From bce634b7a308a7c8d9ad325668a58efea45b65b7 Mon Sep 17 00:00:00 2001 From: luc Date: Sun, 21 Jan 2024 00:44:32 +0100 Subject: [PATCH] Added section geometric optics to optics. --- .../optics/geometric-optics.md | 197 +++++++++++++++++- 1 file changed, 196 insertions(+), 1 deletion(-) diff --git a/docs/en/physics/electromagnetism/optics/geometric-optics.md b/docs/en/physics/electromagnetism/optics/geometric-optics.md index 861c64c..a6fed4e 100644 --- a/docs/en/physics/electromagnetism/optics/geometric-optics.md +++ b/docs/en/physics/electromagnetism/optics/geometric-optics.md @@ -1 +1,196 @@ -# 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. + +
+ +> *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. + +
+ +> *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. + +
+ +## 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. + +
+ +> *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. + +
+ +> *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. + +
+ +> *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. + +
+ +> *Definition*: a lens is defined by two intersecting spherical interfaces with radius $R_1, R_2 \in \mathbb{R}$ respectively. + +
+ +> *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. + +
+ +> *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. + +
+ +> *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.