Analytical model for monochromator performance characterizations under thermal load

Zhengxian Qu; Zhengxian Qu; Yanbao Ma; Yanbao Ma; Guanqun Zhou; Juhao Wu; Juhao Wu

doi:10.1364/OE.394958

1. Introduction

The successful operation of the X-ray free-electron laser (XFEL) [1–4] enables the experimental explorations and measurements in a wide range of fields, such as chemistry [5], life sciences [6] and material sciences [7]. The superior properties of the XFEL beam, such as ultra-short duration [8], high transverse coherence [9] and unprecedentedly high spectral brightness [1], make it possible to probe structures and dynamics [10] with atomic spatial resolution and femtosecond temporal resolution. Compared to the third generation storage ring synchrotron light sources, the XFEL delivers a relatively low average power but an extremely high peak power. The extremely high peak power generates a local hot spot (or thermal bump) at its footprint due to the X-ray-matter interactions, triggering local deviation from the Bragg condition, known as thermal load effects.

Thermal load has been an issue in X-ray optics for several decades [11]. For the X-ray optics serving on the third generation synchrotron beamlines, such as mirrors and monochromators, one fundamental requirement is that the X-ray beam quality should be preserved. However, thermal load results in disturbance (such as thermally induced surface slope and strain) from the Bragg condition. The disturbance is usually non-uniform and leads to the distortion of the crystal rocking curve: its central frequency may shift and its bandwidth may be broadened. Thus, thermal load causes degradation of the X-ray beam quality. Existing solutions to mitigate thermal load include appropriately designed cooling and stiffening strategies [12]. These solutions have been proven successful strategies in suppressing the thermal load effects to an acceptable degree for X-ray optics in synchrotron beamline [13]. However, the extreme peak power provided by XFEL challenges these existing solutions. Compared to synchrotron light sources, the heat flux deposited by XFEL pulses can be several orders of magnitude higher. For example, in the Linac Coherent Light Source (LCLS) of SLAC National Accelerator Laboratory, a typical XFEL has a pulse energy of 1 mJ, transverse spot size (full width half maximum) of 350 $\mu$m, and pulse duration of 100 fs. The energy flux can be calculated as $2.6\times 10^{16}$ W/m$^2$ by averaging the pulse energy over the footprint area (assuming a circular spot with radius of 350 $\mu$m) and pulse duration of 100 fs. For a typical four-bounce silicon monochromator, more than 90% of the energy flux can be absorbed, yielding an absorbed heat flux of $2.3\times 10^{16}$ W/m$^2$. With such extremely high absorbed heat flux, the existing cooling techniques may not be sufficient to dissipate the residual heat within the XFEL footprint. The cooling requirements also become more extreme for applications such as self-seeding [14] and the X-ray free-electron laser oscillator (XFELO) [15], where the transverse spot size is even smaller or the relaxation time for the crystal is very limited (such as at a high repetition rate). Moreover, in a pump-probe mode, the cooling may not work at all due to the short delay time (within tens or hundreds of nanoseconds) between the two pulses.

At present, only an incomplete qualitative estimation of the thermal load is available. For example, the central frequency shift is usually estimated by the maximum thermal strain. However, there is no simple estimation available for bandwidth broadening, even though it is known to be caused by the non-uniformity of the thermal load. There exists a very limited number of studies that quantitatively describe the thermal load effects. Bushuev [16,17] showed that the rocking curves can be strongly distorted by the thermal load. In his model, Bushuev calculated the distorted rocking curves based on analytically solved thermal fields with constant material thermal properties [16] and temperature-dependent thermal properties [17]. However, Bushuev did not provide a separate assessment on how different factors of the thermal load (such as the maximum strain and the non-uniformity of the thermal load) contribute to this distortion. In this study, we derive an analytical model to quantitatively describe the thermal load effects on the distortion of the rocking curve. We show that the central frequency shift and distortion of the rocking curve can be attributed to different factors. The central frequency shift is mainly due to the background disturbance of the crystal under thermal load, while the rocking curve distortion is caused by the non-uniformity of the thermal load. These effects are assessed separately and quantitatively by the analytical model, providing potential directions for designing monochromators and the corresponding cooling schemes.

2. Analytical model derivation

For a perfect crystal without any strain or deformation, the rocking curve is accessible through multiple methods from dynamic diffraction theory [18–20], such as the response function method by Shvyd’ko [21]. When a small perturbation is present (such as weak strain and deformation), a parameter $\alpha$ can be defined [16,17,21,22] to account for local deviation from the exact Bragg condition

(1)$$\alpha\equiv\frac{k^2-\left(\vec{h}+\vec{k}\right)^2}{k^2},$$

where $\vec {k}$ is the wavevector, and $\vec {h}$ is the vector of the reciprocal lattice. Bushuev [16] showed that $\alpha$ can be expressed as

(2)$$\alpha=2\sin{\left(2\theta_B\right)}\left[\delta\theta+\left(\frac{\Omega}{\omega_0}+\frac{\delta d}{d}\right)\tan{\theta_B}\right],$$

where $\theta _B$ is the Bragg angle, $\delta \theta$ is the local incident angle deviation from the Bragg angle, $\Omega = \omega - \omega _0$ is the deviation of the angular frequency $\omega$ from the central angular frequency $\omega _0$ of the Bragg condition, and $d$ is the interplanar distance. The relative change of the interplanar distance is characterized by strain $\varepsilon \equiv \delta d/d$. The parameter $\alpha$ indicates that an effective frequency disturbance can be defined as

(3)$$\frac{\Omega_{\textrm{eff}}}{\omega_0}=\frac{\Omega}{\omega_0}+\cot{\theta_B}\delta\theta+\varepsilon=\frac{\Omega+\Delta\Omega}{\omega_0}.$$

This effective frequency disturbance $\Omega _{\textrm{eff}}$ causes the same disturbance around the Bragg condition without $\delta \theta$ and $\varepsilon$ as an $\Omega$ with $\delta \theta$ and $\varepsilon$ can. Here, $\Delta \Omega$ is the deviation parameter defined as

(4)$$\frac{\Delta\Omega}{\omega_0}\equiv\cot{\theta_B}\delta\theta+\varepsilon.$$

Thus, local deviation from the Bragg condition due to the thermal load is conjointly introduced to the rocking curve as a local effective deviation angular frequency. Hence, the rocking curve at this step is spatial-dependent, as also shown in Eq. (14) of Ref. [16].

However, if a full three-dimensional (3D) spatial-dependent local deviation is considered, an explicit analytical solution is almost impossible. Moreover, a 3D local deviation usually indicates the deviation gradient in the XFEL propagation direction (longitudinal direction). In this case, a complete 3D dynamic diffraction model for the distorted crystal is needed to deal with the longitudinal deviation gradient. To proceed, we assume that the local deviation is two-dimensional (2D). This assumption is applicable in many situations. One example is the transmissive monochromator in hard X-ray self-seeding mode. It is usually a thin CVD diamond plate or film. The thermal strain variation in the thickness direction can be omitted so that the thermal strain field is reduced to 2D [17], especially when the X-ray penetration depth is much larger than the crystal thickness. Another example is a typical silicon reflective monochromator in beamline: when crystal extinction length at the photon energy of the incident pulse is very small compared to the strain variation in the thickness direction, only the thermal slope and the strain field on the surface need to be considered.

Assuming the transverse mode of the FEL beam is Gaussian, a typical self-amplified spontaneous emission (SASE) pulse with $M$ modes ($M$ is an integer) can be described [23] by:

(5)$$E_{\omega}\left(x,y\right)=\sqrt{\frac{I_0}{\pi r_xr_y}}\exp{\left(-\frac{x^2}{2r_x^2}-\frac{y^2}{2r_y^2}\right)}\sum_{j=1}^M\sqrt{\frac{1}{M\sqrt{2\pi}\sigma_{\omega,j}}}\exp{\left[-\left(\frac{\omega-\omega_{0,j}}{2\sigma_{\omega,j}}\right)^2+i\varphi_j\right]},$$

where $I_0$ is the incident SASE pulse energy, $\sigma _{\omega ,j}$ is the rms bandwidth of the $j^{\textrm{th}}$ mode, and $r_x$ and $r_y$ are the intensity transverse radius in $x$ and $y$ direction respectively. The phase for the $j^{\textrm{th}}$ mode, $\varphi _j$, is random and is assumed spatial-independent (due to good transverse coherence [8,9]). Therefore, Eq. (5) can be processed mode by mode and superimposed in the end. For simplicity and without the loss of generality, we select a single mode Gaussian envelope SASE with the same thermal load as a typical multi-mode SASE:

(6)$$E_{\omega}\left(x,y\right)=\sqrt{\frac{I_0}{\sqrt{2\pi}\sigma_{\omega}\pi r_xr_y}}\exp{\left[-\frac{x^2}{2r_x^2}-\frac{y^2}{2r_y^2}-\left(\frac{\omega-\omega_{0}}{2\sigma_{\omega}}\right)^2+i\varphi\right]}.$$

Here, the bandwidth $\sigma _{\omega }$ is the SASE rms bandwidth. The intensity distribution can be calculated as

(7)$$I_{\omega}\left(x,y\right)=\frac{I_0}{\sqrt{2\pi}\sigma_{\omega}\pi r_xr_y}\exp{\left[-\frac{x^2}{r_x^2}-\frac{y^2}{r_y^2}-\left(\frac{\omega-\omega_{0}}{\sqrt{2}\sigma_{\omega}}\right)^2\right]}.$$

For simplification, we normalize the equation by $x^*=x/r_x$, $y^*=y/r_y$ and $\Omega ^*=\Omega /\omega _0$. Thus, Eq. (7) can be written as

(8)$$I_{\omega}\left(x,y\right)=\frac{I_0}{\sqrt{2\pi}\sigma_{\omega}\pi r_xr_y}\exp{\left(-x^2-y^2-\frac{\Omega^2}{2\sigma_{\omega}^2/\omega_0^2}\right)},$$

where the superscript asterisk has been dropped without confusion.

The overall reflectance $R$ (or transmittance $T$) is defined as the weighted average of the local reflectance (or transmittance) of the whole field (according to Bushuev [16])

(9)$$\bar{R} \equiv \frac{\iint_{-\infty}^{\infty} R[\Omega+\Delta \Omega(x, y)] I_{\omega} d x d y}{\iint_{-\infty}^{\infty} I_{\omega} d x d y}$$

The reflectance $R$ is interchangeable with transmittance $T$. Here we take $R$ as an example. Integrating Eq. (9) by parts and applying chain rule lead to the final expression for general 2D cases

(10)$$\begin{aligned} &\bar{R}=R\left(\Omega+\Delta \Omega_{\infty}\right)-\left.\frac{1}{2} \int_{-\infty}^{\infty} \frac{\partial R}{\partial \Delta \Omega} \frac{\partial \Delta \Omega}{\partial x}\right|_{y \rightarrow \infty} \operatorname{erf}(x) d x-\left.\frac{1}{2} \int_{-\infty}^{\infty} \frac{\partial R}{\partial \Delta \Omega} \frac{\partial \Delta \Omega}{\partial y}\right|_{x \rightarrow \infty} \operatorname{erf}(y) d y\\ &+\frac{1}{4} \iint_{-\infty}^{\infty}\left[\frac{\partial^{2} R}{\partial(\Delta \Omega)^{2}} \frac{\partial \Delta \Omega}{\partial x} \frac{\partial \Delta \Omega}{\partial y}+\frac{\partial R}{\partial \Omega} \frac{\partial^{2} \Delta \Omega}{\partial x \partial y}\right] \operatorname{erf}(x) \operatorname{erf}(y) d x d y \end{aligned}$$

Here, $\Delta \Omega _{\infty }$ represents the $\Delta \Omega$ evaluated at infinity, or background deviation. One should be reminded that, unfortunately, the background deviation $\Delta \Omega _{\infty }$ might not vanish, as this infinity is defined based on the intensity distribution length scale. The thermal load typically has a much larger length scale, so that the strain field is not vanishing. In other words, this term represents the deviation at the boundary of the field of view of the XFEL pulse. Outside this range, the deviation does not influence the rocking curve of the current pulse. For example, in a pump-probe situation with short delay time, the temperature increase induced by the first pulse is mostly restricted to the small area around the footprint. The deviation at the boundary of the footprint is negligible. However, for the situations where a quasi-steady state can be reached, a non-zero temperature increase constantly exists everywhere and $\Omega _{\infty }$ is non-zero at the boundary of the footprint.

3. Validations

Bushuev [16,17] analytically solved the temperature field for a thin infinite crystal with multi-pulse incident SASE and provided the thermally distorted rocking curves. Here, we combine our analytical model with this analytically solved temperature field and compare the results for validation.

In his analytical solution, Bushuev [16] considered a 2D infinite domain with constant thermal properties. After $n$ incident SASE pulses, the normalized deviation parameter is given as

(11)$$\Delta\Omega=\varepsilon=\sum_{j=0}^n\Delta\Omega_j\left(x,y,t\right)=\alpha_T\sum_{j=0}^n\frac{\Delta T_j}{\sqrt{\beta_{x,j}\beta_{y,j}}}\exp{\left(-\frac{x^2}{\beta_{x,j}}-\frac{y^2}{\beta_{y,j}}\right)},$$

where $\beta _{x,j} = 1 + \sin ^2{\theta _B}\frac {t-t_j}{\tau _T}$, $\beta _{y,j} = 1 + \frac {t-t_j}{\tau _T}$, $\Delta T_j = \frac {\mu I_0}{\pi \rho c_p a}$ is the instant temperature increase for single shot, $r_x = a/\sin \theta _B$, $r_y = a$, $\tau _T = a^2\rho c_p/4\kappa$ is the characteristic time for heat dissipation, $t_j$ is the arrival time of the $j^{\textrm{th}}$ pulse, $a$ is the spot size, $\alpha _T$, $\mu$, $\rho$, $c_p$ and $\kappa$ are the thermal expansion coefficient, absorption coefficient, mass density, specific heat and thermal conductivity of the crystal, respectively. More details can be found in Ref. [16].

Plugging in the deviation parameter by Eq. (11) into Eq. (10) leads to

(12)$$\begin{aligned} \bar{R}=& R\left(\Omega+\Delta \Omega_{\infty}\right)+\int_{-\infty}^{\infty} \frac{\partial R}{\partial \Delta \Omega}\left(\sum_{j=0}^{n} \frac{\left.\Delta \Omega_{j}\right|_{y \rightarrow \infty}}{\beta_{x, j}}\right) x \operatorname{erf}(x) d x \\ &+\int_{-\infty}^{\infty} \frac{\partial R}{\partial \Delta \Omega}\left(\sum_{j=0}^{n} \frac{\Delta \Omega_{j} \mid x \rightarrow \infty}{\beta_{y, j}}\right) y \operatorname{erf}(y) d y \\ &+\iint_{-\infty}^{\infty} \frac{\partial^{2} R}{\partial(\Delta \Omega)^{2}}\left(\sum_{j=0}^{n} \frac{\Delta \Omega_{j}}{\beta_{x, j}}\right)\left(\sum_{j=0}^{n} \frac{\Delta \Omega_{j}}{\beta_{y, j}}\right) x y \operatorname{erf}(x) \operatorname{erf}(y) d x d y \\ &+\iint_{-\infty}^{\infty} \frac{\partial R}{\partial \Delta \Omega}\left(\sum_{j=0}^{n} \frac{\Delta \Omega_{j}}{\beta_{x, j} \beta_{y, j}}\right) x y \operatorname{erf}(x) \operatorname{erf}(y) d x d y \end{aligned}$$

Equation (12) is now ready to be compared with Figs. 5 and 6 in Ref. [16] for validation, as plotted in Fig. 1. In Fig. 1, the numerical results represent our calculation that exactly follows the process specified in Ref. [16]. Our numerical results are strictly overlapped with our analytical results by Eq. (12), so it is hard to distinguish from each other in Fig. 1. Also, an excellent agreement is observed in the comparison between our results and Bushuev’s.

Fig. 1. Comparison between the results by Bushuev [16] (black and red circles), by our numerical calculation (black and red dash line), and by Eq. (12) (black and red dash dot line). The rocking of undeformed crystal is also plotted using blue solid line as reference. Significant distortion can be observed for rocking curves under thermal load as compared to the original rocking curve for the undeformed crystal.

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As shown in Fig. 1, the thermal load significantly distorts the rocking curve. Severe central frequency shift and distortion can be observed due to thermal load. The peak reflectance is also suppressed, indicating a suppression in peak intensity of the reflected beam. All these effects are captured by Eq. (10). However, even though Eq. (10) predicts the effects of the thermal load in many situations where the assumptions fit, its complicated form still obscures some direct and clear contributions from different factors of the thermal load, as addressed in Sec. 1. To further reveal a clear picture of the thermal load effects, we present an axisymmetric (1D) case.

4. Clear physical insights: an axisymmetric case

In an axisymmetric case, the complicated form of Eq. (10) can be greatly simplified. We start by presenting the 1D axisymmetric form of Eq. (9)

(13)$$\overline{R}\equiv\frac{\int_0^{\infty}RI_{\omega}2\pi rdr}{\int_0^{\infty}I_{\omega}2\pi rdr}=2\int_0^{\infty}R\left(\Omega+\Delta\Omega\right)\exp{\left(-r^2\right)}rdr.$$

Following the similar process for deriving Eq. (10), we have

(14)$$\overline{R}=R\left(\Omega+\Delta\Omega|_{r=0}\right)+\int_0^{\infty}\frac{\partial R}{\partial\Delta\Omega}\frac{\partial\Delta\Omega}{\partial r}\exp{\left(-r^2\right)}dr.$$

Compared to Eq. (10), Eq. (14) is simple, and it offers several straightforward and clear points. The central wavelength shift is strongly related to maximum deviation, as suggested by the first term of Eq. (14). In an axisymmetric case, the maximum deviation is essentially the strain at the center of the footprint. The distortion is caused by the gradient, or the non-uniformity, of the deviation, as suggested by the second term of Eq. (14). For a narrower rocking curve, the first derivative term $\frac {\partial R}{\partial \Delta \Omega }$ is larger, indicating that the rocking curve width will be more sensitive to the thermal load.

Therefore, the maximum thermal strain (estimated by $\alpha \Delta T_{\max }$) is usually a reliable qualitative indicator of the frequency shift [16], though a minor contribution may also come from the second term of Eq. (14). This indicator has been successfully used in the community for quite a long time. On the other hand, it has long been realized [11,12] that the non-uniformity of temperature rise results in a “mapping” error, which can be captured by Eq. (14).

Furthermore, Eq. (14) also suggests that the frequency shift due to thermal load can be compensated by re-orienting the crystal with a small angle

(15)$$\delta\theta_{\textrm{tuning}}=-\Delta\Omega|_{r=0}.$$

However, the distortion due to thermal load can only be minimized by suppressing the non-uniformity. To suppress the non-uniformity, optimal geometric design, or cooling, or even compensated heating can be implemented.

5. Comparison with numerical simulation

To provide a quantitative comparison, we investigate one special axisymmetric case. In this case, we consider incident SASE pulses at 9.83 keV with relative bandwidth (FWHM) of $1.18\times 10^{-3}$, spot size (FWHM) of 47 $\mu$m, and pulse energy of 100 $\mu$J. A 110-$\mu$m-thick diamond plate with (440) orientation is used as the monochromator to produce the seed pulse in a self-seeding mode. The pulse repetition rate is 400 kHz. The Bragg angle here is 90$^{\circ }$, and the absorption length is 1518.3 $\mu$m. Consequently, the 1D axisymmetric assumption is appropriate here because the incidence is normal and the absorption length is much larger than the crystal thickness.

In 1D axisymmetric case, Eq. (11) can be rewritten as

(16)$$\Delta\Omega=\alpha_T\sum_{j=0}^n\frac{\Delta T_j}{\beta_{r,j}}\exp{\left(-\frac{r^2}{\beta_{r,j}}\right)},$$

where $r = \sqrt {x^2+y^2}$ and $\beta _{r,j} = 1 + \frac {t-t_j}{\tau _T}$. In this case, due to the normal incidence of the SASE pulse, the thermal slope is not present in the deviation parameter ($\cot {\theta _B} = 0$) given in Eq. (3). The total strain $\varepsilon$ in Eq. (4) consists of two components: thermal strain and elastic strain. Thermal strain is induced by the temperature change and the corresponding thermal deformation of the material, while elastic strain is generated when a deformation is present due to elasticity. In a quasi-static simulation for a 110-$\mu$m-thick CVD diamond plate [22], elastic strain in the thickness direction is secondary to thermal strain, as will be shown later in the comparison between the results with and without it (only 10% difference). Therefore, the elastic strain component is omitted for now. More details about quasi-static simulation have been described in our previous work [22,24]. Equation (16) also indicates that a quasi-steady state can never be reached because of its divergent harmonic series behavior.

For the forward Bragg diffraction, the transmittance $T$ is considered. Substituting Eq. (16) into the transmittance version of Eq. (14) yields

(17)$$ \overline{T}=T\left( \Omega +{{\alpha }_{T}}\sum_{j=0}^{n}{\frac{\Delta {{T}_{j}}}{{{\beta }_{r,j}}}} \right)-2\underset{0}{\overset{\infty }{\mathop \int }}\,\frac{\partial T}{\partial \Delta \Omega }\left[ \sum_{j=0}^{n}{\frac{{{\alpha }_{T}}\Delta {{T}_{\textrm{j}}}}{\beta _{r,j}^{2}}\exp \left( -\frac{{{r}^{2}}}{{{\beta }_{r,j}}} \right)} \right]\,\exp \left( -{{r}^{2}} \right)rdr. $$

Equation (17) offers the results of the distorted rocking curves under the thermal load.

To provide a concrete comparison and characterize the applicability of our analytical model, we conduct finite element analysis (FEA) using COMSOL with the same SASE properties and crystal orientation and thickness. The crystal we consider is a 110-$\mu$m-thick 2 mm by 2 mm square plate. The temperature on one side surface is fixed with a constant temperature of 300 K (indicated in Fig. 2) to enable a quasi-steady state, where the system behavior is periodic in time. To reduce the computational load, the domain is reduced to one half and symmetric boundary condition is specified as indicated in Fig. 2. Other surfaces are thermally insulated and free of stress. For heat transfer simulation, the quadratic Lagrange element was used, while for mechanical simulation, the quadratic serendipity element was used. The results have been checked and confirmed grid-independent. After around 5000 incident SASE pulses, the system approaches a quasi-steady state. In numerical simulation, the temperature-dependent thermal properties [25] and the boundary effects are both included. In addition, the effect of elastic strain component can be quantified by comparing the results with or without it. With all these effects accounted, we export the strain field obtained by FEA and calculate the field of $\Delta \Omega \left (x,y\right )$ through Eq. (3), and feed it into Eq. (10) to evaluate the rocking curves under thermal load.

Fig. 2. The total strain field from FEA right before the $5001^{\textrm{th}}$ pulse. The crystal is a 110-$\mu$m-thick 2 mm by 2 mm square plate with fixed constraint and constant temperature of 300 K on one side surface. Only half of the domain was simulated due to symmetry. The deformation is magnified for better illustration.

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Figure 3(a) shows the transmittance curves for the 1$^{\rm{st}}$, 10$^{\rm{th}}$, 200$^{\rm{th}}$ and 5000$^{\rm{th}}$ SASE pulse. A significant shift can be observed here, but no significant bandwidth broadening or distortion is captured. However, the transmittance curve deforms noticeably, as the oscillating behavior near the fast-decreasing edges is suppressed when thermal load is present. This suppression can result in the seed pulse energy reduction for self-seeding.

Fig. 3. (a) The distorted transmittance curves under thermal load at different number of pulses calculated by Eq. (17) and (b) the normalized central frequency shift history from $-\alpha _T\Delta T_{\max }$, analytical model Eq. (17) and numerical simulation. "Num. Therm" and "Num. Tot" represent the results calculated with thermal strain only and with total strain (both thermal and elastic).

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On the other hand, Fig. 3(b) summarizes the central frequency shift of the transmittance curve. The central frequency shift is calculated as the deviation of the rocking curve peak location from the designed value. This deviation can be predicted by Eq. (17) as $-\alpha _T\Delta T_{\max }$ when normalized. To show this, we analytically calculate the thermally distorted rocking curve using Eq. (17) and extract its central frequency shift. We take the central frequency at the minimum of the transmittance curve to account for the detuning effects [26]. An outstanding agreement is observed between this calculated deviation (green square line) and $-\alpha _T\Delta T_{\max }$ (red triangle line). These two lines both grow logarithmically, as suggested by the harmonic series of Eq. (16). The other two lines stand for the central frequency shift calculated from the numerical results without the elastic strain (blue diamond line) and with the elastic strain (violet circle line). These two line both converge, indicating the existence of a quasi-steady state. The difference between these two lines is due to the elastic strain component. The elastic strain component results in only about 10% difference, as its magnitude is only about 10% of the thermal strain component.

The numerical result without elastic strain overlaps with the curve of $-\alpha _T\Delta T_{\max }$ in the first 300 pulses. A minor discrepancy between these two curves is due to the temperature-dependence of the thermal properties, which is not considered in the analytical solution. After around 400 pulses, the slope of blue diamond curve starts changing because thermal diffusion reaches the boundary. An estimation of the thermal diffusion characteristic time (about 0.0011 second or 430 pulses) coincides with this observation. This observation indicates that, Eq. (17) is able to provide a good estimation of beam quality degradation in the pump-probe (only two pulses) experiments, or in multi-pulse situations (pulse train mode). However, one may realize that, Eq. (14) (or Eq. (10)) is still valid when equipped with an appropriate surface slope and strain solution, which have been well modeled in thermoelasticity studies.

6. Conclusion

In conclusion, we provide an analytical model for predicting the rocking curve distortion under thermal load. The thermal load (surface slope and strain) can be obtained from either the analytical solution with appropriate assumptions, or numerical simulations. Based on this analytical model, different contributions from the thermal load characteristics (such as maximum temperature increase and temperature gradient) can be quantified, as summarized below:

• The background deviation, or maximum disturbance in an axisymmetric case, results in a shift in frequency.
• The gradient, or the non-uniformity, of the disturbance results in the rocking curve distortion.
• For higher order diffraction, the narrower rocking curve suggests more significant distortion with the same thermal load.

To mitigate the thermal load effects, tuning the angle between the X-ray beam and the crystal should eliminate the frequency shift. However, mitigating the rocking curve distortion, such as bandwidth broadening, can only be achieved through minimizing the thermal load gradient.

Funding

Air Force Office of Scientific Research (FA9550-18-1-0086); National Science Foundation (1637370); U.S. Department of Energy (DE-AC02-76SF00515); U.S. DOE Office of ScienceEarly Career Research Program (FWP-2013-SLAC-100164).

Acknowledgment

The authors thank B. Yang, T.O. Raubenheimer, D. Zhu, L. Zhang, M.H. Seaberg, H. Yavas, F.-J. Decker, A.A. Lutman, Z. Huang, W.J. Corbett of SLAC for stimulating discussions. The authors thank specifically to Boris Fedorov and Ivan Rao for reading and improving the manuscript.

Disclosures

The authors declare no conflicts of interest.

References

1. P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt, J. Bozek, A. Brachmann, P. Bucksbaum, R. Coffee, and F.-J. Decker, “First lasing and operation of an ångstrom-wavelength free-electron laser,” Nat. Photonics 4(9), 641–647 (2010). [CrossRef]

2. T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano, N. Azumi, T. Bizen, H. Ego, K. Fukami, T. Fukui, Y. Furukawa, S. Goto, H. Hanaki, T. Hara, T. Hasegawa, T. Hatsui, A. Higashiya, T. Hirono, N. Hosoda, M. Ishii, T. Inagaki, Y. Inubushi, T. Itoga, Y. Joti, M. Kago, T. Kameshima, H. Kimura, Y. Kirihara, A. Kiyomichi, T. Kobayashi, C. Kondo, T. Kudo, H. Maesaka, X. M. Maréchal, T. Masuda, S. Matsubara, T. Matsumoto, T. Matsushita, S. Matsui, M. Nagasono, N. Nariyama, H. Ohashi, T. Ohata, T. Ohshima, S. Ono, Y. Otake, C. Saji, T. Sakurai, T. Sato, K. Sawada, T. Seike, K. Shirasawa, T. Sugimoto, S. Suzuki, S. Takahashi, H. Takebe, K. Takeshita, K. Tamasaku, H. Tanaka, R. Tanaka, T. Tanaka, T. Togashi, K. Togawa, A. Tokuhisa, H. Tomizawa, K. Tono, S. Wu, M. Yabashi, M. Yamaga, A. Yamashita, K. Yanagida, C. Zhang, T. Shintake, H. Kitamura, and N. Kumagai, “A compact x-ray free-electron laser emitting in the sub-ångström region,” Nat. Photonics 6(8), 540–544 (2012). [CrossRef]

3. H.-S. Kang, C.-K. Min, H. Heo, C. Kim, H. Yang, G. Kim, I. Nam, S. Y. Baek, H.-J. Choi, G. Mun, B. R. Park, Y. J. Suh, D. C. Shin, J. Hu, J. Hong, S. Jung, S.-H. Kim, K. Kim, D. Na, S. S. Park, Y. J. Park, J.-H. Han, Y. G. Jung, S. H. Jeong, H. G. Lee, S. Lee, S. Lee, W.-W. Lee, B. Oh, H. S. Suh, Y. W. Parc, S.-J. Park, M. H. Kim, N.-S. Jung, Y.-C. Kim, M.-S. Lee, B.-H. Lee, C.-W. Sung, I.-S. Mok, J.-M. Yang, C.-S. Lee, H. Shin, J. H. Kim, Y. Kim, J. H. Lee, S.-Y. Park, J. Kim, J. Park, I. Eom, S. Rah, S. Kim, K. H. Nam, J. Park, J. Park, S. Kim, S. Kwon, S. H. Park, K. S. Kim, H. Hyun, S. N. Kim, S. Kim, S.-M. Hwang, M. J. Kim, C.-Y. Lim, C.-J. Yu, B.-S. Kim, T.-H. Kang, K.-W. Kim, S.-H. Kim, H.-S. Lee, H.-S. Lee, K.-H. Park, T.-Y. Koo, D.-E. Kim, and I. S. Ko, “Hard x-ray free-electron laser with femtosecond-scale timing jitter,” Nat. Photonics 11(11), 708–713 (2017). [CrossRef]

4. M. Scholz, “FEL Performance Achieved at European XFEL,” 9th Int. Particle Accelerator Conf. (IPAC’18), Vancouver, BC, Canada, April 29-May 4, 2018 p. 29 (2018).

5. W. Zhang, R. Alonso-Mori, U. Bergmann, C. Bressler, M. Chollet, A. Galler, W. Gawelda, R. G. Hadt, R. W. Hartsock, T. Kroll, K. S. Kjær, K. Kubicek, H. T. Lemke, H. W. Liang, D. A. Meyer, M. M. Nielsen, C. Purser, J. S. Robinson, E. I. Solomon, Z. Sun, D. Sokaras, T. B. van Driel, G. Vankó, T.-C. Weng, D. Zhu, and K. J. Gaffney, “Tracking excited-state charge and spin dynamics in iron coordination complexes,” Nature 509(7500), 345–348 (2014). [CrossRef]

6. M. M. Seibert, T. Ekeberg, F. R. N. C. Maia, M. Svenda, J. Andreasson, O. Jönsson, D. Odic, B. Iwan, A. Rocker, D. Westphal, M. Hantke, D. P. DePonte, A. Barty, J. Schulz, L. Gumprecht, N. Coppola, A. Aquila, M. Liang, T. A. White, A. Martin, C. Caleman, S. Stern, C. Abergel, V. Seltzer, J.-M. Claverie, C. Bostedt, J. D. Bozek, S. Boutet, A. A. Miahnahri, M. Messerschmidt, J. Krzywinski, G. Williams, K. O. Hodgson, M. J. Bogan, C. Y. Hampton, R. G. Sierra, D. Starodub, I. Andersson, S. Bajt, M. Barthelmess, J. C. H. Spence, P. Fromme, U. Weierstall, R. Kirian, M. Hunter, R. B. Doak, S. Marchesini, S. P. Hau-Riege, M. Frank, R. L. Shoeman, L. Lomb, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, C. Schmidt, L. Foucar, N. Kimmel, P. Holl, B. Rudek, B. Erk, A. Hömke, C. Reich, D. Pietschner, G. Weidenspointner, L. Strüder, G. Hauser, H. Gorke, J. Ullrich, I. Schlichting, S. Herrmann, G. Schaller, F. Schopper, H. Soltau, K.-U. Kühnel, R. Andritschke, C.-D. Schröter, F. Krasniqi, M. Bott, S. Schorb, D. Rupp, M. Adolph, T. Gorkhover, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson, H. N. Chapman, and J. Hajdu, “Single mimivirus particles intercepted and imaged with an x-ray laser,” Nature 470(7332), 78–81 (2011). [CrossRef]

7. D. Milathianaki, S. Boutet, G. J. Williams, A. Higginbotham, D. Ratner, A. E. Gleason, M. Messerschmidt, M. M. Seibert, D. C. Swift, P. Hering, J. Robinson, W. E. White, and J. S. Wark, “Femtosecond visualization of lattice dynamics in shock-compressed matter,” Science 342(6155), 220–223 (2013). [CrossRef]

8. I. A. Vartanyants, A. Singer, A. P. Mancuso, O. M. Yefanov, A. Sakdinawat, Y. Liu, E. Bang, G. J. Williams, G. Cadenazzi, B. Abbey, H. Sinn, D. Attwood, K. A. Nugent, E. Weckert, T. Wang, D. Zhu, B. Wu, C. Graves, A. Scherz, J. J. Turner, W. F. Schlotter, M. Messerschmidt, J. Lüning, Y. Acremann, P. Heimann, D. C. Mancini, V. Joshi, J. Krzywinski, R. Soufli, M. Fernandez-Perea, S. Hau-Riege, A. G. Peele, Y. Feng, O. Krupin, S. Moeller, and W. Wurth, “Coherence Properties of Individual Femtosecond Pulses of an X-Ray Free-Electron Laser,” Phys. Rev. Lett. 107(14), 144801 (2011). [CrossRef]

9. C. Gutt, P. Wochner, B. Fischer, H. Conrad, M. Castro-Colin, S. Lee, F. Lehmkühler, I. Steinke, M. Sprung, W. Roseker, D. Zhu, H. Lemke, S. Bogle, P. H. Fuoss, G. B. Stephenson, M. Cammarata, D. M. Fritz, A. Robert, and G. Grübel, “Single Shot Spatial and Temporal Coherence Properties of the SLAC Linac Coherent Light Source in the Hard X-Ray Regime,” Phys. Rev. Lett. 108(2), 024801 (2012). [CrossRef]

10. M. Ware, A. Natan, J. Glownia, J. Cryan, and P. Bucksbaum, “Filming nuclear dynamics of iodine using x-ray diffraction at the lcls,” Bull. Am. Phys. Soc. 62 (2017).

11. D. H. Bilderback, A. K. Freund, G. S. Knapp, and D. M. Mills, “The historical development of cryogenically cooled monochromators for third-generation synchrotron radiation sources,” J. Synchrotron Radiat. 7(2), 53–60 (2000). [CrossRef]

12. M. Howells, “Some fundamentals of cooled mirrors for synchrotron radiation beamlines,” Opt. Eng. 35(4), 1187 (1996). [CrossRef]

13. L. Zhang, W. Lee, M. Wulff, and L. Eybert, “Performance prediction of cryogenically cooled silicon crystal monochromator,” AIP Conf. Proc. 705, 623–626 (2004). [CrossRef]

14. J. Amann, W. Berg, V. Blank, F.-J. Decker, Y. Ding, P. Emma, Y. Feng, J. Frisch, D. Fritz, J. Hastings, Z. Huang, J. Krzywinski, R. Lindberg, H. Loos, A. Lutman, H.-D. Nuhn, D. Ratner, J. Rzepiela, D. Shu, Y. Shvyd’ko, S. Spampinati, S. Stoupin, S. Terentyev, E. Trakhtenberg, D. Walz, J. Welch, J. Wu, A. Zholents, and D. Zhu, “Demonstration of self-seeding in a hard-x-ray free-electron laser,” Nat. Photonics 6(10), 693–698 (2012). [CrossRef]

15. P. Rauer, I. Bahns, W. Hillert, J. Robbach, W. Decking, and H. Sinn, “Integration of an xfelo at the european xfel facility,” Proc. 39th Int. Free. Laser Conf. p. 62 (2019).

16. V. Bushuev, “Effect of the thermal heating of a crystal on the diffraction of pulses of a free-electron x-ray laser,” Bull. Russ. Acad. Sci.: Phys. 77(1), 15–20 (2013). [CrossRef]

17. V. Bushuev, “Influence of thermal self-action on the diffraction of high-power x-ray pulses,” J. Surf. Invest.: X-Ray, Synchrotron Neutron Tech. 10(6), 1179–1186 (2016). [CrossRef]

18. B. W. Batterman and H. Cole, “Dynamical diffraction of x rays by perfect crystals,” Rev. Mod. Phys. 36(3), 681–717 (1964). [CrossRef]

19. A. Authier, Dynamical theory of x-ray diffraction, International Tables for Crystallography pp. 626–646 (2006).

20. V. Bushuev, “Diffraction of x-ray free-electron laser femtosecond pulses on single crystals in the bragg and laue geometry,” J. Synchrotron Radiat. 15(5), 495–505 (2008). [CrossRef]

21. Y. Shvyd’ko and R. Lindberg, “Spatiotemporal response of crystals in x-ray bragg diffraction,” Phys. Rev. Spec. Top.--Accel. Beams 15(10), 100702 (2012). [CrossRef]

22. Z. Qu, Y. Ma, G. Zhou, and J. Wu, “Thermal loading on self-seeding monochromators in x-ray free electron lasers,” Nucl. Instrum. Methods Phys. Res., Sect. A 969, 163936 (2020). [CrossRef]

23. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, The physics of free electron lasers (Springer, 2011).

24. Z. Qu, “Photo-thermo-mechanical analysis and control for high-brightness and high-repetition-rate x-ray optics,” Ph.D. thesis, UC Merced (2020).

25. M. Levinshtein, S. Rumyantsev, and M. Shur, Handbook Series On Semiconductor Parameters, Vol. 1: Si, Ge, C (Diamond), Gaas, Gap, Gasb, Inas, Inp, Insb, vol. 1 (World Scientific, 1996).

26. C. Yang, C.-Y. Tsai, G. Zhou, X. Wang, Y. Hong, E. D. Krug, A. Li, H. Deng, D. He, and J. Wu, “The detuning effect of crystal monochromator in self-seeding and oscillator free electron laser,” Opt. Express 27(9), 13229 (2019). [CrossRef]

Analytical model for monochromator performance characterizations under thermal load

Abstract

1. Introduction

2. Analytical model derivation

3. Validations

4. Clear physical insights: an axisymmetric case

5. Comparison with numerical simulation

6. Conclusion

Funding

Acknowledgment

Disclosures

References

Cited By

Figures (3)

Equations (17)

Optics Express