1. Preparing the microscope slides with fully-grown spherulites

NOTE: See the Table of Materials for details about all materials, reagents, and equipment used in this protocol. All solutions were prepared with deionized water (18.2 MΩ·cm at 25 °C) obtained from the water purification system.

1. Incubate the amyloid spherulites based on the protocol described by Krebs et al¹⁶. with some modifications, as described below.
   1. Weigh 10 mg of insulin powder in a 1.5 mL tube.
   2. Dissolve the powder with a 1 mL aliquot of the deionized H₂O/HCl solution (pH 1.5).
   3. Seal the sample with the tape and put it in a thermal mixer to incubate for 24 h at 70 °C (0 rpm).
      NOTE: To perform imaging of amyloids in their native (i.e., hydrated) environment and prevent sample deformation due to dehydration, it is necessary to prepare microscope slides according to the following description (scheme shown in Figure 2).
2. Wash the microscope glass slide thoroughly with water.
3. Dip the slides in methanol and leave them to dry under ambient conditions on a dust-free wipe.
4. Take a 100 µL aliquot of the spherulite solution from the tube with an automatic pipette (step 1.1.2) and add it to the well located in the central area of the microscope slide.
5. Cover the solution with a coverslip, avoiding air bubble formation.
6. Deposit mountant along the edges of the coverslip on the slide with an automatic pipette to seal the spherulites' native solution.
   NOTE: It is of great importance to deposit slide mountant on both the coverslip as well as slide surfaces. Do this under the fuming hood because of the toxicity of the volatile solvent (xylene). See the inset in Figure 2.
7. Leave the microscope sample under ambient conditions for the mountant to harden.
   NOTE: The hardening process is temperature- and humidity-dependent but should be completed within 12 h.
8. Check if insulin spherulites were formed correctly using a polarized optical microscope (POM) with crossed polarizers. Scan through the sample looking for a characteristic pattern of bright areas, called a Maltese cross, as shown in Figure 3.
   NOTE: The amyloid spherulites can be defined as spherical superstructures characterized by a heterogeneous morphology with a core-shell structure. In detail, they are composed of an amorphous core and radially growing amyloid fibrils¹⁶. Due to the anisotropic character of spherulite structures, they can distinctively interact with polarized light (e.g., by refraction) and alter their phase, which can be easily observed under POM with crossed polarizers. This method was used to study only fully developed spherulites characterized by a model-like Maltase cross pattern. Consequently, aggregates or structurally distorted spherulites were excluded from further investigations.

2. Building and aligning the system

NOTE: A schematic representation of a polarization-sensitive two-photon microscope can be found in Figure 1.

1. Install a femtosecond laser with the output wavelength tunability in the 690-1,080 nm range, for example, operating on a mode-locked Ti: Sapphire laser with ~100 fs pulses of 80 MHz repetition rate.
2. Add a half-wave plate mounted on a rotation stage to the excitation path of the setup to control the polarization of the incident light in the XY microscope sample plane.
3. Install the dichroic mirror to cut off the excitation beam from the detecting optics.
   NOTE: The optical characteristics of a dichroic mirror should allow to reflect the excitation beam (to the specimen) in the NIR range of wavelengths and be simultaneously transparent for the wavelength range corresponding to emission from the samples.
4. Install a piezoelectric scanning stage for raster scans within the XY plane for a chosen Z.
5. Mount the high NA immersion objective, for example, an apochromatic oil immersion objective 100x/1.4 NA.
   NOTE: As the entire setup operates in an epi-fluorescence mode, the incident and emitted signals are passing by the same objective.
6. In the emission path, add a polarizing beam-splitter that splits two-photon excited emission into two orthogonally polarized components (I_X and I_Y)
7. Install two photon-counting avalanche photodiodes (APDs) in a configuration allowing them to collect light transferred and reflected using the beam-splitter, respectively (Figure 1).
8. Mount correct wavelength-dependent cut-off filters on the excitation and emission path of the system, for example, an 800 nm long-pass filter directly in the excitation optical path and a 700 short-pass filter in the emission path.
   NOTE: Analyze the emission spectra from spherulite so that any potential contribution from second harmonic generation (SHG) or laser light could be excluded using the correct emission filters. It is also worth noting that this system should also allow measurements of polarization-sensitive SHG, which could be used to resolve the ordering of biomolecules within the samples. However, the data analysis of the SHG signal differs from fluorescence, which, in more detail, was described by Aït-Belkacem et al.¹⁷.
9. Align the whole system (using the alignment mode built in many lasers) until similar signal intensities are collected using both photodiodes.
10. Check if the excitation beam focused by the high NA objective and imaged on a camera has a concentric circular shape as presented in Figure 4A.
11. Adjust the scan time and corresponding power to minimalize the damage induced by the incident laser on the sample. Exemplary point-like burning is presented in Figure 4B. Measure the nominal power using a digital handheld power meter connected to a photodiode power sensor located at the entrance to the body of the microscope.
    NOTE: Biological specimens can be easily burned due to laser illumination. In this case, the 100 -900 µW power range (at the focal point of the objective lens) was found to be an excellent trade-off between sample stability and intense emission.
12. Before the sample measurement, test the quality of alignment of optics with an isotropic reference sample (e.g., fluorescein embodied in the amorphous polymer). To perform the calibration check, follow the procedure described in section 3 using an isotropic reference instead of an amyloid sample.
    NOTE: With an assumption that the microscopy setup is ideally adjusted for the sample with isotropic properties, both 2P emission components (I_X and I_Y) detected with two APDs should be characterized by the same intensity (Figure 4C).

3. Measurement of the bovine insulin spherulites

NOTE: To perform all the described ps-2PF measurements, hand-written software was used, which controls the positions of the piezoelectric stage and half-waveplate and collects the signal from both photodiodes, allowing for plotting of the XY scans (raster scans) from selected areas on microscope slides as well as polar graphs from specific sample coordinates. Additional notes have also been added to the protocol, which will allow users to perform the measurement without it, since both the piezo-stage and rotation stage used to rotate the half-wave plate could be controlled using their own controllers or corresponding software. Still, it is strongly recommended to write an algorithm combining the rotation angle of a half-wave plate with 2PEF intensity collected with both photodiodes since this correlation (polar graphs) is crucial for proper data analysis resulting in structural information.

1. Mount the specimen with the spherulites on the piezoelectric stage (immobilize the glass slide with tape). Be sure to mount it in a way where a thin glass side (coverslip) is facing the objective as high numerical objectives are characterized by relatively short working distances.
   NOTE: As oil immersion is used, a small drop of mineral oil should be applied to the objective before specimen mounting and focusing.
2. By changing XY and Z positions using microscope knobs, focus the objective on one of the spherulites found in the solution, but be aware that because spherulite diameters are typically in the range of tens of µm, focus within the central area of the entire structure. Look for black and white haloes around the specific spherulite as signs of under- and upper-focus, respectively. Adjust the "Z" axis to be between these extremes.
   NOTE: It is important to find an isolated sample without structural defects. Extremely small spherulites may drift off due to the material stress introduced by the objective while the biggest structures are probably aggregated or highly distorted.
3. Center the spherulite in the field of view of the observed microscopic plane and determine the size of XY scan by looking at how many piezostage steps in X and Y directions (displayed on the piezostage controller or in its software) are needed to cover the area of the whole structure.
   NOTE: It is recommended to set the system in such a way that the beginning of the XY scan is located near one of the corners of the spherulite and coincides with the zero position of the stage (X and Y = 0)
4. Adjust the following scanning parameters:
   1. Adjust the polarization of the excitation beam, and rotate the half-wave plate to obtain the polarization corresponding to the "X" and "Y" axes of the microscope sample plane.
      NOTE: This can be done by measuring the polar graphs of isotropic fluorescent medium such as fluorescein. For such materials, maximum fluorescence intensity is parallel to the excitation beam polarization; thus, rotating the half-wave plate adjusts the polarization with X and Y axis on the observation plane, also denoted as X and Y axis on polar graphs as in Figure 4C. Full polar graphs could be measured by collecting the measured 2PEF intensity on both photodiodes for 180° rotation of half-wave plate (360° rotation of excitation light polarization) and correlating the intensity with the excitation light polarization angle. It could be done manually, by measuring the emission intensity for every half-wave plate angle separately and then assembling it into a polar graph in data analysis software or automatically, using dedicated or self-written software.
   2. Adjust the piezostage parameters: scanning speed, step, and range to cover the area of the whole spherulite.
      NOTE: The scanning range must be higher than the spherulite diameter to fully frame the entire superstructure within the raster scan. Low scanning speed and step allow users to obtain high-quality images; however, this may result in sample burning. Therefore, a trade-off dependent on the spherulite size is necessary. These parameters cannot be universally applied. Exemplary parameters used in Figure 5A: scanning range 45 x 45 µm, scanning step 1 µm, and scanning speed 2 µm/s.
5. Open the shutter, turn on the photodiodes, and collect and 2P emission components for every single step of the selected scanning area-for the excitation beam polarized correspondingly to the X and then, the Y axes. Exemplary two-photon excited autofluorescence (2PAF) raster scans of insulin spherulites are presented in Figure 5A.
   NOTE: Measuring raster scans requires correlating piezo stage coordinates with and collected emission components, which could be done manually, by measuring the intensity in every single point of the selected area and then assembling it into a 2D matrix, or automatically, via self-written software. Raster scans can be presented as a summation of and emission components' intensity or distinctively for a specific emission component. As they are highly structure-dependent, structural distortions within spherulites are easily visible. However, due to the movement of the microscope stage, some samples may drift off from the field of view. Therefore, images need to be screened for artifacts. It is necessary to check the position of the spherulite before and after the scan.
6. To perform full polarization analysis from the specific spot on the sample, turn off the excitation beam.
7. Subsequently, pick specific coordinates of piezostage corresponding to the chosen location on the spherulite where the information about the orientation of molecules is required with the sub-µm resolution.
8. Adjust the piezoelectric stage to center the field of view at indicated (X, Y).
9. Turn on the excitation beam and perform full (360°) polarization analysis of and emission by turning on the rotation of the half-wave plate (180° rotation). Present 2PF Ix and Iy components in the form of a polar graph (as shown in Figure 5B).
   ​NOTE: This step can be repeated in different locations in the sample to collect a sufficient amount of data.

4. Determination of local fibril ordering inside the bovine insulin spherulites

NOTE: All numerical calculations connected to the data analysis were done using the Python programming language and based on the functions available in libraries NumPy and SciPy. Plotting the data requires the Matplotlib library. All calculations are based on formulas presented in supporting information in one of the papers by Obstarczyk et al.⁶.

1. Simulating the intensity of two-photon fluorescence excited for the specified distribution of molecules with a selected direction of the incident light polarization (denoted by α)
   NOTE: The presented formulas are based on the assumption of parallel absorption and emission dipole moments of a fluorophore. For a discussion of other cases (e.g., different directions of the absorption and emission dipole moments, energy transfer between the fluorophores), see the paper by Le Floc'h et al⁸.
   1. Find the parameters accounting for the variation of the electric field by polarization mixing and depolarization effects caused by the dichroic mirror mounted in a setup:
      1. γ represents the amplitude factor between the reflectivity of and polarized light from a dichroic mirror.
      2. δ represents the phase shift (ellipticity) between and polarized light.
         NOTE: Correctly defining these two parameters is crucial for successful structural characterization because of their strong influence on the shape of measured polar graphs^11,18. In the case of the presented system, both parameters were determined during ellipsometry measurements of a dichroic mirror applied in a system.
   2. Calculate the incident electric field vectors propagating in the X and Y axesof every angle measured during the ps-TPFM measurement using equations (1-5).
      (1)
      (2)
      (3)
      (4)
      (5)
      NOTE If the intensity is measured over the whole 360° with step 1°, calculate electric field vectors for all 360°. All angles should be in radians. ρ parameter is connected to optical frequency ω of the simulated electric field (ρ=ω·t) and used integrated from 0 to 2π during the calculations of the incident electric field vectors propagating in X and Y axis for every angle α measured during the ps-TPFM measurement.
   3. Define the functions for transition dipole moments in directions of three axes of a cartesian system according to equations (6-8):
      (6)
      (7)
      (8)
      NOTE: φ is the orientation angle of the amyloid fibril long axis in the XY sample frame. θ and ϕ angles are the polar and azimuthal angles used to define the transition dipole moment orientation of a fluorophore. 
   4. Use the functions defined in step 4.1.3. to define functions for J_x (Φ,θ,ϕ) and J_Y (Φ,θ,ϕ), which account for the contribution of tight light focusing with high numerical aperture objective to the fluorescence polarization detection in X and Y direction using equations (9, 10). 
      (9)
      (10)
      NOTE: K₁, K₂, K₃ factors are related to the microscope objective used during the ps-TPFM measurement. In these experiments, an apochromatic oil immersion objective 100x/1.4 NA was used and K₁, K₂, K₃ factors were 2.945, 0.069, and 1.016, respectively.
   5. Define a function for f(Ω), which is a molecular angular distribution, depending on the half aperture of a fluorophore cone Ψ, with a variable thickness ΔΨ; use equation (11). The graphical representation of all three angles concerning amyloid fibril is presented in Figure 6.
        (11)
      NOTE: Be careful while writing the exponent-the power of 2 works on the function argument, not the whole function!
   6. Define all f_WWIJKL factors as shown in equations (12-21).
      (12)
      (13)
      (14)
      (15)
      (16)
      (17)
      (18)
      (19)
      (20)
      (21)
   7. Calculate two-photon excited fluorescence intensities and for every measured angle (similarly as in point 4.1.2) using equations (22, 23).
      = (22)
      = (23)
   8. Check if the simulation is working correctly using the following variable values (placed in the legend to Figure 7) and compare the obtained results with Figure 7A-C.
      NOTE: All degrees should be written in radians.
2. Fit the simulated intensities into the intensity of two-photon-excited autofluorescence of bovine insulin spherulites collected during ps-2PFM measurements.
   NOTE: Resolving the spherulite structure based on ps-2PFM measurements requires using the equations written in the protocol to simulate the theoretical intensity of two-photon excited fluorescence and fitting all the parameters connected to the ordering of molecular fluorophore. It requires multiple iterations over the various values of Φ, an angle describing the rotation of the fibril in the XY microscope sample, and ΔΨ, aberrations of the conical distribution of the emission dipole Ψ due to the molecular rotations in filaments for fixed Ψ until reaching the highest possible R² coefficient between normalized two-photon excited fluorescence signal intensity collected during measurement and normalized intensities simulated according to step 4.1 of the protocol.
   1. Determine the Ψ half-angle.
      NOTE: For bovine insulin spherulites, it should be equal to Ψ = 29°⁶.
   2. Fitting workflow:
      1. Choose the value of Φ from 0° to 180° (in Figure 8A and 8B Φ = 16° and 127°, respectively).
      2. Choose the value of ΔΨ from 0° to 90° (in Figure 8A,B, ΔΨ = 24° and 1°, respectively).
      3. Calculate (equation 22) and (equation 23) for the whole measured range of θ angles using the chosen values of Φ and ΔΨ.
      4. Compare the intensities calculated during simulations (both normalized to the maximum value of
      5. Calculate ) with normalized intensities of
      6. Calculate and (both normalized to the maximum value of ) measured during ps-2PFM measurement.
         NOTE: In the presented experiments, Pearson product-moment correlation coefficients were calculated using the “corrcoef” function from the NumPy Python library.
   3. In the end, choose the values of Φ and ΔΨ leading to the highest correlation coefficients and convergence between the simulated intensities and measured signal similar to what is shown in Figure 8.