Part XI Optimising K α Sources 333 / 353
K α sources Goals: maximize # photons < 100 fs pulse length minimize spot size (magnification) maximize throughput (ave. power) 334 / 353
Applications of femtosecond K α sources Electron transport diagnostic (Fast Ignitor scheme) Real-time x-ray diffraction Bio-medical imaging (> 50 kev) 335 / 353
Inner shell excitation by fast electrons 2p K α 1s hot e 336 / 353
Model of femtosecond K α generation N Kα (I, Z) = 0 N H (I )f H (E, T H )I K (Z, E)f K (Z, E)dE N H (I ) f H (E, T H ) T H (I ) hot electrons hot electron distribution hot electron temperature I K (Z, E) K α quantum efficiency (# photons/electron) f K (Z, E) K α emission factor I obs /I K self absorption cf: G. H. McCall, J. Phys. D (1982) 337 / 353
Hot electrons Absorbed laser energy: PIC simulations show: U abs = η a U L N H T H = const. η a 40 60%(10 15 < I λ 2 < 10 19 Wcm 2 µm 2 ; θ = 45 0 ; L/λ = 0.3) f (E) = (πet H ) 1/2 exp( E/T H ) T H 100(I 18 λ 2 µ) 1/2 kev N H (I λ 2 µ) 1/2, since U L I focal area = const. 338 / 353
Scaling of hot electron temperature T hot (kev) 5 2 10 3 5 2 10 2 5 2 10 1 5 IOQ-99 Experiments LLNL-00 T h(fkl) T h(w) T h(gb) T RAL-99 h(b) IOQ-97 CELV-96 MBI-00 RAL-96 STA-92 LLNL-99 IOQ-00 CELV-96 MBI-95 MBI-97 LULI-97 IOQ-96 INRS-99 LULI-94 LLE-93 2 10 15 10 16 10 17 10 18 10 19 10 20 2-2 2 I (Wcm m ) 339 / 353
K α emission from solid targets Quantum efficiency Green & Cosslett (1968): ( ) 5/3 E I K = N(Z) 1 E K Replace with fit to Monte-Carlo simulations using monoenergetic electrons: I K 4 10 3 Z 4/3 E 3/2 (196) Emission factor f K I obs I K = f (E/E K ) { 1, EK < E < 20E K 0, otherwise (197) E K = K-shell ionization energy Z 2.2 340 / 353
K α emission factor N em /N gen shows universal behaviour with Z emission factor 10 0 10-1 10-2 10-3 10-4 1 10 100 1000 U scat.depth / abs. length 10 2 10 0 10-2 1 10 100 1000 U Ti Cu Ag Ta Normalised electron energy U E/E k 341 / 353
Solution Normalize energies to K-shell ionization energy: U E E K U H k BT H E K Putting together results (1) (5) gives: 20/UH N Kα (Z, U H ) az 0.6 U 1/2 H [ = az 0.6 e U 1 H e 20U 1 H Ue U/U H du 1/U H ( ) U 1/2 H + U 1/2 H ( U 1/2 H + 20U 1/2 H )] 342 / 353
Photon reabsorption leads to optimal electron energy N(U H ) 10 9 8 7 6 5 4 3 2 1 U H opt = 6.4 no absorption U max = 20 0 0 5 10 15 20 25 30 35 40 45 50 U H = k B T H /E k 343 / 353
Analytical model of intensity optimum scaling Reich, Gibbon, Uschmann, Förster, Phys. Rev. Lett. 84, 4846 (2000) Photon yield: N Kα (I, Z) Z 2.73 I 3/4 20 1 U exp( U/U H )du, where U H is the hot electron temperature normalised to the K-shell ionization energy: Find: U H = k BT hot E k I 1/2 Z 2.2. U opt H 6.4 I opt const Z 4.4 I opt (W/cm 2 ) 10 19 10 18 10 17 10 16 10 15 15 20 30 40 50 70 100 atomic number Z 344 / 353
Particle-in-cell + Monte-Carlo model τ d I, λ, τ p hot electron generation PIC-code 0 01 01 00000 11111 00000 11111 00000 11111 000000 111111 000000 111111 00000 11111 000000 11111100 00000 1111100 001100 0100 0100 1100 01 1 Plasma (n e, L/ λ ) Laser e - e - Kα K α production X-ray image Solid (z) Time dependence f hot(e,t) MC-code K α (x,t) time intensity Total K -yield 345 / 353
Optimal laser intensity for K α yield with constant energy on target yield (photons / sr) 10 10 10 9 Ti Cu Ag Ta E L = 200mJ τ p = 60fs (Ti-Sa) L/λ = 0.3 θ = 45 o 10 15 10 16 10 17 10 18 10 19 laser intensity (W/cm 2 ) 346 / 353
X-ray pulse duration with thick targets photons / fs (a.u.) 10 1 10 0 10-1 10-2 10-3 10-4 10-5 90% of emission over 0 1000 2000 3000 4000 time (fs) Afterglow 90%-pulse duration (fs) 10 4 10 3 10 2 10 15 10 16 10 17 10 18 10 19 laser intensity (W/cm 2 ) 347 / 353
Sub-100 fs pulses with foil targets 10 7 photons sr -1 fs -1 10 8 6 4 2 0 0 100 200 time (fs) 3 10 16 W/cm 2 7 10 15 W/cm 2 1 10 15 W/cm 2 Compromise between high yield & ultrashort duration gives: I opt 2 10 10 Z 2.4 (τ X τ p ) 5/4 Wcm 2 d opt 3Z 1/2 (τ X τ p ) 5/4 nm 348 / 353
Emission region of a 6.4 kev K α burst Fe-Target Ti:Saphir Laser: 200 mj 100 fs 5 10 17 W/cm 2 (Uschmann, Feurer) Intensities [a.u.] 140 120 100 80 60 40 20 0-50 -25 0 25 50 x [ m] K measured K simulated laserprofile 349 / 353
K α yield optimisation using a controlled prepulse Ziener et al., Phys. Rev. E (2002) K signal (V) 3.0 2.5 2.0 1.5 1.0 0.5 Si 5 10 17 1.5 10 17 0.0-300 -200-100 0 Prepulse delay (ps) K signal (V) 1.5 1.2 0.9 0.6 0.3 Ti 0.0-200 -150-100 -50 0 50 Prepulse delay (ps) K signal (V) 0.4 0.3 0.2 0.1 Co 0.0-200 -150-100 -50 0 50 Prepulse delay (ps) E L = 200 mj, λ = 0.8 µm, θ = 45 o I 0 = 5 10 17 Wcm 2, I prepulse = 10 16 Wcm 2 350 / 353
Density scale lengths calculated from isothermal expansion model L/ 8 7 6 5 4 3 2 1 Si Ti Co 0 0 50 100 150 200 d (ps) see : Bastiani et al., PRE 56, 7179 (1999) Schlegel et al., PRE 60, 2209 (1999) 351 / 353
Calculated Kα yield for different density scale-lengths K -photons / sr *10 8 7 6 5 4 3 2 1 Co n/n c =3 0 0.0 0.5 1.0 1.5 2.0 L/ K -photons / sr *10 8 3 2 1 n/n c = 20 n/n c = 3 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 L/ Si 352 / 353
Experimental Kα yields 10-2 10-3 f/electron/sr K/4 sr experiments K Efficiency 10-4 10-5 10-6 0 20 40 60 80 100 Z Overall x-ray conversion efficiency ε K = ε f 4πN h E K U L 353 / 353