Thinking in Waves Episode 44 / Part XV — Making the Medium Ourselves
Designing the dispersion relation
Metamaterials —
Periodic in space gives a gap in frequency; periodic in time gives a gap in momentum
Until now the medium was given. From here we make it.
★ What fixes the sign of a negative index is the presence of loss — the causality condition \(\mathrm{Im}\,n>0\) (Episode 8).
★★ And make the structure periodic in time and the gap opens in \(k\), where the wave grows (rate 0.024993, against the prediction \(hck/4=0.025\)).
Through Episode 43 the wave travelled in a medium that simply was what it was.
★ Now we invert it — choose the dispersion relation first, then build a structure that realizes it. But this series has already decided what may be built.
01Negative index — the sign is fixed by loss
With \(n^2=\varepsilon\mu\) and both \(\varepsilon,\mu\) negative, \(n^2>0\) — two signs are available. The one that is chosen is the one that decays when loss is added (\(\mathrm{Im}\,n>0\)).
| \(\varepsilon\) | \(\mu\) | principal \(\sqrt{\varepsilon\mu}\) | ★ correct \(n\) | State |
|---|---|---|---|---|
| \(+1+0.01i\) | \(+1+0.01i\) | \(1.000+0.010i\) | \(1.000+0.010i\) | ordinary |
| \(-1+0.01i\) | \(+1+0.01i\) | \(0.000+1.000i\) | \(0.000+1.000i\) | ★ evanescent |
| \(-1+0.01i\) | \(-1+0.01i\) | \(1.000-0.010i\) | ★ \(-1.000+0.010i\) | ★ negative index |
| \(-4+10^{-4}i\) | \(-1+10^{-4}i\) | \(2.000-0.0001i\) | ★ \(-2.000+0.0001i\) | ★ negative index |
The point of §01
★★ Only \(\varepsilon<0\) and \(\mu<0\) gives \(\mathrm{Re}\,n<0\). With just one negative, \(n\) is almost purely imaginary — Episode 28's evanescent wave (\(k^2<0\), cannot travel).
★ The sign was chosen by loss. In a lossless ideal medium the two signs are mathematically indistinguishable. Admit even an infinitesimal loss and the decaying branch is unique.
★★ Episode 8's causality is doing the job of picking a sign.
02You cannot build it without dispersion — the energy would go negative
| \(\omega/\omega_p\) | \(\varepsilon=1-\omega_p^2/\omega^2\) | \(d(\omega\varepsilon)/d\omega\) | Verdict |
|---|---|---|---|
| 0.3 | −10.111111 | ★ +12.111111 | ★ OK (even with \(\varepsilon<0\)) |
| 0.5 | −3.000000 | +5.000000 | OK |
| 0.8 | −0.562500 | +2.562500 | OK |
| 1.5 | +0.555556 | +1.444444 | OK |
The point of §02
\(d(\omega\varepsilon)/d\omega=1+\omega_p^2/\omega^2\) is always positive — dispersion is what lets a negative \(\varepsilon\) keep a positive energy.
★★ Conversely, "a negative \(\varepsilon\) without dispersion" cannot exist.
★ That is why a negative-index medium is always narrowband — make \(\varepsilon<0\) over a wide band and \(d(\omega\varepsilon)/d\omega\) must go negative somewhere.
03The sum rule — you cannot win at every frequency
| Quantity | Value |
|---|---|
| numerical integral (Lorentz, \(\omega_0=2,\gamma=0.1\)) | 1.57077133 |
| \((\pi/2)\omega_p^2\) | 1.57079633 |
| ratio | 0.99998 |
The point of §03
★★ The right-hand side depends on neither \(\omega_0\) nor \(\gamma\) — put the resonance anywhere, make it as sharp as you like, the total is the same.
★ "Gain at one frequency costs you another" is now a formula. A designer controls the allocation, never the total.
★★ Like Episode 15's "only dimensionless quantities are physics," this is a ceiling on design itself.
04A 1D photonic crystal — gaps, and slow light
Stack layers of index \(n_1,n_2\) and form the one-period transfer matrix \(M\). The band where \(|\mathrm{tr}\,M/2|>1\) is the gap (no propagation).
| \(n_2/n_1\) | first gap, lower edge | upper edge | ★ relative width |
|---|---|---|---|
| 1.05 | 3.017420 | 3.112540 | 0.031034 |
| 1.2 | 2.691980 | 3.019640 | 0.114735 |
| 1.5 | 2.203420 | 2.815460 | 0.243895 |
| 2.0 | 1.682140 | 2.461920 | 0.376336 |
| 3.0 | 1.135660 | 1.871860 | 0.489573 |
Approach the band edge and the group velocity heads for zero (\(n_2=2\); the first band's top edge is \(\omega=1.68214\)).
| \(\omega\) | \(KL\) (Bloch phase) | ★ \(v_g/c\) |
|---|---|---|
| 0.500 | 0.79254 | 0.627547 |
| 1.400 | 2.29636 | 0.528187 |
| 1.670 | 2.99005 | 0.157780 |
| 1.6820 | 3.12559 (≈\(\pi\)) | ★ 0.017166 |
The point of §04
★ Inside the gap the wave "cannot travel" — exactly Episode 28's \(k^2<0\). Once the Bloch phase reaches \(\pi\) it goes complex.
★★ At the edge \(v_g\to0\) (0.628 → 0.017, a factor of 37) — this is slow light.
★ But the closer to the edge, the narrower the usable band. That is the next section.
05★ Delay and bandwidth can only be traded
| \(\gamma\) | max group delay \(\tau\) | bandwidth \(\Delta\omega\) | ★ \(\tau\cdot\Delta\omega\) |
|---|---|---|---|
| 0.400 | 1.106566 | 0.400 | 0.442627 |
| 0.200 | 1.337111 | 0.200 | 0.267422 |
| 0.100 | 2.195969 | 0.100 | 0.219597 |
| 0.050 | 5.314747 | 0.050 | 0.265737 |
| 0.025 | 15.923708 | 0.025 | 0.398093 |
The point of §05 — ★ honestly
The delay alone can be stretched 1.11 → 15.92, a factor of 14 (dividing \(\gamma\) by 16).
★★ But the product only moves 0.44 → 0.22 → 0.40 — a factor of two.
★ Honestly, it is not monotonic (it is smallest around \(\gamma=0.1\)). Calling it "constant" would be overstating. What can be said is: change \(\gamma\) by 16× and the product stays within 2×.
★★ The design implication is clear all the same — every bit of delay is paid for in bandwidth. Episode 9's bandwidth theorem again.
06★★ Make it periodic in time — a gap in momentum
| \(k\) | \(\det M\) | \(\mathrm{tr}\,M\) | \(|\mu|\) | growth rate | Verdict |
|---|---|---|---|---|---|
| 0.90 | 1.000000 | −1.906769 | 1.000000 | 0 | passes |
| 0.95 | 1.000000 | −1.980778 | 1.000000 | 0 | passes |
| 1.00 | 1.000000 | −2.006168 | 1.081682 | ★ 0.024993 | ★ gap (amplifies) |
| 1.05 | 1.000000 | −1.982314 | 1.000000 | 0 | passes |
| 1.10 | 1.000000 | −1.909803 | 1.000000 | 0 | passes |
The point of §06 — ★★★ the core of this episode
Periodic in space ⟹ a gap in frequency; inside it the wave decays.
Periodic in time ⟹ a gap in momentum; inside it the wave grows.
The gap is centred at \(k=\Omega/2c=1.000\) — exactly Episode 40's parametric resonance condition. The growth rate is 0.024993, against \(hck/4=0.025\).
★★ Decay versus growth is just a question of which direction you cannot travel in. Cannot travel in space, and you shrink with distance. Cannot travel in time — and since time cannot be refused, you grow instead.
★ \(\det M=1.000000\) again — as in Episode 36. A growing solution always comes paired with a decaying one.
07What may not be built — three gatekeepers
| Gatekeeper | Condition | Meaning |
|---|---|---|
| Passivity (Ep. 17, 18) | \(\mathrm{Re}\,Z\ge0\) | no amplification without energy in |
| Ostrogradsky (Ep. 29) | at most two time derivatives | third order and up is always unstable |
| Sum rule (Ep. 8; §03 here) | \(\int\omega\,\mathrm{Im}\,\varepsilon\,d\omega\) fixed | ★ gain here is loss there |
The point of §07
★ Metamaterials are "freely designable" only outside these three.
★★ Negative index breaks none of them — which is why it could be built (dispersive, narrowband, lossy).
★ The "perfect lens" is hard because \(\mathrm{Im}\,\varepsilon\) cannot be made zero — and as §01 showed, that very loss is what fixed the sign of \(n\). Remove the loss and negative index stops being definable at all.
★★ §06's time modulation looks like it breaks passivity, but it does not — wobbling \(\varepsilon(t)\) takes work from outside (as in Episode 40).
08What wave language can and cannot say
| Item | Verdict | Basis |
|---|---|---|
| Loss fixes the sign of \(n\) | ◎ numerics | only \(\varepsilon<0\) and \(\mu<0\) |
| No \(\varepsilon<0\) without dispersion | ◎ numerics | \(d(\omega\varepsilon)/d\omega>0\) |
| The sum rule | ◎ numerics | 4 digits against \((\pi/2)\omega_p^2\) |
| A gap is a "cannot travel" band | ◎ numerics | \(|\mathrm{tr}\,M/2|>1\) |
| \(v_g\to0\) at the band edge | ◎ numerics | 0.628 → 0.017 |
| The delay–bandwidth trade | ○ numerics | ★ not monotonic (16× → within 2×) |
| ★ Time-periodic → momentum gap | ◎ numerics | ★ and it amplifies inside |
| ★ Designing a real device | × no | ★ 1D models only here |
| ★ Feasibility in 3D | × no | ★ isotropic \(\mu<0\) is a separate difficulty |
Exercises
- §06 found the wave growing inside the momentum gap. Where does the energy come from?
Show answer
From whatever is wobbling \(\varepsilon(t)\). Changing the permittivity takes work, and that work goes into the wave.
★ Exactly Episode 40's swing — the person pumping the swing is \(\varepsilon(t)\).
★★ So passivity (Episodes 17–18) is not violated — the system as a whole is externally driven. ★ When you hear "a metamaterial amplified the wave," the correct first question is where the power is being fed in. - §03's sum rule fixes a total. What, then, does negative index pay with?
Show answer
Bandwidth and loss. By §02, \(\varepsilon<0\) always comes with dispersion, and strong dispersion means a narrow band. And strong dispersion means, by Kramers–Kronig, a large \(\mathrm{Im}\,\varepsilon\) nearby — that is, large loss.
★ "Negative index is narrowband and lossy" is not an experimental inconvenience but a consequence of causality.
★★ It is Episode 8's "real and imaginary parts cannot be designed independently" in action. ★ So a design programme aiming at "lossless broadband negative index" is a dead end in principle. - §04 called the inside of a gap "evanescent." How does that differ from Episode 41's topological gap?
Show answer
The gap itself is the same; what differs is what sits at the edge. Both are bands in which the bulk cannot propagate.
★ But in Episode 41 the bulk integer (winding, Chern) is non-zero, so states must appear at the boundary (bulk–boundary correspondence). In an ordinary photonic crystal the integer is zero and nothing appears.
★★ So §04 suffices if you only want a gap, but to get a one-way road at the edge the integer must be non-zero — which requires breaking time-reversal symmetry (a magnetic field, or time modulation as in §06). ★ §06 is also the doorway to that.
Summary — a metamaterial is a designed dispersion relation
The sign of a negative index is fixed by loss. Only \(\varepsilon<0\) and \(\mu<0\) gives \(\mathrm{Re}\,n<0\); with one alone, \(n\) is almost purely imaginary — Episode 28's evanescent wave. In a lossless ideal medium the two signs cannot be told apart.
And it cannot be built without dispersion. The energy condition is \(d(\omega\varepsilon)/d\omega>0\), and for Drude that is \(1+\omega_p^2/\omega^2\), always positive — dispersion is what permits \(\varepsilon<0\). ★ Hence negative index is always narrowband.
And the sum rule \(\int\omega\,\mathrm{Im}\,\varepsilon\,d\omega=(\pi/2)\omega_p^2\) (numerically 1.57077 against 1.57080) — the right side depends on neither the position nor the sharpness of the resonance. A designer controls the allocation, never the total.
In a 1D photonic crystal, \(|\mathrm{tr}\,M/2|>1\) is the gap; at index ratio 2 its relative width is 0.376. At the band edge \(v_g/c\) falls 0.628 → 0.017, a factor of 37 (slow light). But stretching the delay 14× moved the delay–bandwidth product by only a factor of two (★ and not monotonically — calling it constant would overstate it).
★★★ The core —
periodic in space: a gap in frequency, and inside it the wave decays.
Periodic in time: a gap in momentum, and inside it the wave grows.
The time-modulated equation is Episode 40's Mathieu equation exactly: the gap is centred at \(k=\Omega/2c\), the growth rate is 0.024993 (prediction \(hck/4=0.025\)), and \(\det M=1.000000\). ★ Decay or growth is just which direction you cannot travel in — cannot travel in space and you shrink with distance; cannot travel in time and, since time cannot be refused, you grow.
★ And what may be built is decided by three gatekeepers — passivity (Ep. 17–18), Ostrogradsky (Ep. 29), and the sum rule (Ep. 8). Negative index breaks none of them, which is why it exists. The perfect lens is hard because the loss cannot be removed — and that loss is what fixed the sign of \(n\) in the first place.
★ Next time those two meet — shake the vacuum and photons come out of nothing.
★★ The dynamical Casimir effect, and the Unruh effect. Seen as waves, "nothing" is a sea of zero-point motion, and moving a boundary raises ripples in it. Episode 9's uncertainty becomes a source of energy.
This document is Episode 44 of the "Thinking in Waves" series. Computations are in kenshou/meta.js. On the numbers here: the branch selection for the refractive index, \(d(\omega\varepsilon)/d\omega\), the numerical sum-rule integral, the transfer-matrix band structure and group velocities, the delay–bandwidth products, and the Floquet multipliers of the time-modulated medium were all computed here. Negative refractive index (Veselago, Pendry), branch selection via \(\mathrm{Im}\,n>0\), the energy-density condition in a dispersive medium, the f-sum rule, transfer matrices and band gaps in 1D photonic crystals, the delay–bandwidth product, and momentum gaps with amplification in time-modulated media (photonic time crystals) are all well-known results and none are discoveries of this article — what this article did was gather them under the single question "what can and cannot be built," and set the space-periodic and time-periodic cases side by side. The delay–bandwidth product in §05 is not constant: in this model and these parameters a 16-fold change in \(\gamma\) keeps it within a factor of two, but it is not monotonic (smallest near \(\gamma\approx0.1\)) and it is not a universal constant. §04 and §06 are minimal one-dimensional, two-component models and are a different problem from designing a real three-dimensional device (realizing isotropic \(\mu<0\) in particular carries difficulties not treated here). The loss used in §01 is a small value chosen to make the numerics well-defined and is not the loss of any real material. — To print, use your browser's Print and "Save as PDF."