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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Chap 3 Foundations of Scalar
Diffraction Theory
1
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
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Content
3.1
3.2
3.3
3.4
Historical introduction
From a vector to a scalar theory
Some mathematical preliminaries
The Kirchhoff formulation of diffraction by a
planar screen
3.5 The Rayleigh-Sommerfeld formulation of
diffraction
3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld
theories
3.7 Further discuss of the Huygens-Fresnel principle
3.8 Generalization to nonmonochromatic waves
3.9 Diffraction at boundaries
3.10 The angular spectrum of plane waves
2
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.1 Historical introduction
• While the theory discussed here is sufficiently general to
be applied in other field, such as acoustic-wave and radiowave propagation, the applications of primary concern
will be in the realm of physical optics.
• To fully understand the properties of optical imaging and
data processing system, it is essential that diffraction and
the limitation it imposes on system performance be
appreciated.
3
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Area

Small
part
The edge of the aperture is thin enough such
that light maybe regarded as unpolarized.
In addition the area of the aperture can not be
too small. (or be large enough)
4
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Refraction can be defined as the bending of
light rays mat takes place when they pass
through a region in which there is a gradient of
the local velocity of propagation of the wave.
• The most common example occurs when the
light wave encounters a sharp boundary between
two regions having different refractive indices.
5
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• The Kirchhoff and Rayleigh-Sommerdeld theories share
certain major simplifications and approximations.
• Most important light is treated as a scalar phenomenon,
neglecting the fundamentally vectorial nature of the
electromagnetic fields.
6
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.2 From a vector to a scalar theory
• In the case of diffraction of light by an aperture, the


• E and H field modified only at the edges of the aperture
where light interacts with the material which the edges are
composed of , and the effects extend over only a few
wavelengths into the aperture itself.
7
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 1.Wave eq. (in vectorial form)
•
(from Maxwell’s eq.)
     

2
Note
:


a
 ay
 az
x
1  E
2

x

y
z
 E
2
V 2 t 2
2     

x 2


y 2


z 2
• 2. Wave eq. (in scalar forms)
•
.
1 2
2
 u
V 2 t 2
u stands for E x , E y , E z or H x , H y , H z
• 3. Wave eq. (in phasor forms).
1  2U
2
U
V 2 t 2
8
a.
b.
c.
d.
e.
Linear,
Homogeneous,
Isotropic,
Nondispersive,
Nonmagnetic.
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
j ( kr-wt  φ)
• Where the optical disturbance u(p, t) = A  e
r
•
-jwt
•
= U(p) e
A jkr jφ
is called phasor
•
U(p) = e
r
• and represents position variable (i.e. r)
• It follow that
( 2  k 2 ) U  0
(called Helmholtz eq.)
9
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• In general
j (kr-wt )
ψ (r,t )  A  e
r
Secondary
wavelets
Huygens (Huygen’s principle)
Envelop
(new wavefront)
(Young)
Fresnel
Primary
wavefront
two assumption
Kirchhoff
(Fresnel-Kirchhoff formula)
Rayletgh-Summerfled
Fig. 3.1 Huygens’ envelope construction
10
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.3 Some mathematical preliminaries
• 3.3.1 The Helmholtz equation
For a monochromatic wave, the scalar field may be written explicitly
u( P, t)  A( P) cos[2π vt  φ(P)]
(3-1)
where A(P) and  (P ) are the amplitude and phase, respectively, of the
wave at position P, while v is the optical frequency.
If the real disturbance u (P, t) is to represent an optical wave,
it must satisfy the scalar wave equation.
11
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
 2 u-
n 2  2u
c
2
t
2
0
(3-2)
• The complex function U(P) serves as an adequate description of the
disturbance, since the time dependencies known a priori. If (3-1) is
substituted in (3-2), it follows that U must obey the timeindependent equation.
( 2  k 2 ) U  0
(3-3)
12
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.3.2 Green’s theorem
Give U(P), G(p), U, G, 2U, and2G


Let F  UG-GU  F  (U  G  U 2G)-(G  U  G 2U )
According Gauss’s divergence thm.


v   F dv 

 
s F  ds

s (UG-GU )a n ds


G
U
2
2
  v (U G-G U )dv   s (U
-G
)ds
n
n
2
2
 v (U G-G U )dv 
13
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Choice of the Green’s func. G(P)

v
2
2
(U G-G U )dv 


s (UG-GU )an ds
Helmholtz eqs.
( 2  k 2 ) U  0
e jkr
G(P)= r
(from Huygens-Fresnel principle) ,
 ( 2  k 2 )G  0
14
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.3.3 The integral thm. of Helmholtz and Kirchhoff
The Kirchhoff formulation of the diffraction problem is based on a certain
integral theorem which expresses the solution of the homogenous wave
equation at an arbitrary point in terms of the values of the solution and its
first derivative on an arbitrary closed surface surrounding that point.
The problem is to express the optical disturbance at Po in terms of its
values on the surface S. To solve this problem, we follow Kirchhoff in
applying Green’s theorem and in choosing as an auxiliary function a
unti-amplitude spherical wave expanding about the point Po (the so-called
free space Green’s function).
15
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Within the volume V ' , the disturbance G, being simply an
expanding spherical wave, satisfies the Helmholtz equation
(2  k 2 )U  0


2
2
(
U

G-G

U
)
dv

(
U

G-G

U
)
a
'
n ds  0
v
s  s  sε

'
s
 (U
s
 (U
G
U
-G
)
n
n

(U
sε
G
U
-G
)ds  n
n

sε
G
U
-G
)ds
n
n
(U
G
U
-G
)ds
n
n
- 4πU ( Po )
as   0
16
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

n
V’
ε

n
S
P0
Fig. 3.2 Surface of integration
17
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Choice of the adequate volume denoted as V ' and S' satisfying the
requirement of Green thm.
a. The first requirement of Green thm. is that U, U, 2U, G, G and2G
exist in the volume of integration.
v
(U 2G-G 2U )dv 
s

(UG-GU )an ds
b. The second requirement…(see Goodman P.42)
U, U, 2U, G, G and2G are continuous within the volume V '
18
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
1
V  U ( Po ) 
4π
e jkr01 U
 e jkr01
(
)
-U
(
)ds
s r01
n
n r01

(3-4)
This result is known as the integral theorem of Helmholtz and Kirchhoff,
it plays an important role in the development of the scalar theory of
diffraction, for it allows the field at any point P0 to be expressed in terms
of the “boundary value” of the wave on any closed surface surrounding
that point.
19
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.4 The Kirchhoff formula of diffraction
by a planar screen
S2
S1
Σ
R

r01

n
P0
P1
Fig. 3.3 Kirchhoff formula of diffraction by a plane screen
20
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.4.1 Application of the integral theorem
1
U ( Po ) 
4
U
G
G
-U
ds
s
n
n

U
U
[G
-U  ( jkG)]ds 
G(
-jkU ) R 2 dw
s2
Ω
n
n


e jkR
uniformly bounded
R


e
jkR
U
(
-jkU ) Rdw
n
As R→ 
U
(
jkU ) R  0
lim
R   n
21
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.4.2 The Kirchhoff boundary conditions
Kirchhoff accordingly adopted the following assumptions [162]:
U
n
1. Across the surface  , the field distribution U and its derivative
are exactly the same as they would be in the absence of the screen.
2. Over the portion of S1 that lies in the geometrical shadow of the
screen, the filed distribution U and its derivative U are identically
n
zero.
These conditions are commonly knows as the Kirchhoff boundary conditions.
22
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.4.3 The Fresnel-Kirchhoff diffraction formula
P2

r01

r02

n
P0
P1
Fig. 3.4 Point-source illumination a plane screen.
23
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
G(P1 )
1 e jkr01
 
 cos(n, r01)( jk )
n
r01 r01
jkr
• Note：
e
 
 jk cos ( n , r01) 
01
(3-5)
r01
G

   G  an
n
  1 jkr  1 jkr
G  ar  (  e )  ar   e  ( jk )  e jkr  (-1)  (r 2 )
r r
r
1 e jkr

 ar  ( jk- )
1 e jkr

 r r
G  a n  ( a r  a n )( jk- )
r
r
1 e jkr01
 
 cos( a n  a r )( jk)
r01 r01
24
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
1 e jkr01 U
 
U(P0 ) 
[
-jkU
cos
(
n
, r01)]ds (Fowles) (3-6)

4π  r01 n
U ( P0 ) 
1
j


[ jk ( r21  r01 )]
e
r21r01

 
cos(n,r01)- cos(n, r21)
[
]ds
2
Let the illumination source be a point source.
e jkr01
U ( P1 )   U ' ( P1 )
ds
r01

 
 
1 Ae jkr21 cos(n, r01)- cos(n, r21)
U' ( P1 )  [
][
]
jλ r21
2
25
Dr. Gao-Wei Chang
(3-7)
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.5 The Rayleigh-Sommerfeld
formulation diffraction
• It is a well-known theorem of potential theory that if a twodimensional potential function and its normal derivative vanish
together along any finite curve segment, then that potential function
must vanish over the entire plane. Similarly, if a solution of the
three-dimension wave equation vanishes on any finite surface
element, it must vanish in all space.
26
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.5.1 Choice of alternative Green’s function
1
G
U
U ( P0 ) 
)ds
 ( n G-U
4π S
n
1
The conditions for validity of this equation are:
1. The scalar theory holds.
2. Both U and G satisfy the homogeneous scalar wave equation. Helmoltz
equation）
3. The Sommerfeld radiation condition is satisfied.
27
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
1
U
G
U ( P0 ) 
-U
)ds
 (G
4π S
n
n
1
1

4
e jkr
G
r


G1
U
U
G(
G
-U
)ds
(G-U
)ds 

4π S'1
n
n
n
n
(Kirchhoff
(3-8)
spherical wavelet)
jk~
r01
e jkr -e
G- 
 ~
r
r01
(Sommerfeld choose)
28
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
P0

r01

r02

n
P0
P1
(Mirror)
Fig. 3.5 Rayleigh-Sommerfeld formulation of diffraction by a plane screen
29
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
G-1
U ( P0 ) 
ds
 U
4π Σ
n
G - (P1 )
 (P )
2 1
n
n
(3-9)
(refer to eq. (3-5))
jk~r01
e jkr e
 ~
Choose Green func. G  
r
r01
30
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
-1
G
U Ι ( P0 ) 
ds
 U
2π Σ
n
-1
G
U Π ( P0 ) 
ds
 U
2π Σ
n
G-
e
jkr
r

-e
jk~
r01
~
r01
(3-10)
jk~
r01
e jkr e
G 
 ~
r
r01
31
(3-11)
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.5.2 The Rayleigh-Sommerfeld diffraction formula
Rayleigh-Sommerfeld diffraction formula
1
U Ι ( P0 ) 
jλ
1
U Ι ( P0 ) 
jλ

U ( P1 )
e
r01
Σ

U ( P1 )
jkr01
e
jkr01
r01
S'
 
cos(n, r01)ds
(3-12)
 
cos(n, r01)ds
(3-13)
 Fresnel-Kirchhoff diffraction formula
U ( P0 ) 

U ' ( P1 )
e
Σ
jkr01
r01
(3-14)
ds
jkr21

 
cos
(
n
,
r
)
cos
(
n
, r21)
1
Ae
01
U ' ( P1 )  [
][
)]ds
jλ
r21
2
1
U ( P0 ) 
4π

Σ
e
jkr01
r01
[
U
 
-jkU cos(n, r01)]ds
n
32
(3-15)
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.6 Comparison of the Kirchhoff and
Rayleigh-Sommerfeld theories
P0

r01

r02

n
P1
33
P0
Diffraction by a planar screen
∴ ~
r01  r01
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Kirchhoff (one singular point)
G
e
jkr
01
r01
Sommerfeld (multiple singular point)
G- 
G 
e
jkr01
r01
e
jkr
01
r01
-
jk~r 01
e

~r
01
jk~r
e 01
~r
01
34
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Scalar diffraction formula (general form)
U ( P0 ) 
1
U
G
1
(
G
-U
)
ds

[U Ι (P0 )  U Π (P0 )]

4π s 
n
n
2
(Sommerfeld radiation condition)
jkr
G1
1
e 01
1
G
 
U  ( P0 ) 
(
-U
)
ds

U


cos
(
n

r
)
ds

(
-U
)ds
01



4π
n
jλ
r01
2π
n
Σ
2
U Π ( P0 ) 
G
n (Kirchhoff)
Σ
Σ
1
U
1
U
(
G
)
ds

G
ds



4π
n
2π
n
Σ
Σ
2G (Kirchhoff)
35
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• A comparison of the above equations leads us to an interesting and
surprising conclusion: the Kirchhoff solution is the arithmetic
average of the two Rayleigh-Sommerfeld solution.
• In closing it is worth nothing that, in spite of its internal
inconsistencies, there is one sense in which the Kirchhoff theory is
more general than the Rayleigh-Sommerfeld theory. The latter
requires that the diffracting screens be planar, while the former does
not.
36
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.7 Further discussion of the HuygensFresnel principle
• It expresses the observed field U( P0) as a
superposition of diverging spherical Huygens-Fresnel
wavelet exp( jkr01) /r01 origination from secondary
source located at each every point P1 within the
aperture  .
1
U ( P0 ) 
jλ

Σ
e jkr01
U ( P1 )
cosθds
r01
37
(3-16)
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
U(P0 ) 
 h(P0, P1 )U (P1 )ds
(3-17)
Σ
• where the impulse response h(P0 , P1 )
is given explicitly by
1 e jkr01
h( P0 , P1 ) 
cosθ
jλ r01
r(t)
LIT
38
y(t)
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.8 Generalization to nonmonochromatic wave
(Nonomonochromatic time function)
1
U ( P0 ) 
jλ

Σ
e jkr01
U ( P1 )
cosθds
r01
(3-22)
For nonmonochromatic light
U ( P0 ,  ) 

U ( P , )

jV
e
1
Σ
j2 πν (
r01
r01
)
V
cosθds
(3-23)
where the phasor U ( P0 , ν) implies the Fourier transform (FT) of the
disturbance u( P0 , t) with respect to (w, r, t) the temporal
frequency  .
39
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Note:
Free space
(nondispersive medium)
R
L
L
Vs
C
Not e:
T hereal, monochroma
t ic wave
Monochromatic source
u( Po ,t)  Re[U ( Po )e j 2 πνt ]
Vs (t )  Vm cosWot
(orVs (t )  Vm e
Vs (t ) 

Vne
jWo t
or t hecomplex,monochroma
t ic wave
)
u( Po ,t)  U ( Po )e j 2 πνt
For nonmonochr
omat icwaves
jnWo t

u( Po ,t)   U ( Po , )e j 2 πνt d
n  -
40

Dr.
Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
u(P0 ,t) 

Σ
cosθ
[
2πVr01


-j2πv  U ( P1, ν)e
j 2πν(t 
r01
)
V ]ds (3-24)
-

r01
j 2πν(-(τ- ))
cos θ
(3-25)
V
or u(P0 , - ) 
[ -j 2πv  U ( P1, ν)e
]ds
2πVr01
Σ
-


Since

u(P1, - )   U ( P1, ν )e j 2 πνt dν

d
u(P0 , - )   -j2πv  U ( P1, ν )e j 2 πν (- τ )dν
dτ
d
-
dτ
41
-

u(P1, - )   -j2πv  U ( P1, ν )e j 2 πν (- τ ) dν
-
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Eq. (3-25) becomes
u(P0 , - ) 

Σ
or u ( P0 ,t ) 
r
cos θ d
u ( P1, -(-τ - 01 ))ds
2πVr01 dτ
V

Σ
r01
cos θ d
u ( P1, t- )ds
2πVr01 dt
V
(3-26)
V is the velocity of propagation of the disturbance in a medium of
refractive index n (v=c/n), and the relation νλ  V or  V has been

used.
42
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.9 Diffraction at boundaries
• A more physical point-of-view, first qualitatively expressed by
Thomas Young in 1802, is to regard the observed field as consisting
of a superposition of the incident wave transmitted through the
aperture unperturbed, and a diffracted wave originating at the rim of
the aperture. The possibility of a new wave origination in the
material medium of the rim makes this interpretation a more
physical one.
• The applicability of a boundary diffraction approach in more general
diffraction problems was investigated by Maggi [202] and
Rubinowicz [249], who showed that the Kirchhoff diffraction
formula can indeed be manipulated to yield a form that is equivalent
to Young’s ideas. More recently, Miyamoto and Wolf [250] have
extended the theory of boundary diffraction.
43
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.10 The angular spectrum of plane waves
•
•
•
Objective:
To formulate scalar diffraction theory in a framework that closely
resembles the theory of linear, invariant system.
As we shall see,
P1
P0
(1)
if the complex field distribution of a monochromatic disturbance
is Fourier-analyzed across any plane, the various spatial Fourier
components can be identified as plane waves traveling in different
directions always from that plane.
(2)
The field amplitude at any other point (or across any other parallel
plane) can be calculated by adding the contributions of these
plane waves, taking due account of the phase shifts they have
undergone during propagation.
44
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Assume Z



Equivalent to source
Lens
z  z plane
z0
Young’s experiment
Assume Z

Z=0
Source
Screen
45
Dr. Gao-Wei Chang
Screen
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.10.1 The angular spectrum and its physical interpretation
Let the complex field, across z=0 plane, be represented by U(x, y, 0).
Our ultimate objective is to calculate the resulting field U(x, y, z).

u ( P0 , t)  U ( P0 )e j 2πνt
Phasor
Temporal freq.
U ( x,y,0)  A( f x ,f y ,0)
Spatial freq.
The Fourier transform of U(x, y, 0), i.e., its spectrum
 
A( f x ,f y ,0) 
  U ( x,y,0)e
- j2π ( f x x  f y y )
dxdy
(3-27)
-  -
And the inverse Fourier transform of its spectrum
 
U ( x,y,0) 

-  -
A( f x , f y , 0)e
j2π ( f x x  f y y )
46
df x df y
Dr. Gao-Wei Chang
(3-28)
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• To give physical meaning to the function in the integrand of the
above integral, consider the plane wave

j ( k r - 2πνt)
-j 2πνt
p ( x,y,z,t )  1  e
 P( x,y,z )  e
x

k
cos 1 
• Where the phasor
2π
2π

j (αx  βy) j rz
P( x,y,z)  e k r  e λ
e λ
cos 1 
cos 1 
y
 


theposition vector r  a x x  a y y  a z z and the wave vector
 2 


k
(ax x  a y y  az ) and α, β, γ are directioncosines

which are interrelated thru.
47
Dr. Gao-Wei Chang
z
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
γ  1-α 2 -β 2
Thus, across the plane z=0,
β
α
j 2π ( x  y )
λ
λ
P( x,y,0)  e
j 2π ( f x x  f yy )
e
We see that
fx 
α
β
, fy 
, and f z 
λ
λ
1-α 2 -β 2
λ

The angular spectrum of the disturbance U (x,y,0)

α β
A( , ,0) 
λ λ
 
  U ( x,y,0)e
β
α
- j 2π ( x  y )
λ
y
-  -
48
Z=0
Lens
dxdy
Dr. Gao-Wei Chang
z  z plane
z0
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 3.10.2 Propagation of the angular spectrum
Consider the angular spectrum of the disturbance
A(
α β
, ,z) 
λ λ
 

U ( x,y,z)e
β
α
x  y)
λ
y
dxdy
(3-29)
 
d d
 
(3-30)
-  -
 
U ( x,y,z) 
- j 2π (
U (x,y,z)

-  -
α β
A( , ,0)e
λ λ
β
α
j 2π ( x  y )
λ
y
into the Helmholz eq.
49
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
 2U  k 2U  0
gives
α β
2 2
α β
2
2
A( , ,z)  ( ) (1 -  -  ) A( , ,z)  0
2
λ λ

λ λ
dz
d2
(3-31)
where  2   2  1 and  2  1 -  2 -  2 for all true direction cosines.
An elementary solution of Eq. (3-31) can be written in the form
α β
α β
j 2π( 1-α 2 -β 2 )z
A( , ,z)  A( , ,0)e
λ λ
λ λ
50
Dr. Gao-Wei Chang
(3-32)
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Finally, the substitution of Eq. (3-32) into Eq. (3-30) yields
 
U ( x,y,z ) 

α β
j
2
π
(
x  y) α β
α β
j 2π( 1-α -β )z
2
2
λ
λ d d
A( , ,0)e
 circ ( α  β )e
-  -
2
2
λ λ
λ
λ
(3-33)
where the circ function limits the region of integration to the region
within which  2   2  1 is satisfied.
Note:
When  2   2  1,  and  are no longer interpretable as direction
cosines. Eq. (3-32) can be rewritten as
α β
α β
A( , ,z)  A( , ,0) e - z
λ λ
λ λ
Where  
2

 2   2 -1
51
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Incident wave
n1 sin θ1  n 2 sin θ2
Reflected
n1
If θ1  θ2 and n1  n 2
n2
totalinternalrefl.occurs.
Transmitted (or refracted)
(Including obsorbed wave )
52
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.10.3 Effects of a diffracting aperture on the angular spectrum
Define the amplitude transmittance function in the z=0 plane,
t A ( x,y) 
U t ( x,y,0)
U i ( x,y,0)
Then
U t ( x,y,0)  U i ( x,y,0)t A ( x,y)
And by the use of the convolution theorem.
α β
α β
α β
At ( , )  Ai ( , )  T ( , )
λ λ
λ λ
λ λ
where
53
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
β
α
j
2
π
(
x

y)
 
α β
λ
λ dxdy
T( , ) 
t A ( x,y)e
- -
λ λ
 

And is
again the symbol for convolution. For example, if the
incident wave illuminates the diffracting structure normally.
α β
α β
Ai ( , )   ( , )
λ λ
λ λ
α β
α β
α β
α β
At ( , )   ( , )  T ( , )  T ( , )
λ λ
λ λ
λ λ
λ λ
54
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• In this case, the transmitted angular spectrum is found directly by
Fourier transforming the amplitude transmittance function of the
aperture.
• Note that, if the diffracting structure is an aperture that limits the
extent of the field distribution, the is a broadening of the angular
spectrum of the disturbance, from smaller the aperture, the broader
the angular spectrum behind the aperture.
55
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
3.10.4 The propagation phenomenon as a linear spatial filter
• From Eq. (3-33) in Sec. 3.10.2, we have
 
U ( x,y,z ) 

-  -
2
j
( 1- ( λf x ) 2 -( λf y ) 2 ) z
α β
j 2π ( λf x  λf y )
A( , ,0)  circ ( ( λf x ) 2  ( λf y ) 2 )  e 
e
df x df y
λ λ
α
f

where x λ

and f y   , and we have again explicitly introduced the
bandwidth limitation associated with evanescent waves thru the use of a
circ. function.
z
1

2
2
2
2
1 - ( λ fx ) - ( λ f y ) ], f x  f y 
exp[ j 2π
H ( f x ,f y )  
λ
λ

, otherwise
0
56
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
z
or H ( f x ,f y )  exp [ j 2π
1 - ( λf x ) 2 - ( λf y ) 2 ]  circ ( λf x ) 2  ( λf y ) 2
λ
From Eq. (3-32) in Sec. 3.10.2 and Eq. (3-29)
A( f x ,f y ,0)  A( f x ,f y ,0)  H( f x ,f y )
Thus the propagation phenomenon may be regarded as a linear,
dispersive spatial filter with a finite bandwidth:
57
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
•
(1) Within the circular bandwidth, the modulus of the transfer
function is unity but frequency-dependent phase shifts are
introduced.
•
(2) The phase dispersion of the system is most significant at high
spatial frequency and vanishes as both f x and f y approach
zero.
•
•
(3) For any fixed spatial frequency pair the phase dispersion
increases as the distance of propagation z increases.
In closing we mention the remarkable fact that despite the
apparent different of their approaches, the angular spectrum
approach and the first Rayleigh-Sommerfeld solution yield
identical predictions of diffracted fields.
58
Dr. Gao-Wei Chang