## Characterization of photonic crystal microcavities with manufacture imperfections

Optics Express, Vol. 13, Issue 10, pp. 3802-3815 (2005)

http://dx.doi.org/10.1364/OPEX.13.003802

Acrobat PDF (1749 KB)

### 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

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]

## 2. Modelization and simulation of the perturbed photonic crystal

### 2.1. Manufacture errors

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]

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. 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]

*p*,

_{x}*p*(along the

_{y}*x*and

*y*directions), values of the major and minor axis of the ellipse defining the cylinder,

*r*,

_{M}*r*, and orientation of the ellipse itself with respect to the reference frame,

_{m}*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,

*p*,

_{x}*p*,

_{y}*r*,

_{M}*r*}. The values used in this paper are defined as follows:

_{m}*p*

_{x,N}=

*p*,

_{y}*N*the nominal values considered for the perfect microcavity (the pair of these nominal values characterizing the location of each cylinder are different);

*r*is the radius of the cylinders of the perfect case (all the cylinders have the same value but the central one);

_{N}*a*is the lattice constant of the perfect crystal, and

*E*is the manufacture error level; and

*E*=0.01,0.03, and 0.05 respectively). The results recently obtained from the statistical analysis of images of fabricated photonic crystals show that the measured values of imperfection are within the range proposed in this paper [11

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]

*π*). As we will see, this approach introduces an important increment in the information needed to describe a given realization of the photonic crystal. For an unperturbed microcavity we only need three parameters: radius of the cylinders of the grid,

*r*; radius of the central cylinder,

_{N}*r*

_{N,central}; and the spatial period of the crystal arranged in a squared grid,

*a*. Now, each realization of the perturbed photonic crystal needs 125 parameters when an arrangement of 5×5 cylinders are considered. These previous probability distributions have been used to generate an ensemble of 100 realizations of the dielectric permittivity map. In Fig. 1 we present the location and shape of the rods for three realizations having three different levels of manufacture errors (1%, 3%, and 5%, from left to right). In the case treated here, the nominal values are:

*r*=0.2

_{N}*µ*m,

*r*

_{N,central}=0.6

*µ*m, and

*a*=1

*µ*m. The white regions around the rods represent the locations of the rods along the 100 realizations studied in this paper. This white area increases with the manufacture error.

## 2.2. The FDTD simulation

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]

*E*,

_{z}*H*,

_{x}*H*components). Modes are produced because of the central cylinder defect (see Fig. 1). The band-gap encloses frequencies between

_{y}**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

**11**, 1080–1089 (2003). [CrossRef] [PubMed]

**11**, 1080–1089 (2003). [CrossRef] [PubMed]

*t*=5.886×10

^{-17}s). The spatial step, Δs, is 0.025

*µ*m and the Courant factor is

*S*=0.7063. The field is recorded from

*t*

_{1}=40000Δ

*t*to

*t*

_{2}=41000Δ

*t*to form a sequence totaling 101 frames.

## 2.3. The PCA method

## 3. Analysis and results

*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]

**11**, 1080–1089 (2003). [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]

**11**, 1080–1089 (2003). [CrossRef] [PubMed]

*〉. These averages are presented in Fig. 5. It is clear that the first two averaged eigenvalues, 〈PC*

_{k}_{1}〉, 〈PC

_{2}〉, are very close to those coming from the unperturbed microcavity. However, the third, and more clearly the fourth, are heavily perturbed with respect to the third eigenimage obtained for the unperturbed crystal. The deformation increases with the level of imperfection, as it should be expected. Even more, the averaged eigenimages somehow resemble the spatial patterns of modes of the photonic crystal microcavity mixed with the electric field distributions represented by the principal components obtained from the unperturbed crystal (see Fig. 3).

**12**, 2176–2186 (2004). [CrossRef] [PubMed]

*α*is the coefficient of the decomposition, and

_{m}*E*represents the normalized distribution of electric field obtained from the application of the PCA to the unperturbed photonic crystal. In our case, the chosen electric field distributions,

_{m}*E*, are real and have been represented in Fig. 3. The element

_{m}*O*[

_{k}*j*] appearing in the previous equation denotes the part of the principal component, PC

*[*

_{k}*j*], that can not be expanded in the proposed base of functions. The calculation

*[*

_{k}*j*](

*x*,

*y*)|

^{2}

*dxdy*.

*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 〈|cos

*γ*

_{k,m}|〉. They are the ensemble average of the absolute value of the cosine of the angle between the given principal component and

*E*. These cosines have been evaluated as

_{m}_{1}[

*j*] and PC

_{2}[

*j*] on Re[MP] and Im[MP] respectively are larger, in average, for a lower level of manufacture error. For the third principal principal component, PC

_{3}[

*j*], things are a little more complicated. For the lowest level of imperfection (1%) the projection on the Re[SW] spatial distribution is the largest. However, for the other two levels of manufacture errors, PC

_{3}[

*j*], contains a non-negligible portion of the Hexapolar modes (Re[H1], and Re[H2]). This fact can be interpreted following the same reasoning used to justify the appearance of hexapolar modes when some decentration of the excitation source is allowed [8

**12**, 2176–2186 (2004). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowlegments

## 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. |

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. |

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 , |

4. | G. Guida, “Numerical studies of disordered photonic crystals,” Progress in Electromagnetic Research (PIER) , |

5. | W. R. Frei and H. T. Johnson “Finite-element analysis of disorder effects in photonic crystals,” Phys. Rev. B , |

6. | D. F. Morrison, |

7. | J. M. López-Alonso, J. Alda, and E. Bernabéu, “Principal component characterization of noise for infrared images,” Appl. Opt. |

8. | J. M. López-Alonso, J. M. Rico-García, and J. Alda, “Photonic crystal characterization by FDTD and principal component analysis,” Opt. Express , |

9. | S. Guo and S. Albin “Numerical Techniques for excitation and analysis of defect modes in photonic crystals,” Opt. Express , |

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 , |

12. | P. R. Villeneuve, S. Fan, and J. D. Joannopoulos “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B , |

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 , |

14. | A. Taflove and S. Hagness, |

15. | R. Schuhmann and T. Weiland, “The Nonorthogonal Finite IntegrationTechnique Applied to 2D- and 3D-Eigenvalue Problems,” IEEE Trans. on Magnetics , |

16. | J. M. López-Alonso and J. Alda, “Bad pixel identification by means of the principal component analysis,” Opt. Eng. |

17. | J. M. López-Alonso and J. Alda, “Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,” Opt. Eng. |

**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

- 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]
- 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]
- 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]
- G.Guida, �??Numerical studies of disordered photonic crystals,�?? Progress in Electromagnetic Research (PIER), 41, 107-131, (2003). [CrossRef]
- W. R. Frei, H. T. Johnson �??Finite-element analysis of disorder effects in photonic crystals,�?? Phys. Rev. B, 70, 165116-11 (2004). [CrossRef]
- D. F. Morrison, Multivariate Statistical Methods, 3rd ed. (McGraw-Hill, Singapore, 1990) Chap. 8.
- J. M. López-Alonso, J. Alda, 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, J. Alda, �??Photonic crystal characterization by FDTD and principal component analysis,�?? Opt. Express, 12, 2176-2186 (2004). [CrossRef] [PubMed]
- S. Guo, S. Albin �??Numerical Techniques for excitation and analysis of defect modes in photonic crystals,�?? Opt. Express, 11, 1080-1089 (2003). [CrossRef] [PubMed]
- 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).
- 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]
- 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, 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]
- A. Taflove, S. Hagness, Computacional Electrodynamics: The Finite-Difference Time Domain Method , 2nd edition, Artech House (2000).
- 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]
- J. M. López-Alonso, 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, �??Characterization of artifacts in fully-digital image-acquisition systems. Application to web cameras,�?? Opt. Eng. 43, 257-265 (2004). [CrossRef]

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