Three-beam interference with circular polarization for structured illumination microscopy

Hsiao-Chih Huang; Bo-Jui Chang; Li-Jun Chou; Su-Yu Chiang

doi:10.1364/OE.21.023963

1. Introduction

Structured illumination microscopy (SIM) is a wide-field technique that doubles the spatial resolution of a fluorescent image beyond the classical diffraction limit [1–7]. The use of a structured pattern to illuminate samples enables undetectable information of large spatial frequency to be encoded into a detectable region of small frequency, and then computationally extracted to extend the region of the optical-transfer function (OTF) of a wide-field microscope, so doubling the resolution [8]. Although this improvement is less than that of other super-resolution fluorescence microscopic methods [9–13], SIM with its wide-field characteristics requires no special emitters for fluorescence, and enables imaging a live entire cell labeled with multiple colors [14–17]. To improve further the resolution below 100 nm, SIM is combined with other imaging techniques [18–20] or uses a nonlinear structured illumination pattern [21–23]; the scheme for nonlinear illumination can, in principle, provide unlimited resolution. Furthermore, SIM-based contrast imaging methods, developed recently, allow one to observe metal nanoparticles and features without labels inside native samples with improved resolution and image contrast [24–26].

An achievement of the doubled resolution depends greatly on the lateral period and modulation contrast of the structured pattern [8]. The lateral period of that pattern determines mainly the extension of the OTF region and is therefore set near the diffraction limit. To achieve that, the full numerical aperture (NA) of a microscope objective is applied to maximize the interaction angles of the excitation beams; the use of a large NA objective is accordingly crucial. The modulation contrast mainly boosts the components of large spatial frequency to improve the resolution and the image quality. Beams of s-linear polarization (SLP) produce a maximal contrast, and a λ/2-wave plate on a rotating stage or a polarization rotator based on liquid crystal (LC) maintains the modulation contrast at varied orientations − at least three at 0°, 60° and 120° − to achieve isotropically lateral resolution [14–17]. That λ/2-wave plate requires mechanical means and an additional polarizer on a rotating stage to clean the polarization [15]. Although a polarization rotator prevents mechanical motion, it requires two custom-made LC switchable retarders that impede implementation [14].

The original SIM setup is complicated, involving mechanical motions of great precision to shift and to rotate the pattern [8]. Moreover, a small rate of acquisition limits the capability to image live samples. A LC-based spatial light modulator (SLM) has replaced a transmission phase grating [14,15,19,27] to enable a rapid and precise control of the structured pattern in phase and orientation on varying the phases of the LC, which not only prevents mechanical motion but also provides a large rate of imaging to allow study of the dynamics of living cells. 2D- and 3D-SIM images of cellular structures and motions in live cells have been demonstrated [14–17]. To simplify the system further, one can avoid the polarization rotation that maintains the polarization states of beams. One simple approach is to use beams of circular polarization (CP) to generate a structured pattern. We expect the modulation contrast to be independent of the orientation but it becomes weaker; to derive the intensity distribution of the modulation and to understand the influence on image quality requires further investigation.

In this work, we demonstrate the interference of three CP beams to provide a constant modulation contrast independent of its orientation in a 3D-SIM system. This approach requires merely a λ/4-wave plate to alter the interfering beams from linear to circular polarization, without complicated optical components or a mechanical means implemented by previous authors. We derived the resultant intensity distribution of the interference pattern as a function of orientation, and compared the result with those using SLP beams. We evaluated the influence of the decreased modulation contrast on simulated SIM images of 100-nm beads by varying the ratio of signal to noise (SNR). We present experimental results to confirm the simulations. Our 3D-SIM system provides comparable resolution improvement and moderate image contrast, which simplifies the implementation of a 3D-SIM system that requires merely one active component, a SLM.

2. Theory

2.1 Three polarized beams for interference

In a 3D-SIM system, three diffracted beams of orders 0 and ± 1 generate an interference pattern. Their electric fields are described as

{\overset{⇀}{E}}_{j} = {\overset{⇀}{P}}_{j} f_{j} e^{i {\overset{⇀}{k}}_{j} \cdot \overset{⇀}{r}},

with j = 1, 0 and −1, polarization vector

{\overset{⇀}{P}}_{}_{j}

, amplitude

f_{}_{j}

and propagation vector

{\overset{⇀}{k}}_{j}

of beam j ; r(x, y, z) denotes the position within the beam. As the zero-order beam is focused at the center of the back focal plane of the objective and the two first-order beams at opposite edges, we define the incident plane xz at orientation θ = 0°, with the central zero-order beam propagating along the z direction and the two first-order beams propagating in the xz plane to interact respectively with the zero-order beam at angle β from opposite directions in the sample focal plane. Moreover, to generate an orientated interference pattern, the central beam is fixed, but the two first-order beams are rotated with the pattern orientation θ in plane xy. Figure 1 shows (a) side (xz plane) and (b) top (xy plane) views of the propagation vectors, indicated as black arrows, for the three beams propagating from the exit of the microscope objective to the sample focal plane to generate interference patterns at three orientations − 0°, 45° and 90°. The three propagation vectors are accordingly described as

Fig. 1 (a) Side (xz plane) and (b) top (xy plane) views of propagation and polarization vectors of three interfering beams to generate interference patterns in the sample focal plane at θ = 0°, 45° and 90° with interaction angle β = 44°; black arrows represent the propagation vectors, and red arrows and circles represent the polarization states for the fixed linear polarization case; the red circle with a central dot or a cross indicates that the polarization vector is out of the plane in a forward or backward direction.

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{\begin{cases} {\overset{⇀}{k}}_{1} = \cos θ \sin β \hat{x} + \sin θ \sin β \hat{y} + \cos β \hat{z} \\ {\overset{⇀}{k}}_{0} = \hat{z} \\ {\overset{⇀}{k}}_{- 1} = - \cos θ \sin β \hat{x} - \sin θ \sin β \hat{y} + \cos β \hat{z} \end{cases} .

The polarization vector of a beam is perpendicular to its propagation vector, but its direction depends on the polarization state and the orientation of the beam. Two cases for three beams of s-linear polarization relative to each other at orientation θ = 0° are considered, one with and the other without a polarization rotator. The former is called maintained SLP because beams of s-linear polarization are maintained at all pattern orientations. Without a polarization rotator, the linear polarization (LP) states of the beams are not rotated with grating orientation; beams of s-linear polarization occurs only at θ = 0°. Accordingly, we abbreviate this case as fixed LP. The polarization vectors at θ = 0°, 45° and 90° indicated as red arrows and circles for the fixed LP case are shown in Fig. 1. We use arrows to indicate the relative phase and amplitude of the polarization vectors in the plane; the circle with a dot or a cross further indicates that the polarization vector is out of the plane in a forward or backward direction. Accordingly, the polarization vectors of the three beams of s-linear polarization at θ = 0° are in phase on the y axis. At θ = 90°, the two first-order beams are rotated to plane yz and their polarization vectors become out of phase.

To derive the polarization states as a function of orientation, we begin with the use of three elliptically polarized (EP) beams, as a general case for fixed LP and CP. The central zero-order EP beam that is independent of orientation but varies in the xy plane is described as

{\overset{⇀}{P}}_{0} = \frac{1}{\sqrt{2}} (\sqrt{\frac{2}{1 + e^{2}}} \hat{y} - i \sqrt{\frac{2 e^{2}}{1 + e^{2}}} \hat{x}),

with ellipticity e = x/y; x and y correspond to the lengths of the semi-major and semi-minor axes; e = 0 and 1 correspond to fixed LP and circular polarization, respectively. We describe the first-order beam in a general form as

{\overset{⇀}{P}}_{1} = \frac{1}{\sqrt{2}} [\sqrt{\frac{2}{1 + e^{2}}} (x_{r} \hat{x} + y_{r} \hat{y} + z_{r} \hat{z}) + i \sqrt{\frac{2 e^{2}}{1 + e^{2}}} (x_{i} \hat{x} + y_{i} \hat{y} + z_{i} \hat{z})] .

Six unknowns (x_r, y_r, z_r, x_i, y_i, and z_i) require six equations for their evaluation, which we derive based on the normalization and orthogonal properties of the vectors, such as

{\overset{⇀}{P}}_{1} \cdot {\overset{⇀}{k}}_{1} = 0

, and the β dependence of the z components with z_r² + z_i² = sin²β. It is not straightforward to derive the solutions because a cross term results from orthogonality, x_rx_i + y_ry_i + z_rz_i = 0. For simplicity, we derive the z components based on geometric considerations. According to Fig. 1, the two first-order beams are rotated θ clockwise along the z axis to generate a correspondingly orientated interference pattern. The real and imagery parts of the z component thus comprise factors of sinθ and cosθ, respectively. The z component also has a factor of sinβ involving interaction angle β that marks the tilt of the first-order beams from the z axis. Combining both factors, we describe the z component as z_r = -sinθsinβ and z_i = cosθsinβ, and then use them to derive the x and y components from equations. The solution of Eq. (4) is

{\overset{⇀}{P}}_{1} = \frac{1}{\sqrt{2}} {\begin{cases} \sqrt{\frac{2}{1 + e^{2}}} [\cos θ \sin θ (\cos β - 1) \hat{x} + (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y} - \sin θ \sin β \hat{z})] \\ - i \sqrt{\frac{2 e^{2}}{1 + e^{2}}} [(\cos^{2} θ \cos β + \sin^{2} θ) \hat{x} + \cos θ \sin θ (\cos β - 1) \hat{y} - \cos θ \sin β \hat{z})] \end{cases}} .

With a similar procedure we derive

{\overset{⇀}{P}}_{- 1}

except that the interaction angle is -β:

{\overset{⇀}{P}}_{- 1} = \frac{1}{\sqrt{2}} {\begin{cases} \sqrt{\frac{2}{1 + e^{2}}} [\cos θ \sin θ (\cos β - 1) \hat{x} + (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y} + \sin θ \sin β \hat{z})] \\ - i \sqrt{\frac{2 e^{2}}{1 + e^{2}}} [(\cos^{2} θ \cos β + \sin^{2} θ) \hat{x} + \cos θ \sin θ (\cos β - 1) \hat{y} + \cos θ \sin β \hat{z})] \end{cases}} .

According to Eqs. (3)-(6), the polarization states for the fixed LP case are

{\begin{cases} {\overset{⇀}{P}}_{1} = \cos θ \sin θ (\cos β - 1) \hat{x} + (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y} - \sin θ \sin β \hat{z} \\ {\overset{⇀}{P}}_{0} = \hat{y} \\ {\overset{⇀}{P}}_{- 1} = \cos θ \sin θ (\cos β - 1) \hat{x} + (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y} + \sin θ \sin β \hat{z} \end{cases} .

Calculations of the polarization states at θ = 45° and 90° are shown in Fig. 1. The interaction angle β for calculations, about 44°, is calculated from the lateral period 293 nm of the measured 0° interference pattern. As expected, the maximal contrast occurs at θ = 0° and decreases to a minimum at θ = 90° due to the polarization vectors of the two-first order beams becoming out of phase. Analogously, with e = 1, we obtain the polarization states for three CP beams as

{\begin{cases} {\overset{⇀}{P}}_{1} = \frac{1}{\sqrt{2}} [\cos θ \sin θ (\cos β - 1) \hat{x} + (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y} - \sin θ \sin β \hat{z})] \\ - \frac{i}{\sqrt{2}} [(\cos^{2} θ \cos β + \sin^{2} θ) \hat{x} + \cos θ \sin θ (\cos β - 1) \hat{y} - \cos θ \sin β \hat{z})] \\ {\overset{⇀}{P}}_{0} = \frac{1}{\sqrt{2}} (\hat{y} - i \hat{x}) \\ {\overset{⇀}{P}}_{- 1} = \frac{1}{\sqrt{2}} [\cos θ \sin θ (\cos β - 1) \hat{x} + (\sin^{2} θ \cos β + \cos^{2} θ) \hat{y} + \sin θ \sin β \hat{z})] \\ - \frac{i}{\sqrt{2}} [(\cos^{2} θ \cos β + \sin^{2} θ) \hat{x} + \cos θ \sin θ (\cos β - 1) \hat{y} + \cos θ \sin β \hat{z})] \end{cases} .

2.2 Intensity distribution of three-beam interference

The intensity distribution of three-beam interference is described as

I (\overset{⇀}{r}) = ({\overset{⇀}{E}}_{0} + {\overset{⇀}{E}}_{1} + {\overset{⇀}{E}}_{- 1}) {({\overset{⇀}{E}}_{0} + {\overset{⇀}{E}}_{1} + {\overset{⇀}{E}}_{- 1})}^{*} .

Substituting Eqs. (1)-(6) into Eq. (9), we obtain the intensity distribution for the interference of three EP beams as

\begin{array}{l} I {[\overset{⇀}{r} (x, y, z), θ, β]}_{E (e)} = I_{0} + 2 I_{1} \\ + \frac{1}{2} [\begin{array}{l} (\frac{2}{1 + e^{2}}) (\cos^{2} \frac{β}{2} + \cos 2 θ \sin^{2} \frac{β}{2}) \\ + (\frac{2 e^{2}}{1 + e^{2}}) (\cos^{2} \frac{β}{2} - \cos 2 θ \sin^{2} \frac{β}{2}) \end{array}] 4 \sqrt{I_{0} I_{1}} \cos (\begin{array}{l} k x \cos θ \sin β \\ + k y \sin θ \sin β \end{array}) \cos [k z (1 - \cos β)] \\ + \frac{1}{2} [\begin{array}{l} (\frac{2}{1 + e^{2}}) (\cos^{2} β + \cos 2 θ \sin^{2} β) \\ + (\frac{2 e^{2}}{1 + e^{2}}) (\cos^{2} β - \cos 2 θ \sin^{2} β) \end{array}] 2 I_{1} \cos (\begin{array}{l} 2 k x \cos θ \sin β \\ + 2 k y \sin θ \sin β \end{array}), \end{array}

in which intensity I_j = f_j², j = 0, 1 and −1, and I₁ = I_-1. With e = 0, the intensity distribution for the fixed LP case is explicitly described as

\begin{array}{l} I {[r (x, y, z), θ, β]}_{s} = I_{0} + 2 I_{1} \\ + ({cos}^{2} θ + \sin^{2} θ \cos β) 4 \sqrt{I_{0} I_{1}} \cos [k \sin β (x \cos θ + y si n θ)] \cos [k z (1 - \cos β)] \\ + (\cos^{2} θ + \sin^{2} θ \cos 2 β) 2 I_{1} \cos [2 k \sin β (x \cos θ + y \sin θ)], \end{array}

in which

\cos^{2} θ + \sin^{2} θ \cos β

=

\cos^{2} \frac{β}{2} + \cos 2 θ \sin^{2} \frac{β}{2}

. Similarly, with e = 1, the intensity distribution for the CP case is

\begin{array}{l} I {[r (x, y, z), θ, β]}_{c} = I_{0} + 2 I_{1} \\ + (\cos^{2} \frac{β}{2}) 4 \sqrt{I_{0} I_{1}} \cos [k \sin β (x \cos θ + y \sin θ)] \cos [k z (1 - \cos β)] \\ + (\cos^{2} β) 2 I_{1} \cos [2 k \sin β (x \cos θ + y \sin θ)] . \end{array}

According to Eq. (10), the first and second terms correspond to uniform illumination in a wide-field microscope; the third and fourth terms correspond to the interferences of the zero- and first-order beams and the two first-order beams to improve the axial and lateral resolutions, respectively. Although the third term also improves the lateral resolution, the improvement is only one half that of the fourth term and thus becomes unimportant. Figure 2 shows the results of calculations of modulation contrasts for the interferences of (a) the zero- and first-order beams and (b) the two first-order beams in orientations 0°−180° for the cases of fixed LP, CP and maintained SLP. The modulation contrasts of circular polarization are seen to be independent of the pattern orientation. With maintained SLP as a reference, the modulation contrast from the interference of the zero- and first-order beams remains over 85%, but that of the two-first order beams is only about 50%. Regarding the fixed LP case, the modulation contrast varies significantly with the pattern orientation, with a minimum at θ = 90°. Considering an interaction angle 2β = 88° between the two first-order beams, the nearly zero contrast at θ = 90° is rational as the two beams are nearly orthogonal in the sample to make radial polarized interference impossible.

Fig. 2 Modulation contrasts for interferences of the (a) zero- and first-order beams and (b) two first-order beams in orientations 0°−180° for the cases of fixed linear polarization (LP), circular polarization (CP) and maintained SLP; the interaction angle β for the calculations is 44°.

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3. Microscope setup and image reconstruction

3.1 Microscope setup

The setup of our 3D-SIM microscope is similar to that for imaging of light scattering reported previously [23]. Briefly, the system consists of a standard upright microscope (Nikon 80i) with a He-Ne laser (Lasos, LGK7786 P100) operating at 543 nm. Figure 3(a) shows a simple scheme of the optical setup to generate the interference patterns and to detect fluorescence. A phase-only SLM (Hamamatsu, X10468–04) was used to diffract the incident laser beam mainly into zero and ± 1 order beams at a selected orientation. Three lenses collected and focused the three diffracted beams, with the zero-order beam at the center and the two first-order beams near the edges, on the back focal plane (pupil) of a water-immersion objective (Nikon, Plan Apochromat VC, 60x, NA = 1.2) to form an interference pattern on the sample. Because the linear polarization of the laser light must orient along the director axis of the liquid crystal to ensure effective phase modulation, a λ/4-wave plate (CVI Laser, ACWP) was placed after the SLM to alter the three diffracted beams from linear to circular polarization for interference. Fluorescence from the sample was selected with an emission filter (Chroma, HQ600/75) and then detected with an electron-multiplying CCD camera (Andor, DU885). To collect images in a stack at a selected focus step, we controlled the axial position of the sample with a stage driven with a piezoelectric transducer (PI, P-562.3CD). All imaging processes were conducted with programs (LabVIEW).

Fig. 3 (a) Optical setup to generate three diffracted beams of circular polarization for interference in a 3D-SIM system. (b) Portions of 0°, 45°, and 60° SLM patterns and (c) corresponding interference patterns of nearly the same lateral period − 293 nm (0°), 290 nm (45°) and 295 nm (60°); the black pixels indicate no phase retardation and the white pixels indicate a phase retardation 7π/8 of the liquid crystal; the periodicity for each SLM pattern is indicated by coloring one pixel in each unit cell red.

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The interference pattern had its orientation altered or its phase shifted for imaging with a design of SLM patterns. The SLM served 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. According to the nearly linear curve for phase modulation (calibrated by the manufacturer), a π phase shift was set at grey level 120; the 7/8 π phase shift corresponded to level 110. To increase the intensity ratio of the zero- and first-order beams, the white pixels were set at level 110 in our designed SLM patterns. We designed SLM patterns for imaging at orientations 0°, 45°, 90° and 135°, or at orientations 0°, 60°, and 120°. Figure 3(b) shows portions of 0°, 45° and ~60° SLM patterns designed in periods of length 7.0, 5√2 ( = 7.1) and 7.0 pixels and Fig. 3(c) shows corresponding interference patterns with a nearly identical lateral period − 293 nm (0°), 290 nm (45°) and 295 nm (60°); the interference patterns were measured from reflections with a silicon wafer on the sample stage. The SLM patterns are classified into three sets − (0°, 90°), (45°, 135°), and (60°, 120°); those of 90°, 120° and 135° are therefore omitted.

3.2 Image reconstruction

The algorithm for image reconstruction was based on that of Gustafsson et al. [8]. Briefly, the image of objects excited with an interference pattern is described as

D (r) = (S (r) I (r)) \otimes P (r),

with position r, resultant image D(r), object S(r), pattern intensity I(r), point-spread function P(r) and convolution operator ⊗. Accordingly, the corresponding spectrum in frequency space is described with

D (k) = (S (k) \otimes I (k)) O T F (k),

in which D(k), S(k), I(k) and OTF(k) correspond to Fourier transforms of D(r), S(r), I(r) and P(r), respectively. On substitution of the Fourier transform of the pattern intensity in Eq. (10), the spectrum D(k) contains five components − the conventional OTF and four large spatial- frequency components for retrieval. To reconstruct a three-dimensional image at high resolution, we acquired intermediate SI images in stacks at five phases for each pattern orientation, at pattern orientations 0°, 60° and 120° or 0°, 45°, 90° and 135°, and at focus step 100 nm. Accordingly, we retrieved the five components at each pattern orientation and then shifted the four large spatial-frequency components laterally to their correct positions. After fifteen spectra at three pattern orientations or twenty spectra at four pattern orientations were retrieved and shifted, we averaged the conventional OTF spectra and then added the averaged spectrum to the sum of the large spatial-frequency spectra to form a total 3D-SI spectrum. To eliminate the spectral overlap to improve resolution, we performed Wiener deconvolution with the simulated OTF of our 3D-SIM system, OTF_SI(k). The simulation was performed with the experimental parameters, including the excitation wavelength, lateral period of the interference pattern and pixel size of the CCD detector. Using an online program [28], we first obtained the conventional 3D point-spread function (3D-PSF) from a stack of 2D-PSF. We next simulated the 3D-SI spectrum of a point source using the simulated 3D-PSF and then divided the spectrum by the Fourier transform of the point source to obtain OTF_SI(k). The inverse Fourier transform of the Wiener-deconvoluted total 3D-SI spectrum yielded a high-resolution image. We typically chose a Wiener parameter at which the image achieved a doubled lateral resolution but without perceptible image distortion. All images were reconstructed with programs (MATLAB). To present the high-resolution image, we resampled the image two-fold using bicubic interpolation (ImageJ).

4. Results

4.1 Interference patterns

The modulation contrasts for the fixed LP and CP cases were calculated according to Eqs. (11) and (12), respectively, with the system parameters I₀/I₁ = 0.75 and β = 44°. The modulation contrast for maintained SLP is identical to that of fixed LP at θ = 0°. Figure 4(a) shows the lateral modulations at the sample focal plane at θ = 0°, 45°, 60° and 90°, and the corresponding axial modulations projected on the orientated plane for a simple view. The corresponding lateral and axial profiles indicated as yellow lines in Fig. 4(a) are shown in Fig. 4(b); the profiles of CP correspond to those of fixed LP at θ = 45° and are thus omitted.

Fig. 4 Simulations of (a) lateral modulations at the sample focal plane (upper) at θ = 0°, 45°, 60° and 90°, and the corresponding axial modulations (bottom) for fixed linear polarization and circular polarization, and (b) corresponding lateral and axial profiles, indicated in yellow lines in (a). The axial modulation for each orientation is projected on its oriented plane, such as plane x’z at θ = 45° to simplify the view; the parameters for calculations are I₀/I₁ = 0.75 and β = 44°.

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As expected, the maintained SLP corresponding to fixed LP at θ = 0° provides a maximal contrast. For the fixed LP case, the contrast of the lateral and axial modulations varies greatly at the four orientations, and attains a minimum with a greater bandwidth. According to Fig. 2(a), the contrast of the strong lateral modulation at θ = 45°, 60° and 90° relative to θ = 0° decrease about 14%, 21% and 28%, respectively; the decrease of the corresponding weak lateral modulation in Fig. 2(b) is more significant, about 48%, 72% and 96%, respectively. This result indicates that fixed LP provides a highly anisotropic illumination; moreover, the use of four orientations − 0°, 45°, 90° and 135° − to reconstruct a high-resolution image is inappropriate because the component at θ = 90° is too weak to be observed.

In contrast, the use of circular polarization achieves the objective − to avoid rotational components or a complicated polarization rotator, but to produce an identical modulation contrast at varied orientation for imaging. As seen in Fig. 4(a), the modulation contrasts are identical at the four orientations to provide an isotropic illumination. The modulating contrast of CP corresponds to that of fixed LP at θ = 45°, and is therefore greater than those of fixed LP at θ = 60° and 90°. Alternatively, a comparison of fixed LP at θ = 0° and 45° indicates that the modulation contrast from the two-first order beams is only about 50% with circular polarization instead of maintained SLP. This decrease mainly weakens the spatial-frequency components in the lateral direction, and might therefore influence the improvement of lateral resolution and image quality.

4.2 Simulations

To evaluate the influence of modulation contrast on imaging, we simulated the reconstructed SIM component images of 100-nm beads for the fixed LP case at SNR = 10 and 40 as a reference; the results of fixed LP at θ = 0° and 45° correspond to those of circular polarization and maintained SLP, respectively. In the simulation, we generated a raw image stack of 31 layers by adding beads of identical signals but randomly distributed in one layer to a noisy background. The noise in each layer was also randomly generated on a basis pixel by pixel with a mean value about 3, selected based on the experimentally modulated images. Figure 5 shows the reconstructed and normalized SIM component images at the focal plane at θ = (a) 0°, (b) 45°, (c) 60° and (d) 90° with the raw images at SNR about 10. As seen, the image contrast decreases with decreased pattern contrast but the effect is much less at θ = 45° and 60° than that at θ = 90°. The image at θ = 90° reveals indiscernible beads in a large background; this large effect is attributed to a diminutive contrast from the interference of the two first-order beams, as seen in Fig. 4(b). With SNR increased to 40, the image contrasts are improved to become comparable for θ ≤ 60° and are thus not shown. Accordingly, the use of maintained SLP and circular polarization results in merely a small difference in imaging at SNR ≥10.

Fig. 5 Simulations of reconstructed component images of 100-nm beads at the focal plane at θ = (a) 0°, (b) 45°, (c) 60° and (d) 90° for fixed linear polarization with the raw images at the ratio of signal to noise about 10; each 8-bit image is rescaled with the maximal signal to 255 for display.

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We further compared the Wiener-deconvoluted 3D-SIM images of 100-nm beads reconstructed from three orientations − 0°, 60°, and 120°. Because of the severely deteriorated image for fixed LP at θ = 90°, four pattern orientations were not used. Figure 6 shows the lateral (xy) and axial (xz) projections of the reconstructed 3D-SIM images of 100-nm beads with the raw images at SNR about 10 for the cases of (a) fixed LP, (b) circular polarization and (c) maintained SLP. Evidently, the image contrast increases from fixed LP to circular polarization to maintained SLP through an increased modulation contrast. With SNR increased to 40, all image contrasts are improved, as expected, and are thus not shown. To quantify, we calculated the SNR of the lateral and axial projections of 3D-SIM images. The noise is determined from a region 17 by 17 pixels containing no beads; the values of five regions are averaged. For the raw images at SNR about 10 and 40, the SNR of 3D-SIM images from the lateral projections are 13 ± 1 and 27 ± 2 for fixed LP, 16 ± 1 and 34 ± 2 for CP, and 23 ± 2 and 45 ± 2 for maintained SLP, respectively. The corresponding SNR of axial projections are 16 ± 1 and 31 ± 3 for fixed LP, 24 ± 3 and 45 ± 9 for CP, and 31 ± 3 and 56 ± 16 for maintained. Maintained SLP is evidently superior to provide the best image quality. The use of circular polarization provides a satisfactory image quality, about SNR 70−80% SNR. The use of fixed LP is less required with SNR about 50−60%.

Fig. 6 Simulations of lateral and axial projections of reconstructed images of 100-nm beads from three orientations − 0°, 60° and 120° for (a) fixed linear polarization, (b) circular polarization and (c) maintained s-linear polarization with raw images at the ratio of signal to noise about 10; each image is 8-bit with the maximal signal rescaled to 255 for display.

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4.3 Experimental results

To confirm the simulation, we performed measurements of 100-nm fluorescent beads with our 3D-SIM system. We used linear polarization lying on the x axis rather than the y axis because the setup of the SLM for diffractions in our system. We abbreviate this case as fixed LP-x. The result is expected to be the same for that of fixed LP lying on the y axis except a difference 90° in pattern orientation. Figure 7 shows the reconstructed and normalized SIM component images at the focal plane at θ = 0°, 60°, and 120° using beams of (a) fixed LP-x and (b) circular polarization. The raw image stack has 31 layers at z-step 100 nm; the SNR of the raw image is about 250. In Fig. 7(a), the image of fixed PL-x at θ = 0° that reveals fluorescent beads vaguely due to a diminutive contrast is consistent with the simulation of fixed LP at θ = 90° shown in Fig. 5. The image contrasts at θ = 60° and 120° that are nearly the same also confirm that their modulation contrasts are comparable. In Fig. 7(b), the use of circular polarization reveals identical image contrast at the three orientations, confirming that the modulation contrast is independent of the pattern orientation.

Fig. 7 Reconstructed component images of 100-nm fluorescent beads at the focal plane at 0°, 60° and 120° with beams of (a) fixed s-linear polarization on the x axis and (b) circular polarization; the ratio of signal to noise in the measured raw image is about 250.

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To demonstrate the resolution and image contrast with the use of CP, Fig. 8(a) shows the lateral (xy) and axial (xz) projections of the wide-field (WF) and 3D-SIM images of 100-nm fluorescent beads; the corresponding lateral and axial profiles of a single bead are shown in Fig. 8(b), with the solid line for a Gaussian fit. The 3D-SIM image shows improved resolution and image contrast. From the full width at half maximum (FWHM) of a single bead and averaged FWHM of five beads, the lateral and axial resolutions obtained are 280 ± 8 nm and 766 ± 47 nm in the WF image and 127 ± 7 nm and 425 ± 20 nm in the 3D-SIM image. The improving factors are about 2.2 laterally and 1.8 axially.

Fig. 8 (a) Lateral and axial projections of the wide-field (WF) and 3D-SIM images of 100-nm fluorescent beads and (b) corresponding profiles of an individual bead marked in lines; the solid curve corresponds to a Gaussian fit.

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To demonstrate the ability of our 3D-SIM system in biological applications, we acquired a 3D image of Alexa-488-labeled microtubules inside a HeLa cell. The cell was fixed with paraformaldehyde (4%) on the glass slide before immersion in deionized water for imaging. The fluorescence emits at ~520 nm upon excitation at 473 nm. Figure 9 shows the lateral xy view at a selected z position and yz views at three x positions of (a) wide-field and (b) 3D-SIM images of microtubules; the x positions are indicated with dash lines in the xy plane. The wide-field image was obtained from the retrieved conventional OTF spectra. Evidently, the 3D-SIM image reveals the improved resolution and contrast to enable individual microtubules to be well-resolved in a low background. We estimate the resolution from 10 microtubules; the averaged lateral and axial FWHM are about 128 ± 11 and 430 ± 37 nm, respectively.

Fig. 9 Lateral (xy) and yz views of (a) wide-field (WF) and (b) 3D-SIM images of Alexa-488-labeled microtubules inside a HeLa cell; the xy view is at a selected z position; the three dash lines indicate the positions for yz views. The image size is about 16.4 x 16.4 x 5 μm³ in the x, y and z directions.

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5. Conclusion

3D-SIM that improves the image resolution and contrast based on a wide-field scheme offers several advantages, particularly including rapid imaging and a large field of view. To broaden its applications and popularity, the system setup and the algorithm for image reconstruction must be simplified. In previous works, a SLM generated a structured pattern to avoid mechanical motion and to improve the imaging rate. In this work, we generated an interference pattern with three CP beams to provide isotropic illumination. The technique requires merely a λ/4 wave-plate, without mechanical means or a complicated polarization rotator. Based on the derived intensity distribution, we confirmed that the modulation contrast is independent of the orientation and moderate to provide an isotropic image at high resolution with its quality comparable to that using maintained SLP at large SNR. To increase the modulation contrast is feasible. A design of new SLM patterns allows one to vary the intensity ratio of the weak and strong components of the modulation to strengthen the lateral information. Our approach makes the 3D-SIM system simple and cost-effective, to benefit the developments and applications of SI-based systems. Alternatively, for an application to nonlinear SIM to improve resolution further, circular polarization is ineffective because nearly complete modulation contrast is needed; only schemes in which s-linear polarization is maintained for all directions are thus suitable.

Acknowledgments

NSRRC and National Science Council (NSC-99-2113-M-213-005) in Taiwan provided financial support.

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Three-beam interference with circular polarization for structured illumination microscopy

Abstract

1. Introduction

2. Theory

2.1 Three polarized beams for interference

2.2 Intensity distribution of three-beam interference

3. Microscope setup and image reconstruction

3.1 Microscope setup

3.2 Image reconstruction

4. Results

4.1 Interference patterns

4.2 Simulations

4.3 Experimental results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (14)

Optics Express