Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2021; 5(1): 16-22
Published online February 25, 2021 https://doi.org/10.3807/COPP.2021.5.1.016
Copyright © Optical Society of Korea.
Kyu-Won Park^{1} , Jinuk Kim^{1}, Songky Moon^{2}
Corresponding author: ^{*}wind999@snu.ac.kr, ORCID 0000-0002-4933-9799
We present the Shannon entropy as an indicator of the spatial resolutions of the morphologies of the resonance mode patterns in an optical resonator. We obtain each optimized number of mesh points, one of minimum size and the other of maximum one. The optimized mesh-point number of minimum size is determined by the identifiable quantum number through a chi-squared test, whereas the saturation of the difference between Shannon entropies corresponds to the other mesh-point number of maximum size. We also show that the optimized minimum mesh-point increases as the (real) wave number increases and approximates the proportionality constant between them.
Keywords: Shannon entropy, Optical resonator, Boundary element method, Spatial resolution
OCIS codes: (100.5010) Pattern recognition; (140.3410) Laser resonators; (330.6100) Spatial discrimination; (330.6130) Spatial resolution; (350.5730) Resolution
Optical resonators have been considered good candidates for optical sources [1, 2] and have been studied extensively in various theoretical and experimental contexts, such as unidirectional emission [3–5], high quality factors [6, 7], optical sensors [8, 9], and whispering gallery modes [10–12]. They are also regarded as good platforms for studying many fundamental physical phenomena,
The Shannon entropy, first introduced by Shannon, is a functional that measures the average information content of a statistical ensembles of a random variable. It was originally developed and utilized in communication theory [26] and statistical mechanics [27], but recently it has been also exploited in various areas. The Shannon entropy has been used not only for molecular descriptors [28], protein sequences [29] in bio-systems and algorithmic complexity [30] in information theory, but also for avoided crossing in optics: the relation of which to the Shannon entropy has been investigated for microcavities [25], atomic physics [31, 32], and the quantum transition from order to chaos in quantum chaos [33].
In a previous study [25], we showed that the Shannon entropy changes depending on the parameter and is maximized at the center of the interaction, owing to a coherent superposition of wavefunctions (mixing of the wavefunctions), which indicates that the variation of the Shannon entropy is related to the variation of the morphologies of the wavefunctions. This fact suggests that the Shannon entropy can be used to study the morphology of the wavefunction. In this paper, we employ the Shannon entropy to determine the optimal number of mesh points for the morphology of the mode pattern in a two-dimensional elliptical optical resonator.
In the case of the one-dimensional resolution of the boundary-element method, the optimized boundary line element ∆
This paper is organized as follows. In Section II, the Shannon entropy of an elliptical microcavity is introduced. In Section III, we study the difference between Shannon entropies and its saturation. Comparison of two Shannon entropies for spatial resolution is discussed in Section IV. Finally, we summarize our work in Section V.
The eigenvalue trajectories of an elliptical resonator with major axis a and minor axis b are shown in Fig. 1. The eigenvalues and eigenmodes are calculated using the boundary-element method (BEM) [34] with a refractive index of the resonator
with its complex eigenvalues
To introduce the discrete probability distribution, we discretize the area within the resonator into the
We obtain several Shannon entropies by the definition above. In Fig. 1, the Shannon entropies for
We can understand this behavior by considering maximal entropy states. The maximal entropy with
In this section, we consider the Shannon entropies for resonance mode patterns and maximal entropies simultaneously as functions of
To study the behavior of Shannon entropies depending on a specific
Note that Eq. (3) is equivalent to a Kullback-Leibler divergence when one of the two probability distributions is uniform, i.e.,
The Kullback-Leibler divergence is directly related to the difference in Shannon entropies when one of the two probability distributions is uniform, but in general this is not the case. Instead we employ the chi-squared test, which can be defined as an approximation of a Kullback-Leibler divergence [41]. The chi-squared test is used to quantify a difference between expected and observed values in a data set [42]. Its definition is
where
Actually, it is the statistical procedures whose results are evaluated with reference to the chi-squared distribution. However, we take advantage of Eq. 4 to quantify the differences between two data sets, even though our data sets were not evaluated with reference to the chi-squared distribution. Here we suggest that the observed values of the Shannon entropy as a function of a specific
In our case, it should be also noticed that the number of unit cells
The results of the chi-squared test, which are obtained from the observed and theoretical values shown in Fig. 3, are presented in Fig. 4. The red squares for the chi-squared values are shown as a function of the
To manifest this suggestion, we plot the mode patterns at each number of mesh points in Fig. 4. We can expect that the morphologies of the mode patterns become clear as
When the number of mesh points
Intuitively, we can assume that the more massive the morphology of mode pattern becomes, the more mesh points we need for spatial resolution. To check this assumption, we investigate the relation between χ^{2} and the eigenvalue trajectories of increasing Re(
Furthermore, the convergence rate becomes low as Re(
We study the Shannon entropy as an indicator of the spatial resolutions for the morphologies of the resonance-mode patterns in an elliptical resonator, and obtain two types of optimized mesh points of minimum and maximum size.
Using the chi-squared test, the optimized minimum-mesh size for spatial resolution can be confirmed by the identifiable quantum number. On the contrary, the saturation of difference in Shannon entropies can correspond to the optimized maximum-mesh size for spatial resolution, since after saturation the morphology of the mode pattern does not change and the resolution does not increase significantly.
We also investigate the relation between the optimized minimum-mesh point
The Authors thank Kabgyun Jeong for valuable comments and support. This work was supported by Samsung Science and Technology Foundation under Project No. SSTP-BA_{1}502-05, the National Research Foundation of Korea (Grant No. NRF-2020M3E4A_{1}077861 & No. 2016R1D1A_{1}09918326 & No. 2020R1A_{2}C_{3}009299), and the Ministry of Science and ICT of Korea under ITRC program (Grant No. IITP-2019-0-01402).
Curr. Opt. Photon. 2021; 5(1): 16-22
Published online February 25, 2021 https://doi.org/10.3807/COPP.2021.5.1.016
Copyright © Optical Society of Korea.
Kyu-Won Park^{1} , Jinuk Kim^{1}, Songky Moon^{2}
^{1}Department of Physics and Astronomy & Institute of Applied Physics, Seoul National University, Seoul 08826, Korea
^{2}Faculty of Liberal Education, Seoul National University, Seoul 08826, Korea
Correspondence to:^{*}wind999@snu.ac.kr, ORCID 0000-0002-4933-9799
We present the Shannon entropy as an indicator of the spatial resolutions of the morphologies of the resonance mode patterns in an optical resonator. We obtain each optimized number of mesh points, one of minimum size and the other of maximum one. The optimized mesh-point number of minimum size is determined by the identifiable quantum number through a chi-squared test, whereas the saturation of the difference between Shannon entropies corresponds to the other mesh-point number of maximum size. We also show that the optimized minimum mesh-point increases as the (real) wave number increases and approximates the proportionality constant between them.
Keywords: Shannon entropy, Optical resonator, Boundary element method, Spatial resolution
Optical resonators have been considered good candidates for optical sources [1, 2] and have been studied extensively in various theoretical and experimental contexts, such as unidirectional emission [3–5], high quality factors [6, 7], optical sensors [8, 9], and whispering gallery modes [10–12]. They are also regarded as good platforms for studying many fundamental physical phenomena,
The Shannon entropy, first introduced by Shannon, is a functional that measures the average information content of a statistical ensembles of a random variable. It was originally developed and utilized in communication theory [26] and statistical mechanics [27], but recently it has been also exploited in various areas. The Shannon entropy has been used not only for molecular descriptors [28], protein sequences [29] in bio-systems and algorithmic complexity [30] in information theory, but also for avoided crossing in optics: the relation of which to the Shannon entropy has been investigated for microcavities [25], atomic physics [31, 32], and the quantum transition from order to chaos in quantum chaos [33].
In a previous study [25], we showed that the Shannon entropy changes depending on the parameter and is maximized at the center of the interaction, owing to a coherent superposition of wavefunctions (mixing of the wavefunctions), which indicates that the variation of the Shannon entropy is related to the variation of the morphologies of the wavefunctions. This fact suggests that the Shannon entropy can be used to study the morphology of the wavefunction. In this paper, we employ the Shannon entropy to determine the optimal number of mesh points for the morphology of the mode pattern in a two-dimensional elliptical optical resonator.
In the case of the one-dimensional resolution of the boundary-element method, the optimized boundary line element ∆
This paper is organized as follows. In Section II, the Shannon entropy of an elliptical microcavity is introduced. In Section III, we study the difference between Shannon entropies and its saturation. Comparison of two Shannon entropies for spatial resolution is discussed in Section IV. Finally, we summarize our work in Section V.
The eigenvalue trajectories of an elliptical resonator with major axis a and minor axis b are shown in Fig. 1. The eigenvalues and eigenmodes are calculated using the boundary-element method (BEM) [34] with a refractive index of the resonator
with its complex eigenvalues
To introduce the discrete probability distribution, we discretize the area within the resonator into the
We obtain several Shannon entropies by the definition above. In Fig. 1, the Shannon entropies for
We can understand this behavior by considering maximal entropy states. The maximal entropy with
In this section, we consider the Shannon entropies for resonance mode patterns and maximal entropies simultaneously as functions of
To study the behavior of Shannon entropies depending on a specific
Note that Eq. (3) is equivalent to a Kullback-Leibler divergence when one of the two probability distributions is uniform, i.e.,
The Kullback-Leibler divergence is directly related to the difference in Shannon entropies when one of the two probability distributions is uniform, but in general this is not the case. Instead we employ the chi-squared test, which can be defined as an approximation of a Kullback-Leibler divergence [41]. The chi-squared test is used to quantify a difference between expected and observed values in a data set [42]. Its definition is
where
Actually, it is the statistical procedures whose results are evaluated with reference to the chi-squared distribution. However, we take advantage of Eq. 4 to quantify the differences between two data sets, even though our data sets were not evaluated with reference to the chi-squared distribution. Here we suggest that the observed values of the Shannon entropy as a function of a specific
In our case, it should be also noticed that the number of unit cells
The results of the chi-squared test, which are obtained from the observed and theoretical values shown in Fig. 3, are presented in Fig. 4. The red squares for the chi-squared values are shown as a function of the
To manifest this suggestion, we plot the mode patterns at each number of mesh points in Fig. 4. We can expect that the morphologies of the mode patterns become clear as
When the number of mesh points
Intuitively, we can assume that the more massive the morphology of mode pattern becomes, the more mesh points we need for spatial resolution. To check this assumption, we investigate the relation between χ^{2} and the eigenvalue trajectories of increasing Re(
Furthermore, the convergence rate becomes low as Re(
We study the Shannon entropy as an indicator of the spatial resolutions for the morphologies of the resonance-mode patterns in an elliptical resonator, and obtain two types of optimized mesh points of minimum and maximum size.
Using the chi-squared test, the optimized minimum-mesh size for spatial resolution can be confirmed by the identifiable quantum number. On the contrary, the saturation of difference in Shannon entropies can correspond to the optimized maximum-mesh size for spatial resolution, since after saturation the morphology of the mode pattern does not change and the resolution does not increase significantly.
We also investigate the relation between the optimized minimum-mesh point
The Authors thank Kabgyun Jeong for valuable comments and support. This work was supported by Samsung Science and Technology Foundation under Project No. SSTP-BA_{1}502-05, the National Research Foundation of Korea (Grant No. NRF-2020M3E4A_{1}077861 & No. 2016R1D1A_{1}09918326 & No. 2020R1A_{2}C_{3}009299), and the Ministry of Science and ICT of Korea under ITRC program (Grant No. IITP-2019-0-01402).