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The axial shift of the peak in electric energy density as the depth d is altered: solid line l ¼ 488 nm, broken line l ¼ 520 nm. Numerical aperture is 1. 4.
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The aberrations introduced when focusing within a specimen with a refractive index equal to that of water using an oil-immersion objective are investigated theoretically. The peak intensity in the confocal point spread function drops by a factor of two for focusing less than 10 microns into the specimen. The effects on scaling of dimensions in the...
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Context 1
... that the dye molecule also radiates isotropically, these plots give the point spread function for a conventional fluorescence microscope. Figure 4 shows the axial position of the peak in electric energy density as the depth d is altered. The trend is substantially linear for a depth greater than about 30 mm, although the behaviour is nonlinear when the probe is focused closer to the interface. ...
Context 2
... a three-dimensional image is recorded in a confocal microscope by axially moving the stage, the resulting image is stretched in the axial direction as a result of refractive effects. From Fig. 4 we can calculate the axial shift in the peak in electric energy density as the position of the microscope stage is altered. As the curves show some fine structure we can only give an approximate figure for the magnitude of this scaling. ...
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We propose a simple technique for determining the refractive indices of highly absorbing materials. This method uses two transmittance measurements at oblique incidence. A theoretical analysis of the proposed method is presented. The refractive index of water was experimentally determined with a relative error ranging from 0.12% to 1.3%. We discuss...
Citations
... More recently, several scaling theories have been presented that result in re-scaling factors somewhere in between the paraxial and high-angle approximations [4,15,16]. The most accurate method to determine axial scaling is full wave-optics calculations of the microscope's point spread function under RIM [17][18][19]. As these calculations are computationally expensive and complex, they are hardly used by microscopists to calculate the axial re-scaling factor. ...
... [17]. Later, wave-optics calculations showed a non-linear dependence of the focal shift f on the imaging depth for high-NA objectives and large RIMs [18,19]. For instance, Sheppard and Török reported a non-linear dependence of the focal shift f at a depth <30 µm from the coverslip using wave-optics calculations (NA = 1.3, n 1 = 1.52, n 2 = 1.33) [18], where the re-scaling factor was 5% larger close to the coverslip than at large distances. ...
... Later, wave-optics calculations showed a non-linear dependence of the focal shift f on the imaging depth for high-NA objectives and large RIMs [18,19]. For instance, Sheppard and Török reported a non-linear dependence of the focal shift f at a depth <30 µm from the coverslip using wave-optics calculations (NA = 1.3, n 1 = 1.52, n 2 = 1.33) [18], where the re-scaling factor was 5% larger close to the coverslip than at large distances. More recently, the measurements by Petrov et al. showed significant non-linear axial scaling for imaging depths <4 µm [23]. ...
In volume fluorescence microscopy, refractive index matching is essential to minimize aberrations. There are, however, common imaging scenarios where a refractive index mismatch (RIM) between immersion and a sample medium cannot be avoided. This RIM leads to an axial deformation in the acquired image data. Over the years, different axial scaling factors have been proposed to correct for this deformation. While some reports have suggested a depth-dependent axial deformation, so far none of the scaling theories has accounted for a depth-dependent, non-linear scaling. Here, we derive an analytical theory based on determining the leading constructive interference band in the objective lens pupil under RIM. We then use this to calculate a depth-dependent re-scaling factor as a function of the numerical aperture (NA), the refractive indices ${n_1}$ n 1 and ${n_2}$ n 2 , and the wavelength $\lambda$ λ . We compare our theoretical results with wave-optics calculations and experimental results obtained using a measurement scheme for different values of NA and RIM. As a benchmark, we recorded multiple datasets in different RIM conditions, and corrected these using our depth-dependent axial scaling theory. Finally, we present an online web applet that visualizes the depth-dependent axial re-scaling for specific optical setups. In addition, we provide software that will help microscopists to correctly re-scale the axial dimension in their imaging data when working under RIM.
... Later, Mansuripur express the vector diffraction in the form of Fourier transform and developed a method for fast calculation (Mansuripur, 1986;Mansuripur, 1989). Hell et al. (1993), Torok et al. (1995aTorok et al. ( ,b, 1996Torok et al. ( , 1997, Sheppard and Torok (1997) and Haeberlé (2003Haeberlé ( , 2004 and Haeberlé et al. (2003) investigated the vector diffraction of light pass through stratified media. Ref. and are based on Debye approach, Ref. (Hell et al., 1993) is based on Huygens-Fresnel principle. ...
We proposed vector diffraction equations based on 3D Huygens-Fresnel principle and vector decomposition in Cartesian coordinate. Our expression should more accurate than classical Richards-Wolf diffraction, because the deduction of Richards-Wolf diffraction used a number of approximation, while our expression is based on two fundamental physical principles: vector decomposition and Huygens-Fresnel principle. Our expressions mathematical strictly satisfy conservation of energy, while Richards-Wolf’s method is based on the geometrical law of conservation of energy and RCWA(Rigorous Coupled Wave Analysis) is also an approximation. The simulated results of our vector diffraction equations with different NA (numerical aperture) are almost the same as Richards-Wolf diffraction results. Traditional Richards-Wolf vector diffraction equations, which are based on angular spectrum of plane wave, have no clear geometrical meaning, while our vector diffraction equations here have clear geometrical meaning. Our approach of vector diffraction to scalar diffraction is much simpler than Richards-Wolf’s approach. The generation of x/y/z components and their properties in the image plane can be investigated clearly by vector decomposition and synthesis. Their intensity ratios can be explained using paraxial approximation very well. The integral region of exit pupil could be reduced because of the symmetric property of the light vector decomposition and synthesis, thus our expressions have advantages on numerical simulation. Our analysis also indicate that light must ‘polarize’ in other extra dimensions and photon spin is a ‘polarize/resonate’ state in extra dimensions. Also, there is a potential convenient ability to study vector optical transfer function (OTF) analytically in Cartesian coordinate.
... Several theoretical strategies have been developed for calculation of the refractive index mismatch. Török et al [13][14][15][16] developed a valid strategy to address the problem of focusing through a dielectric interface for optical systems with high value of Fresnel number. The method is suitable for conventional microscopy objectives. ...
... The electric field E and the magnetic field H at a point (r, ϕ, z) in focal region of spherical coordinates are defined [13][14][15][16] by: ...
... we see from (11)(12)(13) that along the rotation axis in image space: ...
Refractive index is an important optical constant that characterizes the interaction between light and specimen. A difference in refractive index between specimen and immersion medium introduces the imaging aberration and leads to a problem that the direct thickness measurement of a specimen by optical microscopy is not accurate. However, this aberration correction still requires the exact information of the refractive index of specimen and immersion medium. Herein, we propose an imaging strategy to estimate the refractive index for an unknown specimen. A simplified diffraction model is generated to obtain the relationship between axial scaling factor and refractive index. Then regular fluorescence microscopy is performed to measure the actual axial scaling factors of specimens from mouse muscle and tumor xenograft. Referring to our theoretical plot of axial scaling factor versus refractive index, the refractive index of tissue specimen is determined. For example, we obtain a mean refractive index (n) value of 1.36 for normal muscle tissues, and 1.41 for tumor xenografts. We demonstrate that this diffraction model-based estimation method is an alternative to the current techniques, improving the accurate measurement for refractive index of tissue specimen. The simple instrument requirement with an easy specimen preparation for this estimation method of refractive index may increase the image quality on tissue specimens with less aberration.
... Later, Mansuripur express the vector diffraction in the form of Fourier transform and developed a method for fast calculation [4] [5]. Hell et al. [6],Torok et al. [7][8] [9][10], Sheppard et al. [11] and Heaberle et al. [12] [13] [14] investigated the vector diffraction of light pass through stratified media. Ref. [7] and [8] are based on Debye approach, Ref. [6] is based on Huygens-Fresnel principle. ...
We proposed vector diffraction equations based on 3D Huygens-Fresnel principle and vector decomposition in Cartesian coordinate. Our expression should more accurate than classical Richards-Wolf diffraction, because the deduction of Richards-Wolf diffraction used a number of approximation, while our expression is based on two fundamental physical principles: vector decomposition and Huygens-Fresnel principle.Our expressions mathematical strictly satisfy conservation of energy, while RCWA(Rigorous Coupled Wave Analysis) is an approximation. The simulated results of our vector diffraction equations with different NA(numerical aperture) are almost the same as Richards-Wolf diffraction results. Traditional Richards-Wolf vector diffraction equations, which are based on angular spectrum of plane wave, have no clear geometrical meaning, while our vector diffraction equations here have clear geometrical meaning. Our approach of vector diffraction to scalar diffraction is much simpler than Richards-Wolf's approach. The generation of x/y/z components and their properties in the image plane can be investigated clearly by vector decomposition and synthesis. Their intensity ratios can be explained using paraxial approximation very well. The integral region of exit pupil could be reduced because of the symmetric property of the light vector decomposition and synthesis, thus our expressions have advantages on numerical simulation. Our analysis also indicate that light must 'polarize' in other extra dimensions. Also, there is a potential convenient ability to study vector optical transfer function (OTF) analytically in Cartesian coordinate.
... Even within Q1 the depth range is 25 µm. This underperformance phenomenon is also described in several manuscripts by Hell et al. (1993), Visser & Oud (1994 and Sheppard & Torok (1997), which describe refractive effects. Visser points out that the distance that the light is traveling through the oil is changing as one focuses into the sample, this changes the focal point and enlarges the apparent volume of the object. ...
This article outlines a global study conducted by the Association of Biomedical Resource Facilities (ABRF) Light Microscopy Research Group (LMRG). The results present a novel 3D tissue-like biologically relevant standard sample that is affordable and straightforward to prepare. Detailed sample preparation, instrument-specific image acquisition protocols and image analysis methods are presented and made available to the community. The standard consists of sub-resolution and large well characterized relative intensity fluorescence microspheres embedded in a 120 µm thick 3D gel with a refractive index of 1.365. The standard allows the evaluation of several properties as a function of depth. These include the following: 1) microscope resolution with automated analysis of the point-spread function (PSF), 2) automated signal-to-noise ratio analysis, 3) calibration and correction of fluorescence intensity loss, and 4) quantitative relative intensity. Results demonstrate expected refractive index mismatch dependent losses in intensity and resolution with depth, but the relative intensities of different objects at similar depths are maintained. This is a robust standard showing reproducible results across laboratories, microscope manufacturers and objective lens types (e.g., magnification, immersion medium). Thus, these tools will be valuable for the global community to benchmark fluorescence microscopes and will contribute to improved scientific rigor and reproducibility.
... Later, Mansuripur express the vector diffraction in the form of Fourier transform and developed a method for fast calculation [4] [5]. Hell et al. [6] ,Torok et al. [7] [8] [9][10], Sheppard et al. [11] and Heaberle et al. [12][13] [14] investigated the vector diffraction of light pass through stratified media. Ref. [7] and [8] are based on Debye approximation , Ref. [6] is based on Huygens-Fresnel principle. ...
We proposed a vector diffraction equations based on Huygens-Fresnel principle and vector decomposition in Cartesian coordinate. The simulated result of our vector diffraction equation with different NA(numerical aperture) are almost the same as Richards-Wolf diffraction result. Traditional Richard-Wolf vector diffraction equations, which is based on angular spectrum of plane wave, has no clear geometrical meaning. Our vector diffraction equations here is simple and with clear geometrical meaning. The x, y and z component's intensity distribution and properties in the image plane can be investigated clearly by the vector decomposition and synthesis. Their intensity level can be explained using paraxial approximation very well. Our vector diffraction equations also has advantages on numerical simulation, because of the symmetric property of the light electric vector decomposition and synthesis, the integral region of exit pupil could be reduced. It also has a convenient potential ability to study vector optical transfer function (OTF) analytically in Cartesian coordinate.
... These parameters are covered in this article investigating polymer samples with coplanar smooth phase boundaries and negligible elastic scattering power. Despite these ideal properties, the saturation level falls off, like in many examples of the literature [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] with the in-focal depth position due to the aberration of confocality as a consequence of refractive index mismatch (RIM). The article therefore aims to reduce the drop of intensity as much as possible so that even deeply located objects can be correctly analyzed in terms of chemical composition, size, and shape. ...
... The article therefore aims to reduce the drop of intensity as much as possible so that even deeply located objects can be correctly analyzed in terms of chemical composition, size, and shape. The experimental results are accompanied by theoretical calculations based on literature work on electromagnetic diffraction [20][21][22][23][24][25][26][27], Gaussian beam [26,28], and geometric optics approaches [29,30]. The level of calculation which is necessary to reliably reproduce the experiments depends on the sample thickness and the microscope objective. ...
The article analyzes experimentally and theoretically the influence of microscope parameters on the pinhole-assisted Raman depth profiles in uniform and composite refractive media. The main objective is the reliable mapping of deep sample regions. The easiest to interpret results are found with low magnification, low aperture, and small pinholes. Here, the intensities and shapes of the Raman signals are independent of the location of the emitter relative to the sample surface. Theoretically, the results can be well described with a simple analytical equation containing the axial depth resolution of the microscope and the position of the emitter. The lower determinable object size is limited to 2–4 μm. If sub-micrometer resolution is desired, high magnification, mostly combined with high aperture, becomes necessary. The signal intensities and shapes depend now in refractive media on the position relative to the sample surface. This aspect is investigated on a number of uniform and stacked polymer layers, 2–160 μm thick, with the best available transparency. The experimental depth profiles are numerically fitted with excellent accuracy by inserting a Gaussian excitation beam of variable waist and fill fraction through the focusing lens area, and by treating the Raman emission with geometric optics as spontaneous isotropic process through the lens and the variable pinhole, respectively. The intersectional area of these two solid angles yields the leading factor in understanding confocal (pinhole-assisted) Raman depth profiles.
Graphical abstract
... 25 Fluorescence intensity drops with distance from the cover slip. 26 This could make it difficult to visualize dimmer samples in the orthogonal imaging chamber. While the point spread function becomes more diffuse laterally with depth, 1 the gross shape of object remains unchanged. ...
The x–z cross‐sectional profiles of fluorescent objects can be distorted in confocal microscopy, in large part due to mismatch between the refractive index of the immersion medium of typical high numerical aperture objectives and the refractive index of the medium in which the sample is present. Here, we introduce a method to mount fluorescent samples parallel to the optical axis. This mounting allows direct imaging of what would normally be an x–z cross‐section of the object, in the x–y plane of the microscope. With this approach, the x–y cross‐sections of fluorescent beads were seen to have significantly lower shape‐distortions as compared to x–z cross‐sections reconstructed from confocal z‐stacks. We further tested the method for imaging of nuclear and cellular heights in cultured cells, and found that they are significantly flatter than previously reported. This approach allows improved imaging of the x–z cross‐section of fluorescent samples.
Lay description
Optical distortions are common in confocal microscopy. In particular, the mismatch between the refractive index of the immersion medium of the microscope objective and the refractive index of the sample medium distorts the shapes of fluorescent objects in the x–z plane of the microscope. Here, we introduced a method to eliminate the shape‐distortion in the x–z cross‐sections. This was achieved by mounting fluorescent samples on vertical glass slides such that the cross‐sections orthogonal to the glass surface could be imaged in the x–y plane of the microscope. Our method successfully improved the imaging of nuclear and cellular heights in cultured cells and revealed that the heights were significantly flatter than previously reported with conventional approaches.
... This problem is well documented in confocal microscopy, which tends to use even higher NA optics than those discussed here [11][12][13]. The amount of degradation of the point-spread function (PSF) due to aberration induced at a refractive index interface is dependent not only on the NA of the focusing optics but also on the refractive index and the depth within the aberrating medium [14,15]. Thickness and refractive index data calculated from measurements of the distortion of the PSF have been used to correct for the aberration by introducing a spatial light modulator into the optical system [16]. ...
Confocal scanning combined with low-coherence interferometry is used to provide remote refractive index and thickness measurements of transparent materials. The influence of lens aberrations in the confocal measurement is assessed through ray-trace modeling of the axial point-spread functions generated using optical configurations comprised of paired aspherics and paired achromats. Off-axis parabolic mirrors are suggested as an alternative to lenses and are shown to exhibit much more symmetric profiles provided the system numerical aperture is not too high. The modeled results compare favorably with experimental data generated using an optical instrument comprised of a broadband source and line-scan spectrometer. Refractive index and thickness measurements are made with each configuration with most mirror pairings offering better than twice the repeatability and accuracy of either lens pairing.
... In practice, this so-called focal shift effect produces a stretching of the apparent distances. Although theoretical formulae [50] and experimental protocols [51,52] of various complexities are available to determine the value of the correction factor for different depth ranges, these methods are not sufficient to provide readily usable calibration data suitable for SMLM experiments. ...
... The red circles correspond to the experimental data, and the black line stands for the curvature expected for a sphere of the measured radius. 50 ...
Cell biology relies on imaging tools to provide structural and dynamic information about samples. Among them, fluorescence microscopy offers a compromise between high specificity and low toxicity. Recently, super-resolution methods overcame the diffraction barrier to unlock new fields of investigation. Single molecule approaches prove especially useful for three-dimensional nanoscale imaging, and allow couplings between different detection modalities. Still, their use is hindered by the complexity of the methods as well as the lack of reproducibility between experiments.We propose new methods to render super-localisation microscopy more easily applicable to relevant studies in cell biology, chemistry and material science. First, we introduce dedicated protocols and samples to eliminate sources of error in calibration and performance measurement acquisitions. We also provide examples of uses of three-dimensional super-localisation for state-of-the-art studies in the frameworks of cell adhesion and bacterial resistance to drugs.Then, we focus on the development of a novel optical method that provides unbiased results in three-dimensional single molecule localisation microscopy. This is achieved through the combination of two complementary axial detection strategies: point spread function shaping on the one hand, and supercritical angle fluorescence detection on the other hand. By cross-correlating and merging the lateral and axial positions provided by the different sources, we achieve quasi-isotropic localisation precisions down to 15 nanometres over a 1-micrometre capture range. We demonstrate the insensibility of the method to imaging non-idealities such as axial drift, chromatic aberration and sample tilt, and we propose applications in neurobiology and bacteria labelling.Finally, we introduce two new post-processing approaches for the demixing of simultaneous multi-species acquisitions. They are based respectively on the measurement of the spot sizes, and on the assessment of the dynamic blinking behaviour of molecules. After demonstrating a proof of principle, we assess the impact of the different parameters likely to influence the results. Eventually, we discuss leads to improve the demixing performances, and we discuss the coupling possibilities with complementary single molecule localisation techniques.
