Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 239

The resolving power of an optical lens is restricted by the wavelength of a source. This is known as the diffraction limit of the lens. Fourier optics, i.e. the wave optics adequately explain it in terms of the propagating waves and decaying evanescent waves generated by the object source shown in Fig (5.12a). The object source located at x = 0 generates an arbitrary wave field that can be decomposed in terms of the plane wave spectrum. Their superposition reconstructs the wave field. Thus, the wavefield in the real DPS space, propagating in the x‐direction, is composed of 2D Fourier plane wave components in the Fourier domain, i.e. in the k‐space (propagation vector space). The wave field is expressed through the following 2D Fourier integral:

Figure 5.12 Creation of image using wave optics.

(5.5.27)

The wave propagates in the x‐direction and the spectral field components are in the (k_y − k_z)‐plane. The spectral field amplitudes with phase distribution within the bracket [ ] are Fourier components of the electric field . The propagation constant of the plane wave, given by equation (5.5.27b), is obtained from the dispersion relation (4.5.29d) of chapter 4. The propagation constant of propagating plane waves must be a real quantity, whereas imaginary values of provide decaying evanescent waves in the x‐direction. Thus, Fourier components of the wave originated by an object could be divided into two groups: propagating waves and nonpropagating evanescent waves. The grouping of waves is shown in Fig (5.12b). In a DPS medium, the propagation constants for both groups of waves, from equation (5.5.27b), could be written as follows:

(5.5.28)

The electric fields of both the propagating and evanescent waves in the DPS medium could be written from the above relations as follows:

(5.5.29)

Figure (5.12a) shows both the propagating and exponentially decaying evanescent waves in the DPS medium, generated by an object. The evanescent waves decay fast within a distance under λ. The higher value of transverse components of the wavevector (k_y, k_z) corresponds to the finer spatial details of the object. So at the image-plane, finer spatial details of an image are lost. The DPS based lens cannot recover the lost spatial details in the evanescent waves. The maximum value of the transverse wavevector at the cut‐off wavenumber k_x = 0 determines the limit of the finer spatial details, i.e. the diffraction limit:

(5.5.30)

The maximum resolution Δ is the minimum adjacent distance of the finer spatial details of the object. It called the diffraction limit. The relation Δ × k_t,max = 2π is obtained in subsection (19.1.1) of chapter 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).