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Chapter 5: Superposition of waves

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Superposition principle applies to any linear system At a given place and time, the net response caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.

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in a linear world, disturbances coexist without causing further disturbance

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then the linear combination where a and b are constants is also a solution. Superposition of waves If 1 and 2 are solutions to the wave equation,

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Superposition of light waves 1 2 -in general, must consider orientation of vectors (Chapter 7next week) -today, well treat electric fields as scalars -strictly valid only when individual E vectors are parallel -good approximation for nearly parallel E vectors -also works for unpolarized light

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Light side of life

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Nonlinear optics is another story for another course, perhaps

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What happens when two plane waves overlap?

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Superposition of waves of same frequency propagation distance (measured from reference plane) initial phase (at t =0)

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Superposition of waves of same frequency simplify by intoducing constant phases: thus At point P, phase difference is and the resultant electric field at P is.

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in step Superposition of waves of same frequency out of step constructive interference destructive interference

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In between the extremes: notice the amplitudes can vary; its all about the phase constructivedestructivegeneral superposition

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Simplify with phasors where and Expressed in complex form: General case of superposition (same )

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Phasors, not phasers

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Phasor diagrams projection onto x -axis magnitude angle clock analogy: -time is a line -but time has repeating nature -use circular, rotating representation to track time phasors: -represent harmonic motion -complex plane representation -use to track waves -simplifies computational manipulations

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Phasors in motion http://resonanceswavesandfields.blogspot.com

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Phasor diagrams complex space representation; vector addition from law of cosines we get the amplitude of the resultant field:

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Phasor diagrams taking the tangent we get the phase of the resultant field

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Works for 2 waves, works for N waves -harmonic waves -same frequency

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hence as Two important cases for waves of equal amplitude and frequency randomly phasedcoherent phase differences random hence as in phase; all i are equal

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Light from a light bulb is very complicated! 1 It has many colors (its white), so we have to add waves of many different values of (and hence k -magnitudes). 2Its not a point source, so for each color, we have to add waves with many different k directions. 3Even for a single color along one direction, many different atoms are emitting light with random relative phases. Lightbulb

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Coherent light: - strong - uni-directional - irradiance N 2 Incoherent light: - relatively weak - omni-directional - irradiance N Coherent vs. Incoherent light

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Coherent fixed phase relationship between the electric field values at different locations or at different times Partially coherent some (although not perfect) correlation between phase values Incoherent no correlation between electric field values at different times or locations 1 0 Coherence is a continuum more on coherence next week

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Color mixing intermezzo

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Mixing the colors of light

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Mixing colors to make a pulse of light

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Time Intensity 1. Single mode Supress all modes except one Time Intensity 2. Multi-mode Statistical phase relation amongst modes I N Time Intensity 3. Modelocked Constant or linear phase amongst modes I N 2 T = 2L/c Broadband laser operating regimes

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Boundary condition: Allowed modes: Mode distance: = const. Pulse duration: T (N Peak intensity: N 2 (coherent addition of waves) Modelocking laser cavity

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http://www.physik.uni-wuerzburg.de/femto-welt/ Intermezzo: Femtowelt

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Standing waves - occur when wave exists in both forward and reverse directions - if phase shift = , standing wave is created - when A ( x ) = 0, E R =0 for all t ; these points are called nodes - displacement at nodes is always zero A(x)A(x)

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Standing wave anatomy - nodes occur when A ( x ) = 0 - A ( x ) = 0 when sin kx = 0, or kx = m (for m = 0, 1, 2,...) - since k = 2 x = m - E R has maxima when cos t = 1 - hence, peaks occur at t = mT ( T is the period) where

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Standing waves in action http://www.youtube.com/watch?v=0M21_zCo6UM light water sound http://www.youtube.com/watch?v=EQPMhwuYMy4

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Superposition of waves of different frequencies pp gg kpkp kgkg

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Beats Here, two cosine waves, with p >> g

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Beats beat frequency: The product of the two waves is depicted as:

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2 frequencies4 frequencies 16 frequencies Many frequencies Acoustic analogy

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Here, phase velocity = group velocity (the medium is non-dispersive). In a dispersive medium, the phase velocity group velocity. Phase and group velocity phase velocity: group velocity: envelopemoves with group velocity carrier wave moves with phase velocity

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non-dispersive mediumdispersive medium Superposition and dispersion of a waveform made of 100 cosines with different frequencies

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http://www.youtube.com/watch?v=umrp1tIBY8Q And the beat goes on

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You are encouraged to solve all problems in the textbook (Pedrotti 3 ). The following may be covered in the werkcollege on 28 September 2011: Chapter 5: 2, 6, 8, 9, 14, 18 Exercises (not part of your homework)