Here r0 is the classical electron radius, ν is the frequency of an incident photon and ν′=ν′(θ) is the frequency of the outgoing photon, scattered under the polar angle θ. The aforementioned anisotropy is given by the dependence on ϕ, being the azimuthal angle between the electrical field vector of the incident photon and the propagation direction of the scattered photon. This can be used to reconstruct the degree of linear polarization and the orientation of the polarization axis of a photon beam from the azimuthal emission pattern of Compton scattered photons, as first described by Metzger and Deutsch in 1950.[15] For an extensive overview of Compton polarimetry the reader is referred to the review by Lei, Dean, and Hills.[16]

During the last two decades, several segmented large-volume semiconductor detectors were developed as dedicated Compton polarimeters by the SPARC collaboration. One of these detectors is shown in Figure 1a. It consists of a lithium-diffused silicon crystal allowing an efficient use of the detector for photon energies up to 200 keV. The detector crystal (Figure 1b) has a thickness of 9 mm and an active area of 32 mm× 32 mm which is segmented on front and back side into 32 strips of 1 mm width each with the strips on front and backside being oriented perpendicular to each other. This segmentation of the crystal, leading to an array of 1024 pseudo-pixels, makes it into a high-resolution position-sensitive detector. Each of these strips has its own readout electronics with a cooled preamplification stage leading to an energy resolution of less than 1 keV at a photon energy of 60 keV.[14] The detector crystal serves as Compton scatterer and detector for the scattered photons at the same time. This “monolithic” polarimeter design is much more efficient compared to former polarimeter setups where individual detectors are used as a scatterer and an absorber and typically only a small fraction of the complete azimuthal scattering distribution is covered. If an incident photon is Compton scattered inside the detector crystal and the scattered photon is absorbed inside the crystal as well, information about time, energy and position of both interactions can be used to reconstruct the Compton event.[17] A typical reconstructed scattering profile can be seen in Figure 1c. Here the center of scattering was set to zero while the x–y profile shows the detected Compton scattered photons. From this reconstruction of the scattering events the azimuthal scattering profile, as shown in Figure 1d can be received in order to extract the degree of linear polarization and the orientation of the polarization vector.[18]

2.1 Example Application of the Compton Polarimeter: Polarization Measurements in Elastic Scattering of Hard X-Rays Applying the Compton polarimeter in experimental studies allows in-depth tests of the polarization-dependent features in fundamental processes. One such fundamental process in which polarization studies will allow for more rigorous testing is the elastic scattering of hard X-rays by atoms.[3] Depending on the scattering partner, one can differentiate into separate processes. The scattering by bound electrons is referred to as Rayleigh scattering,[19] while nuclear Thomson scattering[20] and nuclear resonance scattering[21] describe elastic scattering involving the nucleus as scattering partner. Additionally Delbrück scattering describes the scattering of photons by vacuum fluctuations in the presence of a heavy nucleus.[22] The total scattering amplitude A(θ) depends on the angle θ between incident and scattered photon and is given by the coherent sum of individual amplitudes for each process

A(θ)=A(R)(θ)+A(NT)(θ)+A(NR)(θ)+A(D)(θ)(2)

For the scattering by a closed-shell target atom, the absolute value of the scattering amplitude |A(θ)|2 is resolvable into the components |A(θ)⊥|2 and |A(θ)∥|2 describing the parts of the scattered radiation being polarized perpendicular and parallel to the scattering plane, as spanned by the directions of incident and outgoing photons. These quantities can be accessed by state-of-the-art calculations using the S-matrix approach.[3] The (squares of the) amplitudes A∥ and A⊥ can be employed to calculate all the properties of the scattering process. Most naturally this can be done within the framework of the density matrix approach. Since the detailed discussion of the density matrix of incident and scattered radiation has been presented in the literature before, for example, refs. [23, 24], here we restrict ourselves just to the main expressions. For example, the angle–differential cross section of the scattering of incident light with the degree of linear polarization Pl,i is given by

dσdΩ(θ,φ)=14|A∥θ|2+|A⊥θ|2+14Pl,i|A∥θ|2−|A⊥θ|2cos(2φ)(3)

with φ being the azimuthal scattering angle between the polarization plane of the incident beam and the scattering plane. The geometry of the described process of an initially linearly polarized photon beam being scattered on an atomic target explaining the used angles is shown in Figure 2.

The geometry of elastic scattering of a linearly polarized incident photon beam on an atomic target. The polarization plane of the incident beam is spanned by the wave vector k1 and the polarization vector ε1 of the incident beam. The scattered beam, spanning the scattering plane with the incident beam is defined by the polar scattering angle θ between the incident and scattered beam and the azimuthal scattering angle φ between the polarization plane of the incident beam and the scattering plane. The angle between the polarization vector of the scattered beam ε2 and the scattering plane is the polarization angle χ.

The scattered beam will then be polarized with a certain degree of linear polarization Pl,f under the polarization angle χ to the scattering plane. It makes sense to introduce the Stokes parameters P1,f and P2,f that are defined by the fractions of the intensities of the scattered beam polarized under different angles to the scattering plane

P1,f=I0−I90I0+I90=Pl,fcos2χP2,f=I45−I135I45+I135=Pl,fsin2χ(4)

Assuming no occurring circular polarization component, as is valid for most highly linearly polarized radiation sources, the Stokes parameters can be written in terms of the components of the scattering amplitude as[24]

P1,f=|A∥|2−|A⊥|2+Pl,icos2φ|A∥|2+|A⊥|2|A∥|2+|A⊥|2+Pl,icos2φ|A∥|2−|A⊥|2P2,f=2Pl,isin2φReA∥A⊥*|A∥|2+|A⊥|2+Pl,icos2φ|A∥|2−|A⊥|2(5)

For a complete characterization of the scattering process not only the scattering cross section is of interest but also the polarization characteristics need to be regarded. While the cross section measurement only gives the sum and difference of the absolute values of the scattering amplitudes, determining the Stokes parameters provides access to the real value of the (product of) scattering amplitudes. Thus a much more sensitive test of the theoretical models can be achieved.

For photon energies from a few keV to the MeV range Rayleigh scattering is the dominant component of elastic scattering while the other contributions being Delbrück scattering, nuclear Thomson scattering and giant dipole resonance scattering first become relevant at higher photon energies.[3] While polarization characteristics of Rayleigh scattering are theoretically studied in great detail,[24-27] for a long time polarized X-ray sources were not intense and polarimeters not efficient enough to perform an elastic scattering study in which both the polarization of the incident and the outgoing radiation are determined. Before the introduction of powerful sources of hard X-rays, such as the third-generation light-source PETRA III at DESY, incident polarized photon beams were produced from intense gamma sources whose radiation was polarized by scattering from a production target before it could be used in a scattering experiment as shown in Figure 1 of ref. [23]. The need to collimate the emitted photons to form a beam both at the point of initial emission from the gamma source as well as after the polarizer target results in low flux of the incident beam. Combined with the relatively low cross section of the elastic scattering process itself in combination with the need for the scattered photon to subsequently undergo Compton scattering in the polarimeter detector, the lack of sufficiently intense polarized X-ray sources restricted the study of Rayleigh scattering polarization features to scenarios in which either the angular-differential scattering cross section of a linearly polarized incident beam or the polarization of the scattered radiation resulting from an unpolarized incident beam was measured.[23]

With the advent of third-generation synchrotron radiation facilities, the generation of intense, highly-polarized hard X-rays became possible reaching up to energies of a few 100 keV. More, the development of efficient Compton polarimeters, as described above, enabled highly efficient polarimetry measurements for a broad energy regime from a few 10 keV to ≈300 keV. By combining the intense radiation emitted by a third-generation synchrotron source with detection by a highly efficient Compton polarimeter in scattering experiments, the underlying theoretical models of Rayleigh scattering can be tested with unprecedented accuracy.

Utilizing the High Energy Materials Science Beamline P07[28] of the third-generation synchrotron radiation facility PETRA III at DESY and a dedicated highly efficient Compton polarimeter, it was possible for the first time to apply a highly linearly polarized incident photon source (close to 100%) for the measurements of the linear polarization of elastically scattered hard X-rays.[12] The highly linearly polarized synchrotron beam with a photon energy of hν=175 keV was scattered on a thin gold target (see Figure 3 for an example spectrum of such a scattering experiment).

A typical scattering spectrum of hard X-rays with a photon energy of hν=175 keV being scattered on gold foil measured with the dedicated Compton polarimeter under a scattering angle θ=63∘. The spectrum shows the characteristic fluorescence lines Kα and Kβ of the gold target, the angle-dependent Compton peak of X-rays being Compton scattered within the gold target and the Rayleigh scattering peak which has the same energy as the incident beam. Only focusing on detected photons in an energy window around the Rayleigh peak, as shown by the red dashed vertical lines, the polarization features of the elastic scattering peak can be analyzed.

Using the dedicated Compton polarimeter the polarization of the elastically scattered radiation was then analyzed for scattering within the polarization plane of the incident synchrotron beam. For this purpose, only the detected photons in a narrow energy window around the elastic scattering peak were analyzed. The study showed a strong depolarization of the scattered beam, which is most pronounced for scattering angles of θ≈90∘, well in accordance with theoretical predictions when including a small depolarization of the incident synchrotron beam. The exact position of the strongest depolarization of the scattered beam strongly depends on the photon energy of the incident beam and is shifted toward lower polar scattering angles for higher photon energies due to relativistic effects. The depolarization of the scattered beam under a certain polar scattering angle θ depends on the degree of linear polarization of the incident beam and the atomic target. By comparing the measured depolarization of the scattered beam under certain scattering angles with the theoretical predictions for an incident synchrotron beam with a photon energy of 175 keV being scattered on gold atoms, the degree of linear polarization of the used beamline P07 of PETRA III could be determined to be Pl,i=98%.

For a full test of the polarization-dependent features of Rayleigh scattering one has to investigate the scattered radiation not only within the polarization plane of the incident beam but also for nonzero azimuthal scattering angles, φ≠0∘,[24, 27] since this setup enables to analyze the variation of both Stokes parameters P1,f and P2,f. An experiment on Rayleigh scattering for nonzero azimuthal scattering angles was recently performed at beamline P07 of PETRA III and the data analysis is to the moment of this publication still ongoing.

At higher photon energies of hν≳1 MeV also the other constituents of elastic scattering, in particular Delbrück scattering should be testable using this approach. So far however, due to the used detector crystal the detection efficiency of the Compton polarimeter is not sufficient above energies of 300 keV. Further upcoming improvements of the detector setup, as explained in the following chapter, will in future allow for efficient polarimetry for even higher photon energies up to the MeV range.