Pattern Matching

The first step of pattern matching is the calculation of model patterns. This involves the exact wave-optical calculations of excitation patterns for molecules located in a medium of known refractive index, sandwiched between stratified layers of the substrate beneath and above, with their thicknesses, through a high numerical aperture objective. Patterns are calculated for a pre-determined in-planeβ and out-of-plane angles α. The next and final step is to locate the modeled patterns within the measured scan image using a custom written Matlab routine based on the work shown in [120]. Briefly, for each calculated pattern, an error image is calculated according to the equation:

e^(r)_mn = (X²)_mn−(Q^(r)mn)²+ (X_mn)²−2P^(r)Q^(r)mnX_mn

1−(P^(r))² (4.11)

where [m, n] is the index in the scan image and the error is calculated for each pattern p^(r). (X²)mn is the 2-dimensional convolution of the square of the raw image xmn with a background matrix with a finite support s, which is usually the size of the pattern [2L+ 1,2L+ 1] or a circular disk of fixed number of pixels or radius L. The rest of the

4.1. RADIALLY POLARIZED LASER EXCITATION CHAPTER 4. SM ORIENTATION For each pattern, a coefficient Cmn^(r) is calculated by the equation:

C_mn^(r) = (Q^(r)mn)−P^(r)X_mn 1−(P^(r))²

After calculating for all the patterns, the least error ee_mn and the corresponding coef-ficient Cemn for each pixel are obtained and the pattern t responsible for the minimum error is identified. Only if the ratio Ce_mn/√

ee_mn < κ, the pattern t is identified as a good pattern centered at the pixel. If the value κ is predefined close to one, only those intensity patterns having a good signal-to-noise ratio and agreement with the modeled pattern t are given out. Figure 4.14 shows the recognized patterns for figure 4.13.

The fitted image carrying the recognized patterns and the positions is calculated by Imn(xi, yi) = Cemn(xc, yc)×p^(t) where xc, yc is the position recognized by the molecule and x_i, y_i go from x_c−L → x_c+L and y_c−L →y_c+L respectively. Therefore, as a secondary result, one predicts the position of the molecules in the image at the pixel representing the minimum error.

Even though the pattern matching works nicely, there are several limitations. One of the major limitation which is intrinsic to the method is the limited number of the model patterns. One usually calculates the patterns with a step of 10^◦ for α and β, which automatically translates into the resolution of the technique. Fitting with patterns calculated for smaller step sizes greatly slows down the pattern matching. The second disadvantage is that the molecules have to be separated in space in a way that their intensity patterns do not overlap, which leads to false results. Also, molecules which show blinking behavior may not be recognized or may lead to false results as well, which can be seen in figure 4.15 below.

Pattern matching, apart from of course obtaining the orientations of the dipoles, comes as an effective tool for single-molecule data analysis. Since one identifies the

CHAPTER 4. SM ORIENTATION 4.1. RADIALLY POLARIZED LASER EXCITATION

Figure 4.14: Figure showing the raw intensity image of Rhodamine 6G molecules in a thin PVA polymer scanned by a radially polarized laser together with the patterns matched by the least-squares minimization algorithm. In total, 86 molecules were identified. The pixel size of 60 nm, refractive index of the PVA was set to 1.49 and N.A. = 1.49 were used to calculate the model patterns.

Figure 4.15: Left figure shows the recognized intensity patterns against the patterns matched. Every odd row shows the cropped raw image patterns and the row beneath shows the matched patterns. The right image shows the positions of the centers as cyan dots for the matched patterns.

pixels corresponding to each individual molecule, the collected photons from these pixels can be used for estimating decay rates and photophysical properties such as blinking behavior with much more statistics yielding a much higher accuracy than evaluating for each individual pixel. As stated and shown above, one has the position information of the dipoles which can be used for localization microscopy. Performing scans with a smaller pixel size can improve the lateral localization accuracy down to 30 nm. Further, combining the radial scanning with our smMIET method would allow one to localize

4.1. RADIALLY POLARIZED LASER EXCITATION CHAPTER 4. SM ORIENTATION

Figure 4.16: Image showing the fluorescence lifetimes of the identified Rhodamine 6G molecules.

The lifetimes are calculated by collecting the photons from all the pixels belonging to each individual molecule. The fitting is done using a calculated IRF using a parametric model [133]. The intensity patterns which were not recognized by the pattern match algorithm, and therefore do not have a lifetime estimate, are shown in gray scale beneath the lifetime image.

these emitters with nanometer accuracy along the z-axis, which remains a challenge for several superresolution techniques (see chapter 1). Figure 4.16 below shows the lifetime image for the 86 identified molecules from image 4.13.

Figure 4.17: The TCSPC curves and fits of three individual molecules from image 4.16. The curves in the red and blue showing lifetimes of 1.6 and 7.5 ns are the two extreme cases observed.

The fitted lifetime values vary between 1.6 ns to 7.5 ns with an average of 4.5 ns and a standard deviation of 0.9 ns. The variation in the lifetime values can be attributed to two main reasons.

The first reason is the relative position and orientation of a dye molecule with respect to the polymer/air on the top. As was described in the theory chapter, the presence of a dielectric interface, such as in between the polymer (refractive index of 1.5) and air, can lead to a dramatic change in the emission properties of a dipole such as the

CHAPTER 4. SM ORIENTATION 4.1. RADIALLY POLARIZED LASER EXCITATION

Figure 4.18: Average lifetime of a Rhodamine 6G molecule’s excitation transition dipole oriented at an angleαwith respect to the vertical in a thin polymer film (n= 1.5) of thickness∼70 nm with glass below and air above. The variation within the thin polymer layer is neglected and has been averaged for all heights.

angular distribution of the radiation and the total power radiated (see section 2.4.2).

The lifetime values, especially of a vertical dipole, change significantly close to the interface separating the embedding medium and air (see figure 2.23 for example). As the distance from this interface increases, the lifetimes of the dipoles approach to the free space value inside the medium τ0. We collected the lifetime values and the fitted orientations for about 630 Rhodamine 6G molecules from measurements such as shown in figure 4.13. Since the quantum yield of the dye is close to unity in aqueous solutions, we make a simple assumption that only the radiative rates change due to the presence of the molecule in the polymer. Using this assumption, the free space lifetimeτ₀ of the dye and the quantum yield Φ are about 3.6 ns and 0.95, respectively. The blue data in figure 4.18 shows the distribution of the average lifetimes as a function of the inclination angle of the dipoles in the polymer. Assuming a uniform height distribution of the dye molecules throughout the thickness of the polymer film, the red curve shows the theoretical lifetimes averaged over a thickness of 70 nm as a function of the orientation.

Although the data is in good agreement with the theoretical curve, which corroborates with the argument that strong variations in lifetime values in a thin polymer film are observed due to the interface effects, the correct estimates for free space parameters (τ0 and Φ) and the thickness of the polymer can be obtained only through further experiments. One must measure dye molecules inside a thick polymer away from any dielectric interface in order to estimate the free space parameters.

Secondly, the presence of any chemical heterogeneity of the polymer matrix they are embedded in can adversely affect the excited state lifetime values. As was described

4.1. RADIALLY POLARIZED LASER EXCITATION CHAPTER 4. SM ORIENTATION

in detail above, the dye molecules are highly sensitive to properties such as local vis-cosity and charges. The variations in the photophysical properties of single molecules can be attributed to the various possible interactions with the polymer matrix. Of course, changes in the structure of the backbone, the presence of oxygen, and liquid

“pockets” in the vicinity of the molecules alter their properties dramatically. Before performing smMIET experiments, it is mandatory to select an appropriate matrix for immobilization and check for the uniformity of the lifetime values.