An X-ray beam that is stable in position, intensity and energy is a fundamental requirement for the performance of robust and reliable spectroscopic measurements. The variation of the beam position at the sample, fluctuations in the intensity of the radiation and the nonlinearity of the energy scale on the time scale of the experiment are factors that will affect the quality of the data collected.

Although beam drift is an important issue for all spectroscopic beamlines, the delivery of a stable beam is particularly critical in the case of the dispersive variant of the technique (Matsushita & Kaminaga, 1980). This is because it is currently not possible to collect a simultaneous incident-beam intensity measurement while the absorption spectrum is collected on the position-sensitive detector, and thus instabilities in the beam are not as effectively normalized (Pascarelli, Neisius & De Panfilis, 1999; Pascarelli, Neisius, De Panfilis et al., 1999). This means that extra care needs to be taken in the design and implementation of energy-dispersive spectrometers to mitigate the impact of beam drift on the measurements.

1.1. Sources of beam drifts

The drifts observed in the X-ray beam at the sample position have many different origins, but in general can be divided into two main categories: (i) drifts originating at the source and (ii) drifts due to instabilities in the optical components of the beamlines.

1.1.1. Electron-orbit stability

Variations in the electron orbit in the storage ring affect the stability of the beam delivered to the sample. This issue was a very significant factor in the case of second-generation sources, but the advent of third-generation sources considerably improved the situation. On modern synchrotrons the movement of the electron beam in the storage ring should no longer be a significant factor affecting the stability of the X-ray beam at the sample position. State-of-the-art synchrotron sources are nowadays able to achieve submicrometre position (and pointing angle) stability of the electron beam in the vertical plane on timescales from milliseconds to hours, with fluctuations at a level lower than 10% of the root-mean-square beam size (and divergence) (Böge et al., 2002; Decker, 2005; Singh & Decker, 2005; Hubert et al., 2011; Willeke, 2015; Rehm, 2013). This has been achieved thanks to the implementation of stringent engineering practices, such as solid foundations, quiet accelerator vaults, very good temperature control and the use of electron beam-position monitors (EBPMs). The use of global fast and slow orbit feedbacks (Schilcher et al., 2004) also contributes to the stability of the electron beam on timescales from 20 ms to 10 s, while additional stabilization on longer timescales (of up to a week) can be achieved using X-ray beam-position monitors in a feedback loop (Holldack et al., 2001; Singh & Decker, 2001). The stability achieved on long timescales is ultimately limited by the stability of the synchrotron floor.

1.1.2. Beamline instrumentation stability

Movement of the optical elements in the beamlines can cause drifts of the X-ray beam at the sample position. A number of factors are expected to destabilize the optical elements, including mechanical instabilities, varying heat load on the optical components due to the X-ray beam and changes in environmental conditions.

Mechanical instabilities such as motors drifting as a function of time compromise the long-term positional stability of mirrors, monochromator crystals, slits etc. and affect the stability of the X-ray beam (Tucoulou et al., 2008). For example, a drift of the pitch motor of a vertical focusing mirror will create a vertical movement of the beam at the sample. These sorts of effects are nowadays less common, mainly due to good engineering practices but also due to the use of very precise encoders that are able to maintain the position of the optical elements within the required values for long periods of time. Special attention should be paid to the mechanical instabilities that appear when an optical element is scanned. As an example, in the case of a double-crystal monochromator using independent crystals it is essential that the relative angle between the first and second crystals is maintained over the range of the scan. Otherwise, instabilities in the energy, intensity and position of the exit beam will be detected at the sample position. In these cases, the use of feedback mechanisms in intensity and/or position is essential to maintain the stability of the beam. The most commonly used feedback mechanisms will be discussed in the following section.

Control of the temperature stability of the environment in which the optical components are located is also essential to eliminate drift of the X-ray beam at the sample position. Small variations in the temperature of the hutch can readily be translated into movements of the X-ray beam because the frequently used steel vessels that house the optics will expand and contract as a function of ambient temperature (Tucoulou et al., 2008). As an illustration, Fig. 1 shows the correlation between the vertical beam position at the sample and the temperature of the hutch in which the vertical focusing mirror of the beamline is located, 15 m downstream. Nowadays, most instruments achieve good temperature control using state-of-the-art air-handling units that circulate air throughout the room. It is important to highlight that the circulation of air can also introduce noise into the measurements when X-ray beams of nanometre size are used. This is a particular challenge for nanofocus beamlines, and taking the I14 beamline at Diamond Light Source as an example, it was necessary to implement an alternative technology for temperature control based on the use of radiant cooling panels. This technology is frequently used in electron microscopy (Muller et al., 2006).

Figure 1 Correlation between the temperature of the optics hutch and the vertical position of the X-ray beam at the sample point. The data were obtained on beamline I20-EDE at Diamond Light Source before the temperature control of the optics hutch was optimized. The discontinuity at 21 h was caused by a beam loss in the synchrotron, and the vertical displacement of the X-ray beam at the sample position after the beam was recovered is due to the slow stabilization of the white-beam focusing mirror.

The third more common cause of beam drift on the beamline is a change in the heat load on the optical components. The cooling of the optical elements themselves is usually achieved using efficient water cooling. This is the case for most slits, mirrors, attenuators etc. In addition, for higher power components such as monochromators installed on undulator- or wiggler-based beamlines, liquid-nitrogen cooling is generally adopted. Stabilization problems do however continue to arise as the mechanics that hold the optical elements in place are usually not as well cooled as the primary optical elements themselves. The cooling of the supporting mechanical components also generally takes place under ultrahigh-vacuum conditions, where dissipation of heat by convection is not possible. Thus, unless great care is taken, changes in the heat load on the optical elements can be translated into changes in temperature of the mechanics, thus leading to instabilities in the X-ray beam. Improvements in the shielding of the mechanical structure of the optical components and in their cooling are remediation techniques that are increasingly being implemented in the design of modern components for X-ray beamlines. Effort has also been directed towards making the heat load on the optical components independent of the ring current. This issue has been targeted in different ways; for example, Filipponi et al. (2000) developed a feedback system on the horizontal gap of the primary beamline slits to stabilize the average beam power delivered to the first crystal of a monochromator. The size of the slit opening was correlated to the electron beam current in the ring such that as the storage-ring current decayed, the horizontal slits opened. Modern light sources now aim to maintain a constant heat load on the beamline optics by periodically injecting a small amount of current into the storage ring to maintain the overall operating current. This is known as the top-up operating mode. However, it is important to highlight that when using insertion devices working with a variable gap as X-ray sources, the heat load on the optical elements will change. In the case of a gap-scanning undulator, the heat load will change during the energy scan. In these cases, efficient cooling of the optical components and the support mechanisms is particularly critical.

1.2. Type of beam drifts

As mentioned above, the X-ray beam can drift in position, in intensity or in energy as a function of time, and the type of drift that can be found on a typical X-ray absorption spectroscopy beamline depends not only on the origin of the drift but also on the optical components that compose the beamline.

Vertical movements of the source are usually translated into drifts in the focal spot position, as is the case when the beamline is equipped with a double-crystal monochromator device. In this case, the beam not only drifts in position but the energy scale also varies, although the intensity remains constant. However, the same vertical movement of the source would only be translated into a drift in intensity if the beamline made use of a four-bounce monochromator, as is the case for the I20-Scanning beamline at Diamond Light Source (Diaz-Moreno et al., 2009). On these devices the dispersive configuration of the third and fourth crystals prevents movement of the X-ray beam downstream of the monochromator and maintains the energy scale within the angular acceptance of the crystal (Heald, 1988; Kraft et al., 1996; Tolentino et al., 1995). On the other hand, if there is an angular movement of the source, only the intensity is affected regardless of the type of monochromator that the beamline is equipped with. Fig. 2 shows the variation in intensity of the X-ray beam after the double bounce and quad bounce of a four-crystal monochromator when there is a vertical and an angular movement of the X-ray source, in this case a wiggler.

Figure 2 Changes in the intensity of the X-ray beam after the first two crystals (a) and after four bounces (b) of a four-crystal monochromator when there is a vertical movement of the X-ray source. (c, d) As previously, (c) for two bounces and (d) for four crystals, when the movement of the source is angular. The data were obtained on the I20-Scanning beamline at Diamond Light Source.

As we have mentioned above, the drifts of the X-ray beam at modern light sources are typically due to movements of the optical components in the beamline, primary mirrors and monochromators. A slight change in the incident angle of a focusing mirror will be translated into a movement of the beam at the sample position on a typical scanning X-ray spectroscopy beamline. If the mirror is located before a double-crystal monochromator, changes in the energy scale will also be expected unless a four-bounce monochromator is used. In the second case the movement of the mirror will only affect the intensity transmitted by the monochromator. Drifts in the position of the beam at the sample can also be caused by changes of the angle between the two crystals of a double-crystal monochromator, although in this case the intensity will also drift.

It is known that when performing XAS experiments one of the major sources of angular beam drifts occurs during the process of scanning the energy by rotating the fixed-exit double-crystal monochromator. This not only affects the position of the beam but also the intensity.

In the case of Quick-EXAFS monochromators, the height of the exit X-ray beam will change when scanning the energy (Richwin et al., 2001). Although originally this movement was compensated by moving the vertical position of the sample (Frahm, 1988), nowadays the movement is absorbed by the focusing optics placed after the monochromator. This is the strategy adopted both at the Swiss Light Source (Müller et al., 2016) and SOLEIL (Fonda et al., 2012; Briois et al., 2016).

Nowadays, most advanced spectroscopic beamlines correct for beam drifts by using feedback mechanisms based on measuring the intensity or determining the position of the X-ray beam close to the sample position.

The most widely used method to stabilize the beam consists of maintaining its intensity after the monochromator constant by adjusting the parallelism of the two crystals when using a typical fixed-exit double-crystal monochromator. The beam intensity transmitted throughout the monochromator, measured using ion chambers or diodes, is used in a feedback loop to maintain a constant angle between the two crystals (Golovchenko et al., 1981; Greaves et al., 1983). This approach is most frequently used to stabilize the intensity at either side of the rocking curve, where it is relatively simple to create an appropriate feedback-control signal that indicates the way in which and by how much one of the crystals has to move with respect to the other to keep the angle between them constant. Alternatively, stabilization of the X-ray beam intensity can be achieved using extremum-seeking feedback control (ESFC; Krstić & Wang, 2000), allowing the parallelism of the crystals to be maintained at the top of the rocking curve. This is achieved by applying a continuous modulation to the position of the actuator that controls the angular position of one of the crystals in the monochromator. The modulation technique has been used to control the intensity stability of monochromators at several synchrotron-radiation facilities (Krolzig et al., 1984; Mills & Pollock, 1980; Kudo & Tanida, 2007; van Silfhout et al., 2014; Bloomer et al., 2013; Fischetti et al., 2004; Van Mellaert & Schwuttke, 1970), achieving an angular stability of the optical elements of the level of approximately 13 nrad (Stoupin et al., 2010).

Feedback mechanisms are also frequently applied to stabilize the X-ray beam position. The position of the X-ray beam, typically measured using X-ray beam-position monitors (van Silfhout et al., 2011; Sato, 2001; Hignette et al., 2007; Bunk et al., 2005; Xu et al., 2007), is used to adjust the position of the second crystal of the monochromator (Mosselmans et al., 2009) or to steer one of the optical mirrors on the beamline (Zohar et al., 2016; van Silfhout et al., 2014). In the latter case, the stability of the X-ray beam is simultaneously achieved by adjusting the parallelism between the crystals of the monochromator. Van Silfhout et al. (2014) used ESFC to stabilize the intensity, whilst a simple feedback-control loop acting on the position of the focusing mirror placed downstream of the monochromator was used to stabilize the beam in position. A similar concept was used by Zohar et al. (2016). These authors used ESFC to stabilize the X-ray beam intensity by adjusting the angle of the second crystal of the monochromator, and the same feedback strategy was used to stabilize the position by acting on the angle of the vertical focusing mirror placed downstream of the monochromator.

An alternative method to correct for drifts of the X-ray beam is to adjust the sample position. This is the method that was used by van Silfhout (1998) to ensure that the beam was aligned with the sample for long periods of time. The system uses two quadrant photodiodes that measure the position and the angle of the X-ray beam, moving the experimental table to correct for any drift of the X-ray beam.

The approaches described above work well in stabilizing the beam position at most modern beamlines. In the case of beamlines that use X-ray beams of nanometre size, the use of interferometers is becoming routine (Nazaretski et al., 2015). The interferometers monitor the position of the nanofocusing X-ray optics (zone plates or multilayer Laue lenses) and of the stages, and their signals are used in an active feedback-control loop to achieve the required stability (Gofron et al., 2014).

X-rays are transverse electromagnetic waves and therefore the general description of polarized light may be applied. See Detlefs et al. (2012) for a review of the general formalism of polarization and Hart (1978) for a discussion of aspects of polarization related to the dynamical diffraction theory. Hence, the polarization of a photon travelling in the z direction can be expressed on a linear polarization basis by the x and y components of its Jones vector (V1 V2). It is often more convenient to use the Stokes parameters (S0 S1 S2 S3) since this formalism can deal with the degree of coherence between two orthogonal polarization axes (Born & Wolf, 2002; Goldstein, 2011).

In a storage ring, the trajectory of the electron is curved by the magnetic fields that induce acceleration of the electrons radially in the orbit plane. Due to the conservation of angular momentum, the emitted radiation is mostly linearly polarized in the trajectory plane. In contrast, due to the finite projection of the angular momentum L along the X-ray propagation direction, the radiation observed at a finite angle ψ above or below this plane is elliptically polarized. As shown in Fig. 3(a), the polarization has opposite helicities above (anticlockwise, left-handed; Lz = −ħ) and below (clockwise, right-handed; Lz = +ħ) the orbit (Stöhr & Siegmann, 2006). (The sign convention followed here for the helicity refers to the temporal motion of the electric field E about the propagation direction k when looking at the source.) The aforementioned polarization scheme is only valid for bending magnets. For planar undulators and multipole wigglers (with an even number of poles) the radiation is also horizontally polarized when in the orbit plane. Away from the horizontal plane the undulator and wiggler behave differently. For wigglers, the incoherent sum of the left and right circular radiation is equal to zero and, in contrast to a bending magnet, the radiation is not elliptically polarized but naturally polarized. For a planar undulator, the radiation is always linearly polarized regardless of the observation angle, but the inclination of the polarization depends upon the harmonic number and the deflection parameter K as well as the horizontal and vertical observation angles (Clarke, 2004; Hofmann, 2004).

Figure 3 (a) Polarization of synchrotron radiation generated from a bending-magnet source. (b) Normalized flux of the horizontal (σ) and vertical (π) polarization components as a function of the vertical observation angle for different energies. (c) The same as (b) but for right (R) and left (L) circular polarization components. (d) The linear and circular polarization rate and the normalized flux as a function of the vertical observation angle.

The angular spectral power distribution for the two polarization modes (σ, in plane, and π, out of plane) of a bending magnet can be calculated via the Liénard–Wiechert equation (Liénard, 1898; Wiechert, 1900). This equation describes the retarded fields radiated by a charged particle moving with an arbitrary velocity and therefore can be used to calculate the emitted acceleration fields of the synchrotron radiation. A detailed derivation of the field equations as a function of the Airy integrals or the modified Bessel functions of the second kind can be found in Hofmann (2004) and Duke (2000). As shown in Fig. 3(b) for selected values of ɛ/ɛ_C (where ɛ is the photon energy and ɛ_C is the critical energy of the bending magnet), the spatial distribution of the σ-mode is directed mainly in the forward direction, while the π-mode radiation is emitted into two lobes at finite vertical observation angles ψ and zero intensity in the forward direction. Since the distribution of the π-mode radiation overlaps with the σ-mode, the radiation emitted by a bending magnet cannot provide pure vertically polarized X-rays. It is also clearly shown that for energies close to ɛ_C the approximation that the radiation is emitted with a vertical angle of ±1/γ is a good approximation. Upon decreasing the energy, the angular distribution of both modes expands in a similar fashion towards larger ψ angles and the amplitude of the π component gains strength over the σ component. Since the phase difference between the σ and π components is π/2, the polarization state can be equivalently decomposed into two modes of circular polarization of opposite helicity: positive (right-handed) and negative (left-handed). In this case, both angular distributions have the same amplitude for a fixed energy and they do not fully overlap as occurs with the linear modes (see Fig. 3c). Thus, further away from the horizontal plane the degree of circularly polarized light tends to 1 but, as shown in Fig. 3(d), the flux decreases dramatically as ψ increases. Therefore, due to this trade-off between flux and polarization rate, it is necessary to optimize ψ in order to maximize the figure of merit Pc²I for those experiments requiring circularly polarized X-rays at bending magnets (Funk et al., 2005).