3.5.4.2 One example: Michelson interferometer

3.5.4.2 One example: Michelson interferometer mirror 1 mirror 2 light source 1 2 3 beam splitter 4 object (n object ) interference pattern we either observe fringes of same thickness (parallel light) or fringes of same inclination (divergent light) 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 62

3.5.5 Reflection and refraction from wave optics: 3.5.5.1 Reflection n 1 A A' ε ' ε' B B' A'B' = AB ε = ε ' = ' n 2 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 63

3.5.5 Reflection and refraction from wave optics: 3.5.5.2 Refraction: Snellius law n 1 A A' β B' n 1 A'B' = n 2 AB n 1 / n 2 = AB/A'B' A'B'/AB' = sin a AB/AB' = sin b sin a / sin b = n 2 / n 1 n 2 B β 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 64

3.5.6 Interference of reflected beams 3.5.6.1 On a glass plate or on a dielectric film A' n 1 B' A β β n 2 B d β β C D = 2 BC = 2 dn 2 cosβ note: larger tilt angle leads to smaller optical path difference 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 65

3.5.6.1 (alternative 'rays') Interference of reflected beams A' n 1 B' A β β n 2 B d β β C D = 2 BC = 2 dn 2 cosβ 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 66

3.5.6.2 Anti reflective coating λ/(4n h ) λ/(4n l ) high index of refraction n h low index of refraction n l n h n l n h n l substrate 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 67

3.5.7 Young s double slit experiment (introduction) Thomas Young 1802 =r 2 -r 1 =g sin g r 1 r 2 z maxima: sin max = mλ/g m=0, ±1,±2,±3 s x x incident plane wave x x 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 68

3.5.8 Fresnel and Fraunhofer Diffraction 3.5.8.1 Fresnel: divergent light (near field interference) incident plane wave example: double slit blue: = (2m+1) λ/2 red: = m λ minima maxima diffracting object 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 69

3.5.8 Fresnel and Fraunhofer diffraction 3.5.8.2 Fraunhofer: parallel light (far field interference) incident plane wave example: double slit screen diffracting object f lens focusses all parallel rays onto screen (Fraunhofer) but typically: the diffracting object or a defining aperture is focussed onto screen 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 70

3.5.9 Fraunhofer diffraction on a single slit x = b sin b z a f incident plane wave lens 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 71

3.5.9 Fraunhofer diffraction on a single slits and apertures 3.5.9.1 Single slit = b sin b P a incident plane wave lens f x r E S x +b/2 -b/2 b/2 b/2 E P E S exp ik dx E S exp ikxsin dx E S b/2 exp iux iu E S b sinc ub 2 b/2 b/2 E Sb sinc b/2 de P de 0 exp i t kr /kr 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 72 dx E S exp iub/2 exp iub/2 iub/2 bk sin 2 de 0 E s /b dx r with xsin b/2 E S b sinc function (slit function): F sinc sin / de P E s /b exp ik r 0 dx substitute k sin u sin ub/2 ub/2

3.5.9 Fraunhofer diffraction on a single slits and apertures 3.5.9.1 Single slit cont. sinc function electrical field E P sinc E/E 0 I/I 0 diffraction intensity I I 0 sinc 2 with kbsin 2 b sin -3π -2π -π π 2π 3π minima for m sin m with m 1, 2, 3... b note: for 0 we have a maximum! 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 73

3.5.9 Fraunhofer diffraction on a single slits and apertures 3.5.9.2 Circular aperture (dia. D) - Airyfunction Airyfunction: I I 0 2 J 1 2 with 1 kd sin 2 1 0.8 0.6 0.4 0.2-3 -2-1 0 1 2 3-0.2 I 0 first minimum: sin 1. 22 D x D sin 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 74

3.5.10 Resolution of optical instruments min Rayleighcriterium: sin min 1. 22 D 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 75

3.5.11 Fraunhofer diffraction on a grating 3.5.11.1 Schematic i=1 2 3 3 =g sin δ = (2πg/λ) sin b j = (j-1) g sin δ j =(j-1)(2πg/λ) sin g maxima for = m λ incident plane wave 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 76

3.5.11.2 Interference on grating - evaluated 2 g/ sin j j 1 2 g/ sin N E sin exp i j j 1 sin N /2 sin /2 sin Ng 1 sin sin g 1 sin I sin N /2 sin /2 2 but: we have to take slit function into account: F b sin I sin N /2 sin /2 2 F 2 b sin sin2 N g sin sin 2 g sin sinc2 b sin peak intensity increases with N 2 main maxima when denominator 0 sin max m g (where N g sin max Nm 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 77

3.5.11.3 Diffraction from double slit folded with single slit cos 2 (x) sinc 2 (x) x cos 2 (x)sinc 2 (x) x 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 78

3.5.11.4 Diffraction from a grating with single slit folded in (bottom) N=5, b=0.2g x N=20, b = 0.2g N=20, b = 0.5g x x 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 79 x x

3.5.11.5 Summary: diffraction - double slit and grating double slit I F 2 I 0 cos g sin N=2 gratingwith N (illuminated) slits grating constant (slit distance) g grating function: G 2 sin 2 N g sin /sin2 g sin overallintensity from grating I I 0 F 2 G 2 b<<g N=5 N=30 b=0.3g 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 80

3.5.11.6 Resolution of grating Intensity distribution for grating diffraction N=number of slits illuminated coherently g=slit distance, b=slit width = oberserved angle l l m=0 1 2 3 4 order of diffraction I sin2 N g sin sin 2 g sin sinc2 b sin main maxima when denominator 0 sin max m g (where N g sin max Nm adjacent minima for nominator 0 N g sin min Nm n with n 1, 2,.. N 1 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 81

3.5.11.6 Resolution of grating cont. gsin max m and gsin min m n N wavelengths and can be distinguished when maximum at changes to first adjacent minimum at : l l gsin max m! m 1 N resolvingpower (resolution): A mn 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 82

3.5.12 Coherence of electromagnetic waves 3.5.12.1 Longitudinal (temporal) coherence interference of waves with finite coherence length (duration τ) l=c τ l ~ c/δν l 4I 0 I max 2I 0 I min contrast: (I max -I min )/(I max +I min ) 0 for >>l 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 83

3.5.12.2 Lateral (spatial) coherence extended radiation source: coherence -width each photon interferes only with itself! but: interference patterns from different parts of the source may cancel s 1 s = s 4 2 3 interference setup 2I 0 I max I min 4I 0 3+4 1+2 coherence for s = s < λ s < λ / lateral coherence length 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 84

3.5.13 Michelsons stellar interferometer Important application of lateral coherence: measuring the diameter of stars (if distance is known) d s r interference pattern s=1.22(λ / ) =d/r known r d 07.11.01 FU - Physik III - WS 2001/2002 I.V. Hertel 85