# Fluorescence in nonlocal dissipative periodic structures

## Abstract

We present an approach for the description of fluorescence from optically active material embedded in layered periodic structures. Based on an exact electromagnetic Green’s tensor analysis, we determine the radiative properties of emitters such as the local photonic density of states, Lamb shifts, line widths etc. for a finite or infinite sequence of thin alternating plasmonic and dielectric layers. In the effective medium limit, these systems may exhibit hyperbolic dispersion relations so that the large wave-vector characteristics of all constituents and processes become relevant. These include the finite thickness of the layers, the nonlocal properties of the constituent metals, and local-field corrections associated with an emitter’s dielectric environment. In particular, we show that the corresponding effects are non-additive and lead to considerable modifications of an emitter’s luminescence properties.

###### pacs:

42.70.Qs, 73.20.Mf, 78.67.Pt, 42.50.Pq## I Introduction

Modern technology relies more and more on the ability of building microscopic devices based on carefully designed nano-structured materials. For example, engineered stacks of differently doped semiconductors are used in modern transistors and (nano-structured) dielectrics with different indices of refraction are used to guide light in the most exotic ways. A special class of nano-structures consists of alternating metallic and insulating layers arranged into one-dimensional periodic lattices. When carefully designed such combinations of plasmonic and dielectric materials lead – within the effective medium limit – to effective hyperbolic dispersion relations (1); (2); (3) that may be exploited for a number of applications such as subwavelength imaging (4), strong nonlinearities (5), emission engineering (6), and many more. The dispersion relations of these fictitious, spatially uniform hyperbolic meta-materials (HMMs), support radiative modes with infinitely large wave vectors which, in turn, lead to broadband super-singularities of the local photonic density of states DOS (LDOS). It has been recognized that in the actual nano-structure (i.e., without the effective medium description) these singularities become regularized (i) by the fact that for sufficiently large wave-vector values the actual lattice-structure will be resolved (1) and (ii) via the nonlocal properties of the metallic constituents (1); (7). The LDOS itself can be obtained from the electromagnetic Green’s tensor of the relevant structure. In turn, the LDOS modifies the spontaneous decay rate of an emitter and the corresponding so-called Purcell factor can be determined from the electromagnetic Green’s tensor (7).

Most of the above computations, however, have ignored the effects of local-field corrections in the dielectric layers, despite the fact that these corrections, too, may significantly contribute to the decay dynamics of the emitter. Therefore, in the present work, we include these effects into the Green’s tensor formalism and provide a comprehensive study of the non-additive interplay between the above regularization mechanisms and the local-field corrections on the luminescence properties of emitters embedded in finite-sized and infinitely extended layered HMM-type structures. Besides the aforementioned modified decay rates, this also includes frequency shifts that emitters experience when being embedded in such structures. In addition, we study these effects in the context of different material models for the nonlocal optical properties of the plasmonic constituents and identify analogies and differences in the results.

## Ii Decay Rate and Frequency Shift

The dynamical properties of emitters are correlated with the environment surrounding them. In particular, the decay rates from excited states or the intrinsic transition frequencies depend on the LDOS associated with the electromagnetic environment. Since the latter is strongly correlated with the geometry and the optical properties of the objects filling the space around the emitter, it is not surprising that an appropriate choice of these two characteristics can lead to enhancement or suppression of decay rates as well as to frequency shifts of transitions with respect to their vacuum (empty space) values. Even strong non-Markovian effects can be realized (8). In this section we briefly review the general approach for computing decay rates and frequency shifts of emitters that are embedded in an arbitrary electromagnetic environment.

### ii.1 Spontaneous decay and local field corrections

We will focus first on the spontaneous decay and calculate the (Purcell) enhancement factor. Within the standard theory for a simple two-level emitter with transition frequency located at , an emitter’s transition rate is given by (9); (10)

(1) |

Here, and denote, respectively, the vacuum permittivity and the emitter’s dipole operator while represents the free-space wave vector. The quantum average is performed over the emitter’s ground state. The quantity is the electric Green’s tensor, i.e., the solution to the Maxwell equations for an electric point-dipole oscillating with frequency located at subject to the boundary conditions imposed by the structure surrounding the emitter.

In vacuum the spontaneous decay is well defined and has been studied by many authors. In this case the Green’s tensor is

(2) |

In this expression, we have decomposed the vector into a unit vector and a length . Equation (2) is nothing but the electromagnetic field emitted by a dipole radiating in vacuum (11). Using this expression, it is straightforward to show

(3) | ||||

(4) |

Here, we have also averaged over the dipole’s direction so that
.
In the last term we have used the relation
that
connects the ground-state and angle-averaged dipole moment with the static polarizability
. When the dipole is instead embedded in a homogeneous dielectric medium, things are,
however, more complicated.
The corresponding Green’s tensor can be obtained by the formal replacement
where is the dielectric
function describing the medium.
From simple considerations, one would expect that
. In reality, in a naive
application of Eq.(1), with the Green’s tensor given by Eq.(2) for a homogeneous
dielectric medium, even a minute amount of
*dissipation*
would lead to a divergent decay rate. Physically, this difficulty can be understood by thinking
that in a continuum approximation, resulting from a macroscopic average, the emitter can superpose
with an atom of the dielectric leading to a divergent interaction
(16).
The spontaneous decay of an emitter embedded in a dielectric medium thus represents an example of
how dissipation can substantially complicate the theory of quantum phenomena.

For the realistic description of spontaneous decay processes we have to take into
account that the electromagnetic field “seen” by an emitter that is embedded in a medium is *not*
given by solutions of the *macroscopic* Maxwell equations. Rather, the fields provided by
macroscopic electrodynamics are the result of spatial averages where materials are described
as continuous entities characterized by permittivities and permeabilityies. At the microscopic
level the “granular” structure of the medium becomes important and the *local field* felt
by the emitter can be rather different from the result of the macroscopic averaging procedure.
In turn, this may have a very significant impact on the emitters’ dynamics, notably if they are
exposed to multiple scattering effects originating from a complex nano-structured environment.
In order to treat the local-field problem, one can distinguish two different cases
(12); (13); (14); (15); (16):
(a) The emitter is of the same species as the atoms (or molecules) that constitute the host medium
or
(b) the emitter is of a different species as the host medium, i.e., an impurity, a substitution etc.
In both cases (and even for non-dissipative media), the spontaneous decay is non-trivially modified
with respect to the result for vacuum presented above. The first scenario is described within
the virtual-cavity model (13); (14); (15); (16), where the Green’s tensor
is appropriately modified to remove all unphysical divergences. The second scenario is treated within
the Onsager real-cavity model where the emitter is thought to be placed in vacuum at the center of
a spherical cavity carved into the host medium (12); (13); (14); (15); (17); (18),
where the cavity radius is essentially given by the distance to the next atom (molecule) of the dielectric
(see Fig. 1).
In the present work, we will restrict ourselves to systems for which the real-cavity model
may be applied.

Before describing how Eq. (1) has to be modified in order to include the local-field correction, we first consider the impact of multiple scattering effects by complex nano-structured systems on the electromagnetic Green’s tensor. If all media are linear, we have that

(5) |

where the tensor describes the propagation of the electromagnetic field within a homogeneous medium with the same permittivity and permeability as the medium at position . Within the real-cavity model this allows us to identify vacuum as the homogeneous medium required for the homogeneous Green’s tensor, i.e., we have . Further, the tensor represents the scattered Green’s tensor and takes into account the multiple scattering processes due to the material interfaces in the nano-structure. For simple geometries the expressions for are well-known and have been reported in the literature (we refer to (9); (19); (11); (20) and our discussion of layered systems below). In the local-field corrected Green’s tensor within the framework of the real-cavity model, the emitter is placed in vacuum enclosed at the center of a spherical cavity with radius much smaller than the optical wavelength (). In our specific case of alternating metallic and dielectric layers, this spherical local-field cavity with the emitter in its center is located inside a central dielectric layer with permittivity sandwiched between a finite or infinite number of further layers on either side. This means that the scattering Green’s tensor is determined from both, the boundary conditions on the cavity sphere and on interfaces between the layers. Clearly, these two sets of boundary conditions act in a very non-additive way, thus complicating the evaluation of the Green’s tensor. Fortunately, this topic has been extensively discussed in the literature (17); (18); (21); (22) so that we may directly utilize that in this case the scattered Green’s tensor can be written as

(6) |

Here, is the scattered Green’s tensor that exclusively results from the multiple scattering at the layers’ interfaces and the effect of the spherical cavity appears in two terms: An offset that will survive even if we remove all the scattering from all layers and a prefactor to the scattering Green’s tensor that account for the multiple scattering at layer interfaces. This specific structure of the full scattering Green’s tensor makes quite explicit the non-additive character of the local-field and the multiple-scattering corrections. Within the real-cavity model, we have (17); (18); (21); (22)

(7) | ||||

(8) |

where we have introduced the abbreviations and while and denote, respectively, the spherical Bessel function of order one and the spherical Hankel function of the first kind of order one (the prime indicates the derivative with respect to the argument of the corresponding Bessel/Hankel functions). We would like to emphasize that and, consequently, also the wave vector of the dielectric host medium are complex valued due to the fact that the host medium’s permittivity is, in general, complex valued. Also, as a result of the foregoing, the spontaneous decay formally depends on an external parameter, i.e., the cavity radius . As stated above, this parameter is fixed by the microscopic arrangement of the host dielectric atoms and must be experimentally determined. However, for optical frequencies, one may even consider an expansion of the above expressions in the small parameter (see Refs. (17); (18); (21); (22)).

Upon averaging over the emitter’s dipole orientations, the local-field corrected emission rate is given by

(9) |

where we have introduced the tensor

(10) |

The last step in our approach is the determination of , for which we have to specify the arrangement of layers. In this work, we consider one central dielectric layer containing the emitter that is symmetrically sandwiched by a finite or infinite number of identical bilayers that consist of one plasmonic and one dielectric material. For simplicity, we restrict ourselves to the case where all dielectric layers are made from the same material (see Fig. 2). From the emitter’s point of view, the structure resembles a cavity that is formed by two Bragg mirrors and is filled with a dielectric material with permittivity . We align the -direction with the stacking direction of the layers and denote the emitter’s distance from the nearest plasmonic layer with . Further, the left and right Bragg mirror are, respectively, labeled with indices “1” and “2”. Then, we may decompose the scattering Green’s tensor into components parallel and perpendicular to the stacking direction of the layers

(11) |

In turn, the parallel and perpendicular components of the Green’s tensor, and , may be expressed through the structure’s geometrical parameters and the (frequency- and wave-vector dependent) reflection coefficients , and () of the two Bragg mirrors for s- and p-polarized plane waves. Specifically, if is the cavity length, these components of the Green’s tensor read as (20)

(12a) | ||||

(12b) |

where . For our symmetric geometry, the reflection coefficients of the Bragg mirrors are identical so that and . For simplicity, we will subsequently restrict ourselves to the case of so that the emitter is located at the center of the cavity (see Fig. 2). The complete determination of the scattered Green’s tensor components requires the evaluation of the reflection coefficient which will be the subject of the section III.

### ii.2 Frequency shift

In addition to the spontaneous decay rate, modifications of the electromagnetic environment as well as local-field corrections have an impact on the emitter’s energy levels. The treatment of this effect leverages on the same tools as discussed above, i.e., the determination of the Green’s tensor. For a two-level emitter the standard approach relies on the connection between the Casimir-Polder energy () (24); (25) and the change in the emitter’s ground state energy () (9); (19): . (This connection can also be generalized to higher energy levels (9); (19).) If is the ground-state polarizability, the Casimir-Polder energy is given by

(13) |

where we have used the local-field corrected expression (6) for the scattered Green’s tensor. For a two-level emitter and after averaging over the emitter’s dipole direction, we obtain

(14) |

where denotes the unit tensor. Within the framework of second-order perturbation theory (9); (19); (24) the above expression is equivalent to an additional contribution to the vacuum-induced Lamb shift of the ground state energy that is generated by the nano-structure. From Eq.(6), we obtain two distinct contributions. The first is exclusively associated with the local-field correction

(15) |

where

(16) |

The second contribution

(17) |

describes the impact of the Bragg mirrors on the shift and, therefore, depends on the detailed characteristics of the structure such as the distance of the emitter from the dielectric/metal interfaces etc. In the following, we will exclusively focus on this second contribution, since for a specific dielectric it is the only one that can be tuned as a function on the geometric parameters of the surrounding nano-structure (21).

## Iii Periodic Structures: Bloch equation and Scattering Coefficients

The behavior of the electromagnetic field within a periodic structure can be analyzed in the framework of the Bloch theorem (26). If the periodic medium is composed of a stacking sequence of layers, the problem can be tackled analytically with the help of a transfer-matrix approach (27); (28). Specifically, the continuity of the transverse field components leads to boundary conditions across each layer. For the metallic layer, we have (c.f. Fig. 2)

(18) |

while for the dielectric layer, we obtain

(19) |

Here, indicates again the polarization while and represent the transfer matrices associated with each layer. Specifically, as we will exclusively consider spatially local material models for the dielectric, we have introduced the subscript “lc” for dielectric layer transfer matrices. Similarly, as we will mainly focus on spatially nonlocal material models for the metal, we have introduced the subscript “nl” for metal layer transfer matrices. Further, the propagation across a bilayer can be written as

(20) |

where the corresponding transfer matrix is given by

(21) |

For a local dielectric material, the transfer matrix can be written as

(22) |

where, for p- and s-polarization, respectively, we have and . Further, we have defined the local surface impedances

(23) |

In the case of a metallic layer with nonlocal material model the entries of depends on the specific model used to describe the nonlocality (see section IV). Nevertheless, the matrix has a similar structure as in the local case

(24) |

with functions , and that have to be determined from the specific material model. For instance, reciprocity stipulates that both matrices and have a determinant equal to one. While this clearly is fulfilled for , this condition imposes certain restriction to the general form describing the non-local case

(25) |

### iii.1 Infinite Bragg Mirrors

In an infinite periodic structure the Bloch theorem further requires that the field values across a unit cell satisfy

(26) |

This leads to an eigenvalue equation that connects the eigenvalues with the generally complex values of the Bloch vector . Since , the eigenvalues have the form and using the invariance of the trace of a matrix we obtain the generalized Bloch equation

(27) |

where, for simplicity of notation, we have dropped the polarization superscript. For the computation of the scattering Green’s tensor, we require the reflection coefficients from infinitely extended half-spaces and this can be derived in terms of the eigenvector of the transfer matrix (23); (28): Each of the two eigenvectors corresponds to a Bloch mode propagating either to the right or to the left of the unit cell. The components of the mode are the corresponding electric and magnetic fields from which it is possible to derive the surface impedance for the periodic structure (23). We obtain

(28) |

which, upon inserting the specific form of the transfer matrices, explicitly reads as

(29) |

where, again, the polarization superscript has been suppressed. The reflection coefficients for infinite periodic Bragg mirrors can then be written as

(30) |

### iii.2 Finite Bragg Mirrors

The transfer matrix can also be utilized for obtaining the reflection coefficients for structures with a finite number of bilayers (27); (28) embedded into half-spaces of the host dielectric materials. For a single metallic layer (i.e., a slab) with thickness the reflection and transmission coefficients, and , are given by

(31) |

(32) |

Here, again, the polarization superscript has been suppressed. The expression for the reflection coefficients of a finite structure composed of a sequence of can then be written as (27); (28)

(33) |

where, we recall that is the Bloch vector.

## Iv The Description of Nonlocal Media

The description of the nonlocal properties of metals has been the subject of many publications in the past (we refer to (30) for an overview of earlier works) and, in the context of nano-plasmonics, has recently experienced renewed interests. Here, we give a brief review of two different models which have been formulated in the literature and provide the results that are required for our computations.

### iv.1 The SCIB Model

We first consider the approach which goes under the name of *semi-classical infinite barrier* (SCIB) model
(29); (30),
that has been utilized for the description of the anomalous skin effect
(31); (32); (33).
In the SCIB model, the interface is described very crudely through an infinite barrier but it takes into account
the most relevant physical phenomena inside the metal
(29); (30).
Within this approach, the electrons are treated as a classical ideal gas that is governed by the Fermi-Dirac
statistics and whose dynamics is described via the Boltzmann equation. It is further assumed that the electrons
in this nonlocal medium specularly reflect at the interface with another medium
(34).
It has been shown that the fields in the interior of such a nonlocal medium, are identical to the fields that
arise from a current sheet source at the surface. Since in our case, the system is invariant with respect to
translations in the plane and an interface is located at this sheet current has the form (we
follow the notation of Ref. (30))

(34) |

where is the component of the wavevector orthogonal to the -direction ( denotes the unit vector along the -direction). The corresponding electric field has the form

(35) |

and an analoguous expression holds for the magnetic field. If is the three-dimensional wave vector, the dielectric tensor of the metal within SCIB can be written as

(36) |

The Maxwell equations can then be solved in terms of the longitudinal and transverse dielectric functions, and , respectively, and we obtain that the component of the electric and magnetic field parallel to the layer interface can be written as

(37) | ||||

(38) | ||||

(39) | ||||

(40) |

where is the vacuum impedance and the dimensionless functions and are defined as

(41a) | |||

(41b) | |||

(41c) |

At this point, we would like to note that and . Also, it is straightforward to show that .

The above expressions still do not provide an essential piece of information, i.e. the expressions for the dielectric function. In fact, this is where the Boltzmann equation for the dynamics of the electronic fluid (semi-classical approach) has to be employed. In the Boltzmann equation approach, the most important aspect is the treatment of collisions. In the single relaxation-time approximation the corresponding analysis delivers (30); (35); (31); (32)

(42a) | ||||

(42b) |

Here, we have introduced the functions (31)

(43a) | ||||

(43b) |

Further, we have abbreviated where , , and denote, respectively, the plasma frequency, the Fermi velocity, and the dissipation rate of the metal. In addition, the function describes the dielectric function associated with a potential dynamic behavior of the ionic background charge. In the following, we will disregard this contribution for all material models and, consequently, set . Using the above expressions, we may determine the entries of the transfer matrix and obtain

(44a) | |||

(44b) | |||

(44c) |

### iv.2 The Hydrodynamic Model

The SCIB is just one of the possible approaches for treating the nonlocal behavior of metals and alternative, however less realistic (29), descriptions may be based an hydrodynamic models.

The standard approach (for a recent example see, e.g., ref. (38)) describes the metal’s conduction electrons as a compressible fluid and leads to the following equation for the free current in the metal

(45) |

Here, describes the electron fluid’s compressibility. The value of this constant depends on the frequency regime one is interested in. We have is appropriate at low frequency, while should be used in case of a high frequency dynamics (36); (37). Here we chose this second value since we will be interested in phenomena around the plasma frequency. From the above equation, we can directly infer the longitudinal and transverse components of the dielectric tensor. If we follow the above-stated premise that the ionic background does not provide additional contributions to the dielectric behavior from bound charges, we obtain

(46a) | ||||

(46b) |

where, the transverse dielectric constant is identical to the standard (spatially local) Drude dielectric constant . Thus, in the hydrodynamic model, only the longitudinal part of the electric field is actually affected by the nonlocal behavior of the metal.

Following Ref. (23) (see also (7)), inside the nonlocal medium, the field is no longer transverse and is instead given by a superposition of left- and right-propagating waves with transverse and longitudinal wave vectors, and , respectively, where

(47a) | ||||

(47b) |

The wave vector of the longitudinal wave fulfills .

The continuity of the tangential components of and relates two unknown coefficients (per polarization) in the dielectrics with four unknowns (two transverse and two longitudinal) inside the metal. This requires two additional boundary conditions (ABCs) in order to arrive at a well-determined system of equations. In case of the hydrodynamic model, it is reasonable to assume that the density of free carriers in the metal does not create any singularity at the dielectrics/metal interface. An application of Gauss’ theorem immediately yields that the normal component of the displacement field must be continuous across the interface

(48) |

where, and , respectively, denote the electric field in the metal and in the (spatially local) dielectric material. In other words, the normal component of the electric fields exhibits a jump across the metal-dielectric interface, the magnitude of which is governed by the value of . Using the continuity equation of the electric charge, the above relation of the normal component of the electric field is also equivalent to the continuity of the orthogonal component of the current of free carriers across the interface. If in the dielectric there are no free carriers, this is equivalent to a vanishing at the interface of the metal. A second boundary condition that ensures consistent optical properties is obtained by requiring the scalar electric potential to be continuous across an interface (23); (40). With these two ABCs and following a reasoning similar to the one described in the previous subsection, we have the -polarization

(49) |

Here, the transfer matrix may be computed from the interface matrix (23)

(50) |

and the propagation matrix

(51) |

where we have introduced the abbreviations

(52) |

The main difference of these ABCs for the hydrodynamic model with respect to the local case is that in order to take into account the longitudinal waves (bulk plasmons) in the metals we have to add the extra degrees of freedom, and the potential , in the description of the field. These longitudinal waves (bulk plasmons) can only be excited when the electric field in the dielectric exhibits a non-zero component orthogonal to the surface. As this is not the case for s-polarized radiation, we can take over the results of the description of s-polarized waves in local (metallic) media

(53) |

where the transfer matrix is

(54) |

The situation is entirely different in the case of p-polarized radiation: In the local dielectric material just next to the interface, we have

(55) |

The boundary condition at the interface implies that and , leading to the relation which is valid inside the metal just next to the interface (7); (23). This allows us to eliminate and from Eq.(49) so that we finally obtain

(56) |

The entries of the transfer matrix (see eq. (24)) in the hydrodynamic model for p-polarization as thus given by

(57a) | |||

(57b) | |||

(57c) |

### iv.3 Discussion of the Nonlocal Material Models

The two material models described above, display several analogies but also profound differences (29); (30). If we consider the expressions in Eq.(41), the residue theorem allows us to show that can be written as the sum of waves that propagate with wave vectors that are solutions of and . Clearly, these solutions correspond to transverse and longitudinal waves, respectively. This correspondence between the models is, however, only qualitative as the expressions for longitudinal and transverse dielectric functions are quite different. For instance, while the hydrodynamic model only exhibits nonlocal modifications to the longitudinal part of the electromagnetic field, the SCIB predicts nonlocal modifications for both the longitudinal and the transverse part of the field. Probably the most apparent difference between the two models concerns the boundary conditions at the interface. While the SCIB model relies on the symmetries of the Boltzmann equation to determine the behavior of electrons at an interface with dielectric materials, the hydrodynamic description uses the non-locality to implement a finite density of electrons at the interface which removes the discontinuity in the -component of the displacement field.

Despite these differences, the two non-local models provide qualitatively similar results for the reflection coefficients of the infinitely periodic structures and of a thin silver slab embedded in silica matrix. In Fig. 3 we represent the imaginary part of the corresponding reflection coefficients for the p- and s-polarization. The quantities are plotted as functions of the lateral wave-vector for a fixed frequency (). We note that for the periodic structure the usual effective medium approach (EMA) in terms of a local dielectric functions (42); (41) provides a good description of the system for sufficiently small wave-vectors (see Fig.3) (2). As expected, (2) the agreement degrades at larger wave-vectors. The plasmon resonance of the periodic structure (, see top left panel of Fig. 3) coincides with the surface-plasmon-polariton (SPP) of a silica/metal interface

(58) |

The thin single metallic layer exhibits two resonances associated with the symmetric and antisymmetric coupling of the SPPs on the two metal/silica interfaces ( and , see top right panel of Fig. 3) which in other context’s are know as the short-range and the long-range SPP, respectively (39)). In the local description for frequency smaller than and small thickness the resonances’ positions are approximatly given by

(59a) | |||

(59b) |

which correspond to the values for the symmetric (near the light cone) and anti-symmetric SPPs. The anti-symmetric resonance is much stronger than the symmetric resonance. It is also worth noting that for the anti-symmetric SPP the SCIB gives rise to a value which is different from the value for the local and the hydrodynamic description. For s-polarized radiation, the behavior is much simpler and the nonlocality only slightly affects the reflection coefficients. In all cases, a characteristic abrupt change occurs at the light cone, i.e. for .

## V Results

We now apply the above formalism to study the modified radiation dynamics of an emitter embedded in two distinct structures. The first structure consists of a central cavity silica layer () that is symmetrically sandwiched between infinite sequences of bilayers of silver () and silica () as depicted in Fig. 2. The second structure comprises the same central cavity silica layer that is symmetrically sandwiched between two silver layers with thickness and this composite slab-structure is embedded into two half spaces of silica. The dielectric properties of silica are described via a three-oscillator model (43) and we consider the above-discussed and widely used material models for silver, i.e., the local Drude model, the SCIB model and the hydrodynamic model. All these models employ the same plasma frequency eV (nm) and damping constant eV. Additionally, the SCIB model and the hydrodynamic model use the Fermi velocity m/s) of silver (1). In both of the above structures, we position an emitter midway in the cavity layer () and the radius of the real-cavity model for the local field correction is .