Nanophotonics of optical fibers

2.1 Evanescent field of a uniform microfiber

If the optical microfiber diameter is close to or greater than the wavelength of light λ, then the fundamental mode is primarily localized inside the microfiber as shown in Figure 1(A). However, for smaller diameters, considerable fraction of light propagates outside the microfiber, as illustrated in Figure 1(B). The characteristic width of the mode W is determined by the transverse propagation constant γ₀ as W=2/γ₀. For example, for silica microfibers with small radius r₀<<γ and refractive index n=1.46 [24, 25]:

Figure 1

(A) Illustration of the fundamental mode in a microfiber having a diameter close to or greater than the radiation wavelength λ. (B) Illustration of the fundamental mode in a thin microfiber having a diameter much smaller than the radiation wavelength. In this case, the fundamental mode propagates primarily outside the microfiber. (C) The width of the evanescent field calculated as a function of microfiber radius with Eq. (1).

From this equation, the mode width increases dramatically with the decreasing of the microfiber radius. The dependence of the mode width on the microfiber diameter found from this equation is shown in Figure 1(C). From this figure, for a microfiber with radius r₀=100 nm, the fundamental mode width achieves the intuitively unrealistic value of 1 m. Can a microfiber of 100 nm radius support light with the characteristic transverse dimension of a meter? Formal theory of uniform waveguides cannot answer this question, which requires the consideration following below.

2.2 Optics of tunneling from an adiabatically tapered microfiber

Theoretically, the fundamental mode of a uniform dielectric waveguide considered in the previous subsection can be supported by a microfiber independently of its thickness. However, the waveguiding ability of a very low loss silica microfiber is primarily constrained by geometric nonuniformities. Following [24–27] the term “waveguide” is applied here only to such a waveguide that exhibits relatively small losses. In practice, the transmission loss of a waveguide is determined by its own optical properties as well as by the way the input and output of light is performed. In most cases, there exists a way to minimize the input and output losses down to relatively small values. For this reason, these losses can usually be separated from the losses of the waveguide itself. However, for a very thin waveguide and, in particular, for an optical microfiber, the transmission loss is primarily determined by input and output losses, which cannot be reduced significantly [24].

The hypothesis proposed in [24] was that the smallest input/output losses are achieved for a thin microfiber with radius r₀ [Figure 1(B)] if the input/output microfiber [Figure 1(A)] is connected to this microfiber adiabatically. Then, to determine the smallest transmission losses of a thin microfiber, it is necessary to determine losses of an adiabatic taper. As compared to the well-known problem of transmission losses in a bent uniform microfiber which allows approximate separation of variables and is reduced to tunneling through a potential barrier [28], the problem of loses in adiabatically tapered microfibers is more complicated. The optics of light propagation along an adiabatic taper is visualized in Figure 2. Usually, it is determined by the splitting of the fundamental mode near the focal circumferences shown in this figure. In the neighborhood of each focal circumference, the input mode is split into two components. The component which is closer to the microfiber axis continues as the fundamental mode, while another component contributes to the exponentially small radiation loss. The focal circumferences are responsible for splitting off the radiating components. The interference of these components with each other and the guiding mode gives rise to a complex behavior of the evanescent field. As an example, let us consider a taper which is determined by the following variation of the transverse propagation constant along the microfiber axis z [Figure 3(A)]:

Figure 2

Characteristic behavior of the fundamental mode propagating along the tapered microfiber. After passing the focal points indicated by black dots (which are the projections of focal circumferences in 3D space), the mode splits into guiding and radiating components.

Figure 3

Distribution of the electromagnetic field intensity near the microfiber taper defined by Eq. (2) with parameters γ₀=0.2 μm^-1, γ₁=0.4 μm^-1, and L=50 μm, for radiation wavelength λ=1.5 μm [26]. Calculations are performed using the beam propagation method.

Here γ₀ and γ₁ are the transverse propagation constants to the far left and far right hand side of this taper, respectively, and L is the characteristic length of the tapered region. Figure 3(B) shows the surface plot of electromagnetic field intensity which is calculated using the beam propagating method for the taper determined by Eq. (2) with parameters determined in the figure caption [26]. The upper arrow shows the direction of light propagation. The quasiperiodic dips with vanishing field intensity are explained as follows. In the neighborhood of the focal point indicated by a small circle, the radiating component of the field splits off and localizes along a curve starting near the focal point, which is slowly separating from the microfiber axis and going to infinity. From this line down, the radiating component exponentially decreases, while the guiding component exponentially increases. Therefore, at a certain distance from the microfiber axis, these components become equal in magnitude. Since the propagation constants of these components are different, their interference causes strong oscillations of the field.

2.3 How thin can a microfiber be and still guide light?

The theory developed in [24–26] and outlined in the previous subsection allows one to determine the propagation loss of adiabatically tapered microfibers in a relatively simple analytical form. In particular, in the case of a strongly tapered microfiber taper defined by Eq. (2) with γ₁>>γ₀, the propagation loss is independent of the transversal propagation constant γ₁ of the thicker part of the taper [26]:

According to the definition of a waveguide given in Subsection 1.2, a thin microfiber is waveguiding only if P<<1, while the condition P~1 corresponds to the threshold between the waveguiding and non-waveguiding behavior. In Figure 4, the dependencies of the propagation loss on the microfiber radius r₀ are presented for different characteristic taper lengths: small L=100 μm, intermediate L=10 cm, and dramatically large L=100 km. Calculations are performed using Eqs. (1)–(3). This figure demonstrates that, indeed, there is a dramatic threshold behavior of transmission loss (curves 1–3). Mathematically, the threshold is caused by the superfast double-exponential behavior of the transmission power in the microfiber radius [see Eqs. (1) and (3)]. Crucially, while the threshold values of the microfiber radius for a relatively short L=100 μm is 200 nm, it is only two times less than the threshold radius 100 nm for the gigantic L=100 km. Figure 4 demonstrates the fundamental limit for the radius of the waveguiding microfiber at wavelength λ=1.5 μm equal to r_min=100 nm. For the arbitrary wavelength λ and refractive index n of the microfiber, the smallest possible radius of the waveguiding microfiber is found as

Figure 4

Transmission loss as a function of the microfiber diameter r₀ calculated with Eq. (1), (2), and (3) for different characteristic lengths L of the microfiber taper indicated on the figure and radiation wavelength λ=1.5 μm.

From Figure 4, the threshold corresponds to a change in the microfiber radius of only several nanometers, which indicates a significant variation of the transmission spectrum with nanoscale change of the microfiber radius. These predictions have found excellent experimental confirmation both for the silica microfibers considered in the next subsection and for THz fibers [29].

2.4 Transmission threshold of the optical microfiber waveguide: experiment

In this subsection, following Ref. [27], it is shown that the experimentally determined radius of the thinnest possible waveguiding microfiber is in good agreement with predictions of the theory of adiabatic microfiber tapers described in Subsection 2.3. In the experiment considered in [27], the microfiber represents a waist of a biconical fiber taper. It was fabricated using the indirect laser heating method [8]. A conventional 62.5 μm radius optical fiber was placed into a sapphire tube serving as a microfurnace which was heated by an external CO₂ laser beam and drawn down to the submicron cross-section dimensions. The drawing method consists of periodical translations of the fiber into opposite directions with respect to the laser beam and simultaneous pulling in opposite directions [9]. After a cycle of drawing, the fiber radius is set to be reduced by f=0.75 so that after the Nth cycle the microfiber radius was:

The transmission loss of the biconical taper was calculated with the theory described in Subsection 2.3. The tapered sections of fabricated microfiber were approximated by Eqs. (1) and (2) with the characteristic length parameter L~250 μm. This parameter was determined by fitting the experimentally measured variation of the microfiber diameter. The method of measurement, which allows the detection of the radius variation with angstrom-scale precision, is described in Subsection 4.1. In Figure 5(A), the transmission loss defined using Eqs. (1) and (3) is plotted as a function of the microfiber radius for different wavelengths. This plot determines the microfiber radii, in the neighborhood of which the transmission loss starts a noticeable deviation from zero. The solid and dashed lines correspond to the sizes of the tapered region L=250 μm and L=500 μm, respectively. The table below Figure 5(A) determines the closest microfiber radii corresponding to the integer N in Eq. (5). The predictions of Figure 5(A) were verified experimentally by monitoring the transmission power of the microfiber taper in the process of drawing. Figure 5(B) shows the results of measurements [27]. The horizontal axis in this figure is the effective microfiber radius calculated by the continuation of Eq. (5) to the non-integer N. The characteristic wavy behavior of the spectra near the threshold radii is explained by temporal nonuniformity of the drawing process. Comparison of the theoretical calculation in Figure 5(A) with the experimental data in Figure 5(B) demonstrates a remarkable agreement of the values for the threshold radii.

Figure 5

(A) Theoretically calculated transmission loss as a function of the microfiber radius determined for different wavelengths (1230, 1320, 1430, and 1530 nm) from Eqs. (1)–(3). (B) Transmission loss of a microfiber measured in the process of drawing for the same transmission wavelengths as a function of radius. Two measurements for each wavelength illustrate the reproducibility of measurements. The transmission spectra are shifted along the vertical axis for visibility. The table shows the microfiber radii corresponding to the regions separated by the vertical dashed lines (drawing cycles) calculated from Eq. (5) [27].

3.1 “Quantum mechanics” of light at the surface of an optical fiber

We consider slow propagation of WGMs along the surface of an optical fiber with the propagation constant β(λ, z)<<β₀, depending on the radiation wavelength λ and axial coordinate z. Variation of the SNAP fiber radius and refractive index along the fiber axis z is so small and slow that the propagation of light is purely adiabatic. In the cylindrical frame of reference (φ, ρ, z), a WGM can be found as exp(imφ)Ξ_q(ρ)Ψ(z) [28] where m and q are the orbital and radial quantum numbers, and function Ψ(z) satisfies the Schrödinger equation [30, 31]:

where energy is proportional to wavelength variation, and potential is proportional to the effective radius variation:

In this equation, λ_res is the resonant wavelength, and γ_res determines the material attenuation. For silica fibers, parameter γ_res can be as small as 10^-3 pm. Figure 7 illustrates the relation between fiber radius variation and potential in the Schrödinger equation, Eq. (8). Usually, WGMs in an optical fiber are excited by the evanescent field of a transverse microfiber as shown in Figure 7(A). A WGM with wavelength λ has the propagation constant defined from Eq. (8):

Figure 7

(A) WGMs propagating along the surface of an optical fiber with nanoscale radius variation. These modes are excited with a transverse biconical taper having a microfiber waist and connected to the light source and detector. (B) Magnified nanoscale variation of the fiber radius containing a bottle resonator and a tunneling region. (C) Potential V(z) corresponding to the radius variation in (B) with a quantum well and a barrier.

Formally, the propagation constant can be very small (only limited by the value of material losses γ_res). In contrast to the slow light photonic devices, which are conventionally engineered from periodic structures (coupled microresonators [39–44] and photonic crystals [41, 45]), the SNAP fiber exhibits a naturally slow light created in the absence of an optical medium with a periodically modulated refractive index. The simplest derivation of Eqs. (8)–(10) assumes the original cylindrical symmetry of the fiber. However, these equations are also valid for realistic fibers with slightly asymmetric cross-sections. In this case, the local effective fiber radius is calculated by averaging over azimuthal angle.

Figure 7(B) illustrates the magnified fiber radius variation of a SNAP fiber, and Figure 7(C) shows the dependence of the potential in the Schrödinger equation, Eq. (8), proportional to this variation. The position of energy E (proportional to the wavelength variation) in Figure 7(C) exhibits basic building blocks of SNAP corresponding to phenomena well known in elementary quantum mechanics [46]. Region 1 is a bottle resonator [47] in Figure 7(B), which corresponds to a quantum well in Figure 7(C). If the input/output microfiber is situated in this region, it excites the WGMs, which are localized between turning points z_t1 and z_t2 at a discrete series of wavelengths. Region 2 in Figure 7(B) is a very shallow concave fiber waist corresponding to a potential barrier in Figure 7(C). In this region, under the barrier, for E<V(z), the WGM experiences exponential decay. Alternatively, above the barrier, for E>V(z), the WGM is delocalized. Finally, if the input/output microfiber is situated in region 3 has a monotonically increasing fiber radius, then a WGM propagating towards the barrier will reflect from the turning point z_t3 and interfere with itself. It can be shown that, in this case, at discrete wavelengths, when the condition of destructive interference is fulfilled, the distribution of light is fully localized between the turning point z_t3 and microfiber position z₁. Remarkably, in this case, a single point of contact with a microfiber completely halts light propagating along the surface of an optical fiber [30, 31].

The transmission amplitude of a SNAP fiber coupled to a single input/output microfiber illustrated in Figure 7(A) is [35]:

where S₀ is the out-of-resonance amplitude, C and D are the SNAP fiber/microfiber coupling parameters, and G(λ, z₁, z₂) is the Green’s function of the Schrödinger equation, Eq. (8).

Eqs. (8)–(11) fully describe slow propagation of WGMs along the surface of an optical fiber and can be used to create a wide range of SNAP devices. The SNAP fiber can also be coupled to more than one transverse input/output microfiber. In such integrated assemblies, the field distribution along the SNAP device as well as its transmission spectra are described by the general theory developed in [35]. This review is confined to the consideration of SNAP devices coupled to a single input/output microfiber.

Importantly, for slow propagation of WGMs considered here, the characteristic axial wavelength is much larger than the wavelength of light and, typically, has the order of a few tens of microns. This dramatically simplifies fabrication of SNAP devices considered in the next subsection and, in particular, the microresonators considered in Subsections 3.3 and 3.4. In addition, this makes the experimental characterization of SNAP devices described in Subsection 4.2 much easier.

3.2 Methods of fabrication of SNAP devices

A challenging problem of accurate and reproducible modification of the optical fiber effective radius at nanoscale precision is critical for the establishing of SNAP as a practical technology. This problem was solved in [32] using the IR (CO₂ laser) and UV (248 nm excimer laser) beam exposures. It has been shown that it is possible to fabricate coupled SNAP microresonators [32–36] and other devices [37] at the surface of an optical fiber with sub-angstrom precision.

A general method of precise variation of the effective radius of an optical fiber is based on local annealing of the fiber surface performed with a CO₂ laser beam [Figure 8(A)]. It is known that an optical fiber is drawn under certain tension which is frozen in after it cools down [48–50]. It was found that the release of this tension by short-time heating using, e.g., a CO₂ laser beam, causes a nanoscale variation in the fiber effective radius. Depending on the power and duration of the exposure (typically of the order of 1 s), it is possible to reproducibly introduce these variations with sub-angstrom precision (see Subsections 3.3 and 3.4).

Figure 8

(A) Illustration of a method for the variation of the effective radius of an optical fiber based on local annealing with a CO₂ laser beam. (B) Illustration of a method for the variation of the effective radius of an optical fiber by the exposure of UV radiation through an amplitude mask.

Another method for the fabrication of SNAP devices exploits photosensitive fibers. It is known that UV radiation affects both the refractive index and the density of the fiber (see, e.g., [51]). This effect can be used for local variation of the effective fiber radius. In particular, a series of coupled SNAP microresonators can be fabricated along the Ge-doped fiber by exposure to UV radiation through an amplitude mask as illustrated in Figure 8(B). This approach allows the fabrication of very long chains of couple microresonators with sub-angstrom precision [32, 34]. A combination of methods for the local modification of photosensitive optical fiber radii exploiting both UV and CO₂ exposures has been also demonstrated [34].

Both methods illustrated in Figure 8 allow for the fabrication of super-low-loss complex photonic circuits at the optical fiber surface with sub-angstrom precision. In the experiments below, we consider SNAP devices which consist of a series of coupled microresonators (Subsection 3.3) and of a bottle resonator with a parabolic radius variation (Subsection 3.4).

3.3 Fabrication of long coupled microresonator chains with sub-angstrom precision

As described in the previous subsection, the nanoscale modification of the effective radius of an optical fiber can be achieved with a focused CO₂ laser beam. The beam can be translated along the fiber with a varying velocity to introduce the predetermined radius variation. The simplest approach exploits application of multiple discrete laser shots, which can be used both for the introduction of the required variation and its correction. This approach was developed in [36] for the fabrication of coupled microresonator chains.

The experimental setup of Ref. [36] consisted of the SNAP fiber exposure section [Figure 8(A)] and the fiber characterization section described below in Subsection 4.2. The input beam power and the exposure time was controlled by a laser power controller and a shutter with a 1 ms switching time. After the exposure, the SNAP fiber was characterized at the measurement section with the microfiber scanning method [Figure 13(A)] [52, 53]. The spacing between the measurement points was set to 10 μm. The 30 radial bumps shown in Figure 9(A) that were originally introduced were not identical with the desired precision and had a radius deviation of 6 angstroms. The transmission bands and bandgaps (both in the discrete and continuous spectra) are clearly identified in Figure 9(A), while the angstrom-scale nonuniformity give rise to the appearance of localized states in the bandgap region. The uniformity of this microresonator chain was improved by the correction with shorter beam shots corresponding to a much smaller radius variation. First, the calibration of these shots was performed, when the created microresonators were successively exposed to a sequence of laser shots linearly increasing in number. The result of the calibration exposure is shown in Figure 9(B). By comparison of Figure 9(A) and (B), the radius variation introduced by a single shot equal to 0.07 angstrom was determined. Next, to make the microresonators equal, their radii were altered by exposing them to the appropriate numbers of shots. The spectral plot of a remarkably uniform microresonator chain, achieved after two iterations, is shown in Figure 9(C). The theoretical modeling of the created microresonator chain was performed using Eq. (8)–(11). The determined radius variation (solid white curve) is shown in Figure 9(D). The transmission spectrum distribution calculated with this radius variation is in excellent agreement with the experimental spectrum distribution in Figure 9(C).

Figure 9

(A) The surface plot of the transmission spectrum of the originally-fabricated 30 microresonator chain. (B) The transmission spectrum plot of this chain after calibration exposure. (C) The transmission spectrum plot of the corrected chain. (D) The transmission spectrum plot obtained by theoretical modeling of the fabricated microresonator chain. The white solid curve is the calculated effective radius variation.

Analysis of the experimental spectral distribution [Figure 9(C)] shows that, in excellent agreement with theoretical calculations, the chain of 30 coupled SNAP microresonators was fabricated with sub-angstrom precision of 0.7 angstrom and a standard deviation of 0.12 angstrom. This result surpasses the fabrication precision achieved in previously developed photonic technologies [39–45] by two orders of magnitude. An application of the SNAP coupled microresonator chain as a miniature delay line was demonstrated in [54].

3.4 Advanced programmed nanoscale variation of the optical fiber radius

A more general approach to introduce nanoscale variation of the effective fiber radius is performed by the programmed translation of the laser beam along the optical fiber. The beam width at the fiber surface or, more exactly, the axial width of the deformed region, determines the smallest possible axial size of the introduced features. Typically, this size is around 10 μm and is restricted by the CO₂ laser wavelength and fiber radius.

As an example, Figure 10 shows the spectral characterization of a SNAP device, which was fabricated by the translation of the focused CO₂ laser beam along the 19 μm fiber axis with a nonlinearly varied speed to arrive at the parabolic variation of the fiber radius [37]. The deviation of the introduced radius variation from the exact parabolic (white line) was <0.9 angstrom. This SNAP fiber was used in [37] to demonstrate a miniature slow light delay line with a breakthrough performance. In contrast to miniature slow light delay lines considered previously [39–45, 54], this delay line is not based on the periodic photonic crystal or coupled microresonator structures. Instead, it is fabricated of a bottle resonator with a nanoscale parabolic variation of its radius. The footprint of this device is only 0.12 mm² (<0.05 mm²/ns). It exhibits the record large 2.58 ns (3 bytes) dispersionless delay of 100 ps pulses. The intrinsic (0.44 dB/ns) and full (1.12 dB/ns) loss of this device is much smaller than 10–100 dB/ns losses demonstrated for miniature delay lines to date [45].

Figure 10

The surface plot of the experimentally measured transmission amplitude of the SNAP bottle resonator with parabolic radius variation.

In another example shown in Figure 11, the nanoscale fiber radius variation was introduced to imitate the word “NANO.” The height of this word is around 15 nm. This example together with that in Figures 9 and 10 illustrate the flexibility of the developed fabrication methods.

Figure 11

Imitation of the word “NANO” with nanoscale variation of the optical fiber radius.

3.5 Comparison of SNAP with previously developed photonic technologies

The advantages of the SNAP platform compared to previously developed technologies for the fabrication of miniature photonic devices (e.g., planar high and low refractive index, photonic crystal, and silica microresonators [39–45]) are outlined in Table 1. The characteristic dimensions of individual SNAP elements can be potentially as small as 10 μm, while the propagation loss and the fabrication precision of SNAP circuits is two orders of magnitude smaller than that of the previously developed photonic fabrication platforms. The success of SNAP is based on the remarkably low material loss and surface roughness of drawn silica fibers, evidence of strong localization of WGMs caused by nanoscale variations of the fiber radius, and possibility of the introduction and characterization of these variations with sub-angstrom precision.

Table 1

Comparison of the previously developed photonic fabrication platforms with the SNAP platform.

4.1 Characterization of the optical microfiber radius variation with angstrom precision

The characterization of the microfiber radius variation described in this subsection is important for many applications of microfibers and the investigation of their properties. In particular, it was employed for the determination of the taper dimensions in Subsection 2.4. In Ref. [14], a simple measurement tool for the comprehensive characterization of microfiber effective radius variation with sub-nanoscale precision was developed. The measurement setup is illustrated in Figure 12(A). It consists of a partly stripped optical fiber segment used as a probe which was positioned in direct contact with the microfiber under test normal to its axis. Generally, the fiber probe can be replaced by any light-absorbing material with a sharp edge (e.g., a metal razorblade). The microfiber studied in [14] was a waist of biconical fiber taper of 4 mm length and around 0.53 μm radius connected to the light source and detector. The probe was translated along the microfiber and, simultaneously, the transmission power was measured. For relatively small microfiber radius variation Δr_eff(z), the experimentally measured variation of the transmission power in the log scale, log[P(z)], is proportional to the radius variation at the point of the probe-microfiber contact:

Figure 12

(A) Illustration of the setup for characterization of the effective radius variation of a microfiber. (B) Logarithm of transmission power of the microfiber under test as a function of the probe coordinate vs. the radius variation measured by a SEM. (C) Effective radius variation along a 500 μm microfiber segment. Curves 1 and 2 (shifted along the vertical axis for visibility) demonstrate the repeatability of measurements. Curve 3 characterizes the same segment after the microfiber was rotated by 90^o. Inset: magnified curves 1 and 2 along the 100 μm segment.

The calibration constant was determined in Ref. [14] by comparison of the radius variation measured using scanning electron microscope (SEM) with the transmission power variation measured by scanning the probe along the microfiber. Figure 12(B) shows good correspondence between the transmission power and SEM measurements. Curves 1 and 2 in Figure 12(C) show excellent reproducibility of the effective radius variations measured along the 500 μm segment of the microfiber. The inset in Figure 12(C) shows a magnified comparison of these curves along a 100 μm segment. The RMS difference between curves 1 and 2 in Figure 12(C) is 0.37 nm, while in the inset it is only 0.19 nm. Curve 3 in Figure 12(C) characterizes the same segment rotated by 90^o. It is seen that curves 1, 2 and 3 are remarkably similar. This indicates the axial symmetry of the observed microdeformations. Curves 1 and 2 show two pronounced asymmetrically localized defects (sharp peaks), which are absent in curve 3, thus indicating the axial asymmetry of these defects. Using the experimental curves in Figure 12(C), it is possible to estimate the attenuation constant of the microfiber as α~0.01 dB/cm, which was in reasonable agreement with the direct measurements α~0.05 dB/cm [14]. The described technique opens intriguing opportunities for the investigation of optical and physical properties of optical microfibers at nanoscale precision.

4.2 Characterization of conventional optical fiber radius variation with sub-angstrom precision

Characterization of the optical fiber radius along the centimeter-scale length with sub-angstrom precision has been demonstrated in [53]. Aside from general interest and application to the characterization of SNAP devices, accurate measurement of fiber radius variation is important for the improvement of fiber Bragg grating quality [55], understanding the fiber drawing process, and other applications. The previously known methods [56–58] solved this problem with an accuracy at best of a few tens nm.

Figure 13

(A) Illustration of the experimental setup for characterization of the optical fiber radius. (B) Characterization of the effective radius variation along the fiber used for fabrication of SNAP microresonators described in Section 3.

The method of fiber radius characterization [53] develops the idea proposed by Birks and coauthors [52]. Figure 13(A) illustrates the experimental setup used in [53]. A biconical fiber taper with a microfiber waist was connected to a laser source and detector. The microfiber was translated relative to a tested fiber with two linear orthogonal stages. In the process of measurement, the microfiber was translated along the tested fiber and periodically touched it at contact points where the resonant transmission spectrum was measured.

Generally, the fiber radius variation Δr_eff(z) can be determined from the measured transmission amplitude S(λ, z) using the SNAP theory reviewed in Subsection 3.1 and by solving the inverse problem. In several cases, the solution of the inverse problem is simplified. For example, if the fiber radius is changing extremely slowly [53], it is proportional to the shift of the resonant peak [52]

Figure 13(B) shows the surface plot of the resonant transmission amplitude of the 19 μm radius optical fiber used in experiments described in Section 3. The radius of this fiber has the slope of 0.5 nm/mm and is not as uniform as the commercial fiber A considered in [53].

The smooth and elongated shape of a SNAP bottle resonator like that shown in Figure 10 can be determined in the semiclassical approximation [31]. For each microfiber position z, the resonant peaks of amplitude are located in the interval between the turning point wavelengths, λ_Turn(z)=λ_res(1+Δr_eff)(z)/r₀) and the wavelength λ_cont corresponding to the threshold between the discrete and continuous spectra, which is independent of z. Thus, in this approximation,

To determine the fiber radius variation more accurately as well as for more general situations, the inverse problem for the Schrödinger equation, Eq. (8), can be solved numerically.