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Abstract
Axial optical chain (optical bottle beams) beams are widely used in optical micromanipulation, atom trapping, guiding and binding of microparticles and biological cells, etc. However, the generation of axial optical chain beams are not very flexible at present, and its important characteristics such as periodicity and phase shift cannot be easily regulated. Here, we propose a holographic method to achieve the axial optical chain beams with controllable periodicity and phase. A double annular phase diagram is generated based on the gratings and lenses algorithms. The beam incident to the double annular slits was tilted from the optical axis to produce concentric double annular beams. The annular beam with different radius will produce the zero-order Bessel beam with different axial wave vector. Axial optical chain beams is produced by interference of two zero-order Bessel beams with different axial wave vectors. The phase and periodicity of the axial optical chain beams can be changed by changing the initial phase difference and radius of the double annular slits of the double annular phase diagram, respectively. The feasibility and effectiveness of the proposed method are demonstrated by theoretical numerical analysis and experiments. This method will further expand the application of axial optical chain beams in optical tweezers, optical modulation and other fields.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since Durnin et al. proposed Bessel beam [1] in 1987, due to its special properties, Bessel beam has been widely used in laser processing [2], biomedical optical imaging [3,4], micromachining [5], quantum [6] and telecommunications [7] and other fields. The theory of Frozen Waves points out that the interference of Bessel beams can generate arbitrary distribution of optical field in the axial direction [8–11], one of which is the axial optical chain beams [12–14] generated by the interference of two zero-order Bessel beams with different axial wave vectors, also known as optical bottle beams [15]. The optical intensity of the axial optical chain beams shows periodic cosinoidal changes in the axial direction, and the region with zero intensity was surrounded by high-intensity beams. The axial optical chain beams can be used as optical tweezers, which is applied to optical micromanipulation [13], atom trapping [16,17] and other fields, and has a very promising prospect of application.
There are a number of methods to generate axial optical chain beams. Geo M. Philip et al. achieved optical bottle beams by focusing a $\pi$–phase shifted multi-ring hollow Gaussian beam (HGB) using a lens with spherical aberration [18]. A double-negative axicon is produced by chemically etched in the optical fiber tips. A multi-ring hollow Gaussian beam with a large difference in transverse wave vector ${k_r}$ and a small difference in longitudinal wave vector ${k_z}$ can be obtained. By changing the position of the lens relative to the distal end of the fiber, the size and the periodicity of the optical bottle beams are changed. However, this method is relatively complex, which is not flexible enough. Fengtie Wu et al. proposed to use a conical lens to focus Bessel beams to obtain axial optical chain beams [19]. This method requires two conical lenses, one for generating Bessel beams and the other for focusing Bessel beams. Therefore, it is difficult to control the periodicity and phase of the axial optical chain beams with this method. Vladlen G. Shvedov et al. made Gaussian beams pass through uniaxial crystals to obtain axial optical chain beams [20]. In short, the current methods of generating three-dimensional axial optical chain beams are not flexible enough to conveniently control the periodicity and phase.
In this manuscript, we report a holographic method to obtain the axial optical chain beams with controllable periodicity and phase. Firstly, a phase hologram of double annular slits was generated. Phase of grating is applied to the concentric double annular slits to make the beam incident to the double annular slits deviate from the optical axis to produce concentric double annular beams. The double annular beams placed on the front focal plane of the objective lens will produce zero-order Bessel beams with different axial wave vectors near the focal area. Thus, the axial optical chain beams can be achieved by the interference of two Bessel beams with different axial wave vectors. The periodicity and phase shift of the axial optical chain beams can be easily controlled by controlling the radius and phase difference of the inner and outer circular slits of the phase diagram. The feasibility of the method is verified by theoretical numerical analysis, and the effectiveness of the method is verified by experiment. The experimental results are in agreement with the theoretical results.
2. Generation of controllable axial optical chain beams
2.1 Theory of controllable axial optical chain beams
Here we use phase holography to obtain the axial optical chain beams with adjustable parameters. Zero-order Bessel beam is a typical “non-diffractive beam”, which is widely used due to its invariant transmission in space. It is an exact solution $\phi (r,z) = {e^{i{k_z}z}}{J_0}({k_r}r)$ of Helmholtz equation [1]. Where, ${k_z}$ and ${k_r}$ are the axial and radial wave vector, respectively. The zero-order Bessel beam can be regarded as the coherent interference of two plane waves with wave vectors ${\pm} k$.
Zero-order Bessel beam can be generated by annular beam, and zero-order Bessel beams with different axial wave vectors can be generated by concentric double annular slits. Axial optical chain beams can be generated by interference of Bessel beams with different axial wave vectors. According to the theory of Fourier optics, the ideal Bessel beam does not exist in reality due to limited aperture. In practice, the “Bessel beam” that can be realized is “Bessel-Gaussian beam”, which has a finite diffraction-free distance [1]. For an annular slit of finite width, the Fourier transform is the zero-order Bessel function. Thus, the incident light can be turned into Bessel beam by placing an annular slit on the rear focal plane of the lens. If the annular slit is symmetric about the optical axis, the distribution of field intensity $E(r,z)$ near the front focal plane can be known by the scalar diffraction theory [21].
Fig. 1. The axial optical chain beams. (a) Setup for the generation of axial optical chain beams. (b) Axial optical chain beams obtained by simulation. (c) The distribution of the cross-sectional intensity of the brightest region in (b). (d) The distribution of the cross-sectional intensity of the darkest region in (b). NI, normalized intensity.
As mentioned above, the phase method is used to control the periodicity and phase of the axial optical chain beams. The key to this method is to generate a double annular phase diagram ${\phi _h}$ based on the gratings and lenses (GL) algorithms [22]. First, we generate an initial concentric double annular phase diagram ${\phi _{h0}}$, as shown in Fig. 2(a), different gray levels of the inner ring and the outer ring represent different phases. The initial phases of the inner ring and outer ring are ${\phi _1}$ and ${\phi _2}$, respectively. Thus, the initial phase difference $\theta$ is $|{{\phi_2} - {\phi_1}} |$. The initial phase difference can be controlled by changing the gray levels of inner and outer rings. When the phase diagram is loaded into a phase only spatial light modulator, the light incident into the annular slits is phase modulated, while the light incident into other regions is not modulated but directly reflected and superimposed with the modulated light.
Fig. 2. Double annular phase diagram. (a) Initial concentric double annular phase diagram ${\phi _{h0}}$. (b) Phase of gratings ${\phi _{prism}}$, which result in lateral shifts. (c) The double annular phase diagram ${\phi _h} = ({\phi _{prism}} + {\phi _{h0}})\bmod 2\pi$.
In order to separate the modulated light from the unmodulated light, we use the gratings and lenses algorithms to deflect the light incident into the double annular slits from the optical axis to the first-order diffraction. The GL algorithm combines two basic optical elements: the gratings, which result in lateral shifts, and the lenses, which result in axial shifts. Here we only use the phase of gratings ${\phi _{prism}}$ to tilt the double annular beams from the optical axis. When a beam with a flat wavefront is incident into a uniform, constant phase pattern, it will focus on a point after passing through a lens. The focus can be moved laterally by changing the phase of the phase pattern. For example, a point on the focal plane that deviates from the optical axis corresponds to a tilted wavefront on the phase pattern. This is the equivalent of a beam passing through a prism that causes the beam to move laterally. To cause a lateral shift $(\varDelta x,\varDelta y)$ of the position of the focused spot, a linearly increasing phase delay ${\phi _{prism}}({x_h},{y_h}) = \alpha (\Delta x{x_h} + \Delta y{y_h})$ is introduced on the phase plane, as shown in Fig. 2(b). Where, $\alpha$ is a constant related to the wavelength and the imaging characteristics. The phase of grating is introduced into the initial concentric double annular phase diagram ${\phi _{h0}}$, and finally, the double annular phase diagram ${\phi _h} = ({\phi _{prism}} + {\phi _{h0}})\bmod 2\pi$ is obtained, as shown in Fig. 2(c). The double annular phase diagram will cause the light incident into the double annular phase to tilt from the optical axis and generate the initial phase difference $\theta \textrm{ = }|{{\phi_1} - {\phi_2}} |$ between the inner and outer annular beams. It can be seen that the periodicity and phase of the axial optical chain beams can be easily controlled by changing the radius and the initial phase difference of the inner and outer annular slits.
It is worth noting that the efficiency of the beam shaping of double annular phase diagram is mainly determined by the ratio of the light energy incident into the double annular slits to the total incident light energy and the diffraction efficiency $\eta$ (∼ 80%) of spatial light modulator. The energy of the incident light we used is uniform, and the diameter of the incident light spot is equivalent to the size l of the short display axis of the spatial light modulator. Therefore, the efficiency of the beam shaping can be estimated to be ${{\eta (\Delta {r_1}^2 + \Delta {r_2}^2 + 2{r_1}\Delta {r_1} + 2{r_2}\Delta {r_2})} / {{l^2}}}$. Since the original light energy of the laser is much higher than we needed, and the spatial light modulator has high energy threshold, the axial optical chain beams with sufficient optical energy can be obtained by the tight focus of objective lens with high numerical aperture.
2.2 System design
The schematic diagram of the system for generating axial optical chain beams is shown in Fig. 3. A laser (IS8II-E, EdgeWave GmbH) irradiate laser light at the wavelength of 523 nm. First, the laser beam is expanded to about 1 cm in diameter by a beam expander (concave lens with f = 30 mm and f = 200 mm). A 50 µm-diameter pinhole (PH) is placed at the focus as a spatial filter. Then the beam was horizontally polarized and obliquely directed onto the phase only spatial light modulator (SLM, Pluto VIS 006, Holoeye). The SLM has a resolution of 1920 × 1080 pixels and a pixel pitch of 8 µm. The double annular phase diagram is implemented into the SLM before the beam was modulated and focused by a concave lens L1 (f = 180 mm). By placing an iris at the focus of the lens, only the first-order diffracted light (modulated light) is reserved, thus, the double annular beam is achieved. Then the beam is collimated by a concave lens L2 (f = 180 mm). Since a 4 f system is formed by lens L1 and lens L2, the collimated double annular beams is on the rear focal plane of the objective lens (the effective focal length: 9 mm) and is the same size (the radius and the width of the slits) as the double annular slits in the SLM. The double annular beams is focused by the objective lens and interferes near the focus to produce an axial optical chain beams. In the experiment, the light spots near the focus of the objective lens were amplified by a self-made microscope with magnification of 25 X, and recorded by a CCD camera (GS2-GE-20S4M-C, Point Grey). And the light spots at each axial position were recorded by a certain step size.
2.3 Simulation
All the simulations were performed using a 64 bit Intel Core i7-8750H CPU @ 2.20 GHz desktop running windows operating system. MATLAB (MathWorks, Inc., Natick, Massachusetts) is used for programming. The intensity distribution of the axial optical chain beams was simulated according to Equations (1)–(3). First, the system parameters, including the wavelength, the parameters of the double annular slits (radiuses and slit widths of the inner and outer annular slits) and the focal length of the objective lens, are substituted into Eq. (1) and Eq. (3). The integral function “quadgk” in MATLAB (the integral interval is from the inner diameter to the outer diameter of each annular slit) is used to obtain the optical field of the zero-order Bessel beams with different axial wave vectors generated by each annular slit, respectively. Then, the two optical fields of the zero-order Bessel beams are superimposed to obtain the optical field of the axial optical chain beams according to Eq. (3).
3. Results
3.1 Generation of axial optical chain beams
Here, the generation of axial optical chain beams is demonstrated through simulation and experiment. First, a double annular phase diagram is implemented into the SLM. The inner ring has a radius of 320 µm and a slit width of 640 µm, and the outer ring has a radius of 1600 µm and a slit width of 130 µm. The results are shown in Fig. 4. Figure 4(a) shows the distribution of intensity ${I_s}(r,z)$ of the axial optical chain beams calculated according to Eq. (2). The intensity of the axial optical chain beams propagates along the optical axis exhibits a cosinoidal distribution, and the periodicity $\Delta {z_t}$ is calculated as ∼ 36.9 µm. Figure 4(b) shows the axial optical chain beams obtained with a step size of 2.5 µm in the experiment, which is consistent with the results calculated by simulation. The intensity at the center along the optical axis was analyzed. The optical intensity exhibits cosinoidal distribution, and the measured periodicity is about 36.5 µm, which is very close to the theoretical value, as shown in Fig. 4(c). The experimental results demonstrate the effectiveness of the proposed method for generating axial optical chain beams.
Fig. 4. Generation of axial optical chain beams. (a) Axial optical chain beams obtained by simulation. (b) The axial optical chain beams obtained in the experiment. (c) The distribution of optical intensity along the axial direction of the region indicated by the white dashed lines in (b), the optical intensity exhibits a cosinoidal distribution, which is consistent with the theory. (d) and (e) The lateral distribution of optical intensity of theoretical profiles and the corresponding experimental profiles at z = 14.0 µm and z = 31.0 µm respectively. NI, normalized intensity.
The lateral distribution of optical intensity of theoretical profiles and the corresponding experimental profiles was carried out. Figure 4(d) is the lateral distribution of optical intensity at z = 14.0 µm, the optical intensity is almost strongest at the center, the full width at half maximum (FWHM) of the main lobes for theoretical profiles and the experimental profiles are estimated to be 1.60 µm and 1.67 µm, respectively. And the sidelobe 1 (indicated by the arrow 1) is also well identified. Figure 4(e) shows the optical intensity is almost weakest at the center (z = 31.0 µm), and the size of the hollow area for the theoretical profiles and the experimental profiles are estimated to be 2.60 µm and 2.64 µm, respectively. And the sidelobe 2 (indicated by the arrow 2) is also well identified.
In order to better describe the change of the axial optical chain beams, the optical spots of the theoretical profiles and the experimental profiles at different axial positions in a periodicity were obtained, as shown in Fig. 5. Figure 5(a) shows the light spot at the position of $z = 15$µm, which is close to the brightest regions. The optical intensity is strongest at the center, surrounded by two darker side lobes 1 (indicated by white arrows), which is consistent with the simulation results (Fig. 5(b)). When the position is $z = 20$µm, the optical intensity at the center of the spot becomes weaker and a new dark side lobe (indicated by the white dotted circle) appears, as shown in Figs. 5(c) and 5(d). When the position is $z = 25$µm and $z = 30$µm, respectively, the optical intensity in the center of the spot further darkens, and two side lobes appear and the optical intensity increases, as shown in Figs. 5(e)–5(h). When the beam propagates to the position of $z = 32.5$µm, the center of the spot basically disappears, the optical intensity becomes the weakest, and the two side lobes 2 (indicated by yellow arrows) become the strongest (but still much weaker than that of the center of spot in Fig. 5(a)), as shown in Figs. 5(i) and 5(j). As the beam continues to propagate forward, as shown in Figs. 5(k) –5(r), the center and peripheral sidelobes 1 of the spot gradually strengthen, while the sidelobes 2 gradually weaken. When the beam propagates to the position of $z = 50$µm, the intensity of the center of the spot changes back to the strongest. The distribution of optical intensity of theoretical profiles and the experimental profiles were carried out, as shown in Fig. 5(s). The theoretical profiles and the experimental profiles are in good agreement.
Fig. 5. Optical spots at different axial positions. (a) - (r) show the variation of the optical spots of experimental and theoretical results in a periodicity. (s) The distribution of optical intensity of experimental and theoretical profiles and the corresponding experimental profiles in (a) - (r). NI, normalized intensity.
3.2 Control of the periodicity of axial optical chain beams
The average radius of the double annular slits determines the periodicity of the axial optical chain beams. In the experiment, the periodicity of the optical chain beams is controlled by changing the radius of annular slit. Figure 6 shows an axial optical chain beams with a smaller periodicity generated by a double annular phase diagram. The inner ring of the phase diagram has a radius of 320 µm and a slit width of 400 µm, and the outer ring has a radius of 3200 µm and a slit width of 40 µm. Figure 6(a) shows the numerical simulation results, and the theoretical periodicity $\Delta {z_t}$ is calculated as ∼ 9.21 µm. Figure 6(b) shows the axial optical chain beams measured in the focal region with a step size of 0.6 µm. The distribution of the optical intensity is consistent with the theoretical simulation results in both axial and radial directions. The distribution of the axial optical intensity was analyzed, as shown in Fig. 6(c), and the measured periodicity was about 8.78 µm. In order to better analyze the propagation of the axial optical chain beams, the optical spots at several axial positions in a periodicity indicated by the white dashed line in Fig. 6(a) are selected, as shown in Fig. 6(d). When the axial position is $z = 8.4$µm, the optical intensity at the center of the spot is the weakest. When the axial position is $z = 10.8$µm, the optical intensity at the center of the spot becomes stronger and the side lobes becomes weaker. When the axial position is $z = 13.2$µm, the optical intensity at the center of the spot is close to the strongest. When the axial position is $z = 15.6$µm, the optical intensity at the center of the spot decreases. When the axial position is $z = 16.8$µm, the optical intensity at the center of the spot is weakest. Figure 6(e) is the theoretical profiles of different axial positions in a periodicity for Fig. 6(d), and the theoretical results agrees well with the experiment.
Fig. 6. Control of the periodicity of axial optical chain beams. (a) Axial optical chain beams obtained by simulation. (b) The axial optical chain beams obtained in the experiment. (c) is the distribution of optical intensity along the axial direction of the region indicated by the white dashed lines in (b), the optical intensity exhibits a cosinoidal distribution, which is consistent with the theory. (d) and (e) The optical spots of experimental and theoretical results at different axial positions in a periodicity indicated by the white dashed line in (b). Scale, 2 µm. (f) and (g) The distribution of optical intensity of theoretical profiles and the corresponding experimental profiles at z = 13.2 µm (indicated by the red arrow 1) and z = 16.8 µm (indicated by the red arrow 2), respectively. NI, normalized intensity.
To better compare the theoretical and experimental results, The distribution of optical intensity of spot at the axial position of $z = 13.2$µm, and $z = 16.8$µm were carried out, as shown in Fig. 6(f) and Fig. 6(g), respectively. In Fig. 6(f), the optical intensity is almost strongest at the center, the full width at half maximum (FWHM) of the main lobes for theoretical profiles and the experimental profiles are estimated to be 1.01 µm and 1.05 µm, respectively. The lateral position of the lowest point of the main lobes for theoretical profiles and the experimental profiles are estimated to be $y ={\pm} 1.01$ µm and $y ={\pm} 1.05$ µm, respectively. The lateral position of the central point of the side lobes 1 for theoretical profiles and the experimental profiles are estimated to be $y ={\pm} 1.6$ µm and $y ={\pm} 1.72$ µm, respectively. And the lateral position of the outer edge of the side lobes 1 for theoretical profiles and the experimental profiles are estimated to be $y ={\pm} 2.60$ µm and $y ={\pm} 2.64$ µm, respectively. Figure 6(g) shows the optical intensity is almost weakest at the center, and the size of the hollow area (indicated by the arrow) are estimated to be 2.00 µm and 2.11 µm, respectively. We can find that the theoretical profiles and the experimental profiles are in good agreement.
The controllable range of Δz is determined by the resolution of SLM (It determines the maximum radius that the double annular slits can reach) and the axial resolution of objective lens (It determines the minimum periodicity of axial optical chain beams that can be achieved). The number of pixels and the pixel pitch of SLM determines the size of active area, furthermore, the range of variations in radius of double annular slits of the phase diagram is determined. Since the axial optical chain beams is generated alone the optical axis, it will be affected by the axial resolution of the objective lens. If the periodicity is smaller than the axial resolution, the optical chain beams will not be distinguished. In the system, the SLM has a resolution of 1920 × 1080 pixels and a pixel pitch of 8 µm. And the objective lens has a numerical aperture of 1 and an effective focal length of 9 mm. Thus, the maximum radius that the double annular slits can reach is ∼ 4300 µm, and the axial resolution is ∼ 0.523 µm. According to Eq. (3), when the radius of outer annular slit is reach to ∼ 4300 µm (It is assumed that the mean radius of the inner annular slit is fixed at 640 µm), Δz will reach to the minimum value (∼ 5 µm).
The average radius of outer annular slit was increased with steps of 100 µm, and the theoretical evolution Δz as a function of the average radius of outer annular slit ${{({r_2} + \Delta {r_2})} / 2}$ was obtained according to Eq. (2), as shown in Fig. 5(a). It can be seen that the controllable range of Δz is 5 - 700 µm (indicated by the red line in Fig. 7(h)). Three experimental values are also given, the mean radius of the inner annular slit is fixed at 640 µm, and the mean radius of the outer annular slit were 1665 µm, 2440 µm, 3220 µm, respectively, and the Δz were measured (indicated by the blue line in Fig. 7(h)). The theoretical and the experimental results are in good agreement. A wider controllable range of Δz can be obtained by expanding the double annular beams with a higher-resolution SLM or beam-expanding system.
Fig. 7. Control of the phase shift of axial optical chain beams. (a) - (l) are the optical spots of the experimental and theoretical results when the initial phase difference $\theta$ between the two annular slits are 0, ${{2\pi } / 3}$, ${{4\pi } / 3}$, ${{6\pi } / 3}$, ${{8\pi } / 3}$, ${{10\pi } / 3}$ rad, respectively. (m) The theoretical and experimental $\Delta \theta$ as a function of $\theta$. NI, normalized intensity.
3.3 Control of the phase shift of axial optical chain beams
Phase shift is a very important parameter for the axial structured light. According to Eq. (2), the phase shift of the axial optical chain beams can be realized by changing the initial phase difference $\theta$ of the double annular slits. In practice, it is controlled by the difference of the gray level between the inner and outer rings in the double annular phase diagram. The axial position of $z = 21$µm in Fig. 6(b) was selected for phase shift analysis, and the theoretical and experimental result is shown in Fig. 7. First of all, when the initial phase difference $\theta$ of the double annular beam is 0 rad, the optical intensity at the center of the spot is close to the strongest, as shown in Figs. 7(a) and 7(b). When the initial phase difference $\theta$ between the two annular slits is ${{2\pi } / 3}$ rad, phase shift of the axial optical chain beams is occurred, and optical intensity at the center of the spot becomes weak and the strong side lobes appear around it, as shown in Figs. 7(c) and 7(d). When the initial phase difference $\theta$ between the two annular slits is ${{4\pi } / 3}$ rad, optical intensity at the center of the spot is further weakened, as shown in Figs. 7(e) and 7(f). When the initial phase difference $\theta$ between the two annular slits is ${{6\pi } / 3}$ rad, the phase shift of the axial optical chain beams goes through a period, and the distribution of the optical spot is the same as that of the initial phase difference is 0 rad, as shown in Figs. 7(g) and 7(h). Similarly, when the initial phase difference $\theta$ between the two annular slits are ${{8\pi } / 3}$ and ${{10\pi } / 3}$ rad, the distribution of the optical spot is consistent with $\theta = {{2\pi } / 3}$ rad and $\theta = {{4\pi } / 3}$ rad, respectively, as shown in Figs. 7(i)–7(l). The theoretical and experimental phase shift $\Delta \theta$ as a function of initial phase difference $\theta$ were also carried out, as shown in Fig. 7(m). It can be seen that $\Delta \theta$ increases linearly with $\theta$. And the theoretical and the experimental results are in good agreement. The experiment verifies that the double annular phase diagram can well control the phase shift of the axial optical chain beams.
4. Conclusion
In conclusion, we use phase holography to achieve the axial optical chain beams with controllable periodicity and phase. A double annular phase diagram is generated based on the gratings and lenses (GL) algorithms. The beam incident to the double annular slits was tilted from the optical axis to produce concentric double annular beams. The annular beams with different radius will produce the zero-order Bessel beams with different axial wave vectors. Axial optical chain beams is produced by the interference of two zero-order Bessel beams with different axial wave vectors generated by the inner and outer annular beam of the concentric double annular beams. The phase and periodicity of the axial optical chain beams can be controlled by controlling the gray levels and radius of the inner and outer rings of the double annular phase diagram, respectively. The effectiveness of the proposed method is verified by numerical calculation, and experimental results demonstrate the feasibility of the proposed method, which is in agreement with the theory. Axial optical chain beams has a promising prospect of application in the field of optical tweezers and signal modulation in optical imaging.
Funding
Research Foundation for Advanced Talents, Nanchang University.
Disclosures
The authors declare no conflicts of interest
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