The science of light. P. Ewart

Transcription

1 The science of light P. Ewart

2 Lecture notes: On web site NB outline notes! Textbooks: Hecht, Klein and Furtak, Lipson, Lipson and Lipson, Optical Physics Brooker, Modern Classical Problems: Material for four tutorials plus past Finals papers A2 Practical Course: Manuscripts and Experience

3 Structure of the Course 1. Geometrical 2. Physical (Interference) Diffraction Theory (Scalar) Fourier Theory 3. Analysis of light (Interferometers) Diffraction Gratings Michelson (Fourier Transform) Fabry-Perot 4. Polarization of light (Vector)

4 Electromagnetism Electronics Quantum Electronics 10-7 < T < 10 7 K; e - > 10 9 ev; superconductor Quantum Photonics

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

14 Geometrical Ignores wave nature of light Basic technology for optical instruments Fermat s principle: Light propagating between two points follows a path, or paths, for which the time taken is an extremum

15 Ray tracing - revision Focal point axis Focal point

16 Simple magnifier Magnifier: angular magnification = b/a Eyepiece of Telescopes, Microscopes etc. a Object at near point b Virtual image at near point Short focal length lens

17 P 1 Thick lens or compound lens First Principal Plane Location of equivalent thin lens Back Focal Plane

18 Thick lens or compound lens P 2 Front Focal Plane Second Principal Plane

19 Telephoto lens Principal Plane Focal Plane Equivalent thin lens f T

20 Wide angle lens Principal Plane Focal Plane f W

21 Astronomical Telescope f O f E b angular magnification = b/a = f o /f E

22 Galilean Telescope angular magnification = b/a = f o /f E

23 Newtonian Telescope f o angular magnification = b/a = f o /f E b f E

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

25 What size to make the lenses? Aperture stop Image of objective in eyepiece Eye piece ~ pupil size Objective: Image in eye-piece ~ pupil size

26 (a) (b) Field stop

27 ILLUMINATION OF OPTICAL INSTRUMENTS f/no. : focal length diameter

28 Lecture 2: Waves and Diffraction Interference Analytical method Phasor method Diffraction at 2-D apertures

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

30 r 1 r 2 P d q dsinq D

31 Imaginary Oxford Physics: Second Year, Phasor diagram u Real

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

33 Diffraction from a single slit y +a/2 dy q r + ysinq r P -a/2 ysinq D

34 1.0 Intensity pattern from diffraction at single slit 0.8 sinc 2 (b) p p p p p p b

35 +a/2 q r P -a/2 asinq D

36 Phasors and resultant at different angles q q 0 R = P R O q 0 / R P

37 / R sin /2 R R P R

38 Phasor arc to first minimum Phasor arc to second minimum

39 y x q z

40 Diffraction from a rectangular aperture

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

42 Diffraction from a circular aperture

43 Diffraction from circular apertures

44 Dust pattern Diffraction pattern Basis of particle sizing instruments

45 Lecture 3: Diffraction theory and wave propagation Fraunhofer diffraction Huygens-Fresnel theory of wave propagation Fresnel-Kirchoff diffraction integral

46 y x q z

47 Diffraction from a circular aperture

48 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?

49 r r r < / 0 R a a R source R R observing point diffracting aperture

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

51 P O f A B C

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

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

54 Fresnel s Theory of wave propagation

55 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

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

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

58 /2 Phase difference of /2 at edge of 1st HPZ r p q O q P

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

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

61 Babinet s Principle

62 Lectures 1-3: The story so far Geometrical optics No wave effects Scalar diffraction theory: Analytical methods Phasor methods Fresnel-Kirchoff diffraction integral: propagation of plane waves

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

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

65 Fresnel-Kirchoff diffraction integral: Simplifies to: u p i u ds ( r o u p n. r) where b = ksinq A( b ) a u( x) e ibx e ikr dx Note: A(b) is the Fourier transform of u(x) The Fraunhofer diffraction pattern is the Fourier transform of the amplitude function in the diffracting aperture

66 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(b) F.T., of g(x) is G(b) F.T., of h(x) is H(b) H(b) = F(b).G(b) The Fourier transform of a convolution of f and g is the product of the Fourier transforms of f and g

67 Monochromatic Wave.b o p/t T Fourier Transform b

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

69 Comb of -functions V(x) x S x Fourier transform V( b) b

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

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

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

73 Lecture 6: Theory of imaging Fourier methods in optics Abbé theory of imaging Resolution of microscopes Optical image processing Diffraction limited imaging

74 Fresnel-Kirchoff diffraction integral: Simplifies to: u p i u ds ( r o u p n. r) where b = ksinq A( b ) a u( x) e ibx e ikr dx Note: A(b) is the Fourier transform of u(x) The Fraunhofer diffraction pattern is the Fourier transform of the amplitude function in the diffracting aperture

75 Ernst Abbé ( )

76 Fourier plane d a d u(x) q v(x) u f v D

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

78 Diffracted orders from high spatial frequencies miss the objective lens q max defines the numerical aperture and resolution So high spatial frequencies are missing from the image.

79 Image processing Fourier plane Image plane

80 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

81 PIV particle image velocimetry Two images recorded in short time interval Each moving particle gives two point images Coherent illumination of small area produces Young s fringes in Fourier plane of a lens CCD camera records fringe system input to computer to calculate velocity vector

82 PIV particle image velocimetry

83 amplitude object phase object Fourier plane d a d u(x) q v(x) u f v D

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

85 Schleiren photography in i.c.engine Spark plug Schlieren film of autoignition Courtesy of Prof CWG Sheppard University of Leeds

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

87 Lecture 7: Optical instruments and Fringe localisation Interference fringes What types of fringe? Where are fringes located? Interferometers

88 Young s slits Plane waves non-localised fringes

89 8 Oxford Physics: Second Year, Diffraction grating to Plane waves Z q q Fringes localised at infinity: Fraunhofer f

90 P Wedged reflecting surfaces P Point source O

91 P P O Point source q q Parallel reflecting surfaces

92 P Wedged Reflecting surfaces R P R Extended source S O

93 images t source Parallel reflecting surfaces 2t=x Extended source path difference xcosa a 2t=x circular fringe constant a

94 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