this is most general form of diffraction - hanyang
TRANSCRIPT
18. Fresnel Diffraction
• This is most general form of diffraction
– No restrictions on optical layout • near-field diffraction
• curved wavefront
– Analysis difficult
Fresnel DiffractionFresnel Diffraction
Screen
Obstruction
Holmholtz Equation( ) ( ) ( )[ ]PtPAtPu φπν += 2cos,
( ) ( ) ( ){ }tjPUtPu πν2expRe, −=
( ) ( ) ( )[ ]PjPAPU φ−= exp
02
2
2
22 =
∂∂
−∇tu
cnu
( ) 022 =+∇ Uk Holmholtz Equation
λπνπ 22 ==
cnk
Green’s TheoremLet U(P) and G(P) be any two complex-valued functions of
position, and let S be a closed surface surrounding a volume V. If U, G, and their first and second partial derivatives are single-valued and continuous within and on S, then we have
( ) dsnUG
nGUdUGGU
s∫∫∫∫∫
∂∂
−∂∂
=∇−∇ υν
22
Where signifies a partial derivative in the outward normal
direction at each point on S.n∂
∂
Integral Theorem of Holmholtz and Kirchhoff
( ) ( )01
011
exprjkrPG =
Kirchhoff’s G
( ) ( ) ( ) dsr
jkrn
Ur
jkrnUPU
S∫∫
∂∂
−
∂∂
=01
01
01
010
expexp41π
Free-space Green’s function
Kirchhoff’s Formulation of Diffraction
( ) dsnGUG
nUPU ∫∫
∑
∂∂
−∂∂
=π41
0
Kirchhoff boundary conditions
are and On - nU/, U ∂∂∑screen. no were thereif as
0. and 0 ,On - 1 =∂∂= nU/US
Fresnel-Kirchhoff’s Diffraction Formula (I)
( ) ( ) ( )01
01
0101
1 exp1,cosr
jkrr
jkrnnPG
−=
∂∂
( ) ( ) ( )λ>>≈ 0101
0101 forexp,cos r
rjkrrnjk
( ) ( )21
211
expr
jkrAPU =
( ) ( )[ ] ( ) ( ) dsrnrnrr
rrjkjAPU
−+
= ∫∫∑ 2
,cos,cosexp 2101
0121
01210 λ
* Reciprocity Theorem of Helmholtz
Fresnel-Kirchhoff’s Diffraction Formula (II)
( ) ( ) ( )dsrjkrPUPU
01
0110
exp∫∫∑
′=
( ) ( )
=′
21
211
exp1r
jkrAj
PUλ
( ) ( )
−
2,cos,cos 2101 rnrn
• restricted to the case of an aperture illumination consisting of a single expanding spherical wave.
• Kirhhoff’s boundary conditions are inconsistent! : Potential theory says that
“If 2-D potential function and it normal derivative vanish together along any
finite curve segment, then the potential function must vanish over the entire plane”.
Rayleigh-Sommerfeld theory
• the scalar theory holds.
• Both U and G satisfy the homogeneous scalar wave equation.
•The Sommerfeld radiation condition is satisfied.
First Rayleigh-Sommerfeld Solution
( ) dsnGUG
nUPU
S∫∫
∂∂
−∂∂
=1
41
0 π
( ) ( ) ( )01
01
01
011 ~
~expexpr
rjkrjkrPG −=−
( ) dsn
GUPU I ∫∫∑
−
∂∂−
=π41
0
( ) ( )nPG
nPG
∂∂
=∂
∂ − 11 2
( ) dsnGUPU I ∫∫
∑ ∂∂−
=π21
0
Second Rayleigh-Sommerfeld Solution
( ) ( ) ( )01
01
01
011 ~
~expexpr
rjkr
jkrPG +=+
( ) dsGnUPU II +
∑∫∫ ∂
∂=
π41
0
GG 2=+
( ) GdsnUPU II ∫∫
∑ ∂∂
=π21
0
Rayleigh-Sommerfeld Diffraction Formula
( ) ( ) ( ) ( )dsrnrjkrPU
jPU I 01
01
0110 ,cosexp1
∫∫∑
=λ
( ) ( ) ( )dsrjkr
nPUPU II
01
0110
exp21∫∫∑ ∂∂
=π
( ) ( )21
211
exprjkrAPU =
( ) ( )[ ] ( ) sdrnrr
rrjkjAPU I ∫∫
∑
+= 01
0121
01210 ,cosexp
λ
( ) ( )[ ] ( ) sdrnrr
rrjkjAPU II ∫∫
∑
+−= 21
0121
01210 ,cosexp
λ
For the case of a spherical wave illumination,
Comparison (I)
( ) dsn
GUGnUPU K
K∫∫∑
∂∂
−∂∂
=π41
0
( ) ∫∫∑ ∂
∂−= ds
nGUPU K
π21
01
( ) dsGnUPU KII ∫∫
∑ ∂∂
=π21
0
Comparison (II)( ) ( )[ ] ds
rrrrjk
jAPU ψλ
exp
0121
01210 ∫∫
∑
+=
( ) ( )[ ]
( )
( )
−
−
=
21
01
2101
,cos
,cos
,cos,cos21
rn
rn
rnrn
ψ
Fresnel-Kirchhoff theory
First Rayleigh-Sommerfeld solution
Second Rayleigh-Sommerfeld solution
* For a normal plane wave incidence,
[ ]
+
=
1
cos
cos121
θ
θ
ψ
Fresnel-Kirchhoff theory
First Rayleigh-Sommerfeld solution
Second Rayleigh-Sommerfeld solution
Huygens-Fresnel Principle
( ) ( ) ( ) dsrjkrPU
jPU θ
λcosexp1
01
0110 ∫∫
∑
=
( ) ( ) ( )dsPUPPhPU 1100 ,∫∫∑
=
( ) ( ) θλ
cosexp1,01
0110 r
jkrj
PPh =
( ) ( ) ( ) dsr
jkrPUj
PU cosexp1
01
0110 θ
λ ∫∫∑=
01
cosrz
=θ
( ) ( ) ( ) ηξηξλ
ddr
jkrUjzyxU exp,, 2
01
01∫∫∑
=
( ) ( )22201 ηξ −+−+= yxzr
Only two assumptions : scalar theory + λ>>01r
Huygens-Fresnel Principle
Fresnel Approximation
−
+
−
+≈22
01 21
211
zy
zxzr ηξ
( ) ( ) ( ) ( )[ ] ηξηξηξλ
ddyxz
kjUzj
eyxUjkz
2
exp,, 22
−+−= ∫ ∫
∞
∞−
( ) ( ) ( ) ( ) ( )ηξηξ
ληξ
λπηξ
ddeeUezj
eyxUyx
zj
zkjyx
zkjjkz
∫ ∫∞
∞−
+−++
=2
222222
,,
( ) ( ) ( )
zyfzxf
zkjyx
zkjjkz
YX
eUezj
eyxUλλ
ηξηξ
λ/,/
222222
,),(==
++
= F
Positive vs. Negative Phasesy
Wavefront emitted earlier
z Wavefront emitted
later
θ
z
y k
Wavefront emitted
later
Wavefront emitted earlier
( )01exp jkr
( )
+ 22
2exp yx
zkj
( )01exp jkr−
( )
+− 22
2exp yx
zkj
( )yj πα2exp
( )yj 2exp πα−
z=0
z=0
diverging
converging
Accuracy of the Fresnel Approximation
max
2 221 21 21
2 2
21 21 21 2122121
2 201 01 01
2 2
01 01 01 0120101
221 01
21 01
D= x-
(1 ) , since 22
(1 ) , since 22
1 1 12
Let
z r D
D Dz r z rrr
z r D
D Dz r z rrr
Dr r
ξ
λ
∆ = − −
≈ − − ≈ ≅
∆ = − −
≈ − − ≈ ≅
∆ + ∆ = + >
( ) ( )[ ]2max223
4ηξ
λπ
−+−⟩⟩ yxz
• Accuracy can be expected for much shorter distances
Fresnel (near-field) Regime
Fresnel Diffraction between Confocal Spherical Surfaces
( ) ( ) ( )ηξηξ
ληξ
λπ
ddeUzj
eyxUyx
zjjkz +−
∞
∞−∫ ∫=
2
,,
( ){ }zyfzxf
jkz
YXU
zje
λληξ
λ /,/,
=== F
NR Nr
or
N or r N λ = + 2
, , ,
N o o oo o
oN o o N
o
NR r N r r Nr r
rR RSince R Nr f r fr N N
λ λ λ
λ λλ λ
= + − = +
= = = =
22 22 2 2
2 21 1
1
2 4
1
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.
Rectangular symmetric aperture
{ } { } { } { }2 2 2 2*2 1 2 1 2 1 2 1
1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4
I C C S S C C S Sφφ ξ ξ ξ ξ η η η η = = − + − − + −
Fresnel integrals2 2
0 0( ) cos , ( ) sin2 2t tC dt S dtα απ πα α
= =
∫ ∫
Cornu spiral2 2
0 0( ) cos , ( ) sin2 2t tC dt S dtα απ πα α
= =
∫ ∫
Straight edge
Homework:
Plot the Fresnel diffraction patterns
of “your full name” object
at several distances from the object.