13. Fresnel diffraction
Remind! Diffraction regimes
( ) ( )02
exp,
ikrzEE x y d di r
ξ ηλ ∑
= ∫∫
( ) ( )2 22r z x yξ η= + − + −Screen (x,y)Aperture (ξ,η)
r
( ) ( )00
exp( )
ikrEE P F dAi r
θλ ∑
= ∫∫
Fresnel-Kirchhoff diffraction formula
( ) cos zFr
θ θ= =
( ) ( )2 22 2
2 2 2 2
1 112 2 2 2
2 2 2 2
x yx yz zz z z z
x y x yzz z z z z z
ξ ηξ η
ξ η ξ η
⎡ ⎤ − −− −⎛ ⎞ ⎛ ⎞≈ + + = + +⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + + + + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
Obliquity factor :
( ) ( ) ( )
( ) ( )
2 20
2 2
, exp exp2
exp exp 2
E kE x y ikz i x yi z z
k ki i x y d dz z
λ
ξ η ξ η ξ η∑
⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤× + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫∫
Screen (x,y)Aperture (ξ,η)
r( ) ( ) ( )
( ) ( )
( ) ( )
2 20
2 2
2 2
, exp exp2
exp exp 2
exp exp 2
E kE x y ikz i x yi z z
k ki i x y d dz z
k kC i i x y d dz z
λ
ξ η ξ η ξ η
ξ η ξ η ξ η
∑
∑
⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤× + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤= + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
∫∫
∫∫
( ) ( ) ( )2 2, ( , ) exp exp 2k kE x y C U i i x y d dz z
ξ η ξ η ξ η ξ η⎡ ⎤ ⎡ ⎤= + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫∫
( ) ( )2 2
2( , ) ,kjzE x y U e
ξ ηξ η
+⎧ ⎫∝ ⎨ ⎬
⎩ ⎭F
( ) ( )
( )
, ( , ) exp
( , ) exp sin sin
kE x y C U i x y d dz
C U ik d dξ η
ξ η ξ η ξ η
ξ η ξ θ η θ ξ η
⎡ ⎤= − +⎢ ⎥⎣ ⎦
⎡ ⎤= − +⎣ ⎦
∫∫
∫∫
( ){ }( , ) ,E x y U ξ η∝F
Fresnel diffraction
Fraunhofer diffraction
This is most general form of diffraction– No restrictions on optical layout
• near-field diffraction• curved wavefront
– Analysis somewhat difficult
Fresnel Diffraction
Screen
z
Curved wavefront(parabolic wavelets)
Fresnel (near-field) diffraction
( ) ( )2 2
2( , ) ,kjzU x y U e
ξ ηξ η
+⎧ ⎫≈ ⎨ ⎬
⎩ ⎭F
λ16
2Dz ≥
( ) ( )[ ]2max223
4ηξ
λπ
−+−⟩⟩ yxz
• Accuracy can be expected for much shorter distances
( , ) 2 4for U smooth & slow varing function; x D zξ η ξ λ− = ≤
Fresnel approximation
Accuracy of the Fresnel Approximation
In summary, Fresnel diffraction is …
13-7. Fresnel Diffraction by Square Aperture
(b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers NF=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number NF=0.5.
( )dDx /λ=
Fresnel Diffraction from a slit of width D = 2a. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam.
2 / : Fresnel numberFN a zλ=
2w2a
( ) ( ) ( ) ( )[ ] ηξηξηξλ
ddyxz
kjUzj
eyxUjkz
2
exp,, 22
⎭⎬⎫
⎩⎨⎧ −+−= ∫ ∫
∞
∞−
2 / : Fresnel numberFv N a zλΔ = =
Fresnel diffraction from a wire
Fresnel diffraction from a straight edge
From Huygens’ principle to Fresnel-Kirchhoff diffraction
Huygens’ principle
Given wave-front at t
Allow wavelets to evolve for time ∆t
r = c ∆t ≈ λ
New wavefront
What about –r direction?(π-phase delay when the secondary wavelets, Hecht, 3.5.2, 3nd Ed)
Construct the wave front tangent to the wavelets
Every point on a wave front is a source of secondary wavelets.i.e. particles in a medium excited by electric field (E) re-radiate in all directionsi.e. in vacuum, E, B fields associated with wave act as sources of additional fields
secondary wavelets
Secondarywavelet
Huygens’ wave front construction
Incompleteness of Huygens’ principle
Fresnel’s modification Huygens-Fresnel principle
Huygens’ Secondary wavelets on the wavefront surface O
O
P
Spherical wave from the point source SObliquity factor:
unity where θ=0 zero where θ = π/2
Huygens-Fresnel principle
daFer
er
EdaFerr
EE ikr
A
ikrs
rrik
Asp
pp
)(1'
1)('
1 ')'( θθ ∫∫∫∫ ⎥⎦⎤
⎢⎣⎡== +
Kirchhoff modificationFresnel’s shortcomings :
He did not mention the existence of backward secondary wavelets, however, there also would be a reverse wave traveling back toward the source.He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.
Gustav Kirchhoff : Fresnel-Kirchhoff diffraction theoryA more rigorous theory based directly on the solution of the differential wave equation.He, although a contemporary of Maxwell, employed the older elastic-solid theory of light.He found F(θ) = (1 + cosθ )/2. F(0) = 1 in the forward direction, F(π) = 0 with the back wave.
⎟⎠⎞
⎜⎝⎛ <<= ∫∫ 22
, )(1'
1 ' πθπ -daFer
er
EE ikr
A
ikrsp
p
θ
Fresnel-Kirchhoff diffraction formula
( )ππθπ
<<⎭⎬⎫
⎩⎨⎧ +−
= +∫∫ θ -daerr
ikEE rrik
A
sp
p
, '
12cos1
2)'(
Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theoryA very rigorous solution of partial differential wave equation.The first solution utilizing the electromagnetic theory of light.
dar
eEi
EpA
ikr
Op cos1 θλ ∫∫=
This final formula looks similar to the Fresnel-Kirchhoff formula,therefore, now we call this the revised Fresnel-Kirchhoff formula,or, just call the Fresnel-Kirchhoff diffraction integral.
Fresnel-Kirchhoff diffraction integral
λ/2
Z1
Z2 Z3
Spherical wave from source Po
Huygens’ Secondary wavelets on the wavefront surface S
Obliquity factor: unity where χ=0 at C zero where χ=π/2 at high enough zone index
: Fresnel Zones
: Fresnel Zones
The average distance of successive zones from P differs by λ/2 -> half-period zones.Thus, the contributions of the zones to the disturbance at P alternate in sign,
Z1
Z2 Z3
For an unobstructed wave, the last term ψn=0.
Whereas, a freely propagating spherical wave from the source Po to P is
Therefore, one can assume that the complex amplitude of
1 exp( )iksi sλ
⎛ ⎞= ⎜ ⎟⎝ ⎠
(1/2 means averaging of the possible values,more details are in 10-3, Optics, Hecht, 2nd Ed)
: Diffraction of light from circular apertures and disks
(a) The first two zones are uncovered,
(b) The first zone is uncovered if point P is placed father away,
(c) Only the first zone is covered by an opaque disk,
12
1
1
: Babinet principle
RVariation of on-axis irradiance Diffraction patterns from
circular apertures
RP
(consider the point P at the on-axis P)
112ψ≈
Fresnel diffraction from a circular aperture
Poisson spot
Babinet principle
At screen At complementaryscreen
{ }Amplitude of Sψ
{ }Phase of Sψ
{ }Amplitude of CSψ S CS UNψ ψ ψ+ =
{ }Phase of CSψ
Sψ CSψ
without screen
: Straight edge
Damped oscillatingAt the edge
Monotonically decreasing
13-6. The Fresnel zone plateThe average distance of successive zones from P differs by λ/2 -> half-period zones.Thus, the contributions of the zones to the disturbance at P alternate in sign,
Assumeplane wavefronts
22 22 2 2
0 0 00 02 4n
nR r n r r nr r
λ λ λ⎡ ⎤⎛ ⎞⎛ ⎞ ⎢ ⎥= + − = + ⎜ ⎟⎜ ⎟⎝ ⎠ ⎢ ⎥⎝ ⎠⎣ ⎦
( )0 0 nR nr rλ λ≈ >>
R1O
R3
R4
RN
R2
P
0 2r N λ
+
0r
Z1
Z2 Z3
If the even zones(n=even) are blocked 1 3 5( )Pψ ψ ψ ψ= + + + Bright spot at P
It acts as a lens!
2
0nRr
nλ=
( )0 0 nR nr rλ λ≈ >>
R1O
R3
R4
RN
R2
P
0 2r N λ
+
0r2
11 0 ( 1) Rf r n
λ= = =
Fresnel zone-plate lens
0 11
2 2nnR r Rn n
λ= =nR
( )sin sin tan nn m m m
m n
R mR mf R
λθ λ θ θ= ⇒ = =∼
( )( ) ( ) 11
1 12m n nRf R R nR
m mnλ λ⎛ ⎞= = ⎜ ⎟⎝ ⎠
21
mRfmλ
=
1f2f3f
Fresnel zone-plate lens has multiple foci.
Sinusoidal zone plate: This type has a single focal point.
Binary zone plate: The areas of each ring, both light and dark, are equal.It has multiple focal points.
Fresnel zone-plate lens
For soft X-ray focusing
Fresnel lens: This type has a single focal point.Focusing efficiency approaches 100%.