Modulation Transfer Function
Diffraction Effects on Image Contrast
A popular mechanism for interpretation of the modulation transfer function (MTF) of an optical system is to image a precisely defined target having a repeating structure with 100 percent contrast. This interactive tutorial utilizes a periodic line grating as the specimen and simulates images produced with a diffraction-limited optical microscope as a function of spatial frequency.
The tutorial initializes at the lowest spatial frequency (20 line pairs per millimeter) represented as a repeating series of black and white rectangular bars in the periodic grating presented on the left-hand side of the applet window. The resulting image produced in the microscope is shown on the right side of the objective, and appears as a sinusoidal intensity that has reduced contrast, which is plotted in the graph below the image in terms of a relative percentage of the object contrast. One hundred percent contrast represents regular white and black repeating bars, while zero percent contrast is represented by gray bars on a gray background. As the Spatial Frequency slider is moved from left to right, the grating line spacing is decreased (spatial frequency increases) and the image loses contrast. At the highest spatial frequency available with the tutorial (1000 line pairs per millimeter), the image has effectively lost all contrast and appears a soft shade of gray.
In optical microscopy, signal frequency can be equated to a periodicity observed in the specimen, ranging from a metal line grating evaporated onto a microscope slide or repeating structures in a diatom frustule to subcellular particles observed in living tissue culture cells. The number of spacings per unit interval in a specimen is referred to as the spatial frequency, which is usually expressed in quantitative terms of the periodic spacings (spatial period) found in the specimen. A common reference unit for spatial frequency is the number of line pairs per micron or millimeter. As an example, a continuous series of black and white line pairs with a spatial period measuring 1 micrometer per pair would repeat 1000 times every millimeter and therefore have a corresponding spatial frequency of 1000 lines per millimeter.
Modulation of the output signal, the intensity of light waves forming an image of the specimen, corresponds to the formation of image contrast in microscopy. Therefore, a measurement of the MTF for a particular optical microscope can be obtained from the contrast generated by periodic lines or spacings present in a specimen that result from sinusoidal intensities in the image that vary as a function of spatial frequency. If a specimen having a spatial period of 1 micron (the distance between alternating absorbing and transparent line pairs) is imaged at high numerical aperture (1.40) with a matched objective/condenser pair using immersion oil, the individual line pairs would be clearly resolved in the microscope. The image would not be a faithful reproduction of the line pair pattern, but would instead have a moderate degree of contrast between the dark and light bars. Decreasing the distance between the line pairs to a spatial period of 0.5 microns (spatial frequency equal to 2000 lines per millimeter) would further reduce contrast in the final image, but increasing the spatial period to 2 microns (spatial frequency equal to 500 lines per millimeter) would produce a corresponding increase in image contrast.
The limit of resolution with an optical microscope is reached when the spatial frequency approaches 5000 lines per millimeter (spatial period equal to 0.2 microns), using an illumination wavelength of 500 nanometers at high numerical aperture (1.4). At this point, contrast would be barely detectable and the image would appear a neutral shade of gray. In real specimens, contrast depends upon the size, brightness, and color of the image, but the human eye ceases to detect periodicity at very low contrast levels and may not reach the 0.2-micron limit of resolution.
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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