differt.em.fresnel_coefficients

fresnel_coefficients(n_r, cos_theta_i)

Compute the Fresnel reflection and refraction coefficients at an interface.

The Snell’s law describes the relationship between the angles of incidence and refraction:

\[n_i\sin\theta_i = n_t\sin\theta_t,\]

where \(n\) is the refraction index, \(\theta\) is the angle of between the ray path and the normal to the interface, and \(i\) and \(t\) indicate, respectively, the first (i.e., incidence) and the second (i.e., transmission) media.

The s and p reflection coefficients are:

\[r_s = \frac{n_i\cos\theta_i - n_t\cos\theta_t}{n_i\cos\theta_i + n_t\cos\theta_t},\]

and

\[r_p = \frac{n_t\cos\theta_i - n_i\cos\theta_t}{n_t\cos\theta_i + n_i\cos\theta_t}.\]

The s and p refraction coefficients are:

\[t_s = \frac{2n_i\cos\theta_i}{n_i\cos\theta_i + n_t\cos\theta_t},\]

and

\[t_p = \frac{2n_i\cos\theta_i}{n_t\cos\theta_i + n_i\cos\theta_t}.\]

Then, we define \(n_r \triangleq \frac{n_t}{n_i}\) and rewrite the four coefficients as:

\[\begin{split}r_s &= \frac{\cos\theta_i - n_r\cos\theta_t}{\cos\theta_i + n_r\cos\theta_t},\\ r_p &= \frac{n_r^2\cos\theta_i - n_r\cos\theta_t}{n_r^2\cos\theta_i + n_r\cos\theta_t},\\ t_s &= \frac{2\cos\theta_i}{\cos\theta_i + n_r\cos\theta_t},\\ t_p &= \frac{2n_r\cos\theta_i}{n_r^2\cos\theta_i + n_r\cos\theta_t},\end{split}\]

where \(n_t\cos\theta_t\) is obtained from:

\[n_r\cos\theta_t = \sqrt{n_r^2 + \cos^2\theta_i - 1}.\]

Parameters:

• n_r (Inexact[ArrayLike, '*#batch']) –

  The relative refractive indices.

  This is the ratios of the refractive indices of the second media over the refractive indices of the first media.

• cos_theta_i (Float[ArrayLike, '*#batch']) – The (cosine of the) angles of incidence (or reflection).

Return type:

tuple[tuple[Complex[Array, '*batch'], Complex[Array, '*batch']], tuple[Complex[Array, '*batch'], Complex[Array, '*batch']]]

Returns:

The reflection and refraction coefficients for s and p polarizations.

See also

reflection_coefficients

refraction_coefficients

refractive_indices

Examples

The following example reproduces the air-to-glass Fresnel coefficient. The Brewster angle (defined by \(r_p=0\)) is indicated by the vertical red line.

>>> from differt.em import fresnel_coefficients
>>>
>>> n = 1.5  # Air to glass
>>> theta = jnp.linspace(0, jnp.pi / 2)
>>> cos_theta = jnp.cos(theta)
>>> (r_s, r_p), (t_s, t_p) = jax.tree.map(
...     jnp.real,
...     fresnel_coefficients(n, cos_theta)
... )  # Here Fresnel coefficients are purely real numbers
>>> theta_d = jnp.rad2deg(theta)
>>> theta_b = jnp.rad2deg(jnp.arctan(n))
>>> plt.plot(theta_d, r_s, "b:", label=r"$r_s$")
>>> plt.plot(theta_d, r_p, "r:", label=r"$r_p$")
>>> plt.plot(theta_d, t_s, "b-", label=r"$t_s$")
>>> plt.plot(theta_d, t_p, "r-", label=r"$t_p$")
>>> plt.axvline(theta_b, color="r", linestyle="--")
>>> plt.xlabel("Angle of incidence (°)")
>>> plt.ylabel("Amplitude")
>>> plt.xlim(0, 90)
>>> plt.ylim(-1.0, 1.0)
>>> plt.title("Fresnel coefficients")
>>> plt.legend()
>>> plt.tight_layout()

The following example produces the same but glass-to-air interface. The critical angle (total internal reflection) is indicated by the vertical black line.

>>> from differt.em import fresnel_coefficients
>>>
>>> n = 1 / 1.5  #  Glass to air
>>> theta = jnp.linspace(0, jnp.pi / 2, 300)
>>> cos_theta = jnp.cos(theta)
>>> (r_s, r_p), (t_s, t_p) = jax.tree.map(
...     lambda x: jnp.where(jnp.imag(x) == 0, jnp.real(x), jnp.inf),
...     fresnel_coefficients(n, cos_theta)
... )  # Here Fresnel coefficients are purely real numbers before
...    # the critical angle. After the critical angle, they become complex.
>>> theta_d = jnp.rad2deg(theta)
>>> theta_b = jnp.rad2deg(jnp.arctan(n))
>>> theta_c = jnp.rad2deg(jnp.arcsin(n))
>>> plt.plot(theta_d, r_s, "b:", label=r"$r_s$")
>>> plt.plot(theta_d, r_p, "r:", label=r"$r_p$")
>>> plt.plot(theta_d, t_s, "b-", label=r"$t_s$")
>>> plt.plot(theta_d, t_p, "r-", label=r"$t_p$")
>>> plt.axvline(theta_b, color="r", linestyle="--")
>>> plt.axvline(theta_c, color="k", linestyle="--")
>>> plt.xlabel("Angle of incidence (°)")
>>> plt.ylabel("Amplitude")
>>> plt.xlim(0, 90)
>>> plt.ylim(-0.5, 3.0)
>>> plt.title("Fresnel coefficients")
>>> plt.legend()
>>> plt.tight_layout()