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Last Lecture

• Definition of refraction index

• Reflection and refraction

– Law of reflection𝛼 = 𝛽

– Law of refraction (Snell’s law)𝑠𝑖𝑛𝛼

𝑠𝑖𝑛𝛾=𝑛2𝑛1= 𝑛21

Total internal reflection:

𝑤ℎ𝑒𝑛: 𝑛1 > 𝑛2 and 𝛼 ≥arcsin(𝑛21)

𝛼 𝛽

𝛾

𝑛1

𝑛2

Intensity Relationship

• 𝐼𝑟 = 𝑅𝐼𝑟

– Reflected ray becomes polarized

– Refracted ray becomes partially polarized

𝑅𝑝 =𝑡𝑎𝑛2(𝛼 − 𝛾)

𝑡𝑎𝑛2(𝛼 + 𝛾)

𝑤ℎ𝑒𝑛 𝛼 + 𝛾 = 90, 𝑅𝑝 → 0,

Brewster angle:𝛼𝐵 = arctan(𝑛21)

Polarization by Reflection

• Photography

• Polarize sunglasses

• Brewster window

ME557 Experimental Stress Analysis

Light Diffraction

Junlan Wang

Winter 2016

Snow Geese

Diffraction

• Various phenomena which occur when a wave encounters an obstacle.

• In classical physics, diffraction refers to the apparent bending of waves around small particles and the spreading of waves passing small openings

Diffraction vs Interference

• No one has ever been able to define the difference between interference and diffraction satisfactorily. It is just a question of usage. There is no specific, important physical difference between them.

– Richard Feynman

• General suggestions– Interference – only a few sources

– Diffraction – a large # of sources

Explanation by Huygen’s Principle

Each point on a wave front may be regarded as a new source of waves

Single Slit Diffraction

Assumption

• The slit size is small, relative to the wavelength of light.

• The screen is far away.

• Cylindrical waves can be represented in 2D diagrams as circular waves.

• The intensity at any point on the screen is independent of the angle made between the ray to the screen and the normal line between the slit and the screen (this angle is called T below). This is possible because the slit is narrow.

Derivation

ϴ

ϴ

ϴ

𝛿 = 𝑑𝑠𝑖𝑛𝜃 =2

3λ

ϕ =2𝜋

λ𝛿 =

4π

3

d

d

Central Maxima

ϴϴ

𝛿 = 𝑑𝑠𝑖𝑛𝜃 = 0

In phase, bright central band.

d

d

Destructive Interference

• Consider many point sources 1 to k above and k+1 to 2k below all cancel each other at dsin𝜃 =λ

• For the 2nd, 3rd, … mth minima,

dsin𝜃 =mλ

ϴ

ϴ

d

d

ϴ

ϴ

d/2

d/2

Single Slit Diffraction

• Minima d𝑠𝑖𝑛𝜃 = 𝑚λ, m=1, 2, 3 …..

𝜃𝑚 =𝑚λ

𝑑=𝑧𝑚𝐿

a

• Maxima d𝑠𝑖𝑛𝜃𝑚 = 𝑘𝑚λ, m=0, 2, 3 …..

𝑘𝑚 = 0, 1.43, 2.459, 3.471, 4.4711, …..

Alternative Thinking (textbook approach)

s

sθ

P

R

𝑧𝑛

𝐓𝐨𝐩 𝐡𝐚𝐥𝐟: 𝑑𝐸𝑝 𝑠 =𝐾𝑑𝑠

𝑟cos2𝜋

λ(𝑟 − 𝑐𝑡 − 𝑠 𝑠𝑖𝑛𝜃)

𝐁𝐨𝐭𝐭𝐨𝐦 𝐡𝐚𝐥𝐟: 𝑑𝐸𝑝 −𝑠 =𝐾𝑑𝑠

𝑟cos2𝜋

λ(𝑟 − 𝑐𝑡 + 𝑠 𝑠𝑖𝑛𝜃)

Total:

𝐸𝑝 = 0

𝑑/2

𝑑𝐸𝑝 𝑠 + 𝑑𝐸𝑝(−𝑠)

= 0𝑑/2 2𝐾

𝑟cos

2𝜋

λ(𝑟 − 𝑐𝑡)cos

2𝜋

λ(𝑠 𝑠𝑖𝑛𝜃)𝑑𝑠

=𝐾𝑑

𝑟

𝑠𝑖𝑛𝛽

𝛽cos

2𝜋

λ(𝑟 − 𝑐𝑡) where 𝛽 =

𝜋𝑑𝑠𝑖𝑛𝜃

λ

𝐼 =𝐾𝑑

𝑟

2𝑠𝑖𝑛2𝛽

𝛽2

𝑑𝐼

𝑑𝛽= 0

2𝑠𝑖𝑛𝛽

𝛽3𝛽𝑐𝑜𝑠𝛽 − 𝑠𝑖𝑛𝛽 = 0

𝛽 = 𝑛𝜋 𝑛 ≠ 0 → 𝐼 = 0, 𝑛λ = 𝑑𝑠𝑖𝑛𝜃, minima, dark bands

𝛽 = 𝑡𝑎𝑛𝛽 → 𝛽 = 0, 1.43𝜋, 2.459𝜋, 3.471𝜋, 4.477𝜋,𝑘λ = 𝑑𝑠𝑖𝑛𝜃 (𝑘 = 0, 1.432, 2.459, 3.471, …… )

𝐼 = 1,1

21,1

61,1

121…… maxima, bright bands

Multiple Slits

Grating Diffraction

𝑑𝑠𝑖𝑛𝜃 = 𝑚for normal light incidence

Maxima:

White Light Grating Diffraction

dsin 𝜃 = 𝑚

Two Dimensional Grating Diffraction

+

Diffraction from polygonal aperture/obstacle

# of Diffraction Spikes Vs. Polygon Shape

• m𝑠𝑝𝑖𝑘𝑒 = 𝑛𝑒𝑑𝑔𝑒 if 𝑛𝑒𝑑𝑔𝑒 is even number

• m𝑠𝑝𝑖𝑘𝑒 = 2 ∗ 𝑛𝑒𝑑𝑔𝑒 if 𝑛𝑒𝑑𝑔𝑒 is odd number

Pinhole Diffraction

dsin 𝜃 = 𝑘𝑚𝑘𝑚 = 1.22, 2.23, 3.24, 4.24, 5.24

for m =1, 2, 3, 4, 5 𝜃𝑚 =𝑘𝑚λ

𝑑=𝐷/2

𝐿

Summary

• Principles of diffraction

• Diffraction equation (minima or maxima)

– Single slit

– Multiple slit (grating)

– Pinhole

Thursday’s Lab

• To measure the index of refraction of the following media:– Polycarbonate– Acrylic (plexiglass)– Water (or other liquid of student choice)

• Based on the measurement of the angular orientation of the diffracted beams, determine– Frequency of one diffraction grating (line/mm)– Diameter of the pin-hole (µm)