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Imaging with multilayer hyperbolic metamaterials – what are the limits? Tengfei Li1, Vivek Nagal2, David Gracias2 and Jacob B. Khurgin1
1
2
Johns Hopkins University, Department of Electrical and Computer Engineering, Baltimore MD 21218 USA Johns Hopkins University, Department of Chemical and Biomolecular Engineering, Baltimore MD 21218 USA Author e-mail address:
[email protected]
Abstract: Using the Eigen-mode approach we analyze the imaging performance of multilayer hyperbolic metamaterials and show that resolution decreases with the number of layers and amount of metal loss. OCIS codes: (240.6680) Surface plasmons; (110.0110) Imaging systems; (160.3918) Metamaterials.
Conventional optical imaging system resolution is limited to about half of wavelength, due to the lack of high spatial frequency evanescent components, which contain the fine information of object. Among the multitude of attempts to overcome this limit, the one that stands out is the “superlens” incorporating negative index materials first proposed by Pendry in 2000 [1]. While negative index materials in optical range have not become practical, a simplified superlens for one polarization had been realized by using silver slab with negative real part of dielectric permittivity [2] and reaching resolution of about one-sixth of the illumination wavelength [2]. A logical extension of superlens had been the “hyperlens” based on multilayer metal/dielectric structure in cylindrical geometry [3]. Using a multilayer structure (inset of Fig.1a) allows one to increase the distance between the object and image (an important practical consideration) and “rolling” it into the cylinder adiabatically transfers evanescent wave into the propagating one and thus providing magnification. Obviously, in the presence of inevitable metal loss, the overall amount of light passing through the structure decreases, but it is not clear, at least from the existing theoretical descriptions, whether the resolution itself is adversely affected by the increasing number of metal layer. In this work, we address this issue by applying the “eigenmode” model [4] previously developed for a single layer superlens. This model offers a simple and physically transparent explanation of the “superlensing” phenomenon as coupling of the light into surface plasmon polaritons (SPPs) propagating inside metal slab and having large spatial frequencies (k-vectors). Extending the eigen-mode approach to the multilayer structure (Fig.1a) we arrive to the conclusion that increasing the number of layers causes decrease in both amplitude and spatial bandwidth of the Optical Transfer Function (OTF) and provide a clear physical reason for this fall-off in resolution. This conclusion is then confirmed by the exact numerical modeling using transfer matrix (TM) method. As shown in the inset of Fig.1a, according to the eigenmode model the object can be expressed as a sum of the sources (dipoles) with spatial frequencies k x oscillating at the emission frequency , p z , x, t
p( k
x
)eikx x dk x z z0 e it , where z0 is the object distance. The components with | kx | k0 nr / c are
subwavelength and in free space would couple into the rapidly decaying evanescent waves, but placed close to the “hyperlens” they are coupled into the eigenmodes of the hyperbolic structure with normalized field distributions f kn ( z ) and eigen frequencies n ,k x . The total number of these SPPs eigenmodes is equal to N 1 where N is the x
number of metal layers. The coupling strength into n-th SPPs mode is Ckn 2 n2, k 2 j n, k x
x
x
1
pkx f knx z0 ,
where n ,k x is the energy loss rate in the mode, hence the total field in the image space can be expressed as: E ( x, zi )
Nl 1
kx k0
n 1
2 n , kx
2 f kn z0 f kn zi pk eik x dk x Ek zi eik x dk x 2 j n, k k k x
x
x
x
x
x
x
x
(1)
0
where the first term is the contribution of the sub-wavelength confined SPPs modes, and the second term is the contribution of propagation waves, which plays no role in defining the resolution of the lens. Obviously, the term inside the square brackets is the value of OTF for each lateral wavevector k x . The OTF for a typical “flat” hyperbolic lens consisting of N alternating 15 nm thick metal (Ag) and dielectric (Al2O3) layers has been calculated for the wavelength of 600nm, using both eigenmode and TM approaches with similar results, shown in Fig.1a. One can see the sharp resonances associated with SPPs modes whose magnitude drops as the spatial frequency kx increases. The TM method, just as any numerical techniques, sheds no light at the origin of the rapid fall-off, but this origin is immediately elucidated once one looks at the SPPs eigenmode dispersion curves shown in Fig.1b. First, the two closely spaced curves with the smallest spatial frequencies on the left belong to two “surface” SPPs modes extending into the space outside of the layered structure and this having
FTh4H.6.pdf
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large absolute values of f kn z0 f kn zi in Eq. 1. These modes, labeled “1” and “2” in the Fig.1b are responsible for the strongest peak in the OTF in Fig.1a. Since one of these modes is symmetric and the other is anti-symmetric their contributions to the OTF have opposite signs. As the number of layers increases, the eigen frequencies of these two modes become degenerate and the cancellation ensues, causing rapid fall-off of the main peak. Note that the odd and even modes in our structure have opposite symmetries and their contributions cancel each other as the number of layers increases the curves in Fig.1b get more and more dense, hence cancellation gets progressively stronger, especially at the large values of kx. As a result, resolution is expected to decrease with increase of N. This effect can be gauged by first normalizing the OTF curve to its value at the second (broadest) peak and the introducing the cut-off spatial frequency kcut off such that | OTF (kcut off ) | 0.1 | and then plotting kcut off as a function of
x
x
the number of layers, as shown in the Fig. 1c, for the different values of object and image distances. After sharp jump for N=2 associated with the merging of the first two modes, the curves show that the resolution of the lens experiences a steady decrease with the number of layers.
Fig. 1 (a) Absolute value of OTF for N=1,5,9 metal slabs; (b) Dispersion curves of eigen modes for N=5, sp is the surface plasmon frequency; (c) Cutoff lateral wavevector kcut off versus the number of metal layers for three different object and image distances. The wavevectors are nomalized to the wavevector in dielectric k0 nr c .
Fig. 2 (a) PSF for the N 4 hyperlens with two different values of metal loss; (b) Comparison of PSF for two different numbers of layers N 4 and N 5 .
Obviously, the cancellation increases with loss as the resonances get broadened. To study the specific effect of the loss and number of metal slabs, we have calculated the point spread function (PSF). Fig. 2a shows the PSF when the metal damping is chosen 1 2.02 1014 s1 and 2 1 2 respectively. Naturally, the larger loss adversely affects the resolution. Fig. 2b shows the PSF when the structure contains four and five metal slabs, we can see that due to the stronger cancellation, the resolution decreases with the increase of metal slabs number. The main conclusion of this work is that using layered metal dielectric (hyperbolic) structures to increase objectto-image distances, relative to a simple superlens, inevitably leads to the deterioration of resolving power in this structures. Even though only a flat hyperlens was considered in our work, the conclusions are applicable to the cylindrically shaped one. The authors acknowledge the support by NSF Grant # 1507749 and ARO Grant W911NF-15-1-0629 and indispensable help of Dr. P. Noir of JHU. References [1] J. B. Pendry. “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000). [2] Fang, N., Lee, H., Sun, C., and Zhang, X. “Sub–diffraction-limited optical imaging with a silver superlens,” Science, 308, 5721(2005). [3] Z. Jacob, L.V. Alekseyev, E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247 (2006). [4] B. Zhang, and J B. Khurgin. “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons,” Appl. Phys. Lett. 98, 263102 (2011).
