Abstract

Two complementary types of SOI photonic wire based devices, the add/drop (A/D) filter using a racetrack resonator and the Mach-Zehnder interferometer with one arm consisting of an identical resonator in all-pass filter (APF) configuration, were fabricated and characterized in order to extract the optical properties of the resonators and predict the performance of the optical delay lines based on such resonators. We found that instead of well-known waveguide bending and propagation losses, mode conversion loss in the coupling region of such resonators dominates when the air gap between the racetrack resonator and access waveguide is smaller than 120nm. We also show that this additional loss significantly degrades the performance of the optical delay line containing cascaded resonators in APF configuration.

© 2006 Optical Society of America

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References

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  1. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 1998).
  2. R. Taylor and S. R. Forrest, "Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture," IEEE Photon. Technol. Lett. 10, 144-146 (1998).
    [CrossRef]
  3. M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
    [CrossRef]
  4. http://www.littleoptics.com/delay.pdf
  5. S. T. Chu, B. E. Little, V. Van, J. V. Hryniewicz, P. P. Absil, F. G. Johnson, D. Gill, O. King, F. Seiferth, M. Trakalo, and J. Shanton, "Compact full C-band tunable filters for 50 GHz channel spacing based on high order micro-ring resonators, in Proceedings of Optical Fiber Communications Conference, PDP9 (Los Angeles, CA 2004).
  6. Y. A. Vlasov and S. J. McNab, "Losses in single-mode silicon-on-insulator strip waveguides and bends," Opt. Express 12, 1622-1631 (2004),
    [CrossRef] [PubMed]
  7. V. R. Almeida and M. Lipson, "Optical bistability on a silicon chip," Opt. Lett. 29, 2387-2389 (2004).
    [CrossRef] [PubMed]
  8. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, "Optical bistability and pulsating behavior in Silicon-On-Insulator ring resonator structures," Opt. Express 13, 9623-9628 (2005).
    [CrossRef] [PubMed]
  9. A. Yariv, "Universal relations for coupling of optical power between micro resonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000).
    [CrossRef]
  10. J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, "Ultrahigh-quality-factor silicon-on-insulator microring resonator," Opt. Lett. 29, 2861-2863 (2004).
    [CrossRef]
  11. T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
    [CrossRef]
  12. J. Scheuer, G. T. Paloczi, J. Poon, and A. Yariv, "Coupled resonator optical waveguides - toward slowing & storage of light," Opt. Photonics News, 36-40 (2005).
    [CrossRef]
  13. J. B. Khurgin, "Expanding the bandwidth of slow-light photonic devices based on coupled resonators," Opt. Lett. 30, 513-515 (2005).
    [CrossRef] [PubMed]
  14. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
    [CrossRef] [PubMed]
  15. S. J. McNab, N. Moll, and Y. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express 11, 2927-2939 (2003).
    [CrossRef] [PubMed]
  16. E. Dulkeith, F. Xia, L. Sekaric, and Y. A. Vlasov, "Group index and dispersion properties in photonic wire waveguides on SOI substrate," to be published.
  17. J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, and R. W. Boyd, "Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators, " Opt. Lett. 29, 769-771 (2004).
    [CrossRef] [PubMed]
  18. D. Marcuse, Theory of dielectric waveguides (Academic, 1974), Chap. 4.
  19. A. Yariv, Optical electronics in modern communications (Oxford University Press, 1996), chap. 13.
  20. Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, "Experimental demonstration of guiding and confining light in nanometer-size low refractive-index material," Opt. Lett. 29, 1626-1628 (2004).
    [CrossRef] [PubMed]
  21. T. Baehr-Jones, M. Hochberg, C. Walker, and A. Scherer, "High-Q optical resonators in silicon-on-insulator-based slot waveguides," Appl. Phys. Lett. 86, 081101 (2005).
    [CrossRef]

2005 (5)

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

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

J. B. Khurgin, "Expanding the bandwidth of slow-light photonic devices based on coupled resonators," Opt. Lett. 30, 513-515 (2005).
[CrossRef] [PubMed]

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, "Optical bistability and pulsating behavior in Silicon-On-Insulator ring resonator structures," Opt. Express 13, 9623-9628 (2005).
[CrossRef] [PubMed]

2004 (5)

2003 (2)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

S. J. McNab, N. Moll, and Y. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express 11, 2927-2939 (2003).
[CrossRef] [PubMed]

2000 (1)

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

1998 (1)

R. Taylor and S. R. Forrest, "Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture," IEEE Photon. Technol. Lett. 10, 144-146 (1998).
[CrossRef]

Almeida, V. R.

Baehr-Jones, T.

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

Baets, R.

Bogaerts, W.

Bolivar, P. H.

Boyd, R. W.

Cappuzzo, M. A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Chen, E.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Dumon, P.

Forrest, S. R.

R. Taylor and S. R. Forrest, "Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture," IEEE Photon. Technol. Lett. 10, 144-146 (1998).
[CrossRef]

Fukada, H.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Gasparyan, A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Gomez, L. T.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Grange, J. L.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Griffin, A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Heebner, J. E.

Henschel, W.

Hochberg, M.

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

Itabashi, S.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Kasper, A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Khurgin, J. B.

Kippenberg, T. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Kurz, H.

Laskowski, E. J.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Lepeshkin, N. N.

Lipson, M.

Madsen, C. K.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

McNab, S. J.

Moll, N.

Morita, H.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Morthier, G.

Niehusmann, J.

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Panepucci, R. R.

Patel, S. S.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Priem, G.

Rasras, M. S.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Scherer, A.

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

Schweinsberg, A.

Shoji, T.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Takahashi, J.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Takahashi, M.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Tamechika, E.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Taylor, R.

R. Taylor and S. R. Forrest, "Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture," IEEE Photon. Technol. Lett. 10, 144-146 (1998).
[CrossRef]

Tsuchizawa, T.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Van Thourhout, D.

Vlasov, Y. A.

Vörckel, A.

Wahlbrink, T.

Walker, C.

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

Watanabe, T.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Wicks, G. W.

Wong-Foy, A.

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Xu, Q.

Yamada, K.

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

Yariv, A.

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

Appl. Phys. Lett. (1)

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

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]

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

T. Tsuchizawa, K. Yamada, H. Fukada, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. Morita, "Microphotonics devices based on silicon microfabrication technology," IEEE J. Sel. Topics Quantum Electron. 11, 232-240 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

R. Taylor and S. R. Forrest, "Steering of an optically driven true-time delay phased-array antenna based on a broad-band coherent WDM architecture," IEEE Photon. Technol. Lett. 10, 144-146 (1998).
[CrossRef]

M. S. Rasras, C. K. Madsen, M. A. Cappuzzo, E. Chen, L. T. Gomez, E. J. Laskowski, A. Griffin, A. Wong-Foy, A. Gasparyan, A. Kasper, J. L. Grange, and S. S. Patel, "Integrated resonance-enhanced variable optical delay lines," IEEE Photon. Technol. Lett. 17, 834-836 (2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, "Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics," Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

Other (7)

E. Dulkeith, F. Xia, L. Sekaric, and Y. A. Vlasov, "Group index and dispersion properties in photonic wire waveguides on SOI substrate," to be published.

D. Marcuse, Theory of dielectric waveguides (Academic, 1974), Chap. 4.

A. Yariv, Optical electronics in modern communications (Oxford University Press, 1996), chap. 13.

http://www.littleoptics.com/delay.pdf

S. T. Chu, B. E. Little, V. Van, J. V. Hryniewicz, P. P. Absil, F. G. Johnson, D. Gill, O. King, F. Seiferth, M. Trakalo, and J. Shanton, "Compact full C-band tunable filters for 50 GHz channel spacing based on high order micro-ring resonators, in Proceedings of Optical Fiber Communications Conference, PDP9 (Los Angeles, CA 2004).

J. Scheuer, G. T. Paloczi, J. Poon, and A. Yariv, "Coupled resonator optical waveguides - toward slowing & storage of light," Opt. Photonics News, 36-40 (2005).
[CrossRef]

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, 1998).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic view of a racetrack resonator based add/drop (A/D) filter.

Fig. 2.
Fig. 2.

Scanning electron micrograph of an A/D filter

Fig. 3.
Fig. 3.

The transmission spectrum of a racetrack based A/D filter with an air gap width of (103±5) nm

Fig. 4.
Fig. 4.

Transmission spectrum of an A/D filter with an air gap width of 0 nm

Fig. 5.
Fig. 5.

A broad band transmission spectrum of the drop port of an A/D filter with an air gap width of (103±5) nm

Fig. 6.
Fig. 6.

r2a as a function of the air gap width, d

Fig. 7.
Fig. 7.

(r2 + t2) a as a function of the air gap width, d

Fig. 8.
Fig. 8.

A SEM micrograph of a Mach-Zehnder interferometer (MZI) with one arm consisting of a racetrack resonator-based APF.

Fig. 9.
Fig. 9.

Transmission spectrum of a MZI with one arm consisting of a racetrack based all pass filter (APF) with an air gap width of (103±5) nm

Fig. 10.
Fig. 10.

The loss factor, a , deduced from measuring two sets of complementary devices (blue dots) and estimated from the propagation loss (red dashed line).

Fig. 11.
Fig. 11.

A SEM micrograph of a racetrack resonator coupling region with an air gap width of (103±5) nm

Tables (1)

Tables Icon

Table 1. Normalized eigen-mode profiles at cross sections A and B of a racetrack resonator coupling region with the air gap width of (103±5) nm

Equations (21)

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

[ E 4 E 2 ] = [ r it it r ] e jk L C [ E 3 E 1 ]
[ E 8 E 6 ] = [ r it it r ] e jk L C [ E 7 E 5 ]
L C = L s + 2 R arccos [ 1 d C d 2 R + w ]
E 5 = a e jk L P 2 E 4
E 3 = a e jk L P 2 E 6
E 7 = 0
E 2 = r r ( r 2 + t 2 ) a e jk ( 2 L C + L P ) 1 r 2 a e jk ( 2 L C + L P ) e jk L C E 1
E 8 = t 2 a e jk ( L C + L P 2 ) 1 r 2 a e jk ( 2 L C + L P ) e jk L C E 1
k λ ( 2 L C + L P ) = k λ = λ 0 ( 2 L C + L P ) 2 π λ 0 2 n g ( λ λ 0 ) ( 2 L C + L P )
= 2 2 π λ 0 2 n g ( λ λ 0 ) ( 2 L C + L P )
( E 8 ) max ( E 8 ) min 2 = 1 + r 2 a 1 r 2 a 2
r 2 a = ( E 8 ) max ( E 8 ) min 2 1 ( E 8 ) max ( E 8 ) min 2 + 1
( E 2 ) max ( E 2 ) min 2 = 1 + ( r 2 + t 2 ) a 1 ( r 2 + t 2 ) a 2 1 r 2 a 1 + r 2 a 2
( r 2 + t 2 ) a = ( E 8 ) max ( E 8 ) min 2 ( E 2 ) max ( E 2 ) min 2 1 ( E 8 ) max ( E 8 ) min 2 ( E 2 ) max ( E 2 ) min 2 + 1
( 2 L C + L P ) dk d λ λ = λ 1 + λ 2 2 ( λ 2 λ 1 ) 2 π
n g λ = λ 1 + λ 2 2 ( λ 1 + λ 2 ) 2 4 ( 2 L C + L P ) ( λ 2 λ 1 )
[ E 6 E 4 ] = [ r it it r ] e jk L C [ E 5 E 3 ]
E 5 = a e jk ( L C + L P ) E 6
E 4 = r ( r 2 + r 2 ) a e jk ( L C + L P ) 1 ra e jk ( L C + L P ) e jk L C E 3
E 2 = 1 2 { 1 + r ( r 2 + r 2 ) a e jk ( L C + L P ) 1 ra e jk ( L C + L P ) } e jk L TOTAL E 1
( E 2 ) max ( E 2 ) min 2 = 1 + r + ( r 2 + t 2 ) a 1 + ra 1 + r ( r 2 + t 2 ) a 1 ra 2

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