Airy Pattern Basics
The three-dimensional diffraction pattern formed by a circular aperture near the focal point in a well-corrected microscope is symmetrically periodic along the axis of the microscope as well as radially around the axis. When this diffraction pattern is sectioned in the focal plane, it is observed as the classical two-dimensional diffraction spectrum known as the Airy pattern. This tutorial explores how Airy pattern size changes with objective numerical aperture and the wavelength of illumination; it also simulates the close approach of two Airy patterns.
The tutorial initializes with two adjacent Airy patterns and their corresponding radial intensity distributions positioned in the center of the applet window. The patterns, in close approach to each other, are generated by green light (546 nanometer wavelength) passing through a circular aperture. Appearing beneath the Airy patterns is a simulation of the illumination cone projected into the microscope objective the by substage condenser, and a set of sliders that are used to control the tutorial. The Wavelength slider changes the illumination wavelength through a range of 400 nanometers (purple-blue visible light) up to 700 nanometers (red light). Adjacent to the wavelength slider is the Numerical Aperture slider, which is used to modulate the numerical aperture of a virtual objective through which the Airy patterns are generated.
The Separation Distance slider is utilized to translate the Airy patterns and radial intensity distributions back and forth in the image plane (parallel to the browser window). As this slider is moved to the right or left, the distance between Airy patterns becomes either greater or smaller, and the current separation distance, in micrometers, is provided both above the slider knob and in the Airy pattern window. Moving the Separation Distance slider to the right causes the Airy patterns to approach each other, stopping at the limit of resolution. Airy pattern size will decrease with illumination wavelength and with numerical aperture. The simulated light cone increases in size with increasing numerical aperture.
Every point of the specimen is represented by an Airy diffraction pattern in the focused image plane of the microscope. It follows that the resolving power of an objective lens can be determined by examining the size of the Airy pattern formed by that lens. The Airy pattern radius is governed by the wavelength of illumination and the combined numerical apertures of both the objective and condenser.
When the specimen is illuminated by a large-angle cone of light, or for self-luminous objects, the light rays forming adjacent Airy patterns are incoherent and do not interfere with each other. This makes it possible to determine the minimum separation distance that can be resolved with a particular objective by examining the total intensity distribution of closely spaced, or overlapping, Airy patterns in the intermediate image plane. In the case of Airy patterns generated by coherent light waves, adjacent diffraction patterns would interfere with each other, increasing the minimum separation distance necessary to resolve the individual patterns.
The peak-to-peak distance (Separation Distance, D) between adjacent intensity distribution curves equals that between corresponding Airy diffraction patterns. If the Airy pattern radius is defined as the distance r, and when D is greater than r, the sum of intensities of the pair of Airy patterns clearly shows two peaks. In the case where the separation distance (D) equals the Airy pattern radius (r), two overlapping peaks are observed. This condition is known as the Rayleigh criterion, which is the minimum distance between Airy patterns that can be separately resolved. When the separation distance is less than the Airy pattern radius (not illustrated in the tutorial), the intensity distributions merge into a single peak and they are said not to be resolved.
Kenneth R. Spring - Scientific Consultant, Lusby, Maryland, 20657.
Matthew J. Parry-Hill and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.
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