## Chromatic aberration of light focusing in hyperbolic anisotropic metamaterial made of metallic slit array |

Optics Express, Vol. 20, Issue 27, pp. 28586-28593 (2012)

http://dx.doi.org/10.1364/OE.20.028586

Acrobat PDF (1135 KB)

### 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

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

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]

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]

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]

*ε*′

*> 0,*

_{x}*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}## 2. Light focusing in the HAM

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]

*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*

_{z}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]

*ε*′

*< 0,*

_{x}*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}*k*and optical axis is determined by the ratio of Poynting vector components [10

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

*k*modes which are caused by the diffraction at the two edges of the aperture. Combine Eq. (1) and Eq. (2) and let

_{x}*k*→ ∞, the propagation direction of these high-

_{x}*k*modes can be obtained [6

_{x}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]

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

**50**(31), G27–G30 (2011). [CrossRef] [PubMed]

*k*modes from the two edges produces a subwavelength focus [11

_{x}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]

**50**(31), G27–G30 (2011). [CrossRef] [PubMed]

*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}*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

*y*direction are infinite. According to the effective medium theory [6

**74**(11), 115116 (2006). [CrossRef]

*λ*). The effective permittivity in the

*x*and

*z*direction can be written as where

*ε*(

_{m}*ε*) and

_{d}*d*(

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

_{d}*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}*λ*= 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

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

*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}*ε*and

_{m}*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*.

*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*′

*and*

_{x}*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*

_{z}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]

## 4. Numerical simulation and discussion

### 4.1. Simulation results

*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.

_{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*′

*> 0,*

_{x}*ε*′

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

_{z}*ε*′

*< 0,*

_{x}*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}*ε*′

*> 0 and*

_{x}*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

**18**(17), 18151–18157 (2010). [CrossRef] [PubMed]

*ε*″

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

_{z}*ε*″

*increasing.*

_{z}*k*modes. Even though the diffraction at the edges is dominant, some low-

_{x}*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 where For the case of

_{x}*ε*′

*< 0,*

_{x}*ε*′

*> 0 (*

_{z}*ε*′

*> 0,*

_{x}*ε*′

*< 0), |*

_{z}*k*| takes the value in the region of (

_{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

*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

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]

## 5. Conclusion

*ε*′

*< 0,*

_{x}*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

## References and links

1. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

2. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

3. | Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. |

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. |

5. | X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. |

6. | B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B |

7. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

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. |

9. | C. Ma, M. A. Escobar, and Z. Liu, “Extraordinary light focusing and Fourier transform properties of gradient-index metalenses,” Phys. Rev. B |

10. | S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. |

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 |

12. | G. Li, J. Li, and K. W. Cheah, “Subwavelength focusing using a hyperbolic medium with a single slit,” Appl. Opt. |

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 |

14. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B |

15. | H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express |

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 |

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 |

19. | J. T. Costa and M. G. Silveirinha, “Achromatic lens based on a nanowire material with anomalous dispersion,” Opt. Express |

**OCIS Codes**

(160.4760) Materials : Optical properties

(240.6680) Optics at surfaces : Surface plasmons

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: November 5, 2012

Revised Manuscript: December 1, 2012

Manuscript Accepted: December 3, 2012

Published: December 17, 2012

**Citation**

Kai Guo, Jianlong Liu, Yan Zhang, and Shutian Liu, "Chromatic aberration of light focusing in hyperbolic anisotropic metamaterial made of metallic slit array," Opt. Express **20**, 28586-28593 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28586

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### References

- J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [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]
- 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]
- 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]
- X. Zhang and Z. Liu, “Superlenses to overcome the diffraction limit,” Nat. Mater. 7, 435–441 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009). [CrossRef] [PubMed]
- 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]
- 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. 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]
- A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]
- 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]
- E. Palik, ed., “Handbook of optical constants of solids,” (AP, 1985).
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- 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]
- 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]

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