Re: Fresnel equations and metals

In article <45670d2d$0$8759$ed2619ec@xxxxxxxxxxxxxxxxxxxxxxxxxx>,
Enrique Cruiz <jni6l03mdo6nvuu@xxxxxxxxxxx> wrote:

Hi guys,

In most textbooks (at least the one I am reading), the Fresnels
equations given are said ot be valid for dielectrics. But what about
for metals? Do they change? Or it is the same, with the only
differences being the refractive index becoming complex?

Thanks in advance.

Enrique

Ah, you've touched on an interesting topic, and one of considerable
current interest to me.

If you want to understand Fresnel reflection from metals, or more
generally from spatially uniform but lossy or gainy dielectrics, you
have to understand *inhomogeneous* plane waves in their most general
form. "Inhomogeneous plane waves" in this context refers to infinite
plane waves having complex-valued propagation constants in which the
real and imaginary parts are not parallel (and has nothing to do with
spatially inhomogeneous media).

A general treatment of inhomogeneous plane waves is not all that complex
mathematically, although understanding their physical meaning is a bit
more tricky. I've not found a discussion of this concept, however, in
any of the standard modern or classic optics texts, with the possible
exception of Born and Wolf, and even there it's not what I'd call a
really clear presentation. (I'd be very glad to hear of any standard or
current optics texts I've overlooked). To find this concept clearly
discussed you have to turn instead to the classic EM theory texts:

[1] J. M. Stone, Radiation and Optics, McGraw-Hill, 1963.

Notes: Inhomogeneous plane waves on p. 373.

[2] P. C. Clemmow, The Plane Wave Representation of Electromagnetic
Fields, Pergamon Press, 1966.

Notes: Discusses inhomog plane waves in gainy and lossy media in Section
2.1 (not particularly clearly, however) and gives the Poynting vector
result.

[3] J. D. Jackson, Classical Electrodynamics, Wiley, 1975.

Notes: One of the few texts (Stone, Jackson) that discuss IPWs, in
Chapter 7. There is a 3rd edition, 1999.

[4] P. Boulanger and M. A. Hayes, Bivectors and waves in mechanics and
optics, Chapman & Hall, 1993.

Abstract: Bivectors occur naturally in the description of elliptically
polarized homogeneous and inhomogeneous plane waves. The
description of a homogeneous plane wave generally involves a vector
(the unit vector along the propagation direction) and a bivbector (the
complex amplitude of the wave). Inhomogeneous plane waves are
described in terms of two bivectors - the complex amplitude and the
complex slowness. The use of bivectors and their associated
ellipses is essential for the presentation of the 'directional
ellipse' method given in this book, in deriving all possible
inhomogeneous plane wave solutions in a given context.The purpose of
this book is to give an extensive treatment of the properties of
bivectors and to show how these may be applied to the theory of
homogeneous and inhomogeneous plane waves. For each chapter there are
exercises with answers, many of which present further useful
is properties which are referred to afterwards. The material in this
book is suitable for senior undergraduate and first year graduate
students. It will also prove useful for researchers interested in
homogeneous and inhomogeneous plane waves.

Notes: Physics Library QC174.26.B58 B68 1993

[5] M. Born and E. Wolf, Principles of Optics, Seventh Edition,
Cambridge University Press, 1999.

Notes: Defines the Strehl ratio for a uniformly illuminated aperture on
p. 461.

[6] A. E. Siegman, "Propagating modes in gain-guided optical fibers,"
J. Opt. Soc. Am. A, vol. 20, pp. 1617--1628, August 2003.

Abstract: Optical fibers in which gain-guiding effects are
significant or even dominant compared to conventional index guiding may
become of practical interest for future high-power single-mode fiber
lasers. In this paper I derive the propagation characteristics of
symmetrical slab waveguides and cylindrical optical fibers having
arbitrary amounts of mixed gain and index guiding, assuming a single
uniform transverse profile for both the gain and refractive index steps.
Optical fibers of this type are best characterized using a
complex-valued $\vtilde^2$ parameter in place of the real-valued $v$
parameter commonly used to describe conventional index-guided optical
fibers.

You could also try the web page:

<http://www.stanford.edu/~siegman/optics_with_gain.pdf>
.