1 Introduction

Optical fibers has become indispensable for their use in guided wave optical communication systems, free-space optical communication systems and in quantum communication systems due to their ability to guide and transmit optical signals over long distances with minimal attenuation (Ghatak and Thyagarajan 1998). However, when employed in free-space optical communication systems, they are less suitable for directly transmitting the optical beam over a long distance due to the large divergence of the emerging beam from the fiber. As a result, the optical beam needs to be coupled with conventional free-space-to-fiber couplers or collimators before launching the light into free space (Li and Erdogan 2000; Takenaka et al. 2012). These components are comparatively bulky and generally consists of lens assemblies, and can introduce aberrations and distortions to the light beam, particularly when attempting to match the Gaussian phase profile of the fiber mode with the spherical phase of the lenses (Goodman 2005). This mismatch in phase between the ideal Gaussian mode of the fiber and the actual mode at the output of the lens can lead to inefficiencies in coupling and losses in the system. This calls for the design of specialised optics that can generate a Gaussian phase by modulating the phase of the light coming from the fiber.

Metasurfaces consisting of two-dimensional arrays of sub-wavelength scatterers are known to be able to generate complex wavefronts and have excellent focusing and beam shaping properties. With careful design of the geometry, size, and arrangement of these sub-wavelength scatterers, metasurfaces can control the behavior of light in ways that are not possible with natural materials. (Liu et al. 2020; Kim et al. 2021; Khorasaninejad and Capasso 2017; Tseng et al. 2018). Meanwhile, there has also been a growing research focus on developing optical fiber integrated devices owing to their reduced footprint and consolidated functionalities (Pahlevaninezhad et al. 2018; Zaboub et al. 2016; Li et al. 2020; Pisco and Cusano 2020; Yang et al. 2019). Serving as a widely utilized light waveguide, the optical fiber facet inherently serves as a light-coupling platform for a great range of applications. In fact, metalens can be formed on to a fiber facet, by utilizing techniques such as “decal transfer” and “nanoskiving” (Xu et al. 2007; Xiong and Xu 2020) so as to realize a “metalens integrated optical fiber”, which not only collimates the optical beam coming out of the fiber, but also preserves the Gaussian profile of the emerging beam.

The main issue in the design of such metalens integrated optical fiber is the impedance mismatch at the fiber/metasurface interface due to difference in the refractive index (RI) which leads to a lower collimating efficiency (Pahlevaninezhad et al. 2018; Gelfand et al. 2014; Juhl et al. 2019). A study carried out by Ye et al. (2021) has explained the design of the metasurface for collimating the 633 nm beam coming out of a single-mode fiber with the help of a low RI metasurface on the fiber facet. However, optical fibers are known to have a very low attenuation for the 1550 nm wavelength (Henschel et al. 1996; Li et al. 2005) and therefore is the widely used wavelength in most of the optical communication networks. Further, the 1550 nm wavelength is considered to be more suitable for free space daylight quantum communication owing to lower solar irradiance around this wavelength (Liao et al. 2017). To the best of our knowledge, however, there has been no reported work on the design techniques to collimate the 1550 nm beam from a single mode fiber using metasurfaces.

Fig. 1
figure 1

a Schematic diagram of a beam coming out of single mode fiber, b gaussian beam radius of the propagating beam outside the SMF facet, c beam footprint as it propagates outside the fiber, d schematic diagram of a beam coming out of single mode fiber upon modulation with a metalens

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Fig. 2
figure 2

Intensity of the propagating Gaussian beam along the substrate for a low RI material substrate, c SiO\(_{2}\) substrate, normalized intensity at the end of the substrate along the radial cross-section for b low RI material substrate, d SiO\(_{2}\) substrate

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In this paper, we present the optical design of a fiber coupled metalens for collimating the 1550 nm Gaussian beam from a single mode fiber (SMF). The design for the metalens has been carried out using a low RI material (Ye et al. 2021) and also a high RI material (TiO\(_{2}\)), formed on SiO\(_{2}\) substrate. Existing literature has demonstrated the fabrication of TiO\(_{2}\) nanopillars on SiO\(_{2}\). Moreover, the RI of SiO\(_{2}\) (1.44 at 1550 nm) is comparable to the core RI of the SMF, which enables low impedance mismatch at the fiber facet due to minimal RI difference between the two materials. A comparative study between both of these materials for use as metalens is also presented, and it is observed that both these materials are able to achieve diffraction limited collimation. This can pave the way for the formation of fine pointed pencil beams that can be highly beneficial for satellite-to-ground opto-quantum communication payloads with lower footprints. Section 2 of the article presents the design methodology and system considerations for the design of metalens. In Sects. 2.1 and 2.2, the simulation methods for designing and characterizing the unit cell and the full metasurface are presented, respectively. Further, the far field results from the optical fiber after the phase modulation with the metalens are also presented. Finally, the paper is concluded in Sect. 3.

2 Design methodology and simulation

Fig. 3
figure 3

Radial cross section of phase profile of the Gaussian beam out of the substrate for a low RI material, d SiO\(_{2}\). The phase wrapped between [0,2\(\pi\)] for b low RI material, e SiO\(_{2}\). Full wrapped phase of the beam out of the substrate for c low RI material, f SiO\(_{2}\)

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The emitted light beam from a cleaved single-mode optical fiber (SMF) exhibits significant divergence upon exiting the fiber facet. In cases where this divergence is notable, achieving a high Numerical Aperture (NA) metasurface necessitates a substantial refractive index contrast between the scatterers and their immediate surroundings. This is crucial for ensuring that the nanostructures act as discrete scatterers with minimal cross-coupling (Arbabi et al. 2015). In this paper, we have used a single mode fiber SMF-28 made by Corning (Kowalevicz Jr and Bucholtz 2006), whose divergence has already been experimentally verified and is in close match to the divergence of the fundamental Gaussian mode exiting from the fiber facet (Saleh and Teich 2019) which is given by

$$\begin{aligned} \textrm{E}(r, z)=A_0 \frac{\omega _0}{\omega (z)} \cdot \exp {\left[ -\frac{r^2}{\omega ^2(z)}\right] } \cdot \exp {\left( -i\left[ k\left( z+r^2 / 2 R(z)\right) -{\text {arctan}} \left( \frac{z}{z_0}\right) \right] \right) } \end{aligned}$$
(1)

where \(\omega (z)\) is the radius of the fundamental mode Gaussian beam at an axial distance of z upon exiting the fiber, \(\omega\) \(_{0}\) is the beam waist at z = 0 (fiber tip), \(z\) \(_{0}\) is the Rayleigh length and R(z) is the radius of curvature of the Gaussian wavefront. From Fig. 1, it can be seen that the divergence of the Gaussian beam from the core of the cleaved SMF facet is around 4.6 degrees, which is very large and needs to be reduced by collimation. In this work, we use a metalens that is placed at the facet of the optical fiber as shown in Fig. 1d, which acts as a collimator reducing the beam divergence. The core radius of the optical fiber is about 4.1 μm. The spot size of the beam emerging from such a fiber will also have a dimension similar to that of the fiber core radius. If the beam of such spot size is collimated, the diffraction limited divergence will still be large; therefore, initially the beam is propagated through a substrate where it expands and then is collimated using the metalens. It is observed that the thickness of the substrate is crutial as it limits the degree of the diffraction. We have set the substrate thickness such that the beam upon exiting the substrate has a diameter comparable to the fiber cladding (62 μm) thereby limiting the metasurface diameter to be less than the fiber cladding. As a result, the thickness of the SiO\(_{2}\) substrate was obtained to be 0.8 μm while for the low RI substrate, the thickness was 0.83 μm.

Figure 2 shows the profile of the beam inside the substrate and the normalized intensity at the fiber end for both the kinds of substrate used. It is seen that the radius of the beam emerging from the end of the substrate is \(\sim\) 52 μm. The phase of the Gaussian beam at that location is also shown in Fig. 3 for the low RI substrate and SiO\(_{2}\) substrate. The metalens needs to be placed at the end of the substrate to provide an equal and opposite phase to the incoming beam emerging from the substrate and therefore cancel out its Gaussian wavefront.

2.1 Simulation of the unit cell

Fig. 4
figure 4

Top schematic view of the a elliptical, b circular unit cell of the metalens. c Perpetual schematic view of the circular unit cell

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As a part of the design methodology of the metalens, at first, the unit cell of the meta-structure needs to be characterized. We have considered two types of nanorods as the unit cells for the metalens i.e. having circular and elliptical shape as shown in Fig. 4. The characterization is essential in order to determine the phase profile imparted to a plane wave by the different arrangement and dimensions of meta-atoms (Here meta-atom refers to the individual sub-wavelength unit cell having a circular or elliptical shape). This data is later used to design the full metalens. Initially, the metalens in which the meta-atom having a circular shape is characterized. The lattice constant (the spacing between individual meta-atoms) of the nanoposts, in principle, should satisfy two conditions: it should be smaller than the wavelength in the substrate (\(\lambda\)/n) and should be greater than the diffraction condition (\(\lambda\)/2n) Fan et al. (2018). The radius of the unit cell is varied such that the maximum diameter is less than the period of the unit cell. Further, the entire range of the diameters considered should be able to fully modulate the phase of the incoming beam of light from 0 to 2\(\pi\). Therefore, the period for the TiO\(_{2}\) was taken as 0.6 μm while for the low RI material it was 0.98 μm. Moreover, the maximum exit phase that can be provided by the unit cell to the transmitting beam is \(\phi _{max}\)= 2\(\pi\) \((n-1)d\)/\(\lambda\), where d is the height of the unit cell, and \(\lambda\) is the operating wavelength. Hence, to achieve full phase coverage, we set \(\phi\) higher than 2\(\pi\) and fix the height of the nanorod height (d).

Fig. 5
figure 5

Phase imparted by the cicular metalens unit cell with different height and radius for a low RI and e TiO\(_{2}\) material. Transmission level of the metalens unit cell for b low RI and f TiO\(_{2}\). The Phase vs radius profile of the unit cell at c 3.9 μm for Low RI material and g 1.7 μm for TiO\(_{2}\). Transmittance of the unit cell at c 3.9 μm for Low RI material and g 1.7 μm for TiO\(_{2}\)

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Fig. 6
figure 6

Phase imparted by the elliptical metalens unit cell with different radius R1 and R2 for a low RI and b TiO\(_{2}\) material. Transmission level of the metalens unit cell for c low RI material and d TiO\(_{2}\) material

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Table 1 Ranges for parametric sweep for the design of metalens for Low RI and TiO\(_2\) material, for circular and elliptical unit cell
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Optical simulations were performed for the metalens structure using Lumerical FDTD with periodic boundary conditions on all four sides of the unit cell and PML boundary conditions along the top and bottom of the unit cell. A sweep of height and radius of the unit cell of the metalens is performed to obtain the most appropriate height and radius profile of the unit cell; the details of the parametric sweep carried out in simulations is presented in Table 1. It can be seen from Fig. 5a–d that for the metalens having circular nanorod made with low RI material having a height of 3.9 μm and a maximum radius of 0.47 μm, the entire phase can be covered while maintaining a transmittance greater than 90\(\%\). On the other hand, for the metalens made with TiO\(_{2}\) material (Fig. 5e–h), this can be achieved with a post height of 1.7 μm with the maximum radius of 0.28 μm, thereby significantly reducing the amount of material that needs to be used for forming metalens.

Fig. 7
figure 7

Contour plot of the target phase profile for the a metalens made of Low RI material, b metalens made of TiO\(_{2}\) material. Target vs simulated phase profiles along the radial cross section for the c metalens made of Low RI material, d metalens made of TiO\(_{2}\) material

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The radius of the nanorod (R1 and R2) is further varied in the two orthogonal directions so as to realize an elliptically shaped nanorod (nanopost) whose top view is shown in Fig. 4a. The resultant phase profile and transmission obtained for this configuration is plotted in Fig. 6. It can be seen from Fig. 6 that, with elliptic nanoposts, there exists a possibility of increased flexibility in the design as we can get a wider range of phase modulation just by varying the radius along one direction, for both low RI and TiO\(_{2}\) material. Having both radii “R1" and “R2" to be the same in elliptical nanopillars also allow for a full phase sweep needed for metalens functionality as indicated by dotted line in Fig. 6. In the present design for metalens, we have considered only the circular shape (i.e. both R1 and R2 to be same) for the unit cell, due to simplicity as well as considering the ease in fabrication.

2.2 Simulation of the full lens structure

Fig. 8
figure 8

Top view schematic diagram of the full metalens which consists of meta-atoms whose radii is varied depending on the target phase to be imparted to the beam

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Fig. 9
figure 9

The normalized cross-sectional intensity of the optical beam measured at a distance of 5 mm away from the substrate with a no metalens, c metalens made of Low RI material, f metalens made of TiO\(_{2}\) material. The normalized cross-sectional intensity of the optical beam measured at a distance of 10 mm away from the substrate with b no metalens, d metalens made of Low RI material,s, g metalens made of TiO\(_{2}\) material. Intensity profile along the radial cross-section at 10 mm from the substrate for the beam after passing through e low RI Metalens, h TiO\(_{2}\) metalens

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From the simulations carried out for a unit cell, the phase modulation attainable for each nanorod radius is obtained for different nanorod materials. Figure 7 depicts the attainable phase modulation for various radius of the nanowire made of different materials. Using this data, the modulation of the phase offered by the metalens should be designed so as to flatten the phase of the incoming Gaussian wavefront. This can be done by taking the opposite value of the phase of the beam at the top of the dielectric film and use it as the target phase profile for the metalens. The target phase profile for the metalens is given by

$$\begin{aligned} \phi (x, y)=-\left[ k\left( z+\frac{x^2+y^2}{2 R(z)}\right) -\arctan \left( \textrm{z} / z_0\right) \right] \end{aligned}$$
(2)

where (x, y) is the Cartesian coordinate of the spatial phase profile.

Figure 7a and b shows the contour plot of the target phase profile of the metalens made out of low RI material and TiO\(_{2}\), respectively. Figure 7c and d show the target phase and the obtainable phase of the meta lens along the cross section. From Fig. 7c and d, it is seen that the simulated phase is in close agreement to the target phase (for both the materials) which implies that the metalens is able to successfully modulate the phase of incoming beam emerging from the fiber. The top view schematic illustration of the metalens is shown in Fig. 8. The radius of the meta-atoms is varied depending on the target phase to be imparted in that region according to Fig. 6.

In order to test the performance of the metalens, we have used the Fresnel propagation method to evaluate the beam properties after it has propagated through the metalens structure. Figure 9a and b display the cross-sectional intensity of the optical beam after it has propagated a distance of 5 mm and 10 mm away from the substrate, respectively, without the usage of metalens. It can be seen that with no metalens present, the spot size of the beam increases significantly as a result of larger beam divergence. On the other hand, with the presence of the metalens, it is evident that the beam divergence is minimal as can be seen from the spot sizes observed in Fig. 9c, d, f and g. It is noteworthy that the metalens made out of low RI material and well as that made of TiO\(_{2}\) is helpful in minimizing the beam divergence. The normalized intensity plotted along the radial cross-section at 10 mm from the substrate for the beam after passing through the metalens made of low RI material and TiO\(_{2}\) is shown in Fig. 9e and h, respectively. It is seen that the increase in the beam spot size was only due to the diffraction limited divergence which in turn is determined by the wavelength of the beam and the diameter of the metalens. This can be further minimized if a large area metalens is used for collimation of beam from optical fiber to free space.

3 Conclusion

We have carried out FDTD simulations for the theoretical realization of a metalens for collimating 1550 nm Gaussian fundamental mode from a single mode fiber facet. The design has been implemented using both Low RI material and a high RI material (TiO\(_{2}\)) for forming the metalens. It has been observed that both these materials can be used for the collimation of the Gaussian beam. Optimal values for the length and the radius of the meta-atoms for forming metalens using low and high RI materials were determined. The designed metalens can effectively modulate the phase of the Gaussian beam emerging from the fiber, and can achieve diffraction limited divergence, thereby significantly reducing the spot size of the laser beam, which in turn will be highly beneficial for long distance free space optical and quantum communication, and other applications where a collimated beam is desired.