The ray on the wrong side of the normal

The construction that produced every result on this ladder is one circle and one line. The circle is the second medium’s own relation between wavevector and frequency; the line is the tangential wavenumber, which the boundary conserves. Where they cross is the transmitted wavevector.

A line crosses a circle in two places. Every rung so far has taken the upper one without remark, and the reason it works is that the rule which actually selects is energy must leave the interface — and for an ordinary medium the energy travels along the wavevector, so the upper intersection is the one that satisfies it.

The rule and the shortcut come apart when the energy does not travel along the wavevector.

What a negative index is a statement about #

The phrase suggests a material property that could be listed beside a density, and it is better read as a statement about a square root.

The index is $n = \sqrt{\varepsilon\mu}$, and a square root has two branches. When $\varepsilon$ and $\mu$ are both positive, one branch is obviously right and nobody thinks about it. When both are negative their product is still positive, so $\sqrt{\varepsilon\mu}$ is still real — and now the choice of branch is not obvious, because the medium’s response has changed sign in a way the product has forgotten.

Veselago worked out in 1968 what such a medium would do, and the answer is that the correct branch is the negative one. The argument is about energy. Poynting’s vector is $\mathbf{E}\times\mathbf{H}$, and the relation between $\mathbf{H}$ and $\mathbf{B}$ carries a $\mu$: with $\mu$ negative, $\mathbf{H}$ is antiparallel to $\mathbf{B}$, so $\mathbf{E}$, $\mathbf{H}$ and $\mathbf{k}$ form a left-handed set while $\mathbf{E}$, $\mathbf{H}$ and $\mathbf{S}$ remain right-handed. The energy and the phase go opposite ways.

So the physical content is that the group velocity opposes the phase velocity — which this collection has already met as an anomaly near an absorption line. Here it is not an anomaly but the defining property, present at every frequency in the band rather than in a narrow window beside a resonance.

The slab that focuses without curvature #

A lens works by curvature: it is thicker in the middle, so the path through it is longer there, and the wavefront is retarded more. A flat slab of ordinary glass does none of that — it displaces a ray sideways and returns it to its original direction, and that is all.

A flat slab of index $-1$ is different, and it is different in a way that follows immediately from the hero figure.

Three features of that arrangement have no counterpart in ordinary optics.

There is no axis. Nothing about the slab distinguishes one line through it from another, so the “optical axis” of the arrangement is wherever the source happens to be.

There is no focal length. Moving the source moves both foci with it, and the external focus exists only while the source is nearer to the slab than the slab is thick. A slab is a device for transporting an image a fixed distance, not for forming one at a fixed place.

And there is no aberration. An ordinary lens focuses perfectly only for paraxial rays, and every departure from that is a term in a series of corrections that lens designers spend careers on. Here the refraction is exact at every angle by construction, so all rays cross at the same two points — checked to twelve decimal places at four angles.

Everything the sign reverses #

Veselago’s paper is short and most of it is a list, because once the energy runs against the phase every quantity that was defined by comparing the two changes sign. The list is worth having in one place, since each entry is an experiment somebody has since done.

The Doppler shift reverses. A source moving toward an observer inside such a medium produces a lower frequency, because what matters is the motion relative to the phase and the phase is going backwards. It has been measured in a transmission-line metamaterial.

Cherenkov radiation goes backwards. A charge moving faster than the phase speed in the medium emits a cone, and in a left-handed medium the cone points behind the charge rather than ahead of it — the opposite of the forward cone an ordinary medium produces.

Radiation pressure becomes a pull. The momentum carried by the wave is along the wavevector, which is now opposite to the energy flow, so a beam entering such a medium pulls on the interface rather than pushing. That statement runs into the long dispute about what the momentum of light in a medium even is, and it is one of the reasons the dispute became active again.

And a converging beam entering a slab diverges. The lens that focuses is flat, and a shaped lens does the opposite of what its shape suggests: a biconvex piece of negative-index material is a diverging lens.

Not one of those is an additional postulate. Each is what happens to a definition that had a sign in it, and reading the list is a good way to find which of one’s own intuitions about optics were about energy and which were about phase — a distinction that costs nothing while the two agree.

Where the material would have to come from #

Nothing natural does this. A negative permittivity is ordinary — every metal has one below its plasma frequency, which is exactly what makes a metal reflective — but a negative permeability requires a magnetic response, and no material has a useful one above a few gigahertz. Magnetic moments cannot be reoriented fast enough.

The way round is to build a material out of structures rather than atoms. Pendry proposed split-ring resonators in 1999: loops of conductor with a gap, small compared with the wavelength, which respond to a magnetic field by circulating a current and which have a resonance because the loop is an inductor and the gap a capacitor. Above the resonance the response is in antiphase, and the effective permeability is negative.

Smith and colleagues built one in 2000 at microwave frequencies and measured negative refraction through a prism of it the following year. The wavelength was centimetres and the rings millimetres, which is the condition the whole idea needs: the structure has to be small compared with the wavelength, or the material is a periodic structure with a band structure doing something else.

That condition is what has kept the effect out of the visible. An optical metamaterial needs features of tens of nanometres, which is now manufacturable, but it also needs those features to be low-loss resonators — and the loss of a metallic resonator rises steeply toward the visible, because that is where the metal’s own electrons stop keeping up. Optical negative-index materials exist and are lossy enough that the loss, rather than the index, is what limits everything built from them.

The lens that would beat the diffraction limit, and what stops it #

The most consequential claim about negative-index media is Pendry’s, in 2000: a slab of $\varepsilon = \mu = -1$ transmits the evanescent components of a source as well as the propagating ones, so it forms an image with no resolution limit at all.

The claim is correct and the mechanism is not mysterious. The detail an ordinary lens loses is carried by field components whose transverse wavenumber exceeds $k_0$; such components decay rather than propagating, so they are gone before reaching any lens. A negative-index slab amplifies them — not by supplying energy, but because the resonant surface modes on its two faces build up a field that grows across the slab and restores what decayed on the way in.

The restoration is exponential in the thickness, which is what makes it fragile. Any loss at all turns an exponentially large amplification into an exponentially large sensitivity, so the transmission collapses past a wavenumber set by the loss and the thickness. The figure puts the cut-off at 18, 25 and 33 times $k_0$ for losses of a hundredth, a thousandth and a ten-thousandth — a factor of a hundred in the material buying less than a factor of two in resolution.

That logarithm is the practical verdict. A perfect lens is not slightly spoiled by loss; it is a device whose whole advantage is an exponential that loss caps, and improving the material is the slowest possible way to improve the result. What worked instead was to abandon the negative permeability: at a thickness far below a wavelength the electric and magnetic responses decouple, so a thin silver film with $\varepsilon = -1$ alone does the same job for one polarisation. Fang and colleagues demonstrated that in 2005 and imaged 60-nanometre features with 365-nanometre light.

What the resolution actually depends on #

Putting the two superlens figures side by side gives the design rule, and it is not the one the phrase “perfect lens” suggests.

The cut-off wavenumber goes roughly as $\ln(1/\delta)/t$, with $\delta$ the loss and $t$ the thickness. So resolution improves linearly as the slab is thinned and only logarithmically as the material is improved. A factor of a hundred in loss is worth less than a factor of two; a factor of two in thickness is worth a factor of two exactly.

That arithmetic decided the whole experimental programme. Nobody has built a thick negative-index superlens, because there is no material good enough for one to work. What was built instead is the near-field limit of the idea: a silver film thirty-five nanometres thick, with only $\varepsilon = -1$ arranged and $\mu$ left alone, working for one polarisation at one wavelength, imaging onto a photoresist a few tens of nanometres away.

And there is a price the figures do not show. Because the slab must be far thinner than a wavelength, the image is formed within a fraction of a wavelength of it — inside the near field, where the storing part of the field dominates. The image cannot be relayed to a camera, only recorded in place. A superlens is a contact-printing device, which is useful and is not what “lens” ordinarily implies.

The argument the claim had to survive #

Pendry’s perfect lens was disputed within months of publication, and the objections are worth recording because they were good ones and because answering them is what made the result usable.

The first objection was that an amplified evanescent wave violates energy conservation. It does not: an evanescent wave carries no energy in the direction it decays, so amplifying it transports nothing, and the field growing across the slab is a stored resonance rather than a flux. But the objection identified the right worry, because a resonance that grows without bound is a resonance with no steady state.

The second objection was that the steady state does not exist. That is nearly right, and it is where the answer lies. With no loss at all, the surface resonance takes infinitely long to establish; switch the source on and the field grows for ever, approaching the perfect-lens solution but never reaching it. Any loss makes the steady state exist and simultaneously bounds the resolution — so the perfect lens is unreachable for exactly the reason that a nearly-perfect one is reachable.

The third objection was that the whole calculation assumes a continuum, and a real material is atoms. The resolution the ideal lens promises passes below the spacing of any structure the effective $\varepsilon$ and $\mu$ describe, at which point those quantities stop meaning anything. That objection has never been answered so much as accepted: the resolution of a real superlens is bounded by its own granularity as well as by its loss, and which bound bites first depends on the design.

None of the three overturned the result. Together they replaced an unbounded claim with a bounded one that had a design rule attached, which is the normal fate of a striking theoretical statement and is a better outcome than either party wanted at the time.

What it turned into #

The most durable consequence of negative refraction is not any device but a way of writing problems, and it came out of asking what else the freedom to choose $\varepsilon$ and $\mu$ pointwise would buy.

Maxwell’s equations keep their form under a change of coordinates if the change is absorbed into $\varepsilon$ and $\mu$. So a coordinate transformation that distorts space can be replaced by a material that leaves space alone — and a transformation that opens a hole in the middle of a region, pushing all the field lines around it, corresponds to a material that guides light around a volume and returns it to its original path. That is transformation optics, published by Pendry and by Leonhardt independently in 2006, and a cloak is its first example.

The prescriptions it produces are demanding in exactly the ways this essay has already met: the required materials are anisotropic, strongly dispersive, and in places require a component of $\varepsilon$ or $\mu$ below one or below zero. Microwave demonstrations work. Optical ones are narrowband and partial, for the same reason optical negative-index materials are: the loss.

What survived best is the method rather than the cloak. Transformation optics is now a standard way to design a lens, a bend in a waveguide, or a concentrator — cases where the transformation is mild, the materials are ordinary, and the payoff is a device that would have taken a long numerical search to find. A field that began by asking which square root to take ended by supplying a design procedure.

Where this stops being right #

A negative index requires dispersion, and dispersion sets a bandwidth. A medium in which the energy density is to be positive cannot have $\varepsilon$ and $\mu$ constant and negative; both must vary with frequency, and the requirement puts a lower bound on how fast. So the band is narrow by a theorem rather than by a limitation of manufacture, and a negative-index device is a narrowband device.

The slab was taken as $\varepsilon = \mu = -1$ exactly. Away from that, the impedance no longer matches the vacuum, the surfaces reflect, and the aberration-free focusing goes with it. The tolerance is severe: a per cent error in either quantity is enough to lose most of the evanescent restoration.

Everything here is a bulk description of a structure. Effective $\varepsilon$ and $\mu$ are meaningful only when the structure is much smaller than the wavelength, and every optical realisation is only just inside that condition. Where it fails, the object is a diffractive structure whose behaviour has to be computed rather than summarised in two numbers.

And the amplification of evanescent waves takes time to establish. The steady-state solution the figures compute is reached after the surface resonances have built up, and the build-up time rises as the loss falls — the same trade in a different currency. A pulsed superlens has a worse resolution than a continuous one, and the honest figure of merit involves both.

What the pictures cannot show #

The construction figures draw a wavevector and a ray as arrows from a common origin, and the two are not in the same space. The wavevector lives in the space of wavenumbers and the ray in ordinary space; drawing them on one diagram is a convention that works because the medium is isotropic and would mislead in a crystal, where the two spaces are related by a tensor.

The slab figure draws rays, and rays are the one description that cannot express what the superlens does. The evanescent components have no ray direction — they do not propagate — so the resolution the last figure is about is invisible in the first. Two figures in the same essay are describing the same object in languages that do not share a vocabulary.

Where the ladder stands #

Four rungs stand on refraction. The first met the law, the second derived it from phase matching, the third let the index be complex and found two directions where there had been one, and this one finds two intersections where there had been one.

The habit worth carrying away is about the steps a construction takes silently. Every time a geometric argument yields more solutions than there are answers, some rule is doing the selecting, and it is worth finding out what it is before the day it selects differently. Here it is that energy flows away from the boundary — obvious, uncontroversial, and equivalent to the usual rule only while the group and phase velocities agree. The same test applied to a crystal separates the ordinary and extraordinary rays; applied to a plasma it separates the modes that propagate from those that do not.

What is left on this ladder is a boundary that moves. Every rung here has assumed the interface is at rest, which is what makes the frequency conserved and the whole construction possible. A boundary in motion — a shock front, a mirror on a spacecraft, an ionisation front sweeping through a gas — conserves neither frequency nor tangential wavenumber in the same way, and the construction acquires a third dimension.