## Abstract

Wavefront encoding (WFE) with different cubic phase mask designs was investigated in engineering 3D point-spread functions (PSF) to reduce their sensitivity to depth-induced spherical aberration (SA) which affects computational complexity in 3D microscopy imaging. The sensitivity of WFE-PSFs to defocus and to SA was evaluated as a function of phase mask parameters using mean-square-error metrics to facilitate the selection of mask designs for extended-depth-of-field (EDOF) microscopy and for computational optical sectioning microscopy (COSM). Further studies on pupil phase contribution and simulated WFE-microscope images evaluated the engineered PSFs and demonstrated SA insensitivity over sample depths of 30 μm. Despite its low sensitivity to SA, the successful WFE design for COSM maintains a high sensitivity to defocus as it is desired for optical sectioning.

©2011 Optical Society of America

## 1. Introduction

Point-spread function (PSF) engineering, achieved by placing a phase mask at the pupil plane of the imaging lens to encode the wavefront emerging from an imaging system, has been implemented successfully to enhance optical system properties [1,2]. In extended depth-of-field (EDOF) microscopy, wavefront encoding (WFE) with a cubic phase mask (CPM), designed to reduce PSF sensitivity to defocus, produces an intermediate (encoded) image which is then digitally processed to decode the desired information [3,4]. In the final processed image, all structures distributed in a three-dimensional (3D) object are visible and well-focused in the 2D images acquired as the microscope is focused at different depths of the object (regardless of the location of the structures in the 3D volume). In this paper, we present a study that explores PSF engineering with WFE using different CPM-based designs with the goal to render the PSF less sensitive to depth-induced spherical aberration (SA).

Computational imaging plays a significant role in the advances achieved in 3D fluorescence microscopy [5]. Traditional wide-field microscopy has been transformed to quantitative 3D imaging by coupling digital processing to the measured data in order to reduce the impact of aberrations [6,7] and improve optical sectioning [8,9]. The widespread use of computational optical sectioning microscopy (COSM) [8] has motivated the development of new computational methodologies to reduce the impact of depth-induced SA, on the 3D image quality [10–12] by accounting for the fact that 3D microscope imaging is inherently depth variant (i.e., as the imaging depth increases within a sample, the 3D PSF changes due in part to a refractive index (RI) mismatch between imaging layers [13,14]). When the depth variability is significant, the use of multiple depth-variant (DV) PSFs is necessary in data processing to reduce undesirable computation artifacts [10,15,16]. As expected, computational complexity increases with the number of 3D DV-PSFs used in the computations [17] and thus, reducing the sensitivity of the 3D PSF to SA would reduce computational load and processing time, because fewer DV-PSFs would be used in the computations [12,18].

In this study, we investigate the use of WFE with different CPM-based designs as another approach to deal with 3D imaging in the presence of SA. In a previous study, one type of CPM-based design was assessed on its ability to render WFE confocal microscopy less sensitive to SA using simulation [19]. Radial phase masks (quartic and logarithmic) have also been derived to extend DOF and to alleviate SA effects when the system is well focused [20]. In another study, the engineered CPM-PSF has been shown to be susceptible to SA, albeit less than the conventional PSF [21]. The goal of this investigation is to reduce the impact of SA on the 3D CPM-PSF and other WFE-PSF obtained from alternative CPM-based designs (proposed for EDOF purposes in previous studies [1, 22–25]).

Three CPM-based designs [1,22,23] are investigated in this study to determine: a) a phase mask design that renders the PSF insensitive to both defocus and SA, suitable for EDOF microscopy; and b) another phase mask design that renders the PSF insensitive to SA only but not to defocus, suitable for COSM. Because COSM strives to provide the best 3D image of the underlying object intensity by removing from each optical section contributions due to out-of-focus structures, sensitivity to defocus is a desirable imaging characteristic. To our knowledge, this is the first study that investigates integrating WFE to COSM in order to reduce the impact of SA. WFE with a double-helix phase mask integrated to a widefield microscope to improve resolution has recently been presented as a proof of concept for a WFE-COSM system [26].

Previous studies on deriving and optimizing CPM-based designs for EDOF focused on improving the information (or frequency) transfer performance of the WFE system. Consequently, metrics based on the Fisher information [1,22], the optical transfer function and the modulation transfer function [4,22,23] have been used in CPM design investigation and/or optimization. In order to enable PSF engineering that addresses the impact of both defocus and SA for COSM and EDOF microscopy, we propose a metric for the design selection that is directly associated with PSF variability as a function of defocus and SA. We use merit functions based on the mean square error computed between the layers of the WFE-PSF to quantify defocus sensitivity as well as between two 3D DV WFE-PSFs (with different amounts of SA) to quantify SA sensitivity of the WFE-PSF. These merit functions are used to guide the selection of the mask design for both COSM and EDOF microscopy.

The paper is organized as follows: Section 2 reviews image formation theory for a WFE microscope in the presence of SA. Methods used in the evaluation and selection of a suitable phase mask design for EDOF microscopy and WFE-COSM are presented in Section 3. Results from our investigation of WFE-PSF variability as a function of defocus and SA are presented in Section 4, and they are further discussed in Section 5. Simulated WFE-microscope images are also presented in Section 4 to demonstrate the effectiveness of the engineered WFE-PSFs.

## 2. Theory

#### 2.1 Wavefront encoded PSF (WFE-PSF)

In a WFE microscope system a phase mask with phase$\varphi ({f}_{x},{f}_{y})$, inserted at the back focal plane of the imaging lens, modifies the phase of the generalized pupil function or defocused amplitude transfer function (ATF) and thereby changes the properties of the system’s PSF. A system suffering defocus and SA is characterized by a PSF that varies with both the depth $z=zi$ at which the microscope is focused and the depth $z=zo$ at which the light point source is located. Thus, the defocused WFE-PSF can be described as:

*λ*is the emission wavelength, and $W({f}_{x},{f}_{y};{z}_{i},{z}_{o})$is the optical path length error due to defocus and SA as a function of the normalized spatial frequencies${f}_{x}$and ${f}_{y}$. Thus, the phase of the generalized pupil function or defocused WFE-ATF is given by

#### 2.2 Image formation model in the presence of SA

The intensity in the intermediate (i.e., not processed) image formed by a WFE-system characterized by DV 3D PSFs [Eq. (1)] can be represented by the superposition integral:

*O*and$s({x}_{o})$ is the intensity of the underlying object. Equation (4) can be approximated with a strata-based model that requires only a finite number of 3D DV PSFs in the computation of the image formation model [10].

## 3. Methods

#### 3.1 Phase mask design patterns

A family of three CPM-based designs were selected from the literature for evaluation in this study: the CPM [1]; the generalized CPM (GCPM) [23]; and the sinusoidal CPM (SCPM) [22]. The CPM was chosen because it is the first phase mask designed to achieve EDOF through WFE while the recently published GCPM and the SCPM are interesting variations of the CPM, worth investigating, as evident from the mathematical function of the phase $\varphi ({f}_{x},{f}_{y})$for each mask: $\begin{array}{c}\text{CPM}\text{}:\varphi \left({f}_{x},{f}_{y}\right)=\alpha \left({f}_{x}{}^{3}+{f}_{y}{}^{3}\right)\\ \text{GCPM:}\varphi \left({f}_{x},{f}_{y}\right)=\alpha \left({f}_{x}{}^{3}+{f}_{y}{}^{3}\right)+\beta \left({f}_{x}{}^{2}{f}_{y}+{f}_{x}{f}_{y}{}^{2}\right)\\ \text{SCPM:}\varphi \left({f}_{x},{f}_{y}\right)=\alpha \left({f}_{x}{}^{3}+{f}_{y}{}^{3}\right)+\beta \left(\mathrm{sin}(\omega {f}_{x})+\mathrm{sin}(\omega {f}_{y})\right),\end{array}$ where α in radians represents the strength of the CPM [1], and β, also in radians, weighs the contribution of the second term in GCPM and SCPM affecting the deviation of these masks from the CPM. As evident from Eq. (2), if the strength of the CPM is large enough, the phase variation induced by CPM can dominate the phase variation in the pupil plane due to defocus and SA and render the WFE system insensitive to both. In this study, we investigate how the values of the parameters α, β, and ω affect the WFE-PSF’s sensitivity to defocus and SA. Based on reported values of the phase mask parameters in the literature, we chose to investigate the sensitivity of the WFE-PSFs computed by setting the value of α equal to 10, 20, 30, 50, 70, 100, 150 and 200 in all the mask designs; the value of β equal to -α/4, -α/3, -α/2, -α, −2α, −3α and −4α in the GCPM, and to α/3, α/2, α, 2α and 3α in the SCPM; and the value of ω equal to -π/2, -π/4, π/4 and π/2 in the SCPM. Typical and selected patterns of phase mask designs from the study presented in this paper computed within a circular pupil aperture (on a 201 x 201 grid with${f}_{x}$and${f}_{y}$ ranging from [-1 1] over the 201 columns/rows) are shown in Fig. 1 .

#### 3.2 Computation of WFE-PSFs

In this study, 3D DV complex amplitude PSFs for conventional widefield microscopy were computed using the Gibson and Lanni PSF model [13] which is readily available to us via the PSF computation module of our COSM Open Software (COSMOS) package (http://cirl.memphis.edu/COSMOS). The conventional PSFs are computed over a clear circular aperture (CCA) and thus we refer to them as CCA-PSFs when we want to distinguish them from WFE-PSFs based on a phase mask. A total of 13 DV CCA-PSFs were computed on either a 512 x 512 x 300 grid or a 1024 x 1024 x 300 grid with voxels of size 0.1 x 0.1 x 0.1 μm^{3} assuming that: a) the light point source is located at a different depth (${z}_{o}=$0, 5, 10, 15, 20, 25, 30, 40,…, 90 and 100 μm) in water (RI, *n _{water}* = 1.33) below the coverslip; and b) a 60x/1.2 NA oil-immersion objective lens (RI,

*n*= 1.515) and an emission wavelength

_{oil}*λ*= 633 nm are used to image the point source.

_{emission}The DV WFE-PSFs were calculated for all the phase mask designs using Eq. (1) by first computing the 2D Fourier transform of the conventional complex-amplitude PSFs (computed with COSMOS as described above) using Matlab (Mathworks, Natick, MA), thereby obtaining the conventional ATF. The phase patterns (Fig. 1) were resized to a 194 x 194 grid (or 388 x 388) and then padded to a 512 x 512 (or 1024 x 1024) grid so that the circular apertures in the patterns match the circular pupil aperture of the objective lens. A 1024 x 1024 grid was used in the computation of some of the DV WFE-PSFs in order to obtain PSF images without artifacts. Figure 2 shows XY and XZ cut views of WFE-PSFs obtained with different masks without SA [Figs. 2(a-d)] and with SA [Figs. 2(e-h)]. The images of the CPM-PSF with SA [Fig. 2(f)] are consistent with images in a previous study [21]. As evident in Fig. 2, the appearance of the WFE-PSFs changes with SA. This demonstrates the need to investigate and select mask parameters that could reduce the SA impact on WFE-PSFs.

#### 3.3 Simulated 3D images

Simulated intermediate images (i.e., images before any digital processing) from a WFE-widefield microscope (Figs. 10
and 11
) were computed using a strata-based approach [10] that approximates Eq. (4) and 3 computer-generated test objects that consist of 3 small spheres (1 μm in diameter) centered at different depths. In Object 1, the spheres are at depths 0, 5, and 10 μm [Fig. 10(a)] on a 256 x 256 x 300 grid with voxels of size 0.1 x 0.1 x0.1 μm^{3}. The origin of the *x _{o}y_{o}* plane is at the center of the grid, and the origin along the

*z*axis is placed at the 100th

_{o}*x*plane. The (

_{o}y_{o}*x*,

_{o}*y*) coordinates of the three spheres are: (−5 μm, 5 μm), (0 μm, 0 μm), and (5 μm, −5 μm), from top to bottom. Objects 2 and 3 are similar to Object 1 except that the depths of all the spheres are increased by 10 μm and 20 μm, respectively, in order to allow simulation of images with larger amounts of SA. The RI of the 3 objects is assumed to be equal to 1.33, while the immersion medium of the lens is oil (i.e., RI = 1.515).

_{o}In order to simulate the image of Object 1 using the strata model [10], 3 DV WFE-PSFs were computed as described in Section 3.2 assuming the point source is located at depths ${z}_{o}=$ 0, 5, and 10 μm, while PSFs associated with each stratum were approximated using a linear interpolation method, and two PSFs computed at the depths that form the boundaries of a stratum. Additional DV WFE-PSFs were computed at depths ${z}_{o}=$ 15, 20, 25, and 30 μm to generate the simulated images for Objects 2 and 3.

#### 3.4 Metrics for phase mask design evaluation and selection

An appropriate metric for the design evaluation and selection is PSF variability as a function of defocus and SA. To compare the performance of the phase mask designs and select the design that best met our imaging goals we investigated the sensitivity of WFE-PSFs (obtained for different phase mask designs by varying the mask parameters) to both defocus and SA using 2 metrics (Table 1
). To quantify sensitivity to defocus, we proposed a normalized mean square error (NMSE) that measures the difference between the XY layers of a 3D WFE-PSF given by Eq. (5), where ${h}_{{Z}_{k},{z}_{o}}(x,y)$ represents a layer at${z}_{i}=$*Z _{k}* and ${h}_{{Z}_{c},{z}_{o}}(x,y)$ represents the layer with the best focused 2D PSF at ${z}_{i}=$

*Z*. For a 3D WFE-PSF without SA (${z}_{o}=0$μm) the best focused layer (at${z}_{i}=0$) is in the center of the 3D PSF volume. As previously established, in the presence of SA the best focus shifts along the optical axis [19], and thus a new${z}_{i}=$

_{c}*Z*is determined from the conventional PSF. In general, a small NMSE

_{c}_{2D}value over a long range of

*z*indicates that the XY layers of the PSF are similar over the range, (i.e., the PSF is insensitive to defocus) and thus good EDOF is achieved in the WFE microscope.

_{i}Similarly, sensitivity to SA was quantified with another NMSE which measures the difference between 3D DV WFE-PSFs, described by Eq. (6) in Table 1. In Eq. (6), ${h}_{d}(x,y,z)$ is the 3D WFE-PSF with SA due to a point source located at depth ${z}_{o}=d$below the coverslip in the presence of a RI mismatch between imaging layers, and ${h}_{0}(x,y,z)$is the 3D WFE-PSF without any SA (i.e., the point source is located at depth ${z}_{o}=0$ μm below the coverslip at the interface between the sample and the lens’ immersion medium).

In all the WFE-PSF cases that resulted from varying the mask parameters as described in Section 3.1, the NMSE_{2D} and NMSE_{3D} were computed and plotted as a function of defocus and point source location, respectively, in order to investigate the sensitivity of the PSF to defocus and SA. The NMSE analysis was performed using WFE-PSFs on a 512 x 512 grid. PSFs computed on a larger grid were cropped to 512 x 512. Although cropping of the WFE-PSF can affect the values of NMSE_{2D} and NMSE_{3D}, we found that the effect was minimal and the results did not change significantly.

#### 3.5 Merit function for phase mask parameter selection suitable for COSM

For COSM it is necessary to find a phase mask design that reduces the impact of SA without simultaneously extending the DOF of the microscope at the same time. Towards this end, a merit function that combines the two NMSE metrics in Table 1 in a way that reflects this goal was used for the design selection:

_{2D}and the NMSE

_{3D}, respectively, over the investigation range. In this study, ${\overline{\text{NMSE}}}_{\text{2D}}$was averaged over the range ${z}_{i}=$ −5 μm to 5 μm while ${\overline{\text{NMSE}}}_{\text{3D}}$was averaged over the range ${z}_{o}=$0 to 30 μm. For COSM, a large $R$ value is preferred which can be achieved by increasing the NMSE

_{2D}value, (i.e. by increasing the sensitivity to defocus) while decreasing the value of the NMSE

_{3D}, (i.e. by lowering the sensitivity to SA). For all design parameter sets investigated (Section 3.1), Eq. (7) was calculated and the parameter set that yielded the largest $R$value was selected for COSM for each mask type.

#### 3.6 Computation of phase due to defocus and SA from 2D CCA-ATFs

As noted earlier, if the phase variation due to the mask is large enough, it can dominate the 2D ATF phase variation due to defocus and SA [first term in Eq. (2)] and render the WFE microscope insensitive to both aberrations as desirable for EDOF microscopy. For COSM, it is desirable that the mask’s phase variation dominates only the phase variation due to SA. To confirm that the selected phase mask designs for the two microscopy modalities met these design selection goals, their phase values were compared to the phase variation (due to defocus) and SA of the 2D CCA-ATF. CCA-ATFs were obtained by computing the 2D Fourier transform of each layer (at $z={z}_{i}$) of the conventional complex-amplitude 3D PSFs with different amounts of SA.

The phase term due to defocus was obtained from the 2D CCA-ATFs computed from a PSF without SA. The CCA-ATF phase term due to SA was isolated from the CCA-ATF phase term due to defocus by computing the phase difference between CCA-ATFs with SA and CCA-ATFs without SA at the same defocus distance *z _{i}*.

#### 3.7 Computation of 2D Modulation Transfer Function (MTF)

A comparison between 2D defocused MTFs of the conventional (CCA) and CPM-WFE systems has been used in a previous study to investigate and quantify the anticipated SNR reduction in EDOF [4]. Towards this end, 2D MTFs of the CCA and GCPM-WFE systems were computed from PSFs with and without SA to investigate the impact of SA on the anticipated SNR in the final EDOF and WFE-COSM images. In order to study only the effect of SA, 2D XY layers corresponding to the best evident focus were selected (i.e., with zero or minimal defocus due to the PSF discretization). For the comparison, 3 MTFs for each system were generated by computing the 2D Fourier transform of Eq. (3) with: a)${z}_{i}=0$μm and ${z}_{o}=0$μm; b) ${z}_{i}=-1.1$μm and ${z}_{o}=10$μm; and c) ${z}_{i}=1.5$μm and ${z}_{o}=20$μm. To improve accuracy in the computed MTF, the 2D CCA-PSFs were computed using a 4096 x 4096 grid and the CPM-based phase patterns (Sect. 3.1) were computed on a 2001 x 2001 grid. MTF profiles were compared along the normalized spatial frequency${f}_{p}$, which is along the diagonal line ${f}_{x}={f}_{y}$ in the frequency plane.

## 4. Results

#### 4.1 Effect of mask parameter α on the sensitivity of the CPM-PSF to defocus and SA

Results from studying the effect of parameter α on the variability of the CPM-PSF as a function of defocus and SA are summarized in Fig. 3
. The NMSE_{2D} computed for CPM PSFs without SA (${z}_{o}$ = 0 μm) and with SA (${z}_{o}$ = 20 μm) are plotted as a function of defocus (i.e., the distance from the designed best focal plane [FP]) in Fig. 3(a) and Fig. 3(b), respectively, for different α values. Overall, the values of the NMSE_{2D} decrease as the value of α increases indicating that better EDOF is achieved. This result is consistent with a prior reported result [1]. Figure 3(c) plots the NMSE_{3D} computed using PSFs with increasing amounts of SA (denoted by the increasing Z location of the point source which is equivalent to ${z}_{o}$) for different α values. It is evident that using a larger value for α in the CPM pattern renders the PSF less sensitive to both defocus and SA.

#### 4.2 Effect of mask parameters on the sensitivity of the GCPM-PSF to defocus and SA

The effects of the parameters α and β in the GCPM function on the WFE-PSF were investigated using 56 GCPM designs that resulted from the α-β combinations described in Section 3.1. Some of the results in this study are summarized in Fig. 4
. The NMSE_{2D}, computed for GCPM PSFs without SA (${z}_{o}$ = 0 μm), is plotted as a function of defocus for different α values [listed in Fig. 4(a)] and β values. As shown, when β = -α, the change in the α value has little effect on the NMSE_{2D} [Fig. 4(b)] indicating that the resulting 3D GCPM PSFs are similar in this case. When β = −3α [Fig. 4(c)], the NMSE_{2D} plots show greater variability than the one observed when β = -α/3 [(Fig. 4(a)], and they also show a similar trend as the one observed for the CPM design [Fig. 3(a)]. Moreover in this case, the NMSE_{2D} is lower for most α values than in other β cases, indicating a better achieved EDOF. This finding is consistent with prior reported results [23]. The smallest NMSE_{2D} is achieved at α = 150 for β = −3α

among all conditions investigated in this study. Similar NMSE_{2D} plots (not shown) were obtained for PSFs with SA (for ${z}_{o}$ = 20 μm).

Figure 4(d) plots the NMSE_{3D} vs. *z _{o}*, computed using PSFs with increasing amounts of SA obtained for different values of α when β = −3α. Similarly, Fig. 4(e) plots the computed NMSE

_{3D}for different values of β when α = 150. Among all conditions investigated in our study, the smallest NMSE

_{3D}is achieved for α = 150 when β = −3α.

#### 4.3 Effect of mask parameters on the sensitivity of the SCPM-PSF to defocus and SA

The effects of the parameters α, β and ω, in the SCPM function, on the WFE-PSF were evaluated using 160 SCPM designs that resulted from the α-β-ω parameter sets described inSection 3.1. Among the investigated parameter sets, the set α = 150, β = α/3 and ω = -π/2 resulted in the smallest computed NMSE_{2D} and NMSE_{3D}. Some of the results from this study are summarized in Fig. 5
. The NMSE_{2D}, computed from SCPM PSFs without SA, is plotted as a function of defocus for different α values when β = α/3 and ω = -π/2 [Fig. 5(a)]. Similar NMSE_{2D} plots (not shown) were obtained for PSFs with SA (${z}_{o}$ = 20 μm). Figure 5(b) plots the NMSE_{3D} vs. *z _{o}*, computed using PSFs with increasing amounts of SA obtained with the same parameter sets as in the case of the NMSE

_{2D}[Fig. 5(a)]. Overall, the trends observed in these results (Fig. 5) are similar with the ones for the GCPM case (Fig. 4).

#### 4.4 Phase mask design selection for EDOF microscopy

For EDOF microscopy the WFE-PSF is engineered to be insensitive to both defocus and SA. Towards this end, we selected phase mask parameters for each mask type based on the studies presented in Sections 4.1-4.3. Naturally, it is desirable to choose the parameters that yield the smallest NMSE_{2D} values over defocus as this supports the WFE-PSF’s insensitivity to defocus. In all the cases investigated for the CPM, GCPM and SCPM, these parameters also yielded the smallest NMSE_{3D}.

Figure 6
compares the sensitivity to defocus and SA of WFE-PSFs engineered with the 3 selected CPM-based masks. The NMSE_{2D} computed from PSFs without SA (*z _{o}* = 0 μm) and with SA (

*z*= 20 μm) for the 3 selected CPM-based designs is plotted as a function of defocus in Fig. 6(a) and Fig. 6(b), respectively. Figure 6(c) plots NMSE

_{o}_{3D}computed using PSFs with increasing amounts of SA vs.

*z*for the 3 CPM-based designs. Among the 3 designs, the selected SCPM design (with α = 150, β = α/3 and ω = -π/2) achieves the best NMSE

_{o}_{2D}performance, while the selected GCPM design (α = 150 and β = −3α) achieves the best NMSE

_{3D}performance and an acceptable NMSE

_{2D}performance. Thus, from this comparison study, the GCPM design is selected for EDOF microscopy in the presence of SA.

#### 4.5 Phase mask design selection for COSM

Unlike EDOF microscopy, in COSM there is a conflict between optical sectioning and the EDOF characteristic achieved with CPM-based WFE. For COSM the WFE-PSF is engineered to be insensitive to SA without losing its optical sectioning ability (i.e. sensitivity to defocus). Towards this end, we used the merit function $R$ [Eq. (7)] to select the best phase mask designparameters from the studies presented in Sections 4.1-4.3. For comparison purposes, the sensitivity of the CCA-PSF to defocus and to SA is compared to results obtained for the WFE-PSFs that yielded the largest $R$ value among all the designs investigated (Fig. 7
). The NMSE_{2D} computed from PSFs without SA (*z _{o}* = 0 μm) and with SA (

*z*= 20 μm) is plotted as a function of defocus in Fig. 7(a) and Fig. 7(b), respectively. Figure 7(c) plots NMSE

_{o}_{3D}computed using PSFs with increasing amounts of SA vs.

*z*for the 3 CPM-based designs. As it is evident, the CCA-PSF has the largest sensitivity to both defocus and SA among the compared PSFs [Figs. 7(a-c)]. This is also confirmed by comparing the resulting peak $R$ values in Fig. 7(d), where$R$ is plotted as a function of α. The $R$ value (=1.11) for the CCA-PSF is much less than all three peak $R$ values for the selected designs:$R=$ 2.03, 1.70, and 1.67 for the GCPM, the CPM, and the SCPM designs, correspondingly [Fig. 7(d)].

_{o}It is interesting to note that for some parameters, the CPM and SCPM designs yield an $R$ value smaller than the value for CCA [Fig. 7(d)]. This demonstrates that the phase mask must be properly designed to ensure that the use of WFE in COSM is beneficial. The results in Fig. 7 show that the selected GCPM design (α = 50 and β = -α) achieves the best performance for both NMSE_{2D} and NMSE_{3D}, and has the largest $R$value among the 3 CPM-based designs investigated in this study. Therefore, the selected GCPM design was chosen for COSM in the presence of SA.

#### 4.6 Comparison of selected GCPM to the CCA-ATF phase

To further investigate the degree to which selected GCPM designs for the two imaging modalities met the design selection goals, we compared their phases to the phase terms due to defocus and SA of the conventional defocused 2D ATF (or CCA-ATF), computed at different imaging conditions as discussed in Section 3.6. Figure 8 summarizes the CCA-ATF phase analysis results. Figure 8(a) plots profiles of the ATF phase due to defocus only, as a function of the normalized spatial frequency in the back focal plane of the objective lens for different amounts of defocus. Figure 8(b) compares profiles from the ATF phase due to SA only, for different amounts of SA, demonstrating the increase in the phase variation over the back focal plane and the investigated depth range. The phase due to SA was found to be approximately linear with depth.

Figure 8(c) compares profiles of the CCA-ATF phase due to different terms: defocus (${z}_{i}=-15\text{\mu m,}{z}_{o}=0\text{\mu m}$), SA (${z}_{i}=0\text{\mu m,}{z}_{o}=30\text{\mu m}$), the selected GCPM-EDOF phase mask (α = 150, β = −3α), and the selected GCPM-COSM phase mask (α = 50, β = -α). As evident, the selected GCPM-COSM mask has a phase variation which dominates the phase variations due to the SA term only for investigated depths not exceeding 30 μm [Figs. 8(b) and (c)] but it does not dominate the phase due to the defocus term. The latter is important for COSM since optical sectioning requires sensitivity to defocus. On the other hand, the GCPM-EDOF has absolute phase values that dominate all the other phase terms. In addition and as is evident in Fig. 8(d), where the derivatives with respect to${f}_{x}$ of the phase functions shown inFig. 8(c) are compared, the phase variation due to the selected GCPM-EDOF mask dominates those from both the defocus and SA phase terms as is desirable for this imaging modality.

#### 4.7 WFE-PSFs with selected designs

WFE-PSFs with different amounts of SA computed with the selected phase mask designs for EDOF and COSM, respectively, are shown in the different rows of Fig. 9
(the SA term is equal to zero in the first row and it increases in rows 2 and 3 of the figure). Figure 9(a) shows XZ cut-view images of the conventional widefield PSF which is equivalent to a WFE-PSF with CCA. XY (left) and XZ (right) cut-view images of the GCPM-PSF selected for COSM [Fig. 9(b)] and of the GCPM-PSF selected for EDOF microscopy [Fig. 9(c)], respectively. As depicted in Fig. 9, both engineered GCPM-PSFs show reasonable invariance to SA and features with respect to defocus sensitivity consistent with our design selection goals. Images of the GCPM-PSF engineered for COSM show desirable changes with defocus, while the corresponding images of the GCPM-PSF engineered for EDOF show an EDOF over the observed 30-μm *z* range.

#### 4.8 Simulated 3D intermediate images from EDOF microscopy

To evaluate the effect of the selected GCPM on the WFE microscope, simulated intermediate images from EDOF microscopy were generated using the GCPM-PSF in Fig. 9(c) and the three-sphere test objects described in Section 3.3 [Fig. 10(a)]. XZ views of the simulated images for different amounts of SA, achieved by placing the spheres at deeper depths, demonstrate qualitatively the EDOF and SA invariance characteristics achieved with the selected GCPM design and WFE [Fig. 10(b)]. This is also demonstrated by the XY cut-view images from different Z planes (shown by the dotted lines in [Fig. 10(b)] of the 3D image with different amounts of SA [Fig. 10(c)]. This result suggests that even in the presence of SA, with the use of WFE and our selected GCPM, the single 2D-PSF deconvolution approach used in EDOF microscopy [4] may still be adequate for processing these images in order to obtain the final EDOF microscopy images without any processing artifacts.

#### 4.9 Simulated 3D intermediate images from WFE-COSM

To evaluate the effect of the GCPM selected for COSM on the WFE microscope, simulated images from widefield microscopy were generated using the GCPM-PSF in Fig. 9(b) and the three-sphere test objects described in Section 3.3. XZ views from 3 simulated images with different amounts of SA shown in Fig. 11(a) appear very similar, confirming that SA-invariance is achieved with the selected GCPM design and WFE. This is also demonstrated by the XY cut-view images from different planes of the 3D image with different amounts of SA shown in each column of Fig. 11(b). In addition, the resulting WFE-microscope images appear to change with defocus as evidenced both by the intensity distribution of the XZ images and by the change in the appearance of the three beads in the XY images in each row of Fig. 11(b).

Overall results in Fig. 11 show that the use of the selected GCPM renders the WFE microscope less sensitive to SA without making it insensitive to defocus. This result suggests that using a DV stratum-based approach with a single stratum and two 3D WFE-PSFs (one at${z}_{o}=0\text{\mu m}$and one at the largest depth within the sample) may be adequate for processing these intermediate 3D images to obtain the final 3D COSM images without processing artifacts.

## 5. Discussion

Our results show that both parameters α and β have a large impact on the achieved EDOF in the GCPM and SCPM designs. We found that the interaction between α and β can at some times neutralize their impact on the achieved EDOF [Figs. 4(a-c)]. In the GCPM case, we showed that when β = -α, changing the value of α did not affect the achieved EDOF (Fig. 4b). Our study confirms that increasing the value of the parameter α in the CPM-based designs improves the achieved EDOF [Figs. 3(a), 4(c), and 5(a)] of the imaging systems as previously reported for the CPM [1]. Our results also suggest that a large α value should be chosen for EDOF applications.

However, large α values can cause an increase in sampling and manufacturing error. This is because, with a large α value, the mask’s wrapped phase values increase rapidly between 0 and 2π [Fig. 1(d)] and more samples are required in discretizing the phase function in order to represent the phase values accurately in this interval. In simulated experiments with a fixed grid size, it was evident that increasing the α value reduced the number of pixels available to represent one phase wrapping which could result in phase inaccuracies due to sampling error. Additionally, if the error in manufacturing the mask is proportional to the original unwrapped CPM phase term, then it is possible that the error would increase with an increasing α value.

Furthermore, the intensity of the CPM-based WFE-PSFs spreads over a larger area in the *xy* plane [Figs. 2(b) and 9(c)] compared to the CCA-PSF [Fig. 2(a)]. The large extend of the WFE-PSF in the *xy* plane imposes new constraints on image dimensions used for the finite-discrete domain implementation of the forward and inverse imaging problems (due to the crosstalk between multiple fluorescence points in the underlying specimen), which increase computational complexity. The tradeoff between achieved EDOF and computational complexity can be controlled by the chosen α value.

Two well-known issues of CPM-based EDOF microscopy are: (1) image shifting; and (2) low SNR in the final image. We investigated both these issues for several selected designs proposed in this study. As previously established, a lateral shift of the CPM-PSF peak at out-of-focus planes (evident more clearly in Fig. 12(a) than it is in Fig. 2(b) due to the PZ plane view used instead of the XZ plane), introduces artifacts in the final reconstructed image where specimen features away from best focus appear laterally shifted. Our results summarized in Fig. 12 support that these artifacts could be greatly reduced or even eliminated using different mask parameters or designs. For example, a larger α reduces the lateral shift in the CPM-PSF [Fig. 12(b)] while in general, GCPM PSFs do not exhibit a lateral shift within their EDOF range [Figs. 12(c) and (d)].

In previous studies [1,4], a reduction in the CPM-MTF intensity compared to the CCA-MTF intensity was shown which supports, as expected, the reduction of SNR in the final reconstructed images. Our results shown in Fig. 12(e) predicted the same result from the GCPM MTF. Moreover, increasing α from 30 to 150 in the GCPM design caused the corresponding MTF value to drop 36% ± 6% for frequencies in the range of 0.05 to 0.8 along *f _{p}* (Fig. 12e). Thus, it is anticipated that the EDOF-GCPM design will yield final reconstructed images with lower SNR than GCPM designs with a smaller α value. Another finding from Fig. 12(e) is that for the GCPM designs investigated in this study, a limited amount of SA had very little impact on its MTF characteristics, while the α value had some impact on the MTF characteristics as discussed above. For applications with large noise or applications requiring high SNR, the α value must be carefully selected. Furthermore, in some cases, EDOF and SA insensitivity, associated with a high α value, might have to be traded off to reach a required SNR level.

For the COSM application, the merit function *R* [Eq. (7)] was introduced to facilitate the selection of the mask parameters. Increasing the contribution of NMSE_{2D} in Eq. (7) ensures good optical sectioning while increasing the contribution of NMSE_{3D} ensures that better SA-insensitivity is achieved in the WFE-PSF. In most cases, the mask parameters associated with a small NMSE_{2D} also yield a small NMSE_{3D}. Clearly there is a tradeoff in selecting the mask parameters for COSM requiring that some insensitivity to SA must be sacrificed in order for the system to have adequate sensitivity to defocus. In this study, sacrificing insensitivity to SA by reducing the contribution of the NMSE_{3D} as in Eq. (7) yielded a WFE-PSF with desirable characteristics. Other choices to weigh the contributions of the 2 NMSE metrics are possible and could be investigated. The selection of the phase mask parameters for COSM could also be guided by evaluating the mask’s impact on the restored (final) images. The MSE metrics used in this study are directly related to the PSF engineering goals and facilitated the selection of the designs so that the goals could be met. However, determination of optimal phase mask designs requires the development of appropriate optimization methods.

The study presented here is a proof of concept and it provides a methodology for engineering PSFs with reduced variability due to SA using computed theoretical PSFs. Although the CCA-PSF model used in this investigation includes apodisation effects, it does not include vectorial effects [27] which can affect the appearance of the WFE-PSF and consequently the EDOF range for a high NA system predicted by our methodology [4]. Similarly, the predicted depth within the sample over which the WFE-PSF shows reduced sensitivity to SA could be affected by inaccuracies in the PSF model. Based on comparison studies between experimental and theoretical CPM-PSFs [21], it is expected that the presented methodology could provide more accurate predictions if it is applied to WFE-PSFs computed using a model that includes vectorial effects [4,27] instead of the CCA-PSF model used in this study.

## 6. Conclusion

In this study, we evaluated WFE with CPM-based designs with a goal to engineer 3D PSFs that are less sensitive to depth-induced SA (due to a RI mismatch in the imaging layers) than the conventional widefield microscopy PSF for a high-NA lens. Three existing CPM-based designs were investigated using two MSE metrics that quantify the PSF’s sensitivity to defocus and SA. Phase mask designs were selected using these metrics and evaluated for two microscopy applications. Our results show that among the evaluated phase mask designs, the generalized cubic phase mask (GCPM) provided more suitable designs that met our selection goals for EDOF microscopy and for COSM. Achieved EDOF and SA invariance were also evaluated with a comparison of the ATF phase variation (due to SA and defocus) in the pupil plane to the phase variation due to the GCPMs selected for each of the two microscopy applications. The desired imaging characteristics were also confirmed by comparing simulated intermediate images with different amounts of SA and defocus from the resulting WFE-microscope in each case, and were computed using several simple test objects. Our results show that both GCPM designs selected by this study render the WFE-microscope less susceptible to SA for sample depths of 30 μm (when the RI mismatch is between water and oil) and that the GCPM selected for COSM does not render the microscope insensitive to defocus as desired for optical sectioning. Additional studies are currently underway to investigate image restoration of the final 3D images from the intermediate WFE images using the WFE-PSFs. These studies will further confirm the benefits of the engineered PSFs proposed by this study.

## Acknowledgement

This work was supported by the National Science Foundation (NSF CAREER award DBI-0844682 and NSF IDBR award DBI-0852847, PI: C. Preza) and the University of Memphis.

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