Analysis of Statistical Correlations and Intensity Spiking in the Self-Amplified Spontaneous-Emission Free-Electron Laser* S.

Transcription

1 SLAC-PUB-9655 Fruary 3 Analysis of Saisical Corrlaions and Innsiy Spiing in Slf-Amplifid Sponanous-Emission Fr-Elcron Lasr S. Krinsy Sanford Linar Acclraor Cnr, Sanford, CA 9439 R.L. Glucsrn Dparmn of Pysics, Univrsiy of Maryland, Collg Par, MD 74 Asrac narrow and caoic oupu of slf-amplifid sponanous-mission fr-lcron lasr SASE FEL xiis innsiy spis. In linar rgim for sauraion, w us an approac dvlopd y Ric o drmin proailiy disriuions for pa valus of innsiy in o im and frquncy domains. W also find avrag numr of spis pr uni im or frquncy. In addiion, w driv oin proailiis for innsiy in oupu puls o av valus I and I a ims and, and for spcral innsiy o av valus I and I a frquncis and. PACS: 4.6.Cr,.5.-r Sumid o: Pysical Rviw S-AB Prmann addrss: Brooavn aional Laoraory, Upon, Y 973 Wor suppord y Dparmn of Enrgy conracs DE AC3 76SF55 and DE- AC-98CH886.

2 I. IRODUCIO ory of ig-gain singl-pass fr-lcron lasrs as n dvloping sinc la 97 s [-5]. In asnc of an xrnal sd lasr, SASE FEL sars up from so nois in lcron am. Bcaus SASE sars from so nois, a propr ory rquirs a saisical ramn of oupu radiaion [6-7]. Avrag propris of oupu wr sudid in [6-] and flucuaions wr considrd in [3-7]. o dscri so nois, on considrs arrival im of individual lcrons a undulaor nranc o indpndn random varials, and on drmins saisical propris of oupu radiaion y avraging ovr socasic nsml of arrival ims. In linar rgim for sauraion, i follows from Cnral Limi orm [8] a proailiy disriuion dscriing spcral innsiy I im-domain innsiy I, is ngaiv xponnial disriuion [4] and innsiy flucuaion is %., or I / I pi I,. I oupu innsiy as a funcion of im xiis spiing [3] s Fig., and wid of innsiy pas is caracrizd y cornc im [4,5], co /σ, wr σ is SASE gain andwid. spcral innsiy also xiis spis Fig., and wid of spcral pas is invrsly proporional o lcron unc duraion. A a fixd posiion z along undulaor, considr nrgy in a singl SASE puls, W z E, z d,. wr is duraion of an lcron unc aving uniform avrag dnsiy. For z fixd, on can in of dividing puls ino M saisically indpndn im-inrvals of wid co. nrgy flucuaion wiin a singl corn rgion is %, u flucuaionσ W of nrgy in nir puls is rducd and givn y [4,5] W / σw W W W W co..3 M

3 co Figur. Innsiy spiing in im-domain arirary unis. wid of pas is caracrizd y SASE cornc im I /σ. co < I > co Figur.. Innsiy spiing in frquncy-domain arirary unis. In singl-so spcrum sown on lf, wid of pas is invrsly proporional o lcron unc duraion. avrag of many SASE pulss is illusrad on rig, and in is cas wid is proporional o gain andwid σ /. co 3

4 Hr, M is dfind [4,9] o numr of mods in radiaion puls. nrgy pr puls is dscrid y gamma disriuion [4,8,9], M M M W W p W W xp M Γ M..4 W W W z, A z, xp i z i radiad SASE lcric fild, wr L E s s s s c is undulaor rsonan frquncy. In dscripion a fixd z of saisical propris of SASE oupu, wo imporan quaniis [4] ar fild corrlaion funcion and innsiy corrlaion funcion A A g.5 A g A 4 A A A A In linar rgion for sauraion, y ar rlad y..6 g g +..7 nrgy flucuaion σ W in a puls can xprssd in form σw d W d [ g ] d g..8 Comparing Eqs..3 and.8, w s a wn << cornc im can xprssd in rms of fild corrlaion funcion according o [4,9] co co d g..9 In is papr, w advanc saisical dscripion of SASE y applying mamaical analysis of random nois dvlopd y Ric [8]. Som of rsuls of our analysis wr prsnd in rf. [7]. In arlir wor [4] rviwd aov, wo imporan disriuions of a gloal naur wr drivd: xponnial disriuion [Eq..] applis o an arirary im or frquncy, and gamma disriuion applis o

5 oal radiad nrgy. In is papr, w sudy wa mig rmd local saisical propris. In im-domain, w drmin oin proailiy [Eq.3.4] a a a fixd posiion z along undulaor axis, normalizd innsiy I / I in radiaion puls as valus and a ims and. W also find proailiy pr uni im [Eq. 4.] of osrving a spi wi maximum normalizd innsiy. In frquncy domain, w driv oin proailiy [Eq. 3.] a normalizd spcral innsiy as valus and a frquncis and. W also find proailiy pr uni frquncy inrval [Eq. 4.8] of osrving a pa in frquncy spcrum wi maximum normalizd innsiy. In Eq..3, numr of mods M in oupu puls was dfind in rms of puls nrgy flucuaion. Anor quaniy of inrs is numr of innsiy spis s Fig.. Following Ric [8], w caracriz mporal spis y xisnc of local maxima of innsiy. From Eq. 4.4, w find a.. s.7m.7 / co. I is inrsing o no a all nown saisical propris of SASE drivd in prvious wor [4] and in is papr can drivd from Ric s sudy [8] of so nois in lpon sysms, s also rf. [9]. Our papr is organizd as follows: In Scion II, w rviw calculaions of fild and innsiy corrlaions for SASE radiaion. In Scion III, w us Cnral limi orm o drmin oin proailiy disriuions mniond aov. In Scion IV, w find proailiy pr uni im of osrving a spi wi maximum normalizd innsiy. From is rsul, w drmin avrag numr of mporal spis pr uni im. W also find proailiy pr uni frquncy inrval of osrving a pa in frquncy spcrum wi maximum normalizd innsiy. From is rsul, w drmin avrag numr of spcral spis pr uni frquncy. In Scion V, w prsn an approxima calculaion of proailiy pr uni im of osrving a spi wi maximum normalizd innsiy >>. is givs on an inuiiv ali mamaical picur of spis maing up SASE oupu. A summary of our rsuls is givn in Scion VI. 5

6 II. CORRELAIOS W wor wiin classical, on-dimnsional approximaion, in linar rgim for sauraion. uniform dnsiy lcron unc conains lcrons and as lng, L c, long compard o cornc lng of oupu radiaion. W rsric our anion o radiad fild insid lcron unc sufficinly far from ac nd so a corn ffcs [,,] can nglcd. radiaion fild as form, wi ampliud A givn y E i zi z, A z, s s,. is A z, z,.. rsonan wavnumr γ / + a, wr w is undulaor wavnumr, s w w mc γ is iniial lcron nrgy and a w is undulaor srng paramr. SASE grn s funcion is dnod y z undulaor nranc z y, and arrival im of lcron a. indpndn random varials,, ar considrd o uniformly disriud ovr inrval. W us racs o rprsn an avrag ovr arrival ims. For uniform disriuion, f / d f. W also inroduc sorand noaion: A A z, and A A z,. corrlaion of ampliuds a a givn posiion z along undulaor, a wo diffrn ims is givn y: is i s A A d.3 6

7 7 In Eq..3 and in following, w do no xplicily sow z-dpndnc of funcions, sinc w ar woring a fixd z. Wil driving Eq..3, w rain only dominan conriuions caracrizd y asnc of rapid pas variaion. s corrspond o ping pair-wis qual summaion indics from A and A rms. I is ofn usful o inroduc Fourir ransform H, s, via H d i..4 corrlaion of Eq..3 can now xprssd as i H d A A.5 corrlaion of innsiis a wo diffrn ims can calculad as follows: + + lm m l i A A m l s l l l l l + + A A A +..6 is is rsul of Eq..7. Again w av p only os rms wiou rapid pas variaion. Dp ino xponnial gain rgim, a saddl poin analysis yilds Gaussian approximaion o grn s funcion z z i w z i i w w w w z z / 8, + + θ ρ ρ ρ θ,.7 wr z s w s θ + andρ is Pirc paramr [3]. W can wri is asympoic approximaion in form

8 α χ,.8 wrα and χ complx and ral ar funcions of z. From Eqs..7 and.8, w s a i s ρ α..9 4 z w aing Fourir ransform of.8, w find / σ H χ,. αα αα 3 3ρs σ.. α + α z Hr, σ is SASE gain andwid s Fig.. o a H as a dpndnc on s a is no xplicily sown. Using Gaussian approximaion of Eq..8 in Eq..3 w driv w A A χ α + α W dfin normalizd ampliud, σ.. a z, A z, / A z,,.3 and us noaion a az, and a az,. Undr approximaions w ar mploying, a a a a,.4a a a a u + iv,.4 a aa, a + β a,.4c a a u + v β..4d Eqs..4 and.4c provid dfiniions of quaniis u, v, and β. W no a in rms of is noaion, fild corrlaion inroducd in Eq..5 is givn y 8

9 a g a and innsiy corrlaion inroducd in Eq..6 is + g β. In ig-gain rgim for sauraion, i follows from Eqs..,.3 and.4d a / co β,.5 wr co [4,5] is cornc im, wic is rlad o rms andwid σ of oupu SASE radiaion y..6 co σ III. JOI PROBABILIY DISRIBUIOS In Eq.., ampliud A is rprsnd as a sum of indpndn random rms; i follows a proailiy disriuions dscriing oupu radiaion ar drmind from Cnral Limi orm [8]. cnral limi orm s Appndix A sas a disriuion PV of normalizd sum V r + r + r / of indpndn + random vcors approacs normal law as. For simpliciy considr r ; n as, K / P V d M / xp V M V, 3. wr V V,..., V is a K-dimnsional row-vcor suprscrip indicas K ranspos and V corrsponding column vcor. symmric marix M is comprisd of scond momns: µ M µ K wr µ µ µ K K µ KK 3. K µ V V d V V V P V. 3.3 M - is invrs of marix M. o a wn cnral limi orm applis, disriuion is Gaussian and nc is drmind y scond momns. Undr s 9

10 condiions, on nd no compu all igr momns o drmin disriuion a gra simplificaion. As an illusraion, considr spcial cas wn r,...,, r r ar wo-dimnsional vcors wi componns, r x, y. Again, w a x y so indpndnc implis x x y y x y. scond momns of rsulan VX,Y ar: x + + x µ X, 3.4 y + + y µ Y, 3.5 x y + + x y µ µ XY. 3.6 In is cas, cnral limi orm implis a proailiy disriuion PX,Y for approacs normal disriuion µ / + µ µ µ X µ Y µ XY P X, Y xp. 3.7 µ µ µ Rurning o FEL prolm, l us xprss normalizd fild ampliud dfind in Eq..3 as a x + iy iφ. 3.8 Corrlaions of x and y ar drmind from Eqs..4 and on finds: x y / and xy. From Cnral Limi orm, i is sn a proailiy Px,ydxdy for finding x wn x and x+dx, and y wn y and y+dy, is givn y x y P x, y. 3.9 proailiy P,φ ddφ for finding wn and +d, and φ wn φ and φ + dφ, is givn y

11 P, φ. 3. As sown y Ric [8], Cnral Limi orm also nals us o drmin oin proailiy disriuion dscriing fild ampliuds a wo diffrn ims. rquird corrlaions ar drmind from Eqs..4, and w find: x x y y /, x y x y, x x y y /, and u x y y x /. aing vcor V x, y, x,, momn marix M v inroducd in Eq. 3. and is invrs ar givn y y u v v u M, 3. u v v u u v M v u u v u v. 3. v u Also, on finds d M P x, y, x β, y x xp u. I n follows from Eqs. 3. and 3. a y x v y u xx + y y β v x y y x. 3.3 Exprssing x and y in rms of and φ via Eq. 3.8, and ingraing ovr φ and φ, w oain proailiy P, d d for finding normalizd innsiy wn and +d a im, and and +d a im [8]: P, β β β I β β. 3.4 I is Bssl funcion of imaginary argumn of ordr zro. o a wn, n β and P, δ. Also, wn, n β

12 and P,. disriuion of Eq. 3.4 as n usd o dscri narrow and caoic lig []. Som mamaical propris of disriuion av n sudid in [3], wr following xpansion in rms of Lagurr polynomials was drivd n n n, n P β L L. 3.5 From Eq. 3.5, on can calcula momns M M d d P, min M, M!! M!! β. 3.6! M!!! I is now of inrs o drmin condiional avrag of innsiy a, givn innsiy a is. P, + β. 3.7 d corrsponding flucuaion is d P, β β + β. 3.8 W no from Eq. 3.7 a is lss an wn >, and is grar an wn <. is is saisical asis for apparanc of spis in radiaion oupu. analysis us prsnd in im-domain can xndd ino frquncy-domain. L A z, Fourir ransform of Az,, wr s. n A z, i s + H z,, 3.9

13 3 wr H is Fourir ransform of, ] / xp[, σ z H z H. From Eq. 3.9, w find + + i i s s H H A A i H H / / i d H H H H sinc. 3. W dfin,, /, z A z A a, a, and,. oin proailiy, d d P a normalizd spcral innsiis a frquncis and av valus wn and d +, and and d +, rspcivly, is drmind in a mannr similar o drivaion in im-domain lading o Eq W find,, γ γ γ γ γ I P, 3. wr a a γ sinc, 3. and sincxsin x/x. As in im-domain, i is possil o wri Eq. 3. in rms of Lagurr polynomials.

14 IV. IESIY PEAKS W can provid a saisical dscripion of spis in oupu, firs in im-domain and n in frquncy-domain. o prpar for discussion of innsiy pas, w firs rviw rquird mamaics dvlopd y Ric [8]. Considr socasic funcion y F,,..., ; 4. and is drivaiv F y, 4. wr,..., ar random varials. L P ξ, η; dξ dη proailiy of finding y wn ξ and ξ + dξ, and y wn η and η + dη a im. W firs wis o drmin proailiy a r is a zro of funcion F aving posiiv slop somwr in inrval < < + d. Suppos y ξ < a and y somwr in < < + d. If slop a is η >, n F passs roug zro a Hnc, w rquir ξ. η ξ < < + d, i.. ηd < ξ <. rfor, proailiy a η r is a zro of F wi posiiv slop in < < + d is givn y d dξ P ξ, η; d ηd ηdηp, η; η. 4.3 W can now drmin proailiy py ddy a F as a maximum wi valu wn y and y +dy in im inrval << +d. A a maximum, drivaiv of F is zro and is scond drivaiv is ngaiv. From rsul of Eq.4.3, w s a ; ddy ddy dς P y,, ς; p y ς, 4.4 wr P ξ, η, ς; is proailiy dnsiy funcion for varials ξ F,..., ;, F η, and F ς. L a m m a / m and no a m n mn a a i m+ n, 4.5 d m m K z H z,

15 normalizaion Kz is cosn so a /. W wri a x + iy, a x + i y, a x + i y, wr prim dnos diffrniaion wi rspc o im. Using Cnral Limi orm, Ric [8] drmins P x, y, x, y, x, y. aing V x, y, x, y, x, y, marix M of scond momns, Eq. 3., is M Afr compuing invrs of M, P x, y, x, y, x, y is drmind from Eq. 3.. Ric n inroducs x R cosφ, y Rsinφ, and as firs and scond im drivaivs. By ingraing ovr φ, φ, and φ, Ric drmins P R, R, R. H n nos a proailiy p R ddr a a maximum of nvlop R falls wiin lmnary rcangl ddr is givn y [s Eq. 4.4] p R P R,, R R d R. 4.8 indpndn of. W us varial R, and w coos normalizaion Kz suc a / σ K z G z,. 4.9 σ proailiy p dd a a maximum of wi valu wn and +d is found in im inrval d is drmind y [8], n / / 4 3 / n + / p n n 7 co Γ + 4 For >>, sum ovr n in 4. clarly pas nar n. On can ra n as a coninuous varial, convr sum ovr n o an ingral and us mod of sps dscn o oain 5

16 p. 4. co Anor mod for valuaing sum ovr n in 4. for larg is o spara i ino wo pics n + / n 7 4 n / / n 3 4 n / n n n Γ + Γ + Γ + / n n / Summing ovr alrna valus of n and rcognizing a lowr limi n in sums can modifid sligly wiou affcing ovrall limi as asympoic approximaion of Eq. 4. y noing a, on oains m / / m Γ m S W 4 W 4 Figur. 3. p / d is proailiy of a pa in im-domain aving normalizd innsiy wn and +d dimnsionlss varials. numr of pas pr uni im is.7 d p. 4.4 co 6

17 proailiy of finding a maximum wi normalizd innsiy wn and +d is p / d, wic is plod in Fig. 3. avrag pa ig is. 56 and rms pa ig flucuaion is.7. In Eq. 4.4 and Fig. 3, p is drmind from Eq. 4.. In frquncy domain, i + s a. 4.5 Dfining a m m a/ m, w find m n nm a a i m + n, 4.6 / / m m d. 4.7 Following Ric s analysis [8], w find a proailiy p dd a a maximum of a wi valu wn and + d is osrvd in frquncy inrval d is drmind y indpndn of, wi n / n / 3 / 4 / 4 9 / 4 / 5/ 4 An p, n n 7 Γ + 4 m A, 4.9a n / 3/ m / A n n m + 3/ 5 m! m, n. 4.9 For >>, numr of pas pr uni frquncy is p d p.64 /. 4. 7

18 proailiy of finding a maximum wi normalizd spcral innsiy wn and + d is p / d, wic is plod in Fig. 4. avrag spcral pa ig is. 66 and rms spcral pa ig flucuaion is.9. In Eq. 4. and Fig. 4, p is drmind from Eq S a 4 a 4 Figur. 4. p / d is proailiy of a pa in frquncy-domain aving normalizd innsiy wn and +d dimnsionlss varials. V. APPROXIMAE DESCRIPIO OF IESIY SPIKES Moivad y rsul for condiional avrag, Eq. 3.7, i is rasonal o assum a oupu is comprisd of a sris of pas, and in rgion nar ac maximum, innsiy profil can approximad y p p / co, 5. wr p is maximum innsiy of pa cnrd aou p. Rcall from Scion IV a p dd is proailiy of finding in radiaion oupu an innsiy pa of magniud wn and +d, in im inrval d. L us 8

19 considr an inuiiv argumn for drmining larg avior of p. For larg, proailiy of innsiy ing grar an can valuad in wo ways. Firs, i is qual o d. Alrnaivly, i can wrin as d p,, wr, rprsns rgion in, undr a pa aving maximum valu, for wic innsiy is grar an. From Eq. 5., w find " n /, co. Equaing s wo xprssions for proailiy a innsiy grar an, w driv following ingral quaion for p : d " n / p co. 5. For larg, xponnial avior on rig and sid of Eq. 5. rquirs a p F. If on cangs varial of ingraion o y-, n Eq. 5. coms co y y dy F + y " n Clarly, imporan valus of y ar of ordr of. us, for larg, on as y y y y<< and " n +. Expanding F +y for y<< lads o 4 co dy y [ F + yf ] y y 4 / 3 / / 3 / soluion of 5.4 for larg is F. us soluion of Eq for larg is p co co W s a inuiiv argumn lads o a rsul diffring from prcis asympoic rsul of Eq. 4. only in a facor 3/8 sould rplacd y /. 9

20 L us xnd inuiiv picur and suppos oupu innsiy is comprisd of a sum of pas, " p p / co, 5.6 wr now w do no rsric p o larg. n w can approxima avrag valu of y [s Eq. 4.4 and discussion following i] d p d p p p p p co / p co.7.56 co co., 5.7 in rasonal agrmn wi xac valu,, of is avrag. Similarly, l us considr oupu innsiy spcrum o comprisd of a sum of pas sinc " ". 5.8 " W can approxima avrag of y [s Eq. 4. and discussion following i] d d p sinc " " " " " " , 5.9 also in rasonal agrmn wi xac valu,. VI. SUMMARY OF RESULS In is papr, w av usd approac of Ric [8] o xnd our nowldg of SASE saisics. In arlir wor [4], i was sown a proailiy disriuion for innsiy a a givn im or frquncy is xponnial disriuion [Eq..]. fild corrlaion g [Eq..5] and innsiy corrlaion g [Eq..6] wr also drmind [4], as wll as gamma disriuion [Eq..4] for nrgy pr puls. In is papr, w av drivd oin proailiy P, d d for finding normalizd innsiy wn and +d a im, and and +d a im :

21 P, β β β I β β, 6. wr β / co. In addiion, w av drmind oin proailiy P, d d a normalizd spcral innsiis a frquncis and av valus wn and + d, and and + d : P, γ γ γ I γ γ, 6. wr γ sinc. In im-domain Fig., w av drivd proailiy p / d [Eq. 4. and Fig. 3] of finding a maximum wi normalizd innsiy wn and +d. numr of pas pr uni im was found o.7/. 6.3 co In frquncy domain Fig., w av drmind proailiy p / d [Eq. 4.8 and Fig. 4] of finding a maximum wi normalizd spcral innsiy wn and + d. numr of pas pr uni frquncy was found o.64 /. 6.4 Many of saisical propris of radiaion rviwd in inroducion o is papr av n vrifid xprimnally [4]. In paricular, i was osrvd a flucuaions of SASE radiaion nrgy followd gamma disriuion [Eq..4], and oupu of a narrow-and monocromaor avd as xpcd from ory. Spis in spcral disriuion of innsiy av n masurd xprimnally [5-7]. If xprimnal condiions ar sufficinly conrolld so a flucuaions ar dominad y so nois and no variaions in xprimnal paramrs suc as carg, unc lng or mianc, n disriuion of pa igs can masurd and compard wi prdicion of Eq Also, on can compar numr of pas

22 pr uni frquncy wi prdicion of Eq In fuur, if on dvlops mods o dircly masur spis in im domain, n prdicions of Eqs. 4. and 6.3 can compard wi xprimn. ACKOWLEDGEMES is wor was suppord y Dparmn of Enrgy conracs DE AC3 76SF55 and DE-AC-98CH886. REFERECES [].M. Kroll and W.A. McMullin, Pys. Rv. A [] Y.S. Dnv, A.M. Kondrano and E.L. Saldin, ucl. Insrum Mods Pys. Rs A [3] R. Bonifacio, C. Pllgrini, and L.M. arducci, Op. Commun [4] L.H. Yu, S. Krinsy, R.L. Glucsrn, Pys. Rv L [5] M. Xi, ucl. Insrum. M. A445, 59. [6] J.M. Wang and L.H. Yu, ucl. Insrum. M. A [7] K.J. Kim, ucl. Insrum. M. A [8] K.J. Kim, Pys. Rv. L [9] S. Krinsy, AIP Conf. Proc [] S. Krinsy and L.H. Yu, Pys. Rv [] L.H. Yu and S. Krinsy, ucl. Insrum. M. A [] S. Krinsy, Pys. Rv. E [3] R. Bonifacio, L. D Salvo, P Pirini,. Piovlla, and C. Pllgrini, Pys. Rv. L [4] E.L. Saldin, E.A. Scnidmillr, M.V. Yurov, Pysics of Fr Elcron Lasrs, Springr-Vrlag, Brlin,, Capr 6. [5] L. H. Yu and S. Krinsy, ucl. Insrum. M. A [6] P. Caravas al., Pys. Rv. L [7] S. Krinsy and R.L. Glucsrn, in Proc. 3 rd Inrnaional Fr-Elcron Lasr Confrnc, Darmsad, Augus -4, ; ucl. Insrum. M. A483, 57. [8] S.O Ric, Bll Sysm cnical Journal [9] J.W. Goodman, Saisical Opics, Jon Wily & Sons, w Yor 985.

23 [] B.W.J. Mcil, G.R.M. Ro, D.A. Jaroszynsi, Op. Commun [] Z Huang and K.J. Kim, ucl. Insrum. M. A [] G. Vannucci and M.C. ic, J. Op. Soc. Am [3] J.F. Barr and D.G. Lampard, IRE rans.-informaion ory I [4] M.V. Yurov al., ucl. Insrum. M. A [5] Ayvazyan al, Pys. Rv. L. 88, 48. [6] S.V. Milon al, Scinc 9, 37. [7] A. rmain al, ucl. Insrum. M. A APPEDIX A: CERAL LIMI HEOREM Sinc cnral limi orm plays a cnral rol in our analysis, w sall provid a sor urisic drivaion of is imporan rsul. L us considr indpndn random varials x,,x saisfying x, x x σ δ. n in limi, w wis o drmin disriuion of normalizd sum S x A For, i S σ xpi σ / x. A By dfiniion, i S ds i S p S. A3 aing invrs Fourir ransform, i follows from Eqs. A and A3 a p S S / σ, A4 σ a spcial cas of cnral limi orm. 3

24 APPEDIX B. HE SPECRAL DESIY ral lcric fild ε is xprssd in rms of complx ampliud A y i z i z A z s ε,, s + c. c, B Sinc w considr z fixd and ar only concrnd wi im dpndnc, w sall wri ε and A, ignoring funcional dpndnc on z. I n follows a ε ε A A + A A B 4 4 corrlaion of ampliuds can xprssd in rms of Fourir ransform of grn s funcion [Eq..5] as A A Using Eq. B3 in B, w find d H i B3 d ε ε H cos[ s + ] B4 In rms of spcral dnsiy wf dfind y Ric [8], is corrlaion of lcric filds can xprssd as d ε ε df w f cos[ f ] w cos[ ] B5 Comparing Eqs. B4 and B5 w find s + w H + H. B6 As nod afr Eq.., H as a dpndnc on s wic w do no xplicily sow. 4