Applied Optics, Volume 25, No. 9, pp. 1394-1397 (1 May 1986)This work was supported by a contract from the U.S. Army TRADOC Operations Research Activity, White Sands Missile Range, NM

Rockower, Edward B.

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Calhoun: The NPS Institutional ArchiveFaculty and Researcher Publications Faculty and Researcher Publications1986-05Laser beam-quality/aperture-shapescaling relationRockower, Edward B.http://hdl.handle.net/10945/44117Laser beam-quality/aperture-shape scaling relationEdward B. RockowerMany high-energy lasers (HELs) have noncircular output apertures. Some are rectangular in shape with orwithout a central or noncentral (up to 30%) obscuration. However, most high-energy laser propagation codes(especially those developed for systems analysis) model the aperture as either an unobscured circle or as acircle with fixed (e.g., 10%) obscuration. We present a beam-quality/aperture-shape scaling relation whichcan be useful when applying these codes to realistic designs for HELs. Our analysis also yields a generalizedformula for angular size of the Airy disk and definitions of a characteristic aperture length and aperturequality.1. INTRODUCTIONDevelopment of computer codes for nonlinear laserbeam propagation through turbulent atmosphericconditions, including smoke and dust, has required(and received) a large amount of modeling effort (seee.g., Ref. 1). However, other important features ofhigh-energy lasers (HELs) include the rectangular ap-erture and large (possibly up to 30%) central obscura-tion. The latter is characteiisticof unstable resonatordesigns common to most high-energy lasers.Two major categories of computer code have beendeveloped for modeling HEL propagation:(1) Fundamental wave optics codes: these codesstart from the basic microscopic laws of E-M radiationand implement various phenomenological models ofthe atmospheric turbulence and particulates (e.g., Ref.2).(2) Moderate accuracy system-level scaling lawcodes: these codes start from phenomenological andanalytical approximations to the E-M wave equationsand attempt to represent the details of propagationthrough the atmosphere with a few parameters (e.g., aphase integral) based on integrated properties of theatmospheric conditions, light intensity, wave length,etc. These codes are based on experimentally andnumerically (from the fundamental wave optics codes)derived data bases.3One limitation is common to many laser propagationcodes (especially the system's analysis codes), namely,the code may handle only circular laser apertures.This limitation has not stopped application of thecodes outside their range of validity.In attempting to exercise such a propagation code tosimulate performance of rectangular apertur6 lasers, itis reasonable (because of possible nonlinear effects) totry to match the output power and beam intensity ofThe author is with Naval Postgraduate School, Department ofOperations Research, Monterey, California 93943.Received 12 September 1985.the real laser. If the former condition is met, the latteris equivalent to requiring that the area of the circularaperture be the same as for the real laser, and, there-fore,irD2/4 = A, (1)where A is the rectangular aperture area (excludingobscured regions) and D is the circular aperture diame-ter.The following analysis estimates the adequacy ofmodeling realistic HEL apertures as circular aperturesand leads to the derivation of a beam-quality/aper-ture-shape scaling relation. A generalization of theAiry disk size and shape-dependent characteristic ap-erture-length and aperture-quality parameters arepresented based on this analysis. We then apply thescaling relation to one of the system's analysis propa-gation codes to demonstrate an application of the scal-ing relation.11. Derivation of the Scaling RelationAn asymptotic approximation formula for estimat-ing the fraction of encircled energy within a givenradius in a plane perpendicular to the direction ofpropagation has been derived for imaging systems withoddly shaped apertures.4 Modulation transfer func-tion (MTF) techniques were applied to a uniformlyilluminated aperture having arbitrary shape and obs-curations resulting in the following formula:E(r) = 1 - XfR/(27r2r), (2)where X = wavelengthf effective focal length,r = radial dimension in the focal plane,R = HEL aperture perimeter/area ratio, andE = normalized encircled energy, i.e., the frac-tion of the energy transmitted by the aper-ture that falls within a circle of radius rabout the geometrical focal point.The main feature of this result for our purposes isthat, apart from the laser wavelength and system focaldistance, the fraction of encircled energy depends onlyon the ratio R/r. Hence it is independent of details of1394 APPLIED OPTICS / Vol. 25, No. 9 / 1 May 1986the aperture shape apart from the value of R. In thederivation it is assumed that the laser beam propagatesin a linear medium with no effects of atmosphericfluctuations, wind, etc. We see from Eq. (2) that theradius r encircling a given fraction of energy increaseslinearly with R.A related effect results from imperfections of thelaser wave front at the point where the beam leaves thelaser aperture. A degraded beam quality (phase frontdistortion) is often represented with a beam qualityparameter M. The value ofMis always greater than orequal to 1 and is given as times diffraction-limited; i.e.,the beam spread is Mtimes the diffraction-limited rateof beam divergence. Most, if not all, propagationcodes allow the user to set M. If E(r) is the encircledenergy function when M is equal to one, for arbitraryM, the encircled energy becomesBEAM QUALMlY VS RATIO OF RECTANGLE<0qmD -I ...... . - ---1-- ---- --- ........-.. . |4RATIO OF APERTURE DIMENSIONSE'(r) = E(r/M). (3)Combining the two results for beam spread, we obtainE(r) = 1 - (XMR)(f/r)/(2rr 2 ). (4)From Eq. (4) we identify a method of partially com-pensating for rectangular apertures and various sizesand types of obscuration. We see that the asymptoticspread of light emanating from two laser apertureshaving the same value of the product XMR will be thesame. Hence a better approximation to an arbitraryaperture results if we requireMR = M'R'.MamJ0(5)Two lasers with the same value of M*R, power, andaperture area (hence the same intensity) will haveapproximately the same asymptotic beam divergence.This is the main result of this section.This aperture-shape/beam-quality similarity (orscaling) relation is to be distinguished from the so-called similarity law of diffraction, 5 which comparesdiffraction from two apertures of geometrically similarshape but different size.Having two parameters for the equivalent circularaperture at our disposal (M' and D') allows us to matchboth area and ratio R of any HEL aperture. Forexample, we can model a circular aperture with a cen-tral obscuration with another circular aperture of thesame area and output power and with beam quality M'satisfying Eq. (5). These requirements lead toM = M(1 + VF)A/(1-F), (6)D' = DV(1 - F), (7)where F is the fraction obscuration.It is easy to derive similar equations for rectangularapertures with and without (rectangular) obscura-tions. The results for the latter and for circular aper-tures are presented in graphic form in Fig. 1. Eachcurve gives the beam quality of an equivalent uniformcircular aperture with the same area as the rectangularaperture whose length to width (L/W) ratio is shown tothe right of the curve. The curve for a circular aper-ture with obscuration F is also presented.BEAM QUALITY VS FRACTION OBSCURATIONL/W - 4L/W - 3L/W 2SquareCircle0 0.1 0.2 0.3FRACTION OBSCUREDFig. 1. Equivalent beam quality as a function of length/width ratio(LIW) and fraction obscuration F.Note that any aperture can be characterized by theratio R/R'. This is the value of the dimensionlessparameter M' satisfying Eqs. (5) and (1) with M set toone. Since the circle gives the minimum beam spreadfor a given aperture area (the minimum value of R), M'(i.e., R/R') indicates the (inverse of) beaming effective-ness of the aperture, i.e., the actual beam spread rela-tive to this ideal (achieved for a circular aperture andperfect beam quality). We propose calling this quan-tity the aperture quality in close analogy with beamquality. For example, a square aperture has an aper-ture quality of 2/\/fr. The aperture quality, like beamquality, is always greater than one. An alternativeinterpretation of Fig. 1 is as a plot of aperture quality(the ordinate) for various aperture shapes. With thisdefinition of aperture quality, Eq. (5) can be stated inwords as "two apertures having the same product ofbeam quality times aperture-quality will produce thesame asymptotic encircled energy function."11l. Characteristic Length Parameter and DiffractiveBeam SpreadFor an unobscured circle the ratio of perimeter toarea is1 May 1986 / Vol. 25, No. 9 / APPLIED OPTICS 1395F - .3F - .2r .1F --------------- - ----------------------------------- - ---- - ------------- ------ --............................. ...................................................... . .... ........1R = 4/D, (8)and for a square with side a,R = 4/a. (9)This suggests defining a parameter L, the characteris-tic linear dimension of an arbitrarily shaped apertureasL = 4/R. (10)Reference to our scaling relation, Eq. (5), suggests thatfor arbitrary M an effective aperture dimension isL = 4/(MR). (11)For a rectangle of sides a and b, R is found to beR = 2(a + b)/(ab). (12)Hence for a rectangle the characteristic length (if M =1) isL = 2ab/(a + b) (13)or1/L = (1/a + 1/b)/2. (14)In other words, orthogonal dimensions are averagedusing the harmonic mean rather than (for example) thegeometric mean. Similar equations are easily foundfor other aperture shapes.We now quantify more precisely our observationthat the diffractive beam spread will be linearly pro-portional to the product (MR). For a uniformly illu-minated circular aperture the diffraction pattern wasfirst derived by Airy (Ref. 6, p. 396f). In our notation,the half-angular size of the first dark ring in the Airypattern, encircling 0.838 of the transmitted energy, iswell known to be0 = 1.22X/D. (15)Generalizing this, let r now represent the radius of acircle containing 0.838 of the transmitted energy froma general aperture with beam qualityM. Using Eq. (4)we obtain the relation0.838 = 1 - XMR(f/r)l(2ir2 ). (16)Now, solve Eq. (16) for r/f, the half-angle containingalmost 84% of the energy:r/f = 0.31XMR. (17)Using our definition of L and letting 0 now represent(r/f), this can be written approximately as= 1.22XM/L. (18)Equations (10) and (18) generalize Eq. (15) to arbitraryaperture shape and beam quality.If the aperture is an unobscured circle and for Mequal to one, this reduces to Eq. (15). Also, for asquare with side a we obtain= 1.22XM/a. (19)These results are consistent with our use of Eq. (10) fora characteristic aperture length and Eq. (18) for theangular beam spread from an aperture of arbitraryshape. Finally, note that it may be helpful in somecases to calculate the Fresnel number of an opticalsystem using the characteristic length L rather thanthe aperture diameter.IV. Application of the Scaling RelationWe have exercised the HELAWS code (containingBRLPRO and contained in EOSAEL)7 with the followingquestions in mind:(1) How adequate is it to represent a centrally ob-scured and/or rectangular aperture by means of a cir-cular aperture with the same overall area, power, and(therefore) intensity?(2) Given that we would like to represent differentaperture shapes and obscurations with our similaritytransformation, how good a fix can be obtained in thepresence of linear atmospheric effects such as turbu-lence and wind?(3) How good is our transformation in the face ofnonlinear effects, i.e., thermal blooming?Figures 2-4 were generated with HELAWS using vari-ations of the base case parameters listed in Table I. Ineach figure, curve 1 represents the results for a refer-ence uniformly illuminated circular aperture of -mdiameter D, beam quality equal to one. Curve 2 pre-sents the results for a uniformly illuminated circularaperture with a 10% central obscuration but with thesame area and beam quality (i.e., M = 1) as the basecase. Since F is 0.1, Eq. (7) yields an aperture diame-ter D' of 1.054 m. Finally, curve 3 presents the resultsfor a uniformly illuminated circular aperture (no ob-scuration), equal area as before, but with beam quality(1.387) calculated from our scaling formula (or Fig. 1).In other words, curve 2 corresponds to the real HELaperture with 10% obscuration. Curve 1 is what wepredict for an unobscured circular aperture matchingonly the area. Curve 3 results if we also match thescaling parameter M*R. First, refer to Figs. 2 and 3,showing the /e spot diameter and /e peak fluence,respectively, in the absence of thermal blooming.Comparing curves 1 and 2 shows that simply trying torepresent a 10% obscuration with a circular aperture ofequal area would lead to gross errors. It is easy to showthat there is a rectangular aperture (L/W -1.9) withthe same area and value of R as the 10% obscuredcircular aperture. Hence by the encircled energytheorem the rectangle will have the same asymptoticbeam spread as the obscured circle (curve 2). Clearly,if one tries to represent such a rectangle with theunobscured circular aperture (curve 1), the same grosserrors will result, even without thermal blooming.However, for curve 3 (for which M is set using thescaling relation) the match is excellent beyond -4300m. This range defines the beginning of the asymptoticregion for this example. Two things are worth notinghere: (1) the M*R = const similarity relation worksquite well becoming better as the range increases; (2) itworks well even for propagation in the turbulent atmo-sphere.Consulting Fig. 4 in which the thermally bloomed1396 APPLIED OPTICS / Vol. 25, No. 9 / 1 May 19860 -1 1-4000 4100 4200 4300 4400 4500RANGE (m)Fig. 2. Unbloomed spot diameter vs range.-I-Ml=1; D=I90 1 'I.-50 23 t" x b v=1....70 1=; 0I =1.054; lOX obsc460 1=1.307; D= |403020104 0 4 l/e spot diameter is plotted, we see that using thescaling relation (curve 3) gets us only about halfway tothe 10% obscured aperture results (curve 2). Howev-er, the codes handle the 10% obscuration case already,and we are only looking for a means of extending therange of existing propagation codes to include largerobscurations and rectangular apertures.V. ConclusionsThe beam-quality/aperture-shape scaling relationhas been derived under conditions most appropriate tofree-space propagation for lasers with constant pupilfunction. Its extended usefulness has been demon-strated in an application to linear laser propagationwithin the turbulent atmosphere using a representa-tive scaling law code (BRLPRO). In the presence ofthermal blooming the scaling law remains partiallyeffective and may still be useful as a partial fix. Ouranalysis has also provided a generalization of the Airyexpression for the angular spread of light from a laseraperture of arbitrary shape and beam quality. Newdefinitions for a characteristic aperture length and anaperture-quality factor allow comparison of diffrac-tion characteristics of different aperture shapes, inde-pendent of wavelength.4000 4100 4200 4300 4400 4500RANGE m)Fig. 3. Unbloomed peak fluence vs range.SP0T0AETER(cm25 l IlI_ _2 1=1; D=1.054; 1OZ obIc.10120 . | 1C .317 '15 31 1 ' , M I ~; DU1I lI_I104000 4100 4200 4300 4400RANGE m)Fig. 4. Thermally bloomed le spot diameter vs range.Table 1. Input Quantities for HELAWS (BRLPRO) CodeLaser parametersOperating mode Repetitively pulsedWavelength 10.6 umTotal system jitter (1 sigma) 10 AradVariable focal range Equal to target rangeEnergy per pulse 10 kJPulse repetition rate 5 HzPulse duration 20 gsAtmospheric parametersAmbient temperature 100CRelative humidity 85%Turbulence level ModerateRefractive-index structure const 8.40 X 10-14 m- 2/3Magnitude of wind velocity 1.0 m/sThe author would like to express his appreciation toP. Shugart, J. Morris, J. Fleck, and J. Reilly for theirhelp in the study and to L. Ingber and E. Milne forcomments on the manuscript. This work was support-ed by a contract from the U.S. Army TRADOC Opera-tions Research Activity, White Sands Missile Range,NM.References1. F. G. Gebhardt and M. B. Richardson, "Nonlinear Aerosol Vapor-ization and Breakdown Effects Module-NOVAE," EOSAEL 8422 (Draft Report), Atmospheric Sciences Laboratory, WhiteSands Missile Range, NM (Oct. 1984).2. J. A. Fleck, Jr, J. R. Morris, and M. D. Feit, "Time-DependentPropagation of High Energy Laser Beams through the Atmos-phere," Appl. Phys. 10, 129 (1976); J. R. Morris, "Four-D Propa-gation Code for High-Energy Laser Beams: A User's Manual,"UCID-17217, Lawrence Livermore National Laboratory (5 Aug.1976).3. H. J. Breaux, "Simplified Predictive Methodology for NonlinearRepetitive Pulse and CW High Energy Laser Propagation," Proc.Soc. Photo-Opt. Instrum. Eng. 195, 192 (1979).4. P. P. Clark, J. W. Howard, and E. R. Freniere, "AsymptoticApproximation to the Encircled Energy Function for ArbitraryAperture Shapes," Appl. Opt. 23, 353 (1984).5. A. Sommerfeld, Optics (Academic, New York, 1950), pp.220-221.6. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1980).7. P. Beavers, High Energy Laser Weapon Simulation (HELAWS),Analyst's Manual, Vol. 1, U.S. Army Materiel Systems AnalysisActivity, Combat Support Division, Aberdeen Proving Grounds,MD (Sept. 1983).1 May 1986 / Vol. 25, No. 9 / APPLIED OPTICS 1397P0TDAETiERacnIIPEAKFLECE(J/sq cm)

Laser beam-quality/aperture-shape scaling relation

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Laser beam-quality/aperture-shape scaling relation

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