## Abstract

Using acousto-optical deflectors at high deflection speeds via acoustical frequency chirping induces astigmatism, deforming the laser beam in an unfavorable way. Within the paper, we present a method to prevent this effect for an ultrashort pulsed laser beam via acoustical frequency jumps synchronized to the pulse-to-pulse pause. We also demonstrate and give a method to calculate beam shaping capability of acousto-optical deflectors via arbitrary spatial frequency developments during ultrashort laser pulse transit through the deflector. Cylinder-lens-free deflection at >2000 rad/s and beam shaping capability is demonstrated experimentally. In our experiments the switching time between two beam shapes is 1 µs.

© 2013 OSA

## 1. Introduction

Acousto-optical beam deflection is of increasing interest for applications that require high-speed laser beam scanning such as multiphoton confocal microscopy [1–4] and high-power, high-repetition-rate microstructuring with ultrashort pulsed lasers [5]. Since in the current state of technology the laser beam is deflected by inducing an acoustic wave with a constant frequency chirp into the acousto-optical crystal, the so-called cylinder-lensing effect [6] occurs at high scanning speeds, which makes the laser beam astigmatic especially decreasing imaging quality in multiphoton confocal microscopy [1].

Several compensation techniques have been developed to address the cylinder-lensing effect. Of such techniques the use of additional acousto-optical deflectors inducing the inverse amount cylinder lensing proved to be most successful [1,2,4]. In [7,8] a very similar technique is used to realize fast focus position shifting. In this case, not the cylinder lensing, but the deflection is compensated via additional acousto-optical deflectors. Despite the excellent results, the number of acousto-optical deflectors for scanning in two orthogonal directions increases from 2 to 3 or 4. This in turn not only increases investment costs, but decreases total diffraction efficiency, i.e. the laser power, which is deflected by the scanner system, is decreased from η^{2} down to η^{3} or η^{4} with approximately η = 75% … 90%. In microscopy, such decrease of diffraction efficiency is easily compensated by increasing laser power. In microstructuring applications, ablation speed is limited by the available maximum laser power. A decrease of diffraction efficiency would therefore directly impact the ablation efficiency and thus prolong processing time.

Another well-known approach to increase both ablation efficiency and structuring quality is beam profile shaping. Changing the focal beam profile shape from Gaussian to top-hat theoretically increases single pulse ablation efficiency from 37% to 100% [9]. Several techniques have been developed to produce a top-hat beam profile in the focus [10–13]. Generally, beam profile shaping systems are either static or they have slow reaction time (i.e. the time span required to change from one profile to another). Typical reaction times are in the range of ms to several 10 ms with very few exceptions such as the tunable acoustic-gradient lens [14]. For several applications it would be highly beneficial, if beam profile shape could be adapted from pulse to pulse. This would allow for adapting the beam profile to the current geometry to be structured for instance. However, to implement such techniques, reaction times within the period of typical ultrashort laser repetition rates of several MHz are required. We will introduce a novel technique to reach such beam shaping reaction times in this paper.

In the following we will first introduce a technique to deflect an ultrashort pulsed laser beam via acousto-optical deflection without inducing a cylinder-lensing effect. The key enhancement is synchronizing an acoustical frequency jump to the laser repetition rate. As the only adaption is on the controller side, no additional (compensating) optics or optical components are needed to use the proposed technique. Second, further adaption to the controller will be presented, which allows for wave front shaping transverse to the beam axis along the deflection directions. This constitutes a form of basic beam profile shaping. Only the principle of the latter technique will be demonstrated, as the full experimental setup has not been implemented yet. For the presented acousto-optical beam shaping technique a patent is pending.

## 2. Method and experimental setup for cylinder-lens-free acousto-optical beam deflection

Due to the finite acoustic velocity V of the acoustical wave in the acousto-optical deflector (AOD), the gradient dθ/dy of the local deflection angle depends on the gradient df_{y}/dt of the temporal acoustic frequency according to Eq. (1) [6], where λ is the laser wavelength and y is the coordinate along the direction of the propagating acoustic wave, which for low values of deflection angle θ equals the deflection direction.

_{y}/dt = const. = Δf

_{y}/Δt, the AOD will act as a cylinder lens with refraction power D following Eq. (2) [6].A single laser pulse takes the transit time${t}_{T}={d}_{AOD}n/{c}_{0}$to pass the AOD crystal thickness d

_{AOD}(n is the index of refraction of the acousto-optical crystal material and c

_{0}the vacuum speed of light). For typical crystal thicknesses and crystal materials, t

_{T}lies in a range of approximately 20 ps to 200 ps. Thus, it is comparable to or longer than ultrashort laser pulse durations τ. The acoustic wave does propagate approximately d = 20 nm to 600 nm during t

_{T}. The typical acoustical grating spacing of > 5 µm is much wider than d. Accordingly, the acoustical grating remains practically static when an ultrashort laser pulse transmits an AOD. If dθ/dy = 0 is satisfied during t

_{T}, the laser pulse will be deflected, but no cylinder-lens effect occurs.

To fulfill dθ/dy = 0, there has to be enough time to fill the aperture of the AOD with a constant spatial frequency between two laser pulses. Accordingly, the pulse-to-pulse pause has to be longer than the acousto-optical fill time ${t}_{F}={l}_{AOD}/V$, where l_{AOD} is the size of the AOD aperture in acoustic wave propagation direction (if raw beam diameter 2w_{0} is smaller than l_{AOD}, it is save to approximate l_{AOD} by 2w_{0}). If the prerequisites are met (practically static acoustical grating for a single pulse passing through the AOD and pulse-to-pulse pause longer than the acousto-optic fill time), beam deflection as shown in Fig. 1 is feasible.

In Fig. 1, time frame (1) is the moment, when a laser pulse triggers a first acoustic frequency jump from f_{y0} to f_{y0} + f_{y}. As c_{0}/n is much faster than V, the laser pulse is deflected depending on the acoustic frequency before the frequency jump (f_{y0}) during time frame (2). Time frame (3) shows the end of the acoustic fill time t_{F}, shortly before a second laser pulse triggers the second frequency jump at time frame (4) and is deflected depending on the acoustic frequency f_{y0} + f_{y} (5). Time frame (6) is equivalent to (3). It is important to note, that one detected laser pulse triggers the frequency jump for deflection of the next laser pulse. A similar approach is already used in communication technology [15], but was not yet adapted to laser material processing.

To confirm the applicability of the approach we implemented the experimental setup shown in Fig. 2. Table 1 gives specifications of the laser and AODs we used in the experiments. The fs laser beam is reflected by a beamsplitter towards two AODs (Gooch&Housego MD125-5C10T-3), which are arranged for orthogonal beam deflection into x- and y-directions perpendicular to beam propagation axis z. The λ/2 waveplate in between the two AODs is used to optimize the linear polarization angle for highest possible AOD diffraction efficiency. The non-deflected beam (0th order) is dumped, the deflected + 1st order beam is analyzed and used in our experiments. A small portion (~1%) of the raw laser beam passes the beamsplitter and is detected via a photodiode (Thorlabs FDS010). The photodiode signal is used to trigger an FPGA system (National Instruments CRIO9076). The FPGA sends the acoustical frequencies to the AOD-RF-drivers (Gooch&Housego 64100-150-10ADMDFS-A) as digital signals and therefore controls the moments of the frequency jumps. Both frequency jumps in AOD_{x} and AOD_{y} can be delayed in time independently from each other.

## 3. Experimental results of cylinder-lens-free acousto-optical beam deflection

The deflected 1st order laser beam is detected by a beam profile camera (Coherent LaserCam HR), which is positioned 1.7 m behind the AODs. No additional optics are used. Resulting beam profile measurements are depicted in Fig. 3. There is no visually significant cylinder-lensing effect. Note that without proper synchronization but free-running frequency jumps, the captured beam profile will be severely affected, as shown in Fig. 3(c). In Fig. 3(d) the effective deflection angle velocity equals 1800 rad/s (laser repetition rate 1 MHz). To achieve the same deflection angle velocity via frequency chirping, Δf/Δt would have to equal 3⋅10^{13} 1/s^{2}. According to Eq. (1), a cylinder lensing effect with focal distance of 3.0 m would result, which is roughly twice the distance between the beam profile camera and the AODs. The beam profiles shown in Fig. 3(b) would then be strongly elliptic with circularities of approximately 50%.

To demonstrate applicability, we coupled the deflected beam from the AOD-setup to a galvanometer scanner via a Galilean telescope in 4f-setup (magnification 4x). An F-theta objective with f = 32 mm focused the deflected laser beam onto a PMMA-sample for single pulse ablation microstructuring. The position of the focus remained on the sample surface. Figure 4 shows a reflected light microscopy photograph of the produced structures.

The galvanometer scanner deflects the laser beam into y-direction at its highest possible feed rate of 2 m/s (equals 62.5 rad/s). At the same time, the AOD deflects single laser pulses to alternating positions along x-direction. Effectively the feed rate in x-direction is +/−25.8 m/s (+/−806 rad/s), which is more than one order of magnitude faster compared to the galvanometer scanner. As before, the produced structures would be strongly elliptical, if the same deflection was realized via frequency chirping.

## 4. Method and basic demonstration of beam shaping via acousto-optical deflection

By integrating Eq. (1) over dy for one time instant, Eq. (3) results, which quantifies the deflection of an ultrashort laser pulse passing through the AOD while being affected by a grating formed by a standing acoustical wave.

For convenience, f_{y0}and θ

_{y0}designate central values of deflection with optimal diffraction efficiency (i.e. Bragg condition and deflection along z-axis). f

_{y}(y) and θ

_{y}(y) then are the respective offsets from the Bragg condition (i.e. deflections away from the z-axis in y-direction). In the following, only the latter two variables are of interest. Furthermore, a second deflection into x-direction, perpendicular to both y and z, is introduced so that Eq. (4) can be established. For comparison, cylinder-lens-free beam deflection is achieved by θ

_{y}(y) = const. and θ

_{x}(x) = const.Assuming a planar wavefront with optical path difference OPD(x,y) = 0, which is deflected by both AODs, the OPD after deflection is calculated by Eq. (5), see Fig. 5.

_{x}(x) and f

_{y}(y). To demonstrate this technique, we made few adjustments to the experimental setup for cylinder-lens-free beam deflection. We forced the frequency jumps for both AOD

_{x}and AOD

_{y}to not occur during a pulse-to-pulse pause. However, we timed the jumps such, that they were localized in the middle point of the AOD aperture while the laser pulse passed through the AOD. This principle is schematically shown in Fig. 5.

Figure 6 shows resulting beam profiles, which we captured with the same setup that was used in Fig. 3. Different combinations of acoustical frequencies were applied to demonstrate acousto-optical beam shaping as described before and as shown in Fig. 5. In Fig. 6 the unshaped beam profile (a) is forced into shapes (b) through (e) by applying the same frequencies to AOD_{x} and AOD_{y}. For shapes (f) through (j), the applied frequencies differ between AOD_{x} and AOD_{y} .Note, that especially for shapes (d) and (h), the shown profiles represent single laser pulses, which are split up to different directions. Note also, that we obtained the shown profiles from a collimated beam, so that focussing will map these profiles into the corresponding Fraunhofer diffraction patterns.

Switching from one beam shape to another was completed in 1 µs, since we synchronized to the laser repetition rate. The feasible minimum switching time in our experimental setup is approximately t_{F} = 2w_{0}/V = 450 ns. Shorter switching times may be reached by reducing the raw beam diameter 2w_{0}. To our knowledge, such short time spans for switching from one beam shape to another were not demonstrated with other technologies before (the TAG lens achieves 2.6 µs to 7 µs periodicity for resonant beam shaping [16,17]).

## 5. Discussion of beam shaping via acousto-optical deflection

The results shown in Fig. 6 demonstrate the capability of basic beam shaping when using a single frequency jump in AOD_{x} and AOD_{y}. Change of divergence (i.e. focus shifting), astigmatism, beam splitting and self-interference are feasible. When not limiting the frequency change to a step function, but using arbitrary functions instead, Eq. (5) can be rewritten as Taylor-series, for instance, and then Eq. (6) can be established.

^{n}y

^{m}and n,m ≥ 1; values for x

^{0}y

^{0}(piston) are ignored). Additionally, to roughly estimate the capability to create or compensate aberrations, the most right column displays the fraction $\Delta OPD=\mathrm{max}(|\Delta {Z}_{iAOD}|)/\mathrm{max}(\text{\hspace{0.05em}}|{Z}_{i}|)$ for two ranges of r, where r = (x

^{2}+ y

^{2})

^{1/2}. The result quantifies the maximum relative OPD error when applying a Zernike polynomial via AOD beam shaping.

Assuming a Zernike aperture size (i.e. r = 1) equaling the 1/e^{2} beam diameter of a Gaussian laser beam, the range r = 0 … 1 covers 86.5% = 1 – exp(−2) of the total beam power. For the same aperture size, but a 1/e beam diameter instead, the range r = 0 … 2^{-1/2} covers 63.2% = 1 – exp(−1). The difference between the two equals only 23.3%. In order to deflect a significant fraction of the beam power such that a desired aberration is met, it is thus more important to reach low values of the maximum relative OPD error for the smaller range than for the larger range when shaping a Gaussian laser beam.

The results in Table 2 show that some Zernike polynomials (Z_{2}, Z_{3}, Z_{4}, Z_{6} and Z_{12}) are represented fully via Eq. (6) including Defocus, the latter making the AOD beam shaper perfect for focus shifting. Some Zernike polynomials are not representable by Eq. (6) including Z_{5} and Z_{13}. For Zernike polynomials Z_{7}, Z_{8} and especially Z_{11} (spherical aberration) some of the aberration can be represented and max. relative OPD error for the range r = 0…2^{-1/2} is below 1. For Zernike polynomials Z_{9}, Z_{10} and Z_{14} some of the aberration can also be represented, but the max. relative OPD error exceeds 1, which means that a large part of the laser beam wavefront will depart from the aimed aberration.

For Zernike polynomial Z_{11} (spherical aberration), which is a most important aberration in microscopy and laser materials processing, the resulting error ΔZ_{11AOD} equals -√2 ΔZ_{14AOD}. Therefore, when using an AOD beam shaper (Eq. (6)), spherical aberration may be compensated better by accepting a resulting wavefront error Z_{14}. In this case Z_{11} is fully transformed to -√2 Z_{14}, and the max. relative OPD error $\Delta OPD=\mathrm{max}(|\sqrt{2}{Z}_{14}|)/\mathrm{max}(|{Z}_{11}|)$ is further reduced down to {0.5, 2}, which is a workable compromise for inducing or compensating spherical aberration.

## 6. Summary and outlook

In the first section of the paper we introduced cylinder-lens-free acousto-optical beam deflection via frequency jumps that were synchronized to the repetition rate of an ultrashort pulsed laser. In our experiments we used a fs-laser of 355 nm wavelength and 300 fs pulse duration. At 1 MHz repetition rate we achieved a cylinder-lens-free deflection of > 2000 rad/s for a collimated beam diameter of 2.7 mm. In principle, the experimental setup allows for achieving ~9.000 rad/s deflection speed with the same beam diameter but faster laser repetition rate. No additional compensation optics were used.

The second section of the paper introduces a way to shape the beam of the same laser with few adaptations in the experiment. For demonstration purposes we implemented defocus, astigmatism, beam splitting and self-interference. The time span to switch the shape of a beam profile in our experiment was 1 µs short, but could be further shortened to 0.45 µs without adapting the raw beam diameter. Using smaller collimated beam diameters the switching time can be decreased further.

The third section of this paper introduces an analytical description to estimate beam and wavefront shaping capabilities of an acousto-optical beam shaper using arbitrary adjustments of the acoustical frequency. Some wavefront aberrations can be generated/eliminated at perfection, including defocus and 1st and 2nd order astigmatism aligned to the axes of the acousto-optical deflector. Reproduction or generation of other aberrations can either not be accomplished or can only be accomplished in a less perfect way. Regarding the important role of spherical aberration in optics, it was shown that spherical aberration may be fully compensated if Quadrafoil 0° is acceptable as residual aberration. For Gaussian beams this is a workable compromise.

The authors already applied for a patent regarding the beam shaping method presented in this paper (patent pending #10 2013 201 968.8). In the future we plan to demonstrate cylinder-lens-free acousto-optical beam deflection in other applications (especially fluorescence microscopy). We will develop the analytical description of beam shaping via acousto-optical deflection further in order to quantify residual wavefront aberrations in more detail (for both microscopy and laser applications). This will include the calculation of point-spread-functions and diffraction patterns in order to estimate or design focal beam profiles. An experimental setup allowing arbitrary developments of acoustical frequencies is on our agenda in order to further develop, demonstrate and investigate beam shaping capability of acousto-optical deflectors.

## Acknowledgments

The authors would like to thank Wavelight GmbH (part of Alcon Inc.) for providing the fs laser with which the experiments were carried out. Furthermore, the authors gratefully acknowledge the financial support by and scientific exchange with the Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG), the framework of the German excellence initiative. This project is supported by the German Federal Ministry of Education and Research (BMBF), project grant No. 01EX1011C. We acknowledge support by Deutsche Forschungsgemeinschaft and Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding program Open Access Publishing.

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