13. Fresnel's Equations for Reflection and Transmission
13. Fresnel's Equations for Reflection and Transmission
Incident, transmitted, and reflected beams
Boundary conditions: tangential fields are continuous
Reflection and transmission coefficients
The "Fresnel Equations"
Brewster's Angle
Total internal reflection
Power reflectance and transmittance
Augustin Fresnel 1788-1827
Posing the problem
What happens when light, propagating in a uniform medium, encounters a smooth interface which is the boundary of another medium (with a different refractive index)?
k-vector of the incident light
nincident
ntransmitted
boundary
First we need to define some terminology.
Definitions: Plane of Incidence and plane of the interface
Plane of incidence (in this illustration, the yz plane) is the plane that contains the incident and reflected k-vectors.
y zx
Plane of the interface (y=0, the xz plane) is the plane that defines the interface between the two materials
Definitions: "S" and "P" polarizations
A key question: which way is the E-field pointing? There are two distinct possibilities.
1. "S" polarization is the perpendicular polarization, and it sticks up out of the plane of incidence
Here, the plane of incidence (z=0) is the plane of the diagram.
I
R
y zx
The plane of the interface (y=0) T is perpendicular to this page.
2. "P" polarization is the parallel polarization, and it lies parallel to the plane of incidence.
Definitions: "S" and "P" polarizations
Note that this is a different use of the word "polarization" from the way we've used it earlier in this class.
reflecting medium
reflected light
The amount of reflected (and transmitted) light is different for the two different incident polarizations.
Fresnel Equations--Perpendicular E field
Augustin Fresnel was the first to do this calculation (1820's).
We treat the case of s-polarization first:
ki
kr
Ei
Er
Bi
i r
Br
Interface
Beam geometry for
light with its electric field sticking up out of the plane of incidence (i.e., out of the page)
t
Et
Bt kt
ni
y
zx
nt the xz plane (y = 0)
Boundary Condition for the Electric
Field at an Interface: s polarization
y
The Tangential Electric Field is Continuous z x
In other words,
The component of the E-field that lies in the xz plane is continuous as you move across the
ki
kr
Ei
Er
Bi
i r
Br
ni
plane of the interface.
Interface
Here, all E-fields are in the z-direction, which is in the plane of the interface.
t Et
nt
Bt kt
So:
(We're not explicitly writing
Ei(y = 0) + Er(y = 0) = Et(y = 0) the x, z, and t dependence,
but it is still there.)
Boundary Condition for the Magnetic y
Field at an Interface: s polarization
zx
The Tangential Magnetic Field* is Continuous
In other words,
The total B-field in the plane of the interface is continuous.
ki
kr
iBi
Ei
i
i r
Er
Br
ni
Interface
Here, all B-fields are in the xy-plane, so we take the x-components:
t Et
nt
Bt kt
?Bi(y = 0) cosi + Br(y = 0) cosr = ?Bt(y = 0) cost
*It's really the tangential B/, but we're using i t 0
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