Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2023; 7(3): 304-309
Published online June 25, 2023 https://doi.org/10.3807/COPP.2023.7.3.304
Copyright © Optical Society of Korea.
Woongbu Na^{1}, Seung-Yeol Lee^{2}, Hyuntai Kim^{1}
Corresponding author: ^{*}hyuntai@hongik.ac.kr, ORCID 0000-0001-7401-3320
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, a binary dielectric annular nanoring lens is proposed to cover the full range of optical phase. The lens is designed numerically, based on the effective-medium theory. The performance of the proposed lens is verified for the cases of single-focal and dual-focal lenses. The efficiency of a singlefocal lens is improved by 17.19% compared to a binary dielectric lens, and that of a dual-focal lens shows enhancements of 13.11% and 49.41% at the two focal points. This lens design can be applied to other optical components with axially symmetric structures.
Keywords: Effective medium theory, Lens system design, Nanostructures, Subwavelength structures
OCIS codes: (220.3620) Lens system design; (260.2065) Effective medium theory; (310.6628) Subwavelength structures, nanostructures
Optical focusing has been utilized for numerous applications since ancient times. Radially polarized light is particularly useful in focusing, as all radial components disappear and only the longitudinal component is focused, resulting in a small and symmetrical beam size that is advantageous for machining, sensing, and trapping [1–3].
Different types of zone plates have been developed to reduce the thickness of optical components [4–10]. Block-type zone plates such as metallic zone plates are the easiest to fabricate but have the lowest efficiency, as light in the blocked area is not focused [11–14]. Dielectric zone plates have relatively higher efficiency compared to block-type zone plates, but they are sensitive to thickness and cannot act like bulk dielectric lenses, as the phase information is binary [15–18]. A Fresnel lens with continuous correspondence between its height and each phase is more efficient than other binary lenses. However, its fabrication process is challenging because the height must be adjusted for each position [19, 20].
Recently, various phase-controlled metasurface lenses have been researched [21–23]. However, most of their unit cells are square or hexagonal, making it challenging to match the axial symmetry accurately.
In this study, we propose a binary annular nanoslit array lens based on the effective-medium theory (EMT), which corresponds to a continuous phase structure [24–26]. In this research, the first step is to evaluate the characteristics of the annular dielectric nanoslits. Then a complete set of phase-to-structure data is established. To demonstrate the feasibility of the concept, we design and analyze both a single-focal and a dual-focal lens.
EMT provides the effective refractive index of an axially symmetric structure for radially polarized incident light. Equation (1) shows the effective refractive index of a ring-shaped nanoslit, based on the duty-ratio [24–26]:
where
Numerical calculations are performed to match the phase data to the duty-ratio values. The incident light’s wavelength is set to 650 nm, a common red wavelength, and the refractive index of silicon is set to 3.8350 [27, 28]. The period of the structure is set to 100 nm.
The duty ratio is varied for different dielectric thicknesses from 300 to 400 nm. Note that the phase difference between the vacuum and silicon should be 2
The results show that the highest average transmission efficiency is achieved at a thickness of 330 nm. This value is then used as the thickness for further analysis, to create a database linking the phase to the duty ratio. The full data set, linking duty ratio to phase, is selected based on the cycle with the highest average transmission. The established phase-to-structure (duty ratio) data are presented in Fig. 1(c). Note that the duty ratio varies from 0.01 to 0.99, in steps of 0.001. Therefore, the minimal unit size becomes 1 nm, and the resolution of the structure is set to 0.1 nm. However, the value can be adjusted by changing the duty-ratio steps, and also selecting the middle of the range, such as 0.1 to 0.9.
First, we design a single-focal lens to verify our method. To design a lens, we use the virtual-point-source method (VPSM) [29–31], which is an inverse-design technique for optical components. Using the calculated phase from the virtual source as a reference, the duty ratio of each nanoring is obtained from the prepared data. The design schematic, calculated phase, and corresponding binary lens design are shown in Fig. 2.
To design the lens, the VPSM is used. The position of the focal length is selected as 5 μm, and the radius of the lens is set to 10 μm. The phase at the virtual single point is calculated using VPSM, and is varied to optimize the efficiency.
After designing the lens using the VPSM, numerical simulations are performed using COMSOL Multiphysics. The electric field intensity is calculated at the longitudinal axis (
The proposed lens designed using the EMT is compared to a conventional dielectric binary phase lens (also known as a Fresnel zone plate), which considers only positive and negative phases, to evaluate its performance. Note that the binary phase lens has also designed based on VPSM, and the identical tool has been applied for numerical calculations. Table 1 shows the comparison of efficiency and beam spot size. The results show that the EMT lens has a 17.19% improved focal efficiency, compared to the conventional dielectric binary phase lens. This comparison demonstrates the effectiveness of using the EMT method in designing the ring-shaped nanoslit lens. As expected, the EMT lens we propose shows better efficiency, because the binary phase lens approximately modulates the target phase with only two-phase values (in-phase and inverse phase), but the EMT lens modulates the incident light with the continuous phase.
Table 1 Comparison of single-focal lenses
Feature | EMT Lens | Binary Phase Lens |
---|---|---|
Efficiency (%) | 23.31 | 19.89 |
Beam Spot Size (r-axis) (μm) | 0.44 | 0.46 |
Beam Spot Size (z-axis) (μm) | 1.39 | 1.4 |
We also test our methods on another lens, a dual-focal lens [32, 33]. To design the dual-focal lens, the VPSM is used again. To calculate the phase at the lens plane, an artificial source is assumed at focal lengths of 5 and 10 μm, using an inversive propagation method. The process of designing the lens follows the same steps as for the single-focal lens, where the phase of the virtual point is calculated using VPSM and varied to optimize the efficiency. The phase is then used to determine the duty ratio of each annular slit in the lens. The design of the dual-focal lens is then filled according to the corresponding duty ratio. By varying the phase of the virtual source at two different focal positions (hereafter
The results of the numerical calculation show that the designed dual-focal lens is capable of focusing light at two different focal positions
When it comes to a dual-focal lens, there are various considerations when selecting the optimized design. The performance measure that is most important can vary based on the application, with some requiring the overall efficiency to be the top priority, while others prioritize minimizing sublobes.
Out of the many factors to consider, we look at five specific criteria: Maximizing efficiency at either focal point
For the maximum efficiency at the
Table 2 Efficiencies of a dual-focal lens under optimized conditions
Criteria (Phase of Virtual Sources) | Sublobe Efficiency (%) | ||
---|---|---|---|
11.36 | 7.55 | 0.86 | |
7.03 | 10.42 | 0.05 | |
Lobe Min (60°, 30°) | 7.71 | 9.67 | 0.002 |
Minimum Difference (110°, 335°) | 7.60 | 7.60 | 1.15 |
Max Average (220°, 130°) | 10.74 | 8.33 | 0.36 |
Out of the five criteria, the max average case demonstrates good efficiency at both focal points and low sublobe intensity. The electric field distribution for the max average case can be seen in Fig. 5. We compare the EMT lens to a conventional dielectric binary dual focal lens, and find that the efficiency at the
Table 3 Comparison of results for dual-focal lenses
Feature | EMT Lens | Binary Phase Lens |
---|---|---|
10.74 | 7.37 | |
8.33 | 7.19 | |
Beam Size: | 0.44 | 0.44 |
Beam Size: | 1.35 | 1.5 |
Beam Size: | 1.38 | 1.16 |
Beam Size: | 2.41 | 2.13 |
In this paper, utilizing the EMT we have presented a binary lens made up of annular nanoslits that operates as a continuous-phase zone plate. This lens can be used as an alternative to the difficult-to-manufacture continuous Fresnel zone plate. Based on our EMT-based continuous-phase design, single-focal and dual-focal lenses were designed and evaluated through numerical simulations. The results showed that the binary lens using EMT for a single focal point had an efficiency improvement of 17.19%. For the dual-focal case, the efficiencies increased by 13.11% and 49.41% at each focal point, while also exhibiting relatively low sublobe intensity. Our proposed design principle can be effectively applied in various fields, as it utilizes the same fabrication process as conventional binary lenses, and can also be used for various optical elements that require radially polarized light and axially symmetric geometry.
The authors declare no conflicts of interest.
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.
National Research Foundation of Korea (NRF) (2021R1F1A1052193, 2022R1F1A1062278); 2023 Hongik University Research Fund.
Curr. Opt. Photon. 2023; 7(3): 304-309
Published online June 25, 2023 https://doi.org/10.3807/COPP.2023.7.3.304
Copyright © Optical Society of Korea.
Woongbu Na^{1}, Seung-Yeol Lee^{2}, Hyuntai Kim^{1}
^{1}Electrical and Electronic Convergence Department, Hongik University, Sejong 30016, Korea
^{2}School of Electrical and Electronic Engineering, Kyungpook National University, Daegu 41566, Korea
Correspondence to:^{*}hyuntai@hongik.ac.kr, ORCID 0000-0001-7401-3320
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this study, a binary dielectric annular nanoring lens is proposed to cover the full range of optical phase. The lens is designed numerically, based on the effective-medium theory. The performance of the proposed lens is verified for the cases of single-focal and dual-focal lenses. The efficiency of a singlefocal lens is improved by 17.19% compared to a binary dielectric lens, and that of a dual-focal lens shows enhancements of 13.11% and 49.41% at the two focal points. This lens design can be applied to other optical components with axially symmetric structures.
Keywords: Effective medium theory, Lens system design, Nanostructures, Subwavelength structures
Optical focusing has been utilized for numerous applications since ancient times. Radially polarized light is particularly useful in focusing, as all radial components disappear and only the longitudinal component is focused, resulting in a small and symmetrical beam size that is advantageous for machining, sensing, and trapping [1–3].
Different types of zone plates have been developed to reduce the thickness of optical components [4–10]. Block-type zone plates such as metallic zone plates are the easiest to fabricate but have the lowest efficiency, as light in the blocked area is not focused [11–14]. Dielectric zone plates have relatively higher efficiency compared to block-type zone plates, but they are sensitive to thickness and cannot act like bulk dielectric lenses, as the phase information is binary [15–18]. A Fresnel lens with continuous correspondence between its height and each phase is more efficient than other binary lenses. However, its fabrication process is challenging because the height must be adjusted for each position [19, 20].
Recently, various phase-controlled metasurface lenses have been researched [21–23]. However, most of their unit cells are square or hexagonal, making it challenging to match the axial symmetry accurately.
In this study, we propose a binary annular nanoslit array lens based on the effective-medium theory (EMT), which corresponds to a continuous phase structure [24–26]. In this research, the first step is to evaluate the characteristics of the annular dielectric nanoslits. Then a complete set of phase-to-structure data is established. To demonstrate the feasibility of the concept, we design and analyze both a single-focal and a dual-focal lens.
EMT provides the effective refractive index of an axially symmetric structure for radially polarized incident light. Equation (1) shows the effective refractive index of a ring-shaped nanoslit, based on the duty-ratio [24–26]:
where
Numerical calculations are performed to match the phase data to the duty-ratio values. The incident light’s wavelength is set to 650 nm, a common red wavelength, and the refractive index of silicon is set to 3.8350 [27, 28]. The period of the structure is set to 100 nm.
The duty ratio is varied for different dielectric thicknesses from 300 to 400 nm. Note that the phase difference between the vacuum and silicon should be 2
The results show that the highest average transmission efficiency is achieved at a thickness of 330 nm. This value is then used as the thickness for further analysis, to create a database linking the phase to the duty ratio. The full data set, linking duty ratio to phase, is selected based on the cycle with the highest average transmission. The established phase-to-structure (duty ratio) data are presented in Fig. 1(c). Note that the duty ratio varies from 0.01 to 0.99, in steps of 0.001. Therefore, the minimal unit size becomes 1 nm, and the resolution of the structure is set to 0.1 nm. However, the value can be adjusted by changing the duty-ratio steps, and also selecting the middle of the range, such as 0.1 to 0.9.
First, we design a single-focal lens to verify our method. To design a lens, we use the virtual-point-source method (VPSM) [29–31], which is an inverse-design technique for optical components. Using the calculated phase from the virtual source as a reference, the duty ratio of each nanoring is obtained from the prepared data. The design schematic, calculated phase, and corresponding binary lens design are shown in Fig. 2.
To design the lens, the VPSM is used. The position of the focal length is selected as 5 μm, and the radius of the lens is set to 10 μm. The phase at the virtual single point is calculated using VPSM, and is varied to optimize the efficiency.
After designing the lens using the VPSM, numerical simulations are performed using COMSOL Multiphysics. The electric field intensity is calculated at the longitudinal axis (
The proposed lens designed using the EMT is compared to a conventional dielectric binary phase lens (also known as a Fresnel zone plate), which considers only positive and negative phases, to evaluate its performance. Note that the binary phase lens has also designed based on VPSM, and the identical tool has been applied for numerical calculations. Table 1 shows the comparison of efficiency and beam spot size. The results show that the EMT lens has a 17.19% improved focal efficiency, compared to the conventional dielectric binary phase lens. This comparison demonstrates the effectiveness of using the EMT method in designing the ring-shaped nanoslit lens. As expected, the EMT lens we propose shows better efficiency, because the binary phase lens approximately modulates the target phase with only two-phase values (in-phase and inverse phase), but the EMT lens modulates the incident light with the continuous phase.
Table 1 . Comparison of single-focal lenses.
Feature | EMT Lens | Binary Phase Lens |
---|---|---|
Efficiency (%) | 23.31 | 19.89 |
Beam Spot Size (r-axis) (μm) | 0.44 | 0.46 |
Beam Spot Size (z-axis) (μm) | 1.39 | 1.4 |
We also test our methods on another lens, a dual-focal lens [32, 33]. To design the dual-focal lens, the VPSM is used again. To calculate the phase at the lens plane, an artificial source is assumed at focal lengths of 5 and 10 μm, using an inversive propagation method. The process of designing the lens follows the same steps as for the single-focal lens, where the phase of the virtual point is calculated using VPSM and varied to optimize the efficiency. The phase is then used to determine the duty ratio of each annular slit in the lens. The design of the dual-focal lens is then filled according to the corresponding duty ratio. By varying the phase of the virtual source at two different focal positions (hereafter
The results of the numerical calculation show that the designed dual-focal lens is capable of focusing light at two different focal positions
When it comes to a dual-focal lens, there are various considerations when selecting the optimized design. The performance measure that is most important can vary based on the application, with some requiring the overall efficiency to be the top priority, while others prioritize minimizing sublobes.
Out of the many factors to consider, we look at five specific criteria: Maximizing efficiency at either focal point
For the maximum efficiency at the
Table 2 . Efficiencies of a dual-focal lens under optimized conditions.
Criteria (Phase of Virtual Sources) | Sublobe Efficiency (%) | ||
---|---|---|---|
11.36 | 7.55 | 0.86 | |
7.03 | 10.42 | 0.05 | |
Lobe Min (60°, 30°) | 7.71 | 9.67 | 0.002 |
Minimum Difference (110°, 335°) | 7.60 | 7.60 | 1.15 |
Max Average (220°, 130°) | 10.74 | 8.33 | 0.36 |
Out of the five criteria, the max average case demonstrates good efficiency at both focal points and low sublobe intensity. The electric field distribution for the max average case can be seen in Fig. 5. We compare the EMT lens to a conventional dielectric binary dual focal lens, and find that the efficiency at the
Table 3 . Comparison of results for dual-focal lenses.
Feature | EMT Lens | Binary Phase Lens |
---|---|---|
10.74 | 7.37 | |
8.33 | 7.19 | |
Beam Size: | 0.44 | 0.44 |
Beam Size: | 1.35 | 1.5 |
Beam Size: | 1.38 | 1.16 |
Beam Size: | 2.41 | 2.13 |
In this paper, utilizing the EMT we have presented a binary lens made up of annular nanoslits that operates as a continuous-phase zone plate. This lens can be used as an alternative to the difficult-to-manufacture continuous Fresnel zone plate. Based on our EMT-based continuous-phase design, single-focal and dual-focal lenses were designed and evaluated through numerical simulations. The results showed that the binary lens using EMT for a single focal point had an efficiency improvement of 17.19%. For the dual-focal case, the efficiencies increased by 13.11% and 49.41% at each focal point, while also exhibiting relatively low sublobe intensity. Our proposed design principle can be effectively applied in various fields, as it utilizes the same fabrication process as conventional binary lenses, and can also be used for various optical elements that require radially polarized light and axially symmetric geometry.
The authors declare no conflicts of interest.
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.
National Research Foundation of Korea (NRF) (2021R1F1A1052193, 2022R1F1A1062278); 2023 Hongik University Research Fund.
Table 1 Comparison of single-focal lenses
Feature | EMT Lens | Binary Phase Lens |
---|---|---|
Efficiency (%) | 23.31 | 19.89 |
Beam Spot Size (r-axis) (μm) | 0.44 | 0.46 |
Beam Spot Size (z-axis) (μm) | 1.39 | 1.4 |
Table 2 Efficiencies of a dual-focal lens under optimized conditions
Criteria (Phase of Virtual Sources) | Sublobe Efficiency (%) | ||
---|---|---|---|
11.36 | 7.55 | 0.86 | |
7.03 | 10.42 | 0.05 | |
Lobe Min (60°, 30°) | 7.71 | 9.67 | 0.002 |
Minimum Difference (110°, 335°) | 7.60 | 7.60 | 1.15 |
Max Average (220°, 130°) | 10.74 | 8.33 | 0.36 |
Table 3 Comparison of results for dual-focal lenses
Feature | EMT Lens | Binary Phase Lens |
---|---|---|
10.74 | 7.37 | |
8.33 | 7.19 | |
Beam Size: | 0.44 | 0.44 |
Beam Size: | 1.35 | 1.5 |
Beam Size: | 1.38 | 1.16 |
Beam Size: | 2.41 | 2.13 |