is a pre for diffraction

1. Chapter 8 Diffraction
(1) Fraunhofer diffraction
• A coherent line source
• Diffraction by single slit, double slit and many slits
• Rectangular aperature
• Circular aperature
• Diffraction grating
(2) Fresnel diffraction
• Circular aperature
• Circular obstacles
• Rectangular aperature
• Diffraction by a slit, narrow obstacle
(3) Kirchhoff’s scalar diffraction theory

2. Fraunhofer Diffraction

3. Huygens-Fresnel Principle
Every unobstructed point of a wavefront, at a given
instant, serves as a source of spherical secondary
wavelets (with the same frequency as that of the
primary wave). The amplitude of the optical field at any
point beyond is the superposition of all these wavelets
(considering their amplitudes and relative phases).

4. Fraunhofer & Fresnel Diffraction
(Far-field diffraction)
Fraunhofer diffraction
(Near-field diffraction)
Fresnel diffraction
Aperature size
Source (observation) distance
Wavelength
 Phase

5. Aspect of Phase for Fraunhofer Diffraction
R
R
a
S
A
B
phase (SA) – phase (SB) < /4
 R > a2 / 
nm
R
m
a
m
R
mm
a
cm
R
m
a
nm
60
2
.
0
1
1
1
100
633














6. A Coherent Line Source
2
sin
,
sin
2
1
)
(
2
sin
)
2
sin
sin(
sin
,
when
2
2
)
(
2
/
2
/
)
(
)
(
)
(
0












kD
R
D
E
I
e
kD
kD
R
D
E
E
y
R
r
D
R
dy
r
e
E
E
y
e
r
E
E
e
r
E
E
r
t
kR
i
r
D
D
t
kr
i
r
i
t
kr
i
i
r
t
kr
i i

































7. Intensity Profile of a Coherent Line Source
2
2
2
sin
)
0
(
)
(
2
sin
,
sin
2
1
)
(

































I
I
kD
R
D
E
I L
(1) When D >> ,  is large  I (  0)  0
A long coherent line source (D >> ) can be treated as a
single-point emitter radiating (a circular wave)
predominantly in the forward,  = 0, direction.
(2) When D << ,  is small  I()  I(0)
A point source emitting spherical waves.

8. Fraunhofer Diffraction by a Single Slit
y
z
b
2
sin
,
sin
)
0
(
)
(
2





kb
I
I 









,...
46
.
2
,
43
.
1
tan
(max)
sin
0
sin
(min)
0


















m
b
d
dI

9. Fraunhofer Diffraction Pattern of a Single Slit
Diffraction is a kind of
expression of
Uncertainty Principle.

10. a
b
2
sin
),
2
sin
(
cos
sin
4
)
2
sin
cos(
2
sin
)
2
sin
sin(
2
2
2
0
)
2
sin
(
2
/
2
/
)
(
2
/
2
/
)
(












kb
ka
I
I
e
ka
kb
kb
R
bE
dz
e
R
E
dz
e
R
E
E
ka
t
kR
i
L
b
a
b
a
t
kr
i
L
b
b
t
kr
i
L


































Fraunhofer Diffraction
by Double Slits

11. Fraunhofer Diffraction Pattern of Double Slits
)
2
sin
(
cos
4
lim 2
0
0

ka
I
I
kb


(Young’s experiment)
(max) kasin = 2m

12. Fraunhofer Diffraction by Many Slits

13. 2
2
2
2
sin
)
1
(
)
(
1
0
)
sin
(
2
/
2
2
/
2
)
(
2
/
2
/
)
(
2
/
2
/
)
(
sin
sin
sin
2
1
)
(
2
sin
,
sin
sin
sin
2
sin
,
sin
sin
...
































































































N
R
bE
I
ka
e
e
N
R
bE
E
E
kb
e
R
bE
E
z
R
r
dz
e
R
E
dz
e
R
E
dz
e
R
E
E
L
a
N
k
i
t
kR
i
L
N
j
j
kja
t
kR
i
L
j
b
a
b
a
t
kr
i
L
b
a
b
a
t
kr
i
L
b
b
t
kr
i
L
Fraunhofer Diffraction by Many Slits
(Principal maxima) 





m
a
m
N
N
m 



 sin
sin
sin

14. -4 -3 -2 -1 0 1 2 3 4
x 1 0
-3
0
0 .2
0 .4
0 .6
0 .8
1
1 .2
1 .4
1 .6
x 1 0
-7
a=4b, N=4
Comparison between
Experimental and Theoretical Results
Principal maxima Subsidiary maxima



)
2
1
(
1
sin



m
N
N




m
N
N



sin
sin

15. Fraunhofer Diffraction by Rectangular Aperture
Assumption: coherent secondary point sources within the aperture.
Source point (0, y, z), Observation point (X, Y, Z)
2
2
)
(
sin
sin
)
0
(
)
,
(
2
,
2
sin
sin






















































I
Z
Y
I
R
kbY
R
kaZ
R
e
AE
E
t
kR
i
A
]
/
)
(
1
[ 2
)
(
R
zZ
yY
R
r
dS
e
r
E
dE t
kr
i
A









 

16. Fraunhofer Pattern of a Square Aperture
R
kbY
R
kaZ
I
Z
Y
I
2
,
2
sin
sin
)
0
(
)
,
(
2
2






























17. Fraunhofer Diffraction by Circular Aperture















aperture
R
zZ
yY
ik
t
kR
i
A
t
kr
i
A
dS
e
R
e
E
E
R
zZ
yY
R
r
dS
e
r
E
dE
/
)
(
)
(
2
)
(
]
/
)
(
1
[
,


 
 










a
R
q
k
i
t
kR
i
A
d
d
e
R
e
E
E
q
Y
q
Z
y
z
0
2
0
)
cos(
)
/
(
)
(
sin
,
cos
sin
,
cos














18. Bessel Function of the First Kind
)
(
)
(
1
)
(
)]
(
[
2
1
)
(
2
)
(
1
0
0
1
2
0
cos
0
2
0
)
cos
(
u
uJ
u
d
u
J
u
m
u
J
u
u
J
u
du
d
dv
e
u
J
dv
e
i
u
J
u
m
m
m
m
v
iu
v
u
mv
i
m
m





















19.  
 
 
2
1
1
0
2
1
2
2
2
1
2
)
(
0
0
)
(
sin
)
sin
(
2
)
0
(
)
(
2
1
)
(
lim
/
/
2
/
2
)
/
(
2










































ka
ka
J
I
I
u
u
J
R
kaq
R
kaq
J
R
A
E
I
R
kaq
J
kaq
R
a
R
e
E
d
R
kq
J
R
e
E
E
u
A
t
kR
i
A
a
t
kR
i
A
Diffracted Irradiance of a Circular Aperture

20. Airy Disk & Airy Rings
 
 
D
f
q
a
R
q
R
kaq
u
J
R
kaq
R
kaq
J
R
A
E
I
st
A


22
.
1
2
22
.
1
83
.
3
:
ring
dark
1
0
)
83
.
3
(
/
/
2
1
1
1
2
1
2
2
2
















84%

21. Resolution of Imaging Systems
D
f
l

22
.
1
)
( min 

D

 22
.
1
)
( min 

Just resolved when the center of one Airy disk falls on the
first minimum of the Airy pattern of the other.
Angular limit of resolution
Limit of resolution

22. An Interesting Experiment on Image Resolution

23. 



2
/
2
/
sin
)
(
)
sin
(
b
b
iky
t
kR
i
L
dy
e
e
R
E
k
E 


Revision of Fraunhofer Diffraction
Fraunhofer Diffraction functions as “Fourier Transform” from
geometry (space) domain to wave-vector (wave-number)
domain.
 Uncertainty principle
D
R
q

22
.
1


24. The Diffraction Grating


 m
a i
m 
 )
sin
(sin
(Grating equation)
Reflection phase grating
Transmission phase grating

25. Grating Spectroscopy (I)






m
ka
m
a
m
i
i
m





2
sin
0
when
)
sin
(sin
- 4 - 3 - 2 - 1 0 1 2 3 4
x 1 0
- 3
0
0 . 2
0 . 4
0 . 6
0 . 8
1
1 . 2
1 . 4
1 . 6
x 1 0
- 7
(Angular width of a line)
m
Na 


cos
2


(Angular dispersion)
m
a
m
d
d
D



cos




)
2
1
(
2
sin




m
ka
N m
(Max)
(Max)

26. Grating Spectroscopy (II)
(resolving power)











)
sin
(sin
)
(
cos
)
(
)
(
dispersion
angular
cos
)
(
min
min
min
min
i
m
m
m
Na
mN
a
m
Na












)
sin
(sin
)
(
)
(
)
1
(
)
sin
(sin
2
i
m
fsr
i
m
a
m
m
m
a




















finesse
m

 (F-P spectroscopy)
(free spectral range)

27. Resolving power & free spectral range of F-P Cavity
finesse
d
n
m
finesse
d
n
finesse
fsr
f
fsr
f












min
0
0
2
0
0
0
min
0
0
)
(
)
(
2
)
(
2
)
(







Chromatic
resolving power
Free spectral
range

28. Two- and Three-Dimensional Gratings
X-ray difraction pattern for SiO2
Why don’t use visible lights
for diffraction of solid
crystals?

29. Fresnel Diffraction

30. )
cos
1
(
2
1
)
( 
 

K
Obliquity factor
Inclination factor
Fresnel Zones
]
[
0 t
k
i
e
E
E 




Half-period zones

31. Propagation of a Spherical Wavefront
)
sin
(
2
)
cos
1
(
2
1
]
)
(
[








d
dS
K
dS
e
r
E
K
dE t
r
k
i
A



 

)]
(
exp[
)
(
2
)
1
(
2
0
0
1
]
)
(
[
0
1
t
kr
k
i
r
E
K
i
E
dr
e
r
E
K
E
A
l
l
l
r
r
t
r
k
i
A
l
l
l
l






 













0
QE
EA 

32. Optical Disturbance from Fresnel Zones (I)
If m is odd,
2
2
...
2
2
2
...
)
2
2
(
2
2
...
)
2
2
(
)
2
2
(
2
...
1
3
2
1
1
1
4
3
2
2
1
5
4
3
3
2
1
1
3
2
1
m
m
m
m
m
m
m
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E


































If m is even,

33. Optical Disturbance from Fresnel Zones (II)
The last contributing zone occurred at  = 90o
i.e. K() = 0 for /2  | |  
and Km(/2) = 0
]
)
(
[
0
0
2
/
]
)
(
[
0
1
1
0
0
)
(
)
(
2
2
t
r
k
i
i
t
r
k
i
A
m
e
r
E
E
e
e
r
E
K
E
E
E
E


















(Secondary wavelets)
(Primary wave, SP)

34. The Vibration Curve
• The resultant phasor changes in phase by  while the aperture
size increases by one zone.
• K() decreases rapidly only for the first few zones.
Bright fringes
Dark fringes
E1
E2

35. Circular Apertures
Case I: m is even, E  0
Case II: m is odd, E  |E1|

36. Diffraction Patterns for Circular Apertures
of Increasing Size
The intensity at the center
The intensity at the off-axis position

37. Method for Evaluating the Intensity
at Off-Axis Positions

38. Radius of Aperture & Number of Zones







0
2
0
2
0
0
)
(
]
4
)
1
2
(
[
r
R
r
A
R
m
r
r
Am






Number of zones 
>>
Examples:
 = 1 m, r0 = 1 m,  = 500 nm and R = 1 cm
 400 zones
If  and r0 are very large, such that only a small
fraction of the first zone appears in the aperture,
Fraunhofer diffraction occurs.

39. Circular Obstacles
As P moves close to the disk,  increases, K  E
2
...
1
3
2
1











l
m
l
l
l
E
E
E
E
E
E
E

40. Fresnel Zone Plate
2
0
0
0
0
1
1
2
/
)
(
)
(
m
m
m
R
m
r
m
r
r




















For plane wave, 0
 Rm
2 = mr0

41. Near-Field Criterion from aspect of Fresnel Zone Plate
If plane-wave incident, derive from relation between Rm & r0
 Rm
2 = mr0
 D2 > R

42. Rectangular Apertures (I)


 
















2
1
2
2
1
2
0
0
2
1
2
1
2
/
2
/
0
0
]
)
(
[
0
)
(
0
0
0
0
0
0
0
2
2
0
0
]
)
(
[
0
)
(
2
2
)
(
)
(
v
v
v
i
u
u
u
i
t
r
k
i
y
y
z
z
r
ik
t
i
t
r
k
i
dv
e
du
e
r
e
E
E
dydz
e
r
e
E
E
r
r
z
y
r
r
dS
e
r
E
K
dE

















dS
e
r
E
K
dE t
r
k
i
A ]
)
(
[ 
 



43. Rectangular Apertures (II)
   
 
   
 
2
1
2
2
1
2
2
1
2
2
1
2
0
0
2
/
0
2
0
2
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
4
)]
(
)
(
[
)]
(
)
(
[
2
)
(
)
(
)
2
/
sin(
)
(
)
2
/
cos(
)
(
2
1
2
1
2
v
B
v
B
v
A
v
A
u
B
u
B
u
A
u
A
I
I
v
iB
v
A
u
iB
u
A
E
E
w
iB
w
A
w
d
e
w
d
w
w
B
w
d
w
w
A
v
v
u
u
u
w
w
i
w
w




























Example:
u1  -, u2  
v1  -, v2  
 E2 = I0

44. Diffraction Patterns for Increasing Square Apertures

45. The Cornu Spiral
2
2
2
2
0
2
/
)
(
)
(
2
dw
dB
dA
dl
w
iB
w
A
w
d
e
w
w
i









2
1
2
1
and
2
1
2
1
i
B
i
B 



 

• A larger size of aperture corresponds to a larger arc length w.
• For checking the E-field at off-axis positions, move two end
points with constant w.

46. Fresnel Diffraction by a Slit
   
 
   
 
2
0
2
1
2
2
1
2
2
2
0
2
1
]
[
2
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
4
v
v
iB
A
I
v
B
v
B
v
A
v
A
B
B
A
A
I
I















47. Fresnel Diffraction Pattern of a Slit

48. The Semi-Infinite Opaque Screen
   
 
   
 








































2
1
2
1
0
2
1
2
1
2
2
0
)
(
2
1
)
(
2
1
2
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
4
v
B
v
A
I
v
B
B
v
A
A
B
B
A
A
I
I

49. Fresnel Diffraction Pattern of
a Semi-Infinite Opaque Screen

50. Fresnel Diffraction by a Narrow Obstacle
Total E-field is the vector sum.
Total E-field can never be zero.

51. Babinet’s Principle
When the transparent regions on one diffraction screen exactly
correspond to the opaque regions of the other and vice versa,
these two screen are complementary.

52. Fraunhofer Diffraction Patterns of Complementary Screens

53. Kirchhoff’s Scalar Diffraction Theory (I)
dS
n
r
n
r
e
iE
E
e
E
E
dS
r
e
E
dS
E
r
e
E
E
k
E
S
r
ik
P
ik
S
ikr
S
ikr
P























]
2
)
ˆ
,
ˆ
cos(
)
ˆ
,
ˆ
cos(
[
)
(
)
(
4
1
0
)
(
0
0
2
2








Fresnel-Kirchhoff diffraction formula

54. 2
cos
1
)
(
]
2
)
ˆ
,
ˆ
cos(
)
ˆ
,
ˆ
cos(
[
)
(
0










 

K
dS
n
r
n
r
e
iE
E
S
r
ik
P
Kirchhoff’s Scalar Diffraction Theory (II)