Geometrical optics

Menn

1.1.1 Ray optics conventions and practical rules. Real and virtual objects and images

Electro-optical systems are intended for the transfer and transformation of radiant energy. They consist of active and passive elements and sub-systems. In active elements, like radiation sources and radiation sensors, conversion of energy takes place (radiant energy is converted into electrical energy and vice versa, chemical energy is converted in radiation and vice versa, etc.). Passive elements (like mirrors, lenses, prisms, etc.) do not convert energy, but affect the spatial distribution of radiation. Passive elements of electro-optical systems are frequently termed optical systems.

Following this terminology, an optical system itself does not perform any transformation of radiation into other kinds of energy, but is aimed primarily at changing the spatial distribution of radiant energy propagated in space. Sometimes only concentration of radiation somewhere in space is required (like in the systems for medical treatment of tissues or systems for material processing of fabricated parts). In other cases the ability of optics to create light distribution similar in some way 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 points in space (in other words, to create an “image”).

In most cases experienced in practice, imaging systems are based on lenses (exceptions are the imaging systems with curved mirrors).

The functioning of any optical element, as well as the whole system, can be described either in terms of ray optics or in terms of wave optics. The first case is usually called the geometrical optics approach while the second is called physical optics. In reality there are many situations when we need both (for example, in image quality evaluation, see Chapter 2). But, since each approach has advantages and disadvantages in practical use, it is important to know where and how to exploit each one in order to minimize the complexity of consideration and to avoid wasting time and effort.

This chapter is related to geometrical optics, or, more specifically, to ray optics. Actually an optical ray is a mathematical simplification: it is a line with no thickness. In reality optical beams which consist of an endless quantity of optical rays are created and transferred by electro-optical systems. Naturally, there exist three kinds of optical beams: parallel, divergent, and convergent (see Fig. 1.1.1). If a beam, either divergent or convergent, has a single point of intersection of all optical rays it is called a homocentric beam (Fig. 1.1.1b,c). An example of a non-homocentric beam is shown in Fig. 1.1.1 d. Such a convergent beam could be the result of different phenomena occurring in optical systems (see Chapter 2 for more details).

Ray optics is primarily based on two simple physical laws: the law of reflection and the law of refraction. Both are applicable when a light beam is incident on a surface separating two optical media, with two different indexes of refraction, n1 and n2 (see Fig. 1.1.2). The first law is just a statement that the incident angle, i, is equal to the reflection angle, i′. The second law defines the relation between the incident angle and the angle of refraction, r:

i/sinr=n2/n1.

It is important to mention that all angles are measured from the vertical line perpendicular to the surface at the point of incidence (so that the normal incidence of light means that i = i′  = r = 0).

In the geometrical optics approach the following assumptions are conventionally accepted:

(a) radiation is propagated along a straight line trajectory (this means that diffraction effects are not taken into account);

(b) if two beams intersect each other in space there is no interaction between them and each one is propagated as if the second one does not appear (this means that interference effects are not taken into account);

(c) ray tracing is invertable; in other words, if the ray trajectory is found while the ray is propagated through the system from input to output (say, from the left to the right) and then a new ray comes to the same system along the outgoing line of the first ray, but propagates in the reverse direction (from the right to the left), the trajectory of the second ray 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 axisymmetrical, 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 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 positive or negative and obey the following sign conventions and rules:

ray angles are calculated relative to the optical axis; the angle of a ray is positive if the ray should be rotated counterclockwise in order to coincide with OX, otherwise the angle is negative;

vertical segments are positive above OX and negative below OX;

horizontal segments should start from the optical system and end at the relevant point according to the segment definition. If going from the starting point to the end we move left (against propagated radiation), the segment is negative; if we should move right (in the direction of propagated radiation), the corresponding segment is positive.

Examples are demonstrated in Fig. 1.1.3. The angle u is negative (clockwise rotation of the ray to OX) whereas u′ is positive. The object Y is positive and its image Y′ is negative. The segment S defines the object distance. It starts from the point O (from the system) and ends at the object (at Y). Since we move from O to Y against the light, this segment is negative (S < 0). Accordingly, the segment S′ (distance to the image) starts from the system (point O') and ends at the image Y′. Since in this case we move in the direction of propagated light (from left to right) this segment is positive (S′ > 0).

The procedure of imaging is based on the basic assumption that any object is considered as a collection of separate points, each one being the center of a homocentric divergent beam coming to the optical system. The optical system transfers all these beams, converting each one to a convergent beam concentrated in a small spot (ideally a point) which is considered as an image of the corresponding point of the object. The collection of such “point images” creates an image of the whole object (see Fig. 1.1.4).

An ideal imaging is a procedure when all homocentric optical beams remain homocentric after traveling through the optical system, up to the image plane (this case is demonstrated in Fig. 1.1.4). Unfortunately, in real imaging the outgoing beams become non-homocentric which, of course, “spoils” the images and makes it impossible to reproduce the finest details of the object (this is like a situation when we try to draw a picture using a pencil which is not sharp enough and makes only thick lines – obviously we fail to draw the small and fine details on the picture). The reasons for such degradation in image quality lie partially in geometrical optics (then they are termed optical aberrations) and partially are due to the principal limitations of wave optics (diffraction limit). We consider this situation in detail in Chapter 1.2. Here we restrict ourselves to the simple case of ideal imaging.

In performing ray tracing one should be aware that doing it rigorously means going step by step from one optical surface to another and calculating at each step the incident and refraction angles using Eq. (1.1.1). Since many rays should be calculated, it is a time-consuming procedure which today is obviously done with the aid of computers and special programs for optical design. However, analytical consideration remains very difficult (if possible at all). The complexity of the procedure is caused mainly by the...