Abstract |
Since the theoretical prediction and experimental demonstration of optical Airy
beams [Opt. Lett., 32(8):979–981,(2007), Phys. Rev. Lett., 99:213901,(2007)], accelerating
waves have been established as a very important tool in the field of optics. The last
dozen of years accelerating beams have attracted a lot of interest due to their intriguing
properties, and were extensively studied both from a theoretical and an experimental
perspective. Originally, Airy beams were proposed in quantum mechanics in the seminal
work of Berry and Balazs [Am. J. Phys. 47(3):264–267, (1979)], showing that the
potential-free Schrödinger equation admits propagation-invariant solutions in the form
of accelerating Airy wavepackets. Beyond Airy beams, another family of diffraction-free
waves was proposed and experimentally observed by Durnin, the well-known Bessel
beams [J. Opt.Soc. Am. A, 4(4):651–654, (1987), Phys. Rev. Lett., 58:1499–1501, (1987)].
Due to their resilience to diffraction-spreading and the uniformity of their amplitude,
such beams were also exploited in many applications. Furthermore, in the nonparaxial
domain where rays and thus beams can bend at large angles, diffraction free beams accelerating
along circular, elliptical, exponential and general power-law trajectories were
demonstrated. In another concept, abruptly autofocusing waves mainly represented by
Airy beams with radial symmetry, propagate along parabolic trajectories while focusing
most of their energy right before a target.
In this dissertation, we focus on engineering the properties of optical waves. We
focus in the case of propagation-invariant fields of the Airy and Bessel type and on
different classes of accelerating waves. We engineer their fundamental properties such as
their amplitude, their width and their trajectory. Furthermore, we examine the focusing
characteristics of abruptly autofocusing waves. The possibility of optimizing their
focusing features is of our particular interest.
To begin with, we study the generation of accelerating waves in the paraxial domain,
whose propagation defining properties such as trajectory, maximum amplitude and
beam-width will be predesigned. In the case of the power-law trajectories, the propagation
of such beams is described by Airy-type solutions which are directly expressed in
terms of the geometric properties of the preselected path.
Additionally, we investigate the propagation of accelerating beams in the nonparaxial
domain. In this case we study accelerating beams along circular, elliptic and power-law
curves. Our solutions indicate that independently of the trajectory assumed, the dynamics
of the beam near the caustic are described by Airy-type functions. Our formulas
are expressed in an elegant and practical way and highlight the dependence to the
curvature of the predesigned trajectory, among other geometrical features. In particular,
we show that the generation of accelerating beams along nonparaxial trajectories with
pre-engineered amplitude and beam-width, is possible.
Moreover, we consider the propagation of abruptly autofocusing waves in the
paraxial domain. Specifically, we emphasize on the propagation of such beams along
convex but otherwise arbitrary predefined trajectories. Furthermore, in order to optimize
their focusing characteristics, we properly modulate the important parameters such as
the initial amplitude, the curvature of the trajectory, and the distance from the optical
axis on the input plane, in order to achieve higher intensity contrast at the focus along
with damped oscillatory behavior after the focal point.
Beyond accelerating beams, we also study the case of Bessel beams of zeroth order
and higher-order optical vortices of the Bessel-type. We propose a method for generating
such beams, exhibiting pre-engineered maximum amplitude and beam-width or hollowcore
radius along the propagation distance. In both cases, numerical results agree well
with the theoretical model developed.
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