Airy-disc

As an example, consider a planar wavefront from an incoherent source passing through an aberration-free circular lens. When the image is diffraction-limited, an Airy disc pattern is observed (Goodman, 1996). For an aperture of radius 1 / 2n) the pdf for photon arrival is given by... 

Figure 6. Comparison of the (a) Airy disc (oscillating) and Gaussian (monotonically decreasing) distributions, and (b) sinc (oscillating) and Gaussian (monotonically decreasing) distributions.

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). 

If the wavelength in the solid is Ai, then the Airy disc due to diffraction has its first minimum at a distance from the axis (Hecht 2002)... 

If single molecules emit in the sample they show up as bright spots on the camera. The magnification of the microscope is adjusted in such a way that the Airy disc formed on the CCD by a single emitter typically is extended over several pixels which simplifies later analysis. A typical single molecule... 

Fig. 4 Airy discs of two adjacent features with different separations. Fringes of the Airy Disc are not shown...

The radius, r, of the Airy Disc is given by the Rayleigh criterion equation which is shown in Eq. 1. ... 

It is important to have reliable laser diagnostics, preferably on a shot-to-shot basis this is possible for a 10 Hz system. To characterise the temporal profile of the pulses, single-shot spectra and autocorrelation data can establish whether the laser pulses are Fourier-transform limited. For lasers with a sech profile, a product AvAtxO.32 should be achieved. It is also important to monitor the focal spot, its Airy disc and average intensity. Alternatively, a reasonable measure of the focused intensity can be obtained using Xe gas and the known threshold intensities for producing the various stages of ionization [12]. 

A.29.6 The numerical aperture (NA) is a measure of the number of orders of diffraction that can be captured by a lens, NA = n sin u. Since the image information is carried in all of these orders of diffraction, the wider the collechon angle u, the greater the magnitude of NA and the higher the resolving power, i.e. the smaller the distance between two objects set of Airy discs needed to visually separate them. 

The diffraction pattern resulting from a uniformly illuminated circular aperture actually consists of a central bright region, normally known as the Airy disc, which is surrounded by a number of much fainter rings. A circle of zero intensity separates each bright... 

The resolution of the microscope is limited here only by the small aperture halfangle opening a, so that r = a is almost negligible. Resolution is limited now by the aperture diffraction aberration (Airy disc) = 0.61 X/a. 

In the case of amorphous materials, the reciprocal space does not contain localized scattered beams. It is fuU of intensity with some faint maxima (see Section 1.1.2.4 and Figure 1.3). The objective aperture is thus illuminated in any of its positions. It produces bright dots due to the aperture Airy disc combined with the beams scattered by statistical overlapping of atom pairs. The dots are thus not localized in the real space. 

FIGURE 1.23 Effect of the objective aperture on the bright dot size and position (a) Bright dot size versus size of the Airy disc aperture. Curve A corresponds to amorphous carbon film. Curve B corresponds to localized coherent domains, (b) As aperture size increases in the same position (insets), the bright dot size decreases, (c) Effect of variable defocus (-700 A and zero) upon bright dot images (amorphous state), (d) Effect of same defocus in the case of coherent domains (heat-treated fihn). (Adapted from A. Oberlin, M. Oberlin, and M. Maubois. Study of thin amorphous and crystalline carbon films by electron microscopy. Phil. Mag. 32, 833-846 (1975). With permission.)... 

The amount of light passing through the interferometer will remain essentially a constant if averaged over the entire field of view. Only the central (Airy) disc will vary as the interfere-gram of the source. For this reason, one places an aperture at the focus of the interference pattern and only allows the central portion of the fringe pattern to fall on the detector. 

Resolution depends on the distance between two distinguishable radiating points. Here we assume an adequate level of contrast in a mathematical model of Airy discs. However, real optical systems are complex and it is difficult to increase the distance between distinguishable point sources. 

Fig. 8.2 Optical resolution limit difTraction pattern (Airy disc) and transmission function of a single point. Black colored regions denote maximum intensity. The abscissa is given in units of sin a with A the wavelength of the light, a the radius of the aperture...

The diameter of the Airy disc for an imaging lens with focus length / and diameter D is... 

Fig. 8.3 Optical resolution limit diffraction pattern (Airy discs) and transmission function of two neighboring points.

Airy disc Central portion of the diffracted image formed by a circular aperture. Contains 84% of the total energy in the diffracted image formed by an unobstructed aperture. Angular diameter = 2.44 X/D, where A, is wavelength and D is the unobstructed aperture diameter. First determined by G. B. Airy in 1835. 

Diffraction-limited Term applied to a telescope when the size of the Airy disc formed by the telescope exceeds the limit of seeing imposed by the atmosphere or the apparent size of the object itself. 

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