Editing Ray transfer matrix analysis (section)

== Matrices for simple optical components ==
{| border="1" cellspacing="0" cellpadding="4"
|- style="background-color: #AAFFCC"
! Element
! Matrix
! Remarks
|- 
| Propagation in free space or in a medium of constant refractive index
| align="center" |<math>\begin{pmatrix} 1 & d\\ 0 & 1 \end{pmatrix} </math>
| {{mvar|d}} = distance<br/>
|- 
| Refraction at a flat interface
| align="center" | <math>\begin{pmatrix} 1 & 0 \\ 0 & \frac{n_1}{n_2} \end{pmatrix} </math>
| {{math|''n''<sub>1</sub>}} = initial refractive index<br/>
{{math|''n''<sub>2</sub>}} = final refractive index.
|- 
| Refraction at a curved interface
| align="center" | <math>\begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R \cdot n_2} & \frac{n_1}{n_2} \end{pmatrix} </math>
| {{mvar|R}} = radius of curvature, {{math|''R'' > 0}} for convex (center of curvature after interface)<br/>
{{math|''n''<sub>1</sub>}} = initial refractive index<br/>{{math|''n''<sub>2</sub>}} = final refractive index.
|- 
| Reflection from a flat mirror
| align="center" | <math> \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} </math>{{sfnp|Hecht|2002}}
| Valid for flat mirrors oriented at any angle to the incoming beam. Both the ray and the optic axis are reflected equally, so there is no net change in slope or position.
|- 
| Reflection from a curved mirror
| align="center" | <math> \begin{pmatrix} 1 & 0 \\ -\frac{2}{R_e} & 1 \end{pmatrix} </math>
| <math>R_e = R\cos\theta</math> effective radius of curvature in tangential plane  (horizontal direction) <br/>
<math>R_e = R/\cos\theta</math> effective radius of curvature in the sagittal plane (vertical direction)<br/>
{{mvar|R}} = radius of curvature, {{math|''R'' > 0}} for concave, valid in the paraxial approximation<br/>
{{mvar|θ}} is the mirror angle of incidence in the horizontal plane.
|- 
| Thin lens
| align="center" | <math> \begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix} </math>
| {{mvar|f}} = focal length of lens where {{math|''f'' > 0}} for convex/positive (converging) lens. 
Only valid if the focal length is much greater than the thickness of the lens.
|-
| Thick lens
| align="center" | <math>\begin{pmatrix} 1 & 0 \\ \frac{n_2-n_1}{R_2n_1} & \frac{n_2}{n_1} \end{pmatrix} \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R_1n_2} & \frac{n_1}{n_2} \end{pmatrix}</math>
| {{math|''n''<sub>1</sub>}} = refractive index outside of the lens. <br/>
{{math|''n''<sub>2</sub>}} = refractive index of the lens itself (inside the lens). <br/>
{{math|''R''<sub>1</sub>}} = Radius of curvature of First surface. <br/>
{{math|''R''<sub>2</sub>}} = Radius of curvature of Second surface.<br/>
{{mvar|t}} = center thickness of lens.
|- 
| Single prism
| align="center" | <math> \begin{pmatrix} k & \frac{d}{nk} \\ 0 & \frac{1}{k} \end{pmatrix} </math>
| <math>k = (\cos\psi / \cos\phi)</math> is the [[beam expander|beam expansion]] factor, where {{mvar|ϕ}} is the angle of incidence, {{mvar|ψ}} is the angle of refraction, {{mvar|d}} = prism path length, {{mvar|n}} = refractive index of the prism material.  This matrix applies for orthogonal beam exit.<ref name=TLO>{{harvp|Duarte|2003|loc= Chapter 6}}</ref>

|- 
| Multiple prism beam expander using {{mvar|r}} prisms
| align="center" | <math> \begin{pmatrix} M & B \\ 0 & \frac{1}{M} \end{pmatrix} </math>
| {{mvar|M}} is the total beam magnification given by {{math|1= ''M'' = ''k''{{sub|1}}''k''{{sub|2}}''k''{{sub|3}}···''k{{sub|r}}''}}, where {{mvar|k}} is defined in the previous entry and {{mvar|B}} is the total  optical propagation distance{{clarify|date=July 2019}} of the multiple prism expander.<ref name=TLO />
|}