Optics 098 · Near-field imaging

Negative Refraction & Superlens

How does a negative-index slab bend light the wrong way and recover fine detail that normally fades out? In a negative-index slab, Snell's law puts the refracted ray on the same side of the normal as the incident ray, and the phase runs backward relative to the flow of energy. An ideal slab also amplifies evanescent waves, which carry the finest detail and normally die out before reaching an image. Real materials absorb, and that loss limits how sharp such a superlens can be.

Put a negative-index slab between two slits and a near-field probe. Watch rays bend the wrong way, then see two slits closer than a wavelength resolved, or blurred by loss.

Energy direction in the slab for light arriving at 30 degrees $\text{—}$

Phase wavenumber across the slab for straight-on light $\text{—}$

Dip between the two slit images $\text{—}$

Passband edge from mask to probe $\text{—}$

Dip with vacuum instead of the slab $\text{—}$

Passband edge with vacuum instead of the slab $\text{—}$

Smith et al. (2003) estimate for this loss and thickness $\text{—}$

Ideal image plane behind the slab $\text{—}$

Slab constants $\text{—}$

Probe positions recorded $\text{—}$

Two-dimensional wave optics at one frequency, in vacuum wavelengths. The light is transverse magnetic, with its magnetic field along the slits. The mask leaves two Gaussian slits lit in phase, and a slab of infinite width sits half its own thickness behind the mask. Each transverse wavenumber crosses the first gap, the slab and the second gap with the exact single-slab transmission, multiple reflections included; the probe reads the field intensity at its position without disturbing it. The glass slab has a relative permittivity of 2.25 with a fixed absorption of 0.01. The negative-index slab has permittivity and permeability both equal to minus one, each with the same small absorption set by the loss control. The indefinite slab has permittivity plus one along its faces, minus one across them with that absorption, and permeability one. Rays are drawn from the real parts only and follow the direction of energy flow; the dots show which way the phase moves, not the wave spacing. The passband edge is one plus the integral of the transfer magnitude, capped at one, over all evanescent wavenumbers. Not included: the frequency dependence a real negative-index material must have, the finite unit cells of a metamaterial, the probe’s own effect on the field, the other polarization and noise. The transverse integrals use a grid refined around the slab’s guided-mode resonances; a grid four times finer moves no scan sample by more than one percent of the peak.

Physics tutorial

Resolve two slits closer than a wavelength

BackgroundLight leaving a fine object carries its detail in waves that change quickly across the object. Those waves cannot travel: they fade within a fraction of a wavelength, so an ordinary lens never receives them and cannot image anything finer than about half a wavelength.

Why it mattersA slab whose permittivity and permeability are both minus one bends light the wrong way and, in theory, makes the fading waves grow again. The probe behind the slab records whether two slits 0.2 wavelengths apart arrive as two peaks or as one.

Start with the essentials

Focus question: When does the slab bring back the detail that vacuum loses, and what limits how much it can bring back?
One-sentence intuition: Inside a slab with $\varepsilon=\mu=-1$ a fading wave grows by exactly the amount it faded in the gaps on either side, so with the probe at the image plane the transfer from mask to probe is one for every wavenumber. Real materials absorb. A loss $\delta$ caps that growth, and the slab only restores wavenumbers up to roughly $\ln(1/\delta)$ divided by the thickness. A thin slab with low loss resolves the slits; a thick or lossy one does not.

Core mathematical model

Snell’s law with a negative index

\begin{gathered}n_1\sin\theta_1=n_2\sin\theta_2\\n_2=-1\ \Rightarrow\ \theta_2=-\theta_1\end{gathered}

Energy crosses the normal and leaves on the same side it came from. Inside the slab the phase moves back toward the source while the energy moves away from it.

Why fine detail fades in vacuum

\begin{gathered}k_z=\sqrt{k_0^2-k_x^2}=i\kappa\\H\propto e^{-\kappa z}\quad (k_x>k_0)\end{gathered}

A transverse wavenumber $k_x$ larger than the vacuum wavenumber $k_0$ has no real $k_z$ . Its field falls off with distance, faster the finer the detail.

Transmission through a slab

\begin{gathered}T=\frac{1}{\cos(qd)-\frac{i}{2}\left(\eta+\frac{1}{\eta}\right)\sin(qd)}\\\eta=\frac{q}{\varepsilon_x k_z}\end{gathered}

Here $q$ is the normal wavenumber inside the slab of thickness $d$ , and the formula includes every reflection between the faces. For light polarized with its magnetic field along the slits, $\eta$ compares the tangential electric fields inside and outside.

The perfect lens limit

\begin{gathered}\varepsilon=\mu=-1:\quad T=e^{-ik_zd}=e^{\kappa d}\\a+b=d\ \Rightarrow\ e^{ik_z a}\,T\,e^{ik_z b}=1\end{gathered}

The slab undoes the decay of both gaps. With the mask a distance $a$ in front, the image sits $b=d-a$ behind the slab.

How loss limits the restoration

\begin{gathered}|T_{\text{total}}|\approx\frac{1}{1+(\delta/2)^2e^{2\kappa d}}\\k_{\max}\approx\sqrt{k_0^2+\frac{\ln^2(2/\delta)}{d^2}}\end{gathered}

With $\varepsilon=\mu=-1+i\delta$ the transfer at the image plane falls to one half where $\kappa d=\ln(2/\delta)$ . For a loss in the permeability alone, Smith and colleagues (2003) give the same leading form, $k_{\max}\approx-\ln\delta/d$ .

Passband edge used in the readouts

\frac{k_{\text{edge}}}{k_0}=1+\int_{k_0}^{\infty}\min\left(|T_{\text{total}}|,1\right)\frac{dk_x}{k_0}

A path that passes every wavenumber up to some limit and nothing beyond has its edge at that limit. Narrow resonance peaks count no more than a passband of the same width.

Negative refraction without a negative index

\frac{k_x^2}{\varepsilon_z}+\frac{k_z^2}{\varepsilon_x}=\mu k_0^2,\qquad \tan\theta_S=\frac{k_x/\varepsilon_z}{k_z/\varepsilon_x}

With $\varepsilon_x>0$ and $\varepsilon_z<0$ the energy direction $\theta_S$ turns negative while the phase still runs forward. Energy bends the wrong way, yet fading waves are not amplified.

Common difficulties

Negative refraction is not the superlens

Typical misconceptionAny slab that bends light the wrong way will image beyond the diffraction limit.

Better mental modelBending propagating rays only refocuses waves that already travel. Restoring fine detail needs the fading waves to grow, which the indefinite slab here does not do.

The slab does not amplify energy

Typical misconceptionIf the field grows inside the slab, the slab must be adding energy.

Better mental modelA fading wave carries no energy away from the object, so its field can grow without any energy gain, as Pendry (2000) noted. The energy flow of the propagating waves is conserved as usual.

A thicker slab is not a stronger lens

Typical misconceptionA thicker negative-index slab restores more detail.

Better mental modelThe growth needed rises exponentially with thickness while the loss cap does not, so the finest restored wavenumber falls roughly as one over the thickness.

Run the experiment

01
See two slits resolved
Keep the default preset, Near-perfect lens: two slits resolved, and watch the probe sweep.
What to observe: The probe scan shows two peaks with a dip of about 97 % between them. With vacuum in place of the slab, the same probe sees a single peak. The passband edge reads about six times the vacuum wavenumber.
02
Swap in glass
Load Glass slab: the detail fades.
What to observe: The rays bend toward the normal instead of across it, the passband edge drops to about 1.3 and the two slits merge into one peak.
03
Add loss
From the default, drag the material loss up to 0.01, then to 0.1.
What to observe: At 0.01 the dip is still about 85 % and the edge about 4.3. At 0.1 the edge falls to about 2.4 and the two peaks merge, which is the Lossy slab preset.
04
Find the image plane
From the default, drag the probe stage toward the slab and then away from it, or load Probe past the image plane.
What to observe: The dip is largest with the probe about 0.1 wavelengths behind the slab, where the image plane lies, and vanishes by 0.3 wavelengths.
05
Bend energy without restoring detail
Load Negative refraction, forward phase.
What to observe: The energy rays bend the wrong way, by about 24 degrees, while the phase dots move forward. The passband edge stays near 1.4 and the slits still merge.

Negative Refraction & Superlens

Resolve two slits closer than a wavelength

Core mathematical model

Snell’s law with a negative index

Why fine detail fades in vacuum

Transmission through a slab

The perfect lens limit

How loss limits the restoration

Passband edge used in the readouts

Negative refraction without a negative index

Common difficulties

Negative refraction is not the superlens

The slab does not amplify energy

A thicker slab is not a stronger lens

Run the experiment

See two slits resolved

Swap in glass

Add loss

Find the image plane

Bend energy without restoring detail

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