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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8069–8075
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Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk

Victor Liu and Shanhui Fan  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8069-8075 (2013)
http://dx.doi.org/10.1364/OE.21.008069


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Abstract

We demonstrate an extremely compact bend for a photonic crystal waveguide supporting three spatial modes. The bend exhibits nearly 100% transmission over a relative bandwidth of 1% with less than 1% crosstalk. We show that our design is robust with respect to fabrication errors. Our design method is applied to create a structure consisting of dielectric rods, as well as a structure consisting of air holes in a dielectric background.

© 2013 OSA

1. Introduction

Photonic crystal structures have been widely used for creating very compact integrated optical components. In integrated 2D photonic crystal systems, the design of bends for single-mode waveguides has been extensively studied both theoretically [5

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

9

9. Z. Hu and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Commun. 284, 2812–2816 (2011) [CrossRef] .

] and experimentally [10

10. 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] .

15

15. P. Strasser, G. Stark, F. Robin, D. Erni, K. Rauscher, R. Wüest, and H. Jäckel, “Optimization of a 60 ° waveguide bend in inp-based 2d planar photonic crystals,” J. Opt. Soc. Am. B 25, 67–73 (2008) [CrossRef] .

]. In this paper we present the design of a bend for a photonic crystal waveguide supporting three modes with nearly total transmission and almost no crosstalk. The physical size of the bend is less than 3 wavelengths long on a side.

2. The optimized bend structure

Figure 1 shows the optimized device structure, consisting of an input waveguide on the left and an output waveguide on the bottom, highlighted in blue. The structure is embedded within a background photonic crystal of silicon rods (refractive index n = 3.4) of radius 0.2a in air (n = 1) on a square lattice of lattice constant a. The waveguides are formed by removing three rows of rods and support three modes. The dispersion relations of the three modes are shown in Fig. 2. Also shown in Fig. 2 are the out-of-plane electric field profiles for the three modes at the operating frequency.

Fig. 1 Schematic of the optimized bend structure. The blue highlighted regions show the waveguides, and the orange highlighted region shows the bend region where search and optimization were performed. Solid gray circles indicate dielectric rods (refractive index 3.4) surrounded by air. Unlabeled rods have radius 0.2a where a is the lattice constant. The table on the right lists the radii of the labeled rods. The dashed green diagonal line shows the plane of mirror symmetry.
Fig. 2 Band structure of the waveguide modes within the bandgap of the background photonic crystal. The out-of-plane electric field profiles of the three modes at the operating frequency are shown on the right. The profiles are taken along the edge of the waveguide unit cell and are on the same horizontal scale as the cell shown at the bottom right.

The bend region is highlighted in orange in Fig. 1 and consists of rods centered on lattice sites but with varying radii listed in the table on the right of Fig. 1. Although the presented design is an idealized 2D structure, there have been recent efforts to implement such designs using effective index models in 3D for planar fabrication [16

16. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002) [CrossRef] .

].

The structure in Fig. 1 has a mirror symmetry plane along the diagonal dashed line. In general, in our design we considered only structures with such a mirror symmetry. We enforced such a constraint since single-mode photonic crystal waveguide bends with 100% transmission coefficients all have such a symmetry [5

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

]. In [5

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

], it was argued that a single-mode photonic crystal waveguide bend can be described by a resonator model, with a resonance located at the bend region coupled to the two waveguides. Such a resonator model supports 100% transmission only when the resonance couples to the two waveguides with the same strength. The most straightforward way to implement such a symmetric resonator model is to have a structure with mirror symmetry. For multi-mode waveguide bends, there is no strict theoretical basis for enforcing such a symmetry; we simply carried over the intuition that was developed from the case of a single-mode waveguide bend. The success in obtaining a functional structure in such a restricted structural space provides some support of this intuition.

We further restricted the design space by considering only structures with rods centered on the 7×7 possible lattice sites within the bend region. It was assumed that all rods have the same refractive index as the rods in the background photonic crystal, but the radius of the rods in the bend region was allowed to vary. Combined with the mirror symmetry constraint, only 28 of the 49 rods in the bend region were free parameters for design.

3. The design process

All bend structures were evaluated using the following error metric:
J=m=13(1pm)2
(1)
where pm is the power transmission coefficient for the m-th waveguide mode. For the device shown, we selected an operating frequency of ω = 0.4 × 2πc/a.

In the second phase, the initial candidates were fine tuned by adjusting the rod radii in the bend region using a simple gradient descent method to minimize the error metric J in Eq. (1). The gradient of J with respect to rod radius is straightforward to calculate by applying the adjoint variable method throughout the computational process [20

20. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29, 2288–2290 (2004) [CrossRef] [PubMed] .

,21

21. Y. Jiao, S. Fan, and D. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron. 42, 266–279 (2006) [CrossRef] .

]. For the structure shown, the second phase reduced the initial value of J = 0.0155 to a final value of J = 0.000798.

4. Numerical methods

5. Discussion

The transmission spectra of the bend structure shown in Fig. 1 for each of the three modes is shown in Fig. 3. At the design frequency of 0.4 × 2πc/a, the transmission coefficient for all three modes exceeds 98%. The transmission peaks have a sizable relative bandwidth of 1% at a 95% transmission threshold. For operation at λ = 1.55 μm, the bend region has dimensions of 4.34 μm on a side, and a bandwidth of 15 nm. Because the bend structure is lossless, the high transmission coefficients necessarily imply that very little power is reflected or converted into other waveguide modes. For the device shown, the cross-talk parameters (transmission coefficients into other modes) are all less than 1%.

Fig. 3 Transmission spectra for each of the three waveguide modes shown in Fig. 2 for the structure shown in Fig. 1. The inset shows details of the transmission peaks.

Fig. 4 Representative field patterns (out-of-plane electric field) for each waveguide mode of the optimized structure shown in Fig. 1 at the operating frequency of ω = 0.4 × 2πc/a.

The six digits of accuracy presented in Fig. 1 is not actually required for a functioning device. Practical fabrication tolerances limit the precision of the rod radii to two or three significant digits. In Fig. 5 we show the transmission spectra for devices with hole radii rounded to three (dashed lines) and two (solid lines) significant digits. In both cases, the transmission coefficient at the target frequency remains above 98%, suggesting that the design is robust to perturbations in radius.

Fig. 5 Transmission spectra as in Fig. 3, except for the structure in Fig. 1 with rod radii rounded to three (dashed lines) and two (solid lines) significant digits.

The design method is not limited to structures of dielectric rods in air. We have also applied the same method to design an analogous structure in a system with air holes surrounded by dielectric. We consider the TE polarization with an out-of-plane magnetic field. In Fig. 6(a), an optimized structure is shown. The structure is embedded in a background photonic crystal of air holes (n = 1, r = 0.4a) in a dielectric (n = 3.481) background. The holes in the bend region after optimization have the radii listed in the table. The corresponding transmission coefficient for each of the three modes is shown in Fig. 6(b). All three modes achieve greater than 98% transmission at the design frequency of 0.268 × 2πc/a.

Fig. 6 (a) Schematic of an alternate optimized bend structure. Solid white circles indicate air holes (refractive index 1) surrounded by dielectric (refractive index (3.481). Unlabeled rods have radius 0.4a where a is the lattice constant. The table on the right lists the radii of the labeled rods. (b) Transmission spectra of the bend for each of the three modes.

Finally, we comment briefly on the size of the device with respect to fundamental limits. The resonator model of [5

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

] may be used to estimate a lower limit on the size of the bend region required. The ability to pass three spatial modes with total transmission at a single frequency implies the presence of at least three distinct resonances in the bend region. Assuming a typical estimate of one square wavelength per resonant mode, the bend region would have to be at least 3λ2 in area. The bend region we have presented is approximately 8λ2 in area, making it near the theoretical optimum. Note that simply having three resonances is not sufficient to guarantee total transmission of three modes since the field patterns of the resonances must also be compatible with the waveguide spatial modes, likely increasing the minimum device area required.

6. Conclusion

We have described the design of a multi-mode photonic crystal waveguide bend structure exhibiting over 98% transmission, and less than 1% reflection and crosstalk. The structure was shown to be robust to fabrication error with respect to deviations in the rod radii. In addition to a structure of dielectric rods in air, we also demonstrated a second device implemented with air holes in a dielectric background, which may be more experimentally relevant. Finally, we note that the size of these structures is very near the fundamental size limit derived from heuristic arguments.

Acknowledgments

This research was supported in part by the National Science Foundation through XSEDE resources provided by the XSEDE Science Gateways program. This work was also supported in part by the United States Air Force Office of Scientific Research (USAFOSR) grant FA9550-09-1-0704, and the National Science Foundation (NSF) grant DMS-0968809.

References and links

1.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012) [CrossRef] .

2.

X. Yu and S. Fan, “Anomalous reflections at photonic crystal surfaces,” Phys. Rev. E 70, 055601 (2004) [CrossRef] .

3.

P. B. Catrysse and S. Fan, “Routing of deep-subwavelength optical beams and images without reflection and diffraction using infinitely anisotropic metamaterials,” Adv. Mater. 25, 194–198 (2012) [CrossRef] [PubMed] .

4.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3, 1217 (2012) [CrossRef] .

5.

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

6.

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004) [CrossRef] .

7.

A. Miroshnichenko and Y. Kivshar, “Sharp bends in photonic crystal waveguides as nonlinear fano resonators,” Opt. Express 13, 3969–3976 (2005) [CrossRef] [PubMed] .

8.

F. Monifi, M. Djavid, A. Ghaffari, and M. S. Abrishamian, “Design of efficient photonic crystal bend and power splitter using super defects,” J. Opt. Soc. Am. B 25, 1805–1810 (2008) [CrossRef] .

9.

Z. Hu and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Commun. 284, 2812–2816 (2011) [CrossRef] .

10.

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] .

11.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000) [CrossRef] .

12.

E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ = 1.55 μm wavelengths,” Opt. Lett. 26, 286–288 (2001) [CrossRef] .

13.

S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, R. Houdré, and U. Oesterle, “Improved 60 degree bend transmission of submicron-width waveguides defined in two-dimensional photonic crystals,” J. Light-wave Technol. 20, 1198 (2002) [CrossRef] .

14.

L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004) [CrossRef] [PubMed] .

15.

P. Strasser, G. Stark, F. Robin, D. Erni, K. Rauscher, R. Wüest, and H. Jäckel, “Optimization of a 60 ° waveguide bend in inp-based 2d planar photonic crystals,” J. Opt. Soc. Am. B 25, 67–73 (2008) [CrossRef] .

16.

M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002) [CrossRef] .

17.

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett. 30, 141–143 (2005) [CrossRef] [PubMed] .

18.

V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett. 36, 591–593 (2011) [CrossRef] [PubMed] .

19.

V. Liu, D. A. B. Miller, and S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE 101, 484–493 (2013) [CrossRef] .

20.

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29, 2288–2290 (2004) [CrossRef] [PubMed] .

21.

Y. Jiao, S. Fan, and D. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron. 42, 266–279 (2006) [CrossRef] .

22.

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by dirichlet-to-neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006) [CrossRef] .

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(130.5296) Integrated optics : Photonic crystal waveguides

ToC Category:
Photonic Crystals

History
Original Manuscript: February 13, 2013
Revised Manuscript: March 18, 2013
Manuscript Accepted: March 22, 2013
Published: March 27, 2013

Citation
Victor Liu and Shanhui Fan, "Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk," Opt. Express 21, 8069-8075 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8069


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References

  1. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6, 488–496 (2012). [CrossRef]
  2. X. Yu and S. Fan, “Anomalous reflections at photonic crystal surfaces,” Phys. Rev. E70, 055601 (2004). [CrossRef]
  3. P. B. Catrysse and S. Fan, “Routing of deep-subwavelength optical beams and images without reflection and diffraction using infinitely anisotropic metamaterials,” Adv. Mater.25, 194–198 (2012). [CrossRef] [PubMed]
  4. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun.3, 1217 (2012). [CrossRef]
  5. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett.77, 3787–3790 (1996). [CrossRef] [PubMed]
  6. J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett.84, 2022–2024 (2004). [CrossRef]
  7. A. Miroshnichenko and Y. Kivshar, “Sharp bends in photonic crystal waveguides as nonlinear fano resonators,” Opt. Express13, 3969–3976 (2005). [CrossRef] [PubMed]
  8. F. Monifi, M. Djavid, A. Ghaffari, and M. S. Abrishamian, “Design of efficient photonic crystal bend and power splitter using super defects,” J. Opt. Soc. Am. B25, 1805–1810 (2008). [CrossRef]
  9. Z. Hu and Y. Y. Lu, “Improved bends for two-dimensional photonic crystal waveguides,” Opt. Commun.284, 2812–2816 (2011). [CrossRef]
  10. 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,” Science282, 274–276 (1998). [CrossRef] [PubMed]
  11. M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett.76, 952–954 (2000). [CrossRef]
  12. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ = 1.55 μm wavelengths,” Opt. Lett.26, 286–288 (2001). [CrossRef]
  13. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, R. Houdré, and U. Oesterle, “Improved 60 degree bend transmission of submicron-width waveguides defined in two-dimensional photonic crystals,” J. Light-wave Technol.20, 1198 (2002). [CrossRef]
  14. L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express12, 5916–5921 (2004). [CrossRef] [PubMed]
  15. P. Strasser, G. Stark, F. Robin, D. Erni, K. Rauscher, R. Wüest, and H. Jäckel, “Optimization of a 60 ° waveguide bend in inp-based 2d planar photonic crystals,” J. Opt. Soc. Am. B25, 67–73 (2008). [CrossRef]
  16. M. Qiu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett.81, 1163–1165 (2002). [CrossRef]
  17. Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator,” Opt. Lett.30, 141–143 (2005). [CrossRef] [PubMed]
  18. V. Liu, Y. Jiao, D. A. B. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett.36, 591–593 (2011). [CrossRef] [PubMed]
  19. V. Liu, D. A. B. Miller, and S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proc. IEEE101, 484–493 (2013). [CrossRef]
  20. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett.29, 2288–2290 (2004). [CrossRef] [PubMed]
  21. Y. Jiao, S. Fan, and D. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron.42, 266–279 (2006). [CrossRef]
  22. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by dirichlet-to-neumann maps,” J. Lightwave Technol.24, 3448–3453 (2006). [CrossRef]

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