## Direct determination of photonic band structure for waveguiding modes in two-dimensional photonic crystals

Optics Express, Vol. 16, Issue 4, pp. 2461-2468 (2008)

http://dx.doi.org/10.1364/OE.16.002461

Acrobat PDF (550 KB)

### Abstract

We directly determine the experimental photonic band dispersion structure of waveguiding modes under the light line in a two-dimensional photonic crystal (2D PhC) waveguide by using angle-resolved attenuated total reflection spectroscopy. Resonance coupling between the external evanescent wave from total reflection within the prism and the waveguiding modes in the 2D PhC provides clear information on individual band components by resolving the angle (i.e., wave vector *k*) and photon energy. The experimentally determined photonic band structure, which is essential for understanding the novel light propagation properties of PhC systems with many degrees of freedom, agrees well with the band structure predicted by theory. Furthermore, we demonstrate the accuracy and suitability of this method by analyzing field distribution and eigen-photon-energy calculations for a model structure identical to the experimental arrangement of the prism and sample structure.

© 2008 Optical Society of America

## 1. Introduction

12. V. N. Astratov, D.M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B **60**, R16255–R16258 (1999). [CrossRef]

13. S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B **69**, 205109 (2004). [CrossRef]

*ω*=

*ck*) separates the observable region (

*ω*>

*ck*), in which the modes are oscillatory in air, from the region in which modes are evanescent in air and cannot couple with external free photons (

*ω*<

*ck*). However, engineering applications of two-dimensional (2D) PhC waveguides require the use of modes below the light line due to their vertical confinement and long lifetimes. The dispersion characteristics of waveguiding modes below the light line have been studied indirectly by interference measurements of transmission in a 2D PhC slab [14

14. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. **87**, 253902 (2001). [CrossRef] [PubMed]

*n*) dispersion, but it cannot accurately determine the shape of the band dispersion due to the uncertainty in the origin position of the

_{g}*k*-axis. Although determining the photonic band dispersion relation of the waveguiding modes below the light line is indispensable for understanding the unique properties of 2D PhC waveguides, experimental information on the specific shape of these band dispersions has not been obtained directly.

*k*) and photon energy. Moreover, we describe the accuracy and suitability of this method by analyzing it by performing three-dimensional finite-difference time-domain (3D-FDTD) calculations of the field distribution and eigen-photon-energy of the observed photonic band modes for a model structure identical to the experimental arrangement of the prism and sample structure.

## 2. Experiment

15. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. **216**, 398–410 (1968). [CrossRef]

*k*

_{//}>

*ω*/

*c*. When the surface of the 2D PhC waveguide is brought close to (but not touching) the prism, this evanescent wave can couple to the guiding mode. A sharp dip originating from resonance between the evanescent wave and the waveguiding mode can be detected in the reflectance spectrum.

13. S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B **69**, 205109 (2004). [CrossRef]

_{2}/Ar plasmas, optimized to produce straight sidewalls. The detailed techniques for forming the straight sidewalls have been described elsewhere [16

16. S. Inoue, K. Kajikawa, and Y. Aoyagi, “Dry-etching method for fabricating photonic-crystal waveguides in nonlinear-optical polymer,” Appl. Phys. Lett. **82**, 2966–2968 (2003). [CrossRef]

17. S. Inoue and K. Kajikawa, “Inductivity coupled plasma etching to fabricate the nonlinear optical polymer photonic crystal waveguides,” Mater. Sci. Eng. B **103**, 170–176 (2003). [CrossRef]

*n*=1.87. The incident beam was focused onto the sample (100×100 µm

_{p}^{2}) using an achromatic long-focal lens (f=250 mm). The reflectivity spectra were detected using a corrected Czerny–Turner spectrometer (320-mm focal length) and a liquid nitrogen-cooled open-electrode charge coupled device (CCD). The incident and detected angles could be rotated by a stepping motor stage. The external incident angle

*θ*of the beams was scanned in the range 20–50°. For simplicity, the in-plane propagation lattice direction was set along the Γ-X (

_{ext}*ϕ*=0°) direction. The two-dimensional Brillouin zone and the symmetry points for a square lattice are shown in Fig. 1(b). All the measurements were made with transverse electric (TE) polarization of the incident light.

*ε*(

**r**) is the permittivity,

*µ*(

**r**) is the permeability, and

*σ*(

**r**) is the conductivity, and are functions of the spatial coordinates. These equations can be discretized in space and time by a Yee-cell algorithm [18,19]. We used a 10

^{5}discretization grid in the unit cell for the FDTD time-stepping formulas. An initial TE-polarized electric field with a low-symmetry distribution in the core was used to excite the TE-like modes. The evolution of the initial TE-polarized electromagnetic field was calculated using a discrete FDTD method. The model structure used for the calculations consisted of four layers: an air layer, the PMMA patterned core layer, the Ag metallic cladding layer, and the Si substrate layer. The thicknesses and the patterned square lattice of circular air holes were set to be the same as for the specimen used in the experiments. As a first approximation we used the equivalent relative permittivity for a metal (Ag cladding)

*ε*(

*ω*)=

*ε*

_{0}+

*σ/*i

*ω*. This method has a simple and general form and is very stable in FDTD calculations [13

13. S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B **69**, 205109 (2004). [CrossRef]

20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

^{7}S/m. To simulate the 2D PhC waveguide structure in the computational mesh, Mur’s absorbing boundary condition was applied to the top and bottom boundaries normal to the plane of the waveguides. The Bloch boundary conditions over a unit cell of a PhC give the

*k*vector dependence of the eigenmode. All the fields are obtained in the time domain in this method. In order to obtain the spectral information, the calculated fields need to be converted from the time domain into the frequency domain by taking their Fourier transform. The peaks of the spectral intensity correspond to the locations of the eigen-photon-energies of the band modes. The position-dependent field distributions of the band modes are also obtained by taking the Fourier transform of the transient field pattern. By performing the analysis in three dimensions, we were able to evaluate the accuracy and suitability of the technique for analyzing an experimental photonic band structure below the light line.

## 3. Results and discussion

*k*

_{//}=(2

*π*/

*λ*)

*n*sin

_{p}*θ*, where λ is the wavelength of the incident light and

_{int}*θ*=0° is normal incidence. The resonance dip at each resonance angle in the reflectivity is thus directly connected to the experimental dispersion curves, i.e., the experimental photonic band structure. Moreover, in regions of good coupling, the magnitude of the dip, which indicates the extent of the resonance coupling, was very large and had a maximum value of ΔR/R>80%. Since these features are sensitive to the profile of the etched holes and the separation distance between the prism and the PhC surface, it is clear that we have achieved precise nano-lithographical fabrication of the PhC waveguide structure and good coupling between the evanescent field and the photonic band mode for a short separation.

_{int}*d*between the PhC and prism of the lowest photonic energy band indexed as

*A*, the second lowest band indexed as

*B*, and the third lowest band indexed as

*C*, at the X point (see Fig. 3). In this figure, the maximum magnitude of the field is normalized to unity. The figure shows that the fields of the band modes under the light line are strongly confined in the waveguide core for large separation distances. The strength of the evanescent field is stronger at the PhC surface for shorter separation distances, resulting in increased coupling between the incident radiation and the band modes for a small air gap. This can be seen in the data as an oscillating mode in the prism at a separation distance less than 500 nm. Good coupling was observed at about 300 nm in each band mode.

*d*between the PhC and prism for three different energy bands indexed as

*A*,

*B*, and C at the point X (see Fig. 3). The eigen-photon-energy information was obtained by taking a Fourier transform of the fields in the time domain with an energy resolution of 13 meV at a photon energy of 2.0 eV. The photon energy domain for the shorter separation distances shifts towards the high photon energy side for each band. For the high-energy band

*C*, the modulation of the eigen-photon-energy is relativity large. The quantity of the shift in the eigen-photon-energy with decreasing air-gap distance depends mainly on the vertical field distribution of the band mode. The lower energy bands

*A*and

*B*are the lowestorder waveguide mode and its electric field is almost perfectly confined in the waveguide core. On the other hand, the high-energy band

*C*, near the light cone, has a double peak in the z direction, which indicates that the band corresponds to the second-order waveguide mode. This implies that this band mode has a relativity large amplitude in the air side region compared with the bands

*A*and

*B*, and is well coupled with the evanescent wave from the prism surface. This result concerning the differences in the quantity of the eigen-photon-energy shift agrees well with the behavior of the field distributions in Fig. 4. Judging from the field distribution changes in Fig. 4, the separation distance for which good coupling can be obtained is about 300 nm. The eigen-photon-energy shifts between separation distances of 300 nm (good coupling) and 3 µm (isolated) were +13 meV (Δ

*h*ν/

*h*ν=0.6%) for band

*C*and less than the energy resolution for bands

*A*and

*B*. In addition, the energy shifts were comparable with the computational error for our 3D-FDTD calculations, which was roughly estimated to be less than 1% from the convergence behavior of the eigenfrequencies. We found that the fluctuation in the photon energy domain between the band modes of the PhC with the prism and the isolated PhC were barely significant even at the well-coupled separation distance of 300 nm. These results clearly demonstrate that, except in the case of over coupling (

*d*< 100 nm), the angle-resolved attenuated total reflection spectroscopy technique is accurate and suitable for directly determining the experimental photonic band structure below the light line in 2D PhC waveguides.

## 4. Conclusions

## Acknowledgments

## References and links

1. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

2. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

3. | S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science |

4. | P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express |

5. | Y. Akahane, T. Asano, B. Song, and S. Noda, “High- |

6. | S. Foteinopoulou and C. M. Soukoulis, “Negative refraction and left-handed behavior in two-dimensional photonic crystals,” Phys. Rev. B |

7. | A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. |

8. | R. E. Slusher and B. J. Eggleton, |

9. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

10. | S. Inoue and Y. Aoyagi, “Design and fabrication of two-dimensional photonic crystals with predetermined nonlinear optical properties,” Phys. Rev. Lett. |

11. | S. Inoue and Y. Aoyagi, “Ultraviolet second-harmonic generation and sum-frequency mixing in two-dimensional nonlinear optical polymer photonic crystals,” Jpn. J. Appl. Phys. |

12. | V. N. Astratov, D.M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, “Photonic band-structure effects in the reflectivity of periodically patterned waveguides,” Phys. Rev. B |

13. | S. Inoue and Y. Aoyagi, “Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding,” Phys. Rev. B |

14. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. |

15. | A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. |

16. | S. Inoue, K. Kajikawa, and Y. Aoyagi, “Dry-etching method for fabricating photonic-crystal waveguides in nonlinear-optical polymer,” Appl. Phys. Lett. |

17. | S. Inoue and K. Kajikawa, “Inductivity coupled plasma etching to fabricate the nonlinear optical polymer photonic crystal waveguides,” Mater. Sci. Eng. B |

18. | K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. |

19. | A. Taflow, |

20. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(130.2790) Integrated optics : Guided waves

(300.6490) Spectroscopy : Spectroscopy, surface

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(130.5296) Integrated optics : Photonic crystal waveguides

(240.5698) Optics at surfaces : Reflectance anisotropy spectroscopy

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: December 10, 2007

Revised Manuscript: February 1, 2008

Manuscript Accepted: February 4, 2008

Published: February 6, 2008

**Citation**

Shin-ichiro Inoue, Shiyoshi Yokoyama, and Yoshinobu Aoyagi, "Direct determination of photonic band structure for waveguiding modes in two-dimensional photonic crystals," Opt. Express **16**, 2461-2468 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-4-2461

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### References

- E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
- S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
- S. Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, "Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal," Science 282, 274-276 (1998). [CrossRef] [PubMed]
- P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-9-1996. [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003). [CrossRef] [PubMed]
- S. Foteinopoulou and C. M. Soukoulis, "Negative refraction and left-handed behavior in two-dimensional photonic crystals," Phys. Rev. B 67, 235107 (2003). [CrossRef]
- A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, "Negative refraction at infrared wavelengths in a two-dimensional photonic crystal," Phys. Rev. Lett. 93, 073902 (2004). [CrossRef] [PubMed]
- R. E. Slusher and B. J. Eggleton, Nonlinear Photonic Crystals, (Springer, Berlin, 2003).
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- S. Inoue and Y. Aoyagi, "Design and fabrication of two-dimensional photonic crystals with predetermined nonlinear optical properties," Phys. Rev. Lett. 94, 103904 (2005). [CrossRef] [PubMed]
- S. Inoue and Y. Aoyagi, "Ultraviolet second-harmonic generation and sum-frequency mixing in two-dimensional nonlinear optical polymer photonic crystals," Jpn. J. Appl. Phys. 45, 6103-6107 (2006). [CrossRef]
- V. N. Astratov, D.M. Whittaker, I. S. Culshaw, R. M. Stevenson, M. S. Skolnick, T. F. Krauss, and R. M. De La Rue, "Photonic band-structure effects in the reflectivity of periodically patterned waveguides," Phys. Rev. B 60, R16255-R16258 (1999). [CrossRef]
- S. Inoue and Y. Aoyagi, "Photonic band structure and related properties of photonic crystal waveguides in nonlinear optical polymers with metallic cladding," Phys. Rev. B 69, 205109 (2004). [CrossRef]
- M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001). [CrossRef] [PubMed]
- A. Otto, "Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection," Z. Phys. 216, 398-410 (1968). [CrossRef]
- S. Inoue, K. Kajikawa, and Y. Aoyagi, "Dry-etching method for fabricating photonic-crystal waveguides in nonlinear-optical polymer," Appl. Phys. Lett. 82, 2966-2968 (2003). [CrossRef]
- S. Inoue and K. Kajikawa, "Inductivity coupled plasma etching to fabricate the nonlinear optical polymer photonic crystal waveguides," Mater. Sci. Eng. B 103, 170-176 (2003). [CrossRef]
- K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966).
- A. Taflow, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House INC, Norwood, 1995).
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]

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