We present a design method for diffractive fan-out elements generating an array of subdiffraction spots in a predetermined area using propagating light. A specific constraint is introduced to the Gerchberg-Saxton algorithm to control the dispersion of light energy and the phase of spots. In our demonstration, an element generating a 3 × 3 spot array achieves the spot size reduced to 70% comparing to the diffraction limited spot in simulation and about 80% in the experiment. Various layouts of subdiffraction spots can be made with a high signal to noise ratio using our design method. The relationship between the spot size and the power efficiency is also shown based on the simulation.
© 2014 Optical Society of America
A diffractive fan-out element is a simple and compact optical device that splits a light beam into multiple beams propagating to different directions, which generates an array of focused spots with a focusing lens. Various layouts of the spots are obtained based on suitable design methods [1, 2]. In addition, fan-out elements can be implemented with considering about, for example, compensation for chromatic distortion of femtosecond pulses , artifacts arising from a spatial light modulator (SLM) , and usage of a high-power laser source . Generation of a spot array is an effective method for parallelization of various operations in optical applications, including optical tweezers , laser processing [7,8], and microscopy [9,10].
Many types of algorithms have been proposed for designing fan-out elements. For example, an analytical method has been developed to design Dammann gratings that produce a regular array of spots with homogeneous intensities corresponding to the limited number of diffraction orders [11, 12]. Iterative algorithms are applicable more widely because various design conditions can be incorporated into the processing easily. The Gerchberg-Saxton algorithm (GSA)  and its variations, often referred to as iterative Fourier transform algorithms (IF-TAs), are typical ones [14,15]. IFTA optimizes the phase distribution of the input to generate a desired intensity distribution as the output by repeating a cycle of Fourier transform and inverse Fourier transform with constraints on the input and output planes. Effective iterative algorithms also include an algorithm based on the optimal rotation angle method [16,17].
Note that the above mentioned algorithms only design the arrangement of the spots; namely, the shapes and the sizes of the individual spots are not considered in the design method. These properties are governed by the diffraction of light. As a result, the spot sizes are often comparable to or larger than the diffraction limit.
Optical super-oscillation is an interesting phenomenon to achieve energy localization within an arbitrarily small area [18–20]. Even if the spatial frequency of the optical field is limited, the generated optical distribution can spatially oscillate faster than its highest frequency component within the local areas. Optical super-oscillation can be used to generate a hot spot smaller than the diffraction limited spot with high-power side-lobes around it . Super resolution microscopy is a promising application of super-oscillation [22,23]. Most of previous papers on optical super-oscillation treat a single small spot. An exception is found in , which reports optical tweezers using up to two subdiffractoin spots by super resolution holography. The study was, however, based on simulation only and fan-out elements generating more than 2 spots were not discussed. To extend a parallel nature of light, investigation on fan-out elements to generate multiple subdiffraction spots is considered as an important issue.
The functionality of fan-out elements is achieved with a propagating light, not a near-field light. Therefore, it is possible to utilize superior properties of the propagating light, including parallel and non-contact accessibility to objects. In addition, because the function is implemented as a single optical device, optical systems using the element become simplified. Generation of subdiffraction spots improves the resolution, density, and accuracy of the optical operations in a wide range of applications using fan-out elements.
In this paper we present a design method for fan-out elements capable of generating an array of subdiffraction spots. The elements are designed using the GSA with a new constraint on the output plane. The dispersion of light energy and the phase of generated spots are controlled by the constraint for each iteration. Designed fan-out elements are embodied using a spatial light modulator, and the generated optical patterns were observed to confirm the performance. Properties are also evaluated by computer simulation.
2. Design algorithm
The system model used is shown in Fig. 1. A phase-only fan-out element is placed in front of the focusing lens with the focal length of f. The output plane is set on the back focal plane of the focusing lens. The fan-out element is illuminated by a plane light wave with wavelength λ. The intensity distribution I(x, y) on the output plane is written by 
We employed the GSA with a specific constrain on the output plane. The fundamental procedure of the GSA is as follows: and are the amplitude and the phase of the input plane before applying the constraint, respectively, and and are those after applying the contraint. , , , and are the same distributions on the output plane. i = 0, 1,...,imax is the iteration number.
- An initial random phase distribution on the input plane, , is given. The initial amplitude is the amplitude of the illumination light.
- The optical field on the input plane, is Fourier transformed to calculate the optical field on the output plane.
- A constraint on the output plane is applied. (The detail is described later.)
- The optical field is inverse Fourier transformed to calculate the optical field .
- A constraint on the input plane is applied: the amplitude is replaced by that of the illumination light and the phase is held.
- i is increased by 1. If i = imax, the iteration is terminated. Otherwise, steps 2–5 are repeated.
A design result is obtained as and I(x, y) is calculated using Eq. (1).
Two operations are required in the constraint on the output plane to generate subdiffraction spots. One is dispersing the light energy to the area surrounding the spots to remove excess energy from the spots. Although the overall optical pattern spreads to a larger area by this operation, optical patterns smaller than the diffraction limited spot appear within a number of local areas. The other is separating adjacent spots not to merge into a large spot. To arrange spots with a short interval, the phase difference between the individual spots is a crucial attribute to be controlled. When the phase difference between the adjacent spots is zero, the intensity between the spots become higher than the intensity of each spot alone due to interference. The spots can consequently become larger. In contrast, when the phase difference is π, the intensity between the spots is lower than the intensity of each spot alone, and the area where the intensity is approximately zero appears. As a result, they can be separated obviously. This suggests that control of the phase distribution on the output plane is important to generate an array of small spots.
On the basis of the above consideration, a new constraint on the output plane was constructed. The output plane is divided into two areas, the spot-area and the surrounding area, as shown in Fig. 2. The spot-area is further divided into a set of square cells. In the calculation, multiple pixels are assigned to the individual cells. Spots can be placed at the cells colored by red and green in Fig. 2.
Although a phase distribution is held after applying the constraint in the original GSA, our algorithm changes the processing. Any one of operations (I)–(III) is applied to the individual pixels within the spot-area. The same operation is applied to all pixels in each cell. Operation (I) is expressed by Eqs. (2) and (3), where iteration number i and the coordinate (x, y) are eliminated for simplicity.
Operations (I) and (II) are used to move the complex amplitude Aout exp[jϕout] to within the first quadrant and the third quadrant on the complex number plane, respectively. The phase continuity is preserved after applying the constraint. Operation (III) is used to make the intensity 0. In the design, either operation (I) or (II) is applied to the pixels of the cells where a spot is assigned, and operation (III) is applied to the pixels of the other cells as shown in Fig. 2. This ensures that the phases of the four-neighbor spots (above, below, right, and left) are almost inverse to the center spot, and the area whose intensity is approximately zero can appear around the center spot. This is effective to make isolated spots and to reduce the spot-size.
To confirm the validity of our algorithm, several fan-out elements were designed and experimentally demonstrated. Figure 3 shows the experimental setup. A reflection type of the SLM (Hamamatsu Photonics K.K., LCOS-SLM, X10468-01, pixel size: 20 × 20 μm2, number of pixels: 800 × 600) was used to embody the fan-out elements. A Gaussian beam with the 1.64-mm beam-waist created from a He-Ne laser (wavelength: 632.8 nm) through the beam expander consisting of L1 and L2 is modulated by the SLM, and enters the focusing systems via BS2. We observed the output patterns through two focusing systems with a thin lens (L3, focal length: 400 mm) and an objective lens (OL, Olympus, OPLSAPO 4×) as the focusing lens. The effective numerical aperture of the focusing system 1 was 0.0063 and that of the focusing system 2 was 0.056. The output patterns were captured using CCD camera 1 (QImaging, QIClick-F-CLR-12, pixel size: 6.45 μm, sensing area: 8.98 mm × 6.71 mm) and CCD camera 2 (Edmund Optics, EO-10012, pixel size: 1.67 μm, sensing area: 6.41 mm × 4.59 mm). The individual CCDs were chosen to satisfy both that the pixel size was small enough to measure the spot size and that the overall sensing area was large enough to capture the subdiffraction spots and zeroth order light. Using this setup, we can evaluate the performance of fan-out elements used with a low numerical aperture (NA) system and a higher NA system simultaneously.
As an example, Fig. 4 shows the design result for a fan-out element that generates 3 × 3 spots. In the design, we prepared 2048 × 2048 pixels in total on the input plane. Among these pixels, 256 × 256 pixels around the center were used for the fan-out element, and the other pixels were filled by amplitude of zero. The sampling period on the output plane obtained with this configuration of the input plane is 1/8 of that obtained without filling zero, so that this configuration enables to investigate the output pattern in more detail. On the output plane, the spot-area was placed at the upper left part to avoid overlap between the spot-area and the zeroth order light as shown in Fig. 4(a). The number of cells in the spot-area is 17 × 17, the length of the sides of the cells is 0.7 × Ld.l., where Ld.l. is the full width at half maximum (FWHM) of the diffraction limited spot and Ld.l. corresponds to approximately 10 pixels, C in Eq. (9) is 0.75, and imax is 30. Note that the spot size in a spot array strongly depends on the side length of the cells in the design, and the side length should be chosen with considering on an expected spot size. The detailed relationship is shown in the last part of this paper. The designed phase is shown in Fig. 4(b). As seen from Fig. 4(c), a 3 × 3 spot array is generated within the spot-area, and the phase difference between adjacent spots is controlled to be approximately π using our constraint. Furthermore, Fig. 4(d) indicates that the spot sizes, defined as the FWHM of the spots, are smaller than the diffraction limited spot. It is possible to generate an array of subdiffraction spots by an adequate modulation of a single beam.
Figure 5 shows the experimental result for generating 3 × 3 spots. The output patterns obtained with the focusing systems 1 and 2 in the experiment are consistent with the design result. We measured the signal to noise ratio (SNR) of a spot array in the spot-area. In the output pattern, the light around individual spots, from their intensity peaks to the first local minimum towards all directions, is considered as the signal and the other light is considered as the noise. The SNR is calculated by26,27]. In the field of view, the SNR should be high to make an intense needle. Our method can place multiple needles in the field of view as a benefit of generating a high-SNR spot array.
Figure 6 shows the FWHM (spot size) of the individual spots for the case of 3 × 3 spots. The FWHM is normalized by that of the diffraction-limit spot, Ld.l., which is 65 μm for the focusing system 1 and 7.2 μm for the focusing system 2. The FWHMs are reduced to about 70% for all spots in the design. Note that the sizes of all the spots were smaller than Ld.l. also in the experiments. The spot sizes were reduced to 79% comparing to the diffraction limited spot on average when using the focusing system 1 and 80% when using the focusing system 2. The experimental results demonstrate that the designed fan-out elements work well in the focusing systems although small degradation of the performance occurs.
The observed images using another fan-out element are shown in Fig. 7. The element was designed to generate spots arranged in the shape of ”DH2014”. Although the spot layout was more complicated than the layout of 3 × 3 spots, all spots were decreased in size comparing to the diffraction limited spot. Our algorithm is effective to design fan-out elements for implementing various layouts of subdiffraction spots.
To characterize fan-out elements, power efficiency was estimated by computer simulation. The relationship between the power efficiency and the averaged spot-size among 3 × 3 spots is shown in Fig. 8. The power efficiency is defined as the ratio of the power in the spot-area to the total power of the illumination beam. In the design, the side length, Lc, of the cells was varied from 0.9Ld.l. to 0.5Ld.l. with 0.1Ld.l. steps and imax was 50. We investigated with different sets of the parameter C for the individual Lc because the range of C with which subdiffraction spots were able to be generated depended on Lc. We set the minimum value of C as 0.05, 0.04, 0.055, 0.10, and 0.20 for Lc = 0.9Ld.l., 0.8Ld.l., 0.7Ld.l., 0.6Ld.l., and 0.5Ld.l., respectively. When C is smaller than the minimum value of C for each Lc, the excess energy of spots is not dispersed sufficiently, and adjacent spots merge to become a larger spot. Thus subdiffraction spots are not generated. In contrast, as C increases, more light energy disperses to the surrounding area, and the power efficiency decreases. In this evaluation, we set the maximum value of C as 0.3 for all Lc. Figure 8 indicates that we can make the spot size smaller by dispersing the power to the surrounding area. For example, 13% in power efficiency can be achieved with 80% spot-size comparing to the diffraction limit, and about 1% in power efficiency can be achieved with 60% spot-size.
We have presented a design method for fan-out elements to generate an array of subdiffraction spots using propagating light. The GSA with a specific constraint considering the dispersion of the light energy and the phase of the spots is sufficient to design fan-out elements generating various layouts of multiple spots. In the experimental demonstration, a high-SNR array of subdiffraction spots whose size is about 80% of the diffraction limited spot was generated using a thin lens or an objective lens as the focusing lens. In addition, the relationship between the power efficiency and the spot size was shown based on simulation. We believe that the presented approach to making multiple spots beyond the diffraction limit using propagating light is important to utilize parallel and non-contact accessibility to objects in a variety of applications.
This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas ”Nanomedicine Molecular Science” (No. 2306) and that for Scientific Research (c) (No. 26390084) from Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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