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# Diffraction
## Huygens principle
Huygens principle will be used to derive equations for diffraction.
> *Assumption*: According to Huygens principle each point on the wavefront of an electromagnetic wave acts as a source of secondary wavelets. When summed over an extended unobstructed wavefront the secondary wavelets recreate the next wavefront. It is assumed that this principle is valid as it is consistent with the laws of reflection and refraction.
The following theorem follows from Huygens principle.
> *Law*: the net disturbance $E_P: \mathbb{R} \to \mathbb{R}$ at a perceive point $P$ for a wave travelling from source point $S$ travelling a distance $r' \in \mathbb{R}$ to an aperture opening defined for the points in $D \subseteq \mathbb{R}$ and then travelling a distance $r \in \mathbb{R}$ towards $p$ is given by
>
> $$
> E_P(t) = E_0 k e^{-i \omega t} \int_D \frac{1}{2 r r'} (1 + \cos \theta) e^{ik (r+r')} dA,
> $$
>
> for all $t \in \mathbb{R}$ with $E_0 \in \mathbb{R}$, $k \in \mathbb{R}$ the wavenumber of the light, $\omega \in \mathbb{R}$ the angular frequency of the light and $\theta \in [0, 2\pi)$ the angle between the source, aperture and perceive point.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: for two complementary apertures that when taken together form a single opaque screen. Let $E_1$ and $E_2$ be the field at point $P$ for each aperture respectively. Then the combination of these fields must give the unubstructed wave $E_0$. Therefore
>
> $$
> E_0 = E_1 + E_2.
> $$
??? note "*Proof*:"
Will be added later.
## Fraunhofer diffraction
The above law for the diffraction at a perceive point $P$ can be simplified under certain conditions such that the integral can be solved easier.
> *Corollary*: for small angles between the source, aperture and perceive point $\theta$, implying that source and perceive points are far away and the aperture opening is small then in reasonable approximation the net disturbance $E_P: \mathbb{R} \to \mathbb{R}$ at the perceive point may be given by
>
> $$
> E_P = E_0 \int_D e^{ikr}dA,
> $$
>
> with $E_0 \in \mathbb{R}$ and $k \in \mathbb{R}$ the wavenumber. Under the condition that
>
> $$
> r >> \frac{h^2}{2\lambda},
> $$
>
> with $h \in \mathbb{R}$ the height of the aperture and $\lambda \in \mathbb{R}$ the wavelength of the light.
??? note "*Proof*:"
Will be added later.
From this simplification the net disturbance caused by several apertures can be derived, given in the corollaries below.
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a single slit aperture is given by
>
> $$
> E(\theta) = E_0 \text{ sinc } \beta(\theta),
> $$
>
> for all $\theta \in \mathbb{R}$ with $\beta(\theta) = \frac{kb}{2} \sin \theta$ and $E_0, k, b \in \mathbb{R}$ the magnitude of the electric field, the wavenumber and the width of the slit.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a rectangular aperture is given by
>
> $$
> E(\theta, \varphi) = E_0 \text{ sinc } \alpha(\theta) \text{ sinc } \beta(\varphi),
> $$
>
> for all $(\theta, \varphi) \in \mathbb{R}^2$ with $\alpha(\theta) = \frac{ka}{2} \sin \theta$, $\beta(\varphi) = \frac{kb}{2} \sin \varphi$ and $E_0, k, a, b \in \mathbb{R}$ the magnitude of the electric field, the wavenumber, the height and the width of the rectangle.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a circular aperture is given by
>
> $$
> E(\theta) = E_0 \frac{2 J_1(\sigma(\theta))}{\sigma(\theta)},
> $$
>
> for all $\theta \in \mathbb{R}^2$ with $J_1: \mathbb{R} \to \mathbb{R}$ the Bessel function of the first order, $\sigma(\theta) = \frac{kd}{2} \sin \theta$ and $E_0, k, d \in \mathbb{R}$ the magnitude of the electric field, the wavenumber and the diameter of the circle.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: the net disturbance $E: \mathbb{R} \to \mathbb{R}$ of the eletric field for a $N$-slit aperture with $N \in \mathbb{N}$ is given by
>
> $$
> E(\theta) = E_0 \text{ sinc } \beta(\theta) \frac{\sin N \gamma(\theta)}{N \sin \gamma(\theta)}
> $$
>
> for all $\theta \in \mathbb{R}$ with $\beta(\theta) = \frac{kb}{2} \sin \theta$, $\gamma(\theta) = \frac{kd}{2} \sin \theta$ and $E_0, k, d, b \in \mathbb{R}$ the magnitude of the electric field, the wavenumber, the distance between the slits and the width of the slits.
??? note "*Proof*:"
Will be added later.
When taking $N \to \infty$ for the $N$-slits aperture and incidence is normal principal maxima are obtained for $\gamma(\theta) = m \pi$ with $m \in \mathbb{Z}$ therefore
$$
d \sin \theta = m \lambda,
$$
with $d, \lambda \in \mathbb{R}$ the distance between the slits and the wavelength of the light.
When incidence $\theta_i \in \mathbb{R}$ is not normal the principal maxima are given by
$$
d (\sin \theta_i + \sin \theta) = m \lambda,
$$
also known as the grating equation.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: two point sources given by the net disturbances of the eletric field $E_{1,2}: D \to \mathbb{R}$ with $D \subseteq \mathbb{R}$ such that $E_{1,2}$ are bijective can be resolved if they satisfy the Reyleigh criterion given by
>
> $$
> \min E_2^{-1}(E_{02}) \geq \min E_1^{-1}(0),
> $$
>
> $$
> \min E_1^{-1}(E_{01}) \geq \min E_2^{-1}(0),
> $$
>
> with $E_{0(1,2)} \in \mathbb{R}$ the eletric field amplitudes.
This definition will be used in the following propositions.
> *Proposition*: the chromatic resolving power $\mathcal{R}$ of a $N$-slit aperture based on the Reyleigh criterion can be determined by
>
> $$
> \mathcal{R} = N m,
> $$
>
> with $m \in \mathbb{Z}$ the order of the principal maxima and $N \in \mathbb{N}$ the number of slits.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the free spectral range $\text{FSR}$ of a $N$-slit aperture can be determined by
>
> $$
> \text{FSR} = \frac{\lambda}m,
> $$
>
> with $m \in \mathbb{Z}$ the order and $\lambda \in \mathbb{R}$ the wavelength.
??? note "*Proof*:"
Will be added later.

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# Electromagnetic waves
This section is a direct follow up on the section [Maxwell equations](../maxwell-equations.md). Where the Laplacian of the electric field $\mathbf{E}: U \to \mathbb{R}^3$ and magnetic field $\mathbf{B}: U \to \mathbb{R}^3$ in vacuum ($\varepsilon = \varepsilon_0, \mu = \mu_0$) have been determined, given by
$$
\begin{align*}
&\nabla^2 \mathbf{E}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{E}(\mathbf{v}, t) \\\\
&\nabla^2 \mathbf{B}(\mathbf{v}, t) = \varepsilon_0 \mu_0 \partial_t^2 \mathbf{B}(\mathbf{v}, t)
\end{align*}
$$
for all $(\mathbf{v}, t) \in U$.
It may be observed that the eletric and magnetic field comply with the $3 + 1$ dimensional wave equation posed in the section [waves](waves.md). Obtaining the speed $v \in \mathbb{R}$ given by
$$
v = \frac{1}{\sqrt{\varepsilon_0 \mu_0}} = c,
$$
defined by $c$ the speed of light, or more generally the speed of information in the universe. Outside vacuum we have
$$
v = \frac{1}{\sqrt{\varepsilon \mu}} = \frac{c}{n},
$$
with $n = \sqrt{K_E K_B}$ the index of refraction.
> *Proposition*: let $\mathbf{E},\mathbf{B}: U \to \mathbb{R}^3$, a solution for the wave equations of the electric and magnetic field may be harmonic linearly polarized plane waves satisfying Maxwell's equations given by
>
> $$
> \begin{align*}
> \mathbf{E}(\mathbf{v}, t) &= \text{Im}\Big(\mathbf{E}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big) \\ \\ \mathbf{B}(\mathbf{v}, t) &= \text{Im} \Big(\mathbf{B}_0 \exp i \big(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t+ \varphi\big) \Big)
> \end{align*}
> $$
>
> for all $(\mathbf{v}, t) \in U$ with $\mathbf{E}_0, \mathbf{B}_0 \in \mathbb{R}^3$.
??? note "*Proof*:"
Will be added later.
The above proposition gives an example of a light wave, but note that there are much more solutions that comply to Maxwell's equations.
> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ for all solutions of the posed wave equations are orthogonal to the direction of propagation $\mathbf{k}$. Therefore electromagnetic waves are transverse.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ in a electromagnetic wave are orthogonal to each other; $\langle \mathbf{E}, \mathbf{B} \rangle = 0$.
??? note "*Proof*:"
Will be added later.
<br>
> *Corollary*: it follows from the above law that the magnitude of the electric and magnetic fields $E, B: U \to \mathbb{R}$ in a electromagnetic wave are related by
>
> $$
> E(\mathbf{v}, t) = v B(\mathbf{v}, t)
> $$
>
> for all $(\mathbf{v}, t) \in U$ with $v = \frac{c}{n}$ the wave speed.
??? note "*Proof*:"
Will be added later.
## Energy flow
> *Law*: the energy flux density $\mathbf{S}: U \to \mathbb{R}^3$ of an electromagnetic wave is given by
>
> $$
> \mathbf{S}(\mathbf{v}, t) = \frac{1}{\mu_0} \mathbf{E}(\mathbf{v}, t) \times \mathbf{B}(\mathbf{v}, t),
> $$
>
> for all $(\mathbf{v}, t) \in U$. $\mathbf{S}$ is also called the Poynting vector.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: the time average of the magnitude of $\mathbf{S}$ is called the irradiance.
<br>
> *Proposition*: the irradiance $I \in \mathbb{R}$ for harmonic linearly polarized plane electromagnetic waves is given by
>
> $$
> I = \frac{\varepsilon_0 c}{2} E_0^2,
> $$
>
> with $E_0$ the magnitude of $\mathbf{E}_0$.
??? note "*Proof*:"
Will be added later.

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# 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 matrix 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.

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# Interference
> *Definition*: when waves are combined in phase they combine to give a larger amplitude constructive interference occurs. When waves are combined out of phase they tend to cancel, destructive interference occurs.
## Two source interference
For interference between two monochromatic electromagnetic waves given by
$$
\begin{align*}
\mathbf{E}_1(\mathbf{v}, t) = \mathbf{E}_{01} \exp i \big(\langle \mathbf{k_1}, \mathbf{v} - \mathbf{s}_1 \rangle - \omega_1 t + \varphi_1 \big), \\
\\
\mathbf{E}_2(\mathbf{v}, t) = \mathbf{E}_{02} \exp i \big(\langle \mathbf{k_2}, \mathbf{v} - \mathbf{s}_2 \rangle - \omega_2 t + \varphi_2 \big), \\
\end{align*}
$$
for all $(\mathbf{v}, t) \in U$ with $\mathbf{k}_{1,2} \in \mathbb{R}^3$ the wavenumber, $\mathbf{s}_{1,2} \in \mathbb{R}^3$ the position of the sources. Then we have the combined disturbance at $\mathbf{v}$ is given by
$$
\begin{align*}
\mathbf{E}(\mathbf{v}, t) &= \mathbf{E}_1(\mathbf{v}, t) + \mathbf{E}_2(\mathbf{v}, t), \\
&= \mathbf{E}_{01} \exp i \delta_1(\mathbf{v},t) + \mathbf{E}_{02} \exp i \delta_2(\mathbf{v},t),
\end{align*}
$$
for all $(\mathbf{v}, t) \in U$ with $\delta_i$ the phase difference for $i \in \{1,2\}$ given by
$$
\delta_i(\mathbf{v}, t) = \langle \mathbf{k_i}, \mathbf{v} - \mathbf{s}_i \rangle - \omega_i t + \varphi_i.
$$
> *Law*: the irradiance at point $\mathbf{v}$ is then given by
>
> $$
> I(\mathbf{v}, t) = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \Big(\delta_2(\mathbf{v}, t) - \delta_1(\mathbf{v}, t) \Big),
> $$
>
> for all $(\mathbf{v}, t) \in U$ with $I_{1,2} \in \mathbb{R}$ the irradiance for each wave seperately.
??? note "*Proof*:"
Will be added later.
Let $\delta(\mathbf{v}, t) = \delta_2(\mathbf{v}, t) - \delta_1(\mathbf{v}, t)$, then we have for $\delta(\mathbf{v}, t) = 2 m \pi$ with $m \in \mathbb{Z}$ constructive interference and for $\delta(\mathbf{v}, t) = (2m + 1) \pi$ we have destructive interference.
Writing out $\delta$ for plane waves of the same angular frequency $\omega = \omega_1 = \omega_2$ and propation in the $x$-direction gives
$$
\delta(x, t) = k(x_2 - x_1) + (\varphi_2 - \varphi_1) = \frac{2\pi}{\lambda_0} n (x_2 - x_1) + (\varphi_2 - \varphi_1),
$$
for all $(x,t) \in \mathbb{R}^2$ and $n \in \mathbb{R}$ the index of refraction of the medium. The optical path difference is defined as $n (x_2 - x_1)$.
### Double slit interference
Interference is created by plane waves illuminating both slits creating disturbances at both slits that are correlated in time. Assuming the slits are points sources and the waves have the same frequency, we have a superposition point $P$ described vertically with $y \in \mathbb{R}$ and $r_{1,2} \in \mathbb{R}$ the traveling distances from the slits to this point. Obtaining a phase difference
$$
\delta = k(r_2 - r_1) + (\varphi_2 - \varphi_1),
$$
??? note "*Proof*:"
Will be added later.
<br>
If we have $L \in \mathbb{R}$ the horizontal length between the slits and the point $P$ and $d \in \mathbb{R}$ the distance between the slits and assume $L >> d$ and $\varphi_2 - \varphi_1 = 0$ then
$$
\delta(\theta) = kd \sin \theta,
$$
for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $\tan \theta = \frac{y}{L}$.
??? note "*Proof*:"
Will be added later.
## Thin film interference
Interference is created by plane waves illuminating a thin film of thickness $l \in \mathbb{R}$ and index of refraction $n_l \in \mathbb{R}$ under an angle of incidence $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ deposited on a substrate with index of refraction $n_i \in \mathbb{R}$. A phase shift is introduced between the first external and internal reflected rays obtaining a phase difference $\delta$ given by
$$
\delta(\theta) = k 2l \sqrt{n_l^2 - n_i^2 \sin^2 \theta},
$$
for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $k \in \mathbb{R}$ the wavenumber.
??? note "*Proof*:"
Will be added later.
## Michelson interferometer
Interference created by splitting and recombining plane waves that have a difference in optical path. With a setup of two mirrors displaced with lengths $L_1, L_2 \in \mathbb{R}$ from the beam splitter under an angle $\theta$ with respect to the incoming plane wave. Assuming the setup is in *one* medium with index of refraction $n \in \mathbb{R}$. Obtaining a phase difference $\delta$ given by
$$
\delta(\theta) = k 2n(L_2 - L_1) \cos \theta + \pi,
$$
for all $\delta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $k \in \mathbb{R}$ the wavenumber.
??? note "*Proof*:"
Will be added later.
## Fabry-perot interferometer
Interference created by a difference in optical path length with a setup consisting of two parallel flat reflective surfaces seperated by a distance $d \in \mathbb{R}$ If both surfaces have reflection and transmission amplitude ratios $r,t \in [0,1]$ then the phase difference $\delta$ between two adjecent transmitted rays under an angle $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ is given by
$$
\delta(\theta) = 2 kd \cos \theta + 2 \varphi,
$$
for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $\varphi \in [0. 2\pi)$ the phase change due to reflection dependent on the amplitude ratios.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: The finesse $\mathcal{F}$ and the coefficient of finesse $F$ of a Fabry Perot interferometer are defined by
>
> $$
> F = \frac{4R}{(1-R)^2} \quad\text{ and }\quad \mathcal{F} = \frac{\pi \sqrt{F}}{2} = \frac{\pi \sqrt{R}}{1 - R},
> $$
>
> with $R \in [0,1]$ the reflectance. The finesse can be seen as the measure of sharpness of the interference pattern.
<br>
> *Proposition*: the transmitted irradiance $I$ of a Fabry Perot interferometer is given by
>
> $$
> I(\theta) = \frac{I_0}{1 + 4 (\mathcal{F} / \pi)^2 \sin^2 (\delta(\theta) / 2)}
> $$
>
> for all $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ with $I_0 \in \mathbb{R}$.
??? note "*Proof*:"
Will be added later.
### The chromatic resolving power and free spectral range
The chromatic resolving power and free spectral range are measures that define the ability to distinguish certain features in interference or diffraction patterns.
> *Definition*: The full width at half maximum $\text{FWHM}$ for the interference pattern of the Fabry Perot interferometer is defined to be
>
> $$
> \text{FWHM} = \frac{4}{\sqrt{F}},
> $$
>
> with $F \in \mathbb{R}$.
<br>
> *Definition*: the chromatic resolving power $\mathcal{R}$ is defined by
>
> $$
> \mathcal{R} = \frac{\lambda}{\Delta \lambda},
> $$
>
> with $\lambda \in \mathbb{R}$ the base wavelength of the light and $\Delta \lambda \in \mathbb{R}$ the spectral resolution at the wavelength $\lambda$.
<br>
> *Proposition*: the chromatic resolving power $\mathcal{R}$ of a Fabry Perot interferometer based on the $\text{FWHM}$ can be determined by
>
> $$
> \mathcal{R} = \mathcal{F} m,
> $$
>
> with $m \in \mathbb{Z}$ the order of the principal maxima and $\mathcal{F} \in \mathbb{R}$ the finesse.
??? note "*Proof*:"
Will be added later.
<br>
> *Definition*: the free spectral range $\text{FSR}$ is the largest wavelength range for a given order that does not overlap the same range in an adjacent order.
<br>
> *Proposition*: the free spectral range $\text{FSR}$ of a Fabry Perot interferometer can be determined by
>
> $$
> \text{FSR} = \frac{\lambda}{m},
> $$
>
> with $m \in \mathbb{Z}$ the order and $\lambda \in \mathbb{R}$ the wavelength.
??? note "*Proof*:"
Will be added later.

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# Polarisation
If we consider an electromagnetic wave $\mathbf{E}: \mathbb{R}^2 \to \mathbb{R}^3$ with wavenumber $k \in \mathbb{R}$ and angular frequency $\omega \in \mathbb{R}$ propagating in the positve $z$-direction given by
$$
\mathbf{E}(z,t) = \exp i(kz - \omega t + \varphi_1) E_0^{(x)} \mathbf{e}_{(x)} + \exp i(kz - \omega t + \varphi_2) E_0^{(y)}\mathbf{e}_{(y)},
$$
for all $(z,t) \in \mathbb{R}^2$ with $E_0^{(x)}, E_0^{(y)} \in \mathbb{R}$ the magnitude of the wave in the $x$ and $y$ direction. We define $\Delta \varphi = \varphi_2 - \varphi_1$.
> *Definition*: the electromagnetic wave $\mathbf{E}$ is linear polarised if and only if
>
> $$
> \Delta \varphi = \pi m,
> $$
>
> for all $m \in \mathbb{Z}$.
With polarisation angle $\theta \in [0, 2\pi)$ given by
$$
\theta = \arctan \Bigg( \frac{\max E_0^{(y)}}{\max E_0^{(x)}} \Bigg).
$$
??? note "*Proof*:"
Will be added later.
> *Definition*: the electromagnetic wave $\mathbf{E}$ is left circular polarised if and only if
>
> $$
> \Delta \varphi = \frac{\pi}{2} \;\land\; E_0^{(x)} = E_0^{(y)},
> $$
>
> and right circular polarised if and only if
>
> $$
> \Delta \varphi = -\frac{\pi}{2} \;\land\; E_0^{(x)} = E_0^{(y)}.
> $$
For every state in between we have elliptical polarisation with a polarisation angle $\theta \in [0, 2\pi)$ given by
$$
\theta = \frac{1}{2} \arctan \Bigg(\frac{2 E_0^{(x)} E_0^{(y)} \cos \Delta\varphi}{ \big(E_0^{(x)} \big)^2- \big( E_0^{(y)} \big)^2} \Bigg).
$$
??? note "*Proof*:"
Will be added later.
> *Definition*: natural light is defined as light constisting of all linear polarisation states.
## Linear polarisation
> *Definition*: a linear polariser selectively removes light that is linearly polarised along a direction perpendicular to its transmission axis.
We may concretisize this definition by the following statement, considered to be Malus law.
> *Law*: for a light beam with amplitude $E_0$ incident on a linear polariser the transmitted beam has amplitude $E_0 \cos \theta$ with $\theta \in [0, 2\pi)$ the polarisation angle of the light with respect to the transmission axis. The transmitted irradiance $I: [0, 2\pi) \to \mathbb{R}$ is then given by
>
> $$
> I(\theta) = I_0 \cos^2 \theta,
> $$
>
> for all $\theta \in [0, 2\pi)$ with $I_0 \in \mathbb{R}$ the irradiance of the incident light.
??? note "*Proof*:"
Will be added later.
For natural light the average of all angles must be taken, since $\lim_{\theta \to \infty} \cos^2 \theta = \frac{1}{2}$, we have the relation $I = \frac{1}{2} I_0$ for natural light.
## Birefringence
Natural light can be polarised in several ways, some are listed below.
1. Polarisation by absorption of the other component. This can be done with a wiregrid or dichroic materials for smaller wavelengths.
2. Polarisation by scattering. Dipole radiation has distinct polarisation depending on the position.
3. Polarisation by Brewster angle, which boils down to scattering.
4. Polarisation by birefringence, the double refraction of light obtaining two orthogonal components polarised.
??? note "*Proof*:"
Will be added later.
> *Definition*: birefringence is a double refraction in a material (often crystalline) and can be derived from the Fresnel equations without assuming isotropic dielectric properties.
If isotropic dielectric properties are not assumed it implies that the refractive index may also depend on the polarisation and propgation direction of light.
Using the properties of birefringence, wave plates (retarders) can be created. They may introduce a phase difference via a speed difference in the polarisation direction.
* A half-wave plate may introduce a $\Delta \varphi = \pi$ phase difference.
* A quarter-wave plate may introduce a $\Delta \varphi = \frac{\pi}{2}$ phase difference.
## Jones formalism of polarisation
Jones formalism of polarisation with vectors and matrices can make it easier to calculate the effects of optical elements such as linear polarizers and wave plates.
> *Definition*: for an electromagnetic wave $\mathbf{E}: \mathbb{R}^2 \to \mathbb{R}^3$ with wavenumber $k \in \mathbb{R}$ and angular frequency $\omega \in \mathbb{R}$ propagating in the positive $z$-direction given by
>
> $$
> \mathbf{E}(z,t) = \mathbf{E}_0 \exp i(kz - \omega t),
> $$
>
> for all $(z,t) \in \mathbb{R}^2$. The Jones vector $\mathbf{\tilde E}$ is defined as
>
> $$
> \mathbf{\tilde E} = \mathbf{E}_0,
> $$
>
> possibly normalized with $\|\mathbf{\tilde E}\| = 1$.
For linear polarised light under an angle $\theta \in [0, 2\pi)$ the Jones vector $\mathbf{\tilde E}$ is given by
$$
\mathbf{\tilde E} = \begin{pmatrix}\cos \theta \\ \sin \theta\end{pmatrix}.
$$
For left circular polarised light the Jones vector $\mathbf{\tilde E}$ is given by
$$
\mathbf{\tilde E} = \begin{pmatrix} 1 \\ i \end{pmatrix},
$$
and for right circular polarised light
$$
\mathbf{\tilde E} = \begin{pmatrix} 1 \\ -i \end{pmatrix}.
$$
> *Definition*: Jones matrices $M_i$ with $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ may be used to model several optical elements on an optical axis, obtaining the transmitted Jones vector $\mathbf{\tilde E}_t$ from the incident Jones vector $\mathbf{\tilde E}_i$ given by
>
> $$
> \mathbf{\tilde E}_t = M_n \cdots M_1 \mathbf{\tilde E}_i.
> $$
The Jones matrices for several optical elements are now given.
> *Proposition*: the Jones matrix $M$ of a linear polariser is given by
>
> $$
> M = \begin{pmatrix} \cos^2 \theta & \frac{1}{2} \sin 2\theta \\ \frac{1}{2} \sin 2\theta & \sin^2 \theta \end{pmatrix},
> $$
>
> with $\theta \in [0, 2\pi)$ the transmission axis of the linear polariser.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the Jones matrix $M$ of a half-wave plate is given by
>
> $$
> M = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix},
> $$
>
> with $\theta \in [0, 2\pi)$ the fast axis of the half-wave plate.
??? note "*Proof*:"
Will be added later.
<br>
> *Proposition*: the Jones matrix $M$ of a quarter-wave plate is given by
>
> $$
> M = \begin{pmatrix} \cos^2 \theta + \sin^2 \theta & (1 - i) \sin \theta \cos \theta \\ (1 - i) \sin \theta \cos \theta & i(\cos^2 \theta + \sin^2 \theta) \end{pmatrix},
> $$
>
> with $\theta \in [0, 2\pi)$ the fast axis of the quarter-wave plate.
??? note "*Proof*:"
Will be added later.
<br>

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# Reflection and refraction
> *Definition*: light rays are perpendicular to electromagnetic wave fronts.
Reflection and refraction occur whenever light rays enter into a new medium with index of refraction $n \in \mathbb{R}$. Reflection may be informally defined as the change of direction of the rays that stay within the initial medium. Refraction may be informally defined as the change of direction of the rays that transport to the other medium.
> *Law*: the law of reflection states that the angle of reflection of a light ray equals the angle of incidence.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the law of refraction states that the angle of refraction $\theta_t \in [0, 2\pi)$ is related to the angle of incidence $\theta_i \in [0, 2\pi)$ by
>
> $$
> n_i \sin \theta_i = n_t \sin \theta_t,
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium.
??? note "*Proof*:"
Will be added later.
## Fresnel equations
In this section the fractions of reflected and transmitted power for specific electromagnetic waves will be derived.
> *Lemma*: for the electric field perpendicular to the plane of incidence (s-polarisation) the Fresnel amplitude ratios for reflection $r_s \in [0,1]$ and transmission $t_s \in [0,1]$ are given by
>
> $$
> \begin{align*}
> r_s &= \frac{n_i \cos \theta_i - n_t \cos \theta_t}{n_i \cos \theta_i + n_t \cos \theta_t}, \\
> \\
> t_s &= \frac{2 n_i \cos \theta_i}{n_i \cos \theta_i + n_t \cos \theta_t},
> \end{align*}
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium and $\theta_{i,t} \in [0, 2\pi)$ the angle of incidence and refraction.
??? note "*Proof*:"
Will be added later.
<br>
> *Lemma*: for the electric field parallel to the plane of incidence (p-polarisation) the Fresnel amplitude ratios for reflection $r_p \in [0,1]$ and transmission $t_p \in [0,1]$ are given by
>
> $$
> \begin{align*}
> r_p &= \frac{n_i \cos \theta_t - n_t \cos \theta_i}{n_i \cos \theta_t + n_t \cos \theta_i}, \\
> \\
> t_p &= \frac{2 n_i \cos \theta_i}{n_i \cos \theta_t + n_t \cos \theta_i},
> \end{align*}
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium and $\theta_{i,t} \in [0, 2\pi)$ the angle of incidence and refraction.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the fraction of the incident power that is reflected is called the reflectivity $R \in [0,1]$ and is given by
>
> $$
> R = r^2,
> $$
>
> with $r \in [0, 1]$ the Fresnel amplitude ratio for reflection.
??? note "*Proof*:"
Will be added later.
<br>
> *Law*: the fraction of the incident power that is transmitted is called the transmissivity $T \in [0,1]$ and is given by
>
> $$
> T = \bigg(\frac{n_t \cos \theta_t}{n_i \cos \theta_i}\bigg) t^2
> $$
>
> with $t \in [0, 1]$ the Fresnel amplitude ratio for transmission, $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium and $\theta_{i,t} \in [0, 2\pi)$ the angle of incidence and refraction.
??? note "*Proof*:"
Will be added later.
<br>
## Limiting cases
> *Corollary*: we have $r_p = 0$ for an incident angle given by
>
> $$
> \tan \theta_b = \frac{n_t}{n_i},
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. The angle $\theta_b$ is called the Brewster angle.
??? note "*Proof*:"
Will be added later.
Therefore we have for the Brewster angle the reflectivity equal to zero for p-polarisation. Such relation does not exist for s-polarisation.
> *Corollary*: we have $r_s = 1$ or total reflection for $n_i > n_t$ and an incident angle given by
>
> $$
> \sin \theta_i > \frac{n_t}{n_i},
> $$
>
> with $n_{i,t} \in \mathbb{R}$ the index of refraction of the incident and transmitted medium. With
>
> $$
> \sin \theta_c = \frac{n_t}{n_i},
> $$
>
> the critical angle.
??? note "*Proof*:"
Will be added later.
## Phase changes on reflection
> *Proposition*: a reflected light ray may obtain a phase shift if
>
> 1. for all incident angles and $n_i < n_t$ the reflected light ray is phase shifted by $\pi$,
> 2. for incident angles $\theta_i > \theta_c$ and $n_i > n_t$ the reflected light ray is not phase shifted,
> 3. the transmitted light ray is not phase shifted.
??? note "*Proof*:"
Will be added later.
For incident angles $\theta_i < \theta_c$ and $n_i > n_t$ the phase shifts are complex.
## Dispersion
Will be added later.
## Scattering
Will be added later.

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# Waves
> *Definition*: a wave is a propagating disturbance transporting energy and momentum. A $1 + 1$ dimensional wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ travelling can be defined by a linear combination of a right and left travelling function $f,g: \mathbb{R} \to \mathbb{R}$ obtaining
>
> $$
> \Psi(x,t) = f(x - vt) + g(x + vt),
> $$
>
> for all $(x,t) \in \mathbb{R}^2$ and $v \in \mathbb{R}$ the speed of the wave. Satisfies the $1 + 1$ dimensional wave equation
>
> $$
> \partial_x^2 \Psi(x,t) = \frac{1}{v^2} \partial_t^2 \Psi(x,t).
> $$
The derivation of the wave equation can be obtained in section...
> *Theorem*: a right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ with $\lambda, T, A, \varphi \in \mathbb{R}$ the wavelength, period, amplitude and phase of the wave is given by
>
> $$
> \begin{align*}
> \Psi(x,t) &= A \sin \big(k(x-vt) + \varphi\big), \\
> &= A \sin(kx-\omega t + \varphi), \\
> &= A \sin \Big(2\pi \Big(\frac{x}{\lambda} - \frac{t}{T} \Big) + \varphi \Big),
> \end{align*}
> $$
>
> for all $(x,t) \in \mathbb{R}^2$. With $k = \frac{2\pi}{\lambda}$ the wavenumber, $\omega = \frac{2\pi}{T}$ the angular frequency and $v = \frac{\lambda}{T}$ the wave speed.
??? note "*Proof*:"
Will be added later.
A right travelling harmonic wave $\Psi: \mathbb{R}^2 \to \mathbb{R}$ can also be represented in the complex plane given by
$$
\Psi(x,t) = \text{Im} \big(A \exp i(kx - \omega t + \varphi )\big),
$$
for all $(x,t) \in \mathbb{R}^2$.
> *Theorem*: let $\Psi: \mathbb{R}^4 \to \mathbb{R}$ be a $3 + 1$ dimensional wave then it satisfies the $3 + 1$ dimensional wave equation given by
>
> $$
> \nabla^2 \Psi(\mathbf{x},t) = \frac{1}{v^2} \partial_t^2 \Psi(\mathbf{x},t),
> $$
>
> for all $(\mathbf{x},t) \in \mathbb{R}^4$.
??? note "*Proof*:"
Will be added later.
We may formulate various solutions $\Psi: \mathbb{R}^4 \to \mathbb{R}$ for this wave equation.
The first solution may be the plane wave that follows cartesian symmetry and can therefore best be described in a cartesian coordinate system $\mathbf{v}(x,y,z)$. The solution is given by
$$
\Psi(\mathbf{v}, t) = \text{Im}\big(A \exp i(\langle \mathbf{k}, \mathbf{v} \rangle - \omega t + \varphi) \big),
$$
for all $(\mathbf{v}, t) \in \mathbb{R}^4$ with $\mathbf{k} \in \mathbb{R}^3$ the wavevector.
The second solution may be the cylindrical wave that follows cylindrical symmetry and can therefore best be described in a cylindrical coordinate system $\mathbf{v}(r,\theta,z)$. The solution is given by
$$
\Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\sqrt{\|\mathbf{v}\|}} \exp i(k \|\mathbf{v} \| - \omega t + \varphi) \Bigg),
$$
for all $(\mathbf{v}, t) \in \mathbb{R}^4$.
The third solution may be the spherical wave that follows spherical symmetry and can therefore best be described in a spherical coordinate system $\mathbf{v}(r, \theta, \varphi)$. The solution is given by
$$
\Psi(\mathbf{v}, t) = \text{Im}\Bigg(\frac{A}{\|\mathbf{v}\|} \exp i(k\|\mathbf{v}\| - \omega t + \varphi) \Bigg)
$$
for all $(\mathbf{v}, t) \in \mathbb{R}^4$.
> *Principle*: the principle of superposition is valid for waves, since the solution space of the wave equation is linear.
From this principle we obtain the property of constructive and destructive interference of waves.