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-# Geometric optics
\ No newline at end of file
+# Geometric optics
+
+> *Definition*: surfaces that reflect or refract rays leaving a source point $s$ to a conjugate point $p$ are defined as Cartesian surfaces.
+
+<br>
+
+> *Definition*: a perfect image of a point is possible with a stigmatic system. For the set of conjugated points no diffraction and abberations occur, obtaining reversible rays.
+
+<br>
+
+> *Assumption*: in geometric optics use will be made of the paraxial approximation that states that for small angles $\theta$ 
+>
+> $$
+>   \tan \theta \approx \sin \theta \approx \theta,
+> $$
+>
+> and 
+>
+> $$
+>   \cos \theta \approx 1,
+> $$
+>
+> comes down to using the first term of the Taylor series approximation.
+
+<br>
+
+## Spherical surfaces
+
+> *Law*: for a spherical reflecting interface in paraxial approximation the relation between the object and image distance $s_{o,i} \in \mathbb{R}$ and the radius $R \in \mathbb{R}$ of the interface is given by
+>
+> $$
+>   \frac{1}{s_o} + \frac{1}{s_i} = \frac{2}{R}
+> $$
+>
+> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+<br>
+
+> *Definition*: for a object distance $s_0 \to \infty$ we let the image distance $s_i = f$ with $f \in \mathbb{R}$ the focal length defining the focal point of the spherical interface.
+
+Then it follows from the definition that
+
+$$
+    \frac{1}{s_o} + \frac{1}{s_i} = \frac{1}{f}.
+$$
+
+> *Law*: for a spherical refracting interface in paraxial approximation the relation between the object and image distance $s_{o,i} \in \mathbb{R}$ and the radius $R \in \mathbb{R}$ of the interface is given by
+>
+> $$
+>   \frac{n_i}{s_o} + \frac{n_t}{s_i} = \frac{n_t - n_i}{R}
+> $$
+> 
+> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+<br>
+
+> *Definition*: the transverse magnification $M$ for a optical system is defined as
+>
+> $$
+>   M = \frac{y'}{y}
+> $$
+>
+> with $y, y' \in \mathbb{R}$ the object and image size.
+
+<br>
+
+> *Corollary*: the transverse magnification $M$ for a spherical refracting interface in paraxial approximation is by
+>
+> $$
+>   M = - \frac{n_i s_i}{n_t s_o},
+> $$
+>
+> with $s_{o,i} \in \mathbb{R}$ the object and image distance and $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+<br>
+
+> *Definition*: a lens is defined by two intersecting spherical interfaces with radius $R_1, R_2 \in \mathbb{R}$ respectively.
+
+<br>
+
+> *Law*: for a thin lens in paraxial approximation the radii $R_1, R_2 \in \mathbb{R}$ are related to the focal length $f \in \mathbb{R}$ of the lens by
+>
+> $$
+>   \frac{1}{f} = \frac{n_t - n_i}{n_i} \bigg( \frac{1}{R_1} - \frac{1}{R_2} \bigg),
+> $$
+>
+> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. 
+>
+> With the transverse magnification $M$ given by
+>
+> $$
+>   M = - \frac{s_i}{s_o},
+> $$
+>
+> with the object and image distance $s_{o,i} \in \mathbb{R}$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Sign convention
+
+Converging optics have positive focal lengths and diverging optics have negative focal lengths.
+
+Objects are located left of the optic by a positive object distance and images are located right of the optic by a positive image distance.
+
+## Ray tracing
+
+> *Assumption*: using paraxial approximation and assuming that all optical elements have rotational symmetry and are aligned coaxially along a single optical axis.
+
+A ray matrix model may be introduced where the ray is defined according to its intersection with a reference plane.
+
+> *Definition*: a ray may be defined by its intersection with a reference plane by
+>
+> * the parameter $y \in \mathbb{R}$ is the perpendicular distance between the optical axis and the intersection point,
+> * the angle $\theta \in [0, 2\pi)$ is the angle the ray makes with the horizontal.
+
+<br>
+
+> *Proposition*: for the translation of the ray between two reference planes within the same medium seperated by a horizontal distance $d \in \mathbb{R}$ the relation
+>
+> $$
+>   \begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix},
+> $$
+>
+> holds, for $y_{1,2} \in \mathbb{R}$ and $\theta_{1,2} \in [0, 2\pi)$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+<br>
+
+> *Proposition*: for the reflection of the ray at the plane of incidence at a spherical interface of radius $R \in \mathbb{R}$ the relation
+>
+> $$
+>   \begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 2 / R & 1 \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix},
+> $$
+>
+> holds, for $y_{1,2} \in \mathbb{R}$ and $\theta_{1,2} \in [0, 2\pi)$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+This matrix may also be given in terms of the focal length $f \in \mathbb{R}$ by
+
+$$
+    \begin{pmatrix} 1 & 0 \\ f & 1 \end{pmatrix}.
+$$
+
+> *Proposition*: fir the refraction of the ray at the plane of incidence at a spherical interfance of radius $R \in \mathbb{R}$ the relation
+>
+> $$
+>   \begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ - \frac{n_t - n_i}{n_t R} & \frac{n_i}{n_t} \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix}
+> $$
+>
+> holds, for $y_{1,2} \in \mathbb{R}$, $\theta_{1,2} \in [0, 2\pi)$ and $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+This matrix may also be given in terms of the focal length $f \in \mathbb{R}$ by
+
+$$
+    \begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix}.
+$$
+
+> *Law*: the ray mtarix model taken as a linear sequence of interfaces and translations can be used to model optical systems of arbitrary complexity under the posed assumptions.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Abberations
+
+> *Definition*: an abberation is any effect that prevents a lens from forming a perfect image.
+
+Various abberations could be
+
+* Spherical abberation: error of the paraxial approximation.
+* Chromatic abberation: error due to different index of refraction for different wavelengths of light.
+* Astigmatism: deviation from the cylindrical symmetry.