CHAPTER 2 AFOCAL SYSTEMS William B . Wetherell Optical Research Associates Framingham , Massachusetts 2. 1 GLOSSARY BFL back focal length D pupil diameter ER c p eye relief common pupil position ER k eye relief keplerian e exit pupil ; eye space F , F 9 focal points FFL front focal length h , h 9 object and image heights l , l 9 object and image distances M angular magnification m linear , lateral magnification n refractive index OR object relief o entrance pupil ; object space P , P 9 principal points R radius TTL total length tan a slope x , y , z cartesian coordinates D z axial separation 2. 2 INTRODUCTION If collimated (parallel) light rays from an infinitely distant point source fall incident on the input end of a lens system , rays exiting from the output end will show one of three characteristics : (1) they will converge to a real point focus outside the lens system , (2) they will appear to diverge from a virtual point focus within the lens system , or (3) they will 2 .1 2 .2 OPTICAL ELEMENTS emerge as collimated rays that may dif fer in some characteristics from the incident collimated rays . In cases 1 and 2 , the paraxial imaging properties of the lens system can be modeled accurately by a characteristic focal length and a set of fixed principal surfaces . Such lens systems might be called focusing or focal lenses , but are usually referred to simply as lenses . In case 3 , a single finite focal length cannot model the paraxial characteristics of the lens system ; in ef fect , the focal length is infinite , with the output focal point an infinite distance behind the lens , and the associated principal surface an infinite distance in front of the lens . Such lens systems are referred to as afocal , or without focal length . They will be called afocal lenses here , following the common practice of using ‘‘lens’’ to refer to both single element and multielement lens systems . They are the topic of this chapter . The first afocal lens was the galilean telescope (to be described later) , a visual telescope made famous by Galileo’s astronomical observations . It is now believed to have been invented by Hans Lipperhey in 1608 . 1 Afocal lenses are usually thought of in the context of viewing instruments or attachments to change the ef fective focal length of focusing lenses , whose outputs are always collimated . In fact , afocal lenses can form real images of real objects . A more useful distinction between focusing and afocal lenses concerns which optical parameters are fixed , and which can vary in use . Focusing lenses have a fixed , finite focal length , can produce real images for a wide range of object distances , and have a linear magnification which varies with object distance . Afocal lenses have a fixed magnification which is independent of object distance , and the range of object distances yielding real images is severely restricted . This chapter is divided into six sections , including this introduction . The second section reviews the Gaussian (paraxial) image forming characteristics of afocal lenses and compares them to the properties of focusing lenses . The importance of the optical invariant in designing afocal lenses is discussed . The third section reviews the keplerian telescope and its descendants , including both infinite conjugate and finite conjugate variants . The fourth section discusses the galilean telescope and its descendants . Thin-lens models are used in the third and fourth sections to define imaging characteristics and design principles for afocal lenses . The fifth section reviews relay trains and periscopes . The final section reviews reflecting and catadioptric afocal lenses . This chapter is based on an earlier article by Wetherell . 2 That article contains derivations of many of the equations appearing here , as well as a more extensive list of patents illustrating dif ferent types of afocal lens systems . 2. 3 GAUSSIAN ANALYSIS OF AFOCAL LENSES Afocal lenses dif fer from focusing lenses in ways that are not always obvious . It is useful to review the basic image-forming characteristics of focusing lenses before defining the characteristics unique to afocal lenses . Focusing Lenses In this chapter , all lens elements are assumed to be immersed in air , so that object space and image space have the same index of refraction . Points in object space and image space are represented by two rectangular coordinate systems ( x , y , z ) and ( x 9 , y 9 , z 9 ) , with the prime indicating image space . The z - and z 9 -axes form a common line in space , the optical axis of the system . It is assumed , unless noted otherwise , that all lens elements are rotationally symmetric with respect to the optical axis . Under these conditions , the imaging geometry of a focusing lens can be defined in terms of two principal points P and P 9 , two focal points F and F 9 , and a single characteristic focal length f , as shown in Fig . 1 . P , P 9 , F , and F 9 all lie on the optical axis . AFOCAL SYSTEMS 2 .3 FIGURE 1 Imaging geometry of focusing lenses . The focal points F and F 9 , will be the origins for the coordinate systems ( x , y , z ) and ( x 9 , y 9 , z 9 ) . If the origins are at P and P 9 , the coordinates will be given as ( x , y , s ) and ( x 9 , y 9 , s 9 ) , where s 5 z 2 f and s 9 5 z 9 1 f . Normal right-hand sign conventions are used for each set of coordinates , and light travels along the z -axis from negative z toward positive z 9 , unless the optical system has internal mirrors . Figure 1 a illustrates the terminology for finite conjugate objects . Object points and image points are assumed to lie in planes normal to the optical axis , for paraxial computations . Object distance is specified by the axial distance to the object surface , z or s , and image distance by z 9 or s 9 . The two most commonly used equations relating image distance to object distance are 1 1 1 2 5 (1) s 9 s f and zz 9 5 2 f 2 (2) For infinitely distant object points , z 9 5 0 and s 9 5 f , and the corresponding image points will lie in the focal plane at F 9 . To determine the actual distance from object plane to image plane , it is necessary to know the distance sp between P and P 9 . The value of sp is a constant specific to each real lens system , and may be either positive [moving object and image further apart than predicted by Eqs . (1) or (2)] or negative (moving them closer together) . For rotationally symmetric systems , of f-axis object and image coordinates can be expressed by the object height h and image height h 9 , where h 2 5 x 2 1 y 2 and h 9 2 5 x 9 2 1 y 9 2 . Object height and image height are related by the linear magnification m , where h 9 s 9 z 9 1 f m 5 5 5 (3) h s z 2 f 2 .4 OPTICAL ELEMENTS Since the product zz 9 is a constant , Eq . (3) implies that magnification varies with object distance . The principal surfaces of a focusing lens intersect the optical axis at the principal points P and P 9 . In paraxial analysis , the principal surfaces are planes normal to the optical axis ; for real lenses , they may be curved . The principal surfaces are conjugate image surfaces for which m 5 1 1 . 0 . This property makes the raytrace construction shown in Fig . 1 a possible , since a ray traveling parallel to the optical axis in either object or image space must intersect the focal point in the conjugate space , and must also intersect both principal surfaces at the same height . In real lenses , the object and image surfaces may be tilted or curved . Planes normal to the optical axis are still used to define object and image positions for of f-axis object points , and to compute magnification . For tilted object surfaces , the properties of the principal surfaces can be used to relate object surface and image surface tilt angles , as shown in Fig . 1 b . Tilted object and image planes intersect the optical axis and the two principal planes . The tilt angles with respect to the optical axis , u and u 9 , are defined by meridional rays lying in the two surfaces . The points at which conjugate tilted planes intercept the optical axis are defined by s a and s a9 , given by Eq . (1) . Both object and image planes must intersect their respective principal surfaces at the same height y , where y 5 s a tan u 5 s 9a tan u 9 . It follows that tan u 9 s 1 5 a 5 (4) tan u s a9 m a The geometry of Fig . 1 b is known as the Scheimpflug condition , and Eq . (4) is the Scheimpflug rule , relating image to object tilt . The magnification m a applies only to the axial image . The height of f axis of an infinitely distant object is defined by the principal ray angle u p measured from F or P , as shown in Fig . 1 c . In this case , the image height is h 9 5 f tan u p (5) A focusing lens which obeys Eq . (5) for all values of u p within a specified range is said to be distortion - free : if the object is a set of equally spaced parallel lines lying in an object plane perpendicular to the optical axis , it will be imaged as a set of equally spaced parallel lines in an image plane perpendicular to the optical axis , with line spacing proportional to m .