Editing Ray transfer matrix analysis (section)

== Gaussian beams ==
The same matrices can also be used to calculate the evolution of [[Gaussian beam]]s{{sfnp|Rashidian Vaziri|Hajiesmaeilbaigi|Maleki|2013}} propagating through optical components described by the same transmission matrices. If we have a Gaussian beam of wavelength {{nowrap|<math>\lambda_0</math>,}} radius of curvature {{mvar|R}} (positive for diverging, negative for converging), beam spot size {{mvar|w}} and refractive index {{mvar|n}}, it is possible to define a [[complex beam parameter]] {{mvar|q}} by:<ref name=Lei/>
<math display="block"> \frac{1}{q} = \frac{1}{R} - \frac{i\lambda_0}{\pi n w^2} . </math>

({{mvar|R}}, {{mvar|w}}, and {{mvar|q}} are functions of position.) If the beam axis is in the {{mvar|z}} direction, with waist at {{math|''z''{{sub|0}}}} and [[Rayleigh range]] {{mvar|z{{sub|R}}}}, this can be equivalently written as<ref name=Lei>{{cite web|url=http://www.colorado.edu/physics/phys4510/phys4510_fa05/ |author=C. Tim Lei |title=Physics 4510 Optics webpage}} especially [http://www.colorado.edu/physics/phys4510/phys4510_fa05/Chapter5.pdf Chapter 5]{{Self-published source|date=August 2024|expert=y|reason=University course notes, no longer available, not published}}</ref>
<math display="block"> q = (z - z_0) + i z_R .</math>

This beam can be propagated through an optical system with a given ray transfer matrix by using the equation{{explain|date=July 2019}}:
<math display="block"> \begin{bmatrix} q_2 \\ 1 \end{bmatrix} = k \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}q_1 \\ 1 \end{bmatrix} , </math>
where {{mvar|k}} is a normalization constant chosen to keep the second component of the ray vector equal to {{math|1}}.  Using [[matrix multiplication]], this equation expands as
<math display="block">\begin{aligned} q_2 &= k (A q_1 + B) \\
1   &= k (C q_1 + D)\,.\end{aligned}</math>

Dividing the first equation by the second eliminates the normalization constant:
<math display="block"> q_2 =\frac{Aq_1+B}{Cq_1+D} ,</math>

It is often convenient to express this last equation in reciprocal form:
<math display="block">  \frac{ 1 }{ q_2 } = \frac{ C + D/q_1 }{ A + B/q_1 } .  </math>

=== Example: Free space ===
Consider a beam traveling a distance {{mvar|d}} through free space, the ray transfer matrix is
<math display="block">\begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} .</math>
and so 
<math display="block">q_2 = \frac{A q_1+B}{C q_1+D} = \frac{q_1+d}{1} = q_1+d</math>
consistent with the expression above for ordinary Gaussian beam propagation, i.e. {{nowrap|<math> q = (z-z_0) + i z_R</math>.}} As the beam propagates, both the radius and waist change.

=== Example: Thin lens ===
Consider a beam traveling through a thin lens with focal length {{mvar|f}}. The ray transfer matrix is
<math display="block">\begin{bmatrix}A&B\\C&D\end{bmatrix}=\begin{bmatrix}1&0\\-1/f&1\end{bmatrix}.</math>
and so 
<math display="block">q_2 =\frac{Aq_1+B}{Cq_1+D} = \frac{q_1}{-\frac{q_1}{f}+1} </math>
<math display="block">\frac{1}{q_2} = \frac{-\frac{q_1}{f} + 1}{q_1} = \frac{1}{q_1} - \frac{1}{f} .</math>
Only the real part of {{math|1/''q''}} is affected: the wavefront curvature {{math|1/''R''}} is reduced by the [[Optical power|power]] of the lens {{math|1/''f''}}, while the lateral beam size {{mvar|w}} remains unchanged upon exiting the thin lens.