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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  1. Eli Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [Crossref] [PubMed]
  2. Sajeev John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [Crossref] [PubMed]
  3. K. BuschS. LölkesR. B. WehrspohnH. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (Wiley-VCH, Verlag GmbH & Co. KGaA, 2004).
    [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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090
    [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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927
    [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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996
    [Crossref] [PubMed]
  7. M Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. 19, 1970–1975 (2001).
    [Crossref]
  8. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048
    [Crossref] [PubMed]
  9. K. Srinivasan, P. E. Barclay, and O. Painter, “Fabrication-tolerant high quality factor photonic crystal microcavities,” Opt. Express 12, 1458–1463 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1458
    [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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1119
    [Crossref] [PubMed]
  11. S. Boscolo, M. Midrio, and C.G. Someda, “Coupling and decoupling of electromagnetic waves in parallel 2D photonic crystal waveguides,” IEEE J. Quantum Electron. 38, 47–53 (2002).
    [Crossref]
  12. A. Taflove and 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. S. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
    [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, and 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.

2004 (4)

2003 (1)

2002 (2)

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

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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048
[Crossref] [PubMed]

2001 (2)

1996 (1)

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]

1995 (1)

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]

1987 (2)

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]

Asakawa, K.

Barclay, P. E.

Borel, P. I.

Boscolo, S.

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

Chen, J. C.

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]

Cheng, S. -.

Chien, F. S. -.

Fan, Shanhui

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]

Frandsen, L. H.

Hagness, S.C.

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

Harpøth, A.

Hsieh, W. -.

Hsu, Y. -.

Ikeda, N.

Inoue, K.

Jensen, J. S.

Joannopoulos, J. D.

S. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[Crossref] [PubMed]

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]

John, Sajeev

Sajeev John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Johnson, S. G.

Kim, Hyun-Jun

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

Koshiba, M

Kristensen, M.

Kurland, I.

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]

Lee, El-Hang

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

Lee, Seung-Gol

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

McNab, Sharee J.

Mekis, Attila

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]

Midrio, M.

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

Moll, Nikolaj

Nakamura, Y.

O, Beom-Hoan

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

Painter, O.

Park, Dae-Seo

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

Park, In-Su

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

Park, Se-Geun

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

Pennings, E.C.M.

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]

Prather, D. W.

Sharkawy, A. S.

Shi, P.

Shi, S.

Sigmund, O.

Soldano, L.B.

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]

Someda, C.G.

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

Soref, R. A.

Srinivasan, K.

Sugimoto, Y.

Taflove, A.

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

Tanaka, Y.

Villeneuve, Pierre R.

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]

Vlasov, Yurii A.

Yablonovitch, Eli

Eli Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (1)

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

J. Lightwave Technol. (2)

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]

M Koshiba, “Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers,” J. Lightwave Technol. 19, 1970–1975 (2001).
[Crossref]

Opt. Express (7)

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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1048
[Crossref] [PubMed]

K. Srinivasan, P. E. Barclay, and O. Painter, “Fabrication-tolerant high quality factor photonic crystal microcavities,” Opt. Express 12, 1458–1463 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1458
[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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1119
[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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090
[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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927
[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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1996
[Crossref] [PubMed]

S. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

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 (3)

K. BuschS. LölkesR. B. WehrspohnH. Föll, Photonic Crystals: Advances in Design, Fabrication, and Characterization (Wiley-VCH, Verlag GmbH & Co. KGaA, 2004).
[Crossref]

Hyun-Jun Kim, In-Su Park, Dae-Seo Park, Seung-Gol Lee, Beom-Hoan O, Se-Geun Park, and 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.

A. Taflove and 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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