Mode frequency coincidences

Section 6: Mode frequency coincidences and self-imaging

This section tries to clarify a confusing sequence of publications on Gaussian beam modelling, “TEM00-preserving” waveguides, “Talbot” effects and “resonator mode coincidence”. Readers who are not confused, or not interested in IPR manoeuvres, should skip it.

To recap: in a well-aligned dual Case I resonator a single round trip is very nearly equivalent to propagation along an undistorted guide of length 2L, and the resonator transverse modes take essentially the form of the transverse modes of the waveguide itself. The resonant mode frequencies are roughly:

subject to several small corrections. In particular, the difference in frequency between two different transverse modes depends strongly on waveguide bore size, but is constant, or very nearly constant, for different choices of waveguide length.

The “mode coincidence” condition, essentially the Talbot or kaleidoscope condition described by Bryngdahl and Ulrich and applied to hollow waveguide resonators, is: L is an integer multiple of Lc = 16a2/(λΔ), where Δ is a difference of squares of mode numbers, e.g. 22-1 so Lc = 16a2/3λ for the lowest-loss and second-lowest-loss modes, or 32-1 so Lc = 2a2/λ for arbitrary odd-number modes.

Equivalently we say that the round-trip guide propagation phase shift is a multiple of 2π, for any mode (any allowed value of m = 1, 2, 3,…) when L is a multiple of 16a2/λ. Then the frequencies of all resonator transverse modes (up to some reasonable limit consistent with our approximations and physical ability to manufacture the guides – see below) appear in tight clusters spaced by c/2L. If one such frequency is tuned to line centre, all the resonator modes in the cluster near that frequency experience essentially the same frequency-dependent gain: thus there are no gain differentials to assist mode discrimination. All this was clear in the mid-1980s.

Is this situation – or any similar consequence of deliberate guide geometry – desirable? It depends what we want to achieve. There is a significant market for multimode transmission / splitting / combining devices based on these “kaleidoscope” or “Talbot” effects. The basic example above (phase shift = integer multiple of 2π) ensures the regeneration or self-imaging of any linear combination after length 2L, if we accept first-order theory and neglect differential mode attenuation. At intermediate or “fractional Talbot” lengths, copies of fields composed of modes of certain symmetries are obtained (that is, certain subsets of the modes, but not all of them, have their phases aligned in certain ways).

If we are specifically concerned by resonator round-trip effects and lasing, not by propagation alone, a roughly approximate splitting or recombining may be inadequate; we may have to examine the phase shifts and losses in fine detail. That is, we have to consider the possible, plural, self-consistent fields that may emerge after the equivalent of many round trips, and the differences between their round-trip losses and frequencies: these differences may seem very small but, because (in our model) they are the only differences that can arise and influence the real-life laser output, we must treat them with care. In propagation-only devices such as splitters, combiners and interferometers, with just one pass or a few passes, our injected mode or pattern is the only “source”, so other patterns cannot grow to appreciable amplitudes and can (after cautious checks) be considered irrelevant.

Below are some illustrations of “mode coincidences”. It may be best to start with the L = 2a2/λ design that Mike and I discussed, and that I modelled in 1990 and described in the memo below; this 1990 note has the essential points “a mirror phase-matched to a narrow Gaussian…The entering TEM00 beam is very weakly apertured”:

We applied for a patent in 1994 and it was granted in 1997:

In 1990-1992, open publication was delayed while various patents progressed and Mike prioritised the propagation-based devices. These gained, and because of their larger markets probably deserved, more attention and success than our few, specialised, lasers with resonator improvements. By late 1992 I had moved on, promoted to PSO and with new responsibilities in Michael Vaughan’s team, but I kept in touch with Mike and IPR staff.

From early 1986, at the risk of boring everyone from Viv downwards, I tended to say what I planned to do, do it, and then say what I had done. There were jokes about the memo-of-the-month club and discounts for batch orders. Here the 1990 technical memo on “TEM matrix modelling” says “My thanks go to Mike Jenkins for relevant technical discussions. Parts of this work should appear in publications jointly written by Mike and myself”. This reflects the IPR problem. I was avoiding premature public detail of “TEM00-preserving” designs and their Talbot-length waveguides, but recording (while the work was familiar) some account, even if temporarily limited to a small internal RSRE circulation, with eventual publication in view. The CER2/EM5 attributions are, again, linked with our frequent organisational changes. We were in EM5 when I wrote this memo and gave copies to Mike and Andrew, but there was a new cover page in November 1992 when I was moving to fresh work in P Building and tidying my papers – but not quite escaping, as is clear from the patent submission and the Applied Optics version of the February 1992 memo: “revised manuscript received 25 October 1994”.

“Ken Fisher, while at RSRE in summer 1989, coded FORTRAN routines for the monster Bessel files”: Bessel functions were used in models of circular-bore guides, and Ken was a Newcastle student on placement.

I was familiar with switching between sets of basis modes, so that thinking of the waveguide as a “3-D aperture” with its own TEM coupling-and-transmission matrix (just as every other paraxial element in the model had its own matrix, often coded in a separate FORTRAN subroutine) was unremarkable. A “mode-preserving” design usually meant that the top left element (first row and first column, TEM00 to TEM00) was near unity; unfortunately, other elements on the diagonal might also be near unity, and there were off-diagonal ones to consider. I tried to explain this “TEM00 matrix” concept:

“…we extend this idea to a class of resonators with near-TEM00 fundamental modes (of various widths) and “mode-preserving” waveguides…we ignore perturbations such as laser gain and cavity misalignment” – Indeed perturbations were not treated in this simple introduction, but they were mentioned and kept in mind.

Another extract is:

“regions of low fundamental mode loss and reasonable mode discrimination around L = 2a2/λ and L = 4a2/λ, as well as less desirable regions of high loss and/or poor discrimination…TEM00 content of greater than 99 % by energy for the “mode-preserving” lengths…If L = 2a2/λ then (for example) EH11 and EH31/EH13 are in phase at the curved-mirror end (producing the desired single-spot near-TEM00), but exactly π out of phase at the plane-mirror end (producing an interesting symmetric four-spot mode)”

Here reference [23] is Hill, Jackson and Hall (1990).

- Figure 10(b) from 1990 memo on TEM matrix modelling:

- Figure from patent “Waveguide laser” (R M Jenkins and C A Hill) showing the lengths L = 2a2/λ and L = 4a2/λ marked within the favourable regions.

“Resonator designs that favour other w0/a values have been described for the first time, where a distant curved mirror is phase-matched to a narrow Gaussian”

The summary from “Tilts, Kinks and Mode Interference” (July 1992) seems fair:

That is, important insights from HWO propagation studies (where I was part of the team but concentrating on lasers and resonators) did produce “some designs”, low-loss and useful; but those insights did not address the real-life problems of mode discrimination and sensitivity to perturbations. If such problems were not understood there was a strong chance of wasting time and money. I have explained that this happened, more than once, on programs with budgets of millions.

After the submission of our waveguide laser patent application (and others), Jay, Alan and Mike published this L ~ 2a2/λ design in a 1996 conference paper and then a journal version (J. Opt. Soc. Am. B, vol.14, pp. 2378-2380, 1997):

They proposed that an “optimum design” can be based on three conditions:

(1) a guide Fresnel number of 2: that is, L ~ 2a2/λ, where a is the guide half-width and λ is the wavelength;

(2) a plane mirror very near one aperture;

(3) a concave mirror at a distance z ~ 0.5 a2/λ from the other aperture, with radius of curvature R ~ 3a2/λ.

Conditions (1) and (2) preserve any symmetric field pattern after propagation through 2L (a “Talbot” length) in an ideal lossless guide obeying first-order theory. The near-field well-aligned plane mirror is part of this required round-trip 2L in a resonator (that is L, followed by reflection, recoupling, and then the same L in reverse). By condition (3) the curved mirror is phase-matched to a fictitious desired TEM00 beam with waist radius parameter w0 ~ 0.6a. Smaller or larger values (tighter or more generous beam waists at the aperture) are possible, with the mirror curvature adjusted to satisfy R ~ z + b2/z where b = πw20/λ. In the waveguide laser literature, fictitious beams with w0 ~ 0.65-0.7a are often called “approximating Gaussians” because they closely resemble the guide fundamental mode EH11.

Guide attenuation is mentioned, but not always included in the results shown; sometimes, as in Henningsen et al., the only dissipative losses occur at the re-entrance aperture, where light spills outside the guide and is lost. In a real laser we would arrange outcoupling (useful, that is non-dissipative loss): the curved mirror would transmit the quasi-TEM00 beam, and/or the plane mirror would transmit the symmetric four-spot beam into which the re-entering mode is transformed after propagation through L.

This 1997 design encapsulates the previous ideas of mode-preserving guides and phase-matched Gaussians, and depends only on the choices of halfwidth a and wavelength λ. It resembles, but (as explained below) is not quite the same as, the high-quality-single-lobe CO2 waveguide designs demonstrated to UK industry and adapted for beamrider and local oscillator applications; an important unclassified example was the 7 Watt device with “unusually good near-TEM00 transverse mode quality”, mentioned in our February 1992 CER2/EM5 memo and similar to the Howden laser in Figure 1; another version was used in the technical demonstrator as mentioned above. Jay et al. say (understandably citing as their reference [9] our “Effects” paper in its 1995 published version, not the original 1992 one):

“The slight discrepancy that still exists in the Z value for the one proposed earlier9 and for an optimum design based on Fig. 4 can be attributed to the fact that the guide length (20 cm) in the earlier design was slightly longer than 2a2/λ”.

The snag is the claim that this design with L equal to 2a2/λ is “optimum”.

Despite the obvious overlap with my and Mike’s earlier work I had no part in submitting this paper (or the step-profile one discussed below), so I can freely say that I do not think such designs are optimum. The rather pedantic but necessary points are (1) our “Waveguide laser” patent claims a specific design resistant to modest errors in guide length or width; it does not claim resistance to misalignment; and (2) the precise Talbot-length design recommended by Jay et al. is unwise if we do additionally want resistance to misalignment. I had good reason to dispute this “optimum” claim about the precise Talbot-length design, and would have opposed it in 1997 (or at any time after 1989).

The 1992 “Effects” memo, which became the Banerji et al. (1995) open-literature paper, addressed the need for most or all of the following:

(1) Low loss for the fundamental resonator transverse mode

(2) Adequate discrimination against all other resonator transverse modes

(3) Near-TEM00 spatial output

(4) Retention of this near-TEM00 mode quality throughout the laser signature (i.e. regardless of cavity length changes of a few λ)

(5) Adequate tolerance of small manufacturing and alignment errors

(6) Intolerance of the specific “misalignment” associated with an unwanted line of a grating-tuned CO2 laser.

This robust line selection (6) will occur if the resonator loss rises sufficiently when the plane (Case I) mirror is tilted by about 2 mrad (thus causing the reflected light to re-enter the guide with an angular offset of about 4 mrad, which – see above – is typical for adjacent CO2 lines near the peak of the P branch when we use a Littrow-mounted near-field diffraction grating). But at the same time we desire (5) that “small” tilts should not significantly perturb the resonator mode.

Requirement (4) was particularly important for the compact 7 W device described in that 1992 memo; for some applications, such as beamriding where the transmitted beam is detected and used for active guidance, a well-behaved single-lobed beam can be essential, yet the laser may be subject to a harsh military environment with physical and thermal shocks. Thus the design choices for our compact laser favoured a large differential between the fundamental mode loss and the loss of the second-order (two-lobed) transverse mode.

Figures 50 and 51 show relevant predictions:

Figure 50: Predicted round-trip losses for a waveguide laser. In Banerji et al. (1997) guide halfwidth a and wavelength λ are arbitrary, and guide losses are ignored; here for illustration a = 1 mm and the guide attenuation is a few per cent per metre. The mirror 1 is phase-matched to the TEM00 with w0 = 0.6a (confocal parameter b = πw02/λ = 106.7 mm): z1 = 47.2 mm, mirror 1 radius of curvature = z1 + b2/z1 = 288.5 mm.

Figure 51: As Figure 50, with a tilt of the Case I mirror (z2 = 0, φ = 3 mrad). Mode discrimination is much better for L ~ 200 mm than for L ~ 2a2/λ.

I had proposed in 1990 that, if we are concerned about maintaining a loss differential in favour of the fundamental mode, in the presence of perturbations such as misalignment, L should be slightly longer than the Talbot length 2a2/λ (188.7 mm if a = 1 mm and λ = 10.59 μm). That is, there are misalignment-resistant geometries near that precise design (consistent with “modest errors” if you will), and by 1990 I had identified and demonstrated one, which in terms of length L lay within the (usefully wide) region shown in Figure 10(b) of my memo and Figure 6 of our patent. 200 mm looked suitable from the modelling, so I ordered a 200 mm ceramic guide and built the laser. Its behaviour was very encouraging, with 6-8 Watts and a single lobe throughout its signature. Figure 51’s comparison of misalignment (tilt) tolerance for various guide lengths L is not proof, but I think a length L ~ 2a2/λ would have risked poorer transverse mode discrimination and more higher-order modes in the signature.

(Other researchers such as Boulnois and Agrawal (“Mode discrimination and coupling losses in rectangular-waveguide resonators with conventional and phase-conjugate mirrors”, J. Opt. Soc. Am., vol. 72, pp. 853-860, 1982) had previously used CO2 laser waveguides with a ~ 1 mm and L ~ 200 mm, without as far as I know investigating misalignments).

There were adjustable mirror mounts inside the laser enclosure – hence more Allen key rods and O-rings – to permit crude checks on misalignment tolerance in real time. Different emphases in the requirements, such as the need to avoid line-hopping with a diffraction grating, or the need to operate the same laser at different times on widely separated CO2 lines (say 10.25 μm and 10.6 μm), would probably have led to slightly different designs. A summary, with initial experimental data from “mode coincidence” grating-tuned lasers, is in our Hill and Redding 1992 memo.

The details of our invention reports and IPR discussions are not for reproduction here, though obviously I attached importance to mode discrimination and perturbation tolerance – they were not adequately treated in other publications and patents, and offered valuable points of difference in ours. Tenable patents involve dancing around previous work. In the early 1990s I had to tell our IPR team that these ideas, of expressing fields as sums of TEMpq and seeking resonator configurations with high TEM00 content, predated our recent patents and IPR papers and in some cases predated 1985. In particular the resonator round-trip equations with their harmonic EHmn phase relationships were already understood and coded, so that the interfering, multiple, mode amplitudes were necessarily available for inspection at any point in or out of the guide. As I wrote, “there is little distinction between seeking a resonator design with a high TEM00 content and seeking a waveguide that preserves an injected TEM00”. But neither I nor anyone else had fully drawn out the implications and patentable possibilities; and I repeat that Mike, John Heaton and other colleagues deserve the greater share of credit in the area of single-pass multimode interference devices. The 1990 notes and others mentioned in this article do not prove priority in a patent sense for Mike, myself or Malvern, and are not meant to. Showing that X preceded someone’s Y does not show that X was the first occurrence. Little does not mean zero. Making nice distinctions is part of the game.

In any case our local story is moot because many inherited QinetiQ patents covering resonators and other devices were later deemed too costly to maintain. But this “for the first time” in the memo was a claim of “priority”, made to the best of my knowledge, and Mike and I at least advanced it in our patent:

Here – to repeat – the patent mentions guide length 2a2/λ and its multiples, but the plots of predicted resonator loss (published in the 1990 memo and the patent) also show the considerable ranges of L around 2a2/λ and its multiples where the fundamental mode loss and the mode discrimination seem promising. Within these ranges of L, I sought resonator designs which had these acceptable properties when well-aligned and also offered misalignment tolerance that was better than for L = 2a2/λ. This extra feature of misalignment behaviour was key to my 200 mm laser design, but not part of the patent or of the Banerji et al. 1997 paper.

Dual Case I with step-profile mirror

If a narrow well-aligned TEM00 beam enters a square-bore waveguide, it appears as a four-lobe pattern after propagation through 2a2/λ and as a regenerated (self-imaged) single lobe after 4a2/λ. Again the corresponding behaviour in a TEM00-preserving resonator was explicit in the 1990 CER2 memo: “If L = 2a2/λ then (for example) EH11 and EH31/EH11 are in phase at the curved-mirror end (producing the desired single-spot near-TEM00), but exactly π out of phase at the plane-mirror end (producing an interesting symmetric four-spot mode)”. Banerji et al. (2009) therefore proposed a variant which would modify the two mirrors of a dual Case I resonator of length 2a2/λ so that (i) at one end the single lobe is efficiently reflected, with the high-reflectance part of the mirror confined to a small central area (ii) at the other end the four spots are efficiently reflected while the small central area has low reflectance and aids in suppressing unwanted modes.

(Again, though I had no direct involvement with this 2009 paper, I had modelled waveguide resonators with step-profile schemes such as mirror masks and wires intended to increase the losses for unwanted transverse modes. Some were tried in collaboration with Denis Hall and his doctoral student Richard Morley).

Banerji et al. considered reflectance profiles R(x) at one mirror and 1-R(x) at the other. The results below (Figures 52) also use this symmetric case, with the convenient generalisation of a “super-Gaussian” profile:

Amplitude reflectivity = exp(-|x/W|N)

Super-Gaussian mirrors were in vogue in the 1980s and I demonstrated one example with a slab CO2 waveguide resonator in collaboration with Heriot-Watt in early 1990 (reference [20] above).

We choose width parameter W = 0.44 a, and the order is N = 6. In the limit of high-order super-Gaussians (large N) this is equivalent to the Banerji et al. mirror profile with a sharp edge and b = c = 0.44.

Because Case I mirror geometry is assumed (that is, each mirror is very near its guide aperture, so the light has no opportunity to diffract before reflection and recoupling), the mirror tilt is well approximated by a phase term exp(jkφx).

Figure 52: Predicted round-trip losses for the first three resonator modes of a dual Case waveguide laser with a “super-Gaussian” mirror of order 6.

The topmost figure shows, as expected from Banerji et al. and earlier work, that for a well-aligned resonator the guide length L = 2a2/λ is associated with a local minimum in round-trip loss. Further minima occur at length multiples where the round-trip phase shifts for important guide modes are again multiples of 2π; the local minimum around L = 4a2/λ is striking.

In their short paper Banerji et al. (2009) do not discuss misalignment or treat the guide losses quantitatively, but it is worth stressing two points again:

First, as we saw above, fairly significant “kinks” in mode losses can be induced by very small misalignments << 1 mrad.

Second, even the Case III guide-mirror configuration, which offers considerable extra coupling-loss discrimination in favour of the lowest-order guide mode, does not guarantee that the lowest-loss resonator mode is well behaved (and sufficiently favoured above its nearest competitor) in the presence of mirror misalignments of mrad order. Case III was, I think, widely seen as a go-to option when the suppression of higher-order transverse modes was important. Its mode discrimination is excellent with good alignment. But it needs extra space (14-15 cm for a guide halfwidth a ~ 1 mm) that is not otherwise useful (unless we are placing other optical components, such as Brewster windows or modulators, in the path of the light), and seriously reduces the free spectral tuning range c/[2(L + z1 + z2)].

So it was a nasty shock c. 1990 that a configuration often viewed as overkill was no such thing:

Figure 53: Predicted round-trip losses for the first three resonator modes of a waveguide laser with one Case I mirror and one Case III mirror. With a = 1 mm and λ = 10.59 μm the resonator is very sensitive to misalignment for guide lengths L ~ 4a2/λ ~ 37-38 cm.

Validity of some modelling assumptions

We must set “reasonable” upper and lower limits to the mode numbers in the EH and TEM expansions, but how?

First, we might ask whether these cosinusoidal functions accurately represent the physics of the waveguide. If ε is a small but uniform manufacturing error, so that the guide width is not 2a but 2(a + ε), the resulting changes in the single-pass propagation phases βmnL are roughly πLλεm2/(8a3). For typical values a = 1 mm, λ = 10.6 μm and L = 300 mm, and with an assumed error ε = λ, this is roughly 0.013 m2 radians, and our calculations of mode interference effects might be seriously wrong for m ~ 10. And there may be little point in adding more and more modes to a model if (say) the uncertainties in guide smoothness or width mean that the uncertainties in their propagation phases βmnL exceed π.

The mth guide mode involves waves propagating at angles ± mλ/4a, so with λ ~ 10 μm and a ~ 1 mm the paraxial assumption that the angles are “small” becomes suspect if m is a few tens.

I thought that a maximum mode order of 10 or 20 might be reasonable, and was sure that only 3 (as in much of the early literature, and compatible with closed-form solution) was insufficient. But, in any case, how accurately were the guides machined or deposited or assembled? They were neither square nor uniform. A photo cross-section of one of our 1980s alumina guides is in Figure 54. It was nominally 1.4 mm square, and the workmanship of our contractors was admirable, but nobody would expect its “modes” to obey the above equations for orders up to tens and hundreds. Modern methods of depositing and assembling guide structures may have far smaller errors, and this is one advantage of a move from ceramic machining for λ ~ 10 μm to silicon or semiconductor fabrication (e.g. GaAs/AlGaAs) for λ ~ 1 μm.

Figure 54: Left: Measured widths 2a of a ceramic guide with L ~ 190 mm and nominal width 1.4 mm. Right: Photograph of guide cross-section.

Computers have limited precision, speed and memory: better than in the 1960s, and in the 1980s when I wrote my first resonator models in BASIC and FORTRAN, but still finite. Some care is needed when we calculate fields and overlaps and diffraction integrals for Bessel (or Hermite, Laguerre etc.) functions of very high order.

One hopes that, as more modes are added to a model and the computer code is re-run, the results (for losses and phase shifts, eigenvectors and eigenvalues) settle down to fixed values and show no further change, well before any of the above constraints is reached. Realising that this was not guaranteed, I placed caveats and disclaimers in my thesis and our papers. The loss curves (or the spiralling 3-D paths of the eigenvalues) become more and more complicated and jagged as more modes are added. This cannot be helped, within the terms of the mode-expansion and self-consistency problem that we posed. A physically equivalent but mathematically or computationally different formulation, for example with adjustable-mesh diffraction integrals, might solve part but not all of the problem.

Note again that waveguide attenuation losses, in many narrow-bore devices, form a significant (though not necessarily dominant) part of the total loss: if they are ignored, wrong answers for output power and tunability result. Nonzero loss coefficients (scaling as the square of mode order m) imply that single-pass Talbot effects are inexact. Resonator transverse modes (in our sense) exactly repeat, by definition, once per round trip; propagating single-pass fields do not, though they may form approximate “Talbot copies” on the way. Approximate regeneration is still very useful, and the parallel development of non-resonator work is greatly to the credit of colleagues such as Mike Jenkins and John Heaton. But extra losses (or differences in loss between two transverse modes) of a very few per cent per round trip can make a great difference to laser operation, and the neglect of guide attenuation will seriously compromise many resonator predictions.

Conclusions on the history: resonators and other “mode manipulation” devices

It is hard to say when these Talbot-length waveguide lasers first appeared, and when they were first deliberately designed.

The team at the University of Hull (under my Ph. D. supervisor Denis Hall) were operating L ~ 4a2/λ lasers at CO2 wavelengths before 1981, because L values of 37-38 cm were already standard convenient lengths for alumina ceramic, and many examples of square-cross-section guides had 2a ~ 2mm. Having derived the rough expressions for cavity resonances – not complicated, but (as it seemed from talks with experienced UK and US researchers) not well known – I proposed in 1986 these geometries for coincidence of resonator mode frequencies (and thus improved transverse mode behaviour throughout the laser signature, if there was no or very small differential in gain). The idea was described again in Hill (1988), and as far as I know it was first demonstrated with our L ~ (16/3)a2/λ design in Malvern. We tried various carefully machined ceramic guides – see Hill et al. (1990) and Hill and Redding (1992) – for example L ~ 2a2/λ ~ 190 mm with a = 1 mm, and also L ~ (16/3)a2/λ ~ 190 mm with a = 0.65 mm. It made sense to use the same vacuum container, RF amplifiers and optical mounts, keeping the guide length roughly constant and adjusting the phase relationships through choice of guide width. The Bactrian (“double-hump”) and dromedary (“single-hump”) power vs tilt behaviours in e.g. Hill, Redding and Colley (1990) – Figure 47 and the like – were obtained with a ~ 0.65-0.7 mm.

Similarly, regarding the maximisation (see above) of a specific “mode purity”, such as EH11 content within the guide or TEM00 content in free space, I am not aware of work that precedes our various Hull and Malvern technical memoranda and demonstrations. There were other groups with similar interests in (at least) continental Europe, the Soviet Union and the USA, and of course several manufacturers made claims about “mode purity”. Our resonator studies were mostly about finding manufacturable designs, around 10 μm wavelength, with specific advantages (such as power, quasi-TEM00 mode, and tunability) and reasonable tolerances; they did not pay much attention to splitting or recombining field patterns in a single pass. They tried to examine carefully the effects of waveguide loss, and to caution against believing first-order expressions for orthogonal mode fields and propagation constants without good evidence.

For example we can design a half-integer solution where EH21, say, lies halfway in frequency between its two EH11 neighbours and thus experiences maximum discrimination when the laser is appropriately tuned. The per-round-trip difference in gain will ensure – in our model – that EH11 lases in preference to EH21; and this preference is maintained up to a certain offset frequency (which we can estimate). Or we can ensure that EH11 always has the same frequency as EH21 – or indeed that all the frequencies coincide – so that the guide attenuation is the only mechanism of mode discrimination, and EH11 must and will lase in preference to EH21 or any other competitor. This is a nice test of our confidence. Such mode-coincidence designs can work well, but there is a large literature on transverse mode coupling and degenerate cavities (although usually these are open, non-waveguide cavities, as in Klaassen et al. 2005) explaining why in the presence of scattering centres and other imperfections things can go wrong.

Mike and colleagues produced many further papers and patents describing multimode interference in waveguide structures, with and without mirrors. To repeat: after a transition period around 1992-93 I worked almost full-time in our laser systems team at Malvern – in part because someone had to replace the late Ken Hulme. I was less involved in these later publications, but luckily I could still design and redesign the local oscillator, beamrider and higher-power lasers mentioned above.

None of this history reduces the value of Talbot effects and mode preservation, but it stresses again that laser designers may face multiple simultaneous challenges, such as moving to P Building and becoming less up to date with colleagues’ work. The ideas of field-preserving and field-splitting guides, although not new in the 1990s or even the 1970s, brought useful insights (and important mass-manufacture optical components such as splitters and combiners) but did not by themselves solve the additional problems of round-trip conditions, real-life imperfections, and active media.

In summary, the caveats and doubts were cowardly but honest. My multimode modelling was an improvement, and produced worthwhile agreement with experiment, and useful new explanations, and (crucially) resonator designs and redesigns that worked in practice. But it was a close run thing. The necessary blend of laboratory and software skills, thinking time, persistence, and confidence that problems could be overcome, was far from guaranteed. The doctoral and Malvern projects might several times have failed or soured. Managerial patience and funds were limited; manufacturing tolerances were borderline; and at times we just had to say that money and time were needed and might solve a serious problem that so far had escaped everyone.

Hugh Lamberton, and Ken Harrison earlier, cannot have been impressed when, short of a professional project manager for our part in the two-nation CLARA programme, they tried hard and politely to appoint me. I thought my best efforts were needed on the technical side, and was unwilling (having experienced “assignment management”) to add a large nontechnical dose. I do not know how that other career would have gone; perhaps I would have written more memos.

When Hugh had to miss a conference in Paris I took his place, gave his apologies, and explained that instead I would speak about Dundonian waveguide lasers and our lidar trials, as I had led our CO2 lidar anemometer team on three MLRS-related trials in southern France (see C A Hill and M Harris, “Lidar: experiences of wind-sensing trials,” Journal of Defence Science, vol. 10, pp. 238 –245, 2006). Several of the audience left, and our French-to-English simultaneous translator (coping fairly well so far) almost tore off the headphones at Laser Ecosse in a strong east-coast-Scottish accent.

Acknowledgements

It is also hard to track the organisational changes but I have thanked various technical and managerial colleagues above. The teams are too numerous to list, but in addition to all my co-authors I wish to thank (in HWO) Phil Conder, Bob Devereux and Trish Gorton; and (at Hull and Heriot-Watt) Howard Baker, Geoff Pert and Pete Wilson.

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