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Abstract
Polarization and focal length are both critical optical parameters with many applications in many fields, such as optical communications and imaging. The development of metasurfaces provides a new realization of optical systems. In this paper, based on metasurfaces’ powerful electromagnetic modulation capability, we integrate polarization conversion with continuous zoom function and propose a dynamic polarization-regulated metasurface with variable focal length. It realizes the reversible conversion of polarization state, which can convert linearly polarized light into elliptically polarized light and circularly polarized light and convert circularly polarized light to linearly polarized light. At the same time, it achieves a 4.4× zoom range, with a constant focal length variation from 70 µm to 309 µm. The metasurface has the advantages of small size, easy integration, and reconfigurability, providing a new design idea for complex functional optical systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization is an important physical quantity to describe light and has a wide range of applications [1–3] in communication [1], encryption, imaging [2,3], display, remote sensing, and other fields. Conventional polarization-modulated optical systems tend to be complex and bulky. Changing the amplitude ratio and phase difference of the orthogonal components of the incident light allows for conversion between different polarization states. Waveplates based on birefringent crystals, including half-waveplates, quarter-waveplates, etc., are the conventional method of polarization conversion. The development of liquid crystals has allowed more application scenarios for polarization conversion, where dynamic regulation of the polarization state can be achieved by applying an applied electric field [4–6]. However, polarization characteristics are usually used to achieve multifocal. The focal length of an optical zoom system can be changed continuously or discretely within the allowable range. Compared with fixed focus lenses, zoom lenses have more muscular flexibility and adaptability and a wide range of application value in many fields, such as imaging detectors and security monitoring systems. Traditional zoom methods include traditional mechanical zoom [7,8], liquid zoom [9–11], etc. However, the existing zoom methods make it difficult to achieve miniaturization and integration due to the complexity of structure and process [7–11].
In recent years, metasurfaces with powerful electromagnetic wave modulation capability have been developed rapidly and have a wide range of applications in many fields, such as holographic imaging [12,13], beam multiplexer [14–16], structural color [17] and metalens [18–27]. Polarization-regulated metasurfaces [28–33] are popular research directions. Regarding polarization regulation, reflective polarization control metasurfaces [29–31] are proposed in different frequency bands. At the same time, the combination of polarization regulation and focusing also puts forward many applications, such as multi-foci metalens with polarization-rotated focal points [33], all-dielectric trifocal metalens [28], etc. However, polarization characteristics are usually used to achieve multifocal. Dynamically adjustable metasurfaces have been proposed by using the cross-integration of different areas and metasurfaces and realized many achievements, such as dynamic holography [12,13], dynamic beam shaping, and zoom metalens [34–39]. Ways to achieve a zoom metalens include the use of phase change materials [40–43], stretchable flexible material [44–46], and microelectromechanical technology [47], but they often have limited or discontinuous zoom ranges. A continuous zoom lens based on Mohr's principle was implemented, which has reversible continuous zoom characteristics [48]. However, metasurfaces that achieve polarization regulated and zoom control at the same time have hardly been studied.
In this paper, aiming at the two critical optical parameters of polarization and focal length, a dynamic polarization-regulated metasurface with variable focal length is proposed based on the structure of a phase modulation unit with waveplate function and the principle of the Mohr effect. Two all-dielectric metasurface structures with specific phase profiles, metasurface 1 (M1) and metasurface 2 (M2), are constructed. M1 plays the polarization conversion function and realizes the conversion of linearly polarized light to linearly polarized light, circularly polarized light, and elliptically polarized light by changing the polarization angle of linearly polarized light. At the same time, it can also realize the conversion of circularly polarized light to linearly polarized light. The focusing function with reversible variable focal length is realized along with polarization conversion. By changing the relative rotation angles of M2, a continuously reconfigurable zoom metasurface is realized at 10.6 µm. It achieves a 4.4× zoom range, with a constant focal length variation from 70 µm to 309 µm and the numerical aperture from 0.82 to 0.31. The transmittance of the dynamic polarization-regulated metasurface is stable and remains around 0.51 when polarization modulation and focal length change are achieved. Simultaneous modulation of polarization state and focal length is achieved in this paper. The compact and non-cumbersome system provides a new design idea for optical systems.
2. Materials and methods
As shown in Fig. 1(a), the infrared dynamic polarization-regulated metasurface with variable focal length consists of two metasurfaces with corresponding phase profiles, both centered on and perpendicular to the optical axis. They are kept at constant intervals, and the modulation of the polarization state and focal length of the lens is achieved by rotating the relative angles. The M1 realizes the polarization conversion of light. Polarization is one of the fundamental properties of light, and the conversion of polarization states is usually achieved by using the phase delay accumulated in birefringent crystals during light propagation. We realize the polarization conversion function by selecting unit structures with a fixed phase difference (π/2) in the vertical vibration direction. For the unit structure mentioned above, the Jones matrix can be expressed as
when the incident light is linearly polarized and has an amplitude of 1, it can be expressed asAccording to Eq. (3) above, we can find that the outgoing light is linearly polarized, circularly polarized, or elliptically polarized when θp is taken at different angles.
Fig. 1. (a) Schematic of dynamic polarization-regulated metasurface with variable focal length. (b) The phase profile of the M1, φ1 and the phase profile of the M2, φ2, and the corresponding total phase distribution φ, when the rotation angle θr changes.
The rotation of the M2 enables a continuous change in focal length. The following formula can represent the phase profiles on the two metasurfaces:
where r is the coordinate value on the metasurface, θ0 is the azimuth angle at the corresponding position, θr is the angle of rotation of the second metasurface around the optical axis, and a is constant. The following formula can express the total phase profile:From the above Eq. (6), it can be seen that the total phase profile is constantly changing with the change of rotation angle θr.
The following Eq. (7) can express the phase profile of a conventional lens:
Comparing the total phase profile of a zoom metalens (Eq. (6)) with the phase profile of a conventional lens (Eq. (7)), the focal length of a zoom metalens can be approximated by the following equation:
Therefore, the relationship between the rotation angle θr and the focal length f can be obtained. Figure 1(b) shows the phase profile of the M1, the phase profile of the M2, and the corresponding total phase profile when θr changes. When the rotation angle θr is 0, each position on the total phase profile is 0. That is, there is no convergence effect on the incident light. As θr gradually increases, the denser the period in the phase profile, the stronger the convergence effect of the metalens, and the smaller the focal length.
Based on the above design principles, two set structures of the phase control unit with high transmittance are designed at 10.6 µm. A high-transmittance dielectric silicon column with a high refractive index is used as the structure, and high-transmittance dielectric CaF2 is used as the structural substrate. This paper proposes two structures both with a period of 5 µm and a height of 5 µm. M1 utilizes a rectangular column as a unit structure to achieve a phase delay covering 0-2π in the x-direction while ensuring that the phase delay difference in the vertical directions of x and y is kept at about π/2. As shown in Fig. 2(a), we chose eight rectangular columns with different lengths and widths at the same height to realize the phase modulation (the 1st and 9th ones in the figure have the same structure). The array arrangement of the M1 is shown in Fig. 2(b). M2 utilizes cylinders with polarization-independent properties as unit structures. Adjusting the silicon cylinder's radius achieves phase regulation in the 2π range while maintaining a high transmittance, as shown in Fig. 2(c). The array arrangement of the M2 is shown in Fig. 2(d).
Fig. 2. (a-b) Phase control unit structure with a stable phase difference in the vertical direction and M1 composed of it. (Corresponding structure size: L1 = 2.5µm and W1 = 1.6µm, L2 = 2.6µm and W2 = 2.05µm, L3 = 2.8µm and W3 = 2.15µm, L4 = 3.2µm and W4 = 2.2µm, L5 = 3.55µm and W5 = 2.3µm, L6 = 3.7µm and W6 = 2.55µm, L7 = 3µm and W7 = 1µm and L8 = 2.5µm and W8 = 1.6µm) (c-d) Polarization-independent phase control unit structure and M2 composed of it.
3. Results
3.1 Polarization-regulated metasurface
First, we simulate and verify the polarization conversion characteristics of the mirror group. To ensure uniformity and comparability of results, set the rotation angle of the second mirror to θr = 90° (that is when the minimum focal length is obtained). The simulation uses the time-domain finite-difference method, where the z-direction boundary condition is the PML boundary condition, and the x- and y-direction boundary conditions are periodic boundary conditions. When ${\theta _p} = m \cdot {\pi / 2}({m = 1,2,3 \ldots } )$, according to Eq. (3), the outgoing light is still linearly polarized light with the same vibration direction. When the direction of polarization of the incident light is parallel to the x-axis, i.e., θp = 0°, the energy distribution on the focal plane is shown in Fig. 3(a). The |Ex|2 component has a prominent focused spot, and the |Ey|2 component has low energy and no prominent spot. This indicates that the outgoing light is linearly polarized, and the vibration direction is parallel to the x-axis. When the direction of polarization of the incident light is perpendicular to the x-axis, i.e., θp = 90°, the |Ex|2 component has low energy. The |Ey|2 component has a prominent focused spot, as shown in Fig. 3(b). This indicates that the outgoing light is linearly polarized, and the vibration direction is perpendicular to the x-axis.
Fig. 3. (a-b) The energy distribution on the focal plane when the direction of polarization of the incident light is parallel (perpendicular) to the x-axis, i.e., θp = 0° (90°).
When ${\theta _p} = m \cdot \pi \pm {\pi / 4}({m = 1,2,3 \ldots } )$, according to Eq. (3), the outgoing light is circularly polarized. As shown in Fig. 4(b), when the polarization angle of the incident light is 45°, there are apparent focuses on the |Ex|2 component and the |Ey|2 component of energy, and the energy intensities are similar. This indicates that the emitted light is circularly polarized. And when θp is any angle other than the above particular angle, the outgoing light is elliptically polarized. As in Fig. 4(a) and 4(c), the energy distribution is given for the polarization angles of incident light of 30° and 60°, respectively. When θp is 30°, it can be observed that there is a clear focusing spot for both the |Ex|2 component and the |Ey|2 component, but the |Ex|2 component has a higher energy intensity. In contrast, when θp is 60°, the |Ey|2 component has a higher energy intensity.
Fig. 4. (a-c) The intension distribution of the Ex, Ey, ERCP, and ELCP components on the focal plane when the polarization angle of incident light is 30°, 45°, and 60°. (The normalized intensity maxima: |ERCP|230°=0.95, |ELCP|230°=0.086, |ERCP|245°=1, |ELCP|245°=0.069, |ERCP|260°=0.91, |ELCP|260°=0.093.)
To further demonstrate that the outgoing light is circularly polarized or elliptically polarized when θp is 30°, 45°, and 60°, respectively, the energy distributions of the right-handed circularly polarized component and the left-handed circularly polarized component of the outgoing light are calculated based on Ex and Ey for each of the three cases, as shown in Fig. 4. According to the equation for the Stoke matrix [49,50]:
where $\chi$ is the ellipticity angle. And according to Eq. (3), it is known that $\chi$=θp. Thus, the relationship between the left-handed and right-handed circularly polarized components and θp is given by the following equation: when ${\theta _p} = m \cdot \pi \pm {\pi / 4}({m = 1,2,3 \ldots } )$, S3 = 1, this can be achieved when the left-handed polarization component is approximately 0. When θp is 30° (60°), the ratio of the right-handed component's intensity to the left-handed component's intensity is about 13.9 (IRCP/ILCP≈13.9). As shown in Fig. 4, it can be observed that the right-handed component has a more pronounced focused spot with higher intensity in all three cases. The energy of the left-handed component is relatively low and is lowest when θp is 45°. Combined with the energy distribution of Ex and Ey, it can be seen that when θp is 45°, the outgoing light is right-handed circularly polarized light. When θp is 30° or 60°, the energy of the left-handed component is also relatively low, but it is higher than that of the left-handed component when θp is 45°. Combined with the energy distribution of Ex and Ey, it can be seen that when θp is 30° or 60°, the outgoing light is right-handed elliptically polarized but with different major and minor axis distributions.In the above, we analyzed the incidence of linearly polarized light, and we continue to analyze the incidence of circularly polarized light. (To facilitate the analysis of the polarization state of the outgoing light, the metasurface set was rotated by 45° as a whole.) For a unitary structure with a fixed phase difference π/2 in the vertical vibration direction and an azimuthal angle of 45°, the Jones matrix can be expressed as:
when the incident light is circularly polarized, and the amplitude is 1, right-handed and left-handed circularly polarized light can be expressed as:Then, when circularly polarized light passes through the metasurface, the outgoing light can be expressed as:
The two metasurfaces are rotated by 45° simultaneously to polarize the outgoing light in the x-z (y-z) plane, as shown in Fig. 5 (distinguished from Fig. 1(a)). According to Eq. (13), the outgoing light is linearly polarized light that vibrates perpendicular to the x direction when the incident light is right-handed circularly polarized light.
Fig. 5. Schematic of dynamic polarization-regulated metasurface when the two metasurfaces are rotated by 45° simultaneously.
As shown in Fig. 6(a), the |Ey|2 component is clearly focused, and |Ex|2 component has low energy and no visible spots, which indicates that the outgoing light is linearly polarized and the vibration direction is perpendicular to the x-axis. On the contrary, when the incident light is left-handed circularly polarized, the outgoing light is linearly polarized, and the vibration direction is parallel to the x-axis according to the energy distribution in Fig. 6(b).
Fig. 6. (a-b) The energy distribution in the vertical direction on the focal plane when the incident light is right-handed/ left-handed circularly polarized light.
3.2 Continuous zoom characteristics
Next, we simulate and verify the continuous zoom characteristics of the mirror set. To ensure the uniformity and comparability of the results, the polarization state of the incident light is set to be linearly polarized, and the polarization angle is θp = 0° (parallel to the x-axis). The focusing situation of the x-z plane is shown in Fig. 7(a), and the energy intensity curve along the z-axis is shown in Fig. 7(b) when the rotation angle θr of the M2 is 20°-90° (in steps of 10°). Among them, we can observe that when θr decreases gradually from 90°, the position of the focus becomes distant, and the depth of focus tends to become longer as the focal length increases.
Fig. 7. (a) The normalized energy intensity distribution in the x-z plane for different θr. (b) The normalized energy intensity distribution along the z-axis for different θr.
Further analyzing the simulation results, we collate the energy distribution on the corresponding focal plane (Fig. 8(a)) and the energy intensity curve along the x-axis (Fig. 8(b)) at different rotation angles θr. Observing the focal plane energy distribution in Fig. 8(a), it can be found that with the increase of the θr, there are apparent energy concentrations on the focal plane. The focal spot shape is neat, and the focal spot area tends to decrease. A minimum focal length of 70 µm was obtained for a θr of 90°, and a maximum focal length of 309 µm was obtained for a θr of 20°. It achieves a 4.4× zoom range and the numerical aperture from 0.82 to 0.31.
Fig. 8. (a) Focal spot distribution in the focal plane at different θr. (b) Intensity distribution curve on the x-axis at different θr.
To further analyze the focal spot situation, the normalized energy intensity profiles along the x-axis on the focal plane for different θr were compiled (Fig. 8(b)). We can observe that when the θr is 90°, 80°, 70°, and 60°, the position with the highest energy intensity is located at the center (x = 0). However, when the θr continues to decrease, the most robust energy position of the focal spot shifts slightly in the x direction, and there is a tendency for the focal spot to become larger. According to the Airy spot calculation formula, the focused spot size is proportional to the focal length when other parameters are constant. This is the same trend as the simulation results.
The full width at half maximum (FWHM) and focusing efficiencies during the zooming process are shown in Table 1. The FWHM is obtained from the intensity profile of the focal plane along the x-axis. It can be observed that it increases as the focal length increases. When θr is 90°, the focal length is the smallest, and the minimum FWHM of 8.40 µm is obtained, and when θr is 20°, the focal length is the largest, and the maximum FWHM of 24.70 µm is obtained. The focusing efficiency is obtained from the intensity integral in the area three times the diameter of the FWHM when θr is 90° on the focal plane compared to the total intensity integral on the focal plane. It can be observed that the focusing efficiency shows a decreasing tendency with the increase of focal length. When the focal length is the maximum, the efficiency is lowest at 0.622.
Table 1. The FWHM and focusing efficiencies during the zooming process
4. Conclusion
This paper proposes a dynamic polarization-regulated metasurface with variable focal length. A quarter-waveplate-like function is realized by providing a stable phase difference of π/2 in the vertical direction, and linearly-polarized light can be converted to linearly polarized, circularly polarized, and elliptically polarized light. At the same time, circularly polarized light can be converted into corresponding linearly polarized light. While polarization conversion is achieved, a 4.4× zoom is realized by changing the relative rotation angle of the two metasurfaces. The focal length varies continuously in the range of 70-309 µm, and the numerical aperture changes in the range of 0.82-0.31. The metasurface proposed in this paper realizes reversible dynamic adjustment of polarization state and focal length and has more robust flexibility, adaptability, and remodeling. It has broad application value in many fields, such as imaging detectors, security monitoring systems, etc.
Funding
Distinguished Young Scholars of Jilin Province (20230101351JC); Scientific and Technological Development Project of Jilin Province (20220201080GX); National Key Research and Development Program of China (2023YFB3610200); National Natural Science Foundation of China (61735018); Excellent Member of Youth Innovation Promotion Association of the Chinese Academy of Sciences (Y201836); Leading Talents and Team Project of Scientific and Technological Innovation for Young and Middle-aged Groups in Jilin Province (20190101012JH).
Disclosures
The authors declare no competing interests.
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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