Physics
Geometrical Optics
Geometrical optics is a branch of optics that focuses on the behavior of light as it travels through lenses, mirrors, and other optical elements. It simplifies the study of light by treating it as rays that can be traced through these elements using principles such as reflection, refraction, and the formation of images. This approach is particularly useful for understanding the basic properties of optical systems.
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11 Key excerpts on "Geometrical Optics"
eBook - PDF- Daniel Malacara(Author)
- 2001(Publication Date)
- CRC Press(Publisher)
1 Basic Ray Optics ORESTES STAVROUDIS Centro de Investigaciones en Optica, Lean, Mexico 1.1 INTRODUCTION Geometrical Optics is a peculiar science. It consists of the physics of the 17th and 18th centuries thinly disguised by the mathematics of the 19th and 20th centuries. Its contemporary applications are almost entirely in optical design which, like all good engineering, remains more of an art even after the advent of the modern computer. This brief chapter is intended to convey the basic formulas as well as the flavor of Geometrical Optics and optical design in a concise and compact form. I have attempted to arrange the subject matter logically, although not necessarily in histor- icalorder. The basic elements of Geometrical Optics are rays and wavefronts: neither exist, except as mathematical abstractions. A ray can be thought of as a beam oflight with an finitesimal diameter. However, to make a ray experimentally by passing light through a very small aperture causes diffraction to rear its ugly head and the light spreads out over a large solid angle. The result is not a physical approximation to a ray but a distribution of light in which the small aperture is a point source. A wavefront is defined as a surface of constant phase to which can be attributed definite properties such as principal directions, principal curvatures, cusps, and other singularities. But, like the ray, the wavefront cannot be observed. Its existence can only be inferred circumstantially with interferometric methods. However there is in Geometrical Optics an object that is observable and mea- surable: the caustic surface. [1] It can be defined in distinct but equivalent ways: • As the envelope of an orthotomic system of rays; i.e., rays ultimately from a single object point. 2 Stavroudis • As the cusp locus of a wavefront train, or, equivalently, the locus of points where the element of area of the wavefront vanishes.
eBook - PDF- Ariel Lipson, Stephen G. Lipson, Henry Lipson(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
3 Geometrical Optics If this book were to follow historical order, the present chapter should have pre-ceded the previous one, since lenses and mirrors were known and studied long before wave theory was understood. However, once we have grasped the ele-ments of wave theory, it is much easier to appreciate the strengths and limitations of Geometrical Optics, so logically it is quite appropriate to put this chapter here. Essentially, Geometrical Optics, which considers light waves as rays that propagate along straight lines in uniform media and are related by Snell’s law (§2.6.2 and §5.4) at interfaces, has a relationship to wave optics similar to that of classical mechanics to quantum mechanics. For Geometrical Optics to be strictly true, it is important that the sizes of the elements we are dealing with be large com-pared with the wavelength λ . Under these conditions we can neglect diffraction, which otherwise prevents the exact simultaneous specification of the positions and directions of rays on which Geometrical Optics is based. Analytical solutions of problems in Geometrical Optics are rare, but fortunately there are approximations, in particular the Gaussian or paraxial approximation, which work quite well under most conditions and will be the basis of the discussion in this chapter. Exact solutions can be found using specialized computer programs, which will not be discussed here. However, from the practical point of view, Geometrical Optics answers most questions about optical instruments extremely well and in a much simpler way than wave theory could do. For example, we show in Fig. 3.1 the basic idea of a zoom lens (§3.6.4), which today is part of every camera, and is a topic that can be addressed clearly by the methods developed in §3.5. Geometrical Optics fails only in that it cannot define the limits of performance such as resolving power, and it does not work well for very small devices such as optical fibres.
eBook - PDF- Andri M. Gretarsson(Author)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
3 Geometric Optics 3.1 The Geometric Optics Approximation In geometric optics, we assume that the wavelength of light approaches zero. In that limit, successive wavefront normals trace out straight lines in free space, just like a stream of classical particles. For example, laser beams (Gaussian beams) spread out slowly as they travel, due to diffraction. In the limit λ → 0, this spread also becomes zero and all the light moves in a perfectly straight line. (See, for example, Eq. (4.15).) We imagine these straight line paths as “rays” of light that are affected only by the optics placed in their path. In geometric optics, the light waves are also assumed to be incoherent so they will have random phase and amplitude. The mean square field amplitude resulting from a sum of such random fields is the sum of the mean squares of the contributing fields. E 2 tot = E 2 1 + E 2 2 + . . . Just like random errors, random fields add in quadrature. Irradiance (power per unit area) is proportional to the field squared, so this implies that the total irradiance at any point in space is just the sum of the contributing irradiances: I tot = I 1 + I 2 + . . . That’s of course what our intuition tells us should happen. Since we are used to incoherent light in everyday life our intuition is attuned to that case. It is the coherent case that’s less intuitive. When diffraction is insignificant and coherence is low, geometric optics provides a good model for the behavior of light. The most important application of geometric optics is in imaging. Yet, geometric optics is applied in many other contexts such as architectural lighting design, microwave antenna design, spectroscopy, and so forth. 3.2 Refraction Refraction refers to the tendency of light to change direction when there is a change in the index of refraction of the medium in which the wave travels.
eBook - PDFObservational Astronomy
Techniques and Instrumentation
- Edmund C. Sutton(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
8 Optics We begin our treatment of optics by first considering Geometrical Optics. In the limit of small wavelengths, Geometrical Optics describes the direction in which light travels through space as it encounters materials with different indices of refraction. Initially the refractive index will simply be assumed to be a property of a material which describes the speed at which light propagates in that material. That will be sufficient to allow us to treat the theory of aberrations and to look into some basic aspects of telescope design. Next we will look at the physical origins of the refractive index and at the Fresnel coefficients, which are important in a number of contexts including the design of various spectroscopic devices. Then we will consider physical optics, the behavior of light in the regime of finite wavelengths where diffractive effects become important. This will include a look at the Airy pattern. Finally we will introduce the concepts of the point spread function and the modulation transfer function and use them to consider some general properties of imaging. 8.1 Geometrical Optics The properties of light propagation can often usefully be described by geomet- rical optics, an approximation which is valid in the limit of small wavelengths. The wavelength λ is assumed to be small compared with all relevant length scales, including the dimensions of any physical objects present. The media of propaga- tion are described by various values of the refractive index n, which in general is wavelength dependent. In this approximation it is possible to visualize the indi- vidual paths followed by narrow pencils of light, a process known as ray tracing. These rays are considered to have small cross sectional area A and small diver- gence and therefore small étendue. Ray tracing depends on two basic laws: the law of reflection and Snell’s law. 117
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
25 GEOMETRIC OPTICS Figure 25.1 Image seen as a result of reflection of light on a plane smooth surface. (credit: NASA Goddard Photo and Video, via Flickr) Chapter Outline 25.1. The Ray Aspect of Light • List the ways by which light travels from a source to another location. 25.2. The Law of Reflection • Explain reflection of light from polished and rough surfaces. 25.3. The Law of Refraction • Determine the index of refraction, given the speed of light in a medium. 25.4. Total Internal Reflection • Explain the phenomenon of total internal reflection. • Describe the workings and uses of fiber optics. • Analyze the reason for the sparkle of diamonds. 25.5. Dispersion: The Rainbow and Prisms • Explain the phenomenon of dispersion and discuss its advantages and disadvantages. 25.6. Image Formation by Lenses • List the rules for ray tracking for thin lenses. • Illustrate the formation of images using the technique of ray tracking. • Determine power of a lens given the focal length. 25.7. Image Formation by Mirrors • Illustrate image formation in a flat mirror. • Explain with ray diagrams the formation of an image using spherical mirrors. • Determine focal length and magnification given radius of curvature, distance of object and image. Introduction to Geometric Optics Geometric Optics Light from this page or screen is formed into an image by the lens of your eye, much as the lens of the camera that made this photograph. Mirrors, like lenses, can also form images that in turn are captured by your eye. Chapter 25 | Geometric Optics 983 Our lives are filled with light. Through vision, the most valued of our senses, light can evoke spiritual emotions, such as when we view a magnificent sunset or glimpse a rainbow breaking through the clouds. Light can also simply amuse us in a theater, or warn us to stop at an intersection. It has innumerable uses beyond vision. Light can carry telephone signals through glass fibers or cook a meal in a solar oven.
eBook - PDF- Robert Splinter, Brett A. Hooper(Authors)
- 2006(Publication Date)
- CRC Press(Publisher)
The maximal path length will be achieved under gravita-tional lensing; for the conditions pertaining to the material in this book the propagation will strive to satisfy a minimal path length. 3.1.4 Ray Optics Ray optics is the branch of Geometrical Optics that describes light as rays. The rays emanate from a source and are perpendicular to the wavefronts 1.22 D D 78 An Introduction to Biomedical Optics described in Section 2.2 in Chapter 2. The rays can illustrate the Laws of Rectilinear propagation, reflection, and refraction with great accuracy and ease. Examples of Geometrical Optics are presented in image formation with lenses and mirrors in the subsequent chapters. By definition, the light ray at a given point in space is along the gradient of the optical path function. The optical path function L ( r ) is defined as the optical path length from a conve-niently chosen reference wave surface 0 to an arbitrary point P . The point P is identified by the vector r , and this point can be reached by a ray emanat-ing from the wave surface 0, and a point P 0 designates a location on the sur-face 0, with the vector r 0 as position vector for P 0 . The optical path function L, traveling from point A to point B in three-dimensional space can be defined in a medium with speed of light v and index of refraction n as (3.13) or in vector notation (3.14) When there is a disturbance in optical space occurring at time t 0 , the wave-front 0 is affected. At time t , when t t 0 , the wavefront will be at the two-dimensional surface , consisting of all points that reach from 0 after the time interval ( t t 0 ), where the distance to the surface can be defined as (3.15) To obtain the path function L ( r ), when r moves a distance d r toward the loca-tion, P 1 the ray P 0 P P 1 will construct the new wave surface 1 originating from 0 at position P 1 .
eBook - PDF- Naftaly Menn(Author)
- 2004(Publication Date)
- Academic Press(Publisher)
In most cases experienced in practice, 1 2 1 ♦ Geometrical Optics in the Paraxial Area Figure 1.1 Optical beams: (a) parallel, (b,c) homocentric and (d) non-homocentric. 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 thick-ness. 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). If a beam, either divergent or convergent, has a single point of intersection of all optical rays it is called a homocentric beam (Fig. 1.1b,c). An example of a non-homocentric beam is shown in Fig. 1.1d. 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, n 1 and n 2 (see Fig. 1.2). The first law is just a statement that the incident angle, i , is Figure 1.2 Reflection and refraction of radiation.
eBook - PDFTo Measure the Sky
An Introduction to Observational Astronomy
- Frederick R. Chromey(Author)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Chapter 5 Optics for astronomy But concerning vision alone is a separate science formed among philosophers, namely, optics . . . . It is possible that some other science may be more useful, but no other science has so much sweetness and beauty of utility. Therefore it is the fl ower of the whole of philosophy and through it, and not without it, can the other sciences be known. – Roger Bacon, Opus Maius , Part V, 1266 – 68 Certainly Bacon ’ s judgment that optics is the gateway to other sciences is particularly true of astronomy, since virtually all astronomical information arrives in the form of light. We devote the next two chapters to how astronomers utilize the sweetness and beauty of optical science. This chapter introduces the fundamentals. We fi rst examine the simple laws of re fl ection and refraction as basic conse-quences of Fermat ’ s principle, then review the behavior of optical materials and the operation of fundamental optical elements: fi lms, mirrors, lenses, fi bers, and prisms. Telescopes, of course, are a central concern, and we introduce the simple concept of a telescope as camera. We will see that the clarity of the image produced by a telescopic camera depends on many things: the diameter of the light-gathering element, the turbulence and refraction of the air, and, if the telescope uses lenses, the phenomenon of chromatic aberration. Concern with image quality, fi nally, will lead us to an extended discussion of monochromatic aberrations and the difference between the fi rst-order and higher-order ray theories of light. 5.1 Principles of Geometrical Optics This section reviews some results from Geometrical Optics, and assumes you have an acquaintance with this subject from an introductory physics course. Geometrical Optics adopts a ray theory of light, ignoring many of its wave and all of its particle properties. 113 5.1.1 Rays and wavefronts in dielectric media The speed of light in a vacuum is a constant, c , identical for all observers.
eBook - PDF- David A. Boas, Constantinos Pitris, Nimmi Ramanujam, David A. Boas, Constantinos Pitris, Nimmi Ramanujam(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
3 Geometrical Optics .is.the.study.of.light.without.diffraction.or. interference.and.is.based.on. Fermat’s principle . .We.treat.light. as.particles.of.energy.traveling.through.space . .These.particles. follow. trajectories. that. are. called. rays . . Hence,. geometrical. optics.is.often.called. ray optics . .Fermat’s.principle.is.a.concise. statement.that.contains.all.the.physical.laws,.such.as.the. law of reflection .and.the. law of refraction ,.in.geometrical.optics.(Poon. and.Kim.2006) . 1.1 Fermat’s Principle Fermat’s.principle.states.that.the.path.of.a.light.ray.follows.is.an. extremum.in.comparison.to.nearby.paths . .The.extremum.may.be. a.minimum,.a.maximum,.or.stationary.with.respect.to.variations. in.the.ray.path . .However,.the.extremum.is.usually.a.minimum . .For. a.simple.example,.as.shown.in.Figure.1 .1, .the.shortest.distance.(the. minimum.distance).between.two.points.A.and.B.is.along.a.straight. line.(solid.line).in.a. homogeneous medium ,.i .e., .in.a.medium.with.a. constant. refractive index ,.instead.of.taking.the.nearby.dotted.line . . Since.the.speed.of.light.in.a.homogeneous.medium.is.constant,.the. time.it.takes.for.the.ray.to.traverse.the.solid.line.must.be.minimum . . Hence.Fermat’s.principle.is.often.stated.as.a. principle of least time . . Under.this.context,.the.light.ray.would.follow.that.path.for.which. the.time.taken.is.minimum . .For.a.more.complicated.example,.we. show.the.derivation.of.the.well-known.Snell’s.law.of.refraction . .In. Figure.1 .2, . θ i .and. θ t .are.the.angles.of.incidence.and.transmission,. respectively. .The.angles.are.measured.from.the.normal.NN ′ .to.the. interface.MM ′ ,.which.separated.media.1.and.2,.characterized.by. refractive.indexes. n i .and. n t ,.respectively . .The.total.time.taken.to. transit.from.point.A.to.B.is.given.by . t z v v h z v h d z v ( ) ( ) , = + = + + + -AO OB 1 2 1 2 2 1 1 2 2 2 . (1 .1) where. v 1 .and. v 2 .are.the.light.velocities.in.media.1.and.2,.respec-tively.
eBook - PDFPrinciples of Optics
Electromagnetic Theory of Propagation, Interference and Diffraction of Light
- Max Born, Emil Wolf(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
A more powerful approach to the study of the intensity distribution in such regions is offered by methods which will be discussed in the chapters on diffraction. Finally we stress that the simplicity of the Geometrical Optics model arises essentially from the fact that, in general, the field behaves locally as a plane wave. A t optical wavelengths, the regions for which this simple geometrical model is inadequate are an exception rather than a rule ; in fact for most optical problems Geometrical Optics furnishes at least a good starting point for more refined investigations. 3.2 G E N E R A L P R O P E R T I E S OF R A Y S 3.2.1 The differential equation of light rays The light rays have been defined as the orthogonal trajectories to the geometrical wave-fronts S^(x, y, z) = constant and we have seen that, if r is a position vector of a typical point on a ray and s the length of the ray measured from a fixed point on it, then n~ a = grad S/> > (1) * It has been suggested by J. B. K E L L E R [J. Appl. Phys., 28 (1957), 426; also Calculus of Varia-tions and its Application, ed. L . M. GRAVES (New York, McGraw-Hill, 1958), 27] that the behaviour of the contributions represented by the higher-order terms may be studied by means of a model which is an extension of ordinary Geometrical Optics. In this theory the concept of a diffracted ray is introduced, which obeys a generalized FERMAT'S principle. With each such ray an appro-priate field is associated and is assumed to satisfy the same propagation laws as the Geometrical Optics field. Some applications of the theory were described by J. B. K E L L E R , Trans. Inst. Radio Eng., A.P.— 4 (1956), 312 and J. B. K E L L E R , R. M. L E W I S and B. D. SECKLER, J. Appl. Phvs., 28 (1957), 570. See also M. K L I N E and I. W . K A Y , loc. cit.
eBook - PDFDisplays
Fundamentals & Applications, Second Edition
- Rolf R. Hainich, Oliver Bimber(Authors)
- 2016(Publication Date)
- A K Peters/CRC Press(Publisher)
Such “dirty” light, in many cases, cannot be used when it becomes essential that light waves interact in a controlled way (e.g., for holograms, as we will see later). Because of the importance of lasers, we will explain their principles in Section 3.5 at page 81. Wave optics, however, is not always crucial to the optical design of instruments or dis- plays. For the generation or detection of macroscopic image points, which are a lot larger than the wavelength of light (e.g., pixels being projected onto a screen or captured by a regular camera), the wave properties of light can often be ignored. In geometric optics (Section 3.3 at page 57), we assume that light always travels the shortest distance between two points in a homogeneous medium. By completely ignoring its wave properties and using a more abstract representation of light rays optical designs can be simplified a great deal. Instead of complex principles of wave propagation, simple vector notations and operations are sufficient for describing the behavior of optical systems. 53 54 Displays We will explain the most basic concepts of image point formation (Section 3.4 at page 64) based on laws of geometric optics. Various ordinary optical elements, such as lenses, mirrors, and apertures, as well as their effects in optical devices are discussed. Before we end this chapter, we will also introduce the plenoptic function and its relationship to new imaging, display and lighting systems that are based on entire light fields rather than on focused light rays. 3.2 WAVE OPTICS In Chapter 2, we learned that light is an electromagnetic wave. As it is the case for all waves, electromagnetic waves share wave properties such as frequency, amplitude, and phase. For many optical elements, such as holograms, it is necessary to consider all of these wave prop- erties in order to explain particular phenomena. This branch of optics is referred to as wave optics.
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