Chapter 16. Fraunhofer Diffraction
Chapter 16. Fraunhofer Diffraction
Fraunhofer Approximation Fraunhofer Approximation
( ) (
2)
22
01
= z + x − ξ + y − η
r
( ) ( ξ η ) ( ) ξ η
λ r d d
U jkr j
y z x
U exp
,
,
201
∫∫
01∑
=
Huygens-Fresnel Principle
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
) 1 (
) 2 (
1
) 2 (
) 1 1 (
) 2 (
1
2 1 2
1 1
2 2
2 2
2 2
2 2
01
η ξ
η ξ
η ξ
η ξ
y z x
y z x
z
y z z x
y z x
z
z y z
z x r
+
− +
+
≈
+ +
+
− +
+
=
⎥ ⎥
⎦
⎤
⎢ ⎢
⎣
⎡ ⎟
⎠
⎜ ⎞
⎝ + ⎛ −
⎟ ⎠
⎜ ⎞
⎝ + ⎛ −
≈
FT
Fraunhofer Diffraction Fraunhofer Diffraction
Fraunhofer diffraction
• Specific sort of diffraction – far-field diffraction
– plane wavefront
– Simpler maths
(참고) 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 ) ] d ξ d η
z j k z U
j y e
x U
jkz
exp 2 ,
,
2 2⎭ ⎬
⎫
⎩ ⎨
⎧ − + −
= ∫ ∫
∞∞
−
( ) ( ) ( ) ξ η ( ) ( ) ξ η
λ
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 22
,
( )
( ) ( )
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 ξ
λ ξ
/ ,
/ 2
2
2 2 2
2
, )
, (
=
= +
+
⎭ ⎬
⎫
⎩ ⎨
= F ⎧
( ) ( 2 ) 2
2
01 = z + x − ξ + y − η
r
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
16-1. Fraunhofer Diffraction from a Single Slit 16-1. Fraunhofer Diffraction from a Single Slit
• Consider the geometry shown below. Assume that the slit is very long in
the direction perpendicular to the page so that we can neglect diffraction
effects in the perpendicular direction.
Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit
( )
( )
0
0
0 0
exp
0.
P
P
The contribution to the electric field amplitude at point P due to the wavelet emanating from the element ds in the slit is given by
dE dE i kr t
r
Let r r for the source element ds at s Then for any element
dE dE
r
⎛ ⎞ ω
=⎜⎝ ⎟⎠ ⎡⎣ − ⎤⎦
= =
= ⎜⎛⎝ + Δ
{ (
0) }
0
exp
, .
, , .
sin
L L
i k r t
We can neglect the path difference in the amplitude term but not in the phase term
We let dE E ds where E is the electric field amplitude assumed uniform over the width of the slit The path difference s
ω
θ
⎞ ⎡ + Δ − ⎤
⎟ ⎣ ⎦
⎜ ⎟⎠
Δ
=
Δ =
( )
{ } ( ) ( )
( ) ( )
/ 2
0 0 / 2
0 0
/ 2
0
0 / 2
.
exp sin exp exp sin
exp sin
exp sin
L L b
P P b
b P L
b
Substituting we obtain
E ds E
dE i k r s t E i kr t i k s ds
r r
i k s
Integrating we obtain E E i kr t
r i k
θ ω ω θ
ω θ
θ
−
−
⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎡⎣ + − ⎤⎦ =⎜ ⎟ ⎡⎣ − ⎤⎦
⎝ ⎠ ⎝ ⎠
⎡ ⎤
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎢ ⎥
⎝ ⎠ ⎣ ⎦
∫
Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit
( ) ( ) ( )
( ) ( ) ( )
( )
0 0
0 0
0 0
exp exp
exp sin
1 sin 2
exp exp exp
2
exp 2
P L
P L
L
Evaluating with the integral limits we obtain
i i
E E i kr t
r i k
where
k b
Rearranging we obtain
E b
E i kr t i i
r i
E b
i kr t
r i
β β
ω θ
β θ
ω β β
β
ω β
− −
⎡ ⎤
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎢ ⎥
⎝ ⎠ ⎣ ⎦
≡
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎡⎣ − − ⎤⎦
⎝ ⎠
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦
⎝ ⎠
( ) (
0)
0
2 2 2
* 2
0 0 2 0 2 0
0
2 sin exp sin
1 1 sin sin
2 2 sinc
L
P P L
E b
i i kr t
r
The irradiance at point P is given by
I = c E E c E b I I
r
β ω β
β
β β
ε ε β
β β
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦
⎝ ⎠
⎛ ⎞
= ⎜ ⎟ = =
⎝ ⎠
sin β
0sin
2(
21sin θ )
2
0
c I c kb
I
I = =
Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit
2 2 2
* 2
0 0 2 0 2 0
0
0 0
1 1 sin sin
2 2 sinc
sinc 1 0, lim sinc lim sin 1
sin 0, 1 sin 1, 2,
2
L
P P
The irradiance at point P is given by
I = c E E c E b I I
r
The function is for
The zeroes of irradiance occur when or when k b m m
β β
β β
ε ε β
β β
β β β
β
β β θ π
→ →
⎛ ⎞
= ⎜ ⎟ = =
⎝ ⎠
= = =
= = = = ± ± K
( θ )
β sin sin
sin
2 0 2 120
c I c kb
I
I = =
Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit
,
sin , 2 / ,
1 2 2
In terms of the length y on the observation screen y f and in terms of wavelength k we can write
y b y
b f f
Zeroes in the irradiance pattern will occur when
b y m f
m y
f b
The maximum in the irradiance pattern is at β = 0
θ λ π
π π
β λ λ
π π λ
λ
≅ =
= =
= =
2 2
sin cos sin cos sin
0
sin tan cos
. Secondary maxima are found from
d d
β β β β β β
β β β β β
β β β
β
⎛ ⎞ −
= − = =
⎜ ⎟
⎝ ⎠
⇒ = =
1.43π
2.46π
3.47π
0
θ β =
21kb sin
β
2 0
sin c I
I =
Fraunhofer Diffraction from a Single Slit Fraunhofer Diffraction from a Single Slit
Note: x- and y-axes
switched in book, Figs. 16- 5a (here) and Fig. 16-1 do not match.
β
2 0
sin c I
I =
16-2. Beam Spreading 16-2. Beam Spreading
( )
sin
min
1 2
The angular width of the central maximum is defined as the angular
separation Δθ between the first minima on either side of the central maximum,
y f
The first ima in the irradiance pattern will occur when
m f f
y Δθ
b b b
The
θ θ
λ λ λ
= ≅
= = ± ⇒ =
2
width W of the diffraction pattern thus increases linearly with distance from the slit, in the regions far from the slit where Fraunhofer diffraction applies
W = L Δθ L
b
= λ
L
W
16-3. Rectangular Apertures 16-3. Rectangular Apertures
When the length a and width b of the rectangular aperture are comparable, a diffraction pattern is observed in both the x - and y - dimensions, governed in each dimension by the formula we have already developed. The irradiance
pat
(
2)(
2)
0
sinc sinc
, 1 sin 2
tern is
I I
where k a
Zeroes in the irradiance pattern are observed when
m f m f
y or x
b a
α β
α θ
λ λ
=
=
= =
y
Circular Apertures Circular Apertures
xds dA =
∫∫
=
Area A isk
p e dA
r
E E sin θ
0
2 2
2
2 ⎟
⎠
⎜ ⎞
⎝ + ⎛
= x
s
R x = 2 R
2− s
2ds s
R r e
E E R
R A isk p
2 2
sin 0
2 −
= ∫ −
θ
θ γ sin
,
/ R kR s
v = =
{ }
⎭ ⎬
⎫
⎩ ⎨
= ⎧
−
= ∫ − γ
γ
γ 2 π ( )
2 1 1
0 2 1 2
0 1
2 J
r R dv E
v r e
R
E p E A i v A
(the first order Bessel function of the first kind)
Bessel Functions Bessel Functions
832 . 3 2 sin
sin = 1 =
= θ θ
γ kR kD (first zero)
θ γ =
12kD sin
( ) ( )
( )
1 2 1 2
1 2
2 sin
( ) 0
sin J k D
I I
k D θ θ
θ
⎡ ⎤
= ⎢ ⎥
⎢ ⎥
⎣ ⎦
⎭ ⎬
⎫
⎩ ⎨
= ⎧
γ γ π ( )
2
10
2
J
r R E
pE
A0) at
(or, 0 when
2 1 )
1
( → =
⎭ ⎬
⎫
⎩ ⎨
⎧ → γ θ
γ γ J
1 1
min
2 2
sin 3.83 2
2 First minimum in the Airy pattern is at
k D θ k D θ π D θ
λ
⎛ ⎞⎛ ⎞
≅ = = ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
Fraunhofer Diffraction from Circular Apertures:
The Airy Pattern
Fraunhofer Diffraction from Circular Apertures:
The Airy Pattern
) 0 ( I/ I
D θ λ
θ 1 . 22
2
min = Δ
1=
Airy patterns Airy patterns
Airy Disc
Slit and Circular Apertures Slit and Circular Apertures
Circular aperture
Single slit (sinc ftn)
Sin θ
0 λ/D 2λ/D 3λ/D
−λ/D
−2λ/D
−3λ/D
Intensity
16-4. Resolution 16-4. 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
Rayleigh Criterion Rayleigh Criterion
Rayleigh Limit
Rayleigh Limit Rayleigh Limit
Resolution limit of a lens:
(f = focal length) NA NA
D x f
λ λ
λ
61 . 0 22 2
. 1
22 .
min
1
⎟ =
⎠
⎜ ⎞
⎝
≈ ⎛
⎟ ⎠
⎜ ⎞
⎝
= ⎛
Fraunhofer Diffraction from a Double Slit Fraunhofer Diffraction from a Double Slit
Now for the double slit we can imagine that we place
an obstruction in the middle of the single slit. Then all that we have to do to calculate the field from the double slit is to change the limits of
( )
(( ))( )
( )
(( ))( )
( ) ( )
( )
( )
/ 2
0 / 2
0
/ 2
0 / 2
0
/ 2
0
0 / 2
exp exp sin
exp exp sin
exp sin exp
exp sin
L a b
P a b
L a b
a b
a b P L
a b
integration.
E E i kr t i k s ds
r
E i kr t i k s ds
r
Integrating we obtain
i k s i
E E i kr t
r i k
ω θ
ω θ
ω θ
θ
+
−
− −
− +
+
−
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦ +
⎝ ⎠
⎛ ⎞
−
⎡ ⎤
⎜ ⎟ ⎣ ⎦
⎝ ⎠
⎛ ⎞ ⎡ ⎤
=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎢ ⎥ +
⎣ ⎦
⎝ ⎠
∫
∫
( )
( )
( )
( ) ( ) ( )
( ) ( )
( ) ( )
/ 2
/ 2
0
0
0
0
sin sin
exp sin sin
exp exp
sin 2 2
sin sin
exp exp
2 2
exp exp ex
2
a b
a b L
P L
k s i k
i kr t i k a b i k a b
E
r i k
i k a b i k a b
b i kr t
E E i
r i
θ θ
ω θ θ
θ
θ θ
ω α
β
− −
− +
⎧ ⎡ ⎤ ⎫
⎪ ⎪
⎨ ⎢ ⎥ ⎬
⎣ ⎦
⎪ ⎪
⎩ ⎭
− ⎧
⎡ ⎤ + −
⎛ ⎞ ⎣ ⎦ ⎪ ⎡ ⎤ ⎡ ⎤
= ⎜ ⎟ ⎨ ⎢ ⎥− ⎢ ⎥
⎪ ⎣ ⎦ ⎣ ⎦
⎝ ⎠ ⎩
− − − + ⎫
⎡ ⎤ ⎡ ⎤⎪
+ ⎢ ⎥− ⎢ ⎥⎬
⎣ ⎦ ⎣ ⎦⎭⎪
⎡ − ⎤
⎛ ⎞ ⎣ ⎦
= ⎜ ⎟
⎝ ⎠
{
p( )
exp( )
exp( )
exp( )
exp( ) }
sin sin
i i i i i
where k a and k b
β β α β β
α θ β θ
− − + − − −
⎡ ⎤ ⎡ ⎤
⎣ ⎦ ⎣ ⎦
= =
16-5. Fraunhofer Diffraction from a Double Slit 16-5. Fraunhofer Diffraction from a Double Slit
( ) ( )
( ) ( )
( ) ( )( )
( )
0 0
2 2
* 2
0 0 2 0
0
exp exp 2 cos
exp exp 2 sin
exp 2 cos 2 sin
2
1 1 4 sin
4 cos 4 c
2 2 4
L P
L P P
But we know that
i i
i i i
Substituting we obtain
b i kr t
E E i
r i
The irradiance at point P is given by
I = c E E c E b I
r
α α α
β β β
ω α β
β
ε ε α β
β
+ − =
− − =
−
⎡ ⎤
⎛ ⎞ ⎣ ⎦
= ⎜ ⎟
⎝ ⎠
⎛ ⎞ ⎛ ⎞
= ⎜ ⎟ ⎜ ⎟ =
⎝ ⎠
⎝ ⎠
2 2
2
2
0 0
0
os sin
1 2
E b
Lwhere I c
r
α β β ε
⎛ ⎞
⎜ ⎟
⎝ ⎠
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
Fraunhofer Diffraction from a Double Slit Fraunhofer Diffraction from a Double Slit
The irradiance at point P from a double slit is given by the product of the
diffraction pattern from single slit and the interference pattern
from a double slit.
2 2
0 2
4 cos sin
I I α β
β
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
Fraunhofer Diffraction from a Double Slit Fraunhofer Diffraction from a Double Slit
Single Slit
Double Slit
16-6. Fraunhofer Diffraction from Many Slits (Grating)
16-6. Fraunhofer Diffraction from Many Slits (Grating)
Now for the multiple slits we just need to again change the limits of integration. For N even slits with width b evenly spaced a distance a apart, we can place the origin of the coordinate system at t
(
0)
1/ 2{
((22 11)) / 2/ 2( )
0
exp exp sin
j N j a b
L
P j a b
j
he center obstruction and label the slits with the index j
(Note that the diagram does not exactly correspond with this).
E E i kr t i k s ds
r ω = ⎡⎣ − + ⎤⎦ θ
− −
⎡ ⎤
⎣ ⎦
=
⎛ ⎞
=⎜ ⎟ ⎡⎣ − ⎤⎦
⎝ ⎠
∑ ∫
( )
( )
( )
}
( ) ( )
( )
( )
( )
( )
( )
2 1 / 2
2 1 / 2
2 1 / 2
/ 2 0
0 1 2 1 / 2
2 1 /
2 1 / 2
exp sin
exp sin
exp sin
exp sin sin
j a b j a b
j a b j N
P L
j j a b
j a b
j a b
i k s ds
Integrating we obtain
i k s
E E i kr t
r i k
i k s i k
θ
ω θ
θ θ
θ
− − +
⎡ ⎤
⎣ ⎦
− − −
⎡ ⎤
⎣ ⎦
− +
⎡ ⎤
⎣ ⎦
=
= ⎡⎣ − − ⎤⎦
− − +
⎡ ⎤
⎣ ⎦
− − −
⎡ ⎤
⎣ ⎦
+
⎧⎡ ⎤
⎛ ⎞ ⎪
=⎜ ⎟ ⎡⎣ − ⎤⎦ ⎨⎢ ⎥
⎝ ⎠ ⎪⎩⎣ ⎦
⎡ ⎤
+ ⎢ ⎥
⎣ ⎦
∫
∑
( ) ( ) ( )
( ) ( )
2
/ 2 0
0 1
exp 2 1 sin 2 1 sin
exp exp
sin 2 2
2 1 sin 2 1 sin
exp exp
2 2
j N L
j
i kr t i k j a b i k j a b
E
r i k
i k j a b i k j a b
ω θ θ
θ
θ θ
=
=
⎫⎪⎬
⎪⎭
⎧ ⎡ ⎤ ⎡ ⎤
− − + − −
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎛ ⎞ ⎣ ⎦ ⎪ ⎣ ⎦ ⎣ ⎦
= ⎜ ⎟ ⎨ ⎢ ⎥− ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎝ ⎠ ⎪⎩ ⎣ ⎦ ⎣ ⎦
⎡− ⎡⎣ − − ⎤⎦ ⎤ ⎡− ⎡⎣ − − ⎤⎦ ⎤⎫⎪
+ ⎢ ⎥− ⎢ ⎥⎬
⎢ ⎥ ⎢ ⎥⎪
⎣ ⎦ ⎣ ⎦⎭
∑
( ) { ( ) ( ) ( ) ( ( ) ) ( ) ( ) }
( )
/ 2 0
0 1
0
0
sin sin
exp exp 2 1 exp exp exp 2 1 exp exp
2
exp
j N L
P
j
L P
Substuting using k a and k b and rearranging we obtain
b i kr t
E E i j i i i j i i
r i
We can rewrite this as
b i kr t
E E
r
α θ β θ
ω α β β α β β
β
ω
=
=
= =
⎡ − ⎤
⎛ ⎞ ⎣ ⎦
= ⎜ ⎟ ⎡ ⎣ − ⎤ ⎡ ⎦ ⎣ − − ⎤ ⎦ + − − ⎡ ⎣ − − ⎤ ⎦
⎝ ⎠
−
⎛ ⎞ ⎡ ⎣
= ⎜ ⎟
⎝ ⎠
∑
( ) { ( ) ( ) }
( ) { ( ) }
( ) { ( ) ( ) ( ) ( ) }
/ 2
1 / 2 0
0 1
/ 2 0
0 1
2 sin exp 2 1 exp 2 1
2
exp sin Re exp 2 1
exp sin Re exp exp 3 exp 5 exp 1
j N
j j N L
j j N L
j
i i j i j
i
E b i kr t i j
r
E b i kr t i i i i N
r The last te
β α α
β
ω β α
β
ω β α α α α
β
=
=
=
=
=
=
⎤⎦ ⎡ ⎣ − ⎤ ⎦ + ⎡ ⎣ − − ⎤ ⎦
⎛ ⎞ ⎛ ⎞
= ⎜ ⎝ ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦ ⎜ ⎝ ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦
⎛ ⎞ ⎛ ⎞
= ⎜ ⎝ ⎟ ⎠ ⎡ ⎣ − ⎤ ⎦ ⎜ ⎝ ⎟ ⎠ + + + + ⎡ ⎣ − ⎤ ⎦
∑
∑
∑ L
( ) ( ) ( ) ( )
{ }
/ 2
1
2
*
0 0
0
Re exp exp 3 exp 5 exp 1 sin
sin
. int
1 1 sin
2 2
j N
j
L
P P P
rm is a geometric series that converges to
i i i i N N
The details of the last step are outlined in the book The irradiance at po P is given by
I c E E c E b
r
α α α α α
α
ε ε β
β
=
=
+ + + + ⎡ ⎣ − ⎤ ⎦ =
⎛ ⎞
= = ⎜ ⎟
⎝ ⎠
∑ L
2 2 2 2
0
sin sin sin
sin
N N
α I β α
α β α
⎛ ⎞ ⎛ ⎜ ⎞ ⎟ = ⎛ ⎞ ⎛ ⎜ ⎞ ⎟
⎜ ⎟ ⎝ ⎠ ⎜ ⎟ ⎝ ⎠
⎝ ⎠ ⎝ ⎠
Fraunhofer Diffraction from Multiple Slits Fraunhofer Diffraction from Multiple Slits
2 2 2 2 2
*
0 0 0
0
1 1 sin sin sin sin
2 2 sin sin
, sin . , ' '
sin
lim
P P P L
m
The irradiance at point P is given by
E b N N
I c E E c I
r
When m the term N is a maximum For this condition from L Hospital s rule
α
β α β α
ε ε
β α β α
α π α
α
→
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
= = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎝ ⎠ ⎝ ⎠
⎝ ⎠
=
( )
( )
sin
sin cos
lim lim
sin sin cos
1 sin sin 0, 1, 2,
2
, .
m m
d N
N d N N N
d d
The principal maxima the irradiance pattern occur for
p p
ka a m m
N N
For large N the principal maxima are bright and well separated This ana
π α π α π
α
α α α
α α α
α
π π
α θ θ π
λ
→ →
= = = ±
= = = = = = ± ± K
,
sin
lysis gives us the grating equation
a θ = mλ
sin 2
⎟⎟⎠
⎜⎜ ⎞
⎝
⎛ β
β 2
sin sin ⎟
⎠
⎜ ⎞
⎝
⎛ α
α N
N2
λ θ m a sin =
m=1
m=2 m=0
2 2
0
sin sin
sin
I I β N α
β α
⎛ ⎞ ⎛ ⎞
= ⎜ ⎝ ⎟ ⎠ ⎜ ⎝ ⎟ ⎠
Diffraction grating equation Diffraction grating equation
λ θ m
a sin =
m=1m=2 m=0