23PH301 - Optics - Unit 3 - Diffraction.pptx

Optics
Ms Dhivya R
Assistant Professor
Department of Physics
Sri Ramakrishna College of Arts and Science
Coimbatore - 641 006
Tamil Nadu, India
1

U3 – Diffraction - Overview
1. Huygens – Fresnel Theory
2. Fresnel’s Assumptions
3. Rectilinear Propagation of light waves
4. Zone Plate
5. Action of Zone plate on a Spherical wavefront
6. Difference between Zone Plate and Convex Lens
7. Distinction between Interference and diffraction
8. Diffraction pattern due to straight edge
9. Diffraction pattern due to narrow slit
10. Fraunhofer diffraction
11. Fraunhofer Diffraction in single slit
12. Fraunhofer Diffraction in double slit
13. Plane Diffraction Grating
2

Diffraction
• Diffraction refers to various phenomena that occur when a
wave encounters an obstacle or a slit.
• It is defined as the bending of waves around the corners
of an obstacle or through an aperture into the region of
geometrical shadow of the obstacle/aperture.
3

Diffraction
4

Huygen’s Fresnel Theory
Each progressive wave produces a secondary wave, the
envelope of which is secondary wavefront
5

Fresnel’s Assumptions
Augustine Jean Fresnel in 1815, combines Huygen’s wavelets
with the principle of interference and explained Bending
of light.
6

Fresnel’s Assumptions
1. A Wavefront can be divided into a large number of strips
called Zones (Fresnel’s Zones)
2. The affect at any particular point will depend on the
distance of the zone from the point
3. Effect will also depend on the obliquity of the point with
reference to the zone
7

Rectilinear Propagation of Light
• Radii are proportional to the square root of the natural
numbers
ie
• The area of the zones are proportional to
• The wavelength of the light used
• The distance of the point from the wavefront (as discussed in
Fresnel’s assumptions)
8
1𝑏𝜆, 2𝑏𝜆, 3𝑏𝜆, 4𝑏𝜆 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛

• Let m1,m2,m3 be the amplitudes at P due to the
secondary wave from 1st, 2nd, 3rd half period zones
9

• As the amplitudes are of gradually decreasing magnitude,
the amplitude of vibration at P, due to any zone can be
approximately taken as the mean of the amplitudes due to
the zone preceding and succeeding it
10
𝑒𝑔 𝑚2 =
𝑚1 + 𝑚3
2
𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑎𝑡 𝑃 𝑎𝑡 𝑎𝑛𝑦 𝑖𝑛𝑠𝑡𝑎𝑛𝑡 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦
𝐴 = 𝑚1 − 𝑚2 + 𝑚3 − 𝑚4. . . . . . +𝑚𝑛𝑖𝑓𝑛 𝑖𝑠 𝑜𝑑𝑑.
𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑡ℎ𝑒 𝑙𝑎𝑠𝑡 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑖𝑠 − 𝑚𝑛)

11
𝐴 =
𝑚1
2
+
𝑚1
2
− 𝑚2 +
𝑚3
2
+
𝑚3
2
− 𝑚4 +
𝑚5
2
+. . . . . .
𝑏𝑢𝑡 𝑚2 =
𝑚1
2
+
𝑚3
2
𝑎𝑛𝑑 𝑚4 =
𝑚3
2
+
𝑚5
2
𝐴 =
𝑚1
2
+
𝑚𝑛
2
+. . . . . . (𝑖𝑓𝑛𝑖𝑠𝑜𝑑𝑑)
𝐴 =
𝑚1
2
+
𝑚𝑛−1
2
− 𝑚𝑛(𝑖𝑓𝑛𝑖𝑠𝑒𝑣𝑒𝑛)

12
𝐴 =
𝑚1
2
𝑆𝑖𝑛𝑐𝑒 𝑡ℎ𝑒 𝑒𝑓𝑓𝑒𝑐𝑡 𝑑𝑢𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑧𝑜𝑛𝑒𝑠 𝑎𝑟𝑒 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔
𝐼𝛼
𝑚1
2
4
(∵ 𝐼 =
𝐴
2
)

Zone Plate
• Screen – Specially constructed to allow light from either
odd or even numbered zones alone
13

Zone Plate
• If the source is at a larger distance from the zone plate, a
bright spot will be obtained at P
• If the even numbered zones are opaque
• 𝐴 = 𝑚1 + 𝑚3. . . . . . +𝑚𝑛 (brighter)
• 𝐴 = 𝑚1 − 𝑚2 + 𝑚3 − 𝑚4. . . . . . +𝑚𝑛 for unobstructed
• Whitelight source – different colours come to focus at diff
points along OP. ZP Similar to Convex Lens.
14
𝑟1 = 𝑏𝜆 𝑟2 = 2𝑏𝜆 𝑟𝑛 = 𝑛𝑏𝜆
𝑏 =
𝑟𝑛
2
𝑛𝜆
𝑏 = 𝑓𝑛 =
𝑟𝑛
2
𝑛𝜆

Action of Zone plate
15

16

17

18

19

Comparison between Zone Plate
and Convex Lens
20
Sl Zone Plate Convex Lens
1
Forms Real image and the conjugate
distance (fn) is similar to convex lens
Forms Real image and the conjugate
distance (fn) is similar to zone plate
2
Suffers chromatic aberration to a greater
extent
Suffers chromatic aberration to a lesser
extent
3
Acts simultaneously as both convex and
concave lens by forming both real and
virtual image
Forms only real image
4 Works under the principle of diffraction Works under the principle of refraction
5 Multiple foci – intensity of the image is low One focus – intensity of the image is more
6
Each wavefornt suffers a pathdifference of
lambda and phase diffrence of 2pi
All are inphase and zero path diffrence
7
Accepts all wavelengths ranging from
microwave to x-rays
Works only with visible light

Difference between Diffraction
and interference
21
Sl Interference Diffraction
1
Interference is a result of interaction
between waves from two light
sources
Diffraction is a result of interaction
between different parts of same
wave font
2
Interference fringes may or may not
be same width
Diffraction fringes are not of the
same width
3
Regions of minimum intensity are
perfectly dark
Regions of minimum intensity are
not perfectly dark
4 All bright bands are of same intensity
The intensity of the fringes differ,
with the central maximum having
the maximum intensity and
decreasing after

Fresnel diffraction
• Source at a finite distance from the slit
• Diffraction pattern collected on screen without the help of
lens
22

Fraunhoffer diffraction
• Incident wave front must be plane
• Diffraction pattern collected on screen with the help of
lens
• Source must be at infinity from the slit
23

24

25
sin 𝜃 =
𝐴𝐿
𝐴𝐵
=
𝐴𝐿
𝑎
𝐴𝐿 = 𝑎 sin 𝜃
𝑎 sin 𝜃𝑛 = 𝑛𝜆

26
𝑎 sin 𝜃𝑛 = (2𝑛 + 1)
𝜆
2
𝑎 sin 𝜃𝑛 =
(2𝑛 + 1)𝜆
2𝑎

27
sin 𝜃 =
𝑥
𝑓
𝑏𝑢𝑡 sin 𝜃 =
𝜆
𝑎
ℎ𝑒𝑛𝑐𝑒
𝑥
𝑓
=
𝜆
𝑎
𝑜𝑟𝑥 =
𝑓𝜆
𝑎
𝑊𝑖𝑑𝑡ℎ = 2𝑥 =
2𝑓𝜆
𝑎

28

• Diffraction at double slit
29

• Diffraction phenomenon due to two things
• Interference due to secondary waves from the corresponding
points at the two slits
• Diffraction due to the secondary waves from the two slits
individually
• Diffraction angle is denoted as Ѳ (angle between direction of
direct ray and the direction of secondary waves)
30

31

32
sin 𝜃 =
𝐶𝑁
𝐴𝐶
=
𝐶𝑁
𝑎 + 𝑏
𝐶𝑁 = (𝑎 + 𝑏) sin 𝜃
𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑓𝑜𝑟 min 𝑖 𝑚𝑎
𝐶𝑁 = (𝑎 + 𝑏) sin 𝜃 = (2𝑛 + 1)
𝜆
2
sin 𝜃 =
(2𝑛 + 1)𝜆
2(𝑎 + 𝑏)

33
𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑓𝑜𝑟 max 𝑖 𝑚𝑎
𝐶𝑁 = (𝑎 + 𝑏) sin 𝜃 = 𝑛𝜆
sin 𝜃 =
𝑛𝜆
(𝑎 + 𝑏)
𝑖𝑓𝑛 = 1,2,3. . . .
sin 𝜃1 =
𝜆
(𝑎 + 𝑏)
; sin 𝜃2 =
2𝜆
(𝑎 + 𝑏)
; sin 𝜃3 =
3𝜆
(𝑎 + 𝑏)
; . . .
𝑎𝑛𝑔𝑢𝑙𝑎𝑟𝑠𝑒𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑏𝑒𝑡𝑤𝑒𝑒𝑛𝑎𝑛𝑦𝑡𝑤𝑜𝑐𝑜𝑛 sec 𝑢 𝑡𝑖𝑣𝑒 max 𝑖 𝑚𝑎𝑜𝑟 min 𝑖 𝑚𝑎
sin 𝜃2 − sin 𝜃1 =
𝜆
(𝑎 + 𝑏)

Plane diffraction grating
• Large number of narrow slits side by side
• Slits separated by opaque spaces
• Light transmitted through slit and blocked by opaque
regions
• So they are called Plane Transmission Grating
34

35

36

37
𝑃𝑎𝑡ℎ𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝐴𝐶 sin 𝜃
𝐴𝐶 = 𝐴𝐵 + 𝐵𝐶 = (𝑎 + 𝑏) sin 𝜃
𝑀𝑎𝑥𝑖𝑚𝑎𝑎𝑡
(𝑎 + 𝑏) sin 𝜃𝑛 = 𝑛𝜆

• Number of Principal maxima = Number of wavelengths
• Angle of different wavelengths are different
• if there are two wavelengths
38
𝜆𝑎𝑛𝑑𝑑𝜆𝑠𝑜𝑡ℎ𝑒𝑑𝑖𝑓𝑓𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑛𝑔𝑙𝑒𝑠𝑎𝑟𝑒
𝜃𝑎𝑛𝑑𝑑𝜃

39
(𝑎 + 𝑏) sin 𝜃 = 𝜆
(𝑎 + 𝑏) sin( 𝜃 + 𝑑𝜃) = 𝜆 + 𝑑𝜆

23PH301 - Optics - Unit 3 - Diffraction.pptx

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23PH301 - Optics - Unit 3 - Diffraction.pptx