No abstract

The focal region of a beam contains circles of zeros of the beam wavefunction, on which surfaces of different phase meet. The existence of these zeros is topological in origin, and appears to be universal. Two examples of generalised Bessel beams are examined. One of these has zeros only in the focal plane. The other has focal plane zeros but also movement of the zeros away from the focal plane at certain values of a parameter which determines the tightness of the focus, as analysed by Berry in 1998. As tightness of focus increases these two families of beams coalesce into a common most-focused beam. The polarisation properties of both families and of their common limiting form are considered and correlated with the zeros (dislocations) of the beam wavefunctions. We find regions of circular polarisation in beams which are nominally linearly polarised, and rapid variation of the polarisation pattern as the tightness of focus passes through critical values.

Collective effects in the interaction of light with ensembles of identical scatterers play an important role in many fields of physics. However, often the term "identical" is not accurate due to the presence of hyperfine fields which induce inhomogeneous transition shifts and splittings. Here we develop a formalism based on the Green function method to model the linear response of such inhomogeneous ensembles in one-dimensional waveguides. We obtain a compact formula for the collective spectrum, which exhibits deviations from the uniform frequency shift and broadening expected of two level systems. In particular, if the coherent contribution to the collective coupling is large, the effect of inhomogeneous broadening can be suppressed, with the linewidth approaching that of the superradiant value. We apply this formalism to describe collective effects in x-ray scattering off thin-film waveguides for inhomogeneous hyperfine parameters.

Second moment beam widths are commonly used in paraxial optics to define the focal extent of beams. However, second moments of arbitrary beams are not guaranteed to be finite. I propose the focal concentration area as a measure of beam focal area, defined to be the ratio of total radial intensity to radial intensity regulated by a unit area Gaussian distribution. I use the Dirac delta limit of this distribution to establish a rigorous upper bound on the peak intensity of non-paraxial beams of a given total intensity, and show that this is achieved by the recently proposed 'protobeam' solution. I discuss the generalisation to electromagnetic beams, and find the same lower bound as the scalar case. This bound cannot be achieved for any physical beam, and as such the physical lower bound must be higher.

<p>I explore the limits of how tightly a beam can be focused, and derive a focal parameter for scalar beams that can be symbolically evaluated for most beams, and is guaranteed to be convergent for physical beams, that compares peak in- tensity to the total intensity in the beam profile. I argue that this parameter is superior to spot size, and use this to derive a rigorous limit of focusing for scalar beams. A particular beam known as the proto-beam achieves this tight- est focus possible. I show the generalisation of this measure to electromagnetic beams, and place a lower bound on the focal extent of electromagnetic beams. I also propose the use of exponential regulators as alternatives to moment based measures, as a solution to the convergence issues created by the power law decay of exact solutions.  I explore the Doppler shift for finite beams, and how monochromatic beams become polychromatic under a Lorentz boost. The local frequency is also explored, and I show that a deviation of the local frequency from the Doppler frequency will occur due to wavelength broadening near the focus.  Lekner and I examine a beam that closely approximates a paraxial Gaussian beam radially, and examine the phase singularities for optical beams that occur near the zeros of the beams wavefunction. We also investigate attempts to find exact solutions with Gaussian profiles, and show that this is impossible; any such beam will be evanescent and exponentially grow. Finally, I investigate the property of finite classical electromagnetic pulses having a zero momentum frame, and show that for quantum single photon pulses this property holds for the expectation value. I show that any individual measurement however, still measures a light-like four-momentum for the photon.</p>