Nov 12, 2014

The plots below show the optical reflectance, transmittance and coefficient of reflection and transmission phase angles for a single absorbing layer with a plane electromagnetic wave incident from a real-index medium with angle of incidence θ

- exponential propagation factor exp(j[wt - kx]). With this convention, a medium with loss is specified with a complex refractive index having a NEGATIVE imaginary part.
- E-field reflected vector direction for TM (p) case chosen so that r
_{s}= r_{p}at normal incidence

The interface Fresnel reflection and transmission coefficients below, r

where the expressions the 3 region single film are valid for both TE (s) and TM (p) polarizations, with the polarization differences encapsulated in the r12, r23, t12, t23 values. For a nonabsorbing film, R + T=1. If the incident medium is lossy, it is not possible to define reflectance and transmittance values such that R + T = 1 due to field coupling in a lossy medium (MacLeod 1986).

Note that using the E-field vector sign convention above for the TM case, the reflection coefficients rs and rp are identical at normal incidence θ=0°. The actual sign of rs and rp for normal incidence will be reversed if N1 and N2 are interchanged ("internal" verses "external" reflection). However it is easy to show using the interface Fresnel reflection coefficient expressions above that

If the final medium has loss, but the incident medium is lossless, R and T values are well defined and (R + T) =1 . However, the transmittance expressions are more complicated and different for S and P polarizations. The following expressions apply to any angle, including TIR angles, and for any complex indices n2, n3 with the understanding that θ

Macleod (1986) provides alternate expressions for T

The r

Note: For the P polarized (TM) case, it is sometimes convenient to use the Fresnel reflection and transmission coefficients for the Hy field (e.g. Stratton 1941). This is fully equivalent to use of the rp and tp E field coefficients above but the expressions are different, particularly the

Normal Incidence Theta = 0 deg

Born & Wolf "Principles Of Optics" 5th Edn. 1975 p. 631

Note: n2 = 3.5( 1 + j0.1) where k

Fixed t2 = 0.222 um

**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.**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