Abstract

Wavelength filters are among the most important building blocks required for integrated optical circuits. However, existing filter building blocks provide only basic functionality with limited options for controlling the filter function unless sophisticated filter design techniques are employed. Conversely, in the microwave regime, elegant and powerful filter synthesis techniques exist which use coupled resonators. As waveguide ring resonators have emerged, researchers in the optical domain have sought to translate these techniques but the multi-wavelength spacing required to couple optical ring resonator structures severely limits the types of filters that can be realized. In this paper we show how recently reported ridge resonance structures can be arranged as coupled resonators with very close spacing and thus can be harnessed to achieved many of the filter functionalities available in the field of microwave engineering. Our filter is comprised of multiple parallel ridges on a common silicon slab, with each resonator exhibiting a resonant frequency and quality factor which can be controlled through engineering the geometry of the ridge. It is thus possible to choose appropriate combinations of ridge geometries to satisfy the conditions required by filter synthesis prototype. We demonstrate through rigorous simulation how our approach can be used to achieve high order optical bandpass filters at 1.55 $\mu$m center wavelength with Butterworth or Chebychev responses and analyse the impact of non-ideal behaviours on filter performance.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. G. Ren, T. G. Nguyen, and A. Mitchell, “Gaussian beams on a silicon-on-insulator chip using integrated optical lenses,” IEEE Photonics Technol. Lett. 26(14), 1438–1441 (2014).
    [Crossref]

2020 (1)

T. G. Nguyen, A. Boes, and A. Mitchell, “Lateral leakage in silicon photonics: Theory, applications, and future directions,” IEEE J. Sel. Top. Quantum Electron. 26(2), 1–13 (2020).
[Crossref]

2019 (1)

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

2018 (2)

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” Photonics Res. 6(11), 1084–1093 (2018).
[Crossref]

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: Methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

2016 (1)

A.P. Hope, T. G. Nguyen, A. Mitchell, and W. Bogaerts, “Quantitative analysis of TM lateral leakage in foundry fabricated silicon rib waveguides,” IEEE Photonics Technol. Lett. 28(4), 493–496 (2016).
[Crossref]

2015 (2)

Y. Li, Y. Zhang, L. Zhang, and A. W. Poon, “Silicon and hybrid silicon photonic devices for intra-datacenter applications: state of the art and perspectives,” Photonics Res. 3(5), B10–B27 (2015).
[Crossref]

P. P. Absil, P. Verheyen, P. D. Heyn, M. Pantouvaki, G. Lepage, J. D. Coster, and J. V. Campenhout, “Silicon photonics integrated circuits: a manufacturing platform for high density, low power optical I/O’s,” Opt. Express 23(7), 9369–9378 (2015).
[Crossref]

2014 (2)

P. Chen, S. Chen, X. Guan, Y. Shi, and D. Dai, “High-order microring resonators with bent couplers for a box-like filter response,” Opt. Lett. 39(21), 6304–6307 (2014).
[Crossref]

G. Ren, T. G. Nguyen, and A. Mitchell, “Gaussian beams on a silicon-on-insulator chip using integrated optical lenses,” IEEE Photonics Technol. Lett. 26(14), 1438–1441 (2014).
[Crossref]

2013 (1)

2011 (2)

2010 (1)

2009 (1)

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photonics Technol. Lett. 21(7), 486–488 (2009).
[Crossref]

2007 (2)

M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 19(6), 429–431 (2007).
[Crossref]

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(19), 11934–11941 (2007).
[Crossref]

2004 (1)

2002 (1)

2001 (1)

2000 (2)

B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. 25(5), 344–346 (2000).
[Crossref]

G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photonics Technol. Lett. 12(7), 810–812 (2000).
[Crossref]

1999 (1)

Absil, P.

Absil, P. P.

Ahmed, O. S.

Bakr, M. H.

Bezus, E.

L. Doskolovich, E. Bezus, and D. Bykov, “Integrated flat-top reflection filters operating near bound states in the continuum,” Photonics Res.372279 (2019).

Bezus, E. A.

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” Photonics Res. 6(11), 1084–1093 (2018).
[Crossref]

Boes, A.

T. G. Nguyen, A. Boes, and A. Mitchell, “Lateral leakage in silicon photonics: Theory, applications, and future directions,” IEEE J. Sel. Top. Quantum Electron. 26(2), 1–13 (2020).
[Crossref]

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

Bogaerts, W.

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: Methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

A.P. Hope, T. G. Nguyen, A. Mitchell, and W. Bogaerts, “Quantitative analysis of TM lateral leakage in foundry fabricated silicon rib waveguides,” IEEE Photonics Technol. Lett. 28(4), 493–496 (2016).
[Crossref]

D. Vermeulen, S. Selvaraja, P. Verheyen, G. Lepage, W. Bogaerts, P. Absil, D. V. Thourhout, and G. Roelkens, “High-efficiency fiber-to-chip grating couplers realized using an advanced CMOS-compatible silicon-on-insulator platform,” Opt. Express 18(17), 18278–18283 (2010).
[Crossref]

Bykov, D.

L. Doskolovich, E. Bezus, and D. Bykov, “Integrated flat-top reflection filters operating near bound states in the continuum,” Photonics Res.372279 (2019).

Bykov, D. A.

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” Photonics Res. 6(11), 1084–1093 (2018).
[Crossref]

Campenhout, J. V.

Chen, P.

Chen, S.

Chrostowski, L.

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: Methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

Chu, S. T.

Coster, J. D.

Dai, D.

Dalvand, N.

Doskolovich, L.

L. Doskolovich, E. Bezus, and D. Bykov, “Integrated flat-top reflection filters operating near bound states in the continuum,” Photonics Res.372279 (2019).

Doskolovich, L. L.

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” Photonics Res. 6(11), 1084–1093 (2018).
[Crossref]

Greentree, A. D.

Griffel, G.

G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photonics Technol. Lett. 12(7), 810–812 (2000).
[Crossref]

Guan, X.

Heyn, P. D.

Hope, A. P.

Hope, A.P.

A.P. Hope, T. G. Nguyen, A. Mitchell, and W. Bogaerts, “Quantitative analysis of TM lateral leakage in foundry fabricated silicon rib waveguides,” IEEE Photonics Technol. Lett. 28(4), 493–496 (2016).
[Crossref]

Hryniewicz, J. V.

Huang, Y.

Jones, E. M. T.

G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Artech House, 1980).

Knoerzer, M.

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

Koch, T.

M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 19(6), 429–431 (2007).
[Crossref]

Koch, T. L.

N. Dalvand, T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Thin-ridge silicon-on-insulator waveguides with directional control of lateral leakage radiation,” Opt. Express 19(6), 5635–5643 (2011).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photonics Technol. Lett. 21(7), 486–488 (2009).
[Crossref]

Lee, R. K.

Lepage, G.

Li, X.

Li, Y.

Y. Li, Y. Zhang, L. Zhang, and A. W. Poon, “Silicon and hybrid silicon photonic devices for intra-datacenter applications: state of the art and perspectives,” Photonics Res. 3(5), B10–B27 (2015).
[Crossref]

Little, B. E.

Martinelli, M.

Matthaei, G. L.

G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Artech House, 1980).

Melloni, A.

Mitchell, A.

T. G. Nguyen, A. Boes, and A. Mitchell, “Lateral leakage in silicon photonics: Theory, applications, and future directions,” IEEE J. Sel. Top. Quantum Electron. 26(2), 1–13 (2020).
[Crossref]

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

A.P. Hope, T. G. Nguyen, A. Mitchell, and W. Bogaerts, “Quantitative analysis of TM lateral leakage in foundry fabricated silicon rib waveguides,” IEEE Photonics Technol. Lett. 28(4), 493–496 (2016).
[Crossref]

G. Ren, T. G. Nguyen, and A. Mitchell, “Gaussian beams on a silicon-on-insulator chip using integrated optical lenses,” IEEE Photonics Technol. Lett. 26(14), 1438–1441 (2014).
[Crossref]

A. P. Hope, T. G. Nguyen, A. D. Greentree, and A. Mitchell, “Long-range coupling of silicon photonic waveguides using lateral leakage and adiabatic passage,” Opt. Express 21(19), 22705–22716 (2013).
[Crossref]

N. Dalvand, T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Thin-ridge silicon-on-insulator waveguides with directional control of lateral leakage radiation,” Opt. Express 19(6), 5635–5643 (2011).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photonics Technol. Lett. 21(7), 486–488 (2009).
[Crossref]

M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 19(6), 429–431 (2007).
[Crossref]

Mookherjea, S.

Nguyen, T. G.

T. G. Nguyen, A. Boes, and A. Mitchell, “Lateral leakage in silicon photonics: Theory, applications, and future directions,” IEEE J. Sel. Top. Quantum Electron. 26(2), 1–13 (2020).
[Crossref]

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

A.P. Hope, T. G. Nguyen, A. Mitchell, and W. Bogaerts, “Quantitative analysis of TM lateral leakage in foundry fabricated silicon rib waveguides,” IEEE Photonics Technol. Lett. 28(4), 493–496 (2016).
[Crossref]

G. Ren, T. G. Nguyen, and A. Mitchell, “Gaussian beams on a silicon-on-insulator chip using integrated optical lenses,” IEEE Photonics Technol. Lett. 26(14), 1438–1441 (2014).
[Crossref]

A. P. Hope, T. G. Nguyen, A. D. Greentree, and A. Mitchell, “Long-range coupling of silicon photonic waveguides using lateral leakage and adiabatic passage,” Opt. Express 21(19), 22705–22716 (2013).
[Crossref]

N. Dalvand, T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Thin-ridge silicon-on-insulator waveguides with directional control of lateral leakage radiation,” Opt. Express 19(6), 5635–5643 (2011).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photonics Technol. Lett. 21(7), 486–488 (2009).
[Crossref]

Paarmann, L. D.

L. D. Paarmann, Design and Analysis of Analog Filters: A Signal Processing Perspective (Springer, 2001).

Pafchek, R.

M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 19(6), 429–431 (2007).
[Crossref]

Paloczi, G. T.

Pantouvaki, M.

Poon, A. W.

Y. Li, Y. Zhang, L. Zhang, and A. W. Poon, “Silicon and hybrid silicon photonic devices for intra-datacenter applications: state of the art and perspectives,” Photonics Res. 3(5), B10–B27 (2015).
[Crossref]

Poon, J. K. S.

Pozar, D. M.

D. M. Pozar, Microwave Engineering (John Wiley & Sons, Inc., 2005), 3rd ed.

Ren, G.

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

G. Ren, T. G. Nguyen, and A. Mitchell, “Gaussian beams on a silicon-on-insulator chip using integrated optical lenses,” IEEE Photonics Technol. Lett. 26(14), 1438–1441 (2014).
[Crossref]

Roelkens, G.

Rooks, M.

Scherer, A.

Scheuer, J.

Schoenhardt, S.

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

Sekaric, L.

Selvaraja, S.

Shi, Y.

Swillam, M. A.

Thourhout, D. V.

Tummidi, R. S.

N. Dalvand, T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Thin-ridge silicon-on-insulator waveguides with directional control of lateral leakage radiation,” Opt. Express 19(6), 5635–5643 (2011).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photonics Technol. Lett. 21(7), 486–488 (2009).
[Crossref]

Verheyen, P.

Vermeulen, D.

Vlasov, Y.

Webster, M.

M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 19(6), 429–431 (2007).
[Crossref]

Xia, F.

Xu, Y.

Yariv, A.

Young, L.

G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Artech House, 1980).

Zhang, L.

Y. Li, Y. Zhang, L. Zhang, and A. W. Poon, “Silicon and hybrid silicon photonic devices for intra-datacenter applications: state of the art and perspectives,” Photonics Res. 3(5), B10–B27 (2015).
[Crossref]

Zhang, Y.

Y. Li, Y. Zhang, L. Zhang, and A. W. Poon, “Silicon and hybrid silicon photonic devices for intra-datacenter applications: state of the art and perspectives,” Photonics Res. 3(5), B10–B27 (2015).
[Crossref]

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

T. G. Nguyen, A. Boes, and A. Mitchell, “Lateral leakage in silicon photonics: Theory, applications, and future directions,” IEEE J. Sel. Top. Quantum Electron. 26(2), 1–13 (2020).
[Crossref]

IEEE Photonics Technol. Lett. (5)

M. Webster, R. Pafchek, A. Mitchell, and T. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photonics Technol. Lett. 19(6), 429–431 (2007).
[Crossref]

A.P. Hope, T. G. Nguyen, A. Mitchell, and W. Bogaerts, “Quantitative analysis of TM lateral leakage in foundry fabricated silicon rib waveguides,” IEEE Photonics Technol. Lett. 28(4), 493–496 (2016).
[Crossref]

G. Griffel, “Synthesis of optical filters using ring resonator arrays,” IEEE Photonics Technol. Lett. 12(7), 810–812 (2000).
[Crossref]

T. G. Nguyen, R. S. Tummidi, T. L. Koch, and A. Mitchell, “Rigorous modeling of lateral leakage loss in SOI thin-ridge waveguides and couplers,” IEEE Photonics Technol. Lett. 21(7), 486–488 (2009).
[Crossref]

G. Ren, T. G. Nguyen, and A. Mitchell, “Gaussian beams on a silicon-on-insulator chip using integrated optical lenses,” IEEE Photonics Technol. Lett. 26(14), 1438–1441 (2014).
[Crossref]

J. Lightwave Technol. (2)

Laser Photonics Rev. (2)

T. G. Nguyen, G. Ren, S. Schoenhardt, M. Knoerzer, A. Boes, and A. Mitchell, “Ridge resonance in silicon photonics harnessing bound states in the continuum,” Laser Photonics Rev. 13(10), 1900035 (2019).
[Crossref]

W. Bogaerts and L. Chrostowski, “Silicon photonics circuit design: Methods, tools and challenges,” Laser Photonics Rev. 12(4), 1700237 (2018).
[Crossref]

Opt. Express (6)

Opt. Lett. (4)

Photonics Res. (2)

Y. Li, Y. Zhang, L. Zhang, and A. W. Poon, “Silicon and hybrid silicon photonic devices for intra-datacenter applications: state of the art and perspectives,” Photonics Res. 3(5), B10–B27 (2015).
[Crossref]

E. A. Bezus, D. A. Bykov, and L. L. Doskolovich, “Bound states in the continuum and high-Q resonances supported by a dielectric ridge on a slab waveguide,” Photonics Res. 6(11), 1084–1093 (2018).
[Crossref]

Other (4)

D. M. Pozar, Microwave Engineering (John Wiley & Sons, Inc., 2005), 3rd ed.

G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures (Artech House, 1980).

L. Doskolovich, E. Bezus, and D. Bykov, “Integrated flat-top reflection filters operating near bound states in the continuum,” Photonics Res.372279 (2019).

L. D. Paarmann, Design and Analysis of Analog Filters: A Signal Processing Perspective (Springer, 2001).

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Figures (10)

Fig. 1.
Fig. 1. (a) Optical microscope image of an SOI ridge resonator formed by a single ridge on a slab [15]. Parabolic mirrors were used to generate and collect broad TE beams of slab mode in the SOI slab. The arrows illustrate the beam propagation directions; (b) Measured transmission as a function of wavelength showing the resonance; (c) High-order coupled resonators formed by multiple parallel ridges; (d) Simulated reflection and transmission of a coupled resonator structure with 5 identical ridges and spacing equal to an odd multiple of a quarter of wavelength, comparing to the responses of a single ridge resonator.
Fig. 2.
Fig. 2. (a) Normalized LC ladder low-pass filter prototype; (b) Filter implementation using resonators of different Q-factors connecting by quarter-wave transformers.
Fig. 3.
Fig. 3. (a) Cross-section of a ridge resonator with poly silicon loading on either side of the ridge. (b) illustration of the ridge resonator operation: blue arrows are obliquely incident, reflected and transmitted broad beams of vertically guided TE slab mode; the red arrow illustrates the generated guided TM-like mode in the ridge due to the coupling between the TE slab mode and guided TM-like mode in the ridge. (c) The simulated ridge resonance Q-factor (color map) and effective index of the guided TM mode in the ridge (contour lines) as functions of both the silicon core ridge width and the poly silicon loading width.
Fig. 4.
Fig. 4. A high-order filter composing of multiple parallel poly-silicon loading ridge resonators. The filter is excited by a laterally broad but vertically guided TE beam obliquely incident on the ridge array.
Fig. 5.
Fig. 5. (a) Wavelength responses of the three synthesized ridge resonators for the fifth-order Butterworth filter, comparing with the ideal resonator responses; (b) Wavelength responses of the synthesized and ideal third-order and fifth-order Butterworth filters.
Fig. 6.
Fig. 6. Cross-section field distributions of the horizontal ($|E_x|$, TE) and vertical ($|E_y|$, TM) components of the electric field vector when the fifth-order Butterworth filter is excited with a broad TE beam at different wavelengths .
Fig. 7.
Fig. 7. Wavelength responses of synthesized and ideal third-order and fifth-order Chebyshev filters with 20 dB (a) and 40 dB (b) out-of-band attenuation specifications.
Fig. 8.
Fig. 8. Phase response of a quarter-wave transformer connecting two adjacent ridge resonators in the synthesized filters.
Fig. 9.
Fig. 9. Wavelength responses of a fifth-order Butterworth (a) and Chebyshev (b) filter when the ridge resonators are replaced by ideal resonators separated by slab waveguides as the synthesized filters.
Fig. 10.
Fig. 10. Wavelength responses of a fifth-order Butterworth (a) and Chebyshev (b) filter when the the slabs connecting resonators are replaced by ideal quarter-wave transformers.

Equations (4)

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Q k = λ 0 B k = 2 λ 0 g k B ,
g k = 2 sin ( 2 k 1 2 N π )
g 1 = 2 A 1 γ g k = 4 A k 1 A k B k 1 g k 1 ,
γ = sinh ( β 2 N ) A k = sin ( 2 k 1 ) π 2 N B k = γ 2 + sin 2 k π N β = log { coth [ 10 log 10 ( 1 + 1 10 A t / 10 1 ) log 10 40 ] } ,