Abstract

We have integrated lithographically patterned VO2 thin films grown by pulsed laser deposition with silicon-on-insulator photonic waveguides to demonstrate a compact in-line absorption modulator for use in photonic circuits. Using single-mode waveguides at λ = 1550 nm, we show optical modulation of the guided transverse-electric mode of more than 6.5 dB with 2 dB insertion loss over a 2-µm active device length. Loss is determined for devices fabricated on waveguide ring resonators by measuring the resonator spectral response, and a sharp decrease in resonator quality factor is observed above 70 °C, consistent with switching of VO2 to its metallic phase. A computational study of device geometry is also presented, and we show that it is possible to more than double the modulation depth with modified device structures.

© 2010 OSA

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  1. F. J. Morin, “Oxides which show a metal-to-insulator transition at the Neel temperature,” Phys. Rev. Lett. 3(1), 34–36 (1959).
    [CrossRef]
  2. J. B. Goodenough, “The two components of crystallographic transition in VO2,” J. Solid State Chem. 3(4), 490–500 (1971).
    [CrossRef]
  3. A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
    [CrossRef] [PubMed]
  4. H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
    [CrossRef]
  5. B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
    [CrossRef]
  6. C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008).
    [CrossRef]
  7. S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
    [CrossRef]
  8. L. Jiang and W. N. Carr, “Design, fabrication and testing of a micromachined thermo-optical light modulator based on a vanadium dioxide array,” J. Micromech. Microeng. 14(7), 833–840 (2004).
    [CrossRef]
  9. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
    [CrossRef] [PubMed]
  10. A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
    [CrossRef] [PubMed]
  11. E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
    [CrossRef]
  12. T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
    [CrossRef]
  13. G. Gopalakrishnan, D. Ruzmetov, and S. Ramanathan, “On the triggering mechanism for the metal-insulator transition in thin film VO2 devices: electric field versus thermal effects,” J. Mater. Sci. 44(19), 5345–5353 (2009).
    [CrossRef]
  14. G. Stefanovich, A. Pergament, and D. Stefanovich, “Electrical switching and Mott transition in VO2,” J. Phys. Condens. Matter 12(41), 8837–8845 (2000).
    [CrossRef]
  15. M. A. Webster, R. M. Pafchek, A. Mitchell, and T. L. Koch, “Width dependence of inherent TM-mode lateral leakage loss in silicon-on-insulator ridge waveguides,” IEEE Photon. Technol. Lett. 19(6), 429–431 (2007).
    [CrossRef]
  16. R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
    [CrossRef]
  17. J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
    [CrossRef]
  18. L. J. van der Pauw, “A new method of measuring the resistivity and Hall coefficients on lamellae of arbitrary shape,” Philips Tech. Rev. 20, 220–224 (1958).
  19. K. Preston and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009).
    [CrossRef] [PubMed]
  20. R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
    [CrossRef]
  21. M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1522 (2005).
    [CrossRef] [PubMed]
  22. J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
    [CrossRef]
  23. H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
    [CrossRef]

2009 (3)

G. Gopalakrishnan, D. Ruzmetov, and S. Ramanathan, “On the triggering mechanism for the metal-insulator transition in thin film VO2 devices: electric field versus thermal effects,” J. Mater. Sci. 44(19), 5345–5353 (2009).
[CrossRef]

R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
[CrossRef]

K. Preston and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009).
[CrossRef] [PubMed]

2008 (1)

C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008).
[CrossRef]

2007 (1)

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

2006 (1)

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

2005 (3)

B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
[CrossRef]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

M. Borselli, T. J. Johnson, and O. Painter, “Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment,” Opt. Express 13(5), 1515–1522 (2005).
[CrossRef] [PubMed]

2004 (3)

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
[CrossRef]

L. Jiang and W. N. Carr, “Design, fabrication and testing of a micromachined thermo-optical light modulator based on a vanadium dioxide array,” J. Micromech. Microeng. 14(7), 833–840 (2004).
[CrossRef]

2003 (1)

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

2001 (1)

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

2000 (2)

G. Stefanovich, A. Pergament, and D. Stefanovich, “Electrical switching and Mott transition in VO2,” J. Phys. Condens. Matter 12(41), 8837–8845 (2000).
[CrossRef]

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

1996 (2)

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

1994 (1)

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[CrossRef]

1987 (1)

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

1971 (1)

J. B. Goodenough, “The two components of crystallographic transition in VO2,” J. Solid State Chem. 3(4), 490–500 (1971).
[CrossRef]

1959 (1)

F. J. Morin, “Oxides which show a metal-to-insulator transition at the Neel temperature,” Phys. Rev. Lett. 3(1), 34–36 (1959).
[CrossRef]

1958 (1)

L. J. van der Pauw, “A new method of measuring the resistivity and Hall coefficients on lamellae of arbitrary shape,” Philips Tech. Rev. 20, 220–224 (1958).

Ahn, J. S.

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

Attanasio, D. V.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Atwater, H. A.

R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
[CrossRef]

Bennett, B. R.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Borselli, M.

Bossi, D. E.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Briggs, R. M.

R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
[CrossRef]

Carr, W. N.

L. Jiang and W. N. Carr, “Design, fabrication and testing of a micromachined thermo-optical light modulator based on a vanadium dioxide array,” J. Micromech. Microeng. 14(7), 833–840 (2004).
[CrossRef]

Cavalleri, A.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

Chae, B. G.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
[CrossRef]

Chen, M.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Chen, S.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Choi, H. S.

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

Cohen, O.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Donnelly, V. M.

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[CrossRef]

Feldman, L. C.

J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
[CrossRef]

Forget, P.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

Fritz, D. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Goodenough, J. B.

J. B. Goodenough, “The two components of crystallographic transition in VO2,” J. Solid State Chem. 3(4), 490–500 (1971).
[CrossRef]

Gopalakrishnan, G.

G. Gopalakrishnan, D. Ruzmetov, and S. Ramanathan, “On the triggering mechanism for the metal-insulator transition in thin film VO2 devices: electric field versus thermal effects,” J. Mater. Sci. 44(19), 5345–5353 (2009).
[CrossRef]

Haglund, R. F.

J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
[CrossRef]

Hallemeier, P. F.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Han, K. J.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Ido, T.

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

Inoue, H.

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

Jiang, L.

L. Jiang and W. N. Carr, “Design, fabrication and testing of a micromachined thermo-optical light modulator based on a vanadium dioxide array,” J. Micromech. Microeng. 14(7), 833–840 (2004).
[CrossRef]

Johnson, T. J.

Jones, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Jung, J. H.

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

Kang, K. Y.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
[CrossRef]

Ke, C.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Kieffer, J. C.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

Kim, B. J.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Kim, D. H.

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

Kim, H. T.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
[CrossRef]

Kissa, K. M.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Ko, C.

C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008).
[CrossRef]

Koch, T. L.

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

Koizumi, M.

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

Lafaw, D. A.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Lee, Y. W.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Lim, Y. S.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Lipson, M.

K. Preston and M. Lipson, “Slot waveguides with polycrystalline silicon for electrical injection,” Opt. Express 17(3), 1527–1534 (2009).
[CrossRef] [PubMed]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Liu, A.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Lopez, R.

J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
[CrossRef]

Ma, H.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Maack, D.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

McBrien, G. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

McCaulley, J. A.

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[CrossRef]

Mitchell, A.

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

Morin, F. J.

F. J. Morin, “Oxides which show a metal-to-insulator transition at the Neel temperature,” Phys. Rev. Lett. 3(1), 34–36 (1959).
[CrossRef]

Murphy, E. J.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Noh, T. W.

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

Pafchek, R. M.

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

Painter, O.

Paniccia, M.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Pergament, A.

G. Stefanovich, A. Pergament, and D. Stefanovich, “Electrical switching and Mott transition in VO2,” J. Phys. Condens. Matter 12(41), 8837–8845 (2000).
[CrossRef]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Preston, K.

Ráksi, F.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

Ramanathan, S.

G. Gopalakrishnan, D. Ruzmetov, and S. Ramanathan, “On the triggering mechanism for the metal-insulator transition in thin film VO2 devices: electric field versus thermal effects,” J. Mater. Sci. 44(19), 5345–5353 (2009).
[CrossRef]

C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008).
[CrossRef]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Ruzmetov, D.

G. Gopalakrishnan, D. Ruzmetov, and S. Ramanathan, “On the triggering mechanism for the metal-insulator transition in thin film VO2 devices: electric field versus thermal effects,” J. Mater. Sci. 44(19), 5345–5353 (2009).
[CrossRef]

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Sano, H.

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

Scherer, A.

R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
[CrossRef]

Schmidt, B.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Shearn, M.

R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
[CrossRef]

Siders, C. W.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

Soref, R. A.

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

Squier, J. A.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

Stefanovich, D.

G. Stefanovich, A. Pergament, and D. Stefanovich, “Electrical switching and Mott transition in VO2,” J. Phys. Condens. Matter 12(41), 8837–8845 (2000).
[CrossRef]

Stefanovich, G.

G. Stefanovich, A. Pergament, and D. Stefanovich, “Electrical switching and Mott transition in VO2,” J. Phys. Condens. Matter 12(41), 8837–8845 (2000).
[CrossRef]

Suh, J. Y.

J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
[CrossRef]

Suzuki, M.

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

Taha, I.

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[CrossRef]

Tanaka, S.

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

Tao, X.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Tóth, C.

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

van der Pauw, L. J.

L. J. van der Pauw, “A new method of measuring the resistivity and Hall coefficients on lamellae of arbitrary shape,” Philips Tech. Rev. 20, 220–224 (1958).

Vernon, M.

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[CrossRef]

Wang, H.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Webster, M. A.

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

Wooten, E. L.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Xu, Q.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

Yee, K. J.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Yi, X.

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Yi-Yan, A.

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

Youn, D. H.

B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
[CrossRef]

Yun, S. J.

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

C. Ko and S. Ramanathan, “Observation of electric field-assisted phase transition in thin film vanadium oxide in a metal-oxide-semiconductor device geometry,” Appl. Phys. Lett. 93(25), 252101 (2008).
[CrossRef]

R. M. Briggs, M. Shearn, A. Scherer, and H. A. Atwater, “Wafer-bonded single-crystal silicon slot waveguides and ring resonators,” Appl. Phys. Lett. 94(2), 021106 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. A. Soref and B. R. Bennett, “Electrooptical effects in silicon,” IEEE J. Quantum Electron. 23(1), 123–129 (1987).
[CrossRef]

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

E. L. Wooten, K. M. Kissa, A. Yi-Yan, E. J. Murphy, D. A. Lafaw, P. F. Hallemeier, D. Maack, D. V. Attanasio, D. J. Fritz, G. J. McBrien, and D. E. Bossi, “A review of lithium niobate modulators for fiber-optic communications system,” IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

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

J. Appl. Phys. (1)

J. Y. Suh, R. Lopez, L. C. Feldman, and R. F. Haglund., “Semiconductor to metal phase transition in the nucleation and growth of VO2 nanoparticles and thin films,” J. Appl. Phys. 96(2), 1209–1213 (2004).
[CrossRef]

J. Lightwave Technol. (1)

T. Ido, S. Tanaka, M. Suzuki, M. Koizumi, H. Sano, and H. Inoue, “Ultra-high-speed multiple-quantum-well electro-absorption optical modulators with integrated waveguides,” J. Lightwave Technol. 14(9), 2026–2034 (1996).
[CrossRef]

J. Mater. Sci. (1)

G. Gopalakrishnan, D. Ruzmetov, and S. Ramanathan, “On the triggering mechanism for the metal-insulator transition in thin film VO2 devices: electric field versus thermal effects,” J. Mater. Sci. 44(19), 5345–5353 (2009).
[CrossRef]

J. Micromech. Microeng. (1)

L. Jiang and W. N. Carr, “Design, fabrication and testing of a micromachined thermo-optical light modulator based on a vanadium dioxide array,” J. Micromech. Microeng. 14(7), 833–840 (2004).
[CrossRef]

J. Phys. Condens. Matter (1)

G. Stefanovich, A. Pergament, and D. Stefanovich, “Electrical switching and Mott transition in VO2,” J. Phys. Condens. Matter 12(41), 8837–8845 (2000).
[CrossRef]

J. Solid State Chem. (1)

J. B. Goodenough, “The two components of crystallographic transition in VO2,” J. Solid State Chem. 3(4), 490–500 (1971).
[CrossRef]

Nature (2)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Quantum Electron. (1)

S. Chen, X. Yi, H. Ma, H. Wang, X. Tao, M. Chen, and C. Ke, “A novel structural VO2 micro-optical switch,” Opt. Quantum Electron. 35(15), 1351–1355 (2003).
[CrossRef]

Philips Tech. Rev. (1)

L. J. van der Pauw, “A new method of measuring the resistivity and Hall coefficients on lamellae of arbitrary shape,” Philips Tech. Rev. 20, 220–224 (1958).

Phys. Rev. B (2)

J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, “Temperature dependence of the near-infrared refractive index of silicon gallium arsenide, and indium phosphide,” Phys. Rev. B 49(11), 7408–7417 (1994).
[CrossRef]

H. S. Choi, J. S. Ahn, J. H. Jung, T. W. Noh, and D. H. Kim, “Mid-infrared properties of a VO2 film near the metal-insulator transition,” Phys. Rev. B 54(7), 4621–4628 (1996).
[CrossRef]

Phys. Rev. Lett. (3)

F. J. Morin, “Oxides which show a metal-to-insulator transition at the Neel temperature,” Phys. Rev. Lett. 3(1), 34–36 (1959).
[CrossRef]

A. Cavalleri, C. Tóth, C. W. Siders, J. A. Squier, F. Ráksi, P. Forget, and J. C. Kieffer, “Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phase transition,” Phys. Rev. Lett. 87(23), 237401 (2001).
[CrossRef] [PubMed]

H. T. Kim, Y. W. Lee, B. J. Kim, B. G. Chae, S. J. Yun, K. Y. Kang, K. J. Han, K. J. Yee, and Y. S. Lim, “Monoclinic and correlated metal phase in VO(2) as evidence of the Mott transition: coherent phonon analysis,” Phys. Rev. Lett. 97(26), 266401 (2006).
[CrossRef]

Physica B (1)

B. G. Chae, H. T. Kim, D. H. Youn, and K. Y. Kang, “Abrupt metal–insulator transition observed in VO2 thin films induced by a switching voltage pulse,” Physica B 369(1-4), 76–80 (2005).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic of the VO2 modulator test bed, with a grating-coupled through-port waveguide and an evanescently coupled ring resonator. The scanning electron micrographs show the 1-µm coupling gap between the through-port waveguide and the 400 µm-diameter ring resonator (left) and a lithographically defined 2 µm-long polycrystalline VO2 tab (right).

Fig. 2
Fig. 2

(a) Real, n, and imaginary, k, parts of the index of refraction of a 65 nm-thick VO2 film on Si measured by multiple-angle spectroscopic ellipsometry across the near infrared spectrum. (b) X-ray diffraction spectrum of the same polycrystalline VO2 film (scanning electron micrograph shown in inset). (c) Electrical resistivity versus temperature of a 65 nm-thick VO2 film on SOI measured using the four-point van der Pauw method.

Fig. 3
Fig. 3

(a) TE-polarized through-port transmission spectra of a critically coupled Si waveguide ring resonator without a VO2 tab. The resonator Q is unchanged for substrate temperatures between 30 °C and 100 °C; however, grating coupling efficiency is impacted by the thermo-optic effect in Si, resulting in lower off-resonance transmission. (b) Through-port transmission spectra for increasing substrate temperature with the same resonator geometry, but with a 2 µm-long VO2 tab. Modes of the same azimuthal order are indicated with diamond-shaped markers, revealing a thermally induced redshift of 0.08 nm/°C. (c) Round-trip resonator loss near 1550 nm due to the VO2 tab. Upon cooling, thermal hysteresis of over 30 °C is observed.

Fig. 4
Fig. 4

(a) Power profiles of the fundamental TE mode in the VO2-clad Si waveguide at 1550 nm, plotted over the same range of power. The imaginary part of the effective index, n eff, indicates the substantially enhanced modal absorption in metallic-phase VO2. (b) Schematic of the structure used in FDTD calculations, shown with the experimentally realized dimensions.

Fig. 5
Fig. 5

(a) Simulated TE-mode loss due to a 65 nm-thick VO2 tab as a function of tab length calculated using the three-dimensional FDTD method. The predicted losses for a 2-µm tab are within 5% of the measured values. (b) Simulated loss per unit length as a function of VO2 film thickness. The dashed black curve is the modulator figure of merit, defined as the ratio of the modulation depth to the insertion loss.

Equations (3)

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

T ( λ ) = ( a r ) 2 + ( 2 π n g L ) 2 r a ( λ λ 0 ) 2 λ 0 4 ( 1 r a ) 2 + ( 2 π n g L ) 2 r a ( λ λ 0 ) 2 λ 0 4 ,
1 Q ref = 1 Q int + 1 Q coup .
1 Q load = 1 Q int + 1 Q coup + 1 Q tab = 1 Q ref + 1 Q tab .

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