diffraction theory - university of louisville
TRANSCRIPT
Diffraction Theory
1
2
3
ðð1
ðð2
ðžðž ðð, ð¡ð¡ = ðžðž1 ðð, ð¡ð¡
ðžðž ðð, ð¡ð¡
ðžðž ðð, ð¡ð¡ = ðžðž0,1ðð1ðððð ðð ðð1â ðð ð¡ð¡ + ðð1 +
ðžðž0,2
ðð2ðððð ðð ðð2â ðð ð¡ð¡ + ðð2
+ ðžðž2 ðð, ð¡ð¡
ðð3
+ðžðž0,3
ðð3ðððð ðð ðð3 â ðð ð¡ð¡ + ðð3
ðð4
+ ðžðž3 ðð, ð¡ð¡ + ðžðž4 ðð, ð¡ð¡
ðð5
+ðžðž0,4
ðð4ðððð ðð ðð4 âðð ð¡ð¡ + ðð4
+ ðžðž5 ðð, ð¡ð¡
+ðžðž0,5
ðð5ðððð ðð ðð5 â ðð ð¡ð¡ + ðð5
= ï¿œðð
ðžðžðð ðð, ð¡ð¡
+â¯
+ âŠ
= ï¿œðð
ðžðž0,ðð
ðððððððð ðð ðððð âðð ð¡ð¡ + ðððð
4
Huygens-Fresnel Principle
5
ðžðž ðð,ðð, ð¡ð¡ = ï¿œðð
ðžðž0,ðð
ðððððððð ðð ðððð â ðð ð¡ð¡ + ðððð
= ï¿œððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ðð ðŠðŠ, ð§ð§
ðððð ðð ðð ðŠðŠ, ð§ð§ â ðð ð¡ð¡ + ðð ðŠðŠ, ð§ð§ ðððŠðŠ ððð§ð§
ðð ðŠðŠ, ð§ð§ = ð ð 2 + ðð â ðŠðŠ 2 + ðð â ð§ð§ 2
ðð
ð ð
ðð
ðð
ð§ð§
ðŠðŠ
ðð ðŠðŠ, ð§ð§
ï¿œðð
ï¿œððððððððð¡ð¡ðððððð
ðððŠðŠ ððð§ð§
6
ðð ðŠðŠ, ð§ð§ â ð ð 1 +ðð2
2
Fresnel Approximationðð ð ð ðððððððð4
8⪠ðð
= ð ð 1 +ðð2
2âðð4
8+ðð6
16â
5 ðð8
128+ â¯
ðð ðŠðŠ, ð§ð§ = ð ð 2 + ðð â ðŠðŠ 2 + ðð â ð§ð§ 2 = ð ð 1 +ðð â ðŠðŠ 2
ð ð 2+
ðð â ð§ð§ 2
ð ð 2 â¡ ð ð 1 + ðð2
ðð2 â¡ðð â ðŠðŠ 2
ð ð 2+
ðð â ð§ð§ 2
ð ð 2
Fresnel Diffraction
= ð ð +ðð2 + ðð2
2 ð ð â
ðð ðŠðŠ + ðð ð§ð§ð ð
+ðŠðŠ2 + ð§ð§2
2 ð ð
= ð ð 1 +ðð â ðŠðŠ 2
2 ð ð 2+
ðð â ð§ð§ 2
2 ð ð 2
7
ðððððððð ðŠðŠ2 + ð§ð§2
2 ð ð ⪠ðð
ðððððð ðŠðŠ2 + ð§ð§2
ðð ð ð ⪠1
Fraunhofer Approximation
Fraunhofer Diffractionalso known as Far-Field Diffraction
ðð ðŠðŠ, ð§ð§ â ð ð +ðð2 + ðð2
2 ð ð â
ðð ðŠðŠ + ðð ð§ð§ð ð
+ðŠðŠ2 + ð§ð§2
2 ð ð
Fresnel Approximationðð ð ð ðððððððð4
8⪠ðð
8
ðžðž ðð,ðð, ð¡ð¡ = ï¿œððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ðð ðŠðŠ, ð§ð§
ðððð ðð ðð ðŠðŠ, ð§ð§ â ðð ð¡ð¡ + ðð ðŠðŠ, ð§ð§ ðððŠðŠ ððð§ð§
â ï¿œððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ð ð
ðððð ðð ð ð â ðð ðŠðŠ + ðð ð§ð§ð ð âðð ð¡ð¡ + ðð ðŠðŠ, ð§ð§ ðððŠðŠ ððð§ð§
=ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ,ð§ð§ ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
ðð ðŠðŠ, ð§ð§ = ð ð 2 + ðð â ðŠðŠ 2 + ðð â ð§ð§ 2
ð ð 2 â¡ ð ð 2 + ðð2 + ðð2
= ð ð 2 â 2 ðð ðŠðŠ â 2 ðð ð§ð§ + ðŠðŠ2 + ð§ð§2
= ð ð 1 +â2 ðð ðŠðŠ â 2 ðð ð§ð§ + ðŠðŠ2 + ð§ð§2
ð ð 2â ð ð â
ðð ðŠðŠ + ðð ð§ð§ð ð
9
ðžðž ðð,ðð, ð¡ð¡ =ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§ ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
ðð
ð ð
ðð
ðð
ð§ð§
ðŠðŠ
ð ð
Fraunhofer Diffraction
ð ð 2 â¡ ð ð 2 + ðð2 + ðð2
10
Illumination at the Aperture:
In the examples to follow, we will consider a flat wavefront at
normal incidence on the aperture
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§ =ðžðž0
0
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
Inside the aperture
Outside the aperture{
11
Apertures considered here:
1. Single Slit
2. Double Slit
3. Rectangular Aperture
4. Circular Aperture
12
1. Single Slit
ððð ð
ðð
ðð
ð§ð§
ðŠðŠ
ð ð
ð ð â¡ ðð2 + ð ð 2
ðð
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
â ï¿œðð 2
+ ï¿œðð 2
ððâ ðð ðð ððð ð ðŠðŠðððŠðŠ
ðð
ð ð ððð ð ðð =ððð ð
ðŠðŠ
ð§ð§
13
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
1. Single Slit, cont.
ðŒðŒ â¡ ðžðž2
ðŒðŒ ðð,ðð = ðŒðŒ0 ð ð ððð ð ð ð 2ðð ðð ðð
2 ð ð ðŒðŒ0 â¡ðžðž0 2
2 ð ð 2ðð2
ðð = 50 µðð
ðð = 0.6 µððð ð = 1 ðð
ðð ðððð ðð2 ð ð
= ðð ðð
ðð = ±1, ±2, ±3
ð ð â 1 ðð
ðððð = ðððð ð ð ðð
ð ð ððð ð ðððð =ððððð ð
= ðððððð
ðð(ðððð)
ðð1
ï¿œðŒðŒ ðŒðŒ0
zeros at geometrical shadow
ððâ1
with
ðð
ðð
14
Mathematica
15
2. Double Slit
ðð
ðð
ðð
ðŠðŠ
ï¿œðð 2 â ï¿œðð 2
ï¿œðð 2 + ï¿œðð 2
ï¿œâðð2 â ï¿œðð 2
ï¿œâðð2 + ï¿œðð 2
ð§ð§
16
ððð ð
ðð
ðð
ð§ð§
ðŠðŠ
ð ð
ð ð â¡ ðð2 + ð ð 2
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ï¿œâðð2â ï¿œðð 2
ï¿œâðð2+ ï¿œðð 2
ððâ ðð ðð ððð ð ðŠðŠðððŠðŠ + ￜￜðð 2â ï¿œðð 2
ï¿œðð 2+ ï¿œðð 2
ððâ ðð ðð ððð ð ðŠðŠðððŠðŠ
ðð
ð ð ððð ð ðð =ððð ð
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
2 ð ð ððð ð ðð ðð ðð
2 ð ð
17
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
2 ð ð ððð ð ðð ðð ðð
2 ð ð
ðŒðŒ ðð,ðð = 4 ðŒðŒ0 ð ð ððð ð ð ð 2ðð ðð ðð
2 ð ð ð ð ððð ð 2
ðð ðð ðð2 ð ð ðŒðŒ0 â¡
ðžðž0 2
2 ð ð 2ðð2
Mathematica
ðð
ðð
18
3. Rectangular Aperture
ðð
ðð
ðŠðŠ
ð§ð§
19
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ￜￜâðð2
ï¿œðð 2
ððâ ðð ðð ððð ð ðŠðŠðððŠðŠ ￜￜâðð2
ï¿œðð 2
ððâ ðð ðð ððð ð ð§ð§ððð§ð§
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
ðð ð ð ððð ð ð ð ðð ðð ðð
2 ð ð
20
ðð
ðð
ðŒðŒ ðð,ðð = ðŒðŒ0 ð ð ððð ð ð ð 2ðð ðð ðð
2 ð ð ð ð ððð ð ð ð 2
ðð ðð ðð2 ð ð
ðŒðŒ0 â¡ðžðž0 2
2 ð ð 2ðð2 ðð2
21
Emission of Semiconductor Laser
22
4. Circular Aperture
ðððððŠðŠ = ðð ð ð ððð ð ðððð
ð§ð§ = ðð ð ð ððð ð ððð§ð§
ðŠðŠ
23
Observation Plane
Ίðð = ðð ð ð ððð ð Ί
ðð
ðð = ðð ð ð ððð ð Ίðð
ðð
24
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
ðð ðŠðŠ + ðð ð§ð§ = ðð ð ð ððð ð Ί ðð ð ð ððð ð ðð + ðð ð ð ððð ð Ί ðð ð ð ððð ð ðð
= ðð ðð ð ð ððð ð ðð â Ί
ðððŠðŠ ððð§ð§ = ðð ðððð ðððð
ðžðž ðð,Ί, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ0
ðð
ðð ððððï¿œ0
2ðð
ðððð ðð â ðð ðð ðð ðð ðððððð ðð âΊð ð
Ί = 0Due to axial symmetry, we can choose:
= ðð ðð ð ð ððð ð Ί ðð ð ð ððð ð ðð + ð ð ððð ð Ί ð ð ððð ð ðð
25
ðžðž ðð,Ί, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ0
ðð
ðð ððððï¿œ0
2ðð
ðððð ðð â ðð ðð ðð ðð ðððððð ððð ð
A couple of integrals to solve:
26
12 ðð
ￜ0
2ðð
ðððð ðððð ðð ðððððð ðð â¡ ðœðœ0 ð¢ð¢ Bessel function of order zero
ðžðž ðð,Ί, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ0
ðð
ðð ððððï¿œ0
2ðð
ðððð ðð â ðð ðð ðð ðð ðððððð ððð ð
27
ðžðž ðð,Ί, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð 2 ððï¿œ
0
ðð
ðð ðððð ðœðœ0 âðð ððð ð
ðð
ð¢ð¢ â¡ âðð ððð ð
ðð
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð 2 ðð
ð ð ðð ðð
2
ￜ0
âðð ððð ð ðð
α ððα ðœðœ0 α
ðŒðŒ â¡âðð ððð ð
ðð ðð ðððð =ð ð ðð ðð
2
α ððα
28
ￜ0
ðŒðŒ
ðŒðŒ ðœðœ0 ðŒðŒ ðððŒðŒ â¡ ðŒðŒ ðœðœ1 ðŒðŒ
29
ðžðž ðð,Ί, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð 2 ðð
ð ð ðð ðð
2
ￜ0
âðð ððð ð ðð
α ððα ðœðœ0 α
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð 2 ðð
ð ð ðð ðð
2 âðð ðð ððð ð
ðœðœ1âðð ðð ððð ð
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ðð2
2 ðœðœ1ðð ðð ððð ð
ðð ðð ððð ð
30
ðŒðŒ ðð,Ί = ðŒðŒ02 ðœðœ1
ðð ðð ððð ð
ðð ðð ððð ð
2
ðŒðŒ0 â¡ðžðž0 2
2 ð ð 2ðð ðð2 2
ðð ðð ððð ð
ï¿œðŒðŒ ðŒðŒ0
31
zeros at ðð ðð ððð ð
= 3.832, 7.016, 10.173, âŠ
ðð ðð ðð1ð ð
= 3.832
ðð1ð ð
= ð ð ððð ð ðð1 = 3.832ðð
2 ðð ðð= 1.22
ðð2 ðð
first zero at
Light is essentially confined inside the cone: ðððððð ðð1 < ðð.ðððð ðð
ðð ðð
32
Circular Aperture
ð§ð§
ðŠðŠ
ð ð
ðŠðŠ
ðð
ðð
ðð
ð ð ððð ð ðð1 =ðð1ð ð
= 1.22ðð
2 ðð
ð ð 2ðð
Airyâspattern
ðððð1
ðð1ðð1
33
ð§ð§
ðŠðŠ
2ðð
ð ð
ð ð
ðð1} = 0
34
ðŠðŠ
2ðð
ðð1ðð1
ð ð ððð ð ðð1 = 1.22ðð
2 ðð
tan ðð1 =ðð1ðð
ðð1
ðð1 â 1.22ðð ðð2 ðð
ðð
Smallest spot size:ðð1 â 1.22
ðð ððð·ð·ðððððððð
ð·ð·ðððððððð
= 1.22ðððð ððð ð ð·ð·ðððððððð
ð ð
Smallest angular width:ðð1ðð
= 1.22ðððð
ð ð ð·ð·ðððððððð
35
Diameter of primary mirror 2.4 m
Wavelength 0.55 µm
Angular width 0.28 Ã 10-6 rad
36
ð¡ð¡ððð ð ðððððððð â¡ð·ð·ðððððððð2 ðð
ð·ð·ðððððððð
ðððððððð
ðððð â¡ ð ð ð ð ððð ð ðððððððð â ð ð ð·ð·ðððððððð
2 ðð
ðð
ðð#
=ðð
ð·ð·ðððððððð
37
Numerical Aperture
ðððð â¡ ð ð ð ð ððð ð ðððððððð
38
ðð1 = 1.22ðððð
2 ðððð
Smallest spot size from a lens
ðŠðŠ
2ðð = ð·ð·ðððððððð
ðð1ðð1
ðð1
ðð
ð·ð·ðððððððð
ð ð
ðð1 = 1.22ðððð ððð ð ð·ð·ðððððððð
ðððð â¡ ð ð ð ð ððð ð ðððððððð â ð ð ð·ð·ðððððððð
2 ðð
39
Rayleigh Criteria for Resolution
Barely resolved
Resolved
Not resolved
40
ðð1 = 1.22ðððð
2 ðððð
ðððð = 0.55 ðððð
3.36 ðððð 1.34 ðððð 0.52 ðððð 0.27 ðððð
Examples of Diffraction Limit of Objective Lenses
41
ðžðž ðð,ðð, ð¡ð¡ =ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§ ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
ðð
ð ð
ðð
ðð
ð§ð§
ðŠðŠ
ð ð
ð ð â¡ ðð2 + ðð2 + ð ð 2
Fraunhofer Diffraction
ðððððð ðŠðŠ2 + ð§ð§2
ðð ð ð ⪠1
ðððððð ðð â ðŠðŠ 2 + ðð â ð§ð§ 2
ðð ð ð ⪠1
42
In summary, far-field diffraction:
1. Single Slit
2. Double Slit
3. Rectangular Aperture
4. Circular Aperture
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
2 ð ð ððð ð ðð ðð ðð
2 ð ð
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
ðð ð ð ððð ð ð ð ðð ðð ðð
2 ð ð
ðžðž ðð,Ί, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ðð2
2 ðœðœ1ðð ðð ððð ð
ðð ðð ððð ð
43
ðžðž ðð,ðð, ð¡ð¡ =ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§ ððâ ðð ðð ðð ðŠðŠ + ðð ð§ð§ð ð ðððŠðŠ ððð§ð§
ðžðž ðð,ðð, ð¡ð¡ =ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œââ
+â
ðð ðŠðŠ, ð§ð§ ððâ ðð ðððŠðŠ ðŠðŠ +ððð§ð§ ð§ð§ ðððŠðŠ ððð§ð§
ðð ðŠðŠ, ð§ð§ â¡ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§
0
inside apertureopaque obstruction
ðððŠðŠ â¡ðð ððð ð
Fraunhofer Diffraction as a Fourier Transformation
ððð§ð§ â¡ðð ððð ð
{
44
Diffraction Gratings
45
Multiple Slits
ðð
ðð
ðŠðŠ
ðð âðð2
ðð +ðð2
ð§ð§
ðµðµ (infinitely long) slits of width ðð separated by distance ðð
+ðð2
âðð2
ðð â 1 ðð âðð2
ðð â 1 ðð +ðð2
46
ððð ð
ðð
ðð
ð§ð§
ðŠðŠ
ð ð
ð ð â¡ ðð2 + ð ð 2
ðžðž ðð,ðð, ð¡ð¡
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ï¿œ
â ðð2
+ ðð2
+ ￜ
ðð â ðð2
ðð + ðð2
+ ￜ
2 ðð â ðð2
2 ðð + ðð2
+ ⯠+ ￜ
ððâ1 ðð â ðð2
ððâ1 ðð + ðð2
ððâ ðð ðð ððð ð ðŠðŠ ðððŠðŠ
ðð
ð ð ððð ð ðð =ððð ð
47
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
ï¿œðð= 0
ððâ1
ððâ ðð ðð ðð ððð ð ðð
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
1 â ððâðð ðððð ðð ððð ð
1 â ððâðððð ðð ððð ð
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
ððâðð ðððð ðð ðð2 ð ð
ððâðððð ðð ðð2 ð ð
ðð+ðð ðððð ðð ðð2 ð ð â ððâðð ðð
ðð ðð ðð2 ð ð
ðð+ðððð ðð ðð2 ð ð â ððâðð
ðð ðð ðð2 ð ð
=ðžðž0 ðððð ðð ð ð âðð ð¡ð¡
ð ð ðð ð ð ððð ð ð ð
ðð ðð ðð2 ð ð
ððâðð ðððð ðð ðð2 ð ð
ððâðððð ðð ðð2 ð ð
sin ðð ðð ðð ðð2 ð ð
sin ðð ðð ðð2 ð ð
48
ðŒðŒ ðð,ðð = ðŒðŒ0 ð ð ððð ð ð ð 2ðð ðð ðð
2 ð ð
ð ð ððð ð 2 ðð ðð ðð ðð2 ð ð
ð ð ððð ð 2 ðð ðð ðð2 ð ð
ðŒðŒ0 â¡ðžðž0 2
2 ð ð 2ðð2
Intensity Pattern
Mathematica
ðð = 1
ðð = 4
ðð = 1
ð ð = 1
49
ð ð ððð ð ð ð 2ðð ðð ðð
2 ð ð â 1
ðŒðŒ ðð,ðð â ðŒðŒ0ð ð ððð ð 2 ðð ðð ðð ðð2 ð ð
ð ð ððð ð 2 ðð ðð ðð2 ð ð
Small Width Approximation:
ðð = 0.1
ðð = 4
ðð = 1
ð ð = 1
50
ðð ðð ðð2 ð ð
= ðð ðð ðŒðŒ ðð,ðð, ð¡ð¡ = ðð2 ðŒðŒ0
Maxima (intensity peaks)
ðð = 0, ±1, ±2, âŠ
ðð ð ð ððð ð ðððð = ðð ððgrating equation
grating order
51
ðððð ðð ðð
2 ð ð = ðð ðð
ðð = 1, 2, 3, ⊠, (ðð â 1)
Minima (zero intensity)
ðð ðð ðð2 ð ð
=ðððððð
ðð = 0.1
ðð = 4
ðð = 1
ð ð = 10 <
ðð ðð ðð2 ð ð
< ðð
ðð = 0 ðð = 1
10â1 ðð2â2
ðŒðŒ ðð,ðð â ðŒðŒ0ð ð ððð ð 2 ðð ðð ðð ðð2 ð ð
ð ð ððð ð 2 ðð ðð ðð2 ð ð
52
Angular Width
ðð ðð ð ð ððð ð ðððð + âðð2
2= ðð ðð +
1ðððð
ðð ðð ðð2 ð ð
=ðð ðð ð ð ððð ð ðð
2
âðð =2 ðð
ðð ðð ð ð ððð ð ðððð
ðð ðð ð ð ððð ð ðððð ð ð ððð ð âðð2
2â
1ðððð
ðð
53
Spectral Resolution
ðð ð ð ððð ð ðððð = ðð ðð
ðð ð ð ððð ð ðððð ðððð = ðð ðððð
âðððððððð =ðð
ðð ðð
ðððð â¡âðð2
=ðð
ðð ðð ð ð ððð ð ðððððððð â¡ âðððððððð
54
Free Spectral Range
ðð ð ð ððð ð ðð = ðð + 1 ðð = ðð ðð + âððð¹ð¹ð¹ð¹ð ð
âððð¹ð¹ð¹ð¹ð ð =ðððð
55
Oblique Incidence
Normal Incidence
ðð ð ð ððð ð ðð â ðð ð ð ððð ð ðððððððð = ðð ðð
ðð ð ð ððð ð ðððð â ð ð ððð ð ðððððððð = ðð ðð
ðð ð ð ððð ð ðððð = ðð ðð
56
Fresnel Diffraction
Going beyond the Fraunhofer (far-field) approximation
or
getting closer to the aperture
57
ðð ðŠðŠ, ð§ð§ = ð ð 2 + ðð â ðŠðŠ 2 + ðð â ð§ð§ 2
ðð
ð ð
ðð
ðð
ð§ð§
ðŠðŠ
ðð ðŠðŠ, ð§ð§ â ð ð +1
2 ð ð ðð â ðŠðŠ 2 +
12 ð ð
ðð â ð§ð§ 2
ðžðž ðð,ðð, ð¡ð¡ = ï¿œððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ðð ðŠðŠ, ð§ð§
ðððð ðð ðð ðŠðŠ, ð§ð§ â ðð ð¡ð¡ + ðð ðŠðŠ, ð§ð§ ðððŠðŠ ððð§ð§
= ð ð 1 +ðð â ðŠðŠ 2
ð ð 2+
ðð â ð§ð§ 2
ð ð 2
ðð ð ð ðððððð ðð â ðŠðŠ 2 + ðð â ð§ð§ 2 2
ð ð 4⪠ðð
58
ðžðž ðð,ðð, ð¡ð¡ =ðððð ðð ðð â ðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§ ðððððð2 ðð ððâðŠðŠ 2+ ððâð§ð§ 2
ðððŠðŠ ððð§ð§
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ðð â ðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðððððððð ðð ððâðŠðŠ 2+ ððâð§ð§ 2
ðððŠðŠ ððð§ð§
ðžðž0 ðŠðŠ, ð§ð§ ðððð ðð ðŠðŠ, ð§ð§ =ðžðž0
0
Inside the aperture
Outside the aperture{
Flat Wavefront Illumination
59
ðŸðŸ â¡2ðð ð ð
ðð â ðŠðŠ
ðððŠðŠ = âðð ð ð 2
ðððŸðŸ
ð¿ð¿ â¡2ðð ð ð
ðð â ð§ð§
ððð§ð§ = âðð ð ð 2
ððð¿ð¿
ðžðž ðð,ðð, ð¡ð¡ =ðžðž0 ðððð ðð ðð â ðð ð¡ð¡
ð ð ï¿œ
ððððððððð¡ð¡ðððððð
ðððððððð ðð ððâðŠðŠ 2+ ððâð§ð§ 2
ðððŠðŠ ððð§ð§
=ðžðž0 ðððð ðð ðð â ðð ð¡ð¡
ð ð ðð ð ð 2
ï¿œððððððððð¡ð¡ðððððð
ðððððð2 ðŸðŸ2+ ð¿ð¿2 ðððŸðŸ ððð¿ð¿
=ðð ðžðž0 ðððð ðð ðð â ðð ð¡ð¡
2ï¿œðŸðŸ1
ðŸðŸ2
ðððððð2 ðŸðŸ
2ðððŸðŸ ï¿œ
ð¿ð¿1
ð¿ð¿2
ðððððð2 ð¿ð¿
2ððð¿ð¿
60
ï¿œðŸðŸ1
ðŸðŸ2
ðððððð2 ðŸðŸ
2ðððŸðŸ = ï¿œ
ðŸðŸ1
ðŸðŸ2
cosðð2ðŸðŸ2 ðððŸðŸ + ðð ï¿œ
ðŸðŸ1
ðŸðŸ2
sinðð2ðŸðŸ2 ðððŸðŸ
= ðð ðŸðŸ2 â ðð ðŸðŸ1 + ðð ð®ð® ðŸðŸ2 â ð®ð® ðŸðŸ1
ï¿œð¿ð¿1
ð¿ð¿2
ðððððð2 ð¿ð¿
2ððð¿ð¿ = ï¿œ
ð¿ð¿1
ð¿ð¿2
cosðð2ð¿ð¿2 ððð¿ð¿ + ðð ï¿œ
ð¿ð¿1
ð¿ð¿2
sinðð2ð¿ð¿2 ððð¿ð¿
= ðð ð¿ð¿2 â ðð ð¿ð¿1 + ðð ð®ð® ð¿ð¿2 â ð®ð® ð¿ð¿1
ðð ðð â¡ ï¿œ0
ðð
cosðð2ðð2 ðððð ð®ð® ðð â¡ ï¿œ
0
ðð
sinðð2ðð2 ðððð
61
à ðð ðŸðŸ2 â ðð ðŸðŸ1 + ðð ð®ð® ðŸðŸ2 â ð®ð® ðŸðŸ1
à ðð ð¿ð¿2 â ðð ð¿ð¿1 + ðð ð®ð® ð¿ð¿2 â ð®ð® ð¿ð¿1
ðžðž ðð,ðð, ð¡ð¡ =ðð ðžðž0 ðððð ðð ðð â ðð ð¡ð¡
2
ðŒðŒ ðð,ðð =ðŒðŒ04 à ðð ðŸðŸ2 â ðð ðŸðŸ1 2 + ð®ð® ðŸðŸ2 â ð®ð® ðŸðŸ1 2
à ðð ð¿ð¿2 â ðð ð¿ð¿1 2 + ð®ð® ð¿ð¿2 â ð®ð® ð¿ð¿1 2
62
ðð ðð â¡ ï¿œ0
ðð
cosðð2ððâ²2 ððððð¥
ð®ð® ðð â¡ ï¿œ0
ðð
sinðð2ððð¥2 ððððð¥
ðð ðð
ð®ð® ðð
ðð
ðð
ðð
ðð ððð®ð® ðð
63
ðð ðð â¡ ï¿œ0
ðð
cosðð2ðð2 ðððð
ð®ð® ðð â¡ ï¿œ0
ðð
sinðð2ðð2 ðððð
ðððð ðð = cosðð2ðð2 ðððð
ððð®ð® ðð = sinðð2ðð2 ðððð
ð®ð® ðð
ðð ðð
ðððð 2 + ððð®ð® 2 = ðððð 2
ððððððð®ð®ðððð
64
Applications of Fresnel Diffraction1.No obstruction
2.Straight edge
3. Single slit
4. Rectangular aperture
5. Opaque circular disk
65
ðŒðŒ ðð,ðð =ðŒðŒ04 à ðð ðŸðŸ2 â ðð ðŸðŸ1 2 + ð®ð® ðŸðŸ2 â ð®ð® ðŸðŸ1 2
à ðð ð¿ð¿2 â ðð ð¿ð¿1 2 + ð®ð® ð¿ð¿2 â ð®ð® ð¿ð¿1 2
1. No Obstruction
ðŸðŸ â¡2ðð ð ð
ðð â ðŠðŠ
ð¿ð¿ â¡2ðð ð ð
ðð â ð§ð§
ðŠðŠ
ð§ð§
ðŸðŸ2 = ââ
ðŸðŸ1 = +âð¿ð¿2 = ââ ð¿ð¿1 = +â
=ðŒðŒ04
à â0.5 â 0.5 2 + â0.5 â 0.5 2 à â0.5 â 0.5 2 + â0.5 â 0.5 2
= ðŒðŒ0 No surprises here, just the obvious result !!
66
ðŒðŒ ðð,ðð =ðŒðŒ04
à ðð ðŸðŸ2 â ðð ðŸðŸ1 2 + ð®ð® ðŸðŸ2 â ð®ð® ðŸðŸ1 2
à ðð ð¿ð¿2 â ðð ð¿ð¿1 2 + ð®ð® ð¿ð¿2 â ð®ð® ð¿ð¿1 2
ðŸðŸ â¡2ðð ð ð
ðð â ðŠðŠ
ð¿ð¿ â¡2ðð ð ð
ðð â ð§ð§
ðŠðŠ
ð§ð§ðŸðŸ2 = 2
ðð ðððð
ðŸðŸ1 = +âð¿ð¿2 = ââ ð¿ð¿1 = +â
=ðŒðŒ04
à ðð2ðð ð ð
ðð â 0.5
2
+ ð®ð®2ðð ð ð
ðð â 0.5
2
à 2
2. Straight Edge
67
ð®ð® ðð
ðð ðððð = 0
ðð > 0
ðð < 0 ðŒðŒ ðð,ðð, ð¡ð¡ /ðŒðŒ0
ðð
ðð ð ð = 2
ðŒðŒ ðð,ðð =ðŒðŒ02
à ðð2ðð ð ð
ðð â 0.5
2
+ ð®ð®2ðð ð ð
ðð â 0.5
2
68
69
ðŒðŒ ðð,ðð =ðŒðŒ04
à ðð ðŸðŸ2 â ðð ðŸðŸ1 2 + ð®ð® ðŸðŸ2 â ð®ð® ðŸðŸ1 2
à ðð ð¿ð¿2 â ðð ð¿ð¿1 2 + ð®ð® ð¿ð¿2 â ð®ð® ð¿ð¿1 2
ðŸðŸ â¡2ðð ð ð
ðð â ðŠðŠ
ð¿ð¿ â¡2ðð ð ð
ðð â ð§ð§
ðŠðŠ
ð§ð§ðŸðŸ2 =
2ðð ð ð
ðð â ðð2
ðŸðŸ1 =2ðð ð ð
ðð + ðð2ð¿ð¿2 = ââ ð¿ð¿1 = +â
=ðŒðŒ04
à ðð2ðð ð ð
ðð â ðð2 â ðð
2ðð ð ð
ðð + ðð2
2
+ ð®ð®2ðð ð ð
ðð â ðð2 â ð®ð®
2ðð ð ð
ðð + ðð2
2
à 2
3. Single Slit
ðð
70
ð®ð® ðð
ðð ðððð = 0
ðð > 0
ðð < 0
ðŒðŒ ðð,ðð =ðŒðŒ02
à ðð2ðð ð ð
ðð â ðð2 â ðð
2ðð ð ð
ðð + ðð2
2
+ ð®ð®2ðð ð ð
ðð â ðð2 â ð®ð®
2ðð ð ð
ðð + ðð2
2
ðŸðŸ1 â ðŸðŸ2 =2ðð ð ð
ðð
ðŸðŸ1 + ðŸðŸ22
=2ðð ð ð
ðð
71
ðð = 10 ðð
ðð
ððð¹ð¹ â¡ðð2
4 ðð ð ð
ððð¹ð¹ = 10
ððð¹ð¹ = 1
ððð¹ð¹ = 0.5
ððð¹ð¹ = 0.1
ðð = 1
ð ð = 2.5 ðð
ð ð = 25 ðð
ð ð = 50 ðð
ð ð = 250 ðð
Near field
Far field
Fresnel number
72
Mathematica
73
4. Rectangular Aperture
74
5. Circular Objects
Poisson (Arago) spot