The resolution and signal level of confocal microscopy dramatically drops as the focus depth is increased due to the spherical aberrations (SA) induced by refractive index mismatch between immersion and specimen. At a depth of $5\mu \mathrm{m}$ the signal level falls to 40% of the value just above the coverslip; in a depth of $15\mu \mathrm{m}$ only 10% is left.¹ Therefore, in order to gain information from deep within a sample, it is necessary to compensate SA. Different static methods such as alteration of the tube length,² introducing an iris to decrease the pupil area of the objective,³ and dynamic methods such as deformable mirrors⁴ are proposed to compensate the SA. These methods are rather expensive, hard to implement, or need intensive computation. Not only resolution of confocal microscopy but also, e.g., the precision of localization of individual fluorophores in the axial direction is limited by SA.⁵ The simple method presented here can improve confocal microscopy such that information can easily be gained from depths of $100\mu \mathrm{m}$ , a resolution comparable to two-photon microscopy.⁶

The SA appear as a phase in the intensity point spread function⁷:

where ${\Psi }_{\text{tube}}$ stems from the tube length, ${\Psi }_{\mathrm{obj}}$ stems from the objective, and ${\Psi }_{\mathrm{im}∕\mathrm{cg}}$ and ${\Psi }_{\mathrm{cg}∕\mathrm{s}}$ denote the SA introduced by the possible mismatch in refractive indices of the coverglass and the immersion media of the objective and of the media within the sample, respectively. Minimizing ${\Psi }_{\text{total}}$ at any given depth produces the most focused laser beam, the optimal signal, and the best resolution. The idea presented here is to modify both the first, ${\Psi }_{\text{tube}}$ , and third, ${\Psi }_{\mathrm{im}∕\mathrm{cg}}$ , terms of the right-hand side of Eq. 1, such that they optimally cancel the phases introduced by the other terms. The two modifications have been individually reported to have an influence on the stability of an optical trap based on infrared light.^{8, 9, 10}

The experiment was performed using a scanning confocal microscope (Leica SP5) with an infinity tube length corrected (ITLC) objective (Leica HCX PL Apo, 63x, $\mathrm{NA}=1.32,\infty ,0.17$ ) or a finite tube length corrected (FTLC) objective (Leitz, PL APO, 100x, $\mathrm{NA}=1.32,170,0.17$ ). Figure 1 shows a schematic drawing of the sample chamber. Two cover glasses (Menzel Glaser, #1.5) were separated at one side by means of two stripes of double-stick tape to have gold nanoparticles attached to the cover glasses at various heights inside the chamber [Fig. 1a]. The diameter of the gold nanoparticles was $80\mathrm{nm}$ (BBInternational). The size was chosen so that they would provide a reasonable signal level in the nonaberrated case. The inner surface of the lower coverslip was chosen as zero for the depth measurements. The depth of the gold nanoparticles attached to the upper coverslip was measured as the distance traveled by the objective until the particular particle was in focus. To visualize the gold nanoparticles, we used immersion oils used with different refractive indices from Cargille (refractive index liquids set A). After deducting the average background intensity from all pixels, the maximum intensities at the positions of the nanoparticles were measured and the difference between the maximum intensity and the background noise at a given depth can be used as a measure of the confocal visualization efficiency. The gold beads were visualized by the reflection of a $488\text{-}\mathrm{nm}$ laser line (operated at 43% of maximum power) with the following settings: zoom 65, pinhole $600\mu \mathrm{m}$ , and field of view $3.84\times 3.84\mu {\mathrm{m}}^2$ .

Also, we tested the method on a sample containing a dense solution of living Schizosaccharomyces pombe yeast cells. The S. pombe yeast cells expressed green fluorescent protein (GFP) in all membrane parts and were visualized by exciting the GFP by a $514\text{-}\mathrm{nm}$ laser line (using 92% of maximum power). The following settings were used: zoom 3.7, pinhole $152\mu \mathrm{m}$ , and field of view $42\times 42\mu {\mathrm{m}}^2$ . The “set A” immersion oils used for visualizing the gold nanoparticles had autofluorescence in the same interval as GFP emission. Therefore, we used either the standard Leica immersion oil $(n=1.518)$ or a nonfluorescent custom-made Cargille immersion oil ( $n=1.538$ , code 1160, lot 071884) for confocal visualization of the fluorescent yeast cells.

The SA due to the refractive index mismatch at a depth of $d_w$ in the second medium can be written as²

where ${\theta }_1$ , ${\theta }_2$ , $k_0$ , and $d_w$ are the incident angle in first medium, refracted angle in second medium, wave number in vacuum, and nominal depth of focus, i.e., the distance traveled by the objective, respectively, and $n_1$ and $n_2$ are the refractive indices of the first and the second media [see Fig. 1b]. Hence, one way to increase the depth at which SA are minimized, $d_w$ , is to change index of refraction of the immersion media.⁸

If one wishes to image significantly deeper into the sample than can be done by just changing the refractive index of the immersion media, the tube length can be changed.⁹ Changing tube length in a commercial confocal microscope would be very cumbersome. Therefore, we performed another optical change that in effect corresponds to changing the tube length. If the microscope is designed to have parallel light entering the objective, one should use an ITLC objective for optimal visualization. If, instead, one uses an FTLC objective in a such a microscope, this corresponds to changing the tube length. Figure 1c shows the change in optical path of marginal rays corresponding to changing the objective from ITLC to FTLC. Consider a perfect lens, of some thickness, designed to image an axial point-like object $A^{\prime }$ to $A$ [full line in Fig. 1c]. The phase introduced by the lens for the marginal ray (with respect to the axial ray) that crosses the lens a distance $h$ above the optical axis can be written as²:

where the distances $S$ , $S^{\prime }$ , and $h$ are as defined in Fig. 1c, and $n$ is the index of refraction of the immersion media. Likewise, the distances $f$ , $f^{\prime }$ , $X$ , and $X^{\prime }$ are defined in Figure 1c. Knowing the NA, the tube length $\left(X\right)$ , the magnification $\left(M\right)$ of the objective, and considering the relations $f=X∕M$ , $f^{\prime }=nf$ , $X^{\prime }=f^{\prime }∕M$ , $S=X+f$ , and $S^{\prime }=X^{\prime }+f^{\prime }$ , all unknown parameters of Eq. 3 except $h$ can be found. In the present experiments $X=170\mathrm{mm}$ , $M=100$ , and $\mathrm{NA}=1.32$ . Hence, the above parameters become $f=1.7\mathrm{mm}$ , $f^{\prime }=2.58\mathrm{mm}$ , and $X^{\prime }=25.8\mu \mathrm{m}$ , respectively. The only remaining unknown, $h$ , can be calculated as $h=S^{\prime }\tan \theta$ [see Fig. 1c] with $\theta =\arcsin (\mathrm{NA}∕n)$ and $n=1.518$ , giving $h=2.6\mathrm{mm}$ . Substituting these values in Eq. 3 results in ${\psi }_{170}=-4.05k_0$ . If, on the other hand, an ITLC objective is used [dotted line in Fig. 1c] one can rewrite Eq. 3 as ${\Psi }_{\infty }=k_0\left[n(f^{\prime }-\sqrt{f^{\prime 2}+h^2})\right]$ . Inserting values for $f^{\prime }$ and $h$ we get ${\psi }_{\infty }=-4.01k_0$ . Therefore, the phase induced by changing the tube length from $170\mathrm{mm}$ to infinity will be ${\Psi }_{\text{tube}}={\psi }_{\infty }-{\psi }_{170}=0.04k_0$ .

The term ${\Psi }_{\mathrm{cg}∕\mathrm{s}}$ from Eq. 2 gives the contribution from the sample to the SA induced. Assuming that $h$ remains constant, ${\theta }_1=\arctan (h∕f^{\prime })=60.7\mathrm{deg}$ . Substituting this value along with $n_1=1.518$ and $n_2=1.33$ in Eq. 2 results in ${\Psi }_{\mathrm{cg}∕\mathrm{s}}=-0.58k_0{d}_w$ . In the case where the coverglass and the immersion medium are index matched $(n=1.518)$ , ${\Psi }_{\mathrm{im}∕\mathrm{cg}}$ in Eq. 1 vanishes, hence, the optimal microscopy depth is at the point where ${\Psi }_{\text{total}}=\Delta {\Psi }_{\text{tube}}+{\Psi }_{\mathrm{cg}∕\mathrm{s}}=0$ , which corresponds to $d_{w,\text{tube}}=69\mu \mathrm{m}$ . In other words, using the FTLC objective in a microscope designed for an ITLC objective causes the optimal confocal visualization depth to be $69\mu \mathrm{m}$ .

To have a continuous change of imaging depth a change of objective can be combined with a change of refractive index of the immersion media. Following the argumentation of Ref. 8, an increase of refractive index of the immersion medium by $\Delta n=0.01$ implies ${\theta }_0=60.0\mathrm{deg}$ (note that ${\theta }_1=60.7\mathrm{deg}$ ). Thus, the component of the phase factor arising from the oil-glass interface can be written as ${\Psi }_{\mathrm{imm}∕\mathrm{cg}}=0.021k_0{d}_o$ . The thickness of the immersion oil layer, $d_o$ , was measured for the FTLC objective to be $210\pm 9\mu \mathrm{m}$ . Balancing the second and the third terms of Eq. 1 $({\Psi }_{\mathrm{im}∕\mathrm{cg}}+{\Psi }_{\mathrm{cg}∕\mathrm{s}}=0)$ results in $d_w=7.6\pm 0.3\mu \mathrm{m}$ . Hence, if an FTLC objective is used instead of an ITLC objective in a microscope designed for an ITLC objective, an increase of $\Delta n=0.01$ will increase the optimal microscopy depth, $d_w$ , by $7.6\pm 0.3\mu \mathrm{m}$ .

To experimentally test how confocal visualization depends on the refractive index of the immersion oil, $80\text{-}\mathrm{nm}$ gold particles at different depths were imaged using an ITLC objective with two different immersion media. Each data point in Fig. 2a is an average of at least 10 measurements and the error bars show the standard deviation. Figure 2a illustrates that: (1) The maximum intensity for the normal immersion oil $(n=1.518)$ occurs at the glass surface or zero depth. (2) Increasing the refractive index of the immersion media shifts the optimum microscopy depth, as measured by the maximum intensity of the gold nanoparticles, deeper into the sample chamber. Hence, by changing the index of refraction of the immersion media, one can easily shift the optimal imaging depth by $20\text{to}30\mu \mathrm{m}$ . (3) A shift in the immersion media index of refraction from $n=1.518$ to $n=1.57$ causes a shift in optimal microscopy depth of $24.1\pm 0.2\mu \mathrm{m}$ . This shift is in excellent agreement with the predicted value of $\Delta d_w=4.1\pm 0.5\mu \mathrm{m}$ per $\Delta n=0.01$ .⁸ (4) The FWHM (full width at half maximum) of the graph is $\sim 16\mu \mathrm{m}$ . This implies that if a particular immersion oil is chosen, then the confocal visualization is very efficient within an axial distance of $\sim 16\mu \mathrm{m}$ . The right side of Fig. 2 shows confocal images of GFP expressing S. pombe yeast cells $8\mu \mathrm{m}$ from the surface using the normal immersion oil (b1) or the improved method (b2).

Finally, the combination of using an FTLC objective with changing refractive index of the immersion oil was experimentally tested. The inset of Fig. 3 shows confocal images of gold nanoparticles in a setup where an FTLC objective was used in a microscope designed for ITLC objectives. Each row of pictures is taken with a different immersion oil, and the depth increases throughout each row. Noticeable is the fact that the sharpest picture of experiments with $n=1.49$ , 1.518, and 1.57 appear for depths of 45, 65, and $96\mu \mathrm{m}$ , respectively. The larger the index of refraction, the deeper into the sample is the most efficient confocal visualization. Figure 3a illustrates that: (1) Increasing the refractive index of the immersion medium shifts the optimal microscopy deeper into the sample. Efficient confocal visualization is even possible as deep as $100\mu \mathrm{m}$ into the sample. (2) The graphs have FWHMs of $\sim 40\mu \mathrm{m}$ , which is more than twice that of the situation depicted in Fig. 2. (3) The immersion oil recommended by the manufacturer with $n=1.518$ provides an optimal visualization at a depth of $\sim 65\mu \mathrm{m}$ , which is in good agreement with our estimated value of $69\mu \mathrm{m}$ . (4) A 0.01 change in $n$ of the immersion media provides an $\sim 6\text{-}\text{to}7\text{-}\mu \mathrm{m}$ shift in the optimal microscopy depth, in agreement with the theoretically estimated value of $7.6\pm 0.3\mu \mathrm{m}$ . The right part of Fig. 3 illustrates the effect of using an FTLC objective in a microscope designed for ITLC objectives for a dense biological sample of flourescently marked S. pombe yeast cells: There is no visible signal at a depth of $40\mu \mathrm{m}$ if the ITLC objective is used (c1), but a good signal-to-noise ratio results with the FTLC objective (c2).

We presented a simple, easily implementable, and low-cost method to significantly improve confocal microscopy by canceling spherical aberrations. The spherical aberrations can be canceled by changing immersion media, possibly in combination with changing the objective from infinite correction to finite correction. We have shown efficient confocal visualization at any depth up to $100\mu \mathrm{m}$ inside the sample, depths which are comparable to those reachable by two-photon microscopy. In principle, any microscopy based on visible light, including two-photon microscopy, can be similarly improved by this method.