Abstract

Although the theory of photonic crystal cavities has been widely investigated, the interpretation of the experimental emission spectra still creates a challenge because the spontaneous emission enhancement is a spatially and spectrally varying property. We present a comprehensive simulation and analysis of the emission spectrum of photonic crystal cavities, considering the spatially and spectrally varying spontaneous emission enhancement, the material loss, and the coupling efficiency to the detection system. The simulations have been performed with a 3D finite-element Maxwell solver and an efficient mode expansion scheme. They have been compared to measured spectra and show very good agreement.

© 2008 Optical Society of America

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  1. E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681-681 (1946).
    [CrossRef]
  2. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
    [CrossRef]
  3. T. Baba, T. Hamano, F. Koyama, and K. Iga, "Spontaneous emission factor of a microcavity DBR surface-emitting laser," IEEE J. Quantum Electron. 27, 1347-1358 (1991).
    [CrossRef]
  4. M. Francardi, L. Balet, A. Gerardino, C. Monat, C. Zinoni, L. H. Li, B. Alloing, N. L. Thomas, R. Houdré, and A. Fiore, "Quantum dot photonic crystal nanocavities at 1300 nm for telecom-wavelength single-photon source," Phys. Status Solidi C 3, 3693-3696 (2006).
    [CrossRef]
  5. M. L. Adams, M. Loncar, A. Scherer, and Y. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
    [CrossRef]
  6. M. Lee and P. M. Fauchet, "Two-dimensional silicon photonic crystal based biosensing platform for protein detection," Opt. Express 15, 4530-4535 (2007).
    [CrossRef] [PubMed]
  7. P. Alivisatos, "The use of nanocrystals in biological detection," Nat. Biotechnol. 22, 47-52 (2004).
    [CrossRef] [PubMed]
  8. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, "Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity," Phys. Rev. Lett. 81, 1110-1113 (1998).
    [CrossRef]
  9. T. D. Happ, I. I. Tartakovskii, V. D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, "Enhanced light emission of InxGa1−xAs quantum dots in a two-dimensional photonic-crystal defect microcavity," Phys. Rev. B 66, 041303 (2002).
    [CrossRef]
  10. K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, "Definition and measurement of the local density of electromagnetic states close to an interface," Phys. Rev. B 68, 245405 (2003).
    [CrossRef]
  11. W. Lukosz, "Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers," Phys. Rev. B 22, 3030-3038 (1980).
    [CrossRef]
  12. L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).
    [CrossRef]
  13. L.-W. Wang, J. Kim, and A. Zunger, "Electronic structures of [110]-faceted self-assembled pyramidal InAs/GaAs quantum dots," Phys. Rev. B 59, 5678-5687 (1999).
    [CrossRef]
  14. J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1995).
    [CrossRef]
  15. F. L. Teixeira and W. C. Chew, "Complex space approach to perfectly matched layers: a review and some new developments," Int. J. Numer. Model. 13, 441-455 (2000).
    [CrossRef]
  16. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Optimization of the Q factor in photonic crystal microcavities," IEEE J. Quantum Electron. 38, 850-856 (2002).
    [CrossRef]
  17. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005).
    [CrossRef] [PubMed]
  18. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).
  19. J. C. Nédélec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
    [CrossRef]
  20. F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39, 341-352 (2007).
    [CrossRef]
  21. O. Schenk and K. Gärtner, "Solving unsymmetric sparse systems of linear equations with PARDISO," FGCS, Future Gener. Comput. Syst. 20, 475-487 (2004).
    [CrossRef]
  22. G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1989).
  23. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
    [CrossRef] [PubMed]
  24. S. Adachi, Properties of Aluminium Gallium Arsenide (IEE, 1993).
  25. M. Shirane, S. Kono, J. Ushida, and S. Ohkouchi, "Mode identification of high-quality-factor single-defect nanocavities in quantum dot-embedded photonic crystals," J. Appl. Phys. 101, 073107 (2007).
    [CrossRef]
  26. F. Römer and B. Witzigmann, "Investigation of the optical farfield of photonic crystal microcavities," Proc. SPIE 6480, 64801B (2007).
    [CrossRef]
  27. D. H. S. Cheng, "On the formulation of the dyadic Green's function in a layered medium," Electromagnetics 6, 171-182 (1986).
    [CrossRef]
  28. A. Fiore, U. Oesterle, R. P. Stanley, R. Houdré, F. Lelarge, M. Ilegems, P. Borri, W. Langbein, D. Birkedal, J. M. Hvam, M. Cantoni, and F. Bobard, "Structural and electrooptical characteristics of quantum dots emitting at 1.3 μm on gallium arsenide," IEEE J. Quantum Electron. 37, 1050-1058 (2001).
    [CrossRef]

2007

M. Lee and P. M. Fauchet, "Two-dimensional silicon photonic crystal based biosensing platform for protein detection," Opt. Express 15, 4530-4535 (2007).
[CrossRef] [PubMed]

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39, 341-352 (2007).
[CrossRef]

M. Shirane, S. Kono, J. Ushida, and S. Ohkouchi, "Mode identification of high-quality-factor single-defect nanocavities in quantum dot-embedded photonic crystals," J. Appl. Phys. 101, 073107 (2007).
[CrossRef]

F. Römer and B. Witzigmann, "Investigation of the optical farfield of photonic crystal microcavities," Proc. SPIE 6480, 64801B (2007).
[CrossRef]

2006

M. Francardi, L. Balet, A. Gerardino, C. Monat, C. Zinoni, L. H. Li, B. Alloing, N. L. Thomas, R. Houdré, and A. Fiore, "Quantum dot photonic crystal nanocavities at 1300 nm for telecom-wavelength single-photon source," Phys. Status Solidi C 3, 3693-3696 (2006).
[CrossRef]

2005

M. L. Adams, M. Loncar, A. Scherer, and Y. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "Fine-tuned high-Q photonic-crystal nanocavity," Opt. Express 13, 1202-1214 (2005).
[CrossRef] [PubMed]

2004

P. Alivisatos, "The use of nanocrystals in biological detection," Nat. Biotechnol. 22, 47-52 (2004).
[CrossRef] [PubMed]

O. Schenk and K. Gärtner, "Solving unsymmetric sparse systems of linear equations with PARDISO," FGCS, Future Gener. Comput. Syst. 20, 475-487 (2004).
[CrossRef]

2003

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, "Definition and measurement of the local density of electromagnetic states close to an interface," Phys. Rev. B 68, 245405 (2003).
[CrossRef]

2002

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Optimization of the Q factor in photonic crystal microcavities," IEEE J. Quantum Electron. 38, 850-856 (2002).
[CrossRef]

T. D. Happ, I. I. Tartakovskii, V. D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, "Enhanced light emission of InxGa1−xAs quantum dots in a two-dimensional photonic-crystal defect microcavity," Phys. Rev. B 66, 041303 (2002).
[CrossRef]

2001

A. Fiore, U. Oesterle, R. P. Stanley, R. Houdré, F. Lelarge, M. Ilegems, P. Borri, W. Langbein, D. Birkedal, J. M. Hvam, M. Cantoni, and F. Bobard, "Structural and electrooptical characteristics of quantum dots emitting at 1.3 μm on gallium arsenide," IEEE J. Quantum Electron. 37, 1050-1058 (2001).
[CrossRef]

2000

F. L. Teixeira and W. C. Chew, "Complex space approach to perfectly matched layers: a review and some new developments," Int. J. Numer. Model. 13, 441-455 (2000).
[CrossRef]

1999

L.-W. Wang, J. Kim, and A. Zunger, "Electronic structures of [110]-faceted self-assembled pyramidal InAs/GaAs quantum dots," Phys. Rev. B 59, 5678-5687 (1999).
[CrossRef]

1998

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, "Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity," Phys. Rev. Lett. 81, 1110-1113 (1998).
[CrossRef]

1997

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

1995

J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1995).
[CrossRef]

1991

T. Baba, T. Hamano, F. Koyama, and K. Iga, "Spontaneous emission factor of a microcavity DBR surface-emitting laser," IEEE J. Quantum Electron. 27, 1347-1358 (1991).
[CrossRef]

1986

J. C. Nédélec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

D. H. S. Cheng, "On the formulation of the dyadic Green's function in a layered medium," Electromagnetics 6, 171-182 (1986).
[CrossRef]

1980

W. Lukosz, "Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers," Phys. Rev. B 22, 3030-3038 (1980).
[CrossRef]

1946

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681-681 (1946).
[CrossRef]

Electromagnetics

D. H. S. Cheng, "On the formulation of the dyadic Green's function in a layered medium," Electromagnetics 6, 171-182 (1986).
[CrossRef]

FGCS, Future Gener. Comput. Syst.

O. Schenk and K. Gärtner, "Solving unsymmetric sparse systems of linear equations with PARDISO," FGCS, Future Gener. Comput. Syst. 20, 475-487 (2004).
[CrossRef]

IEEE J. Quantum Electron.

A. Fiore, U. Oesterle, R. P. Stanley, R. Houdré, F. Lelarge, M. Ilegems, P. Borri, W. Langbein, D. Birkedal, J. M. Hvam, M. Cantoni, and F. Bobard, "Structural and electrooptical characteristics of quantum dots emitting at 1.3 μm on gallium arsenide," IEEE J. Quantum Electron. 37, 1050-1058 (2001).
[CrossRef]

T. Baba, T. Hamano, F. Koyama, and K. Iga, "Spontaneous emission factor of a microcavity DBR surface-emitting laser," IEEE J. Quantum Electron. 27, 1347-1358 (1991).
[CrossRef]

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, "Optimization of the Q factor in photonic crystal microcavities," IEEE J. Quantum Electron. 38, 850-856 (2002).
[CrossRef]

IEEE J. Sel. Areas Commun.

M. L. Adams, M. Loncar, A. Scherer, and Y. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
[CrossRef]

Int. J. Numer. Model.

F. L. Teixeira and W. C. Chew, "Complex space approach to perfectly matched layers: a review and some new developments," Int. J. Numer. Model. 13, 441-455 (2000).
[CrossRef]

J. Appl. Phys.

M. Shirane, S. Kono, J. Ushida, and S. Ohkouchi, "Mode identification of high-quality-factor single-defect nanocavities in quantum dot-embedded photonic crystals," J. Appl. Phys. 101, 073107 (2007).
[CrossRef]

J. Comput. Phys.

J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys. 114, 185-200 (1995).
[CrossRef]

Nat. Biotechnol.

P. Alivisatos, "The use of nanocrystals in biological detection," Nat. Biotechnol. 22, 47-52 (2004).
[CrossRef] [PubMed]

Nature

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist on light," Nature 386, 143-149 (1997).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003).
[CrossRef] [PubMed]

Numer. Math.

J. C. Nédélec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

F. Römer, B. Witzigmann, O. Chinellato, and P. Arbenz, "Investigation of the Purcell effect in photonic crystal cavities with a 3D finite element Maxwell solver," Opt. Quantum Electron. 39, 341-352 (2007).
[CrossRef]

Phys. Rev.

E. M. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681-681 (1946).
[CrossRef]

Phys. Rev. B

L.-W. Wang, J. Kim, and A. Zunger, "Electronic structures of [110]-faceted self-assembled pyramidal InAs/GaAs quantum dots," Phys. Rev. B 59, 5678-5687 (1999).
[CrossRef]

T. D. Happ, I. I. Tartakovskii, V. D. Kulakovskii, J.-P. Reithmaier, M. Kamp, and A. Forchel, "Enhanced light emission of InxGa1−xAs quantum dots in a two-dimensional photonic-crystal defect microcavity," Phys. Rev. B 66, 041303 (2002).
[CrossRef]

K. Joulain, R. Carminati, J.-P. Mulet, and J.-J. Greffet, "Definition and measurement of the local density of electromagnetic states close to an interface," Phys. Rev. B 68, 245405 (2003).
[CrossRef]

W. Lukosz, "Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers," Phys. Rev. B 22, 3030-3038 (1980).
[CrossRef]

Phys. Rev. Lett.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, "Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity," Phys. Rev. Lett. 81, 1110-1113 (1998).
[CrossRef]

Phys. Status Solidi C

M. Francardi, L. Balet, A. Gerardino, C. Monat, C. Zinoni, L. H. Li, B. Alloing, N. L. Thomas, R. Houdré, and A. Fiore, "Quantum dot photonic crystal nanocavities at 1300 nm for telecom-wavelength single-photon source," Phys. Status Solidi C 3, 3693-3696 (2006).
[CrossRef]

Proc. SPIE

F. Römer and B. Witzigmann, "Investigation of the optical farfield of photonic crystal microcavities," Proc. SPIE 6480, 64801B (2007).
[CrossRef]

Other

S. Adachi, Properties of Aluminium Gallium Arsenide (IEE, 1993).

G. H. Golub and C. F. van Loan, Matrix Computations (Johns Hopkins U. Press, 1989).

L. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).
[CrossRef]

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, 2002).

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Figures (11)

Fig. 1
Fig. 1

Geometry of the optimized H1 cavity. The holes surrounding the defect of the H1 cavity have been scaled down by a factor of 0.7 and displaced by 0.09 a in radial direction. Basic geometric data: slab thickness d = 320 nm , hole distance a = 340 nm , hole radius r = 105 nm .

Fig. 2
Fig. 2

Normalized LDOS of the H1 cavity plotted for the twofold degenerate modes 1H and 2H. Refer to Table 1 for the eigenvalues.

Fig. 3
Fig. 3

Quality factor of the H1 1H cavity mode (only one polarization shown) as a function of the dislocation and diameter of the inner holes. The insets show the angular far field intensity distribution projected onto a unit circle. The center and the edge of the circle depict vertical and lateral emission, respectively.

Fig. 4
Fig. 4

The quality factor of the optimized H1 PCC depends critically on the material loss as shown for the H1 1H and the H1 2H mode. There is a strong decay even for a quite moderate absorption of 1 cm . The loss efficiently cancels the enhancement of the quality factor by extending the photonic crystal around the defect as shown for 11 rows of holes in contrast to eight rows.

Fig. 5
Fig. 5

Spectrally resolved spontaneous emission enhancement in the center of the cavity. The graphs show the contribution of an x- and y-polarized dipole and the total value, respectively. The z component has been omitted. These spectra result from the direct solution of Eq. (7). The field intensity of the modes 2H, 3H, 1P, and 2P vanishes in the center so that they are not present in the spectrum.

Fig. 6
Fig. 6

Spectrally resolved spontaneous emission enhancement at r = ( 0 , 0.18 μ m , 0 ) , direct solution of Eq. (7).

Fig. 7
Fig. 7

Spectrally resolved spontaneous emission enhancement at r = ( 0 , 0.18 μ m , 0 ) computed by mode expansion. The curves show the spontaneous emission enhancement caused by the cavity modes, the radiative modes from a slab, and the sum of both.

Fig. 8
Fig. 8

Spectrally resolved spontaneous emission enhancement in the center of a thin slab ( d = 0.32     μ m ) . The curves show the contribution of radiative modes and the analytic solution compared to the result of the direct solver.

Fig. 9
Fig. 9

Emission spectrum measured at 300 K (labeled “PL”) and simulated (labeled “cavity”) emission spectra of a L3 cavity. The basic geometry data are slab thickness d = 320 nm , hole distance a = 310 nm , hole radius r = 90 nm . The holes defining the cavity on the long axis have been dislocated by 0.17 a and scaled down to 0.465 r . The curve labeled “vacuum” shows the emission spectrum assumed for the QDs.

Fig. 10
Fig. 10

L3 cavity spontaneous emission enhancement in the center and on the long axis at 0.39 μ m distance from the center.

Fig. 11
Fig. 11

Simulated emission spectrum (labeled “cavity”) of the optimized H1 cavity. The curve labeled “vacuum” shows the emission spectrum assumed for the QDs.

Tables (2)

Tables Icon

Table 1 H1 PCC Modes and Optical Efficiency for NA = 0.5

Tables Icon

Table 2 L3 PCC Modes and Optical Efficiency for NA = 0.5

Equations (25)

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

E ( r ) = j k Z 0 V G 11 ( r , r , k ) J ( r ) d 3 r ,
P rad ( R , p ) = V Re { J * ( r ) E ( r ) } d r = k Z 0 Im { p * G 11 ( R , R , k ) p }
P ¯ rad = k Z 0 p 2 4 π 0 π 0 2 π r ̂ Im { G 11 ( R , R , k ) } r ̂ d φ sin ( θ ) d θ = p 2 k Z 0 3 Im { Tr [ G 11 ( R , R , k ) ] } ,
β ( r , k ) = P ¯ rad 6 π k 2 Z 0 p 2 = 2 π k Im { Tr [ G 11 ( r , r , k ) ] } .
× ( μ ̿ 1 × E ) k 2 ϵ ̿ E = j k Z 0 J ,
F ( E ) = Ω ( × E ) μ ̿ 1 ( × E ) k 2 E ϵ ̿ E + 2 j k Z 0 E J d V ,
× ( μ ̿ 1 × G 11 ( r , r ) ) k 2 ϵ ̿ G 11 ( r , r ) = I ̿ δ ( r r ) ,
E i ν ̿ E j = V ( × E i ) μ ̿ 1 ( × E j ) d V ,
E i ϵ ̿ E j = V E i ϵ ̿ E j d V ,
E i J = V E i J d V ,
E i ν ̿ E j i = 0 ,
E i ϵ ̿ E j i = 0
E = j a j E j ,
E i ν ̿ j a j E j k 2 E i ϵ ̿ j a j E j = j k Z 0 E i J
a i ( k i 2 k 2 ) E i ϵ ̿ E i = j k Z 0 E i J
G 11 ( k ) = n 1 k 2 k n 2 E n E n E n ϵ ̿ E n .
β ( r , k ) = 2 π k Im { n 1 k 2 k n 2 E n 2 ( r ) E n ϵ ̿ E n } ,
β ( r , k ) β rad ( k ) + n π k ni k nr 2 1 k ni 2 + ( k k nr ) 2 Re { E n 2 ( r ) E n ϵ ̿ E n } .
d n sp , vac ( r , k ) d k = D 0 ( r ) π τ sp Δ k D Δ k D 2 + ( k k D ) 2 ,
d n sp ( r , k ) d k = β ( r , k ) d n sp , vac ( r , k ) d k .
G ( r ) = n sp ( r ) = d n sp ( r , k ) d k d k .
D 0 ( r ) = G ( r ) τ sp ( β rad ( k D ) + n π k nr 2 Δ k D + k ni ( Δ k D + k ni ) 2 + ( k nr k D ) 2 Re { E n 2 ( r ) E n ϵ ̿ E n } ) 1 .
d N sp , n d k = 1 τ sp Δ k D Δ k D 2 + ( k k D ) 2 k ni k nr 2 1 k ni 2 + ( k k nr ) 2 V D 0 ( r ) Re { E n 2 ( r ) E n ϵ ̿ E n } d V .
ϴ ( k ) = c k ( η rad β rad ( k ) π τ sp Δ k D Δ k D 2 + ( k k D ) 2 V D 0 ( r ) d V + n η n d N sp , n d k ) ,
P ( k f ) = ϴ ( k ) Δ k f 2 Δ k f 2 + ( k k f ) 2 d k ,

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