Abstract

A new finite-difference frequency-domain (FDFD) method based eigenvalue algorithm is developed for analyzing anisotropic optical waveguides with an arbitrary permittivity tensor. Yee’s mesh is employed in the FD formulation along with perfectly matched layer (PML) absorption boundary conditions. A standard eigenvalue matrix equation is successfully derived through considering simultaneously four transverse field components. The new algorithm is first applied to the mode solution of a proton-exchanged LiNbO3 optical waveguide and the results agree with those obtained using a full-vectorial finite-element beam propagation method. Then, the algorithm is used to study modes on a liquid-crystal optical waveguide with arbitrary molecular director orientation. This arbitrary orientation may cause the loss of transverse-axis symmetries of the waveguide with symmetric background structure. Asymmetric mode-field profiles under such situations are clearly demonstrated in the numerical examples.

©2009 Optical Society of America

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References

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  1. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
    [Crossref]
  2. P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
    [Crossref]
  3. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002).
    [Crossref]
  4. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
    [Crossref]
  5. P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
    [Crossref]
  6. N. Thomas, P. Sewell, and T. M. benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2007).
    [Crossref]
  7. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  8. T. Ando, H. Nakayama, S. Numata, J. Yamauchi, and H. Nakano, “Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface,” J. Lightwave Technol. 20, 1627–1634 (2002).
    [Crossref]
  9. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853.
    [PubMed]
  10. C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express,  12, 6165V–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
    [Crossref]
  11. L. Thylen and D. Yevick, “Beam propagation method in anisotropic media,” Appl. Opt. 21, 2751–2754 (1982).
    [Crossref] [PubMed]
  12. C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
    [Crossref]
  13. C. L. D. S. Sobrinho and A. J. Giarola, “Analysis of biaxially anisotropic dielectric waveguides with Gaussian-Gaussian index of refraction profiles by the finite-difference method,” IEE Proc.-H 140, 224–230 (1993).
  14. P. Lüsse, K. Ramm, and H.-G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32, 38–39 (1996).
    [Crossref]
  15. A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26, 1423–1431 (2008).
    [Crossref]
  16. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
    [Crossref]
  17. F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media”, IEEE Microwave Guid. Wave Lett. 8, 223V–225 (1998).
    [Crossref]
  18. R. Mittra and U. Pekel, “A new look an the perfectly matched layer PML concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. 5, 84–86 (1995).
    [Crossref]
  19. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

2008 (2)

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26, 1423–1431 (2008).
[Crossref]

2007 (1)

2004 (1)

C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express,  12, 6165V–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[Crossref]

2002 (4)

2001 (1)

1998 (1)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media”, IEEE Microwave Guid. Wave Lett. 8, 223V–225 (1998).
[Crossref]

1996 (1)

P. Lüsse, K. Ramm, and H.-G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32, 38–39 (1996).
[Crossref]

1995 (1)

R. Mittra and U. Pekel, “A new look an the perfectly matched layer PML concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. 5, 84–86 (1995).
[Crossref]

1994 (3)

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[Crossref]

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
[Crossref]

P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[Crossref]

1993 (1)

C. L. D. S. Sobrinho and A. J. Giarola, “Analysis of biaxially anisotropic dielectric waveguides with Gaussian-Gaussian index of refraction profiles by the finite-difference method,” IEE Proc.-H 140, 224–230 (1993).

1982 (1)

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Ando, T.

benson, T. M.

Brown, T. G.

Chang, H. C.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express,  12, 6165V–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[Crossref]

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
[Crossref]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[Crossref]

Chew, W. C.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media”, IEEE Microwave Guid. Wave Lett. 8, 223V–225 (1998).
[Crossref]

Chiang, P. J.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

Chrostowski, J.

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[Crossref]

Fallahkhair, A. B.

Giarola, A. J.

C. L. D. S. Sobrinho and A. J. Giarola, “Analysis of biaxially anisotropic dielectric waveguides with Gaussian-Gaussian index of refraction profiles by the finite-difference method,” IEE Proc.-H 140, 224–230 (1993).

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

Hadley, G. R.

Huang, W. P.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
[Crossref]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[Crossref]

Koshiba, M.

Li, K. S.

Lüsse, P.

P. Lüsse, K. Ramm, and H.-G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32, 38–39 (1996).
[Crossref]

P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[Crossref]

Mittra, R.

R. Mittra and U. Pekel, “A new look an the perfectly matched layer PML concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. 5, 84–86 (1995).
[Crossref]

Murphy, T. E.

Nakano, H.

Nakayama, H.

Numata, S.

Pekel, U.

R. Mittra and U. Pekel, “A new look an the perfectly matched layer PML concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. 5, 84–86 (1995).
[Crossref]

Ramm, K.

P. Lüsse, K. Ramm, and H.-G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32, 38–39 (1996).
[Crossref]

Saitoh, K.

Schule, J.

P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[Crossref]

Sewell, P.

Sobrinho, C. L. D. S.

C. L. D. S. Sobrinho and A. J. Giarola, “Analysis of biaxially anisotropic dielectric waveguides with Gaussian-Gaussian index of refraction profiles by the finite-difference method,” IEE Proc.-H 140, 224–230 (1993).

Stern, M. S.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
[Crossref]

Stuwe, P.

P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[Crossref]

Teixeira, F. L.

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media”, IEEE Microwave Guid. Wave Lett. 8, 223V–225 (1998).
[Crossref]

Teng, C. H.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

Thomas, N.

Thylen, L.

Unger, H.-G.

P. Lüsse, K. Ramm, and H.-G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32, 38–39 (1996).
[Crossref]

P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[Crossref]

Wu, C. L.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

Xu, C. L.

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
[Crossref]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[Crossref]

Yamauchi, J.

Yang, C. S.

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

Yevick, D.

Yu, C. P.

C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express,  12, 6165V–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[Crossref]

Zhu, Z.

Appl. Opt. (1)

Electron. Lett. (1)

P. Lüsse, K. Ramm, and H.-G. Unger, “Vectorial eigenmode calculation for anisotropic planar optical waveguides,” Electron. Lett. 32, 38–39 (1996).
[Crossref]

IEE Proc.-H (1)

C. L. D. S. Sobrinho and A. J. Giarola, “Analysis of biaxially anisotropic dielectric waveguides with Gaussian-Gaussian index of refraction profiles by the finite-difference method,” IEE Proc.-H 140, 224–230 (1993).

IEEE J. Quantum Electron. (1)

P. J. Chiang, C. L. Wu, C. H. Teng, C. S. Yang, and H. C. Chang, “Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations,” IEEE J. Quantum Electron. 44, 56–66, (2008).
[Crossref]

IEEE Microwave Guid. Wave Lett. (2)

F. L. Teixeira and W. C. Chew, “General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media”, IEEE Microwave Guid. Wave Lett. 8, 223V–225 (1998).
[Crossref]

R. Mittra and U. Pekel, “A new look an the perfectly matched layer PML concept for the reflectionless absorption of electromagnetic waves,” IEEE Microwave Guid. Wave Lett. 5, 84–86 (1995).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

J. Lightwave Technol. (8)

T. Ando, H. Nakayama, S. Numata, J. Yamauchi, and H. Nakano, “Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface,” J. Lightwave Technol. 20, 1627–1634 (2002).
[Crossref]

N. Thomas, P. Sewell, and T. M. benson, “A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides,” J. Lightwave Technol. 25, 2563–2570 (2007).
[Crossref]

P. Lüsse, P. Stuwe, J. Schule, and H.-G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–494 (1994).
[Crossref]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces,” J. Lightwave Technol. 20, 1210–1218 (2002).
[Crossref]

G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
[Crossref]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26, 1423–1431 (2008).
[Crossref]

K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. 19, 405–413 (2001).
[Crossref]

C. L. Xu, W. P. Huang, J. Chrostowski, and S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[Crossref]

Opt. Express (2)

Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853.
[PubMed]

C. P. Yu and H. C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express,  12, 6165V–6177 (2004). http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-25-6165.
[Crossref]

Proc. Inst. Electr. Eng. (1)

C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Electr. Eng. 141, 281–286 (1994).
[Crossref]

Other (1)

P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley and Sons, Inc., New York, 1999).

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Figures (17)

Fig. 1.
Fig. 1. Cross-section of a general optical waveguide and the computational domain with PML regions.

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Fig. 2.
Fig. 2. Two-dimensional Yee’s mesh for the FDFD method.

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Fig. 3.
Fig. 3. Computational domain and structure parameters for the LN waveguide with PE-LN core.

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Fig. 4.
Fig. 4. Calculated effective index as a function of the crystal angle for the LN waveguide of Fig. 3.

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Fig. 5.
Fig. 5. Numerical convergence versus the grid size for the calculated effective index at θC = 50° in Fig. 4.

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Fig. 6.
Fig. 6. Mode-field profiles of (a) ∣Ex ∣, (b) ∣Ey ∣, and (c) ∣Ez ∣ for the TM-like mode of the LN waveguide with θc = 30°.

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Fig. 7.
Fig. 7. Computational domain and structure parameters for the LC-core waveguide.

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Fig. 8.
Fig. 8. Schematic definition of rotation angles for the LC molecular or director.

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Fig. 9.
Fig. 9. ∣Ex ∣ and ∣Ey ∣ mode-field profiles for modes 1 and 2 of the LC waveguide with θc =0°.

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Fig. 10.
Fig. 10. Calculated effective indices versus ϕc for the first four modes of the LC waveguide with θc = 30°.

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Fig. 11.
Fig. 11. ∣Ex ∣ and ∣Ey ∣ mode-field profiles for the first four modes of the LC waveguide with θc = 30° at ϕc = 0°, 30°, 60°, and 90°.

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Fig. 12.
Fig. 12. Calculated effective indices versus ϕc for the first seven modes of the LC waveguide with θC = 60°.

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Fig. 13.
Fig. 13. ∣Ex ∣ and ∣Ey ∣ mode-field profiles for the first four modes of the LC waveguide with θc = 60° at ϕc = 0°, 30°, 60°, and 90°.

Download Full Size | PPT Slide | PDF

Fig. 14.
Fig. 14. ∣Ex ∣ and ∣Ey ∣ mode-field profiles for modes 5–7 of the LC waveguide with θc = 60° at ϕc = 0°, 30°, 60°, and 90°.

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Fig. 15.
Fig. 15. Calculated effective indices versus ϕc for the first seven modes of the LC waveguide with θc = 90°.

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Fig. 16.
Fig. 16. ∣Ex ∣ and ∣Ey ∣ mode-field profiles for the first four modes of the LC waveguide with θc = 90° at ϕc = 0°, 30°, 60°, and 90°.

Download Full Size | PPT Slide | PDF

Fig. 17.
Fig. 17. ∣Ex ∣ and ∣Ey ∣ mode-field profiles for modes 5–7 of the LC waveguide with θc = 90° at ϕc = 0°, 30°, 60°, and 90°.

Download Full Size | PPT Slide | PDF

Equations (61)

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

∇ × E → = − jω [ μ ] H →
∇ × H → = jω D →
− jω μ x H x = ∂ E z ∂ y + jβ E y
− jω μ y H y = − jβ E x − ∂ E z ∂ y
− jω μ z H z = ∂ E y ∂ x + ∂ E x ∂ y
jω D x = ∂ H z ∂ y jβ H y
jω D y = − jβ H x − ∂ H z ∂ x
jω D z = ∂ H y ∂ x − ∂ H x ∂ y .
− jω μ x H x , ( i , j + 1 2 ) = E z , ( i , j + 1 ) − E z , ( i , j ) Δ y + jβ E y , ( i , j + 1 2 )
− jω μ y H y , ( i + 1 2 , j ) = − jβ E x , ( i + 1 2 , j ) − E z , ( i + 1 , j ) − E z , ( i , j ) Δ x
− jω μ z H z , ( i + 1 2 , j + 1 2 ) = E y , ( i + 1 , j + 1 2 ) − E y , ( i , j − 1 2 ) Δ x + E x , ( i + 1 2 , j + 1 ) − E x , ( i , + 1 2 , j ) Δ y
jω D x , ( i + 1 2 , j ) = H z , ( i + 1 2 , j + 1 2 ) − H z , ( i + 1 2 , j − 1 2 ) Δ y + jβ H y , ( i + 1 2 , j )
jω D y , ( i , j + 1 2 ) = − jβ H x , ( i , j + 1 2 ) − H z , ( i + 1 2 , j + 1 2 ) − H z , ( i − 1 2 , j + 1 2 ) Δ x
jω D z , ( i , j ) = H y , ( i + 1 2 , j ) − H y , ( i − 1 2 , j ) Δ x − H x , ( i , j + 1 2 ) − H x , ( i , j + 1 2 ) Δ y
− jω μ 0 [ μ x H x μ y H y μ z H z ] = [ 0 jβ I U y − jβ I 0 − U x − U y U x 0 ] [ E x E y E z ]
jω [ D x D y D z ] = [ 0 jβ I V y − jβ I 0 − V x − V y V x 0 ] [ H x H y H z ] .
U x = 1 Δ x [ − 1 1 − 1 1 ⋱ ⋱ ⋱ ⋱ − 1 1 − 1 ]
U y = 1 Δy [ − 1 1 − 1 ⋱ ⋱ 1 ⋱ − 1 − 1 ]
V x = 1 Δ x [ 1 − 1 1 − 1 ⋱ ⋱ ⋱ − 1 1 − 1 1 ]
V y = 1 Δ y [ 1 1 ⋱ − 1 ⋱ ⋱ 1 − 1 1 ]
D x = ε xx E x + ε xy E y + ε xz E z
D y = ε yx E x + ε yy E y + ε yz E z
D z = ε zx E x + ε zy E y + ε zz E z
− jω μ x H x = jβ E y + U y E z
− jω μ y H y = − jβ E x + U x E z
− jω μ z H z = − U y E x + U x E y
jω ( ε xx E x + ε xy E y + ε xz E z ) = jβ H y + V y H z
jω ( ε yx E x + ε yy E y + ε yz E z ) = − j β H x − V x H z
jω ( ε zx E x + ε zy E y + ε zz E z ) = − V y H x + V x H y .
β [ E x E y H x H y ] = [ A 11 A 12 A 13 A 14 A 21 A 22 A 23 A 24 A 31 A 32 A 33 A 34 A 41 A 42 A 43 A 44 ] [ E x E y H x H y ]
A 11 = − j ( ε zx ε zz ) U x
A 12 = − j ( ε zy ε zz ) U x
A 13 = − ( 1 ωε zz ) U x V y
A 14 = − ( 1 ωε zz ) U x V y + ω μ y I
A 21 = − j ( ε zx ε zz ) U y
A 22 = − j ( ε zy ε zz ) U y
A 23 = − ( 1 ωε zz ) U y V y − ω μ x I
A 24 = − ( 1 ωε zz ) U y V x
A 31 = − ω ε yx I + ( 1 ω μ z ) V x U y + ε yz ω ε zx ε zz I
A 32 = − ω ε yy I − ( 1 ω μ z ) V x U x + ε yz ω ε zy ε zz I
A 33 = − j ( ε yz ε zz ) V y
A 34 = j ( ε yz ε zz ) V x
A 41 = ω ε xx I + ( 1 ω μ z ) V y U y − ε xz ω ε zx ε zz I
A 42 = ω ε xy I − ( 1 ω μ z ) V y U x − ε xz ω ε zy ε zz I
A 43 = j ( ε xz ε zz ) V y
A 44 = − j ( ε xz ε zz ) V x .
[ ε PML ] = [ s y s z s x ε xx s z ε xy s y ε xz s z ε yx s z s x s t ε yy s x ε yz s y ε zx s x ε zy s x s y s z ε zz ]
[ μ PML ] = μ 0 [ s y s z s x 0 0 0 s z s x s y 0 0 0 s x s y s z ]
s j = 1 − j α j
α j = α j , max ( ρ d ) 2
ε xx / ε 0 = n o 2
ε yy / ε 0 = n e 2 cos 2 θ c + n o 2 sin 2 θ c
ε zz / ε 0 = n o 2 cos 2 θ c + n e 2 sin 2 θ c
ε yz / ε 0 = ε zy = ( n o 2 − n e 2 ) cos θ c sin θ c
ε xy / ε 0 = ε yx = ε zx = ε xz = 0
ε xx / ε 0 = n o 2 + ( n e 2 − n o 2 ) sin 2 θ c cos 2 ϕ c
ε xy / ε 0 = ε yx = ( n e 2 − n o 2 ) sin 2 θ c cos ϕ c cos ϕ c
ε xz / ε 0 = ε zx = ( n e 2 − n o 2 ) sin θ c cos θ c cos ϕ c
ε yy / ε 0 = n o 2 + ( n e 2 − n o 2 ) sin 2 θ c cos 2 ϕ c
ε yz / ε 0 = ε zy = ( n e 2 − n o 2 ) sin θ c cos θ c sin ϕ c
ε zz / ε 0 = n o 2 + ( n e 2 − n o 2 ) cos 2 θ c

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