## Introduction

A detailed knowledge of the brain’s nerve fibre architecture and tissue composition is essential to understand its structure and function. Several neuroinflammatory and -degenerative disorders, for example multiple sclerosis1, multiple system atrophy2,3,4, and leukodystrophies5, affect the integrity of neurons, axons, and dendrites. Therefore, it is important to study not only the long- and short-range connections between brain regions, but also the microstructure including axons and their myelin sheaths. In this paper, we show that the recently introduced technique Diattenuation Imaging (DI), which compares measurements of nerve fibre orientations to diattenuation (polarization-dependent attenuation), has the potential to provide new information about microstructural tissue properties.

The myelin content and connectivity patterns can be accessed in vivo by magnetic resonance imaging (MRI) techniques (T1, T26 and diffusion-weighting7,8) in the brains of healthy subjects and patients with a spatial resolution in the millimetre range9. To study the nerve fibre architecture at microscopic resolution as prerequisite for a better interpretation and validation of MRI data, Three-dimensional Polarised Light Imaging (3D-PLI) has been employed10,11,12. In contrast to other microscopy techniques which are limited to smaller tissue samples, 3D-PLI allows to resolve three-dimensional nerve fibre pathways of unstained whole-brain sections at microscopic resolution: The spatial orientations of the nerve fibres are derived by measuring the birefringence of the histological brain sections with a polarimeter13,14. The birefringence is caused by highly ordered arrangements of nerve fibres as well as by the regular molecular structure of the myelin sheath15,16,17 which surrounds most of the axons in the white matter18, leading to negative birefringence with respect to the fibre direction17. In the following, the term nerve fibre will restrictively be used for myelinated axons.

The same anisotropy that causes birefringence (anisotropic refraction) also leads to diattenuation (anisotropic attenuation)19,20, which is caused by anisotropic absorption (dichroism) as well as by anisotropic scattering of light21,22. The intensity of polarised light that is transmitted through a diattenuating medium depends on the direction of polarisation relative to the orientation of the optic axis (symmetry axis) in the medium: the transmitted light intensity becomes maximal for light polarised in a particular direction and minimal for light polarised in the corresponding orthogonal direction.

There exist several studies that investigate the diattenuation of different biological tissue types (biopsy tissue23, skin21,24, heart25, muscle26, tendon26, collagen27, eye28, unmyelinated nerve fibres in the retina29,30). Diattenuation has also been used to distinguish between healthy and pathological tissue (tissue from eye diseases29,30, burned/injured tissue24, cancerous tissue23) and to quantify different tissue properties (e. g. concentration of glucose21,28, thickness29). It can therefore be assumed that diattenuation studies of brain tissue would also reveal valuable additional information, assisting in the future to identify pathological changes.

In our previous study (Menzel et al.31), we have explored the diattenuation of brain tissue for the first time. We have introduced Diattenuation Imaging (DI) as add-on to 3D-PLI: a combined measurement of diattenuation and birefringence compares the diattenuation signals of the brain section to the nerve fibre orientations obtained from 3D-PLI measurements. The developed measurement protocol allows to determine the diattenuation of whole unstained brain sections even with a low signal-to-noise ratio. By performing DI measurements of sagittal rat brain sections, we have found that there exist two different types of diattenuation that are regionally specific: in some brain regions, the transmitted light intensity becomes maximal when the light is polarised parallel to the nerve fibres (referred to as D+), in other brain regions, it becomes minimal (referred to as D). Why this effect occurs and how the type of diattenuation depends on the tissue composition were still open questions.

In the present study, we modelled the observed diattenuation effects and provide an explanation for our previous results. We demonstrate in experimental studies that diattenuation of both types D+ and D can be observed in brain tissue samples from different species (rodent, monkey, and human). By combining finite-difference time-domain simulations and analytical modelling, we explain the observed diattenuation effects and show that they are caused both by dichroism and by anisotropic light scattering. Our studies reveal that the diattenuation signal depends not only on the nerve fibre orientations and embedding time of the sample but also on tissue homogeneity, fibre size, and myelin sheath thickness. Thus, by comparing the diattenuation signals to the nerve fibre orientations, DI measurements allow to distinguish brain regions with different tissue composition. This helps not only to improve the reconstructed nerve fibre architecture but also to identify changes in brain tissue that are not visible in standard 3D-PLI measurements, making Diattenuation Imaging a valuable imaging technique in normal and pathologically altered tissue.

## Results

First, we performed DI measurements on brain sections from different species to investigate how the observed diattenuation effects (D+ and D) depend on the tissue structure. To better understand the observed diattenuation effects and to learn more about how the diattenuation depends on underlying tissue properties, we subsequently simulated the diattenuation signal for different nerve fibre configurations.

### Experimental Studies

The DI measurements31 were performed on 60 μm thick brain sections as shown in Fig. 1 (see Methods for more details): First, a 3D-PLI measurement was performed with a polarimeter (see Fig. 1a) to derive the three-dimensional nerve fibre orientations in each image pixel, i.e. the in-plane orientation (direction) angle φ and the out-of-plane orientation (inclination) angle α, see Fig. 1b (middle). The direction angle corresponds to the phase φP of the measured birefringence signal, see Fig. 1b (left), the inclination angle is related to the amplitude |sin δP| of the birefringence signal via: $${\delta }_{{\rm{P}}}=\mathrm{(2}\pi /\lambda )\,d\,{\cos }^{2}\,{\alpha }_{{\rm{P}}}$$, where δP is the phase shift induced by the birefringent brain section, λ the wavelength, d the thickness of the birefringent brain section, and αP the inclination angle of the nerve fibres. The diattenuation of the brain section was measured with the same polarimeter (using only a rotating polariser, see Fig. 1d). The amplitude (strength of diattenuation) $$|{\mathscr{D}}|$$ and phase φD of the measured diattenuation signal (see Fig. 1b, right) were computed from the maximum and minimum transmitted light intensities (Imax and Imin) via20,22:

$$|{\mathscr{D}}|\equiv \frac{{I}_{{\rm{\max }}}-{I}_{{\rm{\min }}}}{{I}_{{\rm{\max }}}+{I}_{{\rm{\min }}}},\,\,{\phi }_{{\rm{D}}}\equiv \phi (I={I}_{{\rm{\max }}})\,\{\begin{array}{l}{D}^{+}:{\phi }_{{\rm{D}}}\approx {\phi }_{{\rm{P}}}\\ {D}^{-}:{\phi }_{{\rm{D}}}\approx {\phi }_{{\rm{P}}}+{90}^{\circ },\end{array}$$
(1)

where φD denotes the direction of polarisation for which the transmitted light intensity becomes maximal. A pixel-wise comparison of φP and φD (see histogram in Fig. 1c) reveals that some brain regions show diattenuation of type D+ (highlighted in green), i.e. the transmitted light intensity becomes maximal when the light is polarised parallel to the nerve fibre direction (φD ≈ φP), while other brain regions show diattenuation of type D (highlighted in magenta), i.e. the transmitted light intensity becomes maximal when the light is polarised perpendicularly to the nerve fibre direction (φD ≈ φP + 90°). The values $$\{{\phi }_{{\rm{P}}},{\phi }_{{\rm{D}}},|{\mathscr{D}}|\}$$ were used to generate coloured diattenuation images (see middle image in Fig. 1c): all $$|{\mathscr{D}}|$$ values that belong to regions with (φD − φP)  [−20°, 20°] were colourised in green (D+), regions with (φD − φP)  [−70°, 110°] were colourised in magenta (D). The angle ranges account for the uncertainties of φD due to the non-ideal optical properties of the Large-Area Polarimeter (LAP)31 that was used for most DI measurements (see Methods).

#### Diattenuation for different species and nerve fibre orientations

Figure 2 shows the diattenuation images of coronal and sagittal brain sections for three different species: mouse (a), rat (a), and vervet monkey (b). The section planes are oriented perpendicularly to each other. For reference, the coronal (sagittal) section planes are indicated by blue (red) lines in the respective other brain section. The diattenuation measurements were performed with the LAP with an effective object-space resolution of 14 μm/px (a) and 27 μm/px (b). The approximate orientation of the nerve fibres is known from brain atlases32,33 and 3D-PLI measurements34.

Comparing the diattenuation images of coronal and sagittal brain sections to each other reveals that regions with diattenuation of type D (magenta) mostly belong to regions with in-plane fibre structures (coronal: cc, fi; sagittal: aci, cg, CPu), while regions with diattenuation of type D+ (green) mostly belong to regions with out-of-plane fibre structures (coronal: cg, df, sm, CPu; sagittal: cc, fi). Fibre structures that show one type of diattenuation (D+ or D) in one section plane (coronal or sagittal), are likely to show the other type of diattenuation in the other orthogonal section plane.

Measurements with a prototypic polarising microscope with 1.8 μm/px yield similar D+ and D regions (see Supplementary Fig. S1), i.e. the observed diattenuation effects do not depend on the optical resolution of the imaging system.

#### Dependence of diattenuation on embedding time

Prior to the measurements, the brain sections are embedded in glycerine solution (see Methods). To study how the diattenuation of a brain section changes with increasing time after tissue embedding, a coronal section of a vervet monkey brain was measured 1, 8, 18, 22, 30, 37, 51, and 87 days after tissue embedding with the LAP. Figure 3a shows the diattenuation maps 8 days and 51 days after embedding. The size of the regions with diattenuation of type D+ (green) increases with increasing time after embedding, while the size of the regions with diattenuation of type D (magenta) decreases. The yellow arrows mark a region that is already of type D+ directly after embedding (i), a region that is still of type D after 51 days (ii), and a region that changes from type D to D+ (iii). Figure 3b shows the corresponding values for (φD − φP) and $$|{\mathscr{D}}|$$ for all eight measurements.

In regions that show diattenuation of type D+ (D), the strength of the diattenuation signal $$|{\mathscr{D}}|$$ increases (decreases). The same pattern was observed in brain sections from other species including the human brain (see Supplementary Fig. S2).

With increasing embedding time, the diattenuation signal becomes more similar to the measured birefringence signal (cf. Supplementary Fig. S2a,b). Brain sections with very long embedding times (several months) only show diattenuation of type D+ (see Supplementary Fig. S3d).

### Simulation Studies

To model and better understand the observed effects, we simulated the measured diattenuation signals for different artificial nerve fibre configurations. As mentioned in the beginning, diattenuation can be caused both by anisotropic scattering (DS) and by anisotropic absorption (dichroism DK).

Diattenuation caused by anisotropic scattering was investigated by performing finite-difference time-domain (FDTD) simulations and taking the optics of the polarimeter into account (see Methods). The FDTD algorithm computes the propagation of the polarised light wave through the sample by approximating Maxwell’s equations by finite differences35,36,37. In the diattenuation measurement, the polariser is rotated by 18 discrete angles. To save computing time, the diattenuation signal was approximately computed from only two simulation runs: from the transmitted intensity of light polarised along the x-axis (Ix) and from the transmitted intensity of light polarised along the y-axis (Iy), where the x-axis is aligned with the symmetry axis of the sample projected onto the xy-plane:

$${D}_{{\rm{S}}}\equiv \frac{{I}_{{\rm{x}}}-{I}_{{\rm{y}}}}{{I}_{{\rm{x}}}+{I}_{{\rm{y}}}},-\,1\,\le \,{D}_{{\rm{S}}}\,\le \,1\{\begin{array}{l}{D}_{{\rm{S}}}\, > \,0\iff {D}^{+}:{\phi }_{{\rm{D}}}\approx {\phi }_{{\rm{P}}},\\ {D}_{{\rm{S}}}\, < \,0\iff {D}^{-}:{\phi }_{{\rm{D}}}\approx {\phi }_{{\rm{P}}}+{90}^{\circ }.\end{array}$$
(2)

The magnitude of DS is related to the strength of the diattenuation ($$|{{\mathscr{D}}}_{{\rm{S}}}|\approx |{\mathscr{D}}|$$), the sign indicates the phase φD (cf. equation (1)): Positive values (DS > 0 Ix > Iy) correspond to regions with D+ and are shown in green (the transmitted light intensity becomes maximal when the light is polarised parallel to the fibre structure, i.e. in the x-direction). Negative diattenuation values (DS < 0 Ix < Iy) correspond to regions with D and are shown in magenta (the transmitted light intensity becomes maximal when the light is polarised perpendicularly to the fibre structure, i.e. in the y-direction).

The diattenuation caused by dichroism was described by an effective analytical model (see upper panel in Fig. 4c), assuming that birefringence and dichroism can be described by a complex retardance with shared principal axes (see Supplementary Note for derivation): $${D}_{{\rm{K}}}\approx \tanh (\,-\,(2\pi /\lambda )d\,{\rm{\Delta }}\kappa \,{\cos }^{2}\,\alpha )$$, where λ is the wavelength, d the thickness of the dichroic brain section, α the inclination angle of the nerve fibres, and Δκ the anisotropic absorption (dichroism) of the brain tissue. As the birefringence of the brain sections does not change much with increasing embedding time (see Supplementary Fig. S3b), the dichroism is also expected to be mostly time-independent. In contrast, the diattenuation caused by anisotropic scattering is expected to decrease with increasing embedding time because brain sections with long embedding times become transparent, i.e. show less scattering (see Supplementary Fig. S3a). As brain sections with long embedding times only show diattenuation of type D+, dichroism was assumed to cause positive diattenuation: DK > 0  Δκ < 0.

#### Simulated diattenuation for different fibre configurations

To analyse how the diattenuation of brain tissue depends on the out-of-plane orientation (inclination angle) of the enclosed nerve fibres, we simulated the diattenuation images for fibre bundles with different inclination angles (α = {0°, 10°, …, 90°}): a bundle with broad fibre orientation distribution and a bundle of densely grown fibres (see Fig. 4a). Figure 4b shows the diattenuation images obtained from FDTD simulations for the bundle of densely grown fibres for α = 0° and 80°. The diattenuation images for all inclination angles are shown in Supplementary Fig. S4. Regions with maximum and minimum diattenuation values are homogeneously distributed. The diattenuation is mostly negative (magenta) for flat fibres (α≤50°) and becomes more positive (green) for steep fibres (α > 60°). The histograms in Supplementary Fig. S4 show that the diattenuation values are almost symmetrically distributed around the mean value (−17% ± 19% for α = 0° and −22% ± 12% for α = 50°). Thus, the mean value $$\overline{{D}_{{\rm{S}}}}$$ is a good parameter to describe the diattenuation images.

Figure 4c shows the simulated diattenuation curves (mean value of DS plotted against α) for both fibre bundles: The mean diattenuation $$\overline{{D}_{{\rm{S}}}}$$ is negative for non-steep fibres (α≤60°), positive for steep fibres (70° < α < 90°), and almost zero for vertical fibres. For the bundle with broad fibre orientation distribution (red curve), the values range from $$\overline{{D}_{{\rm{S}}}}\approx -\,12.4\, \%$$ to 1.2%. For the bundle of densely grown fibres (orange curve), the range is much larger ($$\overline{{D}_{{\rm{S}}}}\approx -\,22\, \%$$ to 10%).

The fibres were modelled with radii r [0.5, 0.8] μm, consisting of an inner axon and a surrounding myelin sheath with thickness tsheath = 0.35 r (see Methods). To study how the diattenuation depends on the fibre properties, the bundle of densely grown fibres was also simulated for a broader distribution of fibre radii (r [0.3, 1.0] μm, brown curve), and the horizontal bundle (α = 0°) for thicker myelin sheaths (tsheath = 0.6 r, green star) and larger fibres (r [2.5, 4.0] μm, magenta star). While different fibre radii yield similar diattenuation curves, thicker myelin sheaths lead to more negative and larger fibres to less negative diattenuation values for α = 0° (see also Supplementary Fig. S5). Hence, regions with strongly negative diattenuation belong most likely to strongly myelinated, relatively small, straight and horizontal fibres.

To estimate the combined diattenuation effect of anisotropic scattering and absorption, Fig. 4c shows the simulated diattenuation curves $$\overline{{D}_{{\rm{S}}}}$$ in direct comparison to the dichroism DK > 0 plotted against the fibre inclination angle α (see Supplementary Note for derivation): For regions with non-steep fibres (α < 65°), both diattenuation of type D+ and D are observed, depending on whether DK > 0 or DS < 0 dominates. Regions with steep fibres (65° < α < 90°) only show diattenuation of type D+ because both DS and DK are positive. As expected, regions with vertical fibres (α = 90°) show small diattenuation values ($$|{D}_{{\rm{S}}}|,|{D}_{{\rm{K}}}|\ll 1$$).

#### Simulated dependence of diattenuation on embedding time

As mentioned above, scattering is expected to decrease with increasing embedding time, i.e. the refractive indices of the different tissue components become more similar to each other. A possible explanation for this behaviour is that the surrounding glycerine solution (with refractive index of 1.37) soaks into the myelin sheaths and reduces their effective refractive index. So far, the simulations were performed for a myelin refractive index of 1.47, corresponding to literature values of lipids/membranes38. To model the diattenuation of brain tissue with long embedding time, the bundle of densely grown fibres was simulated for a reduced myelin refractive index of 1.39 (see Fig. 4c, blue curve): the strength of the simulated diattenuation signals is much less ($$\overline{{D}_{{\rm{S}}}}\approx -\,\mathrm{2.2 \% }$$ to 1.6%). Supplementary Fig. S5d shows that |DS| decreases with decreasing myelin refractive index. This suggests that anisotropic scattering (DS) decreases with increasing time after embedding the brain section. As dichroism (DK) is expected to remain positive, the net observed diattenuation is expected to become more positive over time.

## Discussion

In previous work (Menzel et al.31), we have shown that brain tissue exhibits two different types of diattenuation: in some regions, the light is minimally attenuated when it is polarised parallel to the nerve fibres (D+), in others, it is maximally attenuated (D). Here, we investigated this effect both in experimental studies and simulations and demonstrated that it depends on nerve fibre orientation, tissue composition, and embedding time.

Our experimental studies show that the same diattenuation effects can be observed in brain sections from different species (mouse, rat, monkey, human) and at different optical resolutions (from 43 μm/px to 1.8 μm/px): regions with out-of-plane fibres show almost exclusively diattenuation of type D+, while regions with in-plane fibres are more likely to show diattenuation of type D (see Fig. 2). With increasing time after embedding the brain sections in glycerine solution, the fraction of regions with diattenuation of type D+ increases (see Fig. 3).

Using a combination of analytical modelling and FDTD simulations, we could explain these experimental observations. The diattenuation caused by anisotropic absorption (dichroism DK) was described by an analytical model (see equation (19) in Supplementary Note): according to this model, the dichroism decreases with increasing out-of-plane inclination angle of the nerve fibres, only causes positive diattenuation (type D+), and does not depend on the time after embedding the brain section. The diattenuation caused by anisotropic scattering (DS) was simulated for various fibre bundles with different inclination angles: regions with flat fibres (with respect to the section plane) show diattenuation of type D while regions with steep fibres show diattenuation of type D+. The strength of the simulated diattenuation signal depends on tissue properties like fibre orientation distribution, fibre size, and myelin sheath thickness, and decreases with increasing embedding time (see Fig. 4c).

To directly compare our simulation results to the experimental observations, we evaluated the diattenuation maps of the coronal vervet brain section in Fig. 3a in regions with different fibre inclinations (see coloured circles in Fig. 4d,e). In order to better compare the type of diattenuation to the fibre inclination, the brain sections on the right separately show the D+ and D regions in the white matter. Regions with diattenuation of type D+ and regions with fibre inclinations >65° are shown in green, while regions with diattenuation of type D are shown in magenta. In freshly embedded brain sections, regions with steep fibres (αP > 65°) show almost exclusively type D+: nearly all green regions in Fig. 4e are also green in Fig. 4d (see red circles). Regions with lower fibre inclinations show both types D+ and D (see yellow circles). Regions with flat fibre inclinations are most likely to show type D (see cyan circles).

All these observations can be explained by the combined model of analytically computed and simulated diattenuation curves (the dashed vertical lines in Fig. 4c mark the inclination angles of the evaluated regions). It should be noted that the analytical model of dichroism only allows a qualitative description as it does not consider any details in the fibre configurations. In future work, a more complex model could be used to study the exact dependence of the diattenuation on the underlying tissue properties.

The dependence of the diattenuation signal on the time after embedding the brain sections could successfully be modelled by reducing the refractive index of the myelin sheaths: the simulations have shown that a reduced myelin refractive index leads to a reduced anisotropic scattering (cf. orange and blue curves in Fig. 4c), i.e. brain tissue with long embedding time shows a smaller fraction of regions with diattenuation of type D (cf. Figure 4d). The same model was used in Menzel et al.35 to explain the increasing transparency of brain tissue samples with increasing embedding time (see also Supplementary Fig. 3a), demonstrating the validity of our model. A possible explanation for the equalisation of refractive indices is that the embedding glycerine solution soaks into the myelin sheaths of the nerve fibres and reduces the effective refractive index of myelin.

While the transmitted light intensity and diattenuation caused by anisotropic scattering are dominated by light scattering and decrease with increasing embedding time, birefringence and dichroism are mostly independent of the embedding time, i.e. they are probably caused by molecular effects. The finding that dichroism is positive (DK > 0) means that the absorption becomes maximal when the light is polarised in the y-direction (Iy < Ix, see equation (2)), i.e. perpendicularly to the nerve fibre axis and in the plane of the lipid molecules in the myelin sheath. This suggests that the dichroism of brain tissue is mainly caused by the myelin lipids.

Our simulations show that the strength and type of diattenuation (D+ or D) depend not only on the out-of-plane inclination angle of the nerve fibres in the investigated brain sections and on the time after tissue embedding, but also on other tissue properties like homogeneity, fibre size, or myelin sheath thickness (cf. Figure 4c and Supplementary Fig. S5b,c). A region with many small fibres, for example, is expected to show a stronger negative diattenuation than a region with few large fibres (see Supplementary Fig. 5c). This allows to distinguish regions with similar myelin densities, i.e. similar birefringence (3D-PLI) signals, but different tissue composition. How the measured type of diattenuation is exactly related to the underlying tissue properties needs to be investigated in future studies.

In conclusion, we could show that the diattenuation of brain tissue provides image contrasts between different tissue types and can therefore be used in addition to other imaging modalities to learn more about the brain’s nerve fibre architecture and tissue composition, and to identify (pathological) changes. This makes Diattenuation Imaging a valuable imaging technique.

## Methods

### Preparation of brain sections

The investigated brain sections were obtained from healthy postmortem brains from different species: mouse (C57BL/6, male, six months old), rat (Wistar, male, three months old), vervet monkey (African green monkey: Chlorocebus aethiops sabaeus, male, between one and two years old), and human (male, 87 years old). All animal procedures were approved by the institutional animal welfare committee at Forschungszentrum Jülich GmbH, Germany, and were in accordance with European Union (National Institutes of Health) guidelines for the use and care of laboratory animals. All methods were carried out in accordance with relevant guidelines and regulations. The postmortem human brain sample was acquired in accordance with the ethics committee at the Medical Faculty of the University of Rostock, Germany. The ethics committee at the Heinrich Heine University Düsseldorf, Germany, confirmed that such postmortem human brain studies do not require any additional approval if a written informed consent of the subject is available. For the human brain used in this study, such a consent is available.

The brains were removed from the skull within 24 hours after death, fixed with 4% buffered formaldehyde for several weeks, immersed in solutions of 10% and 20% glycerine combined with 2% Dimethyl sulfoxide for cryo-protection, and deeply frozen. The frozen brains were cut with a cryostat microtome (Leica Microsystems, Germany) into sections of 60 m. This section thickness ensures both a good quality of the sections (i.e. reduced fissures/deformations) and a sufficient signal strength13,14. The brain sections were mounted on glass slides, embedded in a solution of 20% glycerine, and cover-slipped. The glycerine solution can be used both as cryoprotectant and embedding medium because it does not impair the polarimetric measurements. The brain sections were measured one day after embedding (freshly embedded sections) or after several days to study the dependence of diattenuation on the embedding time.

### Polarimetric measurements

The measurements were performed with the in-house developed Large-Area Polarimeter (LAP) which was also used for the diattenuation studies by Menzel et al.31. The LAP consists of a green light source, a pair of crossed linear polarisers, and a quarter-wave retarder and specimen stage mounted in between the polarisers (see Fig. 1a). The light source contains a matrix of 36 × 36 LEDs (NSPG 510S, Nichia corporation) and a diffuser plate (PMMA) and emits mostly incoherent and unpolarised light with a wavelength λ = (525 ± 25) nm. The polarisers (XP38) and the retarder (WP140HE) were manufactured by ITOS, Germany. During a measurement, the polarisers and the retarder were rotated simultaneously by angles ρ = {0°, 10°, …, 170°} and the transmitted light intensity I(ρ) was recorded by a CCD camera (AxioCam HRc by Zeiss) with the microscanning procedure of the camera sensor, yielding 4164 × 3120 pixels with a resolution down to 14 μm/px.

The in-plane orientation angles φP of the nerve fibres were obtained from 3D-PLI measurements13,14 with the above setup and by performing a discrete harmonic Fourier analysis on the measured light intensities per image pixel (IP(ρ) = a0P + a2P cos (2ρ) + b2P sin (2ρ)) and evaluating the phase of the signal: φP = atan2(−a2P, b2P)/2. The amplitude of the signal |sin δP| is related to the phase shift δP induced by the birefringent brain section17 ($$|\sin \,{\delta }_{{\rm{P}}}|={({a}_{2{\rm{P}}}^{2}+{b}_{2{\rm{P}}}^{2})}^{1/2}/{a}_{0{\rm{P}}},\,{\delta }_{{\rm{P}}}\propto {\cos }^{2}{\alpha }_{{\rm{P}}}$$) and was used to compute the out-of-plane inclination angles αP of the fibres in Fig. 4e, making use of a tiltable specimen stage39 to improve the determination of the inclination angles.

The diattenuation measurements31 were performed by removing the retarder and the second polariser from the light path and measuring the transmitted light intensities as described above (see Fig. 1d). To account for the lower signal-to-noise ratio, the image of the brain section was recorded 20 times for each filter position and averaged. The strength of the measured diattenuation signal $$|{\mathscr{D}}|$$ and the phase φD were determined from the amplitude and phase of the resulting light intensity profile (ID(ρ) = a0D + a2D cos (2ρ) + b2D sin (2ρ)): $$|{\mathscr{D}}|={({a}_{2{\rm{D}}}^{2}+{b}_{2{\rm{D}}}^{2})}^{\mathrm{1/2}}/{a}_{0{\rm{D}}}$$ and φD = atan2(b2D,a2D)/2.

The Fourier coefficients of order zero {a0P, a0D} were used to register the images of the 3D-PLI measurements onto the images of the diattenuation measurements using in-house developed software tools based on the software packages ITK, elastix, and ANTs40,41,42,43,44 which perform linear and non-linear transformations.

The resulting images $$\{{\phi }_{{\rm{P}}},{\phi }_{{\rm{D}}},|{\mathscr{D}}|\}$$ were used to generate the diattenuation images (see Fig. 1c): $$|{\mathscr{D}}|$$ values belonging to regions with (φD − φP)  [−20°, 20°] were colourised in green (referred to as D+), regions with (φD − φP)  [−70°, 110°] were colourised in magenta (referred to as D). The angle ranges account for the uncertainties of φD due to the non-ideal optical properties of the LAP31: max |φD − φP| ≈ 20°(±90°) for $$|{\mathscr{D}}|=\mathrm{1 \% }$$.

### Measurements with prototypic polarising microscope

The prototypic polarising microscope has been described by Wiese et al.45. It consists of a green light source (λ = (532±5) nm), a rotatable polariser, a fixed circular analyser (quarter-wave retarder with linear polariser), and a CCD camera. The polarising filters are of higher quality than in the LAP. The microscope objective has a 4× magnification and a numerical aperture of 0.2, yielding a pixel size in object space of about 1.8 m. The measurements with the prototypic polarising microscope (see Supplementary Fig. S1) were performed as described above. Due to a superior signal-to-noise ratio, repeated diattenuation measurements were not necessary.

### Model of nerve fibre configurations

The fibre bundles used for the FDTD simulations (cf. Figure 4a) were generated by in-house developed software similar to software used in other groups46 allowing to create densely packed fibres without intersections: 700 straight fibres with different radii were randomly uniformly placed in an area of 45 × 30 μm2 and divided into segments of 2–5 μm. The fibre segments were assigned a random displacement in x, y, z: max. 10 μm for the bundle with broad fibre orientation distribution (Fig. 4a, left) and max. 1 μm for the bundle of densely grown fibres (Fig. 4a, right). The resulting fibre segments were split or merged until the length of each segment was again between 2–5 μm, ensuring that the maximum angle between adjacent segments was less than 20°. When a collision between two segments was detected, the segments were exposed to a small repelling force and the previous step was repeated until no more collisions were detected. The mode angle difference between the local fibre orientation vectors and the predominant orientation of the resulting fibre bundle is about 25° for the bundle with broad fibre orientation distribution and less than 10° for the bundle of densely grown fibres. To generate fibre bundles with different inclination angles, the bundles were rotated around the y-axis with respect to the center position and cropped to a volume of 30 × 30 × 30 μm3. To prevent fibres from touching each other after discretisation, all diameters were reduced by 5%.

Each fibre with radius r was modelled by an inner axon (rax = 0.65 r) and a surrounding myelin sheath with thickness tsheath = 0.35 r, which contributes approximately one third to the overall fibre radius47. In brain tissue, the myelin sheath consists of densely packed cell membranes, i.e. alternating layers of lipid bilayers (about 5 nm thickness) and intra- or extracellular space (about 3 nm thickness)18,48. When the brain sections are embedded in glycerine solution (as described above), the extracellular space is expected to become filled with glycerine solution and increases18,49. The myelin (lipid) and glycerine layers were therefore assumed to contribute 3/4 and 1/4 to the overall myelin sheath thickness, respectively. The simulations were performed for a simplified nerve fibre model, consisting of two myelin layers (tm = 3/7 tsheath) and a separating glycerine layer (tg = 1/7 tsheath). The refractive indices of axon, myelin and glycerine layers were estimated from literature values of cytoplasm50 (nax = 1.35), lipids/membranes38 (nm = 1.47), and from refractive index measurements of the glycerine solution used for embedding the brain sections (ng = 1.37, measured with digital refractometer). The surrounding medium was assumed to be homogeneous with a refractive index nsurr = 1.37, corresponding to the refractive index of gray matter51. The absorption coefficients of brain tissue are small52,53 and were therefore neglected. More details about the nerve fibre model can be found in Menzel et al.35.

The fibres shown in Fig. 4a have radii between 0.5 μm and 0.8 μm and a myelin sheath thickness of 0.35 r. The bundle of densely grown fibres was also simulated for different values of r, tsheath, and nm (see Fig. 4c and Supplementary Fig. S5).

### FDTD simulations

The propagation of the polarised light wave through the sample was computed by TDME3DTM–a massively parallel three-dimensional Maxwell Solver54,55,56 based on a finite-difference time-domain (FDTD) algorithm36,57. The algorithm numerically computes the electromagnetic field components by discretising space and time and approximating Maxwell’s curl equations by finite differences37.

The sample contains the fibre configuration (30 × 30 × 30 μm3, see above) and 0.5 m thick layers of glycerine solution on top and at the bottom. The dimensions of the simulation volume are 30 × 30 × 35 μm3, including 1 μm thick uniaxial perfectly matched layer absorbing boundaries. The different components of the sample were modelled as dielectrics with real refractive indices n (defined above). The sample was illuminated by a plane monochromatic wave with linear polarisation (along the x- or y-axis). The simulations were performed on the supercomputer JUQUEEN58 at Forschungszentrum Jülich GmbH, Germany, with 200 periods, a Courant factor of 0.8, and a Yee mesh size of 25 nm. Using an MPI grid of 16 × 16 × 16, each simulation run (calculation of one configuration and one polarisation state) consumed between 7000–8000 core hours, required a minimum memory between 260–360 GB, and lasted between 1:45–2:00 hours.

The polarimeter used for the polarimetric measurements (LAP) has a broad wavelength spectrum and emits mostly incoherent light under different angles of incidence. As each simulation run is performed with coherent light with a certain wavelength and angle of incidence, the LAP is not well suited to be modelled by FDTD simulations. Instead, we performed the simulations for the imaging system of the Polarising Microscope (PM)13,14, which has also been used to model the 3D-PLI measurement35. The PM (manufactured by Taorad GmbH, Germany) contains higher quality components and uses more coherent and less diffusive light with λ = (550 ± 5) nm. As the polarising filters cannot be removed, the PM cannot be used for diattenuation measurements. However, the optics of the PM are similar to those of the prototypic polarising microscope (the microscope objective has a 5× magnification, a numerical aperture of 0.15, and yields an object-space resolution of 1.33 μm/px). Therefore, the diattenuation effects simulated for the optics of the PM are presumably similar to those observed in the measurements (cf. Supplementary Fig. S1).

The imaging system of the PM was simulated as described in Menzel et al.35: The simulations were performed for normally incident light with 550 nm wavelength. The propagation of the light wave through the sample (fibre configuration) was computed by TDME3D, yielding a superposition of monochromatic plane waves with different wave vectors. The numerical aperture (NA ≈ 0.15) was modelled by considering only wave vector angles ≤ arcsin (NA) ≈ 8.6°. The spherical microlenses of the camera sensor were modelled by applying a moving average over the area of the microlens with a diameter of 1.33 μm. The light intensity recorded by the camera was computed as the absolute squared value of the electric field vector (neglecting angle dependencies because of the small aperture). To obtain the intensity at a certain point in the image plane, the electric field vectors for different wave vectors were summed and averaged over time. More details about the simulation model and error analysis can be found in Menzel et al.35.

To model the diattenuation measurement, the simulations were performed for light polarised along the x-axis and along the y-axis. The resulting light intensities (Ix and Iy) were used to compute the diattenuation images: DS = (IxIy)/(Ix + Iy). Positive diattenuation values (DS > 0 Ix > Iy) correspond to regions with D+ effect and were shown in green, regions with negative values (DS < 0 Ix < Iy) correspond to regions with D effect and were shown in magenta (cf. Figure 4b and Supplementary Fig. S4).

### Code availability

Image processing and data analysis were performed with Fiji (https://fiji.sc/Fiji) and in-house developed software tools. The artificial fibre geometries were generated with in-house developed software (using Python 3 and C++) similar to software developed by Altendorf & Jeulin46. The electric field vectors obtained from the Maxwell Solver were processed with in-house developed software using Python 2.7 and Numpy 1.12. All relevant program code, including the geometry parameters of the modelled fibre configurations, are available from the corresponding author upon reasonable request for purposes of academic research. For the FDTD simulations, we used TDME3D, a massively parallel Maxwell solver54,55,56. The software is property of EMBD (European Marketing and Business Development BVBA).