1. Introduction

Over the past twenty years polarization-resolved second harmonic generation microscopy (PSHG) has emerged as a powerful tool for characterizing the structure of collagenous tissues. The high sensitivity of PSHG to both the in (image) plane and out of plane orientation of collagen fibrils enables its use for three-dimensional visualization of collagen distribution in tissues [1]. This is highly desirable for biological imaging as an increase in disorder between fibrils within the extracellular matrix is often associated with cancer and other diseases [2]. Along with the high sensitivity of PSHG to the molecular structure of collagen [3,4], this has led to widespread investigation of the potential of PSHG as a tool for automated and quantitative disease diagnosis [5–8].

The sensitivity of PSHG to the relative out of plane orientation of collagen fibrils has been of particular interest recently [1,9–11]. Two commonly used techniques for determining the relative out of plane polarity of collagen fibrils using PSHG are polarization-in, polarization-out SHG (PIPO-SHG) and circular dichroism SHG (CD-SHG). PIPO-SHG utilizes different orientations of linearly polarized light in excitation and interrogates the linear polarization components of the outgoing SHG to determine parameters related to the disorder and chirality of the structures being imaged. The sign of the chiral parameter $\mathrm{\kappa}$, which can be extracted with this technique, is theoretically related to the relative out of plane polarity of a collagen fibril [1]. It has been shown that for collagen in tendon cut at different angles with respect to the tendon axis, the absolute value of $\mathrm{\kappa}$ increases with cut angle [1,9], however thus far there has been no direct experimental confirmation that the sign of this parameter is related to the relative out of plane polarity of individual collagen fibrils.

CD-SHG utilizes the normalized difference in SHG intensity between right- and left-handed circularly polarized excitation and gives a parameter related to the out of plane angle of collagen fibrils. It has been commonly assumed that the sign of CD-SHG indicates the relative out of plane polarity of a fibril [9,10,12–15] as this is the expected result if the SHG emitter adheres to the electric dipole approximation [9,10]. However, recent results have shown that the sign of CD-SHG is invariant upon flipping the out of plane orientation of the sample for collagen in human cornea, murine fascia, and collagen fibrils assembled in-vitro [11]. This result indicates that CD-SHG may be a product of additional contributions to SHG in collagen fibrils beyond the electric dipole approximation, such as magnetic dipole effects. However, these results have recently been contradicted, with CD-SHG being observed to change sign as relative fibril orientation changes along a rat tail tendon [10].

In order to verify the sensitivity of these two methods to relative out of plane polarity, measurements of individual collagen fibrils oriented off of the image plane are desirable. However, due to the small (∼50-500 nm) diameter of collagen fibrils [16] it is difficult to hold a single fibril at an angle to the image plane in a way that is easily compatible with PSHG. Recently it has been shown that by depositing collagen fibrils onto a pre-strained elastic substrate and allowing it to return to its original length, fibrils parallel to the strain orientation buckle. This buckling leads to a sine wave like shape (buckle) along the fibril, which can oscillate in and out of the image plane if the fibril is dry [17,18]. Here we perform SHG measurements on out of plane buckled collagen fibrils using SHG intensity, PIPO-SHG and CD-SHG. By comparing SHG intensity images to atomic force microscopy (AFM) measurements of identical fibril regions, we show that the change in SHG intensity along a buckled fibril can be used to accurately determine the fibril’s out of plane angle. Through analysis of the dependence of parameters measured using PIPO-SHG and CD-SHG as a function of out of plane angle we demonstrate the ability of PIPO-SHG to determine relative out of plane polarity at the level of individual collagen fibrils. We also find that CD-SHG is not sensitive to the relative out of plane polarity of individual fibrils, validating previous results from collagenous tissues. These results demonstrate the effectiveness of PIPO-SHG as a tool for three-dimensional visualization of collagen structure and are expected to be of high importance in the use of PSHG as a tool for disease diagnosis within collagenous tissues.

2. Materials and methods

Collagen fibril isolation and buckling: The method used for collagen fibril isolation has been previously described [19]. Briefly, collagen fibrils are extracted from ∼2 mm thick tendon sections cut from dissected adult bovine lateral digital extensor tendons obtained from a local abattoir and kept at -80°C before use. The tendon section is hydrated with 1 mL of reverse osmosis water and fibrils are extracted into the surrounding liquid by scraping the tissue with tweezers for approximately 5 minutes. Fibrils are then buckled as described in [17] by depositing ∼2 µL of fibril containing solution onto a ∼120 µm thick, ∼20 mm long strip of polydimethylsiloxane (PDMS, Electron Microscopy Sciences) which is held at 20% strain using a custom-built piezoelectric stretching stage. The sample is then left for 45 minutes to allow fibrils in solution to adhere to the substrate, and gently washed with reverse osmosis water for 2 minutes. The sample is then dried under a stream of nitrogen gas for 5 minutes and then left to air dry overnight. After air drying, the fibrils are buckled by releasing the strain on the PDMS at a rate of 10 µm/s, and then the piece of PDMS containing fibrils is removed from the stretching stage and mounted on a glass coverslip (16004-314, VWR).

AFM imaging: Buckled fibrils were imaged using a Bioscope catalyst (Bruker, USA) atomic force microscope mounted on an inverted optical microscope (Olympus, USA). The images obtained have a size of 20 × 20 µm with a pixel size of 39 nm. Scanning was performed at a rate of 0.5 Hz, and the peak force setpoint was in the range of 5–7 nN depending on sample height. AFM images were flattened and converted to tag image file format (TIFF) images for easy comparison with PSHG images using Gwyddion [20].

PIPO-SHG Imaging: PIPO-SHG imaging was performed using a custom-built laser scanning microscope, previously described in [21,22] and is schematically shown in Fig. 1(a). Briefly, the laser beam was raster scanned across the sample using a pair of galvanometric scan mirrors (ScannerMAX, Pangolin Laser Systems, Inc.) with a pixel dwell time of 6 µs, to create an image with a pixel size of 180 nm and size 18 × 18 or 36 × 36 µm. The beam was focused onto the sample using an air immersion microscope objective lens with a numerical aperture (NA) of 0.8 (Plan-Apochromat 20, Carl Zeiss AG). The sample was placed such that the laser beam would pass through the coverslip and PDMS before encountering the fibrils. SHG signal was collected in transmission geometry using a custom objective lens with an NA of 0.85 (Omex Technologies, Inc.). The ultrafast laser (FemtoLux 3, EKSPLA) had a wavelength of 1030 nm, repetition rate of 5 MHz, and pulse duration of ∼290 fs. The SHG signal was separated from the laser light using an interference filter centered at 515 nm with a 10 nm bandwidth (65-153, Edmund Optics Inc.). SHG signal was measured using a single-photon-counting photomultiplier detector (H10682-210, Hamamatsu Photonics K.K.) and obtained using a data acquisition card (PCIe-6363, NI).

 

Fig. 1. Schematic diagram of the microscope setup (a) and coordinate system used here (b). The following abbreviations are used: L – laser, SM – galvanometric scan mirrors, PSG – polarization state generator, LC – liquid crystal variable retarder, EO – excitation objective, CO – collection objective, PSA – polarization state analyzer, PM – optical power meter, F – filter, PMT – photomultiplier tube. The dashed arrows in (a) indicate components that can be added for calibration and CD-SHG imaging (see below). The fibril is shown in orange in (b) and the imaging plane is shaded in gray.

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The polarization state generator (PSG) consisted of a stationary linear polarizer (LPVIS100, Thorlabs, Inc.) followed by a flat half-wave plate (WPMP2-22-V1030, Karl Lambrecht Corp.) on a motorized rotation stage [see Fig. 1(a)]. The polarization state analyzer (PSA) consisted of a polarizing filter (analyzer, LPVISA 100, Thorlabs, Inc.) on a motorized rotation stage. A PIPO-SHG image stack is produced by obtaining SHG images of the sample at all combinations of eight laser polarization angles and eight analyzer angles at 22.5° increments for a total of 64 images. A 65^th image is subsequently obtained at the same half-wave plate and analyzer angles as the first image as a control for photobleaching and movement of the sample. The PIPO-SHG data is fit using a trust region algorithm in MATLAB (The Mathworks, Inc.) using the following equation which describes SHG intensity (${\textrm{I}_{\textrm{SHG}}}$) as a function of incident polarization angle with respect to the projection of the fibril axis on the image plane ($\mathrm{\theta}^{\prime} = \mathrm{\theta} - \mathrm{\delta}$, for fibril at in plane angle $\mathrm{\delta}$, see angle definitions in Fig. 1) and analyzer angle ($\mathrm{\phi}^{\prime} = \mathrm{\phi} - \mathrm{\delta}$) for a cylindrical sample [1].

Here $\textrm{A}$ is the SHG amplitude, F is the background noise and $\mathrm{\rho}$ and $\mathrm{\kappa}$ are structural parameters defined by the following equations [1].

In these equations $\mathrm{\alpha}$ is the angle between the cylindrical axis of the sample and the image plane, ${\mathrm{\rho }_\textrm{f}}$ is the fibril frame disorder parameter which is determined by the distribution of angles of the peptide backbone in the collagen triple helix [3], ${\mathrm{\kappa }_\textrm{f}}$ is the fibril frame chirality parameter whose sign is determined by the net chirality of all SHG emitters within the focal volume [23] and ${\mathrm{\chi}^{(2 )}}$ is the second order electric susceptibility tensor describing the SHG light-matter interaction of the material. The lab frame coordinate system XYZ and the fibril frame coordinate system xyz are defined according to Fig. 1(b).

CD-SHG imaging: For CD-SHG imaging the PSA is removed and a liquid crystal variable retarder (LC, LCC1223-C, Thorlabs Inc.) is inserted after the PSG. To obtain circular polarization the PSG must be rotated such that the incident laser polarization is at 45° to the axis of the LC and the two voltages of the LC corresponding to left- and right-handed circular polarization must be determined. To do this third harmonic generation (THG) signal is obtained from the glass-air interface of a cover slip (16004-314, VWR) using an interference filter (65129, Edmund Optics Inc.), and the linear polarization angle and LC voltages are selected such that the THG signal is minimized corresponding to circularly polarized excitation [24]. To ensure that the CD-SHG data obtained is reliable the quality of circular polarization obtained must be quantified [25]. Thus an IR linear polarizing filter (LPNIR050, Thorlabs, Inc.) is placed into the motorized stage of the PSA, and an optical power meter (S130C, Thorlabs Inc) is placed just after, and the degree of circular polarization (DOCP) of the laser beam is obtained using the following formula [26].

Here ${\textrm{I}_\mathrm{\sigma }}$ is the measured laser power when the polarizing filter is at $\mathrm{\sigma}$ angles 0, -45, + 45 or 90°. To ensure high quality circular polarization over the entire image the DOCP is measured for two points, the image center and ∼100 µm from the center, to ensure consistent polarization states within our $36 \times 36$ µm scan window. The average of the DOCP values used to obtain CD-SHG images reported here is 0.97 which is in the “quasi perfect” circular polarization range according to recent reports [25,26]. We require that both the left- and right-handed circular polarizations at each point have a value of DOCP > 0.95 to be included here.

Subsequently 10 pairs of SHG intensity images are obtained using left (I_L) and right (I_R) circularly polarized laser excitation and with the PSA removed. The circular dichroism parameter (CD) is then obtained at each pixel of the image using the following equation, and the reported CD is the mean of the CD values from all 10 pairs of images.

Like PIPO-SHG imaging, the first and last images of the CD-SHG data that were obtained using right-handed circular polarization are compared to determine if the sample moved or burned during imaging. This is done by comparing fibril position and SHG intensity between the two images. Only samples with less than one pixel of movement during imaging are included here.

Profile extraction: Profiles of height, SHG intensity, $\mathrm{\rho}$, $\mathrm{\kappa}$, and CD are extracted along the fibril axis using ImageJ [27]. AFM height profiles are manually aligned to the PIPO-SHG data using features such as ends of a fibril, changes in direction, or buckled areas as a reference. To obtain profiles of the $\mathrm{\alpha}$ angle along the fibril the slope is determined between neighboring pixels in the AFM height profile, and the angle is determined by taking the inverse tangent of the slope. For comparison of $\mathrm{\alpha}$ measured with AFM to SHG intensity, the angle is smoothed using a five-pixel moving average, and then the profile is scaled down to the resolution of the SHG images.

To reduce the impact of nonuniformities in focal field polarization near the edge of the focal volume [28] only the most centered (brightest) single pixel across the width of the fibril is included in the profiles of PSHG data.

3. Results and discussion

AFM and PIPO-SHG imaging of buckled fibrils: Five buckled collagen fibrils on the same sheet of PDMS were selected due to their good (± 20°) alignment with the buckling direction, were imaged using PIPO-SHG, and the polarization resolved data was fit using Eq. (1) to extract $\mathrm{\rho }$ and $\mathrm{\kappa }$ maps. The surface heights of the same five fibrils were subsequently measured using AFM to precisely determine the out of plane angle induced along the fibril as a result of buckling at each 39 nm sized pixel. Two different buckling modes are observed, two fibrils have periodic ∼300 nm amplitude modulations in height (Fig. 2(a)) as expected for the buckling instability of a stiff rod attached to a soft substrate [15], and three of the fibrils having a small number of aperiodic high amplitude (>700 nm) buckles (Fig. 2(b)) that correspond to a torsional buckling mode due to the twisted nature of collagen fibrils [17]. SHG scanning results for the two representative fibrils are also shown via SHG intensity images in Fig. 2(c),(d), obtained by summing the 64 polarization images in the PIPO-SHG stack, as well via the fitted values of $\mathrm{\rho }$ (Fig. 2(e), (f)) and $\mathrm{\kappa }$ (Fig. 2(g), (h)). These images show clear changes in relative SHG intensity, $\mathrm{\rho}$ and $\mathrm{\kappa }$ in areas where the fibril is tilted oriented off the image plane ($\mathrm{\alpha} > 0$) (see area indicated by a solid arrow in the top row of Fig. 2 and a dotted arrow in the bottom row of Fig. 2). For both buckling types a variation in $\mathrm{\rho}$ is observed perpendicular to the fibril. This variation in $\mathrm{\rho}$ was previously investigated in individual unbuckled fibrils and altered $\mathrm{\rho}$ values were found to occur when a fibril resided near the edge of the focal volume. The nonuniformity of the polarization within the focal field is a result of high numerical aperture focusing [28], and the effect is only prominent when no SHG emitters are present in the middle of the focal volume [28].

 

Fig. 2. AFM and PIPO-SHG data and fits for buckled fibrils in two modes. AFM images showing fibril height (a), (b), SHG intensity (c), (d), $\mathrm{\rho }$ (e), (f), and $\mathrm{\kappa }$ (g), (h) for a typical periodically buckled, and high amplitude buckled fibrils. Arrows indicate the same low amplitude or high amplitude fibril buckle in all four image types.

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Interestingly values of much greater than 3 are obtained for the fibrils in the high amplitude buckling mode, which contradicts what we expect from Eq. (2a) which is bounded between $\mathrm{\rho}_\textrm{f}$ and 3 (Fig. 2(f)). One reason for high $\mathrm{\rho}$ values could be a particular type of disorder, and in addition to high amplitude buckling mode, these fibrils also have a significant in plane buckling component (see Fig. 2(b),(d)), increasing their disorder. If the disorder within a focal volume occurs due to two fibrils crossing, then $\mathrm{\rho}$ could increase far above 3, as recently shown by a numerical focal spot simulation modeling collagen fibrils [29]. On the other hand, if the disorder is the result of a Gaussian or Lorentzian distribution in the pitch angles of SHG emitters within a molecule then it stays confined to 3 [30]. Another possibility for the $\mathrm{\rho} > 3$ values could be total internal reflection by tilted fibrils. Since dry collagen has an index of refraction of ∼1.5 [31] we expect total internal reflection of the emitted SHG light due to the collagen-air interface at $\mathrm{\alpha}$ = ∼42°. In the case of high amplitude buckling the $\mathrm{\alpha}$ angle is often near or greater than 42°, indicating different transmission of X vs Z polarization components occurs, which alters PSHG data, possibly affecting both $\mathrm{\rho}$ and $\mathrm{\kappa}$. It is also possible that strain during buckling could alter the fibril structure leading to an altered $\mathrm{\rho}$. We note that that eliminating pixels with $|\mathrm{\alpha } |$ > 40° removes all instances of $\mathrm{\rho}$ > 3 for low amplitude buckled fibrils, which suggests the total internal reflection explanation is correct. Therefore, only pixels with $|\mathrm{\alpha } |$ $\le $ 40° are considered for subsequent quantitative analysis to avoid this effect.

In both types of buckled fibrils, there is a clear modulation in SHG intensity along the fibril (Fig. 2(c)). Regions of high SHG intensity correspond to the areas where the fibril is oriented out of the image plane (see areas indicated by arrows in Fig. 2(a),(c) and (b),(d)). The increase in SHG intensity with tilt contradicts previous theoretical results which show that the SHG intensity of an emitter should be proportional to the cosine squared of its $\mathrm{\alpha}$ angle [32]. To understand the reason for this discrepancy it is important to recall that for high numerical aperture focusing the focal volume is much larger axially (Y, see Fig. 1) than laterally (X-Z), e.g., the microscope used here has a focal volume with a size of 0.6 µm laterally, and 2.1 µm axially. Since collagen fibrils are elongated structures with lengths much greater than the lateral size of the focal volume, then more of the fibril is inside the focal volume, as $\mathrm{\alpha}$ increases, from 600 nm when $\mathrm{\alpha} = 0$, and over 3 times that value as $\mathrm{\alpha} \to 90^\circ$. A higher overlap between the fibril and the focal volume means more SHG emitters are in the focal volume. According to recent focal volume simulations, it appears that this additional effect compensates for the decrease in individual emitter intensity with increasing $\mathrm{\alpha}$ in the range up to $0 - 50^\circ $, and then drops back down [33]. A similar increase in signal was observed in THG imaging of canaliculi, thin connective channels between bone cells, with similar diameters to collagen fibrils. The intensity of THG was also observed to increase with increasing $\mathrm{\alpha}$ in the angle range 10-50$^\circ $, and similarly ascribed to the increasing volume overlap between focal volume and the structure [34], as well as in this simulation [35]. Note that when imaging collagen in tissue we still expect a cosine squared relationship between $\mathrm{\alpha}$ and SHG intensity, as the high density of fibrils will result in a near constant number of SHG emitters within the focal volume regardless of the orientation of individual fibrils.

The dependence of SHG intensity on out of plane angles up to $40^\circ $ shows that it may be possible to determine the out of plane angle for each pixel along the fibril through analysis of the changes in SHG intensity rather than using AFM. To confirm this, we extract profiles along each fibril for both height measured with AFM and for normalized SHG intensity (normalized to the maximum intensity along the profile, see Fig. S1 for an example of height and intensity for a typical fibril). We find that height and SHG intensity are well correlated (Pearson’s correlation coefficient 0.77). We then obtain the out of plane angle from each AFM profile as described above (see Materials and Methods). An estimate for $\mathrm{\alpha}$ based on SHG intensity is obtained by taking the inverse tangent of the slope of the normalized intensity profile, similar to how the angle is extracted from AFM data. After binning in increments of 5° in $\mathrm{\alpha }$ estimated from SHG intensity (see Fig. 3), using a linear fit we find a slope of 0.9 ± 0.1 between $\mathrm{\alpha }$ estimated from SHG intensity and $\mathrm{\alpha }$ measured using AFM for the combined data from all five fibrils. This method is able to produce a reasonable estimate of $\mathrm{\alpha}$ based on SHG intensity since the height of the buckled fibrils is typically in the range of 0–1 µm, therefore by normalizing the SHG intensity profile, the height and intensity will be on a similar scale. The high correlation between the angle measured with AFM and angle estimated from SHG intensity validates using SHG intensity to obtain the height, and enables the investigation of the out of plane angle dependence of PSHG parameters without the need for AFM measurements of the angle. This is highly advantageous since AFM measurements can be quite challenging for samples with large variations in height due to the high probability of the tip colliding with the sample.

 

Fig. 3. Comparison of out of plane angle $\mathrm{\alpha }$ measured using AFM and estimated from SHG intensity changes along the fibril. The data is shown as the mean angle in 5° increments with standard error as the error bar. The fitted regression line has a slope of 0.9 ± 0.1 (fit ± 95% confidence bound, R² = 0.97). The combined dataset including all five fibrils contains a total of 321 pixels.

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Comparison of PIPO-SHG and CD-SHG for Buckled Fibrils: PIPO-SHG as well as CD-SHG measurements are performed on three periodically buckled collagen fibrils, and profiles in SHG intensity, $\mathrm{\rho }$, $\mathrm{\kappa }$, and CD are obtained. The $\mathrm{\alpha}$ angle of the fibril was estimated based on SHG intensity as described above (see Fig. 3). This allows us to estimate out of plane angles for these fibrils without the need for AFM. Note that we have no absolute measurement of the fibril polarity, and therefore we define the positive $\mathrm{\alpha }$ direction to be the direction where the majority of pixels have positive $\mathrm{\kappa }.$ Because the pixel size of the SHG images is less than the focal spot size of our microscope, each pixel has significant overlap with the neighboring pixels. To correct for this the profile of $\mathrm{\alpha }$ values along the fibril is smoothed by taking a moving average over five pixels. This results in areas in the vicinity of sharp peaks having $\mathrm{\rho }$, $\mathrm{\kappa }$, and CD values that are either much larger or much smaller than expected. To correct this, datapoints near these peaks are manually removed.

The data is subsequently binned in increments of 5° in the estimated angle, with the standard error in each bin reported as uncertainty. To extract values of ${\mathrm{\rho }_\textrm{f}}$ and ${\mathrm{\kappa }_\textrm{f}}$ for each fibril weighted least squares fits using the inverse square of the size of the error bars as weights are performed using Eqs. (2a) and (b). To evaluate the accuracy of our $\mathrm{\alpha}$ estimate, a scale factor between the estimated angle and the true angle is included as a fitting parameter for the $\mathrm{\rho }$ data, and this is used to correct the angle for each fibril before fitting $\mathrm{\kappa }$. Further, to correct for slight errors in the zero-angle position induced by binning, an offset angle is included in the $\mathrm{\kappa }$ fit. The results of these fits, as well as fits for the combined data set are shown in Table 1. The combined data and fits for $\mathrm{\rho }$, $\mathrm{\kappa }$, and CD are shown in Fig. 4.

 

Fig. 4. Dependence of PSHG parameters on $\mathrm{\alpha}$. Data and fits using Eq. (2) are shown for $\mathrm{\rho }$ (a), $\mathrm{\kappa }$ (b), and CD (c). Graphs show the combined data from all three fibrils, error bars are standard error, and the combined dataset contains a total of 433 pixels.

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Table 1. Fitted values of ${\mathrm{\rho}_{\rm f}}$ and ${\mathrm{\kappa}_{\rm f}}$, as well as associated scale factors and offset angles. The 95% confidence bounds of each fit are reported as the uncertainty.

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For $\mathrm{\rho }$ we see an increase with $|\mathrm{\alpha } |$ as expected based on Eq. (2a) (Fig. 4(a)), similar to previous observations of tendon tissue cut at varying angles [1,9,10]. Note that the scale factors used in the $\mathrm{\rho}$ fits as well as in the slope obtained from the linear fit in Fig. 3 are all within an uncertainty of 1 validating the approach used to estimate the angles. Figure 4(b) shows the sign of the measured $\mathrm{\kappa }$ values have a strong correlation with the sign of the $\mathrm{\alpha }$ angle of the fibril, where the two values match in sign of $\mathrm{\alpha }$ for 77% of image pixels. Although both positive and negative $\mathrm{\kappa }$ values which increase in absolute value with $\mathrm{\alpha}$ have been obtained previously for collagen fibrils in tissues [1,9,36,37], to our knowledge this is the first time the relationship between the sign of $\mathrm{\kappa }$ and individual fibril out of plane polarity has been experimentally demonstrated. This confirms the utility of PIPO-SHG as a technique to map the relative orientations of collagen fibrils in tissues. Importantly these results are obtained for individual collagen fibrils rather than collagen in tissue, thus eliminating the possibility of nonuniform fibril orientation leading to experimental errors [38,39]. This is additionally significant since while a very large number of papers have investigated SHG from collagen fibrils in tissues, very little work has been done so far on the SHG response from individual collagen fibrils [28,40,41]. These results also open the possibility of using PSHG for the analysis of other nanoscale biological structures such as myofibrils and microtubules [42,43].

Note that the fibrils used in this study were buckled after being dried. Changes in hydration are known to result in changes in the $\mathrm{\rho}$ value of the fibril (see supplement S1 in [28]), with native collagen typically being in the hydrated state. Therefore, it would be preferable to perform this experiment on hydrated fibrils for increased relevance to PSHG measurements from tissue. Unfortunately buckling of hydrated fibrils results in a denaturation of the fibril rather than the out of plane buckling used here [17], meaning that only dried fibrils can be used in this experiment.

Our measurement of CD-SHG revealed increases with $|\mathrm{\alpha } |$, and that 91% of all pixels in the 3 measured periodically buckled fibrils have positive values of CD (Fig. 4(c)). Note that $|\mathrm{\alpha } |$ is rarely near zero since there are very few flat regions along the buckled fibrils, meaning that we do not obtain a CD value for fibrils within the image plane from imaging buckled fibrils. To verify that CD is zero for fibrils in the image plane we additionally performed CD measurements of five flat (not buckled) fibrils on a glass coverslip and obtained a mean value of CD = 0.00 ± 0.03 (mean ± standard deviation).

The observation that the sign of CD is invariant to changes in fibril out of plane polarity contradicts the common assumption that the sign of CD is determined by the out of plane polarity of the fibril [9,10,12–15]. The observation supports recent results which suggest that magnetic dipole or electric quadrupole contributions are required to accurately describe CD in collagen fibrils [11]. Importantly our demonstration of buckled isolated collagen fibrils reveals that the invariance in CD under changes in relative out of plane polarity persists at the nano scale, thus removing the possibility that this result is an artifact caused by the presence of multiple fibrils in a focal volume for tissues. Interestingly all three of the fibrils measured here showed predominantly positive values of CD. This is consistent with our previous results showing that fibrils have an apparent imaginary part of ${\mathrm{\kappa }_\textrm{f}}$ which has the same sign for most fibrils [28]. However, both positive and negative values of CD have been measured for collagen in tissue [9,11,12,25,26,44–46], possibly supporting the recently proposed idea that the sign of CD from collagenous tissues is determined by the polarity distribution of multiple fibrils within the focal volume of the microscope [11]. These results also highlight a major gap in our current understanding of collagen SHG, since while the molecular origins of electric dipole allowed SHG have been well characterized for collagen [4] to our knowledge no work has been done so far to investigate the molecular origins of magnetic dipole or electric quadrupole contributions to SHG in collagen. Investigation of the origin of these effects using ab-initio quantum mechanical calculations is a potential future area of research.

The value of ${\mathrm{\rho }_\textrm{m}}$ measured here of 1.95 ± 0.02 agrees with our previously measured value for dried collagen of 1.9 ± 0.2, while our measurement of ${\mathrm{\kappa }_\textrm{f}}$ of 0.042 ± 0.006 disagrees with previously measured values of 0.29 ± 0.04 and 0.39 ± 0.05 for collagen in porcine tendons [1,9]. This result possibly indicates that individual collagen fibrils have only small chiral SHG contributions, and the larger chirality measured in tendons is the result of some higher order structural features. It is also possible that the ${\mathrm{\kappa }_\textrm{f}}$ value is highly dependent on sample conditions such as fixation, or hydration as is the case for ${\mathrm{\rho }_\textrm{m}}$ [28]. This is a potentially important result for the field of nonlinear optical microscopy as the origin of chiral contributions to SHG signal from collagen and other materials has been an area of significant interest [47–50].

4. Conclusions

We have performed the first direct experimental confirmation that PSHG is sensitive to the relative out of plane polarity of collagen at the individual fibril level. We have also confirmed that circular dichroism SHG is not sensitive to the relative out of plane polarity of collagen fibrils, indicating that it originates from effects beyond the electric dipole approximation. These results also demonstrate the potential of PSHG microscopy for the analysis of nanoscale biological structures. We also show that the $\mathrm{\kappa}$ parameter measured using PIPO-SHG is sensitive to the relative out of plane polarity of collagen fibrils, demonstrating its potential for three-dimensional visualization of collagen organization in tissue.