Optics I: lenses and apertures CS 178, Spring 2014 Begun 4/8/14, finished 4/10. Marc Levoy Computer Science Department Stanford University Outline ✦ why study lenses? ✦ thin lenses • graphical constructions, algebraic formulae ✦ thick lenses • center of perspective, lens as 3D → 3D transformation ✦ depth of field ✦ aberrations & distortion ✦ vignetting, glare, and other lens artifacts ✦ diffraction and lens quality ✦ special lenses 2 • telephoto, zoom © Marc Levoy Cameras and their lenses single lens reflex digital still camera (DSC), (SLR) camera i.e. point-and-shoot 3 © Marc Levoy Cutaway view of a real lens Vivitar Series 1 90mm f/2.5 Cover photo, Kingslake, Optics in Photography 4 © 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 ✦ And why is it so complicated? 5 (Canon) © Marc Levoy Panasonic 45-200/4-5.6 Leica 90mm/2.8 Elmarit-M Stanford Big Dish zoom, at 200mm f/4.6 prime, at f/4 Panasonic GF1 $300 $2000 Zoom lens versus prime lens Canon 100-400mm/4.5-5.6 Canon 300mm/2.8 zoom, at 300mm and f/5.6 prime, at f/5.6 7 $1600 $4300 © Marc Levoy Physical versus geometrical optics Christiaan Huygens (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 • λ → 0 • no diffraction • in free space, rays are straight (a.k.a. rectilinear propagation) 8 © Marc Levoy Physical 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, represented on the previous slide by the limit λ → 0 ✦ physical optics is a.k.a. wave optics 9 © Marc Levoy Snell’s law of refraction (Hecht) x sinθ n i = i = t ✦ as waves change x sinθ n speed at an interface, t t i they also change direction speed of light in a vacuum ✦ index of refraction nr is defined as speed of light in medium r 10 © 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) ✦ 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 12 © Marc Levoy Q. What shape should a refractive interface be to make parallel rays converge to a point? (Hecht) A. a hyperbola ✦ so lenses should be hyperbolic! 13 © Marc Levoy Spherical lenses hyperbolic lens (Hecht) spherical lens (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) (Canon) 15 © Marc Levoy Paraxial approximation object image P e P' ✦ assume e ≈ 0 Not responsible on exams for orange-tinted slides 16 © Marc Levoy Paraxial approximation object l image h u P e P' z ✦ 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 17 © Marc Levoy The paraxial approximation is a.k.a. first-order optics φ 3 φ 5 φ 7 ✦ assume first term of sin φ = φ − + − + ... 3! 5! 7! • i.e. sin ϕ ≈ ϕ φ 2 φ 4 φ 6 ✦ assume first term of cos φ = 1 − + − + ... 2! 4! 6! • i.e. cos ϕ ≈ 1 • so tan ϕ ≈ sin ϕ ≈ ϕ these are the Taylor series for sin ϕ and cos ϕ (phi in degrees) 18 © Marc Levoy Paraxial focusing i i' object image P (n) (n') P' equivalent to sinθ n i = t sinθ n Snell’s law: t i n sin i = n' sin i' with n = n for air paraxial approximation: i n' = nt for glass n i ≈ n' i' i, i' in radians θi , θt in degrees 19 © Marc Levoy Paraxial focusing i = u + a Given object distance z, u ≈ h / z i i' what is image distance z’ ? u' ≈ h / z' h r u a u' P (n) (n') P' z z' n i ≈ n' i' 20 © Marc Levoy Paraxial focusing i = u + a a = u' + i' u ≈ h / z i i' a ≈ h / r u' ≈ h / z' h r u a u' P (n) (n') P' z z' n (u + a) ≈ n' (a − u') n (h / z + h / r) ≈ n' (h / r − h / z') n / z n / r n' / r n' / z' n i ≈ n' i' + ≈ − ✦ h has canceled out, so any ray from P will focus to P’ 21 © Marc Levoy Focal length r P (n) (n') P' z z' What happens if z is ∞ ? n / z + n / r ≈ n' / r − n' / z' n / r ≈ n' / r − n' / z' z' ≈ (r n') / (n' − n) ✦ f ≜ focal length = z’ 22 © 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 ⎛ 1 1 ⎞ + = (nl − 1) ⎜ − ⎟ so si ⎝ R1 R2 ⎠ 23 © Marc Levoy Gaussian lens formula ✦ Starting from the lensmaker’s formula 1 1 ⎛ 1 1 ⎞ (Hecht, eqn 5.15) + = (nl − 1) ⎜ − ⎟ , so si ⎝ R1 R2 ⎠ ✦ and recalling that as object distance so is moved to infinity, image distance si becomes focal length fi , we get 1 ⎛ 1 1 ⎞ (Hecht, eqn 5.16) = (nl − 1) ⎜ − ⎟ . fi ⎝ R1 R2 ⎠ ✦ Equating these two, we get the Gaussian lens formula 1 1 1 + = . (Hecht, eqn 5.17) so si fi 24 © Marc Levoy From Gauss’s ray construction to the Gaussian lens formula object image yo yi so si ✦ positive s i is rightward, positive s o is leftward ✦ positive y is upward 25 © Marc Levoy From Gauss’s ray construction to the Gaussian lens formula object image yo yi so si y s i = i yo so y 26 © Marc Levoy From Gauss’s ray construction to the Gaussian lens formula f (positive is to right of lens) object image yo yi so si y s y s f 1 1 1 i i and i i − ..... + = = = s s f yo so yo f o i y 27 © Marc Levoy Changing the focus distance f f ✦ to focus on objects at different distances, sensor move sensor relative to lens 1 1 1 (Flash demo) + = http://graphics.stanford.edu/courses/ cs178/applets/gaussian.html so si f 28 © Marc Levoy Changing the focus distance f f ✦ to focus on objects at different distances, sensor move sensor relative to lens ✦ at s o = s i = 2 f we have 1:1 imaging, because 1 1 1 + = 2 f 2 f f In 1:1 imaging, if the sensor is 1 1 1 36mm wide, an object 36mm + = wide will fill the frame. so si f 29 © Marc Levoy Changing the focus distance f f ✦ to focus on objects at different distances, sensor move sensor relative to lens ✦ at s o = s i = 2 f we have 1:1 imaging, because 1 1 1 + = 2 f 2 f f ✦ can’t focus on objects 1 1 1 closer to lens than its + = focal length f so si f 30 © Marc Levoy Recap ✦ approximations we sometimes make when analyzing lenses • geometrical optics instead of physical optics • spherical lenses instead of hyperbolic lenses • thin lens representation of thick optical systems • paraxial approximation of ray angles ✦ the Gaussian lens formula relates focal length f, object distance so, and image distance si • these settings, and sensor size, determine field of view • 1:1 imaging means so = si and both are 2× focal length • so = f is the minimum possible object distance for a lens Questions? 31 © Marc Levoy Convex versus concave lenses (Hecht) rays from a convex lens converge rays from a concave lens diverge ✦ positive focal length f means parallel rays from the left converge to a point on the right ✦ negative focal length f means parallel rays from the left converge to a point on the left (dashed lines above) 32 © Marc Levoy Convex versus concave lenses (Hecht) rays from a convex lens converge rays from a concave lens diverge ...producing a real image ...producing a virtual image 33 © Marc Levoy Convex versus concave lenses (Hecht) ...producing a real image ...producing a virtual image 34 © Marc Levoy The power of a lens 1 units are meters-1 P = f a.k.a. diopters ✦ my eyeglasses have the prescription • right eye: -0.75 diopters • left eye: -1.00 diopters Q. What’s wrong with me? A. Myopia (nearsightedness) (Pamplona) 35 © Marc Levoy Combining two lenses ✦ using focal lengths 1 1 1 = + ftot f1 f2 ✦ using diopters Ptot = P1 + P2 ✦ example 1 1 1 + = -or- 5.0 + 2.0 = 7.0 diopters 200mm 500mm 143mm 36 © Marc Levoy Close-up filters 1/500mm = +2 diopters (wikipedia) ✦ screw on to end of lens ✦ power is designated in diopters (usually) 37 © Marc Levoy Close-up filters Panasonic 45-200 + ✦ changes longest focal length from 200mm to 143mm 1 1 1 + = -or- 5.0 + 2.0 = 7.0 diopters 200mm 500mm 143mm 38 © Marc Levoy Close-up filters ✦ for a fixed image distance, it reduces the object distance • at f=200mm, this len’s minimum object distance so = 1000mm • at these settings, its effective image distance must be 1 1 s = = = 250mm i 1 1 1 1 − − 3× f so 200mm 1000mm closer! • with the closeup filter and the same settings of focal length and image distance, the in-focus object distance becomes 1 1 s = = = 334mm o 1 1 1 1 − − f si 143mm 250mm 39 © Marc Levoy Close-up filters 200mm lens & 200mm lens & no closeup filter 500D closeup filter so = 1000mm so = 334mm poor man’s macro lens 40 © Marc Levoy Magnification object image yo yi so si yi si M T ! = − yo so 41 © Marc Levoy Close-up filters 200mm lens & 200mm lens & no closeup filter 500D closeup filter so = 1000mm so = 334mm si 250 si 250 M T = − = = − 1: 4 M T = − = = − 3 : 4 so 1000 so 334 42 © Marc Levoy Thick lenses ✦ an optical system may contain many lenses, but can be characterized by a few numbers (Smith) 43 © Marc Levoy Center of perspective (Hecht) • in a thin lens, the chief ray from a point traverses the lens (through its optical center) without changing direction • in a thick