Spatial frequencies

The detectability of critical defects with CT depends on the final image quality and the skill of the operator, see figure 2. The basic concepts of image quality are resolution, contrast, and noise. Image quality are generally described by the signal-to-noise ratio SNR), the modulation transfer function (MTF) and the noise power spectrum (NFS). SNR is the quotient of a signal and its variance, MTF describes the contrast as a function of spatial frequency and NFS in turn describes the noise power at various spatial frequencies [1, 3]. 

FbwO) is the Fourier transformation of effective beam width as a function of spatial frequency / Fuff) is the MTF of the XRll. Because of the XRll windows curvature, projection data must be transformed to obtain uniform pixel spacing, described by Errors in object centre... 

A single echographic "shot" at frequency oj and incidence n gives rise to a point K = -2kn( (with k = co/cq) in the spatial frequency space. 

After propagation into the back focal plane of tire objective lens, the scattered electron wave can be expressed in tenns of the spatial frequency coordinates k as... 

While the spatial resolution in classical microscopy is limited to approximately X/2, where X is the optical wavelength (tlie so-called Abbe Limit [194], -0.2 pm with visible light), SNOM breaks through this barrier by monitoring the evanescent waves (of high spatial frequency) which arise following interaction with an... 

Different values of will result if the integral limits (i.e., band width) or modulation transfer function in the integral change. All surface characterization instruments have a band width and modulation transfer function. If rms roughness values for the same surface obtained using different instruments are to be compared, optimally the band widths and modulation transfer functions would be the same they should at least be known. In the case of isotropic surface structure, the spatial frequencies p and q are identical, and a single spatial frequency (/>) or spatial wavelength d= /p) is used to describe the lateral dimension of structure of the sample. 

The modulation transfer function of the optical scatterometer is nearly unity. The spatial frequency band width, using 0.633-nm photons from a He-Ne laser, is typically 0.014—1.6 jim corresponding to a spatial wavelength band width 70— 0.633 pm. This corresponds to near normal sample illumination with a minimum... 

The first example is the 4-m class William Herschel telescope, at la Pakna, whose optical specifications, drafted by D. Brown, were expressed in terms of allowable wavefront error as a function of spatial frequencies matching those of atmospheric turbulence. 

Another problem with all the dwell time methods is that sinee a speeifie diameter tool is often used to do the polishing, the finished surfaee has a "roughness" of a spatial frequency associated with the tool diameter. This rather eo-herent roughness ean produce diffraction artifacts in the image produeed by the telescope. A partial solution to this problem is to use several tool sizes and do the figuring in stages rather than all at onee. 

The main error sources are noise in the wavefront sensor measurement, imperfect wavefront correction due to the finite number of actuators and bandwidth error due to the finite time required to measure and correct the wavefront error. Other errors include errors in the telescope optics which are not corrected by the AO system (e.g. high frequency vibrations, high spatial frequency errors), scintillation and non-common path errors. The latter are wavefront errors introduced in the corrected beam after light has been extracted to the wavefront sensor. Since the wavefront sensor does not sense these errors they will not be corrected. Since the non-common path errors are usually static, they can be measured off-line and taken into account in the wavefront correction. 

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

The spatial frequency measured by the interferometer is a two-dimensional quantity. It depends on the three-dimensional baseline B between telescopes... 

This equation shows that only limited information is preserved. In particular, depending on the spatial frequency Mo, no information is transferred at all at the zeroes of the phase-contrast function sin(x). The loss of information is even more serious when the phase object approximation holds and for ideal imaging in that case the phase information is completely lost in the Gaussian image of the object and special methods the so-called phase-contrast [94,95] methods should be employed in order to partly recover this information. 

Fourier transform, h(r), does not contain spatial frequencies above a value of / , i.e.,... 

The classical treatment of diffuse SAXS (analysis and elimination) is restricted to isotropic scattering. Separation of its components is frequently impossible or resting on additional assumptions. Anyway, curves have to be manipulated one-by-one in a cumbersome procedure. Discussion of diffuse background can sometimes be avoided if investigations are resorting to time-resolved measurements and subsequent discussion of observed variations of SAXS pattern features. A background elimination procedure that does not require user intervention is based on spatial frequency filtering (cf. p. 140). 

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