Physics 24100 Electricity & Optics Lecture 26 Chapter 33 sec. 1-4 Fall 2017 Semester Professor Koltick
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
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)
δ = 0: Constructive interference δ = π: Destructive interference Amplitude (arbitrary units) Amplitude (arbitrary units) 4 3 2 1 0-1 -2-3 -4 0 20 40 60 80 100 Position (arbitrary units) 2 1.5 1 0.5 0-0.5-1 -1.5-2 0 20 40 60 80 100 Position (arbitrary units) 3 Some other value of δ Amplitude (arbitrary units) 2 1 0-1 -2-3 0 20 40 60 80 100 Position (arbitrary units)
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
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.
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
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
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)
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
Different Path Lengths Small angle approximation: sin θ tan θ θ. tan θ = y/l Maximum intensity occurs at λ L y = m d
Double-Slit Diffraction Spacing between fringes: Δy = λ L d.
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
Reflection Reflection with phase inversion: n > 1 Reflection without phase inversion: n > 1
Reflection mirror light beam projector screen soap film
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.
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
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 How thick should the coating be for destructive interference? 2t = λ 2 t = λ 4 = λ 4n 2 What frequency to use? Visible light: 400-700 nm t
Coating a Glass Lens to Suppress Reflections: For λ= 550 nm and least thickness (m=1) t 4n 550 nm 4 1.38 99.6 nm Note that the thickness needs to be different for different wavelengths.