## Self-guiding in two-dimensional photonic crystals

Optics Express, Vol. 11, Issue 10, pp. 1203-1211 (2003)

http://dx.doi.org/10.1364/OE.11.001203

Acrobat PDF (640 KB)

### Abstract

Dielectric periodic media can possess a complex photonic band structure with allowed bands displaying strong dispersion and anisotropy. We show that for some frequencies the form of iso-frequency contours mimics the form of the first Brillouin zone of the crystal. A wide angular range of flat dispersion exists for such frequencies. The regions of iso-frequency contours with near-zero curvature cancel out diffraction of the light beam, leading to a self-guided beam.

© 2003 Optical Society of America

## 1. Introduction

2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**, 2059 (1987). [CrossRef] [PubMed]

3. S. John, “Strong localization of photons in certain disordered dielectric superlattices.,” Phys. Rev. Lett. **58**, 2486–2489 (1987). [CrossRef] [PubMed]

4. P. St. J. Russell, “Optics of Floquet-Block waves in dielectric gratings.,” Appl. Phys. B **B39**, 231–246 (1986). [CrossRef]

5. R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Optics **34**, 1589–1617 (1987). [CrossRef]

6. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, R10096–10099 (1998). [CrossRef]

7. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A **17**, 1012 (2000). [CrossRef]

6. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, R10096–10099 (1998). [CrossRef]

8. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. **74**, 1212–1214 (1999). [CrossRef]

9. P. Etchegoin and R. T. Phillips, “Photon focusing, internal diffraction, and surface states in periodic dielectric structures,” Phys. Rev. B **53**, 12674–12683 (1996). [CrossRef]

10. D. N. Chigrin and C. M. Sotomayor Torres, “Periodic thin-film interference filters as one-dimensional photonic crystals,” Opt. Spectrosc. **91**, 484–489 (2001). [CrossRef]

*et al.*[6

6. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, R10096–10099 (1998). [CrossRef]

**k**-space. The direct rigorous numerical simulation of the beam propagation in a finite size photonic crystal is given in Section 3. Section 4 summarizes our results.

## 2. Fourier space analysis

*r*=0.15

*d*, where

*d*is the period of the lattice. A band structure calculations was done using the plane wave expansion method [11

11. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

*d*/λ=0.3567 (the first band) and Ω=

*d*/λ=0.5765 (the second band) are depicted.

*d*/λ=0.3567 forms a square with rounded corners, rotated by 45± with respect to the Brillouin zone and centered at M-point of the first Brillouin zone (Fig. 3, left). The iso-frequency contour for the normalized frequency Ω=

*d*/λ=0.5765 consists of two branches, which are plotted in red and blue in Fig. 3, right. Both branches mimic the form of the first Brillouin zone of the crystal being squares with rounded corners. The “red” branch is centered at the Γ-point and the “blue” branch is centered at the M-point of the Brillouin zone (Fig. 3, right).

**A**

_{nk}(

**r**) is dimensionless and satisfies orthogonalization and normalization relations [12

12. J. P. Dowling and C. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A **46**, 612 (1992). [CrossRef] [PubMed]

**A**denotes the vector potential. We assume that a light source is a harmonically oscillating dipole with a frequency ω

_{0}and a real dipole moment

**d**, located at the position

**r**

_{0}inside a photonic crystal. Then the field at the position

**r**in the crystal reads [12

12. J. P. Dowling and C. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A **46**, 612 (1992). [CrossRef] [PubMed]

_{nk}is the Bloch eigenfrequency,

*V*is the volume of the unit cell of the crystal, * denotes the complex conjugate and

*c*is the speed of light in vacuum. Integration is performed over the first Brillouin zone and summation is over different photonic bands, where

*n*is a band index.

**x**=

**r**-

**r**

_{0}the exponential function in the integral (1) will oscillate very rapidly. To evaluate this integral we use the fact that in a typical experiment |

**x**|≫λ, where λ is the wavelength of the electromagnetic wave. Then, by the method of stationary phase, the main contribution to the value of the integral (1) comes from those regions of the iso-frequency surface ω=ω

_{0}, where product

**k**

_{n}

**x**is stationary. We denote the stationary phase points by

13. R. Camley and A. Maradudin, “Phonon focusing at surface,” Phys. Rev. B **27**, 1959 (1983). [CrossRef]

**r**far from the point dipole is given by

**V**

_{nk}=∇

_{k}ω

_{nk}is the group velocity of the eigenwave (

*n*,

**k**),

**k**

_{n}=

**x**·

^{-1/2}and |

**x**|

^{-1}steady-state emission intensity far from a point dipole is given by the time averaged Poynting’s vector

**S**(

**r**)=(

*c*/8π)

*Re*[

**E**(

**r**)×

**H***(

**r**)] and it is proportional to the inverse Gaussian curvature of the iso-frequency surface, ~|

^{-1}and to the inverse square of the distance between a source and an observation point, ~|

**x**|

^{-2}. The asymptotic energy flux shows the necessary amount of decrease with distance (~|

**x**|

^{-2}, providing a finite value of the energy flux in any finite interval of a solid angle, assuming non vanishing Gaussian curvature. A vanishing curvature formally implies an infinite flux along the corresponding observation direction. This phenomenon is known as photon focusing [9

9. P. Etchegoin and R. T. Phillips, “Photon focusing, internal diffraction, and surface states in periodic dielectric structures,” Phys. Rev. B **53**, 12674–12683 (1996). [CrossRef]

10. D. N. Chigrin and C. M. Sotomayor Torres, “Periodic thin-film interference filters as one-dimensional photonic crystals,” Opt. Spectrosc. **91**, 484–489 (2001). [CrossRef]

*d*/λ=0.3567 and Ω=

*d*/λ=0.5765 (Fig. 3). We assume that a point source produces an isotropic and uniform distribution of wave vectors

**k**

_{n}. Then after an averaging of (2) over the dipole moment orientation, the main contribution to the relative radiation intensity far from the point source is given by an inverse Gaussian curvature of the iso-frequency surface, ~|

^{-1}.

*d*/λ=0.3567, Fig. 4, left), the radiation pattern is strongly focused along the [11

11. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

*d*/λ=0.5765, Fig. 4, right) is more complex, due to the presence of two branches of the iso-frequencies contour. While the “red” branch displays very strong focusing along the [10

10. D. N. Chigrin and C. M. Sotomayor Torres, “Periodic thin-film interference filters as one-dimensional photonic crystals,” Opt. Spectrosc. **91**, 484–489 (2001). [CrossRef]

**91**, 484–489 (2001). [CrossRef]

*self-guiding*regime the radiation pattern due to a point source implies that the strongly collimated light propagation is insensitive to the divergence of the initial beam and almost insensitive to the orientations of the beam with respect to the crystal lattice. This property can lead to useful applications of

*self-guiding*for waveguide design.

17. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. **27**, 936, (2002). [CrossRef]

17. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. **27**, 936, (2002). [CrossRef]

## 3. Real space analysis

18. G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A **14**, 3323 (1997). [CrossRef]

*k*sin(θ), β

^{2}=

*k*

^{2}-α

^{2},

*k*=2π/λ. Note that the mean angle of incidence of the beam is α0=

*k*sin(θ0).

*d*/λ=0.3567 (left) and Ω=

*d*/λ=0.5765 (right). The point source is placed in the middle of the crystal, at

*x*

_{0}=0 and

*y*

_{0}=0. The emitted light is guided in channels in the [11

11. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

**91**, 484–489 (2001). [CrossRef]

*d*/λ=0.5765. Figure 6, right shows the field map of the TM polarized incident beam, where the width of the beam is

*W/d*=2.5 and the waist is located at

*x*

_{0}=0 and

*y*

_{0}/

*d*=19.5. The self-guiding in the [10

**91**, 484–489 (2001). [CrossRef]

*et al.*[19

19. A. Talneau, Ph. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett. **27**, 1522 (2002). [CrossRef]

*W/d*=2.5. The upper panel shows the modulus of the incident field when the angle of incidence θ

_{0}=0, the panel second from the top shows the modulus of the transmitted field above the crystal for the same angle of incidence. To see a clear difference on the transmitted field the angle of incidence must be increased up to 5° degrees (see Fig. 7 third curve from the top). For the angle of incidence equal to 10° degrees the shape of the transmitted beam is strongly modified as shown in Fig. 7, bottom.

*d*/λ=0.5765. The first beam impinges from the top of the crystal and the second impinges from the right. The first beam is identical to the incident beam of Fig. 6 but with

*y*

_{0}/

*d*=14.5 and the second one propagates from the right with

*W/d*=2.5. The waist is now located at

*x*

_{0}/

*d*=14.5 and

*y*

_{0}=0. The key feature of the self-guiding illustrated in Fig. 8 is that two beams can cross each other without cross-talk, in a contrast to the case of narrow dielectric waveguides. This effect offers an advantage for applications as it is not trivial to design crossed waveguides with no cross-talk in a comparable size scale.

## 4. Conclusion

## Acknowledgments

## References and links

1. | V. P. Bykov, “Spontaneous emission in a periodic structure,” Sov. Phys. - JETP |

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

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

4. | P. St. J. Russell, “Optics of Floquet-Block waves in dielectric gratings.,” Appl. Phys. B |

5. | R. Zengerle, “Light propagation in singly and doubly periodic planar waveguides,” J. Mod. Optics |

6. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

7. | B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A |

8. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. |

9. | P. Etchegoin and R. T. Phillips, “Photon focusing, internal diffraction, and surface states in periodic dielectric structures,” Phys. Rev. B |

10. | D. N. Chigrin and C. M. Sotomayor Torres, “Periodic thin-film interference filters as one-dimensional photonic crystals,” Opt. Spectrosc. |

11. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

12. | J. P. Dowling and C. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A |

13. | R. Camley and A. Maradudin, “Phonon focusing at surface,” Phys. Rev. B |

14. | D. N. Chigrin, “Radiation pattern of a classical dipole in a photonic crystal,” in preparation. |

15. | A. Yariv and P. Yeh, |

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

17. | A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. |

18. | G. Tayeb and D. Maystre, “Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,” J. Opt. Soc. Am. A |

19. | A. Talneau, Ph. Lalanne, M. Agio, and C. M. Soukoulis, “Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,” Opt. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(260.2110) Physical optics : Electromagnetic optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 2, 2003

Revised Manuscript: May 9, 2003

Published: May 19, 2003

**Citation**

Dmitry Chigrin, Stefan Enoch, Clivia Sotomayor Torres, and Gérard Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express **11**, 1203-1211 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-10-1203

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### References

- V. P. Bykov, �??Spontaneous emission in a periodic structure,�?? Sov. Phys. - JETP 35, 269�??273 (1972).
- E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- S. John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486�??2489 (1987). [CrossRef] [PubMed]
- P. St. J. Russell, �??Optics of Floquet-Block waves in dielectric gratings.�?? Appl. Phys. B B39, 231�??246 (1986). [CrossRef]
- R. Zengerle, �??Light propagation in singly and doubly periodic planar waveguides,�?? J. Mod. Optics 34, 1589�??1617 (1987). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Superprism phenomena in photonic crystals,�?? Phys. Rev. B 58, R10096�??10099 (1998). [CrossRef]
- B. Gralak, S. Enoch, and G. Tayeb, �??Anomalous refractive properties of photonic crystals,�?? J. Opt. Soc. Am. A 17, 1012 (2000). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Self-collimating phenomena in photonic crystals,�?? Appl. Phys. Lett. 74, 1212�??1214 (1999). [CrossRef]
- P. Etchegoin and R. T. Phillips, �??Photon focusing, internal diffraction, and surface states in periodic dielectric structures,�?? Phys. Rev. B 53, 12674�??12683 (1996). [CrossRef]
- D. N. Chigrin and C. M. Sotomayor Torres, �??Periodic thin-film interference filters as one-dimensional photonic crystals,�?? Opt. Spectrosc. 91, 484�??489 (2001). [CrossRef]
- S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell�??s equations in a planewave basis,�?? Opt. Express 8, 173�??190 (2001),<a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a> [CrossRef] [PubMed]
- J. P. Dowling and C. Bowden, �??Atomic emission rates in inhomogeneous media with applications to photonic band structures,�?? Phys. Rev. A 46, 612 (1992). [CrossRef] [PubMed]
- R. Camley and A. Maradudin, �??Phonon focusing at surface,�?? Phys. Rev. B 27, 1959 (1983). [CrossRef]
- D. N. Chigrin, �??Radiation pattern of a classical dipole in a photonic crystal,�?? in preparation
- A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley and Sons, New York, 1984).
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
- A. Yariv, �??Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,�?? 27, 936, (2002). [CrossRef]
- G. Tayeb and D. Maystre, �??Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities,�?? J. Opt. Soc. Am. A 14, 3323 (1997). [CrossRef]
- A. Talneau, Ph. Lalanne, M. Agio, and C. M. Soukoulis, �??Low-reflection photonic-crystal taper for efficient coupling between guide sections of arbitrary widths,�?? Opt. Lett. 27, 1522 (2002). [CrossRef]

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