## Subwavelength imaging with anisotropic structure comprising alternately layered metal and dielectric films

Optics Express, Vol. 16, Issue 6, pp. 4217-4227 (2008)

http://dx.doi.org/10.1364/OE.16.004217

Acrobat PDF (514 KB)

### Abstract

Subwavelength imaging can be obtained with alternately layered metallodielectric films structure, even when the permittivity of metal and dielectric are not matched. This occurs as the effective transversal permittivity tends to be zero or the vertical one approaches infinity, depending on the permittivity value of the utilized dielectric and metal material. Evanescent waves can be amplified through the structure, but not in a manner of fully compensating the exponentially decaying property in dielectric. Numerical illustration of subwavelength imaging is presented for variant configuration of anisotropic permittivity with finite layer number of metallodielectric films.

© 2008 Optical Society of America

## 1. Introduction

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

2. V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

3. N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. **82**, 161–163 (2003). [CrossRef]

7. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express **13**, 2127 (2005). [CrossRef] [PubMed]

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

3. N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. **82**, 161–163 (2003). [CrossRef]

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

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

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

11. Liu Zhaowei, Hyesog Lee, Yi Xiong, Cheng Sun, and Xiang Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science **315**, 1686 (2007). [CrossRef]

12. D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. **90**, 077405(2003). [CrossRef] [PubMed]

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

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

16. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B **73**, 113110 (2006). [CrossRef]

17. K. J. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. **31**, 2130–2132 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-14-2130. [CrossRef] [PubMed]

18. T. Xu, C. Du, C. Wang, and X. Luo, “Imaging with Metallic Nano-slits Array,” Appl. Phys. Lett. **91**, 201501 (2007). [CrossRef]

## 2. Anisotropic media constructed by alternately layered dielectric and metal film

*ε*=

_{x}*ε*=

_{y}*ε*+

_{d}f*ε*(1-

_{m}*f*) and

*ε*

^{-1}

*=*

_{z}*ε*

^{-1}

*+*

_{d}f*ε*

^{-1}

*(1-*

_{m}*f*), respectively.

*f*is the filling factor for dielectric film.

7. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express **13**, 2127 (2005). [CrossRef] [PubMed]

*ε*and

_{x}*ε*

^{-1}

*. The effective permittivity for layered metal and dielectric media are positioned at the line defined by*

_{z}*ε*

^{-1}

_{z}*ε*+

_{d}ε_{m}*ε*=

_{x}*ε*+

_{d}*ε*. The available permittivity is confined by two ends which construct two hyperbolas of

_{m}*ε*

^{-1}

_{z}*ε*=1 in quadrant I and III. The dispersion relation of plane waves in anisotropic media is

_{x}*k*

^{2}

*/*

_{x}*ε*+

_{z}*k*

^{2}

*/*

_{z}*ε*=

_{x}*K*

^{2}

_{0}, where

*k*and

_{x}*k*represent the wave vector. Shown in Fig. 3 are the representative relations between

_{z}*k*and

_{x}*k*in the different quadrants with variant permittivity signs. In quadrant I,

_{z}*ε*>0 and

_{x}*ε*>0, plane wave with

_{z}*ε*and

_{x}*ε*are negative simultaneously (quadrant III), the media behaves like an anisotropic metal and no propagating mode is found here. For another two quadrants, II and IV, it is dramatically different. The media is characterized with no cut off property if

_{z}*ε*and

_{x}*ε*have different signs. That is to say, plane wave with very large

_{z}*k*can propagate in it. To understand this point explicitly, the structure can be viewed as strong anisotropic material which displays metal and dielectric property in the transversal and normal direction, respectively. Thus electromagnetic wave in the form of surface plasmon with very large

_{x}*k*can propagate inside this structure due to the coupling effect between any two adjacent layers of metal and dielectric films.

_{x}**74**, 115116(2006). [CrossRef]

*ε*and

_{x}*ε*

^{-1}

*where*

_{z}*θ*=0. This means that plane wave with large transversal wave vector

*k*propagates in the direction normal to the film. This seems to provide a promise for subwavelength imaging for object positioned closely to the anisotropic media’s surface. As depicted in the permittivity diagram, these points are denoted as A(

_{x}*ε*=0 and

_{x}*ε*

^{-1}

*>0), B(*

_{z}*ε*<0 and

_{x}*ε*

^{-1}

*=0), C(*

_{z}*ε*>0 and

_{x}*ε*

^{-1}

*=0), D(*

_{z}*ε*=0 and

_{x}*ε*

^{-1}

*<0)and O(*

_{z}*ε*=0 and

_{x}*ε*

^{-1}

*=0). Cases A and B are in correspondence with -*

_{z}*ε*>

_{m}*ε*and cases C and D with -

_{d}*ε*<

_{m}*ε*. The dispersion relation turns into

_{d}*K*=0 and

_{z}## 3. OTF for layered dielectric-metal structure

_{m}and ε

_{d}? These questions can be clearly answered with the optical transfer function for effective anisotropic media.

*d*, which is written as

*ε*is the permittivity of dielectric material surrounding the anisotropic slab and

*ε*=∞ and

_{z}*ε*=

_{x}*ε*+

_{m}*ε*, the asymptotic form of Eq. (1) becomes

_{d}*ε*=0 and

_{x}*k*≫

_{x}*K*

_{0}turns to be,

*k*. They promise the enhancement of evanescent waves in a broad range of

_{x}*k*, compared with of evanescent waves’ exponentially decaying behavior in vacuum. The great difference between Eq. (2) and Eq. (3) is the triangular function dependence with

_{x}*d*or the Fabry-Perot effect in the finite depth of anisotropic media C.

*ε*=0 or

_{x}*ε*=∞ does not promise the desired optical transmission for perfect imaging which requires

_{z}*t*(

*k*) being constant valued for all

_{x}*k*component, indicating that the exponentially decaying property of evanescent waves is fully compensated through the structure. But fortunately, it does give rise to the amplification of evanescent wave and helps to implement subwavelength imaging. It is interesting to scompare alternately stacked metal and dielectric film to a single slab of lossless metal with

_{x}*ε*imbedded in dielectric

_{m}*ε*. In this case, the evanescent wave can be amplified but in a much narrowed small

_{d}*k*region. But taking a much thinner metal slab results in greatly extended amplification region of

_{x}*k*. This would help to understand evanescent wave enhancement in the anisotropic media comprising very thin metal and dielectric films.

_{x}*ε*=

_{m}*ε*). Both analytical analysis and numerical calculation [8] show that large

_{d}*k*plane waves can be completely transmitted through the media. But it is worth to note that the perfect transmission does not holds for low valued

_{x}*k*plane waves. Another one happens with the Fabry-Perot resonance condition

_{x}*ε*=∞ and

_{z}*ε*>0 (case C with -

_{x}*ε*<

_{m}*ε*). According to Eq. (2), perfect imaging can be obtained theoretically with the transmitted amplitude for all

_{d}*k*to be 1. In reference [16

_{x}16. P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B **73**, 113110 (2006). [CrossRef]

*ε*=

_{m}*ε*. Both the two types of perfect imaging bear the virtue that imaging characteristics are not affected by the surrounding media.

_{d}*ε*≠

_{m}*ε*). To make the comparison easier, Eqs. (1) and (2) are further simplified according to the permittivity signs of cases A, B, C and D. Another assumption is made that

_{d}*ε*≈

*ε*for the fact that the refractive index difference for variant optical dielectric materials is usually small.

_{d}*ε*>

_{m}*ε*, which results in cases A(

_{d}*ε*=0 and

_{x}*ε*

^{-1}

*>0) and B(*

_{z}*ε*<0 and

_{x}*ε*

^{-1}

*=0). Comparison between Eq. (4) and Eq. (5) yields |*

_{z}*t*(

_{A}*k*≫

_{x}*K*

_{0})|>|

*t*(

_{B}*k*≫

_{x}*K*

_{0})|. If -

*ε*<

_{m}*ε*, the result is reversed with |

_{d}*t*(

_{D}*k*≫

_{x}*K*

_{0})|<|

*t*(

_{C}*k*≫

_{x}*K*

_{0})|. It seems that high transmission occurs in cases A and C, localized at the axis of quadrant I (anisotropic dielectric quadrant) in Fig. 2. In addition to the much lower capability of conversion of evanescent waves, cases B and D also surfer from the phase modulation which are not plotted. This usually results in reduced imaging property for the incorrect restoration of phase information.

_{d}=2 and three representative metal permittivity with ε

_{m}=-5 (for cases A and B), ε

_{m}=-1 (cases C and D), and ε

_{m}=-2 (O). Shown in Fig. 4(a), the transmitted amplitude of evanescent waves decays exponentially with increasing

*k*for anisotropic parameters localized in quadrant I and III (noting the OTF is in logarithmic scales). But for those in quadrant II, obvious oscillating OTF curves with FP effect can be observed. As for the four cases at permittivity diagram coordinates, OTF decrease in a reciprocal way of linear function, not obvious in logarithmic scale. If the permittivity of metal and dielectric are matched in case O, complete conversion of evanescent waves with large

_{x}*k*can be observed. Also can be clearly seen is case A displays superior performance than that of case B and case C much better than D, illustrating the result in the above analysis. According to Eq. (4), the resolution of the anisotropic imaging structure can be approximated as

_{x}*π*(

*ε*+

_{m}*ε*)

_{d}*d*/

*ε*by taking the position with half of the transmission function. So the best resolution occurs with matched metal and dielectric cases.

_{m}## 4. Subwavelength imaging with finite layered absorption metal and dielectric

*k*now. This is mainly attributed to the absorption of light in metal. As shown in paper [8], stacking films of permittivity matched metal and dielectric greatly relieve the great deleterious effect of metal loss on the image resolution. Here, it is illustrated that the same effect happens for stacked metal and dielectric film with unmatched permittivity as well.

_{x}*a*positioned closely at the interface of layered anisotropic structure and surrounding media as depicted in Fig. 1. Note the object is extended uniformly to infinity in the y direction and it can be decomposed into a series of plane waves as

*a*=0.1

*λ*. Clearly, cases A, C and O all have the ability of imaging subwavelength objects, but display different imaging performance. The minimum full width at half maximum (FWHM) of

*E*component image for case A is about 0.25

_{x}*λ*.

*E*Images in cases C and O are narrower, about 0.1

_{x}*λ*wide. The more layer numbers, the narrower image width with a convergence to the effective anisotropic material approximation. This point can also be seen from the OTF curves in Fig. 5. On the other hand, they all exhibit dramatically difference between imaging field

*E*and

_{x}*E*due to the vector nature of transversal magnetic polarized field. Usually, the width of

_{z}*E*field is greatly expanded compared with

_{z}*E*because

_{x}*E*is mainly localized at the edges of object. This would inevitably lower imaging resolution of electric field intensity and bring artifacts to images.

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

*k*slight larger than that in dielectric media, indicating the surface plasmon mode resonance. Similar effect can also be observed for case O with matched permittivity (Fig. 6(e) and Fig. 6(f)), but not so terrible like case A. The difference, we believe, arises from the great localization and propagation loss of surface plasmon in permittivity matched environment. This can also be justified through the OTF plots where the resonance peak is much damped and widened (Fig. 5(c)). Image in case C does not display this effect clearly. Here no transversal wave vector supports evanescent waves and they are all in propagating state (see Fig. 3). The resonant peak in Fig. 4(c) is the Fabry-Perot mode which does not bring obvious extension of images because light propagates in the direction almost normal to the films. Other reasons accounting to the reduced imaging property also include the phase modulation of OTF due to absorption and waveguide modes presence, which is not discussed here any more.

_{x}## 5. Conclusion

*ε*=0 or

_{x}*ε*

^{-1}

*=0. OTF analysis with effective material theory shows transmitted amplitude of plane wave decrease in a reciprocal way of linear function with respect to increased transversal wave vector*

_{z}*k*. In addition, it is shown that condition

_{x}*ε*=0 appreciates metal with larger absolute permittivity than the dielectric and

_{x}*ε*

^{-1}

*=0 just the contrary. Not so good for perfect imaging with almost complete conversion of evanescent waves like super lens, but they do render the capability of subwavelength imaging as illustrated with numerical simulation for finite layer of absorption metal and dielectric films.*

_{z}## Acknowledgments

## References and links

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

2. | V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. |

3. | N. Fang and X. Zhang, “Imaging properties of a metamaterial superlens,” Appl. Phys. Lett. |

4. | P. G. Kik, S. A. Maier, and H. A. Atwater, “Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources,” Phys. Rev. B |

5. | V. A. Podolskiy and E. E. Narimanov, “Near-sighted superlens,” Opt. Lett. |

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

7. | D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express |

8. | S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. |

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

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

11. | Liu Zhaowei, Hyesog Lee, Yi Xiong, Cheng Sun, and Xiang Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science |

12. | D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. |

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

14. | S. Feng and J. Merle Elson, “Diffraction-suppressed high-resolution imaging through metallo- dielectric nanofilms,” Opt. Express |

15. | M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express |

16. | P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B |

17. | K. J. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. |

18. | T. Xu, C. Du, C. Wang, and X. Luo, “Imaging with Metallic Nano-slits Array,” Appl. Phys. Lett. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(230.4170) Optical devices : Multilayers

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: October 23, 2007

Revised Manuscript: January 26, 2008

Manuscript Accepted: January 27, 2008

Published: March 12, 2008

**Citation**

Changtao Wang, Yanhui Zhao, Dachun Gan, Chunlei Du, and Xiangang Luo, "Subwavelength imaging with anisotropic structure comprising alternately layered metal and dielectric films," Opt. Express **16**, 4217-4227 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-4217

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### References

- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- V. Veselago, "The electrodynamics of substances with simultaneously negative values of ? and ?," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- N. Fang and X. Zhang, "Imaging properties of a metamaterial superlens," Appl. Phys. Lett. 82, 161-163 (2003). [CrossRef]
- P. G. Kik, S. A. Maier, and H. A. Atwater, "Image resolution of surface-plasmon-mediated near-field focusing with planar metal films in three dimensions using finite-linewidth dipole sources," Phys. Rev. B 69, 045418 (2004). [CrossRef]
- V. A. Podolskiy and E. E. Narimanov, "Near-sighted superlens," Opt. Lett. 30, 75-77 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-1-75. [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]
- D. O. S. Melville and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127 (2005). [CrossRef] [PubMed]
- S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire and W. J. Stewart, "Imaging the near field," J. Mod. Opt. 50, 1419-1430 (2003).
- Z. Jacob, L. V. Alekseyev, E. Narimanov, "Optical hyperlens: far-field imaging beyond the diffraction limit," Opt. Express 14, 8247 (2006). [CrossRef] [PubMed]
- A. Salandrino and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103(2006). [CrossRef]
- Liu Zhaowei, Hyesog Lee, Yi Xiong, Cheng Sun, Xiang Zhang, "Far-field optical hyperlens magnifying sub-diffraction-limited objects," Science 315, 1686 (2007). [CrossRef]
- D. R. Smith, D. Schurig, "Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors," Phys. Rev. Lett. 90, 077405(2003). [CrossRef] [PubMed]
- B. Wood, J. B. Pendry, and D. P. Tsai, "Directed subwavelength imaging using a layered metal-dielectric system," Phys. Rev. B 74, 115116(2006). [CrossRef]
- S. Feng and J. Merle Elson, "Diffraction-suppressed high-resolution imaging through metallo- dielectric nanofilms," Opt. Express 14, 216 (2006). [CrossRef] [PubMed]
- M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, and C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, "Negative refraction and sub-wavelength focusing in the visible range using transparent metallodielectric stacks," Opt. Express 15, 508 (2007). [CrossRef] [PubMed]
- P. A. Belov and Y. Hao, "Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime," Phys. Rev. B 73, 113110 (2006). [CrossRef]
- K. J. Webb and M. Yang, "Subwavelength imaging with a multilayer silver film structure," Opt. Lett. 31, 2130-2132 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-14-2130. [CrossRef] [PubMed]
- T. Xu, C. Du, C. Wang, and X. Luo, "Imaging with Metallic Nano-slits Array," Appl. Phys. Lett. 91,201501 (2007). [CrossRef]

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