Diffraction Theory - University of Louisville

$: Diffraction Theory - University of Louisville$
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+


𝑠𝑠2

Fresnel Diffraction

= 𝑠𝑠 +𝑌𝑌2 + 𝑍𝑍2

2 𝑠𝑠−

𝑌𝑌 𝑦𝑦 + 𝑍𝑍 𝑧𝑧𝑠𝑠

+𝑦𝑦2 + 𝑧𝑧2

2 𝑠𝑠

= 𝑠𝑠 1 +𝑌𝑌 − 𝑦𝑦 2

2 𝑠𝑠2+


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



≅ �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


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�


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�


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


𝑅𝑅�

− �𝑑𝑑 2

+ �𝑑𝑑 2

𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦𝑑𝑑𝑦𝑦

𝜃𝜃

𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃 =𝑌𝑌𝑅𝑅

𝑦𝑦

𝑧𝑧

13


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�

�−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


=𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑅𝑅 −𝜔𝜔 𝑡𝑡



2 𝑠𝑠𝑐𝑐𝑠𝑠𝑘𝑘 𝑍𝑍 𝑚𝑚

2 𝑅𝑅

17




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


𝑚𝑚

𝑏𝑏

𝑦𝑦

𝑧𝑧

19


𝑅𝑅�




𝑅𝑅��−𝑏𝑏2

�𝑏𝑏 2

𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌𝑅𝑅 𝑦𝑦𝑑𝑑𝑦𝑦 ��−𝑎𝑎2

�𝑎𝑎 2

𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑍𝑍𝑅𝑅 𝑧𝑧𝑑𝑑𝑧𝑧


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


𝑚𝑚𝜑𝜑𝑦𝑦 = 𝜌𝜌 𝑠𝑠𝑖𝑖𝑠𝑠 𝜑𝜑𝜌𝜌

𝑧𝑧 = 𝜌𝜌 𝑠𝑠𝑐𝑐𝑠𝑠 𝜑𝜑𝑧𝑧

𝑦𝑦

23

Observation Plane

Φ𝑌𝑌 = 𝑞𝑞 𝑠𝑠𝑖𝑖𝑠𝑠 Φ

𝑞𝑞

𝑍𝑍 = 𝑞𝑞 𝑠𝑠𝑐𝑐𝑠𝑠 Φ𝑍𝑍

𝑌𝑌

24


𝑅𝑅�



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


𝑅𝑅�0

𝑎𝑎


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


𝑅𝑅�0

𝑎𝑎


2𝜋𝜋

𝑑𝑑𝜑𝜑 𝑒𝑒 − 𝑖𝑖 𝑘𝑘 𝜌𝜌 𝑞𝑞 𝑐𝑐𝑐𝑐𝑐𝑐 𝜑𝜑𝑅𝑅

27


𝑅𝑅2 𝜋𝜋�

0

𝑎𝑎

𝜌𝜌 𝑑𝑑𝜌𝜌 𝐽𝐽0 −𝑘𝑘 𝑞𝑞𝑅𝑅

𝜌𝜌

𝑢𝑢 ≡ −𝑘𝑘 𝑞𝑞𝑅𝑅

𝜌𝜌


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


𝑅𝑅2 𝜋𝜋


2

�0

−𝑘𝑘 𝑞𝑞𝑅𝑅 𝑎𝑎

α 𝑑𝑑α 𝐽𝐽0 α


𝑅𝑅2 𝜋𝜋


2 −𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅

𝐽𝐽1−𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅


𝑅𝑅𝜋𝜋 𝑚𝑚2

2 𝐽𝐽1𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅

𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅

30

𝐼𝐼 𝑞𝑞,Φ = 𝐼𝐼02 𝐽𝐽1



2


2 𝑅𝑅2𝜋𝜋 𝑚𝑚2 2


�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


𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚

𝑁𝑁𝑁𝑁 ≡ 𝑠𝑠 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚𝑎𝑎𝑚𝑚 ≅𝑠𝑠 𝐷𝐷𝑙𝑙𝑝𝑝𝑙𝑙𝑐𝑐

2 𝑓𝑓

𝑓𝑓

𝑓𝑓#

=𝑓𝑓


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


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


𝑅𝑅�



𝑟𝑟

𝑠𝑠

𝑍𝑍

𝑌𝑌

𝑧𝑧

𝑦𝑦

𝑅𝑅

𝑅𝑅 ≡ 𝑌𝑌2 + 𝑍𝑍2 + 𝑠𝑠2

Fraunhofer Diffraction

𝑚𝑚𝑚𝑚𝑚𝑚 𝑦𝑦2 + 𝑧𝑧2


𝑚𝑚𝑚𝑚𝑚𝑚 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2


42

In summary, far-field diffraction:

1. Single Slit

2. Double Slit









2 𝑠𝑠𝑐𝑐𝑠𝑠𝑘𝑘 𝑌𝑌 𝑚𝑚

2 𝑅𝑅




𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠𝑠𝑠𝑘𝑘 𝑍𝑍 𝑚𝑚

2 𝑅𝑅


𝑅𝑅𝜋𝜋 𝑚𝑚2

2 𝐽𝐽1𝑘𝑘 𝑚𝑚 𝑞𝑞𝑅𝑅


43


𝑅𝑅�




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


𝐸𝐸 𝑌𝑌,𝑍𝑍, 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


47




�𝑙𝑙= 0

𝑁𝑁−1

𝑒𝑒− 𝑖𝑖 𝑘𝑘 𝑌𝑌 𝑎𝑎𝑅𝑅𝑙𝑙




1 − 𝑒𝑒−𝑖𝑖 𝑁𝑁𝑘𝑘 𝑌𝑌 𝑎𝑎𝑅𝑅

1 − 𝑒𝑒−𝑖𝑖𝑘𝑘 𝑌𝑌 𝑎𝑎𝑅𝑅




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




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


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


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)


=𝑟𝑟𝑁𝑁𝜋𝜋

𝑏𝑏 = 0.1

𝑚𝑚 = 4

𝑘𝑘 = 1

𝑅𝑅 = 10 <


< 𝜋𝜋

𝑚𝑚 = 0 𝑚𝑚 = 1

10−1 𝑚𝑚2−2

𝐼𝐼 𝑌𝑌,𝑍𝑍 ≅ 𝐼𝐼0𝑠𝑠𝑖𝑖𝑠𝑠2 𝑁𝑁 𝑘𝑘 𝑌𝑌 𝑚𝑚2 𝑅𝑅


52

Angular Width

𝑘𝑘 𝑚𝑚 𝑠𝑠𝑖𝑖𝑠𝑠 𝜃𝜃𝑚𝑚 + ∆𝜃𝜃2

2= 𝑚𝑚 𝜋𝜋 +

1𝑁𝑁𝜋𝜋


=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

𝑠𝑠

𝑍𝑍

𝑌𝑌

𝑧𝑧

𝑦𝑦

𝑟𝑟 𝑦𝑦, 𝑧𝑧 ≅ 𝑠𝑠 +1

2 𝑠𝑠𝑌𝑌 − 𝑦𝑦 2 +

12 𝑠𝑠


𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 = �𝑎𝑎𝑝𝑝𝑝𝑝𝑟𝑟𝑡𝑡𝑝𝑝𝑟𝑟𝑝𝑝



= 𝑠𝑠 1 +𝑌𝑌 − 𝑦𝑦 2

𝑠𝑠2+


𝑠𝑠2

𝑘𝑘 𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚 𝑌𝑌 − 𝑦𝑦 2 + 𝑍𝑍 − 𝑧𝑧 2 2

𝑠𝑠4≪ 𝜋𝜋

58

𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡

𝑠𝑠�


𝐸𝐸0 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖 𝜀𝜀 𝑦𝑦, 𝑧𝑧 𝑒𝑒𝑖𝑖𝑘𝑘2 𝑐𝑐 𝑌𝑌−𝑦𝑦 2+ 𝑍𝑍−𝑧𝑧 2


𝐸𝐸 𝑌𝑌,𝑍𝑍, 𝑡𝑡 =𝐸𝐸0 𝑒𝑒𝑖𝑖 𝑘𝑘 𝑐𝑐 − 𝜔𝜔 𝑡𝑡

𝑠𝑠�


𝑒𝑒𝑖𝑖𝜋𝜋𝜆𝜆 𝑐𝑐 𝑌𝑌−𝑦𝑦 2+ 𝑍𝑍−𝑧𝑧 2


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�


𝑒𝑒𝑖𝑖𝜋𝜋𝜆𝜆 𝑐𝑐 𝑌𝑌−𝑦𝑦 2+ 𝑍𝑍−𝑧𝑧 2


=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


𝑚𝑚

cos𝜋𝜋2𝑚𝑚′2 𝑑𝑑𝑚𝑚𝑥

𝒮𝒮 𝑚𝑚 ≡ �0

𝑚𝑚

sin𝜋𝜋2𝑚𝑚𝑥2 𝑑𝑑𝑚𝑚𝑥

𝒞𝒞 𝑚𝑚

𝒮𝒮 𝑚𝑚

𝑚𝑚

𝑚𝑚

𝑚𝑚

𝒞𝒞 𝑚𝑚𝒮𝒮 𝑚𝑚

63


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


1. No Obstruction





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






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



𝑌𝑌 − 0.5

2


𝑌𝑌 − 0.5

2

68

69


× 𝒞𝒞 𝛾𝛾2 − 𝒞𝒞 𝛾𝛾1 2 + 𝒮𝒮 𝛾𝛾2 − 𝒮𝒮 𝛾𝛾1 2






𝑦𝑦

𝑧𝑧𝛾𝛾2 =

2𝜆𝜆 𝑠𝑠

𝑌𝑌 − 𝑑𝑑2

𝛾𝛾1 =2𝜆𝜆 𝑠𝑠

𝑌𝑌 + 𝑑𝑑2𝛿𝛿2 = −∞ 𝛿𝛿1 = +∞

=𝐼𝐼04


𝑌𝑌 − 𝑑𝑑2 − 𝒞𝒞

2𝜆𝜆 𝑠𝑠

𝑌𝑌 + 𝑑𝑑2

2


𝑌𝑌 − 𝑑𝑑2 − 𝒮𝒮

2𝜆𝜆 𝑠𝑠


2

× 2

3. Single Slit

𝑑𝑑

70

𝒮𝒮 𝑚𝑚

𝒞𝒞 𝑚𝑚𝑌𝑌 = 0

𝑌𝑌 > 0

𝑌𝑌 < 0



𝑌𝑌 − 𝑑𝑑2 − 𝒞𝒞

2𝜆𝜆 𝑠𝑠


2


𝑌𝑌 − 𝑑𝑑2 − 𝒮𝒮

2𝜆𝜆 𝑠𝑠


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


74

5. Circular Objects

Poisson (Arago) spot

Diffraction Theory - University of Louisville

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