Abstract

In this paper we propose analyse the apodisation or windowing of the coupling coefficients in the unit cells of coupled resonator waveguide devices (CROWs) as a means to reduce the level of secondary sidelobes in the bandpass characteristic of their transfer functions. This technique is regularly employed in the design of digital filters and has been applied as well in the design of other photonic devices such as corrugated waveguide filters and fiber Bragg gratings. The apodisation of both Type-I and Type-II structures is discussed for several windowing functions.

© 2007 Optical Society of America

1. Introduction

There has been recently an increasing interest in the field of micro ring resonator based devices. On one hand, this interest is fuelled by the considerable development of different technologies in the field of integrated optics that allow the integration of single and multiple ring cavities in a variety of material substrates and configurations [1]–[5]. On the other hand, devices composed of single or multiple micro ring resonator cavities can be exploited in a wide variety of classical applications including, among others: channel filtering in WDM systems [6], linear and nonlinear digital optics [7], optical buffering [8] and modulation [9], dispersion compensation [10], switching [11]. Furthermore, their range of applications can be extended to encompass the emergent field of quantum information processing [12].

Different architectures integrating multiple rings have been proposed that are capable of performing complex signal processing operations, including the well known side coupled integrated space sequence of resonators (SCISSORs) [13], coupled resonator optical waveguides (CROWs) [14] and 2D structures [15]. Methods for the efficient analysis and synthesis of these structures have been developed by different groups that are based on techniques that are borrowed either from discrete time signal processing [16], microwave engineering [6],[17] or solid state physics [18].

Coupled resonator optical waveguide (CROW) devices are particularly interesting for filtering applications [14],[18] since their operation is similar to that of distributed feedback filters in the so called Type-I configuration [18] and to a stack of dielectric mirrors in the Type-II configuration [18]. In fact, previously published contributions have demonstrated their periodic bandpass filtering behaviour [18]. A common assumption in these works [14],[18] is to take the CROW device as composed of a number of identical sections or unit cells that are cascaded to form the final device. In this case the final filter structure resembles an uniform filter and therefore strong sidelobes are obtained in the bandpass transfer functions. In this paper we propose the apodisation or windowing of the coupling coefficients in the unit cells as a means to reduce the level of secondary sidelobes. This technique is regularly employed in the design of digital filters [19] and has been applied as well in the design of other photonic devices such as corrugated waveguide filters [20] and fiber Bragg gratings [21].

The outline of the paper is as follows: we first describe and analyse in section 2 the apodisation of type-I CROWs giving the basic equations and considerations for their design. Results are provided that illustrate the secondary sidelobe reduction obtained for a variety of apodisation functions and comparison between the performance of several apodisation profiles is established based on the concept of the effective number of rings in the CROW structure. In section 3 a similar analysis is carried for Type-II CROW structures. In section 4 we provide the simulation results obtained using a Finite Difference Time Domain (FDTD) method that confirm the results obtained for a Type-II CROW. Finally section 5 provides the summary and conclusions.

 

Fig. 1. Type I CROW structure layout.

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2. Apodised type-I CROWs

We first consider Type-I CROWs as defined in [18]. The general structure of the apodised Type-I CROWcomposed of N uncoupled rings with equal length Lc each one coupled to an in (upper) and a drop (lower) waveguide is shown in Fig. 1, where the individual unit cells are identified. Here we allow in each unit cell “i” different coupling values to the in and the drop waveguides. Also, the coupling values can change from one cell to another as a result of the apodization. We follow the same nomenclature as that employed in [17] for the cross and direct coupling parameters of each coupling region. Also shown in the figure is the ring separation parameter Lb and the electric field convention employed in the paper where the “+” superscript labels the fields propagating from left to right and the “-” superscript labels the fields propagating from right to left.

We use the transfer matrix method [6]–[14] to analyse the structure which is composed on N-1 unit cells and a closing ring cavity. The layouts of an arbitrary unit cell and the closing cavity are shown in Fig. 2.

The transfer matrices of the arbitrary unit cell MUCi (i=1,2,..N-1) and the closing cavity MCN are given respectively by:

MUCi=1R2i((R1iR2iT1iT2i)ejΔT2ieT1iejΔejΔ)
MCN=1R2N(R1NR2NT1NT2NT2NT1N1)

where:

R1i=t1it2i*(t1i2+κ1i2)τiejδ1τit1i*t2i*ejδ
R2i=t2it1i*(t2i2+κ2i2)τiejδ1τit1i*t2i*ejδ
T1i=κ1i*κ2iτiejδ21τit1i*t2i*ejδ
T2i=κ2i*κ1iτiejδ21τit1i*t2i*ejδ

and the physical parameters related to the cavity round trip phase shift and losses and the intercavity phase shift are given respectively by:

δ=βLc
τ=exp(αLc)
Δ=βLb

With α and β representing the waveguide attenuation and propagation constants.

 

Fig. 2. Type-I CROW unit cell and closing cell.

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The overall transfer matrix of the Type-I structure is then given by:

(EN+EN)=(T11T12T21T22)(E1+E1)
MT=(T11T12T21T22)=MCNi=N11MUCi

from which we get the relevant transmission and reflection transfer functions:

T=EN+E1+EN=0=1T22
R=E1E1+EN=0=T21T22

The apodisation is impressed in this case over the cross-coupled coefficients according to a specified window function [20][21] w(i) i=0,1,2…N-1. For type-I structures several options are possible: apodizing only the cross coupled coefficients in the in bus (κ 1i=κw(i)), apodizing only the cross coupled coefficients in the drop bus (κ 2i=κw(i)) or apodizing the cross coupled coefficients in both buses (κ 1i=κ 2i=κw(i)). In the following we consider the first and third cases and explore the effects of apodisation using standard windowing functions employed in signal processing applications [19].

The formulation above includes the general case of coupling losses. However, the simulations presented are for the case of a loss-less coupler, where t and κ are related through:

t=1K
κ=jK

with K being the power coupling ratio of the coupler.

Fig. 3 shows the results obtained for the reflection transfer function when using a Gaussian apodization function in a 10 ring type-I CROW with Lb=0.5·Lc. The apodisation window is given by [21]:

w(i)=exp(G(iN2N)2)
i=0,1,,N1
K=0.1
G=0,3,4

The left part shows the case of a single bus apodisation while the right part illustrates the effect of apodizing both buses.

The effect of sidelobe reduction due to the apodisation of the cross-coupling coefficients can be observed in both cases as compared to the case of no apodisation (G=0) which is also depicted in blue colour trace for reference. Higher reductions are obtained for the case where the cross coupled coefficients are apodised in both buses. In fact this result is obtained for the different apodisation windows that have been considered.

For example, in Figs. 4 and 5 we plot similar results as those of Fig. 3, when Hamming and Kaiser windowing functions and Lb=0.5 ·Lc are employed.

 

Fig. 3. Type-I CROWreflection transfer function for Gauss window apodisation (parameter G=0, 3 and 4) on (a) one bus and (b) two buses

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Fig. 4. Type-I CROW reflection transfer function for Hamming window apodisation (parameter H=0, 0.15 and 0.3) on (a) one bus and (b) two buses

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For the Hamming window [21], the following apodisation function is implemented:

w(i)=1+Hcos(2πn)1+H
i=0,1,,N1
K=0.1
H=0,0.15,0.3

where the case H=0 is equivalent to no apodisation. Again, sidelobe reduction is observed for both cases with a better performance for the case where the cross coupled coefficients are apodised in both buses. For the Kaiser apodisation window [20]:

w(i)=βksinh(βk)I0(βk14n2)
i=0,1,,N1
n=(iN2)N
K=0.1
βk=1,2,3
 

Fig. 5. Type-I CROWreflection transfer function for Kaiser window apodisation (parameter βk=0, 0.15 and 0.3) on (a) one bus and (b) two buses

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Again, sidelobe reduction is observed for both cases with a better performance for the case where the cross coupled coefficients are apodised in both buses. Apparently, the Kaiser window function provides the best performance regarding sidelobe suppression. However, one must be careful when comparing the performance of the different apodisation functions as it will now be explained. Apodising the coss coupling coefficients is in effect equivalent to reducing the number of rings in the CROW waveguide. The argument is equivalent to that provided in [20] to demonstrate that the effective length of an apodised grating is lower than that of an uniform device. To compare the performance of the different apodisation windows one has to choose a reference metric for the effective number of rings of the CROW device. Then one should compare apodised CROW structures with the same number of effective rings.

We define the effective number of rings in the apodised TypeI CROW structure by:

Neff=NΣi=0N1iw(i)Σi=0N1i

The effective number of rings for the type-I apodised CROWS in Figs. 3, 4 and 5 is displayed in the upper part of each graph. Obviously, for uniform (i.e non apodised) CROWs N=Neff (in this case Neff=10). In each case, as the apodisation parameter is increased the value of Neff decreases.

In Fig. 6, we compare the performance of the three windows for a CROW device with (N=10), Lb=0.5 ·Lc and the same effective number of rings Neff=6.9.

It can be observed that the performance of the three apodisation windows is quite similar when the number of effective rings is the same.

Regarding the delay response, τd of the CROW, it can be obtained using the following expression, where Tc is the round trip time in a single ring:

τdTc=ϕ(δ)δ

where ϕ(δ) is the phase of the reflection response in Eq. 11. The results are shown in Fig. 7 for Gaussian and Kaiser apodisation. As expected, more delay on the centre of the pass band is obtained for smaller bandwidth responses, i.e. a bigger degree of apodisation, which is in good agreement with resonator theory shown elsewhere.

 

Fig. 6. Type-I CROW reflection transfer comparison for Gauss, Hamming and Kaiser window apodisation (effective number of rings 6.9), on (a) one bus and (b) two buses

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Fig. 7. Type-I CROWreflection normalised delay for Gauss and Kaiser window apodisation

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3. Apodised type-II CROWs

In this section, Type-II CROWs as defined in [18] are analyzed. The structure consists on a set of coupled rings between two regular waveguides, as shown in Fig. 8. An N ring Type-II structure is the composed of N-1 unit cells closed by an opening and a closing section which connect them to the input and output waveguides respectively. Figure 9 shows the layouts of the unit cell and the input and output closing sections. The unit cell transfer matrix is, using the same symbol convention as above:

MUCi=1κi(τi12(κi2+ti2)ejδ2ti*tiτi12ejδ2)

The matrix corresponding to the input and output coupling sections of the CROW, (Fig. 9) opening and closing sections respectively, OS and CS, are the following:

MOS=1κ0((κ02+t02)τ014ejδ4t0*τ014ejδ4t0τ014ejδ4τi14ejδ4)
MCS=1κN((κN2+tN2)τN14ejδ4tN*τN14ejδ4tNτN14ejδ4τi14ejδ4)
 

Fig. 8. Type II CROW structure layout.

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Fig. 9. Type-II CROW unit cell, opening and closing sections.

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Hence, the overall transmission and reflection responses can be obtained as for Type-I CROWs, through equations 9 and 11, but using the following transfer matrix instead of equation 10:

MT=MCS[i=N11MUCi]MOS

The response of Type-I and Type-II CROWs are complementary, that is, while for Type-I the transmission and reflection from Eq. 11 are band-reject and band-pass respectively, for Type-II the transmission and reflection are band-pass and band-reject.

In the case of Type-II CROWs the apodisation is impressed on the direct coupling coefficients, ti, therefore ti=tw(i), where the window functions are given by Eqs. 1416. The results are shown in Fig. 10. The apodisation reduces the ripples in the passbands, at a cost of an increase in the filter bandwidth. In Fig. 10-(a) comparison of the three windowing functions for an fixed effective number of rings, (Eq. 16), of 6.6 is shown.

 

Fig. 10. Type-II CROW transmission transfer function for (a) Hamming, (b) Gauss, (c) Kaiser window apodisation (window parameters as in Figs. 35) and (d) comparison for an effective number of rings 6.6.

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4. FDTD simulation vs. analytical expressions

The previous analytical model has been checked against numerical simulations performed with the finite-difference time-domain (FDTD) method [22], using a freely available software package with subpixel smoothing for increased accuracy [23]. The simulations were performed for micro rings on InP deep-etch waveguides technology [24]. Simulations were performed in 2D, by reducing the vertical dimension using the effective index method [25]. Therefore waveguides and rings are of w=0.3 microns width, ring radius is set to R=5 microns and the effective index is neff=3.271. In order to set the precise coupling between the micro rings, and between micro rings and input/output straight waveguides, simulations to determine the lateral distance for a given coupling were performed, and the results are shown in Fig.10. The results show the coupling decreases exponentially with the lateral distance, in good agreement with theoretical analysis [26]. Using the coupling vs. lateral distance relation from Fig. 11, a type II CROW consisting on 6 micro rings, and coupling apodised with a Hamming window (H=0.12), was simulated. The result is compared with the analytical model in Fig. 11. The simulated response matches the model at the location of the peaks on the top of the response. The mismatch is due to radiation losses, that are not accounted for in the model.

 

Fig. 11. Type-II CROW FDTD analysis, (a) power coupling coefficient K vs distance for an InP w=0.3 microns deep-etched waveguide, and (b) model vs FDTD simulation for a 6 ring CROW with Hamming windowing, H=0.12.

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5. Summary and conclusions

We have proposed and analysed the apodisation or windowing of the coupling coefficients in the unit cells of coupled resonator waveguide devices (CROWs) as a means to reduce the level of secondary sidelobes in the bandpass characteristic of their transfer functions. This technique which is regularly employed in the design of digital filters has also been applied as well in the past for the design of other photonic devices such as corrugated waveguide filters and fiber Bragg gratings. The apodisation of both Type-I and Type-II structures have been discussed for several windowing functions and sidelobe suppression in both structures demonstrated. In Type-I structures the cross coupling coefficient is apodised and windowing can be applied either to one bus or the two in the structure with the second option giving an increased performance in terms of secondary sidelobe suppression. The effectiveness of the different windowing functions must be compared using an independent metric and for this purpose the effective number of rings has been defined. Windows with a similar value of effective number of rings in the CROW structure yield a similar performance as far as sidelobe suppression is concerned. Finally, we have presented simulation results for a Type-II structure computed using FDTD method that accurately match the analytical results.

Acknowledgments

This work has been partially funded by the Generalitat Valenciana through project GV/2007/240 APRIL (Active Passive Rings Integrated and Layered), and through the Spanish Plan Nacional de I+D+i projects TEC2004-04754-C03-02 and TEC2007-68065-C03-02. J.D. Domenech wishes to acknowledge a collaboration grant through the project GV/2007/240 APRIL.

References and links

1. T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992). [CrossRef]  

2. 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, 1567–1569 (2003) [CrossRef]   [PubMed]  

3. R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. King, L. C. Calhoun, F. G. Johnson, and P. -T. Ho, “Vertically coupled GaInAsP InP microring resonators,” Opt. Lett. 26, 506–508 (2001) [CrossRef]  

4. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002) [CrossRef]  

5. P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

6. 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]  

7. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002) [CrossRef]  

8. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007). [CrossRef]  

9. H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. 18, 211–213 (2006). [CrossRef]  

10. C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999) [CrossRef]  

11. B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000) [CrossRef]  

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

13. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21, 1818–1832, (2004). [CrossRef]  

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

15. Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005) [CrossRef]  

16. D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures,” J. Opt. Soc. Am. 11, 236–245, (1994). [CrossRef]  

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

18. Y. Landobasa and M. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005). [CrossRef]   [PubMed]  

19. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing. Prentice-Hall (1999)

20. P. S. Cross and H. Kogelnik, “Sidelobe suppression in corrugated-waveguide filters,” Opt. Lett. 1, 43–45 (1977) [CrossRef]   [PubMed]  

21. D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996) [CrossRef]  

22. A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech: Norwood, MA, 2000).

23. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef]   [PubMed]  

24. L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995). [CrossRef]  

25. K. Okamoto. Fundamentals of Optical Waveguides. (Academic Press, 2nd ed, 2005).

26. D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. Optoel. 140, 177–188 (1993). [CrossRef]  

References

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  1. T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
    [Crossref]
  2. 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, 1567–1569 (2003)
    [Crossref] [PubMed]
  3. R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. King, L. C. Calhoun, F. G. Johnson, and P. -T. Ho, “Vertically coupled GaInAsP InP microring resonators,” Opt. Lett. 26, 506–508 (2001)
    [Crossref]
  4. P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002)
    [Crossref]
  5. P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.
  6. 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]
  7. V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
    [Crossref]
  8. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007).
    [Crossref]
  9. H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. 18, 211–213 (2006).
    [Crossref]
  10. C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
    [Crossref]
  11. B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000)
    [Crossref]
  12. K. J. Vahala, “Optical microcavities,” Nature (London) 424, 839–846, (2003)
    [Crossref]
  13. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21, 1818–1832, (2004).
    [Crossref]
  14. J. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004)
    [Crossref] [PubMed]
  15. Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005)
    [Crossref]
  16. D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures,” J. Opt. Soc. Am. 11, 236–245, (1994).
    [Crossref]
  17. A. Yariv, “Universal relations for coupling of optical power between micro resonators and dielectric waveguides,” Electron. Lett. 36, 321–322 (2000)
    [Crossref]
  18. Y. Landobasa and M. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
    [Crossref] [PubMed]
  19. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing. Prentice-Hall (1999)
  20. P. S. Cross and H. Kogelnik, “Sidelobe suppression in corrugated-waveguide filters,” Opt. Lett. 1, 43–45 (1977)
    [Crossref] [PubMed]
  21. D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
    [Crossref]
  22. A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech: Norwood, MA, 2000).
  23. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006).
    [Crossref] [PubMed]
  24. L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
    [Crossref]
  25. K. Okamoto. Fundamentals of Optical Waveguides. (Academic Press, 2nd ed, 2005).
  26. D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. Optoel. 140, 177–188 (1993).
    [Crossref]

2007 (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007).
[Crossref]

2006 (2)

2005 (2)

Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005)
[Crossref]

Y. Landobasa and M. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (2)

2002 (2)

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002)
[Crossref]

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

2001 (1)

2000 (2)

B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000)
[Crossref]

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

1999 (1)

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

1996 (1)

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

1995 (1)

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

1994 (1)

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures,” J. Opt. Soc. Am. 11, 236–245, (1994).
[Crossref]

1993 (1)

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. Optoel. 140, 177–188 (1993).
[Crossref]

1992 (1)

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

1990 (1)

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]

1977 (1)

Absil, P. P.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. King, L. C. Calhoun, F. G. Johnson, and P. -T. Ho, “Vertically coupled GaInAsP InP microring resonators,” Opt. Lett. 26, 506–508 (2001)
[Crossref]

Baets, R.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Beckx, S.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Bermel, P.

Bogaerts, W.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Boyd, R. W.

Bruce, A. J.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing. Prentice-Hall (1999)

Burr, G.

Calhoun, L. C.

Capmany, J.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

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]

Cappuzzo, M. A.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

Chak, P.

Chin, M.

Y. Landobasa and M. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
[Crossref] [PubMed]

Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005)
[Crossref]

Christiaens, I.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Chu, S. T.

B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000)
[Crossref]

Costa, R.

Cross, P. S.

Dalton, L. R.

Darmawan, S.

Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005)
[Crossref]

Demeester, P.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

Dowling, E. M.

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures,” J. Opt. Soc. Am. 11, 236–245, (1994).
[Crossref]

Dumon, P.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Farjadpour, A.

Gomez, L. T.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

Groen, F. H.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

Grover, R.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. King, L. C. Calhoun, F. G. Johnson, and P. -T. Ho, “Vertically coupled GaInAsP InP microring resonators,” Opt. Lett. 26, 506–508 (2001)
[Crossref]

Hagness, S.C.

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech: Norwood, MA, 2000).

Heebner, J. E.

Ho, P. T.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

Ho, P. -T.

Hryniewicz, J. V.

Huang, Y.

Ibanescu, M.

Ibrahim, T. A.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

Joannopoulos, J. D.

Johnson, F. G.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. King, L. C. Calhoun, F. G. Johnson, and P. -T. Ho, “Vertically coupled GaInAsP InP microring resonators,” Opt. Lett. 26, 506–508 (2001)
[Crossref]

Johnson, S. G.

King, O.

Kogelnik, H.

Kokubun, Y

B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000)
[Crossref]

Kominato, T.

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

Landobasa, Y.

Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005)
[Crossref]

Y. Landobasa and M. Chin, “Defect modes in micro-ring resonator arrays,” Opt. Express 13, 7800–7815 (2005).
[Crossref] [PubMed]

Lenz, G.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

Little, B. E.

Love, J. D.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. Optoel. 140, 177–188 (1993).
[Crossref]

MacFarlane, D. L.

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures,” J. Opt. Soc. Am. 11, 236–245, (1994).
[Crossref]

Madsen, C. K.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

Marti, J.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

Martinelli, M.

Melloni, A.

Metaal, E. G.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

Monguzzi, P.

Mookherjea, S.

Muriel, M. A.

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]

Oei, Y. S.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

Ohmori, Y.

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

Okamoto, K.

K. Okamoto. Fundamentals of Optical Waveguides. (Academic Press, 2nd ed, 2005).

Okazaki, H.

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing. Prentice-Hall (1999)

Ortega, D.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

Paloczi, G. T.

Pan, W

B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000)
[Crossref]

Pastor, D.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

Pereira, S.

Poon, J.

Rabiei, P.

Rodriguez, A.

Roundy, D.

Rowland, D. R.

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. Optoel. 140, 177–188 (1993).
[Crossref]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing. Prentice-Hall (1999)

Scheuer, J.

Scotti, R. E.

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

Sekaric, L.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007).
[Crossref]

Sipe, J. E.

Smit, M. K.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

Spiekman, L. H.

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

Steier, W. H.

H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. 18, 211–213 (2006).
[Crossref]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002)
[Crossref]

Taflove, A.

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech: Norwood, MA, 2000).

Takato, N.

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

Tatay, V.

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

Tazawa, H.

H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. 18, 211–213 (2006).
[Crossref]

Vahala, K. J.

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

Van, V.

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

R. Grover, P. P. Absil, V. Van, J. V. Hryniewicz, B. E. Little, O. King, L. C. Calhoun, F. G. Johnson, and P. -T. Ho, “Vertically coupled GaInAsP InP microring resonators,” Opt. Lett. 26, 506–508 (2001)
[Crossref]

Van Thourhout, D.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Vlasov, Y.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007).
[Crossref]

Wiaux, V.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Wouters, J.

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

Xia, F.

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007).
[Crossref]

Yariv, A.

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

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

Yasu, M.

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

Zhang, C.

Electron. Lett. (1)

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

IEE Proc. Optoel (1)

L. H. Spiekman, Y. S. Oei, E. G. Metaal, F. H. Groen, P. Demeester, and M. K. Smit, “Ultrasmall waveguide bends: the corner mirrors of the future?,” IEE Proc. Optoel 142,61–65 (1995).
[Crossref]

IEE Proc. Optoel. (1)

D. R. Rowland and J. D. Love, “Evanescent wave coupling of whispering gallery modes of a dielectric cylinder,” IEE Proc. Optoel. 140, 177–188 (1993).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Landobasa, S. Darmawan, and M. Chin, “Matrix Analysis of 2-D Microresonator Lattice Optical Filters,” IEEE J. Quantum Electron. 41, 1410–1418 (2005)
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

V. Van, T. A. Ibrahim, P. P. Absil, F. G. Johnson, R. Grover, and P. T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. 8, 705–713 (2002)
[Crossref]

IEEE Photon. Technol. Lett. (3)

H. Tazawa and W. H. Steier, “Analysis of ring resonator-based traveling-wave modulators,” IEEE Photon. Technol. Lett. 18, 211–213 (2006).
[Crossref]

C. K. Madsen, G. Lenz, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, and R. E. Scotti, “Integrated all-pass filters for tunable dispersion and dispersion slope compensation,” IEEE Photon. Technol. Lett. 11, 1623–1625 (1999)
[Crossref]

B. E. Little, S. T. Chu, W Pan, and Y Kokubun, “Microring resonator arrays for VLSI photonics,” IEEE Photon. Technol. Lett. 12, 323–325 (2000)
[Crossref]

J. Light. Technol. (1)

D. Pastor, J. Capmany, D. Ortega, V. Tatay, and J. Marti, “Design of apodised linearly chirped fiber gratings for dispersioncompensation,” J. Light. Technol. 14, 2581–2588 (1996)
[Crossref]

J. Lightwave Technol. (3)

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]

T. Kominato, Y. Ohmori, N. Takato, H. Okazaki, and M. Yasu, “Ring resonators composed of GeO2-doped silica waveguides,” J. Lightwave Technol. 12, 1781–1788 (1992).
[Crossref]

P. Rabiei, W. H. Steier, C. Zhang, and L. R. Dalton, “Polymer Micro-Ring Filters and Modulators,” J. Lightwave Technol. 20, 1968–1975 (2002)
[Crossref]

J. Opt. Soc. Am. (1)

D. L. MacFarlane and E. M. Dowling, “Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures,” J. Opt. Soc. Am. 11, 236–245, (1994).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Photon (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photon 1, 65–71 (2007).
[Crossref]

Nature (London) (1)

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

Opt. Express (2)

Opt. Lett. (4)

Other (4)

K. Okamoto. Fundamentals of Optical Waveguides. (Academic Press, 2nd ed, 2005).

A. Taflove and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech: Norwood, MA, 2000).

P. Dumon, I. Christiaens, W. Bogaerts, V. Wiaux, J. Wouters, S. Beckx, D. Van Thourhout, and R. Baets, “Microring resonators on Silicon-on-Insulator,” in Proc. of European Conf. on Integrated Optics, 2005.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing. Prentice-Hall (1999)

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Figures (11)

Fig. 1.
Fig. 1.

Type I CROW structure layout.

Fig. 2.
Fig. 2.

Type-I CROW unit cell and closing cell.

Fig. 3.
Fig. 3.

Type-I CROWreflection transfer function for Gauss window apodisation (parameter G=0, 3 and 4) on (a) one bus and (b) two buses

Fig. 4.
Fig. 4.

Type-I CROW reflection transfer function for Hamming window apodisation (parameter H=0, 0.15 and 0.3) on (a) one bus and (b) two buses

Fig. 5.
Fig. 5.

Type-I CROWreflection transfer function for Kaiser window apodisation (parameter βk =0, 0.15 and 0.3) on (a) one bus and (b) two buses

Fig. 6.
Fig. 6.

Type-I CROW reflection transfer comparison for Gauss, Hamming and Kaiser window apodisation (effective number of rings 6.9), on (a) one bus and (b) two buses

Fig. 7.
Fig. 7.

Type-I CROWreflection normalised delay for Gauss and Kaiser window apodisation

Fig. 8.
Fig. 8.

Type II CROW structure layout.

Fig. 9.
Fig. 9.

Type-II CROW unit cell, opening and closing sections.

Fig. 10.
Fig. 10.

Type-II CROW transmission transfer function for (a) Hamming, (b) Gauss, (c) Kaiser window apodisation (window parameters as in Figs. 35) and (d) comparison for an effective number of rings 6.6.

Fig. 11.
Fig. 11.

Type-II CROW FDTD analysis, (a) power coupling coefficient K vs distance for an InP w=0.3 microns deep-etched waveguide, and (b) model vs FDTD simulation for a 6 ring CROW with Hamming windowing, H=0.12.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

M UCi = 1 R 2 i ( ( R 1 i R 2 i T 1 i T 2 i ) e j Δ T 2 i e T 1 i e j Δ e j Δ )
M CN = 1 R 2 N ( R 1 N R 2 N T 1 N T 2 N T 2 N T 1 N 1 )
R 1 i = t 1 i t 2 i * ( t 1 i 2 + κ 1 i 2 ) τ i e j δ 1 τ i t 1 i * t 2 i * e j δ
R 2 i = t 2 i t 1 i * ( t 2 i 2 + κ 2 i 2 ) τ i e j δ 1 τ i t 1 i * t 2 i * e j δ
T 1 i = κ 1 i * κ 2 i τ i e j δ 2 1 τ i t 1 i * t 2 i * e j δ
T 2 i = κ 2 i * κ 1 i τ i e j δ 2 1 τ i t 1 i * t 2 i * e j δ
δ = β L c
τ = exp ( α L c )
Δ = β L b
( E N + E N ) = ( T 11 T 12 T 21 T 22 ) ( E 1 + E 1 )
M T = ( T 11 T 12 T 21 T 22 ) = M CN i = N 1 1 M UCi
T = E N + E 1 + E N = 0 = 1 T 22
R = E 1 E 1 + E N = 0 = T 21 T 22
t = 1 K
κ = j K
w ( i ) = exp ( G ( i N 2 N ) 2 )
i = 0 , 1 , , N 1
K = 0 . 1
G = 0 , 3 , 4
w ( i ) = 1 + H cos ( 2 π n ) 1 + H
i = 0 , 1 , , N 1
K = 0.1
H = 0 , 0.15 , 0.3
w ( i ) = β k sinh ( β k ) I 0 ( β k 1 4 n 2 )
i = 0 , 1 , , N 1
n = ( i N 2 ) N
K = 0.1
β k = 1 , 2 , 3
N eff = N Σ i = 0 N 1 i w ( i ) Σ i = 0 N 1 i
τ d T c = ϕ ( δ ) δ
M UCi = 1 κ i ( τ i 1 2 ( κ i 2 + t i 2 ) e j δ 2 t i * t i τ i 1 2 e j δ 2 )
M OS = 1 κ 0 ( ( κ 0 2 + t 0 2 ) τ 0 1 4 e j δ 4 t 0 * τ 0 1 4 e j δ 4 t 0 τ 0 1 4 e j δ 4 τ i 1 4 e j δ 4 )
M CS = 1 κ N ( ( κ N 2 + t N 2 ) τ N 1 4 e j δ 4 t N * τ N 1 4 e j δ 4 t N τ N 1 4 e j δ 4 τ i 1 4 e j δ 4 )
M T = M CS [ i = N 1 1 M UCi ] M OS

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