Die optics

The ray transfer matrix analysis is a computing method in paraxial optics , in which the variation of light beam by optical components with the aid of matrices is shown. These are called (beam) transfer matrices or, after their four entries, ABCD matrices .

Basics

Illustration of r, z, ${\displaystyle \alpha }$

One looks at the propagation of light along the optical axis , here defined as the -axis. The state of a ray of light at a point (i.e. at a certain point ) can be described by two values: its distance from the optical axis and the angle it forms with it. The ray can therefore be represented as a vector from these two components: ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle r}$ ${\displaystyle \alpha }$

${\displaystyle {\vec {r}}={\begin{pmatrix}r\\\alpha \end{pmatrix}}}$

Since it represents the inclination of the beam, the angle indicates the change in with . In the context of the paraxial approximation , i.e. after the limit crossing with which and approach zero, the following applies . ${\displaystyle \alpha }$ ${\displaystyle r}$ ${\displaystyle z}$ ${\displaystyle r}$ ${\displaystyle \alpha }$ ${\displaystyle \sin \alpha =\tan \alpha =\alpha }$

If one considers and not as infinitesimal , but finite quantities (in the sense of Gaussian optics ), one must understand the second vector component as the tangent of the angle between the ray and the axis, i.e. as the gradient of the ray, so that there is a linear relationship between and the change in with . ${\displaystyle r}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle r}$ ${\displaystyle z}$

If a ray travels a path in the direction and possibly also passes through imaging elements (lenses, mirrors), the change in the ray vector can be described with a transformation matrix, which is based on the difference between the coordinates and the properties of the elements passed through. The transformation matrix is ​​multiplied from the left to the ray vector, and the resulting vector describes the properties of the ray after it has traversed the path: ${\displaystyle z}$ ${\displaystyle z}$

${\displaystyle {\begin{bmatrix}A&B\\C&D\end{bmatrix}}{\begin{pmatrix}r_{1}\\\alpha _{1}\end{pmatrix}}={\begin{pmatrix}r_{2}\\\alpha _{2}\end{pmatrix}}}$

The usual convention is that the beam direction (i.e. the positive axis) runs from left to right. r is counted positive above the axis and negative below. is positive when the ray is pointing up and negative when it is pointing down. ${\displaystyle z}$ ${\displaystyle \alpha }$

Transfer matrices of important elements

Translation

If a light beam spreads unhindered over the distance along the optical axis without passing through imaging elements, this is described with the following matrix of the optical path, which only depends on the distance and not on the medium passed through: ${\displaystyle d=z_{2}-z_{1}}$

${\displaystyle T={\begin{bmatrix}1&d\\0&1\end{bmatrix}};\quad {\begin{pmatrix}r_{2}\\\alpha _{2}\end{pmatrix}}=T{\begin{pmatrix}r_{1}\\\alpha _{1}\end{pmatrix}}={\begin{pmatrix}r_{1}+d\alpha _{1}\\\alpha _{1}\end{pmatrix}}}$

A beam that simply spreads out does not change its inclination to the axis, but only its distance from it according to its inclination.

Refraction at a plane interface

If a light beam hits a flat boundary surface, the transfer matrix results

${\displaystyle B=\left[{\begin{matrix}1&0\\0&{\frac {{n}_{1}}{{n}_{2}}}\\\end{matrix}}\right]}$

When passing through the interface, the distance to the optical axis does not change, but the angle changes due to the law of refraction . Since the matrix description only applies to the paraxial approximation ( ), the law of refraction is simplified to ${\displaystyle \sin(\alpha )\approx \alpha }$ ${\displaystyle {{n}_{1}}{{\alpha }_{1}}={{n}_{2}}{{\alpha }_{2}}}$

Refraction at the lens surface

If a light beam is refracted on a spherically curved surface, whereby only the direction of the beam and not the coordinate changes, the transfer matrix reads according to the law of refraction ${\displaystyle z}$

${\displaystyle R={\begin{bmatrix}1&0\\\left({\frac {n_{1}}{n_{2}}}-1\right)\cdot \rho &{\frac {n_{1}}{n_{2}}}\end{bmatrix}}}$.

Here and are the refractive indices of the optical media before and after the interface, and is the curvature of the surface at its apex (center of the surface). is positive if the center of curvature lies behind the surface ( convex surface , seen in the positive direction). In a spherical surface with radius is , and is the case of a flat surface (see refraction at a flat interface). ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle \rho }$ ${\displaystyle \rho }$ ${\displaystyle z}$ ${\displaystyle r}$ ${\displaystyle \rho =1/r}$ ${\displaystyle \rho =0}$

Thin lens

By multiplying two surface refraction matrices and using the lens grinding formula , the transfer matrix is obtained for the passage through a thin lens

${\displaystyle L={\begin{bmatrix}1&0\\{\frac {1}{-f}}&1\end{bmatrix}}}$,

where is 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 (diverging lens). ${\displaystyle {\mathit {f}}}$ ${\displaystyle {\mathit {f}}}$

Thick lens

If the thickness of the lens between the lens surfaces with the radii of curvature and is taken into account, the transfer matrix is obtained for the passage through a thick lens ${\displaystyle {\mathit {d}}}$ ${\displaystyle {\mathit {{R}_{1}}}}$ ${\displaystyle {\mathit {{R}_{2}}}}$

${\displaystyle D=\left[{\begin{matrix}1-{\frac {d}{{R}_{1}}}{\frac {{{n}_{L}}-n}{{n}_{L}}}&{\frac {d\,n}{{n}_{L}}}\\({{n}_{L}}-n)\left({\frac {1}{{R}_{2}}}-{\frac {1}{{R}_{1}}}-{\frac {d}{{{R}_{1}}{{R}_{2}}}}{\frac {{{n}_{L}}-n}{{n}_{L}}}\right)&1+{\frac {d}{{R}_{2}}}{\frac {{{n}_{L}}-n}{{n}_{L}}}\\\end{matrix}}\right]=\left[{\begin{matrix}1&{{H}_{2}}\\0&1\\\end{matrix}}\right]\left[{\begin{matrix}1&0\\{\frac {1}{-f}}&1\\\end{matrix}}\right]\left[{\begin{matrix}1&{{H}_{1}}\\0&1\\\end{matrix}}\right]}$,

wherein are the index of refraction of the lens material, the index of refraction of the surrounding medium, and the principal planes of the lens and the focal length measured from the principal planes. ${\displaystyle {\mathit {n}}_{L}}$ ${\displaystyle {\mathit {n}}}$ ${\displaystyle {\mathit {{H}_{1}}}}$ ${\displaystyle {\mathit {{H}_{2}}}}$ ${\displaystyle {\mathit {f}}}$

mirror

For a mirror with the curvature of the apex , the law of reflection gives the matrix ${\displaystyle \rho }$

${\displaystyle K={\begin{bmatrix}1&0\\-2\rho &1\end{bmatrix}}}$,

where describes a plane mirror. is positive for a concave mirror and negative for a convex mirror. For a spherical mirror is the radius . It is important to note the convention that the optical axis corresponds to the general direction of propagation of the light, i.e. reverses its direction on the mirror. ${\displaystyle \rho =0}$ ${\displaystyle \rho }$ ${\displaystyle r=1/\rho }$

Main levels

The equivalent focal length of a thin lens and the main planes of the associated optical system can be determined from a transfer matrix

${\displaystyle \left[{\begin{matrix}A&B\\C&D\\\end{matrix}}\right]\,=\left[{\begin{matrix}1&{{H}_{2}}\\0&1\\\end{matrix}}\right]\ \,\left[{\begin{matrix}1&0\\-{{f}^{-1}}&1\\\end{matrix}}\right]\ \left[{\begin{matrix}1&{{H}_{1}}\\0&1\\\end{matrix}}\right]=\left[{\begin{matrix}1-{\frac {{H}_{2}}{f}}&{{H}_{1}}+{{H}_{2}}-{\frac {{{H}_{1}}{{H}_{2}}}{f}}\\-{\frac {1}{f}}&1-{\frac {{H}_{1}}{f}}\\\end{matrix}}\right]}$

${\displaystyle f=-{\frac {1}{C}}}$

${\displaystyle {{H}_{1}}={\frac {D-1}{C}}}$

${\displaystyle {{H}_{2}}={\frac {A-1}{C}}}$

Thus it becomes possible to disengage an optical system with several lenses by only one equivalent focal length .

Combination of elements

If a beam passes through several optical elements one after the other, the corresponding transfer matrices are successively applied to the beam vector, which is equivalent to multiplying them and then applying the product matrix to the vector. The rules of matrix multiplication apply : if the beam passes through three elements in the sequence , the product is formed in the sequence . ${\displaystyle T_{1},T_{2},T_{3}}$ ${\displaystyle T_{3}\cdot T_{2}\cdot T_{1}}$

The matrices of more complicated systems result as the product of the matrices of the elementary system parts, for example that of a thick lens from those of a lens surface, a translation through the lens glass and another surface, or that of a lens system from a sequence of lens, translation, lens,. .. or surface, translation, area, ... .

Alternative convention

In deviation from the convention used here, some authors define the ray vector as , where n is the refractive index of the medium at the location . As a result, for example, in the matrix for translation through a medium, this additional n has to be corrected; it reads in this convention and is therefore itself explicitly dependent on the medium. The advantage of this convention is that the matrix for refraction on a flat surface becomes the unit matrix . ${\displaystyle {\vec {r}}(z)={\begin{pmatrix}r(z)\\n\alpha (z)\end{pmatrix}}}$ ${\displaystyle (r,z)}$ ${\displaystyle T={\begin{bmatrix}1&{\frac {d}{n}}\\0&1\end{bmatrix}}}$

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

${\displaystyle {\vec {r}}(z)={\begin{pmatrix}n\alpha (z)\\r(z)\end{pmatrix}}}$.

The matrices change accordingly.

Other uses

Gaussian rays

The application of matrix optics is not limited to geometrical optics, it can also be transferred to the concept of Gaussian rays through the transition from matrices to Möbius images . For this purpose, the ABCD matrices and their multiplication rules are completely retained, but they are no longer applied by multiplication to a ray vector, but to the ray parameter according to the following rule: ${\displaystyle q}$

${\displaystyle q_{1}(z)={\frac {Aq_{0}+B}{Cq_{0}+D}}}$.

The beam parameter is calculated using the radius of curvature of the Gaussian beam, the wavelength and the radius of the Gaussian beam (alternative ). ${\displaystyle {1 \over q}={1 \over R}-{i\lambda \over \pi w^{2}}}$ ${\displaystyle R}$ ${\displaystyle \lambda }$ ${\displaystyle w}$ ${\displaystyle q(z)=z+iz_{0}}$

polarization

A method analogous to geometric matrix optics is used to calculate the change in polarization when passing through optical elements. The state of polarization is expressed by Jones vectors and manipulated with Jones matrices.

Technical use

In addition to the mathematical application of the method with z. B. Programs such as MATLAB for calculating beam paths, adaptations of the same are used to anticipate beam paths of moving lens systems and to calculate expected images in advance, such as B. in real-time object tracking or the adjustment of connected lens systems for focusing, such as astronomical mirrors.

literature

• D. Meschede: Optics, light and laser. BG Teubner, Stuttgart / Leipzig 2005, ISBN 3-519-13248-6 .
• F. Pedrotti, L. Pedrotti, Werner Bausch, Hartmut Schmidt: Optics . Prentice Hall, Munich et al. 1996, ISBN 3-8272-9510-6 .

Web links

• Harm Fesefeldt: Fundamentals of paraxial matrix optics. RWTH Aachen.

Individual evidence

1. ↑ E. Hecht: Optics. 4th edition. Oldenbourg, Munich 2005, ISBN 3-486-27359-0 .
2. ↑ W. & U. Zinth: Optics - light rays - waves - photons. 2nd Edition. Oldenbourg, Munich, 2009, ISBN 978-3-486-58801-9 .