Passive alignment stability and auto-alignment of multipass amplifiers based on Fourier transforms
Karsten Schuhmann, Klaus Kirch, Andreas Knecht, Miroslaw Marszalek, Francois Nez, Jonas Nuber, Randolf Pohl, Ivo Schulthess, Laura Sinkunaite, Manuel Zeyen, Aldo Antognini
PPassive alignment stability and auto-alignment of multipassamplifiers based on Fourier transforms
Karsten Schuhmann , Klaus Kirch , Andreas Knecht , Miroslaw Marszalek , Francois Nez ,Jonas Nuber , Randolf Pohl , Ivo Schulthess , Laura Sinkunaite , Manuel Zeyen , and AldoAntognini Institute for Particle Physics and Astrophysics, ETH, 8093 Zurich, Switzerland Paul Scherrer Institute, 5232 Villigen PSI, Switzerland Laboratoire Kastler Brossel, UPMC-Sorbonne Universit´es, CNRS, ENS-PSL ResearchUniversity, Coll`ege de France, 75005 Paris, France. Johannes Gutenberg-Universit¨at Mainz, QUANTUM, Institut f¨ur Physik & ExzellenzclusterPRISMA, 55128 Mainz, Germany * Corresponding author: [email protected] 18, 2019
Abstract
This study investigates the stability to tilts (misalign-ments) of Fourier-based multi-pass amplifiers, i.e., ampli-fiers where a Fourier transform is used to transport thebeam from pass to pass. Here, the stability propertiesof these amplifiers to misalignments (tilts) of their opti-cal components has been investigated. For this purpose, amethod to quantify the sensitivity to tilts based on the am-plifier small-signal gain has been elaborated and comparedwith measurements. To improve on the tilt stability bymore than an order of magnitude a simple auto-alignmentsystem has been proposed and tested. This study, com-bined with other investigations devoted to the stability ofthe output beam to variations of aperture and thermal lenseffects of the active medium, qualifies the Fourier-basedamplifier for the high-energy and the high-power sector.
Multipass laser amplifiers are used to boost the output en-ergy of laser oscillators owing to their smaller losses andhigher damage thresholds [1, 2, 3, 4]. Because of theirapparent simplicity, typically less attention is devoted tothe design of multipass amplifiers compared to oscilla-tors [5, 6, 7].Multipass amplifiers are commonly based on relayimaging (4f-imaging) from active medium to activemedium as the imaging provides identical beam size for each passes at the active medium. The propagation in the4f-based amplifier takes thus the simple form [8] AM − f − AM − f − AM − f .... , where AM indicates a pass on the active medium, and 4fa relay imaging. However, in this design the changes ofthe phase front curvature caused by the thermal lens at theactive medium are adding up from pass to pass, result-ing in output beam characteristics (size, divergence, qual-ity) strongly dependent on the thermal lens of the activemedium.In this study, Fourier-based multipass amplifiers areconsidered, i.e., multipass amplifiers where the propa-gation from active medium to active medium is accom-plished using an optical Fourier transform. The beampropagation in the Fourier-based amplifiers takes the form[9] AM − Fourier − AM − SP − AM − Fourier − AM − SP − AM − Fourier − AM − SP − AM ... , where SP represents a short free propagation and Fourierany propagation that performs an optical Fourier trans-form. In contrast to the 4f-based amplifiers, Fourier-basedamplifiers show output beam characteristics which are in-dependent on variations of the thermal lens ensued by thepumped active medium [9]. This relies on the peculiarproperty of the optical Fourier transform that inverts thephase front curvature, so that the phase front distortions1 a r X i v : . [ phy s i c s . op ti c s ] M a r ccurring in one pass at the active medium are canceled (infirst order) by the successive pass [9]. Another advantageof the Fourier-based design is the transverse mode clean-ing that occurs from the interplay between Fourier trans-form and soft aperture effects at the active medium, favor-ing TEM00-mode operation. Therefore, the Fourier-baseddesign represents an alternative to 4f-based designs espe-cially at the energy and power frontiers where the thermallens becomes one of the paramount limitations.The concept of Fourier-based amplifiers has been in-troduced in [10] and successively elaborated in [11, 12,13, 14, 9], where a detailed comparison between Fourier-based and state-of-the-art multipass amplifiers based onthe 4f-relay imaging is presented. In contrast to these ear-lier considerations, here another crucial aspect of laser de-sign is considered, namely, the stability of the amplifierperformance to misalignments (tilts).This study reveals that Fourier-based designs have alsoexcellent (passive) stability properties for misalignments(tilts). Furthermore, it demonstrates theoretically andpractically, that the Fourier-based design is well apt fora simple auto-alignment system that mitigates the beamexcursion from the optical axis (perfect alignment) causedby the tilts of its various optical components. Hence, thisstudy, combined with the investigations of [9], displays theexcellent performance of the Fourier-based architecturewith respect to thermal lens variations, aperture effects,beam quality, efficiency and alignment stability, qualify-ing it for high-power and high-energy applications. The following procedure is undertaken to evaluate the sen-sitivity of the considered multipass amplifiers to misalign-ments. First, the size of the fundamental Gaussian beam(TEM00 mode) propagating in the amplifier is evaluatedwith the ABCD-matrix formalism applied to the complexGaussian beam parameter q . A complex ABCD-matrix isused to model the aperture effect occurring in the activemedium [6, 14, 9]. As a second step, the excursion ofthe Gaussian beam from the optical axis (the beam prop-agates on the optical axis for perfect alignment) is com-puted making use of the ABCD-matrix formalism appliedto the geometrical ray describing the axis of the Gaussianbeam. For this purpose, another ABCD-matrix must bedefined, describing the aperture effects to this ray propa-gation. The excursion of the laser beam from the opticalaxis can be used as a measure of the sensitivity of the mul-tipass system to misalignment effects. Knowledge of thisexcursion can be exploited also to compute the decrease ofthe multipass amplifier gain caused by the misalignments. The dependency of the gain (transmission) on tilts con-stitutes the sensitivity of the multipass amplifier to mirrortilts.In this study, it is assumed that losses in the multipasssystem occur only at the active medium whose position-dependent gain (and absorption in the unpumped region)can be approximated by an average gain and a position-dependent transmission function (soft aperture) [9]. It isfurther assumed that the soft aperture of the active mediumcan be fairly approximated by a Gaussian aperture [5, 6, 9,14], i.e., an aperture with Gaussian intensity transmissionfunction τ : τ ( x , y ) = e − x + y W , (1)where x and y are the transverse distances from the opticalaxis and W the radius where the intensity is decreased to1 / e . Such an aperture transforms an input Gaussian beamwith 1 / e -radius w in , excursion x in and angle θ in w.r.t. theoptical axis into an output Gaussian beam with 1 / e -radius w out , excursion x out and angle θ out given by1 / w = / w + / W , (2) x out = x in W w + W , (3) θ out = θ in − x in w w + W R , (4)where R is the phase front radius of the beam at the aper-ture position. Note that the phase front radius R remainsunaltered when the beam passes the aperture: R out = R in ≡ R . (5)Figure 1 can be used to deduce these equations relatingthe beam parameters before and after the aperture. Therelations expressed in Eqs. (2)-(5) can be captured intothe ABCD-matrix formalism. Two ABCD-matrices canbe defined to describe the Gaussian aperture at the activemedium: the first applies to the complex parameter q de-fined as [5, 6] 1 q = R − i λπ w , (6)where λ is the wavelength of the laser beam, w and R thelocal 1 / e -radius and phase front radius of the beam, re-spectively. The corresponding ABCD-matrix is obtainedfrom Eqs. (2) and (5). It takes the form M q aperture = (cid:20) − i λπ W (cid:21) . (7) M q aperture is used to compute the size ( w ) and phase front ra-dius ( R ) evolution along the propagation using the relation[5, 6] q out = Aq in + BCq in + D = q in − i λπ W q in + , (8)2igure 1: Scheme showing the effect of a soft aperture onthe beam propagation. The aperture whose transmissionfunction is depicted by the blue area decreases the beamexcursion from x in at the input plane to x out at the out-put plane because it selects only part of the beam. More-over, a non-vanishing excursion of the beam at the impactplane ( x in (cid:54) =
0) results in a change of the output beam an-gle θ out . Geometrical considerations can be used to de-duce that θ out − θ in = ( x out − x in ) / R . In this scheme it isassumed that the input beam is moving parallel to the op-tical axis, i.e. θ in =
0. Yet, the aperture does not affect theposition of the beam focus. The darker and lighter shadedred areas indicate the ± w and the ± w size of the beam.where q in and q out are the q -parameters before and afterthe aperture, respectively.The second ABCD-matrix applies to the geometricalpropagation of the beam axis and is obtained building onEqs. (3) and (4): M geometryaperture = W w in + W − w in w in + W R . (9) M geometryaperture is used to calculate the excursion x out and angle θ out after passing the aperture: (cid:20) x out θ out (cid:21) = W w in + W − w in w in + W R (cid:20) x in θ in (cid:21) . (10)Similar equations are valid for the other transverse direc-tion ( y -direction).As visible from Eq. (2), the passage through an aper-ture reduces the beam size, thereby generating power andintensity losses. For a Gaussian beam on the optical axis,the power transmission through the aperture is given by T alignedaperture = W w + W . (11)This transmission is further decreased when the beam has an offset from the optical axis of x in and y in : T mis − alignedaperture = e − x + y w + W W w + W . (12)The power transmission T tot through the amplifier is ob-tained by multiplying the transmissions of each pass at theactive medium: T tot = N ∏ n = T mis − alignedaperture [ n ] , (13)with N representing the total number of passes at the activemedium (or media) and T mis − alignedaperture [ n ] the transmission atthe n -th pass which depends on the beam size w in , on thedeviation from the optical axis x in and y in , and on the sizeof the aperture W at the n -th pass.It is important to note that this formalism assumes Gaus-sian apertures and Gaussian beams. As mentioned beforea Gaussian aperture transforms a Gaussian beam into aGaussian beam of different size. A non-Gaussian aper-ture, on the contrary, leads to excitation of higher-ordertransverse beam components. Yet, in the Fourier-basedamplifier design presented here, the higher-order compo-nents produced in one pass are filtered out in the next passby the aperture itself. Hence, the TEM00 component dom-inates, validating the use of Gaussian beams. In this paper W = w in is assumed because a Gaussian aperture with W = w in produces a similar reduction of the fundamen-tal mode size and a similar reduction of the fundamentalmode transmission compared to a super-Gaussian aperturefulfilling the relation w in ≈ . R p [15, 14, 16], where R p is the radius of the super-Gaussian pump spot. The latterrelation is a rule of thumb typically applied in the thin-disklaser community as it provides most efficient laser opera-tion in the fundamental mode. The assumption W ≈ w in when w in ≈ . R p is confirmed in Sec. 4 by the measure-ment of the small signal gain versus tilt angles. The investigations of alignment stability of Fourier-basedamplifiers presented in this study are based without lossof generality on the specific thin-disk multi-pass ampli-fier shown in Fig. 2. The working principle of thin-disklasers [17, 18, 19, 20, 21] is not presented in this work asit is of minor relevance for the argumentation exposed. Inthis amplifier, the beam is reflected and amplified 8 times(8-pass amplifier) on the disk while it is propagating be-tween mirror M2 and mirror M1b. The various passes hav-ing slightly different beam paths are realized using an ar-3igure 2: (Top) Scheme of the realized Fourier-based am-plifier with the corresponding beam path. The beam rout-ing in the multipass amplifier is sustained by an array offlat mirrors. (Bottom) Front view on the mirror array andits working principle. The mirrors ◦ mirrors acting as vertical retro-reflector that can assume the functionality of the mirrorM2 in Fig. 2. ray of flat mirrors whose working principle is shown in thebottom part of Fig. 2. L-shaped mirror holders were devel-oped and placed as shown in Fig. 3 to maximize tilt stabil-ity of the individual mirrors while minimizing the arraysize. Commercially available mirror holders with similaralignment stability are significantly larger leading to sub-stantially larger spacing between the mirrors. The smallsize of the mirror array decreases the astigmatism relatedto incident angles and it guarantees that the various pathlengths (especially between elements with non-vanishingdioptric power) are similar for all passes so that the beamsize is reproduced from pass to pass. For the same reason,the multi-pass propagation is designed to have same M1a,M1b, M2 and disk for all passes. In the design presentedin this paper the disk and M1a have a non-vanishing diop-tric power while M1b, M2 and the array mirrors are flat.This choice reduces the complexity and the costs and alsosimplifies the alignment procedure as only the radii of cur-vature of the mirror M1a and the distances M1a-disk andM1a-M1b must be adapted to realize the desired layoutwith the same beam size at each pass.The back and forth propagation between the disk (AM)and M1 is described by an ABCD-matrix approximativelyof the form M AM − M1 − AM ≈ (cid:20) B / B (cid:21) (14)which corresponds to the ABCD-matrix of a Fourier trans-form (see Ref. [9] for more details). Differently, theback and forth propagation from the disk to M2, corre-sponds approximately to a short free propagation so thatits ABCD-matrix takes the form M AM − M2 − AM ≈ (cid:20) L (cid:21) , (15)with a length L short compared with the Rayleigh lengthof the beam, i.e. with L (cid:28) π w / λ , where w is the beamwaist. Note that the beam size between the AM and mirrorM2 is large and that the dioptric power of the mirror M2 iseither zero or a minor correction to the free propagation.Thus, the multipass amplifier depicted in Fig. 2 followsthe scheme AM − Fourier − AM − SP − AM − Fourier − AM − SP − AM − Fourier − AM − SP − AM − ... . In this paper, the focus is on the stability properties of thisFourier-based multi-pass amplifier w.r.t. misalignments(tilts), especially of the active medium. Two amplifierconfigurations based on the scheme of Fig. 2 are inves-tigated: in the first configuration the M2 mirror is a flatback-reflecting mirror (acting almost as a “end-mirror”),in the second configuration, the mirror M2 is replaced bya pair of mirrors oriented at 45 ◦ as shown in Fig. 3 (b) thatacts as a vertical (y-direction) retro-reflector.4 Sensitivity of a multipass amplifierto misalignment
The excursion of the laser beam from the optical axis whilepropagating in the multipass amplifier of Fig. 2 is shownin Fig. 4. It has been calculated using the ABCD-matrixFigure 4: Excursion of the Gaussian beam axis w.r.t. theoptical axis along the propagation in the Fourier-based am-plifier of Fig. 2 computed using the ABCD-matrix formal-ism applied to ray optics and including aperture effects.The blue curve represents the beam excursion for a disktilt of φ = µ rad. The red curve represents the beamexcursion for an input beam with an angle of 100 µ rad,w.r.t. a perfect alignment. As a comparison, the beamsize evolution ± w along the propagation axis z is shownby the grey curves. An aperture width of W =
10 mm hasbeen used in these plots. The vertical lines represent thepositions of the disk (with a focusing dioptric power) andthe positions of the defocusing mirror M1a, respectively.These plots apply for both the horizontal ( x -) and the ver-tical ( y -) directions.formalism applied to the geometric ray representing theGaussian beam axis. Aperture effects at the disk have beenincluded via the ABCD-matrix of Eq. (10) and are visiblein Fig. 4 by the sudden decrease of the beam size at thedisk. The excursion of the propagating beam from the op-tical axis has been evaluated for two different types of mis-alignment and compared to the evolution of the beam sizealong the optical axis. In red is displayed the excursionfor a tilt of the beam in-coupled into the multipass system,in blue for a tilt of the disk. When neglecting aperture ef-fects, at the 8 th pass there is a vanishing beam excursionfor both types of misalignment. Indeed, the multiplicationof the four Fourier transform matrices that take place be-tween the 1 st and the 8 th pass corresponds to the identitymatrix. Hence, disregarding the aperture effects and theshort propagations of Eq. (15), the beam leaves the diskafter the 8 th pass at exactly the same position and with thesame angle w.r.t. the optical axis as in the 1 st pass. In general, the beam cutoffs caused by the apertures atthe active medium damp the beam excursions and anglesrelative to the optical axis. For the special case of the 8 th pass (most stable pass with respect to thermal lens effectsand thin-disk tilt), the beam angle is reduced compared tothe 1 st pass, but the beam position is slightly offset fromthe optical axis. This small excursion decreases with in-creasing aperture size W .Throughout this paper, using the rule of thumb diffusedin the thin-disk laser community and justified in [14] aGaussian aperture with W =
10 mm is assumed for a pumpdiameter of 7 mm at FWHM, and a 1 / e beam radius at thedisk of w = . ◦ mirrors acting as a vertical (y-direction) retro-reflector. This mirror pair inverts the tilt and the excursionof the beam from the optical axis in the y-direction while itdoes not affect the tilt and the excursion in the x-direction.Hence, as already shown in [22], a vertical retro-reflectorsignificantly increases the alignment stability in the verti-cal direction. Here, the evaluation of the alignment stabil-ity of [22] has been advanced to include aperture effects.As can be seen by comparing Fig. 5 to Fig. 4, this modi-fication leads to a significant reduction of the beam excur-sions in particular for the 4 th and the 5 th pass but also toa reduction of the excursion and angle of the out-coupledbeam (leaving the 8 th pass).Note that in the amplifier designs presented in this pa-per (and in the majority of the amplifier designs) the mis-alignment of the beam in the x- and the y-direction canbe treated independently. We chose to implement a retro-reflector in vertical direction to reduce the misalignmenteffect caused by the hot air wedge at the front side of thedisk [3, 23], reducing the sensitivity to variations of thepump power.The use of a corner-cube reflector with 3 mirrors ar-ranged at an angle of 54 ◦ and acting as retro-reflectorswould provide inversion of the beam excursions and an-gles for both the horizontal and the vertical directions, re-sulting in an enhanced alignment stability in both direc-tions.The beam excursions simulated in Figs. 4 and 5 havebeen confirmed by measurements summarized in Fig. 6.These measurements show beam profiles and positions(excursions) for various tilts of the disk for three config-urations: the first column at the 2 nd pass of the Fourier-based multi-pass amplifier of Fig. 2, the second column atthe 8 th pass of the same amplifier, and the third columnat the 8 th pass of the same amplifier but the mirror M2replaced by a pair of 45 ◦ mirrors acting as a vertical retro-reflector. Note that when a beam is reflected at a tilteddisk, it acquires an angle θ w.r.t. the optical axis corre-5igure 5: Similar to Fig. 4, but for a multipass amplifierwhere M2 is replaced by a pair of 45 ◦ mirrors acting asa vertical retro-reflector. The vertical dotted lines indicatethe position of the retro-reflector. The excursion is shownonly for the vertical component. The horizontal compo-nent is not affected by the vertical retro-reflector so that itfollows the evolution shown in Fig. 4.sponding to twice the disk tilt, i.e. θ = φ , where φ is thetilt of the disk axis from perfect alignment.As predicted by the simulations of Figs. 4 and 5 thebeam excursions from the unperturbed position (for 0 µ radtilts) are larger at the 2 nd pass than at the 8 th pass. With in-creasing tilt, the profile at the 8 th pass (second column) isdistorted. These distortions visible for a tilt of φ = µ radoriginate mainly at the 4 th pass, where the beam excur-sions are maximal (see Fig. 4). Indeed, for large ex-cursions the aperture at the disk is poorly approximatedby a Gaussian transmission function resulting in a trans-mitted beam with non-Gaussian (non-symmetric) profile.Larger distortions and even fringe effects occurs for a tiltof φ = µ rad at the 8 th pass (second column). Thesefringes originate from hard apertures and beam cut-offs atthe edge of optical components mostly at, or around the4 th pass, where beam excursions are maximal.A comparison between second and third column dis-closes the improvements in term of stability to misalign-ment yielded by the use of the M2 retro-reflector. It sup-presses beam excursions at the 8 th pass. Even more, beamdistortions originating around the 4 th pass are suppressedas the retro-reflector strongly restrains the beam excur-sions at the intermediate passes (compare Fig. 4 to Fig. 5).As explained in Sec. 2 an alternative way to quantifythe sensitivity to misalignment effects is to measure thegain decrease of the multipass amplifier as a function ofthe disk tilt φ . The total gain through the amplifier canbe readily quantified using Eqs. (11) to (13) and assumingthat there are no other losses than already included in theaperture transmission of the disk. The total gain in this Figure 6: Measured beam profiles for various disk tilts forthree multipass configurations based on Fig. 2. The colorsrepresent the beam intensity. Each picture is normalized toits intensity maximum. These images show how beam ex-cursions and distortions increase with misalignment. Notethat the first row represents the reference point as it showsthe beam position for the aligned amplifier. The first col-umn is the excursion at the 2 nd pass, the second column atthe 8 th pass, and the third column also at the 8 th pass butfor an amplifier where M2 is replaced by a pair of 45 ◦ mir-rors acting as a vertical retro-reflector. The φ = µ radtilt causes such a large beam excursion already at the sec-ond pass that the beam is deviated outside the apertureof the beam profiler used to record the images. At the8 th -pass the beam excursion as shown in Fig. 4 is muchsmaller so the beam enters the profiler. Yet, this beam isfully distorted because of hard aperture effects occurringat the 4 th pass where the excursion is maximal. When theretro-reflector is introduced, the excursion at the 4 th passvanishes so that the beam distortions at the 8 th pass dis-appear. This results in a beam at the 8 th pass with almosta Gaussian profile and a small excursion from the unper-turbed position given in the first row.case takes the form G M2 − single8 − pass ≈ ( G ) (cid:16) W w + W (cid:17) e − . · φ (16)for the 8-pass amplifier with simple M2 mirror, and G M2 − ◦ − pair8 − pass ≈ ( G ) (cid:16) W w + W (cid:17) e − . · φ (17)for the amplifier where M2 is replaced by a retro-reflector.In these equations G is the gain per pass (averaged overthe transverse beam profile) while the approximate symbolaccounts for the small variations of the beam sizes w in atthe various passes and the assumption that the gain doesnot decrease with the number of passes (no saturation).The exponential terms in these equations encompass thelosses caused by the beam excursion from the optical axis.6igure 7: The red curve represents the tilt-dependentpart of G M2 − single8 − pass ( e − . · φ ) for the 8-pass amplifierof Fig. 2 as a function of the disk tilt φ . It correspondsto the tilt-dependent transmission through the multipassamplifier normalized to 1 for vanishing tilts. Similarly,the blue curve is the tilt dependent part of G M2 − ◦ − pair8 − pass ( e − . · φ ) for the same amplifier but M2 replaced by aretro-reflector.A plot of the two functions G M2 − single8 − pass and G M2 − ◦ − pair8 − pass isshown in Fig. 7 in red and blue, respectively.A measurement of the total gain versus disk tilt for thetwo multipass amplifier configurations (M2 simple, M2 asretro-reflector) is shown in Fig. 8. The measured data werealready presented in [22] but here they are compared withthe theoretical predictions of Eqs. (16) and (17). It turnedout that the best empirical fit functions to the measureddata are given by parabola and not by Gaussian functionsas predicted in Eqs. (16) and (17).In our measurement the multipass amplifiers were notperfectly aligned so that the maximal gains were ob-served for non-vanishing disk tilts ( φ = − . µ rad and φ = − . µ rad extrapolated using the parabolic fits).To compare the measurements to the model, the value of ( G ) ( W w + W ) of Eqs. (16) and (17) has been fixed tothe gain maximum obtained from the parabolic fit. Alsothe angle φ in Eqs. (16) and (17) has been redefined toaccount for the offset of the maximum (see parameteriza-tions given in Fig. 8). The resulting functions, shown bythe dashed curves in Fig. 8 that assume W = w in , are notfit to the data, but simply normalized to the maximum ob-tained from the parabolic fits. For small excursions, thereis an excellent agreement between the models expressedby Eqs. (16) and (17) on the one hand and the measure-ments of Fig. 8 on the other hand. This agreement quanti-tatively confirm our assumption that the effective losses atthe active medium can be described by a Gaussian aperturewith W ≈ w in when w in ≈ . R p .For large tilts the measured losses exceed the theoret-ical predictions. Indeed, the Gaussian approximation ofthe aperture becomes increasingly inadequate as the beamaxis approaches the edge of the super-Gaussian aperture. Figure 8: Small signal gain versus disk tilt in vertical di-rection. The red points where measured for the multi-pass amplifier of Fig. 2 with simple M2-mirror. The bluepoints have been obtained from the same amplifier but M2replaced by a vertical retro-reflector. The measurementshave been fitted with parabolic functions (solid lines). Thedashed curves represent the predictions from Eqs. (16) and(17) normalized to match the maxima of the parabolic fits.However, only small misalignments – which are well de-scribed by our model – are relevant for the evaluation of alaser system. Lasers with exceeding fluctuations and driftsof the output power are in fact typically unsuitable for thepractical applications. A significantly higher stability to misalignment can be re-alized by implementing a system actively controlling thealignment (tilt) of mirrors. Auto-alignment units com-prising quadrant detectors (or other devices that measurethe beam position) feedback loops, and motorized mirrorholders are commercially available and simple to imple-ment [24]. However, obtaining the best possible alignmentusing the smallest number of controls requires the correctplacement of quadrant detectors and active mirrors.In this section, a simple active stabilization system forthe Fourier-based multipass amplifier of Fig. 2 is presentedthat compensates for tilts of the in-coupled beam, disk, M1and M2 to overall minimize beam excursions. The atten-tion has been focused primarily to correct for tilts of thedisk, as it is prone to misalignments owing to the pumpingand cooling processes that it undergoes and because of theair wedge at its surface [23]. It is assumed that the diskitself cannot be equipped with actuators to control its tilt.Figure 9 shows the multipass amplifier of Fig. 2 up-graded with an optimal and simple active stabilization sys-tem. The input beam direction is actively controlled bythe feedback loop (C1) acting on the actuators of mir-7or M in with two independent degrees of freedom (verti-cal and horizontal). The beam travels successively to thedisk where it undergoes the first amplification and reflec-tion. For the disk tilted (misaligned) by an angle φ , thebeam acquires an additional angle w.r.t. the optical axis of ∆ θ = φ . The Fourier propagation that takes place in thepropagation disk-M1b-disk transforms the tilt of the beamleaving the disk into a beam excursion when the beam re-turns for the second time (second pass) to the disk. At thedisk, the beam suffers an additional tilt ∆ θ . Because ofthe short propagation length between disk and M2, the ex-cursion of the beam at the mirror M2 is dominated by thetilt of the beam after the first reflection at the disk that isgiven by the tilt of the disk and the tilt of the beam beforeimpinging for the first time on the disk. The excursion ofthe beam at M2 can be measured by placing a quadrant de-tector (Q1) in the vicinity of M2 that measures the beamspuriously transmitted through M2. Any deviation of thebeam position from the set point (when the amplifier isaligned) generates two error signals: one for the horizon-tal, the other for the vertical direction. Through the feed-back loop C1, these error signals act on the actuators ofmirror M in to cancel the beam excursion at M2.After reflection on mirror M2, the beam reaches the diskfor the third time where it again acquires an angle θ = φ .This tilt once again results in an excursion from the setpoint position when the beam reaches M2 for the secondtime. The excursion of the second pass on M2 can be mea-sured with a second quadrant detector (Q2). The error sig-nals generated by Q2 are used to regulate the actuators ofthe mirror M2 through a second feedback loop C2 so thatthe beam excursion at the second pass on mirror M2 isnullified. The same correction (tilt) of mirror M2 auto-matically correct all the successive passes on M2 due tothe repetitive structure of the amplifier (see Fig. 10). Bothloops are sufficiently independent so that their interplaycan compensate for tilts of the in-coupled beam and fortilts of the disk. The regulation can also partially compen-sate for tilts of M1.Only two feedback loops are therefore sufficient to sta-bilize the most critical optical elements of the multipassamplifier for all passes as M1, M2, and active medium arecommon for all passes. Oppositely, it has to be stressedthat individual mirror misalignments within the mirror ar-ray cannot be compensated by this feedback system so thathigh stability is required for the individual holders of thearray mirrors.Both quadrants have to be placed in the vicinity of mir-ror M2. It is possible to detect the position of the firstpass on M2 with Q1 and of the second pass on M2 withQ2 because the two passes impinge on M2 at different an-gles. Thus, a short free propagation from M2 to the quad-rants Q1 and Q2 can be used to separate the two spuriouslytransmitted beams (see Fig. 9). Figure 9: Schematic of the realized multipass amplifierequipped with a simple auto-alignment system. Only twoloops (each with a vertical and a horizontal degree of free-dom) are sufficient to mitigate the excursion of the laserbeam from the optical axis for tilts of the disk, M1a, M1b,M2 and input beam. Each loop (C1 and C2) comprises aquadrant detector (Q1 and Q2) whose error signals act onthe motorized mirrors M in and M2, respectively. Ideally,the quadrants measure the position of the beam at the firstand second pass on the M2 mirror.In Fig. 10, the excursion of the laser beam axis fromthe optical axis is illustrated along the 8-pass amplifier ofFig. 9 equipped with the above-described active stabiliza-tion. Only the propagations for a tilted disk is displayed.The propagation for a tilted in-coupled beam is not shown,as a tilt of the input beam can be exactly canceled by theloop C1 so that the beam would propagate on the opticalaxis ( x = y =
0) of the amplifier. The resulting excursionevolution shown in Fig. 10 can be compared with the ex-cursions in Figs. 4 and 5. Similar overall excursions arevisible but in Fig. 10 the disk tilt has been increased by afactor of 25 compared to the non-active stabilized ampli-fiers. Hence, the active stabilization improves greatly thestability to disk tilts.The auto-alignment system of Fig. 9 has been tested ex-perimentally. The measured gain decrease of the activelycontrolled amplifier for variations of the disk tilt is sum-marized in Fig. 11 and compared with the results fromthe non-actively stabilized amplifiers with the same op-tical layout. The actively stabilized amplifier shows a ma-jor (order of magnitude) decrease of the sensitivity to disktilts. However, the measured stability improvement wassmaller than predicted from simulations. This reduced per-formance can be ascribed to the non-vanishing M2-Q1 andM2-Q2 distances that in the realized amplifier were about1.5 m. This issue could be solved by implementing animaging of M2 on the two quadrants.It is interesting to note that the number of actively con-trolled mirrors can be reduced to one without loss of align-ment stability by eliminating the loop C2 and replacingthe active mirror M2 with a corner-cube reflector with 3mirrors arranged at an angle of 54 ◦ and acting as retro-reflectors.8igure 10: Similar to Fig. 4 but for a multipass amplifierequipped with the simple auto-alignment system depictedin Fig. 9. Because of the efficient compensation producedby the active stabilization, the disk tilt has been increasedto φ = .
25 mrad, (a factor 25 times larger than in Fig. 4).The correction generated by C1 (steering of M in ) is com-pensating the downstream tilts occurring in the first andsecond pass on the disk, so that the beam excursion of thefirst pass on M2 is nullified. Similarly, the second loop C2adjusts the tilt of the mirror M2 so that the beam excur-sions at all successive passes (third, forth etc.) on M2 arenullified. A simple model has been presented to calculate the lossesoccurring at the soft aperture naturally present in a pumpedactive medium and the effects of this aperture for the beampropagation. This knowledge has been used to quantifythe misalignment sensitivity of multipass amplifiers in twoways. The first, by simply tracking the evolution of thebeam excursion w.r.t. the optical axis while the beampropagates in the amplifier for a tilted (misaligned) opti-cal element or input beam. The second, which requires theknowledge of the first, by computing the decrease of theamplifier gain as a function of the tilt of the consideredoptical element.These two methods have been used to investigate thesensitivity of Fourier-based amplifiers to beam tilts. Thechoice of the Fourier-based amplifiers is motivated by thesuperior stability of these multipass amplifiers to varia-tions of thermal lens and aperture effects at the activemedium compared with state-of-the-art amplifiers basedon the 4f-imaging propagation. An extensive comparisonbetween the two multipass architectures can be found in[9]. Three variations of the same Fourier-based amplifierwere investigated in this study: the basic layout of Fig. 2,the basic layout but mirror M2 replaced with a verticalretro-reflector, and the basic layout equipped with an auto-alignment system shown Fig. 9.The simulations were compared with measurements of Figure 11: Measured small-signal gain for three 8-passamplifiers versus the disk tilt φ . The red points havebeen taken for the simple amplifier of Fig. 2, the bluepoints for the same amplifier but M2 replaced by a verticalretro-reflector, the green ones for the amplifier of Fig. 9equipped with the active stabilization system. Polynomialfits have been drawn to guide the eye. For the amplifierwith the active stabilization system the mirror M2 has beenreplaced with a mirror with slightly higher transmission togenerate a robust error signal from Q1 and Q2. This re-duces the overall gain of the amplifier but does not alter itstilt dependence.the amplifier gain and good agreement has been found.The stability of Fourier-based multipass amplifiers hasbeen demonstrated via months long operation of a thin-disk laser based on the layout of Fig. 2 without the need ofrealignment. Furthermore, simulations and observationsshowed that this stability can be improved by a factor of4 in one direction (vertical) when M2 is implemented asa pair of 45 ◦ mirrors acting has vertical retro-reflectors,or in both directions (vertical and horizontal) when M2is implemented as a three-mirror system acting as retro-reflectors in both directions. Alike, more than an orderof magnitude improvement in tilt stability can be obtainedby equipping the multipass amplifier with a simple auto-alignment system comprising only two quadrant detectorsand two motorized mirror holders.These findings combined with the investigation pre-sented in [9] fully qualifies the Fourier-based design to bethe appropriate choice in the high energy and high powersector. We acknowledge the support from the Swiss National Sci-ence Foundation Project SNF 200021 165854, the Eu-ropean Research Council ERC CoG.
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