One of the fundamental limits of classical optical microscopy is the diffraction limit of optical resolution. It results from the finite bandwidth of the optical transfer function (or OTF) of an optical microscope, which restricts the maximum spatial frequencies that are transmitted by a microscope. However, given the frequency support of the OTF, which is fully determined by the used optical hardware, an open and unsolved question is what is the optimal amplitude and phase distribution of spatial frequencies across this support that delivers the ``sharpest'' possible image. In this paper, we will answer this question and present a general rule how to find the optimal OTF for any given imaging system. We discuss our result in the context of optical microscopy, by considering in particular the cases of wide-field microscopy, confocal Image Scanning Microscopy (ISM), 4pi microscopy, and Structured Illumination Microscopy (SIM). Our results are important for finding optimal deconvolution algorithms for microscopy images, and we demonstrate this experimentally on the example of ISM. Our results can also serve as a guideline for designing optical systems that deliver best possible images, and they can be easily generalized to non-optical imaging such as telescopic imaging, ultrasound imaging, or magnetic resonance imaging.
Confocal fluorescent imaging is the de facto standard modality for fluorescence imaging. However, the point-to-point scanning technique leads to a very limited throughput and makes the technique unsuitable for large area and fast multi-focal scanning. We propose an architecture for highly efficient 3D line confocal fluorescence imaging. Our design extends the concept of a line scanning system with continuous ‘push broom’ scanning. Instead of using a line sensor, we use an area sensor that is tilted with respect to the optical axis to acquire image data of multiple depths simultaneously. A multi-line illumination with lines illuminating the specimen at different depths, conjugate to the tilted area sensor, is created by means of a diffractive optical element (DOE). The proposed method is suitable for fast 3D image acquisition with unlimited field of view, it requires no moving components except for the sample scanning stage, has intrinsically low losses, and provides intrinsic alignment of the simultaneously scanned layers of the focal stack.
Confocal scanning microscopy is the de facto standard modality for fluorescence imaging. Point scanning, however, leads to a limited throughput and makes the technique unsuitable for fast multi-focal scanning over large areas. We propose an architecture for multi-focal fluorescence imaging that is scalable to large area imaging. The design is based on the concept of line scanning with continuous 'push broom' scanning. Instead of a line sensor, we use an area sensor that is tilted with respect to the optical axis to acquire image data from multiple depths inside the sample simultaneously. A multi-line illumination where the lines span a plane conjugate to the tilted sensor is created by means of a diffractive optics design, implemented on a spatial light modulator. In particular, we describe a design that uses higher order astigmatism to generate focal lines of substantially constant peak intensity along the lines. The proposed method is suitable for fast 3D image acquisition with unlimited field of view, it requires no moving components except for the sample scanning stage, and provides intrinsic alignment of the simultaneously scanned focal slices. As proof of concept, we have scanned 9 focal slices simultaneously over an area of 36 mm2 at 0.29 µm pixel size in object space. The projected ultimate throughput that can be realized with the proposed architecture is in excess of 100 Mpixel/s.
The Pergamenshchik theory [Phys. Rev. E 48, 1254 (1993)] and Faetti theory [Phys. Rev. E 49, 5332 (1994); 49, 4192 (1994)] of surface elasticity in nematic liquid crystals are tested. Both theories give different predictions concerning the influence of the splay-bend surface elastic constant ${\mathit{K}}_{13}$ on the director profile of nontwisted liquid-crystal cells. The influence of ${\mathit{K}}_{13}$ on the director profile of nontwisted liquid-crystal cells is studied numerically within the framework of the Pergamenshchik theory. The capacitance and optical retardation of thin cells filled with the liquid crystal ZLI 4792 (E. Merck, Darmstadt, Germany) are measured as a function of the applied voltage. The surface tilt as a function of the applied voltage is calculated from these data and compared with numerically calculated curves. The model does not give a complete account of the observed optical behavior. Best agreement with the capacitive measurements is obtained with ${\mathit{K}}_{13}$=0. The effect of ${\mathit{K}}_{13}$ is comparable in magnitude with the effect of variations of the bulk liquid-crystal parameters within their experimental inaccuracy. \textcopyright{} 1996 The American Physical Society.
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.
We compare two recently developed multiple-frame deconvolution approaches for the reconstruction of structured illumination microscopy (SIM) data: the pattern-illuminated Fourier ptychography algorithm (piFP) and the joint Richardson-Lucy deconvolution (jRL). The quality of the images reconstructed by these methods is compared in terms of the achieved resolution improvement, noise enhancement, and inherent artifacts. Furthermore, we study the issue of object-dependent resolution improvement by considering the modulation transfer functions derived from different types of objects. The performance of the considered methods is tested in experiments and benchmarked with a commercial SIM microscope. We find that the piFP method resolves periodic and isolated structures equally well, whereas the jRL method provides significantly higher resolution for isolated objects compared to periodic ones. Images reconstructed by the piFP and jRL algorithms are comparable to the images reconstructed using the generalized Wiener filter applied in most commercial SIM microscopes. An advantage of the discussed algorithms is that they allow the reconstruction of SIM images acquired under different types of illumination, such as multi-spot or random illumination.
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 (Marechal's 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.
A simple, diffraction limited, optical design for a Holographic Data Storage System with a high numerical aperture and large field is presented. A system analysis is performed and the design is compared with different current and future formats for optical data storage.
Modulation enhanced single-molecule localization microscopy (meSMLM) methods improve the localization precision by using patterned illumination to encode additional position information. Iterative meSMLM (imeSMLM) methods iteratively generate prior information on emitter positions, used to locally improve the localization precision during subsequent iterations. The Cramér-Rao lower bound cannot incorporate prior information to bound the best achievable localization precision because it requires estimators to be unbiased. By treating estimands as random variables with a known prior distribution, the Van Trees inequality (VTI) can be used to bound the best possible localization precision of imeSMLM methods. An imeSMLM method is considered, where the positions of in-plane standing-wave illumination patterns are controlled over the course of multiple iterations. Using the VTI, we analytically approximate a lower bound on the maximum localization precision of imeSMLM methods that make use of standing-wave illumination patterns. In addition, we evaluate the maximally achievable localization precision for different illumination pattern placement strategies using Monte Carlo simulations. We show that in the absence of background and under perfect modulation, the information content of signal photons increases exponentially as a function of the iteration count. However, the information increase is no longer exponential as a function of the iteration count under non-zero background, imperfect modulation, or limited mechanical resolution of the illumination positioning system. As a result, imeSMLM with two iterations reaches at most a fivefold improvement over SMLM at 8 expected background photons per pixel and 95% modulation contrast. Moreover, the information increase from imeSMLM is balanced by a reduced signal photon rate. Therefore, SMLM outperforms imeSMLM when considering an equal measurement time and illumination power per iteration. Finally, the VTI is an excellent tool for the assessment of the performance of illumination control and is therefore the method of choice for optimal design and control of imeSMLM methods.
