Phys 2310 Mon. Dec. 11, 2014 Today s Topics. Begin Chapter 9: Interference Reading for Next Time

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1 Phys 30 Mon. Dec., 04 Todays Topics Begin Chapter 9: nterference Reading for Next Time

2 Reading this Week By Wed.: Begin Ch. 9 (9. 9.3) General Considerations, Conditions for nterference, Wavefront-splitting nterferometers

3 Homework this Week Chapter 9 Homework (due Mon. Dec. 7) #4, 5, 7, 0, 6, 3, 34, 37 3

4 Chapter 9: nterference Preliminaries: nterference occurs when two or more M wave overlap in space such that the result is a superposition. The result will depend on both amplitude and phase differences. xamples include Colored bands when light reflects off oil sheen over water Colored bands when light reflects off soap bubbles nterferometers 4

5 5 Chapter 9: nterference General Considerations Principle of Superposition Recall the addition of two plane waves (also applicable to spherical):... 3, 0, when : ) destructive interference (cos Total and... 4, 0, when : ) constructive interference (cos Total cases are : Two special cos we have : / and / but since cos are parallel.n this case : and We will assume polarization issue. a Note though that there remains. and the path length vs.position at any point phases is the phase difference from both the initial ) ( where :, cos that averaging, see text) t can be shown (cosine product and,, with : we might rewrite this as Thus where the second term is the interference term. and we have : ) ( ) ( Thus since : constants). (neglecting some numerical the amplitude : We saw that the irradience at any one point is the square of ) cos( ), ( and ) cos( ), ( min max π π π π ε ε ε ω ε ω ± ± ± ± r r k r k t r k t r t r k t r T T T T

6 Chapter 9: nterference General Considerations Condition for maxim and minima Total constructive interference (cos ) : max Total destructive interference (cos ) : min n the special case where both amplitudes are the same : max 4 f we now consider : k( r max min 0 : ( r : ( r ( cos ) 4 0 r and r r ) ( ε ε ) ) ) min 0 0 when 0, ± π, ± 4π... when 0, ± π, ± 3π... cos and so : [ πm ( ε ε )] [ πm ( ε ε )]/ and so we have : The two equations above simplify further if ( ε ε ) 0, and it is apparent that they are the equations of a hyperbola. / k k So the maxima and minima are distributed along hyperbolic surfaces (see Fig. 9.3) and 6

7 Chapter 9: nterference Conditions for nterference Temporal and Spatial Coherence Recall that atoms emit wave packet and not continuous wave trains Coherence time is only about 0 ns or so depending on how monochromatic the source is. There is a corresponding coherence length (speed of light). See fig. 9.4 White-light fringes can be seen if the path-length differences are small. Lasers make it easy, otherwise we must use a single source Fresnel-Arago Laws Two orthogonal, coherent polarizations cannot interfere since 0. Two parallel, coherent polarizations will interfere Two orthogonal polarizations cannot be made to interfere via rotation since they are incoherent. 7

8 Wavefront-splitting nterferometers Secondary wavelet sources produce interference Youngs -slit xperiment Young showed light could interfere by passing light from a semi-coherent source through two slits (see lab) Difference in OPL from each slit to given point on screen results in interference Chapter 9: nterference y ΔOPL asinθ a (small angle approx.) s and interference requires : ΔOPL asinθ a Δy s a y s mλ and so the separation is : λ (measure s, a, and Δy to measure λ) 8

of interference) Similarly, white light fringes can be seen only at small OPL since they begin to overlap due to the λ dependence of the

9 Chapter 9: nterference Wavefront-splitting nterferometers Youngs -slit xperiment ffects of Finite Coherence Length Finite coherence means that as the path difference increases the fringes loose contrast as due to a lack of temporal coherence (light travel time to the point of interference) Similarly, white light fringes can be seen only at small OPL since they begin to overlap due to the λ dependence of the OPL. Fresnals Double Mirror and Double Prism nterference seen from the differing OPL from two slightly tilted mirrors or very thin prisms 9

10 Chapter 9: nterference Amplitude-splitting nterferometers nterference produced from an original source by splitting path Fringes from Thin Films Fringes from a film of uniform thickness (equal inclination fringes) Circular fringes due to symmetry OPL n f d cos! t! n (AD) but: (AD) (AC)sin! i but substuting using Snell's Law: (AD) (AC) n f n i sin! t and since: (AC) d tan! t we have: (AD) (d tan! t ) n f n i sin! t OPL n f d cos! t (! sin! t ) or: OPL n f d cos! t and since the corresponding phase shift will be: " 4#n f $ 0 d cos! t ± # (note refl. phase shift) d cos! t (m ) $ 0 4n f d cos! t (m) $ 0 4n f (for maxima) (for minima) 0

Chapter 9: nterference Amplitude-splitting nterferometers Fringes from Thin Films Fringes from a film of changing thickness (equal thickness fringes) xamples of Fizeau fringes include soap bubble

11 Chapter 9: nterference Amplitude-splitting nterferometers Fringes from Thin Films Fringes from a film of changing thickness (equal thickness fringes) xamples of Fizeau fringes include soap bubble films (and wedge shown in class), wedge produces parallel bands d x tanα The condition for interference maxima : ( m / ) λ n xα (small angle approx.) αn and so the location of the maxima are : m / xmax λ f where λ f λ0 / n f α Note that the film thickness at each maxima is: d max 0 f d max f x max λ f ( m / ) (odd multiples of λ f / 4)

The wedge thickness varies according to the radius of curvature of the

12 Chapter 9: nterference Newton s Rings are a special case where circular symmetry is involved. The wedge thickness varies according to the radius of curvature of the lens (fig. 9.3) Used in optical testing to determine radius of curvature and surface accuracy. See the derivation in the text

13 Chapter 9: nterference Amplitude-splitting nterferometers Michelson nterferometer One of the simplest and best-known interferometers. Beam-splitter at (O) diverts light down each of two beams. These in turn reflect off mirrors. The beams recombine at (O) before emerging from the interferometer. The compensator plate (C) equalizes the phase difference between the two beams. t also reduces dispersion. Maxima occur when dcosθ m mλ 0 (circular fringes) Useful for precise measurements of small distances (small changes in path length). 3

Chapter 9: nterference Amplitude-splitting nterferometers Mach-Zehnder nterferometer Used to measure the OPL of different materials inserted into

14 Chapter 9: nterference Amplitude-splitting nterferometers Mach-Zehnder nterferometer Used to measure the OPL of different materials inserted into one of the beams. xtremely versatile laboratory instrument. xamples include wind tunnels and shock tubes. Since index of air depends on density. 4

15 Reading this Week By Wed.: Begin Ch. 9 (9. 9.3) General Considerations, Conditions for nterference, Wavefront Splitting nterferometers 5

16 Homework this Week Chapter 9 Homework (due Mon. Dec. 7) #4, 5, 7, 0, 6, 3, 34, 37 6

Electricity & Optics

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