Abstract

Integrated-optic cavity resonators, such as Fabry–Perot microcavities and microrings, are key building blocks of photonics integrated circuits and are used extensively in applications such as optical communications and microwave photonics. For a single, conventional, optical-cavity resonator, resonance peaks appear periodically in frequency and have Lorentzian shapes in nature, which generally cannot be broken. Here, we report on fully tailorable, integrated-optic resonators that allow for independent control of individual resonance or spectral peaks as regards their presence, linewidths and extinction ratios, resonant wavelengths, and shapes and bandwidths. The response shapes can be set to be Lorentzian, Gaussian-like, or square. The resonators are based on chirped waveguide Moiré gratings developed on a silicon-on-insulator platform. We also demonstrate that they can be implemented on compact Archimedean spiral shapes to have sizes comparable to microring and microdisk resonators, with no spectral degradation. The unprecedented spectral flexibility of these resonators makes them attractive for a variety of fields and will enable new avenues for exploration in relevant areas such as optical waveform synthesis and microwave photonics.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (1)

L. Liu, Y. Li, and X. Li, “A broadband tunable laser design based on the distributed Moiré-grating reflector,” Opt. Commun. 458, 124810 (2020).
[Crossref]

2019 (4)

2018 (2)

2017 (2)

2016 (3)

2015 (1)

2014 (2)

2013 (2)

J. R. Ong, R. Kumar, and S. Mookherjea, “Ultra-high-contrast and tunable-bandwidth filter using cascaded high-order silicon microring filters,” IEEE Photon. Technol. Lett. 25, 1543–1546 (2013).
[Crossref]

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

2012 (3)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
[Crossref]

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
[Crossref]

2011 (3)

2008 (1)

2007 (2)

2006 (2)

T. J. Kippenberg, J. Kalkman, A. Polman, and K. J. Vahala, “Demonstration of an erbium-doped microdisk laser on a silicon chip,” Phys. Rev. A 74, 051802 (2006).
[Crossref]

M. A. Popovíc, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. 31, 2571–2573 (2006).
[Crossref]

2005 (1)

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

2004 (1)

2003 (1)

Q. Wang and S. He, “Optimal design of a flat-top interleaver based on cascaded M–Z interferometers by using a genetic algorithm,” Opt. Commun. 224, 229–236 (2003).
[Crossref]

1998 (1)

L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[Crossref]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

1995 (2)

R. Zengerle and O. Leminger, “Phase-shifted Bragg-grating filters with improved transmission characteristics,” J. Lightwave Technol. 13, 2354–2358 (1995).
[Crossref]

L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[Crossref]

1991 (1)

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

1990 (1)

D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Adibi, A.

Alic, N.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

Ataie, V.

E. Temprana, V. Ataie, B. P. Kuo, E. Myslivets, N. Alic, and S. Radic, “Dynamic reconfiguration of parametric frequency comb forsuperchannel and flex-grid transmitters,” in The European Conference on Optical Communication (ECOC) (2014), pp. 1–3.

V. Ataie, E. Temprana, L. Liu, E. Myslivets, B. P. Kuo, N. Alic, and S. Radic, “Flex-grid compatible ultra wide frequency comb source for 31.8 Tb/s coherent transmission of 1520 UDWDM channels,” in OFC (2014), pp. 1–3.

Baets, R.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Baker, N.

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Barwicz, T.

Bayon, F.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Bennion, I.

L. Zhang, K. Sugden, I. Bennion, and A. Molony, “Wide-stopband chirped fibre Moiré grating transmission filters,” Electron. Lett. 31, 477–479 (1995).
[Crossref]

D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Bergman, K.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Bernage, P.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Biberman, A.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Bland-Hawthorn, J.

Bo, F.

Bogaerts, W.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Burla, M.

Buus, J.

D. Reid, C. Ragdale, I. Bennion, J. Buus, and W. Stewart, “Phase-shifted Moire grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Chen, L.

Chen, L. R.

L. R. Chen, “Silicon photonics for microwave photonics applications,” J. Lightwave Technol. 35, 824–835 (2017).
[Crossref]

L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[Crossref]

Chen, M.

S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

Chen, P.

Chen, S.

Chen, W.

Chen, X.

S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
[Crossref]

S. Liu, Y. Shi, Y. Zhou, Y. Zhao, J. Zheng, J. Lu, and X. Chen, “Planar waveguide Moiré grating,” Opt. Express 25, 24960–24973 (2017).
[Crossref]

Cheung, K. C.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Chrostowski, L.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Chu, S. T.

Claes, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Cooper, D. J.

L. R. Chen, D. J. Cooper, and P. W. Smith, “Transmission filters with multiple flattened passbands based on chirped Moiré gratings,” IEEE Photon. Technol. Lett. 10, 1283–1285 (1998).
[Crossref]

Corcoran, B.

Dai, D.

Dai, S.

Dai, T.

De Heyn, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

De Vos, K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Diddams, S. A.

T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011).
[Crossref]

Ding, Y.

Doerfler, M.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Dong, P.

P. Dong, S. F. Preble, and M. Lipson, “All-optical compact silicon comb switch,” Opt. Express 15, 9600–9605 (2007).
[Crossref]

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Douay, M.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Driessen, A.

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Dubé-Demers, R.

Dumon, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon. Rev. 6, 47–73 (2012).
[Crossref]

Edvell, G.

Englund, M.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

Fard, S. T.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Ferdous, F.

V. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6, 186 (2012).
[Crossref]

Fertein, E.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of Moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27, 1945–1947 (1991).
[Crossref]

Flueckiger, J.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophoton. 6, 821–828 (2013).
[Crossref]

Foster, J. D.

A. Biberman, P. Dong, B. G. Lee, J. D. Foster, M. Lipson, and K. Bergman, “Silicon microring resonator-based broadband comb switch for wavelength-parallel message routing,” in LEOS–IEEE Lasers and Electro-Optics Society Annual Meeting Conference Proceedings (2007), pp. 474–475.

Foster, M. A.

Fu, Q.

Gabriel Sengo,

E. J. Klein, D. H. Geuzebroek, H. Kelderman, Gabriel Sengo, N. Baker, and A. Driessen, “Reconfigurable optical add-drop multiplexer using microring resonators,” IEEE Photon. Technol. Lett. 17, 2358–2360 (2005).
[Crossref]

Gaeta, A. L.

Gao, F.

Georges, T.

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O. Gerstel, M. Jinno, A. Lord, and S. J. B. Yoo, “Elastic optical networking: a new dawn for the optical layer?” IEEE Commun. Mag. 50, s12–s20 (2012).
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S. Liu, H. Wu, Y. Shi, B. Qiu, R. Xiao, M. Chen, H. Xue, L. Hao, Y. Zhao, J. Lu, and X. Chen, “High-power single-longitudinal-mode DFB semiconductor laser based on sampled Moiré grating,” IEEE Photon. Technol. Lett. 31, 751–754 (2019).
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Supplementary Material (1)

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» Supplement 1       Supplementary Material

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Figures (12)

Fig. 1.
Fig. 1. (a) Illustration of how superimposing two periodic patterns with the same parameters except for a small difference in their periods can generate a Moiré profile. (b) Schematic representation of a waveguide Moiré Bragg grating. (c) Grating strength and phase profiles; the grating strength follows a Moiré profile along the length, and a $\pi$-phase shift occurs at the crossover point. (d) Spectral responses of the Moiré grating, where a resonance peak is opened at the Bragg wavelength due to the $\pi$-phase shift at the crossover point. For the grating in (c) and (d), ${\Lambda _{G1}}$ and ${\Lambda _{G2}}$ are 311 nm and ${\sim}311.3\;{\rm nm}$, respectively, the grating length, $L$, is ${\sim}0.3\;{\rm mm}$, and the corrugation width, $\Delta W$, is 13 nm, which is defined as the width difference between the inner and outer grating sidewalls on a single side of the waveguide.
Fig. 2.
Fig. 2. Illustration of the situation of a long, un-chirped, waveguide Moiré grating. (a) Sub-grating period profiles; ${\Lambda _{G1}}$ and ${\Lambda _{G2}}$ are 311 and ${\sim}311.9\;{\rm nm}$, respectively, and $L$ is ${\sim}0.84\;{\rm mm}$. (b) Grating Moiré profile, $M(z)$, defined as a complex function whose amplitude and phase [denoted by $|M(z)|$ and $\angle M(z)$, respectively] represent the overall grating strength (i.e., grating coupling coefficient) and phase profiles, respectively. (c) Grating spectral responses; $\Delta W$ is 5 nm.
Fig. 3.
Fig. 3. Illustration of the case where the two sub-gratings of the waveguide Moiré grating in Fig. 2 are applied by the same linear chirp (${\sim}18.5\;{\rm nm/mm}$). (a) Sub-grating period profiles. (b) Bragg wavelength of the overall Moiré grating against the length. (c) Spatial (top $x$ axis) and wavelength (bottom $x$ axis) complex grating Moiré profile. (d) Grating spectral responses.
Fig. 4.
Fig. 4. Resonance peak suppression of a CWMG resonator via applying a compensation phase profile into the overall grating to eliminate the $\pi$-phase shifts at the corresponding crossover points of $M(z)$. (a)–(d) Suppression of the second resonance peak. (a) Compensation phase profile (yellow, right axis) and modified sub-grating phase profiles (red, left axis). (b) Complex Moiré profile of the modified CWMG; the $\pi$-phase shift at the second crossover point has been compensated for/eliminated. (c) Simulated reflection (left) and transmission (right) responses of the original and modified CWMGs. (d) Response comparison of two “$M(\lambda)$-equivalent” CWMGs with different values of $\gamma$, designed based on (a) and (b). (e), (f) Suppression of the second and the fourth resonance peaks. (e) Compensation phase profile (yellow, right axis) and modified sub-grating phase profiles (red, left axis). (f) Simulated reflection (left) and transmission (right) responses of the original and modified CWMGs.
Fig. 5.
Fig. 5. Resonance linewidth and extinction ratio control of a CWMG resonator via grating apodization. (a) $|M(z)|$ and ${\Delta}\!{W}$ distribution of the CWMG and (b) simulated spectral responses of the CWMG designed in (a). (c) Simulated responses of a $M(\lambda)$-equivalent CWMG with a smaller $\gamma$. The dashed lines in (b) and (c) indicate the change in the extinction ratios.
Fig. 6.
Fig. 6. Resonant wavelength control of CWMG resonators. (a) Design of a resonator response containing two four-channel bands, each of which has a different FSR. (b) Design of a resonator response with the fourth resonance eliminated by equivalently doubling the spacing between the corresponding resonance peaks. In each of (a), (b), the top figure shows $|M(z)|/|M(\lambda)|$ (blue, left axis) and sub-grating period profiles (red, right axis), while the bottom plot presents the simulated grating reflection and transmission responses.
Fig. 7.
Fig. 7. (a) Schematic illustration of response shaping of a CWMG resonator, in which (i), (ii), and (iii) illustrate the original case and the cases in which the transmission peaks are shaped to present Gaussian-like and square behaviors, respectively. (b), (c) Designs of five-channel Gaussian and square filters, respectively, via response shaping of CWMG resonators; the upper (i), middle (ii), and lower (iii) figures in (b) and (c) are ${\Lambda _{G1}}(z)$ and ${\Lambda _{G2}}(z)$, $|M(\lambda)|$, and the simulated transmission response of the resonator, respectively. (d), (e) Comparisons of shaped resonator responses with different values of $\Delta W$ for the five-channel Gaussian and square filters, respectively. (f) Response comparison of five-channel square filters developed on two $M(\lambda)$-equivalent CWMGs with different values of $\gamma$. The right insets in (d)–(f) are enlarged views of the first channel responses.
Fig. 8.
Fig. 8. Experimental data of [(a), (b)] two regular CWMG resonators and (c) a CWMG resonator with tailored resonance linewidths and extinction ratios. The top plots in (a)–(c) show ${\Lambda _{G1}}(z)$, ${\Lambda _{G2}}(z)$, and $|M(z)|/|M(\lambda)|$ of the corresponding resonators.
Fig. 9.
Fig. 9. Measurement results of CWMG resonators designed to have (a) the second and (b) second and fourth resonance peaks suppressed, which are designed based on Figs. 4(a)4(b) and Fig. 4(e), respectively.
Fig. 10.
Fig. 10. Measurement results of various resonant wavelength-tailored CWMG resonators. The resonators shown in (a)–(d) are designed based on Figs. 6(a) and 6(b) and Figs. S4(a)–S4(b) in Supplement 1, respectively.
Fig. 11.
Fig. 11. Measurement results of CWMG resonators with the responses shaped to be (a) a five-channel Gaussian and (b)–(d) five-channel square filters. The resonators shown in (a) and (b) are designed based on Figs. 7(b) and 7(c), respectively. (c) Comparison of the fourth channel responses of the five-channel square filters with different values of $\Delta W$, based on the same design as that in (b). The CWMG resonator shown in (d) uses a smaller $\gamma$ compared with that in (b) to demonstrate the potential of achieving ultra-high extinction ratios and roll-off rates of multichannel square filters using CWMGs; the gray curve in (d) is the noise floor of the detector.
Fig. 12.
Fig. 12. (a)–(c) SEM images of the testing circuit for, and an overall and a zoomed-in view of, Spiral CWMG 1, respectively. (d)–(f) Measurement results of Spiral CWMGs 1–3, respectively, which are spiral versions of the previously demonstrated straight CWMGs with their measured data shown in Figs. 8(a), 11(b), and 11(d), respectively. The gray curve in (f) is the noise floor of the detector.

Tables (1)

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Table 1. Average Channel Performances of Straight and Spiral CWMG-Based, Five-Channel, Square Filters

Equations (4)

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Λ M = 2 Λ 1 Λ 2 Δ Λ ,
λ B = λ 0 + γ z ,
γ = 2 C n e f f 2 n g .
F S R = Λ M 2 γ 2 C n e f f 2 Λ G 0 2 n g Δ Λ G ,

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