Analytic solution for double optical metasurface beam scanners

Optical metasurfaces are researched more and more intensively for the possible realization of lightweight and compact optical devices with novel functionalities. In this paper, a new beam-steering system based on double metasurface lenses (metalenses) is proposed and developed. The proposed system is lightweight, small volume, low cost, and easy to integrate. The exact close-form forward and numerical inverse solutions are derived respectively using the generalized Snell’s law of refraction. Given the orientations of the double metalenses, the pointing position can be accurately determined. If the desired pointing position is given, the required metalenses’ orientations can be obtained by applied global optimization algorithms to solve nonlinear equations related to the inverse problem. The relationships of the scan region and blind zone with the system parameters are derived. The method to eliminate the blind zone is given. Comparison with double Risley-prism systems is also conducted. This work provides a new approach to control light beams.


The proposed metalens scanner
Analysis. The schematic of the proposed double-metalens scanner system under the Cartesian coordinate system is shown in Fig. 1. It consists of two metalenses ( GM 1 and GM 2 ) arranged along the z-axis, each of which has a planar structure, a subwavelength thickness, and a pre-determined phase gradient ( G 1 or G 2 ). The two metalenses can rotate around the z-axis independently. The parameters ψ 1 and ψ 2 represent the counterclockwise rotation angles of GM 1 and GM 2 , respectively, with respect to the z-axis. The distances between the two metalenses and the second metalens and the receiving screen are D1 and D2, respectively. The incident light beam is supposed to be along the -z axis, passes Center O of GM 1 and Point M of GM 2 , respectively, and finally arrives at Point P on the receiving screen. The Point M on GM 2 can be positioned with the longitudinal and azimuthal angles (θ, φ) with respect to Center O of GM 1 . The position of Point P is determined by the coordinates (X, Y) on the receiving screen and the longitudinal and azimuthal angles (Θ, Φ) with respect to Point M of GM 2 , respectively.
The relationship between the beam direction and the metalenses' orientations is established based on the generalized Snell's law 20 . In the initial state, the phase gradients G 1 and G 2 are along the x-axis, namely ψ 1 = ψ 2 = 0 . The forward solution is to find out the beam pointing direction or the (X, Y) on the receiving screen with a set of given metalenses' orientations. Assume ψ 1 and ψ 2 is known, the exact formulas can be derived using the generalized Snell's law 20 . First, let us calculate the beam direction of the first metalens GM 1 . The schematic for the 2D generalized Snell's law of refraction is depicted in Fig. 2a. Based on the law, we have where n t and n i are the refractive index of the relative media, 0 is the wavelength of the incident wave, and dφ dx is the phase gradient G 1 , along x axis at the initial time.
As the beam is incident normally on GM 1 , the direction of the leaving or transmitted beam can be expressed as: (1) n t sinθ t − n i sinθ i = 0 2π dφ dx ,  www.nature.com/scientificreports/ where k 0 = 2π 0 is the wavenumber. The metalenses are extraordinarily thin, and the media are air (or n i = 1 and n t = 1 ). The pointing direction (θ, φ) of the beam leaving GM 1 can be found as: Next, we derive the beam direction (Θ, Φ) of GM 2 . From (3), the coordinates of the intersection point M on the second metalens is (D 1 tanθ 1t cosϕ 1t , D 1 tanθ 1t sinϕ 1t , −D 1 ).
For the convenience of the derivation, a new coordinate system is set up. It has the origin at Point M and is rotated by an angle ψ 1 along the z-axis of the original coordinate system. The relative rotation angle of GM 2 in the new system is ψ 2 − ψ 1 because GM 2 rotates by an angle of ψ 2 with respect to the original system. The rotation matrix on the new coordinates, denoted by (x ′ , y ′ , z ′ ) , is then given by: With the phase gradient G 2 initially along the x-axis on GM 2 , corresponding to − → G 2 = (G 2 , 0, 0) T , the rotated phase gradient vector is given by With the 3D generalized Snell's law, whose schematic is shown in Fig. 2b, the following formula is obtained With n 2i = n 2t = 1 and θ 2i = θ , (θ 2t , ϕ 2t ) can be derived from (6) Then by transforming ( θ 2t , ϕ 2t ) into the spherical coordinate (θ 2 , φ 2 ) on the new coordinate system, we have By rotating the new coordinate system back to its original, we have: The coordinate (X, Y) of Point P on the receiving screen is obtained by The scan blind zone. A scan blind zone occurs when the phase gradients of the two metalenses are identical, leading to the target lost within the scan region. Here an example is given to illustrate and discuss the problem. With G 1 =G 2 =4188.8 rad/mm, D 1 = 10 mm , D 2 = 20 mm , and 0 = 500 nm (corresponding to k 0 = 0.0126 nm −1 ), the scan region on the receiving screen is calculated and plotted in Fig. 3a. There is a circular blind zone in the center of the circular scan region. Based on the forward solution above, the blind zone is determined by the phase gradient and occurs when the gradients of GM 1 and GM 2 are identical. Based on the geometrical relationship between the leaving beam of GM 1 and the intersection point M , the radius of the blind zone, |O'M|, is: Since the altitude angle θ increases with the phase gradient G1, the blind zone will increase with D 1 and G 1 . Increasing the phase gradient G 2 of GM 2 is a solution to decrease or eliminate the scan blind zone. From (7) and (8), when |ψ 2 − ψ 1 | = 180 • , we have: www.nature.com/scientificreports/ With the geometrical relationship between the emerging beam of GM 2 and the scan point P on the receiving screen, we get: Combining (11), (12), and (13), we have the increment of G 2 : Figure 3b plots the scan region without a blind zone, where the increment of G 2 is 2187.5 rad/mm, corresponding to an enlarged phase gradient G ′ 2 = 6376.3 rad/mm. The scan region. In Fig. 3, not only the blind zone is eliminated, but also the scan region enlarged when the phase gradient G 2 increases. Therefore, it is necessary to discuss the impact of phase gradients on the scan region. The maximum X and Y of the scan region on the receiving screen can be obtained when |ψ 2 − ψ 1 | = 0 • . From (10), we have: Next, we show how the scan region varies with the parameters of the double-metalens system. For the convenience of the analysis, the phase gradients of GM 1 and GM 2 are set to be identical ( G = G 1 = G 2 ). Figure 4 illustrates the relations among (Xmax, Ymax), D 1 / 0 , D 2 , and G/k 0 . It is obvious that the scan region will increase with D 1 / 0 , D 2 , and G/k 0 .
(13) |NP| = D 2 tan(θ).  In practice, a smaller distance is preferred to reduce the scan blind zone 45 . However, if the distance between GM 1 and GM 2 is too close, mutual coupling between metalenses may become unacceptly large. Hence the compromised distance value should be taken in practice, based on the metasurfaces utilized. In addition, the minimum distance is dependent on the specific characteristics of the metelenses, while the maximum distance on the wavelength of the incident beam, the size of the lens aperture, and the phase gradients of the metalenses.
The distance D 1 also impacts the beam trajectories 12,45 because it affects the scan region. Figure 5 illustrates the differences in trajectories when D 2 is constant. It is seen that the impact of D 1 can be ignored when D 2 /D 1 ratio is larger than 40.

Comparison with the paraxial approximation solution of risley-prism systems.
In the conventional double Risley-prism systems, the first-order paraxial method 8,45 is a typical solution for RPSs with thin prisms that have small deviation angles. However, since the first-order approximation is applied, it only works when the approximation is good enough. Next, comparisons with the first-order method 8,45 with RPSs are conducted to demonstrate the accuracy of the proposed method. The mathematical expressions of the first-order method expressions are: where δ 1 and δ 2 denote the deviation angles by the two prisms, respectively. The relationship between the deviation angle and the phase gradient with the proposed double-metalens scanner system can be obtained from (3): In the comparison, the double-metalens scanner system with two identical metalenses is considered. Let G 1 k 0 and G 2 k 0 be 0.25 and 0.33, respectively, which corresponds to the opening angles of 28.96° and 38.94° of the prisms, respectively, the comparisons are made between the first-order method and the proposed analytic solution. The beam trajectory is acquired by keeping ψ 1 = 90 • and varying ψ 2 from 0 • to 360 • . Figure 6 shows the two comparison results. When the phase gradient is small, corresponding to the small opening angle of the prism, the two results are relatively close (see Fig. 6a). When the phase gradient or the opening angle increases, the differences between the two results increase (see Fig. 6b). The reason is that increasing the opening angle of the prism will introduce a larger error due to the first-order approximation 8,45 while increasing the phase gradient will not change the accuracy of the proposed analytical solution. An increase of the phase gradient of the metalens or the opening angle of the prism in the two systems leads to the increase of the scanning region. Therefore, the proposed metalens system always gives accurate solutions for any pre-determined scanning region without changing the volume of the system. www.nature.com/scientificreports/

Inverse solution and target tracking performance for double-metalens systems
Inverse solution. The inverse solution is to find out the orientation angles of the metalenses for a given pointing position on the receiving screen. It is especially significant in practical applications such as target tracking. Following the exact derivation of the forward solution of the new double-metalens system, an exact inverse solution is found as follows. Equation (8) can be rewritten in the following form when substituting (9) into (8): Then (19) can be further derived as: Let A = G 1 k 0 and B = G 2 k 0 , the new expressions of (7) are: Substituting (21) into (22), we get: Solving (8), (9), and (23) simultaneously, we have: The orientations ψ 1 and ψ 2 can be obtained by solving (10), (24), and (25) through optimization algorithms, for example, genetic algorithm, particle swarm optimization algorithm, differential evolution algorithm, etc. However, they usually converge slowly. For example, with a random initial value, the differential evolution algorithm will take thousands of iterations to converge and find a correct solution for the inverse problem, whereas the other two algorithms need more iterations and times. In this work, we used the MATLAB function vpasolve to get the optimized solutions, which is more efficient for the solution.
Target tracking performance. An asteroid target trajectory is applied to validate the performance of the target tracking using the proposed inverse solution; the expression of the asteroid is: The MATLAB function vpasolve is used for the solution. This function can be used to find all the numerical solutions of algebraic equations with a random starting point. The fitness function or the algebraic equation finding the target trajectory is the sum of the absolute value of (10), (24), and (25).
, 0 ≤ θ ≤ 2π .  Fig. 7a,b, respectively. Substituting the orientations obtained from the inverse solution into the analytic forward solution, we found that the errors are less than 1 μm.

Discussion and conclusions
The optical metalens is currently an important under-explored topic. There are two kinds of metalenses: metallic plasmonic one and all-dielectric one. The latter is free from ohmic losses, leading to a much higher efficiency of operation. The beam-steering systems-based metalenses are lightweight, compact, and easy to be integrated because the volumes of metalenses do not vary much with the change of phase gradients.
The exact forward and inverse solutions for beam-steering systems with double metalenses are successfully derived based on the generalized Snell's law of refraction in this paper, where the forward one is in close form while the inverse one should be obtained by the numerical method. The relationships among the scan region, the blind zone, the beam trajectory, and system parameters are derived or discussed. The approach to eliminating the scan blind zone is presented and numerically demonstrated. Comparison of the new system with the conventional Risley-prism systems is carried out, demonstrating that the proposed method can give effective and accurate results.
There are some limitations with the current double-metalens beam-steering systems. Due to the dispersions of metalenses and the refraction, they are more applicable to monochromatic light beams. The optimization algorithms to solve the inverse solution converge slowly. More efficient methods are needed for practical applications.