Spectroscopic Instruments

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Silas Barber
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1 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

4 Mach-Zehnder Interferometer Interference fringes as a function of δ u 0 2 (e i 2 e i 2 ) = u 0 i sin( /2) 1 p 2 u 0 1 p 2 u 0 e i(kl+ ) r = t =1/ p 2 u 0 2 (e i 2 + e i 2 ) = u 0 cos( /2) 1 p 2 u 0 e i(kl+ ) u 0 1 p 2 u 0 note: π phase jump upon reflection of the denser medium 97 Michelson Interferometer Fourier-transform spectroscopy Michelson-Morley experiment Gravitational wave detection LIGO Hanford 98

6 Michelson Interferometer beam splitter path difference x =2d compensation plate d 99 Fourier-Transform Spectroscopy a ae ikx I out (x) = u u = a + ae ikx 2 = I 0 (1 + cos kx)/2 d = x/

8 Fourier-Transform Spectroscopy F (x) =I(x) I 0 /2 / cos kx a ae ikx x =2d d = x/ Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx S( ) F (x) / cos(k kx 0 x) cos( kx/2) x 0 / k 101-1

9 Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx S( ) F (x) / cos(k 0 x) cos( kx/2) x 0 / k Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx 101-3

10 Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx 101-5

11 Fourier-Transform Spectroscopy scanning Michelson power spectrum F (x) =I(x) I 0 /2 Z S( ) =T F [F (x)](2 ) = k =2 F (x)exp(i2 x)dx Fourier-Transform Spectroscopy I 1 (x) scanning Michelson I 2 (x) x I 1 (x) =I 0 /2(1 + cos k 1 x) x I 2 (x) =I 0 /2(1 + cos k 2 x) x max I(x) I(x) = h u 1 (x)+u 2 (x) 2 i t = u 1 (x) 2 + u 2 (x) 2 /(k 2 k 1 ) x +2hRe(u 1(x)u 2 (x)i t = I 1 (x)+i 2 (x) 102

Instrument Width & Resolving Power The envelope varies slowly in x for at least one period over the full range, we need Instrumental Width cos 1 2 (k 1 k 2

13 Instrument Width & Resolving Power The envelope varies slowly in x for at least one period over the full range, we need Instrumental Width cos 1 2 (k 1 k 2 )x = cos x x max Inst = min =1/x max Resolving Power mean Inst = = x max 103 Finite Coherence Length wave packets of finite duration fringes no interference 104

14 Finite Coherence Length F (x) =I(x) I 0 /2 /ha (x 0 )a(x + x 0 )i x 0 cos kx a ae ikx 1 d = x/ wave trains of finite length l c = c t c Finite Coherence Length F (x) =I(x) I 0 /2 /ha (x 0 )a(x + x 0 )i x 0 cos kx a ae ikx x =2d d = x/2-1 wave trains of finite length l c = c t c 105-2

Autocorrelation Measurements wave packets of

x 0 )i x 0 l c x l c x visibility I min (x) =

ha (x 0 )a(x 0 )i x 0 106 Michelson-Morley

15 Autocorrelation Measurements wave packets of finite coherence length F (x) /ha (x 0 )a(x + x 0 )i x 0 l c x l c x visibility I min (x) = I max = ha (x 0 )a(x + x 0 )i x0 I max + I min ha (x 0 )a(x 0 )i x Michelson-Morley Experiment Light-carrying ether? Speed of light invariant? Michelson-Morley experiment! universal & constant speed of light pattern unchanged! 107-3

19 Fringe Localisation 108 Reflection of Parallel Plates = kx sin = kx cos + = /2 Rotationally symmetric around P-P axis Young s fringe to axis Etalon fringe (Michelson interferometer) to axis 109

22 Parallel Plates Michelson Interferometer d I / cos 2 ( 1 2kx cos ) x =2d equivalent to Michelson! Fringe Localisation point-like (image) sources double slit fringes fringes are not localised extended light sources fringes localised at infinity 111

23 Fringe Localisation point source fringes not localised 112 Fringe Localisation point source fringes not localised 113

Fringe Localisation extended source 114-1 extended

24 Fringe Localisation extended source extended source Fringe Localisation fringes localised at infinity 114-2

25 Fringe Localisation Parallel reflecting surfaces extended source images t source Extended source path difference xcos 2t=x Fringes localized at infinity Circular fringe constant 2t=x circular fringe constant Fringes of equal inclination Parallel reflecting surfaces extended source Fringe Localisation images t source fringes localised at infinity Extended source path difference xcos 2t=x Fringes localized at infinity Circular fringe constant 2t=x circular fringe constant Fringes of equal inclination 115-2

28 Newton s Rings from Wikipedia 116 soap bubbles 117

29 fringes everywhere Fringe Localisation Double Slit Plane waves (Fresnel diffraction) double-slit fringes at infinity (Fraunhofer) 118 Fringe Localisation Grating fringes at infinity Plane waves Z Fraunhofer grating 119

30 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 120 LIGO Laser Interferometric Gravitational Wave Observatory 4 Km 121

31 LIGO Laser Interferometric Gravitational Wave Observatory precision L/L Fabry-Perot Interferometer FPI or etalon 123

35 Fabry-Perot Interferometer multiple beam interference U 0 t 1 t 2 (r 1 r 2 ) 3 e 3i U 0 t 1 t 2 (r 1 r 2 ) 2 e 2i U 0 t 1 r 1 r 2 U0 t 1 t 2 r 1 r 2 e i =2kd cos 124 Fabry-Perot Interferometer Michelson FPI 125

36 Fabry-Perot Interferometer FPI transmission 126 = k2d cos Transmission Fabry-Perot Interferometer F = R 1 R Finesse F = peak separation FWHM of one peak = k2d cos 127

37 Scanning Fabry-Perot Piezo or pressure tuning scan the optical path-length difference 128 FPI as Spectrometer 129

38 FPI as Spectrometer interlacing rings from the Oxford Physics Webpage 130 FPI as Spectrometer 131

FPI Finesse Spectroscopic tool: 10... 100 Laser cavities: 100... 1000 Trapping light: 10,000.

44 FPI Finesse Spectroscopic tool: Laser cavities: Trapping light: 10, ,000,000 (for studying quantum effects at the single-photon level) Chromatic Resolving Power Fabry-Perot: F = FSR FWHM = (2d) 1 p =2d p F = = Grating: p = d sin p dispersion: minimum: p = d p cos p (Np+ 1) = Nksin( p + )d/2 cos p = /(Nd) p N = = =

Where are the Fringes? (in a real system) Div. of Amplitude - Wedged Plates. Fringe Localisation Double Slit. Fringe Localisation Grating

Where are the Fringes? (in a real system) Fringe Localisation Double Slit spatial modulation transverse fringes? everywhere or well localised? affected by source properties: coherence, extension Plane