Zubin Jacob, Leonid V. Alekseyev and Evgenii Narimanov. Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA. Abstract: We ...
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Semiclassical theory of the Hyperlens Zubin Jacob, Leonid V. Alekseyev and Evgenii Narimanov Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
Abstract: We study ray dynamics inside the Hyperlens, a device recently demonstrated as capable of sub-diffraction-limited far-field imaging.The obtained semiclassical result of spiraling rays is confirmed by numerical simulations of gaussian beam scattering from the hyperlens. ©2006 Optical Society of America OCIS codes: (110.0180) Microscopy; (160.1190) Anisotropic optical materials; (080.2720) Geometrical optics, mathematical methods
A conventional lens cannot construct the image of an object with resolution better than λ/2, where λ is the wavelength of the illuminating wave. This is because the conventional lens brings to focus only the propagating waves emanating from the object while the evanescent waves decay away exponentially and are lost in the far field. Such an apparent loss of information gives rise to the diffraction limit of λ/2. It is therefore highly desirable for many practical applications, such as biological imaging, to be able to construct images with feature sizes much below the wavelength of the illuminating light [1]. Recently it was proposed that a hollow core cylinder or half-cylinder made of materials with a strongly anisotropic dielectric response [2,3] can function as a far-field imaging device capable of resolution beyond the diffraction limit [4]. The object is placed inside the hollow core and the magnified image is projected into the farfield which can be processed by conventional optics. The local wavelength at the core of the device, where the object to be imaged is placed, is below the free space wavelength and this leads to the subwavelength resolution[4,5]. The hyperbolic dispersion in strongly anisotropic materials (materials with dielectric permitivitties of opposite signs in two perpendicular directions) is the key to achieving the wavelength compression in the Optical Hyperlens. The advantages of semiclassical approach are two-fold. First, the connection to the underlying ray optics uncovers the physical origin of light propagation and imaging in the device. Second, as opposed to ‘brute-force’ numerical methods that are computationally intensive and suffer from instabilities when treating evanescent fields, the semiclassical approach while quantitatively accurate, is both numerically inexpensive and stable. The accuracy of the semiclassical approach in the hyperlens is due to the wavelength compression in this device. As the light approaches the core of the hyperlens, due to the hyperbolic dispersion relation the radial and tangential momentum increase, leading to substantial suppression of wavelength.
Fig. 1 (a) Schematic of the hyperlens (hollow inner core) with alternating layers of metal and dielectric to achieve εr0 (b) path of 2 rays (shown in red) with different impact parameters impinging from vacuum onto a homogeneous medium with cylindrical anisotropy εr = -1 and εθ = 1 calculated using eq. (1) (c) path of the ray calculated using eq. (1) for εr = -0.01 and εθ = 1. Note that the rays spiral towards the center.
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Fig. 2(a) Schematic of a gaussian beam (blue) scattering from the hyperlens (top view) with N alternating layers of metal (copper colour) and dielectric (black). The center of the incident gaussian beam moves along the dotted line and the impact parameter is p. (b) Absolute value of the field shown in false color for a gaussian beam scattering from the hyperlens with parameters p ~2.4λ, rmin ~ λ, rmax ~ 7 λ, h ~ λ/100, N=600, εm ≈ -0.4, εd ≈ 2.4. The inner and outer boundaries of the hyperlens are shown in red. The ray trajectory shown in black is calculated using eq. (1) and specular reflection at the inner boundary. Note that the center of the beam moves along the calculated ray trajectory.
In the ray-optical approximation the effective Hamiltonian is H eff = c
pr2
εθ
+
pθ2
r 2ε r
. For fixed values of εr
and εθ , Hamilton’s equations of motion can be solved analytically, leading to
r (ϕ ) =
p / εr
ε sinh( r (ϕ − ϕ0 )) εθ
(1)
where p is the impact parameter of a ray impinging on the hyperlens from vacuum and φ0 is a constant related to the impact point on the hyperlens (Fig. 1(b) and (c)). This equation describes a spiral trajectory. To confirm the accuracy of this approach we compare it to exact solutions of Maxwell’s equations obtained numerically. We consider a metamaterial realization of the hyperlens in the cylinder geometry, consisting of alternating layers of metal and dielectric such that the layer thickness h, is much below the operating wavelength (Fig 1(a)). This layered structure can be described as an effective medium yielding the desired dielectric response with cylindrical anisotropy ε θ = (ε 1 + ε 2 ) / 2; ε r = 2ε1ε 2 /(ε 1 + ε 2 ) when h
