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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 17 — Aug. 22, 2005
  • pp: 6354–6375
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Vertically-stacked multi-ring resonator

M. Sumetsky  »View Author Affiliations


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

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

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]

]. The microring-based photonic circuits are usually fabricated lithographically on a planar substrate and therefore have a 2D geometry [2–4

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).

]. 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

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).

]. 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

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

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]

], 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

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).

]. 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).

Fig. 1. Illustration of (a) VMR and (b) PMR

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

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]

]. The most investigated 2D realization of a CROW consists of coupled ring resonators arranged on a plane [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]

,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]

], 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

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]

]. 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

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

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]

]. 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

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

]), 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 Fn (s) = An (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:

dA1ds=κA2
dAnds=κ(An1+An+1),n=2,3N1,
dANds=κAN1
(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

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

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]

]. The condition of closure of each ring has the form

An(0)=An(S)exp(iβS),n=2,3,,N1.
(2)

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

A0(out)=cos(K0)A0(in)+isin(K0)exp(iβS)A1(S)
A1(0)=isin(K0)A0(in)+cos(K0)exp(iβS)A1(S)
(3)
AN+1(out)=isin(K0)exp(iβS)AN(S)
AN(0)=cos(K0)exp(iβS)A1(S)
(4)

where A0(in) and A0(out) are the ingoing and outgoing amplitudes in the first waveguide, AN+1(out) 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

20. A. W. Snyder and J. D. Love, Optical waveguide theory, Chapman & Hall, London, 1983.

]. The coupling parameter K 0 is defined through the local waveguide-ring coupling coefficient κ 0(s)as

K0=Sc(0)κ0(s)ds,
(5)

where Sc(0) 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

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]

) 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 AN+1(in) = 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, Sc(0)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(thru)=A0(out)/A0(in)
T(drop)=AN+1(out)/A0(in).
(6)

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,

T(thru)=ξ+ξ+iμ+,T(drop)=ξ+ξ+iμ+,
(7)

and the electromagnetic field distribution along the mth ring is expressed by the following superposition of spatial harmonics:

Fm(s)=A0(in)sin(K0)n=1NQnsinπmnN+1exp[isS(B+2KcosπnN+1)].
(8)

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

ξ±=(N+1)2(g+)2±16(g)2[(σ+)2(σ)2),μ±=8(N+1)g+gσ±,
g±=1±cos(K0),
(9)
σ±=n=1N(±1)ncot(12B+Kcn)sn2,Qn=2[cot(12B+Kcn)+i](N+1)g++4ig(σ++(1)nσ),
(10)
cn=cos(πnN+1),sn=sin(πnN+1).
(11)

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

B=βS,K=κS.
(12)

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

T(thru)2+T(drop)2=1.
(13)

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:

P=1NSm=10N0SdsFm(s)2=N+12N(A0(in)sin(K0))2n=1NQn2
(14)

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

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

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]

]. 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 Sc . 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 Sc , 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 Bl = 2πl with integer l. Conversely, at k=π(n+12) the rings are fully coupled, forming a closed loop with length NS and equally spaced eigenvalues Bl = 2πl /N.

Fig. 2. Surface plots of the PMR through transmission power, |T(thru) |2, in the (B,K) plane for N = 2,3,5, and 10; (a) - transmission power plots for the ring-waveguide coupling parameter K 0 having sin(K 0)=0.2; (b) - transmission power plots for the ring-waveguide coupling parameter K 0 with sin(K 0)=0.8.

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 tan(12B+Kcn)=0, which can be reduced to the equation

Bln+2Kcn=2πl,n=1,2,,N,
(15)

where l is an integer. Eq. (15) determines straight lines Bln (K) = 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 2π(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 5π/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

Fig. 3. Eigenvalues of the isolated VMR consisting of 5 rings, N=5, in the (B,K) plane. Bold lines correspond to n 1=2 and n 2=5. These lines are tilted with respect to the B-axis by the angles α 1=π/4 and α 2=5π/6, respectively. Line crossing correspond to the eigenvalues which are degenerated in the considered approximation.
Bl1n1,l2n2=2πl2cn1l1cn2cn1cn2,Kl1n1,l2n2=πl1l2cn1cn2,
(16)
Fig. 4. Surface plots of the VMR through transmission power, |T (thru)|2, in the (B,K) plane for N = 2,3,4, and 5; (a) - transmission power plots for the ring-waveguide coupling parameter K 0 having sin(K 0)=0.2; (b) - transmission power plots for the ring-waveguide coupling parameter K 0 with sin(K 0)=0.8.

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

23. L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.

]. The parameters of the pseudo-crossing of eigenvalue trajectories are determined by small terms neglected in the derivation of the coupled wave equations [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]

].

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

Fm(ln)(s)=CsinπmnN+1exp(ilsS),
(17)

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, K0 << 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

23. L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.

]:

T(thru)ΔBΔB+i2Γ,T(drop)(1)n+1i2ΓΔB+i2Γ,P=2A0(in)2K02N(N+1)[ΔB2+14Γ2],
ΔB=BBln(K),Γ=4K02N+1sn2.
(18)

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.

6. Eigenmodes and eigenvalues of a VMR strongly coupled to the waveguides, K0 ~ 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 En -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 oe-13-17-6354-i001.jpg, oe-13-17-6354-i002.jpg, or oe-13-17-6354-i003.jpg-shaped singularities in the vicinity of an eigenvalue, or no singularity (no eigenmode) at all. An oe-13-17-6354-i004.jpg 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 oe-13-17-6354-i005.jpg singularity, the through transmission has a simple Breit-Wigner representation similar to Eq.(18):

Fig. 5. Enlarged surface plots of behavior of transmission power, |T (thru)|2, and average electromagnetic filed intensity, P, near resonances. (a), (a1), and (a2) - N = 4; (b), (b1), and (b2) - N = 5. (a1) and (b1) show the transmission power, while (a2) and (b2) show the average field intensity. The eigenvalues of VMR correspond to the infinite values of P.
T(thru)=i2ΓΔB+i2Γ(ifl1l2odd).
(19)

The expression for T (thru) in the case of an oe-13-17-6354-i006.jpg singularity is

T(thru)=ΔBΔB+i2Γ(ifl1l2isevenandn1+N+12isodd).
(20)

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

Γ=(N+1)g+4g(cn1sn1)2(ΔK)2.
(21)

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 oe-13-17-6354-i007.jpg singularity is localized near two lines crossing each other at the eigenvalue point and determined by the equation

ΔB=θ±ΔK,θ±=±2cn11+2sn12.
(22)

Near these lines we have

T(thru)=i2ΓΔB+θ±ΔK+i2Γ(ifl1l2isevenandn1+N+12iseven),
(23)

where the resonance width is defined as

Γ=(N+1)g+2gcn1sn12(ΔK)2(1+2sn12)2.
(24)

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, oe-13-17-6354-i008.jpg, oe-13-17-6354-i009.jpg, and oe-13-17-6354-i010.jpg, exist. The oe-13-17-6354-i011.jpg 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 oe-13-17-6354-i012.jpg or oe-13-17-6354-i013.jpg 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 oe-13-17-6354-i014.jpg, oe-13-17-6354-i015.jpg, and oe-13-17-6354-i016.jpg singularities in Fig. 6 reproduce the corresponding singular behavior near eigenvalues in Fig. 5(b1).

Fig. 6. Characteristic singular behavior near resonances in the (B,K) plane plotted using (a) Eq. (19) for oe-13-17-6354-i017.jpg singularity, (b) Eq.(20) for oe-13-17-6354-i018.jpg singularity, and (c) Eq.(23) for oe-13-17-6354-i019.jpg singularity (arbitrary units).

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, An (s) = exp(2iκScos(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

B=2πl2Kcos(ξd),
(25)

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

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

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]

]

B=πlarcsin[sin(K)cos(ξd0)],
(26)

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

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]

] (arcsin(sin(K)cos(ξd 0))≈Kcos(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=π(l+12). 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)(2n +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.

Fig. 7. Dispersion relation for (a) VMR and (b) PMR for different interring coupling parameter, K.

8. Discussion and summary

While for weak coupling, all types of CROWs possess similar transmission spectra [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]

], 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

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).

], 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

13. R. Kashyap, Fiber Bragg Gratings, Academic Press, 1999.

,14

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]

], 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 An (s) = Aexp(iλs), which reduces this equation to the inversion of a three-diagonal matrix (see e.g. [26

26. F. Seitz, The modern theory of solids, McGraw-Hill Book Company, New York, 1940.

]). As the result, the general solution of Eq. (1) can be written in the form

As(s)=m=1Namexp(2iκscosπmN+1)sinπmnN+1,
(A1)

where am is an arbitrary coefficient. From Eqs. (2) and (A1), we have

m=1Nam(Em1)sinπmnN+1=0,n=2,3,N1,
(A2)

where Em 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:

x1=m=1Nam(Em1)sinπmN+1
x2=m=1Nam(Em1)sinπNmN+1.
(A3)

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

am=2(N+1)(Em1)(x1sinπmN+1+x2sinπNmN+1).
(A4)

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

am=iA0(in)sin(K0)(Em1)[14(N+1)cos(K0)+i2(1cos(K0))(σ++(1)mσ]sinπmN+1,
(A5)

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

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

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]

], 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 An(±) 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

(An(+)An())=M(An+1(+)An+1()),M=isin(K)(exp(iB)cos(K)cos(K)exp(iB)).
(A6)

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

(A0(in)A0(out))=M0(A1(+)A1()),(AN(+)AN())=M0(AN+1(out)0),M0=isin(K0)(exp(iB)cos(K0)cos(K0)exp(iB)).
(A7)

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

(A0(in)A0(out))=P(AN+1(out)0),P=M0MN1M0,
(A8)

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

T(thru)=P21P11,T(drop)=1P11.
(A9)

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

σ±ΔB,ΔK0νn1n2±+(±1)n12sn12ΔB+2ΔKcn1+(±1)n22ssn22ΔB+2ΔKcn2,
ΔB=BBl1n1l2n2,ΔK=KKl1n1l2n2,
(A10)

where

νn1n2±=n=1Nnn1,n2(±1)ncot(12Bl1n1l2n2+Kl1n1l2n2cos(πnN+1))sn2.
(A11)

From Eq. (A10), we have

(σ+)2(σ)24[1(1)n1+n2]sn12sn22(ΔB+2ΔKcn1)(ΔB+2ΔKcn2)
+4[νn1n2+(1)n1]sn12(ΔB+2ΔKcn1)+4[νn1n2+(1)n2]sn22(ΔB+2ΔKcn2).
(A12)

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

n2=N+1n1.
(A13)

Then, Eq. (16) is simplified to

Bl1n1,l2n2=π(l1l2),Kl1n1,l2n2=πl1l22cn1.
(A14)

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),

σ±(±1)n1cot(πl1+12ΔB+ΔKcn1)sn12
+(±1)n1cot(πl2+12ΔB-ΔKcn1)sn12,
+(±1)N+12cot(π2(l1+l2)+12ΔB)
(A15)

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:

σ±(±1)n1ΔBsn1214(ΔB)2(ΔK)2cn12.
(A16)

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:

σ±(±1)N+1214(ΔB)2(1+2(±1)n1+N+12sn12)(ΔK)2cn1212ΔB[14(ΔB)2(ΔK)2cn12].
(A17)

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

(σ+)2(σ)28sn12[12(ΔB)2(ΔK)2cn12].
(A18)

Appendix 4. Expressions for the transmission amplitudes

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

oe-13-17-6354-i020.jpg-type singularity:

T(thru)=(N+1)g+ΔΛ1(2)(N+1)g+ΔΛ1(2)+8igΔΛ(1)(ifl1l2isodd),
(A19)

oe-13-17-6354-i021.jpg-type singularity:

T(thru)=8ig-ΔΛ(1)(N+1)g+ΔΛ2(2)+8igΔΛ(1)(ifl1l2isevenandn1+N+12isodd),
(A20)

oe-13-17-6354-i022.jpg-type singularity:

T(thru)=(N+1)g+ΔBΔΛ1(2)(N+1)g+ΔBΔΛ1(2)+16igΔΛ2(2)(ifl1l2isevenandn1+N+12iseven),
(A21)

where:

ΔΛ(1)=ΔBsn12,
ΔΛ1(2)=14(ΔB)2(ΔB)2cn12,
ΔΛ2(2)=14(ΔB)2(1+2sn12)(ΔK)2cn12,
(A22)

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 ~ K02 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.

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. 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).

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. 16, 2263–2265 (2004). [CrossRef]

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]

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]

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 19, 722–731 (2002). [CrossRef]

7.

S. Mookherjeaa, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. 84, 3265–3267 (2004). [CrossRef]

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 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]

13.

R. Kashyap, Fiber Bragg Gratings, Academic Press, 1999.

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 CO2 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]

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]

19.

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

20.

A. W. Snyder and J. D. Love, Optical waveguide theory, Chapman & Hall, London, 1983.

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]

23.

L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.

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]

26.

F. Seitz, The modern theory of solids, McGraw-Hill Book Company, New York, 1940.

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

  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).
  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. 16, 2263-2265 (2004). [CrossRef]
  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]
  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]
  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 19, 722-731 (2002). [CrossRef]
  7. S. Mookherjeaa, �??Semiconductor coupled-resonator optical waveguide laser,�?? Appl. Phys. Lett. 84, 3265-3267 (2004). [CrossRef]
  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 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]
  10. 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]
  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]
  13. R.Kashyap, Fiber Bragg Gratings, Academic Press, 1999.
  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 CO2 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]
  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]
  19. K. J. Vahala, �??Optical microcavities,�?? Nature 424, 839-846 (2003). [CrossRef] [PubMed]
  20. A. W. Snyder and J. D. Love, Optical waveguide theory, Chapman & Hall, London, 1983.
  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), <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]
  23. L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.
  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), <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]
  26. F. Seitz, The modern theory of solids, McGraw-Hill Book Company, New York, 1940.

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