A dynamic closed-loop method for focus tracking using a spatial light modulator and a deformable membrane mirror within a confocal microscope is described. We report that it is possible to track defocus over a distance of up to 80 μm with an RMS precision of 57 nm. For demonstration purposes we concentrate on defocus, although in principle the method applies to any wavefront shape or aberration that can be successfully reproduced by the deformable membrane mirror and spatial light modulator, for example, spherical aberration.
©2006 Optical Society of America
The widespread use of confocal and multiphoton microscopy has increased the desire to image at depth into biological material . Unfortunately, with increased depth the quality of the image deteriorates due to increased aberrations [2, 3]. Several approaches have been implemented to overcome this problem, such as, static aberration correction by the introduction of additional lenses or using a microscope objective with a correction collar. Over the last few years adaptive optics has emerged as a new technology for improving image quality and resolution when looking at depth into biological material.
Adaptive optical elements include such technologies as deformable membrane mirrors (DMM)  and spatial light modulators (SLM) [5, 6]. Adaptive optics has been extensively developed by ground-based optical astronomers in order to combat atmospheric turbulence and improve image quality . Using an adaptive optical element the incoming wavefront is altered so as to counteract any aberrations present. Published literature demonstrates how a similar technique can be used in microscopy [8–10].
The key to a successful implementation of adaptive optics in microscopy is determining the wavefront of the incoming beam required in order to overcome the aberrations and improve the final image. In the past this has been achieved either by measuring the aberrations present in the sample  or by using an optimization algorithm that rapidly alters the mirror shape to improve a particular property of the image [8, 11]. Both approaches can be implemented using DMMs and SLMs separately as the adaptive optical element, each having its own advantages and drawbacks. We propose a method that combines the two devices to form a closed-loop system for focus tracking within a confocal microscope, combining the fast response of a DMM with the large effective dynamic range of an SLM.
A DMM is a membrane device and therefore only capable of introducing spatially continuous changes in phase. The DMM employed in this work is used widely in the field and updates rapidly (approximately 1 kHz) compared to a typical nematic-liquid-crystal SLM, but with a limited stroke of about 8 μm. In contrast to a DMM, an SLM is highly pixilated thus capable of introducing changes in phase with a high spatial resolution. Although the phase range of any individual pixel is approximately 2π, using phase-wrapping techniques the overall phase change to a wavefront of many times 2π is possible, limited only by the number of pixels in the SLM. An SLM used in this way acts as a diffractive optical element where the desired phase change occurs mainly in the first diffraction order . Consequently, an SLM has an effective stroke of many tens of wavelengths, with the potential for phase discontinuities. However, the SLM is slower to update than a DMM (typically 15 Hz) and operates with a first-order diffraction efficiency of 30%, compared to a DMM with a reflectivity of > 99% (Ferroelectric SLMs have a higher refresh rate than nematic SLMs, but with lower diffraction efficiency). It is important to note that our approach is best suited to using a single frequency or narrowband source such as a laser. When applied to a broadband source the spectral components are angularly dispersed and appropriate compensation techniques need to be considered [13, 14].
In principle, the focus of the microscope could be maintained during the translation of the objective by applying a defocus correction to either the DMM or SLM, with the DMM offering higher temporal bandwidth and the SLM offering greater dynamic range. Maintaining the returned light signal at a maximum (corresponding to best-focus) is a generic control systems problem. We adopt an approach whereby a high-frequency focus modulation is applied to the DMM, resulting in a corresponding modulation of the returned intensity. This is demodulated using a lock-in amplifier to provide an output voltage which is equivalent to the first derivative of the returned intensity and hence is an error voltage, antisymmetric about the position of best-focus. This error voltage is used within a computer-controlled feedback loop to set the defocus term applied to the SLM. We believe this is the first demonstration of a system combining both the high speed response of a DMM and the equivalent large stroke of an SLM.
2. Experimental details
The optical system (Fig. 1) was based around a laser-scanning confocal microscope, operating in reflection mode, although the same technique could be applied to a single-photon or multiphoton fluorescence system. A He-Ne laser (633 nm, 10-mW maximum output power) was directed into a BioRad MRC 600 scan head. A coverslip and infinity-corrected Nikon air objective with a numerical aperture (NA) of 0.5 and a 20x magnification was used to focus the laser beam onto the sample, consisting of a plane mirror and coverslip. The detection photomultiplier tube (PMT) and confocal aperture were housed within the scan head.
Between the scan head and the objective lens the system incorporated the two adaptive optical components, a DMM (Flexible Optical BV, The Netherlands) and an SLM (LC-R 3000 Holoeye GmbH, Germany). The DMM is comprised of a silicon nitride membrane mounted above 37 electrostatic actuators , each actuator being supplied with up to 175 V to pull the membrane into various shapes with a maximum update rate of 1 kHz. The maximum deformation of the mirror, with 175 V applied to all the actuators, was approximately 7 μm, as confirmed using an interferometer (Zygo, USA). Although the mirror is 15 mm in diameter, because the edge is fixed, to correct for aberrations at the edge of the pupil, it is usual to assume an optimal working aperture of approximately 10 mm diameter .
A custom LabVIEW (National Instruments Inc, USA) software interface was developed allowing various combinations of the first fifteen Zernike polynomials to be applied to the wavefront of the first diffractive order of the SLM. Zernike polynomials are a convenient basis set to use for aberration correction since they are orthogonal over a unit circle and the low order terms can be directly related to specific aberrations in classical optics, for example astigmatism, coma and spherical . Once the phase pattern for a particular Zernike polynomial was calculated it could then be sent to the SLM. The SLM itself works in reflection consisting of 1920 × 1200 pixels and is approximately 20 mm × 12 mm, therefore, also compatible with a 10-mm beam.
Three 4f lens systems were used to re-image the second galvanometer scanning mirror within the scan head, onto the SLM, DMM and finally the back aperture of the microscope objective. The main role of the 4f lens systems was to relate any aberration correction to the object plane of the objective lens and, although not relevant to this experiment which did not involve scanning, they would also produce stationary scan points on the SLM, DMM and object plane of the objective. The 4f lens pairs were also used to expand the 1-mm beam exiting the scan head to 10 mm. This ensured optimal use of the adaptive optics and filled the back aperture of the microscope objective thus utilizing the full NA of the objective. The first diffraction order from the SLM was selected by placing a pinhole, or spatial filter, at the focus of one of the telescope lens pairs. The returned light from the sample followed the reverse optical path into the scan head where a beam splitter directed it to the detector.
The microscope objective was mounted on a piezoelectric translation stage (PZT) (E662 LVPZT Physik Instrumente, Germany) for precise control along the optical axis (z-axis). The axial point-spread function (PSF) was measured by translating the objective, and hence the beam focus, along the z-axis whilst recording the returned intensity on the PMT . The full-width half-maximum (FWHM) of the PSF can be used to quantify the axial resolution of the system.
A 300-Hz modulation was applied to the DMM driving all 37 actuators in parallel. This predominately applied defocus to the system and resulted in modulation of the returned intensity as measured by the PMT. This signal was demodulated using a lock-in amplifier (Brookdeal model 401, USA) to give an output voltage equivalent to the first derivative of the returned intensity. This error voltage was used within a computer-controlled feedback loop to set the defocus term applied to the SLM. Provided that the overall gain and time constants were set appropriately, this combination formed a closed-loop system in which the SLM was continuously adjusted to maintain the position of best-focus.
The components of the closed-loop control system can be seen in Fig. 2, with the majority of the optical components being omitted for clarity. The axial resolution measurement and the SLM feedback loop were both fully automated via LabVIEW programs and an analogue-to-digital and digital-to-analogue converter (DAQPad6015, National Instruments, USA).
Figure 3 shows the PSFs obtained by axially translating the microscope objective whilst differing amounts of defocus are added using either the DMM or SLM. For the DMM, defocus is applied to the system by setting all the actuators to a common voltage. The maximum voltage that can be applied to the actuators is 175 V, giving a focal shift of 21 μm. For the SLM, defocus was applied to the system by addition of the corresponding Zernike term. Figure 3(b) shows the PSFs obtained when shifting the focus by over 30 μm in either axial direction.
For both the DMM and SLM we see that upon defocus both the height and width of the PSFs are degraded. For the DMM, a defocus corresponding to a focal shift of 21 μm reduces the PSF amplitude to 21% of its unshifted value and increases the FWHM by 40%. In the case of the SLM, a defocus corresponding to the same focal shift reduces the PSF amplitude to 45% and increases the FWHM by 3%. In both cases we believe the PSF broadens and the amplitude decreases as there is no longer collimated light entering the back aperture of the microscope objective, further investigations would be needed to fully explore this observation. The performance with the DMM is further degraded since our simple control method does not accurately produce pure defocus, thereby adding small quantities of additional aberrations (particularly spherical aberration).
In order to servo-lock to best-focus whilst translating the microscope objective, as discussed above, it is necessary to derive an error signal and use it to control the SLM. Thus a focus modulation of 300 Hz was applied to the DMM with a depth sufficient to result in a slight increase in the observed FWHM. Figure 4 shows the resulting signals obtained directly from the PMT and from the lock-in amplifiers as the objective lens is axially scanned. The 1F and 2F signals from the lock-in amplifiers (300 ms time constant) correspond to the first and second derivatives of the PSF respectively. The 1F output is the error signal and as expected has a zero crossing corresponding to best-focus. For the case of the 1F signal, a change in amplitude of one arbitrary unit corresponds to a microscope objective movement of 0.8 μm in the axial direction, calculated using the linear region of the 1F output.
Figure 5 shows the output from the PMT and lock-in amplifiers once the feedback loop to the SLM is closed, and the microscope objective is stationary. Measuring the error signal over period of a minute, we used the first derivative to determine that the precision of the lock is better than 57 nm RMS (160 nm peak-to-peak), with little indication of any long term drift.
When translating the 20x, 0.5NA microscope objective, we find that best-focus can be maintained over a maximum axial distance of 80 μm. Beyond 80 μm the PSF deteriorates, as shown in Fig. 3, such that the maximum intensity reduced to over half its initial value and the system loses lock.
4. Discussion and conclusions
An SLM and a DMM have been successfully combined in a confocal microscope system and used to maintain best-focus whilst axially displacing the sample. Defocus is maintained by applying a small, in comparison to the FWHM, modulation to the DMM, such that the system is never far from its optimal position. For a 20x microscope objective with an NA of 0.5, it was possible to maintain best-focus over a sample displacement of 80 μm, with an RMS precision of 57 nm. This range was limited by the degradation of the PSF most likely due to additional aberrations introduced as the microscope objective was used away from its designed conjugates.
Although our system is based on a 20x 0.5NA objective lens, the approach is applicable to any objective lens used within a point-imaging or scanning system. The range of aberration correction and precision of lock will depend upon the detailed characteristics of the objective lens and the extent of the perturbations to which the system is designed to track. Increasing the magnification and NA of the objective would decrease the tracking range but it would seem reasonable to assume that the lock precision as a fraction of the axial PSF FWHM would remain the same. Since the tracking range is limited by degradation of the PSF, as described above, the range could perhaps be extended further by correction of the associated higher-order aberration, which could themselves be pre-calculated. Ultimately the range of correction should only be limited by the spatial resolution of the SLM and hence the maximum spatial variation in corrective phase.
Defocus was used in this demonstration as it is easily produced on the DMM. However, it would be possible using a similar approach to correct any of the Zernike aberrations, for example spherical aberration, provided they could be independently reproduced by the DMM. It should also be possible to simultaneously correct several aberrations using different modulation/demodulation frequencies for each aberration. However, in practice, any cross sensitivity between the different Zernike modes would impact on the precision of the lock. .
The authors gratefully acknowledged financial support from the UK Joint Research Councils Basic Technology Program and the Glasgow and Strathclyde Universities Synergy program.
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