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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 23 — Nov. 9, 2009
  • pp: 21050–21059
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The longitudinal offset technique for apodization of coupled resonator optical waveguide devices: concept and fabrication tolerance analysis

José David Doménech, Pascual Muñoz, and José Capmany  »View Author Affiliations


Optics Express, Vol. 17, Issue 23, pp. 21050-21059 (2009)
http://dx.doi.org/10.1364/OE.17.021050


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Abstract

In this paper, a novel technique to set the coupling constant between cells of a coupled resonator optical waveguide (CROW) device, in order to tailor the filter response, is presented. The technique is demonstrated by simulation assuming a racetrack ring resonator geometry. It consists on changing the effective length of the coupling section by applying a longitudinal offset between the resonators. On the contrary, the conventional techniques are based in the transversal change of the distance between the ring resonators, in steps that are commonly below the current fabrication resolution step (nm scale), leading to strong restrictions in the designs. The proposed longitudinal offset technique allows a more precise control of the coupling and presents an increased robustness against the fabrication limitations, since the needed resolution step is two orders of magnitude higher. Both techniques are compared in terms of the transmission response of CROW devices, under finite fabrication resolution steps.

© 2009 Optical Society of America

1. Introduction

Coupled Resonator Optical Waveguide (CROW) devices [1

1. 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]

] are very relevant structures for the processing of light [3

3. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in gaas micro-ring resonators,” Opt. Lett. 25, 554–556 ( 2000). [CrossRef]

5

5. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon. 1, 65–71 ( 2006). [CrossRef]

], due to their resonant and nonlinear enhancement nature. Both linear and nonlinear operation regimes can be employed to implement a wide range of successful signal processing tasks. CROW based subsystems are envisaged to be critical in several photonics applications and industries. The filtering response of CROW devices is strongly dependent on the coupling constant K for all the couplers involved [6

6. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 ( 2000). [CrossRef]

]. It is known that using the same K value for all the couplers produces filtering responses with significant side-lobes for the side-coupled integrated spaced sequence of resonators (SCISSOR) [2

2. J. E. Heebner, R. W. Boyd, and Q-Han. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 4, 722–731 ( 2002). [CrossRef]

] or significant ripples in the pass-band for the direct coupled microrings (CROW) [1

1. 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]

]. It is also known that the side-lobes/ripples can be reduced, and the pass/reject bands can be made wider, by apodizing the K value of each individual coupler in the structure, starting from a nominal K value (either increasing or decreasing it) [7

7. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 ( 2007). [CrossRef] [PubMed]

].

A technique exists in which the adjustment of the K value is done by changing the distance between the waveguides on the coupler [8

8. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 ( 2007). [CrossRef] [PubMed]

], d in Fig. 1. This is technologically challenging, since the resolution step required in the ’y’ direction [Fig. 1(a)] can be well below the fabrication step resolution which is currently on the order of tenths of nanometers.

2. Coupling constant control through the offset technique

The power exchange between waveguides in a parallel waveguides coupler can be obtained from the power coupling constant [10

10. K. Ebeling, Integrated Optoelectronics (Springer-Verlag, 1993). [CrossRef]

]:

K=sin2(π2LeffLb)
(1)

where Leff is the effective coupler length and Lb=π/(2κ 12) the beat length of the coupler, defined as the physical length where the power is completely transferred from one waveguide to another, and κ 12 the coupling between the waveguides [10

10. K. Ebeling, Integrated Optoelectronics (Springer-Verlag, 1993). [CrossRef]

]. The combination of the coupler length and the distance between the waveguides provides different coupling constants and sets the range of variation of the coupling constant when changing the offset or the gap. The racetrack configuration with long coupling sections allows the employ of gaps big enough to avoid mode conversion losses [11

11. F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express 14,3872–3886 ( 2006). [CrossRef] [PubMed]

].

Fig. 1. Parallel waveguide coupler schematics for (a) gap and (b) offset control of the coupling. Lc is the physical straight section of the coupler, L 0 e is the coupling effective length without offset and Loffe the coupling effective length when a longitudinal offset is applied.

The proposed longitudinal offset technique allows for both decreasing and increasing the value of K, starting from a nominal value. This is shown in Fig. 2(a) and Fig. 2(b) respectively. In both figures, L 0 e is the coupling effective length without offset, as in Fig. 1(a), while Loffe is the coupling effective length with offset, as in Fig. 1(b). In Fig. 2(a) the value of L 0 e is such that the starting value of K is placed in a positive slope region of the coupler transfer function vs. effective length. Therefore, a reduction in the effective length, down to Loffe, will imply an decrease in the K value. In Fig. 2(b) the value of L 0 e is such that the starting value of K is placed in a negative slope region of the coupler transfer function vs. effective length. Therefore, a reduction in the effective length, down to Loffe, will imply an increase in the K value. As the trend in integrated optics is the reduction of the devices footprint, in a normal situation the advice is to employ shorter couplers. However, to obtain high coupling values with short couplers, a small distance d between the waveguides is needed. With the negative slope technique K=1 can be obtained ideally at the cost of incresed device footprint. However if some special applications require higher cavity lengths, i.e. for microwave filter applications, the election of longer coupling sections can be advantageous. Each of these two mentioned techniques is suited to each of the coupled resonator structures, either bus coupled (SCISSOR) or directly coupled (CROW), as already demonstrated in [7

7. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 ( 2007). [CrossRef] [PubMed]

].

3. Coupling constant sensitivity analysis

Fig. 2. Illustration of the coupling constant change as the effective coupling length changes, increasing (a) or decreasing (b) the coupling constant. L 0 e is the coupling effective length without offset and Loffe the coupling effective length when a longitudinal offset is applied.

3.1. Gap coupling control CROWs

The coupling constant K values obtained through BPM simulations of parallel waveguide couplers, with nominal lengths 53.3µm and 20µm, are shown in Figs. 3(a) and 3(b) respectively. The 3D graph shows the value of K versus two parameters, waveguide center-to-center distance, d 0 and waveguide width, w. In photolithographic fabrication processes, the waveguide width can be changed from a nominal value, 500 nm in our study, changing the exposure dose [12

12. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campenhout, P. Bienstman, and D. V. Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–412 ( 2005). [CrossRef]

].

From the graphs and Eq. (1), for a fixed waveguide width, an increase of d 0 (reduction of κ 12), results in a lower value of K for the same coupler length, Lc. The graphs also show, a significant variation in K for changes in d 0 of tenths of nanometers. Approximately a 10 % change in K for 20 nm can be derived from the plots. This argument can be reversed to state that a small change in K, for instance 1 % change, requires a very small change in d 0 (in the order of nanometers), that might not be attainable by some fabrication processes.

On the other hand, for a fixed distance between waveguides, d 0, a variation of the waveguide width, due to an over/down-exposure [12

12. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campenhout, P. Bienstman, and D. V. Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–412 ( 2005). [CrossRef]

], does not result in significant changes in K. This can be explained through Eq. (1) again, since as the waveguide width changes w, the overlap of the fields, κ 12 remains similar, for this very particular case.

Therefore, in this kind of couplers, to accurately set the desired coupling constant changing the gap, a resolution of nanometers in the fabrication process is needed.

Fig. 3. Coupling constant (K) vs waveguide width and d 0 (a) and (b) or longitudinal offset (c) and (d) between the coupler waveguides for a nominal waveguide width w=500nm. The coupler length is set to be 53.3 µm and the gap to 150nm in (a) and (c). The coupler length is set to be 20 µm and the gap to 100nm in (b) and (d).

3.2. Offset coupling control CROWs

A BPM analysis of the parallel waveguides coupler is also performed for couplers in which d 0 is kept constant, but with a longitudinal offset between the coupler arms. The results are shown in Figs. 3(c) and 3(d), for (Lc=53.3µm,gap=0.15µm) and (Lc=20µm,gap=0.10µm) respectively.

In this case, the variation of the coupling constant is controlled by changing the waveguides relative longitudinal position in steps of micrometers or of hundreds of nanometers. From the graph, a variation of 10 % in K requires a step of approximately 2 µm. Reversing the argument, fine control of K, for instance a change of 1 %, can be obtained by an offset step of 200 nm.

Note that depending on the coupler design, an increase or decrease of K is obtained when applying an offset between the coupler arms, as seen in Figs. 3(c) (K increases with offset) and 3(d)(K decreases with offset).

Therefore, in this kind of couplers, to accurately set the desired coupling constant changing the offset, a resolution of hundreds of nanometers in the fabrication process is needed. Moreover with proper design the offset can help to increase or decrease the value of K.

Fig. 4. Transmission for 3 (a) and 5 (b) racetracks CROW. The uniform response (K=0.2 in all couplers) is depicted in continuous line and the apodized response (K values from the table) is depicted with the dashed line.

3.3. Comparison gap vs. offset coupling control

The comparison of both techniques shows that the step size to control K is two orders of magnitude higher in the offset technique. This leads to the following relevant conclusions:

1. To change K a given amount, higher precision is required in the gap technique.

2. Once the minimum achievable gap of a given fabrication process is known, the way to increase K in the gap technique is to layout longer couplers. This approach entails larger ring perimeters and the consequent change in the FSR. The change in the FSR can be avoided employing smaller bend radii but at the cost of increased losses in the bends.

3. With proper coupler design, the offset technique allows for both increasing and decreasing the value of K from a nominal value, for a fixed coupler length and gap.

As a general conclusion, the offset technique presents more relaxed design and fabrication requirements than the gap technique. These conclusions are for the couplers themselves, however it is important to study the impact of these techniques and limitations on the spectral response of CROW devices.

4. Implications in the CROW spectral response

4.1. Spectral response and targeted K values

The frequency response of several CROWs is calculated in this section, using the Transfer Matrix Method [15

15. J. Capmany and M.A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 ( 1990). [CrossRef]

, 16

16. J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled resonator optical waveguides,” Opt. Express 12, 90–103 ( 2004). [CrossRef] [PubMed]

], in order to compare the performance of both presented techniques. Two CROW devices, with 3 and 5 racetracks, for each coupling control technique, are studied. The value of the coupling constants has been extracted from the BPM simulations shown in Fig. 3. The transmission transfer functions, i.e. output on the opposite side of injection, are shown in Fig. 4 for 3 and 5 rings devices, both for the uniform and apodized cases. The spectra shown are limited to a single Free Spectral Range (FSR) in normalized units of δ/2π, being:

δ=βLcav=2πλ·ne·Lcav
(2)

with Lcav the optical path length of a single cavity and ne the mode effective index of the waveguides. The nominal K value, for the uniform device, is K=0.2. This value is changed in the 4 couplers (3 rings device) and 6 couplers (5 rings device), following a Hamming window law [7

7. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 ( 2007). [CrossRef] [PubMed]

] with window parameter H=0.25, for the apodized cases. On CROWdevices the direct coupling coefficient is apodized, therefore after applying the Hamming window the coupling constant values are higher than the nominal value, as detailed in [7

7. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 ( 2007). [CrossRef] [PubMed]

].

Fig. 5. Transmission spectra for 3 ring CROW device, waveguide width 500 nm, and finite step resolution in the process for a coupler length of 20µm (a) and 53.3µm (b). The insets show the ripples in the pass-band.

Table 1. 3 and 5 racetracks CROW gap apodized (a) 3 racetracks CROW

table-icon
View This Table
| View All Tables

The K values are summarized in Tables 1 and 2, along with the gaps/offsets required by design to obtain the apodized responses depicted in Fig. 4 (dashed lines). These K values are for a nominal waveguide width of 500 nm that only allows one mode to propagate. The propagation losses have been estimated to be about 2dB/cm with negligible losses in bends since R>2µm [9

9. F. Ohno, T. Fukazawa, and T. Baba, “Mach-Zehnder Interferometers Composed of µ-Bends and µ-Branches in a Si Photonic Wire Waveguide,” Jpn. J. Appl. Phys 44, 5322–5323 ( 2005). [CrossRef]

]. The offsets and gaps presented have been extracted also from the BPM simulations data for both coupler lengths of 53.3µm and 20µm, Fig 1. The spectra shown in Fig. 4 are given for the Kn values in Tables 1 and 2, regardless of the apodization technique used.

Table 2. 3 and 5 racetracks CROW offset apodized (a) 3 racetracks CROW

table-icon
View This Table
| View All Tables
Fig. 6. Transmission spectra for 5 ring CROW device, waveguide width 500 nm, and finite step resolution in the process for a coupler length of 20µm (a) and 53.3µm (b). The insets show the ripples in the pass-band.

4.2. Sensitivity to finite resolution in fabrication

The sensitivity of the spectral response to finite resolution in the fabrication process, for the couplers analyzed in Section 3, is presented. In practice the fabrication process, either photo or electron-beam lithography, has a finite step resolution. The spectral response was calculated for gaps/offsets rounded in 10 and 20 nm steps. Hence, the Kn values are recalculated using Fig. 3, assuming the discrete values of d 0/offset.

The results are shown in Figs. 5 and 6 for the 3 and 5 racetracks CROWs respectively. In the first example shown in Fig. 5(a), the 3-ring CROW responses where the coupler length is fixed to 20 µm are plotted. The spectra show the ideal response, already depicted in Fig. 4, and the apodized responses where the gaps/offsets have been rounded in 10 and 20 nm steps. The

’ideal’ responses in Figs. 5 and 6 refer to the dashed plots in Fig.4. While the offset apodized response is completely overlapped with the ideal response, the gap apodized responses are deviated from the ideal one. As another example, the 5-ring CROW responses, using couplers of Lc=53.3µm, with a finite step in setting d 0 and the offsets in Table 2(b), are shown in Fig. 6(b). As in all the examples from Figs. 5 and 6, the Kn values used are for gaps and offsets rounded in 10 and 20 nm steps. For instance, if the finite step is 20 nm, the gaps in Table 1(b) will be 120, 160, 200, 200, 160, 120 and Kn (n=1,2,3,4,5,6) 0.7385, 0.4669, 0.1942, 0.1942, 0.4669, 0.7385. Note that the offset technique spectra and the 10 nm step overlap in Fig. 6(b), while a step of 20 nm shows a spectral response already deviated from the targeted.

Fig. 7. Normalized Group Delay for 3 ring CROW device, waveguide width 500 nm, and finite step resolution in the process for a coupler length of 20µm (a) and 53.3µm (b).
Fig. 8. Normalized Group Delay for 5 ring CROW device, waveguide width 500 nm, and finite step resolution in the process for a coupler length of 20µm (a) and 53.3µm (b).

Regarding the delay response, τd of the CROW, it can be obtained using the following expression:

τdTc=ϕ(δ)δ
(3)

where Tc is the round trip time in a single ring and ϕ(δ) is the phase of the transmision response. Figures 7 and 8 show the normalized Group Delay of the examples shown above.

In view of these results, the offset apodization technique appears to be more robust than the gap apodization technique in terms of the required spectral sensitivity of the CROW devices.

5. Conclusions

A novel technique for controlling the coupling between ring resonator in CROW devices has been presented. Employing this technique, an apodized CROWcan be fabricated using identical fixed distances between rings and obtain the final design solely by changing the longitudinal offset. In the simulation examples provided, this technique requires two orders of magnitude less fabrication precision to set the coupling constant values, and under finite resolution steps in fabrication processes, clearly outperforms the gap technique.

Acknowledgements

This work has been funded through the Spanish Plan Nacional de I+D+i 2008-2011 project TEC2008-06145/TEC CROWN and by the Generalitat Valenciana through projects PROME-TEO/2008/092 and GV/2007/240 APRIL.

References and links

1.

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]

2.

J. E. Heebner, R. W. Boyd, and Q-Han. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 4, 722–731 ( 2002). [CrossRef]

3.

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, “Wavelength conversion in gaas micro-ring resonators,” Opt. Lett. 25, 554–556 ( 2000). [CrossRef]

4.

M. A. Preciado and M. A. Muriel, “All-pass optical structures for repetition rate multiplication,” Opt. Express 16, 11162–11168 ( 2008). [CrossRef] [PubMed]

5.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nature Photon. 1, 65–71 ( 2006). [CrossRef]

6.

A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. 36, 321–322 ( 2000). [CrossRef]

7.

J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express 15, 10196–10206 ( 2007). [CrossRef] [PubMed]

8.

F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15, 11934–11941 ( 2007). [CrossRef] [PubMed]

9.

F. Ohno, T. Fukazawa, and T. Baba, “Mach-Zehnder Interferometers Composed of µ-Bends and µ-Branches in a Si Photonic Wire Waveguide,” Jpn. J. Appl. Phys 44, 5322–5323 ( 2005). [CrossRef]

10.

K. Ebeling, Integrated Optoelectronics (Springer-Verlag, 1993). [CrossRef]

11.

F. Xia, L. Sekaric, and Y. A. Vlasov, “Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators,” Opt. Express 14,3872–3886 ( 2006). [CrossRef] [PubMed]

12.

W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campenhout, P. Bienstman, and D. V. Thourhout, “Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology,” J. Lightwave Technol. 23, 401–412 ( 2005). [CrossRef]

13.

“BeamPROP 8.1, RSoft Design Group, Inc.” http://www.rsoftdesign.com.

14.

“ePIXfab, the European Silicon Photonics Platform.” http://www.epixfab.eu/.

15.

J. Capmany and M.A. Muriel, “A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing,” J. Lightwave Technol. 8, 1904–1919 ( 1990). [CrossRef]

16.

J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled resonator optical waveguides,” Opt. Express 12, 90–103 ( 2004). [CrossRef] [PubMed]

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.5750) Optical devices : Resonators

ToC Category:
Integrated Optics

History
Original Manuscript: September 22, 2009
Revised Manuscript: October 23, 2009
Manuscript Accepted: October 24, 2009
Published: November 4, 2009

Citation
Jose D. Domenech, Pascual Muñoz, and Jose Capmany, "The longitudinal offset technique for apodization of coupled resonator optical waveguide devices: concept and fabrication tolerance analysis," Opt. Express 17, 21050-21059 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-23-21050


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References

  1. 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]
  2. J. E. Heebner, R. W. Boyd, and Q-Han. Park, "SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides," J. Opt. Soc. Am. B 4, 722-731 (2002). [CrossRef]
  3. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, "Wavelength conversion in gaas micro-ring resonators," Opt. Lett. 25, 554-556 (2000). [CrossRef]
  4. M. A. Preciado and M. A. Muriel, "All-pass optical structures for repetition rate multiplication," Opt. Express 16, 11162-11168 (2008). [CrossRef] [PubMed]
  5. F. Xia, L. Sekaric, and Y. Vlasov, "Ultracompact optical buffers on a silicon chip," Nat. Photonics 1, 65-71 (2006). [CrossRef]
  6. A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000). [CrossRef]
  7. J. Capmany, P. Munoz, J. D. Domenech, and M. A. Muriel, "Apodized coupled resonator waveguides," Opt. Express 15, 10196-10206 (2007). [CrossRef] [PubMed]
  8. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, "Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects," Opt. Express 15, 11934-11941 (2007). [CrossRef] [PubMed]
  9. F. Ohno, T. Fukazawa, and T. Baba, "Mach-Zehnder Interferometers Composed of μ-Bends and μ-Branches in a Si Photonic Wire Waveguide," Jpn. J. Appl. Phys 44, 5322-5323 (2005). [CrossRef]
  10. K. Ebeling, Integrated Optoelectronics (Springer-Verlag, 1993). [CrossRef]
  11. F. Xia, L. Sekaric, and Y. A. Vlasov, "Mode conversion losses in silicon-on-insulator photonic wire based racetrack resonators," Opt. Express 14,3872-3886 (2006). [CrossRef] [PubMed]
  12. W. Bogaerts, R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campenhout, P. Bienstman, and D. V. Thourhout, "Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology," J. Lightwave Technol. 23, 401-412 (2005). [CrossRef]
  13. "BeamPROP 8.1, RSoft Design Group, Inc." http://www.rsoftdesign.com.
  14. "ePIXfab, the European Silicon Photonics Platform." http://www.epixfab.eu/.
  15. J. Capmany, and M.A. Muriel, "A new transfer matrix formalism for the analysis of fiber ring resonators: compound coupled structures for FDMA demultiplexing," J. Lightwave Technol. 8, 1904-1919 (1990). [CrossRef]
  16. J. Poon, J. Scheuer, S. Mookherjea, G. Paloczi, Y. Huang, and A. Yariv, "Matrix analysis of microring coupled resonator optical waveguides," Opt. Express 12, 90-103 (2004). [CrossRef] [PubMed]

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