Geometrical Optics / Mirror and Lenses Outline Reflection Plane Mirrors Concave/Convex Mirrors Refraction Lenses Dispersion Geometrical Optics In describing the propagation of light as a wave we need to understand: wavefronts: a surface passing through points of a wave that have the same phase and amplitude. rays: a ray describes the direction of wave propagation. A ray is a vector perpendicular to the wavefront. Reflection and Refraction When a light ray travels from one medium to another, part of the incident light is reflected and part of the light is transmitted at the boundary between the two media. The transmitted part is said to be refracted in the second medium. http://www.geocities.com/CapeCanaveral/Hall/6645/propagation/propagation.html *In 1678 the great Dutch physicist Christian Huygens (1629-1695) wrote a treatise called Traite de la Lumiere on the wave theory of light, and in this work he stated that the wavefront of a propagating wave of light at any instant conforms to the envelope of spherical wavelets emanating from every point on the wavefront at the prior instant. From this simple principle Huygens was able to derive the laws of reflection and refraction incident ray reflected ray refracted ray Types of Reflection When light reflects from a smooth surface, it undergoes specular reflection (parallel rays will all be reflected in the same direction). When light reflects from a rough surface, it undergoes diffuse reflection (parallel rays will be reflected in a variety of directions). The Law of Reflection For specular reflection the incident angle θi equals the reflected angle θr: θi =θr (Known since 1000 BC) The angles are measured relative to the normal, shown here as a dotted line. Forming Images with a Plane Mirror A mirror is an object that reflects light. A plane mirror is simply a flat mirror. Plane mirrors are ground to be flat – the flatter the more expensive. (Typically good ones have - where we use visible radiation - no hills or valleys larger than 500nm). Consider an object placed at point P in front of a plane mirror. An image will be formed at point P´ behind the mirror. do = distance from object to mirror di = distance from image to mirror ho = height of object hi = height of image h hi For a plane mirror: o do di do = di and ho = hi vertex Q = do + di Images An image is formed at the point where the rays of light leaving the object either actually intersect or where they appear to originate. If the light rays actually do intersect, then the image is a real image. If the light only appears to be coming from a point, but is not physically there, then the image is a virtual image. We define the magnification, m, of an image to be: image height h d m = = i = − i object height ho do If m is negative, the image is inverted (upside down). The image is called virtual because it does not really exist behind the mirror Real image Plane Mirrors A plane mirror image has the following properties: *The mirror in your bathroom is a piece of plate glass with a coating on the backside so they are second surface mirrors. • The image distance equals the object distance. • The image is unmagnified. • The image is virtual. • The image is not inverted. • Left and right are reversed **The intensity of the reflected beam depends upon the angle of incidence and the indices of refraction and they type of coating. To save expenses, you would like to buy the shortest mirror that will allow you to see your entire body. Should the mirror be (a) half your height (b) two-thirds your height, or (c) equal to your height? Does the answer depend on how far away from the mirror you stand? eye mirror you Spherical Mirrors concave A spherical mirror is a mirror whose surface shape is spherical with radius of curvature R. There are two types of spherical mirrors: concave and convex. **The principal axis (optical axis, vertex) is the straight line between C and the convex midpoint of the mirror We will always orient the mirrors so that the reflecting surface is on the left. The object will be on the left. Focal Point When parallel rays are incident upon a spherical mirror, the reflected rays intersect at the focal point F. For a concave mirror, the focal point is in front of the mirror. For a convex mirror, the The incident rays diverge from the convex focal point is behind the mirror, but they trace mirror. back to the focal point F. Focal Length The focal length f is the distance from the surface of the mirror to the focal point. It can be shown that the focal length is half the radius of curvature of the mirror. Sign Convention: the focal length is negative if the focal point is behind the mirror. For a concave mirror, f = ½R For a convex mirror, f = −½R (R is always positive) Ray Tracing We will use three principal rays to Optical axis determine where an Curvature point image will be located. The parallel ray (P ray) reflects Curvature point through the focal point. The focal ray (F ray) reflects parallel to the axis, and the center-of-curvature ray (C ray) reflects back along its incoming path. Ray Tracing – Examples concave convex Real image Virtual image applet mirror/lens Theorem of intersecting lines h d h0 d0 − R 0 = 0 = − hi R − di − hi di Mirror Equation R − dd with ii= 1 1 1 f= ½ R = + dRdoo− f d0 di The Mirror Equation The ray tracing technique Sign Conventions: shows qualitatively where the image will be located. The do is positive if the object is in front of distance from the mirror to the the mirror (real object) image, di, can be found from do is negative if the object is in back of the mirror equation: the mirror (virtual object) di is positive if the image is in front of 1 1 1 the mirror (real image) + = d is negative if the image is behind do di f i the mirror (virtual image) d = distance from object to o f is positive for concave mirrors mirror f is negative for convex mirrors di = distance from image to mirror m is positive for upright images f = focal length m is negative for inverted images The Refraction of Light The speed of light is different in different materials. We define the index of refraction, n, of a material to be the ratio of the speed of light in vacuum to the speed of light in the material: n = c/v When light travels from one medium to another, its velocity and wavelength change, but its frequency remains constant. http://www.geocities.com/CapeCanaveral/Hall/6645/propagation/propagation.html Snell’s Law In general, when light enters a new material its direction will change. The angle of refraction θ2 is related to the angle of incidence θ1 by Snell’s Law: sinθ sin θ 1 = 2 = constant v1 v2 where v is the velocity of light in the medium. Normal line θ1 Snell’s Law can also be written as: n1sinθ1 = n2sinθ2 Air The angles θ1 and θ2 are Glass measured relative to the line θ normal to the surface 2 between the two materials. Example: Which way will the rays bend? n = 1.4 n = 2 n = 1.6 n = 1.2 Which of these rays can be the refracted ray? Total Internal Reflection When light travels from a medium with n1 > n2, there is an angle, called the critical angle θc, at which all the light is reflected and none is transmitted. This process is known as total internal reflection. The critical angle occurs when θ2= 90 degrees: n2 sinθc = n1 The incident ray is both reflected and Total Internal Reflection refracted. A mirage is created due to the bending of light. The index of refraction of the A pencil in a glass of water looks hot air near the ground is bent due to the light refraction lower than the n of the colder air on the top. Object in the sky appear to be shifted towards the zenith by a small amount. This is due to the refractive effect of the atmosphere. This has been known since the time of Ptlomey in Egypt in 150 BC. ASTRONOMICAL REFRACTION: The displacements of astronomical objects by atmospheric refraction. These effects are many orders of magnitude larger than the accuracy of the best astronomical position measurements, and so large that the mountings of most astronomical telescopes are adjusted to minimize the effects of refraction. http://www.sundog.clara.co.uk/rainbows/primrays.htm RefractionRefraction inin aa TriangularTriangular PrismPrism n=1 n>1 Light always bends toward the base of a triangular prism (if n of the prism is higher than the ambient n). Different colors bend differently. It means that n is different for different colors. The separation of colors is called light dispersion. http://www.wolles-website.de/teste_taeuschungen/taeuschungen_uebersicht.htm Lenses A lens is an object that uses refraction to bend light and form images Light is reflected from a mirror. Light is refracted through a lens. Focal Point The focal point of a lens is the place where parallel rays incident upon the lens converge. converging lens diverging lens Ray Tracing for Lenses Just as for mirrors we use three rays to find the image from a lens. The lens is assumed to be thin. The P ray propagates parallel to the principal axis until it encounters the lens, where it is refracted to pass through the focal point on the far side of the lens.