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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 18 — Aug. 29, 2011
  • pp: 16809–16820
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Subwavelength image manipulation through an oblique layered system

Jin Wang, Hui Yuan Dong, Kin Hung Fung, Tie Jun Cui, and Nicholas X. Fang  »View Author Affiliations


Optics Express, Vol. 19, Issue 18, pp. 16809-16820 (2011)
http://dx.doi.org/10.1364/OE.19.016809


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Abstract

We show in this work an oblique layered system that is capable of manipulating two dimensional subwavelength images. Through properly designed planar layered system, we demonstrate analytically that lateral image shift could be achieved with subwavelength resolution, due to the asymmetry of the dispersion curve of constant frequency. Further, image rotation with arbitrary angle, as well as image magnification could be generated through a concentric geometry of the alternating layered system. In addition, we verify the image mechanism using full wave electromagnetic (EM) simulations. Utilizing the proposed layered system, optical image of an object with subwavelength features can be projected allowing for further optical processing of the image by conventional optics.

© 2011 OSA

1. Introduction

In 1873, Abbe [1

1. E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. 9, 413–468 (1873). [CrossRef]

] determined a physical constraint on the smallest feature resolvable through an ideal optical system, which is known as the diffraction limit. It prevents light from being focused below the order of λ/2. The reason is that high spatial frequency information carried by evanescent waves only exists in the near field of an object, only the propagating light reaches the far-field image plane. So collecting the evanescent information directly in the near field has long been considered as the most straightforward and effective way to overcome the diffraction limit.

Recently, Pendry [2

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

] proposed that a slab of negative index material (NIM) with ɛ = −1 and μ = −1 can form a perfect copy of an object: all details of the object, even smaller than the wavelength of light are reproduced. This proposed lens can couple incident evanescent waves into resonant surface plasmon, therefore amplify and “restore” evanescent components to exhibit perfect focusing. However, there are no natural NIMs and low-loss isotropic artificial NIMs are difficult to fabricate, especially at the infrared and visible frequencies. So a practical scheme [2

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

] was suggested that one can use a thin layer of silver as a superlens to beat the diffraction limit and obtain subwavelength resolution imaging. This idea was confirmed by recent experimental results [3

3. 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]

] which demonstrated the feasibility of subwavelength imaging using silver slabs in optical frequency range. However, the thickness of this silver film has to be very small as compared to the wavelength and the resolution is restricted by losses in the silver.

Subsequently, recent works [4

4. T. J. Cui, D. R. Smith, and R. Liu, eds., Metamaterials—Theory, Design, and Applications (Springer, 2009). [PubMed]

6

6. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101(R) (2003).

, 10

10. P. A. Belov, C. R. Simovski, and P. Ikonen, “Canalization of subwavelength images by electromagnetic crystals,” Phys. Rev. B 71, 193105 (2005).

19

19. R. Kotynski and T. Stefaniuk, “Multiscale analysis of subwavelength imaging with metal-dielectric multilayers,” Opt. Lett. 35, 1133–1135 (2010). [CrossRef] [PubMed]

] have been devoted to the imaging capabilities of multilayered structures. To reduce the influence of material loss, Ramakrishna and Pendry [5

5. 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).

,6

6. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101(R) (2003).

] put forward a realistic structure to improve the subwavelength image in the near-field zone. They cut a metal slab into many thin layers and separated them by alternative metal and dielectric layers. These metal and dielectric layers possess the same thickness, and the real parts of their permittivities have the opposite signs. This structure is equivalent to an array of infinitely conducting wires embedded into the medium with zero permittivity and simply connects object to image point by point [7

7. T. A. Morgado and M. G. Silveirinha, “Transport of an arbitrary near-field component with an array of tilted wires,” New J. Phys. 11, 083023 (2009). [CrossRef]

9

9. A. Rahman, P. A. Belov, Y. Hao, and C. Parini, “Periscope-like endoscope for transmission of a near field in the infrared range,” Opt. Lett. 35, 142–144 (2010). [CrossRef] [PubMed]

]. But the absence of impedance matching between the structure and surrounding medium (generally in the air) causes strong reflection and restricts slab thickness to be much thinner than the wavelength. Belov and Hao [10

10. P. A. Belov, C. R. Simovski, and P. Ikonen, “Canalization of subwavelength images by electromagnetic crystals,” Phys. Rev. B 71, 193105 (2005).

, 11

11. 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]

] suggested a different physical mechanism called as canalization for subwavelength imaging, which does not involve amplification of evanescent modes. It works as a transmission device which delivers all spatial harmonics, produced by the source including evanescent modes, from front interface of the structure to the back one, provided that the structure has a flat isofrequency contour and the thickness of the slab fulfils Fabry-Perot (FP) resonance condition (an integer number of half-wavelengths). In contrast to the case of Ramadrishna’s lens, in the canalization regime the reflection from the slab are absent due to the FP condition which holds for all angles of incidence. However, Li et al. [12

12. X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75, 045103 (2007). [CrossRef]

] questioned the importance of impedance matching in favor of the FP condition. In fact, for a lossless metal and dielectric, the FP resonance is sufficient to entirely eliminate reflections resulting in perfect imaging without impedance matching. Very recently, Jin [13

13. Y. Jin, “Improving subwavelength resolution of multilayered structures containing negative-permittivity layers by flatting the transmission curves,” Prog. Electromagn. Res. 105, 347–364 (2010). [CrossRef]

] further explored multilayered structures to improve subwavelength resolution by flatting the transmission curves. It is found that in the near field imaging, the guided modes inside multilayered structures can amplify some incident evanescent waves for obtaining subwavelength resolution, but usually locally over-amplify them even when material loss exists, which may limit the subwavelength resolution. By choosing appropriate permittivity of dielectric layers, and adding a coating layer to cutoff the corresponding guided modes, they achieved high-subwavelength-resolution imaging. In addition, hyperlens, which resembles Ramakrishna’s lens but in a cylindrical profile, is suggested [20

20. 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]

24

24. M. Yan, W. Yan, and M. Qiu, “Cylindrical superlens by a coordinate transformation,” Phys. Rev. B 78, 125113 (2008). [CrossRef]

] and experimentally [25

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

, 26

26. 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]

], proved to magnify objects beyond the diffraction limit.

In the present paper, we will investigate how to manipulate the subwavelength imaging through an oblique layered metal-dielectric structure. Lateral image shift will be observed through the planar layered structure, and improving subwavelength resolution can be realized by choosing appropriately the permittivity of dielectric layers according to the chosen negative permittivity of metal layers. Image rotation with arbitrary angles, as well as image magnification, can be produced by the concentric geometry of the alternating layered system.

The forthcoming sections of paper are organized as follows: Section 2 describes the oblique layered structures homogenized using effective medium theory (EMT), and their dispersion relation. Section 3 shows guided modes analysis and transmission properties through the oblique layered system. In section 4, we discuss the lateral image shift with subwavelength resolution through properly designed planar layered structure. Section 5 presents the image rotation effect by the designed concentric geometry. Finally, we summarize the work in Section 6.

2. Oblique layered systems

We begin with an alternating metal-dielectric system with two kinds of isotropic materials whose thicknesses are d 1 and d 2 and permittivities are ɛm and ɛd, respectively. We further assume that the magnetic field is perpendicular to xy plane (TM polarization, magnetic field in the direction) and the time harmonic factor is exp(−iωt). According to EMT, when the thickness of the unit cell (d 1 + d 2) is far smaller than the operating wavelength, the effective permittivity of this layered system could be approximated as [27

27. 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]

, 30

30. H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78, 054204 (2008). [CrossRef]

]
ɛx=(d1+d2)ɛmɛdd1ɛm+d2ɛd,ɛy=d1ɛm+d2ɛdd1+d2
(1)
Also, this structure can be regarded as an anisotropic dielectric with a permittivity tensor:
ɛ¯normal=[ɛx00ɛy]
(2)
Here, we consider two-dimensional systems (the direction is homogenous).

Next, we cut the layered system obliquely in the region x 1 < x < x 2 shown in Fig. 1(a), then the normal direction of the alternating layers is oriented at a fixed angle, θ, from the direction. Such an oblique layered structure in x 1 < x < x 2 can be described as
ɛ¯oblique=[ɛxxɛxyɛyxɛyy]=[cosθsinθsinθcosθ][ɛx00ɛy][cosθsinθsinθcosθ]
(3)

Fig. 1 (color online) (a) The geometry of the planar oblique layered system in the xy plane, the û and directions are two principal axes obtained by rotating an angle θ from the and ŷ directions, where ɛu=ɛx and ɛv=ɛy. (b) Dispersion relation between kx and ky for θ = 45°, ω = ω 0 (solid line) and ω′ = 1.05ω 0 (dashed line). The slightly variation of constant frequency contour between ω and ω′ can determine the direction of the group velocity, which therefore points towards the contour at a higher frequency ω′. The length of arrows is proportional to the magnitude of the group velocity.

The behavior of such an oblique layered materials can be understood by considering the dispersion relation between the frequency ω and the wave vector k. We assume that we are dealing with nonmagnetic materials, so that the magnetic permeability μ = 1. Appropriate thickness of the metal and dielectric films in each unit cell could yield positive and negative signs for ɛxx, ɛxy and ɛyy, which represent different forms of dispersion relation
kx2ɛxx+2kxkyɛxy+ky2ɛyy=ω2c2(ɛxxɛyyɛxy2)
(4)
where c is the velocity of light in vacuum, and kx and ky represent the wavevectors in the normal and transversal directions, respectively. Here, typically the parameter ɛxx, ɛxy and ɛyy satisfy the condition ɛxxɛyyɛxy2<0, the dispersion function shows a pattern of hyperbola, and kx is real for a much wider range of values of ky. Even the high spatial frequency components with large |ky|, which would normally be evanescent, now correspond to real values of kx, and hence to propagating waves in the media.

The dispersion relation also provides the key to the preferred propagation direction, which is determined by the group velocity vg = ∇kω(k) [27

27. 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]

, 28

28. B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative refraction,” Phys. Rev. Lett. 105, 266804 (2010). [CrossRef]

]. The preferred propagating direction of the energy flow is normal to the dispersion curve of the constant frequency as shown in Fig. 1(b). It displays two common features. First, the plane wave with large transversal wavevectors, which are evanescent in natural optical materials, could transmit through the metal-dielectric systems. Second, the components with large |ky| propagate almost in one direction, which is defined by the angle α with respect to the x axis as
α=θ±arctanɛyɛx
(5)
Therefore it is clearly seen that α will be approximated as θ, when ɛy ≪ |ɛx|.

It is also worthy of noting that Ref. [29

29. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]

] has shown additional eigenwaves may appear which is completely unpredictable by EMT and caused by the plasmonic nature of the layered metal-dielectric structure. But in the presence of high absorption loss in metal at optical frequencies, all the focusing effects may be destroyed. So in this paper (Sec. 4 and Sec. 5) we compare the results between full-wave simulation and homogenized model to validate the focusing effect is still clear, and EMT is a good approximation to the layered structure as before.

3. Mode analysis and transmission through the oblique layered system

We have seen that we can produce a metamaterial with interesting properties by stacking alternating layers of metal and dielectric. Next, we look at a slab of this effective anisotropic material and examine the guided modes and the transmission properties.

We assume that the slab is embedded in a uniform medium of constant permittivity (which may be unity, representing vacuum). In such a medium, the electromagnetic waves satisfy the dispersion relation
kx2+ky2=ω2c2ɛ
(6)
We write k′x to distinguish the x component of the wave vector in the surrounding medium from that in the slab. Considering the guided wave inside the slab, the fields outside the slab must be evanescent in the longitudinal direction, i.e. k′x = , whereas kx in Eqs. (4) can be real or imaginary which stands for bulk modes and surface modes traveling along the slab, respectively.

Applying the boundary conditions to Maxwell equations, we can obtain the dispersion equations of the slab that define a set of allowed modes:
αɛ=±ɛxxΛɛxxɛyyɛxy2tan±(Λd2)
(7)
where Λ = kx + ɛxyky/ɛxx, and (+) and (−) correspond to symmetric (even) and antisymmetric (odd) guided modes, respectively. In the above equations, α, kx can be expressed in terms of ω and ky. We thereby obtain a set of transcendental equations, which may be solved graphically or numerically to yield the ω vs.ky dispersion curves.

Fig. 2 (a) Dispersion relation with guiding bands. (b) Transmission curves of a lossless anisotropic medium at a given ω = ω 0. The permittivity of the metal is given by (8) with ɛm(∞) = 1.0. The layers are of equal width d 1 = d 2 = λ/40.

To see the functionality of subwavelength-imaging clearly, we usually calculate the transfer function, which is defined as the ratio of the field at the image plane to that at the object plane. For our case, the transfer function has the form
T=2exp[idkyɛxyɛxx]2cos(Λd)+isin(Λd)[(ɛxy2ɛxxɛyy)kxɛɛxxΛ+ɛɛxxΛ(ɛxy2ɛxxɛyy)kx]
(9)

In Fig. 2(b), we plot the transfer function from the object plane to the image plane across a lossless slab for the oblique angle θ = 45°. Due to the excitation of the guided modes, transmission resonances appear. It is worthy noting that transmission curves show a wide flat upheaval for small |ky|, which is very beneficial for subwavelength imaging. Also, it is noticed that due to mismatched impedance, there is some reflection for |ky| < k 0 on the input interface, but not very large. This may reduce the intensity of the image, but influence little the resolution. In addition, due to the translation invariance of the structure along the y direction, the transmission for negative ky is the same as that for positive ky. However, the symmetry will be broken for lossy cases as seen next.

In the above and following analysis, the parameter ɛd we used is larger than −Re(ɛm), otherwise, the incident propagation waves will be reflected strongly, which is not good for imaging. Also we can obtain flat upheavals of transmission curves which can improve the image quality greatly.

4. Shift of subwavelength image

We have seen that this oblique layered system allows to change preferred propagation direction, which is mainly determined by the oblique angle, and also it can provide enhanced transmission of high-spatial-frequency components at certain frequencies. This gives us hope that we may achieve lateral shift of imaging with high resolution by using this system.

As a test, we consider the image of a point source. In this paper, when imaging is carried out, a point source as an object is put on the left (or inner) surface of the structure, and the image plane is defined on the right (or outer) surface. The parameters are used as before, except for the permittivity of the metal. In practice, material loss always exists in a natural negative-permittivity material. To investigate the influence of material loss on the above oblique layered system, we take the metal permittivity ɛm = −3.5 + 0.23i at a particular optical frequency ω = ω 0. Full wave EM simulation based on the two-dimensional finite element method is performed to verify the subwavelength resolution imaging.

Let’s firstly see the image-shifting due to the variation of oblique angles θ. Figs. 3(a)–3(e) show the image of a point source through the effective anisotropic medium, when θ changes from 0° to 15°, 30°, 45°, and 60°. Here, ɛd = 4.3 is unchanged. The source is put at y = 0 at the left of the structure. the imaging point does not locate at y = 0, but makes a displacement, Δy = (x 1x 2) tanθ, due to the designed oblique structure. As we expect, as θ increasing from 0° to 15°, 30°, 45°, 60°, the displacements Δy can be observed as 0, −0.13λ, −0.29λ, −0.51λ, −0.90λ, respectively. To see clearly the image intensity and resolution, we plot in Fig. 4(a) the distribution of magnetic energy density at the image plane. For θ = 0°, the full width at half maximum (FWHM) of image is about 0.08λ, with high intensity. While increasing θ to 60°, the FWHM of image increases to 0.2λ, and the image becomes dimmer. Moreover, strong side-lobes appear and destroy the imaging. This phenomenon of variation of image resolution and intensity as θ increases can be well understood according to transmission curves shown as Fig. 5. With loss introduced, the transmission |T| drops quickly for large |ky|. However, fortunately, such dropping of |T| for large |ky| does not influence significantly high resolution imaging. The amplitudes of evanescent wave with small |ky| are usually larger than those with large |ky|. That is, evanescent waves with small |ky| are more important to generate a subwavelength image and evanescent waves with large |ky| are of little importance. Therefore, for θ = 0°, 15°, 30°, it can still generate a good subwavelength-image through this structure. However, for larger oblique angle, i.e. θ = 45°, 60°, material loss enhances further the quick dropping of the transmission |t|, which influences significantly high-resolution imaging.

Fig. 3 (color online) The distribution of magnetic energy density (a)–(e) for an effective anisotropic medium, and (f)–(j) for an oblique planar layered system in the xy plane, with θ = 0°, 15°, 30°, 45°, and 60°, respectively. Here, the permittivity of metal ɛm = −3.5 + 0.23i, and the permittivity of dielectric ɛd = 4.3. The yellow solid lines indicate the boundaries of the systems.
Fig. 4 (color online) Comparison of magnetic energy density of the image plane along the y direction between (a) the effective anisotropic medium and (b) the planar layered systems for different oblique angle.
Fig. 5 (color online) Transmission curves of a lossy anisotropic medium for different θ (a) 0°, (b) 15°, (c) 30°, (d) 45°, (e) 60°.

Treating the layered system as an effective medium is a helpful simplification and the EMT analysis can give us enough guidance. To confirm the above result, we represent in Figs. 3(f)–3(j) the image of the point source through the oblique layered system for different θ. The distributions of magnetic energy density look quite similar to those shown Figs. 3(a)–3(e) for the corresponding effective anisotropic media. To see clearly, Fig. 4(b) also shows the corresponding distribution at the image plane for the oblique layered system. The comparison between Fig. 4(a) and Fig. 4(b) validates the appropriateness of the EMT, as the layers are made thin enough.

Next, we investigate how the image-shifting is influenced by tuning the dielectric permittivity ɛd of the corresponding multilayered structure. Figs. 6(a)–6(e) show the distribution of magnetic energy density for the effective anisotropic medium, when ɛd increases from 3.5 to 4.0, 4.3, 4.8, to 6.0. Here, θ is fixed, i.e. θ = 30°. For different ɛd, it always indicates the same shift of imaging. However, with ɛd increasing, the imaging quality will be influenced greatly, as clearly seen from Fig. 7(a). For ɛd = −Re(ɛm) = 3.5, the lossy structure can not work well for subwavelength imaging, whereas a high image resolution is obtained for the case of ɛd = 4.0, 4.3, 4.8, and the FWHM is about λ/10 for ɛd = 4.3. However, further increasing ɛd will reduce the image quality, even the location of shift-imaging is changed slightly. The explanation can also be obtained from the transmission curves shown in Fig. 8. From these transmission curves, we can see material loss damps the sharp transmission peaks, but they remain sharp for ɛd = 3.5, and over-amplification of some evanescent waves can not be eliminated completely, which may deteriorate the image quality. Thus it can not work well for subwavelength imaging, due to the existence of sharp transmission. With ɛd increasing and deviating away from 3.5, they display a flat pattern near k 0 for ɛd = 4.0, 4.3 and 4.8, which is beneficial for subwavelength imaging. As ɛd increases further, a new sharp peak is generated near k 0, and material loss decreases the image resolution. The oblique layered structure is used to confirm the above result. In Figs. 6(f)–6(j) we plot the magnetic energy for the oblique layered structure with θ = 30° in case of ɛd = 3.5, 4.0, 4.3, 4.8 and 6.0. The magnetic energy distributions look quite similar to those shown in Figs. 6(a)–6(e). A detailed comparison between Fig. 7(a) and Fig. 7(b) for the energy distribution at the image plane also shows the accuracy of EMT.

Fig. 6 (color online) The distribution of magnetic energy density (a)–(e) for an effective anisotropic medium, and (f)–(j) for an oblique planar layered system in the xy plane, with different permittivity of dielectric ɛd = 3.5, 4.0, 4.3, 4.8, and 6.0, respectively. Here, the permittivity of metal ɛm = −3.5 + 0.23i, and the oblique angle θ = 30°. The yellow solid lines indicate the boundaries of the systems.
Fig. 7 (color online) Comparison of magnetic energy density of the image plane along the y direction between (a) the effective anisotropic medium and (b) the planar layered systems for different permittivity of dielectric.
Fig. 8 (color online) Transmission curves of a lossy anisotropic medium for different permittivity of dielectric ɛd (a) 3.5, (b) 4.0, (c) 4.3, (d) 4.8, and (e) 6.0.

So far, the subwavelength imaging is only considered in the xy plane, which corresponds to one-dimensional imaging. Actually, we can realize two-dimensional subwavelength imaging in the proposed oblique multilayered structure. To see the two-dimensional imaging effect, we show in Figs. 9(a)–9(b) the image of a point source in the xz plane through the effective anisotropic medium for ɛd = 4.3, 4.8, respectively. We take other parameters: the oblique angle θ = 30°, ɛm = −3.5+0.23i in two cases. Obviously from the simulation the good image quality can also be achieved in x–z plane. The FWHM of image is about λ/10 shown respectively in Figs. 9(e)–9(f). Subsequently, tuning the oblique angle θ = 45°, we still achieve the image with subwavelength resolution in Figs. 9(c)–9(d). The corresponding distributions of magnetic energy density at the image plane are shown in Figs. 9(g)–9(h). Thus we can confirm our proposed structure can work well for two-dimensional subwavelength imaging.

Fig. 9 (color online) The distribution of magnetic energy density for an effective anisotropic medium in the xz plane. (a) θ = 30°, ɛd = 4.3, (b) θ = 30°, ɛd = 4.8, (c) θ = 45°, ɛd = 4.3, (d) θ = 45°, ɛd = 4.8. (e)–(h) the corresponding detailed distribution at the image plane along the z direction. The yellow solid lines indicate the boundaries of the systems.

5. Rotation effect of subwavelength image

By using the above concept of the image shifter with oblique planar layers, we then can realize an image rotator using a concentric layered systems. In Fig. 10(a), we illustrate the geometry of the layered system to produce the rotation effect of subwavelength imaging. The detailed shape of each layer is as follow: by using certain curves, we divide the concentric shell (a < r < b, a and b are radius of inner and outer radii of the system, respectively) into N layers for the alternating material of dielectric or metals. For these curves, we have the formula in the cylindrical coordinates (r,ϕ,z) [the direction is homogenous] [30

30. H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78, 054204 (2008). [CrossRef]

]
ϕ=βcot(θ)ln(ar)
(10)
with the starting point in the inner circle, r = a and ϕ = β. Here, θ is the oblique angle, and β = 0, 2π/N, 4π/N,...,2(N – 1)π/N. Starting with these N points, we can produce N curves that will divide the concentric shell (a < r < b) into N fan shaped parts [see Fig. 10(a)]. Then if the number of the alternating layers is large enough, the concentric layered structure with alternating layers of dielectric and metal can be used to mimic the anisotropic properties in the and ϕ^ directions very precisely. We use these alternating layers of dielectric and metallic (ɛd and ɛm) materials the same way as before. According to the EMT, we can obtain the permittivity tensor for an image rotator within the shell region (a < r < b) in the Cartesian coordinates [30

30. H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78, 054204 (2008). [CrossRef]

]:
ɛ¯rotator=[cosθsinθsinθcosθ][cosϕsinϕsinϕcosϕ][ɛx00ɛy]×[cosϕsinϕsinϕcosϕ]×[cosθsinθsinθcosθ].
(11)
Through this image rotator as shown in Fig. 10(b), we then see the image rotation, as well as magnification with subwavelength resolution. It has an inner radius a and an outer radius of b. With the material parameter obtained above, 2D subwavelength objects at the inner surface will be imaged at the outer surface of the structure. For example, point sources at S 1, S 2, and S 3 of the source plane will be restored well at I 1, I 2, and I 3 of the image plane, respectively. Such a structure acts as an optical image component that makes perfect image between source and image plane with the rotation of tan(θ)ln(b/a) [30

30. H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78, 054204 (2008). [CrossRef]

] in the limit of subwavelength resolution. Further, the interesting feature of the image rotator is that the image size is not equal to that of the object with a magnification determined by
IiIjSiSj=ba(i,j=1,2,3,ij)
(12)

Fig. 10 (color online) (a) The geometry of the concentric oblique layered system in the xy plane, the û and directions are two principal axes obtained by rotating an angle θ from the and ϕ^ directions, where ɛu=ɛx and ɛv=ɛy. (b) Schematic of the image rotator configuration. (c)(d) The distribution of magnetic energy density for an effective anisotropic medium and the concentric layered system, respectively. Here, the permittivity of metal ɛm = −3.5 + 0.23i, the permittivity of dielectric ɛd = 4.0, and the oblique angle θ = 30°. The yellow solid lines outline the interior and exterior boundaries of systems.

To verify the performance, we also carry out full wave EM simulation of the proposed the image rotator. Figure 10(c) shows the rotated image of subwavelength resolution can be obtained on the effective anisotropic medium. The image rotator has an inner radius of a = 0.1λ and an outer radius of b = 0.6λ, Three point sources S 1, S 2, and S 3 are located at the inner boundary (−a, 0), (a/2, 3a/2), (a/2, 3a/2), respectively. Here, θ = 30°, ɛd = 4.0 and ɛm = −3.5 + 0.23i. It is clearly demonstrated that well resolved images of the three point sources appear at the outer surface with positions of (−b/2, 3b/2), (b, 0), (−b/2, 3b/2), which confirms the ability of the structure of imaging subwavelength objects. The structure achieves an image with the magnification of b/a = 6, the resolution of image is about λ/10, and the rotation angle of image is nearly 60° as clearly shown in Fig. 11. It is worth noting that small material loss does not influence significantly high-resolution imaging. The key is to transmit evanescent waves in a wide range through the lens structure in appropriate proportion. We also give the imaging through the concentric layered metal-dielectric structure with N = 72 in Fig. 10(d). Appropriateness of the EMT is confirmed by similar distributions of magnetic energy density.

Fig. 11 (color online) Comparison of magnetic energy density of the image plane along the ϕ direction between (a) the effective anisotropic medium and (b) the concentric layered systems for a = 0.1λ, b = 0.6λ, and θ = 30°.

Such magnified image can be further processed by conventional optics and the rotation effect of subwavelength image greatly increases the flexibility of beam control. This design only requires simple isotropic materials without spatial gradient. Since the suggested method is general, it can be applied for other negative permittivities.

6. Conclusion

In this paper, we extend the concept of subwavelength imaging and manipulate subwavelength images flexibly through an oblique layered system. We demonstrate that image shifting could be achieved through an oblique planar layer system, and image rotation with arbitrary angle as well as magnified image could be obtained through an concentric layered system. The theoretical analysis and design procedure of these image processing components have been given, and their performances have been confirmed by full wave EM simulation. The proposed structures can be deployed as a basic element to manipulate light for further optical processing of the image by conventional optics.

Acknowledgments

This work was supported in part by the National Science Foundation of China under Grant Nos. 10647114, 60990320, 60990324, 60871016, and 60901011, in part by the Natural Science Foundation of Jiangsu Province under Grant No. BK2008031, and in part by the 111 Project under Grant No. 111-2-05. J. Wang is grateful for support from China Scholarship Council and Southeast University.

References and links

1.

E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. 9, 413–468 (1873). [CrossRef]

2.

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

3.

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]

4.

T. J. Cui, D. R. Smith, and R. Liu, eds., Metamaterials—Theory, Design, and Applications (Springer, 2009). [PubMed]

5.

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).

6.

S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101(R) (2003).

7.

T. A. Morgado and M. G. Silveirinha, “Transport of an arbitrary near-field component with an array of tilted wires,” New J. Phys. 11, 083023 (2009). [CrossRef]

8.

T. A. Morgado, J. S. Marcos, M. G. Silveirinha, and S. I. Maslovski, “Experimental verification of full reconstruction of the near-field with a metamaterial lens,” Appl. Phys. Lett. 97, 144102 (2010). [CrossRef]

9.

A. Rahman, P. A. Belov, Y. Hao, and C. Parini, “Periscope-like endoscope for transmission of a near field in the infrared range,” Opt. Lett. 35, 142–144 (2010). [CrossRef] [PubMed]

10.

P. A. Belov, C. R. Simovski, and P. Ikonen, “Canalization of subwavelength images by electromagnetic crystals,” Phys. Rev. B 71, 193105 (2005).

11.

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]

12.

X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75, 045103 (2007). [CrossRef]

13.

Y. Jin, “Improving subwavelength resolution of multilayered structures containing negative-permittivity layers by flatting the transmission curves,” Prog. Electromagn. Res. 105, 347–364 (2010). [CrossRef]

14.

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

15.

K. J. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. 31, 2130–2132 (2006). [CrossRef] [PubMed]

16.

B. Wang, L. Shen, and S. He, “Superlens formed by a one-dimensional dielectric photonic crystal,” J. Opt. Soc. Am. B 25, 391–395 (2008). [CrossRef]

17.

B. Zeng, X. Yang, C. Wang, Q. Feng, and X. Luo, “Super-resolution imaging at different wavelengths by using a one-dimensional metamaterial structure,” J. Opt. 12, 035104 (2010). [CrossRef]

18.

R. Kotynski, T. Stefaniuk, and A. Pastuszczak, “Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers,” arXiv:1002.0658v1.

19.

R. Kotynski and T. Stefaniuk, “Multiscale analysis of subwavelength imaging with metal-dielectric multilayers,” Opt. Lett. 35, 1133–1135 (2010). [CrossRef] [PubMed]

20.

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]

21.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Semiclassical theory of the hyperlens,” J. Opt. Soc. Am. A 24, A52–A59 (2007). [CrossRef]

22.

A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32, 3432–3434 (2007). [CrossRef] [PubMed]

23.

W. Wang, H. Xing, L. Fang, Y. Liu, J. Ma, L. Lin, C. Wang, and X. Luo, “Far-field imaging device: planar hyperlens with magnification using multi-layer metamaterial,” Opt. Express 16, 21142–21148 (2008). [CrossRef] [PubMed]

24.

M. Yan, W. Yan, and M. Qiu, “Cylindrical superlens by a coordinate transformation,” Phys. Rev. B 78, 125113 (2008). [CrossRef]

25.

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

26.

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]

27.

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]

28.

B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative refraction,” Phys. Rev. Lett. 105, 266804 (2010). [CrossRef]

29.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]

30.

H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78, 054204 (2008). [CrossRef]

OCIS Codes
(110.0180) Imaging systems : Microscopy
(160.1190) Materials : Anisotropic optical materials

ToC Category:
Imaging Systems

History
Original Manuscript: May 17, 2011
Revised Manuscript: July 5, 2011
Manuscript Accepted: August 1, 2011
Published: August 15, 2011

Virtual Issues
Vol. 6, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Jin Wang, Hui Yuan Dong, Kin Hung Fung, Tie Jun Cui, and Nicholas X. Fang, "Subwavelength image manipulation through an oblique layered system," Opt. Express 19, 16809-16820 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-16809


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References

  1. E. Abbe, “Beitrage zur theorie des mikroskops und der mikroskopischen wahrnehmung,” Arch. Mikrosk. Anat. 9, 413–468 (1873). [CrossRef]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]
  3. 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]
  4. T. J. Cui, D. R. Smith, and R. Liu, eds., Metamaterials—Theory, Design, and Applications (Springer, 2009). [PubMed]
  5. 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).
  6. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101(R) (2003).
  7. T. A. Morgado and M. G. Silveirinha, “Transport of an arbitrary near-field component with an array of tilted wires,” New J. Phys. 11, 083023 (2009). [CrossRef]
  8. T. A. Morgado, J. S. Marcos, M. G. Silveirinha, and S. I. Maslovski, “Experimental verification of full reconstruction of the near-field with a metamaterial lens,” Appl. Phys. Lett. 97, 144102 (2010). [CrossRef]
  9. A. Rahman, P. A. Belov, Y. Hao, and C. Parini, “Periscope-like endoscope for transmission of a near field in the infrared range,” Opt. Lett. 35, 142–144 (2010). [CrossRef] [PubMed]
  10. P. A. Belov, C. R. Simovski, and P. Ikonen, “Canalization of subwavelength images by electromagnetic crystals,” Phys. Rev. B 71, 193105 (2005).
  11. 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]
  12. X. Li, S. He, and Y. Jin, “Subwavelength focusing with a multilayered Fabry-Perot structure at optical frequencies,” Phys. Rev. B 75, 045103 (2007). [CrossRef]
  13. Y. Jin, “Improving subwavelength resolution of multilayered structures containing negative-permittivity layers by flatting the transmission curves,” Prog. Electromagn. Res. 105, 347–364 (2010). [CrossRef]
  14. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103 (2006). [CrossRef]
  15. K. J. Webb and M. Yang, “Subwavelength imaging with a multilayer silver film structure,” Opt. Lett. 31, 2130–2132 (2006). [CrossRef] [PubMed]
  16. B. Wang, L. Shen, and S. He, “Superlens formed by a one-dimensional dielectric photonic crystal,” J. Opt. Soc. Am. B 25, 391–395 (2008). [CrossRef]
  17. B. Zeng, X. Yang, C. Wang, Q. Feng, and X. Luo, “Super-resolution imaging at different wavelengths by using a one-dimensional metamaterial structure,” J. Opt. 12, 035104 (2010). [CrossRef]
  18. R. Kotynski, T. Stefaniuk, and A. Pastuszczak, “Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers,” arXiv:1002.0658v1.
  19. R. Kotynski and T. Stefaniuk, “Multiscale analysis of subwavelength imaging with metal-dielectric multilayers,” Opt. Lett. 35, 1133–1135 (2010). [CrossRef] [PubMed]
  20. 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]
  21. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Semiclassical theory of the hyperlens,” J. Opt. Soc. Am. A 24, A52–A59 (2007). [CrossRef]
  22. A. V. Kildishev and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32, 3432–3434 (2007). [CrossRef] [PubMed]
  23. W. Wang, H. Xing, L. Fang, Y. Liu, J. Ma, L. Lin, C. Wang, and X. Luo, “Far-field imaging device: planar hyperlens with magnification using multi-layer metamaterial,” Opt. Express 16, 21142–21148 (2008). [CrossRef] [PubMed]
  24. M. Yan, W. Yan, and M. Qiu, “Cylindrical superlens by a coordinate transformation,” Phys. Rev. B 78, 125113 (2008). [CrossRef]
  25. H. Lee, Z. Liu, Y. Xiong, C. Sun, and X. Zhang, “Development of optical hyperlens for imaging below the diffraction limit,” Opt. Express 15, 15886–15891 (2007). [CrossRef] [PubMed]
  26. 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]
  27. 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]
  28. B. Stein, J. Y. Laluet, E. Devaux, C. Genet, and T. W. Ebbesen, “Surface plasmon mode steering and negative refraction,” Phys. Rev. Lett. 105, 266804 (2010). [CrossRef]
  29. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]
  30. H. Chen and C. T. Chan, “Electromagnetic wave manipulation by layered systems using the transformation media concept,” Phys. Rev. B 78, 054204 (2008). [CrossRef]

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