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The science of light P. Ewart Oxford Physics: Second Year, Optics • Lecture notes: On web site NB outline notes! • Textbooks: Hecht, Optics Lipson, Lipson and Lipson, Optical Physics Further reading: Brooker, Modern Classical Optics • Problems: Material for four tutorials plus past Finals papers A2 • Practical Course: Manuscripts and Experience Oxford Physics: Second Year, Optics Structure of the Course 1. Physical Optics (Interference) Diffraction Theory (Scalar) Fourier Theory 2. Analysis of light (Interferometers) Diffraction Gratings Michelson (Fourier Transform) Fabry-Perot 3. Polarization of light (Vector) Oxford Physics: Second Year, Optics Astronomical observatory, Hawaii, 4200m above sea level. Oxford Physics: Second Year, Optics Multi-segment Objective mirror, Keck Obsevatory Oxford Physics: Second Year, Optics Hubble Space Telescope, HST, In orbit Oxford Physics: Second Year, HST Deep Field Oldest objects in the Universe: 13 billion years Optics Oxford Physics: Second Year, Optics HST Image: Gravitational lensing Oxford Physics: Second Year, SEM Image: Insect head Optics Oxford Physics: Second Year, Optics Coherent Light: Laser physics: Holography, Telecommunications Quantum optics Quantum computing Ultra-cold atoms Laser nuclear ignition Medical applications Engineering Chemistry Environmental sensing Metrology ……etc.! Oxford Physics: Second Year, Optics CD/DVD Player: optical tracking assembly Oxford Physics: Second Year, Optics Optics in Physics • • • • • • Astronomy and Cosmology Microscopy Spectroscopy and Atomic Theory Quantum Theory Relativity Theory Lasers Oxford Physics: Second Year, Optics Lecture 1: Waves and Diffraction • Interference • Analytical method • Phasor method Oxford Physics: Second Year, Optics T u u t, x Time or distance axis t,z Phase change of 2p Oxford Physics: Second Year, Optics P r1 r2 d dsin D Phasor diagram Optics Imaginary Oxford Physics: Second Year, u Real Oxford Physics: Second Year, Optics Phasor diagram for 2-slit interference up uuo/r o r uuoo/r r Oxford Physics: Second Year, Optics Lecture 2: Diffraction theory • Diffraction at a finite slit - Analytical method - Phasor method • 2-D diffraction at apertures • Fraunhofer diffraction Oxford Physics: Second Year, Optics Diffraction from a single slit +a/2 y dy r+ ysin r ysin -a/2 D P Oxford Physics: Second Year, Optics Intensity pattern from diffraction at single slit 1.0 0.8 0.6 2 sinc () 0.4 0.2 0.0 -10 p p -5 p 0 p 5 p 10 p Oxford Physics: Second Year, Optics P +a/2 -a/2 r asin D Oxford Physics: Second Year, Phasors and resultant at different angles Optics RP= RO / RP Oxford Physics: Second Year, Optics R R sin /2 RP R Oxford Physics: Second Year, Optics Phasor arc to first minimum Phasor arc to second minimum Oxford Physics: Second Year, Optics y x z Diffraction from a rectangular aperture Oxford Physics: Second Year, Optics Diffraction pattern from circular aperture Intensity y x Point Spread Function Diffraction from a circular aperture Diffraction from circular apertures Oxford Physics: Second Year, Optics Dust pattern Diffraction pattern Basis of particle sizing instruments Oxford Physics: Second Year, Lecture 3: Optics Diffraction theory and wave propagation • Fraunhofer diffraction • Fresnel’s theory of wave propagation • Fresnel-Kirchoff diffraction integral Oxford Physics: Second Year, Optics Fraunhofer Diffraction A diffraction pattern for which the phase of the light at the observation point is a linear function of the position for all points in the diffracting aperture is Fraunhofer diffraction How linear is linear? Oxford Physics: Second Year, Optics R source < a R a R R diffracting aperture observing point Oxford Physics: Second Year, Optics Fraunhofer Diffraction A diffraction pattern formed in the image plane of an optical system is Fraunhofer diffraction Oxford Physics: Second Year, Optics P A O B f C Oxford Physics: Second Year, Optics u v Equivalent lens system Diffracted waves imaged Fraunhofer diffraction: in image plane of system Oxford Physics: Second Year, Optics (a) O P (b) O P Equivalent lens system: Fraunhofer diffraction is independent of aperture position Oxford Physics: Second Year, Optics Fresnel’s Theory of wave propagation Oxford Physics: Second Year, Optics dS n r -z P 0 z Plane wave surface unobstructed Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z =0 Oxford Physics: Second Year, Optics n n rn rn P qq Construction of elements of equal area on wavefront Oxford Physics: Second Year, Optics p p Rp Rp (q+/2) (q+/2) q q First Half Period Zone Resultant, Rp, represents amplitude from 1st HPZ Oxford Physics: Second Year, /2 p Optics Phase difference of /2 at edge of 1st HPZ q P O q Oxford Physics: Second Year, Optics Rp As n a infinity resultant a ½ diameter of 1st HPZ Oxford Physics: Second Year, Optics Fresnel-Kirchoff diffraction integral i uo dS ikr up (n, r ) e r Fresnel-Kirchoff diffraction integral Oxford Physics: Second Year, Babinet’s Principle Optics Oxford Physics: Second Year, Optics Lectures 1 - 3: The story so far • Scalar diffraction theory: Analytical methods Phasor methods • Fresnel-Kirchoff diffraction integral: propagation of plane waves Oxford Physics: Second Year, Joseph Fraunhofer (1787 - 1826) Phase at observation is linear function of position in aperture: = k sin y Optics Augustin Fresnel (1788 - 1827) up i Gustav Robert Kirchhoff (1824 –1887) uo dS ikr ( n, r ) e r Fresnel-Kirchoff Diffraction Integral Oxford Physics: Second Year, Optics Lecture 4: Fourier methods • Fraunhofer diffraction as a Fourier transform • Convolution theorem – solving difficult diffraction problems • Useful Fourier transforms and convolutions Oxford Physics: Second Year, Optics Fresnel-Kirchoff diffraction integral: i uo dS up (n.r )eikr r Simplifies to: u p A( ) u ( x)eix dx where = ksin Note: A() is the Fourier transform of u(x) The Fraunhofer diffraction pattern is proportional to the Fourier transform of the transmission function (amplitude function) of the diffracting aperture Oxford Physics: Second Year, Optics The Convolution function: h( x ) f ( x ) g ( x ) f ( x' ).g ( x x' ).dx' The Convolution Theorem: The Fourier transform, F.T., of f(x) is F() F.T., of g(x) is G() F.T., of h(x) is H() H() = F().G() The Fourier transform of a convolution of f and g is the product of the Fourier transforms of f and g Oxford Physics: Second Year, Optics T Monochromatic Wave .o p/T Fourier Transform Oxford Physics: Second Year, -function Optics V V(x) xo Fourier transform Power spectrum V()V()* = V2 = constant x V() Oxford Physics: Second Year, Optics Comb of -functions V(x) xS Fourier transform x V() Oxford Physics: Second Year, Optics Constructing a double slit function by convolution g(x-x’ ) h(x) f(x’ ) Oxford Physics: Second Year, Optics Triangle as a convolution of two “top-hat” functions g(x-x’ ) f(x’ ) h(x) This is a self-convolution or Autocorrelation function Oxford Physics: Second Year, Optics Lecture 5: Theory of imaging • Fourier methods in optics • Abbé theory of imaging • Resolution of microscopes • Optical image processing • Diffraction limited imaging Oxford Physics: Second Year, Optics • Heat transfer theory: - greenhouse effect • Fourier series • Fourier synthesis and analysis • Fourier transform as analysis Joseph Fourier (1768 –1830) Ernst Abbé (1840 -1905) Oxford Physics: Second Year, Optics Abbé theory of imaging (coherent light) Fourier plane d’ d a v(x) u(x) u f v D Fourier plane Oxford Physics: Second Year, Optics The compound microscope Objective magnification = v/u Eyepiece magnifies real image of object Diffracted orders from high spatial frequencies miss the objective lens max defines the numerical aperture… and resolution So high spatial frequencies are missing from the image. Oxford Physics: Second Year, Fourier plane Image plane Optics Image processing Oxford Physics: Second Year, Optics Optical simulation of “X-Ray diffraction” (a) and (b) show objects: double helix at different angle of view a b a’ b’ Diffraction patterns of (a) and (b) observed in Fourier plane Computer performs Inverse Fourier transform To find object “shape” Oxford Physics: Second Year, Optics amplitude or phase object Fourier plane d’ d a v(x) u(x) u f v D Oxford Physics: Second Year, Optics Schlieren photography Refractive index variation Fourier Plane Source Collimating Lens Imaging Lens Knife Edge Image Plane Schlieren photography of combustion Schlieren film of ignition Courtesy of Prof C R Stone Eng. Science, Oxford University Oxford Physics: Second Year, Optics Diffraction pattern from circular aperture Intensity y x Point Spread Function, PSF Oxford Physics: Second Year, Optics Lecture 6: Optical instruments and Fringe localisation • Interference fringes • What types of fringe? • Where are fringes located? • Interferometers Oxford Physics: Second Year, Optics • Interference by: • Division of wavefront - Young’s slits: - N-slit grating: 2 beams multiple beams • Division of amplitude - Michelson: - Fabry-Perot: 2 beams multiple beams What kind of interference fringes and where are they? Oxford Physics: Second Year, Optics What kind of interference fringes and where are they? Depends on: Type of light source – Point source - Extended source Division of wavefront – Point sources Division of amplitude by – Reflection from: Wedged surfaces Parallel surfaces Oxford Physics: Second Year, Optics Division of wavefront Young’s slits non-localised Non-localized fringes fringes Plane waves Usually observed under Fraunhofer conditions – large distance from slits Oxford Physics: Second Year, Division of wavefront Optics Diffraction grating to 8 Fringes localized at infinity Z Plane waves f Fraunhofer condition Oxford Physics: Second Year, P’ Optics Division of Amplitude Point source Wedged reflecting surfaces Non-localized P Fringes of Equal thickness O Oxford Physics: Second Year, Optics Division of Amplitude Point source Parallel reflecting surfaces P’ P O ’ Non-localized Fringes of Equal inclination Oxford Physics: Second Year, Optics Extended source Wedged Reflecting surfaces P’ R’ Division of Amplitude Localized in plane of wedge, near apex P Equal thickness R S O Oxford Physics: Second Year, Optics Division of Amplitude Extended source Parallel reflecting surfaces t images source Localized at infinity Fringes of equal inclination 2t=x path difference xcos 2t=x circular fringe constant Oxford Physics: Second Year, Optics Summary: fringe type and localisation Point Source Extended Source Wedged Parallel Non-localised Equal thickness Non-localised Equal inclination Localised in plane Localised at infinity of Wedge Equal thickness Equal inclination