Wave Incidence at Oblique angles

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Wave Incidenceat Oblique angles

• Sandra Cruz-Pol
• ECE Dept. UPRM

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Quiz VIERNES

• Uno de los problemas asignados en el prontuario
• Enviar contestaciones a preguntas en Word por
  email

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Wave Incidence

• Normal incidence
• Wave arrives at 90o from the surface
• Standing waves
• Oblique incidence (lossless)
• Wave arrives at an angle
• Snells Law and Critical angle
• Two types Parallel or Perpendicular
• Brewster angle

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Oblique incidence

• First, we need to define
• Expression for uniform plane wave traveling in
  any direction
• The Normal , an
• Plane of incidence
• Angle of incidence

x
Medium 2 e2, m2
Medium 1 e1 , m1
qr
qt
y
qi
z
z0
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Oblique incidence

• Uniform plane wave in general form
• the radius or position
  vector
• the wave number or propagation vector
• Where
• For lossless unbounded media, k b

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Oblique Incidence
x
Medium 2 e2, m2
Medium 1 e1 , m1
kr
kt
qr
qt
z
y
qi
z0
ki
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Look at the direction of travel
x
Medium 2 e2, m2
Medium 1 e1 , m1
kr
kt
qr
qt
z
y
qi
kix
z0
ki
kiz
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Expression for Waves
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Tangential E must be Continuous

• From this we know that frequency is a property
  of the wave. So is color.
• So 700nm is not always red!!
• Give example of bathing suit.

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Snell Law

• Equating, we get
• or
• where, the index of refraction of a medium, ni
  , is defined as the ratio of the phase velocity
  in free space (c) to the phase velocity in the
  medium.

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Critical angle, qc All is reflected

• When qt 90o, the refracted wave flows along the
  surface and no energy is transmitted into medium
  2.
• The value of the angle of incidence corresponding
  to this is called critical angle, qc.
• If qi gt qc, the incident wave is totally
  reflected.

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Fiber optics

• Light can be guided with total reflections
  through thin dielectric rods made of glass or
  transparent plastic, known as optical fibers.
• The only power lost is due to reflections at the
  input and output ends and absorption by the fiber
  material (not perfect dielectric).

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• Optical fibers have cylindrical fiber core with
  index of refraction nf, surrounded by another
  cylinder of lower, nc lt nf , called a cladding.
• For total reflection

Figure from Ulaby, 1999
Acceptance angle
Use Snell and critical angle to derive
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Exercise optical fiber

• An optical fiber (in air) is made of fiber core
  with index of refraction of 1.52 and a cladding
  with an index of refraction of 1.49.
• Find the acceptance angle

Answer 17.5o
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Parallel polarization

• Its defined as E is to incidence plane

x
Medium 1 e1 , m1
Er
Et
kt
kr
qr
qt
z
y
z
qi
Ei
z0
ki
Medium 2 e2, m2
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Equating for continuity, the tangent fields

• Which components are tangent to the interface
  between two surfaces?
• y and x
• At z 0 (interface)

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Reflection and Transmission Coefficients
Parallel Incidence

• Reflection

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Reflection and Transmission Coefficients
Perpendicular Incidence
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Java Animation

• http//www.amanogawa.com/archive/Oblique/Oblique-2
  .html

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Exercise

• A plane wave in air with
• Is incident upon planar surface of nonmagnetic
  dielectric material with er4, on zgt0, Find
• The incidence polarization of this wave
• The angle of incidence
• The time-domain expressions for the reflected
  electric and magnetic fields.
• The average power density carried by the wave in
  the dielectric medium.

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Brewster angle, qB

• Is defined as the incidence angle at which the
  reflection coefficient is 0 (all transmission).
• The Brewster angle does not exist for
  perpendicular polarization for nonmagnetic
  materials.

qB is known as the polarizing angle
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Exercise

• A plane wave in free space with
• Is incident upon planar surface of nonmagnetic
  lossless dielectric material with er4, on zgt0,
  Find
• The polarization of the incident wave
• The angle of incidence and transmission
• The reflection and transmission coefficients.
• The E field that is reflected and in dielectric
• Brewster angle

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Reflection vs. Incidence angle.
Reflection vs. incidence angle for different
types of soil and parallel or perpendicular
polarization.
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Exercise (Brewster angle)

• A wave in air is incident upon the flat boundary
  of a nonmagnetic soil medium with er4, at
  qi50o.
• Find reflection and transmission coefficients for
  both incident polarizations, and the Brewster
  angle.
• Answers

EMAG
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Summary wave incidence
Property Normal Incidence Perpendicular Parallel
Reflection coefficient
Transmission coefficient
Relation
Power Reflectivity
Power Transmissivity
Snells Law Snells Law Snells Law Snells Law