Chapter 5: Superposition of waves Chapter 5: Superposition of waves
42
Embed
Chapter 5: Superposition of waves Superposition principle applies to any linear system At a given place and time, the net response caused by two or more.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Slide 1
Slide 2
Chapter 5: Superposition of waves
Slide 3
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.
Slide 4
in a linear world, disturbances coexist without causing further
disturbance
Slide 5
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,
Slide 6
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
Slide 7
Light side of life
Slide 8
Nonlinear optics is another story for another course,
perhaps
Slide 9
What happens when two plane waves overlap?
Slide 10
Superposition of waves of same frequency propagation distance
(measured from reference plane) initial phase (at t =0)
Slide 11
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.
Slide 12
in step Superposition of waves of same frequency out of step
constructive interference destructive interference
Slide 13
Slide 14
Slide 15
In between the extremes: notice the amplitudes can vary; its
all about the phase constructivedestructivegeneral
superposition
Slide 16
Simplify with phasors where and Expressed in complex form:
General case of superposition (same )
Slide 17
Phasors, not phasers
Slide 18
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
Slide 19
Phasors in motion
http://resonanceswavesandfields.blogspot.com
Slide 20
Phasor diagrams complex space representation; vector addition
from law of cosines we get the amplitude of the resultant
field:
Slide 21
Phasor diagrams taking the tangent we get the phase of the
resultant field
Slide 22
Works for 2 waves, works for N waves -harmonic waves -same
frequency
Slide 23
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
Slide 24
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
Slide 25
Coherent light: - strong - uni-directional - irradiance N 2
Incoherent light: - relatively weak - omni-directional - irradiance
N Coherent vs. Incoherent light
Slide 26
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
Slide 27
Color mixing intermezzo
Slide 28
Mixing the colors of light
Slide 29
Mixing colors to make a pulse of light
Slide 30
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
Slide 31
Boundary condition: Allowed modes: Mode distance: = const.
Pulse duration: T (N Peak intensity: N 2 (coherent addition of
waves) Modelocking laser cavity
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)
Slide 34
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
Slide 35
Standing waves in action
http://www.youtube.com/watch?v=0M21_zCo6UM light water sound
http://www.youtube.com/watch?v=EQPMhwuYMy4
Slide 36
Superposition of waves of different frequencies pp gg kpkp
kgkg
Slide 37
Beats Here, two cosine waves, with p >> g
Slide 38
Beats beat frequency: The product of the two waves is depicted
as:
Slide 39
2 frequencies4 frequencies 16 frequencies Many frequencies
Acoustic analogy
Slide 40
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
Slide 41
non-dispersive mediumdispersive medium Superposition and
dispersion of a waveform made of 100 cosines with different
frequencies
Slide 42
http://www.youtube.com/watch?v=umrp1tIBY8Q And the beat goes
on
Slide 43
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)