Interference
Analytical method
Phasor method
Double and single slit
Diffraction at 2-D apertures
Fraunhofer diffraction
Wave propagation (Huygens & Fresnel)
Fresnel-Kirchhoff diffraction integral
Waves and Diffraction
lecture 5
A diffraction pattern for which the phase
of the light at the observation point is a
linear function of the position for all
points in the diffracting aperture is
Fraunhofer diffraction
Fraunhofer Diffraction
eikr −→ eikr0 · ei(!xx+!yy)
Fraunhofer Diffraction
d
δr = |rmax ! d| " a2
8d# λ/8
Fraunhofer Diffraction
!r = !1 + !2 !a2
8
!1ds
+1dp
"" "/8
Fraunhofer Diffraction
! = (k sin ") · y
y
!
aperture
Fraunhofer Diffraction
! = (k sin ") · y
y
!
aperture
f
Fraunhofer Diffraction
! = (k sin ") · y
y
!
aperture
f
illumination
Diffraction in the image plane
A diffraction pattern formed in the
image plane of an optical system is
Fraunhofer diffraction
Fraunhofer Diffraction
what is being imaged?
Fraunhofer
diffraction:
in the image
plane
Fraunhofer Diffraction
Equivalent
lens system:
Fraunhoferdiffractionindependenton aperture position
Fraunhofer Diffraction
Fraunhofer Diffraction & Resolution
! = 1.22 "/D
! ≥ "
peak width !!" f -number
circular
aperture
Rayleigh
criterion
image plane
Fraunhofer Diffraction
• far-field diffraction
• in the image plane
of optical systems
• resolution limit
R! a2/!
!min ! "/a
Bridging the Gap?
• wave propagation
• diffraction in the near field
• Fresnel-Kirchhoff diffraction integral
Fresnel’s Theory of wave propagation
Wave PropagationHuygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...
1 of 1 23.11.2008 19:56 Uhr
Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves, wavelets recombine to produce the propagating wave front. (C) The diffraction of sound around a corner
arising from Huygens' wavelets.
Huygens’ wavelet
print articles
Huygen‘s wavelets recombine to produce
the propagating wavefront
Huygens secondary sources on wavefront at -z
radiate to point P on new wavefront at z = 0
Fresnel‘s Theory of Wave Propagation
plane-to-plane
up = !
!"(#in, #out)
u0
reikrdS
up!= u0
Fresnel‘s Theory of Wave Propagation
!r =!
q2 + "2 ! q " "2
2q
Phase difference of !
at edge of 1st HPZ
Fresnel‘s Theory of Wave Propagation
λ
2=
ρ2π
2q
!!,n =!
"q · n
Phase difference of !
at edge of 1st HPZ
Fresnel‘s Theory of Wave Propagation
λ
2=
ρ2π
2q
!!,n =!
"q · n1
3
5
7
2
4
6
Elements of
equal area
Fresnel‘s Theory of Wave Propagation
!A = "(#2n+1 ! #2
n)
δr = rn+1 ! rn
" δA
2πq
sub-division of HPZ into annuli
First Half Period Zone
Fresnel‘s Theory of Wave Propagation
R! = 2i!u0"
phasor addition
n!! ⇒ resultant ! " diameter of 1st HPZ
Fresnel‘s Theory of Wave Propagation
R! = i!u0"!= u0
Fresnel‘s Theory of Wave Propagation
Fresnel-Kirchhoff diffraction integral
up = ! i
λ
!η(θin, θout)
u0
reikrdS
η(θin, θout) =12(cos θin + cos θout)
obliquity factor
eikr −→ eikr0 · ei(!xx+!yy)
Fraunhofer diffraction is a special case
Fresnel ! Fraunhofer Diffraction
eikr −→ eikr0 · ei(!xx+!yy)
z < dR = a2/!
near field far field
to 1st
minimum
z > dR
slit shadow
Fraunhofer
diffraction
to 1st
minimum
Fresnel Zone Plate24.11.08 01:40http://upload.wikimedia.org/wikipedia/commons/9/97/Zone_plate.svg
Seite 1 von 1
mask out every
second HPZ
in every
transparent
zone, the phase
is running from
0 to !
Fresnel Zone Plate24.11.08 01:40http://upload.wikimedia.org/wikipedia/commons/9/97/Zone_plate.svg
Seite 1 von 1
mask out every
second HPZ
in every
transparent
zone, the phase
is running from
0 to !
acting as
a focussing
lens
Fresnel Lens
1. HPZ
2. HPZ
3. HPZ
4. HPZ
5. HPZ
sub-division into HPZ
Fresnel Lens
phase jump by !
from HPZ to HPZ
1. HPZ
2. HPZ
3. HPZ
4. HPZ
5. HPZ
sub-division into HPZ
Fresnel Lens
phase jump by !
from HPZ to HPZ
section of a lens
in every HPZ
nearly perfect
focussing lens
1. HPZ
2. HPZ
3. HPZ
4. HPZ
5. HPZ
• Geometrical optics
" No wave effects
Geometrical imaging
• " Scalar diffraction theory:" Analytical methods
" Phasor methods
• Fresnel-Kirchhoff diffraction integral:" Fraunhofer diffraction as a limiting case
Propagation of plane waves ... Fresnel-zone plate
Lectures 1-6: The Story so far
lecture 7