7' Fresnel's Equations for Reflection and Refraction

1
7. Fresnel's Equations for Reflection and
Refraction

• Incident, transmitted, and reflected beams
• Boundary conditions--tangential fields are
  continuous
• Calculation of reflection and transmission
  coefficients
• The "Fresnel Equations"
• Brewster's Angle
• Total internal reflection
• Power reflectance and transmittance

2
S and P polarizations

• S polarization is the perpendicular
  polarization, and it sticks up out of the plane
  of incidence
• P polarization is the parallel polarization,
  and it lies parallel to the plane of incidence.

3
Fresnel Equations

• We would like to compute the fraction of a light
  wave reflected and
• transmitted by a flat interface between two media
  with different refrac-
• tive indices. Fresnel was the first to do this
  calculation.

Beam geometry for light with its electric field
per- pendicular to the plane of incidence (i.e.,
out of the page)
4
Boundary Condition for the ElectricField at an
Interface

• The Tangential Electric Field is Continuous
• In other words,
• The total E-field in
• the plane of the
• interface is continuous.
• Here, all E-fields are
• in the z-direction,
• which is in the plane
• of the interface (xz), so
• Ei(x,y0,z,t) Er(x,y0,z,t) Et(x,y0,z,t)

5
Boundary Condition for the MagneticField at an
Interface

• The Tangential Magnetic Field is Continuous
• In other words,
• The total B-field in
• the plane of the
• interface is continuous.
• Here, all B-fields are
• in the xy-plane, so we
• take the x-components
• Bi(x,y0,z,t)cos(qi) Br(x,y0,z,t)cos(qr)
  Bt(x,y0,z,t)cos(qt)
• It's really the tangential B/µ, but we're using
  µ µ0

6
Reflection and Transmission for Perpendicularly
Polarized Light

• Ignoring the rapidly varying parts of the light
  wave and keeping
• only the complex amplitudes

7
Reflection Transmission Coefficientsfor
Perpendicularly Polarized Light
8
Fresnel EquationsParallel electric field
Beam geometry for light with its electric
field parallel to the plane of incidence (i.e.,
in the page)

• This definition of the reflected electric field
  is confusing!
• Consider the case of normal incidence (qi 0)...

9
Reflection Transmission Coefficientsfor
Parallel Polarized Light

• For parallel polarized light, B0i B0r
  B0t
• and E0icos(qi) E0rcos(qr) E0tcos(qt)
• Solving for E0r / E0i yields the reflection
  coefficient, r
• Analogously, the transmission coefficient, t
  E0t / E0i, is
• These equations are called the Fresnel Equations
  for parallel polarized light.

10
Reflection Transmission Coefficientsfor an
Air-to-Glass Interface

• nair 1 lt nglass 1.5
• Note
• 1. Total reflection at q 90
• for both polarizations
• 2. Zero reflection for parallel
• polarization at 56.3
• "Brewster's angle
• (For different refractive indices, Brewsters
  angle will be different.)
• 3. Sign idiocy at q 0

Reflect this curve about the q-axis for
alternative definition of r???
11
Reflection Coefficients for a Glass-to-Air
Interface

• nglass 1.5 gt nair 1
• Note
• Total internal reflection
• above the "critical angle"
• qcrit º arcsin(nt / ni)
• (The sine in Snell's Law
• can't be gt 1!)

Reflect this curve about the q -axis for
alternative definition of r
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Reflectance (R) Transmittance (T)

• Define R º Reflected Power / Incident Power
• T º Transmitted Power / Incident
  Power
• The (average) power per unit area, the
  irradiance,
• So
• And
• The factors of n and cos(q) occur in T because
  the media
• 1) have different velocities of light,
  hence the factors of n
• 2) have different propagation angles,
  hence the factors of cos(q)

13
Reflectance Transmittance for anAir-to-Glass
Interface
Note that R T 1
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Reflection at normal incidence

• When qi 0,
•  
• and
•  
• For an air-glass interface (ni 1 and nt 1.5),
•  
• R 4 and T 96
•  
• The values are the same, whichever direction the
  light travels, from air to glass or from glass to
  air.
•  
• The 4 has big implications for photography
  lenses.