Matrix optics

Matrix optics is a mathematical apparatus for calculating optical systems of varying complexity.

Content

Principle

Let the direction of propagation of the light beam in front of the optical system be known. Let be $y_{1}$ ${\ displaystyle y_ {1}}$ - "height" of the beam above the main optical axis of the system, $v_{1}$ ${\ displaystyle v_ {1}}$ - reduced angle: $v_{1}=n\times \alpha$ ${\ displaystyle v_ {1} = n \ times \ alpha}$ where $\alpha$ ${\ displaystyle \ alpha}$ Is the angle between the direction of beam propagation and the main optical axis of the system, n is the refractive index of the medium at a given point. Then, the corresponding coordinates of the beam after passing through the optical system are connected with the original matrix equation:
${\begin{bmatrix}y_{2}\\v_{2}\end{bmatrix}}={\begin{bmatrix}A&B\\C&D\end{bmatrix}}\times {\begin{bmatrix}y_{1}\\v_{1}\end{bmatrix}}$ ${\ displaystyle {\ begin {bmatrix} y_ {2} \\ v_ {2} \ end {bmatrix}} = {\ begin {bmatrix} A&B \\ C&D \ end {bmatrix}} \ times {\ begin {bmatrix} y_ {1} \\ v_ {1} \ end {bmatrix}}}$ ,

Where ${\begin{bmatrix}A&B\\C&D\end{bmatrix}}$ ${\ displaystyle {\ begin {bmatrix} A&B \\ C&D \ end {bmatrix}}}$ - the matrix of the optical system, also referred to as the beam transfer matrix .

The determinant of the matrix of the optical system is equal to the ratio of the refractive indices at the input and output of the system, usually this ratio is 1. The matrix transformation is an approximate linear description of the system. It works, in particular, when the paraxial approximation is performed.

Matrices of the simplest optical systems

Spherical refractive surface

$M={\begin{bmatrix}1&0\\-\Phi _{1}&1\end{bmatrix}}$ ${\ displaystyle M = {\ begin {bmatrix} 1 & 0 \\ - \ Phi _ {1} & 1 \ end {bmatrix}}}$ , $\Phi _{1}={\frac {n_{2}-n_{1}}{n_{2}*R}}$ ${\ displaystyle \ Phi _ {1} = {\ frac {n_ {2} -n_ {1}} {n_ {2} * R}}}$ where $n_{1}$ ${\ displaystyle n_ {1}}$ and $n_{2}$ ${\ displaystyle n_ {2}}$ - refractive indices of the medium (It is implied that the beam passes from the medium with $n_{1}$ ${\ displaystyle n_ {1}}$ on Wednesday with $n_{2}$ ${\ displaystyle n_ {2}}$ ), R is the algebraic radius of curvature of a spherical surface (R> 0 for a convex surface when the incident ray and radius vector are directed to the center of curvature of the surface, and R <0 for a concave surface).

Spherical Mirror

$M={\begin{bmatrix}1&0\\-\Phi _{2}&1\end{bmatrix}}$ ${\ displaystyle M = {\ begin {bmatrix} 1 & 0 \\ - \ Phi _ {2} & 1 \ end {bmatrix}}}$ , $\Phi _{2}=-{\frac {2\cdot n}{R}}$ ${\ displaystyle \ Phi _ {2} = - {\ frac {2 \ cdot n} {R}}}$ where $n$ ${\ displaystyle n}$ Is the refractive index of the medium, R is the algebraic radius of curvature (see above).

Broadcast

Translation is the direct beam propagation between refractions / reflections, for example, between two lenses.
$M={\begin{bmatrix}1&T\\0&1\end{bmatrix}}$ ${\ displaystyle M = {\ begin {bmatrix} 1 & T \\ 0 & 1 \ end {bmatrix}}}$ , $T={\frac {d}{n}}$ ${\ displaystyle T = {\ frac {d} {n}}}$ , d is the translation length, n is the refractive index.

Application of the method

The final matrix of the optical system is the product of the matrices of the individual elementary elements, moreover, in the order opposite to the order of these elements, i.e. $M=M_{n}\times \cdot \cdot \cdot \times M_{2}\cdot M_{1}$ ${\ displaystyle M = M_ {n} \ times \ cdot \ cdot \ cdot \ times M_ {2} \ cdot M_ {1}}$ where $M_{i}$ ${\ displaystyle M_ {i}}$ - matrix of the i-th optical element, counting from the position of the beam incident on the system.
The optical power of the optical system:
$\Phi =-C$ ${\ displaystyle \ Phi = -C}$
$B=0,y_{2}=A\cdot y_{1}$ ${\ displaystyle B = 0, y_ {2} = A \ cdot y_ {1}}$ - the general condition for image formation at a given point. In this case, A is an increase in the system.

Calculation of the optical power of a thick lens by the matrix method

Let a lens with radii of curvature $R_{1},R_{2}$ ${\ displaystyle R_ {1}, R_ {2}}$ (biconvex for definiteness), of thickness d, from a material with a refractive index n is in air. Then the optical system consists of three simplest elements - two refractive surfaces and translation inside the lens. We have:

$M_{1}={\begin{bmatrix}1&0\\-\Phi _{1}&1\end{bmatrix}}$ ${\ displaystyle M_ {1} = {\ begin {bmatrix} 1 & 0 \\ - \ Phi _ {1} & 1 \ end {bmatrix}}}$
$M_{2}={\begin{bmatrix}1&T\\0&1\end{bmatrix}}$ ${\ displaystyle M_ {2} = {\ begin {bmatrix} 1 & T \\ 0 & 1 \ end {bmatrix}}}$
$M_{3}={\begin{bmatrix}1&0\\-\Phi _{2}&1\end{bmatrix}}$ ${\ displaystyle M_ {3} = {\ begin {bmatrix} 1 & 0 \\ - \ Phi _ {2} & 1 \ end {bmatrix}}}$
The matrix of the entire optical system:
$M=M_{3}\cdot M_{2}\cdot M_{1}={\begin{bmatrix}1&0\\-\Phi _{2}&1\end{bmatrix}}\times {\begin{bmatrix}1&T\\0&1\end{bmatrix}}\times {\begin{bmatrix}1&0\\-\Phi _{1}&1\end{bmatrix}}={\begin{bmatrix}1-T\Phi _{1}&T\\T\Phi _{1}\Phi _{2}-\Phi _{1}-\Phi _{2}&1-T\Phi _{2}\end{bmatrix}}$ ${\ displaystyle M = M_ {3} \ cdot M_ {2} \ cdot M_ {1} = {\ begin {bmatrix} 1 & 0 \\ - \ Phi _ {2} & 1 \ end {bmatrix}} \ times {\ begin {bmatrix} 1 & T \\ 0 & 1 \ end {bmatrix}} \ times {\ begin {bmatrix} 1 & 0 \\ - \ Phi _ {1} & 1 \ end {bmatrix}} = {\ begin {bmatrix} 1-T \ Phi _ {1} & T \\ T \ Phi _ {1} \ Phi _ {2} - \ Phi _ {1} - \ Phi _ {2} & 1-T \ Phi _ {2} \ end {bmatrix}}}$
Hence the optical power of a thick lens:
$\Phi =-C=\Phi _{1}+\Phi _{2}-{\frac {d\Phi _{1}\Phi _{2}}{n}}$ ${\ displaystyle \ Phi = -C = \ Phi _ {1} + \ Phi _ {2} - {\ frac {d \ Phi _ {1} \ Phi _ {2}} {n}}}$
For a thin lens, the third term can be neglected:
$\Phi =-C=\Phi _{1}+\Phi _{2}$ ${\ displaystyle \ Phi = -C = \ Phi _ {1} + \ Phi _ {2}}$
Taking into account $\Phi _{1}={\frac {n-1}{R_{1}}},\Phi _{2}={\frac {n-1}{R_{2}}}$ ${\ displaystyle \ Phi _ {1} = {\ frac {n-1} {R_ {1}}}, \ Phi _ {2} = {\ frac {n-1} {R_ {2}}}}$
, we obtain the well-known formula for the optical power of the lens: $\Phi =(n-1)\cdot ({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}})$ ${\ displaystyle \ Phi = (n-1) \ cdot ({\ frac {1} {R_ {1}}} + {\ frac {1} {R_ {2}}}}$ .

Literature

Gerrard A., Birch J.M. Introduction to matrix optics. M. World 1978 341s
Saleh B.E.A., Teikh M.K. Optics and photonics. Principles and applications. Per. from English: Textbook. In 2 t. Dolgoprudny: Intellect, 2012 .-- 1544 s. - Section 1.4, pages 50-68.