The_Cambridge_Handbook_of_Physics_Formulas

166 Optics
8.4 Fresnel diffraction
Kirchhoff’s diffraction formula a
dA
r
θ
ψ 0
P
source
ρ
y
dS S
ŝ
x
r
z
Source at
infinity
where:
Obliquity
factor
(cardioid)
(source at infinity)
ψ P = − i λ ψ 0
Source at ψ P = − iE 0
finite
distance b
∫
plane
K(θ) eikr
ψ P complex amplitude at P
λ wavelength
r dA (8.45) k wavenumber (= 2π/λ)
ψ 0 incident amplitude
θ obliquity angle
r distance of dA from P (≫ λ)
K(θ)=
2 (1+cosθ) (8.46) wavefront
K obliquity factor
1
dA area element on incident
dS element of closed surface
∮
e ik(ρ+r)
ˆ unit vector
[cos(ŝ· ˆr)−cos(ŝ· ˆρ)] dS s vector normal to dS
λ 2ρr
r vector from P to dS
closed surface
ρ vector from source to dS
(8.47)
amplitude (see footnote)
E 0
a Also known as the “Fresnel–Kirchhoff formula.” Diffraction by an obstacle coincident with the integration surface
can be approximated by omitting that part of the surface from the integral.
b The source amplitude at ρ is ψ(ρ)=E 0 e ikρ /ρ. The integral is taken over a surface enclosing the point P .
Fresnel zones
P
y
source
z 1 z 2
observer
Effective aperture 1
distance a z = 1 + 1 z effective distance
(8.48) z 1 source–aperture distance
z 1 z 2
z 2 aperture–observer distance
n half-period zone number
Half-period zone
y n =(nλz) 1/2 (8.49) λ wavelength
radius
y n nth half-period zone radius
Axial zeros (circular
aperture)
z m = R2
z m distance of mth zero from
(8.50) aperture
2mλ
R aperture radius
a I.e., the aperture–observer distance to be employed when the source is not at infinity.