Optics I:lenses and aperturesMarc LevoyComputer Science DepartmentStanford UniversityCS 178, Spring 2011Begun 4/5/11, finished 4/7. Error on slide 63 corrected 4/12.! Marc LevoyOutline!why study lenses?!thin lenses•graphical constructions, algebraic formulae!thick lenses•center of perspective, 3D perspective transformations!depth of field!aberrations & distortion!vignetting, glare, and other lens artifacts!diffraction and lens quality!special lenses•telephoto, zoom2! Marc LevoyCameras and their lenses3single lens reflex(SLR) cameradigital still camera (DSC),i.e. point-and-shoot! Marc LevoyCutaway view of a real lens4Vivitar Series 1 90mm f/2.5Cover photo, Kingslake, Optics in Photography! Marc LevoyLens 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!Why is it so complicated?5(Canon)Stanford Big DishPanasonic GF1Leica 90mm/2.8 Elmarit-Mprime, at f/4$2000Panasonic 45-200/4-5.6zoom, at 200mm f/4.6$300! Marc LevoyZoom lens versus prime lens7Canon 100-400mm/4.5-5.6zoom, at 300mm and f/5.6$1600Canon 300mm/2.8prime, at f/5.6$4300! Marc LevoyPhysical versus geometrical optics!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(Hecht)! Marc LevoyPhysical versus geometrical optics(contents of whiteboard)!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 optics9! Marc LevoySnell’s law of refraction!as waves changespeed at an interface,they also change direction!index of refraction nt is defined as10(Hecht)xixt=sin!isin!t=ntnispeed of light in a vacuumspeed of light in medium t! Marc LevoyTypical refractive indices (n)!air = ~1.0!water = 1.33!glass = 1.5 - 1.811mirage due to changes in the index of refraction of air with temperature! Marc LevoyRefraction in glass lenses!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 bend12(Hecht)! Marc LevoyQ. What shape should an interface beto make parallel rays converge to a point?A. a hyperbola!so lenses should be hyperbolic!13(Hecht)! Marc LevoySpherical lenses!two roughly fitting curved surfaces ground togetherwill eventually become spherical!spheres don’t bring parallel rays to a point•this is called spherical aberration•nearly axial rays (paraxial rays) behave best14(Hecht)(wikipedia)hyperbolic lensspherical lens! Marc LevoyExamples of spherical aberration15(gtmerideth)(Canon)Canon 135mm soft focus lens! Marc Levoy!assume e ! 016PP 'object imageeNot responsible on examsfor orange-tinted slidesParaxial approximation! Marc LevoyParaxial approximation!assume e ! 0!assume sin u = h / l ! u (for u in radians)!assume cos u ! z / l ! 1!assume tan u ! sin u ! u17PP 'object imageeuzlh! Marc LevoyThe paraxial approximation isa.k.a. first-order optics!assume first term of•i.e. sin " " "!assume first term of•i.e. cos " " 1•so tan " " sin " " "18cos!= 1 "!22!+!44!"!66!+ ...sin!=!"!33!+!55!"!77!+ ...(phi in degrees)these are the Taylor series for sin " and cos "! Marc LevoyParaxial focusing19ii 'PP '(n)(n ')n sin i = n ' sin i 'Snell’s law:n i ! n ' i 'paraxial approximation:object imageequivalent towithsin!isin!t=ntnin = ni for airn ' = nt for glassi, i ' in radians!i,!t in degrees! Marc LevoyParaxial focusing20ii 'auu 'hrPP 'zz 'i = u + au ! h / zu ' ! h / z '(n)(n ')Given object distance z,what is image distance z’ ?n i ! n ' i '! Marc LevoyParaxial focusing!h has canceled out, so any ray from P will focus to P’21ii 'auu 'hrPP 'zz 'i = u + au ! h / zu ' ! h / z 'a = u ' + i 'a ! h / rn (u + a) ! n ' (a " u ')n (h / z + h / r) ! n ' (h / r " h / z ')n / z + n / r ! n ' / r " n ' / z '(n)(n ')n i ! n ' i '! Marc LevoyFocal length! f ≜ focal length = z’22rPP 'zz 'n / z + n / r ! n ' / r " n ' / z '(n)(n ')What happens if z is " ? n / r ! n ' / r " n ' / z 'z ' ! (r n ') / ( n ' " n)! Marc LevoyLensmaker’s formula!using similar derivations, one can extend these results to two spherical interfaces forming a lens in air!as d # 0 (thin lens approximation),we obtain the lensmaker’s formula231so+1si= (nl!1)1R1!1R2"#$%&'(Hecht, edited)siso! Marc LevoyGaussian lens formula!Starting from the lensmaker’s formula!and recalling that as object distance so is moved to infinity,image distance si becomes focal length fi , we get!Equating these two, we get the Gaussian lens formula241so+1si= (nl!1)1R1!1R2"#$%&',1fi= (nl!1)1R1!1R2"#$%&'.1so+1si=1fi.(Hecht, eqn 5.15)(Hecht, eqn 5.16)(Hecht, eqn 5.17)! Marc Levoy!positive is rightward, positive is leftward!positive is upward25object imageyoyisosisisoyFrom Gauss’s ray constructionto the Gaussian lens formula! Marc Levoy26yyiyo=sisoobject imageyoyisosiFrom Gauss’s ray constructionto the Gaussian lens formula! Marc Levoy27yiyo=sisoyoyyisosifandyiyo=si! ffobject image.....1so+1si=1f(positive is to right of lens)From Gauss’s ray constructionto the Gaussian lens formula! Marc LevoyChanging the focus distance!to focus on objectsat different distances,move sensor relative to lens28sensorff1so+1si=1f(Flash demo)http://graphics.stanford.edu/courses/cs178/applets/gaussian.html! Marc LevoyChanging the focus distance!to focus on objectsat different distances,move sensor relative to lens!at = = we have 1:1 imaging, because29sensorff12 f+12 f=1f2 fsosi1so+1si=1fIn 1:1 imaging, if the sensor is36mm wide, an object 36mmwide will fill the frame.! Marc LevoyChanging the focus distance!to focus on objectsat different distances,move sensor relative to lens!at = = we have 1:1 imaging, because!can’t focus on objectscloser to lens than itsfocal length f30sensorff12 f+12 f=1f2 fsosi1so+1si=1f! Marc LevoyConvex versus concave lenses!positive focal length f means parallel rays from the leftconverge to a point on the right!negative focal length f means parallel rays from the leftconverge to a point on the left (dashed lines above)31rays from a convex lens convergerays from a concave lens diverge(Hecht)! Marc LevoyConvex versus concave lenses32rays from a convex