Gaussian Beam Optics

Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 2.1 2.2 2.6 2.10 2.13 2 www.cvimellesgriot.com Gaussian Beam Optics Beam Gaussian Gaussian Beam Optics Gaussian Beam Propagation by Simple Lenses and Magnification Transformation Real Beam Propagation Lens Selection Gaussian Beam Optics Beam Gaussian 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:53 PM Page 2.1 2:53 PM 6/15/2009 2ch_GuassianBeamOptics_Final.qxd 2ch_GuassianBeamOptics_Final.qxd 6/16/2009 2:19 PM Page 2.2

Fundamental Optics In most laser applications it is necessary to focus, modify, or shape the laser For virtually all laser cavities, the propagation of an electromagnetic field, beam by using lenses and other optical elements. In general, laser-beam E(0), through one round trip in an optical resonator can be described propagation can be approximated by assuming that the laser beam has an mathematically by a propagation integral, which has the general form ideal Gaussian intensity profile, which corresponds to the theoretical TEM00 ()1 − jkp ()0 mode. Coherent Gaussian beams have peculiar transformation properties E() x,, y= e∫∫ K() x y, x00, y E() x00, y dx 0 dy 0 which require special consideration. In order to select the best optics InputPlane (2.1) for a particular laser application, it is important to understand the basic where K is the propagation constant at the carrier frequency of the opti- properties of Gaussian beams. cal signal, p is the length of one period or round trip, and the integral is over Unfortunately, the output from real-life lasers is not truly Gaussian (although the transverse coordinates at the reference or input plane. The function K the output of a single mode fiber is a very close approximation). To accom- is commonly called the propagation kernel since the field E(1)(x, y), after M2 (0) modate this variance, a quality factor, (called the “M-squared” one propagation step, can be obtained from the initial field E (x0, y0) factor), has been defined to describe the deviation of the laser beam from through the operation of the linear kernel or “propagator” K(x, y, x0, y0). a theoretical Gaussian. For a theoretical Gaussian, M2 = 1; for a real laser

Gaussian Beam Optics By setting the condition that the field, after one period, will have exactly beam, M2>1. The M2 factor for helium neon lasers is typically less than the same transverse form, both in phase and profile (amplitude variation 1.1; for ion lasers, the M2 factor typically is between 1.1 and 1.3. Collimated 2 across the field), we get the equation TEM00 diode laser beams usually have an M ranging from 1.1 to 1.7. For high-energy multimode lasers, the M2 factor can be as high as 25 or 30. g Exy(),,≡ KxyxyExydxdy(),,() In all cases, the M2 factor affects the characteristics of a laser beam and nm nm ∫∫ 00nm 00, 00 InputPlane (2.2) cannot be neglected in optical designs. where E represents a set of mathematical eigenmodes, and g a In the following section, Gaussian Beam Propagation, we will treat the nm nm corresponding set of eigenvalues. The eigenmodes are referred to as characteristics of a theoretical Gaussian beam (M2=1); then, in the section transverse cavity modes, and, for stable resonators, are closely approx- Real Beam Propagation we will show how these characteristics change as imated by Hermite-Gaussian functions, denoted by TEM . (Anthony the beam deviates from the theoretical. In all cases, a circularly symmetric nm Siegman, Lasers) wavefront is assumed, as would be the case for a helium neon laser or an

argon-ion laser. Diode laser beams are asymmetric and often astigmatic, The lowest order, or “fundamental” transverse mode, TEM00 has a which causes their transformation to be more complex.

Optical Specifications Gaussian intensity profile, shown in figure 2.1, which has the form

Although in some respects component design and tolerancing for lasers is −+()22 , ∝ kx y more critical than for conventional optical components, the designs often Ixy() e (2.3) tend to be simpler since many of the constraints associated with imaging In this section we will identify the propagation characteristics of this low- systems are not present. For instance, laser beams are nearly always used est-order solution to the propagation equation. In the next section, Real Beam on axis, which eliminates the need to correct asymmetric aberration. Propagation, we will discuss the propagation characteristics of higher-order Chromatic aberrations are of no concern in single-wavelength lasers, although modes, as well as beams that have been distorted by diffraction or various they are critical for some tunable and multiline laser applications. In fact, the anisotropic phenomena. only significant aberration in most single-wavelength applications is primary (third-order) spherical aberration.

Scatter from surface defects, inclusions, dust, or damaged coatings is of BEAM WAIST AND DIVERGENCE greater concern in laser-based systems than in incoherent systems. Speckle In order to gain an appreciation of the principles and limitations of

Material Properties content arising from surface texture and beam coherence can limit system Gaussian beam optics, it is necessary to understand the nature of the laser performance. output beam. In TEM00 mode, the beam emitted from a laser begins as a Because laser light is generated coherently, it is not subject to some of the perfect plane wave with a Gaussian transverse irradiance profile as shown limitations normally associated with incoherent sources. All parts of the in figure 2.1. The Gaussian shape is truncated at some diameter either by wavefront act as if they originate from the same point; consequently, the the internal dimensions of the laser or by some limiting aperture in the emergent wavefront can be precisely defined. Starting out with a well- optical train. To specify and discuss the propagation characteristics of a defined wavefront permits more precise focusing and control of the beam laser beam, we must define its diameter in some way. There are two than otherwise would be possible. commonly accepted definitions. One definition is the diameter at which the beam irradiance (intensity) has fallen to 1/e2 (13.5 percent) of its peak, or axial value and the other is the diameter at which the beam irradiance Optical Coatings 2.2 Gaussian Beam Optics Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings ) 2 z ( 2.3 w (2.8) (2.6) (2.7) , =∞would have so that the 1/e for large z z l / 0 www.cvimellesgriot.com w beam, namely, beam, namely, p 0. If a uniform irradiance 00 = Gaussian Beam Optics Beam Gaussian z is called the beam waist radius. is called the beam waist 0 Gaussian Beam Optics , 0, the pattern at z = FWHM diameter 50% of peak 22 rw 2 diameter 13.5% of peak 2 e 0 marks the location of a Gaussian waist, or location of a Gaussian waist, 0 marks the e 2 = 1/ P w 2 p values would have been enormously complicated. would values ) asymptotically approaches is the total power in the beam, is the same at all is the total power in the z z ( . is further increased, asymptotically approaching the asymptotically is further increased, 0 // z l 22 rw p 2 0 ) and P z −− = z Diameter of a Gaussian beam w ( l 0 p contours asymptotically approach a cone of angular radius contours asymptotically approach a cone of angular w itself. The plane z The itself. zw () = = is presumed to be much larger than == wz z w () = () direction of propagation v Ir Ie wz Figure 2.2 Figure toward infinity as toward of z value and w is flat, a place where the wavefront irradiance distribution had been presumed at z function, whereas the been the familiar Airy disc pattern given by a Bessel pattern at intermediate where cross sections of the beam. distribution is a special consequence of the form of the invariance The distribution at of the presumed Gaussian The irradiance distribution of the Gaussian TEM irradiance distribution The Simultaneously, as R Simultaneously, asymptotically approaches the value where ) is z ( (2.4) (2.5) R . z irradiance w mode 2 e 1.5 00 , and rises again z ) is the radius of z ( 0 is the radius of the 1/ 0 w 12 / CONTOUR RADIUS ⎤⎤ ⎥ ⎥ ⎦ 2 ⎤ ⎥ ⎥ ⎦ 2 ⎞ ⎟ ⎠ w laser-beam wavefront were made perfectly flat wavefront laser-beam ⎞ ⎟ ⎠ 2 0 z 2 0 4 w 00 z l w p w l p ⎛ ⎜ ⎝ ⎛ ⎜ ⎝ 1.5 1 ⎡ ⎢ ⎢ ⎣ 4 1 ⎡ ⎢ ⎢ ⎣ Irradiance profile of a Gaussian TEM Irradiance profile 0 0, passes through a minimum at some finite = =+ =+ contour after the wave has propagated a distance z, and R(z) contour after the wave 80 60 40 20 definition. is the wavelength of light, is the wavelength is the distance propagated from the plane where the wavefront 100

13.5 2 2

z () () PERCENT IRRADIANCE PERCENT Rz z wz w the 1/e infinite at z Diffraction causes light waves to spread transversely as they propagate, Diffraction causes light waves collimated beam. The and it is therefore impossible to have a perfectly the predictions of pure spreading of a laser beam is in precise accord with in the present context. diffraction theory; aberration is totally insignificant the beam spreading can be so small it Under quite ordinary circumstances, following formulas accurately describe beam spread- can go unnoticed. The of laser beams. ing, making it easy to see the capabilities and limitations Even if a Gaussian TEM (intensity) has fallen to 50 percent of its peak, or axial value, as shown or axial value, (intensity) has fallen to 50 percent of its peak, second definition is also referred to as FWHM, or full in figure 2.2. This we will be using the remainder of this guide, width at half maximum. For the 1/e and contour at the plane where the wavefront is flat, w contour at the plane where the wavefront Figure 2.1 Figure and radius of curvature after propagating a distance radius of curvature is the wavefront where is flat, l at some plane, it would quickly acquire curvature and begin spreading in and it would quickly acquire curvature at some plane, accordance with 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:54 PM Page 2.3 2:54 PM 6/15/2009 2ch_GuassianBeamOptics_Final.qxd 2ch_GuassianBeamOptics_Final.qxd 6/15/20092:54PMPage2.4

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics range divergence, isthedistancefromwaist atwhichthewavefront to be thedividinglinebetweennear-field divergenceandmid- considered and thenbegintodecrease, asshowninfigure2.4.The Raleighrange, the waist, thewavefront curvature, therefore, mustincreasetoamaximum 2.4 Figure 2.4 approaches theasymptoticlimitdescribedabove. The Raleighrange( divergence angleisextremelysmall;farfromthewaist, thedivergenceangle Near thebeamwaist, whichis typically closetotheoutputoflaser, the Unlike conventionallightbeams, Gaussianbeamsdonotdivergelinearly. Near-Field vsFar-Field Divergence Gaussian TEM This value isthefar-field angularradius(half-angledivergence)ofthe At thebeamwaist (z of defined asthedistanceoverwhichbeamradiusspreadsbyafactor parameter, diameter anddivergencewithdistance It isimportanttonotethat,foragivenvalue of waist, asshowninfigure2.3. at Figure 2.3 propagated awayfrom Gaussianwaist z √ z _ =∞ w 2, is given by is 2, R w w 0 0 profile Gaussian = , thewavefront isplanar[ Gaussian BeamOptics p l w w 0 2 Changes inwavefront radiuswithpropagation distance Growth in1/e 0 laser 00 , thebeamwaist radius. . beam. The vertexoftheconeliesatcenter e 1 2 = irradiance surface 0), thewavefront isplanar[R(0) =∞ 2 R radius withdistance 2 planar wavefront z ( w v ∞ =0 0 ) =∞

]. Asthebeampropagatesfrom asymptotic co z are functionsofasingle l, variations ofbeam

ne w z 0 ]. Likewise, (2.9) z maximum curvature z R = ), 2 z w R v 0

2 Using thisoptimumvalue of for agivendistance, z We canfindthegeneralexpression fortheoptimumstartingbeamradius beam atz radius wouldbe4.5mm.Anyotherstartingvalue wouldresultinalarger best collimation)overadistanceof100m,ouroptimumstartingbeam combination of radius ofabout4.5mm.Therefore, ifwewanted toachievethebest The beamradiusat100mreachesaminimumvalue forastartingbeam l minimum startingbeamdiameterandspread[ratioof specifications) mustbemeasuredatadistancemuchgreaterthanz curvature isamaximum.Far-field divergence(thenumberquotedinlaser uuly>10 (usually equation plottedasafunctionofw at afixeddistance, z equation toseehowfinalbeamradiusvaries withstartingbeamradius to calculatew Typically, onehasafixedvalue forw the far-field distancecanbemeasuredinmeters. can beafewmillimetersorless. For beamscomingdirectlyfromthelaser, focused beam,thedistancefromwaist (thefocalpoint)tothefarfield inaccurate ifnear- ormid-fielddivergencevalues areused.For atightly calculations forspotsizeandotherparametersinanopticaltrainwillbe = zw wz w 632.8 nmandz 0 () () optimum =+ = # 100 m. 0 ( z ⎣ ⎢ ⎢ ⎡ z 1 ) foraninputvalue ofz minimum beamdiameterandspread(or R will suffice).This isaveryimportantdistinctionbecause ⎝ ⎜ ⎛ = p . Figure 2.5showstheGaussianbeampropagation ⎝ ⎜ ⎛ l . Doingsoyields planar wavefront z w = l z =q p 0 2 z 100 m. ⎠ ⎟ ⎞ ⎞ ⎠ ⎟ 2 12 / ⎦ ⎥ ⎥ ⎤ / 12 w 0 will providethebestcombinationof 0 and usestheexpression . However, onecanalsoutilizethis 0 , withtheparticularvalues of Gaussian BeamOptics www.cvimellesgriot.com profile intensity Gaussian (2.10) R Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 0 2.5 w ) = 2 R z ( w R z www.cvimellesgriot.com Gaussian Beam Optics Beam Gaussian 0 w Gaussian Beam Optics 2 beam waist R z 0 w ) = 2 R Focusing a beam expander to minimize beam Focusing a beam expander to minimize z (– w APPLICATION NOTE APPLICATION Do you need . . . BEAM EXPANDERS CVI Melles Griot offers a range of precision beam expanders for better performance than can be achieved with the simple lens combinations shown here. Location of the beam waist is required for most location of the beam waist The CVI Melles Griot lasers are Gaussian-beam calculations. very close to the typically designed to place the beam waist If a more accurate location than this output surface of the laser. the precise is required, our applications engineers can furnish location and tolerance for a particular laser model. beam expander Figure 2.6 Figure over a specified distance radius and spread 0 0 w (2.11) . R z 632.8 nm, w = (mm) 4.5 mm at 100 m, and 0 = w 0 ), z 632.8 nm, we get a value of 632.8 nm, we get a value ( = 100 m and l w 2 factor in overall variation. _ l = √ , as opposed to just R 3.2 mm. Thus, the optimum starting 3.2 mm. Thus, = 0 . For z . For 50 m, and = R = z ), over which the beam radius spreads by a factor R . . mm STARTING BEAM RADIUS STARTING . 4.5 mm, and w 0 0 () = () 2 over a distance of 2z 2 _ ½ 100 m, or Beam radius at 100 m as a function of starting Beam radius at 100 m as a function of ) 2 4.48 mm (see example above). If we put this value for 4.48 mm (see example above). If we put this value 2 0 p = = = / w l = ] over the distance z 012345678910 ) 0 = p R 0 80 60 40 20 (2l R = 100 100 2 4 48 6 3

= () () ()

R FINAL BEAM RADIUS (mm) RADIUS BEAM FINAL ) 2 as _ ) to w z R wz w w wz w √ z z ( ( with By turning this previous equation around, we find that we once again have By turning this previous equation around, we find the Rayleigh range (z Thus, for this example, for this example, Thus, Alternately, if we started off with a beam radius of 6.3 mm, we could Alternately, of w focus the expander to provide a beam waist beam radius is the same as previously calculated. However, by focusing the beam radius is the same as previously calculated. However, expander we achieve a final beam radius that is no larger than our starting while still maintaining the beam radius, a final beam radius of 6.3 mm at 200 m. with Figure 2.5 Figure w (optimum) (optimum) back into the expression for beam radius for a HeNe laser at 632.8 nm beam radius for a HeNe laser at 632.8 w If we use beam-expanding optics that allow us to adjust the position of the we can actually double the distance over which beam divergence beam waist, is minimized, as illustrated in figure 2.6. By focusing the beam-expanding at the midpoint, we can restrict beam spread optics to place the beam waist to a factor of √ of This result can now be used in the problem of finding the starting beam radius result can now be used in the problem of finding This that yields the minimum beam diameter and beam spread over 100 m. Using 2(z 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:54 PM Page 2.5 2:54 PM 6/15/2009 2ch_GuassianBeamOptics_Final.qxd 2ch_GuassianBeamOptics_Final.qxd 6/15/20092:54PMPage2.6

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics $ $ highlighted insuchaplot,canbesummarizedasfollows: The maindifferencesbetweenGaussianbeamopticsandgeometricoptics, and virtualobjectrealimage. correspond torealobjectandimage, realobjectandvirtualimage, figure 2.7.For apositivethinlens, thethreedistinctregionsofinterest equation. Aplotofs/f In thefar-field limitasz In theregularform, terms oftheRayleighrangeinputbeam. waist oftheoutputbeamrepresents theimage. The formulaisexpressedin assuming thatthewaist oftheinputbeamrepresentsobject,and of thelens. For Gaussianbeams, Selfhasderivedananalogousformulaby 2.6 form, indimensionless or, by SimpleLenses Transformation andMagnification $ where The standardlensequationiswrittenas standard lens-maker’s formula. parameters arecalculatedusingaformulaanalogoustothewell-known and beamwaist locationfollowingeachindividualopticalelement.These simple optics, underparaxialconditions, bycalculatingtheRayleighrange Self showsamethodtomodeltransformationsoflaserbeamthrough Self, “Focusing ofSphericalGaussianBeams”). A. was developedbySelf(S. less rigorous, butinmanyways moreinsightful,approachtothisproblem general an unorthodox manner. Siegmanusesmatrixtransformationstotreatthe $ or, indimensionlessform, It isclearfromthepreviousdiscussionthatGaussianbeamstransformin The maximumimagedistanceoccursat Gaussian beams. There isamaximumandminimumimagedistancefor s There isacommonpointintheGaussianbeamexpressionat than ats from zero(i.e., thereisaGaussianfocalshift). A lensappearstohaveashorterfocallengthas beam hasawaist attherearfocus. the incidentbeamhasawaist atthefrontfocusandemerging fsf s sf () zsf s sz / fzf z sf // 11 f +− s // = is theobjectdistance, s problem ofGaussianbeampropagationwithlensesandmirrors. A R Gaussian BeamOptics s + 2 ″ + / 111 / f () ″ ( = = R f 1. For asimplepositivelens, thisisthepointatwhich . 1 = 1 ) + . 2 /R approaches0thisreducestothegeometricoptics // sf versus () ″ sf = ″ − s″/f is theimagedistance, and 1 for various values ofz/R/f + () sf ″ 1 / s = =

1 f . = z z /R, rather /R/f f is thefocallength increases is shownin (2.12) (2.14) (2.13) range oftheoutputbeam: is usefultoknowtheGaussianbeamformulaintermsofRayleigh respect toinputandoutputbeamparameters. For back tracingbeams, it input beam.Unlike thegeometriccase, theformulasare notsymmetricwith All theaboveformulasarewrittenintermsofRayleighrange The Rayleighrangeoftheoutputbeamisthengivenby parameter normalized Rayleighrangeoftheinputbeamas Figure 2.7 Self recommendscalculating consider magnification:w of thebeamcanbecalculated.To carrythisout,itisalsonecessaryto optical elementinthesystemturnsothatoveralltransformation zmz 11 11 s m R + ′′ = IMAGE DISTANCE ( ″/ ) 4 4 4 4 s f

0 1 2 4 5 3 = zsf s sz w 4 3 2 1 4 w +″″− ″ ″ ″+ 0 0 5 ″ 2 Plot oflensformulaforGaussianbeamswith = R R 4 parameter . 2 4 {} () /( ⎣ ⎡ 1 4 − BETDSAC ( OBJECT DISTANCE 3 () fzf z sf // 0 4 ″ ( ) 2 / w z f ⎦ ⎤ 1 z R = 0 2 R 4 . The magnificationisgivenby , + f 1 w ()

0 . , andthepositionofw R 012345 Gaussian BeamOptics 2 . www.cvimellesgriot.com 0.25 0.50 0 s 1 / f ) 2 0 for each (2.17) (2.16) (2.15) Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 2.7 (2.20) www.cvimellesgriot.com , the result is z D Gaussian Beam Optics Beam Gaussian = Gaussian Beam Optics 50% 13.5% intensity intensity l (f-number) . / 12 2.44 ⎤ ⎥ ⎥ ⎦ 2 , and solve for z ⎞ ⎟ ⎠ 0 2 0 z w l ), that is, the range in image space over which the the range in ), that is, . p z 2 0 ⎛ ⎜ ⎝ # 1.05w D w = f/ p 1 0 l ) ⎡ ⎢ ⎢ ⎣ z Airy disc intensity distribution at the image radiation filling the entrance pupil of the lens. For uniform For radiation filling the entrance pupil of the lens. 0 ( . l 032 w =+ .9 .8 .7 .6 .5 .4 .3 .2 .1 is a constant dependent on truncation ratio and pupil illumination, is a constant dependent on truncation ratio and pupil

1.0 ≈± K INTENSITY =×× () z dK wz w D is the wavelength of light, and f/# is the speed of the lens at truncation. is the wavelength Since the depth of focus is proportional to the square of focal spot size, Since the depth of focus is proportional to the (f/#), the depth of focus and focal spot size is directly related to f-number system. is proportional to the square of the f/# of the focusing Figure 2.9 Figure plane DEPTH OF FOCUS DEPTH OF Depth of focus (8 below an arbitrary limit, can be derived from focused spot diameter remains the formula the a depth-of-focus calculation is to set first step in performing The of 5 choose a typical value If we degree of spot size variation. allowable percent, or TRUNCATION the diameter of the image spot is In a diffraction-limited lens, where l intensity profile of the spot is strongly dependent on the intensity The profile of the on the Airy disc intensity profile pupil illumination, the image spot takes shown in figure 2.9. (2.18) (2.19) 0 w 2 / 12 ⎤ ⎥ ⎦ 2 2 0 2 0 2 0 / / . , the equations for image distance and waist size reduce , the equations for image distance and waist / as infinite conjugate objectives, are more effective in are more effective as infinite conjugate objectives, 0 f f fw fw fw = lp lp Concentration of a laser beam by a laser-line lp () () w / + + 1 1 ⎡ ⎢ ⎣ lp beam = = = D 2 ″= ′′ (the input waist is at the front focal point of the optical system). For is at the front focal (the input waist s sf wfw w e 1 0, we get 0 and is at the first principal surface of the lens system) waist (the input f = = = BEAM CONCENTRATION position of a Gaussian beam can be determined spot size and focal The cases of particular interest occur when Two from the previous equations. s s Substituting typical values into these equations yields nearly identical results, Substituting typical values second set of equations can be used. the simpler, and for most applications, a primary aim is to focus the laser to a very small In many applications, lens or a combination spot, as shown in figure 2.8, by using either a single of several lenses. to using a an advantage If a particularly small spot is desired, there is objective to concen- well-corrected high-numerical-aperture microscope objective of the microscope principal advantage trate the laser beam. The aberration. Although over a simple lens is the diminished level of spherical they are not always microscope objectives are often used for this purpose, Suitably optimized lens designed for use at the infinite conjugate ratio. known systems, For the case of s For to the following: beam-concentration tasks and can usually be identified by the infinity symbol on the lens barrel. s and Figure 2.8 Figure focusing singlet and and and 2ch_GuassianBeamOptics_Final.qxd 7/6/2009 2:07 PM Page 2.7 PM 2:07 7/6/2009 2ch_GuassianBeamOptics_Final.qxd 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:54 PM Page 2.8

Gaussian Beam Optics www.cvimellesgriot.com

Fundamental Optics If the pupil illumination is Gaussian in profile, the result is an image spot of Calculation of spot diameter for these or other truncation ratios requires Gaussian profile, as shown in figure 2.10. that K be evaluated. This is done by using the formulas 0. 7125 0. 6445 When the pupil illumination is between these two extremes, a hybrid =+. − KFWHM 1 029 2.179 2.221 (2.22) intensity profile results. ()TT− 0. 2161 ()− 0. 2161 and In the case of the Airy disc, the intensity falls to zero at the point 0. 6460 0. 5320 = #l# − . dzero 2.44 f/#, defining the diameter of the spot. When the pupil =+. 1.821 1.891 (2.23) K 2 1 6449 ()− . ()− . illumination is not uniform, the image spot intensity never falls to zero 1/ e TT0 2816 0 2816 making it necessary to define the diameter at some other point. This is The K function permits calculation of the on-axis spot diameter for commonly done for two points: any beam truncation ratio. The graph in figure 2.11 plots the K factor = vs T(Db/Dt). dFWHM 50-percent intensity point The optimal choice for truncation ratio depends on the relative importance and of spot size, peak spot intensity, and total power in the spot as demon-

Gaussian Beam Optics 2 = d1/e 13.5% intensity point. strated in the table below. The total power loss in the spot can be calculated by using It is helpful to introduce the truncation ratio −2()/ 2 Pe= DDtb (2.24) = D b L T (2.21) D t for a truncated Gaussian beam. A good compromise between power loss and spot size is often a truncation ratio of T = 1. When T = 2 (approximately where D is the Gaussian beam diameter measured at the 1/e2 intensity b uniform illumination), fractional power loss is 60 percent. When T = 1, d 2 point, and D is the limiting aperture diameter of the lens. If T = 2, which 1/e t is just 8.0 percent larger than when T = 2, whereas fractional power loss approximates uniform illumination, the image spot intensity profile is down to 13.5 percent. Because of this large savings in power with approaches that of the classic Airy disc. When T = 1, the Gaussian relatively little growth in the spot diameter, truncation ratios of 0.7 to 1.0 profile is truncated at the 1/e2 diameter, and the spot profile is clearly a are typically used. Ratios as low as 0.5 might be employed when laser hybrid between an Airy pattern and a Gaussian distribution. When T = 0.5, power must be conserved. However, this low value often wastes too much which approximates the case for an untruncated Gaussian input beam, the of the available clear aperture of the lens. Optical Specifications spot intensity profile approaches a Gaussian distribution.

1.0 .9 .8 3.0 .7 2.5

Material Properties .6 2.0 spot measured at 13.5% intensity level 50% .5 intensity 1.5 spot measured at 50% intensity level .4 FACTOR INTENSITY 1.0 .3 K 0.5 .2 spot diameter = K ! l ! f-number 13.5% .1 intensity 0 1.0 2.0 3.0 4.0

T(Db/Dt) 1.83 l (f-number)

Figure 2.10 Gaussian intensity distribution at the image Figure 2.11 K factors as a function of truncation ratio plane Optical Coatings 2.8 Gaussian Beam Optics Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings 2.9 which stage, y-z www.cvimellesgriot.com Optics Beam Gaussian Gaussian Beam Optics FILTERS SPATIAL . . . Do you need and a single-axis translation stage controls the pinhole location, spatial filter mount accepts LSL-series The for the focusing lens. and PPM- objectives, microscope OAS-series focusing optics, series mounted pinholes. precision individual pinholes are for general-purpose The high-energy laser precision pinholes are spatial filtering. The from high- constructed specifically to withstand irradiation energy lasers. spatial filters with CVI Melles Griot offers 3-axis precision SFM 701). These micrometers (07 SFM 201 and 07 that provides access to the devices feature an open-design spatial filter the instrument. The beam as it passes through consists of a precision, differential-micrometer (%) 60 L beam 100 13.5 0.03 P 2 to form e zero — — — 2.44 d be refocused 2 1/e 1.64 1.69 1.83 2.51 d pinhole aperture hence is spatially separated at a lens focal focusing lens FWHM 1.03 1.05 1.13 1.54 and d light Spatial filtering smoothes the irradiance laser the 2.0 1.0 0.5 Infinity Ratio Truncation or about 1.33 times the 99% throughput contour contour at the focus, diameter. 2.12 Figure distribution FILTERING SPATIAL on optical surfaces may Laser light scattered from dust particles residing Such zone planes. produce interference patterns resembling holographic holographic applications patterns can cause difficulties in interferometric and confusing background where they form a highly detailed, contrasting, and filtering is a simple way that interferes with desired information. Spatial a very smooth beam of suppressing this interference and maintaining scattered light propagates in different direc- irradiance distribution. The tions from Power and Fractional Spot Diameters Truncation of Values Loss for Three a collimated beam that is almost uniformly smooth. complete spatial filtering, As a compromise between ease of alignment and times the 1/ it is best that the aperture diameter be about two By centering a small aperture around the focal spot of the direct plane. scattered light while beam, as shown in figure 2.12, it is possible to block result is a cone of light allowing the direct beam to pass unscathed. The can that has a very smooth irradiance distribution and Page 2.9 PM 2:54 6/15/2009 2ch_GuassianBeamOptics_Final.qxd 2ch_GuassianBeamOptics_Final.qxd 6/15/20092:54PMPage2.10

Optical Coatings Material Properties Optical Specifications Gaussian Beam Optics Fundamental Optics of radius(r The propagationequationcanalsobewrittenincylindricalformterms lobes ofthemodeare180degreesoutphase. as showninfigure2.14. functions areidentical,butforhigher-order modes, theydiffersignificantly, non-Gaussian beamspropagateinfreespaceandthroughopticalsystems. to thenumberofnodesinx multimode lasers, theM approximated byLaguerre-Gaussianfunctions, denotedbyTEM series ofaxiallysymmetricmodes, which,forstableresonators, areclosely The sectionIncorporatingM the For atypicalheliumneonlaseroperatinginTEM has comeintogeneraluse. 2.10 Figure 2.13 The fundamentalTEM LASER MODES propagation equation,andinThe PropagationConstant, In LaserModes, wewillillustratethehigher-order eigensolutionstothe factor ofthebeam. To addresstheissueofnon-Gaussianbeams, abeamqualityfactor, higher theorderofmodeis, themorebeamdeviatesfromideal. mechanism (e.g., transversedischarges, flash-lamppumping),andthe the higherpoweroflaseris, themorecomplexexcitation power beamsfromheliumneonlaserscanbeacloseapproximation,but In therealworld,trulyGaussianlaserbeamsareveryhardtofind.Low- Real BeamPropagation neglected inopticaldesigns, andtruncation,ingeneral,increasesthe lowest-order mode, TEM Note thatthesubscriptsn Hermite-Gaussian (rectangular)solutionstothepropagationequation. Propagation. Figure 2.13showsexamplesoftheprimarylower-order satisfy theround-trippropagationcriteriadescribedinGaussianBeam lasers typicallyhaveanM M 2 factor affectsthecharacteristicsofalaserbeamandcannotbe Gaussian BeamOptics ) andangle(f TEM Low-order Hermite-Gaussianresonator modes 00 00 00 2 mode isonlyoneofmanytransversemodesthat ). The eigenmodes(E 2 factor canbeashigh10ormore. Inallcases, and , theHermite-GaussianandLaguerre-Gaussian factor rangingfrom1.1to1.7.For high-energy 2 into thePropagationEquationsdefineshow m and in theeigenmodeTEM y directions. Ineachcase, adjacent TEM rf 01 00 ) forthisequationarea mode, M nm 2 will bedefined. are correlated M 2 rf <1.1. Ion . For the M M TEM 2 2 , 10 the beamfromafewothergaslasers. However, formostlasers(eventhose beam fromawell-controlledheliumneonlasercomesveryclose, asdoes important, on asingle-pointmeasurementisinherentlyinaccurate;and,most it isextremelydifficult,inmanyinstances, tolocatethebeamwaist; relying beam waist. In the beamradiusandwavefront radius, respectively, atdistance where and specified distance(z)fromthebeamwaist, usingtheequations The mode, TEM gular coordinates. Nonetheless, theLaguerre-GaussianTEM or contaminationontheopticstendstodrivesystemtoward rectan- Hermite-Gaussian modespredominatesincestrain,slightmisalignment, diameter, 2 characterization ofabeamcanbemadebysimplymeasuringthewaist its beamwaist diameteroritsfar-field divergence. Inprinciple, full The propagationofapureGaussianbeamcanbefullyspecifiedbyeither THE PROPAGATION CONSTANT neon laserswiththeappropriatelimitingapertures. “bulls-eye” modeisclearlyobservedinwell-alignedgas-ionandhelium TEM osdrdtobe considered zw wz zz Rz 01 () () l modes, locked inphasequadrature. Inreal-worldlasers, the is thewavelength ofthelaser radiation, andw(z)Rare =+ =+ pure Gaussianlaserbeamsdonotexistintherealworld.The w 0 ⎣ ⎢ ⎢ ⎡ 0 1 , orbymeasuringthediameter, 2 ⎣ ⎢ ⎢ ⎡ practice, however, thisapproachisfraughtwithproblems— 01 1 , alsoknownasthe“bagel”or“doughnut”mode, is a superpositionoftheHermite-GaussianTEM ⎝ ⎜ ⎛ ⎝ ⎜ ⎛ p l TEM p w l z w 0 2 z 0 2 ⎠ ⎟ ⎞ 11 ⎠ ⎟ ⎞ 2 ⎦ ⎥ ⎥ ⎤ 2 ⎦ ⎥ ⎥ ⎤ 12 / Gaussian BeamOptics TEM w www.cvimellesgriot.com ( z ), ataknownand 02 10 “target” or z from the 10 and Fundamental Optics Gaussian Beam Optics Optical Specifications Material Properties Optical Coatings z 2.11 (2.25) times greater ) are constant (2v) 2v M M mode) the product of www.cvimellesgriot.com ) times larger than the 2 00 M Gaussian Beam Optics Beam Gaussian 1. (not = 2 times the beam diameter of the Gaussian Beam Optics M M R = ). z ) and the far-field divergence (v divergence ) and the far-field 0 R = )] ) R 0 √2) ( 0 w w w , the beam waist and far-field divergence of the real beam, divergence and far-field , the beam waist (2 [2 R (2 v v v M 0 RR M M 0 w = Gaussian, but it will have the same radius of curvature and the Gaussian, but it will have the same radius of curvature w = and

) = 0R = 0R R w ( 2 w 2 R M 2w respectively, is an accurate indication of the propagation characteristics of indication of the propagation characteristics is an accurate respectively, Gaussian beam, a true the beam. For same Rayleigh range (z than the embedded Gaussian. Consequently the beam diameter of the than the embedded Gaussian. Consequently the be M mixed-mode beam will always embedded embedded Gaussian will propagate with a divergence embedded Gaussian will propagate with a divergence as the beam propagates through an optical system, and the ratio as the beam propagates where EMBEDDED GAUSSIAN shown in figure 2.15, is useful concept of an “embedded Gaussian,” The modeling and laboratory as a construct to assist with both theoretical measurements. A mixed-mode beam that has a waist The concept of a dimensionless beam propagation parameter was developed was propagation parameter of a dimensionless beam concept The for any given based on the fact that, to meet this need, in the early 1970s not operating in the TEM laser beam (even those the beam waist radius (w the beam waist 0 w = 0 2 z 10 TEM * mode), the output contains some com- 01 00 TEM . 00 opagation characteristics of the beam has long been opagation characteristics of the beam has long The embedded Gaussian Low-order axisymmetric resonator modes axisymmetric resonator Low-order embedded Gaussian 00 mixed mode TEM ponent of higher-order modes that do not propagate according to the ponent of higher-order worse for lasers operating problems are even The formula shown above. in high-order modes. need for a figure of merit for laser beams that can be used to The determine pr the the for example, because, recognized. Specifying the mode is inadequate modes and higher-order output of a laser can contain up to 50 percent still be considered TEM Figure 2.15 Figure specifying a fundamental TEM Figure 2.14 Figure 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:54 PM Page 2.11 2:54 PM 6/15/2009 2ch_GuassianBeamOptics_Final.qxd 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:55 PM Page 2.12

Gaussian Beam Optics www.cvimellesgriot.com

Incorporating M2 Into the Propagation Equations Fundamental Optics In the previous section we defined the propagation constant M2 In a like manner, the lens equation can be modified to incorporate M2. The standard equation becomes 2 w v = 0RR 111 M + = w0 v 2 (2.31) + //2 ()− sf′′ szM()R sf where w0R and vR are the beam waist and far-field divergence of the real beam, respectively. and the normalized equation transforms to 2 = 1 1 For a pure Gaussian beam, M 1, and the beam-waist beam-divergence + = 1. 2 (2.32) product is given by ()////+ 2 ()−1 ()′′ / sf() zR Mf sf sf

w0 vlp= /

It follows then that for a real laser beam, Gaussian Beam Optics 2 M l l w0 v => (2.26) RR p p The propagation equations for a real laser beam are now written as Do you need . . . 12/ ⎡ ⎛ 2 ⎞ 2 ⎤ ⎢ zMl ⎥ wz()=+ w0 1 ⎜ ⎟ (2.27) CVI Melles Griot Lasers and RR⎢ ⎝ 2 ⎠ ⎥ ⎣ pw0R ⎦ Laser Accessories and CVI Melles Griot manufactures many types of lasers and laser ⎡ ⎛ 2 ⎞ 2 ⎤ pw0 systems for laboratory and OEM applications. Laser types Rz()=+ z⎢1 ⎜ R ⎟ ⎥ (2.28) R ⎢ ⎝ 22 ⎠ ⎥ include helium neon (HeNe) and helium cadmium (HeCd) lasers, ⎣ zMl ⎦ argon, krypton, and mixed gas (argon/krypton) ion lasers; diode 2 lasers and diode-pumped solid-state (DPSS) lasers. CVI Melles Optical Specifications where wR(z) and RR(z) are the 1/e intensity radius of the beam and the beam wavefront radius at z, respectively. Griot also offers a range of laser accessories including laser beam expanders, generators, laser-line collimators, spatial filters The equation for w0(optimum) now becomes and shear-plate collimation testers.

2 1/2 lzM w0 ()optimum = (2.29) p

The definition for the Rayleigh range remains the same for a real laser beam and becomes

2 pw0R zR = (2.30) M 2 l For M2 = 1, these equations reduce to the Gaussian beam propagation Material Properties equations .

⎡ ⎛ 2 ⎞ 2 ⎤ pw0 Rz()=+ z⎢1 ⎜ ⎟ ⎥ ⎢ ⎝ ⎠ ⎥ ⎣ lz ⎦ and 12/ ⎡ ⎛ ⎞ 2 ⎤⎤ ⎢ lz ⎥ wz()=+ w0 1 ⎜ ⎟ ⎢ ⎝ 2 ⎠ ⎥ ⎣ pw0 ⎦ Optical Coatings 2.12 Gaussian Beam Optics 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:55 PM Page 2.13

Gaussian Beam Optics

www.cvimellesgriot.com Fundamental Optics Lens Selection

The most important relationships that we will use in the process of lens We can also utilize the equation for the approximate on-axis spot size selection for Gaussian-beam optical systems are focused spot radius and beam caused by spherical aberration for a plano-convex lens at the infinite propagation. conjugate:

rd 0. 067 f spot diameter (3 -order spherical aberration) = Focused Spot Radius ()f /# 33 2 l fM

= (2.33) Beam Optics Gaussian wF This formula is for uniform illumination, not a Gaussian intensity profile. pwL However, since it yields a larger value for spot size than actually occurs, its use will provide us with conservative lens choices. Keep in mind that where w is the spot radius at the focal point, and w is the radius of the F L this formula is for spot diameter whereas the Gaussian beam formulas are collimated beam at the lens. M2 is the quality factor (1.0 for a theoretical all stated in terms of spot radius. Gaussian beam). EXAMPLE: OBTAIN AN 8-MM SPOT AT 80 M Beam Propagation Using the CVI Melles Griot HeNe laser 25 LHR 151, produce a spot 8 mm in

12/ diameter at a distance of 80 m, as shown in figure 2.16 ⎡ ⎛ 2 ⎞ 2 ⎤ ⎢ zMl ⎥ wz()=+ w0 1 ⎜ ⎟ The CVI Melles Griot 25 LHR 151 helium neon laser has an output beam RR⎢ ⎝ 2 ⎠ ⎥ ⎣ pw0R ⎦ radius of 0.4 mm. Assuming a collimated beam, we use the propagation and formula

⎡ 2 2 ⎤ 2 1/2 Optical Specifications ⎛ ⎞ lzM ⎢ pw0R ⎥ optimum = Rz()=+ z1 ⎜ ⎟ w0 () R ⎢ ⎝ 22 ⎠ ⎥ p ⎣ zMl ⎦

and to determine the spot size at 80 m: 1/2 2 1// 2 optimum = lzM ⎡ 2 ⎤ w0 () ⎛ 0., 6328×× 10-3 80 000⎞ p ().80 m =+ 0 4⎢ 1 ⎜ ⎟ ⎥ w ⎢ 2 ⎥ ⎝⎜ p()04. ⎠⎟ ⎣⎢ ⎦⎥ where w0R is the radius of a real (non-Gaussian) beam at the waist, and 2 = = 40.3-mm beam radiu wR (z) is the radius of the beam at a distance z from the waist. For M 1, s the formulas reduce to that for a Gaussian beam. w0(optimum) is the beam waist radius that minimizes the beam radius at distance z, and is or 80.6-mm beam diameter. This is just about exactly a factor of 10 larger obtained by differentiating the previous equation with respect to distance than we wanted. We can use the formula for w0 (optimum) to determine the and setting the result equal to zero. smallest collimated beam diameter we could achieve at a distance of 80 m: Material Properties

12/ Finally, ⎛ 0., 6328×× 10−3 80 000⎞ optimum = = 40. mm. 2 w0 ()⎜ ⎟ pw0 ⎝ p ⎠ zR = l

where zR is the Raleigh range.

8 mm 0.8 mm Optical Coatings 45 mm 80 m

Figure 2.16 Lens spacing adjusted empirically to achieve an 8-mm spot size at 80 m

Gaussian Beam Optics 2.13 2ch_GuassianBeamOptics_Final.qxd 6/15/2009 2:55 PM Page 2.14

Gaussian Beam Optics www.cvimellesgriot.com

Fundamental Optics This tells us that if we expand the beam by a factor of 10 (4.0 mm/0.4 mm), In order to determine necessary focal lengths for an expander, we need we can produce a collimated beam 8 mm in diameter, which, if focused at to solve these two equations for the two unknowns. the midpoint (40 m), will again be 8 mm in diameter at a distance of 80 m. In this case, using a negative value for the magnification will provide us with This 10# expansion could be accomplished most easily with one of the a Galilean expander. This yields values of f = 55.5 mm] and f = 45.5 mm. CVI Melles Griot beam expanders, such as the 09 LBX 003 or 09 LBM 013. 2 1 However, if there is a space constraint and a need to perform this task with ff12+=50 mm a system that is no longer than 50 mm, this can be accomplished by using and catalog components. and f Figure 2.17 illustrates the two main types of beam expanders. 2 =−10. f1 The Keplerian type consists of two positive lenses, which are positioned with their focal points nominally coincident. The Galilean type consists of Ideally, a plano-concave diverging lens is used for minimum spherical a negative diverging lens, followed by a positive collimating lens, again aberration, but the shortest catalog focal length available is -10 mm. positioned with their focal points nominally coincident. In both cases, the There is, however, a biconcave lens with a focal length of 5 mm (LDK-5.0- Gaussian Beam Optics overall length of the optical system is given by 5.5-C). Even though this is not the optimum shape lens for this application, the extremely short focal length is likely to have negligible aberrations at this overall length=+ ff12 f-number. Ray tracing would confirm this. and the magnification is given by Now that we have selected a diverging lens with a focal length of 45 mm, we need to choose a collimating lens with a focal length of 50 mm. To magnification = f2 determine whether a plano-convex lens is acceptable, check the spherical f1 aberration formula.

where a negative sign, in the Galilean system, indicates an inverted image The spot spot diameter diameter resulting resulting from from spherical spherical aberration aberration is is (which is unimportant for laser beams). The Keplerian system, with its 0. 067× 50 internal point of focus, allows one to utilize a spatial filter, whereas the = 14 mm. 625. 3 Galilean system has the advantage of shorter length for a given magnification. The pot diameter re ulting from diiffraction (2 ) i Optical Specifications The spots diameter resultings from diffraction (2wo)isw0 s 2 (. 0 6328× 10−3 ) 50 = 5 mm. p40.

Clearly, a plano-convex lens will not be adequate. The next choice would Keplerian beam expander be an achromat, such as the LAO-50.0-18.0. The data in the spot size charts indicate that this lens is probably diffraction limited at this f-number. Our final system would therefore consist of the LDK-5.0-5.5-C spaced about 45 mm from the LAO-50.0-18.0, which would have its flint element facing toward the laser.

f1 f2 Material Properties

Galilean beam expander References A. Siegman. Lasers (Sausalito, CA: University Science Books, 1986). S. A. Self. “Focusing of Spherical Gaussian Beams.” Appl. Opt. 22, no. 5 (March 1983): 658. H. Sun. “Thin Lens Equation for a Real Laser Beam with Weak Lens Aperture f1 Truncation.” Opt. Eng. 37, no. 11 (November 1998). R. J. Freiberg, A. S. Halsted. “Properties of Low Order Transverse Modes in f2 Argon Ion Lasers.” Appl. Opt. 8, no. 2 (February 1969): 355-362. W. W. Rigrod. “Isolation of Axi-Symmetric Optical-Resonator Modes.”Appl. Phys. Let. 2, no. 3 (February 1963): 51-53. M. Born, E. Wolf. Principles of Optics Seventh Edition (Cambridge, UK: Cam- Figure 2.17 Two main types of beam expanders bridge University Press, 1999). Optical Coatings 2.14 Gaussian Beam Optics