142 RADIATION BIOLOGY 



where n is the refractive index and ri and r2 are the radii of curvature 

 of the surfaces. The plano-convex lens has one surface that is a plane, 

 i.e., fi is infinite, and the formula becomes 



1// = (n - \)l/r. (3-9) 



The radii are positive for convex surfaces and negative for concave 

 surfaces. 



Image Defects. The preceding equations are only approximately cor- 

 rect for spherical surfaces. Parallel rays are not brought to a sharp focus 

 by either spherical mirrors or simple spherical lenses ; this imperfection is 

 known as "spherical aberration." Spherical aberration is avoided in con- 

 cave mirrors by the use of a parabolic surface that has the property of 

 bringing all rays parallel to the principal axis to a focus at one point F. 

 The parabolic reflector is used in astronomical mirrors because the objects 

 are at essentially infinite distance and the incident rays are parallel. 

 They are also used in spectroscope collimators for producing parallel flux 

 and as focusing optics for conveying the parallel rays into the focal curve 

 of the spectrum. The elHpse is the ideal curve for condensing mirrors 

 where two conjugate foci are used, since the ellipse causes all rays from 

 one focus to be imaged at the other. 



Spherical aberration is corrected in lenses by the proper choice of radii 

 of curvature and the use of aspherical surfaces. In plano-convex lenses 

 the spherical aberration is at a minimum when the parallel rays are inci- 

 dent to the curved surface (Fig. 3-23a). For this reason plano-convex 

 lenses are often used in condensing systems, with the convex surfaces 

 facing one another. 



A second image-forming defect of both mirrors and simple lenses is 

 astigmatism, which arises when an image is formed by rays that origi- 

 nate off the principal axis. The rays from a point off the axis are brought 

 to a line focus at one point and a second line focus at another point ; the 

 line foci are perpendicular to one another. Between the tw^o line foci is 

 a region of least confusion, which is often taken as the focal point. 



The third image defect, chromatic aberration, is present in lenses but 

 not in mirrors. It arises from the variation in refractive index of the 

 lens material with wave length. If parallel rays of the visible spectrum 

 are incident on a lens, the blue rays will be brought to a focus closer to 

 the lens than the red rays (Fig. 3-3); i.e., the rays are dispersed into a 

 short spectrum along the lens axis. Chromatic aberration may be elimi- 

 nated partially by the use of compound lenses known as "achromats," 

 having components of different dispersions such that the dispersive effects 

 of the two components nullify one another at two selected wave lengths. 

 For the ultraviolet, achromats may be made from quartz-fluorite 

 combinations. 



