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Semi-analytical model for design and analysis of grating-assisted radiation emission of quantum emitters in hyperbolic metamaterials Achiles Mota, Augusto Martins, Heidi Ottevaere, Wendy Meulebroeck, Emiliano Rezende Martins, John Weiner, Fernando Lisboa Teixeira, and Ben-Hur Borges ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.7b01324 • Publication Date (Web): 26 Apr 2018 Downloaded from http://pubs.acs.org on April 26, 2018
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ACS Photonics
Semi-analytical model for design and analysis of grating-assisted radiation emission of quantum emitters in hyperbolic metamaterials Achiles F. da Mota1,2, Augusto Martins1, Heidi Ottevaere2, Wendy Meulebroeck2, Emiliano R. Martins1, John Weiner4, Fernando L. Teixeira3, and Ben-Hur V. Borges1,*. 1
Department of Electrical and Computer Engineering, University of São Paulo, CEP 13566-590, São Carlos, SP, Brazil. 2 Brussels Photonics Team, Department of Applied Physics and Photonics, Vrije Universiteit Brussel, 1050 Brussels, Belgium. 3
ElectroScience Laboratory, The Ohio State University, Columbus, Ohio 43210, USA. Physics Institute of São Carlos, University of São Paulo, CEP 13566-590, São Carlos, SP, Brazil.
4
KEYWORDS. Hyperbolic Metamaterials; Metallic Gratings; Purcell factor; Quantum emitters; Dipole’s Photonic Density of States; Fluorescence Lifetime; Outcoupling optimization; ABSTRACT: We propose a semi-analytical method to model, both in two- and three-dimensions (2D and 3D, respectively), the radiation emission of quantum emitters (QEs) interacting with nano-patterned structures. We then investigate the emission from QEs near a hyperbolic metamaterial (HMM) with a metallic cylindrical grating on its top and a polymethylmethacrylate (PMMA) substrate embedded with QEs on its bottom. The optimization of the cylindrical grating is carried out first using a 2D model (due to its low computational cost), followed by a performance study based on a 3D model. We show that an appropriate choice of grating parameters (period, height and fillfactor) allows not only the control of the QE emission direction but also the increase of both the Purcell factor and the total power coupled from the HMM into free-space. In addition, the proposed method provides a detailed mapping of both the Purcell factor and the radiated power as function of position, enabling us to understand how the QE location affects its behavior. Furthermore, we demonstrate that the QEs with the highest Purcell factor (viz., perpendicularly polarized ones) contribute more to the power radiated into the far field than previously expected. We also show that in addition to a high Purcell factor of about 145, perpendicularly polarized QEs radiate up to two times more power if placed 10 nm from the HMM as they would in free-space.
The study of incoherent light sources for high-speed optical communication has attracted much attention in the past few years due to their lower fabrication and operational costs when compared to coherent sources such as lasers1–3. The most efficient way of generating incoherent light is by means of quantum emitters (QEs), such as quantum dots, quantum wells, fluorescence molecules, among others2,4–7. Even though QEs are cheaper, their longer lifetimes imply lower modulation speeds when compared to their coherent counterparts3. One of the most promising approaches to overcome this limitation involves a new class of metamaterials known as hyperbolic metamaterials (HMM)8,9 that exhibit pronounced anisotropy with permittivity tensor elements with opposite signs for the parallel and perpendicular components8,9. As a result of this feature, a HMM has its isofrequency surface (IS) shaped like a hyperboloid (single- or doublesheeted depending on the signs of the perpendicular and parallel elements of its tensor). This exotic dispersion behavior can be exploited in a wide range of applications such as biosensing10, hyperlenses11,12, and broadband ab-
sorbers13. Two approaches are normally used to create HMMs, namely, a metallic wire medium and planar alternating stack of thin metal and dielectric layers8,9. Both approaches are suitable for controlling the hyperbolic shape of the IS with an appropriate choice of materials and geometry. This unusual IS shape leads to the most outstanding feature of HMMs, namely, the existence of photonic states with wavevectors (k, with modulus k) much larger than those in free-space (k0, with modulus k0), resulting in a large photonic density of states (PDoS). This is because in conventional materials, the IS is an ellipsoid where only finite values of k are possible; in contrast, an ideal hyperboloid-shaped IS has no upper cut-off for k, meaning it supports an infinitely large number of high-k modes8. Unfortunately, this is only valid if spatial dispersion is not taken into consideration14–17. In real HMMs, the high-k values induce spatial dispersion and create an upper cut-off for k that limits the number of possible high-k modes18.
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Assessing the full range of PDoS is important for studying the radiation pattern of the QE. The QE emission can be decomposed into a superposition of propagating (kk0) waves. Therefore, the manipulation of PDoS can be used to control the emission pattern of these sources. Moreover, a large PDoS implies more power being dissipated by the QE, i.e., a higher Purcell factor8,9,19. A higher Purcell factor implies shorter lifetimes and faster modulation speeds. For this reason, some authors take advantage of the HMM’s high-k modes to increase the PDoS of QEs8,9,20–24. The use of HMM for increasing the Purcell factor of QE has been experimentally supported in the literature8,9,20–24, despite claims that metals are better suited for this purpose25–27. Although for particular cases9 it is possible to decrease the QE lifetime by a factor of 100, most of the power dissipated by the QEs is coupled into the HMM as surface plasmon polariton (QE) waves. The high-k states of these waves prevent them from propagating in free-space8,9,20–23. Therefore, the energy is dissipated inside the HMM with a consequent reduction of the sources’ external quantum yield. Some authors have proposed the use of a diffraction grating to solve this problem, since it converts bound SPP waves (high momenta) into propagating waves28–37 (low momenta). This approach considerably increases the power emitted by the QEs as propagating waves maintaining a high Purcell factor. Some of these works have considered QEs either on top of the metamaterials22 or surrounding the gratings28. Remarkably, for these cases, the power coupled into free-space from QEs does not change appreciably when the QEs are moved away from the HMM surface. The Purcell factor, in turn, decreases as the sources are moved away from the HMM’s surface. The net result is that the overall system’s lifetime is increased. Some authors have sought to ameliorate this issue by embedding the QEs in the middle of the HMM31–37. Although this increases the overall Purcell factor, it requires considerable more power to excite the QEs since they are inside the HMM. More importantly, the authors resorted to numerical simulations (as opposed to analytical or semi-analytical simulations) for the grating design optimization that, as shown in the Results and Discussions’ section below, do not provide clear-cut insights on how the grating affects the QEs’ behavior. In this context, we propose here a semi-analytical method that can be successfully applied to the design of diffraction gratings aimed at optimizing the conversion of high-k modes into propagating modes. In addition, this method directly provides the QE’s PDoS, which is a computationally costly task if carried out numerically. Both 2D and 3D configurations are examined, with the former being used for the grating optimization procedure due to its lower computational cost. The structure is implemented in the 3D model only after the optimum grating parameters have been obtained. The proposed HMM design assumes that QEs are embedded in the substrate since the amount of power necessary to excite them is reduced in this case31–37. As a result, the QEs closer to the HMM surface emit more power to the far field, contributing to the decrease of the system’s lifetime. In addition, the pro-
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Figure 1. Proposed geometry in which the QEs are modelled as a dipole embedded in Medium 1 (with permittivity ε1), centered at the origin and at a distance q from the HMM bottom surface. The gratings with periodicity Λx and Λy in x and y directions, respectively, are on top of the HMM and are covered with Medium 2 (with permittivity ε2).
posed method facilitates the grating design, which is useful to control the QEs’ emission directivity. Another advantage of this method is the reduced computational burden to obtain the Purcell factor and the radiated power as function of the QEs and grating relative position, highlighting the behavior of each QE to the system. Although there have been prior generic semi-analytical techniques proposed to calculate the electric field radiated by dipoles inside anisotropic stratified media38,39, as far as the authors are aware, this paper gives the first semianalytical procedure tailored for the detailed and accurate calculation and optimization of the emission pattern and the Purcell factor of QEs under a nano-patterned structure, such as the HMM with grating on its top. The remainder of this paper is organized as follows. The first section of this paper is devoted to introducing the main features of the method. The second section presents the mathematical formalism. We highlight only the most important equations, as the full derivation is provided in the Supporting Material. Next, we apply our optimization procedure to a metallic grating placed on top of an alternating stack of silver (Ag) and titanium dioxide (TiO2) layers. We begin the analysis with the 2D optimization of the grating structure, followed by the 3D simulation carried out with the optimized parameters. The figures-of-merit are the radiation pattern and the Purcell factor, both as function of the QE position. Finally, we present some concluding remarks.
METHOD We consider the geometry illustrated in Figure 1, where the QE is modelled as a dipole embedded in Medium 1 (with permittivity ε1), centered at the origin and at a distance q from the HMM bottom surface. The metallic grating with periodicity Λx and Λy in x and y directions, respectively, is on top of the HMM and is covered with Medium 2 (with permittivity ε2). The dipole source is modeled as a current density (J),
J ( r ) = − jωδ (3) ( x, y, z ) p,
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(1)
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where ω is the angular frequency, () (, , ) is the 3D Dirac delta function, p is the dipole moment and r is the position vector (r = xx + yy + zz). The 2D Fourier transform (FT) of the electric field radiated by the dipole ( ( ∥ , )) can be written as 19:
EFT i ( k || , z ) = j
ωµ1 ξ jk z 1 1 e Ps ( k || ) ⊗ L s ( k || , z ) 4π
(2)
+ P ( k || ) ⊗ L ( k || , z ) p, 1 p
where μ1 is the permeability of Medium 1, = | | = + + is the magnitude of the wavenumber in medium 1, k∥ = kxx+ kyy is the parallel component of k1, ⊗ is the outer product symbol, and ξ = 1 or -1 for wave propagating in z or -z direction, respectively. In (2), the Green’s tensor factor between brackets is written in a convenient fashion as the outer product of the vectors ! and " ! , defined here as
eξ jk zn z
k zn k + k 2 x
Ppn ( k || ) =
k y
kx k x2 + k y2
eξ jk zn z
Lnp ( k || , z ) =
Psn ( k || ) =
2 y
kn
1 k +k 2 x
1
kn
2 y
k y
T
(3)
0 ,
−k x
T
−ξ
ky
k x2 + k y2 , k zn
(4)
T
−k x 0 ,
k x k zn k x2 + k y2
k y k zn
(5)
(
)
T
−ξ kx2 + k y2 . (6)
The vectors are related to the amplitude of where the superscript n denotes the medium where the dipole is embedded and the subscript χ denotes the wave polarization: χ = s for transverse electric (TE) or p for transverse magnetic (TM). The vectors " ! on the other hand effect the decomposition of the electric field into its x-, y- and z-components. A more detailed derivation and physical insight behind the representation (3)-(6) can be found in the Supporting Material. The main goal is to optimize the grating so that the Purcell factor and the power radiated by the dipole into the far field can both be increased. The calculation of these parameters requires the knowledge of the reflected and transmitted electric fields. Since p can be decomposed into x, y, and z components, the solution is separated in two cases: the first case considers the dipole aligned along the z-axis, i.e., perpendicular to the HMM surface (perpendicularly polarized dipole), while the second case considers the dipole aligned along the x- or y-axis, i.e., parallel to the HMM surface (parallel polarized dipole). ,
!
Perpendicularly Polarized Dipole The polarization vector in this case has a component in z-direction, thus p = pzz. In this scenario, the FT of the reflected ( , ) and transmitted ( - ) electric fields are
EFT v ( k || , z ) = j
ω 2µ jk p z V pz, s Psn +V pz, p Ppn e 4π
zn
z − zv
,
(7)
∞
(k ) + (k )
i , m =−∞
0 z1 1
0 x
(k , q ) = − ∑ ||
k x0 = k x − i
z1
1 p
Lns ( k || , z ) =
V
z p ,σ
2
0 y
2
( )
0
e jk z 1q v ip,,mσ k 0 ,
k k
2π = k x − iK x , Λx
k y0 = k y − m
(8) (9)
2π = k y − mK y , Λy
(10)
where v = r or t, V = R or T, n = 1 or 2, and zv = q or zgr, for 1,2 the reflected or transmitted fields, respectively, ./,0 is the reflection (if v=r) or transmission (if v=t) coefficient of the ith diffraction order in x-direction and the mth diffraction order in y-direction for the parallel incident wavevector 8
3∥ , and k 65 = 78 − (6 )8 − :6 ; is the z-component of
the incident wavevector 3 . χ and σ indicate the polarization and are equal to p for TM polarization and s for TE polarization. The reflection and transmission coefficients are calculated using the semi-analytical Rigorous Coupled Wave Analysis (RCWA)40–42 method. It is important to point out that if spatial dispersion effects are to be accounted for, they only need to be incorporated into the RCWA formulation (a version of RCWA including nonlocal effects is avaliable43). For 3D gratings, cross polarization might occur, thus
