High-resolution light-scattering imaging with two-dimensional hexagonal illumination patterns: system implementation and image reconstruction formulations

Chih-Wei Chen; Po-Hsun Wang; Li-Jun Chou; Yin-Yu Lee; Bo-Jui Chang; Su-Yu Chiang

doi:10.1364/OE.25.021652

1. Introduction

Structured illumination microscopy (SIM) is a super-resolution imaging technique that doubles the spatial resolution of conventional wide-field (WF) fluorescence microscopy [1–8]. Its working principle involves modulating a fluorescent image with a structured light pattern with varied phases and orientations, and then computationally retrieving the modulated information at large spatial frequencies to yield an enhanced-resolution image [3, 8]. The structured light pattern is typically sinusoidal and varied at three (minimum) orientations − 0° and ± 60° − to achieve an isotropically lateral resolution improvement [9–12]. Although SIM has less resolution improvement than other super-resolution imaging techniques [13–17], it takes full advantage of WF fluorescence microscopy, especially the larger field of view, less light intensity and more rapid rate of imaging that are essential for the study of live and dynamic samples [9–12, 18]. SIM is also being applied to light-scattering imaging for label-free or nanoparticle-tracking applications. Several SIM-based contrast-imaging methods have been developed to image coherently scattering samples in their native states with improved resolution and image contrast [19–28].

The SIM-based contrast-imaging methods are classified as coherent SIM (CSIM) in terms of their imaging characteristics. In CSIM, the relation between the sample field and the image intensity is nonlinear because the amplitude and phase of a scattering sample are proportional to the coherent transfer function (CTF) of a microscope. CSIM is hence analogous to fluorescent SIM in the use of a structured light pattern, but the formation of a CSIM image requires a different illumination structure or detection scheme. Both intensity- [19–22] and field-based [23–26] CSIM techniques have been developed recently. The field-based CSIM is superior for imaging optically transparent objects. One analogue – structured illumination quantitative phase microscopy (SI-QPM) – was introduced with the use of an off-axis reference wave for reconstruction, and was demonstrated to retrieve sample phases with great accuracy [25, 26]. On the other hand, the intensity-based CSIM is suitable for imaging highly-scattered samples. One introduced analogue is structured oblique-illumination microscopy (SOIM) that uses multiple oblique beams simultaneously to illuminate samples to modulate the multiplexed information at large spatial frequencies into the frequency support of the system [20]. However, due to the nonlinearity between the scattered amplitude and the measured image intensity, such extended frequency support typically does not result in extended imaging resolution beyond the classical Abbe limit [20, 27, 28].

SOIM is easily implemented on a SIM fluorescence system based on a spatial light modulator (SLM), with no hardware modification required. A development of high-resolution SOIM imaging with isotropic resolution enhancement hence allows one to broaden its applications. The SOIM developed previously is simplified in the pattern design and image analysis; the reconstructed image is thus anisotropic and of low resolution [20, 29]. It is more complicated to achieve isotropic resolution improvement; a set of hexagonal SLM patterns at 19 phases and a vectorial analysis of modulated imaging to work out the formulations for image reconstruction are required. Furthermore, it is critical to consider the effect of beam polarizations on the resolution improvement and image quality with the use of a large-NA objective for high-resolution imaging. Simulations of the use of a set of hexagonal-lattice illumination patterns to enhance the isotropy of SOIM imaging were reported [22], but a system to form the illumination pattern from three standing-wave fields would be complicated. Moreover, the simulation results are questionable because some reported patterns of different phases are crushed; a new concept in varying the hexagonal pattern at 19 phases is required to provide robust imaging, and more importantly to reconstruct an image correctly.

In this work, we present high-resolution laterally isotropic SOIM with hexagonal illumination patterns, describing both the system implementation and image reconstruction formulations. We implemented SOIM on an SLM-based SIM fluorescence system, with the design of a hexagonal SLM pattern to generate diffraction beams at 0° and ± 60.3° simultaneously for interference, and undertook calculations to obtain a set of optimal SLM shifts to reconstruct an image correctly. We also derived the distributions of the electrical field of the resultant hexagonal patterns for beams of linear and circular polarizations, and the corresponding image reconstruction formulations. Experimental verifications are presented to show the dependence of hexagonal patterns on polarization, the resolution improvement and the effect of polarization of SOIM imaging on gold nanoparticles (100 nm), and a biological application involving imaging cellular structures of a label-free fixed HeLa cell.

2. Theory

2.1 Hexagonal pattern of six-beam interferences

For isotropic SOIM, three sets of ± 1-order diffracted beams at θ = 0° and ± 60.3° are used to illuminate the sample simultaneously. The electric field $\overset{⇀}{E} (\overset{⇀}{r}, t)$ of resultant 2D hexagonal pattern is

\overset{⇀}{E} (\overset{⇀}{r}, t) = \sum_{n = 1, 2, 3} {\overset{⇀}{P}}_{\pm n} f_{\pm n} e^{i ({\overset{⇀}{k}}_{\pm n} \cdot \overset{⇀}{r} - ω t - ϕ_{\pm n})}

with n = 1, 2, and 3 for the ± 1-order beams at θ = 0° and ± 60.3°, respectively, polarization vectors

{\overset{⇀}{P}}_{\pm n}

, amplitudes

f_{\pm n}

, propagation vectors

{\overset{⇀}{k}}_{\pm n}

, and phases

ϕ_{\pm n}

of beams of n_th ± 1-order, beam position r(x, y, z), and angular frequency ω. Figure 1 shows (a) bottom (xy plane) and (b) side (xz plane) views of the propagation and polarization vectors of three-set beams from the back focal plane (BFP) of the objective to the sample plane; beams of s-linear polarization are parallel at the BFP. In this figure, we define beams at θ = 0° to lie in plane xz and the polarization vectors of all beams to lie along the y axis at the BFP. As the polarization vectors of beams at varied orientations would be refracted by the objective into various directions at the sample plane, we define the polarization vectors of beams at θ = 0° in phase and those at θ = ± 60.3° partially out of phase at the sample plane. We name this case as unidirectional s-linear polarization and abbreviate it as Uni-s for simplicity. The uses of parallel beams of p-linear and circular polarization at the BFP are called unidirectional p-linear and circular polarization, abbreviated as Uni-p and Uni-c, respectively. Similar to Uni-s, the p-linearly or circularly polarized beams at θ = 0° are in-phase at the sample plane.

Fig. 1 (a) Bottom (xy plane) and (b) side (xz plane) views of the propagation and polarization vectors of three sets of ± 1-order beams at θ = 0° and ± 60.3° for Uni-s; the beams are s-linearly polarized with their polarization vectors along the y axis at the BFP of the objective; the interaction angle for each set of ± 1-order beams is 2β = 88°.

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Accordingly, the propagation vectors of ± 1-order beams with interaction angle 2β at the sample plane are

{\overset{⇀}{k}}_{\pm n} = k (\pm \cos θ \sin β \hat{x}, \pm \sin θ \sin β \hat{y}, \cos β \hat{z}) .

with magnitude k = 2π/λ; λ is the wavelength of the incident laser beam. The polarization vectors are

\begin{array}{l} {\overset{⇀}{P}}_{\pm n} = \sqrt{\frac{1}{1 + ε}} (\cos θ \sin θ (\cos β - 1) \hat{x}, (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y}, \mp \sin θ \sin β \hat{z}) \\ - i \sqrt{\frac{ε}{1 + ε}} ((\cos^{2} θ \cos β + \sin^{2} θ) \hat{x}, \cos θ \sin θ (\cos β - 1) \hat{y}, \mp \cos θ \sin β \hat{z}) . \end{array}

with ellipticity ε = L_x/L_y; L_x and L_y correspond to the lengths of the semi-major and semi-minor axes of beams at the BFP; ε = 0, ∞ and 1 correspond to Uni-s, Uni-p and Uni-c, respectively. On substitution of Eqs. (2) and (3) into Eq. (1) and letting

ϕ_{- n} = - ϕ_{n}

and

f_{n} = f_{- n} = 1

(n = 1 − 3) for simplicity, electric fields −

{\overset{⇀}{E}}_{U n i - s} (\overset{⇀}{r}, t)

and

{\overset{⇀}{E}}_{U n i - p} (\overset{⇀}{r}, t)

of Uni-s and Uni-p at the sample plane are

{\overset{⇀}{E}}_{U n i - s} (\overset{⇀}{r}, t) = {(\begin{array}{l} \cos ω t {\frac{\sqrt{3} (\cos β - 1)}{2} [\cos ({\overset{⇀}{k}}_{2} + ϕ_{2}) - \cos ({\overset{⇀}{k}}_{3} + ϕ_{3})]}, \\ \cos ω t {2 \cos ({\overset{⇀}{k}}_{1} + ϕ_{1}) + \frac{1 + 3 \cos β}{2} [\cos ({\overset{⇀}{k}}_{2} + ϕ_{2}) + \cos ({\overset{⇀}{k}}_{3} + ϕ_{3})]}, \\ \sin ω t {- \sqrt{3} \sin β [\sin ({\overset{⇀}{k}}_{2} + ϕ_{2}) - \sin ({\overset{⇀}{k}}_{3} + ϕ_{3})]} \end{array})}_{z = 0} .

{\overset{⇀}{E}}_{U n i - p} (\overset{⇀}{r}, t) = {(\begin{array}{l} \cos ω t {2 \cos β \cos ({\overset{⇀}{k}}_{1} + ϕ_{1}) + \frac{3 + \cos β}{2} [\cos ({\overset{⇀}{k}}_{2} + ϕ_{2}) + \cos ({\overset{⇀}{k}}_{3} + ϕ_{3})]}, \\ \cos ω t {\frac{\sqrt{3} (\cos β - 1)}{2} [\cos ({\overset{⇀}{k}}_{2} + ϕ_{2}) - \cos ({\overset{⇀}{k}}_{3} + ϕ_{3})]}, \\ \sin ω t {- \sin β [2 \sin ({\overset{⇀}{k}}_{1} + ϕ_{1}) + \sin ({\overset{⇀}{k}}_{2} + ϕ_{2}) + \sin ({\overset{⇀}{k}}_{3} + ϕ_{3})]} \end{array})}_{z = 0} .

with

{\overset{⇀}{k}}_{1} = k \sin β \hat{x}

,

{\overset{⇀}{k}}_{2} = k \sin β (\frac{\hat{x} + \sqrt{3} \hat{y}}{2})

, and

{\overset{⇀}{k}}_{3} = k \sin β (\frac{\hat{x} - \sqrt{3} \hat{y}}{2})

. The electric field

{\overset{⇀}{E}}_{U n i - c} (\overset{⇀}{r}, t)

of Uni-c is a combination of s-linearly and delayed p-linearly polarized beams, and described as

{\overset{⇀}{E}}_{U n i - c} (\overset{⇀}{r}, t) = \frac{1}{\sqrt{2}} {[{\overset{⇀}{E}}_{U n i - s} (\overset{⇀}{r}, t) + {\overset{⇀}{E}}_{U n i - p} (\overset{⇀}{r}, t + \frac{π}{2 ω})]}_{z = 0} .

It should be pointed out that the beams of s-linear polarization at the sample plane, not at the BFP of the objective, produce a hexagonal pattern of maximal contrast and greatest symmetry. Such a scheme, however, would complicate the system because it requires one optical component for one beam to control its polarization within a limited space. For reference, we name this case axisymmetric-s linear polarization, with the abbreviation of Axi-s. Its electric field

{\overset{⇀}{E}}_{A x i - s} (\overset{⇀}{r}, t)

is

{\overset{⇀}{E}}_{A x i - s} (\overset{⇀}{r}, t) = {(\begin{matrix} \cos ω t {- \sqrt{3} [\cos ({\overset{⇀}{k}}_{2} + ϕ_{2}) - \cos ({\overset{⇀}{k}}_{3} + ϕ_{3})]}, \\ \cos ω t [2 \cos ({\overset{⇀}{k}}_{1} + ϕ_{1}) + \cos ({\overset{⇀}{k}}_{2} + ϕ_{2}) + \cos ({\overset{⇀}{k}}_{3} + ϕ_{3})], \\ 0 \end{matrix})}_{z = 0} .

2.2 SOIM imaging with hexagonal illumination patterns

In SOIM imaging, the time-averaged intensity-based image $D (\overset{⇀}{r})$ of an object is

D (\overset{⇀}{r}) = 〈 {| \overset{⇀}{H} (\overset{⇀}{r}) \otimes [\overset{⇀}{E} (\overset{⇀}{r}, t) {\overset{⇀}{U}}_{g} (\overset{⇀}{r})] |}^{2} 〉 .

with position r, amplitude point-spread function (APSF)

\overset{⇀}{H} (\overset{⇀}{r})

, electric field

\overset{⇀}{E} (\overset{⇀}{r}, t)

, object

{\overset{⇀}{U}}_{g} (\overset{⇀}{r})

, and convolution operator ⊗. With the use of a large-NA objective for high-resolution imaging,

\overset{⇀}{H} (\overset{⇀}{r})

is a function of beam polarization. We further define

{\overset{⇀}{H}}_{U n i - s} (\overset{⇀}{r})

of Uni-s and

{\overset{⇀}{H}}_{U n i - p} (\overset{⇀}{r})

of Uni-p to be orthogonal and combine them to form

{\overset{⇀}{H}}_{U n i - c} (\overset{⇀}{r})

of Uni-c. According to Eq. (8), the modulated spectrum

D (\overset{⇀}{k})

is

D (\overset{⇀}{k}) = 〈 a c {\overset{⇀}{H} (\overset{⇀}{k}) [\overset{⇀}{E} (\overset{⇀}{k}) \otimes {\overset{⇀}{U}}_{g} (\overset{⇀}{k})]} 〉 .

in which

D (\overset{⇀}{k})

,

\overset{⇀}{H} (\overset{⇀}{k})

,

\overset{⇀}{E} (\overset{⇀}{k})

, and

{\overset{⇀}{U}}_{g} (\overset{⇀}{k})

correspond to Fourier transforms of

D (\overset{⇀}{r})

,

\overset{⇀}{H} (\overset{⇀}{r})

,

\overset{⇀}{E} (\overset{⇀}{r}, t)

, and

{\overset{⇀}{U}}_{g} (\overset{⇀}{r})

, respectively; ac denotes auto-correlation operator. On substitution of the Fourier transform of

\overset{⇀}{E} (\overset{⇀}{r}, t)

in Eqs. (4)-(7) into Eq. (9), respectively,

D (\overset{⇀}{k})

becomes

\begin{array}{l} D (\overset{⇀}{k}) = \sum_{m = x, y, z} 〈 a c {H_{m} (\overset{⇀}{k}) [E_{m} (\overset{⇀}{k}) \otimes U_{g} (\overset{⇀}{k})]} 〉 \\ = \sum_{l = 0 - 18} e^{i Φ_{l}} \sum_{m = x, y, z} C_{l, m} {F^{'}}_{l, m} (\overset{⇀}{k}) . \\ = \sum_{l = 0 - 18} e^{i Φ_{l}} F_{l} (\overset{⇀}{k}) \end{array}

with phase coefficients

Φ_{0 - 18}

, polarization coefficients

C_{0 - 18, x}

,

C_{0 - 18, y}

,

C_{0 - 18, z}

, frequency-shifted components

{F^{'}}_{0 - 18, x} (\overset{⇀}{k})

,

{F^{'}}_{0 - 18, y} (\overset{⇀}{k})

,

{F^{'}}_{0 - 18, z} (\overset{⇀}{k})

, and patterned-excitation components

F_{0 - 18} (\overset{⇀}{k})

. The values of

C_{0 - 18, x}

,

C_{0 - 18, y}

, and

C_{0 - 18, z}

values are associated with

\overset{⇀}{E} (\overset{⇀}{r}, t)

; those of Axi-s and Uni cases are tabulated in Appendix A. The

{F^{'}}_{0 - 18, x} (\overset{⇀}{k})

,

{F^{'}}_{0 - 18, y} (\overset{⇀}{k})

, and

{F^{'}}_{0 - 18, z} (\overset{⇀}{k})

components result from the cross-correlations of

\overset{⇀}{H} (\overset{⇀}{k})

and frequency-shifted

{\overset{⇀}{U}}_{g} (\overset{⇀}{k})

. The correspondences between patterned-excitation components

F_{0 - 18} (\overset{⇀}{k})

and frequency-shifted components

{F^{'}}_{0 - 18, x} (\overset{⇀}{k})

,

{F^{'}}_{0 - 18, y} (\overset{⇀}{k})

,

{F^{'}}_{0 - 18, z} (\overset{⇀}{k})

are as follows.

\begin{array}{l} e^{i Φ_{0}} F_{0} (\overset{⇀}{k}) = \sum_{m = x, y, z} \sum_{n = 1 - 3} C_{0, m - n} [a c (H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} - {\overset{⇀}{k}}_{n})) + a c (H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} + {\overset{⇀}{k}}_{n}))] \\ e^{i Φ_{1, 2}} F_{1, 2} (\overset{⇀}{k}) = e^{\pm i 2 ϕ_{1}} \sum_{m = x, y, z} C_{1, m} / C_{2, m} (H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{1}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{1})) \\ e^{i Φ_{3, 4}} F_{3, 4} (\overset{⇀}{k}) = e^{\pm i 2 ϕ_{2}} \sum_{m = x, y, z} C_{3, m} / C_{4, m} (H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{2}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{2})) \\ e^{i Φ_{5, 6}} F_{5, 6} (\overset{⇀}{k}) = e^{\pm i 2 ϕ_{3}} \sum_{m = x, y, z} C_{5, m} / C_{6, m} (H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{3}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{3})) \\ e^{i Φ_{7, 8}} F_{7, 8} (\overset{⇀}{k}) = e^{\pm i (ϕ_{1} + ϕ_{2})} \sum_{m = x, y, z} C_{7, m} / C_{8, m} (\begin{array}{l} H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{1}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{2}) \\ + H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{2}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{1}) \end{array}) \\ e^{i Φ_{9, 10}} F_{9, 10} (\overset{⇀}{k}) = e^{\pm i (ϕ_{1} - ϕ_{2})} \sum_{m = x, y, z} C_{9, m} / C_{10, m} (\begin{array}{l} H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{1}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{2}) \\ + H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{2}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{1}) \end{array}) \\ e^{i Φ_{11, 12}} F_{11, 12} (\overset{⇀}{k}) = e^{\pm i (ϕ_{1} + ϕ_{3})} \sum_{m = x, y, z} C_{11, m} / C_{12, m} (\begin{array}{l} H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{1}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{3}) \\ + H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{3}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{1}) \end{array}) \\ e^{i Φ_{13, 14}} F_{13, 14} (\overset{⇀}{k}) = e^{\pm i (ϕ_{1} - ϕ_{3})} \sum_{m = x, y, z} C_{13, m} / C_{14, m} (\begin{array}{l} H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{1}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{3}) \\ + H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{3}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{1}) \end{array}) \\ e^{i Φ_{15, 16}} F_{15, 16} (\overset{⇀}{k}) = e^{\pm i (ϕ_{2} + ϕ_{3})} \sum_{m = x, y, z} C_{15, m} / C_{16, m} (\begin{array}{l} H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{2}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{3}) \\ + H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{3}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{2}) \end{array}) . \\ e^{i Φ_{17, 18}} F_{17, 18} (\overset{⇀}{k}) = e^{\pm i (ϕ_{2} - ϕ_{3})} \sum_{m = x, y, z} C_{17, m} / C_{18, m} (\begin{array}{l} H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{2}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \mp {\overset{⇀}{k}}_{3}) \\ + H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{3}) ⋆ H_{m} (\overset{⇀}{k}) U_{g} (\overset{⇀}{k} \pm {\overset{⇀}{k}}_{2}) \end{array}) \end{array}

with

{\overset{⇀}{k}}_{1}

,

{\overset{⇀}{k}}_{2}

, and

{\overset{⇀}{k}}_{3}

defined in Eqs. (4) and (5). Accordingly, a set of

D (\overset{⇀}{r})

acquired with the hexagonal pattern varied at 19 phases is required for

F_{0 - 18} (\overset{⇀}{k})

retrieval.

2.3 Image reconstruction

The illumination of a hexagonal pattern extends $\overset{⇀}{H} (\overset{⇀}{r})$ in the directions of the diffraction beams to improve the resolution. To facilitate the $F_{0 - 18} (\overset{⇀}{k})$ shifts for image reconstruction, we derive the extended CTF ${\overset{⇀}{H}}^{'} (\overset{⇀}{k})$ and the corresponding resolution-enhanced spectrum $D^{'} (\overset{⇀}{k})$ as a reference. The extended ${\overset{⇀}{H}}^{'} (\overset{⇀}{k})$ resultant from the interactions of the beams with $\overset{⇀}{H} (\overset{⇀}{k})$ is

\overset{⇀}{H^{'}} (\overset{⇀}{k}) = \sum_{n = 1 - 3} {\overset{⇀}{C}}_{E, n} \overset{⇀}{H} (\overset{⇀}{k} + {\overset{⇀}{k}}_{n}) + {\overset{⇀}{C}}_{E, - n} \overset{⇀}{H} (\overset{⇀}{k} - {\overset{⇀}{k}}_{n}) .

in which n = 1, 2, and 3 represent the ± 1-order beams at θ = 0° and ± 60.3°, coefficients

{\overset{⇀}{C}}_{E, \pm n}

are the amplitudes of beams in

\overset{⇀}{E} (\overset{⇀}{k})

, and

{\overset{⇀}{k}}_{1}

,

{\overset{⇀}{k}}_{2}

, and

{\overset{⇀}{k}}_{3}

are defined in Eqs. (4)-(7) for Axi-s and Uni cases. The corresponding resolution-enhanced spectrum

D^{'} (\overset{⇀}{k})

is

D^{'} (\overset{⇀}{k}) = a c {{\overset{⇀}{H}}^{'} (\overset{⇀}{k}) \cdot {\overset{⇀}{U}}_{g} (\overset{⇀}{k})} = \sum_{l = 0 - 18} \sum_{m = x, y, z} C_{l, m} {G^{'}}_{l, m} (\overset{⇀}{k}) = \sum_{l = 0 - 18} G_{l} (\overset{⇀}{k}) .

with polarization coefficients

C_{0 - 18, x}

,

C_{0 - 18, y}

,

C_{0 - 18, z}

, frequency-extended components

{G^{'}}_{0 - 18, x} (\overset{⇀}{k})

,

{G^{'}}_{0 - 18, y} (\overset{⇀}{k})

,

{G^{'}}_{0 - 18, z} (\overset{⇀}{k})

, and enhanced-resolution components

G_{0 - 18} (\overset{⇀}{k})

.

On comparison to Eq. (10), the polarization coefficients are the same, both resulting from $\overset{⇀}{E} (\overset{⇀}{r}, t)$ . The ${G^{'}}_{0 - 18, x} (\overset{⇀}{k})$ , ${G^{'}}_{0 - 18, y} (\overset{⇀}{k})$ , ${G^{'}}_{0 - 18, z} (\overset{⇀}{k})$ components are similar to the ${F^{'}}_{0 - 18, x} (\overset{⇀}{k})$ , ${F^{'}}_{0 - 18, y} (\overset{⇀}{k})$ , ${F^{'}}_{0 - 18, z} (\overset{⇀}{k})$ components, except that they are associated with the cross-correlations of frequency-shifted $\overset{⇀}{H} (\overset{⇀}{k})$ and ${\overset{⇀}{U}}_{g} (\overset{⇀}{k})$ , not $\overset{⇀}{H} (\overset{⇀}{k})$ and frequency-shifted ${\overset{⇀}{U}}_{g} (\overset{⇀}{k})$ . In a comparison of the resolution-enhanced spectrum and modulated spectrum, we obtain the correspondences $G_{0 - 18} (\overset{⇀}{k}) = F_{0 - 18} (\overset{⇀}{k} + {\overset{⇀}{K}}_{0 - 18})$ , with shift vectors ${\overset{⇀}{K}}_{0} = 0,$ ${\overset{⇀}{K}}_{1, 2} = \pm 2 {\overset{⇀}{k}}_{1},$ ${\overset{⇀}{K}}_{3, 4} = \pm 2 {\overset{⇀}{k}}_{2},$ ${\overset{⇀}{K}}_{5, 6} = \pm 2 {\overset{⇀}{k}}_{3},$ ${\overset{⇀}{K}}_{7, 8} = \pm ({\overset{⇀}{k}}_{1} + {\overset{⇀}{k}}_{2}),$ ${\overset{⇀}{K}}_{9, 10} = \pm ({\overset{⇀}{k}}_{1} - {\overset{⇀}{k}}_{2}),$ ${\overset{⇀}{K}}_{11, 12} = \pm ({\overset{⇀}{k}}_{1} + {\overset{⇀}{k}}_{3}),$ ${\overset{⇀}{K}}_{13, 14} = \pm ({\overset{⇀}{k}}_{1} - {\overset{⇀}{k}}_{3}),$ ${\overset{⇀}{K}}_{15, 16} = \pm ({\overset{⇀}{k}}_{2} + {\overset{⇀}{k}}_{3}),$ and ${\overset{⇀}{K}}_{17, 18} = \pm ({\overset{⇀}{k}}_{2} - {\overset{⇀}{k}}_{3}) .$ Accordingly, an addition of shifted $F_{0 - 18} (\overset{⇀}{k})$ yields the resolution-enhanced spectrum $D^{'} (\overset{⇀}{k})$ ; the inverse Fourier transform of $D^{'} (\overset{⇀}{k})$ yields a resolution-enhanced image.

3. Algorithm

3.1 Retrieval of patterned-excitation components

The patterned-excitation components $F_{0 - 18} (\overset{⇀}{k})$ are retrieved from 19 modulated images as follows.

[\begin{matrix} D_{0} (\overset{⇀}{k}) \\ D_{1} (\overset{⇀}{k}) \\ \begin{matrix} ⋮ \\ D_{q} (\overset{⇀}{k}) \\ ⋮ \end{matrix} \\ D_{18} (\overset{⇀}{k}) \end{matrix}] = [\begin{matrix} e^{i Φ_{0, 0 - 18}} \\ e^{i Φ_{1, 0 - 18}} \\ \begin{matrix} ⋮ \\ e^{i Φ_{q, 0 - 18}} \\ ⋮ \end{matrix} \\ e^{i Φ_{18, 0 - 18}} \end{matrix}] [\begin{matrix} F_{0} (\overset{⇀}{k}) \\ F_{1} (\overset{⇀}{k}) \\ \begin{matrix} ⋮ \\ F_{l} (\overset{⇀}{k}) \\ ⋮ \end{matrix} \\ F_{18} (\overset{⇀}{k}) \end{matrix}] = A F .

with phase-shift indices q = 0 – 18, phase-shift matrix A, and patterned-excitation matrix F.

3.2 Determining the initial phase of a hexagonal pattern

The initial phase of the hexagonal pattern is critical for $F_{0 - 18} (\overset{⇀}{k})$ retrieval. We determine the initial phase of a hexagonal pattern by extending the one-dimensional calculations developed for a linearly structured light pattern [30] to two-dimensional calculations. Briefly, we guess an initial phase V₀(θ _0,x, θ _0,y) and then calculate the phases V_{q = 1-18}(θ _q,x, θ _q,y) of 18 shifted hexagonal patterns from differences Δθ _q,x = θ _q,x - θ _0,x and Δθ _q,y = θ _q,y - θ _0,y. The summation of the cross-correlations of $F_{0} (\overset{⇀}{k})$ and shifted $F_{0 - 18} (\overset{⇀}{k})$ are

S u m_V_{0} = \sum_{l = 0 - 18} | F_{0} (\overset{⇀}{k}) ⋆ F_{l} (\overset{⇀}{k} + {\overset{⇀}{K}}_{l}) | .

with shift vectors

{\overset{⇀}{K}}_{0 - 18}

defined above. We calculate the summations on varying θ _0,x and θ _0,y with step 0.05π within 0 – 2π, respectively. The maximal summation is obtained at the optimal initial phase. To verify the optimization procedure, Fig. 2 shows (a) a simulated resolution target, (b) 2D correlation map of Sum_V₀, and a comparison of the images reconstructed with (c) V₀(1.68π, 0.58π) and (d) V₀(1π, 1.75π). The two initial phases are at the positions of the maximal summation and a local maximal summation in Fig. 2(b) and are indicated as red and blue cursors, respectively. The image in Fig. 2(c) clearly reveals improved resolution and image contrast relative to that in Fig. 2(d). Because of a poor initial phase, it is also noted that the line structures in Fig. 2(d) are distorted more severely.

Fig. 2 (a) A simulated resolution target, (b) 2D correlation map of Sum_V₀, and images reconstructed with (c) optimized V₀(1.68π, 0.58π) and (d) non-optimized V₀(1π, 1.75π), indicated as red and blue cursors in (b).

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4. Experiments

4.1 SOIM system

Figure 3 shows a schematic of an SLM-based SOIM system, modified from a SIM fluorescence system [31]. Briefly, the mainframe is an upright microscope (Zeiss, Axio Scope, A1) with a diode laser (B&W, BWN-532) operated at 532 nm. The incident laser beam is expanded and then diffracted with a phase-only SLM (Hamamatsu, X10468–04) into beams of orders ± 1 at orientations 0° and ± 60.3° simultaneously. These three-set diffraction beams are collected and focused with five lenses onto the BFP of a water-immersion objective (Zeiss, Plan Apochromat VC 63x, NA = 1.2) and then intersect at the sample plane to form a hexagonal illumination pattern. To alter the polarization states of all beams at the BFP, a λ/2-wave plate (Thorlabs, WPMH05M-532) or λ/4 wave-plate (Thorlabs, AQWP05M-600) is placed after the SLM. The beams at the BFP are parallel and their polarization states shown in Fig. 3 are s-linearly polarized. The scattered light from the sample is collected and then detected with an electron-multiplying CCD camera (Andor, DU-885) with 14-bit digitization. All imaging processes are conducted with programs (LabVIEW).

Fig. 3 Schematic of an SLM-based SOIM microscope; SLM spatial light modulator, L lenses, M mirror, HWP/QWP half/quarter wave plate, BS(50/50) beam splitter of 50% transmission and 50% reflection, BFP back focal plane; length in mm. The polarization states of beams are for the Uni-s case; beams at 0° and ± 60° are indicated with red and blue lines, respectively.

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4.2 SLM pattern design

The SLM serves as a two-level phase grating; the white pixels represent a phase modulation of the liquid crystal at an input grey level whereas the black pixels represent no phase modulation. We designed a hexagonal SLM pattern to generate three-set diffraction beams at 0° and ± 60.3° to form a hexagonal pattern at the sample plane. Figure 4 shows (a) portions of the designed SLM pattern and (b) the Fourier spectrum of the whole SLM pattern. This SLM pattern has periods of length 14 and 13.9 pixels at 0° and ± 60.3° to produce a hexagonal pattern of nearly identical period ratios − 1:0.993:0.993 − in the three directions. The Fourier spectrum in Fig. 4(b) confirms the nearly identical intensities of these diffraction beams.

Fig. 4 (a) Portions of the designed SLM pattern with unit cell 8 x 14 pixels and (b) Fourier spectrum of the entire SLM pattern.

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4.3 Sample preparation

For experimental verification, we used samples of five types. A fluorescent film was prepared on spin-coating diluted colloids onto a coverslip; the colloid solution was a mixture of Rhodamine 6G (R6G, 10⁻² M) dye and PVA (5%) with volume concentration ratio 1:1. The USAF 1951 positive (Edmund Optics, 38-257) and negative (Edmund Optics, 55-622) resolution targets were immersed in deionized water and covered with a coverslip for imaging. To prepare gold nanoparticles (100 nm, BBI, EM.GC100, n ~0.54) on a coverslip, the glass surface was modified with ATPS (SIGMA, A3648-100ML) to attract negatively charged gold nanoparticles in deionized water. The sample of polystyrene beads (100 nm, Polysciences, 00876, n ~1.60) was prepared on dropping the bead solution onto a coverslip. After the solution dried, beads were immersed in deionized water or index-matching oil (Cargille Labs, 16242, n ~1.52) for imaging; the use of an oil medium enhanced the bead signals and minimized light reflected from the coverslip. The cell sample was prepared by incubating cells in a DMEM solution on a coverslip for 24 h, washing the cells with PBS buffer, and then fixing them with paraformaldehyde (4%) before immersion in water for imaging.

5. Results and discussion

5.1 Optimization of a hexagonal SLM pattern at 19 phases

It is essential to shift the designed SLM pattern for robust imaging and to optimize the shifts at 19 phases to reconstruct a resolution-enhanced image correctly. In fluorescent SIM, the linear SLM pattern is typically shifted in the pattern direction nearly equally on the 0 – 2π interval [8, 32]. This equal-shift concept is easy to follow and has been adopted to shift a two-angle orthogonal SLM pattern in previous SOIM work [20, 22], but it becomes challenging to shift a hexagonal SLM pattern because multiple directions are involved in each shift and because the phase correlations among the shifts are undefined. We designed several sets of pattern shifts according to this concept but found most of them unable to yield the reconstructed images correctly.

To obtain a set of optimal shifts, we calculated, as alternative approach, the condition number, cond(A) = ||A||⋅||A⁻¹||. According to Eq. (14), cond(A) predicts the errors of the retrieved $F_{0 - 18} (\overset{⇀}{k})$ ; a small cond(A) thus serves as an effective indicator. As the calculations of cond(A) for all plausible shifts are tedious, for simplicity we divided the hexagonal SLM pattern into four regions in terms of its symmetry, randomly set shifts of six types with their centers within one region, and then imaged the shifts to other regions for calculations. The yellow color indicated in Fig. 4(a) is the region in which we placed the centers of shifts of six types. We obtained the minimal cond(A) = 3.4 for cases of a non-zero determinant. Figure 5(a) shows the shifts – SLM-1 of the minimal cond(A); numbers 0 – 18 indicate the center positions of the original and shifted SLM patterns. The shifts – SLM-2, designed on following the equal-shift concept, are also shown in Fig. 5(a) for comparison. The use of SLM-2 yields a reconstructed image of the best quality among sets that were designed based on the equal-shift concept. As seen, SLM-2 fits the equal-shift concept better, but its cond(A) = 604.6 is much greater than that of SLM-1.

Fig. 5 (a) Comparison of pattern shifts in two sets − SLM-1 and SLM-2, obtained from the calculations of condition number and according to the equal-shift concept, respectively, (b) normalized magnitude maps |A⁻¹| of SLM-1 and SLM-2; the l^th row of |A⁻¹| represents the coefficients of spectra D_0-18 to retrieve the l^th pattern-excitation component F_l (l = 0 – 18), and (c) wide-field image of 100-nm gold nanoparticles and their SOIM images reconstructed with SLM-1 and SLM-2.

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To understand further how SLM-1 and SLM-2 influence the reconstructed images, Fig. 5(b) shows their normalized magnitude maps |A⁻¹|. In each map, each pattern-excitation component F_l is retrieved from the modulated spectra D_0-18 with weighting factors on the l^th row of |A⁻¹|; $\overset{⇀}{k}$ is omitted in the figure for simplicity. The |A⁻¹| of SLM-1 is uniform. This result indicates that F_0-18 are retrieved from equivalently weighted D_0-18. In contrast, the |A⁻¹| of SLM-2 reveals over-weighted F_1-2, F_7-10, and F_13-18, especially F_13-14 that are amplified greatly by D_1–3, D_7–8, D_11–12, and D_16–18. An over-weighting of certain components could deteriorate the reconstructed image in resolution improvement and contrast as the noise tolerance on imaging is decreased by the amplified noises of highly weighted spectra. To confirm further that condition, Fig. 5(c) shows a wide-field image of 100-nm gold nanoparticles and the reconstructed SOIM images with SLM-1 and SLM-2. The use of SLM-1 yields an SOIM image with isotropically improved resolution, whereas that of SLM-2 causes gold nanoparticles to become deteriorated or even indiscernible in the SOIM image. This result demonstrates that the use of cond(A) is a simple and direct method to optimize the shifts of a hexagonal SLM pattern; the equal-shift concept is unsatisfactory.

5.2 Polarization dependence of a hexagonal pattern

To show the polarization dependence of a hexagonal pattern, we calculated $I (\overset{⇀}{r})$ and $I (\overset{⇀}{k})$ from the total electric fields in Eqs. (4)–7), with interaction angle 2β = 88° (for NA ~0.923) determined experimentally for each set of beams and θ − 0° and ± 60.3°. Figure 6 shows the polarization vectors of the interaction beams at the BFP of the objective and calculated $I (\overset{⇀}{r})$ and | $I (\overset{⇀}{k})$ | at the sample plane for (a) Axi-s, (b) Uni-s, (c) Uni-p, and (d) Uni-c, respectively, as well as (e) the horizontal and vertical profiles of $I (\overset{⇀}{r})$ along the blue lines in (a) − (d). The maximal $I (\overset{⇀}{r})$ values of Uni- cases are identical and normalized; that of Axi-s is smaller and scaled accordingly. As expected, the structure and contrast of $I (\overset{⇀}{r})$ vary with the beam polarization. The | $I (\overset{⇀}{k})$ | spectra further reveal the effect of polarization on the 18 components of large frequencies that are associated with resolution improvement.

Fig. 6 Polarization vectors of interfering beams at the BFP of the objective and normalized $I (\overset{⇀}{r})$ and | $I (\overset{⇀}{k})$ | of resultant hexagonal patterns at the sample plane calculated for (a) Axi-s, (b) Uni-s, (c) Uni-p, and (d) Uni-c cases, and (e) horizontal and vertical profiles of $I (\overset{⇀}{r})$ indicated with dash blue lines in (a)-(d). In the calculations, the interaction angle 2β = 88° for each set of diffraction beams at the sample plane was determined experimentally.

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Among the four cases, the Axi-s pattern has superior imaging: its $I (\overset{⇀}{r})$ is uniform with 100% visibility; the visibility, defined as (I_max – I_min) / (I_max + I_min), is calculated from the horizontal profile in Fig. 6(e). The | $I (\overset{⇀}{k})$ | with identical intensities at the 18 large-frequency components further confirm the pattern isotropy. In a comparison of Uni- cases, the $I (\overset{⇀}{r})$ of Uni-s and Uni-p show 100% visibility but poor isotropy, especially in the direction of the strongest modulations. Although the visibility 85% of Uni-c is smaller, its isotropy is superior. We further quantify the anisotropy from the full width at half maximum (FWHM) of $I (\overset{⇀}{r})$ shown in Fig. 6(e). The horizontal and vertical FWHMs are 198 and 279 nm in Uni-s, 278 and 198 nm in Uni-p, and 230 and 230 nm in Uni-c; those values of Uni-s and Uni-p are about 14 – 20% different from that of Uni-c. The | $I (\overset{⇀}{k})$ | confirms the anisotropy of Uni-s and Uni-p at θ = 0° at which the signals are stronger in Uni-s but weaker in Uni-p, as well as the isotropy of Uni-c that shows identical intensities at θ = 0° and ± 60.3°. As a result, the use of Uni-c achieves our purpose – to produce an isotropic illumination without complicating the system.

For experimental verification, Fig. 7 shows the (a) light-scattering images of a negative USAF 1951 resolution target at Group 9, Elements 2 and 3, (b) fluorescence images of hexagonal patterns for Uni-s, Uni-p, and Uni-c cases, and (c) horizontal and vertical profiles of the patterns indicated with dashed yellow lines in (b); the insets in (a) are the enlarged pattern structures reflected from the smooth surface of the target. As seen, the hexagonal patterns in the light-scattering images are identical to the simulated Uni-c pattern shown in Fig. 6(d); no polarization dependence is observed. According to simulations, the polarization effect is negligible at beam interaction angles < 30°. No observation is thus attributed to the small interaction angles, < 1°, of the reflected beams before the CCD detector. In contrast, a polarization dependence is observed in the fluorescence images of Fig. 7(b). The pattern structures agree well with simulations shown in Figs. 6(a)-6(d), respectively. In Fig. 7(c), the periods 1200 and 700 nm of the strong horizontal and vertical modulations in the fluorescence images also agree well with simulations 1152 and 665 nm, respectively. It should be noted that the periods of the fluorescence and light-scattering traces are different by about 9%. This result is mainly due to the difference in wavelength between the emission of R6G dye and the excitation laser; the former is collected in a region of 555 − 585 nm whereas the laser is operated at 532 nm.

Fig. 7 (a) Light-scattering images of a negative USAF 1951 resolution target at Group 9 and Elements 2 and 3, (b) fluorescence images of hexagonal patterns (Uni-s, -p, and -c), and (c) horizontal and vertical profiles of patterns indicated with dashed yellow lines in (b); the insets in (a) are enlarged structures in the red region.

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5.3 Resolution verification and polarization effect

SOIM images of various samples − gold nanoparticles and polystyrenes beads (both 100 nm), and label-free fixed HeLa cells are presented to show the resolution improvement and the effect of polarization. Figure 8 shows the light-scattering WF and SOIM (Uni-s, -p, and -c) images, the enlarged images of a single nanoparticle within a yellow box indicated in the images, and the horizontal and vertical profiles of the single nanoparticle indicated with yellow lines for (a) gold nanoparticles and (b) polystyrene beads (both 100 nm). The gold nanoparticles were immersed in water (n = 1.33) for imaging but the beads were immersed in an index-matching oil (n = 1.52) to enhance the image contrast because of their weak scattering ability. It should be noted that the image of the beads immersed in water is black and is influenced greatly by the strong reflections from the coverslip. As expected, both nanoparticle images confirm the resolution improvement of SOIM. We observed also the effect of beam polarization with SOIM. As seen in the second row of Fig. 8(a), the enlarged gold nanoparticle appears round in Uni-c but becomes oblate in Uni-s and prolate in Uni-p. To further quantify the polarization effect on resolution improvement, we measured the FWHM of a single nanoparticle with the Gaussian fits of the horizontal and vertical profiles. On averaging all nanoparticles in Fig. 8(a), the horizontal and vertical FWHMs of a single gold nanoparticle are 303 ± 10 and 300 ± 13 nm in the WF image, 149 ± 4 and 139 ± 6 nm in Uni-s, 142 ± 10 and 152 ± 6 nm in Uni-p, and 144 ± 6 and 145 ± 4 nm in Uni-c. The FWHM result confirms the improved imaging isotropy with Uni-c and the anisotropy with Uni-s and Uni-p. The improvement factors of WF/SOIM (Uni-s, -p, and -c) in the horizontal/vertical directions are 2.03/2.16, 2.13/1.97, and 2.10/2.07, respectively, agreeing well with an expectation of 2.0. The beads in Fig. 8(b) vary greatly in size and intensity, so we chose ten strongly scattering beads for averaging. The averaged horizontal and vertical FWHM are 296 ± 4 and 307 ± 4 nm in the WF image, 191 ± 6 and 176 ± 6 nm in Uni-s, 179 ± 6 and 189 ± 5 nm in Uni-p, and 187 ± 6 and 182 ± 4 nm in Uni-c, with improvement factors 1.55/1.74, 1.65/1.62, and 1.58/1.69, respectively. The smaller difference in FWHM with Uni-c agrees with that for the gold nanoparticles. The smaller improvement is due mainly to the difference of refractive index of water and oil; a size variation and a weak scattering ability are also plausible factors.

Fig. 8 Light-scattering WF and SOIM (Uni-s, -p, and -c) images, enlarged images of a single nanoparticle within a yellow box indicated in the images, and horizontal and vertical profiles of a single nanoparticle indicated with yellow lines for (a) gold nanoparticles and (b) polystyrene beads (both 100 nm). The gold nanoparticles were immersed in water (n = 1.33) but the beads were immersed in index-matching oil (n = 1.52) to enhance the image contrast.

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To understand the effect of polarization on resolving neighboring particles, Fig. 9 shows (a) WF and SOIM (Uni-s, -p, and -c) images of gold nanoparticles (100 nm) in clusters, (b) four sets of enlarged images numbered 1 − 4 in (a), and (c) profiles indicated with dashed lines in (b). As expected, SOIM images reveal an improved resolution but at slightly different orientations because of the effect of polarization. The two nanoparticles neighboring vertically in the first set are resolved better in Uni-s, indicated with red arrows in Figs. 9(b) and 9(c). Those nearby horizontally in the second set are separated more in Uni-p, indicated with green arrows in the figures. This result agrees with an expectation of the hexagonal patterns shown in Fig. 6. To quantify further the resolving ability of SOIM, Gaussian fits of two curves for the best Uni cases are shown in dotted lines in Fig. 9(c). The separation of the two nanoparticles in the first set in Uni-s is 245 nm, that in the second set in Uni-p is 238 nm, but they remain unresolved. The two nanoparticles in the third and fourth sets are resolved well at separations 280 and 302 nm in Uni-c. The values 280 and 302 nm are about half the FWHM 507 and 625 nm of the two unresolved nanoparticles in the WF image, confirming the maximal enhanced-resolution gain with factor two. However, it should be pointed out that the values 280 and 302 nm are comparable with the FWHM 303 nm of a single gold nanoparticle in the WF image and twice those of FWHM 139 − 152 nm of a single gold nanoparticle in the SOIM images. These results arise because the resolution of SOIM is limited by the CTF region that is about half the OTF region; the image of a single nanoparticle is squared to produce a smaller bandwidth [27].

Fig. 9 (a) WF and SOIM (Uni-s, -p, -c) images of gold nanoparticles (100 nm) in clusters, (b) enlarged images of numbers 1 − 4 in (a), and (c) profiles indicated with dash lines in (b); the arrows indicate the best-resolved nanoparticles among Uni cases and the dotted lines in (c) are the Gaussian fits of two curves.

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To demonstrate the biological applications of SOIM, Fig. 10(a) shows the WF and SOIM (Uni-c) images of cellular structures near the edge of a label-free fixed HeLa cell immersed in water. The cellular structures are black in both WF and SOIM images because of their weak scattering ability, but the SOIM image still reveals improved resolution. Figure 10(b) shows the enlarged images and profiles indicated with lines 1 and 2 in Fig. 10(a) to reveal clearly the improvement. In the first set, the microtubules in the SOIM image are resolved with a separation 300 nm; this value is comparable with the separations 280 − 302 nm of two gold nanoparticles. In the second set, the FWHM of a single microtubule is determined to be 150 nm; four microtubules are resolved in the SOIM image. The FWHM agrees well with that of a single gold nanoparticle. Figure 10(c) shows the WF and SOIM (Uni-c) images of a label-free fixed HeLa cell immersed in an index-matching oil (n = 1.52) to improve contrast as for beads. The cell contrast is enhanced as expected, but the cellular structures in the SOIM image are influenced greatly by grid patterns that prevent individual microtubules from being identified unambiguously. The grid patterns are associated with the hexagonal illumination pattern; its occurrence is likely due to the oil medium that enhances a uniform background from the surface of the coverslip and the paraformaldehyde film that is used to fix cells.

Fig. 10 (a) WF and SOIM (Uni-c) images of cellular structures near the edge of a label-free fixed HeLa cell immersed in water, (b) enlarged images in the yellow regions of (a) and profiles indicated as numbers 1 and 2 in the images, and (c) WF and SOIM (Uni-c) images of cellular structures immersed in an index-matching oil (n = 1.52).

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6. Conclusion

We have developed high-resolution isotropic SOIM imaging with an SLM-based SIM fluorescence system. A hexagonal SLM pattern and its optimal shifts at 19 phases are presented to achieve isotopically lateral resolution. The use of cond(A) is also demonstrated to be a direct method to optimize the shifts of a hexagonal SLM pattern; the equal-shift concept typically adopted in linear SIM is unsatisfactory. We have derived theoretical formulations of image reconstruction from a vectorial analysis of the SOIM imaging. This complicated analysis reveals the polarization dependence of illumination patterns and its influence on resolution; both are crucial for high-resolution SOIM imaging with the use of a large-NA objective. Experimental verifications are presented with the SOIM images of gold nanoparticles and polystyrene beads (both 100 nm). The maximal enhanced-resolution gain = 2.0 is obtained. The resolving power for neighboring gold nanoparticles in Uni-s, Uni-p, and Uni-c cases further reveals the effect of beam polarization on resolution improvement and confirms Uni-c is superior in resolution isotropy. For prospective biological applications, SOIM images of label-free fixed HeLa cells are presented with improved contrast and resolution. This work allows one to perform dual-mode high-resolution imaging − fluorescence and light-scattering, on the same system and is expected to broaden the SOIM applications. Moreover, although the achievable resolution of SOIM is less because of the limit of the CTF, the use of an illumination beam at a shorter wavelength can improve the SOIM resolution further and is simple to implement.

Appendix

Coefficients ${\overset{⇀}{C}}_{0 - 18} = (C_{0 - 18, x}, C_{0 - 18, y}, C_{0 - 18, z})$ of the patterned-excitation components are obtained from the interactions of the diffraction beams in three sets. There are thirty-six combinations − six auto-correlations and thirty cross correlations for each case. Table 1, Table 2, and Table-3 list the coefficients of x, y, and z components for Axi-s, Uni-s, Uni-p, and Uni-c; those of Uni-c are combined from the coefficients of Uni-s and Uni-p, as mentioned in section 2.2. As seen, the coefficients of Axi-s are constants, independent of beam interaction angles and having no z components. Coefficients ${\overset{⇀}{C}}_{0 - 18} = (C_{0 - 18, x}, C_{0 - 18, y}, C_{0 - 18, z})$ of Uni-s, Uni-p, and Uni-c at β ∼ 0 become (0, 1, 0), (1, 0, 0), and (0.5, 0.5, 0), respectively; the values of Uni-s and Uni-p are the same as those derived from scalar fields for the use of an objective of small NA reported previously.

Table 1. x-Components of Coefficients of Patterned-Excitation Components

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Table 2. y-Components of Coefficients of Patterned-Excitation Components

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Table 3. z-Components of Coefficients of Patterned-Excitation Components

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Funding

Ministry of Science and Technology of Taiwan (MOST) (MOST 103-2113-M-213-006, MOST 104-2113-M-213-005); National Synchrotron Radiation Research Center (NSRRC).

Acknowledgments

We thank Mr. Chia-Chun Hsieh for preparing the cell samples.

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Coef.	Axi-s	Uni-s	Uni-p	Uni-c
C_0,x-1	0	0	$\cos^{2} β$	$\frac{\cos^{2} β}{2}$
C_0,x-2,3	$\frac{3}{4}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{{(\cos β + 3)}^{2}}{16}$	$\frac{\cos^{2} β + 3}{8}$
C_1-2,x	0	0	$\cos^{2} β$	$\frac{\cos^{2} β}{2}$
C_3-6,x	$\frac{3}{4}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{{(\cos β + 3)}^{2}}{16}$	$\frac{\cos^{2} β + 3}{8}$
C_7-14,x	0	0	$\frac{\cos^{2} β + 3 \cos β}{4}$	$\frac{\cos^{2} β + 3 \cos β}{8}$
C_15-18,x	$- \frac{3}{4}$	$- \frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{{(\cos β + 3)}^{2}}{16}$	$\frac{- \cos^{2} β + 6 \cos β + 3}{16}$

Coef.	Axi-s	Uni-s	Uni-p	Uni-c
C_0,y-1	1	1	0	$\frac{1}{2}$
C_0,y-2,3	$\frac{1}{4}$	$\frac{{(3 \cos β + 1)}^{2}}{16}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{3 \cos^{2} β + 1}{8}$
C_1-2,y	1	1	0	$\frac{1}{2}$
C_3-6,y	$\frac{1}{4}$	$\frac{{(3 \cos β + 1)}^{2}}{16}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{3 \cos^{2} β + 1}{8}$
C_7-14,y	$\frac{1}{2}$	$\frac{3 \cos β + 1}{4}$	0	$\frac{3 \cos β + 1}{8}$
C_15-18,y	$\frac{1}{4}$	$\frac{{(3 \cos β + 1)}^{2}}{16}$	$- \frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{3 \cos^{2} β + 6 \cos β - 1}{16}$

Coef.	Axi-s	Uni-s	Uni-p	Uni-c
C_0,z-1	0	0	$\sin^{2} β$	$\frac{\sin^{2} β}{2}$
C_0,z-2,3		$\frac{3 \sin^{2} β}{4}$	$\frac{\sin^{2} β}{4}$	$\frac{\sin^{2} β}{2}$
C_1-2,z		0	$- \sin^{2} β$	$- \frac{\sin^{2} β}{2}$
C_3-6,z		$- \frac{3 \sin^{2} β}{4}$	$- \frac{\sin^{2} β}{4}$	$- \frac{\sin^{2} β}{2}$
C_7-8,z		0	$- \frac{\sin^{2} β}{2}$	$- \frac{\sin^{2} β}{4}$
C_9-10,z			$\frac{\sin^{2} β}{2}$	$\frac{\sin^{2} β}{4}$
C_11-12,z			$- \frac{\sin^{2} β}{2}$	$- \frac{\sin^{2} β}{4}$
C_13-14,z			$\frac{\sin^{2} β}{2}$	$\frac{\sin^{2} β}{4}$
C_15-16,z		$\frac{3 \sin^{2} β}{4}$	$- \frac{\sin^{2} β}{4}$	$\frac{\sin^{2} β}{4}$
C_17-18,z		$- \frac{3 \sin^{2} β}{4}$	$\frac{\sin^{2} β}{4}$	$- \frac{\sin^{2} β}{4}$

Coef.	Axi-s	Uni-s	Uni-p	Uni-c
C_0,x-1	0	0	$\cos^{2} β$	$\frac{\cos^{2} β}{2}$
C_0,x-2,3	$\frac{3}{4}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{{(\cos β + 3)}^{2}}{16}$	$\frac{\cos^{2} β + 3}{8}$
C_1-2,x	0	0	$\cos^{2} β$	$\frac{\cos^{2} β}{2}$
C_3-6,x	$\frac{3}{4}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{{(\cos β + 3)}^{2}}{16}$	$\frac{\cos^{2} β + 3}{8}$
C_7-14,x	0	0	$\frac{\cos^{2} β + 3 \cos β}{4}$	$\frac{\cos^{2} β + 3 \cos β}{8}$
C_15-18,x	$- \frac{3}{4}$	$- \frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{{(\cos β + 3)}^{2}}{16}$	$\frac{- \cos^{2} β + 6 \cos β + 3}{16}$

Coef.	Axi-s	Uni-s	Uni-p	Uni-c
C_0,y-1	1	1	0	$\frac{1}{2}$
C_0,y-2,3	$\frac{1}{4}$	$\frac{{(3 \cos β + 1)}^{2}}{16}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{3 \cos^{2} β + 1}{8}$
C_1-2,y	1	1	0	$\frac{1}{2}$
C_3-6,y	$\frac{1}{4}$	$\frac{{(3 \cos β + 1)}^{2}}{16}$	$\frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{3 \cos^{2} β + 1}{8}$
C_7-14,y	$\frac{1}{2}$	$\frac{3 \cos β + 1}{4}$	0	$\frac{3 \cos β + 1}{8}$
C_15-18,y	$\frac{1}{4}$	$\frac{{(3 \cos β + 1)}^{2}}{16}$	$- \frac{3 {(\cos β - 1)}^{2}}{16}$	$\frac{3 \cos^{2} β + 6 \cos β - 1}{16}$

High-resolution light-scattering imaging with two-dimensional hexagonal illumination patterns: system implementation and image reconstruction formulations

Abstract

1. Introduction

2. Theory

2.1 Hexagonal pattern of six-beam interferences

2.2 SOIM imaging with hexagonal illumination patterns

2.3 Image reconstruction

3. Algorithm

3.1 Retrieval of patterned-excitation components

3.2 Determining the initial phase of a hexagonal pattern

4. Experiments

4.1 SOIM system

4.2 SLM pattern design

4.3 Sample preparation

5. Results and discussion

5.1 Optimization of a hexagonal SLM pattern at 19 phases

5.2 Polarization dependence of a hexagonal pattern

5.3 Resolution verification and polarization effect

6. Conclusion

Appendix

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Tables (3)

Equations (15)

Optics Express