RAY OPTICS

CHAPTER RAY OPTICS 1.1 POSTULATES OF RAY OPTICS 1.2 SIMPLE OPTICAL COMPONENTS A. Mirrors B. Planar Boundaries C. Spherical Boundaries and Lenses D. Light Guides 1.3 GRADED-INDEX OPTICS A. The Ray Equation B. Graded-Index Optical Components *C. The Eikonal Equation 1.4 MATRIX OPTICS A. The Ray-Transfer Matrix B. Matrices of Simple Optical Components C. Matrices of Cascaded Optical Components D. Periodic Optical Systems Sir Isaac Newton (1642-1727) set forth a theory of optics in which light emissions consist of collections of corpuscles that propagate rectilinearly. Pierre de Fermat (1601-1665) developed the principle that light travels along the path of least time. 1 Fundamentals of Photonics Bahaa E. A. Saleh, Malvin Carl Teich Copyright © 1991 John Wiley & Sons, Inc. ISBNs: 0-471-83965-5 (Hardback); 0-471-2-1374-8 (Electronic) Light is an electromagnetic wave phenomenon described by the same theoretical 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 wavefunction. This approximate way of treating light is called scalar wave optics, or simply wave optics. 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 speaking, ray optics is the limit of wave optics when the wavelength is infinitesimally small. Thus the electromagnetic theory of light (electromagnetic optics) encompasses wave optics, which, in turn, encompasses ray optics, as illustrated in Fig. 1.0-l. Ray optics and wave optics provide approximate models of light which derive their validity from their successes in producing results that approximate those based on rigorous electro- magnetic theory. Although electromagnetic optics provides the most complete treatment of light within the confines of classical optics, there are certain optical phenomena that are characteristically quantum mechanical in nature and cannot be explained classically. These phenomena are described by a quantum electromagnetic theory known as quantum electrodynamics. For optical phenomena, this theory is also referred to as quantum optics. Historically, optical theory developed roughly in the following sequence: (1) ray optics; + (2) wave optics; + (3) electromagnetic optics; + (4) quantum optics. Not -Quantum optics / \ Electromagnetic Figure 1.0-l The theory of quantum optics provides an explanation of virtually all optical phenomena. The electromagnetic theory of light (electromagnetic optics) provides the most complete treatment of light within the confines of classical optics. Wave optics is a scalar approximation of electromagnetic optics. Ray optics is the limit of wave optics when the wavelength is very short. 2 POSTULATES OF RAY OPTICS 3 surprisingly, these models are progressively more difficult and sophisticated, having being developed to provide explanations for the outcomes of successively more complex and precise optical experiments. For pedagogical reasons, the chapters in this book follow the historical order noted above. Each model of light begins with a set of postulates (provided without proof), from which a large body of results are generated. The postulates of each model are then shown to follow naturally from the next-higher-level model. In this chapter we begin with ray optics. Ray Optics 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 also 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 (as amply attested to by the remaining chapters of this book). Ray optics is concerned with the location and direction of light rays. It is therefore useful in studying image formation-the collection of rays from each point of an object and their redirection by an optical component onto a corresponding 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 energy. Ray bundles can be constructed in which the density of rays is proportional 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 may be traced through an optical system to determine the optical energy crossing a given area. This chapter begins with a set of postulates from which the simple rules that govern the propagation of light rays through optical media are derived. In Sec. 1.2 these rules are applied to simple optical components such as mirrors and planar or spherical boundaries between different optical media. Ray propagation in inhomogeneous (graded-index) optical media is examined in Sec. 1.3. Graded-index optics is the basis of a technology that has become an important part of modern optics. Optical components are often centered about an optical axis, around which the rays travel at small inclinations. Such rays are called paraxial rays. This assumption is the basis of paraxial 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 2 x 2-matrix algebra. Section 1.4 is devoted to this algebraic tool, called matrix optics. 1.1 POSTULATES OF RAY OPTICS 4 RAY OPTICS In this chapter we use the postulates of ray optics to determine the rules governing the propagation of light rays, their reflection and refraction at the boundaries between different media, and their transmission through various optical components. A wealth of results applicable to numerous optical systems are obtained without the need for any other assumptions or rules regarding the nature of light. Figure 1.1-l Light rays travel in straight lines. Shadows are perfect projections of stops. POSTULATES OF RAY OPTICS 5 Plane of incidence la) Figure 1 .l-2 (a) Reflection to prove the law of reflection. from the surface of a curved mirror. (b) Geometrical construction Mirror (bl Reflection from a Mirror Mirrors are made of certain highly polished metallic surfaces, or metallic or dielectric films deposited on a substrate such as glass. Light reflects from mirrors in accordance with the law of reflection: The reflected ray lies in the plane of incidence ; the angle of reflection equals the angle of incidence. The plane of incidence is the plane formed by the incident ray and the normal to the mirror at the point of incidence. The angles of incidence and reflection, 6 and 8’, are defined in Fig. 1.1-2(a). To prove the law of reflection we simply use Hero’s principle. Examine a ray that travels from point A to point C after reflection from the planar -- mirror in Fig. 1.1-2(b). According to Hero’s principle the distance AB + BC must be -- -- minimum. If C’ is a mirror image of C, then BC = BC’, so that AB + BC’ must be a minimum. This occurs when ABC’ is a straight line, i.e., when B coincides with B’ and 8 = 8’. Reflection and Refraction at the Boundary Between Two Media At the boundary between two media of refractive indices n1 and n2 an incident ray is split into two-a reflected ray and a refracted (or transmitted) ray (Fig. 1.1-3). The Figure 1 .I -3 Reflection and refraction at the boundary between two media. 6 RAY OPTICS reflected ray obeys the law of reflection. The refracted ray obeys the law of refraction: The refracted ray lies in the plane of incidence; the angle of refraction 8, is related to the angle of incidence 8 1 by Snell’s law, I I 1 n,sinO, =n,sinO,. 1 (1.1-l) Snell’s Law EXERCISE 1.1-I Proof of Snell’s Law. The proof of Snell’s law is an exercise in the application of Fermat’s principle. Referring to Fig. 1.1-4, we seek to minimize the optical path length nrAB + n,BC between points A and C. We therefore have the following optimization problem: Find 8, and 8, that minimize nrd, set 8t + n,d, set f!12, subject to the condition d, tan 8, + d, tan 8, = d. Show that the solution of this constrained minimization prob- lem yields Snell’s law. nl n2 . . . . . Figure 1.1-4 Construction to prove Snell’s law. dl .‘, ,‘.:‘::.: 1.; ‘:: .‘.:y’,‘: :. ;, .:, ‘, ----- The three simple rules-propagation in straight lines and the laws of reflection and refraction-are applied in Sec. 1.2 to several geometrical configurations of mirrors and transparent optical components, without further recourse to Fermat’s principle. 1.2 SIMPLE OPTICAL COMPONENTS A. Mirrors Planar Mirrors A planar mirror reflects the rays originating from a point P, such that the reflected rays appear to originate from a point P, behind the mirror, called the image (Fig. 1.2-1). Paraboloidal Mirrors The surface of a paraboloidal mirror is a paraboloid of revolution. It has the useful property of focusing all incident rays parallel to its axis to a single point called the - focus. The distance PF= f defined in Fig. 1.2-2 is called the focal length. Paraboloidal SIMPLE OPTICAL COMPONENTS 7 \ \ \ \ \ ‘\ \ ‘\ \ \ \ \ \ \ \ \ \ \ ‘\ 1 ‘\ 4 -------__ p2 Mwror Figure 1.2-1 Reflection from a planar mirror. Figure 1.2-2 Focusing of light by a paraboloidal mirror. mirrors are often used as light-collecting elements in telescopes. They are also used for making parallel beams of light from point sources such as in flashlights. Elliptical Mirrors An elliptical mirror reflects all the rays emitted from one of its two foci, e.g., P,, and images them onto the other focus, P, (Fig. 1.2-3). The distances traveled by the light from P, to P, along any of the paths are all equal, in accordance with Hero’s principle. Figure 1.2-3 Reflection from an elliptical mirror. 8 RAY OPTICS Figure 1.2-4 Reflection of parallel rays from a concave spherical mirror. Spherical Mirrors A spherical mirror is easier to fabricate than a paraboloidal or an elliptical mirror. However, it has neither the focusing property of the paraboloidal mirror nor the imaging property of the elliptical mirror. As illustrated in Fig. 1.2-4, parallel rays meet the axis at different points; their envelope (the dashed curve) is called the caustic curve. Nevertheless, parallel rays close to the axis are approximately focused onto a single point F at distance (- R)/2 from the mirror center C. By convention, R is negative for concave mirrors and positive for convex mirrors. Paraxial Rays Reflected from Spherical Mirrors Rays that make small angles (such that sin 8 = 0) with the mirror’s axis are called paraxial rays. In the paraxial approximation, where only paraxial rays are considered, a spherical mirror has a focusing property like that of the paraboloidal mirror and an imaging property like that of the elliptical mirror. The body of rules that results from this approximation forms paraxial optics, also called first-order optics or Gaussian optics. A spherical mirror of radius R therefore acts like a paraboloidal mirror of focal length f = R/2. This is in fact plausible since at points near the axis, a parabola can be approximated by a circle with radius equal to the parabola’s radius of curvature (Fig. 1.2-5). ---- -- C FP z + (-RI 1 (--RI + -- 2 2 Figure 1.2-5 A spherical mirror approximates a paraboloidal mirror for paraxial rays. SIMPLE OPTICAL COMPONENTS 9 z- *1 I-R) 22 l-R112 0 Figure 1.2-6 Reflection of paraxial rays from a concave spherical mirror of radius R < 0. All paraxial rays originating from each point on the axis of a spherical mirror are reflected and focused onto a single corresponding point on the axis. This can be seen (Fig. 1.2-6) by examining a ray emitted at an angle 0, from a point Pi at a distance zi away from a concave mirror of radius R, and reflecting at angle ( - 0,) to meet the axis at a point P, a distance z2 away from the mirror. The angle 8, is negative since the ray is traveling downward. Since 8, = 8, - 8 and (-0,) = 8, + 8, it follows that (-0,) + 8, = 20,. If 8, is sufficiently small, the approximation tan 8, = 8, may be used, so that 80 = y/(-R), from which (-e,) + 8, = $, (1.2-1) where y is the height of the point at which the reflection occurs. Recall that R is negative since the mirror is concave. Similarly, if 8, and 8, are small, 8, = y/z,, (-e,) = Y/Z,, and (1.2-1) yields y/z1 + y/z, = 2y/(-R), from which 1 1 2 -+-=-R. 21 z2 (1.2-2) This relation hold regardless of y (i.e., regardless of 0,) as long as the approximation is valid. This means that all par-axial rays originating at point P, arrive at P2. The distances zi and z2 are measured in a coordinate system in which the z axis points to the left. Points of negative z therefore lie to the right of the mirror. According to (1.2-2), rays that are emitted from a point very far out on the z axis (zi = 03) are focused to a point F at a distance z2 = (-R)/2. This means that within the paraxial approximation, all rays coming from infinity (parallel to the mirror’s axis) are focused to a point at a distance I i f=$ (1.2-3) Focal Length of a Spherical Mirror 10 RAY OPTICS which is called the mirror’s focal length. Equation (1.2-2) is usually written in the form (1.2-4) Imaging Equation (Paraxial Rays) known as the imaging equation. Both the incident and the reflected rays must be paraxial for this equation to be valid. EXERCISE 1.2- 1 Image Formation by a Spherical Mirror. Show that within the paraxial approximation, rays originating from a point P, = (yl, zl) are reflected to a point P, = (y2, z,), where z1 and z2 satisfy (1.2-4) and y, = -y1z2/zl (Fig. 1.2-7). This means that rays from each point in the plane z = z1 meet at a single corresponding point in the plane z = z2, so that the mirror acts as an image-forming system with magnification -z2/z1. Negative magnifi- cation means that the image is inverted. Figure 1.2-7 Image formation by a spherical mirror. B. Planar Boundaries The relation between the angles of refraction and incidence, 8, and 8,, at a planar boundary between two media of refractive indices n, and n2 is governed by Snell’s law (1.1-1). This relation is plotted in Fig. 1.2-8 for two cases: n External Refraction (~ti < n2). When the ray is incident from the medium of smaller refractive index, 8, < 8, and the refracted ray bends away from the boundary. . Internal Refraction (nl > n2). If the incident ray is in a medium of higher refractive index, 8, > 8, and the refracted ray bends toward the boundary. In both cases, when the angles are small (i.e., the rays are par-axial), the relation between 8, and 8, is approximately linear, n,Bt = yt202, or 8, = (n&z,)&. SIMPLE OPTICAL COMPONENTS 11 External refraction Internal refraction Figure 1.2-8 Relation between the angles of refraction and incidence. Total Internal Reflection For internal refraction (~1~ > its), the angle of refraction is greater than the angle of incidence, 8, > 8,, so that as 8, increases, f12 reaches 90” first (see Fig. 1.2-8). This occurs when fI1 = 8, (the critical angle), with nl sin 8, = n2, so that (1.2-5) Critical Angle When 8, > 8,, Snell’s law (1.1-1) cannot be satisfied and refraction does not occur. The incident ray is totally reflected as if the surface were a perfect mirror [Fig. 1.2-9(a)]. The phenomenon of total internal reflection is the basis of many optical devices and systems, such as reflecting prisms [see Fig. 1.2-9(b)] and optical fibers (see Sec. 1.2D). n2=1 \ (a) (id (cl Figure 1.2-9 (a) Total internal reflection at a planar boundary. {b) The reflecting prism. If n1 > ~6 and n2 = 1 (air), then 8, < 45”; since 8, = 45”, the ray is totally reflected. (c) Rays are guided by total internal reflection from the internal surface of an optical fiber. 12 RAY OPTICS ed a a e Figure 1.2-I 0 Ray deflection by a prism. The angle of deflection 13, as a function of the angle of incidence 8 for different apex angles (Y when II = 1.5. When both (Y and t9 are small 13, = (n - l)(~, which is approximately independent of 13. When cz = 45” and 8 = O”, total internal reflection occurs, as illustrated in Fig. 1.2-9(b). Prisms A prism of apex angle (Y and refractive index n (Fig. 1.2-10) deflects a ray incident at an angle 8 by an angle sin (Y - sin 8 cos CY . I (1.2-6) This may be shown by using Snell’s law twice at the two refracting surfaces of the prism. When cr is very small (thin prism) and 8 is also very small (paraxial approxima- tion), (1.2-6) is approximated by 8, = (n - l)a. (1.2-7) Beamsplitters The beamsplitter is an optical component that splits the incident light beam into a reflected beam and a transmitted beam, as illustrated in Fig. 1.2-11. Beamsplitters are also frequently used to combine two light beams into one [Fig. 1.2-11(c)]. Beamsplitters are often constructed by depositing a thin semitransparent metallic or dielectric film on a glass substrate. A thin glass plate or a prism can also serve as a beamsplitter. (a) (b) (cl Figure 1.2-l 1 Beamsplitters and combiners: (a) partially reflective mirror; (b) thin glass plate; (c) beam combiner. SIMPLE OPTICAL COMPONENTS 13 C. Spherical Boundaries and Lenses We now examine the refraction of rays from a spherical boundary of radius R between two media of refractive indices n, and n2. By convention, R is positive 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 tan 8 = 8, the following properties may be shown to hold: n A ray making an angle 8, with the z axis and meeting the boundary at a point of height y [see Fig. 1.2-12(a)] refracts and changes direction so that the refracted ray makes an angle 8, with the z axis, 82 z “le, - n2 - n1 -Y. 122 n2R (1.2-8) . All paraxial rays originating from a point P, = ( y r, z,) in the z = zr plane meet at a point P2 = (y2, z2) in the z = z2 plane, where nl+-,- n2 112-n, Zl z2 R (1.2-9) 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 h-4 (b) Figure 1.2-l 2 Refraction at a convex spherical boundary (R > 0). 14 RAY OPTICS -(n,ln2)(z2/z,). Again, negative magnification means that the image is inverted. By convention P, is measured in a coordinate system 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 negative). The similarities between these properties and those of the spherical mirror are evident. It is important to remember that the image formation properties described above are approximate. They hold only for paraxial rays. Rays of large angles do not obey these paraxial laws; the deviation results in image distortion called aberration. EXERCISE 1.2-2 Image Formation. Derive (1.2-8). Prove that par-axial rays originating from Pr pass through P2 when (1.2-9) and (1.2-10) are satisfied. EXERCISE 1.2-3 Aberration-Free