As shown by Wirth et al.

Fig 1  Idea.

a Phase patterns and point-spread functions (PSFs) for MINFLUX localization in the x-, y-and z-direction, respectively. The phase φ determines the position of the intensity minimum (see Supplementary Fig. 3). b Theoretical localization precision limit, the Cramér-Rao bound, for 2D MINFLUX for sequential localization with the bilobed x and y patterns (left) and 3D MINFLUX with the bilobed x and y patterns plus the 3D donut (right), in a region at the center of the PSF as denoted in a. Simulation parameters: scan range L = 50 nm in the lateral and L_z = 150 nm in the axial direction and an offset (imperfect zero and background) of 0.5% of the maximum intensity when φ = 0. The calculated localization precisions are normalized by $$\sqrt{N}$$, and thus report the localization precision for a single photon. Circles indicate the positions of the minimum during probing. The calculated localization precisions in the center of the field of view are 22.8 nm$$/\sqrt{N}$$ for 2D and 47.5 nm$$/\sqrt{N}$$ for 3D MINFLUX, respectively. c A fast spatially varying phase plate for MINFLUX excitation comprising an electro-optical modulator (EOM), a spatial light modulator (SLM) and a polarizer. The input beam is polarized under 45° and the output is focused onto the single fluorophore by the microscope objective lens. d Illustration of principle. The vertical polarization component experiences a phase shift of $$\phi$$ by the EOM. The 'on'-pixels of the SLM turn the vertical component into the horizontal component and vice versa, whereas the 'off'-pixels leave the state unchanged. The horizontal state transmitted by the polarizer thus has the phase $$\phi$$ of the EOM for the 'on'-pixels and no additional phase for the 'off'-pixels. Scale bars 500 nm (a), 100 nm (b)

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Phase differences of π give rise to symmetric patterns with the minimum on the optical axis (x, y pattern) or in the focus of the objective lens (z pattern), because here the π phase shift leads to destructive interference. Phase differences different from π displace the position of the minimum because the rays must acquire an additional path difference to reach a phase difference of π (Supplementary Fig. 3). Simulations show that for the bilobed PSFs a true zero at the minimum is preserved even for high-NA systems and large scanning, whereas for the 3D donut the contrast stays high over a z range of a few hundred nm (Supplementary Fig. 4).

The phase patterns can be created with a spatial light modulator (SLM), which, in principle, can scan the position of the intensity minimum by changing the phase. However, SLMs that produce continuous phase delays are too slow (< 1 kHz) for the rapid scanning required for MINFLUX, thus a fast scanner like an electro optical deflector would still be required.

Here we overcome this limitation by combining an EOM, an SLM and a polarizer to generate binary phase patterns with variable phase delay, where the phase delay can be changed on the sub-microsecond time scale by the EOM phase (Fig. 1c, d). The input laser beam is polarized under 45° and enters the EOM for which the polarization axis is oriented vertically under 0°. The EOM can thus cause a phase delay between the vertical and the horizontal polarization component of the beam. The binary SLM is oriented in a way that the 'off'-pixels (dark pixels of the imposed phase pattern) do not change the polarization state, whereas the 'on'-pixels (bright pixels) act as a half-wave plate (HWP) oriented at 45°, which rotate the vertical polarization component into the horizontal polarization. This component carries the extra phase imposed by the EOM. A polarizer (e.g., a polarizing beam splitter) transmits only the horizontal component, which has no extra phase for 'off'-pixels of the SLM and the phase imposed by the EOM for 'on'-pixels.

Using Jones matrices, we can calculate in the horizontal/vertical coordinate system that an input beam of $$E_{\text{in}}=\frac{E_0}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right)$$ acquires a phase of $$E_{\text{EOM}}=\frac{E_0}{\sqrt{2}}\left(\begin{array}{c}1\\ e^{i\phi }\end{array}\right)$$ after the EOM. The 'on'-pixels of the SLM result in $$E_{\text{SLM}}=\frac{E_0}{\sqrt{2}}\left(\begin{array}{c}{e}^{i\phi }\\ 1\end{array}\right)$$, whereas the 'off'-pixels do not change the polarization state. After the polarizer the state is $$E_{\text{off}}=\frac{E_0}{\sqrt{2}}\left(\begin{array}{c}1\\ 0\end{array}\right)$$ for the 'off'-pixels and $$E_{\text{on}}=\frac{E_0}{\sqrt{2}}{e}^{i\phi }\left(\begin{array}{c}1\\ 0\end{array}\right)$$ for the 'on'-pixels. When both components interfere in the vicinity of the focus of the objective lens, the intensity is $$I={|{\mathrm{E}}_{\text{off}}+{\mathrm{E}}_{\text{on}}|}^2={\mathrm{E}}_0^2(1+\cos (\phi -\xi ))$$. $$\xi$$ describes an additional phase difference depending on distance to the focal point.

To experimentally validate our idea without the need of implementing a complete MINFLUX microscope, we designed a beam path in which we replaced the objective lens by an f = 400 mm achromat and the single fluorophore by a small pinhole and photo detector that reports the intensity at the putative single-fluorophore position (Fig. 2a). Additionally, we imaged the MINFLUX PSFs with a CMOS camera.

Fig 2  Experimental demonstration with camera detection.

a Beam path of the test setup. The laser is polarized under 45°, passes the electro-optical modulator (EOM), a halfwave plate (HWP) to compensate for imperfections of the Spatial Light Modulator (SLM) and a beam expander comprising achromatic lenses and a pinhole. After the SLM, it passes another polarizer and is focused by an achromat either onto a CMOS camera or onto a small pinhole with a photo diode that detects the intensity a fluorophore would see. b PSFs recorded with the camera for different phases (as indicated) imposed by the EOM for the bilobed x pattern and the 3D-donut z pattern. The scale bar and values in nm are rescaled corresponding to an implementation in a microscope with an NA 1.35 objective lens. c Profiles through the x-PSF as indicated in b for different EOM phases with a difference of 5° between consecutive profiles. d Intensities at the center pixel of the z-PSF for different EOM voltages. Each profile was measured with the last lens positioned at a different distance from the camera (step size 2 mm) and corresponds to measurements with a fluorophore at different z positions (corresponding to a step size of 18 nm in z when integrated in a microscope). Inset: Cross-section through the PSF as indicated in B for different phases $$\phi$$ imposed by the EOM

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The bisected phase patterns produced the expected bilobed PSFs (Fig. 2b, c) with high contrast. By changing the EOM phase, we could scan these patterns laterally with only a minimal loss in contrast (Fig. 2c). The top-hat phase pattern produced a high-contrast circular illumination pattern in the focus, and a change in the EOM phase led to an increased signal in the center (Fig. 2b). Indeed, when placing the achromat at different distances from the pinhole, which effectively changes the pinhole's z-position, the signal was minimal at different EOM phases, demonstrating that the minimum of the '3D donut' can be scanned in z without degradation (Fig. 2d).

To mimic the excitation signal seen by a single fluorophore, we placed a tiny pinhole (0.03 Airy Units) in the focus of the achromat, which transmits the intensity in the central part of the PSF (Fig. 3a). We measured the intensity with an amplified photo diode while changing the EOM phase. Specifically, we alternated the x, y and z phase patterns on the SLM and produced a linear EOM phase ramp for each pattern. The intensity trace showed pronounced minima for each phase pattern with an excellent contrast (minimum relative to the intensity detected from an Airy PSF created by a flat phase pattern) of 0.2%, 0.3%, and 0.2% for the x, y, and z phase patterns, respectively (Fig. 3b).

Fig 3  Experimetal demonstration of fast scanning using a pinhole, see Fig. 2 for the beam path.

a An experimental x-PSF acqiured by a camera with the position of the pinhole indicated by a circle. b Intensity (normalized to the intensity measured for a flat phase pattern) recorded through the pinhole for a linear ramp in the EOM voltage. The intensity reaches a minimum (contrast value indicated in the graph) when the minimum of the PSF is at the pinhole position. The measurement is repeated for the different phase patterns (insets) for y-and z-localization, respectively. Switching between phase patterns takes 60 µs. c Mimicking of a MINFLUX measurement. By imposing three different voltages, each lasting for 1 µs, the minimum of the phase pattern is positioned at three distinct positions around the fluorophore. x₀ indicates the position of the minimum for the case when the setup is used in conjunction with a microsope objective, L is the diameter of the scan pattern. d By changing the amplitude of the voltage, the scan range L can be reduced to improve the localization precision. The dashed line indicates the zoom region for the subsequent panel. e By using two colinear laser beams of different colors which are alternated (in this case every 9 µs corresponding to 3 scan patterns), dual-color MINFLUX excitation can be realized without further modifications of the setup

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Next, we tested the speed of our MINFLUX excitation module by repeatedly applying three different EOM phases, which positions the x-PSF at 3 positions around the pinhole that mimics the fluorophore. With excitation times as low as 1 µs per position we could reliably detect three different intensity values with high contrast (Fig. 3c, Supplementary Fig. 5). This leads to ~60 scanning cycles within the 180 µs localization time that we use for each coordinate. Note that the 3D localization time is limited by the switching time of the SLM (60 µs).

Multi-color MINFLUX excitation can be realized with the same setup without modifications by using as an input co-linear laser beams. In case of small distances between the fluorophores and low spectral dependence of the phase delay $$\phi$$, both lasers can be used simultaneously. Otherwise, both lasers can be alternated (Fig. 3e). As switching between the lasers takes less than one microsecond, the colors can be alternated many times within one MINFLUX localization time, leading to quasi-simultaneous localization of two or more fluorophores.

The simple schematic in Fig. 1 is based on ideal performance of the optical elements. Here we discuss the impact of non-ideal performance and experimental challenges and present mitigation strategies.

See Fig. 2a. As a light source we used either an iBeam smart 640 nm laser (Toptica) together with an achromatic halfwave plate (AHWP10M-600, Thorlabs) or a multi-color laser engine (MLE HP, Toptica) together with a collimator and beam expander (GBE05-A, Thorlabs) to achieve a beam diameter of approx. 1 mm and a polarization direction of around 45°. The beam was adjusted through a Glan-Thompson polarizer aligned at 45° (GTH10M-A, Thorlabs) and an EOM (EO-AM-NR-C4, Thorlabs) that was mounted under 45° with a custom holder so that its birefringent axis was vertical. As the MgO-doped lithium niobate in this EOM displays a slow negation to an applied DC field for wavelengths shorter than approx. 600 nm, we used a different EOM (LM0202 KD*P 3×3, LINOS) for multi-color measurements, also mounted with its birefringent axis aligned along the vertical direction. The beam then passed an achromatic halfwave plate (AHWP10M-600, Thorlabs) that was used to compensate for imperfections of the SLM and a mode cleaner/beam expander consisting of an f = 40 mm achromat (Thorlabs), a 20 µm pinhole (Thorlabs) and an f = 150 mm achromat (Thorlabs). An iris (SM1D12D, Thorlabs) with a diameter of approx. 2.3 mm resulted in a nearly homogenous beam profile. The beam was then reflected off the SLM (SXGA-R12-STR, Forth Dimension Displays) under an angle of approx. 5°, passed a polarizing beam splitter (PBS251, Thorlabs), and was focused onto a 10 µm pinhole (Thorlabs) with an f = 400 mm achromat (Thorlabs). The intensity after the pinhole was monitored with an amplified photo diode (PDA100A2, Thorlabs). Alternatively, the beam was focused onto a CMOS camera (Chameleon CM3-U3-50S5M-CS, Edmund Optics).

The EOM was driven by a voltage amplifier (HVA200, Thorlabs), which was controlled by a function generator (SDG1062X, Siglent). Intensity signals from the photo diode were acquired with an oscilloscope (SDS2104X-Plus, Siglent). The function generator and the oscilloscope were triggered by the SLM at the start of an image sequence. Additionally, the SLM produces a trigger signal when a valid pattern is established, which was used to switch off the lasers during the pattern switching via a TTL signal. Camera images were recorded asynchronously. For dual-color measurements, we used a custom TTL signal converter in combination with the function generator to alternate between the 561 nm and 638 nm laser line.

Note that our time resolution for 1D probing of 1 μs is currently limited by the electronics for EOM scanning and the bandwidth of the photo diode and can in principle be one order of magnitude faster. Such high speeds might not be necessary for MINFLUX but could be useful for other applications of the variable phase plate.

We used a polarization analyzer (PAX1000VIS/M, Thorlabs) to align the Glen-Thompson polarizer to 45°. Using a flat phase pattern and large iris diameter, the camera was positioned in the focus of the f = 400 mm achromat. The pinhole was then positioned at an equal distance from the achromat.

Using the top-hat phase pattern, the beam was aligned on the SLM to produce a symmetric PSF. Then the iris diameter and EOM phase were optimized to maximize the contrast of the 3D-donut in focus. The contrast was further maximized by aligning the angle of the half wave plate and the EOM phase.

The pinhole, mounted in an x-y translation stage (ST1XY-D/M, Thorlabs), was adjusted in the lateral directions to maximize the signal on the photo diode using a flat SLM phase pattern.

To monitor the phase drift of the EOM (Supplementary Fig. 7), we split off 10% of the laser light after the EOM with a non-polarizing beam splitter, passed that beam through a Glen-Thompson polarizer (GTH10M-A, Thorlabs) oriented orthogonal to the first one before the EOM and monitored the intensity with an amplified photo diode (PDA100A2, Thorlabs). We repeatedly scanned the EOM voltage around the intensity minimum and fitted the minimum with a quadratic function to extract the zero-point voltage V₀.