Figure 1 shows a diagram of our imaging apparatus, which was retro–fitted to a commercial two–photon laser scanning confocal microscope. The devices inserted on the microscope optical path, shown as grey boxes in Figure 1, comprised:

An important constraint in this AO application was to maintain the same adaptive mirror correction action, in terms of impulse response function, over the whole view field explored by the microscope scanning head, avoiding light vignetting. In principle, one could recreate a suitable field invariant pupil after the confocal scanning head using custom designed optics. Unfortunately this proved impossible in our configuration, due to constraints imposed by the underlying commercial architecture. The simplest solution, and the one we adopted, was to intercept the laser beam before it entered the scanning head (Figure 2), at the expense of some criticality in terms of mirror positioning and adjustments due to the resulting long optical leverage. To ease the fitting procedure, the AO module was placed on a sturdy platform with micrometric movements in 3 orthogonal directions. The Gaussian laser beam, with full width at half maximum (FWHM) of about 2 mm, average power of 350 mW and central wavelength of 830 nm, was directed onto the AO module and thereafter reinserted on its ordinary light path. The laser beam cross–section was adapted to the working surface of the EMM (10 mm diameter) by a beam expander (3×) followed by a complementary beam compressor after reflection off the mirror (Figure 2).

A view of the mounted objective with its main specifications is presented in Figure 3. Note that a Zemax model predicts an Airy radius of 2 µm for the PSF of the objective when immersed in water (Figure 3). Therefore, if the Zemax representation of the GRIN element were sufficiently accurate, this would be the order of the expected resolution of the system in the absence of other distortions and aberrations. As a consequence of back–lash between fiber and the capillary tube holder (wall to wall free space of about 0.05 mm) and positioning errors of the capillary on the objective aluminum body, an unavoidable combined optical axis misalignment was present between GRIN fiber and illuminating lens. This introduced a certain amount of optical distortion, as derived from a tolerance analysis made on the Zemax model limited to low order terms, namely: defocusing and spherical aberration, astigmatisms and comas. Theory for deformable, continuous membranes fixed at the border (our electrostatic mirror) [14] shows that these aberrations can be corrected by the AO module, possibly with the only exception of spherical aberration where the mirror can suffer of relatively small modal gain. Likewise, the distance between fiber and illuminating aspheric lens, mounted in the same fiber support (Figure 3), was pre–adjusted during the objective assembly. This lack of mechanisms for further focal optimization was dictated by our aim to construct a compact objective in which most of the fiber length (8 mm) could be inserted in live specimens. Also these distortions should be ultimately taken care of by the AO module. Thus in our view, and in our experience with this peculiar application of GRIN objectives to multiphoton microscopy, AO serves the dual purpose of (i) facilitating system assembly, by correcting static optical distortions due to residual alignment criticalities, and (ii) correcting low order aberrations that are intrinsic in the optical components.

The calibration system was assembled using commercial relay optics and inserted on the fluorescence beam returned by the GRIN objective before the photomultipliers that formed the so called direct detection system (DDS) of the confocal microscope. The calibration system imaged the central portion of the optical field viewed by the objective onto a CMOS camera with a magnification factor of 43.5 (determined using a calibration target). An important aspect of the AO application described here was the pre–calibration of the EMM flexural modes (influence matrix), which was required by the optimization algorithm (described in the next section). This calibration needed to be performed once and for all and provided the full description of the EMM. To perform the calibration we used a Shack–Hartmann camera (SHC) to reconstruct the mirror influence matrix. With reference to Figure 2, the SHC was directly interposed on the collimated beam emerging from the output beam compressor and thus it worked with the same optical configuration used at run time. It should be noted that recent commercial mirrors are directly provided with their peculiar influence matrix.

In a traditional AO application, a classical example is the astronomical case [15], the impinging wavefront phase is retrieved in real–time via a wavefront analyzer (pure phase or modal form) profiting from an in–field reference source (usually a star). Closed–loop control of a deformable optical element is then possible. In contrast, direct wavefront error sensing in microscopy applications is hampered by lack of a suitable reference, as well as lack of a true sub–divisible pupil. Consequently modal wavefront error estimation can be obtained only indirectly, by deliberately introducing known amounts of modal distortion [16]. Alternatively, the whole optical system must be previously optimized on a calibration configuration to be later applied during observation. The latter is the approach used here to minimize the effects of static optical distortions. The procedure we used was based on two off–line steps:

The first operation is detailed in [13] and [17]; a simplified discussion, which is required to introduce our method, is reported hereafter.

After construction, the solicitation matrix was decomposed by singular value decomposition (SVD) in:(1)where:

and with index k spanning the 37 EMM electrodes, whereas x and y are the discrete coordinates (256×256 pixels) of the phase images produced by the SHC.

is usually inverted to create an influence matrix:(2)suitable for a modal correction loop. Instead, we focused on the set of matrices U, S and V^T. It is well known that the matrix U contains the EMM membrane flexure modes in increasing order, the matrix V the corresponding solicitation patterns while the S diagonal represents the transfer ‘gain’ for each flexural mode [14]. In the present case, the resulting modes and corresponding voltage patterns are presented in Figure 4. The flexure modes obtained in this way are orthogonal and their voltage–pattern counterpart can be directly used to optimize the system. This procedure does not require the use of Zernike polynomials [18], [19], with the resulting benefit of a more compact and efficient minimization algorithm.

The second step, the iterative optimization process of the point spread function (PSF), required the immersion of the GRIN objective in a fluorescent solution to retrieve an image of the PSF on the calibration system camera. The key idea is to minimize PSF image size in a loop, whereby low order flexure modes weight factors are perturbed and the resulting PSF is estimated. Iterative minimization was based on a merit factor extracted from each image after a linear fit of a two–dimensional Gaussian to the fluorescence spot profile recorded by the calibration camera:(3)The merit factor ε, a decreasing quantity, was then computed by combining the Gaussian width in the x and y directions, namely the parameters B and C in Equation 3, normalized by the peak value, P_val, of the fitted PSF image intensity:(4)

The optimization metric described by Eq. 4 was preferred because it involves both the intensity and shape of the sampled PSF. The Strehl ratio, the most obvious choice, was considered unsuitable because strictly dependent on PSF intensities, which were relatively unstable due to noise in the detected signal during long optimization trials. Compared to simple ‘spot size’ measurements, the merit factor ε proved to be less susceptible to noise and provided the most stable and fast convergence. In order to minimize the camera noise pattern error, ε was computed on running groups of ten consecutive PSF images. ε was than used by a C code implementation of the classical downhill simplex minimization method [20] working on 8+1 variables, where the first 8 were weight factors for the flexural modes.

Minimizations were tested both using the first eight EMM flexure modes, as well as using just the first four, with similar results; results of a typical PSF minimization trial with four modes are shown in Figure 5. Finally, we tested the stability of the PSF optimization system by reiterating the whole procedure 10 times. Table 1 lists mean ± standard deviation (S.D.) results obtained starting from the same EMM configuration, i.e. with electrodes all fixed at a value of 2¹² √2 = 2895 corresponding to about 226 Volts. Coefficients stabilized typically after 50–70 iterations in a total of approximately 400 s using eight flexure modes. Typical PSF images obtained in the course of one of the 10 trials are presented in Figure 6. The middle panel in Figure 6 highlights the typical degradation undergone by the PSF following insertion of the EMM, mainly due to the natural astigmatism of the membrane at rest. It should also be noted that the PSF shape is never exactly Gaussian. Perhaps it could be better simulated by the sum of two coaxial Gaussian functions with different parameters, i.e. a low and wide background plus a narrow and taller spike. This is visible in Table 2, where the FWHM is estimated both on the PSF shape and on the Gaussian fit. Ellipticity in Table 2 is defined as the ratio between minor and major FWHM of the fitted Gaussian profiles; it demonstrates the action of AO in profile regularization.

As a final test of our imaging apparatus, we imaged fluorescent micro–beads (1.0 µm diameter, peak emission around 515 nm) excited by the two–photon laser tuned at 830 nm with and without AO assist. As shown in Figure 7, the PSF was clearly sharpened and regularized independent of the position in the scanned field (about 80×80 µm). Figure 8 shows a different enlarged field at higher magnification. Notice that PSF sharpening and regularization allowed the detection of the minute spaces between adjacent spheres. The degree of sharpening was compatible with the results reported in Table 2. Tests performed with 0.5 µm beads show that this is close to the resolution limit of the system (Figure S1).