Tool · Optical materials

Fresnel Equations Calculator

How much light does a surface reflect in each polarization, with what phase, and how does a metal differ from glass? From the two refractive indices, including an extinction coefficient for an absorbing medium, and the angle of incidence, the calculator gives the complex reflection coefficients, the reflectance, the phase difference between s and p, and the Brewster and critical angles. Background: Fresnel equations, Brewster angle, total internal reflection, complex refractive index, and reflection.

Interface

Index of the first medium n₁

Second medium, real part n₂

Second medium, extinction coefficient κ₂

Angle of incidence θ₁ (°)

Wavelength (nm)

Light travels from the first medium into the second, whose index is ñ₂ = n₂ + iκ₂. Leave κ₂ at 0 for a transparent material; metals and absorbing semiconductors need it. The wavelength only sets the absorption and evanescent depths.

Presets

Indices at 1550 nm are those in the site’s refractive index table. The metal-like index is illustrative; for a real metal use measured n and κ at your wavelength.

Readouts

Reflectance against angle of incidence

s-polarizedp-polarizedunpolarized

Phase on reflection against angle of incidence

φ_sφ_p

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. Reflectances, Brewster and critical angles, the total-internal-reflection phase difference and the reflectance of an absorbing medium are compared with values worked out by hand and with the figures in the site’s Fresnel equations and complex refractive index entries, and energy conservation is checked with the transmission coefficient.

Check	Expected	Computed	Tolerance

The expected values follow the Fresnel equations as given in Born and Wolf, Principles of Optics, and Hecht, Optics, evaluated by hand for the stated cases. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

Light in a medium of index $n_1$ meets a flat surface of a medium with complex index $\tilde n_2 = n_2 + i\kappa_2$ at angle $\theta_1$ . The amplitude reflection coefficients are

r_s = \frac{n_1\cos\theta_1 - \tilde n_2\cos\theta_2}{n_1\cos\theta_1 + \tilde n_2\cos\theta_2}, \qquad r_p = \frac{\tilde n_2\cos\theta_1 - n_1\cos\theta_2}{\tilde n_2\cos\theta_1 + n_1\cos\theta_2}

with $\tilde n_2\cos\theta_2 = \sqrt{\tilde n_2^2 - n_1^2\sin^2\theta_1}$ taken on the root whose field decays into the second medium. This one expression covers ordinary refraction, total internal reflection, where it is imaginary, and absorbing media, where it is complex. The reflectance is $R = |r|^2$ and the phase on reflection is $\arg r$ . For an absorbing medium $R_p$ never reaches zero; its minimum is the pseudo-Brewster angle, and ellipsometry measures $n_2$ and $\kappa_2$ from the ratio $r_p/r_s = \tan\psi\, e^{i\Delta}$ . The model treats a single flat, clean interface; for coatings and layer stacks see the thin-film reflectance calculator, and for the ray geometry the Snell’s law calculator.

Worked example

Air to glass of index 1.5 reflects 4 % at normal incidence. At 45° it reflects 9.201 % of s-polarized light and 0.8466 % of p, and at Brewster’s angle, 56.31°, the p reflection vanishes while 14.79 % of s is reflected. A metal-like surface with index 0.2 + 3.4i reflects 93.85 % at normal incidence.

References: M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), ch. 1 and 14. E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 4. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).