### Summary

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Wide-field fluorescence microscopy is an essential tool in modern cell biology. Unfortunately the image quality of fluorescence microscopes is often significantly degraded due to aberrations that occur under normal imaging conditions. In this article, we examine the use of adaptive optics technology to dynamically correct these problems to achieve close to ideal diffraction limited performance. Simultaneously, this technology also allows ultra-rapid focusing without having to move either the stage or the objective lens. We perform optical simulations to demonstrate the degree of correction that can be achieved.

### Introduction

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Fluorescent light microscopy of live cells offers cell biologists the opportunity to study molecular and cellular mechanisms in action. Genetic tagging of proteins, especially rare ones, requires imaging with the best three-dimensional resolution using the highest possible numerical aperture objectives, efficient photon collection, minimal photo-damage, sufficient working distances for thick samples, and minimal sample perturbation during fast live data collection. In practice, all of these requirements cannot be met, and cell biologists must compromise on these aspects of fluorescence imaging. We test the incorporation of adaptive optics to come much closer to achieving all of these objectives.

Fluorescence wide-field microscopy provides the optimal methodology for harvesting the most photons in a given optical configuration. Photon efficiency is of critical importance for *in vivo* imaging and requires high numerical aperture objectives, which use an immersion medium, such as oil, glycerol or water. However, the index of refraction of living tissue is different than any of these immersion media. Therefore, imaging into live samples degrades the resolution, contrast and peak intensity of the image very rapidly with depth (Hiraoka *et al.,* 1990; Gibson & Lanni, 1991; Kam *et al*., 1997; Kam *et al*., 2001; Hanser *et al*., 2002). This degradation is caused by depth-dependent aberrations (most significantly, spherical aberrations). Aberrations significantly degrade the performance of deconvolution algorithms routinely applied for three-dimensional wide-field image reconstruction with high NA objectives (Swedlow *et al*., 2002). Using deconvolution algorithms the signal-to-noise ratio and resolution, particularly in the axial direction, are enhanced by pushing the out-of-focus light back to its true three-dimensional source (Swedlow *et al*., 1997; Swedlow & Platani, 2002; Swedlow, 2003). Aberrations make it impossible to faithfully reconstruct the source of the out-of-focus light. The depth aberration problem is fundamental for laser scanning microscopes as well, particularly two-photon microscopy, causing loss of signal intensity (Marsh *et al*., 2003; Tsai *et al*., 2005) and for laser tweezers applications, causing loss of beam holding forces (Theofanidou *et al*., 2004). Finally, the presence of spherical aberrations inhibits the application of structured illumination techniques which otherwise have the ability to double the three-dimensional optical resolution in light microscopy (Gustafsson, 2000; Gustafsson *et al*., 2005).

The image of a point source, known as the point spread function (PSF), provides an effective means to characterize aberration-induced image distortions. Depth-dependent aberrations scale roughly as the difference in the refractive indices of the immersion medium (*n*) and the sample (*n*′) multiplied by the depth within the sample (see Eq. 2 and its Taylor expansion for small (*n*′–*n*) below). Thus with an immersion oil of refractive index, *n*= 1.518, a point image 50 μm below the coverslide in glycerol (*n*= 1.4746, refractive index difference, Δ*n*= 0.0434) has similar distortions as a point image 12 μm inside a volume of a typical buffer (*n*= 1.341, Δ*n*= 0.177). The corresponding peak intensity is about four times lower than would be detected for an identical point source right under the coverslide. For a water immersion objective imaging into tissue (*n*= 1.41, Δ*n*= 0.077), comparable distortions will appear at 28 μm depth. Aberrations can be corrected by adjusting the refractive index of the immersion medium (Hiraoka *et al*., 1990) by adjusting special collars on objectives, or by inserting additional adjustable optics into the microscope imaging tube (Kam *et al*., 1997).

More generally, spherical and other aberrations can be understood to arise from differences in optical path at different locations in the optical wavefront (Ross, 1954). Given a point source imaged at different focal planes, it is possible to reconstruct wavefront amplitude and phase variations in the back aperture of the objective lens. This can then be used in a mixed Fourier/real-space deconvolution algorithm to computationally correct for depth-dependent changes in the PSF (Hanser, 2003). The more general case of sample-induced aberrations was addressed in another paper (Kam *et al*., 2001) using space-variant deconvolution computed from ray-traced PSFs. Aberrations can thus be corrected by modifying optics (Hiraoka *et al*., 1990; Kam *et al*., 1997) or by computational approaches (Kam *et al*., 2001; Hanser *et al*., 2002). Inevitably, these approaches slow down the image acquisition process and pose a serious computational burden for the reconstruction. However, an alternative approach would be to directly correct image distortions by introducing compensating optical path differences into the optical wavefront. The development of fast adaptive optics (AO) elements which can adjust the optical path length over the aperture opens new possibilities for real-time correction of both depth-dependent and sample-induced aberrations in live sample microscopy. Moreover, AO elements can alter the optical wavefront so that the focal plane is swept throughout the sample without having to move either the objective lens or the stage.

AO has had an extraordinary impact in astronomy (Tyson, 1991; Wizinowich & Bonaccini, 2002) and has recently been applied to medical imaging and confocal microscopy (Albert *et al*., 2000; Bartsch *et al*., 2002; Booth *et al*., 2002; Shirai, 2002; Schwertner *et al*., 2004). The correction of aberrations in astronomy (and a recent implementation in scanning confocal microscopy) relies on evaluation of the distortion of point sources (bright stars or the confocal exciting beam) and subsequent modification of the AO element to correct for these distortions. The AO approach described here, in contrast to the astronomy paradigm, deals with the case when we can define and characterize the aberrations beforehand. This approach allows for faster correction and minimizes bleaching because extra images do not need to be acquired. Here we consider two different modes of applying AO to light microscopy of live samples. In the first mode we insert the AO element in a pupil conjugate plane, allowing dynamic focusing without sample perturbation while correcting spherical aberration on the fly for each depth. In the following simulations, we consider imaging into a water-based sample with an oil-immersion lens to demonstrate the correction possible in the extreme case. Through the use of multiple AO elements, the second mode additionally makes possible correction of local refractive index variations within the sample. In this case, we would utilize Differential Interference Contrast (DIC) imaging to empirically provide the required three-dimensional map of index of refraction variations within the sample (Kam 1998; Kam *et al*., 2001).

### Methods

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

The ray tracing program, described in detail before (Kam *et al*., 2001), is based on the analytical solution of ray paths and phases in a gradient refractive index medium. With this algorithm a larger integration step can be used for a given accuracy than in the common method of Sharma *et al.* (1982), therefore cutting the processing time. A fan of homogeneously spread rays within a given aperture cone is generated, traced through a refractive index medium, mapped onto a three-dimensional grid, and the emerging wavefront phases at infinity are used to evaluate the Strehl ratio as a complex amplitude integral over the wavefront (Hardy, 1998). In this work a fluctuating refractive index medium was simulated by adding three-dimensional Gaussians at random displacements from a three-dimensional grid according to:

- (1)

where *G*(*r*) = exp(–|*r*/*w*|^{2}), *r*= (*x*, *y*, *z*), *r*_{n}= (*i*+*p*_{i})1_{x}+ (*j*+*p*_{j})1_{y}+ (*k*+*p*_{k})1_{z}, *i*, *j*, *k* are integers, *p*_{i}p_{j}p_{k} are uniform random variables between 0 and 1 and 1_{x}1_{y}1_{z} are vectors along *xyz* spanning a three-dimensional grid with spacing proportional to *w*, the Gaussian width. Various dimensions and refractive index contrasts were generated and ray traced. The reported results here correspond to about 500 Gaussians at average spacing of 3*w*. To calculate the effects of adaptive optical elements, the rays emerging from the medium are ‘relayed’ by ideal lenses as described in Figures 2 and 4, and treated by one or several adaptive optical devices by tracing to their planes and shifting the phases according to the position they hit.

The presentation of a finite slice of the medium by a single conjugated adaptive optical plane followed one of several options: Parallel-sum means integrating the optical path of the slice along the optical axis *Z* (and perpendicular to the plane) as a function of position *XY*. Fan-sum performs the optical path integration along rays emerging from the origin. For numerical apertures greater than 1.2, light rays inside the sample travel at angles greater than 45° with the optical axis. Thus there is a significant difference between the parallel-sum and the Fan-sum methods when using a small number of conjugates. The Tokovinin option (see Tokovinin *et al*., 2000) ‘blurs’ each slice by an amount increasing with distance from the conjugate plane, and in proportionality to the field of view.

The Zemax ray tracing program (Bellevue, WA 98004-8017) was applied to the Nikon objective described in Yamaguchi (2003). The adaptive element was approximated by expanding Eq. (3) in terms of the first five radial Zernike coefficients.

### Discussion

- Top of page
- Summary
- Introduction
- Methods
- Results
- Discussion
- Acknowledgements
- References

Using ray tracing simulations we have tested two configurations for including AO elements in high-resolution light microscopes to correct aberrations associated with imaging of live thick biological specimens. Using adaptive optics in wide-field microscopy makes possible the correction of parallel data acquisition at different depths and thus is an optimal strategy for collecting live dynamic image information. Because the entire frame is collected at once in wide-field, the response time of the adaptive elements can be much slower compared to the microsecond rates needed for adaptive optics correctors in scanning confocal microscopes (Albert *et al*., 2000; Booth *et al*., 2002). We show that correcting for spherical aberration caused by focussing into sample buffer even for the extreme refractive index mismatch with an oil-immersion objective is straightforward, simple to apply, and yields the double benefit of also providing a mechanism to make very rapid focus changes without perturbing the sample. Because the refractive index of live cells is on the average 1.43, and varies between different tissues, an oil immersion lens is actually the most appropriate to use with adaptive optics. When corrected with AO, it also has the benefit of providing the highest possible numerical aperture and hence the best resolution and light gathering power. Furthermore, complete correction of depth dependent aberrations allows rapid, space invariant computational deconvolution methods to correct for the remaining out-of-focus blurring.

With large stroke AO devices becoming available, it should be possible to image through many cell layers (100 μm or more) in intact tissues. For optimal imaging of such very thick tissues it will also be necessary to develop methods to correct for higher order aberrations, as well as for dealing with light scattering. Two-photon microscopy, which gives good performance in highly scattering tissue, will greatly benefit from adaptive optics correction.

Once an AO element is employed to correct depth-dependent aberrations at the back aperture, it would also be possible to simultaneously correct for additional optical aberrations such as remnant phase shifts and asymmetries existing even in the best-selected objectives. These phase shifts increase dramatically towards the high, peripheral acceptance aperture of the objective, which amounts to high order aberration terms (Juskaitis & Wilson, 1997; Beverage *et al*., 2002; Hanser *et al*., 2002). They show temperature and “age” dependence, especially serious in heat-incubated environmental chambers used for imaging live biological samples. Such objective-dependent phase aberrations can be determined optically (Juskaitis & Wilson, 1997; Beverage *et al*., 2002) or computationally (Hanser *et al*., 2002) but their *in situ* correction using adaptive elements will be fast and will increase image resolution and contrast. It would be also possible to use the same adaptive optic element to correct for the small residual wavelength dependent aberrations that otherwise would degrade multiwavelength imaging. Finally, if AO elements were to become a standard feature in optical microscopes, it might be possible to greatly simplify the design of the objective lenses (e.g. to get rid of correction collars and to reduce number of lenses required for chromatic correction) and to achieve better overall performance and longer working distances using the flexible dynamic characteristics of AO elements to relax the multiple constraints imposed in objective design.

The second mode explored here, tackles the more challenging problem of correcting the smaller, but more complex, spatially variant aberrations arising from the optical properties of the sample itself. In wide-field imaging, a perfect correction would require the construction of an inverse sample having all the resolution and depth of the real sample, but the opposite change in index of refraction. A more practical alternative is to use a small number of multiconjugate AO elements. To explore the feasibility of this approach, we have simulated the effects of using two or four AO elements with a variety of methods for calculating how each should be set given a three-dimensional map of refractive index variation within the sample. These simulations indicate that useful corrections can be obtained using two multiconjugate elements, while using four yields excellent results.

The use of several reflecting adjustable mirrors may be possible, but physically cumbersome, for applications in microscopy. Transmitting adaptive elements (such as liquid crystal spatial phase modulators) capable of introducing position dependent phase shifts through their aperture (Dayton *et al*., 2002; Lee *et al*., 2005) can make this multiconjugate approach practical and effective. They have a sufficiently high pixel density for correction of sample-induced aberrations, and their setting speed, being in the millisecond range, is reasonable for wide-field microscopy. However, the present devices have high transmission losses, imposing a serious constraint for their use in tandem. Thus with current devices, it is most reasonable to consider only the two multiconjugate correction scheme.

The introduction of adaptive optics into light microscopy optimize high-resolution imaging under the less than ideal conditions typical in biological applications, increase three-dimensional sharpness and contrast for thick specimens, and allow the detection of finer details and weaker molecular signals inside live cells. As outlined in the introduction, this last point is the critical issue in live cell imaging. An important aim of this paper is the stimulation and the development of new adaptive optics hardware and software in wide-field microscopy.