Two Models for Electro-Magnetic Wave Amplifier by Utilizing Traveling Electron Beam.

by,Hesham Fares1, Minoru Yamada1, Yuji Kuwamura1

and Masahiro Asada2.1Grad. School of Natural Sci. and Tech, Kanazawa University.2Grad. School of Sci and Eng, Tokyo Inst. of Tech.

2008

Paper No.TuA3

1-General Scheme of Electro-Magnetic Wave Amplifiers.

2-Theoretical Models of Amplification Mechanism.2.A- Coherent Electron Wave (CEW) Model.2.B- Localized Electron (LE) Model.

3-Thermal Effect on the Amplification Gain.

4- Experimental Evidences.

Headlines

What is the Electro-Magnetic wave Amplifiers?

Electro-Magnetic (EM) wave AmplifiersScheme of EM wave amplifier is same from microwave region to X-ray region.

The amplification gain (g) is,

(2)F(z)2g

zF(z) →=∂

∂

F(z) is field amplitude in z-direction and Tz(x,y) is transverse field distribution.

If the electric field component )(),()( )( 1eyxTzFE ztjz →= −βω

EM field

Energy exchange between electron beam and EM-wave.

Dielectric waveguide

Input Output

x

y

zLf

Electron GunElectron Beam

Dielectric Waveguide

Amplification Models“How does the electron see the EM-wave”?

kn: the electron wave number at n-th level .l : the coherent length of electron wave “electron size”.

1m 0.1m 1cm 1mm 0.1mm 10μm 10nm 1nm

MICROWAVES

INFRARED

“SOFT” X-RAYS

VISIB

LE

ULTRAVIOLETRADIO WAVES

1μm 0.1μm

nϕ

z

l=40µm[1]

wavelength (λ)

[1] Y. Kuwamura, M. Yamada, R. Okamoto, T. Kanai and H. Fares, “Observation of TM guided spontaneous emission in high refractive index optical waveguide excited by the traveling electron beam” Proc. 8th IntCLEO/QELS Conf. San Jose, CA, USA, May. 2008.

How does the Electron see the Electromagnetic wave?

(3)e1(r) znkj

3n →=

lϕForm of the electron wave function:

l>>λNon-Localized Electron

l<<λLocalized Electron

Models of amplification gain analysis:

Ez

z

λ

2- Localized Electron Model (l<<λ)

Electron wave size (l) < λ, electrons density modulation is shown.

A- Electron particle picture.B- Classical mechanical trend

1- Coherent Electron Wave Model (l>>λ)λ

nϕz0

z0

Electron wave size (l) > λ.

Electric field

Electron wavel=40µm

Ez

A- Electron wave picture.B- Quantum mechanical trend.

First Model“Coherent Electron Wave Model”

1- Physical interpretation of amplificationThe electron transits basing on the rules of:

Energy diagram of the electron transition

Ei is the electron energy at level i (i=a or b or c).ki is the electron wave number at level i.ω is emitted EM-wave frequency.β is EM-wave propagtion constant

n"consevatio Momentum" βkkn"consevatioEnergy " ωEE

ab

ab

=−=− h

Opticalemission

E

abc

kβ

ωhbE

aE

kbka

Optical absorption

zkkj abe )( −∝

zje β∝The condition of amplification

l

0

aϕ

bϕ

b*aϕϕ

zjkbe∝

zjkae∝

Ezz

z

z

z

0

0

0

Spatial variation coincidence corresponds to momentum conservation

Mixed electron wave

Electric field

Electron wave at final level

Electron wave at initial level

βkk ab =−

2- Gain coefficient in CEW-ModelBy some tools of statistical quantum mechanics,

Finally, the expression of amplification gain in CEW-Model

(5))v,D(vξωnvτJe

εμ

)v,g(v embeff

bo

o

oemb →×=

h t)coefficien(Coupling

dydx|y)(x,T|ξ, s2

z∫∫=

)(

2cωn

)eVωeV(2m

Sinc

2cωn

)ωeVeV(2m

Sinc)v,D(v

effbb

o2

effbb

o2emb

6→

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−+−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−=

lh

h

lh

h

D is the dispersion function controlling the gain profile,

vb is electron velocity influenced by applied voltage Vb.vem=c/neff is EM-wave phase velocity. J0 is average electron beam current density and τ is electron relaxiation time.

)(),( 4eyxTg2

bzj

a →∝ − φφ β

Gain behavior with frequency variation in CEW-Model.

The gain peak is affected by saturation of the dispersion function.

1013 1014 1015 1016

10-2

10-1

100

101

102

103

104

105

0.1mm 10nm

T=0.0Kneff=3.0

ξ=0.1

τ=10-8sec

J0=104A/m2

l=1μm

l=10μm

l=200μm

f [Hz]

g [m

-1]

l=40μm

l=1mml=1cm

λ 10µm 1µm 0.1µm

D=1.0

28380 28384 28388 28392 28396 28400

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

5μm

1.5μm1.25μm

λ=1μmneff=3.0l=1cm

Vb [Volt]

Dis

pers

ion,

D(v

b,vem

)

Variation of gain peak with EM frequency Saturation of dispersion function to 1.

)v,D(vξωnvτJe

εμ

)v,g(v embeff

bo

o

oemb ×=

h

Second Model“Localized Electron Model”

1- Physical interpretation of amplification

Synchronization condition,

l

3 V

Ez

0

0

υφ

νv

νv

λ

z

z

z

One synchronize wave modulates the electron velocity to start the amplification.

Electric field

Electron wave at ν-level

Electron density modulation

Electron velocity modulation

βωvvv ememel =≈ where,

Y(vb,vem) is dispersion function controls gain profile,

(9))v,Y(vm

Jμeξ)v,g(v emb

o

ooemb →×=

(10)ncv,

τωj1v

cn

τω1jRe)v,Y(v

effem

2

beff

emb →=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

Finally, The expression of amplification gain in LE-Model

2- Gain coefficient in LE-Model

The form of velocity-modulation, (same as classical form)

{ } (8)τ

vvc.cey)(x,TF(z)

me

zv

vt

v ννz)βtj(ωz

o

νν

ν →−

−+−=∂∂

+∂

∂ −

From the quantum mechanics point of view,

(Total wave function)

electron-ν of functionthe wave is Ψν

(7)ΨCΨ~ νν

ν →= ∑

Gain behavior with frequency in Localized Electron Model

The gain increases infinitely with frequency, thermal effect limits this behavior.

108 109 1010 1011 1012 1013 1014 1015 1016 101710-210-1100101102103104105106107108109

101010111012

10cm 1mm

τ=10-7 sec

τ=10-9 sec

τ=10-8 sec

T=0kneff=3.0

ξ=0.1

J0=104A/m2

λ

f [Hz]

g [m

-1]

0.1mm 10nm 10µm 1µm 0.1µm 1m 1cm

0.99990 0.99995 1.00000 1.00005 1.00010 1.00015-2.0x109

-1.5x109

-1.0x109

-5.0x108

0.0

5.0x108

1.0x109

1.5x109

2.0x109

ωτ=1.0×104

ωτ=1.5×104

ωτ=5.0×104

1/ωτ ωτ

vb/vem

Dis

pers

ion,

Y(v

b,vem

)

Dispersion function in gain coefficient by the LE-Model.

Variation of gain coefficient with the EM frequency by the LE-Model.

)v,Y(vm

Jμeξ)v,g(v emb

o

ooemb ×=

(ωτ)2

Thermal Effect on the Amplification Gain

Where,

e.temperaturabsolute the is T and constant Boltzmannthe is BK

velocity. electron realthe is bv andvelocity electronaverage the is v

1)vv(

TKVe

expTKπ2

m)v,f(v 2

bB

b

B

ob ⎥

⎦

⎤⎢⎣

⎡−−=

Real gain with thermal effect “velocity broadening around the average value”,

(11)dv)v,g(v)v,f(v)v,vg( bembboem →≈ ∫∞

is the normalized Maxwell-Boltzmann distribution function.)v,f(v b

1.0 at T2

at T1<T2

)v,f(vb

v vb

Illustration of thermal distribution function.

The effect of thermal velocity broadening on gain amplification.

bembboem dv)v,g(v)v,f(v)v,vg( ∫∞≈

)v,f(vb

vvb =

Variation of the peak values of gain coefficient with EM frequency by the CEW-Model and the LE-Model for several temperatures.

108 109 1010 1011 1012 1013 1014 1015 1016 101710-1

100

101

102

103

104

105

106

107

108

109

1010

T=2000K

T=300K

T=0K

neff=3.0ξ=0.1l=40μm

τ=10-8sec

J0=104A/m2

LE_Model

T=2000K

T=300K

T=77K

T=4K

T=0.001K

T=0K

CEW_Model

g [m

-1]

f [Hz]

10cm 1mm λ

0.1mm 10nm 10µm 1µm 0.1µm 1m 1cm

The boundary between two models within THz region.

Illustration for dispersion relationsD

ispe

rsio

n Fu

nctio

ns

Y(vb,vem)

0

D(vb,vem)

vvb >>vvb <<

Without Thermal Effect

Without Thermal Effect

The effect of electron size on the gain amplification

The thermal effect gives same gain peaks for different coherence length.

1012 1013 1014 1015 1016100

101

102

103

2000

τ=10-8secneff=3.0

ξ=0.120002000

300300300

T=77KT=77KT=77K

J0=104A/m2

l=40μm

l=1mm

l=1cm

λ

f [Hz]

g [m

-1]

0.1mm 10nm 10µm 1µm 0.1µm

Gain with different assumed coherence length at different temperature.

Experimental Evidences

Experimental Setup

Focusing lens

InGaAsPhoto-detector

Laser Diode Current Driver

Experiment Setup

Function Generator

Magnetic lens

SOI

Vacuumed Chamber

Monochromator

Lock. In AMPElectron beamElectron gun

Switch

Computer Interface

Computer screen“Signal Express software”

SiO2

100μm

1μm0.32μm

Si

Comparison of the emission profile with theoretical Calculation

3/1

1N

=l

N is electrons density

⎪⎭

⎪⎬⎫

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−+−⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−∝2c

ωn)eVωeV(

2mSinc

2cωn

)ωeVeV(2m

Sinc)v,g(v effbb

o2effbb

o2emb

lh

h

lh

h

EMISSION ABSORPTION

d=0.32μm

1.0μm

n0=1.0n1=3.49

n2=1.44

airSi

SiO2

Emission spectrum for different acceleration voltage

1.2 1.3 1.4 1.5 1.6 1.70.0

0.5

1.0

1.5

2.0

2.5

3.042KV(55μA)

40KV(53μA)

38KV(50μA)

36KV(47μA)

34KV(45μA)

V=32KV(I=42μA)O

ptic

al in

tens

ity [

a.u]

Wavelength [μm]

em

ab

abb

vβ(λ)ω

kkEE1

kE1ation]synchroniz at ([v

==

−−

=∂∂=

hh

THANKS FOR YOUR ATTENTION