Wavefront curvature

Figure 10. A schematic diagram of the wavefront curvature sensor set-up.

A quantitative study of the role played by the laser wavefront curvature and other geometrical/optical parameters on the transmitted far-field intensity profile [3] demonstration of self-defocusing effect for optical limiting application [4]. 

The three different three-level configurations discussed above have been studied experimentally. Problems concerning spectral stab ility, that is line shifts and broadenings as well as transit time and particle wavefront curvature limitations on the obtainable optical resolution have been investigated. 

A nearly perfect diverging wavefront would exit the test plate appearing as though it came from a source 100 m away. A segment would be positioned so that its mean center of curvature was coincident with that virtual source 100 m away. In the worst case in our example, the un-equal air path would be about 4 m rather than 204 m. Interference would take place between the wavefront reflected off the 100 m radius side of the test plate and the segment. The roughly 3 m back to the source and beamsplitter is common path and will not affect the interference pattern. 

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). 

For what is an apparently straightforward problem, wavefront sensing has produced a large number of apparently quite different solutions (for example the Shack-Hartmann (Lane and Tallon, 1992), curvature (Roddier, 1988) and pyramid (Ragazzoni, 1996) sensors). Underlying this diversity is the problem... 

The second problem is how we can obtain a linear relationship between the coefficients describing the wavefront and our measurements. It is how this linear relationship is obtained that differentiates, for example, a Shack-Hartmann and a curvature sensor. In all wavefront sensors to transform a wavefront aberration into a measurable intensity fluctuation it is necessary to propagate the wavefront. As a first approximation this propagation is described by geometric optics, and we discuss the linear relationship between the wavefront slope and the image displacement in Section 24.3. 

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. 

Both the Shack-Hartmann and the curvature sensor measure wavefront slope (Roggemann and Welsh, 1996). Although the latter is more often considered to measure curvature an entirely equivalent analysis based on slope is possi-... 

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. 

With radio wave interferometric imaging of astronomical sources, it is typically assumed that the incoming radiation consists of planar waves. In this limit, any curvature of the incoming wavefronts is negligible. This far-field assumption is used for synthetic imaging and simplifies the imaging reconstruction process. However, most standoff THz applications do not fall in the far-field limit, and the simplified inverse Fourier transform of the electric field correlation - the far-field image reconstruction [99] - must be modified [100] to account for the curvature of the wavefronts in the near-field. 

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. 

Figure 1 Schematic diagram of an absorbing particle in a focussed laser beam The curvature of the wavefront is such that the resulting force can be resolved into a radially inward force and a force in the direction of the beam propagation. The particles are constrained in the vertical direction by the glass microscope slide, and hence trapped.

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. ... 

Taking into account the curvature of the wavefront of the electron emitted by the ionized atom leads to the effective dependence of the scattering amplitude on the interatomic distance. In this review we will not discuss this question in more detail, because it has already been successfully solved in EXAFS spectroscopy [63-68]. 

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