Magnifying perfect lens and superlens design by coordinate transformation

The coordinate transformation technique is applied to the design of perfect lenses and superlenses. In particular, anisotropic metamaterials that magnify two-dimensional planar images beyond the diffraction limit are designed by the use of oblate spheroidal coordinates. The oblate spheroidal perfect lens or superlens can naturally be used in reverse for lithography of planar subwavelength patterns.

Figure 1

(Color online) Electromagnetic fields $\mathbf{E}$ and $\mathbf{H}$ in a physical medium with material constants $\epsilon$ and $\mu$ can be transformed to normalized fields $\stackrel{\mathrm{̃}}{\mathbf{E}}$ and $\stackrel{\mathrm{̃}}{\mathbf{H}}$ that also obey the Maxwell equations, but regard [$u(x,y,z)$, $v(x,y,z)$, $w(x,y,z)$] as Cartesian coordinates in an effective medium with material constants $\tilde{\epsilon }$ and $\tilde{\mu }$.

Figure 2

(Color online) The procedure of perfect lens design by the coordinate transformation technique. First a curved surface $w=C$ in the virtual space (top left) is transformed into a plane (top right). The plane is then mapped onto a slab (bottom right), which is subsequently transformed back to the desired geometry in the physical space (bottom left). The electromagnetic fields propagate through the transformation medium with material constants ${\epsilon }^{\prime }$ and ${\mu }^{\prime }$ in the physical space as if they propagate within an infinitesimal slab in the virtual space. The fields from a point source on the $w^{\prime }=a$ surface propagate like a ray, depicted by yellow arrows, along a $w^{\prime }$ coordinate line inside the transformation medium.

Figure 3

(Color online) Effective anisotropic metamaterial formed by thin films (left) and two possible realizations of the anisotropic perfect lens (center and right). The perfect lens can be formed by pairing positive-refractive-index films with negative-refractive-index films of the same thickness (center), as suggested by Veselago (Ref. 3) and Pendry (Ref. 4), or pairing negative-$\epsilon$ films with negative-$\mu$ films (right), as suggested by Alù and Engheta (Ref. 19).

Figure 4

(Color online) The spherical magnifying perfect lens (center), the corresponding virtual space (left), and the metamaterial implementation of the lens (right). The corresponding regions in the virtual space and the physical space are marked by the same colors in the left and center figures. For clarity, only the $y=0$ cross section is shown. The electromagnetic fields on the inner spherical surface are perfectly mapped onto the outer surface by the lens, enabling far-field detection of subwavelength information. In practice, one can use just half of the spherical lens, so that the object can be placed against the inner spherical surface more conveniently.

Figure 5

(Color online) The oblate spheroidal coordinate system for $0⩽u⩽3\pi ∕2$ and $0⩽v⩽\pi ∕2$. Black lines are coordinate lines, and each color denotes a region between two constant-$w$ surfaces.

Figure 6

(Color online) Sketches of the oblate spheroidal perfect lens (center), the corresponding virtual space (left), and the metamaterial implementation of the lens (right). For clarity, only the $y^{\prime }=0$ cross sections are drawn. The structure is symmetric with respect to rotation about the $z^{\prime }$ axis. Rays propagate along the $w^{\prime }$ coordinate lines and follow hyperbolic trajectories. In practice, one can use just half of the spheroidal lens $(z^{\prime }>0)$ and put the object against the $w^{\prime }=0$ plane.

Figure 7

(Color online) A stack of slanted negative-index thin films can continuously redirect a ray with respect to the normal direction of each interface.

Figure 8

(Color online) TM waves cannot be coupled into a metamaterial slab with zero transverse permittivity, except when the plasmon resonance condition is met because the TM waves inside the metamaterial are actually LE waves in most cases and have a zero magnetic field. By reciprocity, the LE waves also cannot be coupled into TM waves in free space.