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Landreman et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2421

Comparison of parallel and perpendicularpolarized counterpropagating light for suppressing

high harmonic generation

Matt Landreman,1 Kevin O’Keeffe,1,* Tom Robinson,1 Matt Zepf,2 Brendan Dromey,2 and Simon M. Hooker1

1Department of Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK2Department of Physics and Astronomy, Plasma and Laser Interaction Physics,

Queens University Belfast BT7 1NN, UK*Corresponding author: k.okeeffe1@physics.ox.ac.uk

Received February 2, 2007; revised June 28, 2007; accepted July 9, 2007;posted July 16, 2007 (Doc. ID 79601); published August 27, 2007

The use of counterpropagating laser pulses to suppress high harmonic generation (HHG) is investigated ex-perimentally for pulses polarized parallel or perpendicular to the driving laser pulse. It is shown for the firsttime that perpendicularly polarized pulses can suppress HHG. The intensity of the counterpropagating pulserequired for harmonic suppression is found to be much larger for perpendicular polarization than for parallelpolarization, in good agreement with simple models of the harmonic suppression. These results have applica-tions to quasi-phase-matching of HHG with trains of counterpropagating pulses. © 2007 Optical Society ofAmerica

OCIS codes: 190.0190, 190.4160.

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. INTRODUCTIONhen a linearly polarized laser pulse is focused into a low

ensity gas, high-order harmonics of the fundamentalriver frequency are radiated as a coherent, low diver-ence beam [1–4]. This high harmonic generation (HHG)s an attractive source for coherent extreme ultravioletEUV) radiation and has been the subject of intense re-earch for many years. However, the usefulness of HHG isimited by its low conversion efficiency, typically 10−5 [5].he primary cause of poor conversion of laser energy intoarmonic energy is a phase mismatch between harmonicsenerated at different points in the medium. In the casef a laser beam focused into a gas target, each harmonicill experience dispersion due to free electrons, neutraltoms, the Gouy phase shift, and the intensity-dependentelative phase between the driving wave and the localarmonic emission. In the presence of this dispersion thearmonic and the driving field will quickly become out ofhase with each other, preventing further growth of thearmonic signal. Many techniques for overcoming thishase mismatch have been investigated. True phaseatching of HHG (�k=kq−qk0=0, where kq is the k vec-

or of the qth harmonic and k0 is the k vector of the lasereld) has been demonstrated by guiding the driving laserulse in a hollow capillary and balancing the dispersion ofhe capillary waveguide and the partially ionized gas6,7]. However, this approach is limited to low levels ofonization since for higher levels of ionization the anoma-ous dispersion of the plasma and waveguide cannot bealanced by the normal dispersion of the neutral species.At higher ionization levels the problem of phase mis-atch can be overcome using the technique of quasi-

hase-matching (QPM). With this technique the genera-ion of harmonics is suppressed in regions where the

0740-3224/07/092421-7/$15.00 © 2

acroscopic harmonic wave would be out of phase withhe local harmonic emission, with a periodicity, LQPM2� /KQPM, such that �K=�k−KQPM=0. The overall har-onic signal can then be dramatically increased. Previ-

us work on QPM of harmonics used a modulated wave-uide [8] to introduce an intensity modulation along thexis of the waveguide. Since harmonics generated in theut-off region are very sensitive to intensity it was pos-ible to turn the generation of the cut-off harmonics onnd off in a controlled manner, leading to improved effi-iency, even in the presence of substantial ionization. Anlternative approach for QPM was proposed by Peatrosst al. [9], who suggested that a relatively weak parallelolarized counterpropagating laser pulse could be used touppress harmonic production. Further, a train of suchulses could be used to suppress harmonic generation pe-iodically, allowing the possibility of QPM. An initial ex-eriment to demonstrate this effect was performed byoronov et al. [10]. In this experiment a single parallel po-

arized counterpropagating pulse was used to suppressarmonic production in an argon gas jet. The first experi-ent to demonstrate QPM with a train of pulses was re-

ently performed by Zhang et al. [11]. In that experimenttrain of two pulses was used to quasi-phase-match har-onics produced in a hollow waveguide. Using this tech-ique it was shown that in the presence of the counter-ropagating pulses the harmonic signal could benhanced by more than an order of magnitude.

In this paper we further investigate the effect of a coun-erpropagating beam (CPB) on the generation of harmon-cs in a gas cell. In particular, we investigate the effect ofsing a perpendicularly polarized counterpropagatingeam and compare this to results obtained with a parallelolarized beam. We demonstrate that it is possible to sup-

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2422 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Landreman et al.

ress HHG using perpendicularly polarized light, andherefore QPM schemes should also be achievable usingerpendicularly polarized light as well as parallel polar-zed light. We show that for the harmonics studied in thisaper the intensity required to suppress harmonic gen-ration with a perpendicularly polarized beam is approxi-ately ten times higher than for a parallel polarized

eam. Despite the higher energy requirement, a perpen-icularly polarized beam offers a significant practical ad-antage. To realize the full potential offered by QPM withCPB it is necessary to increase the number of phase-atching zones, which requires guiding both the driver

nd CPB. In such cases the CPB will travel exactly alonghe path of the driver beam and into the laser system.locking a perpendicularly polarized beam can bechieved straightforwardly with a linear polarizer; block-ng a parallel polarized CPB requires more complex solu-ions, such as the use of a Faraday isolator that intro-uces significant material into the beam path.

. HARMONIC EXTINCTION BY PARALLELND PERPENDICULARLY POLARIZEDIGHTere we outline how parallel and perpendicularly polar-

zed counterpropagating beams can suppress HHG.

. Parallel Polarizationonsider first the situation in which the counterpropagat-

ng light is linearly polarized along the same axis as theriving field. In this situation, the total harmonic outputs reduced due to phase scrambling of local harmonicmission. This phase scrambling arises from two pro-esses: direct and intensity-dependent phase modulation.o model direct phase modulation Peatross et al. [9] con-idered two plane waves propagating in opposite direc-ions. The electric field in the region where the two beamsverlap is given by

E�z,t� = Re�Edei�kz−�t� + ECPBei�−kz−�t��

= Re�Et�z�ei�kz−�t+��z���, �1�

here Ed is the electric field amplitude of the drivingeam, ECPB is the amplitude of the counterpropagatingeam, and

Et�z� = Ed�1 + �ECPB

Ed�2

+ 2ECPB

Edcos�2kz�, �2�

��z� = − arctan�ECPB

Edsin�2kz�

1 +ECPB

Edcos�2kz� . �3�

he variation in the phase is plotted in Fig. 1 forCPB/Ed=0.1, corresponding to an intensity ratio of:100. It can be seen from this figure that even a rela-ively weak CPB results in a modulation of the fundamen-al intensity and phase. A shift in the phase of the funda-ental by � corresponds to a shift in the phase of the qth

armonic by q�. Hence, while the phase shift may only be

small fraction of � for the fundamental, it will be sig-ificant for harmonics of sufficiently high order. For the

ntensity ratio considered in Fig. 1, harmonics of order q15 will experience a direct phase modulation greater

han �. This fast modulation of the phase prevents thesearmonics from achieving any significant strength. Theet emission of harmonics with q�15 will therefore bereatly reduced compared to the ECPB=0 case.

One can estimate the intensity required in the CPB toxtinguish the harmonics of order q through this mecha-ism. From Eq. (3), the peak-to-peak phase variationaused by the counterpropagating beam, ��P, is

��P = 2 arctan� ECPB/Ed

�1 − �ECPB/Ed�2� 2�ECPB

Ed� , �4�

here the approximation holds for small ECPB/Ed. Har-onics of order q will be extinguished when this phase

ariation is of order � /q. Rearranging this extinction con-ition in terms of the beam intensities yields

ICPB

Id= �ECPB

Ed�2

= � �

2q�2

. �5�

As well as this direct modulation the harmonic phase isodulated indirectly through the modulation of the driv-

ng intensity. According to the quasiclassical theory of Le-enstein et al. [12], the phase of the induced atomic di-ole moment depends on the relevant electronrajectories. For a given electron recollision energy, therere two trajectories—a short and a long trajectory—forach optical cycle. Analytical models suggest that thehase of the harmonic emission varies with intensity12–14] approximately according to

ig. 1. (Color online) Calculated harmonic phase modulationaused by modulation in the phase of the fundamental for har-onic orders q=15 (red solid curve) and q=31 (blue dotted curve)

sing Eq. (3). Intensity-dependent phase modulation using Eqs.2) and (6) is also shown (black dashed curve), assuming k=2�.alculations assume a driver with 1�1014 W/cm2 and a counter-ropagating wave of 1/100 the driver intensity. Figure adaptedrom [10].

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Landreman et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2423

��q

�I= −

Ke2

2�0cme��3 , �6�

here K is a dimensionless constant that differs for eachrajectory. For the long trajectory K2�, and for thehort trajectory K2� /10 [15]. From Eq. (2) the peak-to-eak intensity variation caused by the CPB is given by�ICPBId. Combining this result with Eq. (6), the peak-to-eak phase variation due to this effect, ��1, is approxi-ately

���I� 2Ke2

�0cme��3�ICPBId. �7�

herefore, a phase variation of � or more arises from in-irect phase modulation when

IdICPB �4 � 1026

K2 �W/cm2�2. �8�

For a driving beam intensity of 2�1014 W/cm2 as littles 0.1% of the driving intensity is needed in the CPB for�I�, if the long trajectory is the most relevant path.owever, approximately 100 times more intensity will be

equired in the CPB if the short trajectory is dominant.ccording to Lewenstein et al. [12], the long trajectoryill dominate in the case of a single atom. However, Le-enstein et al. also show that if propagation effects arellowed for, there are substantial regions where the shortath will dominate.

. Perpendicular Polarizationhe physical mechanism responsible for extinction by aerpendicularly polarized pulse is fundamentally differ-nt from that of the parallel polarization case. With a per-endicularly polarized CPB, harmonic generation is ex-inguished due to the ellipticity that results when theeams overlap. Consider the case of colliding cw planeaves with orthogonal polarizations. The total electriceld is the sum of the fields from each beam:

E�z,t� = Re�Ed exp�i�kz − �t���1

0�+ rEd exp�i�− kz − �t���0

1� , �9�

here r=ECPB/Ed is the ratio of the counterpropagatinglectric field amplitude to that of the driver, and the two-omponent vectors represent the two-dimensional spacef possible electric field directions in the transverse plane.t can be shown by construction that any two-componentomplex vector � a

b � can be represented in the form

ei���u

� + i��

− u�� , �10�

here u, v, �, and � are real quantities. Thus, for eachalue of z, values �u ,v ,� ,�� can be found to put E�z , t�nto a form resembling the one in Eq. (9):

E�z,t� = Re�Ede−i�tei��z���u�z�

�z�� + i��z�� �z�

− u�z��� .

�11�

n this representation, it can be seen that the electric fieldill trace out an ellipse with time, where the ellipse wille different for different z. The major and minor axes of

he ellipse are given by �u�z�v�z� � and � v�z�

−u�z� � (not necessarilyn that order). The ellipticity, ��z�, is defined as the ratio ofhe peak field along the minor axis and the peak fieldlong the major axis. Writing Eq. (9) in the form of Eq.11) yields

��z� =1 + r2 − �1 + r4 + 2r2 cos�4kz�

2r sin�2kz� r sin�2kz�,

�12�

here the last step is valid if �r��1. The ellipticity, ��z�, islotted in Fig. 2 for several values of r. A counterpropa-ating beam of perpendicular polarization will thus resultn a position-dependent ellipticity.

The effect of ellipticity on HHG can be understood inerms of the quasiclassical three-step model [16]. For anlliptically polarized electric field the trajectories of thelectrons after ionization in general do not return to thearent ion. Recombination, and harmonic generation, willherefore be suppressed. The ellipticity required to sup-ress harmonic generation can be small. Consider elec-rons ionized at a phase that would yield the maximumeturn energy in the case of linear polarization. If lightith a small ellipticity is considered it may be shown that

hese electrons will miss the ion by roughly.7� eE0 /me�

2. For a driving intensity of 2�1014 W/cm2

he electron will miss the parent ion by at least the Bohradius when the ellipticity ��0.008. The strong depen-ence of the efficiency of HHG on ellipticity has been con-rmed by many experiments, which utilized a single el-

ig. 2. (Color online) Ellipticity, ��z�, as a function of z for a per-endicularly polarized counterpropagating beam with peak elec-ric field amplitude equal to a fraction r of that of the main driv-ng pulse using Eq. (12).

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2424 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Landreman et al.

iptically polarized beam [17–20]. In these experimentshe intensity of the qth harmonic signal varies approxi-ately according to

Iq��� = Iq�0�e−A�q��2. �13�

The experimental results for single-beam HHG effi-iency can be used to predict the efficacy of extinction us-ng perpendicularly polarized counterpropagating light.he parameter A�q� was found from the experimentalingle-beam results [17] to be A�21�32 and A�29�37.he values of A�q� for other values of q are then found by

inear extrapolation.At any particular point, z, the electric field contribu-

ion, , to the total harmonic output will be reduced ac-ording to Eq. (13) and the ellipticity at that location:

= �exp�− A�q���z�2� = exp�−A�q���z�2

2 � , �14�

here ��z� is given by Eq. (12). Averaging over the rel-vant period �� /4�, and squaring to convert fields to inten-ities yields the ratio of the contribution to the harmonicignal from this region with and without the perpendicu-arly polarized CPB:

Iq,CPB

Iq,no CPB= �4

��

0

�/4

exp�−A�q���z�2

2 �dz�2

�exp�−A�q�r2

4 �I0�A�q�r2

4 ��2

exp�−A�q�

2 � , �15�

here I0 is the modified Bessel function, and the approxi-ations are valid for r�1. Figure 3 illustrates the exact

esult, evaluating the integral numerically rather thanaking the approximation. It becomes hard to decrease

he harmonic signal by more than 90% because there are

ig. 3. (Color online) Calculated ratio, �r�2, of the intensity of thePB to that of the driving laser required to suppress harmonicrders q=21 and q=29. Curves are calculated from Eq. (15) usingmpirical A values for argon.

q

lways �=0 points at which harmonic production cannote extinguished. To find the CPB intensity required foralf extinction the approximate formula is reasonable,ielding ICPB,50%1.4/A�q�.

The analysis in this section is based on the interactionf two cw plane waves. However, in a real experiment thentensity will vary with time and position. The analysis istill suitable to describe the effect of a CPB on cut-off har-onics since cut-off harmonics are generated only in a

mall region of space and time—when the laser passeshrough the focus. The driver and CPB intensity will notary significantly in this region, so the cw plane-waveodel is reasonable. Plateau harmonics, on the otherand, are generated over a range of intensities. In thisase it is likely that the overlap between the driver beamnd CPB will be suboptimal in some regions. Therefore,he cw plane-wave model probably underestimates thePB intensity required to extinguish plateau harmonics.

. EXPERIMENTAL SETUPhe laser system used in this experiment was a Ti:sap-hire CPA system capable of delivering 125 mJ, 50 fsulses at a 10 Hz repetition rate with a center wave-ength of 808.8 nm. To create a counterpropagating pulsehe beam was split before the compressor using an R25% beam splitter. The reflected beam was the driver,sed to generate harmonics, and the transmitted beamas used as the counterpropagating beam. The CPB coulde delayed relative to the driver beam using a computer-ontrolled timing slide. A pulse train consisting of up to28 evenly spaced pulses could be produced by passinghe counterpropagating beam through a series of calciterystals. The details of this method of producing a pulserain will be published elsewhere, and consequently onlybrief description will be given here [21]. If a plane po-

arized pulse is sent through a birefringent crystal withts plane of polarization at 45° to the fast axis, pulses willeave the crystal polarized parallel and perpendicular tohe fast axis. These two pulses will be delayed with re-pect to each other owing to the different group velocitiesf the two polarizations within the crystal. The delay be-ween the two pulses is linearly related to the crystalhickness. By using a sequence of crystals of suitablehickness a train of evenly spaced pulses may be created,

ig. 4. (Color online) Configuration of the compressor and tar-et chambers used throughout the experiment. � /2, half-wavelate; T, gas cell target; B, beam dump.

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Landreman et al. Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. B 2425

nd the polarizations of the pulses may be made parallely placing a linear polarizer at 45° to the fast axis of theast crystal. By changing the arrangement of the crystalsn the chain, the number of pulses in the train as well ashe separation between pulses can be altered. In this ex-eriment the total thickness of the calcite crystals in-erted to create the CPB was 52.3 mm. Additional glassas included in the path of the driver beam to balance theispersion experienced by both driver and counterpropa-ating beams. Both beams were then sent through theame compressor and then separated and focused on amm long gas cell from opposite directions using f679 mm singlet lenses. Both beams had a 1/e2 diameterf 13 mm, corresponding to f /52 focusing and giving aheoretical minimum spot size of 27 �m. Each beam pathontained a wave plate-polarizer pair for independent en-rgy control. A half-wave plate was placed in the CPB sohat the polarization could be changed between parallelnd perpendicular orientations. The CPB was steered offhe driver line, so that the beams only overlapped at theocus, to prevent beam propagation back up the laser.his rotation amounted to approximately 2°. This nonco-xial pointing in the gas cell slightly diminishes the over-ap between the two beams, but the change is small com-ared to the total overlap volume [22]. A schematic of thexperimental setup is shown in Fig. 4.

. EXPERIMENTAL COMPARISON OF HHGXTINCTION BY PARALLEL ANDERPENDICULARLY POLARIZED LIGHTseries of scans was made to compare extinction of HHG

y a parallel and perpendicularly polarized CPB. Allcans were performed with 80 mbar argon in the gas cellnd 8.2 mJ in the driver beam. The pulse duration of theriver and each pulse in the CPB were measured by crossorrelation to be approximately 280 fs. The peak intensityf the driver was therefore approximately 2.51015 W/cm2. The CPB used in these experiments com-

rised 16 pulses at the minimum possible spacing. As theuration of each pulse was long compared to the mini-

ig. 5. (Color online) (a) Variation of the high-harmonic spectrandom locations. Each “shot” was a 30 s exposure during whichn the spatial dimension of the CCD. (b) Same spectra, arranged inear a timing slide position of 127.6 mm.

um spacing between the pulses of approximately 160 fs,he superposition of these pulses resulted in a roughlyquare-wave pulse of duration 4 ps. The maximum valuef the peak intensity used in this experiment for eachulse in the CPB was therefore 4�1013 W/cm2, which,ccording to the Ammosov-Delone-Krainov (ADK) ioniza-ion rates [23], is insufficient to cause significant ioniza-ion during the pulse. At each timing slide position a 30 sxposure of the harmonic signal was taken, correspondingo 300 laser shots.

The HHG spectrum was recorded as a function of theelay between the driving laser pulse and the CPB. Toistinguish the effects of the timing slide position fromystematic effects, the scans were performed in a randomrder. Figure 5(a) shows the spectra in the order in whichhey were recorded. It is clear from this graph that thereas no significant signal variation of the recorded spec-

rum with time. Figure 5(b) shows the data sorted by tim-ng slide position. It can be seen that for certain values ofhe timing slide the HHG signal was decreased dramati-

a function of shot number while the timing slide was moved toiving laser pulses were incident on the target. Counts are binnedof timing slide position, showing reproducible extinction of HHG

ig. 6. (Color online) Measured harmonic signal as a function ofhe ratio, �r�2, of the intensity in the CPB to that in the driveream for various harmonic orders, q, and for both parallel anderpendicular polarizations.

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2426 J. Opt. Soc. Am. B/Vol. 24, No. 9 /September 2007 Landreman et al.

ally. At these timing slide positions the CPB collidedith the driver beam in the generating region of the gas

ell, thus disrupting the production of harmonics. It islso apparent that different order harmonics were extin-uished over slightly different ranges of timing slide po-ition. This is due to the fact that different harmonicsere generated in different regions in the gas cell, as a

esult of the spatially varying intensity of the driveream. Figure 5(b) also shows that the CPB does not ionizehe gas since the harmonic signal when the CPB arrivesefore the driver is the same as that when the CPB ar-ives after the driver.

Experiments were then performed to determine the de-endence of the harmonic extinction on the intensity ofhe CPB for both parallel and perpendicularly polarizedight. Harmonic spectra were recorded as a function of theulse energies in the CPB with the timing slide fixed at27.6 mm so that all the recorded harmonic orders visiblen the spectrometer were extinguished. As in the scan de-cribed earlier, data were taken with the independent pa-ameter values ordered randomly to eliminate spuriousrends. Figure 6 shows the dependence of the harmonicignal on the intensity of the CPB for several harmonicsnd for both parallel and perpendicular polarizations. Its clear that much lower intensities in the CPB are re-uired for a given reduction in harmonic signal for paral-el polarization than for perpendicular polarization. Thisoint is illustrated again in Fig. 7, which plots the CPB-river intensity ratio needed to reduce the harmonic sig-al by 50%. The predicted CPB-intensity ratios resultingrom direct phase modulation, indirect phase modulation,nd position-dependent ellipticity according to Eqs. (5),8), and (15), respectively, are also shown.

ig. 7. (Color online) Measured fraction of the intensity in thePB to that of the driving pulse required for half-extinction of

he harmonic signal for several different runs. Solid symbolshow results obtained with a perpendicularly polarized CPB,pen symbols correspond to a parallel polarized beam. The solidreen curve shows the predicted dependence for perpendicularolarization using Eq. (15). The dashed orange curve gives theredicted behavior arising from phase modulation of the driveream by the CPB, Eq. (7). The calculated behavior arising fromntensity modulation of the fundamental, Eq. (5), is shown forhe case of the long (dotted blue curve) and short (dotted–dashedlack curve) trajectories being dominant. The experimental datandicate that in this case the short trajectory dominates.

For both polarizations, the dependence on harmonic or-er is weak, as expected. For the perpendicular polariza-ion scheme, the data accord very well with that predictedy Eq. (15). From Fig. 7 it can be seen that, in the case ofarallel polarization, it is direct phase modulation that isesponsible for extinguishing harmonics. For parallel po-arization the required intensity for extinction agrees wellith that predicted for the direct phase modulation effectiven by Eq. (5). The parallel polarization data also indi-ate that the long trajectory is not the relevant trajectoryere. In Fig. 7 the effect of intensity-dependent phaseodulation agrees with the data only if the short trajec-

ory is dominant. If the long trajectory was dominant itould be expected that approximately 10 times less inten-

ity in the CPB would be required to extinguish harmon-cs due to intensity modulation of the fundamental.

. CONCLUSIONrevious work on extinguishing HHG with counterpropa-ating beams has used light polarized parallel with theriver beam [10,11]. We have succeeded in demonstrating,or what we believe to be the first time, that a perpendicu-arly polarized beam may also be used to extinguishHG. The physical mechanisms for extinction using botherpendicular and parallel polarization were discussed.n particular, it was shown that a perpendicularly polar-zed CPB extinguishes HHG by introducing an ellipticityo the radiation field in regions where the driver and CPBverlap. Using a perpendicularly polarized CPB has theisadvantage that a higher intensity CPB is requiredhan in the case of parallel polarization. However, the per-endicular polarization scheme offers the practical ad-antage that the CPB can easily be prevented from propa-ating back into the laser system, which will bearticularly important if waveguides are used to extendhe distance over which QPM can be achieved. In practice,aveguides are necessary to observe enhancement of har-onic generation due to QPM, since for QPM to be effec-

ive harmonics must be generated over many coherenceengths. For this reason no enhancements such as thoseeported by Zhang et al. [11] were observed during thisxperiment, since focusing into a gas cell did not generateany coherence lengths. In the case of a CPB with paral-

el polarization it was observed that the data were consis-ent with the short, rather than the long trajectory, beinghe relevant trajectory for harmonics production.

CKNOWLEDGMENTShis work was supported by the Engineering and Physi-al Sciences Research Council through grant EP/005449. M. Landreman acknowledges the support of thehodes Trust. M. Zepf is holder of a Royal Society Wolfsonerit Award.

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