Rayleigh criterion of resolution

Two objects are completely resolved if they are separated by 2r, and barely resolved if they are separated by r. The latter condition is sometimes known as the Rayleigh criterion of resolution. The largest numerical aperture that can generally be achieved for a Cassegrainian optic is approximately 0.6, so the diffraction-limited spatial resolution is approximately equal to the wavelength of the Hght when n = 1.0. 

The theoretical definition of the spatial resolution requires a distance of 2r (where r is the Rayleigh criterion of resolution) in order to differentiate two adjacent points completely, and is limited by the wavelength of the IR radiation and the NA of the objective [42, 43] (see Chapter 1, Section 1.2.4). 

The first point is derived from the Rayleigh criterion of resolution, which states that to resolve two spectral lines separated by a distance d, the interferogram must be measured to an optical path length of at least Md 29]. 

Figure 2.6. (a) Resultant of two sinc functions centered 1/A reciprocal centimeters apart, with each of the individual sinc functions shown as a dashed line this condition represented Rayleigh s original definition of resolution for two equally intense lines measured using a diffraction-limited monochromator. (f>) Resultant of two sine functions of equal amplitude centered O.S/A apart note that the features are unresolved in the resultant function, (c) Resultant of two sine functions centered 0.73/A apart the two features are resolved with a dip of approximately 20%, equivalent to the Rayleigh criterion of resolution fm a sinc function. 

This function is shown in Figure 2.7. As we saw from the discussion of the Rayleigh criterion of resolution, this is also the shape of the ILS function of a diffraction-limited grating monochromator. It can be seen that the amplitude of the sidelobes has been considerably reduced from that of the sidelobes for the sine function. Suppression of the magnitude of these oscillations is known as apodization, and functions such as Ai(S), which weight the interferogram for this purpose, are known as apodization functions. 

Using the Rayleigh criterion, the resolution of the mth diffraction order is then given by... 

R /f is the numerical aperture (N.A.) and is marked on all objective lenses, together with the magnification. For example, a x25 objective with N.A. = 0.5 is expected to resolve detail of the order of 2X 1 m, ideally. Equation (2.1) has been derived by considering diffraction by a coherently illuminated periodic object. Rayleigh s well-known criterion of resolution, derived for a nonperiodic object incoherently illuminated, gives nearly the same limit of resolution as that determined from the Abbe approach and need not be considered here (see, e.g., Giancoli 1984). 

Figure 5. The Rayleigh criterion for resolution of two sinc functions of equal amplitude.

Figure 3-10 shows how light from an infinitely narrow slit is focused to cover the face of the prism at the angle of minimum deviation. Two beams emerge of wavelengths X and X + A, which just meet the Rayleigh criterion for resolution. The beams are separated by the angle A0. Thus, A0 = Xja, where a is the beam width. Consequently... 

The resolving power of a grating has the same dimensions as for a prism and use i again made of the Rayleigh criterion for resolution. The numerical value of the resolving power is given by X/AX, where AX meets the Rayleigh definition. 

Lord Rayleigh introduced a criterion of resolution for diffraction-limited line profiles, where two lines are considered to be just resolved if the central diffraction maximum of the profile Ii(X — X ) coincides with the first minimum of h X — X2) [4.3]. 

Lord Rayleigh has somewhat arbitrary introduced a criterion of resolution in connection with prism and grating spectrometers, where the line profiles at the maximum attainable resolution are determined by diffraction and are of the form I(x - Xq) = I(XQ)[sin((x - Xq)/2)(x - Xq)/2] (see below). In this case two lines are considered to be just resolved if the central diffraction maximum of Ij(x - x ) coincides with the first minimum of l2(x - X2) [4.5a]. As shown below, the total intensity distribution I(X) exhibits in this case for I = I2 a dip between the two maxima which drops to (8/tt ) of the maximum intensity 1 (Fig.4.6). Generalizing this Rayleigh criterion to arbitrary line profiles, we may define the resolving power for any dispersing instrument by defining two lines with equal intensities to be just resolved if the dip between the two maxima drops to 2... 

The resolution or "resolving power" of a light microscope is usually specified as the minimum distance between two lines or points in the imaged object, at which they will be perceived as separated by the observer. The Rayleigh criterion [42] is extensively used in optical microscopy for determining the resolution of light microscopes. It imposes a resolution limit. The criterion is satisfied, when the centre of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This minimum distance r can then be calculated by Equation (3). 

Rayleigh criterion The rule-of-thumb for the spectral resolution of two transitions. 

What meaning do these two-point resolution criteria have in describing the deconvolution process, that is, resolution before and after deconvolution Although width criteria may be applied to derive suitable before-after ratios, the Rayleigh criterion raises an interesting question. Because the diffraction pattern is an inherent property of the observing instrument, would it not be best to reserve this criterion to describe optical performance The effective spread function after deconvolution is not sine squared anyway. 

All detection phenomena that use interference or diffraction have a natural limit, known as the Rayleigh criterion The maximum resolution obtainable with light of wavelength X is at best ... 

Dictated by the diffraction limit of light, the lateral resolution of a far-held optical microscope is dehned as the smallest distance between two features that can still be resolved in the image and is given by Rayleigh criterion as [5]... 

When two closely spaced spectral lines of equal intensity satisfy the Rayleigh criterion, the valley between the lines is approximately 19% of the peak intensity. This corresponds to an MTF of about 0.10. Based on the data presented under the discussion of Rayleigh resolution, a MTF of 0.10 would be expected to result from a pattern with about 20 lines/mm. 

If instead of the Rayleigh criterion the full width at half maximum (FWHM) criterium is considered, the angular resolution is AOjei = 1.02A./T) X/D. For an interferometer, two equal brightness sources will be resolved when the fringe contrast goes to zero at the longest baseline b, this is... 

Figure 2.9 shows the fringe pattern for a single telescope and an interferometer for a point source, where the baseline separation isb = 5D. It can be observed that the resolution of the interferometer is 12 times higher than the resolution of the single aperture if the Rayleigh criterion is used. In interferometry, the resolution of the single aperture is the field of view of the interferometer. 

VEM, and in partieular when coupled with PC and DIC optics, permits some bending of the Rayleigh criterion. Jokela et al. (32) experieneed a VEM resolution limit that was about half that stated by the criterion (0.1 um). As described above, the absenee of a magnification limit allows observation of objects smaller than the resolution limit, but the images will appear blurred with lack of detail. However, the contrast-enhaneing ability of VEM, PC, and DIC ean help clarify minute features normally lost owing to the blur-... 

According to the Rayleigh criterion, a diffraction-limited system can distinguish two equally intense points provided that their geometrical images are separated by at least one Airy disk radius. That distance is known as the Rayleigh limit and defines the resolution limit RL of a diffraction-limited system. For a diffraction-limited lens... 

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