## Abstract

The optical microfiber coil resonator with self-coupling turns is suggested and investigated theoretically. This type of a microresonator has a three-dimensional geometry and complements the well-known Fabry-Perot (one-dimensional geometry, standing wave) and ring (two-dimensional geometry, traveling wave) types of microresonators. The coupled wave equations for the light propagation along the adiabatically bent coiled microfiber are derived. The particular cases of a microcoil having two and three turns are considered. The effect of microfiber radius variation on the value of Q-factor of resonances is studied.

©2004 Optical Society of America

## 1. Introduction

Currently, few types of high-Q light confinement structures are known, which can exist in uniformly filled dielectric microcavities. They are the whispering gallery modes (WGM) in cavities with smooth walls ([1–5] and references therein), bouncing-ball modes localized near the stable closed rays experiencing multiple reflection from the cavity wall [4,5], and modes in ring resonators [6]. For these structures, the high-Q light confinement is caused by total internal reflection from the surface of the cavity. Typically, for asymmetric dielectric cavities of this type, both in 2D and 3D, the high-Q modes are confined near a closed geodesic at the surface of the cavity or near a stable closed optical ray.

This work proposes and investigates the type of the uniformly filled dielectric microcavity having the eigenmodes, which are not localized near a closed ray. In practice, these eigenmodes can be formed in a coiled single mode optical microfiber illustrated in Fig. 1 if the diameter of fiber and distance between turns have the order of the wavelength of transmitted radiation, i.e., the order of a micron. In contrast to the WGM and modes in ring resonators, the light confinement in a coil resonator is achieved by an appropriate self-coupling between turns rather than by the presence of a closed optical path. The microcoil resonator has a 3D geometry that cannot be completed by the planar technology. However, similar to the Fabry-Perot and ring resonators, it has a one-dimensional structure of resonances. The important property of the microcoil resonator, which makes it different from other types, is that it can be composed of a single uniform fiber. A similar single-turn macro-device fabricated of a single mode fiber has been suggested and experimentally demonstrated for a 3 m long fiber loop 22 years ago using a fiber coupler [7]. The minimum diameter of the loop, which is allowed for the regular single mode fiber coil resonator, is a few cm. This limitation is mostly due to bending losses of a weakly guiding single mode fiber and dimensions of the fiber coupler. For this reason, the maximum value of the free spectrum range of fiber macro-coil resonator was limited to the order of a gigahertz. Alternatively, using the microfiber allows very small radius of the loop because of relatively large difference between the refractive index of the glass and of the air. Furthermore, the self-coupling in a microfiber coil is generated naturally so that the fiber coupling device is no longer needed.

Previously, the fiber tapers with the diameter of the order of a micron have been used as evanescent couples to microcavities [1,3] and for enhancement of the non-linear effects [8]. In practice, the microfiber coils illustrated in Fig. 1 can be wound on a cylindrical rod having smaller refractive index, e.g., an optical fiber with larger diameter, or can be a ring-shaped node like the one demonstrated in [9]. If the diameter of the microfiber is comparable to the radiation wavelength and the pitch of the coil, then coupling between adjacent turns is essential and may give rise to high-Q resonances. The authors of Ref. [9] suggested a technique for drawing the long silica microfibers having a uniform diameter, which is critical for fabrication of resonant microcoils.

Section 2 of this paper presents derivation of the coupled wave equations for light propagating along the coiled self-coupling optical microfiber with small radius variation. It is assumed that the coil has a shape close to a helix with small pitch and the approximation of parallel transport of electromagnetic field is used. In Section 3 the resonant transmission of electromagnetic field though a self-coupling two-turn microcoil is investigated. It is shown that if the variation of microfiber diameter is small enough then, under the resonant conditions, coupling can ensure the recirculation of electromagnetic wave and the microcoil can behave as a high-Q resonator. In section 4, the structure of resonances in a uniform three-turn microcoil is examined. Section 5 contains the general discussion of microcoil resonators and, in particular, compares the microcoil resonant structures to other types of dielectric microcavities. Section 6 summarizes the results obtained.

## 2. Coupled wave equations for self-coupling single mode fiber helical microcoil

Consider propagation of light along a single mode optical microfiber wound on a dielectric rod having refractive index smaller than the index of the fiber, as illustrated in Fig. 1. The fiber diameter is assumed to be smaller than or comparable to the wavelength of radiation. Introduce the local natural coordinate system (*x*, *y*, *s*), where *s* is the coordinate along the fiber axis and (*x*, *y*) are the coordinates along the normal, **n**, and binormal, **b**, of the axis, respectively. If the characteristic transversal dimension of the propagating mode is much smaller than the characteristic bend radius, then the adiabatic approximation of parallel transport can be applied. In this approximation and for a relatively small pitch of the helical microcoil, the transversal component of the electric field has the form (see Appendix A1 and A2):

Here, the vector function **F**
_{0}(*q*
_{1},*q*
_{2},*s*) defines the local transverse mode corresponding to the propagation constant *β*(*s*). Variation of **F**
_{0}(*q*
_{1},*q*
_{2},*s*) and *β*(*s*) as functions of *s* is assumed to be relatively small and adiabatically slow, in particular:

Below, the input radiation is assumed to have the form defined by Eq. (1), which determines the initial state of polarization. In the case of the weak guiding [10], which generally is not valid for the propagation along the microfiber considered in this paper, **E**
_{t0}(*x*,*y*,*s*) corresponds to the electric field polarized either along the normal, **n**, or along the binormal, **b**, of the fiber axis. Coupling of this state of polarization to the other states is neglected. The latter assumption is justified if the perturbations of refractive index, which break the separation of transversal variables (*x*, *y*), are negligible. For the ideal microfiber with the refractive index independent of *s*, these perturbations have purely geometric nature; they are small for small ratio of the coil’s pitch to its radius and are zero for the circular ring (see Appendix A2). For the non-ideal microfiber, the effect of these perturbations may be small if the unperturbed field distribution across the fiber is asymmetric enough. The asymmetry is caused by geometric bending, bent stress, and penetration into the rod. Then the microcoil will conserve the initial state of polarization similar to polarization maintaining fibers [11]. In further calculations we neglect the dependence of the transverse mode **F**
_{0} (*q*
_{1} ,*q*
_{2},*s*) on *s*.

Let us take into account weak coupling between co-propagating light in the adjacent turns of the microcoil. Coupling is important when the coil pitch *p* is comparable to the characteristic transversal dimension of the propagating mode. Usually this means that the pitch *p* has to be comparable to the fiber radius, *r*, and radiation wavelength, *λ*.

First, consider a coil consisting of two turns shown in Fig. 2. Assume that the lower and upper turns are close to each other near the point *s* of the lower turn. The point of the upper turn adjacent to the point *s* has the coordinate *s*+*S*. Here *S* is the period of the coil. In order to describe the propagating field, let us use the coordinate system of the lower turn, (*x*, *y*, *s*), *s*
_{1} < *s* < *s*
_{1} + *S* along the length *S* as the common coordinate system for both turns (Fig. 2). Point *s*
_{1} can be arbitrary chosen in the beginning of lower turn where coupling is negligible. For simplicity, we assume that the microfiber is wound on the cylindrical rod so that the centers of the upper and lower turns have the same *x*-coordinate. Then the local distance between the fibers can be calculated along the *y*-axis and is determined by the local pitch of the coil, *p*(*s*). Similar to the conventional coupled mode theory [10,12], we look for the solution of the vector wave equation having the transversal electric field component of the form:

where the first and the second term in this expression represent the field in the lower and upper turns, respectively. With this anzatz for electric field, we perform calculations similar to those used in derivation of the coupled wave equations [10,12]. As the result, after neglecting the terms that are exponentially small in *p*(*s*) on the condition of weak coupling and the terms that are small on the condition of relatively slow variation of coefficients *A*_{j}
(*s*), we arrive at the following coupled equations for the coefficients *A*_{j}
(*s*):

where the coupling coefficient *κ*(*s*) is determined by Eq. (A13) of the Appendix. From Eq. (1), the continuity of the field along the fiber axis demands that at a certain common point of the turns, *s*
_{1}, the coefficients *A*
_{1}(*s*) and *A*
_{2}(*s*) satisfy the equation:

The coupled wave equations can be generalized for the case of coil with *M* turns, which takes into account coupling between adjacent turns:

$${\phantom{\rule{3.2em}{0ex}}\chi}_{\mathit{pq}}\left(s\right)={\kappa}_{\mathit{pq}}\left(s\right)\mathrm{exp}\left(2i\underset{{s}_{1}}{\overset{s}{\int}}\left({\beta}_{p}\left(s\right)-{\beta}_{q}\left(s\right)\right)\mathit{ds}\right).$$

Here, the coefficients *A*_{m}
(*s*) are defined at the turns *s*
_{1}<*s*<*s*
_{1}+*S* and satisfy the continuity conditions:

In Eq. (6), the coefficients *κ*
_{m-1,m}(*s*)between the turns *m*-1 and *m* are defined similarly to Eq. (A13) and it is assumed that all the turns have equal length *S*. For the uniform helical microcoil, the coefficients *κ* are independent of *s*.

## 3. Microcoil resonator with two turns

The simplest self-coupling micro-fiber coil shown in Fig. 1a consists of two turns. For the uniform microfiber when

is zero the coupled wave equations, Eq. (4), can be solved analytically for arbitrary *κ*(*s*). In this section we solve Eq. (4) by perturbation theory assuming that the difference between propagation constants in the lower and upper turns is small compared to the coupling coefficient:

As the result, we obtain for the transmission coefficient (see Appendix A4):

$$\Xi =\frac{1}{2}\left[\left(1-{\epsilon}_{1}+{\epsilon}_{2}\right){e}^{-\mathit{iK}}-\left(1+{\epsilon}_{1}^{*}+{\epsilon}_{2}^{*}\right){e}^{\mathit{iK}}\right],$$

where the coupling parameter *K* and small parameters *ε*
_{1}, *ε*
_{2} are determined in Appendix A4. Parameters *ε*
_{1} and *ε*
_{2} are linear and quadratic in *Δβ*(*s*), respectively.

If the propagation losses are ignored, then the propagation constant, *β* is real and |*T*|= 1. In this case, the coil performs as an all pass filter [13] and the resonances of transmission coefficient appear in the group delay, *t*_{d}
, only. The group delay is determined by the derivative of the phase of *T* with respect to radiation wavelength *λ* and is equal to:

where *c* is the speed of light and *n*_{f}
is the refractive index of the fiber. From Eq.(10) and Eq .(11) the resonance condition can be found in the form:

where *m*, *n* are integers and function *δβ*
_{1}(*s*) is equal to *δβ*(*s*) from Eq. (2) in the lower turn. Eq. (12), similar to the theory of fiber couplers [10], defines the condition of complete transmission of the electromagnetic wave from one turn to another through coupling. Eq. (13) is the quantization rule for the propagation constant *β*
_{0}. For the uniform microfiber, when *ε*
_{1} = 0 and *δβ*
_{1}(*s*) = 0 , this equation differs from the similar quantization rule for the ring by additional phase *π*/2 which is acquired in transition between turns through coupling [7]. Near the resonance, the group delay is:

The corresponding Q-factor of the resonance is defined as the ratio of the propagation constant, *β*, to the width of the resonance:

Fig. 3 shows dependencies of the Q-factor as a function of separation of the coupling parameter *K* from the resonant value, which are plotted using Eq. (15) for the uniform fiber (*ε*
_{1} = 0). It is seen that the Q-factors up to 10^{8} are achieved for the difference *K*-*K*_{m}
of ~ 0.01.

In order to estimate the effect of microfiber radius variation, let us approximate coupling between turns by a constant, *κ*(*s*) = *κ*
_{0}, at the interval *Δs*
_{0} and zero elsewhere. Also, let the difference of propagation constants in the region of coupling be constant, *Δβ*(*s*) = *Δβ*
_{0}. Then using Eq. (A18) we find that in Eq. (14) and Eq. (15):

In derivation of Eq. (16) the coupling length *Δs*
_{0} was chosen satisfying the resonance condition, Eq. (12): exp(*iκ*
_{0}
*Δs*
_{0}) =*i*. In the case of a microfiber, when the weak guiding approximation is no longer valid, the propagation constant strongly depends on the fiber radius variation, *Δr*, so that (see, e.g., [14]):

In the experiment [9] the relative variation of the microfiber radius was *Δr* /*r* ~ 10^{-2} per mm. Estimating *Δr*/*r* ~ 10^{-3} along the length of a microcoil turn *S* ~ 100 *μ*m, for *β*
_{0} ~ 4 *μ*m^{-1}, and *κ*
_{0} ~0.4*μ*m^{-1}, from Eq. (16) and Eq. (17) we have (Re*ε*
_{1})^{2} ~10^{-4} . According to Eq (15), the latter corresponds to the maximum quality factor *Q*_{max}
~4 · 10^{6}. The typical variation of the regular single mode fiber radius is *Δr* /*r* ~ 10^{-4} per mm [15]. For this value of radius variation and the same parameters of microcoil we have *Q*_{max}
~4 · 10^{10}.

Similarly to the case of a ring resonator [6], for the finite propagation losses the maximum value of the Q-factor is limited by the value of the attenuation constant *α*: *Q* < *β*
_{0} / *α*. Assuming that the loss in a microfiber *α* is ~ 0.01 cm^{-1} [1], we find that the Q-factors as large as *Q*_{max}
~ 10^{6} are possible. If the losses in a microcoil are similar to the losses observed for the whispering gallery modes in glass microcavities [1–3], then significantly larger Q-factors could be expected for the microfiber with small radius variation.

As an example of a coil with a moderate Q-factor, Fig.4a shows dependences of the group delay for the coil with the diameter 125 μm on the inversed propagation constant, 2*πn*_{f}
/*β*
_{0}, near 2*πn*_{f}
/*β*
_{0} = 1550 nm (here *n*_{f}
= 1.46 is the refractive index of the fiber). The propagation losses are neglected. The dimensionless length of a turn, *β*
_{0}
*S*, is approximately 1600. The resonance dependencies correspond to the cases when coupling parameter *K* deviates from the resonant value *K*
_{1} = *π*/2 by 0.3*K*
_{1} (dashed line) and 0.15*K*
_{1} (solid line). Correspondingly, the off-resonant group delay equal to 2 ps is enhanced at resonances to 36 ps with the Q-factor ≈ 10^{4.5} (dashed line) and to 147 ps with the Q-factor ≈10^{5.1}(solid line).

## 4. Uniform microcoil resonator with three turns

An increase in the number of self-coupling turns leads to a more complex structure of resonances. In this section, we consider an example of a lossless microcoil with three turns shown in Fig. 1b. We assume that the coupling coefficient *κ* is independent of s along the coil and neglect the contribution of transition region at the beginning and the end of the coil where *κ* is assumed to vanish adiabatically. Then the coupling parameter defined by Eq.(A18) is *K* = *κS* and we obtain for the transmission coefficient:

$${Q}_{2}\left(\beta \right)={e}^{-\mathit{i\beta S}}-{2}^{1/2}i\mathrm{sin}\left({2}^{1/2}K\right)-{e}^{-\mathit{i\beta S}}{\mathrm{sin}}^{2}\left({2}^{-1/2}K\right)$$

From Eq. (11) and Eq. (18), we find that the time delay becomes infinitely large if one of the following two resonance conditions are met:

or

where *m* and *n* are integers. Fig.4b shows the group delay dependencies for both types of resonances determined by Eq. (19) (solid) and Eq. (20) (dashed). The corresponding coupling parameters are selected at a distance *ΔK* =0.15 ${K}_{1}^{\left(1\right)}$ and *ΔK* =0.15 ${K}_{01}^{\left(2\right)}$ from the resonant values ${K}_{1}^{\left(1\right)}$ and ${K}_{01}^{\left(2\right)}$ , respectively. The group delay dependence corresponding to the first series of resonances defined by Eq.(19) (solid) demonstrates resonances, which appear twice as frequent as the two-turn coil resonances shown in Fig. 4a. The structure of this series can be interpreted as a structure of a pair of two-turn coils, which are coupling to each other. The second series of resonances of the type described by Eq. (20) has the same frequency as the resonances for the two-turn microcoil. This series exists due to the presence of the plane of symmetry normal to the axis of the microcoil. In the three-turn microcoil, the effect of splitting of group delay peaks as a function of coupling parameter can be observed. Fig. 4b shows the plot of the group delay near the point of splitting (dotted).

## 5. Discussion

The coil resonator complements the well-known Fabry-Perot (standing wave) and ring (traveling wave) types. In the Fabry-Perot resonator, the resonance is formed by interference of light experiencing multiple reflections from the turning points. In the ring resonator, the resonance is formed by interference of light traveling along the closed path. In the coil resonator, the turning points are absent and the light path is open. However, the wave traveling along the open path can be recirculated on resonant conditions. The resonance is formed by the interference of light going from one turn to another along the microfiber and returning back to the previous turn with the aid of weak coupling. According to Eq. (10), the two-turn coil resonator has the same free spectrum range, as the ring resonator. However, the phase shift in the quantization rule for the microcoil in Eq. (10) is affected by coupling.

In general, the complex transmission spectrum of the microcoil can be designed by numerical solution of coupled wave equations. In particular, the microcoil can consist of several sections with a smaller pitch where coupling is significant, which are connected by the sections with a larger pitch where coupling is negligible (Fig. 1(c)). The transmission coefficient of this microcoil is the product of individual transmission coefficients and the total group delay is the sum of delay times of each section.

The coiled-fiber optical microresonators have several advantages compared to the 2D rings and disks, and 3D WGM microcavities. Similarly to the WGM microcavities fabricated by melting, the micro-fiber coil may have the atomically flat surface and generate the high-Q resonances. The microcoil fabricated of a single-mode fiber has a one-dimensional resonant structure and does not have the problem of overpopulation of resonant modes that exist for 3D WGM microcavities. It is significant that coupling into and out of the microcoil is naturally solved for the all-in-one-fiber device where the microcoil is comprised of a waist of a tapered fiber.

## 6. Summary

In this paper, the optical microfiber coil resonator with self-coupling turns is suggested and investigated theoretically. The coupled wave equations, which describe coupling between adjacent turns, are derived and applied to the case of microcoil with two and three turns. Small variation of the propagation constant along the length of microfiber is taken into account. In particular, the effect of microfiber radius variation on the value of Q-factor of resonances is determined.

## Appendix

## A1. Propagation of light along the curved optical microfiber

Propagation of light along the curved optical fiber can be described by the vector wave equation for the electric field:

where *n* is the refractive index, which for simplicity is assumed to be a scalar function, and *λ* is the wavelength of radiation in free space. Let us introduce the local orthogonal coordinates, (*q*
_{1}, *q*
_{2}, *s*), along the fiber axis *s*. It can be shown that, for the curved axis, these coordinates are uniquely defined by the equations [4]:

$${\mathbf{e}}_{1}=\mathbf{n}\phantom{\rule{.2em}{0ex}}\mathrm{cos}\left(\Lambda \left(s\right)\right)-\mathbf{b}\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\Lambda \left(s\right)\right),$$

$${\mathbf{e}}_{2}=\mathbf{n}\phantom{\rule{.2em}{0ex}}\mathrm{sin}\left(\Lambda \left(s\right)\right)-\mathbf{b}\phantom{\rule{.2em}{0ex}}\mathrm{cos}\left(\Lambda \left(s\right)\right),$$

where **r**(*q*
_{1},*q*
_{2},*s*) is the radius-vector of point (*q*
_{1}, *q*
_{2}, *s*), **r**(0,0,*s*) is the radius-vector of the fiber axis, **n** is the normal, and **b** is the binormal of the fiber axis. The rotation angle in Eq. (A2) is defined through the torsion of the fiber axis, *τ*(*s*) as:

The refractive index in Eq. (A1) can be written in the coordinates (*q*
_{1}, *q*
_{2}, *s*):

where the *s*-dependent part of the index, *Δn*(*q*
_{1}, *q*
_{2}, *s*), is small compared to *n*
_{0} (*q*
_{1}, *q*
_{2}) and is a slow function of *s*. In this coordinate system, the differential operators in Eq. (A1) are written as [4]:

$$\nabla =(\frac{1}{G({q}_{1},{q}_{2},s)}\frac{\partial}{\partial s},\frac{\partial}{\partial {q}_{1}},\frac{\partial}{\partial {q}_{2}})$$

$$G({q}_{1},{q}_{2},s)=1-\rho \left(s\right)\left({q}_{1}\mathrm{cos}\Lambda \left(s\right)-{q}_{2}\mathrm{sin}\Lambda \left(s\right)\right)$$

where *ρ*(*s*) is the curvature of the fiber axis at point *s* and *G*(*q*
_{1},*q*
_{2},*s*) is the metric coefficient.

In the case when the metric coefficient and refractive index are adiabatically slow function of *s*, the transversal components *q*
_{1} and *q*
_{2} in Eq.(A1) can be separated. A solution of Eq. (A1) has the form:

where *β*(*s*) is a propagation constant corresponding to the local transversal mode **F**
_{0} (*q*
_{1},*q*
_{2},*s*). The direction and magnitude of this electric field are slowly changing with *s* in the coordinate system (*q*
_{1},*q*
_{2},*s*). This system is rotating in the (**n**, **b**) plane with the speed determined by the torsion of fiber axis according to Eq. (A2) and Eq. (A3). The latter result is the rule of parallel transport of electric field ([4], [16] and references therein). It follows from Eq. (A3) that, if the fiber length is large enough, the rotation angle, *Λ*(*s*), can be finite even for a very weakly curved fiber. After full turn, the solid angle subtended by the fiber (Berry’s phase [4]) is:

where *S* is the length of the turn.

## A2. Metric coefficient *G*(*q*_{1},*q*_{2},*s*) and parallel transport for the helical microcoil with small pitch

Consider the fiber microcoil with the shape close to a helix having the parameters:

where *R* is the coil radius and *p*(*s*) is the coil pitch, which may slowly change as a function of *s*. It is assumed that the coil is wound on a cylinder rod with constant radius. Therefore, the coil radius *R* is independent of *s*.

Notice that in the case of a fiber ring, when *p*(*s*) = 0, the metric coefficient in Eq. (A5) is independent of *s*. Then, in the absence of the *s*-dependent component of the refractive index in Eq. (A4), the transversal coordinates, (*q*
_{1},*q*
_{2}), can be separated and there exists a parallel transport solution of the form defined by Eq. (A6). In order to have essential coupling between the turns, the pitch of the coil should be of the order of the microfiber radius *r*: *p*(*s*) ~ *r* and can be considered as a small parameter in Eq. (A8). Thus, the shape of each turn in the microcoil of our interest is close to a ring. For *p*(*s*) << *R* Eq. (A8) yields:

Let us choose the initial orientation of the coordinate system at *s* = 0 coinciding with the normal and binormal of the fiber axis:

$${\mathbf{e}}_{2}\left(0\right)=\mathbf{b}$$

which also means that *Λ*(0) = 0. For the fundamental mode of the fiber, the characteristic values of coordinates *q*
_{1} and *q*
_{2} have the same order as the fiber radius, *r*, and the pitch, *p*(*s*). Using Eq. (A9) and *p*(*s*)~ *r* ~ *q*_{j}
and neglecting the terms quadratic in *r*/*R*, we obtain the metric coefficient independent of the coordinate *s*:

The term linear in *q*
_{1} in Eq. (A11) contributes to the birefringence of the microfiber and is similar to the effective potential in the cylindrical system of coordinates (compare, e.g., with Ref. [17]). Thus, if only the linear terms in *p*(*s*) are taken into account and the refractive index is independent of s, Eq.(A1) allows separation of the transversal variables in the form of Eq. (A6). The linear dependence on *p*(*s*) is absent in Eq. (A6) and appears after the transformation of variables determined by Eq. (A2) with:

$$\mathrm{sin}\left(\Lambda \left(s\right)\right)=\frac{1}{2\pi {R}^{2}}\underset{0}{\overset{s}{\int}}p\left(s\right)\phantom{\rule{.2em}{0ex}}\mathit{ds}.$$

Similar to the exact *HE*
_{11} modes in the step index fiber [18], the *q*
_{1} and *q*
_{2} components of unperturbed transversal electric field in Eq. (A6) are approximately symmetric or antisymmetric with respect to *q*_{j}
: the *q*
_{1}- (*q*
_{2}-) component is an even (odd) or an odd (even) function of both *q*_{j}
. Then, as it follows from the general perturbation theory, the contribution of the terms linear in *p*(*s*) to the overlap integrals participating in the coupled mode equations considered in Section 2 is canceled. Therefore, the coupling wave equations of Section 2 are valid within the terms or the relative order (*p*/*R*)^{2} ~ (*r*/*R*)^{2}.

## A3. Expression for the coupling coefficient

The expression for the coupling coefficient can be written as follows:

$$+\frac{1}{2{n}_{f}^{2}}\underset{{x}^{2}+{y}^{2}={r}^{2}}{\oint}\frac{\mathit{dl}}{r}\left(x{F}_{0x}\left(x,y\right)+y{F}_{0y}\left(x,y\right)\right)\left(\frac{\partial {F}_{0x}\left(x,y-p\left(s\right)\right)}{\partial x}+\frac{\partial {F}_{0y}\left(x,y-p\left(s\right)\right)}{\partial y}\right]\},$$

where the transversal mode **F**
_{0}(*x*,*y*) is normalized as:

In Eq. (A13), *n*_{f}
and *n*_{e}
are the fiber and external refractive indices, respectively, and the last integral is taken along the circle at the surface of the fiber.

## A4. Solution of the coupled wave equation for the microcoil with two turns

The coupled wave equations, Eq.(4), can be solved exactly for *β*
_{1}(*s*) = *β*
_{2}(*s*). As in section 2, the solutions are considered inside the interval *s*
_{1} < *s* < *s*
_{1}+*S*, where *S* is the period of the coil. At the beginning and at the end of this interval, the turns are away from each other (see Fig. 2) and the coupling is negligible. If

is small compared to *κ*(*s*), solutions of Eq. (4) in lower turn, *A*
_{1} (*s*), and in the upper turn, *A*
_{2}(*s*), can be found by perturbation theory. Let us define these solutions at point *s*
_{1} where *κ*(*s*) = 0 by the boundary condition:

with arbitrary constants *A*
_{0j}. Then, near the point *s*
_{1} + *S* where, again, *κ*(*s*) = 0, within the terms quadratic in *Δβ*(*s*)/ *κ*(*s*) , we have:

$${A}_{2}\left({s}_{1}+S\right)=\frac{1}{2}\left[\left({A}_{01}+{A}_{02}\right)\left(1-{\epsilon}_{1}+{\epsilon}_{2}\right){e}^{\mathit{-}\mathit{iK}}-\left({A}_{01}-{A}_{02}\right)\left(1+{\epsilon}_{1}^{*}+{\epsilon}_{2}^{*}\right){e}^{\mathit{iK}}\right]$$

where:

$${\epsilon}_{1}=i\underset{{s}_{1}}{\overset{{s}_{1}+S}{\int}}\mathit{ds\Delta \beta}\left(s\right)\left[\mathrm{exp}\left(2i\underset{s}{\overset{{s}_{1}+S}{\int}}\mathit{ds\prime}\kappa \left(\mathit{s\prime}\right)\right)-1\right],$$

$${\epsilon}_{2}=2i\underset{{s}_{c1}}{\overset{{s}_{c2}}{\int}}\mathit{ds\kappa}\left(s\right)\left(\underset{{s}_{1}}{\overset{s}{\int}}\mathit{ds}\prime \mathit{\Delta \beta}\left(\mathit{s\prime}\right)\right)[\underset{{s}_{1}}{\overset{s}{\int}}\mathrm{ds}\prime \mathit{\Delta \beta}\left(\mathit{s\prime}\right)\mathrm{exp}\left(2i\underset{\mathit{s\prime}}{\overset{s}{\int}}\mathit{ds\prime \prime}\mathit{\kappa}\left(\mathit{s\prime \prime}\right)\right)].$$

Parameters *ε*
_{1} and *ε*
_{2}, which are linear and quadratic in *Δβ*, respectively, satisfy the relation:

The transmission coefficient is determined as:

From the matching condition at point *s*
_{1}, Eq. (5), we have:

Substitution of Eq. (A17), Eq. (A19), and Eq. (A21) into Eq. (A20) yields Eq. (8).

## Acknowledgments

The author appreciates fruitful discussions with D. DiGiovanni, J. Fini, A. Fotiadi, A. Hale, E. Naumova, and P. Westbrook.

## References

**1. **J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett. **22**, 1129–1131 (1997). [CrossRef] [PubMed]

**2. **V. I. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A **20**, 157 (2003). [CrossRef]

**3. **S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,” Phys. Rev. Lett. **91**, 04902 (2003). [CrossRef]

**4. **V. M. Babich and V. S. Buldyrev, *Short-Wavelength Diffraction Theory* (Springer-Verlag, Berlin, 1991). [CrossRef]

**5. **J. U. NÖckel, “2-d Microcavities: Theory and Experiments,” in *Cavity-Enhanced Spectroscopies*,R. D. van Zee and J. P. Looney, eds. (Academic Press, San Diego, 2002).

**6. **B.E. Little, S.T. Chu, H.A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**, 998–1005 (1997). [CrossRef]

**7. **L.F. Stokes, M. Chodorow, and H.J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. , **7**, 288–230 (1982). [CrossRef] [PubMed]

**8. **T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibres,” Opt. Lett. **25**, 1415–1417 (2000). [CrossRef]

**9. **L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature , **426**, 816–819 (2003). [CrossRef] [PubMed]

**10. **A. Ghatak and K. Thyagarajan, *Introduction to fiber optics* (Cambridge University Press, 1998).

**11. **S. C. Rashleigh, “Origin and control of polarization effects in single-mode fibers,” J. Lightwave Technol. , **1**, 312–331 (1983). [CrossRef]

**12. **A.W. Snyder, “Coupled mode theory for optical fibers,” J. Opt. Soc. Am. **62**, 1267–1277 (1972). [CrossRef]

**13. **G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE Journ. Quant.Electron. **37**, 525–532 (2001). [CrossRef]

**14. **L. Tong, J. Lou, and E. Mazur, “Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides,” Opt. Express , **12**, 1025–1035 (2004). [CrossRef] [PubMed]

**15. **M. Ibsen, M. K. Durkin, R. Feced, M. J. Cole, M. N. Zervas, and R. I. Laming, “Dispersion compensating fibre Bragg gratings,” Proceedings of SPIE , **4532**, 540–551 (2001). [CrossRef]

**16. **M. V. Berry, “Anticipations of the geometric phase,” Physics Today **43**, 34–40 (1990). [CrossRef]

**17. **W.W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, and W.-P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations,” J. Lightwave Technol. **16**, 910–914 (1998). [CrossRef]

**18. **M. J. Adams, *An introduction to optical waveguides* (John Wiley and sons, New York, 1981).