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Geometrical Optics Mirrors and Prisms Mirrors Phys 322 Lecture 14 Chapter 5 Geometrical Optics Mirrors and Prisms Mirrors Ancient bronze mirror Liquid mercury mirror Hubble telescope mirror Planar mirror also called plane, or flat mirrors s = -s i r i o Sign convention: s on the object side is positive, and negative on the opposite side Planar mirror Sign convention: s on the object side is positive, and negative on the opposite side si = -so For a plane mirror, a point source and its image are at the same distance from the mirror on opposite sides; both lie on the same normal line. Image is virtual, up-right, and life-size (MT = +1) yi si The equation for lens works: MT yo so Exercise: plane mirror height How high should be the mirror for a person to see a full image of him/her-self? D Solution: B A C E Triangle ABC is twice as small as ADE BC is half DE (the height of the guy) 1. Mirror (BC) should be at least half of the guy’s height (DE) 2. Its bottom should 1/2 of the height of guy’s eyes from the ground ‘Mirror image’ Mirror image of left hand is a right hand Inversion: converting right-handed coordinate system into left-handed one Even number of mirrors can be used to avoid inversion Applications: steering light DLP projection TV reflex camera (SLR) http://www.plus-america.com/papers.html Atomic force microscope Parabolic aspherical mirror V Make a mirror that will converge plane waves into a point Fermat’s principle: OPL W1A1 A1F W2 A2 A2F Application: headlights, W A A D W A A D 1 1 1 1 2 2 2 2 flashlights, A F A D A F A D radars, 1 1 1 2 2 2 dish antenna, In general: AF AD …. This is the surface of paraboloid: y2 = 4fx (origin at vertext V) Aspherical mirrors off-axis parabolic divergingdiverging convergingdiverging Collects light from one point to another convergingconverging divergingconverging Spherical mirror y2 4 fx Paraboloid and sphere are similar in paraxial approximation y2 x R2 R2 y2 x2 2xR R2 R2 y22xxR2 small when close to axis x Spherical mirror formula SAP is bisected by AC: SC CP SC so R SA PA CP R si Paraxial approximation: SA so PA si s R s R o i s s o i Sign convention: R<0 in real object space so>0 in real object space 1 1 2 s >0 in real image space i f >0 concave mirror so si R Focal lengths: Mirror Formula 1 1 1 2 fooolim1/s 1/f 2/R si so si f R fiiilim1/sf 1/ 2/ R so Spherical mirrors 1 1 1 2 so si f R Note: Both mirror and lens equations are the same, except the real image is in front of mirror, but it is behind the lens Magnification equations are the same as well. Concave mirror: principal axes and image 1 1 1 2 Principal rays for concave mirror: s s f R 1) Parallel to principal axis reflects through F. o i 2) Through F, reflects parallel to principal axis. 3) Through center. S #1 #2 #3 f Image is: C Real (light rays actually cross) si Inverted (Arrow points in opposite direction) Diminished (smaller than object, only if object is further than C) NOTE: Any other ray from object tip which hits mirror will reflect through image tip Convex mirror: principal axes and image Principal rays for convex mirror: 1) Parallel to principal axis appear to originate from F after reflection. 2) Through F, reflects parallel to principal axis. 3) Through center. #1 S #3 #2 P Image is: f F C Virtual (light rays don’t really cross) Upright (same direction as object) Diminished (smaller than object) **For a real object, image is always virtual, upright and diminished Exercise: can a concave mirror form a virtual image? 1 1 1 so si f s o si Concave mirror: so and f are always positive, want to get negative si virtual 1 1 1 0 image si f so so f F An object must be between mirror and its focal plane s o si Spherical mirrors Examples Dispersing prism ni sini nt sint Bending depends on wavelength: dispersing prism, i.e. n=n() Can we use optical flat for dispersing light? - no. Rays emerge parallel to each other Dispersing prism equation Example ni sini nt sint t2 i1 nt=n ni=1 Total deviation is a function of refraction index: 2 2 i1 arcsinsin n sin t1 sini1 cos Minimum deviation min occurs when i1 = t2 sin / 2 n min can use to determine n sin / 2 Spectral analyzer And this arrangement maps position to angle: out x in Prism spectrometers Drawbacks: () - nonlinear dependence Low spectral resolution Small aperture Constant-deviation dispersing prisms Pellin-Broca prism: Abbe prism: o =60o always! min=90 always! min Pellin, Ph. and Broca, André (1899), "A Spectroscope of Fixed Deviation". Ernst Abbe Astrophysical Journal 10 337 1840-1905 Fix input-output at 90o or 60o and rotate prism for different wavelengths Reflecting prism Reflect the beam with no dispersion using total internal reflection If we make t1 = i2 - like in flat glass plate = 2i1 + “achromatic” prism Reflecting prisms The right-angle prism The Porro prism The penta prism The Dove prism.
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