1. Introduction

Second harmonic generation (SHG) microscopy is a non-invasive imaging technique that can visualize ordered molecular structures in biological tissue. SHG efficiency is strongly dependent on the sample symmetry, where, for example, the signal is absent in centrosymmetric materials [1]. Therefore, polarization measurements can be employed in SHG microscopy to obtain molecular organization information. The first SHG polarimetric analysis in biological tissue was demonstrated in rat-tail tendon by Freund, Deutsch and Sprecher [2]. Second-order susceptibility tensor χ⁽²⁾ component ratios have been measured in biological samples such as collagen fibers [3–6], myofibrils [7–10] and starch granules [11], as well as crystalline nanostructures [12,13]. Jones formalism has been used for polarimetric analysis of SHG microscopy images [14, 15]. However, heterogeneous biological samples are strongly scattering materials, therefore, Stokes Mueller polarimetry is better suited for sample characterization due to its ability to separate polarized and unpolarized signal components [16]. Recently, it was demonstrated that degree of linear polarization can be used to measure changes in collagen ultrastructure due to cancer development in thyroid tissue [17]. Linear Stokes polarimetry has been applied to SHG microscopy [18], but it does not capture all the details of the nonlinear interaction, i.e. the obtained linear Mueller matrix assumes a linear relationship between the incoming laser radiation and the outgoing SHG signal, and therefore may only be sufficient to recover some of the nonlinear molecular susceptibility parameters, depending on sample orientation and symmetry. To address these issues we have developed a theoretical formalism of double Stokes Mueller polarimetry (DSMP) [19], which is based on seminal work by Shi et al. [20]. It enables a complete, model-independent SHG polarimetric characterization of scattering samples from a minimum set of 36 polarization measurements. DSMP is capable of calculating all six observable laboratory-frame susceptibility tensor components without prior assumptions of sample symmetry. By selecting a molecular tensor model, molecular susceptibility ratios and the orientation of the molecular frame of reference with respect to the laboratory frame can be extracted for each pixel of the image. In addition, the laboratory-frame susceptibility tensor shows whether Kleinman symmetry can be assumed. Earlier we reported on the preliminary experimental results of double Stokes Mueller polarimetry [21]. Here we present a full implementation of the DSMP microscopy experiment and data analysis.

In this paper, we investigate fruit fly (Drosophila melanogaster) larva wall muscle using DSMP microscopy. The striated SHG pattern in muscle fibers appears due to myosin filaments arranged into anisotropic bands (A-bands) of sarcomeres, and aligned along the muscle fibers. The myosin filaments are composed of two parts with myosin molecules arranged antiparallel to each other, where the middle of the A-band does not contain myosin heads. The isotropic bands (I-bands) are comprised of actin filaments containing globular actin monomers and generate significantly lower SHG signal [22]. The heterogeneous structure of the muscle fibers results in rich polarization-sensitive features of the SHG signal, which makes it suitable for a detailed investigation with the DSMP microscopy technique.

2. Methods

Measurement procedure

Full theoretical treatment of the double Stokes Mueller polarimetry theory can be found in Ref. [19]. Here we present the main steps required for an experimental implementation of the technique. Double Stokes Mueller polarimetry defines a relation between the polarization states of the outgoing SHG signal and the incoming laser radiation via a double Mueller matrix representing the nonlinear properties of the sample [19]:

A DSMP measurement requires a set of nine incoming states to obtain a full polarimetric characterization of an SHG sample (Fig. 1). The chosen polarization states are: horizontally and vertically linearly polarized (HLP and VLP), diagonally linearly polarized (±45°), right-and left-hand circularly polarized (RCP and LCP), as well as linearly polarized at −22.5° and right- and left-hand elliptically polarized (REP and LEP). The Poincaré sphere coordinates of these sates are [16]:

 

Fig. 1 a) Double Stokes Mueller polarimetry microscope setup. POL– polarizers, PSG, PSA– polarization state generator and analyzer, SHG-F – bandpass filter at 514 nm, PMT – single-photon counting photomultiplier tube. b) Graphical representations of the nine input polarization states, and laboratory (XYZ) and molecular (xyz) coordinate systems with a cylindrical myofibril model at an in-plane rotation angle δ. S and s̃ show incoming and outgoing polarization states, respectively.

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The double Stokes vector components of these states can be calculated using the following expression (with s_0...3 denoting the linear Stokes components of the incoming light):

In a DSMP measurement, four Stokes vector s̃_γ components (γ = 0...3) of the outgoing radiation are obtained for each of the Q = 1...9 incoming radiation polarization states, resulting in a 4 × 9 measured Stokes matrix s̃_γ,Q. The double Mueller matrix in the DSMP formalism is defined for pure polarization states, however, due to scattering, background and dark signals, the measured s̃_γ,Q values contain an unpolarized fraction which must be filtered out. This is achieved by setting the degree of polarization (DOP) to unity [23], i.e., replacing s̃₀ components with s′₀:

The absolute values of the 6 observable χ⁽²⁾ components can be extracted from the top left 2 × 3 (NP) double Mueller matrix components as follows:

The final step of the analysis is to relate the obtained laboratory-frame susceptibility components (Eq. 8) to the molecular frame orientation and susceptibilities. For this task, a model of the molecular organization has to be assumed, but it should be noted that a DSMP measurement itself is model-independent, i.e., the obtained polarimetric information is complete and any molecular susceptibility model can be tested against the data. To a good approximation, a cylindrical symmetry model has been used for the description of the nonlinear response of biological samples (e.g., collagen [3], myosin [9], starch [11]). The in-plane orientation (δ) of the cylindrical symmetry axis from the Z axis (Fig. 1(b)), and the ratio of the molecular susceptibility components with parallel and perpendicular excitation to the axis ( $R={\chi }_{zzz}^{(2)}/{\chi }_{zxx}^{(2)}$) are related to the laboratory-frame χ⁽²⁾ components via the following equations [15]:

By substituting Eqs. (9) into the general expression of the double Mueller matrix (Eq. (33) in Ref. [19]) and assuming that the sample is oriented horizontally along the Z axis, i.e. the molecular and laboratory frames of reference coincide, the double Mueller matrix takes on a simple form:

By retrieving R and δ distribution maps in a sample under investigation, variations in the molecular ultrastructure can be studied. For example, changes in collagen R ratio have been observed between healthy and tumor tissue [6]. In this paper, R and δ are obtained by a global fitting routine based on the Levenberg-Marquardt (LM) algorithm with R, δ and amplitude as free parameters to minimize the total root-mean-square error (RMSE) between the measured laboratory-frame χ⁽²⁾ components and components obtained by Eq. (9). The amplitude fit parameter includes any signal collection efficiency factors and absorbs the ${\chi }_{zxx}^{(2)}$ term.

In summary, DSMP analysis is comprised of the following steps: (i) outgoing SHG Stokes vectors s̃ are measured for nine input polarization states; (ii) degree of SHG polarization (DOP) is calculated and filtering is performed to remove the scattering contribution and retrieve s′; (iii) sample double Mueller matrix is calculated; (iv) laboratory-frame χ⁽²⁾ values are obtained from the six NP matrix elements; (v) after choosing a symmetry model for the sample, molecular-frame orientation and χ⁽²⁾ ratios are extracted.

DSMP microscope

A home-built polarimetric microscope (Fig. 1(a)) and a laser oscillator based on a Yb:KGW crystal emitting 400 fs pulses at 1028 nm wavelength with 14 MHz pulse repetition rate were used [24]. The laser power was set to 2 mW at the sample. Incoming and outgoing polarization states were controlled by a polarization state generator (PSG) and analyzer (PSA), implemented by two pairs of motorized rotating waveplates where states were switched serially. Special care was taken to ensure accurate polarization states because of the sensitivity of measured DSMP datasets to systematic errors. The laser was focused with a 0.75 NA objective (Zeiss Plan-Apochromat 20x/0.75), the point spread function of the focal volume was 1 μm laterally and 3.7 μm axially at full-width at half-maximum (FWHM). The sample was raster-scanned by a resonant (EOPC SC30) and galvanometric (Cambridge Technology 6220H) mirror pair. The SHG signal was collected using a home-built objective and detected by a single-photon counting photomultiplier tube (Hamamatsu H7422P-40). The 0.75-NA objective was chosen to minimize axial polarization effects [25], and both objectives were found to have negligible transmission polarization anisotropy. Photon counts were fed into a high-throughput FPGA-based data acquisition card (Innovative Integration X5-210M) running a custom firmware. Recorded data was processed by a combination of custom software written in LabView (National Instruments), MATLAB (Mathworks) and C++, which implements DSMP analysis and visualization.

Sample preparation

A 3^rd instar larva of Drosophila melanogaster was submerged in phosphate buffered saline (PBS) solution and was pinned down on the anterior and posterior ends. The cuticle was cut along the larval body and after removing the internal organs the body was unfurled and pinned to expose the body wall muscles. The larva was then fixed with 4% formaldehyde, placed on a 1 mm-thick microscope slide and covered with a 0.2 mm-thick coverslip without a spacer. After adding a few drops of PBS the coverslip was sealed using clear nail polish. No drying of the sample was observed during imaging.

3. Results and discussion

Stokes vector matrix

The measured 4 × 9 Stokes images at different polarization states for the muscle sample are shown in Fig. 2(a). Each column represents a PSA state, i.e., SHG Stokes vector s̃ parameters (s̃₀, s̃₁, s̃₂, s̃₃), for each of the nine incoming laser radiation PSG states. The size of each image is 79 × 132 pixels, the pixel size is 0.36 μm, and the area is 28 × 48 μm². The image intensities are color-coded as follows: s̃₀ values range from 0 to 1 and correspond to the hue of the map, and s̃₁...s̃₃ range from −1 to 1, where blue and cyan show negative values, yellow and red indicate positive, and black shows zero signal. It can be seen that the row of s̃₀ components has positive values, while the last three rows have positive and negative values depending on the output SHG polarization state.

 

Fig. 2 SHG double Stokes Mueller polarimetric images of Drosophila melanogaster larva wall muscle. a) Measured Stokes component matrix s̃_γ,Q, b) degree of total, linear and circular polarization (DOP, DOLP, DOCP) maps, c) double Mueller matrix elements, normalized by the image-averaged ${\chi }_{ZXX}^{(2)}$ value. Panels a) and b) share the same column labels. Image color mapping is given in the colorbars on the right. Laboratory axes and a 10 μm scale bar are shown in panel a).

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A characteristic striated structure of the muscle [3, 26] can be recognized in Fig. 2(a). The strongest signal originates from the anisotropic bands of sarcomeres containing myofilaments aligned with the laboratory Z axis. The muscle fibers were aligned horizontally with respect to the image frame (laboratory Z axis). Thus, the main features of the measured Stokes components, double Mueller matrix and the laboratory χ⁽²⁾ tensor elements are revealed clearly without the mixing of the values due to the rotation of the molecular tensor relative to the laboratory frame. Consistent with previous publications [7, 9], the largest signal is observed at ±45° linearly polarized input (compare s̃₀ states in the first row of Fig. 2(a)), followed by a lower signal for VLP and almost no signal for HLP polarization states. In addition, the large ${\chi }_{zxx}^{(2)}$ molecular tensor component results in the strongest positive s̃₁ signal for VLP input, while ±45° as well as RCP and LCP have opposite signs for the s̃₂ and s̃₃ components, respectively. The Stokes components of −22.5°, REP and LEP take on intermediate values of the corresponding linear and circular polarization states.

Degree of polarization

Stokes vector components can be used to determine the fraction of light that is fully polarized from scattered light by calculating the degree of polarization (DOP), degree of linear polarization (DOLP) and degree of circular polarization (DOCP) [23]. Figure 2(b) shows these quantities for each incoming polarization state. It can be seen, that similar level of scattering (i.e., comparable DOP) is present for all PSG states. In addition, the A-bands scatter more than the I-bands, as they appear bright red with DOP of ∼ 0.9, compared to dark-red inter-band areas with DOP of ∼ 1, respectively (Fig. 2(b)). Larger A-band scattering most likely occurs due to the high density of myosin. DOP can be used to remove the depolarized signal contribution for a more accurate estimation of the susceptibility ratio, which is important for structural analysis of myofilaments [10]. The apparent difference in RCP scattering compared to LCP could be attributed to the chiral organization of myosin in the myofilaments. DOCP shows that the outgoing circularly polarized fraction is significant when circular and elliptical incoming polarizations are employed for muscle imaging, and therefore, can be potentially used for further structural investigations of myofilaments.

Double Mueller matrix

The double Mueller matrix is obtained from the filtered Stokes matrix using Eq. (5) and is given in Fig. 2(c). The matrix elements are estimated up to a global constant and are normalized to the average value of the laboratory-frame ${\chi }_{ZXX}^{(2)}$ component. Note, that depending on the sample, the ${\chi }_{ZXX}^{(2)}$ value may approach zero in which case another normalization factor should be selected. Since the myofibril axis is oriented horizontally, the relative magnitude of the elements can be compared to Eq. (10). It can be seen that most of the elements correspond well to Eq. (10), e.g., large values are obtained in the upper left 2 × 4 and ${ℳ}_{2,5}^{(2)}$, ${ℳ}_{2,6}^{(2)}$, ${ℳ}_{3,8}^{(2)}$ and ${ℳ}_{3,9}^{(2)}$ elements. In addition, by inspecting ${ℳ}_{0,3}^{(2)}$ and ${ℳ}_{1,3}^{(2)}$ elements in Eq. (10) we can see that they would vanish for R = 1 and become positive for R < 1. By considering the values of these elements in Fig. 2(c), it can be readily seen that the sample has R < 1, consistent with typical muscle values of R = 0.3 – 0.7 [9]. Furthermore, when birefringence is negligible, double Mueller matrix element ratios, for example ${ℳ}_{2,5}^{(2)}/{ℳ}_{2,6}^{(2)}$, will yield R directly, albeit with sub-optimal precision since only a fraction of the acquired data is utilized. The R values estimated from Fig. 2(c) fall within the range of 0.54 – 0.70. As predicted by the theory (Eq. (33) in Ref. [19]), the upper right 3 × 3 values are close to zero, compared with the strong ${ℳ}_{0,1}^{(2)}$ term. This indicates that at 1028 nm, to a good approximation, real susceptibilities can be assumed.

The general double Mueller matrix elements, given in Appendix B of Ref. [19], contain several susceptibility tensor component products. Therefore, a better understanding of the matrix form can be achieved by inspecting the symbolic representation given in Eq. (7). The top-left 2 × 3 (NP) elements contain only squared laboratory-frame χ⁽²⁾ components, and they provide an easy way to extract six absolute susceptibility tensor values. The top-right 2 × 6 (I) elements contain six terms with χ⁽²⁾ phase dependency, providing information about sample retardance related to incoming light. Of these, only ${ℳ}_{0,4}^{(2)}$ and ${ℳ}_{1,4}^{(2)}$ have significant values in accordance with Eq. (10). The six O elements contain tensor component products that have different outgoing polarization orientation for the same incoming polarization. These components approach zero in a sample where the cylindrical symmetry axis is oriented close to the Z or X axis. From the remaining bottom-right 2×6 (OI) elements, ${ℳ}_{2,5}^{(2)}$ and ${ℳ}_{2,6}^{(2)}$ as well as ${ℳ}_{3,8}^{(2)}$ and ${ℳ}_{3,9}^{(2)}$ have significant values as predicted by Eq. (10). Components ${ℳ}_{2,2}^{(2)}$, ${ℳ}_{2,3}^{(2)}$ and ${ℳ}_{3,7}^{(2)}$ become nonzero if the sample is rotated away from the horizontal axis (the deviation was found to be 2.7°, see Fig. 3(c)). A larger than expected ${ℳ}_{3,7}^{(2)}$ indicates a deviation from the cylindrical symmetry. In addition, small nonzero values of ${ℳ}_{2,7}^{(2)}$, ${ℳ}_{2,8}^{(2)}$, ${ℳ}_{2,9}^{(2)}$ and ${ℳ}_{3,4}^{(2)}$, ${ℳ}_{3,5}^{(2)}$, ${ℳ}_{3,6}^{(2)}$ components are related to SHG birefringence, which is a known property of muscle [10].

 

Fig. 3 The laboratory- and molecular-frame nonlinear susceptibility values of the muscle sample. a) Laboratory-frame χ⁽²⁾ tensor elements. b) Extracted molecular-frame tensor ratio R map. c) In-plane orientation angle δ map. d) Horizontal profiles of SHG intensity and R ratio vertically averaged within the box shown in (b). Values in panel (a) denoted by D are image-averages, F gives corresponding best-fit values of a C_6v tensor model. The color mapping of R is given in the colorbar together with the ratio histogram. R values with SNR < 1 are shown in black. δ orientation bars with an SNR > 2 are shown in black, 2 > SNR > 1 are gray to white, respectively. Values with SNR < 2 are omitted when calculating the orientation average. The red points in d) denote measured values connected by a black line to aid viewing, standard errors are shown as gray bars and the blue horizontal line denotes the image-average value of R.

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Laboratory-frame χ⁽²⁾ tensor

Six normalized laboratory-frame χ⁽²⁾ tensor elements can be calculated from the top 2 × 3 (NP) elements of the double Mueller matrix using Eq. (8). These value maps are given in Fig. 3(a) together with image averages, denoted by the label D. Note, that the laboratory-frame susceptibility tensor component images in Fig. 3(a) are obtained after filtering the scattering contribution by replacing s̃₀ with s′₀ according to Eq. (4). Since the myofilaments in the muscle sample are aligned with the horizontal laboratory axis (Z), the laboratory- and molecular-frame tensor components coincide. It can be seen that the highest values are obtained for ${\chi }_{ZXX}^{(2)}$ and ${\chi }_{XXZ}^{(2)}$ elements, followed by ${\chi }_{ZZZ}^{(2)}$, in accordance with the cylindrical symmetry assumption for muscle with a ratio of R ≈ 0.5. The nonzero ${\chi }_{ZXZ}^{(2)}$ and ${\chi }_{XZZ}^{(2)}$ components can be attributed to an offset from the horizontal laboratory axis, out-of-plane sample tilt, and a chiral arrangement of myosin molecules within the muscle fibers, or a deviation from the cylindrical symmetry. To estimate the fraction of non-zero terms ( ${\chi }_{ZXZ}^{(2)}$, ${\chi }_{XZZ}^{(2)}$ and ${\chi }_{XXX}^{(2)}$) that can be explained by a slight in-plane rotation of the whole muscle sample away from the Z axis, the measured image-averaged χ⁽²⁾ values were fit to the C_6v tensor model, assuming Kleinman symmetry (Eq. 9). The best-fit was obtained for R = 0.55 and δ = −1.8° values. The fit tensor components are also given in Fig. 3(a), designated by label F.

The ${\chi }_{ZXX}^{(2)}/{\chi }_{XXZ}^{(2)}=1.06\pm 0.03$ (s.e.) ratio is close to unity in accordance with Kleinman symmetry assumption [1]. The deviation of ${\chi }_{ZXZ}^{(2)}/{\chi }_{XZZ}^{(2)}=3.44\pm 0.44$ ratio from unity is possibly due to the chiral organization and a slight out-of-image-plane tilt of the myofilaments in the muscle. The nonzero ${\chi }_{XXX}^{(2)}$ values indicate a small in-plane deviation of myofilaments from the Z axis. The average ratio of ${\chi }_{ZZZ}^{(2)}/{\chi }_{ZXX}^{(2)}=0.55\pm 0.03$ corresponds to the expected molecular susceptibility ratio R of myosin filaments [7, 9, 10]. Note, however, that the fit and the resulting distribution in Fig. 3(b) is a more accurate estimate of R, which also does not depend on δ.

Molecular-frame χ⁽²⁾ tensor

The per-pixel molecular-frame susceptibility ratio R and the orientation of the cylindrical symmetry axis δ were obtained from the laboratory-frame susceptibilities by global fitting of Eq. (9). The recovered R is independent on the sample orientation, while δ shows the in-plane molecular orientation with respect to the laboratory coordinate system. The extracted molecular parameters are given in Fig. 3, where the ratio values (R) are represented as pixel color in panel (b), and the orientations of cylindrical symmetry axes (δ) are shown as rectangular bars in panel (c). The distribution of R is shown on the same scale as the colorbar. The average value of R is 0.55 and the FWHM range is 0.41 – 0.67. The in-plane orientation bars in Fig. 3(c) are well aligned since a very straight section of the muscle fibril was selected for imaging, and the sample was positioned horizontally. The orientation bars are omitted and ratio pixels are shown as black when the SNR of the SHG intensity gets lower than 1. The SNR of SHG in terms of the number of detected photons N_ph was taken as $1/\sqrt{N_{\mathit{ph}}}$, assuming the signal was shot-noise limited. The average estimated molecular tensor in-plane orientation δ is −2.7°, and the FWHM range is −3.5° to −1.8°.

To better visualize the spatial distribution of R, profiles of total SHG intensity and R, taken horizontally within the box in Fig. 3(b) and averaged over 3 μm (8 px) vertically, are given in Fig. 3(d). The A-band positions correspond to SHG maxima, highlighted in the figure as light gray bands. Intensity minima at the center of the A-bands are highlighted as dark gray bands and correspond to M-lines. The estimated sarcomere size of 7.2 μm compares well to a previously reported value [22]. Fig. 3(d) also shows that the R value varies along the sarcomere axis. The ratio is highest (0.7) at the center of the I-band, while the A-bands have a ratio of ∼ 0.55 with a small dip at the M-line. The decreased ratio values in the middle of the A-band are consistent with our previous investigations [10]. The R ratio values reflect the molecular organization within each focal volume of the sample. This information is important for interpreting observed changes in the molecular susceptibility values from various muscle samples and during muscle contraction.

4. Conclusion

Double Stokes Mueller polarimetry enables a complete characterization of SHG response from the sample with 36 polarization measurements. We have demonstrated how to perform DSMP measurements and obtain Stokes matrix components, degree of polarization maps, double Mueller matrix and laboratory-frame nonlinear susceptibility χ⁽²⁾ tensor elements from the polarimetric data. In addition, molecular ultrastructure parameters, R and δ, were extracted based on a cylindrical model of the molecular susceptibility tensor. The analysis of Drosophila melanogaster larva muscle data revealed a variation of R along the sarcomeres with the highest ratio observed in isotropic bands and the lowest ratio obtained in M-bands. DSMP opens new possibilities to measure nonlinear optical properties of biological samples and to obtain ultrastructure parameters of ordered molecular assemblies inside scattering tissue.