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A Modified Sine Condition For Grazing Incidence Optics

The high quality of some experimental x-ray images formed at grazing incidence by a single reflecting surface raises some questions about the interpretation of Abbe's sine condition. In standard textbooks Abbe's sine condition is usually derived for the refractive case. What if any changes in its form or interpretation occur in the case of reflection and in particular grazing incidence reflection. A commonly encountered form of Abbe's sine condition is n'y' sin θ' - ny sin θ = 0 where n and n' are the refractive indices of object and image space respectively. Since the product of refractive index and geometrical length is defined as an optical path-length, one sees in the statement of Abbe's sine condition essentially a restatement of Fermat's principle. Any application of the zero form of Abbe's sine condition would rule out the single reflector as a good imaging device free of coma at grazing incidence because θ increases as e' decreases. By making use of a geometrical construction dating back to a Thomas Young publication, circa 1807, a modified sine relationship applicable to reflection at grazing incidence can be formulated as n'y' sin θ' - ny sin θ = yy'/f where f is the focal length. Using an error analysis procedure the modified sine relationship can be put into a more practical form and the results tabulated for a range of focal lengths f in cm, objects heights y in microns and magnifications M. Experimental results for a single reflector adjusted to satisfy the modified sine condition at grazing incidence will be shown.