LECTURE 26: Interference ANNOUNCEMENT. Interference. Interference: Phase Differences

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1 ANNOUNCEMENT *Exam : Friday December 4, 0, 8 AM 0 AM *Location: Elliot Hall of Music *Covers all readings, lectures, homework from Chapters 9 through 33. *The exam will be multiple choice. Be sure to bring your student ID card and a handwritten one-page (two sided) crib sheet plus the crib sheets that you prepared for exams and. LECTURE 6: Interference /8/ NOTE THAT FEW EQUATIONS WILL BE GIVEN YOU ARE REMINDED THAT IT IS YOUR RESPONSIBILITY TO CREATE WHATEVER TWO-SIDED CRIB SHEET YOU WANT TO BRING TO THIS EXAM. The equation sheet that will be given with the exam is posted on the course homepage. Click on the link on the left labeled EquationSheet Interference When two waves with the same frequency f and wavelength combine, the resultant is a wave whose amplitude depends on the phase different,. Interference: Phase Differences Constructive Interference Amplitude (arbitrary units) = 0 or # Position (arbitrary units) Destructive Interference Amplitude (arbitrary units) = 0.5 # Position (arbitrary units) /8/ 3 Amplitude (arbitrary units) = 0.3 # /8/ Position (arbitrary units) 4

2 Coherence *If the difference in phase between two (or more) waves remains constant ( i.e., time-independent), the waves are said to be perfectly coherent. Coherence Infinite coherence length *A single light wave is said to be coherent if any two points along the propagation path maintains a constant phase difference. *Coherence length: the spatial extent over which a light wave remains coherent. Finite coherence length Single Photon Only coherent waves can produce interference! /8/ 5 /8/ 6 Intensity of Two Interfering Waves *Two light waves not in phase: E = E o sinωt E = E o sin(ωt + δ ) Intensity of Two Interfering Waves *Maxima occur for: δ = mπ *Minima occur for: δ = (m + )π for m = 0,,, 3. /8/ 7 /8/ 8

3 Interference *Three ways in which the phase difference between two waves can change: Interference: Different Indexes of Refraction *The phase difference between two waves can change if the waves travel through different material having different indexes of refraction.. By traveling though media of different indexes of refraction. By traveling along paths of different lengths 3. By reflection from a boundary n n L N N = L λ (n n ) /8/ 9 /8/ 0 Interference: Different Path Lengths *The phase difference between two waves can change if the waves travel paths of different lengths. Interference: Different Path Lengths *The phase difference between two waves can change if the waves travel paths of different lengths. Thomas Young experiment (80) Interference maxima condition : d sinθ m = mλ, m = 0,,,... m = order number Remember Huygen s principle. /8/ Interference mimima condition : d sinθ m = m λ, m =,,3,... m = order number /8/ 3

4 Interference: Different Path Lengths *The phase difference between two waves can change if the waves travel paths of different lengths. δ Phase difference at a point P: π = d sinθ λ Relationship for distance y m from central point to the mth bright fringe on a screen a distance L away: Interference: Different Path Lengths *The phase difference between two waves can change if the waves travel paths of different lengths. Spacing between fringes: Δ y = y m+ y m = λl d tanθ m = y m L For small angles, tanθ sinθ θ(radians), y m = m λl d /8/ 3 /8/ 4 Interference: Reflection from a Boundary *The phase difference between two waves can change if one or both are reflected. Interference: Reflection from a Boundary *The phase difference between two waves can change if one or both are reflected. DEMO: Soap Bubble mirror light beam projector Reflection Reflection Phase Shift Off lower index zero Off higher index 0.5 # screen soap film /8/ 5 Colored interference fringes will be seen on the screen. /8/ 6 4

5 Interference: Reflection from a Boundary *The phase difference between two waves can change if one or both are reflected. Reflection Reflection Phase Shift Off lower index zero Off higher index 0.5 # Interference: Soap Bubble The phase difference between the two waves is caused by:. Phase shift on reflection for ray of 0.5. Ray does not phase shift on reflection.. Path length difference between ray & ray is t. Ray crosses the film twice. Valid for n > n and n > n 3 For in-phase waves (maxima - bright fringes): t = (m + ) λ n For out of-phase waves (minima - dark fringes): QUESTION: Which ray experiences a shift due to reflection? a) Ray b) Ray c) none d) both /8/ 7 t = m λ n The total difference between ray and ray is the extra difference traveled plus the shift due to reflection. /8/ 8 Interference of Reflections from a film of water Interference of Reflections from a film of water δ from path length difference = t λ ' π, where λ ' = λ n = λ.33 Ray : 80 0 phase shift from reflection off water surface Ray : no phase change going into the water, 80 0 phase change upon reflection from glass. Since both rays are shifted upon reflection the overall difference between the two rays is just the extra path traveled: constructive interference for t = mλ ', m = 0,,, 3,... destructive interference for t = m + λ ', m = 0,,,3... /8/ 9 /8/ 0 5

6 Newton s Rings Newton s Rings: no phase change Fringes from a wedge-shape film of air indicating how optically flat are the glass plates. optical flat ~ zero path length difference 80 0 phase change Poor quality Excellent quality /8/ /8/ Coating a Glass Lens to Suppress Reflections: 80 0 phase change at both a and b since reflection is off a more optically dense medium For destructive interference t = λ' λ' or t = 4 = λ 4n = λ 4.38 What's λ? Engineering decision. Visible λ range : 400 < λ < 700 nm Pick λ 550 nm? vacuum deposited /8/ 3 t Coating a Glass Lens to Suppress Reflections: For = 550 nm and least thickness m = 0. t = λ 4n 550 nm = = 99.6 nm 4.38 Note that the thickness needs to be different for different wavelengths. /8/ 4 6