Julien Roth

Rigidity Results for Geodesic Spheres in Space Forms

Keywords: Hypersurfaces, Space Forms, Rigidity, Pinching

We prove that a hypersurface of a space form with almost constant mean curvature and almost constant scalar curvature is close to a geodesic sphere. In the case of Euclidean space, we deduce new characterizations of geodesic spheres.

Introduction

The well-Known Alexandrov theorem ( 1 ) claims that any compact without boundary hypersurface embedded into the Euclidean space R n+1 with constant mean curvature (CMC) is a geodesic sphere. Later, A. Ros ( 2 ) proved that the result also holds for hypersurfaces of the hyperbolic space H n+1 and the open half sphere S n+1 + . In these results, the assumption that the hypersurface is embedded is crucial. Indeed, the results are false for immersed hypersurfaces. For instance, the so-called Wente's torus (see 3 ) is an example of (non-embedded) immersed surface in R 3 with constant mean curvature which is not a geodesic sphere. Other examples of higher genus are known ( 4 ).

For surfaces in R 3 , Hopf ( 5 ) proved that CMC immersed spheres are geodesic spheres. Here again, the result is not true in general since, we know examples of CMC spheres in higher dimension which are not geodesic spheres (see 6 ).

The goal of this article is to find an alternating assumption to the embedding such that under this assumption, CMC hypersurfaces are geodesic spheres. It is a well-known fact that if the mean curvature H and the scalar curvature Scal are both constant, then the hypersurface is a geodesic sphere. We will show a stability result associated to this assertion. Namely, we have the following result which was proved for δ = 0 in ( 7 ). The cases δ > 0 and δ < 0 are new. Theorem 1.1. Let (M n , g) be a compact without boundary, connected and oriented Riemannian manifold, isometrically immersed into the simply connected space form M n+1 δ of constant sectional curvature δ. Let R be the extrinsic radius of M . If δ > 0, we assume that R < π

4 √ δ . Let h > 0 and θ ∈]0, 1[. Then, there exists ε(n, h, R, θ) > 0 such that if • |H -h| < ε and • |Scal -s| < ε for a constant s, then h 2 -s n(n-1) + δ
Aε, for a positive constant A depending on n, h, δ and R, and M is diffeomorphic and θ-almost isometric to a geodesic sphere in M n+1 δ of radius t -1 δ 1 h , where t δ is the function defined in Section 2.

Remark 1.1. By θ-almost isometric, we mean that there exists a diffeomoprhism F from M into a geodesic sphere of appropriate radius such that Then, from this stability result, we will deduce a new characterization of geodesic spheres in R n+1 with a weaker assumption on the scalar curvature (see Section 4).

|dF x (u)| 2 -1 ≤ θ,

Preliminaries

Let (M n , g) be a n-dimensional compact, connected, oriented Riemannian manifold without boundary, isometrically immersed into the (n + 1)dimensional Euclidean space (R n+1 , can). The (real-valued) second fundamental form B of the immersion is the bilinear symmetric form on Γ(T M ) defined for two vector fields X, Y by

B(X, Y ) = -g ∇ X ν, Y ,
where ∇ is the Riemannian connection on R n+1 and ν a normal unit vector field on M . When M is embedded, we choose ν as the inner normal field.

From B, we can define the mean curvature,

H = 1 n tr (B). Now, we recall the Gauss formula. For X, Y, Z, W ∈ Γ(T M ), R(X, Y, Z, W ) = R(X, Y, Z, W )+ AX, Z AY, W -AY, Z AX, W (1)
where R and R are respectively the curvature tensor of M and M n+1 δ , and A is the Weingarten operator defined by AX = -∇ X ν.

By taking the trace and for W = Y , we get

Ric(Y ) = Ric(Y ) -R(ν, Y, ν, Y ) + nH AY, Y -A 2 Y, Y (2)
Since, the ambient space is of constant sectional curvature δ, by taking the trace a seconde time, we have

Scal = n(n -1)δ + n 2 H 2 -|A| 2 , (3)
or equivalently

Scal = n(n -1) H 2 + δ -|τ | 2 , (4)
where τ = B -HId is the umbilicity tensor. Now, we define the higher order mean curvatures, for k ∈ {1, • • • , n}, by

H k = 1 n k σ k (κ 1 , • • • , κ n ),
where σ k is the k-th elementary symmetric polynomial and κ 1 , • • • , κ n are the principal curvatures of the immersion.

From the definition, it is obvious that H 1 is the mean curvature H. We also remark from the Gauss formula (1) that

H 2 = 1 n(n -1) Scal -δ. (5)
On the other hand, we have the well-known Hsiung-Minkowski formula

M H k+1 Z, ν + c δ (r)H k = 0, (6)
where r(x) = d(p 0 , x) is the distance function to a base point p 0 , Z is the position vector defined by Z = s δ (r)∇r, and the functions c δ and s δ are defined by

s δ (t) =    cos( √ δt) if δ > 0 1 if δ = 0 cosh( √ -δt) if δ < 0 and s δ (t) =      1 √ δ sin( √ δt) if δ > 0 t if δ = 0 1 √ -δ sinh( √ -δt) if δ < 0
Finally, we define the function t δ = s δ c δ . We finish this section of preliminaries by the following recalls about the first eigenvalue of the Laplacian. In ( 8), Heintze proved the following upper bound for λ 1 (∆)

λ 1 (∆) n(||H|| 2 ∞ + δ), (7)
with equality for geodesic sphere. Grosjean ( 9) proved a stability result associated to Heintze's estimate. Precisely, he proved the following Theorem 2.1 (Grosjean, 2007). Let M be compact without boundary, connected and oriented hypersurface of

M n+1 δ . If δ > 0, we assume that M is contained in an open ball of M n+1 δ of radius π 4 √ δ . Let θ ∈]0, 1[, then there exitst a constant C θ (n, ||B|| ∞ , V (M ), δ) > 0 such that if n(||H|| 2 ∞ + δ) -C θ < λ 1 (∆),
then M is diffeomorphic and θ-almost-isometric to a geodesic sphere of radius n λ1 .

Now, we have all the ingredients to prove Theorem 1.1.

Proof of Theorem 1.1

We begin the proof of Theorem 1.1 by the following lemma.

Lemma 3.1. Let h and s be two positive constants. If the mean curvature and the scalar curvature satisfy

• |H -h| < ε and • |Scal -s| < ε,
for some positive ε, then

h 2 - s n(n -1) + δ Aε,
where A is a positive constante depending on h, R, n and δ.

Proof. The proof of this lemma comes directly from the Hisung-Minkowski formula (6). Indeed, the Hisung-Minkowski formula for k = 1 is the following

M H 2 Z, ν + c δ (r)H = 0. ( 8
)
Since we assume that |Scal -s| < ε, we get easily from ( 5) that

H 2 - s n(n -1) -δ < 1 n(n -1) ε. (9)
For more convenience, we will denote h 2 = s n(n-1) -δ. Then, from ( 8)

0 = M H 2 Z, ν + c δ (r)H = M h 2 Z, ν + c δ (r)H + M (H 2 -h 2 ) Z, ν = h 2 h M h Z, ν + M c δ (r)H + M (H 2 -h 2 ) Z, ν = h 2 h M H Z, ν + h 2 h M (h -H) Z, ν + M c δ (r)h + M c δ (r)(H -h) + M (H 2 -h 2 ) Z, ν
Now, we use the Hsiung-Minkowski formula for k = 1, that is

M H Z, ν + c δ (r) = 0, (10)
to get

0 = - h 2 h M c δ (r) + h 2 h M (h -H) Z, ν + M c δ (r)h + M c δ (r)(H -h) + M (H 2 -h 2 ) Z, ν = h - h 2 h M c δ (r) + h 2 h M (h -H) Z, ν + M c δ (r)(H -h) + M (H 2 -h 2 ) Z, ν
Then, since s δ is an increasing function, we deduce

h - h 2 h M c δ (r) h 2 h ε M s δ (R) + ε M c δ (r) + ε n(n -1) M s δ (R)
Using the fact that |H 2 | H 2 , we deduce from the assumptions on h and h 2 that

|h 2 | h 2 + A 1 (n, h, δ)ε,
and then we have

h - h 2 h M c δ (r) ε M c δ (r) + A 2 (n, h, δ) M s δ (R)
We conclude using the fact that V(M) M c δ (r)

A 3 (n, δ, R)V(M), which yields |h 2 -h 2 | A 4 (n, h, R, δ)ε,
which gives the wanted assertion.

Let 0 < ε < 1. We will precise this ε later. We assume |H -h| < ε and |Scal -s| < ε, then from (3) and Lemma 3.1, we have

|τ | 2 = n(n -1)H 2 -Scal (11) A (n, h, R, δ)ε.
This means that M is almost umbilical. Moreover, since |H -h| ε, Inequality ( 11) is equivalent to

|B -hId | A (n, h, R, δ)ε. (12)
This last inequality combining with the Gauss formula (2) yields

Ric(Y ) -(n -1)(h 2 + δ)|Y | 2 A (n, h, R, δ)ε. (13)
Now, we use the Lichnerowicz formula ( 10) to obtain the following lower bound of the first eigenvalue of the Laplacian on M

λ 1 (∆) n(h 2 + δ) -α 1 (ε), (14)
or equivalently

λ 1 (∆) n(||H|| 2 ∞ + δ) -α 2 (ε), (15)
where the positive fonctions α 1 and α 2 depend on n, h, R and δ and tend to 0 when ε tends to 0. Now, since we fix θ ∈]0, 1[. Since α 2 tends to 0 when ε tends to 0, there exists ε 1 (n, h, R, δ) > 0, such that α 2 (ε 1 ) θ 2 . Then, we use Theorem 2.1 to conclude that M is diffeomorphic and θ 2 -quasi-isometric to a geodesic sphere of radius n λ1 . Moreover, because of the pinching of λ 1 (∆), the raddi 1 h and n λ1 are close. So, there exists ε 2 (n, h, R, δ) > 0 such that geodesic spheres of radii 1 h and n λ1 are θ 2 -quasi-isometric. Then, we take ε = inf {ε 1 , ε 2 } and then M is θ-quasi-isometric to a geodesic sphere of radius 1 h .

Rigidity results in the Euclidean space

For the Euclidean case, that is δ = 0, we can obtain form Theorem Proof. This corollary is a direct consequence of Theorem 1.1. Indeed, we know that there exists a diffeomorphism F from M to S n 1 h . But, in the Euclidean space, this diffeomorphism is explicit. Namely,

F (x) = 1 h φ(x) |φ(x)| ,
where φ is the immersion on M into R n+1 . See ( 11) to get this expression. The fact that F is a diffeomorphism implies that the immersion φ is necessarly a one-to-one map, that is an embedding. Since M is embedded with constant mean curvature, it is a geodesic sphere by the Alexandrov theorem. Now, we state a second characterization of geodesic spheres. Corollary 4.2. Let (M n , g) be a compact, connected and oriented Riemannian manifold without boundary isometrically immersed into R n+1 . Let s be a positive constant. Then, there exists ε(n, s) > 0 such that if (1) Scal = s (2) |H -h| ε, for some constant h, then M is a geodesic sphere of radius n(n-1) s .

Proof. The proof is the same, using an Alexandrov type theorem for the scalar curvature due to Ros ( 2 ).

This second corollary gives a partial answer to a conjecture by Yau which states that the only immersed hypersurfaces of the Euclidean space with constant scalar curvature are geodesic spheres.

Remark 1 . 2 .

 12 for any x ∈ M and any unit vector u ∈ T x M . The extrinsic radius of M is the radius of the smallest closed ball in M n+1 δ containing M .

  Note that in this result, ε does not depend on the extrinsic radius R. Indeed, since the mean curvature is constant, from (10), we have

	1.1 new
	characterizations of geodesic spheres. Namely, we have the following
	Corollary 4.1. Let (M n , g) be a compact, connected and oriented Rieman-
	nian manifold without boundary isometrically immersed into R n+1 . Let h
	be a positive constant. Then, there exists ε(n, h) > 0 such that if
	(1) H = h and
	(2) |Scal -s| ε,
	for some constant s, then M is a geodesic sphere of radius 1 h .
	Remark 4.1.

M Z, ν = 1 h V(M). So we do not have to control the extrinsic radius.

Acknowledgments

The author would like to thank the organizers of the VIII International Colloquium in Differential Geometry in Santiago de Compostela.


Sylvain Lefebvre

Li-Yi Wei

Connelly Barnes

Informational Texture Synthesis

come

Introduction

This document is a work in progress and to keep it short and to the point we skip introductory matters and focus instead of the main principle.

The synthesis algorithm described here follows an implementation similar to the wave function collapse (WFC) algorithm https://github.com/mxgmn/ WaveFunctionCollapse, which builds upon the model synthesis algorithm [START_REF] Merrell | Example-based model synthesis[END_REF]. We strongly encourage the interested reader to first get familiar with these methods. In particular [START_REF] Merrell | Example-based model synthesis[END_REF] introduces the key idea of constraint propagation for discarding labels, which is central to the synthesis algorithm and not explained in this document. 1

Previous work

There are a number of previous work considering embedding information in synthesized textures. These works focus on 'stochastic' textures, which content is not globally organized (e.g. images of grass, pebbles, concrete, carpets). [START_REF] Otori | Data-embeddable texture synthesis[END_REF] proposes to encode data in synthesized textures by constraining the color of pixels spread with a specific pattern. Synthesis -inpainting -is then performed to fill-in gaps around these points. [START_REF] Kuo | Steganography using reversible texture synthesis[END_REF] encodes both the message and the exemplar into the final image. The exemplar is stored in pre-positioned square patches which locations are produced from a secret key. Synthesis is performed around these using image quilting [START_REF] Efros | Image quilting for texture synthesis and transfer[END_REF]. Whenever a patch can be added, multiple candidates are considered. The candidates are ranked by how well they fit neighboring patches in the overlap region, and the rank of the selected candidate is used to encode data. [START_REF] Qian | Robust steganography using texture synthesis[END_REF] also relies on image quilting for synthesis around pre-positioned square patches which locations are produced by a secret key. The choice of which patch appears at each location encodes data. The authors note that approaches using patches are more robust to e.g. JPEG compression than those using pixels.

These approaches focus on synthesizing unstructured, stochastic patterns. Our technique synthesizes a different type of texture patterns which are described by tiles (square images) with assembly constraints [START_REF] Merrell | Example-based model synthesis[END_REF]. Instead of predetermining the location of a set of tiles, we hide the data into the pseudo random sequence of choices encountered during synthesis.

Encoding

In the remainder it is to be understood that 'random' means 'pseudo-random using a sequential, deterministic generator'. Therefore, any random sequence produced during the algorithm can be exactly reproduced using the same starting seed.

The core principle is to layout square tiles to form a large texture. Each tile can be seen as a label (the tile id). Tiles cannot be freely placed: constraints define how tiles can be neighboring (e.g. tiles A and B may only be placed with A to the left of B). This lets artists paint pattern pieces onto the tiles which, once assembled while enforcing all constraints form a consistent pattern, e.g. a network of tubes.

Our variant of the algorithm goes as follows:

1. Initialize an array S of size W ×H, where each entry contains a vector of T booleans, with T the number of possible tiles. All vectors are initialized to true. Each boolean represents the possibility that a particular tile appears at a particular location. Initially everything is possible.

2. Visit each location of S in some fixed order. For instance, one can use scanline ordering (left, right then top, bottom).

3. At each location, consider the vector of possibilities. If a choice exists (e.g. more than one tile is possible), we have an opportunity to embed information. We call such a location along the sequence a coding site.

For now let us assume a naive approach which selects the first choice to represent '0' and any other choice to represent '1'. Once the choice is made, set all other entries in the vector to f alse.

4. Propagate the constraints to cancel impossible choices in neighboring cells. This step is unchanged compared to the original algorithm [START_REF] Merrell | Example-based model synthesis[END_REF].

5. Repeat until all cells of S have been visited. After this, a choice has been made everywhere and we can produce the final image by assembling the tiles following S.

This process can encounter a contradiction at step 3: in such cases no possibilities exist in the visited cell, due the propagation of constraints from previously selected tiles. If this happens, the algorithm given above would be stuck: the same message always produces the same choices. We discuss in Section 4.2 how to introduce randomness in the process such that the algorithm can be launched again and make different choices to recover from a contradiction.

Encoding information in choices. The initial proposal of encoding '0' in the first choice and '1' in any other is not a good strategy. First, it is inefficient as multiple choices provide an opportunity to encode more than just a single bit: intuitively, two choices provide one bit, four provide two bits, however an improved scheme will also exploit non power of two cases. We discuss a better approach in Section 4.1. Second, this would bias the algorithm towards always making the same choices. Consider in particular the empty message, represented by a zero bit stream. Selecting consistently the first choice will strongly bias the synthesis process and could quickly run in contradictions that could never be resolved. Instead we randomize which symbol in the stream represents which choice, using a random offset at each coding site. This is discussed in Section 4.2.

Data as a large integer

To encode data efficiently, we take an arithmetic point of view. We transform the message into a large integer. Given a message D of length L such that D[i] is the i-th byte, we compute the large integer

N = ∑ L-1 i=0 D[i] • 2 8•i .
Let us consider the sequence of number of choices in each coding site, denoted as B 0 , B 1 , ..., B i . We encode N onto the varying radix basis formed by the B i . We define N 0 = N . At a coding site j we have B j choices. We choose the symbol as s j = (N j mod B j ) and update N as N j+1 = N j /B j , where all operations are carried with large integers.

Reintroducing randomness

If applied directly, this encoding will introduce strong bias when the message is empty (N=0), which always happens after the last bit of information is encoded. This will degrade synthesis quality, and possibly lead to contradictions.

In addition, the process would be fully determined by the message itself. This provides no randomness: if a contradiction is encountered there would be no possibility to re-run the process.

We introduce randomness by randomly offsetting the symbol encoded at each site: we encode s j = ((N j + P RN G(j)) mod B j ) where P RN G is a pseudo-random number generator.

Changing the seed allows the algorithm to produce a different set of choices, thus affording to explore other potential solutions and attempt to resolve contradictions. The generator seed can also serve as a password, since without it the message cannot be decoded easily (see also Section 7).

Decoding

We assume that the decoder has access to the tileset used for synthesis. The decoding algorithm starts by reconstructing the labels from the input image. This can be done by simple image comparison if the image is unaltered, or with a more elaborate algorithm if the input has been modified (e.g. image compression, noise, etc.). Ultimately we would like to extract the data from a photograph of the texture. We describe in Section 5.1 a scheme robust to lossy compression and noise.

After obtaining the labels, the decoder runs synthesis again. However, when a choice is possible, it uses the input to determine which choice was made during the initial synthesis. This decodes one symbol of the message. At the end the bit stream is fully recovered in the form of a large integer, which is then converted back to a base of radix 256 to recover the bytes of the encoded message D.

If at some point the decoder finds a discrepancy (a set of choices that does not contain the tile selected in the input), then it can conclude that synthesis failed and was re-run. It generates the next seed for the random generator and restarts. Note that this process converges only if the correct initial seed is used, otherwise there is no guarantee this process even terminates.

Robust label extraction

The first step of decoding is, given an image, to reconstruct which tile (label) was selected in each location. Throughout this process we assume the input tileset is known to the decoder, and that all tiles are different (tiles under transforms are seen as different labels, unless they have identical content due to symmetries).

The straightforward approach for label extraction is to split the image into tiles, and then search for corresponding tiles in the tileset. This can be done by comparing the tile images, for instance measuring mean square pixel error. If the image is unaltered, then the difference between an input tile and one of the tileset tiles will be zero. This identifies the tile selected during synthesis.

If the image has been altered, for instance after lossy compression, the tile with smallest difference can be selected. This is however fragile, since under large changes this may no longer be the tile selected during synthesis. Such an error will immediately break the decoding process, as any error cascades into a catastrophic divergence from the original sequence.

To make the process more robust, we propose to exploit the layout constraints. We indeed have a strong a priori knowledge on the input: it enforces the constraints everywhere. The key idea is to first compute matching scores between each input tile and the entire tileset. The input tiles are then sorted according to a confidence score. Intuitively a tile which matches very well a single tile of the tileset will have a high confidence, while a tile which matches equally well the entire tileset has low confidence. We then simulate a synthesis process, visiting the input by decreasing order of confidence. Each time, we select the tile that is best matching and we propagate the constraints. This implies that many potential candidates at neighboring locations are being removed by the constraints, resolving ambiguities. Hence, when we visit the next site, we only consider these candidates which are still available, selecting the one that is best matching.

This process significantly improves robustness. Results are reported in Figures 1 and 2 for both jpeg compression and noise (erasing a percentage of pixels with a random color). As can be seen the improvement is significant for each tileset.

Computing confidence. This is an area of improvement. Right now we use the difference between the second best and best matching across the tileset (intuitively, if this difference is large then the 'best' is a much better match than all others).

Discussion. What makes a tileset more robust than another is a topic of further study. Robustness could also be improved by selecting a better difference metric, as well as a better confidence measure.

Implementation

A proof of concept is available at http://members.loria.fr/Sylvain.Lefebvre/ infotexsyn/. Code will be released when ready, but it should be simple to implement after reading this document. messages than others in a same space. Whether and why a tileset allows for more information space in an intriguing theoretical question that we intend to explore.

Encoding capacity can be measured experimentally, trying to encode increasingly longer random messages with a limited number of tiles. Using this technique, we determined the encoding capabilities presented in Table 1 Messages Another intriguing question is whether pathological messages exist, that would consistently lead to a contradiction regardless of the initial random seed. Such message would be impossible to encode.

Image size As it is difficult to ensure a message will fit, and since the synthesis is fast, we devise the following approach. First, the algorithm determines the size in bits of the message and uses previously computed average capabilities of the tileset to estimate the size of S (assuming some aspect ratio, e.g. 1:1). If the process terminates before all bits are encoded, S is enlarged and the process resumes. If on the contrary a lot of space is left empty, S could be reduced for a tighter fit. Note that messages are always padded with zeros until synthesis completes.

Steganography While this is a form of steganography, it is unclear yet whether our approach is a good or bad steganography approach.

Cryptography It is intriguing to consider how 'secret' the encoding is. In particular, if the PRNG seed is lost, how difficult would it be for an adversary to extract the message?

Embedding the tileset We currently assumed that the decoder has access to the tileset used for synthesis. It would be interesting to consider whether the tile set could be embedded within the texture. This should ensure that both the tiles and constraints are embedded, and that decoding remains robust.

Table 1 :

 1 . It is also interesting to consider this data in light of decoding robustness (Section 5.1). Encoding capabilities of different tiles sets, in bits per tile, measured experimentally.

	Circles Circuit Knots Castle Rooms Summer Squares
	2.39	1.98	1.8	0.76	0.69	0.47	0.24

Tilesets This algorithm works with any tileset that provides choices (e.g. a tileset with a single tile provides none). Some tilesets will allow to encode larger


Annika T Schmitt
email: annika.schmitt@uni-jena.de

Yves Joly

Kai S Schulze

Berit Marx-Glowna

Ingo Uschmann

Benjamin Grabiger

Hendrik Bernhardt

Robert Loetzsch

Amélie Juhin

Jérôme Debray

Hans-Christian Wille

Hasan Yava

Gerhard G Paulus

Ralf And

Röhlsberger
email: ralf.roehlsberger@desy.de

Disentangling X-ray dichroism and birefringence via high-purity polarimetry

come

Introduction

Symmetries in nature are closely related to the fundamental structure of atoms, molecules and solids [1]. Symmetry breaking interactions in condensed matter, for example, form the fundamental basis for macroscopic quantum effects like magnetism, superconductivity, giant magnetoresistance, multiferroicity, and others, rendering the optical properties of such materials anisotropic. Access to symmetries and anisotropies of matter has been provided for centuries by optical effects like dichroism and birefringence [2], in particular by studying how the optical properties of matter depend on the polarization of light and how the polarization of light is affected by the interaction with matter. In recent decades, the use of highly brilliant X-rays from synchrotron radiation sources has provided access to the microscopic origins of magnetic and electronic anisotropies. This is facilitated, amongst others, via a suite of dichroic X-ray absorption spectroscopies in which the polarization dependence of X-ray absorption in the vicinity of atomic inner-shell transitions is monitored [3][4][5][6][7].

Polarization changes of X-rays in the interaction with matter occur not only due to absorptive but also through dispersive effects, leading to dichroism and birefringence, respectively. In the case of linearly polarized X-rays, dichroism causes a rotation of the polarization vector due to an anisotropic absorption cross-section of the sample. X-ray birefringence results from different propagation velocities of two orthogonal polarization components, which leads to a phase shift between those and induces an ellipticity of the light. Both effects constitute sensitive probes for fundamental aspects of the light-matter interaction: In condensed matter physics, the spectral dependencies of X-ray dichroism and birefringence depict a very sensitive fingerprint of the electronic structure of the material. For example, tiny optical anisotropies emerging in the vicinity of phase transitions could reveal precursor mechanisms for structural transformations and electronic ordering in materials [START_REF] Norman | X-ray natural dichroism and chiral order in underdoped cuprates[END_REF][START_REF] Pershoguba | Proposed chiral texture of the magnetic moments of unit-cell loop currents in the pseudogap phase of cuprate superconductors[END_REF][START_REF] Norman | Linear dichroism and the nature of charge order in underdoped cuprates[END_REF]. In quantum electrodynamics (QED) of extremely strong electromagnetic fields it is predicted that even the vacuum becomes optically anisotropic [START_REF] Heyl | Birefringence and dichroism of the QED vacuum[END_REF][START_REF] Heinzl | On the observation of vacuum birefringence[END_REF][START_REF] Karbstein | Vacuum birefringence in strong inhomogeneous electromagnetic fields[END_REF][START_REF] Karbstein | Vacuum birefringence in the head-on collision of x-ray free-electron laser and optical high-intensity laser pulses[END_REF]. The resulting birefringence and dichroism could be sensitively probed by polarization analysis of hard X-rays after interaction with ultraintense light fields [START_REF] Karbstein | Probing vacuum birefringence using x-ray free electron and optical high-intensity lasers[END_REF][START_REF] Karbstein | Enhancing quantum vacuum signatures with tailored laser beams[END_REF][START_REF] Schlenvoigt | Detecting vacuum birefringence with x-ray free electron lasers and high-power optical lasers: a feasibility study[END_REF]. This would be a first test of nonlinear QED since the first considerations on this subject by Euler and Heisenberg in 1936 [START_REF] Heisenberg | Folgerungen aus der Diracschen Theorie des Positrons[END_REF]. It is thus obvious that the precise detection of dichroic and birefringent polarization changes of scattered X-ray radiation would provide fundamental insights into condensed matter physics and QED effects alike. Motivated by these perspectives, very efficient high-purity polarimeters for hard X-rays have been developed with extinction ratios of up to 10 -10 [START_REF] Marx | High-precision x-ray polarimetry[END_REF][START_REF] Bernhardt | Ultra-high precision x-ray polarimetry with artificial diamond channel cuts at the beam divergence limit[END_REF]. They are based on two crossed linear Bragg polarizers between which the X-ray optical activity takes place [START_REF] Hart | X-ray polarization phenomena[END_REF][START_REF] Siddons | X-ray optical activity and the faraday effect in cobalt and its compounds[END_REF][START_REF] Hart | Tunable x-ray polarimeters for synchrotron radiation sources[END_REF][START_REF] Siddons | Time-dependent polarization in mössbauer experiments with synchrotron radiation: Suppression of electronic scattering[END_REF][START_REF] Siddons | Mössbauer spectroscopy using synchrotron radiation: overcoming detector limitations[END_REF][START_REF] Toellner | Polarizer/analyzer filter for nuclear resonant scattering of synchrotron radiation[END_REF][START_REF] Röhlsberger | X-ray optics for 𝜇ev-resolved spectroscopy[END_REF][START_REF] Alp | Polarizer-analyzer optics[END_REF][START_REF] Heeg | Vacuum-assisted generation and control of atomic coherences at x-ray energies[END_REF][START_REF] Heeg | Tunable subluminal propagation of narrow-band x-ray pulses[END_REF][START_REF] Haber | Collective strong coupling of x-rays and nuclei in a nuclear optical lattice[END_REF].

Here we employ high-purity polarimetry to reveal electronic anisotropies in condensed matter with maximum orbital sensitivity, thereby consolidating the research field of spectroscopic polarimetry. We showcase the potential of this analysis technique by presenting a full characterization of the polarization changes across an atomic absorption edge and their separation into dichroic and birefringent contributions. As benchmark systems for correlated materials and parent compounds for cuprate superconductors, two materials have been chosen, CuO and La 2 CuO 4 . We are probing the pronounced electronic anisotropies in these cuprate compounds that result from the particular symmetry of the Cu atom and its hybridizations with the surrounding orbitals in the near-edge region of the Cu K-absorption edge.

Experimental Procedures

High-purity Spectropolarimetry

The experiments were performed at the synchrotron radiation source PETRA III (DESY, Hamburg) with the setup shown in Fig. 1a. A Si(111) heat-load monochromator selected a 1-eV wide energy band out of the incident radiation. The sample was located under ambient conditions between two monolithic Si(440) channel-cut crystals, acting as highly efficient linear polarizers due to multiple reflections under the near-45 • Bragg angle for the X-ray energy of the Cu K-edge. In a 90 • crossed setting, these two crystals form a polarimeter consisting of polarizer and analyzer [START_REF] Hart | Tunable x-ray polarimeters for synchrotron radiation sources[END_REF], providing polarization extinction ratios of up to 10 -10 [START_REF] Marx | High-precision x-ray polarimetry[END_REF] or better [START_REF] Schulze | Fundamental limitations of the polarization purity of x rays[END_REF], which is way superior to what is possible in the regime of optical wavelengths. Any X-ray optical activity occuring between polarizer and analyzer converts a fraction of the highly pure 𝜎 -polarization incident on the sample into the orthogonal 𝜋 -polarization that is then very efficiently transmitted by the analyzer and detected by an avalanche photodiode in single-photon counting mode. For precision angular adjustment relative to the incident beam, the sample was mounted on an Eulerian cradle, providing angular degrees of freedom along 𝜑 and 𝜒.

The energy resolution of the X-ray polarimeter is determined by the Darwin width of the Si(440) polarizer and analyzer rocking curves which translates here into an energy bandpass of 62 meV, thus allowing for the detection of very sharp spectral features. For scanning the polarimeter over the energy range of the Cu K-absorption edge with a constantly high polarization purity, the Bragg angles of polarizer and analyzer crystals have to be varied simultaneously with the Bragg angle of the Si(111) monochromator. To cover the energy range of the Cu-K-edge from 8970 eV to 9010 eV, the Bragg angle on the Si(440) plane has to be varied from 𝜃 𝐵 = 45.78 • to 𝜃 𝐵 = 46.06 • . In this energy range we achieved a polarization purity better than 1.3 • 10 -8 , which was measured at 8970 eV and lies within the range between 1 • 10 -7 and 5 • 10 -9 that is predicted by the dynamical theory of X-ray diffraction.

To accurately determine the intensity of the 𝜎 → 𝜋 scattered photons for a given polarimeter energy and angular setting (𝜑, 𝜒) of the sample (see Fig. 1), the maximum of the rocking curve of the analyzer Bragg angle at each setting was taken. Fig. 2 shows the measured spectra of the 𝜎 → 𝜋 scattered photons for both crystals in the setting 𝜑 = 0 as a function of the angle 𝜒. By comparison with the conventional XANES spectra (see Supplement 1 and the grey lines in Figs. 2e,j), one observes that in case of CuO, the 𝜎 → 𝜋 scattering is maximal at the position of the pre-edge peak at 8984 eV, while in case of La 2 CuO 4 this maximum occurs at the inflection point of the absorption edge at 8994 eV. These energy dependencies can be related to the Cu orbital configuration in the respective compound as will be discussed later.

Modelling the Complex Linear Absorption Coefficient

The dependence of X-ray dichroism and X-ray birefringence on the photon polarization and the sample orientation is ruled by the point group of the crystal [START_REF] Brouder | Angular dependence of x-ray absorption spectra[END_REF]. This is explained in detail in Supplement 1. According to these considerations we chose the orientation 𝜑 = 0 for both sample materials, which corresponds to the electric field vector lying in the 𝑎-𝑐-plane. This allows to detect the full anisotropy of the electric dipole absorption cross-section 𝜎 𝐷 . Accordingly, the single crystals were shaped into (010) -orientated slabs controlled by the Laue method with an accuracy of ≤ 0.3 • . Subsequently they were gently polished to 33 𝜇m (CuO) and 23.5 𝜇m (La 2 CuO 4 ) thin disks. The thickness of the samples was determined by transmission measurements and comparison to Henke data [START_REF] Henke | X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92[END_REF]. Finally, the adjustment of the crystal axes within the 𝑎-𝑐 plane was confirmed again by the Laue method.

For a description of the optical activity of these samples, i.e., the 𝜎 → 𝜋 scattering, we use the complex linear absorption coefficient [START_REF] Collins | X-ray birefringence in highly anisotropic materials[END_REF][START_REF] Joly | Birefringence and polarization rotation in resonant x-ray diffraction[END_REF] Fig. 2. Intensity of the 𝜎 → 𝜋 scattered photons of CuO (top) and La 2 CuO 4 (bottom) normalized to the incident intensity 𝐼 0 on the sample for different angles 𝜒 between the crystal 𝑎-axis and the electric field vector. X-ray birefringence and dichroism were simulated by neglecting the anisotropic part of the real or imaginary part of the complex linear absorption coefficient 𝜇, respectively. The red line in a) to d) and f) to i) marks the 𝜒-position of the line out shown in e) and j), respectively. The corresponding XANES spectra are indicated in grey in e) and j).

𝜇 = 𝜇 + 𝑖𝜇 = 𝜇 𝜎 𝜎 𝜇 𝜎 𝜋 𝜇 𝜋 𝜎 𝜇 𝜋 𝜋 , 𝜇 𝑥 𝑥 ∈ C, (1)
The real part 𝜇 is responsible for dichroism, whereas the imaginary part 𝜇 is related to birefringence.

In order to determine 𝜇 for the spectral region of the Cu K-edge of our samples, ab initio calculations were performed with the FDMNES code following the local density approximation [START_REF] Joly | X-ray absorption near-edge structure calculations beyond the muffin-tin approximation[END_REF]. The relativistic full potential approach was used, including the spin-orbit interaction and the core-hole potential effect. Self-consistent electronic structures around the absorbing atom were calculated in a cluster with a radius of up to 6 Å. The code includes all calculation steps of the polarization changes in a material up to the final transmitted intensity after the analyzer.

For the calculation of polarization changes in the presence of a complex linear absorption coefficient, the Jones matrix formalism is used. The derivation is given in detail in Supplement 1. Consequently, the 𝜋 -polarized X-ray intensity after the sample normalized to the impinging 𝜎 -polarized X-ray intensity on the sample, 𝐼 𝜎 𝜋 , is given by

𝐼 𝜎 𝜋 = 𝑒 -1 2 (𝜇 𝜎 𝜎 +𝜇 𝜋 𝜋 )𝑙 | sinh (𝜏𝑙) | 2 8|𝜏| 2 × | (𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) sin 2𝜒 -2𝜇 𝜎 𝜋 cos 2𝜒| 2 , (2)
where 𝜇 𝜎 𝜋 = 𝜇 𝜋 𝜎 for centrosymmetric crystals, and [START_REF] Joly | Birefringence and polarization rotation in resonant x-ray diffraction[END_REF][START_REF] Lovesey | X-ray birefringence and dichroism obtained from magnetic materials[END_REF]. 𝑙 is the thickness of the sample.

𝜏 = 1 4 √ (𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) 2 + 4𝜇 𝜎 𝜋 𝜇 𝜋 𝜎

Results

Based on this theoretical description, we will now discuss the influence of the symmetry of the complex linear absorption coefficient 𝜇 on the measured 𝜎 → 𝜋 scattered intensity 𝐼 𝜎 𝜋 .

According to equation ( 2), the following behavior is expected: If the non-diagonal tensor elements 𝜇 𝜋 𝜎 are zero, 𝐼 𝜎 𝜋 is maximal at 𝜒 = ±45 • for all energies, since 𝐼 𝜎 𝜋 is then proportional to sin 2 2𝜒. If the non-diagonal tensor elements 𝜇 𝜋 𝜎 are non-zero, the maxima of 𝐼 𝜎 𝜋 as a function of 𝜒 depend on the components of 𝜇 and thus are energy-dependent.

Both symmetry cases of the complex linear absorption coefficient 𝜇 were experimentally verified by investigating the two samples La 2 CuO 4 (𝜇 𝜋 𝜎 = 0) and CuO (𝜇 𝜋 𝜎 ≠ 0), as shown in Fig. 2. In agreement with the theory, all spectral features of La 2 CuO 4 have the same angular dependence for which 𝐼 𝜎 𝜋 is maximum at 𝜒 = ±45 • , whereas for CuO, there is an energydependent shift in 𝜒 for 𝐼 𝜎 𝜋 due to the nonzero 𝜇 𝜋 𝜎 . Furthermore, 𝐼 𝜎 𝜋 has a 𝜒 -periodicity of 𝜋/2 for both crystals, which can easily be explained by their centrosymmetry. The theoretical simulation is in very good qualitative agreement with the experimental data. This is further exemplified by line cuts at selected 𝜒 angles shown in Figs. 2e,j. While the agreement in case of CuO is excellent, the peak intensity for La 2 CuO 4 is overestimated by the simulation.

The theoretical description also allows to calculate the spectra of the 𝜎 → 𝜋 transmission separately for X-ray birefringence and X-ray dichroism. This can be achieved by neglecting the anisotropic part of either the real part or the imaginary part of 𝜇, respectively, which is illustrated in Figs. 2c and2d for CuO and Figs. 2h and 2i for La 2 CuO 4 . It turns out that the total 𝜎 → 𝜋 scattered intensity 𝐼 𝜎 𝜋 is the linear superposition of the 𝜎 → 𝜋 scattered intensities due to X-ray dichroism and birefringence.

To verify this disentanglement of dichroism and birefringence experimentally, we have performed conventional X-ray linear dichroism measurements by taking XANES spectra for two orthogonal sample orientations. This is shown in Fig. 3a. For the calculation of the 𝜎 → 𝜋 scattered intensity due to X-ray dichroism measured at the sample orientation 𝜒, 𝐼 𝐷 𝜎 𝜋 ( 𝜒) = (1/2) sin 2 𝜋/4 -arctan √ 𝑇 + /𝑇 -• (𝑇 + + 𝑇 -), the transmission through the sample, 𝑇 ± = 𝑇 ( 𝜒 ± 𝜋 4 ), measured at the sample orientations 𝜒 ± 𝜋 4 is needed. This formula can be derived very easily by vector superposition as explained in Supplement 1. The 𝜎 → 𝜋 scattered intensity due to X-ray birefringence can now the separated by a simple subtraction 𝐼 𝐵 𝜎 𝜋 ( 𝜒) = 𝐼 𝜎 𝜋 ( 𝜒) -𝐼 𝐷 𝜎 𝜋 ( 𝜒), which is plotted for 𝜒 = 80 • in Fig. 3b, being in excellent agreement with the simulated data shown in Fig. 3c.

The polarization changes due to dichroism and birefringence can be attributed to specific projections of the density of states on the absorbing atoms. In case of investigating a (010)oriented sample with an X-ray beam parallel to the crystal 𝑏 -axis, the density of states 𝛿( 𝑝 𝑥 ), 𝛿( 𝑝 𝑧 ), 𝛿(𝑑 𝑥 𝑦 ) and 𝛿(𝑑 𝑦𝑧 ) of the orbitals 𝑝 𝑥 , 𝑝 𝑧 , 𝑑 𝑥 𝑦 and 𝑑 𝑦𝑧 are involved, where 𝑧 is choosen along the 𝑐 -axis of the crystal. 𝐼 𝜎 𝜋 is related to the difference between the density of states 𝛿( 𝑝 𝑥 ) -𝛿( 𝑝 𝑧 ) and 𝛿(𝑑 𝑥 𝑦 ) -𝛿(𝑑 𝑦𝑧 ). It turned out that 𝐼 𝐷 𝜎 𝜋 is maximal, when 𝛿( 𝑝 𝑥 ) -𝛿( 𝑝 𝑧 ) is maximal, while 𝐼 𝐵 𝜎 𝜋 is maximal for energies where 𝛿( 𝑝 𝑥 ) ≈ 𝛿( 𝑝 𝑧 ). This is illustrated in Supplement 1, Fig. 3.

The X-ray absorption cross section of CuO and La 2 CuO 4 at the Cu K-edge is highly dominated by dipole transitions (E1E1). Consequently, the simulation of the 𝜎 → 𝜋 transmission spectra for different multipole contributions with FDMNES showed that they are mainly due to dipole (E1E1) transitions with a three orders of magnitude weaker quadrupole (E2E2) contribution caused by the 1s→3d pre-peak of the absorption spectrum. This is presented in Supplement 1, Fig. 4. Mixed dipole-quadrupole (E1E2) and dipole electric-magnetic (E1M1) transitions did not play a role at the Cu K-edge for CuO and La 2 CuO 4 .

The analysis of 𝐼 𝜎 𝜋 has considerable advantages over conventional X-ray absorption measurements such as X-ray natural linear dichroism (XNLD): It is essentially background free and allows to monitor X-ray optical activity over a dynamic range of several orders of magnitude. Fig. 4 shows the 𝜎 → 𝜋 transmission of the CuO sample for 𝜒 = 0 • and 𝜒 = 90 • . This measurement not only proves the 90 • periodicity of the anisotropy in this sample, but also highlights the excellent agreement with the theoretical simulation over almost four orders of magnitude. An energy scan that was performed without a sample in the beam illustrates the very low background level of the polarimeter. Thus, a dynamical range of six orders of magnitude and essentially background-free measurements enable unprecedented sensitivity to detect optical activity in the energy range of the Cu K-edge from 8970 eV to 9010 eV. This renders our technique particularly sensitive to higher-order transitions such as weak 1s→3d quadrupolar excitations that are located in the pre-edge region. These transitions often display a pronounced linear dichroism and birefringence due to the symmetry of the orbitals in the excited state [START_REF] Hahn | Observation of an electric quadrupole transition in the x-ray absorption spectrum of a Cu(II) complex[END_REF][START_REF] Glatzel | Resonant x-ray spectroscopy to study K absorption pre-edges in 3d transition metal compounds[END_REF][START_REF] Cabaret | First-principles calculations of x-ray absorption spectra at the K-edge of 3d transition metals: an electronic structure analysis of the pre-edge[END_REF]. Since they are excited from a spherically symmetric 1s ground state, they provide an attractive spectroscopic signature to probe the 3d orbitals of the valence shell. This could be particularly attractive for the study of cuprate superconductors which owe their properties to the electronic structure of their CuO 2 planes. Revealing the anisotropy and occupancy of the corresponding Cu orbitals and their hybridizations, could thus provide clues about the origin of superconductivity in the cuprates. Moreover, adressing the K-edges in the regime of hard x-rays allows one to access samples in absorbing environments like high-pressure cells or buried layers in thin-film systems and induces less radiation damage compared to soft x-rays. In conventional x-ray absorption spectroscopies, pre-edge features in transition metal compounds are often overshadowed by the strong 1s→4p transition, which is the reason why they are only rarely studied. The suppression of any isotropic Fig. 4. Intensity behind the analyzer in crossed position for 𝜒 = 0 • between the 𝑎-axis of the CuO crystal and the electric field vector of the beam. To demonstrate the reproducibility of the measurement, the data for 𝜒 = 90 • are also shown, which is in good agreement for the simulation over many orders of magnitude. The dashed grey line marks the position of the Cu K-edge (8978.9 eV) according to Henke data [START_REF] Henke | X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92[END_REF].

scattering by the crossed polarizers that leads to the exceptional signal-to-noise ratio of this method will alleviate the latter limitation, thus providing a new approach to obtain clear views on charge and orbital anisotropies in the valence shell.

Discussion and Conclusion

This work reports the first comprehensive experimental and theoretical investigation of X-ray birefringence and dichroism at the Cu K-edge for two different crystal systems. By measuring the X-ray dichroism conventionally, the real part of the complex linear absorption coefficient can be determined. The imaginary part, which corresponds to X-ray birefringence, can be determined by subtracting the measured dichroism spectra from the 𝜎 → 𝜋 scattered photon spectra. This is especially interesting for the determination of optical constants of materials that cannot be treated via ab initio calculations. Examples are materials that contain impurities like those that have been modified by ion implantation or doping, and those very strongly correlated systems, for which no suitable theoretical approach is currently capable of simulating their properties.

High purity polarimetry can adress a large part of the K-absorption edges (elements between 𝑍 = 17 and 𝑍 = 43) and L-absorption edges (elements between 𝑍 = 37 and 𝑍 = 92) by a suitable crystal reflection with an Bragg angle near 45 • . Corresponding polarimeters made of silicon, germanium or quartz can reach in most cases theoretically a polarization purity of < 10 -10 (Supplement 1, Tables 1234). This opens a wide field of application potential for the investigation of electronic anisotropies via high purity polarimetry. One example for a highly interesting research field could be the investigation of iron based superconductors at the Fe K-edge with a Si(133) polarimeter.

This method has several advantages over conventional methods for the detection of dichroism, such as XNLD, and can answer questions of fundamental importance. It can avoid the problem of integrating via a finitely measured absorption cross-section using the Kramers-Kronig relation in order to obtain the real part of the refractive index, which is the conventional approach. High polarization sensitivity is particularly suitable for observing small anisotropies as early indicators of phase transitions during or even long before reaching critical parameters. This new approach is especially interesting for investigating very weak anisotropies of quadrupolar or octopolar transitions in the pre-edge region like they were recently detected in Gd 3 Ga 5 O 12 at the Gd L 1 -absorption edge [START_REF] Juhin | Measurement of 𝑓 orbital hybridization in rare earths through electric dipole-octupole interference in x-ray absorption spectroscopy[END_REF]. In contrast to XNLD, this method does not require spectra of orthogonal orientations to be subtracted from each other. Instead, the measurement of anisotropies with a high angular resolution is directly and quickly accessible. This enables single-shot measurements for time-resolved investigations of certain spectral features in a pump-probe setting, for example. Furthermore, in analogy to an optical polarization microscope, in combination with micro-focused x-ray beams it is also possible to map and image X-ray polarization anisotropies with very high spatial resolution.

Disentangling dichroism and birefringence with high sensitivity will also play a pivotal role in future experiments on probing QED in extreme electromagnetic fields with polarized X-rays. The experiments proposed so far [START_REF] Heinzl | On the observation of vacuum birefringence[END_REF][START_REF] Karbstein | Vacuum birefringence in strong inhomogeneous electromagnetic fields[END_REF][START_REF] Karbstein | Vacuum birefringence in the head-on collision of x-ray free-electron laser and optical high-intensity laser pulses[END_REF][START_REF] Karbstein | Probing vacuum birefringence using x-ray free electron and optical high-intensity lasers[END_REF][START_REF] Karbstein | Enhancing quantum vacuum signatures with tailored laser beams[END_REF][START_REF] Schlenvoigt | Detecting vacuum birefringence with x-ray free electron lasers and high-power optical lasers: a feasibility study[END_REF] aim at the detection of the birefringence of the vacuum. A realistic scenario of such an experiment, originally proposed in [START_REF] Heinzl | On the observation of vacuum birefringence[END_REF], has been quantitatively worked out in [START_REF] Karbstein | Vacuum birefringence in the head-on collision of x-ray free-electron laser and optical high-intensity laser pulses[END_REF]: A pulse of linearly polarized hard x-rays (12914 eV) from an x-ray laser traverses a 1𝜇m wide focal waist of a laser beam (𝜆 = 800 nm) with 30 J pulse energy and 30 fs pulse duration. In this interaction, assuming 𝑁 = 10 12 photons in the x-ray pulse, QED calculations predict a small single-digit number of x-ray photons to flip their polarization from 𝜎 to 𝜋. These photons can be detected by a high-purity polarimeter of the kind described here with an extinction ratio in the range of 10 -12 that is reachable today [START_REF] Schulze | Fundamental limitations of the polarization purity of x rays[END_REF]. A dichroic contribution to the signal would imply a correction to QED. It would, therefore, be a direct hint for the existence of particles beyond the standard model such as millicharged or axion-like particles [START_REF] Gies | Polarized light propagating in a magnetic field as a probe for millicharged fermions[END_REF][START_REF] Maiani | Effects of nearly massless, spin-zero particles on light propagation in a magnetic field[END_REF].

Angular dependence of the X-ray absorption cross-section 𝜎

The samples in this experiment belong to two different point groups: Both materials show trichroism, which means that all eigenvalues of the absorption crosssection tensor 𝜎 𝐷 are different. The electronic transitions of CuO and La 2 CuO 4 with the main absorption edge at approximately 8994 eV are mainly of electric dipole origin. The electric dipole absorption cross-section is given by

•
𝜎 𝐷 ( ε) = 𝜎 𝐷 (0, 0) - √ 8𝜋 5 2 ∑ 𝑚=-2 𝑌 𝑚 * 2 ( ε) 𝜎 𝐷 (2, 𝑚) (1)
which depends (a) on the polarization vector ε and the sample orientation relative to the photon wavevector through the spherical harmonics 𝑌 𝑚 𝑙 and (b) on the point group of the crystal and the photon energy through the tensor 𝜎 𝐷 of the dipole absorption cross-section, with the complex tensor components 𝜎 𝐷 (𝑙, 𝑚) [3], where 𝑚 and 𝑙 are the angular momentum quantum numbers.

𝜎 𝐷 (0, 0) is the isotropic part of the electric dipole absorption cross-section which is independent of the polarization. The polarization vector ε = [sin( 𝜒 -𝛽) cos 𝜑, sin( 𝜒 -𝛽) sin 𝜑, cos( 𝜒 -𝛽)] is chosen in accordance with the experimental setup where 𝜒 represents the angle between the 𝑎-axis of the crystal and the electric field vector of the synchrotron beam as shown in Fig. 1b. For 𝜒 = 0 • , the electric field vector is parallel to the 𝑎-axis. 𝛽 is the base angle of the crystal lattice which is ≠ 90 • for the oblique lattice of monoclinic CuO.

In the case of CuO with point group 𝐶 2ℎ (2/𝑚), where 𝜎 𝐷 (2, 1) = 𝜎 𝐷 (2, -1) = 0, the electric dipole absorption cross-section is given by

𝜎 𝐷 ( ε) =𝜎 𝐷 (0, 0) - √ 3 sin 2 ( 𝜒 -𝛽) [cos 2𝜑 Re{𝜎 𝐷 (2, 2)} + sin 2𝜑 Im{𝜎 𝐷 (2, 2)}] - 1 √ 2 3 cos 2 ( 𝜒 -𝛽) -1 𝜎 𝐷 (2, 0). ( 2
)
For La 2 CuO 4 with point group 𝐷 2ℎ (𝑚𝑚𝑚), where 𝜎 𝐷 (2, 1)

= 𝜎 𝐷 (2, -1) = 0 and 𝜎 𝐷 (2, 2) = 𝜎 𝐷 (2, -2), 𝜎 𝐷 ( ε) is simplified to 𝜎 𝐷 ( ε) = 𝜎 𝐷 (0, 0) - √ 3 sin 2 ( 𝜒 -𝛽) cos 2𝜑 Re{𝜎 𝐷 (2, 2)} - 1 √ 2 3 cos 2 ( 𝜒 -𝛽) -1 𝜎 𝐷 (2, 0). (3)

Theoretical description of 𝝈 -𝝅 scattered intensity

We use the (2 × 2) Jones Matrix defined in a (𝜎, 𝜋) basis, 𝜎 horizontal, 𝜋 vertical, and the wave vector 𝑘 perpendicular to this basis.

Transmittance matrix

We use the transmittance matrix under the 𝜎-𝜋 basis where 𝜎 and 𝜋 are 2 directions perpendicular to the wave vector. We note 𝜇 the complex linear absorption coefficient. Following Lovesey and Collins [4], after a distance 𝑙, the transmittance matrix is given by:

𝑇 (𝑙) = 𝑒 -1 4 ( 𝜇 𝜎 𝜎 +𝜇 𝜋 𝜋 )𝑙 cosh(𝜏𝑙) + 𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 4𝜏 sinh (𝜏𝑙) -𝜇 𝜎 𝜋 2𝜏 sinh (𝜏𝑙) -𝜇 𝜋 𝜎 2𝜏 sinh (𝜏𝑙) cosh (𝜏𝑙) + 𝜇 𝜎 𝜎 -𝜇 𝜋 𝜋 4𝜏 sinh (𝜏𝑙) (4)
with

𝜏 = 1 4 √ (𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) 2 + 4𝜇 𝜎 𝜋 𝜇 𝜋 𝜎 = 1 4 (𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) √ 1 + 4𝜇 𝜎 𝜋 𝜇 𝜋 𝜎 (𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) 2

Polarization matrix

In the following we also use the Jones matrix for the polarization:

𝑃 = 1 2 1 + 𝑆 1 𝑆 2 -𝑖𝑆 3 𝑆 2 + 𝑖𝑆 3 1 -𝑆 1 . (5)
When rotating the sample by an angle 𝜒 the linear polarization gets an angle -𝜒 versus its 𝜎 direction. Thus one has:

𝑆 1 = cos 2𝜒, 𝑆 2 = -sin 2𝜒, 𝑆 3 = 0 (6) 𝑃 = 1 2 1 + cos 2𝜒 -sin 2𝜒 -sin 2𝜒 1 -cos 2𝜒 = cos 2 𝜒 -sin 𝜒 cos 𝜒 -sin 𝜒 cos 𝜒 sin 2 𝜒 . (7)

Analyzer matrix

The analyzer matrix (or polarizer after the sample) is given versus its rotation angle, 𝜂, and its Bragg angle, 𝜃, by:

𝐴 = cos 𝜂 -sin 𝜂 cos 2𝜃 sin 𝜂 cos 2𝜃 cos 𝜂 . (8)
For a perfect analyzer crystal cos 2𝜃 = 0. For the polarization flip by 90 • , one then obtains:

𝐴 = cos -𝜋 2 + 𝜒 -sin -𝜋 2 + 𝜒 0 0 = sin 𝜒 cos 𝜒 0 0 . (9)

Transmitted intensity

After the path through the sample, the intensity is given by

𝐼 = 𝑇𝑟 𝐴𝑇 𝑃𝑇 † 𝐴 † = | sin 𝜒 cos 𝜒(𝑇 𝜎 𝜎 -𝑇 𝜋 𝜋 ) + cos 2 𝜒𝑇 𝜋 𝜎 -sin 2 𝜒𝑇 𝜎 𝜋 | 2 (10)
Non-diagonal non-magnetic case 𝑇 𝜎 𝜋 = 𝑇 𝜋 𝜎 :

𝐼 = | sin 𝜒 cos 𝜒(𝑇 𝜎 𝜎 -𝑇 𝜋 𝜋 ) + cos 2𝜒𝑇 𝜋 𝜎 | 2 (11)
𝑇 𝜎 𝜎 𝑇 𝜎 𝜋 𝑇 𝜋 𝜎 𝑇 𝜋 𝜋 = 𝑒 -1 4 ( 𝜇 𝜎 𝜎 +𝜇 𝜋 𝜋 )𝑙 cosh(𝜏𝑙) + 𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 4𝜏 sinh(𝜏𝑙) -𝜇 𝜎 𝜋 2𝜏 sinh(𝜏𝑙) -𝜇 𝜋 𝜎 2𝜏 sinh(𝜏𝑙) cosh(𝜏𝑙) + 𝜇 𝜎 𝜎 -𝜇 𝜋 𝜋 4𝜏 sinh(𝜏𝑙) , (12)
𝐼 =𝑒 -1 2 (𝜇 𝜎 𝜎 +𝜇 𝜋 𝜋 )𝑙 | sin 𝜒 cos 𝜒 𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 2𝜏 sinh(𝜏𝑙) -cos 2𝜒 𝜇 𝜎 𝜋 2𝜏 sinh(𝜏𝑙)| 2 =𝑒 -1 2 (𝜇 𝜎 𝜎 +𝜇 𝜋 𝜋 )𝑙 | sinh(𝜏𝑙)| 2 8|𝜏| 2 |(𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) sin 2𝜒 -2𝜇 𝜎 𝜋 cos 2𝜒| 2 (13)
with

𝜏 = 1 4 √ (𝜇 𝜋 𝜋 -𝜇 𝜎 𝜎 ) 2 + 4𝜇 𝜎 𝜋 𝜇 𝜋 𝜎 (14)

Derivation of the relation between complex linear absorption coefficient, complex refractive index and complex atomic scattering factor

A plane wave described by

𝐸 (𝑧, 𝑡) = 𝐸 0 𝑒 𝑖 (𝑘 𝑧-𝜔𝑡) (15)
passes through a medium with complex refractive index 𝑛 = 1 -𝛿 + 𝑖𝛽 = 𝑐 •𝑘 𝜔 :

𝐸 (𝑧, 𝑡) = 𝐸 0 𝑒 𝑖 [ 𝜔 𝑐 (1-𝛿+𝑖𝛽) 𝑧-𝜔𝑡 ] (16)
= 𝐸 0 𝑒 𝑖 𝜔 ( 𝑧 𝑐 -𝑡) • 𝑒 -2 𝜋 𝜆 (𝑖 𝛿+𝛽) 𝑧 . (17)
By introducing the complex linear absorption coefficient 𝜇 = 𝜇 + 𝑖𝜇 , following S.P. Collins et al. [5], the expression can be written as

𝐸 (𝑧, 𝑡) = Ẽ0 • 𝑒 -𝜇 𝑧 2 ( 18
)
with the vacuum propagation Ẽ0 = 𝐸 0 𝑒 𝑖 𝜔 ( 𝑧 𝑐 -𝑡) . This leads to the Lambert-Beer law for the intensity 𝐼 = |𝐸 | 2 :

𝐼 = 𝐼 0 𝑒 -𝜇 𝑧 . (19)
A comparison of eq. 17 and eq. 18 shows that the complex linear absorption coefficient

𝜇 = 𝜇 + 𝑖𝜇 is connected to the refractive index 𝑛 = 1 -𝛿 + 𝑖𝛽 via 𝜇 = 4𝜋 𝜆 𝛽 and (20)
𝜇 = 4𝜋 𝜆 𝛿. (21)
Since the complex refractive index 𝑛 can also be expressed by using the complex atomic scattering factor 𝑓 = 𝑓 -𝑖 𝑓 ,

𝑛 = 1 - 𝑛 𝑎 𝑟 𝑒 𝜆 2 2𝜋 ( 𝑓 -𝑖 𝑓 ) = 1 -𝛿 + 𝑖𝛽, (22)
where 𝑛 𝑎 is the atomic density and 𝑟 𝑒 is the classical electron radius, there is also a relation between the complex linear absorption coefficient and the complex atomic scattering factor:

𝜇 = 2𝑛 𝑎 𝑟 𝑒 𝜆 𝑓 (23)
𝜇 = 2𝑛 𝑎 𝑟 𝑒 𝜆 𝑓 . ( 24
)
4. X-ray absorption near edge structure of CuO and La 2 CuO 4

The absorption cross-section was simulated for CuO (Fig. 1a) and La 2 CuO 4 (Fig. 1c) with the ab-initio quantum code FDMNES in units of Mbarn (1 Mbarn = 10 -18 cm 2 ) for different angles 𝜒 between the electric field vector and the 𝑎 -axis of the crystal [6]. It was also determined experimentally by measuring the transmission through the sample with ionization chambers for CuO (Fig. 1b) and La 2 CuO 4 (Fig. 1d). The main absorption edge (white line) is at approximately 8994 eV.

Calculation of the 𝜎 → 𝜋 scattered intensity due to X-ray dichroism

The 𝜎 → 𝜋 scattered intensity due to X-ray dichroism can be calculated using experimental XANES data, which are measured by ionization chambers before and behind the sample. Two orthogonal transmission measurements at the sample orientations 𝜒 ± 𝜋 4 are needed to calculate the 𝜎 → 𝜋 scattered intensity due to X-ray dichroism at the sample orientation 𝜒, which is normalized to the incident intensity 𝐼 𝑖𝑛 . The incoming electric field vector 𝐸 𝑖𝑛 is rotated by an angle Δ𝜒 due to dichroism in the sample, i.e. the different transmission of the electric field, 𝑡, at the sample orientations 𝜒 ± 𝜋 4 :

Δ𝜒 = 𝜋 4 -arctan 𝑡 ( 𝜒 + 𝜋 4 ) 𝑡 ( 𝜒 -𝜋 4 ) (25)
The incident electric field vector 𝐸 𝑖𝑛 is 𝜎 -polarized. The outgoing electric field vector

𝐸 𝑜𝑢𝑡 = 𝑡 2 ( 𝜒 + 𝜋 4 ) • 𝐸 2 0 + 𝑡 2 ( 𝜒 - 𝜋 4 ) • 𝐸 2 0 1/2 (26)
is rotated and therefore has also a 𝜋 -polarization component, which can pass the analyzer Fig. 2. The incident electric field vector 𝐸 𝑖𝑛 is rotated by an angle Δ𝜒 due to the different transmission along the two orthogonal axis at the sample orientations 𝜒 ± 𝜋 4 . The outgoing electric field vector 𝐸 𝑜𝑢𝑡 is no longer purely 𝜎 -polarized, but has now a 𝜋 -component, 𝐸 𝐷𝑖𝑐ℎ𝑟 . ( 𝜒), which can pass the X-ray analyzer in crossed position to the polarizer. The 𝜎 → 𝜋 scattered photons due to X-ray dichroism at the sample orientation 𝜒, 𝐸 𝐷𝑖𝑐ℎ𝑟 . ( 𝜒) , can be easily calculated when the transmitted intensity at the sample orientations 𝜒 ± 𝜋 4 , 𝑡 2 ( 𝜒 ± 𝜋 4 ) • 𝐸 2 0 , is measured simultaneously with ionization chambers before and behind the sample, while the sample is rotated.

in crossed position to the polarizer and is measured behind the analyzer as 𝜎 → 𝜋 scattered photons 𝐸 𝐷𝑖𝑐ℎ𝑟 . ( 𝜒). This can be calculated by

𝐸 𝐷𝑖𝑐ℎ𝑟 . ( 𝜒) = sin Δ𝜒 • 𝐸 𝑜𝑢𝑡 (27)
The 𝜎 → 𝜋 scattered intensity due to dichroism, 𝐸 2 𝐷𝑖𝑐ℎ𝑟 . ( 𝜒), is normalized to the incident intensity 𝐼 𝑖𝑛 = √ 2𝐸 0 2 :

𝐼 𝐷 𝜎 𝜋 ( 𝜒) = 𝐸 2 𝐷𝑖𝑐ℎ𝑟 . √ 2𝐸 0 2 ( 𝜒) = 1 2 sin 2 𝜋 4 -arctan √ 𝑇 + 𝑇 -• 𝑇 + + 𝑇 - (28)
with the transmitted intensity 𝑇 ± = 𝑇 ( 𝜒 ± 𝜋 4 ) = 𝑡 2 ( 𝜒 ± 𝜋 4 ).

Density of states of CuO and La 2 CuO 4

The density of states were simulated with FDMNES. 𝑧 was choosen along 𝑐 -axis of the crystal. For a (010) -oriented sample and a beam along the crystal 𝑏 -axis, the involved orbitals are 𝑝 𝑥 , 𝑝 𝑧 , 𝑑 𝑥 𝑦 and 𝑑 𝑦𝑧 . A comparison between the simulation of the 𝜎 → 𝜋 scattered photons and the density of states shows, that the intensity of 𝜎 → 𝜋 scattered photons due to dichroism is maximal for energies with a maximal difference between the density of states of 𝑝 𝑥 and 𝑝 𝑧 . In the other hand, the intensity of 𝜎 → 𝜋 scattered photons due to birefringence is maximal for energies where the density of states of 𝑝 𝑥 and 𝑝 𝑧 are equal.

Contribution of multipole transitions to spectra

The intensity of the 𝜎 → 𝜋 scattered photons and the X-ray absorption near edge structure spectra were simulated with FDMNES for electric dipole (E1E1), quadrupole (E2E2), dipole-qudrupole (E1E2) and dipole electric-magnetic (E1M1) transitions separately. The electric dipole term highly dominates the 𝜎 → 𝜋 scattered photons and the absorption cross section of CuO and La 2 CuO 4 . The electric quadrupole contribution (E2E2) to the 𝜎 → 𝜋 scattered photons is three orders of magnitute weaker than the electric dipole for both elements. Mixed electric dipole-quadrupole transitions and dipole electric-magnetic transitions do not contribute to the spectra.

Absorption edges addressable via high-purity polarimetry

A large part of the X-ray absorption edges can be addressed using high-precision X-ray polarimetry. An overview of possible X-ray polarimeters for the K absorption edges is tabulated in Table 1, and for the L absorption edges in Table 2, 3 and 4. The corresponding polarization purities were determined by calculating the rocking curve for 𝜋-and 𝜎 -polarization components at the corresponding energy using dynamical theory of X-ray diffraction and considering the number of reflections in the channel-cut. The polarization purity is then obtained from the ratio of the 𝜋 and 𝜎 components. For silicon and germanium, which can be realized as conventional channel-cut crystals, 6 reflections per channel-cut were assumed for the calculation of the polarization purity. For quartz and diamond, which can be used as quasi channel-cuts [7], 4 reflections were used. The polarimeter option, which allows the highest possible polarization purity for the specific energy, is tabulated. The calculation does not take into account detour excitations and must be clarified for each individual case. Only crystal reflections with a FWHM of at least 0.1 arcsec were considered.

Fig. 1 .

 1 Fig. 1. a) The sample was mounted on a Eulerian cradle with the 𝑏-axis parallel to the linearly polarized beam (𝜎 -polarization in the horizontal plane). An avalanche photodiode behind the X-ray analyzer in crossed position to the polarizer detected the 𝜎 → 𝜋 scattered photons. Simultaneously ionization chambers (IC) measured the transmitted intensity through the sample. b) Enlarged view of the sample as seen from the direction of the incoming beam.

Fig. 3 .

 3 Fig. 3. Experimentally disentangling dichroism and birefringence. a) The transmission (= 1 -absorption) through the sample at two orthogonal sample orientations 𝜒 ± 𝜋 4 is used to calculate the scattered intensity due to X-ray dichroism, 𝐼 𝐷 𝜎 𝜋 ( 𝜒), which can be subtracted from the total 𝐼 𝜎 𝜋 ( 𝜒) to obtain the 𝜎 → 𝜋 scattered intensity due to birefringence, 𝐼 𝐵 𝜎 𝜋 ( 𝜒), shown in b).c) Simulation with FDMNES. The dashed grey line marks the position of the Cu K-edge (8978.9 eV) according to Henke data [34].

  monoclinic CuO with point group 𝐶 2ℎ (2/𝑚) and space group 𝐶2/𝑐 (𝑎 = 4.6837 Å, 𝑏 = 3.4226 Å, 𝑐 = 5.1288 Å, and 𝛽 = 99.54 • [1]) and • orthorhombic La 2 CuO 4 with point group 𝐷 2ℎ (𝑚𝑚𝑚) and space group 𝐵𝑚𝑎𝑏 (𝑎 = 5.352 Å, 𝑏 = 5.400 Å, 𝑐 = 13.157 Å, and 𝛽 = 90 • [2]).

Fig. 1 .

 1 Fig. 1. X-ray absorption near edge structure (XANES) simulated for CuO (a) and for La 2 CuO 4 (c) for different sample orientations 𝜒. Experimental data for CuO and La 2 CuO 4 are shown in (b) and (d), respectively.

Fig. 3 .

 3 Fig. 3. a) Simulated intensity of the 𝜎 → 𝜋 scattered photons of CuO and b) for La 2 CuO 4 for different angles 𝜒 between the crystal 𝑎-axis and the electric field vector. The intensity of the 𝜎 → 𝜋 scattered photons X-ray due to dichroism b) and f), and due to birefringence c) and g) can be attributed to differences between the density of states 𝛿( 𝑝 𝑥 ) -𝛿( 𝑝 𝑧 ) and 𝛿(𝑑 𝑥 𝑦 ) -𝛿(𝑑 𝑦𝑧 ), respectively. The density of states of the orbitals 𝑝 𝑥 , 𝑝 𝑧 , 𝑑 𝑥 𝑦 and 𝑑 𝑦𝑧 are shown for CuO in d) and for La 2 CuO 4 in h). Maximal difference between the density of 𝑝 -states leads to dichroism (green lines), while birefringence is maximal at energies, where the density of states of both 𝑝 -Orbitals are equal (red lines).

Fig. 4 .

 4 Fig. 4. a) Simulated intensity of the 𝜎 → 𝜋 scattered photons of CuO at 𝜒 = 80 • and b) for La 2 CuO 4 at 𝜒 = 22.5 • c) X-ray absorption near edge structure (XANES) simulated for CuO at 𝜒 = 80 • and for d) La 2 CuO 4 at 𝜒 = 22.5 • . Electric dipole (E1E1), quadrupole (E2E2), dipole-qudrupole (E1E2) and dipole electric-magnetic (E1M1) transitions were calculated separately.

Table 1 .

 1 Possible crystal reflections for an X-ray polarimeter for the energies of the K absorption edge with a polarization purity that can be expected according to dynamic theory with 6 reflections per channel-cut for Si and Ge and 4 reflections per channel-cut for quartz and diamond.

	K absorption edge Element Energy [eV] Polarimeter 17 Cl 2822.4 Si (1 1 1) 19 K 3608.4 quartz (-2 1 0) 20 Ca 4038.5 quartz (0 2 0) 21 Sc 4492 Ge (0 2 2) 22 Ti 4966 quartz (1 1 2) 23 V 5465 Si (1 1 3) 24 Cr 5989 quartz (-1 2 -3) 25 Mn 6539 Si (0 0 4) 26 Fe 7112 Si (1 3 3) 27 Co 7709 Ge (2 2 4) 28 Ni 8333 Si (1 1 5) 29 Cu 8979 Si (4 4 0) 30 Zn 9659 Si (3 1 5) 31 Ga 10367 Ge (3 3 5) 32 Ge 11103 Si (4 4 4) 33 As 11867 Ge (2 4 6) 34 Se 12658 Ge (0 0 8) 35 Br 13474 Ge (2 2 8) 36 Kr 14326 Ge (0 4 8) 37 Rb 15200 Ge (4 4 8) 38 Sr 16105 Ge (0 2 10) 39 Y 17038 Ge(2 4 10) 40 Zr 17998 Ge (0 8 8) 41 Nb 18986 Ge (0 0 12) 42 Mo 20000 Ge (0 4 12) 43 Tc 21044 Ge (4 4 12)	Polarization puriy <1.0E-12 <1.0E-12 3.4E-12 <1.0E-12 1.7E-06 1.1E-10 5.0E-12 9.2E-11 1.5E-12 2.3E-10 <1.0E-12 5.8E-08 5.7E-11 2.0E-10 2.2E-11 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12 <1.0E-12

Acknowledgement

We would like to thank Heike Marschner and Ortrud Wehrhan for sharing their expertise in crystallography, and Claudia Rödl, Martin von Zimmermann, Paul Schenk and Felix Karbstein for helpful and enlightening discussions. Parts of this research were carried out at PETRA III, and we would like to thank Ilya Sergeev, Hlynur Gretarsson, Rene Steinbrügge, Frank-Uwe Dill and Conrad Hagemeister for assistance in using beamline P01.

Funding

This work was funded by the Deutsche Forschungsgemeinschaft (DFG) under Grant No. 416700351 within the Research Unit FOR2783/1, by the Bundesministerium für Bildung und Forschung (BMBF) under Grant No. 05K16SJ2 and by the European Social Fund (ESF) and the Free State of Thuringia in the framework of Forschergruppe 2017FGR0074.

Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content. absorption edge with a polarization purity that can be expected according to dynamic theory with 6 reflections per channel-cut for Si and Ge and 4 reflections per channel-cut for quartz and diamond.

L 1 absorption edge Element Energy [