## Abstract

Recently, V. V. Kotlyar et al. [Opt. Lett. **39**, 2395 (2014)] have theoretically proposed a novel kind of three-parameter diffraction-free beam with a crescent profile, namely, the asymmetric Bessel (aB) beam. The asymmetry degree of such nonparaxial modes was shown to depend on a nonnegative real parameter *c*. We present a more generalized asymmetric Bessel mode in which the parameter *c* is a complex constant. This parameter controls not only the asymmetry degree of the mode but also the orientation of the optical crescent, and affects the energy distribution and orbital angular momentum (OAM) of the beam. As a proof of concept, the high-quality generation of asymmetric Bessel-Gauss beams was demonstrated with the super-pixel method using a digital micromirror device (DMD). We investigated the near-field properties as well as the far field features of such beams, and the experimental observations were in good agreement with the theoretical predictions. Additionally, we provided an effective way to control the beam’s asymmetry and orientation, which may find potential applications in light-sheet microscopy and optical manipulation.

© 2014 Optical Society of America

## 1. Introduction

Bessel modes are the orthogonal nonparaxial solutions of the Helmholtz equation in the cylindrical coordinate system [1, 2]. These modes have been exploited for enormous applications in super-resolution biological microscopy [3–6], optical [7–9] and acoustical [10–12] manipulation, and material processing [13] benefiting from the distinct attributes of non-diffraction over propagation distance and self-healing beyond obstacles. Obviously, any linear combination of the Bessel modes with arbitrary coefficients is also a solution of the wave equation. Recently, such superpositions of Bessel beams have gained considerable interests, and many novel optical beams and unique spatial and topological properties have been observed. For instance, the direct superposition of two high order Bessel beams can form symmetric optical lattice [14] or a rotating helical beam [15]. In addition, propagation invariant Mathieu beams [16] and parabolic beams [17] can also be regarded as a linear combination of Bessel modes, which have peculiar intensity and phase distribution. The two-index Bessel beams, a kind of Bessel superposition modes actually, possess the same radial structure as Laguerre-Gaussian beams and unusual propagation properties [18]. Interestingly, the appropriate superposing of Bessel vortex beams can also create non-diffracting vortex beams with continuous orbital angular momentum order [19] or non-vortex beam of fractional type α [20–22]. Recently, V. V. Kotlyar et al. have introduced a new class of superposition of Bessel modes, termed as “Asymmetric Bessel mode” [23]. Such kind of beams can maintain their intensity profiles during propagation, which can be exploited for light-sheet fluorescence microscopy in deep tissues [5, 24]. While in contrast to conventional Bessel beam, such beam’s characteristic asymmetric excitation pattern may be able to enhance the contrast and resolution, enabling a step change for high-resolution biological microscopy [25].

According to the discussions in [23], the coefficient *c* of the superposition is restricted to positive real numbers, and this number determines the asymmetry of Bessel modes. In this work, we present a more generalized asymmetric Bessel mode in which the parameter *c* is a complex constant. It is verified that the complex parameter *c* determines not only the asymmetry degree of the mode but also the orientation of the optical crescent, and the orientation depends on the angle of complex *c*. As a proof of concept, the high-quality generation of various asymmetric Bessel-Gauss beams were demonstrated by means of a digital micromirror device (DMD). In addition, we experimentally demonstrated the control of the beam’s asymmetry and the rotation of the optical crescent through simple adjustment of the complex parameter, which is also related to the energy distribution and OAM of the beam. We note that control of the asymmetry and orientation of the beam will help enhance the contrast and resolution and control the illumination angle of the light-sheet microscopy. Besides, the modulation of the intensity profiles and OAM will find potential applications in optical manipulation.

## 2. Theoretical considerations

The asymmetric Bessel (aB) mode is a kind of nonparaxial beam with three parameters, and in the cylindrical coordinate system $\left(r,\phi ,z\right)$the complex amplitude of such mode reads [23]:

With the assumption that *c* is a real and positive constant, the diffraction-free modes described by Eq. (2) show asymmetric intensity distribution compared with conventional Bessel modes. Especially for the higher-order aB-modes ($n>0$), the beam presents an elegant form of a “right-convex” crescent at *z* = 0. And as is demonstrated in Figs. 1(a)-1(c), the asymmetry of the crescent aggrandizes in accordance with the increase of *c*. So the parameter *c* is defined as asymmetry degree. In addition, numerous countable isolated intensity zeros can be observed on the horizontal axis of the beam’s cross section, which correspond to optical vortices with unit topological charge (except for the axial one at $r=0$) and opposite sign on opposite sides of 0. Furthermore, an aB-mode generating a “left-convex” optical crescent was also deduced in [23], and the complex amplitude of such beam reads [23]:

We here extend the discussion on the occasion that the parameter *c* is a complex constant value. To obtain the amplitude of such mode, we need to substitute the positive parameter *c* of Eq. (2) with a complex number with a general form of$c={C}_{0}\mathrm{exp}\left(i{\varphi}_{0}\right)$, where the non-negative real value ${C}_{0}$ is the modulus of *c*, and ${\varphi}_{0}$ ranging in (0,$2\pi $) is the angle of *c*. Then this exact non-paraxial solution of the Helmholtz equation reads:

Apparently, when ${\varphi}_{0}=0$ and $\pi $, Eq. (4) can be simplified as Eq. (2) and Eq. (3), respectively. Therefore, Eq. (4) provides a general form for the right convex or left convex crescents. We note that the intensity distribution varies with the value of the complex *c*. When ${C}_{0}$ increases, the aB-mode becomes more asymmetric, which is consistent with the report in [23]. Interestingly, when the value of ${\varphi}_{0}$ increases, the orientation of the beam’s crescent rotates counterclockwise accordingly, exhibiting diverse forms, the “top-convex” crescent, “bottom-convex” crescent, or optical crescents with arbitrary orientations. The crescents with various orientations are simulated with parameters of *n* = 3, $\alpha =0.05$pixel^{−1} (1 pixel = $13.68\mu m$) and different values of *c*, which are shown in Figs. 1(e)-1(h). Hence, Eq. (4) describes a more generalized asymmetric Bessel mode of free space with a complex *c*, where the modulus of complex *c* governs the asymmetry degree of the mode and the angle of argument *c* determines the orientation of the beam crescent, respectively.

Similarly, on the axis within the beam’s cross section whose orientation depends on the angle argument of the complex *c*, a countable number of isolated intensity nulls can be observed. In the beam’s cross section at *z* = 0, the positions of the optical nulls and the main intensity peak governed by the three parameters of *n*,$\alpha $and *c* can be derived from the roots of Bessel function ${\gamma}_{np}$ (${J}_{n}\left({\gamma}_{np}\right)=0$):

*n*, which can be verified by analyzing the phase patterns in Figs. 1(i)-1(l).

Further, the integer *n* and the modulus of complex *c* govern the magnitude of the OAM of the aB mode. The OAM’s projection ${J}_{z}$ onto the optical axis and the beam’s total intensity *I* in a plane perpendicular to the optical axis are given by [23, 27, 28]

*n*and the modulus of complex

*c*but independent on its angle which controls the orientation of the beam. So one can easily adjust the constant

*c*and integer

*n*to obtain OAM with any real values.

## 3. Experimental methodology

As a proof of concept, we demonstrate the experimental realization of the generalized aB beams. Here we take the advantage of a super-pixel method to fully engineer the spatial amplitude and phase of an incident wavefront using a DMD combined with a low pass filter [29]. DMD is a competitive candidate for tailoring light fields due to its fast switchable, wavelength and polarization insensitive merits compared with the liquid crystal spatial light modulator (SLM) commonly designed for visible light. Recently the DMD has been exploited for various spatial modes’ generation [30–33] as well as high-speed focusing and imaging through turbid materials [34–36] and producing reconfigurable novel materials [37]. Upon these applications, the patterns projected onto the DMD need to be either predesigned or generated online with various algorithms. The static pattern is required to shape the asymmetric Bessel beams. We employ the super-pixel method to generate the binary hologram. In this method, the square regions of nearby micromirrors are grouped into super-pixels [29], and each super-pixel can independently modulate the amplitude and phase of light. The advantage of super-pixel method over Lee holography are high modulation fidelity and good robustness. So this method is very appropriate for the generation of complex modes that have a high resolution field in both amplitude and phase using a binary-amplitude DMD.

Conceptually, Figs. 2(a)-2(c) illustrate the principle of the super-pixel method using a binary-DMD and a spatial filter. First, the neighboring pixels within DMD are grouped into various super-pixels, and each super-pixel consisting of $m\times m$ micromirrors defines a complex field in the target plane. The target plane is conjugated with the DMD surface through a 4f configuration which is shown in Fig. 2(g). The key of the super-pixel method has two aspects. On the one hand, the two lenses in the 4f system should be placed a little off-axis with each other, so that the phase prefactors of the target plane responses of the micromirrors within each super-pixel are distributed uniformly between 0 and $2\pi $. On the other hand, an aperture filter is adopted to obstruct the high spatial frequencies in the Fourier plane, so that individual DMD pixels cannot be resolved in the target plane and thus the target plane response of a super-pixel is the sum of the individual pixel responses. For instance, as indicated in Fig. 2(b), the responses ${E}_{pixel}$ (black dots) of the $4\times 4$ pixels defined as a super-pixel are uniformly distributed over a circle in the complex plane with a phase step of$\pi /8$. Then if we make some certain mirrors indicated by green squares in Fig. 2(a) within a super-pixel turned on while the rest turned off, the resulting field ${E}_{\mathrm{superpixel}}$ (blue dot) is determined by the sum of the contributing pixels’ responses (active ${E}_{pixel}$, green dots). So one can construct a great many target fields by turning on different combinations of pixels within a super-pixel, as is displayed in Fig. 2(c).

In order to encode the amplitude and phase of the desired beam with the super-pixel method, a lookup table is employed to calculate and determine which DMD pixels to be turned on for each super-pixel [38]. It can minimize the calculation and optimize the performance. Here we apply this method to encode the complex field of the diffraction-free asymmetric Bessel modes, which have a high resolution field in both amplitude and phase. Figures 2 (d) and 2(e) show the normalized amplitude and phase pattern of a typical aB beam (*n* = 4, $\alpha =0.08$ pixel^{−1} and *c* = −1). The corresponding binary DMD pattern that encodes the desired field information is presented in Fig. 2 (f), in which the white and black represent the on-state micromirrors and off-state ones, respectively. The aB beam will be produced in the target plane when the DMD pattern is illuminated by a collimated laser.

Experimentally, a laser with wavelength 532nm (T532D20, Xi’an Minghui) is expanded through a telescope (L1 = 50mm, L2 = 500mm), as is illustrated in Fig. 2(g). A flat mirror is employed to adjust the direction of the incident beam on the DMD to ensure the incident angle to be 24° with respect to the normal direction. The employed DMD (0.7 in., XGA, Texas Instruments) has a resolution of $1024\times 768$ pixels. To select only the beam of first diffraction order and remove the high frequencies, we adopt a 4-f system (L3 = 250mm, L4 = 100mm) with a pinhole in the Fourier plane. The aperture should be placed at the position of $\left(x,y\right)=\left(-a,ma\right)$ with respect to the 0th diffraction order so that the phase prefactors of the responses of the pixels within each superpixel are distributed uniformly in the target plane [29], where $a=-\lambda f/{m}^{2}d$, $f=250mm$ is the focal length of Lens3, $m=4$ and $d=13.68\mu m$ is the pixel pitch. The experimental alignment is through placing the filter around the first diffraction order, and then fine-tune the position and size of the spatial filter by changing horizontal and vertical gratings to the DMD such that the two diffraction orders exactly pass through at the edge of the filter [29, 38]. Then the intensity pattern of the desired fields will be imaged onto a charge coupled device (CCD) camera placed at the back focal plane of Lens 4. Further, a Fourier lens (Lens 5, f = 250mm, not shown in Fig. 2(g)) placed confocally with Lens 4, produces the far field distribution of the beam. Accordingly, the CCD placed at the back focal plane of Lens 5 records the far field profiles.

## 4. Results and discussions

Theoretical nonparaxial asymmetric Bessel (aB) beam, whose lateral profile is infinite, is not physical. The asymmetric Bessel-Gauss (aBG) beam, being paraxial solution of the Helmholtz equation, has finite energy and can be experimentally produced [39]. Conceptually, the aBG beam can be described by the product of a Gaussian function by the aB beam. So the complex amplitude of the aBG beam at *z* = 0 reads:

#### 4.1 Near-field asymmetric Bessel beams

In order to obtain the desired aB beams with various parameters in the target plane (*z* = 0), the corresponding DMD patterns are designed using the super-pixel method in advance. In Fig. 3(d) we show a typical DMD pattern to create a left-sided aB beam with *n* = 5, $\alpha =0.08$ pixel^{−1} and *c* = −1. From the DMD pattern we can calculate the resulting target field by first applying a fast Fourier transform for the first lens, then a multiplication with a circular mask for the spatial filter and finally a second fast Fourier transform for the second lens [29]. The intensity profile of the calculated field is shown in Fig. 3(b). An excellent match can be observed compared with the profile of the ideal aB beam that is shown in Fig. 3(a). Next, we experimentally generate such beams using a DMD with an aperture. The observed beam is demonstrated in Fig. 3(c). The crescent-shaped lobe as well as the isolated intensity nulls indicated by Eq. (6) are clearly distinguished in the cross-section of the high-order aB beam. Then the one-dimensional transverse intensity profiles at *y* = 0 of the beams were reconstructed. Corresponding curves in Fig. 3(e) are measured transverse intensity profile (red line), theoretical (blue line) and calculated (magenta dashed line) profiles, respectively. Notice that the measured pattern is in good agreement with the theoretical and calculated ones. These results experimentally confirm the main features of an aB beam and the feasibility of high quality generation of such beam using a binary DMD.

Especially, we experimentally examine the influence of the coefficient *c* on the properties of the aB beams. In our discussions, the parameter *c* is a complex number. In order to understand the physical meanings of the parameter, we separately discuss the influence of absolute value and the angle of *c* but other parameters are the same (*n* = 5, $\alpha =0.08$ pixel^{−1}). In Figs. 4(a)-4(d) we show the beam profiles of the theoretical aB beams with different modulus (*C _{0}* = 0, 1, 2, 10) and the same angle (${\varphi}_{0}=0$). When

*C*= 0, the aB mode becomes a conventional Bessel mode whose profile is symmetric. Along with the increase of

_{0}*C*, the beam profiles start to become asymmetric more obviously. All these predictions are verified in our experiments, and the experimental results are demonstrated in Figs. 4(a’)-4(d’). According to the discussion in [23], it is the positive real

_{0}*c*that controls the asymmetry degree of the mode, but here we find it is the modulus of the complex

*c*that characterizes the asymmetry degree in this generalized situation. In addition, we can easily modulate the values of

*C*to obtain various aB beams carrying different OAMs, as indicated by Eq. (9).

_{0}Next, we experimentally explore the role of the angle. We present a supplementary animation (Media 1) to demonstrate the influence of the angle on the orientation of the optical crescent. As can be seen from the animation, the orientation of the beam’s crescent rotates counterclockwise in the same pace as the value of angle increases. Figures 4(e’)-4(h’) show some captured CCD images of the generated aB beams in the target plane with different angles (${\varphi}_{0}=\pi /3,\pi /2,3\pi /4,\pi $) and the same modulus (*C _{0}* = 1). The experimental results agree well with the theoretical predictions displayed in Figs. 4(e)-4(h). So here we theoretically and experimentally demonstrated that the parameter

*c*determined not only the asymmetry degree but also the orientation of the optical crescent. More importantly, we demonstrated a simple but effective way to control these properties of the beam through adjustment of the complex parameter.

We note that control of the asymmetry and orientation of the beam may have great potential in the light-sheet microscopy. Propagation-invariant Bessel beams can create a uniformly thin light sheet and provide an increased field of view (FOV) [3, 24], but the transversal outer ring structure of the Bessel beam produces background fluorescence and in principle precludes high axial resolution [25]. However, in contrast to the conventional Bessel beam, the diffraction-free aB beam presented here possesses asymmetric excitation pattern whose asymmetry can be controlled. Hence, a single aB beam light sheet can suppress the influence of the outer ring structure and thus yield high contrast and resolution over an extended FOV [25]. Besides, the control of the orientation of the light sheet may lend itself to efficient use of the omnibearing illumination and contribute positively to the imaging process.

#### 4.2 Far-field asymmetric Bessel beams

For further study of the aB beam, we investigate the far-field features. The far-field aB beam can be achieved through a Fourier transform of the near-field aB beam. Experimentally, a Fourier lens is adopted to perform the Fourier transform. Through substituting each term of the superposition on the right-hand side of Eq. (1) with the Bessel beam’s angular spectrum${F}_{n}\left(\theta ,\varphi \right)={\left(-i\right)}^{n}\mathrm{exp}\left(in\varphi \right)\delta \left(\theta -{\theta}_{0}\right)$, the Fourier spectrum of the ideal aB mode at *z* = 0 reads [23]:

Equation (11) describes the far-field amplitude of the aB modes, thus we can use it to analyze the characteristics of far-field modes. Let us separate the azimuthal component of the amplitude in respect of different arguments with the form of$A\left(\theta ,\varphi \right)=A\left(\theta \right)A\left(\varphi \right)$:

The angular spectrum of ideal aB mode is confined in a ring $\delta \left(\theta -{\theta}_{0}\right)$ in the frequency space, demonstrating that the aB beam is diffraction-free [40]. Besides, the phase and especially the amplitude undergo a change along the ring, resulting in the asymmetry for the far-field aB beam. The amplitude reaches maximum of $\mathrm{exp}\left({C}_{0}\right)$ at $\varphi =\pi /2-{\varphi}_{0}$ and minimum of $1/\mathrm{exp}\left({C}_{0}\right)$ at $\varphi =3\pi /2-{\varphi}_{0}$, respectively. Hence, the ratio of maximum and minimum is rising with the increase of C_{0}, indicating that the modulus of the parameter *c* governs the asymmetry of far-field aB modes and the energy distribution tends to be concentrated. This is consistent with the near-field aB beam, just with the entire far-field pattern rotated clockwise by $\pi /2$, as shown in Figs. 4(a)-4(d) and 5(a)-5(d). So when the aB beams propagate along optical axis in free space, they rotate clockwise with a rotation of $\pi /2$ from $z=0$ to$z=\infty $but keep the similar asymmetry.

All these predictions have been confirmed by our experiments and calculations as well. The experimental results are shown in Figs. 5(a*)-5(d*) along with the calculated ones in Figs. 5(a’)-5(d’). Indeed, the intensity unevenness of the beam profiles along the ring aggrandizes obviously with the increase of *C _{0}* (

*C*0, 0.5, 1, 2) and the energy of the beam tends to be concentrated. That means the asymmetry of aB beam increases. Further, we observe a good match between the simulation and measurement, apart from some distortion in the center of the profiles which is caused by the aberration in our optical system. As for the error between the simulation and theoretical prediction, it derives from the inherent defects of the super-pixel method using DMD, which is discussed in detail in [29]. Nonetheless, our observations confirmed the theoretical predictions and we experimentally realized the control of the asymmetry of far-field aB beams.

_{0}=Similarly, we realized the continuously regulation of the orientation of the far-field beam’s crescent. As displayed in Figs. 6(a*)-6(d*), the optical crescent is controlled to rotate counterclockwise according to the increasing value of angle (${\varphi}_{0}=0,\pi /3,5\pi /4,3\pi /2$) and with the same modulus (*C _{0}* = 1). The measured intensity distribution is in accordance with theoretical predictions [Figs. 6(a)-6(d)] and simulations [Figs. 6(a’)-6(d’)]. All these theoretical and experimental observations indicate that we provide a simple but effective means using a DMD to generate aB beams and to continuously regulate the asymmetry degree and orientation of such beam through adjustment of the complex parameter, which is also related to the energy distribution and OAM of the modes. Further, one can adjust the modulus and angle simultaneously to obtain desired aB beams with arbitrary asymmetry degree and orientation. The high-quality generation and control of the aB beams with various parameters will benefit future practical researches.

## 5. Summary

In conclusion, we present a more generalized asymmetric Bessel mode in which the parameter *c* is a complex constant. The complex parameter is verified to determine not only the asymmetry degree of the mode but also the orientation of the optical crescent, and affects the energy distribution and orbital angular momentum (OAM). As a proof of concept, the generation and control of various asymmetric Bessel-Gauss beams were demonstrated by means of a DMD and super-pixel based complex field modulation. In addition, we experimentally investigated the influence of the parameter *c* on the near-field properties of such beams as well as the far field features. We find that the absolute value of the complex constant determines the asymmetry and the angle governs the orientation of the crescent. The asymmetry of the aB beam aggrandizes obviously with the increasing value of modulus, while the beam crescent rotates counterclockwise in the same pace with the increasing value of angle. Besides, the far-field aB beam has the similar asymmetry with near-field aB beam but rotates clockwise by $\pi /2$as a whole. All these conclusions are confirmed by experiments, and the agreement between the experimental observations and theoretical predictions are seen to be excellent. More importantly, we provide an effective means to continuously regulate the properties of the beam such as the asymmetry, orientation, or OAM of the beam through simple adjustment of the complex parameter. Benefiting from the novel features of the aB beams and the merits of DMD, the results presented in this paper will enable many potential applications in high-resolution light-sheet microscopy, optical manipulation and microlithography, etc.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11374292 and 11302220).

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