Chapter 1.1 1.1

Chapter 1.1 1.1 Geometrical optics Menn 1.1.1. Ray optics conventions and In most cases experienced in practice, imaging systems are based on lenses (exceptions are the imaging systems practical rules. Real and virtual with curved mirrors). objects and images The functioning of any optical element, as well as the whole system, can be described either in terms of ray Electro-optical systems are intended for the transfer and optics or in terms of wave optics. The first case is usually transformation of radiant energy. They consist of active called the geometrical optics approach while the second and passive elements and sub-systems. In active ele- is called physical optics. In reality there are many situa- ments, like radiation sources and radiation sensors, con- tions when we need both (for example, in image quality version of energy takes place (radiant energy is converted evaluation, see Chapter 2). But, since each approach has into electrical energy and vice versa, chemical energy is advantages and disadvantages in practical use, it is im- converted in radiation and vice versa, etc.). Passive ele- portant to know where and how to exploit each one in ments (like mirrors, lenses, prisms, etc.) do not convert order to minimize the complexity of consideration and to energy, but affect the spatial distribution of radiation. avoid wasting time and effort. Passive elements of electro-optical systems are fre- This chapter is related to geometrical optics, or, more quently termed optical systems. specifically, to ray optics. Actually an optical ray is Following this terminology, an optical system itself a mathematical simplification: it is a line with no thick- does not perform any transformation of radiation into ness. In reality optical beams which consist of an endless other kinds of energy, but is aimed primarily at changing quantity of optical rays are created and transferred by the spatial distribution of radiant energy propagated in electro-optical systems. Naturally, there exist three kinds space. Sometimes only concentration of radiation of optical beams: parallel, divergent, and convergent (see somewhere in space is required (like in the systems for Fig. 1.1.1). If a beam, either divergent or convergent, has medical treatment of tissues or systems for material a single point of intersection of all optical rays it is called processing of fabricated parts). In other cases the ability a homocentric beam (Fig. 1.1.1b,c). An example of of optics to create light distribution similar in some way a non-homocentric beam is shown in Fig. 1.1.1 d. Such to the light intensity profile of an ‘‘object’’ is exploited. Such a procedure is called imaging and the corresponding optical system is addressed as an imaging optical system. Of all the passive optical elements (prisms, mirrors, filters, lenses, etc.) lenses are usually our main concern. It is lenses that allow one to concentrate optical energy or to get a specific distribution of light energy at different Fig. 1.1.1 Optical beams: (a) parallel, (b,c) homocentric and points in space (in other words, to create an ‘‘image’’). (d) non-homocentric. Practical Optics; ISBN: 9780124909519 Copyright Ó 2004 Elsevier Inc. All rights of reproduction, in any form, reserved. SECTION ONE Optical Theory optical rays propagating through the system. Radiation propagates from the left to the right and, consequently, the object space (part of space where the light sources or the objects are located) is to the left of the system. The image space (part of space where the light detectors or images are located) is to the right of the system. All relevant values describing optical systems can be Fig. 1.1.2 Reflection and refraction of radiation. positive or negative and obey the following sign conventions and rules: a convergent beam could be the result of different phe- ray angles are calculated relative to the optical axis; the nomena occurring in optical systems (see Chapter 2 for angle of a ray is positive if the ray should be rotated more details). counterclockwise in order to coincide with OX, Ray optics is primarily based on two simple physical otherwise the angle is negative; laws: the law of reflection and the law of refraction. Both vertical segments are positive above OX and negative are applicable when a light beam is incident on a surface below OX; separating two optical media, with two different indexes horizontal segments should start from the optical system of refraction, n1 and n2 (see Fig. 1.1.2). The first law is and end at the relevant point according to the seg- just a statement that the incident angle, i, is equal to the ment definition. If going from the starting point to reflection angle, i’. The second law defines the relation the end we move left (against propagated radiation), between the incident angle and the angle of refraction, r: the segment is negative; if we should move right (in the direction of propagated radiation), the sinðiÞ=sinðrÞ¼n2=n1: (1.1.1) corresponding segment is positive. Examples are demonstrated in Fig. 1.1.3. The angle u is It is important to mention that all angles are measured negative (clockwise rotation of the ray to OX) whereas u0 from the vertical line perpendicular to the surface at the is positive. The object Y is positive and its image Y0 is point of incidence (so that the normal incidence of light negative. The segment S defines the object distance. It means that i i0 r 0). ¼ ¼ ¼ starts from the point O (from the system) and ends at In the geometrical optics approach the following as- the object (at Y). Since we move from O to Y against the sumptions are conventionally accepted: light, this segment is negative (S < 0). Accordingly, the (a) radiation is propagated along a straight line trajec- segment S0 (distance to the image) starts from the system tory (this means that diffraction effects are not (point O’) and ends at the image Y0. Since in this case we taken into account); move in the direction of propagated light (from left to 0 (b) if two beams intersect each other in space there is right) this segment is positive (S > 0). no interaction between them and each one is prop- The procedure of imaging is based on the basic as- agated as if the second one does not appear (this sumption that any object is considered as a collection of means that interference effects are not taken into separate points, each one being the center of a homo- account); centric divergent beam coming to the optical system. The optical system transfers all these beams, converting each (c) ray tracing is invertable; in other words, if the ray one to a convergent beam concentrated in a small spot trajectory is found while the ray is propagated (ideally a point) which is considered as an image of the through the system from input to output (say, from corresponding point of the object. The collection of such the left to the right) and then a new ray comes to the ‘‘point images’’ creates an image of the whole object (see same system along the outgoing line of the first ray, Fig. 1.1.4). but propagates in the reverse direction (from the An ideal imaging is a procedure when all homocentric right to the left), the trajectory of the second ray optical beams remain homocentric after traveling inside and outside of the system is identical to that of the first ray and it goes out of the system along the incident line of the first ray. Normally an optical system is assumed to be axisym- metrical, with the optical axis going along OX in the horizontal direction. Objects and images are usually located in the planes perpendicular to the optical axes, meaning that they are along the OY (vertical) axis. Ray tracing is a procedure of calculating the trajectory of Fig. 1.1.3 Sign conventions. 4 Geometrical optics CHAPTER 1.1 focal point or just focus). In such a case the only pa- rameter of the lens is its focal length, f 0, measured as the distance between the lens plane and the focus, F0. Each lens has two focuses: the back (F0) and the front (F), the first being the point where the rays belonging to a parallel beam incident on the lens from the left are concentrated and the second being the center of the concentrated rays when a parallel beam comes to the lens from the right. Fig. 1.1.4 Concept of image formation. Obviously, if the mediums at both sides of the lens are identical (for example, air on both sides or the lens being in water) then f 0¼f. In the case when the mediums are through the optical system, up to the image plane (this different (having refractive index n and n0 correspond- case is demonstrated in Fig. 1.1.4). Unfortunately, in real ingly) the relation should be imaging the outgoing beams become non-homocentric which, of course, ‘‘spoils’’ the images and makes it im- nf0 ¼n0f: (1.1.2) possible to reproduce the finest details of the object (this is like a situation when we try to draw a picture using The optical power of a lens, defined as a pencil which is not sharp enough and makes only thick 0 lines – obviously we fail to draw the small and fine details f ¼ 1=f ; (1.1.3) on the picture). The reasons for such degradation in is used sometimes in system analysis, as we shall see later. image quality lie partially in geometrical optics (then Imaging with a simple thin lens obeys the two fol- they are termed optical aberrations) and partially are due lowing equations: to the principal limitations of wave optics (diffraction limit).

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