Plane wavefronts

Ragazzoni, R., 1996, Pupil plane wavefront sensing with an oscillating prism,... 

Figure 14a. Provided the wavefront is not interrupted, it moves out uniformly in a radial direction in accordance with the accepted notion of the rectilinear propagation of light. Suppose we now have a series of plane wavefronts incident on a slit with each successive wavefront being constructed from the envelope of secondary wavelets from the preceeding wave-front as shown in Figure 14b. If the width of the slit is small relative to the wavelength of light, then the portion of the wavefront at the slit may be considered as a single Huygens source radiating in all directions to produce...

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]

As discussed for several decades by a number of authors, the nature of light and photon physics is related not only to the propagation of plane wavefronts but also to axisymmetric wavepackets, the concepts of a rest mass, a magnetic field in the direction of propagation, and an associated angular momentum (spin). 

Actually stable, plane wavefronts do not exist, at least in condensed explosives, as shown by Cook et al. Below are summarized some of the important observed results of these studies. 

Let AB be an incident beam corresponding to a plane wavefront (with 2 wavelength), and let 0 be the angle between this incident beam and the family of planes with interplanar spacing d. From Figure 2 it is clear that a difference exists in the path crossed between the waves scattered at the points Aj and A2 that is equal to BA2 - - A2C = 2d sin 6 (BA2 = A2C and sin 6 = BAi/d). The two waves will combine with maximum positive interference (will be in phase) if this difference in the path is a multiple of X ... 

The reciprocal conductivity tissue yielded isochrones which are shown in Figure 7. Since this is non-physiological there is no experimental evidence for comparison. However, Muler and Markin (1978) determined isochrone shapes for this condition based on the assumption that each isochrone element behaves as if it were part of an infinite plane wavefront. Their result is also a square-like shape but rotated 90°. As noted above there is reason to believe that the plane... 

There are essentially three mechanisms through which a plane wavefront can become destabilized. Two of these are hydrodynamic, stemming from density changes and accompanying buoyancy forces due to nonzero volumes or heats of reaction. These operate to differing extents in gas-phase [33] or condensed-phase [34] flames (usually where there is a phase change in the latter case) and also in solution phase [35-38]. In this section, we concentrate on the third mechanism, that arising from conductive and diffusive processes. 

This is the equation of a plane normal to the vector k and the solution (2.24) therefore represents a plane wavefront travelling with velocity v in the direction k (Problem 2.2) However, the electric and magnetic fields are vector quantities and we therefore assume that the plane-wave solutions of equations (2.19) are of the form ... 

This problem demonstrates the link between the widths of the cavity response and the transmission peaks of this well-known interferometer. However, the fringe pattern observed with a Fabry-Perot etalon should not be confused with the mode pattern of a laser having a plane-parallel resonator. The Fabry-Perot etalon is normally used with the plates so close together that all the transverse modes of the corresponding optical cavity are virtually degenerate in frequency. The plane wavefronts assumed in the discussion of the theory of the etalon are composed of an infinite sum over the transverse modes of the cavity. 

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 ... 

Figure 1 Bragg diffraction. A reflected neutron wavefront (D, Dj) making an angle 6 wKh planes of atoms will show constructive interference (a Bragg peak maxima) whan the difference in path length between Df and (2CT) equals an integral number of wavelengths X. From the construction, XB = d sin 6.

The idea of using propagation of intensity to measure wavefront errors is applied more direetly in the eurvature sensor. Conceptually, the irradiance is measured at two planes which are equally spaced about the telescope focus and the normalized difference in intensity obtained. 

It can be derived from the transport of intensity equation that this signal has two terms at the edges of the pupil it is proportional to the local wavefront gradient in the direction normal to the pupil edge (i.e. radial in the case of a circular pupil) and elsewhere in the pupil it is proportional to the local wave-front curvature (Roddier, 1988). The signal is more intense when the planes are nearer the telescope focus, but diffraction will limit the spatial resolution more. Thus there is a trade-off between resolution and signal-to-noise (see Ch. 24). 

In real curvature sensors, a vibrating membrane mirror is placed at the telescope focus, followed by a collimating lens, and a lens array. At the extremes of the membrane throw, the lens array is conjugate to the required planes. The defocus distance can be chosen by adjusting the vibration amplitude. The advantage of the collimated beam is that the beam size does not depend on the defocus distance. Optical fibers are attached to the individual lenses of the lens array, and each fiber leads to an avalanche photodiode (APD). These detectors are employed because they have zero readout noise. This wavefront sensor is practically insensitive to errors in the wavefront amplitude (by virtue of normahzing the intensity difference). 

The pyramid wavefront sensor has recently been proposed (Ragazzoni, 1996). A four-sided p5n amid is placed with its vertex at the telescope focus, and following the pyramid is some optics to image the pupil plane onto a detec-torSince a four-sided prism is employed, four pupil images will be formed... 

It is the shape of this distribution which imposes limits on the ability to eshmate the wavefront. In the special case where the measurement plane is in fact at the focal plane, the relationship between the aperture and measurement plane simplihes to... 

Unfortunately, there is not a linear relationship between the light intensity in the measurement plane and the wavefront. This is shown in Fig. 2 which shows the intensities measured at the focal plane for a wavefront equal to pure tilt and coma terms individually. It is apparent that scaling the wavefront by a (in this case 5) does not result in a linear increase in the measured intensity by a factor a. The key difference between existing existing wavefront sensors such as the Shack-Hartmann, curvature and pyramid sensors is how they transform the measured intensity data to produce a linear relationship between the measurements and the wavefront. 

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... 

Translating to the ground plane, since the pulse wavefront is perpendicular to the slant range direction, we get the range resolution on the ground to be... 

First, we rewrite expression (5.14), which is valid for plane monochromatic waves, in a more general way, valid for a more general wavefront, as follows ... 

Takagi has shown that the intensity on a projection topograph is given by the integrated intensity and that this is independent of the detailed shape of the wavefront. The phase relations are lost, and we can use the integrated plane wave... 

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. 7.4. Wave model of an acoustic lens focused on a surface the tangent to one of the wavefronts illustrates one of the family of plane waves of which the wavefront is...

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