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Due to its unique properties of nondiffracting propagation, highly-localized intensity distribution, small beam cross-section, and self-healing, a nondiffracting beam is attractive for materials processing, microscopy, and optical research. Various methods have been investigated to generate such beams with conventional optics. However, the transverse size of those nondiffracting beams is restricted by the diffraction-limit. To overcome the diffraction limit, we use the concepts of super-oscillation and the vectorial angular spectrum method to design a phase mask mirror with a focal length of 1 m, radius of 5 mm, and numerical aperture of about 0.005 for a wavelength of 632.8 nm. The phase mask mirror was created with a phase spatial light modulator. Under the illumination of a linearly polarized Gaussian wave, a nondiffracting beam was created with sub-diffraction transverse size. The maximum transverse size of the beam is smaller than the diffraction limit of 0.5λ/NA for a propagation distance greater than 43.3 mm. A nondiffracting beam with smaller transverse size can be realized by further increasing the NA value.
© 2017 Optical Society of America
1. Introduction
Since the first theoretical proposal [1] and experimental demonstration [2] by J. Durnin et al., the generation and application of nondiffracting beams have generated wide interest. For a commonly used Gaussian beam, the smaller M2 factor and larger numerical aperture (NA) give a smaller focal spot at the focus df≈2λM2/πNA [3], where λ is the wavelength; however, this will lead to a shorter Rayleigh range Zr≈πdf2/4λM2 [4]. Therefore, for Gaussian beams, there is a trade-off between a small spot size and long Rayleigh range for applications in materials processing. Compared to Gaussian beams, nondiffracting beams have unique properties of nondiffracting propagation, highly localized intensity distribution, small beam cross-section, and self-healing. They are also predicted to undergo less scattering in the turbulent atmosphere and in non-uniform media. Bessel beams have been successfully used in materials processing [5], such as microdrilling, micro/nano channel fabrication, micromachining, nanopatterning, and photopolyhmerization. Nondiffracting beams are also attractive in optical research [6], including nonlinear optics, microscopy, atom optics, optical micromanipulation, cell sorting, and optoinjecing. Due to its self-reconstructing property, Bessel beams in optical microscopy have demonstrated unexpected robustness against deflection and a reduction in scattering artifacts, which help to increase image quality and penetration depth in dense media [7]. The generation of nondiffracting beams has been extensively studied during last several decades. The first realized nondiffracting beam was a zero-order Bessel beam formed by focusing a plane-wave-illuminated circular slit with a conventional positive lens [2]. Other commonly used approaches include holographic methods [8,9], axicon lenses [10], cemented doublet-lenses [11], spatial-light-modulators (SLM) [12], and digital micromirror devices [13]. The generation of a spiraling zero-order Bessel beam was also proposed by combining a conventional axicon and hologram, which introduces an additional equal spiral phase shift with a constant period [14]. Nonlinear periodic structures were also employed to generate a zero-order Bessel beam with frequency doubling [15]. The m-th order Bessel beam can also be created by illuminating a conventional axicon with an azimuthal phase variation exp(imφ) beam where φ is the azimuthal angle [16]. Unlike the zero-order Bessel beam with its peak intensity at the beam center, higher order Bessel beams have a phase singularity at the beam center, leading to a nondiffracting dark core in the beam. As waves generated in conventional optics, nondiffracting beams formed using the above methods are usually diffraction-limited, or their transverse size is larger than the diffraction limit, 0.5λ/NA. Therefore, it is a topic of interest to develop a method to generate sub-diffraction nondiffracting beams for micro/nano fabrications and manipulation. Super-oscillation is a phenomenon which allows formation of arbitrarily small features in the far-field with superposition of band-limited functions [17,18]. Nondiffracting optical beams with sub-wavelength features can be shaped by superposition of Bessel beams [19,20]; however, their sub-diffraction features are surrounded by undesirable huge side lobes. Super-oscillatory lenses [21–28] are a new class of lens, which can create a sub-diffraction size hot spot far beyond the lens surface. Nondiffracting optical needles with a transverse electrical field and sub-diffraction transverse size have been reported for visible [29] and violet light [30]. A sub-diffraction longitudinally polarized optical needle with a nondiffracting distance of 4λ was theoretically proposed by focusing a radially polarized wave with a binary phase filter and high NA lens [31]. Such a sub-diffraction optical needle with a 5λ propagation distance was experimentally demonstrated by focusing radially polarized light with a single planar binary phase lens [32]. In this paper, for a linearly polarized Gaussian beam at wavelength 632.8 nm we created an ultra-long sub-diffraction nondiffracting optical needle using a pure phase reflective mirror formed with a spatial-light-modulator. The mirror has a 1-meter focal length and 10-mm diameter, and the mirror numerical aperture is about 0.005, corresponding to a diffraction limit (0.5λ/NA) about 63.28 μm (100λ). The nondiffracting beam was observed to have a maximum transverse size about 62 μm (smaller than the diffraction limit) and length about 43.3 mm (68426λ), resulting in a sub-diffraction nondiffracting beam with a very large length-width ratio of 698.
2. Theoretical consideration
As shown in Fig. 1(a), the phase mask is a circular plate with a diameter of 10 mm. It consists of a series of concentric ring belts each with the same width of 20 μm, equal to the SLM pixel size used in the later experiments. On the i-th ring belt the phase change of the reflected wave is an integer multiple of 2π/255, or 2πni/255. For a linearly polarized Gaussian beam with a radius greater than 22 mm the phase delay on the concentric ring belt is optimized with a particle-swarm algorithm [33] to obtain a 50-mm-long nondiffracting beam with maximum transverse size less than the diffraction limit, which is 1 meter away from the mask surface. In the optimization the diffraction pattern is calculated with the vectorial angular spectrum diffraction formulas [34,35] as given in Eq. (1):
where denotes the polarization direction of the incident light, zf is the distance between the mask and the observation plane, g(r) and t(r) are the incident beam profile and transmittance function of the lens, J0 and J1 are the zero- and first-order Bessel functions, q(l) = (1/λ2-l2)1/2, and φ is the angular coordinate with respect to the polarization direction .
Fig. 1 The optimized mirror phase distribution. (a) The gray map of the optimized phase distribution of the mirror; (b)-(f) the phase distribution plotted against radius along the radial direction.
The optimized phase distribution of the mask is plotted against the radial coordinate r in Figs. 1(b)-1(f). The phase mask consists of 254 concentric ring phase belts, whose distribution is quite different from the linear-phase distribution in an axicon for generation of a Bessel beam.
Figures 2(a)-2(f) give the color maps and intensity distribution curves of the diffracted light obtained on the XY planes at different positions on the optical axis at z = 974.4 mm, 985.6 mm, 996.8 mm, 1008.0 mm, 1019.2 mm, and 1030.4 mm respectively. On each of those planes the intensity distribution shows a clear hot spot centered on the optical axis with small side lobes. In the numerical simulation, it was found that the transverse electrical field plays a major role in the formation of those hot spots, while the contribution from the longitudinal component is ignorable on the observation planes. This is why the on-plane optical intensity has good circular symmetry without the azimuthal angle dependence due to the longitudinal component shown in Eq. (1). On those planes the spot full-width-at-half-maximum (FWHM) is 61.6 μm, 60.6 μm, 60.7 μm, 60.9 μm, 60.6 μm, and 58.8 μm respectively. As shown later, all those FWHMs are smaller than the corresponding diffraction limit.
Fig. 2 The theoretically calculated total electric field intensity profiles in the XY plane. (a)-(f) the diffraction patterns obtained at the propagation distances of z = 974.4 mm, 985.6 mm, 996.8 mm, 1008.0 mm, 1019.2 mm, and 1030.4 mm, respectively, within the designed nondiffracting.
Figure 3(a) is the color map of the optical intensity distribution in the XZ plane around the designed focal point at Z = 1000 mm. It shows a nondiffracting beam with a length of more than 50 mm. The peak intensity (red), transverse FWHM (blue) and side lobe ratio (green) (the ratio of maximum side lobe intensity to the central peak intensity) are plotted against z from 965 mm to 1045 mm along the propagation direction in Fig. 3 (b). The left y-axis is for optical intensity, and the right y-axis is for transverse FWHM in μm and side lobe ratio in percentage respectively. The black dash-dot line and the wine colored dash-dot line correspond to the diffraction limit 0.5λ/NA (NA = sin[atan(R/Z)], R and Z being the mirror radius and the diffraction distance on z-axis) and the super-oscillation criteria 0.38λ/NA, respectively. As indicated by the black arrows, the longitudinal FWHM in the z direction is about 46 mm (72693λ) between z = 979 mm and z = 1025 mm. In this area note that the side lobe ratio is less than 14%, smaller than 16%, which is the side lobe ratio of a zero-order Bessel beam, and the transverse FWHM is less than 61.2 μm in the XY plane, which is smaller than the diffraction limit. Actually, between z = 976.3 mm and 1032.0 mm, the side lobe ratio is less than 26% and the transverse FWHM is smaller than the diffraction limit, resulting in a sub-diffraction nondiffracting beam as long as 55.7 mm. As the distance increases, the transverse FWHM almost approaches the super-oscillation criteria at z = 1038.5 mm, but the side lobe also increases to 43%. Figure 3(c) gives the phase distribution (red) and the local spatial frequency (blue) (the derivative of the phase distribution with respect to the radial coordinate) along the radial coordinate on the focal plane at z = 1000mm. The black dashed line denotes the cut-off spatial frequency 2πNA/λ of a conventional mirror with the same NA of 0.005. The phase distribution clearly shows sharp phase change of π at several points along the radial coordinate, and this is quite different from the conventional Bessel beams. It is found that, at those positions with sharp phase change, the local spatial frequency is larger than the cut-off spatial frequency of 2πNA/λ (black dashed line). This is a direct evidence of super-oscillation [18]. In fact, in the curve of local spatial frequency, the two peaks near the center contribute to shape the sub-diffraction transverse size of the designed nondiffracting beam. It should also be noticed that the radial spatial frequency of the zero-order Bessel beam generated by an axicon with the same focal length and numerical aperture is about (2π/λ)/200, which is smaller than the local spatial frequency (red) near the peaks given in Fig. 3(c). To investigate the angular spectrum characteristics of the beam, the angular spectrum amplitude is depicted with respect to radial frequency in Fig. 3(d). Unlike the ideal Bessel beam with only a single radial frequency component, the frequency spectrum of the designed beam is restricted to a comparative narrow band of 0~0.03/λ. This agrees with the definition of nondiffracting beams.
Fig. 3 (a) The theoretically calculated intensity of the diffraction pattern in the propagation plane, (b) the peak intensity, the transverse FWHM, and the side lobe ratio within propagation distance of 96 mm, (c) The phase distribution (blue) and the local spatial frequency (red) of electric field along the radial coordinate on the focal plane at z = 1000mm, where the black dashed line denotes the cut-off frequency corresponding diffraction limit of a conventional mirror with NA = 0.005, (d) the angular spectrum of the designed beam (blue), and the inset shows its zoom-in and the angular spectrum of a zero-order Bessel beam (red dashed line) respectively.
3. Results and discussions
The SLM (LCOS-SLM × 10468, Hamamatsu Corporation) used in the experiment is a pure-phase reflective modulator with pixel pitch of 20 μm, resolution of 792 × 600 pixels, fill factor of 98%, phase modulation range 0~2π for wavelength range from 620 nm to 1100 nm, and phase level of 256. The layout of the experiment setup is illustrated in Fig. 4, where I1-I10 are irises for optical alignment, and M1-M3 are mirrors. The light source is a He-Ne laser with wavelength of 632.8 nm and beam waist of 0.7 mm. A polarizer is used to generate the proper linear polarization for the SLM. M1 and M2 are used to create a 12-m propagation distance to obtain a larger beam size before reaching the beam expander. Then the laser beam size expands to about 22 mm to guarantee a uniform illumination over the whole SLM active area. The incident beam is titled by a small angle of about 1 degree to allow mirror M3 to reflect the modulated wave into a beam profiler (BeamProfiler 3.0, Edmund Optics). The transverse profile of the focused beam is captured by the beam profiler, which is mounted on a translation stage and can move along the z-direction, which is the beam propagation direction.
In Figs. 5(a)-5(f) the experimentally obtained optical intensity on the XY plane is depicted in the color map for different diffraction distances of 979.5 mm, 988.5 mm, 997.5 mm, 1006.5 mm, 1015.5 mm, and 1024.5 mm respectively. The intensity distribution on each plane shows a clear hot spot at the center with comparatively small side lobes. The corresponding intensity distribution is also plotted against the radial coordinate. These results show a good similarity to the theoretical simulations shown in Figs. 2(a)-2(f). Because the experimentally obtained spot does not have perfect circular symmetry, we evaluate the spot size by taking the mean of the FWHMs measured in 10 different directions with an equal angle step of 18 degrees. The obtained average FWHMs are 61.8 μm, 60.6 μm, 59.5 μm, 60.6 μm, 62.2 μm, and 60.5 μm for the focal spot in Figs. 5(a)-5(f), all of which are smaller than the diffraction limit, as shown later.
Fig. 5 The experimentally obtained total electric field intensity profiles in the XY plane. (a)-(f) the diffraction patterns obtained at the propagation distances of z = 979.5 mm, 988.5 mm, 997.5 mm, 1006.5 mm, 1015.5 mm, and 1024.5 mm, respectively, within the designed nondiffracting beam.
In the experiments, a long scan was taken in the range from z = 977 mm to 1040 mm with an equal step of Δz = 0.5 mm along the optical axis. Figures 6(a) and 6(b) illustrate the optical intensity in a color map for the XZ plane and the YZ plane respectively. As shown in the two figures, there is a clear unsymmetrical intensity distribution in the X-direction and Y-direction. We believe this to be caused by the tilted incident beam, which is part of the experiment setup for the reflective SLM. In Fig. 6(c), the peak intensity, the average FWHM, and the side lobe ratio are plotted against the diffraction distance in the beam propagation direction. All three parameters change along the optical axis in a way similar to their theoretical counterparts depicted in Fig. 3. The peak intensity is found at around z = 1000 mm. The intensity distribution shows a longitudinal FWHM of 43.3 mm in the propagation direction from z = 981.5 mm to z = 1024.8 mm, similar to the theoretical simulation. In this area, except for two points the measured transverse FWHM is smaller than the diffraction limit, as indicated by the black dash-dot curve. The side lobe ratio is less than 22% in the range from z = 978 mm to z = 1020 mm. According to the optical intensity obtained on the focal plane, the focused energy is around 5% of the total energy illuminated on the SLM, including about 20% loss due to the higher-order diffraction on the SLM surface.
Fig. 6 The experimentally obtained total electric field intensity profiles in the propagation plane. (a) and (b): the intensity of the diffraction pattern in XZ and YZ plane, respectively. (c) The peak intensity, the average FWHM of the spot in transverse direction, and the side lobe ratio within propagation distance of 64 mm.
Although the vectorial angular spectrum method is adopted in the theoretical design and simulation, as expected the small NA of 0.005 makes the diffraction pattern of the linearly polarized wave closer to that of a scalar wave. The numerical simulation indicates that, at the peak point at z = 1003.2 mm the total optical intensity is dominated by a transverse polarization with an ignorable longitudinal component. This is also true for the whole nondiffracting beam. Therefore, for a sub-diffraction nondiffracting beam generated by a small NA with a linear polarized beam, the scalar diffraction theory will be good enough for the theoretical design and simulation.
4. Conclusions
Although nondiffracting beams and their creation have been extensively studied for a long time, the creation of a nondiffracting beam with sub-diffraction transverse size is still an interesting topic for many applications, such as micro/nano fabrication, microscopy, and optical manipulation. In the sub-wavelength scale, a super-oscillatory lens has been used to generate sub-diffraction optical needles; however, their length-to-width ratio is less than 40. Here, we report the generation of a nondiffracting beam with a sub-diffraction transverse size smaller than 62 μm (98λ) and longitudinal length greater than 43.3 mm (68426λ), resulting in a large length-to-width ratio greater than 698, which is favorable for the application in high depth-to-wide-ratio tunnel fabrication. A smaller transverse size might be achieved by optimizing the phase mask for a larger NA or shorter working distance. The advantage of using SLM for this purpose is its feasible and dynamic control of the creation of nondiffracting beams with sub-diffraction transverse size for different applications. It is also possible to generate nondiffracting beams with sub-diffraction transverse size for multiple wavelengths and even for a certain bandwidth of spectrum by using this approach, taking into account the dispersion property of the SLM.
Funding
China National Natural Science Foundation (61575031, 61177093); National Key Basic Research and Development Program of China (Program 973) (2013CBA01700); Program for New Century Excellent Talent in University (NCET-13-0629); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (University of Electronic Science & Technology of China), P. R. China; Fundamental Research Funds for the Central Universities (106112016CDJZR125503).
Acknowledgments
Authors thank LetPub (www.letpub.com) for their linguistic assistance during the preparation of this manuscript.
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