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Abstract
Coupling characteristics between a single mode fiber (SMF) and a waveguide embedded in a glass chip via a graded index fiber (GIF) tip are investigated at a wavelength of 976 nm. The GIF tips comprise a coreless fiber section and a GIF section. A depressed cladding waveguide in a ZBLAN glass chip with a core diameter of 35 μm is coupled with GIF tips that have a range of coreless fiber and GIF lengths. An experimental coupling efficiency as high as 88% is obtained while a numerical simulation predicts 92.9% for the same GIF tip configuration. Since it is measured in the presence of Fresnel reflection, it can be further improved by anti-reflection coating. Additionally, it is demonstrated that a gap can be introduced between the chip waveguide and the GIF tip while maintaining the high coupling efficiency, thus allowing a thin planar optical component to be inserted. The results presented here will enable miniaturization and simplification of photonic chips with integrated waveguides by replacing bulk coupling lenses with integrated optical fibers.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Graded index fibers (GIFs) have previously been suggested as lenses and their optical characteristics have been extensively studied [1]. GIFs (which are also known as gradient refractive index, or GRIN fibers [2]) have been used for light collimation and collection. One of the most successful applications is in vivo optical coherence tomography [3–5] since GIF lens probes enable minimally invasive imaging of individual cells in deep tissues. Other applications including sensors for physical deformation [6], temperature [7], and strain and temperature simultaneously [8], have been realized because GIF microlenses allow fibers to be used in a small working space. Other reported applications include depth-resolved microendoscopy [9], and on-chip microfluidic spectroscopy using GIF tips [10].
As applications for this technology emerge, more research is being conducted to understand the physical properties of GIFs in various configurations. For example, light propagation in GIFs connecting two single mode fibers (SMF) was studied for optical interconnect applications [11]. Enhancement of fringe visibility in a Fabry-Pérot cavity utilizing GIFs has also been demonstrated [12,13]. Design guidelines for optical coherence tomography were provided by advanced modeling of fiber-optic probes consisting of GIF lenses and phase masks using the Beam Propagation Method (BPM) [14].
In addition to the imaging and sensing applications, a GIF tip has potential to be applied as a microlens that bridges fiber optics to free-space optical components and systems such as waveguides, lenses, and cavities. Replacing bulk lenses with GIF lenses provides advantages in miniaturization and simplification of such systems [2]. Moreover, free-space to fiber coupling is essential for free-space optical communications [15]. With the efforts to integrate fiber and free-space optics, coupling properties of GIF-based optical systems have been previously studied [16,17]. Fabrication of equally-spaced and aligned multiple GIF microlenses have been demonstrated [16], and the coupling efficiency and reflectance of a pair of GIF-terminated fibers facing toward each other have been analyzed [17].
In this paper, we explore using GIF tips to couple light between an SMF and a waveguide inscribed within a ZBLAN glass chip. Thanks to its low phonon energies and high rare-earth solubility, ZBLAN (ZrF$_4$-BaF$_2$-LaF$_3$-AlF$_3$-NaF) is an excellent host for rare-earth ions that emit light from the ultra-violet to the infrared, enabling the development and demonstration of high-efficiency ZBLAN fiber lasers [18,19]. While ZBLAN can emit in a wide range of wavelength bands through the use of different rare-earth dopants, the mechanical fragility of ZBLAN fiber has limited its uptake in applications. In our previous work, we addressed this problem by using depressed cladding waveguides in bulk ZBLAN glass chips and successfully demonstrated waveguide lasers at 1 $\mu m$ [20], 1.5 $\mu m$ [21], and 2.1 $\mu m$ [22], as well as other wavelengths. However, such implementations have relied on bulky free-space optics to couple light into and out of these ‘chip waveguides’.
For more compact integration in such assemblies, it is desirable to replace the bulk lenses used to couple the pump laser to the chip waveguide by GIF tips. The usage of GIF tips provides advantages of structural simplicity and compactness over a free-space coupled optical system. For instance, the alignment becomes simpler and more controllable using two GIF tips than one bulk lens for a dual-waveguide chip laser which has two waveguides in one chip [23]. In addition, the cavity length can be reduced to support fewer longitudinal modes which can be advantageous for some laser applications such as high repetition rate frequency combs. Efficient coupling from optical fibers to chip waveguides to integrate fiber and free-space optics have been studied actively in silicon photonics [24–27]. The concept of matching the mode profiles of the fiber and the chip waveguide is also applicable to ZBLAN waveguides. Couplers in silicon photonics typically reduce the fiber modes to match the guided modes in smaller waveguides which are usually on the order of the wavelength. The waveguides used in our previous ZBLAN chip waveguide laser work, however, have larger diameters than the core of the SMF thus the implemented coupler must expand the beam to match the mode size and also reduce divergence to match the smaller numerical aperture (NA) of the chip waveguide. Although thermally expanded core SMFs offer an alternative approach to customizing beam diameters and reducing the NA [28,29], GIF tips generally provide greater versatility in modifying the divergence angle and beam size. We have previously demonstrated GIF tip coupled frequency comb lasers based on ZBLAN chip waveguides, where we briefly reported coupling properties of a GIF tip to a chip waveguide for that particular case [23]. As the previous report is limited to one specific GIF tip without investigation of coupling mechanism, generalization, and optimization, a comprehensive analysis including various GIF tip configurations is required to extend this work for future applications.
Here, we investigate the coupling from a chip waveguide in ZBLAN glass to an SMF via various GIF tip configurations at the wavelength of 976 nm which is a common pumping wavelength for laser applications, most notably for erbium and ytterbium-doped lasers. The coupling efficiency is measured and simulated for different combinations of coreless fiber and GIF lengths. The divergence of the beam from the GIF tip is experimentally and numerically studied to better understand the relation between the GIF tip configurations and the coupling efficiency. Furthermore, the air gap length between the chip and the GIF tip is varied to observe the dependency on the size of the gap. The coupling in the opposite direction (from SMF to chip waveguide) is also discussed.
2. GIF tips and chip waveguide preparation and optical properties
The GIF tips in this work comprise a coreless fiber section and a GIF section spliced to an SMF as shown in Fig. 1(a). This GIF tip design allows the fiber mode (LP$_{01}$) in the SMF to be expanded in the coreless fiber section followed by collimation in the GIF section so as to match the mode supported by the chip waveguide [13,30]. Thorlabs 1060XP fiber and Thorlabs GIF625 were used for the SMF and the GIF, respectively, while the coreless fiber is pure silica. A Fujikura CT-100 cleaver incorporating appropriate thickness steel shims was used to cleave the fiber sections to the correct lengths and were subsequently verified by optical microscopic imaging. Splicing of the GIF to the coreless fiber and the SMF to coreless/GIF was performed using a Fujikura FSM-100P+ ARCMaster specialty fiber splicer. The typical splice parameters used were an arc current of 15–20 mA and an arc duration of 1–2 s. A microscope image of a representative GIF tip is shown in Fig. 1(c).
Fig. 1. (a) Schematic of the coupling geometry between a chip waveguide and a GIF tip. (b) An optical microscope image of the chip waveguide in ZBLAN glass. Scale bar is 20 $\mu$m. The dark ring is the depressed cladding ($\Delta n = -0.001$). (c) An optical microscope image of a GIF tip. Scale bar is 100 $\mu$m. (d) Coupling efficiency measurement setup. LD: laser diode ($\lambda = 976$ nm); L1 ($f_1=4.55$ mm) and L2 ($f_2=20.0$ mm): lenses; D: power meter.
The ZBLAN substrate for the coupling experiment was chosen from locally available stock for its transparency and guiding of the 976 nm light. It was 0.5 mol% Pr doped ZBLAN glass supplied by the Institute for Photonics and Advanced Sensing (University of Adelaide, Australia) and diced/polished into $12\times 8\times 2$ mm$^{3}$ glass chips. The depressed-index cladding waveguides were fabricated in these substrates along the long axis using an in-house femtosecond laser direct-write setup and technique described in Ref. [20]. In particular, the cladding comprised a series of overlapping rods inscribed around the unmodified waveguide core, inscribed using a $\sim$240 mW, 5 MHz pulse train of $\sim$200 fs long pulses at a wavelength of 515 nm, focused through a 100$\times$ oil immersion objective (Zeiss, N-Achroplan 100$\times$/1.25 NA Oil), and dynamically positioned using a custom air-bearing translation stage (Aerotech, ABL9000). A microscope image of the ZBLAN chip waveguide end facet is shown in Fig. 1(b).
The experimental configuration to measure the coupling efficiency from the chip waveguide to the SMF via GIF tips is shown in Fig. 1(d). The laser beam at $\lambda =976$ nm from a fiber-coupled butterfly laser diode (Thorlabs BL976-PAG900) is imaged into the waveguide by a pair of lenses, L1 and L2, with focal lengths of 4.55 and 20.0 mm, respectively. The GIF tip was aligned in its tilt, rotation, and transverse position, and then brought into contact with the other face of the ZBLAN chip waveguide to couple the guided mode to the SMF. A germanium photovoltaic detector (Thorlabs S112C) was placed at the output end of the SMF to measure the power of the beam averaged over a 4-minute interval.
The core and cladding diameters of the SMF (Thorlabs 1060XP) are 5.8 $\mu$m and 125 $\mu$m, and its refractive indices are 1.4507 and 1.4439, respectively, as shown in Fig. 2(a). The numerical aperture (NA) of the SMF is 0.14. The optical and geometric parameters for the SMF were provided by the manufacturer. The diameter of the coreless fiber is 125 $\mu$m and its refractive index is 1.4507 at the wavelength of 976 nm as given by the Sellmeier equation with the known coefficients for fused silica [31]. The GIF has a power-law refractive index profile given by [14]
Fig. 2. (a) Refractive index profiles of the SMF, coreless fiber, GIF, and chip waveguide at $\lambda =976$ nm. (b) Simulated mode profiles of the SMF and chip waveguide. The calculated mode widths defined by $1/e$ in the $E$-field amplitude are 6.2 $\mu$m and 31.2 $\mu$m for the SMF and the chip waveguide, respectively.
Using the geometric and optical parameters of the SMF and the chip waveguide, the guided modes in the SMF and the chip waveguide are obtained using the Finite Element Method (FEM) [33] of the software package RSoft FemSIM with a resolution of 10 nm, as shown in Fig. 2(b). The widths defined by $1/e$ of the $E$-field amplitude are 6.2 $\mu$m and 31.2 $\mu$m for the SMF and the chip waveguide, respectively. The SMF mode is a bound LP$_{01}$ mode, while the chip waveguide mode is a leaky mode with its tales penetrating into the depressed cladding. Coupling efficiency between two modes, assuming matched NA, can be calculated by the overlap integral [11],
3. Coupling efficiency: chip waveguide to GIF tip
The experimental coupling efficiency is defined as $\eta =P_2 / P_1$, where $P_1$ is the power at the output end of the chip waveguide without the GIF tip while $P_2$ is the power at the SMF end of the GIF tip in the setup shown in Fig. 1(d). $P_1$ ($P_2$) is determined by measuring the power directly after the chip waveguide/air (SMF/air) interface and correcting for the BPM calculated Fresnel reflection losses at the chip waveguide/air and the SMF/air interfaces. As both the chip waveguide and GIF tip faces are not anti-reflection (AR) coated in this experiment, the interaction of the reflections from both of these faces (i.e. an etalon) represents an unaccounted-for modulation in our coupling efficiency measurement. Thus, our observed efficiency is related to the etalon-corrected efficiency by $\eta _{\mathrm {measured}}=\gamma \left (l \right ) \eta _{\mathrm {actual}}$, where $\gamma \left (l \right )$ represents the coefficient of transmission from inside the chip waveguide to inside the GIF that results from this etalon formed by these interfaces separated by gap $l$. BPM simulations discussed later allow us to establish the depth of modulation which is expected to range from 1 to 0.83, and when averaged over a complete period is 0.91. Unless otherwise noted, for the purposes of this work we assume in our measurements that the average effect of the etalon is observed due to mechanical and thermal perturbations of the system over the measurement time. The power measurement and normalization method is described in further detail in Supplement 1.
Fifteen (15) GIF tips were prepared with different lengths of coreless fiber ($0 \leq l_{\rm CL} < 300~\mu$m) and GIF ($0 \leq l_{\rm GIF} < 400~\mu$m) sections as listed in Table 1 based on our previous study [30], and we measured the coupling efficiencies from the chip waveguide to the GIF tips as shown in Fig. 3(a). As a visual aid in Fig. 3(a), the area where $\eta$ has been measured to be $> 60\%$ is shaded, which roughly spans from around a GIF length $l_{\rm GIF} = 110~\mu$m and a coreless fiber length $l_{\rm CL}=190~\mu$m to $l_{\rm GIF}=240~\mu$m and $l_{\rm CL}=40~\mu$m, exhibiting a trend of decreasing $l_{\rm CL}$ with increasing $l_{\rm GIF}$. The highest coupling efficiency of $88\%$ is measured with GIF tip G5, the GIF tip of $l_{\rm GIF} = 185~\mu$m and $l_{\rm CL} = 53~\mu$m. Since it is measured in the presence of Fresnel reflection, it can be further improved by AR coating. In contrast, the coupling of the chip waveguide directly to the SMF (G1) shows only $26\%$ coupling efficiency. It is also observed that the coupling efficiency is as low as $\eta < 10\%$ for the GIF tips with $l_{\rm GIF} > 300~\mu$m and $l_{\rm CL} > 180~\mu$m (G14 and G15), a deterioration of coupling using these configurations. It can be understood that the chip waveguide couples efficiently to G5 while poorly to G14 and G15 due to the beam divergence and the beam size of GIF tips, which will be discussed later in detail.
Fig. 3. (a) Measured and (b) simulated coupling efficiency maps. Datapoints in (a) with black labels are specific GIF tips considered in more detail within the text. Area with efficiency ($\eta$) over 60% is shaded in (a) for visual aid.
Table 1. List of GIF and coreless fiber configurations. $l_{\rm GIF}$: GIF length, $l_{\rm CL}$: Coreless fiber length. The length measurement error is $\pm 2~\mu$m.
As a comparison, the coupling properties are simulated using the Beam Propagation Method (BPM) [34] via RSOFT BeamPROP. The BPM is a wave optics simulation method used to model light propagation in optic and photonic devices including bend loss calculation in a fiber [35] and resonance in a photonic crystal fiber [36]. Coupling efficiencies are calculated for a range of combined coreless fiber ($0 \leq l_{\rm CL} < 300~\mu$m) and graded-index fiber ($0 \leq l_{\rm GIF} < 400~\mu$m) lengths with an increment of 10 $\mu$m. The aforementioned geometry and refractive indices of the fiber segments and the chip waveguide are used for the simulations. The simulation resolutions are 0.05 ${\mathrm {\mu }}$m and 0.25 ${\mathrm {\mu }}$m in the transverse and propagation directions, respectively, as the refractive indices vary rapidly in the transverse direction while they are uniform along the propagation direction except at the interfaces between different fiber components and the chip waveguide where a finer resolution with the grid size of 0.05 ${\mathrm {\mu }}$m is applied in the vicinity of $\pm 5~\mu$m from such interfaces. Circular symmetry is applied to enhance the computation speed as all the components in the simulation are circularly symmetric. The FEM calculated chip waveguide mode shown in Fig. 2(b) is launched at the input end face of the chip waveguide in the BPM simulation. The beam couples to the GIF section, passes the coreless section, and enters the SMF to propagate along the fiber. The simulated power is monitored in the core of the SMF to eliminate the cladding coupling and the coupled power at the interface between the chip waveguide and the GIF tip is obtained by normalizing by the input power.
The simulated coupling efficiency map as a function of GIF and coreless lengths is shown in Fig. 3(b). A region of high coupling efficiency ($\eta > 90\%$) is found in the plotted map, where the GIF length decreases as the coreless fiber length increases which is consistent with the experimental result. The maximum simulated coupling efficiency is found to be $\eta _{\rm max}=97.2\%$ at $l_{\rm GIF}=150~\mu$m and $l_{\rm CL}=110~\mu$m. The experimental and simulated coupling efficiencies exhibit the same trend over the given GIF tip configuration space. It should be noted that the reflection loss is minimal since a perfect contact without a gap between the chip waveguide and the GIF tips is assumed, unlike the experimental situation where a gap between the interfaces cannot be avoided. To more closely simulate the experimental situation, the coupling efficiency including a small gap ($<2 \mu$m) was also explored and the results are presented in Supplement 1. In the presence of the gap, $\eta _{\rm max}=90.1\%$ at $l_{\rm GIF}=150~\mu$m and $l_{\rm CL}=110~\mu$m showing additional $\sim$7% point of reduction in coupling due to the multiple reflections. In the experiment, the GIF tips are contacted to the chip waveguide and the gap is determined by the surface roughness and the cleave angle, which can be a few tens of nm. This specific result demonstrates how an AR coating on the GIF tip and the chip waveguide interfaces would significantly improve the coupling efficiency. Index matching liquid could also be applied to the gap to suppress Fresnel reflections, but was not explored as a part of this work.
For further validation of the experimental results, we individually measured the output beam profiles at the end faces of a GIF tip (G5) and the chip waveguide using a $10\times$ objective (NA=0.25) to image the beam onto a laser beam profiler (Spiricon SP503U) as shown in Fig. 4. The details of the measurement setup and the method are given in Supplement 1. The coupling efficiency ($\eta _\text {mode}$) from the intensity profiles can then be calculated by Eq. (2) assuming perfect alignment and low beam divergence. The calculated coupling efficiency using the experimentally measured mode profiles $\eta _\text {mode}$ is 91% which indicates that the fabricated GIF tip reshaped the chip waveguide mode to a mode with appropriate size and profile for efficient coupling to the LP$_{01}$ mode supported by the SMF.
Fig. 4. Measured intensity profiles of the modes at the end faces of (a) a GIF tip, G5:(185, 53) $\mu$m, and (b) the chip waveguide. The scale bar is 10 $\mu m$.
4. GIF tip: divergence angle and beam width
Coupling of a beam at an interface between two optical devices is dependent on the beam shape, viz. the diameter and the spatial profile, as well as its divergence characteristics. We measured the beam divergence of selected GIF tips in air and compared them with simulations. The beam profiles from the GIF tips (G1, G5, and G13) were captured as a function of position along the propagation direction using the laser beam profiler. G5 and G13 are selected because of their high measured coupling efficiencies (88% and 72%, respectively), and G1 is selected as a control sample with low coupling efficiency (26%). The details of the measurement setup and method are described in Supplement 1. Figure 5(a) plots the beam width (1/$e^{2}$ in intensity) as a function of propagation distance. The beams from G5 and G13 show less divergence than G1 which has neither a coreless fiber nor a GIF segment. The measured radial cross section of intensity as a function of propagation distance of G5 is shown in Fig. 5(c). From this, the far-field divergence angles, $\theta$, as the derivative of the beam radius with respect to the axial position in the far-field, are found to be 5.33$^{\circ }$, 1.56$^{\circ }$, and 1.53$^{\circ }$ for G1, G5, and G13, respectively. As G5 and G13 are the GIF tips which show much higher measured $\eta$ than G1, a small beam divergence can be understood as an important factor for efficient coupling. The BPM simulation results of the beam divergence are shown in Figs. 5(b) and (d) for comparison. G5 and G13 show smaller divergence than G1 in the simulation as well, with the divergence angles of 1.49$^{\circ }$, 1.88$^{\circ }$, and 5.71$^{\circ }$, respectively.
Fig. 5. Beam divergence characteristics of selected GIF tips. (a) Experimental and (b) BPM simulated $1/e^{2}$ beam widths as functions of the propagation distance for GIF tips, G1:(0, 0), G5:(185, 53), and G13:(138, 173), where the numbers in the parentheses are $l_{\rm GIF}$ and $l_{\rm CL}$ in $\mu$m, respectively. G5 and G13 show smaller beam divergence than G1, which makes the coupling more efficient. (c) Experimental and (d) BPM simulated beam profiles (log $|\mathbf {E}|^{2}$) of G5 as functions of the propagation distance. The overlaid white solid lines indicate the beam widths.
To better explore the relation between the beam divergence and the coupling efficiency, the dependence of the beam divergence angle ($\theta$) and the beam width ($w$) on the GIF and coreless fiber lengths were investigated further using the BPM simulation as shown in Fig. 6. In the map of $\theta$, a curve which connects $l_{\rm GIF}$ and $l_{\rm CL}$ combinations giving the smallest divergence angles is found as indicated with a dashed white line. The far-field divergence angle can be smaller than 1.6$^{\circ }$ along the curve, thus a reasonably well collimated beam can be obtained from those GIF tip configurations. The contour plot of the beam width at the end face of GIF tips is also shown, where the white solid line indicates 31.2 $\mu$m which is the beam width of the chip waveguide mode obtained by the FEM simulation. Combining these two maps suggests that high coupling efficiency is expected where $\theta$ is small and $w$ is similar to or slightly less than the chip waveguide mode diameter. It is noted that the crossing point of the dashed white line in Fig. 6(a) and the solid white line in Fig. 6(b) is inside the area of simulated $\eta > 90\%$ in Fig. 3(b). The beam width $w=26.5~\mu$m and the divergence angle $\theta =1.26~^{\circ }$ at $l_{\rm GIF}=150$ and $l_{\rm CL}=110~\mu$m where the highest coupling efficiency was found in the simulation. It is worth noting that a collimated beam with a range of beam widths ($20<w<50~\mu$m) can be generated by selecting corresponding GIF tip configurations which will be useful to efficiently couple from waveguides of different mode sizes.
Fig. 6. (a) A BPM simulated map of divergence angle as a function of the GIF ($l_{\rm GIF}$) and coreless fiber ($l_{\rm CL}$) lengths. The GIF and coreless fiber length combinations which show the smallest divergence angles are connected by the white dashed line. (b) A BPM simulated map of beam diameter at GIF end face as a function of $l_{\rm GIF}$ and $l_{\rm CL}$. The white solid line indicates the combinations of $l_{\rm GIF}$ and $l_{\rm CL}$ that have a beam width of 31.2 $\mu$m which is the FEM simulated beam width of the chip waveguide. The yellow dashed line is the contour line of the simulated coupling efficiency $\eta =90\%$.
Conversely, we measured the beam divergence of the chip waveguide in air and compared it with BPM simulation. The measured and simulated beam divergence angles are $1.39^{\circ }$ and $1.36^{\circ }$, respectively. The measured and simulated beam widths at the chip waveguide end face are $25~\mu$m and $30.9~\mu$m, respectively. Detailed data is presented in Supplement 1. This can be understood that a GIF tip with similar divergence angle and beam width can effectively collect the beam from the waveguide, i.e., an expression of the mode-matching and acceptance angle requirements of the accepting optical component.
5. Gap between the GIF tip and the chip waveguide
Coupling efficiency was also analyzed when there is a gap between the chip waveguide and the GIF tip. In some applications, contacting a GIF tip to a waveguide facet may not be desirable due to the fragility of the surfaces, or so that an additional planar optical component such as a thin film input/output coupler can be inserted between the GIF tip and the waveguide. In such cases, it can be advantageous to have an efficient coupling when such a gap between the chip waveguide and the GIF tip is introduced. Particularly, for a laser formed by a ZBLAN chip waveguide, a planar input/output coupler with a thickness of 100 $\mu$m is often required to be sandwiched between the chip waveguide and the GIF tip. Then the advantages of structural simplicity and compactness by the usage of GIF tips over a free-space coupled optical system can still be achieved in the existence of a gap.
The measured and simulated coupling efficiencies as functions of the gap of 6 selected GIF tips are shown in Figs. 7(c) and (d), respectively. The measurement was conducted by translating the GIF tips in the $z$-direction and measuring the power at each gap position in the setup shown in Fig. 1(d) as previously described. At shorter gap lengths the measurement error is observed to be significant over the time allotted for the measurement when compared to larger gaps, and is attributed to the etalon established by the gap. This is most likely explained by the relative increase in strength (higher finesse) of the etalon at short gap lengths. The simulation was performed by bi-directional BPM to include the multiple reflections between the two interfaces of the gap. This Fabry-Pérot etalon effect is observed in the simulation result as shown in Fig. 7(b), and an average over multiple periods of etalon-induced intensity oscillation (a 2 $\mu$m window of the gap range beginning at the given gap value) is plotted in the simulation results in Fig. 7(d). We do note that the observed measurement error always fell within the depth of intensity modulation anticipated by our simulations.
Fig. 7. (a) A diagram showing the gap. (b) Simulated Fabry-Pérot etalon effect on the coupling efficiency. The solid line shows the average value over the range of the gap. (c) Experimental and (d) simulated coupling efficiencies of selected GIF tips as functions of the gap between the chip waveguide and the GIF tips. The error bars in (c) indicate the maximum and minimum values observed over the measurement duration. The Fabry-Pérot etalon effect is averaged over the range of the gap in the simulation results.
The measured coupling efficiency generally decreases as the gap increases while it is measured to be higher at some non-zero gaps than at the zero gap for G1, G11, and G13. In the simulation result of G13, the coupling efficiency increases as the gap increases from zero to 500 $\mu$m and then starts to decrease. A similar tendency is also observed in the experiment for G13 where $\eta$ is over 65% up to a 600 $\mu$m gap. This can be understood by noting that the emission of G13 is initially converging and actually has its focal plane $\sim 0.5$ mm from the end face of the GIF tip such that it exhibits higher coupling efficiency at a non-zero gap. Its low divergence angle (see Fig. 5) and large depth of focus therefore allow it to maintain efficiency over a large gap range. On the other hand, it is observed that $\eta$ drops rapidly for G2 and G5 both in the simulation and the experiment. It is more obvious in the simulation that G1, G2, G11, and G13 have the maximum coupling efficiencies at non-zero gaps. These results suggest that some GIF tip configurations focus the beam before it diverges such that the optimal gap is at the focal plane. Similar gaps were seen to optimize the coupling efficiency of self-imaging GIF tips in Ref. [37].
Further BPM simulations were performed to understand the coupling efficiency dependence on the gap for the coreless fiber lengths of 0, 100, 200, and 300 $\mu$m as shown in Fig. 8. The gap is scanned to 1 mm for GIF lengths from 0 to 400 $\mu$m. It can be seen that the coupling efficiency decreases as the gap increases in general, yet in some cases such as $l_{\rm CL} = 200~\mu$m, a non-zero gap in the gap range of $200 < g < 700~\mu$m shows $\eta > 92\%$ for the GIF length of 120 $\mu$m. Therefore, a GIF tip can be tailored for coupling at a specific gap for an extended coupling as required for a particular application. Simulation results assuming zero reflections and the average effect of the etalon at near-zero gap are shown in Supplement 1. The same gap dependency holds in the case of zero reflection at the interfaces except with higher coupling efficiencies due to the absence of the Fabry-Pérot etalon effect. The simulation results assuming zero reflection are shown in Supplement 1.
Fig. 8. Simulated coupling efficiency as a function of the gap between the chip waveguide and the GIF tip for different coreless fiber length $l_{\rm CL}$: 0, 100, 200, and 300 $\mu$m.
6. Coupling in the opposite direction: GIF tip to chip waveguide
Coupling is generally nonreciprocal when the supported modes of the two components have different beam widths and divergence angles. As the characteristics of coupling from a chip waveguide to GIF tips (Chip-to-GIF) have been discussed so far in previous sections, coupling efficiencies in the opposite direction, from the GIF tips to the chip waveguide (GIF-to-Chip), are discussed in this section for comparison.
Coupling efficiencies of the GIF-to-Chip coupling are obtained using the BPM simulation. The calculated SMF mode is launched in the simulation and the beam propagates along the GIF tip and enters the chip waveguide. The power is monitored in the 10 mm-long chip waveguide after propagating through it to allow unguided modes to leave the chip waveguide and then the simulated propagation loss ($6.56\times 10^{-2}$ dB/m) is used to correct for these losses to obtain the coupled power at the interface between the GIF tip and the chip waveguide.
The simulated coupling efficiency map of the GIF-to-Chip coupling without a gap between the GIF tips and the chip waveguide is shown in Fig. 9(a). A region of high coupling efficiency ($\eta > 90\%$) is found in the plotted map, where the GIF length decreases as the coreless fiber length increases similarly to the Chip-to-GIF coupling.
Fig. 9. Coupling from GIF tips to the chip waveguide. (a) Simulated coupling efficiency map. (b) Simulated coupling efficiencies of selected GIF tips as functions of the gap between the GIF tips and the chip waveguide. The inset is a diagram showing the direction of the coupling.
It should be noted that the high coupling efficiency area in the GIF-to-Chip coupling is larger than the Chip-to-GIF coupling. This can be understood that underfilling the chip waveguide with a beam from a GIF tip with a small mode diameter suffers less insertion loss than the coupling of a large chip waveguide mode to a GIF tip. Considering the required GIF tip configurations are different depending on the coupling direction, in the case that one GIF tip is used for both the insertion to and the collection from the chip waveguide, an overlapping area of high coupling efficiency for both directions in the coupling efficiency map should be obtained.
The gap dependency is also simulated for the GIF-to-Chip coupling including the Fabry-Pérot etalon effect as shown in Fig. 9(b). They also show higher coupling efficiencies than the Chip-to-GIF coupling shown in Fig. 7(d). In the simulation result of G13, the coupling efficiency increases as the gap increases from zero to 180 $\mu$m and stays almost flat with a high coupling efficiency for a long range of the gap. It has a longer gap range of high coupling efficiency than the Chip-to-GIF coupling of G13, which can be understood that the small divergence angle of G13 allows better coupling for a wide range of the gap particularly in the GIF-to-Chip coupling. Therefore, the GIF-to-Chip coupling has more design flexibility than the Chip-to-GIF coupling in terms of the GIF tip configuration as well as the gap. More simulation data for the GIF-to-Chip coupling is presented in Supplement 1.
7. Conclusion
This study of GIF tip geometries, as they relate to the coupling efficiencies into or from a ZBLAN glass chip waveguide, provides a template to design a GIF tip for optimal coupling efficiencies for a range of situations including coupling from such a chip to an SMF and vice versa, situations where some gap is required between the GIF tip and the target waveguide, and the scenario where light must be coupled in both directions simultaneously. By and large, it is established that the optimal efficiencies observed for coupling from Chip-to-GIF both in simulation and experiment are found by establishing a good mode overlap integral between the modes of the GIF tip and the chip waveguide, and matching the beam divergence from the GIF tip (representing its own acceptance angle for efficient coupling into the SMF) to the low divergence of the chip waveguide. The counter-propagating situation, GIF-to-Chip, offers a similar insight but it becomes evident that the large chip waveguide mode field diameter can accept a wider range of input diameters provided the divergence (acceptance) angle remains well chosen.
It is also clear from our study that the effects of the etalons generated in these optical systems in close-coupling situations where air gaps (or other strong index changes) form are critically important to consider and can have a very strong effect on the overall coupling efficiency. In some sensitive applications such as intra-laser cavity optical elements these effects may represent a potentially unstable loss mechanism which is required to be totally avoided. Ideally one would either ensure such interfaces were minimally reflective such as via anti-reflection coatings, or preferably avoid the problem altogether by direct splicing of GIF tips to optics with similar indices of refraction. Further work will likely focus on minimizing or entirely removing this loss to demonstrate the highest coupling efficiency predicted by the simulation.
Ultimately it is anticipated that this work can easily be extended to other wavelengths and geometries for practical use in designing photonic devices, enabling the miniaturization and simplification of such devices by replacing free-space optics with integrated optical fibers. We recently reported the study on the wavelength dependence of GIF tips where the far-field divergence angle is typically more sensitive to changes in geometry for longer wavelengths [30]. This experimental validation of the performance of a gradient refractive index fiber tip to couple to a waveguide which has a larger guided mode provides a solid foundation to apply a GIF tip as a collimation lens to integrate with other optic and photonic devices. Moreover, the design capability is demonstrated by showing the coupling characteristics with various GIF tip configurations. As the beam divergence angle and the width can be engineered, GIF tips are expected to replace bulk lenses in the field of optical communications where the conversion between the free-space and fiber optics occurs frequently, and opens up the possibility of composite lasers where the cavity includes a chip waveguide and fiber components.
Funding
SmartSat CRC.
Acknowledgments
This work has been supported by the SmartSat CRC, whose activities are funded by the Australian Government's CRC Program.
Disclosures
The authors declare no conflicts of interest.
Data availability
The raw 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.
Supplemental document
See Supplement 1 for supporting content.
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