## Abstract

We demonstrate the capabilities of principal component analysis (PCA) for studying the results of finite-difference time-domain (FDTD) algorithms in simulating photonic crystal microcavities. The spatial-temporal structures provided by PCA are related to the actual electric field vibrating inside the photonic microcavity. A detailed analysis of the results has made it possible to compute the phase maps for each mode of the arrangement at their respective resonant frequencies. The existence of standing wave behavior is revealed by this analysis. In spite of this, some numerical artifacts induced by FDTD algorithms have been clearly detailed through PCA analysis. The data we have analyzed are a given set of maps of the electric field recorded during the simulation.

© 2004 Optical Society of America

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### Equations (9)

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(1)
$$S{e}_{\alpha}={\lambda}_{\alpha}{e}_{\alpha},$$
(2)
$${Y}_{\alpha}=\sum _{i=1}^{N}{e}_{\alpha}\left({t}_{i}\right)F\left({t}_{i}\right).$$
(3)
$${w}_{\alpha}=\frac{{\lambda}_{\alpha}}{{\sum}_{\alpha =1}^{N}{\lambda}_{\alpha}},$$
(4)
$${F}^{a,b}\left(x,y,t\right)={e}_{a}\left(t\right){Y}_{a}\left(x,y\right)+{e}_{b}\left(t\right){Y}_{b}\left(x,y\right),$$
(5)
$${e}_{a}\left(t\right)=\mathrm{cos}\left(\mathit{\omega t}+\varphi \right)=\mathrm{Re}\left[{e}^{-i\left(\mathit{\omega t}+\varphi \right)}\right],$$
(6)
$${e}_{b}\left(t\right)=\mathrm{cos}\left(\mathit{\omega t}+\varphi -\frac{\pi}{2}\right)=\mathrm{Re}\left[{e}^{-\left(\mathit{i\omega t}+\varphi -\frac{\pi}{2}\right)}\right].$$
(7)
$${F}^{a,b}\left(x,y,t\right)=\mathrm{Re}\left[{Y}_{a}\left(x,y\right){e}^{-\mathit{i\omega t}}+i{Y}_{b}\left(x,y\right){e}^{-\mathit{i\omega t}}\right]=\mathrm{Re}\left[\left({Y}_{a}\left(x,y\right)+i{Y}_{b}\left(x,y\right)\right){e}^{-\mathit{i\omega t}}\right],$$
(8)
$${\tilde{Y}}_{a,b}\left(x,y\right)={Y}_{a}\left(x,y\right)+i{Y}_{b}\left(x,y\right)=\mid {\tilde{Y}}_{a,b}\left(x,y\right)\mid \mathrm{exp}\left(\mathit{i\alpha}\left(x,y\right)\right),$$
(9)
$${F}^{a,b}\left(x,y,t\right)=\mid {\tilde{Y}}_{a,b}\left(x,y\right)\mid \mathrm{Re}\left[{e}^{-i\left(\mathit{\omega t}-\alpha \left(x,y\right)\right)}\right]=\mid {\tilde{Y}}_{a,b}\left(x,y\right)\mid \mathrm{cos}\left[\mathit{\omega t}-\alpha \left(x,y\right)\right]$$