Study of Coherent Smith–Purcell Radiation in the Terahertz Region Using Ultra-Short Electron Bunches

Abstract

Smith–Purcell radiation (SPR) can be generated nondestructively, providing valuable applications in light sources and beam monitors. Coherent SPR is expected to enable single-shot measurements of very short bunch lengths on the fs scale. Since the reconstruction of the longitudinal bunch shape from the coherent SPR is based on the reliable SPR spectrum, a more detailed understanding of the properties of the radiation is important in this context. Employing a 100 fs ultrashort electron bunch at the t-ACTS test accelerator, the spectrum, angular distribution, and polarization of the produced coherent SPR were measured in the terahertz frequency region and compared with a model calculation. In addition to the widely known surface current model evaluation, the effect of the geometrical shading effect on induced currents on metal surfaces was evaluated using 3D numerical calculations. The obtained SPR characteristics are also presented. In the evaluation of the grating with a shallow blaze angle, it was found that the shading effect has a non-negligible effect on the generated SPR intensity; the measured angular distribution and polarization results were in good agreement with this result.

1. Introduction

Smith–Purcell radiation (SPR) is produced when electrons pass near metal surfaces with periodic structures [1]. SPR has a dispersion relation between the radiation wavelength ${\mathrm{\lambda }}_m$ and emission angle θ, given by

$${\mathrm{\lambda }}_m=\frac{d}{m}\left(\frac{1}{\beta }-cos\theta \right),$$

(1)

where $\beta$ is the electron velocity normalized to the speed of light, $d$ is the period length of the periodic structure, and $m$ is the order of the radiation (see Figure 1). Radiation emitted from electron bunches that are sufficiently shorter than the radiation wavelength causes “coherent enhancement”, and the degree of enhancement strongly depends on the longitudinal distribution of electron bunches. This has led to attempts to measure bunch length in a single shot by simultaneously measuring the coherent SPR with multiple detectors placed at different radiation angles [2,3,4]. Especially for accelerators with low beam repetition rates, such as laser-plasma acceleration [5], the ability to obtain information on bunches on a beam nondestructively would be a great advantage in terms of efficient beam utilization. A reliable theoretical spectrum corresponding to the measurement conditions must be known when deriving the bunch form factor using coherent radiation. So far, some attempts have been made to measure bunch lengths from a few picoseconds to several hundred femtoseconds using coherent SPR [6,7]. The surface current (SC) model has been used as one of the methods to evaluate the theoretical spectrum of SPR. In this paper, we present coherent SPR measurements with ultrashort electron bunches below 100 fs performed at the t-ACTS (test Accelerator as Coherent THz Source) facility at Tohoku University [8] and discuss the comparison with the SC model evaluation.

2. Expectation by Surface Current Model

2.1. Surface Current Model

The SC model explains SPR through currents induced on the surface of the grating by a charge passing nearby [9,10]. The energy radiated to the far field by the surface current density J [11] is expressed as

$$W\equiv \frac{{\partial }^{2}I}{\partial \omega \partial \mathrm{\Omega }}=\frac{{\omega }^2}{4{\pi }^2{c}^3}{\left|\int dt\int d^{3}x\widehat{\mathit{n}}\times (\widehat{\mathit{n}}\times \mathit{J}\left(\mathit{r},t\right))e^{i\left(\omega t-\mathit{k}·\mathit{r}\right)}\right|}^2,$$

(2)

where $\widehat{\mathit{n}}=\left\{\widehat{\mathit{x}}\sin \theta \cos \varphi +\widehat{\mathit{y}}\sin \theta \sin \varphi +\widehat{\mathit{z}}\cos \theta \right\}$ is the direction of emission, $\mathit{k}=\left(k_x,k_y,k_z\right)=\widehat{\mathit{n}}\omega /c$, and $\omega$ is the radiation frequency. The surface charge is obtained by the electric field caused by the moving charge. Then, the surface current density is given by the product of the surface charge density and its velocity, neglecting the effects of the facet edges and reflections from other facets in the SC model [9]. The SPR energy $dI$ emitted per unit solid angle $d\mathrm{\Omega }$ by a single electron passing at a distance $h$ above the grating is given by

$${\left(\frac{dI}{d\mathrm{\Omega }}\right)}_1=2\pi q^2\frac{Z}{l^2}\frac{m^2{\beta }^3}{{\left(1-\beta cos\theta \right)}^3}{R}^2\exp \left(-\frac{2h}{{\lambda }_e}\right),$$

(3)

where $Z$ is the grating length, q is the electron charge, and ${\lambda }_e$ is “evanescent wavelength”, defined by

$${\lambda }_e\equiv {\left(\frac{4\pi }{\gamma \beta {\lambda }_m}\sqrt{1+{\gamma }^2{\beta }^2{\sin }^2 \theta {\sin }^2 \varphi }\right)}^{-1}.$$

(4)

The factor $R^2$ is referred to as the grating efficiency, which represents the field contribution from each tooth of the grating and can be written as

$$R^2=|\widehat{\mathit{n}}\times (\widehat{\mathit{n}}\times \overline{\mathit{G}})|.$$

(5)

The vector $\overline{\mathit{G}}$ is the vector sum of the contribution from each facet: $\overline{\mathit{G}}={\sum }_{j=1}^N{\overline{\mathit{G}}}_{\mathit{j}}$. The contribution of the jth facet is given by

$$\begin{gathered} {\overline{\mathit{G}}}_{\mathit{j}}=\left(\tan \alpha ,i{k}_y{\lambda }_e\tan \alpha ,1\right)\times \mathrm{e}\mathrm{x}\mathrm{p}\left[\left(\frac{1}{{\lambda }_e}-i{k}_x\right)x_{j1}+i\left(\frac{k}{\beta }-k_z\right)z_{j1}\right]\times \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left[i{D}_j\left(z_{j2}-z_{j1}\right)\right]-1}{i{D}_{j}l}, \\ {D}_j=\frac{k}{\beta }-k_z-k_x\tan {\alpha }_j-i\frac{\tan {\alpha }_j}{{\lambda }_e}, \end{gathered}$$

(6)

where ${\alpha }_j$ is the blaze angle of the jth facet and $\left(x_{j1},z_{j1}\right)$, $\left(x_{j2},z_{j2}\right)$ are coordinates of the leading and trailing facet edges. Because $R^2$ is a complex function of the grating shape and radiation wavelength and cannot be solved analytically, except in very limited cases, it is usually obtained numerically.

An example of the azimuthal ($\varphi$) distribution of the CSPR calculated by the SC model for an echelle-type grating with a period length (d) of 600 μm and braze angle (α) of 6° is shown in Figure 2, with each contribution of the first and second facets. Here, the beam energy, rms bunch length, and observation angle in the θ direction were assumed to be 22 MeV, 100 fs, and 90°, respectively. The surface current on the first facet is oriented approximately in the z-direction, whereas that on the second facet has a large y-component as it spreads out from the center (ϕ = 0). Therefore, the distribution in the $\varphi$ direction shows two peaks, and, correspondingly, an increase in the polarization component parallel to the grating groove is expected off-center. At approximately 22 MeV, where the electron beam energy is not high enough, a peak appears at $\varphi$ = ±2.5°, indicating that to obtain more SPR, it is advantageous to cover a displaced angular range rather than near $\varphi$ = 0. As the electron beam energy increases to as high as 20 GeV, as presented in [2], the two peaks become almost centrally located, making such considerations less necessary.

2.2. 3D Numerical Calculation with Geometrical Shading

When measuring CSPR, there are various constraints that are not present in the model calculations, which may distort the understanding of the measured results. Three-dimensional numerical calculations based on the current density in Equation (2) were performed to evaluate the effects of a finite grating size and neglecting geometrical shading on the SPR intensity. The surface current was derived by dividing the grating surface into meshes and obtaining the induced charge and its velocity at each mesh. The total radiation intensity was then evaluated using Equation (2) by adding the currents at each mesh point over all meshes, considering those phase differences. Figure 3 shows an example of the calculated surface current density when a relativistic electron reaches (x, y, z) = (500 μm, 0 μm, 300 μm) above a grating surface with the same geometry as that for Figure 2. The surface current density of each mesh element on the grating surface is given by the product of the surface charge density at the point and velocity of the charge. Figure 3 shows the current density distribution and the current density vector. The obtained footprint size (FWHM) of the induced charge is consistent with 2b/γ cosα and 2b in the longitudinal (z) and transverse (y) directions, respectively.

In addition, a comparison of the results of the model calculation assuming an infinitely wide grating size and the numerical calculation for a finite size confirmed that for an impact parameter of 500 μm, the disagreement from the model calculation rapidly increases depending on the width of the grating in the y-direction when it is smaller than 5 mm. In this study, the numerical calculation was performed with a grating width of 30 mm for a more detailed comparison with the SC model. Another factor that differs from the actual measurement is the shading effect at the grating edge. As shown in Figure 4, the electric field produced by an electron induces less charge in the area shaded by the top of grating grooves, and the degree of this shading depends on the height of the grating grooves, the impact parameter, and the energy of particle beam. The effect in a grating composed of rod elements was presented in [12]. They pointed out that the effect in SPR depends on the shape of the grating, which may explain why a large discrepancy between experimental results of SPR intensity and theoretical estimations was observed in many studies. In this study, we performed the estimation for the grating with a shallow blaze angle, as shown in Figure 4, where the actual induced charge should be rather small, especially for the second facet due to the shading effect. The shading effect is discussed in Section 5.

3. Experiment

3.1. t-ACTS

The experiment was performed at t-ACTS, and the beam parameters are listed in

Table 1. Beam parameters.

Macro-pulse duration	~2.0 μs
Number of bunches	~5700 (per macro-pulse)
Beam energy	20 MeV
Normalized emittance	~3 πmm mrad
Bunch charge	3 ~10 pC
Bunch length	~100 fs
Transverse beam size at the grating position	~100 μm

. The t-ACTS linac consists of a thermionic RF gun, an alpha magnet with an energy filter, and a 3 m traveling-wave accelerating structure. In the velocity-bunching method, ultrashort electron bunches of less than 100 fs are generated by adjusting the injection phase of the bunch into the accelerating structure. The bunch length was measured using coherent transition radiation generated 1 m downstream of the accelerating structure. The transverse beam sizes and position at the grating location were adjusted by observing the optical transition radiation (OTR) from an aluminum screen fixed on the same stage as the grating, as described later. The typical beam size was approximately 100 μm in both horizontal and vertical directions.

3.2. Experimental Setup

An echelle-type grating with a period length (d) of 600 μm, blaze angle (α) of 6°, and period of 21 was manufactured by machining an aluminum alloy. The grating block had an overall length of 15 mm and width of 10 mm and was installed 2 m downstream of the accelerating structure. The block was mounted on a movable stage and the distance to the beam (impact parameter) was adjusted using a movable linear stage. An Al screen was mounted on the same movable stage to check the beam profile and position using the generated OTR. The typical impact parameter was set to 0.5 mm. Because of the small beam size and impact parameter compared with the block width of 10 mm, in the following discussion, we assume that the grating block is sufficiently wide. In the preliminary measurement performed previously, it was confirmed that the angular distribution of the CSPR intensity has a consistent response to changes in the bunch length [13]. In this study, a grating period of 600 μm was selected, considering the angular region where the SPR intensity variation with the bunch length change is expected to be large for a bunch length of approximately 100 fs. To measure the angular distribution of the SPR intensity, the SPR was extracted from the vacuum chamber through a Z-cut quartz window and detected using a pyroelectric detector (THz10, SLT Sensor and Laser Techniques, Inc., Wildau, Germany) with high sensitivity and wide bandwidth in the terahertz range. As shown in Figure 5, a Michelson interferometer was set up at different observation angles and the SPR spectra were measured at each angle. A vertical linear stage with the detector was installed instead of the interferometer to measure the azimuthal distribution of the SPR. The entire optical system outside the vacuum was purged with dry air to suppress SPR absorption by water. In each measurement, a similar procedure was performed on a dummy block without grooves, and the small contribution of the radiation generated by the structure without grooves was subtracted as the background.

4. Results

4.1. Spectrum Measurement

The Michelson interferometer was moved at a pitch of 2.5° over a range of observation angles from 35° to 60°, and the SPR spectra were measured at each angle. Examples of the measured interferograms and corresponding spectra are shown in Figure 6. The peak frequencies of the fundamental mode measured at radiation angles of 60° and 45° are 0.96 ± 0.07 THz and 1.6 ± 0.1 THz, respectively, and their values are in good agreement with the relationship in Equation (1). Because of the ultrashort bunch and the wide bandwidth of the THz detector, the small peaks of second harmonics were also observed at 1.9 ± 0.1 THz and 3.2 ± 0.2 THz. The uncertainty of the spectrum comes from the maximum optical path length in the interferogram and the finite angular acceptance in an optical system for the SPR measurement. The measured line widths of the spectrum are consistent with the estimated error.

4.2. Polarization Measurement

SPR polarization was measured by rotating the wire grid. The detector was placed at the observation angle normal to the Z-cut window (θ = 90°, ϕ = 0°). A rotation angle of 0° is defined when the wire grid is placed vertically, which coincides with the direction of the grating grooves. The result of the polarization measurement is shown in

Table 2. Results of polarization measurement at the observation angles of θ = 90° and ϕ = 0°.

Rotation Angle (Degrees)	Intensity (mV)
0	136 ± 28
90	20 ± 28
180	147 ± 28
270	19 ± 29
No wire grid	151 ± 19

. Denoting the SPR intensity observed at a wire grid rotation angle of 0° as $I_0$ and the others in the same manner, the polarization given by P = $(I_0-I_{90}+I_{180}{-I}_{270})/(I_0+I_{90}+I_{180}+I_{270})$ was found to be 76 ± 22%. The measured intensity without the wire grid is also listed in the table. The insertion loss was less than 10% of the total intensity and was comparable to the variation in the measurement; therefore, its effect on the polarization measurement was not significant. The comparison with the model calculation is presented in Section 5.

4.3. Azimuthal Distribution

The azimuthal distribution was measured by moving the detector vertically from the same horizontal level as the grating. The maximum vertical displacement was 30 mm from the reference level, which corresponds to an azimuthal angle of approximately 8.2°. The acceptance associated with the linear scan in the vertical direction was corrected to a maximum of 2%. The transmittance of SPR at the Z-cut window was almost constant over the measured angular range but was corrected to account for the measured polarization. The measurement result is shown in Figure 7 for the observation angle of 90°. As shown in Figure 4, the SC model predicts a large dip at an azimuthal angle of 0°; on the other hand, the measurement result shows no decrease there, which is discussed in the next section.

5. Discussion

In addition to the intensity and wavelength of the SPR, there are practical reasons for determining the grating shape, such as the ease of manufacturing. In this study, the echelle-type grating with a shallow blaze angle was used, as shown in Figure 4. In the SC model, the induced current at the grating surface is evaluated using the electric field generated by the electron bunch in a vacuum, which is based on a depiction of the image charge footprint on the grating surface. However, as shown in Figure 4, owing to the presence of a finite braze angle, the shading of the immediately preceding grating near the entrance of the first facet should reduce the magnitude of the time-integrated induced current by that amount. Similarly, the actual contribution to the SPR intensity is even smaller in the second facet corresponding to the back surface because the entire facet is shaded. Therefore, we performed the three-dimensional numerical calculation to evaluate the induced current by considering the geometrical shading effect of the grating.

Concerning the spectrum measurement of the coherent SPR, a strong suppression for the second harmonics observed at 45° may be explained by the small form factor (0.1) for the 100 fs bunch length, while it is expected to be suppressed to ~30% owing to the form factor for the observation angle of 60°. Considering the wavelength dependence of the Z-cut quartz window, it is further reduced by approximately 20%, which results in the expected intensity being a few times larger than the measured intensity. Because of the poor signal intensity in the preliminary measurements performed in this study, it is considered that there was no significant deviation from the measurement results.

According to the measurement obtained by rotating the wire grid, the polarization was measured to be 76 ± 22%. However, the evaluation of polarization components based on the SC model without the shading effect showed significantly lower values of 45%. On the other hand, the polarization from the 3D calculation including the shading effect was also estimated to be more than 99.9%. This result arises from the fact that the contribution from the second facet is very small owing to shading effects, and may be qualitatively consistent with the measured results. However, more accurate polarization measurements are required.

Figure 8 shows the azimuthal distribution of the SPR intensity generated from a single electron, with and without considering the shading effect. The shading effect significantly suppressed the contribution from the second facet in particular, and the radiation from the first facet dominated. Thus, the azimuthal distribution became a single peak, which is in good agreement with the measurement results. The observed polarization of the SPR is dominated by the polarization component orthogonal to the groove of the grating, which is also consistent with the suppression of radiation from the second facet owing to the shading effect. The geometric shading effect is due to the fact that the electric field that should reach the surface on the second facet in a finite time is treated as if it already exists, and qualitatively, the larger the slope of the second facet, the more pronounced the discrepancy with the actual situation.

6. Conclusions

Coherent SPR is expected to be used in a nondestructive bunch length monitor. In this application, the derivation of the longitudinal bunch form factor requires a known radiation spectrum that corresponds to the actual measurement conditions. In the conventional evaluation of SPR using the SC model, the surface current is derived using the electric field caused by the electron bunch in a vacuum, where the geometric shading effect is neglected. However, assuming an echelle-type diffraction grating with a shallow blaze angle, which is easy to manufacture, this leads to an overestimation of the surface current for the grating corresponding to the shading part. To investigate the effect of shading on the SPR corresponding to the measurement conditions, we evaluated the surface current using 3D numerical calculations and compared the results of SPR intensity with those predicted by the SC model. It was shown that the contribution of the second facet, which corresponds to the back side of the incoming electron, is much smaller than that of the first facet on the front side. This was then confirmed by observing the coherent SPR from a 100 fs beam at t-ACTS. The measured results of azimuthal distribution and polarization are in good agreement with the evaluation of the shading effect. Although the SC model provides the advantage of a very short computation time and a physical interpretation of the results, for better utilization of SPR in bunch length measurement, further investigations are needed to study the SPR spectra under more realistic conditions, such as the spatial spread of the electron bunch, as well as the optimization of the grating shape considering shading effects.