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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.

Continuous-phase Lens Design via Binary Dielectric Annular Nanoslits

Woongbu Na1, Seung-Yeol Lee2, Hyuntai Kim1

1Electrical and Electronic Convergence Department, Hongik University, Sejong 30016, Korea
2School of Electrical and Electronic Engineering, Kyungpook National University, Daegu 41566, Korea

Corresponding author: *hyuntai@hongik.ac.kr, ORCID 0000-0001-7401-3320

Received: February 10, 2023; Revised: April 10, 2023; Accepted: April 12, 2023

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 [13].

Different types of zone plates have been developed to reduce the thickness of optical components [410]. 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 [1114]. 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 [1518]. 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 [2123]. 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 [2426]. 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 [2426]:

neff=1xnva2+1xnma2,

where x is the duty ratio of a periodic slit, neff is the effective refractive index, and nva and nma are the refractive indices of the vacuum and material respectively. Based on equation Eq. (1), the effective refractive index in terms of duty ratio is shown in Fig. 1(a).

Figure 1.Numerical calculation results of nanorings. (a) Effective refractive index, in terms of duty ratio, (b) average efficiency for one cycle, in terms of dielectric thickness, and (c) established full phase-to-duty-ratio data.

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 2π to cover one cycle of phase. To find the optimized thickness, the average transmission efficiency is calculated within a single cycle, and the results are presented in Fig. 1(b).

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.

3.1. Single-focal Lens

First, we design a single-focal lens to verify our method. To design a lens, we use the virtual-point-source method (VPSM) [2931], 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.

Figure 2.Virtual-point-source method (VPSM) schematic for a single focus.

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 (r = 0 μm) and the focal plane (z = 5 μm), shown in Figs. 3(a) and 3(b) respectively. In Figs. 3(a) and 3(b) the phase represents that of the virtual source. Figure 3(a) shows the intensity along the longitudinal axis, which is from point (r = 0 μm, z = 1 μm) to point (r = 0 μm, z = 8 μm). Figure 3(b) shows the intensity along the focal plane, which is from point (r = 0 μm, z = 5 μm) to point (r = 5 μm, z = 5 μm). The simulations show that the electric field is focused on the target focal length. The focal efficiency is optimized when the phase of the virtual source is 121 degrees, with a focal efficiency of 23.31%, a full-width at half maximum (FWHM) beam size of 0.44 μm for the r axis (beam waist), and 1.39 μm for the z-axis (depth of focus). The electric field pattern for the optimized case is shown in Fig. 3(c).

Figure 3.Normalized electric field distribution along (a) the longitudinal axis and (b) the focal plane, in terms of phase of the virtual source. (c) Normalized electric field distribution for the optimized case.

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

FeatureEMT LensBinary Phase Lens
Efficiency (%)23.3119.89
Beam Spot Size (r-axis) (μm)0.440.46
Beam Spot Size (z-axis) (μm)1.391.4


3.2. Dual-focal Lens

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 f1 for the 5-μm position and f2 for the 10-μm position), we calculate the efficiency at f1, f2, and the sublobe position, which is 2 μm above the lens. The results are shown in Fig. 4.

Figure 4.Efficiency at the (a) f1 position, (b) f2 position, and (c) sublobe position, in terms of phases of two virtual sources.

The results of the numerical calculation show that the designed dual-focal lens is capable of focusing light at two different focal positions f1 and f2. The calculated focal efficiency at both focal positions is at least 6%, demonstrating the lens’s ability to produce focused light. However, the efficiency distribution at the sublobe position (2 μm) shows that the efficiency is high for certain phases, indicating that it is possible to design the lens to minimize the sublobe intensity. These results can be used to further optimize the lens design and improve its performance for specific applications.

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 f1 or f2, reducing the sublobe efficiency, limiting the discrepancy between the two focal points, and achieving the highest average efficiency overall.

For the maximum efficiency at the f1 position, the virtual-source phases at f1 and f2 are determined to be 270° and 155° respectively, leading to an efficiency of 11.36% at the f1 position. To optimize efficiency at f2, the virtual-source phases are calculated to be 20° and 5°, resulting in an efficiency of 10.42% at the f2 position. The sublobe efficiency is minimized when the phases are 60° and 30°, yielding a sublobe efficiency of 0.002%. The difference between the two focal points is minimized when the virtual-source phases are 110° and 335°, resulting in an efficiency of 7.6% at both positions. The case that is optimized based on the average intensity of the two positions has virtual source phases of 220° and 130°, leading to efficiencies of 10.74% at f1 and 8.33% at f2. These five optimized cases, each based on different criteria, are presented in Table 2.

Table 2 Efficiencies of a dual-focal lens under optimized conditions

Criteria (Phase of Virtual Sources)f1 Efficiency (%)f2 Efficiency (%)Sublobe Efficiency (%)
f1 Max (270°, 155°)11.367.550.86
f2 Max (20°, 5°)7.0310.420.05
Lobe Min (60°, 30°)7.719.670.002
Minimum Difference (110°, 335°)7.607.601.15
Max Average (220°, 130°)10.748.330.36


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 f1 position is 13.11% higher and the efficiency at the f2 position is improved by 49.41%. The results are presented in Table 3.

Table 3 Comparison of results for dual-focal lenses

FeatureEMT LensBinary Phase Lens
f1 Efficiency (%)10.747.37
f2 Efficiency (%)8.337.19
Beam Size: f1 (r-axis) (μm)0.440.44
Beam Size: f2 (z-axis) (μm)1.351.5
Beam Size: f1 (r-axis) (μm)1.381.16
Beam Size: f2 (z-axis) (μm)2.412.13


Figure 5.Normalized electric field distribution for the proposed dual-focal lens.

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.

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.

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Article

Research Paper

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.

Continuous-phase Lens Design via Binary Dielectric Annular Nanoslits

Woongbu Na1, Seung-Yeol Lee2, Hyuntai Kim1

1Electrical and Electronic Convergence Department, Hongik University, Sejong 30016, Korea
2School of Electrical and Electronic Engineering, Kyungpook National University, Daegu 41566, Korea

Correspondence to:*hyuntai@hongik.ac.kr, ORCID 0000-0001-7401-3320

Received: February 10, 2023; Revised: April 10, 2023; Accepted: April 12, 2023

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.

Abstract

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

I. INTRODUCTION

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 [13].

Different types of zone plates have been developed to reduce the thickness of optical components [410]. 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 [1114]. 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 [1518]. 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 [2123]. 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 [2426]. 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.

II. Effective-medium design

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 [2426]:

neff=1xnva2+1xnma2,

where x is the duty ratio of a periodic slit, neff is the effective refractive index, and nva and nma are the refractive indices of the vacuum and material respectively. Based on equation Eq. (1), the effective refractive index in terms of duty ratio is shown in Fig. 1(a).

Figure 1. Numerical calculation results of nanorings. (a) Effective refractive index, in terms of duty ratio, (b) average efficiency for one cycle, in terms of dielectric thickness, and (c) established full phase-to-duty-ratio data.

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 2π to cover one cycle of phase. To find the optimized thickness, the average transmission efficiency is calculated within a single cycle, and the results are presented in Fig. 1(b).

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.

III. Lens design

3.1. Single-focal Lens

First, we design a single-focal lens to verify our method. To design a lens, we use the virtual-point-source method (VPSM) [2931], 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.

Figure 2. Virtual-point-source method (VPSM) schematic for a single focus.

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 (r = 0 μm) and the focal plane (z = 5 μm), shown in Figs. 3(a) and 3(b) respectively. In Figs. 3(a) and 3(b) the phase represents that of the virtual source. Figure 3(a) shows the intensity along the longitudinal axis, which is from point (r = 0 μm, z = 1 μm) to point (r = 0 μm, z = 8 μm). Figure 3(b) shows the intensity along the focal plane, which is from point (r = 0 μm, z = 5 μm) to point (r = 5 μm, z = 5 μm). The simulations show that the electric field is focused on the target focal length. The focal efficiency is optimized when the phase of the virtual source is 121 degrees, with a focal efficiency of 23.31%, a full-width at half maximum (FWHM) beam size of 0.44 μm for the r axis (beam waist), and 1.39 μm for the z-axis (depth of focus). The electric field pattern for the optimized case is shown in Fig. 3(c).

Figure 3. Normalized electric field distribution along (a) the longitudinal axis and (b) the focal plane, in terms of phase of the virtual source. (c) Normalized electric field distribution for the optimized case.

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.

FeatureEMT LensBinary Phase Lens
Efficiency (%)23.3119.89
Beam Spot Size (r-axis) (μm)0.440.46
Beam Spot Size (z-axis) (μm)1.391.4


3.2. Dual-focal Lens

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 f1 for the 5-μm position and f2 for the 10-μm position), we calculate the efficiency at f1, f2, and the sublobe position, which is 2 μm above the lens. The results are shown in Fig. 4.

Figure 4. Efficiency at the (a) f1 position, (b) f2 position, and (c) sublobe position, in terms of phases of two virtual sources.

The results of the numerical calculation show that the designed dual-focal lens is capable of focusing light at two different focal positions f1 and f2. The calculated focal efficiency at both focal positions is at least 6%, demonstrating the lens’s ability to produce focused light. However, the efficiency distribution at the sublobe position (2 μm) shows that the efficiency is high for certain phases, indicating that it is possible to design the lens to minimize the sublobe intensity. These results can be used to further optimize the lens design and improve its performance for specific applications.

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 f1 or f2, reducing the sublobe efficiency, limiting the discrepancy between the two focal points, and achieving the highest average efficiency overall.

For the maximum efficiency at the f1 position, the virtual-source phases at f1 and f2 are determined to be 270° and 155° respectively, leading to an efficiency of 11.36% at the f1 position. To optimize efficiency at f2, the virtual-source phases are calculated to be 20° and 5°, resulting in an efficiency of 10.42% at the f2 position. The sublobe efficiency is minimized when the phases are 60° and 30°, yielding a sublobe efficiency of 0.002%. The difference between the two focal points is minimized when the virtual-source phases are 110° and 335°, resulting in an efficiency of 7.6% at both positions. The case that is optimized based on the average intensity of the two positions has virtual source phases of 220° and 130°, leading to efficiencies of 10.74% at f1 and 8.33% at f2. These five optimized cases, each based on different criteria, are presented in Table 2.

Table 2 . Efficiencies of a dual-focal lens under optimized conditions.

Criteria (Phase of Virtual Sources)f1 Efficiency (%)f2 Efficiency (%)Sublobe Efficiency (%)
f1 Max (270°, 155°)11.367.550.86
f2 Max (20°, 5°)7.0310.420.05
Lobe Min (60°, 30°)7.719.670.002
Minimum Difference (110°, 335°)7.607.601.15
Max Average (220°, 130°)10.748.330.36


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 f1 position is 13.11% higher and the efficiency at the f2 position is improved by 49.41%. The results are presented in Table 3.

Table 3 . Comparison of results for dual-focal lenses.

FeatureEMT LensBinary Phase Lens
f1 Efficiency (%)10.747.37
f2 Efficiency (%)8.337.19
Beam Size: f1 (r-axis) (μm)0.440.44
Beam Size: f2 (z-axis) (μm)1.351.5
Beam Size: f1 (r-axis) (μm)1.381.16
Beam Size: f2 (z-axis) (μm)2.412.13


Figure 5. Normalized electric field distribution for the proposed dual-focal lens.

IV. Conclusions

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.

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.

FUNDING

National Research Foundation of Korea (NRF) (2021R1F1A1052193, 2022R1F1A1062278); 2023 Hongik University Research Fund.

Fig 1.

Figure 1.Numerical calculation results of nanorings. (a) Effective refractive index, in terms of duty ratio, (b) average efficiency for one cycle, in terms of dielectric thickness, and (c) established full phase-to-duty-ratio data.
Current Optics and Photonics 2023; 7: 304-309https://doi.org/10.3807/COPP.2023.7.3.304

Fig 2.

Figure 2.Virtual-point-source method (VPSM) schematic for a single focus.
Current Optics and Photonics 2023; 7: 304-309https://doi.org/10.3807/COPP.2023.7.3.304

Fig 3.

Figure 3.Normalized electric field distribution along (a) the longitudinal axis and (b) the focal plane, in terms of phase of the virtual source. (c) Normalized electric field distribution for the optimized case.
Current Optics and Photonics 2023; 7: 304-309https://doi.org/10.3807/COPP.2023.7.3.304

Fig 4.

Figure 4.Efficiency at the (a) f1 position, (b) f2 position, and (c) sublobe position, in terms of phases of two virtual sources.
Current Optics and Photonics 2023; 7: 304-309https://doi.org/10.3807/COPP.2023.7.3.304

Fig 5.

Figure 5.Normalized electric field distribution for the proposed dual-focal lens.
Current Optics and Photonics 2023; 7: 304-309https://doi.org/10.3807/COPP.2023.7.3.304

Table 1 Comparison of single-focal lenses

FeatureEMT LensBinary Phase Lens
Efficiency (%)23.3119.89
Beam Spot Size (r-axis) (μm)0.440.46
Beam Spot Size (z-axis) (μm)1.391.4

Table 2 Efficiencies of a dual-focal lens under optimized conditions

Criteria (Phase of Virtual Sources)f1 Efficiency (%)f2 Efficiency (%)Sublobe Efficiency (%)
f1 Max (270°, 155°)11.367.550.86
f2 Max (20°, 5°)7.0310.420.05
Lobe Min (60°, 30°)7.719.670.002
Minimum Difference (110°, 335°)7.607.601.15
Max Average (220°, 130°)10.748.330.36

Table 3 Comparison of results for dual-focal lenses

FeatureEMT LensBinary Phase Lens
f1 Efficiency (%)10.747.37
f2 Efficiency (%)8.337.19
Beam Size: f1 (r-axis) (μm)0.440.44
Beam Size: f2 (z-axis) (μm)1.351.5
Beam Size: f1 (r-axis) (μm)1.381.16
Beam Size: f2 (z-axis) (μm)2.412.13

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