a smith-purcell bwo for intense terahertz radiation · a smith-purcell bwo for intense terahertz...
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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)
3KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
≤
4KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
SEM-Based Smith-Purcell Radiator at the U of C,After the Dartmouth Set-Up (O. Kapp, A. Crewe, KJK)
5KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
Heated Specimen Stage and PossibleBlack Body radiation background
6KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
)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
7KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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).
8KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
9KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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−
10KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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 !
11KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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.
12KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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)
13KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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±
14KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
15KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
16KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
17KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
18KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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%
19KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
20KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
Simulation results
Evolution of longitudinal phase space
Electron beam becomes bunched due to SP-FEL interaction
21KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
Outcoupling Maximum efficiency
Outcoupling via
– Mode conversion at entrance
– Bunched beam radiation at exit
( )%1
121 3
≤−
≈γβγλ
ηLeff
22KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
Smith-Purcell FEL is a Backward WaveOscillator
e-beamsurface mode(evanescent)
group velocity
e-beam and phase velocity
23KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
24KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
25KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
26KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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−+=
27KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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
28KJK, Compact SP BWO, The Physics and Applications of High Brightness Electron Beams, Erice, 10/9-14/05
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.