Re: Fresnel equations and metals
- From: Timo Nieminen <timo@xxxxxxxxxxxxxxxxx>
- Date: Tue, 12 Dec 2006 13:25:00 +1000
On Tue, 5 Dec 2006, Enrique Cruiz wrote:
On 2006-12-05 21:26:15 +0000, "Timo A. Nieminen" <timo@xxxxxxxxxxxxxxxxx> said:
It might be useful for you if I just post the correct formula for
complex refractive index/complex permittivity. I'm conferencing at the
moment and am away from office and books. Perhaps next week ...
that would be very useful indeed! Thanks in advance.
OK, here are the results. Explanation of symbols and intermediate
calculations follows the main equations. I recommend that you check that
they give you the correct Fresnel equations when medium 2 is a dielectric
with purely real n_2 (then the angle of refraction gives you k_2z).
Amplitude reflection and transmission coefficients:
TE:
r = ( k_1z k_2 Z_2 - k_2z k_1 Z_1 ) / ( k_1z k_2 Z_2 + k_2z k_1 Z_1 )
t = 2 k_1z k_1 Z_1 / ( k_1z k_2 Z_2 + k_2z k_1 Z_1 )
TM:
r = ( k_1z k_2 Z_1 - k_2z k_1 Z_2 ) / ( k_1z k_2 Z_1 - k_2z k_1 Z_2 )
t = - 2 k_1z k_1 Z_1 / ( k_2z k_1 Z_2 + k_1z k_2z Z_1 Z_2 )
Power reflection coefficients:
TE:
R = [ ( k_1z k_2 Z_2 - k_2z k_1 Z_1 ) ( k_1z k_2* Z_2* - k_2z* k_1 Z_1 ) ]
/ [ ( k_1z k_2 Z_2 + k_2z k_1 Z_1 ) ( k_1z k_2* Z_2* + k_2z* k_1 Z_1 ) ]
= [ k_1z^2 |n_2|^2 |Z_2|^2 + |k_2z|^2 n_1^2 Z_1^2
- 2 k_1z n_1 Z_1 Re( k_2z n_2 Z_2 ) ]
/ [ k_1z^2 |n_2|^2 |Z_2|^2 + |k_2z|^2 n_1^2 Z_1^2
+ 2 k_1z n_1 Z_1 Re( k_2z n_2 Z_2 ) ]
TM:
R = [ ( k_1z k_2* Z_1 - k_2z* k_1 Z_2* )
( k_1z k_2* Z_1 - k_2z* k_1 Z_2* ) ] /
[ ( k_1z k_2* Z_1 + k_2z* k_1 Z_2* )
( k_1z k_2* Z_1 + k_2z* k_1 Z_2* ) ]
= [ k_1z^2 |n_2|^2 Z_1^2 + |k_2z|^2 n_1^2 |Z_2|^2
- 2 k_1z n_1 Z_1 Re( k_2z n_2 Z_2 ) ]
/ [ k_1z^2 |n_2|^2 Z_1^2 + |k_2z|^2 n_1^2 |Z_2|^2
+ 2 k_1z n_1 Z_1 Re( k_2z n_2 Z_2 ) ]
These all assume that n, Z, and k (and all components of k) are purely
real in medium 1. They can be complex in medium 2.
k is the wavenumber, k_z is the z component of the wavevector. n is the
refractive index, Z is the impedance.
To calculate these:
k = 2 pi n / lambda, where lambda is the free-space wavenumber (ie what
you'd have in vacuum).
n = sqrt[ 1 / ( epsilon mu ) ]
Z = sqrt( mu / epsilon )
where epsilon = permittivity, mu = permeability. For most mediums of
interest, mu = mu_0, the permeability of free space, and you can write
n = sqrt( epsilon_r ) where epsilon_r = epsilon / epsilon_0 is the
relative permittivity, also called the dielectric constant.
In medium 1, k_z and k_x can be found from the angle of incidence theta
(you can assume k_y = 0).
k_z = k cos(theta)
k_x = k sin(theta)
Trying to do this for medium 2 gets you into the trouble of complex angles
if n_2 is complex. Stick to wavenumber and wavevectors.
k_x is the same in medium 1 and medium 2.
To find k_2z, use
k_2z = sqrt( k_2^2 - k_x^2 )
and both the real part and imaginary part should be zero or postive
(unless you have a medium with gain, or other exotica).
The power transmission coefficient is equal to 1 - power reflection
coefficient.
The TE wave is polarised with E parallel to the interface, the TM has H
parallel.
RE(..) means the real part.
* means complex conjugate.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
.
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- From: Timo A. Nieminen
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