Nov 14, 2014

A plane wave is reflected at the interface between two homogenous, isotropic, semi-infinite and possibly absorbing media N1 and N2 where absorption is represented by a negative imaginary part of N1 or N2. (N as used below will refer to either N1 or N2):

The plane of incidence is the XZ plane in the orientation shown below. The media interface is the plane X=0:

Within N1, all field vector components have the common factor for the incident and reflected wave respectively:

and in the exit medium N2 all field vector components have the common factor:

where we can write the complex propagation constant vector for medium 1 or 2 as:

The components of the propagation constants k

where:

For a P (TM) polarized wave with Hy being the only nonzero magnetic field component of

where all loss effects (including any conductivity) can be represented by a complex dielectric constant ε or a corresponding complex refractive index where ε = N

For an S (TE) polarized wave, the components of the Poynting vector are:

and using the field expressions above:

The Poynting vector for S polarization thus makes an angle with the normal to the interface of:

For the P (TM) polarized wave, the components of the Poynting vector are:

and using the field expressions above:

The Poynting vector for P polarization makes an angle with the normal to the interface of:

Stratton (1941, pp. 500 - 510) contains an interesting discussion of propogation direction and material properties in a conducting medium at optical and radio frequencies.

E

where

This value will be exactly equal to the power flux computed above for the SINGLE wave in N2, where, for a unit E

Applying exactly the same analysis, the results for the P polarization (TM) are:

where

Note that in all the expressions above for Sx, if there is loss in the incident medium N1, k

The electric field Fresnel interface reflection and transmission coefficients for S polarized (TE) are:

and for P polarization (TM) and with vector direction chosen so that rs and rp are identical at normal incidence:

For the P polarization, it is convenient to use the Hy reflection and transmission coefficients (as used above in the Poynting vector expressions):

The Fresnel reflection coefficients above can easily be written in terms of the normal component of propogation constant, kx in N1 and N2 instead of the (complex) angles above.

- Reflection at a Lossy Incident Medium Interface
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**Optical Properties of Thin Solid Films**, O. S. Heavens, 1965, Dover**Thin-Film Optical Filters**, H. A. Macleod, 2nd Edn., 1986, Adam Hilger Ltd., Bristol pp 28-29**Principles Of Optics**, M. Born and E. Wolf, 5th Edn. 1975, Pergamon Press, pp. 61-63**Electromagnetic Theory**, J. Stratton, 1941, McGraw Hill

**Field Theory of Guided Waves**, R. E. Collin, 1991, IEEE Press

**Fields and Waves in Communication Electronics**, S. Ramo, J. Whinnery, T. Van Duzer, 1984, J. Wiley & Sons