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.
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
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.
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.
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 \(s_{o,i} \in \mathbb{R}\).
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 \(\theta_{1,2} \in [0, 2\pi)\).
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 \(\theta_{1,2} \in [0, 2\pi)\).
Proof:
Will be added later.
This matrix may also be given in terms of the focal length \(f \in \mathbb{R}\) by
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.
Proof:
Will be added later.
This matrix may also be given in terms of the focal length \(f \in \mathbb{R}\) by
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.
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.