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Abstract
A gradient-index optical fiber lens is proposed and fabricated on the tip of a single-mode fiber using focused ion beam milling. Second-order effective medium theory is used to design a gradual change in the fill factor, which ensures a parabolic effective refractive index distribution. The proposed fiber lens design is simulated via the three-dimensional finite-difference time-domain method, and demonstrated through confocal optical measurements. At a wavelength of 1550 nm, the fabricated lenses focus a 10.4 μm mode field diameter exiting the fiber into spot sizes between 3–5 μm, located 4–6 μm away from the fiber tip. Direct coupling into a silicon-on-insulator chip is also demonstrated, where the fabricated gradient-index lens has a coupling efficiency comparable to a commercial lensed fiber.
© 2017 Optical Society of America
1. Introduction
In photonic integrated circuits (PICs), the mode field mismatch between single-mode optical fibers and high-index contrast waveguides is a major contributor to fiber-to-waveguide coupling losses. To overcome this problem, two types of optical couplers are commonly used: out-of-plane (vertical) grating couplers [1–3], which rely on effective phase matching between the fiber and silicon waveguide modes, and in-plane adiabatically tapered edge couplers [4–9]. Both approaches are implemented directly on the PICs, and often require complex fabrication procedures and/or large device footprints. When using grating couplers, coupling losses as low as 1.6 dB have been demonstrated [2,3], when coupled from a standard bare single-mode fiber (∼ 10 μm mode field diameter). With double-stage edge couplers, coupling losses from a similar fiber can be further reduced to about 1 dB [4]. Below 1 dB coupling losses have also been demonstrated in [6–9], when coupling from a fiber with mode field diameters between 2.1 and 4.3 μm. Besides the adiabatic tapers, on-chip mode size converters have also been successfully demonstrated using gradient-index (GRIN) slabs fabricated directly on the PIC [10–12]. Regardless of the choice of the on-chip coupler, reducing the size of the beam exiting the input optical fiber can play an important role in improving the coupling of light into the chip, due to a smaller mismatch between the beam size and the waveguide mode. A common approach is to use commercial optical lensed fibers. By using a small input beam diameter (of a few μm), the on-chip coupler footprint can be significantly reduced and the fabrication procedure might be simplified. As an alternative to tapered lensed fibers, optical fiber lenses using diffractive optical elements [13] and concave fiber tips for mode conversion [14] have also been demonstrated.
In this work, we report on a GRIN optical fiber lens fabricated on the tip of a single-mode fiber, through focused ion beam (FIB) milling. The lens is designed using second-order effective medium theory (EMT) [15,16], for operation at wavelengths around 1550 nm. The GRIN lens can be engineered using two materials with different refractive indices, where the fill factor is gradually changing from period to period, with subwavelength-size features. This particular approach has been demonstrated for a one-dimensional structure defined in a planar silicon-on-insulator (SOI) platform, using electron beam lithography [17]. In our design, the GRIN lens consists of etched concentric rings located at the center of the optical fiber tip. The resulting pattern emulates a parabolic refractive index profile in the radial direction.
2. Lens design
The ability of a radial gradient-index medium to focus light is determined by a parabolic refractive index profile, with a relative permittivity distribution
where n1 is the index of refraction at the optical axis, α the gradient constant, and r the radial distance from the optical axis. This index profile is illustrated in Fig. 1(a). Through effective medium theory, such a lens can be engineered using two alternating materials with different refractive indices n1 and n2. Using second-order EMT [16], the relative permittivities for both polarizations (electric field vector E parallel and perpendicular to the grating vector K) can be calculated as where ε1 and ε2 are the relative permittivities of the two different materials with indices of refraction n1 and n2, respectively, Λ is the period of the grating, and λ is the free-space wavelength of the incident light. fi is the varying fill factor of the structure, defined as a function of the grating period (Λ) and the width (Si) of the i-th region with refractive index n2: In Eqs. (2) and (3), the terms and represent the zeroth-order EMT expressions for both polarizations:
Fig. 1 Illustration of the parabolic gradient-index profile and the second-order EMT lens approximations: (a) GRIN refractive index profile, (b) ellipsoidal-like shape pattern obtained by EMT approximation, and (c) simplified ring-shape design.
Zeroth-order EMT is usually valid for long waves, when the period of the grating is much smaller than the wavelength of light (Λ << λ). When this condition is relaxed, higher-order terms need to be considered in order to correctly describe the effective permittivity. In particular, when the period is comparable to the free-space wavelength (Λ ∼ λ), the use of the second-order EMT approximation is suitable, and has previously been demonstrated [17]. In this work we use second-order EMT, which proves to be a good approximation for our specific values of n1 and n2.
In a single-mode optical fiber under the weakly guiding approximation ((nc − ncl)/nc << 1), the fundamental mode is linearly polarized. If we assume a linear polarization along x, we will need to use both Eqs. (2) and (3), in the x and y directions, respectively, in order to correctly replicate the GRIN lens index profile illustrated in Fig. 1(a). By equating Eqs. (2) and (3) to Eq. (1) for different values of r = iΛ, and numerically solving for fi, we obtain the widths Si of the n2 material, for directions and , respectively. Here, we assume Λ to be constant. An alternative approach would be to assume the width (S) of the n2 material to be constant, and vary the grating period Λ. Once the correct grating dimensions are calculated for the x and y directions, we construct the shape illustrated in Fig. 1(b). While such pattern results in a circular beam spot at the focal plane, it also presents additional fabrication challenges and input polarization sensitivity. Therefore, in our study, we opted for circular patterns, which we obtain by equating Eq. (2) to Eq. (1). This simplifies the fabrication procedure, but results in an elliptical beam spot. The resulting concentric ring pattern is illustrated in Fig. 1(c). In this work, the n2 material is air, defined by etching away material on the tip of an optical fiber, in concentric circular patterns. We fix the outer radii (grating period) of the etched material, while the width of the etched regions, defined by Si, is varied from ring to ring.
3. FDTD simulations
While effective medium theory can be used in different applications, we focus on the GRIN optical fiber lens fabricated on the tip of a single-mode optical fiber. The goal is to focus the fundamental mode exiting the optical fiber into a smaller spot size. The optical properties of the fiber are taken from Corning SMF-28e: the core diameter of the fiber is 8.2 μm, corresponding to a mode field diameter of 10.4 μm. The diameter (D) of the optical fiber lens is chosen to be 10 μm, and α = 0.0162 μm−2. When choosing these values, we considered three main constraints: D needs to be close to the the mode field diameter (ideally larger), the α parameter should be large (resulting in a small spot size at the focal plane), and the resulting features should be easy to fabricate. The selected values for D (which is close to the mode field diameter) and α represent a compromise between spot size, efficiency, and fabrication limitations. For a given α, larger values of D lead to high aspect ratio outermost silica features which are challenging to fabricate. This limitation can be overcome by decreasing the α parameter, which, however, results in a larger spot size. Furthermore, reducing D leads to a smaller lens, with lower efficiency. The upper limit of the value of α was experimentally estimated based on the FIB fabrication process. For values of α larger than 0.0162 μm−2, we observed that the outermost silica feature could not be accurately defined, for the selected D. Smaller values of α present no fabrication issues; however, they result in larger spot sizes. Using α = 0.0162 μm−2, and taking into account that the refractive index on the optical axis matches the refractive index of the core of the fiber (n1 = 1.4678), the effective refractive index at the edge of the fiber lens equals 1.1706. This results in a high variation of the effective refractive index (Δn = 0.2972) over a distance of r = 5 μm. After fixing the parameters D and α, the next important parameter is the outer radii separation (Λ). The smaller the period Λ, the better the match between the engineered lens and the ideal GRIN lens. On the other hand, small values of Λ will lead to high aspect ratio innermost etched features which are difficult to fabricate. In this work, the outer radii separation (Λ) is chosen to be 1 μm − a good compromise between EMT validity and fabrication limitations.
The parameters of the designed lens are summarized in Table 1. The numerical analysis of the lens is performed via the three-dimensional finite-difference time-domain (FDTD) method [18]. In order to observe the focusing effect of the lenses, the intensity distributions along the optical axis (in the xz plane) for different lenses with thicknesses (etch depths) ranging between 1–5 μm are shown in Fig. 2. A fundamental mode at a wavelength of λ = 1550 nm, linearly polarized along the x axis, is launched inside the fiber in the z direction. For visualization purposes, the intensity in each plot is normalized to the maximum value on the optical axis of each fiber lens.
Table 1. Parameters of the GRIN lens structure for FDTD simulations
Fig. 2 Normalized intensity distribution (in dB) in the xz plane, calculated through the FDTD method, for optical fiber lenses with thicknesses of 1–5 μm: (a) 1 μm, (b) 1.25 μm, (c) 1.5 μm, (d) 2 μm, (e) 2.25 μm, (f) 3 μm, (g) 4 μm, and (h) 5 μm.
The performance of the different lenses is assessed by measuring the working distance (WD) and the spot size (the width at 1/e2 intensity points). The WD is identified by determining the distance between the edge of the lens and the position of the maximum field intensity along the optical axis (z axis). It is important to note that the working distance defined here is different from the focal distance of the proposed lens along the x and y axes. In the figures, the edge of the lens is located at z =0 μm. From Fig. 2, the spot size is calculated by extracting the intensity distributions along the x and y axes for the value of z corresponding to the WD, and then fitting them with Gaussian profiles. The working distance and spot size of the resulting lenses are plotted in Fig. 3, as a function of the lens thickness. These results are also compared to those of the ideal GRIN lens described by Eq. (1) and illustrated in Fig. 1(a). From the plots, we can see that as the lens thickness increases, both the WD and spot size decrease. The WD of the proposed EMT-based lens decreases (almost linearly in the studied range) from 9 μm to 3 μm when the lens thickness is increased from 0.75 μm to 5.5 μm, with a maximum WD deviation from the ideal GRIN lens of ∼11% for a 5.5 μm lens thickness. The upper limit of the lens thickness was chosen to be 5.5 μm, which results in an aspect ratio (height/width) of about 10 for the outermost non-etched silica region. From our fabrication experience, it becomes challenging to accurately define these sub-micron structures with aspect ratios greater than 10, in silica. The spot sizes of the proposed and ideal lenses also differ from each other. In particular, for large lens thicknesses, the spot size of the proposed EMT-based lens along the x (y) axis is smaller (larger) than the one of the ideal GRIN lens. This difference is due to the combination of two approximations: the selection of the period Λ (which results in a staircase approximation of Eq. (1)), and the effective medium theory itself. Thus, the spot size has an elliptical shape for the proposed lenses, in contrast to the circular shape of the ideal GRIN lens. For the smaller lens thicknesses, the spot size is closer to circular shape, gradually transforming into an elliptical shape as the lens thickness increases. This elliptical shape is an indication that the focusing power along the x and y axes is different. In particular, the distance at which the beam waist is minimum along the two perpendicular directions is slightly lower and higher than the WD, for the x and y directions, respectively. It is worth mentioning that an elliptical beam may actually be beneficial in some fiber-to-chip coupling applications where the chip input waveguide cross section is rectangular (which will result in an elliptically-shaped mode).
Fig. 3 Performance parameters of the proposed EMT-based and ideal GRIN lenses, as a function of the lens thickness: (a) working distance, and (b) spot size.
4. Fabrication
The fiber lenses were fabricated through focused ion beam micromachining, carried out using an FEI Quanta 3D FEG/FIB Dual Beam system [19]. The protective polymer layer of a single-mode fiber (Corning SMF-28e) was first removed, and the fiber mounted on a metallic sample holder. In order to avoid ion beam induced charging of the sample during the milling procedure, a gold/palladium film of thickness 20 nm was sputtered onto the surface of the fiber (to dissipate the excess charge from the fiber to the metallic sample holder). The milling procedure was carried out over a 10 μm-diameter region centered at the fiber core, using 30 keV accelerated Ga+ ions. A probe current of 10 pA was used for the first two rings and 49 pA for the remaining rings. Before milling the rings, a 10 μm-diameter 50 nm-thick circle was etched away over the fiber core in order to remove the metal, and therefore avoid any plasmonic effects. The ion beam exposure time was varied for the different samples, in order to ensure different milling depths. Four different samples were prepared, with thicknesses of 1.25 μm, 2 μm, 2.25 μm and 4 μm. Scanning electron microscopy (SEM) images of the 1.25 μm-thick fabricated lens are shown in Fig. 4. Before taking the SEM images, a 20 nm-thick gold/palladium film was sputtered onto the surface of the fabricated lens, in order to avoid charging of the sample during imaging.
Fig. 4 SEM images of a 1.25 μm-thick optical fiber lens fabricated on the tip of a single-mode fiber: (a) image of the lens located at the center of the fiber tip, and (b) magnified top view (perpendicular to the optical axis) of the fiber tip, showing five concentric rings with an outer radius spacing of 1 μm.
The different aspect ratios of the rings resulted in a deviation of their pre-set depth values, from one ring to another. Furthermore, the fabricated trenches also vary from the ideal rectangular shape. In order to inspect the depth and shape profile of the milled patterns, FIB cross-section analysis [20] of the structures was conducted. Before cross-sectioning the lens, a platinum (Pt) layer was deposited in order to fill the milled trenches and ensure their protection during the cross-sectioning procedure. A cross-section SEM image of the 2.25 μm-thick lens is presented in Fig. 5. The depth and shape difference between the ideal and the fabricated trenches results in a weaker focusing power of the fabricated lenses. For the ideal lens, the effective refractive index radial profile is parabolic (and therefore has a constant focusing power) throughout the whole thickness of the lens. On the other hand, the fabricated lens shown in the figure can be interpreted as a stack of thinner lenses with decreasing focusing powers as the depth increases. The resulting spot size will therefore be larger than the one predicted from Fig. 3(b).
Fig. 5 Cross-section SEM image of a 2.25 μm-thick lens (angled view with a 52° tilt). The FIB trenches are filled with a Pt layer which helps to protect them during the cross-sectioning process.
One way to avoid the depth difference between the adjacent rings is to fix the etch size of the n2 material (air trenches) and vary the outer radii of the rings, by modifying Eq. (4) to . However, it is important to note that milling high aspect ratio rectangular trenches is not possible through FIB, which will still result in a small mismatch between the lens design and the optical measurements.
5. Optical measurements
The fabricated fiber lenses were tested at a wavelength of 1550 nm, through high-resolution confocal measurements using a high numerical aperture commercial tapered lensed fiber from Nanonics (with a working distance of ∼4 μm and a measured spot size of 1.85 μm). The optical measurement setup is illustrated in Fig. 6.
Fig. 6 Measurement setup used for optical characterization of the optical lensed fibers: PC - polarization controller; S1 and S2 - three-axis micropositioning stages; PD - photodetector; PZT - piezoelectric controllers; IRC - infrared camera; CCD - charge-coupled device camera; BS - beam splitter; O - microscope objective.
Linearly polarized light was coupled to the fabricated lens through a polarization controller (PC), and light collected by the output fiber was measured in a photodetector (PD). The fabricated lenses were mounted on a three-axis micropositioning stage (S1), while the scanning lensed fiber was placed on a high-precision three-axis micropositioning stage (S2) controlled by piezoelectric controllers (PZT). Initial alignment of the fibers was performed via optical inspection through a charge-coupled device (CCD) camera and an infrared camera (IRC). In order to study the beam evolution exiting the fabricated lensed fiber, several cross-section images separated by 0.75 μm (in the z direction) were acquired, with a lateral step size (x and y direction) of 0.2 μm. The measurement results for the 4 fabricated samples (with thicknesses 1.25 μm, 2 μm, 2.25 μm and 4 μm) are shown in Fig. 7. Figs. 7(a) – 7(d) show the beam evolution along the optical axis (xz plane), where the intensity in each plot is normalized to the respective maximum value on the optical axis of each lens. The measured working distance of the lenses are 4–6 μm. It is important to note that the z =0 position in the figures corresponds to the scanning position closest to the EMT lens, which may not coincide with the edge of the lens. This small distance between the scanning position (z =0) and the edge of the EMT fiber lens is due to the alignment which is the primary reason for the small mismatch between the measured and simulated WD results.
Fig. 7 Experimental results for four different samples (lens thicknesses of 1.25 μm, 2 μm, 2.25 μm and 4 μm): (a–d) intensity surface plots (in dB) of the beam evolution, measured from the fiber tip at planes spaced by 0.75 μm in the z direction; (e–h) cross-section of the beams at their focal plane, and (i–l) measured data points (dots) and respective Gaussian fits along x and y directions (dashed line along x and solid line along y).
The beam cross sections at their focal planes are shown in Figs. 7(e) – 7(h), and the respective Gaussian profiles along x and y are shown in Figs. 7(i) – 7(l). At their focal planes, the Gaussian beams have spot sizes of 3.8×4.3 μm (1.25 μm), 3.6×4.2 μm (2 μm), 3.5×3.9 μm (2.25 μm), and 3.1×4.7 μm (4 μm). These values are obtained after deconvolution of the measured cross-section line profiles from the figures with the imaging system response. The FDTD simulation results for those samples, shown in Fig. 3(b), show spot sizes of 3.6×3.7 μm (1.25 μm), 3.2×3.4 μm (2 μm), 3.1 × 3.3 μm (2.25 μm), and 2.6 × 3.1 μm (4 μm), which are lower than the measured values. As discussed, the differences in depth and shape observed in Fig. 5 are the primary reasons for the observed mismatch. However, the obtained measurement results indicate that these EMT lenses, in combination with edge couplers demonstrated in [6,7,9], would still be capable of achieving low coupling losses, since the spot sizes are close to the mode size of the coupler.
The coupling performance of the 4 μm-thick EMT lens was benchmarked against a commercial lensed fiber (same as the scanning fiber) for direct edge coupling into high-index contrast sub-micron waveguides. Both fiber lenses were used to couple light into a standard add-drop ring resonator filter [21] fabricated on a silicon-on-insulator (SOI) platform with a 220 nm-thick top silicon layer. The ring and bus waveguides are 450 nm wide, the ring radius is 20 μm, and the ring-bus gap is 300 nm. At the chip edges, the silicon waveguide is tapered down to 200 nm in order to increase the coupling efficiency. Transverse-electric (TE) light from the tunable laser, controlled by the polarization controller (PC), is coupled into the filter through the input fiber (EMT and commercial), and the light exiting the chip is collected with a standard lensed fiber. The through-port and drop-port responses of the filter are shown in Fig. 8 for both lenses, for wavelengths in the C-band (around 1550 nm). The results are normalized according to the through-port of the commercial lensed fiber (Fig. 8(a)).
Fig. 8 Fiber-to-chip coupling performance in an add-drop ring resonator for TE polarization: (a) commercial lensed fiber, (b) fabricated EMT-based lens.
The results reveal that the coupling performance of the proposed EMT lens is 1–3 dB lower than the commercial lensed fiber. It should be noted that the difference in the shape of the drop-port response below −35 dB is due to residual transverse-magnetic (TM) polarization. A similar coupling performance is observed for TM polarization, with a 1.5–2 dB difference between the two lenses. The primary reason for the measured lower performance is the larger spot size of the EMT lens, which can be reduced by improving the fabrication process and increasing the depth of the milling structures, as indicated by the FDTD simulations in Fig. 3(b). It is also important to note that the bigger spot size of the EMT lens results in a higher alignment tolerance in both x and y directions, which may play an important role in packaging. Optical losses in the EMT lens itself, which were not studied in this work, can also contribute to its lower performance.
6. Conclusion
In this work, we demonstrate a gradient-index fiber lens for efficient fiber-to-waveguide coupling. A radial parabolic effective index profile was achieved through second-order effective medium theory. The lens is fabricated on the tip of a single-mode optical fiber through focused ion beam milling. The performance of the lens is studied through FDTD simulations and validated with optical measurements. At a wavelength of 1550 nm, the fabricated lenses focus a 10.4 μm fiber mode into a spot size of 3.8×4.3 μm, 3.6×4.2 μm, 3.5×3.9 μm, and 3.1×4.7 μm, for thickness values of 1.25 μm, 2 μm, 2.25 μm and 4 μm, respectively, located about 4–6 μm after the fiber tip. The coupling performance of one of the samples (4 μm-thick lens) was compared to a commercial lensed fiber, for direct edge coupling into a silicon photonics chip. The results obtained for the fabricated lens show a 1–3 dB loss, which is due to its larger beam spot size. For coupling into high-index contrast waveguides, the spot size can be reduced by increasing the depth of the milling. In addition, these lenses are suitable for coupling into edge couplers where beam spot sizes of 2–4 μm are required. The flexible design and fabrication procedure presented here can also be easily adapted to other applications, such as coupling at an angle to the chip surface (for grating fiber-to-chip couplers), or matching the focused beam profile with specific non-Gaussian field distributions in edge or grating fiber-to-chip couplers.
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