Abstract

This work analyzes a new simulation approach to the evaluation of the time-domain electromagnetic (EM) field by reducing the number of equations to solve. Scalar Helmholtz-equations are utilized in order to determine the electric and magnetic Hertzian-potentials that yield the EM field. The method is applied to the example of optical waveguide arrays by considering the field-perturbation effect due to high dielectric contrast and dielectric discontinuities. The rigorous Hertzian potentials formulation is extended to bi-dimensional structures.

© 2006 Optical Society of America

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  1. K. S. Yee, "Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
  2. W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 33, 882-893 (1985).
    [CrossRef]
  3. M. Fujii, and W J. R. Hoefer "A three-dimensional Haar-wavelet-based multiresolution analysis similar to the FDTD method-derivation and application," IEEE Trans. Microwave Theory Technol. 46, 2463-2475 (1998).
    [CrossRef]
  4. Y. Chiou and H. Chang, "Analysis of optical waveguide discontinuities using Padè approximants," IEEE Photon. Technol. Lett. 9, 964-966 (1997).
    [CrossRef]
  5. H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
    [CrossRef]
  6. Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
    [CrossRef]
  7. N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
    [CrossRef]
  8. M. Couture, "On the numerical solution of fields in cavities using the magnetic Hertz vector," IEEE Trans. Microwave Theory and Technol. 35, 288-295 (1987).
    [CrossRef]
  9. K. I. Nikoskinen, "Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability," IEEE Trans. Antennas Propag. 39, 698-703 (1991).
    [CrossRef]
  10. T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures, (IEE Electromagnetic wave series 46, London. 1999), Chap. 2.
    [CrossRef]
  11. C. G. Someda, Onde elettromagnetiche, (UTET Ed., Torino 1996), Chap.1.
  12. R.-C. Tyan, A. A. Salvekar, H.-Pu Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, "Design, fabrication, and characterization of form-birefringent multilayer polarizing beam splitter," J. Opt. Soc. Am. A 14, 1627-1636 (1997).
    [CrossRef]
  13. K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998).
    [CrossRef]
  14. T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985).
    [CrossRef]
  15. N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003).
    [CrossRef]
  16. N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004).
    [CrossRef]
  17. N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).
  18. P. Zorabedian, "Axial-mode instability in tunable external-cavity semiconductor lasers," IEEE J. Quantum Electron. 40, 778-783 (2004).
  19. A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
    [CrossRef]
  20. N. Marcuvitz and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides," J. Appl. Phys. 22, 806-820 (1951).
    [CrossRef]
  21. N. C. Frateschi, A. Rubens and B. De Castro, "Perturbation theory for the wave equation and the ‘effective refractive index’ approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
    [CrossRef]
  22. A. Yariv, Quantum Electron., 3rd ed. (John Wiley & Sons, Canada, 1989), Chap. 22.
  23. A. Taflove, and S. C. Hagness, Computational Electrodynamic: the Finite-difference time-domain method, 2nd. ed. (Arthec House Publishers, London 2000), Chaps. 2, 4, and 7.
  24. G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
    [CrossRef]
  25. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 55-62.

2004 (2)

N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004).
[CrossRef]

P. Zorabedian, "Axial-mode instability in tunable external-cavity semiconductor lasers," IEEE J. Quantum Electron. 40, 778-783 (2004).

2003 (2)

N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
[CrossRef]

2002 (1)

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

1999 (1)

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

1998 (3)

M. Fujii, and W J. R. Hoefer "A three-dimensional Haar-wavelet-based multiresolution analysis similar to the FDTD method-derivation and application," IEEE Trans. Microwave Theory Technol. 46, 2463-2475 (1998).
[CrossRef]

K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998).
[CrossRef]

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

1997 (2)

1994 (1)

N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

1991 (1)

K. I. Nikoskinen, "Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability," IEEE Trans. Antennas Propag. 39, 698-703 (1991).
[CrossRef]

1987 (1)

M. Couture, "On the numerical solution of fields in cavities using the magnetic Hertz vector," IEEE Trans. Microwave Theory and Technol. 35, 288-295 (1987).
[CrossRef]

1986 (1)

N. C. Frateschi, A. Rubens and B. De Castro, "Perturbation theory for the wave equation and the ‘effective refractive index’ approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

1985 (2)

W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 33, 882-893 (1985).
[CrossRef]

T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

1981 (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).

1951 (1)

N. Marcuvitz and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Bossert, D. J.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Chang, H.

Y. Chiou and H. Chang, "Analysis of optical waveguide discontinuities using Padè approximants," IEEE Photon. Technol. Lett. 9, 964-966 (1997).
[CrossRef]

Chiou, Y.

Y. Chiou and H. Chang, "Analysis of optical waveguide discontinuities using Padè approximants," IEEE Photon. Technol. Lett. 9, 964-966 (1997).
[CrossRef]

Couture, M.

M. Couture, "On the numerical solution of fields in cavities using the magnetic Hertz vector," IEEE Trans. Microwave Theory and Technol. 35, 288-295 (1987).
[CrossRef]

De Castro, B.

N. C. Frateschi, A. Rubens and B. De Castro, "Perturbation theory for the wave equation and the ‘effective refractive index’ approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

Egan, A.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Fang, D.-G.

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
[CrossRef]

Feng, N.-N.

N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004).
[CrossRef]

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
[CrossRef]

N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

Frateschi, N. C.

N. C. Frateschi, A. Rubens and B. De Castro, "Perturbation theory for the wave equation and the ‘effective refractive index’ approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

Fujii, M.

M. Fujii, and W J. R. Hoefer "A three-dimensional Haar-wavelet-based multiresolution analysis similar to the FDTD method-derivation and application," IEEE Trans. Microwave Theory Technol. 46, 2463-2475 (1998).
[CrossRef]

Hoefer, W J. R.

M. Fujii, and W J. R. Hoefer "A three-dimensional Haar-wavelet-based multiresolution analysis similar to the FDTD method-derivation and application," IEEE Trans. Microwave Theory Technol. 46, 2463-2475 (1998).
[CrossRef]

Hoefer, W. J. R.

W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 33, 882-893 (1985).
[CrossRef]

Huang, W.-P.

N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004).
[CrossRef]

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
[CrossRef]

N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

Indik, R. A.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Lu, Y. Y.

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Marcuvitz, N.

N. Marcuvitz and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

McInerney, J. G.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Mikami, O.

T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

Moloney, J. V.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Mukai, T.

T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

Mur, G.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Muro, K.

K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998).
[CrossRef]

Nikoskinen, K. I.

K. I. Nikoskinen, "Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability," IEEE Trans. Antennas Propag. 39, 698-703 (1991).
[CrossRef]

Ning, C. Z.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Osgood, R. M.

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Rao, H.

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Rubens, A.

N. C. Frateschi, A. Rubens and B. De Castro, "Perturbation theory for the wave equation and the ‘effective refractive index’ approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

Saitoh, T.

T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

Salvekar, A. A.

Scarmozzino, R.

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Schwinger, J.

N. Marcuvitz and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Shiraishy, K.

K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998).
[CrossRef]

Tyan, R.-C.

Wei, S. H.

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Wright, M. W.

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

Xu, C.

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).

Zhou, G.-R.

N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

Zorabedian, P.

P. Zorabedian, "Axial-mode instability in tunable external-cavity semiconductor lasers," IEEE J. Quantum Electron. 40, 778-783 (2004).

IEEE J. Lightwave Technol. (4)

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003).
[CrossRef]

K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998).
[CrossRef]

T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985).
[CrossRef]

N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003).
[CrossRef]

IEEE J. of Quantum Electron. (1)

N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

IEEE J. Quantum Electron. (3)

P. Zorabedian, "Axial-mode instability in tunable external-cavity semiconductor lasers," IEEE J. Quantum Electron. 40, 778-783 (2004).

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998).
[CrossRef]

N. C. Frateschi, A. Rubens and B. De Castro, "Perturbation theory for the wave equation and the ‘effective refractive index’ approach," IEEE J. Quantum Electron. QE-22, 12-15 (1986).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004).
[CrossRef]

Y. Chiou and H. Chang, "Analysis of optical waveguide discontinuities using Padè approximants," IEEE Photon. Technol. Lett. 9, 964-966 (1997).
[CrossRef]

H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999).
[CrossRef]

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

K. S. Yee, "Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).

K. I. Nikoskinen, "Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability," IEEE Trans. Antennas Propag. 39, 698-703 (1991).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

IEEE Trans. Microwave Theory and Technol. (1)

M. Couture, "On the numerical solution of fields in cavities using the magnetic Hertz vector," IEEE Trans. Microwave Theory and Technol. 35, 288-295 (1987).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (2)

W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 33, 882-893 (1985).
[CrossRef]

M. Fujii, and W J. R. Hoefer "A three-dimensional Haar-wavelet-based multiresolution analysis similar to the FDTD method-derivation and application," IEEE Trans. Microwave Theory Technol. 46, 2463-2475 (1998).
[CrossRef]

J. Appl. Phys. (1)

N. Marcuvitz and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (5)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 55-62.

A. Yariv, Quantum Electron., 3rd ed. (John Wiley & Sons, Canada, 1989), Chap. 22.

A. Taflove, and S. C. Hagness, Computational Electrodynamic: the Finite-difference time-domain method, 2nd. ed. (Arthec House Publishers, London 2000), Chaps. 2, 4, and 7.

T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures, (IEE Electromagnetic wave series 46, London. 1999), Chap. 2.
[CrossRef]

C. G. Someda, Onde elettromagnetiche, (UTET Ed., Torino 1996), Chap.1.

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Figures (9)

Fig. 1.
Fig. 1.

General optical multilayer structure.

Fig. 2.
Fig. 2.

General ASR optical multilayer structure.

Fig. 3.
Fig. 3.

Equivalent circuit of the perturbed solution in the multilayered structure.

Fig. 4.
Fig. 4.

Finite difference algorithm of the simulated structure.

Fig. 5.
Fig. 5.

Time evolution of Ey field component.

Fig. 6.
Fig. 6.

Comparison between theoretical and simulated dielectric stack

Fig. 7.
Fig. 7.

Comparison between theoretical and simulated dielectric stack

Fig. 8.
Fig. 8.

Time evolution of normalized Ez field component and contour after 170 time-steps. The simulated structure is drawn on the contour xy-plane.

Fig. 9.
Fig. 9.

Measured and simulated reflectivity of ASR of Fig. 2 for different thickness d2.

Equations (40)

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

Π e ̅ = a ψ e ( x , y , z , t )
Π h ̅ = a ψ h ( x , y , z , t )
E ̅ = Π ̅ e εμ 2 t 2 Π e ̅ μ t ( × Π h ̅ )
H ̅ = Π h ̅ εμ 2 t 2 Π h ̅ + ε t ( × Π e ̅ )
E x = A e + μ t z ψ h
E y = B e + εμ t 2 ψ e
E z = C e μ t x ψ h
H x = A h ε t z ψ e
H y = B h εμ 2 t ψ h
H z = C h + ε t x ψ e
A e , h = x 2 ψ e , h + yx ψ e , h + zx ψ e , h
B e , h = xy ψ e , h + 2 y ψ e , h + zy ψ e , h
C e , h = xz ψ e , h + yz ψ e , h + 2 z ψ e , h
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 = 0
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 μ 2 P pert ( x , y , z , t ) t 2 = 0
P ( x , y , z , t ) = Δ ε ( x , y , z , t ) ψ e , h ( x , y , z , t )
Δε = ε i + 1 ε i i = 1 , m layer .
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 = 0 .
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 μ 2 P pert ( x , y , z , t ) t 2 = 0
ψ n ( i , j ) x = ψ n ( i + 1 2 , j ) ψ n ( i 1 2 , j ) Δ x + O ( Δ 2 x )
ψ n ( i , j ) y = ψ n ( i , j + 1 2 ) ψ n ( i , j 1 2 ) Δy + O ( Δ 2 y )
ψ n ( i , j ) t = ψ n + 1 2 ( i , j ) ψ n 1 2 ( i , j ) Δt + O ( Δ 2 t )
2 ψ n ( i , j ) x 2 ψ n ( i + 1 , j ) 2 ψ n ( i , j ) + ψ n ( i 1 , j ) Δ 2 x
2 ψ n ( i , j ) y 2 ψ n ( i , j + 1 ) 2 ψ n ( i , j ) + ψ n ( i , j 1 ) Δ 2 y
2 ψ n ( i , j ) t 2 ψ n + 1 ( i , j ) 2 ψ n ( i , j ) + ψ n 1 ( i , j ) Δ 2 t
position ( ABC ψ 0 ( j = 1 ) ψ 0 ( j = 0 ) ψ 0 ( j = + 1 ) ABC ABC ψ 1 ( j = 0 ) ABC ABC ABC ABC ABC ABC ABC ABC ψ n ( j = 0 ) ABC ) n x number of nodes time s teps
2 ψ e , h x t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h y 2 = 0 x = 0
2 ψ e , h x t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h y 2 = 0 x = h
2 ψ e , h y t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h x 2 = 0 y = 0
2 ψ e , h y t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h x 2 = 0 y = h
ψ n + 1 ( j ) ( με ( Δt ) 2 ) = ψ n ( j + 1 ) ( 1 ( Δz ) 2 ) + ψ n ( j ) ( 2 με ( Δt ) 2 2 ( Δz ) 2 ) +
+ ψ n 1 ( j ) ( με ( Δt ) 2 ) + ψ n ( j 1 ) ( 1 ( Δz ) 2 )
ψ n + 1 ( j ) ( με ( Δt ) 2 + μΔε ( Δt ) 2 ) = ψ n ( j + 1 ) ( 1 ( Δz ) 2 ) + ψ n ( j ) ( 2 με ( Δt ) 2 2 ( Δz ) 2 + 2 μΔε ( Δt ) 2 ) +
+ ψ n 1 ( j ) ( με ( Δt ) 2 μΔε ( Δt ) 2 ) + ψ n ( j 1 ) ( 1 ( Δz ) 2 )
ψ n + 1 ( j ) = ψ n ( j + 1 ) ( b a ) + ψ n ( j ) ( 2 a 2 b a ) + ψ n 1 ( j ) ( 1 ) + ψ n ( j 1 ) ( b a )
ψ n + 1 ( j ) = ψ n ( j + 1 ) ( b a ' ) + ψ n ( j ) ( 2 a ' 2 b a ' ) + ψ n 1 ( j ) ( 1 ) + ψ n ( j 1 ) ( b a ' )
a = με ( Δt ) 2
a ' = a + μΔε ( Δt ) 2
b = 1 ( Δz ) 2
r ( s ) = ( m 11 + m 12 p l + 1 ) p 0 ( m 21 + m 22 p l + 1 ) ( m 11 + m 12 p l + 1 ) p 0 + ( m 21 + m 22 p l + 1 )

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