## Apodized photonic crystal waveguide gratings

Optics Express, Vol. 14, Issue 10, pp. 4459-4468 (2006)

http://dx.doi.org/10.1364/OE.14.004459

Acrobat PDF (175 KB)

### Abstract

Apodized photonic crystal (PC) waveguide gratings are proposed to suppress sidelobes which appear in reflection spectra of usual PC waveguide gratings. By using specific functions (Gauss and Gauss-cosine functions) for the longitudinal refractive index distribution, it is possible to suppress sidelobes in the reflection spectra of PC waveguide gratings efficiently. The apodization is realized by simply changing diameters of dielectric pillars adjacent to the PC waveguide core. It is shown that by using Gauss-cosine functions for the apodization, Bragg frequency of the waveguide grating becomes insensitive to the magnitude of perturbation leading to the possibility of designing waveguide gratings with arbitrarily Bragg frequency and bandwidth by modulating geometrical parameters only.

© 2006 Optical Society of America

## 1. Introduction

1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature **386**, 143–149 (1997). [CrossRef]

6. T. Fujisawa and M. Koshiba, “Finite-element mode-solver for nonlinear periodic optical waveguides and its application to photonic crystal circuits,” J. Lightwave Technol. **23**, 382–387 (2005). [CrossRef]

8. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

8. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

9. M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. **18**, 102–110 (2000). [CrossRef]

10. T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits,” J. Lightwave Technol. **22**, 684–691 (2004). [CrossRef]

*W*=10,

*R*=10

^{-4}, Δ

*T*=1fs, and

*N*=2×10

_{max}^{4}, where

*W, R*, Δ

*T*, and

*N*stand for the number of perfectly matched layers, the theoretical reflection coefficient, the time interval of calculations, and the number of the calculation steps, respectively.

_{max}## 2. Photonic crystal waveguide gratings with Gaussian apodization

*a*, where the diameter of rods are taken as

*d*=0.5

*a*. By assuming silicon-on-insulator configuration (SOI), refractive indexes of rods and background material are, respectively, taken as 3.4 (Si) and 1.45 (SiO

_{2}). Experiments of the system of pillars have been already demonstrated in Refs. [11

11. M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-µm-wavelength light guiding in waveguides in squarelattice-of-rod photonic crystal slab,” Appl. Phys. Lett. **84**, 4298–4300 (2004). [CrossRef]

13. C.-C. Chen, C.-Y. Chen, W.-K. Wang, F.-H. Fluang, C.-K. Lin, W.-Y. Chiu, and Y.-J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express **13**, 38–43 (2005). [CrossRef] [PubMed]

*a*/λ=0.242 to 0.289 with λ being the free-space wavelength. The propagating mode exists entirely in the PGB region. The waveguide core is formed by eliminating a row of rods, and the rods adjacent to both sides of the waveguide are alternately replaced by rods with different diameters to construct PC waveguide grating. The period is

*N*and the diameters of modulated rods are

*d*(

_{n}*n*=1~

*N*) as shown in Fig. 2. Usual PC waveguide gratings without apodization [7] are formed for

*d*

_{1}=

*d*

_{2}=⋯=

*d*. Here, Gaussian apodization is used for the longitudinal refractive index distribution of PC waveguide gratings. In this case,

_{n}*d*is given by

_{n}*d*is a maximum variation of the rod diameter,

_{c}*ω*

_{0}is a spot size, and Λ=2

*a*is a one period of grating. Figure 2 shows the longitudinal

*d*variation of PC waveguide gratings with Gaussian apodization for

_{n}*d*=0.05

_{c}*a*,

*ω*

_{0}=0.25

*N*Λ, and

*N*=52.

*d*=0.05a,

_{c}*ω*

_{0}=0.25

*N*Λ, and

*N*=52, while the dash-dot line in Fig. 3 shows the reflection spectrum of the PC waveguide grating without apodization for

*d*=0.45

_{n}*a*and

*N*=52. We can see that the sidelobes in the reflection spectra are effectively suppressed by using apodization. However, reflection at the Bragg frequency is reduced. Solid line in Fig. 3 shows the reflection spectrum of PC waveguide gratings with Gaussian apodization for

*d*=0.05

_{c}*a*,

*ω*

_{0}=0.25

*N*Λ, and

*N*=104. It is possible to obtain larger reflection for larger grating periods, however, small sidelobes appear in the lower frequency region.

*d*=0.05

_{c}*a*and

*N*=104) for different values of

*ω*

_{0}. Although larger reflections can be obtained by increasing the value of

*ω*

_{0}, sidelobes in lower frequency region also become larger. This is due to the fact that by increasing the value of

*ω*

_{0}, the longitudinal refractive index distribution comes closer to that of PC waveguide gratings without apodization.

*d*=0.45

_{n}*a*. We can see the anti-crossing of the propagating mode around the Bragg frequency.

## 3. Photonic crystal waveguide gratings with Gauss-cosine apodization

8. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**, 1277–1294 (1997). [CrossRef]

*d*(

_{i}*i*=1~2

*N*) as shown in Fig. 6 which is given by

*d*variation of PC waveguide gratings with Gauss-cosine apodization for

_{i}*d*=0.025

_{c}*a*,

*ω*

_{0}=0.5

*N*Λ, and

*N*=26. In this case, rods with larger and smaller diameters are periodically arranged along with the envelope of the Gaussian function.

*d*=0.025

_{c}*a*, ω0=0.5

*N*Λ, and

*N*=104), with Gaussian apodization (

*d*=0.05

_{c}*a*,

*ω*

_{0}=0.25

*N*Λ, and

*N*=104), and without apodization (

*d*-1=0.5

_{2j}*a*,

*d*=0.45

_{2j}*a*,

*j*=1, 2,⋯,

*N*, and

*N*=52), respectively. The sidelobe appearing in the low frequency region in case of PC waveguide gratings with Gaussian apodization is well suppressed by utilizing Gauss-cosine apodization and the reflection spectrum is symmetric with respect to Bragg frequency.

*d*=0.025

_{c}*a*and

*N*=104) for different values of

*ω*

_{0}. Reflection at the Bragg frequency and sidelobes become larger for larger values of

*ω*

_{0}as in the case of Gaussian apodization because of abrupt change of

*d*. The tolerance of the spot size for the transmission coefficient at the Bragg frequency is shown in Fig. 10.

_{i}## 4. Bragg frequency and bandwidth of photonic crystal waveguide gratings

*d*(

_{c}*ω*

_{0}=0.5

*N*Λ, and

*N*=104). Although larger bandwidth can be obtained for larger perturbation, Bragg frequencies are almost constant. This is quite unusual and can be explained as follows. Figure 14(a) shows one period of PC waveguide grating with Gauss-cosine apodization. This is composed of two PC waveguides, that is, a PC waveguide having rods with diameters

*d*(

_{i-1}*d*-1>

_{i}*d*) adjacent to the core (we call it “waveguide A” hereafter) and a PC waveguide having rods with diameters

*d*(

_{i}*d*<

_{i}*d*) adjacent to the core (we call it “waveguide B” hereafter). Solid, dashed, and dash-dot lines in Fig. 14(b) show dispersion curves of input PC waveguide, waveguide A with

*d*-1=0.525

_{i}*a*, and waveguide B with

*d*=0.475

_{i}*a*, respectively. Suppose

*β*,

*β*, and

_{A}*β*are the propagation constants of input PC waveguide, waveguides A and B, and Δ

_{B}*β*=

_{A}*β*-

_{A}*β*and Δ

*β*=β-

_{B}*β*, Bragg condition is given by

_{B}*d-d*|≈|

_{i-1}*d-*, and therefore, Δ

_{di}|*β*≈Δ

_{A}*β*. In this case, Eq. (4) is reduced to

_{B}*βa/2π*=0.25. This means that Bragg frequency is constant regardless of the magnitude of perturbation and Bragg reflection occurs at the frequency satisfying above condition. From Fig. 14(b), the frequency at which

*βa/2π*=0.25 is

*a/λ*=0.27 and this is consistent with the results of Fig. 13.

*ω*=

_{B}*a/λ*and bandwidth Δ

_{B}*ω*of PC waveguide gratings without apodization (

*N*=52, solid line), with Gaussian apodization (

*ω*

_{0}=0.25

*N*Λ, and

*N*=104, dashed line), and with Gauss-cosine apodization (

*ω*

_{0}=0.5

*N*Λ, and

*N*=104, dash-dot line) as a function of dc. Here,

*λ*is the wavelength at which the reflection is maximum and Δ

_{B}*ω*is defined as |

*ω*+-

*ω*-| where

*ω*± are frequencies at which transmission is 0.5. While Bragg frequency and bandwidth are simultaneously changed with

*d*for PC waveguide gratings without apodization and with Gaussian apodization, Bragg frequency is almost constant for Gauss-cosine apodization. Usually, to obtain arbitrary Bragg frequency and bandwidth, both geometrical parameters and refractive indices of materials have to be modulated. By using Gauss-cosine apodization, PC waveguide gratings having arbitrary Bragg frequency and bandwidth can be simply designed by modulating the geometrical parameters only.

_{c}## 5. Conclusions

14. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slowlight enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002). [CrossRef]

14. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slowlight enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B **19**, 2052–2059 (2002). [CrossRef]

## References and links

1. | J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist of light,” Nature |

2. | M. Koshiba, “Wavelength division demultiplexing and multiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. |

3. | E. A. Camargo, H. M. H. Chong, and R. M. E. L. Rue, “2D photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure,” Opt. Express |

4. | M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E |

5. | Y. Sugimoto, H. Nakamura, U. Tanaka, N. Ikeda, and K. Asakawa, “High-precision optical interference in Mach-Zehnder-type photonic crystal waveguide,” Opt. Express |

6. | T. Fujisawa and M. Koshiba, “Finite-element mode-solver for nonlinear periodic optical waveguides and its application to photonic crystal circuits,” J. Lightwave Technol. |

7. | T. Fujisawa and M. Koshiba, “An analysis of photonic crystal waveguide gratings using coupled-mode theory and finite-element method,” Appl. Opt. to be published. |

8. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. |

9. | M. Koshiba, Y. Tsuji, and M. Hikari, “Time-domain beam propagation method and its application to photonic crystal circuits,” J. Lightwave Technol. |

10. | T. Fujisawa and M. Koshiba, “Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits,” J. Lightwave Technol. |

11. | M. Tokushima, H. Yamada, and Y. Arakawa, “1.5-µm-wavelength light guiding in waveguides in squarelattice-of-rod photonic crystal slab,” Appl. Phys. Lett. |

12. | S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, “Guiding 1.5 µm light in photonic crystals based on dielectric rods,” Appl. Phys. Lett. |

13. | C.-C. Chen, C.-Y. Chen, W.-K. Wang, F.-H. Fluang, C.-K. Lin, W.-Y. Chiu, and Y.-J. Chan, “Photonic crystal directional couplers formed by InAlGaAs nano-rods,” Opt. Express |

14. | M. Soljačić, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slowlight enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(230.1480) Optical devices : Bragg reflectors

**ToC Category:**

Optical Devices

**History**

Original Manuscript: March 21, 2006

Revised Manuscript: May 2, 2006

Manuscript Accepted: May 5, 2006

Published: May 15, 2006

**Citation**

Nobuhiro Yokoi, Takeshi Fujisawa, Kunimasa Saitoh, and Masanori Koshiba, "Apodized photonic crystal waveguide gratings," Opt. Express **14**, 4459-4468 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4459

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### References

- J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, "Photonic crystals: putting a new twist of light," Nature 386, 143-149 (1997). [CrossRef]
- M. Koshiba, "Wavelength division demultiplexing and multiplexing with photonic crystal waveguide couplers," J. Lightwave Technol. 19, 1970-1975 (2001). [CrossRef]
- E. A. Camargo, H. M. H. Chong, and R. M. E. L. Rue, "2D photonic crystal thermo-optic switch based on AlGaAs/GaAs epitaxial structure," Opt. Express 12, 588-592 (2004). [CrossRef] [PubMed]
- M. Soljaèiæ, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601(R), (2002).
- Y. Sugimoto, H. Nakamura, U. Tanaka, N. Ikeda, and K. Asakawa, "High-precision optical interference in Mach-Zehnder-type photonic crystal waveguide," Opt. Express 13, 96-105 (2005). [CrossRef] [PubMed]
- T. Fujisawa and M. Koshiba, "Finite-element mode-solver for nonlinear periodic optical waveguides and its application to photonic crystal circuits," J. Lightwave Technol. 23, 382-387 (2005). [CrossRef]
- T. Fujisawa and M. Koshiba, "An analysis of photonic crystal waveguide gratings using coupled-mode theory and finite-element method," Appl. Opt.to be published.
- T. Erdogan, "Fiber grating spectra," J. Lightwave Technol. 15, 1277-1294 (1997). [CrossRef]
- M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightwave Technol. 18, 102-110 (2000). [CrossRef]
- T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits," J. Lightwave Technol. 22, 684-691 (2004). [CrossRef]
- M. Tokushima, H. Yamada, and Y. Arakawa, "1.5-μm-wavelength light guiding in waveguides in square-lattice-of-rod photonic crystal slab," Appl. Phys. Lett. 84, 4298-4300 (2004). [CrossRef]
- S. Assefa, P. T. Rakich, P. Bienstman, S. G. Johnson, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, E. P. Ippen, and H. I. Smith, "Guiding 1.5 μm light in photonic crystals based on dielectric rods," Appl. Phys. Lett. 85, 6110-6112 (2004). [CrossRef]
- C.-C. Chen, C.-Y. Chen, W.-K. Wang, F.-H. Fluang, C.-K. Lin, W.-Y. Chiu, and Y.-J. Chan, "Photonic crystal directional couplers formed by InAlGaAs nano-rods," Opt. Express 13, 38-43 (2005). [CrossRef] [PubMed]
- M. Soljaèiæ, S. G. Johnson, S. Fan, M. Ibanscu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002). [CrossRef]

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