## Eigenmode symmetry for simple cubic lattices and the transmission spectra

Optics Express, Vol. 3, Issue 1, pp. 19-27 (1998)

http://dx.doi.org/10.1364/OE.3.000019

Acrobat PDF (309 KB)

### Abstract

The existence of uncoupled modes is identified by gaps in the transmission spectra when the density of states is nonzero. We use a group theoretic analysis of the photonic band structure for a simple cubic lattice to tag the symmetry and polarization of each band. The results are compared with transmission spectra calculated by the transfer matrix method.

© Optical Society of America

## 1. Introduction

2. G. Kurizki and J. W. Haus, eds. J. Mod. Opt. **41** (2), (1994). [CrossRef]

7. W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, Phys. Rev. Lett. **68**, 2023 (1992). [CrossRef] [PubMed]

8. W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, J. Opt. Soc. Am. B **10**, 322 (1993). [CrossRef]

9. K. Sakoda, Phys. Rev. B **51**, 4672 (1995). [CrossRef]

12. M. Wada, K. Sakoda, and K. Inoue, Phys. Rev. B **52**, 16297 (1995). [CrossRef]

13. M. Wada, Y. Doi, K. Inoue, and J. W. Haus, Phys. Rev. B **55**, 10443 (1997). [CrossRef]

14. K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J.W. Haus, Japan. J. Appl. Phys. **33**, L1463 (1994). [CrossRef]

9. K. Sakoda, Phys. Rev. B **51**, 4672 (1995). [CrossRef]

20. K. Sakoda, Phys. Rev. B **55**, 15345 (1997). [CrossRef]

21. H. S. Sözüer and J. W. Haus, J. Opt. Soc. Am. B **10**, 296 (1993). [CrossRef]

21. H. S. Sözüer and J. W. Haus, J. Opt. Soc. Am. B **10**, 296 (1993). [CrossRef]

18. C. C. Cheng, V. Arbet-Engels, A. Scherer, and E. Yablonovitch, Phys. Scr. **T68**, 17 (1996). [CrossRef]

24. E. Ozbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas, and K. M. Ho, Appl. Phys. Lett. **64**, 2059 (1994). [CrossRef]

20. K. Sakoda, Phys. Rev. B **55**, 15345 (1997). [CrossRef]

## 2. Results

### 2.1 Group Theory

*a*and a numerical factor including the speed of light, i.e.

*a*/2

*πc*. We devote our attention to waves propagating in the Γ -

*M*direction, where the symmetry is reduced and the lowest bands for the two polarizations are not degenerate. The wavevector is along the (1,1,0) axis. This is a two-fold symmetry direction,

*C*

_{2v}The irreducible representations are :

*A*

_{1},

*A*

_{2},

*B*

_{1}, B

_{2}, which are explained below. At the M-point the irreducible representation of the

*D*

_{4h}symmetry is

*A*

_{1g},

*A*

_{1u},

*B*

_{1g},

*B*

_{1u},

*A*

_{2g},

*A*

_{2u},

*B*

_{2g},

*B*

_{2u},

*E*,

_{g}*E*. The corresponding symmetry of the H-field vector is discussed in Ref. [20

_{u}20. K. Sakoda, Phys. Rev. B **55**, 15345 (1997). [CrossRef]

*M*symmetry contains invariance under two mirror reflection operations. One is the vertical plane defined by the Γ -

*Z*and Γ -

*M*lines; the other is the horizontal plane defined by the Γ -

*X*and Γ -

*M*lines. The eigenfunctions will be symmetric or anti-symmetric with respect to these operations. We define the symmetry with respect to the E-field vector, a complex vector field amplitude.

*B*

_{1}is symmetric with respect to the horizontal plane and anti-symmetric with respect to the vertical plane. It can be coupled to an incident S-polarized wave, which is polarized parallel to the horizontal plane; this mode is colored blue and its a dashed line. By contrast

*B*

_{2}, denoted by the red colored line or the solid line, is symmetric with respect to the vertical plane and anti-symmetric with respect to the horizontal plane; it can couple with a P-polarized wave.

*A*

_{1}(the light-blue colored line or dashed dotted line) is symmetric with respect to both planes and

*A*

_{2}(the green colored line, also a dashed-dotted line) is anti-symmetric in both planes. These modes are not activated by incident plane waves. In the following we apply these assignments for the bands.

### 2.2 Transmission Spectrum

25. J. B. Pendry, J. Mod. Opt. **41**, 208 (1994). [CrossRef]

26. P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, Comput. Phys. Commun. **85**, 306 (1995). [CrossRef]

*M*direction with a crystal that is 32 periods thick is given in Figure 2; the lateral direction is infinite. There is a considerable shift in the width and the depth of the gaps as the sample thickness is increased, but 32 layers provides a clear determination of the gap positions. The oscillations at low frequencies are Fabry-Perot interference due to reflections from opposite surfaces. The interference is strongly affected by the sample thickness; at low frequencies where only one band is found, the number of oscillations is used by us to verify the number of layers. Oscillations occur at the higher frequencies, but they are difficult to interpret becuase of the strong dispersion in the bands and the existence of multiply excited bands with different dispersion.

*M*direction the unit cell is deformed. The separation of the sphere centers is √2 in the propagation direction, but is unity in the transverse directions. We modified the unit cell’s geometry to make the lateral to longitudinal length ratio 10:14. This creates a deformed SC geometry, a contraction of 1% along the longitudinal direction making the lattice parameters 0.99:0.99:1.0. This distortion does not noticably affect the band structure, which is discussed next.

### 2.3 Band Structure

27. H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B **45**, 13962 (1992). [CrossRef]

*N*

^{-1/3}in Figure 3, where

*N*is the number of plane-waves used. The eigenfrequencies from the H-methods (shown on the left side) have a steep slope as they converge. By contrast the E-method eigenfrequencies are nearly flat as a function of

*N*; this is especially true of the lower bands. Throughout the entire Brillouin zone the H-method band structure is qualitatively the same as that given by the E-method. However, even for 1000 plane waves the frequencies are shifted by 10% or more from those of the E-method and the extrapolated values by both methods. Therefore in our analysis we use the E-method band structure calculations. In all cases presented in this paper, the E-method has better convergence. It is interesting to note that the band structure presented for the smallest spheres in this paper exhibited the slowest rate of convergence.

### 2.4 Air-holes

*r*= 0.495 (i.e. they are nearly touching). the symmetries of the lowest 7 bands in the Γ -

*M*direction assigned. A direct gap opens between the second- and third-bands, but there is no common gap over all directions. The direct gap is found for both the E-and H-methods. The Γ -

*M*direction is distinguished by the broken degeneracy of the bands. By examining the symmetry of each band we determine whether an incoming wave will be coupled to it. The validity of our analysis is checked by the structure of the transmission spectra. From the band structure calculations and group analysis two gaps are identified for the P-polarization and one for the S-polarization. The positions of the gaps are indicated in Fig. 1 and their numerical values appear in Table 1.

### 2.5 Dielectric spheres

*r*= 0.495. Non-intersecting spheres are not constructed in the laboratory, but this case is an interesting example of the relation between band symmetry and transmittance. In this case a direct gap opens between the fifth and sixth bands. We also note that the dispersion of the third band in the Γ -

*M*direction is not monotonic; this is a common feature in waveguide dispersion curves. It is of interest since it yields both a negative and positive value for the group velocity at a single frequency. This feaure is also found in two-dimensional photonic crystals, but is absent from one-dimensional ones.

*r*= 0.297. The volume fraction of spheres is low enough that no direct gap is observed, but there is strong dispersion, including the appearance of distinct nonmonotonic bands. The density of states is nonzero over the entire frequency range, which makes this case a good candidate to demonstrate the correspondence between band symmetry and the transmissivity features.

*A*

_{1}or

*A*

_{2}modes spans a portion of the gap region. The lowest two bands for each polarization are in good quantitative agreement with the transmissivity, see Table 1. Although the volume fraction is small, the appearance of large gaps due entirely to predicted uncoupled modes means that the device design parameters based on these features are not stringent; they appear over a wide range of volume fractions.

## 3. Conclusions

20. K. Sakoda, Phys. Rev. B **55**, 15345 (1997). [CrossRef]

*M*direction here, but other directions can likewise be evaluated. The SC lattice transmissivity provided direct confirmation of the symmetry assignments. Our findings demonstrate excellent correspondence between the gaps determined from the symmetry assigned to the bands and the gaps in the transmittance. There were no cases where a gap due to an uncoupled mode was expected, but not found in the transmissivity. The appearance of gaps in the density of states require high volume fractions of air in the structure and as a consequence the structures are extremely fragile and difficult to fabricate. Uncoupled modes on the other hand appear over a wide range of volume fractions yielding mechanically stronger structures.

## Acknowledgments

## References

1. | C. M. Bowden, J. D. Dowling, and H. Everitt, eds. J. Opt. Soc. Am. B |

2. | G. Kurizki and J. W. Haus, eds. J. Mod. Opt. |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

4. | See articles in C. M. Soukoulis, ed. |

5. | J.W. Haus, J. Mod. Opt. |

6. | J.W. Haus, article in |

7. | W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, Phys. Rev. Lett. |

8. | W. M. Robertson, G. Arjavalingam, R.D. Meade, K.D. Brommer, A.M. Rappe, and J.D. Joannopoulos, J. Opt. Soc. Am. B |

9. | K. Sakoda, Phys. Rev. B |

10. | K. Sakoda, Phys. Rev. B |

11. | K. Sakoda, Phys. Rev. B |

12. | M. Wada, K. Sakoda, and K. Inoue, Phys. Rev. B |

13. | M. Wada, Y. Doi, K. Inoue, and J. W. Haus, Phys. Rev. B |

14. | K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi, and J.W. Haus, Japan. J. Appl. Phys. |

15. | K. Inoue, M. Wada, K. Sakoda, M. Hayashi, T. Fukushima, and A. Yamanaka, Phys. Rev. B |

16. | A. Rosenberg, R. J. Tonucci, H-B. Lin, and A. J. Campillo, Opt. Lett. |

17. | H-B. Lin, R. J. Tonucci, and A. J. Campillo, Appl. Phys. Lett. |

18. | C. C. Cheng, V. Arbet-Engels, A. Scherer, and E. Yablonovitch, Phys. Scr. |

19. | K. Ohtaka and Y. Tanabe, J. Phys. Soc. Jpn |

20. | K. Sakoda, Phys. Rev. B |

21. | H. S. Sözüer and J. W. Haus, J. Opt. Soc. Am. B |

22. | H. S. Sözüer and J. D. Dowling, J. Mod. Opt. |

23. | K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. |

24. | E. Ozbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas, and K. M. Ho, Appl. Phys. Lett. |

25. | J. B. Pendry, J. Mod. Opt. |

26. | P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, Comput. Phys. Commun. |

27. | H. S. Sözüer, J. W. Haus, and R. Inguva, Phys. Rev. B |

**OCIS Codes**

(260.2110) Physical optics : Electromagnetic optics

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Focus Issue: Photonic crystals

**History**

Original Manuscript: March 10, 1998

Revised Manuscript: March 23, 1998

Published: July 6, 1998

**Citation**

Zhenyu Yuan, Joseph Haus, and Kazuaki Sakoda, "Eigenmode symmetry for simple cubic
lattices and the transmission spectra," Opt. Express **3**, 19-27 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-1-19

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### References

- C. M. Bowden, J. D. Dowling and H. Everitt, eds. J. Opt. Soc. Am. B 10 (2), (1993).
- G. Kurizki and J. W. Haus, eds. J. Mod. Opt. 41 (2), (1994). [CrossRef]
- J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals, (Princeton Univ. Press, NJ, 1995).
- See articles in C. M. Soukoulis, ed. Photonic Band Gaps and Localization, (Plenum, NY, 1993).
- J.W. Haus, J. Mod. Opt. 41, 195 (1994). [CrossRef]
- J.W. Haus, article in Quantum Optics of Confined Systems, M. Ducloy and D. Bloch, eds., (Kluwer Academic, Dordrecht, 1996), pp. 101-142. [CrossRef]
- W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe and J. D. Joannopoulos, Phys. Rev. Lett. 68, 2023 (1992). [CrossRef] [PubMed]
- W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe and J. D. Joannopoulos, J. Opt. Soc. Am. B 10, 322 (1993). [CrossRef]
- K. Sakoda, Phys. Rev. B 51, 4672 (1995). [CrossRef]
- K. Sakoda, Phys. Rev. B 52, 7982 (1995). [CrossRef]
- K. Sakoda, Phys. Rev. B 52, 8992 (1995). [CrossRef]
- M. Wada, K. Sakoda and K. Inoue, Phys. Rev. B 52, 16297 (1995). [CrossRef]
- M. Wada, Y. Doi, K. Inoue, and J. W. Haus, Phys. Rev. B 55, 10443 (1997). [CrossRef]
- K. Inoue, M. Wada, K. Sakoda, A. Yamanaka, M. Hayashi and J.W. Haus, Jpn. J. Appl. Phys. 33, L1463 (1994). [CrossRef]
- K. Inoue, M. Wada, K. Sakoda, M. Hayashi T. Fukushima, and A. Yamanaka, Phys. Rev. B 53, 1010 (1996). [CrossRef]
- A. Rosenberg, R. J. Tonucci, H-B. Lin, A. J. Campillo, Opt. Lett. 21, 830 (1996). [CrossRef] [PubMed]
- H-B. Lin,R. J. Tonucci, A. J. Campillo, Appl. Phys. Lett. 68, 2927 (1996). [CrossRef]
- C. C. Cheng, V. Arbet-Engels, A. Scherer, and E. Yablonovitch, Phys. Scr. T68, 17 (1996). [CrossRef]
- K. Ohtaka and Y. Tanabe, J. Phys. Soc. Jpn 65, 2670 (1996). [CrossRef]
- K. Sakoda, Phys. Rev. B 55, 15345 (1997). [CrossRef]
- H. S. S"oz" uer and J. W. Haus, J. Opt. Soc. Am. B 10, 296 (1993). [CrossRef]
- H. S. S"oz" uer and J. D. Dowling, J. Mod. Opt. 41, 231 (1994). [CrossRef]
- K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas and M. Sigalas, Solid State Commun. 89, 413 (1994). [CrossRef]
- E. " Ozbay, E. Michel, G. Tuttle, R. Biswas, M. Sigalas and K. M. Ho, Appl. Phys. Lett. 64, 2059 (1994). [CrossRef]
- J. B. Pendry, J. Mod. Opt. 41, 208 (1994). [CrossRef]
- P. M. Bell, J. B. Pendry, L. M. Moreno and A. J. Ward, Comput. Phys. Commun. 85, 306 (1995). [CrossRef]
- H. S. S"oz" uer, J. W. Haus and R. Inguva, Phys. Rev. B 45, 13962 (1992). [CrossRef]

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