Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2021; 5(6): 597-605
Published online December 25, 2021 https://doi.org/10.3807/COPP.2021.5.6.597
Copyright © Optical Society of Korea.
Sungjae Park^{1}, Joonku Hahn^{2}, Hwi Kim^{1}
Corresponding author: ^{*}hwikim@korea.ac.kr, ORCID 0000-0002-4283-8982
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
An extended Fourier modal method (FMM) for optical dipole radiation in three-dimensional photonic structures is proposed. The core elements of the proposed FMM are the stable bidirectional scattering matrix algorithm for modeling internal optical emission, and a novel optical-dipole-source model that prevents numerical errors induced by the Gibbs phenomenon. Through the proposed scheme, the FMM is extended to model a wide range of source-embedded photonic structures.
Keywords: Fourier modal method, Numerical modeling
OCIS codes: (000.3860) Mathematical methods in physics; (050.1755) Computational electromagnetic methods; (050.1960) Diffraction theory
The Fourier modal method (FMM) is one of the most powerful electromagnetic analysis methods in the field of photonics [1]. In FMM, the linear modal theory for light-matter interactions is constructed based on Fourier analysis of Maxwell’s equations. The modal analysis is the crucial feature of FMM, which expands the vectorial optical field into a linear superposition of electromagnetic Bloch eigenmodes. In FMM, the dispersion relation and coupling rate of each eigenmode are extracted completely. On the basis of FMM, the modal spectrum of the vectorial optical field in photonic structures can be manifested [1]. For recent active-photonic applications such as quantum-dot light emitters or photoluminescence in nanowires, where light sources are embedded inside photonic structures, the modal analysis is also of prominent interest. A few previous studies dealt with the numerical modeling of a dipolar point source. The previous studies mainly investigated a periodic dipole arrangement in periodic structures. The discrete dipole approximation in a small-particle grid [2], the one-dimensional integral model of a dipole located within or at the boundary of a planar layered structure [3], periodic arrangement of dipoles [4, 5], RCWA analysis of a central electric dipole in a nonuniform spherical system [6], and more, have been reported. However, generalizing FMM to an aperiodic structure is strongly needed, to analyze recent state-of-the-art light-source-embedding photonic structures, including larger-scale aperiodic active photonic structures such as organic light-emitting diodes (OLEDs). In contrast, our method introduces a generalized FMM technique for aperiodic multiblock structures using a scattering matrix (
Here a novel FMM scheme for optical dipole radiation in a photonic structure is proposed, a scheme that is expected to be able to provide the requisite mode-theoretical analysis framework for light-source-embedded structures not only for the optical field distribution, but also for power flow and energy loss. For our theoretical description, we follow the conventions and symbols used in the previous FMM study [1].
Two contrasting optical field distributions, generated by two dipole emitters with orthogonal polarization embedded in the active block, are presented in Fig. 1. In Fig. 1, the example multiblock structure is schematically shown, with specifications. A two-dimensional dipole emission source of wavelength 532 nm is placed at r = (0, 0, 0) in the active block. The vectorial field distribution for the dipole with polarization P = (1, 0, 0), and that for the dipole with polarization P = (0, 0, 1), are presented in Figs. 1(a) and 1(b) respectively.
The field distribution is strongly dependent on the polarization state of the emission source. These field distributions are calculated by applying the method proposed in this paper to a basic radiative-device structure, to begin the description of the concrete theory more conveniently. This calculation is carried out by FMM, which allows us to analyze the modal structure of a field distribution. The modal-structure analysis provides in-depth insight about the optical power flow and energy loss, and their structural dependency. These topics will be dealt with in our further papers in the near future, but here we focus on the development of the novel extended FMM for optical dipole radiation.
Such a light-source-embedded photonic structure is modeled by the bidirectional multiblock structure shown in Fig. 2(a). The blocks and their interfaces are denoted by
In the proposed scheme, the thickness of the dipole-embedding slab
The optical field in the multiblock structure is represented by the Fourier modal expansions in the respective blocks. Here it is assumed that the complete set of the Bloch eigenmodes
are the electric field and magnetic field of the
where
where
In terms of the
The conventional
The transmission-coefficient operators in the left and right parts, induced by the positive input field, are denoted by
More specifically, we define the internal-light-source operators along the positive
The first term
Also, taking into account the internal field distribution, the coupling-coefficient operators are given by the following algorithm:
Similar to the formation of Eqs. (4a) and (4b), the coupling-coefficient operators of the internal field distributions in the left and right parts are obtained by, respectively,
Consequently, the bidirectional
To represent the dipole-radiation field, a proper source operator model,
The optical emission in
where P and M are the external polarization and magnetization vectors respectively. ε_{0} and μ_{0} denote respectively the electric permittivity and magnetic permeability in free space. ε_{r} is the relative permittivity of
First, we take into account the electric dipole radiation in
where
where
The electric and magnetic fields induced by a dipole source P are obtained respectively from
The radiation power of a dipole, P = (0, 0,
Similarly, the magnetic and electric fields induced by the magnetic dipole can be obtained as
In the case of a magnetic dipole, the radiation power is
The power calculations shown above are rigorous, because the calculation is performed on a continuum support. Since the FMM is a numerical method in the spatial Fourier domain, we need to build up a numerical framework. The core of our discussion is the discretization of the Weyl representation of the Green dyadic function. Direct discretization results in the discrete representation
where
where
where
With the obtained modal spectra, the optical field distribution is reconstructed according to Eqs. (1a)–(1d).
In Fig. 3, the exemplary optical field distribution is calculated by the aforementioned bidirectional
In both Figs. 3(a) and 3(b), the field profiles near
With respect to numerical analysis, when a dipole source of subwavelength dimension is represented by the discrete angular spectrum (
In terms of signal processing, to reduce the Gibbs phenomenon without losing physical meaning, the specific window is wrapped to the angular-spectrum profile of the discrete Weyl representation. We choose the sigma filter [11], σ_{mn}, which is defined by
and try to apply σ_{mn} to the discrete Weyl representation of Eq. (11b) as
The optical field spectra of the discrete Weyl representation refined with the sigma filter, which are comparable to Eqs. (12a)–(12b), read as
Figure 4 compares the field profile of the
As presented in Fig. 5, the optical field distributions obtained with the sigma filter demonstrate that the Gibbs-induced errors have been removed, and continuity in the field distribution is confirmed at r = (
A three-dimensional perfectly matched layer (PML) [12] is used to model this structure, as shown in in Fig. 6(a). Without the sigma filter [Fig. 6(b)], the Gibbs phenomenon occurs across the whole
We have formulated the basis for FMM analysis of general aperiodic source-embedding photonic structures: the bidirectional S-matrix method with the discrete Weyl representation and perfectly matched layer. The proposed scheme can provide an FMM-based modal-analysis framework for the design and analysis of advanced photonic structures with embedded optical sources.
Funding was provided by the National Research Foundation of Korea (NRF) (NRF-2019R1A2C1010243).
Curr. Opt. Photon. 2021; 5(6): 597-605
Published online December 25, 2021 https://doi.org/10.3807/COPP.2021.5.6.597
Copyright © Optical Society of Korea.
Sungjae Park^{1}, Joonku Hahn^{2}, Hwi Kim^{1}
^{1}Department of Electronics and Information Engineering, Korea University Sejong Campus, Sejong 30019, Korea
^{2}School of Electronic and Electrical Engineering, Kyungpook National University, Daegu 41566, Korea
Correspondence to:^{*}hwikim@korea.ac.kr, ORCID 0000-0002-4283-8982
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
An extended Fourier modal method (FMM) for optical dipole radiation in three-dimensional photonic structures is proposed. The core elements of the proposed FMM are the stable bidirectional scattering matrix algorithm for modeling internal optical emission, and a novel optical-dipole-source model that prevents numerical errors induced by the Gibbs phenomenon. Through the proposed scheme, the FMM is extended to model a wide range of source-embedded photonic structures.
Keywords: Fourier modal method, Numerical modeling
The Fourier modal method (FMM) is one of the most powerful electromagnetic analysis methods in the field of photonics [1]. In FMM, the linear modal theory for light-matter interactions is constructed based on Fourier analysis of Maxwell’s equations. The modal analysis is the crucial feature of FMM, which expands the vectorial optical field into a linear superposition of electromagnetic Bloch eigenmodes. In FMM, the dispersion relation and coupling rate of each eigenmode are extracted completely. On the basis of FMM, the modal spectrum of the vectorial optical field in photonic structures can be manifested [1]. For recent active-photonic applications such as quantum-dot light emitters or photoluminescence in nanowires, where light sources are embedded inside photonic structures, the modal analysis is also of prominent interest. A few previous studies dealt with the numerical modeling of a dipolar point source. The previous studies mainly investigated a periodic dipole arrangement in periodic structures. The discrete dipole approximation in a small-particle grid [2], the one-dimensional integral model of a dipole located within or at the boundary of a planar layered structure [3], periodic arrangement of dipoles [4, 5], RCWA analysis of a central electric dipole in a nonuniform spherical system [6], and more, have been reported. However, generalizing FMM to an aperiodic structure is strongly needed, to analyze recent state-of-the-art light-source-embedding photonic structures, including larger-scale aperiodic active photonic structures such as organic light-emitting diodes (OLEDs). In contrast, our method introduces a generalized FMM technique for aperiodic multiblock structures using a scattering matrix (
Here a novel FMM scheme for optical dipole radiation in a photonic structure is proposed, a scheme that is expected to be able to provide the requisite mode-theoretical analysis framework for light-source-embedded structures not only for the optical field distribution, but also for power flow and energy loss. For our theoretical description, we follow the conventions and symbols used in the previous FMM study [1].
Two contrasting optical field distributions, generated by two dipole emitters with orthogonal polarization embedded in the active block, are presented in Fig. 1. In Fig. 1, the example multiblock structure is schematically shown, with specifications. A two-dimensional dipole emission source of wavelength 532 nm is placed at r = (0, 0, 0) in the active block. The vectorial field distribution for the dipole with polarization P = (1, 0, 0), and that for the dipole with polarization P = (0, 0, 1), are presented in Figs. 1(a) and 1(b) respectively.
The field distribution is strongly dependent on the polarization state of the emission source. These field distributions are calculated by applying the method proposed in this paper to a basic radiative-device structure, to begin the description of the concrete theory more conveniently. This calculation is carried out by FMM, which allows us to analyze the modal structure of a field distribution. The modal-structure analysis provides in-depth insight about the optical power flow and energy loss, and their structural dependency. These topics will be dealt with in our further papers in the near future, but here we focus on the development of the novel extended FMM for optical dipole radiation.
Such a light-source-embedded photonic structure is modeled by the bidirectional multiblock structure shown in Fig. 2(a). The blocks and their interfaces are denoted by
In the proposed scheme, the thickness of the dipole-embedding slab
The optical field in the multiblock structure is represented by the Fourier modal expansions in the respective blocks. Here it is assumed that the complete set of the Bloch eigenmodes
are the electric field and magnetic field of the
where
where
In terms of the
The conventional
The transmission-coefficient operators in the left and right parts, induced by the positive input field, are denoted by
More specifically, we define the internal-light-source operators along the positive
The first term
Also, taking into account the internal field distribution, the coupling-coefficient operators are given by the following algorithm:
Similar to the formation of Eqs. (4a) and (4b), the coupling-coefficient operators of the internal field distributions in the left and right parts are obtained by, respectively,
Consequently, the bidirectional
To represent the dipole-radiation field, a proper source operator model,
The optical emission in
where P and M are the external polarization and magnetization vectors respectively. ε_{0} and μ_{0} denote respectively the electric permittivity and magnetic permeability in free space. ε_{r} is the relative permittivity of
First, we take into account the electric dipole radiation in
where
where
The electric and magnetic fields induced by a dipole source P are obtained respectively from
The radiation power of a dipole, P = (0, 0,
Similarly, the magnetic and electric fields induced by the magnetic dipole can be obtained as
In the case of a magnetic dipole, the radiation power is
The power calculations shown above are rigorous, because the calculation is performed on a continuum support. Since the FMM is a numerical method in the spatial Fourier domain, we need to build up a numerical framework. The core of our discussion is the discretization of the Weyl representation of the Green dyadic function. Direct discretization results in the discrete representation
where
where
where
With the obtained modal spectra, the optical field distribution is reconstructed according to Eqs. (1a)–(1d).
In Fig. 3, the exemplary optical field distribution is calculated by the aforementioned bidirectional
In both Figs. 3(a) and 3(b), the field profiles near
With respect to numerical analysis, when a dipole source of subwavelength dimension is represented by the discrete angular spectrum (
In terms of signal processing, to reduce the Gibbs phenomenon without losing physical meaning, the specific window is wrapped to the angular-spectrum profile of the discrete Weyl representation. We choose the sigma filter [11], σ_{mn}, which is defined by
and try to apply σ_{mn} to the discrete Weyl representation of Eq. (11b) as
The optical field spectra of the discrete Weyl representation refined with the sigma filter, which are comparable to Eqs. (12a)–(12b), read as
Figure 4 compares the field profile of the
As presented in Fig. 5, the optical field distributions obtained with the sigma filter demonstrate that the Gibbs-induced errors have been removed, and continuity in the field distribution is confirmed at r = (
A three-dimensional perfectly matched layer (PML) [12] is used to model this structure, as shown in in Fig. 6(a). Without the sigma filter [Fig. 6(b)], the Gibbs phenomenon occurs across the whole
We have formulated the basis for FMM analysis of general aperiodic source-embedding photonic structures: the bidirectional S-matrix method with the discrete Weyl representation and perfectly matched layer. The proposed scheme can provide an FMM-based modal-analysis framework for the design and analysis of advanced photonic structures with embedded optical sources.
Funding was provided by the National Research Foundation of Korea (NRF) (NRF-2019R1A2C1010243).