Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices

Török, Péter; Варга, П.; Németh, Gergely

doi:10.1364/josaa.12.002660

Cited by 88 publications

(41 citation statements)

References 16 publications

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“…Whereas the FREM assumes that the image of a point source is given by an Airy profile, the g-FREM was derived so that more complex image profiles can be analyzed. Such profiles could arise, for example, because of out-of-focus conditions (8), the presence of aberrations (10), or the use of polarized illumination (7). We note that the g-FREM can also be used to calculate the resolution measure for determining the distance of separation between any two (distinct) objects such as cellular organelles, provided the intensities and the image functions of the objects are known.…”

Section: Discussionmentioning

confidence: 99%

“…In many situations the image of the point source significantly differs from the Airy profile, for example, because of the defocus in the objective lens (8), the different orientations of the point-source emission dipole (7,9), or the aberrations present in the imaging setup (10). Moreover, depending on the nature of illumination, the intensity of the point sources can be unequal when their emission dipole orientations are different (7,9).…”

Section: Practical Resolution Measure (Prem)mentioning

confidence: 99%

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Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy

Ram

Ward

Ober

2006

Proc. Natl. Acad. Sci. U.S.A.

245

285

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Rayleigh's criterion is extensively used in optical microscopy for determining the resolution of microscopes. This criterion imposes a resolution limit that has long been held as an impediment for studying nanoscale biological phenomenon through an optical microscope. However, it is well known that Rayleigh's criterion is based on intuitive notions. For example, Rayleigh's criterion is formulated in a deterministic setting that neglects the photon statistics of the acquired data. Hence it does not take into account the number of detected photons, which, in turn, raises concern over the use of Rayleigh's criterion in photon-counting techniques such as single-molecule microscopy. Here, we re-examine the resolution problem by adopting a stochastic framework and present a resolution measure that overcomes the limitations of Rayleigh's criterion. This resolution measure predicts that the resolution of optical microscopes is not limited and that it can be improved by increasing the number of detected photons. Experimental verification of the resolution measure is carried out by imaging single-molecule pairs with different distances of separation. The resolution measure provides a quantitative tool for designing and evaluating single-molecule experiments that probe biomolecular interactions.Cramer-Rao lower bound ͉ photon statistics ͉ Fisher information matrix ͉ fluorescence microscopy A ccording to Rayleigh's criterion, the resolution of an optical microscope is defined as the minimum distance between two point sources such that their presence can be distinguished in the image (1). It is widely believed that the resolution limit imposed by this criterion precludes the single-molecule study of molecular interactions at distances of Ͻ200 nm. Despite the widespread use of Rayleigh's criterion, it is well known that this criterion is based on heuristic notions (2). Formulated within a deterministic framework, Rayleigh's criterion neglects the stochastic nature of the photon-detection process and hence does not consider the total photon count in the acquired data. This formulation is not surprising, because Rayleigh's criterion was developed at a time when the unaided human eye was used as the detector. Therefore, Rayleigh's criterion is not well adapted to current microscope setups, in which highly sensitive photon-counting cameras are used. Recent singlemolecule experiments have shown that Rayleigh's criterion can, in fact, be surpassed in an optical microscope setup (3-5). Thus Rayleigh's resolution limit is an inadequate performance criterion for current quantitative imaging techniques.By adopting a stochastic framework, we propose a resolution measure that overcomes the shortcomings of Rayleigh's criterion and provides a quantitative measure of a microscope's ability to determine the distance between two point sources. Unlike Rayleigh's criterion our resolution measure predicts that the resolution of a microscope can be improved by increasing the number of photons collected from the point sources. The resolution measu...

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Section: Discussionmentioning

confidence: 99%

Section: Practical Resolution Measure (Prem)mentioning

confidence: 99%

Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy

Ram

Ward

Ober

2006

Proc. Natl. Acad. Sci. U.S.A.

245

285

View full text Add to dashboard Cite

show abstract

“…), via Török et al (1995) have used this observation to obtain an analytic expression for A n0 where f (r) takes the form of one of the terms in Eq. (6).…”

Section: The Aberrated Point Spread Functionmentioning

confidence: 99%

Aberration correction for confocal imaging in refractive‐index‐mismatched media

Booth

Neil

Wilson

1998

Journal of Microscopy

229

164

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SummaryA major limitation to the use of confocal microscopes to image thick biological tissue lies in the dramatic reduction in both signal level and resolution when focusing deep into a refractive-index-mismatched specimen. This limitation may be overcome by measuring the wavefront aberration and pre-shaping the input beam so as to cancel the effects of aberration. We consider the images of planar and point objects in brightfield, single-photon fluorescence and twophoton fluorescence imaging. In all cases, the specimens are imaged using an oil-immersion objective through various thicknesses of water.The question of finite-sized pinhole is addressed and it is found, in general, that it is sufficient to correct only the first two or three orders of spherical aberration to restore adequate image signal level and optical resolution, at imaging depths of up to 50-100 wavelengths.

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“…The simulation calculates the field-dependent component of the aberration only. Additional aberration terms due to the refractive index mismatch between immersion medium and embedding medium are not considered here but have previously been calculated (Török et al, 1995;Booth et al, 1998). The simulation assumed diameters of the two concentric spheres of 80 µm and 52 µm as inferred from a transmitted light image of the cell and used the values for the numerical aperture (dry lens, NA = 0.5) and the 'scanned' field (100 µm × 100 µm) as determined experimentally.…”

Section: Modelling Of An Oocyte Cell and Comparison To Experimental Rmentioning

confidence: 99%

“…The spherical aberration components caused by focusing through layers of mismatched refractive index can be calculated analytically if the refractive index and thickness of each layer is known (Török et al, 1995;Booth et al, 1998). Apart from imaging of layered structures, many biological specimens can be approximated by cylindrical or spherical object shapes (e.g.…”

Section: Introductionmentioning

confidence: 99%

Simulation of specimen‐induced aberrations for objects with spherical and cylindrical symmetry

Schwertner

Booth

Wilson

2004

Journal of Microscopy

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SummaryWavefront aberrations caused by the refractive index structure of the specimen are known to compromise signal intensity and three-dimensional resolution in confocal and multiphoton microscopy. However, adaptive optics can measure and correct specimen-induced aberrations. For the design of an adaptive optics system, information on the type and amount of the aberration is required. We have previously described an interferometric set-up capable of measuring specimen-induced aberrations and a method for the extraction of the Zernike mode content. In this paper we have modelled specimen-induced aberrations caused by spherical and cylindrical objects using a ray tracing method. The Zernike mode content of the wavefronts was then extracted from the simulated wavefronts and compared with experimental results. Aberrations for a simple model of an oocyte cell consisting of two spherical regions and for a model of a well-characterized optical fibre are calculated. This simple model gave Zernike mode data that are in good agreement with experimental results.

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Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices

Cited by 88 publications

References 16 publications

Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy

Beyond Rayleigh's criterion: A resolution measure with application to single-molecule microscopy

Aberration correction for confocal imaging in refractive‐index‐mismatched media

Simulation of specimen‐induced aberrations for objects with spherical and cylindrical symmetry

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