Lect 14 Diffraction [호환 모드]
Transcript of Lect 14 Diffraction [호환 모드]
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
a
(1) exp( )A jkRR
(2)(1)y
R
( , )totalE R
siny
(2) exp ( sin )A jk R yR
/2( sin )
/2
y ajk R y
y a
A e dyR
/2sin
/2
y ajkR jky
y a
A e e dyR
Since interference is due to phase difference , ignore the constant phase term
/2sin
/2
( , )y a
jkytotal
y a
AE R e dyR
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
/2sin
/2
Evaluate ( , )a
jkytotal
a
AE R e dyR
2 sin( sin )sin 2
A j akR jk
Let ' siny jky
' sin2
'
' sin2
'( , )sin
ay jk
ytotal
ay jk
A dyE R eR jk
sin( sin )2 2sin
akAR k
sin sin2 21
sin
a ajk jkA e eR jk
=> ' sindy jk dy
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
( )(0)
total
total
EE
( ,0)totalE R
( , ) ?( ,0)
total
total
E RE R
sin( sin )2 2( , )sintotal
akAE RR k
0
0
cos( sin ) cos2 2 2
cos
a ak kA
R k
22
A aR
sin( sin )2 2sin
22
akAR k
A aR
sin( sin )
2sin
2
ak
ak
sin( )2
2
y
y
ak
ak sinc function
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
FT relationship
sin( )( ) 2(0)
2
ytotal
totaly
akEaE k
Sourceo
Diffraction
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
1( ) ( )2
j tf t F e d
f( ) ( )t F
( ) ( ) total yE k A y
/2sin
/2
( , )y a
jkytotal
y a
AE R e dyR
From Signals and Systems
-
1( , ) ( ) y
yjk y
total yy
E R k A y e dyR
sinyk k
Diffraction of A(y) is F.T. of A(y)
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
-
1( , ) ( ) y
yjk y
total yy
E R k A y e dyR
Far-field diffraction
Fraunhoffer Diffraction
Jeseph Ritter von Fraunhoffer
(1787-1826)
German physicist
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
a
a
d.o
a
d
o
A(y)
a a
d
sin( )2
2
y
y
ak
akX
2 2y yd djk jk
e e
sin( )( ) 22cos( )(0) 2
2
y
yy
y
k aE k dk k aE
2cos( )2ydk
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
( , )total x yE k k
- -
~ ( , ) yx
y xjk yjk x
y x
A x y e e dxdy
2 2
- -2 2
yx
b ay x
jk yjk x
b ay x
e e dxdy
2 2
-- -2 2
x x
b ay x
jk y jk x
b ay x
e dy e dx
sin( )
2
2
y
y
bk
bk
sin( )2
2
x
x
ak
ak
( , )(0,0)
total x y
total
E k kE
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
Airy pattern
2
1
0
( ) 2 ( )y y
y
I k J k dI k d
d: diameterJ1: Bessel function of first order
For the first dark ring,
3.83yk d sin3.83
2d
1.22d
is often used for imaging resolution
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
Many imaging systems have circular aperture
Larger d Better resolution
sin 1.22d
Smaller d Large
(But smaller depth of field)
W.-Y. Choi
Lect. 14: Diffraction
Optoelectronics and Photonics (18/2)
Augustin-Jean Fresnel(1788~1827)
French Physicist
- It was assumed R>>(far field)
For R~ (near field, or Fresnel diffraction), FT relation cannot be used.
In reality, observation is made at a flat surface. Consequently, there is
additional phase shifts, requiring more complicated analysis
In our analysis,
- Diffraction is observed at locations having same R