Abstract

Large area photonic crystal cavities are devices of interest for photovoltaics, optoelectronics, and solid-state lighting. However, depending on their dimensions they pose a large computational challenge. Here, we use a local density approach to avoid direct simulation of the device. We capture the effect of both ideal and distorted photonic crystals in an effective mass and an effective potential. We use these to map the problem of calculating the electromagnetic field modes to solving a simple time-independent Schrödinger equation. We show that, in the case that the hole radius varies quadratically as a function of position, the eigenmodes of the photonic crystals can be described by the corresponding eigenmodes of the quantum harmonic oscillator with typical agreements well above 90%.

© 2015 Optical Society of America

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References

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).
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    [Crossref]
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    [Crossref]
  5. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
    [Crossref]
  6. T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
    [Crossref]
  7. A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  13. M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
    [Crossref]
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  15. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
    [Crossref]
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    [Crossref] [PubMed]
  19. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
    [Crossref]
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    [Crossref]
  21. D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, Inc., 2005).
  22. B. R. Johnson, “New numerical methods applied to solving the one-dimensional eigenvalue problem,” J. Chem. Phys. 67, 4086–4093 (1977).
    [Crossref]
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    [Crossref]

2014 (1)

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

2013 (1)

2012 (1)

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

2010 (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

2008 (3)

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

R. Engelen, D. Mori, T. Baba, and L. Kuipers, “Two regimes of slow-light losses revealed by adiabatic reduction of group velocity,” Phys. Rev. Lett. 101, 103901 (2008).
[Crossref] [PubMed]

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

2007 (1)

J. Vigneron and V. Lousse, “Theory of chirped photonic crystals,” Opt. Quantum Electron. 39, 377–385 (2007).
[Crossref]

2006 (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[Crossref]

2005 (3)

2003 (2)

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[Crossref] [PubMed]

A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystals,” Appl. Phys. Lett. 82, 3227–3229 (2003).
[Crossref]

2002 (1)

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

2001 (2)

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[Crossref] [PubMed]

1999 (1)

T. F. Krauss and R. M. D. L. Rue, “Photonic crystals in the optical regime past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[Crossref]

1997 (1)

V. Mandelshtam and H. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

1977 (1)

B. R. Johnson, “New numerical methods applied to solving the one-dimensional eigenvalue problem,” J. Chem. Phys. 67, 4086–4093 (1977).
[Crossref]

Aiello, A.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[Crossref]

Allard, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

Asano, T.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[Crossref]

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

R. Engelen, D. Mori, T. Baba, and L. Kuipers, “Two regimes of slow-light losses revealed by adiabatic reduction of group velocity,” Phys. Rev. Lett. 101, 103901 (2008).
[Crossref] [PubMed]

D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005).
[Crossref] [PubMed]

Beloglasov, V.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Bethge, J.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Bock, M.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Botten, L. C.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Boyd, R. W.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Bruyant, A.

A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystals,” Appl. Phys. Lett. 82, 3227–3229 (2003).
[Crossref]

Burresi, M.

Charbonneau-Lefort, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

de Guise, H.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

de la Hoz, P.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

de Sterke, C. M.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Dossou, K. B.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Engelen, R.

R. Engelen, D. Mori, T. Baba, and L. Kuipers, “Two regimes of slow-light losses revealed by adiabatic reduction of group velocity,” Phys. Rev. Lett. 101, 103901 (2008).
[Crossref] [PubMed]

Erchak, A. A.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Fan, S.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Fischer, D.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Gal, M.

A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystals,” Appl. Phys. Lett. 82, 3227–3229 (2003).
[Crossref]

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, Inc., 2005).

Hall, H. E.

J. R. Hook and H. E. Hall, Solid State Physics (John Wiley and Sons, 1991).

Hook, J. R.

J. R. Hook and H. E. Hall, Solid State Physics (John Wiley and Sons, 1991).

Hradil, Z.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Hugonin, J.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Iliew, R.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Ippen, E. P.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[Crossref]

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

Joannopoulos, J.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Johnson, B. R.

B. R. Johnson, “New numerical methods applied to solving the one-dimensional eigenvalue problem,” J. Chem. Phys. 67, 4086–4093 (1977).
[Crossref]

Johnson, S.

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Karimi, E.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Knight, J.

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[Crossref] [PubMed]

Kolodziejski, L. A.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Krauss, T. F.

T. F. Krauss and R. M. D. L. Rue, “Photonic crystals in the optical regime past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[Crossref]

Kuipers, L.

R. Engelen, D. Mori, T. Baba, and L. Kuipers, “Two regimes of slow-light losses revealed by adiabatic reduction of group velocity,” Phys. Rev. Lett. 101, 103901 (2008).
[Crossref] [PubMed]

Lalanne, P.

Lecamp, G.

Lérondel, G.

A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystals,” Appl. Phys. Lett. 82, 3227–3229 (2003).
[Crossref]

Leuchs, G.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Lousse, V.

J. Vigneron and V. Lousse, “Theory of chirped photonic crystals,” Opt. Quantum Electron. 39, 377–385 (2007).
[Crossref]

Mahmoodian, S.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Mandelshtam, V.

V. Mandelshtam and H. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

McPhedran, R. C.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Mori, D.

R. Engelen, D. Mori, T. Baba, and L. Kuipers, “Two regimes of slow-light losses revealed by adiabatic reduction of group velocity,” Phys. Rev. Lett. 101, 103901 (2008).
[Crossref] [PubMed]

D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005).
[Crossref] [PubMed]

Noda, S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[Crossref]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Petrich, G. S.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Poon, J.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

Poulton, C. G.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Pratesi, F.

Rakich, P.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Reece, P. J.

A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystals,” Appl. Phys. Lett. 82, 3227–3229 (2003).
[Crossref]

Rehácek, J.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Riboli, F.

Ripin, D. J.

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Rue, R. M. D. L.

T. F. Krauss and R. M. D. L. Rue, “Photonic crystals in the optical regime past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[Crossref]

Sánchez-Soto, L. L.

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Sargent, E.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[Crossref]

Sauvan, C.

Sipe, J. E.

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

Skibina, J.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Song, B.-S.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[Crossref]

Steinmeyer, G.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Taylor, H.

V. Mandelshtam and H. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

Vigneron, J.

J. Vigneron and V. Lousse, “Theory of chirped photonic crystals,” Opt. Quantum Electron. 39, 377–385 (2007).
[Crossref]

Vynck, K.

Wedell, R.

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

Wiersma, D. S.

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

Appl. Phys. Lett. (2)

A. Bruyant, G. Lérondel, P. J. Reece, and M. Gal, “All-silicon omnidirectional mirrors based on one-dimensional photonic crystals,” Appl. Phys. Lett. 82, 3227–3229 (2003).
[Crossref]

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, and L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[Crossref]

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

J. Chem. Phys. (2)

V. Mandelshtam and H. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem. Phys. 107, 6756–6769 (1997).
[Crossref]

B. R. Johnson, “New numerical methods applied to solving the one-dimensional eigenvalue problem,” J. Chem. Phys. 67, 4086–4093 (1977).
[Crossref]

Nature (1)

J. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[Crossref] [PubMed]

Nature Mater. (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-q photonic double-heterostructure nanocavity,” Nature Mater. 4, 207–210 (2005).
[Crossref]

Nature Photon. (2)

J. Skibina, R. Iliew, J. Bethge, M. Bock, D. Fischer, V. Beloglasov, R. Wedell, and G. Steinmeyer, “A chirped photonic-crystal fibre,” Nature Photon. 2, 679–683 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

Opt. Express (4)

Opt. Quantum Electron. (1)

J. Vigneron and V. Lousse, “Theory of chirped photonic crystals,” Opt. Quantum Electron. 39, 377–385 (2007).
[Crossref]

Phys. Rev. A (2)

S. Mahmoodian, J. E. Sipe, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Double-heterostructure cavities: From theory to design,” Phys. Rev. A 86, 043802 (2012).
[Crossref]

E. Karimi, R. W. Boyd, P. de la Hoz, H. de Guise, J. Řeháček, Z. Hradil, A. Aiello, G. Leuchs, and L. L. Sánchez-Soto, “Radial quantum number of Laguerre-Gauss modes,” Phys. Rev. A 89, 063813 (2014).
[Crossref]

Phys. Rev. B (1)

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65, 125318 (2002).
[Crossref]

Phys. Rev. Lett. (1)

R. Engelen, D. Mori, T. Baba, and L. Kuipers, “Two regimes of slow-light losses revealed by adiabatic reduction of group velocity,” Phys. Rev. Lett. 101, 103901 (2008).
[Crossref] [PubMed]

Prog. Quantum Electron. (1)

T. F. Krauss and R. M. D. L. Rue, “Photonic crystals in the optical regime past, present and future,” Prog. Quantum Electron. 23, 51–96 (1999).
[Crossref]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[Crossref]

Other (3)

J. R. Hook and H. E. Hall, Solid State Physics (John Wiley and Sons, 1991).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, Inc., 2005).

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Figures (10)

Fig. 1
Fig. 1 Photonic band structure (blue) of a one-dimensional photonic crystal with a constant dielectric slab width of w = 0.45a. In red (dashed), a local parabolic approximation according to Eq. (9) is depicted. The vertical gray dashed line indicates the edge of the first Brillouin zone.
Fig. 2
Fig. 2 MPB calculations for a range of slab widths w. In (a) the energy of the bottom of the band E0(w) (red circles) is depicted. A first order polynomial (blue) is fitted to the data points within the range depicted by the two vertical black dashed lines. In (b) the effective mass m*(w) is indicated by green triangles.
Fig. 3
Fig. 3 FDTD results of the field patterns in one dimension for (a) the first, (b) the second, (c) the third, and (d) the fourth eigenmode. The intensity of the field patterns (blue) are plotted as a function of position x. The absolute squared eigenmodes of the quantum harmonic oscillator (red dashed) are plotted. In the background the dielectric function ε(x) (gray) of the photonic crystal is plotted, its value indicated on the right axis.
Fig. 4
Fig. 4 Obtaining the envelope of an eigenmode of the chirped photonic crystal. In (a) the complex field pattern (blue) of Fig. 3(a) is depicted. In (b) the spatial Fourier transform (blue) of (a) is shown. A mask (red dashed) is centered at the edge of the first Brillouin zone. Multiplying (b) with the complex conjugate of the corresponding plane wave, and taking the inverse Fourier transform results in (c). Here, the resulting envelope Ψ0 is indicated by the green dashed line.
Fig. 5
Fig. 5 Envelope of a single mode Ψn compared to all theoretical harmonic modes ψ n . In (a) the envelope of the third eigenmode Ψ2, (b) the first harmonic ψ0, (c) the second harmonic ψ1, (d) the third harmonic ψ2, (e) the fourth harmonic ψ3, (f) all other harmonic modes ψrest. In the upper right corner of subplots (b) – (f) the contribution | c 2 , n | 2 of ψ n to Ψ2 is depicted.
Fig. 6
Fig. 6 FDTD results of the field patterns in one dimension with a distorted harmonic potential. The intensity of the field patterns (blue) are plotted as a function of position x for (a) the first, (b) second, (c) third, and (d) fourth eigenmode. In each subfigure, the normalized Numerov integration (green dashed) is included. The green dots depict the grid points used for this integration. In the background the dielectric function ε(x) (gray) of the photonic crystal is plotted, its value indicated on the right axis.
Fig. 7
Fig. 7 Photonic band structure with the TE (red) and TM (green) bands of a two-dimensional photonic crystal consisting of a triangular lattice of air holes. The radius of the air holes equals ρ = 0.42a. The blue horizontal region depicts the complete photonic band gap.
Fig. 8
Fig. 8 FDTD results of the field patterns in two dimensions. From white to dark red, the intensity of the field patterns are plotted for (a) the first, (b) the second, (c) the third, and (d) the fourth eigenmode. On top of the field patterns the chirped photonic crystal is plotted, where the dielectric material is depicted in black and the air holes in white.
Fig. 9
Fig. 9 Obtaining the envelope of an eigenmode of the chirped photonic crystal cavity. In (a) the spatial Fourier transform of the field pattern of Fig. 8(a) is shown. A mask (blue dashed) is placed around an intensity maxima. Taking the inverse Fourier transform, and multiplying the result with the complex conjugate of the corresponding plane wave results in the envelope Ψn(r, ϕ) in (b).
Fig. 10
Fig. 10 Envelope of a single mode Ψn compared to all theoretical harmonic modes ψ n . In (a) the envelope of the third eigenmode Ψ2, (b) the first harmonic ψ0, (c) the second harmonic ψ1, (d) the third harmonic ψ2, (e) the fourth harmonic ψ3, (f) all other harmonic modes ψrest. In the upper right corner of subplots (b) – (f) the contribution | c 2 , n | 2 of ψ n to Ψ2 is depicted.

Tables (2)

Tables Icon

Table 1 Simulation parameters for the one-dimensional chirped photonic crystal, where c denotes the speed of light. The value for the excitation point has been chosen such that it does not lie at the origin because this would result in the excitation of only the even (n = 0,2,4,…) eigenmodes.

Tables Icon

Table 2 Simulation parameters for the two-dimensional chirped photonic crystal. The excitation point is given as (x, y). The value for the excitation point has been chosen such that it does not lie on a symmetry axis of the photonic crystal because this would result in the excitation of only the even (n = 0,2,4,…) eigenmodes.

Equations (27)

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× ( 1 μ × E ) ε k 0 2 E = 0 ,
× ( 1 ε × H ) μ k 0 2 H = 0 ,
ε ( x + R ) = ε ( x ) and μ ( x + R ) = μ ( x ) ,
E m , k ( x ) = m , k ( x ) e i k x ,
H m , k ( x ) = m , k ( x ) e i k x ,
E n ( x ) = Ψ n ( x ) a 3 / 2 { k min } m c , k min ( x ) e i k min x ,
H n ( x ) = Ψ n ( x ) a 3 / 2 { k min } m c , k min ( x ) e i k min x ,
V ( x ) = ω 0 ( ρ ( x ) ) .
{ 2 2 2 m + V ( x ) } ψ ( x ) = E ψ ( x ) .
E ( k ) = E 0 + 2 ( k k 0 ) 2 2 m ,
w ( x ) = w 0 ( x η ) 2 ,
2 2 m * d 2 ψ ( x ) d x 2 + 1 2 m * Ω 2 x 2 ψ ( x ) = ( E V ( 0 ) ) ψ ( x ) ,
E n = V ( 0 ) + Ω ( n + 1 2 ) ,
ψ n ( x ) = ( m * Ω π ) 1 / 4 1 2 n n ! H n ( ξ ) e ξ 2 / 2 ,
c n , n = Ψ n ( x ) | ψ n ( x ) ,
H ˜ m c ( k ) = Ψ ˜ n ( k ) ˜ m c , k min ( k ) ,
˜ m c , k min ( k ) = k min , κ ˜ k min , κ δ ( k ( k min + κ ) ) ,
H ˜ mask , m c ( k ) = Ψ ˜ n ( k ) ˜ m c , k min δ ( k k min ) ,
w ( x i ) = w 0 ( x i η ) 2 + Δ w ( x i ) ,
V ( x i ) = 1 2 m * Ω 2 x i 2 + Δ V ( Δ w ( x i ) ) ,
ρ ( x , y ) = ρ 0 + 1 2 ( x 2 + y 2 η 2 ) ,
V ( x , y ) = 1 2 m * Ω 2 ( x 2 + y 2 ) .
{ 2 2 m * [ 2 r 2 + 1 r r + 1 r 2 2 ϕ 2 ] + V ( r ) } ψ n , l = E ψ n , l ,
ψ n , l ( r , ϕ ) = C n , l e ζ 2 / 2 ( ζ ) | l | L p | l | ( ζ 2 ) e i l ϕ ,
ψ n ( r , ϕ ) = l c n , { n , l } ψ n , l ( r , ϕ ) ,
c n , { n , l } = Ψ n ( r , ϕ ) | ψ n , l ( r , ϕ ) .
| c n , n | 2 = l | c n , { n , l } | 2 .

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