Shan-Liang Qiu and Yong-Ping Li, "Q-factor instability and its explanation in the staircased FDTD simulation of high-Q circular cavity," J. Opt. Soc. Am. B 26, 1664-1674 (2009)

The loss of high-Q whispering-gallery modes (WGMs) with lower azimuthal mode number $[m\sim (9\u201312)]$ in a circular cavity have been analyzed by using a two-dimensional finite-difference time domain method (2D FDTD) method employing Cartesian gridding and staircase approximation. The FDTD simulated Q-factors of these WGMs are generally lower than those of theoretical expectations. The variations of FDTD simulated Q-factors with spatial-calculation step size indicate that the FDTD results do not simply approximate to their theoretical expectation but jump unstably under the expectation. A loss estimation method similar to volume current method (VCM) is developed to explain the FDTD results and instability. This method calculates the “incoherent” scattering field of a scattering source under influence of cavity. Theoretical results coincident with the FDTD simulation are obtained, especially for transverse magnetic modes. As based on the developed method, the energy loss is affected by only a few harmonics of boundary fluctuation that cause the FDTD loss instability.

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Comparison between Theoretical Results and FDTD Simulation Results for ${\mathrm{WGH}}_{(9,1)}$^{
a
}

Q-factor

δ (nm)

40

35

32

30

28

25

20

10

${Q}_{\mathrm{FDTD}}$$\left({10}^{4}\right)$

1.088

1.752

0.571

4.253

3.577

1.101

3.608

14.40

${Q}_{\mathrm{sca}}$$\left({10}^{4}\right)$

1.23

2.061

0.580

4.976

4.256

1.122

4.010

2.277

${Q}_{\mathrm{T}}$$\left({10}^{4}\right)$

1.196

1.967

0.573

4.462

3.875

1.094

3.744

14.91

${Q}_{\mathrm{FDTD}}$ denotes FDTD simulated Q-factor. ${Q}_{\mathrm{sca}}$, ${\mathrm{Q}}_{\mathrm{T}}$ denote the scattering-induced Q-factor and theoretical total Q-factor based on our developed method, respectively. The theoretical Q-factor for ${\mathrm{WGH}}_{(9,1)}$ for ideal an CRC is ${Q}_{0}=4.32\times {10}^{5}$, several percent or much less relative error can be found for this mode, and for this mode, degeneracy still exists. Theory coincides with FDTD simulation.

Table 2

Comparison between Theoretical Results and FDTD Simulation Results for ${\mathrm{WGH}}_{(12,1)}$^{
a
}

$Q_{\mathrm{T}}{}^{\mathrm{e}}$, $Q_{\mathrm{T}}{}^{\mathrm{o}}$ denote the theoretical total Q-factors for split even mode and odd mode of initial degenerate ${\mathrm{WGH}}_{(12,1)}$, respectively, and the theoretical Q-factor for ${\mathrm{WGH}}_{(9,1)}$ for ideal CRC is ${Q}_{0}=4.89\times {10}^{7}$. For the split even mode with lower Q-factor, we see more good agreement than for its odd counterpart.

Table 3

Comparison between Theoretical Results and FDTD Simulation Results for ${\mathrm{WGE}}_{(10,1)}$^{
a
}

The theoretical Q-factor for ${\mathrm{WGE}}_{(10,1)}$ for an ideal CRC is ${Q}_{0}=2.7\times {10}^{6}$, and for TE modes the result isn’t as good as for the TM case.

Tables (3)

Table 1

Comparison between Theoretical Results and FDTD Simulation Results for ${\mathrm{WGH}}_{(9,1)}$^{
a
}

Q-factor

δ (nm)

40

35

32

30

28

25

20

10

${Q}_{\mathrm{FDTD}}$$\left({10}^{4}\right)$

1.088

1.752

0.571

4.253

3.577

1.101

3.608

14.40

${Q}_{\mathrm{sca}}$$\left({10}^{4}\right)$

1.23

2.061

0.580

4.976

4.256

1.122

4.010

2.277

${Q}_{\mathrm{T}}$$\left({10}^{4}\right)$

1.196

1.967

0.573

4.462

3.875

1.094

3.744

14.91

${Q}_{\mathrm{FDTD}}$ denotes FDTD simulated Q-factor. ${Q}_{\mathrm{sca}}$, ${\mathrm{Q}}_{\mathrm{T}}$ denote the scattering-induced Q-factor and theoretical total Q-factor based on our developed method, respectively. The theoretical Q-factor for ${\mathrm{WGH}}_{(9,1)}$ for ideal an CRC is ${Q}_{0}=4.32\times {10}^{5}$, several percent or much less relative error can be found for this mode, and for this mode, degeneracy still exists. Theory coincides with FDTD simulation.

Table 2

Comparison between Theoretical Results and FDTD Simulation Results for ${\mathrm{WGH}}_{(12,1)}$^{
a
}

$Q_{\mathrm{T}}{}^{\mathrm{e}}$, $Q_{\mathrm{T}}{}^{\mathrm{o}}$ denote the theoretical total Q-factors for split even mode and odd mode of initial degenerate ${\mathrm{WGH}}_{(12,1)}$, respectively, and the theoretical Q-factor for ${\mathrm{WGH}}_{(9,1)}$ for ideal CRC is ${Q}_{0}=4.89\times {10}^{7}$. For the split even mode with lower Q-factor, we see more good agreement than for its odd counterpart.

Table 3

Comparison between Theoretical Results and FDTD Simulation Results for ${\mathrm{WGE}}_{(10,1)}$^{
a
}

The theoretical Q-factor for ${\mathrm{WGE}}_{(10,1)}$ for an ideal CRC is ${Q}_{0}=2.7\times {10}^{6}$, and for TE modes the result isn’t as good as for the TM case.