Optics 018 · Four lenses that hide a block

Ray Cloaking Challenge

Can a set of lenses steer light around an object and leave the background looking unchanged? Yes, for light near the axis. Four lenses at the right spacings send each ray out at the height and angle it would have after crossing the same length of air, so the background looks untouched. Inside, the light squeezes into a narrow beam along the axis, and an object in the ring around it is not seen. Nothing on the axis can hide, and a steep view or glass that splits colours gives it away.

Hide a block inside Choi and Howell’s four-lens cloak, then look from a steeper angle, swap in plain glass or move it onto the axis until it shows.

View through the cloak showing the wall in its true place $\text{—}$

View through the cloak showing the block $\text{—}$

View through the cloak ending on a black mount $\text{—}$

Largest gap between colours landing on the wall $\text{—}$

View through the cloak showing the wall out of place $\text{—}$

Lens spacings and total length $\text{—}$

Cloak’s ray matrix for blue light at the F line $\text{—}$

Rays traced for this record $\text{—}$

The Lab follows paraxial rays through thin lenses. The design is Choi and Howell’s four-lens cloak (2014): outer lenses of 200 mm focal length, inner lenses of 75 mm, 275 mm between an outer and an inner lens, 330 mm between the two inner lenses and 880 mm in all, with the gridded wall 1.9 m behind and the camera 3.1 m in front, as in their demonstration. Each lens is a thin cemented doublet whose crown and flint parts share the power so that its focal length is the same in red light at the C line and blue light at the F line: N-BK7 with N-SF2 for the outer lenses and BAF11 with N-SF11 for the inner ones, with indices from the SCHOTT and HIKARI catalogue formulas (BAF11 uses HIKARI’s E-BAF11). Plain glass means N-BK7 singlets with the same focal lengths at the d line. Lens thickness is left out, so the cloak is 880 mm long instead of the 910 mm of the real lenses, and only chromatic aberration remains; Choi and Howell’s simulation of the real lenses also gave 7.7 % distortion at −1.5°. All four lenses share one clear diameter, held in black rings 15 mm wide on 12 mm posts. Choi and Howell’s lenses were all 50 mm across; their simulation shows that the second, third and last lenses would need to exceed 33, 51 and 112 mm for no ray to be cut off. The camera is a pinhole aimed at the middle of the last lens. Its image is 150 × 150 pixels, each traced backwards in red, green and blue at the C, d and F lines. A wall point counts as in its true place when all three colours land within 2 mm of where they land with the cloak removed, less than the 2.4 mm width of a grid line. The block is 6 mm square and 24 mm long and is kept 20 mm from each lens. Reflections off the glass, diffraction and the phase of the light are left out; Choi and Howell did not try to preserve the phase either. Their three-lens cloak, and prism and gradient-index cloaks, are not traced.

Physics tutorial

Hide a block inside four lenses

BackgroundIn 2014 Joseph Choi and John Howell at the University of Rochester asked what an optical cloak has to do if light is treated as rays. Their answer: every ray must leave the device at the height and in the direction it would have had after crossing the same length of empty space. They built such a cloak from four ordinary lenses and photographed a gridded wall through it, from four viewing angles.

Why it mattersLenses bend every ray that passes through them, so it seems they could only distort what lies behind them. Four lenses at the right spacings can undo each other’s bending exactly, for rays near the axis, while the light inside takes a narrower path than it would through air. Whatever sits beside that path is not seen.

Start with the essentials

Focus question: Why does the camera see the wall in its true place and not the block between the middle lenses, and what makes the block show again?
One-sentence intuition: Rays near the axis are described by their height $y$ and slope $u$ . The first two lenses form a telescope: light arriving in a parallel beam leaves it $f_{2}/f_{1}=3/8$ as wide, with every tilt made $8/3$ times steeper. The last two lenses undo this, and the spacing between the middle lenses is chosen so that the four together act exactly like 880 mm of empty space. Between the lenses the light runs in a beam about three-eighths as wide as the lenses, and the ring around it carries no light that reaches the camera.

Core mathematical model

What a perfect ray cloak does

\begin{pmatrix}y_{\mathrm{out}}\\u_{\mathrm{out}}\end{pmatrix}=\begin{pmatrix}1&L\\0&1\end{pmatrix}\begin{pmatrix}y_{\mathrm{in}}\\u_{\mathrm{in}}\end{pmatrix}

A ray enters with height $y_{\mathrm{in}}$ and slope $u_{\mathrm{in}}$ and leaves the far end. Across empty space of length $L$ its slope stays the same and its height grows by $Lu_{\mathrm{in}}$ . A cloak whose matrix is the same sends every ray on as if nothing were there, so the background looks unchanged from every direction near the axis (Choi and Howell 2014, Eq. 1).

Each pair of lenses is a telescope

\begin{gathered}\mathbf{M}_{\mathrm{pair}}=\begin{pmatrix}-f_{2}/f_{1}&f_{1}+f_{2}\\0&-f_{1}/f_{2}\end{pmatrix}\\\frac{f_{2}}{f_{1}}=\frac{75\,\mathrm{mm}}{200\,\mathrm{mm}}=\frac{3}{8}\end{gathered}

Two lenses of focal lengths $f_{1}$ and $f_{2}$ placed $f_{1}+f_{2}$ apart turn a parallel beam into a narrower parallel beam, upside down. With 200 mm and 75 mm lenses the beam inside is three-eighths as wide, and every angle is made eight-thirds as large. That squeeze is what leaves room for something to hide.

Spacings that make the cloak perfect

\begin{gathered}t_{1}=f_{1}+f_{2}=275\,\mathrm{mm}\\t_{2}=\frac{2f_{2}(f_{1}+f_{2})}{f_{1}-f_{2}}=330\,\mathrm{mm}\\L=2t_{1}+t_{2}=880\,\mathrm{mm}\end{gathered}

With the outer lenses $t_{1}$ from the inner ones, the gap $t_{2}$ between the inner lenses is the one that makes the four-lens matrix equal that of empty space of length $L$ (Choi and Howell, Eqs. 10, 12 and 13). These are thin-lens lengths; the real doublets made the cloak 910 mm long.

A doublet that focuses red and blue together

\begin{gathered}\phi_{\mathrm{crown}}=\frac{V_{\mathrm{crown}}}{V_{\mathrm{crown}}-V_{\mathrm{flint}}}\,\phi\\\phi_{\mathrm{flint}}=-\frac{V_{\mathrm{flint}}}{V_{\mathrm{crown}}-V_{\mathrm{flint}}}\,\phi\\V=\frac{n_{d}-1}{n_{F}-n_{C}}\end{gathered}

Glass bends blue light more than red. A doublet cements a crown lens to a weaker negative flint lens with more dispersion, splitting the power $\phi=1/f$ so that the two dispersions cancel between the F line (486 nm) and the C line (656 nm). $V$ is the glass’s Abbe number; N-BK7 has 64.2 and N-SF2 33.8.

Common difficulties

The cloak does not bend light around a hollow middle

Typical misconceptionLike a cloak in a story, the lenses steer light around a hidden space in the centre.

Better mental modelThe light inside runs along the axis in a narrow beam, and the hiding place is the ring around it. Choi and Howell note that their designs need the centre of the system to stay clear.

The view is not undisturbed at every angle

Typical misconceptionIf the matrix equals empty space, the cloak works from any direction.

Better mental modelThe matrix holds for small angles, and the inner beam tilts eight-thirds as steeply as the view, so it reaches the hidden ring once the camera rises far enough. Finite lenses also cut off part of the view: with four 50 mm lenses 40 % of it ends on a mount. Choi and Howell call such failures near the lens edges edge effects, and list lenses sized to the design and larger optics among the cures.

Good alignment is not enough

Typical misconceptionAny lenses with the right focal lengths will do.

Better mental modelThe focal length of a plain N-BK7 lens is 1.5 % longer in red light than in blue, so the cloak equals empty space in one colour only. Choi and Howell used achromatic doublets, and found in simulation that moving the outer lens spacing by 1 % changed the magnification by 30 to 40 %.

Rays are not the whole light

Typical misconceptionA cloak that fools a camera hides the object from every measurement.

Better mental modelA camera records how bright the light is where it lands. Choi and Howell kept to ray optics and did not try to preserve the phase of the light, which an interference measurement compares. The Lab also leaves out reflections off the glass.

Run the experiment

01
Look through the cloak
Keep the default preset, Block between the middle lenses: hidden.
What to observe: The camera image in the data row matches the reference without the cloak wherever the wall shows: 60 % of the view through the cloak, with no part of the orange block. The dark ring around it, 40 % of the view, is the lens mounts seen through the lenses. The red, green and blue light of one point lands at most 0.43 mm apart on the wall.
02
Drag the block along and across the cloak
Drag the orange block, or use the two block sliders, keeping it about 20 mm from the axis.
What to observe: It stays hidden from about 70 to 250 mm and from 330 to 805 mm along the cloak, except within 20 mm of the third lens, where the block is not allowed. Near the outer lenses and just past the second lens the light spreads wider and the block shows. At 380 mm it shows once it is less than 17.5 mm above or 11.5 mm below the axis.
03
Put the block on the axis
Load Block on the axis: in plain view.
What to observe: The block now covers 9 % of the view. Inside the cloak the light that reaches the camera always runs along the axis, so the middle of the device can never hide anything; in the view-angle chart the block is seen from every angle.
04
Raise the camera
Load Steeper view: the block shows.
What to observe: At 1.6° the beam inside tilts by about 4.3° and sweeps across the block: 7 % of the view shows it, and half the view now ends on mounts. The chart shows the block first appearing just above 0.8°; Choi and Howell photographed their cloak from −0.65° to 0.95°.
05
Swap the doublets for plain glass
Load Plain glass: the colours split.
What to observe: The block is still hidden, but only 5 % of the view shows the wall in place and 56 % shows it shifted: red, green and blue land up to 12 mm apart, five grid lines’ width. In the colour chart, from 440 to 660 nm, the ray through plain glass lands from 8 mm below to 2 mm above its true place, while the doublets keep it within about 1 mm.
06
Use bigger lenses
Load Bigger lenses hide it again.
What to observe: With 100 mm lenses and the block 40 mm from the axis, it stays hidden at 1.6°, and only 24 % of the view ends on mounts. The inner beam is twice as wide too, so a block 20 mm from the axis would now be seen. The doublets’ leftover colour also doubles at the larger heights: 22 % of the view is shifted by more than 2 mm.

Ray Cloaking Challenge

Hide a block inside four lenses

Core mathematical model

What a perfect ray cloak does

Each pair of lenses is a telescope

Spacings that make the cloak perfect

A doublet that focuses red and blue together

Common difficulties

The cloak does not bend light around a hollow middle

The view is not undisturbed at every angle

Good alignment is not enough

Rays are not the whole light

Run the experiment

Look through the cloak

Drag the block along and across the cloak

Put the block on the axis

Raise the camera

Swap the doublets for plain glass

Use bigger lenses

Related experiments and articles

Interference for measuring length and recording images