3' Maxwell's Equations, Light Waves, Power, and Photons

1
3. Maxwell's Equations, Light Waves, Power, and
Photons

• 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.

4
Waves using complex vector amplitudes

• We must now allow the complex amplitude E0 to be
  a vector

The complex vector amplitude has six numbers that
must be specified to completely determine it!
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The 3D vector wave equation for the electric field

• which has the vector field solution

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Div, Grad, Curl, and all that

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

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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
•  
•  
• The Laplacian tells us the curvature of a
  function.

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Div, Grad, Curl, and 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

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Lemma

• Proof Look at LHS
•  
•  
•  
•  Taking the 2nd yields
•  
• x-component
•  
• y-component
•  
• z-component

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Lemma

• 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!

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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,
•  
•  
• Substituting, we have

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Electric magnetic fields of a light wave

• Suppose a wave propagates in the x-direction and
  has its electric field along the y-direction.
  What is the direction of the magnetic field?
•  
• Using
•  
• and noting that the only non-zero component of
  is
•  
•  
• We have
•  
•  
• So the magnetic field points in the z-direction.

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The magnetic field of a light wave

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

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An Electromagnetic Wave
The electric and magnetic fields are in phase.

• 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
•  
•  
•  
• so
•  
•  
• If B E/c, then

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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 c c e
  E2
• c2 e E B
• And the direction is
  reasonable.

V A c Dt
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The Irradiance

• A light waves average power per unit area is the
  irradiance.
•  
• Substituting a light wave into the expression for
  the Poynting vector,
  
• , yields
•  
• Averaging over time 
• as T goes to , noting that
• the average of cos2 is 1/2

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The Irradiance (continued)

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

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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 it
is 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 distribution 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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Photons--Radiation 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.