Moiré patterns · 01-03-2018  · If 2μ d Cos θ m = m λ then rays 2,3,4, 5, …. are in phase and 1 out of phase. Amplitude of 2+3+4+5 …. = atr’t’(1+ r’ + r’ + r’

Interference with Plane waves and fringe-width

Moiré patterns

Spatial coherence:

Maximum source size

Interference washed out

Temporal coherence

2d=lc

Interference washed out

Temporal coherence

2d < lc

Interference

Stokes’ relations

Thin films: multiple beam interference

Path difference between rays 2 and 1

[(OS + SR)(in film)] – [OM( in air) ]

= [(PS + SR)(in film)] – [OM( in air)]

= [(PR)(in film)] – [OM( in air)]

= μ (PN + NR) – OM

= μ (PN) = μ (OP Cos θ)

Δ= 2μ d Cos θ

If 2μ d Cos θm = m λ

then rays 2,3,4, 5, …. are in phase

and 1 out of phase.

Amplitude of 2+3+4+5 ….

= atr’t’(1+ r’ + r’ + r’ +…) 2 4 6

= atr’t’(1/(1 –r’ )) 2

= atr’t’(1/tt’) = ar’

Amplitude of transmitted beams α, β, γ, δ …

= att’(1+ r’ + r’ + r’ +…) 2 4 6

= a

If 2μ d Cos θm = (m+1/2) λ

then rays 1,2,4, 6, … are in phase

and 3,5,… are out of phase.

Rays α, γ, … in phase and rays β , δ, …

are out of phase

Intensity of fringes

Transmitted beams: Amplitude

= Finesse

Bright fringes Transmitted rays

Dark fringes Reflected rays

Dark fringes Transmitted rays

Bright fringes Reflected rays

Variation of intensities with phase

Fabry-Perot Interferometer

θ

30 o

I0

IR/I IT/I

φ

Transmitted intensity

θ θ

θ θ

30 o

3 o

d/λ = 10 , f = 1

d/λ = 10 , f = 8

d/λ = 10 , f = 2

d/λ = 1000 , f = 7

Fabry-Perot

fringes

Michelson

fringes