Laser beam profile shaping with interlaced binarydiffraction gratings
W. B. Veldkamp
A new C02 laser beam profile shaper was designed and tested. With high-power efficiency, it transformsa fundamental mode laser beam profile into a flattop profile at a focal plane. The shaper uses an interlacedbinary diffraction grating that modulates the E field both in phase and amplitude and generates an apodizedand clipped sinc(x) distribution in the object plane.
1. Introduction
Optical devices that modify the irradiance distribu-tion of a fundamental mode laser into a uniform flattopirradiance distribution are useful in many applications.These include laser fusion,1 semiconductor laser an-nealing systems,2 laser radars that use detector arrays, 3
efficient power extraction in laser amplifiers,4 surfaceheat treatments,5 and optical data processing.6
The redistribution of the irradiance under the beamprofile can be achieved in many ways: by absorptionor apodization 7 ; by beam segmentations; by reflectionsor refractions of beam fluxes; or by diffraction of en-ergy.1' The first technique, using fixed masks or vari-able aperture Pockels cells, is generally very energyinefficient. The second technique, beam segmentation,suffers from high interference and diffraction losses,particularly for small beam diameters. Reflection(axicons) or refractive techniques are energy efficientbut generally require complicated aspheric surfaces, aresometimes nonrealizable, or do not preserve polariza-tion.
We have found that diffractive techniques can beenergy efficient, simple, and flexible. In another pub-lication,3 we discussed the requirement of convertingthe incident beam profile into a 1-D sinc(x) or a 2-DBessinc(r) distribution to generate a flattop distributionat the focal plane. This requirement came from thereciprocal Fourier relationship between the object planeprofile and the necessary far-field or focal-plane profile.We also showed that by modulating only the phase ofthe laser beam between two binary values, we couldachieve a good approximation to a flattop distribution.In this paper we extend this technique to include am-plitude weighting by selective dispersion of energy outof the beam profile.
The author is with MIT Lincoln Laboratory, P. 0. Box 73, Lex-ington, Massachusetts 02173.
The zeros or E-field phase reversals in the requiredsinc(x) field distribution at the object plane are locatedat x = +nA/2, n = ±1, +2,. . ., where A is related to thewidth w of the focal-plane beam profile via A 2F/w.Therefore, any incident field distribution can have itsfield reversed with this spatial frequency by using alamellar grating device with a 7r-phase depth and acentral 7r-phase reversal. The 1-D transmission func-tion of its relief pattern can be described by
f(x) = V/21[exp(jh) - 1](-1)A + [exp(j) + 1, (1)
where A equals integer Ix/A j and 0 is an arbitrary phasedepth of the relief pattern. With an incident E-fieldinput distribution g(x) on such a device [generally aGaussian distribution gaus(x) exp(-7rx2)] the focalplane or far-field irradiance distribution becomes
I(x') = IF(x') 0 (x')12, (2)
where F(x') and G(x') are the Fourier transforms of f(x)and g(x), respectively, and 0 denotes the convolutionoperator. Then E(x) generally exhibits a distributionwith extended flatness and steep edge roll-off resem-bling that of a flattop profile. Such binary phasemodulation on Gaussian field profiles has been provenan effective flattop beam shaper. This concept is shownin Fig. 1.
From the analysis in Ref. 3, we developed two pa-rameters u, 0 that characterized such a beam shaper (a= the ratio of the grating period to the 1le amplitudewidth of the laser beam and 0 = 2irdn/X phase depth)with values that were optimized to approximately a =0.85 and 0 = 0.8r depending quantitatively on unifor-mity requirements. Unfortunately, there are com-peting requirements between flatness and power effi-ciency or sidelobe generation. The parameter de-pends sensitively on this balance. When we optimizeda at -0.85 and used only binary phase modulation,sidelobe ringing in the shaped focal plane intensity wasstrong [see shaded area of Fig. 1(b)]. From a discrete
Fig. 1. Concept of focal plane or far-field beam shaper. The shaperis in the laser beam path where the incident beam profile becomes an
approximate sinc(x) distribution.
NT(a)
o .x) PHASE MODULATIONBT PRIMARY GRATING
o~)()MODULATION STINTERLACED GRATING
Fourier transform (DFT) numerical analysis, it is clearthat these sidelobes are caused by excess energy in theincident beam profile to the right and left of the firstzeros in the required sinc(x) distribution, whereasavailable energy under the third sidelobes is lacking.This lack causes a dip in energy along the optical axisbut can be compensated by a decrease in the indepen-dent phase depth parameter 44.
Let us assume an incident beam with a field profiledistribution g(x) on the beam shaper. After phasemodulation, that distribution becomes g (x) f(x) Ea (x).If we were to introduce a non-negative amplitudeweighting function W(x) = sinc(x)/a(x)j, we wouldhave converted the incident beam into a sinc 2(X) in-tensity profile which is the corresponding exact flattopprofile in the focal plane domain. Unfortunately we arerestricted to passive (no gain) devices which limit theweighting to the 0•< W(x) range. Figure 2(b) showssuch a weighting distribution for an incident Gaussianbeam profile and independently optimized wr-phasemodulation at a- = 0.85. Such an amplitude weightingdistribution can, of course, be achieved conceptuallywith a spatially varying absorber. The precise distri-bution would be difficult to control, and heat dissipationmight be excessive.
To provide localized. attenuation of the beam bydispersing energy at high diffraction angles out of thebeam profile, we used a duty cycle modulated binaryphase grating equally deep as the primary phase-mod-ulation grating. The appropriate weight-fringe widthtransformation is simply W(x) = - y(x)/TI, and-y(x) is the local relief width. The strength of the zerothdiffraction order is linearly proportional to the averageof the transmission function. For example, a uniform50% duty cycle y/T = 0.5 eliminates the zeroth-ordercontribution in a r-phase grating altogether.
The binary weighting mask for the relief pattern hasa variable groove width and covers intervals +n1 T alongthe x direction. The transmission function of this maskis
tOX) = f1 - 0 rect [ _x) n rect f ~ (3)
WEIGHTING UNCTION W(x)' SINCWxo¼)0
(b)-I`--
cm
y/T
LINEWIGTH WEIGHTING
(c)
Fig. 2. Linewidth modulation scheme. The phase modulation ofthe primary grating (top) can be weighted to a clipped sinc(x) dis-tribution by W(x) (center). The weighting functions W(x) can beimplemented with an interlaced binary diffraction grating using
linowidth modulation.
where
JlXj I '/2,
rect(X) = to otherwise.
The fixed period T is determined by T = X/SinOd andthe inequality d »> C with 0 as the laser beam diver-gence. The space variant linewidth y(x) on the pho-tomask is thus inversely proportional to the neededzeroth-order diffraction strength at sampled positions
x,= (n - 2)TThe binary amplitude and phase-modulation func-
tions can be interlaced to form a new binary maskfunction
VW=tw 1x) (J)2
(4)
and the corresponding phase relief transmission func-tion of the weighted beam shaper becomes
Fig. 3. Enlarged transmission pattern of a 1-D beam shaper emulsionmask (top). Primary and secondary modulation patterns (not to
scale) with T = 2500 Am and T = 250 Am, respectively (bottom).
LASER BEAM PROFILE
- SHAPED PROFILE
=0.85, = 0.87,v
Fig. 4. Measured CO2 laser beam profiles ( = 10.6 ,um) with andwithout beam shaper.
NO WEIGHTING
W(x) WEIGHTING
Fig. 5. Calculated beam profiles in the object domains (left and focaldomains (right) by the primary phase-modulation grating (top) and
the amplitude weighted interlaced grating (bottom).
f'(x) = expjl0 t'(x)]. (5)
Since t'(x) is not a simple periodic function in x, ex-pansion of the phase-modulated wave front into a seriesof plane waves with amplitudes that are Bessel func-tions of the first kind is not feasible. In Fig. 3 we showthe photoemulsion mask of the combined t'(x) distri-bution. Areas of black and white in this emulsion maskrepresent the highs and lows of the relief pattern on thesubstrate. Absolute phases or complete contrast re-versals are irrelevant. This mask is then contactprinted onto a transparent substrate that is coated witha positive photoresist, and the developed pattern isetched into the substrate to depth d as explained in Ref.3. In our applications, the primary phase-modulationgrating periodicity was 2.5 mm (for a 3.0-mm CO2 laserbeam diam) interlaced with a 250-mm periodicity am-plitude weighting grating. The poly-GaAs substratewas AR coated with a primary thorium-fluoride layerand a secondary zinc-selenide layer. Clearly this pat-tern also can be etched directly into the laser exit win-dow to become part of its cavity.
The beam shaper was tested with a Sylvania 941SCO2 laser (beam divergence 01/2 = 2 mrad). The shapedfar field was scanned with a phase-locked pyroelectricdetector behind a 15-,um aperture. This scanned far-field profile is shown in Fig. 4 with (solid curve) andwithout (dashed curve) the beam shaper placed in thenear field. Energy efficiency of this beam profile shaperwas -74% as measured with a calibrated thermopile andwith <0.2% energy in the near sidelobes. The remain-ing energy is diffracted, of course, far out of the beamprofile. This can be compared with DFT numericalcalculations for a ir phase-depth binary grating (Fig. 5)where the energy under the main lobe is limited to 66%of the unshaped profile.
111. Iterative Solutions
We have studied the problem of binary phase andamplitude shaping independently; first, the phase dis-tribution was fixed, and then the amplitude weightingdistribution was optimized. The complete problem ofbinary phase modulation with real passive weightingdoes not yield to analytical closed-form solutions be-cause of its complexity. When such an analytical ap-proach fails, it is always possible to solve the problemby a numerical iterative method.12 One can approachthe problem by not assuming an ad hoc sinc(x) phaseand amplitude modulation but by assuming randomreal and imaginary components of a Fourier transformpair that must satisfy the system constraint in an errorreduction scheme.
One iteration (the kth) in the error reduction loopwould proceed as follows: let Ak(x), expUjohk(x)] andBk (x), exp j3qj(x)] represent the amplitude and phasesof the E-field distributions in the object and focal do-mains. The constraints then are a focal plane distri-bution Bk (x) = rect(x) and an object plane distributionclipped between 0 < Ak(x) • gaus(x). The ab initiofocal plane distribution is
Then we calculate the Fourier transform of Bl(x)expfj4'Ax)] and force the magnitude Ak (x) to be clippedto the incident Gaussian beam strength, but the phaseremains unchanged, i.e.,
FTIBi(x') expji(x)J -l A2(x) explq2(x)], (7)
where IAk (x) = gaus(x) if Ak (x) > gaus(x). Next wecalculate the inverse Fourier transform of the forcedinput function and use the phase function as the inputto the next iteration and again force the amplitudedistribution to rect(x). We then transform back andforth between the two forcing functions to satisfy theconstraints in each domain until the mean square errorE
2 defined by
I IAk+l(x) expl0k+i(x)l- A(x) expk a x)l 2 dx
f' Ah (x) expj ok {x) 2dx
is small or does not change anymore. The final Af (x)then defines the amplitude weighting distribution to beimplemented on the beam shaper by linewidth modu-lation via
W(X) = Af(x) (9)gaus(x)
Notice, however, that even in this approach we haveignored the self-imposed constraint of the binary natureof the phase distribution h (x). Fortunately, for flat-top focal plane intensity distributions, this is not a se-rious limitation. As Lee13 discovered and can be seenfrom analytical considerations, the beam shaper'sphasing function is binary in any case, since
sinc(x) (-)+.(10)sinc(x)I
Therefore, by not forcing k (x) to remain binarythrough the various iterations, the technique will pro-duce an amplitude weighting distribution W(x) that isonly marginally different. Ignoring the binary con-straint is only valid, however, for flattop focal planedistributions. Alternatively, one could force a focalplane binary phase solution by clipping the phases andthen continue the loop.
Another useful initial value in the iterative loop wouldbe the partial solution already found in the previoussection:
where 0 is the phase-modulation depth of the grating,and 00 is an arbitrary phase wave-front constraint.
IV. Conclusion
We have presented an efficient flattop beam profileshaper that can easily be constructed as part of the lasercavity. We have shown how a Gaussian CO2 laser beamprofile can be converted to an approximate flattopdistribution at the focal plane, although the techniqueis not limited to this distribution. The shaper we testedused a binary grating that modulated the amplitude andphase of the incident beam and optimized these E-fieldcomponents independently. We also outlined an iter-ative phase-amplitude interdependent solution by usinga least-mean-square-error criterion. For the iterativeapproach we expect only marginal improvement of thispassive device, since its beam shaping quality is nowlimited primarily by the spatial frequency bandwidthof the modulated input beam profile.
I thank C. J. Kastner for help in mask generation andW. T. Brogan for substrate preparation and ion-beamexperiments.
This work was sponsored by the Department of theAir force. The U.S. government assumes no responsi-bility for the information presented.
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