- 1. Ray Optics Light is an electromagnetic wave phenomenon described by the same principles that govern all forms of electromagnetic radiation. Electromagnetic radiation propagates in the form of two mutually coupled vector waves, an electric – field wave and a magnetic – field wave. Nevertheless, it is possible to describe many optical phenomena using a scalar wave theory in which light is described by a single scalar wave function. This approximate way of treating light is called wave op- tics, or simply wave optics. In free space, light waves travel with a constant speed . The range of optical wavelengths contains three bands – ultraviolet (10 to 390 nm), visible (390 to 760 nm), and infrared (760 nm to 1 mm). The corres- ponding range of optical frequencies stretches from . When light waves propagate through and around objects whose dimensions are much greater than the wavelength, the wave nature of light is not readily discerned, so that its behavior can be adequately described by rays obeying a set of geometrical rules. This model of light is called ray optics. Strictly speak- ing, ray optics is the limit of optics when the wavelength is infinitesimally short. However, the wavelength need not actually be equal to zero for the ray – optics theory to be useful. As long as the light waves propagate through and round objects whose dimensions are much greater than the wavelength, the ray theory suffices for describing most phenomena. Because the wavelength of visible light is much shorter than the dimensions of the visible objects en- countered in our daily lives, manifestation of the wave nature of light are not apparent without careful observation. Ray optics is the simplest theory of light. Light is described by rays that travel in different optical media in accordance with a set of geometrical rules. Ray optics is therefore called Geometrical optics. Ray optics is an approximate theory. Although it adequately describes most of our daily experiences with light, there are many phenomena that ray optics does not adequately describe. Ray optics is concerned with the location and direction of light rays. It is there- fore useful in studying image formation – the collection of rays from each 1
- 2. point of an object and their redirection by an optical component onto a cor- responding point of an image. Ray optics permits us to determine conditions under which light is guided within a given medium, such as a glass fiber. In isotropic media, optical rays point in the direction of the flow of optical ener- gy. Ray bundles can be constructed in which the density of rays is proportion- al to the density of light energy. When light is generated isotropically from a point source, for example, the energy associated with the rays in a given cone is proportional to the solid angle of the cone. Rays must be traced through an optical system to determine the optical energy crossing a given area. Optical components are often centered about an optical axis, around which the rays travel at different inclinations. Rays that make small angles (such that ) with the optical axis of a mirror or a lens are called paraxial rays. In the paraxial approximation, where only paraxial rays are considered, a spherical mirror, for example, has a focusing property like that of the parabo- loidal mirror and an imaging property like that of elliptical mirror. The body of rules that results from this approximation forms paraxial optics, also called first – order optics or Gaussian optics. The change in the position and inclination of a paraxial ray as it travels through an optical system can be efficiently described by the use of a . This algebraic tool is called matrix optics. SPHERICAL BOUNDARIES AND LENSES - IMAGE FORMATION The refraction of rays from a spherical boundary of radius R between two me- dia of refractive indices n1 and n2 can be examined. By convention, R is posi- tive for a convex boundary and negative for a concave boundary. By using Snell’s law, and considering only paraxial rays making small angles with the axis of the system so that , the following properties may be shown to hold: A ray making an angle with the z axis and meeting the boundary at a point of height y (see figure 1-a) refracts and changes direction so that the refracted ray makes an angle with the z axis, All paraxial rays originating from a point in the z=z1 plane meet at a point in the z=z2 plane, where: 2
- 3. And The z=z1 and z=z2 planes are said to be conjugate planes. Every point in the first plane has a corresponding point (image) in the second with magnification Negative magnification means that the image is inverted. By convention, P1 is measured in a coordinate sys- tem pointing to the left and P2 in a coordinate system pointing to the right (e.g., if P2 lies to the left of the boundary, then z2 would be nega- tive). Figure 1: Refraction at a convex spherical boundary (R>0) It is worth to indicate here that the image formation properties de- scribed above are approximate. They hold only for paraxial rays. Rays 3
- 4. of large angles do not obey these paraxial laws; the deviation results in image distortion called aberration. A spherical lens is bounded by two spherical surfaces. It is, therefore, defined completely by the radii of curvatures R1 and R2 of its two surfaces, its thickness , and the refractive index n of its material (figure 2). A glass lens in air can be regarded as a combination of two spherical boundaries, air – to – glass and glass– to – air. A ray crossing the first surface at height y and an- gle with the z axis (figure 3-a) is traced by ap- plying equation 1 at the first surface to obtain the inclination angle of the refracted ray, which is ex- tended until it meets the second surface. Then eq- Figure 2: A biconvex spherical lens uation 1 is used once more with replacing to obtain the in- clination angle of the ray af- ter refraction from the second surface. The results are in gen- eral complicated. When the lens is thin, however, it can be as- sumed that the incident ray emerges from the lens at about a the same height y at which it en- ters. Under this assumption, the fol- lowing relations follow: The angles of the re- fracted and incident rays are related by: Where f, the focal length, is giv- b en by: Figure 3: (a) Ray bending by a thin lens. (b) Image forma- tion by a thin lens 4
- 5. All rays originating from a point meet at a point (figure 3-b), where: And This means that each point in the z=z1 plane is imaged onto a corresponding point in the z=z2 plane with a magnification factor The focal length f of a lens therefore, completely determines its effect on paraxial rays. As indicated earlier, P1 and P2 are measured in coordinate system pointing to the left and right, respectively, and the radii of curvatures R1 and R2 are posi- tive for convex surfaces and negative for concave surfaces. For the biconvex lens shown in figure 2, R1 is positive and R2 is negative, so that the two terms of equation 5 add and provide a positive f. Matrix Optics Matrix optics is a technique for tracing paraxial rays. The rays are assumed to travel only within a single angle, so that the formalism is applicable to systems with planar geometry and to meridional rays in circularly symmetric systems. A ray is described by its position and its angle with respect to the optical axis. These variables are altered axis as the ray travels through the system. In the paraxial approximation, the posi- tion and angle at the input Figure 4: A ray is characterized by its coordinate and its angle and output planes of an optical system are related by two linear algebraic equations. As a result, the optical system is described by a matrix called the ray – transfer matrix. The convenience of using matrix methods lies in the fact that the ray – trans- fer matrix of a cascade of optical components (or systems) is the product of the ray transfer matrices of the individual components (or systems). Matrix 5
- 6. optics therefore provides a formal mechanism for describing complex systems in the paraxial approximation. The Ray – Transfer Matrix Consider a circularly symmetric optical system formed by refracting and re- flecting surfaces all centered about the same axis (optical axis). The z axis lies along the optical axis and points in the general direction in which the rays tra- vel. Consider rays in a plane containing the optical axis; say the y-z plane. We proceed to trace a ray as it travels through the system, i.e., as it crosses the transverse planes at different axial distances. A ray crossing the transverse plane at z is completely characterized by the coordinate y of its crossing point and the angle (figure 4). An optical system is a set of optical components placed between two trans- verse planes at z1 and z2, referred to as the input and output planes, respec- y Input plane 1 Output plane 2 y1 y2 Optical Z1 Z2 z axis Optical system Figure 5: A ray enters an optical system at position and angle and leaves at position and angle . tively. The system is characterized completely by its effect on an incoming ray of arbitrary position and direction . It steers the ray so that it has new position and direction at the output plane (figure 5). In the paraxial approximation, when all angles are sufficiently small so that , the relation between and is linear and can generally be written in the form: 6
- 7. Where A,B,C and D are real numbers. Equations 1 and 2 may be conveniently written in matrix form as: The matrix M whose elements are A, B, C, D, characterizes the optical system completely since it permits to be determined for any . It is known as the ray – transfer matrix. It is handy to note that all ABCD matrices, for individual elements or a compli- cated optical system, have the property: This relation can be utilized as a check on one’s calculation after multiplying together a long chain of matrices. Matrices of Simple Optical Components Free – Space Propagation Since rays travel in free space along straight lines, a ray traversing a distance d is altered in accordance with and . The ray – transfer matrix is therefore: d Refraction at a Planar Boundary At a planar boundary between two media of refractive indices n1 and n2, the ray angle changes in accordance with Snell’s law . In the paraxial approximation, . The posi- tion of the ray is not altered, . The ray – transfer matrix is: n1 n2 7
- 8. Refraction at a Spherical Boundary The relation between and for paraxial rays at a spherical boundary be- tween two media is provided by equation 1. The ray height is not altered, . the ray – transfer matrix is: R Convex, R > 0; n1 n2 concave, R < 0 Transmission through a thin lens The relation between and for paraxial rays transmitted through a thin lens of focal length f is given in equation 5. Since the height remains unchanged, f Convex, f> 0 ; concave, f< 0 Reflection from a planar mirror Upon reflection from a planar mirror, the ray position is not altered, . Adopting the convention that the z axis points in the general direc- tion of travel of the rays, i.e., toward the mirror for the incident rays and away from it for the reflected rays, thus , its concluded that . The ray – transfer matrix is, therefore, the identity matrix: z z 8
- 9. Reflection from a spherical mirror Using equation 1, and the convention that the z axis follows the general direc- tion of the rays as they reflect from mirrors, one can similarly obtain: -R Concave, R< 0; convex, R>0 Note the similarity between the ray – transfer matrices of a spherical mirror and a thin lens. A mirror with radius of curvature R bends rays in a manner that is identical to that of a thin lens with focal length . MATRICES OF CASCADE OPTICAL COMPONENTS The ray transfer matrix of an optical system composed of a cascade of optical components whose ray - transfer matrices are M1, M2,…., MN is equivalent to a single optical component of ray – transfer matrix M. Thus: M1 M2 MN M= MN…….M2 M1 …. Note the order of multiplication: The matrix of the system that is crossed by the rays first is placed to the right, so that it operates on the column matrix of the incident ray first. Examples: 1. A Set of Parallel Transparent Plates: consider a set of N parallel planar transparent plates of refractive indices and thicknesses placed in air (n=1) normal to the z axis. Show that the ray – transfer matrix is: 9
- 10. Note that the order of placing the plates does not affect the overall ray – trans- fer matrix. What is the ray – transfer matrix of an inhomogeneous transparent plate of thickness do and refractive index n (z)? Since the ray – transfer matrix of the free space is given by: Here, the light propagates in a transparent plate of thickness d and refractive index n. Thus, the distance of propagation of the light in each plate is equal to the ratio of the optical path to the refractive index of the medium, i.e., The ray – transfer matrix of the given optical system is: In the inhomogeneous medium, the refractive index is function of position (i.e., it does not have the same value in the various parts of the medium). Therefore, the ray – transfer matrix of an inhomogeneous transparent plate of thickness do and refractive index n (z) can be given as follows: 2. A Gap Followed by a Thin Lens: show that the ray – transfer matrix of a distance d of free space followed by a lens of focal length f is : 10
- 11. 3. Imaging With a Thin Lens: Derive an expression for the ray – transfer matrix of a system composed of free space /thin lens/free space, as illu- strated in the figure. Show that if the imaging condition is satisfied, all rays originating from a single point in the input plane reach the output plane at a single point y2, regardless of their angles. Also show that if d2=f, all parallel incident rays are fo- cused by the lens onto a single point in the output plane. To determine the ray – transfer matrix of the thin lens, we refer to the follow- ing figure. For thin biconvex lens: For the above condition to be met (i.e., ), the constant A must be equal to 1 and B must be equal to 0. For ray: 11
- 12. For ray: Therefore, the ray – transfer matrix of the thin biconvex lens will be: If d2=f=image distance, then the ray transfer matrix of the optical system will be: 4. Imaging With a Thick Lens: consider a glass lens of refractive index n, and two spherical surfaces of equal radii of curvatures R (figure 5). De- termine the ray – transfer matrix of the system between the two planes at distance d1 and d2 from the vertices of the lens. The lens is placed in air (refractive index =1). Show that the system is an imaging system (i.e., the input and output planes are conjugate)if: Where: And: 12
- 13. The points F1 and F2 are known as the front and back focal points, respective- ly. The points P1 and P2 are known as the first and second principal points, respectively. The radii of curvatures of the lens’s two spherical surfaces R1 and R2 are taken positive for spherical surface facing to the right (R1) and neg- Figure 5: Imaging with a thick lens. P1 and P2 are the principal points and F1 and F2 are the focal points. ative for spherical surface facing to the left (R2). Show the importance of these points by tracing of rays that are incident parallel to the optical axis. The ray – transfer matrix of the system between the two planes at distance d1 and d2 from the vertices of the lens is given by: The ray – transfer matrix of the thick lens is equal to the product of the ray – transfer matrices of two refracting surfaces of the lens and the distance doptical traversed by light. Thus: 1. Ray – Transfer Matrix of a Lens System: Determine the ray – transfer matrix for an optical system made of a thin convex lens of focal length f 13
- 14. and a thin concave lens of focal length –f separated by a distance f. Dis- cuss the imaging properties of this composite lens. The ray – transfer matrix of the current optical system will be: To demonstrate the imaging properties of such optical system we will consid- er the following input matrix: +f -f When the above Mststem matrix works on Minput the output will be: Output=matrix of the system * the input d=f This means that the image is real, inverted (due to the minus sign), and minia- ture. References 1. D. C. O’Shea, Elements of Modern Optical Design, Wiley, New York, 1985. 2. R. Kingslake, Optical System Design, Academic Press, New York, 1983. 3. Matrix Method in Optics, , Wiley, New York, 1978. 14