## Abstract

Acousto-optic deflection (AOD) devices offer unprecedented fast control of the entire spatial structure of light beams, most notably their phase. AOD light modulation of ultra-short laser pulses, however, is not straightforward to implement because of intrinsic chromatic dispersion and non-stationarity of acousto-optic diffraction. While schemes exist to compensate chromatic dispersion, non-stationarity remains an obstacle. In this work we demonstrate an efficient AOD light modulator for stable phase modulation using time-locked generation of frequency-modulated acoustic waves at the full repetition rate of a high power laser pulse amplifier of 80 kHz. We establish the non-local relationship between the optical phase and the generating acoustic frequency function and verify the system for temporal stability, phase accuracy and generation of non-linear two-dimensional phase functions.

© 2015 Optical Society of America

## 1. Introduction

Spatial shaping of laser beams, by tailoring amplitude and phase of the light field, has major applications in applied optics, including material processing, laser displays, optical communication and microscopy. For high throughput these applications require the technical ability to control the beam shape at high temporal rate. Active phase shaping, often under fast feedback control, is achieved with a variety of technologies including deformable mirrors (DM) and liquid-crystal-based spatial light modulators (LC-SLM), which are either limited in speed to refresh rates of about 1 kHz (LC-SLM) or in the number of pixels in each dimension and thus in the number of discrete scan directions (fast DMs). Acousto-optic deflectors (AODs) [1], which are commonly used for beam steering in fast pointing devices, open the prospect for faster beam control with relatively large dynamic range. In standard use, AODs employ ultrasonic waves of constant frequency to establish a spatially homogeneous density grating in an acousto-optic crystal resulting in beam diffraction at an angle proportional to the acoustic frequency [2]. In principle, it should be possible to extend the application of AODs to generate arbitrary wavefront shapes by varying the spatial frequency of the density grating, and therefore the diffraction angle, along the active axis of the modulator. However, AOD-generated density gratings propagate with the velocity of sound in the crystal and thus fail to generate stationary beam profiles [3]. This issue has been worked around in the simple case of linearly chirped acoustic functions for the implementation of diffractive lenses. In this case, acoustic propagation produces angular scanning, which can be eliminated by adjunctive AODs fed with counter-propagating acoustic waves of the same chirp [4,5]. However, this method does not generalize to the case of acoustic waves carrying arbitrary non-linear frequency modulations. In the present work we propose to solve the non-stationarity issue by synchronizing a sub-nanosecond pulsed laser source to the refresh cycle of the acoustic wave following an earlier and similar proposal by Bechthold et al. [6]. In this configuration, the acousto-optic phase grating is reproduced to the same state each time the laser pulse crosses the AOD optical window, by repeating the same time-locked acoustic function from pulse to pulse [6]. The maximal AOD update rate is given by the time for an acoustic wave to sweep the AOD optical aperture and ranges from 10 kHz to over 200 kHz, depending on AOD aperture size and crystallographic parameters, faster than with any other existing technology.

AOD-based beam shaping may serve many applications demanding fast wavefront control. Notably, AOD update rates match the time scale of gain regeneration in high power laser pulse amplifiers [7]. AODs therefore offer the possibility to implement pulse-to-pulse shaping of amplified laser pulses with ultimate efficiency. In this paper we specifically envision femtosecond amplified laser pulse shaping for applications in non-linear microscopy of bulk biological tissue. By permitting tilt and focus control of individual laser pulses, the modulator will allow addressing points within the accessible tissue volume at laser repetition rate. In the following, we derive the relationship between the acoustic wave pattern in the AOD and the spatial beam shape of the light diffracted from the AOD. As diffraction patterns in single AODs are one-dimensional, we elaborate the symmetry constraints applying to the representation of arbitrary two-dimensional phase functions and quantify the resulting approximation errors using the Zernike polynomial expansion. We demonstrate experimentally the beam shaping capability of this system and characterize its performance in terms of resolution and phase accuracy.

## 2. Methods

#### 2.1 Experimental layout

### 2.1.1 Optical setup

We set up a system [Fig. 1] consisting of two AODs (DTS.XY.400, *L* = 6.7 mm active aperture, 800-1000 nm, 45 mrad deflection range, 35 MHz acoustic bandwidth at 800 nm, AA Optoelectronic, Orsay, France) for beam diffraction in perpendicular directions (referred to as X-AOD and Y-AOD) together with a third AO crystal (AA.MTS.141, *L* = 4.5 mm active aperture, 840 nm, AA Optoelectronic) for compensation of spatial and temporal dispersion (referred to as AOM) [8]. By construction, X- and Y-AODs featured collinear direction of the incident beam and the first order diffracted beam at the central acoustic frequency. The access time of the AODs is *t _{access}* =

*L/v*= 10.3 µs, with

*v*= 650 m/s as acoustic velocity for anisotropic, non-rotated diffraction in TeO

_{2}.

As light source we used a laser system (Pharos SP6, Light Conversion, Vilnius, Lithuania) consisting of a diode-pumped Yb:KGW mode-locked oscillator (1028 nm emission, 76 MHz repetition, 90 fs pulse duration, 1 W power) together with a diode-pumped Yb:KGW chirped-pulse amplifier (80 kHz repetition, 190 fs duration, 75 µJ pulse energy, 6 W power). The amplified pulses were parametrically converted in an optical parametric amplifier (630-2600 nm tuning range, Orpheus OPA, Light Conversion) set to 840 nm and using about 20% of 500 mW available output power. Residual pump light (514 nm) from the OPA was suppressed with a longpass filter (750 nm cut-on, Thorlabs). The laser beam was magnified to fill the AOM aperture and the beam polarization at the AOM and the Y-AOD entrance were optimized using achromatic half-wave plates (690-1200 nm, Thorlabs). Y-AOD and X-AOD were conjugated by an afocal telescope with a magnification of one. The AOM to Y-AOD distance was adjusted to compensate for the pulse temporal dispersion in the acousto-optic systems using a GRENOUILLE autocorrelator (Swamp Optics) and the AOM frequency was set to compensate for the spatial dispersion of the laser pulse in the AODs [8,9]. Finally, the output wavefront was recorded using a Shack-Hartman sensor (HASO 4 FIRST, 3.6x4.6 mm aperture, 32x40 micro lenses, λ/100 accuracy, Imagine Optic, Orsay, France) conjugated to the AODs by a second telescope with magnification of 1, 4/3 or 10, using 100 mm/100 mm, 75 mm/100 mm (as shown in Fig. 1) or 50 mm/500 mm lens tandems, respectively.

### 2.1.2 Generation of the signals controlling the AODs

The radio frequency signals feeding the two AODs were obtained from a two channel direct digital synthesizer (DDS) with integrated power amplifier and digital input interface (DDSPA, AA Optoelectronic) receiving digital command inputs from two 50 MHz digital I/O boards (NI PXIe 6537, National Instruments). The digital commands comprised 32 bits coding for the acoustic frequency (23 bit) and acoustic power (8 bit). Synchronization of the AOD writing cycle to the laser emission was established as follows. First, an 80 kHz laser synchronisation signal was generated with a Schmitt trigger (74VHC123A, Fairchild Semiconductors) from the output of a fast photodiode at the exit of the Pharos pulse compressor. This timing signal was then used to trigger two digital pulse train generators on a multiple timer/counter board (NI PXIe-6612, National Instruments) providing synchronous clock signals for pixel update to the digital I/O and DDS circuits, respectively, in response to single trigger events. The clock signals consisted of 14 MHz trains of 145 pulses, commensurable with the given access time of the AODs (10.3 µs, see above), comprising a programmable post-trigger delay, common to both clocks, and a fixed phase shift to optimize the relative timing of I/O and DDS functions. The AOD update process was configured under LABVIEW 2015 (National Instruments) by supplying the I/O boards, through direct memory access, with update values of acoustic frequency and power, pre-calculated for every cycle of the write clock. The post-trigger delay was experimentally adjusted to shape the following pulse in such a way that the diffracted wavefront was well centred to the optical axis.

### 2.1.3 Optical wavefront analysis

For the analysis of single pulse wavefronts, we synchronized the image capture of the Shack-Hartmann sensor to the laser sync and set the AOD acoustic power to zero for all pulses emitted within the integration time of the camera, except for one pulse. The integration time was set to its minimum (500 µs). To meet the dynamic range of the Shack-Hartmann sensor, the pulse power was attenuated by passing the focalized beam through a 45° dielectric mirror with optical grade rear surface (~99% reflectance 750-1100 nm, BB1-E03P, Thorlabs). Optical wavefronts were reconstructed from the measured wavefront slopes using HASO 3.1 software (Imagine Optic) together with a custom plug-in for processing of large data sets (kindly provided by Imagine Optic) and exported to MATLAB 2013 (Mathworks).

#### 2.2 Phase and amplitude modulation by a single AOD

Let us first consider a single AOD providing phase and amplitude modulation along the x-axis and an incident beam propagating along the z-axis. The beam has a given incident amplitude profile *A _{in}(x,y)*, as e.g. uniform or Gaussian, and a given incident wavefront

*φ*; thus: ${A}_{in}\left(x,y,z\right)={A}_{in}\left(x,y\right){e}^{i\left({k}_{0}z+{\phi}_{0}\left(x,y\right)\right)}$, where

_{0}(x,y)*k*is the wave vector of the incident pulse. At the time of arrival of the laser pulse, the AOD is filled with an ultrasonic pattern of frequency

_{o}*f*and amplitude

_{AOD}(x)*a*at a position

_{AOD}(x)*x*in the AOD, with

*x*= 0 at the beam centre. Hereby it is implicitly assumed that the laser pulse is sufficiently short so that propagation and hence time dependence of the ultrasonic pattern are negligible during the diffraction process. Amplitude and frequency modulation is generated by time-varying command signals for frequency and power,

*f(t)*and

*a(t)*, respectively, applied to the transducer. The time when the commands have to be active to set the frequency and power at position

*x*is

*t = x/v*, with

*v*, the acoustic wave velocity (typically 650 m/s for TeO

_{2}in shear mode), and with the time origin

*t*= 0 chosen as the time when the command for position

*x*= 0 is applied, thus

*f*and

_{AOD}(x) = f(x/v)*a*.

_{AOD}(x) = a(x/v)Therefore, after interaction with the AOD, the wavefront ${A}_{out}\left(x,y,z\right)$ is given by: ${A}_{out}\left(x,y,z\right)={A}_{in}\left(x,y\right)T\left({a}_{AOD}\left(x\right)\right){e}^{i\left({k}_{0}z+{\phi}_{0}\left(x,y\right)+{\phi}_{AOD}\left(x\right)\right)}$, where *T* is the characteristic amplitude transfer function of the AOD, expressing the fraction of the input light transferred into the first diffraction order, as function of the acoustic wave amplitude *a _{AOD}*. The AOD provides therefore a complex spatial modulation

*M*given by:

_{AOD}(x)*T(a*, which is an implicit function of the amplitude modulation of the acoustic wave

_{AOD}(x))*a*and a phase modulation

_{AOD}(x),*φ*which is related to the frequency modulation

_{AOD}(x),*f*in a more complex way.

_{AOD}(x)At each point *x* of the AOD, the rays are deflected by an angle *θ(x)* given by twice the Bragg angle:

*x*, the rays leaving the AOD are perpendicular to the wavefront. Therefore, the optical path difference between the position

*x*and

*x*+

*dx*of the wavefront due to the deflection angle

*θ(x)*is

*θ(x)dx*and the wavefront is related to the acoustic frequency by:

*x = -L*/2 as reference position for the optical phase, where

*L*is the aperture size, and further assuming that the beam is aligned to the crystal centre. This expression shows that the phase profile created at a position

*x*does not only depend on the ultrasonic frequency at the pixel

*x*in the AOD, but non-locally on the entire ultrasonic frequency pattern from the reference position to the point

*x*. Equation (4) can be rewritten as:

*x*position. Equation (4) allows reconstruction of the output wavefront knowing the applied frequency function, while Eq. (3) allows determining the frequency pattern corresponding to a target wavefront

*φ*In the case of a digital wave synthesizer the functions

_{AOD}(x).*f(x)*and

*a(x)*are discretized with a step size set by the pixel writing clock. If the acoustic frequency is changed at

*N*positions

*x*

_{i}_{,}for

*i*= 1 to

*N*, the frequencies

*f*to be applied between

_{i}*x*and

_{i}*x*to create a target wavefront

_{i + 1}*φ*are:

_{AOD}(x)*f*(

_{i}*i*= 1 to

*N*) will create a wavefront

*φ*given by:

_{AOD}(x)#### 2.3 Accuracy of wavefront control

In Eq. (5) and Eq. (7) it is implicitly assumed that a given frequency *f _{i}* is valid exactly between position

*x*(equivalent to the time

_{i}*x*) and

_{i}/v*x*(equivalent to the time

_{i + 1}*x*). A random jitter of standard deviation (STD)

_{i + 1}/v*Δτ*in the electronic control of the AOD frequency pattern (see Discussion) induces a corresponding random phase jitter

*Δφ*at each frequency step

_{AOD}*Δf*[see Fig. 2(a)]. A single frequency step

*Δf*causes an angular deflection of

*Δθ = λΔf/v*of the wavefront, and a temporal jitter

*Δτ*translates to a spatial jitter

*Δx = vΔτ*in the location of the phase change. The resulting phase jitter Δ

*φ*is given by

_{AOD}*Δφ*and thus:

_{AOD}= (2π/λ)Δx Δθ#### 2.4 Phase and amplitude modulation by two orthogonal AODs

2D wavefront shaping can be achieved using two perpendicularly crossed AODs. This configuration will create 2D phase and amplitude modulation according to:

*a*and

_{1}(x)*a*are the ultrasonic amplitudes,

_{2}(y)*T*and

_{1}*T*the AOD transfer functions, and

_{2}*φ*and

_{1}(x)*φ*the phase functions in the X-AOD and Y-AOD, respectively, in analogy to Eq. (3). It follows that two-dimensional phase functions

_{2}(y)*φ(x,y)*can be fully reproduced in the crossed AOD configuration if they are linearly separable, namely obeying

*φ(x,y) = φ*, while functions containing non-separable terms, like crossed polynomial terms of the form

_{1}(x) + φ_{2}(y)*x*, can only be approximately realized. Functions containing only non-separable terms cannot be realized with two orthogonal AODs.

^{a}y^{b}#### 2.5 Diffractive lenses

Wavefront quadratic curvature corresponding to the effect of a spherical lens has the analytical form $\phi \left(x,y\right)=\frac{2\pi}{\lambda}\frac{\left({x}^{2}+{y}^{2}\right)}{2F}$, where F is the effective focal length of the lens, which is linearly separable into ${\phi}_{1}\left(x\right)=\frac{2\pi}{\lambda}\frac{{x}^{2}}{2F}$ and ${\phi}_{2}\left(y\right)=\frac{2\pi}{\lambda}\frac{{y}^{2}}{2F}$. Spherical lenses are therefore represented without error by an orthogonal XY-AOD pair. Using Eq. (3), and considering only the X-AOD, it follows that *φ _{1}(x)* is generated by linearly-chirped frequency signal ${f}_{1}\left(x\right)=\frac{v}{\lambda}\frac{x}{{F}_{AOD}}$ applied to the X-AOD. Recalling than

*t = x/v*, and defining the chirp rate as $\alpha =\Delta f/\Delta t$, the effective focal length

*F*can be expressed as:

_{AOD}#### 2.6 Zernike Modes

Zernike modes often achieve fast convergence in the expansion series of complex optical wavefronts and therefore are a convenient means of assessing beam shaping properties. The Cartesian representation of the Zernike polynomials [10] immediately reveals the decomposition into linearly separable and non-separable polynomial terms. Of the fourteen first modes, five modes are completely reproducible in the XY-AOD configuration, six modes are partially reproducible, while three are not reproducible, see Table 1.

The approximation error is given by the deviation of the AOD-approximated wavefront *Z _{i}^{AOD}* with respect to the corresponding Zernike template mode

*Z*(with

_{i}*i*, the composite Noll index). In our analysis we quantify this error by the RMS score of the relevant coefficient in the expansion series of

*Z*:

_{i}^{AOD}*a*, the coefficients of the Zernike expansion (to the 35th Noll order) of ${Z}_{i}^{AOD}$, that is${Z}_{i}^{AOD}\left(x,y\right)={\displaystyle \sum}_{j=0}^{j=35}{a}_{ij}{Z}_{j}\left(x,y\right)$, giving values between zero (no match) and one (perfect match).

_{ij}## 3. Results

To evaluate the beam shaping capability of the AODs in the synchronized mode, we set up a beam shaper consisting of an XY pair of AODs in 4f configuration and a third acousto-optic element (AOM) for compensation of chromatic and temporal dispersion [8]. We measured the spatial phase of single laser beam pulses after interaction with the acousto-optic phase gratings in the AODs using a Shack-Hartmann phase sensitive camera positioned in an optical plane conjugated to the AODs, as illustrated in Fig. 1. By programming different types of acoustic grating functions we verify the operation of the proposed AOD beam shaper.

#### 3.1 Timing accuracy

Laser and acoustic wave generation involve complex electronic circuits, including proprietary electronics with incomplete timing specifications as well as digital circuits running on asynchronous clocks, altogether challenging synchronization accuracy. We therefore set out to assess the timing properties of the system by characterizing directly the pulse-to-pulse phase variability present in the wavefronts after diffraction in the AODs. For this we defined a simple test protocol, schematized in Fig. 2(a), consisting of a single acoustic frequency jump applied to the Y-AOD producing a roof-shaped planar waveform in the y-direction [Fig. 2(a)]. We determined the statistics of the y-position of the wavefront deflection using single-pulse phase detection as explained in Methods. The centroid of the deflection can be attributed to the extrapolated intersection point of the two emerging planar wavefronts with a spatial precision much higher than defined by the Shack-Hartmann microlens distance. The standard deviation *Δy* of the intersection point is a direct measure of the temporal jitter ∆*t* = ∆*y/v*, with v the acoustic velocity, and of the resulting phase jitter Δ*φ _{AOD}*, according to Eq. (8) and Fig. 2(a). Figure 2(b) depicts the fitted wavefronts of 20 pulses (out of 1000 measured) together with the mean phase in response to 10 MHz acoustic step amplitude. Because the only deflection applied was in y-direction the one-dimensional wavefronts in Fig. 2(b) involved averaging of the phase in x-direction. In this way we measured a space jitter

*Δy*of 5.5 µm, a temporal jitter

*Δτ*of 8.7 ns and a phase jitter

*Δφ*of 2π/10 rad (

_{AOD}*i.e.*0.1λ) [Fig. 2(c)], representing STD. We performed the same measurement for various step amplitudes between −10 and + 10 MHz [Figs. 2(d) and 2(e)] confirming that space and time jitters are independent of the magnitude of frequency changes applied during digital synthesis of the acoustic wave [Fig. 2(f)]. This jitter is much smaller than the clock period

*T*used to write the acoustic wave (

_{write}*T*= 71 ns for a 14 MHz clock) and therefore does not produce significant crosstalk between neighbouring pixels during acoustic wave synthesis.

_{write}#### 3.2 Stable diffractive lenses

Given accurate timing stability we expect the system to produce stable diffractive lenses for jitter-free optical wave defocus. According to Eq. (3) wavefronts of quadratic curvature (Zernike defocus, Table 1) are created from linearly-chirped acoustic gratings (see Methods). To test the defocus we first applied a series of chirped frequency signals with chirp rates between −1.5 and + 1.5 MHz/µs to the Y-AOD [Fig. 3(a)] and measured the resulting wavefronts as before [Fig. 3(b)]. The measured phase was indeed well fitted by parabolic functions [Fig. 3(b)].

The effective focal length calculated from the phase curvature was a linear function of the inverse chirp rate [Fig. 3(c)] in agreement with Eq. (10), exhibiting a slope of 59.2 +/−0.5 cm/(µs/MHz) slightly larger than the theoretical calibration of an ideal Bragg lens of 50.3 cm/(µs/MHz) at 840 nm as given by Eq. (10). Since AOD diffraction gratings are one-dimensional, single AOD-generated lenses are cylindrical. However, since spherical lenses are equivalent to crossed cylindrical lenses of same focal length, spherical diffraction lenses are obtained by feeding synchronously the same acoustic chirp to X- and Y-AOD [Fig. 3(d)]. For the creation of quadratic wavefronts we expect an accumulated phase error of $2\pi \sqrt{N}\Delta \tau \Delta f$with *N*, the pixel number, *Δτ*, the synchronization jitter, and *Δf*, the acoustic frequency change per pixel (see Methods). Since *Δf* is limited by the available acoustic bandwith *f _{BW}* to

*Δf = f*, the upper limit of the phase error is $2\pi {f}_{BW}\Delta \tau /\sqrt{N}$ in quadratic wavefronts. Thus, with

_{BW}/N*Δτ*= 8.7 ns [Fig. 2(f)],

*f*= 35 MHz (see Methods) and

_{BW}*N*= 75, corresponding to the detection pupil in Fig. 3(b), the maximum error expected amounts to 0.035

*λ*. Notably, this error is of the same order of magnitude as the error of phase reconstruction in our phase measurement (see Methods). In fact, we find similar variance in our data of linear (

*N*= 1) and quadratic wavefronts (

*N*= 75). Therefore, within the limits of our measurement, and for practical considerations, phase error due to accumulation of synchronization noise is negligible small.

#### 3.3 Zernike Modes

The limitations of AODs to approximate arbitrary wavefronts stem from the condition that the generating functions must be linearly separable in orthogonal Cartesian coordinates. To elucidate the implications of symmetry constraint we measured the wavefronts of functions approximating Zernike polynomials up to 4th order using the decomposition given in Table 1.

We find good agreement between measured and predicted phase functions [Fig. 4(a)] and associated match indices [Fig. 4(b)]. Thus, tilts (*i* = 1, 2; *m* = 1), focus (*i* = 4; *m* = 1), and 0° astigmatism (*i* = 5, 13; *m* = 0.98) are completely reproduced, while trefoil (*i* = 6, 9; *m* = 0.3), coma (*i* = 7, 8; *m* = 0.6) and spherical aberration (*i* = 12; *m* = 0.46) are partially reproduced. 45° astigmatism (*i* = 3, 11) and quadrafoil (*i* = 10), on the other hand, yield zero match. As expected (see Table 1) symmetry cuts significantly on the representation of higher order modes. In principal, however, it is possible to enhance the representation of individual modes by rotation of the wavefront (or the AOD coordinate system) at the expense of modes of same radial, but different azimuthal order. For instance, 45° rotation obviously transforms 45°-astigmatism (*m* = 0) into 0°-astigmatism (*m* = 1) and vice versa. This suggests that better representation of higher order modes could be achieved by independent modulation in diagonal directions by addition of two AODs as described below (see Discussion).

#### 3.4 Astigmatism compensation

To illustrate the possibility of AOD-implemented adaptive correction of aberrant wavefronts we created virtual astigmatism by adding 0° astigmatism as an offset to the phase detector [Fig. 5(a)]. As vertical astigmatism corresponds to wavefronts with quadratic curvature of same magnitude, but opposite sign, in orthogonal directions, it is completely generated in the XY-AOD configuration as seen in Fig. 4(a). Thus, by supplying the AODs with linear chirps producing astigmatisms opposite to the astigmatism stored in the detector [Fig. 5(b)] it is possible to minimize the overall RMS deviation of the order of λ in the example given [Figs. 5 (c)-5(e)], and recover a nearly flat phase with λ/10 residual deviation [Fig. 5(d)].

## 4. Discussion

We have shown that AODs can perform optical phase modulation of pulsed laser beams with refresh rates close to 100 kHz, which is at least 2 orders of magnitude faster than achievable with current generation LC-SLMs. This performance is achieved using a pulse duration shorter than the period of the acoustic carrier wave (about <10 ns), satisfied by most pulsed lasers, and a pulse rate below the AOD repetition rate (97 kHz in our case) to permit shot-to-shot update of the acousto-optic phase grating. The use of smaller AODs or of faster propagating acoustic modes could adapt the AOD repetition rate to higher frequency lasers, but at the cost of fewer pixels. On the other hand, larger AODs like the 15 mm aperture devices used by Kremer et al. [8], which provide an access time of 23 µs, would yield a resolution of 323 pixels in combination with a 14 MHz update clock (as compared to 145 pixels in our case) at a repetition rate of 43 kHz, still much higher than alternative modulation methods.

AODs handle high-frequency components of the Fourier plane (large deflection angles) directly through the acousto-optic interaction induced by radio-frequency acoustic carrier gratings. By adding low frequency modulation to the carrier wave, AODs allow shaping of localized optical point spread functions while scanning large deflection angles, a combination essential for most applications like two-photon microscopy. In addition to speed, AOD-SLMs, as proposed in this work, offer large angular separation of first and zero order beams (> 9° in our case) and the capability to control the spatial distribution of the beam intensity at the same time, and with same spatial resolution, as its phase. LC-SLMs, in contrast, are pure phase modulators and require immense efforts to clear the modulated beam from un-diffracted zero order light appearing at the centre of the optical axis and forming a high intensity spot in the centre of the Fourier plane, e.g. the focal plane of a microscope objective [11].

While LC-SLMs are true two-dimensional phase modulators, the AOD-implementation equals two one-dimensional modulators in series. In consequence, the total number of pixels in the AOD configuration is the sum of x- and y-pixels versus the product in the case of LC-SLM. This restricts the symmetry of phase patterns that can be reproduced in the AOD-SLM, as we demonstrated in the case of the Zernike modes. These constraints, however, can be more or less limiting depending on the application. 3D focus control relies exclusively on tilt and defocus and therefore is fully achieved in AOD beam shaping mode. Furthermore, in adaptive optics experiments of complex tissue imaging it is often found that local corrections of low order, e.g. tilt, performed on a segmented pupil perform better than global corrections of high order applied to the entire pupil [12,13]. Although AOD-SLMs do not allow segmentation into arbitrary independent sub-pupils as achieved in LC-SLMs, they might still provide useful approximations for segmented phase correction. In this context it is interesting to note that in practice LC-SLM corrections are preferentially performed on overlapping rather than independent sub-pupils in order to increase the signals used for iterative optimization [12]. Moreover, if higher order modes, like primary spherical aberration for instance, are to be fully corrected at high speed one could set up an AOD system comprising more than two AODs. A configuration composed of four AODs, with their active axis separated by 45°, provides an accurate representation of wavefronts separable into directions *x*, *y*, *x + y* and *x-y*, instead of only directions *x* and *y* in the 2-AOD configuration. We expect the 4-AOD configuration to achieve full representation (*m* = 1) of all modes listed in Table 1 including primary spherical aberration (*i* = 12) with the only exception of oblique quadrafoil (*i* = 10) [Figs. 6(a)-6(b)].

Another distinct, and so far neglected [6], aspect of AOD-based systems is that pixels are not independent, as in the LC-SLM, but relate to other pixels through the non-local relationship given in Eq. (4). AODs modulate the optical phase in a fundamentally non-local mode which has important implications. First, pixels can inherit noise from preceding pixels (although these pixels are written later in time, see Methods). We therefore consider it an important outcome of our study to show that synchronization noise can be minimized to a level that the accumulated phase error caused by jitter-induced pixel noise remains actually very small. With regard to the origin of the residual synchronization jitter of about 10 ns in our system we believe that it is dominantly produced by the acoustic wave synthesizer and reflects the asynchrony of the pixel write clock with respect to the phase of the acoustic wave. In our system we use an acoustic carrier frequency of 95 MHz corresponding to an acoustic phase period of 10 ns, in the order of the observed synchronization jitter. A second implication of non-locality is that the AOD-generated phase is always continuous, even though acoustic frequencies are discretized, and free of 2π phase resets. This is a notable difference to LC-SLMs. Therefore AODs are not liable to efficiency loss in the same way as LC-SLMs. In LC-SLMs one major mechanism of efficiency loss, and hence diffraction ghosts, is due to the effective low-pass filtering of the phase reset discontinuity by fringing inter-electrode electric fields [14,15]. In consequence, aberrant diffraction intensity emerges within the angular scanning range of LC-SLMs as function of the phase reset period and therefore principally increases at larger diffraction angles [15]. In AODs, on the other hand, efficiency losses are incurred by aberration of the acoustic waveform in the near field of the piezoelectric wave generator causing deviation from Bragg diffraction [1]. In AODs, therefore, aberrant diffraction originates mostly from off-centre regions and with lesser dependence on diffraction angle.

AOD pulse shaping may offer novel perspectives for non-linear microscopy. Although the AOD update rate is orders of magnitude lower than the repetition rate of standard laser oscillators (80 MHz) used in non-linear microscopy, it matches the repetition rate of fast regenerative amplifiers. The gain in peak power afforded by regenerative amplification may indeed compensate for the lower duty cycle in many applications. This is particularly true for non-linear microscopy in biological tissue where amplified lasers enable deeper imaging depths [16–18]. By adding fast adaptive pulse shaping, AOD laser scanners may further improve deep tissue imaging by adaptive compensation of sample-induced aberrations to enhance the excitation focus beyond the scattering-limited transport length of excitation light in tissue. The most fundamental contribution of active beam shaping to microscopy is certainly the fast defocus. While an AOD lensing function was already successfully implemented into existing 3D microscopes [19–21], these implementations employ the counter-propagating-acoustic-wave configuration in four AODs for beam stabilization [4]. In comparison, the two-AOD plus AOM configuration offers the advantage of higher compactness, less exigent alignment, larger wavelength tunability and general beam shaping function. AOD generated diffraction lenses have a zoom range of $\pm {v}^{2}{t}_{access}/\left(\lambda {f}_{BW}\right)$, according to Eq. (10). With *f _{BW}* = 35 MHz,

*λ*= 840 nm,

*t*= 10.3 µs, the focal range will be −15 to + 15 cm. Therefore the lens can contribute a correction of up to ± 6.8 dioptres, reprogrammable from pulse to pulse, to the optical power of a microscope objective. In combination with an objective of 20x magnification and 9 mm focal length, for instance, this will allow shifting the objective focus up to about ± 520 μm in axial direction when using the full acoustic bandwidth, or ± 260 μm using half of the bandwidth. In this way the focus maintains the high NA of the objective in agreement with the focalization requirements of non-linear microscopy [22,23]. Finally fast defocusing could be used to compensate for fast 3D sample movements during

_{access}*in vivo*optical recordings.

To conclude, we have demonstrated a new AOD-based spatial beam shaper that allows spatial modulation at several tens of kHz up to 200 kHz. Beam shaping at such speed provides opportunities for 3D random-access scanning in non-linear-microscopy, including the possibility to correct aberrations independently for every scanned point.

## Acknowledgments

This work has received support under the program « Investissements d’Avenir » launched by the French Government and implemented by the ANR, with the references: ANR-10-LABX-54 (Memolife), ANR-11-IDEX-0001-02 (PSL* Research University) and ANR-10-INSB-04-01 (France-BioImaging infrastructure). We are grateful to the IBENS Imaging Facility, which received the support of grants from the “Région Ile-de-France” (NERF N°2009-44 and NERF N°2011-45), the “Fondation pour la Recherche Médicale” (N° DGE 20111123023) and the “Fédération pour la Recherche sur le Cerveau - Rotary International France” (2011).

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