Surface nanoscale axial photonics

Dense photonic integration promises to revolutionize optical computing and communications. However, efforts towards this goal face unacceptable attenuation of light caused by surface roughness in microscopic devices. Here we address this problem by introducing Surface Nanoscale Axial Photonics (SNAP). The SNAP platform is based on whispering gallery modes circulating around the optical fiber surface and undergoing slow axial propagation readily described by the one-dimensional Schr\"odinger equation. These modes can be steered with dramatically small nanoscale variation of the fiber radius, which is quite simple to introduce in practice. The extremely low loss of SNAP devices is achieved due to the fantastically low surface roughness inherent in a drawn fiber surface. In excellent agreement with the developed theory, we experimentally demonstrate localization of light in quantum wells, halting light by a point source, tunneling through potential barriers, dark states, etc. This demonstration, prototyping basic quantum mechanical phenomena with light, has intriguing potential applications in filtering, switching, slowing light, and sensing.

optical communications and computing by implementation of superfast all-optical processors of light pulses on a chip and, in particular, miniature optical buffers [1,3] and all-optical switches [2,6].
Yet, despite the remarkable accomplishments, the existing platforms still face a severe limitation: It is necessary but very difficult to fabricate photonic circuits that have both microscopic dimensions and ultra-small losses (required, e.g., for creating the ultra-high quality-factor microresonators). Indeed, the microscopic dimensions of photonic elements can only be achieved with high-index-contrast material interfaces. However, precisely in this regime of high index contrast, the sensitivity to surface roughness and fabrication errors is daunting and potentially fundamentally limiting. To reduce the sensitivity to the interface roughness and uncontrolled attenuation, photonic circuits with low-index contrast can be exploited [9,15]. However, to avoid radiation losses at low-contrast waveguide bends, the size of photonic resonators has to have the millimeter scale [9,10]. These photonic circuits might be low-loss and easier to fabricate, but they are no longer microscopic.
In this Article, we present a new photonic platform with ultra-low loss, flexibility, and elements having microscopic dimensions. Our solution consists in Surface Nanoscale Axial Photonics (SNAP) employing whispering gallery modes (WGMs) [11][12][13][14] that circulate circumferentially around the surface of a thin optical fiber while also undergoing slow propagation along the fiber axis. SNAP elegantly manifests a number of key properties: (a) Remarkably low surface roughness effortlessly results from the fiber draw process. The key benefit of SNAP compared to lithography-based technologies is the orders of magnitude lower loss achieved at a high-index-contrast silica surface. (b) Periodicity and slow light without the periodic modulation of the refractive index (as for photonic crystals): Periodicity is introduced automatically by each revolution around the fiber surface. Axial propagation naturally has features of slow light, since the propagation of slow WGMs is primarily azimuthal. (c) SNAP is readily described by the one dimensional Schrödinger equation: The axial propagation of slow WGMs exhibits turning points, localization in quantum wells, tunneling through barriers, etc. (d) Precise control of light achieved with nanoscale variation of the effective fiber radius, which define the "potential" that steers, localizes, and engineers WGMs. (e) Novel phenomenon of surface WGMs completely halted with a point source of light. (f) Microscopic length scale in each dimension: For slow WGMs, the characteristic axial wavelength is much larger than the wavelength of light and has the order of a few tens of microns. This dramatically simplifies fabrication of SNAP devices and, in particular, the ring microresonators. As opposed to the macroscopic increase of the diameter of ring resonators in low-contrast photonic circuits, we increase the third dimension along the fiber axis and keep it microscopic. The central idea of SNAP is to exploit the sensitivity of WGMs to the extremely small variations of the fiber radius and index near the resonance res λ . Generally, a variation in radius causes coupling between modes and intermodal transitions, a complex problem which is usually addressed with the system of coupled wave equations [15]. In SNAP, this problem is absent: Variation ( ) r z ∆ and ( ) f n z ∆ is so small and smooth that the coupled wave equations become decoupled and a single WGM is defined by a single differential equation. Near the resonance wavelength res λ , this equation takes the form of the stationary Schrödinger equation (see Supplementary Information, Section 1): where the effective energy ( ) E λ is proportional to the wavelength variation and the effective potential Slow WGMs can be excited in an optical fiber using a microfiber (MF) [16], specifically, a micrometer diameter waist of a biconical fiber taper, which is attached normal to the SNAP device and connected to the light source and detector as illustrated in Fig. 1. The MF waveguide acts as a point source in Eq. (1) so that the amplitude of the WGM excited by the MF positioned at The spectrum of excited WGMs can be observed by measuring the transmission amplitude of the MF, Parameter C in Eq. (2) and (3) determines coupling to the MF. It is a weak function of wavelength and, with good accuracy, is constant in the vicinity of resonance res λ . The derivation of Eq. (2) and Eq. (3) based on the formalism of the Lippmann-Schwinger equation [17,18] is given in Section 2 of the Supplementary Information.
SNAP devices include the three basic building blocks illustrated in Fig. 2. The first is the WGM bottle microresonator [19] coupled to a MF ( Fig. 2(a)). This corresponds to a quantum well ( ) V z in Eq. A second building block is a concave fiber waist ( Fig. 2(b)). In this case, for wavelengths in the underbarrier region, ( ) E V z < , the WGM is localized due to the exponential decay of its amplitude away from the MF position 1 z . Alternatively, above the barrier, for ( ) In the particular case of a uniform fiber, Fig. 2(b) offers a simple explanation of the localization of light in a uniform cylindrical microresonator described with a less general semiclassical theory in [20] (see Supplementary Information, Section 3.1).
In the third building block (Fig. 2(c)), the MF is positioned near a turning point t z where the SNAP device has monotonically increasing radius. In this case, the wave that is launched by the MF along the positive axial direction, interferes with the wave that is launched along the negative direction and reflects from the turning point t z . At discrete wavelengths when the condition of destructive interference of these two waves is fulfilled, the distribution of light is fully localized between the turning point t z and MF position 1 z . Fig. 2(c) offers a simple explanation of the remarkable effect when a point contact with a MF source completely halts light propagating along the SNAP device, while the detailed theory of this effect is given in Section 3.2 of the Supplementary Information. In particular, Fig. 2(c) clarifies the appearance of localized conical modes discovered in [21].
Generally, a SNAP device includes a series of these building block elements, which is coupled to one or more transverse MFs. The field distribution along the SNAP device as well as the transmission spectrum, group delay, and dispersion of light transmitted through MFs can be engineered using Eq. (1), (2), and (3). In the experiments below we consider a SNAP device which reproduces the nanoscale radius variation of Fig. 2. In doing this, we experimentally demonstrate all the described basic elements and find excellent agreement of their performance with the developed theory.

Experiment: Localization in quantum wells, tunneling, dark states
The described basic SNAP phenomena, familiar from elementary quantum mechanics [22], are confirmed experimentally in excellent agreement with the developed theory. In our experiments, to arrive at small dimensions of SNAP devices, we fabricated them from a regular optical fiber drawn down to 0 11 m. r µ ≈ The nanoscale variation of the fiber radius, ( ) r z ∆ , was introduced using a simple technique of controlled local heating with CO 2 laser and pulling. Thus we avoid the surface roughness that might result from other methods of fiber post-processing. Other methods include annealing, laser polishing, and UV exposure of photosensitive fibers. Due to the drastic elongation along the fiber axis, the field distribution in SNAP devices is much easier to access, control, and engineer as compared to the WGM-based microdevices demonstrated previously including spherical, toroidal, and ring microresonators [13,14] as well as microresonators directly fabricated by relatively small deformation of the optical fiber surface [23].
We fabricated samples of SNAP devices reproducing the characteristic nanoscale fiber radius variation illustrated in Fig. 2, the shape of an elongated bottle (quantum well) with a neck (barrier). These samples include bottle microresonators with multiple axial localized states ( Fig. 3(b), (c)), three axial localized states ( Fig. 3(d), (e)) and a single axial localized state ( Fig. 3(f), (g)). The samples were experimentally characterized as follows. First, an MF was translated along the test fiber in 20 µm steps where the transmission amplitude spectra (vertical plots in Fig. 3 In excellent agreement with theory, we observed full localization of light in dramatically shallow bottle microresonators. In Fig. 3(b), (c), the fiber shape features an elongated (300 µm in axial length) and extremely shallow (only 7 nm in radius variation) bottle microcavity. In agreement with the developed theory, the bottle resonances in Fig. 3(b) are localized between turning points z t1 and z t2 . Depending on the value of coupling with the MF, the Q-factor of these resonances varies from a relatively low ( 4 3 10 Q ⋅ ) or greater than 10 6 (with the resolution limited by the measurement device). The resonance state can be dark (i.e., practically uncoupled from the MF) if the MF position approaches a node of the localized state or is deep in the underbarrier region (see Supplementary Information, Section 3.3). Approximating the radius variation in Fig. 3(b) and (d), with the quadratic dependence, 2 ( ) / 2 r z z R ∆ = − , we find the axial radius-of-curvature 0.93 R = m of the multi-level bottle microresonator (Fig. 3(c)) and a smaller 0.04 R = m for the three-level bottle microresonator (Fig. 3(e)). The experimental free spectral range of the resonances in Fig. 3(b) and (d) is 0.082 nm and 0.38 nm, which is in good agreement with the theoretical values 0.08 nm and 0.39 nm found from Eq. (1) for the harmonic oscillator, Supplementary Information, Section 4). Finally, Fig. 3(f) and (g) demonstrate a shallow bottle resonator with a single axial state. The length of this resonator is 70 µm and its height is only 5 angstrom. It is remarkable that such a minor deviation from uniformity is able to fully confine light.
Oscillations of spectra outside the quantum wells in Fig. 3 are easily explained. These oscillations result from the interference between light propagating directly through the MF and light, which couples into the test fiber, reflects from a turning point, and then couples back into the MF. Interestingly, on the right hand side of the barriers in Figs. 3(b), (d), and (f) there are high Q-factor resonances, which coincide with the quantum well resonances but are situated outside of the quantum wells. Appearance of these resonances is explained with Fig. 2: Light excited by a MF in the region of Fig. 2(c) propagates in direction of the quantum well, tunnels through the barrier, and resonantly attenuates. Finally, if the MF is positioned deeply in the underbarrier region (bottom of Figs 3(b), (d), and (f)), the outgoing WGMs strongly decay, interference is absent, and the transmission spectra are smooth.

Experiment: Halting light with a point source
We have experimentally demonstrated the remarkable effect of halting light with a point source, as predicted from our theory (see Fig. 2(c) and the description above). This effect takes place takes for the SNAP device illustrated in Fig. 4(a) when light excited at 1 z reflects from t z and destructively interferes at 1 z . The transmission spectra and radius variation of the device, measured as in the previous case, are shown in Fig. 4(b) and (c). The localization of light is confirmed experimentally with two microfibers, MF1 and MF2. MF2 is translated along the test fiber and probes the field excited by MF1 as illustrated in Fig. 4(a). The measurement results are shown in Fig. 4(d), where the vertical axis lines of all spectra correspond to zero transmission. It is seen that at the discrete values of wavelengths indicated by horizontal arrows, the field distribution is fully localized along the fiber segments with a length of less than 100 µm and a radius variation of less than 5 nm. To compare the measurement results with theory, the fiber radius variation is approximated by the quadratic dependence 2 ( ) / 2 r z z R ∆ = , and the Green's function of Eq. (1) is found analytically [26] (see Supplementary Information, Section 4). The axial fiber radius, 3.1 R = m, is found from Fig. 4(c). With this value, the comparison of the transmission spectra measured experimentally and calculated with Eq. (3) and Eq. (A4.2) of the Appendix (respectively, black and blue curves in Fig. 4(b)) shows excellent agreement. A minor deviation near the principal peak maxima is explained by the deviation of the actual radius variation from the quadratic dependence away from its minimum. Finally, for comparison, the surface plot of the theoretical calculation of the WGM field amplitude distribution found from Eq. (A4.2) for the parameters of the experiment, Fig. 4(f), is placed in the background of Fig. 4(d). Good agreement is found both for the positions of the localized states and for the field distribution. A deviation near the principal peaks is, again, due to the assumed quadratic approximation for the radius variation.

Discussion
Nanoscale variation of the effective optical fiber radius (including the variation of the fiber radius and/or refractive index) enables formation and integration of independent or coupled microdevices (e.g., a sequence of microresonators) on a SNAP platform. The axial radiation wavelength of SNAP microdevices is significantly larger than the wavelength of light, which simplifies their operation and broadens the field of potential applications. The further decrease of the axial size of SNAP elements can be achieved with larger variation of the effective fiber radius while retaining adiabatic behavior and adherence to the developed theory. Due to the very low attenuation of light propagation along the optical fiber surface, the SNAP microdevices can exhibit significantly improved performance in filtering, time delay, slowing light, switching, sensing, etc., compared to lithographically fabricated photonic circuits.

Derivation of Eq. (2) for the propagation constant of a slow WGM
The propagation constant β of optical modes in a fiber with radius 0 r and refractive index f n , which is situated in the surrounding medium with index 0 n , is defined by the equation [15] ( ) where p ζ is a root of the Airy function ( 0 2.338 ζ ≈ f f res f res n n r r n This equation coincided with the expression for ( , ) z β λ in Eq. (1).

Derivation of Eq. (2) for the WGM field distribution and Eq. (3) for the resonant transmission amplitude
Generally, the field excited by a MF in the SNAP device is localized in the vicinity of the MF/SNAP fiber contact point and does not have the rotational symmetry. However, near the resonance, res λ λ = , the beam launched by the MF constructively interferes in the process of circulation along the SNAP fiber surface. Then, after a large number of turns, the beam acquires axial symmetry. The axially symmetric component of the beam becomes much greater in amplitude compared to its original asymmetric part, so that the latter can be ignored.
For the nanoscale and adiabatic variation of the SNAP fiber radius and outside the region of coupling with the MF, i.e., in the absence of source, Eq. (1) is the known uncoupled wave equation [15] with the propagation constant defined by Eqs. (A1.9). For a MF with the diameter ~ 1 µm, the axial size of the coupling region is ~1  [17], the Hamiltonian describing the electromagnetic waves in the optical fiber coupled to a MF waveguide is approximated by The total wave function is determined by the Lippmann-Schwinger equation, where ε is a positive infinitesimal number enforcing the outgoing boundary condition and T is the scattering T-matrix. After approximate diagonalization and renormalization, the elements of the Tmatrix, which determine the transition from the incoming wave | 〉 k into the WGM | 〉 c are [ 17,18] , , and the WGM field distribution is found from Eq. (2). From Eq. (A3.1) and Eq. (3), the transmission amplitude through the MF is For a uniform SNAP fiber, 0 ( ) r z r = , the propagation constant is independent of z, γ . In the absence of attenuation, 0 res γ = , in accordance with illustration in Fig. 2(b), the WGM The tree-dimensional problem of resonant transmission through a uniform SNAP device coupled to a MF was solved in [20] by calculation of the sum over the turns of circulating beam excited in the SNAP fiber. Eq. (A3.2) coincides with the expression for the WGM field amplitude obtained in [20]. However, Eq. (A3.4) corrects the expression for the transmission amplitude of Ref. [20], which mistakenly contained the factor i in front of 2 | | C . With this correction, the shape of the transmission resonance appears to be asymmetric rather than symmetric. In particular, for the small MF/SNAP fiber coupling C, the transmission power is The ratio of the Q-factors of these resonators, which is inverse proportional to their FWHMs, is equal to 3.388.

One turning point. Localization enforced by a point contact.
If the radius variation of an optical fiber is monotonic, the presence of a MF may still leads to full localization of a WGM as illustrated in Fig. 2(c). In this case, solution of Eq. (1) exponentially vanishes to the left hand side of the turning point,