The following expressionsĪpply to any angle, including TIR angles, and for any complex indices n2, n3 with the understanding that θ 3 may be complex asĭetermined by the general form of Snell's law above. However, the transmittance expressions are more complicated and different for S and P polarizations. If the final medium has loss, but the incident medium is lossless, R and T values are well defined and (R + T) =1. These results for θ=90° are also true for any multilayer sequence as canīe verified by analyzing the 3 region expression above with r12 being the upper interface Fresnel reflection coefficient. The only exception to this is obviously if N1 and N2 are exactly the same in which case obviously rs=rp=0 and ts=tp=1.0. Independent of the complex N1 and N2 values and, with the sign convention used above it is found that rs(θ=90°) = -1 and rp(θ=90°) = +1 and also ts=tp=0 at θ=90°. However it is easy to show using the interface Fresnel reflection coefficient expressions above that at θ=90° glancing angle of incidence, the reflection coefficients rs and rp are completely The actual sign of rs and rp for normal incidence will be reversed if N1 and N2Īre interchanged ('internal' verses 'external' reflection).

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°. 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). 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. h is the thickness of the medium 2 layer in μm and λ is the vacuum wavelength in μm. kxi = k0*ni*cos(θi) where ni and θi may be complex in general, depending on the media and Kxi are the components of propagation constant, in μm-1 normal to the layers (x direction). Rs, Ts, Rp and Tp are the s and p reflectance and transmittance values. The Fresnel E-field amplitude reflection and transmission coefficients for the 2 region interfaces for s (TE) and p(TM) polarized light respectively. rs, rp, ts and tp are the E-field amplitude reflection and transmission coefficients for the film. Θ is the assumed real angle of incidence in medium 1. Heavens, 'Optical Properties of Thin Solid Films', Dover 1965. These conventions correspond to those of O. E-field reflected vector direction for TM (p) case chosen so that r s = r p at normal incidence.With this convention, a medium with loss is specified with a complex refractive index having a NEGATIVE imaginary part.

exponential propagation factor exp(j).The conventions used in the calculator below are: Loss in the film (n2) and final region (n3). The calculator below can be used to compute the plane wave E-field amplitude complex reflection and transmission coefficients and power reflectance and transmittance values for a 3 region (single film) configuration with arbitrary

Reflections geometry calculator