Electricity & Optics

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Rudolf Owen
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1 Physics Electricity & Optics Lecture 26 Chapter 33 sec. 1-4 Fall 2017 Semester Professor Koltick

2 Interference of Light Interference phenomena are a consequence of the wave-like nature of light Electric field: E x z, t = E 0 sin kz ωt Principle of superposition: E total = E 1 + E 2 Useful trigonometric identity: α + β α β sin α + sin β = 2 sin cos 2 2

3 Interference of Light Consider two waves with the same wavelength and frequency but different phase: E 1 z, t = E 0 sin kz ωt E 2 z, t = E 0 sin kz ωt + δ Their superposition has the same wavelength and frequency, but an amplitude that depends on the phase, δ: E total z, t = 2E 0 cos δ/2 sin kz ωt If δ = π, 3π, 5π, then cos δ/2 = 0 (destructive ) If δ = 0, 2π, 4π, then cos δ/2 = 1 (constructive)

4 δ = 0: Constructive interference δ = π: Destructive interference Amplitude (arbitrary units) Amplitude (arbitrary units) Position (arbitrary units) Position (arbitrary units) 3 Some other value of δ Amplitude (arbitrary units) Position (arbitrary units)

5 Interference of Light To observe interference effects with light, the two interfering waves must be coherent : They must have the same ω and λ They must maintain the same phase difference over relatively large distances Coherence length is the distance over which the phase, δ, remains constant Necessary conditions to observe interference

6 Producing Coherent Light The easiest way to produce coherent light is to produce two waves using the same source. Incandescent lights produce a broad distribution of λ (incoherent). Gas discharge lamps produce spectral lines with the same λ and ω but random phase (also incoherent). However, A given photon has a coherence length of a few millimeters Lasers are a good source of coherent light. Lasers can be coherent over kilometers.

7 Interference of Light Lasers are a good source of coherent light. To observe interference effects you need two waves with different phases. Three ways to produce a phase difference: 1. Travelling through media with different indices of refraction 2. Traveling along paths of different length 3. By reflecting from a boundary between two media. Path length difference can not be longer than the coherence length

8 Different Media Frequency remains constant but the speed of light is lower in media with n > 1: Speed of light, v = c n Wavelength, λ = c nf = λ n L n 1 Phase is φ 1 = 2πL = 2π Ln 1 λ 1 λ n 2 φ 2 = 2πL = 2π Ln 2 λ 2 λ Phase difference is δ = 2π L λ n 1 n 2

9 Different Path Lengths Huygens principle: Light propagates by continuously re-emitting spherical waves The waves emitted from each slit interfere with each other Thomas Young experiment (1801)

10 Interference: Different Path Lengths Interference maxima condition : d sin m m, m 0,1,2,... m order number Interference mimima condition : d sin m m 1, m 1,2,3,... 2 m order number

11 Different Path Lengths Small angle approximation: sin θ tan θ θ. tan θ = y/l Maximum intensity occurs at λ L y = m d

12 Double-Slit Diffraction Spacing between fringes: Δy = λ L d.

13 Reflection Light is partially reflected from an interface between two materials with different indices of refraction (lecture 23): I r = I 0 n 1 n 2 n 1 + n 2 The reflected light has a phase difference of δ = π (180 ) when it enters a material with a greater index of refraction. When light is reflected from an interface into a medium with a lesser index of refraction, there is no change in phase. 2

14 Reflection Reflection with phase inversion: n > 1 Reflection without phase inversion: n > 1

15 Reflection mirror light beam projector screen soap film

16 Interference: Soap Bubble The phase difference between the two waves is caused by: 1. Phase shift on reflection for ray 1 of 180. Ray 2 does not phase shift on reflection. 2. Path length difference between ray 1 and ray 2 is 2t. (Ray 2 crosses the film twice.) For in-phase waves (maxima - bright fringes): 2t (m 1 2 ) n 2 For out of-phase waves (minima - dark fringes): 2t m n 2 The total difference between ray 1 and ray 2 is the extra difference traveled plus the shift due to reflection.

17 Question *The phase difference between two waves can change if one or both are reflected. 1 2 Reflection Reflection Phase Shift Off lower index zero Off higher index 180 QUESTION: Which ray experiences a phase shift due to reflection? a) Ray 1 b) Ray 2 c) none d) both

18 Newton s Rings: Fringes from a wedge-shape film of air indicating how optically flat are the glass plates. Not very flat Very flat

Coating a Glass Lens to Suppress Reflections: 180 0 phase change at both a and b since reflection is off a more optically dense medium 1 2

19 Coating a Glass Lens to Suppress Reflections: phase change at both a and b since reflection is off a more optically dense medium 1 2 How thick should the coating be for destructive interference? 2t = λ 2 t = λ 4 = λ 4n 2 What frequency to use? Visible light: nm t

20 Coating a Glass Lens to Suppress Reflections: For λ= 550 nm and least thickness (m=1) t 4n 550 nm nm Note that the thickness needs to be different for different wavelengths.

PH 222-3A Spring 2010

PH 222-3A Spring 2010 PH -3A Spring 010 Interference Lecture 6-7 Chapter 35 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1 Chapter 35 Interference The concept of optical interference is critical to understanding