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Diffraction Theory
1
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2
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3
𝑟1
𝑟2
𝐸 Ԧ𝑟, 𝑡 = 𝐸1 Ԧ𝑟, 𝑡
𝐸 Ԧ𝑟, 𝑡
𝐸 Ԧ𝑟, 𝑡 =𝐸0,1
𝑟1𝑒𝑖 𝑘 𝑟1−𝜔 𝑡 + 𝜀1 +
𝐸0,2𝑟2
𝑒𝑖 𝑘 𝑟2−𝜔 𝑡 + 𝜀2
+ 𝐸2 Ԧ𝑟, 𝑡
𝑟3
+𝐸0,3𝑟3
𝑒𝑖 𝑘 𝑟3−𝜔 𝑡 + 𝜀3
𝑟4
+ 𝐸3 Ԧ𝑟, 𝑡 + 𝐸4 Ԧ𝑟, 𝑡
𝑟5
+𝐸0,4𝑟4
𝑒𝑖 𝑘 𝑟4 −𝜔 𝑡 + 𝜀4
+ 𝐸5 Ԧ𝑟, 𝑡
+𝐸0,5𝑟5
𝑒𝑖 𝑘 𝑟5 −𝜔 𝑡 + 𝜀5
=
𝑖
𝐸𝑖 Ԧ𝑟, 𝑡
+⋯
+ …
=
𝑖
𝐸0,𝑖𝑟𝑖
𝑒𝑖 𝑘 𝑟𝑖 −𝜔 𝑡 + 𝜀𝑖
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4
Huygens-Fresnel Principle
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5
𝐸 𝑌, 𝑍, 𝑡 =
𝑖
𝐸0,𝑖𝑟𝑖
𝑒𝑖 𝑘 𝑟𝑖 −𝜔 𝑡 + 𝜀𝑖
= ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧
𝑟 𝑦, 𝑧𝑒𝑖 𝑘 𝑟 𝑦, 𝑧 − 𝜔 𝑡 + 𝜀 𝑦, 𝑧 𝑑𝑦 𝑑𝑧
𝑟 𝑦, 𝑧 = 𝑠2 + 𝑌 − 𝑦 2 + 𝑍 − 𝑧 2
𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑖 𝑦, 𝑧
𝑖
ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑑𝑦 𝑑𝑧
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6
𝑟 𝑦, 𝑧 ≅ 𝑠 1 +𝜁2
2
Fresnel Approximation𝑘 𝑠 𝑚𝑎𝑥𝜁4
8≪ 𝜋
= 𝑠 1 +𝜁2
2−𝜁4
8+𝜁6
16−5 𝜁8
128+ ⋯
𝑟 𝑦, 𝑧 = 𝑠2 + 𝑌 − 𝑦 2 + 𝑍 − 𝑧 2 = 𝑠 1 +𝑌 − 𝑦 2
𝑠2+
𝑍 − 𝑧 2
𝑠2 = 𝑠 1 + 𝜁2
𝜁2 ≡𝑌 − 𝑦 2
𝑠2+
𝑍 − 𝑧 2
𝑠2
Fresnel Diffraction
= 𝑠 +𝑌2 + 𝑍2
2 𝑠−
𝑌 𝑦 + 𝑍 𝑧
𝑠+
𝑦2 + 𝑧2
2 𝑠
= 𝑠 1 +𝑌 − 𝑦 2
2 𝑠2+
𝑍 − 𝑧 2
2 𝑠2
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7
𝑟 𝑦, 𝑧 ≅ 𝑠 +𝑌2 + 𝑍2
2 𝑠−
𝑌 𝑦 + 𝑍 𝑧
𝑠+
𝑦2 + 𝑧2
2 𝑠
Fresnel Approximation𝑘 𝑠 𝑚𝑎𝑥𝜁4
8≪ 𝜋
𝐸 𝑌, 𝑍, 𝑡 = ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧
𝑟 𝑦, 𝑧𝑒𝑖 𝑘 𝑟 𝑦, 𝑧 − 𝜔 𝑡 + 𝜀 𝑦, 𝑧 𝑑𝑦 𝑑𝑧
≅ ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧
𝑠𝑒𝑖 𝑘 𝑠 +
𝑌2+𝑍2
2 𝑠 −𝑌 𝑦+𝑍 𝑧
𝑠 +𝑦2+𝑧2
2 𝑠 −𝜔 𝑡 + 𝜀 𝑦, 𝑧 𝑑𝑦 𝑑𝑧
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𝑘𝑚𝑎𝑥 𝑦2 + 𝑧2
2 𝑠≪ 𝜋
𝑚𝑎𝑥 𝑦2 + 𝑧2
𝜆 𝑠≪ 1
Fraunhofer Approximation
Fraunhofer Diffractionalso known as Far-Field Diffraction
𝑟 𝑦, 𝑧 ≅ 𝑠 +𝑌2 + 𝑍2
2 𝑠−
𝑌 𝑦 + 𝑍 𝑧
𝑠+
𝑦2 + 𝑧2
2 𝑠
Fresnel Approximation𝑘 𝑠 𝑚𝑎𝑥𝜁4
8≪ 𝜋
In addition to Fresnel Approximation:
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9
𝐸 𝑌, 𝑍, 𝑡 = ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧
𝑟 𝑦, 𝑧𝑒𝑖 𝑘 𝑟 𝑦, 𝑧 − 𝜔 𝑡 + 𝜀 𝑦, 𝑧 𝑑𝑦 𝑑𝑧
≅ ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧
𝑅𝑒𝑖 𝑘 𝑅 −
𝑌 𝑦 + 𝑍 𝑧𝑅 −𝜔 𝑡 + 𝜀 𝑦, 𝑧 𝑑𝑦 𝑑𝑧
=𝑒𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦,𝑧 𝑒− 𝑖
𝑘 𝑌 𝑦 + 𝑍 𝑧𝑅 𝑑𝑦 𝑑𝑧
𝑟 𝑦, 𝑧 = 𝑠2 + 𝑌 − 𝑦 2 + 𝑍 − 𝑧 2
𝑅2 ≡ 𝑠2 + 𝑌2 + 𝑍2
= 𝑅2 − 2 𝑌 𝑦 − 2 𝑍 𝑧 + 𝑦2 + 𝑧2
= 𝑅 1 +−2 𝑌 𝑦 − 2 𝑍 𝑧 + 𝑦2 + 𝑧2
𝑅2≅ 𝑅 −
𝑌 𝑦 + 𝑍 𝑧
𝑅
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10
𝐸 𝑌, 𝑍, 𝑡 =𝑒𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦, 𝑧 𝑒− 𝑖
𝑘 𝑌 𝑦 + 𝑍 𝑧𝑅 𝑑𝑦 𝑑𝑧
𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑅
Fraunhofer Diffraction
𝑅2 ≡ 𝑠2 + 𝑌2 + 𝑍2
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Illumination at the Aperture:
In the examples to follow, we will consider a flat wavefront at
normal incidence on the aperture
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦, 𝑧 =
𝐸0
0
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑒− 𝑖𝑘 𝑌 𝑦 + 𝑍 𝑧
𝑅 𝑑𝑦 𝑑𝑧
Inside the aperture
Outside the aperture{
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Apertures considered here:
1. Single Slit
2. Double Slit
3. Rectangular Aperture
4. Circular Aperture
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1. Single Slit
𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑅
𝑅 ≡ 𝑌2 + 𝑠2
𝑑
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
− ൗ𝑑 2
+ ൗ𝑑 2
𝑒− 𝑖𝑘 𝑌𝑅 𝑦𝑑𝑦
𝜃
𝑠𝑖𝑛 𝜃 =𝑌
𝑅
𝑦
𝑧
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𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑑 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑑
2 𝑅
1. Single Slit, cont.
𝐼 ≡ 𝐸2
𝐼 𝑌, 𝑍 = 𝐼0 𝑠𝑖𝑛𝑐2
𝑘 𝑌 𝑑
2 𝑅𝐼0 ≡
𝐸02
2 𝑅2𝑑2
𝑑 = 50 µ𝑚
𝜆 = 0.6 µ𝑚
𝑠 = 1 𝑚
𝑘 𝑌𝑚 𝑑
2 𝑅= 𝑚 𝜋
𝑚 = ±1, ±2,±3
𝑅 ≅ 1 𝑚
𝑌𝑚 = 𝑚𝜆 𝑅
𝑑
𝑠𝑖𝑛 𝜃𝑚 =𝑌𝑚𝑅
= 𝑚𝜆
𝑑
𝑌(𝑚𝑚)
𝑌1
ൗ𝐼 𝐼0
zeros atgeometrical
shadow
𝑌−1
with
𝑌
𝑍
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Mathematica
FraunhoferDiffractionAtASingleSlit.cdf
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2. Double Slit
𝑑
𝑑
𝑎
𝑦
ൗ𝑎 2 − ൗ𝑑2
ൗ𝑎 2 + ൗ𝑑2
ൗ−𝑎 2 − ൗ𝑑2
ൗ−𝑎 2 + ൗ𝑑2
𝑧
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𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑅
𝑅 ≡ 𝑌2 + 𝑠2
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
ൗ−𝑎 2− ൗ𝑑2
ൗ−𝑎 2+ ൗ𝑑2
𝑒− 𝑖𝑘 𝑌𝑅
𝑦𝑑𝑦 + න
ൗ𝑎 2− ൗ𝑑2
ൗ𝑎 2+ ൗ𝑑2
𝑒− 𝑖𝑘 𝑌𝑅
𝑦𝑑𝑦
𝜃
𝑠𝑖𝑛 𝜃 =𝑌
𝑅
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑑 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑑
2 𝑅2 𝑐𝑜𝑠
𝑘 𝑍 𝑎
2 𝑅
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𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑑 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑑
2 𝑅2 𝑐𝑜𝑠
𝑘 𝑌 𝑎
2 𝑅
𝐼 𝑌, 𝑍 = 4 𝐼0 𝑠𝑖𝑛𝑐2
𝑘 𝑌 𝑑
2 𝑅𝑐𝑜𝑠2
𝑘 𝑌 𝑎
2 𝑅𝐼0 ≡
𝐸02
2 𝑅2𝑑2
Mathematica
𝑑
𝑎
DoubleSlitDiffractionForParticles.cdf
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3. Rectangular Aperture
𝑎
𝑏
𝑦
𝑧
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𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑒− 𝑖𝑘 𝑌 𝑦 + 𝑍 𝑧
𝑅 𝑑𝑦 𝑑𝑧
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
ൗ−𝑏 2
ൗ𝑏 2
𝑒− 𝑖𝑘 𝑌𝑅 𝑦𝑑𝑦 න
ൗ−𝑎 2
ൗ𝑎 2
𝑒− 𝑖𝑘 𝑍𝑅 𝑧𝑑𝑧
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑏 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑏
2 𝑅𝑎 𝑠𝑖𝑛𝑐
𝑘 𝑍 𝑎
2 𝑅
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𝑌
𝑍
𝐼 𝑌, 𝑍 = 𝐼0 𝑠𝑖𝑛𝑐2
𝑘 𝑌 𝑏
2 𝑅𝑠𝑖𝑛𝑐2
𝑘 𝑍 𝑎
2 𝑅
𝐼0 ≡𝐸0
2
2 𝑅2𝑎2 𝑏2
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Emission of Semiconductor Laser
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4. Circular Aperture
𝑎𝜑
𝑦 = 𝜌 𝑠𝑖𝑛 𝜑𝜌
𝑧 = 𝜌 𝑐𝑜𝑠 𝜑𝑧
𝑦
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Observation Plane
Φ
𝑌 = 𝑞 𝑠𝑖𝑛 Φ
𝑞
𝑍 = 𝑞 𝑐𝑜𝑠 Φ𝑍
𝑌
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𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑒− 𝑖𝑘 𝑌 𝑦 + 𝑍 𝑧
𝑅 𝑑𝑦 𝑑𝑧
𝑌 𝑦 + 𝑍 𝑧 = 𝑞 𝑠𝑖𝑛 Φ 𝜌 𝑠𝑖𝑛 𝜑 + 𝑞 𝑐𝑜𝑠 Φ 𝜌 𝑐𝑜𝑠 𝜑
= 𝜌 𝑞 𝑐𝑜𝑠 𝜑 − Φ
𝑑𝑦 𝑑𝑧 = 𝜌 𝑑𝜑 𝑑𝜌
𝐸 𝑞,Φ, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
0
𝑎
𝜌 𝑑𝜌න
0
2𝜋
𝑑𝜑 𝑒 − 𝑖𝑘 𝜌 𝑞 𝑐𝑜𝑠 𝜑 −Φ
𝑅
Φ = 0Due to axial symmetry, we can choose:
= 𝑞 𝜌 𝑐𝑜𝑠 Φ 𝑐𝑜𝑠 𝜑 + 𝑠𝑖𝑛 Φ 𝑠𝑖𝑛 𝜑
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𝐸 𝑞,Φ, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
0
𝑎
𝜌 𝑑𝜌න
0
2𝜋
𝑑𝜑 𝑒 − 𝑖𝑘 𝜌 𝑞 𝑐𝑜𝑠 𝜑
𝑅
A couple of integrals to solve:
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27
1
2 𝜋න
0
2𝜋
𝑑𝜑 𝑒𝑖 𝑢 𝑐𝑜𝑠 𝜑 ≡ 𝐽0 𝑢Bessel function of order zero
𝐸 𝑞,Φ, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
0
𝑎
𝜌 𝑑𝜌න
0
2𝜋
𝑑𝜑 𝑒 − 𝑖𝑘 𝜌 𝑞 𝑐𝑜𝑠 𝜑
𝑅
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28
𝐸 𝑞,Φ, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 − 𝜔 𝑡
𝑅2 𝜋න
0
𝑎
𝜌 𝑑𝜌 𝐽0 −𝑘 𝑞
𝑅𝜌
𝑢 ≡ −𝑘 𝑞
𝑅𝜌
=𝐸0 𝑒
𝑖 𝑘 𝑅 − 𝜔 𝑡
𝑅2 𝜋
𝑅
𝑘 𝑞
2
න
0
−𝑘 𝑞𝑅 𝑎
α 𝑑α 𝐽0 α
𝛼 ≡−𝑘 𝑞
𝑅𝜌 𝜌 𝑑𝜌 =
𝑅
𝑘 𝑞
2
α 𝑑α
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29
න
0
𝛼
𝛼 𝐽0 𝛼 𝑑𝛼 ≡ 𝛼 𝐽1 𝛼
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𝐸 𝑞,Φ, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 − 𝜔 𝑡
𝑅2 𝜋
𝑅
𝑘 𝑞
2
න
0
−𝑘 𝑞𝑅 𝑎
α 𝑑α 𝐽0 α
=𝐸0 𝑒
𝑖 𝑘 𝑅 − 𝜔 𝑡
𝑅2 𝜋
𝑅
𝑘 𝑞
2−𝑘 𝑎 𝑞
𝑅𝐽1
−𝑘 𝑎 𝑞
𝑅
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝜋 𝑎2
2 𝐽1𝑘 𝑎 𝑞𝑅
𝑘 𝑎 𝑞𝑅
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31
𝐼 𝑞,Φ = 𝐼0
2 𝐽1𝑘 𝑎 𝑞𝑅
𝑘 𝑎 𝑞𝑅
2
𝐼0 ≡𝐸0
2
2 𝑅2𝜋 𝑎2 2
𝑘 𝑎 𝑞
𝑅
ൗ𝐼 𝐼0
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32
zeros at𝑘 𝑎 𝑞
𝑅= 3.832, 7.016, 10.173, …
𝑘 𝑎 𝑞1𝑅
= 3.832
𝑞1𝑅= 𝑠𝑖𝑛 𝜃1 = 3.832
𝜆
2 𝜋 𝑎= 1.22
𝜆
2 𝑎
first zero at
Light is essentially confined
inside the cone: 𝒔𝒊𝒏 𝜃1 < 𝟏. 𝟐𝟐𝝀
𝟐 𝒂
-
33
Circular Aperture
𝑧
𝑦
𝑠
𝑦
𝑌
𝑍
𝑌
𝑠𝑖𝑛 𝜃1 =𝑞1𝑅= 1.22
𝜆
2 𝑎
𝑅2𝑎
Airy’spattern
𝑎𝑞1
𝑞1𝜃1
-
34
𝑧
𝑦
2𝑎
𝑠
𝑅
𝜃1
} = 0
-
35
𝑦
2𝑎
𝜃1𝜃1
𝑠𝑖𝑛 𝜃1 = 1.22𝜆
2 𝑎
tan 𝜃1 =𝑞1𝑓
𝑞1
𝑞1 ≅ 1.22𝜆 𝑓
2 𝑎
𝑓
Smallest spot size:
𝑞1 ≅ 1.22𝜆 𝑓
𝐷𝑙𝑒𝑛𝑠
𝐷𝑙𝑒𝑛𝑠
= 1.22𝜆𝑜 𝑓
𝑛 𝐷𝑙𝑒𝑛𝑠
𝑛
Smallest angular width:
𝑞1𝑓= 1.22
𝜆𝑜𝑛 𝐷𝑙𝑒𝑛𝑠
-
36
Diameter of primary mirror 2.4 m
Wavelength 0.55 µm
Angular width 0.28 × 10-6 rad
-
37
𝑡𝑎𝑛 𝜃𝑚𝑎𝑥 ≡𝐷𝑙𝑒𝑛𝑠2 𝑓
𝐷𝑙𝑒𝑛𝑠
𝜃𝑚𝑎𝑥
𝑁𝐴 ≡ 𝑛 𝑠𝑖𝑛 𝜃𝑚𝑎𝑥 ≅𝑛 𝐷𝑙𝑒𝑛𝑠2 𝑓
𝑓
𝑓
#=
𝑓
𝐷𝑙𝑒𝑛𝑠
-
38
Numerical Aperture
𝑁𝐴 ≡ 𝑛 𝑠𝑖𝑛 𝜃𝑚𝑎𝑥
-
39
𝑞1 = 1.22𝜆𝑜2 𝑁𝐴
Smallest spot size from a lens
𝑦
2𝑎 = 𝐷𝑙𝑒𝑛𝑠
𝜃1𝜃1
𝑞1
𝑓
𝐷𝑙𝑒𝑛𝑠
𝑛
𝑞1 = 1.22𝜆𝑜 𝑓
𝑛 𝐷𝑙𝑒𝑛𝑠
𝑁𝐴 ≡ 𝑛 𝑠𝑖𝑛 𝜃𝑚𝑎𝑥 ≅𝑛 𝐷𝑙𝑒𝑛𝑠2 𝑓
-
40
Rayleigh Criteria for Resolution
Barely resolved
Resolved
Not resolved
-
41
𝑞1 = 1.22𝜆𝑜2 𝑁𝐴
𝜆𝑜 = 0.55 𝜇𝑚
3.36 𝜇𝑚 1.34 𝜇𝑚 0.52 𝜇𝑚 0.27 𝜇𝑚
Examples of Diffraction Limit of Objective Lenses
-
42
𝐸 𝑌, 𝑍, 𝑡 =𝑒𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦, 𝑧 𝑒− 𝑖
𝑘 𝑌 𝑦 + 𝑍 𝑧𝑅 𝑑𝑦 𝑑𝑧
𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑅
𝑅 ≡ 𝑌2 + 𝑍2 + 𝑠2
Fraunhofer Diffraction
𝑚𝑎𝑥 𝑦2 + 𝑧2
𝜆 𝑠≪ 1
𝑚𝑎𝑥 𝑌 − 𝑦 2 + 𝑍 − 𝑧 2
𝜆 𝑠≪ 1
-
43
In summary, far-field diffraction:
1. Single Slit
2. Double Slit
3. Rectangular Aperture
4. Circular Aperture
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑑 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑑
2 𝑅
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑑 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑑
2 𝑅2 𝑐𝑜𝑠
𝑘 𝑌 𝑎
2 𝑅
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑏 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑏
2 𝑅𝑎 𝑠𝑖𝑛𝑐
𝑘 𝑍 𝑎
2 𝑅
𝐸 𝑞,Φ, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝜋 𝑎2
2 𝐽1𝑘 𝑎 𝑞𝑅
𝑘 𝑎 𝑞𝑅
-
44
𝐸 𝑌, 𝑍, 𝑡 =𝑒𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦, 𝑧 𝑒− 𝑖
𝑘 𝑌 𝑦 + 𝑍 𝑧𝑅 𝑑𝑦 𝑑𝑧
𝐸 𝑌, 𝑍, 𝑡 =𝑒𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅ඵ
−∞
+∞
𝜓 𝑦, 𝑧 𝑒− 𝑖 𝑘𝑦 𝑦 +𝑘𝑧 𝑧 𝑑𝑦 𝑑𝑧
𝜓 𝑦, 𝑧 ≡𝐸0 𝑦, 𝑧 𝑒
𝑖 𝜀 𝑦, 𝑧
0
inside aperture
opaque obstruction
𝑘𝑦 ≡𝑘 𝑌
𝑅
Fraunhofer Diffraction as a Fourier Transformation
𝑘𝑧 ≡𝑘 𝑍
𝑅
{
-
45
Diffraction Gratings
-
46
Multiple Slits
𝑏
𝑎
𝑦
𝑎 −𝑏
2
𝑎 +𝑏
2
𝑧
𝑵 (infinitely long) slits of width 𝒃 separated by distance 𝒂
+𝑏
2−𝑏
2
𝑁 − 1 𝑎 −𝑏
2
𝑁 − 1 𝑎 +𝑏
2
-
47
𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑅
𝑅 ≡ 𝑌2 + 𝑠2
𝐸 𝑌, 𝑍, 𝑡
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅න
−𝑏2
+𝑏2
+ න
𝑎 −𝑏2
𝑎 +𝑏2
+ න
2 𝑎 −𝑏2
2 𝑎 +𝑏2
+⋯ + න
𝑁−1 𝑎 −𝑏2
𝑁−1 𝑎 +𝑏2
𝑒− 𝑖𝑘 𝑌𝑅 𝑦 𝑑𝑦
𝜃
𝑠𝑖𝑛 𝜃 =𝑌
𝑅
-
48
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑏 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑏
2 𝑅
𝑛 = 0
𝑁−1
𝑒− 𝑖𝑘 𝑌 𝑎𝑅
𝑛
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑏 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑏
2 𝑅
1 − 𝑒−𝑖 𝑁𝑘 𝑌 𝑎𝑅
1 − 𝑒−𝑖𝑘 𝑌 𝑎𝑅
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑏 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑏
2 𝑅
𝑒−𝑖 𝑁𝑘 𝑌 𝑎2 𝑅
𝑒−𝑖𝑘 𝑌 𝑎2 𝑅
𝑒+𝑖 𝑁𝑘 𝑌 𝑎2 𝑅 − 𝑒−𝑖 𝑁
𝑘 𝑌 𝑎2 𝑅
𝑒+𝑖𝑘 𝑌 𝑎2 𝑅 − 𝑒−𝑖
𝑘 𝑌 𝑎2 𝑅
=𝐸0 𝑒
𝑖 𝑘 𝑅 −𝜔 𝑡
𝑅𝑏 𝑠𝑖𝑛𝑐
𝑘 𝑌 𝑏
2 𝑅
𝑒−𝑖 𝑁𝑘 𝑌 𝑎2 𝑅
𝑒−𝑖𝑘 𝑌 𝑎2 𝑅
sin 𝑁𝑘 𝑌 𝑎2 𝑅
sin𝑘 𝑌 𝑎2 𝑅
-
49
𝐼 𝑌, 𝑍 = 𝐼0 𝑠𝑖𝑛𝑐2
𝑘 𝑌 𝑏
2 𝑅
𝑠𝑖𝑛2 𝑁𝑘 𝑌 𝑎2 𝑅
𝑠𝑖𝑛2𝑘 𝑌 𝑎2 𝑅
𝐼0 ≡𝐸0
2
2 𝑅2𝑏2
Intensity Pattern
Mathematica
𝑏 = 1
𝑎 = 4
𝑘 = 1
𝑅 = 1
MultipleSlitDiffractionPattern.cdf
-
50
𝑠𝑖𝑛𝑐2𝑘 𝑌 𝑏
2 𝑅≅ 1
𝐼 𝑌, 𝑍 ≅ 𝐼0
𝑠𝑖𝑛2 𝑁𝑘 𝑌 𝑎2 𝑅
𝑠𝑖𝑛2𝑘 𝑌 𝑎2 𝑅
Small Width Approximation:
𝑏 = 0.1
𝑎 = 4
𝑘 = 1
𝑅 = 1
-
51
𝑘 𝑌 𝑎
2 𝑅= 𝑚 𝜋 𝐼 𝑌, 𝑍, 𝑡 = 𝑁2 𝐼0
Maxima (intensity peaks)
𝑚 = 0,±1,±2,…
𝑎 𝑠𝑖𝑛 𝜃𝑚 = 𝑚 𝜆grating equation
grating order
-
52
𝑁𝑘 𝑌 𝑎
2 𝑅= 𝑟 𝜋
𝑟 = 1, 2, 3, … , (𝑁 − 1)
Minima (zero intensity)
𝑘 𝑌 𝑎
2 𝑅=𝑟
𝑁𝜋
𝑏 = 0.1
𝑎 = 4
𝑘 = 1
𝑅 = 1
0 <𝑘 𝑌 𝑎
2 𝑅< 𝜋
𝑚 = 0 𝑚 = 1
10−1 𝑚2−2
𝐼 𝑌, 𝑍 ≅ 𝐼0
𝑠𝑖𝑛2 𝑁𝑘 𝑌 𝑎2 𝑅
𝑠𝑖𝑛2𝑘 𝑌 𝑎2 𝑅
-
53
Angular Width
𝑘 𝑎 𝑠𝑖𝑛 𝜃𝑚 +∆𝜃2
2= 𝑚 𝜋 +
1
𝑁𝜋
𝑘 𝑌 𝑎
2 𝑅=𝑘 𝑎 𝑠𝑖𝑛 𝜃
2
∆𝜃 =2 𝜆
𝑁 𝑎 𝑐𝑜𝑠 𝜃𝑚
𝑘 𝑎 𝑐𝑜𝑠 𝜃𝑚 𝑠𝑖𝑛∆𝜃2
2≅1
𝑁𝜋
𝑚
-
54
Spectral Resolution
𝑎 𝑠𝑖𝑛 𝜃𝑚 = 𝑚 𝜆
𝑎 𝑐𝑜𝑠 𝜃𝑚 𝑑𝜃 = 𝑚 𝑑𝜆
∆𝜆𝑟𝑒𝑠 =𝜆
𝑚 𝑁
𝑑𝜃 ≡∆𝜃
2=
𝜆
𝑁 𝑎 𝑐𝑜𝑠 𝜃𝑚𝑑𝜆 ≡ ∆𝜆𝑟𝑒𝑠
-
55
Free Spectral Range
𝑎 𝑠𝑖𝑛 𝜃 = 𝑚 + 1 𝜆 = 𝑚 𝜆 + ∆𝜆𝐹𝑆𝑅
∆𝜆𝐹𝑆𝑅 =𝜆
𝑚
-
56
Oblique Incidence
Normal Incidence
𝑎 𝑠𝑖𝑛 𝜃 − 𝑎 𝑠𝑖𝑛 𝜃𝑖𝑛𝑐 = 𝑚 𝜆
𝑎 𝑠𝑖𝑛 𝜃𝑚 − 𝑠𝑖𝑛 𝜃𝑖𝑛𝑐 = 𝑚 𝜆
𝑎 𝑠𝑖𝑛 𝜃𝑚 = 𝑚 𝜆
-
57
Fresnel Diffraction
Going beyond the Fraunhofer (far-field) approximation
or
getting closer to the aperture
-
58
𝑟 𝑦, 𝑧 = 𝑠2 + 𝑌 − 𝑦 2 + 𝑍 − 𝑧 2
𝑟
𝑠
𝑍
𝑌
𝑧
𝑦
𝑟 𝑦, 𝑧 ≅ 𝑠 +1
2 𝑠𝑌 − 𝑦 2 +
1
2 𝑠𝑍 − 𝑧 2
𝐸 𝑌, 𝑍, 𝑡 = ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧
𝑟 𝑦, 𝑧𝑒𝑖 𝑘 𝑟 𝑦, 𝑧 − 𝜔 𝑡 + 𝜀 𝑦, 𝑧 𝑑𝑦 𝑑𝑧
= 𝑠 1 +𝑌 − 𝑦 2
𝑠2+
𝑍 − 𝑧 2
𝑠2
𝑘 𝑠𝑚𝑎𝑥 𝑌 − 𝑦 2 + 𝑍 − 𝑧 2 2
𝑠4≪ 𝜋
-
59
𝐸 𝑌, 𝑍, 𝑡 =𝑒𝑖 𝑘 𝑠 − 𝜔 𝑡
𝑠ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦, 𝑧 𝑒𝑖
𝑘2 𝑠 𝑌−𝑦
2+ 𝑍−𝑧 2 𝑑𝑦 𝑑𝑧
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑠 − 𝜔 𝑡
𝑠ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑒𝑖𝜋𝜆 𝑠
𝑌−𝑦 2+ 𝑍−𝑧 2𝑑𝑦 𝑑𝑧
𝐸0 𝑦, 𝑧 𝑒𝑖 𝜀 𝑦, 𝑧 =
𝐸0
0
Inside the aperture
Outside the aperture{
Flat Wavefront Illumination
-
60
𝛾 ≡2
𝜆 𝑠𝑌 − 𝑦
𝑑𝑦 = −𝜆 𝑠
2𝑑𝛾
𝛿 ≡2
𝜆 𝑠𝑍 − 𝑧
𝑑𝑧 = −𝜆 𝑠
2𝑑𝛿
𝐸 𝑌, 𝑍, 𝑡 =𝐸0 𝑒
𝑖 𝑘 𝑠 − 𝜔 𝑡
𝑠ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑒𝑖𝜋𝜆 𝑠
𝑌−𝑦 2+ 𝑍−𝑧 2𝑑𝑦 𝑑𝑧
=𝐸0 𝑒
𝑖 𝑘 𝑠 − 𝜔 𝑡
𝑠
𝜆 𝑠
2ඵ
𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒
𝑒𝑖𝜋2 𝛾
2+ 𝛿2 𝑑𝛾 𝑑𝛿
=𝜆 𝐸0 𝑒
𝑖 𝑘 𝑠 − 𝜔 𝑡
2න
𝛾1
𝛾2
𝑒𝑖𝜋2 𝛾
2𝑑𝛾 න
𝛿1
𝛿2
𝑒𝑖𝜋2 𝛿
2𝑑𝛿
-
61
න
𝛾1
𝛾2
𝑒𝑖𝜋2𝛾2 𝑑𝛾 = න
𝛾1
𝛾2
cos𝜋
2𝛾2 𝑑𝛾 + 𝑖 න
𝛾1
𝛾2
sin𝜋
2𝛾2 𝑑𝛾
= 𝒞 𝛾2 − 𝒞 𝛾1 + 𝑖 𝒮 𝛾2 − 𝒮 𝛾1
න
𝛿1
𝛿2
𝑒𝑖𝜋2𝛿2 𝑑𝛿 = න
𝛿1
𝛿2
cos𝜋
2𝛿2 𝑑𝛿 + 𝑖 න
𝛿1
𝛿2
sin𝜋
2𝛿2 𝑑𝛿
= 𝒞 𝛿2 − 𝒞 𝛿1 + 𝑖 𝒮 𝛿2 − 𝒮 𝛿1
𝒞 𝑥 ≡ න
0
𝑥
cos𝜋
2𝑥2 𝑑𝑥 𝒮 𝑥 ≡ න
0
𝑥
sin𝜋
2𝑥2 𝑑𝑥
-
62
× 𝒞 𝛾2 − 𝒞 𝛾1 + 𝑖 𝒮 𝛾2 − 𝒮 𝛾1
× 𝒞 𝛿2 − 𝒞 𝛿1 + 𝑖 𝒮 𝛿2 − 𝒮 𝛿1
𝐸 𝑌, 𝑍, 𝑡 =𝜆 𝐸0 𝑒
𝑖 𝑘 𝑠 − 𝜔 𝑡
2
𝐼 𝑌, 𝑍 =𝐼04× 𝒞 𝛾2 − 𝒞 𝛾1
2 + 𝒮 𝛾2 − 𝒮 𝛾12
× 𝒞 𝛿2 − 𝒞 𝛿12 + 𝒮 𝛿2 − 𝒮 𝛿1
2
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63
𝒞 𝑥 ≡ න
0
𝑥
cos𝜋
2𝑥′2 𝑑𝑥′
𝒮 𝑥 ≡ න
0
𝑥
sin𝜋
2𝑥′2 𝑑𝑥′
𝒞 𝑥
𝒮 𝑥
𝑥
𝑥
𝑥
𝒞 𝑥
𝒮 𝑥
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64
𝒞 𝑥 ≡ න
0
𝑥
cos𝜋
2𝑥2 𝑑𝑥
𝒮 𝑥 ≡ න
0
𝑥
sin𝜋
2𝑥2 𝑑𝑥
𝑑𝒞 𝑥 = cos𝜋
2𝑥2 𝑑𝑥
𝑑𝒮 𝑥 = sin𝜋
2𝑥2 𝑑𝑥
𝒮 𝑥
𝒞 𝑥
𝑑𝒞 2 + 𝑑𝒮 2 = 𝑑𝑥 2
𝑑𝒞
𝑑𝒮𝑑𝑥
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65
Applications of Fresnel Diffraction
1.No obstruction
2.Straight edge
3. Single slit
4. Rectangular aperture
5. Opaque circular disk
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66
𝐼 𝑌, 𝑍 =𝐼04× 𝒞 𝛾2 − 𝒞 𝛾1
2 + 𝒮 𝛾2 − 𝒮 𝛾12
× 𝒞 𝛿2 − 𝒞 𝛿12 + 𝒮 𝛿2 − 𝒮 𝛿1
2
1. No Obstruction
𝛾 ≡2
𝜆 𝑠𝑌 − 𝑦
𝛿 ≡2
𝜆 𝑠𝑍 − 𝑧
𝑦
𝑧
𝛾2 = −∞
𝛾1 = +∞
𝛿2 = −∞ 𝛿1 = +∞
=𝐼04× −0.5 − 0.5 2 + −0.5 − 0.5 2 × −0.5 − 0.5 2 + −0.5 − 0.5
2
= 𝐼0 No surprises here, just the obvious result !!
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67
𝐼 𝑌, 𝑍 =𝐼04× 𝒞 𝛾2 − 𝒞 𝛾1
2 + 𝒮 𝛾2 − 𝒮 𝛾12
× 𝒞 𝛿2 − 𝒞 𝛿12 + 𝒮 𝛿2 − 𝒮 𝛿1
2
𝛾 ≡2
𝜆 𝑠𝑌 − 𝑦
𝛿 ≡2
𝜆 𝑠𝑍 − 𝑧
𝑦
𝑧𝛾2 =
2
𝜆 𝑠𝑌
𝛾1 = +∞𝛿2 = −∞ 𝛿1 = +∞
=𝐼04
× 𝒞2
𝜆 𝑠𝑌 − 0.5
2
+ 𝒮2
𝜆 𝑠𝑌 − 0.5
2
× 2
2. Straight Edge
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68
𝒮 𝑥
𝒞 𝑥
𝑌 = 0
𝑌 > 0
𝑌 < 0 𝐼 𝑌, 𝑍, 𝑡 /𝐼0
𝑌
𝜆 𝑠 = 2
𝐼 𝑌, 𝑍 =𝐼02
× 𝒞2
𝜆 𝑠𝑌 − 0.5
2
+ 𝒮2
𝜆 𝑠𝑌 − 0.5
2
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69
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70
𝐼 𝑌, 𝑍 =𝐼04× 𝒞 𝛾2 − 𝒞 𝛾1
2 + 𝒮 𝛾2 − 𝒮 𝛾12
× 𝒞 𝛿2 − 𝒞 𝛿12 + 𝒮 𝛿2 − 𝒮 𝛿1
2
𝛾 ≡2
𝜆 𝑠𝑌 − 𝑦
𝛿 ≡2
𝜆 𝑠𝑍 − 𝑧
𝑦
𝑧
𝛾2 =2
𝜆 𝑠𝑌 − 𝑑
2
𝛾1 =2
𝜆 𝑠𝑌 + 𝑑2𝛿2 = −∞ 𝛿1 = +∞
=𝐼04
× 𝒞2
𝜆 𝑠𝑌 − 𝑑
2− 𝒞
2
𝜆 𝑠𝑌 + 𝑑
2
2
+ 𝒮2
𝜆 𝑠𝑌 − 𝑑
2− 𝒮
2
𝜆 𝑠𝑌 + 𝑑
2
2
× 2
3. Single Slit
𝑑
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71
𝒮 𝑥
𝒞 𝑥
𝑌 = 0
𝑌 > 0
𝑌 < 0
𝐼 𝑌, 𝑍 =𝐼02
× 𝒞2
𝜆 𝑠𝑌 − 𝑑
2− 𝒞
2
𝜆 𝑠𝑌 + 𝑑
2
2
+ 𝒮2
𝜆 𝑠𝑌 − 𝑑
2− 𝒮
2
𝜆 𝑠𝑌 + 𝑑
2
2
𝛾1 − 𝛾2 =2
𝜆 𝑠𝑑
𝛾1 + 𝛾22
=2
𝜆 𝑠𝑌
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72
𝑑 = 10 𝜆
𝑑
𝑁𝐹 ≡𝑑2
4 𝜆 𝑠
𝑁𝐹 = 10
𝑁𝐹 = 1
𝑁𝐹 = 0.5
𝑁𝐹 = 0.1
𝜆 = 1
𝑠 = 2.5 𝜆
𝑠 = 25 𝜆
𝑠 = 50 𝜆
𝑠 = 250 𝜆
Near field
Far field
Fresnel number
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73
Mathematica
SingleSlitDiffractionPattern.cdf
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74
4. Rectangular Aperture
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75
5. Circular Objects
Poisson (Arago) spot
