1. Introduction

The performance of optical imaging systems, from microscopes to telescopes, is often deteriorated by the presence of temporally or spatially varying optical aberrations, such as those induced by the atmospheric turbulences [1] and the inhomogeneous refractive index distribution within biological specimens [2], which cannot be corrected by a fixed optical design. By dynamically measuring those wavefront distortions and then correcting for them using an active wavefront control device, adaptive optics (AO) can recover diffraction-limited performance of optical imaging systems [1, 2].

According to the way the wavefront is measured over the two-dimensional aperture, adaptive optics systems are classified into two categories: the zonal and modal approaches [1]. The zonal approach divides the wavefront into an array of segments, with the wavefront within each segment represented in terms of its local gradient and phase. The modal approach treats the wavefront as the sum of a series of orthogonal modes (e.g., Zernike polynomials) extending over the whole aperture. Although effective for low-order aberrations, the modal approach is often limited by the challenge of measuring and correcting the increasingly complicated higher-order modes [1]. As a result, the zonal approach, particularly those based on the Shack-Hartman gradient sensors [3], gained popularity for wavefront sensing in adaptive optics.

In a Shack-Hartman gradient sensor, the wavefront is physically divided into independent segments by a lenslet array and generates an array of focal spots on a CCD camera located at the focal plane of the lenslets. The local wavefront gradients for each segment are determined by the displacement of the centroid of each focal spot from its position with plane wave input. To reconstruct wavefront surface from this set of discrete local wavefront gradient measurements, a variety of wavefront reconstruction algorithms have been proposed [4, 5]. For Shack-Hartman sensors, a popular method to determine the wavefront from the gradient data is formulated by Southwell, where the wavefront within each segment is determined by its own gradient and those of its neighbors through numerical methods such as matrix iterations [4, 5].

However, there are several drawbacks for these wavefront reconstruction methods. First, because wavefronts in nearby segments are coupled, gradient measurement error in a single segment would propagate into the wavefront reconstruction of its neighbors. Secondly, not all imaging systems have continuous wavefront: for systems where segmented optics is used (e.g., primary mirrors in Keck telescopes), incorrectly aligned segments lead to discontinuous wavefronts, for which the above reconstruction methods do not apply and other methods have been developed [6–9]. Furthermore, even for a continuous wavefront, if the size of pupil segments is too large to adequately sample the wavefront (e.g., if the segment size of the primary mirror in a telescope is much larger than the atmospheric turbulence coherence length), the best-fit wavefront with the least residue phase error may not be continuous at segment boundaries and using the reconstruction methods described above may lead to sub-optimal performance. For these reasons, it would be valuable to have a phasing method that can directly measure the phase corrections needed for each segment independently after the gradient measurement is achieved. Herein, we reported such a direct phase measurement method and applied it to an AO two-photon fluorescence microscope. We compared its performance with that obtained with phase reconstruction using the zonal matrix iterative method [4, 5] and demonstrated its superiority for correcting large or discontinuous aberrations.

2. Multidither coherent optical adaptive technique for phase measurement in zonal wavefront reconstruction

Our phasing method can be understood with the physical picture that a diffraction-limited focus results from the constructive interference of light rays from all the wavefront segments at the focal spot. In the context of a two-photon fluorescence microscope, where signal and resolution depend solely on the quality of the excitation laser focus, a zonal AO correction is composed of two steps, similar to the process of wavefront sensing with a Shack-Hartman sensor: in the tilt correction step, the local wavefront gradients are measured and used by a wavefront correction device to direct all light rays from all the segments at the back pupil of the microscope objective towards the focal spot; in the phase correction step, to have all rays interfere in phase, we need to add additional phases, obtained by either phase reconstruction or direct measurement, to these rays so that they have identical phase at the focus. We previously demonstrated a direct phase measurement method [10]: assigning one ray as the reference (with unknown phase θ_r), we add additional phase ∆θ to another ray (with unknown phase θ₁) while monitoring the signal at the focal spot, which is determined by focal intensity I = 2 + 2 cos (∆θ + θ₁ - θ_r). Fitting the signal variation curve obtained by stepping the value of ∆θ from 0 to 2π (usually 5 evenly spaced phase values are chosen), we obtain the ∆θ that gives the maximal signal (i.e. ∆θ = θ_r − θ₁). Repeating this process for all other rays relative to the same reference ray, we obtain the phase additions that would allow all rays to be in phase and thus interfere constructively at the laser focus.

However, such serial phase determination for each segment limits the speed of AO correction. In this paper, we employ the concept of multidither coherent optical adaptive technique (COAT) [11–13] to measure the phases of multiple rays (i.e., wavefront segments) in parallel. Consider again the simplified two-ray system described above: if, instead of stopping after one cycle of signal variation, we continue to vary ∆θ at step size ω, laser intensity then varies as I = 2 + 2 cos (ωt + θ₁ − θ_r); to determine the optimal phase ∆θ = θ_r − θ₁ is to find the phase of the function cos (ωt + θ₁ − θ_r). One of the many ways to do this is by Fourier-transforming the time-dependent signal and reading out the phase at the frequency ω/2π. As demonstrated in multidither COAT previously [11–13], one advantage of measuring ∆θ in the frequency domain is that we can now modulate multiple rays at distinct ω_i’s simultaneously and get their respective ∆θ’s in parallel by reading out the phases of the Fourier-transformed signal at distinct ω_i/2π’s. Another advantage is that if the measurement noise power density decreases with frequency (e.g., 1/f noise), better signal-to-noise ratio can be obtained by moving the phase detection away from DC into nonzero modulation frequency.

3. Implementation of multidither COAT-based phase measurement in an AO two-photon fluorescence microscope

Figure 1 illustrates how this multidither COAT-based phase measurement could be implemented in an AO two-photon fluorescence microscope. We first modulate the phases of half of the segments (e.g., grey segments) at a series of distinct frequencies ω_i’s and keep the phases of the other half (e.g., white segments), which serves as the reference, constant, as shown in Fig. 1(a). The modulated rays and reference ray interfere and modulate the intensity of the focus, resulting in two-photon fluorescence signal variation characteristic of an interference fringe in Fig. 1(b). Mathematically, the overall electric field at focus is described by Eq. (1):

 

Fig. 1 Schematics of multidither COAT used to directly measure the phases of pupil segments for an AO two-photon fluorescence microscope. (a) We first modulate the phases of half of the segments (e.g., grey segments) at distinct frequencies ω_i’s and keep the phases of the other half (e.g., white segments), which serves as the reference, constant. (b) The modulated rays and the reference ray interfere and modulate the focal intensity, resulting in two-photon fluorescence signal variation. (c-d) The phase for each modulated segment Δθ_i is directly read out from the Fourier transform of the signal trace at its modulation frequency. (a-d) is repeated to obtain the phases for the other half segments (e.g., modulate white segments with the grey segments as reference). (e) Schematics for the AO two-photon fluorescence microscope. Output from a Ti:Sapphire oscillator reflects off a mirror (M) or a segmented mirror (SM), a spatial light modulator (SLM), a X galvanometer (X galvo), a Y galvanometer (Y galvo), another mirror (M), and is focused by a microscope objective (Obj) into a sample (S). The resulting fluorescence is separated by a dichroic beamsplitter (Dichr), collected by a lens, and detected by a photomultiplier tube (PMT). Achromatic lens pairs are used throughout the beam path for conjugation. A field stop (FS) is used to block the higher diffraction order from the SM when needed.

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The focal intensity then varies as Eq. (2):

Here, M(t) describes the interference between the modulated segments, which, with proper choice of ω_i’s, leads to intensity modulation at frequencies different from ω_i’s. For two-photon fluorescence, the fluorescence signal varies as Eq. (3):

Just like before, the optimal phase ∆θ_i = θ_r – θ_i for each modulated segment can be directly read out from the Fourier transform of S(t) at its modulation frequency ω_i as shown in Figs. 1(c) and 1(d). After the phases for the first half of segments are determined, the reference and the modulated segments are swapped and the same procedure is repeated so that the phases at all segments could be determined. Different from the calculation-based phase reconstruction approaches described above, the phase for each wavefront segment is determined independently in this approach, which doesn’t rely on the assumption that the wavefront is continuous. One obvious problem with this method is that, depending on the initial aberration in the system, the reference phases in the two steps could be different. Several iterations of the phase measurement can be repeated till the reference phases in the two steps are effectively identical.

We implemented this multidither COAT-based phase measurement method in our homebuilt AO two-photon fluorescence microscope described previously [10, 14]. A schematic of our setup is shown in Fig. 1(e). Briefly, a phase-only liquid crystal spatial light modulator (SLM, 512 × 512 pixels, Boulder Nonlinear Systems, Inc.) was used to segment and control the wavefront of the excitation laser (Ti:Sapphire oscillator tuned to 900nm, Ultra II, Coherent Inc.). It was conjugated with lens pairs to the back pupil plane of the microscope objective (Obj, NA 0.8, 16×, Nikon), as well as two galvanometer mirrors (X and Y galvo, 6215HB, Cambridge Technologies) so that the corrective wavefront pattern was stationary at the back pupil during beam scanning. Fluorescence was directed towards a photomultiplier tube (PMT, H7422-40A, Hamamatsu) by a dichroic beamsplitter (Dichr, FF665-Di02-25x36, Semrock Inc.). For the experiment where a segmented mirror (SM, PTT111, Iris AO) was used to introduce discontinuous aberration, the SM was conjugated to the SLM using a lens pair. A field stop (FS) located in the Fourier plane of the first lens after the SM can be used to remove the higher diffraction orders from the SM, if so desired.

For tilt correction (i.e., wavefront gradient sensing), we used the indirect wavefront sensing methods developed previously, since they work for both transparent and scattering samples. In one approach [10], one pupil segment was illuminated at a time and the corresponding two-photon fluorescence images were recorded. Similar to a Shack-Hartman wavefront sensor, the wavefront gradient for each segment can be directly calculated from the image shift relative to a reference image. An alternative approach illuminated the full pupil at all time [14], and wavefront gradients were derived from the additional slope each segment needs to reach maximal interference with the rest of the focus. In either case, wavefront gradients can be presented as a 2D array similar to the foci images from the CCD camera of a Shack-Hartman sensor, where the wavefront tilt in each wavefront segment can be derived from the displacement of the images from center. At the end of tilt correction, phase ramps were added to SLM segments which redirected all the rays to intersect at the same focus. For multidither COAT-based phase measurement, we either modulated the phase of each ray by changing the phase of the SLM segments or the piston of the SM. The phase values obtained from the Fourier transform of the signal were then added to the SLM to obtain the final corrective wavefront patterns.

4. Results

To test the performance of our method, we introduced artificial aberrations to our microscope by overlaying phase patterns to the SLM, and compared how well the corrective wavefronts, obtained either with our direct phase measurement method or the zonal matrix iterative reconstruction approach [4, 5] (referred below as “phase reconstruction”), recovered the image quality of a 2-μm diameter fluorescent bead, as shown in Fig. 2. Figures 2(a)-2(d) show the results from the AO correction of the first artificial aberration. The excitation wavefront was divided into 36 independent segments and a series of two-photon fluorescence images of the bead were taken with one segment illuminated at a time. Those images were arranged in an array according to the relative positions of these segments, as shown in Fig. 2(a), thus generating an output similar to that from a Shack-Hartman sensor. The local wavefront gradients were determined from the shifts of the images (relative to a reference image where the bead is at the center). Figure 2(b) shows the axial images of the bead before AO correction (b1 “No AO”), after applying the corrective wavefront that only corrected the tilt (b2 “Tilt Only”), after applying the corrective wavefront calculated by phase reconstruction (b3 “PR”), after applying the corrective wavefronts obtained after the first (b4 “PM 1st”) and second (b5 “PM 2nd”) iterations of phase measurements, respectively, and finally, the image taken under the ideal, aberration-free condition (b6 “Ideal”). Compared to the image obtained without AO correction, tilt correction alone did not improve the signal, because although the raysintersected at the same spot, they were not in phase and did not constructively interfere. Reconstructed wavefront worked well in recovering the signal to its diffraction-limited ideal value, with 6× signal gain after AO correction. With direct phase measurement, it took two iterations to increase the signal to the same level (see also, the line intensity profile comparison in Fig. 2(c)). Figure 2(d) shows the corrective wavefront patterns obtained by phase reconstruction and direct phase measurement. The AO correction of another artificial aberration was summarized in Figs. 2(e)-2(h), with the wavefront gradients determined the same way as before. Here, Fig. 2(f2) shows that tilt correction improved signal slightly, compared to the signal level when there was no AO correction in Fig. 2(f1). Phase reconstruction in Fig. 2(f3) again recovered the diffraction-limited performance [Fig. 2(f6)] and increased the signal 3× (see also the line intensity profile comparison in Fig. 2(g)). For phase measurement, it only took one iteration [Fig. 2(f4)] for the corrective wavefront to almost achieve the unaberrated signal level, with the second iteration only improving the signal marginally (less than ~8%) as shown in Fig. 2(f5), suggesting that the reference phases used in the two halves of the first-iteration phase measurement are similar to begin with, possibly due to the higher symmetry of the applied artificial aberration pattern. The final corrective wavefront patterns from phase reconstruction and direct phase measurement are shown in Fig. 2(h).

 

Fig. 2 Application of direct phase measurement method to two moderate aberrations. For the first aberration, (a) Wavefront gradients for 36 pupil segments are determined from the centroid shifts. (b) Axial fluorescence images of a 2 µm diameter bead (b1) before AO correction (“No AO”), (b2) after applying corrective wavefront that only corrected the tilt (“Tilt Only”), (b3) after applying the corrective wavefront calculated by phase reconstruction (“PR”), after applying the corrective wavefronts obtained with (b4) the 1st (“PM 1st”) and (b5) 2nd (“PM 2nd”) iteration of phase measurements, and (b6) under ideal, aberration-free condition (“Ideal”), respectively. (c) Line intensity profiles along the dotted yellow line of the images in (b). (d) Final corrective wavefront patterns in units of waves (λ = 900nm) obtained by phase reconstruction and direct phase measurement, respectively. (e-h) same as (a-d), except for the second aberration. Scale bars: (a) 6 μm, (b, f) 2 μm, and (e) 10 μm.

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The above examples show that for continuous aberrations of moderate amplitudes, although additional iterations may be needed, direct phase measurement works similarly well as phase reconstruction. However, for aberrations with high spatial frequencies and/or large amplitudes, the direct phase measurement often works better. In the example shown in Figs. 3(a)-3(d), the artificial aberration has a highly sloped wavefront, as indicated by the large displacements in Fig. 3(a). As a result, the introduction of this aberration into our microscope decreased the signal of the bead by 680× (Fig. 3(b1) was obtained by increasing the excitation laser power by 13 times). Corrective wavefront from phase reconstruction increased the signal by 227× [Fig. 3(b2)]. In contrast, the 1st and 2nd iterations of direct wavefront phase measurement improved the signal 340× [Fig. 3(b3)] and 362× [Fig. 3(b4)], respectively. Here, the 2nd iteration of phase measurement recovered 60% more signal than the phase reconstruction approach. It can be seen from the corrective wavefronts, as shown in Fig. 3(c), that phase reconstruction led to a more continuous wavefront, while the better fit to this aberration obtained via phase measurement was more discontinuous, because the size of the pupil segments is too large given the spatial variation of this aberration. This undersampling can also be appreciated by noting that our best correction only recovered the signal to 53% of the ideal value, as shown in the line intensity profiles in Fig. 3(d).

 

Fig. 3 Application of direct phase measurement method to undersampled aberrations. (a) Wavefront gradients for 36 pupil segments are determined from the centroid shifts. (b) Axial fluorescence images of a 2 µm diameter bead (b1) before AO correction (obtained at a laser power increased by 13 × , thus 170 × increase in signal), (b2) after applying the corrective wavefront calculated by phase reconstruction, after (b3) the 1st and (b4) 2nd iteration of the direct phase measurement, and (b5) under ideal aberration-free condition, respectively. (c) Final corrective wavefront patterns in units of waves (λ = 900 nm) obtained by phase reconstruction and direct phase measurement, respectively. (d) Line intensity profiles along the dotted yellow lines of the images in (b). (e) Lateral (XY) and axial (XZ) fluorescence images of YFP-labeled dendrites at 270 µm depth in mouse brain in vivo. The axial images were taken along the green solid line. (f) Line intensity profiles along the dotted yellow line of the images in (e). (g) Final corrective wavefront patterns in units of waves (λ = 900 nm) obtained by phase reconstruction and direct wavefront phase measurement, respectively. Scale bars: (a) 16 μm, (b) 2 μm, and (e) 5 μm.

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A biologically relevant example is shown in Figs. 3(e)-3(g), where we compared the effectiveness of phase measurement and phase reconstruction in improving the images of YFP-labeled dendrites at 270 µm depth in mouse brain in vivo (Thy1-YFP-H line [15]). Similar to the artificial aberration examples, image signal and resolution were improved by wavefronts obtained with both methods [Fig. 3(g)], as shown by the lateral (XY) and axial (XZ) images in Fig. 3(e). However, phase reconstruction only led to a 2.7× increase in the signal of a dendritic spine, while phase measurement worked better and increased the signal by 4 × [Fig. 3(f)]. This is consistent with our previous observation on the serial phase measurement method, where corrective wavefront obtained from phase measurement gave a better correction than that from phase reconstruction in mouse brain slices [10].

Another condition under which the direct phase measurement approach is superior to the phase reconstruction approach is when dealing with discontinuous aberrations, as commonly seen in telescopes deploying segmented mirrors [7]. In our microscope, a segmented mirror with phasing error was introduced at a plane conjugated to the SLM. Tilt correction from wavefront gradient measurement [Fig. 4(a)] improved the signal 3.2×, as shown in Fig. 4(c2). The direct phase measurement method was implemented using either the SLM or the segmented mirror, both of which led to further signal recovery to 4 × , as shown in Figs. 4(c4) and 4(c5) for segmented mirror and Figs. 4(c6) and 4(c7) for SLM (also see Fig. 4(d) for line intensity profile comparison). In contrast, the phase reconstruction approach failed to improve the signal, as shown in Fig. 4(c3). By attempting to generate a continuous wavefront [Fig. 4(b)] from the gradient data, it introduced large phase error and deteriorated the signal so much that it was only 92% of the signal obtained without any correction [Fig. 4(c1)].

 

Fig. 4 Application of direct phase measurement method to a discontinuous aberration introduced by a segmented mirror. (a) Wavefront gradients for 37 mirror segments were determined from the centroid shifts. (b) Final corrective wavefronts in units of waves (λ = 900 nm) obtained using phase reconstruction, direct phase measurement method implemented on the SM, and direct phase measurement method implemented on the liquid crystal SLM, respectively. (c) Axial fluorescence images of a 2 µm diameter bead (c1) without AO correction, (c2) after applying the wavefront that only corrected the tilt, (c3) after applying the corrective wavefront calculated by phase reconstruction, after (c4) the 1st and (c5) 2nd iteration of the direct phase measurement method implemented on the segmented mirror (SM), after (c6) the 1st and (c7) 2nd iteration of direct phase measurement implemented on a liquid crystal SLM conjugated to the SM, respectively. (d) Line intensity profiles along the dotted yellow line of the images in (c). (e) The discontinuous aberration of the segmented mirror was temporally stable: a previously measured corrective wavefront, when applied 8 hours later, improved the signal by the same amount as a contemporaneously obtained wavefront. Scale bars: (a) 26 μm and (c) 2 μm.

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Since our direct phase measurement method does not rely on the assumption of a continuous wavefront, the resulting corrective wavefronts were discontinuous and corrected the phasing errors on individual mirror segments. We also found that the discontinuous aberration of our segmented mirror was temporally stable for at least 8 hours. As shown in Fig. 4(e), the corrective wavefront obtained 8 hours prior to the time of measurement gave rise to the same signal increase as the corrective wavefront obtained at the time ofmeasurement. Due to its limited capture range ( ± λ/4, see references [6–9]), this multidither COAT-based method cannot prevent segments of the wavefront from being N×2π phase shifted (N being integer), which, if present, would diffract energy away from the excitation beam. However, it is not a problem for two-photon fluorescence microscopy, because the fluorescence signal is limited by photodamage rather than the available laser power, and the nonlinear dependence of the signal to excitation power also makes any contribution from the small diffracted portion insignificant. Furthermore, if so desired, the diffracted energy may be blocked by the field stop in Fig. 1(e). For applications where such phase shift cannot be tolerated, other phasing methods should be applied [6].

5. Discussions and conclusions

Even though we did not demonstrate it in this paper, it should be obvious that, for wavefront measurements where gradient values within a few segments are missing, noisy, or contain large errors, our direct phase measurement method is advantageous in that no special treatment is needed to prevent the lack of data or errors from affecting the wavefronts in nearby segments, because the phase of each segment is determined independently.

The speed of this direct phase measurement method is limited by the maximal modulation frequency of the wavefront correction device with which the algorithm is implemented. For the example in Fig. 4, the same signal improvement was obtained with both the slower liquid-crystal-based SLM (refresh rate 60 Hz) and the much faster segmented mirror (mechanical response < 200 μsec). With the segmented mirror, the data acquisition for each iteration takes 90 ms in our implementation. Both SLM and segmented mirror are ideally suited for the multidither COAT approach because there is no coupling between segments. However, earlier analysis suggested that this approach should work with continuous membrane deformable mirrors as well [13, 16].

The original multidither COAT was developed to correct laser beam distortion by atmospheric turbulence [17], and has since been applied to achieve coherent addition of light in a variety of applications, such as phase-locking optical fiber arrays [18], coherent combination of multiple laser beams [19], and scattering control through random media [20]. In this paper, we used it in the context of adaptive optical wavefront sensing, to obtain the final wavefront after the local wavefront gradients are measured. We compared its performance with that of the traditional phase reconstruction algorithms. Even though it performs similarly to the traditional phase reconstruction algorithms for moderate aberrations, it is superior for large and high-spatial-frequency aberrations as well as discontinuous aberrations. For imaging systems where the nature of aberration is unknown, our direct phase measurement method may be more reliable in obtaining the best-fit wavefront. Even though we demonstrated our method in two-photon fluorescence microscopy, we envision that it may be widely applicable to a number of adaptive optics-enabled imaging systems, where wavefront needs to be obtained from gradient measurements, such as the various adaptive optics systems for vision science [21], biomedical microscopy [2], as well as astronomical telescopes [1].