We describe a high speed 3D Acousto-Optic Lens Microscope (AOLM) for femtosecond 2-photon imaging. By optimizing the design of the 4 AO Deflectors (AODs) and by deriving new control algorithms, we have developed a compact spherical AOL with a low temporal dispersion that enables 2-photon imaging at 10-fold lower power than previously reported. We show that the AOLM can perform high speed 2D raster-scan imaging (>150 Hz) without scan rate dependent astigmatism. It can deflect and focus a laser beam in a 3D random access sequence at 30 kHz and has an extended focusing range (>137 μm; 40X 0.8NA objective). These features are likely to make the AOLM a useful tool for studying fast physiological processes distributed in 3D space
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Two photon microscopy is increasingly being used to study biological processes [1–3]. Applications include imaging morphological structures, monitoring dynamic physiological processes with fluorescent reporters, triggering localized release of biologically active compounds with photolysis and controlling neuronal activity with genetically encoded light activated proteins. Many physiological processes of interest occur in small cellular structures at depths greater than 100 μm within tissue that absorbs and scatters light. The deep tissue penetration and submicrometer resolution that 2-photon microscopy provides has made this approach popular for studying such biological phenomena . However, current 2-photon microscopes, that use galvanometer mirrors to steer the laser beam and build up an image, are too slow to monitor many fast spatially distributed physiological processes, which occur on the 1-100 ms time scale, since they typically take more than 100 ms to form an image . Moreover, most microscopes developed to date are optimized for imaging a single X-Y plane. These constraints are particularly limiting for studying brain function, since information is encoded and transmitted as brief electrical impulses (~1 ms) in groups of neurons distributed in 3D space.
Several strategies have been employed to improve the temporal resolution of 2-photon imaging and monitoring of fluorescent reporters. These include using resonance scanners which speed up scanning and image acquisition , but the inevitable reduction in dwell time limits their utility except for the brightest fluorescent preparations. AODs have also been used because they provide a fast, mass-less scanning solution that is not limited by inertia. The first high speed microscopes using AODs in the linear raster scanning mode used a single AOD for fast X scanning and a galvanometer for the slower Y scanning [6,7]. An additional advantage of AODs is that their fixed and rapid response time (10-25 μs) and high precision control and pointing stability allow them to deflect a focused laser beam from one location to another very rapidly and reproducibly (pointing mode). This enables random access multiphoton (RAMP) measurements to be made from multiple regions of interest (ROI) at high speed [8–13]. Since only ROIs are imaged, rather than the whole plane, RAMP can be orders of magnitude faster than the scanning mode and is particularly suitable for ROIs that are small and sparsely distributed. Such microscopes use 2 (or 3) AODs to make fast 2-photon fluorescent measurements from one 2D XY plane in neural tissue. Similar 2 AOD designs have been used for single photon photolysis  and 2-photon photolysis  in 2D. However, the disadvantage of using AODs is that forming an image is complicated by chromatic and temporal dispersion. An additional problem in the scanning mode is that as the speed of scanning is increased the AOD introduces a cylindrical focusing effect [16,17]. The resulting astigmatism of the illumination point spread function (iPSF) can be corrected with a cylindrical lens, but only for a single scan speed, so most 2 AODs systems use pointing mode for full frame structural imaging. This is slow and can take several seconds per image .
There have been several recent advances in high speed 3D imaging. Sampling many locations (~100 neurons) within a 250 μm cube of cortex at 10 Hz has been achieved by combining fast piezoelectric control of the objective axial (Z) position and sophisticated galvanometer based XY plane scanning . Faster continuous focusing has been demonstrated using a piston mirror, and an ingenious dual objective system that corrects for spherical aberrations . The focusing properties of AODs have also been utilized for high speed focusing. Kaplan et al. (2001) first demonstrated that 2 AODs with counter propagating chirped acoustic waves could be used to produce a high speed (400 kHz) dynamic cylindrical lens  (illustrated in Fig. 1 ). They also proposed that 4 AODs could in principle be used to generate a spherical lens. These ideas were developed further and a proof of principle 3D 2-photon microscope was demonstrated that can focus rapidly and perform RAMP measurements . However, a limitation of this system, which is based on 4 conventional AODs to form the acousto-optic lens (AOL), is that light transmission efficiency drops off rapidly if the light is focused more than ± 25 μm from the natural focal plane of the NA = 1 objective. Fast 3D pointing with this system was demonstrated by measuring calcium transients within neuronal processes  within the octahedral shaped 3D field of view . The temporal dispersion of the AOL increased the 200 fs laser pulse width to the picosecond range, and resulted in reduced 2-photon excitation efficiency and thus increased the laser power required to image by an order of magnitude compared to conventional 2-photon systems. Moreover, structural imaging was slow (seconds per 2D plane) because the AOL was limited to the pointing mode rather than scanning mode . This is problematic because 3D functional optical imaging using RAMP measurements requires the prior acquisition of a stack of high resolution images of the whole volume to be studied so that ROIs can be accurately selected. These and other technical difficulties (section 1.2) have prevented AODs from being widely accepted as suitable deflectors for 2-photon microscopy.
To overcome these technical difficulties we have designed and built a compact spherical AOL that can be used to scan and focus a femtosecond laser beam at high speed. At the core of our AOL are 4 AOD crystals with properties optimized for their function. By incorporating the AOL in a conventional 2-photon microscope we have developed a high speed 3D 2-photon microscope that can image rapidly in 2D raster-scanning mode and perform RAMP measurements in 3D at 30 kHz. Here we describe the crystal design and drive algorithms and quantify the spatial and temporal performance of this prototype AOLM. Our results suggest that this technology is likely to be useful for a range of applications that require high speed imaging and 3D RAMP measurements.
1.1 Principle of operation of cylindrical and spherical AOLs
A dynamic cylindrical lens formed of two AODs is illustrated in Fig. 1. The acoustic transducer on each AOD crystal produces an ultrasonic sound wave that propagates across the optical aperture of the crystal.
The sound wave induces changes in refractive index of the crystal which diffracts the incoming laser beam at an angle determined by the frequency of the sound wave. By changing the sound frequency with time (chirping) the optical wave front can be curved bringing it to a line focus in the XZ plane. A stationary focus requires two counter propagating AODs to cancel movement produced by the curtain of sound as it propagates across the crystal. A pair of AODs can thus form a stationary cylindrical lens in the XZ plane .
Increasing the positive chirp rate moves the line focus up (increasing Z), whilst adjusting the difference between the ramp centre frequencies adjusts the X position. Changing to a negative chirp rate produces a diverging optical wavefront with a virtual focus before the AODs. A second pair of counter propagating AODs orthogonal to the first is required in order to focus in the YZ plane . The 4 AODs can then shape the optical wave front to give a spherically diverging or converging beam. This can be used to focus to a point above and below the natural focal plane of a subsequent fixed lens system.
1.2 Technical difficulties in using AODs for 2-photon microscopy
Although an AOL is potentially attractive for 3D pointing and high speed scanning, there are six significant technical difficulties to overcome for efficient 2-photon microscopy:
- 1. High efficiency AODs are made of tellurium dioxide (TeO2) , which has a high group velocity dispersion (GVD), causing temporal chromatic dispersion of ultra-short Ti-Sapphire laser pulses. Moreover, standard AODs have a large axial depth to aperture ratio so the large aperture (9 mm) AODs required for high resolution imaging have a large axial depth (~30 mm). Four conventional AODs have a GVD of ~80,000 fs2 and stretch femtosecond pulses to picosecond pulses, a level of dispersion that was considered impractical to compensate with a prism-based device . This temporal dispersion contributed significantly to the increase in laser power required for 2-photon imaging from 5 to 10 mW for galvanometer systems to ~100 mW for such an AOD based system .
- 2. The diffraction efficiency of standard AODs is strongly dependent on the incident angle of the laser beam. The range of angles of incidence for the second AOD of each pair is much larger than the first AOD, which receives a collimated incident beam (Fig. 1). As the designed input acceptance angle is increased, the overall AOD efficiency drops. It is important to trade off these factors to maximize the 3D field of view.
- 3. The spectra of the ultra short (100-200 fs) pulses of Ti-Sapphire lasers are typically 10-5 nm wide, respectively. Since the deflection angle is proportional to the wavelength for diffractive deflectors, chromatic aberration can cause significant elongation or distortion of the iPSF. This can be minimized by using a configuration that compensates the aberration at the centre of the field of view [8,22]. However, as the AODs deflect the spot away from the centre, the residual magnification chromatic aberration increases [11,24].
- 4. When AODs are used for high speed scanned imaging (above about 20 frames/second for 100 × 100 voxels) the iPSF becomes distorted as the scan speed increases due to focusing of the faster scanning AOD [16,17,25]. One solution to this is to have a set of astigmatic correction lenses optimized for each of a set of predetermined scan rates. The disadvantages of this are increased mechanical complexity and fixed scan rates.
- 5. The path length of the AOL is over 1 m long due to three telecentric relays that super-impose the output of one AOD grating on the input to the next . This is inconveniently large unless at least three pairs of precision aligned mirrors are used to fold the arrangement. Such a system is still bulky, very challenging to align  and has extra losses from the numerous optical components. It is therefore not ideal for widespread use.
- 6. Remote focusing, in combination with a high NA objective, is likely to be limited by spherical aberration [19,26]. However, it is uncertain how this limits the focal range for 2-photon microscopy, since commercial objective designs are proprietary. It is therefore not clear whether AOL limitations [21,26] or spherical aberration will ultimately limit the axial range of remotely focusing AOLMs.
In the following sections, we show how a high performance AOLM was designed to minimize these technical difficulties. This excludes improvement to residual chromatic aberration, which is beyond the scope of this paper. We also derive the control equations for a new compact AOL and develop a vector based approach to simplify imaging with AOLs.
2. Design of AODs and AOL for 2-photon microscopy
As outlined above, the three most basic problems in designing an AOL for femtosecond 2-photon microscopy are the high temporal dispersion caused by the thickness of the 4 AODs, the sharp fall off of diffraction efficiency with input angle and chromatic aberrations.
2.1 Design of AODs for high resolution imaging and minimum GVD
2.1.1 Speed and resolution
The speed of the AOD is limited by the time, T, it takes to fill the AOD aperture:Fig. 1 shows that the number of resolvable spots is equal to twice the time bandwidth product of each AOD . Equivalently the number of resolvable spots Nr can be expressed in terms of W, λ0 the optical wavelength and ‘s’ the semi-scan angle that the AOD deflects between the centre drive frequency and one of the frequency limits, which typically correspond to the 80% diffraction efficiency limits. These equations are:1]. Based on this, and a geometric optic derivation of the field of view at the same NA, it can be simply estimated that the number of resolvable spots for a 2-photon AOD system Nrtp is approximately:Eq. (1)). However with a 15 mm aperture, the axial thickness of standard 55° acoustic wave walk off angle AOD would cause a large GVD for 4 AODs (~100,000 fs2). This was judged impractically large to compensate  (Fig. 2 ; Technical difficulty 1).
2.1.2 Origins of the acoustic wave ‘walk off’ and design of thin, low GVD, AO crystals
Sound propagation within TeO2 crystals is highly anisotropic: in the <110> crystallographic direction it is V110 = 616 m/s whereas along the <001> crystal axis (optic axis) it is V001 = 2400 m/s. In standard AODs, the high diffraction efficiency, large scan range, uniformity and linear polarization characteristics are achieved with a crystal cut that produces an acoustic wave vector angled at 5-6° to the <110> (Figs. 2a and 2b).
Unfortunately, the 5° off <110> axis acoustic wave vector, coupled to the anisotropy of the acoustic velocity, causes the acoustic wave to walk off at an angle of approximately 55° −65° as shown in Fig. 2a. This has serious side effects for our application in that it causes the crystal to have a thickness much greater than its aperture size (Fig. 2a).To reduce the acoustic walk off, and thus the AOD thickness and GVD, we altered the crystal cut. Reducing the acoustic rotation of the crystal to 2°, produced an acoustic walk off angle to 22° because the acoustic walk off angle is 11-fold greater than the acoustic wave vector rotation from the <110> direction (Fig. 2c). However, if light propagates closer than about 3° to the <001> optical axis, it becomes increasingly circularly polarized. It is therefore difficult to couple external light efficiently to these circular propagation modes. To minimize this problem we optically rotated the crystal by 3° (see inset Fig. 2d) so that the polarization of the input and output waves are >80% linearly polarized. This optically rotated AOD design was first described by Young et al. (1990) . Optically and acoustically rotated AODs were fabricated for us by Gouch & Housego, (Florida, LLC).
The reduced walk off angle of the custom designed AODs reduces the axial depth of 15 mm aperture AODs to 15 mm. This reduced the temporal dispersion of the 4 AODs to 36,000 fs2 at λ0 = 800 nm. This, together with the 13000 fs2 dispersion from other microscope components, was sufficiently low to compensate with a custom prism based pre-chirper, with −50,000 fs2 GVD (APE GmbH, Berlin). This customized AO crystal design therefore overcomes Technical difficulty (1).
2.2 AOD designs for low and high input acceptance angle
The most serious problem to overcome in designing an efficient AOL concerns the steep dependence between diffraction efficiency and input acceptance angle for AODs (Technical difficulty 2). AODs typically have high diffraction efficiency over a wide drive frequency range. This is achieved because the acoustic vector is closely tangential to the inner ordinary index wave number surface that the diffracted output wave moves along as drive frequency is varied (Figs. 2b, 2d). This means that for a fixed direction input optical wave vector and fixed direction acoustic wave, a wide range of output angles can be obtained with only a small error between the sum of the incident and acoustic waves and the diffracted vector, which must remain on the ordinary wave number surface (vector with green polarization ellipse, Figs. 2b, 2d). This efficient acousto-optic coupling is not maintained as the angle of incidence of the input wave vector is varied because the acoustic vector is much less tangential to the index wave number surface (vector with red polarization ellipse). The diffraction efficiency of the AOD thus reduces much more rapidly with variation of incidence angle than with variation of output angle. This is a major problem because, as the laser beam is tilted and curved by the first AOD, transmission through the second AOD in each pair will become inefficient. Unless optimized, this property will severely restrict the focusing range and scan volume of the AOL.
A way around the narrow input acceptance angle problem is to increase the angular spread of the acousto-optic interaction. This can be achieved by lowering the drive centre frequency of the AOD so that acoustic wavelength is increased and by narrowing the width of the acoustic transducer in the direction of optical propagation (Fig. 3 ). The lower centre frequency of our custom AOD (35 MHz), compared to that of a standard commercial AOD (50-70 MHz), accounts for half the increase in acceptance angle shown in Fig. 2, but also reduces output scan angle range proportionately. For this application increasing the input acceptance angle is the limiting issue. To quantify these effects for our custom design we calculated the theoretical relationship between diffraction efficiency and transducer length normalized to 100% at 6 mm  (Fig. 3c). This relationship shows that as the transducer length decreases the acceptance angle increases. But, unfortunately, the diffraction efficiency drops as the transducer length decreases, because the acousto-optic interaction length reduces in proportion. To optimize both diffraction efficiency and uniformity of scanning we chose two different Lithium Niobate transducer lengths for the first and second AODs in each XZ or YZ plane pair. The first AOD of each pair, which receives light at a fixed input angle, had a relatively wide transducer (3.6 mm in our prototype) resulting in approximately 85% peak diffraction efficiency at ~3 W drive. Since we did not know the optimum transducer length for the second crystals, two transducers of 2.4 mm and 1.8 mm width were attached. These give acceptance angles of approximately 4 and 6 mrad at efficiencies of approximately 65% and 50% respectively at ~3 W drive.
In summary, the small input acceptance angle of AODs (Technical Difficulty 2) can be overcome by lowering the drive frequency and reducing transducer length for the second AOD of each pair to 2.4 - 1.8 mm, but this is at the expense of reducing diffraction efficiency as the transducer is narrowed. This trade-off between acceptance angle, which ultimately determines the field of view (FOV) and transmission efficiency, is a key design compromise that can be optimized for a particular AOL application by use of multiple transducer lengths on the second crystal of each pair.
2.3 Aberrations of AOD based deflectors
Unlike a mirror, the angle of deflection of an AOD is proportional to wavelength, introducing chromatic aberration with polychromatic femtosecond pulsed laser light (Technical difficulty 3). Chromatic aberration at the centre of the field of view can be corrected by ensuring that the deflection of the beam in each AOD of a particular pair are opposite in direction (Fig. 1a [22,29]). This is because, at this position, the deflections from the first AOD in each plane are exactly cancelled by the deflection of the second. The extent of the chromatic aberration increases with the radial distance away from the centre of field of view and the spectral width of the source. In the XY plane, a circular diffraction limited iPSF at the centre of field of view becomes an increasingly elongated ellipse as its distance from the centre increases. Its long axis is aligned radially. Equation (4) quantifies the expected dimension of the long axis, lxy, of an imaged bead of width b, at radius r from the centre of field of view (X = Y = Z = 0), for an optical wavelength λ and bandwidth Δλ, assuming a linear convolution of the diffraction limited Gaussian 2-photon PSF width w [1,11,12]:1], z is the axial displacement of the 2-photon PSF from the centre of the field of view, SA is the geometric spherical aberration and AstigZ is the axial distance between the XZ and YZ plane foci. As discussed in Technical difficulty (6), spherical aberration is expected to become significant at sufficiently large AOL focus Z displacement. These equations show that chromatic aberration components in the X, Y and Z dimensions (Technical difficulty 3), can be minimized by choosing a laser with a narrow spectral width . However, since the spectral width and pulse length in time are inversely related, and longer pulse widths reduce 2-photon excitation efficiency, there is likely to be an optimum pulse length.
The astigmatic aberration caused by high speed scanning of a 2 AOD XY scanner (Technical difficulty 4) can be most conveniently quantified by calculating the scan speed induced astigmatism in terms of the drive parameters used to set up the raster scanning. The total scan angle in the X or Y plane is 2s/zoom, where ‘zoom’ is the magnification factor. The additional parameters that define the angular scan rate in the fast X direction of the AOD are the number of voxels making up the image ‘Nvox’ and the dwell time per voxel ‘Dwell’. The angular scan rate is then used to estimate the degree of cylindrical focus from the AOD, which causes the astigmatism. The axial astigmatism ‘AstigZ’ is defined as the axial distance between the two orthogonal elliptical astigmatic beam waists. This can be calculated for a microscope objective in a material of refractive index n using geometric optics based on the assumption that the effective NA of the objective is limited by the width of the AOD W rather than any intermediate apertures. The equation is:
2.4 AOL with telecentric relays and the compact configuration AOL
The equations describing the acoustic drive waveforms necessary to form an AOL  assume zero optical distance between the AODs. A solution to this problem is to use a sequence of three 1:1 telecentric relays that project the image of one AOD onto the next  (Fig. 4 a ). For 15 mm aperture AODs the focal length, F, of each lens is likely to be >100 mm in order to keep the NA sufficiently low for diffraction limited imaging with simple lenses. Since the total length of each relay is 4F, the total length of the AOL is thus >1200 mm. Moreover, telecentric relays with a complex folding mirror arrangement consisting of 6 extra mirrors with 12 degrees of freedom in their adjustment  were used in the proof of principle 3D AOD system . The use of six lenses and six mirrors makes this configuration complex. In order to simplify the AOL we implemented the ‘compact configuration’ (Fig. 4b). This aligns the AODs axially with no intermediate lenses or mirrors so that the overall length of the AOL is reduced to the thickness of 4 AODs plus any air gaps and intermediate optical components (e.g. polarisers; section 3.1). However, this mechanically simple solution complicates AOL control because the drive equations that Kaplan et al., (2001)  derived are no longer directly applicable.
2.5 Drive equations for pointing and scanning for the ‘compact configuration’ AOL
In this section we derive the new equations required for driving the compact AOL in both the pointing and scanning mode.
Consider first the case of AOD1 and AOD2 in Fig. 5a . These are being driven at f1 and f2 respectively with chirped waveforms:Fig. 5b, consider a time t = 0 when the diffracted wave from a position X = 0 on AOD1 is precisely vertical down the Z axis, passing through position X = 0 on AOD2. Next, consider a time Δt later when the frequency of the ramp is shifted by Δf1 = a1Δt so that the angle of diffraction changes by a fixed amount Δθ1. In this case, the focus of the converging beam is displaced in the -X direction as shown. At the fixed point X = 0 on AOD2, the angle of the ray passing through that point on its way to the focus shifts by Δθ2 as shown. It is clear by simple geometry that:Fig. 5b, giving:Fig. 5c).
2.5.1 Deriving the Equations for the pointing mode
In order to calculate the specific drive equations for a particular desired focal spot distance d2′ (Fig. 5a), we used an equation that defines the distance to the focus for a given ramp rate . We also used the fact that in order to keep the focus stationary the curvature of the wavefront entering AOD2 is exactly doubled by the curvature added by AOD2. Simple rearrangement of these equations provides the relationships for setting up and controlling the AOLM:Fig. 5) and negative for diverging rays. It is apparent that if d1 = 0 then a1 equals a2 and the equations simplify to those presented by Kaplan et al. (2001) . For the −1 diffraction order (diffraction towards the transducer) the sign of deflection and curvature is reversed. Note that in the equations and analysis above, the distances are apparent optical thicknesses.
These principles can be extended to a system that uses four AODs to focus in the X, Y and Z dimensions. Figure 6 shows two orthogonal views of a four AOD system. The third and fourth AOD3 and AOD4 are interleaved with AODs 1 and 2 as shown, the distance between AOD3 and AOD4 being d3 and the distance from AOD4 to the focal point being d4′. The ramp rate equations for a3 and a4 are identical to those for a1 and a2 with d3 and d4′ replacing d1 and d2′. This derivation shows that a compact configuration AOL can be used to achieve a stationary focal point at any chosen value of Z within the field of view.
2.5.2 A graphical, vector-based method for finding start and stop frequencies of AODs
Since AODs have a limited acoustic frequency range, and the frequency ramp rates for Z focusing are high, the drive frequencies often reach the acoustic limit and have to be reset. It is therefore necessary to understand how to offset the drive frequencies without causing any movement of the focused spot for pointing mode (or restart an interrupted scan for raster scanning mode). Since the control of 4 AODs is a high dimensional problem, we developed a graphical vector approach to aid calculation of these parameters (Fig. 7 ).
For the case of XZ deflection and focusing with AOD1 and AOD2, the average drive frequencies fav2 and fav1 can be plotted on a graph (Fig. 7a). These ‘f average’ symbols represent the frequency at the centre of the AOD at the particular time being considered. As the ramp rates a1 and a2 are linear in both time and space, the frequencies at the centre are the spatial averages of the frequencies currently in the AOD aperture and in turn determine the average deflection angle of the AOD. Any pair of average frequencies can be represented by a point in the fav2, fav1 plane. A gradual or step change in the pair of drive centre frequencies can be represented by a vector lying in the plane. Its direction determines what happens to the focal spot position. If fav2 and fav1 are changed in such a way that there is no change in the position of the spot, this corresponds to a vector that we refer to as a common mode drive gradient Rcomm (Fig. 7a, blue arrow). A common mode drive frequency is defined as:Figs. 7a and 7b, in fav2, fav1 frequency space the vectors Rcomm = [1, Rcomm] and Rdiff = [1, Rdiff] (where vectors are shown in bold and scalars in normal type face) can be used as the unit vectors of a 2D basis. Any vector in this plane can be analyzed in terms of its common mode and differential mode basis vector components.
In Fig. 7b, the red point ‘P’ represents a particular fav2, fav1 coordinate. This can equally well be represented by the vector equation:
2.5.3 Algorithms for raster scanning mode
These ideas can be extended to raster scanning by simply considering a consecutive series of points in space. The graphical representation of this is shown in Fig. 7b, where the green line represents a time sequence of [fav2, fav1] coordinates that produce a focused beam slowly moving in the X direction. This can also be broken down into the vector components, C × Rcomm + D × Rdiff, as shown by the red and blue arrows forming a triangle with the green arrow. The small differential component reflects the slow speed in X of the focused spot, while the common mode component determines the Z focal plane.
The most commonly used raster scan is to move the focal point in the X direction, keeping the Y and Z values constant. Then the Y position is incremented by some small amount, to perform another scan in the X direction, and so on until a two-dimensional image is built up. The Z direction is then incremented and another two-dimensional grid is scanned until a three dimensional volume has been built up.
Consider now the drive frequency sequence necessary to scan a focal spot along the maximum possible distance in the X direction without exceeding the drive frequency limits of the AODs. Figure 8 shows the two X AOD centre frequencies plotted against time whilst scanning in the X direction. In order to produce the required focusing in Z, the ramp rates on both AODs are high. Since there is a limited frequency range, it is not possible to do a single scan right across the maximum possible X scan range (except at Z = 0). The scanning process therefore has to be broken up into a sequence of ‘miniscans’. Two of these miniscan frequency traces are plotted in the figure showing the transition between them.
It takes one AOD fill time for the sound wave to fill the AOD aperture so that data recording can begin, and the time at which a particular frequency reaches the midpoint of the AOD is half the AOD fill time after it was transmitted by the transducer. Since the frequencies are defined at the centre of the AOD, recording of fluorescence data from a full AOD is only possible for frequencies sent by the transducer more than half an AOD fill time from either the beginning or end of the frequency ramp. This further reduces the available frequency range as shown by the bold dotted lines (Fig. 8). The total time from the end of data gathering of one miniscan to the start of data gathering of the next is thus the AOD fill time plus any reset time for electronically resetting the drive frequencies.
The key point in developing an algorithm for calculating the precise start and stop frequencies of each miniscan is that the position of the focal spot at the start of the first voxel of a new miniscan must be in exactly the same position as it was at the trailing edge of the last voxel of the previous miniscan. This is illustrated in the lower part of Fig. 8 (red line). The frequency offsets Δf1 and Δf2 must therefore be in the exact ratio Rcomm.
Figure 9 illustrates how the fastest possible X scan is built up by a sequence of miniscans in the fav2 fav1 plane. The green arrows show the path in this frequency space of the two drive frequencies at the centre of the AODs. The blue common mode dashed arrows show how the frequencies are reset from the end of one data gathering sequence to the beginning of the next. The start and stop frequencies of the miniscans are then defined by extending these ramps out to the drive frequency limits as shown. Note that as in the pointing mode, the precise direction of the blue common mode unit vector and the red differential mode unit vector are dependent on the chosen focal Z plane according to Eq. (14).
It is now a fairly simple extension of the equations already derived to calculate the full equations for each important parameter. The X scan rate δθ/δ t is set at a rate so that it takes one dwell time to scan across one voxel:Fig. 6, when there are four AODs. In this case, the angular scan rate δθ/δt is that measured about the AOD 2 (which is third in Fig. 6). Since the focal spot is at a finite distance from the AOL, the apparent angular rate as measured about the centre of the last AOD 4, which we use as the reference point for calculating drive parameters, can be obtained by multiplying this scan rate by d2′/d4′.
Referring to Fig. 6, the appropriate equations for the YZ plane are found in a similar manner:
3. Design implementation and experimental Results
In this section we describe the implementation of our compact spherical AOL and the initial experimental results we have obtained. These include a demonstration of focusing at 30 kHz, the use of pointing mode for imaging over a large range of axial focus and a comparison of 2 AOD and 4 AOD imaging at different frame rates.
3.1 Design and assembly of the spherical AOL
Each AOD was mounted on a custom made goniometer that allowed fine control of the crystal orientation (Fig. 10 ). This was important for setting up the AOL because the X and Y deflectors must be orthogonal to a small fraction of a degree in order to avoid measurable astigmatism when focusing or defocusing strongly. Accurate alignment of the input angle is also critical in optimizing the transmission efficiency, because the correct diffraction mode must be selected from multiple diffraction modes on either side of the transmitted zero order mode. For the present application AODs were designed for −1 order mode with a nearly vertical polarization incident laser beam at the Bragg angle and nearly horizontal polarization for the diffracted beam (with reference to a horizontal deflector). To achieve the best alignment we used a polarizer, set near to the polarization of the diffracted beam, and optimized the crystal angle to get the maximum efficiency in the wanted diffraction mode. The same protocol was followed for the subsequent crystals. A half wave plate after the polarizer was used to adjust the polarization of the diffracted beam before it enters the next crystal in the series, enabling maximum efficiency in the desired mode. This combination of polarizer and half wave plate blocked the unwanted zero order mode after every crystal. The first AOD in each XY pair has an efficiency of 85% and the wide acceptance angle of the second AOD (using 2.4 mm transducers) in each XY pair has an efficiency of 65%. The overall efficiency of the AOL with polarizers and half wave plates is approximately 15% (all peak figures).
In order to drive the AOL with the required sequence of high speed radio frequency (RF) ramps a computer controlled digital synthesizer was used (iDDS, Isomet (UK) Ltd) to generate up to 20,000 trigger controlled pre-stored RF ramps. The parameter for each ramp was pre-computed using the algorithms described in section 2.5 in MatLab embedded in 3D imaging software developed in the LabView environment (National Instruments Inc.).
3.2 Speed of Pointing in 3D space
To determine the speed at which the AOL could focus to arbitrary positions in 3D space, we combined it with a single fixed lens (F = 200 mm) and used 3 silicon detectors placed in different focal planes with pinholes in front of each, as illustrated in Fig. 11a .
The location of each of the pinholes was determined by imaging in pointing mode, using the silicon detector signals. Random access pointing mode was then used to focus on each pinhole in turn repeating the sequence four times in different orders. The light intensity through each pinhole increased as focus was attained (Fig. 11b). The plot shows that the AOL focused the laser onto the three detectors in any order within 100 μs, thereby demonstrating random access pointing in 3D at 30 kHz.
3.3 Design and assembly of AOLM
To add the AOL to a standard 2-photon microscope, we relayed the image of the output face of the last AOD of the AOL to form a conjugate image on the galvanometer mirrors of a standard two photon microscope (Ultima, Prairie Technologies Inc., USA). A 4:1 demagnification was required to just fill the galvanometer mirrors. The subsequent field lens and tube lens form a conjugate image of the galvanometer mirrors on the back aperture of the microscope objective. We also used a Pockels cell to control the laser power, a custom designed, prism based pre-chirper with –50,000 fs2 GVD at 800 nm (APE GmbH, Berlin) before the AOL. A diagram of the complete AOLM is shown in Fig. 12 .
3.4 Temporal dispersion compensation
Once all the components of the prototype AOLM were in place, we used a Carpe autocorrelator (APE GmbH, Berlin), introduced in the optical path immediately after the laser (λ = 800 nm, MaiTai, Newport Spectra Physics), and measured the pulse width after the 40X objective with a 2-photon detector. The pulse length on leaving the laser was 100 fs. Without the pre-chirper the large dispersion of the AOL and microscope optics spread the pulse width to 1.52 ps (Fig. 13 ). We then adjusted the pre-chirper to minimize the measured pulse length after the 40X objective at the centre of the field of view.
3.5 3D Structural imaging in the pointing mode
To test the 3D AOL drive algorithms and determine the image quality that can be attained with the AOLM, we used pointing mode to build up 2D images at different focal planes.
To demonstrate 3D imaging we mechanically set the objective lens (water immersion 40X 0.8 NA lens, Olympus) to different focal planes above and below a pollen grain and compensated the focus with the AOL so that the image was of an equatorial section of the same pollen grain.
Figure 14 shows single frame and averaged 125 × 125 voxel images of the same pollen grain at different focal planes over a 137 μm axial focusing range. Visual inspection shows intensity across the pollen grain is reasonably uniform and spines that taper to sub-micron diameter are clearly visible at each focal plane. Indeed, there is relatively little difference in the image quality or the intensity at the different focal planes even though the images were taken at the same laser power and PMT settings. Furthermore, the 8 mW laser power (measured at objective back aperture) and 4 μs dwell time per voxel are comparable to those for a well set up conventional galvanometer based 2-photon microscope. These results show that the AOLM produces high quality 2-photon images at low laser powers over a wide range of focal depths.
3.6 Resolution and aberrations of the point spread function
To examine the optical resolution of the AOLM, we measured the 2-photon PSF with 200 nm fluorescent beads using a water immersion 40X 0.8 NA lens. To compare the predicted distortion of the PSF by chromatic aberration (Eq. (4), Fig. 15a ) and the properties of the PSF of our prototype AOLM, we compared images of beads for two different spectral bandwidths. Figure 15b shows a montage of beads measured at the centre, edges and corners of the field of a 100 μm square at Z = 0 using a Ti-Sapphire laser with a 100 fs pulse length and 10.6 nm FWHM spectral width (MaiTai, Spectra Physics). The red line on each figure indicates the FWHM of an elliptical Gaussian function that best fit the data. Note that the chromatic aberration is approximately radial and centered on X = Y = 0, as predicted (Figs. 15a, 15b). The FWHM for the bead image at the centre was 0.47 × 0.50 μm compared to 0.42 μm in theory (at 800 nm). Beads at the diagonal corners had an average long axis FWHM of 0.82 μm compared to 0.76 μm in theory from Eq. (4).
Figure 15c shows beads imaged with a laser with a longer pulse length (140 fs) and shorter spectral width (7 nm FWHM, Chameleon Ultra II, Coherent). The radial elongation of the bead images is noticeably smaller than for the wider bandwidth laser. Indeed the FWHM for the bead at the centre of the field was 0.55 × 0.58 μm, while the diagonal corner beads had an average long axis FWHM of 0.63 μm compared to 0.608 μm from Eq. (4). These results show that chromatic aberration can be minimized by reducing the laser bandwidth, but this comes at the cost of longer pulse lengths, which require a higher average power to achieve similar 2-photon excitation.
We then examined how the 2-photon PSF dimensions varied along the Z axis at X = Y = 0 by imaging fluorescent beads (Fig. 16 ). The XY dimensions of 200 nm beads were close to the diffraction limit at Z = 0, and the long axis of the elliptical bead image remained below 0.75 μm (FWHM) over the full 160 μm range of the AOL axial focus. The Z dimension of bead images was also close to the diffraction limit at Z = 0 (1.9 μm). The Z dimension of the 2-photon PSF increased with distance from the natural focal plane. The formation of cone shaped structures in addition to the core PSF indicates that spherical aberration is the dominant aberration at large displacements. However, there was also some asymmetry in the relationship between the Z dimension of the 2-photon PSF and axial focal depth, which is presumably due to unidentified residual aberrations. Nevertheless, it remained below 5 μm over the full 160 μm range of the AOL axial focus. These values are suitable for high resolution imaging within the 3D field of view.
3.7 High speed 2D imaging: comparison of a 2 AOD scanner and a 4 AOD AOLM
To compare the properties of our prototype 3D AOLM and conventional 2D 2-AOD scanning, we imaged pollen grains and 1 μm beads at the natural focal plane (Z = 0) at different speeds. The top row in Fig. 17a shows raster scan images of a pollen grain acquired with 2 AODs at 3 different frame rates, with the objective focused at the equatorial plane in each case.
The total light collection time for each image has been kept at 8 μs by integrating multiple frames at higher scan speed. Focusing by the fast scanning AOD becomes more pronounced as the scan rate increases introducing astigmatism and blurring the image. In contrast, raster scan imaging with the 4 AOD AOLM produced identical images at the different scan speeds, with no sign of any change in focus. We quantified the magnitude of the astigmatism by imaging 1 μm fluorescent beads. The measured astigmatism for 2 AOD scanning match that predicted from theory (Eq. (6)) and AstigZ reached 12 μm at 155 Hz (0.4 μs/pixel). There was no measurable astigmatism for the 1 μm bead images for 4 AOD scanning at any frame rate tested (i.e. up to 155 Hz). These results confirm that the scan speed dependent focusing and astigmatic distortion that occurs in 2 AOD scanning systems is not an issue in our prototype 4 AOD AOLM, therefore enabling flexible high speed line and raster scanning.
Here we describe the design of a high performance spherical Acousto-Optic Lens (AOL) that can be used for high speed 3D femtosecond based 2-photon imaging and random access point measurements. The improved performance over a previous AOL microscope design  was achieved by making a number of technical advances. These include using custom designed, thin, optically and acoustically rotated AODs with optimized transducer lengths and a low acoustic centre frequency to form the AOL. This overcame two technical difficulties. Firstly, it substantially lowered the GVD introduced by the 4 AODs allowing femtosecond imaging with a commercial pre-chirper. Secondly, it increased the input acceptance angle by ~3-fold over standard AODs. We have also reduced the complexity and large size of the previously proposed AOL configuration by designing a telecentric relay-free and mirror-free compact design with dimensions of less than 25 × 25 × 20 cm. Lastly, we have derived new control algorithms for the compact AOL and have developed a novel vector based approach that simplifies the calculation of the complex RF waveforms required to drive the 4 AODs during pointing and raster scanning. With our prototype system, we demonstrate random access pointing of the laser beam in 3D at 30 kHz and 2-photon imaging over 137 μm axial focal depth with a 40X 0.8 NA objective. This compares with the 50 μm range for a 1 NA lens previously reported . We show that temporal dispersion of the AOL microscope can be compensated with a prism based pre-chirper, allowing imaging with laser pulses in the 100 femtosecond range. We confirm earlier proposals that the AOL is self compensating against chromatic aberration at the centre of the field and that the observed residual chromatic aberration fits approximately to theory for two lasers with different spectral widths. Moreover, we show that the scan rate dependent astigmatism present in 2 AOD based systems is eliminated in our 4 AOD AOL, and demonstrate for the first time high speed raster scanning (>150 Hz) in 2D that does not exhibit scan rate dependent astigmatism. These results show that our AOLM design has a substantially improved performance over that previously reported  and allows 3D random access and raster scan 2-photon imaging with an order of magnitude lower laser powers.
4.1 Comparison to previous 2D AOD imaging
A new feature of our AOL and drive algorithms is the ability to perform high speed raster scanning without the scan rate dependent astigmatism that has impaired high speed AOD based scanning to date [16,17] (Fig. 17). Astigmatism occurs because the cylindrical focusing property of AODs becomes more pronounced as the scan rate increases. Interestingly, this effect has been used to produce a 3D two-AOD microscope. However, since this microscope’s focusing is cylindrical not spherical (unless the AODs are scanning at the same rate), and since the degree of focus is inextricably linked to the scan rate, capabilities are rather restricted . The scan rate dependent astigmatic effects introduced by AODs can be compensated with fixed astigmatic lenses, but these only compensate properly at one particular scan rate which severely restricts the flexibility of imaging. Although this limitation is rarely discussed, it is sufficiently problematic that most papers using 2 AODs to image [8,10–14,21–23,30] and the first 4 AOD 2-photon system [21,30], use sequential pointing mode, rather than raster scanning to form ‘structural’ images. This takes of the order of 1s for a 200 × 200 pixel 2D image (20 μs/point), which is significantly slower than imaging with conventional galvanometers. Here we develop the appropriate drive algorithms for the AOL to perform high speed raster scanning. We demonstrate, for the first time, variable rate raster scanning with an AOD-based scanning microscope that is free of scan rate dependent astigmatism. Moreover, we show that the astigmatism present with two-AOD scanning at rates up to 155 Hz is absent when the 4 AOD spherical AOL is used. High speed raster scan or line scan imaging is therefore an important design feature of the compact AOL.
4.2 Comparison with previous 3D 2-photon imaging technologies
Experimental results show that our prototype 2-photon AOLM is capable of high resolution 3D imaging and of high speed random access pointing in 3D space at 30 kHz. Indeed, focusing with our compact AOL is three orders of magnitude faster than recently reported for 3D 2-photon imaging using a piezoelectric device to move the microscope objective . However, the focusing range achieved is smaller (250 μm and 137 μm, respectively). Since the AOL can jump from point to point much faster than galvanometer mirrors, a larger fraction of the time can be used to collect photons from sparse regions of interest. These AOLM features should therefore enable considerably faster functional optical imaging application in neuroscience. For example, it should allow recordings from 30 neurons distributed in 3D space at 1 kHz sampling per location, which is considerably faster than current galvanometer based 3D scanning approaches . AOL focusing is also considerably faster and has a larger range than can be achieved with deformable mirrors . A recent approach, involving piston mirrors and the use of two objectives to compensate for spherical aberrations, can also be used to focus rapidly (<1 kHz) over a larger range than can presently be achieved with AOLs . However, like galvanometers, piston mirrors have the disadvantage of being continuous, which limits the duty cycle for sparse ROIs. The random access pointing with an AOLM therefore has some significant advantages over other proposed methods for the problem of high speed sampling of sparse distributed processes in 3D.
One of the key advances reported here is the use of optically and acoustically rotated AODs with less than half the axial length of standard crystals, which reduce the temporal dispersion to a level that can readily be compensated with a prism based pre-chirper, allowing femtosecond imaging. These design features enable 2-photon imaging with 4-10 mW at the back aperture, rather than the 40-100 mW reported for the first 3D 2-photon imaging system . To date such temporal dispersion compensation has only been possible for 1 AOD [6,7] and 2 AOD [8,10–13,15] scanners, but not for a 3D 4 AOD microscope, because the standard AODs employed have been considered too thick, introducing ~80,000 fs2 GVD . Thin AODs are also key in enabling a new compact AOL design, that discards the complex telecentric relays and folding mirrors used previously [15,21], placing the AODs adjacent to one another. Although there is a small increase in the required acceptance angle range for adjacent AODs, compared to telecentric coupled AODs (<8% over the focus range tested here), the imaging results we report demonstrate that the new AOD drive equations that compensate for the physical spacing of the AODs produce high quality images over a wide range of focal planes.
It is perhaps surprising that the remote AOL focusing into a 0.8 NA water immersion objective gives clear high resolution images of the pollen grain spines with little drop-off in imaging sensitivity over the whole 137 μm range. We believe that there are three main factors why our AOLM can focus over a wider range than for the previous design : 1) Assuming the NA of both systems are limited by AOD aperture, reducing the NA at the focal point from 1.0 to the 0.8 used here would be expected to increase the axial range by 50%, thereby increasing the axial focusing range of the previous system  from 50 μm to 75 μm. 2) Under strong focusing conditions, the larger AOD aperture in our system (15 mm compared to 9 mm ) enables lower magnification coupling optics to be used. This reduces the acceptance angle required at the second AOD of each pair for a given axial displacement. This, when combined with the improved acceptance angle of the second AOD in each pair, gives a more uniform aperture illumination. Our analysis agrees with the previous proposal that the acceptance angle is the main factor limiting the axial scan range for the AOL based on standard AODs . 3) Simple geometric optical modeling suggests that under strongly focusing conditions the large acoustic walk off angle (55°) of the conventional AODs previously used introduces a significant coma aberration. Indeed, even if the acceptance angle was increased by narrowing the acoustic transducer , the axial displacement may still be limited to a similar range by the coma aberration caused by the high walk off angle. This effect is substantially reduced with the 22° walk off angle of our AODs. Our imaging results at high axial displacement confirm optically and acoustically rotated AOD design does not cause significant coma, even at an axial displacement of ± 68.5 μm at NA = 0.8. Indeed, the appearance of cone shaped structures above and below the core PSF at such large displacements indicates that spherical aberration is the dominant aberration. Nevertheless, neither spherical aberration nor aberrations introduced by the compact AOL prevent our system from producing high quality images over a 137μm axial range at NA = 0.8.
4.3 Current limitations of the compact AOLM and potential solutions
The features and performance of our 2-photon AOLM are sufficiently advanced to make the current prototype useful for studying fast processes in 2D and 3D. However, a number of limitations still remain. While raster scanning can be performed away from Z = 0, the complex variations in diffraction efficiency that occur during the scanning process make it difficult to produce artifact-free images. Since our solutions to this problem are beyond the scope of this paper, the images showing the 3D scan volume in Fig. 14 were taken in the pointing mode. Increasing the volume over which effective 2-photon excitation can be achieved would also improve the utility of this technology. This is limited by a number of factors, including scan angle, AOL light transmission efficiency at large scan angles and distortion of the iPSF by aberrations. While some of these factors require further developments in AOD design, our predictions and measurement of significant chromatic aberration at the edges of a 100 μm square field of view suggest that the effective 2-photon imaging volume could be significantly increased by reducing this aberration. Although our AOLM compensates for spatial chromatic aberration at the centre of the field of view as previously proposed , we have not corrected for residual magnification chromatic aberration. This could be corrected for further either by reducing the spectral bandwidth of the laser , or by compensating this aberration with a dispersive lens system we have recently proposed .
We present a new design for a 3D 2-photon microscope based on a compact AOL, that has a significantly improved performance over the first proof of principle 2-photon AOLM [21,30]. Its features make it suitable for both high speed 2D imaging applications and 3D pointing applications such as 2-photon functional imaging and photolysis. The small size and mechanical simplicity of our compact configuration AOL, make it possible to simply add it, together with a pre-chirper, to an existing galvanometer based 2-photon microscope thereby enabling high speed 2D and 3D 2-photon imaging.
The work described in this paper was supported by MRC (G0400598), EPSRC (EP/D501199/1), UCL-Capital Infrastructure Fund, UCLB and the Wellcome Trust. PAK was supported by Complex and the Beloe fellowship from the Worshipful Company of Scientific Instrument Makers. RAS is in receipt of a Wellcome Trust Senior Research Fellowship (064413). We thank Stephan Dieudonné for helpful early discussions on AODs, Warren Seal for advice on AOD design, Mike Hillier for advice on iDDS design, Duncan Farquharson and Alan Hogben for building the mechanical mounts for the AOL, Bruno Pichler for early programming help and advice, Thomas Mrsic-Flogel for support and Prairie Technologies for the loan of equipment. We thank Stéphane Dieudonné, David DiGregorio, Michael Szulczewski and Emmanuelle Chaigneau for helpful comments on the manuscript.
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