 CHAPTER 2 2.2

Near and far field lens

It is observed that the phase and intensity in the near field dramatically changes during propagation while its far field (Fourier image) is a fixed pattern which does not change during the propagation in free space. The propagation affects the phase and distribution in the near field due to the phase shift among different plane wave components (the far field). However, the distribution in the far field is unchanged. Note that the far field is only changed after passing through an object such as a lens or prism. For example, a prism has a linearly varying optical thickness in transverse direction given by d(x) ~ x which introduces in the transmitted beam a transversely varying phase with a linear slope A(x)′ = A(x)eix . The far field becomes A(kx)′ = A(kx-) and this effect corresponds to the well-known refraction phenomena. As the surface of the optical element is spherical, the angle of refraction at any positions depends on the local curvature which is how a conventional lens focuses. Therefore, the far field is changed from one to many different plane wave components with varying orientation. It should be emphasized that the flat lens 4 composed of two parallel interfaces cannot change angles of refraction or, equivalently, cannot change the far field distribution. In order to gain insight into the purpose of this work, the comparison between flat lens (near-field lens) and a conventional thin lens (far field lens) is illustrated as shown in Fig. 2.3. The far-field lens means that it modulates the distribution of the far field. It is noted that only modulating the far-field distribution is possible to obtain different size of the focal spots. The field distributions at the focal plane in fact are far fields. For example, the far field is broader or narrower immediately after conventional lenses which results in narrower or broader focal beam waist as shown in Fig. 2.3a and 2.3b respectively. It is known that the conventional phase arrays (similar to Fresnel lens) which manipulate near-field interference are used to produce a desired far-field pattern. Many specialized subwavelength gratings 5 or metasurfaces 6 which modulate the phase of near field can result in the focal distances lying in the near field. It is noted that the principle of focusing of those engineered arrays should be far field lensing like Fresnel lens rather than near field lensing like flat lens. On the contrary, the near-field lens such as flat lenses means that it modulates the distribution of the near field. The transmitted beam behind the flat lens can be broader or narrower as shown in Fig. 2.3c and 2.3d. However, the far field of the beam remains unchanged due to its two parallel flat surfaces, so the focal beam waist is the same with the source beam waist. Remember that, the formula of flat lensing is lobject + limage = F. The difference is that the distance from the flat lens to an image plane is limited by the focal length F of the lens and the image can never be infinitely far away. For example, the limitation of the flat lenses or superlenses is that the projected image only exists near the surface of the

Page 6 | Flat Focusing Mirrors | Yu-Chieh Cheng’s Thesis | January 2015

lens, with no magnification. This fact limits the practical applications for the flat lenses or superlens, since any camera needs to be within the lens’s “near field”. To make a lens useful for far-field imaging below the diffraction limit, the evanescent waves have to be converted into propagating waves, which is what the hyperlens 7 does. The Hyperlens can project the high-resolution image from near field to the far field. Developing a near-field lens with a long focal distance for a observable and measurable phenomenon is always challenging and it is one of the main objectives of this thesis. The ABCD matrix method in optics can also show the difference between far-field lens and near-field lens. A light wave is characterized by the distance x from the axis of the lens and an angle θ with respect to the propagation direction z. These quantities in front and behind the element are related by the ABCD matrix for a far-field lens with a focal length f

 x2   1    q 2   1 f

0   x1    1   q1 

(2.2.1)

For a near-field lens, the angle of incidence and refraction are the same due to the flat interface. However, the distance x is modified. Therefore, the simple ABCD matrix for a near field lens can be obtained as

 x2   1  F   x1       q 2   0 1   q1 

(2.2.2)

and the focal length F of the near-field lens follows lobject+limage = F.

(2.2.3)

These two ABCD matrix also show that the angle (far field) is changed/unchanged and the transmitted field (near field) is unchanged/changed for the far/near field lens . 2.3 Anomalous diffraction The idea of near field lens in the previous section is discussed by geometric optics or ray optics. In mathematics, the focal length can also be calculated from the Maxwell’s equations in free space which has the form called the Helmholtz equation   2  k 2  E (r )  0 , k = n0/c. For the paraxial approximation of the Helmholtz equation :

 2 A  2ik

A 0, z

(2.3.1)

2 2 2 2 2 where      x    y is the transverse part of the Laplace operator. In the paraxial approximation, the complex amplitude of the electric field is E r  A(r )e ikz and A(r ) describes transverse profile with its variation in propagation. In the Fourier domain, Eq. (2.3.1) can be written as

