1. Introduction

For the past few decades, research and development in photonic integrated circuits [1–9] has struggled to build a platform with the miniature dimensions, flexibility, and control needed to deliver breakthrough capabilities in optical computing, communications, and fundamental science. The platform must include waveguides for propagation and routing light, but must also provide means for trapping light. Building microscopic high-quality-factor traps for photons is non-trivial. One cannot simply construct a microscopic version of a traditional optical resonator, since naturally reflecting materials (metals) are highly absorbing. Also, one cannot construct photonic quantum wells using a straightforward analogy to electronics and quantum mechanics. Indeed, this would require total internal reflection of light everywhere at the boundary surrounding the trap, a challenge for the low-loss optical cavities with microscopic dimensions [6].

Two basic platforms for fabrication of photonic circuits with microscale elements have been developed. The first, a ring resonator platform, is built on thin transparent photonic wires that tightly confine light due to the high refractive index contrast between the wire and surrounding material. In this platform, wires form closed rings that are coupled together to provide higher level of functionality [1,2,8,9]. The second platform is based on photonic crystals, materials with periodic modulation of refractive index [5]. Thin waveguides are created using a linear disruption of periodicity while microresonators are formed by localized defects [3–7]. With the advances in lithographic technology and design methods, these two platforms have steadily increased in complexity, achieved lowed loss, and broadened functional capability. The miniature integrated photonic circuits that have been proposed and demonstrated using the ring-resonator and photonic crystals platforms can potentially revolutionize 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 [9,10]. 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 paper, we present a new photonic platform with ultra-low loss, flexibility, and elements having microscopic dimensions [11]. Our solution consists in Surface Nanoscale Axial Photonics (SNAP) employing whispering gallery modes (WGMs) [12–15] that circulate circumferentially around the surface of a thin optical fiber while also undergoing slow propagation along the fiber axis (Fig. 1 ). SNAP elegantly manifests a number of key properties:

 

Fig. 1 Illustration of a SNAP device. The whispering gallery modes (WGMs) are excited in the SNAP fiber with a transverse microfiber (MF) connected to the light source and detector. The nanoscale variation of the effective radius of this fiber determines the distribution of slow WGMs along the fiber surface and also the spectrum of light transmitted through the MF.

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2. SNAP theory: prototyping one-dimensional quantum mechanics with light

Modes in an optical fiber are characterized by the propagation constant $\beta (\lambda ,z)$ which depends on the radiation wavelength λ and variations of the fiber radius, $r(z)=r_0+\Delta r(z)$ and refractive index $n_f(z)=n_{f0}+\Delta n_f(z)$. In conventional optical fibers, light is directed along the interior fiber core and has the propagation constant close to ${\beta }_0(\lambda )=2\pi n_{f0}/\lambda$. In contrast, SNAP employs transverse WGMs wrapped around the fiber surface by total internal reflection. The propagation constant of these modes is much smaller than ${\beta }_0(\lambda )$ and the speed of their axial propagation is much smaller than the speed of light in the fiber material, $c/n_{f0}$. In fact, the axial speed of a WGM and its propagation constant can be zero at the resonance wavelength ${\lambda }_{res}$ defined by the condition of stopped axial propagation $\beta ({\lambda }_{res}+i{\gamma }_{res},z_0)=0$ at a point z₀, where the resonance width ${\gamma }_{res}$ determines the propagation loss.

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 ${\lambda }_{res}$. 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 [16]. In SNAP, this problem is absent: Variation $\Delta r(z)$ and $\Delta n_f(z)$is so small and smooth that the coupled wave equations become decoupled and the dependence of the WGM field on the axial coordinate, A(z), is defined by a single differential equation. Near the resonance wavelength ${\lambda }_{res}$, this equation takes the form of the stationary Schrödinger equation (see Appendix A, Subsection A.1):

Near the resonance wavelength ${\lambda }_{res}$, dramatically small effective radius variations are enough to reflect and confine light. Indeed, it follows from Eq. (1) that for the typical experimentally determined value of ${\gamma }_{res}~{10}^{-7}$μm at wavelength ${\lambda }_{res}~1$μm and fiber radius $r_0~10$μm, light can be confined with an effective radius variation of the order of $r_0{\gamma }_{res}/{\lambda }_{res}~1$ pm. This corresponds to feature perturbations of $\Delta r_{eff}(z)~1$ pm and $\Delta n_f(z)~{10}^{-7}.$ The characteristic axial wavelength in this case is $2\pi /\beta (\lambda ,z)~{\lambda }_{res}^{3/2}{\gamma }_{res}^{-1/2}{n}_f^{-1}~$1mm, i.e., 1000 times greater than the wavelength of light. In the experiments described below in Sections 3 and 4, the characteristic axial wavelength is ~50 μm.

Slow WGMs can be excited in an optical fiber using a microfiber (MF) [17], 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 $z=z_1$ is expressed through the Green’s function $G(\lambda ,z_1,z)$ of Eq. (1),

The spectrum of excited WGMs can be observed by measuring the transmission amplitude of the MF,

Parameter C in Eqs. (3) and (4) determines coupling to the MF. It is a weak function of wavelength and, with good accuracy, is constant in the vicinity of resonance ${\lambda }_{res}$. The derivation of Eqs. (3) and (4) based on the formalism of the Lippmann-Schwinger equation [18,19] is given in Subsection A.2 of Appendix A.

SNAP devices include the three basic building blocks illustrated in Fig. 2 . The first is the WGM bottle microresonator [20] coupled to a MF (Fig. 2(a)). This corresponds to a quantum well $V(z)$ in Eq. (1), and leads to the formation of states localized between turning points $z_{t1}$ and $z_{t2}$ at a discrete series of wavelengths corresponding to resonances in the transmission amplitude $T(\lambda ,z_1)$. From Eq. (4), the resonance transmission amplitude $T(\lambda ,z_1)$ is proportional to the WGM amplitude at the MF position $z=z_1$. In particular, if $z_1$ coincides with a node of the WGM then the coupling to the MF vanishes and the WGM becomes dark, i.e., it practically does not show up in $T(\lambda ,z_1)$ (for details, see Appendix A, Subsection A.3.3).

 

Fig. 2 Basic elements of a SNAP device. (a) – MF coupling to a bottle microresonator. Slow WGMs excited by the MF can be trapped in the quantum well formed by the nanoscale variation of the effective fiber radius. (b) – MF coupling to the neck of the fiber. In this case, variation of the fiber radius forms a potential barrier. (c) – MF in the region of monotonic variation of the fiber radius. In this case, the WGMs emitted by the MF are reflected from the turning point and, under certain conditions, can be completely halted by the MF source.

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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 $z_1$. Alternatively, above the barrier, for $E>V(z)$, the excited WGM is delocalized. In the particular case of a uniform fiber, $\Delta r_{eff}(z)\equiv 0$, 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 Ref. [21]. (see Appendix A, Subsection A.3.1).

In the third building block (Fig. 2(c)), the MF is positioned near a turning point $z_t$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 $z_t$. 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 $z_t$ and MF position $z_1$. This effect is similar to the situation in the Mach-Zehnder interferometer when light propagating along two arms of the interferometer approaches the region of coupling with opposite phases and equal amplitudes so that the waves in two arms cancel each other. Figure 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 Subsection A.3.2 of Appendix A. In particular, Fig. 2(c) clarifies the appearance of localized conical modes discovered in Ref. [22].

These three basic building block elements can be used to create a wide range of densely integrated photonic devices by coupling them together. They can also be coupled to one or more transverse MFs to provide multiple inputs and outputs. In such integrated assemblies, 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), (3), and (4). Recently we have shown that it is possible to reproducibly fabricate SNAP microresonators at the surface of an optical fiber with sub-angstrom accuracy [23]. Further development and application of methods reported in [23] potentially allows creation of low-loss, densely packed devices for filtering, switching, slowing light, and sensing.

In the experiments below we consider SNAP devices which reproduce 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.

3. Experiment: Localization in quantum wells, tunneling, dark states

The described basic SNAP phenomena, familiar from elementary quantum mechanics [24], 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 $r_0\approx 11\mu \text{m}\text{.}$ The nanoscale variation of the effective fiber radius, $\Delta r_{eff}(z)$was introduced using a simple technique of controlled local heating with CO₂ laser and pulling. Thus we avoid the surface roughness that might result from other methods of fiber post-processing. It is known that post-processing of silica by annealing and melting causes variations of both the density and refractive index [25, 26]. In addition, post-processing of an optical fiber result in its refractive index and physical radius variation due to the relaxation of the frozen-in tension, which is introduced by the pulling force in the process of fiber drawing [27, 28]. From Eqs. (1) and (2), only the combined variation of the refractive index and physical fiber radius entering the expression for the effective fiber radius affects the performance of SNAP devices. 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 [14, 15] as well as microresonators directly fabricated by relatively small deformation of the optical fiber surface [29].

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 (Figs. 3(b), (c) ), three axial localized states (Figs. 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 Figs. 3(b), (d), and (f)) were measured with an optical vector analyzer (1.3 pm wavelength resolution). In our experiments, the resonances corresponding to different polarizations of light did not overlap in transmission spectra and only one polarization was considered. The effective fiber radius variations (bold curves in Figs. 3(c), (e), and (g)) are found from these plots by applying the MF scanning method [30, 31] corrected with Eqs. (1) and (4) (see Appendix A, Subsection A.5).

 

Fig. 3 Experimental characterization of the nanoscale effective radius variation and spectra of the fabricated bottle microresonators with a neck. (a) – Illustration of a bottle microresonator with a neck; (b), (d), and (f) – Resonant transmission spectra measured for the MF positions along the SNAP fiber spaced by 20 μm for the microresonator with multiple, three, and one axial state, respectively; (c), (e), and (g) – radius variation of the SNAP fiber in cases (b), (d), and (f).

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In excellent agreement with theory, we observed full localization of light in dramatically shallow bottle microresonators. In Figs. 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 ($Q~3\cdot {10}^4$) or greater than 10⁶ (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 Appendix A, Subsection A.3.3). Approximating the effective radius variation in Figs. 3(b) and (d), with the quadratic dependence, $\Delta r_{eff}(z)=-z^2/2R$, we find the axial radius-of-curvature $R=0.93$ m of the multi-level bottle microresonator (Fig. 3(c)) and a smaller $R=0.04$m for the three-level bottle microresonator (Fig. 3(e)). The experimental free spectral range $\Delta {\lambda }_{FSR}$ of the resonances in Figs. 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, $\Delta {\lambda }_{FSR}={\lambda }_{res}^2{(2\pi n_f)}^{-1}{(r_{0}R)}^{-1/2}$ (see Appendix A, Subsection A.4). At the field maxima, the resonance width is primarily determined by coupling to the MF (see Appendix A, Subsection A.3.3). For example, in Fig. 3(b) the maximum resonance width ${\gamma }_c^{(\max )}~0.1$ nm. Then, using Eq. (A.3.15) the value of coupling to MF is estimated, $|C{|}^2~0.1\mu {\text{m}}^{-1}$. Finally, Figs. 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. In the vicinity of the tip of the barrier, the tunneling rate may affect the Q-factor of resonances. 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.

The characteristic axial length of the low-lying WGMs of the SNAP bottle microresonators, $L_z~50$μm, is surprisingly small for so large axial radii-of-curvature R. This fact is simply explained by the expression for the WGM localized in a quadratic quantum well, Eq. (A.4.6) of Appendix A, yielding $L_z~R^{1/4}$. For example, the axial size L_z increases only by a factor of ten when radius R grows from the typical $R~100$μm [14,15] to the gigantic $R~1$ m considered here. For the same reason, dramatic increase of the axial radius does not significantly affect the value of the WGM volume, an important parameter in cavity quantum electrodynamics [7,14]. The volume $V_{mpq}$of the mode $(m,p,q)$ is usually defined as the ratio of the integral of the field power over the cavity volume and the maximum field power. From Eq. (A.4.6), we find $V_{mpq}={\xi }_{pq}{R}^{1/4}{r}_0^{19/12}{({\lambda }_{res}/n_f)}^{7/6}$ where for the first axial states the dimensionless parameter ${\xi }_{00}=1.78$ ${\xi }_{01}=3.18$, and ${\xi }_{02}=4.14$.

4. 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 $z_1$ reflects from $z_t$ and destructively interferes at $z_1$. The transmission spectra and radius variation of the device, measured as in the previous case, are shown in Figs. 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 $\Delta r_{eff}(z)=z^2/2R$, and the Green’s function of Eq. (1) is found analytically [32] (see Appendix A, Subsection A.4). The axial fiber radius, $R=3.1$ m, is found from Fig. 4(c). With this value, the comparison of the transmission spectra measured experimentally and calculated with Eq. (4) and Eq. (A.4.2) of Appendix A (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. In addition, in our calculations, we ignored the effect of the probe MF2, which can be made relatively small by decreasing the coupling to MF2 (experimentally, this can be achieved by translating MF2 along its length and increasing the MF2 diameter at the contact point). Finally, for comparison, the surface plot of the theoretical calculation of the WGM field amplitude distribution found from Eq. (A.4.2) 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. Note that coupling to the MF2 vanishes together with the field amplitude and, hence, does not change the condition of full localization. A deviation near the principal peaks is, again, due to the assumed quadratic approximation for the radius variation and coupling to the MF2.

 

Fig. 4 Experimental demonstration of the effect of full localization of light between a turning point and a point contact of a SNAP fiber with an MF. (a) – Illustration of the localized WGM field distribution excited by MF1 and scanned by MF2. (b) – Experimental resonance spectra along the SNAP fiber for the MF positions spaced by 100 μm (white curves) compared with theoretically found resonance spectra (blue curves). (c) – Radius variation of the SNAP fiber found from spectra in Fig. 4(b); (d) – Resonant transmission spectra measured for the MF2 positions along the SNAP fiber spaced by 20 μm and MF1 positioned at 240 μm from the minimum of the fiber radius; (f) – Surface plot of the theoretically calculated WGM field distribution as a function of the distance along the SNAP fiber and wavelength for the experimental radius variation found from Fig. 4(c).

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5. 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.