- Email: [email protected]
Gradient–curvature nanolens for nano-imaging
Optics Communications 332 (2014) 103–108
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Gradient–curvature nanolens for nano-imaging Kao-Der Chang a, Yu-Ming Wang a, Pi-Gang Luan b,n a b
Mechanical and Systems Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan Wave Engineering Laboratory, Department of Optics and Photonics, National Central University, Chungli 32054, Taiwan
art ic l e i nf o
a b s t r a c t
Article history: Received 27 March 2014 Received in revised form 18 June 2014 Accepted 29 June 2014 Available online 10 July 2014
In this work, we study the sub-wavelength imaging properties of a plano-convex nano-lens consisted of alternatively arranged dielectric and metallic thin layers. The thickness of one layer is not uniform, and the curvatures of the layer boundaries vary gradually from layer to layer. The image resolution of this nanolens at wavelength 365 nm is about 40 nm, far below the diffraction limit, and its image magnification is much better than that of the original proposed hyperlens of the same size. We also study the imaging properties of the corresponding 3D structures. Numerical simulations reveal that the image recognition ability of the 3D nanolens can be controlled by changing the polarization of the source waves. Devices of this kind may find applications in biological morphology and nano-structured materials researches. & 2014 Elsevier B.V. All rights reserved.
Keywords: Hyperlens Multilayers Metamaterials
1. Introduction Traditional optical devices such like optical microscope are important in many branches of scientific research, for example, biological morphology, pharmaceutics research, microelectronics, and mineralogy. However, their resolution ability is restricted by the diffraction limit, which means the fine features of an object much smaller than one wavelength cannot be resolved using these devices. This limitation is mainly caused by the fact that the subwavelength information of the object encoded in the evanescent waves is lost once they leave the tiny object. Since 2000, this seemed unconquerable restriction has become conquerable, at least in principle, if the traditional devices are replaced by the metamaterial devices such like superlens. According to Pendry's theory, a perfect lens is a slab of metamaterial having negative permittivity and negative permeability, both close to 1 (for the reason of impedance matching). Such a slab also has negative refractive index close to 1, and it not only can compensate the phases of propagating waves of the source but also can recover the amplitudes of the evanescent waves. These unusual properties help to make an almost perfect image of the sub-wavelength sized object, breaking the diffraction limit [1]. Recent developments of metamaterial research indicate that such a slab can indeed be realized by using properly designed structures such as that consisting of parallel conducting wires together with split-ring resonators (SRRs) [2].
n
Corresponding author. Tel: þ 886 3 4227151x65267, fax: þ886 3 4252897. E-mail address: [email protected] (P.-G. Luan).
http://dx.doi.org/10.1016/j.optcom.2014.06.071 0030-4018/& 2014 Elsevier B.V. All rights reserved.
A lot of metamaterial structures based on the original idea of perfect lens have been proposed and tested, and high resolution imaging beyond diffraction limit has indeed been achieved [3,4]. However, in the optical frequency range, it is very difficult to realize negative permeability even in principle. A number of theoretical and experimental works concerning this issue have been done [5–8]. If our purpose is sub-wavelength resolution of nano-object and do not care about whether the image is located at the far field zone, then it can also be achieved by using a thin slab of metal which has negative permittivity only. The most prominent example is a slab of sliver in which surface plasmon modes can be excited due to its negative permittivity [9–12]. However, the material thickness and loss restrict the imaging quality of such a ‘superlens’. In addition, the image formed by a superlens is often of the same size as the object without magnification and it is restricted in the near field zone. An ideal optical imaging system should overcome not only the diffraction barrier but also be compatible with conventional optical system so that post-processing can be easily implemented. The recently proposed hyperlens structure can satisfy this requirement [13,14]. A hyperlens is a cylindrical multilayer structure consisting of alternatively arranged negative permittivity (metallic) and positive permittivity (dielectric) layers. Under the effective medium limit (which means the operating wavelength is much larger than the layer thickness), it leads to anisotropic dielectric tensor of the structure, which means the principal permittivity along the radial and azimuthal directions have opposite signs. This property leads to a hyperbolic dispersion relation in cylindrical coordinates, so it converts the evanescent waves into propagating ones, making some new applications possible [15–17].
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In a hyperlens, the streamlines of the energy flow direct mainly along the radial directions. Usually two sources close to each other with distance smaller than half wavelength are not recognizable from their images if usual lens is used. However, if we put these two sources on the inner surface of the hyperlens, two distinguishable magnified images can be found in the far field zone outside the hyperlens. The image is magnified and the magnifying power (MP) is proportional to the ratio between the outer and inner radii. Although hyperlens magnifies subwavelength objects and makes images in the far field zone if the ratio of outer-to-inner radius is appropriately chosen, the curved inner surfaces brings inconveniences in locating objects on the object surface. For this reason, other methods for obtaining breakingdiffraction-limited resolution such as the oblique lens [18], the planar hyperlens [19] and the pyramid-shaped hyperlens [20] were proposed. The working principle of most of these structures is based on the fact that the energy flows can be guided along certain directions as desired. It reveals that the electromagnetic waves can be controlled under some implementable conditions (canalization regime) if we design the structure appropriately [21–24]. Inspired by these research results, we propose in this paper a gradient curvature (GC) structure of a new plano-convex lens, working near the canalization regime. Our design is similar to that of [19] but we use simpler method to define the layer thickness, which is desirable for fabrication purpose. In addition, we also study the imaging characteristics of the generalized 3D structure, which is lacking in [19]. The simulation results show that the resolution down to 40 nm can be achieved and the magnifying power is more than 5 nm at 365 nm wavelength, much better than the original hyperlens of the same size can do. In addition, we also demonstrate that the corresponding 3D GC lens structure can also work very well, and its imaging quality can be controlled by the source polarization.
2. Theory and models The working principle of the proposed nanolens is based on the highly anisotropic property of the effective permittivity, which can be realized by using multilayered periodic structure made of alternatively arranged dielectric and metallic layers. One period of the structure consists of one dielectric layer of permittivity εd and thickness dd, together with one metallic layer of permittivity εm and thickness dm. The dispersion relation for the propagating mode in this structure is given by [25]. cos ðK d Þ ¼ cos ðK m dm Þ cos ðK d dd Þ
respectively. It is known that if the working frequency ω and thickness ratio dd/dm are chosen appropriately, we can make Re(εt) 40 and Re(εn) o0. This leads to a dispersion curve of hyperbolic type (see Eq. (2)), which implies that an evanescent wave outside can be transformed to a propagating wave inside. This is the underlying working principle for the (original) hyperlens. As mentioned in the introduction, the GC lens structure works near the canalization regime [21–23], which means we choose to design the structure having very large principal permittivity value along the normal direction of the layers (i.e., εn-1, assuming that the imaginary parts of εd and εm are both small enough). According to Eq. (3), this implies that the following condition must be satisfied: dd Reðεm Þ þ dm Reðεd Þ ¼ 0
ð4Þ
For simplicity we set (dd ¼ dm), thus we must choose an appropriate frequency such that Re(εm)E Re(εd). Under this condition the dispersion curve at one single frequency becomes very flat along the direction parallel to the layers, and the streamlines of the energy flow go along almost the same direction, normal to the dispersion curve. This property can be used to design a new kind of nanolens for making images of nano-scale objects by bending the flat multilayer structure in a new way. The information encoded in the propagating and evanescent waves from a tiny object can thus be transferred to the output surface of the nanolens along trajectories locally normal to each dielectric–metal interface. Hereafter we name the nanolens gradient–curvature (GC) lens, which is made of multilayer dielectric–metal films having different curvature in each layer. The proposed GC lens structure is formed by cutting a part of the elliptical shaped reference structure (inside the dashed-line rectangle), as indicated by the left part of Fig. 1. The downward A-axis and the horizontal B-axis define the two axes of the ellipse, and their ratio B/A determines the shape of the reference structure and the curvature of each layer. In the limiting case of A ¼B, the shape of the reference structure becomes circular. The aperture width C of the lens is shorter than the waist width 2B of the ellipse but long enough to make the lens structure work. We assume in this paper that each layer has a fixed local thickness of 10 nm along the A-axis, but the layer thickness is not uniform and it shrinks to zero as we move sideward along the B-axis direction. Specifically, the lower boundary of the nth layer (counting downward) is defined by the equation of the ellipse
y2 x2 þ ð10nÞ 2 B2
¼ 1, here
nanometer is adopted as the length unit. For simplicity, we first consider the 2D structure. The dashed-line rectangle in the right
1 km εd kd εm þ sin ðkm dm Þ sin ðkd dd Þ 2 kd εm km εd
ð1Þ where d ¼dd þ dm is the period of the structure, kd and km are the normal (the direction perpendicular to the layers) components of the wave vectors in the dielectric and metallic layers, and K is the Bloch wave number of the mode. When the wavelength is much longer than the period of this structure, an effective dispersion can be extracted: 2
2
k n k t ω2 þ ¼ εt εn c 2
ð2Þ
where εt ¼
dd εd þ dm εm ; d
εn 1 ¼
1 dd εd 1 þ dm εm d
ð3Þ
are the two principal values of the effective permittivity tensor, ω is the angular frequency of the wave, c is the speed of light in vacuum, and kt and kn ¼ K are the tangential and normal components of the wave vector of the plane wave propagating inside,
Fig. 1. The gradient curvature (GC) lens structure made of multilayer dielectric– metal films having different curvature in each layer. Left: The elliptical shaped reference structure defined by the two axes A and B. Right: The lens is formed by cutting a part of the reference structure with aperture width C. The dashed-line rectangle represents the working area of the lens. Each layer has a fixed local thickness of 10 nm along the A-axis and shrinks to zero sidewardly.
K.-D. Chang et al. / Optics Communications 332 (2014) 103–108
part of Fig. 1 represents the working area of the GC lens, and the top and bottom edges indicate the input (source) plane and output (image) plane, respectively. In our simulation, the permittivity of the metallic layer is approximated by Drude–Lorentz model with plasma frequency 9.01 (eV) and a damping constant 0.048 (eV) [26]. Assuming the working wavelength to be 365 nm, the permittivities of the dielectric and metallic layers are εd ¼ 4.907 and εm ¼ 4.9438þj0.084, satisfying the canalization condition (4) approximately. The structure is embedded in glass and the input side is exposed to water. To study the performance of our device, we adopt COMSOL Multiphysics 3.5 (FEM) software to simulate the imaging characteristics of the GC lens structure. In these simulations, the light sources are located on the flat surface of the GC lens with linearly polarizations. Perfectly matched layers (PML) are used for reducing reflections. The output surface is located at the tangential plane of the GC lens. Based on these simulation results, we can discuss the relations between the image magnification and the geometric parameters used.
3. Numerical results and discussion To test the effect of the axes ratio B/A, we put two point sources at the input plane and record the image–image distance (IID) for every possible B/A ratio from 0.5 to 1.5. The IID curves for A¼ 600 nm, 800 nm and 1000 nm are plotted in Fig. 2, the source–source distance (SSD) is set to be 100 nm. As expected, the light waves from the sources can be guided along the directions locally normal to each of the dielectric–metal interfaces of the multilayer structure. The IID measured at the output plane is longer than the SSD, which confirms that the lens structure magnifies the input object. For A¼600 nm (60 layers) and B/A¼ 0.5, the IID is about 532 nm, corresponding to a magnifying power (MP) exceeds 5. According to these results, if the length of A is fixed, choosing lower B/A ratio gives larger MP. On the other hand, if the B/A ratio is fixed and low enough (less than 0.6), a larger A will give a larger IID. For example, for B/A ¼0.5, the IIDs for A¼600 nm, 800 nm and 1000 nm are 532 nm, 578 nm, and 620 nm, respectively. This tendency can be easily explained by the fact that for a larger CG lens the penetrating lights from the two sources propagate longer distances inside the structure, helping to separate the two images further. When the B/A ratio exceeds 0.7, the three curves merge into each other, and the IID becomes independent of the B/A ratio.
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To make things more concrete, we set A ¼600 nm and compute the electromagnetic fields for three cases: B/A ¼0.5, 1 and 1:5. The field patterns (the intensity of the H field) and the imaging strengths are shown in Fig. 3(a1)–(a3) and (b1)–(b3), respectively. As mentioned before, the low ratio case B/A¼0.5 makes two separate images and the MP is large. However, we also find that for the high B/A ratio cases, although the two sources can indeed make distinct images on the output plane, the two images will merge into a single image if the imaging results are not evaluated at a position close enough to the output plane indicated in Fig. 1. This observation indicates that GC lens is not a different kind of hyperlens for imaging in the far-field zone, but is designed for the purpose of making distinct images in a region smaller than the hyperlens can do. These results also indicate that both the shape and size of the GC lens can affect the MP. Recalling that in the original hyperlens design, the imaging resolution is controlled by the ratio between the outer radius and inner radius of the structure. Our structure provides more free parameters to control the imaging properties. Based on the simulation results, we confirm that it is feasible to design the GC lens proposed in this paper, and its MP can be manipulated under practical conditions. We now compare the magnification ability of the proposed GC lens to that of the original hyperlens, assuming that they have the same aperture size and the same number of layers. The IIDs as functions of SSD for various values of the A-thickness of these two lens structures are plotted as the unmarked curves and marked curves in Fig. 4(a), respectively. The field patterns for the 60-layer structures of GC lens and hyperlens are shown in Fig. 4(b1) and (b2). For all the simulations of the GC lens in Fig. 4, we set B/ A¼0.55. For the original hyperlens (concentric structure), different radius (A) yields the same magnifying power. However, for the GC lenses of fixed axes ratio (B/A ¼0.55), different A show different magnifying ability. As mentioned before, the imaging magnification of the GC lens depends on its A-length. Furthermore, when the aperture size (C) is the same, the GC lens has larger magnifying power than the original hyperlens. Giving SSD ¼100 nm, A¼600 nm, the field distributions for these two structures are plotted in Fig. 4(b1)–(b2). According to these results, both GC lens and the original hyperlens can magnify the objects and form enlarged images at the output plane. However, there are some important distinctions between their imaging behaviors. For example, although the two sources can be imaged at the output plane by the hyperlens, they will merge into each other and form a
Fig. 2. The image–image distance (IID) as functions of the axes ratio B/A for three chosen cases: A ¼ 600 nm, 800 nm, and 1000 nm. The source–source distance (SSD) is fixed as 100 nm.
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Fig. 3. Field strengths and image strengths for the 60-layer (A ¼ 600 nm) GC lenses with B/A ¼ 0.5, 1, and 1.5. Subplots (a1)–(a3) are the field strengths of these three cases, and (b1)–(b3) are the corresponding image strengths on the image plane.
Fig. 4. The magnifying powers and the field distributions for the GC lens and the original hyperlens of the same size (the same A-thickness and the same C-aperture width). (a) The horizontal axis is the source–source distance (SSD) of the two sources on the input plane, whereas the vertical axis is the image–image distance (IID) on the output plane. The distance between the two sources is from 40 nm to 120 nm. The IIDs as the functions of SSD for GC lens and hyperlens are plotted as the unmarked curves and marked curves, respectively. (b1) The field distribution for the A ¼ 60 nm GC lens with B/A¼ 0.55. (b2) The field distribution for the hyperlens with the same thickness and the same aperture width as that of (b1).
single image once leave the output plane. This is caused by the fact that for a hyperlens the farfield imaging ability can be achieved only when the outer-to-inner radius ratio is large enough. The radial propagation reduces the azimuthal component of the wave vector. Thus if the out radius is large enough, the wave vector for the image fields would not get imaginary radial component and can propagate to the far field zone. If we restrict the size of
hyperlens tightly, the propagating waves inside the hyperlens can only become evanescent waves in the output side, and the images can only appear in the near field zone. In contrast, the GC lens can resolve the two images even behind the output plane because two images are separated. This fact reveals that the GC lens has better magnification ability than the original hyperlens of the same size. Such difference disappears at high axes ratio (B/A4 0.7) and the
K.-D. Chang et al. / Optics Communications 332 (2014) 103–108
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Fig. 5. Images of the two magnetic sources with SSD ¼ 40 nm, 60 nm, and 80 nm using the 3D GC lens structures. Subplots (a1)–(a3) are the field strengths of images for the X-polarized sources, whereas (b1)–(b3) are the field strengths for the Y-polarized sources.
imaging behaviors of the GC lens become similar to that of the original hyperlens. Now we turn to the discussion of the imaging behaviours of the corresponding 3D GC lens. The lens is defined as the solid of revolution of the 2D GC lens structure (the rectangle region of the left hand part of Fig. 1) with respect to the A-axis, keeping the material and geometrical parameters unaltered. In the following numerical simulations, two sources with spacing (SSD) from 40 nm to 80 nm are considered, and A¼600 nm, B/A¼0.55 are chosen, which corresponds to the 2D cases having the better performance. First, we set the two sources to have X-polarized magnetic fields. The field distributions of the images for the cases of SSD¼40 nm, 60 nm, and 80 nm are shown in Fig. 5(a1)–(a3). According to theses simulations, the two image spots are indistinguishable if the source spacing is smaller than 40 nm, and the image spots become clearly distinguishable if the source spacing is longer than 60 nm. The IIDs are 302 nm and 405 nm when the SSDs are 60 nm and 80 nm, respectively, which means this 3D GC lens gives MPE5, similar to that of the 2D structure. In addition, we observe that the images of the sources are slightly deformed along the polarization direction of the source fields (the X-direction). Next, we consider the sources having Y-polarized magnetic fields. The imaging results are shown in Fig. 5(b1)–(b3). These simulations reveal that using Y-polarized magnetic sources the two image spots are clearly distinguishable even if we reduce the SSD to 40 nm. The IIDs are 202 nm, 304 nm, and 408 nm for SSD¼40 nm, 60 nm and 80 nm, respectively. Thus for the Y-polarized sources the MP is still close to 5. In addition, the image deformation phenomenon still exists, but now the deformation is along the Y-direction. Since the imaging features for the X- and Y-polarized magnetic sources are different, we conclude that we can modify the image patterns by changing the source polarization. The above mentioned phenomena may probably be explained by considering the phase features of the source fields. Comparing the two kinds of polarization states, they give similar MP when the source spacing is above 60 nm. However, for shorter spacing
(e.g., SSD ¼40 nm), the Y-polarized sources give better imaging results than the X-polarized sources. This might be explained by the fact that the phases of the source fields (the E fields) are opposite in the central line between the two sources when the sources have Y-polarized magnetic fields. In other words, the distinguishable double-image might be caused by the destructive interference of the source fields. However, the detailed analysis of this interference phenomenon and the confirmation of the above mentioned explanation are too complicated and beyond the scope of the present work. We only know at this stage that different polarization state of the sources provides different interference pattern and thus leads to different image recognition ability. In a real imaging system, the image of an arbitrary shaped object should be recognizable. We test this by simulating the imaging features of an N shaped slit illuminated by source fields of chosen polarizations. Fig. 6 shows the imaging patterns of the N font for three different (X-, Y-, and XY-) polarizations of the magnetic source. The simulation results reveal that the 3D CG lens can identify the shape of the font if the characteristic size of the object is larger than 40 nm. Besides, for the Y- and XY- polarized sources, the two vertical edges of the font N can be better observed, whereas for the X-polarized source, only the inclined edge in between can be easily observed. The recognition of the lens is thus dependent on the polarization state of the source. Therefore, we can average the results of different polarizations or utilize un-polarized source to get a better imaging result of the subwavelength sized object. The proposed 2D and 3D GC lens structures in this paper can be fabricated using the Atomic Layer Deposition (ALD) technique, which is a chemical vapor deposition technique suitable for manufacturing inorganic material layers with thickness down to nanoscale [27,28]. Choosing the reactive precursors, temperature gradients and gas flow inside the reactor appropriately, it is possible to control the uniformity and non-uniformity of the layer thickness [29,30].
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Fig. 6. Images of an N-shaped object. The field distributions on the image plane for three different polarizations (X-polarization, Y-polarization, and the XY-polarization) are shown.
4. Conclusion In this paper, we analyze the sub-wavelength imaging features of the 2D and 3D GC lens structures. The numerical simulation results provide feasible solutions to construct these devices for sub-wavelength imaging. To locate an object on the plano-convex GC lens for making its image is not only easier than doing the same thing on the inner surface of the original hyperlens (the concentric structure), but the former also improves magnifying power to about 5, much better than the latter. We have also discussed the polarization effects on the image features of the 3D GC lens structure. In the real applications, the image recognition ability is an important issue. The manipulation of the source polarizations can help us to obtain better imaging results. This kind of new devices is possible to be fabricated by using ALD technique and may have great potential utilities in biological or nano-material related researches. References [1] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [2] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184. [3] J.T. Shen, P.M. Platzman, Appl. Phys. Lett. 80 (2002) 3286. [4] N. Lagarkov, V.N. Kissel, Phys. Rev. Lett. 92 (2004) 077401. [5] S.A. Ramakrishna, Rep. Prog. Phys 68 (2005) 449. [6] D. Melville, R. Blaikie, Opt. Express 13 (2005) 2127. [7] D.O. S. Melville, R.J. Blaikie, J. Opt. Soc. Am. B: Opt. Phys. 23 (2006) 461.
[8] A. Salandrino, N. Engheta, Phys. Rev. B: Condens. Matter 74 (2006) 075103. [9] Z. Liu, S. Durant, H. Lee, Y. Pikus, Y. Xiong, C. Sun, X. Zhang, Opt. Express 15 (2007) 6947. [10] N. Fang, H. Lee, C. Sun, X. Zhang, Science 308 (2005) 534. [11] S. Kawata, Y. Inouye, P. Verma, Nat. Photonics 3 (2009) 388394. [12] X. Zhang, Z. Liu, Nat. Mater. 7 (2008) 435441. [13] Z. Jacob, L.V. Alekseyev, E. Narimanov, Opt. Express 14 (2006) 8247. [14] Z. Liu, H. Lee, Y. Xiong, C. Sun, X. Zhang, Science 315 (2007) 1686. [15] Ivan Iorsh Alexander Poddubny, Pavel Belov, Yuri Kivshar, Nat. Photonics 7 (2013) 948957. [16] Takahiro Minato Zhigang Zang, Paolo Navaretti, Yasuhiro Hinokuma, Marcus Duelk, Christian Velez, Kiichi Hamamoto, IEEE Photonics Technol. Lett. 22 (2010) 721. [17] Keisuke Mukai Zhigang Zang, Paolo Navaretti, Marcus Duelk, Christian Velez, Kiichi Hamamoto, IEICE Trans. Electron.E94-C (2011) 862. [18] Dachun Gan Yanhui Zhao, Jianhua Cui, Changtao Wang, Chunlei Du, Xiangang Luo, Opt. Express 16 (2008) 5697. [19] Xing Hui Wang Wei, Fang Liang, Liu Yao, Ma Junxian, Lin Lan, Wang Changtao, Luo Xiangang, Opt. Express 16 (2008) 21142. [20] Chen Lin, Wang Guo Ping, Opt. Express 17 (2009) 3903. [21] S.A. Ramakrishna, J.B. Pendry, M.C. K. Wiltshire, W.J. Stewart, J. Mod. Opt. 50 (2003) 1419. [22] H.C. Shin, S.H. Fan, Appl. Phys. Lett. 89 (2006) 151102. [23] B. Wood, J.B. Pendry, D.P. Tsai, Phys. Rev. B: Condens. Matter 74 (2006) 115116. [24] K.J. Webb, M. Yang, Opt. Lett. 31 (2006) 2130. [25] Amnon Yariv, Pochi Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley-Interscience, New York, 2002. [26] Aleksandar D. Rakic, Aleksandra B. Djuriic, Jovan M. Elazar, Marian L. Majewski, Appl. Opt. 37 (1998) 5271. [27] A. Syntjoki, T. Alasaarela, A. Khanna, A. Tervonen, S. Honkanen, Proc. SPIE 7606 (2010) 1 (76061G). [28] T. Alasaarela, T. Saastamoinen, J. Hiltunen, A. Syntjoki, A. Tervonen, P. Stenberg, M. Kuittinen, S. Honkanen, Appl. Opt. 49 (2010) 4321. [29] H. Kim, H.B. R. Lee, W.J. Maeng, Thin Solid Films 517 (8) (2009) 2563. [30] R.G. Gordon, D. Hausmann, E. Kim, et al., Chem. Vap. Deposition 9 (2) (2003) 73.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Gradient–curvature nanolens for nano-imaging Kao-Der Chang a, Yu-Ming Wang a, Pi-Gang Luan b,n a b
Mechanical and Systems Research Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan Wave Engineering Laboratory, Department of Optics and Photonics, National Central University, Chungli 32054, Taiwan
art ic l e i nf o
a b s t r a c t
Article history: Received 27 March 2014 Received in revised form 18 June 2014 Accepted 29 June 2014 Available online 10 July 2014
In this work, we study the sub-wavelength imaging properties of a plano-convex nano-lens consisted of alternatively arranged dielectric and metallic thin layers. The thickness of one layer is not uniform, and the curvatures of the layer boundaries vary gradually from layer to layer. The image resolution of this nanolens at wavelength 365 nm is about 40 nm, far below the diffraction limit, and its image magnification is much better than that of the original proposed hyperlens of the same size. We also study the imaging properties of the corresponding 3D structures. Numerical simulations reveal that the image recognition ability of the 3D nanolens can be controlled by changing the polarization of the source waves. Devices of this kind may find applications in biological morphology and nano-structured materials researches. & 2014 Elsevier B.V. All rights reserved.
Keywords: Hyperlens Multilayers Metamaterials
1. Introduction Traditional optical devices such like optical microscope are important in many branches of scientific research, for example, biological morphology, pharmaceutics research, microelectronics, and mineralogy. However, their resolution ability is restricted by the diffraction limit, which means the fine features of an object much smaller than one wavelength cannot be resolved using these devices. This limitation is mainly caused by the fact that the subwavelength information of the object encoded in the evanescent waves is lost once they leave the tiny object. Since 2000, this seemed unconquerable restriction has become conquerable, at least in principle, if the traditional devices are replaced by the metamaterial devices such like superlens. According to Pendry's theory, a perfect lens is a slab of metamaterial having negative permittivity and negative permeability, both close to 1 (for the reason of impedance matching). Such a slab also has negative refractive index close to 1, and it not only can compensate the phases of propagating waves of the source but also can recover the amplitudes of the evanescent waves. These unusual properties help to make an almost perfect image of the sub-wavelength sized object, breaking the diffraction limit [1]. Recent developments of metamaterial research indicate that such a slab can indeed be realized by using properly designed structures such as that consisting of parallel conducting wires together with split-ring resonators (SRRs) [2].
n
Corresponding author. Tel: þ 886 3 4227151x65267, fax: þ886 3 4252897. E-mail address: [email protected] (P.-G. Luan).
http://dx.doi.org/10.1016/j.optcom.2014.06.071 0030-4018/& 2014 Elsevier B.V. All rights reserved.
A lot of metamaterial structures based on the original idea of perfect lens have been proposed and tested, and high resolution imaging beyond diffraction limit has indeed been achieved [3,4]. However, in the optical frequency range, it is very difficult to realize negative permeability even in principle. A number of theoretical and experimental works concerning this issue have been done [5–8]. If our purpose is sub-wavelength resolution of nano-object and do not care about whether the image is located at the far field zone, then it can also be achieved by using a thin slab of metal which has negative permittivity only. The most prominent example is a slab of sliver in which surface plasmon modes can be excited due to its negative permittivity [9–12]. However, the material thickness and loss restrict the imaging quality of such a ‘superlens’. In addition, the image formed by a superlens is often of the same size as the object without magnification and it is restricted in the near field zone. An ideal optical imaging system should overcome not only the diffraction barrier but also be compatible with conventional optical system so that post-processing can be easily implemented. The recently proposed hyperlens structure can satisfy this requirement [13,14]. A hyperlens is a cylindrical multilayer structure consisting of alternatively arranged negative permittivity (metallic) and positive permittivity (dielectric) layers. Under the effective medium limit (which means the operating wavelength is much larger than the layer thickness), it leads to anisotropic dielectric tensor of the structure, which means the principal permittivity along the radial and azimuthal directions have opposite signs. This property leads to a hyperbolic dispersion relation in cylindrical coordinates, so it converts the evanescent waves into propagating ones, making some new applications possible [15–17].
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In a hyperlens, the streamlines of the energy flow direct mainly along the radial directions. Usually two sources close to each other with distance smaller than half wavelength are not recognizable from their images if usual lens is used. However, if we put these two sources on the inner surface of the hyperlens, two distinguishable magnified images can be found in the far field zone outside the hyperlens. The image is magnified and the magnifying power (MP) is proportional to the ratio between the outer and inner radii. Although hyperlens magnifies subwavelength objects and makes images in the far field zone if the ratio of outer-to-inner radius is appropriately chosen, the curved inner surfaces brings inconveniences in locating objects on the object surface. For this reason, other methods for obtaining breakingdiffraction-limited resolution such as the oblique lens [18], the planar hyperlens [19] and the pyramid-shaped hyperlens [20] were proposed. The working principle of most of these structures is based on the fact that the energy flows can be guided along certain directions as desired. It reveals that the electromagnetic waves can be controlled under some implementable conditions (canalization regime) if we design the structure appropriately [21–24]. Inspired by these research results, we propose in this paper a gradient curvature (GC) structure of a new plano-convex lens, working near the canalization regime. Our design is similar to that of [19] but we use simpler method to define the layer thickness, which is desirable for fabrication purpose. In addition, we also study the imaging characteristics of the generalized 3D structure, which is lacking in [19]. The simulation results show that the resolution down to 40 nm can be achieved and the magnifying power is more than 5 nm at 365 nm wavelength, much better than the original hyperlens of the same size can do. In addition, we also demonstrate that the corresponding 3D GC lens structure can also work very well, and its imaging quality can be controlled by the source polarization.
2. Theory and models The working principle of the proposed nanolens is based on the highly anisotropic property of the effective permittivity, which can be realized by using multilayered periodic structure made of alternatively arranged dielectric and metallic layers. One period of the structure consists of one dielectric layer of permittivity εd and thickness dd, together with one metallic layer of permittivity εm and thickness dm. The dispersion relation for the propagating mode in this structure is given by [25]. cos ðK d Þ ¼ cos ðK m dm Þ cos ðK d dd Þ
respectively. It is known that if the working frequency ω and thickness ratio dd/dm are chosen appropriately, we can make Re(εt) 40 and Re(εn) o0. This leads to a dispersion curve of hyperbolic type (see Eq. (2)), which implies that an evanescent wave outside can be transformed to a propagating wave inside. This is the underlying working principle for the (original) hyperlens. As mentioned in the introduction, the GC lens structure works near the canalization regime [21–23], which means we choose to design the structure having very large principal permittivity value along the normal direction of the layers (i.e., εn-1, assuming that the imaginary parts of εd and εm are both small enough). According to Eq. (3), this implies that the following condition must be satisfied: dd Reðεm Þ þ dm Reðεd Þ ¼ 0
ð4Þ
For simplicity we set (dd ¼ dm), thus we must choose an appropriate frequency such that Re(εm)E Re(εd). Under this condition the dispersion curve at one single frequency becomes very flat along the direction parallel to the layers, and the streamlines of the energy flow go along almost the same direction, normal to the dispersion curve. This property can be used to design a new kind of nanolens for making images of nano-scale objects by bending the flat multilayer structure in a new way. The information encoded in the propagating and evanescent waves from a tiny object can thus be transferred to the output surface of the nanolens along trajectories locally normal to each dielectric–metal interface. Hereafter we name the nanolens gradient–curvature (GC) lens, which is made of multilayer dielectric–metal films having different curvature in each layer. The proposed GC lens structure is formed by cutting a part of the elliptical shaped reference structure (inside the dashed-line rectangle), as indicated by the left part of Fig. 1. The downward A-axis and the horizontal B-axis define the two axes of the ellipse, and their ratio B/A determines the shape of the reference structure and the curvature of each layer. In the limiting case of A ¼B, the shape of the reference structure becomes circular. The aperture width C of the lens is shorter than the waist width 2B of the ellipse but long enough to make the lens structure work. We assume in this paper that each layer has a fixed local thickness of 10 nm along the A-axis, but the layer thickness is not uniform and it shrinks to zero as we move sideward along the B-axis direction. Specifically, the lower boundary of the nth layer (counting downward) is defined by the equation of the ellipse
y2 x2 þ ð10nÞ 2 B2
¼ 1, here
nanometer is adopted as the length unit. For simplicity, we first consider the 2D structure. The dashed-line rectangle in the right
1 km εd kd εm þ sin ðkm dm Þ sin ðkd dd Þ 2 kd εm km εd
ð1Þ where d ¼dd þ dm is the period of the structure, kd and km are the normal (the direction perpendicular to the layers) components of the wave vectors in the dielectric and metallic layers, and K is the Bloch wave number of the mode. When the wavelength is much longer than the period of this structure, an effective dispersion can be extracted: 2
2
k n k t ω2 þ ¼ εt εn c 2
ð2Þ
where εt ¼
dd εd þ dm εm ; d
εn 1 ¼
1 dd εd 1 þ dm εm d
ð3Þ
are the two principal values of the effective permittivity tensor, ω is the angular frequency of the wave, c is the speed of light in vacuum, and kt and kn ¼ K are the tangential and normal components of the wave vector of the plane wave propagating inside,
Fig. 1. The gradient curvature (GC) lens structure made of multilayer dielectric– metal films having different curvature in each layer. Left: The elliptical shaped reference structure defined by the two axes A and B. Right: The lens is formed by cutting a part of the reference structure with aperture width C. The dashed-line rectangle represents the working area of the lens. Each layer has a fixed local thickness of 10 nm along the A-axis and shrinks to zero sidewardly.
K.-D. Chang et al. / Optics Communications 332 (2014) 103–108
part of Fig. 1 represents the working area of the GC lens, and the top and bottom edges indicate the input (source) plane and output (image) plane, respectively. In our simulation, the permittivity of the metallic layer is approximated by Drude–Lorentz model with plasma frequency 9.01 (eV) and a damping constant 0.048 (eV) [26]. Assuming the working wavelength to be 365 nm, the permittivities of the dielectric and metallic layers are εd ¼ 4.907 and εm ¼ 4.9438þj0.084, satisfying the canalization condition (4) approximately. The structure is embedded in glass and the input side is exposed to water. To study the performance of our device, we adopt COMSOL Multiphysics 3.5 (FEM) software to simulate the imaging characteristics of the GC lens structure. In these simulations, the light sources are located on the flat surface of the GC lens with linearly polarizations. Perfectly matched layers (PML) are used for reducing reflections. The output surface is located at the tangential plane of the GC lens. Based on these simulation results, we can discuss the relations between the image magnification and the geometric parameters used.
3. Numerical results and discussion To test the effect of the axes ratio B/A, we put two point sources at the input plane and record the image–image distance (IID) for every possible B/A ratio from 0.5 to 1.5. The IID curves for A¼ 600 nm, 800 nm and 1000 nm are plotted in Fig. 2, the source–source distance (SSD) is set to be 100 nm. As expected, the light waves from the sources can be guided along the directions locally normal to each of the dielectric–metal interfaces of the multilayer structure. The IID measured at the output plane is longer than the SSD, which confirms that the lens structure magnifies the input object. For A¼600 nm (60 layers) and B/A¼ 0.5, the IID is about 532 nm, corresponding to a magnifying power (MP) exceeds 5. According to these results, if the length of A is fixed, choosing lower B/A ratio gives larger MP. On the other hand, if the B/A ratio is fixed and low enough (less than 0.6), a larger A will give a larger IID. For example, for B/A ¼0.5, the IIDs for A¼600 nm, 800 nm and 1000 nm are 532 nm, 578 nm, and 620 nm, respectively. This tendency can be easily explained by the fact that for a larger CG lens the penetrating lights from the two sources propagate longer distances inside the structure, helping to separate the two images further. When the B/A ratio exceeds 0.7, the three curves merge into each other, and the IID becomes independent of the B/A ratio.
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To make things more concrete, we set A ¼600 nm and compute the electromagnetic fields for three cases: B/A ¼0.5, 1 and 1:5. The field patterns (the intensity of the H field) and the imaging strengths are shown in Fig. 3(a1)–(a3) and (b1)–(b3), respectively. As mentioned before, the low ratio case B/A¼0.5 makes two separate images and the MP is large. However, we also find that for the high B/A ratio cases, although the two sources can indeed make distinct images on the output plane, the two images will merge into a single image if the imaging results are not evaluated at a position close enough to the output plane indicated in Fig. 1. This observation indicates that GC lens is not a different kind of hyperlens for imaging in the far-field zone, but is designed for the purpose of making distinct images in a region smaller than the hyperlens can do. These results also indicate that both the shape and size of the GC lens can affect the MP. Recalling that in the original hyperlens design, the imaging resolution is controlled by the ratio between the outer radius and inner radius of the structure. Our structure provides more free parameters to control the imaging properties. Based on the simulation results, we confirm that it is feasible to design the GC lens proposed in this paper, and its MP can be manipulated under practical conditions. We now compare the magnification ability of the proposed GC lens to that of the original hyperlens, assuming that they have the same aperture size and the same number of layers. The IIDs as functions of SSD for various values of the A-thickness of these two lens structures are plotted as the unmarked curves and marked curves in Fig. 4(a), respectively. The field patterns for the 60-layer structures of GC lens and hyperlens are shown in Fig. 4(b1) and (b2). For all the simulations of the GC lens in Fig. 4, we set B/ A¼0.55. For the original hyperlens (concentric structure), different radius (A) yields the same magnifying power. However, for the GC lenses of fixed axes ratio (B/A ¼0.55), different A show different magnifying ability. As mentioned before, the imaging magnification of the GC lens depends on its A-length. Furthermore, when the aperture size (C) is the same, the GC lens has larger magnifying power than the original hyperlens. Giving SSD ¼100 nm, A¼600 nm, the field distributions for these two structures are plotted in Fig. 4(b1)–(b2). According to these results, both GC lens and the original hyperlens can magnify the objects and form enlarged images at the output plane. However, there are some important distinctions between their imaging behaviors. For example, although the two sources can be imaged at the output plane by the hyperlens, they will merge into each other and form a
Fig. 2. The image–image distance (IID) as functions of the axes ratio B/A for three chosen cases: A ¼ 600 nm, 800 nm, and 1000 nm. The source–source distance (SSD) is fixed as 100 nm.
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Fig. 3. Field strengths and image strengths for the 60-layer (A ¼ 600 nm) GC lenses with B/A ¼ 0.5, 1, and 1.5. Subplots (a1)–(a3) are the field strengths of these three cases, and (b1)–(b3) are the corresponding image strengths on the image plane.
Fig. 4. The magnifying powers and the field distributions for the GC lens and the original hyperlens of the same size (the same A-thickness and the same C-aperture width). (a) The horizontal axis is the source–source distance (SSD) of the two sources on the input plane, whereas the vertical axis is the image–image distance (IID) on the output plane. The distance between the two sources is from 40 nm to 120 nm. The IIDs as the functions of SSD for GC lens and hyperlens are plotted as the unmarked curves and marked curves, respectively. (b1) The field distribution for the A ¼ 60 nm GC lens with B/A¼ 0.55. (b2) The field distribution for the hyperlens with the same thickness and the same aperture width as that of (b1).
single image once leave the output plane. This is caused by the fact that for a hyperlens the farfield imaging ability can be achieved only when the outer-to-inner radius ratio is large enough. The radial propagation reduces the azimuthal component of the wave vector. Thus if the out radius is large enough, the wave vector for the image fields would not get imaginary radial component and can propagate to the far field zone. If we restrict the size of
hyperlens tightly, the propagating waves inside the hyperlens can only become evanescent waves in the output side, and the images can only appear in the near field zone. In contrast, the GC lens can resolve the two images even behind the output plane because two images are separated. This fact reveals that the GC lens has better magnification ability than the original hyperlens of the same size. Such difference disappears at high axes ratio (B/A4 0.7) and the
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Fig. 5. Images of the two magnetic sources with SSD ¼ 40 nm, 60 nm, and 80 nm using the 3D GC lens structures. Subplots (a1)–(a3) are the field strengths of images for the X-polarized sources, whereas (b1)–(b3) are the field strengths for the Y-polarized sources.
imaging behaviors of the GC lens become similar to that of the original hyperlens. Now we turn to the discussion of the imaging behaviours of the corresponding 3D GC lens. The lens is defined as the solid of revolution of the 2D GC lens structure (the rectangle region of the left hand part of Fig. 1) with respect to the A-axis, keeping the material and geometrical parameters unaltered. In the following numerical simulations, two sources with spacing (SSD) from 40 nm to 80 nm are considered, and A¼600 nm, B/A¼0.55 are chosen, which corresponds to the 2D cases having the better performance. First, we set the two sources to have X-polarized magnetic fields. The field distributions of the images for the cases of SSD¼40 nm, 60 nm, and 80 nm are shown in Fig. 5(a1)–(a3). According to theses simulations, the two image spots are indistinguishable if the source spacing is smaller than 40 nm, and the image spots become clearly distinguishable if the source spacing is longer than 60 nm. The IIDs are 302 nm and 405 nm when the SSDs are 60 nm and 80 nm, respectively, which means this 3D GC lens gives MPE5, similar to that of the 2D structure. In addition, we observe that the images of the sources are slightly deformed along the polarization direction of the source fields (the X-direction). Next, we consider the sources having Y-polarized magnetic fields. The imaging results are shown in Fig. 5(b1)–(b3). These simulations reveal that using Y-polarized magnetic sources the two image spots are clearly distinguishable even if we reduce the SSD to 40 nm. The IIDs are 202 nm, 304 nm, and 408 nm for SSD¼40 nm, 60 nm and 80 nm, respectively. Thus for the Y-polarized sources the MP is still close to 5. In addition, the image deformation phenomenon still exists, but now the deformation is along the Y-direction. Since the imaging features for the X- and Y-polarized magnetic sources are different, we conclude that we can modify the image patterns by changing the source polarization. The above mentioned phenomena may probably be explained by considering the phase features of the source fields. Comparing the two kinds of polarization states, they give similar MP when the source spacing is above 60 nm. However, for shorter spacing
(e.g., SSD ¼40 nm), the Y-polarized sources give better imaging results than the X-polarized sources. This might be explained by the fact that the phases of the source fields (the E fields) are opposite in the central line between the two sources when the sources have Y-polarized magnetic fields. In other words, the distinguishable double-image might be caused by the destructive interference of the source fields. However, the detailed analysis of this interference phenomenon and the confirmation of the above mentioned explanation are too complicated and beyond the scope of the present work. We only know at this stage that different polarization state of the sources provides different interference pattern and thus leads to different image recognition ability. In a real imaging system, the image of an arbitrary shaped object should be recognizable. We test this by simulating the imaging features of an N shaped slit illuminated by source fields of chosen polarizations. Fig. 6 shows the imaging patterns of the N font for three different (X-, Y-, and XY-) polarizations of the magnetic source. The simulation results reveal that the 3D CG lens can identify the shape of the font if the characteristic size of the object is larger than 40 nm. Besides, for the Y- and XY- polarized sources, the two vertical edges of the font N can be better observed, whereas for the X-polarized source, only the inclined edge in between can be easily observed. The recognition of the lens is thus dependent on the polarization state of the source. Therefore, we can average the results of different polarizations or utilize un-polarized source to get a better imaging result of the subwavelength sized object. The proposed 2D and 3D GC lens structures in this paper can be fabricated using the Atomic Layer Deposition (ALD) technique, which is a chemical vapor deposition technique suitable for manufacturing inorganic material layers with thickness down to nanoscale [27,28]. Choosing the reactive precursors, temperature gradients and gas flow inside the reactor appropriately, it is possible to control the uniformity and non-uniformity of the layer thickness [29,30].
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Fig. 6. Images of an N-shaped object. The field distributions on the image plane for three different polarizations (X-polarization, Y-polarization, and the XY-polarization) are shown.
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