Re: Fresnel equations and metals

On Fri, 24 Nov 2006, AES wrote:

"Timo A. Nieminen" <timo@xxxxxxxxxxxxxxxxx> wrote:

Not when considering a single homogeneous plane wave incident on a
z=constant interface. Real k_x, k_y is sufficient on both sides on the
interface since that's what the incident wave has, and thus enough to deal
with general Fresnel coefficients.

I guess we'll have to disagree on this point. My analysis and
conclusions are in the web page I noted.

Take the interface to be at z=constant (ie parallel to the xy-plane). Look at pages 33 and especially 35-38 in your presentation. Note that with the z-axis perpendicular to the interface, all of your diagrams have purely real k_x and k_y, and mostly complex k_z, except for the case of TIR from a lossless/gainless medium, in which case k_z is purely imaginary.

So, I'm not sure what are disagreeing about.

But one question to chew on: Forget Fresnel reflection for the moment.
Just consider a single, linearly polarized, inhomogeneous plane wave
which is propagating in some large volume of space containing a
spatially uniform medium which may be lossless, lossy, or even gainy,
and which satisfies the wave equation in that medium.

[Inhomogeneous, again, means that the real and imag parts of the k
vector of this wave are not parallel. Such a wave is easy to write; in
fact, one can choose from a continuous family of such waves with
varying, but coupled, real and imaginary components.]

My claim (easy to show, I think, and supported by a couple of the EM
theory references I cited): The local Poynting vector associated with
this wave at any point in that space is always parallel to the *real*
part of the k vector, independent of the magnitude of the imaginary part
of the k vector.

Agree or disagree?

Certainly the case for a lossless/gainless medium, and very plausible for the more general case. Without actually checking, I'm relectant to extend the "certainly" to the general case. Very simple to check given the general TE and TM plane wave solutions (ie Morse & Feshbach's M and N wavefunctions for Cartesian coordinates). Sounds like a good exercise to add to the end of the whole TIR/waves in loss/gain media for students.

Anyway, further considering an inhomogeneous plane wave in a uniform isotropic linear medium, if the source is a homogeneous plane wave incident from another medium:

(a) If the interface between the two media is plane, can we tell in which direction it must lie?

(b) Must the interface be plane, for an incident plane wave?

--
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

.

Relevant Pages

  • Re: Fresnel equations and metals
    ... Real k_x, k_y is sufficient on both sides on the interface since that's what the incident wave has, and thus enough to deal with general Fresnel coefficients. ... an arbitrary wave that you might be able to write as a plane wave with complex k_x and k_y can be represented as an integral over plane wave modes with real k_x and k_y. ... If the slab has non-parallel interfaces, then it all beomes a much harder problem. ... In Fresnel reflection from a gainy medium ...
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  • Re: Fresnel equations and metals
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  • Re: Fresnel equations and metals
    ... Not when considering a single homogeneous plane wave incident on a ... interface since that's what the incident wave has, ... Just consider a single, linearly polarized, inhomogeneous plane wave ...
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