## Abstract

We propose a simple method to correct a large amount of spherical aberration caused by a refractive index mismatch. The method is based on inverse ray tracing and can generate correction phase patterns whose peak-to-valley values are minimized. We also demonstrated spherical aberration correction in a transparent acrylic block using a liquid-crystal-on-silicon spatial light modulator (LCOS-SLM). A distorted focal volume without correction was substantially improved with correction. This method is useful in cases where a large phase modulation is needed, such as when employing a high-NA lens or focusing a beam deep inside a sample.

©2009 Optical Society of America

## 1. Introduction

When a light beam is focused deep inside a transparent material, spherical aberration occurs as a result of the refractive index mismatch between different materials. The spherical aberration causes both lateral and longitudinal spreading of the point spread function and decreases the peak intensity. Such spreading and peak intensity reduction degrade the performance of optical instruments used in microscopic applications, such as confocal imaging [1], optical tweezers [2], and microfabrication in transparent materials [3–9]. For example, in microfabrication, the spreading worsens the resolution and positional accuracy, and the reduced peak intensity decreases the energy efficiency. Therefore, there have been many studies examining ways of correcting such spherical aberration.

Utilization of objective lenses with correction collars is one such way. It allows for correction of the spherical aberration by moving a lens group in an objective. However, this movement makes fine tuning difficult. Furthermore, this type of objective must be specially designed.

In recent years, techniques that correct wavefront aberrations by applying a pre-distortion pattern with a wavefront controller, such as a deformable mirror [10] or spatial light modulator [11], have been widely studied. Sherman *et al*. applied a genetic algorithm to generate pre-distortion patterns for multiphoton microscopy [12]. The patterns are iteratively improved so as to maximize a fluorescence signal. Although this method is beneficial for unknown or complicated aberrations, the system response is slow because of the iteration procedure. Booth et al. proposed a correction method in which pre-distortion patterns are determined by reversing the spherical aberration by theoretical calculations [13]. This deterministic approach has the advantage of allowing real-time correction.

In this paper, we propose a new correction method that enables precise correction. Our method uses inverse ray tracing for calculating pre-distortion patterns, instead of the conventional approach of reversing the spherical aberration. We also conducted correction experiments with a spatial light modulator (SLM) employing this approach. Since pre-distortion patterns are calculated so as to minimize the peak-to-valley (PV) value of the phase pattern, this method is suitable for achieving correction with wavefront controllers having a small phase modulation range.

## 2. Calculation of pre-distortion patterns for aberration correction

Figure 1 illustrates the focusing geometry for forming a pre-distortion pattern with inverse ray tracing. The arc indicated in Fig. 1 is a spherical reference plane with radius of curvature *f*. The reference plane passes through the back principal point of an ideal objective lens whose focal length is *f*. The back principal point *C* is located at an intersection of the spherical reference plane and the optical axis in Fig. 1. The point D is the intersection of the optical axis and the interface between different materials. The pre-distortion pattern will be calculated as a distribution of optical path difference on this reference plane. When a plane wave enters the microscope objective and there is no refractive index mismatch, the wavefront outputted from the objective is coincident on the reference plane in shape and converges to an original focal point *O*. Point *O*’ is the desired focal point when a refractive material is set at a distance *f* - *d* from the principal point. Therefore, distance *d*’ means the depth of the desired focal point. We assume that all rays incident on the reference plane are focused at the point *O*’ after being given the pre-distortion. The optical path length of each ray from *O*’ to the spherical reference plane can be calculated by using inverse ray tracing between those two points. In the inverse ray tracing procedure, the distance *d* from *O* to the surface of the material will be used as a parameter to control the PV value of the pre-distortion pattern. From now on, we call the depth *d* the focus depth without refraction and the depth *d*’ the focus depth with refraction.

In Fig. 1, the optical path length of ray *ABO*’, *Φ(θ)*, is given by

$$\phantom{\rule[-0ex]{.2em}{0ex}}=\genfrac{}{}{0.1ex}{}{{n}_{1}\times \left(f\times \mathrm{cos}\theta -d\prime \right)}{\mathrm{cos}{\theta}_{1}}+\genfrac{}{}{0.1ex}{}{{n}_{2}\times d}{\mathrm{cos}{\theta}_{2}},$$

where *θ*
_{1} is the incident angle of the ray after correction, *θ*
_{2} is the refraction angle of the same ray, and *θ* is the ray angle when the refractive material is absent. Here, *n*
_{1} and *n*
_{2} are refractive indices before and after the interface, respectively. A pre-distortion pattern is defined as an optical path difference distribution between ray *ABO*’ and ray *CDO*’, that is *Φ(θ)* - *Φ(0)*.

To calculate Eq. (1), one has to determine an angle *θ*
_{1} corresponding to a specific angle *θ*. The relationship between *θ* and *θ*
_{1} is determined from

An angle *θ*
_{1} for a specific angle *θ* can be obtained by solving these equations. However, we took a different approach. By solving these equations for *θ*, we get

Here, *θ*
_{1} can be obtained by iteratively calculating *θ* while changing *θ*
_{1} until *θ* is a desired angle as follow. Unless point *O*’ is not much deeper than point *O*, *θ*
_{1} is larger than *θ*, as shown in Fig. 1. Therefore, in the search procedure for *θ*
_{1}, we set the initial value of *θ*
_{1} to *θ* and gradually increased the value of *θ*
_{1} until *θ* calculated from Eq. (5) reached the desired value. Then one can calculate a pre-distortion pattern *Φ(θ)* - *Φ(0)* for specific *d*’ and *d*. This pre-distortion pattern, in general, has a large PV value, which has to be minimized. This can be achieved by a search procedure in which *Φ(θ)* - *Φ(0)* is iteratively calculated as *d* is varied until the PV value of the pre-distortion pattern is minimized.

## 3. Experimental set-up

Figure 2 shows the experimental configuration used for aberration correction and focal spot observation. A 660 nm beam from a laser diode (LD, Mitsubishi, ML101J27-01) was first collimated by an achromatic lens (L1) with a focal length of 120 mm and then reflected in the direction of an SLM by a prism mirror (PM). The SLM applied pre-distortion to the laser beam in order to correct the spherical aberration. The SLM used in our setup was a reflection-type SLM based on liquid crystal on silicon (LCOS) technology (LCOS-SLM, X10468-01, Hamamatsu Photonics K.K.) [14]. It had an active area of 16 mm × 12 mm, which was divided into approximately 800 × 600 square pixels (pixel pitch = 20 μm, fill factor = 95%). These pixels were electrically controlled with 8-bit signals from a computer via a digital visual interface (DVI). This type of LCOS-SLM can provide pure phase modulation from 0 to 2π in the visible region (400–700 nm). Furthermore, a larger modulation depth can be obtained by utilizing a phase wrapping technique. The prism mirror and the LCOS-SLM formed a module and were attached to the following relay-lens module by means of a C-Mount.

The relay-lens module, which included 8 light-bending mirrors and a telescope, imaged the wavefront at the LCOS-SLM approximately onto the entrance pupil of a microscope objective (Olympus Corp., LMPlan50IR, N.A. = 0.55, *f* = 3.6 mm). The telescope scaled down the size of the incident beam to match the size of the objective’s pupil diameter of 4 mm. In order to use SLM’s active area effectively, a combination of plano-convex lenses with focal lengths of 200 mm (L2) and 100 mm (L3) was used as the telescope. Thus, the effective area of the LCOS-SLM was imaged approximately onto the objective’s entrance pupil, making an area of 400 pixels in diameter effectively available. The modulated beam was focused into an acrylic block with a refractive index of *n*
_{2} = 1.49 (Mitsubishi Rayon, ShinkoliteL, 302), which was placed in air (*n*
_{1} = 1.0). The focal spot was monitored from the side of the acrylic block. An observation area of 60 μm × 60 μm was expanded onto a CCD camera (Sony, XC-ST30) by using a microscope objective (Mitsutoyo, MPlanNIR50, N.A. = 0.42), an extension tube (5 cm length), and an extender (Pentax, 2-EX). The acrylic block contained blue pigment, which enabled us to observe the focal spot through scattered light.

## 4. Results

In order to quantify the effect of our method, focal spots corresponding to cases with and without correction were evaluated. The focal spots measured without correction at focus depths with refraction of (a) *d*’ = 100 μm, (b) *d*’ = 500 μm, and (c) *d*’ = 1000 μm from the interface of the acrylic block are shown in Fig. 3. These results show that the intensity distributions of the focal spots spread asymmetrically in both radial and longitudinal directions. The asymmetry and the extent of spreading became worse at increasing focusing depths. This spreading, of course, caused the reduction in peak optical density.

The corrected focal spots and the pre-distortion patterns used are shown in Fig. 4 and Fig. 5, respectively. The phase of the pre-distortion patterns was wrapped within a phase range of 2π. Depths without refraction, *d*, were set to 64 μm, 319 μm, and 638 μm for the cases *d*’ = 100 μm, *d*’ = 500 μm, and *d*’ = 1000 μm, respectively. The intensity distributions of the corrected focal spots were symmetrical and similar to each other. Their beam waists were narrow compared with those obtained without correction. For example, the lateral full width at half maximum (FWHM) of the focal spot when *d*’ = 1000 μm was corrected from 2 μm to 1 μm, and the longitudinal FWHM was corrected from 36 μm to 7 μm. From these results, it can be concluded that this correction method worked effectively, even when the beam was focused deep inside the sample.

## 5. Discussion

We compared our method and the conventional method mentioned in Ref. 13. The same experimental conditions mentioned above were used. In the aberration correction procedure described in Ref. 13, the pre-distorted pattern is generated as an infinite sum of Zernike aberration terms. We applied Zernike terms of 4th, 6th, and 8th order and neglected the 2nd-order term that is related to defocus. Negxzlecting the 2nd-order term causes an axial shift of the focus spot. As described in Ref. 13, an axial shift of 𝓏 = 2√3cot(*θ*/2)*d*A_{20} is introduced by neglecting the 2nd-order term, and the theoretical focus depth with refraction, *d*’, becomes *d*+𝓏. The parameter *z* in this conventional method is equivalent to *Δd* in our method. When *d*’ = 1 mm, the focus depth without refraction, *d*, and axial shift, 𝓏 were 735 μm and 265 μm, respectively. The dashed line in Fig. 6 shows the center profile of a pre-distortion pattern generated by the conventional method of Booth *et al*. For comparison, we calculated a pre-distortion pattern with our method for the same focus depth (1 mm). The solid line in Fig. 6 shows the center profile of the resulting pre-distorted pattern. In our method, *d* and *Δd* were 638 μm and 362 μm, respectively. Compared with our pattern, the pattern generated by the conventional method had a PV value 1.5 times larger than that of the pattern generated by our method.

The focal spots corrected by the conventional method are shown in Fig. 7. Compared with the results of our method, the focal spots corrected by the conventional method contained slight asymmetry in their shapes, though they were well concentrated. We also noticed that the positions of the focal spots were displaced from the theoretical ones. We measured the depths of the focal spots in various cases with both our method and the conventional method. The results are shown in Fig. 8. In our approach, the relative displacements of the focal positions from the theoretical ones were approximately 6 %. This small displacement might have occurred because the objective lens used in the experiments was not ideal. On the other hand, the relative displacements in case of the conventional method were approximately 16 %. The origin of these large displacements is unknown.

## 6. Conclusion

We have presented a simple method based on inverse ray tracing to correct the spherical aberration due to a refractive-index mismatch by applying a pre-distortion pattern to the incident wavefront. Additionally, we demonstrated that laser spots focused deep inside a transparent material could be substantially improved through experiments using a phase-only LCOS-SLM as a wavefront controller. The pre-distortion patterns were generated so as to minimize their peak-to-valley (PV) values. The deeper the beam is focused, the larger the spherical aberration becomes. Therefore, this method can be applied to deeper focusing cases than conventional methods. This method is also suitable for high-N.A. focusing lenses or deep focusing positions, which produce large spherical aberration.

## Acknowledgments

The authors would like to thank N. Fukuchi and H. Toyoda of Hamamatsu Photonics K.K. for their technical support. This work was performed as part of a project on “High-efficiency Processing Technology for Three-dimensional Optical Devices”, supported by the New Energy and Industrial Technology Development Organization (NEDO) of Japan.

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