SQUARE45

Let

\mathbf{E}_{i}

and

\mathbf{E}_{r}

be the incident and reflected electric field amplitudes, respectively, at an interface separating two media with relative permittivities

\epsilon_{1}

and

\epsilon_{2}

. Let

\theta_{i}

be the angle of incidence and

\theta_{t}

be the angle of transmission, satisfying Snell's Law. The reflection coefficient for the electric field amplitude,

r

, is defined by the ratio of the reflected field amplitude to the incident field amplitude. For the perpendicular polarization (TE mode,

\mathbf{E} \perp \hat{\mathbf{n}}

), the coefficient is:\n

\mathbf{r}_{\perp} = \frac{\eta_{2} \cos(\theta_{i}) - \eta_{1} \cos(\theta_{t})}{\eta_{2} \cos(\theta_{i}) + \eta_{1} \cos(\theta_{t})}

\nwhere

\eta_{j} = \sqrt{\mu_{j}/\epsilon_{j}}

is the intrinsic impedance of medium

j

. For the parallel polarization (TM mode,

\mathbf{E} \parallel \hat{\mathbf{n}}

), the coefficient is:\n

\mathbf{r}_{\parallel} = \frac{\eta_{2} \cos(\theta_{t}) - \eta_{1} \cos(\theta_{i})}{\eta_{2} \cos(\theta_{t}) + \eta_{1} \cos(\theta_{i})}

\nAlternatively, the reflectance

R

(ratio of reflected intensity to incident intensity) is given by

R = |r|^2

. For the perpendicular case,

R_{\perp} = \left| \frac{\eta_{2} \cos(\theta_{i}) - \eta_{1} \cos(\theta_{t})}{\eta_{2} \cos(\theta_{i}) + \eta_{1} \cos(\theta_{t})} \right|^2

. For the parallel case,

R_{\parallel} = \left| \frac{\eta_{2} \cos(\theta_{t}) - \eta_{1} \cos(\theta_{i})}{\eta_{2} \cos(\theta_{t}) + \eta_{1} \cos(\theta_{i})} \right|^2

Reflection Coefficient