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Slide 2 Special topic: the human eyes Slide 3 The human eyes Eye Components and optical properties Image quality analysis Defocus Diffraction limit Aberration Scattering Adapted from original slide courtesy of Austin Roorda (vision.berkeley.edu/roordalab) Slide 4 Anatomy of the human eye http://science.nasa.gov/headlines/y2004/22oct_cataracts.htm Slide 5 Slide 6 On-axis image formation All rays emanating at y i0 arrive at y t2 irrespective of departure angle i0 Power of the spherical surface [Unit: diopter D, 1D = 1 m -1 ] Slide 7 Object at infinity Image focal length Look in this way: Ambient refractive index of image side 1/Power Slide 8 To simplify image formation in the eye we use the reduced eye. The reduced eye has a single refracting surface n t =4/3 n i =1 Dioptric power = 60 D The reduced eye Slide 9 Cornea (first surface) transition from air (n =1) to front surface of cornea (n = 1.376) radius of curvature = 7.7 mm power: Components of the human optical system cornea Slide 10 Cornea (second surface) Transition from back surface of cornea (n = 1.376) to the aqueous humor (n = 1.336) radius of curvature = 6.8 mm power: total power of cornea ~ +43 D Components of the human optical system cornea Slide 11 Components of the human optical system pupil Function: Govern image quality Depth of focus Control light level? Size affected by: Light conditions Attention Emotion Age Slide 12 The pupil is perfectly located to maximize the field of view of the eye Field of view: the angle subtended from the center of the entrance pupil to the edges of the field stop. Aperture stop Entrance pupil Extremely wide field of view. Cornea Pupil as an aperture stop So you dont feel limited field of view when you bath in sunshine. Slide 13 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 11010 2 10 3 10 4 10 5 10 6 10 7 10 8 10 910 The range of light intensities in the environment is enormous! rod threshhold clear blue sky snow in sunlight solar disc Rodieck, B. The First Steps in Seeing Pupil vs. light intensity Why can you perceive bright and dark environment? Although pupil do equalize light intensity, its effect is only ~ 16 times. The eye perceives light on a logarithmic scale (WeberFechner law) Lumen (cd/sqm 2 ) room light Slide 14 Components of the human optical system - crystalline lens Gradient index of refraction n = 1.385 at surfaces n = 1.375 at the equator n ~ 1.41 at the center Little refraction takes place at the surface but instead the light curves as it passes through. For a homogenous lens to have same power, the overall index would have to be greater than the peak index in the gradient Total power of lens ~ 21 D Slide 15 Accommodation The relaxed eye is under tension at the equator from the ciliary body. This keeps the surfaces flat enough so that for a typical eye distant objects focus on the retina. Components of the human optical system - ciliary body Slide 16 Ciliary body and accommodation In the accommodated eye, the ciliary muscle contracts and relaxes the tension on the equator of the lens. Surface curvature increases. Power of the lens increases. Power of the accommodated lens ~ 32.31 D Slide 17 Images are sampled by millions of rods and cones. Components of the human optical system - retina http://webvision.med.utah.edu/imageswv/scanEMphoto.jpg Slide 18 Retina Fovea: 5 degrees from optical axis Optic disc: 15 deg from fovea, 10 deg from optical axis. studftp.stut.edu.tw/~492f0059/Weekly-Eye.htm Slide 19 optic disc fovea posterior pole 5 deg10 deg Fundus image Left eye image Slide 20 Optic nerve & blind spot Close the left eye, focus the right eye on a single point. Keeping your head motionless, with the right eye about 3 or 4 times as far from the page as the length of the red line, look at each character in turn, until the black circle vanishes. Same for left eye http://ourworld.compuserve.com/homepages/cuius/idle/percept/blindspot.htm Slide 21 Visual angle It is the angle subtended at the second nodal point by the image Equal to the angle subtended at the first nodal point by the object The nodal points are points in the optical system where the light passing through emerges at the same angle The second nodal point in the eye is about 16.5 mm from the retina N N Consider a 1 mm image on the retina Slide 22 Visual angle Rule of finger Your fingertip occupies 1 o with a straightened arm Moon subtends about 0.5 o Typically expressed in radian 1 radian = 57.29 degrees Do you know why radian is preferred? 1 minute ~ 4.8 m on retina 1 foveal cone ~ 2.5 m Slide 23 Image of lunar eclipse on retina at 1 deg from fovea Moon subtends about 0.5 degrees Cones at 1 degree from fovea are about 5 microns in diameter. Moon spans about 144 microns Image is sampled by about 29 cones across (~650 in total) Slide 24 Spatial distribution of rods and cones Slide 25 Spectral response of photoreceptors Slide 26 JW 1 deg nasalJW 1 deg temporal AN 1 deg nasal macaque 1.4 deg nasal S, M, and L cone arrangement Roorda, Nature 397, 520 (1999) Slide 27 Boettner and Wolter, 1962 Transmission of the ocular media Invisible, useful for imaging purpose Slide 28 Effect of pupil size Remember this simple test? Have you ever heard people say that I dont need glass except at night? Can you explain it? Slide 29 2 mm4 mm6 mm Depth of focus is a function of pupil size Hyperopia Myopia Slide 30 2 mm4 mm6 mm In focus Focused in front of retina Focused behind retina Depth of focus is a function of pupil size Slide 31 Computation of geometrical blur l x Wb visual angle D is defocus in diopters Slide 32 Computation of geometrical blur where D is the defocus in diopters 1 D defocus, 8 mm pupil produces 27.52 minute blur size ~ 0.5 degrees A distant star will become the size of the Moon! More introduction to other aberrations will be given next week Slide 33 Depth of focus is a function of pupil size Slide 34 Any deviation of light rays from a rectilinear path which cannot be interpreted as reflection or refraction Sommerfeld, ~ 1894 Diffraction Slide 35 Fraunhofer Diffraction Also called far-field diffraction Occurs when the screen is held far from the aperture. Occurs at the focal point of a lens Slide 36 rectangular aperture square aperture Fraunhofer Diffraction Slide 37 Airy Disc circular aperture The Airy Disc Sir George Biddel Airy: Inventor of spectacles for astigmatism Slide 38 The Airy Disk Slide 39 1 mm2 mm3 mm4 mm 5 mm6 mm7 mm Point spread function vs. pupil size perfect eye Slide 40 Rayleigh resolution limit Unresolved point sources Resolved Resolution Slide 41 0 0.5 1 1.5 2 2.5 12345678 pupil diameter (mm) minimum angle of resolution (minutes of arc 500 nm light) Resolution Larger pupil better resolution Recall the pinhole glass: smaller pupil better resolution Trade off? Slide 42 pupil images followed by psfs for changing pupil size 1 mm 2 mm3 mm4 mm 5 mm6 mm7 mm Point spread function vs. pupil size typical eye More introduction to other aberrations will be given next week Slide 43 Observe your own point spread function Or test with a LED Slide 44 Retinal straylight in the human eye slides courtesy of Tom van den Berg Thomas J. T. P. van den Berg, Michiel P. J. Hagenouw, and Joris E. Coppens The Ciliary Corona: Physical Model and Simulation of the Fine Needles Radiating from Point Light Sources IOVS, 46: 2627-2632 (2005). Slide 45 Tom van den Berg Slide 46 Ciliary corona Actual subjective appearance of straylight: a pattern of very fine streaks, not at all like the circularly uniform (Airy disc-like) scattering pattern of particles of approximate wavelength size Tom van den Berg Slide 47 2 4 3 50 Central diffraction pattern from 2, 3, 4, 50 randomly placed particles Tom van den Berg Slide 48 Diffraction pattern for 1000 particles, as a function of wavelength, including spectral luminosity effect. Tom van den Berg http://www.nin.knaw.nl/ Slide 49 Tom van den Berg Slide 50 Slide 51 Optical illusions Our vision is not solely determined by the ocular imaging system Brain processing is the dominator Slide 52 Intelligent brain Play with your right eye again. What happened to the line? http://ourworld.compuserve.com/homepages/cuius/idle/percept/blindspot.htm Slide 53 Other imaging instruments Slide 54 Single-lens magnifier Photographic camera Microscope Telescope Slide 55 A single-lens magnifier Can we take a photo of this virtual image? What is the magnified physical quantity? Viewing angle Slide 56 Magnifying power Normal viewing distance ~ 254 mm The closest point on which the eye can focus Slide 57 Magnifying power What is the largest MP? Try with a lens very close to your eye Most common case Slide 58 Single lens magnifier A 2.5x lens P = 10 D f = 0.1 m MP = 2.5 ( L =) Slide 59 The camera Pinhole camera Images of a solar eclipse through a leaf canopy Slide 60 Pinhole size vs. clarity Slide 61 Single lens reflex camera Iris diaphragm (aperture stop) Slide 62 Depth of focus vs. f/# Focal length divided by the aperture diameter. f/32 f/5 Slide 63 Retina vs. digital camera Slide 64 Slide 65 The compound microscope Objective magnification Eyepiece magnification Combined magnification Slide 66 The compound microscope Typical value: Normal viewing distance: 254 mm Tube length: L 1 = 160 mm For a 5x objective: f o = 32 mm For a 10x eyepiece: f e = 25.4 mm Combined MP: 50x Slide 67 The telescope Slide 68 Astronomical telescope Infinity conjugates Slide 69 HW 2-1 Ball lens: We intend to use a spherical ball lens of radius R and refractive index n as magnifier in an imaging system, as shown in the figure. The refractive index satisfies the relationship 1 < n < 4/3, and the medium surrounding the ball lens is air (n = 1). a) Calculate the effective focal length (EFL) of the ball lens. Use the thick lens model with appropriate parameters. b) Locate the back focal length (BFL), the front focal length (FFL) and the principal planes of the ball lens. c) An object located at distance d to the left of the back surface of the ball lens, where. Show that the object is one half EFL behind the principal plane, and use this fact to find the location of the image plane. d) Is the image real or virtual? Is it erect or inverted? What is the magnification? Slide 70 HW 2-2 Work out the system matrix for the composite element and use it to answer the following questions. a) What is the optical power of this composite element? b) If a plane wave is incident from the left, where will it focus? c) This system is used to image an object at infinity. Is the image real or virtual? Slide 71 HW 2-3 Mirror-in-a-pool: Consider a perfectly focusing paraboloid mirror filled with a fluid of refractive index n. The mirror surface is described by the equation s(x)= x 2 /4f, where f is the focal length of the mirror. The fluid is present up to a height of f. Light is incident from the top as shown in the figure. You may neglect the slight reflection that occurs when the light rays go from the air into the fluid. a) Calculate the portion of the incoming ray bundle which will exit from the fluid as a divergent ray bundle after focusing. b) Show that the remaining rays will exit as a parallel ray bundle. Slide 72 HW 2-4 It is determined that a patient has a near point at 50 cm. If the eye is approximately 2 cm long. a) How much power does the refracting system have when focused on an object at infinity? When focused at 50 cm? b) What power must the eye have to see clearly an object at the standard near-point of 25 cm? c) How much power should be added to the patients vision system by a correcting lens? d) The preceding calculation overlooks the separation between the correction lens and the eye. Can you find out the best position-power combination of the correction lens?