## Design and modeling of tapered waveguide for photonic crystal slab coupling by using time-domain Hertzian potentials formulation

Optics Express, Vol. 15, Issue 25, pp. 16484-16499 (2007)

http://dx.doi.org/10.1364/OE.15.016484

Acrobat PDF (1484 KB)

### 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

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]

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]

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]

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]

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]

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

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]

## 2. Analysis and design of tapered waveguide.

_{core}(GaAs)=3.408 at the working wavelength λ

_{0}=1.31 µm (waveguide operation wavelength). Figure 3 describes the modal region of the asymmetric slab waveguide and defines the geometrical parameters for a single mode condition. The analysis begins in the transverse z-direction in order to define the effective refractive n

_{effz}index. The dispersion equations [13] are used for a graphical analysis of the single mode region that is useful to evaluate the sensitivity of the solution near the frequency cut-off (of the single TE guided condition) by changing the core thickness. We show in Fig. 4 an example of frequency-cut graphical evaluation concerning the TE and the TM analysis. The used dispersion-equations are [13]

_{1}=1(air), ε

_{2}=11.56(GaAs), ε

_{3}=10.89(AlGaAs), k

_{0}is the wave number in the free space. By using (1),(2),(3) we evaluate k

_{z}=u/d

_{core}and then effective index in the z-direction ε

_{effz}by the wavenumber conservation equation

_{core}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 TE

_{z}mode (only a propagation constant k

_{z}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 TE

_{z}propagated modes is obtained by d

_{core}=0.19 µm (n

_{effz}=3.067 is the corresponding effective refractive index), w

_{2}=0.22 µm.

_{effz}in the x-y plane in order to analyze the behavior of the tapered waveguide by different α slanted-angles, and L lengths, by the time-domain Hertzian formulation [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. 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]

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

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]

**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]

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

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

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

**14**, 2027–2036 (2006). [CrossRef] [PubMed]

**14**, 2027–2036 (2006). [CrossRef] [PubMed]

**14**, 2027–2036 (2006). [CrossRef] [PubMed]

**14**, 2027–2036 (2006). [CrossRef] [PubMed]

_{P}and I

_{P}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

**6183** (2006). [CrossRef]

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]

^{-11}is attained in the simulations. In the 2D case the matrix structure that considers the spatial domain and the ABCs is

*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.

_{e}and ψ

_{h}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:

_{0}=1.31µm,

*T*

_{0}is a constant, and

*dt*is the time step. Figure 9 proves that a tapered waveguide with w

_{1}=5.94µm, w

_{2}=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

^{-7}m.,

*dt*=3.67*10

^{-16}sec., λ

_{0}=1.31µm,

*T*

_{0}=8*10

^{-15}sec.). A spatial monitoring of the E

_{z}field is reported in Fig. 10: we fix the time-step and evaluate the spatial E

_{z}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 w

_{1}=5.94µm, w

_{2}=0.22µm and α=14 deg.

*Sm,n*can be obtained for an impulsive excitation as follows [14]

*Ê*

_{m}is the phasor voltage (DFT of E

_{z}component) at the port

*m*at observation plane

*y*;

_{m}*Ê*is the phasor voltage at the port

_{n}*n*at observation plane

*y*; and

_{n}*Z0,m*and

*Z0,n*are the characteristic impedances of the line connected to these ports defined as

_{ba}=S

_{21}with the port 1 and port 2 defined by the references line a=

*y*and b=

_{1}*y*respectively: the tapered waveguide with w

_{2}_{1}=5.94µm, w

_{2}=0.22µm and α=14 deg, and 45 deg. is characterized by an high transmission coefficient around the working wavelength λ

_{0}=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 S

_{21}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 w

_{2}. 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 n

_{effz}=3.067; the x-period is 3*R, the y-period is 2,5*R, and w

_{2}is the width of the guiding region. We show in Fig.13 the time evolution of the E

_{z}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 :

_{y}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 λ

_{0}=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.

## 5. Conclusion

_{0}=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. |

2. | W. J. R. Hoefer, “The transmission-line matrix method- Theory and applications,” IEEE Trans. Microwave Theory Tech. |

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. |

4. | A. Massaro and T. Rozzi, “Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation,” Opt. Express. |

5. | A. Massaro, L. Pierantoni, and T. Rozzi, “Time-domain modeling and filtering behavior of guided-wave optics by Hertzian potentials,” Proc. SPIE |

6. | M. Couture, “On the numerical solution of fields in cavities using the magnetic Hertz vector,” IEEE Trans. Microwave Theory and Tech. |

7. | K. I. Nikoskinen, “Time-domain study of arbitrary dipole in planar geometry with discontinuity in permittivity and permeability,” IEEE Trans. Antennas Propag. |

8. | T. Rozzi and M. Farina, |

9. | C.G. Someda, |

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. |

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. |

12. | A. Yariv, |

13. | D. Marcuse, |

14. | A. Taflove and S. C. Hagness, |

15. | G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagnetic Compatibility |

**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

Sort: Year | Journal | Reset

### References

- 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).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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.
- 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]
- 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]
- 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.
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 6. |
Fig. 7. |
Fig. 8. |

Fig. 9. |
Fig. 10. |
Fig. 11. |

Fig. 12. |
Fig. 13. |
Fig. 14. |

« Previous Article | Next Article »

OSA is a member of CrossRef.