Abstract

We optimize the threshold gain for cylindrical composite (semiconductor–dielectric–metal) waveguides (WGs) with various metal claddings. We show that the optimal dielectric width is invariant with respect to the imaginary part of the permittivity of the metal, εM, and weakly dependent on the real part, εM. To explain this behavior, we compare optimal geometries of WGs with different semiconductor permittivities, εG. Results from these comparisons indicate that the optimal effective index parallels the optimal threshold gain in its relation to εM. We use our results to heuristically propose an analytical expression for the optimal threshold gain that approximates the numerical solution to within a factor of two over the range of explored εG. Finally, we use data from our optimizations to obtain approximate analytical expressions for the optimal dielectric width and threshold gain as functions of the total WG radius.

© 2013 Optical Society of America

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  1. M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
    [CrossRef]
  2. Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication
  3. J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
    [CrossRef]
  4. K. Ding, M. T. Hill, Z. C. Liu, P. J. van Veldhoven, and C. Z. Ning, Opt. Express 21, 4728 (2013).
    [CrossRef]
  5. A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, Opt. Lett. 33, 1261 (2008).
    [CrossRef]
  6. M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
    [CrossRef]
  7. Q. Ding, A. Mizrahi, Y. Fainman, and V. Lomakin, Opt. Lett. 36, 1812 (2011).
    [CrossRef]
  8. K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
    [CrossRef]
  9. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
    [CrossRef]
  10. M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
    [CrossRef]
  11. H. A. Auda and D. Kajfez, in Dielectric Resonators, D. Kajfez and P. Guillon, eds., (Artech House, 1986).
  12. A. Saleh and J. Dionne, Phys. Rev. B 85, 045407 (2012).
    [CrossRef]
  13. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  14. A. D. Rakic, Appl. Opt. 34, 4755 (1995).
    [CrossRef]
  15. J. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).
    [CrossRef]
  16. K. Kato and N. Umemura, Opt. Lett. 36, 746 (2011).
    [CrossRef]

2013

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

K. Ding, M. T. Hill, Z. C. Liu, P. J. van Veldhoven, and C. Z. Ning, Opt. Express 21, 4728 (2013).
[CrossRef]

2012

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

A. Saleh and J. Dionne, Phys. Rev. B 85, 045407 (2012).
[CrossRef]

2011

2010

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

2009

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

2008

2007

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

1995

1976

J. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Auda, H. A.

H. A. Auda and D. Kajfez, in Dielectric Resonators, D. Kajfez and P. Guillon, eds., (Artech House, 1986).

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

Bondarenko, O.

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

de Vries, T.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

de Waardt, H.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Ding, K.

K. Ding, M. T. Hill, Z. C. Liu, P. J. van Veldhoven, and C. Z. Ning, Opt. Express 21, 4728 (2013).
[CrossRef]

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Ding, Q.

Dionne, J.

A. Saleh and J. Dionne, Phys. Rev. B 85, 045407 (2012).
[CrossRef]

Eijkemans, T. J.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Fainman, Y.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

Q. Ding, A. Mizrahi, Y. Fainman, and V. Lomakin, Opt. Lett. 36, 1812 (2011).
[CrossRef]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, Opt. Lett. 33, 1261 (2008).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Feng, L.

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, Opt. Lett. 33, 1261 (2008).
[CrossRef]

Geluk, E. J.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Gladden, C.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

Gu, Q.

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Hill, M. T.

K. Ding, M. T. Hill, Z. C. Liu, P. J. van Veldhoven, and C. Z. Ning, Opt. Express 21, 4728 (2013).
[CrossRef]

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Kajfez, D.

H. A. Auda and D. Kajfez, in Dielectric Resonators, D. Kajfez and P. Guillon, eds., (Artech House, 1986).

Kato, K.

Katz, M.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Khajavikhan, M.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

Kwon, S.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Lee, J. H.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Lee, Y.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Liu, Z.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Liu, Z. C.

Lomakin, V.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

Q. Ding, A. Mizrahi, Y. Fainman, and V. Lomakin, Opt. Lett. 36, 1812 (2011).
[CrossRef]

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, Opt. Lett. 33, 1261 (2008).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Ma, R.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

McKay, J.

J. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).
[CrossRef]

Mizrahi, A.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

Q. Ding, A. Mizrahi, Y. Fainman, and V. Lomakin, Opt. Lett. 36, 1812 (2011).
[CrossRef]

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, Opt. Lett. 33, 1261 (2008).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Nezhad, M.

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

Nezhad, M. P.

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

A. Mizrahi, V. Lomakin, B. A. Slutsky, M. P. Nezhad, L. Feng, and Y. Fainman, Opt. Lett. 33, 1261 (2008).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Ning, C. Z.

K. Ding, M. T. Hill, Z. C. Liu, P. J. van Veldhoven, and C. Z. Ning, Opt. Express 21, 4728 (2013).
[CrossRef]

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Notzel, R.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Oei, Y.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

Rakic, A. D.

Rayne, J. A.

J. McKay and J. A. Rayne, Phys. Rev. B 13, 673 (1976).
[CrossRef]

Saleh, A.

A. Saleh and J. Dionne, Phys. Rev. B 85, 045407 (2012).
[CrossRef]

Simic, A.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Slutsky, B.

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, Nature 482, 204 (2012).
[CrossRef]

J. H. Lee, M. Khajavikhan, A. Simic, Q. Gu, O. Bondarenko, B. Slutsky, M. P. Nezhad, and Y. Fainman, Opt. Express 19, 21524 (2011).
[CrossRef]

M. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, Nat. Photonics 4, 395 (2010).
[CrossRef]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Slutsky, B. A.

Smalbrugge, B.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Smalley, J. S. T.

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Sub-wavelength semiconductor lasers for dense chip-scale integration,” Adv. Photonics Res., submitted for publication

Smit, M. K.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

Turkiewicz, J. P.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Umemura, N.

van Otten, F. W. M.

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

van Veldhoven, P. J.

K. Ding, M. T. Hill, Z. C. Liu, P. J. van Veldhoven, and C. Z. Ning, Opt. Express 21, 4728 (2013).
[CrossRef]

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

M. T. Hill, Y. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. Kwon, Y. Lee, R. Notzel, and M. K. Smit, Nat. Photonics 1, 589 (2007).
[CrossRef]

Yin, L.

K. Ding, L. Yin, M. T. Hill, Z. Liu, P. J. van Veldhoven, and C. Z. Ning, Appl. Phys. Lett. 102, 041110 (2013).
[CrossRef]

Zentgraf, T.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, Nature 461, 629 (2009).
[CrossRef]

Zhang, X.

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Figures (7)

Fig. 1.
Fig. 1.

Re(sqrt(εR)) and |E| of the TE01 mode as functions of radial distance in an optimized composite WG (CWG) at λ0=1.55μm. The permittivities are εG=11.56+j8.65e4, εD=2.16, and εM=130j3.0, respectively.

Fig. 2.
Fig. 2.

Threshold gain εGth as function of Rcore for two values of εG, with εM parameterized. Blue rectangles and red circles indicate (Rcore,εG)=(Rcore,opt,εGth,opt) for InGaAsP and GaS CWGs, respectively. εD, λ0, and Rtotal are fixed at 2.16, 1.55, and 0.45 μm, respectively.

Fig. 3.
Fig. 3.

neff as function of Rcore for two values of εG, with εM parameterized. Blue rectangles and red circles indicate (Rcore,neff)=(Rcore,opt,neff,opt) for InGaAsP and GaS CWGs, respectively. εD, λ0, and Rtotal are fixed at 2.16, 1.55, and 0.45 μm, respectively.

Fig. 4.
Fig. 4.

ΔD,opt and neff,opt as functions of εG. εM, εD, λ0, and Rtotal are fixed at, 130j3.0, 2.16, 1.55, and 0.45 μm, respectively.

Fig. 5.
Fig. 5.

εGth,opt as function of εG, (open squares) numerical solutions of Eq. (1) and (solid line) analytical approximation of Eq. (2). εM, εD, λ0, and Rtotal are fixed at 130j3.0, 2.16, 1.55, and 0.45 μm, respectively.

Fig. 6.
Fig. 6.

Error in Eq. (2), with εGth and neff substituted for εGth,opt and neff,opt, relative to the numerical solution of Eq. (1) as a function of Rcore with εG parameterized. εD, εM, λ0, and Rtotal are fixed at 2.16, 130j3.0, 1.55, and 0.45 μm, respectively.

Fig. 7.
Fig. 7.

ΔD,opt and log10(εGth,opt) as functions of Rtotal. The red (on red line from lower left to upper right) and blue (on blue line from upper left to lower right) solid triangles correspond to numerical solutions of Eq. (1), whereas the red and blue lines correspond to linear approximations for ΔD,opt and log10(εGth,opt), respectively. εM, εD, εG, and λ0 are fixed at 130j3.0, 2.16, 11.56, and 1.55 μm, respectively.

Equations (2)

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εGth=εMMetal|E|2dAGain|E|2dA=εMRtotal|E(ρ)|2ρdρ0Rcore|E(ρ)|2ρdρ,
εGth,optεM{εG/[εM(neffopt)2]}2.

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