The loss that falls as the frequency rises

The pipe that will not carry a low note found that a wave squeezed sideways acquires a lowest frequency. Below it, the field in a metal pipe goes nowhere; above it, the pipe carries a set of modes, each with its own pattern across the pipe and its own cutoff. That essay treated the walls as perfect conductors. Later arguments replaced the walls by a slower core, by a single surface, and then by two guides lying close enough to share their tails. None of them asked what the walls cost.

A real metal wall is not a perfect conductor. It absorbs a little of every wave that touches it, and the loss usually grows with frequency. This essay follows one mode in a round pipe that does not obey that rule, and the unusual pattern of its field that makes it an exception. For a decade that mode was the best candidate for carrying long-distance communications, and the reason it lost to glass is the other half of the same geometry.

A loss that should always rise #

A current flowing in a metal at high frequency does not fill the conductor. It crowds into a surface layer whose thickness, the skin depth, shrinks as the inverse square root of the frequency. That is the field diffusing into the metal, as how far a field gets into metal derived it. A thinner layer means a larger resistance for the same current, so the power lost to a given surface current grows as the square root of the frequency. Every cable and every waveguide is subject to this. The surface resistance of copper is 8 milliohms per square at 1 gigahertz and 64 at 60.

How much a guided wave loses therefore depends on how much current it drives in the walls for each watt it carries along the guide. For most modes that ratio settles down to something roughly constant far above cutoff, and the loss then rises as the square root of the frequency, like the surface resistance.

The figure plots the loss of three modes in a round copper pipe 60 millimetres across, the size Bell Laboratories used for its long-distance experiments. The TE₁₁ mode, the pipe’s lowest, has its cutoff at 2.93 gigahertz. Its loss is infinite at cutoff, falls steeply just above it as the mode stops bouncing nearly straight across the pipe, reaches a minimum, and then rises. TM₀₁, cut off at 3.82 gigahertz, does the same. By 60 gigahertz TE₁₁ loses 21 decibels per kilometre, so a signal would fall to a hundredth of its power in a kilometre.

TE₀₁, cut off at 6.09 gigahertz, never turns round. Its loss falls from cutoff through the whole range drawn, reaching 0.51 decibels per kilometre at 60 gigahertz and 0.20 at 110. Far above cutoff it falls as the frequency to the power minus three halves, so doubling the frequency cuts the loss by nearly a factor of three. At 110 gigahertz the signal loses half its power only after fifteen kilometres of pipe.

Where the field touches the wall #

The reason lies in the shape of the field.

The figure draws the electric field across the pipe for the three modes, computed from their Bessel-function solutions. TE₁₁ runs straight across the pipe, much like the field between the plates of a capacitor, and its field lines end on the metal at the top and bottom. TM₀₁’s field points radially from the axis out to the wall. Both drive currents that flow along the pipe’s length, in the direction the wave travels, so the wall carries a substantial share of the mode’s current.

TE₀₁’s electric field goes round in circles. It is purely circumferential — tangential to the wall everywhere — and since a tangential electric field must vanish at a perfect conductor, it falls to zero at the wall. None of its field lines end on the metal. The only field TE₀₁ has at the wall is magnetic, and it is the component of the magnetic field that points along the pipe. That drives a current that runs round the pipe’s circumference, not along it, which is a peculiar arrangement: the wall behaves like a long solenoid through which the wave passes.

That wall field is the key to the whole effect. For a TE mode the axial magnetic field is the part that exists because the wave is confined; the transverse part is the part that carries power down the pipe. Far above cutoff the mode’s pattern approaches a plane wave travelling straight along the axis, whose fields are entirely transverse. The axial magnetic field then shrinks in proportion to the ratio of the cutoff to the frequency, and the wall current it drives, for each watt carried, shrinks with it. The power lost goes as the square of the current, times the surface resistance: $(f_c/f)^2 \times f^{1/2}$, which is $f^{-3/2}$. Other modes also have a longitudinal wall current fed by their transverse fields, and that part does not shrink.

The loss formula for TE modes shows the split directly. It carries a bracket $(f_c/f)^2 + n^2/(x^2 - n^2)$, with $n$ the mode’s number of angular oscillations and $x$ the Bessel zero that fixes its cutoff. The first term dies away with frequency; the second does not. For TE₀₁, $n = 0$, and the second term vanishes identically. Among all the modes of a round pipe, only the TE₀ₘ family, with circular electric fields, have that property.

The minimum every other mode has #

A comparison of the modes on a common scale makes the difference plainer.

The figure plots each mode’s loss against frequency measured in units of its own cutoff. All three are infinite at cutoff, where the wave bounces nearly straight across the guide and makes no progress along it. All fall steeply just above. TM₀₁ reaches its minimum at $\sqrt{3}$ times its cutoff and TE₁₁ at a little over twice, and both then rise without limit. TE₀₁ is still falling at ten times its cutoff, where the drawing ends, and in principle keeps falling for ever.

The cutoff therefore does two jobs in this problem. It is the frequency below which a mode cannot travel, as in the essay on the low note, and it is the scale against which the loss is measured. TE₀₁ becomes nearly lossless when it is operated far above its own cutoff, in a pipe much wider than a wavelength. That requirement is what makes the mode both attractive and treacherous.

A wider pipe and seven hundred other ways to go #

Widening the pipe lowers the cutoff and so pushes the operating frequency further above it. The loss falls steeply.

The figure shows both consequences of widening at 60 gigahertz. The loss of TE₀₁ falls roughly as the cube of the radius: 4.1 decibels per kilometre in a pipe 30 millimetres across, 0.51 in one 60 millimetres across and 0.11 in one 100 millimetres across. The number of modes the pipe can carry at that frequency rises as the square of the radius: 175, then 709, then 1,969, counting both orientations of every mode that has two.

Each of those modes is a place where TE₀₁’s power can go. A perfectly straight, perfectly round, perfectly smooth pipe would keep them separate, since modes of an ideal guide do not exchange power. Any real pipe has bends, slight ellipticity, joints and dents, each of which couples modes that have nearly the same propagation constant. Two tails that swap everything showed how two modes with matched propagation constants exchange power completely, and a pipe with seven hundred modes offers many near matches.

The worst was built into the geometry. TM₁₁, one of the pipe’s modes, has exactly the same cutoff as TE₀₁, because the zero of $J_1$ that fixes TM₁₁’s cutoff is the same number, 3.8317, as the zero of $J_0'$ that fixes TE₀₁’s. The two modes are degenerate: they travel at the same speed. At any gentle bend they couple strongly and trade power back and forth, and TM₁₁ has a loss hundreds of times larger. A curve in the pipe that turned TE₀₁ partly into TM₁₁ would lose the signal in the metal. This is the waveguide version of the mode that will not turn a corner, with the difference that the pipe does not radiate at a bend; it scrambles.

Bell Laboratories’ answer was to break the degeneracy. The WT4 system’s pipe was lined with a helix of fine insulated copper wire, wound so that its turns ran round the circumference. TE₀₁’s wall current also runs round the circumference, so the helix carries it with almost no extra loss. Every other mode needs a current along the pipe, which the helix’s gaps interrupt, so those modes are heavily damped and any power converted into them is lost quickly instead of returning to confuse the signal. The helix turned the scrambling from a crosstalk problem into a small, steady loss, and the pipe could then be laid along gentle curves.

A travelling solenoid, and two modes that must not meet #

The circumferential wall current gives the mode a simple mechanical picture. A long solenoid is a wire wound round a cylinder, carrying a current round the circumference, and inside it the magnetic field points along the axis. The field outside the solenoid showed how little of such a winding’s field escapes it, and how that little falls away outside. TE₀₁ is a solenoid made of the pipe itself, with the winding replaced by a sheet of current that circulates round the wall and reverses every half wavelength along the pipe. Inside, the axial magnetic field of each half-wavelength section is closed by the transverse field of the next, and the electric field circles the axis between them. The mode is a train of such solenoids sliding down the pipe at the group velocity.

The picture also explains the loss directly. A solenoid’s winding carries a current proportional to the field it must produce along the axis. At high frequency TE₀₁’s axial field is a small fraction of its total field, because the mode is nearly a plane wave, so its winding carries a small current. A mode with a longitudinal wall current, such as TM₀₁, is more like a coaxial line with the wall as its outer conductor. Its wall current must carry the whole return of the wave’s current, and that does not shrink.

The degeneracy with TM₁₁ has a cleaner description in the same language. Two modes with exactly equal propagation constants are two states of equal energy, and any small perturbation that couples them — a bend, an ellipse — mixes them completely, just as the crossing that never happens found for two levels brought into coincidence. In a perfect pipe the mixing has nothing to act on. In a real one, every departure from roundness and straightness is such a perturbation, and the two modes trade power along the pipe with a beat length set by the perturbation’s strength. The helix lining did not remove the coupling. It made TM₁₁ so lossy that power leaking into it was absorbed before it could return, which turned an exchange back and forth into a small, one-way drain.

That is the same trade engineers of optical couplers face, in reverse. The coupler that does not care about the colour wanted light to move completely from one guide to another and designed a slow sweep through the crossing so that it would. The WT4 designers wanted the opposite: no transfer at all between two modes that nature had made identical, and the only way to guarantee it was to make one of them unable to survive.

The case for a pipe, and the case against it #

The alternative was coaxial cable, and at these frequencies it cannot compete.

The figure plots both. An air-filled coaxial cable 9.5 millimetres across loses 72 decibels per kilometre at 1 gigahertz, 227 at 10 and 557 at 60, rising as the square root of frequency because all of its current runs along its conductors, in a skin that thins as the frequency rises. A cable run of a kilometre at 60 gigahertz would reduce the signal by a factor of more than $10^{55}$. The pipe, over the same kilometre, loses 11 per cent.

In the 1960s that difference justified a large engineering project. Telephone traffic was growing fast, a single TE₀₁ pipe carrying signals from 40 to 110 gigahertz could hold the equivalent of hundreds of thousands of voice circuits, and repeaters could be spaced tens of kilometres apart. Bell Laboratories built and tested the WT4 system through the 1970s, including a field trial in New Jersey with a pipe laid in a trench and bent to follow the terrain. It met its design specifications.

It was never deployed. By the time it was ready, optical fibre — a glass guide in which light is held by the channel with no walls, a core slightly slower than its cladding — had reached losses of a few decibels per kilometre and was heading below one, carried far more information, and could be laid in a trench a fraction the size. Fibre won because it was cheap and flexible, and because its guidance required no metal at all; the loss mechanism the TE₀₁ mode had evaded so cleverly simply does not exist in glass. The same frequency range is now used for short-range wireless links and for radar, and precision circular waveguides are used where their low loss still counts, such as the feeds of large radio telescopes and the transmission lines of high-power microwave sources for heating fusion plasmas.

What the loss formula assumes #

The figures use textbook formulas, and three assumptions in them matter.

The wall is smooth. The surface resistance assumes a flat, polished metal. At 60 gigahertz copper’s skin depth is 0.27 micrometres, and surface roughness comparable with that forces the current to follow a longer path. Measured losses in real pipes run 10 to 30 per cent above the ideal, and the gap grows with frequency as the skin depth shrinks towards the roughness scale.

The pipe is perfectly round and straight. Everything about mode conversion, the reason the helix was needed, is missing from the loss curves. They describe the mode alone, not the pipe as a system, and in the system the loss was set by how well the unwanted modes were suppressed as much as by the copper.

The conductor is classical. At low temperatures and high frequencies the electrons’ mean free path in pure copper exceeds the skin depth, and the simple surface resistance no longer applies — the anomalous skin effect. It raises the loss above the classical prediction, and it limits how much cooling can improve a metal waveguide.

Still open: how far a metal guide can be pushed #

The same arguments apply at terahertz frequencies, between microwaves and infrared light, where there is still no clearly best way to guide a wave over long distances. Metal pipes become lossy there because of roughness and the anomalous skin effect, dielectric fibres absorb strongly, and a range of hybrid designs — hollow metal-coated tubes, bare wires that carry a surface wave, porous plastic fibres — each work in some range and fail in another. Whether a TE₀₁-like mode, with its falling loss, can be preserved at terahertz frequencies in a practical guide, and whether any guide can combine low loss with the tolerance to bends that fibres have, is being investigated, with losses of a few decibels per metre as the current benchmark rather than per kilometre.

The habit worth carrying away is to ask where a wave’s field touches the thing that absorbs it. The loss of a guided wave is set by the current it drives in the walls for each watt it carries, and a mode whose field is arranged to leave the walls alone can escape a rule that binds every other. TE₀₁’s circular electric field ends on nothing and fades as the frequency rises, and the price of that freedom is a wide pipe full of other modes waiting to take its power.