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Abstract
Focused beam shaping is very effective in material laser processing, optical tweezers, and laser microscopy. In particular, size minimization of shaped beams has attracted much attention because it is directly connected to the performance in these applications. In this paper, we demonstrated spatial phase shaping of ultrashort laser pulses to overcome the diffraction limit. In the experiment, it was found that the beam diameter of the sub-diffraction-limit spot was 0.60-times smaller than the diffraction-limit spot. To verify the effectiveness of the proposed method, a sub-diffraction-limit spot was applied to femtosecond laser processing. The diameter of the structure processed by the sub-diffraction-limit spot was reduced to 0.39-times thanks to the nonlinearity of femtosecond laser processing. Furthermore, as the possibility to generate other beam shapes, light patterns formed with combinations of connected and unconnected spots were also demonstrated by the method.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As a result of progress made in the performance and usability of phase-only spatial light modulators (SLMs), computer-generated holograms (CGHs) capable of generating arbitrary wavefronts have revolutionized a wide range of fields, including optical metrology [1], imaging [2], memory [3] and interconnections [4], holographic displays [5], optical tweezers [6–9], fluorescence microscopy [10,11], optogenetics [12], adaptive optics [13], additive manufacturing [14,15], and laser material processing [16–18]. In these applications, focused beam shaping is important because it affects the performance and is directly linked to the resolution, density, accuracy and throughput in the optical operations. In particular, size minimization of shaped beams has attracted much attention in subcellular bioimaging and laser nanofabrication.
The focused spot size is essentially limited by the diffraction of light, which depends on the wavelength of the light and the numerical aperture (NA) of the focusing lens. In the case of laser material processing, an effective NA that depends on the diameter of the beam irradiating the lens may be more practical than the NA of the focusing lens, because the beam diameter is actually set to less than the pupil size of the lens in order to avoid energy loss due vignetting. On the other hand, there is a trade-off between the spot size and the field of view (FOV). The FOV is also important for performing processing of large areas using a high-speed optical scanner. Consequently, the demand for minimizing the spot size while maintaining a wide FOV has increased in industrial applications.
One method of solving the trade-off problem is generating a sub-diffraction-limit spot whose diameter is smaller than the diffraction limit. If the spot size is N (1<N)-times smaller than a diffraction-limit spot without changing the focusing lens, the actual FOV will be essentially N-times larger than that of the diffraction-limit spot. In order to achieve sub-diffraction-limit focusing, a radial polarization has been demonstrated [19], where the sub-diffraction-limit spot by the radial polarization was 0.62-times smaller than the diffraction-limit spot by a linear polarization. The generation of the sub-diffraction-limit spot by the radial polarization requires the use of an objective lens with a high NA. In the alternative method, optical super-oscillations have also been demonstrated. A concept of super-oscillations, which was well known in the microwave community has been firstly suggested in 1952 [20]. The first experimental observation of super-oscillations in optical fields has been reported in 2007 [21], where the sub-diffraction-limit spot by the super-oscillations was 0.50-times smaller than the diffraction-limit spot by the conventional focusing. The super-oscillations has no physical constraints on the size of spot that can be created in principle. Furthermore, it provids super-resolution without evanescent waves. Therefore, the super-oscillations is a method without being in the near-field of the object. However, sidebands always appear around the spot, and the sideband intensity is higher than the spot intensity. In addition, it is difficult to fabricate an ideal phase mask that generates super-oscillations because the masks must have nanoscale accuracy and high density and retardation tolerance. On the other hand, a dynamic mask created with an SLM offers flexible control of the complex amplitude of the optical field [22]; however, the pixel size of the SLM (about 10 µm) becomes a limiting factor. Although metamaterial masks with sub-wavelength elements have recently been demonstrated [23], it still presents significant challenges of fabrication quality.
In order to design the optimal masks for performing super-oscillatory focusing, the optical eigenmode method [24] has been proposed. As an alternative, an algorithm based on an iterative Fourier transform method [25] has also been proposed [26]. In the algorithm, the generation of a sub-diffraction-limit spot was based on a constraint concerning the phase relationship among multiple diffraction spots composed of the center and surrounding spots. The center spot was smaller than the diffraction limit due to a confinement effect from the surrounding spots as a result of destructive interference with a phase difference of $\pi$. Therefore, it is essential to arrange the surrounding spots to generate the sub-diffraction-limit spot. Conversely, however, beam irradiation by the surrounding spots should actually be avoided to ensure high spatial resolution because the surrounding spots are larger than the sub-diffraction limit center spot.
In this paper, we propose a novel algorithm for designing a mask (CGH) for generating a sub-diffraction-limit spot with regulation of the light intensity of the surrounding spots. The CGH design algorithm based on the amplitude optimization around diffraction spots has been demonstrated [27] to avoid a distortion of diffraction spots. Our algorithm was based on proximity complex amplitude optimization between diffraction spots. To demonstrate the effectiveness of the algorithm, a sub-diffraction-limit spot was applied to femtosecond laser processing. To the best of our knowledge, this is the first demonstration of femtosecond laser processing using this kind of sub-diffraction-limit spot. In the experiment, we found that the beam diameter of the sub-diffraction-limit spot was 0.60-times smaller than the diffraction-limit spot. Furthermore, the diameter of the structure processed by the sub-diffraction-limit spot was also reduced to 0.39-times thanks to the nonlinearity of femtosecond laser processing. Fortunately, a certain level of light intensity of the surrounding spots had no effect on the results of laser processing because the intensities of the surrounding spots were lower than the damage threshold of the target sample. Finally, as a supplementary experiment, light patterns formed with combinations of connected and unconnected surrounding spots were demonstrated by using the proposed algorithm. The pattern formed by connected surrounding spots had a spot size comparable to the diffraction limit.
2. CGH design method using complex amplitude optimization
Figure 1(a) shows a schematic diagram of the CGH design algorithm for generating a sub-diffraction-limit spot with regulation of the intensities of the surrounding spots. In the figure, a, $\phi$ and u are the amplitude, phase, and complex amplitude, respectively. The subscript notation means the values at input and output planes. i is an index indicating one particular iteration. The total number of iterarions is set to 30. Figure 1(b) shows a schematic diagram of reconstruction of the CGH and four constraints applied to the reconstruction in the algorithm. The reconstruction is composed of five spots, including center and surrounding (above, below, right, and left) spots, and is arranged in a spot area with 51$\times$51 pixels. The spot area is located at a distance of 75 pixels from the 0-th order light to avoid the influence of the 0-th order.
Fig. 1. (a) Algorithm of CGH design. (b) Schematic diagram of the arrangement of the diffraction spots and constraints of each spot in the algorithm. (c) Generation of sub-diffraction-limit spot by destructive interference between surrounding spots.
An area for calculating the CGH is 2048$\times$2048 pixels, and the size of the CGH is 256$\times$256 pixels. The CGH is arranged in the central region of the area for the calculation. Therefore, the full width at half maximum (FWHM) of the diffraction-limit spot is 8 pixels. Each spot is arranged at the center of a cell of 7$\times$7 pixels. In the first constraint, $c_1$, the complex amplitude of the center spot in the cell is limited in the first quadrant on the complex plane, as shown in Fig. 1(b), by the equation
3. Experimental setup
Figure 2 shows the experimental setup. Our setup was mainly composed of an amplified Ti:sapphire femtosecond laser system (COHERENT, Micra and Legend Elite), a liquid-crystal-on-silicon SLM (LCOS-SLM; Hamamatsu Photonics, X10468-02), laser processing optics, and a personal computer (PC). A femtosecond pulse with a center wavelength of 800 nm, a duration of 110 fs, and a repetition frequency of 1 kHz was radiated on the SLM through a pulse energy controller composed of a half wave plate (HWP) and a polarizing beam splitter (PBS), and beam expanding optics. The illumination was set to the uniform one by clipping the beam passed through the expanding optics using an iris. A size of the clipped illumination was set to be the same as the size of the CGH (5.12 mm). The pulse was diffracted by the CGH displayed on the SLM and was imaged on a pupil plane of an objective lens (OL; Edmund, $\times 40$, NA = 0.65) by a 4f optical system composed of lenses with a focal length of 750 and 400 mm. A size of the imaged CGH on the pupil plane was 2.7 mm, and was smaller than the pupile size of 5.4 mm. Therefore, the effective NA of OL was 0.32. Consequently, the focused spot size was estimated as the Airy disc with the diameter of 2.1 µm (at 1/$e^{2}$ value of intensity profile) by using the Rayleigh criteria. The sample was quartz glass ($ {26}\;\textrm{mm}\times {26}\;\textrm{mm}\times {2}\;\textrm{mm}$) mounted on a three-axis translation motor stage (Thorlabs, ZFS13B). The irradiation pulse energy, E, was the pulse energy of the center spot at the sample plane. A white light emitting diode (LED), a dichroic mirror (DM), an infrared (IR) cut filter, and a complimentary metal oxide semiconductor (CMOS) image sensor (Thorlabs, DCC1240C) were used for monitoring the laser processing. A scanning electron microscope (SEM; Hitachi, S-4500) was used to observe the fabricated structure.
4. Experimental results
Figure 3 shows CGHs together with intensity and phase distributions of their reconstructions. The upper and lower rows indicate computer and experimental reconstructions, respectively. The experimental reconstruction was captured by a charge-coupled device (CCD) image sensor (BITRAN, BU-40L) on the focal plane of the lens after reflection at the SLM. Figure 3(a) shows the result in the case of a conventional diffraction-limit spot, which was used by way of comparison. The full width at half maximum (FWHM) of the diffraction-limit spot in the intensity profile was defined as $\textit {w}_\textrm{DL}$. Figures 3(b), 3(C) and 3(d) show the results in the case of sub-diffraction-limit spots with $\textit {r}_\textrm{n}$ of 0.50, 0.25 and 0.10, respectively. The distance between the center and surrounding spots was defined as $\textit {d}$. The FWHM of the sub-diffraction-limit spot was defined as $\textit {w}_\textrm{SDL}$. $\textit {d}$ and $\textit {w}_\textrm{SDL}$ were normalized by $\textit {w}_\textrm{DL}$ as $\textit {d}_\textrm{norm}$ = $\textit {d}$/$\textit {w}_\textrm{DL}$ and $\textit {w}_\textrm{norm}$ = $\textit {w}_\textrm{SDL}$/$\textit {w}_\textrm{DL}$, respectively, where $\textit {d}_\textrm{norm}$ was fixed to 1.75. From the results, a ratio of the energy contained in the center spot in Fig. 3(b) compared to that in Fig. 3(a) was 0.17 in the simulation. The simulation agreed well with the experiment. From the phase image in the case of the sub-diffraction-limit spot, the phase difference between the center and surrounding spots was successfully fixed to $\pi$.
Fig. 3. CGHs together with intensity and phase images of computer and experimental reconstructions in the case of (a) a diffraction-limit spot and (b) (c) (d) sub-diffraction-limit spots with $\textit {r}_\textrm{n}$ of 0.50, 0.25 and 0.10, respectively.
Figure 4 shows the normalized spot size $\textit {w}_\textrm{norm}$ versus normalized distance $\textit {d}_\textrm{norm}$. The solid lines and points show the simulation and experimental results, respectively. The triangles, squares and diamonds indicate the results in the cases of sub-diffraction limit spots with $\textit {r}_\textrm{n}$ of 0.50, 0.25 and 0.10, respectively. The insets show the optical reconstructions with $\textit {r}_\textrm{n}$ of 0.25 versus $\textit {d}_\textrm{norm}$. From the results, the simulation agreed well with the experiment. When $\textit {d}_\textrm{norm}$ was 2.5, the spot size was slightly larger than that of the diffraction-limit spot. This was due to constructive interference between the center spot and sidelobes of the surrounding spots because the phase of the sidelobes was opposite to that of the mainlobe, as shown in Fig. 3(a). When $\textit {d}_\textrm{norm}$ was 0.5, a sub-diffraction-limit spot was not generated due to constructive interference between the surrounding spots arranged at both sides of the center spot. When $\textit {d}_\textrm{norm}$ = 0.75 and $\textit {r}_\textrm{n}$ = 0.50, $\textit {w}_\textrm{norm}$ was minimized to 0.60. Consequently, the slope of the change of $\textit {w}_\textrm{norm}$ versus $\textit {d}_\textrm{norm}$ was steeper with increasing $\textit {r}_\textrm{n}$ because destructive interference between the surrounding spots was enhanced, as shown in Fig. 1(c). However, $\textit {r}_\textrm{n}$ should be as small as possible from the viewpoint of avoiding laser processing by the surrounding spots. Therefore, the selection of $\textit {r}_\textrm{n}$ is important, depending on the application.
Fig. 4. Diameter of center spot versus distance between surrounding spots with $\textit {r}_\textrm{n}$ of 0.50, 0.25 and 0.10. Solid lines and filled points indicate simulation and experimental results, respectively.
Figure 5 shows the estimated axial intensity distribution of the focused beam along the z-direction in the case of (a) a diffraction-limit spot and sub-diffraction-limit spots with $\textit {r}_\textrm{n}$ of (b) 0.50, (c) 0.25 and (d) 0.10, respectively. The beam passed through from left to right side in the figure. The results show that the intensity distribution was focused by the objective lens and was obtained by using vector diffraction theory in the computer simulation. The intensity in each case was normalized by the maximum one. The FWHMs of the profiles were 14.7, 21.7, 17.3 and 15.3 µm, respectively. From the results, the axial length of the focused beam gradually increased with increasing $\textit {r}_\textrm{n}$ because the light energy was split from the focused position to the front side and back side due to the lateral confinement effect of the surrounding spots. Consequently, the axial length of the focused beam was 1.5-times larger than that of the diffraction-limit spot in the case of $\textit {r}_\textrm{n}$ = 0.50. However, the axial length was comparable to that of the diffraction-limit spot in the case of $\textit {r}_\textrm{n}$ = 0.10 and 0.25.
Finally, femtosecond laser processing with the sub-diffraction-limit spot was demonstrated. Figure 6(a) shows the diameter of the fabricated structure, $\textit {D}$, versus the pulse energy, E, when $\textit {d}_\textrm{norm}$ was fixed to 1.25. The inset indicates an optical reconstruction of the CGH and an SEM image of the fabricated structure in each plot. $\textit {D}$ was measured from an SEM image of the structure. Laser processing was performed by single shot pulse irradiation on the glass surface. The solid line in the figure represents analytical values calculated using
Fig. 5. Intensity distribution of focused beam and its profile along the z-axis in the case of (a) diffraction-limit spot and sub-diffraction-limit spot with $\textit {r}_\textrm{n}$ of (b) 0.50, (c) 0.25 and (d) 0.10, respectively.
Figure 6(b) shows the diameter $\textit {D}_\textrm{norm}$ of the fabricated structure versus the distance $\textit {d}_\textrm{norm}$ when $\textit {E}$ was fixed to 0.17 µJ. $\textit {D}_\textrm{norm}$ was defined as $\textit {D}_\textrm{norm}$ = $\textit {D}_\textrm{SDL}$ / $\textit {D}_\textrm{DL}$, where $\textit {D}_\textrm{DL}$ and $\textit {D}_\textrm{SDL}$ mean the diameters of the structures fabricated by the diffraction-limit and the sub-diffraction-limit spot, respectively. From the result, $\textit {D}_\textrm{norm}$ was minimized to 0.39 (D = 459 nm) when $\textit {r}_\textrm{n}$ and $\textit {d}_\textrm{norm}$ were set to 0.10 and 0.75, respectively. From a comparison with Fig. 4, the minimum $\textit {D}_\textrm{norm}$ was much smaller than the minimum $\textit {w}_\textrm{norm}$ thanks to the nonlinearity of femtosecond laser processing.
Fig. 6. (a) Diameter of fabricated structure versus $\textit {E}$ of the spot with the diffraction-limit spot and sub-diffraction limit spots with $\textit {r}_\textrm{n}$ of 0.50, 0.25 and 0.10, respectively, when $\textit {d}_\textrm{norm}$ was fixed to 1.25. (b) Diameter of fabricated structure versus $\textit {d}_\textrm{norm}$ with $\textit {r}_\textrm{n}$ of 0.25 and 0.10, respectively, when $\textit {E}$ was fixed to 0.17 µJ.
Figure 7 shows focused beams generated by the proposed method. The upper left image indicates the conventional diffraction-limit spot. The other patterns indicate the results obtained by combining connected and unconnected surrounding spots. In the cases of spot connection and disconnection, the phase difference between surrounding spots was set to zero and $\pi$, respectively. By regulating the constraint in the algorithm (Fig. 1(b)), unique patterns with spot sizes comparable to the diffraction limit were arbitrarily generated. The square-shaped patten (center of Fig. 7) was composed of nine (3$\times$3) spots, including 1 center and 8 surrounding spots, where $\textit {d}_\textrm{norm}$, $\textit {r}_\textrm{n}$ and the phase difference between center and surrounding spots were set to 0.88, 1.0 and 0$\pi$, respectively. Therefore, the proposed method will be useful for generating not only sub-diffraction limit spots, but also these kinds of unique patterns.
5. Conclusion
We proposed a new algorithm for designing a CGH for generating a sub-diffraction-limit spot with regulated light intensities of the surrounding spots in order to suppress laser processing by the surrounding spots. The algorithm was based on complex amplitude optimization between surrounding diffraction spots. In the experimental results, the size of a sub-diffraction-limit spot, $\textit {w}_\textrm{norm}$, was reduced to 0.60-times that of the diffraction-limit spot. To verify the effectiveness of the proposed method, a sub-diffraction-limit spot was applied to femtosecond laser processing. Fortunately, a certain level of light intensity in the surrounding spots had no effect on the laser processing because the intensities of the surrounding spots were lower than the damage threshold of the sample. The diameter of the structure processed by the sub-diffraction-limit spot, $\textit {D}_\textrm{norm}$, was also reduced to 0.39-times thanks to the nonlinearity of femtosecond laser processing. In particular, our proposed method will be effective for applications based on nonlinear-optical phenomena, including femtosecond laser material processing and multi-photon microscopy.
Funding
Center of Excellence in Next Generation (UU-COE-Next) Project from Utsunomiya University; Japan Society for the Promotion of Science (JP18K14142); Cross-ministerial Strategic Innovation Promotion Program (SIP) from Cabinet Office, Government of Japan.
Disclosures
The authors declare no conflicts of interest.
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