update efficiency doc

Intern/rayx-core/src/Shader/Efficiency.h

@@ -210,7 +210,7 @@ struct RotationBase {
 // A convention for the up and right vectors is implemented, making this function well defined and for all forward directions
 // TODO(Sven): this convention should be exchanged with one that does not branch
 RAYX_FN_ACC
-inline RotationBase forwardVectorToBaseConvention(glm::dvec3 forward) {
+inline RotationBase forwardVectorToBaseConvention(const glm::dvec3 forward) {
     auto up = glm::dvec3(0, 1, 0);
     glm::dvec3 right;
 
@@ -246,22 +246,22 @@ inline glm::dmat3 rotationMatrix(const glm::dvec3 forward, const glm::dvec3 up)
 }
 
 RAYX_FN_ACC
-inline ElectricField localToGlobalElectricField(LocalElectricField localField, glm::dvec3 forward) {
+inline ElectricField localToGlobalElectricField(const LocalElectricField localField, const glm::dvec3 forward) {
     return rotationMatrix(forward) * ElectricField(localField, complex::Complex{0, 0});
 }
 
 RAYX_FN_ACC
-inline ElectricField localToGlobalElectricField(LocalElectricField localField, glm::dvec3 forward, glm::dvec3 up) {
+inline ElectricField localToGlobalElectricField(const LocalElectricField localField, const glm::dvec3 forward, const glm::dvec3 up) {
     return rotationMatrix(forward, up) * ElectricField(localField, complex::Complex{0, 0});
 }
 
 RAYX_FN_ACC
-inline LocalElectricField globalToLocalElectricField(ElectricField field, glm::dvec3 forward) {
+inline LocalElectricField globalToLocalElectricField(const ElectricField field, const glm::dvec3 forward) {
     return glm::transpose(rotationMatrix(forward)) * field;
 }
 
 RAYX_FN_ACC
-inline LocalElectricField globalToLocalElectricField(ElectricField field, glm::dvec3 forward, glm::vec3 up) {
+inline LocalElectricField globalToLocalElectricField(const ElectricField field, const glm::dvec3 forward, const glm::vec3 up) {
     return glm::transpose(rotationMatrix(forward, up)) * field;
 }
 

docs/src/Model/Efficiency.md

@@ -1,46 +1,104 @@
-# Efficiency 
 
-wiki for efficiency calculations
+# Efficiency
 
-### Snell's law
-A fraction of the light is reflected and another transmitted:
+This wiki documents the usage of efficiency and polarization techniques utilized in RAYX.
+
+## Snell's law
+Snell's law [...] is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air [[1]](#1).
 
 ![refraction_fresnel](https://upload.wikimedia.org/wikipedia/commons/8/89/Fresnel1.svg)<br>
-\\(\theta_i =\\)  (normal) incidence angle <br>
-\\(\theta_r =\\) (normal) reflection angle (same as \\(\theta_i\\))<br>
-\\(\theta_t =\\) (normal) transmittance angle <br>
-\\(N_1 =\\) refraction index of material from which the ray is coming  (left in image)<br>
-\\(N_2 =\\) refraction index of material into which the ray is going (right in image)<br>
+$\theta_i =$ (normal) incidence angle
+$\theta_r =$ (normal) reflection angle (same as $\theta_i$)
+$\theta_t =$ (normal) transmittance angle
+$N_1 =$ refraction index of material from which the ray is coming  (left in image)
+$N_2 =$ refraction index of material into which the ray is going (right in image)
 
-all parameters are potentially complex numbers. The refractive indices are retrieved from files (Palik, Henke, Cromer..)
+All parameters are potentially complex numbers.
+The refractive indices $N_1$ and $N_2$ are retrieved from databases (Palik, Henke, Cromer..).
 
 Snell's law:
-\\[
+$$
 N_1 \sin \theta_i = N_2 \sin \theta_t \rightarrow \sin \theta_t = \frac{N_1}{N_2} \sin \theta_i
-\\]
+$$
 
-\\(\theta_i\\), \\(N_1\\), \\(N_2\\) are known, we are looking for \\(\theta_t\\).  <br>
-We do not calculate the angle specifically but only the cosinus, which is sufficient for further calculations and more efficient/precise than calculating the angle itself because we do not need to use more trigonometric functions.
-We can calculate the incidence angle \\(\theta_i\\) of each ray from its direction and the surface normal. Then we calculate \\(\cos(\theta_i)\\) and from that we can derive \\(\cos(\theta_t)\\) with snell's law:
+With $\theta_i$, $\theta_r$, $N_1$, $N_2$ being known, we are looking for $\theta_t$.
+We can calculate the incidence angle $\theta_i$ of each ray from its direction and the surface normal. Then we calculate $\cos(\theta_i)$ and from that we can derive $\cos(\theta_t)$ with snell's law:
 
-\\[
+$$
 (\sin \theta_i)^2 = 1 - (\cos \theta_i)^2 \\\\
 (\sin \theta_t)^2 = (\frac{N_1}{N_2})^2 (\sin \theta_i)^2 \\\\
 \cos \theta_t = \sqrt{1 - (\sin \theta_t)^2} = \sqrt{1 - \Big(\frac{N_1}{N_2} \sin \theta_i\Big)^2}
-\\]
+$$
 
 The cosine of both angles is then used in the Fresnel equations to calculate the s- and p-polarization
 
-### Fresnel equation
+## Fresnel equation [[2]](#2)
 Any polarization state can be described by two components: one vertical and one horizontal. Or - relative to the plane of incidence - s- and p-polarization. 
 p-polarization (parallel, left image) lies parallel in the plane of incidence and s-polarization (senkrecht, right image) is orthogonal to the plane of incidence.
 
 <img src="https://upload.wikimedia.org/wikipedia/commons/4/4d/Polarisation_p.png" alt="ppol" width="200"/>
 <img src="https://upload.wikimedia.org/wikipedia/commons/3/3c/Polarisation_s.png" alt="spol" width="200"/>
 
-the reflectance of both polarizations is calculated with the fresnel equations:
+the amplitude coefficient of a reflection for both polarizations is calculated with the fresnel equations:
+
+$$r_s = \frac{N_1 \cdot \cos \theta_i - N_2 \cdot \cos \theta_t}{N_1 \cdot \cos \theta_i + N_2 \cdot \cos \theta_t}$$
+$$r_p = \frac{N_2 \cdot \cos \theta_i - N_1 \cdot \cos \theta_t}{N_2 \cdot \cos \theta_i + N_1 \cdot \cos \theta_t}$$
+
+the amplitude coefficient of a refraction for both polarizations is calculated with the fresnel equations:
+
+$$t_s = \frac{2.0 \cdot N_1 \cdot \cos \theta_i}{N_1 \cdot \cos \theta_i + N_2 \cdot \cos \theta_t}$$
+$$t_p = \frac{2.0 \cdot N_1 \cdot \cos \theta_i}{N_2 \cdot \cos \theta_i + N_1 \cdot \cos \theta_t}$$
+
+## Mirror reflection [[2]](#2)
+
+Reflecting a ray on a mirror involves polarization and phase changes of the incident electric field of the ray. The update of the electric field can be described by 3 steps:
+
+1. Rotate incident electric field from global coordinates into local basis of incoming propagation vector $\vec{k_{q-1}}$ using matrix $Q_{q,in}$.
+1. Multiply fresnel amplitude coefficients with components of the electric field ($r_{s,q}$ and $r_{p,q}$ for reflection) using jones matrix $J_{q}$.
+1. Rotate resulting electric field back into global coordinates by rotating into basis of outgoing propagation vector $\vec{k_{q}}$ using matrix $Q_{q,out}$.
+
+In order to obtain rotation matrices $Q_{q,in}$ and $Q_{q,out}$, the basis vectors need to be found.
+
+$$
+\begin{aligned}
+    \vec{s}_{q} &= \frac{\vec{k}_{q-1} \times \vec{\eta}_{q}}{|\vec{k}_{q-1} \times \vec{\eta}_{q}|} \\
+    \quad \vec{p}_{q} &= \vec{k}_{q-1} \times \vec{s}_{q} \\
+    \vec{s}'_{q} &= \vec{s}_{q}, \quad \vec{p}'_{q} = \vec{k}_{q} \times \vec{s}_{q}
+\end{aligned}
+$$
+
+Both incident vector $\vec{k}_{q-1}$ and reflected vector $\vec{k}_{q}$ as well as normal vector $\vec{\eta}_{q}$ lie within the plane of incidence while the normal vector is perpendicular to the plane itself and defines it's front face.
+We define vectors $\vec{s}$ and $\vec{p}$ perpendicular to each other and to propagation vector $\vec{k_{q-1}}$, while $\vec{s}$ is perpendicular to the plane of incidence and $\vec{k}$ lies within the plane.
+
+$$
+\begin{aligned}
+    O_{q,in} &= \begin{pmatrix}
+    s'_{x,q} & p'_{x,q} & k_{x,q} \\
+    s'_{y,q} & p'_{y,q} & k_{y,q} \\
+    s'_{z,q} & p'_{z,q} & k_{z,q}
+    \end{pmatrix} \\
+    J_{q} &= \begin{pmatrix}
+    r_{s,q} & 0 & 0 \\
+    0 & r_{p,q} & 0 \\
+    0 & 0 & 1
+    \end{pmatrix} \\
+    O_{q,out} &= \begin{pmatrix}
+    s_{x,q} & s_{y,q} & s_{z,q} \\
+    p_{x,q} & p_{y,q} & p_{z,q} \\
+    k_{x,q-1} & k_{y,q-1} & k_{z,q-1}
+    \end{pmatrix} \\
+    P_{q} &= O_{q,in} \cdot J_{q} \cdot O_{q,out}
+\end{aligned}
+$$
+
+All operations are composed into the Polarization Ray Tracing Matrix $P_{q}$, encorparating fresnel amplitude coefficients into $J_{q}$.
+
+$$
+E_{q} = P_q E_{q-1}
+$$
 
-\\[r_s = \frac{N_1 \cdot \cos \theta_i - N_2 \cdot \cos \theta_t}{N_1 \cdot \cos \theta_i + N_2 \cdot \cos \theta_t}\\]
-\\[r_p = \frac{N_2 \cdot \cos \theta_i - N_1 \cdot \cos \theta_t}{N_2 \cdot \cos \theta_i + N_1 \cdot \cos \theta_t}\\]
+The incident electric field $E_{q-1}$ can now be multiplied by Polarization Ray Tracing matrix $P_{q}$, resulting in the ray's a new electric field after the intercept with the mirror.
 
-(The transmitted power is then "the rest": \\(t_s = 1 - r_s\\) and \\(t_p = 1 -r_p\\))
+## References
+<a id="1">[1]</a> https://en.wikipedia.org/wiki/Snell%27s_law
+<a id="2">[2]</a> Russel A. Chipman, Wai-Sze Tiffany Lam, Garam Young "Polarized Light and Optical Systems" (2019)