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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
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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)
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Polarization by Reflection
• Photography
• Polarize sunglasses
• Brewster window
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ME557 Experimental Stress Analysis
Light Diffraction
Junlan Wang
Winter 2016
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Snow Geese
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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
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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
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Explanation by Huygen’s Principle
Each point on a wave front may be regarded as a new source of waves
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Single Slit Diffraction
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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.
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Derivation
ϴ
ϴ
ϴ
𝛿 = 𝑑𝑠𝑖𝑛𝜃 =2
3λ
ϕ =2𝜋
λ𝛿 =
4π
3
d
d
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Central Maxima
ϴϴ
𝛿 = 𝑑𝑠𝑖𝑛𝜃 = 0
In phase, bright central band.
d
d
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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
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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, …..
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Alternative Thinking (textbook approach)
s
sθ
P
R
𝑧𝑛
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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
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Multiple Slits
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Grating Diffraction
𝑑𝑠𝑖𝑛𝜃 = 𝑚for normal light incidence
Maxima:
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White Light Grating Diffraction
dsin 𝜃 = 𝑚
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Two Dimensional Grating Diffraction
+
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Diffraction from polygonal aperture/obstacle
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# of Diffraction Spikes Vs. Polygon Shape
• m𝑠𝑝𝑖𝑘𝑒 = 𝑛𝑒𝑑𝑔𝑒 if 𝑛𝑒𝑑𝑔𝑒 is even number
• m𝑠𝑝𝑖𝑘𝑒 = 2 ∗ 𝑛𝑒𝑑𝑔𝑒 if 𝑛𝑒𝑑𝑔𝑒 is odd number
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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
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Summary
• Principles of diffraction
• Diffraction equation (minima or maxima)
– Single slit
– Multiple slit (grating)
– Pinhole
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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)
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