One example: Michelson interferometer

object (n ⇒∆_object) beam splitter

light source

mirror 1

mirror 2

3.5.4.2 One example: Michelson interferometer

∆ interference pattern 1 2 3 4

we either observe fringes of same thickness (parallel light) or fringes of same inclination (divergent light)

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3.5.5 Reflection and refraction from wave optics: 3.5.5.1 Reflection n₁ n₂ A'B' = AB ⇒ ε = ε' ⇒ α = α' α A A' ε α' ε' •B B'

•

3.5.5 Reflection and refraction from wave optics: 3.5.5.2 Refraction: Snellius law

n₁ n₂ n₁A'B' = n₂AB ⇒ n₁/ n₂ = AB/A'B' A'B'/AB' = sin αα AB/AB' = sin ββ ⇒ sin αα / sin ββ = n₂/ n₁ α A A' B' B • β β α

3.5.6 Interference of reflected beams

3.5.6.1 On a glass plate – or on a dielectric film

n₁ n₂ d B' B • • A' • • α β α C A β β β ∆∆ = 2 BC = 2 dn₂ cosβ

3.5.6.1 (alternative 'rays') Interference of reflected beams n₁ n₂ d C B B' α β β • • A A' α β β • • ∆∆ = 2 BC = 2 dn₂ cosβ

3.5.6.2 Anti reflective coating

substrate

high index of refraction n_h

low index of refraction n_l

n_h n_l n_h n_l λ/(4n_h) λ/(4n_l)

x x x

3.5.7 Young‘s double slit experiment (introduction)

Thomas Young 1802

s

incident plane wave

z g r₁ r₂ α ∆ ∆ =r₂-r₁=g sin α maxima: sinα_max = mλ/g m=0, ±1,±2,±3 x

3.5.8 Fresnel and Fraunhofer Diffraction

3.5.8.1 Fresnel: divergent light (near field interference)

incident plane wave

blue: ∆ = (2m+1) λ/2 red: ∆ = m λ

example: double slit

minima maxima

3.5.8 Fresnel and Fraunhofer diffraction

3.5.8.2 Fraunhofer: parallel light (far field interference)

incident plane wave example: double slit

diffracting object

lens

focusses all parallel

rays onto screen (Fraunhofer)

screen

f

but typically: the diffracting object or a defining aperture is focussed onto screen

3.5.9 Fraunhofer diffraction on a single slit

α

∆ = b sin α

incident plane wave

lens f z x a b

dE_P J dE₀ expßiÝgt ? krÞà/kr¶

3.5.9 Fraunhofer diffraction on a single slits and apertures 3.5.9.1 Single slit

∆ = b sin α

incident plane wave

lens f r x a b α P -b/2 +b/2 dx x r E_S with A = xsinJ E_P =E_S b/2 ?b/2

X

b/2

?b/2

X

=E_S expÝ?iuxÞ

iu _?b/2

b/2

= E_S expÝiub/2Þ?expÝ?iub/2Þ

iub/2 b/2 = ESb sinub/2 ub/2 =E_S b sinc ub 2 = ESb sinc bksinJ 2

sinc function (slit function): FÝKÞ = sincÝKÞ = sinÝKÞ/K

dE₀ = ÝE_s/bÞdx¶

3.5.9 Fraunhofer diffraction on a single slits and apertures 3.5.9.1 Single slit cont. – sinc function

I/I₀ E/E₀ -π -2π -3π π 2π 3π electrical field E_{P 9} sincÝKÞ with K = kbsinJ 2 = ^b V sinJ

minima for K = m^ ì sinJ = m V

b with m=±1, ±2, ±3 ...

diffraction intensity I = I₀sinc2ÝKÞ

3.5.9 Fraunhofer diffraction on a single slits and apertures 3.5.9.2 Circular aperture (dia. D) - Airyfunction

-0.2 0 0.2 0.4 0.6 0.8 1 -3 -2 -1 1 2 3 I₀ x = D_V sinJ

first minimum: sinJ = 1. 22 V

D

Airyfunction: IÝLÞ = I₀ 2_L J₁ÝLÞ 2

with L = 1

α_min

3.5.10 Resolution of optical instruments

Rayleigh criterium: sinJ_min = 1. 22 V

incident plane wave

3.5.11 Fraunhofer diffraction on a grating 3.5.11.1 Schematic ∆=g sinα ∆_j = (j-1) g sinα maxima for ∆ = m λ i=1 2 ∆ α α 3 g b δ = (2πg/λ) sinα δ_j =(j-1)(2πg/λ) sinα ∆₃

3.5.11.2 Interference on grating - evaluated N = Ý2^g/VÞsinJ N_j = Ýj ? 1ÞÝ2^g/VÞsinJ EÝsinJÞ = j=1 N

>

sinÝ^gV?1 sinJÞ

but: we have to take slit function into account: FÝ^bsinJÞ

I = sinÝNN/2Þ sinÝN/2Þ 2 I = sinÝNN/2Þ sinÝN/2Þ 2 F2Ý ^b_V sinJÞ = sin2N^ g V sinJ sin2^g V sinJ sinc2Ý^ b_V sinJÞ

peak intensity increases with N2

3.5.11.3 Diffraction from double slit – folded with single slit

x

x

cos2_{(x) sinc}2_(x)

3.5.11.4 Diffraction from a grating –

with single slit folded in (bottom)

x

N=5, b=0.2g

x

N=20, b = 0.5g N=20, b = 0.2g

3.5.11.5 Summary: diffraction - double slit and grating N=5 b<<g N=2 N=30 b=0.3g double slit I = F2I₀ cosÝ^ g_V sinJÞ

grating with N (illuminated) slits grating constant (slit distance) g grating function:

G2 = sin2ÝN^ g_V sinJÞ/sin2Ý^ g_V sinJÞ

overall intensity from grating I = I₀F2G2

3.5.11.6 Resolution of grating

λλ λλ‘

Intensity distribution for grating diffraction N=number of slits illuminated coherently g=slit distance, b=slit width

α= oberserved angle m=0 1 2 3 4 order of diffraction I = sin 2_N^ g V sinJ sin2^g V sinJ sinc2Ý^ b_V sinJÞ

main maxima when denominator = 0 ì sinJmax = m _gV (where N^ g_V sinJmax = ^NmÞ

adjacent minima for nominator = 0 ì ^N g_V sinJ_min = ^ÝNm + nÞ

3.5.11.6 Resolution of grating cont.

λλ λλ‘

maximum at Vv = V + AV changes to first adjacent minimum at V :

wavelengths V and Vvcan be distinguished when

gsinJ_maxv = mÝV + AVÞ ! = mV + 1

N V

gsinJmax = mV and gsinJ_min = mV + _Nn V

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 I_max I_min 2I₀ 4I₀ ∆

contrast: (I_max-I_min)/(I_max+I_min) → 0 for ∆>>l l ~ c/δν

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 α α α ∆s= sα interference setup 1 2 3 4 I_max I_min 2I₀ 4I₀ ∆ 3+4 1+2 coherence for ∆_s= s α < λ

3.5.13 Michelsons stellar interferometer s=1.22(λ / α) α=d/r s d α r interference pattern

Important application of lateral coherence: