research papers
Tunable sidebounce monochromator
^{a}Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439, USA
^{*}Correspondence email: hhong@anl.gov
Sidebounce beamlines with fixedexit angles have been intended to operate with only one selected energy. However, a tunable monochromator in a new geometry is presented here that will make sidebounce beamlines energy tunable. It requires the addition of two more rotations. Analytic solutions for the values of these two rotation angles are provided. The validity of the new concept was checked by ray tracing and twodimensional searches in the additional angles. Operational details on the new scheme, including the exit offset and steering of the beams, were determined. In addition to tunability, the new monochromator will reduce the loss from the polarization factor at low energies.
1. Introduction
Doublecrystal monochromators (DCMs) are used for most of the hard Xray beamlines at synchrotron radiation facilities (Mills & King, 1983; Golovchenko et al., 1981). In some special cases, singlecrystal monochromators (SCMs) are used in a sidebounce mode. The singlebounce monochromator usually has a fixedexit angle, and the available energy is determined by the diffracting crystal and the exit angle. Using a series of different crystals, some selected energies can be made available with a singlebounce monochromator. Here, a continuously tunable monochromator is the goal. A novel scheme for a monochromator is presented here to make a sidebounce (fixedexit) monochromator energy tunable. To make a fixedexit monochromator tunable, it is required to use two crystals as explained below. It ends up somewhere between a vertical DCM and a horizontal DCM, but with two twists on crystals.
When one steers an Xray beam with a monochromator, the steering must hold the Bragg condition between the incoming ray and the diffracting crystal. A rotation of the crystal about a vector normal to the diffraction plane will preserve the Bragg condition. There is no change in the incoming or outgoing rays with this rotation. Another Bragg preserving rotation is the rotation of the entire system around the incoming or outgoing rays. Rotating about the incoming ray will rotate the outgoing ray while preserving the Bragg condition. This is similar to steering the outgoing beam with a `chi motion'. In a usual modern DCM, a χ rotation stage sits on a θ stage. The range of χ that will preserve the Bragg condition is pretty much limited. In a liquid Xray reflectometer with one steering crystal (AlsNielsen & Pershan, 1983), the θ stage sits on the χ stage and it can steer the beam downward through a large range. Furthermore, liquid Xray reflectometers employing two crystals have been developed (Honkimäki et al., 2006; Murphy et al., 2014).
Here are the steps taken when one steers a beam of a particular photon energy to a fixedexit angle, using two new rotations on two crystals.
(1) Start with a vertical DCM as shown in Fig. 1(a). Rotate the whole DCM counterclockwise (watching from upstream) by an angle α about the incoming beam vector q1, as shown in Fig. 1(b). The rotation is by an angle −α about the Z axis in the coordinate system from Fig. 2. This is the standard coordinate system commonly used at the Advanced Photon Source. The exiting ray, q3, is still parallel to the incoming ray, q1. The q2 vector will also rotate and becomes Rq2. All these vectors are assumed to be of unit length for simplicity.
(2) Rotate the second crystal clockwise about Rq2. The q3 vector will rotate around, going down and up. At the same time, the beam acquires a horizontal component. One should rotate q3 until the beam again becomes level. The angle of the second rotation is β. This rotation is shown in Fig. 1(c).
(3) Choose α and β to match the exit angle to the desired one.
The resulting overall beam path through the two crystals is shown in Fig. 2. n1 and n2 are surface normal vectors of the crystals.
2. Solutions for necessary rotation angles
2.1. Analytical solution
One can analytically calculate the necessary rotation angles, α and β. In Fig. 3, OC is an extension of Rq2. When q3 (OA) rotates about OC (or Rq2), the ray makes a circle (or a cone) about the OC axis.
In our coordinate system, the incident beam is along the Z axis and the Y axis points upward.
R(α, (0, 0, 1)) is a transformation matrix for rotation by an angle α about (0, 0, 1). Angle μ is the angle from Rq2 to the YZ plane (∠AOM).
Here, the angle μ is half of the exit angle γ (or the sidebounce angle ∠AOB). Furthermore, the angle ∠OCA = 90°.
Then we can obtain
Now, the rotation angles α and β are determined from the photon energy (2θ) and the exit angle (γ). Table 1 lists various DCM angles (in degrees) for a diamond (111) DCM at selected energies. It is assumed that the fixedexit angle is 23.13° (E = 15.017 keV). In the new geometry, the DCM crystals are not parallel to each other but twisted. Liquid reflectometers steer a monochromatic beam to various downward angles. The sidebounce tunable monochromator has one more constraint. One has to bring the beam to a fixed angle horizontally and parallel to the ground. That is why two diffracting crystals are required.

2.2. Numerical solution
The numerical search for the best rotation angles α and β must satisfy two conditions. The first condition is that the final exit ray after two reflections be horizontal and the second condition is that the final exit ray has the required horizontal γ. The sum of the square of the error in γ and the angle from horizontal was the error function to minimize. The numerical search for a solution consisted of two steps. The first step was to create an array of all possible angle pairs with a coarse step size of ∼12° and to calculate the array with the error function for every (α, β) pair; there were only 225 pairs. The point with the smallest error was then the start for the second step. The second step was a nonlinear leastsquares minimization of the error using a canned routine that only required ∼50 additional error evaluations. This yielded a total error of <10^{−10} degrees, and the corresponding errors in α and β were also <10^{−10} degrees. The angles α and β calculated by this method agreed within the given tolerances with the analytic formulas given in the previous section.
3. Operation of a twisted DCM
3.1. Energy range
A twisted DCM has limits in the energy range. The upper photonenergy limit is about 2E_{0} (E_{0} is the energy where the exit angle γ is the same as 2θ, i.e. the photon energy of an SCM). One can calculate more precise values for the limits. From the equations (10) and (11), one requirement is γ/2 ≤ 2θ. From this, the upper energy limit is given by
where d is the lattice spacing and is 2.0591 Å for a diamond (111) crystal. On a sidebounce beamline designed for 15 keV, a twisted DCM can provide photons up to 29.85 keV.
In this maximum energy, α becomes 90° and β becomes 180°. This situation leads to two crystals sitting in a dispersive condition and reflecting horizontally. When 2θ approaches 180°, the requirement changes to . And this condition gives a lower energy limit of
On a 15 keV sidebounce beamline, the minimum energy a twisted DCM can provide is 3.03 keV.
3.2. Secondcrystal position and offset
The position of the second crystal with respect to the first crystal changes with energy. It is given by
Here d is the distance between the two crystals. gives conventional offsets of crystal planes. n1 is the surface normal of the first crystal. Rq2 and n1 change with θ (photon energy) in a twisted DCM. In conventional DCMs, only n1 changes and one can fix the offset with the second crystal motion, thus d. However, the situation with a twisted DCM is different. The secondcrystal position becomes the exit point of the beam from the monochromator,
ΔD_{γ} is finite and not avoidable, but can be fixed to a constant ΔD_{γ}. Then, the d value should be varied with energy,
This also results in the changing conventional offset, . This forces a secondcrystal translation in the X direction, ΔX = ΔD_{X} − X_{0}. X_{0} is wherever the original X position of the second crystal is located.
The exit point in X and Z position can also be fixed. Here one has to deal with exit points projected to the Y–Z plane (incomingbeam plane, X = 0). At this Y–Z plane, X does not need to be counted. Only the Z position on the incomingbeam plane matters. The virtual exit point at X = 0 is
Fig. 4 shows a top view of the offsets and their projected geometry. To have the exit point fixed, the first crystal can be translated towards the source by the amount from equation (18). At the same time, the second crystal should be translated in the X direction. Table 2 shows the relative positions between two crystals and virtual Z2 positions for two selected energies.

3.3. Steering the beam
If the crystal translations are not desirable or when a fine movement of the beam is required, one can steer the beam by tweaking the angles α and β. The beammotion sensitivities to α and β were estimated by numerical calculation. It was assumed that the photon energy is 10 keV, and the exit angle is 23.13°. The beam positions 15 m from the second crystal were calculated for a 0.01° increase from ideal α and β. The results are shown in Table 3. The Z or X directions are only sensitive to β. Y is more sensitive to α, but β can shift the beam in the Y direction too. Operationally, one would change β first to a desired horizontal point, then adjust α to move in Y. At the same time, the tweaking mechanism ensures that the Bragg condition is maintained during the fine movement of the beam.

4. Ray tracing
Ray tracing was carried out on this new DCM geometry using the SHADOW package (Sanchez del Rio et al., 2011; Rebuffi & Sanchez del Rio, 2016; Rebuffi & Sanchez del Rio, 2017). The source simulation was based on a future superconducting undulator at the Advanced Photon Source after the multibendachromat upgrade (Borland et al., 2018). The superconducting undulator will have 18.5 mm × 70 periods. The monochromator is located 28.3 m from the source. To simplify the simulation, the distance between the two crystals was ignored. The crystal angles were manually set to the calculated values in Table 1 and finetuned to give the maximum intensity. For the first and third harmonics, the undulator is fixed at the same verticaldeflection parameter, Kv. Raytracing results (see Fig. 5 for 5 keV and 15 keV, the first and the third harmonics, respectively) verified that the above steering scheme for two reflecting crystals works as we proposed. It also produces a monochromatic beam with efficiencies comparable with the conventional DCMs. The total from the new DCM is 85% and 75% of that from a vertically deflecting DCM at 5 keV and 15 keV, respectively. On the other hand, these ratios become 370% and 82% compared with a horizontally deflecting DCM at 5 keV and 15 keV, respectively. The gain (or loss) at 5 keV is dominated by the polarization factor, which will be discussed in Section 5. At higher energies (e.g. 15 keV), the reduction is caused by the smaller angular bandwidth of the crystal and the dispersive geometry of the new DCM. The ray tracing was also performed for the second harmonic case. With finite slit beam acceptance, i.e. a slit size, the photonflux maximum appears at a slightly lower energy than the onaxis undulator equation gives. For the second harmonic, this is more pronounced and wide. At the same time, the second harmonic beam is not concentrated on axis but rather spread out, as shown in Fig. 6. One can collect more in the second harmonic tuned to a lower energy but the beam will spread out more in space and some of the photons will spill out of optical elements such as compound refractive lenses or mirrors. To simulate the in the second harmonic, a slightly lower Kv was selected. The majority of the beam with the second harmonic from Kv = 2.315 located mostly within a 1 mmdiameter circle and this ensured the full utilization of optical components with minimal aberration. Fig. 6(c) shows the focusedbeam profile at 43 m using a compound refractive lens located 29.8 m from the source. Table 4 lists the photon fluxes, beam sizes and bandwidth, which were extracted from the raytracing results.

5. Polarization factor and advantage over SCMs
The proposed DCM geometry introduces some polarization changes. However, the effect is not too drastic, but it has a far less severe consequence in intensity reduction for the SCM mode. It is assumed that the polarization (P0) of incoming rays is horizontal. The relation between polarization vectors and crystals is depicted in Fig. 7. Following the (AlsNielsen & McMorrow, 2001), at the first crystal (X1), the polarization factor becomes
Fig. 7 is a view from downstream. The first reflection rotates the polarization, and the angle between the first crystal and the polarization vector (P1) becomes α′,
At the second crystal (X2), the polarization factor becomes
At the second reflection, the polarization rotates and the angle between the resulting polarization vector (P2) and the second crystal becomes β′.
The final total polarization factor is
The total rotation of the polarization vector (counterclockwise) is
The resulting polarization factor is much more favorable than in the SCM case, especially at lower energies. Also, the polarization rotation angle is small. Table 5 shows polarization angles and factors for diamond (111) crystals at selected photon energies. P_{total} is close to unity for a twisted DCM. P_{SCM}, for an SCM, diminishes significantly as the photon energy becomes small.

6. Utilization of twistedgeometry DCMs
Twisted DCMs will provide new possibilities for sidebouncing beamlines. They can be operated over a wide range of photon energies and they can be extended to lowenergy regions without a large polarizationfactor loss. Twisted DCMs can also realize a fixedexit point by translating the first and second crystals. This is almost the same as conventional DCMs in which the second crystal is translated over quite a long stroke, and the first crystal is moved a small amount to maintain a fixedexit point. The widerange tunability is only useful when the experimental station has the full control of the source undulator. In some sidebounce beamlines, one undulator is shared by a few branch lines. This forces the undulator to be deflectionconstant fixed (or gap fixed) during operation. In this case, one can operate a twisted DCM for selected energies, i.e. many different harmonics. For example, if the fundamental energy is 5 keV, each beamline can choose any of the harmonics such as 5, 10, 15, 20, 25 keV, etc. Because of the tunability, one can use this monochromator for Xray absorption fine structure. However, it would be difficult to provide the required precise coupled crystal motions.
7. Summary
The proposed twisted DCM gives significant flexibility to sidebouncing beamlines. It will make a wide range of photon energies available to experimental stations. It also has a polarizationfactor advantage over SCMs at low energies. Therefore, sidebounce beamlines can now extend into the energy range where the polarization factor has prohibited the operation so far. This new monochromator design will be particularly interesting for the new synchrotron sources, which are more coherent and symmetric in source size and divergence. Because of the flexibility, there are more applications for the twisted DCM, which should be investigated in the future.
Acknowledgements
The authors are grateful to Dr G. B. Stephenson of Argonne National Laboratory for the discussion.
Funding information
Work performed at the Advanced Photon Source at the Argonne National Laboratory was supported by the US Department of Energy (DOE), Basic Energy Sciences, under Contract No. DEAC0206CH11357. The work of MH was supported by the US DOE, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division.
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