## Abstract

The generation of multi foci is an established method for high-speed parallel direct laser writing, scanning microscopy and for optical tweezer arrays. However, the quality of multi foci reduces with increasing resolution due to interference effects. Here, we report on a spatial-light-modulator-based method that allows for highly uniform, close to Gaussian spots with diffraction limited resolution using a wavelength of 780 nm. We introduce modifications of a standard algorithm that calculates a field distribution on the entrance pupil of a high numerical aperture objective splitting the focal volume into a multitude of spots. Our modified algorithm compares favourably to a commonly used algorithm in full vectorial calculations as well as in point-spread-function measurements. The lateral and axial resolution limits of spots generated by the new algorithm are found to be close to the diffraction limit.

© 2013 Optical Society of America

## 1. Introduction

The generation of a multitude of spots offers a broad range of applications. Multi foci (MF) have successfully been utilized in the parallelization of multi-photon absorption microscopy [1, 2] and direct laser writing (DLW) [3–14], in optical trapping as well as micro-manipulation [15–20]. Additionally, MF generation is a method to create specific optical or thermal potential landscapes [21, 22].

The splitting of the intensity into multi foci is generally achieved by spatially modulating the field distribution on the entrance pupil of a focussing optic. Various algorithms that determine such a field distribution have been proposed in literature [23–26]. These algorithms calculate a phase or amplitude pattern and can be deterministic or iterative. Experimentally, these patterns are generally encoded into diffractive optical elements or displayed on a spatial light modulator (SLM). A number of algorithms have been tested for uniformity and efficiency [27–29].

When pushing the above mentioned applications to their limits, very accurate knowledge and very flexible control of the isointensity surfaces of the focal volumes is required. However, to the best of our knowledge, no thorough analysis of the three dimensional (3D) intensity distribution of the generated spots has been conducted so far. Furthermore, spot distances below the wavelength of the focussed laser beam have not been demonstrated – in fact, structures fabricated by parallelized DLW at 780 nm wavelength were limited to feature separations of more than 1.75 μm due to interference effects and showed uniformity and thickness variations across the structures [9]. Here, we use full vectorial calculations and point-spread-function (PSF) measurements to show significant differences in the performance of different algorithms approaching the diffraction limit for high numerical aperture objective lenses. Modifications of standard algorithms are introduced to overcome the interference issue and to reach the in many applications, e.g. DLW, demanding requirements on the spots’ uniformity.

This work extends MF generation into the high resolution, highly accurate regime making it relevant for optical trapping, structured microscopy and parallel DLW.

## 2. Setup and methods

The experimental setup is based upon a DLW setup with a phase-only SLM, described in detail in [30]. The light distribution reflected to the first diffraction order by a blazed phase grating displayed on the SLM is imaged onto the entrance pupil of an objective with NA=1.4. Almost arbitrary independent intensity and phase patterns can be generated in the first diffraction order by spatially adjusting the piston (which locally controls the diffraction efficiency) and the bias of the blazed phase grating respectively [31, 32]. All patterns calculated by the algorithms are overlayed with an aberration correction pattern found by the method given in [30]. This is required since the SLM is not perfectly flat and the objective is not used at its design wavelength, causing strong – mostly spherical – aberrations. The resulting pattern P_{SLM} displayed on the SLM therefore consists off the target phase distribution P_{target}, the blazed grating P_{blaze}, the aberration correction term P_{aberr} and the target amplitude distribution A_{target}[32], modulated to a 2*π* phase range:

## 3. Description of algorithms

In [28] Di Leonardo *et al.* describe a number of common phase-only algorithms which they evaluate in their ability to produce highly uniform spot arrays. They introduce an improvement – the weighted Gerchberg-Saxton algorithm (GSW) – which yielded the best results and has in consequence been used by other groups in, e.g., DLW [11]. We compare this algorithm to our modified version which uses amplitude modulation of the pupil function as an additional degree of freedom (amplitude Gerchberg-Saxton: GSA, introduced in Fig. 1). It may also be used in phase-only mode (GSP). In the latter case the algorithm does not differ from the GSW algorithm fundamentally but in the implementation only. To extend the GSA algorithm to 3D space (GSA3D), we make use of the defocus and refocus factors introduced in [33, 34]. The factors take into account the spherical shape of the optical transfer function occurring with high numerical aperture focussing. All algorithms include the same weighing term (from [28]) that increases the uniformity of the spots. Convergence is reached, if the uniformity does no longer improve. 100 iterations were sufficient to ensure this convergence and were thus used for each algorithm.

The GSA algorithm starts with a random amplitude A_{0} and a random phase P_{0}. The pupil function is multiplied by an aperture function (A_{aper}) which is one inside the pupil radius and zero outside (this function is not included in the commonly used GSW algorithm). After Fourier transformation, at the *k*^{th} iteration, the weighing factor (w) for the *m*^{th} spot is calculated and the target function is replaced as introduced in [28]:

_{pup}is set to one and no amplitude modulation is taken into account. The GSA3D additionally multiplies the pupil function with the defocus term given in [33, 34] (

*z*is the axial real space coordinate and

*k*the corresponding Fourier space coordinate): Defoc = exp(i

_{z}*k*). After replacing the target as in the 2D case, a refocus factor (Refoc = exp(−i

_{z}z*k*)) and an averaging procedure is used to find the corresponding 2D pupil function (for details see [33, 34]).

_{z}z## 4. Performance and resolution limits

Figures 2(a)–2(c) shows vectorial calculations (implemented as introduced in [35]) and PSF measurements of three spots laterally 1000 nm apart – well above the used wavelength of 780 nm. Comparing the algorithms, the GSA performs visibly better than the two phase-only algorithms. The difference in performance becomes very prominent when targeting spots 600 nm apart (Figs. 2(d)–2(f)). Here, strong interference effects become visible for the phase-only algorithms while the foci produced by the GSA are still gaussian. Note, that in the calculations the lateral intensity distribution of the phase-only algorithms still looks well shaped – only the axial distribution reveals the lack in performance. Therefore, the former is not necessarily a measure for good focal properties.

Abbe’s criterion gives the lateral two-point resolution of an optical system: *a _{l}* = 0.61

*λ*/NA. Here, this leads to an expected resolution of 340 nm which – in the vectorial calculations –could be produced with the extra degree of freedom of amplitude modulation (Fig. 3(a)). We show that in the experiment spots separated by 390 nm (Fig. 3(b)) are possible (see discussion). Lower separations led to clipping by the aperture resulting in higher intensities of the outer spots. Note, that phase-only algorithms fail at resolutions around the wavelength of light.

The convergence of the 3D phase-only algorithms depended very much on the target pattern. As mentioned in [11], this is due to the weighing factor favoring spots further along the optical axis. Obviously, this is not desirable in applications. In contrast, the combined amplitude and phase modulation of the GSA3D converges well for target patterns down to 1000 nm axial resolution. This is close to the axial resolution limit, given by [36]: ${a}_{z}=\frac{2\pi}{\mathrm{\Delta}{k}_{z}}\ge \lambda /\left(n-\sqrt{{n}^{2}-{NA}^{2}}\right)$, which – in our case – leads to 840 nm. However, the expected resolution could not be reached in the experiment (see discussion). Figures 3(c) and 3(d) show the 3D placement capabilities of the GSA3D. While at 1500 nm spot separation the intensity between the spots dips to 35% of the maximum (Fig. 3(c)), it dips to just under 50% at 1300 nm (not shown) and to 70% at 1100 nm separation (Fig. 3(d)). For the lateral and axial high resolution patterns, differences between the theoretical and the experimental projected amplitude distributions have a strong impact on the uniformity of the foci. Therefore, the intensity distribution hitting the SLM had to be taken into account by dividing the desired distribution by the gaussian distribution of the incoming laser beam.

## 5. Discussion

To evaluate the performance of the algorithms more quantitatively, we take the difference between the maximum intensity of the brightest spot and the maximum intensity of the dimmest spot divided by the average spot intensity as a measure for uniformity. At 1000 nm separation, this leads to experimental (theoretical) max-min deviations of: 20% (16%) for the GSW, 12% (4%) for the GSP and 7% (1%) for the GSA. We did not conduct any further experimental optimization except for the above mentioned basic aberration correction procedure. The max-min deviations for the 600 nm case were: 40% (10%) GSW, 22% (3%) GSP and 9% (2%) GSA. At 390 nm spot separation we found an experimental 10% max-min deviation for the GSA. Since the theoretical calculations of the GSA lead to deviations below 2%, the measured deviations are mostly due to remaining aberrations and errors of the projected phase pattern. Despite the obvious improvement by the GSA the deviations still seem rather high. However, we want to stress that a 10% max-min deviation results in less than 30 nm full-width-half-maximum (FWHM) difference of the foci in our setup, which is tolerable in most applications. Furthermore, the deviation may be lowered if experimental feedback is included in the definition of the target amplitude.

Even after aberration correction the FWHM of a single focus as well as of all multi foci were up to 30% larger than expected from theory. In direct consequence the obtainable resolution was lower by the same percentage. This led to a measured NA of the objective used of about 1.25 instead of the specified NA of 1.4. We attribute this to the fact that we do not use the objective at its design wavelength and that chromatic aberrations are not sufficiently compensated by the objective.

Our method to produce the target amplitude relies on spatially reducing the diffraction efficiency. Therefore, the overall laser power utilization efficiency is reduced when including amplitude modulation. Compared to the laser power reflected by the blazed phase grating only (240 mW), the efficiency in the lateral spot arrangement was reduced to 34% (GSA 1000), 20% (GSA 600) and 16% (GSA 400). In the axial case amplitude modulation led to a reduction to 62% (GSA 1500) and 69% (GSA 1100). Adding the phase-only patterns to the blazed phase grating had no relevant influence on the diffraction efficiency.

## 6. Conclusions

Using three dimensional point-spread-function measurements, multi foci algorithms were tested with emphasis on uniformity as well as axial and lateral resolution limits. We show, that it is not sufficient to only look at the lateral distribution (often done with a CCD image) when evaluating the quality of the obtained foci.

Close to the diffraction limit the performance of the algorithms depends heavily on the degrees of freedom provided. While phase-only modulation allows for good quality spots with a distance of *λ*, independent phase- and amplitude control pushes this down close to the diffraction limit (390 nm experimentally).

Furthermore, added amplitude modulation guarantees the convergence of the algorithm when the target spots are axially distributed. In the experiment, the axial resolution was found to be 1300 nm for independent phase- and amplitude modulation.

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