Open Access
Add to BibSonomyAbstract
Based on a focusing Mikaelian’s lens with small refraction index (0<n<<1), an optical device is designed as a super-thin optical beam compressor (e.g., thickness = 3λ0) with an extremely high beam compression ratio (more than 19:1). This device can also be used as a beam collimator or a cylindrical-to-plane wave convertor with a much higher transmissivity than a zero-index metamaterial slab. The output beam shows good directionality in both near field and far field. A metamaterial structure is also designed to realize this device and verify its performance with finite element method (FEM).
©2013 Optical Society of America
1. Introduction
Metamaterials made by artificial units, which are much smaller than the wavelength, can be engineered to realize nearly arbitrary value of permittivity and permeability [1]. By gradually changing the size of each unit, it can realize a gradual change in complex material parameters [2]. Many interesting optical devices requiring complex materials designed by transformation optics (TO) [3–6] have been successfully realized by metamaterials, such as optical cloaks [3, 4], perfect lens [5–7], optical black holes [8] and beam compressors [9]. Metamaterials with permittivity or/and permeability near to zero give new ways to explore novel phenomena and new devices [10–13].
In 1951 A.L. Mikaelian proposed a gradient structure whose refraction index changes with the radius in the form of a hyperbolic secant [14]. This structure is widely used in information optics due to its focusing effect and is called Mikaelian’s lens (ML) [15, 16]. Many studies have been made on ML’s focusing properties [14–16]. ML is an aberration-free lens [17] which can also be used as a coupler between waveguides [18]. The light propogation in ML has been studied in geometrical optics [14–16] and wave optics [19, 20]. The focusing properties of ML have been verified by experiments in microwave band [10]. ML can also be realized by photonic crystals [20]. Previous studies on the ML require the refraction index of the whole device to be larger than 1 due to the fact that almost all natural materials’ refraction indices are larger than 1 at microwave and optics wavelengths (except that of dielectrics at X-ray and some metals at some optical frequency ranges). Little has been reported on an ML with refraction index lower than 1 [21].
With the help of metamaterials, we propose in this paper an optical device based on the ML with the refraction index close to 0, which can be used as either a compressor or a collimator. This low index ML gives rise to many additional advantages: extremely high beam compression ratio (more than 19:1), super-thin thickness (e.g., 3λ0) and good directionality of the output beam. Note that one can also use TO to design light beam compressors and optical collimators [9, 22, 23]. It is very complicated to design the corresponding metamaterials (to realize such devices designed by TO), which are often anisotropic and inhomogeneous magnetic materials. The device proposed in this paper is isotropic and nonmagnetic, which is much easier to realize by metamaterials. We also design a metamaterial structure composed of metal wires to realize our ML with 0<n<<1 and use the finite element method (FEM) [24] to confirm its performance. The designed structure shows a good beam compression effect at frequencies of interest.
2. Design method and simulation results
The ML is a cylindrical lens, whose refraction index decreases gradually from the center to the edge in the radial direction in the following form [14]:
where n0 is the refraction index on the optic axis, r is the radial distance, g is a gradient constant, d = π/2g is the thickness (also focal length) of the ML and sech(x) is the hyperbolic secant function [17]. For a two-dimensional (2D) case, the refraction index can be written as (assuming the optical axis is along the x-axis):In the literature n0 is chosen to be larger than 1 to ensure n>1 in the whole device. Utilizing the advantages of metamaterials, we propose to realize an ML with 0<n<1. When a light wave normally impinges on an ML with 0< n0<<1, two optical effects can be expected: (i) The size of the focused spot should be larger than that in a traditional case, as the spot size is inversely proportional to the local refraction index [25]; (ii) The output beam is no longer a focused beam but a collimated beam. According to Snell’s law, the output beam should be almost perpendicular to the lens’ surface, if the refraction index of the lens approaches zero. Thus, it can work as an adapter to compress/expand an illumination beam. It should be noted that the spot size of the output beam is limited by the diffraction limit [25], since the ML of inhomogeneous positive refraction index can neither amplify nor convert the evanescent waves [7].
We use finite-element-method (FEM) simulation to verify this idea as shown in Fig. 1 . Our simulation is in a 2D space for TE polarization, and the refraction index of the ML is set by Eq. (2). The thickness of the lens is d = π/2g = 6λ0 and the height is h = 16λ0 (λ0 is the free-space wavelength). The incident wave is a Gaussian beam with waist radius w = 5λ0. We can see the performance of the ML changes from a focusing lens to a beam-size compressor, as we change n0 from 1 to 0.2 (see Fig. 1). To verify our FEM simulation results, we choose one example (Fig. 1(c)) to verify with an alternative technique – FDTD method (see Fig. 2(d) below). One sees that the FDTD simulation result in Fig. 2(d) agrees well with our FEM simulation result in Fig. 1(c).
Fig. 1 The 2D TE polarization FEM simulation result. The thickness of the ML is d = π/2g = 6λ0 and the height is h = 16λ0. The incident wave is a Gauss beam with waist radius w = 5λ0. The absolute value of electric fields in z direction are shown with (a) n0 = 1, (b) n0 = 0.6, (c) n0 = 0.2 and (d) n0 = 0.1. In order to show clearly the output beam, we rescale the color bar. The white regions mean the amplitude of the electric field is larger than the maximum value in the color bar.
Fig. 2 The 2D FEM simulation result (TE polarization) when we set a line current with distance λ0 away from the front surface of an ML or a zero-index slab for (a), (b) and (c). We only plot the absolute value of the electric field after the slab. (a) The ML’s case with thickness d = π/2g = 6λ0, height h = 16λ0 and n0 = 0.2. (b) A zero-refraction index slab with thickness d = 6λ0, height h = 16λ0. (c) The ML’s case with thickness d = π/2g = 6λ0, height h = 16λ0 and n0 = 3. (d) The 2D FDTD simulation result (TE polarization) for the amplitude of the electric field. Both the structure and the incident wave in this simulation are the same as those for Fig. 1(c). The FDTD simulation is based on Meep (http://ab-initio.mit.edu/wiki/index.php/Meep).
If we put a line current in front of an ML with 0<n0<<1, it can also be used as a collimator due to its focusing property [15]. This has also been verified by FEM as shown in Fig. 2(a). A homogenous slab with refraction index n = 0 can also be used as a collimator [26]. When we set a line current with unit amplitude in front of an ML with n0 = 0.2 and a zero-refraction index slab separately, the transmissivity of the ML (about 5.582%) in Fig. 2(a) is higher than that of the zero-index slab (about 0.011%) in Fig. 2(b). Here the transmissivity is defined as the ratio of the integration of power flux on the output surface with the device to the one without the device. We should note that an ML with n0>1 does not work as a collimator but a focusing lens (see Fig. 2(c)) and the focusing length of this lens decreases as n0 increases.
Zero-refraction index material is used as a cylindrial-to-plane wave convertor in [27]. A device made by four MLs with 0<n0<<1 has a similar functionality. The device is composed of four regions labeled I, II, III, and IV (see Fig. 3(a) ). The material of regions I and III gradually changes according to Eq. (2) and we can replace y in Eq. (2) with x to obtain the refraction index in regions II and IV. The refraction index distribution in the whole device is shown in Fig. 3(b). If we set a line current at the center of this new device, the output beam is no longer a cylindrical wave but four collimated beams (see Fig. 3(c)). If we fill regions I, II, III and IV with a zero-index material, it can also be a cylindrical-to-plane wave convertor (see Fig. 3(d)). Obviously the advantage of our device is that the intensity of the output field is much higher (one order higher) than that of the device (made by a zero-index material) proposed in [27].
Fig. 3 (a) The four regions of the proposed device. (b) the refraction index distribution of this new device based on ML’s refraction index profile. The white regions in (a) and (b) represent the free space. When we set a line current at the center of the device, the absolute value of the electric field distribution outside the device is shown in (c) and (d) (calculated with FEM simulation for the 2D TE polarization case). (c) The device with refraction index profile in (b). (d) The four regions of the device are filled with a zero-index material. The white regions in (c) and (d) are where the amplitude of the electric field is larger than the maximum value in the color bar.
In this paper, our attention is mainly on the beam compression property of the ML with 0<n0<<1 and the method to implement the ML using metamaterials. We found this compression function very robust against the lens thickness. As shown in Fig. 4(a) , even if the thickness of an ML with n0 = 0.2 is reduced to 3λ0, it still gives a good beam compression performance. However, as the thickness decreases, the transmissivity also decreases (see Fig. 5(a) ). To keep the focusing performance of the ML, we should ensure the medium varies gradually so that the following condition can be fulfilled [28]:
where λ = λ0/n. For a 2D ML, by substituting Eq. (2) into Eq. (3), the above condition can be written as:where sinh(x) is a hyperbolic sine function and g = π/2d . If the thickness d of the ML decreases, g will increase and then |1/sinh(gy)| decreases. For a fixed height, when the ML is much thinner than the wavelength (e.g., less than λ0/10), condition (4) cannot be satisfied any more. This means the ML will no longer be a graded inhomogeneous medium but some effective medium for the incident wave. Consequently, the focusing property of the ML will disappear. When the thickness of the ML is comparable to the wavelength (e.g., the thickness is λ0), it is the transition state between a graded inhomogeneous medium and an effective medium.Fig. 4 FEM simulation results (for TE polarization) when the incident wave is a plane wave with unit amplitude for a 2D ML with n0 = 0.2 [(a), (b) and (c)], and for a beam compressor designed by TO [(d) and (e)]. All devices have the same thickness d = 3λ0, height h = 90λ0 and a compression ratio of 19:1. (a) Distribution of the absolute value of the electric field in the ML with transmissivity 7.15%. The white regions are where the amplitude of electric field is larger than the maximum value in the color bar. (b) The normalized absolute value of the output beam from the ML in far field which shows a good directionality of the output beam. (c) The absolute value of the electric field in the impedance matched ML with ε = μ = n0sech(gy) with transmissivity 99.12%. (d) Distribution of the absolute value of electric field behind the TO-based compressor. (e) Far field pattern of the output beam from the TO-based device. The compression ratio is defined as the full width of null-to-null magnitude (FWNM) of the incident beam to the FWNM of the output beam. (f) A plane wave with full width 90λ0 incident on a diaphragm with a hole width 4.77λ0 (equivalent to the full width of the output beam in (a)).
Fig. 5 The 2D FEM simulation results (for TE wave) when a plane wave impinges on an ML with n0 = 0.2. (a) The relation between the transmissivity and the thickness of an ML with fixed height h = 90λ0. (b) The relation between the size of the output beam and the size of the incident beam for an ML with fixed thickness d = 3λ0. One sees that the spot size of the output beam has a nearly constant FWHM around 3.4λ0. FWHM means the full width of half magnitude. (c) The relation between the size of the output beam (λ0 after the back surface of the ML with fixed height h = 90λ0) and the thickness of the ML. One sees that the spot size of the output beam has a nearly constant FWHM around 3.4λ0.
The beam compression ratio of an ML with 0<n0<<1 can be extremely high. For example in Fig. 4(a), when the full width of the incident plane wave is 90λ0, the full width of the output beam is 4.77λ0 (the compression ratio is almost 19:1). If we increase the size of the input beam, the size of the output beam is almost unchanged (see Fig. 5(b)). Thus, we can obtain an extremely high beam compression ratio simply by increasing the height of the ML. A single aperture (e.g., a diaphragm) can also be used as a beam compressor with an extremely high compression ratio [25]. Compared with the present ML with 0<n0<<1, a single aperture has two drawbacks: (i) The energy efficiency is quite low. An aperture simply blocks the edge part of the rays, while the ML can guide almost all the rays (including the edge part) to the optical axis. (ii) The directionality of the output beam is not good due to the diffraction of the light beam when passing through an aperture, while the output beam of an ML with n0 = 0.2 shows a very good directionality in both near-field (see Fig. 4(a)) and far-field regions (see Fig. 4(b)). For a fair comparison, we also simulate the case when a plane wave with a full width of 90λ0 impinges on a diaphragm with a hole width of 4.77λ0 (see Fig. 4(f)). In this case, the output beam does not have good directionality or uniformity as compared with our ML in Fig. 4(a).
The relation between the thickness of the ML and its transmissivity is shown in Fig. 5(a). As the thickness varies, many transmissivity peaks appear periodically like for a FP resonator [25]. As the thickness of the ML increases, the transmissivity (including the transmissivity peaks) also increases. If the ML’s thickness increases further, peak transmissivity tends to be stable (e.g., nearly 91.2% in this case). We can explain this phenomenon in this way: If the thickness of the ML is large enough, the refraction index varies continuously and slowly for a lightwave with wavelength λ0 and thus Eq. (4) is satisfied. In this case, if a lightwave impinges on the ML, the reflections only occur at its front and back surfaces. Since the local reflection coefficients at the ML’s front and back surfaces are different (unlike the FP resonance of a homogenous dielectric slab), the peak transmissivity is not 100% in this case. If the thickness of the ML is too small and the wavelength λ0 remains unchanged (then Eq. (4) cannot be satisfied), the refraction index will no longer vary gradually. In this case, ML can be treated as some effective medium with very low refraction index, and thus the transmissivity is very low. As the ML’s thickness d increases, the region with the refraction index not very close to zero also increases. Thus, the transmissivity increases with the increasing thickness. While the output beam is mainly focused on the optical axis, the size of the output beam is mainly determined by the refraction index n0 on the optic axis. That is the reason why the size of the output beam changes little as the thickness changes (see Fig. 5(c)). With increasing thickness of the ML, the transmissivity increases and the compression ratio keeps unchanged. In practice, we can choose a resonant frequency as the working frequency or use a thick device to obtain a high transmissivity. For a super-thin device, we can improve its transmissivity by using impedance matched materials with ε = μ = n (see Fig. 4(c)).
As the ML’s height increases, the region with the refraction index very close to zero also increases, and more edge part of the incident beam will be reflected. To check further whether the edge part of the incident wave can contribute to the output beam, we make some comparisons by adding a diaphragm with hole width 6λ0 before the ML (see Fig. 6 ). For a ML with height 90λ0 and width 3λ0, the edge part of the incident beam contributes little to the output beam (cf. Figure 6(a) and 6(b)). For the impedance matched ML with height 90λ0 and width 3λ0, the edge part of the incident beam really contributes and makes an obvious difference to the output beam (cf. Figure 6(c) and 6(d)).
Fig. 6 FEM simulation results (for TE polarization) when the incident wave is a plane wave with unit amplitude and full width 90λ0 incident on an ML with n0 = 0.2 n = n0*sech(gy), thickness d = 3λ0, height h = 90λ0 [(a) and (b)] and an impedance matched ML with n0 = 0.2 ε = μ = n0*sech(gy), thickness d = 3λ0, height h = 90λ0 [(c) and (d)]. We add a diaphragm with width 6λ0 before the ML in (b) and the impedance matched ML in (d). To illustrate whether the edge part of the incident beam really contributes to the output beam, we only plot the electric field amplitude of the output beam.
3. Experimental design
In this section, as an example we will show how to implement such a low-index ML using metamaterials. We design a metamaterial structure composed of copper wires to realize a 2D ML with n0 = 0.3, thickness d = 3λ0 and height h = 10λ0 for TE polarization working at 10 GHz (λ0 = 30 mm). The whole ML is divided into 30 square units in height (y direction), 9 square units in thickness (x direction) and is infinitely long in the z direction. The size of each square unit is 10mm × 10mm (about λ0/3<λ/10) to make the effective medium theory valid. An infinitely long cylindrical copper wire with diameter Dr is set at the center of each square unit (see Fig. 7(a) ). We use standard commercial software CST-microwave Studio to extract each unit’s S parameters and use these S parameters to retrieve the effective refractive index n and impedance z [29]. We obtain a relation between the diameter of each copper wire and the effective refractive index n of each metamaterial unit. Then we choose different diameters of cylindrical wires in different units (see Fig. 7(b)) according to the ML’s refraction index distribution. The radii of the cylindrical wires are identical along the x direction and change gradually along the y direction. FEM shows a good beam compression performance of the designed metamaterial structure (see Fig. 8 ). To reduce the fabrication work, we make some approximation (with step changes) of this ML. The approximate parameters are also shown in Fig. 7(b). The reduced device also performs well the beam compression (see Fig. 8 (c)). As the refraction index is close to zero in the whole device, the local wavelength λ = λ0/n is much larger than the size of each unit, and thus a slight reduction in unit cell can hardly influence the performance of the device. The designed wavelength is 30 mm, while FEM simulation results show that both the non-reduced and reduced devices can work in the band between λ0 = 29mm to λ0 = 31mm.
Fig. 7 (a) A unit cell in the designed metamaterial: we set a cylindrical copper wire with diameter Dr and infinitely long in the z direction at the center of a square unit of 10 mm x10 mm. The square region is filled with air. (b) The diameter of each cylindrical copper wire and the effective refraction index of each metamaterial unit in different rows of the whole device. From the center y = 0 to two edges of the ML, we label rows 1 to 15.
Fig. 8 The 2D FEM simulation results (for TE polarization) for the Mikaelian’s lens. The distribution of the absolute value of the electric field in the ML structure with the designed material parameters [(a) and (b)] and the reduced material parameters (c) with n0 = 0.3, thickness d = 3λ0 and height H = 10λ0. The input beam is a plane wave with width w = 5λ0 and a wavelength of λ0 = 30 mm (a), λ0 = 29 mm (b), and λ0 = 30 mm (c). Here we use the PEC (perfect electric conductor) model for the copper.
4. Discussion and conclusions
In conventional optics, an aperture or a telescope system [30, 31] is used to compress the size of the light beam. A comparison between our ML with 0<n0<<1 and a single aperture has been discussed in section 2. Compared with an ML, a telescope system used as a compressor has some drawbacks: (i) The whole device is not compact. The thickness of a telescope system is often much larger than the working wavelength (e.g., the thickness is about a million of the wavelength). (ii) The compression ratio is low (e.g., 5:1). When used as a collimator, our ML has higher alignment accuracy due to its aberration-free feature. Compared with a zero-index slab for a collimator, the transmissivity of our ML with 0<n0<<1 is much larger.
It is easy to design a beam compressor using TO [23] by choosing the following 2D transformation:
where (x, y, z) is a coordinate point in the reference space and (x’, y’, z’) is a coordinate point in the physical space. d is the thickness of the device and α is the compression ratio. The corresponding material parameters can be determined by TO [3–6, 23]:whereandThe parameters of the beam compressor designed by TO are often complicated. Although quasi-conformal mapping can be used to simplify the device [32], the quasi-conformal mapping theory can only be applied to 2D device for TE polarization [33]. The compression ratio of the TO-based device is determined by material parameters: a larger α requires more complicated materials. For our ML used as a compressor, the size of the output beam hardly changes when the size of the incident beam increases (see Fig. 5(b)), and thus it can obtain an extremely high compression ratio by increasing the height of the device. The performance of the TO-based device described in Eq. (6) is also verified by FEM as shown in Fig. 4(d). Under the same condition, the output beam of our ML shows a better directionality in both near field and far field, compared with the TO-based device (cf. Figure 4 (b) and 4(e)).
In this paper we have designed a lens device which can be used as either a beam compressor or a beam collimator, and FEM simulations have been carried out to verify its performance. Compared with traditional beam compressors and collimators, our Mikaelian’s lens device with 0<n0<<1 has many advantages: super-thin thickness, extremely high beam compression ratio, and very good directionality of the output beam. We have also designed a metamaterial structure composed of copper wires to realize such a 2D ML with n0 = 0.3. The present idea can be extended to terahertz and infrared bands. The ML with 0<n0<<1 may have many potential applications in integrated optics, aligning optical instruments (e.g., binoculars), optical couplers for planar lightwave circuits, free space optical communication, etc.
Acknowledgments
This work is partially supported by the National High Technology Research and Development Program (863 Program) of China (No. 2012AA030402), the National Natural Science Foundation of China (Nos. 61178062, 61271085 and 60990322), the Program of Zhejiang Leading Team of Science and Technology Innovation, Swedish VR grant (# 621-2011-4620) and AOARD.
References and links
1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef] [PubMed]
2. D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036609 (2005). [CrossRef] [PubMed]
3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
4. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006). [CrossRef] [PubMed]
5. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8(10), 247 (2006). [CrossRef]
6. U. Leonhardt and T. G. Philbin, Geometry and Light: Science of Invisibility (Dover, 2010).
7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
8. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]
9. H. Ma, S. Qu, Z. Xu, and J. Wang, “Using photon funnels based on metamaterial cloaks to compress electromagnetic wave beams,” Appl. Opt. 47(23), 4193–4195 (2008). [CrossRef] [PubMed]
10. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046608 (2004). [CrossRef] [PubMed]
11. Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18(16), 16587–16593 (2010). [CrossRef] [PubMed]
12. V. Mocella, S. Cabrini, A. S. P. Chang, P. Dardano, L. Moretti, I. Rendina, D. Olynick, B. Harteneck, and S. Dhuey, “Self-collimation of light over millimeter-scale distance in a quasi-zero-average-index metamaterial,” Phys. Rev. Lett. 102(13), 133902 (2009). [CrossRef] [PubMed]
13. A. Alù, M. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern,” Phys. Rev. B 75(15), 155410 (2007). [CrossRef]
14. A. L. Mikaelian, “Application of stratified medium for waves focusing,” Dokl. Akad. Nauk SSSR 81, 569–571 (1951).
15. A.L. Mikaelian, “Self-focusing medium with variable index of refraction,” in Progress in Optics XVII, 283–346 (1980).
16. A. L. Mikaelian, “General method of inhomogeneous media calculation by the given ray traces,” Dokl. Akad. Nauk 83(2), 219 (1952).
17. R. Ilinsky, “Gradient-index meniscus lens free of spherical aberration,” J. Opt. A, Pure Appl. Opt. 2(5), 449–451 (2000). [CrossRef]
18. M. I. Kotlyar, Y. R. Triandaphilov, A. A. Kovalev, V. A. Soifer, M. V. Kotlyar, and L. O’Faolain, “Photonic crystal lens for coupling two waveguides,” Appl. Opt. 48(19), 3722–3730 (2009). [CrossRef] [PubMed]
19. V. V. Kotlyar, A. A. Kovalev, and V. A. Soifer, “Subwavelength focusing with a Mikaelian planar lens,” Opt. Mem. Neural. Networks 19(4), 273–278 (2010). [CrossRef]
20. Y. R. Triandaphilov and V. V. Kotlyar, “Photonic crystal Mikaelian lens,” Opt. Mem. Neural. Networks 17(1), 1–7 (2008).
21. D. V. Nesterenko, “Metal-dielectric Mikaelian’s lens,” Computer Optics 35(1), 47 (2011) (in Russian; not accessible to the authors, but mentioned by a reviewer).
22. H. Ma, S. Qu, Z. Xu, and J. Wang, “General method for designing wave shape transformers,” Opt. Express 16(26), 22072–22082 (2008). [CrossRef] [PubMed]
23. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100(6), 063903 (2008). [CrossRef] [PubMed]
24. The finite element simulation is conducted by using commercial software COMSOL Multiphysics. http://www.comsol.com/. The number of pixels per wavelength is larger than 20 in our simulation.
25. M. Born and E. Wolf, Principles of Optics, 5th edition (Pergamon, 1975).
26. Y. Jin and S. L. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17(03), 349–355 (2008). [CrossRef]
27. J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express 18(1), 244–252 (2010). [CrossRef] [PubMed]
28. Y. A. Kravtsov and Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, 1990).
29. R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(2), 026606 (2007). [CrossRef] [PubMed]
30. F. J. Duarte and L. W. Hillman, Dye Laser Principles (Elsevier, 1990).
31. F. J. Duarte, Tunable Laser Optics (Elsevier, 2003).
32. W. X. Tang, Y. Hao, and F. Medina, “Broadband extraordinary transmission in a single sub-wavelength aperture,” Opt. Express 18(16), 16946–16954 (2010). [CrossRef] [PubMed]
33. J. Li and J. B. Pendry, “Hiding under the carpet: A new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef] [PubMed]
