CHAPTER 1 Fig. 1.1 Two different concepts of flat focusing mirrors are considered in my PhD: (a) the structure with a flat surface which keeps the transversal invariance and (b) the subwavelength periodic structure which also has the transversal invariance in a larger-than-l scale. As a comparison, (c) the engineered structure with laterally varying periods breaks the transversal invariance.

(a)

(b)

(c)

x z

The implementations of flat focusing mirrors with the transversal invariance can be conceived from two different geometrical approaches:

1)

Multilayer thin film structures The first idea of flat focusing mirror is based on the modulation in the normal-to-surface direction (z-axis) to keep the transversal invariance (x-axis) simultaneously as shown in Fig. 1.1a. The physical process of light focusing can be understood under the idea of â€œbringing into correct phaseâ€?. One of the examples of correcting phase is the technique for solving a chromatic dispersion of optical pulses. The common solution is using one-dimensional (1D) chirped mirrors (CMs), consisting in multiple layered structures with two alternating materials of high and low refractive indices 25. The interesting perspective of such 1D CM is that its longitudinal periods dz(z) vary along the propagation distance z so that the different frequencies of the optical pulse reflect at different depths inside the CM. The phase shift of different frequency components can be manipulated in reflections. Therefore, a pulse compression can be obtained from CMs 26. The same general principle can be applied for a monochromatic beam. The monochromatic beam has only one frequency component but many angular components. The dephasing between the angular components makes a monochromatic beam spreads or diverges. To obtain a change of the shape of the beam upon reflection, the spatial components of the beam should be modulated. In this case, a CM is used to compensate the phase delays among different angular components for a monochromatic beam. It is a similar way as a CM which fix the problem of the chromatic dispersion for an optical pulse. If the phase delays of the angular components of beam could be also compensated or even over-compensated in the mirror, the original beam is restored or imaged at the focal plane after reflecting from the flat focusing mirror. One of the limitations is that the flat focusing mirror can only perform a compensation of phase delay so that the focal beam waist is the same as the source one. In addition, the flat focusing mirror works only for a beam and never for a plane wave which, for example, only can be focused by spherical lens or curved mirrors.

Page 2 | Flat Focusing Mirrors | Yu-Chieh Chengâ€™s Thesis | January 2015

2) Subwavelength periodic structures Another possibility is to modulate the incoming light using a surface grating with a transverse period dx(x) as shown in Fig.1.1b. The modulation in the x direction should be periodic in order to keep the transversal invariance in a large space scale (larger than a modulation period). For the operation of the focusing gratings, only the zero-order reflection mode is needed and this can be achieved by working in the subwavelength range. In this regime, the diffractive angle of the high-order modes becomes very large so the high-order modes decay exponentially as evanescent waves. These high-order modes can still interact with a small number of grating periods. The phase shifts of angular components of a beam can be manipulated between the first and adjacently second scatters. As a result, the beam can be diffused (diverging) or anti-diffused (converging) depending on the ratio between the transverse period and the wavelength 27. It is noted that the principles of both flat focusing devices with the transversal invariance are similar with the near field lensing effects of flat lenses. They are very different from the far field lenses/mirrors such as conventionally spherical lenses/mirrors, Fresnel lenses, gradient-index material (GRIN) 28-30 lenses and engineered gratings [Fig.1.1c]. The difference is that the far field lenses/mirrors never achieve transversal invariance because their spatial modulations break the transversal invariance. These non-invariant focusing mirrors are not considered in the thesis for three reasons. First, the compact devices can be applied generally in photonic integrated circuits. However, the fabrication tolerances are more stringent because the desired phase response is sensitive to the local periods which must be accurately patterned in nanoscale range. Second, although these mirrors are flat, the existence of optical axes also limits their applications. Third, these specially designed SWGs are hard to be designed for narrow beams because large phase variation is required in the latter case which results in much more complicated structures 4.

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