Numerical calibration of the spatial overlap for subtraction microscopy

Nan Wang; Takayoshi Kobayashi

doi:10.1364/OE.23.013410

1. Introduction

Superresolution microscopic techniques, such as stimulated emission depletion (STED) microscopy [1], structured illumination microscopy (SIM) [2], photoactive localization microscopy (PALM) [3], stochastic optical reconstruction microscopy (STORM) [4], and others [5–10], have achieved great success in breaking diffraction barrier of optical microscopy with lateral resolution beyond 100 nm or even 20 nm [11]. Meanwhile, subtraction microscopy [12–15] developed recently attracts interests for providing a simpler way to obtain superresolution by mathematically processing a group of fluorescence images. The concept is to subtract the acquired two fluorescence images of the sample that are excited with single wavelength but two different spot shapes. The produced image gets an enhanced resolution compared with the initial confocal images. Because of the problems of negative signals and over subtraction in such microscopes, several progressive ideas have also been proposed. To reduce the nonlinearity brought by negative sidebands, excitation lights with radial and azimuthal polarizations and with extended solid and hollow spot shapes have been put forward [16, 17]. Further to achieve appropriate subtraction, concept of subtraction threshold has also been provided [18]. However, inevitable displacements of the two excitation light spots could exist in the experiment. It may result from imprecise experimental adjustments or inaccurate localization of the excitation spot centers during data processing. In STED or pump-probe microscopy, the effects of this imperfect overlap are small or even negligible due to the quenching characteristic in STED or insensitivity of the product profile of pump-probe spots. In contrast in subtraction microscopy, the resolution enhanced images are derived from the direct subtraction of two fluorescent images. The displacements may induce the reduction of resolution and intensity of the signal, or even distortions in the subtracted images. In these cases, the effects of the displacements on the subtracted results have to be found out. However, there is neither such a report on explaining these effects nor a model to realize analysis on the quality of subtracted images. A fast, precise approach for inspecting the overlap of spot centers or digitally calibrating the acquired fluorescence images are also desired for reliable subtraction microscopes. In this paper, a common model to evaluate the quality of subtracted images is established based on vector beam diffraction theory (VBDT). Parameters of peak intensity, negative sidebands or peak position shift are numerically described. With this model, the effects of imperfect overlaps in lateral and axial directions on the resolution, signal intensity and optimum subtraction threshold are performed. It quantitatively clarifies the importance of the overlap to improve the performance of subtraction microscopes. With the calculations, lateral displacement of 0.1λ for the excitation light spots is found to have about 3% deterioration of image resolution and 10% loss of signal peak intensity in the case of sample size of 0.1λ, focused numerical aperture of 1.4 and subtraction factor of unity. The peak position shift versus subtraction factors can be used for conveniently and simply calibrating the overlap. The two spots aligned with an axial displacement of 0.3λ can suppress the negative sidebands by 20% compared with complete overlap while the corresponding resolution and signal intensity loss are 10% and 15%, respectively. It provides a strategy to reduce the nonlinearities brought by negative sidebands without introducing special optics. Besides, the simulated results with different sample sizes, numerical aperture of the objective lens and polarization combination for excitations are also investigated and discussed for calibration of the spatial overlaps.

2. Principle and numerical models

The schematics of the spatial displacements in subtraction microscope and their effects on the subtracted results are shown in Fig. 1. When the two excitation spots have lateral displacement, illustrated in Fig. 1(a), the subtracted images could generate inconspicuous or notable deformations as the images in Figs. 1(b) and 1(c). Figure 1(d) shows the horizontal cross sectioning profiles of the images, in which asymmetrical distortion can be observed. Figures. 1(e)–1(h) show the results with displaced excitation spots in the axial plane. Although shapes of the subtracted spots are not deformed, the resolution degradation in subtracted images can be clearly seen.

Fig. 1 Schematics of spatial displacements in subtraction microscope. (a)–(d) Excitation spots with lateral misalignment of δx, subtracted images with factor of 0.3 while misaligned values are δx₁=0.05λ and δx₂=0.2λ, and the cross sectioning profiles. (e)–(f) Excitation spots with axial misalignment of δz, subtracted images with factor of 0.3 while misaligned values are δz₁=0.05λ and δz₂=0.2λ, and their cross sectioning profiles.

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To quantitatively analyze the effects of the spatial displacements on the quality of the images and to investigate the appropriate parameters for inspecting the overlap in the subtraction microscope, a numerical calculation is performed based on the following conditions. The location of point spread function (PFS) of the excitation beam is parallelly shifted in space rather than tilted light incidence for simulating the displacements and simplifying the calculation. The sample is assumed to be thin and have a fluorophore concentration distribution of Gaussian shape. The confocal detection and the fluorescence Stokes shift are not taken into account. It is to avoid the complexity introduced by the inhomogeneous fluorescence collection efficiency and various detection PSFs. The optical collection efficiencies and the photoelectric conversion efficiencies for the two excitation light spots are considered to be the same. Then, the theoretical model can be written with a group of mathematical formulas. The focused excitation point spread function can be obtained from VBDT [19–23]:

\begin{array}{l} E_{x, y, z} (x, y, z) & = & \frac{i C}{λ} \int_{0}^{2 π} \int_{0}^{arcsin (NA / n)} A_{AMP} \cdot A_{Phase} \cdot A_{L} (\begin{array}{l} P_{x} \\ P_{y} \\ P_{z} \end{array}) \\ sin θ \cdot e^{- i k {z cos θ + \sqrt{x^{2} + y^{2}} sin θ cos [φ - arctan (y / x)]}} d θ d φ \end{array}

I_{G, D} (x, y, 0) = {| E_{x} |}^{2} + {| E_{y} |}^{2} + {| E_{z} |}^{2}

Here C is the normalization constant; λ is the wavelength of incident light; NA is the numerical aperture of the objective lens; n is the refractive index of the contact medium between the objective and the cover glass for the sample, the value of which is 1.518 for the immersion oil; k is the wave number; A_AMP is the amplitude factor; A_PHASE is the phase factor. The set values of A_AMP, A_PHASE and the polarization unit vector matrix for I_G and I_D are shown in Table 1. The I_G and I_D with radial and azimuthal polarizations are only used in the simulations in Section 3.5.

Table 1. Set values of I_G and I_D.

View Table

A_L is the vector weight matrix of the aplanatic objective lens, calculated as follows:

A_{L} = \sqrt{cos θ} (\begin{matrix} 1 + (cos θ - 1) {cos}^{2} φ & (cos θ - 1) cos φ sin φ & - sin θ cos φ \\ (cos θ - 1) cos φ sin φ & 1 + (cos θ - 1) {sin}^{2} φ & - sin θ sin φ \\ sin θ cos φ & sin θ sin φ & cos θ \end{matrix})

The sample has a preset full width at half maximum (FWHM) value of r, express as:

O b (x, y, r) = C_{0} e^{- (4 ln 2) \frac{x^{2} + y^{2}}{r^{2}}}

where C₀ is the intensity normalization constant.

The subtracted images are calculated as

\begin{array}{l} \begin{array}{l} I_{RAW} (x, y, z, δ x, δ z, r, γ) & = & I_{G} (x + δ x / 2, y, z + δ z / 2) * Ob (x, y, r) \\ - γ \cdot I_{D} (x - δ x / 2, y, z - δ z / 2) * Ob (x, y, r) \end{array} \\ I (x, y, z, δ x, δ z, r, γ) = {\begin{array}{l} I_{RAW} (x, y, z, δ x, δ z, r, γ) & I_{RAW} (x, y, z, δ x, δ z, r, γ) \geq 0 \\ 0 & I_{RAW} (x, y, z, δ x, δ z, r, γ) < 0 \end{array} \end{array}

where δx and δz are the displacements in X component (or lateral) and Z component (or axial) in the Cartesian coordinates, γ is the subtraction factor, * is the convolution. After the convolution and subtraction, the generated negative sidebands are removed and the final image I(x, y, z, δx, δz, r, γ) is obtained.

Additionally, we give definitions of other parameters to quantitatively model the quality of subtracted images. The evolution of peak intensities, or Ipeak, against subtraction factors are used to evaluate the loss of signal intensities. The negative sidebands, which may bring nonlinearity into the subtraction microscope, are also defined by the following.

I_{peak} = \frac{MAX [I (x, y, z, δ x, δ z, r, γ)]}{MAX [I (x, y, z, δ x, δ z, r, 0)]}

| I_{Neg} | = {\begin{array}{l} 0 & MIN [I_{RAW} (x, y, z, δ x, δ z, r, γ)] \geq 0 \\ \frac{| MIN [I_{RAW} (x, y, z, δ x, δ z, r, γ)] |}{MAX [I (x, y, z, δ x, δ z, r, γ)]} & MIN [I_{RAW} (x, y, z, δ x, δ z, r, γ)] < 0 \end{array}

The subtraction threshold line, or r₀ − γ₀ line, describing an appropriate subtraction, is defined as the resolved FWHM value equals to the preset FWHM value. The threshold peak intensity factor or I_P factor, defined as the normalized peak intensity subtracted with the factor at the threshold, describes the threshold signal intensity with proper subtraction. They are expressed as:

{FWHM}_{Ob (x, y, r_{0})} = {FWHM}_{I (x, y, z, δ x, δ z, r_{0}, γ_{0})}

I_{P} = \frac{MAX [I (x, y, z, δ x, δ z, r_{0}, γ_{0})]}{MAX [I (x, y, z, δ x, δ z, r_{0}, 0)]}

The peak position shifted values in lateral plane are defined as the relative shifted values of peak positions versus subtraction factors according to the initial position without subtraction.

(Δ x, Δ y) = {(X, Y)}_{MAX [I (x, y, z, δ x, δ y, r, γ)]} - {(X_{0}, Y_{0})}_{MAX [I (x, y, z, δ x, δ y, r, 0)]}

With these defined parameters, the FWHM resolution, signal intensity, maximum values of negative signals after subtraction and the subtraction thresholds can be investigated and evaluated with different spatial displacements as well as the sample size, NA of objective lens and the polarization of incident beams.

3. Calculation results

3.1. Lateral displacement

Firstly, effects of lateral position shifts with the amount of δx=0, 0.05λ, 0.1λ, 0.2λ are analyzed. As are shown in Figs. 2(a) and 2(b), when the cross sectioning lines of excitation spots are shifted from each other with δx, the subtracted lines with factors of γ=0.3, 1 exhibit resolution degradation, signal intensity decrease, and peak position shifts. Calculating with the defined formulas, the evolutions of the FWHM values, normalized peak intensities, and maximum absolute values of the negative signal are plotted against the subtraction factors shown in Fig. 2(c)–2(e) for different amounts of displacement. It can be found that, if the displacement value is smaller than 0.1λ, the FWHM is about 3% larger than the case of perfect overlap; the relative peak intensity decreases quickly with increasing misalignment higher than 0.1λ; the absolute values of the negative signal after subtraction increase with subtraction factors and with δx. The results indicate that although slight lateral displacement within 0.1λ does not degrade the FWHM values more than 3%, the loss of signal intensity by 10% and the increased negative sideband by 20% may deteriorate the image quality significantly. In this case, it is necessary to inspect whether or not there is offset between the two fluorescence images or excitation spots.

Fig. 2 (a) Cross sectioning lines of lateral shifted excitation spots. (b) Subtracted results with γ=0, 0.3, 1. (c)–(f) Evolution of subtracted FWHM values, normalized peak intensities, maximum values of negative sidebands and the subtracted peak position shifts versus the subtraction factors with excitation displacements of δx=0, 0.05λ, 0.1λ, 0.2λ. (g)–(h) Subtraction threshold lines and the threshold peak intensity factors with excitation displacements of δx=0, 0.05λ, 0.1λ, 0.2λ. (i) schematic of key parameters in the subtracted microscopes. The preset sample size is r=0.1λ.

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Here the peak position shift is found to be effective parameters. As is shown in Fig. 2(f), when the misalignment exists, the subtracted peak positions are shifted with the increasing subtraction factors. The larger the slope of this peak position shift against subtraction factors, the larger is the misalignment. It is also more convenient and rapid to realize the algorithm of subtracting the two fluorescence images and locating the position of the peak rather than locating the centers of excitation spots [12], where fitting the donut spot is much difficult and requires accuracy. In addition, the fluorescence peak intensity or the value of negative sidebands can be a secondary parameter used for calibrating the misalignments even though it is less convenient to apply. Shifting the confocal images, subtracting them with the same factor and then checking if the peak intensity in the subtracted image is increased or if the value of negative sidebands is decreased is essential. The signal intensity is maximal or the value of negative sidebands is minimal when the overlap is optimized.

There is a third strategy, i.e., combining the peak intensity curves with the subtraction threshold lines to inspect the overlap of the excitation spots. The concept can be simply introduced as follows. Get the subtraction factor value when the peak intensity after subtraction drops to 0.6, corresponding to the proper subtraction without displacement [18]. Find the sample size (or FWHM value) from the subtraction threshold line with this factor. Compare the FWHM value obtained in the subtracted image with this predicted value. If they cannot be matched, there is substantial misalignment existing. Optimize the displacement and repeat the same procedure to check. However, this method is not recommended because it is too time consuming and because of the error which may be induced in the process of matching the FWHM values. Moreover, the ellipticity of the spot shape after subtraction can also be used to analyze the misalignments. Due to its lower effectiveness and possible incorrect ellipticity values caused by the same amounts of displacements in the two axes, it is not discussed here further.

The subtraction threshold lines and threshold peak intensity factors with separate amounts of displacement are shown in Figs. 2(g) and 2(h). They describe the obtained FWHM values in subtracted images equal to the preset FWHM values of the sample. To resolve the image of sample with small sizes of 0.05λ, subtraction factor as large as 15 is needed. It can be found that the larger displacement can have a smaller subtraction threshold factor and smaller threshold peak intensity. It indicates that displaced fluorescence images would suffer from signal degradation and are vulnerable to over subtraction for reaching appropriate subtraction. The parameters of FWHM, peak position shift, signal loss and negative sidebands are illustrated in Fig. 2(i).

3.2. Axial displacement

Similarly, the simulations with axial shifts of δz can also be realized, shown in Fig. 3. After subtracting the cross sectioning lines of excitation PSFs, the obtained peak intensities are reduced without peak-position shifts as shown in Figs. 3(a) and 3(b). Figure 3(c) shows the FWHM value becomes pretty large for position shifts larger than 0.3λ. It is interesting to see that the peak intensity after subtraction decreases more slowly and the absolute negative values are reduced by about 20% in the cases of displaced excitations, as displayed in Figs. 3(d) and 3(e). It means that small axial position shift can reduce the nonlinearity brought by negative sidebands at the cost of small amount of reduction in resolving power and signal intensity. The peak intensity seems to be apparently increased with displacements. However, because the focused spot is shifted from the center of the focal plane, the signal loss in subtracted images contains two parts. One is from the signal decrease in subtraction; the other is from the signal decrease due to the focal spot shifting from the center. Therefore, the actual signal after subtraction is the multiplication of the I_peak values in Fig. 3(d) with the focused PSF beam profile along the axial direction, shown with a blue curve in the top right figure in Fig. 3(d). The subtraction threshold lines and the threshold peak intensity factors are shown in Figs. 3(f) and 3(g), respectively. The corresponding subtraction threshold factors are larger and the peak intensity factors are smaller in the cases of axial position shifts. It indicates that subtraction results with axial displacements are relatively insensitive to over subtraction while the signal intensity reduction cannot yet be avoided at the subtraction threshold. The subtraction threshold lines and peak intensity factors for displacements with Gaussian spot only and donut spot only shift from the central focal plane are shown in Figs. 3(h) and 3(i). The only Gaussian-spot shift can readily lead to over subtraction rather than the case of only donut-spot shift. However, further calculations find that the only donut-spot shift generates larger negative signals than the case of only Gaussian-spot shift. It indicates a trade-off between tolerance of over subtraction and production of negative side bands.

Fig. 3 (a) Intensities along the axial shifted excitation spots. (b) Subtraction results of them with γ=0, 0.3, 1. (c)–(e) Evolution of subtracted FWHM values, normalized peak intensities and maximum values of negative signal versus the subtraction factors with excitation displacements of δz=0, 0.2λ, 0.3λ, 0.4λ. (f)–(g) Subtraction threshold lines and the peak intensity factors with excitation displacements. (h)–(i) Subtraction threshold lines and the peak intensity factors with only I_G or I_D shifted from central focal plane. The preset sample size is r=0.1λ.

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3.3. Sample size

Simulating with a larger sample size of r=0.3λ, the subtracted FWHM, peak intensities, and maximum values of negative signal are not sensitive to the amount of displacement as illustrated in Figs. 4(a)–4(c). With lateral displacement of δx=0.1λ, the resolution is degraded by 3% and the peak intensity is reduced by 6%. Compared with former results obtained with sample size of r=0.1λ, it demonstrates that the sample with a larger size is less sensitive to the negative effects of lateral misalignments. In contrast, the peak position shifts are still effective and sensitive to investigate the good overlap, the curves of which with different amount of lateral displacement are distinguishable and shown in Fig. 4(d). The subtracted results with axial displacements are displayed in Figs. 4(e)–4(g). The maximam negative values can be significantly reduced by almost 50% according to axial displacement of 0.3λ. It indicates that the reduction of the negative sidebands with axially shifted excitations is more effective for the sample with larger sizes.

Fig. 4 Simulation results with preset sample size of r=0.3λ. (a)–(d) FWHM values, normalized peak intensities, normalized absolute values of maximum negative signals and peak position shifts versus subtraction factors with lateral displacements of δx=0, 0.05λ, 0.1λ, 0.2λ. (e)–(g) FWHM values, normalized peak intensities and normalized absolute values of maximum negative signals versus subtraction factors with axial displacements of δz=0, 0.2λ, 0.3λ, 0.4λ. (h)–(k) Corresponding subtraction threshold lines and peak intensity factors with lateral and axial displacements.

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3.4. Numerical aperture

Results of simulation with an objective of NA=0.9 describe that the loose focus in subtraction microscopy permits a higher robustness against the displacements, shown in Fig. 5. Compared with the values with a perfect overlap, the FWHM values after subtraction vary within 6% even with lateral displacement of 0.2λ or with axial displacement of 0.5λ as depicted in Figs. 5(a) and 5(e). Figures 5(b) and 5(f) show that the subtracted peak intensities corresponding to the above cases are 10% weaker than or almost the same (effects of axial PSF excluded) with the cases of perfect overlaps, respectively. The related maximum values of the negative signal after subtraction in Figs. 5(c) and 5(g) result in 30% increase or are almost no change. The peak position shift is still sensitive to the subtraction factors with lateral shifts, as shown in Fig. 5(d). Therefore it is proved to be an effective parameter to calibrate the misalignment. In Figs. 5(h) and 5(j), the subtraction threshold lines are similar to the ones without misalignments. It means the required subtraction factors are approximately the same in both cases with and without misalignments. However, the misalignment in the lateral plane leads to the peak intensity loss at subtraction threshold, while it in the axial plane almost does not change as shown in Figs. 5(i) and 5(k).

Fig. 5 Results of simulation with objective NA=0.9. (a)–(d) FWHM, peak intensities, Maximum negative values and peak position shifts versus the subtraction factors with lateral displacements. (e)–(g) FWHM, peak intensities, maximum negative value versus the subtraction factors with axial displacements. (h)–(k) Subtraction threshold lines and peak intensity factors with lateral and axial displacements, respectively. The preset sample size is r=0.4λ.

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3.5. Polarization

Simulated results with polarization combination of radial and azimuthal polarizations and focused objective of NA=1.4 are shown in Fig. 6. Similar with the results in Figs. 2 and 3, spatial misalignment of δx=0.1λ or δz=0.3λ is acceptable for the subtracted FWHM value and the corresponding peak intensity decrease are within 10% or almost the same (effects of axial PSF excluded) compared with the results without misalignments, as shown in Figs. 6(a), 6(b), 6(e) and 6(f). However, the absolute negative values are increased by 50% or decreased by 10% compared with the results without displacements, displayed in Figs. 6(c) and 6(g). Figure 6(d) describes the peak intensity shifts versus the subtraction factors with different misalignments in lateral direction. It can be seen that it is also useful to estimate the misalignments. The subtraction threshold lines in Figs. 6(h)–6(k) show similar rules with former polarizations. However, the subtracted line shapes versus the subtraction factors are not as straight as the lines in Fig. 2(h) and Fig. 3(g). The FWHM curve with axial displacement of δz=0.4λ is also not smooth. It is due to appeared fringes of excitation PSFs with a large axial displacement. Furthermore, a large shift in axial direction leads to asymmetric spot deformation after subtraction. It results in the slightly increase of FWHM for δx=0.2λ in Fig. 6(a).

Fig. 6 Simulation results with radial and azimuthal polarizations. (a)–(d) FWHM, peak intensities, Maximum negative values and peak position shifts versus the subtraction factors with lateral displacements. (e)–(g) FWHM, peak intensities, maximum negative value versus the subtraction factors with axial displacements. (h)–(k) Subtraction threshold lines and peak intensity factors with lateral and axial displacements, respectively. Sample size is r=0.1λ.

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3.6. A simulation with 2D excitation spot shifts

Four groups of subtracted images with corresponding 2D evolution of peak position shifts and the peak intensity value of beads sample are simulated and shown in Fig. 7. The wavelengths of the light sources are both 488 nm. The objective lens is oil-immersion type with NA=1.4. The sample is a combination of beads with four sizes of 200 nm, 150 nm, 100 nm and 50 nm. The pixel sizes in the images are 2 nm. The illumination light with Gaussian spot shape is spatially position-shifted with different amounts in the lateral plane. The first groups results with position shifts δx=12 nm and δy=20 nm are shown in Figs. 7(a)–7(c). From the evolution of the peak position shifts, the directions of the displacement can be estimated while the amount of the displacement can be compared. Both of the positive shift values in X and Y directions indicate that the I_G has relative positive shifts to I_D images in both X and Y directions. Mean-while, the shifted amount in Y direction increase rapidly than it in the X direction. It indicates that the I_G images have larger shifted value in Y direction than in X direction. In contrast, the shifts cannot be directly resolved from the subtracted images with this small amount of displacements. Similarly, the trends of position shifts in the second and third groups also agree with the preset displacement values of δx=40 nm, δy=−24 nm and δx=−52 nm, δy=−56 nm as shown in Figs. 7(d)–(i). Compared with non-displaced results shown in Figs. 7(j)–(l), it can be seen that the images of beads in the third groups are distorted after subtraction. The observed shifted directions agree with the directions analyzed from the Figs. 7(g) and Figs. 7(h). From the peak position shifts of the former three groups of displacements, it can also be found that the larger sample sizes correspond to a smaller position shift. It can be a potential parameter to compare the relative sample sizes.

Fig. 7 Simulations with 2D position shifts with beads of FWHM=200, 150, 100 and 50 nm (from left to right, top to bottom in the subtracted images). (a)–(c) Evolution of peak position shifts in X, Y direction and the peak intensity values with I_G displaced with δx=12 nm and δy=20 nm. Similarly, (d)–(e) Results with δx=40 nm and δy=−24 nm. (g)–(i) Results with δx=−52 nm and δy=−56 nm. (j)–(l) Results without misalignment.

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4. Discussion

The calculation in this paper describes the numerical calibration method of spatial overlap in the subtraction microscope. When there is misalignment in the lateral plane, it reduces the resolving power, signal intensity, and also introduces larger negative values which bring in nonlinearity. By checking the evolution of peak position shifts versus subtraction factors, the misalignment can be evaluated. The overlap without misalignment gets no peak position shifts. What is more, the algorithm of checking the peak of the image is easier to realize than it of fitting the donut spot, it can be easily integrated in the data acquisition software, such as Labview. Then the acquired data can be analyzed in real time for optimizing the alignments and improving the efficiency. If the misalignments appear in the axial plane, the resolution becomes worse and the peak intensity also drops. However, it is not necessary to optimize the axial overlap to reach perfect overlap because the slight misalignment can reduce the negative sidebands while it does not seriously affect the resolution and the signal intensity.

The calculations in this paper do not take the confocal detection and Stokes-shifts of the fluorophores into consideration. In case of a small confocal pinhole for detection, the fluorescence emitted from displaced excitation spot could deviate from the confocal detection position and give rise to a low collection efficiency or even image distortion. Therefore, a more accurate result can be obtained with specific detection parameters and fluorescent labels included. In addition, the misalignment is simulated with the simple shifting of the two focal spots. In the experiment, the misalignments may be caused by unintentionally tilted illumination. That is, the incident light is not perpendicular to the focal plane. In this case, detection inhomogeneous could generate aberrations in the fluorescence images. Generally, if the amount of tilt is too large, the obtained images are highly distorted. For the experimental conditions of tiny tilt, which generates slight position shift, the excitation spots can be considered to be homogeneous and parallel position shifted. The noise effect is not included in the present simulation. Usually, the effect of the background noise and the fluctuation of the fluorescence intensity can be ignored if the signal noise ratio (SNR) of the microscope is higher than ten in the experiments. For subtraction microscopes, the raw images are taken with common confocal microscopes. It may be easy to obtain common confocal images with SNR larger than ten. If the quality of the images is really not good, eg., the fluorophore has weak emission efficiency or the detector is very noisy, in this case, fitting the curves to get the peak can also be used for calibration. Furthermore, the excitation saturation and biological samples with complex structures are not discussed here temporally. Meanwhile, the resolution of the subtraction microscope, relating with the implemented subtraction factors and sample sizes, is still the main problem of subtraction microscope. In this paper, the smallest sample we used for simulation has preset FHWM of 0.05λ. With corresponding subtraction factor of around 15, the resolved FWHM reaches 0.05λ. However, if this factor of 15 is used, the samples with sizes larger than 0.05λ are over-subtracted. Because of the rapid progress in the imaging technology, concepts to solve the problem of size-dependant and factor-dependant subtraction are hopeful to be realized in the near future.

5. Conclusion

In summary, a common model to analyze the quality of subtracted images for subtraction microscopy is established based on vector beam diffraction theory. With this model, the effects of lateral and axial displacements on the resolution, signal intensity, nonlinearity, subtraction threshold and peak intensity factor are investigated quantitatively combining with the sample sizes, numerical aperture of the objective lens and the polarization combination for excitations. The lateral misalignments of the excitation spots should be inspected and calibrated for they can reduce the image resolution, signal intensity and increase the nonlinearity. The threshold subtraction factors and peak intensity factors also decrease with the displacement, indicating easier over-subtractions and inevitable signal loss to reach the appropriate subtraction. The peak position shift after subtraction versus the subtraction factors is found to be convenient and sensitive to check the amounts and the directions of misalignments. It can also be a potential parameter for estimating the relative sample sizes. The larger sample size has a smaller amount of peak position shift. Dependances of peak intensity after subtraction and the value of negative sidebands on the subtraction factors can be secondary effective parameters to inspect the overlap. The signal intensity is maximal or the negative sideband intensity is minimal when the overlap is optimized.

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I_PSF	Polarization	A_AMP	A_PHASE	[P_x P_y P_z]
I_G	Right Circular	1	1	$[\begin{array}{l} 1 & - i & 0 \end{array}] / \sqrt{2}$
I_D	Left Circular	1	e^iφ	$[\begin{array}{l} 1 & i & 0 \end{array}] / \sqrt{2}$
I_G	Radial	1	1	[cosφ sinφ 0]
I_D	Azimuthal	1	1	[−sinφ cosφ 0]

Numerical calibration of the spatial overlap for subtraction microscopy

Abstract

1. Introduction

2. Principle and numerical models

3. Calculation results

3.1. Lateral displacement

3.2. Axial displacement

3.3. Sample size

3.4. Numerical aperture

3.5. Polarization

3.6. A simulation with 2D excitation spot shifts

4. Discussion

5. Conclusion

References and links

Cited By

Figures (7)

Tables (1)

Equations (10)

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