Conference Proceedings 12th International

Conference Proceedings 12th International Conference on Laser and Fiber-Optical Networks Modeling LFNM’ 2013

Sudak, Ukraine 11-13 September 2013

Organized and sponsored by IEEE Photonics Society Chapter Ukraine IEEE AP/MTT/ED/AES/GRS/NPS/EMB East Ukraine Joint Chapter University of Guanajuato V. N. Karazin National University Taurida National V. I. Vernadsky University Kharkiv National University of Radio Electronics

in cooperation with IEEE AP/MTT/ED/AES/GRS/NPS/EMB East Ukraine Joint Chapter Institute of Physics, National Academy of Sciences of Ukraine IEEE Photonics Society Ukraine Student Chapter Kharkov Nat. Univ. of Radio Electronics SPIE Student Chapter Kharkov Nat. Univ. of Radio Electronics OSA Student Chapter

Editors: Shulika O.V., Sukhoivanov I. A.

2013 International Conference on Laser and Fiber-Optical Networks Modeling LFNM 2013

IEEE Catalog Number:

CFP13502-CDR

ISBN:

978-1-4799-0158-6

Copyright and Reprint Permission: Abstracting is permitted with credit to the source. Libraries are permitted to photocopy beyond the limit of U.S. copyright law for private use of patrons those articles in this volume that carry a code at the bottom of the first page, provided the per-copy fee indicated in the code is paid through Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923. For other copying, reprint or republication permission, write to IEEE Copyrights Manager, IEEE Operations Center, 445 Hoes Lane, Piscataway, NJ 08854. All rights reserved. Copyright ©2013 by IEEE

LFNM*2013 program committee Vasiliy A. Svich I. A. Sukhoivanov W. Freude L. A. Ageev M. O. Azarenkov G. Belenky P. Chamorro-Posada N. Dogru S. F. Dyubko M. I. Dzubenko N. N. Elkin V. I. Grygoruk S. O. Iakushev V. A. Katrich V. K. Kononenko L. N. Lytvynenko M. Marciniak V. A. Maslov V. K. Miloslavsky A. I. Nosich V. E. Privalov Yu. S. Shmaly S. N. Shulga M. O. Szymanski R. Vlokh V. G. Volostnikov V. M. Yakovenko I. I. Zalyubovsky

Conference Co-Chair, V. N. Karazin National University, Kharkiv, Ukraine Conference Co-Chair, Conference PC Co-Chair, University of Guanajuato, Mexico Conference PC Co-Chair, University of Karlsruhe, Germany V. N. Karazin National University, Kharkiv, Ukraine V. N. Karazin National University, Kharkiv, Ukraine State University of New York, USA Universidad de Valladolid, Valladolid, Spain University of Gaziantep, Gaziantep, Turkey V. N. Karazin National University, Kharkiv, Ukraine Institute of Radiophysics and Electronics of National Academy of Sciences of Ukraine, Kharkiv, Ukraine State R&D Center TRINITI, Troitsk, Russia Taras Shevchenko National University of Kyiv, Kyiv, Ukraine National University of Radio Electronics, Kharkiv, Ukraine V. N. Karazin National University, Kharkiv, Ukraine Institute of Physics, Minsk, Belarus Institute of Radio Astronomy, National Academy of Sciences of Ukraine, Kharkiv, Ukraine National Institute of Telecommunications, Poland V. N. Karazin National University, Kharkiv, Ukraine V. N. Karazin National University, Kharkiv, Ukraine Institute Radiophys and Electronics of National Academy of Sciences of Ukraine, Kharkiv, Ukraine BSTU, St. Petersburg, Russia University of Guanajuato, Mexico V. N. Karazin National University, Kharkiv, Ukraine Institute of Electron Technology, Warsaw, Poland Institute of Physical Optics, Ukraine Samara branch of P. N. Lebedev Physical Institure of RAS Usikov Institute of Radiophysics and Electronics, National Academy of Sciences of Ukraine, Kharkov, Ukraine V. N. Karazin National University, Kharkiv, Ukraine

LFNM*2013 organizing committee I. A. Sukhoivanov V. A. Maslov I. V. Dzedolik O. V. Shulika A. V. Kublik A. V. Degtyarev O. V. Gurin V. I. Fesenko T. B. Gryschenko S. G. Gryschenko I. V. Guryev S. O. Iakushev M. V. Klymenko A. Levchenko T. F. Ruban A. N. Topkov A. V. Vasyanovich V. Boiko V. Gryaznova I. Khomenko M. Pikula A. Shevchenko C. Shimko

Organization Chair, University of Guanajuato, Mexico Co-chair, Local Management, V. N. Karazin National University, Kharkiv, Ukraine Local Chair, Taurida National V. I. Vernadsky University, Simferopol, Crimea, Ukraine Coordinator, Publication Chair, Web-maintenance, University of Guanajuato, Mexico Secretary Local Organization, V. N. Karazin National University, Kharkiv, Ukraine Local Organization, V. N. Karazin National University, Kharkiv, Ukraine Local Organization, Institute of Radio Astronomy of National Academy of Sciences of Ukraine, Kharkiv, Ukraine Social Program, National University of Radio Electronics, Kharkiv, Ukraine Local Organization, National University of Radio Electronics, Kharkiv, Ukraine Web- maintenance, University of Guanajuato, Mexico Local Organization, National University of Radio Electronics, Kharkiv, Ukraine Local Organization, University of Liege, Liege, Belgium Local Organization, V. N. Karazin National University, Kharkiv arkov, Ukraine Local Organization, V. N. Karazin National University, Kharkiv, Ukraine Local Organization, V. N. Karazin National University, Kharkiv, Ukraine Local Organization, National University of Radio Electronics, Kharkiv, Ukraine Local Organization, Taurida National V.I.Vernadsky University, Simferopol, Ukraine Local Organization, Taurida National V.I.Vernadsky University, Simferopol, Ukraine Local Organization, Taurida National V.I.Vernadsky University, Simferopol, Ukraine Local Organization, Taurida National V.I.Vernadsky University, Simferopol, Ukraine Local Organization, Taurida National V.I.Vernadsky University, Simferopol, Ukraine Local Organization, Taurida National V.I.Vernadsky University, Simferopol, Ukraine

I

Contents (Invited) Designing optical filters with electrical models of photonic crystal defects R. K. Shevgaonkar, Preeti B. Patil .........................................................................................................................................................................1 (Invited) Influence of the moderate-strong pulsed laser field at the quantum electrodynamics processes S. P. Roshchupkin ....................................................................................................................................................................................................4 Role of periodicity in the scattering by a cloud of randomly located plasmonic nanowires D. M. Natarov, M. Marciniak, R. Sauleau ..............................................................................................................................................................10 Population dynamics in GaN/AlGaN quantum cascade structure under femtosecond pumping S. V. Gryshchenko, V. V. Lysak ..............................................................................................................................................................................13 Pair production to the lowest Landau levels by an electron in a magnetic field O. Novak ....................................................................................................................................................................................................................15 Class D lasers vs. class B lasers: Dynamical spectra analysis N. S. Ginzburg, E. R. Kocharovskaya, A. S. Sergeev.............................................................................................................................................17 Dumping effect on the forming of negative dielectric properties in crystal S.G. Felinskyi, P. A. Korotkov, G. S. Felinskyi ......................................................................................................................................................20 Interference of counter-propagating electromagnetic pulses on a dielectric slab B. A. Kochetov ...........................................................................................................................................................................................................23 Nonrelativistic electron scattering on a nucleus in the field of a bichromatic laser pulse A. A. Lebed', S. P. Roshchupkin .............................................................................................................................................................................26 Modeling of correctional amendments for pulse laser distance meter S. V. Tyurin, V. S. Tyurin, I. S. Morozov ................................................................................................................................................................29 One-photon emission of electron in the field of two pulsed laser waves A. I. Voroshilo, S. P. Roshchupkin, V.N.Nedoreshta ............................................................................................................................................32 Gain coefficient in the course of the electron scattering by ions in a weak electromagnetic field: general relativistic case V. A. Tsybul'nik, S.P. Roshchupkin .......................................................................................................................................................................35 Accurate calculation of hole bound states in quantum wells with polynomial potential profile R.F. Polupanov, S.N. Evdochenko .........................................................................................................................................................................38 Formation of an electron-positron pair by a photon in the field of two pulsed laser waves O. B. Lysenko, S. P. Roshchupkin, A.I.Voroshilo ..................................................................................................................................................41 Fiber attenuation irregularities and simulation of Raman amplification band M.Y. Dyriv, G.S. Felinskyi, P.A. Korotkov .............................................................................................................................................................44 Laser-modified compton scattering in the middle-intensity pulsed field V.N. Nedoreshta, S.P. Roshchupkin, A.I. Voroshilo .............................................................................................................................................47 Interference electron-muon scattering in two-mode pulse-wave laser field O. Denisenko, S. Roshchupkin ................................................................................................................................................................................50 Energy loss of a charged particle in an electron gas in the second Born approximation M. Diachenko ............................................................................................................................................................................................................53 Calculation of the electron thermal plasma permittivity O.V.Khelemelia .........................................................................................................................................................................................................55 Resonance interference scattering of a lepton by a lepton in the bichromatic pulsed laser field E. A. Padusenko, S. P. Roshchupkin, Alexandr A. Lebed ....................................................................................................................................58 Soliton-like behavior of electrons in the electron cooling O. Novak, R. Kholodov ............................................................................................................................................................................................61

II

The effective interaction force between positron and electron in a pulsed laser field S.S. Starodub, S.P. Roshchupkin .............................................................................................................................................................................64 Multi-GPU-accelerated FDTD method for PhC fibers characterization I.V. Guryev, I.A. Sukhoivanov, N.S. Gurieva, J.A. Andrade Lucio, E. Vargas Rodriguez, O.G. Ibarra Manzano ........................................68 Properties of prismatic structures consisting dielectric waveguide H.A.Petrovska, V.M. Fitio, I.Ya. Yaremchuk, Ya.V.Bobitski................................................................................................................................71 High transmission of light through metallic grating limited by dielectric layers I.Ya. Yaremchuk, V.M. Fitio, Ya.V. Bobitski ..........................................................................................................................................................74 Free-volume entities in thick-film nanostructures studied with PAL spectroscopy H.Klym, A.Ingram, I.Hadzaman, O.Shpotyuk ......................................................................................................................................................77 Thermal characterization of light-emitting sources of Cree types A.S.Vaskou, V.K. Kononenko, V.S. Niss, A.L. Zakgeim, A.E. Chervaykov .......................................................................................................79 Pulse low-energy electron beam pumped IR-lasers based on InGaAs/AlGaAs/GaAs nanoheterostructures N.A.Gamov, E. V. Zhdanova, M. M. Zverev, A. A. Marmalyuk, M. A. Ladugin, I.A. Anishchenko, T.A. Bagaev, D. V. Peregoudov ...................................................................................................................82 Dispersion and nonlinearity engineering in nanocrystal slot waveguides E.V.Borisov, E.A.Romanova ....................................................................................................................................................................................84 Design optimization of light emitting diode module В«chip-on-boardВ» for light extraction increase S.N.Lipnitskaya, K.D.Mynbaev, L.A.Nikulina, V.V. Kramnik, V.E.Bougrov, A.R.Kovsh, M.A.Odnoblyudov, A.E.Romanov .................87 Generation of drifting optical pulses A. Nerukh, D. Nerukh ..............................................................................................................................................................................................90 Oscillations in fine structure of radiation spectrum of sequence of electrons moving in spiral in medium Aurel V. Konstantinovich, Ivan A. Konstantinovich ............................................................................................................................................92 Dynamics of a Charged Particle in a Ramp Magnetic Field M. Raya-Armenta, E. Alvarado-Méndez, J.F. Gómez-Aguilar ............................................................................................................................95

III

LFNM*2013 International Conference on Laser & Fiber-Optical Networks Modeling, 11-13 September, 2013, Sudak, Ukraine

Designing Optical Filters with Electrical Models of Photonic Crystal Defects R. K. Shevgaonkar1, Fellow IEEE, Preeti B. Patil2 2

1 Indian Institute of Technology, Delhi 110016, INDIA Electronics Dept., K. K.Wagh Institute of Engg. Education and Research, Nasik 422003, INDIA

(Invited Paper)

Abstract A systematic approach for designing photonic crystal based band-pass filters with different filter characteristics has been presented in this paper. The filter characteristics are manipulated by varying the location of the defects in a photonic crystal waveguide. The photonic crystal waveguide is modeled as a dispersive transmission line and the defects are modeled by equivalent electrical T-networks. A filter then is visualized as a cascade of the T-networks with suitable interconnecting sections of transmission line. S-parameters are used to find the transmission and reflection frequency response of the filter. An optimization approach is used to obtain the most optimal locations of the defects which give the desired frequency response. It is shown that more than one defect combinations are possible for getting the desired filter characteristics.

I. INTRODUCTION Photonic Crystal (PC) based optical filters are receiving consideration because substantial demand exists for frequency selective circuits in optical communication systems. It is now well known that introduction of a defect inside a photonic crystal waveguide alters the frequency response of the waveguide and therefore by choosing proper combination of defects a desired filter characteristic can be realized. Various types of filters like band-pass filter [1], third order Chebyshev filter [2] etc., have been obtained by fabricating artificial defects in a photonic crystal waveguide (PCW). Generally resonator models have been used for predicting the filter frequency. However, for complex filter response the simple resonator model may not be adequate to obtain the filter characteristics. Since there are no closed form analytical expressions for obtaining the spectral characteristics of a PCW with defects, for each PC defect combination, the frequency characteristics have to be computed using numerical methods like FEM or FDTD. Many combinations of defects have to be tried till we get the satisfactory frequency response. This trial and error approach of filter design is not only computationally inefficient but is also suboptimal as it may not provide optimal filter parameters. In this paper we propose a systematic filter design approach using electrical models of photonic crystal defects. First we create electrical models for photonic crystal defects. A filter is then designed as a cascade of the equivalent electrical models. The use electrical model for defects has been demonstrated in [3] to design a narrow band-pass filter. Although this approach is not as accurate as the numerical simulations, the advantage of

this method is that the frequency response can be computed quickly. Further the frequency response can be optimized with the equivalent circuit parameters. Once the desired frequency response is realized, the defect parameters corresponding to those circuit parameters can be obtained from the defect library. For fine tuning the defect parameters one can then run the full numerical simulation on the filter designed with the electrical models. This paper therefore presents a filter design approach based on optimization of defect parameters. It is found that there may be multiple sets of defects which may give desired frequency response within the acceptable accuracy. II. FILTER DESIGN APPROACH The filter design approach consists of two steps namely the electrical modeling of defects, and the optimization of the defect parameters like their size, location etc. for the desired frequency response. For developing electrical models, full numerical simulations are carried out on defects of different sizes to extract the frequency dependent electrical parameters. Since the defect size may have size comparable to the wavelength, an S-parameter model is more accurate than a simple electrical circuit model [4]. A. Electrical Modeling of Defects Let us consider a two dimensional photonic crystal with square lattice. Let the dielectric pillars of refractive index 3.4 be immersed in a dielectric substrate of refractive index 1.5. For lattice pitch 0.46µm and radius of rod 0.1µm, the PC exhibits a TE band gap over a wavelength range of 1.46µm to 1.74µm. Removal of a line of rods creates a PCW. For well confined model field the waveguide can be modeled as a transmission line. Introduction of an additional pillar (defect) in the PCW alters the energy flow and therefore a defect can be considered equivalent to electrical impedance connected across the transmission line. Further assuming that the defect is lossless, the defect can be modeled as purely reactive impedance. The electrical parameters of a defect are computed using FDTD over the photonic band. It is shown that a defect can be modeled as a single shunt element which could be capacitive or inductive depending upon whether the refractive index of the defect rod is higher or lower than that of the PCW substrate. The single reactive element model is accurate enough for small defects. But for a defect rod of size comparable to the lattice rods, a generalized T model for a PCW defect

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has been proposed [4]. For defect rods with refractive index greater than that of the PCW substrate, the T section model consists of a shunt capacitive element and series inductive elements. For defects with refractive index lower than that of the substrate, the inductive and capacitive elements of the equivalent T section are interchanged. With the help of FDTD simulations the electrical values of the T-network are computed at closely spaced discrete frequencies. The frequency sampling is chosen fine enough to give satisfactory interpolation between the sampled frequencies. The parameters of Tnetwork can also be equivalently preserved in the form of frequency dependent S-parameters of the two-port Tnetwork. One time exercise is carried out to generate electrical parameter space for all possible defects over the desired frequency band. B. Optimization of Defect Parameters Having created the electrical parameter space for the defects the filter design is treated as an optimization problem where the square of the difference between the desired spectrum and the modeled spectrum is minimized with respect to the electrical parameters of the defects. In general the defect parameters may be number of defects, sizes and shapes, refractive index, and locations, and the true optimization would be over all these parameters. However the design procedure presented here shows that even with just two parameters i.e. number of defects and the separations between them, one can practically design a filter with wide desired frequency characteristics. Practically this option is attractive because the defect is of same size and refractive index as that of the lattice and therefore the fabrication process is same as that needed to create the PCW. Even with restricted parameter space there are N optimization parameters for N defects, the number of defects, N and (N-1) spacings between them. Let the N defects, D1, D2,..,DN be located with spacings d1, d2,.., dN-1, as shown in Fig. 1(a). Let the desired spectrum be given as Fd(), and let the synthesized spectrum for the N defects be given as Fs(). The optimization function to be minimized is then given by



2

 F ( )  F ( )  d

1

S

2

d

(1)

Where 1 and 2 are lower and upper cutoffs of the photonic band respectively. A gradient search algorithm can be used to minimize  with respect to the distance vectord  [d1, d2,.....,dN-1]. The frequency response of PCW with defects is computed using scattering parameters for the equivalent cascaded T sections as shown in Fig. 1(b). For this, first the S-parameters are converted to the [ABCD] parameters. If the [ABCD] matrix for the nth defect is [An], and the [ABCD] matrix of the section of the waveguide of length dn is [Ln], then the [ABCD] matrix of the filter is (2) [A] = [A1][L1]……[An-1][Ln-1][AN]

2

Fig. 1. (a) Cascade of N defects. (b)Equivalent Transmission line model. From the [ABCD] matrix, we can obtain the transmission and reflection spectrum of the filter. III. FILTER DESIGN EXAMPLES Let us now demonstrate the design approach presented above. We show here two cases of band-pass filters, one a narrowband filter and other a wideband filter. As discussed above, for designing a filter first number of defects is to be decided. One can start with only two defects and then slowly increase the number to meet the filter bandwidth requirement. A. Narrowband Gaussian Shaped Filter Let us design a PCW based filter with a Gaussian spectral response with center frequency of 192.3THz and 3dB bandwidth of 1.5THz. With the optimization approach defined above, let us consider a simple case of two defects separated by a distance d1. Since we do not have any a priory estimate of d1, we can take initial guess as 0.46µm same as that of the lattice period. Gradient search method is employed to minimize the error function. The minimum least square error giving the best match between the optimized and the desired spectrum corresponds to d1 = 1.6µm. Comparison of desired spectrum with that obtained using modeling for the two defect cavity with d1 = 1.6µm shows that although there is a good agreement between the peak frequencies, the filter with just two defects has much lower Q than expected. The next logical step in the design may be to add one more defect to the above configuration. However the study of the effect of number of defects and their spacing on the spectral parameters like the peak frequency and bandwidth shows that to obtain band-pass spectral response, the defect combination required at least four defects. Hence let us consider a filter with four defects with three spacings to optimize. Again we start with equispaced defects with pitch period spacing 0.46µm. Note that the initial choice of distance between defects is arbitrary and may affect only the number of iterations for convergence. The computed spectrum for the first iteration is practically at zero level indicating that any incident wavelength is always blocked due to the photonic band-gap effect of the four consecutive defects placed with inter-defect spacing same as the pitch period. However, within 8-10 iterations the spectrum converges to the desired spectrum. It is observed that during the optimization iterations, distance d2 changes significantly


from initial guess of 0.46µm to 2.305µm whereas d1 and d3 change marginally from 0.46µm to 0.5µm. This is in good agreement with the general understanding that the peak frequency is primarily decided by the central cavity and outer elements are for shaping of the spectral response. Fig. 2(a) shows desired and the optimized spectra obtained using the electrical T-network model. For sake of comparison the FDTD spectrum for the optimized distances is also shown. The excellent agreement between the three spectra demonstrates that using electrical models one can design PCW based filters fairly accurately. Although it is not necessary, one might prefer a symmetrical defect configuration. By placing an additional restriction during the optimization process, of maintaining d1 and d3 identical in order to preserve the symmetry of the cavity, different solutions for the same filter specifications can also be obtained. We show that there are two other defect configurations (and may be more) with identical d1 and d3 which give match with the desired spectral characteristics. For the first configuration d1 = d3 = 0.46µm and d2 = 1.60µm whereas for the second configuration d1 = d3 = 0.55µm and d2 = 1.575µm. The comparison of the FDTD spectrum and the desired spectrum for both solutions is shown in Fig. 2(b) and Fig. 2(c) respectively. This clearly demonstrates that the optimization approach using electrical models of the defects is not only computationally efficient but also provides flexibility in choice of filter configuration.

Fig. 2. Comparison of desired spectra (dotted line) for a PCW with four defects with spectra obtained using FDTD (dashed line) and the electrical model (thick line). Separation between defects is (a) d1 = 0.505µm, d2 = 2.305µm, d2 = 0.524µm, (b) d1 = d3 = 0.46µm and d2 = 1.60µm (c) d1 = d3 = 0.55µm and d2 = 1.575µm. B. Wideband Flat Top Filter As a second example we design a wideband filter with flat pass-band. The center frequency of the filter is 189THz and the half power bandwidth is 6THz. Again we take a four defect configuration. Following the steps mentioned above for the earlier example, we obtain optimized spacings d1, d2, d3 between four defects as 0.92μm, 0.66μm, 0.92μm respectively. Figure 3 shows the comparison of the desired spectral characteristics and that obtained using FDTD and equivalent T network model for the optimized distances. The designed filter exhibits a center frequency of 189.3THz and a 3dB bandwidth of 6THz.

Fig. 3. Comparison of spectral characteristics of a wideband band-pass filter. Desired spectral response (thick line), Spectral response obtained using FDTD (dashed line) and equivalent T network model (dotted line). Filter parameters are d1 = d3 = 0.92μm and d2 = 0.66μm. IV. CONCLUSIONS This paper describes a numerically efficient design approach for systematically designing optical filters using defects in a photonic crystal waveguide. Generally the photonic crystal based filters are designed by trial and error method and computationally intensive numerical technique are used for simulation. In this paper it is shown that if a photonic crystal waveguide and its defects are modeled by electrical circuits, the filter design can be made more efficient. With electrical modeling the problem of filter design can be treated as an optimization problem. Two cases presented in the paper amply demonstrate the strength of the approach. The main attraction of this approach is that the filter is first designed quickly using electrical models and the detailed FDTD simulations can be carried out only on the predesigned selective filters to improve the design if needed. The optimization approach also shows that there could be multiple defect configurations which can meet the desired filter specifications. REFERENCES [1] R. Costa, A. Melloni and M. Martinelli, “Bandpass resonant ﬁlters in photonic-crystal waveguides,” IEEE Photon. Technol. Lett., vol. 15, no. 3, pp. 401– 403, Mar. 2003. [2] D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightw. Technol., vol. 23, no. 5, pp. 1923–1928, May 2005. [3] Preeti. B. Patil, Sarang Pendharker and R. K. Shevgaonkar, “Electrical Modeling of Photonic Crystal Defects,” Microwave and Optical Technol. Lett., vol. 54, No. 11, pp.2523-2528, Nov. 2012. [4] P. B. Patil and R. K. Shevgaonkar, “Improved SParameter Model for Photonic Crystal Defects,” SPIE Select Proceedings of Photonics 2010, 8173 (2011).

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Influence of the moderate-strong pulsed laser field at the quantum electrodynamics processes (Invited Paper)

Sergei P. Roshchupkin Institute of Applied Physics, National Academy of Sciences of Ukraine 58, Petropavlovskaya Str., Sumy, 40030 Ukraine, e-mail: [email protected]

Abstract: The review on the quantum electrodynamics (QED) processes proceeding in the moderate-strong pulsed light fields, realized in modern powerful pulsed lasers is presented. Resonant and nonresonant processes of quantum electrodynamics in strong laser fields are considered. Following QED processes of the second order in the fine structure constant in the pulsed laser field are considered: resonant and nonresonant spontaneous bremsstrahlung by an electron scattered by a nucleus, resonant and nonresonant scattering of a lepton by a lepton. The resonant peak’s altitude and width are defined by the external pulsed wave properties. It is demonstrated that the resonant cross sections may be several orders of magnitude greater than the corresponding cross sections in the absence of an external field. Results obtained may be experimentally verified, for example, by the scientific facilities at the SLAC National Accelerator Laboratory and FAIR (Facility for Antiproton and Ion Research, Darmstadt, Germany).

I. INTRODUCTION Use of a powerful coherent light source in modern applied and fundamental research has stimulated study of the external strong field influence on quantum electrodynamics (QED) processes [1-2]. A characteristic feature of electrodynamics processes of second order in the fine-structure constant in a laser field is associated with the fact that such processes may occur under both nonresonant and resonant conditions [1-7]. The resonant character relates to the fact that lower-order processes, such as spontaneous emission and one-photon creation and annihilation of electron-positron pairs, may be allowed in the field of a light wave. Therefore, within a certain range of energy and momentum, a particle in an intermediate state may fall within the mass shell. Then the considered higher-order process is effectively reduced to two sequential lower-order processes [1-7]. The appearance of resonances in a laser field is one of the fundamental problems of QED in strong fields. As a result of laser technology development different types of coherent light sources have become available, with intensities that have increased up to 1022 W cm 2 in recent years. The new experimental conditions have required constant improvements in calculations and model development. The amplitude of the field intensity of powerful ultrashort pulsed lasers changes greatly in space and time. In the description of QED processes in the presence of a pulsed laser the external field is usually modeled as a plane nonmonochromatic wave, when a characteristic pulse width τ obeys the condition [1-7] (1) ωτ >> 1 .

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Following QED processes of the second order in the fine structure constant in the pulsed laser field are considered: resonant and nonresonant spontaneous bremsstrahlung by an electron scattered by a nucleus [1,3-5], resonant and nonresonant scattering of a lepton by a lepton [1,3,4,7]. There are two characteristic parameters in these processes of QED in the field of a pulsed electromagnetic wave. The first is the classical relativistic-invariant parameter [1-7], eF  (2) η0 = 0 2 , mc which in the pulse peak equals numerically the ratio of work done by the field within the distance equal to a wavelength to the electron rest energy ( e and m are the charge and the mass of an electron, F0 and  = c ω are the strength and the wavelength of an electric field in the pulse peak). The Bunkin-Fedorov quantum parameter is specified [1-7]: mv c (3) γ i = η0 i . ω ( vi is the electron speed). We treat these problems of QED within the range of moderate-strong-field intensities, when (4) η0 > Z 137 , Z is the nucleus charge number). The process of electron-nucleus SB in the presence of pulsed light wave may occur under resonant conditions when the fourmomentum of an intermediate electron lies near the mass surface ( kqi ) m (5)  ω ; otherwise, ωres  ω . The process of resonant electron–nucleus SB in the field of a pulsed light wave can be effectively reduced to two sequential processes of the first order in the fine-structure constant: emission of a photon with a four-momentum k ′ by an electron pi in a pulsed light wave and scattering of an electron qi by a nucleus in the field of the pulsed wave (see Fig. 1). The resonant differential cross section of electron–nucleus SB in the field of pulsed light wave for moderately strong intensities when the electron is scattered by a large angle is: 2 dσ res 1 E ω ′ qi ( kpi ) (6) = 2⋅ i ⋅ Pres ⋅ dσ s dW (1) . 2 d Ω′ π ′ ( kk ) pi Here dσ s is the differential cross section of scattering of an intermediate electron with a four-momentum qi by a nucleus in the field of the wave, dW (1) is the probability that an electron with a four-momentum pi = ( Ei , pi ) absorbs one photon from the external field and spontaneously emits a photon with a fourmomentum k ′ = (ω ′, k ′ ) ,

ρ

iβ  ∫− ρ dφ ⋅ erf  φ + 2

2

  +1 , 

(7)

qi2 − m 2 ωτ  1 . 4 ( kqi )

(8)

The parameter β (8) specifies how close the four-momentum of an intermediate electron coincides with the value on the mass surface in the resonant conditions. The dependence of the function Pres on the parameter β defines a magnitude and a shape of the resonant peak in the cross section of electron-nucleus SB process in the pulsed light field. The parameter ρ is the relation between observation time and pulse-width. The function Pres can be easily written in the form a1 2 ( kqi ) (9) Pres ≈ , . Γ = τ 2 a2 m (ωτ ) ( q 2 − m2 ) + ( 2mΓ )2 i

τ

Here coefficients a1 and a2 weakly depend on the parameter ρ . A transit resonant width Γτ arose from the finite time of particle-field interaction. lg Rres

vi Fig. 2. The ratio Rres as a function of the electron velocity for preset orientations of the electron momentum in the initial and final states and fixed orientation of the spontaneous photon ( θ ′ = 120 and ϕ ′ = 10 solid line; θ ′ = 120 and ϕ ′ = 60 dashed line). Let’s consider the relation between resonant differential cross section of electron-nucleus SB and the cross-section of electron-nucleus SB in an absence of external field. The following parameters: the laser wave frequency ω = 2.35eV ; the laser pulse-width τ = 1.5ps ; the field strength in pulse peak

F0 = 6 ⋅109 V cm were chosen for the calculation. Fig. 2 displays ratio Rres as a function of the initial velocity of the electron. As can be seen from Fig. 2, within the range of relativistic electron energies, the resonant differential cross section of electron–nucleus SB may be five orders of magnitude higher than the corresponding cross section in the absence of the external field. Within the range of ultrarelativistic electron energies, this ratio drastically decreases.

5


III. NONRESONANT BREMSSTRAHLUNG OF AN ELECTRON SCATTERED BY AN ION IN A PULSED LASER FIELD

Here we describe a theory of nonresonant spontaneous bremsstrahlung (SB) produced by the scattering of an electron by a Coulomb center in the presence of pulsed external electromagnetic field (see Fig.3) [1,4].

probability in this range towards lower values for a pulsed wave model is considered. In this case the most probable processes are the processes with a photon number l  γ i in contradistinction to the monochromatic wave case, when the most probable processes are the processes with l  γ i . In addition, the distribution of probability of emission (absorption) over a number of wave photons for a pulsed wave is more uniformly and has not reasonably sharp jumps for neighboring values of l , specific for the case of a monochromatic wave.

Fig. 3. Nonresonant SB related to the scattering of an electron by a nucleus in the field of a pulsed light wave. Double lines correspond to the Volkov function of an electron in the initial and final state, the inner line corresponds to the Green function of an electron in a pulsed field. Wavy lines corespond to the four-momenta of a spontaneous photon and a photon of nucleus recoil. The nonresonant process of electron-nucleus SB in the presence of pulsed light wave may occur under nonresonant conditions when the parameter β (8) satisfies to following conditions

β=

qi2, f − m 2 4 ( kqi , f

)

ωτ  1 .

(10)

Fig. 4. Distribution of the probability of emission (absorption) Wl (14) over a number of wave photons when the parameter γ i = 50 . The solid line corresponds to the case of a pulsed wave, the dotted line corresponds to the case of a monochromatic wave.

For relativistic energy the nonresonant partial SB cross section assumes the form: dσ =

∞

∑ dσ

l = −∞

l

,

(11)

= dσ l dσ BH ⋅ Wl ,

Here dσ BH field,

Wl =

(12) is the cross section in the absence of an external

τ 2T

Tτ

∫ τ L  χ l

−T

2

p f pi

, γ p f pi (φ ) , β p f pi (φ )  dφ

(13)

We choose the circular polarization of an external wave ( δ = ±1 ) and the envelope in the form of the Gaussian function. Thus, the probability Wl (13) assumes the form: = Wl

1

ρ

ρ ∫0

J l2 γ fi ⋅ exp ( −φ 2 )  dφ .

(14)

Consequently, for nonresonant SB of an electron scattered by a nucleus in a moderately strong pulsed field the probability of multiphoton processes is determined by the Bunkin-Fedorov parameter γ fi ~ γ i (3). Figs. 4−5 display the diagrams of the probability of emission (absorption) for various values of the parameter γ i as function of a number of wave photons. One can see from Figs. 4−5 that the essential range of a number of emitted wave photons is determined by the relationship l  γ i as for a monochromatic wave and for a pulsed wave. Redistribution of

6

Fig. 5. Distribution of the probability of emission (absorption) Wl (14) over a number of wave photons when the parameter γ i = 250 . The solid line corresponds to the case of a pulsed wave, the dotted line corresponds to the case of a monochromatic wave We consider nonresonant SB of a nonrelativistic electron scattered on a nucleus in a pulsed light field, if the velocity of electron oscillation in the pulse peak is in order with the velocity of electron translational movement (15) η0  vi  1 Energy conservation in this case assumes the form:


mv 2f

mvi2 1 (16) + ω ′ + lω  ⋅ω ωτ 2 2 The differential cross section may be easily integrated over dϕ f d Ω′ . We choose the direction of initial electron movement −

along external wave propagation. The partial cross section scattering into the interval of polar angles [θ , θ + dθ ] with emission of a spontaneous photon with the energy in the interval [ω ′, ω ′ + dω ′] assumes the form:

16 2 2 1 dω ′ , Z α re 2 f l (ω ′, θ ) dθ 3 ω′ vi ρl (ω ′ ) ⋅ sin θ ⋅ Wl (ω ′, θ ) , 2 1 + ρl (ω ′ ) − 2 ρl (ω ′ ) ⋅ cos θ

dσ l (ω ′, θ ) = f l (ω ′, θ )

= Wl (ω ′, θ )

1

2

∫J 2

2 l

0

ton energy ω ′ = 1,9 keV the maximum of the total cross sec-

tion is determined by the scattering angle θ max = 21 and the total cross section exceeds the conventional cross section for all electron scattering angles (the dash-dotted line on Fig. 6). Such a character of dependence of ratio of cross sections is due to fact that a hard photon can be emitted at the expense of a great number of external field photons. Also we underscore that SB spectrum in the absence of an external field has a boundary, determined by the value of the initial electron kinetic energy. For SB in an external field such a principal boundary is absent.

(17) (18)

 mvi 2  η0 ω ρl (ω ′ ) ⋅ sin θ ⋅ exp ( −ξ )  d ξ (19)  

pf 2 (ω ′ + lω ) (20) = 1− pi mvi2 The question how the total differential cross section (11), (17) summed over all partial processes differs from the corresponding cross section in the absence of an external field is of experimental interest. Fig. 6 presents dependence of the ratio of the total differential cross-section, summed over all possible processes of emission (absorption) of wave photons to the conventional cross-section dσ / dσ ∗vi 1 on the electron scattering angle for different values of the spontaneous photon energy when the field intensity and frequency are fixed. One can see from Fig. 6 that the total cross-section differs substantially from the corresponding conventional cross-section except the case of the scattering angles close to π-value (the backscattering). Herewith, the character of the dependence changes qualitatively on the spontaneous photon energy. Thus, when the soft photon ω ′ = 3,53 keV  ω is emitted then the total cross section represents monotone dependence and is less than the cross section in the absence of an external field for all scattering angles of an electron (the solid line on Fig. 6). When the hard photon ω ′  mvi2 2 is emitted the character of dependence changes qualitatively. The strongly pronounced maximum for a certain angle occurs in the total cross section. The magnitude and width of this maximum increase with increasing of the spontaneous photon energy. Consequently, for the spontaneous photon energy ω ′ = 1 keV (see the dotted line on Fig. 6) the maximum appears for the scattering angle θ max = 10 . Herewith, there are the region where the total cross

ρ= l

section exceeds the conventional cross section ( θ < 250 ), and the region where the total cross section is less than the conventional cross section ( θ > 250 ). Father greater spontaneous pho-

Fig. 6. Dependence of the ratio of the total section to the cross section of SB in the absence of an external field dσ / dσ ∗vi 1 (11), (17) on the electron scattering angle for different values of the spontaneous photon energy. The initial electron moves along the direction of wave propagation with the velocity vi = 0,1 in a field with the intensity in the pulse peak η0 = 0,1 and the frequency ω = 2,35 eV. The solid lines corresponds to the spontaneous photon energy ω ′ = 3,53 eV, the dotted line - ω ′ = 1 eV, the dash-dotted line ω ′ = 1,9 eV. IV. RESONANT SCATTERING OF A LEPTON BY A LEPTON IN A PULSED LASER FIELD

Here we describe a theory of resonant scattering of a lepton by a lepton in a pulsed laser field (see Fig.7) [1,3,6]. The scattering of a lepton by a lepton in the presence of pulsed light wave may occur under resonant conditions when the four-momentum of an intermediate photon lies near the mass surface ( kq1′ ) ω 2 (21) q1′2   0)

H(l1 ,l2 ) . (ωf pf,− )

Here dΩf is an element of solid angle of the final photon, we introduced the notation H

(l1 ,l2 )

˜ (0,0) 2 = −2 Fl1 l2 + 1 +

u2 2(1 + u)

×

(1,0) 2 (1,0) 2 (0,0) (2,0)∗ + × η12 Fl1 −1,l2 + Fl1 +1,l2 − 2< Fl1 l2 Fl1 l2

+η22

(0,1) 2 (0,1) 2 (0,0) (0,2)∗ + Fl1 ,l2 −1 + Fl1 ,l2 +1 − 2< Fl1 l2 Fl1 l2

i (1,0) (0,1)∗ (1,0)∗ (0,1) +2η1 η2 < Fl1 −1,l2 Fl1 ,l2 −δ1 δ2 + < Fl1 +1,l2 Fl1 ,l2 +δ1 δ2 , (n ,n2 )

Fl1 l21 ˆ∞ n1

=

dφ1 g (φ1 )g −∞

n2

=

τ1 φ1 Il1 ,l2 (φ1 )eiGl1 l2 (φ1 ,φ1s ) , (4) τ2

Gl1 l2 (φ1 , φ1s ) = ϕ0

υ × ul1 l2


2 P j=1

ηj2

h√

2π 8

τj τ1

√ i Erf 2 2φ1 ττ1j − g 2 φ1s ττ1j φ1 ττ1j

2 υ P 1+ ηj2 g 2 φ1s · 2ul1 l2 j=1

Il1 l2 = e−i[l1 χ1 +l2 χ2 ] ×

! ,

τ1 τj

×

(5) where η = |e|aj /m is the wave intensity parameter in the pulse peak; e and m are the charge and the mass of an electron; Erf(φ) is the error function; pf,− = pi,− − kf,− , pi,− = ~ E i − pi,z , kf,− = ωf (1 − cos ∠(k, kf ), ϕ0 = l1 ϕ10 + l2 ϕ20 · τ1 1 and the method of a stationary phase should be τ2 (n ,n )

used to calculate the function Fl1 l21 2 (4); the law of fourmomentum conservation is satisfied for each partial process (with fixed l1 , l2 ) in the points of a stationary phase φ1s : p˜i + (l1 k1 + l2 k2 ) = p˜f + kf ,

∞ X

eisδ1 ∆ Js (z−δ1 δ2 )Jl1 −s (y1 )Jl2 +δ1 δ2 s (y2 ).

Here Js (y) is Bessel function, the quantity ∆ in (8) stands for the angle between the polarization vectors ~e1x and ~e2x of the considered light waves and parameters yj and z−δ1 δ2 in expression (8) are given by m l1 ω1 + l2 ω2 ηj g (φj ) yj = 2 ωj m ˜ z−δ1 δ2 = 2

where p˜i,f is the four-quasi- momentum   2 2 X m  η 2 g 2  k1 , p˜i,f = pi,f + 2(k1 pi,f ) j=1 j j

(8)

s=−∞

χ1 = δ1 ψ,

s

u

1−

u ˜l1 l2

u u ˜ l1 l2

,

u l1 ω1 + l2 ω2 η1 η2 g (φ1 ) g (φ2 ) , ω1 − δ1 δ2 ω2 ul1 l2 ~⊥ , ψ = ∠ ~e1x , Q

(9)

(10)

χ2 = δ2 (ψ − ∆) .

The process is described by invariant parameters



p˜2i,f = m ˜ 2,

  2 X m ˜ 2 = m2 1 +  ηj2 gj2  .

υ=

j=1

(l1 ω1 + l2 ω2 )(nnf ) , (pi nf )

The frequency of the photon emitted from stationary points (ωf ≡ ωf (φ1s )) ωf =

(l ω + l2 ω2 )(npi ) # ! , (6) " 1 1 2 1 P τ (pi nf ) 1 + 1 + η 2 g 2 j φ1s υ ul1 l2 j=1 j j τ1 ωf,min ≤ ωf ≤ ωf,max ,

(0) ωf,max =

2

+ (Ei /m) ∠(P~ , ~nf )2

,

1 + ul1 l2 +

2 P j=1

, 2

ηj2 + (Ei /m) ∠(P~ , ~nf )2

We note that, at optical frequencies

(12)

υ 1+

υ 2ul1 l2

2 P j=1

,

ηj2 1 + δj2 gj2

(nkf ) = υ˜, (npf )

υ˜ = 1+

υ 2ul1 l2

υ . (13) 2 P ηj2 gj2 1 + δj2

j=1

When m/Ei 1, but (l1 ω1 + l2 ω2 ) (Ei /m2 ) & 1 we obtain: ul1 l2 , 1 + δf2

δf =

Ei θf m

(14)

I NTERFERENCE REGION The interference region called the area in which yj = 0 [1]. It is equivalent to the condition: u=u ˜l1 l2 ⇒ cos ∠ ~j, ~nf = 1,

θf = ∠(~ pi , ~nf ).

(l1 ω1 + l2 ω2 ) sin θi ∠(~ pi , P~ ) = m

(11)

m 2 2(l1 ω1 + l2 ω2 )(npi ) = u , l l 1 2 m ˜2 m ˜

υ≈ q pi , P~ )2 + θf2 − 2∠(~ pi , P~ )θf · cos ψf , ∠(P~ , ~nf ) = ∠(~

2(l1 ω1 + l2 ω2 )(npi ) , m2

(l1 ω1 + l2 ω2 )(nnf ) = (p˜i nf )

u ˜l1 l2 =

u=

Ei ul1 l2

(0)

ωf,min =

υ˜ =

(7)

Ei ul1 l2 1 + ul1 l2

ul1 l2 =

0 ≤ υ ≤ ul1 l2 ,

m Ei

m . Ei

The functions Il1 l2 depend on five parameters (two of these parameters χ1,2 influence only the phase factors in the argument) and can be expressed as

where ~j = p~i − m2 /2(npi ) ~n, ~nf is the direction of propagation a final photon. Thus in the interference region a final photon is emitted along the vector ~j (see fig. 1). The behavior of the parameters z−δ1 δ2 , yj near the interference region:

33


Fig. 1.

Interference region: the final photon is directed along the vector ~j

z−δ1 δ2 = 2l1 η1 η2

yj = 2

m 2 m ˜

(0,0) 2 Fig. 2. Dependence F1,−1 on δf for φ1s = 0 and waves intensities: I1 = 1.88 · 1022 W/cm2 ( η1 = 4, ω1 = 3.90 · 1016 s−1 ), F2 = 1.20 · 1021 W/cm2 ( η2 = 5, ω2 = 7.89 · 1015 s−1 )

1 , 1 + δf2

l1 [ω1 − δ1 δ2 ω2 ] m ηj δf . ωj m ˜

III. C ONCLUSION

In the interference region the function Il1 l2 is simplified Il1 l2 = Jl1 (z−δ1 δ2 ) · δl2 ,−δ1 δ2 l1 ,

(15)

where δl2 ,−δ1 δ2 l1 is the Kronecker delta. The expression (15) indicates that the probability of one photon emission in the interference region differs from zero for partial processes when second wave emits or absorbs equal numbers of photons from the first laser wave. The corresponding frequency of a final photon is ωf =

=

R EFERENCES

l1 (ω1 − δ1 δ2 )(npi ) (pi nf ) 1 + υ +

υ

2 P

ul1 ,−δ1 δ2 l1

j=1

ηj2 gj2

τj τ1 φ1s

! . (16)

The character behaviors probability of process depends on (0,0) 2 behaviors functions Fl1 l2 . For low-intensity laser waves (η1,2 1), multiphoton processes involve a small number of photons, the interference region is not expressed as the parameter z−δ1 δ2 within it is much smaller than the parameters y1,2 outside. For intensities η1,2 ∼ 1 the probabilities of such processes are of the same order within and outside the interference region. For high- intensity waves (η1 ∼ η2 > 1), multiphoton processes involving a large number of photons occur, the probability of multiphoton processes in the interference region may considerably exceeds the probability of partial processes outside this region. The figure 2 demonstrates that in the interference region (δf 1 ) the probability of multiphoton processes in the interference region may considerably exceeds the probability of partial processes outside this region.

34

The performed investigation of the emission of photon by an electron in the field of two pulsed laser waves which propagate in the same direction demonstrates that the emission of a photon may occur in different kinematic domains. Specifically, when a photon is emitted outside the interference region, the classical parameter y1,2 plays the dominant role. However, when a photon is emitted in the plane of the initial momentum and the wave vector, the number of photons involved in multi photons processes is determined by classical parameters z−δ1 δ2 . For high-intensity fields, emission mainly occurs in the interference region. The results can be tested in the experiments on the verification of quantum electrodynamics in the presence of strong fields (SLAC and FAIR).

[1] A. I. Voroshilo and S. P. Roshchupkin, Interference Effect in the Emission of a Spontaneous Photon by an Electron in the Field of Two Light Waves, Laser Physics, Vol. 7, No. 2, 873 (1997). [2] 1. S. P. Roshchupkin and A. I. Voroshilo, Resonant and Coherent Effects of Quantum Electrodynamics in the Light Field (Naukova Dumka, Kiev, 2008), in Russian. [3] S. P.Roshchupkin and A. A. Lebed’, Effects of Quantum Electrondynamics in the Strong Pulsed Laser Field (Naukova Dumka, Kiev, 2013), in Russian. [4] S. P. Roshchupkin, A. A. Lebed’, E. A. Padusenko, and A. I. Voroshilo, Quantum Optics and Laser Experiments, edited by S. Lyagushyn (Intech, Rijeka, 2012), pp. 107–156. [5] S. P. Roshchupkin, A. A. Lebed’, E. A. Padusenko, Nonresonant quantum electrodynamics processes in a pulsed laser field // Laser Physics, 2012, Vol. 22, No. 10, pp. 1513-1546 [6] S. P. Roshchupkin, A. A. Lebed’, E. A. Padusenko, A. I. Voroshilo, Quantum electrodynamics resonances in a pulsed laser field // Laser Physics, Vol. 22, No. 6 , pp. 1113-1144.


Gain coefficient in the course of the electron scattering by ions in a weak electromagnetic field: general relativistic case V.A. Tsybul’nik and S.P. Roshchupkin Institute of Applied Physics, National Academy of Sciences of Ukraine, 58, Petropavlovskaya St., Sumy, 40030 Ukraine, e-mail: [email protected]

Abstract: For weak electromagnetic field the gain coefficient in the F (1) A (ϕ )= ex cos ϕ + δ ⋅ ey sin ϕ , ϕ= ω ( t − z ) . scattering of electrons by ions in the elliptically polarized light wave ω is theoretically studied in the general relativistic case. Simple analytical expression for a field amplification constant in logarithmic = Here, ex (= 0, e x ) , ey 0, e y are 4-vectors of the wave poapproach is obtained. It is shown for the ultrarelativistic electron energies the gain coefficient depends on energy as a cubic degree of larization and F , ω , and δ are the wave field strength, freenergy and can be major enough. Results obtained may be experi- quency, and polarization, respectively. In particular, the linear mentally verified, for example, by the scientific facilities at the and circular polarizations correspond to δ = 0 and δ = ±1 , reSLAC National Accelerator Laboratory and FAIR (Facility for An- spectively. We also assume that the classical relativistic invariant tiproton and Ion Research, Darmstadt, Germany). parameter η is much less than unity:

(

)

(

I.

INTRODUCTION

The analysis of the amplification of electromagnetic radiation in the course of the electron scattering by ion in the presence of the field of a plane electromagnetic wave was started in [1] and has a long history (see, for example, [1-3]. In the quasi-classical approximation with respect to the electron interaction with the ion field (for relatively slow electrons with vi c > Z 137 ), the amplification of radiation was studied in [13]. Let's note that the simple analytical formula for the gain in case of the arbitrary elliptic polarization of an electromagnetic wave and nonrelativistic electron energies has been obtained in [1,3] for weak and medium fields. It is important to emphasize that in article [3] the anomalous amplification effect of an electromagnetic radiation in the scattering of the nonrelativistic electrons by ions in the elliptically polarized light wave for the moderately-strong fields is theoretically predicted. In the present paper the simple analytical formula for the gain coefficient of the arbitrary elliptic polarization of a weak electromagnetic wave in a general relativistic case is obtained. We employ the system of units in which = c= 1 . II.

GAIN COEFFICIENT

We study the scattering of relativistic electrons by ion in the presence of the field of a plane electromagnetic wave. We choose the 4-vector potential of the external field as the elliptically polarized (in the ( xy ) plane) electromagnetic wave that propagates along the z axis:

)

eF (2) 0 ) or absorption ( l < 0 ) of l wave pho= η

tons by electron is represented as [1-3] 0, E f − Ei + lω =

(3)

where Ei and E f are the electron energies prior to and after the scattering. Below, we assume that the wave photon energy is significantly less than the electron energy:  ω > m ) and small angles (θi d (see e.g. [12]). For definiteness but without loss of generality let us next consider the case when they are constant at all values of z and conditions 2) are satisfied. Since A(z) is polynomial matrix function (3) at 0 ≤ z ≤ d then the following formal power series for each particular solution satisfying (2) ∞

y(z) =

∑

yk,z k ,

(4)

k= 0

(k+1)yk+1 = A0yk + A1yk-1 + ... + Amyk-m; k = 0,1, ... , where yk are constant (z-independant) 2n-components vectors, converge at |z| < ∞ and give exact solutions to (2) (immediate corollary of [8, Theorem 4]). Here y0 is some given 2n-vector that is one of 2n linear-independent 2n-vectors corresponding to particular solutions of the fundamental system of solutions of (2). It is clear that in actual calculations one should truncate summation in (4) at sufficiently large value k = kmax. Since A(z) = A = const at z ≤ 0, z ≥ d solutions of (2) satisfying the definite conditions at z → ± ∞ are easy to find: they are superpositions of 2n-vectors (eigenvectors of the matrix A) multiplied by exponential functions with generally speaking complex arguments (eigenvalues of the matrix A). In case of the eigenvalue problem one should consider such range of E-values that eigenvalues of A have nonzero real parts and one should retain only those exponentials in solutions which vanish as z → ± ∞.


Let 2n×2n matrix A = A(E) has n eigenvalues with positive real parts and n eigenvalues with negative real parts in some range of E-values: λ1, λ2 , …, λn: Re λi > 0; λn+1, λn+2 , …, λ2n: Re λj < 0. At z ≤ 0 we have, as a matrix of n solutions of (2) tending to zero as z → -∞, the following 2n×n matrix function F(z) ≡ (u1exp(λ1z), u2exp(λ2z), …, unexp( λnz))

(5)

Here n-elements vectors u1,...,un are eigenvectors of A(E) that correspond to its eigenvalues λ1,…,λn with positive real parts. Using each of n columns of 2n×n constant matrix F(0) as a vector y0 in (4) we derive a 2n×n matrix function Y (z) which is built from n particular solutions yi(z) of (2) at 0 ≤ z ≤ d ∞

Y(z) = (y1(z), …, yn(z)) =

∑

Yjz j,

(6)

k= 1

(k+1)Yk+1 = A0Yk + A1Yk-1 + ... + AmYk-m; k = 0,1, ... ; Y0 = F(0) At z ≥ d we have, as a matrix of n solutions of (2) tending to zero as z → +∞, the following 2n×n matrix function Γ(z) = ( un+1 exp( λn+1z ), …, u2n exp( λ2nz ) ),

(7)

Where un+1,…,u2n are eigenvectors of the matrix A(E) which correspond to its eigenvalues λn+1,…,λ2n with negative real parts. One can easily compute 2n×n matrix functions Y(z) (6) and Γ(z) (7) when z = d at any value of the parameter E. It is obvious that for existence of the eigenfunction of (2) it is necessary and sufficient that such non-nil n-dimensional vectors of the matching Ξ1 = Ξ1(E) and Ξ2 = Ξ2(E) exist that the following condition is valid Y (d) Ξ1 = Γ(d) Ξ2.

(8)

The light- and heavy-hole states in semiconductors are described by the 4×4 Luttinger Hamiltonian matrix [9]. We choose z direction to be perpendicular to interfaces in a quantum-well structure (the direction of crystal growth). The potential V(z) breaks the translational symmetry along z axis but components of quasi-momentum parallel to the interfaces (kx, ky) remain good quantum numbers. Within the axial approximation a unitary transformation [10] block diagonalizes the Hamiltonian matrix into two 2×2 blocks. Then, after substitution kz → -i d/dz in the Hamiltonian, the Schrödinger equation is reduced to (1) with n = 2, and with matrices a, b, c that depend on the Luttinger parameters of the valence band and the value kl (kl2 = kx2 + ky2) of the lateral component of quasi-momentum (in the case of symmetric quantum wells one should consider only one of two blocks of the Hamiltonian). Note that when kl = 0 (1) is a system of decoupled equations and is of the form of two one-dimensional Schrödinger equations describing purely heavy- and light-hole states (identical to the equation describing electron states with respective masses). At kl ≠ 0 (1) become coupled equations and we designate hole states conventionally as a heavy- and lighthole states according to their origin. Hole bound states energies and corresponding wave functions for finite symmetric parabolic potential of GaAs QW, namely V(z) = -U0[1-(2z-1) 2] at 0 ≤ z ≤ d , V(z) = 0 at z ≤ 0, z ≥ d were calculated as functions of the lateral quasimomentum component and parameters of the potential. The Luttinger parameters were taken as γ1=6.85, γ2=2.10, γ3=2.90. For illustrative purposes results of calculations in spherical approximation of the dispersions of all hole bound states (U0 = 0.085 eV, d = 10 nm), and wave functions (kl = 0.3 nm-1) of the ground and the third excited state are demonstrated in Fig. 1-3. Note, that no oscillatory theorem for the wave function is valid in the matrix case (i.e. if n >1 in (1)). Owing to admixture of heavy- and light-hole states at kl ≠ 0 there is always light-hole exponential in asymptotic of solutions like in the ‘Coulomb’ case [11].

This implyies that some value E = E0 is the eigenvalue of (2) if the following equality is satisfied det M(E ) = 0,

(9)

where M = M(E) is 2n×2n matrix M ≡ (Y(d), - Γ(d)). Thus the problem of finding eigenvalues of (2) is reduced to numerical solution of (9). After computation of the eigenvalue E = E0, n-dimensional vectors Ξ1 and Ξ2 from (8), and normalization we derive the following expressions for the corresponding eigenfunction y(z) = F(z) Ξ1(E0), z ≤ 0, y(z) = Y(z) Ξ1(E0), 0 ≤ z ≤ d, y(z) = Γ(z) Ξ2(E), z ≥ d .

(10)

Note that (10) gives exact analytical expression for the eigenfunctions of (2) and hence eigenfunctions and their first derivatives of (1). Normalizing of the wave functions is performed analytically.

Fig.1. Dispersion of hole bound states in a single parabolic GaAs QW. d = 10 nm, U0 = 0.085 eV.

39


REFERENCES [1] [2]

[3]

[4]

[5] Fig. 2. Wave functions of the hole ground state (hh1) in parabolic GaAs QW. U0 = 0.085 ev, d = 10 nm, kl = 0.3 nm –1, E0 = -58.058 meV

[6] [7]

[8]

[9] [10] [11] [12]

Fig.3. Wave functions of the third excited state (hh3). Parameters are those as in Fig.2. E0 = -0.5947 meV

40

Y. Ando, T. Itoh, “Calculation of transmission tunneling current across the arbitrary potentials”, J. Appl. Phys., vol. 61, pp. 1497-1501, 1987. W.W. Lui, M. Fukuma, “Exact solution of the Schrodinger equation acroaa an arbitrary onedimensional piecewise-linear potential barrierJ. Appl. Phys., vol. 60, pp.1555 –1559, 1986. G. Bastard, E.E. Mendez, L. Chang, L. Esaki “Variational calculation on aquantum well in an electric field”, Phys. Rev., vol. B28, pp. 3241-3244, 1983 D. Ahn, S.L. Chuang, Y.-C. Chang, “Valence-band mixing effects on the gain and the refractive index change of quantum-well lasers”, J. Appl. Phys., vol. 64, pp. 4056- 4063, 1988. P. Harrison Quantum Wells, Wires and Dots. Theoretical and Computational Physics: J. Wiley & Sons, 2011. Z. Ikoniĉ, V. Milanoviĉ, “Hole-bound-state calculation for semiconductor quantum wells”, Phys. Rev., vol. B45, pp. 8760-8762, 1992, A.F. Polupanov, “Energy spectrum and wave functions of an electron in a surface energy well in a semiconductor”, Sov. Phys. Semicond., vol. 19, pp. 1013- 1015, 1985. V.I. Galiev, A.F. Polupanov, I.E. Shparlinski, “On the construction of solutions of systems of linear ordinary differential equations in the neighbourhood of a regular singularity”, J. of Compututational and Applied Mathematics, vol. 39, pp. 151 – 163, 1992. J.M. Luttinger, “Quantum theory of cyclotron resonance in semiconductors: General theory”, Phys. Rev., vol. 102, pp. 1030-1041, 1956. D.A. Broido, L.J. Sham, “Effective masses of holes at GaAs-AlGaAs heterojunctions”, Phys. Rev. B, vol. 31, pp. 888-892, 1985. V.I. Galiev, A.F. Polupanov, “Accurate solutions of coupled radial Schrödinger equations”, J. Phys. A: Math. Gen., vol. 32, pp. 5477-5492, 1999. A.F. Polupanov, V.I. Galiev, A.N. Kruglov. “The over-barrier resonant states and multi-channel scattering by a quantum well”, Int. J. of Multiphysics, vol. 2, pp. 171-177, 2008.


1

Formation of an Electron–Positron Pair by a Photon in the Field of Two Pulsed Laser Waves Oleg B. Lysenko, Sergei P. Roshchupkin, Alexey I. Voroshilo Institute of Applied Physics of National Academy of Sciences of Ukraine 58, Petropavlivska St., 40000 Sumy, Ukraine

Abstract—The process of a formation of an electron–positron pair by a photon in the field of two pulsed circularly polarized laser waves was studied theoretically. The probability of studied process was obtained. It is shown that the probability process of formation of an electron-positron pair in an interference domain may be greater then out it.

The following condition is satisfied for pulse durations: ϕ0j = ωj τj 1. Thus, the spectral density of four-potential (2) represents a sharp peak with an amplitude of about ϕ0j and a width of about ϕ−1 0j . Therefore, it is expedient to consider ωj as the frequency of the quasi-monochromatic field. and for the theoretical analysis, we choose the wave envelope given by (see [2]–[5])

I NTRODUCTION The processes of interaction of laser radiation with matter are very important problems in quantum electrodynamics (QED) of strong fields. Overview of the most important results in this area can be found in [1-3]. A number of important processes of QED in the field of one pulse wave was studied in detail in [1]-[5]. Experimental verification of the nonlinear equations of quantum electrodynamics of strong fields was made possible by the creation of compact laser optical frequency pulse with power W ∼ 1018 − 1019 W/cm2 . This capacity corresponds to a field in the focus F ∼ 1011 V/cm or the value of the classical relativistic-invariant parameter η ∼ 1. New experimental opportunities arise in connection with the construction of a fundamentally new laser facility in Russia (Nizhny Novgorod) with a pulse power of up to W ∼ 1022 W/cm2 . The relativistic system of units, where ~ = c = 1, and the standard metric for 4-vectors (ab) = a0 b0 − ab will be used throughout this paper.

g (φj ) = exp −4φ2j .

(3)

The relativistic system of units, where ~ = c = 1, and the standard metric for 4-vectors (ab) = a0 b0 − ~a~b will be used throughout this paper. The amplitude of the formation of an electron– positron ¯f ) pair with four-momenta pi = (Ef , pf ) and p¯f = (E¯f , p by a photon with a four-momentum ki = (ωi , ki ) and the polarization vector ε in the field (1) is given by the expression ˆ exp (−i(ki x)) ¯ p (x)ˆ √ , (4) Sf i = −ie d4 xΨ εΨ−p¯(x) f 2ωi where Ψp (x) is the wave functions of an electron with a momentum p in the field (1) (Volkov functions), εˆ = (εµ γ µ ), γµ (µ = 0, 1, 2, 3) are the Dirac matrices. The amplitude (4) can be represented as a sum of partial amplitudes: X (l l ) Sf i = Sf i1 2 . (5) l1 l2

Here I. A MPLITUDE

¯ f,⊥ ) × S (l1 l2 ) = Cδ (2) (ki,⊥ − pf,⊥ − p

The 4-potential this field was choose in form A = A1 (ϕ1 ) + A2 (ϕ2 ),

(1)

i h ×δ (ki,− − pf,− − p¯f,− ) εµ u ¯pf Oµ(l1 l2 ) u−p¯f .

Aj = aj g(φj ) (ejx cos ϕj + δj ejy sin ϕj ) ,

(2)

where up is the bispinor of a free electron with a fourmomentum p; ki,− = ωi − nki , pf,− = Ef − npf , (l l ) ¯f − n¯ p¯f,− = E pf ; C is the normalization constant; Oµ 1 2 is determined by expressions m ˆ (0,0) Olµ1 ,l2 = Fl1 l2 γ µ + Dl l nµ + 2¯ pf,− 1 2

where ϕj = (kj r) = ωj t − kj r is the wave phase (j = 1, 2); φj = ϕj /ϕ0j ; ϕ0j = ωj τj ; τj is the pulse duration in the laboratory frame of reference; r = (t, r) is the four radius vector; aj = Fj /ωj ; Fj and ωj are the field strength at the center of pulse and wave frequency in the laboratory system; δ = ±1 corresponds to the circularly polarized wave ejx,y = (0, ejx,y ) and kj = (ωj , kj ) are the four polarization vector and four momentum of the external field photon such that k 2 = 0, e2x,y = –1, and (ex,y k) = 0; g(φj ) is the envelope of potential that satisfies the conditions g(0) = 1 and decreases exponentially with |φj | 1 (ϕj ϕ0j ) . 978-1-4799-0159-3/13/$31.00 ©2013 IEEE

+

m2 m Bl1 l2 nµ − Dlµ1 l2 n ˆ+ 8¯ pf,− pf,− 2¯ pf,− m 1 1 µ ˆl l n + D + 1 2 ˆγ , 4 pf,− p¯f,−

(6)

(7)

41


(n ,n ) Fl1 l21 2

ˆ∞ = −∞

u=

=

τ1 Il1 ,l2 (φ1 )eiϕ0 Gl1 ,l2 (φ1 ) , (8) dφ1 g n1 (φ1 ) · g n2 φ1 τ2 ϕ0 = l1 ϕ01 + l2 ϕ02

Gl1 ,l2 (φ1 ) =

2 u X

ul1 l2

ηj2

j=1

τ1 τ2

,

(9)

"√ √ 2π τ1 τ1 erf 2 2φ1 − 8 τj τj

τ1 −g 2 φ1 φ1 , τj

(10)

(1,0)

(−)

(1,0)

(+) (0,1) (−) (0,1) +η2 ε2 Fl1 ,l2 −1 + ε2 Fl1 ,l2 +1 . (±)

(12)

= ejx ± iδj ejy .

In the expressions (4)-(12) n = (1, n); η = |e|aj /m is the wave intensity parameter in the pulse peak; e and m are the charge and the mass of an electron; erf(φ) is the error function. In (8) the functions Il1 l2 have the same form as in the case of a monochromatic wave. They depend on five parameters (two of these parameters χ1,2 influence only the phase factors in the argument) and can be expressed as Il1 l2 (φ1 ) = e−i[l1 χ1 +l2 χ2 ] ×

×

∞ X

eisδ1 ∆ Js (z12 )Jl1 −s (y1 )Jl2 +δ1 δ2 s (y2 ),

(13)

Here Js (y) is Bessel function, the quantity ∆ in (13) stands for the angle between the polarization vectors e1x and e2x of the considered laser waves and classical parameters yj and z12 in expression (13) are given by m ˜p u (˜ ul1 l2 − u), m

z12 = 2

χ1 = δ1 ψ,

42

κj =

(kj ki ) ηj , m2

l1 ω1 + l2 ω2 u η1 η2 , ω1 − δ1 δ2 ω2 ul1 l2

~⊥ , ψ = ∠ ~e1x , Q

 q −1/3 2 iG (0)  · Ai(γ), φ∗1 ≤ ϕ0 ;  e l1 ,l2 3 G000 l1 ,l2 (0) r × 8π cos Gl l (φ∗ ) + π , φ∗ > ϕ−1/3 .  00 1 1 1 2 0  4 ∗ (φ ) G 1

Here the position of the stationary phase; γ = G0l1 ,l2 (0) · 3 2/G000 l1 ,l2 (0), Ai(γ) is the Airy function. At the points of stationary phase, the law of conservation of 4momentum is satisfied: ˜¯f , ki + (l1 k1 + l2 k2 ) = p˜f + p

(19)

  2 2 X m τ 1   k1 , (20) p˜f = p˜f (φ∗1 ) = pf + η 2 g 2 φ∗1 2(k1 pf ) j=1 j τj   2 2 X τ m 1   ˜¯f = p ˜¯f (φ∗1 ) = p¯f + η 2 g 2 φ∗1 p k1 , (21) 2(k1 p¯f ) j=1 j τj

2

˜¯f = m p˜2f = p ˜ 2,

v u 2 X u τ1 m ˜ = m(φ ˜ ∗1 ) = mt1 + ηj2 g 2 φ∗1 . τj j=1 (22)

II. T HE INTERFERENCE REGION

s=−∞

yj = 4ηj2 κ−1 j

m 2 (l1 ω1 + l2 ω2 )(nki ) , u ˜ = u . (18) l l l l 1 2 1 2 2m2 m ˜ Thus the condition ϕ0 1 is satisfied the method of a stationary phase should be used to calculate the function (n ,n ) Fl1 l21 2 (8): ) ( 2Il1 l2 (0) (n1 ,n2 ) ∗ × Fl1 l2 (φ1 ) = g n1 (φ∗1 ) · g n2 φ∗1 ττ12 Il1 l2 (φ∗1 )

˜¯f is the four-quasi- momentums where p˜f , p

Dl1 l2 = η1 ε1 Fl1 −1,l2 + ε1 Fl1 +1,l2 +

εj

(17)

φ1 =qφ∗1

h i (1,1) (1,1) +η1 η2 e−iδ1 ∆ Fl1 +1,l2 −δ1 δ2 + eiδ1 ∆ Fl1 −1,l2 +δ1 δ2 , (11) (+)

1≤u≤u ˜l1 l2 ,

ul1 l2 =

l1 ,l2

(2,0) (0,2) Bl1 l2 = 4 η12 Fl1 l2 + η22 Fl1 l2 +

2 ki,− , 4pf,− p¯f,−

2

(14)

It is of considerable interest to analyze the kinematic region where a pair is produced in such a way that the parameters yj are equal to zero and the pair is created through the correlated emission and absorption of wave photons. This area will be referred to as the interference region (e.g., see [1], [6]). A pair can be produced in this region if the four-momenta of the emerging particles meet the condition yj = 0

⇒

u=u ˜l1 l2

⇒

2pf,− (ki pf ) = m2 ki,− , 2¯ pf,− (ki p¯f ) = m2 ki,− .

(23)

It is equivalent to the condition (15)

χ2 = δ2 (ψ − ∆) , (16)

cos ∠ (j, ni ) = cos ∠ ¯j, ni = 1, ¯ f − m2 /(2¯ where j = pf − m2 /(2pf,− ) n, ¯j = p pf,− ) n, ni is the direction of propagation of a incident photon. Thus in


3

where V is the normalization volume. For low-intensity laser waves (η1,2 1), multiphoton processes involve a small number of photons, the interference region is not expressed as the parameter z12 within it is much smaller than the parameters y1,2 outside. For intensities η1,2 ∼ 1 the probabilities of such processes are of the same order within and outside the interference region. For high- intensity waves (η1 ∼ η2 > 1), multiphoton processes involving a large number of photons occur, the probability of multiphoton processes in the interference region may considerably exceeds the probability of partial processes outside this region. In the interference region the probability of multiphoton processes in the interference region may considerably exceeds the probability of partial processes outside this region. Figure 1. Interference region: the incident photon is directed along the vectors j , ¯j

the interference region momentums of emerging particles lie in the plane formed by the vectors n, ki (see fig. 1).Therefore, pf , p¯f can be represented as a combination of the vectors n and ni . Coefficient of expansion we find using equations (23) and the conservation law for stationary points (19): 0 0 ni ; ni , p ¯ f = E∓ n + E∓ pf = E± n + E± 0 0 ¯f = E∓ + E∓ . E Ef = E± + E± , Here E± =

1 m 2 ∆± · (l1 ω1 + l2 ω2 ) , 2 m ˜ ∆± = 1 ±

0 E± =

1 ωi , 2˜ ul1 l2 ∆±

q 1−u ˜−1 l1 l2 ,

where signs ± correspond to the two possible solutions for the coefficient of expansion. The ejection angles of the pair θf = ∠(pf , n) and θ¯f = ∠(¯ pf , n) can be found from the expression !−1/2 sin2 θi , cos θf = 1 + 0 + cos θ 2 E± /E± i cos θ¯f =

1+

0 + cos θ E∓ /E∓ i

2

.

In the interference region the function Il1 l2 is simplified Il1 l2 = Jl1 (z12 ) · δl2 ,−δ1 δ2 l1 ,

Analysis of the formation of an e+ e– pair in the field of two pulsed laser waves has demonstrated that this process may occur in different kinematic regions—interference and noninterference ones. In the noninterference regions, pair formation mainly involves the independent interaction with the waves, and multiphoton effects are characterized by classical parameters y1 , y2 (14). In the interference region, pair formation mainly occurs due to the interaction with the interference wave field, and multiphoton processes are characterized by classical interference parameters z12 (15). Importantly, pair formation occurs in the plane defined by the directions of the incident photon and the direction of wave propagation. The wave photons involved in the interaction resulting in pair formation are strongly correlated in this regime. As a result, the total probability of wave photons involved in pair formation is equal to zero or ±1. For high-intensity waves, the probability of pair formation in the interference region may considerably exceed the probability of pair formation in any other kinematic region. The results can be tested in the experiments on the verification of quantum electrodynamics in the presence of strong fields (SLAC and FAIR). R EFERENCES

!−1/2

sin2 θi

III. C ONCLUSION

(24)

where δl2 ,−δ1 δ2 l1 is the Kronecker delta. The expression (24) indicates that the probability of one photon emission in the interference region differs from zero for partial processes when second wave emits or absorbs equal numbers of photons from the first laser wave. For processes l2 = −δ1 δ2 l1 the differential probability of pair formation by a nonpolarized photon in the interference region is given by 2 dE dW (l1 ,−δ1 δ2 l1 ) 2π 3 e2 m2 (0,0) f = (˜ ul1 l2 − 1) Fl1 ,−δ1 δ2 l1 , dΩf ωi V |pf | p¯f,−

[1] S.P. Roshchupkin and A.I. Voroshilo, Resonant and Coherent Effects of Quantum Electrodynamics in the Light Field (Naukova Dumka, Kiev, 2008), in Russian. [2] S.P.Roshchupkin and A.A. Lebed’. Effects of Quantum Electrondynamics in the Strong Pulsed Laser Fields (Naukova Dumka, Kiev, 2013), in Russian. [3] S.P. Roshchupkin, A.A. Lebed’, E.A. Padusenko, and A.I. Voroshilo, Quantum Optics and Laser Experiments, edited by S. Lyagushyn (Intech, Rijeka, 2012), pp. 107–156. [4] S.P. Roshchupkin, A.A. Lebed’, E.A. Padusenko. Nonresonant quantum electrodynamics processes in a pulsed laser field // Laser Physics, 2012, Vol. 22, No. 10, pp. 1513–1546 [5] S. P. Roshchupkin, A. A. Lebed’, E. A. Padusenko, A. I. Voroshilo. Quantum electrodynamics resonances in a pulsed laser field // Laser Physics, Vol. 22, No. 6 , pp. 1113-1144. [6] O. I. Voroshilo and S. P. Roshchupkin. Formation of an Electron–Positron Pair by a Photon in a Multifrequency Electromagnetic Field // Laser Physics. – 2000. – Vol. 10, No 5. – P. 1078-1085.

43


Fiber attenuation irregularities and simulation of Raman amplification band M.Y. Dyriv, G.S. Felinskyi, P.A. Korotkov

Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Abstract: Analytical expression of wavelength dependence of signal attenuation coefficient in the range of S+L+C telecommunication window is obtained using polynomial approximation in single-mode silica fibers: DCF, TrueWaveRSTM and pure SMF. It allowed correcting the threshold conditions of Raman amplification. It is shown that Raman gain threshold increased adjusted for wavelength irregularity of attenuation in comparison with threshold at constant attenuation of optical wave. It led to significant narrowing of Raman amplification band byturn, especially up to a few THz in DCF. Presented technique of simulation of Raman amplification band is useful for designing of wideband fiber Raman amplifiers.

Such parameter as amplification band is a result of Raman threshold condition of light amplification and plays an important role in synthesis of fiber Raman amplifiers (FRAs) based on effect of stimulated Raman scattering. Proposed Raman gain threshold [1, 2] is suitable for single-pass laser generation, which arises from Stokes noise for lack of outside optical signals. It is used for analysis of Raman lasers [3, 4]. Such way of approach isn’t applicable for FRAs [5, 6], in which outside optical wave’s amplification without own noises self-excitation is considered. Modeling of Raman pump power threshold without optical losses irregularity was proposed in [3-7]. However, this irregularity (or wavelength dependence of signal attenuation coefficient) effects on amplification bandwidth of FRA. That’s why the purpose of given paper is the development of methodology for estimation of Raman pump power threshold and the analysis of influence of threshold pumping change on wideband Raman amplification of optical signals in the main types of quartz fibers, namely dispersion compensating fiber (DCF), non-zero dispersion shifted fiber (TrueWaveRSTM) and pure quartz single-mode fiber (SMF). The differential equations set [5, 6] takes the single pumpsignal interaction into consideration. From this set the Raman pump power threshold has the next view under the conditions that Pp = const and dPs ( z ) = 0 on fixed frequency (or dz

wavelength):

on effective interaction fiber area Aeff . The expression (1) is a limiting condition between wave extinction and amplification in quartz fiber in the issue. It’s necessary to obtain analytical expressions for quantities, which are the part of equation (1) for calculating of threshold pump power. We use LevenbergMarquardt nonlinear regression algorithm with Gauss spectral decomposition method (for g R (ω ) ) and polynomial approximation (for α s (ω ) ) reproducing these coefficients directly from experimental data. We can estimate the accuracy of approximation of experimental curves by analytical curves due to mean-square error in standard least-square method: n

δ=

∑ (F (x ) − y ) i =1

i

n +1

i

2

,

(2)

where F ( xi ) – calculated function, yi – experimental function,

n – the total number of experimental points.

Experimental RGE profiles for SMF, DCF and TrueWaveRSTM are digitized from [8]. We realized RGE approximation with highest possible precision: < 0.5 % (DCF), < 0.7 % (TrueWaveRSTM) and < 1.8 % (SMF) using 10-component Gauss spectral decomposition. Wavelength dependence of signal attenuation coefficient is given in digital form by [9] for investigated types of single-mode fiber similarly. The analytical expression of polynomial approximation of function dependence α s = α s (λs ) is: n   α s (λs ) = α smin 1 + ∑ α k (λs − λs ,α min ) k , s  k =1 

where

(3)

α smin і λs ,α – minimal value of signal attenuation min s

coefficient and correspondent signal wavelength, α k – constant approximation coefficients, λs – operating signal wavelength. So as quantity of α s (λs ) changes within range 1450-1630 nm

almost the same for both fibers SMF and TrueWaveRSTM the least order of polynomial, which satisfactorily approaches (approximation accuracy is δ = 0.23·10 −3 1/km) digital data where α s (ω ) – optical attenuation coefficient of signal, α s = α s (λ s ) , is the third (Fig. 1). In the case of DCF (dispersion g R (ω ) – Raman gain efficiency (RGE) coefficient, normalized compensating fiber) polynomial order is the fourth (Fig. 2) with respective MSE δ = 0.25·10 −3 1/km.

Ppth =

α s (ω ) , g R (ω )

978-1-4799-0159-3/13/$31.00 ©2013 IEEE

44

(1)

0.060

Difference between curves becomes > 60 mW at wavelength 1462 nm; obtained curves almost match at 1576 nm. Such results are explained by that that coefficient α s at short

Experimental Calculated

wavelengths from the range 1460-1620 nm is larger than at 1550 nm, respectively threshold pump power value is larger in

0.056

min

TM

0.052

TrueWaveRS

0.048

SMF

1440

1480

1520

1560

1600

1640

Signal wavelength, nm Fig. 1. Signal attenuation coefficient in dependence of signal wavelength for SMF and TrueWaveRSTM. TABLE 1 Parameters of polynomial approximation of experimental curves of signal attenuation coefficient in typical quartz fibers Fiber types

DCF

TWRSTM

SMF

-1

0.102

0.050

0.047

, nm

1574

1570

1570

1.11

1.59

1.69

2.35

2.25

2.37

2.54

0.52

0.55

2.39

–

–

α

min s

, km

λs ,α min s

-4

-1

α1 ·10 , nm -5 -2 α 2 ·10 , nm -7 -3 α 3 ·10 , nm -9 -4 α 4 ·10 , nm

If we will use 3-order polynomial for approximation of −3 α s (λs ) in DCF the standard error will be δ > 1.1·10 1/km, which is not adequate as seen from Fig. 2. Parameters of polynomial approximation α s = α s (λ s ) (equation (3)) for investigated fibers are presented in Table 1. We calculated the value of threshold pump power in dependence of signal wavelength substituting obtained expressions for g R (λs ) and α s (λs ) in equation (1). Function dependences P = P (λs ) th p

th p

for all types of quartz fiber are plotted in Fig. 3. We observe signal amplification in the wide range of wavelengths (above upgraded curve pointed by black triangles): roughly up to 20.3 THz in dependence of pump power quantity exceeding pump power threshold. We corrected threshold data taking into consideration wavelength dependence of signal attenuation coefficient: its value increases within investigated range. Minimal pump threshold in DCF is ≈ 33 mW at wavelength 1552 nm. For TrueWaveRSTM and SMF at the same wavelength least threshold values are: dB/km) and

comparison with the case when α s = const = α s (at 1550 nm). Consequently, Raman pumps power threshold increasing leads to signal amplification bandwidth decreasing at constant pump power. Thereby, we obtained next corrections to amplification band for given types of quartz fibers: bandwidth reduces on 2.6 THz as maximum for DCF at pump power 125 mW, on 1.2 THz – for TrueWaveRSTM at pump power 250 mW and on 0.9 THz – for SMF at pump power close by 500 mW. Given values of pumping powers are chosen taking into consideration that threshold pump power is defined by RGE of each fiber type (see Fig. 3). Our simulation results show that taking wavelength irregularity of signal attenuation causes the growing of Raman pump power threshold close by edges of studied spectral range. It causes, by-turn, the narrowing of Raman amplification band in typical silica fibers. The technique of determination of threshold pump power in fiber-optical Raman amplifiers bases on analytical expressions of RGE and signal attenuation coefficients. Wideband threshold pump power correction is needed because any set pump power, which exceeds pumping threshold, clearly defines scope of amplification band. Accomplished quantitative analysis is adapted for creating generalized model of dynamics operation of fiber Raman amplifier and solution of synthesis tasks of such devices.

Attenuation coefficient, km-1

Attenuation coefficient, km-1


Experimental 3-th order polynomial 4-th order polynomial

0.14 0.13 0.12 0.11

DCF

0.10

1440

1480

1520

1560

1600

1640

Signal wavelength, nm Fig. 2. Signal attenuation coefficient in dependence of signal wavelength for DCF.

Ppthmin (TWRS ) ≈ 71 mW ( α s = 0.21

Ppthmin ( SMF ) ≈ 119 mW ( α s = 0.21 dB/km). Major

discrepancy between obtained curves is observed close by short wavelengths, as seen from Fig. 3.

45


[9]

Threshold pump power, W

1.4 SMF αs= const

1.0

αs=αs(λs)

TrueWaveRSTM

0.6

DCF 0.2

1480

1520

1560

1600

Signal wavelength, nm Fig. 3. Threshold pump power updating with consideration of wavelength dependence of signal attenuation coefficient in the edges of range 1460 – 1620 nm in fiber DCF. REFERENCES [1]

[2] [3] [4]

[5]

[6]

[7]

[8]

46

C. Lin, R.H. Stolen, W.G. French and T.G. Malone, “A cw tunable near-infrared (1.085-1.175-µm) Raman oscillator”, Opt. Lett., vol. 1, No. 3, pp. 96-97, 1977. R.H. Stolen, C. Lin, J. Shan and R.F. Leheny, “A Fiber Raman Ring Laser”, IEEE J. of Quant. Electron., vol. 14, No. 11, pp. 860-862, 1978. A. Bertoni, G.C. Reali, “1.24-μm cascaded Raman laser for 1.31-μm Raman fiber amplifiers”, Appl. Phys. B 67, 5-10. S.D. Jackson, P.H. Muir, “Theory and numerical simulation of nth-order cascaded Raman fiber lasers”, J. Opt. Soc. Am. B, vol. 18, No. 9, pp. 12971306, 2001. G.S. Felinskyi, P.A. Korotkov, “Raman threshold and optical gain bandwidth in silica fibers”, Sem. Phys., Quantum Electron. & Opt., vol. 11, No. 4, pp. 360-363, 2008. G.S. Felinskyi, P.A. Korotkov, “Lasing threshold for stimulated Raman generation of monochrome optical wave in single mode fibers”, Proc. 7th Int. Conf. on Laser and Fiber-Optical Networks Modeling, 29 Jun.- 1 Jul., Kharkiv, pp. 110-112, 2006. A.S. Samra, H.A.M. Harb, “Wide band flat gain Raman amplifier for DWDM communication systems”, Proc. IFIP Int. Conf. on Wireless and Optical Comm. Networks, Apr. 28-30, Cairo, pp. 15, 2009. J. Bromage, K. Rottwitt and M.E. Lines, “A method to predict the Raman gain spectra of germanosilicate fibers with arbitrary index profiles”, IEEE Photonics Techn. Lett., vol. 14, No. 1, pp. 24-26, 2002.

C. Headley, G.P. Agrawal, “Raman amplification in fiber optical communication systems”, Elsevier Academic Press, San Diego, CA, 2005.


Laser-modified Compton Scattering in the Middle-intensity Pulsed Field V.N. Nedoreshta, S.P. Roshchupkin and A.I. Voroshilo Institute of Applied Physics, National Academy of Sciences of Ukraine Sumy, Ukraine Telephone: 380 (542) 22 27 94 Fax: 380 (542) 22 37 60 Email: [email protected]

Abstract— The scattering of photon by electron in the presence of the field of the middle-intensity circularly-polarized pulsed laser wave is analyzed at a laser pulse duration that is significantly greater than the characteristic oscillation time. The probability of such process is calculated for a range of external field intensities I ∼ 1017 Wcm−2 . It is demonstrated that the resonance probability of laser-modified process can be significantly greater than the probability of the Compton effect in the absence of the external field. In the range of the optical frequencies for nonrelativistic energies this excess relates amounts to four orders-of-magnitude.

kf

kf

p i

+

q

f

ki

p f

p i

ki

p f

Fig. 1. Feynman diagrams for the Compton effect in the presence of the field of the pulsed optical wave

I. I NTRODUCTION High-power pulsed lasers whose field cannot be simulated using a plane monochromatic wave are used in the experiments on the verification of quantum electrodynamics. Thus, the theoretical works widely employ the model of the pulsed electromagnetic field that represents the four-potential with envelope (see, for example, [1]–[4])

In this work, we study the scattering of photon by electron in the presence of the pulsed light field for the resonance of direct diagram. In this paper we use relativistic units ~ = c = 1.

A (ϕ) = ag (φ) (ex cos ϕ + δey sin ϕ) ,

The scattering amplitude of the photon with the fourmomentum ki = (ωi , ki ) by electron with four- momentum pi = (Ei , pi ) in the presence of external field (1) is given by (Fig. 1) R 4 4 0 µ 0 ν 0 2 ¯ S f i∗= −ie d rd0 r Ψp∗f (r)γ 0 G(r, r )γ Ψpi (r )× (5) Aµ (kf r)Aν (ki r ) + Aν (kf r )Aµ (ki r) ,

(1)

where ϕ = (kr) = ωt − kr is the wave phase; r = (t, r) is the four-radius vector; φ = ϕ/ϕ0 , ϕ0 = ωτimp ; a = F/ω, F and ω are the field strength at the center of pulse and wave frequency in the laboratory coordinates; ex,y = (0, ex,y ) and k = (ω, k) are the four-polarization vector and four-momentum of the external field photon, such that k 2 = 0, e2x,y = −1, and (ex,y k) = 0; δ = ±1 corresponds to the circularly polarized wave; τimp is the pulse duration in the laboratory frame of reference. For the theoretical analysis, we choose the wave envelope given by g (φ) = exp −4φ2 . (2) The following condition is satisfied in the range of the optical frequencies and picosecond pulse durations: ϕ0 = ωτimp 1.

II. A MPLITUDE OF THE PROCESS

where pf = (Ef , pf ) and kf = (ωf , kf ) are the fourmomenta of the final electron and photon, γ ν (ν = 0, 1, 2, 3) are Dirac matrices, Ψpi (r0 ) is wave function of electron in the presence of rapidly oscillating field [7], G(r, r0 ) is the Green function of electron in the presence of field (1) [5], Aµ (ki r0 ) = p 2π/ωi eµ exp{−i(ki r0 )} is the wave function of photon, eµ is the four-polarization vector of photon. Amplitude (5) accurate to terms ∼ η 2 is written as

(3)

Therefore, it is expedient to consider ω as the frequency of the quasimonochromatic field. Nonlinear effects of the process of is governed by classical relativistic-invariant parameter η [5], [6], which determines the wave intensity at the center of the pulse: η = |e| a/m. (4)

Sf i ≈ Bδ (2) (pi,⊥ +ki,⊥ − pf,⊥ − kf,⊥ )× δ(pi,− + ki,− − pf,− − kf,− )e0 eµ · u ¯pf Tfvµ i upi , Tfνµ i = η2

P

j [η

0

(j)νµ T˜0,0 +

(j)νµ

T0,0

P

+η

l,l0 ∈(|l−l0 ||l0 |=1) !

P l,l0 ∈(|l−l0 |+|l0 |=2)

(j)νµ

(6)

(j)νµ

Tl−l0 ,l0 + (7)

Tl−l0 ,l0 ], j = e, d,

978-1-4799-0159-3/13/$31.00 ©2013 IEEE

47


−1 2 −2 p 3 where B = −ie2 (2π) 2 ωi Ei ωf Ef V 2 ϕ0 ω is the normalization factor; pi,⊥ , ki,⊥ , pf,⊥ and kf,⊥ are the projections of the corresponding vectors along the wave polarization plane; pi,− = Ei −pi,z , ki,− = ωi −ki,z , pf,− = Ef −pf,z , and kf,− = ωf −kf,z are differences between the zero components of the corresponding four-momenta and their projections along the wave propagation direction; l, l0 = 0, ±1, ±2, l0 are integers, corresponding to the difference between the number of absorbed and emitted wave photons during the whole process (d)νµ (d)νµ and the beginning of the process respectively; Tl−l0 ,l0 , T˜0,0 (e)νµ are matrix elements that relate to the direct diagram, Tl−l0 ,l0 = (d)νµ (e)νµ (d)νµ Tl−l0 ,l0 , T˜0,0 = T˜0,0 (q → f, kf ↔ −ki ) – to exchange diagram; q and f are the momenta of the intermediate particles that correspond to the direct and exchange diagrams in Fig. 1, respectively: q = pi + ki ,

f = pi − kf .

−∞

Z∞ f1 (a−ζl0 ) =

β=

ql20 − m2 ϕ0 4(kql0 )

(10)

is resonant parameter [1], [3], [8]. Therefore, we can conclude that the corrections to the probability related to the effect of wave is ∼ η and hence for its small values are insignificant. However, the situation changes dramatically when virtual electron come to the mass shell, ie one of the following conditions is satisfied: β (ql0 ) . 1 (ql20 = m2 ),

β (fl0 ) . 1 (fl20 = m2 ).

(11)

In this case the corrections to the probability related to the effect of wave are significantly greater than the probability in the absence of external field. We take into account that the resonance of direct diagram is possible only when one photon of field is emitted at the beginning and one photon is absorbed at the end of the process [8]. Moreover, when the condition for the resonance of direct diagram is satisfied, the contribution of exchange diagram to the total probability of scattering is proportional to ∼ ϕ−1 1 and, hence, can be neglected. 0 Thus, in (7) should take into account only bispinor matrix (d)νµ T1,−1 , which is given by  ∞  Z qˆ + m (d)νµ µ f1 2 T1,−1 = M1ν  f−1 dqz  M−1 , (12) q − m2 −∞

√

g (φ) eiϕ0 [(a−ζl0 )φ+

2π 2 u0 8 η u ˜

√ 2φ)]

erf(2

dφ, (15)

−∞

when parameters u,u0 belong to the interval [0,˜ u1 ]: u=

(kki ) 0 (kkf ) 2(kq) , , u= , u ˜1 = (kpi ) (kpf ) m2

(16)

a, ζl0 are invariant parameters determined from the equations: pi + ki = q − (1 + ζl0 )k, (17) pi + ki − ak = pf + kf . In the area of the wave intensity u ˜1 1 u ˜1 1 η2 . , 0 , u ϕ0 u ϕ0

(18)

the terms ∼ η 2 can be disregarded in the argument of the exponential function in expressions (14), (15). Then, the simple analytical representation of function f±1 is written as 1 √ −ϕ20 (a−ζl0 )2 16 . πe 2 (19) In the case of middle intensity when a parameter η satisfies u 1 ϕ0 η 2 . 20, η 2 1, (20) u ˜1 f−1 (ζl0 ) =

where

(13)

Here y0 (pf , q), x ≡ x(pf , q) are kinematic parameters. Functions f±1 in (12) can be written as follows Z∞ √ √ 2π 2 u f−1 (ζl0 ) = g (φ) eiϕ0 [ζl0 φ− 8 η u˜1 erf(2 2φ)] dφ, (14)

(8)

Note that, in amplitude (6), the terms proportional to the zero power of η determine the amplitude of the Compton effect in the absence of external field. The terms that are proportional to the first power of parameter η determine the corrections related to the process involving one photon of wave. The terms that are proportional to the second power of parameter η 2 correspond to the involvement of two photons. An order-ofmagnitude estimation of the probability of such processes is given by 1, β . 1, (l−l0 ,l0 ) 2 2(|l−l0 |+|l0 |) W ∼ φ0 η · (9) β −2 , β 1,

48

m 2 h (−) ν ˆ (−)ν i εˆ k − kε + 4 (kq) m 1 1 ˆ ν + y0 (pf , q) e−ix γ ν . − εˆ(−) kγ 4 (kpf ) (kq) 2 M1ν (pf , q) =

1 √ −ϕ20 ζl20 πe 16 , 2

f1 (a−ζl0 ) =

the terms ∼ η 2 can not be neglected in the argument of the exponential function in expressions (14), (15). Therefore, using the expansion of exponent inp the Fourier series in the interval −∆ ≤ φ ≤ ∆, where ∆ = ln(ϕ0 )/4, function f±1 take the form: √ nX 2 ϕ2 π 1 max π 0 (ζl0+n1 λ) 16 , λ= , (21) f−1 (ζl0 ) = an1 e− 2 n =n ϕ0 ∆ 1

1 min

√ f1 (a − ζl0 ) =

π 2

nX 2 max

bn2 e−

2 ϕ2 0 (a−ζl0 +n2 λ) 16

.

(22)

n2 =n2 min

Substitute the expressions for f±1 in (12): I(β, a) =

nX 2 max

nX 1 max

bn2 an1 I (n1 ,n2 ) (βn1 ,a0 ),

(23)

n2 =n2 min n1 =n1 min

where 2 02 0 ϕ2 π 0 a +8(βn1 −ϕ0 a /4) 16 I (n1 ,n2 ) (βn1 , a0 ) = e− × 4(kql0 ) ( ! ) √ 2 (βn1 − a0 ϕ0 /4) erfi +i , 2

(24)


of the contributions of the front and posterior segments of the pulse. Also the resonant frequency range of the initial photon increase with the intensity increasing. In contrast to the area (18) function Pres (β) depends on the parameter u0 and the integration in (27) impossible to make analytically. The ratio of the total resonance probability of the scattering of photon by electron via direct diagram (27) to the total probability of the Compton effect in the absence of external field is Wfresi τimp ≈ R(u, u ˜1 , β), (31) WComp T R(u, u ˜1 , β) = Zu˜1

2η 4 ϕ20 2 π u ˜1 F (˜ u1 )

[f (u0 ,˜ u1 )f (u,˜ u1 )−g(u0 ,˜ u1 )g(u,˜ u1 )]

×

Pres (β) du0 , (1 + u0 )2

(32)

0

Fig. 2. Dependence of the resonance profile on the resonant√parameter β in the intense pulsed light field (ω = 2.35 eV, T /τimp = 2). Field intensity in pulse peak is (solid line) I = 7 × 1016 W cm−2 , I = 2 × 1017 W cm−2 (dashed line), I = 4.4 × 1017 W cm−2 (dash-dotted line), I = 6.3 × 1017 W cm−2 (dotted line).

a0 = a + (n1 + n2 ) λ,

βn1 = β − (ϕ0 /2)n1 λ.

(25)

The characteristic range of variation of parameters a, β: 0 . β . ϕ0 η 2

u u0 u , 2η 2 . a . 2η 2 . u ˜1 u ˜1 u ˜1

(26)

III. R ESONANT PROBABILITY In the resonance approximation q 2 = m2 (except for function I(β, a)) the probability of direct diagram averaged over the initial and summed over final polarizations and integrated over the azimuthal angle of scattering photon is given by Wfresi ≈ Zu˜1

2e4 η 4 m2 2 ϕ τimp × πωi Ei V u ˜1 0

Pres (β) du0 . [f (u0 ,˜ u)f (u,˜ u1 )−g(u0 ,˜ u1 )g(u,˜ u1 )] (1 + u0 )2

(27)

0

g(u, u ˜1 ) =

u 1− u ˜1

(2 + u)(˜ u1 − 2u)u . 2˜ u1 (1 + u)

,

IV. C ONCLUSION We draw the following conclusion based on the analysis of the laser-modified Compton scattering in the middle-intensity pulsed field. The resonance probability of the Compton effect in the presence of the intense laser field can be significantly greater than the probability in the absence of the external field. The results can be tested in the experiments on the verification of quantum electrodynamics in the presence of strong fields (SLAC and FAIR). R EFERENCES

Here, we use the notation: u u2 −4 f (u, u ˜1 ) = 2 + 1+u u ˜1

where T is the observation time (T & τimp ), which is determined by the experimental conditions, 4 8 1 1 8 F (˜ u1 ) = 1− − 2 ln(1+˜ . (33) u1 )+ + − u ˜1 u ˜1 2 u ˜1 2(1+ u ˜1 )2 Calculations showed that resonance probability can be significantly greater than the probability of the Compton-effect in the absence of external field. For example, the probability ratio is R ∼ 104 under the condition η = 0.25 and u 1 in the range of the optical frequencies Ei /m m/ω ∼ 105 .

(28) (29)

Behavior of the probability in the resonance area is determined by the resonance profile Pres (β) (Fig. 2): Z∞ 1 2 Pres (β) = |I(β, a)| da. (30) 2π −∞

Fig 2 shows that increasing of the field intensity involves the appearance of new peaks. This is the result of the interference

[1] S. P. Roshchupkin and A. A. Lebed Effects of quantum electrodynamics in the strong pulsed laser field, Kiev, Ukraine: Naukova Dumka, 2013. [2] N. B. Narozhny and M. S. Fofanov, ”Photon emission by an electron colliding with a short focused laser pulse”, Zh. Eksp. Teor. Fiz., vol. 110, pp. 26-47, 1996. [3] S.P. Roshchupkin, A.A. Lebed, E.A. Padusenko, and A.I. Voroshilo, ”Quantum electrodynamics resonances in a pulsed laser field”, Laser Phys., vol. 22, pp. 1113-1144, 2012. [4] S. P. Roshchupkin, A. A. Lebed, and E. A. Padusenko, ”Nonresonant quantum electrodynamics processes in a pulsed laser field”, Laser Phys., vol. 22 pp. 1513-1546, 2012. [5] V.I. Ritus and A.I. Nikishov, Quantum electrodynamics phenomena in the intense field, (Tr. FIAN vol 111), Moscow: Nauka, 1979. [6] S.P. Roshchupkin and A.I. Voroshilo, Resonant and coherent effects of quantum electrodynamics in the light field, Kiev, Ukraine: Naukova Dumka, 2008. [7] D.M. Volkov, ”Electron in the field of plane unpolarized electromagnetic waves from the standpoint of the Dirac equation”, Zh. Eksp. Teor. Fiz., vol. 7, pp. 1286-1289, 1937. [8] A.I. Voroshilo, S.P. Roshchupkin, and V.N. Nedoreshta, ”Resonant scattering of photon by electron in the presence of the pulsed laser field”, Laser Phys., vol. 21, pp. 1675-1687, 2011.

49


Interference e − µ Scattering in Two-mode Pulse-wave Laser Field O. Denisenko and S. Roshchupkin Institute of Applied Physics 58, Petropavlovskaya, St., Sumy, 40030 Ukraine Email: [email protected]

Abstract— Nonresonant process of electron scattering at the muon in the two-pulse field circularly polarized moderately strong electromagnetic waves in the interference region (IO) has been studied theoretically. A significant increase of the partialscattering cross section of the process is found even in case of moderate laser pulse intensities.

II. A MPLITUDE Let’s write the two-mode pulse-wave laser four-vector potential A for circular polarization as A=

2 X

Aj (ϕj ),

(4)

ϕ j aj · [ejx cos ϕj +ejy sin ϕj ], ωj τj

(5)

j=1

I. I NTRODUCTION The rapidly evolving laser technology makes it possible to achieve the extremely intense laser fields as the field strength accelerates an electron up to the speed of light within the optical cycle duration. The intensities are provided with essentially short laser pulses [1], [2]. In such fields the new interesting nonlinear effects are expected to occur [3], [4] . In the present paper we consider the nonrelativistic electronmuon scattering in a two-mode pulse-wave field in the special kinematic region [5]. We define the two-mode field as a superposition of two plane-waves laser pulses with frequencies ω1 and ω2 , propagating in the same direction. We also considered a situation when the driving laser pulses are still sufficiently long to be interpreted as the quasi-monochromatic plane waves. In such a case we assume that for pulse duration τj (j = 1, 2) the condition ωj τj 1

(1)

is fulfilled. The probabilities of the process modified by the intensive electromagnetic field are regulated by two invariant parameters whose orders of magnitude are determined by the relations eE η= (2) mω and γ=η

mv , ω

(3)

where E is the amplitude of the wave field, ω is the frequency of the wave. The electron mass is denoted by m, and the electron charge by e. The reduced strength of the e.m. field (2) becomes η ∼ 1 for the optical-frequency lasers with the pulse amplitude E ∼ 1011 V/cm. The Bunkin-Fedorov quantum parameter (3) arises in the processes with the Coulomb interaction. Throughout the paper, we use relativistic natural units such that ~ = c = 1 978-1-4799-0159-3/13/$31.00 ©2013 IEEE

50

Aj = gj

where ϕj = kj x, kj = (ωj , kj ) is the 4-vector of the j-th wave pointed the z direction out. We take the four-vectors ej(x,y) as a basis for polarization that satisfy the conditions kj2 = 0,

kj · ej(x,y) = 0,

(6)

And the gj (ϕj /ωj τj ), which we require to equal gj (0) = 1 at the center of the pulse, and to decrease exponentially for |ϕj | ωj τj is the envelope of the potential slowly varying with ϕj . Then the quantity τj can be regarded as the pulse duration. The Volkov solution in the field of a circularly polarized pulse-waves with the potential (4)–(5) has the form   2 ˆ ˆ X e kj ·Aj (kj x)  up iSp (x) p e , (7) ψp (x|A) = 1+ · 2 j=1 kj p 2Ep where e Sp (x) = −p·x− κp

ξ(x) Z

e p·A− A2 dξ, 2

(8)

−∞

is the classical action of an electron in the field with the potential (4)–(5) and p is the 4-momentum of a free electron. The element of the S matrix for the process has the form ZZ 2 Sf i =ie d4 x1 d4 x2 Dµν (x1 −x2 ) ·[ψ¯10 (x1 |A)γ µ ψ1 (x1 |A)] ·[ψ¯0 (x2 |A)γ ν ψ2 (x2 |A)]. 2

(9)

Here, Dµν (x1 −x2 ) denotes the propagator function of the free photon Z d4 q 4πeiq(x1 −x2 ) µν Dµν (x1 −x2 ) = g (10) (2π)4 q2 and γ µ are the Dirac matrices.


The integrand in Eq. (9) is a linear combination of the harmonic functions, to within a slow dependence of the envelope of the potential, so it can be expanded in a Fourier series over the interval [ϕ, ϕ + 2π]. ∞ X

Gp (ϕ1,2 ) =

Hpn1,2 (ϕ1,2 )ei(n1 ϕ1 +n2 ϕ2 ) ,

(11)

The quantum parameters αp±

1 1 − 0 κp κp

(21)

are the main multiphoton parameters outside the BunkinFedorov kinematic region.

n,r=−∞

III. I NTERFERENCE REGION

where Hpn1,2 (ϕ1,2 )

1 |d∓ | = η1 η2 · m 2 2 ω1 ± ω2

1 = (2π)2

Z Zπ

Lets consider the situation when parameters γjp (19) are negligibly small. Such a situation occurs when the vector gpp0 (20) is directed along (or vise verse) the wave vector n.

dϕ01 dϕ02 Gp (ϕ01,2 , ϕ1,2 )

−π 0

0

· e−i(n1 ϕ2 +n2 ϕ2 )

(12)

Hence, for moderately strong fields η 1, the element of the S matrix (9) is used in the following form X Sf i = S (`1 `2 ) , (13) `1 `2

where the partial amplitude S (`1 `2 ) with absorption (`1,2 < 0) or emission (`1,2 > 0) of the `1 photons of the first mode and the `2 photons of the second one are given by the expression: ie2 (2π)4 δ(~q⊥ )δ(q− ) Eµ0 Eµ Ee0 Ee X 1 Z 1 d(τ2 φ)e 2 q+ (τ2 φ) [¯ u0µ γj uµ ][¯ u0e γ j ue ] · 2 q ss0 1 τ2 τ2 I`1 −s,`2 −s0 φ, φ Is,s0 φ, φ . τ1 τ1

Fig.1: Interference plane

S (`1 `2 ) = − p

(14)

For the whole process, we have both momentum and energy conservation, as given by the Dirac δ-function in Eq. (14) q = p0e + p0µ − pe − pµ + `1 k1 + `2 k2 ,

(16)

Ω± = ω1 ± ω2 .

−π

can be expressed in terms of ordinary Bessel function. Its argument is now a slow function of ϕ and Sp0 = γ1p sin(ϕ1 −χ1 ) + γ2p sin(ϕ2 −χ2 )− αp+

αp−

sin(ϕ1 +ϕ2 −τ− ) − sin(ϕ1 −ϕ2 −τ+ ), m γjp = ηj [(ejx · gpp0 )2 + (ejy · gpp0 )2 ]1/2 . ωj

(22)

and the scattering in the two-mode field retains the same form as in the monochromatic field but with absorption or emissions the photons of combinational frequencies

Here, as is typical for occurrence of two waves, the functions I`s determining the multiphoton processes Zπ Zπ 0 1 I`1 `2 (p) = dϕ1 dϕ2 ei(Sp −`1 ϕ1 −`2 ϕ2 ) , (17) (2π)2 −π

I`1 `2 → ein± χ± Jn± (α± )

(15)

And the propagator momenta are given as q1 = p0µ −pµ +(`1 −s)k1 +(`2 −s0 )k2 ,

In fact, this is true only if an electron (and a muon) is scattered in the plane formed by the initial electron (and muon) momentum and the wave vector. This plane is called the interference region. In this region the functions (17) are considerably simplified

(18) (19)

Note that the quantum parameter γjp (19) is the same BunkinFedorov quantum parameter as in Eq. (3) with the kinematic factor (ejy · gpp0 ) p0 p tan χj = , gpp0 = − . (20) (ejx · gpp0 ) κp0 κp

(23)

In that plane with azimuthal angles of the electron in the initial and final states being equal ϕf = ϕi ,

(24)

the polar angles and the energies are linked by the following relation pf sin θf pi sin θi = , (25) κf κi from which one can obtain the expression for scattering angle in the interference plane q θint = 2 arctan(ρn± ± ρ2n± − sin2 θi )/ sin θi (26) where ρn±

pf ≡ = pi

s 1−

2n± Ω± mvi2

(27)

In Fig.2 we plot the n+ dependence of the interference angle for the set of the initial angles θi

51


V. C ONCLUSION We have considered the partial cross section of electronmuon scattering propagating through a two-mode laser pulse. The process takes place in the special interference region for non relativistic energies and moderately strong fields. Under these conditions the interference region increases the cross section up to two orders of magnitude relative to the Bunkin-Fedorov region. We found a significant dependence of the scattering angle in the interference plane θint from the amount of n+ photons of combinational frequencies. For the fixed initial angles the interference conditions is performed for a limited number of partial processes. In the interference region the partial probabilities of the processes of emission and absorption at a fixed scattering angle acquire additional broadening due to pulsed character of the external laser field. The received results can be tested at FAIR project.

Fig.2: Interference angle θint as a function of n+ for different θi . IV. C ROSS SECTION For partial cross section in the interference region and the center-of-mass system, we get dσ (n± ) 4m2 = α2 4 ρn± Wn± , dΩ qn±

m1 m2 , m= m1 + m2

(28)

and the partial probability is proportional to √

Wn+

Z 2 1/ 2 ∼ Jn+ [α(φ)] exp[i(ζωτ )φ] dφ .

(29)

√ −1/ 2

The broadening effect due to the Gaussian envelope of the potential ∼ exp(−φ2 ) is illustrated in Fig.3. The partial probability Wn+ Eq. (29) as a function of θf calculated in close proximity to the interference angle (26) The parameters employed are ω ≡ Ω+ = 5 eV, ωζ = Const·(θf −θint ), ωτ = 50, θint = 143.696◦ , η1 = η2 = 0.01 (or 5.2×108 V/cm, or 7×1014 W/cm2 ), θi = 30◦ , and v = 0.1.

Fig.3: Probability Wn+ vs θf for n+ = 1. The relatively fast oscillations with θf are caused by occurrence of cosine function in Eq. (29).

52

R EFERENCES [1] S. V. Bulanov, G. A. Mourou, and T. Tajima, “Optics in the relativistic regime,” Rev.Mod.Phys., vol. 78, pp. 310–371, 2006. [2] A. D. Piazza, C. Mller, K. Z. Hatsagortsyan, and C. H. Keitel, “Extremely high-intensity laser interactions with fundamental quantum systems,” Rev. Mod. Phys. vol., vol. 84,, p. 1177, Nov. 2011. [Online]. [3] S. Roshchupkin and A. Voroshilo, Resonant and Coherent Effects of Quantum Electrodynamics in the Light Field. Naukova Dumka, Kiev, 2008. [4] S. Roshchupkin and A. Lebed, Effects of Quantum Electrodynamics in the Strong Pulsed Laser Field. Naukova Dumka, Kiev, 2013. [5] S. Roshchupkin, “The interference effect in the scattering of an electron by a nucleus in the field of two plane electromagnetic waves,” J. Exp. Theor. Phys., vol. 106, no. 1(7), pp. 102–118, 1994.


1

Energy loss of a charged particle in an electron gas in the second Born approximation Mykhailo Diachenko Institute of Applied Physics, NAS of Ukraine, Sumy 40000, Ukraine Email: [email protected]

Abstract—The scattering of a charged particle moving in an electron gas by quantum field theory methods has been reviewed. Probability of the process in the first Born approximation has been obtained (without the use of the Green’s function). The analysis of the general expression for the probability in the second Born approximation has been done.

I. I NTRODUCTION In modern high energy physics for experiments with colliding beams of heavy and light charged particles is very important to compress the beam, to reduce its size and momentum spread. The best known method of cooling the charged particles is electron cooling. It is used in modern particle accelerators. The idea of the electron cooling method suggested by Budker in 1966 [1]. The idea of electron cooling is simple. A charge particle beam is passed through parallel intensive an electron beam with the same average velocity in straight part of storage ring. The electron beam has smaller momentum spread in comparison with the charge particle beam. Another words the electron beam is colder. Then hot charge particle gas is cooled by cold electron gas because of Coulomb interaction. As a result, the beam phase volume is decreased over all degrees of freedom and the beam is compressed. The first experiments carried out in Novosibirsk in 1974 demonstrated the high efficiency of this method and triggered all the development of heavy particle cooling methods and made an important step towards the realization of experiments with proton-antiproton colliding beams. Despite the fact that electron cooling is widely used in particle accelerators now, there are several theoretical problems: 1) account of the temperature and the temperature anisotropy of the electron gas in the energy loss of charged particles moving through the gas; 2) diference in friction force for the positively and negatively charged particles moving in magnetized electron plasma [2], [3]. For the first time the difference between the energy loss of positively and negatively charged particles was observed at ”MOSOL” facility [2], [3]. This experiment gave important information about the frictional force of charged particles moving in a magnetized electron gas (magnetic field up to 4 kG) (Fig.1) and dependence of the energy loss of particles with different signs of charges on the magnetic field (Fig.2). Experiments showed that there is a significant dependence of 978-1-4799-0159-3/13/$31.00 ©2013 IEEE

the friction force for the positively and negatively charged particles on the magnetic field. For a long time for the theoretical description of electron cooling the pair collisions method was used but it is not enough to address contemporary theoretical issues in electron cooling. Alternative theories are dielectric model and quantum field theory methods [4]-[9]. In this paper we used quantum-field approach to explain the differences between the forces of friction positive and negative charged particles.

Fig.1. Change of energy of ions of different signs as a function of electron energy [2]

Fig.2. Maximum friction force as a function of magnetic field [2]

II. S CATTERING OF CHARGED PARTICLE IN AN ELECTRON GAS

The system of interacting electrons and a charged particle passing through an electron gas is considered. Hamiltonian of the system of interacting electrons has the form ∑ 1∑ + (1) V⃗k a+ H= εp⃗ a+ ⃗+ p ⃗ ap⃗′ ap⃗′ −⃗ p ⃗ ap k ap ⃗+⃗ k, 2

53


and hamiltonian interaction of charged particle with the electron gas can be written as ∑ + Hi = V⃗k a+ (2) p ⃗α ⃗ αp⃗1 ap ⃗−⃗ k, p⃗1 −k

a+ ⃗ p ⃗ , ap

where are creation and annihilation operators of an electron with momentum p⃗, V⃗k is the Fourier component of the interaction potential, αp+ ⃗ are creation and annihilation ⃗ , αp operators of a charged particle with momentum p⃗. The charged particle is fast enough (e2 /¯ hv ≪ 1). Then the probability of transition in the first Born approximation is expressed by the well-known formula of quantum mechanics w⃗k = 2π| < m, p⃗1 − ⃗k|Hi |n, p⃗1 > |2 × × δ(Em − En − εp⃗1 + εp⃗1 −⃗k ),

(3)

where n, m are the initial and final ∑states of the electron gas, | < m, p⃗1 − ⃗k|Hi |n, p⃗1 > | = V⃗k ( p⃗ ap+ ⃗ ap ⃗−⃗ k ). After averaging over the initial states of the electron gas and summing over the final states the full probability has the form W⃗k = 2πV⃗k2 Φ⃗k (εp⃗1 − εp⃗1 −⃗k ), (4) where Φ⃗k (ω) =

∑ ∑ 2 β(Ω+µNn −En ) |( a+ × p ⃗ ap ⃗−⃗ k )mn | e mn

p ⃗

(5)

× δ(Em − En − ω). In the works of Larkin and Akhiezer [4], [5] the communication function (5) with the Green’s function was used ˜ ⃗k, ω) ImK( , Φ⃗k (ω) = π(1 − e−βω )

(6)

where ΦT (⃗r, τ ) = Sp[eβ(Ω+µN −H) ψ + (r⃗1 , τ1 )ψ(r⃗1 , τ1 )×

(10) × ψ + (r⃗2 , τ2 )ψ(r⃗2 , τ2 )], ∑ where ψ(⃗r, τ ) = eHτ p⃗ ap⃗ ei⃗p⃗r e−Hτ , ξs is the Matsubara frequency. The function (10) can be written as a series in degrees of the interaction 1 {Sp[eβH0 ψ¯D (r⃗1 τ1 )× ΦT (⃗r, τ ) = βH Sp[e 0 σ(β)] (11) × ψ D (r⃗1 , τ1 )ψ¯D (r⃗2 , τ2 )ψ D (r⃗2 , τ2 )]},

where σ(β) is the temperature scattering matrix and it is given by ∫ ∫ β ∞ ∑ (−1)n β dτ1 ... dτn Tτ {H1 (τ1 )...H1 (τn )}. σ(β) = n! 0 0 n=0 (12) Function (11) in one-loop approximation can be represented in a diagram Feymana (Fig. 3).

Fig.3. Feynman diagram for the temperature function ΦT (⃗k, ξs ) in one-loop approximation

Then the Dyson equation has the form ΦT (⃗k) = Πn (⃗k) + Πn (⃗k)V⃗ ΦT (⃗k). n

k

n

The total transition probability has the following form 2V⃗k2 Π(⃗k, ω) W⃗k = . Im −βω 1−e 1 − V⃗k Π(⃗k, ω)

(13)

(14)

˜ ⃗k, ω) is a spectral Green’s function. Also the comwhere K( munication between the temperature Green’s function (for which can be applied diagrammatic technique) and time Green’s function was used in [4], [5] which has the form

Finally, the energy loss of the charged particle in the electron gas has the form ∫ dE (15) = (2π)−3 (εp⃗1 − εp⃗1 −⃗k )W⃗k d3 k. dt

˜ ⃗k, ξn ). Kn (⃗k) = K(

III. C ONCLUSION In this paper a new method of calculation of energy loss of a charged particle moving in an electron gas is shown. The expression for the function Φ(⃗k, ξs ) agrees well with Eq.(6) which was obtained in Larkin and Akhiezer [4], [5]. Analysis of the general expression for the probability of the process in the second Born approximation shows that cross-term accounts for the dependence on the sign of the charge of the particle.

(7)

It is necessary to consider the second Born approximation to solve the problem differences friction force for positively and negatively charged particles, then the probability has the form w⃗ = 2π| < m, p⃗1 − ⃗k|Hi |n, p⃗1 > − k

−

∑ < m, p⃗1 − ⃗k|Hi |l >< l|Hi |n, p⃗1 > |2 × El − En − εp⃗1

(8) R EFERENCES

l

× δ(Em − En − εp⃗1 + εp⃗1 −⃗k ), where l is an intermediate state. The dependence on the sign of the charge of the particle is determined by the cross-term. In this case a connection between probability and the temperature Green’s function is not simple, so it is necessary to calculate by the direct method. In this paper we use the Fourier transform and Mattsubara technique. Then the function (5) has the following form Φ(⃗k, ω = iξs ) =

54

2

1 ImΦT (⃗k, ξs ), ξs > 0, π(1 − e−iβξs )

(9)

[1] G.I. Budker, A.N. Skrinsky, Physics-Uspekhi, 124, 561-595, (1978). [2] N.S. Dikanskii, N.Kh. Kot, V.I. Kudelainen, V.A. Lebedev, V.V. Parkhomchuk, Zh. Eksp. Teor. Fiz., 94, p.65-73, (1988) [3] I.N. Meshkov, Phys. Elementery Part. Atomic Nucl., 25, 1487 (1994). [4] A.I. Larkin, Zh. Eksp. Teor. Fiz., 37, 264-272, (1959). [5] I.A. Akhiezer, Zh. Eksp. Teor. Fiz., 40, 954-962, (1961). [6] V.A.Balakirev, V.I.Miroshnichenko, V.E.Storizhko and A.P.Tolsoluzhsky, Problem of Atomic Science and Technology, 53, 181-185, (2010). [7] V.V. Parkhomchuk, A.N. Skrinsky, Physics-Uspekhi, 43, 433-452, (2000). [8] V.V. Parkhomchuk, Nuclear Instruments and Methods in Physics Research A 441, 9-17, (2000). [9] M.M. Dyachenko, V.I. Miroshnichenko, R.I. Kholodov, Reports of National Academy of Sciences of Ukraine 10, 70-76 (2012)


Calculation of the Electron Thermal Plasma Permittivity Oleksii Khelemelia Institute of Applied Physics, Sumy, Ukraine Email: [email protected]

Abstract—The expression for the permittivity of the electron gas with considering the temperature in the framework of quantum field theory is numerically calculated . The results are compared with analytical expressions that take into account temperature of the electron gas in the linear approximation. Index Terms—plasma permittivity, energy loss, MPI, electron cooling, HESR.

I. I NTRODUCTION In 1967 G.I. Budker [1] proposed the electron cooling method of charge massive particle beam (damp volume space). One combines of moving “hot” ion and “cold” electron beams at a some location of the storage ring. Due to the Coulomb interaction beams temperatures equalized then ”warmed up” electrons are removed from the system drive. The cooled ion beam continues to move in a storage ring. Charmonium spectroscopy, which is one of the main items in the experimental program of HESR (modern facilities High Energy Storage Ring, Germany, FAIR Collaboration), requires antiproton momentum up to 8.9 GeV/c with a resolution ∆p/p ∼ 10−5 . This can be achieved only with electron cooling. Originally Coulomb theory of binary collisions is used to describe the process of electron cooling. The advantage of this method is evident in the construction of the theory [2]–[4]. Also, the friction force can be obtained by the methods of plasma physics (dielectric model). Here the properties of the medium are given by the dielectric constant of the plasma, which determines the form of the expression for the energy loss of ions moving in the electron gas [5]. The quantum field theory method is devoid of a disadvantage related with the introduction of empirical values which previous theories used such as the cutoff parameters in the Coulomb logarithm. Within the quantum field theory the Coulomb logarithm contains values determined from the first principles [6], [7]. But in frame of these models results can be obtained analytically only in some limited cases [1]–[7]. In the general case the dependencies must be evaluated numerically. In first section basic principles of calculation of projectile particle energy losses in electron gas are considered. The second section is devoted to the numerical computation. II. E NERGY L OSSES OF A PARTICLE IN A E LECTRON G AS Let consider the basic steps of quantum field theory to the description of electron cooling.

The Hamiltonian of the system of charged particles interacting by Coulomb’s law with the passing through of the particles can be written as [6], [7], [10] H = H0 + H ′ (t),

(1)

where H0 is the main Hamiltonian of nonperturbative system of plasma particles, the second term H ′ (t) , the Hamiltonian of interaction, describes the perturbation introduced by the projectile particle: ∫ ′ H (t) = d⃗rJ0 (⃗r, t)a0 (⃗r, t), (2) ∫ −1 a0 (⃗r, t) = (4π)−1 d⃗r′ |⃗r − ⃗r′ | j0 (⃗r′ , t). a0 is operator of a scalar potential, j0 and J0 are operators of the charge density of the system and of the projectile particle, respectively, the operators are taken in the interaction representation. The elements of the scattering matrix    ∫∞  S = T −i H ′ (t) dt (3)   −∞

connects the various states of the original system and external particle. These states are characterized by the quantum numbers α, n, where α ≡ (p, q) are quantum numbers of the projectile particle and n is set of quantum numbers describing the state of the environment with a certain energy En and a certain number of particles Nn . We assume ( the speed of )the moving particle V is large enough that e2 V −1 ¯h−1 ≪ 1 its interaction with the particles of the medium can be considered by perturbation theory. In the linear approximation in H ′ (t) probability of transition from the initial state α, n to the final state α′ , n′ has the form Wq⃗ =

2Vq2 Π(⃗q, ω) ℑ 2 , 1 − exp (−βω) q − 4πΠ(⃗q, ω)

(4)

where ¯hω = Ep⃗i − Ep⃗i −¯hq⃗ , β = 1/T is the factor of inverse ∫ p2 1 temperature, Ep⃗i,e = 2mi,ei,e , n = (2π¯ np d3 p, Π(⃗q, ω) is h)3 the polarized operator. The graphic technique is applied to calculate the polarization operator. In the one-loop approximation (first Born approximation) the polarization operator are described by the Feynman diagram shown in Figure 1.

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55


( )2 f+ (w, ⃗ w⃗ ⃗ n, ⃗g ) = w⃗ ⃗ n + δ ⋆ (1 − m⋆ ) w2 /2 − δ ⋆⃗g w ⃗ + ν2.

( )2 f− (w, ⃗ w⃗ ⃗ n, ⃗g ) = w⃗ ⃗ n − δ ⋆ (1 + m⋆ ) w2 /2 − δ ⋆⃗g w ⃗ + ν2. There are dimensionless parameters on eq. (9) and (10) (frequency ω/ωP = w⃗ ⃗ n − w2 δ ⋆ m⋆ /2, temperature 2T /me Vi2 = 2 ⃗i /Vi , wave vector and momenτ , unit velocity vector ⃗n = V tum ⃗q ωVPi = w, ⃗ p⃗ h¯VωiP = ⃗g , mass ratio m⋆ = me /mi . The h ¯ ωP 2 parameter δ ⋆ contains a temperature δ ⋆ = mh¯eωVP 2 = m 2τ . ev

Fig.1. The Feynman diagram of the polarization operator in the one-loop approximation.

Π(⃗q, ω) =

e2 (2π¯h)3

∫∞ d3 p −∞

np⃗ − np⃗−¯hq⃗ , Ep⃗ − Ep⃗−¯hq⃗ − ¯ hω − iν

i

(5)

e

One can analytically to obtain a series of approximations for verifying numerical calculation. In the low-temperature approximation (small parameter τ ≪ 1) the real part is (

) 3 w2 2 1+ 2 τ + ... . (11) 2 (w⃗ ⃗ n)

where additional imaginary part ν is determined as infinitesimal value from the replacement of frequency ω by expression ω − iν|ω|/ω to avoid singularity [11]. The dielectric susceptibility κ (⃗q, ω) can be determine through the polarized operator: 4π κ(⃗q, ω) = − 2 Π(⃗q, ω). (6) q The energy losses per unit of time of the particle is connected with the probability [6], [7], [10] ∫ dEi 1 − = (Ep⃗i − Ep⃗i −¯hq⃗ ) Wq⃗ d3 q. (7) dt (2π)3

In dimensional terms in Eq. (10) corresponds to the classical case non-magnetized cold plasma [5], [11].

III. N UMERICAL C ALCULATION OF E LECTRON T HERMAL P LASMA P ERMITTIVITY

The analytically possible for the imaginary part to obtain the expression in the case w ⃗ = (w, 0, 0)

At first the behavior of the main variables included in the expression for the energy loss of a charged particle moving in the electron gas must be analyzed. The polarization operator consists of real and imaginary parts

( [( )2 √ e2 n π 1 1 δ ⋆ m⋆ 2 ℑΠ(w, ⃗ w⃗ ⃗ n) = exp − 2 2 w⃗ ⃗n − w + ¯hωP wτ τ w 2 (13) ( ⋆ )2 ]) ( ⋆( )) ⋆ ⋆ δ 2 δ δ m 2 + w × sinh w⃗ ⃗ n − w . 2 τ2 2

Π(w, ⃗ w⃗ ⃗ n) = ℜΠ(w, ⃗ w⃗ ⃗ n) + iℑΠ(w, ⃗ w⃗ ⃗ n),

(8)

they can be written as 3

ℜΠ(w, ⃗ w⃗ ⃗ n) =

e2 (ωP ) ¯hωP (2πVi )3

∫∞ −∞

e2 n w 2 ⋆ Π(w, ⃗ w⃗ ⃗ n) ≈ 2δ ¯hωP (w⃗ ⃗ n)

ε(⃗q, ω) ≈ 1 −

ωP2 ω2

( ) T q2 1+3 + ... . m ω2

(12)

d3 gfre (w, ⃗ w⃗ ⃗ n, ⃗g )ng , f+ (w, ⃗ w⃗ ⃗ n, ⃗g )f− (w, ⃗ w⃗ ⃗ n, ⃗g ) (9)

[( fre =

( ) ⋆ 2 2δ − w + 2 ] 2 + (δ ⋆⃗g w) ⃗ − ν 2 δ ⋆ w2 .

δ ⋆ m⋆ 2 w w⃗ ⃗n − 2

3

e2 (ωP ) ℑΠ(w, ⃗ w⃗ ⃗ n) = ¯hωP (2πVi )3

∫∞ −∞

)2

d3 gfim (w, ⃗ w⃗ ⃗ n, ⃗g )ng , ⃗ w⃗ ⃗ n, ⃗g ) f+ (w, ⃗ w⃗ ⃗ n, ⃗g )f− (w, (10)

( fim = 2νδ ⋆ w2 w⃗ ⃗n −

56

⋆

⋆

)

δ m 2 w − δ ⋆⃗g w ⃗ . 2

Fig.2. The imaginary part of the polarization operator, normalized to the

(ωP )3 e2 h ¯ ωP (2πVi )3

as a function of the wave vector wz . The direction of movement of projectile particles is fixed ⃗ n = (0, 0, 1). w1,2 = ∓1/2δ ⋆ (1 ∓ m⋆ ).


Fig.3. The real part of the polarization operator, normalized to the (ωP )3 e2 h ¯ ωP (2πVi )3

as a function of the wave vector wz . The direction of movement of projectile particles is fixed ⃗ n = (0, 0, 1).

One can be seen from the figures 2, 3 in case ⃗n = (0, 0, 1) real and imaginary parts of polarization operator are pair function in the approximation that the mass of the electron is much smaller than the projectile particle one. This property turns out from Landau rule for poles. IV. C ONCLUSION The program for calculation was written with support for MPI technology [12]. It makes possible to use parallel algorithms and at times reduce the 3-dimensional integral computation time. Computer calculations were performed on a cluster of the Institute of Applied Physics in Sumy. After a more detailed examination of the behavior of the integrand for the energy loss of the moving particle across electron gas we plan to calculate expression of energy loss which consists of 6-dimensional integral. R EFERENCES [1] G.I. Budker. Atomnaya Energiya. 1967, vol. 22, 346. [2] G.I. Budker, A.N. Skrinsky. ”Electron cooling and new possibilities in elementary particle physics” . UFN 1978, v.21, 277–296 [3] V.V. Parkhomchuk, A.N. Skrinsky. ”Electron cooling: physics and prospective application”. Rep.Prog.Phys. 1991, vol.54 , 919-947 [4] I. N. Meshkov. ”Electron Cooling: Status and Perspectives”. Physics of Elementary Particles and Atomic Nuclei. 1994, vol. 25, 6, 1487-1560 [5] A.I. Akhiezer, R.V. Polovin, et al. Plasma Electrodynamics. Oxford: Pergamon Press, 1967 [6] A.I. Larkin, ”Passage of particles through plasma”. Sov. Phys. JETP 1960, vol.10, 186-191 [7] I. A. Akhiezer Sov. Phys. JETP 1961, vol.13, 667 [8] V.V. Parkhomchuk. ”Physics of fast electron cooling”. Proc. Workshop on Electron Cooling and Related Applications (Karlsruhe, 1984) ed H. Poth (Karlsruhe: KfK) [9] G. I. Budker, N. S. Dikansky, V. I. Kudelaineen, I. N. Meshkov, V. V. Parkhomchuk, A. N. Skrinsky and B. N. Sukhina. ”First experiments on electron cooling”. Proc IV All-Union Meering an Accelerators of Charged Parricles (Moscow, 1974) (Moscow: Nauka) vol.2, p.309: 1975 IEEE Trans. Nucl. Sci. VS-22 2093-7 [10] M.M. Dyachenko, V.I. Miroshnichencko, R.I. Kholodov. Reports of the National Academy of Sciences of Ukraine. 2012, vol.10, 70-76. [11] S. Ichimaru. Basic principles of plasma physics: a statistical approach. Moskva, 1975. [12] V.D. Korneev. Parallel programing in MPI. 2nd ed. Novosibirsk, Russian, 2002.

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RESONANCE INTERFERENCE SCATTERING OF A LEPTON BY A LEPTON IN THE BICHROMATIC PULSED LASER FIELD Elena A. Padusenko, Sergey P. Roshchupkin, Alexandr A. Lebed’ Institute of Applied Physics, National Academy of Sciences of Ukraine, 58, Petropavlovskaya Str., Sumy, 40030 Ukraine Phone: 380 (542) 22 27 94, 28 04 64, Fax: 380 (542) 22 37 60, e-mail:[email protected] Abstract – Resonance scattering of a lepton by a lepton in the field of two codirectional pulsed laser waves within the interference kinematical region is theoretically studied. Stimulated emission and absorption of waves’ photons by lepton is correlated in the interference region. Resonance conditions in the interference region are specified. Analytical expressions for amplitude and differential cross section for circular polarization are obtained. The cross section contains a resonant peak, its altitude and width are determined by external waves parameters. Considerable effect of wave polarization on scattering of a lepton by a lepton is specified. The obtained results can be verified at the facility FAIR (Darmstadt, Germany).

INTRODUCTION Improvement of a model of description of electromagnetic pulsed field is required by development and introduction to the experiment practice of powerful pulsed lasers (pico- and femtosecond ones with power reaching the 10 22 W/cm 2 value in the pulse peak). The basic quantum electrodynamics (QED) processes in the field of a single pulsed wave were studied in [1-5]. The interference quantum effect in the field of two or more plane monochromatic waves was discovered [1, 4]. The essence of this effect is that emission and absorption of photons of external waves are correlated at a certain kinematics of the process. It is specified that probability of stimulated processes in the interference kinematics is generally greater than in other one. Studying of nonlinear effects when leptons scatter by each other in an external electromagnetic field is one of the fundamental directions of QED, therefore there is interest to study interference effect in the field of two or more pulsed waves in such processes. Also, one of the fundamental effects of QED in the light wave field is the resonances associated with phenomenon when a particle in an intermediate state may fall within the mass shell (for the processes of the second and higher order in the fine structure constant). As a result, the considered scattering process of the higher order effectively splits into two lower-order process. At that value of resonance scattering cross sections may exceed the corresponding ones in the external field absence in several orders of magnitude. Nonresonant scattering of an electron by a muon was considered in Refs. [5, 7, 8], the resonant one – in Refs. [6, 9]. Studying of influence of the bichromatic pulsed field on resonant scattering of a lepton by a lepton in the interference region is a subject of theoretical and experimental interest.

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The external field is modeled customary as plane quasimonochromatic wave for description of processes of interaction of leptons with the pulsed laser field. Characteristic pulsewidth τ n satisfies the condition [2, 3, 5-10]

ωnτ n ≫ 1 , n = 1, 2 ,

(1)

where ωn is the characteristic frequency of wave field oscillation (the index n labels the wave), ωnτ n 2π is the characteristic number of oscillation of the external field amplitude in the electromagnetic pulse. The condition (1) allows to apply derived differential cross-section for the real experiment description [10]. The characteristic duration of a pulse τ n may be in order of the magnitude with tens of femtosecond even for the external field within the optical range of frequency. .Nonlinear effects in processes of interaction of leptons with the wave field and with each other are specified by the following parameters. The first one is the relativistically invariant classical parameter [1, 2] eF0 n p , j = 1, 2 , (2) η n( j ) = m j cωn which numerically equals to the ratio of the work done by the field within the wavelength to the electron rest energy (here, e is the electron charge and m j is the mass of a lepton (the index j labels the particle); F0 n is the strength of the wave electric field. The parameter which determines the multiplicity of multiphoton processes in nonresonant scattering of leptons by each other is the quantum Bunkin-Fedorov parameter [1, 2] p p m jυ j c γ n( j ) = η n( j ) (3) ℏωn (here, ℏ is the Planck constant). Within the optical fre(p ) (p ) quency range the correlation γ n j ≫ η n j is correct. Consequently, such problems are studied usually within the range of moderately strong field, for which considered parameters obey the following conditions p η n( j ) ≪ 1 , (4) p γ n( j ) ≳ 1 . (5) The parameter (3) becomes a classical one in the case of resonant scattering and is of the order with the parameter

LFNM*2013 International Conference on Laser & Fiber-Optical Networks Modeling, 11-13 September, 2013, Sudak, Ukraine p η n( ) [1-3, 5, 6, 9]. Therefore, resonant scattering of a lepton j

by a lepton is studied within the intensity range (4). The rather narrow specific region, namely the interference one, appears in the field of two electromagnetic (p )

waves. The Bunkin-Fedorov quantum parameter γ n j doesn’t reveal [1, 4]. The interferential quantum parameter appears when processes in the field of two laser waves, propagating in the same direction, are studying. It is the general parameter which determines the multiplicity of multiphoton processes out of the Bunkin-Fedorov region: ( pj )

α±

( pj ) ( pj )

= η1 η 2

pj ⋅c

 m j c2    ℏ (ω1 + ω2 )  E j 

2

(6) ( pj )

(here, E j - is the lepton energy). Parameters α ±

(6) de-

termine correlated emission and absorption of waves’ photons in the interference region. The relativistic system of units, where ℏ = c = 1 , and the standard metric for 4-vectors ( ab ) = a0 b0 − ab will be used throughout this paper. We consider the bichromatic pulsed laser field as two plane codirectional pulsed waves. At that their planes of polarization belong to the plane xy , and the waves propagate along the z -axis. The four-potential of such a field has the form [2, 3, 10] A (ϕ ) = A1 (ϕ1 ) + A2 (ϕ 2 ) , (7)

 ϕ  An (ϕ ) = A0 n ⋅ g n  n  ,  ωnτ n  F A0 n (ϕ ) = 0 n ( enx cos ϕ n + δ n eny sin ϕ n ) ,

ωn

ϕ n = kn xk = ωn ( tk − zk ) ,

Subsequent analysis will be performed for the circularly wave polarization δ n = ±1 and the Gaussian function

{

g (ϕ n ωnτ n ) = exp −2 (ϕ n ωnτ n )

2

}.

We consider resonances for direct Feynman diagrams of the scattering type exceptionally (Fig. 1). The exchange diagrams for identical leptons and the annihilation ones of scattering of a lepton by an antilepton are outside of attention. Such a problem statement is possible due to fact that the resonances for the direct diagrams of scattering type and the resonances for the exchange (annihilation) diagrams within the intensity range (4) occur within the essentially different nonoverlapping kinematical regions [1, 4]. Consequently, the direct amplitude is sufficiently to be calculated. For the direct scattering amplitude within the field range (4) resonant scattering of lepton by a lepton occurs when the leptons scatter forwards into the small angles in the frame of the reference related to the center of inertia of initial particles and effectively decomposes into two processes of the first order similar to Compton scattering of a wave by a lepton.

p1′

p1

q ′2 ≈ 0

(8)

p2′

p2

(9) (10)

where ϕ is the wave phase; kn = (ωn , k n ) are the wave

Fig. 1. The Feynman diagram of resonance scattering of a lepton lepton

l1 by a

l2 in the field of two pulsed light waves. The external incoming

vectors; δ n is the wave ellipticity parameter ( δ = 0 corresponds to the linear polarization case, δ = ±1 corresponds to

and outgoing double lines correspond to lepton wave functions in the initial and final states in the field of the plane wave (the Volkov functions), and the inner dashed line corresponds to the Green function of a free photon.

four-vectors of wave polarization, meeting the standard conditions: enx2 , ny = −1 , ( enx , ny kn ) = k 2 = 0 .

CROSS-SECTION The interferential region of scattering of a lepton by a lepton in the field of two light waves is determined by the conditions, which can be met exceptionally if particles scatter in the plane formed by lepton initial momenta and the wave vector (within the frame of reference related to the center of inertia of initial particles): p γ n( j ) = 0 . (11) In this region real processes of correlated emission and absorption of the equal number of photons of the both waves take place. For the case of wave circular polarization the process formally looks like the case of one wave with emission and absorption of photons with combinational frequency Ω ± = ω1 ± ω2 (here, the upper sign corresponds to case of

the circular polarization case); enx , ny = ( 0, e nx , ny ) are the

The function g n (ϕ n ωnτ n ) in the expression (8) is the

envelope function of the four-potential of external wave that allows to take into account the pulsed character of the laser field. Generally the envelope function is chosen to be equal to unity in the center of a pulse and to decrease exponentially when ϕ n ≫ ωnτ n . It is assumed that the duration of waves’ pulses exceeds the characteristic period of their oscillation [10] (see, the condition (1)). We note, that envelope functions g n (ϕ n ωnτ n ) are chosen as functions of variables ϕ n , consequently the electromagnetic field with the four-potential (8) represent the plane wave. Therefore the exact solutions of the Dirac’s equation for an electron in the plane-wave field of arbitrary spectral composition (the Volkov functions) can be employed for description of lepton states in the field of the quasimonochromatic wave.

waves’ different polarization ( δ1 = −δ 2 ), the bottom one – to the case of the waves’ same polarization ( δ1 = δ 2 )). Within the frame of reference related to the center of inertia of initial particles the resonance appears if leptons scatter by each other into the resonant region of angles:

59


θ = θ nres = 2

ωn

sin θ i ~

p

ωn p

≪1,

(12)

where θ = ∠ ( p′, p ) is the angle of particle scattering in the frame of reference related to the center of inertia, θ res is the resonant scattering angle, where p and p′ are relative momenta of initial and final particles, respectively; after scattering the momentum p changes only the direction, i.e.

p ′ = p , θi = ∠ ( k , p ) is the angle between directions of wave propagation and initial relative momentum. It is shown that the squared four-momentum of intermediate photon q′2 is founded within the very narrow region near zero 2 ω2 1 q ′2 ≈ p 2 (θ nres ) ∼ n ≪ ωn2 . (13)

ω1τ1

ω1τ 1

Using expressions (11) and (13) it is easy to ascertain: 2  1 p  int res θ = θn = θn  2 + 2  , (14)  sin θ i m j    where θ nint is the scattering angle in the interference region. Resonant scattering of a lepton by a lepton in the interference region obviously appears if input initial angles come close to π 2 . The differential cross-section of resonant scattering of nonpolarized leptons by each other in the interference region in the field of bichromatic pulsed circularly polarized wave may be represented in the form: 2 2 2 2 dσ res ( 2π ) re me m1 m2 res = f . d Ω′ 4p 4 E1 E2 4

(15)

Here, re is the classical electron radius.

f res = f1res ( ρ , β1 ) + f 2res ( ρ , β 2 ) ,

(16)

f1res ( ρ , β1 ) ≡ (ω1τ 1 θ ⋅ θ1res ) ⋅ f1 ( ρ , β1 ) ,

(17)

2

1 2ρ

f1 ( ρ , β1 ) =

ρ

∫ρ dφ c ( β ) 1

1

−

(

{b (η η )} 2

+ kδ b22 η1( p1 )

( p1 ) ( p2 )

2 1

1

η1

1 2ρ

∫ρ dφ c ( β ) 2

2

−

(

{b (η η )}

+ kδ b12 η1( p2 )

c1 ( β1 ) = e

−

β12 4

c2 ( β 2 ) =

( p1 ) ( p2 )

2 2

2

η2

2

− ( 2φ )

2

, b2 = e

τ 2 − 4 2i β φ  e e  erf τ1  2 2

 τ  −  2 1 φ   τ2 

(19)

)

2

+

2

( p1 )

2

e 2i β1 φ ( erf ( 2φ + i β1 2 ) + 1) ,

b1 = e β 22

2

+

(18)

2

f2 ( ρ , β2 ) =

2

2

( p2 )

2

f 2res ( ρ , β 2 ) ≡ (ω1τ1 θ ⋅ θ 2res ) ⋅ f 2 ( ρ , β 2 ) , ρ

)

(20) (21)

2

 τ2 β2 2 φ +i τ 2  1

,

(22)

   + 1 , (23)  

1 2 2 (1 + δ1δ 2 ) cos 2 ∆ + (δ1 + δ 2 ) sin 2 ∆  . (24) 4 Resonant parameters β n have the form kδ =

60

1 2



β n = ω1τ 1 1 − 

θ   ≲1. θ nres 

(25)

The ratio ρ = T τ 1 expresses a relationship between the observation time and the pulse duration. Thus, the cross section depends considerably on whether directions of rotation of vectors of the field strength of laser waves or not. The dependence of the function f res (17), (19) on parameters β n (25) determines the magnitude and form of the resonant peak in the cross section. Using the expression (15) one can estimate the ratio of the resonant cross section of scattering of a lepton by a lepton in pulsed bichromatic field (in the interference region) to the cross section in the absence of the external field. Thereby, for the nonrelativistic case and moderately strong field (within the optical frequency) and the number of oscillation of the field amplitude at the characteristic pulse duration ω1τ1 ∼ 102 this ratio is one order of the magnitude. The obtained results can be verified at the facility FAIR (Darmstadt, Germany). REFERENCES [1]. S.P. Roshchupkin and A.I. Voroshilo, Resonant and Coherent Effects of Quantum Electrodynamics in the Light Field (Naukova Dumka, Kiev, 2008), in Russian. [2]. S.P.Roshchupkin and A.A. Lebed’. Effects of Quantum Electrondynamics in the Strong Pulsed Laser Field (Naukova Dumka, Kiev, 2013), in Russian. [3]. S.P. Roshchupkin, A.A. Lebed’, E.A. Padusenko, and A.I. Voroshilo, Quantum Optics and Laser Experiments, edited by S. Lyagushyn (Intech, Rijeka, 2012), pp. 107–156. [4]. Roshchupkin S.P. Resonant Effects in Collisions of Relativistic Electrons in the Field of a Light Wave // Laser Physics. – 1996. – V.6, № 5. – P. 837-858. [5]. S.P. Roshchupkin, A.A. Lebed’, E.A. Padusenko. Nonresonant quantum electrodynamics processes in a pulsed laser field // Laser Physics, 2012, Vol. 22, No. 10, pp. 1513–1546 [6]. S. P. Roshchupkin, A. A. Lebed’, E. A. Padusenko, A. I. Voroshilo. Quantum electrodynamics resonances in a pulsed laser field // Laser Physics, Vol. 22, № 6 , pp. 1113-1144. [7]. Padusenko E.A., Roshchupkin S.P., Voroshilo A.I. Nonresonant scattering of relativistic electron by relativistic muon in the pulsed light field // Laser Physics Letters. – 2009. – Vol. 6, № 3. – P. 242-251. [8]. E.A. Padusenko, S.P. Roshchupkin, and A.I. Voroshilo. Nonresonant scattering of nonrelativistic electron by nonrelativistic muon in the pulsed light field // Laser Physics Letters. – 2009. – Vol. 6, № 8. – P. 616-623. [9]. Padusenko E.A. Resonant scattering of a lepton by a lepton in the pulsed light field / Padusenko E.A., Roshchupkin S.P. // Laser Physics. – 2010. – Vol. 20, № 12. – P. 2080-2091. [10]. N.B. Narozhny and M.S. Fofanov, Zh. Eksp. Teor. Fiz. 110, 26 (1996).


1

Soliton-like behavior of electrons in the electron cooling Oleksandr Novak, Roman Kholodov Institute of Applied Physics, NAS of Ukraine, Sumy 40000, Ukraine Email: [email protected]

Abstract— Simulation of electron movement on a string in the presence of a charged particle is carried out. A sharp peak is found in dependence of the transferred energy on the impact parameter for the scattering of an electron on a negatively charged particle in the accompanying reference frame. Movement of the electron has soliton-like character near the peak.

I. I NTRODUCTION Electron cooling is one of the main methods for obtaining beams of charged particles with small phase space (emittance) that necessary for the performance of modern highenergy physics. The electron cooling method was proposed by G. I. Budker in 1960, after which the method was developed at the Novosibirsk Institute of Nuclear Physics, both in the experimental and the theoretical aspects[1,2]. Influence of the sign of an ion on the friction force in electron cooling was studied in 1988 [3,4]. Scheme of the facility ”MOSOL” for study of electron cooling is shown in Fig. 1.

Fig. 1. Scheme of facility ”MOSOL” [3,4]. In these experiments it was found that the frictional force for negative ions is several times greater than for the positive one (Fig. 2.).

Fig. 2. Change of energy of ions of different signs as a function of electron energy [3,4].

The difference in the friction force of positively and negatively charged particles arises due to the presence of a strong magnetic field. In a strong magnetic field, when the Larmor’s radius of electron is less than the average distance between the electrons, the motion of the electrons are magnetized. The movement that is transverse to the magnetic field is suppressed, and the longitudinal one is free. An electron is scattered by a positively charged particle always ahead without changing the electron momentum. Scattering on a negative charged particle is very different. The electron is reflected back when the impact parameter is less than rmin = 2e2 /mv 2 . As a result electron momentum transfer 2mv gives an additional correction to the friction force. Novosibirsk scientists refer this behavior as a “bulldozer effect”. It should be noted that there is still no complete analytical description of the influence of the sign of the charged particles on the friction force for arbitrary value of magnetic field. There are numerical calculations and simulations only. In this paper, simulation of electron motion on a string in the presence of a charged particle is carried out. The dependence of the transferred energy on the impact parameter is analyzed both in the laboratory and in the accompanying reference frame. The equation of motion of the electron near the charged particle is studied.

II. M OVEMENT OF THE ELECTRON ON A STRING

Let the magnetic field be very strong, so that the transverse motion of an electron is missing. Thus, there is a one-dimensional electron motion along the axis x (Fig. 3.). Nevertheless, we assume that the magnetic field has no effect on the movement of heavy charged particle. Heavy particle interacts only with the electron. The equation of motion of the electron on a string in the presence of the charge is solved numerically. After the collision, the change of the particle energy is calculated to plot the dependence of the transferred energy on the impact parameter. Energy is measured in units of the projectile initial energy. A distance is measured in units of rmin = e2 /E0 , where E0 = mv02 /2 (m and v0 are the mass and the velocity of the electron at the initial moment of time in the accompanying reference frame).

978-1-4799-0159-3/13/$31.00 ©2013 IEEE

61


Fig. 3. The motion of the electron in a very strong magnetic field is equivalent to a particle on the string. Figure 4 shows the dependence of the dimensionless transferred energy on the dimensionless impact parameter in the laboratory reference frame (at t = 0: ve = 0, vq = v0 ; the ion mass is decreased to 10m for clarity).

2

corresponding curves in Fig. 5. The solid curve in Fig. 5 has a sharp peak. The peak has the place when the impact parameter is nearly equal to rmin (really it is slightly less then rmin due to recoil effect). Electron stops in front of the negatively charged particle when the initial energy of the electron is equal to the potential energy at the nearest distance between the particles. The magnetic field does not allow the electron to move in the transverse direction. The time of interaction between particles at close distance increases. It leads to the maximum transferred energy. It should be noted that the behavior of the electron in the region of the peak is similar to the behavior of the nonlinear pendulum near the separatrix. The Fig. 6 shows an analogy between the electron near a negatively charged particle and a nonlinear pendulum.

Fig. 6. Analogy between the nonlinear pendulum and the electron on a string (up case).

Fig. 4. Transferred energy as function of the impact parameter (laboratory frame of reference). The solid (eq > 0) and dashed (eq < 0) curves correspond to the scattering of the electron on negatively and positively charged particles, respectively. The solid curve is similar to Heaviside function while the transferred energy for the case eq < 0 is insignificant, that coincides with the results [3,4]. The results of similar calculations in the accompanying reference frame (at t = 0: ve = v0 , vq = 0) are shown in Fig. 5.

At initial time the pendulum is fixed in horizontal position and has a speed upward (up case). If the pendulum moves along the separatrix, it stops at the upper position in unstable equilibrium. It corresponds to the peak in Fig. 5. If at initial time the pendulum is also fixed in horizontal position but has a speed downward (down case), such the pendulum is similar to the electron in a string near positive charge particle (Fig. 7).

Fig. 7. Analogy between the nonlinear pendulum and the electron on a string (down case).

Fig. 5. Transferred energy as function of impact parameter (accompanying frame of reference). By using the Galilean transformation for energy and momentum px = p′x + mv0 , E = E ′ + p′x mv0 + mv02 /2,

(1)

it is easy to show that the curves in Fig. 4 transform into the

62

The pendulum in the lowest position has a maximum speed, as well as the electron, which runs near the positively charged particle. Note that the nonlinear pendulum movement along the separatrix has soliton-like character. Movement of magnetized electron near a charged particle has the same properties when impact parameter ρ is equal rmin . The equation of motion of the electron on the string near the fixed charged particle has the form d2 x e2 x m 2 =δ 2 , (2) dt (x + ρ2 )3/2 where δ is −1(+1) for scattering on the positively (negatively) charged particle. Near the origin of coordinates (x

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