Field curvature

A further result of Sadler s 2D-simulation was a relation between the step density and growth rate on the one hand and the inclination of the surface with respect to the principal axes on the other. From this relation crystal shapes were derived which show considerable curvature. This result of an exact treatment stands in contrast to Frank s mean-field curvature expression which gives essentially flat profiles. We will return to the discussion of curved edges in Sect. 3.6.3. 

In this equation, r and z are cylindrical coordinates (z = 0 being the axis of symmetry of the field), C is a constant, k is field curvature, and Rm is the characteristic radius. The... 

The object of an ideal optical configuration is that there are no optical aberrations, no stray light, and no energy loss through the system. In practice, various compromises have to be made to minimize the effect of the various variables. Within a practical optical system, five types of optical aberration are possible which give rise to image distortion, namely primary spherical aberration, coma, astigmatism (anamorphism), field curvature, and distortion. 

Let us assume that a liquid is incompressible, B oo, and discuss orientational (or torsimial) elasticity of a nematic. In a solid, the stress is caused by a change in the distance between neighbor points in a nematic the stress is caused by the curvature of the director field. Now a curvature tensor dnjdxj plays the role of the strain tensor ,y. Here, indices i,j = 1, 2, 3 and Xj correspond to the Cartesian frame axes. The linear relationship between the curvature and the torsional stress (i.e., Hooke s law) is assumed to be valid. The stress can be caused by boundary conditions, electric or magnetic field, shear, mechanical shot, etc. We are going to write the key expression for the distortion fi-ee energy density gji, related to the director field curvature . To discuss a more general case, we assume that gji t depends not only on quadratic combinations of derivatives dnjdxj, but also on their linear combinations ... 

We will not give details of the aberrations of quad-rupoles and prisms here. Quadrupoles have more independent aberrations than round lenses, as their lower symmetry leads us to expect, but these aberrations can be grouped into the same families aperture aberrations, comas, field curvatures, astigmatisms, and distortions. Since the optic axis is straight, they are third-order aberrations, like those of round lenses, in the sense that the degree of the dependence on Xo, x, yo, and y is three. The primary aberrations of prisms, on the other hand, are of second order, with the axis now being curved. 

Even in the absence of other aberrations, the image surface of a lens is not truly a plane but rather a curved surface. The aberration is known dis field curvature, and the focal surface is called the Petzval surface. The Petzval surface curves toward a thin, positive lens. Its radius of curvature is the reciprocal of the Petzval sum where the... 

Novel planar permanent magnet, MR microscopy lenses , were presented, where only a point region of the sample located in the focus spot of the structures become resonant. The high magnetic field curvatures of the lenses ensure that the regions outside of the focus spot are inactive or filtered out in the resonance detection process. Other concepts and technologies for available data storage are presented. 

Limitations Low definition images Field curvature artifacts Low-definition image Image artifacts Moderate throughput at... 

All optical components are imperfect. Lenses suffer from different so-called aberrations chromatic (light of different wavelengths is refracted differently), spherical, coma, astigmatism, field curvature (the sharp image falls on a curved surface) and distortion. Modem lenses correct for some of the existing aberrations achromats and apochromats correct for chromatic aberration. Lenses corrected for field... 

Generally optical lenses are very well corrected they have been able to obtain resolutions very close to the theoretical diffraction limit for a century or more. However, full correction of chromatic aberration, spherical aberration, and field curvature to give a Plan Apochromat will be expensive. In contrast, lenses for electron microscopy are generally... 

The off-axis aberrations (off-axis astigmatism, coma, field curvature, and distortion) all vanish for rays parallel to the optical axis. All rays in the TEM are at such small angles to the optical axis that these aberrations are not usually important. The axial astigmatism can be corrected by adjustment of stigmators that cancel the residual non-circularity of the objective. This can be difficult, but assuming that it is done correctly, the remaining effective aberrations are chromatic and spherical. 

Field curvature" Qff-axis images are not focused on the ideal surface, usually a plane. 

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