Imaging with a longitudinal electric field in confocal laser scanning microscopy to enhance spatial resolution

Yuichi Kozawa; Ryota Sakashita; Ryota Sakashita; Yuuki Uesugi; Shunichi Sato

doi:10.1364/OE.396778

1. Introduction

The generation of a strong longitudinal electric field is a distinctive feature emerging at the focus of a radially polarized beam in focusing conditions using a high numerical aperture (NA) lens [1]. This feature enables producing a small focal spot, compared with that of a linearly or circularly polarized beam [2]. The focal spot size can be further decreased by focusing radially polarized beams with specific intensity and phase distributions, e.g., an annularly shaped beam [3,4] and a higher-order radially polarized Laguerre–Gaussian (RP-LG) mode beam [5,6]. Such a focusing property was successfully applied to laser scanning fluorescence microscopy to achieve enhanced spatial resolution for biological samples [7–9]. In principle, this is possible because the spatial resolution of laser scanning microscopy (LSM) for fluorescent signals primarily depends on the intensity distribution of an excitation focal spot, i.e., the point spread function (PSF), provided that detected signals are essentially unpolarized, i.e., incoherent [10,11]. However, for LSM that detects scattered or reflected light and LSM that exploits nonlinear responses, including second-harmonic generation (SHG), emitted light is generally polarized, depending on the polarization of an illumination or excitation focal spot. Longitudinally polarized emissions from a sample are collected as radially polarized light by an objective lens and form a doughnut-shaped image at the detector plane. This propagation characteristic causes a significant reduction in signal intensity in the presence of a small confocal pinhole because the signals with the doughnut-shaped image are largely blocked by the pinhole, which degrades the image quality in LSM using radially polarized illumination [12].

Conversion of radial to linear polarization in a detection path in confocal LSM (CLSM) is an effective approach for improving the detection efficiency for longitudinally polarized emission under a small confocal pinhole [12–16]. Utilization of such a “polarization converter” was first proposed by Tang et al. [12] and subsequently applied to surface-plasmon-coupled emission microscopy [13], far-field mapping of the longitudinally polarized field [14], and spatial resolution enhancement in CLSM [15–17]. Most of these studies have targeted the optical response of a longitudinally oriented dipole illuminated by a focused radially polarized beam, where only the longitudinal component of an illumination focal spot contributed optical responses. However, tight focusing of a radially polarized beam produces a weak but non-negligible transverse component and a strong longitudinal component at the focus, which will result in complicated optical responses for more general objects including nanoparticles, molecules, and surfaces with fine structures. Thus, for CLSM with polarization conversion, a more detailed and comprehensive analysis is necessary to fully elucidate the optical responses and the image formation of an arbitrarily oriented dipole imaged by a tightly focused radially polarized beam.

In this paper, we examine image formation in CLSM with polarization conversion using a tightly focused beam with radial polarization. The image of a small scatterer, acquired by a radially polarized beam, is numerically simulated using vector diffraction theory under tight focusing conditions. We evaluate in detail the dependence of pinhole size on the resultant confocal images. Numerical simulations reveal that the focusing of a higher-order RP-LG beam with a multi-ring-shaped intensity distribution can produce a point image approaching 100 nm in lateral size with reduced side lobes, which offers a favorable property for confocal imaging using a longitudinal electric field. We experimentally verify the imaging capability for scattering samples observed by a radially polarized beam in a confocal microscope setup equipped with a polarization converter. The present study provides a novel method for improving the spatial resolution in various types of laser microscopy in which the signal intensity strongly relates to the polarization direction of the excitation and/or emitted light.

2. Numerical simulation

To investigate image formation in confocal detection with polarization conversion, we calculated the image of an isolated small scatterer such as a metallic nanoparticle illuminated by a focused light as illustrated in Fig. 1. For a point-like small scatterer, the scattered light can be represented as radiation from a dipole, the orientation of which coincides with the polarization of the focused light. Without polarization conversion in confocal detection, i.e., in conventional detection, an image of an isolated dipole can be calculated according to the theoretical framework presented in [18,19]. By modifying this theory, we can obtain the mathematical representation of a dipole image with polarization conversion in confocal detection, which is briefly described in the Appendix to this paper. Note that the equivalent integral representation with slightly different forms was also derived in [16], which addressed only the response of a dipole induced by the longitudinally polarized field in CLSM. Our study provides the generalized responses of a dipole induced by a focal spot with three-dimensionally polarized fields.

Fig. 1. Schematic diagram of confocal detection with polarization conversion. (a) Geometrical representation of the detected path of the optical system, where ê_x, ê_y, ê_r, and ê_ϕ represent unit vectors along the x, y, radial, and azimuthal directions, respectively; (b) conceptual diagram of the image formation with polarization conversion for longitudinally and transversely oriented dipoles.

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Figure 1(a) is a schematic of a confocal microscope setup with a polarization converter and illustrates only a detection path for the sake of simplicity. Numerical apertures for an objective lens and a tube lens are given as NA₁ (= nsinα₁) and NA₂ (= sinα₂), respectively, where n is the refractive index of a medium at the objective plane, and α₁ and α₂ are the maximum focusing angles of the objective and tube lenses, respectively. A polarization converter was placed at the back focal plane (pupil plane) of the objective lens with a focal length f_o, where the tube lens was located at its focal length f_t from the pupil plane. As implemented in a segmented half-wave plate [7,20] and a q-plate [21,22], the polarization converter changed the incident linear polarization to radial or azimuthal polarization and vice versa. Because of this polarization conversion, an emitted field from a longitudinally oriented dipole was changed into a linearly polarized field, resulting in a bright, spot-shaped image at the image plane [Fig. 1(b)]. In contrast, the transversely oriented dipoles were imaged as a doughnut-shaped pattern [Fig. 1(b)]. This is a key principle that can relatively increase the transmission of signals from longitudinally oriented dipoles, compared with transmission of signals from transversely oriented dipoles, in the presence of a small confocal pinhole at the image plane [12–16].

According to the mathematical representation (see Appendix), we were able to calculate the electric field of an image at the image plane for a dipole oriented in an arbitrary direction with polarization conversion as ${\textbf{e}_\textrm{d}}({x_\textrm{d}},{y_\textrm{d}})$, where (x_d, y_d) is a position at the image plane and the dipole is located at the focus (x_s = y_s = 0). In LSM, a sample (or, equivalently, an illumination focal spot) is raster-scanned on the focal plane. Thus, when a dipole is located at (x_s, y_s) on the object plane, the intensity distribution of the image can be given as follows:

(1)$${I_\textrm{d}}({{x_\textrm{d}},{y_\textrm{d}};{x_\textrm{s}},{y_\textrm{s}}} )= {|{{\textbf{e}_\textrm{d}}({{x_\textrm{d}} - M{x_\textrm{s}},{y_\textrm{d}} - M{y_\textrm{s}}} )} |^2},$$

where $M = {f_\textrm{t}}/{f_\textrm{o}}$ is a transverse magnification of the imaging system. In the presence of a confocal pinhole with radius R, the measured signal intensity in LSM is obtained by integrating the signal within the pinhole as follows:

(2)$${I_{\textrm{sig}}}({{x_\textrm{s}},{y_\textrm{s}}} )= \int\limits_0^{2\pi } {\int\limits_0^R {{I_\textrm{d}}({{r_\textrm{p}}\cos {\varphi_\textrm{p}},{r_\textrm{p}}\sin {\varphi_\textrm{p}};{x_\textrm{s}},{y_\textrm{s}}} ){r_\textrm{p}}d{r_\textrm{p}}d{\varphi _\textrm{p}}} } ,$$

where (r_p, φ_p) is a cylindrical coordinate defined inside the pinhole.

The image of a small scatterer illuminated by a tightly focused laser beam is then obtained by assuming that a local dipole moment p(x_s, y_s) is represented as follows:

(3)$$\textbf{p}({{x_\textrm{s}},{y_\textrm{s}}} )= \left( {\begin{array}{ccc} {{\alpha_{xx}}}&0&0\\ 0&{{\alpha_{yy}}}&0\\ 0&0&{{\alpha_{zz}}} \end{array}} \right){\textbf{e}_{\textrm{ill}}}({x_\textrm{s}},{y_\textrm{s}}),$$

where e_ill is the electric field of the illumination at the focus and α_i (i = xx, yy, zz) is a polarizability component of a scatterer. To examine an isolated small spherical scatterer such as a metallic nanoparticle, we set ${\alpha _{xx}} = {\alpha _{yy}} = {\alpha _{zz}} = 1$ for the sake of simplicity. Hence, the scattered field at (x_s, y_s) is represented by dipole radiation as $\textbf{p} = {\textbf{e}_{\textrm{ill}}}$. Equations (1)–(3) indicate that the intensity distribution of a scattered field at the image plane varied under the local polarization of an illumination focal spot, in contrast to a space-invariant PSF such as an Airy pattern, which is commonly assumed in fluorescent microscopy [23].

Using the abovementioned procedure, we were able to calculate images of an isolated small scatterer in CLSM. In our calculation, we assumed an oil-immersion objective lens (NA = 1.4 and n = 1.52) and a transverse magnification of M = 100. The electric field e_ill at the focus was calculated using vector diffraction theory [24,25]. Figures 2(a) and 2(b) show the cases of conventional detection (without polarization conversion) for a linearly polarized Gaussian (LP-Gauss) beam and an RP-LG_p_,1 beam with a radial mode index p = 0, i.e., the lowest-order doughnut-shaped RP-LG_0,1 mode. For an LP-Gauss beam polarized along the x-axis [Fig. 2(a)], we assumed that the incident Gaussian beam radius had twice the pupil radius of the objective lens. For an RP-LG_0,1 beam [Fig. 2(b)], the beam size parameter, denoted as β₀ in [5], was set to 1. As shown in the top row in Fig. 2 (corresponding to |e_ill|²), the focal spot of the LP-Gauss beam illumination shows an elliptical shape elongated along the x-axis, whereas the RP-LG_0,1 beam produces a circular focal spot. These focusing characteristics were due to the generation of the longitudinal component at the focus (plotted as red lines in the second row in Fig. 2), which appeared only in high NA focusing conditions [24,25].

Fig. 2. Numerical simulations of a small scatterer image obtained by CLSM with and without polarization conversion. Conventional detection using (a) LP-Gauss and (b) RP-LG_0,1 beams; detection with polarization conversion using (c) LP-Gauss, (d) RP-LG_0,1, and (e) RP-LG_5,1 beams. The first and second rows show the intensity distributions of the focal spot (the sum of all polarization components) and its intensity profiles along the x-axis across the center spot, respectively. The black, blue, and red solid lines of the intensity profiles in the second row are the total, transverse, and longitudinal components, respectively. The calculated images with different pinhole diameters (as indicated on the left) are shown in the third to sixth rows. The scale bar represents 1λ_m (= λ / n). The calculated region is 4λ_m × 4λ_m. The color scale is normalized to the maximum value of each condition. Finally, NA = 1.4 and n = 1.52.

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In conventional detection, the calculated image of a small scatterer (point image) obtained by the LP-Gauss beam reflected the illumination focal spot and gradually shrank in size with a decreasing pinhole size; this was measured by Airy unit (AU), where 1 AU = 1.22λM/NA, with λ denoting the illumination wavelength. The intensity distribution of the image obtained by an RP-LG_0,1 beam, however, strongly depends on the pinhole size as shown in Fig. 2(b). For a confocal pinhole smaller than ∼1 AU, the image changed into a doughnut-like shape. This behavior originated from the fact that the detection of the longitudinally polarized radiation was significantly suppressed by a small confocal pinhole in conventional detection because its signal at the image plane was radially polarized and formed a doughnut-shaped distribution, for the most part, blocked by the pinhole [12]. In such circumstances, the resultant doughnut-shaped image [shown in the sixth row in Fig. 2(b)] is predominantly composed of the signals produced by the radial (transverse) component at the focus [plotted as blue lines in the second row in Fig. 2(b)]. Thus, for conventional detection in CLSM, a single isolated scatterer is inaccurately imaged by a radially polarized beam for a small confocal pinhole.

Using polarization conversion, the point image was changed in small pinhole conditions as shown in Figs. 2(c) and 2(d). A spot-shaped image was formed for the RP-LG_0,1 beam [Fig. 2(d)], whereas a two-lobe-shaped image arose for the LP-Gauss beam [Fig. 2(c)] in small pinhole conditions. This difference observed in the small pinhole conditions indicated the relative enhancement of the detection efficiency for longitudinally polarized emission compared with transversely polarized emission, converted into radial and/or azimuthal polarization by the converter. Thus, the calculated images strongly depended on the distribution of the longitudinal electric field of a focal spot. Noticeably, however, a side lobe around the central bright spot remained, even for a pinhole of 0.4 AU in the RP-LG_0,1 beam illumination with polarization conversion [see the sixth row of Fig. 2(d)]. The residual side lobe was attributed to the transverse component of the RP-LG_0,1 beam at the focus. The ratio of the peak intensity of the transverse component to that of the longitudinal component was 0.31 [see the second row in Fig. 2(d)]. Because of the transverse component of the illumination focal spot, the side lobe appeared in the resultant image even for an infinitely small pinhole. Such image formation behavior essentially differed from PSFs anticipated in confocal imaging for (incoherent) fluorescent signals, in which side lobes were simply suppressed by a small pinhole [11].

Next, we considered the focusing property and its image formation by a higher-order RP-LG beam, characterized by a smaller focal spot property [5,6]. Figure 2(e) shows the case for an RP-LG_5,1 beam with a six-ring-shaped intensity distribution. In our calculation, we set the incident beam parameter β₀ to 3.6746 to maximize the generation of the longitudinal component at the focus. The calculated intensity distribution at the focus exhibited a sharp center spot with extended side lobes. The size of the center spot was 0.489λ_m, which had been reduced by 33% from that of the LP-Gauss focus (along the x-axis). As shown in the third to sixth rows in Fig. 2(e), the calculated scatterer images that were taken by the RP-LG_5,1 beam presented apparent side lobes around the center spot for large pinhole conditions, but the side lobes were effectively suppressed for smaller pinholes. Notably, the scatterer image obtained for a pinhole of 0.4 AU showed an almost single point image, which differed from the image obtained by the RP-LG_0,1 beam. The reason for this difference was that the transverse component around the center focal spot of the RP-LG_5,1 beam [see the second row in Fig. 2(e)] was much smaller than that of the RP-LG_0,1 beam. For the focal spot of the RP-LG_5,1 beam, the ratio of the peak intensity of the nearby transverse component to that of the center longitudinal component was 0.139, reduced by 55% compared with the case of RP-LG_0,1 beam focusing. It is notable that, under the same illumination power at the focus, the peak intensity at the focus of RP-LG_p_,1 beams was always smaller than that of an LP-Gauss beam due to the energy spreading to the side-lobes. The actual peak intensities for the RP-LG_0,1 and RP-LG_5,1 beams (top row in Fig. 2) were 59% and 14% of that of the LP-Gauss beam. Thus, for RP-LG beam illumination, in principle, the reduction in size of its center focal spot was accomplished at the cost of low peak (signal) intensity with expanding side lobes, which in turn, reduced the scattered image size.

Figure 3 plots the full width at half maximum (FWHM) size of the calculated point image as a function of the confocal pinhole diameter. As with confocal imaging for fluorescent signals [26], the lateral FWHM size of a point image decreased when the pinhole diameter D was smaller than ∼1 AU for the conventional detection using an LP-Gauss beam and for detection with polarization conversion for RP-LG beams. Importantly, the FWHM size obtained by the RP-LG_5,1 beam was always smaller than that obtained by the LP-Gauss beam in conventional detection for D < ∼0.8 AU, and approached 0.286λ_m for D = ∼0, corresponding to 92 nm for λ = 488 nm with n = 1.52. These results suggested that even for scattering signals, a spatial resolution of less than 100 nm can be expected when using an RP-LG_5,1 beam in confocal imaging with polarization conversion under small pinhole conditions. Thus, a higher-order RP-LG beam is advantageous for markedly enhancing spatial resolution with reduced side lobes for imaging scattering objects, e.g., metallic nanoparticles.

Fig. 3. Lateral FWHM size of a small scatterer image as a function of the pinhole diameter. The size was measured in units of λ_m (= λ / n). NA = 1.4 and n = 1.52.

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3. Experimental results

To verify the imaging capability of an RP-LG beam with polarization conversion, we built a confocal laser scanning microscope equipped with a liquid-crystal-based polarization converter as illustrated in Fig. 4. We used a laser diode with a wavelength of 488 nm as an illumination laser source. The output beam from a polarization-maintaining single-mode fiber was directed at a liquid crystal on silicon spatial light modulator (LCOS-SLM, X10468-01, Hamamatsu Photonics) to control the phase distribution of an incident beam. Then, the modified laser beam was projected on the pupil plane of an oil-immersion objective lens with NA = 1.4 (UPLSAPO 100XO, Olympus) using relay optics. In front of the objective lens, we inserted a duodecimally segmented half-wave plate implanted by a liquid crystal device [7] that operated as a polarization converter. As test samples, gold nanoparticles sparsely distributed on a coverslip and embedded by immersion oil were used. These sample conditions enabled us to avoid reflections on the coverslip interface, which may cause undesired interference [27] and the enhancement of directional radiation [28]. Under the illumination of the focused beam, the sample was raster-scanned using a piezo stage (Nano-LPQ, Mad City Labs) with a scanning pitch of 10 nm and a pixel dwell time of either 0.2 or 0.267 ms/pixel. At each scanning position, scattered light was collected by the same objective lens and detected by a photo-multiplier tube (H10721, Hamamatsu Photonics) after passing through a confocal pinhole with various pinhole sizes. An illumination laser power at the sample position was set to about 10 µW. To obtain similar signal levels for each experimental condition, we adjusted a signal gain of the detector in the range of 5500 to 7500. When the polarization converter was operated using this setup, the polarization of both an illumination beam and scattered light was converted, depending on the incident polarization direction. For RP-LG_5,1 beam illumination, we operated the LCOS-SLM to impose concentric 0-π phase flips, the radii of which corresponded to the nodes of an RP-LG_5,1 mode on the incident beam.

Fig. 4. Experimental setup. LCOS-SLM, liquid crystal on a silicon spatial light modulator; HWP, half-wave plate.

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Figure 5 shows the measured images of an isolated 100-nm gold particle acquired by the confocal microscope (pixel dwell time: 0.267 ms/pixel) with different pinhole sizes using LP-Gauss (polarized along the x-axis), RP-LG_0,1, and RP-LG_5,1 beams. For conventional detection using an LP-Gauss beam, the acquired images exhibited an elliptical shape along the x-axis without a confocal pinhole [the top row, denoted by “Open,” in Fig. 5(a)] and the image size decreased for small confocal pinholes (the second to fourth rows). Likewise, the images that were taken by RP-LG_0,1 [Fig. 5(b)] and RP-LG_5,1 [Fig. 5(c)] beams, accompanied by detection with polarization conversion, were changed and shrank in size as the pinhole size decreased. Additionally, under a pinhole of 0.4 AU, the side lobes observed in the image taken by the RP-LG_0,1 beam illumination was effectively reduced for the RP-LG_5,1 beam illumination. These behaviors corresponded to the results anticipated in the numerical simulations [see Figs. 2(a), 2(d), and 2(e)]. This also indicated that an isolated gold particle used in our experiments behaved like a single dipole that emitted radiation (as scattered light) polarized parallel to the local polarization of the illumination focal spot. In these measurements, the detector gain for the RP-LG_0,1 and RP-LG_5,1 beam illumination was increased about 1.3 times higher than that for the LP-Gauss beam illumination to obtain sufficient signal levels.

Fig. 5. The measured images of an isolated 100-nm gold particle acquired with different confocal pinhole sizes (indicated on the left). (a) Conventional detection using an LP-Gauss beam; detection with polarization conversion using (b) RP-LG_0,1 and (c) RP-LG_5,1 beams. The scale bar in (a) represents 250 nm. (d–f) Intensity profiles (red dashed lines) along the x-axis of the images corresponding to (a)–(c), acquired by 0.4 AU, and the simulation results (blue solid lines).

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The intensity profiles along the x-axis of the particle images in Figs. 5(a)–5(c), obtained for a confocal pinhole of 0.4 AU, are shown in Figs. 5(d)–5(f), respectively. The measured intensity profiles showed good agreement with those calculated in our simulation for all conditions. The FWHM sizes of the measured particle image shown in Figs. 5(d)–5(f) were determined as 120, 104, and 99 nm, whereas the corresponding values predicted by the simulations were 126, 105, and 101 nm, respectively. These evaluated results demonstrate that the smaller image size for a single scattering object was achieved by the detection with polarization conversion using RP-LG beam illumination compared with the conventional detection using LP-Gauss beam illumination.

Next, we imaged clusters of gold particles with a nominal diameter of 150 nm to examine the imaging capability for more complicated structures. Figure 6 shows images of the same clustered region fixed on the coverslip acquired by each condition, with a confocal pinhole of 0.4 AU and a pixel dwell time of 0.2 ms/pixel. For the conventional detection condition [Fig. 6(a)], individual particles inside the cluster were blurred and almost indistinguishable. For the detection with polarization conversion, the acquired images presented entirely different characteristics between RP-LG_0,1 and RP-LG_5,1 beams as can be seen in Figs. 6(b) and 6(c), respectively. Even though both beams yielded a small-sized isolated particle image compared with that by the LP-Gauss beam (as noted above), only the RP-LG_5,1 beam illumination could enhance the spatial resolution and the contrast for the clustered particle image. This was due to the residual side lobes remaining, even in the small pinhole condition for RP-LG_0,1 beam illumination, which deteriorated the image contrast. This feature is also highlighted in the comparison between the magnified images [Figs. 6(d)–6(f)] and the corresponding intensity profiles [Fig. 6(g)]. Each gold particle in the image could be resolved and distinguished only for RP-LG_5,1 beam illumination, implying the superior ability of higher-order RP-LG beams for the imaging of intricately structured objects with enhanced spatial resolution.

Fig. 6. Images of clusters of gold particles with a nominal diameter of 150 nm acquired with a confocal pinhole of 0.4 AU. (a) Conventional detection using an LP-Gauss beam; detection with polarization conversion using (b) RP-LG_0,1 and (c) RP-LG_5,1 beams. The scale bars in (a)–(c) represent 500 nm. (d–f) Magnified images of the region indicated by red rectangles in (a)–(c); scale bars in (d)–(f) represent 200 nm. (g) Normalized intensity profiles along the red dashed lines in (d)–(f).

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4. Discussion and conclusion

The present work revealed an optical response and image formation for a small scatterer illuminated by a tightly focused radially polarized beam in CLSM with polarization conversion. A small pinhole condition enabled increasing the detection efficiency of the longitudinally polarized emission, induced by the longitudinal component of an illumination focal spot, relatively compared with that of a transversely polarized one. However, even in sufficiently small pinhole conditions, the transverse component of the focal spot for a radially polarized beam was also able to contribute to the image formation in the form of a side lobe for an isolated scatter image. The use of a higher-order RP-LG beam enhanced the generation of the longitudinal component at the focus, resulting in favorable imaging capability with an improved spatial resolution when imaging with scattering signals. The spatial resolution can be further enhanced by incorporating PSF engineering, which aims to design the intensity and polarization distribution of a focal spot such as superoscillation focusing [9].

In addition to the polarization direction of an illumination focal spot, light scattering from small objects and the far-field radiation characteristic are strongly associated with anisotropic structure, orientation, and surrounding substrates. Our method is particularly advantageous for imaging nanostructures, which will primarily respond to longitudinally polarized illumination, e.g., perpendicularly oriented nanorods [29], nanopillars [30], sharpened tips [31,32], and molecules [33]. Additionally, this method can be readily extended to other microscope techniques utilizing coherent nonlinear processes including SHG and higher-harmonic generation, sum-frequency generation, and stimulated Raman scattering, as these processes strongly rely on the polarization between excitation and emission. In terms of numerical consideration, this can be achieved by modifying the form of an induced dipole moment, expressed by Eq. (3), to represent a specific optical process.

In summary, we investigated image formation in CLSM with polarization conversion using an RP-LG beam. The mathematical representation of the optical response for a small scatterer under a small confocal pinhole was presented and experimentally verified by imaging a small gold particle. Both numerical simulation and experimental measurements demonstrated that the use of a higher-order RP-LG beam illumination enabled enhancing the spatial resolution and the contrast of imaging for scattering objects. Our technique can be applied to other microscopes that detect coherent signals.

Appendix: dipole images with polarization conversion

In the optical system shown in Fig. 1(a), we first considered an electric dipole moment with Cartesian components $\textbf{p} = ({{p_x},{p_y},{p_z}} )$, located at the focus (x_s = y_s = 0) of the objective lens and induced by a focused light with a wavelength given as λ. We assumed that the emitted radiation was collected and collimated by the objective lens. The electric field of the radiation at the pupil plane had only transverse components E_cx and E_cy, which can be expressed as follows [19]:

(4)$$\left( {\begin{array}{c} {{E_{cx}}}\\ {{E_{cy}}} \end{array}} \right) = \frac{1}{{\sqrt {\cos {\theta _1}} }}\left( {\begin{array}{c} {\frac{{{p_x}}}{2}[{(1 + \cos {\theta_1}) - (1 - \cos {\theta_1})\cos 2\phi } ]- \frac{{{p_y}}}{2}(1 - \cos {\theta_1})\sin 2\phi - {p_z}\sin {\theta_1}\cos \phi }\\ { - \frac{{{p_x}}}{2}(1 - \cos {\theta_1})\sin 2\phi + \frac{{{p_y}}}{2}[{(1 + \cos {\theta_1}) + (1 - \cos {\theta_1})\cos 2\phi } ]- {p_z}\sin {\theta_1}\sin \phi } \end{array}} \right),$$

where (θ₁, ϕ) are spherical coordinates representing the pupil plane with 0 < θ₁ < α₁ and 0 < ϕ < 2π. In conventional detection (without any polarization conversions), the image of the dipole is formed at the focus of the tube lens, and its electric field ${\textbf{e}_\textrm{d}} = ({{e_{\textrm{d},x}},{e_{\textrm{d},y}},{e_{\textrm{d},z}}} )$ at the image plane $({{x_\textrm{d}},{y_\textrm{d}},{z_\textrm{d}} = 0} )$ is obtained on the basis of vector diffraction theory as [18]:

(5)$$\left( {\begin{array}{c} {{e_{\textrm{d},x}}}\\ {{e_{\textrm{d},y}}}\\ {{e_{\textrm{d},z}}} \end{array}} \right) ={-} \frac{{iA}}{\pi }\int\limits_0^{2\pi } {\int\limits_0^{{\alpha _2}} {{\textbf{E}_2}\exp [{ik{r_d}\sin {\theta_2}\cos ({\phi - {\phi_d}} )} ]\exp ({ik{z_d}\cos {\theta_2}} )\sin {\theta _2}d{\theta _2}d\phi } } ,$$

where A is a constant, k = 2π / λ, ${r_\textrm{d}} = \sqrt {{x_\textrm{d}}^2 + {y_\textrm{d}}^2}$, ${\phi _\textrm{d}} = {\tan ^{ - 1}}({{y_\textrm{d}}/{x_\textrm{d}}} )$, and

(6)$${\textbf{E}_2} = \frac{1}{2}\left( {\begin{array}{c} {{E_{cx}}[{1 + \cos {\theta_2} - ({1 - \cos {\theta_2}} )\cos 2\phi } ]- {E_{cy}}({1 - \cos {\theta_2}} )\sin 2\phi }\\ { - {E_{cx}}({1 - \cos {\theta_2}} )\sin 2\phi + {E_{cy}}[{1 + \cos {\theta_2} + ({1 - \cos {\theta_2}} )\cos 2\phi } ]}\\ { - 2{E_{cx}}\cos \phi \sin {\theta_2} - 2{E_{cy}}\sin \phi \sin {\theta_2}} \end{array}} \right).$$

Next, we considered the polarization conversion at the pupil plane. The polarization converter changed the incident linear polarization along the x or y-axis to radial or azimuthal polarization following the incident polarization direction as ê_x → ê_r and ê_y → −ê_ϕ [see Fig. 1(a)]. Taking into account the relationship between Cartesian and cylindrical coordinates, this conversion generated the converted electric field $({{E_{cx}}^{\prime},{E_{cy}}^{\prime}} )$ at the pupil plane, i.e.,

(7)$$\begin{aligned} \left( {\begin{array}{{c}} {{E_{cx}}^\prime }\\ {{E_{cy}}^\prime } \end{array}} \right) &= \left( {\begin{array}{{c}} {{E_{cx}}\cos \phi + {E_{cy}}\sin \phi }\\ {{E_{cx}}\sin \phi - {E_{cy}}\cos \phi } \end{array}} \right)\\ &= \frac{1}{{\sqrt {\cos \theta_{1} } }}\left( {\begin{array}{{c}} {{p_x}\cos \theta_{1} \cos \phi + {p_y}\cos \theta_{1} \sin \phi - {p_z}\sin \theta_{1} }\\ {{p_x}\sin \phi - {p_y}\cos \phi } \end{array}} \right). \end{aligned}$$

To apply this conversion, E_cx and E_cy in Eq. (6) were replaced by the converted electric field E_cx′ and E_cy′, respectively, and E₂ was subsequently written as follows:

(8)$${\textbf{E}_2} = \frac{1}{{\sqrt {\cos {\theta _1}} }}\left( {\begin{array}{c} {\frac{{{p_x}}}{4}({A\cos \phi + B\cos 3\phi } )+ \frac{{{p_y}}}{4}({C\sin \phi + B\sin 3\phi } )- \frac{{{p_z}}}{2}({D - E\cos 2\phi } )}\\ {\frac{{{p_x}}}{4}({A^{\prime}\sin \phi + B\sin 3\phi } )- \frac{{{p_y}}}{4}({C^{\prime}\cos \phi + B\cos 3\phi } )+ \frac{{{p_z}}}{2}E\sin 2\phi }\\ { - \frac{{{p_x}}}{2}({D^{\prime\prime} - E^{\prime\prime}\cos 2\phi } )+ \frac{{{p_y}}}{2}E^{\prime\prime}\sin 2\phi + {p_z}F\cos \phi } \end{array}} \right),$$

where

(9)$$\begin{aligned}A &={-} 1 + \cos {\theta _2} + ({1 + 3\cos {\theta_2}} )\cos {\theta _1},\\ A^{\prime} &= 1 + 3\cos {\theta _2} - ({1 - \cos {\theta_2}} )\cos {\theta _1},\\ B &= ({1 - \cos {\theta_2}} )- ({1 - \cos {\theta_2}} )\cos {\theta _1},\\ C &= 1 - \cos {\theta _2} + ({3 + \cos {\theta_2}} )\cos {\theta _1},\\ C^{\prime} &= 3 + \cos {\theta _2} + ({1 - \cos {\theta_2}} )\cos {\theta _1},\\ D &= ({1 + \cos {\theta_2}} )\sin {\theta _1},\\ D^{\prime\prime} &= ({1 + \cos {\theta_1}} )\sin {\theta _2},\\ E &= ({1 - \cos {\theta_2}} )\sin {\theta _1},\\ E^{\prime\prime} &= ({1 - \cos {\theta_1}} )\sin {\theta _2},\\ F &= \sin {\theta _1}\sin {\theta _2}. \end{aligned}$$

Finally, after substituting Eq. (8) into Eq. (5), we obtained the following:

(10)$${\textbf{e}_d} = \left[ {\begin{array}{c} {\frac{i}{2}{p_x}K_1^I\cos {\phi_d} + \frac{i}{2}{p_y}K_1^{II}\sin {\phi_d} - \frac{i}{2}({{p_x}\cos 3{\phi_d} + {p_y}\sin 3{\phi_d}} )K_3^I - {p_z}K_0^I - {p_z}K_2^I\cos 2{\phi_d}}\\ {\frac{i}{2}{p_x}K_1^{III}\sin {\phi_d} - \frac{i}{2}{p_y}K_1^{IV}\cos {\phi_d} - \frac{i}{2}({{p_x}\sin 3{\phi_d} - {p_y}\cos 3{\phi_d}} )K_3^I - {p_z}K_2^I\sin 2{\phi_d}}\\ { - {p_x}K_0^{II} - ({{p_x}\cos 2{\phi_d} + {p_y}\sin 2{\phi_d}} )K_2^{II} + 2i{p_z}K_1^V\cos \phi } \end{array}} \right].$$

In Eq. (10), the integral representations $K_n^N$ with indices N (= I, II, III, IV, and V) and n ( = 0, 1, 2, and 3) are expressed as follows:

(11)$$K_n^N({r_d},{z_d}) = \int\limits_0^{{\alpha _2}} {\sqrt {\frac{{\cos {\theta _2}}}{{\cos {\theta _1}}}} } \sin {\theta _2}O_n^N{J_n}({k}{r_d}\sin {\theta _2})\exp (i{k}{z_d}\cos {\theta _2})d{\theta _2},$$

where J_n is the Bessel function of the first kind of order n and

(12)$$\begin{aligned}O_0^I &= ({1 + \cos {\theta_2}} )\sin {\theta _1},\\ O_0^{II} &= ({1 + \cos {\theta_1}} )\sin {\theta _2},\\ O_1^I &={-} 1 + \cos {\theta _2} + ({1 + 3\cos {\theta_2}} )\cos {\theta _1},\\ O_1^{II} &= 1 - \cos {\theta _2} + ({3 + \cos {\theta_2}} )\cos {\theta _1},\\ O_1^{III} &= 1 + 3\cos {\theta _2} - ({1 - \cos {\theta_2}} )\cos {\theta _1},\\ O_1^{IV} &= 3 + \cos {\theta _2} + ({1 - \cos {\theta_2}} )\cos {\theta _1},\\ O_1^V &= \sin {\theta _1}\sin {\theta _2},\\ O_2^I &= ({1 - \cos {\theta_2}} )\sin {\theta _1},\\ O_2^{II} &= ({1 - \cos {\theta_1}} )\sin {\theta _2},\\ O_3^I &= ({1 - \cos {\theta_1}} )({1 - \cos {\theta_2}} ). \end{aligned}$$

Equation (10) represents the electric field components of an image for a dipole located at the focus of an objective lens after the polarization conversion. Note that the electric field at the image plane for dipole radiation without polarization conversion (conventional detection) corresponds to Eq. (17) in [18] and Eq. (74) in [19].

Funding

Research Foundation for Opto-Science and Technology; Murata Science Foundation; Precursory Research for Embryonic Science and Technology (JPM JPR15P8); Japan Society for the Promotion of Science (15H05953, 19H02622); Japan Agency for Medical Research and Development (Brain Mapping by Integrated Neurotechnologies).

Acknowledgments

We thank Citizen Watch Co., Ltd., for the provision of the liquid crystal device.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Imaging with a longitudinal electric field in confocal laser scanning microscopy to enhance spatial resolution

Abstract

1. Introduction

2. Numerical simulation

3. Experimental results

4. Discussion and conclusion

Appendix: dipole images with polarization conversion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (6)

Equations (12)

Optics Express