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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 10 — May. 16, 2005
  • pp: 3802–3815
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Characterization of photonic crystal microcavities with manufacture imperfections

José M. Rico-García, José M. López-Alonso, and Javier Alda  »View Author Affiliations


Optics Express, Vol. 13, Issue 10, pp. 3802-3815 (2005)
http://dx.doi.org/10.1364/OPEX.13.003802


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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

1. Introduction

Manufacture imperfections in real photonic crystal structures are unavoidable. In fact, the disorder provoked by them might be a drawback in the performance of any photonic device. Alterations in the high-symmetry structure of the crystal may lead to unexpected deviations from theoretical designs. Changes in the band-gap of filters and resonant cavities have been reported, both numerically and experimentally [1

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]

]. Modes of photonic crystal fibers are modified if a random perturbation is introduced into the fiber cladding [2

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]

]. Then, a symmetry-breaking mechanism drives a modification in the higher-order modes characteristics, whereas the fundamental mode remains unaffected. Crystals made of dielectrics materials show a different behaviour than their metallic counterparts under equal degree of disorder [3

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]

]. The reality is that a certain amount of disorder in a photonic crystal can be important, and the net effect in its working parameters could be noticeable. However, it is a tough task to get randomness into numerical computations [4

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

]. Extensive simulations are required to achieve statistically meaningful results, regardless the method used to compute crystal parameters. Thus, the amount of data generated can be very large, and extracting useful information is challenging. Furthermore, a realistic simulation of manufacture errors involves structures that keep periodicy on the average, the so-called extended defect crystals [4]. Here, disorder stems from site displacements and size randomness of the fundamental units of the crystal [5

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]

].

2. Modelization and simulation of the perturbed photonic crystal

2.1. Manufacture errors

The photonic crystal analyzed in this paper is a microcavity showing defect modes within a near infrared bandgap [9

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]

]. The geometrical structure has been perturbed by allowing a controlled amount of change in the geometric parameters of the cylinders of the photonic crystal (see Fig. 1). The values of the electric permittivity of the materials composing the microcavity (GaAs rods in air) remain unchanged. These cylinders are characterized by their location and their radii. For an ideal microcavity, all the cylinders are equal but the central one (the defect) which is larger than the rest of them. The cylinders are arranged in a regular rectangular grid, having 5×5 cylinders. This number of rods is enough to enclose radiation efficiently [9

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]

, 12

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]

]. At the same time, this arrangement is computationally manageable. This geometry is perturbed to simulate the manufacture imperfections. In this paper the centers of the cylinders can be moved from their nominal position. The shape of the rods changes from a circular shape to a more general elliptic form. Then, each cylinder is characterized by five parameters: location of the center of the cylinder, px, py (along the x and y directions), values of the major and minor axis of the ellipse defining the cylinder, rM, rm, and orientation of the ellipse itself with respect to the reference frame, θ. The probability distribution functions that generate these five parameters for each rod have been Gaussian distributions for the dimensional parameters (location and axis of the ellipse), and an uniform distribution for the angular parameter. The Gaussian probability distribution function for each variable can be written as follows,

PDFZ=1σZ2πexp[(ZμZ)22σZ2],
(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.

2.2. The FDTD simulation

As it has been explained elsewhere [12

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

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]

], the photonic structure has a band-gap for TMz polarized fields (Ez,Hx,Hy components). Modes are produced because of the central cylinder defect (see Fig. 1). The band-gap encloses frequencies between 0.29ca and 0.42ca [9

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]

] (c is the speed of light). We look for spatial-temporal changes in the evolution of the modes when the dielectric structure is randomly perturbed. Hence, we supply the microcavity with energy in the same way as for the unperturbed case. For the sake of simplicity, we have payed attention to the monopolar mode of the unperturbed crystal[9

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]

]. A “soft” [14

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

] dipole source is placed in the center of the cavity. It evolves in time quasimonochromatically, oscillating at the original monopolar frequency[9

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]

]. The Maxwell equations are solved for TMz polarization in a 2D grid. The equations read as:

tHx=1μ0yEz
(2)
tHy=1μ0xEz
(3)
tEz=1ε(xHyyHx)
(4)

The grid contains 221×221 nodes. An Uniaxial Perfect Matched Layer (UPML) surrounds the computational domain [14

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

]. It absorbs the outgoing waves running away from the photonic crystal. We have used a UPML having a thickness of 10 cells. The source is switched off after 5000 temporal steps (approximately 30 periods of the quasi-harmonic excitation). The rise and fall of the monochromatic train of the source is smoothed through a continuous wave ramped excitation. Then, the fields advance in time freely. The electric field is recorded over the whole grid each 10Δt, where Δt is the temporal step of the algorithm (Δt=5.886×10-17s). The spatial step, Δs, is 0.025 µm and the Courant factor is S=0.7063. The field is recorded from t 1=40000Δt to t 2=41000Δt to form a sequence totaling 101 frames.

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].

Due to the rectangular grid employed in the computation, staircasing is unavoidably present and noticeable in the border of the elliptical cylinders. In this case, the spatial step is small enough (25 nm) to be sure that mode computation is not affected by this issue. However, when the minimum feature size is comparable with the spatial step, the results would be significatively distorted. A more refined approach using non-orthogonal grids or Finite Integration techniques would improve the performance of the algorithm [15

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]

].

The electromagnetic fields are computed for each realization of the perturbed map of dielectric permittivity. Maxwell equations have been solved for the 100 realizations of each ensemble to get a reliable PCA analysis. The output of the simulations are spatial-temporal maps of the electric and magnetic fields, resembling the monopolar mode of the unperturbed crystal. A complete simulation of an ensemble takes ~7 hours in a Pentium 4 with 1 Gb of RAM memory and with a clock frequency of 2 GHz. The results of the FDTD algorithm for the realizations shown in Fig. 1 are presented in Fig. 2. The distortion of the mode clearly increases with the error.

2.3. The PCA method

3. Analysis and results

PCA is applied after the fields have been calculated. Nevertheless, it should be noticed that we have excited the perturbed cavity as if it was unperturbed, because a real crystal works with the nominal parameters provided by theoretical studies. Thus, if the manufacture tolerances can be estimated and introduced into the FDTD simulation, PCA puts into new light the statistical deviations from the expected performance and makes easier the analysis of the FDTD output. In this case, we are interested in the stability of the mode inside the cavity when the permittivity map does not correspond with the perfect photonic crystal.

In the case of the excitation of the monopolar mode of a perfect photonic crystal microcavity we found two quasi-monochromatic processes. Each one is composed of two electromagnetic distributions having the same temporal frequency and shifted π/2 in time one with respect to the other. They can be added to form a complex electromagnetic distribution. The first and second principal component have been identified with the real and imaginary part of a complex electromagnetic distribution corresponding with the monopolar mode (see [8

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]

] and [9

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]

]). The third and fourth principal components configure the second quasi-monochromatic process. We interpreted this second process as a standing wave having a temporal frequency outside the bandgap and surviving in the structure at a very low level (although its contribution explains only 0.013%of the total variance of the data, the PCA method was able to show it and quantify its temporal evolution within the data). When the manufacture imperfections are included in the photonic crystal, the permittivity map losses its symmetry and periodicity. This will affect the electromagnetic distribution within the microcavity for a given excitation. In our case, the excitation is not centered anymore. We saw that an almost negligible decentering of the excitation source in the case of the monopolar mode was able to excite some other modes, even in the perfect structure [8

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]

]. The perturbed photonic crystal is excited with a source located in the same place used to excite the monopolar mode in the perfect photonic crystal. As far as all the rods are allowed to move and deform within some given range, the centration is lost for every realization. Therefore, we expect the excitation of a variety of electromagnetic distributions within the structure, and not only those expected for the unperturbed crystal. The collection of basic electric field distributions in this paper has been taken from the PCA decomposition of the perfect photonic crystal excited with sources that generate the five modes described in reference [9

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]

]. We took the first four PCA for the monopolar excitation because we use the same excitation in the example shown in this paper. The first and the second principal components correspond with the monopolar mode of the structure. For the other four excitations we only took the first two principal components (they describe the modes of the microcavity). Each pair of principal components forms a quasi-monochromatic process. The distribution of the electric field for the selected principal components are shown in Fig. 3. All these electric field distributions will be used later to expand the results obtained from the PCA method.

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.

Table 1. Relative contribution of λ1 and λ2 to the variance of the data

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An important application of the PCA is the filtering of the data set by taking those principal components with a clear physical meaning. At the level of imperfection studied in this paper, only the first two principal components seem to be in good accordance with the expected results obtained for the unperturbed case. In Fig. 7 we have generated the spatial-temporal evolution of the filtered data using only the first two principal components. Besides, we have generated the evolution of the difference between the original and the filtered data for the selected realization (for the sake of simplicity we only show it for the case of 5% of imperfection). The filtered data resembles very much the shape of the monopolar mode. The evolution of the difference should be compared with the plot of Fig. 4 of reference [8

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]

] where the same kind of filtering was applied to the results obtained for the unperturbed crystal.

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.

From the results obtained through the PCA we have performed another analysis of the data. In this case we have been interested in the description of the principal components in terms of the principal components produced by those excitations that generate the modes of the unperturbed microcavity applied to the perfect photonic crystal. These principal components were presented in Fig. 3. These electric field distributions are orthogonal and constitute a suitable non-complete base for the expansion of the results obtained for the actual realizations of the photonic crystal. The decomposition can be written as follows

PCk[j]=mαk,m[j]Em+Ok[j],
(5)

of these coefficients is given by

αk,m[j]=PCk[j](x,y)Em(x,y)dxdy.
(6)

In Table 2 we have evaluated the average for all the realizations of the percentage of energy of the principal component that is not described by the proposed non-complete base. We assume that each principal component represents a real electric field distribution having an energy given by ∬|PCk[j](x, y)|2 dxdy.

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

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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The results of this decomposition is shown in Fig. 8. The plots on the left of the figure are for the absolute value of the 〈|α k,m|〉 coefficients for k=1,2, and 3, and m corresponding to the electric field distribution of Fig. 3 denoted as Re[MP], Im[M], Re[SW], Im[SW], Re[Q1], Re[Q2], Im[Q2], Re[H1] and Re[H2] (Re[] and Im[] denote the real and imaginary part of the complex electric field distribution of the quasi-monochromatic process). The coefficients for the remaining Im[Q1], Im[H1], and Im[H2] are negligible for the data considered in these plots. The error bars represent the standard deviation of the plotted data. In the right of this figure we have plotted 〈|cosγ k,m|〉. They are the ensemble average of the absolute value of the cosine of the angle between the given principal component and Em. These cosines have been evaluated as

cosγk,m[j]=PCk[j](x,y)Em(x,y)dxdyPCk[j](x,y)2dxdyEm(x,y)2dxdy.
(7)

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).
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.
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).

4. Conclusion

In this paper we have modeled the manufacture errors in the geometry of the elements of a photonic crystal microcavity formed by dielectric rods immersed in vacuum and having a central defect. A collection of permittivity maps have been generated using several probability distribution functions for the location and shape of the rods. The rods are allowed to be elliptic. The level of imperfection is included in the variance of the Gaussian probability distributions used to model the dimensional parameters. We have analyzed three levels of error with respect to the nominal value: 1%, 3% and 5%. For each error we have generated 100 photonic crystal microcavities. Each statistical realization has been excited with the same source. The characteristics and location of the excitation is the same used to generate the so-called monopolar mode in the ideal photonic crystal. The response of the crystal has been analyzed using an FDTD algorithm. A sequence containing 101 frames was recorded for each realization. Each one of the sequences has been analyzed using the PCA method. Although the analysis shown here is applied to a very specific case, we should recall that the method outlined in this paper can be extended to any other permittivity map, by defining the appropriate probability distribution functions and analyzing the FDTD results by using the PCA method.

As a final remark, we should emphasize that the results shown here are only possible when analyzing the FDTD data by using the PCA method. Although the method is blind in nature, it is able to extract enough information to justify and conclude important relations using the knowledge of the nature of the analyzed problem.

Acknowlegments

This work has been partially supported by the project TIC2001-1259 of the Ministerio de Ciencia of Tecnología of Spain, and by the project GR/MAT/0497/2004 of the Comunidad of Madrid, Spain. One of authors, J. M. Rico-García, acknowledges the financial support by Universidad Complutense de Madrid through a post-graduated fellowship.

References and links

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]

OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(230.3990) Optical devices : Micro-optical devices
(350.3950) Other areas of optics : Micro-optics

ToC Category:
Research Papers

History
Original Manuscript: April 7, 2005
Revised Manuscript: May 5, 2005
Published: May 16, 2005

Citation
José Rico-García, José López-Alonso, and Javier Alda, "Characterization of photonic crystal microcavities with manufacture imperfections," Opt. Express 13, 3802-3815 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3802


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References

  1. G. Guida, T. Brillat, A. Amouche, F. Gadot, A. De Lustrac, 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, 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, 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, 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, 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, J. Alda, �??Photonic crystal characterization by FDTD and principal component analysis,�?? Opt. Express, 12, 2176-2186 (2004). [CrossRef] [PubMed]
  9. S. Guo, 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, 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, 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, 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, S. Hagness, Computacional Electrodynamics: The Finite-Difference Time Domain Method , 2nd edition, Artech House (2000).
  15. R. Schuhmann, T. Weiland, �??The Nonorthogonal Finite Integration Technique Applied to 2D- and 3D-Eigenvalue Problems,�?? IEEE Trans. on Magnetics, 36, 897-901 (2000). [CrossRef]
  16. J. M. López-Alonso, J. Alda, �??Bad pixel identification by means of the principal component analysis,�?? Opt. Eng. 41, 2152-2157 (2002). [CrossRef]
  17. J. M. López-Alonso, J. Alda, �??Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,�?? Opt. Eng. 43, 257-265 (2004). [CrossRef]

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