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We present a novel optical configuration that allows for generation of ultra-high speed dynamic Bessel beams and dynamic bottle beams. The method is based on combination of the axisymmetric acousto-optic device and the spatial filtering enabled by a mask or a digital micromirror device. Selected features of dynamic non-diffracting beams and bottle beams are investigated using time-resolved approach with stroboscopic pulsed illumination, including spatial intensity distribution, spatial modulation factors, MHz-range temporal modulation, and scalability. The numerical simulations based on Fourier optics as well as experimental realizations are demonstrated.
© 2016 Optical Society of America
1. Introduction
Non-diffracting optical fields have been proposed in the late 1980’s by Durnin et al. who demonstrated that a set of solutions of Helmholtz equation can be propagation-invariant [1, 2]. The simplest representation of diffraction-free beams has transverse profile described by the Bessel function of the first kind of zero order J0. Due to their specific features, electromagnetic (optical), acoustic as well as electron Bessel beams were intensively investigated over last decades [3–8]. Firstly, the central spot size of the Bessel beam is not altered during propagation (non-spreading). Secondly, Bessel beam has the ability to re-establish its transverse intensity profile after passing through an obstacle (self-healing, self-reconstructing) [7]. Thirdly, group velocity can exceed the speed of light since such optical beams can be interpreted as the result of interference of many plane waves propagating on a conic surface (superluminal propagation) [9]. Bessel beams were exploited in various research areas, and the main fields of application include: optical micromanipulation (e.g. optical tweezers), microfabrication, trapping, non-linear optics etc [10–13]. Recently, advanced microscopic modalities utilized Bessel beams in biomedical imaging [14–19].
The properties of Bessel beams can be also used to generate structured optical fields such as the optical bottle beams. Introduced by Artl and Padgett, an optical bottle beam is characterized by a low-intensity focus surrounded by the regions of higher intensity [20]. Interference of various optical fields can produce three-dimensional (3-D) intensity distribution that can be regarded as a chain or an array of optical bottles and it is often used in effective optical trapping [21, 22]. Different configurations of superimposed Bessel beams were studied theoretically and experimentally [21, 23–27].
Although the ideal Bessel beam cannot be practically realized, different methods have been proposed to generate experimental approximations of Bessel beams that exhibit the desired properties at limited propagation distance [28]. Those techniques use i.e. an axicon, an annular slit, a resonant cavity, holograms, a conical mirror, a spatial light modulator (SLM), a digital mirror device (DMD) etc [1, 9, 13, 29–32]. In particular, the techniques based on light diffraction aim at creating ring(s) (annular illumination) that can be further transformed by the lens [33, 34]. It has to be emphasized that in most cases those methods are rather static and hard to be reconfigured. An interesting solution to that research issue is the application of axisymmetric acousto-optic modulator (AOM).
Acousto-optic devices usually utilize planar ultrasonic transducers that generate plane acoustic waves [35]. However, non-standard ultrasound of axial symmetry in AOMs forms a platform which is well-suited due to light intensity distribution in many optical systems. The light interaction with axially symmetric (cylindrical) ultrasonic wave can be described in terms of diffraction or refraction. Diffraction of light by cylindrical ultrasound manifests as a set of concentric rings (diffraction orders) in the far field [36, 37]. In contrast, refraction of light by cylindrical ultrasound leads to high-speed control of the tunable focus that was demonstrated in laser processing and in imaging modalities like confocal microscopy and optical coherence tomography [37–42]. Multiscale Bessel beam formed in the near field of light diffraction, where light amplitude is modulated, was also shown [43, 44]. Generation of temporally modulated Airy beams using the phenomenon of light refraction by standing acoustic wave was later demonstrated [45].
In this paper, we present a novel configuration that enables generation of ultra-high speed dynamic Bessel beams and dynamic bottle beams. The method is based on the axisymmetric acousto-optic phenomenon and the spatial filtering of diffraction orders with masks or DMD. Selected features of dynamic non-diffracting beams and bottle beams are studied using time-resolved approach with stroboscopic pulsed illumination. The numerical simulations based on Fourier optics and experimental realizations are demonstrated.
2. Theoretical description
2.1. Acousto-optic effect with cylindrical symmetry of acoustic wave
Let us consider the AOM that is composed of cylindrical piezoelectric transducer (shell) with an acoustic medium inside, as shown in Figs. 1(a) and 1(b). Vibrations of the piezoelectric shell walls enable generation of standing ultrasonic wave of axial symmetry [36]. If the amplitude of ultrasound Δp is not too high, proportional variations of the refractive index of the medium can be observed. Then, the distribution of the refractive index inside the acousto-optic cell can be written as:
where n0 is the refractive index of the medium, n1 is the amplitude of the refractive index variations, J0 is the Bessel function of the first kind of zero order, K and Ω are the wavenumber and the circular frequency of the acoustic wave, respectively, , where cs is the speed of ultrasound in the acoustic medium). Equation (1) indicates that the highest variations can be given at the axis of the AOM, where ultrasound focusing appears (function J0 has the maximum).
Fig. 1 Operating principle of the cylindrical acousto-optic modulator. (a) AOM geometry. (b) Image of the far field of light diffraction. (c) Modulated wavefront behind the AOM measured with Hartmann-Shack sensor. Three-dimensional plot of the reconstructed wavefront, color map of the wavefront and central profile showing agreement between theoretical model and the experimental data (F = 3.4 MHz, vmax = 12; see Visualization 1).
Propagation of the light beam (of wavenumber k) through such a modulator will cause spatio-temporal modulation of phase (and amplitude). In the Raman-Nath regime, the distribution of the refractive index in Eq. (1) acts as a pure phase diffraction grating. When the plane electromagnetic wave passes through the modulator of the length L, the electric field distribution at the exit plane of the AOM can be described as:
whereis the Raman-Nath parameter which represent the maximum phase change of the light beam. We visualized the wavefront exiting the AOM using Hartmann-Shack sensor (CLAS-2D; AMO Advanced WaveFront Sciences LLC, USA). The results of the test shown in Fig. 1(c) confirms standing nature of the axisymmetric ultrasound as well as the profile of the modulation amplitude in the form of Bessel function. Additionally, temporally resolved acquisition allows for visualization of the wavefront at different time instances within a period of ultrasound (Visualization 1). The effect of the spatio-temporal modulation of the wavefront can be also observed in the far field of light diffraction, i.e. when transforming lens with the focal length f is inserted. The example of the diffraction pattern is shown in Fig. 1(b). The symmetry of the modulator determines the symmetry of the diffraction pattern, in which the diffraction orders have a form of concentric rings. The far field distribution at the point Q(w) can be calculated based on Fresnel-Kirchhoff theory:whereis a dimensionless distance, and ρ is the geometric distance of the observation point Q from optical axis. The integral in Eq. (3) has an analytic solution only if vmax = 0 (Airy diffraction pattern). The diffraction field is modulated with ultrasound frequency since our axisymmetric ultrasound is in fact a standing wave.Beyond the Raman-Nath regime of acousto-optic interaction, not only the wavefront is modulated but also the amplitude of the electric field. Therefore, both Eqs. (2) and (3) become much more complicated. As a result, numerical approach (based on beam propagation method and Fourier optics) can be applied to perform theoretical simulations of the studied effects [42, 46].
2.2. Generation of Bessel beam
The method of generation of dynamic Bessel beams uses the phenomenon of light diffraction by cylindrical ultrasound. As shown in Section 2.1, the diffraction orders have the form of concentric rings, and filtering the particular ring enables annular illumination of the lens, which creates an interferometric pattern behind the lens that is described by the zero-order Bessel function J0. Figure 2(a) presents basic geometry used in the description of Bessel beam generation. The wavevector k of the converging annular beam can be decomposed into both radial (transverse) and longitudinal components (kr and kz). The electric field is therefore represented by the following relation:
where A0 is the amplitude of the electric field, , , , d stands for the diameter of the annular slit, r and z are the radial and longitudinal coordinates, respectively. One can estimate the central spot size Δx and propagation distance zmax of the Bessel beam:where: R stands for the semi-diameter of the lens [12].Fig. 2 Generation of Bessel beam using annular illumination (a) with single ring and (b) with two annular rings.
2.3. Generation of bottle beam (interference of Bessel beams)
Availability of more diffraction orders in the Fourier space (far field) of acousto-optic interaction allows for other spatial filtration cases when more annular beams (orders) are selected. In such scenarios, superposition (interference) of Bessel beams form structured three-dimensional bottle beam. Figure 2(b) shows the case when two ring-shaped beams pass through the lens. When both Bessel beams are described by the relations:
where i = 1, 2, the superposition of both beams generate the following interferometric pattern:where. This equation indicates that the obtained optical field is more complex since light intensity is modulated along beam propagation direction. The axial periodicity is given by: .Light intensity in the diffraction orders on standing ultrasonic wave is also modulated with the frequency corresponding to ultrasound frequency. Therefore, when two selected annular beams come from such diffraction orders, one needs to take into account also amplitude and phase evolution in the chosen diffraction orders.
3. Experimental setup
We developed the setup that is shown in Fig. 3. The light source used in the experiments was a modulated diode laser (LDM473.100.500; Omicron-Laserage Laserprodukte GmbH, Germany) emitting the light with the wavelength of λ = 473 nm. The light was coupled into a single mode fiber to obtain circularly symmetric Gaussian beam at the end of the fiber. The collimated Gaussian beam of diameter 3 mm was incident on the acousto-optic modulator (AOM). The custom-designed AOM was actually a cylindrical piezo-electric transducer (shell) closed at both ends with the optical glasses and filled with water (length L = 50 mm, outer diameter 30 mm, inner diameter 26 mm) [42]. The axis of the AOM was aligned to coincide with the optical axis of the system. The AOM was driven by a sinusoidal signal from a two-channel function generator (TGA12102; TTI, UK). Vibrations of the walls of the piezoelectric transducer generated the ultrasonic standing wave inside the water medium. The calibration procedure was applied to obtain the Raman-Nath parameter [42]. The frequency of driving signal was F = 3.4 MHz, and the signal amplitude provided the Raman-Nath parameter either vmax = 4 (driving voltage 3.6 V) or vmax = 14.4 (driving voltage 13.0 V). According to Eq. (2), this gave n1 values of 0.75⋅10−5 and 2.71⋅10−5, respectively. The operation parameters of the AOM determined the Klein-Cook parameter of the interaction Q = 0.465 [47].
Fig. 3 Configuration of experimental setup for generation of dynamic Bessel beams and dynamic bottle beams using spatial filtering with (a) standard circular beam stop (BS) or (b) digital micromirror device (DMD). (c) Stroboscopic illumination scheme. AOM – axially symmetric acousto-optic modulator, L1-L5 – lenses, OL – objective lens, MLM – motorized linear module, DDG – digital delay generator, CCD – array detector, PC - computer.
The stroboscopic illumination was implemented to be able to perform the time-resolved experiments. The AOM was illuminated with 10 ns light pulses with the repetition rate corresponding to the ultrasound frequency. Therefore, the TTL signal from the function generator triggered a digital delay generator (P400; Highland Technology Inc., USA) to provide appropriate signal to the digital modulation input of the diode laser. Additionally, the particular time, at which the AOM was illuminated, was controlled by the phase delay ϕ between the ultrasound and the illumination pulses. The phase delay could be set in the range from 0 to 360 degrees, which corresponded to the period of ultrasound T, as shown in Fig. 3(c).
The light field exiting the AOM was modulated in space and time, and the far field diffraction pattern occurred in the focal plane of the lens L1 with f1 = 100 mm (plane Σ in Fig. 3). The diffraction pattern was later magnified 10 × by a 4f lens system (f2 = 19 mm and f3 = 200 mm) to obtain acceptable spatial resolution of the image. The spatial filtering was implemented in the plane Σ’ by inserting the annular custom-made beams stops that cut off selected diffraction order(-s). This enabled achieving high-speed modulated annular illumination. The image passing through the beam stop was further demagnified by a reducing system of two lenses (f4 = 200 mm and f5 = 19 mm), and the dynamic Bessel beam was formed behind the objective lens OL (fOL = 19mm). The beam profiles were acquired by the CCD camera (uEye UI-1240SE-M-GL; Imaging Development Systems GmbH, Germany) on a motorized linear module (MLAS-0600-4EK-001; Wobit, Poland), as shown in Fig. 3(a). The measurement was performed automatically, and the PC computer synchronized the CCD acquisition, movement of the linear module and phase delay ϕ.
In order to investigate the impact of the side lobes of the diffraction order on Bessel beam quality, we used also the DMD (DMD DLP6500FQL; Texas Instruments Inc., USA) as a spatial filter in the plane Σ’. The DMD was an array of 1980 × 1020 micro-mirrors, each of 7.56 μm in size. The principle of DMD operation required setting the incident angle to maximize the light reflection efficiency, which was schematically presented in Fig. 3(b). Moreover, additional diffraction grating and a pair of lenses were inserted between the planes Σ and Σ’ to compensate the angular dispersion introduced by the DMD (not indicated in Fig. 3). The optical elements behind the DMD were the same as in the previous configuration.
4. Results and discussion
4.1. Generation of dynamic Bessel beam
Firstly, we demonstrate generation of the dynamic Bessel beam using acousto-optic light diffraction and compare it with standard beams. Our set-up enabled also visualization of three types of beams: Gaussian beam, Bessel beam using standard annular illumination and Bessel beam using filtration of acousto-optic diffraction orders. We used the DMD to project appropriate pattern with experimental configuration presented in Fig. 3(b). The pattern was an annulus of the diameter of 0.777 mm and the width of 76 μm (10 pixels in the DMD array). The Gaussian beam was made with the system, in which lens L1 was removed and the AOM was switched off (vmax = 0). Generation of both Bessel beams required projection of a thin annulus on DMD. We performed the three-dimensional beam profiling when continuous illumination scheme was applied. The images of the beam were recorded for different axial positions of the CCD detector. The stack of images enabled 3-D reconstruction of light intensity.
Figure 4 presents the sagittal cross-sections of intensity in the plane containing the optical axis, the transverse sections as well as corresponding intensity profiles. Additionally, we included the results of numerical simulations to demonstrate the agreement of experimental results with the theoretical considerations.
Fig. 4 Three-dimensional profiling of (a) Gaussian beam, (b) standard Bessel beam made by annular illumination, (c) dynamic Bessel beam made by filtering central ring of the first diffraction order (continuous illumination, F = 3.4 MHz, vmax = 4). The rows demonstrate sagittal cross-sections (numerical simulation results and experimental data), central axial intensity profiles (dots – experiment, lines – simulation) and the transverse profiles showing the beam spots.
Bessel beams generated using both annular illumination methods have the lateral profiles described by the Bessel function. Therefore, the central spot is surrounded by side lobes, which do not appear in Gaussian beam. What is more, non-diffractive Bessel beams are also characterized by effectively longer propagation distance zmax (that achieves almost 100 mm) compared to the depth of focus (DOF) of the Gaussian beam. Optical configuration used in the experiment provided zmax at least ~10 × longer than DOF of standard Gaussian beam.
4.2. Properties of dynamic Bessel beams
In the next experiments, we explored different features of the dynamic Bessel beams. It is inherent property of light diffraction by axisymmetric grating that the diffraction orders possess the fine structure, i.e. the main ring is accompanied by side rings. This can be also observed in the magnified diffraction image in Fig. 1(b). In the case of light diffraction by cylindrical ultrasound, both shape and structure of the diffraction orders are influenced by the amplitude of ultrasound, the width of the incident light beam, and the fact that that the Bessel function J0 is not strictly a periodic function. We assessed the impact of the fine structure of the first diffraction order (ring) on the quality of Bessel beam. Beam profiles were acquired when the DMD in plane Σ’ filtered the entire first diffraction order or only the central ring of that order. Accordingly, we projected the annulus of the diameter of 0.777 mm and the width of 450 μm or 76 μm, respectively.
Comparison of the plots and axial profiles shown in Figs. 5(a) and 5(b) demonstrates that the presence of additional rings close to the main ring in the diffraction order contributes to the low-frequency modulation of time-averaged light intensity along the axial direction. However, when only a single ring passes through the filter, the axial profile of the dynamic Bessel beam is similar to the regular Bessel beam made by annular illumination. In addition to that, the propagation distance at which Bessel beam intensity achieves maximum values is moved outside the objective lens when side rings are not filtered out. This might be useful when one needs the Bessel beam at larger working distance.
Fig. 5 Dynamic Bessel beam properties. Impact of fine structure of diffraction order on beam quality. Light intensity cross-section, time-averaged intensity profile and time-resolved intensity profiles when (a) the central ring of the first diffraction order is filtered or (b) the entire first diffraction order is filtered (F = 3.4 MHz, vmax = 4). (c) Modulation of light intensity in the point indicated by red arrow in (a).
Another important observation is that the light intensity in the diffraction orders is temporally modulated since the standing ultrasound is involved in acousto-optic interaction. Therefore, the difference between standard and acousto-optic generation of Bessel beams comes from the fact that the latter method enables high-speed MHz-range modulation of light intensity. The ability to control the temporal dynamics of light intensity led us to call the generated non-diffracting light fields as dynamic Bessel beams. The plots in the right column in Figs. 5(a) and 5(b) demonstrate the instantaneous axial intensity profiles acquired at different time instances. The phase difference between the central ring and the side rings shifts the maximum intensity in time, which is not observed when a single central ring of the diffraction order is used. Figure 5(c) shows also temporal modulation at the central point of the beam, the position of which is indicated by the arrow in Fig. 5(a). The results confirm that the intensity is modulated with the frequency 2F ( = 6.8 MHz).
Diffraction of light by axisymmetric ultrasound enables generation of multiple diffraction orders. The diffraction order can be characterized by its particular diameter, as shown in Fig. 6(a). Consequently, spatial filtering allows to control the parameters of the Bessel beam such as the central spot size and the maximum propagation distance, which have a direct impact on the performance of imaging modalities, according to Eq. (5). Therefore, in the next experiment, different diffraction orders from the same pattern were selected and served as annular illumination for the lens L4. We used the annular masks of the diameters 1.2 mm, 2.5 mm, 3.3 mm, 4.6 mm and 5.9 mm (the width of each was 1.35 mm) to consecutively filter out the first five diffraction orders in the plane Σ’. We measured the transverse profile of the formed Bessel beam and extracted the beam widths. The results presented in Fig. 6(b) contain the Bessel beam profiles for the diffraction orders from n = 1 to n = 5 (i.e. the order diameter in plane Σ ranged from d = 0.22 mm to d = 1.06 mm). The higher order is used, the narrower beam is generated. The spot size Δx and the beam propagation distance zmax are inversely proportional to the diameter d of the diffraction order. Figures 6(c) and 6(d) show that Δx ranged from 29.5 μm to 6.5 μm, and zmax from 44.5 mm to 2.8 mm.
Fig. 6 Scalability of dynamic Bessel beams. (a) Far field of light diffraction by axisymmetric ultrasound. (b) Intensity profiles of the Bessel beams for different size of annular illumination (diffraction orders from n = 1 to 5). (c) Dependence of the central spot size Δx on the diameter d of the diffraction order. (d) Dependence of the beam propagation distance zmax on the diameter d of the diffraction order (continuous illumination, F = 3.4 MHz, vmax = 14.4).
4.3. Interference (superposition) of dynamic Bessel beams
The superposition of two or more dynamic Bessel beams can be practically realized when appropriate number of diffraction orders passes through the spatial filter. Interfering Bessel beams form a 3-D chain of bottle beams. Accordingly, we designed custom beam stops using high-resolution printing, which were schematically presented in Fig. 3(a). The beams stops enabled filtration of the annuli of the diameters of 1.2 mm, 2.5 mm, 3.3 mm, 4.6 mm and 5.9 mm (the width of each was 1.35 mm). The DMD was not used since it had limited dimensions and introduced additional aberrations for larger annuli. The diffraction pattern in the far field contained at least 7 diffraction orders. To demonstrate generation of the dynamic bottle beams, we compared the intensity field when different combinations of diffraction orders were used, and synchronic illumination was applied. The plots given in Fig. 7(a) indicate that the intensity modulation along propagation direction occurs when two (or more) Bessel beams are superposed. The interference pattern depends on the difference between the diameters of selected annuli, which defines Δkz in Eq. (7). The transverse profiles of the periodic bottle beam are presented in Fig. 7(b). Visualization 2 is a fly-through showing the transverse bottle beam profiles at different axial positions. The propagation distance determines if the intensity in the central spot achieves maximum or is surrounded by a ring of high intensity. We also extracted the periodicity for different combinations of the diffraction orders by measuring the distance between adjacent intensity maxima. As shown in Fig. 7(c), the periods are inversely proportional to the diameter of the second selected order while the first order is kept the same. In addition to that, the experimental results confirm the data based on numerical simulations as well as the theoretical considerations based on Eq. (7).
Fig. 7 Generation of bottle beams with interfering Bessel beams (stroboscopic illumination, F = 3.4 MHz, vmax = 14.4, t = 0). (a) Sagittal cross-sections of light intensity distribution for different combinations of diffraction orders. Visualization 2 shows propagation evolution of transverse bottle beam profiles when Bessel beams from diffraction orders n = 1 and n = 4 were superposed. (b) Transverse profiles extracted from 3-D intensity distribution at the propagation distances indicated by arrows in (a). (c) Dependence of the axial periodicity of the bottle beams on the diameter of one of the diffraction orders involved in the interference.
Finally, we performed time-resolved study of the obtained bottle beam. The simulation and the experimental results given in Fig. 8(a) show that the intensity distribution depends on the illumination instance of the period. Visualization 3 demonstrates that the maximum intensity becomes minimum when we shift illumination by a half of the ultrasound period. This observation is more clear when we analyze the extracted axial intensity profiles shown in Fig. 8(b).
Fig. 8 Dynamics of bottle beams (stroboscopic illumination, F = 3.4 MHz, vmax = 14.4). (a) Magnified sagittal cross-sections of light intensity distribution for two time instances. Bessel beams from diffraction orders n = 1 and n = 3 were superposed (see Visualization 3). (b) Axial intensity profiles of the dynamic bottle beams.
5. Discussion and conclusions
This study demonstrates the ability to generate diffraction-less optical Bessel beams and optical bottle beams using the combination of the acousto-optic diffraction and the spatial filtering. In fact, the principle of the method is based on generating annular illumination. Light diffraction by axisymmetric standing ultrasound is well suited to the standard symmetry of many optical systems, which makes this application feasible. Furthermore, the interaction of light with such ultrasound is very efficient, i.e. relatively low voltage is required to observe the effect in a form of diffraction orders. This is the result of a specific ultrasound focusing properties of the transducer at the axis of symmetry, which we directly proved using wavefront sensing.
Both types of non-diffracting optical fields generated by our method are easily reconfigurable [4, 41]. Firstly, the central core width (the beam spot size) is determined by the annulus diameter and the numerical aperture of the objective lens. However, the diameters of ring-shaped diffraction orders depend on the frequency of ultrasound. Although the acousto-optic modular can operate efficiently only at specific resonant conditions, the facts that the shell ultrasound transducer can operate also at higher harmonics and different orders can be selected during spatial filtering makes the method flexible in generating the beams of desired parameters. We showed that higher diffraction orders generate narrower Bessel beams with shallower propagation distance, which confirms the theoretical relations for standard Bessel beams made by the annular illumination. As an example, we demonstrated the ability to scale down the beam spot size from 29.5 μm to 6.5 μm, and the beam propagation distance from 44.5 mm to 2.8 mm. The beams with reduced propagation diffraction are characterized by much longer maximum propagation distance when compared with the Rayleigh range of the Gaussian beam of equivalent spot size. In addition, the access to many annular diffraction orders enables selecting two or more orders. This in turn determines the spatial intensity modulation periodicity and modulation depth in bottle beams.
Secondly, the obtained optical fields are highly dynamic with the modulation frequency in the MHz range. This unique property comes from the fact that there is actually a standing ultrasonic wave inside vibrating piezoelectric shell. The ultrasound wave, which can be regarded as a temporally modulated diffraction grating, disappears twice during its period. When the grating amplitude drops, most of light is lost during spatial filtration. Therefore, the temporal characteristics (dynamics) of the obtained non-diffracting beams can be also controlled by choosing the appropriate Raman-Nath parameter (proportional to the voltage applied to the transducer electrodes) and / or the illumination instant of time. The possibility of ultrafast reconfiguration of the optical fields can be advantageous in the advanced optical micromanipulation, where dynamically changed parameters of the system of interest must be provided.
Thirdly, the ultrasonic field that provides the refractive index distribution in Eq. (1) represents the simplest case of cylindrical piezoelectric shell vibrations. Higher vibrational modes of the transducer are also possible [48]. The light diffraction by such ultrasound fields would impact the far field pattern, which indicates that the diffraction orders do not necessarily have a form of rings. If this is combined with more sophisticated filtering scenarios and the control of the phase between the orders, higher order optical non-diffracting beams can be formed.
The method of non-diffractive beam generation is based on using the annular slit, which in principle is much less efficient than the regular methods based on the axicon. The efficiency of the presented set-up depends on the diffraction order that passes through the annular slit. The numerical simulations shown in Fig. 9 illustrate that it is possible to transfer up to 40% of incident light into the first diffraction order. Taking into account the fact that the DMD can reflect up to 65% of light, the total efficiency of the system achieves up to 28%. Consequently, application of the AOM as a diffractive element enables higher light efficiency than illuminating the comparable annular slit with a Gaussian beam.
Fig. 9 Diffraction efficiency of the AOM. Total power in the diffraction orders vs. the Raman-Nath parameter (F = 3.4 MHz).
It has to be emphasized here that the acousto-optic interaction has already been used to generate the Bessel beams [43–45]. In the near field of light diffraction and beyond the Raman-Nath regime, the light amplitude is spatially modulated according to the ultrasonic wave. Therefore, it was possible to obtain the light beam, whose amplitude is described by the Bessel function J0 since this function is generated inside vibrating piezoelectric shell. One of those previous reports [44] adapted refraction-based approach to explain the observed phenomena qualitatively. However, we used the far field of light diffraction, and performed spatial filtering of annular diffraction orders in the current study. The numerical approach applied here was based on calculation of light propagation through non-homogenous media. Consequently, this technique allowed to describe more complex phenomena (light interaction in / beyond the Raman-Nath regime as well as light refraction) although no analytical formula could be derived.
The experiments show that the intensity modulation dynamics can be revealed with pulsed illumination synchronous with ultrasound. Stroboscopic illumination scheme can be also utilized in the application studies to control the time of illumination. This scenario is compulsory when the bottle beams is generated. Moreover, the DMD used in the experiments plays a role of high-resolution spatial filter. Application of the DMD enabled understanding slow modulation effects observed in the experiments due to the fine structure of the selected diffraction order. Thus, it allowed achieving the quality of non-diffracting beams similar to those formed by the standard methods. It has to be pointed out here that the trade-off between the spatial resolution and the size of the active area of the spatial modulator had to be taken into account. This made it impossible to use the DMD technology for filtering higher diffraction orders (with larger annulus diameter). Moreover, using larger area in the spatial modulator introduced the distortions due to fact that the focal plane Σ’ did not coincide with the DMD plane [29]. We also noticed that the DMD caused chromatic dispersion and other geometric aberrations, which were also reported earlier [16].
Generally, liquid crystal based SLM technology can be also used instead of DMD. Although both technologies offer comparable resolution (ca. 2-10 μm) and active area (diagonal ca. 0.7-0.9”) of modulation, the DMD technology allows for much higher temporal light modulation than the SLM. Moreover, both technologies operate in a reflection mode. However, the SLMs cause usually phase modulation, which implies that pure amplitude modulation can be obtained with the aid of polarization optics. On the other hand, the DMD operates truly at the principle of light reflection, which indicates that the spatial modulation can be directly obtained by projecting a desired pattern. The advantage of using the DMD technology is that it allows for controlling the optical beams with higher power with no need for thermal stabilization.
The experiments were in a good agreement with the theoretical predictions calculated numerically with the beam propagation method. Consequently, it provides a feasible platform for designing the optical fields for applications such as: selective excitation, dynamic optical trapping, light sheets with large working distance etc.
In conclusion, we demonstrated an acousto-optic method for generation of the ultra-high speed dynamic Bessel beams and the dynamic bottle beams. We investigated the spatial intensity distribution of dynamic non-diffracting beams and bottle beams, their quality, the dynamics and the ability to be reconfigured by the application of different filtration scenarios. We also showed that the numerical simulations are confirmed by the experimental data.
Funding
Polish National Science Center (NCN) (SONATA-BIS; #2014/14/E/ST7/00637).
Acknowledgments
Scholarship of the Polish Ministry for Science and Higher Education is acknowledged (IG).
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