Revised Dove prism formulas

H. Z. Sar-EI E.-Op Electro-Optic Industries, Ltd. P.O. Box 1165, Reho-vot 71660, Israel. Received 25 May 1990. 0003-6935/91/040375-02$5.00. © 1991 Optical Society of America.

Examination of the known formulas of the Doυe prism led to their revision. The variation of the revised length and weight of the Dove prism as a function of prism base angle is depicted for BK7 and SF6 glasses and interpreted.

The Dove prism formulas, as found in accepted reference books,1-3 are all similar in shape except for missing brackets in the denominator of one of them. References 1-3 present the same numerical example for the Dove prism made of glass with refractive index n = 1.5170 and α = 45° base angle. Attempts to rederive the known formulas failed, and instead we obtained a different set. Both sets appear to fit for the α = 45° case only. A comparison between the old and the new reveals an apparent reason for this single fit. Adopting the nomenclature of Ref. 3, the new formula set has the following expressions:

where A is the width and the height, B is the length, α is the base angle, and n is the refractive index of the prism; T stands for the path length of the center ray in the prism. The length of the prism in units of A, according to Eq. (1), is plotted in Fig. 1 as a function of its base angle for two different optical glasses: BK7(n = 1.51680) and SF6(n = 1.80518).4

As is clearly seen from Fig. 1, there is a base angle for which the prism length is minimum. The same attribute has not been found for weight W given by the relation

where p is the glass density. Equation (3) is plotted in Fig. 2 for the above-mentioned glasses, where W is given in units of ρA3.

The fact that the Dove prism has a minimum length for a certain base angle is not new. Vaughan5 reported on α ≈ 27°

Fig. 2. Variation of the Dove prism weight with its base angle.

Fig. 1. Variation of the Dove prism length with its base angle.

as the base angle for which the length of a Dove prism having n = 1.517 is minimum. As seen in Fig. 1, the optimum base angle is ≈32.5°. The different behavior of the length in Fig. 1, in comparison to the weight curves in Fig. 2, can be simply explained by Eqs. (1) and (3). The prevailing term that brought about the minimum length is sin(2α) in the denomi-

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nator in Eq. (1). For bot'h extreme values, α = 0 and α = π/2, Eq. (1) is singular and thus a minimum is achieved. This is not the case in Eq. (3) where the cotangent term balances the minus term for α = 0 but not for the other extreme value α = π/2. Thus, the asymmetric shape of the weight curve is obtained.

References 1. Military Standardization Handbook, Optical Design, MIL-

HDBK-141 (5 Oct. 1961), pp. 13-34 and 13-35.

2. L. Levi, Applied Optics (Wiley, New York, 1968), p. 388.

3. W. G. Driscoll, Ed., Handbook of Optics (McGraw-Hill, New York, 1978), p. 2-46.

4. Schott pocket catalog of optical glasses (Schott GmbH, Optics Division, P.O.B. 2480, D-6500 Mainz, Germany, 1986), pp. 48 and 82.

5. W. Vaughan, "Dove Prisms Properties and Problems," Opt. Spectra 15, 68-70 (Oct. 1981).

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