approaches were calibrated to achieve the desired permittivity values at 450 THz where the total number of inclusion particles used are 15533, 16699 and 16783, respectively. The highest volume fractions for the three optical lens designs are 0.15, 0.22 and 0.20, respectively. The final achieved permittivity curves are shown in Fig. 7. Similar to the microwave lens examples, homogenization in the inner layers is still difficult for the two uniform lattice approaches. The averaged permittivity profile of the random approach remains accurate with a maximum standard deviation value of 0.01. The electric field map of the proposed optical Luneburg lenses are demonstrated in Fig. 8, with a larger area than the microwave example for better visualization. It can be noticed that all three approaches successfully transformed the spherical wave into plane wave, showing the effectiveness of the deign scheme in the optics regime. However, the random approach, although achieves the most accurate permittivity profile, shows much stronger scattering effects around the lens than the other two approaches. The scattering difference can also be observed in the corresponding far-field beam steering patternsas demonstrated in Fig. 9 where the random approach shows higher side lobes and back lobes. The achieved mean antenna gain as the source was varied in the azimuthal plane are 18.60 dBi (max: 18.89, min: 18.38), 19.45 dBi (max: 20.12, min: 18.25) and 17.86 dBi (max: 18.68, min: 17.48) with standard deviations of 0.27 dB, 0.46 dB and 0.44 dB for the three approaches, respectively. This results suggests that accurately achieved effective permittivity values based on the MG formula does not necessarily yield the best performance. The geometrical arrangement of particles can have a huge impact on the accuracy of the true homogenization process.

 

Fig. 8 Component y of electric field values when excited by a y-polarised 450 THz Hertzian dipole placed at the focal point (x = 0, y = 0, z = −1.34) in micrometre. Three calibrated approaches by (a) simple cubic packing, (b) hexagonal close packing and(c) random packing.

Download Full Size | PDF

 

Fig. 9 Far-field gain plot of optical Luneburg lens based on the (a) simple cubic packing, (b) hexagonal close packing and (c) random packing. A y-polarized dipole is varied on the focal curve in 7 steps (from 180° to 360°) in the azimuthal plane. Labels on the outer circle indicate the ϕ variation (in degrees) in the azimuthal plane, gain labels on circles are in dBi, and the gain values below 0 have been suppressed.

Download Full Size | PDF

As mentioned in the previous section, the permittivity of a plasmonic material is frequency dependent. An optical lens designed for a certain frequency will probably have a distinct response at other frequencies due to shifted effective permittivity values. To study the dispersion property of the optical lens, we varied the operating frequency from 300 THz to 450 THz and plotted the corresponding achieved effective permittivity curves where the inclusion relative permittivity is calculated from the Drude model. As can be seen from Fig. 10(a), the achieved permittivity values decrease as the frequency decreases. This linear variation is in agreement with the permittivity change of the plasmonic nanoparticles as shown in Fig. 10(b). Moreover, the permittivity changing rate varies across six layers. The maximum permittivity shift is 0.18 at layer 1 while the minimum permittivity shift is 0.02 at layer 6. This is caused by the volume fraction difference in different layers. As the inner layer has higher scatterer volume fraction than outer layers, it is obviously more sensitive to the permittivity change of nanoparticles. Overall, considering the MG formula is only an approximation, the permittivity shift is minor at this frequency band. The resulted far-field gain patterns across the frequency band are demonstrated in Fig. 11. The dipole gain decreases as the frequency decreases just like a conventional solid dielectric lens. Higher frequency above 450 THz are not simulated due to dramatically increased memory requirement as each nanoparticle has to be subdivided into smaller micro-domain basis functions to ensure accuracy.

 

Fig. 10 Effective permittivity change of a randomly packed lens due to dispersion of the gold nanopartiles from 300 THz to 450 THz. The lens radius is fixed to 2λ at 450 THz. The inclusion scatterer radius is 46.5 nm; the inclusion relative permittivity is calculated from the Drude model. (a) Calculated permittivity values of the random lens across 6 layers and (b) effective permittivity of gold nanoparticles against frequency.

Download Full Size | PDF

 

Fig. 11 Dipole gain (from 300 THz to 450 THz in 7 equal steps) in the presence of a randomly packed optical Luneburg lens. A y-polarized dipole is placed in the azimuthal plane at ϕ = 360°.

Download Full Size | PDF

Unlike the Luneburg lens manifestation based on the additive manufacturing techniques, the proposed scheme is independent of the polarisation in the tangent plane of the lens and can scan any direction in space. It should be noted that although we assumed free-space as host medium in all the designs here, the proposed scatterer-host volume-fraction-based scheme is universal and can be easily tweaked to perform a different EM function, for instance, a Luneburg lens with a flat focal surface for electronic beam scanning application [35] and even for an all-dielectric TO cloak realisation [36].

4. Conclusions

We demonstrated a design scheme that allows for effective control over homogenization parameters so as to realize controlled guidance of electromagnetic waves in medium. Three packing approaches were discussed and utilized for the geometric arrangement of inclusion scatterers. The three approaches were first applied to design and simulate microwave Luneburg lens at 10 GHz. With the MEDM solver, we demonstrated the applicability of the design scheme and compared the scatterer based lens electric field in a plane to that of homogeneously discretised layers in CST, and observed a good agreement between the two. The performance of the three packing approaches were also discussed and evaluated in terms of achieved effective permittivity accuracy. The three design approaches were then extended to design and simulate optical Luneburg lens in the visible frequencies, which demonstrated a similar performance to the microwave counterpart in terms of gain and side lobes. Further, the dispersion property of the optical lens was analyzed and shown to be applicable over a broad frequency of operation.