• Occurs when light encounters obstacle.
• Different parts of wavefront interfere with each other.
• Two sorts of diffraction – Fresnel diffraction – Fraunhofer diffraction
Diffraction
16. Fraunhofer Diffraction
( ) ( ) ( )
r ds P jkr
j U P
U 1 exp cos
01 01 1
0 θ
λ
∫∫
∑
=
01
cos r
= z θ
( ) (
ξ η) ( )
ξ ηλ r d d
U jkr j
y z x
U exp
,
, 2
01
∫∫
01∑
=
( ) (
2)
22
01 = z + x − ξ + y −η
r
Only two assumptions : scalar theory + r01 >>
λ
Huygens-Fresnel Principle
Fresnel Approximation
+ −
+ −
≈
2 2
01 2
1 2
1 1
z y z
z x
r ξ η
( )
λ( )
ξ η[ (
x ξ) (
y η) ]
dξdηz j k z U
j y e
x U
jkz
exp 2 ,
, 2 2
− + −
=
∫ ∫
∞∞
−
( )
( )( )
ξ η ( ) ( ) ξ ηλ
η λ ξ
η π
ξ e d d
e U
z e j y e
x
U z j z x y
j k y
z x j k
jkz
∫ ∫
∞∞
−
+ + −
+
= 2 2 2 , 2 2 2 2
,
( )
( ) ( )
z y f z x f z
j k y
z x j k jkz
Y X
e U
z e j y e
x U
λ λ
η
η
ξλ ξ
/ ,
/ 2
2
2 2 2
2
, )
, (
=
= +
+
= F
Fresnel Diffraction
• This is most general form of diffraction – No restrictions on optical layout
• near-field diffraction
• curved wavefront – Analysis difficult
Fresnel Diffraction Fresnel Diffraction
Screen
Obstruction
A very historically important example of Fresnel Diffraction is Poisson’s Spot .
Fraunhofer Approximation
( ) ( ) ( ξ η ) ξ η
λ η π
λ U ξ j z x y d d
z j e y e
x U
y z x
j k jkz
− +
= ∫ ∫
∞∞
−
+
2
exp ,
,
) 2 (
2 2
( )
2
max 2
2
η
ξ
+〉〉 k z
( )
{ }
f x z f y zy z x
j k jkz
Y X
z U j
e e
λ
η
λλ ξ
/ , /) 2 (
,
2 2
=
= +
= F
Fraunhofer Diffraction Fraunhofer Diffraction
Fraunhofer diffraction
• Specific sort of diffraction – far-field diffraction
– plane wavefront – Simpler maths
Fraunhofer diffraction from a
rectangular aperture. The central lobe of the pattern has half-angular widths
y y
x
x λ / D andθ λ / D
θ = =
The Fraunhofer diffraction
pattern from a circular aperture produces the Airy pattern with the radius of the central disk
subtending an angle θ =1.22λ /D
Fraunhofer Diffraction – examples
Fraunhofer diffraction - Single slit
• Divide slit into two parts
– Extra distance for each pair of points is sinθ 2
b
θ θ
b b
θ
⋅sin s
s
• Destructive interference when
• dark fringe for
• Dividing slit into 2, 4, 6, 8, ...
– dark fringes at
b b
2 2 sin sin θ θ = = λ λ 2 2
sin sin θ θ = = λ λ b b
sin sin θ θ = = 2 2 λ λ , , 3 3 λ λ , , 4 4 λ λ , ,
b b b b b b K K
Analysis
s b b
s s sin sin θ θ
θ θ
For Fraunhofer, screen is at infinity(or, at focal plane), so all rays have the same angle.
P
f
Fraunhofer Diffraction
• Achieved by using parallel light
– produced by lens.
• Lens
– Fourier transform of aperture on screen
screen
Focal length
aperture
θ
Analysis
• Displacement at point P, dE
P– from wave passing through small length ds is proportional to
ω θ
) sin
( = + ⋅
= e − , r r s r
dEp dEo i kr t o
• From Hoygens principle, we assume that
waves spherical
: r
E 1
∝
Analysis
( ) ( )
[ ]
) ( sin sin
sin sin
sin 1 sin
sin :
2 2
2
2 1
2 / sin 2
/ sin 2
2 sin
) (
) 2 (
2
sin
) sin
(
β β β
θ β β
β
θ θ
θ
θ θ θ
ω θ ω
ω θ
c I
I I
kb
r , b E E
e ik e
r E ik
e r
E E
e E e
ds r e
E E
r s
since
, r e
ds dE E
length unit
per amplitude E
, ds E dE
o o
o R L
ikb ikb
o L b
b iks
o R L
t kr R i
t kr b i
b
iks o
p L
t o ks
kr i o
p L
L L
o
o o
o
≡
=
∴
≡
=
−
=
=
≡
=
<<
=
=
−
−
−
−
−
− +
∫
Analysis
( )
(
m)
b if 1ely, Approximat
, 3
, 2
, 1.43
graphic by
solved
tan
d d
, when I
I
f ( y
b m f y
/ 2 k
b
m
2, 1,
m with m
kb
when, c
I I
21
1 2
1
max 21
o
>>
≅ +
=
=
=
→
=
− =
=
=
⊕
≅
=
=
=
±
±
=
=
=
=
=
⊕
β θ
λ
π β
π β
π β
β β
β
β β
β β
β β
λ θ
λ π θ
λ
π θ
β
β
sin
47 . 46
.
sin 0 cos
sin
) sin
sin
, sin
0 )
( sin
2 2
L L
0 0
0.10.1 0.20.2 0.30.3 0.40.4 0.50.5 0.60.6 0.70.7 0.80.8 0.90.9 11
θ β = kb2 sin
π 2π 3π
−3π −2π −π 0
I
oI /
Circular aperture
• For a circular aperture
– Diffraction pattern given by
– J
1( )is a first order Bessel function – D is diameter of aperture
( ( ) )
2211
θ θ θ θ
∝ ∝
sin sin sin sin
½ ½ kD kD
½ ½ kD kD J J
I I
Circular Aperture
Circular aperture
Single slit (sinc ftn)
Sin θ
0 λ/D 2λ/D 3λ/D
−λ/D
−2λ/D
−3λ/D
Intensity
Airy Disc
• Similar pattern to single slit – circularly symmetrical
• First zero point when ½kD sinθ = 3.83 – sinθ = 1.22λ/D
– Central spot called “Airy Disc”
• Every optical instrument
• microscope, telescope, etc.
– Each point on object
• Produces Airy disc in image.
Resolution
• Ability to discern fine details of object
– Lord Rayleigh in 1896
• resolution is function of the Airy disc.
– Rayleigh: Limit of resolution
• Two light sources must be separated by at least the diameter of first dark band.
• Called Rayleigh Criterion
Resolution
∆ϕ
Rayleigh Limit
∆ϕ
minAngular limit of resolution = ∆ϕ
min=1.22λ/D
∆l
minLimit of resolution = ∆l
min=1.22f λ/ D
f = focal length
Rayleigh Criterion
Light distribution of a cross section of respective airy disc.
Rayleigh Limit
Resolving Power
• Defined as
– 1/∆ϕ
min• or
– 1/ l
min• To increasing resolving power
– increase diameter of lens
– decrease wavelength
Double-slit diffraction
Double-slit diffraction
( ) ( ) ( ) ( )
[ ]
[ ]
β α α β
β ε β
ε
β α β β
θ α
θ β
θ
β β
α β
β α
θ θ
θ θ
θ θ
2 2
2 2 2
2
2 1 2
1
2 / sin ) ( 2
/ sin ) ( 2
/ sin ) (
2 / sin ) (
2 ) (
2 ) ( 2 sin
) (
2 ) (
sin
sin cos 4
sin cos 2
2 2
sin cos ) 2
( )
2 (
sin ,
sin sin
1
=
=
=
=
− +
−
=
≡
≡
− +
−
=
+
=
−
−
−
− +
−
− +
−
+
−
−
− +
− ∫
∫
o o
L o
o R
o i L
i i i
i i o
R L
b a ik b
a ik b
a ik b
a ik o
L
b a
b a
iks o
b L a
b a
iks o
R L
r I b E c
irradiance
; c E
I
r b e E
e e
e e
i e b r
E E
ka
kb
e e
e ik e
r E
ds r e
ds E r e
E E
a b
b (a-b)/2 (a+b)/2
Double-slit diffraction
m b a p
: (2)
&
(1) satisfying or
orders, missing
for Condition
2, 1, 0, m , θ b
m : minima n
diffractio
(2) - - - - - - - - - b
b
: term envelope n
diffractio The
2, 1, 0, p , θ a
p : maxima ce
interferen
(1) - - - - - - - - - a
: term ce interferen The
r b E I c
I
I
o L o o
o
=
±
±
=
=
±
±
=
=
=
=
=
L L
sin sin sin sin
sin cos sin
cos , 2 sin cos
4
2 2
2
2 2
2
λ λ
θ π λ
θ π
λ
λ θ α π
α ε β
β
Multiple-slit diffraction
[[ ]]
[[ ]]
{ }
2 2 2 /
1
2 ) ) 1 2 (
2 ) ) 1 2 (
2 ) ) 1 2 (
2 ) ) 1 2 (
sin sin
sin sin
sin
=
+
= ∑ ∫ ∫
=
+
−
−
−
−
−
+
−
−
−
α α β
β
θ θ
I N I
ds e
ds r e
E E
o
N
j
b a j
b a j
b a j
b a j
iks iks
o R L
integers other
all p
for occur minima
2N, N,
0, p for occur maxima
principal
2, 1, 0, p , θ N a
p or N , : p
minima
N N N
: N magnitude
2, 1, 0, m , θ a
m or , m :
maxima principal
(1) - - - - - - - - - N
: term ce interferen The
lim
limm m
2
=
±
±
=
⇒
±
±
=
=
=
±
=
=
±
±
=
=
→
→
→
L L
L
sin
cos cos sin
sin
sin sin
sin
λ α π
α α α
α
λ π
α α
α
π α π
α
