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A novel technique of symmetric type quasi-linear electron pulse duration modulation is proposed. The salient feature from the conventional photoelectron gun is the introduction of the alternating electric field resonator. The electric field that results is synchronously controlled to generate the desired quasi-linear differential energy modulation on the electron pulse passing through. The effect resulted directly is that the leading electrons undergo negative energy modulation and decelerate, while the rear ones positive energy modulation and accelerate, which eventually leads to electron-pulse-duration modulation. The technical details are demonstrated.

Electron-optical methods based on ultrafast manipulation of ultra-short electron pulse have become firmly established as the most advanced in the area of time-resolved events recording and investigation [

In this contribution, we propose the method of symmetric type quasi-linear electron pulse duration modulation. The salient feature is the introduction of an alternating electric field resonator into the conventional photoelectron gun. The electric field that results is synchronously controlled to provide the desired quasi-linear differential energy modulation on the electron pulse. With the result that the leading electrons undergo negative energy modulation and decelerate, while the rear ones positive energy modulation and accelerate, the initial energy modulation eventually leads to electron-pulse-duration modulation. The technical details are demonstrated.

The proposed electron-pulse-duration modulation technique is illustrated in _{1}, upon which the voltage U is applied. The differential energy modulation on the electron pulse is achieved in energy modulation region of axial length d_{2} between the two mesh electrodes, while the electron-pulse-duration modulation is finished in the field-free drifting region of axial length L between energy modulation electrode and target plane. The energy modulation region and field-free drifting region altogether constitute the electron pulse modulation region. An axial uniform magnetic field B is incorporated to confine electrons pulse radically throughout the whole system.

During the transition process of electron pulse in the electron-optical system, the two factors that contribute much to its pulse broadening are well known as the initial energy spread of photoelectron from photocathode and the inherent space charge effect. As for the latter, the electrons in leading part are subject to an overall accelerating force, while the rear ones decelerating force. The joint influence from both is the transition time dispersion between electrons of different part in the electron pulse, which is the essence of electron pulse broadening. In regards of the electron pulse broadening from space charge effect, the analytic or relatively simple calculating program is not yet available. But it has been proved, both theoretical and experimentally, that this kind of broadening possesses the characteristics of pulse center symmetry in comparison with the case without space charge effect, i.e., the leading part and the rear almost have equal broadening magnitude [

In the electron-optical system of

d 2 z d t 2 = η ϕ ′ ( z ) , (1)

in which, origin of coordinate axis Z, also the system symmetric axis, is fixed at center of photocathode and points along the motion direction of electron pulse, z indicates the electron position at t, ϕ ( z ) the potential at position of z, and η = e / m e the ratio of electron charge to mass. Expression (1) steps further as following

( d z d t ) 2 = 2 η [ ϕ ( z ) − ϕ ( 0 ) ] + ( d z d t ) 2 | t = 0 . (2)

Considering the square electron pulse just emitted from photocathode, with initial energy spread of Δ ε 0 and initial pulse duration of τ 0 , the initial axial

length l_{0} and the axial velocity difference Δ v 0 between the electrons in the most leading and the most rear can be given respectively as

l 0 = τ 0 2 Δ ε 0 m e + η U τ 0 2 2 d 1 , (3)

Δ v 0 = 2 Δ ε 0 m e + η U τ 0 d 1 . (4)

So the electron initial axial energy of the two portions mentioned is approximated as ε l a s t = 0 and ε f i r s t = 0.5 m e Δ v 0 2 . Using the same settings as that in practical engineering, l_{0} is of micrometer scale and can be neglected for electron transition time in photocathode-anode region since it is much smaller than the distance of d_{1} in millimeter scale. Using expression (2), the time for the electron with initial axial energy to fly through the photocathode-to-anode acceleration region turns out to be

t 1 ( ε i ) = m e d 1 e U [ 2 ( ε i + e U ) m e − 2 ε i m e ] , (5)

so the electron pulse duration at the anode mesh is

τ = t 1 ( ε l a s t ) − t 1 ( ε f i r s t ) . (6)

Apparently, the following relation is established

τ > τ 0 , (7)

which indicates the inevitable temporal broadening of electron pulse during the photocathode-to-anode acceleration region, just like that in streak camera or UED.

The alternating voltage of energy modulation electrode relative to anode in the resonator can be assumed to take the general form of U m ( t ) = − U max sin ( 2 π t / T m + φ 0 ) . For simplicity the initial phase φ 0 is chose to be 0, without any hurt on the method detail understanding. The overall transition time for electron with initial axial energy ε i to fly through the system in

t ( ε i ) = t 1 ( ε i ) + t m ( ε i ) , (8)

t m ( ε i ) = t 2 ( ε i ) + t 3 ( ε i ) , (9)

t 2 ( ε i ) = m e d 2 e U m ( t 1 ( ε i ) ) [ 2 { ε i + e U + e U m [ t 1 ( ε i ) ] } m e − 2 ( ε i + e U ) m e ] , (10)

t 3 ( ε i ) = m e L 2 2 { ε i + e U + e U m [ t 1 ( ε i ) ] } , (11)

in which t 1 ( ε i ) , t 2 ( ε i ) and t 3 ( ε i ) are the transition time of electrons respectively in acceleration region, energy modulation region and field-free drifting region. Obviously t m ( ε i ) is the transition time in electron pulse modulation region. Considering two electrons with initial axial energy relation of ε 1 < ε 2 , with the following conditions on energy-modulation voltage met,

| U ( ε 1 ) | , | U ( ε 2 ) | > ε 1 , ε 2 , (12)

U ( ε 1 ) > 0 , (13)

U ( ε 2 ) < 0 , (14)

the following relation is always established

t m ( ε l a s t ) < t m ( ε f i r s t ) , (15)

which means that the compensation for electron pulse broadening emerges.

t m i d = ( k + 1 2 ) T m , (16)

t 1 ( ε f i r s t ) ≥ ( k + 5 1 2 ) T m . (17)

Here k = 0 , 1 , 2 , ⋯ . The parameter k is called “alternating voltage nodes”, which defines the corresponding time interval for differential energy modulation. Combining (16) with (17), one can obtain the restriction on k.

k ≤ 6 t 1 ( ε f i r s t ) − 5 t m i d 12 [ t m i d − t 1 ( ε f i r s t ) ] . (18)

The template is used to format your paper and style the text. All margins, column widths, line spaces, and text fonts are prescribed; please do not alter them. You may note peculiarities. For example, the head margin in this template measures proportionately more than is U m ( t ) = U max sin ( 2 π t / T m ) . This measurement and others are deliberate, using specifications that anticipate your paper as one part of the entire journals, and not as an independent document. Please do not revise any of the current designations.

To quantitatively define the modulation effect, the transition time for electrons to travel through the pulse modulation region, in the case of no energy modulation, is given

t 0 ( ε i ) = m e ( L + d 2 ) 2 2 ( ε i + e U ) . (25)

With quasi-linear energy modulation, the changes of transition time for electrons in the most leading and the most rear portions can be got as the following, respectively

Δ t ( ε f i r s t ) = t 0 ( ε f i r s t ) − t m ( ε f i r s t ) = m e ( L + d 2 ) 2 2 ( ε f i r s t + e U ) − m e d 2 e U m ( t min ) [ 2 [ ε f i r s t + e U + e U m ( t min ) ] m e − 2 ( ε f i r s t + e U ) m e ] − m e L 2 2 [ ε f i r s t + e U + e U m ( t min ) ] , (26)

Δ t ( ε l a s t ) = t 0 ( ε l a s t ) − t m ( ε l a s t ) = m e ( L + d 2 ) 2 2 ( ε l a s t + e U ) − m e d 2 e U m ( t max ) [ 2 [ ε l a s t + e U + e U m ( t max ) ] m e − 2 ( ε l a s t + e U ) m e ] − m e L 2 2 [ ε l a s t + e U + e U m ( t max ) ] . (27)

So the overall modulation magnitude of electron pulse duration

Δ t = Δ t ( ε l a s t ) − Δ t ( ε f i r s t ) . (28)

Using the practical parameter setting of d 2 ≪ L , expressions (27) and (28) reduce to

Δ t ( ε f i r s t ) = L U m ( t min ) 2 U 1 2 η U , (29)

Δ t ( ε l a s t ) = L U m ( t max ) 2 U 1 2 η U . (30)

With U m ( t max ) = − U m ( t min ) of the prerequisite for symmetric type modulation, expression (28) gives

Δ t = L U m ( t max ) U 1 2 η U . (31)

An example of a cylindrically symmetric photoelectron gun system is presented here to verify the method. The geometric parameters as illustrated in

With these settings, the parameters of the electron pulse in acceleration region can be got as t min = 5.8107 × 10 11 s ， t max = 5.8407 × 10 11 s and t mid = 5.8207 × 10 11 s . Using relations determined by expressions (16), (17) and (18) for the case of U m ( t ) = − U max sin ( 2 π t / T m ) , k ≤ 31 can be got and the related parameters, e.g., T m , can be achieved for different alternating voltage nodes scope k ranging from 0 to 31. The case of k = 31 only is used in the following simulation.

The modulation voltages on both the most leading and the most rear electrons thus is U m ( t max ) = 0.50 U max and U m ( t min ) = − 0.50 U max respectively. With expression (29), solving the inequality of Δ t ≥ 200 fs gives U max ≥ 120 V .The compression effect on electron pulse duration is shown in

We have shown that the proposed technique can provide symmetric type

quasi-linear compensation for electron pulse broadening. This is accomplished by introducing an alternating electric field resonator, which renders the desired quasi-linear differential energy modulation on the electron pulse, with both the leading electrons undergoing negative energy modulation to decelerate and the rear ones positive energy modulation to accelerate. The technical details are demonstrated and the example gives the further illustration.

The work was supported by the National Natural Science Foundation of China (Grant No. 11675258, 11505289).

The authors declare no conflicts of interest regarding the publication of this paper.

Wang, C. and Kang, Y.F. (2018) Compressing Electron Pulse with Alternating Electric Field. Journal of Applied Mathematics and Physics, 6, 2313-2321. https://doi.org/10.4236/jamp.2018.611192