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Abstract
In this paper, soft-edge toroidal amplitude filter (STAF) and soft-edge toroidal complex amplitude filter (STCAF) are designed according to the principle that soft-edge structures can eliminate diffraction. Based on the Mach-Zehnder interference principle, a double optical path compound interference modulation method that can generate soft annular beams is proposed by using STAF and STCAF. The 1/e2 radius and peak-to-average ratio (PAR) were used to evaluate the ring width and uniformity of the annular beam. Compared with the annular beams generated by STAF and hard-edge toroidal amplitude filter (HTAF), it can be found that the soft annular beam generated by this proposed method has the advantages of high uniformity, small ring width increment, and smooth edges. By analyzing the influence of the number and height of the sawtooth structures on the annular beam propagation performance, the relationship between the PAR and the structure parameters of the STAF was established. Moreover, three kinds of toroidal filters were designed by lithography processing, and an experimental system was built to generate the soft annular beam. In the experiment, the average value of the ring width increment of the soft annular beam is 0.0125, the PAR is less than 1.5, and the root mean square error of the PAR curve is 0.0865, which indicates that the soft annular beam maintains high uniformity during propagation.
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1. Introduction
Annular beams have high impedance and are widely used as incident beams in systems that generate optical needles. Optical needles have wide applications in many fields, such as photolithography [1], material processing [2], optical data storage [3], particle acceleration [4–7], and super-resolution microscopy [8,9]. The amplitude, phase and polarization state of the incident beams are the main factors that determine the performance of the optical needles [10–16]. In the past two decades, with the development of lithography, modulation devices represented by amplitude filters, phase filters, and complex amplitude filters have appeared one after another. Among them, the hard-edge toroidal amplitude filter (HTAF, for consistency with the nomenclature of soft-edge toroidal amplitude filter (STAF) and soft-edge toroidal complex amplitude filter (STCAF), the annular diaphragm or annular aperture with the center blocked is referred to as HTAF in this paper.) has attracted much attention due to its simplicity in design and processing. For example, a focal spot of 100 nm was achieved at an incident beam wavelength of 488 nm by using HTAF, which is used in microscopy to improve resolution [11]. In 2010, Kitamura et al. obtained an optical needle (FWHM = 0.4λ, DOF = 4λ) by tightly focusing the radially polarized annular beam generated by an HTAF through a high numerical aperture (NA) objective lens. It has great potential for applications in optical systems requiring high tolerance and high resolution [14]. In 2011, M. Gu. et al. used an HTAF to improve the effective numerical aperture of high NA objectives, and the focus obtained by tight focusing broke the diffraction limit. This finding can achieve an enhanced photoreduction threshold effect in the two-photon induced three-dimensional optical data storage [17]. However, the HTAFs utilized in these studies all have low annular factors, and thus the influence of diffraction effects on the propagation of the annular beam in space is neglected. After that, L. Yang. et al. [18] found that the resolution and depth of focus of the optical needle were not proportional to the annulus factor when the annulus factor was large in the experiment, and it is impossible to get a clearer spot. This is due to the diffraction effect when the annular beam propagates in the free-space medium, which leads to an increase in the ring width and a decrease in the uniformity of the annular beam.
However, the soft-edge amplitude filter (SAF) formed by adding some sawtooth structures on the edge of the hard-edge amplitude filter (HAF) can adjust the intensity distribution of the incident beam, limit the shape of the beam, and improve the uniformity of the beam. SAF was originally used to reduce the nonlinear self-focusing caused by Fresnel diffraction [19]. The influence of SAF with various structures on the beam has been analyzed at present [20–25]. Recently, L. Shi. et al. analyzed the performance of SAF with six different structures [26]. The research results show that SAF can effectively suppress the diffraction effect, improve uniformity, and energy efficiency. However, compared with HAF, the sawtooth structures of SAF also produce diffraction images, which makes the edge of the beam have burrs. Therefore, adding sawtooth structures directly to the inner and outer rings of the HTAF results in burrs at the edges and increases the ring width of the annular beam.
In order to solve this problem, we design a soft-edge toroidal amplitude filter (STAF) and a soft-edge toroidal complex amplitude filter (STCAF) in this paper. Based on the principle of Mach-Zehnder interference, we develop a double optical path compound interference modulation method, aiming at eliminating diffraction to improve the uniformity and eliminate the ring width increment of the annular beam in the space propagation process. Among them, STAF and STCAF are used to modulate the two branches of the interference optical path, respectively. They are respectively located at 3/4 position of the optical path of each branch. The amplitude transmittance functions of the two filters are the same, while the phase transmittance functions differ in some areas by π. The annular beam created by this method is called the soft annular beam. In this work, the low ring width increment and high uniformity (PAR < 0.0865) of the soft annular beam have been demonstrated theoretically and experimentally.
2. Method
2.1. Optical system
The schematic diagram of the optical path is shown in Fig. 1(a). The source is a semiconductor laser with a wavelength of 656nm. Lenses L1 and L2 form a collimated beam expander, which expands the beam from the laser to be 8mm width. Both BS1 and BS2 are 1:1 beam splitters, and R1 and R2 are reflectors. The structures of the HTAF, STAF, and STCAF are shown in Fig. 1(b-d), respectively. To ensure the equal length of the two branches, the distance between the beam splitter (BS) and the reflector (R) satisfies d1 = d2 = d3 = d4. In addition, the STAF and STCAF are located at d3/2 and d4/2, respectively. The amplitude distribution is the same for both filters, except that the yellow sawtooth region of the STCAF has a phase of π, while the other regions have the same phase as the STAF (both are 0). According to the principle of interference constructive destructiveness, when the phase difference of two coherent beams is an integer multiple of 2π, constructive interference occurs. Conversely, when the phase difference between the two coherent beams is an odd multiple of π, interference cancellation occurs.
Fig. 1. (a) The schematic diagram of the optical path. (b) Hard-edge toroidal amplitude filter. (c) Soft-edge toroidal amplitude filter. (d) Soft-edge toroidal complex amplitude filter.
After collimation and expansion, the incident beam is first split by BS1. The two split beams are adjusted by STAF and STCAF, respectively, and then combined by BS2. Finally, a soft annular beam can be observed at C1. It is usually chosen to observe at C1 rather than C2. The reason for this is that when the two split beams meet at the C1 position, they are both reflected twice. In contrast, when the split beams on the two branches converge at position C2, the split beam on the lower branch is reflected three times, while the split beam on the upper branch is reflected once. The intensities of the split beams on the two branches may be slightly different, which often results in stray light in the soft annular beam.
2.2. Physical model of the beam propagation in space
Based on the Fresnel diffraction theory, the physical model of the beam propagating in space after passing through the toroidal filter is established. For an incident beam with a wavelength of λ, the electric field distribution of the diffracted light field in the cartesian coordinate system can be expressed as Eq. (1) when the diffraction distance satisfies the Fresnel diffraction condition.
The function model of the sawtooth shape of STAF and STCAF is shown in Eq. (3).
According to the function model of STAF and STCAF, the influence of the structural parameters (sout, sin, Mout, Min) on the uniformity and ring width of the annular beam should be comprehensively considered in the optimization design.
Fig. 2. The intensity distributions of the annular beams were obtained by the HTAF at different diffraction distances. The diffraction distances of the black, green, and blue curves are 0mm, 100mm, and 200mm, respectively.
3. Parameter analysis and structure design
Based on the established physical model of Fresnel diffraction, the intensity distributions of the annular beam generated by different toroidal filters during the propagation in free space are analyzed. In order to characterize and evaluate the quality of the annular beam at a certain diffraction position, the 1/e2 radius and peak-to-average ratio (PAR) are used to measure the ring width (RW) and uniformity, respectively. The ring width (RW) is the distance within the range of 1/e2 of the peak intensity of the annular beam. The calculation of PAR is shown in Eq. (6), where Imax and Iave are the maximum and average intensity with 1/e2 of the peak intensity as the boundary range, respectively. Therefore, under the condition of no diffraction, PAR = 1. In addition, as shown in Eq. (7), the root mean square error (RMSE) of the peak-to-average ratio curve is used to evaluate the uniformity of the beam during the entire propagation distance. Among them, m is the number of sampling points, Pideal is the ideal peak-to-average ratio, and Pactual(n) represents the actual peak-to-average ratio of the nth sample.
3.1. Propagation characteristics of the annular beam generated by HTAF in free space
Taking the HTAF with parameters r = 0.85R and R = 4mm as an example, the spatial propagation characteristics of the generated annular beam are analyzed. The distance between the filter and the CCD is defined as the diffraction distance, which is denoted as d. When the diffraction distances are 0mm, 100mm, and 200mm, the normalized intensity distributions of the annular beams along the white dashed line are shown as black, green, and blue curves in Fig. 2, respectively. When d = 0mm, the PAR and RW are 1.0 and 0.5957, respectively. With the increase of the diffraction distance, when d = 100mm, PAR and RW are 1.6816 and 0.6445, respectively. Furthermore, when d = 200mm, the PAR and RW are 1.77 and 0.6836, respectively. Obviously, HTAF can realize the amplitude modulation from the incident plane wave to the annular beam. However, the diffraction effect leads to oscillations in the light intensity, and the longer the diffraction distance, the more pronounced the oscillations will be. Therefore, the uniformity of the annular beam decreases with the increase of the diffraction distance. In addition, it is easy to see that the RW broadens as the diffraction distance increases. It is worth noting that when the diffraction distance is increased to 200mm, the nonlinear self-focusing of the beam caused by Fresnel diffraction results in a larger intensity at the center point. As we all know, it is disadvantageous for optical devices. The nonlinear self-focusing effect is essentially caused by the interference superposition of higher-order terms of the incident beam produced by diffraction.
3.2. Propagation characteristics of the annular beam generated by STAF in free space
The following analysis is the performance of the annular beam obtained after the beam passes through the STAF as shown in Fig. 1(c). The parameters of the STAF are: Mout = Min = 72, Sout = Sin = 0.125, r = 0.85R, R = 4 mm. When the diffraction distances are 0mm, 100mm, and 200mm, the intensity distributions of the annular beams along the white dashed line are shown in the black, green, and blue curves of Fig. 3, respectively. When the diffraction distance is 0 mm, the PAR of the annular beam is 1.0 and the RW is 1.4941. When d = 100 mm, the PAR and RW are 1.7245 and 1.2598, respectively. In addition, when d = 200 mm, the PAR and RW are 1.9833 and 1.0254, respectively. Compared with HTAF, the ring width increment of the annular beam obtained by STAF is 1.1288mm, 0.6153mm, and 0.3418mm when d is 0mm, 100mm, and 200mm, respectively. The main reason for the increase in RW is that the sawtooth structures on the edges of the inner and outer rings of STAF have a certain height. Affected by the diffraction images of the sawtooth structures, the edges of the annular beam obtained by STAF have burrs. Moreover, comparing the PAR values, it is found that the uniformity of the annular beam obtained by STAF is poor, which is mainly since the diffraction images of the sawtooth structures are considered in the calculation of PAR. Finally, comparing the blue curves in Fig. 2 and Fig. 3, the annular beam obtained by STAF has no intensity near the zero point when the diffraction distance is 200mm, which indicates that the sawtooth structures of the STAF can change the complex amplitude distribution of the incident beam on the observation screen, suppress the diffraction effect, and reconstruct the light field.
Fig. 3. The intensity distributions of the annular beam were obtained by STAF at different diffraction distances. The diffraction distances of the black, green, and blue curves are 0mm, 100mm, and 200mm, respectively.
3.3. Propagation characteristics of the soft annular beam in free space
From the above simulation and analysis, it can be found that the uniformity of the annular beam obtained by HTAF decreases and the RW increases with the increase of diffraction distance due to the diffraction effect. On the contrary, the STAF can suppress the diffraction effect. However, the sawtooth structures at the inner and outer rings lead to burrs at the edges of the obtained annular beam, which also leads to an increase in RW and a decrease in uniformity. Therefore, it is impossible to obtain a soft annular beam with high uniformity, small ring width increment and smooth edge using only HTAF or STAF.
Using the double optical path compound interference modulation method proposed in this paper, the normalized intensity distributions of the obtained soft annular beams along the white dashed line are shown in the black, green and blue curves of Fig. 4 when the diffraction distances are 0mm, 100mm, and 200mm, respectively. The RW of the soft annular beam remains unchanged at 0.5957 when the diffraction distance increases. In addition, when the diffraction distance is 0 mm, 100 mm, and 200 mm, the PAR is 1.0, 1.2285, and 1.3009, respectively. Compared with the annular beam produced in Fig. 2 and Fig. 3, the RW of the soft annular beam obtained by this method is not affected by the diffraction distance. Moreover, under the same diffraction distance, the uniformity of the soft annular beam is the best. The diffraction suppression effect of STAF and STCAF can keep the soft annular beam with high uniformity, and the double optical path compound interference modulation method based on the principle of interference constructive destructiveness can make the soft annular beam with low ring width increment and smooth edge. Therefore, the soft annular beam obtained by this method has the characteristics of low ring width increment, high uniformity and smooth edge.
Fig. 4. The Intensity distributions of the soft annular beams under different diffraction distances. The diffraction distances of the black, green, and blue curves are 0mm, 100mm, and 200mm, respectively.
Fig. 5. The relationship between PAR and diffraction distance. The black and green curves represent the relationship between the PAR and the diffraction distance of the annular beam generated by the HTAF and STAF, respectively. The blue curve denotes the relationship between the PAR and the diffraction distance of the soft annular beam obtained by the double optical path compound interference modulation method.
3.4. Analysis of the uniformity of the annular beams during the entire propagation process
As shown in Fig. 5, to further evaluate the uniformity of the annular beams during the entire propagation process, the PARs under different diffraction distances are simulated. The RMSE is used to measure the fluctuation of the uniformity of the annular beam over the entire propagation distance. The RMSE of the annular beams generated by the HTAF and STAF are 0.49 and 0.8452, respectively. However, the RMSE of the soft annular beam is 0.1047. The RMSE of the soft annular beam obtained by the double optical path compound interference modulation method is the smallest, which indicates that the soft annular beam can maintain high uniformity in spatial propagation.
3.5. Optimization of STAF and STCAF parameters
The number and height of sawtooth structures on the inner and outer rings of the STAF and STCAF are the main factors affecting the uniformity of the soft annular beam. Since the amplitude distributions of STAF and STCAF are the same, the following takes STAF as an example to study the influence of the sawtooth structure parameters (sout, sin, Mout, Min) of the inner and outer rings on the annular beam.
When sout and sin are determined, the influence of Mout and Min on the annular beam is studied. The case of Mout = Min is analyzed first. As shown in Fig. 6, the black, green, and red curves represent the relationship between PAR and diffraction distance when the number of the sawtooth is 72, 108, and 144, respectively. The corresponding RMSE is 0.1047, 0.0609, and 0.0530, respectively. The results show that increasing the number of sawtooths on the inner and outer ring at the same time can improve the uniformity of the annular beam to a certain extent. It is mainly because the amplitude distribution through the STAF will have a smoother gradient with the increase of the number of sawtooths, thus making the diffraction effect weaker. However, as the number of sawtooths further increases, the uniformity remains essentially constant, which only makes the processing of STAF and STCAF more difficult. In addition, by comparing the blue and magenta curves, the black and blue curves, or the red and magenta curves in Fig. 6, respectively. It can be found that an increase in the number of sawtooth structures on the outer ring is more effective than an increase in the number of sawtooth structures on the inner ring for the uniformity of the annular beam. Therefore, appropriately increasing Mout can improve the uniformity of the annular beam.
Fig. 6. The relationship between the PAR and diffraction distance when the number of sawtooths on the inner and outer rings of STAF is different.
Reference [25] shows that the uniformity of the beam is optimal when the structural parameter S of the SAF is 0.25. Therefore, the outer ring structure parameter sout of the STAF we designed selects 0.25. The effect of the inner ring structure parameter sin on the uniformity of the annular beam was investigated under the condition that the number of sawtooth structures of both inner and outer rings was 72. As shown in Fig. 7, when sin increases from 1/32 to 1/4, the RMSE drops from 0.0656 to 0.0479. It is because when sin is small, the inner ring formed by the sawtooth structures can be approximated as a circular hard-edged inner ring, which makes its diffraction effect increased and thus leads to poor uniformity of the annular beam. In addition, it is worth noting that the RMSE is 0.508 when sin is 1/2, which indicates that sin cannot be too large. Comparing the RMSE under different sin, the uniformity of the annular beam is the best when sin = 1/4, which is consistent with the outer ring structure parameter sout.
Fig. 7. The relationship between the PAR and diffraction distance when the sin of STAF is different.
4. Experiment and discuss
4.1. Processing of HTAF, STAF, and STCAF
The lithography process flow chart for processing HTAF and STAF is shown in Fig. 8. The quartz is selected as the substrate material, and a layer of Cr with a thickness of 100 nm was evaporated on the substrate surface after pretreatment of the substrate surface. The photoresist (AZ1500) was spin-coated on the Cr layer surface at a speed of 5000rad/min. The coating time, pre-baking temperature, pre-baking time, and resist thickness were 30s, 100°C, 5min, and 1µm, respectively. The illumination source selects a laser with a wavelength of 365nm. The exposure time is 30s. Then the development process was carried out, and the development time, post-baking temperature and post-baking time were 3min, 120°C, and 30min, respectively. AZ1500 is a positive photoresist, which undergoes a photolysis reaction with the developer, and the photoresist at the exposed position is dissolved in the developer. The substrate was placed into the Cr removal solution, and the exposed Cr layer was removed after 40s. Finally, the photoresist layer above the Cr layer is removed with anhydrous ethanol to obtain the desired HTAF or STAF structure.
STCAF also uses quartz as the substrate for processing, and its refractive index is 1.45637@λ=656nm. Different phases can be achieved by changing the thickness of the quartz through the etching process. As calculated by Eq. (8), the phase values 0 and π correspond to etching thicknesses of 0nm and 718.7nm, respectively.
Compared with STAF, the processing technology of STCAF is relatively complicated. However, the main process parameters are basically the same. The lithography process for processing STCAF is shown in Fig. 9. Firstly, the sawtooth structures of the inner and outer rings are exposed, developed, and wet etched. It should be noted that the etching depth here includes the depth corresponding to the Cr layer (100nm) and the π phase (718.7nm). Then, after the secondary coating, alignment exposure, development and etching processes, the middle circular part is aligned and exposed, and the Cr layer is removed. Finally, the surface photoresist is removed to obtain the desired STCAF structure. Among them, the overlay accuracy is about 1µm.
The fabricated results of HTAF, STAF are shown in Fig. 10(a) and 10(b), respectively. The fabricated result of STCAF is shown in Fig. 10(c), and a magnified view of the rectangular position under the microscope is shown on the right. The radius R of the outer ring is 4mm, the sawtooth height is 1mm, and the number of the sawtooth is 144. The radius r of the inner ring is 3.4mm, the sawtooth height is 0.85mm, and the number of the sawtooth is 72. Due to the use of hot-melt surface morphology, the structure on the photoresist during development is eliminated, and the accuracy of the sawtooth tip and the sawtooth root can reach 1µm, which ensures that the details of the sawtooth structures can be well reflected. The thickness of the Cr layer is 100nm, and the height difference of STCAF measured by a profilometer is 808.6nm. Therefore, the etching depth is 708.6nm, and the machining error is 1.405%.
4.2. Verification and analysis
The performance of the soft annular beam generated by the double optical path compound interference modulation method was verified by experiments. The experimental optical configuration is shown in Fig. 11. According to the geometric principle that two points determine a straight line, two coaxial circular variable diaphragms with equal heights are used as auxiliary tools in the optical path adjustment process to ensure that the upper and lower arms of the optical path are at the same height and collinear. A 656nm semiconductor laser is selected. The beam emitted by the laser forms a wide-aperture plane wave after the spatial filter and the collimating beam expander. After passing through the diaphragm(D), the spot diameter becomes 8mm. It is then split into two beams by BS1 (beam splitting ratio: 1:1): the transmitted beam and the reflected beam. The transmitted beam is reflected by R2 and then vertically incident on the STAF. At the same time, the reflected beam is reflected by R1 and then vertically incident on the STCAF. The modulated transmitted beam and reflected beam are merged and interfered by BS2 to obtain a soft annular beam, which is recorded by the CCD and transmitted to the computer.
It is worth noting that the distance from BS1 to R2 and R1 and the distance from R1 and R2 to BS2 are both 20cm. In addition, the distance from STAF or STCAF to CCD is the diffraction distance, which can be adjusted by the 3D long-travel stage. The strict distance requirement is mainly to ensure that the phase difference caused by the optical path is equal.
The experimental results are shown in Fig. 12. The intensity distributions of the annular beams obtained by HTAF when the diffraction distances are 100mm and 200mm are shown in Fig. 12(a1) and (a2), respectively. Moreover, the intensity distributions of the soft annular beams obtained by the double optical path compound interference modulation method proposed in this paper when the diffraction distances are 100mm and 200mm are shown in Fig. 12(a3) and (a4), respectively. Comparing Fig. 12(a1) and (a2), it can be seen that with the increase of the diffraction distance, the ring width of the annular beam obtained by HTAF increases significantly, the uniformity deteriorates, and the self-focusing phenomenon caused by diffraction becomes obvious. In contrast, the soft annular beams can effectively suppress diffraction, which not only have high uniformity, but also maintains low ring width increments and smooth edges. In addition, as shown in Fig. 12(b), compared with the annular beam obtained by STAF and HTAF, the ring width increment of the soft annular beam obtained by simulation is 0, and the average value of the ring width increment of the soft annular beam in the experiment is 0.0125. The experimental results are close to the theoretical results. The annular beam obtained by STAF is affected by the sawtooth shapes, and there are burrs on the edge, which leads to a large increase in ring width. Furthermore, the RW of the annular beam obtained by HTAF basically increases with the increase of diffraction distance. From Fig. 12(c), it can be seen that the light intensity distribution on the observation surface varies with the propagation distance, leading to fluctuations in the PAR curve. In addition, external environmental factors also affect the distribution of light intensity. The RMSEs of the PAR curves for annular beams generated by STAF and HTAF were 1.9901 and 0.4900, respectively. However, the RMSE of the PAR curve of the soft annular beam obtained by the double optical path compound interference modulation method proposed in this paper is only 0.0865, which indicates that this method can keep the annular beam maintaining a high uniformity during propagation and has a high beam quality.
Fig. 12. (a1-a2) The intensity distributions of the annular beams were obtained by HTAF when the diffraction distances are 100mm and 200mm, respectively. (a3-a4) The intensity distributions of the soft annular beams when the diffraction distances are 100mm and 200mm, respectively. (b) The relationship between RW and diffraction distance. (c) The relationship between PAR and diffraction distance. The green and blue curves represent the results of the experiment of STAF and HTAF, respectively. The red and purple curves denote the simulation and experiment results of the soft annular beam, respectively.
5. Conclusions
In summary, we designed and fabricated three kinds of toroidal filters: hard-edge toroidal amplitude filter, soft-edge toroidal amplitude filter and soft-edge toroidal complex amplitude filter. PAR and RW are used to evaluate the uniformity and ring width of the annular beams produced by the toroidal filters. According to the simulation and experimental results, the soft annular beam generated by the double optical path compound interference modulation method proposed in this paper can not only maintain a high uniformity in a certain propagation range, but also solve the problem of ring width increase caused by diffraction. This method can effectively improve the beam quality of the annular beam, thus improving the resolution and depth of focus of the optical needle produced by tightly focusing the annular vector beam with a high numerical aperture objective, which is expected to be widely used in laser processing, super-resolution imaging systems and other fields.
Funding
The Instrument Development of the Chinese Academy of Sciences (No.YJKYYQ20200060, No.YJKYYQ20210041); Sichuan Province Science and Technology Support Program (No.2021JDRC0089, No.2022102, No.2022YFG0223, No.2022YFG0249).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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