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Generation of single cycle attosecond pulse via Thomson scattering of Gaussian beam interacting with a single electron in the electrostatic field environment
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Generation of single cycle attosecond pulse via Thomson scattering of Gaussian beam interacting with a single electron in the electrostatic field environment
T
⁎
Guodong Tonga, , Meng Zhanga, Changquan Xiab, Qianru Zhanga, Guoqing Lia a b
National ASIC System Engineering Technology Research Center, Southeast University, Nanjing, China Department of Physical Science and Technology, Yangzhou University, Yangzhou, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Attosecond pulse Gaussian beam Thomson scattering Single electron
The new way of generating the attosecond pulses via Thomson scattering of linear polarization Gaussian beam propagating on solid surface is investigated. It is found that a single attosecond laser pulse is generated which renders the appropriate strength electrostatic field to the environment. After studying the influence of the strength of the electrostatic field on it, the results show that a single attosecond pulse is generated when intensity of Gaussian beam is 5 and intensity of the electrostatic field is Es = 5 ×10−4, which is more intense and direction simplification.
1. Introduction At present, attosecond pulses can be generated in a number of ways, such as high-harmonic generation [1,2], undulator radiation [3], or Thomson scattering [4], and so on. High-harmonic generation is capable of producing single-cycle attosecond pulses [5–7]. But they have not yet been proven to control the pulse shape. This is also true of radiation sources based on relativistic electrons [8]. There are also single-cycle or multi-cycle attosecond pulses that are proposed to produce carrier-envelope waveform control by coherent undulators, but this technique requires very high energy [9–11]. In addition, there is also a way of generating of single-cycle attoseond pulses based on Thomson scattering of terahertz pulse [12]. And attosecond pulse generation in Thomson scattering with phase-controlled few-cycle laser pulse. However, this method can also produce a single-cycle attosecond pulse, it dose not produce a single-cycle pulse. And it is more difficult to achieve. In this paper, we propose a method of applying the new way of generating the single-cycle attoseond pulse by Gaussian beam acting with an electrons in the appropriate electrostatic. So we studied the effect of uniform electrostatic field on electron emission in the linear polarization. It is found that this method can obtain the attosecond pulse, and the uniform electrostatic field is more likely to produce a single attosecond pulse. The intensity of the attosecond pulse becomes stronger and radiation direction tends to be singular. Electron acceleration diagram shows in Fig. 1. It is assumed that an electron are a beam of Gaussian beams through a solid surface, the electrons run out to a solid, electrons in the presence of Gaussian beam, electrons are accelerated to generate attosecond pulse radiation. 2. Formulation We studied that an electron in the absence of the electrostatic field and the presence of a uniform electrostatic field is driven by ⁎
Corresponding author. E-mail address: [email protected] (G. Tong).
https://doi.org/10.1016/j.ijleo.2019.04.139 Received 24 April 2019; Received in revised form 29 April 2019; Accepted 30 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 1. Acceleration of Gaussian beam and an electron interaction in the uniform electrostatic field.
Gaussian beam acceleration [13,14]. As shown in Fig. 1. We assume that the initial position of the electron is at the origin (that is, the Gaussian beam waist, x = 0, y = 0, z = 0), Its initial speed is zero, and uniform electric field propagates along the positive direction of the y-axis, Gaussian beam along the positive direction of the x-axis [15,16]. We use the expression for Gaussian beam:
w r2 r2 u 0 (x , y, z ) = ⎧ 0 exp ⎡− 2 ⎤ ⎫ × exp ⎡− 2 ⎤ cos(ωt ) ⎢ ⎥ ⎢ ⎨ ⎬ w z w z w ( ) ( ) (z ) ⎥ ⎣ ⎦⎭ ⎣ ⎦ ⎩ λz ⎞ ⎤ ⎫ kr 2 ⎤ ⎧ × exp −i ⎡ × exp ⎡−i kz − arctan ⎜⎛ 2 ⎟⎥⎬ ⎢ ⎨ ⎢ R 2 (z ) ⎥ πw ⎣ ⎦ ⎝ 0 ⎠⎦⎭ ⎩ ⎣
(1) λz
where w0 is the Gaussian beam amplitude, λ is the wavelength, kz is described as the phase shift, arctan πw 2 described the additional phase of the phase shift relative to the spatial distance z-axis a Gaussian beam, and
kr 2
0
represents a phase shift relative to the
2R (z ) eE0 mg cω
direction. In the description of the intensity of the laser field, we use the dimensionless q =
to represent, and E0 is the intensity of
the laser field. According to the theory of quantum electrons absorb multiple photons and emit high frequency photos [17,18]. With the enhancement of the field strength, this nonlinear effect is improved and the energy is increased. The study of the interaction between lasers and electrons how to first consider a basic physical problem in laser energy conversion electrons is the field of single electron motion in electromagnetic fields [19,20]. The most important parameter in a strong laser is the intensity of the laser beam, 1 which is normalized to obtain a normalization factor. a0 = 0.856(Iλ2) 2 , where the unit of laser intensity is 1018 Wcm−2, the laser wavelength is λ = 1.0 μm, the expression of relativistic factors in the case of linear polarization and circular polarization respectively is γ = is:
1+
a02 2
and γ =
1 + a02 . Under the action of Lorentz force, the equation of motion of electrons in the electromagnetic field
→ → 1 dp = e ⎛E + → v × B⎞ dt c ⎝ ⎠
(2)
p = γm 0→ v , relativistic kinetic energy is T = (λ − 1)m0c2. In the electromagnetic field, since only the electric field has an where → effect on the electron energy increment, the relational expression can be described as: → dT d (γm 0 c 2) = −e (→ v ·E ) = dt dt
(3)
without loss of generality, on the basis of formula (3), we assume that t0, where the single electron is located in the middle of the waist of the Gaussian beam (x = 0, y = 0, z = 0) from the solid ionization, and from the t = t0 we begin to calculate, where our electron field and magnetic field expression is Ey = a · upl. Where a is the laser amplitude intensity. Ey = a · upl is the initial y-direction intensity of the laser beam. When the intensity of the electric field is added, the strength is Ey = Ey + E0 = a · dpl + Es, where Es is the uniform electric field propagating along the y-direction. The initial position of the electron is at (x = 0, y = 0, z = 0). We set the initial time is t0, and sum time ts is express as:
nx × → x −→ ny × → y −→ nz × → z ts = t0 + X − →
(4)
→ → → where n x , n y and n z are denoted as components in the x, y and z directions, respectively. X is electron's initial position. Our point of view is along the y-direction. The attosecond pulse will be generated from the nonlinear Thomson scattering of the electrons from the Gaussian beam. In the process, according to the x–y movement trajectories, we calculated the time t in the Gaussian beam in the electric area, according to (△t ) =
△y πa
, where we set the △y = 3, then it calculates △t = 13.82, it is in the context of this study. 264
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 2. Pulsed radiation images of different intensity Es electrostatic field. (a) electrostatic field Es = 0 pulse intensity time image; (b) electrostatic field Es = 5 × 10−6 pulse intensity time image; (c) electrostatic field Es = 5 ×10−5 pulse intensity time image; (d) electrostatic field Es = 5 ×10−4 pulse intensity time image. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
3. Numerical results and discussions It is proposed that an electron is accelerated and it produce radiation with Gaussian beam under Thomson scattering, according to dP(t ′) dΩ
e2 16π 2ε0 c
−(1 + β ) β + β˙ β˙
x y y x ⎤, where t′ ⎡ (1 + βx )3 ⎦ ⎣ o X − r (t ′) nˆ v ˆ β = n is the delay time of the electron motion t = t ′ + , the normalized electron velocity is , is the direction of observation. c c According to the above formula, we take the non-linear Thomson scattering of the Gaussian laser pulse with single electron scattering in the case of its linear polarization. This process so that the electrons move at the moment close to the speed of light, attosecond dP(t ) pulse is thus generated radiation. By dΩ = |A (t )|2 , Formula can be derived A(t) of the equation:
classical electrodynamics theory. The expression for angular distribution of radiation power is
A (t ) =
˙ e2 ⎡ nˆ × [(nˆ − β ) × β ] ⎤ ⎥ ˙ )3 16π 2ϵ 0 c ⎢ ˆ − nβ (1 ⎣ ⎦ret
=
(5)
It is as shown that the intensity of the NVP (Normalized Vector Potential) images produced by the Gaussian beam and single electrons under linear polarization in Fig. 2(a)–(d). The red line represents the attosecond pulse chain obtained in the y-axis direction. The blue line is an image of the pulse radiation in the x-axis direction. Gaussian beam parameters are intensity a = 5, pulse radius r0 = 5, waist width w0 = 3, continuous cycle time is 200 ms, P = 1 (linear polarization), we can obviously get from Fig. 2, no electrostatic field generation pulse, when the addition of different intensity of the electrostatic field, it will product different attosecond pulse. From Fig. 2(a)–(d), it can be obtained that four different intensity of the electrostatic field pulse width, from the pulse width, and did not find any big difference. It is concluded that when the electrostatic field is less than Es = 5 ×10−6, from Fig. 3. It can be obtained that the effect of the Gaussian beam on the electrons is almost small and does not change the radiation direction of Ay. When the electrostatic field Es = 5 ×10−5 and Es = 5 ×10−4, the radiation direction of Ay is changed, but the Guassian beam can be not used to produce a single attosecond pulse with a single electron. When the electrostatic field is larger than Es = 5 ×10−4, the Gaussian beam and single electron can change the radiation direction of Ay, and can produce a single attosecond pulse. From Fig. 2(e) about the pulse diagram of the relationship between strength and electrostatic field, electrostatic field can be obtained for Es = 5 ×10−4, pulse intensity is the strongest. Based on the above findings, we continue to study the causes of Ay radiation changes. According to A(t) of formula (5), we draw 265
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 3. Pulse strength linear curve of different intensity electrostatic field.
image of the numerator and denominator and velocity components. It is found that in Fig. 4 whether there is an electrostatic field, the value of the denominator is greater than zero, and molecular changes in positive and negative directions. From Fig. 5, the results show that both Es = 0 and Es = 5 ×10−4, molecular radiation curve changes consistent with the red line Ay radiation curves. And they are shown that numerator and determines the direction of Ay radiation changes, namely the denominator is negative changes affecting the Ay pulse curve. According to the formula (5) of A(t), v ˆ x, m ˆ y, m ˆ z ) , get nˆ = (nˆ x , nˆ y , 0) , analyzing speed β = (βˆx , βˆy , 0) , β = c ,where β is a normalized velocity. And by (nˆ − βˆ) × βˆ = (m nˆ × mz = (nˆ x − βˆ ) × β˙ − (nˆ y − βˆ ) × β˙ , mx = 0, my = 0, mz = 0, which is derived, x
y
y
x
ˆ y , −nˆ x m ˆ z , −nˆ y m ˆ x ) = nˆ y × [(nˆ x − βˆx ) × β˙ y − (nˆ y − βˆy ) × β˙x ] xˆ − nˆ x × [(nˆ x − βˆx ) × β˙x − (nˆ y − βˆy × β˙x )] yˆ . Among (nˆ x − βˆx ) × β˙ y = (nˆ y m
Fig. 4. The corresponding relationship to the NVP and with the electric field of Es = 5 ×10−4. 266
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G. Tong, et al.
Fig. 5. NVP corresponds to the numerator and denominator. (a) The electrostatic field intensity is Es = 5 × e−4. (b) The electrostatic field intensity is E0 = 0. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
them, according to period analysis of pulse in Fig. 4, we take the point where the maximum value that ts (formula (4)) is 4.2. Through calculating, the result is: nˆ x − βˆx = −0.0093, nˆ y − βˆy = −0.0483, β˙ y ≈ 1290 , βˆx ≈ −0.00036. So (nˆ x − βˆx ) × β˙ y ≈ −11.99, (nˆ y − βˆ ) × β˙ ≈ 1.76 × 10−5 , where (nˆ y − βˆ ) × β˙ is so a little, it can be ignored. We also took other points, the results of the analysis y
x
y
x
is the same. So we came to the conclusion that β˙ y determines the direction of the value of Ay. For the above situation, we continue to study the direction of Ay movement, from the time domain and space were analyzed. We know that this may be related to the radiation direction of the photon when Thomson scattering occurs, because the value of the x-direction is always positive, and the value of the y-direction is positive or negative. Where the different size Es radiation pattern is show below. In Fig. 6(a) and (b), radiation direction and angle are the same, the radiation of the observation point corresponds to 0 to π is nˆ = [0.9029 0.4298 0], and the angle with the x-axis radiation is θ = 0.436 rad, for π to 2π, nˆ = [0.8738 − 0.4862 0], its angle with xaxis uadiation is θ = 5.573 rad. Where the radiation values of 0 to π are found to be the largest, so this angle corresponds to the Ay radiation region of the enlarged area in Fig. 6. For Fig. 6(c), the radiation of the observation point corresponds to 0 to π is nˆ = [0.8738 0.4862 0], and the angle with the x-axis radiation is θ = 0.506 rad. From π to 2π, nˆ = [0.9029 − 4298 0], the maximum radiation direction of the radiation pulse generated in the process is changed from the first quadrant to fourth quadrant, and the size 267
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 6. The pulse radiation direction image with different size electrostatic field (a) Es = 0; (b) Es = 5 ×10−6; (c) Es = 5 ×10−5; (d) Es = 5 ×10−4.
of the radiation changes. By Thomson scattering the principle we can see, electronically generated radiation direction is usually symmetrical along the x-axis (that is, zero-degree angle), indicating that the radiation direction in x-axis radiation is θ = 5.826 rad. For Fig. 6, the radiation of the observation point corresponds to 0 to π is nˆ = [0.7961 0.6182 0], and the angle with the x-axis radiation is θ = 0.663 rad. For π to 2π, nˆ = [0.9284 − 0.3717 0], its angle with x-axis radiation is θ = 5.86 rad. It is found that when the magnitude of the size of the electrostatic field increases from Es = 5 ×10−6 Fig. 6(d) should also be two, for Es = 5 ×10−4. It can be seen from Fig. 2, the change in the radiation angle corresponds to the radiation direction of Ay in Fig. 2. In which the direction of the radiation angle in addition to the static field generated by changing, each angle does not vary greatly not due to the electrostatic field. In addition, the principle of Thomson scattering as known. The radiation direction of electrons is usually symmetrical along the xaxis(that is, the zero angle, the radiation direction in Fig. 6(d) should also be two. Therefore, it is shown the radiation pattern of 0 to π and π to 2π for Es = 5 ×10−4. It is found from Fig. 7 that the magnitude of the radiation in the upper half and the magnitude of the radiation in the lower half are order of magnitude 5 and 7 respectively. Thus, it is explained that the radiation pattern of the upper half is almost invisible in Fig. 7. Therefore when Es = 5 ×10−4, the Gaussian beam is more upper half is more inclined in one direction with the direction of the isolated electrons, another radiation direction is compared to the pulse radiation is very small. Therefore we conclude that the direction of the radiation generated by the Es = 5 ×10−4 is relatively simple. 4. Conclusion In this paper, when superimposing the field and electrons in the y-direction electrostatic field, theoretical analysis analysis and numerical calculation are carried out. The results show that the linearly polarized situation, the applied electrostatic field of appropriate size for easier regulation attosecond pulse is generated, it is easier to produce a single attosecond pulse, enhanced strength, and a single radiation direction. In addition, the larger the laser intensity for a0, the greater the intensity of the attoseond pulse generated by the laser. And the uniform static field to be adjusted is such that the attosecond pulse capable of generating a single cycle is proportional to the laser intensity a0. When the laser intensity reaches the order of negative power of four, a single-cycle attosecond pulse can be generated. When the magnitude of the negative fifth power is reached, the radiation Ay direction can be changed. On the order of the negative power of six, the Ay radiation direction cannot be changed. The magnitude of the negative fifth 268
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 7. Es = 5 ×10−4 of the upper and lower half of the radiation pattern.
power and the negative sixth power cannot produce a single attosecond pulse, and the radiation direction is in two directions. Finally, when the electrostatic field strength is increased to 5 × 10−4, a single attosecond pulse with a pulse width of 16 as is generated, and the electrostatic field intensity has little effect on the width of each of the attosecond pulse in the pulse chain. Compared to a normal Gaussian beam, the attosecond pulse width obtained by the superposition is smaller. And it can produce highperformance single attosecond pulse, which has the advantages of simple structure and easy implementation. References [1] F. Bornath, R. Reinholz, Röpke, X-ray Thomson scattering cross-section in strongly correlated plasmas, Laser Particle Beams 27 (2) (2009) 311–319. [2] K. Kovács, V. Toşa, B. Major, E. Balogh, K. Varjú, High-efficiency single attosecond pulse generation with a long-wavelength pulse assisted by a weak nearinfrared pulse, IEEE J. Sel. Topics Quantum Electron. 21 (5) (2015) 1–7. [3] D.B. Milo[sbreve]ević, W. Becker, Attosecond pulse generation by bicircular fields: from pulse trains to a single pulse, J. Modern Optics 52 (2–3) (2005) 233–241. [4] B. Bódi, E. Balogh, V. Tosa, E. Goulielmakis, K. Varjú, P. Dombi, Attosecond pulse generation with an optimization loop in a light-field-synthesizer, Optics Express 24 (19) (2016) 21957. [5] S. Xiao, A. Hu, C. Hui, X. Han, T. Wang, Q. Zang, J. Zhao, Analysis of baffles for stray light reduction in the Thomson scattering diagnostic on east, Fusion Eng. Des. 105 (2016) 33–38. [6] J. Lu, K. Hill, M. Bitter, N. Pablant, L. Delgado-Aparicio, P. Efthimion, H. Lee, U. Zastrau, Characterization of X-ray imaging crystal spectrometer for highresolution spatially-resolved X-ray Thomson scattering measurements in shock-compressed experiments, J. Quant. Spectrosc. Radiat. Transf. 187 (2017) 247–254. [7] A.T. Amatuni, Nonlinear Thomson scattering of two electromagnetic waves on the relativistic electron, Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 455 (1) (2000) 60–62. [8] G. Tóth, Z. Tibai, A. Sharma, J.A. Fülöp, J. Hebling, Single-cycle attosecond pulse generation by thomson scattering of terahertz pulses, CLEO: Science and Innovations, Optical Society of America, 2018 pJTh2A-190. [9] L. Feng, H. Liu, Generation of the ultrabroad bandwidth with kev by three-color low intense mid-infrared inhomogeneous pulse, Optics Laser Technol. 81 (2016) 7–13. [10] C. Fortmann, A. Wierling, G. Röpke, Influence of local-field corrections on Thomson scattering in collision-dominated two-component plasmas, Phys. Rev. E 81 (2) (2010) 026405. [11] S.K. Nielsen, H. Bindslev, L. Porte, J. Hoekzema, S.B. Korsholm, F. Leipold, F. Meo, P. Michelsen, S. Michelsen, J. Oosterbeek, et al., Temporal evolution of confined fast-ion velocity distributions measured by collective Thomson scattering in textor, Phys. Rev. E 77 (1) (2008) 016407. [12] G. Tóth, Z. Tibai, A. Sharma, J.A. Fülöp, J. Hebling, Generation of single-cycle attosecond pulses based on Thomson scattering of terahertz pulses, Mid-Infrared Coherent Sources, Optical Society of America, 2018, p. JT5A-3. [13] B. Bódi, E. Balogh, V. Tosa, E. Goulielmakis, K. Varjú, P. Dombi, Genetic optimization of attosecond-pulse generation in light-field synthesizers, Phys. Rev. A 90 (2) (2014) 1185–1188. [14] E. Balogh, K. Kovacs, P. Dombi, J.A. Fulop, G. Farkas, J. Hebling, V. Tosa, K. Varju, Single attosecond pulse from terahertz-assisted high-order harmonic generation, Lasers & Electro-optics Europe, (2011). [15] T. Hammond, E. Balogh, K. Taec Kim, Isolation of attosecond pulses from the attosecond lighthouse, J. Phys. B At. Mol. Phys. 50 (1) (2017) 014006. [16] D. Kroon, D. Guénot, M. Kotur, E. Balogh, E.W. Larsen, C.M. Heyl, M. Miranda, M. Gisselbrecht, J. Mauritsson, P. Johnsson, Attosecond pulse walk-off in highorder harmonic generation, Optics Lett. 39 (7) (2014) 2218–2221. [17] I. Yamada, K. Narihara, H. Funaba, H. Hayashi, T. Kohmoto, H. Takahashi, T. Shimozuma, S. Kubo, Y. Yoshimura, H. Igami, Improvements of data quality of the LHD Thomson scattering diagnostics in high-temperature plasma experiments, Rev. Sci. Instrum. 81 (10) (2010) 4607. [18] H. Du, L. Luo, X. Wang, B. Hu, Attosecond ionization control for broadband supercontinuum generation using a weak 400-nm few-cycle controlling pulse, Optics Express 20 (24) (2012) 27226–27241. [19] H. Tojo, A. Ejiri, J. Hiratsuka, T. Yamaguchi, Y. Takase, K. Itami, T. Hatae, First measurement of electron temperature from signal ratios in a double-pass Thomson scattering system, Rev. Sci. Instrum. 83 (2) (2012) 023507. [20] G.T. Zhang, Isolated ultrashort attosecond pulse generation from a coherent superposition state in a few-cycle chirped laser pulse, Mol. Phys. 111 (20) (2013) 3117–3125.
269
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Generation of single cycle attosecond pulse via Thomson scattering of Gaussian beam interacting with a single electron in the electrostatic field environment
T
⁎
Guodong Tonga, , Meng Zhanga, Changquan Xiab, Qianru Zhanga, Guoqing Lia a b
National ASIC System Engineering Technology Research Center, Southeast University, Nanjing, China Department of Physical Science and Technology, Yangzhou University, Yangzhou, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Attosecond pulse Gaussian beam Thomson scattering Single electron
The new way of generating the attosecond pulses via Thomson scattering of linear polarization Gaussian beam propagating on solid surface is investigated. It is found that a single attosecond laser pulse is generated which renders the appropriate strength electrostatic field to the environment. After studying the influence of the strength of the electrostatic field on it, the results show that a single attosecond pulse is generated when intensity of Gaussian beam is 5 and intensity of the electrostatic field is Es = 5 ×10−4, which is more intense and direction simplification.
1. Introduction At present, attosecond pulses can be generated in a number of ways, such as high-harmonic generation [1,2], undulator radiation [3], or Thomson scattering [4], and so on. High-harmonic generation is capable of producing single-cycle attosecond pulses [5–7]. But they have not yet been proven to control the pulse shape. This is also true of radiation sources based on relativistic electrons [8]. There are also single-cycle or multi-cycle attosecond pulses that are proposed to produce carrier-envelope waveform control by coherent undulators, but this technique requires very high energy [9–11]. In addition, there is also a way of generating of single-cycle attoseond pulses based on Thomson scattering of terahertz pulse [12]. And attosecond pulse generation in Thomson scattering with phase-controlled few-cycle laser pulse. However, this method can also produce a single-cycle attosecond pulse, it dose not produce a single-cycle pulse. And it is more difficult to achieve. In this paper, we propose a method of applying the new way of generating the single-cycle attoseond pulse by Gaussian beam acting with an electrons in the appropriate electrostatic. So we studied the effect of uniform electrostatic field on electron emission in the linear polarization. It is found that this method can obtain the attosecond pulse, and the uniform electrostatic field is more likely to produce a single attosecond pulse. The intensity of the attosecond pulse becomes stronger and radiation direction tends to be singular. Electron acceleration diagram shows in Fig. 1. It is assumed that an electron are a beam of Gaussian beams through a solid surface, the electrons run out to a solid, electrons in the presence of Gaussian beam, electrons are accelerated to generate attosecond pulse radiation. 2. Formulation We studied that an electron in the absence of the electrostatic field and the presence of a uniform electrostatic field is driven by ⁎
Corresponding author. E-mail address: [email protected] (G. Tong).
https://doi.org/10.1016/j.ijleo.2019.04.139 Received 24 April 2019; Received in revised form 29 April 2019; Accepted 30 April 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 1. Acceleration of Gaussian beam and an electron interaction in the uniform electrostatic field.
Gaussian beam acceleration [13,14]. As shown in Fig. 1. We assume that the initial position of the electron is at the origin (that is, the Gaussian beam waist, x = 0, y = 0, z = 0), Its initial speed is zero, and uniform electric field propagates along the positive direction of the y-axis, Gaussian beam along the positive direction of the x-axis [15,16]. We use the expression for Gaussian beam:
w r2 r2 u 0 (x , y, z ) = ⎧ 0 exp ⎡− 2 ⎤ ⎫ × exp ⎡− 2 ⎤ cos(ωt ) ⎢ ⎥ ⎢ ⎨ ⎬ w z w z w ( ) ( ) (z ) ⎥ ⎣ ⎦⎭ ⎣ ⎦ ⎩ λz ⎞ ⎤ ⎫ kr 2 ⎤ ⎧ × exp −i ⎡ × exp ⎡−i kz − arctan ⎜⎛ 2 ⎟⎥⎬ ⎢ ⎨ ⎢ R 2 (z ) ⎥ πw ⎣ ⎦ ⎝ 0 ⎠⎦⎭ ⎩ ⎣
(1) λz
where w0 is the Gaussian beam amplitude, λ is the wavelength, kz is described as the phase shift, arctan πw 2 described the additional phase of the phase shift relative to the spatial distance z-axis a Gaussian beam, and
kr 2
0
represents a phase shift relative to the
2R (z ) eE0 mg cω
direction. In the description of the intensity of the laser field, we use the dimensionless q =
to represent, and E0 is the intensity of
the laser field. According to the theory of quantum electrons absorb multiple photons and emit high frequency photos [17,18]. With the enhancement of the field strength, this nonlinear effect is improved and the energy is increased. The study of the interaction between lasers and electrons how to first consider a basic physical problem in laser energy conversion electrons is the field of single electron motion in electromagnetic fields [19,20]. The most important parameter in a strong laser is the intensity of the laser beam, 1 which is normalized to obtain a normalization factor. a0 = 0.856(Iλ2) 2 , where the unit of laser intensity is 1018 Wcm−2, the laser wavelength is λ = 1.0 μm, the expression of relativistic factors in the case of linear polarization and circular polarization respectively is γ = is:
1+
a02 2
and γ =
1 + a02 . Under the action of Lorentz force, the equation of motion of electrons in the electromagnetic field
→ → 1 dp = e ⎛E + → v × B⎞ dt c ⎝ ⎠
(2)
p = γm 0→ v , relativistic kinetic energy is T = (λ − 1)m0c2. In the electromagnetic field, since only the electric field has an where → effect on the electron energy increment, the relational expression can be described as: → dT d (γm 0 c 2) = −e (→ v ·E ) = dt dt
(3)
without loss of generality, on the basis of formula (3), we assume that t0, where the single electron is located in the middle of the waist of the Gaussian beam (x = 0, y = 0, z = 0) from the solid ionization, and from the t = t0 we begin to calculate, where our electron field and magnetic field expression is Ey = a · upl. Where a is the laser amplitude intensity. Ey = a · upl is the initial y-direction intensity of the laser beam. When the intensity of the electric field is added, the strength is Ey = Ey + E0 = a · dpl + Es, where Es is the uniform electric field propagating along the y-direction. The initial position of the electron is at (x = 0, y = 0, z = 0). We set the initial time is t0, and sum time ts is express as:
nx × → x −→ ny × → y −→ nz × → z ts = t0 + X − →
(4)
→ → → where n x , n y and n z are denoted as components in the x, y and z directions, respectively. X is electron's initial position. Our point of view is along the y-direction. The attosecond pulse will be generated from the nonlinear Thomson scattering of the electrons from the Gaussian beam. In the process, according to the x–y movement trajectories, we calculated the time t in the Gaussian beam in the electric area, according to (△t ) =
△y πa
, where we set the △y = 3, then it calculates △t = 13.82, it is in the context of this study. 264
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
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Fig. 2. Pulsed radiation images of different intensity Es electrostatic field. (a) electrostatic field Es = 0 pulse intensity time image; (b) electrostatic field Es = 5 × 10−6 pulse intensity time image; (c) electrostatic field Es = 5 ×10−5 pulse intensity time image; (d) electrostatic field Es = 5 ×10−4 pulse intensity time image. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
3. Numerical results and discussions It is proposed that an electron is accelerated and it produce radiation with Gaussian beam under Thomson scattering, according to dP(t ′) dΩ
e2 16π 2ε0 c
−(1 + β ) β + β˙ β˙
x y y x ⎤, where t′ ⎡ (1 + βx )3 ⎦ ⎣ o X − r (t ′) nˆ v ˆ β = n is the delay time of the electron motion t = t ′ + , the normalized electron velocity is , is the direction of observation. c c According to the above formula, we take the non-linear Thomson scattering of the Gaussian laser pulse with single electron scattering in the case of its linear polarization. This process so that the electrons move at the moment close to the speed of light, attosecond dP(t ) pulse is thus generated radiation. By dΩ = |A (t )|2 , Formula can be derived A(t) of the equation:
classical electrodynamics theory. The expression for angular distribution of radiation power is
A (t ) =
˙ e2 ⎡ nˆ × [(nˆ − β ) × β ] ⎤ ⎥ ˙ )3 16π 2ϵ 0 c ⎢ ˆ − nβ (1 ⎣ ⎦ret
=
(5)
It is as shown that the intensity of the NVP (Normalized Vector Potential) images produced by the Gaussian beam and single electrons under linear polarization in Fig. 2(a)–(d). The red line represents the attosecond pulse chain obtained in the y-axis direction. The blue line is an image of the pulse radiation in the x-axis direction. Gaussian beam parameters are intensity a = 5, pulse radius r0 = 5, waist width w0 = 3, continuous cycle time is 200 ms, P = 1 (linear polarization), we can obviously get from Fig. 2, no electrostatic field generation pulse, when the addition of different intensity of the electrostatic field, it will product different attosecond pulse. From Fig. 2(a)–(d), it can be obtained that four different intensity of the electrostatic field pulse width, from the pulse width, and did not find any big difference. It is concluded that when the electrostatic field is less than Es = 5 ×10−6, from Fig. 3. It can be obtained that the effect of the Gaussian beam on the electrons is almost small and does not change the radiation direction of Ay. When the electrostatic field Es = 5 ×10−5 and Es = 5 ×10−4, the radiation direction of Ay is changed, but the Guassian beam can be not used to produce a single attosecond pulse with a single electron. When the electrostatic field is larger than Es = 5 ×10−4, the Gaussian beam and single electron can change the radiation direction of Ay, and can produce a single attosecond pulse. From Fig. 2(e) about the pulse diagram of the relationship between strength and electrostatic field, electrostatic field can be obtained for Es = 5 ×10−4, pulse intensity is the strongest. Based on the above findings, we continue to study the causes of Ay radiation changes. According to A(t) of formula (5), we draw 265
Optik - International Journal for Light and Electron Optics 188 (2019) 263–269
G. Tong, et al.
Fig. 3. Pulse strength linear curve of different intensity electrostatic field.
image of the numerator and denominator and velocity components. It is found that in Fig. 4 whether there is an electrostatic field, the value of the denominator is greater than zero, and molecular changes in positive and negative directions. From Fig. 5, the results show that both Es = 0 and Es = 5 ×10−4, molecular radiation curve changes consistent with the red line Ay radiation curves. And they are shown that numerator and determines the direction of Ay radiation changes, namely the denominator is negative changes affecting the Ay pulse curve. According to the formula (5) of A(t), v ˆ x, m ˆ y, m ˆ z ) , get nˆ = (nˆ x , nˆ y , 0) , analyzing speed β = (βˆx , βˆy , 0) , β = c ,where β is a normalized velocity. And by (nˆ − βˆ) × βˆ = (m nˆ × mz = (nˆ x − βˆ ) × β˙ − (nˆ y − βˆ ) × β˙ , mx = 0, my = 0, mz = 0, which is derived, x
y
y
x
ˆ y , −nˆ x m ˆ z , −nˆ y m ˆ x ) = nˆ y × [(nˆ x − βˆx ) × β˙ y − (nˆ y − βˆy ) × β˙x ] xˆ − nˆ x × [(nˆ x − βˆx ) × β˙x − (nˆ y − βˆy × β˙x )] yˆ . Among (nˆ x − βˆx ) × β˙ y = (nˆ y m
Fig. 4. The corresponding relationship to the NVP and with the electric field of Es = 5 ×10−4. 266
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Fig. 5. NVP corresponds to the numerator and denominator. (a) The electrostatic field intensity is Es = 5 × e−4. (b) The electrostatic field intensity is E0 = 0. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
them, according to period analysis of pulse in Fig. 4, we take the point where the maximum value that ts (formula (4)) is 4.2. Through calculating, the result is: nˆ x − βˆx = −0.0093, nˆ y − βˆy = −0.0483, β˙ y ≈ 1290 , βˆx ≈ −0.00036. So (nˆ x − βˆx ) × β˙ y ≈ −11.99, (nˆ y − βˆ ) × β˙ ≈ 1.76 × 10−5 , where (nˆ y − βˆ ) × β˙ is so a little, it can be ignored. We also took other points, the results of the analysis y
x
y
x
is the same. So we came to the conclusion that β˙ y determines the direction of the value of Ay. For the above situation, we continue to study the direction of Ay movement, from the time domain and space were analyzed. We know that this may be related to the radiation direction of the photon when Thomson scattering occurs, because the value of the x-direction is always positive, and the value of the y-direction is positive or negative. Where the different size Es radiation pattern is show below. In Fig. 6(a) and (b), radiation direction and angle are the same, the radiation of the observation point corresponds to 0 to π is nˆ = [0.9029 0.4298 0], and the angle with the x-axis radiation is θ = 0.436 rad, for π to 2π, nˆ = [0.8738 − 0.4862 0], its angle with xaxis uadiation is θ = 5.573 rad. Where the radiation values of 0 to π are found to be the largest, so this angle corresponds to the Ay radiation region of the enlarged area in Fig. 6. For Fig. 6(c), the radiation of the observation point corresponds to 0 to π is nˆ = [0.8738 0.4862 0], and the angle with the x-axis radiation is θ = 0.506 rad. From π to 2π, nˆ = [0.9029 − 4298 0], the maximum radiation direction of the radiation pulse generated in the process is changed from the first quadrant to fourth quadrant, and the size 267
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Fig. 6. The pulse radiation direction image with different size electrostatic field (a) Es = 0; (b) Es = 5 ×10−6; (c) Es = 5 ×10−5; (d) Es = 5 ×10−4.
of the radiation changes. By Thomson scattering the principle we can see, electronically generated radiation direction is usually symmetrical along the x-axis (that is, zero-degree angle), indicating that the radiation direction in x-axis radiation is θ = 5.826 rad. For Fig. 6, the radiation of the observation point corresponds to 0 to π is nˆ = [0.7961 0.6182 0], and the angle with the x-axis radiation is θ = 0.663 rad. For π to 2π, nˆ = [0.9284 − 0.3717 0], its angle with x-axis radiation is θ = 5.86 rad. It is found that when the magnitude of the size of the electrostatic field increases from Es = 5 ×10−6 Fig. 6(d) should also be two, for Es = 5 ×10−4. It can be seen from Fig. 2, the change in the radiation angle corresponds to the radiation direction of Ay in Fig. 2. In which the direction of the radiation angle in addition to the static field generated by changing, each angle does not vary greatly not due to the electrostatic field. In addition, the principle of Thomson scattering as known. The radiation direction of electrons is usually symmetrical along the xaxis(that is, the zero angle, the radiation direction in Fig. 6(d) should also be two. Therefore, it is shown the radiation pattern of 0 to π and π to 2π for Es = 5 ×10−4. It is found from Fig. 7 that the magnitude of the radiation in the upper half and the magnitude of the radiation in the lower half are order of magnitude 5 and 7 respectively. Thus, it is explained that the radiation pattern of the upper half is almost invisible in Fig. 7. Therefore when Es = 5 ×10−4, the Gaussian beam is more upper half is more inclined in one direction with the direction of the isolated electrons, another radiation direction is compared to the pulse radiation is very small. Therefore we conclude that the direction of the radiation generated by the Es = 5 ×10−4 is relatively simple. 4. Conclusion In this paper, when superimposing the field and electrons in the y-direction electrostatic field, theoretical analysis analysis and numerical calculation are carried out. The results show that the linearly polarized situation, the applied electrostatic field of appropriate size for easier regulation attosecond pulse is generated, it is easier to produce a single attosecond pulse, enhanced strength, and a single radiation direction. In addition, the larger the laser intensity for a0, the greater the intensity of the attoseond pulse generated by the laser. And the uniform static field to be adjusted is such that the attosecond pulse capable of generating a single cycle is proportional to the laser intensity a0. When the laser intensity reaches the order of negative power of four, a single-cycle attosecond pulse can be generated. When the magnitude of the negative fifth power is reached, the radiation Ay direction can be changed. On the order of the negative power of six, the Ay radiation direction cannot be changed. The magnitude of the negative fifth 268
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Fig. 7. Es = 5 ×10−4 of the upper and lower half of the radiation pattern.
power and the negative sixth power cannot produce a single attosecond pulse, and the radiation direction is in two directions. Finally, when the electrostatic field strength is increased to 5 × 10−4, a single attosecond pulse with a pulse width of 16 as is generated, and the electrostatic field intensity has little effect on the width of each of the attosecond pulse in the pulse chain. Compared to a normal Gaussian beam, the attosecond pulse width obtained by the superposition is smaller. And it can produce highperformance single attosecond pulse, which has the advantages of simple structure and easy implementation. References [1] F. Bornath, R. Reinholz, Röpke, X-ray Thomson scattering cross-section in strongly correlated plasmas, Laser Particle Beams 27 (2) (2009) 311–319. [2] K. Kovács, V. Toşa, B. Major, E. Balogh, K. Varjú, High-efficiency single attosecond pulse generation with a long-wavelength pulse assisted by a weak nearinfrared pulse, IEEE J. Sel. Topics Quantum Electron. 21 (5) (2015) 1–7. [3] D.B. Milo[sbreve]ević, W. Becker, Attosecond pulse generation by bicircular fields: from pulse trains to a single pulse, J. Modern Optics 52 (2–3) (2005) 233–241. [4] B. Bódi, E. Balogh, V. Tosa, E. Goulielmakis, K. Varjú, P. Dombi, Attosecond pulse generation with an optimization loop in a light-field-synthesizer, Optics Express 24 (19) (2016) 21957. [5] S. Xiao, A. Hu, C. Hui, X. Han, T. Wang, Q. Zang, J. Zhao, Analysis of baffles for stray light reduction in the Thomson scattering diagnostic on east, Fusion Eng. Des. 105 (2016) 33–38. [6] J. Lu, K. Hill, M. Bitter, N. Pablant, L. Delgado-Aparicio, P. Efthimion, H. Lee, U. Zastrau, Characterization of X-ray imaging crystal spectrometer for highresolution spatially-resolved X-ray Thomson scattering measurements in shock-compressed experiments, J. Quant. Spectrosc. Radiat. Transf. 187 (2017) 247–254. [7] A.T. Amatuni, Nonlinear Thomson scattering of two electromagnetic waves on the relativistic electron, Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 455 (1) (2000) 60–62. [8] G. Tóth, Z. Tibai, A. Sharma, J.A. Fülöp, J. Hebling, Single-cycle attosecond pulse generation by thomson scattering of terahertz pulses, CLEO: Science and Innovations, Optical Society of America, 2018 pJTh2A-190. [9] L. Feng, H. Liu, Generation of the ultrabroad bandwidth with kev by three-color low intense mid-infrared inhomogeneous pulse, Optics Laser Technol. 81 (2016) 7–13. [10] C. Fortmann, A. Wierling, G. Röpke, Influence of local-field corrections on Thomson scattering in collision-dominated two-component plasmas, Phys. Rev. E 81 (2) (2010) 026405. [11] S.K. Nielsen, H. Bindslev, L. Porte, J. Hoekzema, S.B. Korsholm, F. Leipold, F. Meo, P. Michelsen, S. Michelsen, J. Oosterbeek, et al., Temporal evolution of confined fast-ion velocity distributions measured by collective Thomson scattering in textor, Phys. Rev. E 77 (1) (2008) 016407. [12] G. Tóth, Z. Tibai, A. Sharma, J.A. Fülöp, J. Hebling, Generation of single-cycle attosecond pulses based on Thomson scattering of terahertz pulses, Mid-Infrared Coherent Sources, Optical Society of America, 2018, p. JT5A-3. [13] B. Bódi, E. Balogh, V. Tosa, E. Goulielmakis, K. Varjú, P. Dombi, Genetic optimization of attosecond-pulse generation in light-field synthesizers, Phys. Rev. A 90 (2) (2014) 1185–1188. [14] E. Balogh, K. Kovacs, P. Dombi, J.A. Fulop, G. Farkas, J. Hebling, V. Tosa, K. Varju, Single attosecond pulse from terahertz-assisted high-order harmonic generation, Lasers & Electro-optics Europe, (2011). [15] T. Hammond, E. Balogh, K. Taec Kim, Isolation of attosecond pulses from the attosecond lighthouse, J. Phys. B At. Mol. Phys. 50 (1) (2017) 014006. [16] D. Kroon, D. Guénot, M. Kotur, E. Balogh, E.W. Larsen, C.M. Heyl, M. Miranda, M. Gisselbrecht, J. Mauritsson, P. Johnsson, Attosecond pulse walk-off in highorder harmonic generation, Optics Lett. 39 (7) (2014) 2218–2221. [17] I. Yamada, K. Narihara, H. Funaba, H. Hayashi, T. Kohmoto, H. Takahashi, T. Shimozuma, S. Kubo, Y. Yoshimura, H. Igami, Improvements of data quality of the LHD Thomson scattering diagnostics in high-temperature plasma experiments, Rev. Sci. Instrum. 81 (10) (2010) 4607. [18] H. Du, L. Luo, X. Wang, B. Hu, Attosecond ionization control for broadband supercontinuum generation using a weak 400-nm few-cycle controlling pulse, Optics Express 20 (24) (2012) 27226–27241. [19] H. Tojo, A. Ejiri, J. Hiratsuka, T. Yamaguchi, Y. Takase, K. Itami, T. Hatae, First measurement of electron temperature from signal ratios in a double-pass Thomson scattering system, Rev. Sci. Instrum. 83 (2) (2012) 023507. [20] G.T. Zhang, Isolated ultrashort attosecond pulse generation from a coherent superposition state in a few-cycle chirped laser pulse, Mol. Phys. 111 (20) (2013) 3117–3125.
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