Introduction To Modern Planar Transmission Lines. Anand K. Verma

19 from the relation between the direct space period (Δ) and the propagation vector in the k‐space. The diffraction limit for a very wide lens aperture could be reduced to λ/2 [J.13].

      The DNG planar slab acts as a flat lens [J.2, J.13–J.15, J.22]. The DNG slab provides exponentially increasing evanescent waves inside the slab. So the DNG can compensate for the decaying evanescent waves that reach the image plane creating a high‐resolution image with finer details. It is shown in Fig (5.12c). It is also examined in the next subsection (5.5.6).

      5.5.6 DNG Flat Lens and Superlens

      The focal length (f) of a thin lens is related to its radius of curvature (R) by the expression f = R/(n − 1), where n is the refractive index of a lens. For n = +1, the lens does not refract, and the focusing of the EM‐wave at the image plane does not occur. However, for a DNG lens with n = −1, refraction occurs. Veselago [J.3] has shown that due to negative refraction in a DNG slab, even a slab acts as a flat lens. Pendry has further shown that such a lens is a perfect lens in the near‐field region as it enables recovery of the decaying evanescent field at the image plane [J.2]. Such recovery of the evanescent waves is not possible with a normal lens, irrespective of the size of its aperture. The perfect lens, also called the superlens, has a subwavelength resolution, breaking the diffraction limit barrier. The superlens creates an image in the near‐field very close to the lens, so it is difficult to use it in practice. However, anisotropic hyperbolic DNG lens creates a perfect image in the far‐field region, by converting the evanescent waves into the propagating waves. Such a lens is called the hyperlens. The proof of concept has been demonstrated experimentally for both the superlens and hyperlens in optical and microwave range. The proper functioning of these lenses is limited by the losses associated with a DNG medium. A brief theory of three lenses is presented in this section.

      Veselago Flat Lens

Figure (5.13) shows the ray diagram of a DNG slab acting as a lens due to the negative refraction [J.3]. It has two foci N and Q. The focal length f₁ of focus N is located inside the DNG slab of thickness d. The focal length f₂ of focus Q is located outside the slab. To avoid mismatch at the interface, the DNG slab has μ_r = ε_r = − 1 and . Figure (5.13) shows the angles of incidence θ_i and refraction θ_r and other geometrical dimensions. Snell's law for the DNG slab is written below:

(5.5.31)

      The first focal length f₁ is determined as follows:

(5.5.32)

      The incident ray OM and the output ray P₁ Q are parallel, i.e. θ_i = θ_o.

Figure 5.13 Ray diagram of a DNG flat lens.

The second focal length f₂ is computed as follows:

(5.5.33)

For d > f₁, equations (5.5.32c) and (5.5.33c) provide a relation between two focal lengths:

      (5.5.34)

Equation (5.5.31) shows a matched DNG slab has θ_i = θ_r. For this case, both focal lengths, and also distance D between the object and image at the second focus are given below:

      (5.5.35)

      Pendry Superlens Lens

In the case of a DNG‐based matched flat superlens, the image is reconstructed at the second focus by amplification of the evanescent electric fields. Thus, the DNG slab provides an opportunity to design subdiffraction lens, i.e. practically a lens without any diffraction limit. The refractive index of the DNG, matched to air medium, is . The longitudinal wavevector k_x, given by equation (5.5.28), is rewritten for the backward wave supporting DNG slab as follows [J.2, J.13, J.22]:

      (5.5.36)

The electric fields of both the propagating and evanescent waves, inside the DNG slab, are obtained using the above equations with equation (5.5.29a,b):

(5.5.37)

Equation (5.5.37a) shows that the propagating waves acquire the leading phase (β_xd) at the end of the DNG slab of thickness x = d. The evanescent waves are amplified exponentially by α_xd at the end of the slab. Figure (5.12c) shows such behavior of the DNG based superlens. At the output of the DNG slab, the wave is still evanescent waves in the near‐field region; although at the image plane its magnitude level is the same as that of at the object plane. So at the second focus, the image has high resolution beyond the diffraction limit. It is noted that the evanescent waves still propagate at the interface along the y‐axis in the (y − z)‐plane as the surface waves. The surface wave is bounded in the x‐direction. The amplification of the evanescent wave is also treated through the concept of coupled quasi‐particles called plasmon‐polariton, especially in the optical frequency range [J.13, B.11].

      HyperLens

The anisotropic DNG medium‐based hyperlens converts the evanescent waves to the propagating waves by reducing the values of the transverse wavenumber below the wavenumber in the medium. It is realized by the hyperbolic dispersion relation of the uniaxial anisotropic medium discussed in section (4.7.5) of chapter 4. Thus, a hyperlens projects the high‐resolution image in the far‐field region. It is shown in Fig (5.12d).