PH 222-3A Spring 2010

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1 PH -3A Spring 010 Interference Lecture 6-7 Chapter 35 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1

2 Chapter 35 Interference The concept of optical interference is critical to understanding many natural phenomena, ranging from color shifting in butterfly wings to intensity patterns formed by small apertures. These phenomena cannot be explained using simple geometrical optics, and are based on the wave nature of light. In this chapter we explore the wave nature of light and examine several key optical interference phenomena.

3 Light as a Wave Huygen s Principle: All points on a wavefront serve as point sources of spherical secondary wavelets. After time t, the new position of the wavefront will be that of a surface tangent to these secondary wavelets. Fig. 35-3

$Fig. 35-3 Law of Refraction λ1 λ λ1 v1 t = = = v1 v λ v λ1 sin θ1 = (for triangle hce) hc λ sin θ = (for triangle hcg) hc sinθ1 λ1$

4 Fig Law of Refraction λ1 λ λ1 v1 t = = = v1 v λ v λ1 sin θ1 = (for triangle hce) hc λ sin θ = (for triangle hcg) hc sinθ1 λ1 v1 = = sinθ λ v c c n1 = and n = v1 v sinθ1 cn1 n sinθ = cn = n 1 n Index of Refraction: Law of Refraction: 1 1 n = sinθ = n sinθ 4 c v

$Wavelength and Index of Refraction λn v v λ λn λ λn λ = c = c = n v c n c f = n f λ = λ n = λ = The frequency of light in a medium is the same as it is in vacuum. n Fig.$

5 Wavelength and Index of Refraction λn v v λ λn λ λn λ = c = c = n v c n c f = n f λ = λ n = λ = The frequency of light in a medium is the same as it is in vacuum. n Fig Since wavelengths in n 1 and n are different, the two beams may no longer be in phase. L L Ln1 Number of wavelengths in n1: N = 1 λ = n1 λ n = 1 λ L L Ln Number of wavelengths in n: N = λ = n λ n = λ Ln Ln L Assuming n > n1: N N1 = = ( n n1) λ λ λ 5 N N1 = 1/ wavelength destructive interference

Rainbows and Optical Interference Fig. 35-5 The geometrical explanation of rainbows given in Ch. 34 is incomplete.

6 Rainbows and Optical Interference Fig The geometrical explanation of rainbows given in Ch. 34 is incomplete. Interference, constructive for some colors at certain angles, destructive for other colors at the same angles, is an important component of rainbows. 6

7 Diffraction For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting. Fig

$interfere and form an interference (diffraction) pattern. Fig.$

8 Young s Experiment For waves entering two slits, the emerging waves interfere and form an interference (diffraction) pattern. Fig

9 Locating Fringes The phase difference between two waves can change if the waves travel paths of different lengths. What appears at each point on the screen is determined by the path length difference ΔL of the rays reaching that point. Path Length Difference: Δ L= dsinθ Fig

10 Locating Fringes Fig ( )( ) If Δ L= dsinθ = integer λ bright fringe Maxima-bright fringes: dsin θ = mλ for m= 0,1,, ( )( ) If Δ L= dsinθ = odd number λ/ dark fringe 1 Minima-dark fringes: θ ( ) dsin = m+ λ for m = 0,1,, m 1 λ 1 1.5λ = bright fringe at: θ = sin m = 1 dark fringe at: θ = sin d d 10

11 Coherence Two sources can produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E 0 cos(ωt+φ). Coherent sources: Phase φ must be well defined and constant. When waves from coherent sources meet, stable interference can occur. Sunlight is coherent over a short length and time range. Since laser light is produced by cooperative behavior of atoms, it is coherent of long length and time ranges. Incoherent sources: φ jitters randomly in time, no stable interference occurs, 11

12 Intensity in Double-Slit Interference ( ) E1 = E0sin ωt and E = E0sin ωt+ φ E 1 1 π d I = 4I0 cos φ φ = sinθ λ 1 π d φ = mπ m= φ = mπ = θ λ θ λ Maxima when: for 0,1,, sin dsin = m for m= 0,1,, (maxima) ( ) ( ) Minima when: φ = m+ π dsin θ = m+ λ for m= 0,1,, (minima) E I = I avg 0 Fig

13 Proof of Eqs. 35- and 35-3 Eq. 35- ( ) 0 ω 0 ( ω φ) = ( cos β) = 1 cos φ E t = E sin t+ E sin t+ =? E E E E = 4E cos φ I I E = = 4cos φ I = 4I cos E φ Fig β + β = φ Eq phase path length difference difference = π λ phase π path length difference = λ difference π φ = ( d sinθ) λ 13

14 In general, we may want to combine more than two waves. For example, there may be more than two slits. Procedure: Combining More Than Two Waves 1. Construct a series of phasors representing the waves to be combined. Draw them end to end, maintaining proper phase relationships between adjacent phasors.. Construct the sum of this array. The length of this vector sum gives the amplitude of the resulting phasor. The angle between the vector sum and the first phasor is the phase of the resultant with respect to the first. The projection of this vector sum phasor on the vertical axis gives the time variation of the resultant wave. E 4 E 3 E E 1 E 14

15 Interference from Thin Films φ 1 =? θ 0 Fig

16 Reflection Phase Shifts n 1 > n n 1 n n 1 < n Off lower index 0 Reflection Reflection Phase Shift n 1 n Off higher index 0.5 wavelength Fig

Fig. 35-17 Equations for Thin-Film Interference λ 0 Three effects can contribute to the phase difference between r 1 and r. 1. Differences in reflection conditions.

17 Fig Equations for Thin-Film Interference λ 0 Three effects can contribute to the phase difference between r 1 and r. 1. Differences in reflection conditions.. Difference in path length traveled. 3. Differences in the media in which the waves travel. One must use the wavelength in each medium (λ / n) to calculate the phase. 17

Thin film interference 3.General description of constructive and destructive interference. Constructive: t f f f + Δ = λ,λ,3λ... An observer would see a uniform bright film Destructive: t + Δ = 4.

18 Thin film interference 3.General description of constructive and destructive interference. Constructive: t f f f + Δ = λ,λ,3λ... An observer would see a uniform bright film Destructive: t + Δ = 4. For a particular gasoline film floating on a puddle of water Constructive: Destructive: t = t f f f = λ,λ,3λ... where t is extra distance traveled by wave mλ f λ f 3λ f, 5λ f, λ air... no phase change 180 phase change λ film = λ air /n film 1. Because of reflection and refraction, two light waves, represented by rays 1 and, enter the eye when light shines on a film of gasoline floating on a thick layer of water.. When the index of refraction is increasing (as at an air-gasoline interface), a wave suffers a phase change Δ=180 =λ film / in reflection. When the index of refraction decreases, no phase change occurs.!!! The wavelength that is important for thin film interference is the wavelength within the film λ film 18

19 19

20 Film Thickness Much Less Than λ r 1 r If L is much less than λ, for example L < 0.1λ, then phase difference due to the path difference L can be neglected. Phase difference between r 1 and r will always be ½ wavelength destructive interference film will appear dark when viewed from illuminated side. 0

They will do so repeatedly for every λ/ translation of adjustable mirror.

21 The Michelson Interferometer If D A -D F =d the extra distance traversed by the beam A, includes distance d twice, once before and once after reflection. If, in addition, d=mλ, the two beams interfere constructively. They will do so repeatedly for every λ/ translation of adjustable mirror. Precise distance measurements can be made with the Michelson interferometer by moving the mirror and counting the interference fringes which move by a reference point. The distance d associated with m fringes d=mλ/. 1

Chapter 35. Interference

Chapter 35. Interference Chapter 35 Interference The concept of optical interference is critical to understanding many natural phenomena, ranging from color shifting in butterfly wings to intensity patterns formed by small apertures.