Multimode theory vs experiment

Section 5: Resonators sensitive and insensitive to perturbations: multimode theory vs experiment

The original motivation for studying tilted-mirror cavities was not that we expect to see real CO2 waveguide lasers whose mirrors are out of true alignment by several milliradians. Such large errors are conceivable, but not likely in robust well-built devices. I reasoned rather that grating-tuned CO2 waveguide lasers, in their simplest and commercially popular embodiments, replace one near-case I plane mirror with a “Littrow configuration” grating which approximates a well-aligned plane mirror for the desired CO2 transition but a tilted plane mirror for the neighbouring transitions. The tilt is roughly 2-4 mrad for the nearest neighbours, twice that for the next-nearest neighbours, and so on.

So we designed bespoke mechanisms for in situ (mis)alignment of laser mirrors which sat within the typical large vacuum chambers in our Hull and Malvern labs. Expensive computer-controlled electric and piezoelectric optics mounts would have been nice, but we used Allen key rods, spring-supported adjustment plates, and reflected HeNe laser spots (on ruled graph paper) to obtain power vs tilt data. Colleagues such as Paul Jackson, Alan Colley, Paul Monk and the Malvern team led by Mike Jenkins deserve thanks for their patient help.

It was soon clear that, despite many caveats, our results improved on the existing single-mode or few-mode approaches:

Figure 45: Output power as a function of mirror 2 tilt, P(φ), for z1 ≈ z2 ≈ 2 mm. Symbols refer to three separate experimental data sets, adjusted to give equal peak powers ≈ 1.55 W at φ = 0. Solid line: distributed-loss Rigrod prediction for λ = 10.59 μm, based on multimode matrix model with 15 x 15 EHmn modes and 30 x 20 TEMpq modes. Regions where lines of the R branch appeared are marked; elsewhere the laser preferred the peak of the P branch. From Hill and Colley (1990).

Figure 46: As Figure 45 for z2 = 11.2 mm and z2 = 21.4 mm, with λ = 10.59 μm and 10.25 μm, respectively, in the model. The P branch dominates for z2 = 11.2 mm (regions of R branch operation marked). The R branch dominates for z2 = 21.4 mm (regions of P branch operation marked), as expected from the matrix model results, but the agreement of measured and predicted powers is poor. From Hill and Colley (1990).

We were at least partly explaining the output power, line-hopping and mode-hopping of grating-tuned devices. The switching points, and the “hooting” behaviour near those points, were predicted only roughly, but this was still a sharp advance. “Hooting” was coined to describe the output of a laser on more than one resonating mode simultaneously; it should not occur in a well-behaved laser subject to textbook homogeneous CO2 line broadening and competition, but obviously it occurred in our research lasers and in commercial devices, had not previously been explained, and needed to be understood.

(“Rigrod” equations, named after William Rigrod and his 1960s research, are simplified predictions for laser output power given the cavity mirror reflectances, dissipative losses, and laser gain properties. Typical waveguide resonators need care because there is significant dissipative loss distributed nonuniformly along the guide, not just at the end mirrors; and there are other complications (not discussed here) with the spatial variations of gain and saturation intensity in RF-excited discharges, and the mode-interference effects including variation of transverse profile in both directions along the guide. The work by Henningsen et al. (1990), for instance, is correct and valuable but limited to well-aligned lasers for which “propagation losses in the [circular-bore Pyrex] waveguide are insignificant”. With the tilted-mirror dual Case I studies, I began on the opposite simplifying assumption that the dissipative loss was due solely to propagation. Section 4 has shown that this assumption – restrictive, but accurate for many real lasers with mirrors very near the guide ends or indeed butted against them – still allows interesting results. The initial near-Case I experiments at Hull were just sufficiently simple yet productive, and encouraged me to continue at Malvern).

Some strange, even pathological laser behaviour can be described only briefly here.

The sharp local minima in loss for well-aligned resonators (e.g. Figure 16) were interesting, because low loss is generally the first thing one seeks. But their sharp local maxima suggested to me that, other things being equal in the complicated space of resonator parameters, a laser could be designed to operate in a small region so unfavourable that a small perturbation such as a mirror tilt – something that would not normally help – might reduce the loss. We described how such a laser was designed, built, and shown to agree with theory: as the mirror position was changed, the curve of output power vs. mirror tilt showed a single symmetric peak (the usual form); or showed a dip (a local minimum) with an increase in power when the mirror was tilted; or simply stayed flat – the laser giving roughly the same power for any tilt, within a range (several mrad) that is enormous in terms of conventional alignment.

Figure 47: Extracts from Hill, Redding and Colley (1990). Top: Evidence that the multimode model correctly identifies the resonator modes responsible for undesirable “hooting”; these modes are supposed to be “sub-threshold”, prevented from lasing by the CO2 homogeneous broadening physics, but sometimes they do lase. Reference [13], mentioned here in the context of “Talbot” lengths and 2π phase shifts, is Hill (1988). Bottom: Output power as a function of tilt, P(φ), for z2 ~ 21 mm in a 2a ~ 1.42 mm square-bore laser.

This 1990 paper says “The main and expected result is that both 1.4 mm lasers give a ‘double hump’ or Bactrian P(φ) curve…Clearly, if a tilted (dispersed) wavelength offers more power than the Littrow-selected one, the implications for grating-tuned line selection are amusing”.

This was lighthearted but serious. We had predicted and demonstrated why some grating-tuned waveguide lasers show a reduced (or even zero) range of operation at wavelengths for which the cavity has been built and aligned with all possible care. It is well aligned – but it does not obey single-mode theory.

We now had a combination of understanding and computer codes that offered a chance to design – with far less trial and error – lasers that were stable against modest perturbations. I made the point to colleagues in RSRE/DRA/DERA, UK industry, and the NATO RSG-13 panel on laser technology (see insert below). Already in the 1980s and into the 1990s much time and money had been spent on badly behaved lasers before their perturbation behaviour was appreciated and corrected. I do not say that UK lasers in the programmes to which I had access were particularly poor, or worse than those made elsewhere.

In the specific area of this article, one of our 1990s lidar technical demonstrators called for two CO2 waveguide lasers, one large and one small. The proposed large laser had been developed through some years of “VX” contract work, initially aimed at a 20 W upgrade for RSRE’s FINDER lidar. It had a folded cavity with an electro-optic crystal modulator; it was intended to run quasi-TEM00 in pulsed and CW modes, but suffered from power loss and mode distortion.

In 1990, alarmed, I was not always right or tactful, but time was short and I felt bound to speak up. The memo on TEM matrix modelling says “even resonators with EH11-favouring Case III reflectors are prone to mode/alignment problems if N is chosen incorrectly”. Then at the end of October 1990 “Recent work on waveguide resonator theory suggests that the Case III reflector is no guarantee of good mode quality, contrary to generally accepted opinion. There are ranges of guide Fresnel number…where the resonator mode losses and profiles are highly sensitive to misalignment…With this in mind I inspected [the currently proposed design] and found, as expected, that the resonator lay in a region of some risk…During September and October 1990, a detailed resonator model was run by Dr Banerji of Royal Holloway’s Computer Science Department, and his work confirmed my suspicion. If, as seems reasonable but is unproved, the crystal distortion effects are dominated by misalignment (or any process coupling the guide modes of odd and even parity), our [redesign] should raise the threshold at which the fundamental mode is seriously threatened”.

(I was the RSRE technical authority, from late 1987, for this Royal Holloway contract on laser modelling. I make no attempt here at a history of the wider SERL/Baldock/RSRE/DRA/DERA programmes, including Mike’s dedicated work as Hollow Waveguide Optics leader. He and others are far more qualified).

The problems were addressed (to the point of allowing successful flight tests) through a perturbation-insensitive resonator design and, later, an understanding of the polarisation behaviour of lasers with multiple Brewster-angled surfaces [Hill et al. 1996]. The powers obtained in a reasonable low-distortion mode were increased by perhaps 50-70 %. In general, laser weak points tended to be power supplies and cooling systems rather than optics. Obviously I remember the nervous time-pressured decisions about this mid-90s ceramic guide geometry, betting our project funds and future results on my resonator modelling, but it may have been just as important that the thermal sinks were upgraded.

The second laser, used as a local oscillator when the first was in pulsed mode, was designed for reliable near-TEM00 output (as previously demonstrated, in 1988, for a beamrider laser design); it gave very little trouble.

Many will recall this as another period of reorganisation, changing priorities, Severe Restraint in funding, and long delays in contract approval and financing, before “agreed” technical tasks could start. We and UK industry and universities muddled through. In its more commercial guise, DRA showed confidence in my work and won a small contract for my redesign of another beamrider laser. Andrew Scott led our group at the time and helped to negotiate the redesign payment. Also, during the 1990 RSG-13 meeting in Québec / Valcartier I had promised that 1991 would see a UK talk in French, in Paris. This was bold, but Andrew approved some French conversation classes, and soon there were benefits in the MLRS trials and other collaborations with ONERA and French government scientists.

And colleagues will agree that sometimes in the 1990s, as we increased our proportion of bidding and collaborating, issues of etiquette and good practice arose – what should one do? I was now working on Doppler lidar systems for UK and international trials, as well as (with intermittent support) the resonator modelling legacy. Our partners (from the UK and at least two other countries) had, I thought, gone wrong. For instance, the frequency-stabilisation method for an important technical demonstrator lidar source (from one company) was incompatible with its Doppler algorithms (from another); the latter were faulty in any case; nobody noticed until I read manuals for both. Later there was annoyance when I said that the impressively low frequency noise of an expensive lidar source was being swamped by other real-life noises and was irrelevant in practice.

So our large collaborative projects ran bumpily through various administrative and technical challenges, but I must add my appreciation of the managers who allowed essential time and resource for concentrated thinking and calculating: Jayne Ackroyd, Richard Fletcher, Norm Geddes, Kevin Welford and others as well as those mentioned in Section 1.

Interlude: Output beam patterns observed in 1983-4 from Hull University CO2 waveguide lasers

Figure 48: “TEM00-like” burn pattern (left), and burn pattern from a misaligned laser (right).

Figure 49: Five burn patterns. The main lobes can be widely separated, suggesting a high content of modes of order 4 or more. The leftmost burn pattern has just captured two faint lobes that are “interior” i.e. nearer the beam axis. From left to right there are then two attempts to capture the (predominantly) two-lobe emission; then (on different paper or card) the two lobes strikingly separated; finally a splodge of two or three overlapping lobes.

My doctoral work at Hull included the design and testing (greatly helped by Paul Monk) of a computer-controlled scanning beam profiler that could accommodate such unusually divergent beams.

Interlude: NATO Research Study Group 13 on laser technology (RSG-13)

I followed Dave Willetts in 1990, and Phil Soan followed me in 1998, as a UK national representative on RSG-13. This post sounded impressive but involved much legwork and, at each yearly meeting, the presentation of various UK achievements where colleagues trusted me to describe their work after a steep short learning curve. Conversely I had to absorb the other representatives’ presentations, and circulate summaries to UK government and industry. Viv Roper announced the RSG-13 appointment (to my surprise) at one of our group Christmas meals.

RSG-13 in Germany, May 1990.

P H (Hywel) Davies is at far right, front row. He was a composer, singer in Malvern Festival Chorus, and PSO at RSRE. We shared an office when I moved to P Building.

RSG-13 chairman Robert T (Bob) Hintz is fourth from left, front row.

Beside him, fifth from left, is Dieter Höhn our host at FfO Tübingen; his successor Maurus Tacke had published the curved-wavefront paper that helped me towards an explanation of RSRE’s anomalous results (see above). Maurus’s colleague Reinhard Ebert is third from left, back row, between Alain Le Dortz and Lee Pratt.

Sandy Smith from the US Air Force labs is second from left, middle row; at his invitation I gave a tutorial talk on resonators at Wright-Patterson in 1991. Other national representatives include Knut Stenersen (Norway), Pierre Mathieu (Canada) and Bob Lerou (Netherlands). All are wearing jackets.

All these countries had waveguide CO2 laser programmes. I attended RSG-13 lidar trials as a UK delegate (at Captieux and Huntsville). No longer a delegate, but invited by the US, I attended 2001 multisensor trials at China Lake and analysed the results. Helping the US team while European shipments of technical kit were delayed in US Customs, I had interesting experience of a lidar with furred cooling pipes and tricky rf matching in Carl Buczek’s workshop; see comment above on power supplies and cooling.