## Abstract

An experimental implementation of the nonlinear Stokes-Mueller polarimetric (NSMP) microscopy in third-harmonic generation modality is presented. The technique is able to extract all eight 2D-accessible *χ*^{(3)} components for any sample from 64 polarization measurements, and can be applied to noninvasive ultrastructural characterization. The polarization signature of an isotropic glass coverslip is presented, and carotenoid crystallites in the root of orange carrot (*Daucus carota*) are investigated, showing complex *χ*^{(3)} components with a significant chiral contribution.

© 2017 Optical Society of America

## 1. Introduction

Harmonic generation microscopy is a noninvasive optical technique suitable for visualizing biological structures *in vivo* with three-dimensional (3D) sectioning, endogenous contrast and low out-of-focus phototoxicity [1,2]. Second-harmonic generation (SHG) occurs in noncentrosymmetric media [3], including many semicrystalline biopolymers, such as collagen [4–6], myosin (in muscle) [7–10] and amylopectin (in plant starches) [11,12], with applications in diagnostic medicine [13] and in basic studies of molecular structure and function [14,15]. Third-harmonic generation (THG) occurs at interfaces and complements the SHG contrast mode [4, 16–20]. Focused beams do not produce THG in homogeneous media due to the Gouy phase shift [21]. Therefore, in microscopy samples, THG highlights interfaces or sub-resolution structures where there is a change in refractive index or nonlinear susceptibility. In addition, harmonic generation is a highly polarization-sensitive nonlinear process. Using polarization-resolved microscopy, SHG- and THG-active regions can be visualized, and their nonlinear susceptibility tensor components can be determined which in turn can be used to infer structural details of the sample. For example, polarization-resolved SHG imaging was used to infer the stroke position and molecular conformation of the myosin micromotor in muscle [9], while simultaneous SHG and THG imaging was used to study the distribution of collagen, and the composition and structural anisotropy of human cornea [4,17], as well as its response under dynamic pressure changes [6].

The polarization of the generated harmonic signal depends on the polarization state of the incoming light and *χ*^{(n)}, the nonlinear susceptibility tensor of the sample [3]. Typically in SHG and THG polarimetric microscopy, samples are investigated by assuming a certain material symmetry and often considering only linearly-polarized light [5,7,9–12,19]. Hence, these measurements are model-specific and the assessment of the validity of the chosen symmetry model can be difficult. A recently-developed framework called nonlinear Stokes-Mueller polarimetry (NSMP) [22] takes a more generalized approach, in which the nonlinear interaction is described without any prior knowledge of the underlying nonlinear susceptibility tensor, decoupling the measurement from the analysis and giving access to all of the polarimetric information contained in the nonlinear process. According to the NSMP theory, the minimum number of required measurements for SHG and THG is 36 and 64, respectively, from which the accessible *χ*^{(n)} tensor components can be reconstructed.

An experimental implementation of SHG NSMP has been reported previously [23]. This paper describes an experimental setup for THG polarimetry and demonstrates the technique by recovering the *χ*^{(3)} tensor components of two common symmetry classes found in biological samples: an isotropic case measured at a glass-air interface of a microscopy coverslip, and a cylindrical case obtained in carotenoid-rich chromoplasts in the root of orange carrot (*Daucus carota*). The *χ*^{(3)} tensor components are first recovered directly from the measured data, without assuming a tensor model, and then compared to the isotropic and *C*_{6} models, demonstrating the decoupling of the *χ*^{(3)} tensor measurement and modeling steps. The presented analysis is applicable to other THG-active samples as well.

## 2. Theory

The NSMP theory is developed by Samim et al. in [22], and the details pertaining to the THG case are given in [24]. In this section, the experimentally-relevant aspects are briefly summarized.

The purpose of a THG NSMP experiment is to retrieve the *χ*^{(3)} tensor components of the sample from the smallest number of polarization-resolved measurements. Each such measurement records the Stokes vector of the outgoing THG radiation for each of a complete set of incoming laser polarization states. The incoming set forms an orthogonal basis and consists of sixteen states: seven linearly-polarized (LP), two circularly-polarized (right- and left-handed, RCP and LCP) and seven elliptically-polarized (five right- and two left-handed, REP and LEP, respectively, all with ellipticity of
$\sqrt{2}/2$). The order of the states is as follows: 0°, 90°, 45°, −45°, RCP, LCP, −22.5°, REP (major axis at 90°), LEP (45°), 22.5°, 67.5°, REP2 (22.5°), LEP2 (90°), REP3 (45°), REP4 (0°) and REP5 (−22.5°). The Poincaré coordinates of the states are given in [24].

The outgoing THG Stokes vector *s′* is expressed as a matrix product of the sample Mueller matrix *ℳ* and the incoming triple Stokes vector *S*:

*s′*has four components:

*s′*

_{0}is the total intensity, while

*s′*

_{1},

*s′*

_{2}and

*s′*

_{3}give the differences in intensities between the horizontal and vertical, 45° and −45°, and RCP and LCP polarization states, respectively. To represent the nonlinear interaction in matrix notation, the Stokes vector of the incoming light is rewritten in a triple Stokes form

*S*with sixteen components, where each is a function of the four linear Stokes parameters of the incoming light (Eq. (9) in [24]). In a THG NSMP experiment, four outgoing

*s′*Stokes parameters are recorded for each of the sixteen incoming states, which can be expressed in the same form as Eq. (1), where the (4 × 16)

*s′*and the (16 × 16)

*S*matrices represent the Stokes measurement matrix, and the incoming laser state matrix, respectively. The matrix

*S*is comprised of the 16 incoming polarization state triple Stokes column vectors. In the normalized form, all Stokes matrix elements fall in the ±1 range, except

*s′*

_{0}which is always positive.

The (4*×*16) Mueller matrix *ℳ* is obtained by inverting Eq. (1) (invertibility of *S* is guaranteed by the orthogonality of the chosen input states). The nonlinear Mueller matrix elements determine the THG polarization properties of the sample in the laboratory frame, i.e., differently oriented identical samples will have different Mueller matrices. For any sample with a real-valued *χ*^{(3)} tensor, the bottom-left 1 × 10 and the top-right 3 × 6 Mueller matrix elements vanish. The elements of *ℳ* can be positive or negative depending on the relative component values of the *χ*^{(3)} tensor. All elements of the Stokes and Mueller matrices are measured quantities and, therefore, are real-valued even when the sample *χ*^{(3)} is complex-valued. The laboratory- and molecular-frame *χ*^{(3)} tensor components are represented by uppercase (*IJKL*) and lowercase (*ijkl*) indices, respectively. The first index refers to the Cartesian component of the outgoing THG radiation, while the last three refer to the Cartesian components of the thrice degenerate incoming laser electric field, and, therefore, they can be freely permuted. The orientation of the molecular frame with respect to the laboratory frame in terms of the in-plane (*δ*) and out-of-plane (*α*) angles is shown in Fig. 1(b), and the two tensors are related via *χ _{IJKL}* =

*R*(where

_{Ii}R_{Jj}R_{Kk}R_{Ll}χ_{ijkl}*R*are the components of the rotation matrix by

_{mn}*δ*and

*α*angles, corresponding to Fig. 1(b)) [25]. By assuming plane-wave approximation, the axial incoming electric field can be neglected (

*E*= 0). Tensor components involving the

_{y}*Y*index are inactive, and only the following

*χ*

^{(3)}components are accessible:

*ZZZZ*,

*ZXXX*,

*ZZXX*,

*ZZZX*,

*XZZZ*,

*XXXX*,

*XXZZ*,

*XXXZ*. The

*χ*

^{(3)}components in the laboratory frame are obtained from the (8 × 8) X correlation matrix, whose elements are 64 pairwise products of the accessible

*χ*

^{(3)}components (indexed by

*i*and

*j*): ${X}_{ij}={\chi}_{i}{\chi}_{j}^{*}$ (Eq. (3) in [26]). Since microscopy measurements are relative to a global constant, individual tensor components were normalized to

*χ*= 1. All these operations are closed-form matrix multiplications and do not require any

_{ZZZZ}*a priori*knowledge about the sample. Molecular-frame tensor components can be estimated by appropriately rotating the laboratory-frame tensor so that the primary molecular

*z*-axis aligns with the laboratory

*Z*-axis, or by using nonlinear fitting [27], decomposition [26], maximum-likelihood estimation [28] and similar techniques. In this study, isotropic and

*C*

_{6}

*χ*

^{(3)}tensor models were considered. For these two cases, the relationship between the laboratory and molecular frames is given in Appendix A.

## 3. Materials and methods

The microscope setup and the SHG NSMP experiment has been described previously [23]. This study extends the setup for THG polarimetry. Briefly, the system consists of a home-built inverted laser-scanning nonlinear microscope (Fig. 1(a)) powered by a Yb:KGW laser oscillator emitting 400 fs pulses at 1028 nm wavelength with 14 MHz pulse repetition rate [29]. Incoming and outgoing polarization states are controlled and analyzed by a polarization state generator (PSG) and a polarization state analyzer (PSA), respectively. The PSG is comprised of a fixed polarizer (Thorlabs LPVIS100), followed by motorized low-order half- and quarter-wave plates (EKSMA 461–4208 and 461–4408), and in the PSA zero-order half- and quarter-wave plates (EKSMA 460–4241 and 460–4441) are followed by the polarizer (Meadowlark VLM-100-UV). The beam is focused using a 0.75 NA objective (Zeiss Plan-Apochromat 20×/0.75), with a point-spread function of 0.75 × 3 μm laterally and axially respectively, at full-width at half-maximum (FWHM). Pulse energies in the range of 400 – 750 pJ are used for imaging, and the typical scan field size is 100 × 100 μm^{2}. The forward THG signal is collected by a custom-built UV-transmitting objective, sent through a 343 nm bandpass filter and detected by a single-photon-counting photomultiplier tube (Hamamatsu H7422P-40). The use of a medium-NA excitation objective results in a negligible deviation from the plane-wave approximation [30], and beam scanning was determined to have a negligible effect on the fidelity of the polarization states.

Model tensor parameters were estimated by using the Trust-Region Reflective algorithm of the
`lsqnonlin` fitting routine in MATLAB (Mathworks). When comparing multiple models and selecting the free parameters in a given model, best fits were chosen by their Akaike information criterion (AIC) weights [31]. Fit parameter errors were estimated by a Monte-Carlo process [27], in which the Stokes data was randomly sampled from Gaussian distributions, whose means and standard deviations (SD) were the measured Stokes values and their errors, respectively. Distributions of Mueller matrix values, *χ*^{(3)} tensor components and fit model parameters were constructed from the randomly sampled data, and the final values were taken as the mean and the SD, respectively, of these distributions.

Fresh orange carrots (*Daucus carota*) were obtained from the local market and stored in the dark at 20 °C. Microscopy samples were prepared by slicing thin wedges of the carrot root phloem and mounting them in polyacrilamide gel between two No. 1 borosilicate coverslips (VWR SuperSlips) each having a thickness of 130 – 170 μm. The thin apex of the wedge-cut was imaged to minimize scattering. Isotropic glass *χ*^{(3)} measurements were performed using the same type of coverslip. Signal acquisition time for each dataset was about 10 minutes.

The reported total measurement uncertainties are given as ±2 SD, spanning 95% confidence intervals. They were estimated from three borosilicate glass coverslip NSMP measurements, taken on different days, as the residual differences between the expected and obtained THG Stokes parameters of an isotropic *χ*^{(3)} tensor. Using these estimates proportional and offset Stokes parameter uncertainty factors were derived, which were then applied to all measurements to yield the reported values. The errors account for photon-detection shot noise, 1% long-term laser intensity variation and residual polarimeter errors, which include optical misalignment and imperfections as well as other undesired polarization effects in the microscope. The residual error is defined as the error between the expected and measured Stokes matrices of a glass coverslip, averaged over multiple measurements.

## 4. Results

#### 4.1. Isotropic glass coverslip

A sample with isotropic *χ*^{(3)} symmetry has the simplest THG polarimetric response, characterized by a rotation-invariant tensor with only a single unique component, which acts as the overall amplitude of the THG process. Therefore, the obtained normalized dataset is parameterless, and thus can be directly compared to the theoretical model. This property also makes an isotropic sample useful for polarimetric calibration of the microscope. In the following sections, the measured Stokes matrix, the calculated Mueller matrix, and the complex susceptibility tensor *χ*^{(3)} values for the isotropic sample are presented.

### Stokes matrix

The measured isotropic Stokes matrix, recorded from the glass-air interface of a borosilicate coverslip, is shown in Fig. 2(a). The signal from the uniform surface had negligible lateral variation and, therefore, the images were averaged to a single value for each input-output polarization state combination. The following analysis, however, can be performed over an entire image on a per-pixel basis. A theoretical isotropic matrix is shown for comparison in Fig. 2(b).

By comparing the measured and expected Stokes matrix values it can be seen that all elements match within error. In particular, there is no *s′*_{3} signal (< 0.01) for LP input states, indicating that negligible circularly-polarized THG is detected for a linearly-polarized laser beam, which is expected for a real-valued *χ*^{(3)} tensor. It will be shown later that this is not the case for the carotenoid-containing chromoplasts, suggesting that their THG response is governed by a complex-valued *χ*^{(3)} tensor.

As predicted by the theory (Eq. (2)), no THG signal is emitted from an isotropic sample when the incoming light is circularly polarized. Furthermore, for elliptically-polarized input light (REP and LEP columns in Fig. 2), the emitted THG has the same ellipticity as the input beam.

### Mueller matrix

The measured and theoretical isotropic (4 × 16) Mueller matrices are given in Fig. 2(c) and (d). The elements are best inspected by comparing to a theoretical isotropic model. Note also that the isotropic THG case is not the same as the operation of an identity-matrix (i.e., where the input and output states are identical), because of the lack of THG from circularly-polarized input.

Compared to the Stokes matrix, the Mueller matrix is more informative for evaluating the polarization properties of the medium. An isotropic sample has an orientation-invariant Mueller matrix, which is relatively easy to interpret. In particular, the absence of signal in the bottom-left 1 × 10 or the top-right 3 × 6 segments suggests a real-valued tensor. By comparison, the Mueller matrices of other symmetries change depending on the sample orientation; the elements become coupled and quickly lose this apparent simplicity.

It should be noted, that several Mueller matrix elements (namely *ℳ*_{1,2}, *ℳ*_{1,3}, *ℳ*_{2,7} and *ℳ*_{3,13}) have inherently low values in the isotropic case, which is consistent with the theoretical expectation (Fig. 2(d)), while the *ℳ*_{0,11} element has the same magnitude as *ℳ*_{3,13}, and it is determined to be zero within the error. Therefore, it is important to accurately estimate measurement uncertainties in order to compare Mueller matrix elements.

While the Stokes matrix errors are proportional to their respective values, the Mueller matrix errors exhibit vastly different noise sensitivities. For example, columns 5 and 11 have four times greater error than columns 1 and 4, despite having similar values. This is because each Mueller matrix element is a different weighted-sum of the Stokes components (i.e., a matrix product of Eq. (1)), and, therefore, the final error strongly depends on elements that are summed and their respective weighting coefficients. This should be taken into consideration when analyzing NSMP results, especially when the measurement errors become large, since in such a case some Mueller matrix elements may have a too low signal-to-noise ratio (SNR) for useful analysis.

*χ*^{(3)} tensor components

The extracted and theoretical isotropic *χ*^{(3)} tensor components are given in Fig. 2(e) and (f). Since the isotropic symmetry is rotationally-invariant, values measured in any frame of reference are identical. The theoretical values are all real: *χ _{zzzz}* =

*χ*= 1,

_{xxxx}*χ*=

_{zzxx}*χ*= 1/3, and zero for all the other components. The measured values correspond very well with the theory: the real parts of

_{xxzz}*χ*and

_{zzzz}*χ*agree within 2 – 3%;

_{xxxx}*χ*and

_{zzxx}*χ*agree within 6%, while

_{xxzz}*χ*,

_{zxxx}*χ*,

_{zzzx}*χ*and

_{xxxz}*χ*are at their expected zero values within 0.02. All imaginary parts are zero within 0.01 – 0.04. The ratio

_{xzzz}*χ*= 0.97 ± 0.08 indicates that the Kleinman ratio is measured to be unity within 8%.

_{xxzz}/χ_{zzxx}#### 4.2. Carotenoid-containing chromoplasts

Carotenoid pigments can be found in plants as homogeneous dispersions in lipid droplets or membrane-bound crystallites, and their bioaccessibility depends on the microstructure [32]. Therefore, NSMP microscopy presents itself as a valuable tool for *in vivo* structural investigations of carotenoid-containing structures. The obtained polarization signatures of carotenoid-rich chromoplasts in orange carrot root showed a large variability, and two distinct types were identified, with either a near-isotropic or an ordered crystalline polarization signature. The main difference between the two types is that for the near-isotropic case, the *χ _{zzzz}* and

*χ*components are close to 3

_{xxxx}*χ*and 3

_{zzxx}*χ*, while in the crystalline case, the

_{xxzz}*χ*component is dominant, with a secondary contribution from a complex-valued

_{zzzz}*χ*chiral component. These polarization signatures can only be observed by using a polarimetry technique which includes linearly- and circularly-polarized states.

_{xyzz}### Near-isotropic chromoplast

Polarization-resolved Stokes and Mueller images for a near-isotropic case are shown in Fig. 3(a) and (d). Orange carrot root chromoplasts have been shown to be spatially homogeneous [33], and the object in Fig. 3(a) exhibits only an overall intensity variation when the polarization states are changed. Therefore, each image can be averaged to a single value to improve the SNR and simplify the discussion. The average Stokes and Mueller matrices are shown in Fig. 3(b) – (f), and the extracted molecular-frame tensor components are given in Fig. 4.

Although the overall pattern of the Stokes values resembles the isotropic case, several Stokes states are significantly different from it (compare HLP, −22.5° and REP2 columns in Figs. 2(a) and 3(b)); hence, this polarimetric signature is referred to as near-isotropic. Such a polarimetric signature can be described by a *χ*^{(3)} *C*_{6} model with nonzero complex susceptibilities *χ _{zzzz}* = 1 (fixed),

*χ*= (0.78 ± 0.03) − (0.19 ± 0.08)

_{xxxx}*i*,

*χ*= 0.27 ± 0.01,

_{zzxx}*χ*= (0.29 ± 0.01) − (0.09 ± 0.01)

_{xxzz}*i*and an in-plane orientation angle

*δ*= 73 ± 4°. The resulting model Stokes and Mueller matrices are shown in Fig. 3(c) and (f), and the de-rotated

*χ*

^{(3)}tensor is given in Fig. 4. The similarity between the

*C*

_{6}and isotropic models is justified because when

*χ*=

_{zzzz}*χ*= 3

_{xxxx}*χ*= 3

_{zzxx}*χ*the two symmetry cases become indistinguishable (see Eq. (4) in Appendix A). The appearance of nonzero

_{xxzz}*s′*

_{3}terms for linearly-polarized input is consistent with a complex

*χ*

^{(3)}tensor. Circularly-polarized incoming states (RCP and LCP) should not produce any signal for an isotropic sample; therefore, it is interesting to note that the object in Fig. 3 generates a small (0.05 – 0.06) but significant nonzero circularly-polarized signal, indicating a presence of chiral susceptibility components (e.g.,

*χ*).

_{xyzz}The nonlinear Mueller matrix values further highlight the deviation from the isotropic model as can be seen from the nonzero *ℳ*_{0,4}, *ℳ*_{3,4} *ℳ*_{3,8} and *ℳ*_{2,15} elements, which should be zero in the isotropic case. In addition, nonzero terms in the bottom-left 1 × 10 and top-right 3 × 6 segments indicate a complex-valued *χ*^{(3)} tensor.

Note that the estimated model parameters have smaller error ranges compared to the measured Stokes values due to the reduced number of free parameters (there are 64 matrix elements, but only at most 12 free fit parameters). Techniques that take into account all of the input data and reduce the SNR requirements are particularly useful for microscopic polarimetry, where high-SNR data may be difficult to obtain.

The *χ*^{(3)} components in the laboratory-frame were obtained directly from the measured values via the X susceptibility correlation matrix [24]. The measured tensor was then de-rotated back into its molecular frame by the best-fit in-plane angle *δ* = 73°. Alternatively, *δ* can be estimated by minimizing the tensor components that are expected to be zero in the molecular frame (e.g., *χ _{ZXXX}*,

*χ*), or it can be inferred from sample morphology (e.g., for ordered fibers). The recovered tensor component images in the molecular-frame, their average values and the corresponding best-fit values are shown in Fig. 4(a) – (c), respectively. Five susceptibility components (

_{XZZZ}*χ*,

_{zxxx}*χ*,

_{zzzx}*χ*,

_{zzxx}*χ*and

_{xzzz}*χ*; with

_{xxxz}*χ*set to unity) correspond to the theoretical values of either the isotropic or

_{zzzz}*C*

_{6}symmetry. The preference for the

*C*

_{6}model in the polarimetric response of the near-isotropic chromoplast is revealed only in the

*χ*and

_{xxxx}*χ*components (

_{xxzz}*χ*= (0.7 ± 0.1) − (0.3 ± 0.2)

_{xxxx}*i*,

*χ*= (0.31 ± 0.05) − (0.12 ± 0.05)

_{xxzz}*i*vs.

*χ*= 3

_{xxxx}*χ*=

_{xxzz}*χ*= 1 in the isotropic case).

_{zzzz}### Ordered crystallite

The NSMP Stokes and Mueller images of a single 6-μm-long carotene crystallite are shown in Fig. 5(a) and (b), respectively. The crystallite is comprised of two homogeneous domains, labeled A and B (see Fig. 5(a)). Molecular-frame *χ*^{(3)} tensor components for the two domains are given in Fig. 5(c) – (f). Average Stokes and Mueller matrix element values are omitted for brevity.

The *χ*^{(3)} tensor of the carotenoid crystallites is dominated by the *χ _{zzzz}* molecular-frame component [18,19]. Incoming states that couple to this component produce a strong THG signal: e.g., for any incoming polarization state the outgoing polarization is predominantly determined by the

*χ*tensor element, and its in-plane orientation in the laboratory frame. This interaction results in a repeated pattern in the outgoing Stokes vector component columns in Fig. 5(a), with the overall intensity of each column depending on the THG efficiency for that incoming state. The nonzero

_{zzzz}*s′*

_{3}components for LP input states, as well as the differences between the total intensities of the RCP and LCP columns are indicative of a complex-valued

*χ*

^{(3)}tensor. Furthermore, the fact that the

*s′*

_{3}states do not change sign for different incoming polarizations suggests a presence of a chiral susceptibility component. Domains A and B in Fig. 5(a) have different

*s′*

_{3}values, indicating differences in the complex values of their chiral susceptibility components. According to theory, in the absence of a chiral contribution, the

*s′*

_{3}values vanish and the CP columns become identical.

The molecular-frame *χ*^{(3)} components of the crystallite can be estimated by de-rotating the measured laboratory-frame tensor to match its molecular-frame by maximizing the *χ _{ZZZZ}* term. The recovered molecular-frame

*χ*

^{(3)}tensor images for the A and B domains are given in Fig. 5(c) and (d), and their respective average values are shown in panels e and f. The second largest term in the resulting de-rotated tensor of domain A is

*χ*which can be attributed to a chiral

_{XZZZ}*χ*molecular-frame component (Eq. (4) in Appendix A). This can be verified by fitting the data to a

_{xyzz}*χ*

^{(3)}

*C*

_{6}model. For domain A the fit yields a small, but essential effective

*χ̂*= (−0.04 ± 0.01) + (0.07 ± 0.01)

_{xyzz}*i*component (normalized to

*χ̂*= 1). The effective chiral component ratio

_{zzzz}*χ̂*measured in 2D polarimetry is related to the 3D molecular ratio via an out-of-plane tilt factor sin

_{xyzz}/χ̂_{zzzz}*α*/cos

^{2}

*α*(see Appendix A). Without prior knowledge of either

*α*or the

*χ*ratio, only their product can be determined; however, when

_{xyzz}/χ_{zzzz}*α*≈ 65° the effective and the true ratios are equal. The

*C*

_{6}fit does not account for all measured

*χ*

^{(3)}features in domain A, indicating that a more detailed modeling may be required. The de-rotated tensor of domain B is less reliable compared to domain A due to a lower signal, which manifests in the high error estimate of

*χ*= 1.0 ± 0.5. Region B shows nonzero

_{ZZZZ}*χ*and

_{ZXXX}*χ*components, in addition to the

_{ZZXX}*χ*term, indicating a deviation from the

_{XZZZ}*C*

_{6}model as well.

Overall, the Mueller matrix (Fig. 5(b)) has most of the significant values in its upper-left side, and has several nonzero elements in the bottom-left 1 × 10 segment, consistent with a complex-valued tensor. These features are well replicated by the *C*_{6} model with only two nonzero components: *χ̂ _{zzzz}* and

*χ̂*. However, some of the signs of the elements in columns 11 – 16 are inverted, suggesting a reversal of the signs in one of the

_{xyzz}*χ*or

_{ZZZX}*χ*components in the underlying

_{XZZZ}*χ*

^{(3)}tensor.

### Multiple chromoplasts

The polarimetric response for two representative cases of the carotenoid-containing chromoplasts found in orange carrot root was presented in the preceding section. Similar analyses were performed on a total of 80 chromoplasts from twelve 100 × 100 μm^{2} scanned areas in three carrot samples, and 25 well-defined chromoplasts with high signal strength and relatively good stability (no more than 40% decay) were selected for further investigation. Most chromoplasts showed a near-isotropic (13 cases) or a highly-ordered crystalline (6 cases) polarization signature, while 6 chromoplasts could not be assigned to either of the two cases. The average degree of total polarization (
${p}^{2}=\left({s}_{1}^{2}+{s}_{2}^{2}+{s}_{3}^{2}\right)/{s}_{0}^{2}$) for the near-isotropic chromoplasts was *p* = 0.78 ± 0.08 (spread is given as ±1 SD), and the degrees of linear (
${p}_{L}^{2}=\left({s}_{1}^{2}+{s}_{2}^{2}\right)/{s}_{0}^{2}$) and circular (*p _{C}* = |

*s*

_{3}/

*s*

_{0}|) polarization were

*p*= 0.7 ± 0.1 and

_{L}*p*= 0.3 ± 0.1, respectively. The degrees of polarization for the highly-ordered crystallites were

_{C}*p*= 0.8 ± 0.1,

*p*= 0.6 ± 0.2,

_{L}*p*= 0.4 ± 0.2, respectively. For comparison, in the glass coverslip calibration datasets these values were

_{C}*p*= 0.9 ± 0.1,

*p*= 0.94 ± 0.06 and

_{L}*p*= 0.06 ± 0.07, respectively. The relative susceptibility values for the near-isotropic cases are: 0.1 < Re(

_{C}*χ̂*) < 0.8, 0.2 < Re(

_{xxxx}*χ̂*), Re(

_{zzxx}*χ̂*) < 0.6 with median values of Re(

_{xxzz}*χ̂*), Re(

_{xxxx}*χ̂*), Re(

_{zzxx}*χ̂*) ≈ 0.3 for the real parts, and |Im(

_{xxzz}*χ̂*)| < 0.7, |Im(

_{xxxx}*χ̂*)|, |Im(

_{zzxx}*χ̂*)| < 0.3 with median values of |Im(

_{xxzz}*χ̂*)| ≈ 0.25 and |Im(

_{xxxx}*χ̂*)|, |Im(

_{zzxx}*χ̂*)| ≈ 0.1 for the magnitudes of the imaginary parts. All ordered crystalline cases and 30% of the near-isotropic cases benefited from the inclusion of a chiral tensor term into the model. The magnitudes of the real parts of the chiral term are in the range of |Re(

_{xxzz}*χ̂*)| < 0.06 with a median value of |Re(

_{xyzz}*χ̂*)| = 0.05. The magnitudes of the imaginary parts are in the range of 0.04 < |Im(

_{xyzz}*χ̂*)| < 0.18 with a median value of |Im(

_{xyzz}*χ̂*)| = 0.07. The real and imaginary parts are positive in 20% and 70% of the cases, respectively. The distribution of

_{xyzz}*χ̂*values in the near-isotropic and highly-ordered crystalline cases is similar. The Kleinman ratio in the molecular frame (i.e.,

_{xyzz}*χ̂*) significantly deviates from unity by 10 to 30% in half of the near-isotropic cases; the ratio cannot be reliably determined in the ordered crystalline cases because both

_{xxzz}/χ̂_{zzxx}*χ̂*and

_{zzxx}*χ̂*are close to zero. Collectively, the differences in susceptibility values among the individual crystallites indicate a rich variation in the polarimetric signatures, suggesting variations in the underlying structure, orientation and microenvironment of the carotenoid crystallites, which can be accessed using NSMP on a per-pixel basis.

_{xxzz}The orientation angle of the molecular-frame *χ*^{(3)} tensor (*δ*) deviates from the long axis orientation angle of the crystallites. The difference between these two angles is called the slip angle *κ* and it corresponds to a stacking rather than to an end-to-end arrangement of the carotene molecules in the crystalline aggregates [19]. The values of the *κ* angle were generally in the 10° < *κ* < 25° range, but there were cases with *κ* ≈ 0 and *κ* > 25°, indicating a large structural variability between the crystallites.

## 5. Discussion

The third-harmonic NSMP modality can be implemented in a nonlinear microscope by adding the required THG polarization-control components (Fig. 1). It should be stressed, that a proper calibration is an essential step for accurate polarimetry [34], and especially crucial for the nonlinear harmonic-generation case. Simple homogeneous materials, such as a glass coverslip for THG and *z*-cut quartz for SHG, are excellent benchmark tests, for which near-perfect polarimetric responses are attainable (Fig. 2).

Using NSMP theory, polarization information contained in the harmonic generation signals can be extracted, and materials with various symmetry classes can be identified at diffraction-limited spatial resolution. The NSMP method can be applied to study biological samples as demonstrated for the carotenoid crystallites in the chromoplasts of orange carrot root.

Electron microscopy studies of carotenoid crystallites have found diverse structures, such as straight-faceted rhomboids, multilamellar sheets and twisting ribbons as well as multilayer and helical tubes, often encompassing the entirety of the chromoplast plasma membrane [35]. The arrangement of the carotene molecules in these assemblies is not completely understood. Since harmonic signal generation is very sensitive to molecular ordering, well-aligned carotene molecules will produce strong THG. Consequently, densely-packed tubular formations will likely produce bright THG images [18, 19]. This is also suggested by the rod-like appearance of the crystallites (Fig. 5). The largest nonlinear second hyperpolarizability of *β*-carotene, which is the main carotene in orange carrots, is oriented along the polyene backbone [36], and, therefore, tubular structures are expected to have a *χ*^{(3)} tensor with a dominant component close to the axis of symmetry, and a secondary chiral or transverse component. Interestingly, such crystallites convert linearly-polarized laser excitation into circularly-polarized THG emission. This phenomenon may be related to other circular polarization effects, such as first-order optical activity in carotenoid aggregates [37], nonlinearly-enhanced circular anisotropy in chiral films [38,39], and circularly-polarized luminescence [40]. In the THG case, circularly-polarized effects can be attributed to complex-valued *χ*^{(3)} components. The invariant circularly-polarized fraction (*s′*_{3}/*s′*_{0}) for all incoming states also indicates that these complex-valued terms are chiral. The value of the effective chiral components of the ordered crystalline chromoplasts can be estimated from a *C*_{6} model to be in the range of 0.04 < |Im(*χ̂ _{xyzz}*)| < 0.2, which could be attributed to the aforementioned helical or twisted multilamellar sheet crystallite structure. The measured

*cĥi*and

_{ZZZX}*χ*components in the domain A of Fig. 5(c) and (e) are simultaneously positive with a ratio of

_{XZZZ}*χ*= 2 ± 1. The sign of the ratio is a deviation from the

_{XZZZ}/χ_{ZZZX}*C*

_{6}model, which requires that

*χ*= −3 (see Eq. (4) in Appendix A, with

_{XZZZ}/χ_{ZZZX}*δ*= 0). According to the

*C*

_{6}model, positive

*s′*

_{3}values and a stronger THG response to right-hand polarized input light is expected (if

*χ*is positive; for negative

_{xyzz}*χ*the trend is reversed). However, in Fig. 5(a),

_{xyzz}*s′*

_{3}is positive, but LCP input gives a stronger THG response. This trend has been observed in all six investigated ordered crystalline cases but not in the near-isotropic cases. Therefore, future studies may employ a model with more degrees of freedom for describing chiral behavior (i.e., separate parameters affecting input and output CP states), or a detailed molecular-scale structural modeling of the bulk

*χ*

^{(3)}tensor may be required to better describe the polarimetric response of the carotenoid crystallites.

It should be noted that birefringence can also produce nonzero *s′*_{3} values in the measured THG, and would result in an apparent chiral tensor contribution. However, given the small size of the crystallites and the overall noncrystalline nature of the carrot tissue, birefringence effects were assumed to be negligible.

The heterogeneous polarization response of the two domains in Fig. 5 suggests a twisted ribbon, a broken helix or some other nonuniform cylindrical-like structure, and, therefore, a more detailed study may need to consider such inhomogeneous mixing of the *χ*^{(3)} tensor. Since the chromoplast membranes can contain several crystallites of various sizes, it is possible that the observed near-isotropic tensor response comes from a near-homogeneous collection of a few sub-1-μm crystallites within the focal volume [26]. When many small crystallites are replaced with a few large ones, or their orientation anisotropy increases, the effective tensor smoothly transitions from the isotropic to ordered crystalline signature, with near-isotropic intermediate cases.

Finally, when comparing the three presented THG polarimetry cases (Figs. 2 – 5), it is worth mentioning that the Stokes and Mueller matrices, and the *χ*^{(3)} tensor components, are not equal proxies for evaluating how a particular model fits the experimental data. In the isotropic case, all three representations were comparably consistent with the theory; in the near-isotropic case the Mueller matrix representation seemed the most informative, whereas for the ordered crystalline case the Stokes matrix was the most straightforward representation. In addition, the molecular-frame *χ*^{(3)} tensor components clearly highlighted the differences between several domains within a single ordered crystallite. To conclude, the reason behind the differences in the representations is twofold: first, different symmetries (e.g., *C*_{6} vs. isotropic) and polarization effects (e.g., sample bleaching, scattering) affect the three representations differently; second, the experimental noise is not distributed equally. Consequently, it is worthwhile to inspect all three representations when analyzing NSMP data.

## 6. Conclusion

The presented THG polarimetric analysis of carotene crystallites is readily applicable to other samples suitable for harmonic-generation microscopy. It can be used to measure all available complex-valued tensor elements (up to a global constant), irrespective of the underlying symmetry of the susceptibility tensor. Using an NSMP dataset, different structural symmetry models can be tested, and per-pixel molecular structure details can be inferred. In addition, birefringence, validity of the Kleinman symmetry and the presence of chiral tensor elements can be evaluated. In principle, any effect which has a nondegenerate *χ*^{(n)} signature in the elements accessible with 2D polarimetry can be revealed using NSMP. Other coherent nonlinear processes such as SFG (sum-frequency generation) or CARS (coherent anti-Stokes Raman scattering) can also be analyzed by applying the NSMP theory in its general from [22]. Since several nonlinear signals can often be produced simultaneously, for example, SHG and THG polarimetry can be performed in parallel to increase the amount of accessible structural information. With further improvements in experimental and analysis techniques, nonlinear Stokes-Mueller polarimetry is well positioned to simplify and standardize polarization-resolved nonlinear microscopy measurements, and provide access to the full available polarization information content, increasing the usefulness of nonlinear microscopy for biomedical and materials science applications.

## Appendix A: Tensor expressions in the laboratory frame

This section provides the expressions for the relationship between the molecular- and laboratory-frame *χ*^{(3)} tensors for the isotropic and *C*_{6} symmetry cases.

## Isotropic tensor

The isotropic *χ*^{(3)} tensor is orientation-invariant and has only one unique element. The 2D-accessible components are related as [3]: *χ _{zzzz}* =

*χ*= 3

_{xxxx}*χ*= 3

_{zzxx}*χ*.

_{xxzz}The Stokes vector of the outgoing THG radiation can be obtained by carrying out the multiplication *s′* = *ℳ*_{ISO} *S*, where *ℳ*_{ISO} is the triple Mueller matrix for the isotropic case (Eq. (18) in [24]). Assuming the input light is fully polarized, the outgoing Stokes vector is:

*s*are the incoming Stokes components of the laser beam, normalized to

_{i}*s*

_{0}= 1, and

*p*is the degree of linear polarization ( ${p}_{L}^{2}=\left({s}_{1}^{2}+{s}_{2}^{2}\right)/{s}_{0}^{2}$). It can be seen that the THG Stokes vector

_{L}*s′*has the same state as the input Stokes state

*s*, and it scales with the incoming peak intensity ${I}_{0}^{3}$, the degree of linear polarization ${p}_{L}^{2}$ and a characteristic susceptibility component ${\chi}_{zzzz}^{2}$. Note that for circularly-polarized light

*p*= 0 and no THG is generated.

_{L}## C_{6} tensor

A medium with *C*_{6} symmetry has six unique *χ*^{(3)} components in the molecular frame: *χ _{zzzz}*,

*χ*,

_{xxxx}*χ*,

_{zzxx}*χ*,

_{xxzz}*χ*,

_{xyzz}*χ*. The laboratory-frame tensor is obtained by rotating the molecular-frame tensor by the

_{xyyy}*δ*and

*α*angles (with rotation matrices

*R*defined according to Fig. 1(b)):

_{mn}*𝒞*

^{2}= (1 − sin

^{2}

*δ*cos

^{2}

*α*),

*𝒮*

^{2}= (1 − cos

^{2}

*δ*cos

^{2}

*α*), so that for the in-plane case

*α*= 0,

*C*

^{2}= cos

^{2}

*δ*and

*S*

^{2}= sin

^{2}

*δ*.

If *χ _{zzzz}* =

*χ*= 3

_{xxxx}*χ*= 3

_{zzxx}*χ*≫

_{xxzz}*χ*,

_{xyzz}*χ*for any

_{xyyy}*α*or if

*χ*≫

_{xxxx}*χ*for

_{xyyy}*α*=

*π*/2, the

*C*

_{6}tensor reduces to the isotropic case. If tensor parameters are close to these conditions, a

*near-isotropic*polarimetric signature is observed. If

*χ*is dominant, an

_{zzzz}*ordered crystalline*polarimetric signature is observed.

In 2D polarimetry, axial electric fields along the propagation direction are not considered. If the molecular tensor is tilted out-of-plane (i.e. |*α*| > 0), it can be shown that the true *χ*^{(3)} molecular-frame tensor is indistinguishable from an effective *χ̂*^{(3)} tensor, whose elements absorb the *α* dependency using the following expressions, obtained by setting *δ* = 0 in Eq. (4):

*χ*components the

*α*angle has to be known, and it is common to assume

*α*≈ 0.

## Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) (06923, CHRPJ 462842-14) and the Canadian Institutes of Health Research (CIHR) (CPG-134752).

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