Transcript 投影片 1

Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Chap 4 Fresnel and Fraunhofer
Diffraction
1
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Content
4.1 Background
4.2 The Fresnel approximation
4.3 The Fraunhofer approximation
4.4 Examples of Fraunhofer diffraction patterns
2
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Full wave
equation
z~λ
Rayleigh-Sommerfield &
Fresnel
Fersnel-Kirchhoff
(near field)
3
z>>λ
z   / 4[( x   )2  ( y  )2 ]2max
Beamprop
BPM CAD
Gsolver
Fraunhoffer
(far field)
z  k ( 2   2 ) / 2
光線追跡
DOE CAD
( , )
λ= 850 nm
λ= 1550 nm
ZEMAX
Code V
OSLO
ASAP
(x,y)
850 nm
1550 nm
966 um
791 um
4.6 mm
2.5 mm
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4.1 Background
• These approximations, which are commonly made in many fields
that deal with wave propagation, will be referred to as Fresnel and
Fraunhofer approximations.
• In accordance with our view of the wave propagation phenomenon
as a “system”, we shall attempt to find approximations that are valid
for a wide class of “input” field distributions.
4
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.1.1 The intensity of a wave field
• Poynting’s thm.
  
S  EH

2
1
S  ( ε 0  E )V
2
2 1
1
 ε0  E 
2
με

S  I  E2
When calculation a diffraction pattern, we will general regard the intensity
of the pattern as the quantity we are seeking.
5
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.1.2 The Huygens-Fresnel principle in rectangular coordinates
• Before we introducing a series of approximations to the HuygensFresnel principle, it will be helping to first state the principle in
more explicit from for the case of rectangular coordinates.
• As shown in Fig. 4.1, the diffracting aperture is assumed to lie in the
plane, and is illuminated in the positive z direction.
• According to Eq. (3-41), the Huygens-Fresnel principle can be
stated as
1
U ( P0 ) 
jλ
 U ( P1 )
Σ
e
jkr01
r01
cosds
(1)

where is theangle between the outward normalnˆ and the vectorr01
pointingfrom P0 to P1.
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

y

P0




X
Z
P1
Fig. 4.1 Diffraction geometry
7
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
T he termcos is given exactlyby
cosθ 
z
r01
and therefore the Huygens-Fresnel principle can be rewritten
U ( x,y) 
z
jλ

Σ
U ( , )
e
jkr01
2
dd
(2)
r 01
where thedistancer01 is given exactlyby
r01  z 2  ( x-ξ 2 )  ( y-η 2 )
(3)
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
There have been only two approximations in reaching this expression.
1.One is the approximation inherent in the scalar theory
2. T hesecondis theassumptionthat theobservation distance
is many wavelengthsfrom theaperture, r01  λ.
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4.2 Fresnel Diffraction
• Recall, the mathematical formulation of the Huygens-Fresnel , the
first Rayleigh- Sommerfeld sol.
1
e jkr01
U ( po ) 
U ( p1 )
cos(r01 , a n ).ds

j z
r01
• The Fresnel diffraction means the Fresnel approximation to
diffraction between two parallel planes. We can obtain the
approximated result.
j ( x  ) 2  ( y  ) 2 
e jkz 
2z
U ( x, y) 
U
(

,

)
e
d d




jz
k
10
Dr. Gao-Wei Chang
(1)
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU

y

"" j

K
x   2   y  2
2z
e

x
(Why?)
(wave propagation)
z
wave propagation z
Aperture Plane
Observation Plane
Corresponding to
The quadratic-phase exponential with positive phase
k

j
( x  ) 2  ( y  ) 2 
i.e, ,for
z>0
2z
e
11
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Note:
The distance from the observation point to an aperture point

r01  z  ( x   )  ( y   )
2
2
2

1
2
x  2
y  2 

 z 1  (
) (
) 
z
z


1
2
=b
Using the binominal expansion, we obtain the approximation to
1 x   2 1 y  2 

r01  z 1  (
)  (
) 
2
z
2
z


1
 z
(x   )2  ( y )2
2z


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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• as the term
 x   2 y  2 
( z )  ( z ) 
is sufficiently small.
The first Rayleigh Sommerfeld sol for diffraction between two
parallel planes is then approximated by
1
e
U ( x, y)   U ( , )
j 
jk[ z 
1
( x  ) 2 ( y  ) 2 ]
2z
2
r01
13
z d d
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
z
 cos(r01 , a n ) 
• (
r01 ) , the r01 in denominator of the
integrand is supposed to be well approximated by the first term only
in the binomial expansion, i.e, r01  z
•
In addition, the aperture points and the observation points are
confined to the (  ,  ) plane and the (x,y) plane ,respectively. )
• Thus, we see
j ( x  ) 2  ( y  ) 2 
e jkz
2z
U ( x, y) 
U
(

,

)
e
d d

jz 
k
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Furthermore, Eq(1) can be rewritten as
U ( x, y )  


 U ( , )h( x   , y   )d
d
(2a)
• where the convolution kernel is
e jkz
jk
h( x, y) 
exp[ ( x 2  y 2 )]
jz
2z
(2b)
•
• Obviously, we may regard the phenomenon of wave propagation as
the behavior of a linear system.
15
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Another form of Eq.(1) is found if the term e
is factored outside the integral signs, it yields
k
k
j
k
(x2  y2 )
2z
2
j
( x  y )
j ( 2  2 )
e jkz j 2 z ( x 2  y 2 ) 
z
2z
U ( x, y) 
e
[U ( , )e
]e
d d




jz
(3)
which we recognize (aside from the multiplicative factors) to be the
Fourier transform of the complex field just to the right of the aperture
and a quadratic phase exponential.
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
•
We refer to both forms of the result Eqs. (1) and (3), as the Fresnel
diffraction integral . When this approximation is valid, the observer
is said to be in the region of Fresnel diffraction or equivalently in the
near field of the aperture.
Note:
In Eq(1),the quadratic phase exponential in the integrand
e

k
j
( x  ) 2  ( y  ) 2
2z
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
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
do not always have positive phase for z>0 .Its sign depends on the
direction of wave propagation. (e.g, diverging of converging
spherical waves)
In the next subsection ,we deal with the problem of positive or
negative phase for the quadratic phase exponent.
18
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.2.1 Positive vs. Negative Phases
• Since we treat wave propagation as the behavior of a linear system
as described in chap.3 of Goodman), it is important to descries the
direction of wave propagation.
• As a example of description of wave propagation direction, if we
move in space in such a way as to intercept portions of a wavefield
(of wavefronts ) that were emitted earlier in time.
19
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
f (t )  f ( z, t )
f ( z, t )
z
t
t
f ( z  zc , t )
c
z
t
tc
zc
f (t  2tc )  f ( z, t  2tc )
f ( z  2 zc , t )
t
2 zc
2tc
In the above two illustrations, we assume the wave speed v=zc/tc
where zc and tc are both fixed real numbers.
20
Dr. Gao-Wei Chang
z
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• In the case of spherical waves,
Diverging spherical wave
Converging spherical wave

r

r

k

k
21
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Consider the wave func.
e j k r
r
and r >0 and k  a k  k  a k
If
,where
r  ar r
2

ak  ar ,then
1 j k r 1  j k r
e
 e
r
r
(Positive phase)
implies a diverging spherical wave.
Or if
a k  a r
22
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
1 j k r 1  j k r
e
 e
r
r
(Negative phase)
implies a converging spherical wave.
Note：
For spherical wave ,we say they are diverging or converging ones
instead or saying that they are emitted “earlier in time ” or “later in
time”.
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Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Specifically, for a time interval tc >0, we see the following relations,
“Earlier in time ”
e
e  j 2v (t tc )  e  j 2vt
 j 2v ( t tc )
Positive phase
 j 2vt
The term e
standing for the time dependence of a traveling
wave implies that we have chosen our phasors to rotate in the clockwise
direction.
24
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Therefore, we have the following seasonings：
• “Earlier in time ”
Positive phase
(e.g., diverging spherical waves)
• “Later in time”
Negative phase
(e.g., converging spherical waves)
Note：
“Earlier in time ” means the general statement that if we move in space
in such a way as to intercept wavefronts (or portions of a wave-field ) that
were emitted earlier in time.
25
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
To describe the direction of wave propagation for plane waves, we cannot
use the term diverging or “converging” .Instead .we employ the general
statement ,for the following situations.
ay
Propagation direction
c  0
az
Spatial distribution of
wavefronts
26
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The phasor of a plane wave,
e
, (where  >0)
 j 2 y
multiplied by the time dependence gives
e
 j 2y
e
 j 2vt
e
 j 2v ( t tc )
1
, where t c  y c
v
We may say that ,if we move in the positive y direction , the argument of
the exponential increases in a positive sense, and thus we are moving to a
portion of the wave that was emitted earlier in time.
27
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
In a similar fashion , we may deal with the situation for   0 or c  0
c  0
Propagation direction
28
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Note：
Show that the Huygens-Fresnel principle can be expressed by
1
e jkr01
 
U ( p0 ) 
U ( p1)
 cos(r01, an )ds
j 
r01

<pf>
Recall the wave field at observation point P0
1
U ( p0 ) 
4
u
G
 (G n  U n )ds
29
(1)
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
For the first Rayleigh –Sommerfeld solution ,the Green func.
jkr01
jk~
r01
e
e
G 
 ~
r01
r01
(2)
Note we put the subscript “-”, i.e, G- to signify this kind of Green
func.
Substituting Eq(2) into Eq.(1) gives
30
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
G
1
U ( p0 ) 
(U
)ds

4
n
(3)
or
Gk
1
U ( p0 ) 
(U
)ds

2
n
(4)
where the Green func. proposed by Kirchhoff
e jkr01
Gk 
r01
31
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The term in the integrand of Eq.(4)
GK
 (GK )  an
n
 e jkr01
 ar01
(
) an
r01 r01
 1
jkr01
jkr01
2 
 cos(r01, an )   jke
e
 (1)  r 
01
 r01

1 e jkr01
 cos(r01, an )( jk 
)
r01
r01
32
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
as K 
2


1
or
r01
r01  
GK
2 e jKr01
 j

 cos( r01, an )
n

r01
(5)
Finally, substituting Eq.(5) into Eq.(4) yields
1
e jkr01
 
U ( p0 ) 
U ( p1)
 cos(r01, an )ds
j 
r01

33
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4.2.2 Accuracy of Fresnel Approximation
Recall Fresnel diffraction integral
Parabolic wavelet
 x  2   y  2 
j
e jkz
2z
U x, y  
U  , e
dd

jz  

k
…(4.14)
Aperture point (varying withΣ)
observation point (fixed)
We compare it with the exact formula
1
U x, y  
j
z
r

01
e jkr01
 
 U  ,  r01 cosr01andd
Spherical wavelet
where
(or
  x   2  y  2 
r01  z 1  
 
 
z

  z  

 1  x 
r01  z 1   
 2   z

2
1
2

 y  
 


 z 
2

  1   x      y  
 8  z 
 z 


34
2

  




2
Dr. Gao-Wei Chang
)
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
since the binomial expansion
1  b  12
1
1
 1 b  b2  
2
8
where
 x     y  
b
 

 z   z 
2
2
The max.approx.error (i.e.,( 1  b
1
2
2
1 2 1  x   
 y   
b  
 
 
8
8  z 
 z  
2
1 

 1  b 
2 

)max)
2
and the corresponding error of the exponential
e
1 
jkz b 2 
8 
is maximized at the phase

2
(or approximately 1 radian)
35
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
A sufficient condition for accuracy would be
 2  1  x  
z 
   
   8  z
2
2 2

 y  
 
  max <<1

 z  


(x,y)
(x,y)

x   2   y   2
 z 
4
3
2
max
For example
1cm
(ξ,η)
36
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU



2 2 
3.14 110
 2


z3 
4  0.5 10 6
2
 6.28 m3
η
or z  0.4 m
y
(x,y)
ξ
x
(x,y)
(ξ,η) is variable
observation point
(fixed)

az
z
This sufficient condition implies that the distance z must be
relatively much larger than


 x   2   y   2
4
37

2
max
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Since the binomial expansion
1  b  12
 1
1
1
1
b  b 2    1  b  HOT
2
8
2
 x     y  
where b  
 

 z   z 
2
(high order term)
2
we can see that the sufficient condition leads to a sufficient small
value of b
However, this condition is not necessary. In the following, we will
give the next comment that accuracy can be expected for much
smaller values of z (i.e., the observation point (x , y) can be located
at a relatively much shorter distance to an arbitrary aperture point
on the (ξ,η) plane)
38
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
We basically malcr use of the argument that for the convolution
integral of Eq.(4-14), if the major contribution to the integral comes
from points (ξ,η) for which ξ≒x and η≒y, then the values of
the HOTs of the expansion become sufficiently small.(That is, as
(ξ,η) is close to (x , y)
 x     y  
b
 

 z   z 
2
Consequently, 1  b
1
2
2
gives a relatively small value
1
can be well approximated by 1  b. )
2
39
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
In addition it is found that the convolution integral of Eq.(4-14),

 x  2   y  2 
j
e jkz
z


U x, y  
U

,

e
dd

jz  


jkz 
or U x, y   e
U  , e


jz  
  x 
j  
  z
2

 y 
  

 z



2



dd
e jkz
j X 2 Y 2 
U  , e
dd
jz 

where
X 
x 
z
and
Y
y 
z
,
40
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
can be governed by the convolution integral of the function
e j X Y with a second function (i.e., U(ξ,η)) that is smooth and
slowly varying for the rang –2 < X < 2 and –2 < Y < 2. Obviously,
outside this range, the convolution integral does not yield a
significant addition.
2
2
( Note
For one dimensional case


e

jX 2
dX  1 is governed by
we can see that




e

j X 2  Y 2

2
e
2
jX 2
dX
dXdY  1
is well approximated by
2
 
2
e j  X
2 2
2
Y 2
dXdY
41
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Finally, it appears that the majority of the contribution to the
convolution integral for the range -∞ < X < ∞ and -∞ < Y < ∞
or the aperture area Σ comes from that for a square in the (ξ,η)
4 centered
z
plane with width
and
on the point ξ= x,η= y
x 
y 
(i.e., the range –2 <
<2 and
–2<
<2 or
z
z
x   < 2 z and y   < 2 z )
As a result within the square area, the expansion
η
1  b  12
1
1
 1 b  b2  
2
8
(x,y)
as well approximated, since
 x     y  
b
 

 z   z 
is small enough.
2
ξ
y
(x,y)
4 z
2
42
x

az
z
Dr. Gao-Wei
Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
From another point of view, since the Fresnel diffraction
integral
 x  2   y  2 
j
e jkz
z
U x, y  
U  , e
dd


jz 


j
 x  2   y  2 
e jkz
z


U

,

e
dd


jz Corresponding square area

yields a good approximation to the exact formula
1
U P0  
j
e jkr01
 



U
P
cos
r
01 , a n ds
  1 r01

where r01  z 1   x      y   
  z   z 
2
43
2



1
2
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
we may say that for the Fresnel approximation (for the aperture area
Σ or the corresponding square area) to give accurate results, it is not
necessary that the HOTs of the expansion be small, only that they do
not change the value of the Fresnel diffraction integral significantly.
Note：
From Goodman’s treatment (P.69 70), we see that

X
jX 2
e
dX
can well approximate
X


e

jX 2
dX
or


e
jX 2
dX
Where the width of the diffracting aperture is larger than the
length of the region –2 < X < 2
44
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
For the scaled quadratic-phase exponential of Eqs.(4-14) and
Eq.(4-16), the corresponding conclusion is that the majority of the
contribution to the convolution integral comes from a square in the
(ξ,η) plane, with width 4 z and centered on the point (ξ=
x ,η= y)
In effect,
1. When this square lie entirely within the open portion of the
aperture, the field observed at distance z is, to a good
approximation, what it would be if the aperture were not
present. (This is corresponding to the “light” region)
45
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
2. When the square lies entirely behind the obstruction of the
aperture, then the observation point lies in a region that is, to a
good approximation, dark due to the shadow of the aperture.
3. When the square bridges the open and obstructed parts of the
aperture, then the observed field is in the transition (or gray)
region between light and dark.
For the case of a one-dimensional rectangular slit, boundaries
among the regions mentioned above can be shown to be
parabolas, as illustrated in the following figure.
46
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4 z
2 z
Aperture stop
Dark
Consider the
rectangular slit
with the width 2w
Transition
(Gray)
x
2 z
x
z
Light
y
Incident
wavefront
Dark
47
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
The light region
W – x ≧ 2 z, x ≧ 0
W + x ≧ 2 z , x ＜ 0
Thus, the upper (or lower) boundary between the transition
(or gray) region and the light region can be expressed by
x  w2  4z
(or
x  w
2
48
 4z )
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.2.3 The Fresnel approximation and the Angular Spectrum
• In this subsection, we will see that the Fourier transform of the
Fresnel diffraction impression response identical to the transfer func.
of the wave propagation phenomenon in the angular spectrum
method of analysis, under the condition of small angles.
• From Eqs.(4-15)and (4-16), We have
U ( x, y)  


U ( , )h(   , y  )dd
Where the convolution kernel (or impulse response) is
h ( x, y ) 
e
jkz
jz
e
k 2
( x  y 2)
z
49
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• The FT of the Fresnel diffraction impulse response becomes
jkz
F[h( x, y)] 

 e
H F ( f x , f y) 
j
jz 
k 2 2
(x  y )
2z
 j 2 ( f x x  f y y )
e
dxdy
The integral term

  e
j
 2 2
(x  y )
z
 j 2 ( f x x  f y y )
e
dxdy
can be rewritten a
-j
e
 2 2 2
( f x ( z )  f y (z ) 2 ) 
z
  e
j
 2 2
( p q )
dpdq
z
where
p  x  z
f
x
and
q  y  z
50
f
y
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• (because the exponents

2
j [ x  2 x ( z )
z
f
x
p  x  z
where
2
j
[ y  2 y ( z )
z
f
f
y
x z )
( f
2
j
]
( x  z
z
f x)
2
f y)
2
P
x
 ( f y z )
2
j
( y  z
]
z
q
where q  y  z
f
y
as a result,
HF (
f
x
,
f
y
)
e
jkz
1 

jz 

 e
j
e
j

( f 2x ( z ) 2  f 2y ( z ) 2)
z
 2 2
( p q )
dpdq=1
z
51
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• so
jkz  jz ( f 2x  f 2y )
H F ( f x , f y)  e  e
On the other hand, the transfer function of the wave propagation
phenomenon in the angular spectrum method of analysis is expressed by
H a( f x, f
 jkz 1-(λ( )2 -( 1-λλ )2 , f 2  f 2  1
x
y
x
y
e
λ
)
y
0
, ot herwise

under the condition of small angles (as noted below the term)
e
jkz 1
( f x )  ( f y )
2
2
can be approximated by
jkz(1
1 ( )2 1 ( f )2
fx 
y )
2
2
e
jkz  jz (
e e
(because k 
f 2x  f 2y )
52
2

)
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• (Note: because
r
z
o1
[1 (
1
2
x 
y 

) (
)]
z
z
2
2
For Fresnel approximation, the sufficient condition ma be
z
3


2
2
[

]max
(
x


)
(
y


)
4
The obliquity factor cos(an, r o1) then approaches 1
That is, 
-1  
 cos (an , r01) is small angle
53
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Therefore, we have shown that the FT of the Fresnel diffraction
impulse response
H F ( f x , f y)  H a ( f x , f y)
• Which is the transfer function of the wave propagation phenomenon
in the angular spectrum method of analysis under the condition of
small angles.
54
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4.2.4 Fresnel Diffraction between Confocal Spherical surfaces.
y

x

r
Paraxial region
o1
r
r
o1
o1
55
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
1 x 
1 y 
) 2(
))
r o1  z (1  2 (
2
2
z
z
 2x  x  z
2
 z (1 
2z
as x, y,  ,
2
2

y 
2z
 2y 
2
2
2
)
are all very close to zero, (i.e, the paraxial condition)
x
 y
r o1  z  z  z
Recall the Rayleigh Sommerfeld sol, (for the paraxial condition
U ( x, y ) 

1
1
U ( , )

j

U ( , ) e

j z 
j
e
r o1
2

jk r o1
e
r a n)dd
cos( o1,
k
 j (x y )
dd
z
56
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• as a result, for the paraxial region,
U ( x, y )  e
jkz
j
U ( , ) e

j z

2
(x y )
z
dd
(including the paraxial representation of spherical phase)
This Fresnel diffraction eq. expresses the field U ( , )
observed on the right hand spherical cap as the FT of the filed
U(x,y) on the left-hand spherical cap.
Comparison of the result with Eq(4-17),the Fresnel diffraction
integral (including Fourier-transform-like operation)
57
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
jkz
U ( x, y ) 
e e
jz
j
k 2 2
(x  y )
2z
  [U ( , ) e
j
k 2 2
(  )
]
2z
e
j
2
( x  y )
dd
z
quadratic phase parabolic phase
Note: Recall
sphere
Parabola
Paraxial region
58
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• The two quadratic phase factors in Eq(4-17)are in fact simply
paraxial representations of spherical phase surfaces, (since the
Rayleigh Sommerfeld sol. can be applied only to the planar screens),
and it is therefore reasonable that moving to the spheres has
eliminated them.
• For the diffraction between two spherical caps, it is not really valid
to use the Rayleigh-Sommerfeld result as the basis for the
calculation (only for the diffraction between two parallel planes).
• However, the Kirchhoff analysis remains valid, and its predictions
are the same as those of the Rayleigh-Sommerfeld approach
provided paraxial conditions hold.
59
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4.3 The Fraunhofer approximation
• From Eq(4-17), We see
jkz
U ( x, y ) 
e e
jz
j
k 2 2 
(x  y )
2z

  [U ( , ) e
j
k 2 2
(  )
]
2z
e
j
2
( x  y )
dd
z
If the exponent
k
2
2
[ (  )]
2z
 1
max
We have

z  [(

  )]max
or
2
k
z 
2
2
[(  )]max
2
2
60
Dr. Gao-Wei Chang
(4-17)
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• The observed filed strength U(x,y) can be found2 directly
from a FT
2
k
j0
of the aperture function itself (because j 2 z (  ) )
e 1
e
That is, Eq.(4-17)with the Fraunhofer approximation becomes
jkz
e
e
U ( x, y ) 
j
k 2 2
(x  y )

2z
 
j z
U ( , )
e
 j 2 ( f x  f y )
dd
(4-25)
(Aside from the multiplicative phase factors, this expression is simply
the FT of the aperture distribution)
where
fx

x
and
z
f
y

y
z
(4-26)
61
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Note：
•
Recall the different forms of Fresnel diffraction integral
jkz
U ( x, y ) 
U ( x, y ) 
e  U ( , ) e
j z 
j

[ ( x  ) 2  ( y  ) 2]
dd..................(4 - 14)
z

  U ( , )h( x   , y  )dd.............................(4 -15)
where the Fresnel diffraction impulse response
h ( x, y ) 
e
jkz
j z
e
j
k
( x 2  y 2)
2z
(4-16)
and that of Eq(4-17)
62
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Comparison of Eqs(4-15)and (4-16) with Eqs.(4-25)and (4-26) tell
us that there is no transfer function for the Fraunhofer (or far-field)
diffraction since Eqs(4-25) and (4-26) do not include impulse
response.
•
Nonetheless, since Fraunhofer diffraction is only a special case of
Fresnel diffraction, the transfer function Eq(4-21) remains valid
throughout both the Fresnel and the Fraunhofer regimes. That is, it is
always possible to calculate diffracted field in the Fraunhofer region
by retaining the full accuracy of the Fresnel approximation.
Treating the wave propagation phenomenon as a linear system
63
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
4.4 Examples of Fraunhofer diffraction patterns
• 4.4.1 Rectangular Aperture
• If the aperture is illuminated by a unit-amplitude, normally incident,
monochromatic plane wave, then the field distribution across the
aperture is equal to the transmittance function .Thus using Eq.(4-25),
the Fraunhofer diffraction pattern is seen to be
rect(x)
U ( x, y ) 
jkz
e e
j
k 2 2
(x  y )
2z
j z
1
F{U ( , )}
-Wx
f X  x / z
fY  y /  z
64
Wx
2Wx
Dr. Gao-Wei Chang

Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.4.2 Circular Aperture
Suggests that the Fourier transform of Eq.(4-25) be rewritten as a
Fourier-Bessel transform. Thus if  is the radius coordinate in the
observation plane, we have
U ( )
e jkz
k 2

exp( j
)  {U (q)}
jz
2z
p  r / z
65
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.4.3 Thin Sinusoidal Amplitude Grating
• In practice, diffracting objects can be far more complex. In accord
with our earlier definition (3-68),the amplitude transmittance of a
screen is defined as the ratio of the complex field amplitude
immediately behind the screen to the complex amplitude incident on
the screen . Until now ,our examples have involved only
transmittance functions of the form
1

t A ( , )  
0


in the aperture
out the aperture
Binary transmission
(amplitude grating)
66
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• Spatial patterns of phase shift can be introduced by means of
transparent plates of varying thickness, thus extending the realizable
values of tA to all points within or on the unit circle in the complex
plane.
• As an example of this more general type of diffracting screen,
consider a thin sinusoidal amplitude grating defined by the
amplitude transmittance function
1 m

  
 
t A  ,     cos2f 0  rect
rect

2 2

 2w 
 2w 
(4-33)
where for simplicity we have assumed that the grating structure is
bounded by a square aperture of width 2w. The parameter m represents
the peak-to-peak change of amplitude transmittance across the screen,
and f0 is the spatial frequency of the grating.
67
Dr. Gao-Wei Chang
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
• 4.4.4 Thin sinusoidal phase grating
( or x)
Binary phase grating
m
j[ sin( 2πf0 ξ )]
U (ξ,y )  e 2
rect (
68
ξ
η
)rect ( )
2w
2w
Dr. Gao-Wei Chang