## Dielectric metamaterial magnifier creating a virtual color image with far-field subwavelength information

Optics Express, Vol. 18, Issue 11, pp. 11216-11222 (2010)

http://dx.doi.org/10.1364/OE.18.011216

Acrobat PDF (1155 KB)

### Abstract

We propose an approach for far-field optical subwavelength imaging by using a dielectric metamaterial magnifier with gradient refractive index. Different from previous superlens and hyperlens that form a real image with subwavelength features within narrowband, this magnifier creates a virtual color image with sub-100 nm resolution over broadband that can be captured directly by a conventional microscope in the far field. Because the magnifier is made of isotropic dielectric materials, the fabrication will be greatly simplified with existing metamaterial technologies.

© 2010 Optical Society of America

## 1. Introduction

2. S. W. Hell, “Far-Field Optical Nanoscopy,” Science **316**, 1153–1158 (2008). [CrossRef]

3. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

6. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

4. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science **308**, 534–537 (2005). [CrossRef] [PubMed]

8. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects,” Science **315**, 1686 (2007). [CrossRef] [PubMed]

9. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying Superlens in the Visible Frequency Range,” Science **315**, 1699–1701 (2007). [CrossRef] [PubMed]

10. A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. **33**, 43–45 (2008). [CrossRef]

11. D. P. Gaillot, C. Croenne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal-dielectric planar hyperlens,” New J. Phys. **10**, 115039 (2008). [CrossRef]

## 2. Design of magnifier with gradient refractive index

*r*,

*ϕ*), where the

*E*field is along the out-of-plane direction of

*z*, and the distribution of refractive index satisfies

*a*and

*b*are the inner and outer radii, respectively. We can think of this geometry as an immersion lens with a tailored GRIN shell. We now proceed to justify this selection of GRIN shell using transformation optics [12–14

12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

*r*

^{′},

*ϕ*

^{′}= 0) and (

*r*

^{′},

*ϕ*

^{′}=

*π*), respectively. For each point source at

*r*

^{′}and

*ϕ*

^{′}, its radiation is typically

*E*=

_{z}*H*

^{(1)}

_{0}(

*k*∣

*r̅*−

*r*̅

^{′}∣), where

*H*

^{(1)}

_{0}is the zeroth order Hankel function of the first kind. From the addition theorem, we know

*J*is the

_{m}*m*th order Bessel function,

*r*

_{<}= min{

*r*,

*r*

^{′}} and

*r*

_{>}= max{

*r*,

*r*

^{′}}. We can see that the radiation is decomposed into cylindrical waves with different angular variations.

*r*,

*ϕ*,

*z*) to (

*,*

*x̃**,*

*ỹ**), where*

*z̃**= ln*

*x̃**r*,

*=*

*ỹ**ϕ*and

*=*

*z̃**z*. According to the formal invariance of Maxwell’s equations, in the transformed space of (

*,*

*x̃**,*

*ỹ**z̃*),

*ε*̿/

*ε*

_{0}=

*μ*̿/

*μ*

_{0}=

^{2x̃}), where

*J*̿ =

12. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

*μ*

_{x̃},

*μ*

_{ỹ}and

*ε*

_{z̃}enter into Maxwell’s equations, and thus the refractive index in the transformed space can be written as

*n*(

*x̃*) =

*e*

^{x̃}, i.e. a one-dimensional (1D) inhomogeneous medium with refractive index varying along

*x̃*. The original

*m*th order cylindrical wave becomes

*E*

_{z̃}=

*H*

^{(1)}

_{m}(

*k*

_{0}

*e*

^{x̃})

*e*

^{imỹ}in this transformed space. So now the propagation of a cylindrical wave away from the origin in the original space is transformed to propagation of a plane wave from left to right in a 1D inhomogeneous medium with refractive index

*n*=

*e*

^{x̃}, as shown in Fig. 1(b). The spatial frequency

*k*

_{ỹ}of this plane wave equals

*m*, the angular momentum of the original cylindrical wave. Similar 1D exponential profiles that lead to solutions with expressions of Hankel functions were discussed previously in [15].

*m*from left to right, because of the low starting value of refractive index, it is evanescent. After propagating for some distance, the refractive index is sufficiently increased, and converts the evanescent wave to a propagating one. The reflected wave, although very small, is still important in explaining the physical picture. For example, the explanation of the energy tunneling of evanescent waves to propagating waves in the original cylindrical space will be difficult, if one approximates the cylindrical wave as a single plane wave locally [6

6. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

*a*and ln

*b*, respectively (

*a*<

*b*), the refractive index profile for the left half part now is

*n*(

*x̃*) =

*e*

^{x̃+(lnb-lna)}=

*e*

^{x̃}. Assuming CP=BQ, a propagating plane wave with

*k*

_{ỹ}=

*m*excited at point B in Fig. 1(c) looks to an observer in the far field similar to a plane wave excited at point C in Fig. 1(b), except there is a phase shift due to traveling through the additional layer of QP. In the paraxial approximation, all propagating waves in the newly extended layer of QP are almost parallel. Hence, they experience nearly the same phase shift after traveling though QP. In other words, a virtual image will appear at point C of the true source at B for the observer in the far field. Accordingly, in the physical space of cylindrical coordinates, the spacing between the two point sources is magnified. The magnification can also be clearly seen in Fig. 1(d), where the new profile of refractive index is plotted as the red dotted line. For the two point sources at B in the extended-layer medium at coordinate of

*e*

^{x̃}=

*r*

^{′}, the virtual images will appear as if they are located at point C with coordinate

*e*

^{x̃}=

*r*

^{′}in the unextended medium.

*x̃*,

*ỹ*,

*z̃*) back to the physical space (

*r*,

*ϕ*,

*z*). The extended and shifted stratified medium then becomes a concentric cylindrical structure with inner radius

*a*and outer radius

*b*as shown in Fig. 1(a) and described by Eq. (1). The magnification that this magnifier can provide is

*b*/

*a*in the core converts some originally evanescent waves into propagating waves. The concentric shell with the GRIN profile

*b*/

*r*is able to deliver all propagating waves to the far field because this concentric shell is transformed from a homogenous slab which guarantees that a propagating wave on one side will still be propagating on the other side after it traverses this slab. The GRIN profile inside the shell can be achieved by constructing a multi-layer structure of several different materials or by drilling air holes in a high refractive index material according to some location-dependent ratio that can be calculated in a straightforward fashion [16

16. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Mater. **8**, 568–571 (2009). [CrossRef]

## 3. Numerical demonstration of subwavelength resolution

*b*= 4

*a*= 2

*μ*m and the free space wavelength

*λ*

_{0}= 500nm, so the refractive index in the core

*r*<

*a*is 4, close to the refractive index of silicon, and the magnification is 4. In Fig. 2(a), two coherent and in-phase point sources are located along the

*x*axis and separated by

*λ*

_{0}/4. We can see that they are difficult to distinguish. In Fig. 2(b), these two point sources are now placed inside the magnifier, and they generate a recognizable pattern in the far field. Fig. 2(c) shows the

*E*field with these two sources in free space but with the separation increased to 2

*λ*

_{0}. The far field in Fig. 2(c) is almost the same as that in Fig. 2(b), except there is an evident phase shift which is caused by the insertion of a finite layer in Fig. 1(c). It also confirms the magnification of 4 of this magnifier. In practice, a complete circular structure is difficult to use. Here we keep a half of the magnifier and study its performance by finite-element-analysis (FEA) simulation as in Fig. 2(d). The two sources now are attached to the bottom interface at

*y*= 0 with separation of

*λ*

_{0}/4. The radiation pattern has been largely preserved in the upper half space.

*y*axis as the optical axis and suppose a well-corrected microscope with numerical aperture of 0.95 (receiving angle of 144° subtended by the origin in the object plane). To simulate the imaging function of the microscope, we set a finite plane above the magnifier where the

*E*field is recorded with 144° receiving angle, collect the propagating plane wave components with

*k*smaller than

_{x}*k*

_{0}, and finally propagate back these same plane wave components to reconstruct an image incoherently, i.e. superimposing the point image intensities. To emulate the GRIN profile dictated by Eq. (1) in the realistic case of silicon operating at broadband, we set up the following procedure: we assume a distribution of air-filled holes of negligible diameter compared to the shortest wavelength and of gradient density

*f*according to Eq. (1) with the ideal values

*b*= 4

*a*. We then scale the effective complex refractive index in the region

*a*<

*r*<

*b*according to

*n*

^{2}

_{eff}= (1 −

*f*)

*n*

^{2}

_{si}+

*fn*

^{2}

_{air}using

*n*

_{si}(506nm) = 4.26 + 0.0439i,

*n*

_{si}(605nm) = 3.93+0.0191i, and

*n*

_{si}(709nm) = 3.76+0.00992i for the green, orange and red colors, respectively [17]. Figure 3 provides the intensities of the reconstructed images of two incoherent sources with different separations. We can see that resolution below 100 nm can be achieved for our entire chosen wavelength range.

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | S. W. Hell, “Far-Field Optical Nanoscopy,” Science |

3. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

4. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science |

5. | T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-Field Microscopy Through a SiC Superlens,” Science |

6. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

7. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. E |

8. | Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects,” Science |

9. | I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying Superlens in the Visible Frequency Range,” Science |

10. | A. V. Kildishev and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. |

11. | D. P. Gaillot, C. Croenne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal-dielectric planar hyperlens,” New J. Phys. |

12. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science |

13. | U. Leonhardt, “Optical Conformal Mapping,” Science |

14. | U. Leonhardt and T. G. Philbin, “Transformation Optics and the Geometry of Light,” Prog. Opt. |

15. | J. R. Wait, |

16. | J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nature Mater. |

17. | E. D. Palik, |

**OCIS Codes**

(110.0180) Imaging systems : Microscopy

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(160.3918) Materials : Metamaterials

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: March 29, 2010

Revised Manuscript: May 10, 2010

Manuscript Accepted: May 10, 2010

Published: May 12, 2010

**Virtual Issues**

Vol. 5, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Baile Zhang and George Barbastathis, "Dielectric metamaterial magnifier creating a virtual color image with far-field subwavelength information," Opt. Express **18**, 11216-11222 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11216

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### References

- M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999).
- S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316, 1153–1158 (2008). [CrossRef]
- J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 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, 534–537 (2005). [CrossRef] [PubMed]
- T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-Field Microscopy Through a SiC Superlens,” Science 313, 1597 (2006). [CrossRef]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef] [PubMed]
- A. Salandrino, and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74, 075103 (2006).
- Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-Field Optical Hyperlens Magnifying Sub-Diffraction-Limited Objects,” Science 315, 1686 (2007). [CrossRef] [PubMed]
- I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Magnifying Superlens in the Visible Frequency Range,” Science 315, 1699–1701 (2007). [CrossRef] [PubMed]
- A. V. Kildishev, and V. M. Shalaev, “Engineering space for light via transformation optics,” Opt. Lett. 33, 43–45 (2008). [CrossRef]
- D. P. Gaillot, C. Croenne, F. Zhang, and D. Lippens, “Transformation optics for the full dielectric electromagnetic cloak and metal-dielectric planar hyperlens,” N. J. Phys. 10, 115039 (2008). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, “Optical Conformal Mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
- U. Leonhardt, and T. G. Philbin, “Transformation Optics and the Geometry of Light,” Prog. Opt. 53, 69 (2009). [CrossRef]
- J. R. Wait, Electromagnetic Waves in Stratified Media, (IEEE Press, New York, 1996).
- J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8, 568–571 (2009). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, Orlando, 1997).

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