Basic theory

As highlighted in the preceding section, owing to its inherent advantages, including cost-effectiveness, system stability, and simplicity, the Q-scanning method has been extensively used in emittance measurements across various beam injectors.

As shown in Fig. 1, a Q-magnet is used to focus the beam in either the horizontal or vertical direction onto a downstream Fluorscreen [34]. The beam distribution on the Fluorscreen is measured by capturing the optical transition light generated when the beam passes through the Fluorscreen. Through iterative adjustments of the focusing strength parameter K of the Q-magnet and simultaneous measurement of the corresponding beam cross-sectional sizes intercepted by the Fluorscreen, it is possible to use the beam transport principles to calculate the transverse emittance. In practical measurements employing a particle statistical model, the subsequent analysis describes the beam parameters using root-mean-square (rms) values, denoted as $\sigma (\mathrm{...})$, while the average values are represented as $\langle \mathrm{...}\rangle$. The beam spot size on the Fluorscreen $\sigma \left(x_1\right)$ can be expressed in terms of the initial beam parameters and transfer matrix elements R₁₁ and R₁₂ as follows: $\begin{array}{c}{\sigma }^2\left(x_1\right)=\langle x_1^2\rangle \\ =R_{11}^2{\sigma }^2\left(x_0\right)+R_{12}^2{\sigma }^2\left({x^{\prime }}_0\right)+2R_{11}{R}_{12}\langle x_0{x}_0{}^{\prime }\rangle \\ ={\epsilon }_0\left[R_{11}^2{\beta }_0-2R_{11}{R}_{12}{\alpha }_0+R_{12}^2{\gamma }_0\right],\end{array}$ (1) where x and x’ denote variations in the transverse position and divergence relative to the central particle, respectively. In the statistical model, $\sigma \left(x_0\right)$ denotes the initial transverse size of the beam at the entrance of the measuring system and $\sigma \left({x^{\prime }}_0\right)$ and $\sqrt{\langle x_0{x^{\prime }}_0\rangle }$ denote the initial transverse divergence and correlation, respectively. Simultaneously, the initial Twiss parameters are α₀, β₀ and γ₀, where ε₀ denotes the transverse emittance to be determined. $\{\begin{array}{l}{R}_{11}=\cos \left(\sqrt{k}{L}_Q\right)-L_d\sqrt{k}\sin \left(\sqrt{k}{L}_Q\right)\hfill \\ R_{12}=\sin \left(\sqrt{k}{L}_Q\right)/\sqrt{k}+L_d\cos \left(\sqrt{k}{L}_Q\right)\hfill \end{array}$ (2) where LQ and L_d denote the effective length of the Q-magnet and downstream drifting distance, respectively. The Q-magnet’s focusing parameter is defined as $k=\frac{1}{B\rho }\frac{\partial B}{\partial r}\left({\text{1/m}}^{\text{2}}\right)=\frac{299.792g(\text{T/m})}{E(\text{MeV})}$, where Bρ denotes the magnetic rigidity, E denotes the beam energy, and g represents the magnetic field gradient determined by the inherent properties of the Q-magnet and its current value.

Fig. 1Full-screen View

Emittance measurement system

In accordance with Eqs. (1) and (2), by measuring the beam spot sizes corresponding to three different sets of K values, the equation set can be solved to determine the Twiss parameters, subsequently allowing the emittance calculation.

Thin-lens approximation

Although the equation set given in the previous subsection provides a means to measure the beam emittance, it is noteworthy that this method is susceptible to relatively large errors. In particular, when K = kLQ is close to zero, the emittance measurement system can employ the thin-lens approximation, where the transfer matrix elements, specifically R₁₁ and R₁₂, are approximated as $\{\begin{array}{l}{R}_{11}=1-K{L}_{\text{d}}\hfill \\ R_{12}=L_{\text{d}}\hfill \end{array}$ (3) Based on the aforementioned statements, Eq. (1) can be reformulated as Eq. (4) by substituting Eq. (3). Subsequently, this can be simplified to obtain Eq. (5) as follows: $\begin{array}{l}\begin{array}{c}{\sigma }^2\left(x_1\right)={\epsilon }_0\left[{\left(1-K{L}_{\text{d}}\right)}^2{\beta }_0-2\left(1-K{L}_{\text{d}}\right)L_{\text{d}}{\alpha }_0+L_{\text{d}}^2{\gamma }_0\right]\end{array}\hfill \end{array},$ (4) $\begin{array}{l}\begin{array}{c}{\sigma }^2\left(x_1\right)=a{\left(K-b\right)}^2+c=a{K}^2-2abK+a{b}^2+c\end{array}\hfill \end{array}.$ (5) A parabolic relationship exists between σ²(x₁) and K, and the corresponding coefficients are as follows: $\begin{array}{lll}a={\epsilon }_0{L}_{\text{d}}^2{\beta }_0\hfill & b=\frac{\left({\beta }_0{L}_{\text{d}}-L_{\text{d}}^2{\alpha }_0\right)}{{\beta }_0{L}_{\text{d}}^2}\mathrm{,}\hfill & c=\frac{{\epsilon }_0{L}_{\text{d}}^2}{{\beta }_0}\hfill \end{array}.$ (6) Hence, the transverse emittance along with the Twiss parameters at the entrance of the Q-scanning system can be succinctly characterized by the following parabolic coefficients: $\begin{array}{lll}{\epsilon }_{\text{0}}=\frac{\sqrt{ac}}{L_{\text{d}}^2}\mathrm{,}\hfill & {\alpha }_0=\sqrt{\frac{a}{c}}\left(-b+\frac{1}{L_{\text{d}}}\right)\mathrm{,}\hfill & {\beta }_0=\sqrt{\frac{a}{c}}\hfill \end{array}.$ (7) Therefore, as a common practice to mitigate measurement errors and improve the precision, rather than solving the equation set, it is customary to employ a parabolic fitting approach. This involves acquiring 10–15 sets of beam size data [27] and employing parabolic fitting techniques to determine the emittance.

Corrections via fixed parabola vertex

As discussed in the previous subsection, when employing the thin-lens approximation, determining the emittance through parabolic fitting is feasible. However, various fitting errors may arise across the diverse ranges of independent variables, leading to inconsistent measurement outcomes. The conventional strategy involves collecting multiple data points in proximity to the focal point for fitting, requiring the initial identification of the focal point.

For convenience, let us define the focus size as σ_f,min(x), with independent variable $K_0=f_{\min }^{-1}$. Here, f_min denotes the focal length. Then, Eq. (4) can be expressed as $\begin{array}{l}\begin{array}{c}{\sigma }^2\left(x_1\right)={\sigma }_{\text{f}\mathrm{,}\min }^2\left(x\right)+\frac{L_{\text{d}}^4}{{\sigma }_{\text{f}\mathrm{,}\min }^2\left(x\right)}{\epsilon }_0^2{\left(K-K_0\right)}^2\end{array}\hfill \end{array}.$ (8) Accordingly, the coefficients in the original parabolic fitting equation correspond to $\begin{array}{l}\begin{array}{c}a=\frac{L_{\text{d}}^4}{{\sigma }_{\text{f}\mathrm{,}\min }^2\left(x\right)}{\epsilon }_0^2\mathrm{,}b=\frac{1}{f_{\min }}\mathrm{,}c={\sigma }_{\text{f}\mathrm{,}\min }^2\left(x\right).\end{array}\hfill \end{array}$ (9) Subsequently, the emittance can be calculated using a much simpler formula, as follows: $\begin{array}{l}\begin{array}{c}{\epsilon }_{\text{0}}=\frac{\sqrt{ac}}{L_{\text{d}}^2}\end{array}\hfill \end{array}.$ (10) To examine and contrast the influence of the independent variable range for K on the emittance measurement results, the beam parameters of the HUST injector [35], as listed in Table 1, serve as an illustrative example. The setting parameters of the Q-scanning system employed in the HUST injector are also listed in Table 1.

To streamline the analysis and concentrate on examining the errors arising from the thin-lens approximation, the beam parameters were directly substituted into the complete transfer matrix of the Q-scanning system of the HUST injector. By varying K, multiple sets of beam-spot sizes were generated at the system exit. This approach allowed us to comprehensively understand the impact of the thin-lens approximation on the system’s performance.

Three parabolas are shown In Fig. 2(a), each fitted to a distinct range of K values. Notably, the fitted parabolic vertices do not coincide when K assumes values in different ranges, suggesting that the K values corresponding to the focus points are different. For comparison, Fig. 2(b) shows the parabolic fitting curves under a fixed vertex, maintaining the approach of selecting points for K as in Fig. 2(a). It can be seen that, although the overlap of the fitted parabolas in Fig. 2(b) is enhanced compared with that in Fig. 2(a), the parabolas exhibit inconsistencies because they deviate from their vertices. Consequently, this inevitably introduces disparities in the emittance measurements determined by their coefficients.

Fig. 2Full-screen View

Parabolic fitting results under different ranges of K values: (a) Arbitrary range (Case I: K=-3.29~1.39 m^-1, Case II: K=-1.65~3.08 m^-1, Case III: K=0.04~4.77 m^-1); (b) Fixed vertex, range identical to that in (a); (c) Fixed vertex with symmetric sampling (Case I: K=-0.76~2.20 m^-1, Case II: K=-1.20~2.64 m^-1, Case III: K=-1.65~3.08 m^-1); (d) Measurement error curves

To further elucidate this issue, the measurement errors induced by the two fitting methods are shown in Fig. 2(d), where the abscissa $K_{\text{range}}=K_{\text{max}}-K_{\text{min}}$ denotes the maximum K scanning range. The vertical axis in Fig. 2 indicates the absolute error of the measured emittance. By examining the measurement errors represented by the red dotted line in Fig. 2(a), it is evident that despite setting K at equally spaced values in accordance with operational practices during actual measurements, varying K_range contributes to inconsistencies in the measurement errors, with an increasing trend as K_range increases.

For the method illustrated in Fig. 2(b). The errors and their inconsistencies, as denoted by the dotted blue line in Fig. 2(d), are notably reduced. In other words, fixing the parabolic vertex can significantly reduce both the measurement errors and their inconsistencies in the Q-scanning method. Nevertheless, it is noteworthy that the measurement error exhibits a gradual increase with K_range.

To address this issue more effectively, considering the distinctive characteristics of parabolic fitting errors and keeping the vertex fixed, we opted to select points symmetrically on both sides of the vertex. The data fitted using this method resulted in a series of parabolas, as shown in Fig. 2(c). Notably, the parabolas obtained through this approach exhibited substantial overlap, and their measurement error consistency was significantly enhanced, as shown in Fig. 2(d).

To analyze the impact of a small-size resolution, for beam spots near the vertex smaller than 0.1 mm, we set their sizes to 0.1 mm in the parabola fitting process (the optical system resolution for beam spot measurements is generally better than 0.05 mm), as shown in Fig. 3(a). The corresponding error analyses of the different data processing methods for small and indistinguishable vertex sizes are shown in Fig. 3(b). When the beam spot size resolution was 0.1 mm, the proposed fixed vertex fitting method was suitable for beams with an initial normalized emittance greater than 2 mm mrad.

Fig. 3Full-screen View

(a) Parabola fitting curves and (b) error analysis for different data processing methods under small and indistinguishable vertex sizes

Chromatic effect correction

In the realm of beam injectors designed for diverse applications, the energy spread exhibits a considerable range, from a fraction of a percent to a few percent. When a beam characterized by significant energy spread traverses a Q-magnet, the quadrupole kick undergoes variations within the bunch itself, inducing transverse emittance changes [31]. Consequently, this chromatic effect introduces additional errors into the emittance measurements. Therefore, it is advantageous to correct the transfer matrix of the Q-scanning system, leading to rewriting Eq. (3) as follows: $\{\begin{array}{l}{R}_{11}=1-K(1-\delta )L_{\text{d}}\hfill \\ R_{12}=L_{\text{d}}\hfill \end{array}$ (11) where δ = Δ p / p represents the energy deviation relative to the central particle, and σ(δ) denotes the energy spread within the statistical model. Therefore, substituting Eq. (11) into Eq. (1), the beam spot size at the downstream Fluorscreen yields $\begin{array}{c}{\sigma }^2\left(x_1\right)={\sigma }_{\delta =0}^2\left(x_1\right)+{\left(K{L}_{\text{d}}\right)}^2{\sigma }^2\left(\delta x_0\right)+2K{L}_{\text{d}}\left(1-K{L}_{\text{d}}\right)\langle \delta x_0^2\rangle \\ +2K{L}_{\text{d}}^2\langle \delta x_0{x^{\prime }}_0\rangle ,\end{array}$ (12) where ${\sigma }_{\delta =0}^2\left(x_1\right)$ denotes the beam spot size and neglects the energy spread. We assumed that there is no correlation between the longitudinal and transverse properties, namely, $\langle \delta x_0^2\rangle =\langle \delta x_0{x^{\prime }}_0\rangle =0$ and $\langle {\delta }^2{x}_0^2\rangle =\langle {\delta }^2\rangle \langle x_0^2\rangle$. Then Eq. (12) can be simplified by using the Twiss parameters as follows: $\begin{array}{l}\begin{array}{c}{\sigma }^2\left(x_1\right)={\sigma }_{\delta =0}^2\left(x_1\right)+{\left(K{L}_{\text{d}}\right)}^2{\epsilon }_0{\beta }_0{\sigma }^2\left(\delta \right)\end{array}\hfill \end{array}.$ (13) Therefore, the coefficients obtained from the fitting parabolic formula are $\{\begin{array}{l}a={\beta }_0{\epsilon }_0\left(1+{\sigma }^2\left(\delta \right)\right)L_{\text{d}}^2\mathrm{,}\hfill \\ b=\frac{-\left(L_{\text{d}}{\alpha }_0-{\beta }_0\right)}{{\beta }_0\left(1+{\sigma }^2\left(\delta \right)\right)L_{\text{d}}}\mathrm{,}\hfill \\ c={\epsilon }_0\left[{\beta }_0-2L_{\text{d}}{\alpha }_0+L_{\text{d}}^2{\gamma }_0-\frac{{\left(L_{\text{d}}{\alpha }_0-{\beta }_0\right)}^2}{{\beta }_0\left(1+{\sigma }^2\left(\delta \right)\right)}\right]\hfill \end{array}$ (14) Subsequently, by combining ${\gamma }_0=\frac{1+{\alpha }_0^2}{{\beta }_0}$, the emittance considering the influence of the energy spread can be solved.

Furthermore, to ascertain the efficiency of this correction method, beam dynamics simulations were performed using the HUST injector system parameters listed in Table 1. These simulations were executed in tandem with ASTRA [36]. It is worth noting that for practical relevance, in subsequent discussion, the magnetic field of the scanning Q-magnet at the HUST injector is incorporated into the beam dynamics software ASTRA. The corresponding simulation results are shown in Fig. 4.

Fig. 4Full-screen View

Measurement error variations according to K_range for different σ(δ): (a) Traditional method corresponding to Eq. (8); (b) Correction method corresponding to Eq. (13)

Note that the correction method with fixed vertices and symmetric K values at equal intervals, as stated in Sect. 3.1, is conventionally adopted during the simulation process. As shown in Figs. 4(a) and 4(b), the measurement errors remain consistent, regardless of the chromatic effect correction. Under this correction, a significant reduction in the influence of σ(δ) on the measurement errors is observed. Furthermore, a larger energy spread corresponds to a more pronounced correction effect. For example, when the energy spread is 6%, the relative measurement error decreases from approximately 12.5% to 7%, highlighting a significantly improved correction performance. In summary, the theoretical and simulation results collectively indicate a substantial reduction in emittance measurement errors and their inconsistencies by applying the proposed correction methods.

Theoretical analysis

As discussed in Sect. 2, the traditional Q-scanning method relies on thin-lens approximation, where $\sqrt{k}{L}_Q\to 0$, and only the linear terms of the system transfer matrix elements are considered. However, for beam injectors operating at approximately 10 MeV, shorter beam transport lines are preferred to mitigate the space-charge effect impact. Consequently, it is imperative to rectify the errors that stem from the aforementioned approximation. Therefore, considering the second-order terms R₁₁ and R₁₂, the following equations can be formulated: $\{\begin{array}{l}{R}_{11}=1-K\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)\hfill \\ R_{12}=L_{\text{d}}+L_Q-\frac{1}{2}K{L}_Q{L}_{\text{d}}\hfill \end{array}$ (15) After inserting Eq. (15) into Eq. (1), the beam spot size can be expressed as in Eq. (16), the parabolic equation coefficients adhere to the conditions outlined in Eq. (17). Subsequently, the emittance is determined by incorporating the Twiss parameter relation ${\gamma }_0=\frac{1+{\alpha }_0^2}{{\beta }_0}$. $\begin{array}{c}{\sigma }^2\left(x_1\right)={\left[1-K\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)\right]}^2{\epsilon }_0{\beta }_0+{\left(L_{\text{d}}+L_Q-\frac{1}{2}K{L}_Q{L}_{\text{d}}\right)}^2{\epsilon }_0{\gamma }_0\\ -2\left[1-K\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)\right]\left(L_{\text{d}}+L_Q-\frac{1}{2}K{L}_Q{L}_{\text{d}}\right){\epsilon }_0{\alpha }_0\end{array}$ (16) $\{\begin{array}{l}a={\epsilon }_0\left[{\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)}^2{\beta }_0-\left(\frac{1}{2}{L}_Q^2{L}_{\text{d}}+L_Q{L}_{\text{d}}^2\right){\alpha }_0+\frac{1}{4}{L}_Q^2{L}_{\text{d}}^2{\gamma }_0\right]\hfill \\ -2ab={\epsilon }_0\left[-\left(L_Q+2L_{\text{d}}\right){\beta }_0+2\left(2L_Q{L}_{\text{d}}+\frac{1}{2}{L}_Q^2+L_{\text{d}}^2\right){\alpha }_0-\left(L_Q{L}_{\text{d}}^2+L_Q^2{L}_{\text{d}}\right){\gamma }_0\right]\hfill \\ a{b}^2+c={\epsilon }_0\left[{\beta }_0-2\left(L_{\text{d}}+L_Q\right){\alpha }_0+\left(L_{\text{d}}^2+L_Q^2+2L_{\text{d}}{L}_Q\right){\gamma }_0\right]\hfill \end{array}$ (17) Section 3.2 provides an in-depth examination of the effects of σ(δ); Eq. (15) can be rewritten as $\{\begin{array}{l}{R}_{11}=1-K\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)+K\delta \left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)\hfill \\ R_{12}=L_{\text{d}}+L_Q-\frac{1}{2}K{L}_Q{L}_{\text{d}}+\frac{1}{2}K\delta L_Q{L}_{\text{d}}\hfill \end{array}$ (18) The beam spot size equation can be decomposed into two components: ${\sigma }^2\left(x_1\right)={\sigma }_{\delta =0}^2\left(x_1\right)+{\sigma }_{\delta }^2\left(x_1\right)$, where ${\sigma }^2\left(x_1\right)={\sigma }_{\delta =0}^2\left(x_1\right)+{\sigma }_{\delta }^2\left(x_1\right)$ denotes the spot size in the absence of a chromatic effect, which can be directly calculated using Eq. (16). Meanwhile, ${\sigma }_{\delta }^2\left(x_1\right)$ represents the contribution of σ(δ) to ${\sigma }^2\left(x_1\right)$. By substituting Eq. (18) into Eq. (1) and integrating it with Eq. (16), a new expression for ${\sigma }^2\left(x_1\right)$ is obtained, as in Eq. (19). Subsequently, neglecting the longitudinal and transverse coupling enables the derivation of Eq. (20). Leveraging the parabolic fitting equation, the first coefficient in Eq. (17) is refined using Eq. (21), while the other two equations remain unchanged. $\begin{array}{c}{\sigma }_{\delta }^2\left(x_1\right)=K^2{\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)}^2{\sigma }^2\left(\delta x_0\right)\\ +\frac{1}{4}{K}^2{L}_Q^2{L}_{\text{d}}^2{\sigma }^2\left(\delta x_0{}^{\prime }\right)+K^2{L}_Q^2{L}_{\text{d}}^2\langle {\delta }^2{x}_0{x}_0{}^{\prime }\rangle \\ +K\left(L_Q+2L_d\right)\left[1-K\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)\right]\langle \delta x_0^2\rangle \\ +K{L}_Q{L}_{\text{d}}\left(L_{\text{d}}+L_Q-\frac{1}{2}K{L}_Q{L}_{\text{d}}\right)\langle \delta x_0{{}^{\prime }}^2\rangle \\ +K\left\{L_Q{L}_{\text{d}}\left[1-K\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)\right]\right\}\\ +\left(L_Q+2L_{\text{d}}\right)\left(L_{\text{d}}+L_Q-\frac{1}{2}K{L}_Q{L}_{\text{d}}\right)\}\langle \delta x_0{x}_0{}^{\prime }\rangle \end{array}$ (19) $\begin{array}{c}{\sigma }^2\left(x_1\right)={\sigma }_{\delta =0}^2\left(x_1\right)+{\sigma }^2\left(\delta \right)K^2{\epsilon }_0[{\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)}^2{\beta }_0\\ +\frac{1}{4}{L}_Q^2{L}_{\text{d}}^2{\gamma }_0-L_Q^2{L}_{\text{d}}^2{\alpha }_0]\end{array}$ (20) $\begin{array}{c}a={\epsilon }_0\left[{\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)}^2{\beta }_0-\left(\frac{1}{2}{L}_Q^2{L}_{\text{d}}+L_Q{L}_{\text{d}}^2\right){\alpha }_0+\frac{1}{4}{L}_Q^2{L}_{\text{d}}^2{\gamma }_0\right]\\ +{\epsilon }_0{\sigma }^2\left(\delta \right)\left[{\left(\frac{1}{2}{L}_Q+L_{\text{d}}\right)}^2{\beta }_0+\frac{1}{4}{L}_Q^2{L}_{\text{d}}^2{\gamma }_0-L_Q^2{L}_{\text{d}}^2{\alpha }_0\right]\end{array}$ (21)

Verification via multi-particle simulations

Similar to the procedure discussed in Sect. 3.2, the system parameters of the HUST injector were fed into the ASTRA software for the beam dynamics calculations. By employing virtual measurements to assess the second-order correction effects, curves depicting the fluctuation of the measurement errors according to K_range under different energy-spread conditions were obtained, as shown in Fig. 5.

Fig. 5Full-screen View

Measurement errors under second-order correction with different σ(δ) settings: (a) without chromatic effect correction (Eq. (16)), (b) with chromatic effect correction (Eq. (20))

Fig. 5(a) shows the measurement errors with second-order correction, excluding the chromatic effect. These errors are associated with the emittance derived by solving the coefficient equations Eq. (17) and post-parabolic fitting using Eq. (16). Compared with the measurement results shown in Fig. 4(a), which do not undergo second-order correction, it is discernible that the measurement errors are generally reduced after second-order correction, even across various σ(δ) settings.

Furthermore, Fig. 5(b) shows the measurement errors after chromatic effect correction, aligned with the emittance determined via fitting using Eq. (20). Notably, building upon the second-order correction foundation, there is a further reduction in the measurement errors, especially when the energy spread is significant. Moreover, after all corrections outlined in this context, the relative beam emittance errors within the 0.1%-6% energy spread range were reduced to approximately 1%. These results mark a significant advancement compared with the initial relative errors shown in Figs. 4(a) and 4(b).

Additionally, to further highlight the versatility of the proposed correction methodology, additional beam dynamics simulations and virtual measurements were carried out for various beam emittances while maintaining a specific σ(δ) (0.27%), which was also obtained from the simulated beam parameters of the HUST injector, as listed in Table 1. The resulting measurement errors are shown in Fig. 6. Notably, compared with the emittance determined by the fitting coefficients of Eq. (13), the measurement approach with second-order correction corresponding to Eq. (20) demonstrates a relative-error-reduction capability of more than two-thirds within the normalized emittance range of 2-25 mm mrad. For clarity, Fig. 7 shows curves illustrating the measurement error fluctuations under a fixed K_range regarding the energy spread and emittance for various correction methods.

Fig. 6Full-screen View

Measurement errors under chromatic effect correction with different emittance settings: (a) traditional method (Eq. (13)), (b) with second-order (Eq. (20)), where, ε_n,0 denotes the normalized emittance

Fig. 7Full-screen View

Measurement error variations under different correction methods: (a) along with σ(δ) at a specific emittance of ε_n,0 = 4.9 mm mrad, (b) along with ε_n,0 at σ(δ) = 0.27%

As shown in Fig. 7(a), without chromatic effect correction (depicted by the dashed red and black lines corresponding to Eqs. (8) and (16), respectively), the emittance measurement errors exhibit a monotonically increasing trend as σ(δ) increases. The traditional Q-scanning method exhibited improved error consistency after chromatic effect correction (depicted by the dotted blue line corresponding to Eq. (13)); however, the measurement error remained at approximately 7% for different σ(δ). Nonetheless, after second-order correction and considering chromatic effect elimination (depicted by the dotted green line corresponding to Eq. (20)), the absolute relative error decreased to less than 1.15%. Similarly, Fig. 7(b) shows a comparison between the measurement errors for different correction methods across various emittances. It is evident that the measurement errors are reduced by more than two-thirds after second-order correction, aligning with the discussion of the results shown in Fig. 6.

Validation for the HUST injector

The schematic of the HUST injector is shown in Fig. 8(a). Transverse sizes can be measured on Fluro-screens using 16-bit digital cameras. Two fast current transformers (FCTs) were employed to separately measure the macroscopic beam current and length. An energy analysis system, consisting of a magnetic spectrometer (AM) and two Fluro-screens settled upstream (Flag) and downstream (Screen1), was used to measure the beam energy and energy spread.

Fig. 8Full-screen View

(a) Schematic of the HUST injector; (b) Experimental emittance measurement data for the HUST injector

Note that although there are three Q-magnets originally used for beam transport in the beam line, to reduce interference items in the emittance measurement, only Q1 associated with Screen2 was selected, while the other two Q-magnets were not in the working state. Because all simulations were performed using the beam parameters and system configurations of the HUST injector, it is reasonable to extend the application of the proposed methods to the experimental results obtained from the HUST injector. The corresponding beam spot size data and fitted curves are shown in Fig. 8(b). The measurement results are presented in Table 2.

It is worth noting that the dotted red line in Fig. 8(b) represents the original fitting curve, and the corresponding emittance measurement value determined based on it is listed in Table 2 as “original result,” from which the energy spread influence is already excluded, as reported in [35]. Furthermore, leveraging the use of fixed vertices, as discussed in Sect. 3, the result is reduced to 8.6 mm mrad. Although this improvement appears modest, the comprehensive application of the proposed correction methods, corresponding to Eq. (20), yields the corrected result of 7.9 mm mrad, which is determined from the fitting curve depicted by the dashed blue line in Fig. 8(b). This value is close to the simulation value of 4.9 mm mrad, as listed in Table 1. The main contribution of this study is the emittance measurement accuracy and result consistency improvement by improving both the pre- and post-processing methods for the experimental data. Such an enhanced approach could be used as a general processing method for measuring emittance based on Q-scanning techniques. The remaining discrepancy might be attributed to the effects of the space charge and long-range wake fields, which have already been discussed previously [35] and are beyond the scope of the current study. Further investigations will be undertaken to analyze the measurement effects induced by factors such as the beam spot size and regularity.