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Abstract
White-light non-diffraction beams such as Airy beam and Bessel beam have potential applications in multispectral imaging and micromanipulation. Generation of white-light Airy beam and Bessel beam with high quality and high efficiency still remains challenging for conventional diffractive or refractive optics which suffers from significant chromatic dispersion. In this paper, both high-quality white-light Airy beam and Bessel beam are generated using a deformable mirror by modulating the incident LED beam with tunable cubic and conical wavefronts. The main lobe of the generated white-light non-diffraction beams does not suffer from chromatic dispersion along the propagation. The results also show that the generation of the white-light Bessel beam has higher requirements for spatial coherence than white-light Airy beams. Our work expands the understanding of the white-light non-diffraction beams and paves the way for the applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Non-diffracting beams, such as Bessel beam and Airy beam, receive considerable attentions since the transverse intensity distribution of these beams remain invariant during the propagation [1,2]. The Bessel beams have non-diffraction and self-healing properties, while Airy beams have additional self-accelerating property. Additionally, some new types of Bessel-like beams that have self-accelerating property have also been demonstrated [3,4]. Non-diffracting beams have important applications in optical microscopy [5,6], optical particle manipulation [7], material processing [8], and plasma channel generation [9].
So far, the fundamental and applied research of the non-diffracting beams is predominantly performed using the coherent light (e.g. laser beam). Recently, several groups carry out relevant research of the non-diffracting beams generated using light with different degrees of coherence (e.g. broadband light and white light) which have potential applications in multicolour imaging and micromanipulation [10–12]. Basano et al. firstly demonstrated that the white-light Bessel beam can be created using an annular slit and a lens with a halogen lamp [13]. Zhu et al. generated white-light Bessel-like beams using miniature all-fiber device [14]. Fischer et al. generated Bessel beams with white-light sources passing through an axicon and concluded that spatial coherence is a key criterion while temporal coherence is not significant important [15]. They further found that the propagation distances, the size of central spot and the radii of surrounding rings are wavelength dependent since the refractive index of the axicon is a function of the wavelength [16]. The refractive index dispersion also occurs in the generation of the white-light Airy beams using a refractive cubic phase element. Valdmann et al. [17] and Cai et al. [18] found the main lobe of the Airy beam is laterally dispersed significantly since the deflection coefficient is wavelength dependent. Valdmann et al. further adopted a reflective cubic phase element to solve the chromatic dispersion problem. The main lobe of the ultra-broadband Airy beam did not disperse significantly [19]. Mansour et al. proposed a reflective cylindrical beam expander, which can be properly adjusted to perform as a tunable continuous cubic phase modulation device, to generate the white-light Airy beams [20]. However, the tunability is limited and the reflective cylindrical mirrors require extremely precise alignment. Spatial light modulator (SLM) is a flexible diffractive device and is commonly used to generate both tunable Bessel beam and Airy beam by programming the phase change of each pixel. For generating white-light Bessel beams, an additional prism [21] or grating [22] is needed to compensate the chromatic dispersion. Stoyanov et al. generated an achromatic broadband Bessel beams by initially nesting and subsequently annihilating multiply charged optical vortices and then Fourier-transforming the resulting ring-shaped beam by a thin lens [23]. However, the wavelength dependent deflection of the white light Airy beam is hard to be compensated, resulting in the significant spectral separation of the main lobe during propagation [24]. Up to now, there are no suitable devices to the best of our knowledge that can generate both high-quality non-dispersion white-light Bessel beams and Airy beams, thus further limiting their applications.
Here, we show that both white-light Airy beams and Bessel beams can be generated in high quality using a deformable mirror (DM) and a broadband LED source. As a key component of adaptive optics system, DM is an adaptive device with a controllable reflective mirror surface [25]. Recently, it has been proved that the DM can not only correct the aberrations in the optical system, but also generate a high-precision conical or cubic mirror surface for generating tunable Bessel beams and Airy beams [26–29]. In this paper, the propagation behaviors of the generated white-light non-diffracting beams are investigated. It is observed that the main lobe of the generated white-light non-diffraction beams does not disperse along the propagation. It is also verified that rather than temporal coherence, spatial coherence is a critical prerequisite for generation of the white-light non-diffraction beams. Furthermore the generation of the white-light Bessel beam has higher requirements for spatial coherence than white-light Airy beams.
2. Experimental setup
The schematic diagram of the experimental setup is shown in Fig. 1(a). A phosphor-generated warm white LED source (Jingyi Optoelectronics JYL-2000LED) with a spectrum range of 400-750 nm was used. The spectrum of the white LED source is shown in Fig. 1(b). The white LED light is temporally and spatially incoherent. According to Fischer’s research, spatial coherence is important for generating white-light non-diffracting beams [15]. Hence the white light is coupled into a single mode fiber with a core diameter of 9 µm to increase the spatial coherence. Reducing fiber diameter helps to improve the spatial coherence at the expense of reducing the intensity of the output light. The light emitted from the fiber is collimated by a lens L1 with a focal length of 120 mm. Then the collimated beam reaches the mirror surface of the DM and is modulated with a cubic or a conical wavefront by the DM. In the experiment, a homemade 61-element unimorph DM comprising of a piezoelectric layer and a silicon layer was employed [30]. The modulated beam passes through a 0.2× telescope system consisting of the lenses L2 and L3. The beam after the telescope is divided into two parts by a plate beam splitter (BS). The wavefront deformation (optical path difference) of the beam is measured by a Shack-Hartmann wavefront sensor (WFS, Thorlabs WFS150-7AR). The measured wavefront deformation is used as the feedback to control the mirror surface of the DM for compensating the aberrations of the optical system and generating the target modulation wavefront.
Fig. 1. Experimental setup. (a) Schematic of the experimental system, (b) the spectrum of the white LED source, and (c) the spectral sensitivity of the CCD camera.
Compared with the counterpart refractive or diffractive devices by which the generated wavefront deformation strongly depends on the wavelength, the wavefront deformation generated by a reflective DM is twice the mechanical deformation of the mirror surface and does not depend on the wavelength of the incident beam. The relationship between phase ϕ(x, y, λ) and wavefront deformation G(x, y) can be described as:
The white-light Airy beams are generated by performing optical Fourier transformation using a lens L4 with a 100 mm focal length. The back focal plane of the lens L4 is defined as the original point (z = 0 mm). The intensity distribution of the generated Airy beams is measured using a color CCD camera (Basler acA1300-30gc) with a pixel size of 3.75 µm. The spectral sensitivity of the CCD camera is shown in Fig. 1(c). The CCD is mounted on a slide block and can slide freely on a linear guide rail. For generating white-light Bessel beam, the lens L4 is removed and the back focal plane of the lens L3 (the conjugate plane of the DM mirror) is defined as the original point (z = 0 mm). In order to eliminate the influence of chromatic dispersion, all the lenses used in this system are achromatic.
The mirror surface of the DM is accurately controlled in closed loop [30]. The deformable mirror can reproduce cubic and conical wavefronts with different amplitudes accurately. Figure 2 shows the comparison of the experimentally generated and the theoretical cubic and conical wavefronts. The generated wavefronts agree with the ideal wavefronts quite well. For generation of cubic wavefront with a peak-to-valley (PV) value of 11.3 µm, corresponding to a root-mean-square (RMS) value of 2.23 µm and 17.9 waves for a wavelength of 632 nm, the RMS wavefront error between the targeted and the measured wavefronts is about 50 nm, with a relative error of less than 2.5%. The generated conical wavefront exhibits a rounded tip which is much smaller comparing to the generated axicon. For generation of a conical wavefront with a PV value of 10 µm, corresponding to an RMS value of 2.36 µm and 15.8 waves for a wavelength of 632 nm, the RMS wavefront error is about 60 nm, with a relative error of less than 3%, which mainly comes from the rounded tip.
Fig. 2. Comparison of the experimentally generated and target wavefronts. (a) and (d) are the experimentally generated cubic and conical wavefronts respectively, (b) and (e) are the cross-section profiles of the cubic and conical wavefronts respectively, (c) and (f) are the wavefront errors of the cubic and conical wavefronts respectively.
3. Generation of white-light non-diffracting beams
3.1 White-light Airy beam
The generation of Airy beam can be considered as the optical Fourier transformation of Gaussian beam modulated with a cubic phase which is described as [1]:
Since the cubic phase is generated by the reflective DM, the wavefront deformation is expressed as:
where g is a scale factor related to the physical deformation of the DM and is wavelength independent. According to the relationship between the phase and the wavefront deformation (Eq. (2)), β can be expressed using g:Substituting Eq. (6) into Eq. (4), w0 can be rewritten using g:
This formula indicates w0 is proportional to λ2/3, which means the width of the main lobe of the Airy beam increases with the wavelength.
It is known that the main lobe follows a parabolic trajectory as the beam propagates. The transverse deflection in x direction and y direction of the main lobe with different wavelengths can be calculated using [20]:
Substituting Eq. (7) into Eq. (8), the transverse deflection is rewritten as:
From this formula, the deflection of the white-light Airy beam is independent of the wavelength, which means the main lobe of the white-light Airy beam will not disperse during the propagation. This result is different from the white-light Airy beam generated by refracting or diffractive phase modulators which suffer from significant lateral dispersion [24].
The simulated and experimentally measured white-light Airy beam and monochromatic Airy beams are shown in Fig. 3 and Fig. 4. The monochromatic beams with wavelength of 632 nm, 532 nm and 440 nm were generated by filtering the white LED light through an interference filter with a bandwidth of 10 nm. These monochromatic lights are temporally coherent since temporal coherence is inversely proportional to the bandwidth of the beam. These Airy beams are generated by the same cubic surface of DM as shown in Fig. 2. The light in simulation is spatially coherent. The figure shows that DM can generate both white-light and monochromatic Airy beams with high quality. The measured Airy beams coincide with the simulation ones. The main lobe of the white light Airy beam is clear without chromatic dispersion. The results indicate that temporal coherence is not a strong prerequisite for Airy beam generation. The size of the main lobe as well as the lobe space slightly increases with the wavelength, as predicted by Eq. (7), resulting in the color separation of side lobes for white light Airy beam though the main lobe is free of chromatic dispersion. This phenomenon can be seen in the generated white light Airy beam (the partial enlarged view in Fig. 3).
Fig. 3. The simulated and experimentally measured intensity distribution of white light Airy beams at z = 0.
Fig. 4. The simulated and experimentally measured intensity distribution of the monochromatic Airy beams at z = 0.
The cross-section intensity profiles of the simulated and the measured Airy beams are compared in Fig. 5. It shows that the side lobes of the measured Airy beams are a little fuzzy and decay faster compared with simulation results for both white light Airy beam and monochromatic Airy beam. The contrast of the side-lying peaks of the Airy beams is also reduced. In our previous research [26], the monochromatic Airy beam generated using a DM and a diode laser (Thorlabs HLS635) has much clearer side lobes. The diode laser has much higher spatial coherence than the LED light used in this work. This means that spatial coherence is a key criterion for generation of Airy beam. The DM can generate white-light Airy beam with higher performance by using a supercontinuum source with high spatial coherence.
Fig. 5. The simulated and experimentally measured cross-section intensity profiles of the white-light Airy beam and the monochromatic Airy beams corresponding to Fig. 3: (a) white light, (b) 632 nm, (c) 532 nm, and (d) 440 nm
Furthermore, the intensity distributions of the generated white light Airy beam along the propagation were measured from -50 mm to 50 mm with a 2 mm interval, as shown in Fig. 6. The generated white-light Airy beam has a high quality. The full width at half-maximum (FWHM) of the main lobe varies slightly from 56 µm to 71 µm when the Airy beam propagates from -20 mm to 20 mm. The diffraction half-angle of the central lobe is about 180 μrad. The center positions of the main lobe along propagation were extracted from each measured intensity distribution image. Then the axial propagation trajectories were drawn in Fig. 6(b) and (c). The propagation follows a parabolic trajectory, indicating that the generated white light Airy beam maintains expected lateral acceleration. The trajectories of different spectral components Airy beam are wavelength independent as predicted in Eq. (9), thus the main lobe of the Airy beam does not disperse, unlike the beam generated using the refraction or diffraction phase modulators which suffers from strong lateral dispersion. For example, Morris et al. generated a white light Airy beam using a SLM and a supercontinuum laser. They compensated the chromatic dispersion at z = 0 using an additional prism. However, the deflection of the white light Airy beam generated using SLM is wavelength dependent, the spectrum of the main lobe still separates significantly along the propagation [24].
Fig. 6. The propagation trajectory of the white-light Airy beam generated using the DM. (a) intensity distribution of the white-light Airy beam at different propagations, and the trajectory of the main lobe in (b) horizontal direction and (c) vertical direction.
3.2 White-light Bessel beam
The zero-order Bessel beam can be considered as the self-interferce of the light beam with conical wavefront which can be expressed as:
where k is a scale factor related to the physical deformation of the DM and is wavelength independent. The tangent value of cone angle α of the conical wavefront can be calculated by dividing the peak to valley (PV) value of the wavefront by the radius of the beam, this means α is also wavelength independent. The parameters of the Bessel beam, such as the core size and the propagation distance, are determined by the cone angle, the wavelength and the size of the incident beam [2]: where d is the beam diameter. Thus the main spot size as well as the radii of the surrounding rings are wavelength dependent, while the propagation distance is wavelength independent.The white-light Bessel beam with a cone angle of 5 mrad was generated by the DM experimentally. The intensity distribution of the Bessel beam is measured from 70 mm to 350 mm with an interval of 2 mm. The results are shown in Fig. 7. It is shown that a Bessel-like beam with bright central lobe along the propagation is obtained. The Bessel beam formed by the white light source can be divided into three different regions along the propagation. At the beginning, a bright spot with little surrounding rings is observed at the propagation distance from 70 mm to 90 mm. It is affected by the rounded tip of the conical wavefront generated by the DM. The rounded tip generates a partial focusing of the wave, which affects the spatial intensity distribution along the propagation. The outset of the Bessel beam is also shifted backward. Then, a bright central lobe with distinct concentric ring structure is observed at the propagation distance from 90 mm to 220 mm. Color separation appears at the surrounding rings since the radii of the surrounding rings are proportional to the wavelength. As the generated Bessel beam further propagates, it progressively evolves into a light pipe with no fringe structure after the propagation distance of 220 mm. The FWHM of the main lobe along the propagation were calculated as shown in Fig. 8, which increases with the propagation distance. The diffraction half-angle of the generated Bessel beam is approximately 220 μrad. The experimental results are consistent with Fischer’s work where the white Bessel beams are made by using BK7 axicon and Halogen Bulb with both poor spatial and temporal coherence. The propagation distance is wavelength independent in theory for DM, while that is wavelength dependent for axicon [15,16].
Fig. 7. The propagation of the experimentally generated white-light Bessel beams. The insets are the intensity distributions of the Bessel beam at different propagations.
In order to investigate the influence of spatial coherence on the Bessel beam, monochromatic Bessel beams were further generated using a laser beam with a wavelength of 635 nm and a monochromatic beam with wavelength of 632 nm filtered from the LED light using an interference filter. The laser beam has a much higher spatial coherence than the LED light. Figure 9 shows the intensity profiles and propagation trajectory of generated Bessel beam with the two types of light. Notably, for the Bessel beam formed by laser beam, the size of the main lobe remains invariant during the propagation with a FWHM of about 50 µm. For the Bessel beam formed by the filtered LED light, the FWHM of the main lobe gradually increases from ∼50 µm to ∼150 µm, and the fringe structure becomes blurry and disappears gradually. This phenomenon is consistent with the generated white-light Bessel beam. This means that temporal coherence is not of significant importance while spatial coherence is a critical factor. The spatial coherence of the collimated beam formed from LED source decreases with radial direction. As the beam with conical wavefront propagates, the outer beam with lower coherence overlaps and interferes. This causes the broadening of the central lobe and the smoothening of the fringe structure. Compare with the generation of the white-light Airy beam, the generation of the white-light Bessel beams has higher requirements for spatial coherence.
Fig. 9. Propagation of Bessel beams generated by (a) laser beam and (b) filtered monochromatic light. (c) and (d) are the cross-section intensity profiles of the Bessel beams generated by laser beam and filtered monochromatic light at the locations indicated by the dotted line in (a) and (b).
4. Conclusions
We have generated white-light non-diffraction beams (both Airy beam and Bessel beam) using a deformable mirror by modulating the incident beam with high-precision tunable cubic and conical wavefronts. The main lobe of the generated white-light non-diffraction beams does not disperse along the propagation, solving the chromatic dispersion problem in generation of white-light non-diffraction beams using refracting or diffractive phase modulators. The results prove that spatial coherence is a critical factor for generation of the white-light non-diffraction beams and temporal coherence is not. The generation of the white-light Bessel beam has higher requirements for spatial coherence than white-light Airy beams. The limited spatial coherence of the LED light caused the broadening of the central lobe and the smoothening of the fringe structure for Bessel beams, while the side lobes of the Airy beams are only a little fuzzy. Furthermore, the diffraction half-angle of the generated white Bessel beam is larger than white Airy beam. The generation method can be extended to the cases of fully spatial coherence such as supercontinuum source, and the quality of the generated white-light Airy beam or Bessel beams is anticipated to be much better. Considering the correction function of the DM for optical aberrations of the system, the proposed generation method of the white-light non-diffraction beams is more practical for potential applications in multispectral imaging and micromanipulation [10–12].
Funding
Zhejiang Province Public Welfare Technology Application Research Project (LGG22E050002); State Key Laboratory of Industrial Control Technology (ICT2021B44); The Key Laboratory for Metallurgical Equipment and Control of Ministry of Education in Wuhan University of Science and Technology (MECOF2019B03); Science and Technology Innovation 2025 Major Project of Ningbo (2018B10005).
Disclosures
The authors report no conflicts of interest in this work.
Data Availability
Data underlying the results presented in this paper are available from the corresponding author upon reasonable request.
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