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Abstract
We discuss the effects of image scanning microscopy using doughnut beam illumination on the properties of signal strength and integrated intensity. Doughnut beam illumination can give better optical sectioning and background rejection than Airy disk illumination. The outer pixels of a detector array give a signal from defocused regions, so digital processing of these (e.g., by simple subtraction) can further improve optical sectioning and background rejection from a single in-focus scan.
© 2023 Optica Publishing Group
1. INTRODUCTION
Image scanning microscopy (ISM) is rapidly growing in importance as a method that offers a modest degree of superresolution, coupled with a strong image signal [1–6]. The basic approach is to replace the pinhole of a confocal microscope with a detector array, situated in the image plane of the sample, followed by digital processing of the measured signals. The simplest form of digital processing is summing up the signals from the array elements after pixel reassignment in the spatial domain [1,3]. In some cases, processing by pixel reassignment is achieved optically [7–10]. The resolution of ISM may not be quite as good as in structured illumination microscopy (SIM), but the spatial frequency bandwidth is the same, and the confocal optical sectioning is retained, allowing background rejection and penetration into a scattering medium [11].
In a recent paper, we discussed the properties of integrated intensity, optical sectioning, and background in both confocal microscopy and ISM [12]. Integrated intensity, ${I_{{\rm{int}}}}$, is the transverse integral over the point spread function (PSF), equal to the flux in the paraxial approximation. It is also equal to the intensity in the image of a uniform fluorescent sheet (plane) [13], given by the integral over the intensity in the detector plane within the pinhole/detector area for confocal/ISM, respectively. For ISM, it is thus equal to that in confocal microscopy with a pinhole equal in size to the detector array, and independent of any pixel reassignment. The variation in ${I_{{\rm{int}}}}$ with defocus determines the strength of optical sectioning. Conventionally, this is specified by ${u_{1/2}}$ [13], the normalized axial distance for ${I_{{\rm{int}}}}$ to fall to one half its in-focus value, where $u$ is the axial optical coordinate, which is $u = 8\pi nz{\sin}^2 ({\alpha /2})/\lambda$ [14], where $n\sin \alpha$ is the NA (in which $n$ is the refractive index of the immersion medium and $\alpha$ is the angle of convergence of the objective lens), and $z$ is the true axial distance. The variation in the integrated intensity with a position in the detector plane and with defocus is what we call the axial fingerprint ${F_{{\rm{axial}}}}({\textbf{x}},u)$. The axial fingerprint determines the optical sectioning properties for an offset detector pixel. The background ${B_{{\rm{vol}}}}$ is the signal recorded for a uniform fluorescent volume object. For a single pixel, it is equal to the axial integral over the axial fingerprint. For the complete detector, it is equal to the integral over the intensity in the detector plane for a fluorescent volume. Again, it is independent of the pixel reassignment.
The peak intensity in the image of a fluorescent point object does, however, differ for confocal microscopy and ISM, and also depends on the pixel reassignment factor. In fact, it can be greater than the peak in a conventional image, as the collected energy is squeezed into a smaller PSF (superconcentration or superbrightness) [3,15]. In the recent paper, we also showed that the axial fingerprint demonstrates that the outer regions of the detector array tend to detect the signal from the defocused regions of the sample [12]. This is an explanation for why axial resolution in confocal microscopy decreases with the pinhole size. By processing the signals from different regions of the detector, the optical sectioning can be improved [12,16].
In an earlier paper, we showed theoretically and experimentally that ISM can produce good images when the illumination beam is a doughnut-shaped vortex (dISM) [17]. As the PSF describes the probability of the origin of a detected photon, and pixel reassignment has the effect of maximizing the peak of the PSF, this can be regarded as an example of maximum liklihood restoration. Optical sectioning was found to be better than for ISM with Airy disk illumination (AISM) for detector arrays larger than 1.1 Airy units (AU). Doughnut illumination also improves the dynamic range, as the light distribution on the detector is more spread out. Doughnut vortex beams have widespread applications in microscopy, including optical trapping [18], subtractive imaging [19,20], stimulated emission depletion (STED) microscopy [21–24], and MINFLUX microscopy [25]. Vortices can also be used to correct for radial polarization [26], and as models to focus different polarization distributions when a polarized beam is focused by a lens [22–24,27].
The aim of the present paper is to investigate further integrated intensity, optical sectioning, background, and also axial resolution, in dISM. The scalar paraxial approximation is used for simplicity and also to stress the fundamental optics of the system.
2. PSF AND OPTICAL TRANSFER FUNCTION FOR A DOUGHNUT BEAM
In this paper we considered the case when the pupil of the illuminating lens is overfilled by a scalar Laguerre–Gauss mode vortex, so that the focused illumination spot is a doughnut (vortex model 1). This geometry was analyzed in our previous papers, and has the advantage of giving analytic results for some properties [17,28]. We assumed the illuminating lens has an aberration-free circular aperture, radius ${r_0}$, with a vortex pupil function in cylindrical coordinates $r,\phi$ given by
where $\rho = r/{r_0}$ is a normalized cylindrical radius and $m$ is the charge of the vortex. Note this model is different from the vortex formed by the phase modulation of a plane wave by a vortex phase plate (vortex model 2), where the factor ${\rho^m}$ is absent [29].The in-focus intensity PSF for the illuminating lens for the vortex model 1 is [28]
where ${J_m}$ is a Bessel function of the first kind of order $m$, the dimensionless optical coordinate $v = 2\pi n\sin \alpha /\lambda$, and the NA is $n\sin \alpha$. Then, $v \approx 3.83$ corresponds to 1 Airy unit (1AU), the first zero of the Bessel function ${J_1}$. Here, the absolute magnitude has been chosen so that the total power of the PSF (the integrated intensity) is equal to $4\pi$ and independent of $m$. If $m = 0$, the doughnut beam degenerates to an Airy disk, with height unity. The doughnut beam for $m = 1$ exhibits a zero in intensity on axis: a first bright ring, height 0.259, at $v = 2.30 \equiv 0.600\rm AU$; and a first dark ring at $v = 5.14 \equiv 1.34\rm AU$.In contrast, the in-focus intensity PSF for vortex model 2 and $m = 1$ is
For $m = 0$, the in-focus 2D optical transfer function (OTF) (with ${C_{2D}}(0) = 1$) is
Fig. 1. 2D defocused OTF for (a) a circular pupil and (b) for a vortex doughnut (vortex model 1), charge 1. The normalized transverse spatial frequency is $l$, and the normalized defocus distance is $u$. The color is only to distinguish the different curves.
The 3D OTF for the Airy disk is [12,30,31]
Fig. 2. 3D plots of the 3D OTFs for (a) a circular pupil (a) and (b) a vortex doughnut (vortex model 1), charge 1. The normalized transverse and axial spatial frequencies are $l$ and $s$, respectively. The OTF for the vortex doughnut exhibits negative values. Color represents lighting from the top left.
The defocused 2D OTFs can be calculated efficiently by an axial Fourier (cosine) transform of the 3D OTF, and are shown in Fig. 1. The more well-known approach to calculate the defocused 2D OTF for the Airy disk case is the method used in [32], a single integral over the area of overlap of two displaced circles.
3. SIGNAL LEVELS AND DETECTABILITIES FOR CONFOCAL MICROSCOPY
For Airy disk illumination, the intensity in the plane of the detector for a centrally illuminated point object is ${I_{{\rm{point}}}}(v) =\def\LDeqbreak{} {H_2}(v) = [2{J_1}(v)/v{]^2}$, the intensity PSF of the collection lens. The signal ${S_{{\rm{point}}}}$ from a centrally illuminated point object with a pinhole of radius ${v_d}$ is
The normalized intensity variation in the plane of the detector, plotted against distance in AUs, for a centrally illuminated point object with Airy disk illumination, is shown in Fig. 3, and the signal level ${S_{{\rm{point}}}}$ in Fig. 4. For small ${v_d}$, ${S_{{\rm{point}}}} \approx v_d^2/4$. With doughnut beam illumination, for central illumination it is, of course, zero. The maximum in the image of a point is 0.259, but the origin of coordinates is offset, so that if the point object is placed on the ring of maximum illumination intensity, the intensity in the plane of the detector is no longer axially symmetric. The peak signal is different for a confocal microscope and ISM, and also depends on the pixel reassignment factor. However, the integrated intensity and the background from uniform planar (sheet) or volume objects do not suffer from these complications.Fig. 3. Normalized intensity variation $I$ in the plane of the detector, with offset measured in AU, for a centrally illuminated point object with Airy disk illumination, a uniform fluorescent sheet, and a featureless volume object. The variations for charge 1 vortex, doughnut illumination of a sheet or a volume are shown with dashed lines.
Fig. 4. Signal levels for a finite-sized circular detector, with radius in AUs, for a centrally illuminated point object with Airy disk illumination ${S_{{\rm{point}}}}$ and a uniform fluorescent sheet ${S_{{\rm{sheet}}}}$. The integrated intensity for a point object is also given by ${S_{{\rm{sheet}}}}$. ${S_{{\rm{sheet}}}}$ for doughnut illumination is shown with a dashed line.
In a previous paper, we gave expressions and plots for the intensity in the plane of the detector and the signal (or background) for a uniform fluorescent sheet object and a uniform fluorescent volume object with Airy disk illumination [12]. The normalized intensity in the plane of the detector for a defocused fluorescent sheet and doughnut illumination can be calculated from the defocused 2D OTFs [12,33] by
The signal (or background) from a fluorescent sheet, equal to the integrated intensity from a point object and to the value of the 2D OTF at zero spatial frequency with a pinhole of radius ${v_d}$ is [12,33,34]
where it is normalized to unity for large ${v_d}$, as shown in Fig. 4 (colored green). As the intensity in the detector plane is broader for doughnut illumination, the signal for a fluorescent sheet rises more slowly with detector size (dashed curve) than for illumination with a circular pupil (solid curve). For small ${v_d}$, ${S_{{\rm{sheet}}}} \approx 1.69 v_d^2$ for Airy disk illumination, and $\approx 0.035 v_d^2$ for doughnut illumination.The normalized intensity in the detector plane for a featureless fluorescent volume is independent of defocus and is [34]
The background from a featureless volume object, again obviously independent of defocus, is obtained by integrating over the pinhole radius ${v_d}$ so that for doughnut illumination,
Fig. 5. Variation in the background from a featureless fluorescent volume object ${B_{{\rm{vol}}}}$ with a detector radius measured in AUs for Airy disk illumination (solid curve). The variation for doughnut illumination is shown with a dashed line, indicating that the background is weaker for doughnut illumination.
Gan and Sheppard proposed three measures of SNR called detectabilities [36]. The 2D detectability ${D_{2D}}$ is the peak signal from a point object divided by the square root (to calculate the Poisson noise) of the background from a uniform fluorescent sheet. The 3D detectabilty ${D_{3D}}$ is the signal from a point object divided by the square root of the background from a featureless fluorescent volume. The axial detectabilty ${D_{{\rm{axial}}}}$ is the signal from a fluorescent sheet divided by the square root of the background from a featureless fluorescent volume.
The peak intensity for ISM with an optimum reassignment factor for doughnut illumination was shown in a previous paper [17]. Because the peak signal from a point object depends on whether the system is a confocal microscope or ISM and also on the pixel reassignment factor, ${D_{{\rm{axial}}}}$ is the most useful measure of axial imaging noise performance. Further, central illumination of a point object with doughnut illumination gives no signal. The variation in the axial detectability for doughnut illumination is shown by a dashed line in Fig. 6. For doughnut illumination, it exhibits a maximum value at 1.33AU. The three detectabilities for Airy disk illumination are also shown as solid lines: The axial detectability exhibits a maximum value at 0.944AU.
Fig. 6. Variation in the axial detectability for Airy disk illumination ${D_{{\rm{axial}}}}$ with the detector size, measured in AU. The behavior for doughnut illumination is shown with a dashed line. For Airy disk illumination, ${D_{2D}}$ and ${D_{3D}}$ are also shown.
4. OPTICAL SECTIONING FOR AN OFFSET POINT DETECTOR
Optical sectioning is quantified by the variation in integrated intensity with defocus, which is equal for confocal microscopy or ISM, regardless of the pixel reassignment factor.
The intensity in the detector plane for a defocused fluorescent sheet can be calculated from Eq. (9) using the defocused 2D OTF, efficiently found as an axial Fourier transform of the 3D OTF. Figure 7 shows the variation in intensity in the detector plane with defocus for a fluorescent sheet object and doughnut illumination. The maximum intensity is reached for a distance from the optical axis of about 0.5AU.
Fig. 7. Variation in intensity $I$ in the detector plane (with the in-focus and on-axis case normalized to unity), measured in AU, for a fluorescent sheet with doughnut illumination, for different values of normalized defocus distance $u$.
Figure 8 shows the variation in integrated intensity with defocus (the axial fingerprint) for doughnut illumination. The behavior is qualitatively similar to that for Airy disk illumination, except that the in-focus intensity increases slightly as the offset is increased from zero to about 0.5AU. For no detector offset, the defocus for the integrated intensity to drop to one half is smaller than for Airy disk illumination; $u = 4.26$ compared to $u = 5.53$. The distance from the optical axis for the intensity variation near focus to be flat is increased from 0.854AU (${v_d} \approx 3.27$) for Airy disk illumination to about 1.20AU (${v_d} \approx 4.67$) for the doughnut case. For an offset of 2AU, the maximum integrated intensity occurs at about $u = 8.34$, compared to $u = 8.7$ for Airy disk illumination.
Fig. 8. Variation in integrated intensity with normalized defocus distance $u$, with the in-focus and on-axis case normalized to unity, (the axial fingerprint) for an offset point detector and doughnut illumination. The offset is specified in Airy units. Integrated intensity is equivalent to the intensity from a uniform fluorescent sheet object.
Figure 9 shows the variation in integrated intensity with defocus for rings of detectors, for radii ${\ge} 0.8\rm AU$ and doughnut illumination. The elemental width of the ring is assumed constant, so its area increases linearly with radius. For a ring of detectors with a radius up to 0.8AU, the signal increases with the ring radius, which compares to a radius of 0.5AU for Airy disk illumination. Plots were given in [17,35], but were calculated by integration over the PSF, which is a less accurate method.
Fig. 9. Variation in integrated intensity ${I_{{\rm{int}}}}$ (arbitrary units) with normalized defocus distance $u$ for doughnut illumination, for rings of offset point detectors, with radius in Airy units, and radius $\ge 0.8\rm AU$.
The variation in integrated intensity with defocus for doughnut illumination and a disk-shaped detector was shown in [17], so it is not repeated it here. Figure 10 shows the axial distance for the integrated intensity drops to one half for doughnut or Airy disk illumination. For doughnut illumination, for small detector sizes the optical sectioning strength increases with detector radius, reaching an optimum value at a radius of 0.56AU; for a small detector, ${u_{1/2}} \approx 5.528 {-} 0.619{{\rm{AU}}^2}$. For a detector size in the domain 1.7-2AU, ${u_{1/2}}$ increases linearly, as $\approx 1.43 + 3.77\rm AU$. For detector radii greater than 1.06AU, doughnut illumination gives better optical sectioning than Airy disk illumination.
Fig. 10. Normalized axial distance ${u_{1/2}}$ for the integrated intensity to fall to half the in-focus value, as a function of detector array size in AUs, for doughnut or Airy disk illumination. For detector radii greater than 1.06AU, doughnut illumination gives better optical sectioning than Airy disk illumination.
Figure 11 shows the variation in integrated intensity with a disk-shaped detector size for doughnut illumination and defocus as the parameters. Unlike the plot in Fig. 4, the curves are not normalized. Similar curves for Airy disk illumination were presented in [35]. The integrated intensity converges to a value 1.756 as the detector size tends to infinity, but since there is then no optical sectioning, the integrated intensity is independent of the defocus, which is consistent with the fact that the background from a volume object diverges for large detector size.
Fig. 11. Variation in integrated intensity ${I_{{\rm{int}}}}$ (unnormalized) with disk-shaped detector size for doughnut illumination and normalized defocus $u$ as parameters. The integrated intensity tends to a value of 1.756, independent of the defocus, for an infinite detector.
5. DEFOCUSED PSF FOR AN OFFSET POINT DETECTOR
In ISM, each point of the detector array acts as a confocal microscope in which the detector is offset from the optical axis, for which the PSF equals the product of the illumination PSF and the offset detection PSF [37]. In our previous paper, we took the pixel reassignment factor as necessary to bring the maximum peak of the PSF for each offset point detector to the same position, and studied the performance for disk-shaped detector arrays of different sizes [17]. For a thick object, this maximum can be related to a defocused peak [12], so that the axial resolution can be degraded, as seen in [35]. Here. we chose to limit the size of the detector array for the doughnut case to 1.6AU.
Cross-sections through the 3D PSF in the plane containing the direction of offset ($x$) and the optical axis ($z$) for different offsets are shown in Fig. 12. For no offset, there are two equal peaks that result from a cross-section through a doughnut. As the offset is increased, one of these peaks becomes relatively stronger, and the strength of the defocused regions becomes weaker. For offsets between about 0.5 and 1AU, the PSF is compact in all three directions. At an offset of about 1.58AU the defocused peak becomes of equal height to the in-focus peak. In general, the PSF has less structure than for Airy disk illumination, mainly because the illumination and detection PSFs have different shapes [12].
Fig. 12. 3D plots of intensity values $I$ of the 3D PSF on cross-sections through the $x,z$ plane, where $x$ is the direction of offset for a confocal microscope with doughnut illumination and an offset point detector. The coordinate $v$ is now a normalized distance in the $x$ direction and can be negative. Color represents illumination from the top left. Top: No offset. Middle, left: 0.5AU offset. Middle, right 1AU offset. Bottom, left: 1.5AU offset. Bottom, right: 2AU offset.
For an elemental ring-shaped detector, after integration over all orientations for a particular pixel reassignment factor, the PSF is rotationally symmetrical about the optical axis. The relative contribution from the central peak is boosted relative to other peaks, so the PSF is more compact than for a single element at the same distance from the optical axis. For a disk-shaped array size of 1.6AU, the transverse FWHM of the PSF is 0.7 times narrower than for a conventional microscope with Airy disk illumination, which is slightly better than for a confocal microscope with point detector, while the peak of the PSF is about 1.3 times that in a conventional microscope [17].
6. SUBTRACTIVE IMAGING
Because detector elements further from the optical axis preferentially image defocused regions, this suggests that digital processing can be used to remove the background and improve optical sectioning [12,16]. A simple form of processing is to subtract the signal from an annulus made up of outer rings of detectors from that of a central disk [12]. As the integrated intensity decays with an inverse square law behavior, the subtraction can be arranged to give a positive result for all values of defocus, thus avoiding a major disadvantage of subtractive imaging. If the subtraction is such that the inverse square law terms cancel, then the integrated intensity decays as $1/{u^4}$. As an example, we consider subtraction of $c$ times the signal from an annulus, inner radius 1.6AU and outer radius 2AU from a disk of radius 1.6AU. As seen in Fig. 13, for a subtraction of 2.2 times the signal from the annulus, the value of ${u_{1/2}}$ is reduced to 4.50 (cf. 4.28 for true confocal with Airy disk illumination), but it also decays very quickly for large $u$. Another way of considering optical sectioning is in terms of the background from a uniform volume, ${B_{{\rm{vol}}}} = 0.71$ for the present example (compared to 2.41 for a detector disk of 1.6AU).
Fig. 13. Log–log plot of the integrated intensity after subtracting $c = 2.0,2.1,2.2$ times the signal from an annular detector array, inner radius 1.6AU and outer radius 2AU, from that from a disk detector, radius 1.6AU (black lines). The integrated intensity for disk detectors of different radii in AUs are also shown for comparison.
These effects on optical sectioning are independent of the pixel reassignment. We propose to reassign the central disk as usual, but to reassign the annulus to maximize the defocused peaks. As shown in Fig. 12, the defocused peaks that correspond to $u \approx \pm 5$ almost coincide in the transverse direction with the in-focus maximum. Therefore, instead of adding signals from the outer annulus after reassignment to those from the central disk, we subtracted them because we recognized that they primarily originated from defocused regions of the sample. We found that the resulting in-focus PSF was almost unchanged in shape. For $c = 2.2$, the width of the PSF is just 3% wider; i.e., a factor 1.41 better than conventional Airy disk fluorescence compared to 1.44 better than conventional Airy disk fluorescence for dISM with 1.6AU detector array. There are also slight negative values (${-}0.3\%$) in the PSF, but the peak of the PSF is still 10% stronger than in a conventional image, and the optical sectioning is much improved. The axial resolution after subtraction is 1.45 times better than conventional Airy disk fluorescence, compared to 1.28 times better than conventional Airy disk fluorescence for dISM with 1.6AU detector array. There are slight negative values (${-}0.8\%$) in the axial PSF. Note that in Fig. 9 in [17], the scale of the horizontal axis is labeled incorrectly: The figures shown for $u$ should be halved. This, however, does not affect Fig. 10 in [17]).
Other alternative strategies might be advantageous. Because the PSF represents the probability of the origin of a detected photon, the signals from some radii could be reassigned partly to both in-focus and defocused reconstructions. We envisage that more advanced processing strategies, such as in focus-ISM, should improve overall imaging properties [16].
7. DISCUSSION
The significance of integrated intensity, optical sectioning, and background for a microscope with doughnut illumination has been discussed. Integrated intensity is equal to the signal from a uniform fluorescent sheet (or, after normalization, from any 2D object), and is independent of pixel reassignment. Similarly, the background from a uniform volume object is independent of pixel reassignment. The background from a volume and the intensity distribution across the detector are expressed as analytic expressions. Pixel reassignment can be considered a form of maximum liklihood reconstruction, which corrects for a doughnut beam illumination, giving a PSF compact in 3D. Optical sectioning is better than for Airy disk illumination for detector arrays larger than about 1.1AU. As outer regions of the detector array, for a single in-focus scan, image defocused regions of the object, digital processing using these signals can substantially improve the optical sectioning and background rejection.
Disclosures
M.C., P.B., A.D., and G.V. have personal financial interests (as co-founders) in Genoa Instruments, Italy.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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