Abstract

This work introduces a new simulation approach to the evaluation of the time-domain electromagnetic (EM) field useful in the modeling of tapered waveguide for the Photonic Crystal Slab (PCS) coupling. Only solutions of two scalar Helmholtz-equations are used in the evaluation of electric and magnetic Hertzian-potentials that yields the EM field and the frequency response of the tapered waveguide. By considering simultaneously an analytical and a numerical approximation it is possible to reduce the computational burden. In order to compare the computational time we analyze the 2D structure by also using the Finite Difference Time Domain (FDTD) method and by the 3D Finite Element Method (FEM). The method is applied by starting from design criteria of the tapered structures in order to set the correct geometrical and physical parameters, and considers the field-perturbation effect in proximity of the dielectric discontinuities by generators modeling.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K.S. Yee, "Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media," IEEE Trans. Antennas Propagat. AP-14, 302-307 (1966).
  2. W. J. R. Hoefer, "The transmission-line matrix method- Theory and applications," IEEE Trans. Microwave Theory Tech. MTT-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 Tech. 46, 2463-2475 (1998).
    [CrossRef]
  4. A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express. 14, 2027-2036 (2006).
    [CrossRef] [PubMed]
  5. A. Massaro, L. Pierantoni, and T. Rozzi, " Time-domain modeling and filtering behavior of guided-wave optics by Hertzian potentials," Proc. SPIE 6183 (2006).
    [CrossRef]
  6. M. Couture, "On the numerical solution of fields in cavities using the magnetic Hertz vector," IEEE Trans. Microwave Theory and Tech. MTT-35, 288-295 (1987).
    [CrossRef]
  7. 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]
  8. T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures, (IEE Electromagnetic wave series 46, London. 1999), Ch.2.
    [CrossRef]
  9. C.G. Someda, Onde elettromagnetiche, (UTET Ed., Torino 1996), Ch. I.
  10. N. Marcuvitz, and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in wave guides," J. Appl. Phys. 22, 806-820 (1951).
    [CrossRef]
  11. N. C. Frateschi, A. Rubens, 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]
  12. A. Yariv, Quantum Electronics, (John Wiley & Sons, 3rd ed. Canada 1989), Ch. 22.
  13. D. Marcuse, Theory of Dielectric Opt. Waveguides, (Academic Press, New York 1974), Ch. I.
  14. A. Taflove, S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London 2000), Ch. 2,3,4,7.
  15. G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagnetic Compatibility 23, 377-382 (1981).
    [CrossRef]

2006 (1)

A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express. 14, 2027-2036 (2006).
[CrossRef] [PubMed]

1998 (1)

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 Tech. 46, 2463-2475 (1998).
[CrossRef]

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 Tech. MTT-35, 288-295 (1987).
[CrossRef]

1986 (1)

N. C. Frateschi, A. Rubens, 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 (1)

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

1981 (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagnetic Compatibility 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 Propagat. AP-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 wave guides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Couture, M.

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

De Castro, B.

N. C. Frateschi, A. Rubens, 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]

Frateschi, N. C.

N. C. Frateschi, A. Rubens, 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 Tech. 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 Tech. 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 Tech. MTT-33, 882-893 (1985).
[CrossRef]

Marcuvitz, N.

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

Massaro, A.

A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express. 14, 2027-2036 (2006).
[CrossRef] [PubMed]

Mur, G.

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

Rozzi, T.

A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express. 14, 2027-2036 (2006).
[CrossRef] [PubMed]

Rubens, A.

N. C. Frateschi, A. Rubens, 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]

Schwinger, J.

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

Yee, K.S.

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

IEEE J. Quantum Electron. (1)

N. C. Frateschi, A. Rubens, 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 Trans. Antennas Propag. (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]

IEEE Trans. Antennas Propagat. (1)

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

IEEE Trans. Electromagnetic Compatibility (1)

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

IEEE Trans. Microwave Theory and Tech. (1)

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

IEEE Trans. Microwave Theory Tech. (2)

W. J. R. Hoefer, "The transmission-line matrix method- Theory and applications," IEEE Trans. Microwave Theory Tech. MTT-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 Tech. 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 wave guides," J. Appl. Phys. 22, 806-820 (1951).
[CrossRef]

Opt. Express. (1)

A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express. 14, 2027-2036 (2006).
[CrossRef] [PubMed]

Other (6)

A. Massaro, L. Pierantoni, and T. Rozzi, " Time-domain modeling and filtering behavior of guided-wave optics by Hertzian potentials," Proc. SPIE 6183 (2006).
[CrossRef]

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

C.G. Someda, Onde elettromagnetiche, (UTET Ed., Torino 1996), Ch. I.

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

D. Marcuse, Theory of Dielectric Opt. Waveguides, (Academic Press, New York 1974), Ch. I.

A. Taflove, S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London 2000), Ch. 2,3,4,7.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1.
Fig. 1.

SEM image of the fabricated photonic crystal with the tapered waveguide.

Fig. 2.
Fig. 2.

Cross section of tapered waveguide and analytical approach (α=atan((W1/2)-(W2/2))/L).

Fig. 3.
Fig. 3.

Modal map by varying the thickness of the core dcore and width w2. The working region is the single TEz mode-region (λ0=1.31µm, ncore(GaAs)=3.408).

Fig. 4.
Fig. 4.

Example of graphical approach of the dispersion equations solution (λ0=1.30µm, ncore(GaAs)=3.408, nsub(AlGaAs)=3.042, dcore=0.56µm+20nm). TE dispersion equation (above), TM dispersion equation (below).

Fig. 5.
Fig. 5.

Finite difference time-domain Hertzian potentials algorithm.

Fig. 6.
Fig. 6.

(a). Above :computational domain Ω=(x,y) and dielectric grid-mask of 45 degree tapered profile with transmission line perturbed modeling; below: PCS with guiding region and perturbed modeling of air hole (radius R).

Fig. 6.
Fig. 6.

(b). Left: geometrical construction of the unit cell of dielectric tapered profile (α=atan(x/y)=18.449 deg) ; right: slanted angle of 45 deg.

Fig. 7.
Fig. 7.

Transmission line modeling of dielectric profile.

Fig. 8.
Fig. 8.

Source: time-evolution in air (pulse excitation).

Fig. 9.
Fig. 9.

Time evolution of Ez components after 350 and 400 time-steps in spatial domain (x,y). After 400 time steps the field is coupled in the guide with thickness w2).

Fig. 10.
Fig. 10.

Ez field component after 380 time-steps for different cross-section (reference of y-position).

Fig. 11.
Fig. 11.

Frequency responses of a tapered waveguide with w1=5.94µm, w2=0.22µm,α=14 deg; a and b are the reference section of the S21 transmission coefficient.

Fig. 12.
Fig. 12.

FDTD and Hertzian Potentials S21 transmission coefficient of a tapered waveguide with w1=5.94µm, w2=0.22µm, α=14 deg., α=45 deg.

Fig. 13.
Fig. 13.

Time evolution of Ez field component after 400, 410, 420, and 430 time-steps inside the PC after the coupling with the tapered waveguide (with thickness w2). After about 400 time-steps the wave arrives at the PCS input.

Fig. 14.
Fig. 14.

Coupling efficiency for different alpha angles.

Tables (1)

Tables Icon

Table I. Comparison between CPU time for the 2D Hertzian Potenzials Method (HPM), FDTD 2D, and 3D FEM method.

Equations (27)

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

w 1 ( u ) = ( u 2 tan ( 2 u ) u v 1 2 u 2 ) u + tan ( 2 u ) · v 1 2 u 2
w 1 ( u ) = ( u 2 ε 1 ε 2 tan ( 2 u ) u ε 2 ε 3 v 1 2 u 2 ) u ε 1 ε 3 + ε 2 2 tan ( 2 u ) · v 1 2 u 2
v 1 = k 0 d core ε 2 ε 1
v 3 = k 0 d core ε 2 ε 3
w 2 ( u ) = v 3 2 u 2
k 0 = ω ε 0 μ 0
k 0 2 ε effz = k 0 2 ε 2 k z 2
Π e ¯ = a ψ e ( x , y , z , t )
Π h ¯ = a ψ h ( x , y , z , t )
E ¯ = · Π e ¯ ε μ 2 t 2 Π e ¯ μ t ( × Π h ¯ )
H ¯ = · Π h ¯ ε μ 2 t 2 Π h ¯ + ε t ( × Π e ¯ )
2 ψ e , h ( x , y , z , t ) μ ε 2 ψ e , h ( x , y , z , t ) t 2 = 0
2 Ψ e , h ( x , y , z , t ) μ ε 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 )
Δ ε = ε i + 1 ε i i = cell position in x direction .
Δ ε = ε j + 1 ε j j = cell position in y direction .
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 = μ ε ( Δ t ) 2
a ' = a + μ Δ ε ( Δ t ) 2
b = 1 ( Δ z ) 2
i , j space domain position
( ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ψ n ( i , j ) ψ n ( i + 1 , j ) ψ n ( i + 2 , j ) ABC ABC ψ n ( i + 1 , j + 1 ) ψ n ( m , j + 1 ) ABC ABC ABC ABC ψ n ( i + 1 , j ) ABC ABC ABC ABC ABC ABC ABC ABC ABC ABC ) m x 1
Ψ source e , h = exp ( ( t · d t T 0 ) 2 ) · cos ( ω · t · d t )
S m , n ( ω , y m , y n ) = E ̂ m ( ω , y m ) E ̂ n ( ω , y n ) Z 0 , n ( ω ) Z 0 , n ( ω )
Z 0 ( ω , y i ) = DET ( E z , y i ) DET ( H x , y i ) .
η = W output ( y ; t ) W input ( y ; t ) = w 2 P y ( x , y , t ) d x w 1 P y ( x , y , t ) d x = w 2 ( E x H z + ( E z H x ) ) d x w 1 ( E x H z + ( E z H x ) ) d x

Metrics