Ray transfer matrix analysis

The Matrizenoptik is a computational method in paraxial optics, in which the variation of light beams through optical devices with the help of matrices is presented. This is called a ( Ray ) transfer matrices or even any of their four entries, ABCD matrices.

• 5.1 Gaussian beams
• 5.2 polarization

Basics

Looking at the propagation of light along the optical axis, defined herein as the z- axis. The state of a light beam at a point (that is, for a given z) can be described by two values: its distance r from the optical axis and the angle it forms with it. One can therefore represent the beam with a z- dependent vector of these two components:

The angle of doing this, since it is the slope of the beam, the change from r to z. Within the paraxial approximation, that is, with only small distances r from the optical axis and the small inclination, is considered. Thus, there is between two state vectors, the same beam at different z- values ​​is a linear relationship which can be described by a matrix. Is multiplied to the time determined by the characteristics of an optical element in the transformation matrix of the first vector and is replaced by the second, so the properties of the beam after passing through the element:

The usual convention is that the beam direction (the positive Z -axis) extending from left to right. r is above the axis positive, negative counted below. is positive when the beam is pointing upwards, and negative when the beam is pointing downwards.

Transfer matrices of important elements

Translation

A light beam propagates freely across the distance d along the optical axis, this is described by the following matrix of the optical path:

The translation matrix is independent of the medium through which the beam propagates. It follows the vector:

So a simply propagating beam changes its inclination to the axis not, but only according to its initial inclination, his distance from her.

Refraction at the lens surface

A light beam is refracted at a surface, that is the transfer matrix according to the law of refraction

In this case, and the refractive indices of the optical media before and after the boundary surface, and the curvature of the surface at its apex ( center of area ). is positive when the center of curvature lies behind the surface ( convex surface as viewed in the positive direction). On a spherical surface of radius, and in the case of a flat surface.

Thin lens

From the lens grinder formula or by multiplication of two surfaces - Brechungsmatrizen obtained for the passage through a thin lens

Wherein the focal length of the lens. is greater than 0, if the lens has a focusing effect ( converging lens ), and less than 0 for a defocusing lens ( negative lens ).

Mirror

For a mirror of the vertex curvature is obtained using the law of reflection, the matrix

Which describes a plane mirror. is positive for a concave mirror, and negative for a convex mirror. For a spherical mirror is the radius.

Combination of elements

Going through a beam multiple optical elements in succession, one after the other as the corresponding transfer matrices are applied to the ray vector, which is equivalent to this is to multiply them and then apply the product matrix on the vector. The rules of matrix multiplication apply: through which the beam passes three items in the order, the product is made in the order.

Thus, the matrices sophisticated systems produce a product of matrices of elementary parts of the system, such as a thick lens from which a lens surface, a translation of the lens surface and another surface, or a lens system of a series of lens, translation, lens. or surface .., translation, surface ....

Alternative Convention

Some authors defined differently for the convention used here, the ray vector as, where n is the refractive index of the medium is in place. This has the consequence that must be corrected for translation through a medium for this additional n about in the matrix, it is in the Convention and is thus itself explicitly depending on the medium. The advantage of this convention is that the matrix for refraction at a plane surface to the unit matrix.

Some authors also swap the two entries of the ray vector, so that it is defined as follows:

The matrices change accordingly.

Other applications

Gaussian beams

The application of Matrizenoptik is not limited to geometrical optics, it can be transmitted through the transition matrices to Möbius pictures on the concept of Gaussian beams. To this end, the ABCD matrices and their multiplication rules are properly preserved, but they are no longer used by multiplying a vector of blasting, but on the beam parameter q according to the following rule:

The beam parameter is calculated in this case by R the radius of curvature of the Gaussian beam, the wavelength and w is the radius of the Gaussian beam.

Polarization

An analog to Matrizenoptik geometrical method is used to calculate the change in polarization when passing through optical elements. The polarization state is expressed by Jones vectors and manipulated with Jones matrices.

Technical Terms

In addition to the mathematical application of the method with example Programs such as MATLAB to calculate ray paths are adaptations of the same used to in order to anticipate beam paths of moving lens system and predict expected pictures such as in the real-time object tracking or adjustment of associated lens systems for focusing, as astronomical mirrors.