The eigenvalue trajectories and their corresponding Shannon entropies. (a) Real part of eigenvalues kR as the eccentricity ε is varied. (b) Imaginary part of eigenvalues kR as ε is varied. The Shannon entropies for N = 98, 212, 398, 596, 810, and 1040 are shown in (c), (d), (e), (f), (g), and (h), respectively.|@|~(^,^)~|@|Saturation of Shannon entropy. (a) The Shannon entropies for mode patterns and maximal entropies as the number of mesh point N is varied. The curved lines are Shannon entropies for the probability density of the resonance mode pattern whereas the straight lines are maximal entropies. The red circles are for N = 504, the blue up-ward triangles are for N = 1040, the green down-ward triangles are for N = 1480, the orange left-facing triangles are for N = 2020, the cyan right-facing triangles are for N = 2480, the violet diamonds are for N = 3008, and the pink stars are for N = 3492. (b) Differences between the two types of entropies in (a) are shown. D_SE shows a tendency toward saturated behavior past N₍₄₎ = 2020. The inset clearly shows this behavior.|@|~(^,^)~|@|Observed and expected values for the chi-squared test. Red curves of the observed values and blue curves of the expected values for Shannon entropies, depending on each N value: (a) 98, (b) 212, (c) 398, (d) 596, (e) 810, and (f) 1040. The two curves (red and blue) become more similar to each other as N increases.|@|~(^,^)~|@|The red squares for chi-squared test values are shown as N is varied. We set B (N = 810) to be the optimized minimum-mesh point for spatial resolution. Inset: A_1,2,3 is the mode pattern for N = 212 at ε = 0.0, 0.25, and 0.38, B_1,2,3 is the mode pattern for N = 810, and C_1,2,3 is the mode pattern for N = 3492. We can identify the quantum number (l = 5, m = 3) for B₁ as well as C₁.|@|~(^,^)~|@|Spatial resolution and increasing wave number. (a) χ² as a function of N. The red circles are for χ² Re(kR)~2.8, the blue upward triangles are for χ² Re(kR)~5.8, the green downward triangles are for χ² Re(kR)~11.0, and the pink left-facing triangles are for χ² Re(kR)~16.0. The absolute values of χ² increase as Re(kR) increases. The speed of convergence slows as the Re(kR) increases. The optimized minimum-mesh point for spatial resolution in (b) is N = 212, that in (c) is N = 810, that in (d) is N = 2952, and that in (e) is N = 6180. These optimized minimum-mesh points N_O are approximated as 2.2 × (nkR)².

Curr. Opt. Photon. 2021;5:16~22 https://doi.org/10.3807/COPP.2021.5.1.016