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Abstract
We demonstrate a new electromagnetic mode which is formed by the dynamic interaction between a magnetic quadrupole mode and an electric monopole mode in a two-dimensional electromagnetic Helmholtz cavity. It is termed a magnetic symmetric dipole mode since it shares similarity with a magnetic dipole mode in the sense that their radiation is both overwhelmingly dominant in the forward and backward directions with respect to the incident wave. However, the phase distribution in the two radiation directions is symmetric, in stark contrast to the antisymmetry of magnetic dipole modes. When the Helmholtz cavities are arranged in a line, the incident wave will be reflected back to the source, in other words, retroreflection occurs because of the peculiar properties of magnetic symmetric dipole modes. We show that the retroreflection is quite robust against the disorder of the orientation angle of Helmholtz cavities and there exists a wide tolerance for wavelength and the outer radius of the cavity. With low fabrication demands, this might offer a feasible solution for the design of ultrathin retroreflectors towards device miniaturization and the realization of multiplexing holography.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Retroreflection refers to the phenomenon in which an incident wave is redirected back to the source when striking a surface. It has important applications in information [1] and sensing [2,3], laser ranging [4], the production of a standing wave, etc. In particular, retroreflectors are the only option for target detection and tracking in the dark environment. A practical example is concomitant satellites which are responsible to monitor and take photo of a master satellite at all times [5]. A corner reflector structured by three mirrors mutually perpendicular to one another is a retroreflector used extensively in ordinary applications [6]. Besides, phase conjugate mirrors composed of nonlinear media [7], random media due to coherent backscattering [8,9], Luneburg lens and a blazed reflection grating in a Littrow geometry [10] all can function as a retroreflector. However, the device miniaturization and high-density integration require the size and weight shrinking of optical devices with improving performance. This promotes the demand for ultrathin retroreflectors. Recently several approaches based on metamaterials and metasurfaces have been proposed to realize retroreflection [11–14]. In particular, retroreflection and other high-order diffractions are stimulating the development of multiplexing holographic metasurfaces [15–21] based on the fact that the diffraction angles are remarkably sensitive to wavelengths and angle of incidence.
Metamaterials are usually several-wavelength thick so that homogenization can be carried out to define effective refractive indices [22–29]. Graded metasurfaces are a single-layer array of nano-antennas and allow for the design of ultrathin subwavelength optical components [30,31]. However, it suffers from inherent limits on conversion efficiency. In addition, the varying geometric configure and size of nanoantennas in graded metasurfaces increase the fabrication difficulty. Here we achieve retroreflection by inheriting the subwavelegth merits of graded metasurfaces but without the need for the gradient change of size and shape of constitutive particles. The single-layer material is a linear array consisting of the identical subwavelength Helmholtz cavities. Its extraordinary abilities come from the excitation of an unconventional electromagnetic (EM) mode, strikingly different from metamaterial retroreflectors and graded metasurface retroreflectors in physics. It has been shown that some unconventional EM modes are able to inhibit the EM scattering in the undesired diffraction directions in gratings and engineer the majority of the scattering field to certain desired directions [32,33]. In such materials, as a collective the particle arrays only specify all the possible grating diffraction orders according to the ratio of wavelength to the spacing between adjacent particles, whereas the true outgoing direction is chosen by the anomalous EM modes excited inside the individual particles.
Here the unconventional mode we will demonstrate is a magnetic symmetric dipole mode (MSDM) which can lead to the occurrence of retroreflection in a single particle layer. Its electric counterparts, namely electric symmetric dipole modes, have been successfully generated in dielectric rods for a wide range of permittivities [34]. But a MSDM which is strong enough to dominate light scattering behaviors of particles is not found in dielectric rods. This raises the question whether there exists such a mode in nature. On the other hand, the existence of a magnetic counterpart helps to develop magnetic retroreflectors based on a single layer of particles. Here we verify that a MSDM can be definitely excited in a two-dimensional (2D) EM Helmholtz cavity when illuminated by a transverse-magnetic polarized wave with its electric (E) field along the axis of the cavity. Our studies for electric/magnetic symmetric dipole modes show that they are one of general mechanisms of retroreflection.
2. Magnetic symmetric dipole mode
We need to first handle the EM scattering problem of a 2D Helmholtz cavity with infinite thin perfect electric conductor (PEC) wall rigorously. Here the generalized dual series approach developed in Ref. [35–40] is adopted. Figure 1(a) indicates the radius of the Helmholtz cavity $r_s$, the angular width of the opening $\theta _o$ and the orientation angle of the cavity $\theta _r$ which is defined as the angle between the center of the opening and the $x$ axis. The scattering field can be written as
where $A_m$ is the scattering field coefficient and $H_{|m|}^{(2)}$ is the mth order Hankel function of the second kind. For a system of multiple cavities, we use the multiple scattering theory [33] to calculate the spatial field distribution. The Helmholtz cavity with finite wall thickness (the outer radius $r_s$ and the inner radius $r_i$) is also displayed in this figure and will be numerically solved latter by Lumerical FDTD.
Fig. 1. (a) Schematic illustration of an EM Helmholtz cavity and retroreflection by a linear array of these cavities. $\theta _o$ and $\theta _r$ is the angular width of the opening and the orientation angle of the Helmholtz cavity, respectively. The wall thickness is $t=r_s-r_i$ with $r_s$ and $r_i$ the outer and inner radius, respectively. (b) The absolute value of the scattering field coefficients $A_0$, $A_1$ and $A_2$ versus wavelength $\lambda$. The H field vectors produced by the electric monopole mode (c) and the MQ mode(d) around a Helmholtz cavity with $r_s=60$ mm, $\theta _o=130^\circ$ and $\theta _r=180^\circ$ at $\lambda =185$ mm. (e) The scattered E field by the same Helmholtz cavity as in panels (c) and (d), which represents an excited MSDM. (f) The scattered E field contributed by the angular momentum channels $n=\pm 1$ which represent an excited MD mode with conventional antisymmetric nature. The incident wave propagates along the x axis in panels (c)-(f).
Based on the generalized dual series approach, we first investigate the EM scattering of an isolated Helmholtz cavity with $r_s=60$ mm, $\theta _o=130^\circ$ and $\theta _r=180^\circ$ when a plane wave with wavelength $\lambda =185$ mm is incident along the $x$ direction. The scattering coefficients $A_0$, $A_1$ and $A_2$ correspond to the electric monopole, the magnetic dipole (MD) and the magnetic quadrupole (MQ) mode according to the Mie scattering theory, respectively. The absolute value of the first three coefficients is given as a function of wavelength in Fig. 1(b) since the higher order coefficients are negligible in the plotted wavelength range. It shows that $|A_0|$ is nearly equal to $|A_2|$ with a very small $|A_1|$ at the wavelengths ranging from $185$ mm to $230$ mm. Accordingly, an electric monopole and a MQ with equal magnitude are simultaneously excited in the cavity, whereas the MD mode is hardly excited. As a result, the scattering behaviors of the cavity are determined together by an electric monopole and a MD mode. As is known to us, the magnetic field vectors arising from an electric monopole mode will rotate about the center of the cavity, whereas those from a MQ mode have to be identified in four regions, as respectively shown in Fig. 1(c) and (d). The magnetic fields are parallel to those from an electric monopole mode in the regions 2 and 4 and antiparallel in the other both regions. When the MQ has the same magnitude as the electric monopole, namely $|A_0|=|A_2|$, the electric fields will be completely eliminated in the two antiparallel regions and at the same time strengthen each other in the two parallel regions. Consequently, the interaction between the electric monopole and the MQ modes produces the radiation which is overwhelmingly dominant in two opposite directions, as demonstrated in Fig. 1(e). The radiation fields are located in the forward and backward directions with respect to the incident wave, similar to that of a MD mode in Fig.1(f). However, the phase distribution is symmetric in the two opposite directions, strikingly different from the antisymmetry nature of a MD mode. So we term it a magnetic symmetric dipole mode.
3. Retroreflection
Next we investigate what optical effect the unconventional EM mode will give rise to in a single layer of the Helmholtz cavities. The cavities studied in Fig. 1(e) but with $\theta _r=45^\circ$ is arrayed along the x axis and the adjacent cavities has the identical spacing $d$ which satisfies the Littrow mounting for the given wavelength $\lambda =185$ mm and angle of incidence $\theta _i=45^\circ$. Moreover, as a grating it only supports the 0th and -1st diffraction orders and thus the far-field outgoing waves from it may travel along four different directions, as indicated in Fig. 2(a). Above all, the spacing should be $d=\lambda /2$sin$\theta _i$ according to the grating equation. Our simulation shows that the linear array channels all the incident wave into the -1st-order reflected direction, in other words, retroreflection occurs. When the radiation feature of the MSDM is considered, the fact that the grating chooses the -1st-order reflection as the ultimate outgoing direction can be easily understood. In the viewpoint of the scattering theory, the ordinary (0th-order) reflected wave, the retroreflected wave and the -1st-order transmitted wave are the contribution of the total scattered waves by all the cavities, whereas the ordinary transmitted wave is the consequence of the superposition of the total scattered waves and the incident wave. Now no wave is scattered by any cavity in the ordinary reflection direction and the -1st-order transmission direction when $\theta _i=45^\circ$ since the two directions are perpendicular to the incident wave while the radiation field of the MSDM locates in the forward and backward directions with respect to the incident wave. So the ordinary reflection and the -1st-order transmission will not occurs in the linear array. In the ordinary transmission direction, the total scattered field by all the cavities and the incident field are exactly out of phase and will cancel each other due to the destructive interference. Thus, near-perfect retroreflection can be obtained in the array.
Fig. 2. (a) Retroreflection when a Gaussian beam with $\lambda =185$ mm strikes a linear array of the Helmholtz cavities at angle of incidence $\theta _i=45^\circ$. (b) Retroreflectance versus the orientation angle of the Helmholtz cavities at the fixed angle of incidence $\theta _i=45^\circ$. The scattered E field when the opening of the isolated Helmholtz cavities deviates from the incident wave by $10^\circ$ (c) and $20^\circ$ (d). The orientation angle $\theta _r$ is kept to be $0^\circ$ and the wave vector of the incident wave makes an angle of $10^\circ$ (c) and $20^\circ$ (d)with the x axis, as indicated by the arrows $k_i$ in panels (c) and (d). The scattering fields exhibit obvious symmetric nature of MSDMs. The same Helmholtz cavities and wavelength as those in Fig. 1(e) are used here.
Note that the retroreflection cannot be owed to the strong inter-cavity interaction. This is because the scattered field by a cavity can not nearly arrive at the adjacent cavities and so the inter-cavity interaction in the array is extremely weak. In short, the linear array, as a grating, only gives all possible diffraction orders and is not able to determine which will be eventually chosen. It is the MSDM that chooses the -1st order reflected direction (namely, retroreflection) as the eventual outgoing direction.
In Fig. 2(a), the opening of the cavities exactly faces the incident wave. It is evident that the wave should not strike the back of the cavity since the excitation of the cavity mode needs the entry of the wave into the cavity. On the other hand, once the incident wave enters the cavity the cavity mode will be excited even though they do not exactly face each other. Figures 2(c) and (d) display the scattering field distribution of an isolated cavity when the opening deviates from the incident wave by $10^\circ$ and $20^\circ$. They still exhibit the obvious feature of MSDMs. Our calculation shows that a strongly MSDM can be always excited under the deviation of $25^\circ$. On the contrary, once the deviation is beyond $25^\circ$ the MSDM will be dramatically weakened. In Fig. 2(b) the retroreflectance is given as a function of the orientation angle while the angle of incidence is fixed at $\theta _i=45^\circ$. One can see that retroreflectance is beyond $90\%$ between $25^\circ$ to $70^\circ$, which coincides with the strength of the excited MSDMs. This shows again that the retroreflection is closely relative to the MSDM.
With the wide orientation angle tolerance, the retroreflection is probably quite robust against the disorder of the orientation angle. To this end, we study the array in Fig. 2(a) at $\theta _i=45^\circ$, but the orientation angle of the cavities is designed to be different from each other. The deviation of the orientation angle with respect to the incident direction is expressed as $\Delta \varphi =\zeta \cdot \varphi _{m}$ where $\zeta$ is the random numbers uniformly distributed between -1 and 1 and $\varphi _{m}$ is the maximum variation in the orientation angle. In Figs. 3(a) and (b), the E field distribution is respectively demonstrated when $\varphi _{m}$ is equal to $15^\circ$ and $20^\circ$ with the random orientation angle between $30^\circ$ and $60^\circ$ and between $25^\circ$ and $65^\circ$. One can see that the disorder does not cause a significant negative impact on retroreflectance and the retroreflection shows the rather robust behavior to the orientation angle. Moreover, the comparison between Fig. 3(a) and (b) shows that the lower the degree of disorder is, the higher the retroreflectance is. It corroborates the significance of MSDMs in retroreflection since the deviation of the orientation angle impacts the quality of MSDMs. The robust behavior significantly lowers the fabrication demand and favors the practical realization of retroreflectors.
Fig. 3. Retroreflection when the orientation angle of the Helmholtz cavities in the linear array randomly changes between $30^\circ$ and $60^\circ$ (a) and between $25^\circ$ and $65^\circ$ (b) at the fixed angle of incidence $\theta _i=45^\circ$. The same Helmholtz cavities and wavelength as those in Fig. 1(e) are used here.
We also examine the sensitivity of the MSDM to the incident wavelength for the same cavity as in Fig.1(e). The results show the cavity can support the MSDM in a wide wavelength range. Thus one can expect a broadband retroreflector realized by the EM Helmholtz cavity arrays. In Fig.4 retroreflectance is given as a function of wavelength. Here the incident angle remains $45^\circ$ and the spacing between the adjacent cavities varies with the wavelength according to $d=\lambda /2$sin$\theta _i$ so that the reflection in the -1st reflected order always results into retroreflection. One can see that retroreflectance above $90\%$ can be achieved for the wavelength ranging from 185 mm to 230 mm, above $80\%$ from 180 mm to 250 mm. The broad operating bandwidth also enables it a promising platform for multiplexed broadband holograms.
In addition to the incidence angle of $45^{\circ }$, the array can also produce retroreflection at other incident angles. Figure 4(b) shows that near-perfect retroreflection efficiency can be achieved for the incident angles ranging from $43^{\circ }$ to $50^{\circ }$ only by adjusting the orientation angle of the cavities $\theta _r$. The optimal orientation angles at different incident angles are given at the same time in Fig. 4(b). In all cases, the MSDM always dominates the scattering behaviors of the cavities in these arrays.
Fig. 4. (a) Retroreflectance versus wavelength $\lambda$ at the fixed angle of incidence $\theta _i=45^\circ$. The retroreflectance above $90\%$ can be achieved for the wavelength ranging from 185 mm to 230 mm. (b) Retroreflectance and the optimal orientation angle $\theta _r$ versus the incident angle $\theta _i$. The retroreflectance above $90\%$ can be achieved for the incident angle between $43^{\circ }$ and $50^{\circ }$. The spacing is changed with wavelength (the incident angle) according to $d=\lambda /2$sin$\theta _i$ in panel (a)((b)).
In the above simulations, we have assumed that the PEC wall of the Helmholtz cavities is infinitely thin. However, the finite wall thickness has to be considered in a practical design. We assume the cavity of finite thickness has the outer and inner radius $r_s$ and $r_i$, respectively, and the wall thickness is $t=r_s-r_i$, as shown schematically in Fig.1(a). The other parameters of the cavity are the same as the cavity in Fig. 1(e). Figure 5 simulates the retroreflection in the different wall thicknesses using Lumerical FDTD. In Figs. 5(a) and (b), the outer radius is $r_s=60$ mm which equals the radius of the infinite thin cavity in Fig. 1(e). But their inner radius $r_i$ is respectively 59 mm and 55 mm with the wall thickness $t$ equal to 1 mm and 5 mm. The obvious ordinary reflection in Fig.5(b) implies that the enhancement of the wall thickness significantly lowers the retroreflection efficiency. In Figs. 5(c) and (d), the fixed inner radius is $r_i=60$ mm and the enhancement of the wall thickness is caused by the change of the outer radius. The outer radius $r_s$ is respectively 61 mm and 65 mm with the wall thickness $t=1$ mm and $t=5$ mm in Figs. 5(c) and (d). The results in Figs. 5(c) and (d) show that the enhancement of the outer radius does not bring a substantial impact on the retroreflection effect, in contrast to the cases in Fig. 5(a) and (b). It shows that though the cavities in Figs. 5(b) and (d) have the same wall thickness, the retroreflection efficiency of the linear arrays composed of them are strikingly different. Physically, the EM modes excited at a particular ratio of wavelength to the size of the interior hollow region dominate the EM scattering behavior of a Helmholtz cavity. Different from the outer radius, the enhancement of the inner radius changes the size of the interior hollow region. So the change of the inner radius impacts the excitation of the MSDM and thus lowers the retroreflectance. Whereas the interior hollow region is not changed when the outer radius is enhanced, and as a result, the MSDM and the retroreflection are hardly affected. It shows that a realistic EM Helmholtz cavity with finite wall thickness can be designed to excite the MSDM and achieve high-efficiently retroreflection.
Fig. 5. Retroreflection when the Helmholtz cavities have the different wall thickness $t=r_s-r_i$ with $r_i=59$ mm and $r_s=60$ mm (a), $r_i=55$ mm and $r_s=60$ mm (b), $r_i=60$ mm and $r_s=61$ mm (c), and $r_i=60$ mm and $r_s=65$ mm (d). The incident angle and the orientation angle both are $45^\circ$ in all the cases.
4. Conclusion
In this work, we find a MSDM in EM Helmholtz cavities and demonstrate a retroreflection behavior of waves in a linear array composed of these Helmholtz cavities. The retroreflection shows the rather robust behavior to the orientation angle of the Helmholtz cavities. The Helmholtz cavities with finite wall thickness support the MSDMs of the same quality as infinite thin one when the interior hollow region is not changed. The single layer allows the realization of a high-efficiency and broadband retroreflector with low fabrication demand. The compact design favors device miniaturization and offers a promising platform to realize multiplexing holography.
Funding
National Natural Science Foundation of China (11474098).
Disclosures
The authors declare no conflicts of interest.
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