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Toroidal Mirrors(Definitions)

Toroidal mirrors are focusing devices having two different radii whose axes are oriented perpendicularly. They are utilized in instances where a beam must be focused and folded. Rather than using both a spherical mirror and a plane mirror for this purpose, both functions may be combined in one element. Toroidal mirrors also correct for the astigmatism that result when a spherical mirror is used off axis.

α – the angle of incidence (to Normal); one half the included angle of the incident and reflected rays;
S – Object conjugate distance; S’ – Image conjugate distance;
S’s – for the sagittal plane; S’t – for the tangential plane;
r – the Radius whose axis of revolution is oriented in the sagittal plane; also called the cylinder curve.
R – the Radius whose axis of revolution is in the tangential plane; also called the base curve.
1/S +1/S’s = 2cosα /r =1/fsag; 1/S + 1/S’t = 2/Rcosα=1/ ftan; r/R=cos 2α; ftan=fsag , if cosα r/R

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