Although evanescent fields from tapered optical fibers with small diameters (<5 μm) enable efficient coupling to microresonators, fabrication and handling difficulties motivate the use of half-tapered fibers. As a prelude to microresonator coupling, however, we quantify just the coupling between two half-tapered fibers using fabrication, theory, design analysis, simulation and experimental verification. The dependence of coupling strength on taper diameter, fiber separation and coupling length are highlighted.
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Whispering gallery mode microresonators have found wide range of applications due to their high quality factor, small mode volume and morphology dependent resonance. Many examples exist of tapered fiber coupling to microresonators [1–4]. The coupling efficiency is dependent on phase matching of the modes, including phase synchronism and spatial overlap, the polarization and the coupling regime dictated by the separation between the fiber and the microresonator [5,6]. The tapered fiber in majority of cases is a bitaper, meaning a single fiber is used to both couple light into and out of the resonator. When used to create a comb spectrum, the light coupled out will contain pump intensity ~20 dB or more than other comb lines. Since measurment of the output includes light transmitted by the tapered fiber and that coupled back from the resonator, a true measurement of the internal field of the resonator is not obtained. Furthermore, depending on the microresonator properties, the taper waist requirement can be as low as one micron making fabrication and handling non-trivial. A half-taper, by contrast, allows true measurement of the internal field without the effect of the input pump. While fabrication and handling is easier, the downside is that alignment of the two half-tapers needs to be accurate.
A single resonator can be dual pumped and support two modes that can yield low-noise microwaves . In order to develop a high-stability microresonator comb source using rugged and connectorized components that can be readily assembled, we wish to understand details of coupling to the microresonator using half-tapered fiber. As a prelude, a much simpler geometry of symmetrically coupled fibers is attractive, since it is both tractable and repeatable. The main goal in this work is to identify the limitations of using tapered fibers, especially half-tapered fibers in coupling power. We expect to use 1 to 5 μm diameter tapered fibers to couple to microresonators and so is important to understand coupling efficiency for fibers with tapers down to 1 micron. This work helps identify how well the fibers have to be aligned with each other for strong coupling.
Alignment is critical when using half-tapered fibers to couple to microresonators. We address fabrication of half-tapered fibers and simulate the associated loss, using analytical solutions to point out limitations. In particular, the taper creates higher-order modes, and this is simulated and supported by experiment. The higher order modes generated include not only the azimuthal LP0,2 mode as expected but also the radial mode LP1,1, probably resulting from fiber asymmetries.
The uniqueness of this work is that it addresses all aspects of coupled fibers, from fabricated taper issues to coupling mismatch from a practical standpoint, with experiments backed by simulations and theory. Furthermore, we have done this by quantifying coupling of both fundamental and higher order modes.
Total attenuated reflection inside the fiber core results in an evanescent field outside of the core that decays exponentially. The coupling of this evanescent field with whatever is external to the core results in power transfer. The extent of the evanescent field outside the core is critical. The penetration depth (dp) is given by Eq. (1)
Phase matching is critical for efficient coupling of the fiber mode to external optical elements. Phase matching and the need for large evanescent fields requires fiber diameters of 1 to 5 µm, making them extremely fragile and difficult both to fabricate and to handle. This has motivated half-tapered fibers. We seek to understand their evanescent field behavior and the efficiency of coupling by coupling two half-tapered fibers under different taper diameters, coupling lengths and separation.
Section 2 models the profile of a tapered fiber and is a function of the fabrication process. The tapered fiber causes the fundamental mode shape to change with length and this can lead to loss in higher order modes, depending on the rate of taper. Minimizing this loss is achieved by satisfying the adiabaticity criteria detailed in Section 3. Normal local mode analysis used to quantify the coupling between two tapered fibers is described in Section 4. Beam propagation simulations are presented in Section 5. The experimental setup is in Section 6 and a comparison between experiment and simulation is in Section 7, which is followed by a summarizing figure in the conclusions of Section 8.
2. Fabrication and shape of optical fiber
The two common heating techniques for tapering are using a flame or a focused CO2 laser [8,9]. Flame heating requirements scale inversely with fiber radius and are able to produce submicron sized fibers. Disadvantages include the difficulty in controlling flame instabilities caused by gas regulation and air flow variations, and also contamination with combustion byproducts. CO2 laser based heating has an inverse-square relationship with fiber radius and the smallest tapers are about 1 μm diameter. Coupling between half tapers with diameters greater than 1 μm are investigated in this work so the use of a CO2 laser is appropriate. CO2 lasers offer the advantage of a controlled source of heat, high spatial resolution, and no residue.
Birks  and Xue  model the shape of tapered optical fibers formed by stretching.a fiber in variable length heat source. Grellier et. al.  numerically calculate the final diameter of tapered fiber fabricated using a CO2 laser. The process is self regulating since below a certain diameter insufficient heat is absorbed to reach the softening temperature. They model the heat transfer from a CO2 laser in  taking into account the effects of conduction, convection, radiation, energy storage and heat generation. The heat generation is based on the intensity of a beam with an elliptical TEM00 mode and the efficiency of absorption is derived from Mie theory. The fiber temperature and the diameter of the waist are thus determined as a function of the laser power. However, they conclude by requiring more accurate fiber parameters to accurately predict the diameter.
In practice, the fiber is in the near field of the laser, and calculation of the heat absorbed by the fiber is not trivial. In this work, the fabricated taper profile is compared to the theoretical adiabatic shape to get a qualitative estimate of loss of the fundamental mode. A quantitative loss is obtained from simulations.
3. Optimal shape from adiabaticity condition
In a tapered fiber, the fundamental mode shape is determined by the local radius and is thus continually changing in the direction of the taper. The total field is unable to change rapidly, resulting in power transfer to the closest higher modes [13,14]. This loss is minimized by keeping the taper angle low; the fiber is considered adiabatic if this loss is negligible. Figure 1(a) shows how the taper angle should be varied along the length of the taper to minimize losses. The curve in Fig. 1(a). separates steeper taper angles that lead to lossy tapers from the gradual taper angles that form adiabatic tapers. This delineation is not rigorous but we use it to determine, qualitatively, if the fabricated taper is lossy. The core and cladding guidance regions are shown as asymptotic curves. The normalized radius of about 0.4 is where the transition occurs; this is about 3 μm for the fibers used. Figure 1(b) shows the taper angles along the length of the fabricated fiber. The fiber is adiabatic towards the end of the taper, but does have losses at the larger diameters. This is quantized by simulation to be about 10-15% of input power. The adiabaticity criteria are determined using two different techniques, a length scale criterion and a weak power transfer criterion [13,14]. Results from the analysis in  and  are summarized here.
The length scale criterion in Eq. (2) states that the taper length scale must be much larger than the coupling length between the fundamental mode and the dominant mode it couples to. Thus,
Where, ρ and Ω are the local radius and taper angle respectively, are the propagation constants for the fundamental and second order mode, is the taper length scale and is the beat length. Stating this condition in terms of the taper angle results in Eq. (3).Figs. 1(a) and 1(b) are obtained using the propagation constants of the local normal modes and the above taper angle condition reduced to an equality . The normalized fiber frequency parameter V(z) given by Eq. (4). is used to understand the change in the modal behavior.
Another delineating criterion is obtained by considering the field to be a superposition of modes. The predominant loss is assumed to be from the fundamental to the second mode and is evaluated using a coupled mode analysis similar to the one in the following section on coupling between two tapered fibers .
4. Theory and analysis of coupled tapered fibers
Full power transfer between fibers can be obtained if the process is adiabatic . For an adiabatic process the external perturbation varies slower compared to the internal dynamics, causing the system to remain in one eigenmode throughout. For coupled fibers this leads to the condition that the phase mismatch should vary slowly compared to the square of the coupling and phase mismatch must be large at the beginning and end of coupling. The tapered coupled analysis in  is used to evaluate the power coupled between two half tapers as a function of the individual diameters, the separation between the fibers and the coupling length. As the fiber tapers, the mode pattern continually changes. If the taper is gradual, however, then Maxwell’s equations for a untapered fiber can be used within a local region. The fields of a uniform cylindrical fiber with diameter equal to the local radius are used at each location along the axis of the fiber. An overlap integral gives the local coupling strength. Coupled mode equations are then solved to give the amount of power coupled. The local fields account for the effect of the fiber diameter, the overlap integral accounts for the separation between the fibers and the coupled mode equations account for the coupling length effect.
4.1 Fiber field equations
The electric and magnetic fields will satisfy the Helmholtz equations as in Eq. (5)Eq. (6)Eq. (7).Eq. (7) results in Eq. (8)Eq. (9) and Eq. (10).Equations (11) and (12) in the core and cladding region are:
The solutions are Bessel functions and the fundamental fields are given as in Eqs. (13) and (14).Eq. (15).
4.2 Coupled mode analysis
We achieve strong coupling by generating large evanescent fields from tapers down to 1 μm. We maximize field overlap by minimizing the separation of the fibers. This geometry is laid out in Fig. 2(a). Based on the geometry and material properties a coupling coefficient is defined as an overlap integral in Eq. (16).Figure 2(b) shows the sensitivity of the normalized coupling coefficient to the above geometrical and material parameters. Here is the profile height parameter  and r1 is the fiber radius. These are used since for a given wavelength, coupling is dependent only on the geometry and index profile. As seen, coupling drops rapidly with separation and improves for a smaller receiving tip. The magnitude of this coupling coefficient agrees well with previously cited value in literature .
Coupled mode theory is used to evaluate the interaction strength between two optical fibers . The fields profiles calculated are used to determine the amount of overlap in the form of an integral. The separation between the fibers is So. The below analysis uses r1 for the radius of excited fiber (f1) and r2 the radius of the receiving fiber (f2).
The evanescent field from f1 couples into f2. Substituting the fiber fields the coupling coefficient is given as Eq. (17):17] are valid for this analysis as well. The strongest overlap will occur on the surface of f2, the receiving fiber. Hence a Taylor series expansion of the f2 field at the radius r2 is used. The circular section of f2 is approximated by a parabolic function: and the ‘y’ dependence is projected along the perimeter (p) of fiber f2 close to f1.Thus, With these approximations Eq. (17) simplifies to Eq. (18), which is evaluated using integration by parts and standard formulae in Eqs. (19) and (20). The integral is only valid in the proximity of the perimeter of the receiving fiber. As long as the limits of integration cover this region the result should have low error, since the field from the exciting fiber decays exponentially beyond this point.
4.3 Theory of mismatched coupling16] assumes the propagation occurs at the average of the local phase velocities, the variations in,, and are slow and the WKB (Wentzel-Kramers-Brillouin) approximation holds. In this work, single mode power is fed as input to fiber f1. Thus ‘U’ is taken to represent the fundmental mode. The mode in fiber f2 that most strongly couples to ‘U’ is taken as V. Eliminating ‘V’ from Eq. (21) by using Eq. (22) yields a harmonic equation that is solved to give the amplitude. The power in each fiber is obtained as the square of the amplitude The power in each fiber is given by Eqs. (23) and (24) :Equations (23) and (24) reduce to perfect squares. Solving these for B+ and B- in terms of a + and a- we obtain, andNumerical Matlab analysis is done for two tapered fibers matching the experimentally-used fibers. The varying propagation constants and coupling coefficients along the length of the tapered fibers are numerically calculated and the above coupling equations are used to obtain the power transferred to the receiving fiber. The analysis is repeated by translating the receiving fiber along the axial direction to obtain the plot in Fig. 3(a). The X-axis is the ratio of receiving fiber radius to the excited fiber tip radius, at the location of the very tip of the excited fiber as shown in Fig. 3(b). When the tip of the excited fiber is coupled to the receiving fiber cross-section with a larger radius than the tip, there is a section when the radii of both fibers match and power is coupled. Beyond this region the radius of the receiving fiber increases and it carries the total power, as seen for ‘x’ values > 1 in Fig. 3(a). On the contrary, if the receiving fiber radius is always smaller than the excited fiber during the entire coupling length, very poor coupling is obtained and this is seen in Fig. 3(a) for ‘x’ values < 1.
Another parameter of interest is the separation between the fibers. As seen in Fig. 3(a)., the curve for the fibers separated by normalized distance of 0.66 shows very poor coupling. Coupling of power occurs in the region where the cosine terms in Eqs. (23) and (24) and not negligible. The sharp slope of the curve for the poorly coupled case indicates that the cosine term is negligible throughout the axial translation. Hence the unity power transfer as seen for this case is in error and show the limitation of this analysis. The analysis does give a qualitative idea of the amount of separation that will still result in efficient coupling.
This analysis is also useful in estimating the coupling length. The actual power transfer occurs when the fibers are matched in diameter. From Fig. 3(a), we see that this happens within 1% of the diameter change, which for a measured taper angle of 2° as shown in Fig. 1(b), is 0.01*0.75 micron/tan(2°) = 0.2 micron. Based on geometry and using a taper angle of about 2°, a minimum of 7 μm of coupling length is needed to align the 1 μm receiving fiber with the 1.5 μm excited fiber so that there is a region of matched diameters. This agrees well with simulation and experiment.
In practice for the fibers used in the experiments, multiple modes are generated in the receiving fiber, since the tapered tip has cladding-guided modes at the air interface. As the receiving taper diameter increases, the modes are transformed back to core guided modes. However, only the lower order modes LP0,1, LP1,1 and LP0,2 with the closest propagation constants will be preserved, while higher-order modes will be lost. These competing higher-order modes would have to be included for a more complete analysis. Beam propagation simulations described in the next section confirm this behavior.
5. Beam propagation simulations showing multi-mode behavior
Whereas Fig. 3(a) shows the maximum theoretically possible coupled power, Figs. 4(a) and 4(b) shows the simulation results that predict coupled power using the fabricated tapers in this work. Figure 4(a) shows the power entering fiber f1 from the bottom. The dotted line shows the separation of the two fibers. The near horizontal lines indicate the field as it propagates along the fibers. The fibers are simulated at the tapered ends which are cladding guided and are multi-moded. As seen there is significant conversion to higher order modes LP0,2 and LP1,1. As the receiving fiber tapers back to its original diameter, these modes transition to the single mode. The algebraic addition of the power in these modes corresponds to the total coupled power measured in the experiments.
Beam propagation simulations are done for tapered fibers matching the fabricated tapered fibers with tips close to 1.5 μm and 1 μm. The fiber shape observed under a microscope is used for the simulation. The separation between fibers and the coupling length is varied with best coupling obtained with the fibers in contact. The fiber smoothness as observed with white light interferometry is about 0.1 nm. The adiabaticity is confirmed by plotting the the total and fundamental mode power in each fiber. A loss of about 10% power in one fiber and about 17% in the other fiber is observed with simulations done on individual tapers in air. Next, simulations were done with the fibers coupled. Simulations conditions used are close to experimentally possible fiber geometries and separation. From the multiple modes captured, only the lower order modes LP0,1, LP1,1 and LP0,2 are summed to obtain the power to compare with the results is in the next section (see Table 1). The total power in Fig. 4(b). includes higher order modes that will not be efficiently converted to the final single mode. Thus there is significant amount of power in modes higher than LP1,1 and LP0,2. This is shown as curve labeled “2, Power” in Fig. 4(b). The gradual increase in power in each of the modes shows effect of coupling as compared to the theoretical analysis that predicts a more abrupt increase in power. The simulation parameters are: Taper length = 30 μm, Taper tip diameters: 1 μm and 1.5 μm, Separation at 0.02 μm.
6. Description of fabrication and experiments
A CO2 laser based fiber puller from Sutter (Sutter 2000)  is used for the tapering. The control parameters are the laser power (upto 10W), the pull force (upto 53 lbf), the start time and duration of the pull force and the length of the fiber that is heated. The goal is to obtain an adiabatic taper while simultaneously generating a small tip. The parameters are optimized by iteration and the dimensions obtained viewed under a microscope. The results are compared with the analytic solution of the adiabatic tapers and beam propagation simulations to confirm adiabatic operation. The fiber alignment is achieved using two three-axis positioners with the experimental setup shown in Fig. 5. Each pair of tapered fibers are aligned at different coupling lengths and the coupled power is recorded as a percentage based on the power measured before the fiber was tapered.
7. Comparison of experiments with simulations
The tapered fibers are brought in contact so their separation, which would be of the order of the surface roughness, does not significantly contribute to the loss. The geometrical parameters that determine the plot are the absolute diameters of the fibers and also the relative difference in the diameters, at each point of contact. Coupling increases with decreasing diameter and with better matching between the fibers. Only the lower order modes LP0,1, LP1,1 and LP0,2 from the simulation are summed to obtain the total power. These modes have similar propagation constants and must get converted to LP0,1 mode at the untapered single mode end of the receiving fiber. The strongest coupling is about 30% with very short coupling length of 10 μm.
Based on geometry resulting from different tip diameters of 1.5 μm and 1 μm and using a taper angle of about 2°, a minimum of 7 μm of coupling length is needed to align the 1 μm receiving fiber with the 1.5 μm excited fiber so that there is a region of matched diameters. Hence the coupled power starts increasing after about 7 μm. As the coupling length is increased there is more length that corresponds to phase matched region and the coupled power continues to increase, until phase matching can only occur at larger diameters. For larger diameter fibers coupling is weaker since evanescent field is weaker and hence the coupled power starts dropping as seen beyond about 10 μm of coupling. The fiber parameters used in Fig. 6. are, for fiber 1, the tip is 1.5 μm and taper length is 1.15 mm and for fiber 2, the tip is 1 μm and taper length is 3.3mm.
The power flow in the coupled fibers using experimental measurements of output power is summarized in Fig. 7. The uniqueness of this work is that we have quantified coupling of fundamental and higher order modes using simulations confirmed by experiment with fabricated tapers. We survey many parameters contributing to evanescent coupling between two half-tapered fibers, including the shape of the individual tapers and their coupling geometry. Fabrication of tapered fibers using CO2 lasers has a theoretically known shape; adiabaticity criteria are used to determine an optimal taper shape for minimal loss. The experimental shape is compared to this and shows expected low loss. This loss is confirmed with a beam propagation simulation.
Coupled tapered fibers are analyzed and simulated to estimate the coupling coefficient and the effect of fiber taper size, separation and coupling length. A fixture was devised to accomplish this using two three axis positioners. The experiments demonstrate the high sensitivity of coupling to mismatch in the taper diameters and separation. Coupling lengths of a few tens of microns are sufficient. Tapers lengths of the order of a mm are typical for most used fabrication methods and coupling lengths much larger than a few 10’s of microns leads to a large mismatch in the coupled fiber diameters and reduced coupling. The experiments show that for tapers of about 1.5 μm and coupling length of 10 μm, about 30% power was coupled.
In future work, the fabrication parameters can be optimized to better meet the adiabaticity criteria. Also this should produce the smallest tapers possible with close to perfect matching. These fibers can then be used to couple power into and out of a microresonator. This study helps isolate tapered fiber related limitations to coupling and in future will aid understanding of limitations when fiber is coupled with microresonator.
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