1. Introduction

Smith-Purcell effect is widely known as a source of electromagnetic radiation in various frequency ranges, since 1953, when Smith and Purcell revealed that the electrons moving above a grating without crossing it, emit the electromagnetic radiation [1]. As most radiation processes accompanying the motion of charged particles, it can be explained as a result of scattering of the Coulomb field of the moving charged particles on the irregularities of media. This radiation can be described as well as a result of dynamical polarization of periodically inhomogeneous target by the field of charged particles beam. In this paper we shall appeal to later way of description, though both explanations are correct and lead to identical results.

The characteristic feature of Smith-Purcell radiation (SPR), considering its spectral-angular distribution, is a strong link between the wavelength $\lambda$, angle of emission $\theta$, period of the grating $d$ and the reduced velocity of the electrons $\beta$:

This dispersion law is universal, it does not depend on the material or profile of the grating, and can be obtained directly from optical analogies and laws of conservation [2]. The radiation peaks are rather narrow, having the width inversely proportional to the number of rulings of the grating; all the peaks are distributed in the plane containing the electrons trajectory and perpendicular to the surface of the grating, see the plane $xz$ in Fig. 1 (to be precise, this is the case only when the particles trajectory and the grating rulings cross under the right angle; when the angle is arbitrary, SPR is distributed over some conical surface, see [2]).

 

Fig. 1 Layout for generation of Smith-Purcell radiation.

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It is important, that in SPR the electrons are not scattered in the material of the target. For this reason the flow of charged particles does not damage the target, providing the reliability and long survival time for practical SPR based radiation sources. Along with that, SPR provides very attractive opportunity of non-destructive diagnostics, because the beams are not spoiled in the process of radiation bringing the information about its parameters. Also, SPR offers excellent prospects for realization of the so-called single-shot measurements [3–5], not only in the picosecond range, but potentially in the femtosecond range as well.

In practice SPR is used most frequently in coherent regime, when all the electrons of the beam radiate coherently, so that the radiation intensity is proportional to the squared number of electrons $N^2$. The coherent regime takes place if the beams are separated into short bunches or modulated with period of modulation shorter than a wavelength of radiation. As SPR is quasi-monochromatic according to its dispersion relation, then one can expect that if SPR is generated by the beam having its own periodicity (train of bunches, long modulated beam), these two mechanisms can overlap [6, 7], enhancing each other.

The modulated or microbunched beams are obtained and used very often in practice due to impulsiveness of generation processes. For instance, very common photo-injector, or photo-gun, is operated by an impulsive laser, pico- or femtosecond. Producing such beams is a problem requiring taking into account strong Coulomb interaction on the stage of injector when the electrons are nonrelativistic; the details of this process the interested reader can find in [8]. Modulated beams are operated in the Free Electron Laser (FEL) as well. In FELs pre-bunching (the external field acts upon the beam) [9, 10] or self-amplification (the beam is split by its own radiation field or by the field of surface waves running on the surface of the grating) [9, 11–13] is absolutely necessary, since it is the most significant feature providing coherent mode and differing FEL by itself from undulators and wigglers. A new and promising scheme of obtaining pre-bunched beams is to use the transverse-to-longitudinal phase exchange technique [14, 15]. In addition, though the modulation with very short period is a problem by itself, being once realized it could become the key to enhance drastically the intensity of the inverse Compton source, the most promising source of X-ray radiation for medical and other applications; using it in coherent mode would be decisive for bringing this very promising device to life, with accompanying strong social impact.

In SPR using the trains of bunches also leads to a powerful effect of changing the angular distribution of SPR, which was called “frequency-locked SPR” [16, 17]. This effect has been a key for explaining the experimental observation [18] of another, so-called conical effect in SPR ([2], see also [19]), which, in its turn, proves to be responsible for the unusual and reach diffraction pattern of SPR from 2D photon crystals [20]. The authors of the cogent experiment published recently in [21] have demonstrated clearly that the inner structure of such microbunched beams can be detected with high accuracy.

While Smith-Purcell radiation from modulated and pre-bunched beams has been considered in many papers (besides those referred above, we, not intending to be exhaustive, would mention the initial paper [11], very important ones [9, 12] and recent [21–25]), more or less general theory of this phenomenon has not yet been constructed. We here fill this gap, presenting consecutive consideration of SPR, embracing cases of modulated beams with arbitrary depth of modulation and train of bunches, and obtaining the expressions for the form-factors from the first principles, what allows describing both coherent and incoherent properties of the radiation.

The paper is organized as follows. In Section 2 we generalize the universal expressions describing SPR from bunches of charged particles. In Section 3 we derive from the first principles the form-factors for Gaussian-shaped beam with sinusoidal modulation of arbitrary depth and in Section 4 the same for train of identical microbunches. In Section 5 we analyze numerically the spectral and angular characteristics of SPR in THz and X-ray ranges, and suggest some new practical applications proceeding from the numerical analysis performed before. Section 5.3 contains examination of real beam parameters, such as beam emittance, angular divergence, energy spread, which are important in experiments; in that section we prove that for practically interesting parameters our suggestions are feasible. In Summary we sum up the results and suggest new opportunities for control of the radiation characteristics.

2. Fundamentals of Smith-Purcell radiation from bunches

Let us consider the beam of $N$ electrons arranged in $N_b$ microbunches forming the internal structure with period ${\lambda }_0$. The beam moves at a constant distance $h$ above a grating surface with constant velocity $v=\left(v,0,0\right)$. The period of the grating is $d$, $N_{st}$ is the number of the strips in the grating, and the single strip width is $a$. Speaking of strips, we would like to stress that the type of the grating is important only for calculation of a single-electron emission, while the general results concerning the radiation of a multi-bunched structure, remain the same for any grating. The qualitative scheme for generation of SPR is shown in Fig. 1.

The general expression for the spectral-angular distribution of SPR has the form [26, 27]:

In Eq. (3) $F_{inc}$ and $F_{coh}$ are the incoherent and coherent form-factors. Note that for SPR $F_{inc}\ne 1$ [4, 27–30]. For the modulated beam $F$ can be derived like in [28] from the formulas:

The factor $G$ is known to have a simple form (see, e.g., [2]):

In this paper we consider the radiation from the beams modulated in two ways, i.e. we describe the longitudinal distribution of the particles in the beam by two different functions: the sine and the sum of Gaussians.

3. Gaussian-shaped beam with sinusoidal modulation

Let us consider the bunch having the Gaussian distribution with the periodic inner structure, for example harmonic periodicity. Such a beam charge distribution is typical for the electron beam passed through a FEL undulator [31, 32] where the fundamental radiation harmonic has the wavelength ${\lambda }_0$. The bunch structure can be described as a set of harmonics

The contribution from higher harmonics is small if $K<1$ ($K$ is the FEL undulator field strength parameter) [33] and one can consider the first harmonic microbunching only. The longitudinal profile of the beam in this case can be approximated by the function

 

Fig. 2 The longitudinal profile of the beam $f_l\left(x\right)$ described by Eq. (7) for different parameters.

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The transversal distribution $f_{tr}\left(x\right)$ is assumed to be

All distributions written here and below are normalized to unity.

In practice, most beams are better described by Gaussian distributions. The uniform distribution, however, can be used as well, and not only for the so-called elliptical distribution [34, 35] (“water-bag”), but for any beams when the type of the distribution does not play a crucial role for the effect being considered. For the sake of simplicity, here we use the uniform distribution in $z$ direction.

Integrating in Eq. (4) for the beam with the function of distribution described in Eqs. (7), (8) for independent distributions, i.e.$P\left(r\right)=f_l\left(x\right)f_{tr}\left(y,z\right)$, one can easily find the expression for the incoherent form-factor and for the coherent form-factor

From Eq. (12) the condition of strong enhancement of radiation intensity follows:

For ultrarelativistic particles, i.e. for the Lorentz factor$\gamma >>1$, $\beta =\sqrt{1-{\gamma }^{-2}}\approx 1$ the condition in Eq. (14) goes to ${\lambda }_0\approx \lambda$. The condition in Eq. (14) was obtained proceeding from the assumption ${\sigma }_x>>{\lambda }_0$, i.e. $N_b>>1$. Figure 3 presents the dependence of the longitudinal form-factor $F_l^f$ from Eq. (12) on the wavelength. It is evident that the bandwidth of the spectral line is determined by a microbunch number$N_b\approx 4{\sigma }_x/{\lambda }_0$. From Eq. (12), it can be shown that for $\gamma >>1$ the dependence of $F_l^f$ on the Lorentz factor disappears.

 

Fig. 3 The longitudinal form-factor $F_l^f$ described by Eq. (12) for $\mu =0$ and $\gamma =2\cdot {10}^4$.

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The width of the peak in Fig. 3 can be estimated from Eq. (12) as:

This value defines also the “area” of SPR enhancement due to beam modulation. For the parameters of Fig. 3 we can estimate, that if ${\lambda }_0={\sigma }_x/1000$ then $\Delta \lambda /{\lambda }_0=6.3\cdot {10}^{-4}$; if ${\lambda }_0={\sigma }_x/500$then $\Delta \lambda /{\lambda }_0=13\cdot {10}^{-4}$. These estimations are in accordance with Fig. 3.

4. Train of identical microbunches

The second way is convenient to describe the beam with rather big spacing between microbunches. In this case the Gaussian distribution of each microbunch should be taken into account. Last decade different techniques to produce a train of ultrashort electron bunch with subpicosecond spacing were developed [21, 36]. Such kind of the beam can be produced, for example, using so-called transverse-to-longitudinal phase exchange technique [14]. The other way to produce such beam is based on a laser photogun – in this case the number of microbunches in the “train” is of the order of $10$ [37, 38].

The longitudinal distribution can be written as:

Here ${{\sigma }^{\prime }}_x$ is the length of a single microbunch, unlike ${\sigma }_x$ in Eq. (7). Function in Eq. (16) describes a fully modulated beam at ${\lambda }_0>4{{\sigma }^{\prime }}_x$. This function is plotted in Fig. 4 for different parameters.

 

Fig. 4 The longitudinal profile of the beam described by $g_l\left(x\right)$ from Eq. (16).

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The transversal distribution $g_{tr}\left(y,z\right)$ is the same as in Eq. (8). Integrating in Eq. (4) for the beam with the function of distribution described in Eqs. (8), (16) one can find the expression for the coherent form-factor:

The resonant condition similar to Eq. (14) can be found from Eq. (19)

Figure 5 presents the dependence of the longitudinal form-factor $F_l^g$ from Eq. (19) on the wave length for different parameters.

 

Fig. 5 The longitudinal form-factor of the beam $F_l^g$ described by Eq. (19) at $\gamma =16$ (a) for $N_b=16$, (b) for ${{\sigma }^{\prime }}_x={\lambda }_0/7$.

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The widths of the peaks in Fig. 5 can be estimated from Eq. (19) as

This value defines also the “area” of SPR enhancement due to beam modulation. For the parameters of Fig. 5(a) we can estimate the widths of the highest two peaks as $\Delta \lambda /{\lambda }_0=0.13$ for $s=1$, and $\Delta \lambda /{\lambda }_0=0.06$ for $s=2$. The estimations are in accordance with Fig. 5. Note that for rather small number of microbunches $N_b\approx 2÷4$ there is some range of parameters

5. Characteristics of the radiation

Let us consider how the obtained form-factors influence on the spectral-angular distribution of radiation. To illustrate the general results, below we shall use the parameters of real facilities: LUCX (KEK, Japan) for moderately relativistic beams, Lorentz factor $\gamma$ is between 16 and 50 (energy of electrons are between $E_e=8MeV$and $25MeV$), and FLASH (DESY, Germany) for ultrarelativistic beams, $\gamma =2\cdot {10}^3$ ($E_e=1GeV$).

For LUCX we shall use the least of possible energies 8 MeV; we shall take the beam parameters as follows: the microbunch length ${{\sigma }^{\prime }}_x=100\mu m$, its transversal size ${\sigma }_{y,z}=250\mu m$ [21], so impact-parameter for THz region can be easily taken $h=0.7mm$, the period of the beam ${\lambda }_0=0.8mm$ [21], and is assumed that the number of bunches in the train can be $N_b=2,4,16$.

We would like to stress, however, that the expressions obtained above are valid for wide range of parameters, so the results are not depleted by the parameters for specific facilities like LUCX or FLASH.

For considered beam parameters the incoherent radiation is suppressed in comparison with the coherent one. For example, the comparison between coherent and incoherent parts of form-factor $F_{inc}^g$ and $N{F}_{coh}^g$ for particles with $\gamma =16$ is shown in Fig. 6. In Fig. 6 and other figures it is denoted:

 

Fig. 6 The comparison between $F_{inc}$(red dashed curve) and $N{F}_{coh}^g$ (black solid curve) for the beam of size ${{\sigma }^{\prime }}_x=100\mu m$, ${\sigma }_{y,z}=250\mu m$, ${\lambda }_0=0.8mm$, $\varphi =0$, $N_b=4$, $N/N_b={10}^8$, $\gamma =16$.

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One can see that the coherent part of radiation dominates starting from wavelengths of the order of the bunch length. For longer wavelengths $F_{inc}\approx 1$. For this reason below we shall consider only coherent radiation. What is important, is that the behaviour of the coherent form-factor is not monotonic. As is seen in the insert, there can be regions where incoherent radiation can dominate even if the wavelength exceeds the length of the bunch. The peaks in Fig. 6 are caused by the last factor in Eq. (19), i.e. it is a multibunched structure of the beam that is responsible for these peaks.

5.1 THz Smith-Purcell radiation

There are many sub-mm and THz radiation sources, most powerful of which are based on electron beams. This is natural, because the energy stored in the beams of charged particles with population of ${10}^7-{10}^{11}$ is extremely large. The most powerful among them are FELs, as well as gyrotrons (tens of kW, which requires, however, extremely high (more than 10 Tesla) magnetic fields, and working frequency achieves only 1.3 THz), while others, like orotrons, are less powerful (from one hundred of mW at 1 THz to 1 W at 0.3 THz).

The first experiment with SPR [11] demonstrated that SPR at the wavelength $\lambda =0.49mm$ can be obtained with help of nonrelativistic beam of a scanning electron microscope, with currents 0.1 – 1 mA. Another experimental result was reported in [40], where the authors demonstrated that a pulsed electron beam of 1.44 MeV energy with mean currents in tens of $\mu A$ can provide an SPR based THz source in the range 0.1-1 mm. Recent particle-in-cell simulations of SPR [41] promise the power up to 1 kW with help of 10 A currents, which means that SPR based source can be rather powerful. Comparing it with the closest competitor, gyrotron, we should like to stress that in contrast to it, SPR based source is tunable, pulsed and can easily cover all THz range.

Thus, SPR-based THz source occupies the intermediate position in the list of radiation sources: it is rather compact (3-5 metres even for the sources of relativistic electrons), which is more than the sizes of, e.g., gyrotrons, but much less in size than a FEL or a synchrotron, to say nothing of the price; its characteristics like wide working range, tunability and comparatively high power make it very promising for practical applications.

There are only few principal analytical models in theory of SPR (if the details are of interest, see, e.g., discussions in [2, 31], and also in [42]) valid in THz frequency domain. One of the most elaborated descriptions of SPR proceeds from the exact model solution obtained by Kazantsev and Surdutovich [43] for an ideally conducting infinitely thin target. The spectral-angular distribution of SPR from a single electron was adapted to the concerned coordinate system and geometry in [2]:

As a result, the spectral-angular distribution of the radiation at optical and lower frequencies, including THz ones, has the form of Eq. (2) with Eqs. (9) - (13), (28), (25), (5).

The relation between the parameters of the grating and those of the modulated beam, for which the enhancement of the radiation intensity is maximal, can be obtained from Eqs. (21) and (1) for the train of microbunches:

It may seem that condition in Eq. (26) is satisfied for all parameters. Actually, we can always find two integer numbers for any parameters. For real problems, however, it is of interest to enhance the radiation of more intensive SPR peaks, which are usually thought to be associated with small value $m$.

It follows from the Eq. (26) that the maximal SPR yield for the fixed value of ${\lambda }_0$ we shall observe at the angles

The enhancement of SPR at the angles from Eq. (27) is shown in Fig. 7. It shows that for the train of identical bunches for the fixed period of beam modulation ${\lambda }_0$ there are several peaks of enhanced radiation for each diffraction order of SPR ($m=const$).

 

Fig. 7 Spectral-angular distribution of SPR at THz frequencies for the trains of microbunches with population of each microbunch $N/N_b={10}^8$. Black solid curve is plotted for $\lambda ={\beta }^{-1}{\lambda }_0\approx 0.8mm$ (see Eq. (21) for $s=1$), red dashed curve is plotted for $\lambda ={\beta }^{-1}{\lambda }_0/2\approx 0.4mm$ (see Eq. (21) for $s=2$). For all curves ${\lambda }_0=0.8mm$, ${{\sigma }^{\prime }}_x=100\mu m$, ${\sigma }_{y,z}=250\mu m$, $\varphi =0$, $N_b=4$, $N_{st}=7$, $\gamma =16$, $h=0.7mm$, $d=2mm$, $a=1mm$.

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Figure 8 demonstrates the strong dependence of the spectral-angular distribution of the radiation on the wavelength of radiation for different number of the microbunches. The population of each microbunch was supposed to be the same for all curves. It is seen that the more $N_b$ is, the higher and narrower the peaks are. Thus, the modulated electron beam consisting of a set of bunches makes it possible to improve significantly the SPR spectral line monochromaticity for short gratings with comparatively small number of strips/rulings.

 

Fig. 8 Spectral-angular distribution of SPR at THz frequencies for the trains of microbunches with population of each microbunch $N/N_b={10}^8$ depending on the wavelength. Black solid curve is plotted for $N_b=2$, red curve $N_b=4$, blue curve $N_b=16$. For all the curves ${\lambda }_0=0.8mm$, ${{\sigma }^{\prime }}_x=100\mu m$, ${\sigma }_{y,z}=250\mu m$, $\varphi =0$, $N_{st}=7$, $\gamma =16$, $h=0.7mm$, $d=2mm,$ $a=1mm$, $\theta =0.2rad$.

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Note that not one only, but several peaks of Smith-Purcell radiation can be enhanced simultaneously due to the condition in Eq. (21) for each $s$. It depends on the ration between the “area of enhancement” defined by Eq. (22) and the width of the SPR peak $\Delta {\lambda }_{SP}\propto N_{st}^{-1}$, see Figs. 9(a) and 9(b). The black curves correspond to the spectral-angular distribution of radiation. The blue curves describe the factor $G$ from Eq. (5), which defines the position and the width of SPR peaks $\Delta {\lambda }_{SP}$, multiplied by $2\cdot {10}^{-2}$. The red curves correspond to $F_l^g$ from Eq. (19) multiplied by 10, they define the “area” of radiation enhancement $\Delta \lambda$.

 

Fig. 9 Spectral-angular distribution of SPR at THz frequencies (a) for $N_b=2$, $N_{st}=9$, $d=11mm$; (b) for $N_b=16$, $N_{st}=9$, $d=11mm$; (c) for $N_b=16$, $N_{st}=9$, $d=2.2mm$. For all the curves ${{\sigma }^{\prime }}_x=100\mu m$, ${\sigma }_{y,z}=250\mu m$, ${\lambda }_0=0.8mm$, $\varphi =0$, $\theta ={50}^0$, $\gamma =16$, $h=0.7mm$, $a=d/2$, $N={10}^9$. The blue curve is $2\cdot {10}^{-2}G$ with $G$ from Eq. (5), the red curve is the longitudinal form-factor multiplied by 10, i.e. $10F_l^g$, with $F_l^g$ from Eq. (19).

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If the number of the strips/rulings of the grating $N_{st}$ is much more than the number of the microbunches $N_b$, then $\Delta \lambda >>\Delta {\lambda }_{SP}$ and several SPR peaks can be enhanced, see Fig. 9(a). If $\Delta \lambda \approx \Delta {\lambda }_{SP}$, which happens when the number of the microbunches $N_b$ exceeds the number of strips of the grating $N_{st}$, then only one SPR peak can be enhanced, see Fig. 9(b). Moreover, the parameters of the grating can be chosen in such way, that $\Delta \lambda <<\Delta {\lambda }_{SP}$ and the narrowing of the spectral line is observed, as is shown in Fig. 9(c).

The depth of beam modulation can be detected by measuring the ratio between the THz radiation distribution of the different values of $s$, i.e. different wavelengths. This turns to be possible because the value ${{\sigma }^{\prime }}_x/{\lambda }_0$ defines the depth of beam modulation. For the ratio of the spectral-angular distributions of THz radiation for $s=1$ and $s=3$ for the third diffraction order, the dependence on the parameter ${{\sigma }^{\prime }}_x/{\lambda }_0$ is shown in Fig. 10. The curve has the specific maximum that makes the defining of the depth of beam modulation accurate.

 

Fig. 10 Dependence of $R=\frac{dW\left(m=3,s=1\right)}{dW\left(m=3,s=2\right)}$ on the depth of the beam modulation for $\theta =1.77\text{}rad$. The other parameters are as in Fig. 7.

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5.2 X-ray Smith-Purcell radiation

In the paper [44] the authors came to the conclusion that SPR would be a viable X-ray source being able to compete with magnetic undulators only if the emittance would be orders of magnitude less than it was available at that time. The emittance, however, usually decreases with increase in the beam energy, so, in contrast to the opinion of the authors of [44], nowadays it is asserted that SPR can be rather effective radiation source [45]. On the other hand, SPR from ultra-relativistic electrons with energies 20 GeV (e.g., LCLS (20 GeV) and European XFEL (17,5 GeV)) and more can be of interest not only as an EUV/X-ray source, for some beam parameters being able to compete with FELS, but as an instrument for noninvasive beam diagnostics with unattainable today submicron accuracy, which can be provided owing to the short wavelength.

For the period ${\lambda }_0\le 10nm,$ Smith-Purcell radiation is generated in soft X-ray region and the spectral-angular distribution of radiation from a single electron $d^2{W}_1\left(n,\omega \right)/d\omega d\Omega$ can be obtained proceeding from the theory developed in [2, 25]:

The relation between the parameters of the grating and those of the modulated beam, for which the enhancement of the radiation intensity is maximal, can be obtained from Eqs. (14) and (1) for the Gaussian-shaped beam with sinusoidal modulation:

It follows from the Eq. (31) that the maximal SPR yield for the fixed value of ${\lambda }_0$ we shall observe at the angles

The enhancement of SPR at the angles from Eq. (32) is shown in Fig. 11(a). In is seen that there is only one peak of enhancement for each peak of Smith-Purcell radiation. Notice that the wavelength in the Fig. 11(a) does not satisfy Eq. (14). If it did, the black peak for $m=1$ would have the height of the order of $2\cdot {10}^8$, while the blue peak for $m=1$ would not be observable.

 

Fig. 11 Spectral-angular distribution of SPR for the Gaussian-shaped beam with sinusoidal modulation at X-ray frequencies, according to Eqs. (2) and (28). (a) Black solid curve: radiation from fully modulated beam $\mu =0$; red dashed curve: partially modulated beam $\mu =0.2$; blue dotted curve for the single not modulated bunch. For all the curves ${\sigma }_x=20\mu m$, ${\lambda }_0=10nm$, $N_b\approx 2000$, $h=0.05mm$, $d=6.5\mu m$, $\lambda =10.005nm$. (b) Black solid curve: $d=6.4\mu m$, which corresponds to Eq. (32), $m=3$; blue dashed curve: $d=6.6\mu m$. For all curves: $\mu =0$, ${\sigma }_x/{\lambda }_0=500$, ${\lambda }_0=3nm$, $\theta =53\text{}mrad$, $h=10\mu m$. For both figures: $\gamma =2\cdot {10}^3$, ${\sigma }_{y,z}=10\mu m$, $\varphi =0$, $a=d/2$, $N={10}^{10}$, $\hslash {\omega }_p=26.1eV$(beryllium), $N_{st}=7$.

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The enhancement of radiation is very sensitive to the parameters of the grating. Figure 11(b) demonstrates the dependence of the spectral-angular distribution of SPR at X-ray frequencies on the wavelength of radiation for different periods of the grating. It is seen that the difference between black curve plotted for $d=6.4\mu m$ corresponding to the Eq. (32) for $m=3$, and the blue dashed curve plotted for $d=6.6\mu m$, is essential.

The depth of the beam modulation can be detected by measuring the X-ray radiation distribution for the different diffraction orders. This turns to be possible because of the parameter $\mu$, see Eq. (12). The dependence of the spectral-angular distribution of X-ray radiation on the parameter $\mu$ for the different diffraction orders is shown in Fig. 12.

 

Fig. 12 Dependence of the spectral-angular distribution of the radiation on the depth of beam modulation $\mu$. The black curve corresponds to the first SPR peak $m=1$, $\theta =55.5mrad$, the blue dashed curve corresponds to the fifth SPR peak $m=5$, $\theta =124.1mrad$. Other parameters as in Fig. 11(a).

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5.3 Influence of real beam parameters

Now let us discuss beam emittance, angular divergence, energy spread, and how these parameters affect the bandwidth of SPR produced by modulated beams.

As was shown before in transition radiation theory [46], the beam emittance can be described by two multipliers in the form-factor in equations like Eq. (3): the first one allows for the transversal distribution of the electrons in the beam, and the second describes the angular divergence of the beam. We investigated this problem in [47] and found that the structure of these multipliers remains the same also for diffraction radiation. Since SPR is described as diffraction radiation from periodical structures [31, 32], this conclusion holds true for SPR as well.

The transversal part of the form-factor has been taken into account already in Eq. (18) and hence is reflected in the figures above. Indeed, for the plane $n_y=0$ where SPR is maximal, the dependence on the transversal size of the beam ${\sigma }_z$ is negligible when

The contribution of the divergence can be estimated proceeding from the dispersion relation for SPR taken for non-zero angle ${\theta }_t$ between trajectory and the grating surface [48]:

Here it has been taken into account that for ultrarelativistic beams the angle of divergence is much less than unity, i.e. ${\theta }_t<<1$. For ${\theta }_t=0$ this expression turns into Eq. (1). The SPR peak broadening then can be estimated as

Meaning the real parameters, one can take for FLASH $\theta =0.05,{\theta }_t={10}^{-4}$, and therefore $\Delta \lambda /{\lambda }_t=4\times {10}^{-6}$, and for LUCX $\theta =1,{\theta }_t={10}^{-2}$, i.e. $\Delta \lambda /{\lambda }_t={10}^{-4}$.

Influence of beam divergence on form-factor in X-ray diffraction radiation was calculated in [36] and shown to depend on the parameters $D_{inc},\text{}\text{}{D}_{coh}$ for incoherent and coherent radiation modes, correspondingly:

Therefore, the contribution of the divergence is negligible if

It can be satisfied or not, depending on the real beam parameters; the influence of the factors $D_{inc},D_{coh}$, however, is not reflected in the width or position of the radiation peaks, and therefore it can lead just to a changing of intensity.

The contribution of the energy spread to the spectral width of SPR can be estimated as

For SPR, using the dispersion relation Eq. (1) for $\gamma >>1$ one gets

To sum up, the theory constructed in ultrarelativistic case works very well for real parameters of modern beams, as the beam emittance, angular divergence, and energy spread do not change significantly nor the form of the radiation peaks, neither their positions. The corrections due to divergence and energy spread can only widen the SPR peaks slightly and make them less intensive (decrease their height), but this effect depends neither on the depth of modulation nor on the number of microbunches, i.e. cannot change qualitatively the main results of our research.

6. Summary

Above we have demonstrated that the increase in Smith-Purcell radiation intensity caused by both periodicity of the target and periodicity of the beam depends strongly on the depth of the beam modulation. The measurements of Smith-Purcell radiation therefore can be used for detecting the depth of beam modulation both in X-ray and THz frequency ranges. For the THz frequencies this effect has been proved to be more pronounced. This might apparently be explained by the fact that for real parameters, which we used for numerical estimates, for T-rays the model for train of bunches was used (which can be considered as a fully modulated beam), while for X-rays the model of partially modulated beams is more adequate. So, we suppose, the depth of modulation plays here a key part.

Further, the microbunched beam after a FEL undulator (provided bending magnets remain the beam structure the same, i.e. not turned) can be used as an effective supplementary source of radiation, in a wide range from THz to X-rays, e.g. in FLASH, European XFEL, or other accelerator complexes. The width of the spectral line of Smith-Purcell radiation for a non-modulated beam can be estimated as

The width of a modulated beam is defined not only by the number of the grating strips $N_{st}$but also by the “area” of radiation enhancement $\Delta \lambda \approx \frac{8{\lambda }_0}{\beta \pi N_b}$ (Gaussian-shaped beam with sinusoidal modulation) or $\Delta \lambda =\frac{2}{\left(N_b^2{s}^2-1\right)}\frac{N_b{\lambda }_0}{\beta }$ (train of microbunches). The resulting width of the spectral line is then defined by the smallest one of these three. In particular, if $\Delta \lambda <<\Delta {\lambda }_{SP}$, then the narrowing of the spectral line of the radiation occurs.

The authors of the work [49] proposed to use a train of short electron bunches (modulated electron beam in our terms) to produce a narrowband THz source based on coherent TR. The train with 4 bunches enables achieving 25% bandwidth from the initially continuous TR spectrum. We believe that the same technique can provide the SPR line monochromaticity better than 1% even for a 8-bunches train, which is feasible, e.g. for LUCX (KEK, Japan), or other facilities of this type.

To sum up, in this paper we present new analytical expressions describing coherent and incoherent form-factors in radiation of electron bunches, and suggest new opportunities for experiment:

All these opportunities have not been realized yet; their realization can open new ways for realization of pulse sources of monochromatic electromagnetic radiation based on electron and other type charged particle beams.

The analytical expressions obtained are rather simple, and on the other hand are capable of giving a strong impact for a wide range of applications, because the coherent and incoherent form-factors in radiation contain key information vitally important for most applications: undulator radiation in FELs, promising X-ray radiation source based on inverse Compton scattering, other sources of radiation using charged particle beams (based on Cherenkov, transition, diffraction, parametric radiations), in a wide range from THz to X-rays.