OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 16 — Aug. 2, 2010
  • pp: 16546–16560
« Show journal navigation

Analysis of the Purcell effect in photonic and plasmonic crystals with losses

Hideo Iwase, Dirk Englund, and Jelena Vučković  »View Author Affiliations


Optics Express, Vol. 18, Issue 16, pp. 16546-16560 (2010)
http://dx.doi.org/10.1364/OE.18.016546


View Full Text Article

Acrobat PDF (2326 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We study the spontaneous emission rate of emitter in a periodically patterned metal or dielectric membrane in the picture of a multimode field of damped Bloch states. For Bloch states in dielectric structures, the approach fully describes the Purcell effect in photonic crystal or spatially coupled cavities with losses. For a metal membrane, the Purcell factor depends on resistive loss at the resonant frequency of surface plasmon polariton (SPP). Analysis of an InP-Au-InP structure indicates that the SPP’s Purcell effect can exceed a value of 50 in the ultraviolet. For a plasmonic crystal, we find a position-dependent Purcell enhancement with a mean value similar to the unpatterned membrane.

© 2010 OSA

1. Introduction

The surface plasmon polariton (SPP) is a collective motion of electrons in metal that confines electromagnetic modes to the vicinity of the metal-dielectric interface. The excitonic spontaneous emission (SE) rate into a SPP cavity mode is enhanced over the vacuum decay rate Γ0 through its spatially confined photon energy density (PED), as described by the Purcell effect [1

1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

6

6. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006). [CrossRef] [PubMed]

]. Due to the spatial and spectral distribution of PED, the SE enhancement rate of a cavity mode, called Purcell factor Fcav, is proportional to the ratio of the mode’s quality factor Q to its mode volume V [7

7. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999). [CrossRef]

]. Plasmonic cavities with large Q/V values are therefore of great interest in a wide range of applications.

Even in the absence of a cavity, metallic structures can provide a large density of optical states, and hence a large SE rate enhancement. For instance, a large density of states occurs at the metal-dielectric interface, where a field-electron resonance leads to small group velocityvg [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

] and a large enhancement in the radiative decay rate Γ [9

9. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004). [CrossRef] [PubMed]

12

12. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999). [CrossRef]

]. The enhancement of Γ in the metal structure is given by F = Γ/nΓ0, ignoring non-radiative recombination, where nΓ0 is the SE rate in a bulk material with the index n. We refer to F as the Purcell enhancement factor [12

12. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999). [CrossRef]

]. F is the enhancement into all photonic states and is therefore distinguished from the cavity Purcell factor Fcav for the radiation into a cavity mode. F is a measure of the Purcell effect at a quasi-infinite metal-dielectric interface, where a number of SPP modes are involved in the SE enhancement.

Several approaches have been taken to analyze F for an exciton coupled to a number of propagating waves in a quasi-infinite structure [13

13. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]

19

19. V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007). [CrossRef]

]. In one approach, the 1D-confined PED is evaluated in the picture of quantum electrodynamics (QED). In analogy to the single-cavity mode volume V, the mode length L of a traveling SPP at a uniform metal-dielectric interface is considered as the normalized 1D-integral of the PED: L = (ωε)/ω|E(z)|2dz/max[(ωε)/ω|E(z)|2] (the length is calculated in the z-direction, perpendicular to the interface) [6

6. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006). [CrossRef] [PubMed]

,12

12. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999). [CrossRef]

]. Evaluating L of an SPP mode with momentum k provides the SE rate enhancement into SPP modes with a specific k-regime, and numerically reveals the relation between the frequency-dependence of F and the SPP’s dispersion. However, in the QED picture the effect of the resistive loss on the SE into the propagating waves has not been clarified. The resistive loss strongly depends on the mode’s k value, which determines the mode’s overlap with the metal region and its group velocity [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

,20

20. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

22

22. M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007). [CrossRef] [PubMed]

]. The loss in the metal results in a spectral linewidth for a particular mode which has to be included in the F calculation. The total SE rate, Γ, is thus obtained by summing these SE spectra over all SPP and non-SPP modes. At the band edge of a periodic metal structure or at the SPP’s resonant frequency [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

,9

9. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004). [CrossRef] [PubMed]

], the SE rate into a SPP mode is particularly strongly influenced because of a high density of photonic states (slow vg) [23

23. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]

,24

24. S. Nojima, “Enhancement of Optical Gain in Two-dimensional Photonic Crystals with Active Lattice Points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]

]. Our analysis of SE in terms of k-states with absorption-dependent linewidth represents a simple and intuitive approach for designing SPP structures to control the excitonic radiation.

2. Purcell enhancement at a uniform metal-dielectric interface

Consider the geometry of Fig. 2(a)
Fig. 2 (a) SPP’s electromagnetic field with a momentum k at a uniform metal (z ≤ 0) -dielectric (z > 0) boundary with an area l 2. Ekz is an out-of-plane component of the electric field, and Ek// and Hkare in-plane components of the electric and magnetic fields. Ekz and Ek// are parallel to z-direction and k, respectively. An emitter with an electric dipole μ lies at the distance of zA from the metal surface (positioned at z = 0). (b) Time-evolution of Ekz and Ek// oscillating in the z-k plane. ek is a unit vector composed of the amplitudes of Ekz and Ek//.
where an emitter with a frequency υ and an electric dipole μ lies in the dielectric medium of index n, a distance zA from the metal surface with an area l 2. An electromagnetic mode guided along the interface has an energy which decays as e-αx, where α is a decay constant and x is a distance traveled. In terms of its quality factor Q, the mode energy decays as e-ωt /Q, where ω is the mode frequency. By equating the last two expressions, we obtain Q = ω/αvg, where v g is the group velocity of the wavepacket [22

22. M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007). [CrossRef] [PubMed]

,25

25. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, New Jersey, 1984).

,26

26. L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

]. The electric field of SPP mode k can be represented as:
Ekwp(r,t)ΔkAΔk2[fk+Δk(r)eiωkt+fk+Δk*(r)eiωkt] eωkt/2Qk.
(1)
Here, fk(r) is a normalized time-independent part of SPP’s electric field with a frequency ω k, and fk(r)=uk(z)eikx, where k and x are a SPP’s momentum and in-plane position vector on metal surface, respectively. In dispersive media, the electric field energy density is not ε E 2/2 but [∂(εω)/∂ω]E 2/2 [27

27. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuum Media (Pergamon, New York, 1984).

]. Therefore, the normalization is represented as follows (εω)/ω|uk|2dr = 1. If ω k is smaller enough than plasma frequency ωp with ε=1ωp2/ω2, ∂(εω)/∂ω ≈|ε| and then (εω)/ωfk(r)|ε|fk(r) is considered to be a normalized orthogonal field for the modes with different k’s (it should be noted that in general εEof SPP is not necessarily precisely orthogonal for an arbitrary structure but approximately is, due to the frequency-dependence of ε in metal) [28

28. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991). [CrossRef] [PubMed]

]. In the microscopic picture, the field quantization in metal is required to analyze a motion of free-electron’s polarization P, and find orthogonal canonical conjugate variables, which generally relies on an effective Hamiltonian [29

29. T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988). [CrossRef] [PubMed]

,30

30. P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999). [CrossRef]

]. However, by taking the time-average of the field energy, the motion of P is included in an electric field energy density [∂(εω)/∂ω]E 2/2 with a frequency-dependent permittivity ε [27

27. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuum Media (Pergamon, New York, 1984).

,31

31. Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998). [CrossRef]

]. Hence, we start with the time-averaged field energy to find canonical conjugate operators. The Hamiltonian and results of the field quantization are summarized in Appendix I.Ignoring non-radiative processes, the Purcell enhancement factor F is defined as a sum of the SE rate into SPP modes, Γsp, and into non-SPP modes, Γnon-sp, normalized by nΓ0: F(Γsp+Γnonsp)/nΓ0. Here Γ0 is a free space SE rate, Γ0=μ2υ3/3πε0c3 [2

2. M. O. Scully, and M. S. Zubairy, Quantum Optics (Cambridge University Press,1997), Chap. 9.

,7

7. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999). [CrossRef]

,32

32. E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).

]. By adding the decay rates for individual k-states (which can be obtained from the solution to the Jaynes-Cummings Hamiltonian or Fermi’s golden rule, with the electric field shown in Appendix I [2

2. M. O. Scully, and M. S. Zubairy, Quantum Optics (Cambridge University Press,1997), Chap. 9.

,7

7. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999). [CrossRef]

,11

11. A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002). [CrossRef]

,32

32. E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).

]), we obtain the radiative decay rate Γsp into SPP modes, normalized by nΓ0:
Fsp(υ)ΓspnΓ0=k2πnΓ0(υ)|gk(zA)|2Dk(υ) ,
(2)
where gk(zA) is a coupling factor between a k-mode and the exciton; gk=g0ψ(z)cos(ϑ) with g0=(μ/)υ/[(1+Θk)n2l2Lk], ψ(z)=|Ek(r)|/max|Ek(r)|, and cos(ϑ)=ekeμ where eμ is a unit vector parallel to the excitonic dipole, and ek[uk(z)+uk*(z)]/|uk(z)+uk*(z)|, (see Fig. 2(b) for the geometry of ek) [12

12. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999). [CrossRef]

,28

28. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991). [CrossRef] [PubMed]

]. The mode length L k is defined by L k(εω)/ω|Ek(r)|2dz/max[(εω)/ω|Ek(r)|2], and 1/(1+Θk)describes the ratio of the electric field energy to the total field energy (Θk (1/μ0)|×(fk/iωk)|2dr, as derived in the Appendix I). The introduction of the 1/(1+Θk) term is critical in this case, and is the result of the electromagnetic field quantization, as the energy is not equally distributed between electric and magnetic field (as described in the Appendix I). Dk(υ) in Eq. (2) is the Lorentzian function describing the density of optical states (DOS):
Dk(υ)=1πωk/2Qk(ωkυ)2+(ωk/2Qk)2  .
(3)
To analyze the Purcell enhancement of state k, we introduce the distributed Purcell Factor, Fdis(υ,k), defined as follows:
Fdis(υ,k)1ΔkxΔky2πnΓ0(υ)|gk(zA)|2Dk(υ) ,
(4)
where ΔkxΔky(2π)2/l2is the reciprocal-space area of one SPP mode. If the real-space area l 2 is large enough that k can be considered continuous, then we can approximate Eq. (4): Fdis(υ,k)=2Fsp(υ)/kxky. In this form, the normalized SE rate into a small reciprocal-space area Δk is expressed as Fdis(υ,k)Δk. From Eq. (4), the definition of g k , and the expression for Γ0, Fdis(υ,k) becomes:
Fdis(υ,k)=321n3c3υ211+Θkhk(zA)Lk(ekeμ)2Dk(υ) ,
(5)
where hk(z)=n2|Ek(r)|2/max[(εω)/ω|Ek(r)|2]. LkeffLk/[hk(zA)(ekeμ)2] is an effective mode volume for 1D-field confinement (effective mode length) [6

6. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006). [CrossRef] [PubMed]

].

Now we consider a uniform metal-dielectric interface. By integrating Fdis(υ,k) in polar coordinates k = (k cosφ, k sinφ, 0), we rewrite Fsp(υ) = 02πdϕ0Fdis(υ,k) kdk = 0f(υ,k)dk, where

f(υ,k)=3πn3c3υ211+Θkς hk(zA)LkkDk(υ) .
(6)

The subscript kk indicates the values independent of φ. The factor ς results from the angular dependence of the coupling strength and averaging of (ekeμ)2 over φ withek=(ek//cosϕ, ek//sinϕ, ekz): ς=1/2×(ek//)2 when the dipole μ is parallel to the metal surface, and ς=(ekz)2 when μ is normal to it.

In Fig. 3
Fig. 3 (a) Distributions of the SPP spontaneous emission enhancement spectrum i(υ,k) by classical electrodynamics analysis (left-side panel), following References [1316] and by quantum electrodynamics analysis, f(υ,k) (right-side panel). The electric dipole lies normal to the Au surface at a distance of 10 nm from it. ωp and τp are set 1.21 × 1016 sec−1 and 1.05 × 10−14 sec, respectively. (b) Spectra of f(υ,k) and i(υ,k) at k = 0.1 nm−1 along the dashed lines in Fig. 3(a), showing an excellent match between the two analyses. The Δυk shows a spectral half-width. (c) Distributions of i(υ,k) above the light line. The distances of μ from the Au surface are set to 200 nm and 400 nm.
we compare the distributed spontaneous emission rate enhancement f(υ,k) for a dipole near a uniform metal-dielectric interface obtained using the approach we presented and the classical analysis, i(υ,k), (from the power radiated by a classical dipole in the modified and unmodified electromagnetic environment [13

13. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]

16

16. R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]

]). For the numerical calculation, we consider an exciton close to a uniform Au/InP interface: the dipole is normal to the Au surface (for example, conduction-to-light hole band transition in a quantum well lying beneath the membrane [10

10. J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000). [CrossRef]

,26

26. L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

]), the InP has index n = 3.2, and Au has plasma frequency ωp = 1.21 × 1016 sec−1 (2πc/ωp = 156 nm) and plasma life-time τp = 1.05 × 10−14 sec (τp = 3.15 μm/c). A decay constant was evaluated by the imaginary part of propagation constant k of SPP: α2k, while vkg and k are obtained by solving Maxwell’s equations under the condition Imεp<< |Reεp| with εp1ωp2/(ω2+iω/τp), as in Ref. [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

]. The expressions for i(υ,k) for a uniform metal-dielectric boundary were also summarized by R. Chance et al. [16

16. R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]

] and W. L. Barnes [13

13. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]

,14

14. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998). [CrossRef]

]. The resulting i(υ,k) and the f(υ,k) by Eq. (6) are plotted in Figs. 3(a) and 3(b). The area of i(υ,k) above the light line corresponds to SE into non-SPP modes. When the emitter is moved away from the metal surface, then i(υ,k) develops oscillations above the light line. This is evident in Fig. 3(c), which graphs i(υ,k) for emitter distances of 200 and 400 nm above the metal surface. These oscillations in the emission rate are attributed to reflections from the Au surface [13

13. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]

16

16. R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]

,33

33. H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53(1), 101–108 (1970). [CrossRef]

]. Below the light line, which corresponds to the radiation into SPP modes, the spectral half-width Δυk of i(υ,k) is results from damp of SPP mode k with resistive loss at the dipole’s position. In the spectra plotted in Figs. 3(a) and 3(b), the spectral distributions f(υ,k) are closely identical to the Lorentzian spectra of i(υ,k) in the same k-range, 0.02 nm−1 < k < 0.3 nm−1.

Figure 4
Fig. 4 Purcell enhancement factors at various frequencies from quantum analysis - Fsp (black lines) and classical analysis - Isp (red lines) for an exciton zA = 10 nm and 30 nm away from the Au/InP boundary, estimated by summing up f(υ,k) and i(υ,k) over k ≤ 0.3 nm−1, respectively. The values of f(υ,k) and i(υ,k) for zA = 10 nm are plotted in Fig. 3(a). Blue lines show the Purcell enhancement factor Fnon-absp for non-absorbing media. The electric dipole lies normal to the Au surface. Non-SPP modes were ignored in the calculation of these plots.
compares the classical and quantum mechanically derived SE enhancement, Isp(υ) and Fsp(υ), respectively. These were obtained by summing i(υ,k) and f(υ,k) over all SPP states, i.e., over the k-states below the light line. We additionally plot the enhancement Fnonabsp(υ) for non-absorbing media (τp → ∞). For the exciton lying 30 nm below the Au surface, these three estimates provide similar values. However, for a small exciton spacing of 10 nm, Fnonabsp(υ) gives a higher peak around υpωp/(1+n2)1/2, as absorption losses matter more for small dipole separation from the metal surface. The deviation of Fnonabsp(υ) shows that it is crucial to include the spectral spreading due to resistive loss in the slow group velocity region. On the other hand, at sufficiently low frequencies (υ/c < 0.01 nm−1), the SPP dispersion approaches the light line, where the spectral spreading of Dk(υ) is averaged in the integration over k and results in a Purcell modification that is nearly independent of resistive loss.

3. Purcell enhancement in a photonic or plasmonic crystal

We will now consider 2D-plasmonic or photonic crystal structures. We take the same approach but note that the structure with a periodic permittivity ε(r) folds the dispersion diagram into the first Brillouin zone. The normalized electric fields of SPP modes fK,j(r) in a 2D-plasmonic crystal are represented as Bloch states [34

34. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, New Jersey, 1995).

]:
fK,j(r)eiωt=uK,j(r)eiKxeiωt ,
(7)
where uK,j(r) with dr[(εω)/ω]|uK,j|2 = 1 is a normalized Bloch function, and K denotes an in-plane wave vector in the first Brillouin zone. Since the number of unit cells in the xy-space (with dimensions l × l) is l2/Scell, where Scell is the area taken by one unit cell, the mode volume VK,j for the (K, j)- mode is expressed as VK,j=VK,jcell(l 2/Scell) where:
VK,jcellunit celldr(εω)/ω|uK,j|2/max[(εω)/ω|uK,j|2]  .
(8)
The mode’s quality factor Q K , j is defined by the ratio of the energy damping rate and stored energy of the mode: QK,j=ωK,jWK,j/(ΔK,jab+ΔK,j) where WK,j, ΔK,jab, ΔK,j, and ωK,j are the stored energy, energy absorption rate in metal, out-of-plane leakage rate, and angular frequency of the mode. Replacing V k, Q k and k in Eqs. (2)-(6) with V K , j, Q K , j and jK, we obtain the Purcell enhancement factorFcry(υ) and distributed Purcell factor Fcrydis(υ, K,j) for the exciton at position r A in a periodic structure:
Fcry(υ)=ΔKxΔKyjKFcry-dis(υ, K,j)+ΔKxΔKyjKOK,j(υ) ,
(9)
Fcry-dis(υ, K,j)=321n3c3υ211+ΘK,jScellVK,jcellhK,j(rA)(eμeK,j)2DK,j(υ) ,
(10)
where,
hK,j(rA)=n2|uK,j(υ,rA)|2/max[(εω)/ω|uK,j(υ,r)|2]  .
(11)
In the above formulae, eK,j(u+K,juK,j*)/|u+K,juK,j*|, ΘK,jdr(1/μ0)|×(fK,j/iωK,j)|2, and DK,j(υ) is defined by Eq. (3) with Qk, ωkQK,j, ωK,j. The second term in Eq. (9), shown by the summation of OK,j(υ), represents the emission into the modes that are not bound to the metal-dielectric interface. The emission into non-SPP modes or highly leaky SPP modes can be included in the second term. We only used the properties of 2D-Bloch functions to derive Eqs. (9)-(11). Therefore, these expressions are valid for any modes described as 2D-Bloch states, no matter if the structure is a plasmonic or photonic crystal (PhC). For planar photonic crystals with negligible absorption, the Q-factor is defined by the radiation loss in the direction perpendicular to the PhC slab, i.e., QK,jQK,jωK,jWK,j/ΔK,j. In this case, QK,j for the modes located under the light line approaches infinity, and the spectrum DK,j(υ) is expressed by a Dirac’s δ-function [35

35. A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005). [CrossRef]

,36

36. Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000). [CrossRef]

].

3a Purcell enhancement in a coupled photonic crystal cavity array

As the first example of the use of Eqs. (9)-(11), we will now analyze the Purcell effect for an exciton located in an array of coupled high-Q resonators embedded in photonic crystal [37

37. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000). [CrossRef]

,38

38. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007). [CrossRef]

]. Figure 5(a)
Fig. 5 (a) 1D-array of high-Q resonators, composed of defects in 2D-photonic crystal with a periodic permittivity ε(r). The defects are aligned in x-direction with a lattice vector R e x. An exciton lies at the position r A in a defect. (b) A single defect in a 2D-photonic crystal with a permittivity ε 0(r).
shows a 1D-array of resonators, described by the permittivity ε(r). The spacing of the resonators is large enough for a small mode overlap that results in weak, nearest-neighbor coupling. The resonators themselves consist of single defects in a 2D-photonic crystal, given by ε0(r) as shown in Fig. 5(b), and have resonant fields Ej0(r). The field of the array is then described as the superposition of the single-defect modes, EK,j(r) = Aexp(iωK,jt)meimKREj0(rmRex), where j denotes the mode of the single defect cavity, and Rex and K are a lattice vector and a mode’s momentum, respectively. The frequency of the 1D array differs from the single defect resonance Ωj due to the coupling as follows: ωK,j=Ω j[1Δα/2+Δκcos(KR)] where Δκ = dr[ε0(rRex)ε(rRex)][Ej0(r)Ej0(rRex)] is a coupling coefficient between the neighboring defect modes and Δα = dr[ε(r)ε0(r)][Ej0(r)]2 [37

37. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000). [CrossRef]

]. Assuming that the overlap of the nearby resonators is weak enough that drε0(r)Ej0(r)Ej0(rRex)<<drε0(r)[Ej0(r)]2, and hK,j(rA)(eμeK,j)2 = 1, we obtain the Purcell enhancement factor shown for the modes of the branch j:
Fj(υ)=2πΔKK341n3c3υ2RVK,jcellDK,j(υ) ,
(12)
where ΔK = 2π/l. To obtain Eq. (12) for PhC in dielectric media, 1/(1+Θk) is set equal to 1/2 [28

28. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991). [CrossRef] [PubMed]

]. Equation (12) shows that the deviation of the frequencies ω K , j from Ωj due to the interactions causes wider spectral spreading of F j(υ) than the SE spectrum of a single resonator [38

38. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007). [CrossRef]

]. Taking the limit of R → ∞, F j(υ) will approach the Purcell factor for the single defect cavity shown in Fig. 5(b). Hence, in the weak coupling condition, F j(υ) can be understood as a Purcell factor of a high-Q resonator which is affected by the energy exchange to the neighboring defects.

3b Purcell enhancement in a plasmonic crystal

We also investigate the effect of periodic patterning on the Purcell effect in a gold membrane with hexagonally arranged dielectric holes, sandwiched by half-infinite InP layers, as shown in Fig. 6(a)
Fig. 6 (a) The plasmonic crystal consists of hexagonally arranged InP-filled holes in the Au membrane. The thickness of the Au layer is 20 nm, and the periodicity of the crystal a and radius r of the InP holes are determined by a = 450 nm and r/a = 0.2. (b) Dispersion diagrams of antisymmetric modes in the plasmonic crystal, obtained by FDTD simulation (dots). Dashed lines show the dispersion branches of the unpatterned InP/Au/InP structure.
. (The Purcell enhancement in such a structure without any patterning is studied in the Appendix II). The thickness of the Au layer is 20 nm, and the periodicity of the crystal and radius of the InP holes are a = 450 nm and r/a = 0.2, respectively. We call this structure a “plasmonic crystal”. For simplicity, we considered only antisymmetric SPP modes around the Γ-point. These modes are particularly interesting in plasmonic devices because of low resistive losses and vertical emission from the metal surface (The effect of resistive losses on the Purcell effect in an unpatterned InP/Au/InP structure is summarized in Appendix II). Figure 6(b) shows the dispersion diagram for antisymmetric modes around the Γ-point, obtained by a Finite Difference Time Domain (FDTD) simulation. The dispersion of the antisymmetric modes in the unpatterned InP/Au/InP structure is shown in the dashed lines, which are folded into the first Brillouin zone of the hexagonal lattice [7

7. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999). [CrossRef]

]. The plasmonic crystal dispersion is represented by the dots, which was verified to match the dashed line in the limit of vanishing hole radii. Figure 7(a)
Fig. 7 (a) Field patterns on surface of Au layer in the plasmonic crystal, belonging to the upper edge (top figures) and the lower edge (bottom figures) of the plasmonic band gap shown in Fig. 6(b), obtained by FDTD simulation. The field of each band edge consists of three orthogonal modes (monopoles in the left figures and dipoles in the right figures). At the band edge, three dispersion branches overlap so close that their eigen-modes could not be separated by our FDTD simulation. (b) The sum of the density of photonic states jKDK,j(υ) for the plasmonic crystal (red line) and unpatterned structure (black line). For computation, we take the summation over the region K ≤ 0.0005 nm−1, considering the degeneracy of each branch. The Q-factor for the plasmonic crystal and unpatterned structure equals 96 and 98 by FDTD simulation, respectively. In the estimation, the upper branches were ignored because of the large leakage loss.
shows the components of the electrical field belonging to the lower and upper edges of the plasmonic band gap [39

39. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996). [CrossRef] [PubMed]

]. The lower and upper bands both consist of three orthogonal modes (a monopole and two dipoles), and their field maxima are located beneath the Au layer and the dielectric holes, respectively. The electric field at the lower band (metal band) is confined out-of-plane, but the fields in the upper band (dielectric band) have large leakage losses. The Q-factors at the Γ-point are 96 for the monopole and 80 for the dipoles in the metal band, and less than 10 in the dielectric band. The Q-factors of the metal band are close to the one for an unpatterned structure: Qk|k = G = 98 with G = 4π/3a.

In the studied structure, only SPP modes around the Γ-point are strongly affected by the periodical patterning, because of weak scattering by the holes in a thin metal membrane; the modes far from the Γ-point are not affected significantly relative to unpatterned structure. We here investigate the change of SE rate of coupling to the modes around the Γ-point, which is observable as the vertical emission from the membrane. From the field patterns shown in Fig. 7(a), we estimate the values of VK,jcell/Scell for the monopole and dipole components in the metal band at Γ-point to be 56 nm and 65 nm, respectively. On the other hand, for the modes belonging to the dielectric band, the values of VK,jcell/Scell could not be accurately evaluated because of their large leakage loss. This indicates that the coupling strength into the dielectric bands is much weaker than that into the metal band around the Γ-point. Hence, although the field components shown in Fig. 7(a) overlap with the whole surface of the membrane (both on the metal and dielectric holes), the difference of the field confinements in the metal and dielectric bands produces the position-dependence of SPP’s coupling strength for the observable vertical emission [40

40. H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008). [CrossRef] [PubMed]

].

To estimate the effect of the patterning on the Purcell enhancement, we compare the sum of the density of optical states in the periodic plasmonic crystal jKDK,j(υ) relative to an unpatterned structure (the structure in the limit of vanishing hole radii, studied in the Appendix II) around Γ-point (K ≤ 0.0005 nm−1). We ignored dielectric bands because of their weaker coupling strengths at Γ-point. This comparison is shown in Fig. 7(b). It indicates that the change of the density of states is small around Γ-point, and thus the enhancement at the band edge is not visible because of large spectral width (induced by losses) and a small region of v g ≈0. Therefore, in the studied frequency range, the development of the low group velocity regions in the plasmonic crystal is not leading to an enhancement in the spontaneous emission rate relative to an unpatterned metal-dielectric structure. However, we should point out that patterning can certainly help in improving the light extraction from the structure, so the collected emission at the output can be larger.

It should be noted that for a structure with strong patterning induced perturbation such as holes engraved in both metal and dielectric layers, a wide band gap and a wider small-v g region can appear in a dispersion diagram. In this case, it is required to consider all SPP modes in an interested frequency range, following the formulae in Eqs. (9) and (10).

4. Conclusion

We evaluated the full Purcell enhancement factor F(υ) by summing contributions to the spontaneous emission rate enhancement f(υ, k) over various points of the photonic/plasmonic band diagram k. F(υ) for a uniform metal layer has large resistive loss-dependence at the SPP resonant frequency of υp=ωp/(1+n2)1/2, where ωp is the metal’s plasma frequency. Around the frequencies where the group velocity vg vanishes, it is essential to consider resistive loss. Otherwise, the width of the photon energy spectrum is neglected and F(υ) overestimated. A large Purcell enhancement in the slow-vg regime is therefore achievable only in extremely low-resistive metal (i.e., a metal with less defect and surface-roughness at low temperature). On the other hand, the Purcell enhancement in the infrared due to the exciton-SPP coupling is almost independent of resistive loss because the spectral width is greatly diminished in summing f(υ,k) over k. Hence, the approximation of non-absorbing media reliably estimates F(υ) in the infrared.

Appendix I: EM field quantization

The electromagnetic field of a mode with a particular k-vector can be described as follows [28

28. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991). [CrossRef] [PubMed]

]:
E(r,t)=12[Dfk(r)η(t)+c.c.]  ,
(13)
H(r,t)=121μ0[Dωk×fk(r)χ(t)+c.c.] ,
(14)
and
η(t)=q(t)+ip(t) ,
(15a)
χ(t)=p(t)iq(t) ,
(15b)
where q(t) ≡ cos(ωk t) and p(t) ≡ −sin(ωk t) describe time-varying (oscillating) parts, μ0 is the magnetic permeability of free space, and D is a constant. As described previously, fk(r) is a normalized time-independent part of electric field: fk(r)=uk(z)eikx with (εω)/ω|uk|2dr = 1. The total energy can then be expressed as [27

27. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuum Media (Pergamon, New York, 1984).

]:
W=12[(εω)/ωE(r,t)2+μ0H(r,t)2]dr
=14D2(p2+q2)+14D2Θk(p2+q2)=12D21+Θk2(p2+q2) ,
(16)
where and 1/(1+Θk)describes the ratio of the electric field energy to the total field energy (Θk(1/μ0)|×(fk/iωk)|2dr). The choice of D=i2ωk/(1+Θk) leads to satisfying of the Hamilton's equations:
Wq=ωkq=p˙ ,
(17a)
Wp=ωkp=q˙  .
(17b)
Then the mode can be represented as a harmonic oscillator and the quantized Hamiltonian and electromagnetic fields are:

H=ωk(a+a+12) ,
(18)
E(r,t)=iωk1+Θkfk(r)a+H.C. ,
(19)
H(r,t)=(1+Θk)ωk1μ0×fk(r)a+H.C.  .
(20)

The Fig. 8
Fig. 8 Values of 1/(1+Θk), WEz/W, WE///W, and WH/W plotted for different k’s for a SPP mode at Au/InP interface, where WE//12(ωε)/ω|Ek//|2dr, WEz12(ωε)/ω|Ekz|2dr, WH12μ0|Hk|2dr, and W is a k-mode’s total field energy. The lines show the values estimated by analytically solving Maxwell’s equations [21], and the marks by FDTD. The index and plasma frequency at Au/InP interface are set n = 3.2, and 2πc/ωp = 156 nm, respectively.
below shows the 1/(1+Θk) and contributions of the perpendicular and parallel electric field and magnetic field components to total energy for SPP's at Au/InP interface.

Appendix II: Purcell enhancement in a uniform semiconductor-metal-semiconductor structure

We consider the Purcell enhancement in a semiconductor-metal-semiconductor structure following the classical analyses by R. Chance et al. [16

16. R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]

] and W. L. Barnes [13

13. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]

,14

14. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998). [CrossRef]

].

The structure is shown in Fig. 9(a)
Fig. 9 (a) An InP/Au/InP structure with an electric dipole lying on the dielectric side (top), and electric field components parallel to the Au surface of an antisymmetric and symmetric modes (bottom). The Au membrane is 10 nm thick. The electric dipole lies normal to the Au surface at a distance of 10 nm apart from it. (b) The distribution of ilayer(υ,k) estimated for the structure shown in Fig. 9(a). The υv=0is a frequency with vg=0 for the antisymmetric modes, and υpωp/(1+n2)1/2. (c) Dissipation spectrum ilayer(υ,k) at k = 0.1 nm−1 along the dashed line in Fig. 9(b). The ilayer(υ,k) below the light line is expressed by the sum of two Lorentzian spectra, ilayer(υ,k) = ianti(υ,k) + isym(υ,k), corresponding to SE into the antisymmetric and symmetric modes. The Δυk is a half-width of the spectrum.
and consists of a 10 nm thick Au membrane sandwiched by InP. In this structure, the dispersion diagram of SPP is split into symmetric and antisymmetric modes [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

,20

20. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

,21

21. E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

], whose electric fields parallel to Au surface are shown in the bottom of Fig. 9(a). The antisymmetric modes have less resistive loss than symmetric modes because of their smaller overlap with a metal membrane. The dissipation spectrum ilayer(υ,k) for this structure has been derived by R. Chance et al. [16

16. R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]

], and is graphed in Figures 9(b) and 9(c) for an exciton 10 nm from Au surface. In these plots, the ilayer(υ,k) below the light line is expressed as the sum of two Lorentzian spectra, which correspond to SE into symmetric and antisymmetric modes, respectively: ilayer(υ,k) = isym(υ,k)+ianti(υ,k). Figure 10(a)
Fig. 10 (a) Quality factor υk0/Δυk of symmetric and antisymmetric modes plotted with different k’s, estimated by fitting ilayer(υ,k) at RT (shown in Fig. 9(c)) and 77 K with Lorentzian spectra. The τp is set 1.05 × 10−14 sec at RT and 3.72 × 10−14 sec at 77 K, respectively, and ωp = 1.21 × 1016 sec−1. The circles show the quality factors, estimated by υ/2kvg for antisymmetric modes at RT [8]. (b) Purcell enhancements for the symmetric and anti-symmetric mode, Isym and Ianti, at RT and 77 K, estimated by summing up isym and ianti shown in Figs. 9(b) and 9(c) over k ≤ 0.3 nm−1, respectively.
shows the Q-factor υk0/Δυk for the symmetric and antisymmetric modes at room temperature (RT) (τp = 1.05×10−14 sec) and 77 K (τp = 3.27×10−14 sec), estimated by fitting ilayer(υ,k) with Lorentzian spectra. The values of υ/2kvkg for the antisymmetric modes, analytically evaluated as in Reference [8

8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

], are also plotted in Fig. 10(a) (blue circles) and in close agreement with the corresponding υk0/Δυk. As expected, the values of υk0/Δυk for the antisymmetric modes are larger than those for symmetric modes, due to less absorption in the Au membrane. Fig. 10(b) shows the sums of the dissipation spectra, Isym(υ)kisym(υ,k) and Ianti(υ)kianti(υ,k). According to the comparison between QED and CED analyses, which are made in Section 2 for a single metal-dielectric boundary, Isym(υ) and Ianti(υ) are considered to be the enhanced SE rate into symmetric and antisymmetric modes, normalized by nΓ0, respectively. Comparing Ianti(υ) shown in Fig. 10(b) with ianti(υ,k) shown in Fig. 9(b), it is evident that the small-vg regime of the antisymmetric mode’s dispersion diagram results in a peak in Ianti(υ) at υ=υv=0, where a large number of ianti(υ,k) overlap at the same frequency. As for a single metal-dielectric boundary, a high susceptibility to resistive loss is observed in the slow group velocity region at υ=υv=0.

Acknowledgements

This work has been supported by Canon Inc. and the IFC.

References and links

1.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

2.

M. O. Scully, and M. S. Zubairy, Quantum Optics (Cambridge University Press,1997), Chap. 9.

3.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006). [CrossRef] [PubMed]

4.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002). [CrossRef] [PubMed]

5.

Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). [CrossRef]

6.

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006). [CrossRef] [PubMed]

7.

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999). [CrossRef]

8.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

9.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004). [CrossRef] [PubMed]

10.

J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000). [CrossRef]

11.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002). [CrossRef]

12.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999). [CrossRef]

13.

W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]

14.

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998). [CrossRef]

15.

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974). [CrossRef]

16.

R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]

17.

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000). [CrossRef]

18.

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007). [CrossRef] [PubMed]

19.

V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007). [CrossRef]

20.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]

21.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]

22.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007). [CrossRef] [PubMed]

23.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]

24.

S. Nojima, “Enhancement of Optical Gain in Two-dimensional Photonic Crystals with Active Lattice Points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]

25.

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

26.

L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

27.

L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuum Media (Pergamon, New York, 1984).

28.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991). [CrossRef] [PubMed]

29.

T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988). [CrossRef] [PubMed]

30.

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999). [CrossRef]

31.

Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998). [CrossRef]

32.

E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).

33.

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53(1), 101–108 (1970). [CrossRef]

34.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, New Jersey, 1995).

35.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005). [CrossRef]

36.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000). [CrossRef]

37.

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000). [CrossRef]

38.

J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007). [CrossRef]

39.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996). [CrossRef] [PubMed]

40.

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008). [CrossRef] [PubMed]

41.

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008). [CrossRef] [PubMed]

OCIS Codes
(130.2790) Integrated optics : Guided waves
(130.3120) Integrated optics : Integrated optics devices
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Photonic Crystals

History
Original Manuscript: May 24, 2010
Revised Manuscript: July 3, 2010
Manuscript Accepted: July 4, 2010
Published: July 22, 2010

Citation
Hideo Iwase, Dirk Englund, and Jelena Vučković, "Analysis of the Purcell effect in photonic and plasmonic crystals with losses," Opt. Express 18, 16546-16560 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16546


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  2. M. O. Scully, and M. S. Zubairy, Quantum Optics (Cambridge University Press,1997), Chap. 9.
  3. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006). [CrossRef] [PubMed]
  4. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002). [CrossRef] [PubMed]
  5. Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007). [CrossRef]
  6. S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006). [CrossRef] [PubMed]
  7. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999). [CrossRef]
  8. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).
  9. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004). [CrossRef] [PubMed]
  10. J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000). [CrossRef]
  11. A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002). [CrossRef]
  12. I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999). [CrossRef]
  13. W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999). [CrossRef]
  14. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998). [CrossRef]
  15. R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974). [CrossRef]
  16. R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978). [CrossRef]
  17. R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000). [CrossRef]
  18. G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007). [CrossRef] [PubMed]
  19. V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007). [CrossRef]
  20. D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981). [CrossRef]
  21. E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]
  22. M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007). [CrossRef] [PubMed]
  23. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994). [CrossRef]
  24. S. Nojima, “Enhancement of Optical Gain in Two-dimensional Photonic Crystals with Active Lattice Points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]
  25. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, New Jersey, 1984).
  26. L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).
  27. L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuum Media (Pergamon, New York, 1984).
  28. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991). [CrossRef] [PubMed]
  29. T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988). [CrossRef] [PubMed]
  30. P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999). [CrossRef]
  31. Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998). [CrossRef]
  32. E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).
  33. H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53(1), 101–108 (1970). [CrossRef]
  34. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, New Jersey, 1995).
  35. A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005). [CrossRef]
  36. Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000). [CrossRef]
  37. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000). [CrossRef]
  38. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007). [CrossRef]
  39. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996). [CrossRef] [PubMed]
  40. H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008). [CrossRef] [PubMed]
  41. S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited