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 [15]. 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 [69]. Combining the HAM and diffraction gratings or zone plates, the subwavelength focusing has been studied [1012]. 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 εx > 0, εz < 0 and εx < 0, εz > 0 are studied, respectively. εx (εz) denotes the real part of the permittivity of the HAM in the transverse (longitudinal) direction.

Despite of many discussions about HAM [1115], 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

kx2εz+kz2εx=k02
where k0 is the wavenumber in vacuum, kx(kz) is the wave vector in the x(z) direction, and εx(εz) is the complex permittivity of the medium in the x(z) direction: εx,z = εx,z + x,z. The dispersive relation could be hyperbolic if εx · εz < 0. The materials with such combination of εx and εz 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].

Dispersion relation of two types of ideal HAM are plotted in Figs. 1(a) (εx < 0, εz > 0) and 1(d) (εx > 0, εz < 0). Examples of the light focusing in HAM, whose permittivities are εx = −3, εz = 4 (Fig. 1(b)) and εx = 3, εz = −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.

 

Fig. 1 (a) and (d): Dispersion relation of two types of ideal HAM; (b) and (e): focusing behavior; (c) and (f): normalized magnetic field distribution in focus plane.

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In anisotropic medium, the angle between propagation direction of light with kx and optical axis is determined by the ratio of Poynting vector components [10]:

θ(kx)=arctan(εxkxεzkz)

The subwavelength focusing ability of the HAM originates from the subdiffraction of the high-kx modes which are caused by the diffraction at the two edges of the aperture. Combine Eq. (1) and Eq. (2) and let kx → ∞, the propagation direction of these high-kx modes can be obtained [6, 1012]:

θ=arctan(εxεz)
The interference of the high-kx modes from the two edges produces a subwavelength focus [11, 12]. The interference position or focal length can be written as
f=w2|tanθ|=w2εzεx
where 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 kx → ∞. The deviation brought by this approximation will be discussed in Sec. 4.2.

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 εz/εx. After the aperture width 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

εx=εd+ηεm1+η,1εz=11+η(1εd+ηεm)
where
η=dmdd
εm(εd) and dm(dd) are the relative permittivities and thicknesses, respectively, of metal (dielectric) layer.

 

Fig. 2 Schematic of single aperture and HAM which are formed by alternately stacking metal and dielectric layers.

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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 εm, εd can be treated as a constant. When incident wavelength λ is shorter than the plasma wavelength of metal, Re(εm) is positive [16] and HAM can’t be realized. As Re(εm) is negative and monotonically decreases with the wavelength below the plasma frequency, the dispersion relation changes. First, we assume that εx(λ1) = 0 and 1/εz(λ2) = 0, where λ1 and λ2 are the threshold wavelengths. According to Eq. (5), the relationship between λ1 and λ2 can be divided into three situations: 1. If η = 1, εx and εz change signs at the same wavelength, which means λ1 = λ2; 2. If η < 1, then λ1 > λ2; 3. If η > 1, then λ2 > λ1. In the region between λ1 and λ2, εx ·εz > 0 and HAM can not be realized, and we called it blind zone. The analytical results are summarized and listed in Table 1.

Tables Icon

Table 1. Focusing property of HAM in different region

Figure 3(a) plots the effective permittivity of the HAM with different thickness ratios. Silver, whose plasma wavelength is around λp = 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 εx and εz, respectively. The permittivity of the dielectric layer is set as εd = 7, and the thickness ratio is 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 εx ·εz < 0 can be satisfied at almost all wavelengths (except threshold wavelength). The results prove the theoretical analysis about the blind zone.

 

Fig. 3 Real parts of εx and εz, as functions of λ, for a multilayered metamaterial with (a) εm = εAg, εd = 7 and η = 0.5 (green), η = 1 (blue) and η = 2 (red); (b) εm = εAg, η = 1 and εd = 4 (green), εd = 7 (blue). Solid and dashed lines plot εx and εz, respectively.

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Equation (5) also indicates that after εm and η being fixed, the increasing of εd causes red shift of both threshold wavelength λ1 and λ2. Figure 3(b) shows the cases of η = 1 and two different dielectric permittivity with εd = 4 (green lines) and εd = 7 (blue lines). The curves confirm that εd increasing causes red shift of the threshold 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 λ0 is got. On the two sides of λ0, there exists two broad regions in which εx and εz 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.

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 dm = dd = 10 nm is used. The aperture diameter 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 (|Hy|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 λ0 = 456 nm and the effective relative permittivities satisfy εx > 0, εz < 0. As shown in the figure, the focal length increases as the incident wavelength gets larger.

 

Fig. 4 Distribution of the intensity of |Hy|2 for the ideal HAM (a)–(c) and the silver-SiC multilayered structure (d)–(f) with incident wavelength of 350, 370, 390 nm, respectively.

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Figure 5 show the results for the same structure under the condition of εx < 0, εz > 0. The wavelengths are set to be larger than the threshold value λ0, which leads to a result of εx < 0, εz > 0. On the contrary, the focus becomes nearer as the incident wavelength getting larger under the condition of εx < 0, εz > 0.

 

Fig. 5 Distribution of the intensity of |Hy|2 for the ideal HAM (a)–(c) and the silver-SiC multilayered structure (d)–(f) with incident wavelength of 550, 650, 850 nm, respectively.

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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 εx > 0 and εz < 0, the focus shifts away from the aperture as incident wavelength becoming larger; in the region of εx < 0 and εz > 0, the focus get closer to the aperture when the wavelength getting larger.

 

Fig. 6 Relation between the incident wavelength λ and the focal length. The plots express the theoretical focal length (blue solid line), simulation results of lossy (red circle), lossless (black asterisk) ideal medium and silver-SiC structures (green square).

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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. εz, brings about focal shift. The deviation increases with the value of εz increasing.

Second, the theoretical values correspond to the interference position of high-kx modes. Even though the diffraction at the edges is dominant, some low-kx 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 kz is positive from the physical point of view. For the case of εx < 0, εz > 0 (εx > 0, εz < 0), |kx|/kz decreases (increases) when |kx| increases, as indicated in Fig. 1(a) ((b)). The absolute value of θ monotonically decreases (increases) with |kx|. In order to give mathematical analysis, the derivation of Eq. (2) with respect to kx is given

tanθkx=εxεzkz+εx2kx2/kzεzkz2
where
k=εx(k02kx2/εz)
For the case of εx < 0, εz > 0 (εx > 0, εz < 0), |kx| takes the value in the region of ( εzk0, +∞) ((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-kx modes propagation angle is smaller (bigger) than low-kx modes and the focal shift caused by the low-kx modes is positive (negative) in the case of εx < 0, εz > 0 (εx > 0, εz < 0).

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

tanθ=εxεz=1(1+η2)(1+η2+εη)
where ε and η denote εm/εd + εd/εm and dm/dd, respectively. Equation (9) tells that tanθ′ 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].

 

Fig. 7 (a) Relation between εm/εd +εd /εm and the wavelength for silver-SiC structure. (b) Focal length varies with thickness ratio under λ = 365 nm and λ = 673 nm.

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

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).

References and links

1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef]   [PubMed]  

2. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef]   [PubMed]  

3. Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007). [CrossRef]   [PubMed]  

4. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009). [CrossRef]   [PubMed]  

5. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435–441 (2008). [CrossRef]   [PubMed]  

6. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]  

7. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef]   [PubMed]  

8. X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006). [CrossRef]   [PubMed]  

9. C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011). [CrossRef]  

10. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009). [CrossRef]   [PubMed]  

11. G. Ren, Z. Lai, C. Wang, Q. Feng, L. Liu, K. Liu, and X. Luo, “Subwavelength focusing of light in the planar anisotropic metamaterials with zone plates,” Opt. Express 18(17), 18151–18157 (2010). [CrossRef]   [PubMed]  

12. G. Li, J. Li, and K. W. Cheah, “Subwavelength focusing using a hyperbolic medium with a single slit,” Appl. Opt. 50(31), G27–G30 (2011). [CrossRef]   [PubMed]  

13. C. Wang, Y. Zhao, D. Gan, C. Du, and X. Luo, “Subwavelength imaging with anisotropic structure comprising alternately layered metal and dielectric films,”Opt. Express 16(6), 4217–4227 (2008). [CrossRef]   [PubMed]  

14. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]  

15. H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007). [CrossRef]   [PubMed]  

16. E. Palik, ed., “Handbook of optical constants of solids,” (AP, 1985).

17. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

18. Y. Gao, J. Liu, X. Zhang, Y. Wang, Y. Song, S. Liu, and Y. Zhang, “Analysis of focal-shift effect in planar metallic nanoslit lenses,” Opt. Express 20(2), 1320–1329 (2012). [CrossRef]   [PubMed]  

19. J. T. Costa and M. G. Silveirinha, “Achromatic lens based on a nanowire material with anomalous dispersion,” Opt. Express 20(13), 13915–13922 (2012). [CrossRef]   [PubMed]  

References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [CrossRef] [PubMed]
  2. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [CrossRef] [PubMed]
  3. Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
    [CrossRef] [PubMed]
  4. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
    [CrossRef] [PubMed]
  5. X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435–441 (2008).
    [CrossRef] [PubMed]
  6. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
    [CrossRef]
  7. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
    [CrossRef] [PubMed]
  8. X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
    [CrossRef] [PubMed]
  9. C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011).
    [CrossRef]
  10. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
    [CrossRef] [PubMed]
  11. G. Ren, Z. Lai, C. Wang, Q. Feng, L. Liu, K. Liu, and X. Luo, “Subwavelength focusing of light in the planar anisotropic metamaterials with zone plates,” Opt. Express 18(17), 18151–18157 (2010).
    [CrossRef] [PubMed]
  12. G. Li, J. Li, and K. W. Cheah, “Subwavelength focusing using a hyperbolic medium with a single slit,” Appl. Opt. 50(31), G27–G30 (2011).
    [CrossRef] [PubMed]
  13. C. Wang, Y. Zhao, D. Gan, C. Du, and X. Luo, “Subwavelength imaging with anisotropic structure comprising alternately layered metal and dielectric films,”Opt. Express 16(6), 4217–4227 (2008).
    [CrossRef] [PubMed]
  14. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
    [CrossRef]
  15. H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
    [CrossRef] [PubMed]
  16. E. Palik, ed., “Handbook of optical constants of solids,” (AP, 1985).
  17. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [CrossRef]
  18. Y. Gao, J. Liu, X. Zhang, Y. Wang, Y. Song, S. Liu, and Y. Zhang, “Analysis of focal-shift effect in planar metallic nanoslit lenses,” Opt. Express 20(2), 1320–1329 (2012).
    [CrossRef] [PubMed]
  19. J. T. Costa and M. G. Silveirinha, “Achromatic lens based on a nanowire material with anomalous dispersion,” Opt. Express 20(13), 13915–13922 (2012).
    [CrossRef] [PubMed]

2012

2011

G. Li, J. Li, and K. W. Cheah, “Subwavelength focusing using a hyperbolic medium with a single slit,” Appl. Opt. 50(31), G27–G30 (2011).
[CrossRef] [PubMed]

C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011).
[CrossRef]

2010

2009

S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
[CrossRef] [PubMed]

2008

2007

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

2006

X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[CrossRef]

2005

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
[CrossRef] [PubMed]

Chan, C. T.

X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Cheah, K. W.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Costa, J. T.

Du, C.

Durant, S.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Escobar, M. A.

C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011).
[CrossRef]

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
[CrossRef] [PubMed]

Fan, X.

X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Fang, N.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Feng, Q.

Gan, D.

Gao, Y.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

Lai, Z.

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
[CrossRef] [PubMed]

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Lee, J.

X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Li, G.

Li, J.

Liu, J.

Liu, K.

Liu, L.

Liu, S.

Liu, Z.

C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011).
[CrossRef]

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435–441 (2008).
[CrossRef] [PubMed]

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
[CrossRef] [PubMed]

Luo, X.

Ma, C.

C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011).
[CrossRef]

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

Pikus, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

Podolskiy, V. A.

Ren, G.

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

Silveirinha, M. G.

Song, Y.

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
[CrossRef] [PubMed]

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Thongrattanasiri, S.

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[CrossRef]

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
[CrossRef] [PubMed]

Wang, C.

Wang, G. P.

X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Wang, Y.

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[CrossRef]

Xiong, Y.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
[CrossRef] [PubMed]

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
[CrossRef] [PubMed]

Zhang, X.

Y. Gao, J. Liu, X. Zhang, Y. Wang, Y. Song, S. Liu, and Y. Zhang, “Analysis of focal-shift effect in planar metallic nanoslit lenses,” Opt. Express 20(2), 1320–1329 (2012).
[CrossRef] [PubMed]

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435–441 (2008).
[CrossRef] [PubMed]

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15(24), 15886–15891 (2007).
[CrossRef] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Zhang, Y.

Zhao, Y.

Appl. Opt.

Nano Lett.

Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007).
[CrossRef] [PubMed]

Nat. Mater.

X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435–441 (2008).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[CrossRef]

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006).
[CrossRef]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[CrossRef]

C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B 84(19), 195142 (2011).
[CrossRef]

Phys. Rev. Lett.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902(2009).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[CrossRef] [PubMed]

X. Fan, G. P. Wang, J. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Science

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[CrossRef] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[CrossRef] [PubMed]

Other

E. Palik, ed., “Handbook of optical constants of solids,” (AP, 1985).

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Figures (7)

Fig. 1
Fig. 1

(a) and (d): Dispersion relation of two types of ideal HAM; (b) and (e): focusing behavior; (c) and (f): normalized magnetic field distribution in focus plane.

Fig. 2
Fig. 2

Schematic of single aperture and HAM which are formed by alternately stacking metal and dielectric layers.

Fig. 3
Fig. 3

Real parts of εx and εz, as functions of λ, for a multilayered metamaterial with (a) εm = εAg, εd = 7 and η = 0.5 (green), η = 1 (blue) and η = 2 (red); (b) εm = εAg, η = 1 and εd = 4 (green), εd = 7 (blue). Solid and dashed lines plot εx and εz, respectively.

Fig. 4
Fig. 4

Distribution of the intensity of |Hy|2 for the ideal HAM (a)–(c) and the silver-SiC multilayered structure (d)–(f) with incident wavelength of 350, 370, 390 nm, respectively.

Fig. 5
Fig. 5

Distribution of the intensity of |Hy|2 for the ideal HAM (a)–(c) and the silver-SiC multilayered structure (d)–(f) with incident wavelength of 550, 650, 850 nm, respectively.

Fig. 6
Fig. 6

Relation between the incident wavelength λ and the focal length. The plots express the theoretical focal length (blue solid line), simulation results of lossy (red circle), lossless (black asterisk) ideal medium and silver-SiC structures (green square).

Fig. 7
Fig. 7

(a) Relation between εm/εd +εd /εm and the wavelength for silver-SiC structure. (b) Focal length varies with thickness ratio under λ = 365 nm and λ = 673 nm.

Tables (1)

Tables Icon

Table 1 Focusing property of HAM in different region

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

k x 2 ε z + k z 2 ε x = k 0 2
θ ( k x ) = arctan ( ε x k x ε z k z )
θ = arctan ( ε x ε z )
f = w 2 | tan θ | = w 2 ε z ε x
ε x = ε d + η ε m 1 + η , 1 ε z = 1 1 + η ( 1 ε d + η ε m )
η = d m d d
tan θ k x = ε x ε z k z + ε x 2 k x 2 / k z ε z k z 2
k = ε x ( k 0 2 k x 2 / ε z )
tan θ = ε x ε z = 1 ( 1 + η 2 ) ( 1 + η 2 + ε η )

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