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Abstract
Inspired by the concept of phase-gradient metasurfaces (PGMs), we present a way to design a multi-functional PGM-based light beam splitter (LBS) operating in the optical regime by engineering the anomalous diffraction properties. As an example of a proof of concept, the designed LBS is a purely metallic slit array with gradient slit width, termed metagrating. It is shown that the designed LBS can simultaneously achieve high-efficiency light beam splitting on both energy and polarization, and it has broadband and wide-angle response. In addition, we also show that the Ohmic loss of metals plays an important role in determining the diffraction efficiency of each diffraction order, which is the physics for designing the LBS that can deliver the incident energy equally into the reflection and refraction sides. Our work enriches the existing methods of designing LBSs and particularly provides a route for the design of multi-functional LBSs with high performance.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light beam splitters (LBS), which can split an incident light beam into two parts, are indispensable optical elements in modern advanced optical technology, playing an important role in many applications and various optical devices, such as optical switches [1], photopolarimeters [2,3], quantum photonics integrated circuits [4], and communication devices [5,6]. Conventionally, LBSs are typically gratings or semi-reflection speculars, which are heavy and bulky so that they are hard to integrate into the compact optical devices [7]. As the development of integrated photonics, the compact high efficiency LBS is extremely desired, which has motivated great efforts to achieve this goal with diverse means [8–11]. Optical phase-gradient metasurfaces (PGMs) emerging in nanophotonics provide us a new paradigm to design compact and flat LBS with high performance [12–15]. PGM are periodic arrangements of subwavelength meta-atoms that can effectively manipulate the intrinsic properties of electromagnetic (EM) waves in terms of their amplitude, phase and polarization by properly engineering the interaction between light and meta-atoms, leading to various functionalities [16–26], such as ultrathin cloaking [16], metalens [17–19], retroreflection [20–22] and asymmetric propagation [23,24]. However, most of metasurface-based LBSs reported thus far can only achieve either polarization splitting or only energy separation for a fixed-polarized light. Few studies have been reported to realize above-mentioned two functions simultaneously in one design. The multi-functional high-efficiency LBSs are desired in photonic integration systems due to the flexible and diversified capability of controlling the flow of light.
In this work, based on the concept of PGMs, we design and study a multi-functional PGM-based LBS operating in the optical regime by exploring and manipulating their diffraction properties. As known, the PGMs are periodic gratings with a supercell containing with m unit cells of different optical responses that discretely introduce an abrupt phase shift (APS) of fully covering 2π. The introduced APS gives rise to a phase gradient (i.e. an additional wave vector), which modifies the fundamental law of reflection and refraction of light occurring at the interface [27]. The diffraction effect is ubiquitous in PGMs [28–30], with higher order diffraction described by parity-dependent diffraction law [31]. Therefore, the free control of diffraction effect in PGMs and their efficiency is the key to improving the performance of PGM-based devices, including the LBSs. As an example of a proof of concept, the designed LBS is a purely plasmonic PGM, and the required APS along the transmitted interface is introduced by adjusting the width of air slit that determines the propagating wave vector of surface plasmon passing through it. We will show that the designed LBS can simultaneously achieve high-efficiency light beam splitting on both energy and polarization, and has broadband and wide-angle response. For instance, when both transverse-electric (TE) and transverse-magnetic (TM) polarized light are incident simultaneously with ${\theta _i} ={-} {30^ \circ }$, the designed LBS can make the TE polarized light totally reflected, with reflected angle of ${\theta _r} ={-} {30^ \circ }$, i.e., it is happened the perfect specular reflection. While for TM polarized incidence, due to diffraction effect in the designed PGM, high-efficiency negative refraction taking the lowest diffraction order is seen in designed LBS, and the refracted angle is ${\theta _t} = {30^ \circ }$. In this way, the polarization splitting is obtained. Moreover, we will show that the Ohmic loss of metals, plays an important role in determining the diffraction efficiency of each diffraction order on both reflection and transmission sides for TM polarization. With these physics in hand, the LBS also can deliver the incident energy of TM polarization equally into the reflection and refraction sides. Our work is fundamentally significant in the study of PGM-based LBSs and the proposed design shows great potential in applications such as integrated optical communication or optical measurement.
2. Metagrating design
Figure 1(a) schematically shows the geometry of the designed light beam splitter (LBS), a gradient metallic (silver) grating with a phase gradient at the transmitted interface. For the sake of easy, it is termed as a metagrating. Consider two independent linearly polarized light, i.e., TE and TM light, incident from above to it. Figure 1(b) shows the cross-sectional view of the metagrating consisting of a periodically repeated supercell with a period length of p and a thickness of d. Each supercell includes m air slits with identical depths of d whose widths are denoted by w1, . . ., w5 from left to right. The center distance between two adjacent air grooves is a = p/m. Due to the subwavelength size of these air grooves, i.e., $w \ll \lambda $, only the fundamental mode exists inside the air grooves for TM polarized light, with its propagation constant ${\beta _i}$ given by [32]
Fig. 1. (a) Schematic diagram of the polarization beam splitter based on the metallic metagrating. (b) Schematic plot of a supercell for the metagrating, composed of 5 air slits with widths of w1,…, w5. The thickness and period length are d and p, respectively. The distance between two adjacent slits is a. (c) The transmission phase retardation $\phi$ in air slits versus the width w. The triangles indicate the corresponding widths of the air slits for designed metagrating.
In this work, the working wavelength is set to be $\lambda = 650\textrm{ nm}$, where ${\varepsilon _m} ={-} 17.36 + 0.715i$ [32]. For demonstration, we consider the supercell containing five air grooves (i.e., m = 5), with the period of $p = \lambda $ and the thickness of $d = 1.5\lambda $. Figure 1(c) displays the relationship between the required ${\phi _i}$ and ${w_i}$ based on Eq. (1). In particular, the corresponding widths of air slits are w1 = 120 nm, w2 = 68 nm, w3 = 46 nm, w4 = 34 nm and w5 = 27 nm. In this way, when the incident TM light passes through the metagrating, a phase gradient $\zeta = \nabla \phi (x) = {k_0}$ at transmitted side is introduced, which will steer the directions of the outgoing wave. Based on the diffraction law [33], the direction of each diffracted light is given by angularly asymmetric diffraction
where ${\theta _i}$ and ${\theta _t}$ are the angles of incidence and refraction, respectively, and $G = {{2\pi } / p}$ is the reciprocal lattice vector. Due to $\zeta = G$, the Eq. (2) can be re-expressed as ${k_0}\sin {\theta _i} = {k_0}\sin {\theta _t} + nG$ with $n = v - 1$. Note that $v = 0\textrm{ (}n ={-} 1\textrm{)}$ is the lowest diffraction order (i.e. corresponding to the generalized Snell's law [27]), which predicts a critical angle of ${\theta _c} = {0^ \circ }$ for the emergence of higher order diffraction. Therefore, when the incident angle is less than the critical angle, i.e., ${\theta _i} < {0^ \circ }$, the outgoing wave follows the order of n=−1, while for ${\theta _i} > {0^ \circ }$, the outgoing wave will be governed by the higher order of n = 1 [33]. On the other hand, TE incident light, owing to subwavelength slits, will be totally reflected by the designed metagrating.3. Metagrating-based polarization splitting
Next, we will show the performance of polarization splitting in the designed metagrating. Figure 2(a) shows the numerically calculated diffraction efficiency of each diffraction order for TM incident light. It is clear that when ${\theta _i} < {0^{\circ}}$, the transmission of the lowest diffraction order is dominant (i.e., n=−1, blue solid line), with the transmission efficiency of approximately 70% at ${\theta _i} ={-} {30^ \circ }$. When ${\theta _i} > {0^ \circ }$, the diffraction is dominated by the reflection of a higher order of n = 1 (red dotted line), which is due to the odd parity of m (m = 5 here) [31]. The incident light will efficiently couple to the reflected light of n = 1 order, which means that the retroreflection occurs in this case. In particular, when ${\theta _i} = {30^ \circ }$, the corresponding reflection angle is ${\theta _r} ={-} {30^ \circ }$ and the reflection efficiency is ${R_1} \approx 40\%$. The more dissipation or lower diffraction efficiency in higher order diffractions is caused by the multiple total internal reflections inside the metagratings [33]. Therefore, considering the diffraction characteristics of different orders and the parity of the metagrating, anomalous transmission or retroreflection can be achieved for TM polarized light. In addition, Fig. 2(b) show the diffraction efficiency for TE incident light. Only the reflection of the n = 0 order is left and dominant, which is due to the subwavelength air slits that are far below the cut-off frequency for supporting TE light passing through them. Therefore, for TE polarized incident light, the specular reflection occurs with ${\theta _i} = {\theta _r}$ for full incidence ranging from $- {90^ \circ }$ to ${90^ \circ }$. In particular, when ${\theta _i} ={\pm} {30^ \circ }$, the corresponding reflection efficiency is R0 = 96%. These diffraction results of different orders are the physical basis of polarization splitting.
Fig. 2. The beam splitting for two orthogonal polarizations. The diffraction efficiency (transmission T or reflection R) of different diffraction orders of metagrating for TM incident light (a) and TE incident light (b). (c) Magnetic field patterns at ${\theta _i} ={-} {30^ \circ }$ for TM polarized light. (d) Electric field patterns at ${\theta _i} ={-} {30^ \circ }$ for TE polarized light. (e) Magnetic field patterns at ${\theta _i} = {30^ \circ }$ for TM polarized light. (f) Electric field patterns at ${\theta _i} = {30^ \circ }$ for TE polarized light.
In order to demonstrate the polarization splitting characteristics of the designed metagrating, the electromagnetic field patterns at incident angle ${\theta _i} ={\pm} {30^ \circ }$ are given by performing COMSOL Multiphysics. Figures 2(c) and (d) show the numerically calculated total magnetic and electric field patterns at incident angle ${\theta _i} ={-} {30^ \circ }$, corresponding to the TM and TE polarized light, respectively. The black arrows indicate the directions of the incidence and the specular reflection of n = 0 order. The blue arrows indicate the directions of the anomalous transmission of n=−1 order. Clearly, it is shown that high-efficient negative refraction occurs for TM polarized light and perfect specular reflection occurs for TE polarized light. At this time, the separated TM and TE polarized light is located on both sides (i.e., reflection and transmission areas) of the metagrating. For comparison, Figs. 2(e) and (f) show the total magnetic and electric field patterns when the incident angle is ${\theta _i} = {30^ \circ }$, corresponding to TM and TE polarized light, respectively. For TE polarized light, it appears specular reflection, regardless of the incident angle. However, for TM polarized light, the outgoing light is governed by the higher order of n = 1, and due to the number of unit cells m of the metagrating is odd (m = 5), the retroreflection effect can be observed (shown by the white arrow). At this time, the separated TM and TE polarized light is located on the same side (reflection area) of the metagrating. Therefore, by considering the diffraction characteristics of different orders and the parity of the metagrating, the designed metagrating can achieve high-efficiency light beam splitting on polarization.
The extinction ratio is usually an important parameter used to evaluate the performance of the polarization beam splitting device. The extinction ratio can be divided into reflection extinction ratio (ERTE) and transmission extinction ratio (ERTM) [34], which is
The reflection extinction ratio refers to the ratio of the reflection efficiency of the TE light to the reflection efficiency of the TM light, and the transmission extinction ratio refers to the ratio of the transmission efficiency of the TM light to the transmission efficiency of the TE light. We calculate the extinction ratio (ERTE and ERTM) of the metagrating at different incident angles and wavelength. Figure 3(a) shows the relationship between the ERTE and the incident angle. For ${\theta _i} \in ( - {74^ \circ },\; - {7^ \circ })$, the reflection extinction ratios are all above 10 dB, and when ${\theta _i} ={-} {62^ \circ }$, they are as high as 18 dB. Figure 3(b) shows the transmission extinction ratio, which is relatively high (ERTM > 130 dB) among the whole angle range. Generally, when the reflection extinction ratio and the transmission extinction ratio are greater than 10dB [35], the device is considered to have a good polarization beam splitting effect. Therefore, the designed metagrating for polarization splitting has a wide angle response. Moreover, Fig. 3(c) and (d) display the frequency response of the designed LBS when ${\theta _i} ={-} {30^ \circ }$. Although the LBS is designed at a target wavelength of $\lambda = 650\textrm{ nm}$, it still has a broadband response for polarization splitting due to the tolerance in PGM design [31]. For ERTE >10dB, the band width is approximately 78 nm, with wavelength ranging from 590 nm to 668 nm, where ERTM is greater than 132 dB.
Fig. 3. Performance of polarization splitting. (a-b) The relationship between the extinction ratio (ERTE and ERTM) and the angle of incidence for $d = 1.5\lambda$. (c-d) The relationship between the extinction ratio (ERTE and ERTM) and the incident wavelength when ${\theta _i} ={-} {30^ \circ }$.
4. Energy splitting performance analysis
Moreover, we will show that the designed metagrating can also realize the energy splitting for only TM polarized incident light. As shown in Fig. 2(a) or 2(c), the incident energy is split into three parts, corresponding to the diffraction of R0, T0 and T−1 order, respectively. In fact, in this design, due to the interplay between two loss of metal structure itself and the SPs passing through the slits, the performance of energy splitting can be further improved by controlling the thickness of the metagrating that determines the diffraction efficiency of each diffraction orders. To show this point, Fig. 4(a) and (b) illustrate the diffraction efficiency of T−1 and R0 as a function of incident angle for different thickness, i.e., $d = 0.6\lambda$, $\lambda$, $\textrm{1}\textrm{.5}\lambda$ and $\textrm{2}\textrm{.4}\lambda$. Note that in all these cases, the phase gradient remains unchanged, i.e., $\zeta = \nabla \phi (x) = {k_0}$. However, as the thickness changes, ${\beta _i}(w)$ should change to obtain the unchanged APS (i.e., ${\phi _i} = {\beta _i}(w)d - \delta$), which can be met by choosing suitable widths of air slits according to Eq. (1). As shown in Fig. 4(a), as the d increases from $0.6\lambda$ to $\textrm{2}\textrm{.4}\lambda$, the efficiency of T−1 firstly increases and then decreases. The efficiency of R0 decreases gradually with increasing thickness (see Fig. 4(b)). The absorption of the entire structure increases gradually with increasing thickness (Fig. 4(c)). This is because the decrease in thickness will narrow the width of the air slits, and the duty cycle of metals in the metagrating will increase accordingly, which will result in a decrease in transmission and an increase in reflection. However, on the other hand, the thickness cannot be too large. This is because the increase in thickness will cause more loss when light (i.e., SPs) propagates in the air slits. Consequently, the absorption efficiency will increase and the corresponding transmission efficiency will decrease. Therefore, there is a critical thickness for transmission due to loss. In particular, when $d = 0.6\lambda$ and ${\theta _i} ={-} {30^ \circ }$, the transmission efficiency of the n=−1 order is almost equal to the reflection efficiency of n = 0 order, which are 43% and 39%, respectively. Figure 3(d) shows the corresponding simulated magnetic field pattern. It is clearly seen that the incident light is divided into two beams (reflection and transmission), and the two beams are in a straight line. Thereby the designed metagrating can realize multi-functions for light splitting.
Fig. 4. Energy separation for TM polarized light. The relationship between the anomalous transmission efficiency (a), the normal reflection efficiency (b), and the absorption efficiency (c) of four metagratings with different thicknesses and the angle of incidence for TM polarized light. (d) The magnetic field patterns for the TM polarized light with incident angle ${\theta _i} ={-} {30^ \circ }$ when $d = 0.6\lambda$.
5. Conclusion
In this work, we have showed how a multi-functional high-efficiency light beam splitter can be designed based on the concept of the metagrating. A LBS of purely plasmonic metagrating operating in the optical regime has been proposed, which can simultaneously achieve high-efficiency light beam splitting on both energy and polarization, and has a broadband response. The underlying mechanism is the diffraction physics in designed metagrating. In addition, we have also shown that the Ohmic loss of metals plays an important role in determining the performance of energy splitting. Subsequently, this result is helpful for the study of light beam splitter based on non-Hermitian system and behind physics, such as exceptional points. This work provides new insight for designing multi-functional LBS based on PGMs. The proposed LBS offers an alternative way to control light flow and some possibilities for imaging systems and optical communications.
Funding
National Natural Science Foundation of China (11974010, 12104331); China Postdoctoral Science Foundation (2018T110540, 2020M681701); Postdoctoral Science Foundation of Jiangsu Province (2021K276B); Priority Academic Program Development of Jiangsu Higher Education Institutions; Key Lab of Modern Optical Technologies of Education Ministry of China.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data used to support the findings of this study are available from the corresponding author upon request.
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