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Abstract
Alvarez lenses, a kind of passive zoom lenses with reconfigurable focus, have been widely applied in optics but very few at lower frequencies such as in a microwave band, where the phase approximation for Alvarez lenses becomes inaccurate. In this article, we propose a design of a modified Alvarez lens with phase compensation for microwave, which consists of a pair of transmissive metasurfaces with high efficiency. The proposed metasurface consists of miniaturized units with the capability of full 2π phase modulation. We further analyze the phase distribution principle of the Alvarez lens and proposed a phase compensation scheme. The simulation results confirm that the proposed modified Alvarez lens has a very good dynamic focal length with theoretical expectation and can be continuously adjusted from 100 to 200 mm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the rapid development of microwave and millimeter-wave technologies, precise regulation of electromagnetic wavefronts has gradually become a popular and challenging research direction in electromagnetics [1–3]. One of the important ways to realize the control of electromagnetic wavefront is the use of electromagnetics lens. Metasurfaces are periodic structures composed of subwavelength resonant units, which can achieve unique capabilities for the manipulation of electromagnetic waves. It plays an important role in the fields of flat lens [4–8], beamforming [9,10], holographic imaging [11,12], and encrypted communication [13,14]. Different from the lens in traditional optical devices, metasurface lenses realize phase variation through different subwavelength units. Such highly efficient transmissive metasurfaces have been well studied, including Huygens metasurfaces [15,16], PB phase metasurfaces [9,17], polarization conversion metasurfaces [18,19], etc. Compared with traditional lenses, metasurface lenses have the advantages such as compact size, ease of fabrication, and integration.
The functionalities of most metasurface lenses cannot be changed after fabrication, lacking dynamical adjustability. Recent years, a variety of focus variable metasurface lenses have been proposed. The implementation methods mainly include: using multiple metasurfaces for phase combination [20–25], introducing physical deformation such as substrate stretching [26–30], controlling the phase response of materials by voltage bias [31–34] and so on. In 1967, Alvarez [35] proposed a zoom lens scheme, which offers a simple zoom method by simply manipulating the relative displacement of a pair of lens. Alvarez lenses have been gradually applied in the field of optics in recent years [36–39]. For example, Dutterer et al. [40] proposed an optical Alvarez lens fabricated by a single-crystal diamond milling process. Through ultra-precision multi-axis machining, single-crystal diamond milling can be used to fabricate these previously difficult freeform surfaces. Han et al. [41] proposed an optical Alvarez lens with integrated MEMS, which has the advantages of low power consumption and fast driving. At the same time, the entire lens is fully compatible with modern semiconductor manufacturing technology, making it possible to be mass-produced at lower cost. Martin et al. [42] proposed an improved Alvarez lens, instead of translational movement of the sub-lenses, it is mutually imaged by means of a 4f configuration, and the zoom effect can be achieved by rotating. The refocusing frequency can reach 1 kHz or higher. It is worth noting that all the above applications of Alvarez lenses are in optics, whiles no Alvarez lenses have been reported at microwave frequencies.
In this article, we proposed a focus-variable Alvarez lens working in microwave band. By analyzing the electromagnetic wave regulation principle of metasurfaces, metasurface units with 2π full phase regulation capability are proposed. Based on Alvarez lens principle, we realized a metasurface Alvarez lens in microwave band through proposed unit cells. It is found that the simulated focal length of this lens has a significant deviation from the theoretical calculation values. In view of this, we have analyzed the reasons for the deviation and made corresponding corrections to subsequent lens phase distribution by adding phase compensation, so that the simulated values can be well matched with the theoretical values, and thus accurate zooming can be effectively achieved. Our design could expand the application prospect of Alvarez lens in microwave band and provide a new idea for the design of microwave zoom lenses.
2. Unit cell design
Metasurface lens is essentially a discretized phase distribution function implemented by two-dimensional array of unit cells. The requirements for unit cells are as follows: full 2π phase control capability, as small unit period as possible, and as high transmission efficiency as possible. In order to obtain metasurface units that meet the above requirements, we firstly used CST Microwave Studio to simulate the two most basic metasurface units of metal patch type and metal ring type, both of which are composed of single-layer dielectric and double-layer metal patterns. The x-direction and y-direction are set as periodic boundary conditions, and the z-axis direction is set as Floquet boundary condition.
Figure 1(a) shows the structural diagram of the square ring type element, where P is the unit period length and w is the width of metal square ring. When the plane wave is incident vertically, the double-layer metal ring structure exhibits inductive characteristics. The thin dielectric layer in the middle can be equivalent to a short transmission line, and the entire circuit can be equivalent to that shown in Fig. 1(d). Figure 1(b) shows the structural diagram of the square patch type element, where P is the unit period length and s is the width of square metal patch. For a plane wave with normal incidence, the double-layer metal patch exhibits capacitive characteristics during the transmission process, and the thin dielectric layer in the middle can also be equivalent to a short transmission line. Therefore, the entire circuit can be equivalent to that shown in Fig. 1(e). The equivalent capacitance and equivalent inductance of these elements can be effectively controlled by adjusting their geometric sizes. The combination of these two structures can integrate the equivalent capacitance and equivalent inductance into a transmission network. Therefore, the two elements can be combined into a single meta-atom with a wide range of phase regulation while maintaining high transmittance. The structure is shown in Fig. 1(c) and its equivalent circuit is shown in Fig. 1(f).
Fig. 1. Schematic of (a) square ring element, (b) square patch element and (c) compound element; Equivalent circuit of (d) square ring element, (e) square patch element and (f) compound element.
In order to ensure that the unit cells could cover 2π full phase adjustment capability, the structure shown in Fig. 1(c) is cascaded. According to transmission array constraints [43], a metasurface unit capable of 2π phase transmission with -3 dB efficiency requires at least 3 layers. The metasurface optimized in this paper is composed of three layers of dielectric and four layers of metal, working at 15 GHz and the period P is 6 mm. The dielectric layer is made of Rogers 4350B with a thickness of 0.762 mm, and the metal layer is made of copper with a thickness of 0.018 mm, as shown in Fig. 2(a). By adjusting the parameters of the metal layer, 2π full-phase control of the transmitted wave can be achieved.
Fig. 2. (a) Schematic of phase gradient metasurface element (b) Simulated transmission magnitude (c) Simulated transmission phase.
When using metasurface units to form a lens, in order to ensure a good focusing effect, we hope that the transmittances of these units are as same as possible, and the requirements for phase gradient can be appropriately reduced. Therefore, in subsequent simulations, we actually use four cells with a phase gradient of 90° and basically the same transmittance. Figure 2(b) and Fig. 2(c) show the transmission amplitudes and phases of the four units, respectively. The transmittances of these four units are -0.81 dB, -1.08 dB, -1.36 dB, -0.89 dB, all around -1 dB, and the transmission phases are 120°, 30°, -60°, -150°. Table 1 shows the specific parameters of these four units, where W is the width of metal square ring, and S1, S2, S3, and S4 are the lengths of each metal patch.
Table 1. Unit parameters
3. Metasurface lens design
Alvarez lens is a zoom lens composed of two sub-lenses, and its zoom function is realized by controlling the relative displacement of the two sub-lenses. The phase distribution of the two sub-lenses is shown in Eq. (1) [35], where φ0 is a constant term and A is a parameter about the phase distribution.
Except for the constant term, their phases are opposite to each other. So, when the two sub-lenses are aligned, the overall phase of the system is a constant, and the propagation direction of transmission wave will not be affected. Assuming that the relative displacement of the two sub-lenses in the x direction is d, then the phase distribution of the two sub-lenses is shown as:
Figure 3(a) is a phase diagram of an Alvarez lens. Some areas that were originally aligned still remain overlapped in the direction of wave propagation after the movement, while some areas are no longer overlapped. The overlapping area of the two sub-lenses is the part that plays the role of focusing, and its equivalent phase expression is:
where -Ad(x2+y2) can represent a sphere lens with positive optical power. Its focal length is:It is seen from the above equation that the focal length f is inversely proportional to the displacement d. By adjusting the displacement d, the equivalent phase distribution of the entire system will be changed to achieve the zoom function.
To implement Alvarez lenses by metasurfaces, we need to discretize the phase distribution first. After discretization, the lens can be replaced by a metasurface array, and the required phase at specific position will be provided by a metasurface unit which has full phase regulation capability. The phase distribution after discretization is shown as:
There are two parameters that need to be considered when designing the lens: one is the variation range of displacement parameter d. According to Eq. (4), the focal length f is inversely proportional to displacement d. Therefore, expanding its variation range can effectively expand the zoom range. However, it can be seen from Fig. 3(a) that when the displacement parameter d is increased, the overlapping area that can focus incident wave will decrease. This condition limits the value range of d to a certain extent. Considering the actual situation, it is a suitable compromise to set the variation range of displacement parameter d to be one quarter to one half of the size of the metasurface. The parameter A is another key factor. In order to avoid large errors due to undersampling in the process of phase discretization, the relationship between the phase gradient and the size of metasurface unit must satisfy Eq. (6). Furthermore, the maximum value of parameter A is given by Eq. (7).
After determining the size of metasurface and the value of the parameter A, the discrete phase distribution can be obtained according to Eq. (5). Then, we can use the proposed metasurface units to realize the Alvarez lens. As shown in Fig. 3(b), the lens consists of upper and lower metasurfaces, where each metasurface consists of 60 × 40 cells, and the overall size is 360 mm × 240 mm. The variation range of displacement parameter d is set to be 90 mm∼180 mm, the focal length variation range is set to be 200 mm∼100 mm, and the working frequency is 15 GHz. Different from most optical zoom lenses in the visible waveband, the displacement amount and the focal length change amount is comparable in our design. Considering that the focal length is inversely proportional to the parameter A, the designer can increase the focal length to a certain extent by reducing the value of the parameter A when necessary.
Figure 4 shows the simulation results obtained with different displacements d. When the displacement parameter d is set to 90 mm, 120 mm, 150 mm, and 180 mm, the simulated focal length f is 157 mm, 118 mm, 95 mm, and 75 mm, respectively. Table 2 shows the difference between the simulated focal length and the theoretical focal length.
Table 2. Comparison of simulated focal length and theoretical focal length
As can be seen from the above table, the focal length obtained by the simulation has a large error compared with the theoretical focal length, which is basically between 20% and 30%. In the process of using metasurface to realize Alvarez lens, there are two main sources of systematic error: first, the error caused by undersampling when the continuous phase distribution is discretized in space domain; second, the process of using metasurface unit to realize discrete phase distribution is equivalent to phase value discretization with a certain gradient, which will also generate errors.
For the errors generated in phase distribution discretization process, when we determined the value of parameter A, we have avoided excessive errors due to undersampling by satisfying the conditions of Eq. (7). The error generated in the phase value discretization process can be quantified by the root mean square of the Wave Aberration Function (WAF). When WAFrms satisfies Eq. (8), the aberration caused by phase value discretization can be ignored [44].
Table 3 shows the ratio of WAFrms to the wavelength λ under different phase gradients. It can be seen that WAFrms under 90° phase gradient satisfies Eq. (8), that is, the phase value discretization will not produce large errors. Therefore, although the systematic errors caused by the phase distribution discretization process and phase value discretization process exist, it is difficult to use them to explain the 20%∼30% deviation between simulation results and theoretical values.
Table 3. Ratio of WAFrms to wavelength λ
4. Phase compensation scheme
To achieve beam focusing, the optical path from each point on lens to the focal point needs to be equal. The corresponding phase distribution is shown in Eq. (9), where f is the focal length, r is the distance from each point to the center, and φ0 is the phase of center point.
However, in Alvarez lens, we achieve a phase distribution as:
When the focal length f satisfies the condition of f ≫r:
It can be seen from the above equations that under the condition of f ≫r, φ and φAlvarez are approximately equal, and the focus of Alvarez lens satisfies the theoretical value given by Eq. (4). However, in the application we implemented above, the variation range of f is 100 mm∼200 mm, and the lens size is 360 mm × 240 mm, which does not satisfy this condition obviously. The difference between φ and φAlvarez is unneglectable. Phase distribution φ corresponds to the theoretical focus calculated by Eq. (4), and phase distribution φAlvarez corresponds to the focus that actually appears in the simulation, which explains the reason for the excessive error above.
Based on the above discussion, the traditional Alvarez lens design method is not applicable in the microwave band, which is manifested as a large error between the theoretical focal length and the actual focal length. The main reason is that there is a large gap between the equivalent phase of the system and the phase required to complete the theoretical zoom range. Therefore, in order to reduce this gap, we need to add a phase compensation term so that over the entire zoom range, the equivalent phase of the system is again consistent with the phase required to achieve the theoretical focal length. This is the difference of the design method between the metasurface Alvarez lens in the microwave band and that in the usually discussed visible or infrared band.
Concretely, we add a phase compensation term φα to the original phase distribution φAlvarez, so that the phase distribution after compensation satisfies φAlvarez +φα≈φ. The form of φα is shown in Eq. (13), where k1, k2, and k3 are undetermined parameters, and their values can be determined by minimizing the root mean square of φAlvarez+φα-φ.
It is worth noting that it can be seen from Eq. (9) and Eq. (13) that the calculations of φ and φα both involve the focal length f. It means that the calculation of phase compensation term φα depends on the determined focal length f, which is contrary to the characteristic of zoom lens. Therefore, we can calculate the compensation term φα at both ends of the zoom range, and take their average value as the phase compensation term for the entire zoom range. This problem is the inherent limitation of the proposed scheme. But it can be seen from the final effect that for a focal length variance from 200mm to 100mm, the error caused by this trade-off approximation is within an acceptable range. What's more, the improvements brought by this scheme greatly outweighed its error.
We simulate the modified metasurface Alvarez lens again in CST Microwave Studio. For the convenience of comparison, its size and zoom range remain same as the unmodified lens: it consists of upper and lower metasurfaces, where each metasurface consists of 60 × 40 cells. The displacement variation range is set to be 90 mm∼180 mm, the zoom range is 200 mm∼100 mm, and the working frequency is 15 GHz.
Figure 5(a-d) shows the simulation results of this zoom lens with four different displacements (90 mm, 120 mm, 150 mm, 180 mm). The focal length f is 213 mm, 141 mm, 115 mm, and 92 mm, respectively. According to Table 4, the simulated focal length of compensated Alvarez lens has an error of less than 10% compared with theoretical values, which is significantly improved compared to the error of more than 20% before phase compensation. In order to show the effect more clearly, the theoretical focal length, simulated focal length before phase compensation and after compensation are compared in Fig. 5(e). Before phase compensation, although the simulated focal length has the same trend as the theoretical focal length, there is a significant difference in value between them. After phase compensation, the simulated focal length is in good agreement with the theoretical focal length, which verifies the rationality of the error analysis about the excessive system error and the effectiveness of the proposed phase compensation scheme.
Fig. 5. (a-d) Modified metasurface lens simulation results with d = 90, 120, 150, 180 mm, (e) Focal length comparison.
Table 4. Comparison of theoretical and simulated focal length after phase correction
In addition, focal efficiency is also an important factor for lenses. Figure 6 introduces the change in the focal efficiency of Alvarez lens before and after phase compensation. It can be seen that the average focal efficiency of the Alvarez lens before compensation is about 30%, and the average focal efficiency of the Alvarez lens after compensation is about 43%, which shows that the proposed phase compensation scheme also has a contribution on the focal efficiency.
Fig. 6. (a, b) The power flow distribution at cross section in the propagation direction of the unmodified/modified Alvarez lens with d = 180 mm(c) Focal efficiency comparison.
5. Conclusion
In summary, we have proposed a modified metasurface Alvarez lens, which can work in microwave band and has the ability to zoom accurately. We have investigated a series of metasurface units that can achieve full 2π phase modulations, which are the key building blocks for a metasurface lens. Then, we realized a metasurface Alvarez lens through implementing discretized phase distribution function by these miniaturized metasurface units. In view of the excessive error of this microwave Alvarez lens, we further analyzed the phase distribution principle of Alvarez lens, and proposed a corresponding phase compensation scheme. The simulation results verified the effectiveness of this scheme. Our work not only gives a new direction for the application of Alvarez lenses, but also provides a new approach for microwave-band zoom lens design.
Funding
National Natural Science Foundation of China (62071291).
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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