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Abbe Limit - Ernst Abbe's specification for the limit of resolution of a diffraction-limited microscope. According to Abbe, a detail with a particular spacing in the specimen is resolved when the numerical aperture (NA) of the objective lens is large enough to capture the first-order diffraction pattern produced by the detail at the wavelength employed.
In order to fulfill Abbe's requirements, the angular aperture of the objective must be large enough to admit both the zeroth and first order light waves. Because the angular aperture (a) is equal to the arcsine of the wavelength (l) divided by the resolution (R), the finest spatial frequency that can be resolved equals:
In the numerical aperture equation, n is the refractive index of the imaging medium (usually air, oil, glycerin, or water). Ernst Abbe demonstrated that in order for a diffraction grating image to be resolved, at least two diffraction orders (usually the zeroth and the first) must be captured by the objective and focused at the rear focal plane. As the numerical aperture increases, additional higher-order rays are included in the diffraction pattern, and the integrity of the specimen (line grating) becomes clearer. When only the zeroth and first orders are captured, the specimen is barely resolved, having only a sinusoidal intensity distribution within the image.
Abbe Number (or V Number) - The value (often symbolized by the Greek letter n or the letter V) expressing the dispersion of an optical medium, which is directly proportional to the chromatic quality of a lens. Its value is given by the expression:
Where n(d), n(F) and n(c) are the refractive indices pertaining to the wavelengths of the Fraunhofer lines 587.6, 486.1, and 656.3 nanometers respectively. The Abbe number of a glass mixture is carefully monitored and controlled by manufacturers to achieve target dispersion values, which usually range between 20 and 60. A typical plot of Abbe number versus refractive index is presented in the figure below. The glass map is subdivided into regions that represent various types of glass.
There are over 250 glass formulations that have differing refractive index and Abbe numbers, many of which are represented by points in the glass map illustrated above. Refractive indices range between 1.46 and 1.97 and have corresponding V-numbers from 20 to 85 (although the glass map illustrated above ends at V = 70). Utilizing glass maps, optical designers can locate the closest match to a glass made by another company or determine the range of V-number formulations available for a particular refractive index.
Only a fraction of the large number of known glasses are employed in optical design on a routine basis, with about 50 glass formulations being useful based on cost, ease of optical fabrication, thermal stability, and uniqueness. In the sub regions of the glass map, individual formulations have designations that end in K (referring to a crown glass) or F (flint glass). The border between crown and flint glasses (at V = 50) for the entire refractive index spectrum has been included in the glass map above. Another important boundary in the map extends from the low refractive index and high V-numbers on the lower left to high refractive index, low V-number formulations on the upper right. When all available glass formulations are included on the map, about one third cluster around this boundary.
Abbe Test Plate - A device for testing the chromatic and spherical aberration of microscope objectives. When testing for spherical aberration, the cover glass thickness for which the objective is best corrected is also found. The test plate consists of a slide on which is deposited an opaque metal layer in the form of parallel strips arranged in groups of different widths. The edges of these strips are irregularly serrated to allow the aberrations to be judged more easily. The slide is covered with a wedge-shaped cover glass, whose increasing thickness is marked on the slide.
Presented in the figure above is an illustration of a typical Abbe test plate used to test for spherical aberration in microscope objectives. The plate is constructed with multiple coverslips (an alternative to the wedge-shaped cover glass) having a thickness range varying between 0.09 and 0.24 millimeters (greatly exaggerated in the drawing), each covering a series of straight lines engraved through a silver coating leaving alternating clear and opaque areas. A practical application of the test plate is to determine the optimum coverslip thickness for a given objective, and to ascertain the degree of image deterioration due to thickness variations. The plate can also be employed to calibrate the amount of tube length adjustment that must be made to compensate for deviations in coverslip thickness in older microscopes.
Abbe Theory (of Image Formation) - An explanation of the mechanism by which the microscope image is formed. It is based on the necessity for the light rays diffracted by the specimen to be collected by the objective and allowed to contribute to the image; if these diffracted rays are not included, the fine details which give rise to them cannot be resolved.
Ernst Abbe demonstrated that if a grating is employed for a specimen and its conoscopic image is examined at the rear aperture of the objective lens with the condenser aperture closed to a minimum, an orderly series of images of the condenser iris opening is observed. These images are arranged in a row at right angles to the periodic line grating. For gratings with broad spacings, several condenser iris images appear within the aperture of the objective lens, perhaps overlapping each other depending on the side of the iris diaphragm opening. Gratings having narrowing spacings display a greater degree of separation between the iris diaphragm images, and fewer become visible. Thus, there is a reciprocal relationship between the line spacings in the specimen and the separation of the conoscopic image at the aperture plane. What is actually observed is the diffraction pattern from the specimen, or the image of the condenser iris diffracted by the periodic spacing of the line grating.
The figure presented above illustrates a schematic drawing of a microscope optical system consisting of a condenser iris diaphragm, condenser, and objective with a periodic grating representing the specimen. The periodic specimen diffracts a collimated beam (arising from each point of the condenser aperture), giving rise to first-order, second-order, and higher order diffracted rays on both sides of the undeviated zeroth-order beam. The diffracted rays occur by constructive interference at a specific angle (f). Each diffracted-order ray (including the zeroth) is focused at the rear focal plane of the objective. The period (s) between the focused diffraction orders is proportional to the numerical aperture of the ray entering the objective. The situation is governed by the equation:
Where f is the focal length of the aberration-free objective that fulfills the sine condition, l is the wavelength of light in the specimen plane, and f is the angle between the lens axis and the diffracted wave.
The most surprising fact about Abbe's experiments is that when the first-order diffraction pattern is masked at the objective rear aperture, so that only the zero- and second-order diffraction patterns are transmitted, the image of the specimen appears with twice the spatial frequency, or with only half the spacing between the lines. In the absence of what is really the first-order diffraction pattern, the image is now generated by interference between the zero- and second-order diffraction patterns, the latter of which is masquerading as the first-order pattern. The observation proves that the waves making up the diffraction pattern at the aperture plane converge and interfere with each other in the image plane and generate the orthoscopic image.