A Smith-Purcell BWO for Intense Terahertz Radiation · A Smith-Purcell BWO for Intense Terahertz...

Argonne National Laboratory is managed byThe University of Chicago for the U.S. Department of Energy

A Smith-Purcell BWO forIntense Terahertz Radiation

Kwang-Je Kim and Vinit Kumar

ANL and The University of Chicago

The Physics and Applications of High BrightnessElectron Beams

Erice, Sicily

October 9-14, 2005

2KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05

Non-Linear Behavior in Smith-PurcellRadiation ? (J. Urata et al., PRL 80 (1998) 516-519)


SEM-Based Smith-Purcell Radiator

β= 0.35 (35 keV)

Ι 1 mΑ

λg = 173 µm, d = 100 mm,w = 62 µm,

b = 10 µm, L = 12.7 mm

≤


SEM-Based Smith-Purcell Radiator at the U of C,After the Dartmouth Set-Up (O. Kapp, A. Crewe, KJK)


Heated Specimen Stage and PossibleBlack Body radiation background


)cos1( θββ

λλ −= g

*S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953)

Waves on a Grating: Propagating andEvanescent Modes

propagating mode

surface mode(evanescent)

_

_

_g

current-induced field

electron

Metal grating


Sheet Current

Consider a sheet electron beam having current density*

∑∑ −Δ

=−Δ

=i

ii

ziz zttxy

qvzzx

y

qtzxJ ))(()()()(),,( δδδδ

Fourier transform of this current density is given by

)exp()exp(Äy

q)(),,( 0zikixzxJ

iiz ∑= ωξδω

),( ωzK ←slowly varying function in z

)exp()(0 zK µωµα

βξβω

ik

cztck ii

−=

−==

00

0 /,/ ↓

)exp()()(),,( 00 ziKxzxJ z αωδω =

*K.-J. Kim and S. B. Song, Nucl. Instrum. Methods Phys. Res. A 475, 158 (2001).


EM Fields Induced by a Sheet Current

Solving the Maxwell equations with proper symmetry, we get

0

])(exp[)()(2

1

0

0

000

===

−

∂

∂=

∂

∂−=

Γ−=

Iz

Ix

Iy

z

IyI

z

IyI

x

Iy

HHE

Jx

HiE

z

HiE

xxziKxH

ωε

ωε

εαωε ε(x) = -1 for x < 0

+1 for x > 0

βγω

ωα

c

c

/

/ 22200

=

−=Γ

These are slow plane waves, propagating along z-axis with speed v, butdecaying along x-axis with decay constant Γ0. These are non-radiating,zeroth order evanescent wave.

Hy


E- Field, Energy Modulation, and Bunching;Three-Fold Way for FELs

Ez-Field gives rise to energy modulation

Energy modulation gives rise to bunching

Bunching gives rise to surface mode

Quadratic equation for growth rate if e00 is a smooth function*

However, e00 is singular !

*K.-J. Kim and S. B. Song, Nucl. Instrum. Methods Phys. Res. A 475, 158 (2001).

),(2 tzEmcq

dzd

zγη=

0

0

γγγ

η−

=

23γβηξ

cdz

d−=

( ) zibz eKee

iE 00 )(1

2 02

000

0 αωωε

−Γ

= Γ−


Singularity in e00 and Freely PropagatingSurface Mode

• The reflection coefficient e00 diverges atλ=690 m.

• Freely propagating surface mode at this λ.

• For a non-zero growth rate (µ) it has asimple pole

100 )( χµχ

µ +−

=i

e

ψ

βγχ

µ ibsur

sur eey

IZdzdE

E −Γ−

Δ== 020

2

Thus we recover cubic equation !


Surface Mode at λ=690 m

Scattering coefficients from mth to nth spatial modes

There is a singularity in e00, indicating that a free-propagating surfacemode

Due to linear relation between different emn, em0 are in general singular

The mth spatial waves combine to satisfy the grating BC

A surface mode of a perfectly conducting grating does not couple toany propagating modes…If it did, the singularity cannot be infinitelynarrow.


Surface Mode Has Negative GroupVelocity*

Phase velocity =ω/kz=βc

,ν dω/dkz < 0

ν Thus SP-FEL is a Backward Wave Oscillator (BWO)

ν Optical energy accumulates exponentially to saturationwithout feedback mirrors

*H.L. Andrews et al., Phys. Rev. ST Accel. Beams. 8, 050703 (2005)


Including Time Dependence via

∂∂±∂∂⇒ // vtµ

Time-dependent Maxwell equation:

( ) ψ

ψ

χβγ

βγ

χ

ibsc

ibgg

eey

iIZE

eey

vIZ

z

Ev

t

E

−Γ−

−Γ−

−Δ

−=

Δ−=

∂

∂−

∂

∂

0

0

21

0

20

12

2

Lorentz equation:

( )( )

p

pisii

iscii

zv

t

cceEEmc

ev

zv

ti

γ

γγ

γβωψψ

γγ ψ

−=

∂

∂+

∂

∂

++=∂

∂+

∂

∂

22

2..

*First obtained for microwave circuit by N. S. Ginzburg et al., Sov. Radiophys. Electron., 21, 728(1979), See also B. Levush et al., IEEE Trans. Plasma Sci., 20, 263 (1992).

According toforward andbackward±


Maxwell-Lorentz Equations

Dimensionless variables:

( )

b

pp

s

A

spp

ss

pp

s

pipp

si

gpp

eLk

yI

IJ

Ec

Lk

mc

e

Ec

Lk

mc

e

Lk

Lvvv

z

Lz

0244

3

33

2

2

33

2

2

33

1

2

111

/

Γ−

−

Δ=

=

=

−=

+

−=

=

γβχ

π

γβε

γβε

γγγβ

η

ττ

ς

( )

( ) ψ

ψ

ψ

χχ

ε

ηςψ

εεςη

ςε

τε

ibsc

ii

isci

i

eeL

Ji

cce

eJ

i

−Γ

−

−=

=∂

∂

++=∂

∂

−=∂

∂−

∂

∂

021

..

Maxwell-Lorentz equations indimensionless variables:

Boundary conditions:

),0(),,0(),,1( τςητςψτςε === ii

should be known for all τ


Boundary Conditions for a BWO

0),0( == τςψ i

0),0( == τςηi

•No bunching at the entrance of the grating:

0),1(.,.

0),0(/),1(

==

===

τςε

τςετςε

ei

•Oscillation starts when field at the exit vanishesrelative to the field at the entrance:

•No energy modulation at the entrance


Analytic Solution in the Linear Regime J.A. Swegle, Phys. Fluids 30, 1201 (1987)

QBP

,PB

,Bi +=ς∂

∂=

ς∂∂

=ς∂∂

−τ∂∂ EJEE

023 =+ν+κ−νκ−κ JiQQ

•Collective variables a la Bonifacio00 ,, ψψ δηδψ ii ePeB −− ==E

•Solution of the form exp(ντ)exp(ηζ)

•General solution:

( ) [ ]ςκςκςκνττς 321321, eAeAeAe ++=E

•Boundary conditions

B = 0, P = 0 at ζ = 0, E = 0 at ζ = 1


Analytic Solution in the Linear Regime (cont’d)

• Nontrivial solution if

• This is a transcendental equation on ν. Find that there is a threshold value of J above which ν has a positive real part.

⇒ Start current condition

bA

s eL

I.yI

0232

44

26857 Γ

χπλγβ

=Δ

( )( ) ( )( ) ( )( ) 032121

2313

2232

21 =κ−κ−κ+κ−κ−κ+κ−κ−κ κκκ eQeQeQ


Simulation Results:Start Current and Saturation

For I/Δy = 50 A/m, at saturation, P/Δy = 13.7 mW/µm

Power e-folding time = 0.2 ns (simulation)

0.17 ns (analytic formula)

Lasing wavelength = 694.5 µm (simulation)

694 µm (analytic formula)

I/Δy = 50 A/m

I/Δy = 36 A/m

After saturation

@ z = 0 Energy conversion efficiency = 0.8%


Simulation Results:Start Current as a Function of Gap Distance

For b = 10 µm,

Ist/Δy = 37.5 A/m (simulation)

= 36 A/m (analytic formula)

• If we maintain an rms averagebeam radius of 10 µm over theentire interaction regime, thestart surface current density is37.5 A/m


Simulation results

Evolution of longitudinal phase space

Electron beam becomes bunched due to SP-FEL interaction


Outcoupling Maximum efficiency

Outcoupling via

– Mode conversion at entrance

– Bunched beam radiation at exit

( )%1

121 3

≤−

≈γβγλ

ηLeff


Smith-Purcell FEL is a Backward WaveOscillator

e-beamsurface mode(evanescent)

group velocity

e-beam and phase velocity


Reference case:

λg = 173 µm, β = 0.35 (35 keV)

d = 100 µm, w = 62 µm,

b = 10 µm, L = 12.7 mm

λg

L

φ

w

yx

z

2a

h

d

E-Beam and Grating Parameters


Beam Design for SP-FELs

For clarity, assume KV distribution

σx = a/2

σy = b/2

Choose β* = L at the grating center (beam size variation is small)

For a good overlap of evanescent wave with e-beam

Diffraction condition in y-direction

These conditions are satisfied by sheet beam (a << b). Thus the theory forsheet beam developed in the above can be used for practical SP-FEL design

L,L yyXX ε=σε=σ

y

a

b

πλβγ

≤≤4

ha

πλβ

ε <

4~y


Beam Design Examples Start current condition

For Dartmouth parameter the coupling parameter _ = 10/cm

(Case A) A set of beam parameters satisfying these conditionsa = 20 µ, b = 500 µ, εx = 0.8 × 10-8 m-r, εy = 5 × 10-6 m-r, Is = 65 mA

ν Condition that space change force is less than the emittance force in the beamenvelope equation:

ν Case A violates the space change condition by a factor of 5.

( ) hA

s oeL

I.dydI

dydI Γ

χπλβγ

=> 232

4

277

( ) ( )yxAx

x

I

I

σ+σβγ≥

σ

ε33

3


Phase Velocity, Group Velocity, and Diffraction

A wave evanescent in the x-direction and diffracting in y, with waist σy atz=0:

This satisfies free space wave equation if k2=kz2-Γ2

The phase velocity and diffraction property are determined by the operatingvalue of k and kz. For example, the diffraction angle σy’=1/2kzσy , the phasefront curvature R=(z2+ZR

2)/z, etc.

The group velocity, including its sign, is determined by how kz changes as afunction of k near the operating point.

For example let Γ=gk(1-αk), thus . The group velocityis negative if αk=3/4.

]41

)21

1(exp[ 2222 φσφφφ yzzz kyikxzikikctd −+Γ−−+−∫

22 )1(1 kgkkz α−+=


Beam Designs Satisfying Also the Condition ThatSpace Charge Emittance Growth Is Small

(Case B) Increase the depth of groove d:100_ 150 µ.

⇒ _ increases from 10 to 100 /cm ⇒ Is reduced by a factor of 10.The wavelength increases also, but only by about 10%.

(Case C) Increase L:1.25 _ 5 cm.a = 20 µ, b = 200, εx = 2.0 × 10-9 m-r

εy = 1.25 × 10-6 m-r, Is = 0.36 mA


Conclusions

• We have developed a theory of SP-FELs driven by sheet beams operating asa BWO, using Maxwell-Lorentz equations.

• Simple formula for start current is derived from linear analysis .

• Results from a simulation code based on Maxwell-Lorentz equations agreewith linear theory where applicable and give saturation behavior.

• The sheet beam theory can be used for designing a portable SP FEL forTHz radiation.

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