## Vertically-stacked multi-ring resonator

Optics Express, Vol. 13, Issue 17, pp. 6354-6375 (2005)

http://dx.doi.org/10.1364/OPEX.13.006354

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### 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

1. C. K. Madsen and G. Lenz, “Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,” IEEE Photon. Technol. Lett. **10**, 994–996 (1998). [CrossRef]

2. C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, “An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,” Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002).

9. M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express **12**, 2303–2316 (2004),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

10. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express **13**, 4331–4340 (2005),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331. [CrossRef] [PubMed]

11. M. Raburn, B. Liu, K. Rauscher, Y. Okuno, N. Dagli, and J. E. Bowers, “3-D Photonic Circuit Technology,” IEEE J. Sel. Top. Quant. Electron. **8**, 935–942 (2001). [CrossRef]

12. P. Koonath, K. Kishima, T. Indukuri, and B. Jalali, “Sculpting of three-dimensional nano-optical structures in silicon,” Appl. Phys. Lett. **83**, 4909–4911 (2003). [CrossRef]

14. M. Sumetsky, B. J. Eggleton, and P. S. Westbrook, “Holographic methods for phase mask and fiber grating fabrication and characterization,” Proc. SPIE **4941**, 1 (2003). [CrossRef]

15. S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. **76**, 2668–2670 (2000). [CrossRef]

16. G. Kakarantzas, T.E. Dimmick, T.A. Birks, R. LeRoux, and P.St.J. Russell “Miniature all-fiber devices based on CO_{2} laser microstructuring of tapered fibers,” Opt. Lett. **26**, 1137–1139 (2001). [CrossRef]

17. M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. **29**, 8–10 (2004). [CrossRef] [PubMed]

*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

5. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

5. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

18. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665–1672 (2004). [CrossRef]

5. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

9. M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express **12**, 2303–2316 (2004),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

10. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express **13**, 4331–4340 (2005),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331. [CrossRef] [PubMed]

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

*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

19. K. J. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

*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:

*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

9. M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express **12**, 2303–2316 (2004),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

10. M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express **13**, 4331–4340 (2005),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331. [CrossRef] [PubMed]

*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

*N*+1 is equal. Eqs. (3) and (4

4. J. Niehusmann, A. VÖrckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. **29**, 2861–2863 (2004). [CrossRef]

*N*+1 and ring

*N*, respectively. In Eq. (4), it is assumed that the ingoing amplitude

*S*, which is large compared to the radiation wavelength.

*m*

^{th}ring is expressed by the following superposition of spatial harmonics:

*B*, and the coupling parameter,

*K*, are introduced:

*β*is real and

*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

21. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

22. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express **12**, 3353–3366 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353. [CrossRef] [PubMed]

*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.

*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

*NS*and equally spaced eigenvalues

*B*

_{l}= 2

*πl*/

*N*.

*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

*K*

_{0}→ 0. From Eqs. (14), (9), and (10), the eigenvalues of an isolated COR are determined by the condition

*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

**12**, 2303–2316 (2004),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303. [CrossRef] [PubMed]

*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)).

*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

*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]:

*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}.

18. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665–1672 (2004). [CrossRef]

*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

24. M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots”, J. Phys. Condens. Matter **3**, 2651–2664 (1991). [CrossRef]

25. M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**, 381–391 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381. [CrossRef] [PubMed]

*K*

_{0}≪1, Eqs. (A19), (A20), and (A21) can be simplified by substituting

*g*

^{+}=2 and

*g*

^{-}=

*ΔB*,

*ΔK*≪

*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

*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):

*T*

^{(thru)}in the case of an singularity is

*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}.

*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

*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

*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

18. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B **21**, 1665–1672 (2004). [CrossRef]

21. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

*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

**24**, 711–713 (1999). [CrossRef]

*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*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

**24**, 711–713 (1999). [CrossRef]

*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.

2. C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, “An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,” Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002).

14. M. Sumetsky, B. J. Eggleton, and P. S. Westbrook, “Holographic methods for phase mask and fiber grating fabrication and characterization,” Proc. SPIE **4941**, 1 (2003). [CrossRef]

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

*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

*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}:

## Appendix 2. Transmission amplitudes of the PMR

21. J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E **53**, 4107–4121 (1996). [CrossRef]

22. Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express **12**, 3353–3366 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353. [CrossRef] [PubMed]

*n*is defined as

*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

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

^{±}in Eq.(10) is singular and can be expressed as

*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.

*l*

_{1},

*n*

_{1}) and (

*l*

_{2},

*n*

_{2}) are symmetrical with respect to the vertical line and

*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),

*ΔB*,

*ΔK*and less.

*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:

^{+})

^{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:

^{+})

^{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

*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}~

*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

## References and links

1. | C. K. Madsen and G. Lenz, “Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,” IEEE Photon. Technol. Lett. |

2. | C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, “An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,” Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002). |

3. | B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very High-Order Microring Resonator Filters for WDM Applications,” IEEE Photon. Technol. Lett. |

4. | J. Niehusmann, A. VÖrckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. |

5. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

6. | J. E. Heebner, R. W. Boyd, and Q-H. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B |

7. | S. Mookherjeaa, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. |

8. | P. C. Sercel, K. J. Vahala, D. W. Vernooy, G. Hunziker, and R. B. Lee, “Fiber ring optical resonators,” United States Patent Application Publication, US 2002/0041730 A1 (2002). |

9. | M. Sumetsky, “Optical fiber microcoil resonator,” Opt. Express |

10. | M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express |

11. | M. Raburn, B. Liu, K. Rauscher, Y. Okuno, N. Dagli, and J. E. Bowers, “3-D Photonic Circuit Technology,” IEEE J. Sel. Top. Quant. Electron. |

12. | P. Koonath, K. Kishima, T. Indukuri, and B. Jalali, “Sculpting of three-dimensional nano-optical structures in silicon,” Appl. Phys. Lett. |

13. | R. Kashyap, |

14. | M. Sumetsky, B. J. Eggleton, and P. S. Westbrook, “Holographic methods for phase mask and fiber grating fabrication and characterization,” Proc. SPIE |

15. | S. Shoji and S. Kawata, “Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,” Appl. Phys. Lett. |

16. | G. Kakarantzas, T.E. Dimmick, T.A. Birks, R. LeRoux, and P.St.J. Russell “Miniature all-fiber devices based on CO |

17. | M. Sumetsky, “Whispering-gallery-bottle microcavities: the three-dimensional etalon,” Opt. Lett. |

18. | J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, “Designing coupled-resonator optical waveguide delay lines,” J. Opt. Soc. Am. B |

19. | K. J. Vahala, “Optical microcavities,” Nature |

20. | A. W. Snyder and J. D. Love, |

21. | J. M. Bendickson, J. P. Dowling, and M. Scalora, “Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,” Phys. Rev. E |

22. | Y. Chen and S. Blair, “Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,” Opt. Express |

23. | L. D. Landau and E. M. Lifshitz, |

24. | M. Sumetskii, “Modeling of complicated nanometer resonant tunneling devices with quantum dots”, J. Phys. Condens. Matter |

25. | M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express |

26. | F. Seitz, |

**OCIS Codes**

(060.2340) Fiber optics and optical communications : Fiber optics components

(190.0190) Nonlinear optics : Nonlinear optics

(230.5750) Optical devices : Resonators

(230.7370) Optical devices : Waveguides

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 8, 2005

Revised Manuscript: August 4, 2005

Published: August 22, 2005

**Citation**

M. Sumetsky, "Vertically-stacked multi-ring resonator," Opt. Express **13**, 6354-6375 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-17-6354

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### References

- C. K. Madsen and G. Lenz, �??Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,�?? IEEE Photon. Technol. Lett. 10, 994-996 (1998). [CrossRef]
- C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, �??An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,�?? Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002).
- B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, �??Very High-Order Microring Resonator Filters for WDM Applications,�?? IEEE Photon. Technol. Lett. 16, 2263-2265 (2004). [CrossRef]
- J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, �??Ultrahigh-quality-factor silicon-on-insulator microring resonator,�?? Opt. Lett. 29, 2861-2863 (2004). [CrossRef]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711-713 (1999). [CrossRef]
- J. E. Heebner, R. W. Boyd, and Q-H. Park, �??SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,�?? J. Opt. Soc. Am. B 19, 722-731 (2002). [CrossRef]
- S. Mookherjeaa, �??Semiconductor coupled-resonator optical waveguide laser,�?? Appl. Phys. Lett. 84, 3265-3267 (2004). [CrossRef]
- P. C. Sercel, K. J. Vahala, D. W. Vernooy, G. Hunziker, and R. B. Lee, �??Fiber ring optical resonators,�?? United States Patent Application Publication, US 2002/0041730 A1 (2002).
- M. Sumetsky, �??Optical fiber microcoil resonator,�?? Opt. Express 12, 2303-2316 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303</a>. [CrossRef] [PubMed]
- M. Sumetsky, �??Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,�?? Opt. Express 13, 4331-4340 (2005), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331</a>. [CrossRef] [PubMed]
- M. Raburn, B. Liu, K. Rauscher, Y. Okuno, N. Dagli, and J. E. Bowers, �??3-D Photonic Circuit Technology,�?? IEEE J. Sel. Top. Quant. Electron. 8, 935-942 (2001). [CrossRef]
- P. Koonath, K. Kishima, T. Indukuri, and B. Jalali, �??Sculpting of three-dimensional nano-optical structures in silicon,�?? Appl. Phys. Lett. 83, 4909-4911 (2003). [CrossRef]
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- M. Sumetsky, B. J. Eggleton, and P. S. Westbrook, �??Holographic methods for phase mask and fiber grating fabrication and characterization,�?? Proc. SPIE 4941, 1 (2003). [CrossRef]
- S. Shoji and S. Kawata, �??Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,�?? Appl. Phys. Lett. 76, 2668-2670 (2000). [CrossRef]
- G.Kakarantzas, T.E.Dimmick, T.A.Birks, R.LeRoux, and P.St.J.Russell "Miniature all-fiber devices based on CO2 laser microstructuring of tapered fibers," Opt. Lett. 26, 1137-1139 (2001). [CrossRef]
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- J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, �??Designing coupled-resonator optical waveguide delay lines,�?? J. Opt. Soc. Am. B 21, 1665-1672 (2004). [CrossRef]
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- Y. Chen and S. Blair, �??Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,�?? Opt. Express 12, 3353-3366 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353</a> [CrossRef] [PubMed]
- L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.
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- M. Sumetsky and B. J. Eggleton, �??Modeling and optimization of complex photonic resonant cavity circuits,�?? Opt. Express 11, 381-391 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381</a>. [CrossRef] [PubMed]
- F. Seitz, The modern theory of solids, McGraw-Hill Book Company, New York, 1940.

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