## Analysis of light propagation in slotted resonator based systems via coupled-mode theory |

Optics Express, Vol. 19, Issue 9, pp. 8641-8655 (2011)

http://dx.doi.org/10.1364/OE.19.008641

Acrobat PDF (1569 KB)

### Abstract

Optical devices with a slot configuration offer the distinct feature of strong electric field confinement in a low refractive index region and are, therefore, of considerable interest in many applications. In this work we investigate light propagation in a waveguide-resonator system where the resonators consist of slotted ring cavities. Owing to the presence of curved material interfaces and the vastly different length scales associated with the sub-wavelength sized slots and the waveguide-resonator coupling regions on the one hand, and the spatial extent of the ring on the other hand, this prototypical system provides significant challenges to both direct numerical solvers and semi-analytical approaches. We address these difficulties by modeling the slot resonators via a frequency-domain spatial Coupled-Mode Theory (CMT) approach, and compare its results with a Discontinuous Galerkin Time-Domain (DGTD) solver that is equipped with curvilinear finite elements. In particular, the CMT model is built on the underlying physical properties of the slotted resonators, and turns out to be quite efficient for analyzing the device characteristics. We also discuss the advantages and limitations of the CMT approach by comparing the results with the numerically exact solutions obtained by the DGTD solver. Besides providing considerable physical insight, the CMT model thus forms a convenient basis for the efficient analysis of more complex systems with slotted resonators such as entire arrays of waveguide-coupled resonators and systems with strongly nonlinear optical properties.

© 2011 OSA

## 1. Introduction

3. G. W. Pan, *Wavelets in Electromagnetics and Device Modeling*, Microwave and Optical engineering, (Wiley Interscience, 2003). [CrossRef]

7. K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express **17**(17), 14934–14947 (2009). [CrossRef] [PubMed]

*Q*) that allow for the efficient trapping of electromagnetic energy. This unique property may be exploited for a number of applications. For instance, the discrete nature of these resonances lend themselves for the realization of compact filtering and multiplexing devices [8

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

11. A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli, “Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,” Opt. Lett. **28**(17), 1567–1569 (2003). [CrossRef] [PubMed]

12. C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Capuzzo, L. T. Gomez, T. N. Nielsen, and I. Brener, “Multistage dispersion compensator using ring resonators,” Opt. Lett. **24**(22), 1555–1557 (1999). [CrossRef]

13. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett. **31**(12), 1896–1898 (2006). [CrossRef] [PubMed]

14. S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. **2**, 181–190 (2004). [CrossRef]

15. V. R. Almeida, Q. Xu, C. A. Barios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. **29**(11), 1209–1211 (2004). [CrossRef] [PubMed]

16. C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics **2**(3), 216–219 (2009). [CrossRef]

17. T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. **86**, 081101 (2005). [CrossRef]

18. T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express **13**(14), 5216–5226 (2005). [CrossRef] [PubMed]

19. K. R. Hiremath, “Analytical modal analysis of bent slot waveguides,” J. Opt. Soc. Am. A **26**(11), 2321–2326 (2009). [CrossRef]

*Q*-factors of the resonator. From a modeling point of view, coupled waveguide-resonator systems, where the resonators consists of slotted ring cavities thus provide a particularly challenging class of optical systems. For instance, a numerical solver must keep under control not only any spatial discretization errors in the representation of curved surfaces and efficiently resolve the sub-wavelength slot, but must also be able to consistently handle the discontinuities in the fields. We will demonstrate below that a Discontinuous Galerkin Time-Domain (DGTD) method [5

5. J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. **7**(1), 2–11 (2009). [CrossRef]

7. K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express **17**(17), 14934–14947 (2009). [CrossRef] [PubMed]

20. X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin Time Domain (DGTD) methods for the study of 2-D waveguide-coupled microring resonators,” J. Lightwave Technol. **23**(11), 3864–3874 (2005). [CrossRef]

21. K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. **257**, 277–297 (2006). [CrossRef]

19. K. R. Hiremath, “Analytical modal analysis of bent slot waveguides,” J. Opt. Soc. Am. A **26**(11), 2321–2326 (2009). [CrossRef]

## 2. CMT model of slotted resonators

### 2.1. Overall device description

23. M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in F. Michelotti, A. Driessen, and M. Bertolotti, editors, *Microresonators as building blocks for VLSI photonics*, volume 709 of AIP conference proceedings, pages 48–71. American Institute of Physics, Melville, New York (2004).

*N*

_{b}guided modes, and the straight waveguides support

*N*

_{s}guided modes and that these modes are power normalized [24

24. K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. **37**(1–3), 37–61 (2005). [CrossRef]

_{vw}represent the coupling of mode w ∈ {b

*,*s} to mode v ∈ {b

*,*s} and the indices b

*,*s label the different classes of modes, i.e., the index b refers to the modes of the bent slotted waveguide segments and the index s refers to the modes of the straight input- and output-port waveguides.

19. K. R. Hiremath, “Analytical modal analysis of bent slot waveguides,” J. Opt. Soc. Am. A **26**(11), 2321–2326 (2009). [CrossRef]

*N*

_{b}× N

_{b}diagonal matrices with entries 𝖦

*= exp (−iγ*

_{p,p}_{b}

*),*

_{p}L*= exp(−iγ*

_{p,p}_{b}

*̃) for*

_{p}L*p*= 1

*,…, N*

_{b}, and

*L*and

*L*̃ are the lengths of the bent slotted waveguide segments (cf. Fig. 1). Here γ

_{b}

*=*

_{p}*β*

_{b}

*– i*

_{p}*α*

_{b}

*,*

_{p}*p*= 1

*,…, N*

_{b}, denote the complex-valued propagation constants of the bent slotted waveguide modes with corresponding phase and attenuation constants,

*β*

_{b}

*and*

_{p}*α*

_{b}

*, respectively.*

_{p}*A*and

_{q}*Ã*,

_{q}*q*= 1

*,…, N*

_{s}, we can regard Eqs. (1), (2), and (3) as a system of linear equations for the amplitudes of the output waveguide,

*B*and

_{q}*B̃*,

_{q}*q*= 1

*,…, N*

_{s}. In particular, if only the In-port waveguide feeds the system, i.e.,

*Ã*≡ 0,

_{q}*q*= 1

*,…, N*

_{s}, we obtain for the amplitudes in the Through- and Drop-port waveguides where Ω = 𝖨 – 𝖲

_{bb}𝖦

_{bb}

_{b}

*,*

_{p}*p*= 1

*,…, N*

_{b}are available as a function of the wavelength λ. By using semi-analytic arguments based on modal descriptions of the bent slotted and straight waveguides, we can determine these parameters from first principles as described in the following subsections.

*xz*-plane as indicated in Fig. 1 without any variation of the material and structural parameters along the

*y*-axis. We comment on the extension to the fully three-dimensional setting in Sec. 4.

### 2.2. Bent slotted waveguide modes

24. K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. **37**(1–3), 37–61 (2005). [CrossRef]

*y*-component of the magnetic field (

*H*) and the in-plane components of the electric field (

_{y}*E*,

_{x}*E*) — in the Cartesian co-ordinate system — or (

_{z}*E*,

_{r}*E*) — in the polar co-ordinate system — have to be considered. Clearly, as the normal component of the dielectric displacement field is continuous across a material interface, we have for a bent slotted waveguide that the radial component of the electric field (

_{θ}*E*) exhibits concurrent jumps from low to high values as the radial position is scanned across jumps from high-index to a low-index materials and vice versa (see Fig. 4).

_{r}**26**(11), 2321–2326 (2009). [CrossRef]

_{b}

*and the associated field distribution {*

_{p}*E*

_{b}

*,*

_{p}*H*

_{b}

*} of the bent slotted waveguide modes for a given vacuum wavelength*

_{p}*λ*. Owing to the lossy nature of these modes, their propagation constants are complex-valued and so are the associated effective indices which are defined as

*n*

_{eff, b}

*= γ*

_{p}_{b}

*/*

_{p}*k*, where

*k*represents the wave number

*k*= 2

*π*/

*λ*(see Ref. [19

**26**(11), 2321–2326 (2009). [CrossRef]

### 2.3. Bent slotted - straight waveguide couplers

21. K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. **257**, 277–297 (2006). [CrossRef]

*λ*, the spatial coupled-mode approach constructs the fields in the coupler region by using the modal solutions of the uncoupled constituent waveguides. Consequently, for the straight waveguides we use the well-known modal solutions [26] and for the bent slotted waveguide we employ the analytical model as described in Sec. 2.2 above together with an appropriate transformation from polar (

*r, θ*) to Cartesian (

*x,z*) co-ordinates that are better suited for representing the coupler geometry. Here

*x*=

*r*cos

*θ*, and

*z*=

*r*sin

*θ*. Explicitly, we use the modal solutions for the fields {

*E*

_{s}

*,*

_{q}*H*

_{s}

*} and associated propagation constant*

_{q}*β*

_{s}

*for straight waveguides with refractive index profiles*

_{q}*n*

_{s}(

*x*) (or, equivalently, relative permittivity profiles

*E*

_{b}

*,*

_{p}*H*

_{b}

*} and propagation constants γ*

_{p}_{b}

*of bent slotted waveguides with refractive index profiles*

_{p}*n*

_{b}(

*x, z*) (or, equivalently, relative permittivity profiles

*R*is the bend radius as defined in Fig. 1, and

*θ*= tan

^{−1}(

*z/x*) is the polar angle corresponding to the radial position

24. K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. **37**(1–3), 37–61 (2005). [CrossRef]

*z*-direction.

*x*,

_{l}*x*]

_{r}*×*[

*z*,

_{i}*z*] (see Fig. 2). Outside this region the fields propagating in each waveguide are assumed to be uncoupled so that the individual modes propagate undisturbed according to their respective harmonic dependence. Within the coupler region, the corresponding fields {

_{o}*E,H*} are approximated by a superposition of the uncoupled modal fields, {

*E*

_{b}

*,*

_{p}*H*

_{b}

*} and {*

_{p}*E*

_{s}

*,*

_{q}*H*

_{s}

*} of the forward propagation modes of the uncoupled bent slotted and straight waveguides, respectively. Explicitly, this reads as with the*

_{q}*a priori*unknown,

*z*-dependent amplitudes

*C*. Here, the index

_{vi}*v*∈ {b

*,*s} identifies the type of waveguide (b: bent slotted waveguide, s: straight waveguide) and the index

*i*runs over the corresponding number of guided modes, i.e.,

*i*= 1,…,

*N*and

_{v}*v*∈ {b,s}.

*C*(

*z*) = {

*C*(

_{vi}*z*)} by combining the above ansatz (7) with the variational formulation of Maxwell’s equations In the above equation, the stationarity of the functional ℱ(

*E,H*) implies that

*E*and

*H*satisfy Maxwell’s curl equations ∇ ×

*E*=

*–*iωμ

_{0}

*H*and ∇ ×

*H*= iωε

_{0}ε

*E*. Following the usual variational arguments [21

21. K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. **257**, 277–297 (2006). [CrossRef]

*j*= 1

*,…,N*, and

_{w}*w*∈ {b, s}. In the above equation, we have introduced the abbreviations

*e*denotes the unit vector in

_{z}*z*-direction, the superscript ‘*’ stands for complex conjugation, and ε(

*x,z*) represent the full relative permittivity profile for the entire coupler region. For a given value of

*z*, the integration is carried out over the pre-defined coupler region in

*x*-direction [

*x*,

_{l}*x*].

_{r}*z*) at

_{j}*z*=

*z*which connects the unknown amplitudes

_{j}*C*at the position

*z*=

*z*with the amplitudes at position

_{i}*z*=

*z*according to

_{j}*C*(

*z*

_{j}) = 𝖳(

*z*)

_{j}*C*(

*z*

_{i}). By exploiting the linearity of Eq. (10) with respect to

*C*, we may thus reformulate the problem in terms of the transfer matrix 𝖳(

*z*) as a set of coupled differential equations with initial condition 𝖳 (

*z*

_{i}) = 𝖨. Here, 𝖨 denotes the identity matrix. Upon solving Eq. (11) on an

*a priori*defined domain [

*x*,

_{l}*x*]

_{r}*×*[

*z*,

_{i}*z*] via standard integrators (we typically use a fourth-order Runge-Kutta method), we determine the transfer matrix 𝖳(

_{o}*z*), which gives the coupler’s output amplitudes

_{o}*C*(

*z*) at position

_{o}*z*=

*z*in terms of the coupler’s input amplitudes

_{o}*C*(

*z*

_{i}) at position

*z*=

*z*

_{i}according to

*C*(

*z*

_{o}) = 𝖳(

*z*)

_{o}*C*(

*z*

_{i}).

#### Projection corrections

*a priori*defined coupler domain [

*x*,

_{l}*x*]

_{r}*×*[

*z*,

_{i}*z*]. For a meaningful simulation of these coupler regions, it is essential to make sure that the coupler domain is appropriately selected. By examining the spatial extent of the mode profiles of the constituent waveguide modes in the direction normal to the waveguides, we can straightforwardly determine an appropriate length of the coupler domain in

_{o}*x*-direction [

*x*,

_{l}*x*] (see the mode profiles depicted in Fig. 4).

_{r}*z*-direction is less straightforward. A choice of [

*z*,

_{i}*z*] will be justified, if the corresponding simulation results validate the assumption that the modes of the constituent waveguides are uncoupled outside the coupler domain. In order to verify this numerically, we consider a system where only the fundamental mode of the straight waveguide is excited, i.e.,

_{o}*C*(

*z*

_{i})

*=*0 except for

*C*

_{s0}(

*z*

_{i}) = 1. Then by matching the fields at the output position

*z*=

*z*, we determine the amplitudes of the straight waveguide modes

_{o}*B*=

*C*(

*z*

_{o}). Keeping

*z*=

*z*fixed and varying the other end

_{i}*z*=

*z*, allows us to monitor the evolution of the straight waveguide’s fundamental mode amplitude

_{o}*B*

_{s0}. If the assumption of uncoupled modes is satisfied, we expect that this amplitude’s absolute value |

*B*

_{s0}| = |

*C*

_{s0}(

*z*

_{o})| attains a stationary value.

*B*

_{s0}| even for relatively large values of

*z*(dashed lines). Although, these oscillations will eventually be damped out for large values

_{o}*z*, this would lead to undesirably large coupler domains. Therefore, the ‘naive’ field-matching approach described above is rather unsatisfactory, and we should not directly use the transfer matrix 𝖳 (

_{o}*z*) as the required coupler scattering matrix 𝖲 for the CMT model described in Sec. 2.1.

_{o}*B*

_{s0}and all other amplitudes. In Fig. 3, we also display the results of this projection technique (solid line) and observe a superior performance relative to the naive field matching technique (dashed line). For a detailed explanation of this projection technique and reasoning behind its effectiveness we would like to refer the reader to Ref. [21

**257**, 277–297 (2006). [CrossRef]

*z*).

_{o}### 2.4. Evaluation of the spectral response

**257**, 277–297 (2006). [CrossRef]

## 3. Simulation results

*R*= 5

*μ*m and width

*w*

_{tot}= 1

*μ*m which is made of silicon nitride (refractive index

*n*

_{c}= 2.1) is micro-structured to exhibit a slot of width

*w*

_{slot}= 0.2

*μ*m that is filled with air (refractive index

*n*

_{slot}= 1). As depicted in Fig. 1, the position of the slot inside the ring resonator is controlled by the asymmetry parameter

*η*. This slotted ring resonator is coupled to two identical straight silicon-nitride waveguides (refractive index

*n*

_{s}= 2.1) that are

*w*

_{s}= 0.4

*μ*m wide. The minimal separation between these bus waveguides and the ring resonator is

*g*=

*g*̃ = 0.4

*μ*m. Finally, we assume that the entire device is placed in an air background (refractive index

*n*

_{b}= 1) and study its spectral response in the wavelength range [1.5

*μ*m, 1.6

*μ*m]. For these settings, the uncoupled straight waveguide is mono-modal, whereas for the uncoupled bent slotted waveguide we have to take into account all the modes which exhibit a real part of their effective index that is above ‘cutoff’, i.e., for which ℜ(

*n*

_{eff}) ≥

*n*

_{b}. We have found that for the above settings this corresponds to the fundamental and the first-order mode.

*η*. The corresponding properties of the bent slotted waveguide modes strongly depend on the slot’s position. In particular, the first-order TM

_{1}mode is moderately lossy only if the slot is symmetrically positioned within the ring (

*η*= 0.5). For instance, at

*λ*= 1.55

*μ*m, we obtain its effective index

*n*

_{eff}= 1.19039 – i 1.69572 × 10

^{−3}. For the first-order mode, asymmetric positions of the slot either lead to a very lossy mode or a mode located close to ‘cutoff’. For instance, at

*λ*= 1.55

*μ*m for TM

_{1}we obtain for

*η*= 0.4 an effective index

*n*

_{eff}= 1.09129 – i 9.96678 × 10

^{−3}and for

*η*= 0.7 we find

*n*

_{eff}= 1.13048 – i 1.49073 × 10

^{−2}.

*x*,

_{l}*x*] = [1

_{r}*μ*m, 8

*μ*m] and [

*z*,

_{i}*z*] = [−3

_{o}*μ*m, 3

*μ*m] by using discretizations of

*h*= 0.005

_{x}*μ*m and

*h*= 0.1

_{z}*μ*m along the

*x*- and

*z*-direction, respectively. Following the procedure outlined in Sec. 2.4, we calculate the spectral response of the slotted resonator over the entire wavelength range [1.5

*μ*m, 1.6

*μ*m] via a cubic interpolation technique that utilizes computations at the nodal wavelengths 1.5

*μ*m, 1.55

*μ*m, and 1.6

*μ*m.

*η*. In particular, each dip in the through-power

*P*

_{T}corresponds to a particular TM

*cavity mode resonance of the slotted resonator, where the indices*

_{n,m}*n*and

*m*denote radial and angular mode number, respectively. The prominent sharp dips correspond to the resonances of the fundamental cavity modes TM

_{0,}

*and they exhibit wildly varying quality factors (see discussion below). In addition, for the symmetric case*

_{m}*η*= 0.5, we also find further secondary dips, which, in general, correspond to higher-order cavity modes — in the particular case we are considering — they correspond to the first-order TM

_{1,}

*modes. Consistent with our discussion above, we find that in the case of*

_{m}*η*= 0.4 and

*η*= 0.7, the TM

_{1}bent slotted waveguide modes are very lossy and, therefore, fail to appreciably contribute to the overall spectral response of the device.

7. K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,, Opt. Express **17**(17), 14934–14947 (2009). [CrossRef] [PubMed]

6. J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonator using DGTD and FDTD,” IOP J. Opt. A: Pure Appl. Opt. **11**, 114015:1–10 (2009). [CrossRef]

*p*= 6 (see Ref. [5

5. J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostr.: Fundam. Appl. **7**(1), 2–11 (2009). [CrossRef]

27. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. **181**(3), 687–702 (2010). [CrossRef]

*η*= 0.5, the CMT and DGTD results differ significantly for the lossy first-order resonances.

*a priori*unknown coupled mode amplitudes

*C*in Eq. (7) are assumed to be

_{vi}*z*-dependent only. These assumptions are violated if the modes of the bent slotted waveguides are not anymore well confined to the waveguide core (e.g., the TM

_{1}bent slotted waveguide mode displayed in Fig. 4) and the constituent waveguides in the coupler are not sufficiently separated. Therefore, we expect that increasing the minimal coupler separations,

*g*and

*g*̃, leads to an improvement of the CMT model’s accuracy, notably regarding the secondary resonances associated with the first-order modes. In order to validate this reasoning, we have computed the spectral response of the symmetric slotted resonator device (

*η*= 0.5) for larger values of the minimal separations

*g*and

*g*̃, and display the results in Fig. 7. Indeed, we find that the discrepancy between the results of the CMT model and the DGTD computations are strongly reduced for larger values of the separation. These results provide a very reliable method for determining the validity and/or accuracy of the adiabatic coupling assumption of the coupled-mode approach.

_{0,30}resonance as the asymmetry parameter

*η*is varied and display the results in Fig. 8. We observer that, for

*η*= 0.4, the principal field component

*H*is neatly localized in the outer high-index layer of the ring cavity. When the slot is placed symmetrically within the core of the ring, i.e., when

_{y}*η*= 0.5, we still observe a significant localization of the magnetic field

*H*in the outer high-index ring. However, we cannot anymore neglect the magnetic field in the inner ring. For even higher values of, say,

_{y}*η*= 0.7, the magnetic field is predominantly localized in the inner high-index layer of the ring. This strong confinement of the field is the primary reason for the sharp resonances which we have observed for

*η*= 0.7 as compared to the resonance for

*η*= 0.4, 0.5.

*Q*, which is defined as a ratio of the central resonance wavelength to the full-width-at-half-maximum value. The corresponding

*Q*-values of this TM

_{0,30}resonance for

*η*= 0.4, 0.5, and 0.7 are approximately 900, 1200, and 26000 respectively. This demonstrates that a careful positioning of the slot is of particular importance when cavities with high quality factors are desired. Finally, in the far-right panel of Fig. 8 we depict the field distribution that corresponds to the (weak) secondary resonance of the slotted resonator’s TM

_{1,25}mode for

*η*= 0.5. Due to the lossy nature of this mode, the field is only weakly confined and a considerable amount of the field resides outside the actual cavity.

## 4. Conclusions

28. R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. **256**, 46–67 (2005). [CrossRef]

## Acknowledgment

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10. | H. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, “High-Q channel-dropping filters using ring resonators with integrated SOAs,” IEEE Photon. Technol. Lett. |

11. | A. Melloni, R. Costa, P. Monguzzi, and M. Martinelli, “Ring-resonator filters in silicon oxynitride technology for dense wavelength-division multiplexing systems,” Opt. Lett. |

12. | C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Capuzzo, L. T. Gomez, T. N. Nielsen, and I. Brener, “Multistage dispersion compensator using ring resonators,” Opt. Lett. |

13. | A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett. |

14. | S. Pereira, P. Chak, S. E. Sipe, L. Tkeshelashvili, and K. Busch, “All-optical diode in an asymmetrically apodized kerr nonlinear microresonator system,” Photon. Nanostr.: Fundam. Appl. |

15. | V. R. Almeida, Q. Xu, C. A. Barios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. |

16. | C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics |

17. | T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, “High-Q optical resonators in silicon-on-insulator-based slot waveguides,” Appl. Phys. Lett. |

18. | T. Baehr-Jones, M. Hochberg, G. Wang, R. Lawson, Y. Liao, P. Sullivan, L. Dalton, A. Jen, and A. Scherer, “Optical modulation and detection in slotted silicon waveguides,” Opt. Express |

19. | K. R. Hiremath, “Analytical modal analysis of bent slot waveguides,” J. Opt. Soc. Am. A |

20. | X. Ji, T. Lu, W. Cai, and P. Zhang, “Discontinuous Galerkin Time Domain (DGTD) methods for the study of 2-D waveguide-coupled microring resonators,” J. Lightwave Technol. |

21. | K. R. Hiremath, R. Stoffer, and M. Hammer, “Modeling of circular integrated optical microresonators by 2-D frequency domain coupled mode theory,” Opt. Commun. |

22. | K. R. Hiremath, Coupled mode theory based modeling and analysis of circular optical microresonators, PhD thesis, University of Twente, The Netherlands (2005). |

23. | M. Hammer, K. R. Hiremath, and R. Stoffer, “Analytical approaches to the description of optical microresonator devices,” in F. Michelotti, A. Driessen, and M. Bertolotti, editors, |

24. | K. R. Hiremath, M. Hammer, S. Stoffer, L. Prkna, and J. Čtyroký, “Analytic approach to dielectric optical bent slab waveguides,” Opt. Quantum Electron. |

25. | T. Tamir, editor, |

26. | A. Ghatak and K. Thyagarajan, |

27. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. |

28. | R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(130.3120) Integrated optics : Integrated optics devices

(130.6010) Integrated optics : Sensors

(230.5750) Optical devices : Resonators

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 13, 2011

Manuscript Accepted: March 11, 2011

Published: April 19, 2011

**Citation**

Kirankumar R. Hiremath, Jens Niegemann, and Kurt Busch, "Analysis of light propagation in slotted resonator based systems via coupled-mode theory," Opt. Express **19**, 8641-8655 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8641

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### References

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- C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photonics 2(3), 216–219 (2009). [CrossRef]
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- R. Stoffer, K. R. Hiremath, M. Hammer, L. Prkna, and J. Čtyroký, “Cylindrical integrated optical microresonators: Modeling by 3-D vectorial coupled mode theory,” Opt. Commun. 256, 46–67 (2005). [CrossRef]

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