The spatial resolution characteristics in confocal laser scanning microscopy (LSM) and two-photon LSM utilizing a higher-order radially polarized Laguerre–Gaussian (RP-LG) beam are numerically analyzed. The size of the point spread function (PSF) and its dependence on the confocal pinhole size are compared with practical LSM using a circularly polarized Gaussian beam on the basis of vector diffraction theory. The spatial frequency response in terms of the optical transfer function (OTF) is also evaluated for LSM using the RP-LG beam. The smaller focal spot characteristics of higher-order RP-LG beams contribute to a dramatic enhancement of the lateral spatial resolution in confocal LSM and two-photon LSM.
© 2015 Optical Society of America
When a radially polarized beam is focused, it generates a longitudinal electric field at the focal point and can produce a smaller focal spot than linearly or circularly polarized beams  under tight focusing conditions using a high numerical aperture (NA) lens. An annular mask placed at the pupil plane is one of the most effective and simplest ways to enhance the longitudinal component of a radially polarized beam at the focus [2–6]. In principle, using an annular mask of infinitely thin width, the lateral intensity profile of the focal spot is represented by [J0(kNAr)]2 for the longitudinal component , where J0 is a Bessel function of the first kind of order zero, k is the wavenumber of the focused light, and r is the lateral position from the focus. This intensity profile forms the smallest focal spot, approaching a full-width at half-maximum (FWHM) size of 0.36 λ/NA , where λ is the wavelength of the focused beam.
For laser scanning microscopy (LSM), such a small focal spot is quite attractive since it can directly lead to an improvement of the spatial resolution. However, a narrow annular mask placed at the pupil plane is practically undesirable due to the considerably low transmittance of both illumination (excitation) light and signal (fluorescent) light. A concentric binary phase shifter is an alternative method to produce a smaller focal spot formed by the longitudinal component of a radially polarized beam [7–10]. Although various designs for phase shifters have been proposed, analytical optimization of their designs under tight focusing conditions still remains a matter of debate.
We have reported that a higher-order radially polarized Laguerre–Gaussian (RP-LGp,1) beam is capable of producing a smaller focal spot as the radial mode order p increases [11,12]. This smaller focal spot of RP-LG beams can be attributed to quasi-J0-Bessel beam formation caused by the characteristic amplitude and phase patterns of RP-LG beams . We have recently applied this preferable focusing property of RP-LG beams to confocal LSM  and two-photon LSM  to improve the lateral spatial resolution. In addition to these demonstrations, other research groups have employed the smaller focal spot characteristics of a radially polarized beam to improve the spatial resolution of LSM [15–18]. However, at present, detailed discussions of the spatial resolution in terms of commonly used evaluation criteria, such as the dependence of the confocal pinhole size on the size of the point spread function (PSF)  and the spatial frequency characteristics [20, 21], have not been fully elucidated for LSM using a radially polarized beam. These investigations are particularly important to reveal the ideal performance and the limitation of LSM using a radially polarized beam and to further develop LSM with a significantly enhanced spatial resolution.
In this paper, we numerically analyze the spatial resolution improvement in confocal LSM and two-photon LSM using a higher-order RP-LG beam. The size of the PSF and its dependence on the confocal pinhole size are evaluated in detail. The spatial frequency characteristics are also investigated in terms of the optical transfer function (OTF). The spatial resolution of LSM using a radially polarized beam having an infinitely narrow ring as a limiting case of higher-order RP-LG beam is also discussed. The remaining parts of the paper are organized as follows. In Section 2, we present the focusing property of higher-order RP-LGp,1 beams focused by a high NA objective lens. PSFs and OTFs in confocal LSM and two-photon LSM using RP-LGp,1 beams are evaluated in Sections 3 and 4, respectively. A summary and some concluding remarks are given in Section 5.
2. Focusing of higher-order, radially polarized Laguerre–Gaussian beams
Figure 1 illustrates a simplified model of a (confocal) LSM using a higher-order RP-LG beam. An excitation beam is focused by an objective lens. The fluorescence signal emitted from the fluorescent specimen is collected by the same objective lens and then focused on a detector (image plane) by another lens (tube lens). In confocal imaging, a confocal pinhole is placed in front of the detector. In two-photon excitation imaging, the confocal pinhole is removed. In our numerical study, for the sake of simplicity in the comparison, we employed a focusing condition of NA = 0.95 in air for the objective lens and the wavelength of the fluorescence signal was assumed to be identical to that of the excitation laser beam. These conditions allow us to compare our analysis with a conventional model of confocal LSM argued by scalar diffraction theory [19–21].
We supposed that the beam waist of a higher-order RP-LGp,1 beam is located at the pupil plane of the objective lens. The electric field of an RP-LGp,1 beams at the pupil plane is expressed as [11, 12, 22]11] and an optical cage .
The intensity distribution at the focus corresponding to the excitation PSF in confocal LSM was calculated based on vector diffraction theory [24, 25]. Figures 2(a)–2(d) summarize the beam patterns at the pupil and the calculated intensity distributions near the focus (on the xy- and xz-planes) for circularly polarized (CP) Gaussian (for comparison) and RP-LGp,1 beams with p = 1, 5, 10. For focusing of the CP-Gaussian beam shown in Fig. 2(a), the Gaussian beam radius was set to the pupil radius. On the other hand, since the focusing property of RP-LG beams strongly depends on the value of NA and the incident beam size with respect to the pupil size , the values of w0 in Eq. (1) were chosen so as to generate the strongest longitudinal component. The size parameters, defined as the ratio of the pupil radius to the value of w0, for RP-LG1,1, RP-LG5,1, and RP-LG10,1 beams used in these calculations were 1.9978, 3.6530, and 4.9334, respectively.
As shown in Figs. 2(b)–2(d), the intensity distribution at the focus for the RP-LG beams is characterized by a sharp center spot with many side lobes around the center on the xy-plane. In addition, extension of the focal spot along the z-axis is also apparent for the higher-order RP-LG beam. These behaviors become noticeable as the radial mode index p increases. It is important to note that in a tight focusing condition, such as NA = 0.95, the generation of the longitudinal component is significant even for linearly and circularly polarized beams. In particular, as is well known, the focal spot of a linearly polarized beam elongates along the polarization direction mainly due to the longitudinal component . For a CP-Gaussian beam, the focal spot exhibits a perfectly circular spot, as shown in Fig. 2(a), because the polarization-dependent elongation is averaged over the lateral directions. However, for CP-Gaussian beam focusing, this unavoidable longitudinal component results in an increase in the focal spot size (FWHM of 0.64λ). This spot size is relatively larger than the FWHM value of an Airy pattern (0.51λ/NA = 0.54λ), which is expressed as [2J1(kNAr)/kNAr]2, derived by scalar theory.
Figure 2(e) shows the FWHM values of the center focal spot estimated for the total intensity distributions and the longitudinal component of higher-order RP-LGp,1 beams with p = 1 to 25. The black dashed line in Fig. 2(e) indicates the spot size obtained from the longitudinal component of a radially polarized beam with an infinitely narrow ring (a limiting case of annular beam). As the value of p increases, the focal spot size decreases. Furthermore, the focal spot size measured for the longitudinal component of higher-order RP-LG beams seems to approach the smallest spot size of 0.36λ/NA ( = 0.38λ). This may be attributed to the quasi-J0-Bessel beam formation of the longitudinal component . The FWHM values of the center focal spot size evaluated for the total intensity distributions of RP-LG beams with p = 1, 5, 10 are 0.50λ, 0.46λ, and 0.44λ, respectively. For the longitudinal component, the FWHM values are 0.43λ, 0.41λ, and 0.40λ. Thus, for example, the focal spot size of the RP-LG5,1 beam is 28% smaller than that of the CP-Gaussian beams, which can directly contribute to the lateral resolution enhancement in LSM .
3. Confocal laser scanning microscopy using an RP-LG beam
3.1 Point spread function in confocal microscopy
In order to quantitatively evaluate the spatial resolution in confocal LSM, a PSF in the confocal system using an RP-LG5,1 beam, which has been employed in our experimental demonstrations , is numerically studied. As will be discussed in section 4, although a much smaller focal spot can be formed by using higher-order RP-LGp,1 beams with p >5, it is obtained at the cost of further lowering the peak intensity and expanding the side lobes at the focus. Therefore, in the practical use of RP-LG beams, this trade-off relation between the smaller spot size and the weakened peak intensity must be considered.
A confocal PSF, which is here expressed as PSFconf, is obtained by the relation PSFconf = PSFex × PSFdet, where PSFex and PSFdet are the PSFs for the excitation and detection optics in the confocal system . PSFex is identical to the intensity distribution at the focus of an excitation beam as shown in Fig. 2(c). PSFdet is given by the formula PSFdet = PSFem ⊗ D, where PSFem represents the image of the fluorescent emission from an infinitely small fluorescent molecule, the symbol ⊗ denotes the convolution operation, and D is the sensitivity function of the detector restricted by a circular confocal pinhole, i.e., D = 1 within the confocal pinhole and D = 0 otherwise. In our calculation, for the rigorous simulation of the confocal LSM system using a high NA lens, we adopted fully vectorial theory to calculate PSFem based on the mathematical formula derived in [27–30]. We supposed that the fluorescent emission is randomly polarized with respect to the excitation light. The magnification M of the imaging system is assumed to be 100, yielding 1 Airy unit (1 AU = 1.22λM/NA) corresponding to 128λ at the detector plane of this system.
Figures 3(a)–3(d) show the lateral PSFconf using the RP-LG5,1 beam without a confocal pinhole and with pinholes of 4, 1, and 0.5 AU. The profiles along the lateral direction are plotted in Fig. 3(e). As shown in Fig. 3, the outer side lobes of the RP-LG5,1 beam are sufficiently suppressed by using the small confocal pinhole. For the confocal pinhole of 1 AU, which is commonly used in a confocal LSM setup, the outer side lobes almost disappear, as confirmed by the previous experimental result .
Figure 4 shows the lateral and axial FWHMs of the PSFconf using the CP-Gaussian and RP-LG5,1 beams estimated as a function of the confocal pinhole size. In both beams, the lateral PSF sizes decrease when the confocal pinhole is approximately less than 1 AU. Owing to the smaller focal spot characteristics of the RP-LG5,1 beam, the lateral PSF size is always smaller than that of the CP-Gaussian beam. For the confocal pinhole of 1 AU, the FWHM values for the CP-Gaussian and RP-LG5,1 beams are 0.60λ and 0.44λ. By contrast, the axial PSF size for the RP-LG5,1 beam is larger than that of the CP-Gaussian beam due to the extended depth of focus, as shown in Fig. 2. However, this extension along the z-axis can be noticeably reduced by the confocal pinhole. The FWHM values of axial PSFs for the CP-Gaussian and RP-LG5,1 beams are 1.25λ and 2.26 λ for the confocal pinhole of 1 AU, whereas FWHM values of 1.44 λ and 3.98 λ are obtained for the CP-Gauss and RP-LG5,1 beams without the confocal pinhole. As a result, the axial extension ratio, defined as the ratio of the axial FWHM of the spot for RP-LG5,1 beam to that for the CP-Gaussian beam, reduces from 2.8 (without a confocal pinhole) to 1.8 (with a confocal pinhole of 1 AU), suggesting that the effect of a confocal pinhole is remarkable for the axial PSF of the RP-LG5,1 beam.
3.2 Optical transfer functions
The spatial frequency characteristics of the imaging system can be evaluated by means of OTF, which is obtained by the Fourier transform of PSF. The cut-off frequency of the OTF for confocal LSM depends on the confocal pinhole size. For an infinitely small confocal pinhole, the cut-off frequency of the OTF along the lateral direction is derived as 4 NA/λ  and is twice that of wide-field microscopy. Figure 5 shows the lateral OTF profiles calculated for confocal LSM with confocal pinholes of 4 AU, 1 AU, 0.5 AU, and infinitely small (0 AU) for a CP-Gaussian beam, an RP-LG5,1 beam, and the longitudinal component of the focal spot for an RP beam with an infinitely narrow ring. For comparison, the OTF profiles of confocal LSM and wide-field microscopy calculated based on scalar theory are also plotted in Fig. 5. In these calculations, a circular window with a radius of 100λ was assumed in computing the Fourier transform of each PSF.
In Fig. 5, the OTF profile of the CP-Gaussian beam (blue solid curve) in the high frequency range from 2 NA/λ to 4 NA/λ is dominated by a very weak tail even for an infinitely small pinhole. Moreover, for any pinhole conditions, the OTF profiles of the CP-Gaussian beam are always smaller than that calculated based on scalar theory (dotted and dashed curve) in almost the entire frequency range. This deterioration of the OTF can be mainly attributed to the increase of the PSF size due to the generation of the longitudinal component, as mentioned in Section 2. Consequently, the spatial frequency response of the practical confocal LSM employing a CP-Gaussian beam is not as high as that predicted by scalar theory when a high NA objective lens is used.
The shape of the OTF profile for the RP-LG5,1 beam (red solid curve in Fig. 5) drastically varies according to the pinhole size. For the pinhole size of 4 AU, the OTF value of the RP-LG5,1 beam in the low and middle frequency ranges is considerably lower than that of the CP-Gaussian beam because of the presence of the side lobes as shown in Fig. 3(b). When the confocal pinhole is less than 1 AU, this depression in the OTF profile is mostly recovered due to the suppression of the side lobe by the pinhole. Furthermore, the OTF of the RP-LG5,1 beam in the higher frequency range (around 2 NA/λ) is strongly enhanced compared to that of the CP-Gaussian beam, resulting in a resolution improvement in the confocal LSM. This behavior is further pronounced when we consider only the longitudinal component of an RP beam with an infinitely narrow ring that produces a “J0-Bessel beam” at the focus, as shown by the green solid curve in Fig. 5. For a sufficiently small confocal pinhole, the OTF of the J0-Bessel beam significantly exceeds the conventional OTF characteristics and ultimately gives a limiting case of the frequency response of higher-order RP-LG beams.
3.3 Typical example in confocal laser scanning microscopy
It is worth mentioning the spatial resolution in the practical conditions commonly used in confocal LSM. For example, suppose that an excitation wavelength of 488 nm, an oil-immersion objective lens of NA = 1.4 with a refractive index of 1.52, a magnification of 100, and a confocal pinhole of 1 AU are employed in confocal LSM and that the center wavelength of fluorescence emission is 1.05 times longer than that of the excitation. Then, the lateral FWHM values of PSFconf using a CP-Gaussian beam and an RP-LG5,1 beam having a size parameter of 3.6726 are estimated to be 195 nm and 151 nm, whereas the size of PSFconf calculated based on scalar theory is 172 nm. Thus, the RP-LG5,1 beam can provide a superior spatial resolution in confocal LSM, which surpasses the spatial resolution of the confocal LSM using a CP-Gaussian beam and also that predicted by scalar theory. Besides, the lateral size of the PSFconf for the longitudinal component of an RP beam with an infinitely narrow ring is 122 nm, which is reduced by 37% from that of the CP-Gaussian beam.
4. Two-photon laser scanning microscopy using an RP-LG beam
We consider the spatial resolution and the spatial frequency characteristics of two-photon LSM using an RP-LG beam. In two-photon LSM, the PSF is proportional to the square of the intensity distribution of the excitation focal spot ; therefore, excitation of fluorescent specimens occurs only at the focus. This technique is advantageous for optical sectioning and three-dimensional imaging without a confocal pinhole. Another advantage for two-photon imaging is that the square of the intensity distribution of a focal spot corresponding to a two-photon PSF leads to enhancement of the focal spot sharpness while the weak side lobes are expected to be further reduced.
Figures 6(a) and 6(b) display two-photon PSFs using CP-Gaussian and RP-LG5,1 beams, respectively. The side lobes caused by the focusing of the RP-LG5,1 beam are effectively suppressed in the two-photon PSF, as shown in Fig. 6(b). Moreover, the focal spot along the z-axis is remarkably extended, which has recently been utilized for novel imaging with a long depth of field [14, 18]. Figure 6(c) shows the lateral intensity profiles of the PSFs shown in Figs. 6(a) and 6(b) as well as the two-photon PSF obtained by the square of a J0-Bessel beam given by [J0(kNAr)]4, as a limiting case of the longitudinal component of a RP-beam with an infinitely narrow ring. Additionally, the intensity profile of the square of an Airy pattern, which corresponds to the two-photon PSF based on scalar theory, is plotted in Fig. 6(c). The FWHM values of the lateral PSFs for the CP-Gaussian beam and RP-LG5,1 beam are estimated to be 0.46λ and 0.33 λ, whereas those for the PSFs predicted by scalar theory and the J0-Bessel beam are 0.39 λ and 0.27 λ. Also in two-photon microscopy, due to the small focal spot characteristics of the RP-LG beam, the lateral PSF size for the RP-LG5,1 beam is reduced by 28% compared to that for the CP-Gaussian beam.
The OTF profiles corresponding to the PSFs are shown in Fig. 6(d). In all cases, the theoretical cut-off frequency of OTF is 4 NA/λ. However, in the frequency range around 1 NA/λ, the OTF profiles for the RP-LG5,1 beam is relatively lower than that of the CP-Gaussian beam. This decline of the OTF profile in the low and middle frequency ranges arises from the weak but non-vanishing side lobes spreading around the center focal spot. On the other hand, in the higher frequency range from 2NA/λ to 4NA/λ, an enhancement of the OTF profile is clearly observed compared to the CP-Gaussian beam. As a result, the greater OTF values in the higher frequency range for the RP-LG5,1 beam greatly contribute to the improvement of the spatial resolution in the lateral direction for two-photon imaging. For the J0-Bessel beam as a limiting case, the OTF profile can obviously attain the theoretical cut-off frequency of 4NA/λ. These OTFs imply that the smaller focal spot produced by these RP beams can effectively extract the information for high spatial frequency within a bandwidth which cannot be acquired by conventional two-photon microscopy. It should be noted that owing to the oscillating and slowly decaying behavior of the J0-Bessel function in an infinite region, the shape of the OTF profile, normalized to the value at the origin, depends on the integral window.
Although the OTF profile for much higher-order RP-LGp,1 beams, for example, p = 10 provides slightly better characteristics at the high frequency range than that of the RP-LG5,1 beam, the influences of the extensive side lobes and the relatively weakened peak intensity of the center spot may hinder the superior frequency response in two-photon imaging. Therefore, the suitable mode order p of an RP-LGp,1 beam depends on the signal-to-noise ratio of the systems, which is determined by such factors as the sensitivity of the detector and the fluorescence efficiency of the samples.
We numerically analyzed the spatial resolution characteristics in confocal LSM and two-photon LSM utilizing a higher-order RP-LG beam. In confocal LSM, a confocal pinhole plays an essential role in the forming of the confocal PSF when an RP-LG beam is used for the excitation beam. A confocal pinhole of 1 AU, which is commonly used in practical confocal LSM setups, is sufficient to suppress the side lobes appearing at the focus of an RP-LG beam. In this condition, owing to the smaller focal spot characteristics of RP-LG beams, the lateral size of the PSF for an RP-LG5,1 beam is reduced by 28% compared to that for the CP-Gaussian beam. This superior performance in confocal LSM was also investigated by evaluating the spatial frequency response in terms of the OTF. Compared to the OTF of conventional LSM using a CP-Gaussian beam, the OTF of the RP-LG5,1 beam at the high spatial frequency range is strongly enhanced. This behavior is further pronounced when we consider a focal spot composed of the longitudinal component of an RP beam with an infinitely narrow ring as a limiting case of higher-order RP-LG beam focusing. In two-photon LSM, since the corresponding PSF is obtained from the square of the intensity distribution of the focal spot of excitation beam, the influence of the side lobes produced by a higher-order RP-LG beam can be weakened without a confocal pinhole while the center focal spot of the RP-LG beam is further sharpened. As a result, the lateral spatial resolution in two-photon LSM is also effectively enhanced when a higher-order RP-LG beam is used.
The authors thank T. Nemoto, T. Hibi, and H. Yokoyama for useful discussions. This work was supported in part by Core Research for Evolution Science and Technology (CREST), the Japan Science and Technology Agency (JST), and by JSPS KAKENHI Grant Nos. 25600106 and 25790062.
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