Abstract

A numerical study of threshold gain and modal dispersion in integrated semiconductor laser optical frequency comb sources is presented. We consider an example device where one of the cleaved facets of the laser is replaced by a short Bragg grating section and show that as many as 16 modes can be selected at the first harmonic of the underlying Fabry–Perot cavity. An intracavity approach to limiting the grating-induced dispersion that can be implemented directly through the grating profile is demonstrated.

© 2011 Optical Society of America

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References

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  1. A. Alduino and M. Paniccia, Nat. Photonics 1, 153 (2007).
    [CrossRef]
  2. X. Yi, N. K. Fontaine, R. P. Scott, and S. J. B. Yoo, J. Lightwave Technol. 28, 2054 (2010).
    [CrossRef]
  3. A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
    [CrossRef]
  4. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
    [CrossRef] [PubMed]
  5. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
    [CrossRef]
  6. F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
    [CrossRef]
  7. Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
    [CrossRef]
  8. S. O’Brien, S. Osborne, D. Bitauld, N. Brandonisio, A. Amann, R. Phelan, B. Kelly, and J. O’Gorman, IEEE Trans. Microwave Theory Tech. 58, 3083 (2010).
    [CrossRef]
  9. D. Bitauld, S. Osborne, and S. O’Brien, Opt. Lett. 35, 2200(2010).
    [CrossRef] [PubMed]
  10. S. O’Brien, A. Amann, R. Fehse, S. Osborne, E. P. O’Reilly, and J. M. Rondinelli, J. Opt. Soc. Am. B 23, 1046 (2006).
    [CrossRef]
  11. G. Adolfsson, J. Bengtsson, and A. Larsson, J. Opt. Soc. Am. B 27, 118 (2010).
    [CrossRef]

2011 (1)

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

2010 (5)

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

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

G. Adolfsson, J. Bengtsson, and A. Larsson, J. Opt. Soc. Am. B 27, 118 (2010).
[CrossRef]

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

X. Yi, N. K. Fontaine, R. P. Scott, and S. J. B. Yoo, J. Lightwave Technol. 28, 2054 (2010).
[CrossRef]

2009 (1)

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
[CrossRef]

2007 (2)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

A. Alduino and M. Paniccia, Nat. Photonics 1, 153 (2007).
[CrossRef]

2006 (1)

2004 (1)

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Adolfsson, G.

Alduino, A.

A. Alduino and M. Paniccia, Nat. Photonics 1, 153 (2007).
[CrossRef]

Amann, A.

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

S. O’Brien, A. Amann, R. Fehse, S. Osborne, E. P. O’Reilly, and J. M. Rondinelli, J. Opt. Soc. Am. B 23, 1046 (2006).
[CrossRef]

Arcizet, O.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Bengtsson, J.

Besnard, P.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Bitauld, D.

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

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

Borgne, E.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Bramerie, L.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Brandonisio, N.

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

Del’Haye, P.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Delfyett, P. J.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
[CrossRef]

Fehse, R.

Fontaine, N. K.

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

Gaeta, A. L.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

Gee, S.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
[CrossRef]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

Holzwarth, R.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Ilchenko, V. S.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Kelly, B.

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

Kippenberg, T. J.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

LaRochelle, S.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Larsson, A.

Le, Q. T.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Lelarge, F.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

Lipson, M.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

M’Sallem, Y. B.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Maleki, L.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Matsko, A. B.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Mohageg, M.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Nguyen, Q.-T.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

O’Brien, S.

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

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

S. O’Brien, A. Amann, R. Fehse, S. Osborne, E. P. O’Reilly, and J. M. Rondinelli, J. Opt. Soc. Am. B 23, 1046 (2006).
[CrossRef]

O’Gorman, J.

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

O’Reilly, E. P.

Osborne, S.

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

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

S. O’Brien, A. Amann, R. Fehse, S. Osborne, E. P. O’Reilly, and J. M. Rondinelli, J. Opt. Soc. Am. B 23, 1046 (2006).
[CrossRef]

Ozharar, S.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
[CrossRef]

Paniccia, M.

A. Alduino and M. Paniccia, Nat. Photonics 1, 153 (2007).
[CrossRef]

Phelan, R.

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

Quinlan, F.

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
[CrossRef]

Rondinelli, J. M.

Rusch, L. A.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Savchenkov, A. A.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Schliesser, A.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Scott, R. P.

Shen, A.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Simon, J.-C.

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

Strekalov, D.

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

Wilken, T.

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Yi, X.

Yoo, S. J. B.

IEEE Photon. Technol. Lett. (1)

Y. B. M’Sallem, Q. T. Le, L. Bramerie, Q.-T. Nguyen, E. Borgne, P. Besnard, A. Shen, F. Lelarge, S. LaRochelle, L. A. Rusch, and J.-C. Simon, IEEE Photon. Technol. Lett. 23, 453 (2011).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

J. Lightwave Technol. (1)

J. Opt. A (1)

F. Quinlan, S. Ozharar, S. Gee, and P. J. Delfyett, J. Opt. A 11, 103001 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Photonics (2)

A. Alduino and M. Paniccia, Nat. Photonics 1, 153 (2007).
[CrossRef]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

Nature (1)

P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki, Phys. Rev. Lett. 93, 243905(2004).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

One-dimensional optical potential that we consider. There are N index step features in the interior region of the cavity, and one external mirror is replaced by a short Bragg grating section comprising N MR index step features (indicated by a larger value of the index step). The cavity length is L c , the effective index is n 1 , and the index step features have effective index n 2 . The matrix T relates the left and right moving fields at the cavity mirrors as shown.

Fig. 2
Fig. 2

(a) Feature density function (solid line). Inset: Exact calculation of the modal threshold gain with N = 160 and an index step of 0.0025 . (b) Calculation of the modal threshold gain with an index step of 0.020 . Inset: Variation of the end mirror reflectivity in the case where N MR = 20 . (c) Calculated frequency shift. Primary modes are indicated with filled circles.

Fig. 3
Fig. 3

(a) Calculated modal threshold gain, (b) frequency shift, and (c) primary mode spacing for the cavity with N = 160 , Δ n = 0.02 , and a correction included to increase the threshold of the nonprimary modes at the comb edges. (d) Calculated modal threshold gain, (e) frequency shift, and (f) primary mode spacing for the same device but with corrections included to limit the dispersion in the primary mode spacing. In (b) and (e), primary modes are indicated with filled circles. In (c) and (f), the dashed curve is the mode spacing at the first harmonic of a plain FP device.

Equations (5)

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1 =     r 1 r 2 exp ( 2 i Σ θ i ) + i Δ n n j sin θ 2 j [ r 1 exp ( 2 i ϕ j ) + r 2 exp ( 2 i ϕ j + ) ] ( Δ n n ) 2 k > j = 1 N sin θ 2 j sin θ 2 k exp ( 2 i ϕ j k ) .
γ m ( 1 ) = 1 L c r 1 r 2 n k 1 cos Δ m π k A ( ϵ k 0 ) a k cos 2 Δ m π ϵ k 0 ,
δ m ( 1 ) = 1 2 r 1 r 2 n k 1 cos Δ m π k A + ( ϵ k 0 ) a k sin 2 Δ m π ϵ k 0 ,
δ m ( 2 ) = 1 2 r 1 r 2 γ m ( 1 ) L c n k 1 k ϵ k 0 A ( ϵ k 0 ) a k sin 2 Δ m π ϵ k 0 + 1 r 1 r 2 δ m ( 1 ) n k 1 k ϵ k 0 A + ( ϵ k 0 ) a k cos 2 Δ m π ϵ k 0 + 1 2 n k 2 k > k = 1 n k 1 exp ( ϵ k 0 ϵ k 0 ) γ m ( 0 ) L c × a k a k sin 2 Δ m π ( ϵ k 0 ϵ k 0 ) .
Δ δ m 2 C 1 2 ( Δ n n ) 2 n k 1 p = 1 1 2 n k 1 [ A p exp p n k 1 γ m ( 0 ) L c + A n k p exp ( 1 p n k 1 ) γ m ( 0 ) L c ] sin 2 Δ m π p n k 1 ,

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