Dielectric Boundary — Fresnel Reflection
When a wave crosses the boundary between two dielectrics it partly reflects and partly refracts. The Fresnel equations give the reflected fraction for the two polarisations (TE/s and TM/p) as a function of the incidence angle. This tool also finds the Brewster angle (zero TM reflection) and, for a wave entering a less-dense medium, the critical angle for total internal reflection.
Equations & Parameters ▸
\(n_1\sin\theta_i = n_2\sin\theta_t,\quad n=\sqrt{\varepsilon_r}\)
\(r_{\text{TE}} = \dfrac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t},\quad r_{\text{TM}} = \dfrac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}\)
\(\theta_B = \arctan\dfrac{n_2}{n_1},\qquad \theta_c = \arcsin\dfrac{n_2}{n_1}\ (n_1>n_2)\)
\(r_{\text{TE}} = \dfrac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t},\quad r_{\text{TM}} = \dfrac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}\)
\(\theta_B = \arctan\dfrac{n_2}{n_1},\qquad \theta_c = \arcsin\dfrac{n_2}{n_1}\ (n_1>n_2)\)
| εr1, εr2 | Relative permittivities of the incidence and transmission media (n = √εr). |
| θi | Angle of incidence from the surface normal (degrees). |
| RTE, RTM | Power reflectance for the two polarisations = |r|². Transmittance = 1 − R. |
| θB, θc | Brewster angle (RTM = 0) and critical angle (total internal reflection). |
References: D. K. Cheng, Field and Wave Electromagnetics, 2nd ed., Addison-Wesley, 1989. · J. D. Kraus & D. A. Fleisch, Electromagnetics, 5th ed., McGraw-Hill, 1999.
Inputs
Incidence side
Transmission side
deg
From normalResults
Reflection & refraction
Refraction angle θt—
R (TE / s-pol)—
R (TM / p-pol)—
Special angles
Brewster angle—
Critical angle—
Diagram