1. Introduction

Optical fiber tapers are useful tools for evanescently coupling light to whispering gallery mode (WGM) resonators [1–3]. The fiber taper typically consists of off-the shelf single mode fiber that has been geometrically transformed to a step index dielectric waveguide of significantly smaller diameter. Fabrication of a taper with a specific diameter is equivalent to fabricating a waveguide with specific dispersion characteristics. In the context of coupling to WGM resonators, a fiber taper of appropriate diameter acts as a highly efficient coupler to specific high-Q mode families.

In isolation, the step index portion of the fiber taper will support multiple modes of propagation since the distinction between the core and cladding regions of the single mode fiber from which the taper was made is lost. However, within the appropriate wavelength range there is only one mode in the un-tapered SMF28 fiber. So long as the transition from the single mode fiber to the tapered region is adiabatic, only one mode is predicted to couple to the taper [4]. Numerous experimental studies have verified this prediction [5].

To enhance the coupling efficiency to a specific WGM while suppressing the coupling to all other resonator modes, the interaction length between the uniform fiber taper and the WGM resonator must be increased. As the interaction length between the resonator and the fiber taper is increased, a larger proportion of light will be coupled to a specific spatial resonator mode. This is analogous the concentration of the polar radiation distribution of a dipole antenna as the separation between nodes is increased.

In practice, the extended contact described above requires that the uniform taper must bend around the resonator edge. For coupling to WGM and related resonances, it is critical that the mode structure within the taper subject to bending is known in order to ensure coupling to the appropriate mode family. If there is mode transformation within the fiber taper as a result of bending, the wrong WGM family could be coupled to. In this paper, we theoretically predict and experimentally demonstrate modal transformation in a fiber taper that is subject to a 90° bend about a circular arc.

2. Fabrication of the cleaved taper

The two ends of a stripped and cleaned length of SMF28 optical fiber were clamped to a manual and a motorized linear translation stages. A hydrogen torch was ignited and held in the vicinity of the exposed optical fiber. Only one thermal maximum of the flame was close enough to the fiber to cause heating. If two thermal maxima are in the vicinity of the fiber, two tapered regions will be formed. The temperature of the torch was high enough to soften the glass fiber, though not too high to melt it outright. When the glass was sufficiently softened, the motorized translation stage (a stepper motor) was pulled across several mm to create the taper structure. When the desired taper diameter was achieved, the manual translation stage was abruptly pulled to snap the optical fiber at the position of minimum diameter of the taper (Fig. 1). Orthogonality of the cleave with respect to the taper axis was verified by observing the direction of light emission from the cleaved end of the taper. This process resulted in a smooth cleave orthogonal to the taper axis in 3 out of 5 times.

 

Fig. 1. (a). Fiber taper before cleaving. The diameter of this taper varies by less than 2% over 300 μm. b) 4 μm diameter cleaved taper. c) 10 μm diameter cleaved taper. The cleaved face is orthogonal to the taper axis. d) A sub-micron diameter taper that is too small to be resolved by our microscope.

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Uniformity in diameter of the fiber taper was measured using an optical microscope. The minimum diameter of the tapers fabricated was smaller than one micron. The taper diameter at various distances from the position of minimum diameter was measured. Assuming a linearly increasing diameter, we estimate that the taper diameter varies by as little as 5% over a 300 micron distance from the center. Ultimately, the shape of the taper formed in the method described above should resemble a catenary, not a line. The 5% deviation is an overestimate; the true variation is most likely about 2%. This assertion could be verified by SEM or related microscopy.

In applications involving extended coupling, the diameter of the taper over the length of the interaction region should be uniform to ensure coupling to a single WGM or related resonance. Specifically, the effective frequency variation in the taper due to the change in taper diameter should not exceed the line width of the WGM. For high-Q modes [6], the achieved 2–5% variation over 300 microns is sufficient.

The cleaved taper was placed transverse to a 600 micron diameter glass tube that was stripped and cleaned. By pushing the taper with a Teflon screw, the taper was forced to bend around the glass tube by as much as 120°. We observed neither significant power losses nor deformation of the taper during the bending process.

3. Physical model and theoretical prediction

The optical field distributions of the various existing taper modes are well known [7]. To determine which of these modes are occupied due to bending the taper requires a model of the field distribution within the bend region. By inspection, the bent portion of the fiber taper strongly resembles a section of a toroidal WGM resonator (Fig. 2.).

Since the modal structure within the taper before the bend possesses azimuthal and radial symmetry, we assume transformation to a similarly symmetric toroidal WGM. For modes possessing this type of symmetry, the WGM field distribution in a toroidal resonator of large internal radius is similar to the field distribution of a spherical WGM [8]. Thus, our physical model requires the numerical evaluation of the overlap integral between the fundamental WGM of a sphere with radius corresponding to the fixed bend radius, and the various modes of the multimode step index taper.

Where Ψ_WGM is the WGM wave function [8], and Ψ_TAPER is the taper wave function [7]. The coordinates r, θ, and φ are the radial, polar, and azimuthal coordinates with respect to the center of the taper. Alternatively, these coordinate could be taken with respect to the center of the effective WGM resonator. For a numerical evaluation of the overlap integral, it is more reasonable to implement the summation using the first approach over a cross sectional slice of the taper-WGM interaction region (set θ = constant.) This overlap integral was solved numerically for fixed A over several taper modes.

In our calculations, we fixed the wavelength of light at 1550 nm. The bend radius A was also fixed to 300 microns. The taper radius was varied from 0.5 to 30 microns. Using spherical WGM wave functions in Eq. (1) is valid when the ratio of the taper radius to A is on the order of the angular index of the WGM. The numerical approximation of the overlap integral predicts that light is preferentially coupled to the TE₀₁ mode for tapers with radius less than three microns. The model predicts coupling to multiple modes for tapers with radius larger than ten microns. Similarly, if light with a shorter wavelength were sent through a smaller bent taper, multiple taper modes would be coupled to. For a taper with radius between six to ten microns, the model predicts preferential coupling to the TE₁₁ mode (Fig. 3.)

 

Fig. 2. Motivation for the physical model. Region I is an unmodified SMF28 fiber. Region II is a transition to a step index taper of radius b. The optical mode in region II possesses the same spatial symmetry as the single mode in region I. A circular bend of radius A is applied to the taper in region III. In our model, we treat the mode in this region as a WGM associated with a sphere (toroid) of radius A. The eventual mode in region IV is possibly determined by the bend in region III.

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Fig. 3. (a). Results of numerical computation of Eq. (1) for the first four TE taper modes. For large diameters, bending couples light to multiple taper modes. For small diameters, the bend couples to a single symmetric mode. For intermediate diameters, bending causes preferential coupling to the TE₁₁ mode. b) Diagram illustrating physical overlap between the bend mode (colored fill) and the taper TE₁₁ mode (blue contour lines) in an 8 μm diameter taper subject to a 300 μm radius bend.

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4. Observations

A video microscope was used to image the intensity distribution at the cleaved end of the taper. An aspheric lens of short focal length (12 mm) was adjusted to project an image of the surface intensity profile of an four micron radius taper onto a CCD detector 0.5 m away, giving magnification of 44×. The evanescent field of the taper interacts with contaminants that scatter light and occlude the far field emission. In addition to providing magnification of the far field pattern, using a microscope with such a long arm causes the light scattered from surface features to diverge.

Laser light of 635 nm and 1550 nm was sent through the fiber taper. Without a bend applied, the far field pattern captured by the CCD camera is in both cases symmetric and consistent with the predictions from [3]. A 90° bend was applied to the fiber and the measurements were repeated. With the bend, the intensity pattern observed with 1550 nm light resembles the TE₁₁ mode of the taper, consistent with the predictions from the physical model outlined in this paper. At 635 nm, the far field pattern is a superposition of many modes, which is also consistent with our predictions (Fig. 4). By rotating a polarizing film along the output path and consistently monitoring the output pattern, we were able to verify that the taper mode is not a mode with mixed polarization.

 

Fig. 4. Results using a 4 μm radius taper. a) Far field at 635 nm, no bend. The mode is spatially symmetric, with some structure that could be attributed to multimode propagation in the SMF28 fiber. b) Far field at 1550 nm, no bend. The mode is spatially symmetric. c) Far field at 635 nm, 90o bend. This is a superposition of many modes, with at least 9 visible peaks. d) Far field at 1550 nm, 90° bend. There are two apparent peaks, consistent with the predictions of Eq. (2).

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5. Conclusion

The particular mode structure within a fiber taper is not important for most applications using tapers to couple to WGM resonances, so long as the taper supports a single mode. However, in the extended interaction-length scheme described above, the propagation constant of the taper mode plays an important role in determining the particular WGM that will be coupled.

Radiative losses associated with bending [9] are small for the large bend radius discussed in this paper. In the extended coupling example, the taper could wrap around a WGM resonator several times. Even in this case, the total loss due to repeated bending is not expected to significantly affect taper coupler performance [10].

Fiber tapers are also promising candidates for use as a coupling mechanism to related resonances, such as Bessel modes in a dielectric rod [11]. In such an application, the distinction between particular modes in the coupler is of paramount importance. Generally speaking, increasing the interaction length between a coupler and a resonant cavity will eventually overload the cavity and cause a reduction in the Q-factor of the resonance. In the non-resonant case discussed in Ref. [11], overloading is no longer a limitation and the interaction length should be made as long as possible to couple primarily to a single mode in the dielectric rod.