Sjoerd Stallinga

Measuring image resolution in optical nanoscopy

Resolution in optical nanoscopy (or super-resolution microscopy) depends on the localization uncertainty and density of single fluorescent labels and on the samples spatial structure. Currently there is no integral, practical resolution measure that accounts for all factors. We introduce a measure based on Fourier ring correlation (FRC) that can be computed directly from an image. We demonstrate its validity and benefits on two-dimensional (2D) and 3D localization microscopy images of tubulin and actin filaments. Our FRC resolution method makes it possible to compare achieved resolutions in images taken with different nanoscopy methods, to optimize and rank different emitter localization and labeling strategies, to define a stopping criterion for data acquisition, to describe image anisotropy and heterogeneity, and even to estimate the average number of localizations per emitter. Our findings challenge the current focus on obtaining the best localization precision, showing instead how the best image resolution can be achieved as fast as possible.

Fast multicolor 3D imaging using aberration-corrected multifocus microscopy.

Conventional acquisition of three-dimensional (3D) microscopy data requires sequential z scanning and is often too slow to capture biological events. We report an aberration-corrected multifocus microscopy method capable of producing an instant focal stack of nine 2D images. Appended to an epifluorescence microscope, the multifocus system enables high-resolution 3D imaging in multiple colors with single-molecule sensitivity, at speeds limited by the camera readout time of a single image.

Accuracy of the Gaussian Point Spread Function model in 2D localization microscopy

The gaussian function is simple and easy to implement as Point Spread Function (PSF) model for fitting the position of fluorescent emitters in localization microscopy. Despite its attractiveness the appropriateness of the gaussian is questionable as it is not based on the laws of optics. Here we study the effect of emission dipole orientation in conjunction with optical aberrations on the localization accuracy of position estimators based on a gaussian model PSF. Simulated image spots, calculated with all effects of high numerical aperture, interfaces between media, polarization, dipole orientation and aberrations taken into account, were fitted with a gaussian PSF based Maximum Likelihood Estimator. For freely rotating dipole emitters it is found that the gaussian works fine. The same, theoretically optimum, localization accuracy is found as if the true PSF were a gaussian, even for aberrations within the usual tolerance limit of high-end optical imaging systems such as microscopes (Marechals diffraction limit). For emitters with a fixed dipole orientation this is not the case. Localization errors are found that reach up to 40 nm for typical system parameters and aberration levels at the diffraction limit. These are systematic errors that are independent of the total photon count in the image. The gaussian function is therefore inappropriate, and more sophisticated PSF models are a practical necessity.

Information-theoretic security analysis of physical uncloneable functions

We propose a general theoretical framework to analyze the security of Physical Uncloneable Functions (PUFs). We apply the framework to optical PUFs. In particular we present a derivation, based on the physics governing multiple scattering processes, of the number of independent challenge-response pairs supported by a PUF. We find that the number of independent challenge-response pairs is proportional to the square of the thickness of the PUF and inversely proportional to the scattering length and the wavelength of the laser light. We compare our results to those of Pappu and show that they coincide in the case where the density of scatterers becomes very high.Finally, we discuss some attacks on PUFs, and introduce the Slow PUF as a way to thwart brute force attacks.

Re-scan confocal microscopy: scanning twice for better resolution

We present a new super-resolution technique, Re-scan Confocal Microscopy (RCM), based on standard confocal microscopy extended with an optical (re-scanning) unit that projects the image directly on a CCD-camera. This new microscope has improved lateral resolution and strongly improved sensitivity while maintaining the sectioning capability of a standard confocal microscope. This simple technology is typically useful for biological applications where the combination high-resolution and high-sensitivity is required.

The lateral and axial localization uncertainty in super-resolution light microscopy.

A study of the uncertainty of localizing single-molecule emitters for super-resolution light microscopy is presented. Maximum likelihood estimation (MLE) is found to be superior to least-squares fitting for low background levels, but the performance difference between the two methods decreases to a few percent for practical background levels. It is shown that the performance limit of MLE, the Cramér-Rao lower bound, is well described by a concise analytical formula with only spot width and signal and background photon count as input parameters. These predictions for the lateral localization uncertainty are compared with the localization error obtained from repeated localizations of the same single-molecule emitter. Agreement within a few percent is found, thus verifying the validity of the fitting model and the concise analytical approximation. The analysis is extended by novel analytical results for the dependence of the axial localization uncertainty on background level for the astigmatic, bifocal, and double-helix methods.

Axial birefringence in high-numerical-aperture optical systems and the light distribution close to focus

The effects of birefringence on the light distribution in the focal region of a high-NA optical system are investigated with use of the Debye approach to vector diffraction theory. The attention is limited to uniaxially birefringent media with symmetry axis along the optical axis of the imaging system. The radially (p) and tangentially (s) polarized fields in the exit pupil produce spots in the focal region that are defocused with respect to each other. For small birefringence values the relative defocus causes a distortion and broadening of the spot; for larger values the two spots separate completely. As a corollary to the theory it is shown that there is a tangential tornadolike flow of energy in the focal region when the polarization in the entrance pupil is elliptical.

Position and orientation estimation of fixed dipole emitters using an effective Hermite point spread function model

We introduce a method for determining the position and orientation of fixed dipole emitters based on a combination of polarimetry and spot shape detection. A key element is an effective Point Spread Function model based on Hermite functions. The model offers a good description of the shape variations with dipole orientation and polarization detection channel, and provides computational advantages over the exact vectorial description of dipole image formation. The realized localization uncertainty is comparable to the free dipole case in which spots are rotationally symmetric and can be well modeled with a Gaussian. This result holds for all dipole orientations, for all practical signal levels, and for defocus values within the depth of focus, implying that the massive localization bias for defocused emitters with tilted dipole axis found with Gaussian spot fitting is eliminated.

Compact description of substrate-related aberrations in high numerical-aperture optical disk readout

Optical disks are read out by focusing a beam of high numerical aperture (NA) through the substrate. Deviations of the thickness from the nominal value result in spherical aberration; tilting the substrate results in coma. Exact analytical expressions for the rms aberration per micrometer thickness mismatch (for spherical aberration) and per degree tilt (for coma) are derived. The paraxial estimates for these sensitivities proportional to NA4 (spherical aberration) and NA3 (coma) underestimate the exact values by a factor of approximately 2 for the value NA = 0.85, corresponding to the new Blu-ray disk format. Expansion of the aberration function in Zernike aberrations shows that the exact aberration functions are well described by the lowest-order Zernike spherical aberration (A40) and coma (A31) term for all but the very highest NA values.

Berreman 4×4 matrix method for reflective liquid crystal displays

The Berreman 4×4 matrix method is extended in order to calculate the optical behavior of reflective liquid crystal displays. Expressions for the reflection matrix and the reflection coefficient are derived. Possibly present uniaxial and/or biaxial retarders can be included in the calculation. Conservation of energy and time reversal invariance of Maxwell’s laws imply an important symmetry relation. This relation is used to reduce the computational time with a factor of 2. The new numerical method is illustrated by calculations of a direct-view twisted nematic effect, without and with a compensating biaxial retarder.