optoelectronic systems lab., dept. of mechatronic tech., ntnu dr. gao-wei chang 1 chap 4 fresnel and...
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Chap 4 Fresnel and Fraunhofer Diffraction
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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
Dr. Gao-Wei Chang
Full wave equation
z~λ
Rayleigh-Sommerfield &Fersnel-Kirchhoff
z>>λ
Fresnel(near field)
max2223 ])()[(4/ yxz
Fraunhoffer(far field)
2/)( 22 kz光線追跡
BeampropBPM CAD
GsolverDOE CAD
ZEMAXCode VOSLOASAP
(x,y)),( λ = 850 nmλ = 1550 nm
850 nm1550 nm
966 um791 um
4.6 mm2.5 mm
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 4.1.1 The intensity of a wave field
• Poynting’s thm.
HES
μεEε
VEεS
1
2
1
)2
1(
20
20
2EIS
When calculation a diffraction pattern, we will general regard the intensity of the pattern as the quantity we are seeking.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
scos)(1
)(01
1001
dr
ePU
jλPU
jkr
Σ
• 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 positive z direction.
• According to Eq. (3-41), the Huygens-Fresnel principle can be stated as
. to from pointing
vector theand ˆ normal outward ebetween th angle theis where
1 0
01
PP
r n
(1)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Fig. 4.1 Diffraction geometry
X
Z
y
1P
0P
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
byexactly given is cos termThe
01cos
r
zθ
and therefore the Huygens-Fresnel principle can be rewritten
ddr
eU
jλ
zx,yU
jkr
Σ 012
01
),()(
byexactly given is distance thewhere 01 r
)()( 22201 y-ηx-ξzr
(2)
(3)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
.01 λ r
There have been only two approximations in reaching this expression.
1.One is the approximation inherent in the scalar theory
aperture, thefrom engthsmany wavel is
distancen observatio that theassumption theis second The 2.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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
011
01
ddeU
zj
eyxU
yxz
kjjkz 22 )()(2),(),( (1)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
z
x
y
22
2"" yx
z
Kj
e (Why?) (wave propagation)
wave propagation z
Aperture PlaneObservation Plane
Corresponding to
The quadratic-phase exponential with positive phasei.e, ,for z>0 22 )()(
2 yx
z
kj
e
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Note: The distance from the observation point to an aperture point
Using the binominal expansion, we obtain the approximation to
2
1
22
21
22201
)()(1
)()(
z
y
z
xz
yxzr
=b
22
2201
)()(2
1
)(2
1)(
2
11
yxz
z
z
y
z
xzr
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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
ddzr
eU
jyxU
yxz
zjk
2
])()(2
1[
01
22
),(1
),(
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• ( ) , 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 are confined to the ( , ) plane and the (x,y) plane ,respectively. )
• Thus, we see
0101 ),cos(
r
zar n
zr 01
ddeU
zj
eyxU
yxz
kjjkz 22 )()(2),(),(
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• Furthermore, Eq(1) can be rewritten as
(2a)
• where the convolution kernel is
(2b)
•
• Obviously, we may regard the phenomenon of wave propagation as the behavior of a linear system.
ddyxhUyxU ),(),(),(
)](2
exp[),( 22 yxz
jk
zj
eyxh
jkz
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• Another form of Eq.(1) is found if the term
is factored outside the integral signs, it yields
)(2
22 yxz
kj
e
ddeeUe
zj
eyxU
yxz
jz
kjyx
z
kjjkz )(
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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
•
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
22 )()(2
yxz
kj
e
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
),2( tzzf c
),( tzzf c
),( tzf
cz
cz2
z
z
z
),()( tzftf
ct
)2,()2( 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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• In the case of spherical waves,
r
k
r
k
Diverging spherical wave Converging spherical wave
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Consider the wave func. r
e rkj
,where rar r
and r >0 and2
kk akak
If rk aa ,then
rkjrkj er
er
11
(Positive phase)
implies a diverging spherical wave.
Or if rk aa
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
rkjrkj er
er
11
implies a converging spherical wave.
(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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The term standing for the time dependence of a traveling wave implies that we have chosen our phasors to rotate in the clockwise direction.
“Earlier in time ”
Positive phase)(2 cttvje
vtjttvj ee c 2)(2
vtje 2
Specifically, for a time interval tc >0, we see the following relations,
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
0cza
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 general statement ,for the following situations.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The phasor of a plane wave, yje 2 , (where
multiplied by the time dependence gives
)(222 cttvjvtjyj eee , where cc yv
t 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.
>0)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
0c
Propagation direction
In a similar fashion , we may deal with the situation for 0 or 0 c
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Note:
Show that the Huygens-Fresnel principle can be expressed by
dsarr
epU
jpU n
jkr),cos()(
1)( 01
0110
01 <pf>
Recall the wave field at observation point P0
dsn
GU
n
uGpU
)(4
1)( 0 (1)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
For the first Rayleigh –Sommerfeld solution ,the Green func.
01
~
01~
0101
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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
(3)
dsn
GUpU k
)(2
1)( 0 (4)
or
where the Green func. proposed by Kirchhoff
01
01
r
eG
jkr
k
dsn
GUpU
)(4
1)( 0
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The term in the integrand of Eq.(4)
010101
2
0101
0101
01
01
0101
01
01
)1
)(,cos(
)1(1
),cos(
)(
)(
r
e
rjkar
rejker
ar
ar
e
ra
aGn
G
jkr
n
jkrjkrn
n
jkr
r
nKK
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
as 01
12
rK
or 01r
n
GK ),cos(2
0101
01
n
jKrar
r
ej
Finally, substituting Eq.(5) into Eq.(4) yields
dsarr
epU
jpU n
jkr),cos()(
1)( 01
0110
01
(5)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
4.2.2 Accuracy of Fresnel Approximation
Recall Fresnel diffraction integral
ddeU
zj
eyxU
yxz
kjjkz 22
2,,
observation point (fixed)Aperture point (varying withΣ)
Parabolic wavelet
…(4.14)
We compare it with the exact formula
ddnar
r
eU
jyxU
jkr 10
01
cos,1
,01
Spherical wavelet
01r
z
where 2
122
01 1
z
y
z
xzr
(or )
2222
01 8
1
2
11
z
y
z
x
z
y
z
xzr
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
since the binomial expansion
221
8
1
2
111 bbb
where
22
z
y
z
xb
The max.approx.error (i.e.,( )max)
bb
2
111 2
1
2222
8
1
8
1
z
y
z
xb
and the corresponding error of the exponential
2
8
1bjkz
eis maximized at the phase (or approximately 1 radian) 2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
A sufficient condition for accuracy would be
max
222
8
12
z
y
z
xz
<<1
max2223
4
yxz
For example
(ξ,η)
(x,y)1cm
(x,y)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
6
222
3
1050.4
210114.3
z or 3m28.6 m 0.4z
(x,y)(x,y)
(ξ,η) is variable
ξ
η
z
x
y
observation point (fixed)
za
This sufficient condition implies that the distance z must be relatively much larger than
max222
4
yx
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Since the binomial expansion
HOTbbbb 2
11
8
1
2
111 22
1 (high order term)
where 22
z
y
z
xb
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)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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)
22
z
y
z
xb
gives a relatively small value
Consequently, can be well approximated by . )
21
1 b b2
11
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
In addition it is found that the convolution integral of Eq.(4-14),
ddeUzj
eyxU
yxz
jjkz 22
,,
ddeUzj
eyxU
z
y
z
xjjkz
22
,,or
ddeU
zj
e YXjjkz 22
,
where and , z
xX
z
yY
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
can be governed by the convolution integral of the function 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.
22 YXje
( Note For one dimensional case
12
dXe Xj is governed by
2
2
2
dXe Xj
we can see that 122
dXdYe YXj
is well approximated by
2
2
2
2
22
dXdYe YXj
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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) plane with width and centered on the point ξ= x,η= y (i.e., the range –2 < <2 and –2< <2 or
< and < )
z4
z
x
z
y
x z2yz2
As a result within the square area, the expansion
221
8
1
2
111 bbb
as well approximated, since
22
z
y
z
xb
is small enough.
(x,y)(x,y)ξ
η
z
x
y
zaz4
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
From another point of view, since the Fresnel diffraction integral
ddeUzj
eyxU
yxz
jjkz 22
,,
ddeUzj
e yxz
jjkz 22
,Corresponding square area
yields a good approximation to the exact formula
dsarr
ePU
jPU n
jkr ,cos
101
0110
01
where 2
122
01 1
z
y
z
xzr
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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
X
Xj dXe2
can well approximate
dXe Xj 2
or dXe Xj 2
Where the width of the diffracting aperture is larger than the length of the region –2 < X < 2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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 and centered on the point (ξ= x ,η= y)
z4
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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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
z4
z2
x
y
z
Aperture stop
Incidentwavefront
z2
x
Dark
Dark
Light
Transition(Gray)
Consider the rectangular slit
with the width 2w
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Thus, the upper (or lower) boundary between the transition (or gray) region and the light region can be expressed by
zwx 42 (or ) zwx 42
The light region
W – x , x 0 ≧ ≧W + x ≧ , x < 0
z2z2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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
ddyhUyxU ),(),(),(
Where the convolution kernel (or impulse response) is
ee yxz
kjkz
zjyxh
)( 22),(
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• The FT of the Fresnel diffraction impulse response becomes
dxdy
jyxhF eeffH
yxjk
jjk
yxFf yf xyx )(2)(
z2
z22
z),()],([
The integral term
dxdyeeyxjj f yf xyx
)(2)(
z22
can be rewritten a
dpdqeeqpf yf x j
zj )(
z))z((- 2222)z( 22
where
fx
zxp fy
zyq and
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• (because the exponents
)()(222
])(2[ fzxzffx xz
jxzx
zj
x
where fx
zxp
)()(222
])(2[ fzyzffy yz
jyzy
z
jy
where fy
zyq
as a result,
eeff zf yzf xz
jjkz
yxFH)( )( 22)( 22
),(
dpdq
zj eqp
zj )( 221
=1
P
q
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• so eeffH
f yf xzjjkz
yxF)( 22
),(
On the other hand, the transfer function of the wave propagation phenomenon in the angular spectrum method of analysis is expressed by
otherwise , 0
),(
111z 2222
λff,)-λλ-()-(λ(jk
yxa
yxyxeffH
under the condition of small angles (as noted below the term)
ef yf xjkz )()(
221
can be approximated by
eee
f yf x
f yf x
zjjkz
jkz
)(
)2
1
2
11(
22
)( 2)( 2
(because
2
k )
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• (Note: because
])()(1[22
2
1
1z
yz
xr z
o
For Fresnel approximation, the sufficient condition ma be
])()([ 224 max
3
yxz
The obliquity factor ),cos(1ra on then approaches 1
That is, ),(cos 01-1 ran
is small angle
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
•
• Which is the transfer function of the wave propagation phenomenon in the angular spectrum method of analysis under the condition of small angles.
),(),(aF ffHffH yxyx
Therefore, we have shown that the FT of the Fresnel diffraction impulse response
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
4.2.4 Fresnel Diffraction between Confocal Spherical surfaces.
ro1
ro1
x
y
Paraxial region
ro1
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
)2
2
2
21(
)2
1
2
11(
2
22
2
22
22
1 )()(
zy
zzx
z
y
z
xr
yxz
zo
as ,,, yx are all very close to zero, (i.e, the paraxial condition)
z
y
z
xzr o
1
Recall the Rayleigh Sommerfeld sol, (for the paraxial condition
ddUzj
ddUj
yxU
ee
arre
yxzk
jj
noo
jk ro
)(2
11
),(1
),cos(),(1
),(1
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• as a result, for the paraxial region,
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)
ddUzj
yxU ee yxz
jjkz
)(2
),(),((including the paraxial representation of spherical phas
e)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
ddU
zjyxU eeee yx
zj
zk
jz
kj
jkzyx )(
2)(
2)(
2 ]),([),(2222
quadratic phase parabolic phase
Note: Recall
sphere
Parabola
Paraxial region
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
4.3 The Fraunhofer approximation
• From Eq(4-17), We see
ddU
zjyxU eeee yx
zj
zk
jz
kj
jkzyx )(
2)(
2)(
2 ]),([),(2222
If the exponent
122 )](2
[max
z
k
We have
)][(
)][(22
22
max
max
2
k
zor
z
(4-17)
61
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• The observed filed strength U(x,y) can be found directly from a FT of the aperture function itself (because ) e z
kj )(2
22
10 e
j
That is, Eq.(4-17)with the Fraunhofer approximation becomes
ddUzj
yxU eee f yf x
yxjz
kjjkz
)(2)(
2),(),(
22
(Aside from the multiplicative phase factors, this expression is simply the FT of the aperture distribution)
where z
yand
zff
yx
x(4-26)
(4-25)
62
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• Note:• Recall the different forms of Fresnel diffraction integral
)14-4........(..........),(),(][ )( 2)( 2
ddUzj
yxU ee yxz
jjkz
)15-4.........(....................),(),(),(
ddyxhUyxU
where the Fresnel diffraction impulse response
ee yxz
kj
jkz
zjyxh
)(2
22),(
(4-16)
and that of Eq(4-17)
63
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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
64
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
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
kjjkz
UFzj
eeyxU
//
)(2
)},({),(
22
Wx-Wx
1
2Wx
rect(x)
65
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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
zrp
jkz
qUz
kj
zj
eU
/
2
)( )}({)2
exp(
66
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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
aperturetheout
aperturetheint A
0
1),(
Binary transmission (amplitude grating)
67
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 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
wrect
wrectf
mt A 22
2cos22
1, 0
(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.
68
Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
• 4.4.4 Thin sinusoidal phase grating
or x) (Binary phase grating
)2
()2
()()]2(sin
2[ 0
w
ηrect
w
ξrecteξ,yU
ξπfm
j