18. Fresnel Diffraction
• This is most general form of diffraction
– No restrictions on optical layout
• near-field diffraction
• curved wavefront
– Analysis difficult
Fresnel Diffraction Fresnel Diffraction
Screen
Obstruction
Holmholtz Equation
( ) P t A ( ) P [ t ( ) P ]
u , = cos 2 πν + φ
( ) P t { U ( ) P ( j t ) }
u , = Re exp − 2 πν
( ) P A ( ) P [ j ( ) P ]
U = exp − φ
2
0
2 2
2
2
=
∂
− ∂
∇ t
u c
u n
( ∇ 2 + k 2 ) U = 0 Holmholtz Equation
λ π π ν 2
2 =
= n c
k
Green’s Theorem
Let 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
( ) ds
n G U n
U G d
U G
G U
s
∫∫
∫∫∫ ∇ − ∇ υ = ∂ ∂ − ∂ ∂
ν
2 2
Where signifies a partial derivative in the outward normal direction at each point on S.
∂n
∂
Integral Theorem of Holmholtz and Kirchhoff
( ) ( )
01 01 1
exp r P jkr
G =
Kirchhoff’s G
( ) ( ) ( )
r ds jkr U n
r jkr n
P U U
S
∫∫
∂
− ∂
∂
= ∂
01 01 01
01 0
exp exp
4 1
π
Free-space Green’s function
Kirchhoff’s Formulation of Diffraction
( ) ds
n U G
n G P U
U ∫∫
∑
∂
− ∂
∂
= ∂ π 4
1
0
Kirchhoff boundary conditions
are and
On
- ∑ , U ∂ U/ ∂ n
screen.
no were
there if
as
0.
and 0
, On
- S
1U = ∂ U/ ∂ n =
Fresnel-Kirchhoff’s Diffraction Formula (I)
( ) ( ) ( )
01 01 01
01
1
1 exp
,
cos r
jkr jk r
r n n
P
G
−
∂ =
∂ G G
( ) ( ) ( >> λ )
≈
0101 01
01
exp for
,
cos r
r r jkr
n jk G G
( ) ( )
21 21 1
exp r
jkr P A
U =
( ) [ ( ) ] ( ) ( )
r ds n r
n r
r
r r
jk j
P A
U
−
= ∫∫ +
∑
2
, cos ,
cos
exp
01 2101 21
01 21
0
G G G
G λ
* Reciprocity Theorem of Helmholtz
Fresnel-Kirchhoff’s Diffraction Formula
( ) ( ) ( (II) )
r ds P jkr
U P
U
01 01 1
0
∫∫ exp
∑
= ′
( ) ( )
=
′
21
21 1
exp 1
r
jkr A
P j
U λ
( ) ( )
−
2
, cos ,
cos n G r G
01n G r G
21• 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
( ) ds
n U G n G
P U U
S
∫∫ ∂ ∂ − ∂ ∂
= 4
11
0
π
( ) ( ) ( )
01 01 01
01
1
~
exp ~ exp
r r jk r
P jkr
G
−= −
( ) ds
n U G P
U
I∫∫
∑
−
∂
∂
= − π 4
1
0
( ) ( )
n P G n
P G
∂
= ∂
∂
∂
− 1 12
( ) ds
n U G P
U
I∫∫
∑
∂
∂
= − π 2
1
0
Second Rayleigh-Sommerfeld Solution
( ) ( ) ( )
01 01 01
01
1
~
exp ~ exp
r r jk r
P jkr
G
+= +
( ) G ds
n P U
U
II +∫∫
∑∂ ∂
= 4 π 1
0
G G
+= 2
( ) Gds
n P U
U
II∫∫
∑
∂
= ∂
π 2
1
0
Rayleigh-Sommerfeld Diffraction Formula
( ) ( ) ( ) ( ) n r ds
r P jkr
j U P
U
I 0101 01 1
0
1 exp cos G , G
∫∫
∑= λ
( ) ( ) ( )
r ds jkr n
P P U
U
II01 01 1
0
exp 2
1 ∫∫
∑
∂
= ∂ π
( ) ( )
21 21 1
exp r A jkr
P
U =
( ) [ ( ) ] ( )
s d r r n
r
r r
jk j
P A
U
I∫∫
∑
= +
0101 21
01 21
0
exp cos G , G
λ
( ) [ ( ) ] ( )
s d r r n
r
r r
jk j
P A
U
II∫∫
∑
− +
=
2101 21
01 21
0
exp cos G , G
λ
For the case of a spherical wave illumination,
Comparison (I)
( ) ds
n U G
n G P U
U ∫∫
K K∑
∂
− ∂
∂
= ∂ π 4
1
0
( ) ∫∫
∑
∂
− ∂
= ds
n U G
P
U
Kπ 2
1
0 1
( ) G ds
n P U
U
II∫∫
K∑
∂
= ∂ π 2
1
0
Comparison (II)
( ) [ ( ) ]
r ds r
r r
jk j
P A
U ψ
λ
exp
01 21
01 21
0
∫∫
∑
= +
( ) ( )
[ ]
( )
( )
−
−
=
21 01
21 01
, cos
, cos
, cos
, 2 cos
1
r n
r n
r n r
n
G G
G G
G G G
G
ψ
Fresnel-Kirchhoff theory
First Rayleigh-Sommerfeld solution Second Rayleigh-Sommerfeld solution
* For a normal plane wave incidence,
[ ]
+
= 1
cos
cos 2 1
1 θ
θ ψ
Fresnel-Kirchhoff theory
First Rayleigh-Sommerfeld solution
Second Rayleigh-Sommerfeld solution
Huygens-Fresnel Principle
( ) ( ) ( )
r ds P jkr
j U P
U θ
λ cos
exp 1
01 01 1
0
∫∫
∑
=
( ) P h ( P P ) ( ) U P ds
U
0∫∫
0,
1 1∑
=
( ) ( ) θ
λ cos
exp , 1
01 01 1
0
r
jkr P j
P
h =
( ) ( ) ( )
r ds P jkr
j U P
U 1 exp cos
01 01 1
0
θ
λ ∫∫
∑
=
01
cos r
= z θ
( ) ( ξ η ) ( ) ξ η
λ r d d
U jkr j
y z x
U exp
,
,
201
∫∫
01∑
=
( ) (
2)
22
01
= z + x − ξ + y − η
r
Only two assumptions : scalar theory + r
01>> λ
Huygens-Fresnel Principle
Fresnel Approximation
+ −
+ −
≈
2 2
01
2
1 2
1 1
z y z
z x
r ξ η
( ) λ ( ) ξ η [ ( x ξ ) ( y η ) ] d ξ d η
z j k z U
j y e
x U
jkz
exp 2 ,
,
2 2
− + −
= ∫ ∫
∞∞
−
( ) ( ) ( ) ξ η ( ) ( ) ξ η
λ
η λ ξ
η π
ξ
e d d
e U
z e j y e
x
U
z j z x yj k y
z x j k
jkz
∫ ∫
∞∞
−
+ + −
+
=
2 2 2 ,
2 2 2 2,
( )
( ) ( )
z y f z x f z
j k y
z x j k jkz
Y X
e U
z e j y e
x U
λ λ
η
η
ξλ ξ
/ ,
/ 2
2
2 2 2
2
, )
, (
=
= +
+
= F
Positive vs. Negative Phases
y
Wavefront emitted
earlier
z Wavefront emitted
later
θ
z y
k G
Wavefront emitted
later
Wavefront emitted
earlier
(
01)
exp jkr
( )
2+
2exp 2 x y
z j k
(
01)
exp − jkr
( )
−
2+
2exp 2 x y
z j k
( j 2 πα y )
exp
( j 2 y )
exp − πα
z=0
z=0 diverging
converging
Accuracy of the Fresnel Approximation
max
2 2
21 21 21
2 2
21 21 2 21 21
21 21
2 2
01 01 01
2 2
01 01 2 01 01
01 01 2
21 01
21 01
D= x-
(1 ) , since 2 2
(1 ) , since 2 2
1 1 1
2 Let
z r D
D D
z r z r
r r
z r D
D D
z r z r
r r r r D ξ
λ
∆ = − −
≈ − − ≈ ≅
∆ = − −
≈ − − ≈ ≅
∆ + ∆ = + >
( ) ( )
[ ]
2max2 3 2
4 ξ η
λ
π − + −
〉〉 x y
z
• Accuracy can be expected for much shorter distances
Fresnel (near-field) Regime
Fresnel Diffraction between Confocal Spherical Surfaces
( ) ( ) ξ η ( ) ξ η
λ
η λ ξ
π
d d e
z U j
y e x
U
j z x yjkz ∞ − +
∞
∫ ∫
−=
2
, ,
( )
{ }
f x z f y zjkz
Y X
z U j
e
λ
η
λλ ξ ,
= / , = /= F
R
Nr
Nr
oN o
r = + r N λ
2
, , ,
N o o o
o o
o
N o o N
o
R r N r r N N
r r
R R r
Since R Nr f r f
r N N
λ λ λ
λ λ
λ λ
= + − = +
= = = =
2 2 2
2 2 2
2 2
1 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.
( D)d
x = 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 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
I = φφ = 4 C ξ − C ξ + S ξ − S ξ C η − C η + S η − S η
Fresnel integrals
2 2
0 0
( ) cos , ( ) sin
2 2
t t
C α =
α π dt S α =
α π dt
∫ ∫
Cornu spiral
2 2
0 0