Nov 12, 2014

This note discusses plane wave reflection at an interface in which the incident medium is lossy.

In the example above, the field in the x direction normal to the layers will exponentially increase into medium N2 for ANY angle of incidence, if medium N2 is lossless. However, if medium N2 is lossy, the field amplitude in N2 may exponentially grow or decay depending on the relative values of complex refractive index N1 and N2 and the angle of incidence, as in the external reflection case above:

and both real and imaginary parts of the indices affect this behaviour. As the angle is varied, the real or imaginary part of kx2 will have a zero crossing, as seen in the curves below, at an incident angle θz given by (provided a real angle exists, i.e. n2κ2/(n1κ1) <1):

The behaviour is somewhat complex and nonintuitive. To show the range of possible behaviour, the graphs below show the real (blue) and imaginary (red) parts of kx2 versus angle of incidence as the loss of N2 is varied with the loss in the incident medium held fixed. The transition from a zero crossing for the imaginary part of kx2 (similar to the external reflection case) to a zero crossing for the REAL (phase parameter) part of kx2 is determined by the condition below for a full complex zero of kx2:

Note that if N1 and N2 satisfy this exact zero condition for kx2, the wave in N2 will be a simple homogeneous attenuating plane wave moving exactly in the z direction parallel to the interface with a fixed amplitude (zero decay or growth) extending infinitely far into N2, exactly like the well-known TIR angle for lossless media. In this case, the S (and P) reflectivity will be exactly 1.0 and the power flow into N2 normal to the interface is exactly zero. A contour plot of this is shown below:

For perspective, the graph below shows kx2 for the lossless case with a total internal reflection angle of 41.81°. With loss in N1 and/or N2, there is no true "total internal reflection" at any angle for the two region interface, but the graphs above show that the rapidly changing structure in kx2 is close to the lossless TIR angle. In a sense, loss in N1 or N2 "washes out" the sharpness of the TIR angle effect, similar to classic resonant behaviour damping.

and assuming a homogeneous lossless plane wave in N1, θ is a real angle of incidence:

The standard complex square root (with positive or zero real part) is:

The alternate root is the negative of that above.

There may be a zero crossing at some angle of incidence in either the real or imaginary part of kx2, depending on the complex values of N1 and N2. A zero crossing will exist if there is a real angle such that Z

at an angle:

By inspection of the complex square root expression above, the zero crossing will occur for kx2_real if Z

and the zero will occur for kx2_imag if the expression is positive. A full zero in kx2 (i.e. a simulaneous zero in both kx2_real and kx2_imag) will occur if the expression above is identically zero.

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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