Convergent measure of focal extent, and largest peak intensity for non-paraxial beams

Abstract

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.

Some exact solutions for light beams

“…It is easy to see from the z-derivative of (12) that the expression for the proto-beam in the focal plane z = 0 is proportional to J 1 (k 0 r ⊥ ) /r ⊥ , which is a feature of the proto-beam discussed in [2,23,22]. The proto-beam was found in [24] to meet a criterion for largest intensity increase from focusing. We will discuss examples based on the basic beam solution (12) and derivatives thereof, but none of our examples will have a z-derivative of (12) and so they will not be in Lekner's class of solutions.…”

“…We will discuss examples based on the basic beam solution (12) and derivatives thereof, but none of our examples will have a z-derivative of (12) and so they will not be in Lekner's class of solutions. We refer the reader to [22,24,25] for many interesting details of the proto-beam and electromagnetic beams built from it. Lekner [22] also points out that further solutions can be generated by the method of displacing z by a complex constant, which was used to generate the "complex-sources" waves described in the Introduction.…”

“…We will discuss examples based on the basic beam solution (12) and derivatives thereof, but none of our examples will have a z-derivative of (12) and so they will not be in Lekner's class of solutions. We refer the reader to [22,24,25] for many interesting details of the proto-beam and electromagnetic beams built from it.…”

Focal extent of scalar beams

“…In the  kb 0 limit, there is just the one length parameter,k . 1 Hence, both the longitudinal and transverse extents of the focal region will be proportional tok . 1 The focal plane and axial forms are…”

“…The longitudinal and transverse extents of the beams are different in their dependence on k and b. The Andrejic [1,2] focal extent measures are used. We shall illustrate localization first in the approximate (paraxial) Gaussian beam, and then compare and contrast these results with those for the exact solutions.…”