Fresnel Equations: What are they? (Derivation & Explanation)

Contents

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Key learnings:

Fresnel Equations Definition: Fresnel Equations describe the ratios of electric fields of reflected and transmitted waves to the incident wave.
Light Reflection and Transmission: These equations explain how light reflects and transmits at the boundary between two different mediums.
Historical Insight: Augustin-Jean Fresnel developed these equations, understanding light as a transverse wave.
Polarizations: Light polarization can be S (perpendicular) or P (parallel) to the plane of incidence.
Fresnel Equations Derivation: The detailed derivation shows how to calculate reflection and transmission coefficients for both S-polarization and P-polarization.

What are the Fresnel Equations?

The Fresnel equations give amplitude reflection and transmission coefficients at a flat interface. Each coefficient is a ratio between the reflected or transmitted electric field amplitude and the incident amplitude, under a stated polarisation and sign convention. A complex coefficient describes relative amplitude and phase.

These equations describe a plane wave at the interface between two homogeneous optical media. Augustin-Jean Fresnel developed the relations through the transverse-wave theory of light; modern derivations apply Maxwell’s boundary conditions.

At a dielectric interface, the reflected angle equals the incident angle. The transmitted angle follows Snell’s law. The field amplitudes also depend on polarisation, refractive indices and angle.

Reflection from a smooth water surface is a familiar example: glare changes with viewing angle and polarisation. On a rough surface, each small facet has its own local normal, so Fresnel reflection combines with surface scattering.

For most dielectric interfaces, reflectance rises as incidence approaches grazing. P-polarised reflection falls to zero at Brewster’s angle for ideal lossless non-magnetic media, then rises again.

The angle of incidence is measured between the incident ray and the surface normal, not between the ray and the surface. Any diagram or calculation must use the same normal-based convention.

S and P Polarizations

The plane of incidence contains the incident wavevector and the surface normal.

The plane of incidence defines the two independent linear-polarisation cases. Here, polarisation refers to the orientation of the electric-field oscillation.

Resolve the incident electric field into these two components:

S-Polarization
P-Polarization

For S-polarisation, the electric field is perpendicular to the plane of incidence. The letter S comes from the German senkrecht, meaning perpendicular. This case is also called transverse electric (TE) relative to the plane of incidence.

For P-polarisation, the electric field lies parallel to the plane of incidence. Its magnetic field is perpendicular to that plane, so this case is also called transverse magnetic (TM).

The figure identifies the incident, reflected and transmitted waves for S- and P-polarisation.

Fresnel Equations Complex Index of Refraction

Fresnel amplitude coefficients may be real or complex. Their magnitude and phase describe the field response, but they are not themselves power fractions. Complex values arise with absorbing media and total internal reflection.

The symbols r and t denote amplitude reflection and transmission coefficients. Their signs depend on the chosen field directions, especially for P-polarisation, so equations from different references can differ by a sign while predicting the same reflectance.

Here, ‘r’ is the reflected-to-incident electric-field amplitude ratio and ‘t’ is the transmitted-to-incident ratio. For lossless non-magnetic media, power reflectance is the squared magnitude of r, while power transmittance also needs the refractive-index and angle factor.

Let θ_i be the incident angle, θ_r the reflected angle and θ_t the transmitted angle, all measured from the interface normal.

N_i is the refractive index of the incident medium and N_t is the index of the transmitted medium. The following form assumes isotropic, homogeneous, non-magnetic media and no free surface current.

There are four amplitude coefficients: r_p and r_s for reflection, plus t_p and t_s for transmission.

Fresnel Equations Derivation

First consider S-polarised incidence at the planar boundary.

With no free surface current or charge, the tangential electric and magnetic fields satisfy Maxwell’s boundary conditions. For S-polarisation, applying those conditions to the incident, reflected and transmitted plane waves gives the following scalar relations.

Write the boundary equations for the E and B amplitudes using the directions shown:

(1) $\begin{equation*}E_i + E_r = E_t\end{equation*}$

(2) $\begin{equation*}B_i \cos(\theta_i) - B_r \cos(\theta_r) = B_t \cos(\theta_t)\end{equation*}$

For a plane wave in the assumed non-magnetic medium, use the following relation to eliminate B:

$B = \frac{nE}{c_0}$

The law of reflection also gives:

$\theta_i = \theta_r$

Substitute this value into equation 2:

(3) $\begin{equation*} \frac{n_i E_i}{c_0} \cos(\theta_i) - \frac{n_i E_r}{c_0} \cos(\theta_i) = \frac{n_t E_t}{c_0} \cos(\theta_t) \end{equation*}$

(4) $\begin{equation*}n_i \cos(\theta_i) [ E_i - E_r ] = n_t E_t \cos(\theta_t) \end{equation*}$

(5) $\begin{equation*}n_i \cos(\theta_i) [ E_i - E_r ] = n_t [ E_i + E_r ] \cos(\theta_t) \end{equation*}$

(6) $\begin{equation*}n_i E_i \cos(\theta_i) - n_i E_r \cos(\theta_i) = n_t E_i \cos(\theta_t) + n_t E_r \cos(\theta_t)\end{equation*}$

(7) $\begin{equation*}n_i E_i \cos(\theta_i) - n_t E_i \cos(\theta_t) = n_t E_r \cos(\theta_t) + n_i E_r \cos(\theta_i) \end{equation*}$

(8) $\begin{equation*}E_i [ n_i \cos(\theta_i) - n_t \cos(\theta_t) ] = E_r [n_t \cos(\theta_t) + n_i \cos(\theta_i)]\end{equation*}$

(9) $\begin{equation*}r_s = \frac{E_r}{E_i} = \frac{n_i \cos(\theta_i) - n_t \cos(\theta_t)}{n_t \cos(\theta_t) + n_i \cos(\theta_i)}\end{equation*}$

Now solve for transmission coefficient t using equations 1 and 4:

(10) $\begin{equation*}n_i \cos(\theta_i) [ E_i - (E_t - E_i) ] = n_t E_t \cos(\theta_t) \end{equation*}$

(11) $\begin{equation*}n_i \cos(\theta_i) [ 2E_i - E_t ] = n_t E_t \cos(\theta_t) \end{equation*}$

(12) $\begin{equation*} 2E_i n_i \cos(\theta_i) - E_t n_i \cos(\theta_i) = n_t E_t \cos(\theta_t) \end{equation*}$

(13) $\begin{equation*} 2E_i n_i \cos(\theta_i) = E_t n_i \cos(\theta_i) + n_t E_t \cos(\theta_t) \end{equation*}$

(14) $\begin{equation*}t_s = \frac{E_t}{E_i} = \frac{2 n_i \cos(\theta_i)}{ n_i \cos(\theta_i) + n_t \cos(\theta_t)} \end{equation*}$

These are the amplitude coefficients for S-polarised light under the stated assumptions.

Next, consider light with parallel (P) polarisation.

For P-polarisation, resolve the fields along the interface and apply the same boundary conditions:

(15) $\begin{equation*}E_i \cos(\theta_i) + E_r \cos(\theta_i) = E_t \cos(\theta_t)\end{equation*}$

(16) $\begin{equation*}B_i - B_r = B_t\end{equation*}$

Use the same non-magnetic plane-wave relation to eliminate B:

$B = \frac{nE}{c_0}$

(17) $\begin{equation*}n_i E_i - n_i E_r = n_t E_t\end{equation*}$

$n_i [E_i - E_r] = n_t E_t$

$\frac{n_i}{n_t} [E_i - E_r] = E_t$

Substitute this value into equation 15:

(18) $\begin{equation*}E_i \cos(\theta_i) + E_r \cos(\theta_i) = \frac{n_i}{n_t} [E_i - E_r] \cos(\theta_t)\end{equation*}$

(19) $\begin{equation*}n_t [E_i \cos(\theta_i) + E_r \cos(\theta_i)] = {n_i} [E_i - E_r] \cos(\theta_t)\end{equation*}$

(20) $\begin{equation*}n_t E_i \cos(\theta_i) + n_t E_r \cos(\theta_i) = n_i E_i \cos(\theta_t) - n_i E_r \cos(\theta_t)\end{equation*}$

(21) $\begin{equation*} n_t E_i \cos(\theta_i) - n_i E_i \cos(\theta_t) = -n_t E_r \cos(\theta_i) - n_i E_r \cos(\theta_t) \end{equation*}$

(22) $\begin{equation*}E_i [n_t \cos(\theta_i) - n_i \cos(\theta_t)] = -E_r [n_t \cos(\theta_i) + n_i \cos(\theta_t)] \end{equation*}$

(23) $\begin{equation*}E_i [ n_i \cos(\theta_t) - n_t \cos(\theta_i)] = E_r [n_t \cos(\theta_i) + n_i \cos(\theta_t)] \end{equation*}$

(24) $\begin{equation*}r_p = \frac{E_r}{E_i} = \frac{ n_i \cos(\theta_t) - n_t \cos(\theta_i)}{n_t \cos(\theta_i) + n_i \cos(\theta_t)}\end{equation*}$

Now solve for transmission coefficient t using equation 17:

$n_i E_i - n_t E_t = n_i E_r$

$E_i -\frac{n_t}{n_i} E_t = E_r$

Substitute this value into equation 15:

(25) $\begin{equation*}E_i \cos(\theta_i) + [ E_i -\frac{n_t}{n_i} E_t] \cos(\theta_i) = E_t \cos(\theta_t)\end{equation*}$

(26) $\begin{equation*}E_i \cos(\theta_i) + E_i \cos(\theta_i) - \frac{n_t}{n_i} E_t \cos(\theta_i) = E_t \cos(\theta_t) \end{equation*}$

(27) $\begin{equation*}2 E_i \cos(\theta_i) = \frac{n_t}{n_i} E_t \cos(\theta_i) + E_t \cos(\theta_t) \end{equation*}$

(28) $\begin{equation*}2 E_i n_i \cos(\theta_i) = n_t E_t \cos(\theta_i) + {n_i} E_t \cos(\theta_t) \end{equation*}$

(29) $\begin{equation*}2 E_i n_i \cos(\theta_i) = E_t [n_t \cos(\theta_i) + {n_i} \cos(\theta_t)] \end{equation*}$

(30) $\begin{equation*} t_p = \frac{E_t}{E_i} = \frac{2 n_i \cos(\theta_i)}{ n_t \cos(\theta_i) + {n_i} \cos(\theta_t)} \end{equation*}$

The four field-amplitude coefficients derived with this article’s sign convention are:

$r_s = \frac{n_i \cos(\theta_i) - n_t \cos(\theta_t)}{n_t \cos(\theta_t) + n_i \cos(\theta_i)}$

$t_s = \frac{2 n_i \cos(\theta_i)}{ n_i \cos(\theta_i) + n_t \cos(\theta_t)}$

$r_p = \frac{ n_i \cos(\theta_t) - n_t \cos(\theta_i)}{n_t \cos(\theta_i) + n_i \cos(\theta_t)}$

$t_p = \frac{2 n_i \cos(\theta_i)}{ n_t \cos(\theta_i) + {n_i} \cos(\theta_t)}$