1. Introduction

Over the past years, numerous strategies for manipulating the intensity, amplitude, or phase distribution near the focal plane of a lens have been proposed. Such three-dimensional (3D) point-spread function (PSF) engineering is particularly of importance in optical trapping and various imaging modalities [1–3]. Typically, the manipulation of a 3D PSF is performed by placing an optical element at the back pupil plane of the focusing lens, which achieves the desired transfer function of the optical system. This scheme has been employed in many applications, including super-resolution imaging [4, 5], adaptive optics [6, 7], bending PSF generation [8], and wavefront coding [9].

One of the main challenges in PSF engineering is to extend the depth-of-focus (DoF) without compromising the lateral resolution [10]. Non-diffracting beams, such as a Bessel beam [11] or Airy beam [12], have been exploited to extend the DoF in imaging systems [8, 13–20]. To generate the Bessel beam, one may place an annular aperture in the back pupil of a lens, but this approach compromises the output efficiency. The use of an axicon and relay optics is a possible alternative, but the setup is complex compared to the case with an annular aperture, and the beam profile is asymmetric along the axial direction [21]. Although this asymmetry can be addressed by shaping the wavefront of the input beam prior to the axicon, it requires a specially-designed diffractive optical element (DOE) or spatial light modulator (SLM), which would add complexity and cost [21]. An Airy beam can be generated using a cubic phase mask [12], but its complex phase profile hinders its widespread use. Cubic phase masks are generated using an SLM or multi-level lithography [22, 23]. The exploitation of aberrations is also an alternative method for achieving a DoF extension. For instance, Palillero-Sandoval et al. generated a phase-only DoF extending filter by combining four Zernike polynomial terms, associated with coma and trefoil [24]. Axisymmetric aberrations, i.e., defocus and primary spherical aberrations, have also been investigated to extend the DoF [25]. However, to generate such “beneficial” aberrations, precise wavefront-control devices such as a deformable mirror or phase-only SLM are required. Recently, Tomer et al. induced a strong spherical aberration in the sample plane of a light sheet microscope by placing a thick glass slab or layer of index matching gel between the sample and objective lens [26]. Although it does not require dedicated wavefront-control devices, it requires sufficient space for inserting the glass slab between a sample and objective, limiting the use of objectives with a high numerical aperture (NA) and short working distance. Such requirement for space also would enlarge the footprint of a sample chamber.

Considering the small footprint and simplicity in realization, the use of a binary optical element can be an attractive approach for PSF engineering. An axisymmetric binary phase mask (BPM), which is composed of concentric circles with phases of 0 and π rad (1, −1 in amplitude) have been utilized to realize DoF extension [27–29]. The DoF extension of these BPMs results from complex interference among the light passing through each phase ring of the mask. Therefore, it is important to determine the radii of the rings in order to obtain the desired focal behavior. The associated design procedures typically involve either exhaustive searching [28, 30] or optimization algorithms [31,32]. However, these procedures are time-consuming, and the solutions can be trapped in local minima. The processing time for the design-searching algorithm would exponentially increase as the number of concentric rings (i.e., phase-transition points) increases [28]. Sheppard et al. proposed an analytical approach by approximating the axial focal response of the BPM to Butterworth filters in one dimension (1D) [33] and two dimensions (2D) [29]. This method involves solving multivariable, high-order (≥ 4th order) nonlinear simultaneous equations. The designs made by these methods provide flat-top beam profiles along the axial direction, as with the Butterworth filter, but it is difficult to obtain solutions for cases with more than five rings owing to the increasing complexity of the nonlinear equations [34].

In this article, we propose a novel design methodology for a BPM, which is based on the binarization of phase-only continuous pupil functions. By converting the axisymmetric continuous pupil to a BPM with a phase of 0 and π rad, the response of the BPM is observed to be similar to that of its continuous original, with the exception of twin foci generation in the axial direction. We note that re-positioning and coherently summing two foci result in an elongated focus with a significantly longer DoF. The re-positioning of two foci in the axial direction can be achieved by multiplying the defocus term (i.e., quadratic phase term) in the continuous phase functions. As exemplary designs of such BPMs, we have chosen two representative axisymmetric phase functions, i.e., phase axicon and spherical aberration. The binary phase filters of the defocused axisymmetric phase functions enable DoF extension, but with different lateral PSFs; the former generated a Bessel-like needle focus and the latter generated an elongated focus without sidelobes. We describe our design method and present results that support the validity of our design strategy.

2. Method and results

The design procedures and validities of our BPMs are demonstrated through numerical simulation (MathWorks, Inc., Natick, MA, USA) based on scalar diffraction theory [35]. In the simulation, we considered an optical setup in which a BPM in the back pupil plane of a 0.2-NA lens is coherently illuminated by a plane wave of 0.47 μm light. The sampling rate for pupil plane was 1024 × 1024 pixels. Our simulation may be inaccurate for an optical arrangement with a high-NA lens (i.e., NA>0.5). In that case, vectorial calculations [36] can be employed to correctly estimate 3D PSF [27].

Our design aim is to obtain BPM structures that can produce uniform axial intensity distribution over the extended DoF. These BPM filters can be applied to various deep tissue imaging modalities that require high spatial resolution over larger DoF. The examples include optical coherence tomography (OCT) and light sheet fluorescence microscopy (LSFM). As in Ref [34, 37, 38], the DoF was defined as the axial distance between two positions that correspond to 90% of the maximum axial intensity.

2.1. Binarization of axisymmetric phase function

We first examine the focal behaviors of axisymmetric continuous phase functions and their binarization. For a complex-valued pupil function $P(\rho )$, we define the binarization as:

Figures 1(a)-1(c) shows the phase distribution of the three pupils and the corresponding 3D focal behaviors. Note that the pupil coordinates ${\overline{k}}_{\text{x}}$ and ${\overline{k}}_{\text{y}}$ are the spatial frequencies normalized with$NA/\lambda$. The $u$ and $v$ represent the dimensionless optical coordinates in the axial and lateral directions, respectively, as defined in [39]. For a phase axicon with a small slope parameter (e.g., $\alpha$ = 5) (Figs. 1(a1)-1(a2)), the light would first generate a Bessel beam right behind the axicon, and transmits through the lens, producing an annular ring pattern in the Fourier plane. One notable feature in this case is the formation of Bessel-like beam before the focal plane. This Bessel-like beam can be explained by the constructive interference of multitudes of spherical waves with their sources in the annular ring [40]. While the positive axicon generates the Bessel beam prior to the focal plane, as shown in Fig. 1(a2), the negative axicon would generate Bessel beam after the focal plane. A more detailed description of this behavior is provided in the Appendix. Figures 1(b1)-1(b2) shows the quadratic phase distribution of the defocus pupil of $\psi$ = 3.15 and its focal response. As can be imagined, the optical focus is observed to be shifted in the axial direction (shift distance $\Delta u$ = −40 in this case) without significant alteration of its overall shape compared with the case of the clear aperture. The spherical aberration in the pupil plane generates an elongated focus along the axial direction owing to the different focal lengths along the radial direction in the pupil domain (i.e., light from each radial position of the pupil generates focus at different axial positions). Figure 1(c1) shows the phase map of the positive SA pupil ($\gamma$ = 4). The corresponding focal response is characterized by the formation of the aberrated focus before the focal plane (Fig. 1(c2)).

 

Fig. 1 (a-c) Computer-generated pupil phase maps of axicon, defocus, and spherical aberration, along with the corresponding three-dimensional (3D) focal responses. (d-f) Binarized version of the pupil functions (a1, b1, c1), and their 3D focal responses. The pupil coordinates ${\overline{k}}_x$ and ${\overline{k}}_y$ are the spatial frequencies normalized with$NA\text{/}\lambda$. The u and v represent the dimensionless optical coordinates in the axial and lateral directions, respectively, as defined in [39]. Note that the binary phase filter generates two foci symmetric to the focal plane (u = 0 plane) in the optical axis.

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Binarization of these three pupils using Eq. (1) generates the BPMs, as depicted in Figs. 1(d1), 1(e1) and 1(f1). An examination of their 3D focal responses (Figs. 1(d2), 1(e2) and 1(f2)) reveals that these BPMs generate two foci that are symmetric about the focal plane in the axial direction. Because the axial response of the axisymmetric pupil can be expressed as the Fourier transform of the pupil function, $P(t)(t={\rho }^2)$ [39], the formation of two mirrored foci can be explained as the Fourier transform of a real-valued function, resulting in two mirrored signals in the Fourier domain.

The two foci generated with the BPMs are symmetrically separated by a certain distance around the focal plane. Therefore, the elongated focus can be obtained by repositioning and coherently summing the two foci. The shift of the two foci can be handled by adding the defocus term in the original continuous phase functions. It is observed that the defocus function (i.e., quadratic phase term) shifts each aberrated focus without changing its overall PSF shape. Therefore, we superimpose the defocus phase function with appropriate parameter value to the aberration function, and binarize the resultant phase function. In the subsequent sections, we present multiple designs of the DoF-extension BPM based on the proposed concept.

2.2. BPM of defocus + axicon (BPM-DF-AXI)

It is well understood that an axicon placed in the back focal plane of a lens would generate an annular ring in the focal plane [41]. However, for an axicon with a small slope coefficient $\alpha$($<N{A}^2\cdot f/\lambda$), the Bessel-like beam can be obtained either before or after the focal plane, resulting from the interference of multiple spherical waves with their sources located in the annular ring in the focal plane. The beam propagation simulation [42] for this particular case was performed, as detailed in the Appendix.

As noted previously, the BPM of the phase axicon generates two Bessel-like beams that are separated by a certain distance. Hence, by shifting and coherently summing the twin Bessel-like beams near the focal plane, an elongated focus can be obtained. The shift of the optical focus along the axial direction can be performed by multiplying a quadratic phase (or defocus) term and the axicon pupil function. The transmission function of the BPM of the defocused axicon is expressed as:

Figure 2 presents exemplary optimal BPMs with different$\alpha$values. We note that as$\alpha$increases, the number of phase transitions (or rings) and the optimal defocus parameter also increase. The separation distance between two symmetric Bessel-like beams is proportional to the slope of the axicon, $\alpha$. Therefore, the axial shift by $\psi$ should be increased accordingly to shift and combine two symmetric Bessel beam foci.

 

Fig. 2 BPMs through binarization of defocused phase axicon. (a-e) computer-generated BPM phase maps with optimal defocus parameters for given α’s. (f-j) Focal intensity profiles in a meridional plane generated by the corresponding pupil functions (a-e).

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The lateral and axial PSF profiles of the BPMs are shown in Fig. 3. The BPMs of the defocused axicon increase the DoF with a larger $\alpha$, and the flat-top axial response could be obtained (Fig. 3(a)). For the lateral response, PSF profiles exhibit narrower main lobes than for the case with a clear aperture. However, the pronounced sidelobes are observed, as in the Bessel beams from the annular aperture configuration [11]. The DoF increases by a factor of 3.2 × –7.1 × compared to the clear aperture. On the other hand, the full-width at half-maximum (FWHM) in the lateral dimension becomes smaller by a factor of 0.82 × –0.76 × . This indicates that increasing $\alpha$ leads the focal field towards that of the “true” Bessel beam. On the other hand, it should be noted that the Bessel-like beam generated by our BPM-DF-AXI does not completely follow the characteristics of the Bessel beam generated by an annular aperture. An annular aperture with a narrowing width would exhibit a lateral PSF with a smaller main lobe, and the distance between the lobes decreases, which is similar to the results in Fig. 3(b). However, the magnitudes of the sidelobes relative to the main lobe become larger for a narrower annular aperture [11]. For our BPMs, on the other hand, as $\alpha$ increases, the sidelobes are more suppressed, which can be beneficial in imaging systems that require an extended DoF. Such a sidelobe suppression effect for larger $\alpha$ may be an evidence of a Laguerre-Gaussian beam with a 0 azimuthal order [43], but further discussion about the beam characteristics is beyond the scope of this paper.

 

Fig. 3 Normalized (a) axial and (b) lateral intensity profiles for the BPM-DF-AXIs. The inset in (b) shows the magnified version of (b). In (a) and (b), the black line denoted by “Clear” represents the focal responses with a clear aperture.

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It has been noted that the maximum axial intensity decreases, as the DoF increases [29]. We examined Strehl ratio ($S$), namely the ratio of the maximum intensities obtained with the BPMs and a clear aperture, as a function of DoF gain (${\text{G}}_{\text{DoF}}$) (Fig. 4). The ${\text{G}}_{\text{DoF}}$ is defined as the ratio of DoFs obtained with our BPMs (${\text{DoF}}_{\text{BPM-DF-AXI}}$) and the clear aperture (${\text{DoF}}_{\text{Clear}}$), i.e, ${\text{G}}_{\text{DoF}}{\text{=DoF}}_{\text{BPM-DF-AXI}}/{\text{DoF}}_{\text{Clear}}$. One can observe that $S$ decreases, as the DoF extends. The result is consistent with the previous studies [29, 34, 44].

 

Fig. 4 Strehl ratio ($S$) vs. DoF gain (${\text{G}}_{\text{DoF}}$). It can be noted that $S$ decreases as ${\text{G}}_{\text{DoF}}$ increases. The BPM designs in the insets correspond to the ones in Fig. 2.

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2.3. BPM of defocus + spherical aberration (BPM-DF-SA)

The transmission function of the BPM-DF-SA can be expressed as:

Examples of the optimal BPM-DF-SAs with different $\gamma$ values are shown in Fig. 5. As $\gamma$ increases, the number of phase transitions gradually increases. However, unlike the BPM-DF-AXI case, the value of $\psi$ that is required for an elongated PSF formation did not vary markedly. It is likely that a larger spherical aberration elongates the focus to a larger extent along the optical axis, without a significant shift of the optical focus. Therefore, the amount of defocus is small, i.e., within the range of 1.3~1.5.

 

Fig. 5 BPMs through binarization of DF and spherical aberration. (a-e) computer-generated BPM phase maps with optimal defocus parameters for a given $\gamma$. (f-j) Intensity in a meridional plane generated by the corresponding pupil functions (a-e).

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Figure 6 shows the lateral and axial PSF profiles of the BPM-DF-SAs considered in Fig. 5. One can see that the BPM-DF-SAs produce a uniform intensity distribution over an extended axial range, as compared to the case with a clear aperture (Fig. 6(a)). As $\gamma$ increases, the intensity ripple is observed, and this may be due to the interference between two long and aberrated foci. Still, in our BPM-DF-SAs, the ripples stay above 90% of the maximum intensity. Figure 6(b) shows the lateral PSFs of the BPM-DF-SAs. It can be seen that while a BPM-DF-SA provides an extended DoF, it compromises the lateral resolution. Compared to the case with a clear aperture, the BPM-DF-SAs increase the DoF by 2.8 × –14.8 × , but broadens the lateral FWHM by a factor of 1.27 × –2.6 × .

 

Fig. 6 Normalized axial (a) and lateral (b) intensity profiles for the BPM-DF-SAs. The inset in (b) shows the magnified version of (b). In (a) and (b), the black line denoted by “Clear” represents the focal responses with a clear aperture.

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As in the case of BPM-DF-AXIs, the relationships between $S$ and ${\text{G}}_{\text{DoF}}$ ($\text{=}{\text{DoF}}_{\text{BPM-DF-SA}}{\text{/DoF}}_{\text{Clear}}$) were examined for the BPM-DF-SA designs in Fig. 5. The $S$ for the BPM-DF-SA filters also decays as ${\text{G}}_{\text{DoF}}$ increases (Fig. 7), as the case with the BPM-DF-AXI.

 

Fig. 7 Strehl ratio ($S$) vs. DoF gain (${\text{G}}_{\text{DoF}}$). It is observed that $S$ decreases as${\text{G}}_{\text{DoF}}$increases. The BPM designs in the insets correspond to the ones in Fig. 5.

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The BPM-DF-SAs provide the uniform intensity distribution over the extended DoF, while offering better lateral resolution compared to the case with an apodization-free low-NA lens. For instance, a 0.05-NA lens without apodization provides a DoF of ~144 μm, which is similar to the DoF obtained with the combination of 0.2-NA lens and the BPM-DF-SA ($\gamma =10,\psi =1.488$). However, the focal spot size with this combination (~3.21 μm) is 0.68 × narrower than that from the 0.05-NA lens only (~4.72 μm). It should also be noted that, in some applications where the undistorted PSF is required [45], the small sidelobes produced by BPM-DF-SAs are highly advantageous compared to the beams with larger sidelobes, such as the Bessel beam or Airy beam.

2.4. Design of BPM-DF-AXI and BPM-DF-SA for desired DoF

In Sec. 2.2-2.3, the features of two axisymmetric BPM structures for DoF extension, i.e., BPM-DF-AXI and BPM-DF-SA, were presented. For certain values of $\alpha$ and $\gamma$, the optimal value of $\psi$ could be readily obtained, which enables uniform axial intensity distribution over the extended DoF for each BPM structure. In practice, however, the desired DoF gain (${\text{G}}_{\text{DoF}\text{.d}}$) is specified for a certain optical system, and one may wish to design the BPMs that can provide ${\text{G}}_{\text{DoF}\text{.d}}$. Here, we describe the design procedures for such practical applications. Our method obtains two parameters, $\alpha$ and $\psi$ for BPM-DF-AXI and $\gamma$ and $\psi$ for BPM-DF-SA, which achieves desired DoF gain. Our design problem is stated as:

Equation (6) can be solved using various optimization methods such as stimulated annealing algorithm [46], or particle swarming optimization [32]. However, it has been recognized that direct search scheme is more efficient for the cases with a small number of parameters to be obtained [47]. In our case, the number of parameters is only two, and therefore we solved the problem via direct search algorithm. In solving Eq. (6), multiple $p$’s, which achieve the desired DoF gain, may be obtained. In that case, among those $p$’s, we chose the $p$ vector that maximizes the Strehl ratio.

To validate our design strategy, we designed the BPM-DF-AXI and BPM-DF-SA for a desired DoF gain of 5 via the proposed method. The axial and lateral intensity distributions of the obtained BPM-DF-AXI and BPM-DF-SA are shown in Fig. 8. The obtained BPM parameters are presented in the insets in Figs. 8(b) and 8(d). Note that the DoF gains for the obtained BPM structures are 5.04 (BPM-DF-AXI) and 5.03 (BPM-DF-SA), respectively. The Strehl ratios of the obtained BPM-DF-AXI and BPM-DF-SA were measured to be ~19.5% and ~15.6%, respectively.

 

Fig. 8 Axial and lateral focal responses of BPM-DF-AXI (a, b) and BPM-DF-SA (c, d) for the DoF gain of 5 obtained with the proposed design strategy. The BPM parameters were found via direct searching, and specified in the insets of (b) and (d). The insets in (a) and (c) are the phase maps of the BPM-DF-AXI and BPM-DF-SA designs.

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2.5. Responses of BPMs to broadband light illumination

Our BPM designs were conducted for a 0.47-μm monochromatic light. However, one may consider the use of the BPMs in optical systems that employ a broadband light source. Exemplary applications include optical coherence tomography [30] and various nonlinear microscopy techniques [41]. In order to assess the robust operation of our BPM designs to polychromatic systems, we examined the 3D focal responses of exemplary BPM-DF-AXI and BPM-DF-SA, designed at a monochromatic light at $\lambda =0.47\text{μm}$. The design parameters for the BPM filters are $\alpha =10$ and $\psi =4.808$ for BPM-DF-AXI and, $\gamma =10$ and $\psi =1.488$ for BPM-DF-SA. For simulation, a broadband light source centered at 0.47-μm with Gaussian spectral distribution, spanning 100 nm, was assumed. The 3D focal responses were obtained by summing the amplitude responses of the BPM at each wavelength. Figure 9 shows the axial and lateral focal responses of the BPM-DF-AXI (Figs. 9(a) and 9(b)) and BPM-DF-SA (Figs. 9(c) and 9(d)) under the illumination of the monochromatic and broadband light sources. It can be noted that no significant degradation both in the axial and lateral intensity profiles was observed as compared to the monochromatic case, albeit the small intensity ripples in the axial focal responses. These results indicate that our BPMs can readily be exploited in the optical systems employing broadband light sources, such as optical coherence tomography [48] and two-photon microscopy [41].

 

Fig. 9 Axial and lateral focal responses of representative BPM-DF-AXI ($\alpha \text{=10,}\psi =4.808$) (a-b) and BPM-DF-SA ($\gamma \text{=10,}\psi =1.488$) (c-d), under the monochromatic ($\text{λ=0}\text{.47μm}$) and broadband light ($\text{λ=0}\text{.47μm},\Delta \lambda =100nm$) illuminations. No significant deteriorations in the focal responses of the BPMs to the broadband light were observed as compared to the monochromatic cases. The insets in (a) and (b) are the BPM designs employed in the simulation.

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3. Discussion

Our design strategy for the DoF extension BPMs is based on the concept that the Fourier transform of a real-valued pupil generates two foci that are symmetrically distributed around the focal plane. Combining the two foci coherently using the defocus term results in a DoF extension. The idea that exploits such a “quasi-bifocus” for DoF extension has been suggested conceptually in [49], and a similar method using multiple foci was introduced in the 80s-90s [50]. However, the pupil filter designed with these methods modulates the amplitude; in particular, in [50], the pupil function is similar to an annular aperture, which is poor in terms of light throughput. Furthermore, the filters based on these methods exhibit continuous amplitude profiles, which is difficult to realize in practice. Our method considers a binary phase filter, and it can therefore be manufactured with various mass-production processes such as thin-film deposition [51], photomask lithography [52], and nanoimprinting technology [30]. The BPMs would find numerous applications, ranging from fiber-based miniature probes [30, 48] to wavefront coding masks for microscopy [53].

Previous design methods for the DoF-extension BPMs mostly employ optimization algorithms involving a large number of variables. Because the 3D PSF response of a BPM is a result of complex interactions among the light passing through multiple segments in the pupil filter, the objective function for the optimization of a BPM is usually nonconvex and has a number of local minima [31]. Therefore, it is critical to set the correct direction for the algorithm to converge rapidly. Many methods rely on iterative procedures that involve extensive field-based calculations and long searching time [28]. On the other hand, our design strategy utilizes Zernike phase functions related to aberrations, of which 3D PSF responses are well characterized. Therefore, the design is intuitive, does not suffer from increasing computational time for BPMs with a large number of rings, and the associated parameters can be easily specified.

The topology of our BPM designs is also worth further discussion. For the BPM-DF-AXI, the concentric phase rings are distributed with a near equal width, except for the one around the pupil perimeter. The lateral PSF becomes narrower as the DoF is extended. However, the rings of the BPM-DF-SA are concentrated in the outer part of the pupil. It generates an elongated focus along the axial direction, but the lateral FWHM also increases. Such a tendency was reported in the previous publications. Sheppard et al. [34] reported a DoF-extension BPM by approximating the axial focal response to a Butterworth filter. In their design, the pupils with the smallest lateral PSF were found to have near-equally spaced rings, with the exception of the one with the largest width around the pupil perimeter. This design is similar to that of our BPM-DF-AXI. The pupil designs with the largest lateral PSF found in [34] are characterized by the rings concentrated at the outer part of the pupil, which is also the same feature in our BPM-DF-SA design. Based on these observations, we suggest that the DoF-extension BPM designs reported in [34] and our proposed designs may all be related to the binarization of the Zernike polynomial pupil composed of axisymmetric terms. It should be noted that the continuous Zernike pupil has already been proposed as a DoF extension filter [24]. By performing the binarization of such Zernike pupils with a weighted sum of multiple aberration terms, we may be able to realize interesting features as with the continuous Zernike pupil function, beyond our demonstrations that have utilized only two Zernike terms.

On a final note, one should note that our design strategy can be extended to incoherent optical systems. Our simulation and analysis have been performed under the assumption of a coherent optical system. In other words, the BPMs placed in the back focal plane of a lens were illuminated by 0.47-μm coherent light such as a laser, and the lateral and axial focal responses were examined. The focus generation has been explained by the coherent sum of light passing through the BPMs. On the other hand, the BPM structures have also been proposed for DoF extension in incoherent optical systems. For instance, Zalevsky et al. [54] and Ben-Eliezer et al. [47] have proposed axisymmetric BPM structures for DoF extension in photography applications. Zhai et al. [55] investigated extended focus through an axicon in incoherent systems. In their studies, the incoherent light is represented by a combination of mutually incoherent monochromatic components that spans over a range of frequencies. Each component produces its own focal response, and the axially elongated focus is regarded as the sum of the intensities in these monochromatic responses. Our design method can be employed in such systems with proper intensity-based analysis for incoherent optical systems.

4. Conclusion

We proposed a simple design approach for BPMs that extend the DoF of a lens. The viability of our design was demonstrated with two BPMs: a BPM with a defocused phase axicon (BPM-DF-SA) and a BPM of spherical and defocus aberrations (BPM-DF-AXI). It was shown that for given parameters related to the axicon slope and SA strength, the defocus parameter can be easily determined, leading to a uniform axial response. In our numerical simulation, DoF gains were estimated in the range of 3.2 × –7.1 × for BPM-DF-AXIs and 2.8 × –14.8 × for BPM-DF-SAs, respectively. We anticipate that BPMs with stronger aberration and defocus would further extend the DoF. The BPMs designed using our method can be realized with conventional fabrication processes, and are expected to have wide applications, including optical trapping [56], focus-scanning microscopy [18], wavefront coding [47], and light-sheet microscopy [15, 31].

Appendix

Here, we examine the light propagation and three-dimensional optical focus distribution for an optical system composed of an axicon and positive lens. The simulation was performed based on the methods in [42]. First, we consider a positive axicon placed in the back focal plane of a lens (Fig. 10(a)). If a plane wave is incident on the axicon, the light would refract and produce a Bessel beam right behind the axicon. The generation of the Bessel beam can be explained as the constructive interference of multitudes of plane waves that are refracted toward the optical axis via the rear facet of the axicon. The plane waves propagate further and are transformed into the spherical waves through a lens of focal length, $f$, which are subsequently focused on the focal plane. The trajectories of these foci form an annular ring pattern, as depicted in Fig. 10(a). For an axicon with a slope parameter $\alpha$ smaller than $N{A}^2\cdot f/\lambda$, the converging spherical waves after the lens interfere on the optical axis, generating a Bessel-like beam (see the beam denoted by the dashed rectangle in Fig. 10(a)). In a similar manner, a negative axicon in the back focal plane of a lens leads to the generation of a “virtual” Bessel beam before the axicon (Fig. 10(b)). The plane waves emanating from the “virtual” Bessel beam propagate and are transformed into the spherical waves via the lens, which are again focused in the focal plane. These spherical waves interfere on the optical axis, generating a Bessel-like beam after the focal plane (see the region denoted by the dashed rectangle in Fig. 10(b)). Note that the Bessel-like beams can be observed using both positive and negative axicons at the back focal plane of the lens; the positive axicon produces the Bessel-like beam before the focal plane, while the negative axicon generates the Bessel-like beam after the focal plane.

 

Fig. 10 Beam-propagation simulations of axicon-lens systems. Positive (a), negative (b) and binary phase axicons (c) are placed in the back focal plane of the lens. Shown in the Insets are the x-y intensity profiles for each axial position. The intensities of the insets are normalized for better visualization.

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It is then straightforward to imagine light distribution for the case of a binary phase axicon (Fig. 10(c)). As noted in Sec. 2.1, the binarization of either positive or negative axicon would generate Bessel-like beams before and after the focal plane. The two beams are marked with dashed rectangles in Fig. 10(c). It should be noted that in the focal plane, a double-ring structure with different intensities is observed, which is consistent with the previous observation [41]. Therefore, the shift and coherent superposition of these two beams produce an optical focus with an elongated DoF.

Funding

National Research Foundation of Korea (NRF) (NRF-2015R1A1A1A05001548).

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