## Abstract

A vertically-stacked multi-ring resonator (VMR), which is a sequence of ring resonators stacked on top of each other, is investigated. The light in the VMR propagates horizontally in the plane of rings and at the same time propagates vertically between the adjacent rings due to evanescent coupling. If fabricated, the VMR may be advantageous compared to the conventional planar arrangement of coupled rings due to its dramatic compactness and more flexible transmission characteristics. In this paper, the uniform VMR, which consists of N rings coupled to the input and output waveguides, is studied. The uniform VMR is a 3D version of a coupled resonator optical waveguide (CROW). Closed analytical expressions for the transmission amplitudes and eigenvalues are obtained by solving coupled wave equations. In the approximation considered, it is shown that, in contrast to the conventional planar ring CROW, a VMR can possess eigenmodes even when interring coupling as well as coupling between rings and waveguides is strong. For the isolated VMR, the eigenvalues of the propagation constant are shown to change linearly with the interring coupling coefficient. The resonance transmission near the VMR eigenvalues is investigated. The dispersion relation of a VMR with an infinite number of rings is found. For weak coupling, the VMR dispersion relation is similar to that of a planar ring CROW (leading, however, to a much smaller group velocity), while for stronger coupling, a VMR does not possess bandgaps.

© 2005 Optical Society of America

## 1. Introduction

Microrings are often considered as building blocks for the microphotonic circuits. A variety of photonic devices based on microring resonators has been suggested and explored both theoretically and experimentally [1–7]. The microring-based photonic circuits are usually fabricated lithographically on a planar substrate and therefore have a 2D geometry [2–4]. However, arrangement of microrings in 3D is feasible as well. For example, instead of a planar substrate utilized in conventional lithography, one can use a cylindrical substrate having a radius comparable to the radius of microrings [8]. A photonic circuit created at the surface of a cylinder has a 3D geometry and can be much more compact than a planar circuit. So far, there has been little investigation on photonic structures that are specific to this 3D geometry. An example of this type of structure is a coil optical resonator (COR) [9,10], which can be created by wrapping a microfiber around a cylindrical substrate. Alternatively, a COR could be fabricated lithographically on a cylindrical substrate with the technology of Ref. [8]. Another example, illustrated in Fig. 1(a), is the sequence of ring resonators wrapping around the surface of an optical rod and coupling to each other along their lengths. The latter structure represents the simplest case of a 3D arrangement of microrings. We will call this structure a vertically-stacked multi-ring resonator (VMR).

In practice, a few vertically-stacked rings can be created with 3D photonic circuit technology (see e.g. Refs. [11,12]). The cylindrical substrate method is, however, more advantageous when the number of rings in the VMR is large. In particular, VMR-based waveguides containing many rings can be created interferometrically on the surface of a cylindrical substrate. The latter technique is similar to fiber Bragg gratings, diffraction gratings, and photonic crystal interferometric fabrication methods [13,14,15] and is advantageous for the fabrication of periodic structures like the one considered in this paper. With the interferometric technique, the refractive index and/or shape of the surface layer of the substrate cylinder is modulated along the length of the cylinder following the interference pattern of applied radiation. In another approach, a VMR can be created by periodic modulation of the optical cylinder radius mechanically, (e.g., by melting and pulling) [16]. Then VMR is formed by coupling between the whispering gallery modes, which exist in this structure. The theory of these modes can be developed using the method of Ref. [17].

This paper considers a uniform VMR consisting of *N* identical rings shown in Fig. 1(a). A uniform VMR is a 3D version of a uniform sequence of resonators, which are locally coupled to each other and are often referred to as coupled resonator optical waveguide (CROW) [5]. The most investigated 2D realization of a CROW consists of coupled ring resonators arranged on a plane [5,18], shown in Fig. 1(b). In this paper, the latter type of CROW will be called a planar multi-ring resonator (PMR). In the case of weak coupling between identical resonators, all types of CROW structures have qualitatively identical spectra [5]. However, for strong coupling, the qualitative behavior of a CROW spectrum depends on the mutual positions of rings. This paper investigates the transmission spectrum of a VMR and compares it with the spectrum of a PMR.

Expressions for the transmission amplitudes of a uniform VMR coupling to the input and output microfibers are derived in section 2. The calculation method used below has been developed previously in application to a COR [9,10]. This method consists in the solution of coupled wave equations, which describe coupling between adjacent rings, and continuity equations, which determine the condition of the ring closure. Section 3 briefly addresses the spectrum characteristics of a PMR, which is compared to a VMR in subsequent sections. In section 4, the calculation of eigenvalues and eigenmodes of an isolated VMR is performed. In section 5, simple Breit-Wigner formulae for the transmission amplitudes of the VMR weakly coupled to waveguides are presented. In section 6, it is shown that the VMR can possess eigenmodes even when interring coupling and coupling between rings and waveguides is strong. Simple expressions for the eigenvalues of a VMR are obtained, and the behavior of transmission amplitudes near eigenvalues is studied. Section 7 presents the dispersion relation of a VMR with a large number of rings and compares it to that of a PMR. For weak coupling, the VMR dispersion relation is shown to be similar to that of a PMR (leading, however, to much smaller group velocity), while for stronger coupling, a VMR does not possess bandgaps. In the final section, the obtained results are summarized and discussed.

## 2. Solution of the coupled wave equations for a uniform VMR

Consider the sequence of *N* evenly spaced identical rings positioned on top of each other as shown in Fig. 1(a). For determinacy, it is assumed that the rings are composed of single mode waveguides. The upper ring (*n*=1) and the lower ring (*n*=*N*) are coupled to the input and output waveguides (e.g., fiber tapers [19]), which are labeled by numbers 0 and *N*+1, respectively. At each ring or waveguide, *n*, the spatial component of the stationary electromagnetic field, which depends on the coordinate along the ring, *s*, is defined as *F*_{n}
(*s*) = *A*_{n}
(*s*)exp(*iβs*), where *β* is the propagation constant. For the rings, we assume 0 < *s* < *S*, where *S* is the length of a ring. The propagation of light along the sequence of coupled rings is described by the coupled wave equations:

$$\frac{d{A}_{n}}{\mathit{ds}}=\kappa \left({A}_{n-1}+{A}_{n+1}\right),\phantom{\rule{.2em}{0ex}}n=\mathrm{2,3}\cdots N-1,$$

$$\frac{d{A}_{N}}{\mathit{ds}}=\kappa {A}_{N-1}$$

where *κ* is the coupling coefficient between adjacent rings, which is assumed to be independent of *s*. The detailed discussion and derivation of Eqs. (1) can be found in Refs. [9,10]. The condition of closure of each ring has the form

The boundary conditions for Eqs. (1) and (2) are determined by the following equations of coupling between waveguides and adjacent rings:

$${A}_{1}\left(0\right)=i\mathrm{sin}\left({K}_{0}\right){A}_{0}^{\left(\mathit{in}\right)}+\mathrm{cos}\left({K}_{0}\right)\mathrm{exp}\left(\mathit{i\beta S}\right){A}_{1}\left(S\right)$$

$${A}_{N}\left(0\right)=\mathrm{cos}\left({K}_{0}\right)\mathrm{exp}\left(\mathit{i\beta S}\right){A}_{1}\left(S\right)$$

where ${A}_{0}^{\left(\mathit{\text{in}}\right)}$ and ${A}_{0}^{\left(\mathit{\text{out}}\right)}$ are the ingoing and outgoing amplitudes in the first waveguide, ${A}_{N+1}^{\left(\mathit{\text{out}}\right)}$ is the outgoing amplitude in the second waveguide, and *K*
_{0} is the coupling parameter between a waveguide and an adjacent ring. Eqs. (3), (4) are obtained by solving the coupled wave equations near the coupling region, similar to the theory of directional couplers [20]. The coupling parameter *K*
_{0} is defined through the local waveguide-ring coupling coefficient *κ*
_{0}(*s*)as

where ${S}_{c}^{\left(0\right)}$ is the full coupling length between a waveguide and an adjacent ring. Eqs. (3) and (4) assume that the coupling to waveguides 0 and *N*+1 is equal. Eqs. (3) and (4) describe lossless coupling between waveguide 0 and ring 1 and between waveguide *N*+1 and ring *N*, respectively. In Eq. (4), it is assumed that the ingoing amplitude ${A}_{N+1}^{\left(\mathit{\text{in}}\right)}$ = 0. Coupling between rings is supposed to be uniformly distributed along their lengths. Conversely, coupling between waveguides and adjacent rings is assumed to be localized at the small interval, ${S}_{c}^{\left(0\right)}$≪ *S*, which is large compared to the radiation wavelength.

The through and drop transmission amplitudes corresponding to the input and output waveguides, respectively, (Fig. 1(a)) are determined as follows:

$${T}^{\mathit{\left(}\mathit{drop}\right)}={A}_{N+1}^{\mathit{\left(}\mathit{out}\right)}/{A}_{0}^{\left(\mathit{in}\mathit{\right)}}.$$

We have solved Eqs. (1)-(4) analytically as follows. The general analytical solutions of the three-diagonal Eq.(1) were substituted into Eqs. (2)-(4). Then, the free parameters of these solutions were determined using the discrete Fourier transform (for more details, see the Appendix 1). As a result, the expressions for the transmission amplitudes are obtained,

and the electromagnetic field distribution along the *m*^{th}
ring is expressed by the following superposition of spatial harmonics:

In Eqs. (7),(8), the following notations are used:

$${g}^{\pm}=1\pm \mathrm{cos}\left({K}_{0}\right),$$

Here, the dimensionless propagation constant, *B*, and the coupling parameter, *K*, are introduced:

In the absence of losses considered below, the propagation constant *β* is real and

The useful parameter, which characterizes the magnitude of the field trapped by the resonator, is the value of electromagnetic field intensity averaged over the length of VMR:

In particular, the eigenvalues of VMR can be determined from the equation *P* = ∞. It is seen from Eqs. (7)-(11) and (14) that there are only three dimensionless parameters that determine the VMR transmission amplitudes and the average field. Those parameters are *B*, *K*, and *K*
_{0}. In addition, the spatial distribution of the electromagnetic field, Eq. (8), depends on the dimensionless length, *s*/*S*. Sections 4 and 5 investigate the behavior of transmission amplitudes and field distribution of a VMR as a function of dimensionless parameters *B* and *K*. The spectrum of a VMR will be compared with the spectrum of a PMR, which is described in the next section.

## 3. Structure of PMR eigenvalues

In the following sections, the VMR illustrated in Fig. 1(a) is compared with the PMR, which is illustrated in Fig. 1(b) and briefly considered in this section. The transmission amplitudes of a PMR are calculated with the transfer matrix method [21,22]. The coupling parameter between a waveguide and an adjacent ring, *K*
_{0}, is defined similarly to that of the VMR, by Eq. (5). The interring coupling parameter, *K*, is defined for the PMR by the equation *K* = ∫_{Sc}*κ*(*s*)*ds*, where the integral is calculated along the full interring coupling length *S*_{c}
. Thus, parameters *K* and *K*
_{0} have the same physical meaning as those of a VMR. The difference is that the *K* parameter of a VMR is determined by uniform coupling, while the *K* of a PMR is determined by localized coupling along the relatively short length *S*_{c}
, where adjacent rings are close to each other. The PMR through and drop transmission amplitudes, *T*
^{(thru)} and *T*
^{(droP)} , are defined through the parameters *B* and *K* in Appendix 2.

If coupling between waveguides and adjacent rings is small (*K*
_{0} ≪ 1), the transmission resonances are narrow and correspond to the eigenvalues of the isolated PMR. Fig. 2(a) shows the behavior of the through transmission resonances of the PMR in the (*B*, *K*) plane for small ring-waveguide coupling (*K*
_{0} =0.2) and *N* = 2,3,5,10. The dark lines of the resonance maxima correspond to the positions of the eigenvalues of an isolated PMR (*K*
_{0} → 0). At *K* = *πn* , *n* = 0,1,2,⋯, the rings are completely decoupled and have eigenvalues corresponding to those of an isolated ring, determined from the equation *B*_{l}
= 2*πl* with integer *l*. Conversely, at $k=\pi (n+{\scriptscriptstyle \frac{1}{2}})$ the rings are fully coupled, forming a closed loop with length *NS* and equally spaced eigenvalues *B*_{l}
= 2*πl* /*N*.

In the case of strong coupling to the waveguides (*K*
_{0} ~ 1), the resonance lines in the (*B*,*K*) plane broaden. As an example, Fig. 2(b) shows the distribution of the drop transmission amplitude for *K*
_{0} = 0.8. Here, the only remaining sharp resonances are located near points of complete interring decoupling, where *K* = *πn* . At these points, eigenmodes exist because all rings except for those adjacent to the waveguides are isolated. For this reason, at *K*
_{0} ~ 1, the 2-ring PMR (*N* = 2) cannot possess eigenmodes because it does not contain rings that are not adjacent to the waveguides. In fact, in Fig. 2(b), only the plot for *N* = 2 contains no sharp resonances.

## 4. Eigenvalues and eigenmodes of an isolated VMR

An isolated VMR corresponds to infinitesimal coupling to the input and output waveguides, *K*
_{0} → 0. From Eqs. (14), (9), and (10), the eigenvalues of an isolated COR are determined by the condition $\mathrm{tan}\left(\frac{1}{2}B+K{c}_{n}\right)=0,$ which can be reduced to the equation

where *l* is an integer. Eq. (15) determines straight lines *B*_{ln}
(*K*) = *Sβ*_{ln}
(*κ*) in the plane (*B*, *K*). The angle between these lines and the *B*-axis in the (*B*, *K*) plane equals *πn* /(*N* + 1) radian. The lines with numbers *l*
_{1} and *l*
_{2} are spaced along the *B*-axis by 2*π*(*l*
_{2} -*l*
_{1}). Fig. 3 shows the set of lines defined by Eq.(15) for *N* = 5 . The bold lines in Fig. 3 correspond to *n*
_{1} = 2 and *n*
_{2} = 5 , which are tilted with respect to axis *B* by angles *π*/4 and 5*π*/6, respectively. Figure 4(a) shows the profile of *T*
^{(thru)} for *K*
_{0} = 0.2 and the number of rings *N* = 2, 3, 4, and 5. The dark lines, which correspond to the transmission resonances, coincide with the straight trajectories determined by Eq. (15) (compare Fig. 3 and Fig. 4(a) for *N* = 5 ). While the behavior of VMR and PMR eigenvalues is similar for small coupling parameters *K*, it is qualitatively different for the larger *K*. In fact, as opposed to the PMR, the VMR exhibits the crossing of eigenvalue trajectories (compare Fig. 2 and Fig. 4(a)). At the crossing points (*B*
_{l1n1,l2n2} , *K*
_{l1n1,l2n2}), which are determined by the equation

the eigenmodes become degenerate. The present analysis, based on the approximation of the coupled wave equations, cannot conclude if it is a real crossing or the more common pseudo-crossing [23]. The parameters of the pseudo-crossing of eigenvalue trajectories are determined by small terms neglected in the derivation of the coupled wave equations [9].

If the eigenvalues determined by Eq. (15) are not degenerate, then only a single term in Eq. (9) contributes to the value of the electromagnetic field. This term defines the eigenmode of the isolated VMR in the form

where *C* is an arbitrary constant. Eq. (17) determines the eigenmode profile along the VMR. According to this equation, the amplitude of the eigenmode is constant along each ring. The amplitude varies with the ring number *m* as sin(*πmn* /(*N* +1)).

At the points where the lines cross each other (Fig. 3), the eigenmodes are degenerate. They are doubly degenerate if only two lines cross. This results in the existence of two pairs, (*l*
_{1},*n*
_{1}) and (*l*
_{2},*n*
_{2}), so that *β*
_{l1n1} (*K*) = *β*
_{l2n2}(*κ*). For the odd number of rings, *N*, the eigenmodes can also be triply degenerated. The latter happens when a vertical line intersects two lines that are symmetric with respect to this vertical line (see cases *N*=3 and *N*=5 in Fig. 4(a)). In the case of infinitesimal interring coupling (*K* → 0), the modes in the rings become independent and therefore *N*-fold degenerated. At the crossing points, the eigenmodes are determined as a linear combination of solutions defined by Eq. (17).

## 5. Transmission amplitude of a VMR weakly coupled to the waveguides, *K*_{0} << 1

Consider the case of weak coupling to the waveguides when *K*
_{0} << 1 and *K*
_{0} << *K*. The expressions for transmission amplitudes, *T*
^{(thru)} and *T*
^{(drop)}, as well as for the average field intensity, *P*, can be simplified near resonances determined by Eq. (15), (*B*
^{(ln)} (*K*),*K*), where they have the characteristic Breit-Wigner form [23]:

$$\mathit{\Delta B}=B-{B}_{\mathit{ln}}\left(K\right),\phantom{\rule{.2em}{0ex}}\Gamma =\frac{4{K}_{0}^{2}}{N+1}{s}_{n}^{2}.$$

From these equations, the resonance width, *Γ*, is independent of the coupling parameter, *K*. The latter result confirms the uniformity of line widths, which can be seen in Fig. 4(a). For *n* ~ 1, the resonance width, *Γ*, decreases with the growth of *N* as *N*
^{-3} , and the field intensity, *P*, grows as *N*
^{4}. Consequently, the dwell time at resonance, which is inversely proportional to *Γ*, grows as *N*
^{3}.

At the crossing points, the eigenvalues are degenerate, and Eq.(18) is no longer valid. This paper considers only the case of symmetric crossing, which corresponds to the condition *n*
_{2} = *N* + 1-*n*
_{1} in Eq. (16). In this case, the crossing lines determined from Eq.(15) are symmetrical with respect to the vertical line *B* = *B*
_{l1n1,l2n2}. The expressions for the through transmission amplitude in the case of symmetric crossing of two and three resonance lines are given in Appendix 4. These expressions, Eqs. (A19), (A20), and (A21), represent particular cases of the generalized Breit-Wigner formula [24,25], corresponding to the resonance transmission through the doubly and triply degenerate eigenvalues. For *K*
_{0} ≪1, Eqs. (A19), (A20), and (A21) can be simplified by substituting *g*
^{+} =2 and *g*
^{-}=${K}_{0}^{2}$ /2. For *ΔB*,*ΔK* ≪ ${K}_{0}^{2}$, Eqs. (A19), (A20), and (A21) can be simplified further to take the form of standard Breit-Wigner formulae, which are valid for *K*
_{0} ~ 1 as well. They will be considered in the next section.

## 6. Eigenmodes and eigenvalues of a VMR strongly coupled to the waveguides, *K*_{0} ~ 1

From Eq. (18), the widths of resonances grow with *K*
_{0} and, as seen in Fig. 4(b), the resonance lines blur for *K*
_{0} ~ 1. In fact, for finite values of *K*
_{0}, the VMR is open and generally does not contain eigenmodes. However, this section shows that eigenmodes still exist for discrete series of the coupling parameter, *K*. From Eqs. (8) and (9), the points that could be suspected for eigenvalues are those that satisfy the equation *E*_{n}
-1 = 0, i.e., lie on the resonance lines determined by Eq. (15). From Fig. 4(b), it follows that if eigenvalues exist, they should correspond to the crossing of resonance lines determined by Eq. (15). At different crossing points, resonances may have different local behaviors on the (*B*,*K*) plane. The local behavior of the through transmission amplitude near resonances is shown in Fig. 5 for *N* = 4 and 5. This figure compares the local behavior of the transmission amplitude and the corresponding average field intensity. At the eigenvalue points, the average intensity *P* → ∞, while it is finite elsewhere. It can be shown that not all of the crossing points correspond to the eigenvalues of the VMR. Particularly, the crossing points correspond to the eigenvalues only if they are defined by numbers *n*
_{1} and *n*
_{2} of the same parity (see Appendix 3). As an example, comparison of behavior of the through transmission amplitude and average field intensity for *N* = 4 in Figs. 5(a1) and (a2) shows that the crossing of lines corresponding to *n*
_{1} = 1 and *n*
_{2} = 4, which have different parities, does not correspond to a VMR eigenvalue. However, in the same figures, the crossing of lines with *n*
_{1} = 1 and *n*
_{2} = 3 corresponds to an eigenvalue. From Figs. 4(b) and 5, symmetric crossing may result in , , or -shaped singularities in the vicinity of an eigenvalue, or no singularity (no eigenmode) at all. An singularity may occur only if three lines (two symmetric and one vertical) cross. No eigenmode exists for the numbers *n*
_{1} and *n*
_{2} with different parities. Here we present particular examples of VMR resonance behavior near its eigenvalues. For an odd *N* and for *k*
_{0} ~ 1, Eqs. (A19), (A20), and (A21) of Appendix 4 can be simplified as follows. In the case of an singularity, the through transmission has a simple Breit-Wigner representation similar to Eq.(18):

The expression for *T*
^{(thru)} in the case of an singularity is

In Eqs.(19), (20) the resonance width is

Using Eq. (11), it can be found that for *n*
_{1} ~ 1, as opposed to the case *K*
_{0} ≪ 1 considered in section 5, the resonance width decreases inversely to *N*, *Γ* ~ *N*
^{-1}.

The singularity is localized near two lines crossing each other at the eigenvalue point and determined by the equation

Near these lines we have

where the resonance width is defined as

Consider an example of a VMR with five turns, *N*=5, depicted in Figs. 5(b), 5(b1), and 5(b2). Here, all types of singularities, , , and , exist. The singularity is situated at the crossing of the pairs of resonance lines, which do not cross the vertical line. These pairs correspond to the numbers *n*
_{1} = 1 and *n*
_{2} = 5 and to the numbers *n*
_{1} = 2 and *n*
_{2} = 4. If the crossing point of the same pairs of lines lies on the vertical line, then the crossing point corresponds to the or singularity. Fig. 6 shows characteristic profiles of the through transmission amplitudes on the plane (*B.K*) plotted using simple relations given by Eqs. (19), (20), and (23). It is seen that the characteristic plots of , , and singularities in Fig. 6 reproduce the corresponding singular behavior near eigenvalues in Fig. 5(b1).

## 7. Dispersion relation for a VMR with a large number of rings, *N*≫1

The dispersion relation for a VMR with an infinite number of rings can be obtained from the partial solution, *A*_{n}
(*s*) = exp(2*iκS*cos(*λ*)+*inλ*), of the coupled wave equations, Eq. (1). By applying conditions of ring closure, Eq. (2), to the latter solution, we find the dispersion relation in the form

where *l* is an integer, *d* is the pitch of the VMR, and *ξ* is the effective propagation constant along the vertical direction determined from the Bloch condition for the electromagnetic field *F*(*s* + *S*) = *F*(*s*)exp(*iξd*). In comparison, the dispersion relation of the infinite PMR is [18,21]

where *d*
_{0} is the period of the PMR, and *ξ* is the effective propagation constant along the PMR length. The plots of the dispersion relation for a relatively small and a large *K* are shown in Figs. 7(a) and 7(b), respectively. If *K* is small enough, then, similar to the PMR, the VMR exhibits bandgaps as shown in Figs. 7(a) and (b) for *K* = 0.5 For weak coupling (*K* ≪1), Eq. (25) is similar to the dispersion relation of the PMR, Eq. (26), obtained in the tight binding approximation [5] (arcsin(sin(*K*)cos(*ξd*
_{0}))≈*K*cos(*ξd*
_{0})). Notice, however, the significant difference in the value of the spatial period entering the dispersion relations, Eqs. (25) and (26), for a VMR and a PMR. For a VMR the spatial period, *d*, usually has the order of ring thickness, i.e., ~ 1 micron. For a PMR, the spatial period, *d*
_{0}, is close to the diameter of a ring, i.e., it may be orders of magnitudes greater than for a VMR. Respectively, the group velocity of the VMR is smaller than that of the PMR by a factor equal to the ratio of their spatial periods. The VMR dispersion relation, Eq. (25), is a “tight-binding” form of the dispersion relation, which, as opposed to the PMR, holds for the large *K* as well. Then, as follows from Eq. (25) and as illustrated in Fig.7(a) for *K* = 3, the VMR may have no bandgaps at all. The absence of bandgaps can also be explained by examining the eigenvalue behavior for a finite *N*. In fact, it is seen from Fig. 2 and Fig. 4(a) that if *K* is small enough, then for both the PMR and the VMR, a bandgap exists and remains empty for the large *N*. For the larger *K* and *N* → ∞ , the eigenvalue lines of the PMR do not cross except for the discrete values of $K=\pi \left(l+\frac{1}{2}\right).$ In contrast, for the VMR, if *K* is large enough (specifically, *K* > *π* for *N* → ∞), the eigenvalue lines cross and, intuitively, do not leave space for bandgap regions. For the PMR, the bandgap is closed only at discrete values of the coupling parameter, *K* = (*π* /2)(2*n* +1). For the VMR, the same coupling parameters correspond to the crossover between the condition with the existence and the condition with the absence of bandgaps.

## 8. Discussion and summary

While for weak coupling, all types of CROWs possess similar transmission spectra [5], their spectra may be qualitatively different if one of the structures is essentially non-one-dimensional and if coupling between resonators, of which it consists, is strong. The results obtained demonstrate quite different transmission characteristics of a uniform VMR, compared to those of the PMR. Closed analytical expressions for the transmission amplitudes and the electromagnetic field, which were obtained in this paper for a uniform VMR, allow for a rather comprehensive analysis of the VMR performance. The major differences in behavior of optical properties of a VMR and the PMR are evident in both their resonant spectra and their dispersion relations. The interesting feature of aVMR is the linear behavior of eigenvalues in the (*B*,*K*) plane and the existence of eigenvalues, which are degenerate for an isolated VMR. We have found conditions when the latter eigenvalues correspond to the eigenvalues of an open VMR strongly coupled to the input and output waveguides. Different types of singular behavior of transmission amplitudes near the VMR eigenvalues are investigated. Another interesting finding was the fact that the VMR dispersion relation has a simple form, which, for the general type of a CROW, is valid only for weak coupling. With this form of the dispersion relation, the closure of bandgaps with the growth of interring coupling is simply explained.

Having a 3D structure, a VMR can be much more compact than a PMR. For example, the characteristic diameter of ring resonators used in telecommunications is of the order 1 mm [2–4], which is ~ 1000 times greater than the ring thickness and the pitch of a VMR. The length of a VMR, compared to that of a PMR with the same number of rings, is ~ 1000 times smaller. Respectively, the group velocity of a VMR, compared to that of a PMR with similar rings and coupling parameters, is ~ 1000 times smaller as well.

The most appropriate technique for the fabrication of a uniform VMR seems to be the holographic method. In fact, in holographic fabrication, the periodicity of the pattern produced by the interference of coherent beams is ensured by the nature of the optical interference. The holographic method, which is well developed for the fiber Bragg grating and diffraction grating fabrication [13,14], can be naturally expanded to the fabrication of VMR structures.

The uniform VMR considered in this paper does not exhaust possible realizations and applications of more general types of a VMR. For example, a VMR may consist of rings with changing diameters and interring coupling coefficients. Because of the 3D nature of interring coupling, many of the VMR arrangements are not possible in 2D. Their optical properties may be significantly different from those of conventional planar photonic ring resonator circuits. For this reason, they are worth investigating.

## Appendix 1. Solution of the system of coupled wave equations

The solution of the three-diagonal Eq. (1) at the *n*
^{th} ring can be found by substitution *A*_{n}
(*s*) = *A*exp(*iλs*), which reduces this equation to the inversion of a three-diagonal matrix (see e.g. [26]). As the result, the general solution of Eq. (1) can be written in the form

where *a*_{m}
is an arbitrary coefficient. From Eqs. (2) and (A1), we have

where *E*_{m}
is defined by Eq. (8). Eq. (A2) has the form of a discrete Fourier transform, which omits two sums corresponding to *n*=1 and *n*=*N*. Let us define these sums as unknowns *x*
_{1} and *x*
_{2}:

$${x}_{2}=\sum _{m=1}^{N}{a}_{m}\left({E}_{m}-1\right)\mathrm{sin}\frac{\mathit{\pi Nm}}{N+1}.$$

Then, inversing the discrete Fourier transform determined by Eqs. (A2) and (A3), we find:

The values *x*
_{1} and *x*
_{2} are determined from a system of two linear equations, which are obtained by substituting Eqs. (A1) and (A2) into Eqs. (3) and (4). This results in

where σ^{+} and σ^{-} are defined by Eq. (10). From Eqs. (A1), (A4), and (A5), it is now simple to derive Eqs. (6) and (7) for the transmission amplitudes as well as to determine the electromagnetic field distribution along the rings.

## Appendix 2. Transmission amplitudes of the PMR

Following Refs. [21,22], the transmission amplitudes of a PMR are defined using the transfer matrix method. The spatial component of the electromagnetic field amplitude along the ring with number *n* is defined as ${A}_{n}^{(\pm )}$ exp(*iβs*), where + and - refer to the waves going in and out of the coupling region, respectively (see Fig. 1(b)). The relation between amplitudes along the adjacent rings is defined by the equation

Similarly, the relation between the amplitudes at the waveguides and at the adjacent rings is defined as

From Eqs. (A6) and (A7), we have

and, from this equation and Eq. (6), the through and drop transmission amplitudes are defined as

## Appendix 3. Calculation of σ^{±} near the crossing points

In the vicinity of the crossing points, the behavior of σ^{±} in Eq.(10) is singular and can be expressed as

$$\mathit{\Delta B}=B-{B}_{{l}_{1}{n}_{1}{l}_{2}{n}_{2}},\phantom{\rule{.2em}{0ex}}\mathit{\Delta K}=K-{K}_{{l}_{1}{n}_{1}{l}_{2}{n}_{2}},$$

where

From Eq. (A10), we have

$$+\frac{4\left[{\nu}_{{n}_{1}{n}_{2}}^{+}-{\left(-1\right)}^{{n}_{1}}\right]{s}_{{n}_{1}}^{2}}{\left(\mathit{\Delta B}+2\mathit{\Delta K}{c}_{{n}_{1}}\right)}+\frac{4\left[{\nu}_{{n}_{1}{n}_{2}}^{+}-{\left(-1\right)}^{{n}_{2}}\right]{s}_{{n}_{2}}^{2}}{\left(\mathit{\Delta B}+2\mathit{\Delta K}{c}_{{n}_{2}}\right)}.$$

It is seen from Eq. (A12) that if *n*
_{1} and *n*
_{2} have different parities, then, for *ΔB*,*ΔK* → 0 , the function (σ^{+})^{2} -(σ^{-})^{2} has a higher order singularity than the functions σ^{+} and σ^{-}. In this case, Eqs. (7)-(11) and (14) show that the electromagnetic field and the transmission amplitudes do not have singularities at the crossing points. On the contrary, if *n*
_{1} and *n*
_{2} have the same parity, the first term in Eq. (A12) becomes zero. Then, for *ΔB*,*ΔK* → 0 , the
singularities of (σ^{+})^{2} - (σ^{-})^{2} and σ^{±} have, in general, the same order, and the singularity exists.

Let us consider in detail the symmetric crossing, i.e., the situation in which the lines corresponding to (*l*
_{1},*n*
_{1}) and (*l*
_{2},*n*
_{2}) are symmetrical with respect to the vertical line and

Then, Eq. (16) is simplified to

In the case of an even number of turns, *N*, symmetric crossing does not lead to the singularity of the field and the transmission amplitude because, from Eq. (A13), *n*
_{1} and *n*
_{2} are numbers of different parities. Therefore, let us consider the case of an odd number of turns, *N*. In the vicinity of the crossing point defined by Eqs. (A13) and (A14), only three terms in the sum for σ^{±} in Eq. (10) can be very large. They correspond to the numbers *n* = *n*
_{1}, *n*
_{2} and (*N* + 1)/2. Due to the symmetry with respect to the number *n* = (*N* +1)/2, the rest of the terms in the sum for σ^{±} vanish for *ΔB*, *ΔK* → 0. Thus, near the crossing point defined by Eqs. (A13) and (A14),

$$+{(\pm 1)}^{{n}_{1}}\mathrm{cot}\left(\pi {l}_{2}+\frac{1}{2}\mathit{\Delta B}-\mathit{\Delta K}{c}_{{n}_{1}}\right){s}_{{n}_{1}}^{2},$$

$$+{(\pm 1)}^{\frac{N+1}{2}}\mathrm{cot}\left(\frac{\pi}{2}\left({l}_{1}+{l}_{2}\right)+\frac{1}{2}\mathit{\Delta B}\right)$$

where we neglected small terms of the order *ΔB*,*ΔK* and less.

For the odd *N*, the singular behavior of the transmission amplitudes and the electromagnetic field determined from Eqs. (8)-(11) and (A15) depends on the parity of *l*
_{1}-*l*
_{2}. Assume first that *l*
_{1} and *l*
_{2} have different parities. In this case, the last term in Eq. (A15) vanishes with *ΔB*. It corresponds to the situation in Fig. 4, when two symmetric resonance lines cross each other, but there is no vertical resonance line. With substitution 1/ *x* for cot(*x*), Eq. (A15) yields:

From this equation, (σ^{+})^{2} -(σ^{-})^{2} ≈ 0. Otherwise, if *l*
_{1} and *l*
_{2} have the same parity, then all three terms in Eq. (A15) contribute to the singularity. In this case, three resonance lines (two symmetric and one vertical) cross in Fig. 4. Then Eq. (A15) is simplified as follows:

From this equation, we find (σ^{+})^{2} - (σ^{-})^{2} ≈ 0 for the even *n*
_{1} + (*N* +1)/2. If *n*
_{1}+(*N* +)/2 is odd, then

## Appendix 4. Expressions for the transmission amplitudes

Substituting Eqs. (A16)-(A18) into Eqs. (7)-(10) yields the following expressions for symmetric crossing:

-type singularity:

-type singularity:

-type singularity:

where:

$$\Delta {\Lambda}_{1}^{\left(2\right)}=\frac{1}{4}{\left(\mathit{\Delta B}\right)}^{2}-{\left(\mathit{\Delta B}\right)}^{2}{c}_{{n}_{1}}^{2},$$

$$\Delta {\Lambda}_{2}^{\left(2\right)}=\frac{1}{4}{\left(\mathit{\Delta B}\right)}^{2}\left(1+2{s}_{{n}_{1}}^{2}\right)-{\left(\mathit{\Delta K}\right)}^{2}{c}_{{n}_{1}}^{2},$$

and *g*
^{±} =(1 ± cos(*K*
_{0})). Notice that the terms in the denominator of Eqs. (A19), (A20), (A21) are of the same order only for the small *K*
_{0}, when (*ΔB*)^{2} ~(*ΔK*)^{2} ~ ${K}_{0}^{2}$
*ΔB*. For *K*
_{0} ~ 1, one can neglect terms ~ (*ΔB*
^{2}) in Eqs. (19) and (22) but not in Eq. (A21). Similar expressions can be obtained for *T*
^{(drop)}.

## Acknowledgments

Thanks to Natasha Sumetsky for help in preparation of the manuscript.

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