- Email: [email protected]
Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals
Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals S.V. Anishchenko ⇑, V.G. Baryshevsky Research Institute for Nuclear Problems, Bobruiskaya str., 11, 220030 Minsk, Belarus
a r t i c l e
i n f o
Article history: Received 17 November 2014 Received in revised form 13 March 2015 Accepted 19 March 2015 Available online xxxx Keywords: Cooperative radiation Terahertz X-rays Quasi-Cherenkov radiation Photonic crystals
a b s t r a c t We study the features of cooperative parametric (quasi-Cherenkov) radiation arising when initially unmodulated electron (positron) bunches pass through a crystal (natural or artificial) under the conditions of dynamical diffraction of electromagnetic waves in the presence of shot noise. A detailed numerical analysis is given for cooperative THz radiation in artificial crystals. The radiation intensity above 200 MW/cm2 is obtained in simulations. The peak intensity of cooperative radiation emitted at small and large angles to particle velocity is investigated as a function of the current density of an electron bunch. The peak radiation intensity appeared to increase monotonically until saturation is achieved. At saturation, the shot noise causes strong fluctuations in the intensity of cooperative parametric radiation. It is shown that the duration of radiation pulses can be much longer than the particle flight time through the crystal. This enables a thorough experimental investigation of the time structure of cooperative parametric radiation generated by electron bunches available with modern accelerators. The complicated time structure of cooperative parametric (quasi-Cherenkov) radiation can be observed in crystals (natural or artificial) in all spectral ranges (X-ray, optical, terahertz, and microwave). Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The generation of short pulses of electromagnetic radiation is a primary challenge of modern physics. They find applications in studying molecular dynamics in biological objects and charge transfer in nanoelectronic devices, diagnostics of dense plasma and radar detection of fast moving objects. The advances in the generation of short pulses of electromagnetic radiation in infrared, visible, ultraviolet, and X-ray ranges of wavelengths are traditionally associated with the development of quantum electronic devices — lasers. Radiation in lasers is generated via induced emission of photons by bound electrons. Electrovacuum devices, operating in a cooperative regime [1,2], have recently become considered as an alternative to short-pulse lasers, whose active medium is formed by electrons bound in atoms and molecules. These are free electron lasers, cyclotronresonance masers, and Cherenkov radiators, whose active medium is formed by initially unmodulated electron bunches propagating in complex electrodynamical structures (undulators, corrugated ⇑ Corresponding author. E-mail addresses: [email protected] (S.V. Anishchenko), [email protected] (V.G. Baryshevsky).
waveguides and others) and being much greater than the radiation wavelength. The feature of the cooperative operation regime lies in the fact that the peak radiation power scales as the squared number of particles in the bunch. (This allows calling this regime ‘‘superradiance’’ by analogy with the phenomenon predicted by Dicke in quantum electronics [1]). To avoid misunderstanding, let us note that initially, the coherent part of electromagnetic radiation is highly suppressed, because the length of an electron bunch is much greater than the radiation wavelength. The electromagnetic wave emission starts as incoherent spontaneous process, becomes coherent through the nonlinear interaction of the electrons and the electromagnetic field. In free electron lasers, the initial phases of charged particles in the electromagnetic wave are homogeneously distributed. As a result, bremsstrahlung produced by oscillating electrons starts from incoherent spontaneous emission. This is true even if the bunch length is much smaller than the radiation wave length. In contrast to bremsstrahlung, Cherenkov (quasi-Cherenkov) radiation starts from coherent spontaneous emission when such a short-length bunch is injected into a slow-wave structure, i. e. the radiation power is proportional to the squared number of particles.
http://dx.doi.org/10.1016/j.nimb.2015.03.054 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
2
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Fig. 1. The two-wave (left) and three-wave (right) diffraction geometries.
This paper considers cooperative quasi-Cherenkov radiation emitted by electron bunches when charged particles pass through crystals (natural or artificial) under the conditions of dynamical diffraction of electromagnetic waves. The concept of parametric (quasi-Cherenkov) radiation was originated in the works devoted to the interaction of charged particles with natural crystals (see, for example [3,5,6]). According to [3,5], the dynamical diffraction of photons emitted by a relativistic charged particle in a crystal leads to a change in the refractive index of X-ray quanta. In accordance with [3,5], diffraction of virtual photons in a crystal kÞ, some of which may possesses a set of refractive indices nl ðx; ~ k is the photon wave vector). appear to be greater than unity (~ Particularly, in the case of two–wave diffraction, the refractive indices are n1 ðx; ~ kÞ > 1 and n2 ðx; ~ kÞ < 1, and accordingly, two types of waves propagate in the crystal: a fast wave (n2 < 1) and a slow one (n1 > 1). The Cherenkov condition can be fulfilled for a slow wave, but not for a fast one. The former gives rise to spontaneous quasiCherenkov radiation, called the parametric X-ray radiation (PXR). The latter is emitted at the vacuum–crystal boundary. The considered phenomenon has a universal character and can be observed in different spectral ranges (microwave, terahertz, optical, etc.) for particles passing through varied two- and three-dimensional spatially periodic structures [9], often called the photonic crystals. A detailed analysis of the features of incoherent spontaneous radiation of electrons passing through crystals in both frequency [3] and time [4] domains has been carried out. This radiation, emitted at both large and small angles with respect to the direction of electron motion. The problems of amplification of induced parametric X-ray radiation and microwave (optical) quasi-Cherenkov radiation have also been thoroughly studied in the literature [7], and the threshold current densities providing lasing in crystals have been calculated [8]. Coherent spontaneous radiation produced by modulated electron bunches in crystals has been analysed in [10–12]. The paper is arranged as follows: In the beginning, a nonlinear theory of interaction of relativistic charged particles and the electromagnetic field in crystals is set forth, followed by the results of numerical calculations of the parametric radiation pulse. The dependence of the radiation intensity on the current density of an electron bunch and the geometrical parameters of the system is considered. 2. Nonlinear theory of cooperative radiation A theoretical analysis of radiation can be performed only by means of a self-consistent solution of a nonlinear set of the Newton–Maxwell equations:
pa d~ ~~ Eð~ ¼ qe ~ v a Hð ra ; tÞ þ ~ ra ; tÞ=c ; dt
ð1Þ
~ 1 @H ; c @t ~ ~ ¼ 1 @ D þ 4p~j; rH c @t c D ¼ 4pq; r~ ~ ¼ 0; rH
ð2Þ
E¼ r ~
~ describing the electron motion in the electric ~ E and magnetic H P P ~ v a dð~r ~ra Þ and q ¼ qe a dð~r ~ra Þ are the curfields. Here j ¼ qe a ~ rent and charge densities, respectively. Since the crystal is a periodic linear medium with frequency dispersion, the Fourier Dð~ r; xÞ relates to the transform of the electric displacement field ~ ~ ~ electric field Eðr; xÞ as
~ Dð~ Eð~ r; xÞ ¼ ð~ r; xÞ~ r; xÞ ¼
1 þ v0 ðxÞ þ
X
! Eð~ 2v~s ðxÞ cosð~ s~rÞ ~ r; xÞ;
ð3Þ
~ s
where the summation is made over all reciprocal lattice vectors. This permits to reduce Maxwell’s Eqs. (2) to the equation of the form:
1 @2 c2 @t 2
Z
t
ð~r; t t1 Þ~Eð~r; t1 Þdt1 þ rðr ~EÞ D~E ¼
1
4p @~j : c2 @t
ð4Þ
Let’s simplify the Eq. (4) for the case when jv0;s j 1 and two strong waves are excited in the crystal: the forward wave and the diffracted wave (the so-called two-wave diffraction case). The k0 ) is emitted at small forward wave (its wave vector is denoted by ~ angles with respect to the particle velocity, while the diffracted ks ¼ ~ k0 þ ~ s, is emitted at large angles one, having the wave vector ~ to it (Fig. 1). Under the conditions of dynamical Bragg diffraction, k2 ~ k2 x2 =c2 . the following relation is fulfilled: ~ s
0
Let us perform the following simplifications: First, we shall D ! 0) fields of the bunch. Second, neglect the longitudinal (r ~ E using the method of slowly we shall seek for the electric field ~ varying amplitudes. Third, we shall assume that a transversally infinite bunch executes one-dimensional motion along the OZ-axis (this is achieved by inducing a strong axial magnetic field in the system). E can be Under the conditions of two-wave diffraction, the field ~ presented as a sum
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx ~ ~ ~ E ¼~ e0 E0 ðz; tÞeiðk0~rxtÞ þ ~ es Es ðz; tÞeiðks~rxtÞ ;
ð5Þ
where the amplitudes of the forward E0 and diffracted Es waves are slowly varying variables. This means that for the distances comparable with the wavelength and the times comparable with the oscillation period, the values of E0 and Es remain practically the same. Substituting (5) into (1) and (4) and then averaging them over the length l which is equal to the integer number of the wavelength and is smaller than the characteristic lengths appearing in the problem (geometrical dimensions, the beam instability length), we obtain
dpza dt
¼ 2qe e0z Re E0 eiðk0z za xtÞ ;
ð6Þ
1 @E0 @E0 iv0 x iv x þ c0 þ E0 þ s Es c @t @z 2c 2c Z 2p zþl=2 iðxtk0z zÞ ¼ e0z jz e dz=l; c0 ¼ k0z =k0 ; c zl=2 1 @Es @Es iv0 x iv x þ cs þ Es þ s E0 c @t @z 2c 2c Z 2p zþl=2 ¼ esz jz eiðxtksz zÞ dz=l; cs ¼ ksz =ks : c zl=2
ð7Þ
z ¼ 0. In the case of gaussian beams (jb ðtÞ ¼ j expðc2 t 2 =L2b Þ), the electrons are injected into the crystal continuously at 1 < t < þ1. The difference between the two diffraction schemes is not merely kinematic, but radical. In the Bragg case, there is a synchronous wave moving against the electrons of the beam, which gives rise to the internal feedback and absolute instability. In Laue diffraction geometry, a backward wave is absent, and as a result absolute instability does not evolve. It may seem that electromagnetic radiation is not generated. However, fluctuations of the electron current (the shot noise), which always occur in real beams, are amplified when the beam enters the crystal (due to convective instability, excited in the beam). In analyzing multiparametric problems, to which the problem of cooperative parametric (quasi-Cherenkov) radiation refers, it is convenient to write Eqs. (6) and (7) in a dimensionless form. This procedure enabled transferring the calculation results from one set of parameters to another. The substitution of variables xt ! t; xL=c ! L, mcxE0;s =qe ! E0;s then gives
ð8Þ
X hv eiðtk0z za þ/a Þ @Es @E0 @E0 iv0 iv ba þ c0 þ E0 þ s Es ¼ ; @t @z 2 2 2 Nl @t j @Es iv0 iv þ Es þ s E0 ¼ 0: @z 2 2
vb ¼ vb0 expðz2 =L2b Þ;
ð10Þ
where the bunch length Lb is further assumed to be equal to 0:1L=c. Obviously, in the three-diffraction case, the set of equations analogues to (9) should be rewritten as follows:
@E2 @E2 iv0 iv þ c2 þ E2 þ s ðE3 þ E1 Þ ¼ 0; @t @z 2 2 @E3 @E3 iv0 ivs þ c3 þ E3 þ ðE1 þ E2 Þ ¼ 0: @t @z 2 2
ks , vanishes in the parthat of the wave Es , emitted in the direction ~ ticles’ exit plane (z ¼ L). In Laue geometry, both waves E0 and Es are emitted into the half-space z > 0, and both of them vanish in the particles’ entry plane (z ¼ 0). The time t ¼ 0 corresponds with the maximum value of the current density jb ðtÞ of the beam injected into the crystal in the plane
þ cs
corresponding electron density, h is the angle between the particle k0 ; /a is the initial phase of the ath velocity and the wave vector ~ particle, and N l is the number of particles over the length l. The set of equations with boundary conditions contains four independent parameters: v0;s ; xL=c; h that define the geometry of the system. In addition to these parameters, we need to specify the beam profile. Let
X hv eiðtk1z za þ/a Þ @E1 @E1 iv0 iv ba þ c1 þ E1 þ s ðE2 þ E3 Þ ¼ ; @t @z 2 2 2 Nl a
Now let us complete the set of Eqs. (6) and (7) with boundary conditions. The boundary conditions imposed on the fields follow from the form of the equations governing E0 Es and are standard boundary conditions used in the dynamical theory of diffraction of electromagnetic waves in crystals at jv0 j; jvs j 1 [13,14]. In Bragg geometry, the amplitude of the wave E0 , emitted in the k0 , vanishes in the particles’ entry plane (z ¼ 0), while direction ~
dpza ¼ 2hRe E0 eiðtk0z za þ/a Þ ; dt
3
ð9Þ
Here the quantity vba ¼ 4pq2e na =mx2 is determined at the moment when the ath particle enters the system, na is the
ð11Þ
3. Simulation results Let us see now how the dynamical diffraction of electromagnetic waves affects the cooperative radiation in crystals. How the Bragg diffraction case is different from the case of Laue diffraction? We will try to answer this question with the help of the numerical code [15] based on the particle-in-cell method [16], which enables studying kinetic phenomena. Let us note that in most of the existing codes (see, e.g. [17,18]) used for simulating the interaction of charged particles and a synchronous wave, the motion of charged particles is considered within the framework of the hydrodynamic approximation. Let us start our consideration with the Bragg case. In this case, along with the electromagnetic wave emitted in the forward direction, one can observe the electromagnetic wave that is emitted by charged particles in the diffraction direction and leaves the crystal through the bunch entrance surface. We shall assume that c ¼ 3:0; h ¼ 0:33 rad, v0 ¼ 0:2; ¼ vs ¼ 0:1; L ¼ 6 cm and j > 10 kA/cm2. The beams with current densities as high as j > 10 kA/cm2 are obtained at relativistic high-current accelerators [19]. According to the analysis [20], at electron energies less than 3 MeV, the artificial structures are tolerant of the current pulses lasting for t pulse if jtpulse < 4:5 105 C/cm2 (at larger jtpulse , plasma is formed on metal surfaces, disturbing the conditions of electromagnetic-wave propagation). Let j ¼ 30 kA/cm2, t pulse ¼ 1 ns, then jt pulse ¼ 3 105 C/cm2 and satisfies the set requirements. According to the estimates [21], under the electron beam of current density j ¼ 30 kA/cm2 and electron energy of 1:5 MeV (c ¼ 3), the temperature in the tungsten wires of a photonic crystal built from metallic threads rises by less than 300 K. The peak intensity of cooperative radiation emitted at small and large angles to particle velocity is investigated as a function of the peak current density j. The peak radiation intensity appeared to increase monotonically until saturation is achieved (Fig. 2). At saturation, the shot noise causes strong fluctuations in the intensity of cooperative parametric radiation. The dispersion of radiation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensity d0 ¼ j < I0 >2 < I20 > j as a function of j is presented on Fig. 2 (the brackets < :: > denote average values). The results of computation (Fig. 3) show that the cooperative radiation emitted at large angles lasts much longer (t rad 0:6 ns) than the particle flight time through the crystal (tp ¼ 0:2 ns), though that is much lower than the radiation intensity emitted in forward direction. We would like to note that the long duration
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
4
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Fig. 2. The Bragg case. Quasi-Cherenkov radiation at small angles to particles’ velocities: radiation intensity (left), intensity dispersion (right) [hB ¼ 67:5o ; m ¼ 0:1 THz].
Fig. 3. The Bragg case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [hB ¼ 67:5o ; j ¼ 10 kA/cm2,
m ¼ 0:1 THz].
Fig. 4. The Bragg case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [hB ¼ 67:5o ; m ¼ 1:0 THz].
Fig. 5. The Laue case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [hB ¼ 22:5o ; j ¼ 10 kA/cm2,
of parametric radiation can be observed in spontaneous processes too [4]. Figs. 2 and 3 correspond to the frequency m ¼ 0:1 THz and current density j ¼ 10 kA/cm2. If we increase m in ten times leaving the current density unchanged the peak radiation intensity will be 240 MW/cm2 (Fig. 4).
m ¼ 0:1 THz].
Now, let us consider the Laue case. In this case, the electromagnetic waves emitted by charged particles in the forward and diffraction directions leave the crystal through the same surface. Under Laue diffraction conditions (Fig. 5), the pulses of parametric radiation emitted in forward and diffracted directions have comparable amplitudes and durations.
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
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S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Fig. 6. The Laue case. Quasi-Cherenkov radiation in the absence of shot noise (left) and in the presence of shot noise (right), respectively [hB ¼ 22:5o ; Lb =L ¼ 1:0; j ¼ 10 kA/ cm2, m ¼ 0:1 THz, vb ðzÞ ¼ const].
Fig. 7. The three-wave diffraction case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [j ¼ 10 kA/cm2,
We should point out that the shot noise results in strong fluctuations in radiation intensity at saturation. Under Laue diffraction conditions, the shot noise leads to an appreciable change in the pulse form due to the convective character of instability: in the absence of noise, the generation occurs only at the ends of the bunch of charged particles. As a result, the cooperative pulse posses a two-peak structure (Fig. 6). The presence of noise leads to an appreciable change in the pulse form: the interval between the two pulses is filled with a chaotic signal (Fig. 6). Under three-wave diffraction conditions (Fig. 1), the results of computation are very similar to those of the Bragg case (Fig. 7). Namely, the intensity of cooperative radiation emitted at large angles lasts much longer than the particle flight time through the crystal, though that is much lower than the radiation intensity emitted in forward direction. 4. Conclusion This paper studies the features of parametric (quasi-Cherenkov) cooperative radiation emitted at both large and small angles to the particle velocity direction in two- and three-wave diffraction cases. A detailed numerical analysis is given for cooperative THz radiation in artificial crystals. The intensity of THz radiation above 200 MW/cm2 is obtained in simulations. The peak intensity of cooperative radiation emitted at small and large angles to particle velocity is investigated as a function of the peak current density. The peak radiation intensity appeared to increase monotonically until saturation is achieved. At saturation, the shot noise causes strong fluctuations in the intensity of cooperative parametric radiation. It is shown that, the intensity of cooperative radiation emitted at large angles can last much longer than the particle flight time through the crystal. This enables a thorough experimental investigation of the time structure of cooperative parametric
m ¼ 0:1 THz].
radiation generated by electron bunches available with modern accelerators. The complicated time structure of cooperative parametric (quasi-Cherenkov) radiation can be observed in crystals (natural or artificial) in all spectral ranges (X-ray, optical, terahertz, and microwave). References [1] R. Bonifacio et al., Rivista Nuovo Cimento 13 (9) (1990) 1–69. [2] S.D. Korovin et al., Phys. Rev. E 74 (2006) 016501. [3] V.G. Baryshevsky, I.D. Feranchuk, A.P. Ulyanenkov, Parametric X-ray Radiation in Crystals: Theory, Experiment and Applications, Springer-Verlag, Berlin Heidelberg, 2005. [4] S.V. Anishchenko, V.G. Baryshevsky, A.A. Gurinovich, Nucl. Instrum. Methods B293 (2012) 35–41. [5] V.G. Baryshevsky, High-Energy Nuclear Optics of Polarized Particles, World Scientific Publishing, Singapore, 2012. [6] P. Rullhusen, X. Artru, P. Dhez, Novel Radiation Sources using Relativistic Electrons: From Infrared to X-rays, World Scientific Publishing, Singapore, 1998. [7] V.G. Baryshevsky, I.D. Feranchuk, Phys. Lett. A 102 (1984) 141–144. [8] V.G. Baryshevsky, Available from: 1211.4769, 2012. [9] V.G. Baryshevsky et al., Dokl. Akad. Nauk SSSR 299 (6) (1988) 1363–1365 [in Russian]. [10] V.G. Baryshevsky, Vesti AN BSSR 1 (1984) 31–37 [in Russian]. [11] V.G. Baryshevsky, Avaible from: 1101.0783v1, 2011. [12] K.A. Ispirian, Nucl. Instrum. Methods B309 (2012) 4–9. [13] Z.G. Pinsker, Dynamical Scattering of X-rays in Crystals, Springer, Berlin, 1978. [14] S. Chang, Multiple Diffraction of X-rays in Crystals, Springer, Berlin, 1978. [15] S.V. Anishchenko, V.G. Baryshevsky, Available from: 1411.2960v1, 2014. [16] I.J. Morey, C.K. Birdsall. Memorandum No. UCB/ERL M89/116. Electronics Research Laboratory. University of California, 1989. [17] T.M. Antonsen et al., Proc. IEEE 87 (1999) 804–839. [18] N.S. Ginzburg, S.P. Kuznecov, T.H. Fedoseeva, Izv. vuzov. Radiofizika 21 (1978) 1037–1052. [19] G.A. Mesyats, Pulsed Power, Kluwer Academic/Plenum Publishers, New York, 2004. [20] S.P. Bugaev, et al. in: Relyativistskaya vysokochastotnaya electronika. Gorky, pp. 5–75. 1979. [in Russian]. [21] V.G. Baryshevsky, et al. THAAU03. Proceedings of FEL 2007.
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals S.V. Anishchenko ⇑, V.G. Baryshevsky Research Institute for Nuclear Problems, Bobruiskaya str., 11, 220030 Minsk, Belarus
a r t i c l e
i n f o
Article history: Received 17 November 2014 Received in revised form 13 March 2015 Accepted 19 March 2015 Available online xxxx Keywords: Cooperative radiation Terahertz X-rays Quasi-Cherenkov radiation Photonic crystals
a b s t r a c t We study the features of cooperative parametric (quasi-Cherenkov) radiation arising when initially unmodulated electron (positron) bunches pass through a crystal (natural or artificial) under the conditions of dynamical diffraction of electromagnetic waves in the presence of shot noise. A detailed numerical analysis is given for cooperative THz radiation in artificial crystals. The radiation intensity above 200 MW/cm2 is obtained in simulations. The peak intensity of cooperative radiation emitted at small and large angles to particle velocity is investigated as a function of the current density of an electron bunch. The peak radiation intensity appeared to increase monotonically until saturation is achieved. At saturation, the shot noise causes strong fluctuations in the intensity of cooperative parametric radiation. It is shown that the duration of radiation pulses can be much longer than the particle flight time through the crystal. This enables a thorough experimental investigation of the time structure of cooperative parametric radiation generated by electron bunches available with modern accelerators. The complicated time structure of cooperative parametric (quasi-Cherenkov) radiation can be observed in crystals (natural or artificial) in all spectral ranges (X-ray, optical, terahertz, and microwave). Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction The generation of short pulses of electromagnetic radiation is a primary challenge of modern physics. They find applications in studying molecular dynamics in biological objects and charge transfer in nanoelectronic devices, diagnostics of dense plasma and radar detection of fast moving objects. The advances in the generation of short pulses of electromagnetic radiation in infrared, visible, ultraviolet, and X-ray ranges of wavelengths are traditionally associated with the development of quantum electronic devices — lasers. Radiation in lasers is generated via induced emission of photons by bound electrons. Electrovacuum devices, operating in a cooperative regime [1,2], have recently become considered as an alternative to short-pulse lasers, whose active medium is formed by electrons bound in atoms and molecules. These are free electron lasers, cyclotronresonance masers, and Cherenkov radiators, whose active medium is formed by initially unmodulated electron bunches propagating in complex electrodynamical structures (undulators, corrugated ⇑ Corresponding author. E-mail addresses: [email protected] (S.V. Anishchenko), [email protected] (V.G. Baryshevsky).
waveguides and others) and being much greater than the radiation wavelength. The feature of the cooperative operation regime lies in the fact that the peak radiation power scales as the squared number of particles in the bunch. (This allows calling this regime ‘‘superradiance’’ by analogy with the phenomenon predicted by Dicke in quantum electronics [1]). To avoid misunderstanding, let us note that initially, the coherent part of electromagnetic radiation is highly suppressed, because the length of an electron bunch is much greater than the radiation wavelength. The electromagnetic wave emission starts as incoherent spontaneous process, becomes coherent through the nonlinear interaction of the electrons and the electromagnetic field. In free electron lasers, the initial phases of charged particles in the electromagnetic wave are homogeneously distributed. As a result, bremsstrahlung produced by oscillating electrons starts from incoherent spontaneous emission. This is true even if the bunch length is much smaller than the radiation wave length. In contrast to bremsstrahlung, Cherenkov (quasi-Cherenkov) radiation starts from coherent spontaneous emission when such a short-length bunch is injected into a slow-wave structure, i. e. the radiation power is proportional to the squared number of particles.
http://dx.doi.org/10.1016/j.nimb.2015.03.054 0168-583X/Ó 2015 Elsevier B.V. All rights reserved.
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
2
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Fig. 1. The two-wave (left) and three-wave (right) diffraction geometries.
This paper considers cooperative quasi-Cherenkov radiation emitted by electron bunches when charged particles pass through crystals (natural or artificial) under the conditions of dynamical diffraction of electromagnetic waves. The concept of parametric (quasi-Cherenkov) radiation was originated in the works devoted to the interaction of charged particles with natural crystals (see, for example [3,5,6]). According to [3,5], the dynamical diffraction of photons emitted by a relativistic charged particle in a crystal leads to a change in the refractive index of X-ray quanta. In accordance with [3,5], diffraction of virtual photons in a crystal kÞ, some of which may possesses a set of refractive indices nl ðx; ~ k is the photon wave vector). appear to be greater than unity (~ Particularly, in the case of two–wave diffraction, the refractive indices are n1 ðx; ~ kÞ > 1 and n2 ðx; ~ kÞ < 1, and accordingly, two types of waves propagate in the crystal: a fast wave (n2 < 1) and a slow one (n1 > 1). The Cherenkov condition can be fulfilled for a slow wave, but not for a fast one. The former gives rise to spontaneous quasiCherenkov radiation, called the parametric X-ray radiation (PXR). The latter is emitted at the vacuum–crystal boundary. The considered phenomenon has a universal character and can be observed in different spectral ranges (microwave, terahertz, optical, etc.) for particles passing through varied two- and three-dimensional spatially periodic structures [9], often called the photonic crystals. A detailed analysis of the features of incoherent spontaneous radiation of electrons passing through crystals in both frequency [3] and time [4] domains has been carried out. This radiation, emitted at both large and small angles with respect to the direction of electron motion. The problems of amplification of induced parametric X-ray radiation and microwave (optical) quasi-Cherenkov radiation have also been thoroughly studied in the literature [7], and the threshold current densities providing lasing in crystals have been calculated [8]. Coherent spontaneous radiation produced by modulated electron bunches in crystals has been analysed in [10–12]. The paper is arranged as follows: In the beginning, a nonlinear theory of interaction of relativistic charged particles and the electromagnetic field in crystals is set forth, followed by the results of numerical calculations of the parametric radiation pulse. The dependence of the radiation intensity on the current density of an electron bunch and the geometrical parameters of the system is considered. 2. Nonlinear theory of cooperative radiation A theoretical analysis of radiation can be performed only by means of a self-consistent solution of a nonlinear set of the Newton–Maxwell equations:
pa d~ ~~ Eð~ ¼ qe ~ v a Hð ra ; tÞ þ ~ ra ; tÞ=c ; dt
ð1Þ
~ 1 @H ; c @t ~ ~ ¼ 1 @ D þ 4p~j; rH c @t c D ¼ 4pq; r~ ~ ¼ 0; rH
ð2Þ
E¼ r ~
~ describing the electron motion in the electric ~ E and magnetic H P P ~ v a dð~r ~ra Þ and q ¼ qe a dð~r ~ra Þ are the curfields. Here j ¼ qe a ~ rent and charge densities, respectively. Since the crystal is a periodic linear medium with frequency dispersion, the Fourier Dð~ r; xÞ relates to the transform of the electric displacement field ~ ~ ~ electric field Eðr; xÞ as
~ Dð~ Eð~ r; xÞ ¼ ð~ r; xÞ~ r; xÞ ¼
1 þ v0 ðxÞ þ
X
! Eð~ 2v~s ðxÞ cosð~ s~rÞ ~ r; xÞ;
ð3Þ
~ s
where the summation is made over all reciprocal lattice vectors. This permits to reduce Maxwell’s Eqs. (2) to the equation of the form:
1 @2 c2 @t 2
Z
t
ð~r; t t1 Þ~Eð~r; t1 Þdt1 þ rðr ~EÞ D~E ¼
1
4p @~j : c2 @t
ð4Þ
Let’s simplify the Eq. (4) for the case when jv0;s j 1 and two strong waves are excited in the crystal: the forward wave and the diffracted wave (the so-called two-wave diffraction case). The k0 ) is emitted at small forward wave (its wave vector is denoted by ~ angles with respect to the particle velocity, while the diffracted ks ¼ ~ k0 þ ~ s, is emitted at large angles one, having the wave vector ~ to it (Fig. 1). Under the conditions of dynamical Bragg diffraction, k2 ~ k2 x2 =c2 . the following relation is fulfilled: ~ s
0
Let us perform the following simplifications: First, we shall D ! 0) fields of the bunch. Second, neglect the longitudinal (r ~ E using the method of slowly we shall seek for the electric field ~ varying amplitudes. Third, we shall assume that a transversally infinite bunch executes one-dimensional motion along the OZ-axis (this is achieved by inducing a strong axial magnetic field in the system). E can be Under the conditions of two-wave diffraction, the field ~ presented as a sum
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx ~ ~ ~ E ¼~ e0 E0 ðz; tÞeiðk0~rxtÞ þ ~ es Es ðz; tÞeiðks~rxtÞ ;
ð5Þ
where the amplitudes of the forward E0 and diffracted Es waves are slowly varying variables. This means that for the distances comparable with the wavelength and the times comparable with the oscillation period, the values of E0 and Es remain practically the same. Substituting (5) into (1) and (4) and then averaging them over the length l which is equal to the integer number of the wavelength and is smaller than the characteristic lengths appearing in the problem (geometrical dimensions, the beam instability length), we obtain
dpza dt
¼ 2qe e0z Re E0 eiðk0z za xtÞ ;
ð6Þ
1 @E0 @E0 iv0 x iv x þ c0 þ E0 þ s Es c @t @z 2c 2c Z 2p zþl=2 iðxtk0z zÞ ¼ e0z jz e dz=l; c0 ¼ k0z =k0 ; c zl=2 1 @Es @Es iv0 x iv x þ cs þ Es þ s E0 c @t @z 2c 2c Z 2p zþl=2 ¼ esz jz eiðxtksz zÞ dz=l; cs ¼ ksz =ks : c zl=2
ð7Þ
z ¼ 0. In the case of gaussian beams (jb ðtÞ ¼ j expðc2 t 2 =L2b Þ), the electrons are injected into the crystal continuously at 1 < t < þ1. The difference between the two diffraction schemes is not merely kinematic, but radical. In the Bragg case, there is a synchronous wave moving against the electrons of the beam, which gives rise to the internal feedback and absolute instability. In Laue diffraction geometry, a backward wave is absent, and as a result absolute instability does not evolve. It may seem that electromagnetic radiation is not generated. However, fluctuations of the electron current (the shot noise), which always occur in real beams, are amplified when the beam enters the crystal (due to convective instability, excited in the beam). In analyzing multiparametric problems, to which the problem of cooperative parametric (quasi-Cherenkov) radiation refers, it is convenient to write Eqs. (6) and (7) in a dimensionless form. This procedure enabled transferring the calculation results from one set of parameters to another. The substitution of variables xt ! t; xL=c ! L, mcxE0;s =qe ! E0;s then gives
ð8Þ
X hv eiðtk0z za þ/a Þ @Es @E0 @E0 iv0 iv ba þ c0 þ E0 þ s Es ¼ ; @t @z 2 2 2 Nl @t j @Es iv0 iv þ Es þ s E0 ¼ 0: @z 2 2
vb ¼ vb0 expðz2 =L2b Þ;
ð10Þ
where the bunch length Lb is further assumed to be equal to 0:1L=c. Obviously, in the three-diffraction case, the set of equations analogues to (9) should be rewritten as follows:
@E2 @E2 iv0 iv þ c2 þ E2 þ s ðE3 þ E1 Þ ¼ 0; @t @z 2 2 @E3 @E3 iv0 ivs þ c3 þ E3 þ ðE1 þ E2 Þ ¼ 0: @t @z 2 2
ks , vanishes in the parthat of the wave Es , emitted in the direction ~ ticles’ exit plane (z ¼ L). In Laue geometry, both waves E0 and Es are emitted into the half-space z > 0, and both of them vanish in the particles’ entry plane (z ¼ 0). The time t ¼ 0 corresponds with the maximum value of the current density jb ðtÞ of the beam injected into the crystal in the plane
þ cs
corresponding electron density, h is the angle between the particle k0 ; /a is the initial phase of the ath velocity and the wave vector ~ particle, and N l is the number of particles over the length l. The set of equations with boundary conditions contains four independent parameters: v0;s ; xL=c; h that define the geometry of the system. In addition to these parameters, we need to specify the beam profile. Let
X hv eiðtk1z za þ/a Þ @E1 @E1 iv0 iv ba þ c1 þ E1 þ s ðE2 þ E3 Þ ¼ ; @t @z 2 2 2 Nl a
Now let us complete the set of Eqs. (6) and (7) with boundary conditions. The boundary conditions imposed on the fields follow from the form of the equations governing E0 Es and are standard boundary conditions used in the dynamical theory of diffraction of electromagnetic waves in crystals at jv0 j; jvs j 1 [13,14]. In Bragg geometry, the amplitude of the wave E0 , emitted in the k0 , vanishes in the particles’ entry plane (z ¼ 0), while direction ~
dpza ¼ 2hRe E0 eiðtk0z za þ/a Þ ; dt
3
ð9Þ
Here the quantity vba ¼ 4pq2e na =mx2 is determined at the moment when the ath particle enters the system, na is the
ð11Þ
3. Simulation results Let us see now how the dynamical diffraction of electromagnetic waves affects the cooperative radiation in crystals. How the Bragg diffraction case is different from the case of Laue diffraction? We will try to answer this question with the help of the numerical code [15] based on the particle-in-cell method [16], which enables studying kinetic phenomena. Let us note that in most of the existing codes (see, e.g. [17,18]) used for simulating the interaction of charged particles and a synchronous wave, the motion of charged particles is considered within the framework of the hydrodynamic approximation. Let us start our consideration with the Bragg case. In this case, along with the electromagnetic wave emitted in the forward direction, one can observe the electromagnetic wave that is emitted by charged particles in the diffraction direction and leaves the crystal through the bunch entrance surface. We shall assume that c ¼ 3:0; h ¼ 0:33 rad, v0 ¼ 0:2; ¼ vs ¼ 0:1; L ¼ 6 cm and j > 10 kA/cm2. The beams with current densities as high as j > 10 kA/cm2 are obtained at relativistic high-current accelerators [19]. According to the analysis [20], at electron energies less than 3 MeV, the artificial structures are tolerant of the current pulses lasting for t pulse if jtpulse < 4:5 105 C/cm2 (at larger jtpulse , plasma is formed on metal surfaces, disturbing the conditions of electromagnetic-wave propagation). Let j ¼ 30 kA/cm2, t pulse ¼ 1 ns, then jt pulse ¼ 3 105 C/cm2 and satisfies the set requirements. According to the estimates [21], under the electron beam of current density j ¼ 30 kA/cm2 and electron energy of 1:5 MeV (c ¼ 3), the temperature in the tungsten wires of a photonic crystal built from metallic threads rises by less than 300 K. The peak intensity of cooperative radiation emitted at small and large angles to particle velocity is investigated as a function of the peak current density j. The peak radiation intensity appeared to increase monotonically until saturation is achieved (Fig. 2). At saturation, the shot noise causes strong fluctuations in the intensity of cooperative parametric radiation. The dispersion of radiation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi intensity d0 ¼ j < I0 >2 < I20 > j as a function of j is presented on Fig. 2 (the brackets < :: > denote average values). The results of computation (Fig. 3) show that the cooperative radiation emitted at large angles lasts much longer (t rad 0:6 ns) than the particle flight time through the crystal (tp ¼ 0:2 ns), though that is much lower than the radiation intensity emitted in forward direction. We would like to note that the long duration
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
4
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Fig. 2. The Bragg case. Quasi-Cherenkov radiation at small angles to particles’ velocities: radiation intensity (left), intensity dispersion (right) [hB ¼ 67:5o ; m ¼ 0:1 THz].
Fig. 3. The Bragg case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [hB ¼ 67:5o ; j ¼ 10 kA/cm2,
m ¼ 0:1 THz].
Fig. 4. The Bragg case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [hB ¼ 67:5o ; m ¼ 1:0 THz].
Fig. 5. The Laue case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [hB ¼ 22:5o ; j ¼ 10 kA/cm2,
of parametric radiation can be observed in spontaneous processes too [4]. Figs. 2 and 3 correspond to the frequency m ¼ 0:1 THz and current density j ¼ 10 kA/cm2. If we increase m in ten times leaving the current density unchanged the peak radiation intensity will be 240 MW/cm2 (Fig. 4).
m ¼ 0:1 THz].
Now, let us consider the Laue case. In this case, the electromagnetic waves emitted by charged particles in the forward and diffraction directions leave the crystal through the same surface. Under Laue diffraction conditions (Fig. 5), the pulses of parametric radiation emitted in forward and diffracted directions have comparable amplitudes and durations.
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054
5
S.V. Anishchenko, V.G. Baryshevsky / Nuclear Instruments and Methods in Physics Research B xxx (2015) xxx–xxx
Fig. 6. The Laue case. Quasi-Cherenkov radiation in the absence of shot noise (left) and in the presence of shot noise (right), respectively [hB ¼ 22:5o ; Lb =L ¼ 1:0; j ¼ 10 kA/ cm2, m ¼ 0:1 THz, vb ðzÞ ¼ const].
Fig. 7. The three-wave diffraction case. Quasi-Cherenkov radiation at small (left) and large (right) angles to particles’ velocities [j ¼ 10 kA/cm2,
We should point out that the shot noise results in strong fluctuations in radiation intensity at saturation. Under Laue diffraction conditions, the shot noise leads to an appreciable change in the pulse form due to the convective character of instability: in the absence of noise, the generation occurs only at the ends of the bunch of charged particles. As a result, the cooperative pulse posses a two-peak structure (Fig. 6). The presence of noise leads to an appreciable change in the pulse form: the interval between the two pulses is filled with a chaotic signal (Fig. 6). Under three-wave diffraction conditions (Fig. 1), the results of computation are very similar to those of the Bragg case (Fig. 7). Namely, the intensity of cooperative radiation emitted at large angles lasts much longer than the particle flight time through the crystal, though that is much lower than the radiation intensity emitted in forward direction. 4. Conclusion This paper studies the features of parametric (quasi-Cherenkov) cooperative radiation emitted at both large and small angles to the particle velocity direction in two- and three-wave diffraction cases. A detailed numerical analysis is given for cooperative THz radiation in artificial crystals. The intensity of THz radiation above 200 MW/cm2 is obtained in simulations. The peak intensity of cooperative radiation emitted at small and large angles to particle velocity is investigated as a function of the peak current density. The peak radiation intensity appeared to increase monotonically until saturation is achieved. At saturation, the shot noise causes strong fluctuations in the intensity of cooperative parametric radiation. It is shown that, the intensity of cooperative radiation emitted at large angles can last much longer than the particle flight time through the crystal. This enables a thorough experimental investigation of the time structure of cooperative parametric
m ¼ 0:1 THz].
radiation generated by electron bunches available with modern accelerators. The complicated time structure of cooperative parametric (quasi-Cherenkov) radiation can be observed in crystals (natural or artificial) in all spectral ranges (X-ray, optical, terahertz, and microwave). References [1] R. Bonifacio et al., Rivista Nuovo Cimento 13 (9) (1990) 1–69. [2] S.D. Korovin et al., Phys. Rev. E 74 (2006) 016501. [3] V.G. Baryshevsky, I.D. Feranchuk, A.P. Ulyanenkov, Parametric X-ray Radiation in Crystals: Theory, Experiment and Applications, Springer-Verlag, Berlin Heidelberg, 2005. [4] S.V. Anishchenko, V.G. Baryshevsky, A.A. Gurinovich, Nucl. Instrum. Methods B293 (2012) 35–41. [5] V.G. Baryshevsky, High-Energy Nuclear Optics of Polarized Particles, World Scientific Publishing, Singapore, 2012. [6] P. Rullhusen, X. Artru, P. Dhez, Novel Radiation Sources using Relativistic Electrons: From Infrared to X-rays, World Scientific Publishing, Singapore, 1998. [7] V.G. Baryshevsky, I.D. Feranchuk, Phys. Lett. A 102 (1984) 141–144. [8] V.G. Baryshevsky, Available from: 1211.4769, 2012. [9] V.G. Baryshevsky et al., Dokl. Akad. Nauk SSSR 299 (6) (1988) 1363–1365 [in Russian]. [10] V.G. Baryshevsky, Vesti AN BSSR 1 (1984) 31–37 [in Russian]. [11] V.G. Baryshevsky, Avaible from: 1101.0783v1, 2011. [12] K.A. Ispirian, Nucl. Instrum. Methods B309 (2012) 4–9. [13] Z.G. Pinsker, Dynamical Scattering of X-rays in Crystals, Springer, Berlin, 1978. [14] S. Chang, Multiple Diffraction of X-rays in Crystals, Springer, Berlin, 1978. [15] S.V. Anishchenko, V.G. Baryshevsky, Available from: 1411.2960v1, 2014. [16] I.J. Morey, C.K. Birdsall. Memorandum No. UCB/ERL M89/116. Electronics Research Laboratory. University of California, 1989. [17] T.M. Antonsen et al., Proc. IEEE 87 (1999) 804–839. [18] N.S. Ginzburg, S.P. Kuznecov, T.H. Fedoseeva, Izv. vuzov. Radiofizika 21 (1978) 1037–1052. [19] G.A. Mesyats, Pulsed Power, Kluwer Academic/Plenum Publishers, New York, 2004. [20] S.P. Bugaev, et al. in: Relyativistskaya vysokochastotnaya electronika. Gorky, pp. 5–75. 1979. [in Russian]. [21] V.G. Baryshevsky, et al. THAAU03. Proceedings of FEL 2007.
Please cite this article in press as: S.V. Anishchenko, V.G. Baryshevsky, Cooperative parametric (quasi-Cherenkov) radiation produced by electron bunches in natural or photonic crystals, Nucl. Instr. Meth. B (2015), http://dx.doi.org/10.1016/j.nimb.2015.03.054