Diffraction Theory

1

2

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

4

Huygens-Fresnel Principle

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

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

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

8

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:

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

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

11

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{

12

Apertures considered here:

1. Single Slit

2. Double Slit

3. Rectangular Aperture

4. Circular Aperture

13

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

14

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

15

Mathematica

FraunhoferDiffractionAtASingleSlit.cdf

16

2. Double Slit

𝑑

𝑑

𝑎

𝑦

ൗ𝑎 2 − ൗ𝑑2

ൗ𝑎 2 + ൗ𝑑2

ൗ−𝑎 2 − ൗ𝑑2

ൗ−𝑎 2 + ൗ𝑑2

𝑧

17

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

18

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

19

3. Rectangular Aperture

𝑎

𝑏

𝑦

𝑧

20

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

21

𝑌

𝑍

𝐼 𝑌, 𝑍 = 𝐼0 𝑠𝑖𝑛𝑐2

𝑘 𝑌 𝑏

2 𝑅𝑠𝑖𝑛𝑐2

𝑘 𝑍 𝑎

2 𝑅

𝐼0 ≡𝐸0

2

2 𝑅2𝑎2 𝑏2

22

Emission of Semiconductor Laser

23

4. Circular Aperture

𝑎𝜑

𝑦 = 𝜌 𝑠𝑖𝑛 𝜑𝜌

𝑧 = 𝜌 𝑐𝑜𝑠 𝜑𝑧

𝑦

24

Observation Plane

Φ

𝑌 = 𝑞 𝑠𝑖𝑛 Φ

𝑞

𝑍 = 𝑞 𝑐𝑜𝑠 Φ𝑍

𝑌

25

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

26

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:

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

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

29

න

0

𝛼

𝛼 𝐽0 𝛼 𝑑𝛼 ≡ 𝛼 𝐽1 𝛼

30

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

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

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

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

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

65

Applications of Fresnel Diffraction

1.No obstruction

2.Straight edge

3. Single slit

4. Rectangular aperture

5. Opaque circular disk

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 !!

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

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

69

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

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

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

73

Mathematica

SingleSlitDiffractionPattern.cdf

74

4. Rectangular Aperture

75

5. Circular Objects

Poisson (Arago) spot