I ,

- l r. I (j) APPRo·.~ .... ¥ -r~M:;

JUSI ..,1 \,i- ... ~ ... . L

I' II ) :- I t: \ I. H I \ I I 1\ " \ ' 0 I. I' \1 I :: u . \ I ' \I Ill II ;. \0 \ ' I 11 11 II I fJ 7 <l

Ul'!.nnauc fluor seen e in a wrak racli :1ti 111 fi ltl \>il h arbitrary pcr tral distrib ution

, 1 (i I t ,, ~ tn ··r • :1nd J. l'clO j•t· r

)~._t 1 'it fn tt•tur. j:Jr l .al\.lr"'turr .1 \tr;'f ·'• '\1 \, l fdr\ o· ,,j ( .; • uln IJ fl ,f • .. :, . ., Jl llllft'U fl t• StilfaJ,z ,ch, lb uldl'r, Cul.mhlo 'f)j () c)

(lt"''CI>C I l b July t•l 'I)

The :.utiHH ' htn-. ti:.H t1 ,c 1.''\' f and ttlth r!kd 10 n :'t.''III,Hh.'l' Outl h '\4."1.'1\("C :11 l,lw tuh'thtiiC"ii ,. ,, 1 h"· tf e~h'l..l II Hh l r('3d tl ) II\ h,'rtH\ 1,\f 'llll'"lc \'t'U\1 !ut .·11' , .wd furt'l : r th .H thl\ ,lp('hl U.. h ,til l". '(lllf' 1 l Ulll'-tdt' l ,\I I

i u iJl' l1 t fi ~oiJ \ '. l( h ,HI ;ul •t l f f) \l'(.~t. lr:tf 0 1,1'1t,._lttl , \•o!t h l.lf \\lt1 uH l tit.: J'll.- 'h.' ' ~f Jt )1, I ' (t .UC · IJrn:tdCII IUt,)

colh 1on<

I. INTHOIJUCliO.

In a r cent p.1p r by Kn i •hl , . lo l.wrl<' r, and r 0ud 1 (Kr.l ) an inl<'r :,li ~ p da;.:o~ic al dls rus ­

sion wa ~~ i\· <'n o r the t'f f cts o f las r l.J.mdwtdth n a tont ic r son:mc • nuor · c •nc -!WClt'.l a t low in-t n iti s. 1\ ~~umin;.: a Lo t· nt7 t.tn- <:hapc·d la s 1·

s p ctnun (I hasr - cl iff u ion mo 1'•1) l hLy ·how d

t ha t t h f111ite l.Ja ndwtdlh '1\' l'S n ' <' l o a n as •mm -t r in t c ... call c r ed spectrum when lhe inc 1 ·nl fiel d i delunc fr om t h a tonnc t· s n:-tnc . They offer ·d a phys ir:-t l ~: pl:lt k"l ti ot for til asymmNry, fit·st found by Kimule ::\lid :-.1 nde l, 1 in l crms of a contmuillt; r eintt i:L tion of l h lt·:-tn!>t nt respons o f t he :t lom owin': to l h fl uct uati ons in t he l.t st' r held . \'.'l11le thi xolat a ti on is mo l ce r l.lin ly corr ct, one poinl r nl.l i:-~ed uncl <'a l' , I ha t c nrcrnin~ th !'Ole f l hc COil\' luliOII . n t hi. pr bl 01 1 whic h iS ::l

:nti cub r case o f li t :-tr response l J.eoJ·y. The 111rpose of lhi "l m nt is twofold : to show

l hal l he Ia !> r bandwidth l'ffc <:l s in r sonan ce flu o ­rt'S ence a t low intcn: iti l'S can be treated mos l

<':tdily in t e rms or a simple com·olution and lo

where w is the frequ ncy of the scatt red l ig ht. The ang ular lwackets ~() indica te bolh a quantum exp eta ion \'a lue a.ndanensemble averageov rthc stalistic5 of the dri\·ing fie ld.

Fm· a monoch r oma t ic laser the field-correlation

T his r esu lt can be evaluated to t;ive

rr n~ gN 8 (w,wL)=-2 ( )' ,6(w-wL),

W 0 -w 1• i·yN (5i

which ~hows th:1l onh· ela s tic scallerin~ occurs fo r a n isol:l l ed a tom suff

20

FEB 1 1

H

J !:how ful'l he t· lha llhis appr oach all ows one to con­

I c r· an in c idl• nt fiel d w ith :1 n ::u ·btl J·a r y s pcdra l di s tr ibuti on, with or WI h ut lh pr senc c o f al om­ie ( l ine -broaden in~) colli ~ i ons .

II. LFI' I CT Of L '> Lil Ll 1:. SIIAI'E:

For e:1 se o f compari son we will follow K IS a nd use, where possible, t he ir no tati on. We s lal'l wilh the low -intensity oper ::~tor Dloch quation for l he posit i\' e-fr cquency d ipol operator ii ., . .

a.(f) = (iw0 - yN)o.(l) - (id/ h)E '(I) 1 (1)

wh re 0 is lh alom ic resonance fr <'quency, }. is one-ha lf lhe nalural cl cav wtdth, dis th lram.i­lion dipo l mom nl, and 17:·(1) i the nc ;alivP-fre­quency elt'rtl'ic field opt'rato r for th cldvin~ fi eld wh ich •· ci ll a t s with ft• qucncy w r.· T hP s e::~tl c t · rj

SJ CC'trUtn ,I.'( W,w L) iS !;1\'ell lll S( ady !t:lle b· the Four i r tra nsform of lhe dipol a ulocorrelalion functio n , cr,(/)& .(1 +r)» , which is obtain ed by for­mally solvin t; l::q. (1) . Thi s yie lds

(2)

functi on is

giving for lhe natTow-ba nd spect rum r; 118(w,w L)

KMS l r :t l the IJro:td -b:l!ld 1:\ s r by assuming the ph.,se-d iffusion model2

• 3 fo r the field

«,~-(/ ' )/~" (!")»

= I 1::'0 12 expliw L (I' -I") I •xp(- ) LIt' - 1" 1) , (G)

22311 @ 1979 The Amcn~an l'h)~ical Socicry

8 2 02 10 Ot 8

:.'II lti':O.CI;\ \ ;\c'l I 1. 1 IIHI' "Cl. \t:l : J\ ,\ \',J , ,\1, tt ,\IIL\TICI\ 1'11.1.11 ...

wht' rt' ) 1 i •; o:w-h:t tr tho · l. t ··• ·r lo.nulwHith. l ':.inl: thi s in E (:!) k .td" dlrt•t·l ,. 111 l ht' 111.1111 " " '' llll o!

t\MS ' :

(7)

wh I ' th cll•tunin!! is ~ =w0 - "" 1 :111d t hC' H.tiJt !I· -qn ncr ~ ~ n = 2d l £: II . Thi s spt•c trum c1 :1 rly show: two m:t.\int:t: b " t ic ~ . c-:tl l rin nt r d :t t

int n ~alur:~t inl! inc1d nt field s. II ow n ·r, w

will show h ,. t h:l t d w."-• t ) in Eq. (7) , v:tlid !or

1 w ! i ld., , ~~ o t:~inaiJ ~ lly .\ ~ imp l onvol uti on. This is :IS P~.p~ I din lin .1r r ' I n-;e t h •o•·y.

WP will solve th pr IJl m in :1 diU t'C'nt, m or r evea l in,! , \\',1)'. The pow r sp ch·um I ' ~. <w~) of the incid nt fi lu is !; t'll r :t lly c •v n I.JyG

1 ~- a a l'~.(w~l = 2 dr cxp(i ~r) E-(/)E.(t +T)}) , :; -· (8)

and thl' inv ~r se can be written as

(E-(t')E.(t " )}) = J: dt..•;. expfi c. ... ~(t '-t'"))I'"( {). (!l)

lnseJ•tinr. thi s form for till' correlation function into Eq. (2) !or the spectrum g( ,w~,), :1nd com­par i ll [: t h result in:; qu:~hon with the narrow -b:J nd spectr um c A 0 (161, w ~, ) in 1::'1 . (4), it become'S appar­ent that I he spectrum ~:(w, w ~.> can be written as

g(w, w~, ) = IE0 1, ~~ d161;. cNn(w,w~)J'L(w~). (10)

T his i s t h m ain result or thi s paper and shows that, fc,r an arbilmn· l.l f. <' r sp ctnuu 1' ~.(w~ ), the scatter d spectrunl);(w,w~,), at low intensity, is Ci\'en by the n:trrow -lland scattered !;pecl&·um c 11 11 (w, w 1 ) con\'olv<'d wil h the l:tsPr spectrum. Us­Ing lhC' 6 -(unction form or .l.' .vs(w , wL) fronl Eq. (5) in Eq. (10) results iol

( ) - l f'l..,rr f22 f'() /: W t W L - , '0 2 ( )" • L W t

W - wn - "Yit (11)

which shows th:~l lhC' scatter d li r, lal •·elains a dis­tort.. cl \ ' l'Sion O( th iqC'Idl•nt lit:ht l; (l('Cirum P~.(w~).

F01' the speci:ll C.1SC or the phase-diffusion mod-

<' 1, ll .' t·tl ltv 1'. 1 :; ·" ' I dt•se nlw cl in 1·:<1. (G). t h • l:t ­t<! I' ~po · l'lnun b t' ( ' • ll l ii ' S I hy ins<'rlint: Eq. ( ti) into l :fl . (Ulj

(12)

a I.o•·cnt 7.i:tn, :1s :.l at <'c1 <' :tl'l i t·r. Thi s t h n r: i l' f' ~; ,

fr om Eq . (II ), I Itt' SC:tltCI'<'cl SJWCti'Ulll a

II fl: y./n J:(w,w~, ) =2!'(;:;-:-Jl .. ) ;,]l(w -w~.)' +) tJ , (13)

:t r suit wh ich is simplt• in form , l>ut i n fad (artcr s :11c a l ~ br:t) i s idcnlit':tl to l h mon · complcx­lool: in • <'xpn· ~s i un t:ivl' n lly Kl\tS in Eq. {7 ). This show t h:lt t hC' .· r:t lt l'<'d spe ·trum i s u·. ' product of two L or nl zians , orw p al d. t t h<' l:J . <' 1' !re­qucn · · •~, :~ml th oth C' r :~ t the :l l omic r son:1 nc f r qu ncy w 0 •

Ill . CO!\tM ENTS

WP h:wc shown I Eq. (I 0) J that , Co•· l ow- intensity reson:tnr c !luor<'sccnc<' in the abs nc of colli­sions, lh scattl"r ed . pcctrum is given by tll:lt <le­riv<'d l.i ' :tssumint-: a monochrom:ttic drlvina la ser convolv d with th ·spectrum o r t he actna l drivinr. la ser . Thi s I' suit holds for a1·bitmn la ser line sh:~p sand, in pJrticular, for the L r C'nt zian line shape deriVl'c1 from the phase-diffusion modet2· 3

and used by KMS. The convolution is no l ont;er V:Jiid when an int nse, saturalins drivm~ laser is consid r ed, :ts the r esponse of the atom is th £>n nonlincar. 4 The con lusion or KMS, that the in­e1:u;tlc component of the scattet·l'd l ight (centered on ~o~0 ) is due to the continuin~ reiniliat ion of the transi nl response or the atom by fluctuati ons in the drivin(~ fi eld, is strengthened by the pr<'SI'nt result. The c nclusion holds regardless of the form of the power spectrum or the fluctuations.

As a Curth<'l ' demon stration of the generality of the convolution result, we con sidC'r the effect of atomic line -bJ'O:ldenin~; collisions on the sc:~tter d spectrum. These include dephasinc; (u1· el:~stic) collisi ons :1nd quenching colli sions, as well :~s alignnl<'nt- and orientation-changing co lli s ions. The effect or colli s ions h:ts been studied in detail (or the case o f a l ow-intensity m onochromatic driving fi<'ld . , _ ln particui:Jr, it i s known th:~t col­li sions reclistribu!(' t he h·<'quenq· or the scattc>r d lir,ht and tl ou~ E:q . (5), !or the sca tt l't•d spcctrun1, Is not appropri:ll<' in thi s c. se. llowevC'I' , denoting the :~pprop•· i:ttc n:trrow-h:tnd sc:Jttered spcct rum 10

by G 118 (~A~,w 1 ), the scatt<'rC'd spectrum G(~A~,u• 1) for broad-hand •·xcitation can lie shown still t o lJ • a simple convolution

rmmmmmmm MMM

mmum"^'^^ ,.IVV1>>P II» - —,—^ ^^,^-~

^;' Kl M. (;. II A \ M IIS AM) J. IM) or 1.11 an

wlu rc üc.nin l',<^',) in HIP (:u l)itr;iiy) spiulniüi nf Hu' Uiw-iiiliMiMly ilrivmi; (icld. [In;; rrsult IMII lii< vrrifird t>v inspcclin', l-'.qa. (■!) ;iiul (Ti) of Uef. 7, which civi- llic MMlliicil spi'cliuni i:sl,W,*',) :is im iiilri;ral ovtr w.-vcral Icrm.s incliulinj'. the factor

/Ia:.ro(c1.i.,)cxi)i-«wt(/1-^,)i,

whore /, Is Ihi' intensity of thr inciilont laser wiMi |)olari7iit)on vector <, and jT is the atomic ilipole operator. For an arbitrary «IriviiiR field spectrum this factor sliould be replaced by

where /,,(/) is the electric field of the laser, and the field averar.e has l>ein perfDrnud. This field- correlatitm fuiu'tuui is analogous to the one ap- pcarini; in the integrand in Kq. {.!) for ^(u,',w,), where culliaiims were i(;n(ired. 'rims the deriva- tion of (.'(10, ui;.) in Kq. (M) follows identically to that of ^(ui.ui,) in Kq. (10). The convohilion re- mill Kq. (l-t) for the sraltercd spcclruin Giui.u,) holds for arbitrary collisional line shapes, both inside and outside the impact rcpion.

This research was supported by the Office of Naval Research and the National Science Founda- tion.

•Present address; Tho Institute of Optics, Univ. of noelicsler, lloclmslor, N.V. 1'1G27.

'p. L. Knlßht, W. A. Molandcr, and C. n. Stnnid. Jr., Phys. Hev. A ]7, 1547 (1978).

'j. II, Ebcrly, fiiys. HCV. Lett. a7, 1387 (1970). JM. 0. Haymcr, J. Mostowskl, and J. L. Carlstcn, Phys.

Hcv. A JJ), 2301 (1979), ^I. J. Klmblo and I-. Mandel, Phys. Rov. A 15, C89

(1977). sTlic sniiarc »n llio fls was inadvertently left out of tho

equation in Kef 1.

•Seo, for example, U. R. Mollow, Phys. Itcv. A 188, 196<> (1909).

TA. Omont, E. W. Smith, and J. Cooiier, Astrophys. J. 175, 185 (1972).

•j. Cooper, Astrophys. J. 228, 339 (1079). "M. G. Riiymcr, J. L. Carlsten, and G. Piehlor, J. Phys.

B 12, Mill (1970); J. L. Carlsten, A. Szöko, and M. G. Raymer, Phys. Rev. A 15, 1029 (19T7), and references therein.

"Sec, for example, Eq. (03) of Hcf. 7 and Eq. (88) of Rcf. 8.

Acc»sslon For

NTIS GRAil DTIC TA3

Juötificatlaa.

^q Py Dlstributica/

'■Avail on^/u.- ' Spocial Dlat

■m

m