## Abstract

We present an approach for point spread function (PSF) engineering that allows one to shape the optical wavefront *independently* in both polarisation directions, with two adjacent phase masks displayed on a single liquid-crystal spatial light modulator (LC-SLM). The set-up employs a polarising beam splitter and a geometric image rotator to rectify and process both polarisation directions detected by the camera. We shape a single-lobe (“corkscrew”) PSF that rotates upon defocus for each polarisation channel and combine the two polarisation channels with a relative 180° phase-shift on the computer, merging them into a single PSF that exhibits two lobes whose orientation contains information about the axial position. A major advantage lies in the possibility to measure and eliminate the aberrations in the two polarisation channels independently. We demonstrate axial super-localisation of isotropically emitting fluorescent nanoparticles. Our implementation of the single-lobe PSFs follows the method proposed by Prasad [Opt. Lett. **38**, 585 (2013)], and thus is to the best of our knowledge the first experimental realisation of this suggestion. For comparison we also study an approach with a rotating double-helix PSFs (in only one polarisation channel) and ascertain the trade-off between localisation precision and axial working range.

© 2014 Optical Society of America

## 1. Introduction

The 3D localisation of fluorescence particles or single molecules in a single exposure has moved into focus in recent years, as this allows for precise tracking of their movement. Several methods have been developed to achieve this task, e.g. biplane imaging [1] and astigmatic imaging [2], or imaging with helical point spread functions (PSFs) that exhibit one or two lobes which revolve around the optical axis under defocus [3,4]. The latter method aroused particular interest as it offers high 3D localization precision over a large axial range. Single- or double-helix point spread functions in principle can be implemented by displaying specially designed phase masks on a liquid crystal spatial light modulator (LC-SLM) in a microscope’s imaging path. A disadvantage of using LC-SLMs is that they typically affect only one polarisation direction and thus – assuming unpolarised fluorescence light – 50% of the light is effectively lost, which is a major problem in photon-limited applications. One way to overcome this drawback is to manufacture the phase mask in a transparent material such as glass [5], which will modulate both polarisation directions, but does not posses the dynamic capabilities of SLMs. Another approach is to use a polarising beam splitter to separate both polarisation directions and to rotate one either with a half-wave-plate [4] or geometrically [6, 7] to match the input polarisation required by the SLM. So far, in all experiments based on the latter approach, both beams were guided onto the same phase mask on the SLM and finally lead to two adjacent images on the camera (one for each polarisation direction). Such a polarisation sensitive detection can contain additional information about the sample, such as information about the orientation of fluorescent dipoles [4, 6, 7].

Here we apply an alternative set-up which exploits the non-quadratic panel shape of available LC-SLMs to display two *separate* phase masks simultaneously. Our modification causes no extra costs, as it only exploits areas on the SLM that would not be used otherwise. The resulting gain in flexibility can be advantageous for imaging applications where polarisation control is desired, such as imaging of birefringent samples, but also allows for independent PSF engineering on both polarisation directions.

We would like to focus on particular advantages our set-up offers in view of the precise *z*-localisation of fluorescent microparticles. For the imaging of isotropic emitters, our implementation improves the obtainable localisation precision compared to other SLM-based systems that have been introduced so far. This can be achieved by the freedom to shape two single-lobe PSFs that build a single double-lobe PSF when the two orthogonally polarised images on the greyscale camera are merged on the computer. To preserve the polarisation information, the images are not added, but each assigned to a different colour channel in the final image. In this manner the same signal to noise ratio (SNR) can be achieved as if the two images were physically recombined on the camera because noise (e.g. readout noise) is not enhanced as would be the case if the two images are added. At the same time, our set-up still offers the freedom of polarisation sensitive detection and does for instance support measurements of dipole orientations as described in Refs [4, 6–8].

A further advantage of our implementation is that it allows for correcting aberrations introduced by optical components or the sample itself. The aberration correction functions can be different for both polarisation directions, which may also be useful for compensating potential effects of birefringence introduced by the specimen or plastic dishes.

In the following, we would like to sketch our set-up and demonstrate the advantages mentioned above by means of experiments with fluorescent beads. We design a single-helix (“corkscrew”) PSF using a method that was recently introduced by Prasad [9]. Furthermore, we show how this method can be generalised to produce also a PSF that exhibits two lobes and discuss a modification in design that influences the tradeoff between achievable localisation precision and axial working range.

## 2. Experimental implementation

The experimental implementation (Fig. 1) consists of an inverted epi-fluorescence microscope and relay lenses, where the first two achromatic lenses TL and L1 (*f*_{TL}, _{1} = 200mm) are used to image the back-focal plane of the objective (Olympus ZPlanFL N 40× 0.75 NA) onto the phase-only SLM (Holoeye Pluto VIS, 1920×1080 pixels), which is mounted with the longer side along the vertical axis and therefore will only modulate vertically polarised light. Directly after lens L1, a dielectric polarising beam splitter is placed such that the horizontally polarised light is reflected downwards. At both exits of the beam splitter, right angle prism mirrors are placed to reflect the light in a direction parallel to the table (see inset in Fig. 1). This leads to a geometric rotation by 90° of the polarisation direction of the beam reflected by the beam splitter and therefore both beams will have the same polarisation direction. Note that also the image itself will be rotated by 90° and that a flip between the two images is introduced, as the number of their respective reflections differs by one. The geometric rotation has the advantage over using a half-wave plate of being achromatic. The two beams are sent to separate areas on the SLM by two mirrors, where the vertical angle between the beams was chosen such that they overlap roughly at the center of lens L2 (*f*_{2} = 300mm). This leads to two separated images on a CCD camera (MatrixVision Bluefox 224G, 1600×1200 pixels, cooled to about 15°C with a Peltier element). An iris at the intermediate image plane between lenses TL and L1 is used to limit the field of view to avoid any overlap of the images on the camera. The alignment is relatively straightforward since no beam overlap at the SLM or camera is required.

The broad emission bandwidth (emission filter band: *λ _{c}* = 560 nm ± 20 nm) required the SLM to be used in an on-axis configuration in order to avoid dispersion. This was possible since the zeroth and higher diffraction orders were sufficiently weak (on the order of 5% compared to the first order).

#### 2.1. Aberration correction

An important advantage of using two separate phase masks on the SLM is that it allows not only for correcting aberrations introduced by the SLM, but also for aberrations introduced by the optics and the sample. Contrary to this, in set-ups that use a common phase mask for both polarisation directions, only spherical components of these aberrations can be compensated, since both beams show a mutual rotation and mirror inversion when they arrive at the SLM.

Before imaging, we measured and corrected the aberrations in our microscope using modal wavefront sensing [10] based on an image sharpness metric. We considered Zernike modes up to spherical aberration *Z*_{11} (using the indexing scheme of Neil et al. [10]), as coefficients for higher Zernike polynomials were negligibly small. The measured mode coefficients can be seen in the bar plot of Fig. 2(a), where the different colours represent the two phase masks. Each bar represents the mean over 12 measurement repetitions with the corresponding standard deviation. Figs. 2(b) and 2(c) show an uncorrected and a corrected fluorescence image of three fluorescent 500 nm spheres (Invitrogen TetraSpeck Fluorescent Microspheres, dried on a cover slip), respectively. The aberration correction improves the peak intensity by 27%. Note that the corrected image of a spherical particle looks elliptical, which is an effect of high NA imaging with linearly polarised light. We simulated the image of an unpolarised sample, a 500 nm fluorescent bead, for our 0.75 NA objective and a polariser placed behind it (see Fig. 2(d)) and found the ratio between the short and long axis to be 0.90, which is in reasonably good agreement with the experimentally found value of 0.84. Related simulations are for example given in [11]. Note that there is a difference to the well studied case of focusing a linearly polarised laser with a high NA objective [12, 13].

The measured correction pattern is added to the phase pattern on the SLM, but these corrections are not included in the phase masks displayed in Section 3.

#### 2.2. Calibration procedure

The two adjacent images on the camera need to be accurately overlapped on the computer such that the two orthogonally polarised single-helix PSFs of an individual fluorescent particle together build a PSF which looks similar to a double-helix (DH) PSF, whose orientation contains the information about the axial position of the particle. For this sake a calibration image of several spheres distributed in the whole field of view was taken. The two images on the CCD were mapped onto each other using control point image registration from the Image Processing Toolbox in Matlab. The mapping procedure outputs a transformation function that will be later applied to images taken in experiments with engineered PSFs. Note that the mapping procedure does not simply add the images but assigns them separate colour channels in the combined image in order to enable a separate image analysis. In our set-up, the transformation includes a mirror inversion along a single axis as well as a rotation close to 90°. This information is vital for combining the orthogonally polarised images taken with modified point spread functions and also for calculating the phase masks correctly, such that the two corkscrew PSFs exactly face each other (with a mutual rotation of 180°) and show the same direction of rotation (this changes with an image flip).

## 3. Results

In the following, after discussing and comparing different helical PSFs, the experimental results of the polarisation-coded axial super-localisation are presented.

#### 3.1. Comparison of helical phase masks

We created our corkscrew phase masks following a recent suggestion by Prasad [9]. The aperture of the mask is segmented into several rings containing phase vortices of increasing helical charge when counting from the centre to the *n*-th ring of the phase mask:

*P*denotes the phase pattern of ring number

_{n}*n*and

*ϕ*the azimuthal coordinate. The outer radius of each ring is determined by a simple square root law: where

*N*is the total number of rings which can be chosen by the user and

*R*the radius of the phase mask.

_{mask}The amplitude PSF of each individual phase ring is an optical vortex, the radius of which is influenced by its helical charge and the radius of the phase ring on the SLM. The method of Prasad now aims at superposing the vortex-PSFs of all phase rings. This will lead to a single interference maximum (the “lobe”) along their common circumference, which can be seen from the fact there exists only one radial slice with constant phase in the phase mask.

Moreover we found that Prasad’s method for generating corkscrew phase masks can be extended to produce phase masks creating double-helix PSFs: If one creates a similar vortex phase mask with two radial lines along which the phase is constant, the corresponding PSF will exhibit two lobes, as long as the prerequisite of reasonably matching vortex sizes and rotational rates is fulfilled. Such a phase mask can be generated by using exclusively even or odd numbered helicities for the phase annuli (see Fig. 3(a)). Intensity images of such a DH PSF for different values of defocus are shown in Fig. 3(b). The rotational speed of a double-helix PSF shaped by *n* phase rings is comparable to that of a corkscrew shaped by 2*n* − 1 rings. For the double-helix with 3 and 4 rings we found rotational rates of 35.6°/μm and 27.44°/μm, respectively, which is in good agreement with 39.9°/μm of the 5 ring corkscrew and 29.36°/μm of the 7 ring corkscrew.

To achieve common rotational rates (and diameters) of all optical vortices, which together form the rotating PSF, the phase rings of higher helical charge have to be focused with higher NA, which is achieved by assigning them larger ring diameters. Choosing them according to the square root law of Eq. (2) indeed leads to approximately equal rotational rates for all optical vortices, as was pointed out by Prasad [9]. This can be understood if one considers what happens under defocus, where – according to the parabolic phase function describing a lens – each ring approximately experiences a phase offset that is proportional to its squared ring radius: Θ(*n*) ∝ *R _{out}* (

*n*)

^{2}. For an azimuthal phase mask of helical charge

*n*as described in Eq. (1), applying a phase offset of Θ is identical to a rotation of the ring by the angle

*α*= Θ/

*n*. The square root law of Eq. (2) now leads to a common rotation angle

*α*for all rings if defocus is applied:

*n*. The above considerations are made for a corkscrew PSF, but are analogous for a DH-PSF. The square-root law also helps to equalise the light power contained in each optical vortex. To match their diameters, however, the ring radii should be chosen according to a different power law, the exponent of which is approximately 3/4. As was found by numerical simulations, this exponent will lead to a tighter and intenser lobe when the particle is in-focus, at the cost of shape invariance upon rotation, i.e. the lobe will disperse more quickly when the particle gets out of focus. By tuning the function defining the ring radii one can thus find a tradeoff between localisation precision and axial working range that is optimal for the requirements defined by the application. Fig. 4 illustrates this principle by comparing two DH-PSFs that are shaped by two different phase masks. Both masks are segmented into 7 rings the outer diameters of which are determined by the power functions (

*n/N*)

^{1/2}and (

*n/N*)

^{3/4}, respectively. As expected, the exponent of 1/2 leads to “rotational invariance” but less compact lobes, and the exponent of 3/4 to compacter lobes but increased “spread” under defocus. This is evident from the image series in the middle showing simulations of both PSFs at different values for defocus, and also from the graph on the right, where the maximum intensity of the PSF was plotted against defocus. The simulation assumes a NA of 1.4 and a wavelength of 550 nm. The plot suggests that for a 7-ring mask the 3/4-power law is a better choice if the axial working range is less than 2 μm, because the lobes are more compact within this range.

#### 3.2. Experimental results

We started with two identical corkscrew phase masks. Each phase mask has a size of 844×844 pixels, which exactly matched the image of the objective back aperture on the SLM. One of the phase masks was mirrored and rotated by about 90° (Figs. 5(b) and 5(c)). The exact angle was optimised on the basis of the calibration images to compensate for non perfect alignment. That led in the combined image to two lobes lying on opposite sides of the original centre of a bead and rotating in the same direction under defocus. After numerically overlapping the two images using the known transformation from the image registration process described previously, the total image resembles a double-helix PSF. Axial stacks of rotating PSFs created by 3, 5 and 7 rings were recorded using a piezo focus positioner (Piezosystem Jena MIPOS 100 PL SG RMS) with an axial step size of 50 nm. Fig. 5(d) shows images from the *z*-stack of a 3-ring-corkscrew. The axial interspacing is 1 μm. Note that the two orthogonally polarised images are not simply added (which would decrease the SNR), but assigned different colour channels within a RGB image. Each colour channel was min-max scaled for enhanced contrast. On each channel, a 2D Gaussian fit was performed to infer the centre coordinates of the contained lobe. From these data, the rotation angle was calculated. This was done for each frame in every stack. The resulting curves are shown in Fig. 5(a). The error bars correspond to the standard deviation for a single measurement, estimated from four independent measurements. As expected, the relation between defocus and rotational angle is linear around the actual *z*-position of the fluorescence emitter. Also, as expected from the considerations above, the slope gets steeper with reduced number of rings. According to theory, the slopes should be inversely proportional to the number of displayed phase rings (see Eq. (3)). We tested this prediction by calculating the ratios of the three slopes. The results are (with the theory values in brackets): *A*_{2}/*A*_{1} = 0.64(0.60), *A*_{3}/*A*_{1} = 0.47(0.43) and *A*_{3}/*A*_{2} = 0.74(0.71).

The advantage of using a larger number of rings is the extended accessible range of defocus. Note that the shown precision and range are only for comparison and do not represent “hard” limits as these are determined by the imaging NA, the SNR of the system and by the efficiency of the fitting method. With our SNR (see Fig. 6), a NA of 0.75 and a Gaussian fit for determining the center of each lobe, we achieved a *z*-localisation precision of ±25 nm using a 5-ring-corkscrew. We also found that the other corkscrews we tested (shaped by 3 and 7 rings) provided almost equal position accuracy. Responsible for this is the fact that for a large number of phase rings the drawback of a low rotation rate is partly compensated by the large distance between the lobes, which increases the accuracy of the rotational angle estimation.

The total light efficiency, i.e. the ratio between the light reaching the camera with and without our additional components, of the set-up is 58%, where 26% are contained in the beam transmitted by the polarising beam splitter and 32% in the reflected beam. We attribute the difference of the efficiencies of the two paths to imperfections of dielectric beam splitters, which implies that the reflected channel contains also light with a polarisation direction that cannot be modulated by the SLM. The losses are mainly due to diffractive losses and absorption at the SLM.

In Fig. 6, a comparison of the achieved SNR is shown for different PSFs. The SNR was calculated by averaging over the 5 highest pixels of the image divided by the standard deviation of the background. As expected, a pair of aberration corrected corkscrews achieves the best SNR, followed by a pair of corkscrews where aberrations were not corrected. Note that the SNRs for both polarisation directions are shown as they are slightly different due to uneven light efficiency. The lowest SNR is obtained for a DH-PSF in a single imaging channel.

## 4. Conclusions

We have presented an efficient way of dynamic point spread function engineering in a fluorescence microscope set-up, where both polarisation directions of unpolarised light can be manipulated by an SLM. In contrast to previously presented implementations, our approach allows for arbitrary phase manipulations on each of the two polarisation-coded channels. This feature enables to correct aberrations introduced by the optics and the sample and may thus also be used to compensate effects of birefringence introduced by the specimen or plastic dishes. Since our SLM panel has an aspect ratio of 16:9 it is possible to display the two round phase masks next to each other on the same phase modulator.

Our approach is based on shaping a single-lobe PSF in each polarisation channel and performing image analysis on each image separately. Merging these two images on the computer gives an output image resembling a double-lobe PSF (without deteriorating the SNR) that finally allows for an accurate localisation of particles in three dimensions. We have applied this to image isotropic, i.e. unpolarised, emitters such as fluorescently labelled 500 nm beads.

Besides giving an experimental demonstration of Prasad’s single lobe PSF [9], we have extended this approach to generate rotating PSFs that exhibit two lobes and have compared the various PSFs with respect to the trade-off between localisation precision against axial working range.

With this kind of polarisation-coding it should also be possible to implement a variety of other methods, e.g. polarised biplane or astigmatic imaging.

## Acknowledgments

This work was supported by the ERC Advanced Grant 247024 catchIT.

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