1. INTRODUCTION

Light microscopy in biological tissue can be hampered by a degradation of the focus due to refractive index inhomogeneities within the specimen [[1], [2], [3], [4], [5]]. Using adaptive wavefront correction the wavefront of the illumination light can be shaped in such a way that the specimen-introduced distortions are precompensated and a diffraction-limited focus is restored. Two fundamentally different approaches to adaptive wavefront correction can be distinguished [[6]]: (1) iterative optimization of the fluorescence signal using trial wavefront perturbations on the incident light, and (2) direct wavefront sensing and subsequent wavefront correction using the phase-conjugation approach. The first approach can be slow and requires the excitation of fluorescence light, which causes photobleaching and photodamage [[1], [2], [3], [7], [8]].

In contrast, coherence-gated wavefront sensing (CGWS) implements direct wavefront sensing and can measure the specimen-introduced distortions even in strongly scattering samples [[5], [9]]. CGWS is based on a low-coherence interferometer, which can select that portion of the backscattered sample light that originates from a region close to the focus. In combination with phase-shifting interferometry [[10]] and a real [[11]] or virtual Shack–Hartmann sensor (vSHS [[9], [12]]), the aberrations are reconstructed and can then be used for wavefront correction. The CGWS approach allows fast wavefront correction since the appropriate information about the distortions in the optical path can be obtained with a single set of measurements [[5]]. Here we investigate CGWS theoretically and experimentally and characterize CGWS performance, in particular with respect to coherence length, density of scatterers, coherence-gate (CG) position, and the polarization of illumination and reference light. Monte Carlo simulations (MCSs) of the model are compared with experimental results on test samples.

2. MODEL OF CGWS

In the following a model is developed to derive the characteristic properties of CGWS by simulating all experimentally implemented steps [[5], [9]]. While our model resembles models [[13], [14], [15]] of optical coherence tomography (OCT [[16]]), we pay particular attention to the phase of the backscattered light instead of its intensity. Based on ray tracing, the model determines the optical travel distances of light scattered by a random distribution of discrete scatterers within the geometrical focal (double) cone. Only singly scattered light is taken into account.

The electric field of the backscattered sample light ${\mathbf{E}}_S$ (complex-valued vector) interferes with that of the reference light ${\mathbf{E}}_R$ on a spatially resolving detector, whereupon the location on the detector is parameterized by u and v. The intensity of the interferogram is

The averaging over time can be carried out by using the (complex) degree of self-coherence $\gamma \left(\tau \right)$, which is also called the normalized self-coherence function of the scalar electric field E [[17]],

The first two terms in Eq. (3) are the total intensities of the light backscattered from the sample and of the reference light, respectively, both of which need to be subtracted to isolate the interference (third) term. This can be done, for example, by using phase-shifting interferometry [[10]]. The self-coherence function $\mid \gamma \left({\tau }^{\left(k\right)}\right)\mid$ weights the interference term and thus can select backscattered light that arrives within a certain time window [defined by $\mid \gamma \left({\tau }^{\left(k\right)}\right)\mid$] at the detector.

Usually, the detector is located in a plane that is optically conjugate to the back focal plane (BFP) of the objective [[5], [9]]. If one assumes a flat reference wavefront, a simple analytical expression for ${\tau }^{\left(k\right)}$ can be derived. The path difference between light scattered at $\mathrm{P}(x,y,z)$ [light path A–P–B, Fig. 1a ] and the reference light is

Because of the finite width of $\mid \gamma \left({\tau }^{\left(k\right)}\right)\mid$ for low-coherence light, mainly light scattered within a certain region, which we call the coherence volume (CV), contributes to the interference term in Eq. (3). Since $\mid \gamma \left({\tau }^{\left(k\right)}\right)\mid$ never becomes strictly zero, the definition of the CV is somewhat arbitrary. We use the region where $\mid \gamma \left[\tau (\mathbf{r},\mathbf{w})\right]\mid >0.5$. It is important to realize that, since $\tau [\mathbf{r},\mathbf{w}(u,v)]$ depends on the detection position $(u,v)$, the CV depends on the location in the BFP [Figs. 1b, 1c] and is therefore different for different sublenses.

The electric field of the coherence-gated sample light $E_{\mathit{CGWS}}(u,v)$ can be directly obtained from the interference term in Eq. (3),

3. METHODS

Our Monte Carlo simulation (MCS) is based on sets of discrete scatterers randomly sampled from a uniform distribution, closely follows the model derived in the previous section, and is similar to an approach [[24]] commonly used to analyze OCT.

First one ensemble of scatterers was established. Then the sum in Eq. (5) was calculated separately for each pixel on the detector. The time delay, τ depends essentially only on the location, r, of the scatterer and on the scattering direction, w, [Eq. (4), Fig. 1a]. Phase (time) shifts due to scattering (depending on the scattering angle) are included in $W_P^{\left(k\right)}$, but can be neglected, in particular for Rayleigh scatterers (see Section 4). By keeping track of the propagation of the polarization, which is changed by a generic objective [[25]] and by Mie scattering [[26]], the weight function $W_P^{\left(k\right)}$ can be calculated for each scattering event (for a detailed description see Appendix A). Since it is computationally expensive to keep track of the polarization for all light rays, a simplified version of the MCS was also implemented with $W_{\mathit{Pol}}^{\left(k\right)}=1$; i.e., the polarization of light is neglected and the Mie scattering phase functions are replaced by an isotropic scattering function. The simplified version of the MCS was used except when the polarization dependence was to be investigated.

For the calculation of $E_{\mathit{CGWS}}$ further simplifications were made. First, in the limit of geometrical optics the amplitude of the backscattered light depends only on z (as $\propto 1∕\mid z\mid$) for scatterers inside the illumination cone, assuming laterally uniform illumination. This is justified because MCSs showed the same results for both the uniform and the, experimentally used [[5]], Gaussian profiles. Second, the self-coherence function was assumed to be Gaussian, which is computationally simpler but very similar to a squared hyperbolic secant, which best describes the self-coherence function of passively mode-locked lasers, such as the Ti:sapphire oscillator used in our experiments [[27]].

To save computational time, the detector used for MCS calculations had $105\text{\hspace{0.17em} pixels}$ across the aperture diameter even though the detector used in the actual experiments sampled the aperture using $375\text{\hspace{0.17em} pixels}$ across the diameter. As long as the speckles are larger than $2\text{\hspace{0.17em} pixels}$ in diameter, this restriction does not lead to significant deviations [[12]]. Furthermore, scatterers were distributed uniformly but only up to 4 coherence lengths from the center of the CG position in both axial directions; Gaussian and squared hyperbolic secant functions have very small weight outside of this volume.

After the coherence-gated electric field was calculated, the wavefront was reconstructed by using the vSHS. The circular aperture was covered by 37 lenslets, each containing $15\times 15\text{\hspace{0.17em} pixels}$. For precise detection of the focus displacement for each lenslet the complex electric field was zero padded to a field of $65\times 65\text{\hspace{0.17em} pixels}$ before Fourier propagation. In addition to the centroid estimation, which was used unless noted otherwise, the peak position of the diffraction patterns was in some cases determined by least-squares fitting a Gaussian distribution to the central part of each diffraction pattern. Then the wavefront was reconstructed by least-squares fitting a linear combination of 21 Zernike modes (⩽ fifth order [[28]]) to the peak- or centroid-position displacements in all lenslets. For ensemble averaging at least three different ensembles of scatterers were used.

For the experimental measurements 441 lenslets, covering a circular aperture with a diameter of $375\text{\hspace{0.17em} pixels}$, were used. Before Fourier transformation of the coherence-gated electric field the $15\times 15\text{\hspace{0.17em} pixels}$ region of each lenslet was zero padded to $65\times 65\text{\hspace{0.17em} pixels}$ [[5], [12]]. Centroid estimation was used, except for the measurements with linearly polarized illumination light, where peak fitting was used [[9]]. The use of peak fitting had an effect only on rotationally symmetric aberrations, such as defocus (data not shown). The wavefront, averaged over 20 different ensembles of scatterers, was fitted by a linear combination of 28 Zernike modes (⩽ sixth order). Details can be found in [[5]].

4. RESULTS

4A. Properties of the Lenslets’ Diffraction Patterns

We first investigated how CGWS-measured wavefronts and their errors due to speckle depend on CG position, density of scatterers, and coherence length. The speckle error increases with decreasing speckle size in the BFP [[12]], because then the peaks in the virtual detection plane of the vSHS become broader. The speckle size scales inversely [[29]] with the number of phase singularities in CGWS-measured electric fields, which we determined by detecting whether the phase integrals along closed paths were zero or not [[30]]. We found (by MCS) that the number of singularities did not change with the density of scatterers but changed with CG position and coherence length [Figs. 2a, 2c, 2e ]. This is because the number of singularities depends linearly on the solid angle under which the CV is seen from the BFP (speckle size scales inversely with the solid angle [[31]]). Thus, with a smaller coherence length or with proximity of the CG position to the focus, the CV becomes smaller and the speckle error for a given number of ensemble averages is reduced.

First, no distortions were included in the optical path for these MCSs, and only the defocus component of the wavefront (Zernike coefficient $c_4$) was investigated. When the wavefront was determined by centroid estimation, $c_4$ remained constant as the coherence length or the density of scatterers was varied, but changed linearly with CG position [Figs. 2b, 2d, 2f]. Interestingly, $c_4$ was always smaller when determined by peak fitting than by centroid estimation with the difference increasing with increasing coherence length [Fig. 2d]; $c_4$ also changed linearly with CG position [with a smaller slope than centroid estimation, Fig. 2b] but remained constant as the density of scatterers was varied [Fig. 2f]. This means that centroid and fitted peak position diverge as the CG moves farther from the focus and the coherence length becomes longer [Figs. 2b, 2d]. This is likely due to the fact that the diffraction patterns of the vSHS (which are the projections of the CVs) are, in particular for peripheral lenslets, rather asymmetrical, and as a consequence that peak position and centroid are rather different [Figs. 3a, 3b ].

For a single scatterer located at the center of the CG on the optical axis, the change of $c_4$ with CG position should be $102\mathrm{nm}∕\mu \mathrm{m}$. For MCS the slopes found by centroid estimation and by peak fit were $119\pm 2$ and $78\pm 3\mathrm{nm}∕\mu \mathrm{m}$, respectively (Fig. 2b).

The centroid-estimation slope (MCS) is in good agreement with the experimental values of $117\pm 1$ and $101\pm 1\mathrm{nm}∕\mu \mathrm{m}$ for a scattering phantom containing $110\mathrm{nm}$ scattering beads (for preparation see [[5]]) and a chemically fixed organotypic rat hippocampus slice [[32]], respectively [Fig. 2g].

4B. CGWS-Measured Wavefront Aberrations

Next we investigated whether the distortions present in the optical path are accurately reflected in the wavefront measured by CGWS. This is crucial for fast and complete wavefront correction. A systematic bias might occur, for example, because the light encounters the distortions in the sample arm twice, on the way to the focus and on the way back (double pass), while for the preemptive wavefront correction in two-photon microscopy only the information about the distortions on the way to the focus is needed. The behavior of the wavefront sensor can depend strongly on the sample properties, as becomes very apparent when comparing a mirror and a single scatterer at the focus location. For a mirror double passing doubles point-symmetric aberrations, such as astigmatism or defocus, but eliminates point antisymmetric aberrations, such as coma, which therefore cannot be detected at all. For a single scatterer, from which a spherical wavelet emanates, the aberrations of the illumination light are lost completely, and only inhomogeneities encountered during backpropagation are seen (single-pass aberrations). Therefore, with a single scatterer single-pass aberrations are correctly detected and, as a consequence of optical reciprocity [[33]], are equal to the aberrations encountered by the incoming light on the way to the focus.

For a collection of randomly distributed scatterers we used MCSs with distortions introduced by phase plates to investigate whether the single-pass aberrations are measured correctly by CGWS. To account for the change in travel time caused by the distortions, we replaced $\tau (\mathbf{r},\mathbf{w})$ in Eq. (5) with

As an example, we inserted at the BFP of the objective a phase plate carrying astigmatism, $c_6=-0.3\mu \mathrm{m}$ $(c_5=0\mu \mathrm{m})$, and coma, $c_8=-0.5\mu \mathrm{m}$ $(c_7=0\mu \mathrm{m})$, which was passed by the incoming and the scattered light. For all coherence lengths tested the wavefronts determined by MCS (the density of scatterers was $10\mu {\mathrm{m}}^{-3}$) were those expected for the single-pass aberrations [Fig. 4b]. As in the single-scatterer case, aberrations encountered by the incoming light are lost completely, and therefore the wavefront detected corresponds to the incoherent superposition of the coherence-gated light and is the amplitude-weighted average of the wavefronts independently emanating from all the scatterers within the CV. This result does not depend on the type and amount of aberration and is consistent with measurements described below and in [[5], [9]]. However, the assumption of randomly distributed scatterers is essential, since the backscattered light from, for example, a dense layer of scatterers in a single plane (acting as a mirror) would, of course, be sensitive to distortions in the incoming light path.

Another assumption needed for the incoherent superposition to hold [Eq. (9)] is that the number of scatterers within the focal cone is lower than ${\rho }_{\lim }=\mid (\alpha -\beta )∕\beta \mid$, beyond which the coherent term would dominate the incoherent term, even if the scatterers are randomly distributed. To roughly estimate α and β, we assumed an objective with a low numerical aperture (NA). Then, because $x,y⪡z$, only the axial coordinates of the scatterers need to be taken into account. Furthermore, we neglect polarization effects $(W_{\mathit{Pol}}=1)$ and assume that the detection points are ${\mathbf{x}}_1={\mathbf{x}}_2=0$, which, from Eq. (5), yields

It was also verified by MCS for an experimentally realistic coherence length of $50\mu \mathrm{m}$ [[5]] that for scatterer densities tested up to $6\times {10}^5\mu {\mathrm{m}}^{-3}$ coherent effects do not affect wavefront measurements.

All distortions (Zernike modes higher than $c_5$) are therefore correctly measured by CGWS, but can focus displacements due to spatially varying tip–tilt and defocus, which are not accompanied by a change in focus shape but cause image distortions, also be measured? For episcopic illumination focus displacements due to tip and tilt cannot be sensed [[36]], since a conjugate displacement is introduced on the way back to the objective. An axial displacement of the focus due to a defocus introduced into the optical path is, however, detectable, since CGWS is sensitive to the travel time of the scattered light.

4C. Aberrations Close to Focus

A bias toward a flat wavefront may, however, be introduced if the distortion layer is located close to the focus rather than far above. Then the wavefront as measured by CGWS no longer accurately reflects the actual specimen-introduced distortions. The reason is that in this case, due to the lateral extent of the CV, light scattered at different locations is affected by refractive index inhomogeneities differently, which then leads to spatial averaging of distortions [Fig. 4a]. We explored (using MCS) this effect by inserting phase plates at various distances above the focus. We chose for the phase plates the same nominal distortion ($c_6=-0.5\mu \mathrm{m}$ and $c_8=-0.3\mu \mathrm{m}$) across the diameter of the focal cone at the point of insertion. As a result the same aberrations are caused in a spherical wave emanating directly from the focus, but note that phase plates closer to the focus contain aberrations with higher spatial frequencies.

This is in contrast to real samples, where distortions with a certain refractive index variation would typically cause less aberration closer to the focus.

MCSs show that nearly correct distortion values are obtained as long as the phase plate is sufficiently far from the focus with substantial deviations visible only at distances below $50\mu \mathrm{m}$ for a coherence length of $50\mu \mathrm{m}$ and for a CG position at $5\mu \mathrm{m}$ [Fig. 4c]. This deviation toward a flat wavefront is due solely to spatial averaging of the distortions and not to scatterers lying above the phase plate, because those were not included in the MCS. For a CG position centered at the focus no significant deviations are observed, presumably because of the smaller lateral extent of the CV [Fig. 4c].

4D. Aberrations Caused by a Tilted Glass Plate

Next we tested, using experimental measurements and MCSs, whether CGWS correctly detects the distortions due to a tilted glass plate (BK7, thickness $145\mu \mathrm{m}$, 10° tilt angle) inserted between the scattering sample and the objective [Fig. 5a ]. To model this situation realistically, the time delays for illumination and scattered light due to retardation and refraction by the glass plate need to be taken into account, which can be done by ray tracing using straightforward geometrical considerations. Since the effect of the glass plate on a transmitted ray is invariant under lateral and axial translations, a displacement of the CG position will not change the measured aberrations in this particular case. Note, however, that for an objective that meets the sine condition [[33]] all rotationally symmetric aberrations, such as, e.g., $c_4$ (defocus) or $c_{11}$ (first-order spherical aberration) vary with CG position even when no actual aberrations are present, albeit with steeply declining coefficients as the order increases.

The aberrations caused by the tilted glass plate and calculated by ray tracing for a single scatterer on the optical axis were mainly astigmatism, $c_6=-90\mathrm{nm}$ ($c_5$ was zero because the tilt was along one of the principal axes, in this case the y axis), coma, $c_8=-220\mathrm{nm}$ $(c_7=0\mathrm{nm})$, and spherical aberration, $c_{11}=-170\mathrm{nm}$. For a random collection of scatterers MCS also showed astigmatism, $c_6$, and coma, $c_8$, which remained almost constant when the CG position was varied [Fig. 5b], with average values $c_6=-100\pm 20\mathrm{nm}$ and $c_8=-220\pm 10\mathrm{nm}$. The defocus, $c_4$, and the spherical aberration, $c_{11}$, changed linearly with slopes of $108\pm 5$ and $6.0\pm 0.5\mathrm{nm}∕\mu \mathrm{m}$, respectively, with the CG position at the focus $c_{11}=-167\pm 5\mathrm{nm}$. All other Zernike coefficients tested (up to $c_{11}$) were below the speckle-noise level of about $10\mathrm{nm}$.

Using a scattering sample with $110\mathrm{nm}$ beads [[5], [9]] we measured the aberrations caused by a tilted glass plate experimentally [Fig. 5c] and found $c_6=-90\pm 20\mathrm{nm}$ and $c_8=-230\pm 10\mathrm{nm}$ (averaged over all measured CG positions), in agreement with the values found by MCS. For the measured value of $c_{11}=-120\pm 10\mathrm{nm}$ (CG position centered at the focus) there is a substantial discrepancy with the value obtained by MCS $(c_{11}=-167\pm 5\mathrm{nm})$, possibly because the refractive index of the scattering sample is slightly different from that of water. The slope of $c_4$ versus CG position was $120\pm 5\mathrm{nm}∕\mu \mathrm{m}$, and that for $c_{11}$ was $4\pm 1\mathrm{nm}∕\mu \mathrm{m}$, both in good agreement with slopes found with MCSs (see above). Experimentally, $c_6$ changed slightly with CG position, possibly due to the illumination-light polarization’s being not strictly circularly (see below).

These aberrations obtained experimentally, by MCS, and calculated for a single scatterer are in agreement, showing that single-pass aberrations caused by the tilted glass plate are correctly measured.

4E. Polarization Effects

The phase and amplitude of the coherence-gated electric field in the BFP are affected not only by the self-coherence function $\gamma \left[{\tau }^{\left(k\right)}(u,v)\right]$ and the amplitude distribution across the focal cone $E_S^{\left(k\right)}(u,v)$ but also by the polarization dependence of scattering as described [Eq. (5)] by $W_P^{\left(k\right)}(u,v)$, which can be calculated exactly for spherical scatterers by using Mie scattering theory [[37]], which, in addition to providing polarization-resolved amplitudes also provides the phase delay for any scattering angle. Here, the effects of the incident- and reference-light polarizations on CGWS-measured wavefronts is investigated (preliminary results have been published in [[38]]).

First, measurements with linearly polarized light, with parallel directions for the incident sample and reference light, were performed by using the optical setup described in [[9]]. Two scattering samples (with beads of $100\mathrm{nm}$ and $1\mu \mathrm{m}$ diameter) with distinct polarization-dependent scattering properties were examined. As the CG position was changed, not only did the defocus $c_4$, change (not shown), which was expected, but also the astigmatism $c_6$ [Figs. 6a, 6b ], which was not expected. The slope for $c_6$ ($c_5\approx 0\mu \mathrm{m}$ for linear polarization along the x axis) depended on the bead size with values of $370\pm 3$ and $8\pm 3\mathrm{nm}∕\mu \mathrm{m}$ for larger and smaller beads, respectively. None of the other Zernike modes varied with CG position. The measured astigmatism cannot be due to actual distortions, because it depends on the bead size.

The origin of spurious astigmatism is somewhat complicated. It is not caused by the polarization dependence of the phase shift during scattering, which MCSs show to be too small to account for the observed astigmatism. This is obvious for scatterers that are small compared with the wavelength (Rayleigh scatterers), where the phase delays are almost independent of scattering angle within the acceptance cone of the objective (NA of 0.9). What is responsible instead for the spurious astigmatism is an angle-, polarization-, BFP-position-, and scatterer-position-dependent weight, which can be written as $W_P({\mathbf{r}}^{\left(k\right)},u,v,{\mathbf{p}}_R,{\mathbf{p}}_I)$, where ${\mathbf{r}}^{\left(k\right)}=(x,y,z)$ is the position of the scatterer k, $(u,v)$ is the detection position, and ${\mathbf{p}}_R$ and ${\mathbf{p}}_I$ are the polarizations of the reference and incident sample light, respectively [see Eq. (5)]. Note that the time delay ${\tau }^{\left(k\right)}(u,v)=\tau ({\mathbf{r}}^{\left(k\right)},u,v)$ does not change when both the scatterer, $(x,y)$, and the detection point, $(u,v)$, rotate around the optical axis, but $W_P({\mathbf{r}}^{\left(k\right)},u,v,{\mathbf{p}}_R,{\mathbf{p}}_I)$ will generally change.

For linearly polarized light rotational symmetry is obviously broken, but $W_P$ still shows two mirror symmetries, parallel and perpendicular to the direction of the linear polarization. If the linear polarization is along either the x or the y axis

Symmetry arguments do not, however, allow quantitative predictions. For those we used MCSs. To model the effects that a generic objective lens has on polarization, refraction by a prism was used [[25]]. The steps performed to keep track of the polarization state are detailed in Appendix A. Because keeping track of the polarization is computationally expensive, a coherence length of $2\mu \mathrm{m}$ and a density of scatterers of $1\mu {\mathrm{m}}^{–3}$ were chosen to keep the number of scatterers small. As shown above for the polarization-insensitive case, wavefronts calculated by the MCSs are, except for the size of the speckle error, rather insensitive to the coherence length and density of scatterers. We confirmed this for the polarization-sensitive case, where we performed MCSs with a coherence length of $8\mu \mathrm{m}$ for some CG positions and found that the results were unchanged (data not shown).

For linearly polarized illumination and reference light (polarizations parallel), MCSs showed a linear change of $c_6$ with CG position [Figs. 6c, 6d]. The slopes of $24\pm 4$ and $50\pm 5\mathrm{nm}∕\mu \mathrm{m}$ for $100\mathrm{nm}$ and $1\mu \mathrm{m}$ beads, respectively, were, however, significantly larger than the experimental values of $8\pm 3$ and $37\pm 3\mathrm{nm}∕\mu \mathrm{m}$. In addition we found, by MCS but not in our experiments, for $1\mu \mathrm{m}$ beads a spurious $c_{12}$ with a slope of $-6\pm 1\mathrm{nm}∕\mu \mathrm{m}$.

Possible explanations for these discrepancies are, first, that only singly scattered light is taken into account for the MCSs, but experimentally there was no way to exclude coherence-gated multiply scattered light. Multiple scattering should reduce the observed spurious aberrations, since then polarization effects are averaged over a range of scattering angles. Another explanation could be the depolarization of the incident sample light caused by polarization-dependent transmission losses [[39]] at individual lens components within the objective [[40], [41]], which was not taken into account in MCSs.

Important for the application of CGWS to wavefront correction is that all spurious aberrations vanish when the CG is centered at the focus. But polarization-mediated spurious aberrations can be avoided completely (for all CG positions) if circularly polarized illumina tion and reference light is used, since then $W_P(x,y,z,u,v,{\mathbf{p}}_R,{\mathbf{p}}_I)$ is rotationally symmetric.

For the experimental measurements we used the optical setup described in [[5]], where the incident sample light was circularly polarized, but the reference light was linearly polarized. However, since the backscattered sample light passes a quarter-wave plate, this is equivalent to circularly polarized reference light. We found only a very small amount of spurious astigmatism, $c_6$, with slopes of $2\pm 2$ and $6\pm 3\mathrm{nm}∕\mu \mathrm{m}$ for $100\mathrm{nm}$ or $1\mu \mathrm{m}$ beads, respectively [Figs. 7a, 7b ]. For the $1\mu \mathrm{m}$ beads CG position-dependent $c_{12}$ was significant, with a slope of $3\pm 1\mathrm{nm}∕\mu \mathrm{m}$. No significant CG position-dependent $c_6$ and $c_{12}$ were seen by MCS [Figs. 7c, 7d]. The slopes of $c_4$ (defocus), obtained by MCS for circularly polarized light, were $99\pm 4$ and $104\pm 4\mathrm{nm}∕\mu \mathrm{m}$ for $100\mathrm{nm}$ and $1\mu \mathrm{m}$ beads, respectively, very similar to the slope obtained by polarization-insensitive MCSs where point scatterers had been assumed ($119\pm 2\mathrm{nm}∕\mu \mathrm{m}$; see above).

5. DISCUSSION

Experimentally measured wavefronts using CGWS were compared with wavefronts obtained by Monte-Carlo simulations (MCS) and, in some cases, to direct calculations. All experimentally observed properties of CGWS-measured wavefronts, such as the change of the Zernike defocus with the variation of the CG position and that the measured wavefront is not affected by aberrations in the incident beam, are consistent with MCS results. Only for the dependence of the CGWS-measured wavefronts on the incident-light polarization did we find a discrepancy between experiment and MCS. However, in the model a number of additional simplifications were made, which need to be discussed. First, attenuation of the illumination intensity with depth due to scattering and absorption was ignored. Attenuation results in more weight being given to scatterers near the surface of the specimen and thus shifts the centroid of the CV toward the surface, in particular if the attenuation length is comparable to the axial extent of the CV. Such a shift will significantly affect rotationally symmetric Zernike modes.

Second, our model neglects multiply scattered light, which can have a total travel time that allows it to pass the CG while having been scattered by scatterers outside the single-scattering CV. In strongly scattering specimens multiple scattering can be neglected only for low probing depths. To estimate at which focus depth multiply scattered light might begin to dominate singly scattered light, results from OCT can be used. For OCT the maximum probing depth is, even with confocal detection [[13], [15]], which strongly suppresses multiply scattered light, limited by imaging contrast to several (5–8) mean free path lengths, depending on the scattering properties of the tissue and on the optical configuration [[14], [24], [42]]. For CGWS, however, a pinhole, which acts as a spatial filter, in the sample arm could lead to an underestimation of actual distortions. A subresolution pinhole size, for example, would not only suppress the signal size by orders of magnitude but would also completely filter out all aberrations. However, a properly sized pinhole should suppress multiply scattered light without much affecting the singly scattered light, reducing both bias and noise. Even if aberrations are underestimated, proper correction might still be possible either by correcting, for a known degree of underestimation or by using a larger number of measure correction iterations [[5]].

Since the coherence-gated multiply scattered light that is high-angle scattered more than once is predominantly scattered above the CV (of the singly scattered light) and therefore originates from a laterally more extended volume, the CGWS is affected in two ways. First, the speckle size of the coherence-gated backscattered light is reduced, leading to a larger speckle error (see above), which then requires averaging over more ensembles of scatterers to achieve a certain wavefront error. Second, lateral averaging of distortions deeper inside the sample (closer to the focus) is more severe (see above) leading to an underestimation of the actual distortions.

A third assumption we made is that of discrete, randomly located scattering particles. This assumption relies on studies that show that the scattering properties of biological specimens can be mimicked by distributions of discrete particles of different sizes [[35], [43], [44]]. Most of the extinction by scattering is caused by sizes $2\lambda –4\lambda$ [[35], [43], [44]], but high-angle scattering (backscattering) is due mainly to sizes $\lambda ∕4–\lambda ∕2$ [[34], [35]]. Cellular organelles, in particular mitochondria ($0.3–0.7\mu \mathrm{m}$ in diameter), lysosomes $(0.2–0.5\mu \mathrm{m})$, and structures within the nucleus thus contribute most to backscattering [[34], [45]]. For particles in the range of $\lambda ∕4–\lambda ∕2$ backscattering is largely independent of size and thus behaves as pointlike.

The assumption that the distribution of scatterers is spatially homogenous is likely to be violated to a varying degree in real tissue.

6. CONCLUSION

We have demonstrated both experimentally and by Monte Carlo simulation that wavefronts measured by CGWS represent single-pass specimen-introduced distortions correctly. Since for a two-photon microscope [[46]] only the distortions for the incident light need to be compensated, the necessary wavefront aberrations are directly measured by CGWS and can thus be used for wavefront correction in a single step. This allows fast feed-forward wavefront correction.

APPENDIX A: PROPAGATION OF POLARIZED LIGHT IN THE SAMPLE ARM

The interference of reference and sample light in terms of polarization is described by $W_P^{\left(k\right)}={\mathbf{p}}_R^{*}\bullet {\mathbf{p}}_S^{\left(k\right)}$ [Eq. (5)], where ${\mathbf{p}}_R$ and ${\mathbf{p}}_S^{\left(k\right)}$ are the polarizations of reference light and of light backscattered at the scatterer k, respectively. The effects by the objective on polarization can be locally approximated by that of a prism [[25], [47]], and the polarization dependence of scattering is contained in Mie’s theory [[37]].

First, the light rays were traced through the sample arm, and at each change of direction a new local basis $({\mathbf{e}}_{\vartheta },{\mathbf{e}}_{\phi },{\mathbf{e}}_r)$ was defined (Fig. 8 ) relative to a fixed basis $({\mathbf{e}}_x,{\mathbf{e}}_y,{\mathbf{e}}_z)$ with ${\mathbf{e}}_x\times {\mathbf{e}}_y={\mathbf{e}}_z$. For each local basis ${\mathbf{e}}_{\vartheta }^{\left(i\right)}\times {\mathbf{e}}_{\phi }^{\left(i\right)}={\mathbf{e}}_r^{\left(i\right)}$, whereby ${\mathbf{e}}_r^{\left(i\right)}$ points in the current direction of propagation and ${\mathbf{e}}_{\vartheta }^{\left(i\right)}$ lies in the plane that contains both the current and the previous direction of propagation. An exception is the first basis, which essentially corresponds to the fixed basis, but ${\mathbf{e}}_r^{\left(1\right)}$ is in the direction of propagation for the incident sample light (see Fig. 8).

Basis 1. For the incoming light rays

$${\mathbf{e}}_r^{\left(1\right)}=-{\mathbf{e}}_z;{\mathbf{e}}_{\phi }^{\left(1\right)}=-{\mathbf{e}}_y;{\mathbf{e}}_{\vartheta }^{\left(1\right)}={\mathbf{e}}_x.$$

Basis 2. Rotation of basis 1 about ${\mathbf{e}}_r^{\left(1\right)}$ such that ${\mathbf{e}}_{\vartheta }^{\left(1\right)}$ lies in the plane of refraction (containing both ${\mathbf{e}}_r^{\left(1\right)}$ and ${\mathbf{e}}_r^{\left(3\right)}$) caused by the objective

$${\mathbf{e}}_r^{\left(2\right)}={\mathbf{e}}_r^{\left(1\right)};{\mathbf{e}}_{\phi }^{\left(2\right)}={\mathbf{e}}_{\phi }^{\left(3\right)};{\mathbf{e}}_{\vartheta }^{\left(2\right)}={\mathbf{e}}_{\phi }^{\left(2\right)}\times {\mathbf{e}}_r^{\left(2\right)}.$$

Basis 3. After passing the objective but before scattering at $\mathrm{P}(x,y,z)$,

$${\mathbf{e}}_r^{\left(3\right)}=-\frac{\mathrm{sgn}\left(z\right)}{\sqrt{x^2+y^2+z^2}}(x{\mathbf{e}}_x+y{\mathbf{e}}_y+z{\mathbf{e}}_z);$$

$${\mathbf{e}}_{\phi }^{\left(3\right)}=\frac{{\mathbf{e}}_r^{\left(1\right)}\times {\mathbf{e}}_r^{\left(3\right)}}{\mid {\mathbf{e}}_r^{\left(1\right)}\times {\mathbf{e}}_r^{\left(3\right)}\mid };{\mathbf{e}}_{\vartheta }^{\left(3\right)}={\mathbf{e}}_{\phi }^{\left(3\right)}\times {\mathbf{e}}_r^{\left(3\right)}.$$

Basis 4. Rotation of basis 3 about ${\mathbf{e}}_r^{\left(3\right)}$ such that ${\mathbf{e}}_{\vartheta }^{\left(3\right)}$ lies in the scattering plane (containing ${\mathbf{e}}_r^{\left(3\right)}$ and ${\mathbf{e}}_r^{\left(5\right)}$)

$${\mathbf{e}}_r^{\left(4\right)}={\mathbf{e}}_r^{\left(3\right)};{\mathbf{e}}_{\phi }^{\left(4\right)}=\frac{{\mathbf{e}}_r^{\left(4\right)}\times {\mathbf{e}}_r^{\left(5\right)}}{\mid {\mathbf{e}}_r^{\left(4\right)}\times {\mathbf{e}}_r^{\left(5\right)}\mid };{\mathbf{e}}_{\vartheta }^{\left(4\right)}={\mathbf{e}}_{\phi }^{\left(4\right)}\times {\mathbf{e}}_r^{\left(4\right)}.$$

Basis 5. After scattering in the direction w,

$${\mathbf{e}}_r^{\left(5\right)}=w_x{\mathbf{e}}_x+w_y{\mathbf{e}}_y+w_z{\mathbf{e}}_z;{\mathbf{e}}_{\phi }^{\left(5\right)}={\mathbf{e}}_{\phi }^{\left(4\right)};{\mathbf{e}}_{\vartheta }^{\left(5\right)}={\mathbf{e}}_{\phi }^{\left(5\right)}\times {\mathbf{e}}_r^{\left(5\right)}.$$

Basis 6. Rotation of basis 5 about ${\mathbf{e}}_r^{\left(5\right)}$ such as ${\mathbf{e}}_{\vartheta }^{\left(5\right)}$ lies in the plane of refraction (containing ${\mathbf{e}}_r^{\left(5\right)}$ and ${\mathbf{e}}_r^{\left(7\right)}$) caused by the objective

$${\mathbf{e}}_r^{\left(6\right)}={\mathbf{e}}_r^{\left(5\right)};{\mathbf{e}}_{\phi }^{\left(6\right)}={\mathbf{e}}_{\phi }^{\left(7\right)};{\mathbf{e}}_{\vartheta }^{\left(6\right)}={\mathbf{e}}_{\phi }^{\left(6\right)}\times {\mathbf{e}}_r^{\left(6\right)}.$$

Basis 7. After passing the objective on the way back ($f_{\mathit{\text{back}}}$ is the back focal length),

$${\mathbf{e}}_r^{\left(7\right)}=(x-\frac{z}{w_z}{w}_x){\mathbf{e}}_x+(y-\frac{z}{w_z}{w}_y){\mathbf{e}}_y+f_{\mathit{\text{back}}}{\mathbf{e}}_z,$$

$${\mathbf{e}}_{\phi }^{\left(7\right)}=\frac{{\mathbf{e}}_r^{\left(5\right)}\times {\mathbf{e}}_r^{\left(7\right)}}{\mid {\mathbf{e}}_r^{\left(5\right)}\times {\mathbf{e}}_r^{\left(7\right)}\mid };{\mathbf{e}}_{\vartheta }^{\left(7\right)}={\mathbf{e}}_{\phi }^{\left(7\right)}\times {\mathbf{e}}_r^{\left(7\right)}.$$

In the next step, the polarization, described by using the Jones formalism [[48]], is traced for each change of the local basis. The polarization of the incoming light, expressed in the fixed basis, is ${\mathbf{p}}_{\mathit{ini}}=a_{\mathit{ini}}{\mathbf{e}}_x+b_{\mathit{ini}}{\mathbf{e}}_y$, which is the same as the polarization of the reference light ${\mathbf{p}}_R$.

Fixed $\mathit{\text{Basis}}\to \mathit{\text{Basis}}$ 1. Transformation into the local basis

$$\left(\begin{array}{c}{a}_1\\ b_1\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\bullet \left(\begin{array}{c}{a}_{\mathit{ini}}\\ b_{\mathit{ini}}\end{array}\right)$$

with ${\mathbf{p}}_{\mathit{ini}}=a_{\mathit{ini}}{\mathbf{e}}_x+b_{\mathit{ini}}{\mathbf{e}}_y$ and ${\mathbf{p}}_k^{\left(1\right)}=a_1{\mathbf{e}}_{\vartheta }^{\left(1\right)}+b_1{\mathbf{e}}_{\phi }^{\left(1\right)}$.

Basis $\mathit{1}\to \mathit{\text{Basis}}$ 2. Rotation into the plane of refraction

$$\left(\begin{array}{c}{a}_2\\ b_2\end{array}\right)=\left(\begin{array}{cc}\cos \left(\beta \right)& \sin \left(\beta \right)\\ -\sin \left(\beta \right)& \cos \left(\beta \right)\end{array}\right)\bullet \left(\begin{array}{c}{a}_1\\ b_1\end{array}\right)$$

with $\cos \left(\beta \right)={\mathbf{e}}_{\vartheta }^{\left(2\right)}•{\mathbf{e}}_{\vartheta }^{\left(1\right)}$ and $\sin \left(\beta \right)={\mathbf{e}}_{\vartheta }^{\left(2\right)}•{\mathbf{e}}_{\phi }^{\left(1\right)}$.

Basis $\mathit{2}\to \mathit{\text{Basis}}$ 3. Change due to the objective [[25]]

$$\left(\begin{array}{c}{a}_3\\ b_3\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\bullet \left(\begin{array}{c}{a}_2\\ b_2\end{array}\right).$$

Basis $\mathit{3}\to \mathit{\text{Basis}}$ 4. Rotation into the scattering plane

$$\left(\begin{array}{c}{a}_4\\ b_4\end{array}\right)=\left(\begin{array}{cc}\cos \left(\phi \right)& \sin \left(\phi \right)\\ -\sin \left(\phi \right)& \cos \left(\phi \right)\end{array}\right)\bullet \left(\begin{array}{c}{a}_3\\ b_3\end{array}\right)$$

with $\cos \left(\phi \right)={\mathbf{e}}_{\vartheta }^{\left(4\right)}•{\mathbf{e}}_{\vartheta }^{\left(3\right)}$ and $\sin \left(\phi \right)={\mathbf{e}}_{\vartheta }^{\left(4\right)}•{\mathbf{e}}_{\phi }^{\left(3\right)}$.

Basis $\mathit{4}\to \mathit{\text{Basis}}$ 5. Mie scattering functions $S_1\left(\vartheta \right)$ and $S_2\left(\vartheta \right)$ [[26]]

$$\left(\begin{array}{c}{a}_5\\ b_5\end{array}\right)=\left(\begin{array}{cc}{S}_2\left(\vartheta \right)& 0\\ 0& S_1\left(\vartheta \right)\end{array}\right)\bullet \left(\begin{array}{c}{a}_4\\ b_4\end{array}\right)$$

with $\cos \left(\vartheta \right)={\mathbf{e}}_r^{\left(4\right)}•{\mathbf{e}}_r^{\left(5\right)}$.

Basis $\mathit{5}\to \mathit{\text{Basis}}$ 6. Rotation into the plane of refraction

$$\left(\begin{array}{c}{a}_6\\ b_6\end{array}\right)=\left(\begin{array}{cc}\cos \left(\epsilon \right)& \sin \left(\epsilon \right)\\ -\sin \left(\epsilon \right)& \cos \left(\epsilon \right)\end{array}\right)\bullet \left(\begin{array}{c}{a}_5\\ b_5\end{array}\right)$$

with $\cos \left(\epsilon \right)={\mathbf{e}}_{\vartheta }^{\left(6\right)}•{\mathbf{e}}_{\vartheta }^{\left(5\right)}$ and $\sin \left(\epsilon \right)={\mathbf{e}}_{\vartheta }^{\left(6\right)}•{\mathbf{e}}_{\phi }^{\left(5\right)}$.

Basis $\mathit{6}\to \mathit{\text{Basis}}$ 7. Change due to objective [[25]]

$$\left(\begin{array}{c}{a}_7\\ b_7\end{array}\right)=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\bullet \left(\begin{array}{c}{a}_6\\ b_6\end{array}\right).$$

Basis $7\to \mathit{\text{Fixed}}$ Basis.

$$\left(\begin{array}{c}{a}_f\\ b_f\end{array}\right)=\left(\begin{array}{cc}{\mathbf{e}}_{\vartheta }^{\left(7\right)}\bullet {\mathbf{e}}_x& {\mathbf{e}}_{\phi }^{\left(7\right)}\bullet {\mathbf{e}}_x\\ {\mathbf{e}}_{\vartheta }^{\left(7\right)}\bullet {\mathbf{e}}_y& {\mathbf{e}}_{\phi }^{\left(7\right)}\bullet {\mathbf{e}}_y\end{array}\right)\bullet \left(\begin{array}{c}{a}_7\\ b_7\end{array}\right)$$

with ${\mathbf{p}}_S^{\left(k\right)}=a_f{\mathbf{e}}_x+b_f{\mathbf{e}}_y$. Since $a_f$ and $b_f$ are related to $a_{\mathit{ini}}$ and $b_{\mathit{ini}}$, the polarization weight $W_P^{\left(k\right)}={\mathbf{p}}_R^{*}\bullet {\mathbf{p}}_S^{\left(k\right)}$ can be calculated for each scattered light ray.

ACKNOWLEDGMENTS

We thank Manfred Hauswirth, Michael Müller, and Jürgen Tritthardt for technical support, and Jonas Binding, Marcus Feierabend, and Marcel Lauterbach for helpful discussions and comments on the manuscript. The work was supported by the Max-Planck Society.

 

Fig. 1 (a) Calculation of ${\tau }^{\left(k\right)}$ for sample light scattered by a scatterer at $\mathrm{P}(x,y,z)$ based on ray tracing. The incident ray passes the BFP at point A, is refracted by the objective lens toward the focus F, scatters at P in the direction w, and crosses the BFP at B on its return path (red online). For the CG centered at the focus the reference light takes a path that equals in length C-F-C (blue online) and A-F-B. Typical CVs, shown in cross section, for detection points (b) in the center and (c) at the edge of the BFP of the objective, respectively. The dotted curves correspond to $\mid \gamma \left(\tau \right)\mid =0.5$ and delineate the CV. The coherence length was $50\mu \mathrm{m}$ (FWHM, normalized Gaussian self-coherence function), corresponding to a CG length of $18.8\mu \mathrm{m}$ within the specimen, and the CG position was $+5\mu \mathrm{m}$.

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Fig. 2 (a), (c), (e) Number of phase singularities and (b), (d), (f) size of $c_4$ as they change with (a), (b) CG position, (c), (d), coherence length, (e), (f), and density of scatterers. Plotted are $c_4$ values as determined by estimating the centroid and those determined by fitting the peak by a Gaussian distribution. When not varied, the CG position, the density of scatterers, and the coherence length were $+5\mu \mathrm{m}$, $100\mu {\mathrm{m}}^{-3}$, and $30\mu \mathrm{m}$ (FWHM), respectively. Linear fits to the data in (b) yield slopes of $119\pm 0.002\mathrm{nm}∕\mu \mathrm{m}$ (centroid) and of $78\pm 0.003\mathrm{nm}∕\mu \mathrm{m}$ (peak fit). Error bars are not shown in (b) because they are too small. (g) Experimentally measured $c_4$ as a function of the CG position for a scattering sample with $110\mathrm{nm}$ scattering beads (crosses, black) and for a chemically fixed organotypic rat hippocampus slice (dots, red). Also shown are linear fits to the data giving slopes of $101\pm 0.001$ and $117\pm 0.001\mathrm{nm}∕\mu \mathrm{m}$ (very close to the theoretical expected value of $119\pm 0.002\mathrm{nm}∕\mu \mathrm{m}$). Note that the vertical displacement between the data series is not meaningful, since only relative CG positions were measured. Except for (b) and (g), the data points were connected by straight lines.

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Fig. 3 Asymmetric diffraction patterns for off-axis vSHS lenslets. (a) Illustration of how the CV is projected through a sublens. Peak fit (P) and centroid estimation (C) of the diffraction pattern differ. (b) Diffraction pattern numerically calculated for a edge lenslet and averaged over nine different scatterer ensembles for a CG position, density of scatterers, and coherence length (FWHM) of $+5\mu \mathrm{m}$, $100\mu {\mathrm{m}}^{-3}$, and $24\mu \mathrm{m}$, respectively. Note that, for better visibility, CV and focal cone are not drawn to scale.

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Fig. 4 MCSs showing the influence of distortions on the CGWS-measurement process. (a) The CV is deformed by distortions, and due to the extended CV distortions are averaged laterally, depending on the distance of the distortion layer to the focus. (b) The aberrations (Zernike coefficients $c_5$ to $c_8$) as a function of the coherence length with the phase plate located in the BFP. The dashed lines show the expected aberrations due to a single pass through a phase plate with $c_6=-0.5\mu \mathrm{m}$ and $c_8=-0.3\mu \mathrm{m}$, respectively. The density of scatterers was $10\mu {\mathrm{m}}^{-3}$, and the CG was at $+5\mu \mathrm{m}$. (c) Phase plates with $c_6=-0.5\mu \mathrm{m}$ and $c_8=-0.3\mu \mathrm{m}$ located at varying distances above the focus and simulated for CG positions at 0 and $5\mu \mathrm{m}$. The coherence length was $50\mu \mathrm{m}$, and the density of scatterers was $1\mu {\mathrm{m}}^{-3}$. Only Zernike coefficients $c_6$ and $c_8$ are shown. The dashed lines indicate the expected aberrations for $c_6$ and $c_8$.

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Fig. 5 (a) Distortions due to a tilted glass plate (10°, $145\mu \mathrm{m}$ thick) inserted between the objective and the $100\mathrm{nm}$ bead sample. (b) Aberrations (Zernike coefficients $c_4$ to $c_{11}$) determined by MCS for a coherence length of $6\mu \mathrm{m}$ and a density of scatterers of $1\mu {\mathrm{m}}^{-3}$ for CG positions in steps of $2\mu \mathrm{m}$. (c) Experimentally measured aberrations for the tilted glass plate with the same parameters for CG positions in steps $1\mu \mathrm{m}$. The data points were connected by straight lines.

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Fig. 6 Detected wavefronts using linearly polarized light. Measured wavefronts using scattering samples with (a) $100\mathrm{nm}$ and (b) $1\mu \mathrm{m}$ sized beads. The CG position was varied in steps of 2 and $3\mu \mathrm{m}$, respectively. The absolute CG position was not determined, only relative positions are depicted. For comparison, wavefronts determined by MCSs using (c) $100\mathrm{nm}$ and (d) $1\mu \mathrm{m}$ sized beads are shown. A scatterer density of $1\mu {\mathrm{m}}^{-3}$ and a coherence length of $2\mu \mathrm{m}$ were used for the calculation. The CG position was varied from $-8$ to $8\mu \mathrm{m}$ in steps of $4\mu \mathrm{m}$. Only spurious (varying with CG position) Zernike modes are shown: $c_6$ and $c_{12}$. The slopes, obtained by linear fitting, are (a) $8\pm 3$, (b) $37\pm 3$, (c) $24\pm 4$, and (d) $50\pm 5\hspace{0.17em}\mathrm{nm}∕\mu \mathrm{m}$ for $c_6$ and (a) $-1\pm 1$, (b) $1\pm 1$, (c) $2\pm 1$, and (d) $-6\pm 1\mathrm{nm}∕\mu \mathrm{m}$ for $c_{12}$.

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Fig. 7 Detected wavefronts using circularly polarized light. Experimentally measured wavefronts using scattering samples with (a) $100\mathrm{nm}$ (b) $1\mu \mathrm{m}$ sized beads. The CG position was varied in steps of $1\mu \mathrm{m}$. For comparison, wavefronts determined by MCSs using (c) $100\mathrm{nm}$ and (d) $1\mu \mathrm{m}$ sized beads. A scatterer density of $1\mu {\mathrm{m}}^{-3}$ and a coherence length of $2\mu \mathrm{m}$ were used. The CG position was varied from $-8$ to $8\mu \mathrm{m}$ in steps of $4\mu \mathrm{m}$. Only spurious (varying with CG position) Zernike modes are shown: $c_6$ and $c_{12}$. Linear fitting gave slopes of (a) $2\pm 2$, (b), $6\pm 3$, (c) $1\pm 3$, and (d) $-2\pm 3\hspace{0.17em}\mathrm{nm}∕\mu \mathrm{m}$ for $c_6$, and (a) $1\pm 1$, (b) $3\pm 1$, (c) $0\pm 3$, and (d) $1\pm 2\hspace{0.17em}\mathrm{nm}∕\mu \mathrm{m}$ for $c_{12}$.

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Fig. 8 Propagation of the polarization through sample arm. The local bases are depicted as small arrows (red online). The fixed basis is shown at the focus. The scattering angle is ϑ.

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