## Abstract

The dispersion of a hyperbolic anisotropic metamaterial (HAM) and the chromatic aberration of light focusing in this kind of HAM are studied. The HAM is formed by alternately stacking metal and dielectric layers. The rules of materials and filling factors affecting the optical property of HAM are given. The chromatic aberration of light focusing is demonstrated both theoretically and numerically. By comparing the theory with the simulation results, the factors influencing the focal length, including the heat loss of material and low spatial frequency modes, are discussed. The investigation emphasizes the anomalous properties, such as chromatic aberration and low spatial frequency modes influencing focus position, of HAM compared with that in conventional lens. Based on the analysis, the possibility of using HAM to focus light with two different wavelengths at the same point is studied.

© 2012 OSA

## 1. Introduction

To break diffraction limit in conventional lens, many structures which can recover or enhance the evanescent waves have been demonstrated both theoretically and experimentally [1–5]. Recent studies on metal-dielectric multilayered structures have proposed that such structures can be treated as hyperbolic anisotropic metamaterial (HAM). And using HAM, one could control the behavior of the electromagnetic waves and achieve subwavelength resolution [6–9]. Combining the HAM and diffraction gratings or zone plates, the subwavelength focusing has been studied [10–12]. As shown in Refs. [11] and [12], subwavelength resolution have been achieved when light passes though a single subwavelength slit into the HAM. In these two papers, the cases of *ε*′* _{x}* > 0,

*ε*′

*< 0 and*

_{z}*ε*′

*< 0,*

_{x}*ε*′

*> 0 are studied, respectively.*

_{z}*ε*′

*(*

_{x}*ε*′

*) denotes the real part of the permittivity of the HAM in the transverse (longitudinal) direction.*

_{z}Despite of many discussions about HAM [11–15], it confuses readers that authors give the chosen materials and filling factors directly without any explanation. Consequently, rules of materials and filling factors affecting the optical property of HAM are beneficial. And in the common sense, almost all metals and dielectrics are dispersive. So in practical application, chromatic aberration of the HAM can not be ignored. As far as we known, previous studies were confined to single or discrete wavelength illumination. Investigation about the detailed focusing property over a broad wavelength range is expected. Motivated by these issues, we present in this work the rules of controlling the optical property of HAM and show the anomalous chromatic aberration of light focusing from ultraviolet to infrared.

This paper is organized as follows: First, an overview of light focusing in HAM is given. After studying the dispersion of multilayered structure, regulars about controlling the optical property by choosing materials and tuning filling factors are summarized. Simulations and comparisons between theoretical and simulated results about the focal length are done. Based on the comparisons, the influences to the focus position are analyzed. At last, possibility of using HAM to focus light with two different wavelengths at the same point is analyzed.

## 2. Light focusing in the HAM

In the 2D anisotropic medium [6], the dispersion of TM-polarized wave takes the form of

where*k*

_{0}is the wavenumber in vacuum,

*k*(

_{x}*k*) is the wave vector in the

_{z}*x*(

*z*) direction, and

*ε*(

_{x}*ε*) is the complex permittivity of the medium in the

_{z}*x*(

*z*) direction:

*ε*

_{x,z}=

*ε*′

_{x,z}+

*iε*″

_{x,z}. The dispersive relation could be hyperbolic if

*ε*′

*·*

_{x}*ε*′

*< 0. The materials with such combination of*

_{z}*ε*′

*and*

_{x}*ε*′

*have been demonstrated to have the ability of focusing light into a deep-subwavelength spot when illuminated by a TM-polarized plane wave from a narrow aperture [11, 12].*

_{z}Dispersion relation of two types of ideal HAM are plotted in Figs. 1(a) (*ε*′* _{x}* < 0,

*ε*′

*> 0) and 1(d) (*

_{z}*ε*′

*> 0,*

_{x}*ε*′

*< 0). Examples of the light focusing in HAM, whose permittivities are*

_{z}*ε*= −3,

_{x}*ε*= 4 (Fig. 1(b)) and

_{z}*ε*= 3,

_{x}*ε*= −4 (Fig. 1(e)), are shown. The incident wavelength is 365 nm, and the width of the aperture is set to be 100 nm. It is clear that the diffraction field through the single slit shrinks to a focal point with subwavelength size. The distance between the focal point and the aperture is 57 nm and 58 nm, respectively. The intensity distributions on the focal plane are extracted and shown in Figs. 1(c) and 1(f). The size of the focal spot (full-width at half-maximum) is 20 nm and 26 nm which is deeply lower than the wavelength.

_{z}In anisotropic medium, the angle between propagation direction of light with *k _{x}* and optical axis is determined by the ratio of Poynting vector components [10]:

The subwavelength focusing ability of the HAM originates from the subdiffraction of the high-*k _{x}* modes which are caused by the diffraction at the two edges of the aperture. Combine Eq. (1) and Eq. (2) and let

*k*→ ∞, the propagation direction of these high-

_{x}*k*modes can be obtained [6, 10–12]:

_{x}*k*modes from the two edges produces a subwavelength focus [11, 12]. The interference position or focal length can be written as

_{x}*w*is the diameter of the aperture. The analytic focal length calculated using Eq. (4) for the two cases in Fig. 1 are both 57.7 nm, which agrees well with the simulation results. It is worth noting that Eq. (4) is based on the approximation

*k*→ ∞. The deviation brought by this approximation will be discussed in Sec. 4.2.

_{x}It can be deduced from Eq. (4) that the focal length *f* can be tuned by changing the aperture width *w* or the permittivity ratio *ε*′* _{z}*/

*ε*′

*. After the aperture width*

_{x}*w*is determined and fixed, the aberration only comes from the dispersion of the HAM itself. In this paper, we will focus on an unique realization of hyperbolically dispersive metamaterial: metallic nanolayer, because of their structural simplicity and relatively low loss.

## 3. Dispersion of layered metamaterial

Figure 2 shows the schematic diagram of the structure to be studied. It consists of a half-infinite stack of metal and dielectric layers, covered by an opaque mask with a single aperture. All components in the *y* direction are infinite. According to the effective medium theory [6], the whole area can be treated as an effective anisotropic medium if each layer is homogeneous, isotropic and thin enough (≪ *λ*). The effective permittivity in the *x* and *z* direction can be written as

*ε*(

_{m}*ε*) and

_{d}*d*(

_{m}*d*) are the relative permittivities and thicknesses, respectively, of metal (dielectric) layer.

_{d}Apparently, the dispersive relation of such structure is highly dependent on the metal and dielectric materials and can also be tuned by varying the thickness ratio of the two components.

Compared to *ε _{m}*,

*ε*can be treated as a constant. When incident wavelength

_{d}*λ*is shorter than the plasma wavelength of metal, Re(

*ε*) is positive [16] and HAM can’t be realized. As Re(

_{m}*ε*) is negative and monotonically decreases with the wavelength below the plasma frequency, the dispersion relation changes. First, we assume that

_{m}*ε*′

*(*

_{x}*λ*

_{1}) = 0 and 1/

*ε*′

*(*

_{z}*λ*

_{2}) = 0, where

*λ*

_{1}and

*λ*

_{2}are the threshold wavelengths. According to Eq. (5), the relationship between

*λ*

_{1}and

*λ*

_{2}can be divided into three situations: 1. If

*η*= 1,

*ε*′

*and*

_{x}*ε*′

*change signs at the same wavelength, which means*

_{z}*λ*

_{1}=

*λ*

_{2}; 2. If

*η*< 1, then

*λ*

_{1}>

*λ*

_{2}; 3. If

*η*> 1, then

*λ*

_{2}>

*λ*

_{1}. In the region between

*λ*

_{1}and

*λ*

_{2},

*ε*′

*·*

_{x}*ε*′

*> 0 and HAM can not be realized, and we called it blind zone. The analytical results are summarized and listed in Table 1.*

_{z}Figure 3(a) plots the effective permittivity of the HAM with different thickness ratios. Silver, whose plasma wavelength is around *λ _{p}* = 330 nm, is chosen as the metal layer because of its relative low loss from ultraviolet to visible region. The permittivity data of silver is from Refs. [16, 17]. The solid and dashed lines represent

*ε*′

*and*

_{x}*ε*′

*, respectively. The permittivity of the dielectric layer is set as*

_{z}*ε*= 7, and the thickness ratio is

_{d}*η*= 0.5 (green lines),

*η*= 1 (blue lines) and

*η*= 2 (red lines), respectively. It is clear that, only when

*η*= 1, the condition

*ε*′

*·*

_{x}*ε*′

*< 0 can be satisfied at almost all wavelengths (except threshold wavelength). The results prove the theoretical analysis about the blind zone.*

_{z}Equation (5) also indicates that after *ε _{m}* and

*η*being fixed, the increasing of

*ε*causes red shift of both threshold wavelength

_{d}*λ*

_{1}and

*λ*

_{2}. Figure 3(b) shows the cases of

*η*= 1 and two different dielectric permittivity with

*ε*= 4 (green lines) and

_{d}*ε*= 7 (blue lines). The curves confirm that

_{d}*ε*increasing causes red shift of the threshold

_{d}*λ*.

We conclude this section reminding that we’d better choose *η* = 1 to avoid blind zone and get wider visible zone which can realize HAM. Through appropriate choices of materials, a threshold *λ*_{0} is got. On the two sides of *λ*_{0}, there exists two broad regions in which *ε*′* _{x}* and

*ε*′

*take opposite signs. Calculating the dispersion of Eq. (4) (not shown), we believe this HAM has anomalous chromatic abberation behavior compared to conventional lens [18]. In the following section, simulation results and analysis about the chromatic abberation of light focusing will be given.*

_{z}## 4. Numerical simulation and discussion

#### 4.1. Simulation results

To verify the chromatic aberration properties of the HAM, we give some numerical simulation results obtained using COMSOL MultiPhysics. Silver and SiC layered structure with *d _{m}* =

*d*= 10 nm is used. The aperture diameter

_{d}*w*is set to be 100 nm. A normal plane wave with TM-polarization is illuminated from the left of the mask as shown in Fig. 2.

Figures 4(a)–4(c) give the distributions of the normalized magnetic field intensity (|H_{y}|^{2}) of the ideal HAM whose relative permittivity is calculated using Eq. (5). Figures 4(d)–4(f) show the corresponding results of the real structure which is constructed using Silver-SiC layers. The wavelengths are all smaller than the threshold wavelength *λ*_{0} = 456 nm and the effective relative permittivities satisfy *ε*′* _{x}* > 0,

*ε*′

*< 0. As shown in the figure, the focal length increases as the incident wavelength gets larger.*

_{z}Figure 5 show the results for the same structure under the condition of *ε*′* _{x}* < 0,

*ε*′

*> 0. The wavelengths are set to be larger than the threshold value*

_{z}*λ*

_{0}, which leads to a result of

*ε*′

*< 0,*

_{x}*ε*′

*> 0. On the contrary, the focus becomes nearer as the incident wavelength getting larger under the condition of*

_{z}*ε*′

*< 0,*

_{x}*ε*′

*> 0.*

_{z}The simulation results of the focal length at different wavelengths and the theoretical results obtained using Eq. (4) are all summarized and plotted in Fig. 6. From the figure, we can see that the simulation results agree well with theoretical prediction. In conclusion, the HAM made of metal-dielectric layers possesses such chromatic aberration: in the region of *ε*′* _{x}* > 0 and

*ε*′

*< 0, the focus shifts away from the aperture as incident wavelength becoming larger; in the region of*

_{z}*ε*′

*< 0 and*

_{x}*ε*′

*> 0, the focus get closer to the aperture when the wavelength getting larger.*

_{z}#### 4.2. Analysis about the difference between theory and simulation

As seen from Fig. 6, there exists difference between theory and simulation. Difference between theory (blue lines) and silver-SiC structures (green square) comes from the errors of effective medium theory. However, the results for ideal HAM still have deviation from theory. This can be explained as following.

First, the heat loss in metal will reduce the resolution [11]. In Fig. 6, the difference of the simulated focal length between lossy and lossless material indicates that heat loss influences the position of the focus. By simulation and analysis, we find the imaginary part of the effective permittivity in the propagating direction, i.e. *ε*″* _{z}*, brings about focal shift. The deviation increases with the value of

*ε*″

*increasing.*

_{z}Second, the theoretical values correspond to the interference position of high-*k _{x}* modes. Even though the diffraction at the edges is dominant, some low-

*k*modes still propagate in the HAM and cause the difference between ideal lossless HAM (black asterisk) and the theory. The propagation direction of these modes abides by Eq. (2). Since we only care about the light propagating forward, the sign of

_{x}*k*is positive from the physical point of view. For the case of

_{z}*ε*′

*< 0,*

_{x}*ε*′

*> 0 (*

_{z}*ε*′

*> 0,*

_{x}*ε*′

*< 0), |*

_{z}*k*|/

_{x}*k*decreases (increases) when |

_{z}*k*| increases, as indicated in Fig. 1(a) ((b)). The absolute value of

_{x}*θ*monotonically decreases (increases) with |

*k*|. In order to give mathematical analysis, the derivation of Eq. (2) with respect to

_{x}*k*is given

_{x}*ε*′

*< 0,*

_{x}*ε*′

*> 0 (*

_{z}*ε*′

*> 0,*

_{x}*ε*′

*< 0), |*

_{z}*k*| takes the value in the region of ( $\sqrt{{\epsilon}_{z}^{\prime}}{k}_{0}$, +∞) ((0, + ∞)), as indicated in Fig. 1(a) and 1(d). Through simple algebraic analysis, it can be deduced that the sign of the derivation is always positive (negative). Based on the graphical and mathematical analysis, we can draw the conclusion: the absolute value of high-

_{x}*k*modes propagation angle is smaller (bigger) than low-

_{x}*k*modes and the focal shift caused by the low-

_{x}*k*modes is positive (negative) in the case of

_{x}*ε*′

*< 0,*

_{x}*ε*′

*> 0 (*

_{z}*ε*′

*> 0,*

_{x}*ε*′

*< 0).*

_{z}#### 4.3. Focusing light with two different wavelengths at the same point

Focusing light with different wavelength into the same point is difficult in conventional lens. Figure 6 shows that this kind metamaterial has potential application in focusing light with two different wavelengths at the same position. To study this possibility, we insert the Eq. (5) into Eq. (3) and get

*ε*and

*η*denote

*ε*/

_{m}*ε*+

_{d}*ε*/

_{d}*ε*and

_{m}*d*/

_{m}*d*, respectively. Equation (9) tells that tan

_{d}*θ*′ and the focal length only depend on the value of

*ε*for a certain structure. Figure 7(a) plots the dispersion of

*ε*for silver-SiC structure and shows the phenomenon of different

*λ*with the same

*ε*value, for example

*λ*= 365 nm and

*λ*= 673 nm. Figure 7(b) plots focal length as function of thickness ratio

*η*under the chosen

*λ*and indicates that light with

*λ*= 365 nm and

*λ*= 673 nm focus at the same point in an arbitrary silver-SiC multilayered structure. To focus light with more than two wavelengths at the same spot, materials with opposite dispersions may be used to correct the chromatic abberation [19].

## 5. Conclusion

In conclusion, we summarize rules of controlling the optical property of HAM (metal-dielectric multilayered structures) and present some anomalous properties of the structure which consists of a subwavelength aperture and HAM from ultraviolet to infrared. It is found that in the cases of *ε*′* _{x}* < 0,

*ε*′

*> 0 and*

_{z}*ε*′

*> 0,*

_{x}*ε*′

*< 0, the chromatic aberration are opposite. Two factors influencing the focus position are investigated: the heat loss of material and the low-*

_{z}*k*modes. At last, we show the potential application of this structure focusing light with two different wavelengths at the same spot. Because these multilayered structures are essentially one dimensional, they can be fabricated layer by layer using deposition technology. We hope the results shown in this paper could offer a better understanding of the effective metal-dielectric medium which has been widely studied for many applications such as super-resolution imaging, nanolithography and so on.

_{x}## Acknowledgments

This work was supported by the National Basic Research Program of China (Grant No. 2011CB301801), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2010003) and the Heilongjiang Postdoctoral (Grant No. LBH-Z10142).

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