Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2021; 5(5): 491-499
Published online October 25, 2021 https://doi.org/10.3807/COPP.2021.5.5.491
Copyright © Optical Society of Korea.
Soobin Kim^{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.
A scalar Fourier modal method for the numerical analysis of the scalar wave equation in inhomogeneous space with an arbitrary permittivity profile, is proposed as a novel theoretical embodiment of Fourier optics. The modeling of devices and systems using conventional Fourier optics is based on the thin-element approximation, but this approach becomes less accurate with high numerical aperture or thick optical elements. The proposed scalar Fourier modal method describes the wave optical characteristics of optical structures in terms of the generalized transmittance function, which can readily overcome a current limitation of Fourier optics.
Keywords: Electromagnetic theory, Fourier modal method, Numerical modeling
OCIS codes: (000.3860) Mathematical method in physics; (050.1755) Computational electro-magnetic methods; (050.1960) Diffraction theory
In conventional scalar wave optics [1], optical elements are approximated using a thin-element model that is characterized by a simple transmittance function, a process which is referred to as the thin-element approximation (TEA) [2]. A major limitation of the TEA model is its directional invariance, meaning that the transmittance function of a TEA element is invariant to the direction of an incident optical wave denoted by the two-dimensional complex function
In previous research, the beam-propagation method (BPM) [3–5] and wave-propagation method (WPM) [6] have been proposed to deal with nonparaxial diffraction fields and thick elements, but these approaches struggle with the complex optical structures of high-contrast or gradient inhomogeneous index profiles [6]. To characterize complex optical systems consisting of thick, volumetric inhomogeneous elements, we propose a frequency-domain method capable of calculating a full transmittance function with directional
In this paper, we investigate the essence of the Fourier modal method (FMM) [14–16], and propose the scalar Fourier modal method (SFMM) as a new numerical framework for the analysis of the scalar wave equation:
where
This paper is organized as follows. Section 2 establishes the numerical framework for the proposed SFMM, and numerical examples are presented in Section 3 with the introduction of absorbing boundaries. Concluding remarks are provided in Section 4.
The construction of the SFMM follows steps similar to those in the conventional vectorial FMM [16]. In the first step, the model function of the optical field based on the pseudo-Fourier series converts the scalar wave equation of Eq. (1) to an algebraic eigenvalue equation.
According to the conventional structure-modeling method, we take a staircase representation of the 3D permittivity profile ε_{r}(r) along the optical axis, in the form of a multilayered structure, as illustrated in Fig. 1 [16]. Details of the propositions related to the structural modeling are presented in Section 3. For a single thin layer with permittivity profile ε_{r}(
The models of the permittivity profile and the field representation take the form of a Fourier series and a pseudo-Fourier series respectively:
where ε_{m,n} and U_{m,n} are the Fourier coefficients of the two-dimensional relative permittivity profile and the wave function respectively.
Here the longitudinal wave-vector component
Equation (4) is set so that it is satisfied for any
The above eigenvalue equation can be expressed in a compact matrix form:
The detailed expressions for
The Bloch eigenmodes obtained for each region can be classified into two groups of positive (
where
where
To determine the coupling coefficients, we employ the scattering-matrix (
Under the scalar approximation, the energy of an optical wave is completely conserved at a lossless dielectric interface, stating that the sum of the squares of the reflection and transmission coefficients is equal to the square of the input field coefficient, when the loss in both regions can be ignored. The coupling coefficients
The substitution of Eqs. (9a), (9b), and (9c) into Eqs. (10a) and (10b) leads to
and
The term
where
The corresponding matrix form of the boundary condition is given by
By solving Eqs. (12) and (14), the reflection coefficient
where and
The associative rules enable the parallel computation of the internal coupling coefficients:
For instance, when the incident field is coming from the left side, the reflection field in the left half-infinite block and the transmitted field in the right half-infinite block can be calculated respectively by
The total field in the left half-infinite block, where z is less than 0, can be visualized by
and the fields of the
where
Because we already have the same coefficients for the opposite direction, it is straightforward to visualize the right-to-left case. This outcome is the key result of this paper, and we hereby refer to this overall approach as the SFMM.
In the modeling of thick optical elements with smooth surfaces, the conventional staircase approximation of the structure can generate physical errors, and may be limited by computational inefficiency. Figure 2(a) compares the conventional and proposed staircase approximation methods for a smooth optical element. Using the conventional method, unexpected physical phenomena can be observed, such as diffraction at the edges of the slabs, which will lead to errors if the number of slabs is not high enough to model the curved surface.
Figure 3 presents a numerical experiment for the calculation of the field, in which the number of staircase layers in a convex lens profile is varied. The diameter of the lens aperture is 1 mm, the focal length of the lens is 12 mm, and the curvature of the lens is 24 mm. The refractive index of the lens is 1.5, and that of the surrounding material is 1.0. The number of Fourier harmonics is set to 301 in each direction, which means that the truncation numbers are set to 150. A plane wave with a wavelength of 633 nm is normally incident from the left.
The staircase structure with only 10 layers exhibits a significant edge-diffraction effect, losing the essential optical properties of the convex lens [Fig. 3(a)]. In the 140-layer model of the same convex lens, some defects caused by the edge diffraction are still observable [Fig. 3(b)]. More than 200 layers are required to model the lens with convergent results [Fig. 3(c)]. A staircase approximation with sharp edges requires significant computational resources for smooth surface modeling because the
The proposed approximation is based on the assumption that a smoothly varying permittivity profile in the transverse direction at the edges of the slabs,
where
The TEA can only approximate light transmittance, while the SFMM can model light absorption, diffraction, and multiple reflections, for an arbitrarily complex refractive index. In Fig. 5, a cylindrical lens in free space is modeled using the SFMM with various materials. In this example, the diameter of the lens is 5 µm, and 152 layers and 201 Fourier harmonics are used for the simulation. The angle of the incident plane wave is set to 30°, and the wavelength is 633 nm. In Figs. 5(b) and 5(c), the absorption due to the complex refractive index is observable by the field inside the lens, and the diffraction is also observable by the field behind the lens. On the front face of the lens, the interference patterns between the incident light and the reflected light on the glass-air boundary is shown in Figs. 5(b) and 5(c). The TEA is often used to model optics based on the scalar diffraction theory, such as in the angular spectrum method. The TEA does not consider multiple reflections, which is essential for some applications. The SFMM is thus an efficient tool for calculating a scalar field with the rigorous consideration of multiple internal reflections.
Another advantage of the SFMM is that it can obtain the directionally variant transfer function, unlike the TEA method. As mentioned in the introduction, practical optics often have a high NA and nonnegligible thickness, leading to directionally variant characteristics that need to be modeled accurately. To confirm these characteristics of the SFMM, a bi-convex lens of sufficient thickness and a high NA is modeled using the SFMM and TEA. The curvature radius of the bi-convex lens is 66.67 µm, and the focal length of the approximated thin lens is also 66.67 µm. Because the diameter of the lens is 100 µm, the NA of the lens is 0.60. To realize the simulation of the aperiodic optical structure, the additional numerical technique of an absorbing boundary condition (ABC) [16] is used on both side boundaries. Figure 6(a)–6(c) show field distribution by SFMM calculation, and (d)–(f) show field distribution by TEA. Comparing Figs. 6(c) and 6(f), the difference between the SFMM and TEA is noticeable. In Fig. 6(c), a different focal length for the various angles of the incident wave is confirmed. The simulated convex lens has a field-curvature aberration whose focal length varies depending on the direction of the incident plane wave, which the SFMM can model. Because the transfer function of TEA is directionally invariant, the focal length cannot vary with the angle of incidence.
In conclusion, we have proposed the SFMM for wave-optic optical-element modeling. The SFMM is an appropriate method for the calculation of a full transfer matrix, for practical applications and for fundamental research. In addition to thick optical elements, HOEs (which have been spotlighted as an optical element for AR displays [12, 13]) and light scattering and transportation through disordered media [8, 9] may benefit from the application of the SFMM.
For matrix operations, the Toeplitz matrix is means of
where
where
and,
This study was supported by the National Research Foundation of Korea (NRF) (NRF-2019R1A2C1010243).
Curr. Opt. Photon. 2021; 5(5): 491-499
Published online October 25, 2021 https://doi.org/10.3807/COPP.2021.5.5.491
Copyright © Optical Society of Korea.
Soobin Kim^{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.
A scalar Fourier modal method for the numerical analysis of the scalar wave equation in inhomogeneous space with an arbitrary permittivity profile, is proposed as a novel theoretical embodiment of Fourier optics. The modeling of devices and systems using conventional Fourier optics is based on the thin-element approximation, but this approach becomes less accurate with high numerical aperture or thick optical elements. The proposed scalar Fourier modal method describes the wave optical characteristics of optical structures in terms of the generalized transmittance function, which can readily overcome a current limitation of Fourier optics.
Keywords: Electromagnetic theory, Fourier modal method, Numerical modeling
In conventional scalar wave optics [1], optical elements are approximated using a thin-element model that is characterized by a simple transmittance function, a process which is referred to as the thin-element approximation (TEA) [2]. A major limitation of the TEA model is its directional invariance, meaning that the transmittance function of a TEA element is invariant to the direction of an incident optical wave denoted by the two-dimensional complex function
In previous research, the beam-propagation method (BPM) [3–5] and wave-propagation method (WPM) [6] have been proposed to deal with nonparaxial diffraction fields and thick elements, but these approaches struggle with the complex optical structures of high-contrast or gradient inhomogeneous index profiles [6]. To characterize complex optical systems consisting of thick, volumetric inhomogeneous elements, we propose a frequency-domain method capable of calculating a full transmittance function with directional
In this paper, we investigate the essence of the Fourier modal method (FMM) [14–16], and propose the scalar Fourier modal method (SFMM) as a new numerical framework for the analysis of the scalar wave equation:
where
This paper is organized as follows. Section 2 establishes the numerical framework for the proposed SFMM, and numerical examples are presented in Section 3 with the introduction of absorbing boundaries. Concluding remarks are provided in Section 4.
The construction of the SFMM follows steps similar to those in the conventional vectorial FMM [16]. In the first step, the model function of the optical field based on the pseudo-Fourier series converts the scalar wave equation of Eq. (1) to an algebraic eigenvalue equation.
According to the conventional structure-modeling method, we take a staircase representation of the 3D permittivity profile ε_{r}(r) along the optical axis, in the form of a multilayered structure, as illustrated in Fig. 1 [16]. Details of the propositions related to the structural modeling are presented in Section 3. For a single thin layer with permittivity profile ε_{r}(
The models of the permittivity profile and the field representation take the form of a Fourier series and a pseudo-Fourier series respectively:
where ε_{m,n} and U_{m,n} are the Fourier coefficients of the two-dimensional relative permittivity profile and the wave function respectively.
Here the longitudinal wave-vector component
Equation (4) is set so that it is satisfied for any
The above eigenvalue equation can be expressed in a compact matrix form:
The detailed expressions for
The Bloch eigenmodes obtained for each region can be classified into two groups of positive (
where
where
To determine the coupling coefficients, we employ the scattering-matrix (
Under the scalar approximation, the energy of an optical wave is completely conserved at a lossless dielectric interface, stating that the sum of the squares of the reflection and transmission coefficients is equal to the square of the input field coefficient, when the loss in both regions can be ignored. The coupling coefficients
The substitution of Eqs. (9a), (9b), and (9c) into Eqs. (10a) and (10b) leads to
and
The term
where
The corresponding matrix form of the boundary condition is given by
By solving Eqs. (12) and (14), the reflection coefficient
where and
The associative rules enable the parallel computation of the internal coupling coefficients:
For instance, when the incident field is coming from the left side, the reflection field in the left half-infinite block and the transmitted field in the right half-infinite block can be calculated respectively by
The total field in the left half-infinite block, where z is less than 0, can be visualized by
and the fields of the
where
Because we already have the same coefficients for the opposite direction, it is straightforward to visualize the right-to-left case. This outcome is the key result of this paper, and we hereby refer to this overall approach as the SFMM.
In the modeling of thick optical elements with smooth surfaces, the conventional staircase approximation of the structure can generate physical errors, and may be limited by computational inefficiency. Figure 2(a) compares the conventional and proposed staircase approximation methods for a smooth optical element. Using the conventional method, unexpected physical phenomena can be observed, such as diffraction at the edges of the slabs, which will lead to errors if the number of slabs is not high enough to model the curved surface.
Figure 3 presents a numerical experiment for the calculation of the field, in which the number of staircase layers in a convex lens profile is varied. The diameter of the lens aperture is 1 mm, the focal length of the lens is 12 mm, and the curvature of the lens is 24 mm. The refractive index of the lens is 1.5, and that of the surrounding material is 1.0. The number of Fourier harmonics is set to 301 in each direction, which means that the truncation numbers are set to 150. A plane wave with a wavelength of 633 nm is normally incident from the left.
The staircase structure with only 10 layers exhibits a significant edge-diffraction effect, losing the essential optical properties of the convex lens [Fig. 3(a)]. In the 140-layer model of the same convex lens, some defects caused by the edge diffraction are still observable [Fig. 3(b)]. More than 200 layers are required to model the lens with convergent results [Fig. 3(c)]. A staircase approximation with sharp edges requires significant computational resources for smooth surface modeling because the
The proposed approximation is based on the assumption that a smoothly varying permittivity profile in the transverse direction at the edges of the slabs,
where
The TEA can only approximate light transmittance, while the SFMM can model light absorption, diffraction, and multiple reflections, for an arbitrarily complex refractive index. In Fig. 5, a cylindrical lens in free space is modeled using the SFMM with various materials. In this example, the diameter of the lens is 5 µm, and 152 layers and 201 Fourier harmonics are used for the simulation. The angle of the incident plane wave is set to 30°, and the wavelength is 633 nm. In Figs. 5(b) and 5(c), the absorption due to the complex refractive index is observable by the field inside the lens, and the diffraction is also observable by the field behind the lens. On the front face of the lens, the interference patterns between the incident light and the reflected light on the glass-air boundary is shown in Figs. 5(b) and 5(c). The TEA is often used to model optics based on the scalar diffraction theory, such as in the angular spectrum method. The TEA does not consider multiple reflections, which is essential for some applications. The SFMM is thus an efficient tool for calculating a scalar field with the rigorous consideration of multiple internal reflections.
Another advantage of the SFMM is that it can obtain the directionally variant transfer function, unlike the TEA method. As mentioned in the introduction, practical optics often have a high NA and nonnegligible thickness, leading to directionally variant characteristics that need to be modeled accurately. To confirm these characteristics of the SFMM, a bi-convex lens of sufficient thickness and a high NA is modeled using the SFMM and TEA. The curvature radius of the bi-convex lens is 66.67 µm, and the focal length of the approximated thin lens is also 66.67 µm. Because the diameter of the lens is 100 µm, the NA of the lens is 0.60. To realize the simulation of the aperiodic optical structure, the additional numerical technique of an absorbing boundary condition (ABC) [16] is used on both side boundaries. Figure 6(a)–6(c) show field distribution by SFMM calculation, and (d)–(f) show field distribution by TEA. Comparing Figs. 6(c) and 6(f), the difference between the SFMM and TEA is noticeable. In Fig. 6(c), a different focal length for the various angles of the incident wave is confirmed. The simulated convex lens has a field-curvature aberration whose focal length varies depending on the direction of the incident plane wave, which the SFMM can model. Because the transfer function of TEA is directionally invariant, the focal length cannot vary with the angle of incidence.
In conclusion, we have proposed the SFMM for wave-optic optical-element modeling. The SFMM is an appropriate method for the calculation of a full transfer matrix, for practical applications and for fundamental research. In addition to thick optical elements, HOEs (which have been spotlighted as an optical element for AR displays [12, 13]) and light scattering and transportation through disordered media [8, 9] may benefit from the application of the SFMM.
For matrix operations, the Toeplitz matrix is means of
where
where
and,
This study was supported by the National Research Foundation of Korea (NRF) (NRF-2019R1A2C1010243).