Maxwell's Equations and Light Waves

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Maxwell's Equations and Light Waves

• Longitudinal vs. transverse waves
• Div, grad, curl, etc., and the 3D Wave equation
• Derivation of wave equation from Maxwell's
  Equations
• Why light waves are transverse waves
• Why we neglect the magnetic field
• Photons and photon statistics

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Longitudinal vs. Transverse waves

Motion is along the direction of Propagation
Longitudinal
Motion is transverse to the direction
of Propagation
Transverse
Space has 3 dimensions, of which 2 directions are
transverse to the propagation direction, so there
are 2 transverse waves in ad- dition to the
potential longitudinal one.
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Vector fields

• Light is a 3D vector field.
• A 3D vector field assigns a 3D vector
  (i.e., an arrow having both direction and length)
  to each point in 3D space.

A 2D vector field
A light wave has both electric and magnetic 3D
vector fields
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The 3D vector wave equation for the electric field
Note the vector symbol over the E.
This is really just three independent wave
equations, one each for the x-, y-, and
z-components of E.

• which has the vector field solution

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Waves using complex vector amplitudes

• We must now allow the complex field and its
  amplitude to be vectors

Note the arrows over the Es!
The complex vector amplitude has six numbers that
must be specified to completely determine it!
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Div, Grad, Curl, and all that

• Types of 3D vector derivatives
•  
•  
• The Del operator
•  
•  
• The Gradient of a scalar function f
•  
• The gradient points in the direction of steepest
  ascent.

If you want to know more about vector calculus,
read this book!
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Div, Grad, Curl, and all that

• The Divergence of a vector function

The Divergence is nonzero if there are sources or
sinks.
A 2D source with a large divergence
Note that the x-component of this function
changes rapidly in the x direction, etc., the
essence of a large divergence.
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Div, Grad, Curl, and more all that

•  The Laplacian of a scalar function
•  
•  
•  
• The Laplacian of a vector function is the same,
• but for each component of f

The Laplacian tells us the curvature of a vector
function.
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Div, Grad, Curl, and still more all that
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A function with a large curl
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The equations of optics are Maxwells equations.

• where is the electric field, is the
  magnetic field, r is the charge density, e is the
  permittivity, and m is the permeability of the
  medium.

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Derivation of the Wave Equation from Maxwells
Equations

• Take of
•  
• Change the order of differentiation on the RHS

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Derivation of the Wave Equation from Maxwells
Equations (contd)

• But
•  
• Substituting for , we have
•  
•  
• Or

assuming that m and e are constant in time.
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Lemma

• Proof Look first at the LHS of the above
  formula
•  
•  
•  
• Taking the 2nd yields
•  
• x-component
•  
• y-component
•  
• z-component

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Lemma (contd)

• Proof (contd)
•  
• Now, look at the RHS

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Derivation of the Wave Equation from Maxwells
Equations (contd)

• Using the lemma,
•  
• becomes
•  
• If we now assume zero charge density r 0,
  then
•  
• and were left with the Wave Equation!

where me 1/c2
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Why light waves are transverse

• Suppose a wave propagates in the x-direction.
  Then its a function of x and t (and not y or z),
  so all y- and z-derivatives are zero
•  
•   
• Now, in a charge-free medium,
•  
• that is,

and
Substituting the zero values, we have
So the longitudinal fields are at most constant,
and not waves.
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The magnetic-field direction in a light wave

• Suppose a wave propagates in the x-direction and
  has its electric field along the y-direction so
  Ex Ez 0, and Ey Ey(x,t).
• What is the direction of the magnetic field?
•  
• Use
•  
• So
•  
• In other words
•  
• And the magnetic field points in the z-direction.

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The magnetic-field strength in a light wave

• Suppose a wave propagates in the x-direction and
  has its electric field in the y-direction. What
  is the strength of the magnetic field?

and
Take Bz(x,0) 0
Differentiating Ey with respect to x yields an
ik, and integrating with respect to t yields a
1/-iw.
So
But w / k c
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An Electromagnetic Wave
The electric and magnetic fields are in phase.
snapshot of the wave at one time

• The electric field, the magnetic field, and the
  k-vector are all perpendicular

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The Energy Density of a Light Wave

• The energy density of an electric field is
• The energy density of a magnetic field is
• Using B E/c, and , which
  together imply that
• we have
• Total energy density
• So the electrical and magnetic energy densities
  in light are equal.

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Why we neglect the magnetic field

• The force on a charge, q, is
•  
•  
•  
• Taking the ratio of the magnitudesof the two
  forces
•  
•  
• Since B E/c

So as long as a charges velocity is much less
than the speed of light, we can neglect the
lights magnetic force compared to its electric
force.
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The Poynting Vector S c2 e E x B

• The power per unit area in a beam.
• Justification (but not a proof)
• Energy passing through area A in time Dt
• U V U A c Dt
• So the energy per unit time per unit area
• U V / ( A Dt ) U A c Dt / ( A Dt )
  U c c e E2
• c2 e E B
• And the direction is
  reasonable.

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The Irradiance (often called the Intensity)

• A light waves average power per unit area is
  the irradiance.
•  
• Substituting a light wave into the expression for
  the Poynting vector,
  
• , yields
•  
• The average of cos2 is 1/2

real amplitudes
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The Irradiance (continued)

• Since the electric and magnetic fields are
  perpendicular and B0 E0 / c,

becomes
where
Remember this formula only works when the wave
is of the form
that is, when all the fields involved have the
same
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Sums of fields Electromagnetism is linear, so
the principle of Superposition holds.

• If E1(x,t) and E2(x,t) are solutions to the wave
  equation,
• then E1(x,t) E2(x,t) is also a solution.
•  
• Proof and
•  
•  
• This means that light beams can pass through each
  other.
•  
• It also means that waves can constructively or
  destructively interfere.

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The irradiance of the sum of two waves

• If theyre both proportional to
  , then the irradiance is

Different polarizations (say x and y)
Intensities add.
Same polarizations (say )
Note the cross term!
Therefore
The cross term is the origin of interference!
Interference only occurs for beams with the same
polarization.
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The irradiance of the sum of two waves of
different color
We cant use the formula because the ks and ws
are different. So we need to go back to the
Poynting vector,
This product averages to zero, as does
Intensities add.
Different colors
Waves of different color (frequency) do not
interfere!
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Irradiance of a sum of two waves
Different polarizations
Same polarizations
Same colors
Different colors
Interference only occurs when the waves have the
same color and polarization.
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Light is not only a wave, but also a particle.

• Photographs taken in dimmer light look grainier.

Very very dim
Very dim
Dim
Bright
Very bright
Very very bright
When we detect very weak light, we find that its
made up of particles. We call them photons.
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Photons

• The energy of a single photon is hn or
  (h/2p)w
• where h is Planck's constant, 6.626 x 10-34
  Joule-sec.
• One photon of visible light contains about 10-19
  Joules, not much!.
• F is the photon flux, or
• the number of photons/sec
• in a beam.
• F P / hn
• where P is the beam power.

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Counting photons tells us a lot aboutthe light
source.

• Random (incoherent) light sources,
• such as stars and light bulbs, emit
• photons with random arrival times
• and a Bose-Einstein distribution.
• Laser (coherent) light sources, on
• the other hand, have a more
• uniform (but still random) distribution
  Poisson.

Bose-Einstein Poisson
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Photons have momentum

• If an atom emits a photon, it recoils in the
  opposite direction.

If the atoms are excited and then emit light, the
atomic beam spreads much more than if the atoms
are not excited and do not emit.
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PhotonsRadiation Pressure

• Photons have no mass and always travel at the
  speed of light.
• The momentum of a single photon is h/l, or
• Radiation pressure Energy Density
  (Force/Area Energy/Volume)
• When radiation pressure cannot be neglected
• Comet tails (other forces are small)
• Viking space craft (would've missed Mars by
  15,000 km)
• Stellar interiors (resists gravity)
• PetaWatt laser (1015 Watts!)

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Photons

• "What is known of photons comes from observing
  the
• results of their being created or annihilated."
• Eugene Hecht
• What is known of nearly everything comes from
  observing the
• results of photons being created or annihilated.