Diffraction Theory 1 2 , 2 4 3 5 1 + 2 - - PowerPoint PPT Presentation

Diffraction Theory

1

2

3

𝑠

1

𝑠2 𝐹 Ԧ 𝑠, 𝑢 = 𝐹1 Ԧ 𝑠, 𝑢 𝐹 Ԧ 𝑠, 𝑢

𝐹 Ԧ 𝑠, 𝑢 =

𝐹0,1 𝑠1 𝑓𝑗 𝑙 𝑠1− 𝜕 𝑢 + 𝜁1 + 𝐹0,2

𝑠2 𝑓𝑗 𝑙 𝑠2− 𝜕 𝑢 + 𝜁2

+ 𝐹2 Ԧ 𝑠, 𝑢 𝑠3

+ 𝐹0,3 𝑠3 𝑓𝑗 𝑙 𝑠3 − 𝜕 𝑢 + 𝜁3

𝑠

4

+ 𝐹3 Ԧ 𝑠, 𝑢 + 𝐹4 Ԧ 𝑠, 𝑢 𝑠5

+ 𝐹0,4 𝑠

4

𝑓𝑗 𝑙 𝑠4 − 𝜕 𝑢 + 𝜁4

+ 𝐹5 Ԧ 𝑠, 𝑢

+ 𝐹0,5 𝑠5 𝑓𝑗 𝑙 𝑠5 − 𝜕 𝑢 + 𝜁5

= ෍

𝑗

𝐹𝑗 Ԧ 𝑠, 𝑢

+ ⋯

+ …

= ෍

𝑗

𝐹0,𝑗 𝑠𝑗 𝑓𝑗 𝑙 𝑠𝑗 −𝜕 𝑢 + 𝜁𝑗

4

Huygens-Fresnel Principle

5

𝐹 𝑍, 𝑎, 𝑢 = ෍

𝑗

𝐹0,𝑗 𝑠

𝑗

𝑓𝑗 𝑙 𝑠𝑗 − 𝜕 𝑢 + 𝜁𝑗 = ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑠 𝑧, 𝑨 𝑓𝑗 𝑙 𝑠 𝑧, 𝑨 − 𝜕 𝑢 + 𝜁 𝑧, 𝑨 𝑒𝑧 𝑒𝑨

𝑠 𝑧, 𝑨 = 𝑡2 + 𝑍 − 𝑧 2 + 𝑎 − 𝑨 2 𝑠 𝑡 𝑎 𝑍 𝑨 𝑧

𝑗 𝑧, 𝑨

෍

𝑗

ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑒𝑧 𝑒𝑨

6

𝑠 𝑧, 𝑨 ≅ 𝑡 1 + 𝜂2 2 Fresnel Approximation 𝑙 𝑡 𝑛𝑏𝑦 𝜂4 8 ≪ 𝜌 = 𝑡 1 + 𝜂2 2 − 𝜂4 8 + 𝜂6 16 − 5 𝜂8 128 + ⋯ 𝑠 𝑧, 𝑨 = 𝑡2 + 𝑍 − 𝑧 2 + 𝑎 − 𝑨 2 = 𝑡 1 + 𝑍 − 𝑧 2 𝑡2 + 𝑎 − 𝑨 2 𝑡2 = 𝑡 1 + 𝜂2 𝜂2 ≡ 𝑍 − 𝑧 2 𝑡2 + 𝑎 − 𝑨 2 𝑡2

Fresnel Diffraction

= 𝑡 + 𝑍2 + 𝑎2 2 𝑡 − 𝑍 𝑧 + 𝑎 𝑨 𝑡 + 𝑧2 + 𝑨2 2 𝑡 = 𝑡 1 + 𝑍 − 𝑧 2 2 𝑡2 + 𝑎 − 𝑨 2 2 𝑡2

7

𝑠 𝑧, 𝑨 ≅ 𝑡 + 𝑍2 + 𝑎2 2 𝑡 − 𝑍 𝑧 + 𝑎 𝑨 𝑡 + 𝑧2 + 𝑨2 2 𝑡 Fresnel Approximation 𝑙 𝑡 𝑛𝑏𝑦 𝜂4 8 ≪ 𝜌

𝐹 𝑍, 𝑎, 𝑢 = ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑠 𝑧, 𝑨 𝑓𝑗 𝑙 𝑠 𝑧, 𝑨 − 𝜕 𝑢 + 𝜁 𝑧, 𝑨 𝑒𝑧 𝑒𝑨

≅ ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑡 𝑓𝑗 𝑙

𝑡 + 𝑍2+𝑎2 2 𝑡 − 𝑍 𝑧+𝑎 𝑨 𝑡 + 𝑧2+𝑨2 2 𝑡 − 𝜕 𝑢 + 𝜁 𝑧, 𝑨

𝑒𝑧 𝑒𝑨

8

𝑙 𝑛𝑏𝑦 𝑧2 + 𝑨2 2 𝑡 ≪ 𝜌 𝑛𝑏𝑦 𝑧2 + 𝑨2 𝜇 𝑡 ≪ 1 Fraunhofer Approximation

Fraunhofer Diffraction

also known as Far-Field Diffraction 𝑠 𝑧, 𝑨 ≅ 𝑡 + 𝑍2 + 𝑎2 2 𝑡 − 𝑍 𝑧 + 𝑎 𝑨 𝑡 + 𝑧2 + 𝑨2 2 𝑡 Fresnel Approximation 𝑙 𝑡 𝑛𝑏𝑦 𝜂4 8 ≪ 𝜌 In addition to Fresnel Approximation:

9

𝐹 𝑍, 𝑎, 𝑢 = ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑠 𝑧, 𝑨 𝑓𝑗 𝑙 𝑠 𝑧, 𝑨 − 𝜕 𝑢 + 𝜁 𝑧, 𝑨 𝑒𝑧 𝑒𝑨 ≅ ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑆 𝑓

𝑗 𝑙 𝑆 − 𝑍 𝑧 + 𝑎 𝑨 𝑆 − 𝜕 𝑢 + 𝜁 𝑧, 𝑨

𝑒𝑧 𝑒𝑨 = 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧,𝑨 𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨

𝑠 𝑧, 𝑨 = 𝑡2 + 𝑍 − 𝑧 2 + 𝑎 − 𝑨 2 𝑆2 ≡ 𝑡2 + 𝑍2 + 𝑎2 = 𝑆2 − 2 𝑍 𝑧 − 2 𝑎 𝑨 + 𝑧2 + 𝑨2 = 𝑆 1 + −2 𝑍 𝑧 − 2 𝑎 𝑨 + 𝑧2 + 𝑨2 𝑆2 ≅ 𝑆 − 𝑍 𝑧 + 𝑎 𝑨 𝑆

10

𝐹 𝑍, 𝑎, 𝑢 = 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨

𝑠 𝑡 𝑎 𝑍 𝑨 𝑧 𝑆

Fraunhofer Diffraction

𝑆2 ≡ 𝑡2 + 𝑍2 + 𝑎2

11

Illumination at the Aperture:

In the examples to follow, we will consider a flat wavefront at normal incidence on the aperture

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 = 𝐹0

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨

Inside the aperture Outside the aperture

{

12

Apertures considered here:

• 1. Single Slit
• 2. Double Slit
• 3. Rectangular Aperture
• 4. Circular Aperture

13

• 1. Single Slit

𝑠 𝑡 𝑎 𝑍 𝑨 𝑧 𝑆 𝑆 ≡ 𝑍2 + 𝑡2 𝑒

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

− ൗ 𝑒 2 + ൗ 𝑒 2

𝑓− 𝑗 𝑙 𝑍

𝑆 𝑧𝑒𝑧

𝜄 𝑡𝑗𝑜 𝜄 = 𝑍 𝑆 𝑧 𝑨

14

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 𝑒 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑒 2 𝑆

• 1. Single Slit, cont.

𝐽 ≡ 𝐹

2

𝐽 𝑍, 𝑎 = 𝐽0 𝑡𝑗𝑜𝑑2 𝑙 𝑍 𝑒 2 𝑆 𝐽0 ≡ 𝐹0 2 2 𝑆2 𝑒2

𝑒 = 50 µ𝑛 𝜇 = 0.6 µ𝑛 𝑡 = 1 𝑛

𝑙 𝑍

𝑛 𝑒

2 𝑆 = 𝑛 𝜌

𝑛 = ±1, ±2, ±3 𝑆 ≅ 1 𝑛

𝑍

𝑛 = 𝑛 𝜇 𝑆

𝑒 𝑡𝑗𝑜 𝜄𝑛 = 𝑍

𝑛

𝑆 = 𝑛 𝜇 𝑒 𝑍(𝑛𝑛) 𝑍

1

ൗ 𝐽 𝐽0 zeros at

geometrical shadow

𝑍

−1

with 𝑍 𝑎

15

Mathematica

16

• 2. Double Slit

𝑒 𝑒 𝑏 𝑧

ൗ 𝑏 2 − ൗ 𝑒 2 ൗ 𝑏 2 + ൗ 𝑒 2 ൗ −𝑏 2 − ൗ 𝑒 2 ൗ −𝑏 2 + ൗ 𝑒 2

𝑨

17

𝑠 𝑡 𝑎 𝑍 𝑨 𝑧 𝑆 𝑆 ≡ 𝑍2 + 𝑡2

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

ൗ −𝑏 2− ൗ 𝑒 2 ൗ −𝑏 2+ ൗ 𝑒 2

𝑓− 𝑗 𝑙 𝑍

𝑆 𝑧𝑒𝑧 +

න

ൗ 𝑏 2− ൗ 𝑒 2 ൗ 𝑏 2+ ൗ 𝑒 2

𝑓− 𝑗 𝑙 𝑍

𝑆 𝑧𝑒𝑧

𝜄 𝑡𝑗𝑜 𝜄 = 𝑍 𝑆

= 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑒 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑒 2 𝑆 2 𝑑𝑝𝑡 𝑙 𝑎 𝑏 2 𝑆

18

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑒 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑒 2 𝑆 2 𝑑𝑝𝑡 𝑙 𝑍 𝑏 2 𝑆 𝐽 𝑍, 𝑎 = 4 𝐽0 𝑡𝑗𝑜𝑑2 𝑙 𝑍 𝑒 2 𝑆 𝑑𝑝𝑡2 𝑙 𝑍 𝑏 2 𝑆

𝐽0 ≡ 𝐹0 2 2 𝑆2 𝑒2

Mathematica

𝑒 𝑏

19

• 3. Rectangular Aperture

𝑏 𝑐 𝑧 𝑨

20

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

ൗ −𝑐 2 ൗ 𝑐 2

𝑓− 𝑗 𝑙 𝑍

𝑆 𝑧𝑒𝑧

න

ൗ −𝑏 2 ൗ 𝑏 2

𝑓− 𝑗 𝑙 𝑎

𝑆 𝑨𝑒𝑨

= 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑐 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑐 2 𝑆 𝑏 𝑡𝑗𝑜𝑑 𝑙 𝑎 𝑏 2 𝑆

21

𝑍 𝑎

𝐽 𝑍, 𝑎 = 𝐽0 𝑡𝑗𝑜𝑑2 𝑙 𝑍 𝑐 2 𝑆 𝑡𝑗𝑜𝑑2 𝑙 𝑎 𝑏 2 𝑆

𝐽0 ≡ 𝐹0 2 2 𝑆2 𝑏2 𝑐2

22

Emission of Semiconductor Laser

23

• 4. Circular Aperture

𝑏 𝜒

𝑧 = 𝜍 𝑡𝑗𝑜 𝜒

𝜍

𝑨 = 𝜍 𝑑𝑝𝑡 𝜒

𝑨 𝑧

24

Observation Plane

Φ

𝑍 = 𝑟 𝑡𝑗𝑜 Φ

𝑟

𝑎 = 𝑟 𝑑𝑝𝑡 Φ

𝑎 𝑍

25

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨

𝑍 𝑧 + 𝑎 𝑨 = 𝑟 𝑡𝑗𝑜 Φ 𝜍 𝑡𝑗𝑜 𝜒 + 𝑟 𝑑𝑝𝑡 Φ 𝜍 𝑑𝑝𝑡 𝜒

= 𝜍 𝑟 𝑑𝑝𝑡 𝜒 − Φ 𝑒𝑧 𝑒𝑨 = 𝜍 𝑒𝜒 𝑒𝜍

𝐹 𝑟, Φ, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

𝑏

𝜍 𝑒𝜍 න

2𝜌

𝑒𝜒 𝑓 − 𝑗 𝑙 𝜍 𝑟 𝑑𝑝𝑡 𝜒 − Φ

𝑆 Φ = 0 Due to axial symmetry, we can choose:

= 𝑟 𝜍 𝑑𝑝𝑡 Φ 𝑑𝑝𝑡 𝜒 + 𝑡𝑗𝑜 Φ 𝑡𝑗𝑜 𝜒

26

𝐹 𝑟, Φ, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

𝑏

𝜍 𝑒𝜍 න

2𝜌

𝑒𝜒 𝑓 − 𝑗 𝑙 𝜍 𝑟 𝑑𝑝𝑡 𝜒

𝑆

A couple of integrals to solve:

27

1 2 𝜌 න

2𝜌

𝑒𝜒 𝑓𝑗 𝑣 𝑑𝑝𝑡 𝜒 ≡ 𝐾0 𝑣

Bessel function

• f order zero

𝐹 𝑟, Φ, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

𝑏

𝜍 𝑒𝜍 න

2𝜌

𝑒𝜒 𝑓 − 𝑗 𝑙 𝜍 𝑟 𝑑𝑝𝑡 𝜒

𝑆

28

𝐹 𝑟, Φ, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 2 𝜌 න

𝑏

𝜍 𝑒𝜍 𝐾0 − 𝑙 𝑟 𝑆 𝜍 𝑣 ≡ − 𝑙 𝑟 𝑆 𝜍 = 𝐹0 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 2 𝜌 𝑆 𝑙 𝑟

2

න

−𝑙 𝑟 𝑆 𝑏

α 𝑒α 𝐾0 α

𝛽 ≡ −𝑙 𝑟 𝑆 𝜍 𝜍 𝑒𝜍 = 𝑆 𝑙 𝑟

2

α 𝑒α

29

න

𝛽

𝛽 𝐾0 𝛽 𝑒𝛽 ≡ 𝛽 𝐾1 𝛽

30

𝐹 𝑟, Φ, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 2 𝜌 𝑆 𝑙 𝑟

2

න

−𝑙 𝑟 𝑆 𝑏

α 𝑒α 𝐾0 α = 𝐹0 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 2 𝜌 𝑆 𝑙 𝑟

2 −𝑙 𝑏 𝑟

𝑆 𝐾1 −𝑙 𝑏 𝑟 𝑆 = 𝐹0 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 𝜌 𝑏2 2 𝐾1 𝑙 𝑏 𝑟 𝑆 𝑙 𝑏 𝑟 𝑆

31

𝐽 𝑟, Φ = 𝐽0 2 𝐾1 𝑙 𝑏 𝑟 𝑆 𝑙 𝑏 𝑟 𝑆

2

𝐽0 ≡ 𝐹0 2 2 𝑆2 𝜌 𝑏2 2

𝑙 𝑏 𝑟 𝑆 ൗ 𝐽 𝐽0

32

zeros at

𝑙 𝑏 𝑟 𝑆 = 3.832, 7.016, 10.173, …

𝑙 𝑏 𝑟1 𝑆 = 3.832 𝑟1 𝑆 = 𝑡𝑗𝑜 𝜄1 = 3.832 𝜇 2 𝜌 𝑏 = 1.22 𝜇 2 𝑏 first zero at Light is essentially confined inside the cone: 𝒕𝒋𝒐 𝜄1 < 𝟐. 𝟑𝟑 𝝁

𝟑 𝒃

33

Circular Aperture

𝑨 𝑧 𝑡 𝑧 𝑍 𝑎 𝑍 𝑡𝑗𝑜 𝜄1 = 𝑟1 𝑆 = 1.22 𝜇 2 𝑏 𝑆 2𝑏

Airy’s pattern

𝑏 𝑟1 𝑟1 𝜄1

34

𝑨 𝑧 2𝑏 𝑡 𝑆 𝜄1

} = 0

35

𝑧 2𝑏 𝜄1 𝜄1 𝑡𝑗𝑜 𝜄1 = 1.22 𝜇 2 𝑏 tan 𝜄1 = 𝑟1 𝑔 𝑟1 𝑟1 ≅ 1.22 𝜇 𝑔 2 𝑏 𝑔

Smallest spot size:

𝑟1 ≅ 1.22 𝜇 𝑔 𝐸𝑚𝑓𝑜𝑡 𝐸𝑚𝑓𝑜𝑡 = 1.22 𝜇𝑝 𝑔 𝑜 𝐸𝑚𝑓𝑜𝑡 𝑜

Smallest angular width:

𝑟1 𝑔 = 1.22 𝜇𝑝 𝑜 𝐸𝑚𝑓𝑜𝑡

36

Diameter of primary mirror 2.4 m Wavelength 0.55 µm Angular width 0.28 × 10-6 rad

37

𝑢𝑏𝑜 𝜄𝑛𝑏𝑦 ≡ 𝐸𝑚𝑓𝑜𝑡 2 𝑔 𝐸𝑚𝑓𝑜𝑡 𝜄𝑛𝑏𝑦

𝑂𝐵 ≡ 𝑜 𝑡𝑗𝑜 𝜄𝑛𝑏𝑦 ≅ 𝑜 𝐸𝑚𝑓𝑜𝑡 2 𝑔

𝑔

𝑔 # = 𝑔 𝐸𝑚𝑓𝑜𝑡

38

Numerical Aperture

𝑂𝐵 ≡ 𝑜 𝑡𝑗𝑜 𝜄𝑛𝑏𝑦

39

𝑟1 = 1.22 𝜇𝑝 2 𝑂𝐵

Smallest spot size from a lens

𝑧 2𝑏 = 𝐸𝑚𝑓𝑜𝑡 𝜄1 𝜄1 𝑟1 𝑔 𝐸𝑚𝑓𝑜𝑡 𝑜 𝑟1 = 1.22 𝜇𝑝 𝑔 𝑜 𝐸𝑚𝑓𝑜𝑡 𝑂𝐵 ≡ 𝑜 𝑡𝑗𝑜 𝜄𝑛𝑏𝑦 ≅ 𝑜 𝐸𝑚𝑓𝑜𝑡 2 𝑔

40

Rayleigh Criteria for Resolution

Barely resolved Resolved Not resolved

41

𝑟1 = 1.22 𝜇𝑝 2 𝑂𝐵 𝜇𝑝 = 0.55 𝜈𝑛 3.36 𝜈𝑛 1.34 𝜈𝑛 0.52 𝜈𝑛 0.27 𝜈𝑛

Examples of Diffraction Limit of Objective Lenses

42

𝐹 𝑍, 𝑎, 𝑢 = 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨 𝑠 𝑡 𝑎 𝑍 𝑨 𝑧 𝑆 𝑆 ≡ 𝑍2 + 𝑎2 + 𝑡2

Fraunhofer Diffraction

𝑛𝑏𝑦 𝑧2 + 𝑨2 𝜇 𝑡 ≪ 1 𝑛𝑏𝑦 𝑍 − 𝑧 2 + 𝑎 − 𝑨 2 𝜇 𝑡 ≪ 1

43

In summary, far-field diffraction:

• 1. Single Slit
• 2. Double Slit
• 3. Rectangular Aperture
• 4. Circular Aperture

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑒 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑒 2 𝑆 𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑒 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑒 2 𝑆 2 𝑑𝑝𝑡 𝑙 𝑍 𝑏 2 𝑆 𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑐 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑐 2 𝑆 𝑏 𝑡𝑗𝑜𝑑 𝑙 𝑎 𝑏 2 𝑆 𝐹 𝑟, Φ, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝜌 𝑏2 2 𝐾1 𝑙 𝑏 𝑟 𝑆 𝑙 𝑏 𝑟 𝑆

44

𝐹 𝑍, 𝑎, 𝑢 = 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 𝑓− 𝑗 𝑙 𝑍 𝑧 + 𝑎 𝑨

𝑆

𝑒𝑧 𝑒𝑨 𝐹 𝑍, 𝑎, 𝑢 = 𝑓𝑗 𝑙 𝑆 − 𝜕 𝑢 𝑆 ඵ

−∞ +∞

𝜔 𝑧, 𝑨 𝑓− 𝑗 𝑙𝑧 𝑧 +𝑙𝑨 𝑨 𝑒𝑧 𝑒𝑨 𝜔 𝑧, 𝑨 ≡ 𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 inside aperture

• paque obstruction

𝑙𝑧 ≡ 𝑙 𝑍 𝑆

Fraunhofer Diffraction as a Fourier Transformation

𝑙𝑨 ≡ 𝑙 𝑎 𝑆

{

45

Diffraction Gratings

46

Multiple Slits

𝑐 𝑏 𝑧

𝑏 − 𝑐 2 𝑏 + 𝑐 2

𝑨 𝑶 (infinitely long) slits of width 𝒄 separated by distance 𝒃

+ 𝑐 2 − 𝑐 2 𝑂 − 1 𝑏 − 𝑐 2 𝑂 − 1 𝑏 + 𝑐 2

47

𝑠 𝑡 𝑎 𝑍 𝑨 𝑧 𝑆 𝑆 ≡ 𝑍2 + 𝑡2

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 න

− 𝑐 2 + 𝑐 2

+ න

𝑏 − 𝑐 2 𝑏 + 𝑐 2

+ න

2 𝑏 − 𝑐 2 2 𝑏 + 𝑐 2

+ ⋯ + න

𝑂−1 𝑏 − 𝑐 2 𝑂−1 𝑏 + 𝑐 2

𝑓− 𝑗 𝑙 𝑍

𝑆 𝑧 𝑒𝑧

𝜄 𝑡𝑗𝑜 𝜄 = 𝑍 𝑆

48

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑐 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑐 2 𝑆 ෍

𝑜 = 0 𝑂−1

𝑓− 𝑗 𝑙 𝑍 𝑏

𝑆 𝑜

= 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑐 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑐 2 𝑆 1 − 𝑓−𝑗 𝑂 𝑙 𝑍 𝑏

𝑆

1 − 𝑓−𝑗 𝑙 𝑍 𝑏

𝑆

= 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑐 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑐 2 𝑆 𝑓−𝑗 𝑂 𝑙 𝑍 𝑏

2 𝑆

𝑓−𝑗 𝑙 𝑍 𝑏

2 𝑆

𝑓+𝑗 𝑂 𝑙 𝑍 𝑏

2 𝑆 − 𝑓−𝑗 𝑂 𝑙 𝑍 𝑏 2 𝑆

𝑓+𝑗 𝑙 𝑍 𝑏

2 𝑆 − 𝑓−𝑗 𝑙 𝑍 𝑏 2 𝑆

= 𝐹0 𝑓𝑗 𝑙 𝑆 −𝜕 𝑢 𝑆 𝑐 𝑡𝑗𝑜𝑑 𝑙 𝑍 𝑐 2 𝑆 𝑓−𝑗 𝑂 𝑙 𝑍 𝑏

2 𝑆

𝑓−𝑗 𝑙 𝑍 𝑏

2 𝑆

sin 𝑂 𝑙 𝑍 𝑏 2 𝑆 sin 𝑙 𝑍 𝑏 2 𝑆

49

𝐽 𝑍, 𝑎 = 𝐽0 𝑡𝑗𝑜𝑑2 𝑙 𝑍 𝑐 2 𝑆 𝑡𝑗𝑜2 𝑂 𝑙 𝑍 𝑏 2 𝑆 𝑡𝑗𝑜2 𝑙 𝑍 𝑏 2 𝑆 𝐽0 ≡ 𝐹0 2 2 𝑆2 𝑐2

Intensity Pattern Mathematica

𝑐 = 1 𝑏 = 4 𝑙 = 1 𝑆 = 1

50

𝑡𝑗𝑜𝑑2 𝑙 𝑍 𝑐 2 𝑆 ≅ 1

𝐽 𝑍, 𝑎 ≅ 𝐽0 𝑡𝑗𝑜2 𝑂 𝑙 𝑍 𝑏 2 𝑆 𝑡𝑗𝑜2 𝑙 𝑍 𝑏 2 𝑆

Small Width Approximation:

𝑐 = 0.1 𝑏 = 4 𝑙 = 1 𝑆 = 1

51

𝑙 𝑍 𝑏 2 𝑆 = 𝑛 𝜌 𝐽 𝑍, 𝑎, 𝑢 = 𝑂2 𝐽0

Maxima (intensity peaks)

𝑛 = 0, ±1, ±2, … 𝑏 𝑡𝑗𝑜 𝜄𝑛 = 𝑛 𝜇 grating equation grating

• rder

52

𝑂 𝑙 𝑍 𝑏 2 𝑆 = 𝑠 𝜌 𝑠 = 1, 2, 3, … , (𝑂 − 1)

Minima (zero intensity)

𝑙 𝑍 𝑏 2 𝑆 = 𝑠 𝑂 𝜌

𝑐 = 0.1 𝑏 = 4 𝑙 = 1 𝑆 = 1 0 < 𝑙 𝑍 𝑏 2 𝑆 < 𝜌 𝑛 = 0 𝑛 = 1 1 −1 𝑛 2 −2 𝐽 𝑍, 𝑎 ≅ 𝐽0 𝑡𝑗𝑜2 𝑂 𝑙 𝑍 𝑏 2 𝑆 𝑡𝑗𝑜2 𝑙 𝑍 𝑏 2 𝑆

53

Angular Width

𝑙 𝑏 𝑡𝑗𝑜 𝜄𝑛 + ∆𝜄 2 2 = 𝑛 𝜌 + 1 𝑂 𝜌 𝑙 𝑍 𝑏 2 𝑆 = 𝑙 𝑏 𝑡𝑗𝑜 𝜄 2 ∆𝜄 = 2 𝜇 𝑂 𝑏 𝑑𝑝𝑡 𝜄𝑛 𝑙 𝑏 𝑑𝑝𝑡 𝜄𝑛 𝑡𝑗𝑜 ∆𝜄 2 2 ≅ 1 𝑂 𝜌

𝑛

54

Spectral Resolution

𝑏 𝑡𝑗𝑜 𝜄𝑛 = 𝑛 𝜇 𝑏 𝑑𝑝𝑡 𝜄𝑛 𝑒𝜄 = 𝑛 𝑒𝜇 ∆𝜇𝑠𝑓𝑡 = 𝜇 𝑛 𝑂 𝑒𝜄 ≡ ∆𝜄 2 = 𝜇 𝑂 𝑏 𝑑𝑝𝑡 𝜄𝑛 𝑒𝜇 ≡ ∆𝜇𝑠𝑓𝑡

55

Free Spectral Range

𝑏 𝑡𝑗𝑜 𝜄 = 𝑛 + 1 𝜇 = 𝑛 𝜇 + ∆𝜇𝐺𝑇𝑆 ∆𝜇𝐺𝑇𝑆 = 𝜇 𝑛

56

Oblique Incidence Normal Incidence

𝑏 𝑡𝑗𝑜 𝜄 − 𝑏 𝑡𝑗𝑜 𝜄𝑗𝑜𝑑 = 𝑛 𝜇 𝑏 𝑡𝑗𝑜 𝜄𝑛 − 𝑡𝑗𝑜 𝜄𝑗𝑜𝑑 = 𝑛 𝜇 𝑏 𝑡𝑗𝑜 𝜄𝑛 = 𝑛 𝜇

57

Fresnel Diffraction

Going beyond the Fraunhofer (far-field) approximation

• r

getting closer to the aperture

58

𝑠 𝑧, 𝑨 = 𝑡2 + 𝑍 − 𝑧 2 + 𝑎 − 𝑨 2 𝑠 𝑡 𝑎 𝑍 𝑨 𝑧 𝑠 𝑧, 𝑨 ≅ 𝑡 + 1 2 𝑡 𝑍 − 𝑧 2 + 1 2 𝑡 𝑎 − 𝑨 2

𝐹 𝑍, 𝑎, 𝑢 = ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑠 𝑧, 𝑨 𝑓𝑗 𝑙 𝑠 𝑧, 𝑨 − 𝜕 𝑢 + 𝜁 𝑧, 𝑨 𝑒𝑧 𝑒𝑨

= 𝑡 1 + 𝑍 − 𝑧 2 𝑡2 + 𝑎 − 𝑨 2 𝑡2 𝑙 𝑡 𝑛𝑏𝑦 𝑍 − 𝑧 2 + 𝑎 − 𝑨 2 2 𝑡4 ≪ 𝜌

59

𝐹 𝑍, 𝑎, 𝑢 = 𝑓𝑗 𝑙 𝑡 − 𝜕 𝑢 𝑡 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 𝑓𝑗 𝑙

2 𝑡 𝑍−𝑧 2+ 𝑎−𝑨 2 𝑒𝑧 𝑒𝑨

𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑡 − 𝜕 𝑢 𝑡 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑓𝑗 𝜌

𝜇 𝑡 𝑍−𝑧 2+ 𝑎−𝑨 2 𝑒𝑧 𝑒𝑨

𝐹0 𝑧, 𝑨 𝑓𝑗 𝜁 𝑧, 𝑨 = 𝐹0 Inside the aperture Outside the aperture

{

Flat Wavefront Illumination

60

𝛿 ≡ 2 𝜇 𝑡 𝑍 − 𝑧 𝑒𝑧 = − 𝜇 𝑡 2 𝑒𝛿 𝜀 ≡ 2 𝜇 𝑡 𝑎 − 𝑨 𝑒𝑨 = − 𝜇 𝑡 2 𝑒𝜀 𝐹 𝑍, 𝑎, 𝑢 = 𝐹0 𝑓𝑗 𝑙 𝑡 − 𝜕 𝑢 𝑡 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑓𝑗 𝜌

𝜇 𝑡 𝑍−𝑧 2+ 𝑎−𝑨 2 𝑒𝑧 𝑒𝑨

= 𝐹0 𝑓𝑗 𝑙 𝑡 − 𝜕 𝑢 𝑡 𝜇 𝑡 2 ඵ

𝑏𝑞𝑓𝑠𝑢𝑣𝑠𝑓

𝑓𝑗 𝜌

2 𝛿2+ 𝜀2 𝑒𝛿 𝑒𝜀

= 𝜇 𝐹0 𝑓𝑗 𝑙 𝑡 − 𝜕 𝑢 2 න

𝛿1 𝛿2

𝑓𝑗 𝜌

2 𝛿2 𝑒𝛿 න 𝜀1 𝜀2

𝑓𝑗 𝜌

2 𝜀2 𝑒𝜀

61

න

𝛿1 𝛿2

𝑓𝑗 𝜌

2 𝛿2 𝑒𝛿 = න 𝛿1 𝛿2

cos 𝜌 2 𝛿2 𝑒𝛿 + 𝑗 න

𝛿1 𝛿2

sin 𝜌 2 𝛿2 𝑒𝛿 = 𝒟 𝛿2 − 𝒟 𝛿1 + 𝑗 𝒯 𝛿2 − 𝒯 𝛿1 න

𝜀1 𝜀2

𝑓𝑗 𝜌

2 𝜀2 𝑒𝜀

= න

𝜀1 𝜀2

cos 𝜌 2 𝜀2 𝑒𝜀 + 𝑗 න

𝜀1 𝜀2

sin 𝜌 2 𝜀2 𝑒𝜀 = 𝒟 𝜀2 − 𝒟 𝜀1 + 𝑗 𝒯 𝜀2 − 𝒯 𝜀1 𝒟 𝑦 ≡ න

𝑦

cos 𝜌 2 𝑦2 𝑒𝑦 𝒯 𝑦 ≡ න

𝑦

sin 𝜌 2 𝑦2 𝑒𝑦

62

× 𝒟 𝛿2 − 𝒟 𝛿1 + 𝑗 𝒯 𝛿2 − 𝒯 𝛿1 × 𝒟 𝜀2 − 𝒟 𝜀1 + 𝑗 𝒯 𝜀2 − 𝒯 𝜀1 𝐹 𝑍, 𝑎, 𝑢 = 𝜇 𝐹0 𝑓𝑗 𝑙 𝑡 − 𝜕 𝑢 2 𝐽 𝑍, 𝑎 = 𝐽0 4 × 𝒟 𝛿2 − 𝒟 𝛿1

2 + 𝒯 𝛿2 − 𝒯 𝛿1 2

× 𝒟 𝜀2 − 𝒟 𝜀1

2 + 𝒯 𝜀2 − 𝒯 𝜀1 2

63

𝒟 𝑦 ≡ න

𝑦

cos 𝜌 2 𝑦′2 𝑒𝑦′ 𝒯 𝑦 ≡ න

𝑦

sin 𝜌 2 𝑦′2 𝑒𝑦′

𝒟 𝑦 𝒯 𝑦 𝑦 𝑦 𝑦 𝒟 𝑦 𝒯 𝑦

64

𝒟 𝑦 ≡ න

𝑦

cos 𝜌 2 𝑦2 𝑒𝑦 𝒯 𝑦 ≡ න

𝑦

sin 𝜌 2 𝑦2 𝑒𝑦 𝑒𝒟 𝑦 = cos 𝜌 2 𝑦2 𝑒𝑦 𝑒𝒯 𝑦 = sin 𝜌 2 𝑦2 𝑒𝑦

𝒯 𝑦 𝒟 𝑦

𝑒𝒟 2 + 𝑒𝒯 2 = 𝑒𝑦 2

𝑒𝒟 𝑒𝒯 𝑒𝑦

65

Applications of Fresnel Diffraction

1.No obstruction 2.Straight edge

• 3. Single slit
• 4. Rectangular aperture
• 5. Opaque circular disk

66

𝐽 𝑍, 𝑎 = 𝐽0 4 × 𝒟 𝛿2 − 𝒟 𝛿1

2 + 𝒯 𝛿2 − 𝒯 𝛿1 2

× 𝒟 𝜀2 − 𝒟 𝜀1

2 + 𝒯 𝜀2 − 𝒯 𝜀1 2

• 1. No Obstruction

𝛿 ≡ 2 𝜇 𝑡 𝑍 − 𝑧 𝜀 ≡ 2 𝜇 𝑡 𝑎 − 𝑨

𝑧 𝑨 𝛿2 = −∞ 𝛿1 = +∞ 𝜀2 = −∞ 𝜀1 = +∞

= 𝐽0 4 × −0.5 − 0.5 2 + −0.5 − 0.5 2 × −0.5 − 0.5 2 + −0.5 − 0.5 2

= 𝐽0

No surprises here, just the obvious result !!

67

𝐽 𝑍, 𝑎 = 𝐽0 4 × 𝒟 𝛿2 − 𝒟 𝛿1

2 + 𝒯 𝛿2 − 𝒯 𝛿1 2

× 𝒟 𝜀2 − 𝒟 𝜀1

2 + 𝒯 𝜀2 − 𝒯 𝜀1 2

𝛿 ≡ 2 𝜇 𝑡 𝑍 − 𝑧 𝜀 ≡ 2 𝜇 𝑡 𝑎 − 𝑨

𝑧 𝑨 𝛿2 =

2 𝜇 𝑡 𝑍

𝛿1 = +∞ 𝜀2 = −∞ 𝜀1 = +∞

= 𝐽0 4 × 𝒟 2 𝜇 𝑡 𝑍 − 0.5

2

+ 𝒯 2 𝜇 𝑡 𝑍 − 0.5

2

× 2

• 2. Straight Edge

68

𝒯 𝑦 𝒟 𝑦 𝑍 = 0 𝑍 > 0 𝑍 < 0 𝐽 𝑍, 𝑎, 𝑢 /𝐽0 𝑍 𝜇 𝑡 = 2 𝐽 𝑍, 𝑎 = 𝐽0 2

× 𝒟 2 𝜇 𝑡 𝑍 − 0.5

2

+ 𝒯 2 𝜇 𝑡 𝑍 − 0.5

2

69

70

𝐽 𝑍, 𝑎 = 𝐽0 4 × 𝒟 𝛿2 − 𝒟 𝛿1

2 + 𝒯 𝛿2 − 𝒯 𝛿1 2

× 𝒟 𝜀2 − 𝒟 𝜀1

2 + 𝒯 𝜀2 − 𝒯 𝜀1 2

𝛿 ≡ 2 𝜇 𝑡 𝑍 − 𝑧 𝜀 ≡ 2 𝜇 𝑡 𝑎 − 𝑨

𝑧 𝑨 𝛿2 = 2 𝜇 𝑡 𝑍 − 𝑒

2

𝛿1 = 2 𝜇 𝑡 𝑍 + 𝑒

2

𝜀2 = −∞ 𝜀1 = +∞

= 𝐽0 4

× 𝒟 2 𝜇 𝑡 𝑍 − 𝑒

2

− 𝒟 2 𝜇 𝑡 𝑍 + 𝑒

2 2

+ 𝒯 2 𝜇 𝑡 𝑍 − 𝑒

2

− 𝒯 2 𝜇 𝑡 𝑍 + 𝑒

2 2

× 2

• 3. Single Slit

𝑒

71

𝒯 𝑦 𝒟 𝑦 𝑍 = 0 𝑍 > 0 𝑍 < 0 𝐽 𝑍, 𝑎 = 𝐽0 2

× 𝒟 2 𝜇 𝑡 𝑍 − 𝑒

2

− 𝒟 2 𝜇 𝑡 𝑍 + 𝑒

2 2

+ 𝒯 2 𝜇 𝑡 𝑍 − 𝑒

2

− 𝒯 2 𝜇 𝑡 𝑍 + 𝑒

2 2

𝛿1 − 𝛿2 = 2 𝜇 𝑡 𝑒 𝛿1 + 𝛿2 2 = 2 𝜇 𝑡 𝑍

72

𝑒 = 10 𝜇 𝑒 𝑂𝐺 ≡ 𝑒2 4 𝜇 𝑡 𝑂𝐺 = 10 𝑂𝐺 = 1 𝑂𝐺 = 0.5 𝑂𝐺 = 0.1 𝜇 = 1 𝑡 = 2.5 𝜇 𝑡 = 25 𝜇 𝑡 = 50 𝜇 𝑡 = 250 𝜇 Near field Far field Fresnel number

73

Mathematica

74

• 4. Rectangular Aperture

75

• 5. Circular Objects

Poisson (Arago) spot