Higher-order moments and overlaps of Cartesian beams

We introduce a closed-form expression for the overlap between two different Cartesian beams. In the course of obtaining this expression, we establish a linear relation between the overlap of circular beams with azimuthal symmetry and the overlap of Cartesian beams such that the knowledge of the former allows the latter to be calculated very easily. Our formalism can be easily applied to calculate relevant beam parameters such as the normalization constants, the M2 factors, the kurtosis parameters, the expansion coefficients of Cartesian beams, and therefore of all their relevant special cases, including the standard, elegant, and generalized Hermite–Gaussian beams, cosh-Gaussian beams, Lorentz beams, and Airy beams, among others

Exponential beams of electromagnetic radiation

We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann-Silberstein vector and the Whittaker representation of the solutions of the Maxwell equations in terms of just one complex function. The connection between the Bessel beams and the exponential beams is made explicit by constructing the exponential beams as wave packets of Bessel beams.Comment: Dedicated to the memory of Edwin Powe

Rhomboid prism pair for rotating the plane of parallel light beams

An optical system is described for rotating the plane defined by a pair of parallel light beams. In one embodiment a single pair of rhomboid prisms have their respective input faces disposed to receive the respective input beams. Each prism is rotated about an axis of revolution coaxial with each of the respective input beams by means of a suitable motor and gear arrangement to cause the plane of the parallel output beams to be rotated relative to the plane of the input beams. In a second embodiment, two pairs of rhomboid prisms are provided. In a first angular orientation of the output beams, the prisms merely decrease the lateral displacement of the output beams in order to keep in the same plane as the input beams. In a second angular orientation of the prisms, the input faces of the second pair of prisms are brought into coincidence with the input beams for rotating the plane of the output beams by a substantial angle such as 90 deg

Influence of section depth on the structural behaviour of reinforced concrete continuous deep beams

YesAlthough the depth of reinforced concrete deep beams is much higher than that of slender beams, extensive existing tests on deep beams have focused on simply supported beams with a scaled depth below 600 mm. In the present paper, test results of 12 two-span reinforced concrete deep beams are reported. The main parameters investigated were the beam depth, which is varied from 400 mm to 720 mm, concrete compressive strength and shear span-tooverall depth ratio. All beams had the same longitudinal top and bottom reinforcement and no web reinforcement to assess the effect of changing the beam depth on the shear strength of such beams. All beams tested failed owing to a significant diagonal crack connecting the edges of the load and intermediate support plates. The influence of beam depth on shear strength was more pronounced on continuous deep beams than simple ones and on beams having higher concrete compressive strength. A numerical technique based on the upper bound analysis of the plasticity theory was developed to assess the load capacity of continuous deep beams. The influence of the beam depth was covered by the effectiveness factor of concrete in compression to cater for size effect. Comparisons between the total capacity from the proposed technique and that experimentally measured in the current investigation and elsewhere show good agreement, even though the section depth of beams is varied