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Spatial Frequency and Image Resolution

When a line grating is imaged in the microscope, a series of conoscopic images representing the condenser iris opening can be seen at the objective rear focal plane. This tutorial explores the relationship between the distance separating these iris opening images and the periodic spacing (spatial frequency) of lines in the grating.

A method often utilized in defining the limiting resolution of an optical microscope is to observe the conoscopic image of periodic line gratings as a function of spatial frequency and numerical aperture. Conoscopic images of the condenser iris opening, which represent a diffraction pattern induced by periodic spacing in the specimen, are oriented at right angles to the long axis of lines in the periodic grating. If the grating has very large spacings between adjacent lines, several images of the condenser iris appear within the objective lens aperture when the rear focal plane is viewed with a focusing telescope or Bertrand lens. In cases where the iris is not closed to its smallest size, these images can overlap with each other.

To operate the tutorial, use the Spatial Frequency slider to change the periodicity (spatial frequency) of the line grating, measured in cycles per millimeter (or line pairs/mm). When the tutorial initializes, the slider is set to the highest spatial frequency (2000 line pairs/mm) for the line grating, which corresponds to the largest spacing between adjacent images of the condenser iris diaphragm observed in the objective rear focal plane. Illustrated next to the schematic microscope diagram is an orthoscopic view of the line grating whose spatial frequency is dependent upon the slider position. Moving the slider causes the spatial frequency (number of lines) of the line grating to increase or decrease, resulting in a simultaneous decrease or increase in the spacing (S) of condenser iris images at the objective rear focal plane. From this tutorial, it is evident that a reciprocal relationship exists between the periodicity of line spacings in the specimen (line pairs/mm) and the separation distance (S) of diffracted conoscopic images at the objective aperture plane.

The schematic microscope drawing (on the left-hand side of the tutorial) depicts the zeroth and higher-order diffracted light waves that are focused at the objective rear focal plane. Both the condenser and objective are represented by a single lens element, and the condenser iris diaphragm opening is shown at the bottom of the figure. The specimen (a line grating) is depicted as a dashed line through which illuminating light rays pass from the condenser iris. Orientation of the diffracted rays are governed by the equation:

S/f @ l/D = sin(f)

where f is the objective focal length, l is the wavelength of light in the specimen plane, and f is the angle between the lens axis and the diffracted wave. The conoscopic image period between focused diffraction orders, (S), is proportional to the numerical aperture of the light rays entering the objective lens:

S @ n(sinf)

where n is the refractive index of the imaging medium. Ernst Abbe demonstrated that in order for the diffraction grating image to be resolved, at least two diffraction orders (usually the zeroth and the first) must be captured by the objective lens and be 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.

Contributing Authors

Kenneth R. Spring - Scientific Consultant, Lusby, Maryland, 20657.

Brian O. Flynn 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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