Lecture 2 Qualitative explanation of x-ray sources based. on Maxwell equations

Transcription

1 Lcur Qualiaiv xplanaion of x-ray sourcs basd Oulin on Maxwll quaions Brif qualiaiv xplanaion of x-ray sourcs basd on Maxwll quaions From Maxwll quaions o Wav quaion (A and (B. Th fild E radiad by a currn dnsiy J. Fourir mods E T (k,w via J T (k,w. Fourir dnsiy J T (k,w of an lcron. Elcric filds by 1 and N acclrad lcrons. Poyning s nrgy flux S of an EM fild of x-rays. Rlaions bwn E, H, S and <S> in vacuum. Powr dp 1 /dw radiad by 1 acclrad lcron. dp N /dw and ds N /dw of x-rays radiad by cohrn moion of h N acclrad lcrons. Toal powrs P 1 and <P 1 > radiad by 1 lcron. P N, <P N >, S N and <S N > of x-rays radiad by h cohrn moion of N acclrad lcrons. X-rays by acclrad fr or bound lcrons. Opraion rgims of x-ray sourcs. Angl divrgnc of incohrn x-ray sourc. From incohrnc o ransvrs cohrnc. Th angl divrgnc of cohrn x-ray sourc. Problms of Maxwll xplanaion of x-ray lasrs. Qualiaiv xplanaion of x-ray sourcs. From qualiaiv xplanaion o quaniaiv dscripion of x-ray sourcs. Problms as hom assignmns Rfrncs TÁMOP C-1/1/KONV-1-5 projk 1

2 Brif qualiaiv xplanaion of x-ray sourcs basd on Maxwll s quaions Lcur 1 (Idas how x-ray sourcs opra Lcur (qualiaiv xplanaion r j X-ray radiaion mdium L i r i EM wav (X-ray r T (oupu aprur Fig. 1 Sourc of incohrn or cohrn EM wavs (x-rays Brif qualiaiv-xplanaion basd on Maxwll s quaions: W rmmbr in Elcrodynamics E i (r, ~ i a Ti ( r j /c + ϕ j /r j Thus h suprposiion of EM wavs (x-rays E(r, = Σ i E i (r, E(r, ~ Σ i i a Ti ( r j /c + ϕ j / /r j by h incohrn (ϕ i = ϕ j, a Ti = a Tj, r i = r j or cohrn (ϕ i = ϕ j, a Ti = a Tj, r i = r j or r i = r j moion of acclrad and/or d-acclrad ( a Ti = lcrons of h radiaion mdium L us show ha E i (r, ~ i a Ti ( r j /c + ϕ j /r j TÁMOP C-1/1/KONV-1-5 projk

3 From Maxwll quaions o Wav quaion (A A qualiaiv xplanaion of x-ray sourcs (lasrs using Maxwll quaions Maxwll s quaions Wav quaion? E i (r, ~ i a Ti ( r j /c + ϕ j /r j Cohrn EM wavs (X-rays E (r, ~ Σ i i a T ( - r/c + ϕ / r Maxwll s quaionsbefaraday s law: = D JAmpr s law: = + ( (1DWav quaion? Coulomb s law: = H ρ (3 B = (4 T c (7 rj Er1(, T = (, - Marial quaions DεE= ε (5 H= μ (6 B TÁMOP C-1/1/KONV-1-5 projc 3

4 From Maxwll quaions o Wav quaion (B W firs ak h curl of Eq. 1 E = E = + E Thn by using h vcor qualiy ( ( E (9B (1 Jr re (, c r 1 c (, = + ρ( ε ε -w g h Wav quaion (WE, (11 whr J(r, and ρ(r, ar rspcivly h currn and 1charg dnsiis, and h vacuum ligh vlociy is givn by c = ε μ (1( In h cas of ransvrs wavs, h Wav quaion is simplifid o h ransvrs WE 1rJ E W should T (, c (, = show ha rt -ε (13 E T ~ dj T /d ~ a T and E i (r, ~ i a Ti ( r j /c + ϕ j / /r j TÁMOP C-1/1/KONV-1-5 projc 4

5 Th fild E radiad by currn dnsiy J E T ~ dj T /d ~ a T? Th fild E T of x-rays radiad by h currn dnsiy J T is drmind by Eq. 13, which w solv by using h Fourir-Laplac ransformaion mhod. Th mhod algbrizs h and / opraors and ransforms h diffrnial quaion (13 o an ingral form. Th Fourir dcomposiions of fild E T (r, and currn dnsiy J T (r, yild rspcivly, E T(, ω r( 4 kik( ω d d ω kertjrt = (14 k(π Jkkr( ω d dω Ti, ω (15 (π (, ( 4 = whr h ampliuds of h im-harmonic plan wavs in (14 and (15 ar drmind by h invrs Fourir ransforms JkrETkEiE( Tr i( ω r, ω = (, d d krtkjtr i( ω r(, ω (, d d = Noic, on may us Grn funcion formalism for h mor dirc soluion of feq. 13. By subsiuing h fild (14 and currn dnsiy (15 ino Eq. 13 w g h algbraic quaion for h ampliuds of h im-harmonic plan wavs -k Eki ω Jk( ω c T(, ω T(, ω ε = (19 projc itámop c-1/1/konv (16 (17 (18

6 Fourir mods E T (k,ωvia J T (k,ω Equaion (19 indicas ha currn dnsiy J T (r, is a sourc of h fild E T (r, of x-rays. Th EM wavs inducd by currn dnsiy hn xis and propaga in h vacuum wihou any hlp of h sourc (sourc-lss propagaion. In such a cas, w hav -k Ek( ω (, ω = Tc Equaion ( yilds h wll-known disprsion rlaion ( ω = ck(1 wih h wav numbr k=π/λ. Th Fourir mods E T (k,ω mdiad by h componns J T (k,ω ar givn by Eq. 19 as Ek(, ω Jkiω (, ω = k( ε ω Tω c- ( c Th us of ( in h Fourir dcomposiion (14 yilds a Grn-lik funcion soluion ( (, (, ω Ji TkkkErω d ω 4 ε ( ω (π JJ d i rt ω = (3 -kc J T (k, ω? TÁMOP C-1/1/KONV-1-5 projc 6

7 JJTFourir dnsiy J T (k,ω of an lcron J T (k, ω? In Lcur 1, w hav undrsood ha x-rays ar gnrad by an acclrad and/or d-acclrad lcron. In Elcrodynamics an lcron in a poin r is dscribd by h poin-lik charg dnsiy and h charg currn dnsiy Jrr( ρ = n = δ ( (4 rrv(, = n(, (, = δ ( ( by using h 3-dimnsional Dirac dla funcion δ(r = δ(xδ(yδ(z. Th us of Eq. (5 for h ransvrs (kj = currn yilds rrv(5 krkrv i( ω rviω (, ω = [ δ ( ( ] d d = [ ( ] d (6 TFrom Eq (6 w g h Fourir dnsiy From Eq. (6 w g h Fourir dnsiy kv(, ω = ( ω (7 TÁMOP C-1/1/KONV-1-5 projc 7

8 EiBTElcric filds by 1 and N acclrad lcrons Th us of h Fourir currn dnsiy (7 in Eq. (3 yilds E(, rtv( ( ω ω ki ( ω 4 d T i d rk ω = (7 kε ( ω c- ( π For calculaion of ingral (7 w us h sandard mahmaical procdur shown in Fig., which yilds ErT(, v= ( iω 4πε c r T ( ω iω( rc dω π y calculaing (7 and using dv/d= a, w finally g (8(a k kr, r Pol a k = - ω/c k Im Pol a k= ω/c Fig. For calculaion of ingral (7 E i (r, ~ i a Ti ( r j /c + ϕ j /r j k R whr ( r d Ti ( rc 4πε c r d v, = =T (8(b Incohrn or cohrn EM wavs (X-rays E N (r, ~Σ in E i (r, ~ Σ in a Ti ( r j /c + ϕ j / r j Maxwll s quaions Wav quaion For cohrn EM wavs (X-rays (9 E (r, = E N (r, ~ N E T1 Cohrn EM wavs (X-rays by E N (r,~σ in E i (r,~σ in i a T1 (-r/c+ϕ/r TÁMOP C-1/1/KONV-1-5 projc 8

9 Poyning s nrgy flux S of an EM fild of x-rays E (r, = E N (r, ~ N E T1 S (r, = S N (r, =? Th Poyning nrgy flux S of an EM fild of x-rays is found via gnral formulaion of Poyning s HEq.( 1 EEq.( (3 horm. L us calcula h valu ( ( Th calculaion yilds h qualiy EEHH E E( ( = Th us of marial quaions (5 and (6 and h vcor qualiy yilds Poying s horm in h diffrnial form E BDHEEHE( ( = ( H J (31 H (3 J μ E( = ε HEH Th horm ingral form is obaind by ingraing Eq. (31 ovr a volum and using h Gauss-Osrogradsky divrgnc horm surfac HEHs μ d = dv ε E volum E ( ( dv whr h Poying nrgy flux is givn by TÁMOP C-1/1/KONV-1-5 projc 9 SE= ( H(35 volum J(33 (34

10 Rlaions bwn E, H, S and <S> in vacuum Th Poyning nrgy fluxs S and <S> of x-rays inducd by an lcron ar found via sablishing h gnral rlaions bwn E, H, S and <S> in vacuum. Th Maxwll Hquaion (1 wih h marial quaion (6 yilds ETh us of h Fourir dcomposiions yilds h qualiy Er = μ Ek krkd dω k(π d dω (π i( ω (, (, ω 4 = rhikh (, (, ω 4 = keki, = kh( ω iωμ (, ω i ( ω ω (36 (37 (38 Th us of h disprsion rlaion ω=kc and c=1/(ε μ 1/ in Eq. (37 yilds h rlaion rhk re(, = ε (,, (4 μ k Using h Fourir invrs ransform and Eq. (33 yilds ε ε μ k k E (, = Sr Th on-cycl im avraging yilds TÁMOP C-1/1/KONV-1-5 projc S1 ε 1 = μ 1 k ke (4 (39 (41

11 Powr dp 1 /dω radiad by 1 acclrad lcron Th powr of x-rays pr uni solid angl (dp 1 /dω radiad by an acclrad/dacclrad lcron is calculad by using Eqs. (8 and (41. Subsiuing E Trar(43 ino S( = ( kε T μ, T c 16π ε c k E (, = rw firs g h insananous powr pr uni ara radiad by an acclrad/dacclrad lcron ars 3 16 π ε c r T (, = k whr a T = a sinθ, and Θ is h angl bwn a/a and k/k. Thn Eq. (45 is prsnd as (, = kr (44 rak sin Θ ( π ε c r k S Using h qualiis S=(dP/ds (k/k and ds=r dω, w obain h powr pr uni solid angl a Th lcron as a sourc of x-rays Fig. 3 Th angl disribuion sin of radiaion dscribd by dp Θ Θ 1 Eq. (47 ak/k = 3 (47 Ω 16 π ε c dω ~sin Ω TÁMOP C-1/1/KONV-1-5 projc 11

12 dp N /dω and ds N /dω of x-rays radiad by cohrn moion of h N acclrad lcrons W alrady dmonsrad ha Maxwll s quaions Wav quaion Cohrn EM wavs (X-rays E (r, ~ Σ in i a T ( - r/c + ϕ / r E (r=e (r, E N (r~n (r, E 1 From h on-lcron quaion w g dp1 = d sin 16π ε c 3 Θ and h N-lcron quaion Ω dp N /dω = N in h cohrn x-ray radiaion of h supr- alik dp 1 /dω (48(a (48(b Thn w finally g lik in h cohrn x-ray radiaion of h suprradiaion opraion-mod or supr-fluorscnc opraion- mod of an x-ray lasr ds N /dω = N ds 1 /dω TÁMOP C-1/1/KONV-1-5 projc 1

13 Toal powrs P 1 and <P 1 > radiad by 1 lcron Th oal powr P 1 of x-rays radiad by an acclrad/dacclrad lcron is calculad by ingraing h powr pr uni solid angl (dp 1 /dω ovr h oal solid angl as a a π π sin Θ sin Θ P1 dω = sin ΘdΘdϕ 3 3 (49 = oal solid angl 16π ε c 16π ε c whr w usd h rlaion dω=sinθdθdϕ. Th ingraion yilds P 1 = a(5 3 πε 6 c In h cas of h im-harmonic i moion dscribing by a( = a sin(ω, h avrag ovr on cycl yilds aap 1 = a sin( ω = sin ( ω = πε c 6 πε c πε c 1 (51 Thn TÁMOP C-1/1/KONV-1-5 projc P a= = ( πε c 13

14 P N, <P N >, S N and <S N > of x-rays radiad by h cohrn moion of N acclrad lcrons W alrady dmonsrad ha Maxwll s quaions Wav quaion Cohrn EM wavs (X-rays E (r, ~ Σ in i a T ( - r/c + ϕ / r E (r, = E N (r, ~ N E 1 From h on-lcron quaions w g P 1 P 1 6πεa= c 3 1πεa= c 3 and h N-lcron quaion P N = N P 1 (53 (a Thn w finally g <P N > = N <P 1 > (54 (a S N = N S 1 (54 (b lik in h cohrn x-ray radiaion of supr-radiaion or <S N > = N <S 1 > (54 (b supr-fluorscnc of an x-ray lasr TÁMOP C-1/1/KONV-1-5 projc 14

15 X-rays by acclrad fr or bound lcrons re, a( 16 r = T π ε c r ( 1 c E N N N (r, = Σ in E i (r, ~ Σ in a Ti ( - r j /c + ϕ j / r j, Incohrn x-rays by incohrn r ϕ ϕ i i j moion of fr or bound lcrons a Ti a Tj r j E N (r, = Σ in E i (r, ~ Σ in i a T1 ( - r /c + ϕ / r = NE 1 Cohrn x-rays by cohrn a a moion of fr or bound lcrons ϕ ϕ Ti Tj i j r r L << λ?? i j in a sub-wavlngh rgion E N (r, = Σ in E i (r, ~ Σ in a Ti ( - r j /c + ϕ j / r j, r i r i ϕ i ϕ j and r r ϕ m ϕ m n n a Ti a Tj a Tm a Tn Parially-cohrn x-rays by parially-cohrn moion of fr or bound lcrons E (r=σ N (r N N (r, Σ in E i (r, ~ Σ in i a /r T1 ( r j /c + ϕ j ~ NE 1 Cohrn x-rays by cohrn a r r ϕ ϕ moion of fr or bound Ti a Tj i j i j lcrons in a macroscopic rgion TÁMOP C-1/1/KONV-1-5 projc 15

16 Opraion rgims of x-ray sourcs re, a( 16 r = T π ε c r ( 1 c E N N N (r, = Σ in E i (r, ~ Σ in a Ti ( - r j /c + ϕ j / r j, Incohrn x-rays: Crooks r r ϕ ϕ ub, vacuum ub, ho plasma, a Ti a Tj i j i j synchroron E N (r, = Σ in E i (r, ~ Σ in i a T1 ( - r /c + ϕ / r = NE 1 Cohrn x-rays: Subwavlngh x-ray lasr by a a ϕ ϕ Ti Tj i j r i r L << λ?? j supr-radiaion!? E N (r, = Σ in E i (r, ~ Σ in a Ti ( - r j /c + ϕ j / r j, r i r i ϕ i ϕ j and r r ϕ m ϕ m n n a Ti a Tj a Tm a Tn Parially-cohrn x-rays: X-ray amplifir by amplifid sponanous mission (ASE E (r=σ N (r N N (r, Σ in E i (r, ~ Σ in i a /r T1 ( r j /c + ϕ j ~ NE 1 Cohrn X-rays: X-ray lasr r r ϕ ϕ by supr-fluorscnc a Ti a Tj i j i j (saurad ASE, FEL, HHG TÁMOP C-1/1/KONV-1-5 projc 16

17 Angl divrgnc of incohrn x-ray sourcs E (r ~Σ N (r ~ N N (r, Σ i E i (r, Σ i a Ti ( r j /c + ϕ j /r j dp N /dω ~ N dp 1 /dω 1 1 dp N (Θ /dω ~ Nsin Θ Θ X-ray Lambrian surfac-sourc of x-rays (black body, ho plasma, Crookrs ub, vacuum ub, synchroron Incohrn mdium L Θ ~ π (oupu aprur r T Diffracion limi a Incohrn mdium L (oupu aprur r T 1 X-ray Θ L fap. Θ ~ r T / L fap If Θ ~ λ / r T, hn incohrn cohrn surfac sourc of x-rays Wha is h cohrnc? L. 3 Fig. 4 Th divrgnc (Θ: (a for 1 aprur and (b for aprurs TÁMOP C-1/1/KONV-1-5 projc 17

18 From incohrnc o ransvrs cohrnc 1 X-ray Incohrn mdium Θ L r T Θ ~ r T / L fap (oupu aprur radius r T L fap. 1 A B Incohrn mdium L L fap. An incohrn sphrical wav ~ Θ ~ an axial plan wav Fig. 5 From incohrnc o ransvrs cohrnc of x-ray sourcs TÁMOP C-1/1/KONV-1-5 projc 18 If Θ ~ λ / r T (diffracion limi, hn an incohrn surfac sourc of x-rays cohrn on If h fron dlay AB < λ/ Th ransvrs cohrnc If AB ~ λ/ Th parially, ransvrs cohrnc How do w dscrib ransvrs cohrnc? van Zir-Crnik horm in Lcur 3

19 Angl divrgnc of cohrn x-ray sourcs E N (r, = Σ in E i (r, ~ Σ in i a T1 ( r j /c + ϕ / r j ~ NE 1 Incohrn radiaion dp N /dω ~ N dp 1 /dω 1 Parially cohrn radiaion (ASE dp N (Θ /dω ~ N sin Θ Cohrn radiaion = = supr-fluorscnc = = saurad ASE X-ray lasr Volum sourc of cohrn x-rays (x- ray lasr, FEL, HHG Cohrn mdium Θ L r T cohrn wavs (X-rays r (oupu aprur Θ ~ r T /L Diffracion limi if Θ ~ λ / r T 1. Why x-rays insad of h EM wavs of x-rays?. How do w calcula angl divrgnc? Diffracion in Lcur 3 Fig. 6 Divrgnc angl (Θ of a cohrn x-ray sourc TÁMOP C-1/1/KONV-1-5 projc 19

20 Problms of h Maxwll xplanaion of X-lasrs A BSc lvl h qualiaiv Aciv mdium xplanaion of x-ray sourcs by Maxwll quaions is appropria. Θ Θ ~ r /L L T r Diffracion limi if r T (oupu aprur cohrn wavs (X-rays Θ ~ λ / r T Cohrn x-rays: E N (r, = Σ in E i (r, ~ Σ in i a T1 ( r j /c + ϕ / r j ~ NE 1 a r r ϕ ϕ Ti a Tj i j i j Fig. 7 Qualiaiv xplanaion of cohrn x-ray sourcs (x-ray lasr, FEL, HHG An x-ray lasr is basd on cohrn acclraion of bound lcrons in ions (non-classical, quanum objcs Problms: How can w cra h condiions Lcurs 5-1 a Ti a Tj r i r j in ions (quanum objcs of an x-ray lasr? ϕ i ϕ j TÁMOP C-1/1/KONV-1-5 projc

21 Qualiaiv-xplanaion of x-ray sourcs Qualiaiv xplanaion Maxwll s quaions Wav quaion Incohrn and cohrn x-ray sourcs E N (r, = Σ in E i (r, ~ Σ in a Ti ( - r j /c + ϕ j / r j, Incohrn x-rays: Crooks r r ϕ ϕ ub, vacuum ub, ho plasma, a Ti a Tj i j i j synchroron E N (r, = Σ in E i (r, ~ Σ in a Ti ( - r j /c + ϕ j / r j, r i r i ϕ i ϕ j and r r ϕ m ϕ m n n a Ti a Tj a Tm a Tn Parially-cohrn x-rays: X-ray lasr by amplifid sponanous mission (ASE E N (r, = Σ in E i (r, ~ Σ in i a T1 ( r j /c + ϕ / r j ~ NE 1 a Ti a Tj Cohrn x-rays: An x-ray lasr r r ϕ ϕ by supr-fluorscnc i j i j (saurad ASE; FEL, HHG TÁMOP C-1/1/KONV-1-5 projc 1

22 From qualiaiv xplanaion o quaniaiv dscripion of x-ray sourcs Qualiaiv xplanaion of x-ray sourcs is basd on Maxwll s quaions?? (a MSc lvl (a BSc lvl Quaniaiv dscripion of x-rays producd by fr-lcrons of synchrorons and FELs is basd on Maxwll s quaions and Einsin spcial rlaiviy for fr-lcrons Quaniaiv dscripion of x-rays producd by boundd lcrons in Crooks ub, vacuum ub, ho plasma, amplifirs, lasrs and HHG is basd on h us of Quanum Mchanics for lcrons boundd in aoms and ions + Plasma Physics TÁMOP C-1/1/KONV-1-5 projc

23 Problms as hom assignmns (A 1. Dmonsra (ω k c E(k,ω =(1/ε [-iωj(k,ω+ ic kρ(k,ω] by using h Fourir ransformaions of E(r,, J(r, and ρ(k,ω..explain how h quaion (ω k c E(k,ω =(1/ε [-iω(k,ω +ic kρ(k,ω ] lads o h fr-spac vacuum disprsion rlaion ω=kc in h absnc of currn sourcs. 3. Show ha h im-avragd powr pr uni-ara is givn by <P> = (1/R(E x H* 4. Exprss h quaion S(r, = (ε /μ 1/ E (k/k as h wav innsiy I in rms of h lcric fild. 5. Using h valu I of h prvious xampl calcula h lcric fild in h focus of a lasr wih 8 nm, 3 fs puls duraion if h lasr focal innsiy is 1 14 W/cm. 6. Compar h valu I of h prvious xampl wih h lcric fild in a Hydrogn aom. 7. Wha nrgy could an lcron ak in h focus of h lasr of h prvious xampl by assuming h lcron o b a harmonic oscillaor? 8. Wha innsiy is rquird o ioniz a hydrogn aom? 9. Confirm h scaling E ~ NE 1 of h lcric fild and h scaling P~N P 1 of h powr of h cohrn X-ray sourc of N cohrnly acclrad lcrons. TÁMOP C-1/1/KONV-1-5 projc 3

24 Problms as hom assignmns (B 1. Esima h angl of divrgnc of h incohrn x-ray sourc having h wavlngh λ << r T (s, Fig. 5, Lcur radiaing h sof x-rays. Th aprur dimnsion r T =.1mm and L ap =.5m. 11. For h incohrn sourc of Fig. 5 (Lcur radiaing h sof x-rays wih h wavlngh λ = 5nm sima h aprur dimnsion r T, whn h incohrn sourc will b ransvrsally cohrn (l L ap =.5m. 1. For h sourc of cohrnly acclrad lcrons of Fig. 7 (Lcur radiaing h sof x-rays wih h wavlngh λ = 5nm sima h aprur dimnsion r T, whn h incohrn sourc will b ransvrsally cohrn (l L ap =.5m. TÁMOP C-1/1/KONV-1-5 projc 4

25 Rfrncs 1. J.D. Jackson, Classical lcrodynamics (Wily, Nw York, For addiional informaion s: 1. R. C. Elon, X-ray lasrs, Acadmic Prss, David Awood, Sof-X-rays and Exrm Ulraviol Radiaion, Cambridg Univrsiy Prss, ; David Awood, Sof-X-rays and Exrm Ulraviol Radiaion ( brkly.du. 3. J.J. Rocca, Rviw aricl. Tabl-op sof x-ray lasrs, Rv. Sci. Insr. 7, 3799 ( H. Daido, Rviw of sof x-ray lasr rsarchs and dvlopmns, Rp. Prog. Phys. 65, 1513 ( 5. A.V. Vinogradov, J.J. Rocca, Rpiivly pulsd X-ray lasr opraing on h 3p-3s ransiion of h N-lik argon in a capillary discharg, Kvan. Elcron., 33, 7 (3 6. S. Suckwr, P. Jagl, X-Ray lasr: pas, prsn, and fuur, Lasr Phys. L. 6, 411 (9. TÁMOP C-1/1/KONV-1-5 projc 5