Magnification of Small Ray Deviations by Laser Rods with Brewster-Angle Ends James N. Bradford

U.S. Naval Research Laboratory, Washington, D.C. Received 7 June 1965.

Occasionally, in the rush to develop the modern field of quan­tum optics, some important phenomenon in classical optics is lost sight of. We recently had occasion to observe this when we examined some discrepancies in measurements on the beamspread introduced by refractive inhomogeneities in large ruby rods. The spread of a gas laser beam by the ruby had been measured independently and precisely by two groups, and the measure­ments disagreed. We discovered that the discrepancy resulted from the fact that the two sets of measurements had been made with different rod geometries. Measurements had first been made through parallel end surfaces normal to the rod axis. Then the rods had been refinished with ends canted for inci­dence at Brewster's angle and the second set of measurements were made.

The discrepancy of course arose from the introduction of angular magnification by the Brewster-angle geometry. This type of classical prismatic magnification seems first to have been treated by Lord Rayleigh in a demonstration based on Fermat 's principle.1 In view of the fact that a lack of knowledge of this concept has already caused a certain amount of bewilder­ment (and expense) in some laser circles, it seems appropriate to note here a simple analysis which may be helpful to those who design and evaluate laser systems in which beam divergence is critical.

Let θ' be the angle of incidence within a medium of refractive index n, and 0 the angle of refraction in air. Then sinθ = n sinθ' and by differentiation: cosθ dθ = n cosθ' dθ', or

where the primary deviation dθ' lies in the plane of incidence. In general, the primary incident ray and the deviated incident ray determine a plane at a dihedral angle γ to the primary plant of incidence (Fig. 1). If we denote the general deviation angle by dα,

and cosθ' cosdθ' - sinθ' sindθ' = cosθ' cosdα - sinθ' sindα COSΓ. In the limit, cosθ' - sinθ' dθ' = cosθ' - sinα' COSΓ dα, and

Substituting this into Eq. (1), we have dθ = n(cosθ'/cosθ) cos Γ dα, or

Fig. 1. Angular relationships of primary and deviated rays.

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If θ' = 0, as in a laser rod whose end faces are normal to the rod axis, do' = da and

We may now define an angular magnification for any angle of incidence 0',

For ruby rods (η 1.765) with Brewster-angle ends, tan0 = 1.765, θ = 60° 28', and θ' = 29° 33'. Then m = (0.8699 ÷ 0.4929) COST = 1.765 cosγ. The average or mean magnifica­tion then is m = (1.765 ÷ π/2) = 1.12. This value is in sub­stantial agreement with the discrepancies which had to be re­solved.

References 1. R. W. Wood, Physical Optics (Macmillan, New York,

1934), 3rd ed., p. 77.

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