Fully Open Ring Microcavities Based on Metagratings

Open optical microcavities are crucial to precise manipulation of light–matter interaction in solid‐state and biological systems. However, only open linear microcavities are practically realized, limiting the application of open microcavities in more scenarios. Herein, an ease‐fabrication fully open ring microcavity is theoretically demonstrated that supports simultaneously multiple concentric cavity modes with coupling interaction. It comprises two orthogonal dielectric metagratings, which are a single layer of silicon rods and enable transmitted waves to travel in a negative angle. Except for fully open configuration, the open microcavity also incorporates the advantages of low absorption loss and the compatibility to the current CMOS process. It offers a unique platform for controlling the interaction between light and atoms, molecules, nanoparticles, or biomolecules and will find a broad range of applications.

In this article, negative transmission of optical waves in metagratings, another way of enabling light to propagate in a negative way, will be employed to impetus a beam of light to circulate in a closed path. Different from negative refraction, transmitted waves in a negative angle and incident waves travel in the same media since negative transmission does not originate from the peculiar properties of the media in the transmitted region. When the medium in both regions is the air, we may realize an fully open ring microcavity with the entire cavity domain accessible to external environments. Negative transmission can be achieved by metagratings [29][30][31] and metasurfaces. [32,33] Metagratings have lower fabrication difficulties and higher operating efficiency, and the in-plane operation on light waves allows a more convenient realization of an open cavity, Our fully open cavity therefore will be constructed by silicon metagratings which can produce negative transmission behaviors with the compatibility to the current complementary metal-oxide semiconductor (CMOS) process.

Results and Discussion
Our open cavity is designed as shown in Figure 1a, where two metagratings are placed orthogonally to form a cross-shaped structure with four identical arms. The negative transmission by metagratings guides optical waves to propagate along the closed path A ! B ! C ! D ! A, resulting in the occurrence of the cavity mode. The negative transmission is related to the dipole scattering resonance of the constituent rods in the metagratings under the transverse electric (TE) polarization with its magnetic field along the axis of the rod. In the Mie scattering theory, the scattering resonances of rods are manifested by the maximum of j tan η n j as a function of frequency, [34,35] where tanη n ¼ m s J n ðk s r s ÞJ 0 n ðk 0 r s Þ À J n ðk 0 r s ÞJ 0 n ðk s r s Þ m s J n ðk s r s ÞN 0 n ðk 0 r s Þ À N n ðk 0 r s ÞJ 0 n ðk s r s Þ and η n is the scattering phase shift of the nth angular momentum channel. Here k s = ω ffiffiffiffiffiffi ε s μ p is the wave number, m s ¼ ffiffiffiffiffiffiffiffiffiffi ε s =ε 0 p , J n and N n are the Bessel and Neumann functions and J 0 n and N 0 n their derivative. To derive Equation (1), we have supposed that the incident and scattered magnetic (H) fields are expanded as and H s ðr, ϕÞ ¼ where H ð1Þ n is the nth-order Hankel function of the first kind. The Mie scattering coefficient α n ¼ b n =p n is related to tanη n by α n ¼ À1=ð1 þ i cot η n Þ. The dipole resonance state of rods, namely the resonance in the first angular momentum channel, corresponds to the frequency at which jtanη 1 j reaches its maximum. , respectively. c) Two concentric cavity modes excited by an external source and d) their corresponding eigenmodes. Each metagrating arm is composed of 15 rods. www.advancedsciencenews.com www.adpr-journal.com A silicon metagrating operating at telecom wavelength of λ = 1550 nm [29] is used here, whose negative transmission has been verified in experiment. [30] The silicon rods have the relative dielectric constant ε = 12 and a negligible imaginary part associated to absorption loss. [36] Negative transmission occurs when the rods have radius r s = 255 nm with a sharp peak of jtanη 1 j at λ = 1550 nm; in other words, the rods are on the dipole resonant state. The separation between the adjacent rods is set to be a 0 ¼ λ= ffiffi ffi 2 p so that the array acts as a first-order grating which only supports the 0th and À1st diffraction orders at the angle of incidence θ i ¼ 45 ∘ . The ordinary (the 0th) reflection and transmission diffraction order can be suppressed completely with rotating (circular) dipole modes excited inside the rods because of the strong dipole-dipole interaction between the rods. The outgoing wave in the À1st transmitted diffraction order lies in the same side of the normal as the incident that, and as a result, the metagrating behaves like a mirror placed along the its own normal, as shown in Figure 1b. When such two metagratings are orthogonally arranged, the four arms will force optical waves to travel along a closed loop. In Figure 1a, we mark the mth rod in the left upper arm as rod A, numbered from the origin O, and rod B, C, and D in the corresponding positions in the other arms, respectively. The distance between rod A and rod B is d AB ¼ ffiffi ffi 2 p ma 0 ¼ mλ and the total distance of a round trip d ABCDA ¼ 4mλ. The optical path of integer wavelengths implies that a cavity mode can be supported in this loop. The total optical path in the closed loop is a nonzero integral number of wavelengths, whereas the negative-index-based open cavities have the zero optical path that can be seen as a particular case of an integral number of wavelengths. It is interesting to notice that the diffraction in all diffraction orders contributes to the formation of the cavity modes though the metagrating doesn't support perfect negative transmission with about 79% transmissivity in the -1st transmitted diffraction order and 21% reflectivity in the À1st reflected diffraction order. This is because the reflected light in the À1st reflected diffraction order travels along the same optical circuit dictated by the cavity modes and thus also supports their formation.
In Figure 1c we use the finite-difference time-domain (FDTD) method [37] to simulate the field distribution when a beam with its source at the location (À6λ,0) strikes one of arms along the þy direction. Here each arm is composed of 15 rods. The propagation path of the wave will form a closed loop since the transmission occurs in the negative way each time it encounters an arm. We see that two concentric cavity modes are simultaneously excited in the inner part of the cross-shaped array in Figure 1c. The simultaneous excitation of the multiple cavity modes is caused by the transfer of EM energy along the array because of the strong coupling between resonant particles. [38] The eigenmodes of the cavity are calculated in Figure 1d by shutting off the launch field at the instant of time t ¼ 100 cT, before which the cavity modes were stably excited. Here 1 cT denotes the required time for the wavefront of light to move forward 1 μm in free space. We see that the inner mode has much smaller mode volume than the outer one. The inner cavity mode is constructed by only four rods closest to the center, whereas the outer one is realized by multiple rods in each arm. The mode volume of the inner mode is λ 2 approximately, nearly the same as the other optical microcavities.
To verify that the induced concentric modes in Figure 1c are a standing wave when the external source is on, the H field is respectively calculated on the lines y = 3.8 μm and y = 0.7 μm, the location of two induced modes, at the instant of time t 0 , t 0 þ T=8, t 0 þ T=4, t 0 þ 3T=8. Here T denotes the period of the incident wave. Figure 2a,b show the existence of antinodes and nodes which always have the zero field amplitude and the distance between the adjacent nodes or antinodes is λ/2. This indicates that they are the standing waves and the cavity modes are formed.
Next we investigate a cavity composed of a longer metagrating with each arm consisting of 25 rods. When the source is situated at (À4λ,0), Figure 3a displays that two cavity modes are excited, the same as the case in Figure 1c. When the source is moved outward to (À8λ,0), three cavity modes can be obtained in the same structure, as shown in Figure 3b. The comparison between Figure 3a,b shows that the excitation source should be placed in the position far away from the center in order to simultaneously excite multiple cavity modes. With the increment of length of the array, it is even possible to achieve arbitrary number of concentric cavity modes. The simultaneous existence of multiple cavities in the same structure may be a new addition to the cavity quantum electrodynamics and strongly correlated system toolbox. [39][40][41][42][43][44] The quality (Q ) factor of each cavity mode is calculated to analyze the feature of the open cavity. We know that the degradation of the electromagnetic (EM) energy density U in an optical cavity follows U ¼ e Àω r t=Q where ω r is the resonant angular frequency, so the quality factor Q can be calculated by fitting the U-t curve. Based on the FDTD method, we calculate the U of cavity modes at different time t to obtain a U-t curve after shutting off the external source at some time when the stable cavity modes have been excited. We respectively examine Q factors of the inner and outer modes for the cavity structures of 15-and 25-rod arms. Figure 4a,b shows the total EM energy density U of as a function of t for the excited modes in Figure 1d and 3a, respectively. The inner and outer modes in the cavity structures of 15-rod arm have the Q factors of 1844 and 1629, respectively, while for the case of 25-rod arm they are 2626 and 2186, respectively. We see that the inner cavity modes have the larger Q factor than the outer those. It shows that the outer cavity leaks the EM energy faster than the inner one. Quality factor Q and mode volume V are two most important parameters to characterize the performance of optical microcavities. The conventional closed microcavities, whether linear Fabry-Perot cavities or ring cavities such as microdisks, microspheres, and photonic crystal defect microcavities, usually have higher quality factor in contrast to open ones, while ring cavities generally have smaller mode volumes than linear cavities. In general, standard closed cavities are used when the strong interaction between light and matters is required because of their higher Q and smaller mode volume, whereas open cavities are needed when to implement a control over matters inside the cavities. As an open cavity, the Q factor of our cavity is smaller than the conventional closed ones, but it is completely comparable to other open cavities despite its fully open configuration. Especially, its inner mode has the mode volume of λ 2 which is in the same order as the conventional closed cavities.
Moreover, with the number of nanoparticles increasing, the EM energy in the both inner and outer cavities begins to oscillate with time, as shown in Figure 4a,b. The inset of Figure 4b shows an enlarged view of the oscillation of the inner and outer eigenmode with time. In the same cavity structure, the inner mode has the minimum EM energy when the outer mode has the maximum energy and vice versa. This implies that the inner cavity has energy exchange with the outer and they have obvious coupling interaction. Coupled cavities have shown important applications for the transfer of photon and quantum states [40][41][42] and for the observation and control of quantum phase transitions in strongly correlated systems. [43,44] Note that the inner and outer modes have nearly the same vibration amplitude. The vibration of the outer mode doesn't appear pronounced only because the total energy of the outer cavity mode (marked by blue longitudinal scale on the left) shows exponential decay with a decay amplitude much greater than the oscillation amplitude in one time period.
When multiple cavity modes are excited, the outer and inner cavity modes are clearly separated by a small distance but their distribution along the arm cannot be predicted. Here we give an empirical formula to easily find the position of cavity modes in the 25-rod-arm cavity structure. To do this, we first plot the field intensity of the eigenmodes in Figure 3a,b on the line y = 0 in Figure 5a,b, respectively. The peaks in Figure 5 manifest the central location of each mode. We provide a formula ðax 2 À bjxjÞ 2 e Àcjxj that fits the field intensity curve in Figure 5a well, and the obtained fitting curve from the formula is shown with the red dashed line. The coefficients a, b, and c in the formula are a = 1.378, b = 3.17 and c = 1.277, respectively. The field intensity in Figure 5b is fitted by the formula ðajxj 3 À bx 2 À cjxjÞ 2 e Àdjxj where the coefficients are a = 0.2, b = 1.84, c = 3.14, and d = 0.864. The empirical formulas fit the field intensity well and the position of the excited eigenmodes can be exactly found by them.
The silicon-based fully open cavity has negligible absorption loss and is also realizable experimentally at the same time.
(b) (a) Figure 3. The eigenmodes in the cavity structure of 51-rod arm with a) two and b) three concentric cavity modes excited. The two and three cavity modes are excited when the source is located at (À4λ,0), (À8λ, 0), respectively.   www.advancedsciencenews.com www.adpr-journal.com The metagrating which constitutes this cavity has been experimentally demonstrated in Ref. [30] where SOI wafer with the top silicon layer of 5.15 μm and the buried oxide layer of 1.5 μm was used as starting material. In fabrication process, a 270 nm-thick thermal silicon oxide layer was first grown on the top as the hardmask. After E-beam lithography, the hardmask was etched in a RIE equipment to transfer the patterns from the photoresist, and then the silicon posts were sculpted through an elaborate Bosch-process DRIE etching. As a reference, the fully open cavity can be achieved through the same fabrication strategies. As a single-particle-layer material, the utilization of metagratings greatly lowers the manufacturing difficulty. In contrast, photonic-crystal defect microcavities and negativerefractive-based open microcavities are based on the artificial structured bulk materials, preparation of which often challenges current manufacturing limits. The upright posts used in our work are also much easier to precisely manufacture than microdisks and microspheres, and moreover, slight deviations in geometric shape and size can cause the dramatic decay of Q factor of the latter due to the sensitivity of whispering gallery modes to the geometric parameters. Our open cavity is not only easier to be implemented, but also easier to obtain the target Q factor at the target wavelength.
Such a fully open microcavity can also be made from other dielectric materials because a wide range of permittivity can lead to negative transmission of metagratings over a wide frequency range. [45] Negative transmission and open microcavities can be scaled to the visible light range since it is enough to use dielectric materials with permittivity as small as 6 that are common in the visible light range. Other kinds of open cavities are less open than ours and accordingly, the way the cavity interacts with the environment is greatly limited. The fully open configuration in our microcavity allows to freely apply electric, magnetic, or optical fields to matters that are placed in the cavity and tune their states such as quantum states of atoms and molecules or electric/magnetic and optical behaviors of nanoparticles. For example, when applying an optical field with a frequency different from the excitation frequency of cavity modes at any desired angle, one may acquire information in the cavities by the change of polarization states, intensity, phase, or even frequency of the applied optical field. Simultaneously, the change of the applied optical field means conveying of its information to microcavities. Information exchange between microcavities and the external environment will become easier when such an open cavity is utilized.

Conclusion
We have exploited the remarkable ability of metagratings-to manipulate transmitted optical waves in a negative way when the both reflection and transmission regions are air, to achieve a fully open optical microcavity. A fully open configuration provides a way to directly observe the physical process occurring in microcavities and allows to conveniently exchange information between cavities and the external environment. The support for multiple concentric cavity modes with the coupling interaction may make it a useful tool for cavity quantum electrodynamic and strong correlated systems. Our cavity structure also has the advantage of the low absorption loss and simple design with the compatibility to the current CMOS process. The design strategy applies equally to other classical waves and particularly to acoustic waves. www.advancedsciencenews.com www.adpr-journal.com