## Resonant transmission of self-collimated beams through coupled zigzag-box resonators: slow self-collimated beams in a photonic crystal |

Optics Express, Vol. 20, Issue 8, pp. 8309-8316 (2012)

http://dx.doi.org/10.1364/OE.20.008309

Acrobat PDF (1251 KB)

### Abstract

The resonant transmission of self-collimated beams through zigzag-box resonators is demonstrated experimentally and numerically. Numerical simulations show that the flat-wavefront and the width of the beam are well maintained after passing through zigzag-box resonators because the up and the down zigzag-sides prevent the beam from spreading out and the wavefront is perfectly reconstructed by the output zigzag-side of the resonator. Measured split resonant frequencies of two- and three-coupled zigzag-box resonators are well agreed with those predicted by a tight binding model to consider optical coupling between the nearest resonators. Slowing down the speed of self-collimated beams is also demonstrated by using a twelve-coupled zigzag-box resonator in simulations. Our work could be useful in implementing devices to manipulate self-collimated beams in time domain.

© 2012 OSA

## 1. Introduction

1. 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]

5. D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express **11**, 1203–1211 (2003). [CrossRef] [PubMed]

6. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

10. T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Ring-type Fabry-Perot filter based on the self-collimation effect in a 2D photonic crystal,” Opt. Express **18**, 17106–17113 (2010). [CrossRef] [PubMed]

11. Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **85**, 4834–4386 (2004). [CrossRef]

6. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. **83**, 3251–3253 (2003). [CrossRef]

8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of selfcollimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. **87**, 181106 (2005). [CrossRef]

## 2. Results and discussion

*a*= 5 mm and

*r*= 0.4

*a*= 2 mm, respectively. Two parallel aluminum plates with periodically drilled holes are used to hold the alumina rods vertically.

*E*-polarized microwaves (the electric-field parallel to the rod axes) of frequencies around 12.5 GHz can propagate with almost no diffraction along the ΓM direction inside the PC [9

9. T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express **18**, 5384–5389 (2010). [CrossRef] [PubMed]

10. T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Ring-type Fabry-Perot filter based on the self-collimation effect in a 2D photonic crystal,” Opt. Express **18**, 17106–17113 (2010). [CrossRef] [PubMed]

*x*= ▵

*y*=

*a*/32. The discrete time step is set to ▵

*t*=

*S*▵

*x*/

*c*, where the Courant factor

*S*is chosen to be 0.5 for stable simulations. To obtain transmission spectra, a Gaussian pulse with a waist of

*w*= 4

*a*is launched into the PC and the transmitted power

*P*(

_{t}*ω*) is computed at the end of the PC.

*P*(

_{t}*ω*) is normalized to an incident power

*P*(

_{i}*ω*) calculated at near a source plane without the PC structure. Antireflection structures are employed to eliminate unwanted reflection at the PC-air interfaces [14

14. S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic crystal interfaces,” Opt. Express **16**, 4270–4277 (2008). [CrossRef] [PubMed]

15. T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. **95**, 011119 (2009). [CrossRef]

*f*

_{0}= 12.575 GHz (12.559 GHz). The measured frequency is slightly lower than the simulated one. The discrepancy between them may come from the small uncertainty of dielectric constant of the alumina rod since the simulated frequency was matched to the measured one when the dielectric constant of the alumina rod is 9.68, which is slightly smaller than 9.7. The Q-factor estimated from the relation of

*f*

_{0}/Δ

*f*in the measurement (the FDTD simulation), where Δ

*f*is a full width half maximum, is about 607 (718). The simulated electric-field distribution of the beam at

*f*

_{0}= 12.559 GHz represented in Fig. 2(c) shows that the flat-wavefront and the width of the resonant self-collimated beam are well maintained after passing through the zigzag-box resonator because the up and the down zigzag-sides prevent the beam from spreading out and the wavefront is perfectly reconstructed by the output zigzag-side of the resonator.

_{1}= 12.518 GHz (12.502 GHz) and Ω

_{2}= 12.630 GHz (12.615 GHz). The simulated electric-field distributions of the resonant modes with resonant frequencies of Ω

_{1}and Ω

_{2}are represented in Fig. 3 (b) and 3(c), respectively. One can see that the resonant mode with Ω

_{1}(Ω

_{2}) mimics an odd (even) symmetry mode, even though it has been known that a coupling between two identical resonant modes in coupled high dielectric cavities makes their frequency split into a lower frequency of even mode and a higher frequency of odd mode. Kee and Lim have demonstrated that the parity of split resonant modes in two-coupled resonators in a 2D PC could be switched due to the correlation of the inter-distance between resonators and the period of oscillatory decaying evanescent fields of resonant modes. [16

16. C.-S. Kee and H. Lim, “Coupling characteristics of localized photons in two-dimensional photonic crystals,” Phys. Rev. B **67**, 073103 (2003). [CrossRef]

*α*and

*β*are TB parameters and

*f*

_{0}is a resonant frequency of single resonator. The physical meanings of the TB parameters are described in detail in Ref. [17

17. A. Yariv, Y Xu, R. K. Lee, and A. Scherer, “Coupled- resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

*f*, the dispersion relation

*f*(

*k*) and the group velocity

*v*(

_{g}*k*) of propagation modes are determined by coupling coefficient

*κ*between the nearest resonators, Δ

*f*≃ 2

*f*

_{0}|

*κ*|,

*f*(

*k*) ≃

*f*

_{0}[1 +

*κ*cos(

*ka*)] and

*v*(

_{g}*k*) ≃ −2

*πf*

_{0}

*aκ*sin(

*ka*), where the coupling coefficient is defined as

*κ*=

*β*–

*α*[17

17. A. Yariv, Y Xu, R. K. Lee, and A. Scherer, “Coupled- resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

18. M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. **84**, 2140–2143 (2000). [CrossRef] [PubMed]

*f*

_{0}, Ω

_{1}and Ω

_{2}, we obtained the TB parameters and coupling coefficient

*α*= −0.0085 (0.0022),

*β*= −0.0174 (−0.0067), and

*κ*= −0.0089, respectively. From the TB parameters, one can expect that the group velocities of self-collimated beams should be less than 2

*πf*

_{0}

*a*|

*κ*| ≃ 0.05

*c*in the whole transmission band (Δ

*f*≃ 0.22 GHz from 12.45 to 12.67 GHz) and approach to zero at band edges, provided that the TB model is valid. To check the validity of the TB model of the coupled zigzag-box resonator system for self-collimated beams, we compared theoretical resonant frequencies of a three-coupled zigzag-box resonator predicted by the TB model, Γ

_{2}≃

*f*

_{0},

19. T. F. krauss, “Why do we need slow light?” Nat. Photon. **2**, 448–450 (2008). [CrossRef]

21. J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. **2**, 287–318 (2010). [CrossRef]

*v*(

_{g}*f*) = 2

*π*[

*df*(

*k*)/

*dk*], where

*k*is a wave vector in the media.

*ϕ*between a light propagating through a PC with thickness

*L*and air is given by (

*k*–

_{pc}*k*)

_{air}*L*, where

*k*and

_{pc}*k*are wave vectors in a PC and air, respectively [22

_{air}22. E. Ozbay, A. Abeyta, G. Tuttle, M. Tringides, R. Biswas, C. T. Chan, C. M. Soukoulis, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B **50**, 1945–1948 (1994). [CrossRef]

*ϕ*(

_{air}*f*) for air, (2)

*ϕ*(

_{pc}*f*) for a PC without a resonator and (3)

*ϕ*(

_{res}*f*) for a PC with the twelve-coupled zigzag-box resonator, the dispersion relation was determined by the relation,

*v*(

_{g}*f*) = 2

*π*/[

*dk*(

*f*)/

*df*] and the results were plotted in Fig. 4(c) with the theoretical curve from the TB model. The calculated group velocities (black-solid square) oscillate around the theoretical curve (red-thin line) and has local minimum values at resonant frequencies where the transmission exhibits peak values. The oscillatory behavior of the group velocity will disappear and the group velocity curve will be close to the theoretical curve, if the number of cavities becomes infinite [23

23. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**, 741–747 (2008). [CrossRef]

*c*/100 as a frequency approaches to edges of the transmission band.

## 3. Conclusion

## Acknowledgments

## References and links

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

2. | P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačiv́, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. |

3. | Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental Demonstration of Self-Collimation inside a Three-Dimensional Photonic Crystal,” Phys. Rev. Lett. |

4. | S.-H. Kim, T.-T. Kim, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S Kee, “Experimental demonstration of self-collimation of spoof surface plasmons,” Phys. Rev. B |

5. | D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express |

6. | X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. |

7. | D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. |

8. | S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of selfcollimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. |

9. | T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express |

10. | T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Ring-type Fabry-Perot filter based on the self-collimation effect in a 2D photonic crystal,” Opt. Express |

11. | Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett. |

12. | A. Taflove, |

13. | |

14. | S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic crystal interfaces,” Opt. Express |

15. | T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett. |

16. | C.-S. Kee and H. Lim, “Coupling characteristics of localized photons in two-dimensional photonic crystals,” Phys. Rev. B |

17. | A. Yariv, Y Xu, R. K. Lee, and A. Scherer, “Coupled- resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

18. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. |

19. | T. F. krauss, “Why do we need slow light?” Nat. Photon. |

20. | T. Baba, “Slow light in photonic crystals,” Nat. Photon. |

21. | J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon. |

22. | E. Ozbay, A. Abeyta, G. Tuttle, M. Tringides, R. Biswas, C. T. Chan, C. M. Soukoulis, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B |

23. | M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(260.5950) Physical optics : Self-focusing

(230.4555) Optical devices : Coupled resonators

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 20, 2012

Revised Manuscript: March 14, 2012

Manuscript Accepted: March 15, 2012

Published: March 26, 2012

**Citation**

Sun-Goo Lee, Seong-Han Kim, Teun-Teun Kim, Jae-Eun Kim, Hae Yong Park, and Chul-Sik Kee, "Resonant transmission of self-collimated beams through coupled zigzag-box resonators: slow self-collimated beams in a photonic crystal," Opt. Express **20**, 8309-8316 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8309

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### References

- 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. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačiv́, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater.5, 93–96 (2006). [CrossRef] [PubMed]
- Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental Demonstration of Self-Collimation inside a Three-Dimensional Photonic Crystal,” Phys. Rev. Lett.96, 173902 (2006). [CrossRef] [PubMed]
- S.-H. Kim, T.-T. Kim, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S Kee, “Experimental demonstration of self-collimation of spoof surface plasmons,” Phys. Rev. B83, 165109 (2011). [CrossRef]
- D. Chigrin, S. Enoch, C. Sotomayor Torres, and G. Tayeb, “Self-guiding in two-dimensional photonic crystals,” Opt. Express11, 1203–1211 (2003). [CrossRef] [PubMed]
- X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett.83, 3251–3253 (2003). [CrossRef]
- D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. J. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett.29, 50–52 (2004). [CrossRef] [PubMed]
- S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of selfcollimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett.87, 181106 (2005). [CrossRef]
- T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express18, 5384–5389 (2010). [CrossRef] [PubMed]
- T.-T. Kim, S.-G. Lee, S.-H. Kim, J.-E Kim, H. Y. Park, and C.-S. Kee, “Ring-type Fabry-Perot filter based on the self-collimation effect in a 2D photonic crystal,” Opt. Express18, 17106–17113 (2010). [CrossRef] [PubMed]
- Z. Li, H. Chen, Z. Song, F. Yang, and S. Feng, “Finite-width waveguide and waveguide intersections for self-collimated beams in photonic crystals,” Appl. Phys. Lett.85, 4834–4386 (2004). [CrossRef]
- A. Taflove, Computational Electrodynamics : The Finite-Difference Time-Domain Method (Artech House, Boston, 1995).
- http://ab-initio.mit.edu/wiki/index.php/Meep .
- S.-G. Lee, J.-S. Choi, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Reflection minimization at two-dimensional photonic crystal interfaces,” Opt. Express16, 4270–4277 (2008). [CrossRef] [PubMed]
- T.-T. Kim, S.-G. Lee, M.-W. Kim, H. Y. Park, and J.-E. Kim, “Experimental demonstration of reflection minimization at two-dimensional photonic crystal interfaces via antireflection structures,” Appl. Phys. Lett.95, 011119 (2009). [CrossRef]
- C.-S. Kee and H. Lim, “Coupling characteristics of localized photons in two-dimensional photonic crystals,” Phys. Rev. B67, 073103 (2003). [CrossRef]
- A. Yariv, Y Xu, R. K. Lee, and A. Scherer, “Coupled- resonator optical waveguide: a proposal and analysis,” Opt. Lett.24, 711–713 (1999). [CrossRef]
- M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett.84, 2140–2143 (2000). [CrossRef] [PubMed]
- T. F. krauss, “Why do we need slow light?” Nat. Photon.2, 448–450 (2008). [CrossRef]
- T. Baba, “Slow light in photonic crystals,” Nat. Photon.2, 465–473 (2008). [CrossRef]
- J. B. Khurgin, “Slow light in various media: a tutorial,” Adv. Opt. Photon.2, 287–318 (2010). [CrossRef]
- E. Ozbay, A. Abeyta, G. Tuttle, M. Tringides, R. Biswas, C. T. Chan, C. M. Soukoulis, and K. M. Ho, “Measurement of a three-dimensional photonic band gap in a crystal structure made of dielectric rods,” Phys. Rev. B50, 1945–1948 (1994). [CrossRef]
- M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics2, 741–747 (2008). [CrossRef]

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