Automated aberration correction of arbitrary laser modes in high numerical aperture systems

Julian Hering; Erik H. Waller; Georg Von Freymann

doi:10.1364/OE.24.028500

1. Introduction

Even in times of digital microscopy and powerful image correction algorithms it is still the point-spread-function (PSF) of an optical system defining the best possible performance [1]. While exact knowledge of an even strongly aberrated PSF allows for perfect image reconstruction in conventional microscopy, there are applications strongly depending on an aberration-free PSF. Aberration correction enables better resolved two- and three-dimensional structures in lithography [2] and micro-manipulation [3]. Also, modern stimulated-emission-depletion (STED) based microscopy [4–6] and lithography [7–9] rely on doughnut and bottlebeam modes, which are only realizable in high quality without aberrations. The creation of specific optical and thermal landscapes [10–12], multi foci generation for parallelized multi photon absorption microscopy [13,14], direct laser writing (DLW) [15–20], and optical tweezers [21–27] are further examples requiring an aberration-free PSF.

Correcting aberrations would be straightforward, if amplitude and phase could be directly measured throughout the PSF. As in the above mentioned applications only acess to the intensity distribution in the focal plane is provided, one has to infer the required phase information from the data at hand. To address this problem, several experimental approaches have recently been presented [2,18,20,28,29] pointing the path towards a possible automated aberration correction for optical systems, providing the same level of comfort as already implemented in electron microscopy [30]. All of these approaches are more or less based on the Gerchberg-Saxton algorithm (GSA) [31–35], which principially allows for retrieving the phase of an intensity distribution by transferring phases and amplitudes iteratively between the focal and pupil plane (an overview of the most important publications is given in [29]). A complete treatment of the problem of aberration correction for high NA fluorescence microscopy is presented in [29]. Images collected from fluorescent beads in different focusing depth are collected. A Gerchberg-Saxton based algorithm is used for retrieval of the pupil function. The authors also discuss the proper treatment of the vectorial nature of light. However, for the actual phase retrieval the influence of vectorial treatment versus scalar treatment is shown to be approximately on the same order as the noise in the measurements. An active correction of the calculated aberrations is not performed, but the approach demonstrates promising results for image correction. Correcting aberrations for a doughnut mode in a low NA system employed for optical tweezers using a Gerchberg-Saxton algorithm has been investigated in [28], here correction has been performed for a single image plane only, neglecting defocus effects not so important for low NA systems. Aberration correction for a high NA system is presented in [18] but only correcting defocus and spherical aberrations via estimated refractive indices and distances, without actual measuring the PSF of the system. In our previous work we extended this active aberration correction for high NA systems by taking into account higher order Zernike coefficients [2] and recently presented a modified version of the Gerchberg-Saxton algorithm, in which we not only modified the phase but additionally the amplitude distribution of the pupil function, allowing for the first time the generation of diffraction limited multi foci modes [20].

Here, we present a Gerchberg-Saxton based phase retrieval algorithm for automated aberration compensation of arbitrary laser modes in a high NA system. Using a spatial-light-modulator (SLM) based setup, we test our approach not only for the Gaussian TEM₀₀ mode but also generate several different modes like doughnut, slit, and multi foci modes. While from a rigorous treatment as the one presented in [29] we would have expected that correcting the TEM₀₀ should already contain all relevant information, we find that the corrections further improve, if the retrieval is done directly for the desired mode pattern.

2. Materials and methods

A simplified scheme of the experimental setup is shown in Fig. 1. An optically pumped continous wave semiconductor-laser operating at 532 nm (Verdi G2, Coherent Inc.) illuminates the active window of a phase-only spatial-light-modulator (SLM, X10468-01, Hamamatsu Photonics). The laser beam is circularly polarized by a quarter-wave plate to average out the polarization-induced asymmetry found by focusing through high NA objective lenses. The laser beam is diffracted by a blazed phase grating displayed on the SLM: The diffraction efficiency within the first diffraction order is locally controlled by spatially adjusting the piston [2,36,37]; the resulting phase distribution is independently controlled via the bias. The thus shaped first diffraction order contains modulated light only and is imaged via a 4f-system onto the entrance pupil of a galvanometer mirror system (Nanoscribe GmbH).

Fig. 1 Sketch of the experimental setup. The beam path, its main optical components and the scanning procedure are depicted.

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Unwanted remaining light from the zeroth order of the SLM is blocked by an iris aperture in the back focal plane of the first lens (L₁). Finally, the laser beam is imaged onto the entrance pupil of a high NA oil-immersion objective (NA = 1.4, HC PL APO 63x/1.40–0.60 OIL, Leica Microsystems GmbH). The objective focuses the laser beam through the immersion oil and a 170 µm thick glass substrate (Gerhard Menzel B.V. & Co. KG) onto an 80 nm wide gold nano-sphere embedded in immersion oil. Since this gold bead acts as a point scatterer, the back scattered intensity is approximately proportional to the incident intensity and is collected by the same objective also used for focusing. The scattered intensity is measured with a photodetector (PD, DET36A/M, Thorlabs Inc.). Scanning the gold bead laterally by deflecting the laser beam with the galvanometer mirror system and axially by moving the substrate with a 3-axis piezo stage (P-563.3CD, Physik Instrumente (PI) GmbH & Co. KG) allows for measuring the 3D PSF. Since the entire focal region is index-matched (objective to oil, oil to glass and glass to oil, see Fig. 1, bottom right side), the aberrations present in the system are not affected by moving the sample or deflecting the laser beam. Focal volumes of about (2 × 2 × 2) µm³ with 50 nm step sizes in each dimension may be scanned and imaged within less than one minute. To reduce the influences of measurement errors, each PSF is scanned ten times for averaging. Additionally, a tracking algorithm using the TEM₀₀ mode of the same laser beam is implemented to keep the center of the focal volume fixed and thus, to account for possible drifts of the gold nano-sphere.

3. Description of the algorithm

The algorithm for high NA aberration compensation of arbitrary laser modes is based on our modified Gerchberg-Saxton algorithm [2]. Additionally, as already presented in [29], we also perform the phase retrieval for a number of slices with positive and negative defocus to account for higher order aberrations. Figure 2 shows a sketch of the operation principle, which is described in detail in the following.

Fig. 2 Scheme of the algorithm. The algorithm starts with the initial amplitude and phase pattern of a lateral-double-spot. Replacing the magnitude in Fourier- and real-space, respectively, retrieves the phase information P_GSA containing all aberrations of the system. In this example, seven equidistant slices, separated by 75 nm and centered around the focal plane, are sufficient for aberration compensation.

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We initialize the algorithm by setting the amplitude A_start- and phase P_start-patterns to generate the starting electrical field of the wave within the pupil plane for an ideal aberration-free system. These initial values form the pupil function:

E_{pup} (k_{x}, k_{y}) = A_{start} (k_{x}, k_{y}) \cdot \exp [i P_{start} (k_{x}, k_{y})],

with k_x and k_y as lateral Fourier space coordinates and [k_x] = [k_y] = 1/lenght [38]. In the next step, the defocus factor is implemented for focusing the pupil function by a distance z. This factor can be expressed as exp(ik_zz) [29,38], with z being the axial real-space coordinate and

k_{z} (k_{x}, k_{y}) = \sqrt{{(2 π n / λ)}^{2} - (k_{x}^{2} + k_{y}^{2})}

being the corresponding axial Fourier-space coordinate. Thus, the two-dimensional (2D) pupil function is translated to a sliced 3D field distribution. The subsequent Fourier-transformation generates the 3D ideal magnitude

A_{foc} = \sqrt{I_{foc}} = | E_{foc} |

and phase P_foc of the initial wave function. The amplitude of each slice is then replaced by the square-root of the experimentally scanned intensity distribution

A_{scan} = \sqrt{I_{scan}}

, normalized to A_foc. The following inverse Fourier-transformation

(ℱ^{- 1})

converts the modified light wave back into the pupil plane, which coincides with the plane of the SLM. Multiplying each slice with a corresponding refocus factor exp(−ik_zz) accounts again for the propagation and ideally results in identical distributions. To increase data quality all resulting phase and amplitude distributions are averaged over the slices. This provides the corresponding 2D pupil function. Now, in addition to the procedure described in [29] the 2D pupil magnitude A_pup is replaced by the start amplitude A_start. In the last step the aperture function F_aper, which is unity inside the objective’s pupil and zero outside, is applied and completes the first iteration of the algorithm.

After each iteration, the absolute value of the correlation coefficient |CC| between the scanned $A_{scan}^{m n}$ and the ideal $A_{ideal}^{m n}$ focal amplitudes is calculated and serves as abort criterion:

| C C | = | \frac{Σ_{m} Σ_{n} (A_{scan}^{m n} - {\bar{A}}_{scan}) (A_{ideal}^{m n} - {\bar{A}}_{ideal})}{\sqrt{[Σ_{m} Σ_{n} {(A_{scan}^{m n} - {\bar{A}}_{scan})}^{2}] [Σ_{m} Σ_{n} {(A_{ideal}^{m n} - {\bar{A}}_{ideal})}^{2}]}} |,

with mn indicating the respective matrix entries at A_ideal ≠ 0 and Ā_scan, Ā_ideal being the average values of the respective quantities averaged over the whole matrix. If the correlation coefficient improves less than 0.01% after the last iteration, the algorithm has converged and the best phase pattern P_GSA is found. Usually, this takes at most 15 iterations. The resulting phase pattern allows for calculating the compensating phase distribution of the hologram

H_{SLM}^{j}

for the next cycle of the above described process [37]:

H_{SLM}^{j} = {\mod [P^{j - 1} - (P_{GSA}^{j - 1} - P_{start}) + P_{blaze}, 2 π] - π} s {inc}^{2} ((1 - A_{start}) π) + π .

Here, the index j denotes the current cycle number, P^j⁻¹ is the phase pattern of the previous cycle, $P_{GSA}^{j - 1}$ the latest GSA pattern and P_blaze the blazed phase pattern. Within the first cycle P^j⁻¹ and $P_{GSA}^{j - 1}$ do not exist and thus, Eq. (4) simplifies to the commonly used equation for the resulting phase distribution of the hologram [37]:

H_{SLM}^{1} = [\mod (P_{start} + P_{blaze}, 2 π) - π] s {inc}^{2} ((1 - A_{start}) π) + π .

Usually, no significant further improvement can be observed after 10 cycles.

4. Results and discussion

We demonstrate the performance of our approach by choosing six differently shaped PSFs (Fig. 3), all of relevance for three-dimensional laser lithography: a) a TEM₀₀ mode, mostly used as excitation mode in optical microscopy/lithography, b) a doughnut mode, established in STED applications, c) a slit mode to receive an aspect ratio of one within one plane, utilized for writing of waveguides with circular cross-section [39], and d) – f) three different lateral and/or axial multi-spot modes, possibly useful in optical tweezer arrays and STED applications as well. Phase and amplitude patterns of the latter three modes are impossible to receive without a weighted Gerchberg-Saxton algorithm, using amplitude modulation as additional degree of freedom [20].

Fig. 3 Experimental results. Major aberrations present in a) a TEM₀₀ mode, b) a doughnut mode, c) a slit mode, d) a lateral-double-spot mode, e) an axial-double-spot mode, and f) a six-spots mode are compensated. The gray, blue (dotted) and red (dashed) colored lines indicate the intensity cross-sections shown in Fig. 4. The respective phase patterns used are shown in the upper left corner of each laser mode. The scale bars correspond to 500 nm, respectively.

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Fig. 4 Intensity cross-sections of the above described laser modes (Fig. 3 a) – f)). The gray color indicates the ‘ideal PSF’-case, blue (dotted) and red (dashed) represent the ‘initial’-and ‘corrected PSF’-cases, respectively.

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In Fig. 3, ‘ideal PSF’ represents the numerical expectations of the aforementioned laser modes, whereas the resulting respective experimentally scanned focal intensity distributions are called ‘initial PSF’. As expected, aberrations caused by the complex high NA setup and the non-perfect SLM-surface distort the ideal shape of each laser mode. To compare and quantify the quality of the laser modes, we use again the absolute value of the correlation coefficients (compare Eq. (3)), averaged over both, the lateral (p = 1) and axial (p = 2) PSF-images depicted in Fig. 3:

| \bar{C C} | = \frac{1}{2} \sum_{p = 1}^{2} | C C^{p} | .

The individual correlation coefficients between ‘ideal PSF’ and ‘initial PSF’ reach only up to 86.9% on average for all laser modes. These values are presented in Table 1. In addition to any obvious distortions the lateral and axial full widths at half maximum (FWHM) and the resulting aspect ratios of the TEM₀₀ and the slit mode differ strongly from the ideal case (see Table 1). Within ten correction cycles of the algorithm (j ≤ 11), major distortions of all modes are compensated, as depicted in ‘corrected PSF’ of Fig. 3. The final phase patterns are indicated in the upper left corner of each laser mode. The wavefront correction utilizes the maximal possible phase-shift of the SLM, as can be best seen for the TEM₀₀ mode. For further comparison and quantification, intensity cross-sections along the most crucial directions of each single laser mode are shown in Fig. 4, whereby the colors gray, blue (dotted) and red (dashed) represent the ideal, initial and corrected PSFs, respectively. For the latter ones, the correlation coefficient increases to 98.2% on average (compare the right side of Fig. 5 as an exemplary trend for the correlation coefficient) and the aforementioned FWHM and aspect ratio improve close to the theoretical values (Table 1, Fig. 4).

Table 1. Criteria of mode quality. The most crucial quality parameters for the above described laser modes (Fig. 3 a)–f)) are depicted. ‘Aspect ratio’ represents the ratio of the axial and lateral FWHM, whereas ‘min. intensity’ means the minimal intensity in the center of the STED-relevant laser modes. Here, the initial PSF values are not meaningful but listed in brackets for the sake of completeness.

View Table

Fig. 5 Intensity mean square error as function of the algorithm’s iterations within the first cycle of the doughnut mode’s aberration compensation (left). On the right side the improvement (correlation coefficient) of this mode is shown as function of the amount of cycles.

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The minimal intensity at the center of the doughnut, lateral, axial-double-spot, and six-spot mode is crucial for applications in STED-microscopy/lithography. Hence, we also show them in Table 1. These values are very close to the theoretical expectation. Deviations from the expected values are due to minimal intensity values not distinguishable from general measurement noise and, thus, might stem from the averaging procedure in the algorithm.

From theoretical considerations [29], we originally expected that phase-retrieval for the TEM₀₀ mode should suffice to describe the aberrations present in the system. However, we find in our experiments that the quality of the retrieval considerably improves if the target mode is also used for the phase retrieval. This fact can be explained in the presence of experimental noise: For the TEM₀₀ mode intensity rapidly falls off with increasing distance from the focal plane. Higher order spherical aberrations just barely effect the focal region of the TEM₀₀ mode in which intensity is high. With intensity levels reaching the experimental noise level, phase retrieval does not lead to proper determination and subtle changes to the TEM₀₀ mode are not properly accounted for.

This explanation is supported by the correlation coefficients reached with the correct mode compared to the ones where we used the retrieved phase from the TEM₀₀ mode only. Also, the amount and distance of slices required for an optimal fit supports this assumption: The axial FWHM of the TEM₀₀ mode is about 520 nm, hence, seven slices with 75 nm step size (∆z) are sufficient for the phase retrieval. The axial-double-spot mode claims much more space along the optical pathway and, thus, nine slices with a larger spacing (∆z = 200 nm) are necessary for aberration compensation. Again, from theoretical considerations it is well known that the vectorial nature of light should not be neglected dealing with high NA systems [29, 40, 41]. The so called ’wavefront compression’ is accounted for by the proper definition of the pupil function [29]. Effects resulting from different polarizations are averaged out, as we use circularly polarized light. However, even with all corrections employed, corresponding effects are shown to be small compared to the influence of experimental noise and other deviation not accounted for by the model used [29]. We find that taking into account the amplitude of the pupil function as well as correcting aberrations for the desired target mode directly has a much more pronounced effect on the correlation reached. The most critical mode with respect to the polarization of light is the doughnut mode. A correlation coefficient of 97.6% is reached for this mode after only very few cycles. Furthermore, the intensity mean square error, calculated in analogy to the method given in [29], converges after only 5 iterations to values as small as 6 · 10⁻⁴ which is significantly smaller than the values published in [29] (compare Fig. 5) The very small rise between iteration number five and twelve is possibly caused by a non-perfectly centered PSF during the scanning procedure, measurement noise or similar experimental faults, and can be neglected.

The vortex template for the doughnut mode provides another challenge, as it requires an orientation-dependent phase pattern. In one of 20 cases, our non-vectorial algorithm changed the phase pattern’s orientation from clockwise to counterclockwise (or vice versa), which dramatically effects the resulting PSF for circularly polarized light [42]. While the aberrations are indeed correctly taken care of, the ‘flipped’ phase pattern has to be multiplied by −1 and then modulated to 2π to be useful. This is realized by an automated control of the vortex’s slope sign along a predefined direction: a reversion of the sign indicates the phase pattern’s flip.

Finally, the entire duration for the aberration compensation can be summarized as follows: Measurement of all slices including the ten-fold averaging takes less than one minute. Considering the algorithm’s computation time of less than 30 seconds per cycle, the complete automated aberration compensation requires less than 15 minutes. All aberration compensation patterns retrieved by the algorithm can be used until the experimental setup changes.

5. Conclusions

We demonstrated a modified Gerchberg-Saxton algorithm for automated aberration correction in high NA systems, especially suitable for three-dimensional laser lithography. Correlation coefficients between the targeted intensity distribution and the experimentally achieved one are better than 99.0% for some of the presented modes. This improvement over other approaches is mainly due to correcting aberrations with the target mode directly and not with the more general Gaussian TEM₀₀ mode. Due to different characteristics of the selected modes, the sensitivity to aberrations is different for those modes, leading to the better overall correction. Especially, crucial parameters like FWHM, aspect ratio and center intensity of STED-relevant laser modes receive values very close to the theoretical expectation. Our approach might pave the way to improved performance and ease-of-use of SLM based systems for such diverse applications as microscopy, lithography and optical tweezers.

Funding

German Research Foundation (DFG) (Collaborative Research Center CRC 926 “Microscale Morphology of Component Surfaces” project B11).

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	correlation coefficient. / %		aspect ratio / nm:nm			min. intensity / %			number of cycles j / #
	initial	corr. (by TEM₀₀)	ideal	initial	corr.	ideal	initial	corr. (by TEM₀₀)	number of cycles j / #
a)	90.1	99.2 (–)	518:213 = 2.43	>750:225 > 3.41	542:221 = 2.46	–	–	–	6
b)	82.1	97.6 (94.7)	–	–	–	0.0	(5.2)	3.7 (8.5)	9
c)	93.6	99.4 (99.0)	553:554 = 1.00	1016:703 = 1.45	596:606 = 0.98	–	–	–	9
d)	92.0	98.1 (98.7)	–	–	–	2.0	(4.3)	0.0 (0.0)	6
e)	80.0	96.6 (96.6)	–	–	–	3.9	(4.7)	1.3 (5.0)	10
f)	83.4	98.4 (97.8)	–	–	–	0.9	(10.2)	0.8 (3.9)	11

Automated aberration correction of arbitrary laser modes in high numerical aperture systems

Abstract

1. Introduction

2. Materials and methods

3. Description of the algorithm

4. Results and discussion

5. Conclusions

Funding

References and links

Cited By

Figures (5)

Tables (1)

Equations (6)

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