Abstract

We show that the self-imaging principle still holds true in multi-mode photonic crystal (PhC) line-defect waveguides just as it does in conventional multi-mode waveguides. To observe the images reproduced by this self-imaging phenomenon, the finite-difference time-domain computation is performed on a multi-mode PhC line-defect waveguide that supports five guided modes. From the computed result, the reproduced images are identified and their positions along the propagation axis are theoretically described by self-imaging conditions which are derived from guided mode propagation analysis. We report a good agreement between the computational simulation and the theoretical description. As a possible application of our work, a photonic crystal 1-to-2 wavelength de-multiplexer is designed and its performance is numerically verified. This approach can be extended to novel designs of PhC devices.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. Sajeev John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486�??2489 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length," Opt. Express 12, 1090-1096 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090<a/>
    [CrossRef] [PubMed]
  5. Sharee J. McNab, Nikolaj Moll, and Yurii A. Vlasov, �??Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,�?? Opt. Express 11, 2927-2939 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927</a>
    [CrossRef] [PubMed]
  6. P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996</a>
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    [CrossRef]
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  9. K. Srinivasan, P. E. Barclay, and O. Painter, "Fabrication-tolerant high quality factor photonic crystal microcavities," Opt. Express 12, 1458-1463 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1458">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1458</a>
    [CrossRef] [PubMed]
  10. F. S. -. Chien, Y. -. Hsu, W. -. Hsieh, and S. -. Cheng, "Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides," Opt. Express 12, 1119-1125 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1119">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1119</a>
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    [CrossRef]
  12. A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, second ed., (Artech House, Boston, 2000).
  13. L.B. Soldano and E.C.M. Pennings, �??Optical multi-mode interference devices based on self-imaging: principles and applications,�?? J. Lightwave Technol. 13, 615-627 (1995
    [CrossRef]
  14. 8S. 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]
  15. 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]
  16. Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, El-Hang Lee, �??Multi-Mode Interference-Based Photonic Crystal Waveguide Devices: Demultiplexer and Power Splitter,�?? presented at the Frontiers in Optics 2004/Laser Science XX, Rochester, New York, USA, 10-14 Oct. 2004.

Frontiers in Optics 2004

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, El-Hang Lee, �??Multi-Mode Interference-Based Photonic Crystal Waveguide Devices: Demultiplexer and Power Splitter,�?? presented at the Frontiers in Optics 2004/Laser Science XX, Rochester, New York, USA, 10-14 Oct. 2004.

IEEE J. Quantum Electron.

Boscolo, S., Midrio, M., Someda, and C.G., �??Coupling and decoupling of electromagnetic waves in parallel 2D photonic crystal waveguides,�?? IEEE J. Quantum Electron. 38, 47-53 (2002).
[CrossRef]

J. Lightwave Technol.

L.B. Soldano and E.C.M. Pennings, �??Optical multi-mode interference devices based on self-imaging: principles and applications,�?? J. Lightwave Technol. 13, 615-627 (1995
[CrossRef]

Koshiba M, �??Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,�?? J. Lightwave Technol. 19, 1970-1975 (2001).
[CrossRef]

Opt. Express

A. S. Sharkawy, S. Shi, D. W. Prather, and R. A. Soref, "Electro-optical switching using coupled photonic crystal waveguides," Opt. Express 10, 1048-1059 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048</a>
[CrossRef] [PubMed]

K. Srinivasan, P. E. Barclay, and O. Painter, "Fabrication-tolerant high quality factor photonic crystal microcavities," Opt. Express 12, 1458-1463 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1458">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1458</a>
[CrossRef] [PubMed]

F. S. -. Chien, Y. -. Hsu, W. -. Hsieh, and S. -. Cheng, "Dual wavelength demultiplexing by coupling and decoupling of photonic crystal waveguides," Opt. Express 12, 1119-1125 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1119">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1119</a>
[CrossRef] [PubMed]

Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length," Opt. Express 12, 1090-1096 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090<a/>
[CrossRef] [PubMed]

Sharee J. McNab, Nikolaj Moll, and Yurii A. Vlasov, �??Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,�?? Opt. Express 11, 2927-2939 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927</a>
[CrossRef] [PubMed]

P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996</a>
[CrossRef] [PubMed]

8S. 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]

Phys. Rev. Lett.

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]

Eli Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Sajeev John, �??Strong localization of photons in certain disordered dielectric superlattices,�?? Phys. Rev. Lett. 58, 2486�??2489 (1987).
[CrossRef] [PubMed]

Other

K. Busch (Editor), S. Lölkes (Editor), R. B. Wehrspohn (Editor), H. Föll (Editor), Photonic Crystals: Advances in Design, Fabrication, and Characterization ( Wiley-VCH, Verlag GmbH & Co. KGaA, 2004).
[CrossRef]

A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, second ed., (Artech House, Boston, 2000).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic illustration of a multi-mode waveguide. Input image is reproduced at x=Lm and at x=Ld .

Fig. 2.
Fig. 2.

Computational setup for observation of self-imaging phenomena. The black dots represent dielectric rods (n=3.4) in air and their radius is 0.18a, where n is the refractive index of the rods and a is the lattice constant of the PhC.

Fig. 3.
Fig. 3.

(a) The dispersion curve for the access PCW and the computational super-cell (inset). The access PCW ensures single-mode operation from 0.312(a/λ) to the top of band gap. (b) The dispersion curve for the multi-mode PCW and the computational super-cell (inset). The multi-mode PCW supports 4 guided modes at 0.37(a/λ) and 5 guided modes at 0.43(a/λ).

Fig. 4.
Fig. 4.

Modal patterns of electric field z-component for each mode at the operation frequency 0.37(a/λ) presented in Fig. 3. (a) Input image for the access PCW, (b) the 0th mode, (c) the 1st mode, (d) the 2nd mode, and (e) the 3rd mode at 0.37(a/λ).

Fig. 5.
Fig. 5.

(a) Steady-state electric field distribution at 0.37(a/λ). (b) Time-averaged Poynting vector distribution at 0.37(a/λ).

Fig. 6.
Fig. 6.

The designed 1-to-2 PhC de-multiplexer.

Fig. 7.
Fig. 7.

Steady-state electric field distributions in the designed PhC wavelength de-multiplexer (a) at 0.37(a/λ) and (b) at 0.43(a/λ).

Tables (3)

Tables Icon

Table 1. Parameters Used to Calculate Lm at 0.37(a/λ)

Tables Icon

Table 2. Parameters Used to Calculate Ld at 0.37(a/λ)

Tables Icon

Table 3. Output Power Normalized to Total Input Power

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

Ψ ( x , y ) = n = 0 p 1 c n φ n ( y ) e j β n x ,
Ψ ( L m , y ) = n = 0 p 1 c n φ n ( y ) e j β n L m
= c 0 φ 0 ( y ) e j β 0 L m + c 2 φ 2 ( y ) e j β 2 L m + c 4 φ 4 ( y ) e j β 4 L m + ,
+ c 1 φ 1 ( y ) e j β 1 L m + c 3 φ 3 ( y ) e j β 3 L m + c 5 φ 5 ( y ) e j β 5 L m + · ·
= Ψ ( 0 , y ) e j Δ m
Ψ ( 0 , y ) e j Δ m = n = 0 p 1 c n φ n ( y ) e j Δ m
= c 0 φ 0 ( y ) e j Δ m + c 2 φ 2 ( y ) e j Δ m + c 4 φ 4 ( y ) e j Δ m + .
c 1 φ 1 ( y ) e j Δ m c 3 φ 3 ( y ) e j Δ m c 5 φ 5 ( y ) e j Δ m
φ n ( y ) = { φ n ( y ) for n even φ n ( y ) for n odd
β n L m = { 2 k n π + Δ m for n even ( 2 k n 1 ) π + Δ m for n odd with k n = 1 , 2 , 3 .
β n L d = 2 k n π + Δ d with k n = 1 , 2 , 3 ,

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