Optics 013 · Glass that bends light with no surfaces inside

Gradient-Index Optics Observatory

How can a smoothly varying refractive index guide and focus light with no surfaces inside it? A refractive index that changes smoothly bends rays all along their path, so no hidden surface is needed. In a slab whose index falls off parabolically from the center, rays swing back and forth and keep re-forming the image. A Luneburg sphere focuses a parallel beam from any direction onto its far surface. A Maxwell fisheye sends every ray from one point to the point opposite it.

Send a flash of light through a Luneburg lens, Maxwell’s fish-eye, graded slabs and a laser-written spot, and watch where its rays meet again.

Narrowest flash: this profile / uniform glass $\text{—}$

Optical path to the focus $\text{—}$

Spread of optical path at the focus: this profile / uniform glass $\text{—}$

Refractive index in the medium $\text{—}$

Position of the focus $\text{—}$

Largest drift of the optical momentum’s length from the local index $\text{—}$

Largest drift of the conserved angular or axial momentum $\text{—}$

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

Rays are traced in a plane by geometrical optics: no diffraction, dispersion, absorption or polarization. A real lens a few wavelengths across blurs its focus by diffraction; the SCRIBE Luneburg lenses of Ocier et al. (Light: Science & Applications 9, 196, 2020) focus to spots about 0.4 µm wide. Lengths are in units of R, the radius of the lens or mirror, the half-width of a slab’s core or the unit of the written plate; ray optics has no length scale of its own, so the same rays hold at any size. The Luneburg lens has an index falling from the square root of 2 at its centre to 1 at its rim, in air; the contrast control scales how far the squared index rises above 1. Maxwell’s fish-eye is closed by a mirror at its rim, as in Leonhardt (New Journal of Physics 11, 093040, 2009), so the image of a point forms at its mirror point through the centre. Both slabs have index 1.6 on the axis and the same curvature there; the hyperbolic-secant profile brings every ray in the plane back to an exact image each half period, while the parabolic one does so only for rays near the axis, the effect Gloge and Marcatili (Bell System Technical Journal 52, 1563, 1973) computed for graded fibres. A slab stops getting steeper where the index at the edge of its core reaches 1. The written spot is a Gaussian rise of 0.25 above an unfilled porous-silicon scaffold of index 1.28, inside the range from 1.28 to 1.85 that Ocier et al. reached by filling its pores with laser-set polymer; it is a single spot, not a pattern a visitor can paint. Each record traces 121 rays through the graded medium and the same 121 through uniform glass of the same peak index, which bends rays only at its surface; every fifth ray is drawn. A fourth-order Runge–Kutta integration keeps the length of each ray’s optical momentum equal to the local index, and its conserved angular or axial momentum constant, to better than one part in a billion.

Physics tutorial

Focus light with glass that has no surfaces inside

BackgroundLight travels more slowly in glass of higher refractive index. An ordinary lens has one index inside and bends light only at its two surfaces. Gradient-index glass changes its index smoothly from point to point instead.

Why it mattersIn such glass a ray bends all the way along, towards the higher index. Luneburg and Maxwell found index profiles that bring every ray from one point to another; graded-index fibres use a profile like these to keep light pulses short. Since 2020 a femtosecond laser can also set the index point by point inside a porous silicon film, by choosing how much polymer it sets in the pores (Ocier et al. 2020).

Start with the essentials

Focus question: A beam of parallel light enters a sphere whose index falls from the square root of 2 at the centre to 1 at the rim. Why does every ray of the beam meet at one point on the far rim?
One-sentence intuition: Let a flash of light leave the source at one moment. Without dispersion, after a time $t$ every ray has covered the same optical path $L=\int n\,\mathrm{d}s$ , so the points at equal $L$ are the flash. A medium images perfectly when the flash closes to a single point: every ray reaches the image after the same optical path. In the Luneburg lens the axial ray takes the shortest route but through the densest glass; outer rays go farther through thinner glass, and the two effects cancel exactly.

Core mathematical model

Rays in a graded medium

\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}\sigma}=\mathbf{p},\qquad\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}\sigma}=\tfrac12\nabla n^2,\qquad|\mathbf{p}|=n

The optical momentum $\mathbf{p}=n\hat{\mathbf{t}}$ points along the ray, and the parameter $\sigma$ advances by $\mathrm{d}s/n$ . The ray bends towards higher $n$ . These are the equations of a particle in the potential $-n^2/2$ at zero energy, which is why a ray rolls into regions of high index like a marble into a bowl. The optical path is $L=\int n^2\,\mathrm{d}\sigma=\int n\,\mathrm{d}s$ .

Luneburg lens

\begin{gathered}n(r)=\sqrt{1+c\,(1-r^2)}\quad(r\le1)\\L_{\mathrm{lens}}=\int_{-1}^{1}\sqrt{2-x^2}\,\mathrm{d}x=1+\frac{\pi}{2}\end{gathered}

With $c=1$ the index is $\sqrt{2-r^2}$ : the square root of 2 at the centre and 1 at the rim, so light enters from air without reflecting. Inside, $n^2=1+c-c\,r^2$ is a harmonic potential and every ray is an ellipse. A parallel beam meets at the rim opposite its entry, and the axial ray’s path through the lens is $1+\pi/2$ times the radius; every other ray matches it. A steeper profile $c>1$ still focuses perfectly, at $1/\sqrt{c}$ from the centre.

Maxwell’s fish-eye with a mirror

\begin{gathered}n(r)=\left(\frac{2}{1+r^2}\right)^{c}\\z_0\;\longrightarrow\;-z_0\quad(L=\pi)\end{gathered}

With $c=1$ every ray from a point $z_0$ is a circle through $z_0$ and through $-z_0/|z_0|^2$ . A mirror at $r=1$ folds that second point back inside, so every ray from $z_0$ passes $-z_0$ after an optical path of $\pi$ and returns to $z_0$ after $2\pi$ (Leonhardt 2009).

Two slabs with the same axis

\begin{aligned}n&=n_0\,\mathrm{sech}(gy)\\\sinh(gy)&=a\sin(gx+\varphi)\\[6pt]n^2&=n_0^2\,(1-g^2y^2)\\y&=\frac{\sin\theta_a}{g}\,\sin\!\left(\frac{g\,x}{\cos\theta_a}\right)\end{aligned}

Both slabs have $n_0=1.6$ and the same curvature on the axis. In the sech slab $\sinh(gy)$ swings like a pendulum with period $2\pi/g$ for every ray, steep or shallow, so the source is re-imaged exactly every half period. In the parabolic slab a ray crossing the axis at angle $\theta_a$ has period $2\pi\cos\theta_a/g$ : steep rays turn back sooner, and the image blurs.

A written index spot

n=n_{\mathrm{b}}+\delta\,e^{-r^2/w^2},\qquad f\approx\frac{w\,n_{\mathrm{b}}}{2\sqrt{\pi}\,\delta}

A weak Gaussian rise $\delta$ of width $w$ in glass of index $n_{\mathrm{b}}$ deflects a ray at small height $h$ by $2\sqrt{\pi}\,\delta h/(w\,n_{\mathrm{b}})$ , which makes it a thin lens of focal length $f$ . The Lab’s spot rises by 0.25 above porous silicon of index 1.28; the written range there is 1.28 to 1.85 (Ocier et al. 2020).

Common difficulties

Light does not need a surface to bend

Typical misconceptionLight travels straight until it meets a surface.

Better mental modelLight travels straight only where the index is uniform. Where the index changes across the ray, the ray curves towards the higher index all the way along, as the relief in the scene shows.

Rays do not take the shortest route

Typical misconceptionEvery ray to the focus follows the shortest path.

Better mental modelRays to a perfect focus have equal optical path, not equal length. In the plot of the index along five rays, the outer rays are longer but run through lower index, and the area under every curve is the same.

A parabolic profile is not a perfect lens

Typical misconceptionA parabolic index profile images every ray exactly.

Better mental modelIt images exactly only in the limit of rays close to the axis. The sech profile with the same axis does image every ray in the plane, and the parabolic slab’s small error is the residual delay spread that Gloge and Marcatili (1973) computed for graded fibres.

Run the experiment

01
Watch the flash close
Keep the default preset, Luneburg lens: a beam focuses on the far rim, and watch the gold flash cross the lens.
What to observe: The flash bows forward in the middle, then closes to a point on the far rim. All 121 rays meet there after the same optical path, 3.17 R; the narrowest flash is less than a millionth of a millionth of R wide, the precision of the integration. A uniform ball of the same peak index, 1.41, spreads them over 0.027 R.
02
Turn the beam
Tilt the beam to 25°, or drag it across the axis.
What to observe: The focus slides round the rim to the point straight ahead along the beam’s new direction, and the optical path stays 3.17 R. The lens is the same from every direction, so it focuses every direction.
03
Steepen the profile
Load Steeper profile: the focus moves inside.
What to observe: At contrast 1.5 the rays still meet at one point, now 0.82 R from the centre, inside the glass. Lower contrast instead, to 0.6, and the focus moves out to 1.36 R and blurs to 0.005 R.
04
Image a point in the fish-eye
Load Maxwell’s fish-eye: every ray reaches the image.
What to observe: Rays from the source at (−0.45, 0.30) R fan out in every direction, bounce off the mirror and all pass through (0.45, −0.30) R after an optical path of 3.14 R, then return to the source. A uniform disc of index 2 never brings them closer than 0.22 R.
05
Compare the two slabs
Load Sech slab: the source re-forms exactly, then Parabolic slab: steep rays refocus early.
What to observe: In the sech slab every ray reaches the image 4.19 R along after the same optical path, and reaches the end face together; in a step-index slab of the same peak index the paths across the end face differ by 14 %. In the parabolic slab steep rays turn back sooner: the narrowest flash is 0.081 R wide, 3.54 R along, and the end-face paths differ by 5.1 %.
06
Write a lens
Load Laser-written spot: a lens with no surface.
What to observe: A Gaussian spot of extra index bends the beam into a focus 1.48 R behind its centre, 0.050 R wide. Uniform glass of the spot’s peak index shaped as a disc does slightly worse, 0.057 R; neither is perfect, because a single spot is not a Luneburg profile.

Gradient-Index Optics Observatory

Focus light with glass that has no surfaces inside

Core mathematical model

Rays in a graded medium

Luneburg lens

Maxwell’s fish-eye with a mirror

Two slabs with the same axis

A written index spot

Common difficulties

Light does not need a surface to bend

Rays do not take the shortest route

A parabolic profile is not a perfect lens

Run the experiment

Watch the flash close

Turn the beam

Steepen the profile

Image a point in the fish-eye

Compare the two slabs

Write a lens

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