Rodney Loudon (1934-2022)

In the 1990s at RSRE Malvern I had joined the laser systems group of Michael Vaughan, having initially worked there with Mike Jenkins on hollow waveguide devices. Our optical sources, including some with “my” hollow-waveguide-resonator designs, were only parts of larger sensors, and I now had to learn something of optical detection, interferometric designs, signal processing and muddy field trials. These sensors included CO2 (10 micron wavelength) laser Doppler radars, anemometers, and multifunction demonstrators such as the UK/France CLARA pod that flew on both helicopters and fixed-wing aircraft.

RSRE studied various other lasers at wavelengths above, below and within the visible band. To explain how I joined some studies of optical heterodyning and cavity dynamics, I backtrack to my Ph. D. supervised by Denis Hall at the University of Hull.

There are several often-used approaches to optical resonator analysis, and different readers will prefer one or other. A classic 1960s approach made popular by Fox & Li takes an initial field and repeatedly propagates it round and round the resonator (using more or less simplified diffraction integrals for each step of propagation or reflection) until it shows no significant further change. Such a settled field is then identified as a self-consistent “resonator transverse mode”. There are refinements where information is retained about several round trips and used to infer the properties of several such resonator transverse mode profiles (which are in general all present to some extent in any randomly chosen initial field, and which betray their presence by circulating with different round-trip phase shifts and decay rates).

A compact expression of the self-consistency condition is ME = γE: that is, the optical field E is “operated on” by the round-trip operator M, and self-consistency means that the result is the same optical field to within a multiplying factor γ. The round-trip phase shift (or precise resonating frequency) is found from the phase of γ, and the round-trip loss is 1-|γ|².

This expression for “resonator eigenfunctions” is indeed compact and may be pursued in different ways through large literatures. In a sense, not perhaps a very enlightening sense, the resonator is a clever and fast solver of complicated matrix equations. Do self-repeating solutions necessarily exist for a given optical cavity? How can we know? What do we do with cases where self-repeating solutions appear, but their precise form is sensitive to very small changes in some elements of M? What about “degenerate” or “coalesced” cases, where two or more modes want to occupy the same eigenvalue "territory"?

In a corner of this large subject, I wrote “multimode waveguide resonator” code that:

- expressed any relevant optical field as a linear sum of free-space propagating modes (in free space), and then as a linear sum of waveguide propagating modes (within the waveguide);

- “changed the basis”, i.e. switched between these two forms whenever the light entered or left a waveguide, with the help of appropriate coupling coefficients;

- multiplied all the propagation and coupling coefficients to form the overall (sometimes large) “round-trip matrix” M;

- and sent this to some convenient numerical computation package which produced its eigenvectors and eigenvalues.

I wrote such code in various languages - Apple II BASIC, FORTRAN 77, HP BASIC, further mainframe and PC FORTRAN, and MATLAB.

Certain problems were already well known to manufacturers and users of 10 micron waveguide lasers, if not widely admitted. A smooth, single-lobe, near-Gaussian spatial profile was usually desired: it would offer a clearly defined beam “centre” during examination of a target, and reduce ambiguities in “beamrider” detector schemes for a military projectile. But imperfect spatial profiles, with asymmetries and multiple peaks and even distantly separated lobes, were sometimes seen. A stable single-frequency laser emission was also desired, especially in heterodyne detection lidars; but strong beat notes could appear from the detector, perhaps at several MHz and obstructing the measurement of target Doppler shifts. Reliable selection of a single CO2 laser “line” – there are many possible lines, or specific molecular transitions, around 9 to 11 microns – was supposed to be ensured by setting a dispersive element (usually a ruled metal grating) at a known angle to the cavity axis. But this sometimes yielded an undesired line and/or an undesired beam profile. These faults did not seem to occur predictably, or in particular manufacturing batches. Researchers were annoyed.

I proposed that a multimode waveguide resonator model, even one restricted because of computer limits to low mode orders (typically 5 – 20), and neglecting most waveguide and gas discharge nonuniformities and how they distort the amplification and propagation of light, might help. Some interesting discoveries (and parts of my Ph. D. thesis, and book chapters) emerged. The cavity eigenvalues and eigenvectors (and hence the spatial shapes and round-trip losses and frequencies) could be found for several transverse mode orders, and for waveguides and cavities that were perturbed in various realistic ways: with mirrors and gratings set back some distance from the guide and/or misaligned, and with guides that were not perfectly machined.

A curious and important feature was that our various bright ideas for improving mode “quality” with the help of mirrors matched to a desired near-Gaussian beam, or for designing self-imaging resonators, seemed less bright – or, at least, insufficient by themselves – once the need for misalignment tolerance was appreciated. In self-imaging designs, several waveguide modes (basis modes) have the same round-trip phase shift (modulo 2π) so that, for example, a desired near-Gaussian beam or any other assembly of basis modes – though it might split into multiple lobes during a resonator round trip – automatically reforms and “self-repeats” in a waveguide version of “Talbot” imaging. In a simple plane/plane resonator with diagonal mirror coupling matrices, this is equivalent to the “mode coincidence” condition [1] where a “competitor” waveguide mode (the presumably unwanted mode that is nearest in loss to the desired mode, and that may be “bubbling under”) experiences the same gain as the desired mode but has higher loss, and thus always loses the competition no matter how the exact laser length changes (because, for example, of temperature drifts or deliberate tuning). Alternatively, a “competitor” can be placed π out of phase so it lies halfway between the desired resonances on each side, and is disadvantaged as far as possible when either of those resonances is tuned to “line centre” where the laser gain is maximum. I considered both tactics for manipulating the “signatures” of waveguide lasers; the design decisions might depend on one’s confidence in the tolerances for manufacture and alignment, but one would generally want some quantitative understanding of how the mode frequencies and losses would vary for typical imperfections.

The early 1980s waveguide structures for λ ~ 10 μm at Hull satisfied a Talbot condition L ~ 4a2/λ, because alumina/metal guides with RF CO2 discharges worked well with guide widths 2a ~ 2 mm, and convenient “shoebox” size constraints and standard ceramic manufacturing encouraged guide lengths L ~ 40 cm. This may have been an amusing, unintentional coincidence; as far as I know the above manipulations of mode frequencies were not specifically suggested till the mid 1980s, and by the time they were implemented and published I was working at Malvern with Mike Jenkins, Bob Devereux, Jim Redding and others. In the mid 1990s, the discovery that an MoD lidar technical demonstrator source had been made inadvertently sensitive to misalignment was less amusing; I recall long days of FORTRAN, and important support from the late Hugh Lamberton who led our division, while I redesigned it.

Thus I gained experience in optical resonator theory, and kept in mind that apparently sensible single-frequency laser sources might have other potential modes of oscillation that “bubbled under” and could – with modest encouragement in the form of tweaks or misalignments – join or replace the desired mode.

This turned out to be just the thing needed for my colleagues’ work in sensitive heterodyne probing of sub-threshold mode behaviour. Rodney, Mike Harris, Terry Shepherd, Chris Shackleton and others were using a visible argon-ion laser – a different wavelength and a non-waveguide cavity – but I applied the same resonator modelling approach and included the perturbations due to deliberately introduced apertures (or obstructions). Sliding a knife edge into the beam path between the mirrors, for example, meant a new set of coupling coefficients for the basis Gaussian beams; I could calculate the expected variations of the losses and phase shifts for the “bubbling under” modes that were nearly but not quite lasing and whose resonances could be measured with a faint, easily tuned probe beam. And so I became a minor co-author in publications with Rodney.

C A Hill, P Monk and D R Hall, “Tunable RF-excited CO2 waveguide laser with variable guide width”, IEEE Journal of Quantum Electronics, vol. QE-23, no. 11, pp. 1968-1973, 1987.

C A Hill, J R Redding and A D Colley, “Multimode treatment of misaligned CO2 waveguide lasers”, J. Modern Optics, vol. 37, no. 4, pp. 473-481, 1990.

H Kogelik and T Li, “Laser beams and resonators”, Applied Optics, vol. 15, pp. 1550-1567, 1966.

C J Shackleton, R Loudon, C A Hill, T J Shepherd, M Harris and J M Vaughan, “Transverse modes above and below threshold in a single-frequency laser”, Physical Review A, vol. 52, no. 6, pp. 4908-4920, 1995.

Chris Hill, “Waveguide laser resonators”, in Handbook of Laser Technology and Applications, ed. C Webb and J Jones, vol. 1, CRC Press, 2003.

[1] “Coincidence” in the sense that the frequencies νmn may be made equal for different m and/or different n (m, n = the transverse mode numbers) by suitable choice of L, a, b and the axial mode number j. We take advantage of the distinctively strong dependence of waveguide mode phase shifts on the squares of guide width and of mode number, which contrasts with the weaker dependence in more familiar open resonators (Kogelnik and Li 1966).

An example of a perturbed waveguide resonator (revisited)

Recently while writing this note I returned to the 1987 paper, and of course confirmed that some hundreds of hours of student or civil servant work can now be recreated in a day of MATLAB:

We can illustrate such “sharp kinks” – for example by inspecting once again the resonator behaviour “near a = 0.97 mm”.

The tilt of 0.5 mrad has caused a significant disturbance. Making the tilt progressively smaller, we see:

For a very small tilt, we have just-discernible effects in a small range around a ~ 0.945 mm. This is indeed one of the guide widths for which the first and fourth x-dimension modes (m = 1, m’ = 4) are frequency-degenerate, as we see from inserting our specified L = 90 mm, λ = 10.59 μm and m’2-m2 = 42 – 12 = 15:

In this equation for the ν (Hill et al. 1987), we need not for our present purposes consider the y-axis behaviour in detail, since the y-axis term involving n and b merely adds a constant to the mode frequency, and does not affect the difference in frequency between two modes of different order in the x dimension.

Another way to express this degeneracy is that the phase shifts of the first and fourth eigenvalues (ranked in order in the x dimension) are equal (modulo 2π). There is a well-known tendency of two coupled oscillators to interact most strongly when their natural frequencies lie close together, i.e. near “resonance” in another sense of the word; here we have a variation where the “oscillators” are eigenvectors sharing a single optical cavity. I think the argument in this early paper (whatever faults are present in the computations and approximations) correctly notes the condition of destructive interference between the light that remains in the fundamental mode and the light that experiences one round trip of length 2L in the other mode (as well as two tilted-mirror couplings with the same sign of tilt; an alternating-sign or zigzag tilt sequence is another matter). “Destructive interference” means a phase difference of π, and each of the two coupling phase shifts is π/2, and the net result (zero difference modulo 2π) is the same as the condition for frequency degeneracy.

The extract above is reasonably clear that the computer code, although its propagation and coupling terms were precisely those used to express M, performed only an iterative calculation (for the eventual lowest-loss resonator mode), not the full matrix manipulation that extracts all the eigenvectors and eigenvalues (or at least the practically useful ones, say the first five or ten). As I recall, I was still writing the fuller version.

Moreover, neither this 1987 argument, nor the full matrix solution for multiple self-repeating transverse modes, explicitly describes any dynamics of the resonator (e.g. time evolution of fields in a Fox & Li calculation): we obtain only steady-state fields, with no details of mode-to-mode, to-and-fro, energy exchanges over time. A computer package may “iterate” towards an approximate solution for multiple eigenvectors, but this has no necessary parallel with the optical iteration (repeated circulation) of laser fields.

Although the νmn equation immediately produces the Talbot-type resonator designs, i.e. configurations where sets of transverse modes or indeed all of them have the same round-trip phase shifts (modulo 2π), the point in 1985/86 was rather that we needed a better appreciation of multimode effects in theory and practice. The basic Talbot-type resonator design (meaning, broadly, a guide with L ~ 4a2/λ plus two mirrors suited to a desired transverse profile) was and is interesting – and the non-resonator splitting/recombining devices, developed not by me but by Mike Jenkins and our colleagues, are of great importance – but from a certain point of view the designs with several or many mode degeneracies may not be ideal; in fact they may be asking for trouble if misalignments or other mode-coupling perturbations are likely. It depends on the environment, and the manufacturing tolerances, and the customer’s needs – as usual.