Impedance Matching – can it tell us anything about fast electron transport? (or Why undergrads need to learn electronics!) Roger Evans Imperial College London

Impedance of Free Space 377  In SI units E is Volt/m and H is Ampere/m So has units of Ohm In a plane E-M wave And Poynting vector = E x H is W m -2 so I = E 2 /Z o cf P=V 2 /R In order to develop the analogy I will take the ponderomotive scaling for T hot

Running out of electrons In the relativistic regime the laser will interact at the corrected critical density Suppose all the electrons at n 1 acquire an average energy equal to the ponderomotive energy E pond = a 0 mc 2 In the relativistic case they will all travel at speed c so the power flux is The power flow in the electromagnetic wave of the laser is E x H = E 2 /Z 0 So there are only just enough electrons to absorb the laser energy (or 50% if we allow for a neutralising return current. Is this a coincidence? No - it happens because at the critical density the group velocity becomes zero and the quiver energy (also a 0 mc 2 ) then exactly matches the electromagnetic energy

Impedance of the laser driven fast electron source For the ponderomotive scaling of T hot the ‘voltage’ associated with the electron source is If the focal spot radius is r then incident power is Absorption fraction f implies “V” x “I” = f P Typically Z = 0.5 Ohm for a 7 micron spot and 50% absorption

Impedance of Alfven Current Jonathan Davies (PRE )gives the convenient form In the relativistic case take Compare with Z~0.5  for the laser source

2-D Simulation gives ~ N 1/2 filaments

Vacuum Transmission lines d w Need w/d ~ 600 to get 0.5 

K-  decay length matches 1 – 2 MeV electrons Skin layer appears hotter than bulk of wire Cone micron wire

Conventional argument for skin depth relies on energy flow (Poynting Vector) in vacuum. Impedance argument appears to rule this out 0.5  vs ~100 

Conventional argument for skin depth relies on energy flow (Poynting Vector) in vacuum. Impedance argument appears to rule this out But collimation argument of Robinson acts in reverse in this geometry - electrons deviated out of wire. Maybe energy flux in wire is so small that vacuum flux can dominate

Micro Z-Pinch The 'voltage' of the ejected electrons is given by V = (Il 2 / ) 1/2 x 511kV The current density follows from absorbed power - assume 50% j = 0.5 x I / V and the current from j and the spot size: J = 0.5 x I/V x d 2 We obtain a characteristic impedance Z = V / J = 25 (l/d) 2 Ohm For a spot size of 3 wavelengths the impedance is around 2.5 Ohm and falls even lower for larger focal spots. There is no wavelength dependence if the focal spot remains the same number of wavelengths. There is no intensity dependence as long as we stay in the regime a 0 > 1 A wire of length l and diameter d has an inductance of approximately L = 2 l (2.3log(4l/d) (d/2l)) nH or L ~ 20 l nH A 100 micron by 10 micron wire has an inductance of 0.2nH and with a 2 Ohm voltage drive the current rise time is t = L/Z = 100psec The current rise time is much longer than the laser pulse duration in existing experiments! Laser

Conclusions Most of this could be formulated without any use of impedance but it may help to bring several different concepts together Other uses of undergraduate physics eg equipartition of energy? Our energy balance / group velocity argument also implies that a saturated instability could generate E and B comparable to the laser fields - could this be a mechanism to inhibit energy flow?

Conclusions Most of this could be formulated without any use of impedance but it may help to bring several different concepts together Other uses of undergraduate physics eg equipartition of energy? Our energy balance / group velocity argument also implies that a saturated instability could generate E and B comparable to the laser fields - could this be a mechanism to inhibit energy flow? Energy / power: A 2GW(e) power station is 6GW(thermal) target gain of 150 means average laser power is 40MW 400kJ laser would have 100Hz rep rate