Airy disk

The traditional analysis has then used a Gaussian to approximate both the Airy disk and sinc distribution as in Fig. 6 (Primot et al., 1990 Welsh and Gardner, 1989). The value of a in Eq. 41 is determined by matching the Gaussian approximation and the actual intensity at both the peak and the exp[—1] points (Welsh and Gardner, 1989). When this is done we find... 

An interesting result occurs when instead of using the Gaussian approximation the Airy disk is used directly in the calculations. In this case the variance goes to infinity,... 

Figure 5. Simulation of photon arrival for 100 photons for (a) the Airy disk and (b) the Gaussian approximation.

Point spread function (PSF) If a tiny population of 100 nm fluorescent beads sandwiched between a coverslip and a microscope slide are examined at high resolution (i.e. at 100x objective magnification, 1.4 NA. and in a correctly matched refractive index of oil), it can actually show a tiny set of rings in the horizontal (XY) view (also called an airy disk (see Fig. below). This airy disk cannot be avoided due to diffraction and the wave nature of light. If a specimen is optically sectioned and projected in a vertical (XZ) view (see Fig. xx), a set of concentric rings will flare from the center. When a three-dimensional image of this specimen is collected, a complete point spread function is said to be recorded for each bead. The (PSF)... 

In this figure airy disks of fluorescent beads can be seen in the XY (horizontal) plane (left) and the XZ (vertical) plane (right). Courtesy of Applied Precision, Inc., Issaquah, WA, USA. 

An Airy disk (named after George Biddell Airy) is the central bright circular region of the pattern produced by light diffracted when passing through a small circular aperture. The central disk is surrounded by less intense concentric rings. 

The factor 1.22 in Eq. 2.1 was empirically derived by Rayleigh. It may be derived from the radius of the circle, known as the Airy disk, from the optical transfer function. In 1873, the German physicist Ernst Karl Abbe (1840-1905) showed that the numerical... 

Figure 1.4 Intensity distribution of two Airy disks with a distance. Ii indicates the maximum intensity of each point and I2 represents overlap intensity.

Figure 1.6 Geometric relation among the depth of field (If). the half angle entering the objective lens (a) and the size of Airy disk (d).

For diffraction-limited microscopy, it may be thought that the image should be focused on the CCD such that the Airy disk fills one pixel. However, Adar et al. [41]... 

Light is focused through a pinhole in order to create a point source of light (or, more appropriately, an Airy disk of light), which is then refocused down onto the sample with a lens. Recall that information does not have to be exactly at the focal point of an optical lens for the lens to transmit the light information. Therefore, it is necessary to utilize a second pinhole just before the detector in order to exclude light information that is not exactly at the focal point. Hence, the name confocal has been used to describe this microscopy technique. 

Lord Rayleigh considered this problem and concluded that in any optical system an unwanted pathlength of A/4 or less can be tolerated, as having an extremely small affect on the size of the Airy disk. 

The objective lens aperture diffracts the incident beam, enlarging the focal point into an Airy disk with half-width = 0.611a. For this effect, the larger the value of a, the smaller contribution of dj. Thus, spherical aberration and aperture diffraction vary in opposite directions with a. This leads to the need to find an optimum aperture angle, aopt, that is a balance between these two effects. For perfect lenses the diffraction error forms the physical limit to the minimum obtainable probe size. For field-emission microscopes this term becomes significant. The electron wavelength, 1, is given approximately by 1 = 1.226/ sJU (U in V, /I in nm). 

Airy disk The diffraction pattern of a circular aperture, as seen in the image plane of a well-corrected lens. The radius of the Airy disk is a measure of the resolution limit of a well-corrected lens. 

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... 

According to the wave theory of aberrations the converging wave that propagates toward an image point should in principle have exactly spherical wavefronts. The maximum distance between the wavefront and a true sphere is called the wavefront aberration. If the wavefront aberration exceeds one-quarter wavelength, then the lens is aberration-limited. The point spread function is broader than the Airy disk, and the intensity of the peak is correspondingly reduced. The ratio of the peak intensity of the point spread function to that of the Airy disk is called the... 

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