The_Cambridge_Handbook_of_Physics_Formulas

172 Optics
8.7 Coherence (scalar theory)
Mutual
coherence
function
Complex degree
of coherence
Combined
intensity a
Γ 12 (τ)=〈ψ 1 (t)ψ ∗ 2(t+τ)〉 (8.97)
γ 12 (τ)= 〈ψ 1(t)ψ ∗ 2(t+τ)〉
[〈|ψ 1 | 2 〉〈|ψ 2 | 2 〉] 1/2 (8.98)
=
Fringe visibility V (τ)= 2(I 1I 2 ) 1/2
if |γ 12 (τ)| is a
constant:
Γ 12 (τ)
[Γ 11 (0)Γ 22 (0)] 1/2 (8.99)
I tot = I 1 +I 2 +2(I 1 I 2 ) 1/2 R[γ 12 (τ)]
(8.100)
I 1 +I 2
|γ 12 (τ)| (8.101)
V = I max −I min
I max +I min
(8.102)
if I 1 = I 2 : V (τ)=|γ 12 (τ)| (8.103)
Γ ij
τ
ψ i
mutual coherence function
temporal interval
(complex) wave disturbance
at spatial point i
t time
〈·〉 mean over time
γ ij complex degree of coherence
∗ complex conjugate
I tot
I i
R
combined intensity
intensity of disturbance at
point i
real part of
I max max. combined intensity
I min min. combined intensity
Complex degree
of temporal
coherence b
γ(τ)= 〈ψ 1(t)ψ ∗ 1(t+τ)〉
〈|ψ 1 (t) 2 (8.104)
|〉
∫
I(ω)e −iωτ dω
= ∫ (8.105)
I(ω)dω
γ(τ) degree of temporal coherence
I(ω) specific intensity
ω radiation angular frequency
c speed of light
Coherence time
and length
Complex degree
of spatial
coherence c
∆τ c = ∆l c
c ∼ 1 ∆ν
(8.106)
〈ψ
γ(D)=
1 ψ ∗ 2〉
(8.107)
[〈|ψ 1 | 2 〉〈|ψ 2 | 2 〉]
∫ 1/2 I(ŝ)e ikD·ŝ dΩ
= ∫ (8.108)
I(ŝ)dΩ
Intensity
〈I 1 I 2 〉
correlation d [〈I 1 〉 2 〈I 2 〉 2 ] 1/2 =1+γ2 (D) (8.109)
∆τ c
∆l c
∆ν
coherence time
coherence length
spectral bandwidth
γ(D) degree of spatial coherence
D spatial separation of points 1
and 2
I(ŝ) specific intensity of distant
extended source in direction ŝ
dΩ differential solid angle
ŝ
unit vector in the direction of
dΩ
k wavenumber
Speckle
intensity
distribution e pr(I)= 1
〈I〉 e−I/〈I〉 (8.110) pr probability density
∆w
Speckle size
c characteristic speckle size
(coherence ∆w c ≃ λ λ wavelength
(8.111)
width)
α
α source angular size as seen
from the screen
a From interfering the disturbances at points 1 and 2 with a relative delay τ.
b Or “autocorrelation function.”
c Between two points on a wavefront, separated by D. The integral is over the entire extended source.
d For wave disturbances that have a Gaussian probability distribution in amplitude. This is “Gaussian light” such as
from a thermal source.
e Also for Gaussian light.