deviation of light incident at the brewster angle on a wedged window
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Deviation of light incident at the Brewster angle on a wedged window Eric W. Young
United Technologies Optical Systems, Inc., Optics and Applied Technology Laboratory, P.O. Box 109660, West Palm Beach, Florida 33410-9660. Received 27 July 1987. 0003-6935/88/010019-02$02.00/0. © 1988 Optical Society of America.
Dispersing prisms have been investigated since the days of Newton. They have been used in precision spectrometers since last century and in many applications are preferred over gratings. A window with a small wedge can be thought of as a section of a prism with a small apex angle. We return to this comparison below.
The general case of light incident on a wedged window is treated first. The results are then applied to the case in which the angle of incidence is the Brewster angle. Snell's law of refraction, applied to the first surface, yields the following result (Fig. 1):
The angle of incidence is 0t; the transmitted angle is θt; and n is the index of refraction. The light transmitted through a window with wedge angle θω will emerge from the second surface slightly deviated by dθ from the original direction. Snell's law applied at the second surface yields
This equation is next expanded using well-known trigonometric formulas. The following small-angle approximations are used:
Fig. 1. Light transmitted through a slightly wedged window.
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(For high precision work, higher-order approximations can be developed.) The result, for general angle of incidence, is
When light is incident at the Brewster angle, θi is replaced by θb, for which
Substituting results in the following,
This interesting result hints at the quadratic dependence of the deviation angle on the index of refraction. This result can also be derived from the standard equations for a refracting prism.1
Summarizing: An intriguing and very simple result has been derived for the deviation of light through a wedged window placed at the Brewster angle.
References 1. R. Kingslake, "Dispersing Prisms," in Applied Optics and Opti
cal Engineering, Vol. 5 (Academic, New York, 1969), p. 12.
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