Reflection and transmission of beams at a curved interface

Calculation of Wideband Ultrasonic Fields Radiated by Immersed Transducers into Solids Through Curved Interfaces

“…Let us assume a spherical interface which radius of curvature is 9t The sign convention for the curvature of the surface is as follows : 9t = + l9t I for an interface which is concave when viewed from the source and 9t = -l9t I for a convex interface. lt can be easily checked that, as 9t tends to the infinity, that is, as the Explicit analytical expression of DF for cylindrical interfaces can be found in the Iiterature only in the 2-D case (in [5] for example). In the 3-D case of a ray in an arbitrary plane relatively to the cylindrical interface, the angle made by the plane of incidence (the same plane as the plane containing the refracted ray) and the axis of symmetry of the cylinder must be taken into account ( on the other hand, such an angle is nonsensical for plane or spherical interfaces).…”

“…Attenuation in both media is neglected. Figure 1 displays the field radiated at field-points of a 2-D zone defined by: z [5,30] mm and x (or y) [-15, 15] …”

Reflection and Transmission of Light Beams at a Curved Interface: Coherent State Approach

“…At first, the incident beam field is expanded into coherent states, representing elementary Gaussian beams with axis displacement and tilt. Note, that in a sense such Gaussian beams are similar to the complex rays used for simulation of reflection and transmission of beams at a curved interface in [10]. Coherent states (CS) form a full set of functions ( )…”

Quasi-ray Gaussian beam algorithm for time-harmonic two-dimensional scattering by moderately rough interfaces

“…2 generates a Gaussian beam that is launched from -indexed locations and tilted according to -indexed locations. The radiated incident field in the half-space [see (2)] therefore admits via (8) a similar discretized representation (we follow the notation in [1]; however, there are some sign changes with respect to [1] since here we assume propagation into the negative half-space ) (13) where the beam functions are expressed by Gaborweighted line-source superposition (14) with being defined in (4). By saddle point methods, the integral in (14) (or its spectral counterpart) can be evaluated asymptotically in the beam paraxial far zone, yielding the following complex source point approximation [2], [3]: (15) with representing the complex distance between the observer at and the CSP…”

“…In Section II, we summarize the rigorous, self-consistent Gabor-based Gaussian beam algorithm for a general aperture field distribution and the ensemble of paraxially approximated narrow-waisted CSP beams that this excitation generates [1]. Section III deals with the preliminary (canonical) problem of beam reflection from, and transmission through, a curved interface between two homogeneous dielectrics [14]. These constituents have been used previously for beam tracking through planar and curved layered dielectric configurations [2], [3].…”