Sorting OAM modes with metasurfaces based on raytracing improved optical coordinate transformation

Zhibing Liu; Jiahui Zou; Zhaoyu Lai; Jiajing Tu; Shecheng Gao; Weiping Liu; Zhaohui Li; Zhaohui Li

doi:10.1364/OE.435923

1. Introduction

In recent years, the ever-growing demand for information bandwidth has posed a huge challenge to the current optical communication system. Since the time, frequency, wavelength and polarization dimensions of light have been deeply researched before, exploiting the space dimension of light, also known as space division multiplexing (SDM), is considered to be the next-generation communication technology [1,2]. Among SDM, one of the most representative subsets is orbital angular momentum (OAM) multiplexing, which has given rise to special research interest in mode division multiplexing (MDM). Generally, a light beam with spiral phase wavefront factor ${e^{il\theta }}$, where $\theta $ is the azimuthal coordinate, l is the topological charge (TC), carries $l\hbar $ OAM per photon [3]. In principle, l is an unbounded integer, OAM beams with different l are orthogonal and thus have great application prospects in optical communication [4]. Apart from communication, OAM beams have also been applied to fields such as optical manipulation [5], optical tweezer [6], imaging [7], and quantum information [8]. In all of these applications, l is a determining parameter since it is directly related to the quantity of OAM. Owing to its great importance, measuring l or demultiplexing different OAM beams has attracted extensive researches [9–15].

Although there have been many studies on OAM measurement, simultaneously demultiplexing multiple OAM beams without shrinkage of efficiency is still challenging. To cope with this problem, in 2010, Berkhout et al. proposed a scheme that is based on optical coordinate transformation (OCT). In their way, an annular OAM beam with spiral phase is firstly transformed into a rectangular beam with plane wave phase and then can be sorted by a focused lens, this method is also known as log-polar coordinate transformation (LPCT) [16]. Due to the simple implementation and unitary efficiency, the LPCT method has received widespread concerns [17–22]. Inspired by the LPCT, some other versions of OCT for OAM manipulation have also been proposed, such as circular-sector transform [23] and spiral transform [24]. Though the OCT has its great advantage in sorting OAM, its performance could suffer from severe distortion when used for higher-order OAM state, this is because the beam needs to be assumed to be normally incident while the higher-order OAM beam has large skew angle [16]. To relieve this problem, Lavery et al. have analyzed the distortion and used two large phase masks to achieve up to 28 orders of OAM measurement in 632nm [19]. However, though we know the distortion could be introduced by non-normal incident light rays, there are still no quantitive analysis of the distortion and no corresponding solutions to improve the performance of the OAM sorter that works under specific configurations.

In this work, through analyzing the ray’s trajectory among the coordinate transformation process, we will explain the rationality to use the OCT beyond normal incident assumption. It should be noted that the ray deviated from normal incidence will go through incorrect deflection in the second phase plate. Then we can further provide a quantitative result of the distortion in OCT, and for the case of LPCT, we give an upper bound of l that can be measured under certain distortion. In addition, to alleviate the performance degradation caused by the distortion, we present a scheme to improve the distortion by raytracing assisted optimization, which is related to Zernike polynomial based phase compensation. After the optimization, we use three phase masks realized by metasurfaces to build the experimental setup and implement the measurement.

2. Theoretical analysis of distortion

Generally, the OCT maps the points of two separate planes, let $({x,\; y} )$ denote the first plane and $({u,v} )$ denote the second. Assuming light is normally incident, to achieve the map, from the view of ray optics and the paraxial approximation condition, a phase mask $\varphi $ should be put on the first plane and its gradient satisfied

(1)$$\frac{{\partial \varphi }}{{\partial x}} = \frac{{2\pi }}{\lambda }\frac{{u - x}}{f}, \qquad \frac{{\partial \varphi }}{{\partial y}} = \frac{{2\pi }}{\lambda }\frac{{v - y}}{f},$$

where $\lambda $ is the wavelength, and f is the distance between the two planes. When the ray arrives at the second plane, similarly, we need a phase mask $\psi $ to compensate the ray’s deflection, and its gradient satisfied

(2)$$\frac{{\partial \psi }}{{\partial u}} = \frac{{2\pi }}{\lambda }\frac{{x - u}}{f}, \qquad \frac{{\partial \psi }}{{\partial v}} = \frac{{2\pi }}{\lambda }\frac{{y - v}}{f}.$$

In general, given a reasonable relationship between $({u,v} )$ and $({x,y} )$, we can figure out the corresponding $\varphi $ and $\psi $ from above equations to achieve the transformation.

As shown in Fig. 1, a ray is normally incident on the $({x,y} )$ in the first mask, then passes through the device and arrives at the $({u,v} )$, and normally emerge from the second mask. However, if the ray is non-normally incident, its trace will deviate from the position $({u,v} )$, and also experience unexpected phase gradients. In this situation, further discussion is needed.

Fig. 1. Coordinate transformation with normal incidence and non-normal incidence.

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For simplicity, assuming the incident ray can be described by phase distribution P, then the ray’s deviation in the second mask caused by its phase gradient can be described as

(3)$$\overrightarrow d = ({{d_u},{d_v}} )= \frac{{\lambda f}}{{2\pi }}\nabla P,$$

where $({{d_u},\; {d_v}} )$ is the coordinate representation of $\vec{d}$. This means that if the incident light is non-normal incidence, or in other words, $\nabla P \ne 0$, it will cause the distortion of position of OCT. Since OAM beam has skew angle, $|{\vec{d}} |\ll r$ is supposed in reported works [19,23] for better performance, which causes a restriction on the l of OAM beam with the given size r.

In above, the position after transformation is considered within the scope of ray optics. However, for the application of OAM (de)multiplexing or manipulation, the distortion of position after OCT is not the only problem, because the phase transformation is the main purpose. To better evaluate the distortion, the ray’s direction after OCT is needs to be considered. Due to the position distortion, the ray will obtain a corresponding phase gradient change in the second plane, and by using Eq. (2) the gradient change can be described as

(4)$$\varDelta \frac{{\partial \psi }}{{\partial u}} = \frac{{2\pi }}{{\lambda f}}({ - {d_u} + \varDelta x} ), \qquad \varDelta \frac{{\partial \psi }}{{\partial v}} = \frac{{2\pi }}{{\lambda f}}({ - {d_v} + \varDelta y} ),$$

where $\Delta x = x({u + {d_u},v + {d_v}} )- x({u,v} )$, $\Delta y = y({u + {d_u},v + {d_v}} )- y({u,v} )$, since the ray’s deflection caused by the first mask can be offset by the second mask, its ultimate phase gradient accumulated by the transformation process can be expressed as

(5)$$\overrightarrow G = ({{G_x},{G_y}} )= \nabla P + \left( {\varDelta \frac{{\partial \psi }}{{\partial u}},\varDelta \frac{{\partial \psi }}{{\partial v}}} \right)\textrm{ = }\frac{{2\pi }}{{\lambda f}}({\varDelta x,\varDelta y} ).$$

Eq. (5) shows the phase gradient that represents the ray’s direction. In order to explore the ray’s property, we perform Taylor expansion on $\vec{G}$ and the gradient can be written as

(6)$$\overrightarrow G = \frac{{2\pi }}{{\lambda f}}({\overrightarrow d \cdot \nabla^{\prime}x,\overrightarrow d \cdot \nabla^{\prime}y} )\textrm{ + }\varDelta \overrightarrow G ,$$

where $\nabla ^{\prime}$ is vector differential operator of u-v coordinate, $\Delta \vec{G}$ is higher-order residuals. From Eq. (2) and using $\partial x/\partial v = \partial y/\partial u$, the ray’s phase gradient can be further written as

(7)$$\overrightarrow G = \nabla ^{\prime}P + \varDelta \overrightarrow G .$$

From the right hand of Eq. (7), we can see that the input ray with phase gradient $\nabla P$ has transformed into the output ray with gradient $\nabla ^{\prime}P$ after OCT, it is also mean that the phase distribution in the first plane has transformed into the second plane. In this way, the rationality of OCT to transform the phase distribution is justified by the first-order Taylor approximation. Meanwhile, the residual term in the equation is actually a distortion that causes the diffusion of the ray and can be calculated from Eq. (5) and Eq. (7). For convenience, this distortion can also be approximated by the second-order Taylor expansion

(8)$$\varDelta \overrightarrow G \approx \frac{\pi }{{\lambda f}}\overrightarrow d \overrightarrow d :[{({\nabla^{\prime}\nabla^{\prime}x} )\overrightarrow i + ({\nabla^{\prime}\nabla^{\prime}y} )\overrightarrow j } ],$$

where “:” denotes the scalar product of tensors. In most situations, the coordinate transform is conformal, or more specifically, it is an antianalytic function [25]

(9)$$x\textrm{ + }iy = F({u\textrm{ - }iv} ),$$

where F is an analytic function. The distortion can also be expressed as

(10)$$\varDelta {G_x}\textrm{ + }i\varDelta {G_y} = \frac{\pi }{{\lambda f}}{({{d_u} - i{d_v}} )^2}{F^{(2 )}}({u - iv} ).$$

From Eq. (10), we can conveniently calculate the distortion of various OCT. From Eq. (3) we can see the position deviation is proportional to $|{\vec{d}} |$, while the phase gradient distortion $|{\Delta \vec{G}} |$ is proportional to ${|{\vec{d}} |^2}$.

Next, we start to consider the special case of LPCT that has been widely used for OAM sorting. In this case, $u ={-} a\ln ({r/b} )$, $v = a\theta $, where a, b are positive parameters of the transformation, then we can derive the corresponding relationship

(11)$$x + iy = b{e^{ - {{({u - iv} )} / a}}}.$$

In addition, for an OAM beam with given radius r and phase wavefront $P = l\theta $, from Eq. (3) the deviation caused by its skew angle can be described as

(12)$$\overrightarrow d = \frac{{\lambda f}}{{2\pi }}\nabla ({l\theta } )= \frac{{\lambda fl}}{{2\pi r}}({ - \sin \theta ,\cos \theta } ).$$

Then from Eq. (10), the distortion of LPCT can be expressed as

(13)$$\varDelta \overrightarrow G = \frac{{\lambda f{l^2}}}{{4\pi r{a^2}}}({ - \cos \theta , \sin \theta } ).$$

Here, we postulate the distortion is smaller than the difference between the adjacent OAM modes in order to suppress the crosstalk. Since the ideal gradient difference of the adjacent OAM modes after LPCT is $2\pi /2\pi a = 1/a$, the inequality $|{\Delta \vec{G}} |/({1/a} )\le 1$ can be obtained, from which, we can get

(14)$$|l |\le \sqrt {{{4\pi ar} / {\lambda f}}} .$$

This is the upper bound of the OAM states that can be sorted by the LPCT under given distortion. For other type of coordinate transformations such as spiral transformation [24], similar constraints can also be obtained through the same treatment.

3. Raytracing assisted distortion improvement

In order to improve the distortion of LPCT derived in the above section, we introduce raytracing to simulate the process. Since raytracing is a common tool for the geometrical optics design, it should also be suitable for analyzing OCT. Here, to implement the LPCT, we use three phase masks to work as transformer, compensator, and a lens respectively to separate or sort the input OAM beams. According to Eq. (1) and Eq. (11), generally, the transformer can be derived through solving the partial differential equations. However, there is a much simpler method to calculate the phase masks by using the analytical function corresponding to the mapping. For convenience, we redefine the relationship as follows

(15) $$iu + v = h({x + iy} ).$$

This is also an antianalytic function and is equivalent to Eq. (9). In addition, since Eq. (1) represents the transformation system without lens, it is more convenient to derive the case with lens firstly, and then put the phase of the lens on the result [26]. So, we adjust Eq. (1) to the following form

(16)$$\frac{{\partial \phi }}{{\partial x}} = \frac{{2\pi }}{\lambda }\frac{u}{f}, \qquad \frac{{\partial \phi }}{{\partial y}} = \frac{{2\pi }}{\lambda }\frac{v}{f}.$$

Eq. (16) can be further rewritten as follows

(17)$$i\frac{{\partial \phi }}{{\partial x}} + \frac{{\partial \phi }}{{\partial y}} = \frac{{2\pi }}{\lambda }\frac{{iu + v}}{f} = \frac{{2\pi }}{{\lambda f}}h({x + iy} ).$$

Since h is an analytic function, we can find analytic function $H = A + iB$, which satisfies

(18)$$H^{\prime} = \frac{{2\pi }}{{\lambda f}}h = \frac{{\partial A}}{{\partial x}} + i\frac{{\partial B}}{{\partial x}} = i\frac{{\partial B}}{{\partial x}} + \frac{{\partial B}}{{\partial y}}.$$

Obviously, here B is the phase required by Eq. (16). Then after putting the phase of the lens on B, we can derive ${\varphi }$ that satisfy Eq. (1) as follow

(19)$$\varphi ({x,y} )= B - \frac{\pi }{{\lambda f}}({{x^2} + {y^2}} ).$$

Finally, using the relationship h of LPCT, we can get

(20)$$\varphi ({x,y} )= \frac{{2\pi }}{{\lambda f}}[{xu(r )+ yv(\theta )+ ax - {{({{x^2} + {y^2}} )} / 2}} ].$$

Moreover, given ${\varphi }({x,y} )$ satisfies Eq. (1), it is easy to prove that the phase distribution $\mathrm{\psi }$ of the compensator can be directly described as following

(21)$$\psi ({u,v} )= \frac{{2\pi }}{{\lambda f}}({xu\textrm{ + }yv} )- \frac{\pi }{{\lambda f}}({{x^2} + {y^2} + {u^2} + {v^2}} )- \varphi ({x,y} ).$$

$\mathrm{\psi }({u,v} )$ can satisfy Eq. (2) and can be rewritten as

(22)$$\psi ({u,v} )\textrm{ = } - \frac{{2\pi }}{{\lambda f}}[{ax({u,v} )\textrm{ + }{{({{u^2} + {v^2}} )} / 2}} ].$$

The above derivation can be also used as a calculation method or proof of the existence of ${\varphi }$ and $\mathrm{\psi }$ of any other OCT represented by h. It is worth noting that the LPCT is still valid if we slightly change the relationship between $({x,y} )$ and $({r,\theta } )$. For instance, we can set $r{e^{i\theta }} = {e^{ - i\delta }}({x + iy} ),\; \; \theta \in ({ - \pi ,\pi } )$, the azimuth of unwrapping OAM beam will rotate angle $\delta $ with the negative half axis of x-axis. For the sake of simplicity, here we set $\delta = 0$. The corresponding phase masks determined by Eqs. (20–22) are shown in Figs. 2(a-b) respectively. Besides, we also added a lens with focal length f. Since a Fourier lens can be directly added to the compensator, the purpose of adding an extra lens is to increase the degree of freedom so as to facilitate the optimization in subsequent contexts. The third phase mask of the lens is shown in Fig. 2(c). In addition, the input OAM beam used here is LG beam, of which beam size r is varied according to l and follow $r = w\sqrt {|l |} $, where w is the brightest radius of the LG₀₁ mode. Under this condition, to limit the interference on adjacent order as derived in the above section, the range of TC should satisfy

(23)$$|l |\le \sqrt[3]{{{{({{{4\pi aw} / {\lambda f}}} )}^2}}}.$$

The raytracing then is done according to Eq. (1–2) where the paraxial condition is used. Figure 2(d) shows the raytracing schematic diagram of sorting multiple OAM beams, different OAM beam has corresponding skew angle determined by its order and radius, and the rays will go through the three masks and convergence at the focal plane. As is shown in Fig. 2(e), the rays of OAM beams with different order l, where $- 10 \le l \le $10, have different radius. The radius w of the LG₀₁ beam is set to be 0.18 $mm$, and the parameters of the LPCT respectively set as, $\lambda = 1.55\,\mu m$, $a = 1.5/2\pi \,mm$, $b = 0.45\,mm$, and $f = 15\,mm$. For simplicity, the focal length of the convergence lens is also set to be f. Then from Eq. (23) we can find the critical TC is about $l = 8$. Accordingly, Fig. 2(f) shows the results of focused rays based on the installation. It can be seen that the ray will go to adjacent focus when the order is larger than 8. This ray optics simulation result confirms our theoretical analysis and shows good accuracy to approximate the distortion of LPCT using second-order Taylor expansion.

Fig. 2. Raytracing simulation of LPCT. (a)-(c) The three phase masks used in the raytracing simulation. (d) Schematic diagram of raytracing to simulating LPCT. (e) The start points of the rays in the input plane and (f) The endpoints of rays in the focal plane.

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Next, to overcome the huge distortion of LPCT resulted from the skew angle of the input OAM beam, we further use raytracing to optimize the phase masks. In the field of optical design, Zernike polynomials are often used to describe aberrations or to fit freeform surfaces. In mathematics, they are a sequence of polynomials that are orthogonal. In view of these advantages, so we define a Zernike polynomials based phase compensation to implement the optimization. For every mask, the first 45 order Zernike polynomials are added to express the compensated phase, fewer terms of polynomials may lead to worse results while more terms would increase the difficulty of optimization. The modified phase mask thus can be described as following

(24)$$Q^{\prime}({x,y} )= Q({x,y} )+ \sum\limits_{k = 1}^{45} {{a_k}{Z_k}({x,y} )} ,$$

where Q is the original phase, ${Z_k}$ is the $k$-order Zernike polynomial, ${a_k}$ is the corresponding coefficient, and $Q^{\prime}$ is the modified phase. When a ray goes through the mask, its traces will be regulated by an extra phase denoted by Zernike polynomial, and the coefficients ${a_k}$ thus can be used as optimizing variables. Besides, we define the root mean square error (RMSE) as the extent of distortion for the optimization, this can be denoted by

(25)$$FoM = \sqrt {\frac{1}{N}\sum\limits_{n = 1}^N {[{{{({{x_n} - \widetilde {{x_n}}} )}^2}\textrm{ + }{{({{y_n} - \widetilde {{y_n}}} )}^2}} ]} } ,$$

where $FoM$ needs to be minimized, N is the number of rays, $({x,y} )$ and $({\widetilde {x,}\tilde{y}} )$ are the ray’s arrival position and ideal destination in the focal plane. After setting up this raytracing model, we use an evolutionary algorithm named covariance matrix adaptive moment estimation (CMA-ES) to regulate the optimizing parameters or the Zernike coefficients [27], which is a commonly used global optimization algorithm. Other methods such as genetic algorithm (GA) [28], or particle swarm optimization (PSO) [29] may be also valid here. The simulation was implemented by using MATLAB (R2020a) and a computer (AMD Ryzen Threadripper 2970WX CPU @3.00 GHz and 64 GB of RAM, GPU was not used to speed up the program). In the optimization, the number of optimizing variables is 135 and the total number of rays N is 2112. The CMA-ES algorithm executed a total of 250000 iterations to achieve the optimized phase masks and the simulation took about 3.5 hours.

Figures 3(a-c) show the phase masks after Zernike polynomial based phase compensation. Figure 3(d) shows the corresponding convergence of rays in the focal plane. There is a great deal of diminution of distortion especially for the higher-order OAM, and the RMSE has dropped from the original value of $15.87\mu m$ to $3.27\mu m$ while the spacing of adjacent TC is $15.50\mu m$. In Figs. 3(e-f) and Figs. 3(g-h), we show the physical optics based simulation results before and after optimization. Figure 3(e) and 3(f) show the intensity distribution of multiple OAM beams in the focal plane and corresponding normalized intensity distribution along the central axis. From these focused spots we can find that with the increasing l, there is also increasing distortion. The critical order is $l = 8$ when the distortion goes to heavily influences the adjacent spots, which is in good agreement with the theoretical prediction. Correspondingly, the simulation results after optimization are shown in Fig. 3(g) and 3(h). From the figure, we can see a huge improvement of the sorting performance. The spots in Fig. 3(g) have roughly uniform size and less overlap with adjacent spots. Meanwhile, the intensity distribution of the sorted OAM beams along the central axis in Fig. 3(h) also indicates less interference to other beams. These simulation results signify that the distortion improvement is effective.

Fig. 3. Simulation results after raytracing optimization. (a)-(c) The phase masks after raytracing assisted optimization. (d) The ray’s arrival position in the focal plane after optimization. (e-f) Sorting results before optimization. (g-h) Sorting results after optimization.

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Further, to better demonstrate the LPCT before and after raytracing optimization, we have shown the transformation process in Fig. 4. For every mode, the sorting process includes four pictures, which represent the light field distribution at four positions transformer, compensator, focal lens, and the final focal plane. Clearly, by comparing the original transformation and the optimized transformation, we can find that to reduce the distortion, we have optimized the phase mask that was originally used as a lens. Meanwhile, both the intensity and phase of the light beam in the transformation process have been changed obviously after optimization. More importantly, due to the distortion improvement, the energy of the focused spot becomes more concentrated. Though there are still some residual distortions, it is possible to use our method to further optimize the spots for coupling with optical fiber by improving the algorithm used here or adding more optimization degrees of freedom.

Fig. 4. Simulation of the transformation process before and after raytracing optimization. The original and optimized phase masks used in LPCT are corresponding to Figs. 2(a-c) and Figs. 3(a-c) respectively.

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4. Device preparation and characterization

In order to prepare the above optimized phase masks for experiment, we used three metasurfaces to provide the corresponding phase modulation. Here the function of metasurfaces is mainly concentrated on $2\mathrm{\pi }$ phase coverage and high transmittance, a most favorable selection is the dielectric metasurface that has been demonstrated in reported researches [30–34]. As shown in Fig. 5(a), the metaunit of our dielectric metasurface is composed of amorphous silicon (a-Si) cylinder and SiO₂ substrate, the metaunit thus is polarization insensitive. In addition, to prevent the cylinder from collapsing in the fabrication, 20nm a-Si has remained in the substrate. Under the given structure, to maximize the transmittance while cover 2π phase, we have done a parameter sweep to determine the metaunit and respectively set $p = 760\; nm$ and $h = 890\; nm$ according to the sweeping results. The phase and transmittance response of the metaunit are shown in Fig. 5(b) and the working wavelength is set to 1.55 $\mu m$. We select 96 structures with phase difference $2\pi /96$ through interpolation and have marked them with the asterisk in the Fig. 5(b). The diameter of these structures is from $175\; nm$ to $520\; nm$, and the aspect ratio of these pillars is less than 5. Based on the metaunit, coverage of $2\pi $ phase can be achieved and the average transmittance of these structures is 90.77%.

Fig. 5. Metasurfaces design and fabrication. (a) The schematic diagram of metaunit with $p = 760nm$ and $h\; = \; 890nm$ respectively. (b) The phase and transmittance response of the metaunit with diameter range from $175nm$ to $520nm$. (c-e) The fabricated metasurfaces were captured in the microscope. (f-g) The partial fabrication details of the device in SEM.

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After the metaunit design, we generated the layout to meet the required phase masks in the above section and fabricated the device accordingly. The fabricated process can be briefly described as follows. Firstly, a layer of 890-nm-thick a-Si was evaporated in the SiO₂ substrate by using chemical vapor deposition (CVD). After pretreatment of the sample, the AR-P6200.13 (ARP) resist was spin-coated in the a-Si film. Before electron beam lithography (EBL), a layer of Al was deposit in the ARP resist. The device was then put into the EBL device to write the mask. After exposure, we remove the Al layer and developed the resist to get the patterned mask. Afterward, we use inductive coupled plasma (ICP) to etch the a-Si layer. Finally, the ARP resist was removed and the device was fabricated. In Figs. 5(c-e), we show the fabricated metasurfaces that are captured in a microscope, which are consistent with Figs. 3(a-c). Figures 5(f-g) show the partial fabrication detail in the scanning electron microscope (SEM). From these images, we can see the device is well fabricated and the pillars have been etched appropriately.

Based on the metasurfaces described above, we built the experiment measurement setup as shown in Fig. 6. A laser with 1550nm wavelength was used as the light source and the generated beam was collimated into SLM, where we use a PC to regulate its polarization. After the reflection and modulation of the SLM, the OAM-carrying beam was then contracted by a pair of lenses L1 and L2 to match the size of the metasurfaces. To diminish the unexpected diffraction order of SLM, an iris was placed between the two lenses. The OAM beam then arrived at the metasurface MS1 and undergo the modulation of MS1-3. Finally, since the focal spot is small, we used lenses L3 and L4 to expand it and the CCD was used to detect the results.

Fig. 6. Experiment setup. PC: polarization controller. COL: collimator. SLM: space light modulator. L1-L4: the lens with focal length, f1 = 15cm, f2 = 7.5cm, f3 = 5cm, f4 = 12.5cm. MS1-MS3: the fabricated metasurfaces. CCD: charge coupled device.

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At last, we carried out the experiment, and the sorting results of OAM beams with $- 16 \le l \le 16$ are shown in Fig. 7. As a comparison, Figs. 7(a-b) show the simulation results before and after optimization. It can be seen that the simulation results have severe distortion before optimization when $l > 8$, but after optimization, the divergence of the focal spot is improved significantly. As shown in Fig. 7(c), the experimental detection of OAM beams up to ${\pm} 16$ order is achieved, and the distortion of higher-order OAM beams also have been significantly suppressed. Further, in order to quantify the effect of distortion improvement, the transfer matrix of simulation and experiment needs to be determined. Here the relative power of the detected transfer matrix T is defined by

(26)$${T_{mn}} = \mathop{\int\!\!\!\int\limits_{{S_n}}} {{I_m}({x,y} )dxdy} ,\;{S_n} = \{{({x,y} )|{{x^2} + {{({y - {y_n}} )}^2} \le {R^2}} } \},$$

where $R = \lambda f/4\pi a$, ${y_n} = \lambda f{l_n}/2\pi a$, and ${I_m}$ is the focal intensity distribution of ${l_m}$ order OAM beam. The calculation of the transfer matrix here is different from the ordinary form of which the integral is based on dividing the y-axis into equally spaced areas. Because the focal spot is slightly inclined in this work, which may result from the bending of the beam during propagation. Therefore, the beam will be easier to run to adjacent areas, but the real interference may not reach the same level. In addition, detection or coupling beams are generally achieved through circular cross-section. So, we think it is more reasonable to evaluate the improvement of the spot divergence with circular aperture described as Eq. (26).

Fig. 7. Sorting results of OAM beams with $- 16 \le l \le 16$. (a) Simulation results before raytracing optimization. (b) Simulation results after raytracing assisted optimization. (c) The experimental detection results.

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Further, the corresponding transfer matrix are shown in Fig. (8) and the crosstalk can be defined as

(27)$$X{T_k} = 10{\log _{10}}\left( {{{\sum\limits_{m \ne k} {{T_{mk}}} } / {\sum\limits_m {{T_{mk}}} }}} \right),$$

where $X{T_k}$ is the crosstalk on the channel ${l_k}$. From Fig. 8(a) we can see that the crosstalk is very high before optimization, the range of $XT$ is between −2.07 dB and −0.05 dB when l$> 8$, so the higher-order OAM beams will be indistinguishable from the interfering light. This problem can be alleviated by ray tracing optimization, as shown in Figs. 8(b-c), both the simulation and experimental results show improved distinction even for $l ={\pm} 16$. The maximal $XT$ of simulation after raytracing optimization and experiment are −5.90 dB and −2.00 dB respectively. There are some differences between the simulation and experiment results, which may be due to the manufacturing errors and the imperfect experimental setup. On the whole, the performance of LPCT has been improved greatly, and the experimental results can well support the reliability of simulation and indicate good effectiveness of the raytracing assisted distortion improvement.

Fig. 8. The detected transfer matrices of the OAM modes sorter. (a) Simulation before raytracing assisted optimization. (b) Simulation results after optimization. (c) Experimental results.

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5. Conclusions

In conclusion, we have provided a quantitative analysis of the distortion in OCT, and also clarified the rationality of optical coordinate transformation for non-normally incident beams in the view of geometrical optics. For the case of LPCT, we give an upper bound of l that can be measured under certain distortion. As the OCT can only work well in the first-order Taylor expansion approximation, the sorting distortion will increase as the TC of OAM beam increases. Therefore, to alleviate this problem, we further provide a method based on the raytracing and Zernike polynomial based phase compensation. After raytracing optimization, the RMSE of the focused rays can be reduced to 1/5 of the original and the physical optic simulation also shows obvious distortion improvement. At last, to confirm the simulation results, we implemented the experiment using three fabricated metasurfaces. The experimental results are in good agreement with the simulation and show obvious performance improvement after raytracing optimization. Results in this paper may also help to further improve other OCT related applications and have the potential to extend by adding more masks.

Funding

National Key Research and Development Program of China (2019YFA0706300); National Natural Science Foundation of China (61775085, 61875076, 61935013, 62035018, U1701661, U2001601); Guangzhou Science and Technology Program key projects (201904020048); Local Innovation and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121); Science and Technology Planning Project of Guangdong Province (2017B010123005, 2018B010114002); Leading Talents Program of Guangdong Province (00201502); State Key Laboratory on Integrated Optoelectronics (IOSKL2019KF12).

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 maybe obtained from the authors upon reasonable request.

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Sorting OAM modes with metasurfaces based on raytracing improved optical coordinate transformation

Abstract

1. Introduction

2. Theoretical analysis of distortion

3. Raytracing assisted distortion improvement

4. Device preparation and characterization

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (27)

Optics Express