1. Introduction

Application of subwavelength optical microfibers for fabrication of microphotonic devices is the subject of significant interest and technical challenge. An important advantage of the microfiber circuit is that its building block – a microfiber – has an atomically smooth surface and, therefore, its losses can be significantly smaller than losses introduced by lithographically fabricated photonic devices [1–3]. Though the loss in the microfibers fabricated so far [2,3] limits the Q-factor of the microfiber resonators to the value of ~10⁶ [4], it is possible that the glass microfibers could possess losses as small as those of the glass microcavities [5]. For this reason, the Q-factor of microresonators made of microfibers could potentially compete with the extremely high-Q whispering gallery mode microcavities, i.e. Q~10¹⁰ [5]. Recent experiments with silica microfibers [1,2,3] stimulated research aimed at understanding the future development of microfiber photonics, which resulted in creating the high-Q microfiber loop optical resonator [6,7].

 

Fig. 1. (a) Illustration of a COR; (b)illustration of a layered structure and a CROW; (c) illustration of a SCISSOR. Arrows show possible directions of light propagation.

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The all-pass microfiber coil optical resonator (COR) [4] is a three-dimensional generalization of the loop and ring resonators. This resonator can be created by wrapping an optical microfiber on a dielectric rod with smaller refractive index as illustrated in Fig. 1(a). Unique properties of the COR and feasibility of its fabrication in the near future offer a potential of using this device as a basic functional element for the microfiber-based photonics. Similarly to the two-dimensional resonant microring structures [8,9,10,11], CORs can perform complex filtering, time delay, switching, and lasing functions. In addition, the three-dimensional COR-based optical devices have two significant advantages over planar devices: smaller losses and greater compactness.

If the distance between the turns of a COR is small, the light propagating along the coiled microfiber can also move in the forward and backward directions through the interturn evanescent coupling and, under certain conditions, can be completely trapped by the COR. As a result, a COR, which appears to be geometrically opened to the incident light, can possess eigenmodes. Figure 1 shows a COR (Fig. 1(a)) along with two known types of periodic resonator waveguides: a coupled resonator optical waveguide (CROW) [12,13] (Fig. 1(b)) and a side-coupled integrated spaced sequence of resonators (SCISSOR) [14] (Fig. 1(c)). As opposed to a uniform COR, in which the interturn coupling is constant along the microfiber length, coupling in CROW and SCISSOR is localized near relatively short segments along the direction of light propagation. In the CROW and layered structures (Fig. 1(b)), the resonances are formed by an interference of waves propagating in opposite directions along the length of the structure. In an all-pass SCISSOR (Fig. 1(c)), the light propagates in one direction and the resonances are determined by propagation of light in a single ring. Comparison of the directions of propagation of light in these devices (Fig. 1) suggests that the COR resonances are created by combination of the interference effects specific to CROW and SCISSOR.

This paper investigates the transmission characteristics of a uniform COR with N turns. It is shown that the spectral and transmission properties of the uniform COR are unexpectedly different from those of the known types of resonators and photonic crystal structures. The analytical expression for the transmission amplitude of a uniform COR is derived in section 2. In section 3, the transmission time delay of a COR is described as a function of dimensionless propagation constant, B, and interturn coupling parameter, K. It is shown that, for certain discrete sequences of these parameters, the light is completely trapped by the resonator. For N→∞, the transmission time delay profile in the plane (B, K) experiences a fractal collapse to the points with coordinates K=½ and B=π(2n-½), where n is an integer. In section 4, the dispersion relation and transmission properties of a long COR are considered. It is shown that the COR with many turns (N→∞) has no stop bands. It is also shown that the coupling parameter K=½ corresponds to the crossover between two regimes of propagation: with (K>½) and without (K<½) zeroing of the group velocity. At the crossover point K=½, the group velocity has a second order zero (i.e. the group velocity becomes zero simultaneously with the inversed group velocity dispersion). At K<½, the behavior of a COR waveguide resembles that of a SCISSOR, whereas at K>½, it resembles the behavior of a CROW.

2. Solution of coupled wave and continuity equations for a uniform COR

Let us consider the stationary electromagnetic field propagating along the microfiber in the form A(s) exp(iβs)F(x, y) exp(iωt), where s is the coordinate along the microfiber, x and y are the transversal coordinates, t is the time, β is the propagation constant along the microfiber, and ω is the frequency. It is convenient to define the amplitude of the field at a turn m as A_m (s) and to consider s as the common coordinate along turns, so that 0<s<S, where S is the length of the turn. The propagation of light along the coil is described by the coupled wave equations [4]:

where κ is the coupling coefficient between adjacent turns. In Eq. (1), A ₁ (0) represents the input wave at the beginning of the COR and AN (S) represents the output wave at the end of the COR. In addition to Eq. (1), the following continuity condition should be taken into account:

Solution of Eqs. (1) and (2) is:

The transmission amplitude is defined as:

and the time delay is:

where n_eff is the effective refractive index of the microfiber and c is the speed of light. For the lossless COR considered below, |T|=1. Eq. (3) shows that the optical properties of COR depend on three dimensionless parameters: the number of turns, N, the dimensionless propagation constant, B=βS, and the coupling parameter, K=κS. The parameters B and K, which define the eigenmodes of the COR can be determined from the condition t_d =∞.

3. Time delay resonances and spectral behavior of a COR

The transmission spectrum of a uniform COR with two turns (N=2) is physically similar to the transmission spectra of the loop and ring resonators [4]. In this case, the eigenmodes exist at two conditions: (1) full interturn light transfer, κS=(π/2)(2m-1), and (2) constructive interference between the electromagnetic field at the adjacent turns, βS=(π/2)(4n-2m+1), where m and n are integers. If these conditions are met, the light in the input and output waveguides of the COR is completely separated from the light in its coil-shaped section. During the roundtrip, the light transfers from one turn to another 2m-1 times and fully returns to the original turn with the same phase. In Ref. [4] it was shown that the two-turn COR exhibits spectral behavior similar to that of conventional loop and ring resonators.

For the uniform COR with N>2, more than two turns are coupling simultaneously leading to complex interplay of propagation along the length of the microfiber and propagation through interturn coupling. Nevertheless, under certain conditions, the COR with N>2 possesses eigenmodes. Figure 2 shows the surface relief of the time delay t_d (B,K) for the CORs with different numbers of turns, N. Because t_d (B+2π,K)=t_d (B,K), only one full period of t_d (B,K), 2πn<B<2π(n+1), where n is an integer, is shown. From these plots, the spectrum of the COR with the fixed coupling parameter, K, is determined along a vertical line with K=const, as illustrated in Fig. 3. Figure 3(a) shows the time delay dependences on B of the COR with N=4 for K=0.5, 1.5, 2.5, and 3.5. Fig. 3(b) and (c) compare these dependencies with the characteristic time delay spectrum of a CROW with 4 turns and a SCISSOR. The spectra of an N-ring CROW and a SCISSOR have, respectively, the characteristic N peaks and a single peak in the interval of periodicity. However, the shape of the COR spectrum qualitatively depends on the value of K. The COR time delay spectral dependencies shown in Fig. 3(a) can be understood by comparing these dependences with the corresponding cross-sections of the N=4 plot in Fig. 2.

 

Fig. 2. Surface plots of the time delay in the plane (B,K) for the number of turns N equal to 2 through 7, 10, and 20. Brighter points correspond to larger time delay. For N=2,3,4,5, the points corresponding to the COR eigenmodes are marked by black dots. Upper row of plots show the time delay for 0<K<20, while the lower plots show it for 0<K<2.5 with higher resolution. In the lower plots, the blue and green circles mark similar features, V-shaped and W-shaped, respectively. For N→∞, all similar features tend to the collapse point. The ordinate of collapse points, K=0.5, is marked by a dotted red line.

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The top row of plots in Fig. 2 shows that the spectrum behavior is periodic as a function of coupling parameter, K, only for N=2 and N=3. The periodicity for N=2 follows from the discussion in the beginning of this section, whereas for N=3 an intuitive explanation of the periodicity has not been found. For N≥4, the dependence on coupling parameter, K, is no longer periodic. In all plots, the dark (infinitely small) points breaking the light lines correspond to the eigenvalues of COR. For each N the eigenvalues ues ($B_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ , K ^(N)_g) can be described by a pair of integers g=(g ₁, g ₂), where g ₁ counts the eigenvalues along the B-axis and g ₂ - along the K-axis. For any number of turns, N, the COR with fixed coupling parameter, K, has no more than two eigenvalues. This means that the first number of an eigenmode can have only two values, g ₁=1,2. However, for the fixed K, the number of resonances in the time delay may be as high as N. The second number of an eigenmode, g ₂, grows with K from 1 to infinity. The analytical behavior of the all-pass transmission amplitude in the neighborhood of an eigenvalue, ${\mathit{(}B}_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$, $K_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$), is described as T(B,K)≈(B-${\Gamma }_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$(K))/(B-Γ ^(N)* _g (K)), where ${\Gamma }_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$(K)=$B_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ +$c_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$${\mathit{(}K\mathit{-}K}_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$)+d(N)_g(K-$K_{\mathrm{g}}^{\left(\mathrm{N}\right)}$)² and $c_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ and $d_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ are constants. From the analyticity of T(B,K) and the causality arguments, it follows that the imaginary component of ${\Gamma }_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$(K) does not contain a term linear in K-$K_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ , i.e., ${\text{Im}\mathit{(}c}_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$)=0. Consequently, the singularities of the time delay, t_d (B,K), in the neighborhood of the COR eigenvalues have the form:

 

Fig. 3. Comparison of the time delay dependencies on the propagation constant for COR, CROW, and SCISSOR. (a) time delay spectrum for a COR with N=4 turns for K=0.5, 1.5, 2.5, and 3.5; (b) characteristic time delay spectrum of a CROW consisting of 4 rings; (c) characteristic time delay spectrum of a SCISSOR.

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It is seen from Eq. (6) that the singular behavior of the time delay is non-uniform as a function of B and K. In particular, the condition ${K\mathit{=}K}_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ corresponds to the absence of the resonance and to decoupling of an eigenmode from the input and output parts of the COR. Equation (6) shows that the resonances have qualitatively similar singular behavior that is independent of the number of turns N.

Notice the interesting evolution of the spectrum with growth of N, which is shown in the lower rows of plots in Fig. 2. All features of the spectrum including the eigenvalues which appear for some values of N in the lower row, do not disappear for larger N but rather move along the straight axial lines towards the point of spectral collapse, (B_c,K_c )=(π(2n-1/2), 1/2), (n is an integer). The size of these features shrinks proportionally to the distance from the collapse point. Figure 2 indicates the fractal behavior of spectrum for N→∞, which, in contrast to the spectrum behavior of other types of periodic structures for N→∞ [12,13], fully collapses at (B_c,K_c ). Consequently, for N→∞ any series of the eigenvalues, ($B_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ ,$K_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$), with fixed g tends to (B_c,K_c ).

 

Fig. 4. Surface plots of time delay for N=5,10,15,20,25 and 30. Each plot has different scale enhanced proportionally to N ². The lower value of coupling parameter for all plots is K=1/2. Lines connect the upper left eigenvalue points of the V-shaped features at each plot.

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For large number of turns, N≫1, the scaling law of the collapse can be determined as follows. Note that the number of identical features in the lower row in Fig. 2 along the K-axis is proportional to N. At the same time, the length of an individual feature, $w_q^{\mathit{\left(}N\mathit{\right)}}$ , which is counted from the point of

Table 1. Smallest eigenvalue coupling parameters of the COR, $K_{\mathit{(}\mathit{1}\mathit{,}\mathit{1}\mathit{)}}^{\mathit{\left(}N\mathit{\right)}}$ , as a function of number of turns, N

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collapse, (B_c,K_c ), grows proportional to its number q: $w_q^{\mathit{\left(}N\mathit{\right)}}$ ~C^(N)q. Here C^(N) is the characteristic size of the smallest feature near the collapse point. Therefore, the total length of identical features in Fig. 2 is proportional to C^(N) ${\sum }_{q=1}^N$ q~C^(N)N² . On the other hand, Fig. 2 also suggests that C ^(N) ${\sum }_{q=1}^N$ q~1. The latter yields the scaling law for characteristic size of features in the neighborhood of the collapse point as C^(N)~N^-2 . In particular, all eigenvalues approach the collapse point 2 K_c =½ according to the law $K_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$-Kc~N^-2 . Fig. 4 numerically confirms the N ^-2 law of collapse. It shows the behavior of time delay in the neighborhood of a collapse point rescaled with the scaling factor of C/N² for N=5,10,15,20,25, and 30. For better visualization, the upper left points of the V-shaped features are connected with the lines. The N^-2 law is manifested by the fact that, for large N, these lines become horizontal. Table 1 shows the magnitudes of the smallest eigenvalue coupling parameters, $K_{(1,1)}^{\mathit{\left(}N\mathit{\right)}}$, as a function of N. The N ^-2 law is specified for this series as $K_{(1,1)}^{\left(N\right)}-\frac{1}{2}\underset{N\to \infty }{\approx }3.25N^{-2}$.

 

Fig. 5. Spatial distribution of the electromagnetic filed for the first COR eigenmodes corresponding to g ₁=1 through 5 for N=2,3,4, and 10. For each N, the surface plot shows the corresponding time delay profile, similar to the ones in Fig. 2. The arrows link eigenvalues on the (B,K) plane and the corresponding eigenmodes. In case of N=10, for better visualization, the scale of the surface plot is enhanced in the neighborhood of K=1/2. The enhanced region is indicated by curved arrows. The numbers shown in each eigenmode plot is the corresponding value of the coupling parameter, K.

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The spatial distribution of the eigenmodes with smallest coupling parameters, $K_{\mathbf{g}}^{\mathit{\left(}N\mathit{\right)}}$ , having g ₁=1,2 and g ₂=1, 2, 3, 4, 5 for the COR with N=1,2,3,4,10 is shown in Fig. 5. For each N, the plot of eigenmodes is places next to the corresponding distribution of the time delay (compare with Fig. 2). The arrows in Fig. 5 connect the eigenmodes with the corresponding eigenvalues. Interestingly, the spatial distribution of the eigenmodes is smooth and has no correlation with the length of the turn, S. The eigenvalues having equal values of coupling parameter, K, correspond to the eigenmodes with similar spatial dependences.

 

Fig. 6. Characteristic dispersion relation for a COR, a CROW, and a SCISSOR. (a) dispersion relations for a COR having the coupling parameter K=0.25 (a1), K=0.5, (a2), and K=1.5 (a3); (b) dispersion relation of a CROW; (c) dispersion relation of a SCISSOR.

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3. Dispersion relation for infinite COR

For the COR with infinite number of turns, N→∞, the first and last equations in Eq. (1) can be ignored. Then Eq. (1) has a partial solution $A_m^{(\pm \mathrm{k})}$ (s)=exp[2iκcos(ξS)s±iξSm], where the integer m is a number of a turn and ξ is an effective propagation constant. The continuity condition, Eq. (2), applied to this solution leads to the dispersion relation:

where ω=cβ/n_eff is the frequency of electromagnetic field. Eq. (7) indicates that the coil optical waveguide does not have stop bands. From this equation, three qualitatively different situations occur depending on the value of coupling parameter, K. If K<½, the dispersion relation is monotonic. This case is illustrated in Fig. 6(a1) and is qualitatively similar to the dispersion relation for a SCISSOR, Fig. 6(c) [14]. The crossover situation occurs when K=½. The corresponding dispersion relation is shown in Fig. 6(a2). In this case, the function ω(ξ) has inflection points at ${\xi }_n=\pi \left(2n-\frac{1}{2}\right)⁄S,$, where n is an integer. In the vicinity of these points, ω(ξ)≈ω(ξ_n )+α(ξ-ξ_n )³ and the group velocity is zeroing simultaneously with the inverse group velocity dispersion. Similar situation has been recently investigated in photonic crystal waveguides [15]. A pulse with the spectrum in the neighborhood of these points experiences dramatic distortion. Finally, in the case of strong coupling (Fig. 6(a3)), when K>½, the dispersion relation has the minima and maxima points corresponding to zero group velocity. Propagation of light in the neighborhood of these points is similar to propagation near the band edges of photonics crystals and CROWs. This dispersion relation is illustrated in Fig. 6(b) [12,13]. In the vicinity of the local maxima and minima, the COR behavior is similar to the behavior of photonic crystals and CROWs near a band edge. However, in contrast to the CROW, the COR is the all-pass and no bandgap structure.

4. Summary

It is shown herein that the time delay spectrum of the uniform COR is qualitatively different from the resonators and waveguides studied previously. The behavior of the time delay resonances and eigenmodes of this resonator are investigated in the plane of its dimensionless propagation constant and coupling parameters (B,K). For N→∞, the spectral characteristics of the COR experience the fractal collapse in the plane (B,K) to the points (π(2n-½),½), where n is an integer. The dispersion relation of an infinitely long COR shows that the group velocity has the second order zero in the collapse points. Depending on the value of the coupling parameter, the COR may behave similarly to the CROW (K>½) or SCISSOR (K<½).

The subject of this paper was limited to the case of a uniform COR. Recent progress in fabrication of uniform microfibers [1,2,3] and microfiber loop resonators [6,7] suggests that the fabrication of a uniform COR with a few numbers of turns is feasible. Theoretical estimates impose strong limits on the nonuniformity of the COR [4]. For this reason, fabrication becomes more challenging as the number of turns increases. In general, the COR parameters can vary along the length of the microfiber. For example, it is of interest to consider a chirped COR, in which the length of the turns is gradually varying along the length of the microfiber coil. In another case, the COR may have a varying coupling coefficient, which can be localized near certain points of the microfiber. Generally, by choosing the appropriate geometrical parameters of the microfiber coil, one can design a COR with predefined optical transmission characteristics. If fabricated, this structure will serve as a flexible multifunctional device for the future microfiber-based photonics.