Nicholas J. Giordano www.cengage.com/physics/giordano Chapter 24 Geometrical Optics Marilyn Akins, PhD Broome Community College
Optics The study of light is called optics Some highlights in the history of optics Study of optics dates to at least third century BC Eyeglasses invented around 1300 Microscopes and telescopes invented around 1600 Applications depend on the ability of lenses and mirrors to focus light Light is an electromagnetic wave and its wave nature needs to be accounted for Introduction
Geometrical Optics Applies to the regime where light travels in straightline paths Effects involving wave interference are not important Describes cases in which the wavelength of the light is much smaller than the size of the objects in the light s path Wavelength of visible light is less than 1µm Describes many everyday applications Including the behavior of mirrors and lenses Introduction
Rays Rays indicate the path and direction of propagation of the light wave In A, the waves pass through a large opening and, to a very good approximation, follow straight lines that pass through the opening In B, the opening is about the same size or smaller than the wavelength of the light and wave optics is needed to explain the spreading of the light Section 24.1
Wave Fronts Wave front surfaces are determined by the crests and troughs of the wave They are always perpendicular to the associated rays The shape of a wave front depends on how the wave is generated and the distance from the source Section 24.1
Two Properties of Light The motion of light along a light ray is reversible If light can travel in one direction along a ray that connects point A to point B, light can also propagate in the reverse direction, from B to A The perpendicular distance between two wave fronts is proportional to the speed of light Because of the way wave fronts are related to crests and troughs of a wave Section 24.1
Ray Tracing Light from an object is used by your eye to form an image of the object When your eye combines the rays to form an image, your brain extrapolates the rays back to their origin The method of following the individual rays as they travel from an object to some other point is called ray tracing Ray tracing involves the use of geometry Section 24.1
Ray Tracing, cont. The figure shows a few rays from the object There are an infinite number of actual rays The light waves associated with all the rays contribute to the image formed by your eye In most ray diagrams, we draw just a few rays from the top and bottom of the image Section 24.1
Image Formation Two questions must be considered to understand how images are formed What happens to light rays when they reflect from a surface such as a mirror or a piece of glass What happens to light rays when they pass across a surface from one material to another such as when they pass from air into a piece of glass You must also distinguish between a flat surface and a curved surface Section 24.1
Reflection from a Plane Mirror Light rays travel in straight lines until they strike something After striking an object, the rays may be reflected Rays may be reflected from a plane mirror A flat surface that reflects all or nearly all the light that strikes it If the light is a plane wave, all the rays are parallel and strike a surface at many different points Section 24.2
Reflection from a Plane Mirror, cont. Characterize the reflection by a single ray The normal (vertical dashed line in fig. B) is perpendicular to the mirror The direction of the incoming and outgoing rays are measured relative to the normal Section 24.2
Law Of Reflection The incoming ray is called the incident ray The angle it makes with the normal is called the angle of incidence, θ i The outgoing ray is called the reflected ray The angle it makes with the normal is called the angle of reflection, θ r The Law of Reflection says θ i = θ r Reflection from a perfectly flat mirror is called specular reflection Section 24.2
Diffuse Reflection If the reflecting surface is rough, the reflections from each individual piece of the surface must be analyzed An incident plane wave will give rise to many reflected rays propagating outward in many different directions This is called diffuse reflection Section 24.2
Image Formation Plane Mirror An image formed by a plane mirror is shown Two representative rays are shown coming from the object There is an infinite number of rays emanating from each point on the object The rays that reflected from the mirror and reached your eyes form the image Section 24.2
Image Formation, cont. To your eye, the location of the image is the point from which these rays appear to emanate This point can be found by ray tracing Extrapolate the rays back in a straight line to their apparent point of origin Each ray obeys the law of reflection Applying geometry will allow the location of the image to be found Section 24.2
Image Formation, final Characteristics of the image The distance from the object to the mirror is the same as the distance from the image to the mirror The size (height) of the image, h i, is the same as the size (height) of the object, h o The image is virtual The image point is located behind the mirror The light does not actually pass through the image The same analysis can be applied to multiple mirrors Section 24.2
Reflection at a Surface The material of a surface will determine if it does or does not reflect light The electric and magnetic fields of the light wave exert forces on electrons and ions near the surface A typical mirror consists of a metal surface The motion of the electrons and ions induced by the fields produces the reflected light ray Section 24.2
Refraction When a light ray strikes a transparent material, some of the light is reflected and some is refracted The reflected ray obeys the Law of Reflection The refracted ray passes into the material The incident angle is now denoted as θ 1 Section 24.3
Angle of Refraction The direction of the refracted ray is measured by using θ 2 (refer to fig. 24.10) The value of this angle depends on the incident angle and the speed of light in the material The speed of light in a vacuum is 3 x 10 8 m/s When the light travels through a material substance, its interactions with the atoms of the material slows down the wave Section 24.3
Snell s Law The change in the speed of light from the vacuum to the material changes the direction of the wave From the geometry of the waves in the material, c sinθ1 = sinθ2 v Section 24.3
Snell s Law, cont. The ratio c/v is called the index of refraction and is denoted by n n = c / v n is unitless Then, sin θ 1 = n sin θ 2 This assumes the wave is incident in a vacuum A more general statement can be applied to any two materials with indices of refraction n 1 and n 2 n 1 sin θ 1 = n 2 sin θ 2 This relationship is called Snell s Law Section 24.3
Speeds and n s for Various Materials Section 24.3
Applying Snell s Law Refraction is also reversible Snell s Law applies whether light begins in the material with the larger or smaller index of refraction Possible angles of refraction are always between 0 and 90 The side with the larger index of refraction has the smaller angle
Direction of Refracted Ray Light is refracted toward the normal when moving into the substance with the larger index of refraction Light is refracted away from the normal when moving into the substance with the smaller index of refraction Shown here Section 24.3
Total Internal Reflection When light is incident from the side with a higher index of refraction, it is bent away from the normal As the incident angle gets larger, the refracted angle also increases Eventually, θ 2 will reach 90 The angle of incidence for which the angle of refraction is 90 is called the critical angle If the angle of incidence is increased beyond the critical angle, Snell s Law has no solution for θ 2 Physically, there is no refracted ray Section 24.3
Total Internal Reflection, cont. This behavior is called total internal reflection This is only possible when the light is incident from the side with the larger index of refraction Section 24.3
Critical Angle From Snell s Law, with θ 2 = 90, θ 1 = θ crit θ crit n 1 2 = sin n1 When the angle of incidence is equal to or greater than the critical angle, light is reflected completely at the interface Section 24.3
Fiber Optics Total internal reflection is used in fiber optics Optical fibers are composed of specially made glass and used to carry telecommunication signals These signals are sent as light waves They are directed along the fiber using internal reflection Section 24.3
Dispersion When light travels in a material, the speed depends on the color of the light This dependence of wave speed on color is called dispersion Since the index of refraction is slightly different for each color, the angle of refraction will be different for each color Section 24.3
Dispersion and Prisms Dispersion is used by a prism to separate a beam of light into its component colors There are two refractions with the prism The red and blue show the extremes of the incident beams Section 24.3
Curved Mirrors A curved mirror can produce an image of an object that is magnified The image can be larger or smaller than the object Magnified images are used in many applications Telescopes Car s review mirror Many others Section 24.4
Ray Tracing Curved Mirror A spherical mirror is one in which the surface of the mirror forms a section of a spherical shell The radius, R, of the sphere is the radius of curvature of the mirror The mirror s principal axis is the line that extends from the center of curvature, C, to the center of the mirror Section 24.4
Concave Spherical Mirror Properties of concave spherical mirrors Incoming rays that are close to and parallel to the principal axis reflect through a single point F F is the focal point It is located a distance ƒ, the focal length, from the mirror Rays that originate at the focal point reflect from the mirror parallel to the principal axis From reversibility of light Section 24.4
Image From a Concave Mirror Examples Section 24.4
Image from Concave Mirror Ray Diagram Trace rays emanating from the top of the object The rays all intersect at a single point This is the top of the image A similar result would be found with rays from other parts of the object Section 24.4
Drawing A Ray Diagram Three rays are particularly easy to draw There are an infinite number of actual rays The focal ray From the tip of the object through the focal point Reflects parallel to the principal axis Section 24.4
Drawing A Ray Diagram, cont. The parallel ray From the tip of the object parallel to the principal axis Reflects through the focal point The central ray From the tip of the object through the center of curvature of the mirror Reflects back on itself The three rays intersect at the tip of the image Section 24.4
Properties of an Image Magnification is the ratio of the height of the image, h i, to the height of the object, h o By convention, the image height of an inverted image is negative Therefore, the magnification is also negative Images smaller than the object are said to be reduced Section 24.4
Real vs. Virtual Images If the rays that form the image all pass through a point on the image, the image is called a real image Real images and virtual images differ Light rays only appear to emanate from a virtual image, they do not actually pass through the image For a real image, the light rays do actually pass through the image An object and its real image are both on the same side the mirror A virtual image is located behind the mirror while the object is in front Section 24.4
Concave Mirror and Virtual Images Use ray tracing to find the image when the object is close to the mirror Closer than the focal point Use the same three rays The rays do not intersect at any point in the front of the mirror Section 24.4
Virtual Images Extrapolate the rays back behind the mirror They intersect at a single image point The rays appear to emanate from the image point behind the mirror The image is virtual because light does not actually pass through any point on the image The object and its image are on different sides of the mirror The image is upright and enlarged Section 24.4
Rules for Ray Tracing Mirrors Construct a figure showing the mirror and its principal axis The figure should also show the focal point and the center of curvature Draw the object at the appropriate point One end of the object will often lie on the principal axis Draw three rays that emanate from the tip of the object The focal ray passes through the focal point and reflects parallel to the principal axis Section 24.4
Rules, cont. Three rays, cont. The parallel ray is parallel to the principal axis and reflects through the focal point The central ray passes through the center of curvature of the mirror and reflects back through the tip of the object The point where the three rays intersect is the image point This point may be in front of the mirror giving a real image This point may be in back of the mirror giving a virtual image Found by extrapolation of the rays behind the mirror Section 24.4
Rules, final This ray-tracing procedure can be repeated for any desired point on the object This allows you to find other points on the image It is usually sufficient to consider just the tip of the image Section 24.4
Ray Tracing Convex Spherical Mirrors A mirror that curves away from the object is called a convex mirror The center of curvature and the focal point lie behind the mirror After striking the convex surface, the reflected rays diverge from the mirror axis The parallel rays converge on an image point behind the mirror This is the focal point, F Section 24.4
Ray Tracing Convex Mirrors, cont. The same three rays are used as were used for concave mirrors The focal ray is directed toward the focal point but is reflected at the mirror s surface, so doesn t go through F The three rays extrapolate to a point behind the mirror Produces virtual image Section 24.4
Mirror Equation Geometry can be used to find the characteristics of the image quantitatively The distance from the object to the mirror is s o The distance from the image to the mirror is s i The given rays produce similar triangles Section 24.4
Mirror Equation and Focal Length From the similar triangles, For an object at (approximately) infinity, 1/s o = 0 But an infinite object will produce parallel rays Parallel rays all intersect at the focal point Therefore, the focal length can be found from the radius of curvature of the mirror Section 24.4
Mirror Equation and Magnification The mirror equation can be written in terms of the focal length The magnification can also be found from the similar triangles shown in fig. 24.30 Section 24.4
Sign Conventions All diagrams with mirrors should be drawn with the light ray incident on the mirror from the left The object distance, s o, is positive when the object is to the left of the mirror and negative if the object is to the right of (behind) the mirror The image distance, s i, is positive when the image is to the left of the mirror and negative if the image is to the right of (behind) the mirror The image distance is positive for real images and negative for virtual images Section 24.4
Sign Conventions, cont. The focal length is positive for a concave mirror and negative for a convex mirror For a concave mirror, ƒ = R / 2 For a convex mirror, ƒ = - R / 2 The object and image heights, h o and h i, are positive if the object/image is upright and negative if it is inverted Section 24.4
Sign Convention, Summary Section 24.4
Other Mirror Shapes The derivation of the equations assumed a spherical mirror Other shapes give better focusing properties for rays that are far from the principal axis Parabolic mirrors Used in many high-performance telescopes Shape gives smaller aberrations than a spherical mirror More difficult and expensive to make than a spherical shape Section 24.4
Lenses A lens uses refraction to form an image Typical lenses are composed of glass or plastic The refraction of the light rays as they pass from the air into the lens and then back into the air causes the rays to be redirected Although refraction occurs at both surfaces of the lens, for simplicity the rays are drawn to the center of the lens Section 24.5
Lenses, Focal Point Parts B and C show the simplification of the single deflection of the rays Parallel rays close to the principal axis intersect at the focal point This is true for incident rays from either side of the lens The focal points are at equal distances on the two sides of the lens Section 24.5
Spherical Lenses The simplest lenses have spherical surfaces The radii of curvature of the lenses are called R 1 and R 2 The radii are not necessarily equal Section 24.5
Types of Lenses Converging lenses All the incoming rays parallel to the principal axis intersect at the focal point on the opposite side Diverging lenses All the incoming rays parallel to the principal axis intersect at the focal point on the same side as the incident rays Section 24.5
Focal Point Diverging Lens The parallel incident rays from the left are refracted away from the axis The rays diverge after passing through the lens The rays on the right appear to emanate from a point F on the left side of the lens This point F is one of the focal points of the lens Section 24.5
Image from a Converging Lens Ray tracing analysis can be used to find the image An infinite number of rays emanate from the object For simplicity, choose three rays that are easy to draw Start at the tip of the object Section 24.5
Rays for a Converging Lens The parallel ray is initially parallel to the principal axis Refracts and passes through the focal point on the right (FR) The focal ray passes through the focal point on the left (FL) Refracts and goes parallel to the principal axis on the right The center ray passes through the center of the lens, C Section 24.5
Rays, cont. If the lens is very thin, the center ray is not deflected by the lens These three rays come together at the tip of the image on the right of the lens In this case, the image is inverted The image is real The rays pass through the image Section 24.5
Rules for Ray Tracing Lenses Construct a figure showing the lens and its principal axis The figure should also show the focal points on both sides of the lens Draw the object at the appropriate point One end of the object will generally lie on the principal axis Draw three rays that emanate from the tip of the object The parallel ray is initially parallel to the principal axis and after refraction passes through one of the focal points Section 24.5
Rules, cont. Three rays, cont. The focal ray is directed at the other focal point and after refraction the ray is parallel to the principal axis The central ray passes through the center of the lens and is not deflected The point where the three rays or their extrapolation intersect is the image point If the rays actually pass through the lens, the image is real If the rays do not pass through the lens, the image is virtual Section 24.5
Rules, final Real image When a lens forms a real image, the object and image are on opposite sides of the lens Virtual image When a lens forms a virtual image, the object and image are on the same side of the lens All other rays that pass through the lens will also pass through the image point Section 24.5
Ray Tracing Diverging Lens Follow the rules for ray tracing for lenses Since the refracted rays do not intersect on the right side of the lens, extrapolate the rays back to the left side of the lens The extrapolations do intersect The point of intersection is the image point at the tip of the image Section 24.5
Sign Conventions for Lenses Assume light travels through the lens from left to right The object will always be located to the left of the lens The object distance, s o, is positive when the object is to the left of the lens According to the first convention, the object distance will always be positive The image distance, s i, is positive when the image is to the right of the lens and negative if the image is to the left of the lens Section 24.5
Sign Conventions, cont. The focal length ƒ is positive for a converging lens and negative for a diverging lens The object height, h o, is positive if the object extends above the axis and is negative if the object extends below The image height, h i, is positive if the image is extends above the axis and is negative if the image extends below Section 24.5
Sign Conventions for Lenses, Summary Section 24.5
Thin-Lens Equation Geometry can be used to find a mathematical relation for locating the image produced by a converging lens The shaded triangles are similar triangles Section 24.5
Thin-Lens Equation and Magnification The thin-lens equation is found from an analysis of the similar triangles The magnification can also be found from the similar triangles shown These results are identical to the results found for mirrors Section 24.5
The Eye Compared to a Lens Section 24.6
The Eye The eye is a complicated structure but can be understood using the principles of refraction and the properties of light Path of light in the eye Light enters through the cornea Passes through a liquid region called the aqueous humor Goes through a lens Enters a gel-like region called the vitreous humor Finally to the retina The retina contains light-sensitive cells that convert light into electrical signals carried to the brain by the optic nerve Section 24.6
Corneal Refraction The indices of refraction are different for different parts of the eye Vary from about 1.33 to 1.40 The biggest index change and the largest refraction occur when light first enters the cornea Snell s Law and geometry can be used to determine the focal length of the corneal lens in the eye Section 24.6
Properties of the Eye For the simplified model of the eye shown in fig. 24.47C, the focal length of the eye is approximately 3.5 R This is about 1.5 R behind the retina In reality, the cornea protrudes slightly from the eye The eye is not truly spherical R is really the radius of curvature of the cornea Section 24.6
Lens of the Eye The lens is inside the eye Light passes through the lens after being refracted by the cornea To simplify the model, assume the lens is just outside of the eye The lens is a converging lens For a normal person, ƒ lens ~ 25 cm Section 24.6
Lens Maker s Formula The focal length of a lens is related to the radii of curvature of the two surfaces forming the lens The relationship is given by the lens maker s formula This form applies only to the double convex lens shape and assumes the lens is in air More general forms of the formula can be used for other situations Section 24.6
Lens Maker s Formula and the Eye Assume the radii of curvature are equal and a focal length of 25 cm is needed R = 0.21 m The radius of curvature of the eye is several times larger than the diameter of the eye This is consistent with a real eye The front surface of the lens in the eye is rather flat, indicating a large radius of curvature The actual lens in the eye can change shape to allow for objects at different distances to focus on the retina Section 24.6
Rainbow An incident ray from the Sun is refracted when it enters a water droplet The refracted angle depends on the color of the light Rays for different colors travel at different angles inside the droplet Section 24.7
Rainbow, cont. When light reaches the back surface, a portion is reflected The rest refract and propagate out of the back of the droplet The reflected rays are refracted again when they leave the droplet The outgoing rays emerge over a range of angles Different colors of a rainbow appear at different positions (angles) in the sky Section 24.7
Twinkling Stars The index of refraction of air depends on its temperature and pressure The index of refraction varies from place to place in the atmosphere Each small region acts like a small lens or prism Section 24.7
Twinkling Stars, cont. These lenses change from moment to moment The deflection of rays through these lenses also change from moment to moment This causes the stars to twinkle A similar effect is found with light from the Moon and planets, but their size limits the twinkling effect Astronomers build telescopes on tops of mountains or in space to lessen or eliminate the effects of the atmosphere Section 24.7
Aberrations The ray diagrams and equations used assumed all the rays from an object intersect at a single image point The images formed by real mirrors and lenses suffer from effects called aberrations Aberrations are deviations from the ideal images encountered so far Section 24.8
Chromatic Aberration Is related to dispersion Different colors are actually diffracted by different amounts The focal length of a lens is different for each color Multiple lens can be used to minimize the effect Section 24.8
Spherical Aberration In real situations, rays that are not exactly on the axis will not focus at the ideal image points found for lenses and mirrors These deviations are called spherical aberrations since they are due to the spherical shape of the mirror or lens Using a parabolic shape can eliminate spherical aberration Section 24.8