Optics I lenses and apertures CS 178 Spring 2011 Begun 4 5 11 finished 4 7 Error on slide 63 corrected 4 12 Marc Levoy Computer Science Department Stanford University Outline why study lenses thin lenses thick lenses 2 graphical constructions algebraic formulae center of perspective 3D perspective transformations depth of field aberrations distortion vignetting glare and other lens artifacts diffraction and lens quality special lenses telephoto zoom Marc Levoy Cameras and their lenses single lens reflex SLR camera 3 digital still camera DSC i e point and shoot Marc Levoy Cutaway view of a real lens 4 Vivitar Series 1 90mm f 2 5 Cover photo Kingslake Optics in Photography Marc Levoy Lens quality varies Why is this toy so expensive EF 70 200mm f 2 8L IS USM 1700 Why is it better than this toy EF 70 300mm f 4 5 6 IS USM 550 5 Why is it so complicated Canon Marc Levoy Stanford Big Dish Panasonic GF1 Panasonic 45 200 4 5 6 zoom at 200mm f 4 6 300 Leica 90mm 2 8 Elmarit M prime at f 4 2000 Zoom lens versus prime lens 7 Canon 100 400mm 4 5 5 6 zoom at 300mm and f 5 6 1600 Canon 300mm 2 8 prime at f 5 6 4300 Marc Levoy Physical versus geometrical optics Hecht light can be modeled as traveling waves the perpendiculars to these waves can be drawn as rays diffraction causes these rays to bend e g at a slit geometrical optics assumes 0 no diffraction in free space rays are straight a k a rectilinear propagation 8 Marc Levoy Physical versus geometrical optics contents of whiteboard 9 in geometrical optics we assume that rays do not bend as they pass through a narrow slit this assumption is valid if the slit is much larger than the wavelength physical optics is a k a wave optics Marc Levoy Snell s law of refraction Hecht 10 as waves change speed at an interface they also change direction xi xt sin i sin t nt ni speed of light in a vacuum index of refraction nt is defined as speed of light in medium t Marc Levoy Typical refractive indices n air 1 0 water 1 33 glass 1 5 1 8 mirage due to changes in the index of refraction of air with temperature 11 Marc Levoy Refraction in glass lenses Hecht 12 when transiting from air to glass light bends towards the normal when transiting from glass to air light bends away from the normal light striking a surface perpendicularly does not bend Marc Levoy Q What shape should an interface be to make parallel rays converge to a point Hecht A a hyperbola 13 so lenses should be hyperbolic Marc Levoy Spherical lenses hyperbolic lens spherical lens Hecht wikipedia two roughly fitting curved surfaces ground together will eventually become spherical spheres don t bring parallel rays to a point this is called spherical aberration nearly axial rays paraxial rays behave best 14 Marc Levoy Examples of spherical aberration Canon 135mm soft focus lens gtmerideth 15 Canon Marc Levoy Paraxial approximation object image e P P assume e 0 Not responsible on exams for orange tinted slides 16 Marc Levoy Paraxial approximation object image l u h e P P z 17 assume e 0 assume sin u h l u for u in radians assume cos u z l 1 assume tan u sin u u Marc Levoy The paraxial approximation is a k a first order optics 3 5 7 assume first term of sin 3 5 7 i e sin 2 4 6 assume first term of cos 1 2 4 6 i e cos 1 so tan sin these are the Taylor series for sin and cos phi in degrees 18 Marc Levoy Paraxial focusing object P i i n n Snell s law n sin i n sin i paraxial approximation n i n i 19 image equivalent to P sin i nt sin t ni with n ni for air n nt for glass i i in radians i t in degrees Marc Levoy Paraxial focusing i u a u h z u h z Given object distance z what is image distance z i i h u r a P n z u P n z n i n i 20 Marc Levoy Paraxial focusing i u a u h z u h z a u i a h r i i h u r a P n z u P n z n u a n a u n h z h r n h r h z n i n i 21 n z n r n r n z h has canceled out so any ray from P will focus to P Marc Levoy Focal length r P n P n z What happens if z is z n z n r n r n z n r n r n z z r n n n 22 f focal length z Marc Levoy Lensmaker s formula using similar derivations one can extend these results to two spherical interfaces forming a lens in air so si Hecht edited as d 0 thin lens approximation we obtain the lensmaker s formula 1 1 so si 23 1 1 nl 1 R2 R1 Marc Levoy Gaussian lens formula Starting from the lensmaker s formula 1 1 so si 1 1 nl 1 R2 R1 Hecht eqn 5 16 Equating these two we get the Gaussian lens formula 1 1 so si 24 Hecht eqn 5 15 and recalling that as object distance so is moved to infinity image distance si becomes focal length fi we get 1 fi 1 1 nl 1 R2 R1 1 fi Hecht eqn 5 17 Marc Levoy From Gauss s ray construction to the Gaussian lens formula object image yo yi so 25 si positive si is rightward positive so is leftward positive y is upward Marc Levoy From Gauss s ray construction to the Gaussian lens formula object image yo yi so si yi si yo so y 26 Marc Levoy From Gauss s ray construction to the Gaussian lens formula f positive is to right of lens image object yo yi so yi si yo so y 27 and si yi si f yo f 1 1 1 so si f Marc Levoy Changing the focus distance to focus on objects at different distances move sensor relative to lens Flash demo http graphics stanford edu courses cs178 applets gaussian html 28 f f sensor 1 1 1 so si f Marc Levoy Changing the focus distance to focus on objects at different distances move sensor relative to lens f f sensor at so si 2 f we have 1 1 imaging because 1 1 1 2f 2f f In 1 1 imaging if the sensor is 36mm wide an object 36mm wide will fill the frame 29 1 1 1 so si f Marc Levoy Changing the focus distance to focus on objects at different distances move sensor relative to lens f f sensor at so si 2 f we have 1 1 …