Airy diffraction

Fig. 8j4 Definition of the angle ec, which is important for the numerical aperture NA, and two examples for the variation of the Airy diffraction image as a function of NA. The effective angle, under which the object is seen, amounts to 37° (left-hand side) and 58° (right-hand side).

The numerical aperture (more exactly the square of the numerical aperture) describes the light collection efficiency of an objective and thus determines resolution and sensitivity. The values of numerical apertures above unity can be reached only if one takes advantage of total internal reflection. For that purpose, the objective and object holder are connected by a medium with high index of refraction (e.g., an immersion oil or water). In this way light is refracted via the immersion medium towards the normal of the objective (Fig. 8.4). Hence a larger number of Airy diffraction orders reach the imaging system and thus the contrast increases. 

For the simplest of mirrors, circular apertures, the effects of diffraction cause the diffraction-limited image to be an Airy pattern, and for large distances this pattern falls as 1/0 and will be azimuthally symmetric. 

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... 

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)... 

Figure 3.1 X Calculated intensity profiles for a simple, full aperture objective (Airy pattern, left) and a Schwarzschild objective with central obscuration (right). The same NA and wavelength were used for both calculations. Note the large first-order diffraction ring for die Schwarzschild objective.

Figure 3.12 Calculated sensitivity as a function of radial distance from the center of an objective s diffraction pattern, comparing the nonconfocal case for a Schwarzschild and a conventional objective (no obscuration). The plateau near 5 p,m distance for Schwarzschild corresponds to the first diffraction minima. Thus, only j of the sensitivity is located in the diffraction pattern s central peak. Contrast this to the Airy pattern where more than 80% is contained in the central maximum.

Airy formula, intensity distribution of the Fraunhofer diffraction pattern, 148... 

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)... 

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. 

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. 

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]... 

Figure 3. Confocal optical detection channel demonstrating the concept of spatial filtering. A microscope objective lens collect the light emitted from a point light source or a single molecule. The image appears as a diffraction pattern (Airy pattern, see insert). The diameter of the pinhole placed in the image plane is such that only light from the bright central spot can pass onto the detector. Radiation from an out-of-focus light source in the sample is efficiently discriminated.

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). 

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