## Arbitrary-order interface conditions for slab structures and their applications in waveguide analysis

Optics Express, Vol. 18, Issue 5, pp. 4088-4102 (2010)

http://dx.doi.org/10.1364/OE.18.004088

Acrobat PDF (222 KB)

### Abstract

We extend the continuity relations of field derivatives across an abrupt interface to arbitrary orders for transverse electric and magnetic waves in slab structures. Higher-order finite-difference formulation is then obtained by combining the systematically-obtained interface conditions with Taylor series expansion. Generalized Douglas scheme is also adopted to further enhance the convergence of truncation errors by two orders. We apply the derived finite-difference formulation, up to nine-points in this paper, to solve the guided modes in simple a slab waveguide and multiple quantum well waveguides. The results shows the truncation error is much higher, up to tenth order, as expected. Using those higher-order schemes, accurate results are obtained with much fewer sampled points, and hence with tremendously less computation time and memory.

© 2010 Optical Society of America

## 1. Introduction

2. C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Elect. Eng. J. **141**, 281–286 (1994). [CrossRef]

*et al*. have extended this technique to full-vectorial waveguide mode solver and photonic crystal analysis [7–9

7. Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. **20**, 1609–1618 (2002). [CrossRef]

*O*(

*h*

^{2})), since only two adjacent sampled points are referenced to approximate second-order derivative in Helmholtz equation. Higher-order convergence is desired in certain situations to achieve higher accuracy and less computation. One effective method to increase the order is using generalized Douglas (GD) scheme. Hadley [10

10. G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation I: beam propagation,” J. Lightwave Technol. **16**, 134–141 (1998). [CrossRef]

11. J. Yamauchi, J. Shibayama, and H. Nakano, “Application of the generalized Douglas scheme to optical waveguides analysis,” Optic. Quantum Eletron. **31**, 675–687 (1999). [CrossRef]

*et al*. [12

12. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. **18**, 243–251 (2000). [CrossRef]

*O*(

*h*

^{4})) for mode solving of step-index structures. Such formulation is adopted in waveguide discontinuities [13

13. H. Zhang, Q. Guo, and W. Huang, “Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme,” J. Lightwave Technol. **25**, 556–561 (2007). [CrossRef]

14. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” Photon. Tech. Lett. **18**, 2535-2537 (2006). [CrossRef]

15. B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, “Fourth-order accurate subsampling for finite-difference analysis of surface plasmon metallic waveguides,” Microwave Opt. Technol. Lett. **50**, 995–1000 (2008). [CrossRef]

*O*(

*h*

^{4}) may be desired when we want to achieve even higher accuracy or even less computation. Hadley proposed a finite-difference formulation magically free of truncation error [16

16. G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation II: vertical-cavity surface-emitting lasers,” J. Lightwave Technol. **16**, 142–151 (1998). [CrossRef]

*n*̄

^{2}from ∂

^{2}/∂

*z*

^{2}in Helmholtz equation is inside the trigonometric functions. The derivation of finite-difference formulation is back to the commonly used Taylor series expansion. To achieve convergence of higher orders, we may include more sampled points to evaluate higher-order derivatives. However, the formulation is usually derived in homogeneous region, and it does not consider the discontinuity across abrupt interfaces. Therefore, when there are abrupt interfaces, the convergence is still of low order. It might not be problematic when the number of interfaces is small, since it is still of high order in the rest homogeneous regions. When the number of interfaces are large, like multiple quantum well structures, the convergence will be lowered, especially in structures of higher index difference. Therefore, a generalized higher-order finite-difference formulation considering the interface conditions is required in such cases to improve the numerical performance.

## 2. Derivation of Higher-Order Interface Conditions

*k*

_{0}= ω/

*c*. For an interface in a slab structure, we assume electric permitivity ε and magnetic permeability are homogeneous in

*y*- and

*z*-directions, and the discontinuity is along

*x*-direction as in Fig. 1. We also assume the propagation is along

*z*-direciton. A propagating mode with propagation constant β and modal field distribution

*E̅*=

*E̅*(

*x*) and

*H̅*=

*H̅*(

*x*), the electric and magnetic fields can be expressed as

*E*, and

_{y}, E_{z}, H_{y}*H*, and the continuity of normal components of electrical displacement

_{z}*D*= ε

_{x}*E*and magnetic flux density

_{x}*B*= μ

_{x}*H*as in Fig. 1. From the continuity of the field components, we will derive the continuity of their derivatives.

_{x}### 2.1. TE case

*E*, and

_{y}, H_{x}*H*are not zero. From the continuity of the tangential electrical and magnetic fields, we have

_{z}*L*and

*R*represent infinitesimally close to the interface on the left and right, respectively, as shown in Fig. 1. From (1), we have

*H*is continuous across the interface, we have continuity relation for the first-order derivative

_{z}*n*= 0 from (11). Second, we assume the statement is true for

*n*=

*k*, that is,

*n*=

*k*+ 1. Therefore, by the principle of mathematical induction, the statement holds for all non-negative integers.

*N*-th order can be written in matrix form as

*a*= μ

_{R}/μ

_{L}and

*b*= ω

_{k}^{2}(μ

_{L}ε

_{L}− μ

_{R}ε

_{R}). We will use this equation to drive higher-order finite-difference formulation.

### 2.2. TM case

*H*and

_{y}, E_{x}*E*are nonzero in TM mode. Continuity relations of the tangential fields

_{z}*a*= ε

_{R}/ε

_{L}in TM case.

## 3. Implementation in Mode Solvers

### 3.1. Normal Finite-Difference Scheme

*ψ*

_{i+1}and its derivatives with

*ψ*and its derivatives up to 2

_{i}*N*-th order. Sampled points in homogeneous medium can be described by Taylor series expansion up to 2

*N*-th order, and we have the relation between

*ψ*and

_{L}*ψ*as

_{i}*ψ*

_{i+1}and

*ψ*as

_{R}_{i+1}and Ψ

_{i}is thus established as

*ψ*

_{i+1}and

*ψ*. For example, if there are

_{i}*T*interfaces with left and right fields represented as

*ψ*and

_{Lt}*ψ*at

_{Rt}*t*th interface, respectively, we can have the relation between Ψ

_{i+1}and Ψ

_{i}as

**M**

_{Rt∙Lt}’s are from matching the interface conditions, and

**M**

_{i+1∙RT},

**M**

_{L1:i}, and

**M**

_{R(t+1)∙Lt}’s are from Taylor series expansion. Applying the same procedure recursively, we can obtain the relation between Ψ

_{i+j}and Ψ

_{i},

*j*= 1, 2, ,⋯ ,

*N*. Similarly, we can have relation for negative

*js*.

*ψ*

_{i+j}in terms of

*ψ*and its derivatives from the first row in Ψ

_{i}_{i+j}=

**M**

_{i+j:i}Ψ

_{i}

*O*(

*h*

^{2N+1}). After applying similar procedure on all the (2

*N*+ 1) sampled points, we have the relation between the field at sampled points and

*ψ*and its derivatives up to 2

_{i}*N*th order as

*D*, we have

_{x}^{j}ψ_{i}**D**

_{i}= [

*ψ*,

_{i}*D*,

_{x}ψ_{i}*D*

^{2}

*, ⋯ ,*

_{x}ψ_{i}*D*, ⋯ ,

_{x}^{j}ψ_{i}*D*

^{2N}

_{x}

*ψ*]

_{i}^{T}. Finally, we obtain the finite-difference approximation of derivatives up 2

*N*-th order considering the interface conditions at of

*ψ*with (2

_{i}*N*+ 1) sampled points

**B**

_{i}=

**U**

^{-1}

_{i}. The third row in (43) is the approximation of second derivative expressed in terms of the sampled points, which is

*O*(

*h*

^{2N+1})/

*h*

^{2}=

*O*(

*h*

^{2N-1}) accurate. We use centered-difference scheme here, but it is not compulsory. We can also use forward-difference or backward-difference scheme. The advantage of using centered-difference scheme is that the truncation error can be enhanced by one order, from

*O*(

*h*

^{2N-1}) to

*O*(

*h*

^{2N}), when the medium is homogeneous and the grids are uniformly positioned [19].

^{2}

*ψ*/∂

*x*

^{2}in (9) and (10) with

*D*

^{2}

_{x}

*ψ*. Then we have the discretized Helmholtz equation

*D*

^{2}

_{x}

*ψ*is

*O*(

*h*

^{2N}) if the medium is homogeneous between the (2

*N*+ 1) points and the grids are uniformly positioned [19]. If the grids are nonuniformly positioned, the truncation error is usually reduced to

*O*(

*h*

^{2N-1}). If there is an abrupt interface or interfaces, the truncation error is usually

*O*(

*h*

^{2N-1}). In waveguide problems, most grids are usually in homogeneous regions. Therefore, the eigen problem in (44) is

*O*(

*h*

^{2N}) accurate in global sense.

### 3.2. Generalized Douglas Finite-Difference Scheme

**G**

_{1}and

**G**

_{2}are (2

*N*+ 1) × 1 vectors. Multiplying the inverse matrix

**B**

_{i}=

**U**

^{-1}

_{i}as in (43), we have improved formulation

*v*is the third element of the vector

_{j}**V**=

**B**

_{i}

**G**

_{j}in (46). We further approximate

*D*

^{2N-1}

_{x}and

*D*

^{2N}

_{x}in (43), and we have the generalized Douglas scheme to improve the accuracy of second-order derivative

*v*

_{1}

*D*

^{2N-1}+

*v*

_{2}

*D*

^{2N}

_{x}), we finally have improved discretized Helmholtz equation

*O*(

*h*

^{2N+2}) if the medium is homogeneous between the (2

*N*+ 1) points and the grids are uniformly positioned. If the grids are nonuniformly positioned, the truncation error is usually reduced to

*O*(

*h*

^{2N+1}). If there is an abrupt interface or interfaces, the truncation error is

*O*(

*h*

^{2N+1}). In waveguide problems, most grids are usually in homogeneous region. Therefore, the eigen problem in (50) is usually

*O*(

*h*

^{2N+2}) accurate in global sense.

## 4. Applications in Slab Waveguides

### 4.1. Slab Waveguide

*n*= √11.088 and

_{core}*n*= √11.044, respectively. Waveguide width is 2 μm and wavelength is 1.55 μm. The computation domain is 40 μm in width and transparent boundary condition (TBC) is adopted on the outer boundary. The relative error of propagation constant β is defined as

_{clad}_{calculated}is calculated from finite-difference method, and β

_{exact}is from the analytical solution of dispersion relation.

*x*. The thinnest black and blue lines are from three-point formulation based on graded-index (GI) formulation as in [1] with or without index averaging. Intermediate thick and thickest lines are from calculation considering the interface conditions without and with generalized Douglas (GD) scheme as in Section 3, respectively. The approximation of a step index interface with graded index is only

*O*(

*h*

^{0}). The higher index difference the larger deviation of the approximation is. In this case, however, only two interfaces with small index difference exist, and most sampled points are in homogeneous region. Therefore, using conventional graded-index (GI) approach can still yield

*O*(

*h*

^{2}) accuracy. The red lines are calculated from three-point formulation without and with GD scheme as proposed in [12

12. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. **18**, 243–251 (2000). [CrossRef]

*O*(

*h*

^{1}) and

*O*(

*h*

^{3}) accurate, respectively. However, most sampled pointed are in homogeneous region, the accuracy are still

*O*(

*h*

^{2}) and

*O*(

*h*

^{4}) accurate, respectively. Furthermore, if the interfaces are at the center between two sampled points or at one sampled point, the three-point formulation in (44) and (50) can be

*O*(

*h*

^{2}) and

*O*(

*h*

^{4}) accurate, respectively.

*N*+1)-point calculation with boundary condition matching, the convergence is approximately

*O*(

*h*

^{2N}) as the intermediate thick lines, and they can be further increased up to

*O*(

*h*

^{2N+2}) by applying GD scheme as in the thickest lines. Overall, error analysis quite matches theoretical expectation. Including more points in evaluating the second derivative in Helmholtz equation, we can reach desired accuracy with much fewer grid points. For example, if we want to reach relative error ε

_{r}= 10

^{-8}, the grid number in nine-point formulation with GD scheme is about 1/1000 of that in conventional three-point formulation. The required memory in the former is only about 3/1000 of that in the latter. The computation time in finding the eigenvalue in the former is almost negligible as compared with that in the latter.

*O*(

*h*), decreases polynomially with grid size

^{P}*h*, Round-off error is due to finite digits of the floating point numbers in computers. It accumulates with the number of operations and increases with grid number polynomially. Therefore, the lower bound of of the accuracy is limited by the round-off error. As we can see in the figure, lower-order formulations require more operations and hence result in larger round-off error. For example, the lower bound of the relative error in three-point formulation is about 10

^{-9}and that in nine-point formulation with GD scheme is about 10

^{-14}.

### 4.2. Multiple-Quantum-Well Waveguides

12. Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. **18**, 243–251 (2000). [CrossRef]

*n*

_{covr}= 3.2224,

*n*

_{well}= 3.3704

*n*

_{well}= 3.3704,

*n*

_{barr}= 3.2874, and

*n*

_{clad}= 3.2224, respectively. The relative error of propagation constant is defined as

*et al*. proved that the MQW region could be approximately replaced with a homogeneous layer with an averaged or effective index [17

17. S. Ohke, T. Umeda, and Y. Cho, “Optical waveguides using GaAs-Al_{x}Ga_{1-x}As multiple quantum well,” Opt. Commun. **56**, 235–239 (1985). [CrossRef]

18. S. Ohke, T. Umeda, and Y. Cho, “TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide,” Opt. Commun. **70**, 92–96 (1989). [CrossRef]

*w*is the thickness of

_{i}*i*-th layer with refractive index

*n*.

_{i}*n*

_{well}or

*n*

_{barr}depending on the point position. The convergence is relatively slow and roughly

*O*(∆

*x*) with relatively large fluctuations. As mentioned, the graded index approximation is only

*O*(

*h*

^{0}) accuracy. The results do not converge well since there are more than one-hundred layers.

17. S. Ohke, T. Umeda, and Y. Cho, “Optical waveguides using GaAs-Al_{x}Ga_{1-x}As multiple quantum well,” Opt. Commun. **56**, 235–239 (1985). [CrossRef]

18. S. Ohke, T. Umeda, and Y. Cho, “TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide,” Opt. Commun. **70**, 92–96 (1989). [CrossRef]

*O*(∆

*x*

^{2}) in TE polarization. It is worse in TM polarization and close to

*O*(∆

*x*). The electric field and its derivative in TE polariation are continuous across the interface, therefore, the index averaging gives a better approximation. However, the derivative of magnetic field in TM polarization is discontinuous across the interface, the index averaging does not working so well as that in TE polarization.

**18**, 243–251 (2000). [CrossRef]

*O*(

*h*

^{1}) and

*O*(

*h*

^{3}) accurate, respectively. As mentioned in [12

**18**, 243–251 (2000). [CrossRef]

*O*(

*h*

^{2}) and

*O*(

*h*

^{4}) accurate, respectively. Besides, the truncation error may be positive or negative depending on the interface position. Since there are many layers, the truncation error from different layers may be compensate each other to reach higher accuracy. Therefore, in large grid size, the accuracy fluctuates between

*O*(

*h*) and

*O*(

*h*

^{2}) for the formulation without GD scheme, and it fluctuates between

*O*(

*h*

^{3}) and

*O*(

*h*

^{4}) for that with GD scheme. When the grid is smaller, more grids are in homogeneous region. The fluctuation becomes smaller with finer grid size and the accuracy is approximately

*O*(

*h*

^{2}) and

*O*(

*h*

^{4}) for the formulation without and with GD scheme, respectively.

*O*(

*h*

^{2N-1}) and

*O*(

*h*

^{2N}) as the intermediate thick lines, and that with GD scheme is between

*O*(

*h*

^{2N+1}) and

*O*(

*h*

^{2N+2}) as in the thickest lines. In seven-point formulation with GD scheme and nine-point formulation, when the round-off error dominates, the grid size is still larger than layer thickness of wells or barriers. The lines still fluctuate and no straight section exists.

*n*

_{well}= 3.4,

*n*

_{barr}= 1.6, and

*n*

_{cover}=

*n*

_{clad}= 1.5. Layer number and thickness of wells and barriers are the same. Relative error for TE and TM modes are illustrated in Figs. 6 and 7, respectively. The truncation error with (2

*N*+ 1)-point formulation is still between

*O*(

*h*

^{2N-1}) and

*O*(

*h*

^{2N}), and it is between

*O*(

*h*

^{2N+1}) and

*O*(

*h*

^{2N+2}) with generalized Douglas scheme. When the grid size is smaller, most points are in homogeneous region, the truncation error is almost

*O*(

*h*

^{2N}) and

*O*(

*h*

^{2N+2}) for formulation without and with GD scheme. The transverse variation is much larger due to larger index difference. The propagation constant diverges for grid size larger than 0.1 μm.

### 4.3. Some Remarks

*β*

^{2}with ∂

^{2}/∂

*z*

^{2}and using

*ψ*(

*x,z*) = Ψexp(-

*jβz*), we have

^{2}Ψ/∂

*z*

^{2}is dropped and Crank-Nicolson scheme is adopted. The resultant matrices are (2

*N*+ 1) banded instead of tridiagonal in three-point formulation. Note that wide angle techniques based on Padé approximants and multistep method [20

20. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. **17**, 1743–1745 (1992). [CrossRef] [PubMed]

14. J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” Photon. Tech. Lett. **18**, 2535-2537 (2006). [CrossRef]

15. B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, “Fourth-order accurate subsampling for finite-difference analysis of surface plasmon metallic waveguides,” Microwave Opt. Technol. Lett. **50**, 995–1000 (2008). [CrossRef]

## 5. Conclusion

*N*+ 1)-point formulation are

*O*(

*h*

^{2N}), and it is enhanced to

*O*(

*h*

^{2N+2}) with generalized Douglas scheme. In multiple quantum well waveguides, the truncation error with (2

*N*+ 1)-point formulation without and with generalized Douglas scheme are, at worst,

*O*(

*h*

^{2N-1}) and

*O*(

*h*

^{2N+1}), respectively. Adopting higher-order formulation, we can reach desired accuracy with much fewer grid points. Therefore, it can significantly reduce the computation time and memory. Furthermore, the ultimate accuracy is also lower due to smaller round-off error. Such higher-order formulation may find its applications in various optical and electromagnetic slab structures, especially when high accuracy is essential in their analysis. The formulation can be directly applied to beam propagation analysis. Three-dimensional semivectorial analysis is also possible by extending one-dimensional transverse direction to two-dimensional transverse directions.

## Acknowledgement

## References and links

1. | M. S. Stern, “Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles,” Inst. Elect. Eng. Proc. J. |

2. | C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” Proc. Inst. Elect. Eng. J. |

3. | M. S. Stern, “Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles,” Proc. Inst. Elect. Eng. J. |

4. | P.-L. Liu and B.-J. Li, “Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides,” IEEE J. Quantum Electron. |

5. | W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method: analysis and assessment,” J. Lightwave Technol. |

6. | C. Vassallo, “Improvement of finite difference methods for step-index optical waveguides,” Proc. Inst. Elect. Eng. J. |

7. | Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles,” J. Lightwave Technol. |

8. | Y.-P. Chiou, Y.-C. Chiang, C.-H. Lai, C.-H. Du, and H.-C. Chang, “Finite-difference modeling of dielectric waveguides with corners and slanted facets,” J. Lightwave Technol. |

9. | Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, “Finite-difference frequency domain analysis of 2-D photonic cry-tals with curved dielectric interfaces,” J. Lightwave Technol. |

10. | G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation I: beam propagation,” J. Lightwave Technol. |

11. | J. Yamauchi, J. Shibayama, and H. Nakano, “Application of the generalized Douglas scheme to optical waveguides analysis,” Optic. Quantum Eletron. |

12. | Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. |

13. | H. Zhang, Q. Guo, and W. Huang, “Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme,” J. Lightwave Technol. |

14. | J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, “A three-dimensional multistep horizontally wide-angle beam-propagation method based on the generalized Douglas scheme,” Photon. Tech. Lett. |

15. | B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, “Fourth-order accurate subsampling for finite-difference analysis of surface plasmon metallic waveguides,” Microwave Opt. Technol. Lett. |

16. | G. R. Hadley, “Low-truncation-error finite difference equations for photonics simulation II: vertical-cavity surface-emitting lasers,” J. Lightwave Technol. |

17. | S. Ohke, T. Umeda, and Y. Cho, “Optical waveguides using GaAs-Al |

18. | S. Ohke, T. Umeda, and Y. Cho, “TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide,” Opt. Commun. |

19. | R. L. Burden and J. D. Faires, |

20. | G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. |

21. | J. Yamauchi, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(040.4200) Detectors : Multiple quantum well

(130.2790) Integrated optics : Guided waves

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: January 11, 2010

Revised Manuscript: February 11, 2010

Manuscript Accepted: February 11, 2010

Published: February 16, 2010

**Citation**

Yih-Peng Chiou and Cheng-Han Du, "Arbitrary-order interface conditions for slab structures and their applications in waveguide analysis," Opt. Express **18**, 4088-4102 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4088

Sort: Year | Journal | Reset

### References

- M. S. Stern, "Semivectorial polarised finite difference method for optical waveguides with arbitrary index profiles," Inst. Elect. Eng. Proc. J. 135, 56-63 (1988).
- C. L. Xu, W. P. Huang, M. S. Stern, and S. K. Chaudhuri, "Full-vectorial mode calculations by finite difference method," Proc. Inst. Elect. Eng. J. 141, 281-286 (1994). [CrossRef]
- M. S. Stern, "Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles," Proc. Inst. Elect. Eng. J. 138, 185-190 (1991).
- P.-L. Liu and B.-J. Li, "Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides," IEEE J. Quantum Electron. 28, 778-782 (1992). [CrossRef]
- W. Huang, C. Xu, S.-T. Chu, and S. K. Chaudhuri, "The finite-difference vector beam propagation method: analysis and assessment," J. Lightwave Technol. 10, 295-305 (1992). [CrossRef]
- C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng. J. 139, 137-142 (1992).
- Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002). [CrossRef]
- Y.-P. Chiou, Y.-C. Chiang, C.-H. Lai, C.-H. Du, and H.-C. Chang, "Finite-difference modeling of dielectric waveguides with corners and slanted facets," J. Lightwave Technol. 27, 2077-2086 (2009). [CrossRef]
- Y.-C. Chiang, Y.-P. Chiou, and H.-C. Chang, "Finite-difference frequency domain analysis of 2-D photonic crytals with curved dielectric interfaces," J. Lightwave Technol. 26, 971-976 (2008). [CrossRef]
- G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation I: beam propagation," J. Lightwave Technol. 16, 134-141 (1998). [CrossRef]
- J. Yamauchi, J. Shibayama, and H. Nakano, "Application of the generalized Douglas scheme to optical waveguides analysis," Opt. Quantum Eletron. 31, 675-687 (1999). [CrossRef]
- Y.-P. Chiou, Y.-C. Chiang, and H.-C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightwave Technol. 18, 243-251 (2000). [CrossRef]
- H. Zhang, Q. Guo, and W. Huang, "Analysis of waveguide discontinuities by a fourth-order finite-difference reflective scheme," J. Lightwave Technol. 25, 556-561 (2007). [CrossRef]
- J. Shibayama, T. Takahashi, J. Yamauchi, and H. Nakano, "A three-dimensional multistep horizontally wideangle beam-propagation method based on the generalized Douglas scheme," Photon. Tech. Lett. 18, 2535-2537 (2006). [CrossRef]
- B. Hu, P. Sewell, J. G. Wykes, A. Vukovic, and T. M. Benson, "Fourth-order accurate subsampling for finitedifference analysis of surface plasmon metallic waveguides," Microwave Opt. Technol. Lett. 50, 995-1000 (2008). [CrossRef]
- G. R. Hadley, "Low-truncation-error finite difference equations for photonics simulation II: vertical-cavity surface-emitting lasers," J. Lightwave Technol. 16, 142-151 (1998). [CrossRef]
- S. Ohke, T. Umeda, and Y. Cho, "Optical waveguides using GaAs-AlxGa1−xAs multiple quantum well," Opt. Commun. 56, 235-239 (1985). [CrossRef]
- S. Ohke, T. Umeda, and Y. Cho, "TM-mode propagation and form birefringence in a GaAs-AlGaAs multiple quantum well optical waveguide," Opt. Commun. 70, 92-96 (1989). [CrossRef]
- R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks Cole, 2004).
- G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett. 17, 1743-1745 (1992). [CrossRef] [PubMed]
- J. Yamauchi, Propagating Beam Analysis of Optical Waveguides, (Exeter, UK: Research Studies Press, 2003).

## 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.