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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 25 — Dec. 10, 2007
  • pp: 16484–16499
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Design and modeling of tapered waveguide for photonic crystal slab coupling by using time-domain Hertzian potentials formulation

A. Massaro, M. Grande, R. Cingolani, A. Passaseo, and M. De Vittorio  »View Author Affiliations


Optics Express, Vol. 15, Issue 25, pp. 16484-16499 (2007)
http://dx.doi.org/10.1364/OE.15.016484


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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

1. Introduction

Fig. 1. SEM image of the fabricated photonic crystal with the tapered waveguide.
Fig. 2. Cross section of tapered waveguide and analytical approach (α=atan((W1/2)-(W2/2))/L).
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. 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).

2. Analysis and design of tapered waveguide.

w1(u)=(u2tan(2u)uv12u2)u+tan(2u)·v12u2
(1)

for the TE modes, and

w1(u)=(u2ε1ε2tan(2u)uε2ε3v12u2)uε1ε3+ε22tan(2u)·v12u2
(2)

for the TM modes, with

v1=k0dcoreε2ε1
v3=k0dcoreε2ε3
w2(u)=v32u2
k0=ωε0μ0
(3)

and ε1=1(air), ε2=11.56(GaAs), ε3=10.89(AlGaAs), k0 is the wave number in the free space. By using (1),(2),(3) we evaluate kz=u/dcore and then effective index in the z-direction εeffz by the wavenumber conservation equation

k02εeffz=k02ε2kz2
(4)

With the help of map shown in Fig. 3 we derived the core-thickness dcore and the width w2 (see Fig. 2) in order to obtain only one TE propagated-mode at the PCS input. In Fig. 3 we show the working region in which only TEz mode (only a propagation constant kz will be solution of the dispersion equation) will propagates in the waveguide. In this map is also reported a security region (in to gray lines) by considering also a real possible error of ±20nm in the core thickness fabrication. In our case the single TEz propagated modes is obtained by dcore=0.19 µm (neffz=3.067 is the corresponding effective refractive index), w2=0.22 µm.

3. Time-domain Hertzian potentials modeling of tapered waveguide.

In Fig. 5 is shown the time-domain Hertzian Potentials algorithm used in this work[4

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

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]

]

Fig. 5. Finite difference time-domain Hertzian potentials algorithm.

The Hertzian electric and magnetic vectors are [6

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]

]–[8

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]

] in rectangular coordinates represented by

Πe¯=aψe(x,y,z,t)
Πh¯=aψh(x,y,z,t)
(1)

where a is unit vector. From (1) it is possible to evaluate all the components of the EM field as

E¯=·Πe¯εμ2t2Πe¯μt(×Πh¯)
H¯=·Πh¯εμ2t2Πh¯+εt(×Πe¯)
(2)

Ψe,h(x,y,z,t) represents the solution of the homogeneous wave equation for a non-dissipative medium [4

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]

]

2ψe,h(x,y,z,t)με2ψe,h(x,y,z,t)t2=0
(5)

It is known that the scalar wave equation may lead to inconsistencies because, in an inhomogeneous medium, it is, in general, not equivalent to Maxwell’s equations. Electromagnetic scattering problems, including free space, involve the calculation of the fields produced in the presence of geometrical discontinuities by arbitrary currents. Such discontinuities may be replaced by equivalent generators [4

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]

], (see Fig. 6(a) and Fig. 7), giving an accurate solution of the EM field for structures with high dielectric contrast. In fact the scalar wave equation (5) for a non-dissipative medium can be rewritten as [4

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]

]

2Ψe,h(x,y,z,t)με2Ψe,h(x,y,z,t)t2μ2Ppert(x,y,z,t)t2=0
(6)

where

P(x,y,z,t)=Δε(x,y,z,t)Ψe,h(x,y,z,t)
(7)

represents the dielectric polarization and for the bi-dimensional case:

Δε=εi+1εii=cellpositioninxdirection.
(8)
Δε=εj+1εjj=cellpositioninydirection.
(9)

Therefore we solve (6) in proximity of the dielectric interfaces, and (5) in the homogenous region. The difference between the parametric solution of (5), and (6) in the iterative form (by using the finite difference FD discretisation [14

14. A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London2000), Ch. 2,3,4,7.

]),is in the coefficients [4

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]

]; in fact for one propagation direction

ψn+1(j)=ψn(j+1)(ba)+ψn(j)(2a2ba)+ψn1(j)(1)+ψn(j1)(ba)
(10)
ψn+1(j)=ψn(j+1)(ba')+ψn(j)(2a'2ba')+ψn1(j)(1)+ψn(j1)(ba')
(11)

with

a=με(Δt)2
a'=a+μΔε(Δt)2
b=1(Δz)2
(12)
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. (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. Transmission line modeling of dielectric profile.

In Fig. 6(a), Fig. 6(b) and Fig. 7 we show the discretisation and the modeling of the dielectric profiles (dielectric tapered profile and circular air hole profile). The algorithm define the sub-domain boundaries (profiles) by inserting the generators around a reference cell when the perturbed conditions (8) and (9) are satisfied. We observe that the generators VP and IP of Fig.7 represent the variation of the coefficients (12), and the dielectric mask is loaded as a bidimensional vector of the x-y spatial domain, so it is possible to define the perturbed region (inhomogeneous region) by considering all the cells around the reference cell. The flow chart of this procedure is reported in Fig. 4. Therefore we solve (6) in proximity of the dielectric interfaces (inhomogeneous region), and (5) in the uniform space (homogenous region) by considering the different parametric solutions given by (12). The absorbing boundary conditions (ABCs) [5

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]

],[14

14. A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London2000), Ch. 2,3,4,7.

],[15

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]

], around the computational domain Ω=(x,y) (see Fig. 6), permits all outward-propagating numerical waves to exit Ω as if the simulation were performed on a computational domain of infinite extent. Extremely small local reflection coefficient of the order of 10-11 is attained in the simulations. In the 2D case the matrix structure that considers the spatial domain and the ABCs is

i,jspacedomainposition
(ABCABCABCABCABCABCABCABCABCABCψn(i,j)ψn(i+1,j)ψn(i+2,j)ABCABCψn(i+1,j+1)ψn(m,j+1)ABCABCABCABCψn(i+1,j)ABCABCABCABCABCABCABCABCABCABC)mx1
(13)

where m x l is the node number of the spatial domain, and time-step is fixed. By approximation (10),(11),(12) a small node number is requested for the numerical convergence.

4. Time-domain Hertzian potentials results.

As shown in Fig. 8 (source time-evolution) the source is decomposed in two components ψe and ψh that construct the wavefront in the spatial domain (x,y). We use in both directions the source as a carrier modulated by an exponential signal:

Ψsourcee,h=exp((t·dtT0)2)·cos(ω·t·dt)
(13)

where ω is the angular frequency referred to an operative wavelength λ0=1.31µm, T 0 is a constant, and dt is the time step. Figure 9 proves that a tapered waveguide with w1=5.94µm, w2=0.22µm,α=30 deg., after some time-steps, guides the single TE mode at the PCS input (the parameters used in the simulation are: dx=dy=0.11*10-7m., dt=3.67*10-16sec., λ0=1.31µm, T 0=8*10-15sec.). A spatial monitoring of the Ez field is reported in Fig. 10: we fix the time-step and evaluate the spatial Ez profile for different cross-sections. Also in this case is possible to evaluate the coupled-field inside the structure and the radiated field in air (the field confined inside the waveguide is about one order higher than the radiated field). For the simulation of Fig.10 we set w1=5.94µm, w2=0.22µm and α=14 deg.

The evaluation of transmission coefficient completes the analysis of the tapered waveguides. In order to define the frequency response we consider the Discrete Fourier Transform (DFT). The scattering parameters Sm,n can be obtained for an impulsive excitation as follows [14

14. A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London2000), Ch. 2,3,4,7.

]

Sm,n(ω,ym,yn)=Êm(ω,ym)Ên(ω,yn)Z0,n(ω)Z0,n(ω)
(14)

where Ê m is the phasor voltage (DFT of Ez component) at the port m at observation plane ym; Ên is the phasor voltage at the port n at observation plane yn ; and Z0,m and Z0,n are the characteristic impedances of the line connected to these ports defined as

Z0(ω,yi)=DET(Ez,yi)DET(Hx,yi).
(15)

Figure 11 and Fig. 12 show the scattering transmission coefficient Sba=S21 with the port 1 and port 2 defined by the references line a=y1 and b=y2 respectively: the tapered waveguide with w1=5.94µm, w2=0.22µm and α=14 deg, and 45 deg. is characterized by an high transmission coefficient around the working wavelength λ0=1.31µm. In particular in Fig. 11 we show the comparison between the frequency response of: a tapered waveguide with dx=dy=0.11µm with generators, of a tapered waveguide with dx=dy=0.05µm without generators, and of a 3D tapered waveguide with tetrahedron length of 0.01µm (FEM simulation). We observe that the solution with dx=dy=0.05µm is characterized by a numerical error oscillations (absence of generators). It is evident by the graph that the solution with dx=dy=0.11µm (with generators) converges better without oscillations. In order to verify the convergence solution we show in the same graph the 3D FEM solution. Moreover in Fig. 12 is reported the convergence between the S21 Hertzian potentials coefficient and the 2D FDTD one: HPM results converge with greater unit cell grid dimension as compared to the FDTD method (in FDTD with dx=dy=0.01µm, instead in HPM by using dx=dy=0.11µm). In order to observe the coupling effect on a PC structure we simulate the whole structure (PC slab with a guide coupled to a tapered waveguide). This kind of PC (with a guiding region) is important because it is possible to increase the quality Q-factor of the whole structure. After about 400 time-steps the PCS will couple with the waveguide with width w2. By the single mode condition the maximum energy will be transmitted to the PC that will generate PC modes characterized by the group velocity defined by the geometry of the crystal. In this simulation we analyze the coupling at the PC input by observing also the propagation inside a PC structure in order to confirm the presence of the field in the PC guiding region. We consider in our numerical example air holes with radius R=0.11µm in an effective medium with neffz=3.067; the x-period is 3*R, the y-period is 2,5*R, and w2 is the width of the guiding region. We show in Fig.13 the time evolution of the Ez field component inside the PCS by proving the efficacy of the tapered waveguide in terms of field confinement. We also evaluate the coupling efficiency at the PC input by :

η=Woutput(y;t)Winput(y;t)=w2Py(x,y,t)dxw1Py(x,y,t)dx=w2(ExHz+(EzHx))dxw1(ExHz+(EzHx))dx
(16)

where Py represents the Poynting vector along the y propagation direction. Figure 14 shows how the efficiency decreases when the α-angle increases. For large α values most energy will be irradiate in the air region. By comparing Fig. 12 with Fig.14 it is evident that at working wavelength of λ0=1.31 µm the transmission is high α=14 deg. and α=45 deg., but a good coupling efficiency is performed by a tapered waveguide with α=14 deg. The results is validated also by Fig. 10 in which, for α=14 deg., the EM field is well confined at the output of the tapered waveguide.

We validate the model presented in this work (analytical model by using effective index and numerical generators model) by comparing the Central Processing Unit (CPU) time for different numerical methods. The computation time comparison is reported in the Table I for different tapered waveguides with L1=L2=5 µm (see Fig.2), and for a 1-GHz PC, 512/M-RAM. In this comparison we consider the 2D FDTD method (with effective index approach and dx=dy=0.01µm), and the 3D FEM method. In order to compare the 2D Hertzian Potentials solution with the 3D FEM one, we refine the length of tetrahedral elements (FEM mesh) below the specified value of 0.01 µm. The length of a tetrahedron is defined as the length of its longest edge. In the Hertzian Potential method all the field components depend only on the Ψe,h scalar potentials [4

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]

], instead in the FDTD algorithm the components are interrelated and so the computational cost increases.

Fig. 8. Source: time-evolution in air (pulse excitation).
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. Ez field component after 380 time-steps for different cross-section (reference of y-position).
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. 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. 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. Coupling efficiency for different alpha angles.

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

table-icon
View This Table

5. Conclusion

This Hertzian potentials formulation with effective refractive index approach provides the 3D behavior of the tapered waveguide with a low computational cost and a good numerical convergence solution. The numerical model shows the field confinement and field PCS-coupling for different slanted α angles at the working wavelength of λ0=1.31 µm. The proposed time-domain model is used to evaluate the single TE transmitted field at the PCS input by considering waveguide coupling, and the frequency response around the working wavelength. We complete the coupling analysis by comparing the method with the 2 FDTD and 3D FEM results, and by evaluating the field coupled inside the PCS along the guiding region. The numerical and the analytical approximations used in this work give the same convergent results of the 3D FEM model with less computational cost.

References and links

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, (UTETEd., Torino1996), 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, 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]

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 York1974), Ch. I.

14.

A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London2000), 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]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(310.0310) Thin films : Thin films

ToC Category:
Photonic Crystals

History
Original Manuscript: October 4, 2007
Revised Manuscript: November 15, 2007
Manuscript Accepted: November 19, 2007
Published: November 28, 2007

Citation
A. Massaro, M. Grande, R. Cingolani, A. Passaseo, and M. De Vittorio, "Design and modeling of tapered waveguide for photonic crystal slab coupling by using time-domain Hertzian potentials formulation," Opt. Express 15, 16484-16499 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-25-16484


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References

  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]

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