1. Introduction

The diffractive lens, as an alternative to the traditional plano-convex lens for many optical applications, offers significantly reduced thickness with its flat profile, thus possessing the merits of easy fabrication and small volume [1]. However, the tradeoff for such a low profile lens, based on the principle of wave diffraction instead of refraction, is its narrow band operation.

Given a Fresnel lens [2–4] with a design wavelength ${\lambda }_0$, a focal length F, and a lens aperture D, the successive radii of every Fresnel zone and the lens thickness can be calculated as

 

Fig. 1 Configuration of the quarter wave Fresnel lens. The demonstration phase-correcting Fresnel lens working at 30 GHz is chosen from [3], with $D=63.5\text{\hspace{0.17em} mm}$, $F=D/4$, $t=9\text{\hspace{0.17em} mm}$,${\epsilon }_1=2.48$,${\epsilon }_2=5.8$,${\epsilon }_3=45.4$,${\epsilon }_4=3.43$. (a) Top view. (b) Lateral view. (c) Ray tracing through every sub-zone of the phase-correcting Fresnel lens (magnified picture of the blue dashed box in (b)) .

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There are existing attempts using diffractive doub/trip-let or hybrid refractive-diffractive components to minimize the aberration due to chromatic effects [5]. However, those schemes involving an optical system rather than a simple lens unavoidably increase the overall size and deteriorate the radiation efficiency. Several other studies addressed this problem with the proposal of fractal zone plates [6–9] which improves the classical Fresnel zone plates in certain applications where multiple foci are needed. In fact, the chromatic aberration is a typical distortion which is unavoidable even for the traditional refractive plano-convex lens. However, it can be argued that refractive optics is better placed to deal with this problem, compared with diffractive optics. Unfortunately, refractive lenses always have a large volume with a curved surface which makes them impractical in many applications of electromagnetics. Therefore, it is desirable to converge the incoming light based on a refractive approach while retaining the benefits, such as low weight and low cost, from the diffractive method.

Transformation optics as a practical strategy to control lights has been intensively studied and widely adopted recently [10–24]. It provides a straightforward connection between the geometrical optics of lenses and the gradient refractions of electromagnetic materials. Based on this conceptual design, the conversion of conventional plano-convex lenses into flat systems becomes possible [21–24]. To avoid the loss and narrow bandwidth issues typically present in metamaterials, we introduce a selective-sampled transformation optics technique from which a broadband zone plate lens is designed based on its hyperbolic prototype.

2. Theory and numerical results

The design procedure is shown in Fig. 2 . We start with a two-dimensional (2D) coordinate transformation [10,11], and then rotate the 2D flat lens into a three-dimensional (3D) zone plate lens [19]. The permittivity and permeability tensors in virtual space ($\overline{\overline{\epsilon }}$,$\overline{\overline{\mu }}$) and physical space ($\overline{\overline{\epsilon }}\text{'}$,$\overline{\overline{\mu }}\text{'}$) have the relation of

 

Fig. 2 Schematic showing of the transformed zone plate lens design using the selective-sampled transformation technique. (a) 2D hyperbolic lens with nearly orthogonal mapping. The parameters are: $D=63.5\text{\hspace{0.17em} mm}$, $F=D/4$, $t=9\text{\hspace{0.17em} mm}$. (b) 2D flat lens with the permittivity map consisting of 110 × 20 blocks. (c) 2D flat lens with the permittivity map consisting of 22 × 4 blocks. The four layers have heights of 1.4mm, 3mm, 3mm and 1.6mm respectively in the$\widehat{z}$direction, and the 1st to 10th blocks have the width of 3mm in$\widehat{x}$direction with the 11th block having a width of 1.75mm (d) 3D transformed zone plate lens.

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It has been proven [14] that the effective refractive index for E-polarization could be dependant only on the permittivity value as long as the grid is carefully generated into nearly orthogonal cells in the virtual space as shown in Fig. 2(a). Figure 2(b) illustrates the orthogonal grid of Cartesian cells with the permittivity map consisting of 110 × 20 blocks. So far, a conventional hyperbolic lens in the virtual space has been completely compressed into the flat lens in the physical space. However, it is noted that the boundary of the original structure in Fig. 2(a) is different to that of the transformed one in Fig. 2(b), hence there are supposed to be additional transformed layers between the flat surface and the curved surface. Unfortunately, such layers require that the permittivity be less than one which can only be realized using metamaterials. Therefore, we only select the area within the lens aperture in the original space to perform the transformation, as shown in the procedure moving from Fig. 2(a) to Fig. 2(b). Such a selective transformation of the original lens can be viewed as the removal of the metamaterial layer on the transformed flat lens, which will unavoidably cause refraction discontinuities in the ray trace. However, the total reflection for the transformed lens can be reduced by simply removing the metamaterial layer, since such epsilon less than one material will increase the mismatch of the transformed lens [25].

A further step to approximate and simplify the transformation optics is based on a recent design approach used in an implementation of a simplified carpet cloak made of only a few blocks of all-dielectric isotropic materials [17]. When applied to the present work, the high resolution permittivity map shown in Fig. 2(b) can thus be approximated using a relatively low resolution sampling map. It is demonstrated below that the sampled 22 × 4-block in the transformed lens, shown in Fig. 2(c) and Table 1 , can perform as well as the original convex lens, while it is easy to fabricate and has a low profile.

Table 1. Relative Permittivity Values of the Transformed Zone Plate Lens

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To complete the lens design, we rotate the simplified 2D flat lens around the$\widehat{z}$axis to generate a 3D transformed zone plate lens. The 22 samples in the $\widehat{x}$direction thus become 11 zones in the horizontal plane, whilst the 4 samples in the $\widehat{z}$direction become the 4 layers of the lens in vertical plane as shown in Fig. 2(d). Up to now, through a selective-sampled transformation optics process, and a rotation around the symmetric axis, we reach our final design. Even though the present lens in Fig. 2(d) has the similar appearance to the conventional phasing-correcting Fresnel lens and we still use the terminology as zone plate lens, but note that the proposed lens has no repetition of the full wave zone.

A full-wave finite-element simulation (Ansoft HFSS v12) is then performed to verify the proposed design. When we place a point source on the focal points of the conventional hyperbolic lens, the phase-correcting Fresnel lens and the transformed zone plate lens, all the three lens have high directivity and the considerably matched radiation patterns as expected at the design frequency of 30 GHz. The respective patterns for this scenario are shown in Fig. 3(b) . Compared with the original hyperbolic lens, the side lobes of the transformed lens become a slightly higher due to the simplification and approximation from the selective-sampled transformation. However, it is noted that the transformed zone plate lens has superior performance in terms of operational bandwidth. It is clear that the phase-correcting Fresnel lens has a severely degraded directivity at both 20 GHz and 40 GHz due to the phase error as shown in Fig. 3(a) and Fig. 3(c). Figure 3(d) further explores the bandwidth property of the refractive and diffractive lenses in terms of radiation directivity which refers to power density the lens radiates in the direction of its strongest emission, relative to the power density radiated by an ideal isotropic radiator radiating the same amount of total power. The transformed zone plate lens has a steady performance over 20-40 GHz, while the phase-correcting Fresnel lens only has 5 GHz bandwidth from 30 GHz to 35 GHz.

 

Fig. 3 The radiation patterns of the conventional 3D hyperbolic lens, 3D phase-correcting Fresnel lens in [3] and 3D transformed zone plate lens at (a) 20 GHz, (b) 30 GHz, (c) 40 GHz. (d) The comparison of the bandwidth performance of 3D phase-correcting Fresnel lens and 3D transformed zone plate lens from 20GHz to 40GHz.

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3. Potential realization

This kind of zone plate lens is practically manufacturable in many ways. Constraints from metamaterials have been completely removed since the required permittivity value over the lens aperture is ranging from 2.62 to 15.02. Therefore, materials based on non-resonant elements can be used to achieve such permittivity values which often have low losses cross a wide range of frequencies. One well known method is to drill subwavelength holes of different sizes along the $\widehat{z}$ direction in dielectric host medium. This is the same technology used in the developments of the conventional Fresnel lenses [3,4] and the recent 3D carpet cloak [19]. Alternatively, we can use a periodic array of high-index dielectric cylinders to manipulate material properties as the approach for designing a one-directional free space cloak [18].

These schemes can be regarded as a homogeneous medium replacing the air and the dielectric region. Generally, Perforations in the host medium (${\epsilon }_{r1}$) lead to a lower relative permittivity value ${\epsilon }_{eff}$ ranging from ${\epsilon }_{r2}^{air}$to ${\epsilon }_{r1}$. Obviously, several host media will be needed to fulfill our design due to such a limited effective permittivity range. An efficient solution to this problem is to insert high-index dielectric cylinders (${\epsilon }_{r2}^{high}$) into the drilled holes, in place of air. Therefore, effective permittivity is expanded to ${\epsilon }_{r2}^{air}<{\epsilon }_{eff}<{\epsilon }_{r2}^{high}$.

Figure 4 gives the graphical representation of the relationship between the effective permittivity and the perforation. The host materials we employ here are ${\epsilon }_{r1}=5.8$ and the filling high index dielectric is ${\epsilon }_{r2}^{high}=\text{36 .7}$. These specific permittivity values are chosen because of their availability of dielectric materials, and have been used in the conventional Fresnel lens fabrications [3,4] and unidirectional free space cloak design [18]. The effective permittivity can be estimated from the area ratio between the holes ($\pi d^2/4$) and the unit cell ($l^2$) [3,4]. A more accurate way is to retrieve the permittivity value from S parameters [19,26]. Given$l=1\text{\hspace{0.17em} mm}$, as observed in Fig. 4 (b), different d will lead to different effective permittivity. As d varies, the effective permittivity ranges from 1.68 to 24.11. In addition, we can see that the effective permittivity is for our purposes dispersionless, and does not vary much with the frequency as long as the unit cell is subwavelength, as shown in Fig. 4(c), Fig. 4(d), Fig. 4(e), and Fig. 4(f). Table 2 gives the details of the hole size to realize the required permittivity profile with the side length of the unit cell $l=1\text{\hspace{0.17em} mm}$. If the required relative permittivity is larger than 5.8, we drill the holes and then fill the high-index dielectric cylinders. Therefore, our proposed design for transformed zone plate lens, with permittivity value ranging from 2.62 to 15.02, is practically realizable with this way.

 

Fig. 4 Graphical representation of the relationship between the effective permittivity and the perforation. The host medium chosen here is ${\epsilon }_{r1}=5.8$, and the filling materials is air ${\epsilon }_{r2}^{air}=1$ or ${\epsilon }_{r2}^{high}=36.7$. d refers to the hole diameter, and l refers to the length of side of the unit cell. $l=1\text{\hspace{0.17em} mm}$ is subwavelength scaled in order to make the effective media more homogeneous. (a) Perforated host medium with square unit topology. (b) The effective permittivity achieved from the square unit topology at 30 GHz as d is varied. The maximum value is 24.11, while the minimum value is 1.68. (c) Frequency response from 20 to 40 GHz of the effective perforated medium. (d) Magnified picture of the frequency response in (c) around the effective permittivity equal to 2.62. (e) Frequency response from 20 to 40 GHz of the perforated medium filled with high dielectric material. (f) Magnified picture of the frequency response in (e) around the effective permittivity equal to 15.02. (The curves at different frequencies are very similar, and in (c), (d), (e), (f) clearly overlapped. This represents, for our purposes, dipersionless effective media. The parameter retrieval in (b), (c), (d), (f), and (e) assumes the unit cell to be periodically infinite. In practice, the finiteness of the structure will result in slight changes in effective permittivity.)

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Table 2. Perforation Size on Every Unit with Unit Side Length l = 1 mm^a

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4. Conclusion

In conclusion, a broadband zone plate lens has been proposed and discussed in detail based on the transformation optics. Material simplification is obtained through a selective-sampled transformation process. We have shown that the conventional hyperbolic lens and the transformed zone plate lens have a considerable agreement over a broad frequency band, thus providing a superior alternative approach to the conventional band limited phase-correcting Fresnel lens. We expect that this conceptual design based on the sampling of selectively transformed dielectrics, would potentially lead to innovation in the practical design of broadband optical devices.