## Performance analysis of three dimensional high index contrast dielectric waveguides

Optics Express, Vol. 12, Issue 4, pp. 633-643 (2004)

http://dx.doi.org/10.1364/OPEX.12.000633

Acrobat PDF (127 KB)

### Abstract

This paper presents the implementation of a parallelized Finite-Difference Time-Domain method, based on the Message Passing Interface (i.e. MPI), which is used to study the modal properties of three-dimensional (3D) dielectric waveguide structures. To this end, we also use the least-square method to obtain the wave vector, *β*, along the axis of propagation. Lastly, bending losses in arbitrary-angle waveguides are also discussed.

© 2004 Optical Society of America

## 1. Introduction

*E*⃑(

*x, y, z*)=

*E*⃑(

*y, z*)

*e*

^{-jkx}(where

*x*is the propagation direction) 3D waveguides, can be reduced to 2D using the compact FDTD method [1,2

2. A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” *Electron. Lett*. , **vol. 28**, pp. 1451–1452 (1992). [CrossRef]

13. J.H. Greene and A. Taflove, “Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack,” Opt. Lett. **28**, 1733–1735 (2003). [CrossRef] [PubMed]

*β*, in straight waveguides and the study of losses in waveguide bends.

## 2. Brief discussion of the FDTD method and absorbing boundary conditions

3. K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. **14**, 302 (1966). [CrossRef]

5. J.P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Physics **114**, 185–200 (1994). [CrossRef]

^{-6}to 10

^{-8}. For brevity, further details concerning the FDTD method and the PML absorbing boundary conditions are referred to in [6]. As indicated in Fig. 1(a), to terminate the infinite extension of straight waveguides, proper PML parameters are chosen in the inhomogeneous media region to maintain a stable and accurate formulation. Also, as the FDTD method is a solution to an initial-value problem, it requires an excitation scheme to start the computation. In this study, the guided wave source is excited near the left-end boundary of waveguides, see Fig. 1 (a), which was obtained by the Marcatili method [7]. In the following section, we present an overview of our implementation of the FDTD method using MPI.

## 3. Parallel computation using MPI

*n*=3.5 and dimensions of 0.2286

*λ*×0.2286

*λ*×20

*λ*, with

*λ*=1

*µm*, and a sampling rate of

*λ*/56, it would take over 7 days using a single computer (assuming a 2 GHz CPU and 1 GB RAM). The high index of refraction of the silicon requires the higher sampling rate (i.e. 56) in the FDTD computation to minimize the dispersion error.

15. V. Varadarajan and R. Mittra, “Finite-Difference Time Domain (FDTD), Analysis Using Distributed Computing,” IEEE Microwave and Guided Wave Letters , **4**, 144–145 (1994). [CrossRef]

*N*, where

*N*is the number of subsets. In this paper, the computational region is divided into several parts along the propagation direction of the waveguide, each of which is assigned to a different processor, see Fig. 2. As such, the computational aspect of the analysis on each processor is the same and only the grid values are different. However, to calculate the field along the boundary cells, one needs to know the field values in the adjacent cells, which belong to the neighboring subset. This is the result of the spatial central finite-difference expression, where the value at one point is not only dependent on the information at this point, but also the information at the nearest four adjacent points. As such, the new electric field component (at each discrete point), such as

*E*, is a function of the electric field component at the point from the previous time step and the magnetic field components surrounding the point from the current time step. As a result, the values along the border points in each processor’s computational region are dependent on the values at the points in the neighboring processor’s region. To address this, MPI [8] is used to realize the communication between neighboring processors. The details of its implementation can be found in [8,14]. In the next section, we present our numerical results obtained by using the parallelized FDTD method described above.

_{z}## 4. Numerical results and discussion

*β*, in 3D straight waveguides. A non-linear mean-square fitting algorithm is used to obtain the wave vector in both single-mode and multi-mode waveguides, combined with the parallelized FDTD algorithm. The second case is the study of losses in 3D waveguide bends, shown in Fig. 1(b), as a function of the bend angle and the length of the diagonal waveguide region. Lastly, this section also presents the analysis of an optimal curved bend that is shown to overcome the losses encountered in sharp bends.

### 4.1 Calculation of wave vector -- β for straight waveguides

*β*is to apply the discrete Fourier transform on the electric, or magnetic, field component values along the propagation direction. However, this requires relatively large sampling rates,

*N*, to ensure that

*dβ*, which is equal to

*β*. For this reason, we use a non-linear mean-square fitting algorithm [9

9. M.A. Hernandez-Lopez and M. Quintillan, “Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method,” J. Electromagnetic Waves and Applications **14**,1707–1722 (2000). [CrossRef]

*E*, are assumed to fit into a theoretical modal expansion as indicated in Eq. (1):

_{z}*E*(

_{M}*z*) is the modal function,

_{i}, a_{k}*A*is the value of mode amplitudes,

_{k}*β*is the propagation constants, {

_{k}*z*} are the values of the sampling points along the propagation direction, and

_{i}*M*is the number of modes.

*β*, of each mode satisfies:

_{k}*E*(

_{z}*z*)(

_{i}*i*=1,..,

*n*) are the values of the

*E*component, which are sampled along the propagation direction.

_{z}*β*in Eq. (1). In our implementation,

_{k}*β*is obtained by applying the FFT on the electric or magnetic field component along the propagation direction. In this case, the longer the waveguide, the more accurately one can determine the initial values of the wave vector.

_{initial}*λ*, which is necessary to ensure single mode operation [12

12. A. Kumar, K. Thyagarajan, and A.K. Ghatak, “Analysis of rectangular -core dielectric waveguides - A accurate perturbation approach,” Opt. Lett. **8**, 63–65 (1983). [CrossRef] [PubMed]

*λ*×0.1789

*λ*(

*λ*=1

*µm*). In this case, the propagation constant is equal to 9.324×10

^{6}, which was obtained by Marcatili’s method [7]. Using the parallelized FDTD method, we numerically analyzed this structure, which had a waveguide length of 10

*λ*. The results are compared in Table 1, which represent the theoretical result obtained from the Marcatili method, the FFT result, and the optimized result using our least-square method (parallelized FDTD). The FFT result is obtained by applying a discrete Fast Fourier transform to the sampling points along propagation direction. From Table 1, we can see that the result using the FDTD method in combination with the least-square method is very close to the theoretical result (Marcatili method).

*λ*×0.257

*λ*(

*λ*=1

*µm*). For this structure, there exist two modes with theoretical propagation constants of 1.63201×10

^{7}and 7.2658×10

^{6}. This waveguide was analyzed for a length of 10

*λ*using our MPI FDTD method. Table 2 shows the comparison between the theoretical result, again obtained using the Marcatili method, the FFT result, and the optimized result from the least-square method. As shown, the FDTD method in combination with the least-square method provides very accurate results. However, in the case of

*β*we can see that the FFT method was more accurate. We attribute this to the selection of initial values of the mode amplitudes

_{1}*A*.

_{k}*a⃑*=(

*x*

_{0},

*y*

_{0},

*z*),

_{i}*z*∈[

_{i}*z*

_{1},

*z*

_{2}].

### 4.2 Study of waveguide bending

*L*) and its angle with respect to the input and output direction of propagation (

*θ*). For each of the structures analyzed, the transverse dimensions of the waveguides are 0.1789

*λ*×0.1789

*λ*(

*λ*=1

*µm*) and a core with a refractive index of

*n*=3.5 surrounded by air cladding. Using the 10-node Beowulf cluster, we studied the loss, which is computed as the fraction of input power coupled into the fundamental mode of output waveguide, with respect to the variation of L and

*θ*. The calculations of power and loss require reference planes, wherein we used one at the input waveguide and one at the output waveguide. The Poynting vector gives the power at each point, thus, the power in the

*x*-direction at the entire reference plane is:

*P*is measured in the reference plane at the output of the waveguide, shown in Fig. 1(a), and

_{out}*P*is measured in the reference plane at the input of the waveguide. To accurately determine

_{in}*P*, a straight waveguide with same dimensions is considered. The same source is used to excite the waveguide modes at the left section. Using this method,

_{in}*P*can be measured in a reference plane located near the extreme right boundary of the straight waveguide.

_{in}*θ*=30°, and varied

*L*according to

*mλ*, where

*m*varies from 0.5 to 10. From Fig. 4 we can see that the loss initially fluctuates with respect to

*L*and tends to become flat while

*L*is large. The smaller the length of

*L*, the more electromagnetic coupling between the two bending corners. In this case, a resonance may occur and the propagation loss will vary with

*L*. From the figure, we also can see that there are three resonance occurring at points, where the length of

*L*is around 0, 2.3

*λ*and 4.6

*λ*respectively. When

*L*increases, the two bends can become decoupled in which case the propagation loss can be approximated by considering two separate bends. However, there may exist some optimal length at which relatively low loss can be achieved. In this study, we have found such an optimal length, which is

*L*=2.5

*λ*.

*L*=4

*λ*, and evaluated the propagation loss while

*θ*changed from 15° to 40°. Figure 5 shows the propagation loss versus bending angle. From the figure, we can see that in this case the loss increases significantly as the angle increases. It agrees with the trend of coupling losing analyzed in a 2-D optical waveguide bends [17

17. S.M. Lee, W.C. Chew, S.L. Chuang, and J.J. Coleman, “Bending loss in optical waveguides for nonplanar laser array applications,” J. Appl. Phys. **71**, 2513–2520 (1992). [CrossRef]

*θ*equal to 4

*λ*and 40°, respectively. The sharp bends were replaced by curved bends with 1.43

*µm*radius of curvature, to soften the impact of discontinuity. The two-dimensional

*H*field component in the middle plane is shown in Fig. 6. The propagation loss is equal to -0.96dB, which is 59% lower than the loss in the case of the sharp bends, shown in Fig. 5. Figure 7 shows two movies that recorded the wave propagation of

_{z}*H*component through the sharp-bend waveguide (Fig. 7(a)) and curve-bend waveguide (Fig. 7(b)) respectively. From Fig. 7(a), we found there was scattering loss around two sharp bends and strong reflection in the input waveguide. However, from Fig. 7(b), the back reflection in the input waveguide significantly reduced and it has smaller scattering loss in the curved bands than in the sharp bends.

_{z}## 5. Conclusion

## Reference and links

1. | T. Tamir, |

2. | A. Asi and L. Shafai, “Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2D-FDTD,” |

3. | K.S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

4. | G. Mur, “Absrobing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-filed equations,” IEEE Trans Electromagn. Compat. |

5. | J.P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Physics |

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

7. | E.A.J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. |

8. | Willam Gropp, Ewing Lusk, and Anthony Skjellum, |

9. | M.A. Hernandez-Lopez and M. Quintillan, “Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method,” J. Electromagnetic Waves and Applications |

10. | W.H. Press, S.A. Teulcoolsky, W.T. Vetterling, and B.P. Flannery, |

11. | J.E. Goell, “A circular harmonic computer analysis of rectangular dielectric waveguides,” Bell Syst. Tech. J. |

12. | A. Kumar, K. Thyagarajan, and A.K. Ghatak, “Analysis of rectangular -core dielectric waveguides - A accurate perturbation approach,” Opt. Lett. |

13. | J.H. Greene and A. Taflove, “Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack,” Opt. Lett. |

14. | Ao Jiang and Finite “Difference Time Domain Method on Paralle Architecture using Message Passing Interface,” M.S. Thesis, Dept. of Ece, University of Delaware (2003). |

15. | V. Varadarajan and R. Mittra, “Finite-Difference Time Domain (FDTD), Analysis Using Distributed Computing,” IEEE Microwave and Guided Wave Letters , |

16. | A.M. Liu, A.S. Mohan, T. A. Aubrey, and W.R. Belcher, “Techniques for Implementation of FDTD Method on CM-5 Parallel Computer,” IEEE Antennas and Propagation Magazine |

17. | S.M. Lee, W.C. Chew, S.L. Chuang, and J.J. Coleman, “Bending loss in optical waveguides for nonplanar laser array applications,” J. Appl. Phys. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(220.0220) Optical design and fabrication : Optical design and fabrication

(230.0230) Optical devices : Optical devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 3, 2004

Revised Manuscript: February 11, 2004

Published: February 23, 2004

**Citation**

Ao Jiang, Shouyuan Shi, Ge Jin, and Dennis Prather, "Performance analysis of three dimensional high index contrast dielectric waveguides," Opt. Express **12**, 633-643 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-4-633

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### References

- T. Tamir, Integrated Optics (Berlin: Springer-Verlag, 1975), Chap.2.
- A. Asi, and L. Shafai, �??Dispersion analysis of anisotropic inhomogeneous waveguides using compact 2DFDTD,�?? Electron. Lett., vol. 28, pp. 1451-1452 (1992). [CrossRef]
- K.S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propag. 14, 302 (1966). [CrossRef]
- G. Mur, �??Absrobing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-filed equations,�?? IEEE Trans Electromagn. Compat. EMC-23, pp.377-382 (1981). [CrossRef]
- J.P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comp. Physics 114, 185-200 (1994). [CrossRef]
- A. Taflove and S.C. Hagness, Computational Electromagnetics: The Finite-Difference Time Domain Method, 2nd ed. (Artech House, Norwood MA, 2000).
- E.A.J. Marcatili, �??Dielectric rectangular waveguide and directional coupler for integrated optics,�?? Bell Syst. Tech. J. 48, 2071-2102 (1969).
- Willam Gropp, Ewing Lusk, and Anthony Skjellum, Using MPI: Portable Parallel Programming with the Message-Passing Interface (The MIT Press, Cambridge MA, 1994).
- M.A. Hernandez-Lopez and M. Quintillan, �??Propagation characteristics of modes in some rectangular waveguides using the finite-difference time-domain method,�?? J. Electromagnetic Waves and Applications 14,1707-1722 (2000). [CrossRef]
- W.H. Press, S.A. Teulcoolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in FORTRAN(Cambridge University Press, 1992).
- J.E. Goell, �??A circular harmonic computer analysis of rectangular dielectric waveguides,�?? Bell Syst. Tech.J. 48, 2133-2160 (1969).
- A. Kumar, K. Thyagarajan, and A.K. Ghatak, �??Analysis of rectangular �??core dielectric waveguides �?? A accurate perturbation approach,�?? Opt. Lett. 8, 63-65 (1983). [CrossRef] [PubMed]
- J.H. Greene and A. Taflove, "Initial three-dimensional finite-difference time-domain phenomenology study of the transient response of a large vertically coupled photonic racetrack," Opt. Lett. 28, 1733-1735 (2003). [CrossRef] [PubMed]
- Ao Jiang, Finite �??Difference Time Domain Method on Paralle Architecture using Message Passing Interface,�?? M.S. Thesis, Dept. of Ece, University of Delaware (2003).
- V. Varadarajan and R. Mittra, �??Finite-Difference Time Domain (FDTD), Analysis Using Distributed Computing,�?? IEEE Microwave and Guided Wave Letters, 4, 144-145 (1994). [CrossRef]
- A.M. Liu, A.S. Mohan, T. A. Aubrey, and W.R. Belcher, �??Techniques for Implementation of FDTD Method on CM-5 Parallel Computer,�?? IEEE Antennas and Propagation Magazine 37, 64-71 (1995). [CrossRef]
- S.M. Lee, W.C. Chew, S.L. Chuang, and J.J. Coleman, �??Bending loss in optical waveguides for nonplanar laser array applications,�?? J. Appl. Phys. 71, 2513-2520 (1992). [CrossRef]

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