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 = E2/Zo cf P=V2/R

In order to develop the analogy I will take the ponderomotive scaling for Thot

H

EZ 0

c

EHB 0

00

2 1

c

0

00

Z

Running out of electrons

In the relativistic regime the laser will interact at the corrected critical density

Suppose all the electrons at n1 acquire an average energy equal to the ponderomotive energy Epond = a0 mc2

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 = E2/Z0

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 a0mc2) then exactly matches the electromagnetic energy

2

20

001 e

manan c

0

22

03

01 Z

EcEmcanPhot

cm

eEa

0

Impedance of the laser driven fast electron source

For the ponderomotive scaling of Thot 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

220""

Ec

e

mca

eV pond

2

0

2

rZ

EP

2

0

"" rcZ

fEI

2

2

0

""

""

rwc

f

Z

I

VZ

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

pe

I A 4

2mceVE

c

VI

0

4

3044

1 0

Z

I

V

A

2-D Simulation gives ~ N1/2 filaments

Vacuum Transmission lines

d

w

w

dZZ 0

1

20 ln r

rZZ

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

10 - 20 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 = (Il2/2.1018)1/2 x 511kV

The current density follows from absorbed power - assume 50%j = 0.5 x I / Vand the current from j and the spot size: J = 0.5 x I/V x d2

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 a0 > 1

A wire of length l and diameter d has an inductance of approximately

L = 2 l (2.3log(4l/d) - 1 + (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