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

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 Optical Society of America OCIS codes: (240.6680) Surface Plasmons; (130.3120) Integrated optics devices; (130.2790) Guided waves 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. 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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 0  Γ through its spatially confined photon energy density (PED), as described by the Purcell effect [1][2][3][4][5][6].Due to the spatial and spectral distribution of PED, the SE enhancement rate of a cavity mode, called Purcell factor cav F , is proportional to the ratio of the mode's quality factor Q to its mode volume V [7].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 velocity g v [8] and a large enhancement in the radiative decay rate Γ [9-12].The enhancement of Γ in the metal structure is given by F = Γ/nΓ 0 , ignoring non-radiative recombination, where nΓ 0 is the SE rate in a bulk material with the index n.We refer to F as the Purcell enhancement factor [12].F is the enhancement into all photonic states and is therefore distinguished from the cavity Purcell factor cav F 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][14][15][16][17][18][19].In one approach, the 1D-confined PED is evaluated in the picture of quantum electrodynamics (QED).In analogy to the singlecavity 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: (the length is calculated in the zdirection, perpendicular to the interface) [6,12].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,[20][21][22].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 [8,9], the SE rate into a SPP mode is particularly strongly influenced because of a high density of photonic states (slow v g ) [23,24].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.
In this paper, we study the Purcell enhancement factor F for the exciton coupled to SPP modes both at a uniform and periodically patterned absorptive metal-dielectric interface (or periodically patterned lossy photonic crystal), as illustrated in Figs.1(a) and 1(b).To investigate the effect of resistive loss in a periodically patterned metal surface (plasmonic crystal), we expand the emission rates over the damped plasmonic Bloch states with different in-plane momenta K and quantum numbers of dispersion branches j.These states indexed by K and j have different quality factors Q K,j .Our analysis yields the SE enhancement spectrum dis F in reciprocal space, called the distributed Purcell factor, while the total spontaneous emission enhancement factor F is described as a sum of dis F over all photonic states.The values of dis F provide us with the Purcell enhancement ratio for the SE into a specific kmode, with resistive loss considered.Compared to the previous works, our approach reveals how the overlap of SE spectrum with the modes affects the total SE rate; this overlap has a crucial role in a slow-v g regime.Hence, it is a useful tool to study SE enhancement in the structures, such as a patterned metal membrane or nanowire, arranged metal particles, and coupled cavities in lossy photonic crystal.

Purcell enhancement at a uniform metal-dielectric interface
Consider the geometry of Fig. 2(a) where an emitter with a frequency υ and an electric dipole µ lies in the dielectric medium of index n, a distance z A 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/αv g , where v g is the group velocity of the wavepacket [22,25,26].The electric field of SPP mode k can be represented as: Here, ( ) is a normalized time-independent part of SPP's electric field with a frequency ω k , and ( ) ( ) , 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 εE 2 /2 but [∂(εω)/∂ω]E 2 /2 [27].Therefore, the normalization is represented as follows , ∂(εω)/∂ω ≈|ε| and then ( ) is considered to be a normalized orthogonal field for the modes with different k's (it should be noted that in general E ε of SPP is not necessarily precisely orthogonal for an arbitrary structure but approximately is, due to the frequency-dependence of ε in metal) [28].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 [29,30].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 frequencydependent permittivity ε [27,31].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, sp where is a coupling factor between a k-mode and the exciton; , and where µ e is a unit vector parallel to the excitonic dipole, and , (see Fig. 2(b) for the geometry of k e ) [12,28].The mode length L k is defined by describes the ratio of the electric field energy to the total field energy ( , as derived in the Appendix I).The introduction of the 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).2) is the Lorentzian function describing the density of optical states (DOS): To analyze the Purcell enhancement of state k, we introduce the distributed Purcell Factor, ) , ( , defined as follows: ( ) ( ) ( ) ( ) where is 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): . In this form, the normalized SE rate into a small reciprocalspace area ∆ k is expressed as 4), the definition of g k , and the where effective mode volume for 1D-field confinement (effective mode length) [6].Now we consider a uniform metal-dielectric interface.By integrating ) , ( The subscript k → k indicates the values independent of φ.The factor ς results from the angular dependence of the coupling strength and averaging of  In Fig. 3 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][14][15][16]).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 [10,26]), the InP has index n = 3.
, as in Ref. [8].The expressions for i(υ,k) for a uniform metal-dielectric boundary were also summarized by R. Chance et al. [16] and W. L. Barnes [13,14].The resulting i(υ,k) and the f(υ,k) by Eq. ( 6) are plotted in Figs.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 is averaged in the integration over k and results in a Purcell modification that is nearly independent of resistive loss.

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 ) ( , r f K j in a 2D-plasmonic crystal are represented as Bloch states [34]: for the (K, j)-mode is expressed as The mode's quality factor Q K, j is defined by the ratio of the energy damping rate and stored energy of the mode:  2)-( 6) with V K, j , Q K, j and ∑ ∑ j K , we obtain the Purcell enhancement factor , , , where, In the above formulae, ω .The second term in Eq. ( 9), shown by the summation of ) ( , υ j O K , 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., for the modes located under the light line approaches infinity, and the spectrum is expressed by a Dirac's δ-function [35,36].

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 [37,38].Figure 5(a) shows a 1D-array of resonators, described by the permittivity ) (r ε .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 2Dphotonic crystal, given by ) ( 0 r ε as shown in Fig. 5(b), and have resonant fields ) ( 0 r E j .The field of the array is then described as the superposition of the single-defect modes, , where j denotes the mode of the single defect cavity, and x Re and K are a lattice vector and a mode's momentum, respectively.The frequency of the 1D array differs from the single defect resonance j Ω due to the coupling as follows: , and = 1, we obtain the Purcell enhancement factor shown for the modes of the branch j: where ∆K = 2π/l.To obtain Eq. ( 12) for PhC in dielectric media, is set equal to 1/2 [28].Equation (12) shows that the deviation of the frequencies ω K, j from j Ω due to the interactions causes wider spectral spreading of F j (υ) than the SE spectrum of a single resonator [38].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).(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).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].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) shows the components of the electrical field belonging to the lower and upper edges of the plasmonic band gap [39].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: . 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 cell cell , / S V j K 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 cell cell , / S V j K 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].
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 ∑ ∑ ) ( , 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).  ) ( , 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.

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 , where ω p is the metal's plasma frequency.Around the frequencies where the group velocity v g 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-v g 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.For 2D-periodic structures, our approach links F(υ) to Bloch states and ) ( cav υ F for a single unit cell (or single cavity inside of the cavity array).The Purcell enhancement for antisymmetric SPP in a Au membrane with a plasmonic crystal shows that the positiondependence of the Purcell enhancement depends on the leakage loss of the SPP and its field redistribution.In a hexagonal plasmonic crystal, large leakage loss of the dielectric (upper) band weakens the SPP's coupling of the emitters beneath the dielectric holes, while the emitters beneath the metal region strongly radiate into SPP.This result agrees with our experimental observation of the Purcell effect for wave-guided antisymmetric SPP modes [40].Although the precise form of the dispersion and field patterns is required to estimate the total Purcell enhancement factor, our approach enables an analysis of the Purcell effect in plasmonic crystals with respect to SE spectra, as opposed to CED analyses relying on the total power dissipation.In this work, we focused on plasmonic crystals, but our approach is also applicable to emitters coupled to nanowires, in which traveling waves are two-dimensionally confined [41].The theoretical platform discussed in this paper is a useful tool to fully describe the Purcell effects in photonic and plasmonic crystals, and spatially coupled cavities.

Appendix I: EM field quantization
The electromagnetic field of a mode with a particular k-vector can be described as follows [28]: and where q(t) ≡ cos(ω k t) and p(t) ≡ −sin(ω k t) describe time-varying (oscillating) parts, µ 0 is the magnetic permeability of free space, and D is a constant.As described previously, The total energy can then be expressed as [27]: where and describes the ratio of the electric field energy to the total field energy leads to satisfying of the Hamilton's equations: Then the mode can be represented as a harmonic oscillator and the quantized Hamiltonian and electromagnetic fields are: The Fig. 8 below shows the and contributions of the perpendicular and parallel electric field and magnetic field components to total energy for SPP's at Au/InP interface.for the antisymmetric modes, analytically evaluated as in Reference [8], are also plotted in Fig. 10

Fig. 1 .
Fig. 1.(a) A metal membrane with uniform metal-dielectric interfaces and the electrical field cross-section of a SPP mode.(b) A plasmonic crystal fabricated on top of a metal membrane.

Fig. 2 .
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 .z k E is an out-of-plane component of the electric normal to it.

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 [13-16] 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 × 10 16 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 ∆υk shows a spectral halfwidth.(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.

Fig. 4 .
Fig. 4. Purcell enhancement factors at various frequencies from quantum analysis -sp F (black

Figure 4 F
Figure 4 compares the classical and quantum mechanically derived SE enhancement, ) ( sp υ I and ) ( sp υF, 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 ) ( sp

S
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 the area taken by one unit cell, the mode volume j V , K 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. ( at position r A in a periodic structure: . Assuming that the overlap of the nearby resonators is weak enough that

Fig. 5 .
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 Rex.An exciton lies at the position rA in a defect.(b) A single defect in a 2D-photonic crystal with a permittivity ε 0 (r).

Figure 6 (
Figure 6(b) shows the dispersion diagram for antisymmetric modes around the Γ -point,

Fig. 6 .
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.

Fig. 7 .
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 ∑ ∑ j j D K K

Fig. 8
Fig. 8. Values of ) 1 /( 1k Θ + , W W Ez / , W W E / // (a) (blue circles) and in close agreement with the corresponding 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, the comparison between QED and CED analyses, which are made in Section 2 for a single metal-dielectric boundary, be the enhanced SE rate into symmetric and antisymmetric modes, normalized by nΓ 0 , respectively.Comparing ) (υ anti I shown in Fig. 10(b) with ) , ( k i anti υshown in Fig.9(b), it is evident that the small-g v regime of the antisymmetric mode's dispersion diagram results in a peak in ) 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

Fig. 9 .
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 ) , ( k i layer υ estimated for the structure