Depth-extended, high-resolution fluorescence microscopy: whole-cell imaging with double-ring phase (DRiP) modulation

We report a depth-extended, high-resolution fluorescence microscopy system based on interfering Bessel beams generated with double-ring phase (DRiP) modulation. The DRiP method effectively suppresses the Bessel side lobes, exhibiting a high resolution of the main lobe throughout a fourto five-fold improved depth of focus (DOF), compared to conventional wide-field microscopy. We showed both theoretically and experimentally the generation and propagation of a DRiP point-spread function (DRiP-PSF) of the imaging system. We further developed an approach for creating an axially-uniform DRiP-PSF and successfully demonstrated diffraction-limited, depth-extended imaging of cellular structures. We expect the DRiP method to contribute to the fast-developing field of non-diffractingbeam-enabled optical microscopy and be useful for various types of imaging modalities. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

of Bessel beams allow robust light propagation over a longer distance in highly heterogeneous biological specimens, effectively overcoming heterogeneity-induced distortions in deep-tissue imaging [19,21,22].
Although highly promising, the challenge for broader applications of Bessel beams remains due to the influence of Bessel side lobes [27]. While the side lobes are crucial for the non-diffracting and self-healing properties of the main lobe, their extended profiles contribute substantial out-of-focus background, leading to degraded image contrast and resolution. Existing methods to circumvent this limitation include two-photon excitation [18,25], structured illumination [18,28,29], and confocal line detection [30,31]. However, these methods induce complexity in instrumentation. Alternatively, several recent studies have demonstrated that the destructive interference between two harmonic cosine-Gaussian beams can largely suppress the side lobes without deteriorating the main lobe [32,33]. These methods allow for simplified system design while maintaining high controllability of the Bessel waveforms, thus implying a promising imaging scheme for Bessel-beam-facilitated optical microscopy.
We introduce here a DRiP method that enables depth-extended, high-resolution fluorescence microscopy using interfering Bessel beams. Compared to conventional widefield microscopy, the method effectively suppresses the Bessel side lobes, exhibiting a high resolution of the main lobe throughout a substantially extended DOF. We showed both theoretically and experimentally the generation and propagation of the DRiP point-spread function (DRiP-PSF) of the imaging system. Lastly, we optimized the DRiP-PSF and successfully demonstrated diffraction-limited, depth-extended imaging of cellular structures. The imaging system used in the study is shown in Fig. 1 In practice, Bessel beams can be generated by propagating a Gaussian beam through an axicon lens or a narrow annular aperture at the Fourier plane [3]. In this work, the DRiP modulation was obtained by creating two concentric annular apertures on the SLM with average radii (R 1 = 1500 µm and R 2 = 3000 µm) as illustrated in Fig. 1(b). Each annular aperture on the SLM is a Bessel-Gauss approximation using finite thickness ( 1 ΔR = 500 µm and 2 ΔR = 1000 µm) to resemble the ideal Bessel beam solution without compromising its axial extent [18,34]. The radii and thicknesses of the two annuli were determined based on screening varying SLM phase patterns for an optimized DRiP beam performance, including the main lobe width, the ratio between the main and side lobes, the axial extent, as well as the photon efficiency provided by the finite thicknesses.

Theory of the 3D DRiP waveform
An ideal Bessel beam solution to the scalar Helmholtz equation can be considered as a composition of plane waves propagating on a cone [2], which opening angle is described as where j represents the constituent Bessel beams (j = 1,2). Considering the Bessel-Gauss approximation in practice, the superposition of the two beams in three dimensions are given as [34]: sin , e x p t a n 2 where A j is the amplitude, 0 w and ( ) w z are the Gaussian beam widths at waist and at propagation distance z, respectively, and ( ) µm. Notably, unlike a single Bessel beam, the interference of the two Bessel beams leads to a remarkable spatial sequence of alternating bright and dark regions along the axial dimension at a period 1 2 2 / cos cos d λ η η = − [9,35]. In this study, to form an axially-uniform profile throughout the DOF, we developed an approach to creating two additional DRiP beams that were axially translated by / 2 d ± , respectively, using opposing Fresnel propagation generated by the SLM. As a result, the final beam, computationally created by averaging the three DRiP beams in three dimensions, exhibited a uniform profile for imaging the entire DOF without missing axial ranges (detailed in Section 3).

Numerical modeling for the 3D DRiP-PSF
The wavefunction of the 3D propagation of a point emitter imaged at the intermediate image plane of a high-NA microscopy system can be described by the scalar diffraction theory [36]: where obj f is the focal length of the objective lens, and 0 J is the zeroth-order Bessel function of the first kind. The variables v and u represent normalized radial and axial coordinates; the two variables are defined by Next, the fluorescence emission can be modulated into Bessel beams using the 4-f imaging system as shown in Fig. 1(a). The spatial modulation of annular apertures in the Fourier plane is described as ). Thus, the final wavefunction on the camera sensor can be described as: represents the Cartesian coordinates on the camera sensor. The 3D PSF of the system for a point emitter can thus be described as x p . In this study, we utilized the above numerical model for simulating the propagation of the 3D DRiP-PSF in a high-resolution imaging system, which results agreed well with the experimental results (detailed in Section 3).

Characterization of the DRiP-PSF
To characterize the 3D PSF of the imaging system, we first used sub-diffraction-limit 200-nm fluorescent beads as point emitters and recorded their images at different axial positions (Fig.  2). We compared the DRiP-PSF with the standard Gaussian PSF (GAU, or Airy disk, to be exact) and the Bessel PSFs generated with each individual annulus (represented as inner-ring (IR) or outer-ring (OR) PSFs). We noted three main features. First, as the sample was translated in z, the standard Gaussian PSF, recorded without phase modulation on the SLM, became significantly expanded due to diffraction beyond 0.5-1 µm from the focal plane ( Fig.  2(a)). In contrast, the DRiP-PSF maintains a largely non-diffracting profile at a diffractionlimited FWHM of 250-300 nm over a >4-µm range, showing >4-5 × improvement in the DOF (Fig. 2(a)-(c)). Secondly, the extended lateral profiles of both the IR and OR-PSFs induced stronger background compared to the DRiP-PSF ( Fig. 2(a)). The interference of the two Bessel beams in the DRiP-PSF substantially suppressed the side lobes, exhibiting >2-3 × improvement in the ratio between the main lobe and the first side lobe. In addition, the DRiP-PSF showed a slightly narrower beam profile compared to the IR-PSF and a further extended DOF compared to the OR-PSF ( Fig. 2(b),(c)), consistent with the theoretical prediction for Bessel beams generated with annular apertures of different opening angles. It should be noted that the FWHM values were obtained using Gaussian fitting of the main lobes of each PSF, which were affected by the lower peak-to-background ratio in the experimental results, especially for the OR-PSF. Theoretically, the main lobe width of a Bessel beam could reach the sub-diffraction limit compared to that of the corresponding Gaussian beam [37] (e.g. the OR-PSF in Fig. 2(d)-(f)). However, in practice, the experimental OR-PSF was broadened due to the fluorescent signal and stronger background ( Fig. 2(b),(c)). Thirdly, the bright and dark regions alternating along the axial dimension in the DRiP-PSF were clearly noticed, leading to variations of both main lobe intensity and lateral size. The spacing between the two dark regions was measured to be 1.20 µm, compared to the theoretical value  It should also be noted that the photon efficiency of the current DRiP design is measured as 39.0%, compared to the theoretical value of 43.5%. The deviation from the theory is mainly because the theoretical model considered uniform distribution of photons on the pupil, which is practically Gaussian-distributed. Numerical simulations on the performance of the DRiP PSF as a function of double-ring parameters have been demonstrated in Fig. 5 in Appendix 1. The axially alternating pattern of the main lobe of the DRiP-PSF leads to missing axial information in those dark regions. To address this limitation, we introduced two additional DRiP beams, axially translated in opposing directions by applying two corresponding phase masks of Fresnel propagation using the SLM (Fig. 3). The bright and dark regions of the original DRiP-PSF can thus be compensated by the two oppositely translated beams. Practically, the corresponding Fresnel phase pattern used on the SLM is described as

Creating axially-uniform DRiP-PSF
, where the axial spatial frequency z ν , represented as a function of the lateral spatial frequencies (i.e. Cartesian coordinates on the Fourier plane) ( ) , x y ν ν can be described as ( ) [38]. Such phase pattern allows the incident beam to propagate by 0 z after Fourier transform on the camera plane, which is effectively 2 0 / z M in the object domain. The distance of translation could first be estimated using the experimentally measured spacing between the dark regions, i.e. / 2 d ± = 0.6 ± µm. We then adjusted about this predicted value and identified 0 z = ± 7 mm, i.e. ± 0.7 µm in the object domain, so that the original DRiP beam ( 0 z = 0) could be optimally compensated ( Fig. 3(a)-(c)). Next, we computationally averaged the three DRiP beams in three dimensions, which exhibited an axially-uniform DRiP-PSF, maintaining a high resolution of the main lobe throughout a substantially extended DOF over > 4 µm (Fig. 3(d),(e)). By removing the alternating broadening, this method makes the DRiP PSF practical for imaging. The use of the SLM allows for automatic creation and acquisition of the translated PSFs and generation of the axially-uniform DRiP-PSF in a timely manner up to 60 Hz without the need for any mechanical scanning. In addition, the numerical simulation using the Fresnel propagation agreed well with the experimental measurements (Fig. 3). Finally, we imaged immune-labelled mitochondria in mammalian cells using the axiallyuniform DRiP-PSF and compared with conventional images taken using the Gaussian PSF without any modulation on the SLM (Fig. 4(a),(b)). Protocols for sample preparation were detailed in Appendix 2. For better visualization without epi-fluorescent background, similar comparison was also made between the corresponding deconvolved images using the experimental Gaussian and DRiP-PSFs, respectively (Fig. 4(c),(d)). Due to the limited DOF, the Gaussian images were recorded at a step size of 1 µm over 5 layers and stacked with different colors for comparison ( Fig. 4(a),(c)). Remarkably, the sub-micrometer mitochondrial structures spanning a ~4-µm axial range were clearly observed using the DRiP-PSF (Fig. 4(e)-(h)). The DRiP images not only maintained a high resolution over an extended range without the need for any sample or focal-plane scanning, but also captured mitochondria that were completely undetectable in the conventional images due to diffraction of the Gaussian PSF ( Fig. 4(g),(h)). The cell imaging results including the resolution, DOF and image contrast were consistent with the DRiP-PSF measurement (Figs. 2 and 3). The high-controllability of the DRiP waveform using the SLM, or other faster devices like deformable mirrors or digital micromirror device (DMD), allows for further optimization to visualize many other types of cellular structures that expand across a large volume in cells.

Conclusion
In this work, we have developed a DRiP method for depth-extended, high-resolution fluorescence microscopy. Using two interfering Bessel beams, the method not only maintains the non-diffracting property of Bessel beams, but also effectively suppresses the Bessel side lobes, substantially improving image contrast and resolution. Compared to conventional wide-field microscopy, the DRiP-PSF exhibits a diffraction-limited FWHM value of ~300 nm with four-to five-fold improvement of the DOF, allowing for imaging across a significant volume of whole cells without the need for scanning. The method represents a new method for fast projection of whole-cell dynamics for wide-field microscopy. Beyond wide-field microscopy, the method can be extended to various types of imaging modalities, such as particle tracking, light-sheet microscopy and one or two-photon microscopy. Furthermore, the DRiP method can be realized by implementing a fabricated double-annulus mask to the back pupil of the objective (i.e. the Fourier plane), advancing the method as a plug-in device compatible with most commercial optical microscopes. Transforming conventional Bessel beams, the method can also be used for optical manipulation, micro-machining, as well as generation of non-optical waveforms such as electron beams, acoustic and plasmonic waves. Fig. 5. Performance of the DRiP PSF as a function of double-ring parameters. (a) an increase in the radius of the inner ring, while maintaining all other parameters, will include more high spatial frequencies, thus leading to a decrease in the beam width; (b) an increase in the thickness of the inner ring causes the DRiP PSF close to a Gaussian; (c) an increase in the radius of the outer ring will continue to decrease the beam width at the beginning until the outer ring exceeds the pupil size of the objective, when gaining only low spatial frequencies leads to an increase in the beam width; (d) an increase in the thickness of the outer ring introduces more high spatial frequencies, leading to an decrease in the beam width.