## Transmission characteristics of finite periodic dielectric waveguides

Optics Express, Vol. 14, Issue 8, pp. 3263-3272 (2006)

http://dx.doi.org/10.1364/OE.14.003263

Acrobat PDF (233 KB)

### Abstract

Transmission properties of the periodic dielectric waveguide (PDWG) formed by aligning a sequence of dielectric cylinders in air are investigated theoretically. Unlike photonic crystal waveguides (PCWs), light confinement in a PDWG is due to total internal reflection. Besides, the dispersion relation of the guided modes is strongly influenced by the dielectric periodicity along the waveguide. The band structure for the guided modes is calculated using a finite-difference time-domain (FDTD) method. The first band is used for guiding light, which makes PDWG single mode. Transmission is calculated using the multiple scattering method for various S shaped PDWGs, each containing two opposite bends. When PDWG operates in appropriate frequency ranges, high transmission (above 90%) is observed, even if the radius of curvature of the bends is reduced to three wavelengths. This feature indicates that the guiding ability of PDWG can be made better than the conventional waveguide when used in an optical circuit. In addition, PDWG has the advantage that it can be bent to any arbitrary shape while still preserves the high transmission, avoiding the geometric restriction that PCW is subject to.

© 2006 Optical Society of America

## 1. Introduction

4. Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. **77**, 3787–3790 (1996). [CrossRef] [PubMed]

5. A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, and C. M. Soukoulis, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 *μ*m,” Appl. Phys. Lett. **80**, 547–549 (2002). [CrossRef]

6. A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. **80**, 1698–1700 (2002). [CrossRef]

7. Amnon Yariv, Yong Xu, Reginald K. Lee, and Axel Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

8. Shayan Mookherjea, “Dispersion characteristics of coupled-resonator optical waveguides,” Opt. Lett. **30**, 2406–2408 (2005). [CrossRef] [PubMed]

7. Amnon Yariv, Yong Xu, Reginald K. Lee, and Axel Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

9. S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one-and two-dimensional photonic crystals,” Phys. Rev. B **65**, 165208 (2002). [CrossRef]

10. Shanhui Fan, N. Winn, Adrian Devenyi, J. C. Chen, Robert D. Meade, and J.D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B **12**, 1267–1272 (1995). [CrossRef]

11. Dmitry N. Chigrin, Andrei V. Lavrinenko, and Clivia M. Sotomayer Torres, “Nanopillars photonic crystal waveguides,” Opt. Express **12**, 617–622 (2004). [CrossRef] [PubMed]

*et.al*. [12

12. M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. **87**, 8268–8275 (2000). [CrossRef]

*multiple scattering method*[14

14. Bikash C. Gupta, Chao Hsien Kuo, and Zhen Ye, “Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems,” Phys. Rev. E **69**, 066615 (2004). [CrossRef]

## 2. System description and numerical results

**E**=

*E*

**z**̂ , and

**z**̂ is the direction of the cylinder axis. We begin with the consideration of an

*x*-propagating mode.

**k**=

*k*

**x**̂ is the Bloch wavevector (-

*π*/

*a*<

*k*≤

*π*/

*a*), and

**r**= (

*x*,

*y*) is the position vector. The vector function

**U**(

**r**) =

*U*(

*x*,

*y*)

**z**̂ satisfies the periodic boundary condition along

*x*

*y*= 0

*a*is the lattice spacing between two successive cylinders.

12. M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. **87**, 8268–8275 (2000). [CrossRef]

*a*× 9

*a*(See Fig. 1(a)). The Bloch boundary condition Eq.(2) is imposed in the

*x*direction, whereas the perfectly matched layers (PML) are in the |

*y*| ≥ 4.5

*a*regions. The dielectric constant of the cylinders is

*ε*= 11.56 (GaAs), and the radius of the cylinders is

*r*= 0.2

*a*. The calculated results for the first and second bands are shown in Fig. 1(a). The fact that the first band curve is below the light line indicates that the modes in this band is extended in the

*x*direction and localized in the

*y*direction, thus they are indeed guided modes. Since the first band is wider than the second, it provides a wider frequency range for guiding light. In addition, since for a first band mode the local field pattern around each cylinder is nearly the monopole type, and the second band does not overlap in frequency with the first band, thus the waveguide is single mode for the first band. Hereafter we study only the first band modes.

*v*=

_{g}*dω*/

*dk*can be derived, which is the pulse-propagation velocity for a narrow-bandwidth pulse. As is shown in Fig. 1(b), the group velocity goes to zero as

*k*approaches

*π*/

*a*. The slow group velocity behavior near the band edge can be understood as follows. A mode with Bloch wavevector

*k*≈

*π*/

*a*is almost a standing wave [1], thus the field energy density (energy per unit length along

*x*)

*YZ*plane per unit time)

*v*of a Bloch mode is equal to the energy velocity [2], defined as

_{g}*d*

^{2}

*ω*/

*dk*

^{2}< 0) there implies that photon becomes massive, with a negative effective mass

*m*=

_{eff}*α*(

*d*

^{2}

*ω*/

*dk*

^{2})

^{-1}. Here

*α*is a nonzero constant. In that region light is guided via the photon hopping mechanism [13

13. M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: A waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B **61**, R11 855–858 (2000). [CrossRef]

8. Shayan Mookherjea, “Dispersion characteristics of coupled-resonator optical waveguides,” Opt. Lett. **30**, 2406–2408 (2005). [CrossRef] [PubMed]

*ω*, we calculate the electromagnetic field using

*multiple scattering method*[14

14. Bikash C. Gupta, Chao Hsien Kuo, and Zhen Ye, “Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems,” Phys. Rev. E **69**, 066615 (2004). [CrossRef]

14. Bikash C. Gupta, Chao Hsien Kuo, and Zhen Ye, “Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems,” Phys. Rev. E **69**, 066615 (2004). [CrossRef]

**z**̂) with frequency

*ω*. The source is located one lattice spacing apart from the first cylinder of the waveguide.

*ω*= 0.25(2

*πc*/

*a*) and calculate the

**E**field for a long enough (120 cylinders) straight PDWG. A snapshot of the calculated

**E**-field (the real part of the complex amplitude) is shown in Fig. 2(a). As one can see, the field energy is indeed localized in the

*y*direction. According to the dispersion relation, the Bloch wavevector corresponding to this frequency is

**k**= 0.4(2

*π*/

*a*)

**x**̂, having a mode wavelength

*λ*= 2

*π*/

*k*= 2.5

*a*. However, the

*x*dependence of the field pattern we obtained does not seem like a sine curve of wavelength 2.5

*a*. The reason for this is that the amplitude function

*U*(

*x*) is not a constant but a periodic function of period

*a*, and 2.5

*a*is not large enough when compared to

*a*, so the field pattern becomes more complicated. To check if it is indeed the

**k**= 0.4(2

*π*/

*a*)

**x**̂ mode been excited, we make the following transformation. We define a new “effective wavenumber”

*k*′ as

^{n}

*E*(

*na*,0)] (the field evaluated at the

*n*th cylinder center, times (-1)

^{n}) follows a sine curve. For the present case, we get |

*k*′| = 0.1(2

*π*/

*a*), and the corresponding new effective wavelength is

*λ*′ = 10

*a*. Figure 2(b) shows the field pattern in (a), multiplied by the factor (-1)

^{n}and evaluated at the center of the

*n*th cylinder. Obviously, the modified field pattern now fits to a sinusoidal wave of wavelength 10

*a*.

*N*-cylinder PDWG we insert two planes of width 6a (between

*y*= ±3

*a*) at

*x*= 20.5

*a*and

*x*= (

*N*- 0.5)

*a*to evaluate the input power

*P*and output power

_{i}*P*. The transmission is then defined as

_{o}*T*=

*P*/

_{o}*P*(See Fig. 3(a1)). Three cases of transmission calculation for straight PDWGs are shown in Fig. 3(a2), they correspond to

_{i}*L*= 40

*a*, 81

*a*, and 100

*a*. Here

*L*= (

*N*- 1)

*a*stands for the length of the PDWG. The low frequency part of the these cases are slightly different, which is caused by the finite size effect. As we increase the frequency, the transmission curves merge, which implies they approach the result of infinite long PDWG. In this plot we also observe that at high frequency the transmission is slightly lower than that at low frequency, with some fluctuations in between. We believe this result is caused by the numerical restrictions of the method we used, such as the cross section planes are finite, and the sum of infinite Bessel series are replaced by a sum of only finite number of terms.

*a*, and the radius of curvature of the bend is

*R*. The bend angle is given by

*θ*= (

*N*-1)

_{b}*δ*, where

*N*is the number of cylinders in the bend, and

_{b}*δ*= 2sin

^{-1}(

*a*/2

*R*) is the angle corresponding to one lattice spacing. Beyond this region are the input and output parts of the waveguide. In the following simulations about one single bend the we assume the input and output region contains 21 and 2 cylinders, respectively.

*R*. Four cases are studied and the results are shown in Fig. 3(b2). One lattice spacing in the bend region for the

*R*= 57.3

*a*, 19.1

*a*, 11.5

*a*, 5.7

*a*waveguides correspond to

*δ*= 1°,3°,5°, and 10°. The total number of cylinders

*N*of the four waveguides are 114, 54, 42, and 33, respectively. These results reveal that the transmission in the low frequency region can be dramatically reduced when a bend is present, whereas the bend effect diminishes if the frequency becomes high enough. At frequency

*ω*= 0.25(2

*π*/

*a*) the calculated transmissions for the four cases all exceed 90%. In the following simulations for the

*θ*≠ 90° cases we always choose

*R*= 11.5

*a*, corresponding to

*δ*= 5°. With this choice the transmission can keep higher than 0.85 if the frequency

*ω*is chosen to be larger than 0.2(2

*πc*/

*a*).

*θ*. Such a structure is chosen to replace the one-bend PDWG so as to prevent possible numerical errors due to non-parallel cross section planes when we evaluate the energy flows. The S shaped PDWG contains five sections: the input region, the first bend, the connection part, the second bend, and the output region. We fix the number of cylinders of the whole PDWG to be

*N*= 80. Both the input and output parts contain 21 cylinders. For each bend we choose

*δ*= 5° and

*R*= 11.5

*a*, and it contains

*N*=

_{b}*θ*/

*δ*+ 1 cylinders. The remaining

*N*- 2

*N*- 2 × 21 = 38 - 2

_{b}*θ*/

*δ*cylinders are in the connection region.

*θ*< 10° region. When the bend angle is larger than 10°, the bend effect becomes important, and high frequency leads to high transmission. There are still some oscillatory behaviors, especially for the low frequency cases. This can be understood intuitively by the fact that the long wavelength propagating modes are more sensitive to the left-right asymmetry caused by bending the PDWG to one side. The results for the two cases with reduced frequencies

*a*/

*λ*= 0.225 and 0.25 are almost the same, which imply the transmission behaviors become stationary at high frequency region, consistent with the results in Fig. 5(a).

*R*must be large enough in order to reduce the loss. Typically the bend region has at least a size of millimeter order. For a PDWG, if the working wavelength is

*λ*= 1.55

*μm*, corresponding to

*a*/

*λ*= 0.25, then the bend radius

*R*= 11.5

*a*< 3

*λ*can be reduced to 5

*μm*, much smaller than that of the conventional dielectric waveguide. Furthermore, since it can be bent arbitrarily, it can avoid the geometric restriction that must be obeyed by the PCW.

*ε*= 11.56) CWGs using FDTD method. The width and length of the CWG are chosen as 0.4

*a*and 40

*a*, respectively. The simulation results are compared with the results of a PDWG of the same size, shown in Fig. 7. The source is a point source located one lattice spacing apart from the insertion edge of the waveguides. In Fig. 7(a) the effective index of the waveguides, defined as [15]

*P*is the incident power when the waveguide is absent (evaluated by integrating the normal component of the Poynting vector over a cross section plane 0.5

_{inc}*a*from the source), and

*P*is the output power when the waveguide is present. From Fig. 7(a), we find that when the reduced frequency is lower than 0.23, the effective index of PDWG is smaller than that of CWG, which is consistent with the lower reflection in Fig. 7(b). The oscillatory behavior of the reflection curve for the PDWG might be caused by the discrete structure of the PDWG. As the frequency increases further, the effective index of PDWG grows to a very high value, corresponding to the very slow group velocity of the high frequency (near the band edge) guided modes in the PDWG. In Fig. 7(c), the transmission as functions of bend radius for the two kinds of waveguides in the 90° bent situation are compared. We find that if the reduced frequency is chosen to be higher than 0.18, than for any case with

_{out}*R*≥ 11.5

*a*, the guiding ability of PDWG is better than CWG. We also find that if the bend radius is too small, for instance,

*R*= 5.7

*a*, the guiding ability of CWG becomes better than that of PDWG. For practical applications, the bend radius should be larger than 11.5

*a*, and the operating (reduced) frequency should be chosen between 0.18 and 0.23.

## 3. Conclusion

*λ*and the frequency is chosen high enough. PDWG has the advantage that it can be bent arbitrarily in a small region, which might be helpful to make a more compact optical circuits. At the same time it can avoid the geometric restriction that must be obeyed by a photonic crystal waveguide.

## Acknowledgments

## References and links

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

2. | Kazuaki Sakoda, |

3. | C. Kittel, |

4. | Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J.D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. |

5. | A. Talneau, L. Le Gouezigou, N. Bouadma, M. Kafesaki, and C. M. Soukoulis, “Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 |

6. | A. Chutinan, M. Okano, and S. Noda, “Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. |

7. | Amnon Yariv, Yong Xu, Reginald K. Lee, and Axel Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

8. | Shayan Mookherjea, “Dispersion characteristics of coupled-resonator optical waveguides,” Opt. Lett. |

9. | S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one-and two-dimensional photonic crystals,” Phys. Rev. B |

10. | Shanhui Fan, N. Winn, Adrian Devenyi, J. C. Chen, Robert D. Meade, and J.D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B |

11. | Dmitry N. Chigrin, Andrei V. Lavrinenko, and Clivia M. Sotomayer Torres, “Nanopillars photonic crystal waveguides,” Opt. Express |

12. | M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. |

13. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Propagation of photons by hopping: A waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals,” Phys. Rev. B |

14. | Bikash C. Gupta, Chao Hsien Kuo, and Zhen Ye, “Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems,” Phys. Rev. E |

15. | Katsunari Okamoto, |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: February 17, 2006

Revised Manuscript: March 31, 2006

Manuscript Accepted: March 31, 2006

Published: April 17, 2006

**Citation**

Pi-Gang Luan and Kao-Der Chang, "Transmission characteristics of finite periodic dielectric waveguides," Opt. Express **14**, 3263-3272 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3263

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### References

- J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals-Molding the Flow of Light (Princeton University Press, 1995).
- Kazuaki Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2001).
- C. Kittel, Introduction to Solid State Physics, 7th ed., (John Wiley & Sons, Inc., 1996).
- Attila Mekis, J. C. Chen, I. Kurland, Shanhui Fan, Pierre R. Villeneuve, and J.D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996). [CrossRef] [PubMed]
- A. Talneau, L. Le Gouezigou, N. Bouadma,M. Kafesaki, and C.M. Soukoulis, "Photonic-crystal ultrashort bends with improved transmission and low reflection at 1.55 μm," Appl. Phys. Lett. 80, 547-549 (2002). [CrossRef]
- A. Chutinan, M. Okano, and S. Noda, "Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 80, 1698-1700 (2002). [CrossRef]
- Amnon Yariv, Yong Xu, Reginald K. Lee, and Axel Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999). [CrossRef]
- Shayan Mookherjea, "Dispersion characteristics of coupled-resonator optical waveguides," Opt. Lett. 30, 2406-2408 (2005). [CrossRef] [PubMed]
- S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, "Analysis of defect coupling in oneand two-dimensional photonic crystals," Phys. Rev. B 65, 165208 (2002). [CrossRef]
- Shanhui Fan, N. Winn, Adrian Devenyi, J. C. Chen, Robert D. Meade, and J.D. Joannopoulos, "Guided and defect modes in periodic dielectric waveguides," J. Opt. Soc. Am. B 12, 1267-1272 (1995). [CrossRef]
- DmitryN. Chigrin, Andrei V. Lavrinenko, Clivia M. Sotomayer Torres, "Nanopillars photonic crystal waveguides," Opt. Express 12, 617-622 (2004). [CrossRef] [PubMed]
- M. Qiu and S. He, "A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions," J. Appl. Phys. 87, 8268-8275 (2000). [CrossRef]
- M. Bayindir, B. Temelkuran, and E. Ozbay, "Propagation of photons by hopping: A waveguiding mechanism through localized coupled cavities in three-dimensional photonic crystals," Phys. Rev. B 61, 855-858 (2000). [CrossRef]
- BikashC. Gupta, Chao Hsien Kuo, and Zhen Ye, "Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems," Phys. Rev. E 69, 066615 (2004). [CrossRef]
- Katsunari Okamoto, Fundamentals of Optical Waveguides (Academic Press, first Edition, 2000).

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