ESSESA~ZA REVISTA MEXICANA DE FíSICA..I7 (6) 562-567

Gallssiall beams as wave packetsP. Halcvi

InWltutn Nacional de Astrojúica. (Í¡Hica y Eleetr(Ílllca, Cl'lltm de Im'estigacián ('1/ FÚlcaApartado postal 51. 72000 PI/eh/a. /J/le'.. Mexico

1\LA. e~rvantt'sUIIÍI'l'rsit/{/(lde SOI/01'(/.

AIWttac/o postal 5-088, 831 <)0 Hcrmosillo, SOIl" Me.ric()

RL'cihidt) el 17 de may() de 2()()1: ;K'eptadtl el 14 de septiembre de 2()()I

[)ICIEMBRE lOO]

Two CtlIH:ep'"01 l'urrcnt usage in modcrn science are thosc nI' ~;¡ll..•..•ian bcams ami wavc packets. In tlu ..•work \liCprcscm a dme conneelionbl'{WCCllIhell1alld ..•how hu\\' il Is pm•..•iblc 10derive a mathemaul'aJ rcpre ..•enlatlOrl l'or ~aussian be;¡lm !'mm the idea 01'a W¡¡VCpackel. In facl,\Veprovc Ihal a ¡!.lll..•sian bcam is essclllially a f,upcrposition 01'plarle wavcs who ..•c pmpagation dlrccllon IS rC.'>lnc[cLÍto a very narrow cune1•."l'lltcrcd on lile hl'<lm axis. \Ve prcscnt an mi/lOe approach ha..•ed on physical lIOllons conlrastin~ wi[h lhe usual mathematical dcrivation.This anil'le ¡¡hu l'eviews the csscntial properties 01'this simple ..•l fmm uf ,1hcalll 01'ligll1. ,llld is wcllsuilcd rOl'dassroorll prescnt;]ti01l.

1\1'.1'11'01"1"": Curricula: le,lching mcthods; slfatcgies and evaluation: hl'am charactcri"'lic ..•: profile: IlHCllSltyand power

DI)'"L'lll\L:Cpl\l'".l((lJ;lImellle llliliz.¡dos en cicllcia modcrna sun lo...Je ha/. gaussiano y paquete de onda ....En ('..•Ie trahajo prcscntamo~ unac...lredlil (lllh.>\IÚnl'l\lre ellos y dcmoslramos cómo cs posihle del'i\':lr una repre:--cnl;ll'Íóll matemática para los ha.:cs gaussianos a partir de1;1iJea dc Ull paquete dc (l[JJa~. DL' he.:ho del11o.'>lfamo.'"que UIl ha/. gaussiallu es e..•eneialmentc Ulla superposi.:ión de ondas planas cuyadirección de propa!!;lciún e~tú rc ..•tringida a un angosto cOllilccntrado cn el eje del ha/ .. Prescntamos un enfoquc od ¡IOC hasado en nocionesrí..•iC<l'"que dilicrl' de la Jerivación matcm.ítica usual. Este trahajll 1,1Il1hiéllrcvisa la..•propicdaJe:-- escl1l:i¡dc:--del haz luminoso m;í..•simple yn:-..•ulta lllUY"pruplado para fine:--did:Íl"licos Cllel aula.

j)~,\cril'tor/'.\: ClIl'rínlla: mélOdos de ensl'i'lanza; estrategias y ('valuación: l'aracterístll'a ..•de] haz ¡!au..•..•iano: perfiL intensidad y potencia

1. 1utmdurti()u

In the cla ...•sroolll. an al'l'O\\' is frcquently referred to as a h('ol1l,

while. in ICíllÍly it jllst reprcscnts a planc wav~. Hen:, \vcIm.'sl'nt a :--implc. physically appealing, treatment ol" thc sim.plesl C;¡"'C01' a tme heam. namely the gaussian bcam.

A gaus:-.ian beam lI] is basically <lwave neld whosc CIH~r.gy dcnsit)' is strongly localized alung i.1particulardirection in~pac('. The dislriblltion 01' 11Iminolls cncrgy in a planc Iran~.\'l'rse \O thc propagation dircction varies according \O lh~normal. or gaussian distribution. 'lOday, mos{ \l,.'"clldcsigneJlasers ~l'nerall' an Olltput bcalll that is very c10se to an idealgau~sian prolik. This is so for rcasons al' convenicncc: Sumenf the mosl importanl aspects of laser beams are rclaled lOlheir Jirectiollality. Ihat is. propagation in a straight line wilha lllinilllum div~rgellce. This Icads to importa .t applicationsin prccision alignlllem, surveying, aiming, and the Iik~; ofradiation iJ1lo very small far-ncld ~lIlgles which produce tinyilluminatcd arcas 01",under t¡ght focllsing at close distances.lo.cxlrclllely smal! spots \I¡'ilh very large cnergy dcnsity perunil arca.

Lasers are an appealing tool in the endeavor 01' populariz-íllg sciclll'L'. ¡¡lid I'rom the cducational point of vicw, the topicnI' la:-.t'rs is :-.tL'adily increasing (sec for example Refs. 2 .. ~and :)). The IISL'of helilllll-neon gas lascrs is \videly spread inlIl:lny lIndcrgradualc [aboratorics. Diode lasers are massivclyL1scd. fOl' instanL'c. as pointers or in check-out cash registers.¡hus pnwiding lively examplcs 01' the prcsencc 01' the abovelIlentiuned heallls in dady life.

Unlikc Ihe case uf olhl'r wavc helcls. Ihis is .\ particularIypc of hcam \vhosc fundalllL'nwl nature is not altercd by freeprop.¡gation in vaCUUlI1, or by Ihe passagc through unabber-rated Ienscs: l.£' .• ,',.'hile Ihe width as wcll as the radius 01' cur-vature may change, as it propagales, the beam profile remains

gaussian.

Dile 10 its simplicity, Ihe normal distribution is convenientin Illany applicalions where high-power t1ux is nOI an essen-tial rcquirclllent. The understanding of the charactcristics oflhese heams is thus a prcrequisitc 10 the study 01' many laser-relatcd phcllomena.

Abundant spccialized litcrature is available about this sortof beallls. Th~ non-spccialisl rcader is rcferred 10 Ref. 4 01' 10Siegl1lan's bool.: [G] as a slarting point. and the extensivc listof rdcrcnL'cs thereill.

Tht: qucstion of why or how the emission of such ligh!sources turns out lO be gaussian is normally answered in spe.cialized hooks whose leve! nf explanation ¡s, in general. tootcchnical rol' non-specialists.

Ikre \\le present an alternativc uerivation to that providedin more adV<lnced lexts 171. The purpose 01' this papel' is to",hll\V that a gaussian bC:llll is ;1 packet (01' superposition) ofphll1e \vavcs ",hose dircctiolls 01' propagalion are restrictedlo. esseJlIially. acolle 01 sJl1al\ salid angle areuno a particulardireclion (lhe heam dircction). This approach stresses a pe-dagogical point of vie\v. ha sed Vil physical CorlCL'pts Ihat COIl-trasls with a mathematical approach which relies on severalapproxilllations (7].

P.HALEVI ANO r-.1.A.CERVANTES 563

\vhere

2.1. (;aussian heam

(21

ir .fo

[ (z ) '] 1/2 '.'-'1+ _2k

1" (:)

Here, :!...,), -= /II(()), J..- == :!<T/"\. and :: = () is taken in tile planeof lile bC:llll wais!. '/;1 i:-.a COll:-.tant proportional to lhe oll-axi:-.ilncnsily villllC of the bealll. ilS \vill be shown shortly.

A:-. ,: v¡¡r¡es. the beam profile givcn by the aboye Eqs. (1)c!l.lllgl.:." acconling to Ihe givcn funcliol1s of z. giving rise tothc hehavior abm-e outlined. 11e.m be shown that. w¡th thecxceplion of Ihe illllllcdiale vielnity of the pl;.me z = O. thewavefronts are parabolic. and that for /1 « =2 the distinclionbetwcell par.tboJic and :-.phericaJ ."urfaces is unimportant.

Prom Eq. (2), at lar~c .:, the ilSYIllPlotic representalian01"11'(::) i.'ia slraight Jine.nalllely,

ami

zzOLAa:R

.\¡ain conccpts

';re;l' 1{l. ,. Sdll'matlc 01 a la"L'r pn1ducing a gall~si~m hcam propa-~;lllll~ alpll1! lhe +>:lXI"'. The Illain paramelers are lIldicalcd anddl'lincd: The rauiu .••nf lhe .••pOI w, lhe lilr lield sprcad <Ingle (Jh. ~llldlile ]{;Iyk'i~h rangc (. Thc neal' and far ¡¡cid rcgions are dclillCt!\\1[11rc:-.pcc[ lo paramclcr ( a.••di"'Cll!'>\cd in lhe [ex\. Tht: alllplilullcdi .•.lrihulillJl i .....••IlO\vn Oll lile Idl h,llld .••idc.

ro 11lL' naKL'd eyc. Ihe gmlssian oeam of everyday life i.....•.lIai~hl <tlld has some width that V,'Cdesignalc by Ihe pa-ralllL'lef 2J']~. Actually. lhe inlcnsity is grcatcst 011Ih~ axisuf lhe hi.,'alll. aud decreases radially as l'Xp( -f':!. /'2~:!), whc-Il' (/ = J.,.:!' + y'!. i...(he radial dislance lO rhe propagalioll;1\ I~ .:.

This ,,,illlpJe behavior is valid only in the ne¡u' ticld I.Ol1e,llall1cll', 1'01'distances :: (I"rolll the laser ollttd) that arc subs.l;llllially ~lllaIJl:r than ,he: paramcter .J;r...,),:.'/,.\ WhL'l"l',.\ is lhe\\d\'l;'lcn~lh (aboLlt IJ..-) IIIll in the \'¡sibk re~ionJ. Thl' \aluetll llii." parilllleler i~ typlcdJy several meter~. and lhe bcam,rar'" di\"l'rgill!l onJv for di~lanccs .: > 10 m! In lhe far lleJdl\'f!ioll,: » -liT...,),:.'./,.\. "file bchaviOl:;s dominated by diver-:,.'CllCl'which. Ilevenhekss, is vcry smal!.

¡:i~lln: I l.s a represcntatioll 01' a gaussian bcalll produccdby ;1 laser poillting in the +z.dircction. In acldition to Ihebl'lJ ~h;lpl'd intensity distribution, the beam is characterizcclh~ 1\\0 Illain parameters, Ilamely, the radius of curvature ofllll' IIl'arly ~pherical wavefronts. R. and the 1;'lteral e:xtent ofthe healll. abo referred lo as Ihe beam spot size 1/'.

11."llally. hut nOI al\Vays. the: \Vaist of the bcalll is malletI) l'oIJll'idc \\iith the out pUl mirTO!"01' the laser cavity. As lhe:hc;lI11lran:l.s. both the spot size and the radius of cUl"valurt: 01'lhe W;l\'l'IÚlllt chan~c according to a prescribccl manner ex-plaillcd helow. The radius 01"curvature is measurcd fmm Ih~;"ial point (/1 :::::.: = UJ whcre the spot size IS minilllllllL this1" ~oll1clillle:-. called {JIl' mli.\"I (~lr"{'h(,wll.

rile "urface obtained by COlHlccling POllltS of cqllal alll-plilude ilre hypcrboloids of revoJutian arollnd lhe direclioll ofpnlp¡¡~i\lillll. o:.

l'lll' illtcl1~ity di:-.tribulion. in cylindrical coordinate:-., lIscdIn repn'''l'lll a ~al1SSiall be:llll as dcscribcd abo\'e is

1.1. "'an' packt,(

This angJe defines the divergence 01"Ihe beam in the far zone.Thcll. 1"01"arhilrary.:. Eq. (2) lIlay he: rcwritten as

(3)

(4)

(7)

(5)

11'(:) = (_.\ ) o.liT ...,),

1"0 - f < 1.. < 1"0 + f.f(l.) el- 11.

Thi!'opl"O¡la,t:ariofllill(, pa:-.~es lhrollgh lhe origin with an incli-nation, wilh respcct lO the z-axis. givcn by the angle

.\H" = 2",-, (<< 1).

(;(1. f) = l: f(k) p,p[i(h - ",t) Jik. (6)

\Ve kno\\' h()m quanttlIll mechanics that a particle withsharply dl'lilll'd lllome:nllllll is not localized in spacc. Thewave llle:chanieal analogue that pcrhaps most dosel y rcsern-bIes a c!as:-.ic;I[ Iwrticle is a \Va ve p¡lcket. a slIpcrposition of agroup 01"planc \Van:s of nearly lhe samc \vavelength that in-terferc dcstnKlively cverywhcre excepl at the ¡ocation af lheparticlc and il.'i vicinily.

\Vc a~:-'lll1lCth;l1lhc principlt' of linear sllperposilion holelsfOl"\\'-'<lVl'S.In ~lIch a ca:-.c. a wave packel can be represcllleelby the ~lIpl'lp(l:-.itloll uf a ll11lllbcr 01"planc waves. Then 1"0l"a\\'ave pad,el prop¡¡g¡lling along Ihc.r axis one can wrilt:

Herc fU,') l." l!le ¡¡mplitucle of the plane wave exp(ik.r);lml.,.: = .....-'(/;).Ir l!lis is to represcnl a wave group travel-ing \vitll characleristic group vclocity, il is nccessary that theI"<Ingeof propagatioll vcetors 1.. included in the slIperposilionbe quilc slllal!. In other wO!"ds. il is assumed that the fune-lion f(l;) i~IlOll/cm l'0l"only a ~llIall range al" k about a par-ticular \;tiUL'¡"11 of 1,'. Thi~ conLlilionll1cans l!lar

(1 I[-2f/][(f'. '.) = I,,(.:),'xp -,- .11'- (o)

564 (jAUSSIAN BEAMS AS WAVE PACKETS

(9)

(10)

1.'..1 + 1.'." + 1.'." = Ikl" = 1.'."r~: -

k,., k" « k,.

where f is Ihe dicleclric con",tant of the mcdiul1l, usually airoTbis is wlly in tht: integral aboye we do not integralc

over 1.:;. The depcndclll'c nI" f( k) on Ihe direclJun .of k deter-mines the fOl"m 01' F( •.). For a hCilllI of lighl direcled in lhe zdireclion this shollld be thc pn:doll1inant dirccllon of k. How.ever, small transverse compollents Á:r and kl¡ are expected 10limit lhe infinite cxtensiOll of a pl,me wave. Thus. we willilssllme that

Here. il is important lo realizc that the Ihrec components of kare not independenl bUI are relaled by

,

/(k)(j(_xl__ D,X ----l

-1 i-X kola) (b)

Fl(ilfKI: 2. ;1)Wavcpackcl rcprc!ooenlalion .100 h) Ihe k vcctoruí~(ri.huiion IIMt proJllCC~il.

This ¡¡pproximation corresponds to a small divergence of thencam which. is the very essence of a well detlned beam.

A distribution 01' transverse components of the k vector.lh;]t can approxilllillcly realizable in practice. is a gaussian:

Here I / ~ delínes the width of the distribution. Thus,Iransverse \'lave veclors are in Ihe mnge

It is .llso lIsually assulllcd that Ihe function w(k) can be ex-panded in a power series ayer a small range of values in lhevicinily ol' 1.:0- lf this expansion is used. Eq. (6) can he pUlin lhe fOl"m of i.l product 01'an envclope fUlletion nnd ti plancW.IVl.;'ICilding 10 a reprcscntatioll which is illustrated schcma-t¡cally in Fig. 2. FUI" a more detailed discussion 01' wavc-packcts Ihe rcader is referred 10 lhe standard literature [8.9].

In this subscction \Ve have assumcd that lhe magnitudeol' k vafies according to sorne dispersion law w = w(k).This. of (ourse, implics dealing with polychromatic waves.On Ihe olller hand. for monochromalie waves Ihe magnitudcof k is l:onstant. Henee. tlle only way to fonn a wavepacketi", hy varying the dircction of k. This. precisely. leads 10 theforlll<.llion of a gaussian beam which is the topie of lile next

seclioll.

Thcrcfore.

[1 1.'." + 1.'.1]

k. = JI.'.' - 1.'." - 1.'." :o: l. I - -' ".- '" :l J•• "J

[1.'." 1;1] 1/1 < .!c'+'1 "'~.

( I 1l

( 12)

the

( 13)

J. F•.••m wave \lackcls lo gaussian bcams

•..In a hOlllogeneous. isolropic medilll1l with real dielectricl'on",I,lIlt f, a monochrolllalic wave has a wave vector k whosemagnitlldc rcmains constant bUI whose direction may change.

Con",ider the rcpresentation of wave packets in spaee. As-slIllling l1lonochromatic waves. lhe general form is

JI dk,dk" f(k) ('xp(ik. r-wt) '" F(r),,-ó." (8)

I

The k veclors lic within a COIlC. sllch that lhe small salid an.gk tlU forlllcd by thcm is given by

., 'ir 1f >.:.!k- dI! - ~1 --> tll! - 1.1~1 = 47f~1' (14)

Hen: /\ is the wi.lvelength in the lllcdiul11. Thcn. one must rc-quire..\:2 « ~:.!in arder lo have a weJl dctlned beam. In thevisible region typically rlO¡'lrr •......'2 x lO-ti. Thus. the spreadis extremely slll,,11.

Then, by Eqs. (8), (12), and (11 l,

( 15)

Bóth inlcgrals are of the s<tme fonn wilh :r -t y. and they are 01"Ihe typc enCOllJ1lered for gaussian pulses. The rcsults are

ó,. 7f ( ,,1/.1)F(r):o: f,," . ~1 + ;z/U: "Xl' - ~1 + ;.:/21.'. .

(16)

\Ve C;lll write Ihis as

(/7)

lú'\: "'kx. Fú. '¡7 (6) (2(X)I) 562-567

P.HALEVI AND M.A. CERVANTES 565

( 18)

( 19)

(20)

bell shaped fOl'ln. is l,,(z) "xp[ -p' /w(z)l. This, 01' course, isg<lllssian for any value ol':::. The remaining factor in Eq. (17)has the fOl'ln "xp[i,,(.:.I)I, \Vith the phase" given by

kp"2"(0. f) : kz - 'P(z) + 211(z) - wt. (28)

At any gi\'cn instal1t 0'- timc t. the phase of the wave has thesame valuc il'

Frum ,he last equation for F(r) we can obtain lhe ¡("ra.diilllCC (01"in!cllsity) distribution associated with lhe energyc;lrricd hy Ihe heam:

;l1ld

,,)(..) = l,all-I (_.Z~)'1 - '2kD..2 (21)

This equation then determines surfaces of constant phasenI" wavefronts. They have. of course. cylindrical symmetryarolllld lile .: axis. however their form is very complicatedfOI":; ••..••(. Simpl illcation occurs for distances ;; thm are eitherIlluch slllaller. or Illuch greater, than the Rayleigh range (.

4.1. Ncar '¡cid wne. z «(

.t DisClIssioll

No", \Ve Imn to the physical interprclation of the resuh¡:,q. lI7). I \ere the amplitude of the wave thar accounts for its

1(".:) : IF(/'. 0)1' : If,,(z)J' exp [- -+-( "']. (22)1V- z)

(30)

In this region f,,(o) e: rrf,,/Ll.' and III(Z) e: 2Ll.. The am-plitude 01' the \Vave becomos ("fo/Ll.')exp[-p'/4L'.'], in.deJlcndent of Z (over distances of sevcral meters, in practice).The intensity gradually decreases away from the axis fJ = O.md vanishes as p -+ oo. Thus. lhe radius of the beam is un-ddincd. Ne\'crlhcless. lhe cleclromagnetic energy is mostlyronnned to a distance ";;11I.2t:i from the beam axis. Thisqumllily. 2~, rcpresents the lateral uncertainty, Dop. in theposilioll nf a pilotan in lhe beam, or ~p = 2Do. In a simi.lar fashioll. wc.:sce from Eq. (12) that the uncel1ainty in theIransverse W<lvcvector is ~/.:f' :::;:1/ ~ and the corresponding1I1lccrtainty in lhe 1l10mcntlllll of the pilotan is fJ.P(1 :::;: II/.ó..Hcnce, the product of the uncertainlies of these conjugatevariables (1)./',,) is """ . Ll./'¡, : (2Ll.) . (h/ Ll.) : 2h. Thus,we essentially connrlll the uncertainty principie for a photonin ¡¡ bealll of Iight.

In the same limit. ¡/J(z) e: z/( and n(z) e: (' /z -> oo.Thereforc. unless IJ is c.\ll"cll1cly large. according to Eq. (29).Ihc surfaces nf constant phase reduce simply to z = eOllst.This gives rise lo lhe planc wave hehavior exp(ikz); oneshould rClllclIlbcr. hO\\:cver, that this is not an illjillitl' planewave. hut olle whose w<lvcfront has an arca "V 1r(2'ó'):.!.

llere fll(;;) e: rrfll(/""'o allll '!V(z) e: 2Ll.z/(: (.\/2rrLl.)z.This is to say that the width of the beam increases in directpropol1ionality 10 the dislallce z, The spread angle is

".2. Far t¡del zonc. z » (

.\tll, : 11'(0)/.:: -,-

2 rr '"as noted belore in Eq. (4). Typically, ti" ~ 11)-,1 rad e: 10-'dc.:grces. a r;'llhc.:rslllall divergence. We also get, in this limit,Ihat ,I>(z) ?! rr/2 allll /I(z) "" z. Thcn. Eq. (29) reduceslo z + t':!' /'2;;. :::;:nHlst. It i~ rcadily scen that Ihis cquationdescribes a spheric;'ll wavdront. For tlle locus of such a wave-frolll is (,,.:!. + .ti:!. + .::!.) :::;: ¡/- + .:;:!. = ,."2 and if z » p lhisapproximates to l' ~ .: + ,,:!. /2:::. Hence we can identify Ihe

(27)

(23)

(24)

(26)

(25)

( =l1rLl.'/.\.

w(O):a[l+ (z)'] 1/'.

II(Z):z[I+(Dl

Thi ....",olution agrees witll Eq. (1) of Seco 2.1, and is idcntical10 IhOltgiven in lhe theory of aplical resonators [7,10-14].Thcse det1nilions imply propagation in a transparent medium:hO\ ••..cver. il is no! diftlcult to generalize them lo allow fOI"¡I<lmping.

1\ •.•Ihe distancc:: incrcases. a point is rcached al which Iheare;l of tlll" spOI is t\Vice as largc as tha! al z = n. Thb defineslhe so callcd Rayleiglz mllg£'. Using Eq. (J 8) we nnd tl1allhisparticular disl<lnCC is

In Seco 2.1 v.-e already noted that lhe condilions:; « (~llld: » (, i.lt:tinc. respectivcly. the lIe;'lr and far t1eld zones.

The set (JI' Eqs. (18)-(21) complctcly describe the beam;l~ ji propagatc.:s, U~illg Ihe aboye definilioll of the Rayleighrange Ihey can be put as

f(n. Mi'.\". Fí.\'. '¡7 (6) (2(KlI) 562-567

S6t) GAUSSIAN BEAMS AS \VAVE PACKETS

R(z/W? w(zf?)/2ó.

rp(zm.. .

FHiIlKF. 4. Tlle túnclioll R(;:) uf ~Igausstal1 be;¡m, The quanti-ly U(:/O/C is p!t)lh:d I'I'rSils:j( tu relll.lcr a lltli\wS;IJ l'ur\'e. Theslralgl11 tille is Ihe ••...ympltlIIC 11Illlllur.: » (,.The positions 01"lheRaylcigh range allllmillilllUITI r;ldiu ..•an: inJicalet.! hy gridlincs .

12 Zj?

20

2 t, El 1 (l

f.1(iURE ,1, The quanlilY 111/2¡). IS plottcd 1'{,/,SlfS z/( lO ren-del' il lI11i\'l'r,,~dcurVl' for Ihe wid!h of lhe heam, Thc Raylcighr~lll~Cpmltlllll 1" IlldicalcJ hy Ihe.:vertical lineo The minillllllll "POI•.•i/l.'. l...J.. ¡'CClIr" al ¡he origlll. The curvc approachcs asymploli-L'all) IIlL'1;11't¡cld divel',!!CllCCan~[e represclllcd hy Ihe slraigbl hile01 ..•tupe '2J./(, This [inl' fonns;ln angk 01"45° \vith Ihe ;::-axisin (JIII'ditlll'll ..•íOllleSSrcprCSelllalioll. Al a dislance of OtlCRaylcighl'angc unil. Ihe ¡¡fCaof lhe spOI is l\Vil'e as Iargc as Ihal al :; ;;;;O.

o .•

•••"consl:' ;¡ho\'e as the radius ofthe wavcfront.ln addilioll. Iy-pically l' ::S U'(z) o:; (A/2rr :::")z. In Ihis case the terlll/" /2z 1.2

lllay be llc~.!Iecled. ami Ihe surfaces 01' constant phasc are.agalll.: =: nJllsI, jllst as in !he near-tlcld zone.

~.3. :\rhilrary z 0.6

o .•

FI(iIIRE 5, The phasc tUllclion o \'('/',\/1.1' ,:/(. The phasc angle in-ercases lincdrly iJl the Ile~lrticld rc~iull. ~lllJ lcnt.!s lO lhe asymplOli(.'[¡mil 7r/'2 in lhc far 1ielJ rc~illlc,

dircctions are limitt:d to a cone 0'- very small solid <Ingle (cen-lered along the be~1l11axis). Thc derivatioll is slraightforwardand relics only 011 a ...•ingle approx.imation. namely Ihat thedi\'ergence of Ihe beam is \'CT)' smal!. Frolll Ihe cducationalpoint of vie\\'. Ihis approach sl10uld ser\'e to f1l1 an existingconceptual gap helwC'cn aplane wave amI a beam 01' light.

Additionally. a classical simi!e 10 the uncertainty princi-pie is conveyed in Ihis approach, This can be helpl'ul for ahetter Ilnder~lal1dillg 0'- Ihis imporlant principie ol' quantumIIlcchanics,

The hch"y;or of Il'.Ii, "nd <J; for arhitrary z [E'Is. 24). (25)amI {27ll. a~ functions ol' the dimensionless paramcter z/(is shown in Figs. 3-5. In !hese universal plots. we recognizeIhe !lear tleld region as olle for which z/( « 1 whereas thefar tic Id cOITesponds lo :;/ ( » 1. Figun: :3 shows thc beamwaisl :2~ at :: = ()am\ll1(: asymptotic approach lo Ihe line ol'",lope ;,ltl~Jc given by Eq. (30). NOle lhe slow varialion nearIhe waisl of Ihe beam.

In Fig. -l. note thal (he beam atlains a mini mal \'aluenf 1:(;;) precisely al Ihe Rayleigh rangc (: l!lis valllei:-.H =: '2(" ¡\\Vay fmm the mínimulll it increases very rapidlyrol' : <•••• (, ;md Illuch more slo\\"I)' ror z > (. In bOlh lilll-ib .. : --t tJ and :; ---+ x. H -t oc, \\'hich is to say that Ihewa\'efronts are planc very near and also very far frolll thelasl..'r apl..'r1ure. i-=igurc .5 shows that the phase rj.J(z) exhibits,111 esscnlially linear ¡ncrease in the near field ami a constan!\alul' 01' ii' /'2 in Ihe far-field regimc.

Figu1'l:':-'3-) cOllhnn our near anJ far-tield limils.

0.2

10 " 20 zll;

5. SUlllln.u'y

In this papeL we have presenled .1I1alternalive approach forIhe dcri\'<llion 01' a gallssian beam. based on the nOlion of awan~p;¡cJ...cl 01' Iransversal components of the wa\'e-vector.Essenlially we have shown Ihat a gaussian be.1I11may bev¡cwed as a superpositioll of plane waves whose propagation

A cknow Ied !(lIIcn ts

Tile authors (P.l1. ;Illd f\1.A.C.) gratefully acknowledge thelinancial support oí' CONACyT (J\Jcx.ico) undcr grants No.;~:lI!JI-E anel 2500;);). respectively.

R('I'.Me,\'.Ff~. 47 (6; (2001) 562-567

P.HALEVI AND ~1.A. CERVA:-.ITES 567

J. ,\I"'D l'allcd;l !!;lll......•all ...phcncal wavc lielJ. In the litcratlll"C it 1 •••

1111('11rL'lern..J lo a •••a hcam produeed hy a la ...er opcrallllg III {he.•.i1l1plc .•.t Iranwcr ...c clcctromagm:tie moJe. (termed TEt\loo).

'2. r.A. Arn;luJ. ,\11I . .1. ¡'''ys. -H (1973) 5.t9.

:;. "LA Wi!!gin ...and R.1\1. Herman. AII1 . ./. I'h)'.\'. 55 (19X7) .120.

1. .1. Ikcht anJ D .. 'i..n: ...i TA. UIlt1cr.\'/allt1illg 1-oi1.1'('I",\'. (Dovel". NewYorl\. Il)X-t).

d. 1'.11.BordlCHh. ,.\/11 . .1. /'lly.l. JI) ( 1971 ) 6XO.h. .\.1:. Sicgman. ,\lIl1ltmt1ltc/ioll ro iÁl.\('rs aJ/(i M{uas. (i\lc(jra\\"

Ilill. i\cw York. 1971 J.

l. A. Yariv. IlIlmt1IWlioll lo Opli("tli l:"/('("(mnics. Ololt Rcinhandlld Win .•.tull. 1\'cw York. 1(76)

H. I.D. bcbol1. ('/m.\t('u{ U('('/roti\"//lIl/1in. (\Víley. New York.191)\), Jrd. cJitioll.

!J. P.A. Tipler FOIfllt1afulI/,\ o/Mot/e,." j'/¡r,\'l('.\". (\Vonh Puhli ...hcr ...•!':c\v York. 1l)()1) J. p. 20 l.

lO. 11. Ko!!c1nikalHlT.Li./'ml"./hEl:"54(jt)(¡(») 1312.

11. (i.D. Bnyd alld J .1'. (ionloll. llell .\'y.\fl'III Téch. 1. .JO ( 1()(l6) 4X<).

I:!. 11. Kilgclnil\. "1'1'1.011(.'" ([965) 1562.

I:L G.D. Boyo ami H. Ko~cln¡k. Ud/.\'.\'.\'I('11I Tech. 1. .H (1962)1347.

1.1. A.( •. Fox and T. Lí./k/l .\'\'S/(,III T('ch. 1..t1) (1961) 453.