Metamaterials with gradient negative index of ... - OSA Publishing

A. O. Pinchuk and G. C. Schatz

Vol. 24, No. 10 / October 2007 / J. Opt. Soc. Am. A

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Metamaterials with gradient negative index of refraction Anatoliy O. Pinchuk* and George C. Schatz Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois, 60208-3113, USA *Corresponding author: [email protected] Received January 19, 2007; revised July 2, 2007; accepted July 5, 2007; posted July 6, 2007 (Doc. ID 78856); published September 6, 2007 We propose a new metamaterial with a gradient negative index of refraction, which can focus a collimated beam of light coming from a distant object. A slab of the negative refractive index metamaterial has a focal length that can be tuned by changing the gradient of the negative refractive index. A thin metal film pierced with holes of appropriate size or spacing between them can be used as a metamaterial with the gradient negative index of refraction. We use finite-difference time-domain calculations to show the focusing of a plane electromagnetic wave passing through a system of equidistantly spaced holes in a metal slab with decreasing diameters toward the edges of the slab. © 2007 Optical Society of America OCIS codes: 260.2110, 240.6680, 240.0310.

1. INTRODUCTION Recently there has been a growing interest in so-called left-handed metamaterials—artificial composite materials with simultaneous negative permittivity ⑀ ⬍ 0 and permeability ␮ ⬍ 0. With these two prerequisites, the refractive index of the material is negative, and many phenomena traditional in geometrical optics, e.g., refraction of light at the interface between positive and negative index materials, work in a reverse manner. In 1968, left-handed metamaterials were first suggested by Veselago [1], who revealed their unusual physical properties: reverse Doppler effect, reverse refraction of light at the interface between right-handed and left-handed metamaterials, reverse Cherenkov effects [1,2], etc. The concept of a “perfect lens” was introduced more than 30 years later by Pendry [3], who showed that a point-like source of light on one side of a slab of negative index material can be focused by the slab to a point on the opposite side. Pendry also noted that a silver slab (roughly 40 nm thick) can produce a “superlens” effect provided that the point source is within the near-field region of the slab. The silver-slab result has recently been experimentally verified [4,5] in the optical frequency range. A scanning nearfield optical microscope was used as the optical source, and fluorescent molecules embedded in a polymer matrix were used as the detector system. Although subdiffraction focusing is of great fundamental and technological importance, the perfect lens cannot be used as a normal optical element because it does not possess a focal length [6,7]. A parallel beam of light coming from distant objects remains parallel after passing through such a slab, as shown in Fig. 1(a), and only a point-like source of light, e.g., a molecular dipole, can be focused on the opposite side of the slab if placed the appropriate distance from the surface of the film in the nearfield zone; see Fig. 1(a). Optical materials with a gradient index of refraction (GRIN) have been known since 1850 [8]. A parallel slab of 1084-7529/07/100A39-6/$15.00

a transparent material with a gradient of the refractive index can focus a parallel beam of light, like the usual focusing of a convex lens. Focusing of the parallel beam is achieved by a phase correction that arises from the different optical lengths of the rays as they traverse parts of the slab with different refractive indices (usually, for a focusing slab, the refractive index increases toward the edges of the slab). Gradient negative index materials have been recently proposed and experimentally tested in the microwave frequency region [9,10]. The isotropic eikonal equation was used to design the profile of a gradient negative index metamaterial [11], and the focusing of an electromagnetic wave 共17.4 GHz兲 was demonstrated by a gradient negative index metamaterial [12]. In this paper we propose to extend the idea of gradient index metamaterials to the optical frequency region and to use a structured metal film to design a slab having a gradient negative index of refraction. Such a slab should focus a parallel beam of light coming from a distant object, in contrast to the usual slab with a macroscopically homogeneous negative index of refraction, as shown in Fig. 1(b). Focusing a paraxial beam of light would be of interest for the development of optical elements based on negative refraction metamaterials, ultrasensitive molecular sensors, etc. [13].

2. GRADIENT NEGATIVE INDEX OF REFRACTION In this section we derive a formula for the gradient negative index of refraction (NGRIN) following Fermat’s principle for light propagation in inhomogeneous media. The idea is to design a lens with a gradient negative index of refraction that can focus a plane electromagnetic wave. Suppose we have a slab of a metamaterial with a negative index of refraction n共r兲, which has a gradient of the refractive index; see Fig. 1(b). We implicitly assume that the index of refraction can be negative, n共r兲 ⬍ 0, and explicitly © 2007 Optical Society of America

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2(a) and 2(b)], Fermat’s principle determines the behavior of the gradient of the index of refraction [Figs. 2(c)–2(e)]. The modulus of the gradients of both positive and negative index of refraction [Eq. (3)] increases toward the edges of the slabs, but the difference between the two is in the focusing property: A slab with the gradient of the positive index should act as a concave lens; see Fig. 2(d), while a slab with the gradient given by Eq. (3) should act as a convex lens; see Fig. 2(e). Notice that the usual positive GRIN lens has n共r兲 = N − r2/共2df兲;

共4兲

that is, the index of refraction decreases toward the edges of the lens, and this slab works as a convex lens; see Fig. 2(c). Consider a parallel slab of gradient negative index of refraction metamaterial having a width d = 0.1 (all numbers are dimensionless) and an aperture rmax = ± 1. Figure 3 depicts a profile of the gradient negative index of refraction for three different focal distances f calculated according to Eq. (2). For the closest focal distance f = 0.2, the slab has the highest gradient of the index of refraction n共r兲. The dashed curve depicts the gradient of the positive inFig. 1. (a) Macroscopically homogeneous left-handed metamaterial can focus a light source located near a plane slab (1) and cannot focus a light coming from distant objects (2). (b) Slab of a metamaterial with gradient negative index of refraction (NGRIN in figure) which can focus a parallel beam of light.

denote the index of refraction at the axis of the lens to be negative, −N. Let us consider two optical paths ray 1 and ray 2 in Fig. 1(b). The optical length of ray 1 is n共r兲d + l and of ray 2 is −Nd + f, where the length l = 冑r2 + f2. The function of a lens is phase compensation; thus these two paths must be equal: n共r兲d + l = − Nd + f.

共1兲

Solving this equation for the negative index of refraction n共r兲, we obtain 1 n共r兲 = − N −

d

共冑r2 + f2 − f兲.

共2兲

If the focal distance is much larger than the lens aperture f r, then, taking into account that 冑共1 + r2 / f2兲 ⬇ 1 + r2 / 2f2, we obtain r2 n共r兲 = − N −

共2df兲

.

Fig. 2. (a) Slab of a homogeneous positive refractive material, (b) slab of homogeneous negative refractive material, (c) slab with a gradient of a positive refractive material [given by Eq. (4)], (d) slab with an inverse gradient of positive refractive material, (e) slab with a gradient of negative refractive index material [given by Eq. (3)].

共3兲

Here we explicitly keep the negative sign to indicate that we consider negative index materials. In the next section we derive a formula for the profile of a structured metal thin film by using the relation between the index of refraction and the dielectric permittivity n2 = ⑀. This might lead to confusion between the negative index of refraction given by Eq. (3) and the gradient of the positive index of refraction. To clarify when we should understand the gradient as referring to positive or negative index of refraction, we refer to the Fermat’s principle [2]. While Snell’s law gives an idea of how the slab works for a given positive or negative homogeneous index of refraction [Figs.

Fig. 3. (Color online) Solid curves, gradient negative index of refraction (NGRIN in figure) as a function of the distance from the optical axis of the lens r for three different focal lengths f. Dotted curve, gradient of the usual GRIN lens with the index of refraction given by Eq. (4); dashed curve, gradient of a positive index of refraction.



dex of refraction and is characterized by concave focusing properties. The dotted curve corresponds to the usual convex GRIN lens, with index of refraction given by Eq. (4). Left-handed metamaterials can be manufactured by combining interlocking parallel thin metal wires and split ring resonators, which simultaneously gives rise to a negative effective permittivity and a negative magnetic permeability [14]. Another possibility is to use a photonic crystal. It has been demonstrated experimentally [15] as well as theoretically [16] that a photonic crystal slab can behave like a slab of negative refractive index metamaterial. A slab of parallel aluminum [15] or silver [16] rods having particular dimensions and spatial arrangement can focus a point light source located in the vicinity of the slab. Experiments were carried out in the radio frequency range, but theory predicts the same effect in the optical frequency range. Asymmetric layered structures with a gradient negative index metamaterial placed between two positive index materials with different index of refraction may have near-perfect lens properties [17], although index and impedance matching problems may lead to some distortions [13]. The effective permittivity and permeability of a metamaterial can be written as [18]

⑀eff共␻兲 = 1 −

␮eff共␻兲 = 1 −

␻p2 ␻2

共5兲

,

profile of the index of refraction such as the gradient index of refraction, Eq. (3). The real effective dielectric permittivity, Eq. (7), of the structured film is an approximation to a complex one: The film dissipates electromagnetic waves and should be characterized by the complex dielectric permittivity ⑀ = ⑀⬘ + i⑀⬙, which is related to the complex index of refraction nc = n + ik by ⑀⬘ = n2 − k2 and ⑀⬙ = 2nk. While it is a complicated problem to explicitly derive a formula for the profile of a film characterized by the complex dielectric permittivity, we solved the problem numerically by considering the system of nonlinear equations relating the complex dielectric permittivity and the complex index of refraction [20]: n2 =

k2 =

␻2 − ␻02 + i␻⌫

共6兲

,

where 0 ⬍ ␩ ⬍ 1 is the filling factor associated with elements of a composite material. Varying the distance between the elements constituting the composite metamaterial (e.g., thin metal wires and split ring resonators), we can change gradually the filling factor of the inclusions ␩ and thus the effective permeability ␮eff. The effective index of refraction is related to the effective permeability 2 = ⑀eff␮eff共␩兲; hence, choosing the gradient of the effecneff tive permeability we can design the profile using Eq. (2), for a given focal length f.

冑⑀⬘2 + ⑀⬙2 + ⑀⬘ 2

冑⑀⬘2 + ⑀⬙2 − ⑀⬘ 2

,

共8兲

.

共9兲

To introduce the complex effective dielectric permittivity for a patterned metal film, we modified the model proposed by Pendry [3] by introducing a damping constant ␭d into the equation for the effective dielectric permittivity of the film:

⑀eff共␭兲 =

␩␻2

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␲ 2b 2⑀ h 8a2

冢

冉冊冣 ␭2

1−

4a ⑀h 1 + i 2

␭

.

共10兲

␭d

The parameter ␭d is analogous to the scattering rate ␥ in the Drude-like formula for the dielectric permittivity of free-electron metals ␭d = l⬁c / vF, where vF is the Fermi velocity of the conduction electrons, and l⬁ is the mean free path of the conduction electrons. Solving Eqs. (8) and (10) numerically for the diameter of the equidistantly spaced holes in a thin metal film, we obtain a profile for the patterned metal film shown in Fig. 4. Assuming that the distance between the centers of the holes is b = 0.1⬎ amax, we have chosen ␭ = 0.2 and the damping parameter ␭d = 0.05

3. STRUCTURED METAL FILM WITH GRADIENT NEGATIVE INDEX OF REFRACTION One of the possible structures that can be used for designing metamaterials with gradient negative index of refraction is a metal film pierced with square or circular holes. Such a film can mimic delocalized surface plasmon polariton waves and has an effective dielectric permittivity similar to that of a Drude metal [17,19]:

⑀eff共␭兲 =

␲ 2b 2⑀ h 8a

2

冉

1−

␭2 4a2⑀h

冊

,

共7兲

where a is the diameter of the holes, b is the distance between the centers of the holes, and ⑀h is the dielectric permittivity of a medium occupying the space inside the holes. The dependence of the effective dielectric permittivity on the diameter of the holes and on the distance between them, Eq. (7), makes it possible to design a specific

Fig. 4. (Color online) Profile of the diameters of the holes a as a function of the distance from the optical axis r for three given focal lengths f calculated with the complex effective dielectric permittivity of the film [Eqs. (8) and (10)]. Refractive index at the optical axis is N = 1.12, dielectric constant ⑀h = 1, width of the slab is d = 0.4.

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lens. Figure 5(d) depicts another possibility, involving a photonic crystal structure with varying distance between the rods.

4. FOCUSING OF A PLANE WAVE BY A STRUCTURED METAL FILM

Fig. 5. (Color online) Sketch of a thin metal film pierced by circular holes with decreasing radii. (a) Side view; (b) top view, where b is the distance between the centers of the holes and a is the diameter of the holes; (c) circular disc made of a thin metal film pierced with the holes having different diameters; this structure should focus a spatial 3D beam of light; (d) photonic crystal structure with varying distance between the rods.

for the design of a structured film with the complex effective dielectric permittivity. The wavelength of the incident light should be larger than either the largest diameter of the holes in the film or the distance between them, ␭ b ⬎ amax. However, the wavelength should not be too large, because the electromagnetic waves passing through such a system decay exponentially. Therefore, the choice of the wavelength is in fact a trade-off between the requirement of the electrostatic approximation ␭ b and the damping of the electromagnetic wave. Given the profile of the structured film, we propose three simple possible configurations for the design of a gradient negative index of refraction lens, as shown in Fig. 5. A slab of metal, pierced by an array of holes with decreasing radii, is shown in Fig. 5(a) (side view) and in Fig. 5(b) (top view). A circular disk made of a thin metal film, pierced with the holes as shown in Fig. 5(c), should focus a beam of light in three dimensions, like a normal

In this section we outline numerical finite-difference time-domain (FDTD) calculations with appropriate periodic boundary conditions, which we use to prove the focusing of a plane electromagnetic wave by a structured metal film designed to have a gradient effective dielectric permittivity based on Eq. (10). Here we use a thin metal film pierced with equidistantly spaced circular holes as schematically depicted in Figs. 5(a) and 5(b). The diameters of the holes, calculated using Eqs. (8) and (10), are given in Table 1 and depicted in Fig. 4. The distribution of the intensity of the electric field in the xz plane simulated using a two-dimensional FDTD method is shown in Fig. 6 for two different focal distances f and three wavelengths of the incident light. All the parameters of the structured metal film a , b , d, and the wavelength of the incident light ␭ were normalized by the same coefficient in order to make our simulations applicable to an arbitrary region of the electromagnetic spectrum. Thus, in order to get the absolute dimensions of the parameters of the film and the wavelength of the light beam, one should multiply the system parameters by a coefficient with dimensions of length, say, 10 mkm. The bulk refractive index of the metal used for the structured film is n0 = −50+ i300 [19]. The electromagnetic wave was assumed to be linearly polarized in the y direction. The field inside the holes is confined by conductive boundaries. Because the size of the holes is smaller than half the wavelength of the incident wave, the electromagnetic wave decays exponentially while passing through the system. Figure 6(a) depicts the focusing of a ␭ = 0.2 plane electromagnetic wave coming from the left side of the picture and passing through the structured film. The enhanced intensity of the electric field, manifested as a spot (red online) on the optical axis of the lens, corresponds to the focal point of the lens, thus confirming the focusing properties of the structured film. If we tune the wavelength of the incident light close to ␭ = 0.25 and ␭ = 0.3, we observe a shift of the focal point along the focal axis close to the center of the film; see Figs. 6(b) and 6(c). Next we carried out numerical simulations for a structured film designed to have a focal length f = 0.5. Figures 6(d)–6(f) depict the intensity distribution of the electric field of the plane wave passing through the system. The

Table 1. Diameters a of Equidistantly Spaced Holes Pierced in a Thin Metal Film for Three Focal Lengths f a ␦r f = 0.2 f = 0.5 f=1 a

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.096 0.096 0.096

0.09 0.093 0.094

0.08 0.087 0.091

0.07 0.079 0.086

0.063 0.072 0.08

0.057 0.066 0.075

0.052 0.06 0.069

0.048 0.055 0.064

0.045 0.052 0.06

0.043 0.048 0.056

0.04 0.046 0.053

The values of the diameters correspond to that of Fig. 4.



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to new prospective for the development of lenses with negative index of refraction. Our goal was to demonstrate the possibility of focusing a collimated beam of light by a structured metal film. We did not attempt to optimize the structure of the film to achieve “perfect focusing.” Further theoretical investigations aimed at relating the damping constant ␭d to the parameters of the film and full 3D FDTD simulations should improve the quality of the focusing. One possible realization of the “superlens” is based on SiC thin semiconductor film [21]. A gradient negative index of refraction could be achieved by inhomogeneous doping of the semiconductor, thereby producing an inhomogeneous GRIN free from the effective averaging problem.

ACKNOWLEDGMENT We gratefully acknowledge support from Defence Advance Research Agency.

REFERENCES 1. 2. Fig. 6. (Color online) Focusing of a plane electromagnetic wave by a structured metal film: 2D FDTD simulations of a paraxial light beam coming from a distant object (from the left side of the image) and passing through the structured metal film. The electric field is polarized in the y direction; the aperture of the lens is rmax = 2; the refractive index of the film is n0 = −50+ 300i; the thickness of the film is d = 0.1; the effective refractive index in the middle of the film, i.e., on the optical axis, is N = −1.12. (a)–(c) The focal length is f = 0.2, and the normalized wavelength is (a) ␭ = 0.2, (b) ␭ = 0.25, (c) ␭ = 0.3. (d)–(f) The focal length is f = 0.5, and the normalized wavelength is (d) ␭ = 0.2, (e) ␭ = 0.25, (f) ␭ = 0.3.

3. 4. 5.

6. 7.

focal point for three different wavelengths in these simulations shifted from the results in Figs. 6(a)–6(c) as expected, and the focal spot turns out to be more smeared in this case. Also, the focus is located slightly closer to the surface of the film, i.e., at f ⬇ 0.4, which is in disagreement with that used to design the film 共f = 0.5兲; this is probably due to approximations made in the design of the film. Thus, these simulations show that the structured metal film focuses a plane electromagnetic wave coming from a distant object. The focal point shifts as expected for different configurations of the holes pierced in the film. The convex-like behavior of the lens confirms that the structured film mimics a gradient of the negative index of refraction.

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11.

12. 13.

5. CONCLUSIONS In conclusion, we have shown that a metamaterial with a gradient of the effective negative index of refraction can focus a plane electromagnetic wave coming from distant objects. A thin metal film pierced with equidistantly spaced holes with decreasing diameters toward the edges of the film was proposed as a metamaterial that shows gradient index of refraction focusing. FDTD calculations confirm the focusing properties of this system, giving rise

14.

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