Abstract

We present a design method and numerical results describing the construction of distributed feedback grating filters that support discrete combs of transmission resonances. These filter designs define open superstructure grating resonators with transmission channels that can be placed at predetermined frequencies, such as those defined by the wavelength division multiplexing grid or by a secondary frequency comb source. Focusing on a specific example with 40 GHz channel spacing, we optimize an active structure that defines three low-threshold lasing modes. How our design approach relates to filter synthesis techniques based on cascaded grating resonators is also discussed.

© 2014 Optical Society of America

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  1. B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).
  2. A. W. Fang, E. Lively, Y.-H. Kuo, D. Liang, and J. E. Bowers, Opt. Express 16, 4413 (2008).
    [CrossRef]
  3. O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
    [CrossRef]
  4. A. W. Fang, B. R. Koch, K.-G. Gan, H. Park, R. Jones, O. Cohen, M. Paniccia, D. J. Blumenthal, and J. E. Bowers, Opt. Express 16, 1393 (2008).
    [CrossRef]
  5. L. Agazzi, J. D. B. Bradley, M. Dijkstra, F. Ay, G. Roelkens, R. Baets, K. Worhoff, and M. Pollnau, Opt. Express 18, 27703 (2010).
    [CrossRef]
  6. H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).
  7. J. Kim and P. J. Delfyett, Electron. Lett. 40, 1182 (2004).
    [CrossRef]
  8. T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).
  9. D. V. Strekalov and N. Yu, Phys. Rev. A 79, 041805 (2009).
    [CrossRef]
  10. N. A. Cholan, M. H. Al-Mansoori, A. S. M. Noor, A. Ismail, and M. A. Mahdi, Opt. Express 21, 6131 (2013).
    [CrossRef]
  11. M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
    [CrossRef]
  12. A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).
  13. S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” arXiv:1305.3262 (2013).
  14. S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).
  15. D. Bitauld, S. Osborne, and S. O’Brien, Opt. Express 19, 13989 (2011).
    [CrossRef]
  16. D. Bitauld, S. Osborne, and S. O’Brien, Opt. Lett. 36, 2985 (2011).
    [CrossRef]
  17. D. Bitauld, S. Osborne, and S. O’Brien, Opt. Lett. 35, 2200 (2010).
    [CrossRef]
  18. X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).
  19. H.-C. Kim, K. Ikeda, and Y. Fainman, Opt. Lett. 32, 539 (2007).
    [CrossRef]
  20. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  21. H. A. Haus and Y. Lai, IEEE J. Quantum Electron. 28, 205 (1992).
    [CrossRef]
  22. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
    [CrossRef]

2013

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

N. A. Cholan, M. H. Al-Mansoori, A. S. M. Noor, A. Ismail, and M. A. Mahdi, Opt. Express 21, 6131 (2013).
[CrossRef]

2012

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

2011

2010

D. Bitauld, S. Osborne, and S. O’Brien, Opt. Lett. 35, 2200 (2010).
[CrossRef]

S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).

L. Agazzi, J. D. B. Bradley, M. Dijkstra, F. Ay, G. Roelkens, R. Baets, K. Worhoff, and M. Pollnau, Opt. Express 18, 27703 (2010).
[CrossRef]

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

2009

D. V. Strekalov and N. Yu, Phys. Rev. A 79, 041805 (2009).
[CrossRef]

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

2008

2007

2004

J. Kim and P. J. Delfyett, Electron. Lett. 40, 1182 (2004).
[CrossRef]

1997

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

1992

H. A. Haus and Y. Lai, IEEE J. Quantum Electron. 28, 205 (1992).
[CrossRef]

Agazzi, L.

Al-Mansoori, M. H.

Amann, A.

S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).

Ay, F.

Baets, R.

Barry, L. P.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Bitauld, D.

Blumenthal, D. J.

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

A. W. Fang, B. R. Koch, K.-G. Gan, H. Park, R. Jones, O. Cohen, M. Paniccia, D. J. Blumenthal, and J. E. Bowers, Opt. Express 16, 1393 (2008).
[CrossRef]

Bondarenko, O.

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

Bowers, J. E.

Bradley, J. D. B.

Byun, H.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Cholan, N. A.

Chrostowski, L.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).

Chu, S. T.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Cohen, O.

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

A. W. Fang, B. R. Koch, K.-G. Gan, H. Park, R. Jones, O. Cohen, M. Paniccia, D. J. Blumenthal, and J. E. Bowers, Opt. Express 16, 1393 (2008).
[CrossRef]

Del’Haye, P.

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” arXiv:1305.3262 (2013).

Delfyett, P. J.

J. Kim and P. J. Delfyett, Electron. Lett. 40, 1182 (2004).
[CrossRef]

Diddams, S. A.

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” arXiv:1305.3262 (2013).

Dijkstra, M.

Fainman, Y.

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

H.-C. Kim, K. Ikeda, and Y. Fainman, Opt. Lett. 32, 539 (2007).
[CrossRef]

Fang, A. W.

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Frolov, S.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Gaeta, A. L.

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

Gan, K.-G.

Gu, Q.

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

Habruseva, T.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Hanjani, A.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

H. A. Haus and Y. Lai, IEEE J. Quantum Electron. 28, 205 (1992).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

Hegarty, S. P.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Huyet, G.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Ikeda, K.

Ippen, E. P.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Ismail, A.

Jaeger, N. A. F.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).

Johnson, A. R.

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

Jones, R.

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

A. W. Fang, B. R. Koch, K.-G. Gan, H. Park, R. Jones, O. Cohen, M. Paniccia, D. J. Blumenthal, and J. E. Bowers, Opt. Express 16, 1393 (2008).
[CrossRef]

Kartner, F. X.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Kelly, B.

S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).

Kim, H.-C.

Kim, J.

J. Kim and P. J. Delfyett, Electron. Lett. 40, 1182 (2004).
[CrossRef]

Koch, B. R.

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

A. W. Fang, B. R. Koch, K.-G. Gan, H. Park, R. Jones, O. Cohen, M. Paniccia, D. J. Blumenthal, and J. E. Bowers, Opt. Express 16, 1393 (2008).
[CrossRef]

Kuo, Y.-H.

Lai, Y.

H. A. Haus and Y. Lai, IEEE J. Quantum Electron. 28, 205 (1992).
[CrossRef]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Lamont, M. R. E.

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

Levy, J. S.

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

Liang, D.

Lipson, M.

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

Little, B. E.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Lively, E.

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

A. W. Fang, E. Lively, Y.-H. Kuo, D. Liang, and J. E. Bowers, Opt. Express 16, 4413 (2008).
[CrossRef]

Mahdi, M. A.

Morandotti, R.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

Moss, D. J.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

Noor, A. S. M.

O’Brien, S.

O’Donoghue, S.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

O’Gorman, J.

S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).

Okawachi, Y.

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

Osborne, S.

Paniccia, M.

Papp, S. B.

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” arXiv:1305.3262 (2013).

Park, H.

Park, Y.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

Pasquazi, A.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

Peccianti, M.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

Phelan, R.

S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).

Pollnau, M.

Pudo, D.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Rachinskii, D.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Rebrova, N.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Reid, D. A.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

Roelkens, G.

Shi, W.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).

Shmulovich, J.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

Simic, A.

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

Slutsky, B.

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

Strekalov, D. V.

D. V. Strekalov and N. Yu, Phys. Rev. A 79, 041805 (2009).
[CrossRef]

Vafaei, R.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).

Wang, X.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).

Worhoff, K.

Yu, N.

D. V. Strekalov and N. Yu, Phys. Rev. A 79, 041805 (2009).
[CrossRef]

Appl. Phys. Lett.

O. Bondarenko, Q. Gu, A. Simic, B. Slutsky, and Y. Fainman, Appl. Phys. Lett. 103, 043105 (2013).
[CrossRef]

Electron. Lett.

J. Kim and P. J. Delfyett, Electron. Lett. 40, 1182 (2004).
[CrossRef]

IEEE J. Quantum Electron.

H. A. Haus and Y. Lai, IEEE J. Quantum Electron. 28, 205 (1992).
[CrossRef]

IEEE Photon. Technol. Lett.

H. Byun, D. Pudo, S. Frolov, A. Hanjani, J. Shmulovich, E. P. Ippen, and F. X. Kartner, IEEE Photon. Technol. Lett. 21, 3763 (2009).

IEEE Photonics Technol. Lett.

T. Habruseva, S. O’Donoghue, N. Rebrova, D. A. Reid, L. P. Barry, D. Rachinskii, G. Huyet, and S. P. Hegarty, IEEE Photonics Technol. Lett. 22, 359 (2010).

IEEE Trans. Microwave Theory Tech.

S. O’Brien, S. Osborne, D. Bitauld, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).

J. Lightwave Technol.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, J. Lightwave Technol. 15, 998 (1997).
[CrossRef]

Lasers Photonics Rev.

B. R. Koch, A. W. Fang, E. Lively, R. Jones, O. Cohen, D. J. Blumenthal, and J. E. Bowers, Lasers Photonics Rev. 3, 355 (2009).

Nat. Commun.

M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, Nat. Commun. 3, 765 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Photon. Technol. Lett.

X. Wang, W. Shi, R. Vafaei, N. A. F. Jaeger, and L. Chrostowski, Photon. Technol. Lett. 23, 290 (2011).

Phys. Rev. A

D. V. Strekalov and N. Yu, Phys. Rev. A 79, 041805 (2009).
[CrossRef]

Other

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

A. R. Johnson, Y. Okawachi, M. R. E. Lamont, J. S. Levy, M. Lipson, and A. L. Gaeta, “Microresonator-based comb generation without an external laser source,” arXiv:1305.1901 (2013).

S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” arXiv:1305.3262 (2013).

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Figures (3)

Fig. 1.
Fig. 1.

One-dimensional optical potential that we consider. There are N index steps in the interior of the cavity. The cavity effective index is n1, and the index steps have effective index n2. The matrix T relates the left and right moving electric fields at the cavity boundaries as shown. The period of a first-order grating section is Λ, and there is a half-wavelength subcavity at the device center as shown.

Fig. 2.
Fig. 2.

(a) Effective coupling constant of the 40 GHz spaced filter. (b) Calculations of the modal threshold gain with Δn=0.020 and N=1280. First iteration (open circles), 40 GHz spaced iteration (filled circles). Inset: mode indices, m0, and device lengths, Lc. (c) Feature spacings for the 40 GHz filter. (d) Transmission of the passive 40 GHz filter.

Fig. 3.
Fig. 3.

(a) Effective coupling constant. Vertical lines indicate the subdivision of the cavity used for temporal coupled-mode theory (TCMT) calculations. (b) Feature spacings. (c) Calculation of the modal threshold gain with N=1280 and Δn=0.020 (filled circles) and Δn=0.020+0.00012i (open circles). Inset: mode index m0. (d), (e) Intensity field patterns of the resonant lasing modes with Δn=0.020. (f) Transmission of the lossless filter calculated using the transmission matrix (solid line) and TCMT (dashed line).

Equations (4)

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

γm(1)=1Lcr1r2cosΔmπjA(ϵj)sjcos2Δmπϵj.
0.50.5A(ϵ)sin(2m0πϵ)jδ(ϵϵj)cos2Δmπϵdϵ,
κ(q)1q2+r2(q1)2κ(q=1),
a1=2τ1eiΔω+1τ1e+K122iΔω+K232iΔω+1τ3+e.

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