Abstract

We revisited the fundamental problem of bending loss calculation in optical waveguide theory using transformation optics (TOs). Due to the fact that TOs is based on the form invariance property of Maxwell equations, this new approach provides more accurate calculation of waveguide bending loss compared to the conventional refractive index conformal mapping method, especially for small bending radii typical for plasmonic waveguides or photonic waveguides with high-index contrast. We believe our results provide a simple yet reliable way of bending loss calculation for highly confined optical (including plasmonic) waveguides.

© 2013 Optical Society of America

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[CrossRef]

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[CrossRef]

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2007

T. Holmgaard and S. I. Bozhevolnyi, Phys. Rev. B 75, 1 (2007).
[CrossRef]

2006

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

2004

L. Prkna, M. Hubálek, and J. Ctyroký, IEEE Photon. Technol. Lett. 16, 2057 (2004).
[CrossRef]

1998

1975

M. Heiblum and J. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

Bozhevolnyi, S. I.

T. Holmgaard and S. I. Bozhevolnyi, Phys. Rev. B 75, 1 (2007).
[CrossRef]

Chen, H.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

Ctyroký, J.

L. Prkna, M. Hubálek, and J. Ctyroký, IEEE Photon. Technol. Lett. 16, 2057 (2004).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Gabrielli, L. H.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, Nat. Commun. 3, 1217 (2012).
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Gao, L.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

Harris, J.

M. Heiblum and J. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

Heiblum, M.

M. Heiblum and J. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

Hirono, T.

Holmgaard, T.

T. Holmgaard and S. I. Bozhevolnyi, Phys. Rev. B 75, 1 (2007).
[CrossRef]

Hou, B.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
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Huang, W.

Hubálek, M.

L. Prkna, M. Hubálek, and J. Ctyroký, IEEE Photon. Technol. Lett. 16, 2057 (2004).
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Johnson, S. G.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, Nat. Commun. 3, 1217 (2012).
[CrossRef]

Justice, B. J. J.

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Lai, Y.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

Li, J.

Liang, Z.

Lipson, M.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, Nat. Commun. 3, 1217 (2012).
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Liu, D.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, Nat. Commun. 3, 1217 (2012).
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Lui, W. W.

Luo, J.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

Pendry, J. B. B.

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Prkna, L.

L. Prkna, M. Hubálek, and J. Ctyroký, IEEE Photon. Technol. Lett. 16, 2057 (2004).
[CrossRef]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

Smith, D. R. R.

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Starr, A. F. F.

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

Sun, X.

Xiao, J.

Xu, P.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

Yokoyama, K.

Appl. Phys. Lett.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

IEEE J. Quantum Electron.

M. Heiblum and J. Harris, IEEE J. Quantum Electron. QE-11, 75 (1975).
[CrossRef]

IEEE Photon. Technol. Lett.

L. Prkna, M. Hubálek, and J. Ctyroký, IEEE Photon. Technol. Lett. 16, 2057 (2004).
[CrossRef]

J. Lightwave Technol.

Nat. Commun.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, Nat. Commun. 3, 1217 (2012).
[CrossRef]

Opt. Express

Phys. Rev. B

T. Holmgaard and S. I. Bozhevolnyi, Phys. Rev. B 75, 1 (2007).
[CrossRef]

Science

J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. J. Justice, S. A. Cummer, J. B. B. Pendry, A. F. F. Starr, and D. R. R. Smith, Science 314, 977 (2006).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic (top view) of an optical waveguide forming a 90 deg bend with radius R from the origin to the center of the waveguide, (b) top view of the transformed straight waveguide, (c) cross section of the original waveguide shown in (a) at the plane z=0, and (d) cross section of the transformed straight waveguide.

Fig. 2.
Fig. 2.

Calculated mode profiles of the TE-polarized mode for the SOI wire waveguide when the bending radius is 1 μm. (a) in the uvw coordinates, (b) in the xyz coordinates calculated with the field data shown in (a), and (c) from 3D simulations.

Fig. 3.
Fig. 3.

(a) Calculated light transmission through a 90 deg DLSPPW bend as a function of the bend radius. Inset is a schematic of the DLSPPW. (b) Amplitude of electric field at the upper gold surface in the 3D FEM simulation.

Tables (1)

Tables Icon

Table 1. Comparison of the Mode Effective Index for an SOI Wire Waveguide for Different Bending Radii Using Four Different Methods

Equations (5)

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

T:u+jw=Rlnx+jzR,v=y
εi=εQuQvQw/Qi2,μi=μQuQvQw/Qi2,
Qi2=(x/i)2+(y/i)2+(z/i)2(i=u,v,w),
εuvw=ε·diag(1,e2u/R,1)μuvw=μ·diag(1,e2u/R,1).
T=exp(n2k0Luvw),

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