Abstract

Miniaturized tunable lasers are important as light sources in optical networks and sensing systems. However, challenges remain in realizing a low-cost on-chip tunable laser with high tuning accuracy and wide tuning range. Here, an external-cavity tunable laser integrated by using a pair of coupled-ring resonators as a reflector onto a single silicon chip is designed and demonstrated. The silicon coupled-ring reflector provides simultaneous tuning of the central reflective wavelength and the free spectral range of the entire cavity to guarantee phase matching. The effective optical length of the coupled-ring reflector is controlled by the complementary thermo-optic and free-carrier dispersion effects, thus enabling high-resolution wavelength fine-tuning while maintaining wide wavelength tuning range. Experimentally, the tunable laser has high external-cavity efficiency (22%) and high wavelength tuning accuracy (2 pm) throughout the 41.6 nm tuning range. Potential applications of the external-cavity tunable laser include optical networks, optical sensing, and on-chip optoelectronic systems.

© 2015 Optical Society of America

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References

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  1. F. J. Duarte, Tunable Lasers Handbook (Academic, 1995), pp. 349–442.
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  8. H. Cai, A. Q. Liu, X. M. Zhang, J. Tamil, D. Y. Tang, J. Wu, and Q. X. Zhang, Appl. Phys. Lett. 92, 051113 (2008).
    [Crossref]
  9. V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
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  12. T. Creazzo, E. Marchena, S. Krasulick, P. K. L. Yu, D. V. Orden, J. Y. Spann, C. C. Blivin, L. He, H. Cai, J. M. Dallesasse, R. J. Stone, and A. Mizrahi, Opt. Express 21, 28048 (2013).
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  16. B. Liu, A. Shakouri, and J. E. Bowers, IEEE Photon. Tech. Lett. 14, 600 (2002).
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  17. S. Matsuo and T. Segawa, IEEE J. Sel. Top. Quantum Electron. 15, 545 (2009).
    [Crossref]
  18. I. Chremmos and N. Uzunoglu, IEEE Photon. Tech. Lett. 17, 2110 (2005).
    [Crossref]
  19. T. Komljenovic, M. Davenport, S. Srinivasan, J. Hulme, and J. E. Bowers, “Narrow linewidth tunable laser using coupled resonator mirrors,” in Proceedings of the Optical Fiber Communication Conference, Los Angeles, California, March22–26, 2015, paper W2A.52.

2013 (2)

2012 (2)

T. Mansuripur, S. Menzel, R. Blanchard, L. Diehl, C. Pflugl, Y. Huang, J. H. Ryou, R. D. Dupuis, M. Loncar, and F. Capasso, Opt. Express 20, 23339 (2012).
[Crossref]

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

2009 (1)

S. Matsuo and T. Segawa, IEEE J. Sel. Top. Quantum Electron. 15, 545 (2009).
[Crossref]

2008 (2)

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, Nat. Photonics 2, 180 (2008).
[Crossref]

H. Cai, A. Q. Liu, X. M. Zhang, J. Tamil, D. Y. Tang, J. Wu, and Q. X. Zhang, Appl. Phys. Lett. 92, 051113 (2008).
[Crossref]

2007 (1)

A. Q. Liu and X. M. Zhang, J Micromech. Microeng. 17, R1 (2007).
[Crossref]

2006 (3)

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Y. D. Jeong and Y. H. Won, Opt. Lett. 31, 2586 (2006).
[Crossref]

B. Jalali and S. Fathpour, J. Lightwave Technol. 24, 4600 (2006).
[Crossref]

2005 (2)

I. Chremmos and N. Uzunoglu, IEEE Photon. Tech. Lett. 17, 2110 (2005).
[Crossref]

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

2004 (2)

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[Crossref]

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

2002 (2)

B. Liu, A. Shakouri, and J. E. Bowers, IEEE Photon. Tech. Lett. 14, 600 (2002).
[Crossref]

H. Ma, A. K.-Y. Jen, and L. R. Dalton, Adv. Mater. 14, 1339 (2002).
[Crossref]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[Crossref]

Andersen, K. N.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Baets, R.

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

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[Crossref]

Barton, E.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Bienstman, P.

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

Binsma, H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Bjarklev, A.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Blanchard, R.

Blivin, C. C.

Bogaerts, W.

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

Borel, P. I.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Bowers, J. E.

B. Liu, A. Shakouri, and J. E. Bowers, IEEE Photon. Tech. Lett. 14, 600 (2002).
[Crossref]

T. Komljenovic, M. Davenport, S. Srinivasan, J. Hulme, and J. E. Bowers, “Narrow linewidth tunable laser using coupled resonator mirrors,” in Proceedings of the Optical Fiber Communication Conference, Los Angeles, California, March22–26, 2015, paper W2A.52.

Busico, G.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Cai, H.

M. Ren, J. Huang, H. Cai, J. M. Tsai, J. Zhou, Z. Liu, Z. Suo, and A. Q. Liu, ACS Nano 7, 1676 (2013).
[Crossref]

T. Creazzo, E. Marchena, S. Krasulick, P. K. L. Yu, D. V. Orden, J. Y. Spann, C. C. Blivin, L. He, H. Cai, J. M. Dallesasse, R. J. Stone, and A. Mizrahi, Opt. Express 21, 28048 (2013).
[Crossref]

H. Cai, A. Q. Liu, X. M. Zhang, J. Tamil, D. Y. Tang, J. Wu, and Q. X. Zhang, Appl. Phys. Lett. 92, 051113 (2008).
[Crossref]

Capasso, F.

Carter, A. C.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Chang-Hasnain, C. J.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, Nat. Photonics 2, 180 (2008).
[Crossref]

Chremmos, I.

I. Chremmos and N. Uzunoglu, IEEE Photon. Tech. Lett. 17, 2110 (2005).
[Crossref]

Claes, T.

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

Creazzo, T.

Dallesasse, J. M.

Dalton, L. R.

H. Ma, A. K.-Y. Jen, and L. R. Dalton, Adv. Mater. 14, 1339 (2002).
[Crossref]

Davenport, M.

T. Komljenovic, M. Davenport, S. Srinivasan, J. Hulme, and J. E. Bowers, “Narrow linewidth tunable laser using coupled resonator mirrors,” in Proceedings of the Optical Fiber Communication Conference, Los Angeles, California, March22–26, 2015, paper W2A.52.

De Heyn, P.

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

De Vos, K.

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

De Vries, T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Den Besten, J. H.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Diehl, L.

Dorren, H. J. S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Duarte, F. J.

F. J. Duarte, Tunable Lasers Handbook (Academic, 1995), pp. 349–442.

Duck, J. P.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Dumon, P.

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

Dupuis, R. D.

Fage-Pedersen, J.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Fathpour, S.

Frandsen, L. H.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Hansen, O.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

He, L.

Hill, M. T.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Huang, J.

M. Ren, J. Huang, H. Cai, J. M. Tsai, J. Zhou, Z. Liu, Z. Suo, and A. Q. Liu, ACS Nano 7, 1676 (2013).
[Crossref]

Huang, M. C. Y.

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, Nat. Photonics 2, 180 (2008).
[Crossref]

Huang, Y.

Hulme, J.

T. Komljenovic, M. Davenport, S. Srinivasan, J. Hulme, and J. E. Bowers, “Narrow linewidth tunable laser using coupled resonator mirrors,” in Proceedings of the Optical Fiber Communication Conference, Los Angeles, California, March22–26, 2015, paper W2A.52.

Jacobsen, R. S.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Jalali, B.

Jen, A. K.-Y.

H. Ma, A. K.-Y. Jen, and L. R. Dalton, Adv. Mater. 14, 1339 (2002).
[Crossref]

Jeong, Y. D.

Khoe, G.-D.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Komljenovic, T.

T. Komljenovic, M. Davenport, S. Srinivasan, J. Hulme, and J. E. Bowers, “Narrow linewidth tunable laser using coupled resonator mirrors,” in Proceedings of the Optical Fiber Communication Conference, Los Angeles, California, March22–26, 2015, paper W2A.52.

Krasulick, S.

Kristensen, M.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Lavrinenko, A. V.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Leijtens, X. J. M.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[Crossref]

Liu, A. Q.

M. Ren, J. Huang, H. Cai, J. M. Tsai, J. Zhou, Z. Liu, Z. Suo, and A. Q. Liu, ACS Nano 7, 1676 (2013).
[Crossref]

H. Cai, A. Q. Liu, X. M. Zhang, J. Tamil, D. Y. Tang, J. Wu, and Q. X. Zhang, Appl. Phys. Lett. 92, 051113 (2008).
[Crossref]

A. Q. Liu and X. M. Zhang, J Micromech. Microeng. 17, R1 (2007).
[Crossref]

Liu, B.

B. Liu, A. Shakouri, and J. E. Bowers, IEEE Photon. Tech. Lett. 14, 600 (2002).
[Crossref]

Liu, Z.

M. Ren, J. Huang, H. Cai, J. M. Tsai, J. Zhou, Z. Liu, Z. Suo, and A. Q. Liu, ACS Nano 7, 1676 (2013).
[Crossref]

Loncar, M.

Ma, H.

H. Ma, A. K.-Y. Jen, and L. R. Dalton, Adv. Mater. 14, 1339 (2002).
[Crossref]

Mansuripur, T.

Marchena, E.

Matsuo, S.

S. Matsuo and T. Segawa, IEEE J. Sel. Top. Quantum Electron. 15, 545 (2009).
[Crossref]

Menzel, S.

Mizrahi, A.

Moulin, G.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Oel, Y.-S.

M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijtens, J. H. Den Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, Nature 432, 206 (2004).
[Crossref]

Orden, D. V.

Ou, H.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[Crossref]

Peucheret, C.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, Nature 441, 199 (2006).
[Crossref]

Pflugl, C.

Ponnampalam, L.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Reid, D. C. J.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Ren, M.

M. Ren, J. Huang, H. Cai, J. M. Tsai, J. Zhou, Z. Liu, Z. Suo, and A. Q. Liu, ACS Nano 7, 1676 (2013).
[Crossref]

Robbins, D. J.

A. J. Ward, D. J. Robbins, G. Busico, E. Barton, L. Ponnampalam, J. P. Duck, N. D. Whitbread, P. J. Williams, D. C. J. Reid, A. C. Carter, and M. J. Wale, IEEE J. Sel. Top. Quantum Electron. 11, 149 (2005).
[Crossref]

Ryou, J. H.

Segawa, T.

S. Matsuo and T. Segawa, IEEE J. Sel. Top. Quantum Electron. 15, 545 (2009).
[Crossref]

Selvaraja, S. K.

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

Shakouri, A.

B. Liu, A. Shakouri, and J. E. Bowers, IEEE Photon. Tech. Lett. 14, 600 (2002).
[Crossref]

Smalbrugge, B.

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Opt. Express (2)

Opt. Lett. (1)

Other (2)

T. Komljenovic, M. Davenport, S. Srinivasan, J. Hulme, and J. E. Bowers, “Narrow linewidth tunable laser using coupled resonator mirrors,” in Proceedings of the Optical Fiber Communication Conference, Los Angeles, California, March22–26, 2015, paper W2A.52.

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Figures (6)

Fig. 1.
Fig. 1. Schematic illustration of (a) the integrated silicon–photonic tunable laser using a pair of coupled-ring resonators, with its lasing wavelength electrically tuned by the complementary free-carrier dispersion and thermo-optic effects and (b) the equivalent model of the external-cavity tunable laser with a coupled-ring reflector.
Fig. 2.
Fig. 2. SEM images of the integrated silicon–photonic tunable laser using a pair of coupled-ring resonators. The gain chip is bonded by flip-chip packaging processes.
Fig. 3.
Fig. 3. Transmission and reflection spectra of the coupled-ring reflector. The reflection spectrum is measured using a fiber optical circulator with a 1.2 dB insertion loss.
Fig. 4.
Fig. 4. Experimental results of (a) the demonstration of accurate wavelength tuning due to free-carrier injection, denoted by the improvement of the SMSR and the output power and (b) the response speed of the free-carrier dispersion effect; the rise and fall times are 0.78 and 0.72 ns, respectively.
Fig. 5.
Fig. 5. Mode maps of (a) the SMSR as a function of VT and VPIN applied on Ring 2 and (b) the lasing wavelength as a function of the thermal voltages applied on the two ring resonators.
Fig. 6.
Fig. 6. Optical spectra of a series of single-frequency lasing ranging from 1564.18 to 1522.54 nm; it obtains over 40 dB SMSR and maximal 2 mW output power due to the fine tuning of the free-carrier injection.

Equations (3)

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

reff(nring,λ)=rc(nring,λ)·t2exp(j4πλnbusLbus),
Δλtuneλ0=RΔR·pΔnsinring,
Δnsi=AVT2B(VPINV0).

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