We propose a new way to realize a microfiber optical resonator by implementing the topology of a reef knot using two microfibers. We describe how this structure, which includes 4 ports and can serve as an add-drop filter, can be fabricated. Resonances in an all-silica reef knot are measured and good fits are obtained from a simple resonator model. We also show the feasibility of assembling a hybrid silica-chalcogenide reef knot structure.
©2009 Optical Society of America
Optical resonators have been extensively studied for over a century and have found many applications since their inception . Recently, much effort was dedicated to their miniaturization and various microresonator platforms and structures have been proposed [2, 3]. Microfibers appear to be a particularly simple platform providing low loss, easy connection to fiber based systems . Several resonant structures such as loops , knots , coils , and Fabry-Perot based structures based on microfiber Sagnac loop mirrors have already been fabricated with microfibers .
While these structures have two ports, more ports may be needed as in add-drop filters. It is possible to add extra ports to existing resonant structures as has been demonstrated in Ref. 9. In this work three microfibers were used: the first one to form a simple knot, and the two others to collect the power transmitted through and within the knot. Both collecting microfibers were just overlapping the knot’s microfiber and solely held to it through electrostatic and van der Waals forces. They could easily be displaced and would not hold upon immersion in a liquid, where these forces are considerably reduced.
Here we propose to use a different knot topology, the reef knot, to form a stable cavity out of two intertwined microfibers, which naturally provide four ports. This knot topology is well known in knot theory and is frequently used by sailors to join two ropes, but has not yet been implemented in optics. Apart from its size and material, our structure differs from the sailor’s reef knot in that we do not tighten the knot. This allows for the formation of a ring structure, in which light can resonate. This reef knot has a nearly planar geometry (This feature is emphasized in the French name of the reef knot: “noeud plat” meaning flat knot.) and is anchored in space by four arms, which makes it mechanically stable.
The fabrication of microfiber reef knots requires only basic equipment and consumables: an alcohol burner, four holders of which at least two are mounted on micropositioning stages, and optical fibers. A visible laser, a lamp, and a stereomicroscope can also be used for visualization purpose.
Figure 1 shows the main steps followed for the fabrication of a silica microfiber reef knot. First a standard optical fiber is spliced to two optical cables for connection to sources and detectors. Then its polymer coating is removed over a few centimetres. This uncoated section of the fiber is stretched by the flame brushing technique over an ethanol burner to form a biconical taper with its thinnest diameter reaching micron to submicron size (Fig. 1(a)). The biconical taper is bent into a U-turn (Fig. 1(b)), and the two thick legs (coated fiber sections) of the “U” are placed on holders. A simple taper ending as a microfiber approximately the same diameter as in the U-turn is drawn (Fig. 1(c)) and its thick end is placed on a third holder in front of the U-turn and parallel to the two legs of the “U”. The microfiber end is guided from the bottom in between the two legs of the “U”, passed above and below one of the legs, then below and above the other leg, and passed from below again in between the “U” (Fig. 1(d)). The microfiber end is then placed and glued on a tapered fiber fixed to the fourth holder. By moving the four holders, the shape and the size of the knot can be varied.
Figure 2 shows a photograph of the simple setup used. The reef knot is hung horizontally thanks to the four holders, of which two are mounted on micropositioning stages. A black cloth underneath as well as visible red light launched into the knot are used to improve visualization. A stereomicroscope with zoom magnification 7 to 56 is focussed on the reef knot plane. The reef knot is illuminated from the side with a fibre bundle connected to a white light source. Photographs are taken through the stereomicroscope with a commercial digital camera.
Figure 3(a) shows a photograph of an all-silica microfiber reef knot, whose spectral resonance curves are shown in Fig. 5. We also demonstrate the feasibility of assembling a hybrid silica-chalcogenide reef knot structure, whose photograph appears in Fig. 3(b). The motivation in developing hybrid devices consists in the following. It is well known that microfibers can be drawn from a large range of materials , but combination between them has not been much investigated. Chalcogenide glasses are a family of highly nonlinear glasses , from which microfibers have been recently drawn in order to enhance Kerr nonlinearity . The prospect of combining the good connectivity of silica microfibers with the high nonlinearity of chalcogenide microfibers into a resonating structure appears as a challenging perspective: nonlinear resonators operating with moderate input powers could then be developed .
The chalcogenide glass used for the elaboration of the hybrid ring was a sulfide one, from the As2S3 composition. The fabrication process for this chalcogenide microfiber is detailed in Ref. 14. So far, we have been limited to diameters not smaller than 5 micrometers for the As2S3 microfiber, a size too large for good coupling via the evanescent optical field, and no resonance could be observed. However, we tested the mechanical feasibility of the structure. Although the bulk chalcogenide glass can break very easily as compared to silica, the manipulation of the chalcogenide microfiber was not an issue. The reddish coloration of the chalcogenide microfiber (see Fig. 3(b)) made its visualization easier than for the silica microfiber. The difference of refractive index between the two glasses can cause unwanted reflections in the coupler regions unless the two microfiber diameters are carefully chosen to equalize the effective index of the propagating modes. The microfibers should also be drawn thinner than for the structure shown in Fig. 3(b) or should be clad by a liquid or a solid [18–20] to increase the extent of the evanescent field and thereby achieve efficient cross-coupling. We are currently addressing these issues.
3. Modeling, measurements, and discussion
Figure 4 is a schematic illustration of the operation of an add-drop filter showing the parameters at play. We can derive the basic properties of this resonator by assuming steady-state operation and matching fields, leading to the following set of equations:
where Ei is the electric field at the location “i” indicated in Fig. 4, K1 and K2 are the power cross coupling coefficients of the two couplers, A is the total power transmission coefficient through the coupler (assumed to be the same in both couplers), Φ12 and Φ34 are phase terms whose sum is Φ:
where l = l12 + l 34 is the circumference of the resonator, λ is the wavelength, and neff is the effective index of the propagating mode.
Solving this system of equation for the fields and using the complex conjugate product to calculate the powers we obtain:
For the characterization of our reef knots we used a Super K supercontinuum source from Koheras A/S as input and an optical spectrum analyzer model AQ6315 from Ando Corp. to measure the spectra transmitted at the through and drop ports. Fig. 5 shows the through and drop port spectra measured using the reef knot shown in Fig. 3(a), together with fits. The free spectral range (FSR) is 1.1 nm which would correspond to a diameter of 450 μm if the resonator were circular. In a smaller reef knot we obtained resonances with an FSR of 2 nm  and we should be able to go down to the FSR of 15 nm measured in a simple knot . From Fig. 3(a) we see that the geometry of the knot departs from a circle, and we measure a long axis of 470 μm and a short axis of 340 μm, with a circumference within 10% of the value calculated for a circular geometry.
Strictly speaking the ratio of the wavelength to the full-width at half-maximum (FWHM) of a resonance peak is a measure of the Q-factor only if the FWHM is much smaller than the FSR. Although we depart from this condition we use this ratio as an indication of the Q-factor. Using this approximation the Q-factor measured at the through port is around 10 000. This value is close to the 8 000 measured in a polymer add-drop filter fabricated by photolithography . However, the Q-factor measured at the drop port is much lower, with an average value of 3 500. We attribute this difference between the Q-factors measured at the through and drop ports to the different coupling coefficients in the two couplers. Indeed, from the fits we obtain K1 = 20.8% and K2 = 46.2%. We believe that this dissymmetry is due to the difference of diameters between the two microfibers used, and to non-optimal coupling lengths. We have assumed the same value of A in both couplers, but note that we could systematically obtain good fits despite this simplification. The value of the power transmission through the couplers is fitted to A=0.46. This relatively modest value could be improved (we found 0.92 in another knot), as it is likely to be caused by contamination.
The condition for critical coupling is that the square of A equals the ratio of K1 over K2 . From the numerical value we see that we are far from fulfilling this condition and the extinction ratio measured at the through port is only 12 dB. Values of A, K1, and K2 better approaching the critical coupling condition are desirable for practical add-drop filter applications.
Finally we would like to mention two methods, CO2 laser processing  and embedding in a solid matrix [18–20], which may improve the performance and increase the robustness of microfiber reef knots.
We have proposed and fabricated reef knot structures made of two microfibers. In one knot the two microfibers were made of the same material, silica, while in the hybrid knot both a silica and a chalcogenide microfiber were used. The latter was motivated by the prospect of high nonlinearity with silica fiber connectivity. The hybrid knot could also incorporate other materials such as a laser active glass. Resonance spectra showed that the all-silica reef knot could act as an add-drop filter, although fits to a simple model showed that the couplers should be improved. We expect that a better control of the microfiber diameters and of the knot size, as well as post-processing steps, will allow improvement over this initial demonstration.
This work is supported by the National Natural Science Foundation of China (grant No 60678039), by the EGIDE program ‘Programme de Recherches Avancées Franco-Chinois’ (project 15660SL), and by the Agence Nationale de la Recherche (project ANR05-BLAN-0152-01).
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