We report our numerical and experimental studieswith regard to the use of a transmissive mode-selecting diffractive element in a regenerative ringresonator to generate a desired output beam shape.We aim to show that the use of a transmissive ele-ment in a ring resonator delivers superior perfor-mance and is easier to handle than a Fabry–Perotstyle resonator with a diffractive mirror.
High-power solid-state laser systems employ amaster oscillator power amplifier configuration toachieve maximum focused power. The initial mas-ter oscillator generates relatively low power but awell-defined laser pulse that is amplified with highfidelity in the power amplifiers. Generally, the mas-ter oscillator functions in the TEM00 fundamentaltransverse mode and generates a Gaussian radialintensity distribution. To avoid diffraction from theedges of apertures, the intensity of the beam must falloff to a value much lower than the peak intensity,typically below 1%. Because of the bell-shaped in-tensity profile of the TEM00 mode, this leads to inef-ficient use of the gain medium. It is thus desirablefor this pulse to have a constant intensity profileacross the full beam width to extract maximum gainfrom the subsequent amplification stages, i.e., a
The authors are with the Department of Physics, Heriot-WattUniversity, Edinburgh EH14 4AS, UK. The e-mail address for K.Balluder is [email protected].
Received 31 March 1999; revised manuscript received 1 June1999.
5768 APPLIED OPTICS y Vol. 38, No. 27 y 20 September 1999
flat-top beam profile needs to be generated, which isa rectangular beam profile that maintains a highintensity across the beam profile and rapidly falls offon the sides. This can be achieved with existingsetups by overfilling an aperture, resulting in an ap-proximately 90% loss in intensity. Although the ap-plication for a master oscillator is our mainmotivation, our approach is not limited to this. Sev-eral uses for non-Gaussian beam profiles exist forindustrial applications and the same setup can beused to produce beam shapes other than flat-tops.The use of customized mirrors and diffractive opticalelements to alter the fundamental mode of a Fabry–Perot resonator to produce a certain profile has beendemonstrated in recent years, most notably by Legeret al.12 We previously designed and tested suchmode-selecting diffractive element ~MSE! designs for
master oscillator with a Fabry–Perot style resona-tor.3,4
However, use of MSE’s for a Fabry–Perot resonatorposes some problems5 that could be overcome by useof a ring resonator. For a linear resonator one hasthe option of using either a diffractive mirror or atransmissive MSE together with the existing resona-tor mirror. Use of a diffractive mirror is difficult athigh-power levels because of the low damage thresh-old of the reflective coating. The use of a transmis-sive MSE together with an existing mirror is notmuch better, because it is not easy to create goodcontact between the MSE and the mirror, and theresulting gap between them degrades the perfor-mance. This result is demonstrated in Fig. 1, whichshows the design for a diffractive mirror imple-mented as a transmissive element together with aplane mirror. The additional propagation distance
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between the two passes through the element intro-duces phase changes so that the wave is no longerprecisely phase conjugated. This then producesstrong deviations from the desired flat profile, includ-ing a deep central intensity minimum.
We propose to overcome these problems by using aring resonator that is not vulnerable to the problem ofMSE positioning and reshapes the output of the mas-ter oscillator into the desired flat-top profile with onlyminor loss in intensity. If we use a ring resonator asa regenerative ring resonator to reshape an incomingGaussian beam, modifications to the generally care-fully adjusted master oscillator can be avoided. TheMSE also offers the additional advantage of beingable to compensate for thermal lensing in the rod.
2. Design
Positioning a MSE in one arm of a ring resonatorallows us to form arbitrary mode profiles at the op-posite side of the resonator. The MSE is designed tofunction as a phase conjugator for a mode that pro-duces the desired beam profile. A transverse modewith this intensity profile, therefore, undergoes min-imal loss in the resonator as it reproduces itself at theoutput coupler, after the steady state has beenreached, and thus it becomes the fundamental trans-verse mode of the resonator, determining its outputbeam profile. Since the phase profile of the MSE isstatic, in contrast with phase-conjugating mirrors, itallows only the desired mode to reproduce itself.When the mode is being built from noise, the compo-nent of the desired mode in the superposition ofmodes that propagate through the cavity increasesuntil the steady state has been reached and the out-put has assumed the desired mode shape.
Our simulations are based on an existing ring res-onator that is used as a regenerative amplifier for ahigh-power laser system. Currently this resonatoris being used in its unmodified form to regenerate adistorted input pulse into a beam with a clean Gauss-
Fig. 1. Graph showing both the experimental data and the sim-ulation results for the output of a linear resonator in which atransmissive MSE was used in conjunction with an existing mir-ror. Inasmuch as the MSE and the mirror could not be broughtinto direct contact, they had to be mounted separately to allow foradjustment, leading to a 2.5-cm gap that caused the distortedintensity distribution. The measurements are from a scanthrough the center of the beam profile.
20
ian shape. For the simulations presented in thispaper we have modified the original setup for usewith the MSE shown in Fig. 2. The setup consists ofa four-mirror ring resonator with an approximately4-m circumference. The input beam enters throughthe mirror at lower right. After a predeterminednumber of round trips the beam exits the resonatorthrough the output coupler at lower left. For phaseconjugation to work, the resonator must be symmet-ric with respect to the MSE. This means that theaperture of all additional components must be largeenough so that the beam can pass through unclipped.The rod has a 3-mm diameter; the input beam has aGaussian intensity profile with a half-width of v0 5 1mm. To avoid diffraction from the rod’s aperture, itmust be positioned as close as possible to the centralaperture because this is the area of minimum beamwidth. We chose to position the rod just in front ofthe central aperture, because at that location thebeam is sufficiently small to be unaffected by the rod.
The MSE modifies the resonator in such a way as tocreate a fundamental mode that produces an nth-order super-Gaussian intensity profile given by
I~x, y! 5 expF2Sx2 1 y2
v02 DnG . (1)
For our simulations we used values for n up to 20.The results in this paper were calculated for n 5 8,which gives a sufficiently flat profile but avoids toomany high spatial frequency contributions thatwould require larger apertures in the setup to avoidclipping of the beam. ~A detailed discussion ofGaussian beams and their properties can be found inRef. 6, Chap. 16 and 17.!
If one follows the guideline that the intensity ataperture edges should be no more than 1% of the peakintensity, one can compare the total intensity carriedby super-Gaussian beams of different order. Withincreasing order the width of the beam can be in-creased and still fulfill that criterion. In Table 1 weshow the comparison of a normalized Gaussian beam~n 5 1, v0 5 1 in arbitrary units! with two higher-rder super-Gaussian beams. All these intensitiesre integrated over the same aperture ~r 5 2.1455! at
which the Gaussian beam’s intensity has dropped to
Fig. 2. Setup used for the simulations.
September 1999 y Vol. 38, No. 27 y APPLIED OPTICS 5769
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Table 1. Comparison of Total Intensity Carried in a Beam through a
a
5
1%. An eighth-order super-Gaussian can thus carry3.6 times more energy through that aperture than asimple Gaussian.
Beginning with such a real mode at the centralaperture, one must propagate the mode to the posi-tion of the MSE. Phase conjugating the mode to theMSE and propagating it back to the aperture willrestore the original wave.7 Thus this mode is notonly an eigenmode of the resonator but also under-goes minimum loss and becomes the fundamentaltransverse mode of this resonator, determining theoutput beam profile.
To calculate the MSE, a real mode of the desiredintensity profile propagates numerically through theresonator to calculate the desired MSE profile. Tooptimize the mode width and resonator dimensionswe performed a Fox–Li analysis8 to produce valuesor the loss of the individual modes. We used scalariffraction theory and propagation of the angularpectrum for all the calculations.To evaluate the performance of the proposed designe performed a number of simulations with varyingarameters for the flat-top beam width and the out-ut coupler aperture. From these we optimized theelative width of the super-Gaussian to the inputeam and the size of the central aperture. We main-ained the size of the rod diameter as well as theircumference of the total resonator. The rod had a-mm diameter and a 10-cm length. The half-widthf both the input beam and the output beam was v0 5
1 mm.Figure 3 shows beam profiles for a number of round
trips through the resonator. The most remarkableresult is that a single trip through the resonator pro-duces a relatively good flat-top output. As expected,the quality of the flattop improves when the numberof round trips increases, but the improvements forn . 3 are minor. The loss for this fundamentalmode is 1.9%, compared with 84.1% for the nexthigher-order mode, giving excellent mode discrimina-tion. The deviations from the desired intensity pro-file are less than 7%.
Obviously, this resonator can be used in the cwmode to reshape an input beam into a flat-top beam,or as an oscillator to build such a mode from noise.Our aim is to use it to reshape pulses with a durationof up to 10 ns. The time for the light to make asingle round trip through the resonator is 13.6 ns.
Given Circular Aperturea
n v0 I InyI1
1 1.00 3.11 18 1.95 11.25 3.60
20 2.07 12.31 3.96
aThe aperture radius has been fixed to r 5 2.1455, where theintensity of the normalized Gaussian beam ~v 5 1! has fallen to 1%of the peak intensity. With increasing order n of the super-Gaussian beam, the value for the half-width v0 can be increasednd still fulfill this criterion.
770 APPLIED OPTICS y Vol. 38, No. 27 y 20 September 1999
This sets an upper limit for the duration of pulsesthat can be reshaped with this setup. Longer pulseswould overlap with themselves in the resonator, re-sulting in beam profile degradation.
To demonstrate that this approach is not limited tocircular super-Gaussian output profiles, we modifiedthe design to produce a ring-shaped profile and asquare output profile. These can be seen in Fig. 4.These two additional designs were not optimized likethe flat-top design but were generated in exactly thesame setup with the only change being the calcula-tion of the MSE. Thus by just exchanging the MSE,one can generate a range of different output profilesfrom the same experimental setup. The square pro-file is significantly worse than the other two becausethe input beam is circular and that coupled with theexisting parameters creates an extremely poor dis-crimination against the next higher-order mode.
Fig. 3. Beam intensity profile evolution. The pictures show thebeam intensity profiles after n round trips through the resonator.The graphs for n 5 0 show the input beam of the Gaussian inten-sity profile before it enters the resonator.
Fig. 4. Other possible beam shapes generated from a circularGaussian input beam after n 5 10 round trips through the reso-nator.
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The fundamental mode losses for the ring-shapedand square profiles as calculated by the Fox–Li anal-ysis are 1.83% and 4.45% and for the next higher-order modes they are 12.7% and 22.3%, respectively.As one would expect, the larger the difference be-tween the output profile and the input beam, theworse the mode discrimination. For a square beamthis is evident from the intensity picture, whereas thering-shaped beam looks almost perfect because boththe fundamental and the next higher-order modesare ring shaped and overlap.
3. Experimental Verification
We fabricated the elements on a fused-silica sub-strate by using a standard photolithographic processwith electron-beam written chrome masks andreactive-ion etching. During this process the highoptical quality of the fused-silica surface and its highlaser damage threshold were preserved, which is im-portant for their use in high-power systems. By us-ing multiple binary chrome masks and repeating theprocess several times, one can obtain multilevelstructures that approximate the ideal continuousphase profile.
For the designs presented in this paper we used 16discrete phase levels. The use of fewer levels wouldincrease the quantization error and reduce the per-formance of the elements. Also, with the currenttechnology it does not make sense to produce morephase levels because the error that is due to the align-ment of individual masks outweighs the benefits offiner quantization. For a comparison of experimen-tal and numerical results the number of phase levelsis not important, because we used the actual quan-tized profile of the manufactured element for the sim-ulations.
The elements were calculated with a resolution of1024 3 1024 pixels of which the central 512 3 512
ixel section was used for the fabrication. The fab-icated element had a width of 25 mm.
We have been able to test the MSE for the flat-toprofile only in a setup that builds a mode from noise.he MSE for the ring-shaped output has been testednly in a one-pass configuration. Figures 5 and 6how the output of the resonator mode built fromoise. The beam clearly has the desired super-aussian intensity profile, as predicted by the nu-erical simulations. However, the beam still shows
ome intensity variations, typically around 620%,ith one pronounced peak reaching 150% deviation
from the desired intensity. We believe that this iscaused by several factors. One might be a smallbackreflection from the element. Coating the sur-faces of the element already produced a significantimprovement compared with the performance of theuncoated element. However, inasmuch as the ele-ment has been coated on both surfaces, the influenceof coated surfaces should be minor. We intend toinvestigate this and the surface quality of the coatingin the near future. We also plan to investigate theeffects of self-focusing in the amplifier, although it isbeing operated at threshold level and so the focusing
20
should not be a major contribution. The asymmetryin the beam profile, which produces the pronouncedpeak on its right edge, is believed to be caused by anasymmetry in the pump head of the laser setup thatwe could not completely compensate for. We alsoexamined the elements in a separate test setup with-out an amplifier and the beam did not exhibit thisasymmetry. In general, the experimental perfor-mance of this ring cavity when a mode is built fromnoise is similar to the Fabry–Perot style resonatorsthat we examined previously, which typically showmaximum deviations of 615% from the desired pro-file when built with a diffractive mirror and muchlarger deviations of from 30% to as high as 50% whentransmissive MSE’s were used.3–5,9,10
The results from the single-pass setup without anamplifier, starting with a clipped Gaussian inputbeam, can be seen in Figs. 7 and 8. Although thissetup produces much noisier results than the otherone, it nevertheless demonstrates the reshaping ef-fect of the element even after only a single round trip.In addition, this test does not reproduce the asym-metry that is visible in Figs. 5 and 6, confirming that
Fig. 5. Output of the actual ring resonator with the MSE thatbuilds a mode from noise. The beam asymmetry is caused byasymmetry in the pump head.
Fig. 6. Line scan through the center of the beam as shown in Fig.5 and a comparison with the desired flat-top profile.
September 1999 y Vol. 38, No. 27 y APPLIED OPTICS 5771
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it is not an effect inherent in the MSE design itself.The same setup without an MSE produces a broad-ened Gaussian beam profile, which is identical to theincoming Gaussian beam that propagates in freespace.
We also used this same test setup to observe theoutput from the MSE that was designed to produce aring-shaped profile. In this case the desired beamshape could not be achieved within one round trip,which is consistent with the simulations that alsoshow that we need multiple passes to obtain the ringprofile. However, the profile obtained after one passthrough the MSE ~Fig. 9! and its numerical simula-ion are quite similar ~Fig. 10!, showing a centraleak ~that disappears during successive passes! sur-ounded by two rings. The numerical simulationhows a stronger central peak, but this feature istrongly dependent on the input beam radius of cur-ature ~which has been estimated to be r 5 4.5 m forhis setup and is included in the simulation!, so amall variation of it could account for this difference.his feature is also distinctively different from therofiles after one pass through the flat-top profile
Fig. 7. Output from the test setup showing a Gaussian profilebeam after a single pass through the MSE.
Fig. 8. Line scan through the image from Fig. 7, showing thebeam after a single pass through the MSE. This test setup has noamplifier rod and does not show the asymmetry caused by thepump head as is visible in Figs. 5 and 6.
772 APPLIED OPTICS y Vol. 38, No. 27 y 20 September 1999
hat generates elements ~figs. 7 and 8! that do nothow such a pronounced ring structure.A possibly important advantage of a transmissiveSE over diffractive mirrors could be observed dur-
ng the experiments. Resonators with diffractiveirrors are generally considerably more difficult to
et up and adjust than conventional resonators, be-ause they are sensitive to precise alignment with theiffractive mirror. Unlike a conventional resonator,hey do not start to lase until the element is close tots ideal position. Resonators employing transmis-ive MSE’s do not show this behavior and are quiteasy to set up and align. First the resonator isligned with the plane mirror and then the MSE isdded. When the rest of the resonator is properlyligned, it is then easy to move the MSE to its correct
Fig. 9. Beam observed with the second MSE designed to producea ring-shaped output. This picture shows the beam after one passthrough the MSE and features a central maximum surrounded bytwo rings.
Fig. 10. Simulated beam produced by the second MSE after onepass. This picture relates to the experimental result from Fig. 9.
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position. We observed that when the MSE was any-where near the correct position, it produced a beamthat clearly shows the misalignment as an asymmet-ric structure on the beam, indicating in which direc-tion the MSE had to be moved ~Fig. 11!.
4. Gain Saturation
We carried out additional simulations to examine theinfluence of gain saturation on the resonator’s per-formance. For these simulations we used a simplemodel to simulate the gain saturation based on themodel suggested in Ref. 11. We modeled the beamtraveling in one direction through the ring resonator,with two gain sheets located at either end of theresonator. For each round trip we recalculated thegain that was due to the intensity of passing throughthe medium:
g~x, y! 5g0
1 1 I~x, y!yIs. (2)
Equation ~2! expresses the spatially resolved gain~x, y! as a function of the initial small-signal gain0, the saturation intensity Is, and the beam inten-
sity I~x, y! of the previous round trip. Beginningith uniform gain g0, we continuously adjusted the
Fig. 11. Beam distortion as observed when the MSE was mis-aligned horizontally. The dark shadow disappeared when theMSE was being moved back into the central position.
Fig. 12. Simulated total energy output from the resonator withthe inclusion of a simple model for gain saturation.
20
gain. The total output energy ** dxdy I~x, y! as aunction of time ~expressed in number of roundrips! can be seen in Fig. 12. This graph displayshe typical pulse evolution as expected from such aystem, with a pulse duration ~full width at half-eight! of approximately eight round trips or 109s. The output from the resonator, after it hadeen seeded with a low-intensity clipped Gaussiannput pulse, is shown in Fig. 13, which displays theutput beam profile after 4, 7, and 11 round tripshat correspond to half-height and full height of theulse. Although this flat-top profile is consider-bly noisier than the one from the setup withoutain saturation effects, it shows that the resonatortill produces a reasonably flat beam profile that isufficient to extract much higher energy from theod than would be possible with a Gaussian beam.s expected, the quality of the flat-top beam im-roves when the number of round trips increases.hese simulations suggest that it should be possibleo use diffractive elements to increase the energyutput from a high-energy, pulsed laser systemith gain saturation, if the pulse is built from a
ufficiently flat initial seed pulse. Trying to buildflat-top beam with the same setup from noise failsecause the random intensity distribution destroyshe initial gain uniformity. It is, therefore, essen-ial to seed the resonator with a small, sufficientlyat pulse to avoid this problem.
5. Conclusions
We have demonstrated the performance of a regen-erative ring resonator that employs a mode-selectingdiffractive element to reshape an incoming beam ofGaussian intensity profile into the more desirablesuper-Gaussian beam. We have experimentally ex-amined the performance of the MSE’s in building amode from noise and after one round trip in a nonam-plifying resonator.
Using a master oscillator with a super-Gaussianbeam intensity profile allows us to carry almost fourtimes as much energy through the system than witha Gaussian profile, greatly increasing the energy ex-traction in the amplification stages. We have pre-sented a design that produces a high-quality super-
Fig. 13. Simulated beam profiles at half-height and full pulseheight of the pulse output from the resonator with the inclusion ofgain saturation.
September 1999 y Vol. 38, No. 27 y APPLIED OPTICS 5773
for mode shaping of a Nd:YAG Laser,” Opt. Lett. 19, 108–110
1
5
Gaussian output within only a few round trips,making it suitable for use as either a master oscilla-tor or a regenerative resonator for an existing system.The loss per round trip for this system was calculatedto be 1.9%, with excellent mode discriminationagainst higher-order modes. The deviation from thedesired flat-top profile is less than 7%. We thus be-lieve that the use of a regenerative ring resonatorpromises to be a better alternative to a modifiedFabry–Perot style master oscillator. To demon-strate that this approach is not limited to circularsuper-Gaussian output profiles, we have presentedadditional designs that can be used to produce ring-shaped and square super-Gaussian profiles.
Our numerical studies of the gain evolution in thissystem also suggest that the MSE can be used inhigh-power pulsed systems in which gain saturationplays an important role.
We acknowledge that this research has been sup-ported by a grant from the German Academic Ex-change Service as part of the gemeinsamesHochschulsonderprogramm 3 von Bund und Land-ern. The authors thank Valeri Kovalev and NickHopps for their help with the experimental tests.
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774 APPLIED OPTICS y Vol. 38, No. 27 y 20 September 1999
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