Spherical wavefront

The divergence factor (DF) introduced by the asymptotic expansion, accounts for the deformation of the refracted wavefront (initially spherical in the coupling medium). It ensures, under the GO approximation, the energy conservation of a ray-pencil propagating... 

We first supposed that the field radiated into the piece by the transducer is known, thanks to the Champ-Sons model. Then, the main approximation used consists in making far field assumptions in the beam defect interaction area. In the case of a focused transducer we assume that the incident wavefronts on the defect are plane. This is equivalent to say that the defect is located in or near the transducer focal area and that a defect located outside this zone does not cause a significant echo. In the case of planar contact transducer, the incident wavefronts on the defect are assumed to be spherical The incident field on the defect is therefore approximated by the product of a spatial function gfp,0,z)describing the amplitude distribution in the beam and a time-delayed waveform < ) ft) representing the plane or spherical propagation in the beam. The incident field on the defect can therefore be approximated for ... 

There is another simple type of wavefront in three dimensions where the wave propagation is restricted to one direction. The disturbance at a given instant is now the same at all points of a plane perpendicular to this direction. This is called a plane wave. For both spherical and plane waves the propagation is described as... 

The distribution of light intensity in Figure 13 can be computed by application of Huygens principle which allows us to calculate the shape of a propagating wavefront provided the wavefront at an earlier instant is known. According to this principle, every point of a wavefront may be considered as a source of secondary waves (often called a wavelet) which spread out in all directions, i.e., all points on a wavefront are point sources for the production of spherical secondary wavelets. The new wave front 2 is then found by constructing a surface tangent to all the secondary wavelets as shown in... 

In a phase contrast microscope, an incident wavefront present in an illuminating beam of light becomes divided into two components upon passing through a phase specimen. The primary component is an undeviated (or undiffracted) planar wavefront, commonly referred to as the surround (S) wave. It passes through and around the specimen but does not interact with it. In addition, a deviated or diffracted spherical wavefront (D-wave) is also produced. It becomes scattered in many directions. After leaving the specimen plane, sur-... 

Fig. 2.1. A lens for high-resolution acoustic microscopy in reflection. The central transparent part is a single crystal of sapphire, with its c-axis accurately parallel to the axis of the cylinder. The sandwich structure at the top is the transducer, with the yellow representing an epitaxially grown layer of zinc oxide between two gold electrodes. The pink shaded areas within the sapphire represent the plane-wavefronts of an acoustic pulse they are refracted at the lens cavity so that they become spherical in the coupling fluid. A lens for use at 2 GHz would have a cavity of radius 40f[Pg.8]

A simplified picture of the transducer, lens surface, and reflecting object is shown in Fig. 7.4. The waves radiated by the transducer are refracted by the lens so as to form a spherical wavefront centred on the focal point of the lens. Each point on this wavefront can be described by its angular coordinates from the focus let these be 6 for the zenithal angle (i.e. the angle to the lens axis, again taken to be normal to the specimen surface) and for the azimuthal angle. Thus the spherical wave emerging from the lens can be described by... 

We will first concentrate on the illumination cone. A spherical wavefront is emitted in all directions from a point source (A) and expands with the speed of light. In our chosen coordinate system, which is limited to only two space coordinates and one time coordinate, this phenomenon is represented by a cone with its apex at A, expanding in the direction of the positive time axis. The passing of time is represented by cross sections of the cone by planes parallel to the x-y plane at increasing ct values (Fig. 2). These intersections will, when projected down to the x-y plane, produce circles of increasing radius that in our 3D world represent the expanding spherical wavefront from the point source at A. As time increases, the circles around A expand, while those around B contract. The ellipses caused by the intersection of those circles stay fixed and unchanged as explained later, in Section IV. 

The curvature of a wavefront appears transformed into the curvature of a mirror surface shaped so that it would focus the total wavefront into the point of ohservation.The reason is that a focusing mirror reflects light in such a way that the total wavefront arrives to the focal point at one point of time. Thus, a small flat wavefront that passes by will appear tilted at 45°. A larger flat wavefront will not only appear tilted but will also be transformed into a paraboloid whose focal point is the point of observation. A spherical wavefront appears transformed into an ellipsoid, where one focal point is the point source of light (A) and the other is the point of observation (B). This configuration represents one of the ellipsoids of the holodiagram. 

The moire of two cross sections of the hyperboloids represents the difference of two interference patterns. If the two cross sections are identical but one is displaced, the moire fringes also represent the loci of constant resolution in the displacement direction. If one focal point is fixed, and the other one is displaced, the resulting moire pattern forms a new set of hyperboloids whose foci are the two positions of the displaced focal point. This new pattern is independent of the position of the fixed focal point (a rotation of one of the original spherical wavefronts of Fig. 5 produces no moire effect). Therefore hologram interference fringes are independent of the position of the point source of the fixed reference beam. 

The near-field correction is calculated conceptually by repositioning the detector positions from a linear arrangement to a spherical arrangement that matches the curvature of the incoming wavefront. The theoretical phase delay from a point source at normal incidence is subtracted from the measured phase. As shown in those Figures, the curvature of the phase (indicating a curved wave front) is removed by the near-field correction, yielding a linear dependence of phase on detector position. The slopes of the near-field-corrected phase versus detector position plots indicate the direction to the source. 

The situation can be analysed more closely by considering a source of monochromatic light located at S in Fig. 8.4 from which two wavefronts propagate, one with a spherical surface and the other with an ellipsoidal surface. A wavefront is the locus of points of equal phase, i.e. the radii (e.g. SO and SE), which are proportional to the ray speeds and inversely proportional to the refractive indices. Figure 8.4 is a principal section of the wavefront surfaces because it contains the optic axis SY. 

The above discussion implicitly obeys Huygens 97 principle, that each point on a spherical wavefront can be regarded as the source of a secondary wavelet (another spherical wave), as well as Fermat s98 principle of least time. 

All points on a wavefront can be considered as point sources of secondary spherical wavelets after a time t, the new position of the wavefront is given by the surface that is tangent to these secondary wavelets. 

The concepts of coherence and incoherence are related to the way in which the neutron, both as a wave and as a particle, interacts with the scattering sample. Wave-like representations of the neutron view its interaction with solids as occurring simultaneously at several atomic centres these atoms become the sources of new wavefronts. Since the scattering occurs simultaneously from all of these atoms the new wavefronts will spread out spherically from each new source and remain in phase. Provided the lattice is ordered, the coherence of the incident wave has been conserved. Constmctive interference between the new wavefronts leads to the generation of distinctive diffraction patterns with well-defined beams, or reflections, appearing only in certain directions in space and no intensity in other directions. 

It was assumed that any small portion of the wavefront may be considered spherical with a local radius of curvature r and that the hydrodynamic equations appropriate to these conditions, with the exception of the equation of continuity, are, to a good approximation, the same as for a plane wave. The equation of continuity... 

The radius of curvature R of the spherical wavefront for point initiation of a cylindrical charge increases at first geometrically (i = L) but quickly settles down to a constant or steady-state value significantly at L L. ... 

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