Based on the recently-introduced Surface Nanoscale Axial Photonics (SNAP) platform, we demonstrate a chain of 30 coupled SNAP microresonators spaced by 50 micron along an optical fiber, which is fabricated with the precision of 0.7 angstrom and a standard deviation of 0.12 angstrom in effective microresonator radius. To the best of our knowledge, this result surpasses those achieved in other super-low-loss photonic technologies developed to date by two orders of magnitude. The chain exhibits bandgaps in both the discrete and continuous spectrum in excellent agreement with theory. The developed method enables robust fabrication of SNAP devices with sub-angstrom precision.
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A remarkable fabrication precision (several nm) and low propagation loss (typically, in excess of 0.1 dB/cm, limited by the surface roughness) of miniature photonic integrated circuits have been demonstrated with modern fabrication technologies [1–4]. Despite the notable progress, the achieved high precision and low loss are still insufficient for the fabrication of long chains of coupled microresonators and other photonic structures proposed to date for efficient slowing and manipulation of light. In fact, the remaining nanoscale-small randomness of the microresonator dimensions and interface roughness lead to significant attenuation of light and collapse of the transmission band of long coupled microresonator chains [1,5]. In addition, methods developed for the reduction of fabrication errors by trimming exhibit temporal instability  and/or attenuation of light  unacceptable for super-low-loss photonic circuits.
Recently, we introduced a technological platform enabling fabrication of microphotonic circuits at the atomically-smooth surface of a drawn optical fiber with unprecedented angstrom accuracy [8–13]. This technology allowed us to fabricate a series of ultra-low-loss coupled whispering gallery mode (WGM) microresonators along the optical fiber by nanoscale variation of its effective radius (which includes both the variation of the physical fiber radius and of its refractive index). Similar to the other types of silica microresonators fabricated by melting (spherical, toroidal, bottle, etc .), the Q-factor of these microresonators can exceed a billion. The surface nature of WGMs , their slow axial propagation described by a one-dimensional Schrödinger equation [8,9,13], as well as the simplicity and accuracy of the nanoscale variation of the fiber radius [10–13] motivated us to coin the name Surface Nanoscale Axial Photonics and its abbreviation “SNAP” for this technology.
We have demonstrated the fabrication of SNAP microresonators with angstrom-scale precision by focused CO2 laser annealing of silica fibers [10,11,13] and also by the UV radiation exposure of photosensitive Ge-doped optical fibers through an amplitude mask [10,12]. However, the chains of angstrom-precise coupled SNAP microresonators fabricated so far contained a relatively small number of microresonators due to fabrication errors growing with the length of the SNAP device. While a few angstrom precision in fabrication of multiple coupled microresonator chains is typical, a sub-angstrom precision  was achieved after multiple tries only. This was particularly caused by the original nonuniformity of the fiber radius, surface contamination, imperfect system alignment, and fluctuations of the beam power control system. Evidently, the future success of SNAP depends on the solution of the problem of fabrication errors.
In this paper, we demonstrate the fabrication of a SNAP device with multiple laser shots and consistently reduce the fabrication errors by iterations. We explore a CO2 laser annealing method applicable to general types of optical fibers. A similar iterative approach for photosensitive fibers, where the effective fiber radius is treated with UV and visible light, could also be feasible [12,15,16]. The developed method allowed us to achieve a 0.7 Å precision and 0.12 Å standard deviation in the effective heights of 30 coupled microresonators. It is suggested that the demonstrated precision and standard deviation can be improved for chains of microresonators and other SNAP devices having similar and larger axial lengths.
2. Experimental setup
Our experimental setup is illustrated in Fig. 1 . It consists of the SNAP fiber (SF) exposure section (Fig. 1(a)) and the fiber characterization section (Fig. 1(b)). The nanoscale effective radius variation (ERV) of the optical fiber is introduced with a CO2 laser beam focused onto the fiber by the parabolic lens (Fig. 1(a)). The input beam power is controlled by a laser power controller and a shutter with a 1 ms switching time. Annealing the optical fiber relaxes the tension, which is frozen-in during fiber drawing  and causes the nanoscale ERV . After the exposure, the SF is characterized at the measurement section of our setup with the microfiber scanning method (Fig. 1(b)) [18,19]. In this method, the microfiber is scanned along the SF touching it at contact points where the spectra of the microfiber coupled to the SF are measured. The spacing between the contact points (set to 10 μm) determines the axial resolution of measurements. At each contact point we measure the resonant transmission amplitude spectrum with 1.3 pm resolution using the Luna Optical Vector Analyzer. The characterization step, including measurement, saving data, and translation of the microfiber, takes 20 s. The spectra are used for the evaluation of SF performance and its ERV.
3. Fabrication and correction of the microresonator chain
In our experiment, a chain of 30 strongly coupled microresonators is created along a 19 μm radius fiber. Each microresonator is introduced by a sequence of 20 CO2 laser 0.1s/0.1s on/off exposures using the beam shutter. The spacing between microresonators is set to 50 μm. The SF is characterized by the microfiber scanning method. Figure 2(a) shows the surface plot of the resonant microfiber/SF transmission amplitude spectra. The spectra exhibit the fundamental transmission band near the wavelength nm followed by a bandgap. The wavelength variation of resonances, (left vertical axis), is related to the characteristic ERV, Δreff (right vertical axis), by a simple scaling relation, Δλ/λres = Δreff /r0 [8,9,18], where λres = 1.56 μm is the resonance wavelength and r0 = 19 μm is the SF radius. The full characterization time of the SF along the 1.9 mm length was 1 hour. During this time, the temperature was maintained constant within ΔT = 0.1°C. The ERV due to this temperature change was less than δr = r0(n−1dn/dT + dl/l)ΔT = 0.16Å, where for silica we set the refractive index n = 1.46, dn/dT = 1.2·10−5/°C, and the coefficient of expansion dl/l = 0.5·10−6. To evaluate the radial nonuniformity of the microresonator chain, we determine the positions of blue-color apexes (BCAs) corresponding to each of the microresonators in Fig. 2(a). The method is justified by the theoretical calculations below. The coordinates of these apexes are plotted in Fig. 3(a) . From Fig. 3(a), the radial nonuniformity of the microresonator chain in Fig. 2(a) is equal to Å.
To improve the uniformity of the microresonator chain, we perform its correction. We start with the calibration of the laser shots. To this end, using the beam shutter (Fig. 1(a)), the microresonator of the chain (except for the last one) is exposed to m 54 ms long shots (the length of shots is determined experimentally to arrive at the resultant local ERV of less than 0.1 Å). The spectral plot of the so-modified SF is shown in Fig. 2(b). To evaluate the change, curve 1 and 2 in Fig. 3 show the positions of BCAs corresponding to the microresonators in Fig. 2(a) and Fig. 2(b), respectively. The calculated resonance shifts (curve 3) are approximately proportional to the number of shorts, which allows us to find the average shift in the effective radius per shot as 0.07 Å. Next, we alter the radii of microresonators in order to make them equal by exposing the microresonators to the appropriate numbers of shots. The resultant spectral plot of remarkably uniform microresonator chain achieved after two iterations is shown in Fig. 2(c).
4. Theoretical analysis
The introduced ERV of the microresonator chain can be determined by fitting the spectral surface plot in Fig. 2(c) to the analytical model, which, for the uniform chain, is taken in the form:13]. Specifically, we fit the bandgap and fundamental transmission band dimensions in Fig. 2(c) to those of the spectrum of the Schrödinger equation with the potential defined by the ERV of Eq. (1) [8,9,13],Fig. 2(d) (white curve). The height and modulation amplitude of the introduced ERV are nm and nm, respectively. We found the remarkably good correspondence of positions of individual resonances below bandgap 1, while the actual fine spectrum of the fundamental transmission band above bandgap 1 was not well-resolved with the resolution of our measurements (1.3 pm in wavelength corresponding to 0.15 Å in ERV). The theoretical spectral plot shown in Fig. 2(d) is determined using the expression for the transmission amplitude ,Eq. (2) and parameters are determined by fitting the experimental data in Fig. 2(c): , , and .
Similar to the original microresonator chain shown in Fig. 2(a), the nonuniformity of the corrected chain is estimated from the variation of the vertical coordinates of BCAs at the fundamental transmission band in Fig. 2(c), which is magnified in Fig. 2(e). From Fig. 2(e), due to the smooth spatial and wavelength spectral behavior in the regions of BCAs and interpolation incorporated into the plots of the experimental data, we are able to determine BCA positions with an accuracy of ~0.01 Å. Consequently, the variation of BCA coordinates of the fabricated microresonator chain is estimated at 0.56 Å and their standard deviation at 0.12 Å. The precision of our measurements is estimated from the nonuniformity (deviation from the horizontal direction) of the discrete spectral lines situated below bandgap 1 in Fig. 2(c) and found to be close to the measurement resolution 0.15 Å. Generally, the nonuniformity of these lines determines the spectral shift due to the measurement errors and temperature variation, which can be calibrated out. The summarized fabrication precision of the microresonator chain is estimated at 0.7 Å.
Our approach for the evaluation of microresonator nonuniformity is validated by sample calculations shown in Fig. 2(f), which illustrates the effects of perturbation of the ERV of the uniform microresonator chain defined by Eq. (1) (white curve in Fig. 2(d)). In the first example, we perturb the center of this ERV by an angstrom-high and relatively wide Gaussian bump with Å and μm (white curve 1 in Fig. 2(f)). Comparison of this bump with the corresponding spectral plot of the fundamental transmission band in Fig. 2(f) confirms that wide (adiabatic) ERV perturbation envelopes the introduced shifts of BCAs. For narrower bumps, the effect of the apex shift is smaller. In the second example, we numerically correct the height of the BCA corresponding to the first microresonator of the chain, which, for the uniform chain, is shorter than others by 0.53 Å. This is accomplished using the Gaussian bump with Å and μm applied to the center of the first microresonator of the chain (white curve 2). This perturbation makes the height of the first BCA close to those in the middle of the chain.
In summary, a chain of 30 coupled SNAP microresonators is fabricated along the optical fiber with sub-angstrom precision of 0.7 angstrom and a standard deviation of 0.12 angstrom in excellent agreement with theoretical calculations. To the best of our knowledge, this result surpasses those achieved in other super-low-loss photonic technologies developed to date by two orders of magnitude. The fabrication is accomplished by application of iterative sequences of CO2 laser shots, where each shot introduces a small ~0.1 Å change in the local fiber radius. Generally, laser shots of appropriate durations and powers can be used for the nanoscale variation of the fiber radius and its iterative modification with sub-angstrom accuracy. The achieved fabrication precision is the result of sub-angstrom sensitivity of the fiber radius variation in response to the applied annealing power and repeatability of this variation. The demonstrated iterative method is applicable to the fabrication of a broad class of SNAP devices with sub-angstrom precision including uniform and nonuniform chains of coupled microresonators with predetermined spectral characteristics. Generally, this method allows the creation of photonic circuits with smaller microresonators, down to 10 μm in axial length . We suggest that with the improved fabrication the demonstrated sub-angstrom precision can be achieved without iterations. In addition, an order-of-magnitude improvement in fabrication precision, i.e., below 0.1 Å, is feasible with smaller iteration powers and more accurate fabrication and characterization processes.
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