research papers
A cantilevered liquidnitrogencooled silicon mirror for the Advanced Light Source Upgrade
^{a}Lawrence Berkeley National Laboratory, 1 Cyclotron Rd, Berkeley, CA 94720, USA
^{*}Correspondence email: gdcutler@lbl.gov
This paper presents a novel cantilevered liquidnitrogencooled silicon mirror design for the first optic in a new soft Xray beamline that is being developed as part of the Advanced Light Source Upgrade (ALSU) (Lawrence Berkeley National Laboratory, USA). The beamline is optimized for photon energies between 400 and 1400 eV with full polarization control. Calculations indicate that, without correction, this design will achieve a Strehl ratio greater than 0.85 for the entire energy and polarization ranges of the beamline. With a correction achieved by moving the focus 7.5 mm upstream, the minimum Strehl ratio is 0.99. This design is currently the baseline plan for all new ALSU insertion device beamlines.
Keywords: cantilevered mirrors; wavefront propagation; cryogenics; silicon mirrors; liquidnitrogen cooling; high heatload.
1. Introduction
A project to upgrade the Advanced Light Source (ALS) is currently underway. This project (known as the ALSUpgrade or ALSU) includes a new soft Xray `FLEXON' beamline (FLuctuation and EXcitation of Orders in the Nanoscale) optimized for photon energies between 400 and 1400 eV with full polarization control. The current design of this beamline incorporates a 4 mlong apple Xtype undulator (Schmidt & Calvi, 2018), a horizontally deflecting planar first mirror (referred to in this article as M1), a vertically deflecting monochromator, a horizontally deflecting refocusing mirror and an exit slit. This photon delivery system will serve ambitious research programs relying on highly coherent beams, requiring preservation of the wavefront under conditions of exceptionally high power density in the soft Xray range.
The fundamental challenge in the mechanical design of M1 is to integrate the cooling and mounting system such that the distortion of the mirror – quantified in this paper as the rootmeansquare (r.m.s.) height error – is acceptably small for the broad range of heat loads that correspond to the operating range of the undulator. The advantageous material properties of cryogenically cooled silicon and germanium have been well known in the synchrotron light source community since at least 1986 (Rehn, 1986; Bilderback, 1986). For high power applications it has been estimated that the slope error for liquidnitrogencooled silicon is approximately 100 times smaller than for watercooled silicon (Zhang, 1993). In the 1990s, this technology was tested in monochromator crystals (Comin, 1995; Knapp et al., 1995; Rogers et al., 1995; Meron et al., 1997) and it is now used routinely.
A variety of cryogenically cooled silicon monochromator designs have been developed which generally fall into one of two categories: directly or indirectly cooled. Indirectly cooled crystals are clamped between liquidnitrogencooled copper blocks with a layer of indium foil (Carpentier et al., 2001; Lee et al., 2001; Mochizuki et al., 2001; Tamasaku et al., 2002; Zhang et al., 2003; Chumakov et al., 2004; Zhang et al., 2013; Huang et al., 2014). In a directly cooled crystal the liquid nitrogen is in direct contact with the crystal, which is clamped to a coolant manifold, with the fluid being contained against the silicon by a compressed metal seal (Lee et al., 2000; Rowen et al., 2001; Liu et al., 2014).
Silicon mirrors cooled with liquid nitrogen are significantly less common, but do exist (Polack et al., 2010; Brookes et al., 2018). Considerations in the design of a successful liquidnitrogencooled silicon mirror system include, but are not limited to, carbon contamination of the optical surface, thermal strain between the mounting system and the mirror, mounting system stiffness, and thermal control, the latter three of which we will now briefly discuss. For a discussion of carbon contamination the reader is referred to YaoLeclerc et al. (2011), Risterucci et al. (2012), Pellegrin et al. (2014) and Toyoshima et al. (2015). We plan to clean the goldcoated mirror while in operation, using the proven oxygen flow technique (Risterucci et al., 2012) in which the mirror is exposed to oxygen continuously, and the action of the undulator light is to create reactive radical species that prevent carbon formation. As an alternative, or for implementation of the cryogenically cooled mirror with coatings that are not compatible with oxygen, the mirror chamber can be equipped with in situ RF plasma cleaning ports. In this case the mirror temperature would be raised for cleaning and the cleaning gas mixture would depend on the mirror coating (Pellegrin et al., 2014).
The thermal strain in silicon on cooling from 295 to 125 K is approximately 2.5 × 10^{−4} m m^{−1}. To the extent that the mirror mounting structure applies a reaction force to the mirror in response to this contraction, the mirror will deform. Typically, synchrotron beamline optics are kinematically mounted to permit this thermal strain with minimal reaction force. Kinematic mounting can be accomplished with spheres and Vgrooves or with flexures; in either case the six of the optic are exactly constrained and rigidbody motion is prevented. Additionally, differential can be managed by controlling the temperature of the mounting system (Saveri Silva et al., 2017). However, the stiffness of the mounting system is also important because it partially determines the positional stability of the mirror. The system has to be designed so that vibration does not cause significant intensity noise. The criterion used is that the angular deflection of the mirror is less than 2.5% of the FWHM of the angular source size at the highest energy of the beamline, corresponding to an amplitude noise of 0.1%. For the worst case at ALSU, this corresponds to an angular amplitude of ∼27 nrad r.m.s. We estimate that, to minimize sensitivity to environmental noise and stay below these vibration limits, the M1 assembly must have a first natural frequency (FNF) above 200 Hz. While this FNF requirement does not necessarily preclude kinematic mounting, designing a sufficiently stiff nonkinematic overconstrained mounting scheme is much more straightforward Another fundamental problem in the design of a kinematically mounted cooled mirror are the forces applied by the coolant lines to the mirror system. These forces vary in magnitude, direction and time with coolant pressure, temperature, flow rate and potentially also mirror alignment, and therefore are not easily characterized. To prevent unwanted motion or distortion of the mirror, the sphere and Vgroove based kinematic mounts are often springloaded with sufficiently high preload that friction at the sphere to Vgroove interface renders the mount nonkinematic. Intentionally nonkinematic designs also exist, one example being the watercooled cantilever mirror designed for use on an NSLS 1 beamline (Ice & Sparks, 1988).
Regarding thermal control, ideally the temperature of a cryogenically cooled silicon mirror or crystal is held near 125 K, where the instantaneous coefficient of K, and therefore the temperature drop from the peak mirror temperature to the coolant would also vary by a factor of approximately five. One solution is to use electric heaters near the coolant manifold; ideally the extra heat load would be applied on the reflecting surface of the mirror, which could be achieved with the incident Xray beam itself, by opening an upstream aperture as the K of the undulator is reduced. However, as we shall show, in the case of the current M1 neither heaters nor variable apertures are needed, as the optical performance is adequate even at temperatures significantly below 125 K.
(CTE) of silicon is approximately zero. To a firstorder approximation, the temperature drop from the hottest part of the mirror to the coolant is proportional to the absorbed heat load. In the case of M1 the heat load varies by a factor of approximately five (45 to 220 W), depending on the undulator deflection parameterIn this paper we present the design of a novel liquidnitrogencooled silicon mirror (Fig. 1); this endcooled cantilever design addresses the fundamental challenges of thermal strain, mounting stiffness, unknown coolant line forces and thermal control, as described in the previous paragraph. This paper is divided into two main parts. In the first part we describe our design, outline our analytical approach to thermal tuning and present finiteelement calculations of the thermoelastic distortion of the mirror. In the second part we first describe our method for calculating the r.m.s. height error, phase error and Strehl ratio from the finiteelement results, and then present wavefront propagation simulations using the deformed mirror shape. Based on these calculations we predict that with a fixed pitch adjustment – but without any higherorder (for example circular) correction – our design will achieve a Strehl ratio greater than 0.85 for the entire operating range of the beamline for two polarization modes. With a correction achieved by adjusting the focal length by 7.5 mm, the minimum Strehl ratio is calculated to be 0.988.
2. Description of design
In the ALSU FLEXON M1 (Fig. 1), one end of the silicon mirror substrate is clamped to a manifold made from a nickel–iron Invar alloy. Heat is transferred through the mirror substrate, through a layer of indium foil, across an array of pins machined into the manifold and into the flow of liquid nitrogen. The mirror is clamped to the manifold by a single screw and barrel nut. The clamping preload is set to achieve the required thermal contact conductance and is maintained at cryogenic temperatures by a spring washer. Translation of the substrate relative to the manifold is prevented by a hollow dowel pin in a hole concentric with the clamping screw, while rotation is prevented by a second pin in a slot.
The idea behind this design is to confine the deformation of the mirror substrate caused by thermal strain of the mirror and its mounting system to an optically insignificant part of the mirror. In other words, the center of the Xray beam is located sufficiently far from the manifold that strain at the manifold–substrate interface does not significantly affect the shape of the reflecting area. In so doing we are freed from the need to mount the mirror substrate kinematically and can use a comparatively stiff overconstrained mounting system. The stiffness of this mounting system, combined with the tapered shape of the mirror, means that the first natural frequency of the mirror–manifold system is at 402 Hz, a factor of two greater than the design target for ALSU optics. Thermal control is achieved by tuning the
between the mirror substrate and the coolant, and is described in the next section.2.1. Source considerations
In the case of M1, both the magnitude and spatial distribution of the absorbed power vary with the undulator deflection parameter K and polarization mode (Fig. 2). To maximize the from 400 to 1400 eV, the first harmonic of the undulator radiation would be used from K = 2.1 to K = 1, with a switch to the third harmonic at ∼874 eV and thereafter using the range K = 2.6 to 1.9. For some applications, the beamline will be used down to 230 eV, reached by using the first harmonic to a maximum K of 3. The peak power density ranges from approximately 0.2 W mm^{−2} at K = 1 to 1 W mm^{−2} at K = 3. Over this same K range the total absorbed power ranges from 45 to 220 W (assuming a fixed aperture dimension corresponding to ±3 standard deviations of the spatial distribution of 230 eV photons). Additionally, the polarization mode of the undulator can be changed in approximately 3 s. Storage ring, undulator and M1 parameters are given in Table 1.

2.2. Thermal tuning
We tuned the ). In this model the of conduction in the mirror substrate R_{s} is related to the length L_{s}, κ_{s} and crosssectional area A_{s} of the substrate by
of our mirror system using a simple onedimensional thermalresistor model (Fig. 3The contact resistance at the substrate–manifold interface is found from
where ν is the thermal contact conductance and A_{i} is the interface area. Note that we assume the resistance of conduction across the indium foil to be negligible. The conduction resistance in the manifold is
where L_{m}, κ_{m} and A_{m} are, respectively, the length, and crosssectional area of the manifold between the interface and the coolant. Finally, the convection resistance is
where h is the convection film coefficient and A_{h} is the total manifold–coolant interface area. The temperature drop across the mirror substrate, T_{1} − T_{2}, is
The temperature drop across the substrate–manifold interface is
that across the manifold to the coolant interface is
and that from the coolant interface to the coolant bulk is
To these `design equations' we also add the temperature rise of the coolant flowing through the manifold at a mass flow rate ,
where C_{p} is the of the coolant and T_{6} is the temperature of the coolant at the manifold outlet. We can choose up to 14 of the 19 variables in these five equations; for the purpose of designing this mirror, it is convenient to choose T_{1}, T_{2}, T_{4}, T_{5}, T_{6}, q, κ_{s}, L_{s}, ν, A_{i}, κ_{m}, A_{m}, A_{h} and C_{p}, and solve for A_{s}, T_{3}, L_{m}, h and . If we assume that the coolant is liquid nitrogen at T_{5} = 77 K, we can choose manifold–coolant interface and manifold outlet temperatures that limit the pressure necessary to prevent vaporization of the coolant, for example T_{4} = 80 K and T_{6} = 79 K. Because the instantaneous coefficient of of silicon is approximately zero at 125 K, we choose T_{1} = 125 K, and because we would like to limit the thermal gradient in the substrate we choose a similar value for the minimum temperature, T_{2} = 120 K. For a given undulator K and aperture size we know the power q. Assuming the substrate is silicon and the manifold is an Invar alloy, we know κ_{s} and κ_{m}. Based on published measurements of the thermal contact conductance (Yu et al., 1992; Asano et al., 1993; Khounsary et al., 1997; Marion et al., 2004) we assume a conservatively low value of ν = 1500 W m^{−2} K^{−1}. Once the and convection film coefficient are found, we find the dimensions of the pin array using the model developed by Zukauskas (1972),
where C1, C2, m and n depend on the pinarray geometry, Re is the Reynolds number, Pr is the Prandtl number, Prs is the pin surface Prandtl number, κ_{c} is the of the coolant and d is the pin diameter. Despite its simplicity, this onedimensional thermalresistor model agrees with the threedimensional finiteelement model discussed in the next section.
2.3. Finiteelement calculations of thermoelastic distortion
To evaluate the performance of our design we calculated the thermoelastic distortion using the finiteelement code ANSYS (ANSYS^{®} Mechanical APDL, Release 19.0). In this calculation, the full threedimensional geometry of the design is modeled, along with temperaturedependent and orthotropic material properties for singlecrystal silicon. We modeled the strain at the substrate–manifold interface by fixing the positions of nodes at the interface, as if the substrate were `welded' to a manifold with zero coefficient of This assumption is conservative because it overpredicts the strain in the substrate. In reality, in cooling from room temperature the manifold contracts, the substrate slides at the interface, and while the indium layer contracts it also permits some internal shear, all of which combine to reduce the overall strain and the resulting thermoelastic distortion of the mirror. To model the strain at the substrate–barrelnut interface we first computed the local contact pressure with a finely meshed model of the barrel nut and substrate split at the symmetry planes, and then used the resulting pressure distribution as a boundary condition in the substrate model without the meshed barrelnut geometry. This modeling sequence reduces computational cost and increases accuracy compared with a finemeshed contact model for the full substrate and nut geometry. We computed the thermoelastic distortion of the mirror for various load steps that simulate the assembly and operation of the mirror. In the first load step we applied a preload tension to the screw, pulling the barrel nut against the substrate. Next we applied gravity, and then cooled the assembly from 295 K to a uniform 77 K. After that we `turned on' the Xray beam and computed the steadystate temperature distribution and thermoelastic distortion for a range of heat loads corresponding to undulator deflection parameter K values between 1 and 3, for both linear horizontal and vertical polarization modes. An example of the temperature distribution is plotted in Fig. 4. The total power absorbed by M1 as a function of undulator deflection parameter K and the temperature of the mirror as a function of total absorbed power are plotted in Fig. 5.
From the results of these simulations we conclude that the thermoelastic distortion is dominated by the cooling step from 295 to 77 K. In other words, not only do clamping and gravity contribute relatively little to the distortion (Fig. 6), the effect of the Xray beam power is also small compared with the effect of differential between the manifold and substrate (Figs. 7 and 8). This strain causes a `cooldown' pitch in the reflecting portion of the mirror of −0.6 µrad, and this can be corrected either by a rotational stage or by the initial orientation of the mirror during beamline assembly.
3. Estimation of height error and Strehl ratio
We postprocessed the finiteelement results to calculate the height error, phase error and Strehl ratio. The height error g is the rootmeansquare (r.m.s.) value inside a window on the mirror's surface. From the height error, the grazing angle θ and the wavelength λ, we compute the phase error φ from
and the Strehl ratio S from
Based on the wavefront propagation simulations described in the next section, we determined that the correct window size for computing the r.m.s. height error is 6σ (six standard deviations) of the (approximately Gaussian) spatial distribution of photons of wavelength λ. This window is larger than the 2 × FWHM (or 4.7σ) window discussed elsewhere (Goldberg & Yashchuk, 2016; Cocco & Spiga, 2019) because the height error of this mirror is not random, but instead has a particular profile (Fig. 9) which mostly affects the spherical and defocus aberrations. The shape of the height error is an important factor in assessing the window size, as also pointed out by Herloski (1985). In other words, different aberrations should be weighted over different apertures. For example, for coma and astigmatism, 4.8 and 4.7σ are, respectively, the correct window sizes. For spherical aberration the size goes up to 5.6σ. The derivation from Herloski is based on a twodimensional radially symmetric distribution and provides a guideline for defining the proper window to use for calculating the shape error, phase error and Strehl ratio.
The calculated height error and Strehl ratio are plotted in Fig. 10. After removing the constant cooldown pitch angle of −0.6 µrad, the height error range is 0.5 to 3.5 nm, and the Strehl ratio range is 0.837 to 0.997. However, to reach the maximum energy of the beamline (1400 eV) the third harmonic is used at K = 1.9, where the Strehl ratio is 0.85. The peak Strehl ratio is at K = 3, which is to be expected as the mirror system is thermally tuned to be near the zero CTE temperature of silicon at this operating point. At lower K values the mirror is colder and the shape error is concave.
3.1. Wavefront propagation simulations
To validate the Strehl ratio calculations and visualize the effect of the thermally induced mirror deformation on the spot size, we simulated a few cases with the wavefront propagation code WISEr (Raimondi & Spiga, 2015) on the opensource platform OASYS (Rebuffi & Sanchez del Rio, 2017; Sanchez del Rio & Rebuffi, 2019). WISEr is a physical optics simulation package which computes the complex electromagnetic field downstream of optical elements. It works across the Xray spectrum and with grazing angles of incidence, using spatially and temporally fully coherent sources.
We simulated the cases with the lowest Strehl ratios (K = 1.5 in first and third harmonics, both polarization modes) and a case that gave a relatively high Strehl ratio (K = 2.5, first harmonic and linear horizontal polarization). For the purpose of understanding the effect of mirror deformation on the spot, we performed the simulation in the tangential direction only. Therefore we only considered two mirrors: M1 and the downstream focusingplane elliptical mirror. M1 is located 13.73 m from the undulator source, while the elliptical mirror is 15.75 m downstream of the flat mirror (source to mirror = 28.752 m) with a focal distance of 4.775 m. Because WISEr works with diffractionlimited beams, the dimension at the source was adapted to give the same footprint on M1 as was used in the finiteelement model.
For the five cases we calculated the spot at the focal location of the elliptical mirror for three conditions: a perfectly flat M1, a thermoelastically distorted M1, and a thermoelastically distorted M1 with the focal distance of the elliptical mirror corrected to minimize the spot profile. Because M1 is concave at low K values, the minimum spot dimensions are 5 to 7.5 mm upstream of the ideal focus. We then calculated the Strehl ratio for the thermoelastically distorted M1 with and without correction by comparing the peak intensity with that of the perfectly flat mirror (Fig. 11). We also compared the FWHM of the intensity distribution – the spot size – for the perfectly flat mirror with that of the distorted M1 with and without focus correction. In all calculated cases, the ratio of the distorted FWHM to the ideal FWHM is the same as the Strehl ratio. The height errors, Strehl ratios, spot sizes and focus corrections are summarized in Figs. 12 and 13.
4. Summary and conclusions
In this paper we have presented the novel cantilevered liquidnitrogencooled silicon mirror design that is being developed as the baseline M1 (or first mirror) for the Advanced Light Source Upgrade (ALSU). Our calculations indicate that, without correction, this design will achieve a Strehl ratio greater than 0.85 for the entire energy and polarization ranges of the beamline. With a correction achieved by moving the focus 7.5 mm upstream, the minimum Strehl ratio is 0.99. This focal distance change corresponds to about 0.16% of its original value and, if required, can be accomplished with a singleactuator mechanical bender.
Several important conclusions can be made from the results presented in this paper. First, temperatures in the mirror system can be accurately calculated from a onedimensional thermalresistor model, which facilitates tuning the system for specified heat loads. Second, in this case it is not strictly necessary to operate the mirror at the socalled `sweetspot' temperature of silicon (∼125 K), which means that additional heaters or variable apertures are not necessary. Third, the appropriate window size for calculating the r.m.s. height error (and Strehl ratio) for this particular deformation is 6σ of the spatial distribution of photons of the wavelength of interest. Fourth, the Strehl ratio and the increase in spot size (the ratio between the FWHM of the ideal spot and the thermoelastically deformed spot) are in agreement, as expected, because the wavefront aberration is mostly spherical and does not change the Gaussian distribution of the beam. Fifth, although the lowest uncorrected Strehl ratio is 0.85, in reality a Strehl ratio close to or in excess of 0.9 is more than adequate for all the situations we currently envision for the ALSU FLEXON beamline, especially because the spherical aberration of the mirror does not produce beam striation out of focus. Even for a beamline using imaging techniques, which requires a uniform beam in and out of focus, the presented cryogenically cooled mirror is the ideal solution.
Acknowledgements
The authors would like to thank Lin Zhang, Ali Khounsary, Thomas Rabedeau, Andrew Ringwall, Ken Goldberg, and Antoine Wojdyla for helpful suggestions, and Gilles Eggenspieler, Michael Rife, Christine Taylor, Thanakorn Siriaksorn and Julian Ware at ANSYS Inc.
Funding information
The following funding is acknowledged: Director, Office of Science, Office of Basic Energy Sciences, US Department of Energy (contract No. DEAC0205CH11231).
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