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Abstract
Optical fibers integrated with metasurfaces have drawn tremendous interest in recent years due to the great potential for revolutionizing and functionalizing traditional optics. However, in most cases, metasurfaces have been placed on the fiber end-facet where the area is quite limited. Here, by dressing a series of identical dielectric rings around the side surface of the microfiber and adjusting their positions along the microfiber axis, we extracted guided waves into free-space radiation with continuously controllable phase shift and achieved circular-arc-shaped line focusing. We demonstrated that the off-fiber foci could be rotated around the fiber axis by tuning the polarization of the guided waves. In addition, we demonstrated that the shape of the focus could be further tuned by introducing symmetry breaking into the dielectric rings. Our study provides a new dimension for the design of optical fiber devices decorated with metasurfaces.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical metasurfaces are generally composed of judiciously designed artificial structures with feature sizes much smaller than light wavelength [1–4]. Metasurfaces provide a powerful approach to control the phase, amplitude and polarization state of light waves reflected or scattered from them. The remarkable light manipulation ability of metasurfaces has been utilized to realize complex functions such as focusing [5–6], mode conversion [7], holographic image construction [8], and vortex beam generation and detection [9–10]. In the past few years, various metasurfaces-based devices, such as metalens [11–16], meta-waveguides [17–18], photonic circuits [19–20] and topological photonics [21], have been reported, which could effectively address the requirements of miniaturization, multi-function, configurability, and compatibility of optoelectronic devices.
Benefiting from the development of the “lab-on-fiber”, the integration of metasurface with optical fibers has drawn tremendous interest in the last decade [22–23]. The metasurface-integrated fiber devices have found applications in sensing [24], biomedicine [25], and clinical diagnosis [26]. In most cases, metasurfaces have been placed on the fiber end-facet to manipulate the output mode profile. It should be noted that the core of single-mode or few-mode fibers usually has a diameter in the range of several to tens of micrometers. Thus, the area of the fiber end-facet which is available for the integration is greatly limited. Although using coreless fibers [27–29], photonic crystal fibers or multimode fibers [6,30–31] with relatively large core diameters could expand the area available for metasurfaces integration, those fibers themselves are large, which may hinder the development of the miniaturized all-fiber devices.
Recently, Ni et al. demonstrated a concept of guided wave-driven metasurfaces consisting of an array of metal-dielectric-metal nanoantennas on top of a dielectric rectangular waveguide, in which meta-atoms could be directly driven by guided wave and the phase of the extracted light could be tuned by varying the size of the meta-atoms [32,33]. Instead of using different sized meta-atoms, the change in the coordinate position of the meta-atoms along the direction of guided wave propagation could be utilized to modulate the phase of the extracted light in the guided wave-driven metasurfaces [34,35]. Similar mechanism of the phase modulation has been previously reported to shape the spatial and spectral far-field distribution of the Smith-Purcell radiation from one-dimensional gratings, where a shift function in the grating groove position was introduced [36]. By analogy with those previous works on the guided wave-driven metasurfaces, the metasurfaces could be integrated on the side surface instead of the end-facet of the optical fiber to couple the fiber modes into out-plane desired mode in the free space, which is expected to remove the restriction of the limited core area in the conventional design of metasurfaces-decorated fibers. Unlike the planar waveguides used in the previous works where only the optical fields in the top half-space could be manipulated because only the top surface is allowed to be decorated with meta-structures [32–35], the entire side surface of the optical fiber is available to be decorated with meta-atoms and thus the optical fields in the whole space around the fiber could be controlled. Furthermore, compared with the planar waveguides, the polarization of the guided modes in the optical fibers could be easily tuned, which offers an ability to dynamically control the out-plane mode in the free space.
In this work, we proposed an integrated microfiber (MF) device for light focusing. An array of identical dielectric rings has been placed around the side surface of the MF to extract fiber modes into free-space off-fiber radiation. By adjusting the position of each meta-ring along the MF axis, the phase modulation within a range from 0 to 2π could be achieved. We showed that the off-fiber foci with circular-arc line shape could be realized and dynamically rotated around the MF axis by simply tuning the polarization of the guided modes. Moreover, we also demonstrated that the shape of the focus could be further tuned by introducing symmetry breaking into the dielectric rings. We hope our study provides a new dimension for the design of metasurfaces-decorated optical fiber devices.
2. Fundamentals and design principles
Microfibers have attracted extensive concerns in the past decade owing to their unique properties such as strong light confinement, large evanescent fields, great mechanical strength, and low-loss connection [37]. We selected MF as waveguide and nanoscale rings as the meta-atoms to form a ‘guide wave-driven meta-fiber’. Assume that the waveguide lays straight along the x-direction and the propagation constant is β. The total phase shift of the extracted wave provided by meta-atoms along the waveguide can be divided into two parts: (i) the abrupt and spatially variant phase shift induced by each meta-atom Δφ and (ii) the phase accumulation from the propagation β(x). Hence, the phase profile of the extracted wave along the x-direction can be formulated as:
Next, the two phase contributors are discussed. Different from the research which turns the phase (approximate 2π phase shift range) by varying the geometrical parameters of meta-atoms [33–34], we use all the same meta-atoms and adjust each position to focus the incident-guided wave to the free space. All the subwavelength-sized meta-atoms embedding in the waveguide share the same shape and material, making the abrupt phase shift caused by each element equal, so that the influence of the resonance phase need not to be considered. Therefore, we just need to calculate the phase accumulated from the propagation of guided waves. According to the effective refractive index (neff) of the waveguide, the period of each atom can be set as P = 2π/β. Since β is equal to the free-space light wavevector κ multiplied by neff (β = neff × κ), the phase accumulation from the propagation of the guided wave for each period length is 2π (βP = 2π). When the distance between the focal point and each unit is much larger than the wavelength, the periodic block can be regarded as a single-pixel point. Then, the phase (Δϕ) provided by the shift (Δxi) of each meta-atom in the range of the period can be expressed as:
Fig. 1. Schematic and working principle of the guided wave-driven rings. (a), (b) Structure of a unit of wave-driven MF. r: the radius of MF. h: height of the ring. w: width of the ring. The incident-guided wave is along the x-axis. (c) Illustration of the wavefront formation of the extracted wave. Each meta-atom located at a different position of its own period will carry the corresponding phase from the guided wave propagation Δxi: the shift of each meta-atom in the range of the period. The arrows schematically denote the propagation directions of the electromagnetic waves. (d) The extracted light converged at the designed focal point by arrays of circular-ring or broken-ring atoms in free space.
3. Results and discussion
3.1 Driven by meta-rings
As the symmetry of the MF on the y-z plane, we firstly investigated the wave-driven MF with circular rings to demonstrate the relationship between the intensity distribution of the extracted beam and the electric field vectors of the waveguide. We excite the guided wave to propagate along the x-direction at the working wavelength of 1.55µm. In the cylindrical fiber modal, the mode with 0 cut-off frequency is defined as the fundamental mode, which has both electric and magnetic field components in the direction of propagation. Under the same normalized frequency, the energy binding factor of the fundamental mode is the highest, indicating that the fundamental mode is the most easily to be excited. In the MF, the fundamental mode is HE11 mode. Therefore, we will focus on the performance of meta-rings driven by HE11 mode. The radius of MF is 1.5µm and neff of HE11 mode is 1.3996. Silicon is selected as the material of meta-rings. In our work, simulations were performed using commercial finite-difference time-domain (FDTD) software LUMERICAL. The boundary conditions of x-, y-, and z-directions are all set to perfectly matched layers (PML). HE11 guided mode was selected using the mode source and input from the end-facet of the microfiber. The position of each meta-ring on microfiber was determined according to Eqs. (2) and (3). The mesh grid size was set to 50 nm. The size of the nanostructure is carefully designed, which is determined by the disturbance and the radiation to the waveguide. The larger the size is, the more disturbance will be brought to the waveguide which affects the data of neff. Meanwhile, the array of rings acts like a diffraction grating extracting radiation of the optical power from the waveguide into free space, which will lead to the wave decaying [39]. Excessive decaying of waves is not conducive to phase accumulation. In addition, as we use all the same meta-atoms and arrange each position to adjust the phase instead of varying the geometrical parameters, each atom needs to emit the energy into free space as much as possible. Combining the points above, we need to find the meta-atoms array that can effectively extract the wave and only introduce a weak perturbation and very limited attenuation to the waveguide mode. As a result, the meta-ring with 0.1µm width and 0.4µm height is selected to create an array to extract the wave. By controlling the amount of spatial mode overlap between the antenna mode and the guided mode, we can flexibly adjust the amplitude of the extracted wave.
According to formula (3), we design an array of 69 units on MF with a focal length f = 10µm (we chose a shorter focal length to reduce the demand for computational resources). We set the coordinate of the focus on the x-axis as the initial point (x = 0). Hence, we first discuss the influence of the polarization direction of HE11 on the beam steering. The array can convert the HE11 guided mode of MF into free space, shown in Fig. 2. Figures 2(a)(I) and 2(a)(III) illustrate the electric field distribution of input modes and output mode respectively. The color-maps represent out-of-plane electric field component (Ex). The arrows indicate the direction of in-plane electric field components (Ey and Ez). We observed that electric field distributions are almost the same and does not cause the mode conversion of the guided wave in MF. Thus, the metasurface has little influence on the waveguide mode in the optical fiber. Figure 2(a)(II) demonstrates the foci distribution in the x-z plane. The array extracts wave into free space and forms two symmetrical foci, where the brightest strip in the middle represents the intensity field distribution of MF modes. Figure b(I) exhibits the intensity distribution of foci in the y-z plane (the tangent plane along the blue dotted line marked in (a)(II)). The meta-rings extract wave into free space and form a pair of symmetrical foci. Both foci are like semi-circular rings surrounding the MF on the y-z plane. The centers of foci are at z=±11.54µm. The focus distributions on the x-y plane at z = 11.54µm (the tangent plane along the white dotted line marked in Fig. 2(a)(II)), are shown in Fig. (b)(II). To quantitatively analyze the performance of guided wave-driven meta-rings, we also calculated the electric field intensity distribution in the focus plane. The curve is conducted in Fig. (b)(III). The full width at half maximum (FWHM) of the focus is 0.6010µm.
Fig. 2. Demonstration of off-fiber light focusing with guided-wave-driven meta-rings. (a)(I), (a)(III) The color-maps represent out-of-plane electric field component (Ex). The arrows indicate the direction of in-plane electric field components (Ey and Ez). (a)(II) The intensity distribution of transmission. (b)(I) The intensity distribution of foci in the x-z plane (the tangent plane along the blue dotted line marked in (a)(II)). The array extracts wave into free space and forms two symmetrical foci, where the brightest strip in the middle is the intensity field distribution of MF. The centers of foci are at z=±11.54µm. (b)(II) The focus distributions on the x-y plane at z = 11.54µm (the tangent plane along the white dotted line marked in (a)(II)). (b)(III) Electric-field intensity distribution of the focus plane at y = 0 (along the chain-dotted line marked in (b)(II)). The FWHM of the focus points is 0.6010µm.
To further demonstrate the ability of beam modulation, we perform a comparison of the focusing performance with different mode polarization in Fig. 3. Figures 3(a)(I), 2(b)(I), and 2(c)(I) show the electric field distribution Ex of guided mode with different polarization directions, while Figs. 3(a)(II), 3(b)(II), and 3(c)(II) display the corresponding intensity distributions of foci on the y-z plane in free space. We can find that the distributions of the two foci vary with the polarization of the transmission mode. Due to the high symmetry of MF with meta-rings in the y-z plane, the unique advantage is that it can focus the extracted wave into free space to any position and direction by changing the polarization of the incident waveguide. Compared with the waveguide with an array of rectangular meta-atoms, MF integrated with meta- rings are more flexible for light manipulation.
Fig. 3. Demonstration of off-fiber light focusing with different polarization direction. (a)(I), (b)(I), (c)(I) The arrows indicate the direction of in-plane electric field components, and the color-maps represent out-of-plane electric field component in the y-z plane. (a)(II), (b)(II), (c)(II) The corresponding intensity distributions of foci on the y-z planes in free space.
3.2 Driven by broken meta-rings
Based on the quasi-circular focus surrounding MF being realized by an array of meta-rings, we can also design other shaped focal by changing the structure of the atoms from a complete ring to a broken ring. Before shaping the focus, we need to further understand the specific relationship between the electric field distribution and the characteristics of broken rings. The electric field distribution of the extracted waves from four different broken meta-rings are displayed in Fig. 4. The input mode is also HE11 mode, shown in Fig. 4(a). The small figures in the upper right corners of Figs. 4(b), 4(c), 4(d), and 4(e) represent the schematic diagrams of MF integrated with different broken rings in the y-z plane, respectively. The light blue solid circle is the cross-section of MF, and the red broken rings are different meta-atoms. We name these four broken rings structure 1 (S1), structure 2 (S2), structure 3(S3), and structure 4 (S4). The waveguide mode is coupled to the free space by a single atom. When the tangent direction of the broken meta-ring is perpendicular to the electric field vector, more waves can be extracted into free space, as can be seen in Figs. 4(b) and 4(d). On the contrary, when the tangent direction parallels the electric field vector, it will have less effect on the radiation phase, as shown in Figs. 4(c) and 4(e). Light fields with directed spatial distribution can be obtained by the adjustment of the waveguide polarization.
Fig. 4. Electric field distribution of the extracted waves from four different broken meta-rings. (a) The electric field distribution of input mode. The color-maps represent out-of-plane electric field component (Ex). The arrows indicate the direction of in-plane electric field components (Ey and Ez). (b), (c), (d), (e) Electric field distribution of the extracted waves. The small figures in the upper right corners represent the schematic diagrams of MF integrated with different broken rings in the y-z plane, respectively. These four broken rings are defined as S1, S2, S3, and S4.
The distribution of the antenna radiation in the waveguide will also affect the characteristics of focus. Foci constructed by different guided wave-driven meta-rings are represented in Fig. 5. The array also consists of 69 units. Figures 5(a)(I), 5(b)(I), 5(c)(I), and 5(d)(I) display the intensity distributions of foci formed by S1, S2, S3, and S4 in the y-z plane, separately. The dark red circle in the middle is the intensity field distribution of MF. Figures 5(a)(II), 5(b)(II), and 5(c)(II) depict the corresponding focus points in the x-y plane. As the tangent directions of S1 and S3 are perpendicular to the mode polarization, both foci can be formed well. Slightly different from the focus constructed by S3, the electric-field intensity distribution of the focus which is extracted by S1 is wider. Since the tangential directions of both ends of S2 are perpendicular to the mode polarization, it is divided into two weaker focal points in Fig. 5(b). By contrast, as the tangential direction of S4 is parallel to mode polarization, it is hard to focus well. In addition, we define the distance from the center of the focus to the center of MF as R. The centers of the foci formed by S1 and S3 are at R = 11.1µm while the centers of the foci formed by S2 and S4 are at R = 11.3 µm.
Fig. 5. Demonstration of off-fiber light focusing with different broken meta-rings. (a)(I), (b)(I), (c)(I), and (d)(I) The intensity distributions of foci surrounding MF on the y-z plane. The dark red circle in the middle is the intensity field distribution of MF. R: Distance from the center of focus to the center of MF. (a)(II), (b)(II), (c)(II), (d)(II) Representations of focal plane (the tangent plane along the white dotted line marked in (a)(I), (b)(I), (c)(I), and (d)(I)).
In order to quantitatively analyze the performance of different broken meta-rings, we also calculated the electric-field intensity distribution in the focus plane and the comparison is conducted in Fig. 6. The FWHM of the four foci is 0.6603µm, 0.7304µm, 0.6678µm, and 1.4796µm. The values of FWHM show that S1, S2, and S3 can form focus in free space due to part of them being perpendicular to the mode polarization. Meanwhile, S4 is difficult to keep perpendicular to the mode polarization, hence the focal point is relatively diffused. Moreover, we can find that the focus formed by S3 becomes shorter in length and larger in intensity when the input waveguide mode is consistent. It demonstrates the depth and intensity of focus are related to the position and shape of meta-rings. Similarly, when the structure of meta-atom is determined, we can also dynamically adjust the focus characteristics by changing the polarization of the incident mode.
Fig. 6. Electric-field intensity distribution in the focus plane of meta-fiber with four different rings. The FWHM of the three focus points is 0.6603µm, 0.7304µm, 0.6678µm, and 1.4796µm.
4. Conclusion
MF integrated with wave-driven metasurface is proposed in this paper to realize complex functions in free space. Different nanoscale rings and broken rings are selected as atoms to construct the antenna array along the propagation direction of MF. We investigate these two types of rings separately to explore the performance of phase radiation and light manipulation. When the array consists of circular rings, we can form two symmetrical semicircular ring-shape foci surrounding the MF. The dynamic regulation of a pair of symmetrical focus distributions can be achieved by changing the polarization direction of the transmission mode. We can also control the shape and intensity of focus by broken meta-rings. Our work demonstrates that it can focus the extracted wave into free space to the designed position and direction by changing the polarization of the incident modes, while the characteristics of the focus will change accordingly. These detailed investigations offer different alternate ways for the applications of the wave-driven metasurface, which is not confined to the two-dimensional plane but diverges into the three-dimensional cylindrical coordinate space. Our study provides a new dimension for the application of fiber-metasurface devices.
Funding
National Key Research and Development Program of China (2021YFA1401103, 2018YFA030620); National Natural Science Foundation of China (11834007, 12174189).
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 may be obtained from the authors upon reasonable request.
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