The science of light. P. Ewart

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1 The science of light P. Ewart

2 Oxford Physics: Second Year, Optics Parallel reflecting surfaces t images source Extended source path difference xcos 2t=x Fringes localized at infinity Circular fringe constant 2t=x circular fringe constant Fringes of equal inclination

3 Oxford Physics: Second Year, Optics 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

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

5 Optical Instruments for Spectroscopy Some definitions: Dispersion: dq/dl, angular separation of wavelengths Resolving Power: ldl, dimensionless figure of merit Free Spectral Range: Instrument width: Instrument function: extent of spectrum covered by interference pattern before overlap with fringes of same l and different order width of pattern formed with monochromatic light. shape of the interference fringe Étendu: or throughput measure of how efficiently the instrument uses available light.

6 The Diffraction Grating Spectrometer Calculate the shape of fringe pattern Fringe formation phasors Effects of groove size Fourier methods Angular dispersion Resolving power Free Spectral Range Practical matters - brightness

7 Oxford Physics: Second Year, N-slit grating Optics 4 N = 2 I ~ N 2 I 0 N-1 minima Peaks at = n2

8 N-slit grating I 9 4 N = 3 0 Peaks at n2, N-1 minima, I ~ N 2 1 st minimum at 2/N

9 N-slit grating N 2 I 4 0 N N mn

10 N-slit diffraction grating (a) Order x 2, (b) I(q) = I(0) sin 2 {N/2} sin 2 {/2} sin 2 0 {/2} {/2} 2 (c) = (2l) d sinq (l) a sinq. slit separation slit width Order 2

11 l ldl l ldl 2n q p q D min 2 N Dq min Dq l (a) = (2l)dsinq (b)

12 Reflected light Reflected light Diffracted light Diffracted light (a) q Reflected light Diffracted light q (b)

13 (a) Order (b) Unblazed 0 (c) Blazed Order 2

14 Diffraction Grating Spectrographs Transmission gratings not used Lenses have chromatic aberration

15 Diffraction Grating Spectrographs Reflection grating used Mirrors have no chromatic aberration

16 f 1 f 2 Dx s (a) f 1 grating Dx s q f 2 (b)

17 Instrument function Instrument width (Grating spectrometer): Instrument width = Resolving Power Dn Inst = 1. Maximum path difference = ldl Inst = ndn Inst = 2Wl 1 2W Maximum path difference in units of wavelength n

18 Optical Instruments for Spectroscopy Interference by division of wavefront: The Diffraction Grating spectrograph Interference by division of amplitude: 2-beams - The Michelson Interferometer

Albert Abraham Michelson 1852 1931 Michelson interferometer

19 Albert Abraham Michelson Michelson interferometer Fringe properties interferogram Resolving power Instrument width

20 M 2 Michelson Interferometer M / 2 M 1 t CP BS Detector Light source

21 Oxford Physics: Second Year, Optics images t source path difference xcos 2t=x Fringes of equal inclination Localized at infinity 2t=x circular fringe constant

22 I(x) ½ I 0 n Input spectrum x x = 2t Detector signal Interferogram I(x) = ½ I 0 [ 1 + cos 2nx ]

23 (a) I( n ) 1 x (b) I( n ) 2 x x max (c) I( n) x

24 Michelson Interferometer Dv- Inst = 1/x max Instrument width = 1. Maximum path difference Size of instrument Resolving Power = Maximum path difference in units of wavelength

25 LIGO, Laser Interferometric Gravitational-Wave Observatory 4 Km

26 LIGO, Laser Interferometric Gravitational-Wave Observatory Vacuum ~ atmosphere Precision ~ m

27 Michelson interferometer x = pl Path difference: measure l by reference to known l calibration Instrument width: Dv Inst = 1/x max WHY? Fourier transform interferometer Fringe visibility and relative intensities Fringe visibility and coherence Albert Abraham Michelson

28 (a) I( n ) 1 x (b) I( n ) 2 x x max (c) I( n) x

29 Coherence Longitudinal coherence Coherence length: l c ~ 1/Dn _ Transverse coherence Coherence area: size of source or wavefront with fixed relative phase

30 Transverse coherence d sin < l w s d r >> d r < ld Interference Fringes d defines coherence area

31 Michelson Stellar Interferometer Measures angular size of stars

32 Division of wavefront Young s slits (2-beam) Diffraction grating (N-beam) Sharper fringes Division of amplitude Michelson (2-beam) Fabry-Perot (N-beam)? fringes cos 2( /2) sin 2 (N/2) sin 2 (/2) fringes cos 2 (2)

33 d t r r t 1 t r r t r r t1r2 r1 tr 1 2 q 3 3 -i3 E o t1t2r1 r2 e 2 2 -i2 E o t1t2r1 r2 e E o t t r r e-i E o tt 12 E o t 1 t 2

34 d E o tr 2 6 e -i3 tr 4 tr 2 tr 5 tr 3 tr q E o tr 2 4 e -i2 E o tr 2 2 e -i E o t 2 E o t

35 d E o tr 2 6 e -i3 tr 4 tr 2 tr 5 tr 3 tr q E o tr 2 4 e -i2 E o tr 2 2 e -i E o t 2 E o t Airy Function

36 The Airy function: Fabry-Perot fringes I( ) m2 (m+1)2 (l)d.cosq

37 Fabry-Perot interferometer Lens Screen d (l)2d.cosq Extended Source Lens Fringes of equal inclination Localized at infinity

38 I( ) The Airy function: Fabry-Perot fringes D FWHM m2 (m+1)2 Finesse = D FWHM

39 Finesse = 10 Finesse = 100

40 Fabry-Perot Interferometer Multiple beam interference: Fringe sharpness set by Finesse F = R (1-R) Instrument width Dv Inst Free Spectral Range FSR Resolving power Designing a Fabry-Perot

41 I( ) The Airy function: Fabry-Perot fringes D FHWM m2 (m+1)2 Finesse, F = D FHWM D Inst = F

42 Fabry-Perot Interferometer: Instrument width Dv- Inst = 1. 2d.F = 1/x max effective Instrument width = 1. Maximum path difference

43 I( ) The Airy function: Fabry-Perot fringes D Inst Free Spectral Range m2 (m+1)2 Finesse = D

44 n ndn I( d) m th (m+1) th d FSR

45 n ndn R I( d) Dn Inst Resolution criterion: Dn R Dn Inst d

46 Centre spot scanning q m-1 r m-1 m th fringe (on axis) Aperture size to admit only m th fringe Typically aperture ~ r m-1 10

47 Maxwell s equations t E E r o r o t H H r o r o ). ( r k t i E o e E H n H E o o r o r o 1 E n H constant E.M. Wave equations index refractive n

48 E 1 E 2 E 1 n 2 n 1.n 2 Boundary conditions: Tangential components of E and H are continuous

49 (a) Layer (b) E o E 1 k o.k 1 E T E o E 1.k T k o.k 1 n 2.n o.n 1..n T l

50 n O E O E O n cos k n 1 l sin 1 T i k1 ( E O E O ) isin k1 cos l E l n T k1 l T E T (9.7) (9.8) 1 A cos k 1 ; B i sin k1; n l l C in 1 sin k 1 l ; D cos k 1 l 1 EO AnO BnOnT C DnT r (9.10) E An Bn n C Dn O O O T T E E T O t An O Bn 2n O n O T C Dn T (9.11)

51 E E O O r An An O O Bn Bn O O n n T T C C Dn Dn T T l l 4, Quarter Wave Layer: A=0, B=-i/n 1, C=-in 1, D=0 R = r 2 n n O O n n T T n n Anti-reflection coating, n 1 2 ~ n O n T, R 0 e.g. blooming on lenses

52 layer Uncoated glass Reflectance 2 layer 1 layer wavelength nm

53 E o k o k o E o n 2.n o.n H n L.n n 2 H n L.n T Multi-layer stack

54 n O n EO EO cos k n 1l sin 1 ( E O E O ) isin k1l cos T i k1 n T k1 l E l T E T (9.7) (9.8) 1 A cos k 1 ; B i sin k1; n l l C in 1 sin k 1 l ; D cos k 1 l r = ( A + Bn T )t n O ( 1 r ) = (C + Dn T )t 1 1 A B 1 + r = t n O -n O C D n T

55 .n H n L.n H n L.n H n L.n H.n H.n H n L n L.n T l Interference filter; composed of multi-layer stacks

56 Polarized Light Production Analysis Manipulation

57 z E oz E x E oy y

58 z Source y x

59 Circularly polarized light: E z = E o sin(kx o t) ; E y = E o cos(kx o t) z z E z = Eosin( o) E kx Source Source E = 0 z x x o E y = Eocos( kxo) y x x o E E = y y E o (a) x = x o, t = 0 (b) x = x o, t = kx / o

60 z Source y x Looking towards source: Clockwise rotation of E Right circular polarization

61 E L E L E P E R E R Superposition of equal R & L Circular polarization = Plane polarized light Superposition of unequal R & L Circular polarization = Elliptically polarized light

62 z z E oz E E oy y E q y (a) (b)

63 Polarization states defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t - ) = 0 0 Linearly Polarized Elliptically Polarized

64 Polarization states defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t - ) = 0 Linear = + /2 E oy = E oz Left/Right Circular /2 E oy E oz Left/Right Elliptical axes along y,z /2 E oy E oz Left/Right Elliptical axes at angle to y,z

65 Unpolarized light defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t (t) = (t) Phase varies randomly in time unpolarized light

66 Plane Right circular Left circular Looking towards source Right elliptical Left elliptical

67 = 0, 5 Note: if E oy = E oz these ellipses become circles

68 Polarization states defined by: E y = E oy cos(kx - t) E z = E oz cos(kx - t - ) = 0 0 Linearly Polarized Elliptically Polarized = 0, 5

69 Uniaxial Crystals z n ex z n o n o n ex x,y x,y (a) Optic Axis (b) Positive Negative

70 Uniaxial Crystals E z n ex n o propagation z n o n ex e-ray x,y Phase shift: x,y = (2l)[n o -n e ]l o-ray (a) (b) Positive Negative

71 Linear Input change Output q= 45 O, E oy =E oz, l Left/Right Circular q 45 O, E oy E oz, l L/R Elliptical q 45 O, E oy E oz,l Elliptical tilted axis q 45 O, E oy E oz, l Linear at q to input

72 z z E z q E P E P q E z E y x E y y x y Half-wave plate: introduces -phase shift

73 e-ray e-ray e-ray o-ray Optic axis o-ray Optic axis o-ray Prism Polarizers

74 A d 1 d 1 B d 2 d 2 (a) Babinet (b) Babinet-Soleil

75 Analysis of polarized light z z y z z y E E q y y Elliptically polarized light Linearly polarized light after passage through the l/4 plate (a) (b)

76 Analysis of polarized light z z E y q y Find E oz :E oy and z z E y q y (a) Elliptically polarized light Find ratio of major:minor axes and angle q - approximately by rotating linear polarizer E oz /E oy = tan (b) Linearly polarized light after passage through the l/4 plate Make linear polarization with l/4 plate Find angle of linear polarization using linear polarizer - precisely (q) q, tanq cos

77 A B A B C (a) (b) A B C (c). D

Spectroscopic Instruments

Spectroscopic Instruments 95 Spectroscopic Instruments by division of amplitude Mach-Zehnder (division of amplitude) Michelson Fringe localisation LIGO Fabry-Perot (FPI) Multi-layer coatings 96 Mach-Zehnder