These equations for waves at the interface of two insulators describe the reflected and transmitted amplitudes in and out of the plane of incidence, along with the special case of Brewster's angle.  Please note that these equations only solve for the amplitude, not the polarization (which way the electric field points). If you need a quick refresher about reflected/transmitted angles or Snell's law, please see the Angles of Reflection and Refraction Calculator.  

We use complex amplitudes so that they also contain the phase.  If you don't like the complex amplitude notation, feel free to treat the amplitudes as real numbers and store the phase information somewhere else.

This calculator assumes a monochromatic plane wave of the following format:

• `tilde vec E_I (vec r, t) = tilde E_(0_I) e ^ ((vec k_1 * vec r - omega t)) hat n`

• `tilde vec B_I (vec r, t) = 1/v_1 (hat k_1 times tilde vec E_I)`

where `hat n` is the polarization of the incident wave.

Now to address the calculator tabs in reverse order:

`alpha`, `beta`, and `theta_i`

• `alpha` stands for `cos(theta_t)/cos(theta_i)`, and can be rewritten as `sqrt(1 - (n_1/n_2 sin(theta_i)^2))/ cos(theta_i)` using Snell's law.

• `beta = (mu_1 v_1)/(mu_2 v_2) = (mu_1 n_2)/(mu_2 n_1)`, where "`v`" is the speed of the wave in a given media, and `v = c/n`

• `theta_t = sin^(-1) (n_1/n_2 sin(theta_i))`

"Perpendicular" (S-Polarization)

The EM waves we're considering are purely transverse, so we can describe their polarization with two components, one perpendicular to the plane of incidence (the plane containing the incident, reflected, and transmitted waves), and the other parallel to the plane of incidence.

To describe the amplitude of the waves perpendicular to the plane of intersection (a.k.a. s-polarization), use can use Fresnel's Equations for Polarization Perpendicular to the Plane of Incidence¹:

• `tilde E_(0_R) = (1 - alpha beta)/(1 + alpha beta)tilde E_(0_I)`

• `tilde E_(0_T) = (2)/(1 + alpha beta)tilde E_(0_I)`

"Parallel" (P-Polarization)

The "Parallel" tab describes the amplitudes of waves that lie in the plane of intersection (a.k.a. p-polarization).  Here we use Fresnel's Equations for Polarization in the Plane of Incidence² :

• `tilde E_(0_R) = (alpha - beta) / (alpha + beta) tilde E_(0_I)`

• `tilde E_(0_T) = 2 / (alpha + beta) tilde E_(0_I)`

It's worth paying attention to the first equation for a second here.  Notice that when `alpha = beta`, the reflected amplitude is simply 0, i.e. there is no reflected.  The incident angle that causes `alpha = beta` is called "Brewster's Angle"³  `(theta_B)

• `theta_B =  sin^(-1)  (  sqrt(  (1-beta^2)/( (n_1/n_2)^2 - beta^2) )  )`

That's not the world's most pleasant equation, but when `mu_1` and `mu_2` are much closer than `n_1` and `n_2` are, it simplifies to:

•  `tan(theta_B) approx n_2/n_1`
  Which is much nicer.

On a more subtle note, the phase and sign of the reflected amplitude can be vague since multiplying by -1 is the same as shifting the phase by 180 degrees.  Please keep in mind that these equations were derived using the convention that the reflected amplitude in the plane of incidence is positive if it points towards the interface.

1. ^  Jackson, John David. "Plane Electromagnetic Waves and Wave Propagation." Classical Electrodynamics. New York: Wiley, 1962. 281. Print.
2. ^ Griffiths, David J. "Electromagnetic Waves in Matter." Introduction to Electrodynamics. 4th ed. N.p.: Prentice Hall, 2013. 409. Print.
3. ^ Griffiths, David J. "Electromagnetic Waves in Matter." Introduction to Electrodynamics. 4th ed. N.p.: Prentice Hall, 2013. 410. Print.