## Abstract

In modern high-NA optical scanning instruments, like scanning microscopes, the refractive-index mismatch between the sample and the immersion medium introduces a significant amount of spherical aberration when imaging deep inside the specimen, spreading out the impulse response. Since such aberration depends on the focalization depth, it is not possible to achieve a static global compensation for the whole 3D sample in scanning microscopy. Therefore a depth-variant impulse response is generated. Consequently, the design of pupil elements that increase the tolerance to this aberration is of great interest. In this paper we report a hybrid technique that provides a focal spot that remains almost invariant in the depth-scanning processing of thick samples. This invariance allows the application of 3D deconvolution techniques to that provide an improved recovery of the specimen structure when imaging thick samples.

© 2009 Optical Society of America

## 1. Introduction

The acquisition of high-resolution images of 3D biological samples by optical microscopes requires the production of tightly focused spots, which are usually achieved by high numerical aperture (NA) immersion objectives. The refractive-index mismatch between the immersion liquid and the solution in which the sample is embedded introduces a significant amount of spherical aberration (SA) when imaging deep inside the specimen, spreading out the impulse response. Since such aberration depends on the focusing depth, the impulse response is degraded in a different manner for different slices inside the specimen. Thus, it is not possible to achieve a global compensation for the whole sample. Some methods were developed, in paraxial context, for the reduction of SA impact [1–6]. These proposals are not directly applicable to optical microscopy since they do not take into account the tight focusing. More recent new proposals, based on non-paraxial equations, have been reported. The idea of altering the objective tubelength [7], or the use of adaptive optics [8–10], are very interesting. Another interesting approach is the use of a depth-variant stratum-based algorithm [11]. This interesting approach allows the acquisition of aberration-free images taking advantage of the fact that it is possible to compute the deconvolution by using a small number of spatially invariant PSFs, each computed at different depth.

Here we focus on a technique that is very simple, but that is able to stabilize the SA impact over a large range of imaging depths. The technique is known as wavefront coding. This technique was pioneeringly suggested, in a different context, by Ojeda-Castañeda [12] and mainly developed by Cathey’s group [13,14]. Wavefront coding was developed for increasing the depth-of-field of imaging systems. A phase pupil mask is used to encode the wavefront emerging from the imaging system while the digital restoration produces the decoding. In this paper we take advantage of the fact that the axial values of the intensity spot obtained after focusing, through a plane interface, the light proceeding from a high-NA objective is proportional to the Fresnel transform of a properly mapped version of the objective aperture stop. This fact allows us to adapt the wavefront coding technique to the aim of reducing the impact of the SA induced during the axial scanning of the sample. The hybrid technique we report here provides a focal spot that remains almost invariant in the depth-scanning processing of thick samples. This invariance allows the application of 3D deconvolution techniques with the purpose of obtaining an improved recovery of the specimen structure when imaging thick samples.

## 2. Spherical aberration induced in the axial scanning

As is well known, the imaging features of an optical microscope can be accurately described through the 3D intensity point-spread-function (PSF), namely, by the 3D intensity distribution that is obtained in the neighborhood of the focal point of the microscope objective when it is illuminated by a monochromatic plane wave. Since, to obtain optical images with good resolution, the microscope objectives need to have high NA, the PSF cannot be calculated by means of paraxial diffraction formulas, but through the non-paraxial Debye’s equation. According to the scalar Debye’s formulation, and assuming that sine condition and axial symmetry hold, the intensity PSF is [15].

In this equation, P(*θ*) accounts for the amplitude transmittance at the objective exit pupil, and *α* is the maximum value for the aperture angle *θ*. Lateral and axial positions in the focal region are expressed through cylindrical coordinates (*r*, *z*) as measured from the Gaussian focus, *F*. The phase factor *W*(*θ*) accounts for potential phase distortions occurred during the focusing.

In many microscope observations a significant mismatch between refractive indices of the sample medium and the immersion liquid occurs. This mismatch induces a significant amount of SA. To analyze this effect, we consider the simple geometry in which the spherical wave emerging from the microscope objective is focused deeply through a planar interface between two media of different refractive indices (see Fig. 1) [16]. To evaluate the phase distortions, we take into account that, according to Debye’s formulation, the spherical wave can be seen as the superposition of plane waves whose directions fall inside the cone defined by the focus, *F*, and the exit pupil. Each plane-wave component of the field emerging from the objective obeys Snell’s law, *n*
_{1} sin*θ*=*n*
_{2} sin*θ*’, when refracted at the interface. The resulting field is reconstructed as the superposition of refracted plane waves.

In Fig. 1 we have represented a plane-wave component of the field by a light-ray normal to the wavefront. We have named as the focusing depth, *d*
_{0}, the distance between the interface and the focus, *F*. On the contrary, we have named as the penetration depth, *d*′_{0}, the distance between the interface and the place where the wave effectively focuses.

In a typical microscopy procedure, the image of the different sections of the sample are obtained after axial scanning of the microscope. If at a given stage of the axial scanning the microscope is displaced a distance *z*
_{S} towards the sample, the focusing depth is *d*
_{S}=*d*
_{0}+*z*
_{S}. In such case the penetration depth is *d*′_{S}=*d*′_{0}+*z*′_{S}(*z*′_{S} being the axial displacement suffered by the effective focus). If we assume that by use of the correction collar [17] the spherical aberration is statically compensated for the focusing depth *d*
_{0}, the wave distortions *W*(*θ*;*z*
_{S}) induced at a focusing depth *d*
_{S} can be easily evaluated by [18,19]

The above equation has been obtained after expanding *W*(*θ*;*z*
_{S}) into a power series and making a fourth-order approximation. Besides, we have introduced the well-known defocus and SA coefficients, which are given, respectively, by

In Fig. 2 we show the results of a numerical simulation of the 3D intensity distribution in the focal region of a microscope objective when typical axial-scanning induced SA occurs. In our calculations we have considered a light microscope equipped with an immersion objective of NA=1.4 (oil; *n*
_{1}=1.52), and a specimen whose refractive index is close to *n*
_{2}=1.33, which is the index of the water solution where the specimen is embedded. We have assumed that the SA has been compensated, with a correction collar, for a penetration depth *d*′_{0}=10*µm*, which corresponds to a focusing depth *d*
_{0}=11.7*µm*.

In the second step of the simulation we have calculated the images of a spoke target set perpendicular to the optical axis of the microscope at the penetrations depths used in Fig. 2. The images were obtained by convolving the target with the lateral response at the best image plane (marked with a yellow line in the figure). Note that in contrast with what is commonly assumed [20], we have not set the best focal plane at *w*
_{20}=-*w*
_{40}, but at the axial position of the PSF main peak. The images are shown in Fig. 3. A significant feature comes out from this figure: there is only a very small degradation of images of 2D samples. This is because, although the 3D PSF is strongly spread, the lateral PSF at the best image plane still has a sharp peak even in case of significant amount of SA.

## 3. Effect of the spherical aberration in 3D scanning microscopy

We have just seen that the SA does not degrade the images of 2D objects significantly, provided that one selects the penetration depth adequately. However we have seen that SA strongly degrades the 3D PSF and therefore we infer that images of 3D objects will be degraded as well. To analyze this fact in a simple way it is convenient to study the influence of the SA in the axial PSF. To this end, we perform some mathematical manipulations of Eq. (1). Then by defining the normalized coordinate

and after performing the nonlinear mapping

we straightforwardly obtain the axial response of the system as [21]

Note that Eq. (7) establishes a Fresnel-transform relationship between the axial response of the system and the mapped pupil *q*(*ζ*). The same result can be encountered in the analysis of 1D paraxial focusing systems when considering the case of a cylindrical lens illuminated by a monochromatic plane wave. When the 1D aperture stop (of transmittance *p*(*x*
_{0})) is placed, as usually, at the front focal plane of the cylindrical lens of focal length *f*
_{x}, the intensity distribution on planes parallel to the back focal plane is given by

where we have introduced the well-known 1D defocus coefficient, defined as

Δ being the width of the slit.

As pointed out in [22] in a paraxial context, it is apparent that the 1D defocus coefficient *w ^{x}*

_{20}plays in Eq. (8) a role similar to the one played by coefficient

*w*

_{40}in Eq. (7). Thus, one can conclude that the axial intensity distribution produced by a high-NA scanning optical microscope in which certain amount of SA is induced, is the same as the transverse intensity distribution obtained at a defocused plane

*z*=-2

*λf*

^{2}

_{x}w_{40}/Δ

^{2}in the 1D paraxial focusing experiment. Since very efficient solutions have been given for decreasing the effect of defocusing in the 1D case (i.e., for increasing the

*depth-of-field*of such systems), we can now apply the analogous techniques to efficiently desensitize the axial response of the high-NA objective to the sample-induced SA.

## 4. Application of the wavefront coding technique

A well-known method for extending the depth-of-field of imaging systems is *wavefront coding* (WFC). In the WFC technique a pupil mask designed for increasing the depth-of-field is used in combination with a digital image-restoration process. Thus, as a first stage in the WFC method, the defocused patterns are uniformly blurred over a large axial range by effect of the mask. The almost-constant transverse impulse response in this case allows a sharp reconstruction inside this extended range by a single step deconvolution. A successful proposal in this sense is the 1D cubic phase filter proposed by Dowski and Cathey [13]. To adapt this technique for the problem of reducing the SA impact, we propose the use of a properly scaled radial version of the cubic phase filter, namely *q*(*ζ*)=exp-(-*i*2*πAζ*
^{3}). Following the suggestions made in [13], we have set a value for *A* much higher than 5. Thus we set *A*=50.

Since SA is of particular impact only in 3D imaging experiments, we have performed a simulation in which we consider a confocal scanning microscope. To perform the calculations we assume, again, that the spherical aberration is compensated, by the correction collar, for a penetration depth *d*′_{o}=10*µm*. We calculated, as the squared modulus of Eq. (7), the confocal PSFs corresponding to seven equally-spaced scanning steps (see Fig. 4). In the video we compare, for varying penetration depth, the focal spots produced by the circular aperture (CA) and by the cubic filter (CF). We see that the PSFs obtained with the cubic filter are very different from the ones obtained with the circular aperture. In fact they are much more spread. But, in contrast with the circular-aperture case, they do not change significantly as the penetration depth increases. So, we can affirm that the cubic phase filter produces a spread focus that is insensitive to the amount of SA.

In Table 1 we compare the values of *d*′_{S} and *d*
_{S}, and show the corresponding value of the SA coefficient. Note that there is not any proportional relation between the penetration depth and the induced spherical aberration [23].

d′_{S} (mm) | 10.0 | 12.5 | 15.0 | 17.5 | 20.0 | 22.5 | 25.0 |

d
_{S} (mm) | 13.2 | 16.5 | 19.7 | 22.9 | 26.3 | 29.7 | 33.1 |

w_{40}
| 0.0 | -0.6 | -1.1 | -1.6 | -2.1 | -2.6 | -3.2 |

To illustrate the utility of the proposed wavefront coding method we have performed a numerical simulation of a confocal imaging experiment. As the 3D object we have used a synthetic object composed by 8 plane plates set at *z*′_{S}=7.5*µm*+*iδ*, with *i*=0,…,7 and *δ*=1.25*µm*. The plates are shown in Fig. 5.

Any 2D section of the 3D image, for example the one corresponding to a given scanning depth *z*′_{S}, is calculated as the incoherent superposition of eight 2D images. Each of these eight images is obtained as the 2D convolution between the corresponding plate and the defocused transverse section of the 3D PSF associated to the selected *z*′_{S}.

In Fig. 6(a) we show five lateral sections of the 3D image obtained with the circular aperture as the objective aperture stop. We clearly see that the deeper the scanning position the bigger the induced SA and therefore the higher the influence of adjacent plates in the 2D sections of the 3D image.

We have applied a typical inverse filtering procedure to the 3D confocal image obtained with the circular aperture. We have used for the 3D deconvolution algorithm the PSF corresponding to an intermediate depth, *z*′_{S}=10.00 *µm*. In Fig. 6(b) we show five sections of the deconvolved 3D image. These images are, or course much worse than the original. This is because it is in fact nonsense to apply deconvolution to systems whose impulse response varies very fast.

In Fig. 6(c) we show the sections of the 3D confocal image but obtained by placing the cubic phase filter in the aperture stop of the objective. As expected, the images are, mainly for low values of the SA, much worse than the ones obtained with the circular aperture. However, we can infer from the figure that the influence of adjacent plates keeps fairly constant. Note that due to the non-symmetrical form of the PSFs, each section of the 3D image is influenced mainly by less deep plates.

Since the PSF obtained with the cubic-phase filter remains practically invariant over a wide range of focusing depths, the microscope in which the filter is inserted can be considered, in quite good approximation, as 3D linear and shift invariant, and therefore a good candidate for the application of deconvolution techniques. Thus we have applied a typical inverse filtering procedure to the 3D confocal image obtained with the cubic filter. As the 3D PSF of the system we have selected the 3D PSF corresponding to an intermediate focusing depth, namely *z*′_{S}=10.00 *µm*. After performing the 3D deconvolution we have obtained a new 3D image of the synthetic object. In Fig. 6(d) we show five 2D sections of the said object. We conclude from the figure that a wavefront coding technique provides 3D images that are highly immune to the SA induced by axial scanning.

## 5. Conclusions

We have presented a hybrid technique for the reduction of impact of spherical aberration induced in the axial scanning inherent to microscopy observations. Note that, in general our design model should have taken into account that the radiation emitted by distant object points might be scattered or reflected by nearer object points. However, we have assumed that the first-order Born approximation holds, so that the multiple scattering and depletion of the incident beam are negligible [24]. The main feature of the proposed method is that with a minimum modification of the objective architecture one can mitigate the SA induced during the axial scanning inherent to any 3D imaging procedure. We have illustrated the utility of our method by means of a numerically simulated 3D microscopy experiment. A question that remains open is how to reduce the noise artifacts inherent to the use of wavefront coding technique. In this sense, in further research we will consider the use of non-linear deconvolution procedures, as suggested in [25,26].

## Acknowledgements

This work has been funded in part by the Plan Nacional I+D+I (grant DPI2006-8309), Ministerio de Educación y Ciencia, Spain. We are very indebted to an anonymous reviewer for her/his fine comments and suggestions, which have really helped to improve the quality of this paper.

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