optoelectronic systems lab., dept. of mechatronic tech., ntnu dr. gao-wei chang 1 chap 5 wave-optics...
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Chap 5 Wave-optics Analysis of Coherent Systems
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Outline
• 5.1 A thin lens as a phase transformation
• 5.2 Lens as a Fourier transform
• 5.3 Image formation ： monochromatic illumination
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
5.1 A thin lens as a phase transformation
x
y
),( yx
0
Figure 5.1The thickness function. (a) Front view, (b) side view
(a) (b)
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The total phase delay suffered by the wave at coordinates
in passing through the lens may be written
Where n is the refractive index of the lens material.
is the phase delay introduced by the lens.
is the phase delay introduced by
the remaining region of free space
between the two planes.
),( yx
)],([),(),( 0 yxkyxknyx
),( yxkn
)],([ 0 yxk
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Equivalently the lens may be represented by a
multiplicative phase transformation of the form
The complex field across a plane immediately
behind the lens is then related to the complex field
incident on a plane immediately in front of the lens by
)],()1(exp[]exp[ 0 yxnjkjktl
),(' yxU l
),( yxU l
),(),(),(' yxUyxtyxU lll
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
222
11 yxRR
01
1R),( yx
02
),( yx2R
03
222
22 yxRR
(a) (b)
(c)Figure 5.2Calculation of the thickness function. (a) Geometry for , (b) geometry for , and (c) geometry for
01 0203
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Total thickness function as the sum of three individual
thickness functions
The thickness function is given by
),(),(),(),( 321 yxyxyxyx
),(1 yx
)11(
)(),(
2
1
22
101
222
11011
R
yxR
yxRRyx
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The second component of the thickness function comes
from a region of glasses of constant thickness .
The third component is given by
02
)11(
)(),(
2
2
22
203
222
22033
R
yxR
yxRRyx
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The total thickness is seen to be
where
)11()11(),(2
2
22
22
1
22
10 R
yxR
R
yxRyx
0302010
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
We consider only values of x and y sufficiently small to
allow the following approximations to be accurate
With the help of these approximation, the thickness
function becomes
2
2
22
2
2
22
2
1
22
2
1
22
211
211
R
yx
R
yx
R
yx
R
yx
)11
(2
),(21
22
0 RR
yxyx
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Substitution into
yields the following
approximation to the lens transformations ：
Note ： For a thin lens
)],()1(exp[]exp[ 0 yxnjkjktl
)11
(2
),(21
22
0 RR
yxyx
)]11
(2
)1(exp[]exp[),(21
22
0 RR
yxnjkjknyxtl
00
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
The physical properties of the lens (that is, n, and )
can be combined in a single number f called the focal
length, which is defined by )11
)(1(1
21 RRn
f
1R 2R
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Neglecting the constant phase factor, which we shall drop
hereafter, the phase transformation may now be written
This equation will serve as our basic representation of the
effects of a thin lens on an incident disturbance.
)](2
exp[),( 22 yxf
kjyxtl
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
5.2 Lens as a fourier transform
),(' yxU l
),( vuU
),( U
),( yxU l
Eq (4-17) Eq (4-17)
),( vu
Lens as a phasor transformer
)(2'
22
),(),(yx
f
kj
ll eyxUyxU
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Pupil function ：
otherwise ,0
aperture lens theinside ,1),( yxP
)](2
exp[),(),(),( 22' yxf
kjyxPyxUyxU ll
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
0
(x,y))115(),(),( yxAtyxU Al
),(' yxU l
Object Image),( ),( vu
Transparency
),( vuU f
fbfl lt),( U Z
lbffZ
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
dxdyeyxUfj
evuU
vyuxf
j
l
vuf
kj
f
)(2)(
2
),(),(
22
is similar to
where is called Fraunhofer (or far-field) diffraction
formula (or FT of )
ddeUzj
eeyxU
yxz
jyx
z
kj
jkz )(2)(
2
),(),(
22
Quadratic phase
),( U
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Recall Fresnel diffraction formula
ddeeUezj
eyxU
yxz
jz
kjyx
z
kjjkz )(
2)(
2)(
2 ]),([),(2222
dxdyeeyxUe
fj
evuU
vyuxf
jyxf
kj
l
vuf
kjjkf
f
)(2
)(2'
)(2 ]),([),(
2222
f is constant.It is neglected.
),(),( yxPyxU l If Lens is enough big,then P(x,y) is
equal 1.
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
is neglected.
dxdyeyxUfj
evuU
vyuxf
j
l
vuf
kj
f
)(2)(
2
),(),(
22
Object
),( yxP
Fraunhofer diffraction formula → )],([),( yxUFvuU lf
focal length
Fraunhofer diffraction patten
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Lens as a Fourier transformer ：
dxdyeyxUfj
evuU
yvxuf
j
l
vuf
kj
f
)(2)(
2
),(),(
22
neglected)],([ yxUF l
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
5.3 Image formation ： monochromatic illumination
),(' yxU l
),( U
),( yxU l
),;,(or ),(),( vuhvuhvuLi
1Z
),(),(0 U
Object
2Z
Image
Linear system
v)(u,U i
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
By the shifting property, we see
),(
),(),;,(
vuh
MvMukvuh
ddvuhUvuU i ),,,(),(),( 0
Output Intput Impulse response
Space invariant ),( vuh
Ideally),(0 MvMukU
Perfect image
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
),(),( vuvuh
),( yxP),()],([ vuhyxPF
(Actually, it will be derived later )],([),( yxPFvuh
Lens coordinate
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
To find the impulse response (or called point spread
function) h, let and substituting it to
gives
),(),(0 U
ddeUzj
eyxU
yxz
kjjkz 22 )()(2),(),(
]})()[(2
exp{1
),( 22
11
yxz
jk
zjyxU l
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
After passage through the lens (focal length f), the filed
distribution becomes
)](2
exp[),(),(),( 22' yxf
jkyxPyxUyxU ll
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Finally, using the Fresnel diffraction equation
to account for
propagation over distance , we have
ddeUzj
eyxU
yxz
kjjkz 22 )()(2),(),(
2z
dxdyyvxuz
jkyxU
zjvuh l ]})()[(
2exp{),(
1),;,( 22
2
'
2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Where constant phase factors have been dropped.
Combining ，
and
and
again neglecting a pure phase factor, yields the formidable
result
]})()[(2
exp{1
),( 22
11
yxz
jk
zjyxU l
)](2
exp[),(),(),( 22' yxf
jkyxPyxUyxU ll
dxdyyvxuz
jkyxU
zjvuh l ]})()[(
2exp{),(
1),;,( 22
2
'
2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Let the magnification (Ref. geometrical optics), we
see
dxdyyz
v
zx
z
u
zjk
yxfzz
jkyxP
z
jkvu
z
jk
zzvuh
]})()[(exp{
)])(111
(2
exp[),(
)][(2
exp[)][(2
exp[1
),;,(
2121
22
21
22
1
22
2212
1Const.
0
Classical lens law of geometrical opticals
1
1
2
Z
ZM
dxdyyMvxMuz
jyxPzz
vuh ]})()[(2
exp{),(1
),;,(221
2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
As it currently stands, the impulse response is that of a
linear space-variant system, so the object and image are
related by a superposition integral but not by a convolution
integral. This space-variant attribute is a direct result of the
magnification and image inversion that occur in the
imaging operation. To reduce the object-image relation to a
convolution equation, we must normalize object-plane
variables be introduced ：
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
in which case the impulse response of
reduces to
M~ M~
dxdyyMvxMuz
jyxPzz
vuh ]})()[(2
exp{),(1
),;,(221
2
dxdyyvxuz
jyxPzz
vuh ]})~()~
[(2
exp{),(1
)~,~
;,(221
2
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
which depends only on the differences of coordinates
A final set of coordinate normalizations simplifies the
results even further. Let
)~,~
( vu
2
~z
xx
2
~z
yy
h
Mh
1~
ydxdyvxujyzxzPvuh ~~)]~~(2exp[)~,~(),(~
22
or
Impulse response or point spread function. (PSF) )],([),( yxPFvuh
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
Then the object-image relationship becomes
or
Where
is the geometrical-optics prediction of the image.
~~)]
~,
~(
1[)~,
~(
~),( 0 dd
MMU
MvuhvuU i
),(),(~
),( vuUvuhvuU gi (convolution form)
)~
,~
(1
),( 0 MMU
MvuU g
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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU
Dr. Gao-Wei Chang
There are two main conclusions from the analysis and
discussion above.
• The ideal image produced by a diffraction-limited optica
l system (i.e. a system that is free from aberrations) is a s
caled and inverted version of the object.
• The effect of diffraction is to convolve that ideal image
with the Fraunhofer diffraction pattern of the lens pupil.
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