colour.phenomena.polarised_light_transmission_coefficient#

colour.phenomena.polarised_light_transmission_coefficient(n_1: ArrayLike, n_2: ArrayLike, theta_i: ArrayLike, theta_t: ArrayLike) → NDArrayComplex[source]#

Compute Fresnel transmission power coefficients (transmittance).

This function computes the transmission power coefficients, which represent the fraction of incident power that is transmitted through a dielectric interface for both s-polarisation (perpendicular) and p-polarisation (parallel) components.

Parameters:

  • n_1 (ArrayLike) – Refractive index of the incident medium \(n_1\).

  • n_2 (ArrayLike) – Refractive index of the transmitted medium \(n_2\).

  • theta_i (ArrayLike) – Incident angle \(\theta_i\) in degrees.

  • theta_t (ArrayLike) – Transmitted angle \(\theta_t\) in degrees.

Returns:

Fresnel transmission power coefficients (transmittance) for s and p polarisations stacked along the last axis. The array contains \([T_s, T_p]\).

Return type:

numpy.ndarray

Notes

The Fresnel transmission power coefficients (transmittance) are given by:

\[\begin{split}T_s &= \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} |t_s|^2 = \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} \left|\frac{2n_1 \cos \theta_i}{n_1 \cos \theta_i + n_2 \cos \theta_t}\right|^2 \\ T_p &= \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} |t_p|^2 = \frac{n_2 \cos \theta_t}{n_1 \cos \theta_i} \left|\frac{2n_1 \cos \theta_i}{n_2 \cos \theta_i + n_1 \cos \theta_t}\right|^2\end{split}\]

Where:

  • \(T_s\): s-polarisation transmittance (fraction of incident power transmitted)

  • \(T_p\): p-polarisation transmittance (fraction of incident power transmitted)

  • \(t_s, t_p\): complex transmission amplitude coefficients

  • The s-polarisation electric field is perpendicular to the plane of incidence

  • The p-polarisation electric field is parallel to the plane of incidence

The refractive index factor \(\frac{n_2 \cos \theta_t}{n_1 \cos \theta_i}\) accounts for the change in beam cross-section and energy density in the transmission medium.

Energy Conservation: For non-absorbing media: \(R_s + T_s = 1\) and \(R_p + T_p = 1\), where \(R_s, R_p\) are the corresponding reflectance coefficients.

The transmittance values satisfy: \(0 \leq T_s, T_p \leq 1\).

References

[Byr16]

Examples

>>> polarised_light_transmission_coefficient(1.0, 1.5, 0.0, 0.0)
array([ 0.96+0.j,  0.96+0.j])