---
title: "Compound Optics Workbench Interactive Lab | Sandbox Physics"
description: "How does one matrix tell what two lenses do to a laser beam?"
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language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/compound-optics-workbench/"
last_modified: "2026-10-29"
---

Optics 023 · One matrix for a pair of lenses

# Compound Optics Workbench

**How does one matrix tell what two lenses do to a laser beam?** Multiply the matrices of lens 1, the gap and lens 2 in order, and the pair becomes one matrix of four numbers. Its lower-left entry is minus the pair’s power. When the gap equals the sum of the two focal lengths, that entry is zero: a parallel laser beam leaves parallel, wider by the size of the upper-left entry, and spreads less by the same factor, so its spot on a distant wall is smaller.

Slide two lenses in a He-Ne beam and read their single ray-transfer matrix: when its lower-left entry is zero, the beam leaves parallel, wider and less divergent.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The two lenses are thin lenses in air, combined with the paraxial ray-transfer matrices of Kogelnik and Li (1966), with lengths between their principal planes; real lenses have thickness, aberrations and reflection losses that are left out. The laser beam is an ideal Gaussian beam in its fundamental mode at 632.8 nm, carried through the same matrices by the ABCD law for its complex beam parameter. Its waist is taken to sit on the laser’s output face, 0.80 mm across between the points where the irradiance falls to 13.5 % of its peak, which gives the 1.00 mrad full divergence listed for a 05-LHP-151 He-Ne laser; every beam width on the page is measured the same way. Every lens is 25.4 mm across; when the beam at lens 2 is wider than that, the page says so but does not model the light the rim cuts off. A matrix entry a thousand times smaller than any the sliders’ steps can produce is rounding and reads as zero. The wall is square to the beam and the room is still air. Widths in the scene are drawn three times larger than lengths, and the flight to the wall is drawn at its true length.

Physics tutorial

## Read a beam expander from its matrix

**Background** A laser beam is narrow, but it spreads: the 0.8 mm beam of a small He-Ne laser is about 10 mm across on a wall 10 m away. Two lenses in a row, a beam expander, make it wider first so that it spreads less. What the pair does can be read from one matrix of four numbers, the product of the matrices of lens 1, the gap and lens 2.

**Why it matters** Laser pointers for surveying, laser rangefinders and the beams sent into telescopes are expanded this way. The same four numbers tell you the focal length of any lens group, where its principal planes are, and whether it keeps a parallel beam parallel.

Start with the essentials

- **Focus question**: How does one matrix tell what two lenses do to a laser beam?
- **One-sentence intuition**: The lower-left entry $C$ is minus the pair’s power. When the gap equals the sum of the focal lengths, $C=0$: a parallel beam leaves parallel, $|A|$ times wider, and by the ABCD law for a laser beam it also spreads $|A|$ times less. On a wall far enough away, the wider beam makes the smaller spot.

### Core mathematical model

#### A ray-transfer matrix

$$
\begin{pmatrix}y_{2}\\\theta_{2}\end{pmatrix}=\begin{pmatrix}A&B\\C&D\end{pmatrix}\begin{pmatrix}y_{1}\\\theta_{1}\end{pmatrix}
$$

A ray is its height $y$ above the axis and its slope $\theta$. Free space of length $L$ has $A=D=1$, $B=L$, $C=0$; a thin lens has $A=D=1$, $B=0$, $C=-1/f$ (Kogelnik and Li, 1966, Table I). The matrix of several parts is their product, the last part on the left, and its determinant $AD-BC$ is 1. The Lab shows $B$ in metres and $C$ in $\mathrm{m^{-1}}$, so that $C$ reads in dioptres like the lens sliders.

#### The pair’s power

$$
C=-\frac{1}{f_{1}}-\frac{1}{f_{2}}+\frac{d}{f_{1}f_{2}}
$$

Multiplying lens 1, a gap $d$ and lens 2 gives this entry, Kogelnik and Li’s Table I No. 4 with nothing before lens 1. The pair acts as one lens of focal length $f=-1/C$, measured from its rear principal plane, which sits $(A-1)/C$ back from lens 2 and need not lie between the lenses. For 20 D and −40 D lenses 30 mm apart, $f=250\,\mathrm{mm}$ and that plane is 150 mm in front of lens 2.

#### Two foci together

$$
d=f_{1}+f_{2}:\quad A=-\frac{f_{2}}{f_{1}},\ D=-\frac{f_{1}}{f_{2}}
$$

With the gap equal to the sum of the focal lengths, $C$ vanishes and the pair is afocal. $A$ is the change in a ray’s height: −4 for 25 mm then 100 mm, the minus meaning the beam is turned upside down; +4 for −25 mm then 100 mm. Edmund Optics gives the same magnifying power, $-f_{2}/f_{1}$, and spacing for Keplerian and Galilean expanders.

#### The ABCD law for a laser beam

$$
q_{2}=\frac{A\,q_{1}+B}{C\,q_{1}+D}
$$

A Gaussian beam is described by one complex number $q$, whose real part says how far the beam is from its waist and whose imaginary part, $\pi w_{0}^{2}/\lambda$ at the waist, says how narrow the waist is (Kogelnik and Li, Eqs. 16 to 18 and 43). Through an afocal pair, $q_{2}=A^{2}q_{1}+AB$: the waist grows by $|A|$, and since the full spread is $2\lambda/(\pi w_{0})$, it falls by the same factor.

### Common difficulties

#### A wider beam spreads less

**Typical misconception** Making a laser beam wider makes its spot bigger everywhere.

**Better mental model** Only nearby. One metre from the laser the expanded beam is 3.27 mm across against the bare 1.29 mm, but it spreads four times less, and at 10 m it is 4.34 mm against 10.1 mm. A beam’s spread is set by its narrowest width, so a wider waist spreads less.

#### Parallel is not the smallest far spot

**Typical misconception** To get the smallest spot on a distant wall, the beam must leave exactly parallel.

**Better mental model** Parallel light gives the smallest spread at great distances. At a finite distance the smallest spot comes from focusing the wider beam onto the wall: at 10 m, with lens 2 1.2 mm further out, the spot is 2.40 mm instead of 4.34 mm.

#### An afocal pair has no focal length

**Typical misconception** An expander is a lens with a very short or very long focal length.

**Better mental model** Its $C$ entry is zero, so $f=-1/C$ does not exist and there are no principal planes. It changes a parallel beam’s width and spread without bringing it to a focus; the Lab shows its focal length as infinite.

#### The focal length is not measured from the lenses

**Typical misconception** The focal length of two lenses is the distance from the last lens to the focus.

**Better mental model** That distance is the back focal distance, $-A/C$. The focal length $-1/C$ is measured from the rear principal plane, which in the telephoto lies 120 mm in front of lens 1, outside both lenses: 100 mm back focal distance, 250 mm focal length.

### Run the experiment

1. 01

   #### Read the matrix

   Keep the default preset, Keplerian expander, and read the matrix and the wall spot.

   **What to observe:** $A=-4$ and $C=0$: the beam leaves 3.24 mm wide, parallel and upside down, after passing through a focus between the lenses. On the wall it is 4.34 mm across, against 10.1 mm for the laser alone, and the grey dashed outline in the data row shows the bare beam overtaking it.

2. 02

   #### Drag lens 2

   Drag lens 2 along the rail a few millimetres each way.

   **What to observe:** The pair’s power in the process row crosses zero at 125 mm. The wall spot is smallest, 2.40 mm, with lens 2 at 126.2 mm: there the beam leaves slightly converging and comes to its waist near the wall.

3. 03

   #### Put lens 2 5 mm out

   Load Lens 2 off by 5 mm: the far spot swells.

   **What to observe:** $C=2\,\mathrm{m^{-1}}$ now: the pair is a −500 mm lens. The beam narrows to a waist 2.1 m beyond lens 2, then spreads faster than the bare laser, and the wall spot grows to 12.2 mm.

4. 04

   #### Use a diverging lens first

   Load Galilean expander: no focus inside.

   **What to observe:** With a −25 mm lens 1, the lenses are 75 mm apart and $A=+4$: the same width, upright, from a shorter bench. The light seems to spread from a point 25 mm in front of lens 1, and the wall spot is 4.48 mm.

5. 05

   #### Turn the expander round

   Load Turned round: the beam narrows and spreads.

   **What to observe:** With the 100 mm lens first, $A=-0.25$. The beam leaves 0.219 mm wide and spreads 4.03 mrad, and the wall spot is 39.1 mm across.

6. 06

   #### Relay the beam

   Load Relay: same width, upside down.

   **What to observe:** Two 100 mm lenses 200 mm apart give $A=D=-1$. The beam keeps its 0.80 mm width and is turned upside down; its waist is carried 400 mm down the bench, so the wall spot is 9.70 mm, slightly smaller than the laser alone’s.

7. 07

   #### Make a telephoto

   Load Telephoto: a long focal length in a short space.

   **What to observe:** A 50 mm lens and a −25 mm lens 30 mm behind it act as one 250 mm lens. Its rear principal plane is 120 mm in front of lens 1, so parallel light focuses only 130 mm from lens 1, half the focal length.

Physical conclusion

The default pair, a 25 mm lens then a 100 mm lens with 125 mm between them, has $A=-4$ and $C=0$. The beam arrives 0.81 mm wide and leaves 3.24 mm wide, spreading 0.252 mrad instead of 1.01 mrad, and on the wall 10 m from the laser its spot is 4.34 mm across instead of 10.1 mm, 5.4 times brighter at the centre. Moving lens 2 5 mm too far makes the pair a −500 mm lens: the beam comes to a waist 2.1 m beyond it and the wall spot grows to 12.2 mm. A −25 mm diverging lens in place of lens 1 widens the beam just as much in 75 mm, with no focus between the lenses. The same two lenses turned round make the beam four times narrower and the spot 39.1 mm.

## Related experiments and articles

Probes and measurement · Reading path

### Molecules, forces and ultrafast dynamics

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Optical tweezers: from holding a bead to measuring force**](https://sandboxphysics.com/stories/probe-evidence/optical-tweezers-force-calibration/)
2. [Lab **Optical Tweezers: Brownian Calibration** Calibrate a bead trap from thermal motion](https://sandboxphysics.com/labs/optical-tweezers-calibration/)
3. [Lab **Laser Cooling & MOT** Compare light forces on atoms](https://sandboxphysics.com/labs/laser-cooling-magneto-optical-trap/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)