Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure

Bai, Qiang; Perrin, Mathias; Sauvan, Christophe; Hugonin, Jean‐Paul; Lalanne, Philippe

doi:10.1364/oe.21.027371

Cited by 202 publications

(296 citation statements)

References 13 publications

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“…In a full 3D calculation the use of a PML may lead to more significant errors. For instance, using the approach of [8], a deviation of about 2% of the PF from the direct numerical evaluation of the GF was found [8,18] for an optical mode in a gold rod with cylindrical symmetry. It is also important to note that the derivation of the normalization in [8] uses a frequency-independent (nondispersive) PML, while in finite-difference time domain methods typically a dispersive PML is used to reduce numerical complexity.…”

Section: Appendix D: Comparison With Sauvan Et Almentioning

confidence: 99%

“…(6) can be solved numerically by replacing the δ-like source term with a finite-size dipole at a given position r . The mode volume can then be evaluated by calculating numerically the EM field emitted at frequencies close to the pole of the GF under consideration using an iterative algorithm converging towards the pole, as has been recently shown [18]. Our approach is instead based on the exact form of the solution of Eq.…”

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Exact mode volume and Purcell factor of open optical systems

2016

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The Purcell factor quantifies the change of the radiative decay of a dipole in an electromagnetic environment relative to free space. Designing this factor is at the heart of photonics technology, striving to develop ever smaller or less lossy optical resonators. The Purcell factor can be expressed using the electromagnetic eigenmodes of the resonators, introducing the notion of a mode volume for each mode. This approach allows an analytic treatment, reducing the Purcell factor and other observables to sums over eigenmode resonances. Calculating the mode volumes requires a correct normalization of the modes. We introduce an exact normalization of modes, not relying on perfectly matched layers. We present an analytic theory of the Purcell effect based on this exact mode normalization and the resulting effective mode volume. We use a homogeneous dielectric sphere in vacuum, which is analytically solvable, to exemplify these findings. We furthermore verify the applicability of the normalization to numerically determined modes of a finite dielectric cylinder. DOI: 10.1103/PhysRevB.94.235438 In his short communication [1] published in 1946, Purcell introduced a factor of enhancement of the spontaneous emission rate of a dipole of frequency ω resonantly coupled to a mode in an optical resonator, which is now known as the Purcell factor (PF). He estimated this factor aswith the speed of light c, the quality factor Q n of the optical mode n, and its effective volume V n , the latter being evaluated as simply the volume of the resonator. This rough estimate of V n has subsequently been refined [2,3] towhere r d is the position of the dipole and e the unit vector of its polarization. In this expression, the electric field of the modewhere ε(r) is the permittivity of the resonator. The integration is performed over the "quantization volume" V. However, for an open system this volume is not defined, and simply extending V over the entire space leads to a diverging normalization integral since eigenmodes of an open system grow exponentially outside of the system due to their leakage. This issue was mostly ignored in the literature and patched by phenomenologically choosing a finite integration volume. Such an approach can result in relatively small errors when dealing with modes of high Q n , as we will see later. However, the fundamental problem of calculating the exact mode normalization and thus of the mode volume remained.Recently, a solution to this problem has been suggested. Kristensen et al. [4,5] have used the normalization which was introduced by Leung et al. [6] for one-dimensional (1D) optical systems and later applied [7] to three dimensions. In this approach, the volume integral in Eq. (3) is complemented by a surface term and the limit of infinite volume V is taken:where ω n is the complex eigenfrequency of the mode and S V is the boundary of V. It was found [4] that for high-Q modes, the surface term was leading to an approximately converging value of the normalization with increasing V, for the limited ...

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Section: Appendix D: Comparison With Sauvan Et Almentioning

confidence: 99%

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Exact mode volume and Purcell factor of open optical systems

2016

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“…Due to the outgoing radiation conditions, the Hamiltonian of the system is non-hermitian and the normal modes of the system have complex eigenfrequencies ω µ [26,31,32]. Let us notice that a normalization condition of the kind V |E µ (r, ω µ )| 2 dV cannot be applied to normal modes with resonant complex frequencies ω µ because E µ (|r| → ∞, ω µ ) → ∞ [20,[26][27][28][29]33]. In this work, we define the normal modes associated to the eigenfrequencies ω µ following the normalization condition given in [28,34]:…”

Section: Introductionmentioning

confidence: 99%

Purcell factor of spherical Mie resonators

Zambrana‐Puyalto

Bonod²

2015

Phys. Rev. B

139

112

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We present a modal approach to compute the Purcell factor in Mie resonators exhibiting both electric and magnetic resonances. The analytic expressions of the normal modes are used to calculate the effective volumes. We show that important features of the effective volume can be predicted thanks to the translation-addition coefficients of a displaced dipole. Using our formalism, it is easy to see that, in general, the Purcell factor of Mie resonators is not dominated by a single mode, but rather by a large superposition. Finally we consider a silicon resonator homogeneously doped with electric dipolar emitters, and we show that the average electric Purcell factor dominates over the magnetic one.arXiv:1501.07452v2 [physics.optics]

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“…However, this scattered field is not the same field as the QNM and it cannot be properly normalized for use in quantum optics, e.g., for obtaining the Purcell factor and effective mode volume [12]-two well known quantities that help describe the underlying physics of cavity light-matter interactions. While some frequency-domain techniques exist for computing the QNMs of MNRs [18,19], it is highly desirable to be able to compute the QNMs using the commonly employed and general FDTD technique.The FDTD method is already widely used by the plasmonics community, and its accuracy for obtaining the enhanced field has been verified against other numerical techniques such as the multipole expansion technique [20]. In addition, the LDOS can be calculated by employing a dipole excitation source [21][22][23], which can also model local field effects, e.g., associated with finite-size photon emitters inside a MNR [21].…”

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confidence: 99%

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confidence: 99%

Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach

Hughes

2014

Opt. Lett.

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We describe how the finite-difference time-domain (FDTD) technique can be used to compute the quasinormal mode (QNM) for metallic nano-resonators, which is important for describing and understanding light-matter interactions in nanoplasmonics. We use the QNM to model the enhanced spontaneous emission rate for dipole emitters near a gold nanorod dimer structure using a newly developed QNM expansion technique. Significant enhanced photon emission factors of around 1500 are obtained with large output β-factors of about 60%. [7]. While the optical properties of MNRs are being actively pursued, numerically modelling of the basic cavity physics is extremely demanding, and analytical solutions of the modes only exists for very simple structures such as spheres. For resonant cavity structures, the natural modes of the system are called quasinormal modes (QNMs) [8,9], defined as the frequency domain solutions to the wave equation with open boundary conditions (the Silver-Müller radiation condition). Kristensen et al. [10] first used the QNMs to introduce a rigorous definition of the "generalized effective mode volume" and Purcell factor [11], and applied these results to photonic cavity structures. For MNRs, the QNMs also form the natural starting point for developing analytical theories of lightmatter interactions in nanoplasmonics [12][13][14].One of the most common numerical techniques for obtaining the cavity mode for dielectric cavities is the finite-difference time-domain (FDTD) technique [15]. The FDTD technique allows one to simulate open boundary conditions with "perfectly matched layers" (PMLs) located at the leaky mode region outside the cavity. For dielectric cavities, this open-boundary FDTD approach has been shown to yield excellent agreement with direct integral equation methods [10]. Other timedomain techniques such as the discontinuous Galerkin time-domain approach can also use PMLs [16]. For metals, a major problem occurs when using the usual mode calculation approach with a dipole excitation source, since the extracted mode depends sensitively on the dipole position [17], and is therefore incorrect. Thus it has been common practice to excite the MNR with a plane wave source and obtain the scattered field. However, this scattered field is not the same field as the QNM and it cannot be properly normalized for use in quantum optics, e.g., for obtaining the Purcell factor and effective mode volume [12]-two well known quantities that help describe the underlying physics of cavity light-matter interactions. While some frequency-domain techniques exist for computing the QNMs of MNRs [18,19], it is highly desirable to be able to compute the QNMs using the commonly employed and general FDTD technique.The FDTD method is already widely used by the plasmonics community, and its accuracy for obtaining the enhanced field has been verified against other numerical techniques such as the multipole expansion technique [20]. In addition, the LDOS can be calculated by employing a dipole excitation source [21][22][23], wh...

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Efficient and intuitive method for the analysis of light scattering by a resonant nanostructure

Cited by 202 publications

References 13 publications

Exact mode volume and Purcell factor of open optical systems

Exact mode volume and Purcell factor of open optical systems

Purcell factor of spherical Mie resonators

Design of an efficient single photon source from a metallic nanorod dimer: a quasi-normal mode finite-difference time-domain approach

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