Abstract

The manufacture of a photonic crystal always produce deviations from the ideal case. In this paper we present a detailed analysis of the influence of the manufacture errors in the resulting electric field distribution of a photonic crystal microcavity. The electromagnetic field has been obtained from a FDTD algorithm. The results are studied by using the Principal Component Analysis method. This approach quantifies the influence of the error in the preservation of the spatial-temporal structure of electromagnetic modes of the ideal microcavity. The results show that the spatial structure of the excited mode is well preserved within the range of imperfection analyzed in the paper. The deviation from the ideal case has been described and quantitatively estimated.

©2005 Optical Society of America

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References

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  1. G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
    [Crossref]
  2. N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
    [Crossref]
  3. M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
    [Crossref]
  4. G. Guida, “Numerical studies of disordered photonic crystals,” Progress in Electromagnetic Research (PIER),  41, 107–131, (2003).
    [Crossref]
  5. W. R. Frei and H. T. Johnson “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B,  70, 165116–11 (2004).
    [Crossref]
  6. D. F. Morrison, Multivariate Statistical Methods, 3rd ed. (McGraw-Hill, Singapore, 1990) Chap. 8.
  7. J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal component characterization of noise for infrared images,” Appl. Opt. 41, 320–331 (2002).
    [Crossref] [PubMed]
  8. J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Photonic crystal characterization by FDTD and principal component analysis,” Opt. Express,  12, 2176–2186 (2004).
    [Crossref] [PubMed]
  9. S. Guo and S. Albin “Numerical Techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express,  11, 1080–1089 (2003).
    [Crossref] [PubMed]
  10. J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Numerical artifacts in finite-difference time-domain algorithms analyzed by means of Principal Components,” IEEE Trans. Antennas and Propagation (in press) (2005).
  11. M. Skorobogatiy, G. Bégin, and A. Talneau, “Statistical analysis of geometrical imperfections from the images of 2D photonic crystals,” Opt. Express,  13, 2487–2502 (2005).
    [Crossref] [PubMed]
  12. 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]
  13. M. Qiu and S. He “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B,  61, 12871–12876 (2000).
    [Crossref]
  14. A. Taflove and S. Hagness, Computacional Electrodynamics: The Finite-Difference Time Domain Method, 2nd edition, Artech House (2000).
  15. R. Schuhmann and T. Weiland, “The Nonorthogonal Finite IntegrationTechnique Applied to 2D- and 3D-Eigenvalue Problems,” IEEE Trans. on Magnetics,  36, 897–901 (2000).
    [Crossref]
  16. J. M. López-Alonso and J. Alda, “Bad pixel identification by means of the principal component analysis,” Opt. Eng. 41, 2152–2157 (2002).
    [Crossref]
  17. J. M. López-Alonso and J. Alda, “Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,” Opt. Eng. 43, 257–265 (2004).
    [Crossref]

2005 (1)

2004 (4)

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Photonic crystal characterization by FDTD and principal component analysis,” Opt. Express,  12, 2176–2186 (2004).
[Crossref] [PubMed]

J. M. López-Alonso and J. Alda, “Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,” Opt. Eng. 43, 257–265 (2004).
[Crossref]

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

W. R. Frei and H. T. Johnson “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B,  70, 165116–11 (2004).
[Crossref]

2003 (2)

G. Guida, “Numerical studies of disordered photonic crystals,” Progress in Electromagnetic Research (PIER),  41, 107–131, (2003).
[Crossref]

S. Guo and S. Albin “Numerical Techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express,  11, 1080–1089 (2003).
[Crossref] [PubMed]

2002 (2)

J. M. López-Alonso and J. Alda, “Bad pixel identification by means of the principal component analysis,” Opt. Eng. 41, 2152–2157 (2002).
[Crossref]

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal component characterization of noise for infrared images,” Appl. Opt. 41, 320–331 (2002).
[Crossref] [PubMed]

2001 (1)

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

2000 (3)

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

M. Qiu and S. He “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B,  61, 12871–12876 (2000).
[Crossref]

R. Schuhmann and T. Weiland, “The Nonorthogonal Finite IntegrationTechnique Applied to 2D- and 3D-Eigenvalue Problems,” IEEE Trans. on Magnetics,  36, 897–901 (2000).
[Crossref]

1996 (1)

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]

Albin, S.

Alda, J.

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Photonic crystal characterization by FDTD and principal component analysis,” Opt. Express,  12, 2176–2186 (2004).
[Crossref] [PubMed]

J. M. López-Alonso and J. Alda, “Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,” Opt. Eng. 43, 257–265 (2004).
[Crossref]

J. M. López-Alonso and J. Alda, “Bad pixel identification by means of the principal component analysis,” Opt. Eng. 41, 2152–2157 (2002).
[Crossref]

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal component characterization of noise for infrared images,” Appl. Opt. 41, 320–331 (2002).
[Crossref] [PubMed]

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Numerical artifacts in finite-difference time-domain algorithms analyzed by means of Principal Components,” IEEE Trans. Antennas and Propagation (in press) (2005).

Amouche, A.

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

Bayindir, M.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

Bégin, G.

Bernabéu, E.

Brillat, T.

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

Bulu, I.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

Cubukcu, E.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

De Lustrac, A.

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

Fan, S.

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]

Folkenberg, J. R.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

Frei, W. R.

W. R. Frei and H. T. Johnson “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B,  70, 165116–11 (2004).
[Crossref]

Gadot, F.

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

Guida, G.

G. Guida, “Numerical studies of disordered photonic crystals,” Progress in Electromagnetic Research (PIER),  41, 107–131, (2003).
[Crossref]

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

Guo, S.

Hagness, S.

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

Hansen, K. P.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

He, S.

M. Qiu and S. He “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B,  61, 12871–12876 (2000).
[Crossref]

Joannopoulos, J. D.

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]

Johnson, H. T.

W. R. Frei and H. T. Johnson “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B,  70, 165116–11 (2004).
[Crossref]

Lgsgaard, J.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

López-Alonso, J. M.

J. M. López-Alonso and J. Alda, “Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,” Opt. Eng. 43, 257–265 (2004).
[Crossref]

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Photonic crystal characterization by FDTD and principal component analysis,” Opt. Express,  12, 2176–2186 (2004).
[Crossref] [PubMed]

J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal component characterization of noise for infrared images,” Appl. Opt. 41, 320–331 (2002).
[Crossref] [PubMed]

J. M. López-Alonso and J. Alda, “Bad pixel identification by means of the principal component analysis,” Opt. Eng. 41, 2152–2157 (2002).
[Crossref]

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Numerical artifacts in finite-difference time-domain algorithms analyzed by means of Principal Components,” IEEE Trans. Antennas and Propagation (in press) (2005).

Morrison, D. F.

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

Mortensen, N. A.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

Nielsen, M. D.

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

Ozbay, E.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

Priou, A.

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

Qiu, M.

M. Qiu and S. He “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B,  61, 12871–12876 (2000).
[Crossref]

Rico-García, J. M.

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Photonic crystal characterization by FDTD and principal component analysis,” Opt. Express,  12, 2176–2186 (2004).
[Crossref] [PubMed]

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Numerical artifacts in finite-difference time-domain algorithms analyzed by means of Principal Components,” IEEE Trans. Antennas and Propagation (in press) (2005).

Schuhmann, R.

R. Schuhmann and T. Weiland, “The Nonorthogonal Finite IntegrationTechnique Applied to 2D- and 3D-Eigenvalue Problems,” IEEE Trans. on Magnetics,  36, 897–901 (2000).
[Crossref]

Skorobogatiy, M.

Soukoulis, C.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

Taflove, A.

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

Talneau, A.

Tut, T.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

Villeneuve, P. R.

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]

Weiland, T.

R. Schuhmann and T. Weiland, “The Nonorthogonal Finite IntegrationTechnique Applied to 2D- and 3D-Eigenvalue Problems,” IEEE Trans. on Magnetics,  36, 897–901 (2000).
[Crossref]

Appl. Opt. (1)

IEEE Trans. on Magnetics (1)

R. Schuhmann and T. Weiland, “The Nonorthogonal Finite IntegrationTechnique Applied to 2D- and 3D-Eigenvalue Problems,” IEEE Trans. on Magnetics,  36, 897–901 (2000).
[Crossref]

J. App. Phys. (1)

G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, and A. Priou “Dissociating the effect of different disturbances on the band gap of a two dimensional photonic crystal,” J. App. Phys. 88, 4491–4497 (2000).
[Crossref]

J. Opt. A Pure Appl. Opt. (1)

N. A. Mortensen, M. D. Nielsen, J. R. Folkenberg, K. P. Hansen, and J. Lgsgaard “Small-core photonic crystal fibers with weakly disordered air-hole claddings,” J. Opt. A Pure Appl. Opt. 6, 221–223 (2004).
[Crossref]

Opt. Eng. (2)

J. M. López-Alonso and J. Alda, “Bad pixel identification by means of the principal component analysis,” Opt. Eng. 41, 2152–2157 (2002).
[Crossref]

J. M. López-Alonso and J. Alda, “Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,” Opt. Eng. 43, 257–265 (2004).
[Crossref]

Opt. Express (3)

Phys. Rev. B (4)

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis. “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals,” Phys. Rev. B,  64, 195113–7 (2001).
[Crossref]

W. R. Frei and H. T. Johnson “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B,  70, 165116–11 (2004).
[Crossref]

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]

M. Qiu and S. He “Numerical method for computing defect modes in two-dimensional photonic crystals with dielectric or metallic inclusions,” Phys. Rev. B,  61, 12871–12876 (2000).
[Crossref]

Progress in Electromagnetic Research (PIER) (1)

G. Guida, “Numerical studies of disordered photonic crystals,” Progress in Electromagnetic Research (PIER),  41, 107–131, (2003).
[Crossref]

Other (3)

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

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

J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Numerical artifacts in finite-difference time-domain algorithms analyzed by means of Principal Components,” IEEE Trans. Antennas and Propagation (in press) (2005).

Supplementary Material (5)

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Figures (8)

Fig. 1.
Fig. 1. Permittivity maps for three realizations of the photonic crystal microcavity. The error increases from left to right (1%, 3%, and 5%). The white portion around the rods represent the possible locations of the rods for the statistical realizations analyzed in this paper. This portion grows as the manufacture imperfection increases.

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Fig. 2.
Fig. 2. Temporal evolution of the electric field component, Ez , for three realizations of the photonic crystal microcavity having manufacture errors (the realizations are the same than those presented in Fig. 1. The error increases from left to right: 1% (video file 1.78 Mb), 3% (video file 1.83 Mb), and 5% (video file 1.78 Mb). The unperturbed case can be seen in Fig. 7.c of reference [9].

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Fig. 3.
Fig. 3. Plot of the basic electric field distributions obtained from the PCA method for several excitations and for the unperturbed photonic crystal microcavity. The columns [MP] and [SW] are for the excitation of the monopolar mode. Only the column [MP] is describing the monopolar mode. These four plots are the first four eigenimages obtained from PCA (see Fig. 6). The columns [Q1] and [Q2] are the first two eigenimages for the two possible quadrupolar excitations. The columns [H1] and [H2] are for the hexapolar excitations. The eigenimages located in the same column correspond with eigenvalues having the same frequency but shifted π/2 in time. This temporal shift justifies their interpretation as Real and Imaginary parts of a complex mode (see reference [8]). The normalized frequency is shown below the column.

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Fig. 4.
Fig. 4. Plot of the logarithm of the first ten eigenvalues obtained from the PCA decomposition. The unperturbed case (green) can be compared with the those cases showing a 1% (black), 3% (red), and 5% (blue) of error. The dots are for the ensemble average, 〈λk 〉. The bars represent the range comprised within the 5% percentile and 95% percentile of the λk [j] distribution. Please note that the scale is logarithmic and the error bars are asymmetric. The horizontal location of the plotted points have been displaced to improve the representation.

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Fig. 5.
Fig. 5. Spatial distribution of the averaged principal components 〈PC1〉, 〈PC2〉, 〈PC3〉, and 〈PC4〉, for the three level of imperfection analyzed in this paper.

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Fig. 6.
Fig. 6. Plot of the electromagnetic field distributions obtained from the first 4 principal components for three realization of the permittivity map (the same realizations presented in Fig.s 1 and 2).

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Fig. 7.
Fig. 7. On the left of this Fig. we present the spatial temporal evolution of the filtered version of the original data set at 5% level of imperfection (video file 1.09 Mb). The filtering has been performed by taking into account only the first two principal components. The difference between the original data and the filtered one is also presented for comparison on the right of the Fig. (video file 1.31Mb). This difference takes into account only 1.9 % of the variance of the data set.

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Fig. 8.
Fig. 8. Plots of the average and standard deviation (error bars) of the coefficients (left column) and cosines (right column) obtained when projecting the first three principal components on the basis of electromagnetic distributions obtained from the PCA method applied to the unperturbed photonic crystal. The labels on the horizontal axis denote the modes presented in Fig. 3. The three manufacture imperfections are presented with different colors 1% (black), 3% (red), and 5% (blue).

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Tables (2)

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Table 1. Relative contribution of λ 1 and λ 2 to the variance of the data

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Table 2. Ensemble average of the percentage of energy explained by Ok , i. e., that is not described by the proposed non-complete base. The values are for the three first principal components and the three levels of imperfections analyzed in this paper.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

PDF Z = 1 σ Z 2 π exp [ − ( Z − μ Z ) 2 2 σ Z 2 ] ,
∂ t H x = − 1 μ 0 ∂ y E z
∂ t H y = − 1 μ 0 ∂ x E z
∂ t E z = 1 ε ( ∂ x H y − ∂ y H x )
PC k [ j ] = ∑ m α k , m [ j ] E m + O k [ j ] ,
α k , m [ j ] = ∬ PC k [ j ] ( x , y ) E m ( x , y ) d x d y .
cos ⁡ γ k , m [ j ] = ∬ PC k [ j ] ( x , y ) E m ( x , y ) d x d y ∬ ∣ PC k [ j ] ( x , y ) ∣ 2 d x d y ∬ ∣ E m ( x , y ) ∣ 2 d x d y .

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