Transcript of Chapter 5: Superposition of waves Superposition principle applies to any linear system At a given...
- 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
- Slide 32
- http://www.physik.uni-wuerzburg.de/femto-welt/ Intermezzo:
Femtowelt
- Slide 33
- 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)