Fundamental Behavior of Light

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Fundamental Behavior of Light

• Light AS A Wave
• gtLight can be treated as an Electromagnetic Wave
  with time varying electric (E(x)) and magnetic
  fields (B(y)).
• E and B have an orthogonal relationship
  propagating through space (homogeneous medium).
• Assuming a sinusoidal traveling wave along the z
  axis the general mathematical form of this event
  can be stated as

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Electromagnetic Fields and Light

• The surface with which a wave has constant face
  is referred to as a Wavefront.
• Faradays Law states that time varying magnetic
  fields result in time varying electric fields.
• The two fields always accompany one another with
  the same frequency and propagation constant but
  transverse to one another in direction
• Typically it is the electric field component that
  is referred to, when dealing with non conductive
  matter as the electric field is responsible for
  displacing electrons in molecules or ions in a
  crystal and therefore determines the Polarization
  of matter.
• The Optical Field refers to the electric field.

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Phase Velocity (Constant Phase)

• There is a relationship in time and space for a
  given phase.
• A surface where the phase of a wave is constant
  is called a wavefront.
• When the all electric fields within a wavefront
  are in phase in the x,y plane a phase velocity
  can be determined by

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Phase Velocity (Constant Phase)

• It is of importance to know the phase difference
  at a given point in time between two points on a
  wave.
• If 2 points are separated by then phase
  difference is because is the same
  for each point.
• If the phase difference is 0 or multiples of 2 pi
  then the two points are in phase. Therefore

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Maxwells Wave Equation

• A perfect plane wave propagates without
  divergence. The amplitude of plane wave is not
  dependant on distance from a reference point and
  is the same at all points on a given (x,y) plane
  perpendicular to k. As planes travel to infinity
  they have infinite energy. Ideal with infinitely
  large source.
• A perfect spherical wave a traveling field
  emerging from a point EM source where the
  amplitude decays with distance r. k vectors
  diverge out as wave propagates and constant phase
  surfaces become larger.Optical divergence for the
  spherical wave is 360 degrees.Point source. Real
  EM source has finite size and power.

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Maxwells Wave Equation

• The field amplitude at any point r from the
  source is given by
• May require transformation from cartesian to
  spherical coordinates.
• Divergent Beam Wavefronts slowly bend away
  spreading the wave. Plane waves are typically
  used for explaining behavior of light rays as the
  nature of discussion is for a point far away from
  a source that approximates a small spatial
  region and the wavefront appears as a plane in
  that immediate area.

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Beam Divergence

• Consider a Gaussian beam travelling along the
  z-axis.
• The light intensity distribution across the beam
  cross-section is Gaussian.
• Beam diameter 2 increases as the beam travels
  along z with the cross sectional area at
  that point contains 85 of the beam power.
• The waist and spot size is where the wavefronts
  are parallel at 2 and the waist radius is
  .
• Beam divergence is given by

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Index of Refraction

• The oscillating electric field of an EM wave
  travelling in a dielectric medium polarizes
  molecules of the medium at the frequency of
  operation.
• The field and induced molecular dipoles become
  coupled.
• Result? The polarization mechanism delays the
  propagation of the EM wave with respect to a
  vacuum.
• The relative permittivity ( ) is a measure of
  the ease with which a medium becomes polarized.

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Index of Refraction

• An EM wave traveling in a nonmagnetic medium of
  relative permittivity will have a phase velocity
  given by

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Index of Refraction

• Polarization and optical frequency range ( )
• gtInfrared frequencies and below is
  due to
• both electronic and ionic
  polarization.
• gt Balance of optical range is due to
• electronic polarization as ionic
  polarization
• too slow to respond to the field.
• Refractive Index is the ratio of C in free space
  to C in a medium.

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Index of Refraction

• Isotropic non crystalline solids such as glass,
  water and cubic crystals ( one refractive index
  in all directions
• Anisotropic- all crystal except for cubic
  crystal.
• Note The relative permittivity for many
  materials can vary considerably over a frequency
  range.

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Relative Permittivity

• Relative permittivity (dielectric constant)
  of materials is frequency dependant of the EM
  wave.
• Relative permittivity can vary over a great range
  for many materials due to polarization
  mechanisms.
• For optical only electronic polarization responds
  to the oscillating field.

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Group Velocity

• In practice monochromatic waves do not exist.
• Groups of waves can differ in wavelength as they
  travel along a z direction.
• Consider two waves interfering and generating a
  wave packet.
• An oscillating field develops at a mean frequency
  that is amplitude modulated by a more slowly
  varying field at a frequency of .
• Emax moves with a wavevector (group
  velocity)

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Group Velocity

• Group velocity defines the speed with which the
  enevelope propagates.
• The amplitude variations of the envelope (Emax)
  travel with velocity while the phase
  variations of the electric field travel at the
  phase velocity V.

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