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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References

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Appl. Opt.

J. Multivar. Behav. Res.

R. B. Cattell, �??The scree test for the number of factors,�?? J. Multivar. Behav. Res. 1, 245-276 (1966).
[CrossRef]

Opt. Commun.

C. Lixue, D. Xiaxou, D. Weiqiang, C. Liangcai, and L. Shutian, �??Finite-difference time-domain analysis of optical bistability with low treshold in one-dimensional non-linear photonic crystal with Kerr medium,�?? Opt. Commun. 209, 491-500 (2002).
[CrossRef]

Opt. Eng.

J. M. López-Alonso, and J. Alda, �??Operational parametrization of the 1/f noise of a sequence of frames by means of the principal components analysis in focal plane arrays,�?? Opt. Eng. 42, 427-430 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, �??Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,�?? Phys. Rev. B 54, 7837-7842 (1996).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Science

A. Tredicucci, �??Marriage of two device concepts,�?? Science 302, 1346-1347 (2003).
[CrossRef] [PubMed]

Other

A. Taflove and S. C. Hagness, Computacional Electrodynamics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, Boston, 2000).

D. F. Morrison, Multivariate Statistical Methods, 3rd ed. (McGraw-Hill, Singapore, 1990), Chap. 8.

D. D. Ghiglia and M. D. Pritt, Two Dimensional Phase Unwrapping (Wiley, New York, 1998), Chap. 4.

Supplementary Material (4)

» Media 1: AVI (512 KB)     
» Media 2: AVI (495 KB)     
» Media 3: AVI (809 KB)     
» Media 4: AVI (975 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Sequence of frames obtained for the photonic crystal analyzed in this paper and excited to generate the monopolar mode. In all the figures of this paper, the grid nodes coincides with the centers of the cylinders of the microcavity (video file, 512 KB)

Fig. 2.
Fig. 2.

Semilog plot of the eigenvalues of each principal component after application of the PCA method to the sequence of frames shown in Fig. 1. The classification and grouping technique of the principal components can be applied by grouping together those principal components that have consecutive overlapping in their respective uncertainties. We have plotted the images corresponding to four principal components (numbers 20, 40, 60, and 80) embedded in the same noise process.

Fig. 3.
Fig. 3.

The temporal evolution of the principal component is described by the associated eigenvectors. In this figure we plot the first three eigenvectors obtained from the PCA method. They show an harmonic dependence. In (a) we plotted the first two eigenvectors, which show the same temporal frequency but shift π/2. Image (b) corresponds to the third eigenvector. When representing e 2 versus e 1, in (c), we can check that the frequency is the same and that the relative phase delay is π/2.

Fig. 4.
Fig. 4.

On the left-hand side we show the sequence of frames (video file 495 KB) obtained by reconstructing the data with only Y 1(x,y) and Y 2(x,y) for the original data presented in Fig. 1. This result can be considered as a filtered version of the FDTD output that contains 99.98701% of the total variance of the original data. The mode observed here is the monopolar one. On the right-hand side we present the difference between the original and the filtered version (video file 809 KB).

Fig. 5.
Fig. 5.

In this figure we have plotted the spatial maps of the electric field associated with the first two principal components, Y 1(x,y) (top left) and Y 2(x,y) (top right), obtained from the simulation carried out for the monopolar mode. It is possible to evaluate the modulus (bottom left) and the phase(bottom right) of the complex eigenimage associated with these principal components.

Fig. 6.
Fig. 6.

Spatial distribution of the modulus and the phase associated with the “quasi-monochromatic” processes obtained for the hexapolar (left column) and one of the quadrupolar (right column) modes of the photonic crystal analyzed in this paper.

Fig. 7.
Fig. 7.

On the left we show the temporal evolution of the frames rectified by using only the standing wave principal component excited when the monopolar mode is fed at the center of the structure (video file 975 KB). The modulus of the instantaneous Poynting vector is displayed in the central map. The plot on the right corresponds to the standing wave obtained for the hexapolar mode excitation.

Fig. 8.
Fig. 8.

Spatial distribution of one of the principal components obtained for a decentered excitation of the monopolar mode. The spatial distribution closely resembles much the hexapolar mode that is generated when the excitation is not centered. The temporal evolution given by the associated eigenvector also coincides with the hexapolar mode frequency. The relative contribution to the total variance of this principal component is 0.0017%.

Equations (9)

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S e α = λ α e α ,
Y α = i = 1 N e α ( t i ) F ( t i ) .
w α = λ α α = 1 N λ α ,
F a , b x y t = e a ( t ) Y a x y + e b ( t ) Y b x y ,
e a ( t ) = cos ( ωt + ϕ ) = Re [ e i ( ωt + ϕ ) ] ,
e b ( t ) = cos ( ωt + ϕ π 2 ) = Re [ e ( iωt + ϕ π 2 ) ] .
F a , b x y t = Re [ Y a x y e iωt + i Y b x y e iωt ] = Re [ ( Y a x y + i Y b x y ) e iωt ] ,
Y ˜ a , b x y = Y a x y + i Y b x y = Y ˜ a , b x y exp ( x y ) ,
F a , b x y t = Y ˜ a , b x y Re [ e i ( ωt α x y ) ] = Y ˜ a , b x y cos [ ωt α x y ]

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