Optical Sensing Techniques and Signal Processing-3
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Transcript of Optical Sensing Techniques and Signal Processing-3
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang1
Chap 4 Fresnel and FraunhoferDiffraction
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang2
Content
4.1 Background
4.2 The Fresnel approximation
4.3 The Fraunhofer approximation
4.4 Examples of Fraunhofer diffraction patterns
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
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max2223 ])()[(4/ L\PT "" yxz 2/)( 22 L\ "" kz
),( L\
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang4
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.
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4.1.1 The intensity of a wave field
Poyntings thm.
HESXXX
v!
E
VES
1
2
1
)2
1(
20
20
!
!X
XX
2EIS w
X
When calculation a diffraction pattern, we will general regard the intensity
of the pattern as the quantity we are seeking.
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Dr. Gao-Wei Chang6
scos)(
1
)( 0110
01
dr
e
PUjPU
jkr
U!
4.1.2 The Huygens-Fresnel principle in rectangular coordinates
Before we introducing a series of approximations to the Huygens-Fresnel 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 positivezdirection.
According to Eq. (3-41), the Huygens-Fresnel principle can be
stated as
.toropointing
ectortheandnor alout ardebet een thangletheishere
10
01
PP
rn X
U
(1)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang7
Fig. 4.1 Diffraction geometry
y
y
L
1P
0P
\
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang8
byexactlygiveniscostermThe U
01
cosr
z !
and therefore the Huygens-Fresnel principle can be rewritten
L\L\ ddr
eU
j
zx,yU
jkr
012
01
),()( !
byexactlyentancethewhere 01r
)()( 22201 y-x-zr !
(2)
(3)
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.01 r ""
There have been only two approximations in reaching this expression.
1.One is the approximation inherent in the scalar theory
apert retheromengthsmany wavelis
istancenobservatiothat theass mptiontheisseconThe.
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4.2 Fresnel Diffraction
Recall, the mathematical formulation of the Huygens-Fresnel , the
first Rayleigh- Sommerfeld sol.
The Fresnel diffraction means the Fresnel approximation to
diffraction between two parallel planes. We can obtain the
approximated result.
!
z
n
jkr
o dsarr
epU
jpU ).,cos()(
1)( 01
01
1
01
P
? A
g
g
! L\L\
P
L\ddeU
zj
eyxU
yxz
kj
jkz 22 )()(2),(),( (1)
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z
x
y
\
L
? A222
"" L\ yxz
Kj
e (Why?)
(wave propagation)
wave propagation z
Aperture PlaneObservation Plane
Corresponding to
The quadratic-phase exponential with positive phase
i.e, ,for z>? A22 )()(2
L\ yxz
kj
e
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N
ote: The distance from the observation point to an aperture point
Using the binominal expansion, we obtain the approximation to
? A2
1
22
21
222
01
)()(1
)()(
!
!
z
y
z
xz
yxzr
L\
L\
=b
? AL\
L\
!
!
yxz
z
z
y
z
x
zr
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as the term
is sufficiently small.
The first Rayleigh Sommerfeld sol for diffraction between two
parallel planes is then approximated by
22 )()(
z
y
z
x L\
L\L\P
L\
ddzr
eU
jyxU
yxz
zjk
7
!2
])()(2
1[
01
22
),(1
),(
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( ) , the r01 in denominator of the
integrand is supposed to be well approximated by the first term only
in the binomial expansion, i.e,
In addition, the aperture points and the observation points areconfined to the ( , ) plane and the (x,y) plane ,respectively. )
Thus, we see
),cos( r
z
ar n !3
zr !01
\ L
? A
! L\L\
P
L\ddeU
zj
eyxU
yxz
kj
jkz 22 )()(2),(),(
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Furthermore, Eq(1) can be rewritten as
(2a)
where the convolution kernel is
(2b)
Obviously, we may regard the phenomenon of wave propagation asthe behavior of a linear system.
g
g L\L\L\ ddyxhUyxU ),(),(),(
)](2
exp[),( 22 yxz
jk
zj
eyxh
jkz
!P
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Dr. Gao-Wei Chang16
Another form of Eq.(1) is found if the term
is factored outside the integral signs, it yields
)( yxz
kj
e
g
g
4
! L\L\P
L\P
L\
ddeeUezj
e
yxU
yxz
jz
kjyx
z
kj
jkz)(
2)(
2)(
2
]),([),(
2222
(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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We refer to both forms of the result Eqs. (1) and (3), as the Fresneldiffraction 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
? A22 )()(2 L\ yxzkje
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do not always have positive phase for z> .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.
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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.
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),2( tzzf c
),( tzzf c
),( tzf
cz
cz2
z
z
z
tzftf !
ct
),()( cc ttzfttf !
ct2
ct
t
t
t
In the above two illustrations, we assume the wave speed v=zc/tc
where zc and tc are both fixed real numbers.
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In the case of spherical waves,
r
k
r
k
Diverging spherical wave Converging spherical wave
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Consider the wave func. r
e rkj
,where rar r!
and r > andP
Tkk
akak !!
If rk aa ,then
rkjrkj er
er
!11
(Positive phase)
implies a diverging spherical wave.
Or ifrk
aa !
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rkjrkj
erer
!
11
i lies a c er i s erical a e.
(Negative phase)
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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The term standing for the time dependence of a traveling
wave implies that we have chosen our phasors to rotate in the clockwisedirection.
Earlier in time
Positive phasecttvj
e T
vtjttvjee c
TT 2)(2
vtje
T2
Specifically, for a time interval tc > , we see the following relations,
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Dr. Gao-Wei Chang25
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.
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0"cU
za
ya
Propagation direction
Spatial distribution of
wavefronts
To describe the direction of wave propagation for plane waves, we cannot
use the term diverging or converging .Instead .we employ the generalstatement ,for the following situations.
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The phasor of a plane wave,yj
eTE2
, (whereE
multiplied by the time dependence gives
(222 cttvjvtjyj eee
!TTTE , where cc y
vE
1!
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.
> )
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0cU
Propagation direction
In a similar fashion , we may deal with the situation for or cUE
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Note
Show that the Huygens-Fresnel principle can be expressed by
dsarr
epU
jpU n
jkr
),cos()()(XX
!P
Recall the wave field at observation point P
dsn
GU
n
uGpU
x
x
x
x! )(
4)(
T( )
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For the first RayleighSommerfeld solution ,the Green func.
1
~
1~
11
r
e
r
eG
rjkjkr
!
Note we put the subscript -, i.e, G- to signify this kind of Green
func.
Substituting Eq(2) into Eq.(1) gives
(2)
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( )
dsn
p k xx! )(1)( 0 T (4)
or
where the Green func. proposed by Kirchhoff
01
01
r
eG
jkr
k !
dsn
G
UpU xx
!
)(4
1
)( 0 T
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The term in the integrand of Eq.(4)
010101
2
0101
0101
01
01
0101
01
01
)1)(o (
)1(1
)o (
)(
)(
re
rjar
rej er
ar
a
r
e
r
a
aGn
G
j r
n
j rj rn
n
j r
r
nKK
!
!
x
x!
!x
x
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as1
12
rK ""
PT or P""r
!x
x
n
GK ),cos(2
1
1
1
n
j r
r
r
ej
P
T
Finally, substituting Eq.(5) into Eq.(4) yields
dsarrpjp n
jkr
),cos()(
1
)( 010110
01 XX
! P
(5)
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4.2.2 Accuracy of Fresnel Approximation
Recall Fresnel diffraction integral
? A
g
g
! L\L\
P
L\ddeU
zj
eyxU
yxz
kj
jkz 22
2,,
observation point (fixed)Aperture point (varying with)
Parabolic wavelet
(4.14)
We compare it with the exact formula
g
g
d! L\L\P
ddnarr
eU
jyxU
jkrXX
10
01
cos,1
,01
Spherical wavelet
01r
z
where
!
z
y
z
xzr L\
(or )
! .
2222
018
1
2
11
z
y
z
x
z
y
z
xzr
L\L\
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since the binomial expansion
.! 2218
1
2
111 bbb
where22
!z
y
z
x L\
The max.approx.error (i.e.,( )max)
bb2
111 2
1
222
2
8
1
8
1
!z
y
z
x L\
and the corresponding error of the exponential
8bjkz
e
is maximized at the phase (or approximately 1 radian)T
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A sufficient condition for accuracy would be
a
z
y
z
xz L\
P
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6
222
3
1050.4
210114.3
vv
v
z or6 v! m0.4z
za
This sufficient condition implies that the distance z must be
relatively much larger than
? AmaL\P
T yx
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Since the binomial expansion
HOTbbbb !! 21
18
1
2
111 22
1
. (high order term)
where22
!
z
y
z
xb
L\
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 (,) plane)
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We basically malcr use of the argument that for the convolutionintegral of Eq.(4-14), if the major contribution to the integral comes
from points (,) for whichx andy, then the values of
the HOTs of the expansion become sufficiently small.(That is, as
(,) is close to (x , y)
!z
y
z
xb
L\gives a relatively small value
Consequently, can be well approximated by . ) 2
1
1b
b2
1
1
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In addition it is found that the convolution integral of Eq.(4-14),
? A g
g
! LLP
L
P
T
ddeUzj
eyxU
yxz
jjkz22
,,
g
g
! L\L\P
P
L
P
\T
ddeUzj
eyxU
z
y
z
xjjkz
22
,,or
7 L\L\
PT ddeU
zj
e YXjjkz
22
,
where and ,z
x
X P
\
! z
y
Y P
L
!
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can be governed by the convolution integral of the function
with a second function (i.e., U(,)) 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.
22 YXje T
( Note
For one dimensional case
12
!g
gdXe XjT is governed by
dXe XjT
we can see that
! g
g
dXdYe YXj
T
is well approximated by
dXdYeYXjT
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Finally, it appears that the majority of the contribution to the
convolution integral for the range - < X < and - < Y < or the aperture area comes from that for a square in the (,)
plane with width and centered on the point = x,= y
(i.e., the range 2
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From another point of view, since the Fresnel diffraction
integral
? A
7
! L\L\
L\
T
ddeUzj
eyxU
yxz
jjkz 22
,,
? A
L\L\
P
L\PT
ddeUzj
e yxz
jjkz 22
,
Corresponding square area
yields a good approximation to the exact formula
7! dsarre
PU
j
PUn
j rXX
,cos1
01
01
10
01
Pwhere
z
y
z
xzr
L\
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we may say that for the Fresnel approximation (for the aperture area
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.
NoteFrom Goodmans treatment (P. 9 7 ), we see that
X
X
XjdXe
2T
can well approximate
g
gdXe Xj
2Tor
7dXe Xj
2T
Where the width of the diffracting aperture is larger than the
length of the region 2 < X < 2
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For the scaled quadratic-phase exponential of Eqs.(4-14) and
Eq.(4-1 ), the corresponding conclusion is that the majority of the
contribution to the convolution integral comes from a square in the
(,) plane, with width and centered on the point (=
x ,= y)zP4
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)
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2. When the square lies entirely behind the obstruction of the
aperture, then the observation point lies in a region that is, to agood 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.
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zP4
zP2
zP2
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Thus, the upper (or lower) boundary between the transition
(or gray) region and the light region can be expressed by
zwx P4! (or ) zwx P4!
The light region
W x , x
W + x , x
zP2zP2
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4.2.3 The Fresnel approximation and the Angular Spectrum
In this subsection, we will see that the Fourier transform of theFresnel 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 (4-1 ), We have
g
g! L\L^EL^ ddyhUyxU )()()(
Where the convolution kernel (or impulse response) is
ee yxk
jk
jyxh
)()(
x
x
!
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The FT of the Fresnel diffraction impulse response becomes
g
g
!! dxdy
jyxhF eeffH
yxjkj
jk
yxF
fyfxyx )()(z
z
z),()],([
T
The integral term
dxdyee yxjj fyfxyx g
g )(
2)(z
22
T
T
can be rewritten a
g
g
dpdqee
qpfyfx jj )(
z
))z((- )z(P
TP
P
TP
where
fx
zxp P! fy
zyq P!and
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( eca se t e e e ts
(( ]([ fzxzffx xzj
xzxzj x PP P
T
PP
T
!where f
xzxp P!
)()(222
])(2[ fzyzffy yz
j
yzy
z
j
yPP
P
TP
P
T!
where fy
zyq P!
as a result,
eeffzfyzfx
zjjkz
yxFH
)( )()(
),(PP
P
T
dpdq
zj eqp
zj )(1P
T
P=1
P
q
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soeeH
y
xzjjkz
yxF
)( 22),(
!
TP
On the other hand, the transfer function of the wave propagation
phenomenon in the angular spectrum method of analysis is expressed by
!
otherwise,
, ,)--()-((jk
yxa
yxyx
eH
under the condition of small angles (as noted below the term)
e yxjkz )()( PP
can be approximated by
ee
efyfx
fyfx
zjj z
j z
)(
)2
1
2
11(
22
)( 2)( 2
TP
PP
(becauseP
T!k )
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(Note: because
)()(122
2
1
1
z
y
z
xr zo
L\
!
For Fresnel approximation, the sufficient condition ma be
][22
4 maxL\
P
T
"" yxz
The obliquity factorco ra on then approache
That i co ran XX!U is small angle
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Which is the transfer function of the wave propagation phenomenon
in the angular spectrum method of analysis under the condition of
small angles.
a ffHffH yxyx !
Therefore, we have shown that the FT of the Fresnel diffraction
impulse response
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4.2.4 Fresnel Diffraction between Confocal Spherical surfaces.
\
ro1
ro1
L
ro1
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)2
2
2
21(
)2
1
2
11(
2
22
2
22
22
1 )()(
zz
z
zz
r
yxz
zo
L
L\
L\
!
$
as L\yx are all very close to zero, (i.e, the paraxial condition)
z
y
z
xzro
L\ $
1
Recall the Rayleigh Sommerfeld sol, (for the paraxial condition
!
!
L\L\P
L\L\P
L\P ddU
zj
ddUj
yxU
ee
arre
yxz
kjj
noo
jkro
)(2
),(
),cos(),(),(
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as a result, for the paraxial region,
This Fresnel diffraction eq. expresses the field ,L\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)
! L\L\
PL\
PT
ddUzj
yxU eeyx
zj
jkz)(
),(),(
(including the paraxial representation of spherical phase)
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g
! L\L\
P
L\P
TL\
dd
zj
yx eeee yx
zj
z
kj
z
kj
jkzyx )(
2)(
2)(
2 ]),([),(2222
quadratic phase parabolic phase
Note: Recall
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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 validto 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 approachprovided paraxial conditions hold.
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4.3 The Fraunhofer approximation
From Eq(4-17), We see
g
g
! L\L\
P
L\P
TL\
ddUz
yxU eeee yx
zzz
zyx )(
2)(
2)(
2 ]),([),(2222
If the exponent
22)](
2[
max
z
k
We have
a
a
L\
L\P
T
""
""
zor
z
(4-17)
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Dr. Gao-Wei Chang61
The observed filed strength U(x,y) can be found directly from a FT
of the aperture function itself (because )e zk
j )(2
22
L\
1
0
ej
That is, Eq.(4-17)with the Fraunhofer approximation becomes
g
g
! \\P
\T
ddUzjyxU eee fyfx
yx
jz
kjjkz
)(2)(
2
),(),(
22
(Aside from the multiplicative phase factors, this expression is simply
the FT of the aperture distribution)
where z
y
andz ff yx PP !!x
(4-2 )
(4-25)
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Note
Recall the different forms of Fresnel diffraction integral
! )14-4........(..........),(),(
][ )()(L\L\
P
L\
PT
ddUzj
yxU ee yx
zj
jkz
)15-4.........(....................),(),(),(
g
g ! ddyxhUyxU
where the Fresnel diffraction impulse response
ee yx
z
kj
jkz
zjyxh
!
P
(4-1 )
and that of Eq(4-17)
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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
zY
zXyfxf
yxz
kj
jkz
UFzj
ee
yxU
PP
L\P
//
)(2
)},({),(
22
!!
! \
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4.4.2 Circular Aperture
Suggests that the Fourier transform of Eq.(4-25) be rewritten as a
Fourier-Bessel transform. Thus if Kis the radius coordinate in the
observation plane, we have
zrp
jkz
qUz
kjzj
eUP
FP /
2
)( )}({)2
exp(!
!
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4.4.3 Thin Sinusoidal Amplitude Grating
In practice, diffracting objects can be far more complex. In accord
with our earlier definition (3- ),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
ape tu etheut
ape tu ethein
tA0
1
),( L\
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Spatial patterns of phase shift can be introduced by means of
transparent plates of varying thickness, thus extending the realizable
values oftA to all points within or on the unit circle in the complexplane.
As an example of this more general type of diffracting screen,
consider a thin sinusoidal amplitude grating defined by the
amplitude transmittance function
!
w
rectw
rectfm
tA
222cos
22
1 L\\TL\ (4-33)
where for simplicity we have assumed that the grating structure isbounded by a square aperture of width 2w. The parameter m represents
the peak-to-peak change of amplitude transmittance across the screen
andf0
is the spatial frequency of the grating.
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4.4.4 Thin sinusoidal phase grating
or x)(\Binary phase grating
)2
()2
()()]2(sin
2[ 0
w
rect
w
recte,yU
fm
j !