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Diffraction Theory
1
2
3
𝑟𝑟1
𝑟𝑟2
𝐸𝐸 𝑟𝑟, 𝑡𝑡 = 𝐸𝐸1 𝑟𝑟, 𝑡𝑡
𝐸𝐸 𝑟𝑟, 𝑡𝑡
𝐸𝐸 𝑟𝑟, 𝑡𝑡 = 𝐸𝐸0,1𝑟𝑟1𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟1− 𝜔𝜔 𝑡𝑡 + 𝜀𝜀1 +
𝐸𝐸0,2
𝑟𝑟2𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟2− 𝜔𝜔 𝑡𝑡 + 𝜀𝜀2
+ 𝐸𝐸2 𝑟𝑟, 𝑡𝑡
𝑟𝑟3
+𝐸𝐸0,3
𝑟𝑟3𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟3 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀3
𝑟𝑟4
+ 𝐸𝐸3 𝑟𝑟, 𝑡𝑡 + 𝐸𝐸4 𝑟𝑟, 𝑡𝑡
𝑟𝑟5
+𝐸𝐸0,4
𝑟𝑟4𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟4 −𝜔𝜔 𝑡𝑡 + 𝜀𝜀4
+ 𝐸𝐸5 𝑟𝑟, 𝑡𝑡
+𝐸𝐸0,5
𝑟𝑟5𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟5 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀5
= �𝑖𝑖
𝐸𝐸𝑖𝑖 𝑟𝑟, 𝑡𝑡
+⋯
+ …
= �𝑖𝑖
𝐸𝐸0,𝑖𝑖
𝑟𝑟𝑖𝑖𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟𝑖𝑖 −𝜔𝜔 𝑡𝑡 + 𝜀𝜀𝑖𝑖
4
Huygens-Fresnel Principle
5
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 = �𝑖𝑖
𝐸𝐸0,𝑖𝑖
𝑟𝑟𝑖𝑖𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟𝑖𝑖 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀𝑖𝑖
= �𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧𝑟𝑟 𝑦𝑦, 𝑧𝑧
𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟 𝑦𝑦, 𝑧𝑧 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝑟𝑟 𝑦𝑦, 𝑧𝑧 = 𝑠𝑠2 + 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2
𝑟𝑟
𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑖𝑖 𝑦𝑦, 𝑧𝑧
�𝑖𝑖
�𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
6
𝑟𝑟 𝑦𝑦, 𝑧𝑧 ≅ 𝑠𝑠 1 +𝜃𝜃2
2
Fresnel Approximation𝑘𝑘 𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝜃𝜃4
8≪ 𝜋𝜋
= 𝑠𝑠 1 +𝜃𝜃2
2−𝜃𝜃4
8+𝜃𝜃6
16−
5 𝜃𝜃8
128+ ⋯
𝑟𝑟 𝑦𝑦, 𝑧𝑧 = 𝑠𝑠2 + 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2 = 𝑠𝑠 1 +𝑌𝑌 − 𝑦𝑦 2
𝑠𝑠2+
𝑍𝑍 − 𝑧𝑧 2
𝑠𝑠2 ≡ 𝑠𝑠 1 + 𝜃𝜃2
𝜃𝜃2 ≡𝑌𝑌 − 𝑦𝑦 2
𝑠𝑠2+
𝑍𝑍 − 𝑧𝑧 2
𝑠𝑠2
Fresnel Diffraction
= 𝑠𝑠 +𝑌𝑌2 + 𝑍𝑍2
2 𝑠𝑠−
𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑠𝑠
+𝑦𝑦2 + 𝑧𝑧2
2 𝑠𝑠
= 𝑠𝑠 1 +𝑌𝑌 − 𝑦𝑦 2
2 𝑠𝑠2+
𝑍𝑍 − 𝑧𝑧 2
2 𝑠𝑠2
7
𝑘𝑘𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦2 + 𝑧𝑧2
2 𝑠𝑠≪ 𝜋𝜋
𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦2 + 𝑧𝑧2
𝜆𝜆 𝑠𝑠≪ 1
Fraunhofer Approximation
Fraunhofer Diffractionalso known as Far-Field Diffraction
𝑟𝑟 𝑦𝑦, 𝑧𝑧 ≅ 𝑠𝑠 +𝑌𝑌2 + 𝑍𝑍2
2 𝑠𝑠−
𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑠𝑠
+𝑦𝑦2 + 𝑧𝑧2
2 𝑠𝑠
Fresnel Approximation𝑘𝑘 𝑠𝑠 𝑚𝑚𝑚𝑚𝑚𝑚𝜃𝜃4
8≪ 𝜋𝜋
8
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 = �𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧𝑟𝑟 𝑦𝑦, 𝑧𝑧
𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟 𝑦𝑦, 𝑧𝑧 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
≅ �𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧𝑅𝑅
𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 − 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 −𝜔𝜔 𝑡𝑡 + 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
=𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦,𝑧𝑧 𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝑟𝑟 𝑦𝑦, 𝑧𝑧 = 𝑠𝑠2 + 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2
𝑅𝑅2 ≡ 𝑠𝑠2 + 𝑌𝑌2 + 𝑍𝑍2
= 𝑅𝑅2 − 2 𝑌𝑌 𝑦𝑦 − 2 𝑍𝑍 𝑧𝑧 + 𝑦𝑦2 + 𝑧𝑧2
= 𝑅𝑅 1 +−2 𝑌𝑌 𝑦𝑦 − 2 𝑍𝑍 𝑧𝑧 + 𝑦𝑦2 + 𝑧𝑧2
𝑅𝑅2≅ 𝑅𝑅 −
𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅
9
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝑟𝑟
𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑅𝑅
Fraunhofer Diffraction
𝑅𝑅2 ≡ 𝑠𝑠2 + 𝑌𝑌2 + 𝑍𝑍2
10
Illumination at the Aperture:
In the examples to follow, we will consider a flat wavefront at
normal incidence on the aperture
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 =𝐸𝐸0
0
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
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
𝑟𝑟𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑅𝑅
𝑅𝑅 ≡ 𝑌𝑌2 + 𝑠𝑠2
𝑑𝑑
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
− �𝑑𝑑 2
+ �𝑑𝑑 2
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦𝑑𝑑𝑦𝑦
𝜃𝜃
𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃 =𝑌𝑌𝑅𝑅
𝑦𝑦
𝑧𝑧
13
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑑𝑑 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑑𝑑2 𝑅𝑅
1. Single Slit, cont.
𝐼𝐼 ≡ 𝐸𝐸2
𝐼𝐼 𝑌𝑌,𝑍𝑍 = 𝐼𝐼0 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠2𝑘𝑘 𝑌𝑌 𝑑𝑑
2 𝑅𝑅 𝐼𝐼0 ≡𝐸𝐸0 2
2 𝑅𝑅2𝑑𝑑2
𝑑𝑑 = 50 µ𝑚𝑚
𝜆𝜆 = 0.6 µ𝑚𝑚𝑠𝑠 = 1 𝑚𝑚
𝑘𝑘 𝑌𝑌𝑚𝑚 𝑑𝑑2 𝑅𝑅
= 𝑚𝑚 𝜋𝜋
𝑚𝑚 = ±1, ±2, ±3
𝑅𝑅 ≅ 1 𝑚𝑚
𝑌𝑌𝑚𝑚 = 𝑚𝑚𝜆𝜆 𝑅𝑅𝑑𝑑
𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 =𝑌𝑌𝑚𝑚𝑅𝑅
= 𝑚𝑚𝜆𝜆𝑑𝑑
𝑌𝑌(𝑚𝑚𝑚𝑚)
𝑌𝑌1
�𝐼𝐼 𝐼𝐼0
zeros at geometrical shadow
𝑌𝑌−1
with
𝑌𝑌
𝑍𝑍
14
Mathematica
15
2. Double Slit
𝑑𝑑
𝑑𝑑
𝑚𝑚
𝑦𝑦
�𝑚𝑚 2 − �𝑑𝑑 2
�𝑚𝑚 2 + �𝑑𝑑 2
�−𝑚𝑚2 − �𝑑𝑑 2
�−𝑚𝑚2 + �𝑑𝑑 2
𝑧𝑧
16
𝑟𝑟𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑅𝑅
𝑅𝑅 ≡ 𝑌𝑌2 + 𝑠𝑠2
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
�−𝑎𝑎2− �𝑑𝑑 2
�−𝑎𝑎2+ �𝑑𝑑 2
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦𝑑𝑑𝑦𝑦 + ��𝑎𝑎 2− �𝑑𝑑 2
�𝑎𝑎 2+ �𝑑𝑑 2
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦𝑑𝑑𝑦𝑦
𝜃𝜃
𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃 =𝑌𝑌𝑅𝑅
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑑𝑑 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑑𝑑2 𝑅𝑅
2 𝑠𝑠𝑐𝑐𝑠𝑠𝑘𝑘 𝑍𝑍 𝑚𝑚
2 𝑅𝑅
17
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑑𝑑 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑑𝑑2 𝑅𝑅
2 𝑠𝑠𝑐𝑐𝑠𝑠𝑘𝑘 𝑌𝑌 𝑚𝑚
2 𝑅𝑅
𝐼𝐼 𝑌𝑌,𝑍𝑍 = 4 𝐼𝐼0 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠2𝑘𝑘 𝑌𝑌 𝑑𝑑
2 𝑅𝑅𝑠𝑠𝑐𝑐𝑠𝑠2
𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅 𝐼𝐼0 ≡
𝐸𝐸0 2
2 𝑅𝑅2𝑑𝑑2
Mathematica
𝑑𝑑
𝑚𝑚
18
3. Rectangular Aperture
𝑚𝑚
𝑏𝑏
𝑦𝑦
𝑧𝑧
19
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅��−𝑏𝑏2
�𝑏𝑏 2
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦𝑑𝑑𝑦𝑦 ��−𝑎𝑎2
�𝑎𝑎 2
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑍𝑍𝑅𝑅 𝑧𝑧𝑑𝑑𝑧𝑧
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑏𝑏 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑏𝑏2 𝑅𝑅
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠𝑘𝑘 𝑍𝑍 𝑚𝑚
2 𝑅𝑅
20
𝑌𝑌
𝑍𝑍
𝐼𝐼 𝑌𝑌,𝑍𝑍 = 𝐼𝐼0 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠2𝑘𝑘 𝑌𝑌 𝑏𝑏
2 𝑅𝑅𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠2
𝑘𝑘 𝑍𝑍 𝑚𝑚2 𝑅𝑅
𝐼𝐼0 ≡𝐸𝐸0 2
2 𝑅𝑅2𝑚𝑚2 𝑏𝑏2
21
Emission of Semiconductor Laser
22
4. Circular Aperture
𝑚𝑚𝜑𝜑𝑦𝑦 = 𝜌𝜌 𝑠𝑠𝑖𝑖𝑠𝑠 𝜑𝜑𝜌𝜌
𝑧𝑧 = 𝜌𝜌 𝑠𝑠𝑐𝑐𝑠𝑠 𝜑𝜑𝑧𝑧
𝑦𝑦
23
Observation Plane
Φ𝑌𝑌 = 𝑞𝑞 𝑠𝑠𝑖𝑖𝑠𝑠 Φ
𝑞𝑞
𝑍𝑍 = 𝑞𝑞 𝑠𝑠𝑐𝑐𝑠𝑠 Φ𝑍𝑍
𝑌𝑌
24
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧 = 𝑞𝑞 𝑠𝑠𝑖𝑖𝑠𝑠 Φ 𝜌𝜌 𝑠𝑠𝑖𝑖𝑠𝑠 𝜑𝜑 + 𝑞𝑞 𝑠𝑠𝑐𝑐𝑠𝑠 Φ 𝜌𝜌 𝑠𝑠𝑐𝑐𝑠𝑠 𝜑𝜑
= 𝜌𝜌 𝑞𝑞 𝑠𝑠𝑐𝑐𝑠𝑠 𝜑𝜑 − Φ
𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧 = 𝜌𝜌 𝑑𝑑𝜑𝜑 𝑑𝑑𝜌𝜌
𝐸𝐸 𝑞𝑞,Φ, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�0
𝑎𝑎
𝜌𝜌 𝑑𝑑𝜌𝜌�0
2𝜋𝜋
𝑑𝑑𝜑𝜑 𝑒𝑒 − 𝑖𝑖 𝑘𝑘 𝜌𝜌 𝑞𝑞 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 −Φ𝑅𝑅
Φ = 0Due to axial symmetry, we can choose:
= 𝑞𝑞 𝜌𝜌 𝑠𝑠𝑐𝑐𝑠𝑠 Φ 𝜌𝜌 𝑠𝑠𝑐𝑐𝑠𝑠 𝜑𝜑 + 𝑠𝑠𝑖𝑖𝑠𝑠 Φ 𝑠𝑠𝑖𝑖𝑠𝑠 𝜑𝜑
25
𝐸𝐸 𝑞𝑞,Φ, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�0
𝑎𝑎
𝜌𝜌 𝑑𝑑𝜌𝜌�0
2𝜋𝜋
𝑑𝑑𝜑𝜑 𝑒𝑒 − 𝑖𝑖 𝑘𝑘 𝜌𝜌 𝑞𝑞 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑𝑅𝑅
A couple of integrals to solve:
26
12 𝜋𝜋
�0
2𝜋𝜋
𝑑𝑑𝜑𝜑 𝑒𝑒𝑖𝑖 𝑝𝑝 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑 ≡ 𝐽𝐽0 𝑢𝑢 Bessel function of order zero
𝐸𝐸 𝑞𝑞,Φ, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�0
𝑎𝑎
𝜌𝜌 𝑑𝑑𝜌𝜌�0
2𝜋𝜋
𝑑𝑑𝜑𝜑 𝑒𝑒 − 𝑖𝑖 𝑘𝑘 𝜌𝜌 𝑞𝑞 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑𝑅𝑅
27
𝐸𝐸 𝑞𝑞,Φ, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅2 𝜋𝜋�
0
𝑎𝑎
𝜌𝜌 𝑑𝑑𝜌𝜌 𝐽𝐽0 −𝑘𝑘 𝑞𝑞𝑅𝑅
𝜌𝜌
𝑢𝑢 ≡ −𝑘𝑘 𝑞𝑞𝑅𝑅
𝜌𝜌
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅2 𝜋𝜋
𝑅𝑅𝑘𝑘 𝑞𝑞
2
�0
−𝑘𝑘 𝑞𝑞𝑅𝑅 𝑎𝑎
α 𝑑𝑑α 𝐽𝐽0 α
𝛼𝛼 ≡−𝑘𝑘 𝑞𝑞𝑅𝑅
𝜌𝜌 𝜌𝜌 𝑑𝑑𝜌𝜌 =𝑅𝑅𝑘𝑘 𝑞𝑞
2
α 𝑑𝑑α
28
�0
𝛼𝛼
𝛼𝛼 𝐽𝐽0 𝛼𝛼 𝑑𝑑𝛼𝛼 ≡ 𝛼𝛼 𝐽𝐽1 𝛼𝛼
29
𝐸𝐸 𝑞𝑞,Φ, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅2 𝜋𝜋
𝑅𝑅𝑘𝑘 𝑞𝑞
2
�0
−𝑘𝑘 𝑞𝑞𝑅𝑅 𝑎𝑎
α 𝑑𝑑α 𝐽𝐽0 α
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅2 𝜋𝜋
𝑅𝑅𝑘𝑘 𝑞𝑞
2 −𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
𝐽𝐽1−𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝜋𝜋 𝑚𝑚2
2 𝐽𝐽1𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
30
𝐼𝐼 𝑞𝑞,Φ = 𝐼𝐼02 𝐽𝐽1
𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
2
𝐼𝐼0 ≡𝐸𝐸0 2
2 𝑅𝑅2𝜋𝜋 𝑚𝑚2 2
𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
�𝐼𝐼 𝐼𝐼0
31
zeros at 𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
= 3.832, 7.016, 10.173, …
𝑘𝑘 𝑚𝑚 𝑞𝑞1𝑅𝑅
= 3.832
𝑞𝑞1𝑅𝑅
= 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃1 = 3.832𝜆𝜆
2 𝜋𝜋 𝑚𝑚= 1.22
𝜆𝜆2 𝑚𝑚
first zero at
Light is essentially confined inside the cone: 𝒔𝒔𝒔𝒔𝒔𝒔 𝜃𝜃1 < 𝟏𝟏.𝟐𝟐𝟐𝟐 𝝀𝝀
𝟐𝟐 𝒂𝒂
32
Circular Aperture
𝑧𝑧
𝑦𝑦
𝑠𝑠
𝑦𝑦
𝑌𝑌
𝑍𝑍
𝑌𝑌
𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃1 =𝑞𝑞1𝑅𝑅
= 1.22𝜆𝜆
2 𝑚𝑚
𝑅𝑅2𝑚𝑚
Airy’spattern
𝑚𝑚𝑞𝑞1
𝑞𝑞1𝜃𝜃1
33
𝑧𝑧
𝑦𝑦
2𝑚𝑚
𝑠𝑠
𝑅𝑅
𝜃𝜃1} = 0
34
𝑦𝑦
2𝑚𝑚
𝜃𝜃1𝜃𝜃1
𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃1 = 1.22𝜆𝜆
2 𝑚𝑚
tan 𝜃𝜃1 =𝑞𝑞1𝑓𝑓
𝑞𝑞1
𝑞𝑞1 ≅ 1.22𝜆𝜆 𝑓𝑓2 𝑚𝑚
𝑓𝑓
Smallest spot size:𝑞𝑞1 ≅ 1.22
𝜆𝜆 𝑓𝑓𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
= 1.22𝜆𝜆𝑐𝑐 𝑓𝑓𝑠𝑠 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
𝑠𝑠
Smallest angular width:𝑞𝑞1𝑓𝑓
= 1.22𝜆𝜆𝑐𝑐
𝑠𝑠 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
35
Diameter of primary mirror 2.4 m
Wavelength 0.55 µm
Angular width 0.28 × 10-6 rad
36
𝑡𝑡𝑚𝑚𝑠𝑠 𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚 ≡𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐2 𝑓𝑓
𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚
𝑁𝑁𝑁𝑁 ≡ 𝑠𝑠 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚 ≅𝑠𝑠 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
2 𝑓𝑓
𝑓𝑓
𝑓𝑓#
=𝑓𝑓
𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
37
Numerical Aperture
𝑁𝑁𝑁𝑁 ≡ 𝑠𝑠 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚
38
𝑞𝑞1 = 1.22𝜆𝜆𝑐𝑐
2 𝑁𝑁𝑁𝑁
Smallest spot size from a lens
𝑦𝑦
2𝑚𝑚 = 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
𝜃𝜃1𝜃𝜃1
𝑞𝑞1
𝑓𝑓
𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
𝑠𝑠
𝑞𝑞1 = 1.22𝜆𝜆𝑐𝑐 𝑓𝑓𝑠𝑠 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
𝑁𝑁𝑁𝑁 ≡ 𝑠𝑠 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚 ≅𝑠𝑠 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐
2 𝑓𝑓
39
Rayleigh Criteria for Resolution
Barely resolved
Resolved
Not resolved
40
𝑞𝑞1 = 1.22𝜆𝜆𝑐𝑐
2 𝑁𝑁𝑁𝑁
𝜆𝜆𝑐𝑐 = 0.55 𝜇𝜇𝑚𝑚
3.36 𝜇𝜇𝑚𝑚 1.34 𝜇𝜇𝑚𝑚 0.52 𝜇𝜇𝑚𝑚 0.27 𝜇𝜇𝑚𝑚
Examples of Diffraction Limit of Objective Lenses
41
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝑟𝑟
𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑅𝑅
𝑅𝑅 ≡ 𝑌𝑌2 + 𝑍𝑍2 + 𝑠𝑠2
Fraunhofer Diffraction
𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦2 + 𝑧𝑧2
𝜆𝜆 𝑠𝑠≪ 1
𝑚𝑚𝑚𝑚𝑚𝑚 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2
𝜆𝜆 𝑠𝑠≪ 1
42
In summary, far-field diffraction:
1. Single Slit
2. Double Slit
3. Rectangular Aperture
4. Circular Aperture
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑑𝑑 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑑𝑑2 𝑅𝑅
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑑𝑑 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑑𝑑2 𝑅𝑅
2 𝑠𝑠𝑐𝑐𝑠𝑠𝑘𝑘 𝑌𝑌 𝑚𝑚
2 𝑅𝑅
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑏𝑏 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑏𝑏2 𝑅𝑅
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠𝑘𝑘 𝑍𝑍 𝑚𝑚
2 𝑅𝑅
𝐸𝐸 𝑞𝑞,Φ, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝜋𝜋 𝑚𝑚2
2 𝐽𝐽1𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅
43
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑅𝑅 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�−∞
+∞
𝜓𝜓 𝑦𝑦, 𝑧𝑧 𝑒𝑒− 𝑖𝑖 𝑘𝑘𝑦𝑦 𝑦𝑦 +𝑘𝑘𝑧𝑧 𝑧𝑧 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝜓𝜓 𝑦𝑦, 𝑧𝑧 ≡𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧
0
inside apertureopaque obstruction
𝑘𝑘𝑦𝑦 ≡𝑘𝑘 𝑌𝑌𝑅𝑅
Fraunhofer Diffraction as a Fourier Transformation
𝑘𝑘𝑧𝑧 ≡𝑘𝑘 𝑍𝑍𝑅𝑅
{
44
Diffraction Gratings
45
Multiple Slits
𝑏𝑏
𝑚𝑚
𝑦𝑦
𝑚𝑚 −𝑏𝑏2
𝑚𝑚 +𝑏𝑏2
𝑧𝑧
𝑵𝑵 (infinitely long) slits of width 𝒃𝒃 separated by distance 𝒂𝒂
+𝑏𝑏2
−𝑏𝑏2
𝑁𝑁 − 1 𝑚𝑚 −𝑏𝑏2
𝑁𝑁 − 1 𝑚𝑚 +𝑏𝑏2
46
𝑟𝑟𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑅𝑅
𝑅𝑅 ≡ 𝑌𝑌2 + 𝑠𝑠2
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅�
− 𝑏𝑏2
+ 𝑏𝑏2
+ �
𝑎𝑎 − 𝑏𝑏2
𝑎𝑎 + 𝑏𝑏2
+ �
2 𝑎𝑎 − 𝑏𝑏2
2 𝑎𝑎 + 𝑏𝑏2
+ ⋯ + �
𝑁𝑁−1 𝑎𝑎 − 𝑏𝑏2
𝑁𝑁−1 𝑎𝑎 + 𝑏𝑏2
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦 𝑑𝑑𝑦𝑦
𝜃𝜃
𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃 =𝑌𝑌𝑅𝑅
47
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑏𝑏 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑏𝑏2 𝑅𝑅
�𝑙𝑙= 0
𝑁𝑁−1
𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑎𝑎𝑅𝑅𝑙𝑙
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑏𝑏 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑏𝑏2 𝑅𝑅
1 − 𝑒𝑒−𝑖𝑖 𝑁𝑁𝑘𝑘 𝑌𝑌 𝑎𝑎𝑅𝑅
1 − 𝑒𝑒−𝑖𝑖𝑘𝑘 𝑌𝑌 𝑎𝑎𝑅𝑅
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑏𝑏 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑏𝑏2 𝑅𝑅
𝑒𝑒−𝑖𝑖 𝑁𝑁𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅
𝑒𝑒−𝑖𝑖𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅
𝑒𝑒+𝑖𝑖 𝑁𝑁𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅 − 𝑒𝑒−𝑖𝑖 𝑁𝑁
𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅
𝑒𝑒+𝑖𝑖𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅 − 𝑒𝑒−𝑖𝑖
𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡
𝑅𝑅𝑏𝑏 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠
𝑘𝑘 𝑌𝑌 𝑏𝑏2 𝑅𝑅
𝑒𝑒−𝑖𝑖 𝑁𝑁𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅
𝑒𝑒−𝑖𝑖𝑘𝑘 𝑌𝑌 𝑎𝑎2 𝑅𝑅
sin 𝑁𝑁 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
sin 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
48
𝐼𝐼 𝑌𝑌,𝑍𝑍 = 𝐼𝐼0 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠2𝑘𝑘 𝑌𝑌 𝑏𝑏
2 𝑅𝑅
𝑠𝑠𝑖𝑖𝑠𝑠2 𝑁𝑁 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
𝑠𝑠𝑖𝑖𝑠𝑠2 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
𝐼𝐼0 ≡𝐸𝐸0 2
2 𝑅𝑅2𝑏𝑏2
Intensity Pattern
Mathematica
𝑏𝑏 = 1
𝑚𝑚 = 4
𝑘𝑘 = 1
𝑅𝑅 = 1
49
𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠2𝑘𝑘 𝑌𝑌 𝑏𝑏
2 𝑅𝑅≅ 1
𝐼𝐼 𝑌𝑌,𝑍𝑍 ≅ 𝐼𝐼0𝑠𝑠𝑖𝑖𝑠𝑠2 𝑁𝑁 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
𝑠𝑠𝑖𝑖𝑠𝑠2 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
Small Width Approximation:
𝑏𝑏 = 0.1
𝑚𝑚 = 4
𝑘𝑘 = 1
𝑅𝑅 = 1
50
𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
= 𝑚𝑚 𝜋𝜋 𝐼𝐼 𝑌𝑌,𝑍𝑍, 𝑡𝑡 = 𝑁𝑁2 𝐼𝐼0
Maxima (intensity peaks)
𝑚𝑚 = 0, ±1, ±2, …
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 = 𝑚𝑚 𝜆𝜆grating equation
grating order
51
𝑁𝑁𝑘𝑘 𝑌𝑌 𝑚𝑚
2 𝑅𝑅= 𝑟𝑟 𝜋𝜋
𝑟𝑟 = 1, 2, 3, … , (𝑁𝑁 − 1)
Minima (zero intensity)
𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
=𝑟𝑟𝑁𝑁𝜋𝜋
𝑏𝑏 = 0.1
𝑚𝑚 = 4
𝑘𝑘 = 1
𝑅𝑅 = 10 <
𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
< 𝜋𝜋
𝑚𝑚 = 0 𝑚𝑚 = 1
10−1 𝑚𝑚2−2
𝐼𝐼 𝑌𝑌,𝑍𝑍 ≅ 𝐼𝐼0𝑠𝑠𝑖𝑖𝑠𝑠2 𝑁𝑁 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
𝑠𝑠𝑖𝑖𝑠𝑠2 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
52
Angular Width
𝑘𝑘 𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 + ∆𝜃𝜃2
2= 𝑚𝑚 𝜋𝜋 +
1𝑁𝑁𝜋𝜋
𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅
=𝑘𝑘 𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃
2
∆𝜃𝜃 =2 𝜆𝜆
𝑁𝑁 𝑚𝑚 𝑠𝑠𝑐𝑐𝑠𝑠 𝜃𝜃𝑚𝑚
𝑘𝑘 𝑚𝑚 𝑠𝑠𝑐𝑐𝑠𝑠 𝜃𝜃𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 ∆𝜃𝜃2
2≅
1𝑁𝑁𝜋𝜋
𝑚𝑚
53
Spectral Resolution
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 = 𝑚𝑚 𝜆𝜆
𝑚𝑚 𝑠𝑠𝑐𝑐𝑠𝑠 𝜃𝜃𝑚𝑚 𝑑𝑑𝜃𝜃 = 𝑚𝑚 𝑑𝑑𝜆𝜆
∆𝜆𝜆𝑟𝑟𝑝𝑝𝑐𝑐 =𝜆𝜆
𝑚𝑚 𝑁𝑁
𝑑𝑑𝜃𝜃 ≡∆𝜃𝜃2
=𝜆𝜆
𝑁𝑁 𝑚𝑚 𝑠𝑠𝑐𝑐𝑠𝑠 𝜃𝜃𝑚𝑚𝑑𝑑𝜆𝜆 ≡ ∆𝜆𝜆𝑟𝑟𝑝𝑝𝑐𝑐
54
Free Spectral Range
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃 = 𝑚𝑚 + 1 𝜆𝜆 = 𝑚𝑚 𝜆𝜆 + ∆𝜆𝜆𝐹𝐹𝐹𝐹𝑅𝑅
∆𝜆𝜆𝐹𝐹𝐹𝐹𝑅𝑅 =𝜆𝜆𝑚𝑚
55
Oblique Incidence
Normal Incidence
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃 − 𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑖𝑖𝑙𝑙𝑐𝑐 = 𝑚𝑚 𝜆𝜆
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 − 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑖𝑖𝑙𝑙𝑐𝑐 = 𝑚𝑚 𝜆𝜆
𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 = 𝑚𝑚 𝜆𝜆
56
Fresnel Diffraction
Going beyond the Fraunhofer (far-field) approximation
or
getting closer to the aperture
57
𝑟𝑟 𝑦𝑦, 𝑧𝑧 = 𝑠𝑠2 + 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2
𝑟𝑟
𝑠𝑠
𝑍𝑍
𝑌𝑌
𝑧𝑧
𝑦𝑦
𝑟𝑟 𝑦𝑦, 𝑧𝑧 ≅ 𝑠𝑠 +1
2 𝑠𝑠𝑌𝑌 − 𝑦𝑦 2 +
12 𝑠𝑠
𝑍𝑍 − 𝑧𝑧 2
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 = �𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧𝑟𝑟 𝑦𝑦, 𝑧𝑧
𝑒𝑒𝑖𝑖 𝑘𝑘 𝑟𝑟 𝑦𝑦, 𝑧𝑧 − 𝜔𝜔 𝑡𝑡 + 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
= 𝑠𝑠 1 +𝑌𝑌 − 𝑦𝑦 2
𝑠𝑠2+
𝑍𝑍 − 𝑧𝑧 2
𝑠𝑠2
𝑘𝑘 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2 2
𝑠𝑠4≪ 𝜋𝜋
58
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡
𝑠𝑠�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖𝑘𝑘2 𝑐𝑐 𝑌𝑌−𝑦𝑦 2+ 𝑍𝑍−𝑧𝑧 2
𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡
𝑠𝑠�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑒𝑒𝑖𝑖𝜋𝜋𝜆𝜆 𝑐𝑐 𝑌𝑌−𝑦𝑦 2+ 𝑍𝑍−𝑧𝑧 2
𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 =𝐸𝐸0
0
Inside the aperture
Outside the aperture{
Flat Wavefront Illumination
59
𝛾𝛾 ≡2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑦𝑦
𝑑𝑑𝑦𝑦 = −𝜆𝜆 𝑠𝑠2
𝑑𝑑𝛾𝛾
𝛿𝛿 ≡2𝜆𝜆 𝑠𝑠
𝑍𝑍 − 𝑧𝑧
𝑑𝑑𝑧𝑧 = −𝜆𝜆 𝑠𝑠2
𝑑𝑑𝛿𝛿
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡
𝑠𝑠�
𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑒𝑒𝑖𝑖𝜋𝜋𝜆𝜆 𝑐𝑐 𝑌𝑌−𝑦𝑦 2+ 𝑍𝑍−𝑧𝑧 2
𝑑𝑑𝑦𝑦 𝑑𝑑𝑧𝑧
=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡
𝑠𝑠𝜆𝜆 𝑠𝑠2
�𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝
𝑒𝑒𝑖𝑖𝜋𝜋2 𝛾𝛾2+ 𝛿𝛿2 𝑑𝑑𝛾𝛾 𝑑𝑑𝛿𝛿
=𝜆𝜆 𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡
2�𝛾𝛾1
𝛾𝛾2
𝑒𝑒𝑖𝑖𝜋𝜋2 𝛾𝛾
2𝑑𝑑𝛾𝛾 �
𝛿𝛿1
𝛿𝛿2
𝑒𝑒𝑖𝑖𝜋𝜋2 𝛿𝛿
2𝑑𝑑𝛿𝛿
60
�𝛾𝛾1
𝛾𝛾2
𝑒𝑒𝑖𝑖𝜋𝜋2 𝛾𝛾
2𝑑𝑑𝛾𝛾 = �
𝛾𝛾1
𝛾𝛾2
cos𝜋𝜋2𝛾𝛾2 𝑑𝑑𝛾𝛾 + 𝑖𝑖 �
𝛾𝛾1
𝛾𝛾2
sin𝜋𝜋2𝛾𝛾2 𝑑𝑑𝛾𝛾
= 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 + 𝑖𝑖 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1
�𝛿𝛿1
𝛿𝛿2
𝑒𝑒𝑖𝑖𝜋𝜋2 𝛿𝛿
2𝑑𝑑𝛿𝛿 = �
𝛿𝛿1
𝛿𝛿2
cos𝜋𝜋2𝛿𝛿2 𝑑𝑑𝛿𝛿 + 𝑖𝑖 �
𝛿𝛿1
𝛿𝛿2
sin𝜋𝜋2𝛿𝛿2 𝑑𝑑𝛿𝛿
= 𝒞𝒞 𝛿𝛿2 − 𝒞𝒞 𝛿𝛿1 + 𝑖𝑖 𝒮𝒮 𝛿𝛿2 − 𝒮𝒮 𝛿𝛿1
𝒞𝒞 𝑚𝑚 ≡ �0
𝑚𝑚
cos𝜋𝜋2𝑚𝑚2 𝑑𝑑𝑚𝑚 𝒮𝒮 𝑚𝑚 ≡ �
0
𝑚𝑚
sin𝜋𝜋2𝑚𝑚2 𝑑𝑑𝑚𝑚
61
× 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 + 𝑖𝑖 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1
× 𝒞𝒞 𝛿𝛿2 − 𝒞𝒞 𝛿𝛿1 + 𝑖𝑖 𝒮𝒮 𝛿𝛿2 − 𝒮𝒮 𝛿𝛿1
𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝜆𝜆 𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡
2
𝐼𝐼 𝑌𝑌,𝑍𝑍 =𝐼𝐼04 × 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 2 + 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1 2
× 𝒞𝒞 𝛿𝛿2 − 𝒞𝒞 𝛿𝛿1 2 + 𝒮𝒮 𝛿𝛿2 − 𝒮𝒮 𝛿𝛿1 2
62
𝒞𝒞 𝑚𝑚 ≡ �0
𝑚𝑚
cos𝜋𝜋2𝑚𝑚′2 𝑑𝑑𝑚𝑚𝑥
𝒮𝒮 𝑚𝑚 ≡ �0
𝑚𝑚
sin𝜋𝜋2𝑚𝑚𝑥2 𝑑𝑑𝑚𝑚𝑥
𝒞𝒞 𝑚𝑚
𝒮𝒮 𝑚𝑚
𝑚𝑚
𝑚𝑚
𝑚𝑚
𝒞𝒞 𝑚𝑚𝒮𝒮 𝑚𝑚
63
𝒞𝒞 𝑚𝑚 ≡ �0
𝑚𝑚
cos𝜋𝜋2𝑚𝑚2 𝑑𝑑𝑚𝑚
𝒮𝒮 𝑚𝑚 ≡ �0
𝑚𝑚
sin𝜋𝜋2𝑚𝑚2 𝑑𝑑𝑚𝑚
𝑑𝑑𝒞𝒞 𝑚𝑚 = cos𝜋𝜋2𝑚𝑚2 𝑑𝑑𝑚𝑚
𝑑𝑑𝒮𝒮 𝑚𝑚 = sin𝜋𝜋2𝑚𝑚2 𝑑𝑑𝑚𝑚
𝒮𝒮 𝑚𝑚
𝒞𝒞 𝑚𝑚
𝑑𝑑𝒞𝒞 2 + 𝑑𝑑𝒮𝒮 2 = 𝑑𝑑𝑚𝑚 2
𝑑𝑑𝒞𝒞𝑑𝑑𝒮𝒮𝑑𝑑𝑚𝑚
64
Applications of Fresnel Diffraction1.No obstruction
2.Straight edge
3. Single slit
4. Rectangular aperture
5. Opaque circular disk
65
𝐼𝐼 𝑌𝑌,𝑍𝑍 =𝐼𝐼04 × 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 2 + 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1 2
× 𝒞𝒞 𝛿𝛿2 − 𝒞𝒞 𝛿𝛿1 2 + 𝒮𝒮 𝛿𝛿2 − 𝒮𝒮 𝛿𝛿1 2
1. No Obstruction
𝛾𝛾 ≡2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑦𝑦
𝛿𝛿 ≡2𝜆𝜆 𝑠𝑠
𝑍𝑍 − 𝑧𝑧
𝑦𝑦
𝑧𝑧
𝛾𝛾2 = −∞
𝛾𝛾1 = +∞𝛿𝛿2 = −∞ 𝛿𝛿1 = +∞
=𝐼𝐼04
× −0.5 − 0.5 2 + −0.5 − 0.5 2 × −0.5 − 0.5 2 + −0.5 − 0.5 2
= 𝐼𝐼0 No surprises here, just the obvious result !!
66
𝐼𝐼 𝑌𝑌,𝑍𝑍 =𝐼𝐼04
× 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 2 + 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1 2
× 𝒞𝒞 𝛿𝛿2 − 𝒞𝒞 𝛿𝛿1 2 + 𝒮𝒮 𝛿𝛿2 − 𝒮𝒮 𝛿𝛿1 2
𝛾𝛾 ≡2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑦𝑦
𝛿𝛿 ≡2𝜆𝜆 𝑠𝑠
𝑍𝑍 − 𝑧𝑧
𝑦𝑦
𝑧𝑧𝛾𝛾2 = 2
𝜆𝜆 𝑐𝑐𝑌𝑌
𝛾𝛾1 = +∞𝛿𝛿2 = −∞ 𝛿𝛿1 = +∞
=𝐼𝐼04
× 𝒞𝒞2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 0.5
2
+ 𝒮𝒮2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 0.5
2
× 2
2. Straight Edge
67
𝒮𝒮 𝑚𝑚
𝒞𝒞 𝑚𝑚𝑌𝑌 = 0
𝑌𝑌 > 0
𝑌𝑌 < 0 𝐼𝐼 𝑌𝑌,𝑍𝑍, 𝑡𝑡 /𝐼𝐼0
𝑌𝑌
𝜆𝜆 𝑠𝑠 = 2
𝐼𝐼 𝑌𝑌,𝑍𝑍 =𝐼𝐼02
× 𝒞𝒞2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 0.5
2
+ 𝒮𝒮2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 0.5
2
68
69
𝐼𝐼 𝑌𝑌,𝑍𝑍 =𝐼𝐼04
× 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 2 + 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1 2
× 𝒞𝒞 𝛿𝛿2 − 𝒞𝒞 𝛿𝛿1 2 + 𝒮𝒮 𝛿𝛿2 − 𝒮𝒮 𝛿𝛿1 2
𝛾𝛾 ≡2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑦𝑦
𝛿𝛿 ≡2𝜆𝜆 𝑠𝑠
𝑍𝑍 − 𝑧𝑧
𝑦𝑦
𝑧𝑧𝛾𝛾2 =
2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑑𝑑2
𝛾𝛾1 =2𝜆𝜆 𝑠𝑠
𝑌𝑌 + 𝑑𝑑2𝛿𝛿2 = −∞ 𝛿𝛿1 = +∞
=𝐼𝐼04
× 𝒞𝒞2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑑𝑑2 − 𝒞𝒞
2𝜆𝜆 𝑠𝑠
𝑌𝑌 + 𝑑𝑑2
2
+ 𝒮𝒮2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑑𝑑2 − 𝒮𝒮
2𝜆𝜆 𝑠𝑠
𝑌𝑌 + 𝑑𝑑2
2
× 2
3. Single Slit
𝑑𝑑
70
𝒮𝒮 𝑚𝑚
𝒞𝒞 𝑚𝑚𝑌𝑌 = 0
𝑌𝑌 > 0
𝑌𝑌 < 0
𝐼𝐼 𝑌𝑌,𝑍𝑍 =𝐼𝐼02
× 𝒞𝒞2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑑𝑑2 − 𝒞𝒞
2𝜆𝜆 𝑠𝑠
𝑌𝑌 + 𝑑𝑑2
2
+ 𝒮𝒮2𝜆𝜆 𝑠𝑠
𝑌𝑌 − 𝑑𝑑2 − 𝒮𝒮
2𝜆𝜆 𝑠𝑠
𝑌𝑌 + 𝑑𝑑2
2
𝛾𝛾1 − 𝛾𝛾2 =2𝜆𝜆 𝑠𝑠
𝑑𝑑
𝛾𝛾1 + 𝛾𝛾22
=2𝜆𝜆 𝑠𝑠
𝑌𝑌
71
𝑑𝑑 = 10 𝜆𝜆
𝑑𝑑
𝑁𝑁𝐹𝐹 ≡𝑑𝑑2
4 𝜆𝜆 𝑠𝑠
𝑁𝑁𝐹𝐹 = 10
𝑁𝑁𝐹𝐹 = 1
𝑁𝑁𝐹𝐹 = 0.5
𝑁𝑁𝐹𝐹 = 0.1
𝜆𝜆 = 1
𝑠𝑠 = 2.5 𝜆𝜆
𝑠𝑠 = 25 𝜆𝜆
𝑠𝑠 = 50 𝜆𝜆
𝑠𝑠 = 250 𝜆𝜆
Near field
Far field
Fresnel number
72
Mathematica
73
4. Rectangular Aperture
74
5. Circular Objects
Poisson (Arago) spot