The science of light. P. Ewart

Transcription

1

2 The science of light P. Ewart

3 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

4 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)

5 Astronomical observatory, Hawaii, 4200m above sea level.

6 Multi-segment Objective mirror, Keck Obsevatory

7 Hubble Space Telescope, HST, In orbit

8 HST Deep Field Oldest objects in the Universe: 13 billion years

9 HST Image: Gravitational lensing

10 SEM Image: Insect head

11 Coherent Light: Laser physics: Holography, Telecommunications Quantum optics Quantum computing Ultra-cold atoms Laser nuclear ignition Medical applications Engineering Chemistry Environmental sensing Metrology etc.!

12 CD/DVD Player: optical tracking assembly

13 Optics in Physics Astronomy and Cosmology Microscopy Spectroscopy and Atomic Theory Quantum Theory Relativity Theory Lasers

14 Lecture 1: Waves and Diffraction Interference Analytical method Phasor method

15 u u T Time or distance axis t, x t,z Phase change of 2p

16 r 1 r 2 P d dsin D

17 Imaginary Oxford Physics: Second Year, Optics Phasor diagram u Real

18 Phasor diagram for 2-slit interference u p u /r uo o r u o u o/r r

19 Lecture 2: Diffraction theory Diffraction at a finite slit - Analytical method - Phasor method 2-D diffraction at apertures Fraunhofer diffraction

20 Diffraction from a single slit y +a/2 dy r + ysin r P -a/2 ysin D

21 1.0 Intensity pattern from diffraction at single slit 0.8 sinc 2 ( ) p p p p p p

22 +a/2 r P -a/2 asin D

23 Phasors and resultant at different angles R = P R O / R P

24 R sin /2 R R P R

25 Phasor arc to first minimum Phasor arc to second minimum

26 y x z

27 Diffraction from a rectangular aperture

28 Diffraction pattern from circular aperture Intensity y x Point Spread Function

29 Diffraction from a circular aperture

30 Diffraction from circular apertures

31 Dust pattern Diffraction pattern Basis of particle sizing instruments

32 Lecture 3: Diffraction theory and wave propagation Fraunhofer diffraction Fresnel s theory of wave propagation Fresnel-Kirchoff diffraction integral

33 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?

34 < R a a R source R R observing point diffracting aperture

35 Fraunhofer Diffraction A diffraction pattern formed in the image plane of an optical system is Fraunhofer diffraction

36 P O f A B C

37 u v Equivalent lens system Diffracted waves imaged Fraunhofer diffraction: in image plane of system

38 (a) O P (b) O P Equivalent lens system: Fraunhofer diffraction is independent of aperture position

39 Fresnel s Theory of wave propagation

40 ds n r P -z 0 z Plane wave surface unobstructed Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z = 0

41 n r n r n q P Construction of elements of equal area on wavefront

42 p (q+ /2) (q+ /2) R p R p q q First Half Period Zone Resultant, R p, represents amplitude from 1 st HPZ

43 /2 Phase difference of /2 at edge of 1st HPZ p q O q P

44 R p As n a infinity resultant a ½ diameter of 1 st HPZ

45 Fresnel-Kirchoff diffraction integral u p i u o ds (n,r) r e ikr Fresnel-Kirchoff diffraction integral

46 Babinet s Principle

47 Lectures 1-3: The story so far Scalar diffraction theory: Analytical methods Phasor methods Fresnel-Kirchoff diffraction integral: propagation of plane waves

48 Joseph Fraunhofer ( ) Augustin Fresnel ( ) Gustav Robert Kirchhoff ( ) Phase at observation is linear function of position in aperture: = k sin y u p i uods (n,r) e r ikr Fresnel-Kirchoff Diffraction Integral

49 Lecture 4: Fourier methods Fraunhofer diffraction as a Fourier transform Convolution theorem solving difficult diffraction problems Useful Fourier transforms and convolutions

50 Fresnel-Kirchoff diffraction integral: Simplifies to: u p i u ds ( r o u p n. r) where = ksin A( ) u( x) e i x e ikr dx 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

51 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

52 Monochromatic Wave. o p/t T Fourier Transform

53 -function V V(x) x o x Fourier transform Power spectrum V( )V( )* = V 2 = constant V( )

54 Comb of -functions V(x) x S x Fourier transform V( )

55 Constructing a double slit function by convolution g(x-x ) f(x ) h(x)

56 Triangle as a convolution of two top-hat functions g(x-x ) f(x ) h(x) This is a self-convolution or Autocorrelation function

57 Lecture 5: Theory of imaging Fourier methods in optics Abbé theory of imaging Resolution of microscopes Optical image processing Diffraction limited imaging

58 Heat transfer theory: - greenhouse effect Fourier series Fourier synthesis and analysis Fourier transform as analysis Joseph Fourier ( )

59 Ernst Abbé ( )

60 Abbé theory of imaging (coherent light) Fourier plane d a d u(x) v(x) u f v D Fourier plane

61 The compound microscope Objective magnification = v/u Eyepiece magnifies real image of object

62 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.

63 Image processing Fourier plane Image plane

64 Optical simulation of X-Ray diffraction (a) and (b) show objects: double helix at different angle of view a b Diffraction patterns of (a) and (b) observed in Fourier plane Computer performs Inverse Fourier transform To find object shape a b

65 amplitude or phase object Fourier plane d a d u(x) v(x) u f v D

66 Schlieren photography Refractive index variation Fourier Plane Source Collimating Lens Imaging Lens Knife Edge Image Plane

67 Schlieren photography of combustion Schlieren film of ignition Courtesy of Prof C R Stone Eng. Science, Oxford University

68 Diffraction pattern from circular aperture Intensity y x Point Spread Function, PSF

69 Lecture 6: Optical instruments and Fringe localisation Interference fringes What types of fringe? Where are fringes located? Interferometers

70 Interference by: Division of wavefront - Young s slits: 2 beams - N-slit grating: multiple beams Division of amplitude - Michelson: 2 beams - Fabry-Perot: multiple beams What kind of interference fringes and where are they?

71 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

72 Division of wavefront Young s slits Plane waves non-localised fringes Non-localized fringes Usually observed under Fraunhofer conditions large distance from slits

73 8 Oxford Physics: Second Year, Optics Division of wavefront Diffraction grating Fringes localized at infinity Plane waves to Z f Fraunhofer condition

74 Division of Amplitude P P Point source Wedged reflecting surfaces Non-localized Fringes of Equal thickness O

75 Division of Amplitude P P O Point source Parallel reflecting surfaces Non-localized Fringes of Equal inclination

76 Division of Amplitude R R P P Extended source Wedged Reflecting surfaces Localized in plane of wedge, near apex Equal thickness S O

77 images t Division of Amplitude source Extended source Parallel reflecting surfaces Localized at infinity path difference xcos 2t=x Fringes of equal inclination 2t=x circular fringe constant

78 Summary: fringe type and localisation Wedged Parallel Point Source Non-localised Equal thickness Non-localised Equal inclination Extended Source Localised in plane of Wedge Equal thickness Localised at infinity Equal inclination