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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
Important features of toroidal mirrors

Ideal toroidal shape mirrors have a big aberration erased from ideal toroidal form reflection at off-axis beams. Toroidal shape without aberrations is an ellipsoid which production is much more difficult and more expensive. For achievement of the minimum distortion “ZILTA” adjust (fine tune works) the CA by means of inclusion small Aspherical areas of the highest order accordance to tested 2D topography, when the quality of the image is indicated as final result. “ZILTA” changes manufacturing techniques for ordered mirrors and the test scheme for certification to reach the best quality of images.

Slope error

According to the definition this parameter determines the angular tolerance for the deviation of the reflected rays by the mirror in comparison with the ideal surface. Slope Error characterizes the error of manufacturing the reflective surface of the mirror only- “Height Error”. This parameter characterizes the mirror itself, not the image it creates. Using the Slope Error parameter to characterize image quality is not correct, although there is a some dependency: Slope Error is the cause and image is the consequence. Very often the Image haves the good quality by good Slope Error (by good “Height Error”) with the exception of the commonly used Toroidal mirrors because of aberration distortion and low image quality. Even for theoretical surfaces, spherical aberrations can significantly distort the image when the rays fall on the toroidal mirror at grazing angles. To eliminate misunderstandings and help end-users the most effective form of the mirror, below is a simplified transformation of the toroidal surface into an elliptical mirror through an intermediate stage - "Aspherized toroidal surface".


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