Towards precise diagnosis time profile of ultrafast electron bunch trains using orthogonal terahertz streak camera

Yang Xu; Yifang Song; Cheng-Ying Tsai; Jian Wang; Zhengzheng Liu; Hong Qi; Kuanjun Fan; Jinfeng Yang; O. I. Meshkov

doi:10.1364/OE.488132

1. Introduction

Ultrafast electron bunch trains with Terahertz (THz) repetition rates have been of great interest in recent decades for many applications [1–9], such as high-power sources of THz radiation [10–13] and plasma wakefield accelerators [5–8]. These applications require accurate knowledge of the time profile of the bunch train, including individual bunch duration and each bunch-to-bunch spacing. For example, in the tunable of THz radiation, such as Smith-Purcell radiation [12–18], the bunch length determines the width of the spectrum, and the bunch-to-bunch spacing determines the center frequency of the radiation.

There are several techniques available to generate ultrashort bunch trains, such as energy modulation by self-excited wakefield [12], transverse-longitudinal phase space exchange using a mask [19,20], and the more commonly used approach of direct generation from a photocathode driven by a comb-like laser pulse train [21]. In these cases, the time profile of the electron bunch train can be precisely controlled at the initial time. For example, the initial temporal structure of electrons emitted from a photocathode is determined by the driving laser pulses. However, after acceleration and drift, the time profile of the bunch train will change due to factors such as the space charge effect, velocity bunching, and the time jitter between laser pulses and the RF field. These cause the time profile of the bunch train to deviate from its initial values. Therefore, accurately diagnosing the time profile of the bunch train is an essential prerequisite for precise manipulation. For instance, by adjusting the bunch-to-bunch spacing in Smith-Purcell radiation, the center frequency of radiation can be controlled [15]. Consequently, accurately measuring the individual spacing is crucial for optimizing the radiation spectrum.

Measuring the individual bunch length and bunch-to-bunch spacing accurately remains a challenge. Conventional streak cameras [22–25] are typical methods for measuring electron bunch length, but they face difficulties when applied to bunch trains due to the high streaking frequency required to track picosecond-spaced individual bunches. Spectrum analysis of coherent Smith-Purcell radiation [26] can be used for bunch-to-bunch spacing measurement in the case of equal spacing. However, diagnosing unevenly-spaced bunch trains is considerably more complex, requiring analysis of the complete radiation spectrum.

Currently, the most popular techniques used to characterize ultrashort electron bunches are the autocorrelation functions method [1,12–14] and the zero-phasing method [27]. The autocorrelation functions method requires multi-shot measurements, and the temporal resolution is in the order of hundred femtoseconds. The zero-phasing method can measure both the individual bunch length and bunch-to-bunch spacing in one shot. However, these methods involve the implementation of traditional RF techniques [22,28], which have difficulties improving the resolution further because of the limited field gradient and the phase jitter.

THz-driven subwavelength structures [28–35], such as Split Ring Resonator (SRR) [36–38], have been studied for precision electron bunch diagnosis. This method has two prominent features that overcome the traditional RF techniques. Firstly, the THz field can provide 2$\sim$3 orders higher gradient streaking THz field, improving the diagnosis precision significantly. Secondly, the driving THz field and the incident electron bunch originate from the same laser system, guaranteeing an exact synchronization to manipulate picosecond-spaced electron bunches exactly. Using THz-driven SRR to measure an electron bunch length was first proposed by Fabianska et al. [36], and sub-10 fs temporal resolution has been achieved in Ref. [28]. However, current research on SRR is limited to diagnosing a single bunch. For a bunch train, one SRR can streak the individual bunches, but it will imprint each bunch on the detector at the same position. Since the spacing between multiple bunches is too short, the detector cannot acquire information ultimately before the succeeding bunches come, overlapping bunch information.

In order to obtain both the individual bunch length and bunch-to-bunch spacing in single-shot measurement, this paper proposes an orthogonal Terahertz streak camera to diagnose the bunch train time profile with femtosecond resolution. This all-optical method uses two SRRs to manipulate the electron bunches. The upstream SRR provides a high-gradient streaking THz field to streak electron bunches vertically for high-resolution bunch length and arrival time measurements. In contrast, the downstream SRR generates a decaying field to sweep bunches horizontally at each peak field from the axis. Consequently, the bunches are separated from each other, making clear images on the detector screen, and thus, the length information and the bunch-to-bunch spacing can be obtained simultaneously.

A bunch train consisting of six bunches has been studied, and the beam tracking result agrees with the analytical model well. The temporal resolution of each bunch is within 2.5 fs. At the same time, the bunch-to-bunch spacing is obtained with a temporal resolution of nearly 1 fs. With this method, we can comprehensively understand the bunch train’s longitudinal information.

2. Orthogonal terahertz streak camera principle

The principle of the orthogonal Terahertz streak camera for bunch train diagnosis is illustrated in Fig. 1. The system uses two SRRs to provide orthogonal deflecting electric fields that manipulate the incident bunch sequentially. The first SRR (streaking-SRR) streaks the individual electron beam in the y-direction, where the electrons in the bunch experience a linearly varying electric field and obtain a deflecting angle. If the bunch passes through the SRR at each zero-crossing phase, the deflecting angle is proportional to the longitudinal position of the particle in the bunch. The bunch acquires no net deflection but rotates vertically and is symmetrically imprinted on the downstream detector. The field time derivative is maximum at the zero-crossing phase, providing the strongest correlation between the longitudinal and transverse positions and ensuring the highest temporal resolution. If the bunch has a small phase shift from the zero-crossing, it will experience a net centroid kick, leading to an off-axis imprinting on the screen.

Fig. 1. Schematic of the orthogonal Terahertz streak camera. The electron beams are produced in an RF gun with a UV laser. The laser also drives two synchronized optical-rectification stages, generating single-cycle terahertz pulses for two SRRs.

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The second SRR (sweeping-SRR), which provides decaying orthogonal THz fields, is responsible for sweeping the electron beam in the x-direction. For a single electron bunch, the wavelength of the terahertz field ($\sim$3000 fs) is much greater than the bunch length or time jitter ($\sim$100 fs), resulting in little difference in the effect of terahertz waves on the electrons at the bunch head and tail. Therefore, this only causes the entire electron beam to shift in the x-direction. For multiple bunches, the degree of displacement gradually decreases due to the decay of the applied electric field, resulting in the separation of the bunch imprintings on the detector.

It is worth emphasizing that the information of both the bunch length and the arrival time of the electron beam is achieved by the streaking-SRR. In the orthogonal Terahertz streak camera, the sweeping-SRR only serves to separate the imprintings of multiple electron bunches. As its THz electric field has only an x-direction component, it does not affect the distribution of the electron beam in the y-direction. Adjusting the distance between two SRRs and the driving THz phase allows each electron bunch to experience the THz field with a maximum gradient in the streaking-SRR while seeing the peak THz field in the sweeping-SRR.

Precise control of the phase between the THz fields and the electron bunch is essential to achieve high temporal resolution in diagnosing the bunch train. As shown in Fig. 1, the UV pulse is used to produce the electron beams from a Ti: Sapphire laser. This laser also drives two synchronized optical-rectification stages, generating driving THz pulses for two SRRs. Both the electron bunches from the photocathode and the THz field in the SRRs gap originate from the same laser. As a result, the synchronization between the electron bunches and the THz fields is intrinsically guaranteed, which enables us to control the phases accurately.

Two fundamental processes are involved in THz manipulating electron bunch trains, the streaking and sweeping process. To simplify the analysis, several reasonable assumptions are made in the created analytical model: 1) When an electron bunch passes through the SRR, it is deflected by the THz E-field only, and the magnetic field is neglected; 2) The space charge effect is not considered during the streaking process, which is promised by installing a collimator before the streaking SRR; 3)The field does not change during the transition time.

2.1 Streaking process

When an electron bunch train passes through the first SRR, where a streaking THz field has been set up, an electron with a longitudinal position $\zeta _n$ will gain a transverse momentum (y-direction),

(1)$$\Delta P_y={-}q \int_{-\infty}^{+\infty} E_y(z) \sin \left(\omega t+\varphi_0\right) \exp{\left(-\frac{\zeta_n-\tau_0}{\tau}\right)} d t$$

(2)$$\zeta_n \in[-\Delta T / 2+n T / 2,+\Delta T / 2+n T / 2], n \in N$$

where $q$ is the elementary charge, $\varphi _0$ is the phase between the electron and the streaking THz electric field, $\omega =2\pi {f_0}$, $f_0$ is the THz frequency. $E_y(z)$ is the streaking THz field along the z-direction, $n$ is the sequence of each bunch in the bunch train where $n=0$ corresponds to the first bunch, and $\zeta _n=0$ corresponds to the bunch center. The streaking THz field period is $T=1/f_0$, which is also twice the bunch-to-bunch spacing of the bunch train. $\Delta {T}$ is the full width of a single electron bunch. And $\exp {\left [-\left (\zeta _n-\tau _0\right ) /\tau \right ]}$ is the decaying term of the streaking THz field, which can be simplified and separated from the integral term under the third assumption.

Then, the y-direction transverse momentum received by each bunch is

(3)$$\begin{aligned} \Delta P_y\left(\zeta_n\right)&={-}q \int_{\zeta_n-T_p / 2}^{\zeta_n+T_p / 2} \bar{E}_y \sin \left(\omega t+\varphi_0\right) d t \cdot \exp{\left(-\frac{\bar{\zeta}_n-\tau_0}{\tau}\right)}\\ &={-}q \int_{\zeta_n-T_p / 2}^{\zeta_n+T_p / 2} \bar{E}_y \sin \left(\omega t+\varphi_0\right) d t \cdot \exp{\left(-\frac{n T / 2}{\tau}\right)} \end{aligned}$$

(4)$$\begin{aligned} \bar{E}_y&=\frac{\omega}{2} \frac{\int_{-\infty}^{+\infty} E_y\left(\beta_z c t\right) \cos (\omega t) d t}{\sin \left(\omega T_p / 2\right)}\\ &=\frac{\omega}{2} \frac{\int_{-\infty}^{+\infty} A\left(\beta_z c t\right) \cos (\omega t) d t}{\sin \left(\omega T_p / 2\right)} E_{\text{max }}=C_s E_{\text{max }} \end{aligned}$$

where $\bar {\zeta }_n$ is the longitudinal position of the central electron. Since the electron transit time is extremely short, the electric field $E_y(z)$ does not change and can be replaced by a constant field $\bar {E}_y$ to simplify the calculation and $E_{\max }$ is the peak electric field. The parameter $C_s$ is only determined by the dimensions of SRR when the beam energy is fixed.

When the initial transverse momentum of the bunch is negligible, the trajectory angle of the electron is

(5)$$y^{\prime}\left(\zeta_n\right)=\frac{\Delta P_y}{P}=\frac{-2 q \bar{E}_y}{P} \sin \left(\frac{\omega}{2} T_p\right)\left(\frac{1}{\omega} \sin \varphi_0+\zeta_n \cos \varphi_0\right) \exp{\left(-\frac{n T / 2}{\tau}\right)}$$

where $P$ is the momentum of the bunch. Take the first moment of the above equation, and it gives the mean trajectory angle of the bunches in Eq. (6). Assuming the distance from SRR to the screen is $D$, we have the y-direction displacement of the beam centroid shown in Eq. (7).

(6)$$\bar{y^{\prime}}(n)=\frac{-2 q \bar{E}_y}{\omega P} \sin \left(\frac{\omega}{2} T_p\right) \sin \varphi_0 \exp{\left(-\frac{n T / 2}{\tau}\right)}$$

(7)$$\bar{y}(n)=\bar{y^{\prime}}(n) D$$

Considering the uniform distribution, the RMS divergence of the bunch is

(8)$$\begin{aligned} \sigma_{y^{\prime}}(n)&=\sqrt{\int_{-\Delta T / 2+n T / 2}^{\Delta T / 2+n T / 2} \frac{\left[y^{\prime}\left(\zeta_n\right)-\bar{y^{\prime}}(n)\right]^2}{\Delta T} d \zeta}\\ &=\frac{2 q \bar{E}_y}{P} \sin \left(\frac{\omega}{2} T_p\right)\left|\cos \varphi_0\right| \sqrt{\int_{-\Delta T / 2+n T / 2}^{\Delta T / 2+n T / 2} \frac{\zeta_n^2}{\Delta T} d \zeta} \cdot \exp{\left(-\frac{n T / 2}{\tau}\right)}\\ &=\frac{q \bar{E}_y}{\sqrt{3} P} \sin \left(\frac{\omega}{2} T_p\right)\left|\cos \varphi_0\right| \Delta T \cdot \exp{\left(-\frac{n T / 2}{\tau}\right)} \end{aligned}$$

In practice, we set a collimator before the SRR for two purposes, limiting the bunch charge to minimize the space charge effects during bunch length diagnosing and restricting the beam transverse size to match the small acceptance of the SRR gap. Approximately, the electrons are assumed to be uniformly distributed in a bunch. After passing from the SRR, the electrons will imprint on the downstream detector, and the RMS vertical beam size can be calculated by Eqs. (9) and (10),

(9)$$\sigma_y=\sqrt{\sigma_{\mathrm{un}}^2+\left(\sigma_y D\right)^2}$$

(10)$$\sigma_{\mathrm{un}}=\sqrt{\sigma_{y 0}^2+\left(\sigma_{y^{\prime} 0} D\right)^2}$$

where $\sigma _{\mathrm {un}}$ is the unstreaked beam size on the detector, $\sigma _{y 0}$ and $\sigma _{y^{\prime } 0}$ are the initial RMS electron beam size and divergence, respectively.

The streaking speed is given by Eq. (11)

(11)$$\begin{aligned} \omega_s(n)&=\frac{\left|\Delta \theta_s\right|}{\Delta T}=\frac{\left|y^{\prime}(\Delta T / 2)-y^{\prime}(-\Delta T / 2)\right|}{\Delta T}\\ &=\frac{2 q \bar{E}_y}{P} \sin \left(\frac{\omega}{2} T_p\right) \exp{\left(-\frac{n T / 2}{\tau}\right)} \end{aligned}$$

where $\left |\cos \varphi _0\right |=1$ since the streak effect is most potent at the zero-crossing phase.

The streaking speed determines the bunch length and arrival time measurement accuracy. The temporal resolution of the bunch length measurement, related to the size of the unstreaked beams, is defined as

(12)$$\tau_s=\frac{\sigma_{\text{un }}}{\omega_s(n) D}=\frac{\sigma_{\mathrm{un}} P}{2 q \bar{E}_y \sin \left(\omega T_p / 2\right) \exp{\left(-\frac{n T / 2}{\tau}\right)} D}$$

Usually, the initial vertical beam size is negligible, combining Eqs. (9) and (10), (12) is reduced to Eqs. (13). That is commonly used for temporal resolution evaluation.

(13)$$\tau_s=\frac{\sigma_{y^{\prime} 0}}{\omega_s(n) }=\frac{\sigma_{y^{\prime} 0} P}{2 q \bar{E}_y \sin \left(\omega T_p / 2\right) \exp{\left(-\frac{n T / 2}{\tau}\right)}}$$

The bunch-to-bunch spacing is determined by the arrival time of the bunch, which is attained by the time-dependent beam image centroid on the detector. The temporal resolution of the measurement is mainly affected by the shot-to-shot fluctuation of the beam centroid divergence $\sigma _{\text {y-centroid}^{\prime }}$ without SRR, thus defined as

(14)$$\tau_{\text{s-arrival}}=\frac{\sigma_{\text{y-centroid}^{\prime} }}{\omega_s(n) }$$

Since we can adjust the phase between the electron bunch and the THz field precisely, the first electron bunch in a bunch train has the optimal resolution because of the highest electric field gradient of the THz field at the zero-crossing phase. However, The succeeding bunch will see a decaying streaking THz field. Furthermore, the phases of the succeeding bunch will shift away from the zero-crossing phase for some reasons, such as space charge effects, leading to lower temporal resolution. The more the bunch number is in the bunch train, the lower the resolution will be obtained. In principle, we can further improve temporal resolution by increasing the THz pulse strength, reducing the beam size, and developing a new interaction structure [32].

2.2 Sweeping process

When the electron bunch train passes through the sweeping-SRR, the succeeding bunch will be alternatively swept in the x-direction by the sweeping THz electric fields, which does not affect the beam in the y-direction. Suppose each electron inside a bunch is subjected to the sweeping THz field in the x-direction. The x-direction position of the beam can be expressed by

(15)$$\bar{x}(n)=\frac{q E_\text{max} T_p D}{P} \cdot \exp{\left(-\frac{n T / 2}{\tau}\right)}$$

Since the subsequent electron beam will experience the decaying sweeping THz field, each bunch will consequently be imprinted on the detector in the x-direction with decreasing separation distance. Obviously, the larger the sweeping THz field strength, the bigger the separation, enabling more bunches to be measured in a single shot. In this case, the sweeping THz field amplitude and its decaying speed determine the total number of bunches we can measure.

The separation of the vertical beam size is essentially due to the decaying of the deflecting field in the x-direction. Obviously, the number of bunches that can be measured depends on the difference in the strength of the deflecting field between the last two bunches.

At first, we assume that the sweeping THz frequency $f_\text {swp}$ is half the repetition frequency ($f$) of the bunches, and in this case, the number of bunches that can be diagnosed is maximum. Actually, the orthogonal Terahertz streak camera works as long as $f_\text {swp}=n / 2 f$. The resonant frequency of the sweeping-SRR can be adjusted to match the bunch separation for better temporal resolution. In the following beam tracking simulation, we also discuss the applicability of the orthogonal Terahertz streak camera when there is an inevitable fluctuation of the bunch spacing.

3. SRR model and its generated THz field

The two SRRs have different functions. Consequently, they have different designed parameters, as shown in Fig. 2 and Table 1. The streaking-SRR is optimized for bunch length measurements ranging from 50 to 500 fs [38]. After passing through the first SRR, the streaking THz field elongates the electron bunch in the y-direction. Since the sweeping-SRR needs to sweep the entire electron bunch evenly, it is designed with a wide gap in the y-direction to accommodate the vertically rotating electron bunch. This makes the SRR resemble a parallel-plate capacitor at a high resonance frequency. Both SRRs are made of copper with a resonance frequency of 0.3 THz, and the second SRR is located 10 mm downstream to avoid the interference of the two exciting Terahertz pulses.

Fig. 2. The geometry of the two SRRs. a) Streaking-SRR, b) Sweeping-SRR.

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Table 1. Parameters of two SRRs.

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The performance of the SRR was investigated numerically using the CST simulation code [39]. Driving THz pulses with a center frequency of 0.3 THz can be efficiently generated in nonlinear optical crystals, such as LiNbO3. The typical shape of such a THz pulse is shown in Fig. 3(a), consisting of only one quasi-single cycle, and the corresponding spectrum is relatively broad, about 0.3 THz (shown in Fig. 3(b)). The driving THz gradient is normalized to 1 MV/m, allowing us to compare the performance of the two SRRs. The actual generated THz field in the SRR can be scaled by adjusting the driving THz for a given SRR.

Fig. 3. (a) Incident THz pulse used for simulations. (b) The corresponding spectrum with a center frequency of 0.3 THz.

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Figure 4 depicts the resulting THz field in the streaking-SRR gap that decays over time. The fields in the two SRRs are quite similar, except that their field enhancement factors differ due to their structures [40], which are approximately 23 for the streaking-SRR and 15 for the sweeping-SRR, respectively. If the driving THz field strength is higher than 50 MV/m, the streaking THz field of $\sim$ GV/m can be generated in the streaking-SRR gap [41], demonstrating the great potential of the Terahertz-driven resonator for ultrafast electron manipulation.

Fig. 4. The in-gap field in the streaking-SRR / sweeping-SRR (a) and corresponding spectrum (b).

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As shown in Fig. 5, the on-axis electric field longitudinal distribution depends on the SRR structure. The field will be further implemented for particle tracking to explore the bunch train behavior after the interaction length in detail.

Fig. 5. Normalized distribution of electric field in the gaps. The shaded part represents the gap area.

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4. Bunch train behavior in ideal cases

In order to assess the potential streaking performance of the orthogonal Terahertz streak camera and further study the bunch train behavior, numerical simulations based on beam tracking code ASTRA [42] have been performed, in which the three-dimensional electromagnetic fields are from the CST model. An electron bunch train consisting of 6 identical bunches with an equal spacing of 1.667 ps is studied using ASTRA to depict the streaking process. The selection of the parameters of the electron bunches is based on the typical parameters of THz electron bunch trains. In an ideal case, the Coulomb interaction between the electrons was not considered. To match the bunch-to-bunch spacing, the SRR is designed with a resonant frequency of 0.3 THz so that the individual bunch can experience the maximum streaking THz field. The beam is set to be 3 MeV with no initial divergence and energy spread. Each ideal electron bunch has the same full width of 100 fs, which represents the range of bunch length for a single bunch in the THz electron bunch train. In practice, longer bunches require streaking-SRR with lower resonance frequency for measurement to ensure that the electron beam is subjected to the linear region of the electric field. Shorter bunches require stronger streaking effects, that is, higher temporal resolution.

As shown in Fig. 1, the sweeping-SRR is positioned 10 mm downstream of the streaking-SRR, while the detector is located 250 mm further downstream from the streaking-SRR. In the absence of the space charge effect, a drift length of 250 mm is sufficient for the beam image to be resolved on the detector. The field gradients in the two SRR gaps are set to 100 MV/m and 200 MV/m, respectively, which correspond to driving THz pulse amplitudes of 4 MV/m and 13 MV/m. Assuming a Gaussian pulse with an FWHM duration of 3.33 ps, the driving THz pulse energy is approximately 95 nJ and 833 nJ. With a laser-to-THz conversion efficiency of about 0.1%, the laser energy required for two driving THz pulses is approximately 1 mJ. The streaking THz field strength in the streaking-SRR is compromised to ensure that the incident electron bunch is sufficiently streaked and that all the rotating electrons do not exceed the acceptance of the sweeping-SRR, thereby ensuring that the entire bunch is swept.

Accurately controlling the timing of each electron bunch is essential to ensure that it experiences the streaking THz field at the zero-crossing phase. When this is achieved, the entire bunch acquires zero net deflection while being vertically streaked at maximal speed, providing the highest possible temporal resolution. However, if the bunch train misses the zero-crossing phase, each bunch will experience a phase-dependent deflecting angle, as predicted in Eq. (6). As a result, the entire bunch will exhibit a net centroid deflecting angle, leading to a phase-dependent bunch center position on the detector.

In the ideal case, the behavior of an electron bunch train is studied using the ASTRA code. To characterize a bunch train’s temporal structure precisely, we need to find the zero-crossing phase of the streaking THz field for the incident electron bunch, which is one of the most important parameters determining the temporal resolution. It can be realized by phase scanning in ASTRA, in which the electron bunch enters into the SRR gap at a phase that can be controlled precisely from 0 to 360 degrees. Consequently, each bunch has a phase-dependent deflecting angle, leading to a phase-dependent bunch centroid position on the downstream detector. Fig. 6(a) depicts the simulated individual centroid positions of a bunch train as a function of the interaction phase, which agrees well with the analytical prediction. It can be seen that at the zero-crossing phase, the bunch centroid position is at the beam axis, zero. Beam centroid deflection as a function of the time delay between the electron beam and streaking THz field is shown in Fig. 6(b), the first bunch in a bunch train will get the highest streaking force at the zero-crossing phase and thus has the most extended imprinting length on the detector, leading to the highest temporal resolution. Fig. 6(c) compares the bunch imprinting size on the detector with the streaking THz field is "ON and OFF." The streaked beam’s vertical size is $\sim$20 times larger than the unstreaked spot. As Fig. 6(b) shows, the streaking speed at the first zero-crossing phase is the maximum, as high as 5.97 $\mathrm{\mu}$rad/fs. In this case, the temporal resolution of an electron beam with an initial beam divergence of 10 $\mathrm{\mu}$rad can be estimated to be 1.68 fs based on Eq. (13).

Fig. 6. (a) The centroid position of each bunch as a function of the streaking THz field phase. The black dotted curve is obtained from the equation. The six lines correspond to six bunches in a bunch train. (b) Beam centroid deflection as a function of the time delay between the electron beam and streaking THz field. (c) Comparison of the electron beam size on the detector with the streaking THz field on/off.

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The curve of the beam centroid position versus the interaction phase of the streaking THz field can map the arrival time of each bunch, determining the bunch-to-bunch spacing (Fig. 6(a)). Assuming the beam centroid divergence is 3.0 $\mathrm{\mu}$rad, the temporal resolution for bunch-to-bunch spacing measurement is 0.50 fs based on Eq. (14).

Through particle tracking, the images of the six bunches streaked at the zero-crossing phase can be clearly obtained on the detector. As shown in Fig. 7, the rightmost stripe corresponds to the first bunch, and the leftmost stripe is the second bunch, and so on. The exponential decaying of the THz fields in both SRRs results in the stripe length shrinking and the stripe separation reducing. A higher sweeping THz field provided by the 2nd SRR would increase stripe intervals and thus enable more bunches in a bunch train to be diagnosed simultaneously. In our simulation, the entire bunch train contains 6 bunches, and the final temporal resolution of bunch length measurement is determined by the last bunch, which is better than 2.5 fs in our case. At the same time, the temporal resolution of bunch-to-bunch spacing is 1 fs.

Fig. 7. Particle tracking simulation results of orthogonal Terahertz streak camera. (a) The imprinting particle distribution on the detector. (b) The x-direction position of six bunches.

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The individual length of six bunches is calculated based on the imprinting size on the detector, as shown in Fig. 8. The results show that the analytical model agrees well with the bunch length simulation, except for a deviation of less than 1.2 %, which proves the method is reliable.

Fig. 8. Bunch length of 6 bunches and the error compared to 100 fs. The results are derived from projections on the detector.

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5. Practical influences on bunch train behavior

In order to evaluate the performance of the orthogonal THz camera in a more realistic situation, simulations of electron bunches with more practically realistic parameters have been performed. Besides, several critical parameters that significantly impact the measurement results must be investigated further in this paper. Three effects that compromise the precision of the measurement will be analyzed, including the space charge effect, the effects of the deviation phase from zero-crossing, and the effects of frequency mismatching of two SRRs.

5.1 Space charge effect

During the transmission of the streaked electron beam to the detector, the beam experiences nonlinear space charge forces that distort its distribution, making it difficult to compensate accurately and leading to inaccurate beam length measurements. Since the space charge effect is a cumulative process resulting from Coulomb interactions among electrons, two methods have been implemented to alleviate its impact: reducing the beam charge intensity and shortening the beam flying time.

To reduce the beam charge intensity, we install a beam slit in front of the streaking-SRR to scrape the external charge of the incident beam without sacrificing the beam length information. Figure 9 compares the streaking results with and without installing the beam slit. The severe space charge effects deform the imprint beam shape on the detector, obscuring the bunch length information. After scraping, the space charge effect reduces significantly, so it can maintain the beam shape roughly except for linear expansion, which can be evaluated and compensated based on particle tracking or analytical calculation. The beam slit is designed to confine the charge of a single bunch to roughly 0.1 fC, which keeps the bunch length information and ensures the beam can pass through both SRR gaps without further beam loss.

Fig. 9. The space charge effects on the beam imprinting shape on the detector (a)with beam scraping (0.1 fC) and (b)without beam scraping (0.05 pC).

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To minimize the beam drifting distance to the detector and reduce the flying time, a high streaking THz field is preferable. It speeds up the expansion of the beam and quickly dilutes the charge intensity, thus reducing the space charge force. The fast streaking speed also improves resolution by avoiding long-distance drifts, shortening the working time of the Coulomb force and reducing the accumulated effects.

For a bunch train, the space charge force will change the individual bunch spacing, leading to an increasing phase shift from the zero-crossing phase for each consecutive bunch in the streaking process. As a result, the bunch train will experience changing streaking force. Figure 10 shows the space charge effects on the measurement of an electron bunch train. Clearly, the head and tail bunches in a bunch train deviate most, but they are all within 0.4% deviation if the bunch charge remains within 0.1 fC. The phase shift of each bunch can be estimated by the centroid offset. Compared to the bunch train without the space charge effect, the first bunch experiences phase advancing, hence a higher THz field. The last bunch in the bunch train experiences phase delaying, hence a lower THz field. The phase shift of the bunch in the middle of the train is relatively small. As discussed above, the deviation can be mitigated by increasing the streaking THz field or reducing the bunch charge by optimizing the beam slit size.

Fig. 10. Bunch length of each bunch in the case of space charge effect and no space charge effect and the phase shift of each bunch. The bunch charge is 0.1 fC.

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5.2 Arrival time jitter effects

Ideally, an electron bunch should pass through the streaking THz field at the zero-crossing phase to achieve the most significant streaking effect. However, in practice, an electron bunch always has a phase shift to the zero-crossing phase due to its arrival time uncertainty, as discussed earlier. There is a limited time window in which the electron bunch also interacts with the linear THz field, and the bunch train can pass through the sweeping-SRR entirely. As a result, we can obtain the imprintings of the entire bunch train on the detector, even if the beam centroid is off-axis, and deduce the bunch train’s time structure.

Using ASTRA particle tracking, we can investigate the effects of time delay on the streaking process. For instance, assume the single electron bunch has an arrival time delay of 0 fs, 40 fs, 80 fs, 120 fs, 160 fs, and 200 fs, with corresponding phase shifts to the zero-crossing phase of 0, 4.32, 8.64, 12.96, 17.28, and 21.60 degrees, respectively. Figure 11 displays the particle distribution on the detector. As the time delay increases, the beam interaction phase shifts further from the zero-crossing phase. Consequently, the beam imprinting centroid deviates further from the beam axis, and the beam shape becomes severely distorted. If the time delay is less than 100 fs, the bunch length remains almost unchanged, and the bunches can imprint on the detector deflected by linear orthogonal THz fields in the y-direction.

Fig. 11. The particle distribution on the detector when the time delay of the beam is 0 fs, 40 fs, 80 fs, 120 fs, 160 fs and 200 fs.

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If the time delay exceeds 100 fs, the bunch length increases significantly because the beam will encounter the nonlinear THz field, leading to beam distortion as depicted in Fig. 11 and Fig. 12. When the time delay increases beyond 150 fs, the electron beam after streaking will exceed the sweeping-SRR acceptance, and part of the electron beam goes to the detector without deflection in the x-direction. Therefore, the acceptable time window for diagnosing the bunch length is approximately $\pm$150 fs. However, the beam centroid position in the y-direction increases linearly with the time delay shown in Fig. 12, from which the actual arrival time of the bunch can be determined. For a bunch train, the orthogonal THz fields enable the images of each bunch not to affect each other and can be determined simultaneously.

Fig. 12. The bunch lengths and central positions with a varying time delay of the beam. If the x-coordinate is expanded to negative, the beam centroid curve is still linear, and the bunch length curve is symmetric along the y-axis.

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For a bunch train consisting of 6 electron bunches, if each individual electron bunch has a random time delay within a time window of $\pm$150 fs, the entire bunch train can be imprinted completely on the detector, as shown in Fig. 13. We performed four sets of simulations where the initial time delay of the bunches is set to have a Gaussian distribution with random jitter (mean of 0 and standard deviation of 50 fs). From the images obtained, we can deduce the time structure of the bunch train. The schematic diagrams of the deduced bunch spacing are also shown in the figure, where we define a positive value of time delay to indicate a delay and a negative value to indicate an advance.

Fig. 13. The electron bunch distribution on the detector when the initial time delays of six bunches are (a) +5 fs, -67 fs, -21 fs, -35 fs, +11 fs, +38 fs, (c) +31 fs, +87 fs, +47 fs, -14 fs, +10 fs, +27 fs, (e) -36 fs, +66 fs, +18 fs, +58 fs, -88 fs, -19 fs and (g) -28 fs, -42 fs, -11 fs, +34 fs, -16 fs, +5 fs. The deduced bunch length and time delay from the images are shown in (b), (d), (f), and (h). The slight discrepancies with the original set values are mainly due to the space charge effects between multiple bunches.

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5.3 Frequencies mismatching effects

Considering the practical cases, the two SRRs may have different resonance frequencies because of fabrication errors, which introduces the out-synchronization between the two SRRs and leads to measurement inaccuracy.

In order to simplify the analysis, assuming only the streaking-SRR frequency match the bunch repetition frequency. Since the sweeping-SRR resonance frequency deviates from the designed value, the streaked electron bunch from streaking-SRR no longer passes through the peak of the sweeping THz field. Consequently, the entire electron bunch is imposed a reducing electric force from the sweeping THz field, decreasing the bunch imprintings separation. Moreover, the electron bunch head and tail will see different sweeping THz fields, broadening the bunch imprinting size in the x-direction on the detector, which leads to image overlapping between the succeeding electron bunches, invalidating the bunch train diagnosis. The frequency matching of the orthogonal Terahertz fields is critical to guarantee that more electron bunches in a bunch train can be diagnosed.

The x-position of the head and tail electrons of the nth bunch on the detector is given:

(16)$$x_{{\pm}}(n)=\frac{q T_P D E_\text{max}}{P} \sin \left[2 \pi\left(f_0+\Delta f\right)\left(\frac{n}{f_0}+\frac{1}{4 f_0} \pm \frac{\Delta T}{2}\right) \right]\exp{\left(-\frac{n T / 2}{\tau}\right)},$$

where $\Delta$f is the frequency difference of the orthogonal THz field. In order to ensure the bunch imprinting separation, the detuning of the sweeping-SRR cannot exceed 10% if more than six bunches are measured at a design frequency of 0.3 THz. Corresponding to the geometric of SRR, the parameter that has the greatest influence on the resonance frequency is $l$, so its machining accuracy should be controlled within 20 $\mathrm{\mu}$m. The higher the frequency, the higher the temporal resolution of the beam length measurement. But the impact of detuning will be greater, so it is necessary to balance the measurement accuracy and the number of bunches in a bunch train.

6. Summary and discussion

Focusing on measuring the bunch length and bunch-to-bunch spacing in a bunch train, this paper presents a two-SRRs model based on a THz-driven streak camera. The proposed analytical model describing the streaking process of the bunch train is verified by particle tracking simulation. The space charge effect is not included in the analysis model but is assessed to contribute less than 0.4 percent to the measurements. The simulation results achieved six beam length measurements with resolutions better than 2.5 fs. From the practical point of view, the limitations of the orthogonal Terahertz streak camera are analyzed considering the variation of bunch spacing. This method can study the longitudinal information of the bunch train and greatly benefit the development of narrow-band Terahertz sources and compact accelerators.

The method proposed in this paper is mainly aimed at the condition that the target period of the bunch train is fixed. Measuring the bunch train at the tunable period is difficult because the frequency of SRR is challenging to change once it is completed. If the bunch train at another period needs to be measured, the geometric size of SRR needs to be redesigned to match the resonance frequency. Fortunately, the resonant frequencies of different SRR geometric sizes can be easily obtained by theory or simulation [38]. The parameters of SRRs in this paper are derived from a streak camera used for single beam length measurement in MeV ultrafast electron diffraction (UED) [38,43,44]. The proposed orthogonal Terahertz streak camera is more suitable for ten or fewer electron bunches to maintain high temporal resolution. To fully benefit from this novel design, the parameters of the SRR structure will be optimized to cope with the demand of measuring a larger number of bunches, and beamline spatial distribution must be carefully designed in experiments. At the same time, it is hoped that this method can be applied in UED to determine the diffraction pattern’s time accuracy by accurately measuring the length and arrival time of the bunches. We expect this method to have wide applications in many areas of research.

Funding

National Natural Science Foundation of China (12235005, 12275094); Fundamental Research Funds for the Central Universities (2021GCRC006); Science and Technology Project of State Grid (5400-202199556A-0-5-ZN).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Towards precise diagnosis time profile of ultrafast electron bunch trains using orthogonal terahertz streak camera

Abstract

1. Introduction

2. Orthogonal terahertz streak camera principle

2.1 Streaking process

2.2 Sweeping process

3. SRR model and its generated THz field

4. Bunch train behavior in ideal cases

5. Practical influences on bunch train behavior

5.1 Space charge effect

5.2 Arrival time jitter effects

5.3 Frequencies mismatching effects

6. Summary and discussion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (1)

Equations (16)

Optics Express

Parameters	Streaking-SRR	Sweeping-SRR
$l$ / $μ$ m	95	80
$w$ / $μ$ m	10	10
$g$ / $μ$ m	10	10
$h$ / $μ$ m	100	100
$s$ / $μ$ m	-	30