mathematics-physics-wiki/docs/en/physics/electromagnetism/optics/geometric-optics.md

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 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.