Aperture function

H(u) is the Fourier Transform of h(r) and is called the contrast transfer function (CTF). u is a reciprocal-lattice vector that can be expressed by image Fourier coefficients. The CTF is the product of an aperture function A(u), a wave attenuation function E(u) and a lens aberration function B(u) = exp(ix(u)). Typically, a mathematical description of the lens aberration function to lowest orders builds on the Weak Phase Approximation and yields the expression ... 

Equation (1.18) applies to a uniform aperture over which the amplitude of the diffracted wave is the same for all points (x, J7). For a nonuniform aperture, we can introduce the function g(x, j7) which gives the amplitude of the diffracted wave originating from an element of area dxdy of the aperture. g(x,y) is usually called the aperture function, but because this nonuniform aperture is our object, we call g(jc,j ) the object function. 

EELS and detector aperture function For analysis by electron energy loss, however, the situation is rather different. Here, the detector is usually limited by a collector aperture in order to reduce energy resolution degradation. This has two effects firstly, only beams spread by an amount t are collected, where fc is the solid angle subtended by the collector aperture. Thus an estimate of A, the resolution obtainable by EELS, is... 

Let us define the objective lens pupil (aperture) function P for an ideal lens as the portion of light that enters the lens it is 1 inside the aperture and 0 outside " ... 

Figure 3. Stress vs. aperture function (Equation 4) fitted to laboratory experimental results...

Such an expression can only be solved directly for a few specific aperture functions. To account for an arbitrary aperture, we must approximate, simplify and restrict the regions in which we evaluate the diffracted pattern. If the point P is reasonably coaxial (close to the z axis, relative to the distance R) and the aperture A(x, y) is small compared to the distance R, then the lower section of (1.2) for dE can be assumed to be almost constant and that for all intents and purposes, r = R. The similar expression in the exponential term in the top line of (1.2) is not so simple. It cannot be considered constant as small variations are amplified through the exponential. To simplify this section we must consider only the far field or Fraunhofer region where. 

Hence the far field diffraction pattern at the point P is related to the aperture function A x, y), by the Fourier transform. The final step is to remove the scaling effect of R in the equation, as it does not affect its structure, just its size. The coordinates [a, jS] are absolute and are scaled by the factor R. For this reason, we normalise the coordinates and define the Fourier transform of the aperture in terms of its spatial frequency components [u, v],... 

In the paraxial approximation, the aperture function A is simply a mathematical device defining the area of integration in the aperture plane A = 1 inside the pupil and A = 0 outside the pupil. If we wish to include the effect of geometric aberrations, however, we can represent them as a phase shift of the electron wave function at the exit pupil. Thus, if the lens suffers from spherical aberration, we write... 

We find from diffraction theory that, E x,y) is in fact the analytical Fourier transform of A(x,y). The pattern of E x,y) is called the far field diffraction pattern of the original aperture function. Hence we have the relationship between E x,y) and A x,y) linked by the Fourier transform... 

Far field region = focal plane of a positive lens = FT(aperture function)... 

Amatore, C., Oleinick, A. I., Svir, 1. 2010. Reconstruction of aperture functions during full fusion in vesicular exocytosis of neurotransmitters. ChemPhysChem 11 159-174. 

Aperture The aperture functions to lower background signal by rejecting most of the uncollimated light incident on the paraboloid. 

Equation 36 must be corrected for changes in the drop shape and for the effects of the inertia of Hquid flowing through the orifice, viscous drag, etc (64). As the orifice or aperture diameter is increased, d has less effect on the drop diameter and the mean drop si2e then tends to become a function only of the system properties ... 

Resonant Sound Absorbers. Two other types of sound-absorbing treatments, resonant panel absorbers and resonant cavity absorbers (Helmholtz resonators), are used in special appHcations, usually to absorb low frequency sounds in a narrow range of frequencies. Resonant panel absorbers consist of thin plywood or other membrane-like materials installed over a sealed airspace. These absorbers are tuned to specific frequencies, which are a function of the mass of the membrane and the depth of the airspace behind it. Resonant cavity absorbers consist of a volume of air with a restricted aperture to the sound field. They are tuned to specific frequencies, which are a function of the volume of the cavity and the size and geometry of the aperture. 

The discharge coefficient for the screen C with aperture D, is given as a function of screen R nolds number Re = D,(V/d) /[L in Fig. 6-16 for plain square-mesfi screens, Ot = 0.14 to 0.79. This cui ve fits most of the data within 20 percent. In the laminar flow region, Re < 20, the discharge coefficient can be computed from... 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

Since A x,y) can really be any function bounded by the aperture of the system it is best to use as general description as possible. One such description of this function is to expand A x, y) = A p, 0) about (p, 0) in an infinite polynomial series. One set of polynomials that are frequently used are Zernike polynomials. Thus one can write A(p, 0) = Y.m,n CmnFmn p, 0). 

Figure 14. One of the three contrasts (left) and phase closure (right)evolution as functions of the optical delay to be generated between the telescopes of a three aperture stellar interferome-ter(OPD between T1 - T2 and T2 - T3 reported in the horizontal plane).

With the advent of extremely large aperture telescopes, there is a growing interest in the statistical properties of the field emitted by astronomical sources (Dravins, 2001). The goal is to obtain important physical information concerning the source by looking at the statistical characteristics of the light it emits. This domain was pioneered by two radio astronomers, Hanbury Brown and Twiss (1956) who measured the intensity correlation function + r))... 

The wavefront is represented over a finite region, usually the instrument aperture, as a sum of basis functions, k(u, v). There is some flexibility in how we choose these functions, but essentially they should be chosen so that we can represent an arbitrary wavefront distortion, (p u, v), by... 

For a circular aperture a typical set of basis functions are the Zemike polynomials (Noll, 1976), but for other geometries alternative basis functions may be more appropriate. The objective of most wavefront sensors is to produce a set of measurements, m, that can be related to the wavefront by a set of linear equations... 

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