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.

Matrix of the pair, lens 1 to lens 2, lengths in metres $\text{—}$

Effective focal length of the pair $\text{—}$

Beam width leaving lens 2 $\text{—}$

Spot width on the wall $\text{—}$

Spot width on the wall with the laser alone $\text{—}$

Peak brightness on the wall, against the laser alone $\text{—}$

Rear principal plane, from lens 2 (negative: before it) $\text{—}$

Focus of a parallel beam, from lens 2 $\text{—}$

Narrowest point of the beam, from lens 2 $\text{—}$

Full spreading angle after lens 2 $\text{—}$

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

BackgroundA 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 mattersLaser 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 misconceptionMaking a laser beam wider makes its spot bigger everywhere.

Better mental modelOnly 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 misconceptionTo get the smallest spot on a distant wall, the beam must leave exactly parallel.

Better mental modelParallel 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 misconceptionAn expander is a lens with a very short or very long focal length.

Better mental modelIts $C$ 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 misconceptionThe focal length of two lenses is the distance from the last lens to the focus.

Better mental modelThat 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

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.
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.
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.
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.
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.
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.
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.

Compound Optics Workbench

Read a beam expander from its matrix

Core mathematical model

A ray-transfer matrix

The pair’s power

Two foci together

The ABCD law for a laser beam

Common difficulties

A wider beam spreads less

Parallel is not the smallest far spot

An afocal pair has no focal length

The focal length is not measured from the lenses

Run the experiment

Read the matrix

Drag lens 2

Put lens 2 5 mm out

Use a diverging lens first

Turn the expander round

Relay the beam

Make a telephoto

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