Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation

doi:10.1364/OE.14.002027

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Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation

A. Massaro and T. Rozzi »View Author Affiliations

Optics Express, Vol. 14, Issue 5, pp. 2027-2036 (2006)
http://dx.doi.org/10.1364/OE.14.002027

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Abstract

This work analyzes a new simulation approach to the evaluation of the time-domain electromagnetic (EM) field by reducing the number of equations to solve. Scalar Helmholtz-equations are utilized in order to determine the electric and magnetic Hertzian-potentials that yield the EM field. The method is applied to the example of optical waveguide arrays by considering the field-perturbation effect due to high dielectric contrast and dielectric discontinuities. The rigorous Hertzian potentials formulation is extended to bi-dimensional structures.

1. Introduction

Numerical simulation has become a powerful and important technique for solving electromagnetic problems. This is due to increased computer performance as well as to growing complexity of problems that must be solved. Time domain methods such as finite-difference time-domain (FDTD) [1

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

], transmission line matrix (TLM) [2

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

], and wavelet- Galerkin [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]

] methods are gaining importance by virtue of their versatility and the natural way in which they simulate what happens in reality. Nevertheless, at present these methods are limited by available computer memory and computational time required in the implementation of Maxwell’s equations. Several beam propagation method (BPM) algorithms have been proposed [4

Y. Chiou and H. Chang, “Analysis of optical waveguide discontinuities using Padè approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997). [CrossRef]

]–[7

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, “A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method,” IEEE J. Lightwave Technol. 21, 1996–2001 (2003). [CrossRef]

] for handling dielectric structures. However, all those algorithms are based on formulations in the frequency domain and are therefore not directly applicable for the simulation of pulse propagation and reflection. FDTD can be used naturally for simulation of full-wave propagation and reflection in the time domain, but it is time- and memory-intensive and not suitable for structure of large optical dimensions. In this direction we propose an efficient numerical algorithm to solve the EM field by using the Hertz vectors [8

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]

]–[10

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

] starting from the Helmholtz scalar equation. [10

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

]–[11

C. G. Someda, Onde elettromagnetiche , (UTET Ed., Torino 1996), Chap.1.

]. The number of equations to solve is reduced to two scalar equations instead of six in the one-dimensional case. Subsequently all the EM components are obtained by the Hertzian potentials [8

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]

]–[10

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

]. In our work, we analyze the example of one and bi-dimensional waveguide dielectric arrays at high frequency. The first structure is a polarizing beam splitter (see Fig. 1) [12

R.-C. Tyan, A. A. Salvekar, H.-Pu Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, “Design, fabrication, and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997). [CrossRef]

]–[13

K. Muro and K. Shiraishy, “Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°,” IEEE J. Lightwave Technol. 16, 127–133 (1998). [CrossRef]

] ,in which two type of thin dielectric films, with high (n₁) and low (n₂) refractive index alternate periodically (i.e., grating period is much smaller than the wavelength of the incident field). The second structure analyzed is an antireflection (AR) coatings for photonic devices [14

T. Saitoh, T. Mukai, and O. Mikami “Theoretical analysis and fabrication of antireflection coatings on laser-diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985). [CrossRef]

]–[17

N.-N. Feng and W.-P. Huang, “Time-domain reflective beam propagation method,” IEEE J. of Quantum Electron. 30, 1542–1552 (1994).

] (see Fig. 2). AR coatings are required for devices as the semiconductor optical amplifiers (SOA), the tunable external –cavity semiconductor laser, and the master oscillator power amplifiers (MOPAs) [18

P. Zorabedian, “Axial-mode instability in tunable external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 40, 778–783 (2004).

]–[19

A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, “Dynamic instabilities in master oscillator power amplifiers semiconductors,” IEEE J. Quantum Electron. 34, 166–170 (1998). [CrossRef]

]. In the frequency domain, the transmission and reflection properties of dielectric discontinuities may be derived by means of an equivalent circuit [20

N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides,” J. Appl. Phys. 22, 806–820 (1951). [CrossRef]

], [21

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]

], [22

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

] that automatically ensures continuity of the fields and their first derivatives along the axis z. If potentials are used, instead, second derivatives are involved and generators are necessary at each dielectric interface (see Fig. 3). In this work we (i) describe the time-domain field solution using Hertzian Potentials approach: one of the reasons for introducing the Hertzian potentials is the possibility of reducing the number of equations to be solved, in fact, only two scalars are needed to represent the field; (ii) we simulate one-dimensional dielectric multilayer optical waveguides by considering the accurate transmission line model, inclusive of generators, of Fig. 3 in the time domain, and finally (iii) we extend the model to an ASR bi-dimensional structure.

Fig. 1. General optical multilayer structure.

Fig. 2. General ASR optical multilayer structure.

Fig. 3. Equivalent circuit of the perturbed solution in the multilayered structure.

2. Hertzian field produced by dielectric discontinuities

The Hertzian electric and magnetic vectors are [8

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]

]–[10

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

] in rectangular coordinates represented by

\overset{̅}{Π_{e}} = a ψ_{e} (x, y, z, t)

\overset{̅}{Π_{h}} = a ψ_{h} (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

\overset{̅}{E} = \nabla \nabla ∙ {\overset{̅}{Π}}_{e} - εμ \frac{\partial^{2}}{\partial t^{2}} \overset{̅}{Π_{e}} - μ \frac{\partial}{\partial t} (\nabla \times \overset{̅}{Π_{h}})

\overset{̅}{H} = \nabla \nabla ∙ \overset{̅}{Π_{h}} - εμ \frac{\partial^{2}}{\partial t^{2}} \overset{̅}{Π_{h}} + ε \frac{\partial}{\partial t} (\nabla \times \overset{̅}{Π_{e}})

(2)

By writing (2) explicitly in the Cartesian coordinates all the electromagnetic field components are given by

E_{x} = A^{e} + μ \partial_{t} \partial_{z} ψ^{h}

E_{y} = B^{e} + εμ \partial_{t}^{2} ψ^{e}

E_{z} = C^{e} - μ \partial_{t} \partial_{x} ψ^{h}

H_{x} = A^{h} - {ε \partial}_{t} \partial_{z} ψ^{e}

H_{y} = B^{h} - {εμ \partial^{2}}_{t} ψ^{h}

H_{z} = C^{h} + {ε \partial}_{t} \partial_{x} ψ^{e}

(3)

where

A^{e, h} = \partial_{x}^{2} ψ^{e, h} + \partial_{yx} ψ^{e, h} + \partial_{zx} ψ^{e, h}

B^{e, h} = \partial_{xy} ψ^{e, h} + {\partial^{2}}_{y} ψ^{e, h} + \partial_{zy} ψ^{e, h}

C^{e, h} = \partial_{xz} ψ^{e, h} + \partial_{yz} ψ^{e, h} + {\partial^{2}}_{z} ψ^{e, h}

(4)

Ψ^{e, h}(x, y, z, and t) represents the solution of the homogeneous wave equation for a non-dissipative medium [7

], [8

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]

]

\nabla^{2} ψ_{e, h} (x, y, z, t) - με \frac{\partial^{2} ψ_{e, h} (x, y, z, t)}{\partial t^{2}} = 0

(5)

In the next section, we report the discretised solution of the scalar potentials Ψe, h(x, y, z, and t) and the implementation formulae. 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 [20

N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides,” J. Appl. Phys. 22, 806–820 (1951). [CrossRef]

] (see Fig. 3), giving an accurate solution of the EM field for structures with high dielectric contrast. In fact, the scalar wave Eq. (5) for a non-dissipative medium can be rewritten as [22

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

]

\nabla^{2} ψ_{e, h} (x, y, z, t) - με \frac{\partial^{2} ψ_{e, h} (x, y, z, t)}{\partial t^{2}} - μ \frac{\partial^{2} P_{pert} (x, y, z, t)}{\partial t^{2}} = 0

(6)

where

P (x, y, z, t) = {Δ ε (x, y, z, t) ψ}_{e, h} (x, y, z, t)

(7)

represent the dielectric polarization and for the one-dimensional case:

Δε = ε_{i + 1} - ε_{i} i = 1, m layer .

(8)

Therefore we solve Eq. (6) in the dielectric array and in proximity of the dielectric interfaces, and Eq. (5) in the homogenous region. In appendix A, we report the iterative solution of Eq. (6) and the parametric differences with the solution of Eq. (5) for the mono-dimensional case.

3. Algorithm description: finite-difference (FD) method implementation and ABC Mur condition.

In this section, we describe the algorithm of Fig. 4 used for the simulations. In order to reduce the computational time before the starting of the FD kernel we set the permittivity mask of the structure and establish the discretized region in which we solve the homogeneous Helmhotz scalar equation for a non-dissipative medium [7

], [8

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]

]

\nabla^{2} ψ_{e, h} (x, y, z, t) - με \frac{\partial^{2} ψ_{e, h} (x, y, z, t)}{\partial t^{2}} = 0 .

(9)

Or the inhomogeneous scalar equation

\nabla^{2} ψ_{e, h} (x, y, z, t) - με \frac{\partial^{2} ψ_{e, h} (x, y, z, t)}{\partial t^{2}} - μ \frac{\partial^{2} P_{pert} (x, y, z, t)}{\partial t^{2}} = 0

(10)

For the solution of Eq. (9) and Eq. (10) we use the FD discretisation [9], [10], that for a general bi-dimensional space is represented by

\frac{\partial ψ^{n} (i, j)}{\partial x} = \frac{ψ^{n} (i + \frac{1}{2}, j) - ψ^{n} (i - \frac{1}{2}, j)}{Δ x} + O (Δ^{2} x)

\frac{\partial ψ^{n} (i, j)}{\partial y} = \frac{ψ^{n} (i, j + \frac{1}{2}) - ψ^{n} (i, j - \frac{1}{2})}{Δy} + O (Δ^{2} y)

\frac{\partial ψ^{n} (i, j)}{\partial t} = \frac{ψ^{n + \frac{1}{2}} (i, j) - ψ^{n - \frac{1}{2}} (i, j)}{Δt} + O (Δ^{2} t)

(11)

and for the second order by

\frac{\partial^{2} ψ^{n} (i, j)}{\partial x^{2}} ≃ \frac{ψ^{n} (i + 1, j) - {2 ψ}^{n} (i, j) + ψ^{n} (i - 1, j)}{Δ^{2} x}

\frac{\partial^{2} ψ^{n} (i, j)}{\partial y^{2}} ≃ \frac{ψ^{n} (i, j + 1) - {2 ψ}^{n} (i, j) + ψ^{n} (i, j - 1)}{Δ^{2} y}

\frac{\partial^{2} ψ^{n} (i, j)}{\partial t^{2}} ≃ \frac{ψ^{n + 1} (i, j) - {2 ψ}^{n} (i, j) + ψ^{n - 1} (i, j)}{Δ^{2} t}

(12)

where i and j are the geometrical coordinates and n represents the time. We use for the implementation of Eq. (9

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]

) and Eq. (10

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

) a matrix structure

\begin{matrix} \to position \\ {(\begin{matrix} ABC & • & • & ψ^{0 (j = - 1)} & ψ^{0 (j = 0)} & ψ^{0 (j = + 1)} & • & • & ABC \\ ABC & • & • & • & • & ψ^{1 (j = 0)} & • & • & ABC \\ ABC & • & • & • & • & • & • & ABC \\ ABC & • & • & • & • & • & • & • & ABC \\ ABC & • & • & • & • & • & • & • & ABC \\ ABC & • & • & • & ψ^{n (j = 0)} & • & • & • & ABC \end{matrix})}^{n x number of nodes} ↓ time s teps \end{matrix}

(13)

Where the first and the last column represents the ABCs Mur boundary condition [23

A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference time-domain method , 2nd. ed. (Arthec House Publishers, London 2000), Chaps. 2, 4, and 7.

], [24

G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981). [CrossRef]

] that considers the wave equation:

\frac{\partial^{2} ψ_{e, h}}{\partial x \partial t} - \frac{1}{c} \frac{\partial^{2} ψ_{e, h}}{\partial t^{2}} + \frac{c}{2} \frac{\partial^{2} ψ_{e, h}}{\partial y^{2}} = 0 x = 0

\frac{\partial^{2} ψ_{e, h}}{\partial x \partial t} - \frac{1}{c} \frac{\partial^{2} ψ_{e, h}}{\partial t^{2}} + \frac{c}{2} \frac{\partial^{2} ψ_{e, h}}{\partial y^{2}} = 0 x = h

\frac{\partial^{2} ψ_{e, h}}{\partial y \partial t} - \frac{1}{c} \frac{\partial^{2} ψ_{e, h}}{\partial t^{2}} + \frac{c}{2} \frac{\partial^{2} ψ_{e, h}}{\partial x^{2}} = 0 y = 0

\frac{\partial^{2} ψ_{e, h}}{\partial y \partial t} - \frac{1}{c} \frac{\partial^{2} ψ_{e, h}}{\partial t^{2}} + \frac{c}{2} \frac{\partial^{2} ψ_{e, h}}{\partial x^{2}} = 0 y = h

(14)

where x=0, h and y=0, h represents the boundary-coordinates of a bi-dimensional domain (x, y). We observe that for a bi-dimensional structure matrix Eq. (13) is three-dimensional. After the temporal loop, we use the values of Ψ^{e, h} to evaluate the EM field by Eqs. (2), (3), and (4). By performing the Discrete Fourier transform (DFT), we obtain the reflectivity by considering the Fourier transform of the reflected electric field. In fact, we evaluate the reflection coefficient by using the ratio Ey reflected/ Ey homogeneous (transmission without the layers), where the reflected field Ey reflected is given by the difference between Ey and Ey homogeneous.

4. Iterative solution of inhomogeneous scalar equation

In order to highlight the parametric differences between the homogeneous and inhomogeneous solution of the wave Eq. (9) and Eq. (10) we use Eq. (11) and Eq. (12). In this manner, the solution of Eq. (5) in the iterative form can be expressed as:

ψ^{n + 1} (j) (\frac{με}{{(Δt)}^{2}}) = ψ^{n} (j + 1) (\frac{1}{{(Δz)}^{2}}) + ψ^{n} (j) (\frac{2 με}{{(Δt)}^{2}} - \frac{2}{{(Δz)}^{2}}) +

+ ψ^{n - 1} (j) (- \frac{με}{{(Δt)}^{2}}) + ψ^{n} (j - 1) (\frac{1}{{(Δz)}^{2}})

(15)

Instead, the inhomogeneous solution of Eq. (10) is expressed by:

ψ^{n + 1} (j) (\frac{με}{{(Δt)}^{2}} + \frac{μΔε}{{(Δt)}^{2}}) = ψ^{n} (j + 1) (\frac{1}{{(Δz)}^{2}}) + ψ^{n} (j) (\frac{2 με}{{(Δt)}^{2}} - \frac{2}{{(Δz)}^{2}} + \frac{2 μΔε}{{(Δt)}^{2}}) +

+ ψ^{n - 1} (j) (- \frac{με}{{(Δt)}^{2}} - \frac{μΔε}{{(Δt)}^{2}}) + ψ^{n} (j - 1) (\frac{1}{{(Δz)}^{2}})

(16)

In the parametric form, it is possible to highlight the differences in the coefficients. In fact, Eqs. (15) and (16) become respectively

ψ^{n + 1} (j) = ψ^{n} (j + 1) (\frac{b}{a}) + ψ^{n} (j) (\frac{2 a - 2 b}{a}) + ψ^{n - 1} (j) (- 1) + ψ^{n} (j - 1) (\frac{b}{a})

(17)

ψ^{n + 1} (j) = ψ^{n} (j + 1) (\frac{b}{a^{'}}) + ψ^{n} (j) (\frac{2 a^{'} - 2 b}{a'}) + ψ^{n - 1} (j) (- 1) + ψ^{n} (j - 1) (\frac{b}{a^{'}})

(18)

with

a = \frac{με}{{(Δt)}^{2}}

a' = a + \frac{μΔε}{{(Δt)}^{2}}

b = \frac{1}{{(Δz)}^{2}}

(19)

We observe that the generators V_P and I_P of Fig. 3 represent the variation of the coefficients of the solution Eq. (15).

Fig. 4. Finite difference algorithm of the simulated structure.

5. Results

The first simulated structure is the dielectric multilayer structure of Fig. 1. In particular Fig. 5 shows the E_y component evaluated one cell before the multilayer structure, for 15-layer structure of Si and SiO₂ materials, in particular n₁=3.48, d₁=0.11μm, n₂=1.44, d₂=0.26μm. The structure is discretized by 70 domain cells with Δz=4∙10^-8 m, the first five cells and the last five ones are filled by air, Δt =1.33∙10^-16 sec. The input signal is a sinusoidal signal with wavelength 1.523 μm modulated by a Gaussian pulse. We compare in Fig. 6 (20-layer structure with n₁=3.48, d₁=0.1μm, n₂=1.465, d₂=20 nm) and Fig. 7 (20-layer structure with n₁=3.48, d₁=0.1μm, n₂=1.465, d₂=30 nm) the theoretical values of the reflection coefficient with the numerical ones (“FDP” finite difference perturbed solution). We observe that the reflection coefficient of a dielectric multilayer is obtained from

r (s) = \frac{(m_{11} + m_{12} p_{l + 1}) p_{0} - (m_{21} + m_{22} p_{l + 1})}{(m_{11} + m_{12} p_{l + 1}) p_{0} + (m_{21} + m_{22} p_{l + 1})}

(20)

where l is the number of the layers, p_j = n_j cos (θ_j) (j=0, 1, 2… l+1), and m ₁₁, m ₁₂, m ₂₁, m ₂₂ are the elements of the characteristic matrix M=M₁∙M₂…M_j…M_l [25

M. Born and E. Wolf, Principles of Optics , 5th ed. (Pergamon, New York, 1975), pp. 55–62.

]. We then extend the solution of (10) to a bi-dimensional structure: the ASR simulated structure of Fig. 2 is characterized by n_s=3.17, n_c=3.524, d_c=0.11μm, d₁=0.1816 μm, and λ0=1.54 μm. We use 80×80 cells of domain, 11x35 cells for the central slab, 18 cells of the thickness of the first layer, 5 cells of the thickness of the second layer, Δx=Δy=0.01μm, and Δt=3.335∙10^-17sec. In Fig. 8 is shown the time evolution and the contour plot of the Ez-field component. In Fig. 9, we report the comparisons between the simulated reflectivity and the experimental one by varying the thicknesses of the second layer d₂. In this case, we have a good agreement with the measured data [14

T. Saitoh, T. Mukai, and O. Mikami “Theoretical analysis and fabrication of antireflection coatings on laser-diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985). [CrossRef]

Fig. 5. Time evolution of E_y field component.

Fig. 6. Comparison between theoretical and simulated dielectric stack

Fig. 7. Comparison between theoretical and simulated dielectric stack

Fig. 8. Time evolution of normalized Ez field component and contour after 170 time-steps. The simulated structure is drawn on the contour xy-plane.

Fig. 9. Measured and simulated reflectivity of ASR of Fig. 2 for different thickness d2.

5. Conclusion

In this work, we suggest an alternative approach to the numerical simulation of behavior of open waveguides. The reduction of the number of equations to solve is very useful in obtaining low computational costs. The model of the dielectric discontinuities provides accurate EM field solutions close to the real behavior of the optical dielectric array. The proposed algorithm can be also extended to 3D cases and non-linear dielectric layers.

References and links

1.	K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media,” IEEE Trans. Antennas Propag. 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.	Y. Chiou and H. Chang, “Analysis of optical waveguide discontinuities using Padè approximants,” IEEE Photon. Technol. Lett. 9, 964–966 (1997). [CrossRef]
5.	H. Rao, R. Scarmozzino, and R. M. Osgood, “A bi-directional beam propagation method for multiple dielectric interfaces,” IEEE Photon. Technol. Lett. 11, 830–832 (1999). [CrossRef]
6.	Y. Y. Lu and S. H. Wei, “A new iterative bi-directional beam propagation method,” IEEE Photon. Technol. Lett. 14, 1533–1535 (2002). [CrossRef]
7.	N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, “A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method,” IEEE J. Lightwave Technol. 21, 1996–2001 (2003). [CrossRef]
8.	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]
9.	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]
10.	T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures , (IEE Electromagnetic wave series 46, London. 1999), Chap. 2. [CrossRef]
11.	C. G. Someda, Onde elettromagnetiche , (UTET Ed., Torino 1996), Chap.1.
12.	R.-C. Tyan, A. A. Salvekar, H.-Pu Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, “Design, fabrication, and characterization of form-birefringent multilayer polarizing beam splitter,” J. Opt. Soc. Am. A 14, 1627–1636 (1997). [CrossRef]
13.	K. Muro and K. Shiraishy, “Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°,” IEEE J. Lightwave Technol. 16, 127–133 (1998). [CrossRef]
14.	T. Saitoh, T. Mukai, and O. Mikami “Theoretical analysis and fabrication of antireflection coatings on laser-diode facets,” IEEE J. Lightwave Technol. LT-3, 288–293 (1985). [CrossRef]
15.	N.-N. Feng, G.-R. Zhou, and W.-P. Huang, “Space mapping technique for design optimization of antireflection coatings in photonic devices,” IEEE J. Lightwave Technol. 21, 281–285 (2003). [CrossRef]
16.	N.-N. Feng and W.-P. Huang, “An efficient computation scheme for time-domain reflection at optical waveguide discontinuities,” IEEE Photon. Technol. Lett. 16, 461–463 (2004). [CrossRef]
17.	N.-N. Feng and W.-P. Huang, “Time-domain reflective beam propagation method,” IEEE J. of Quantum Electron. 30, 1542–1552 (1994).
18.	P. Zorabedian, “Axial-mode instability in tunable external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 40, 778–783 (2004).
19.	A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, “Dynamic instabilities in master oscillator power amplifiers semiconductors,” IEEE J. Quantum Electron. 34, 166–170 (1998). [CrossRef]
20.	N. Marcuvitz and J. Schwinger, “On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides,” J. Appl. Phys. 22, 806–820 (1951). [CrossRef]
21.	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]
22.	A. Yariv, Quantum Electron., 3rd ed. (John Wiley & Sons, Canada, 1989), Chap. 22.
23.	A. Taflove and S. C. Hagness, Computational Electrodynamic: the Finite-difference time-domain method , 2nd. ed. (Arthec House Publishers, London 2000), Chaps. 2, 4, and 7.
24.	G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. 23, 377–382 (1981). [CrossRef]
25.	M. Born and E. Wolf, Principles of Optics , 5th ed. (Pergamon, New York, 1975), pp. 55–62.

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

ToC Category:
Physical Optics

History
Original Manuscript: November 15, 2005
Manuscript Accepted: February 3, 2006
Published: March 6, 2006

Citation
A. Massaro and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express 14, 2027-2036 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-5-2027

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References

K. S. Yee, "Numerical solution of initial boundary value problems involving maxwell’s equation in isotropic media," IEEE Trans. Antennas Propag. 14, 302-307 (1966).
W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 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 Technol. 46, 2463-2475 (1998). [CrossRef]
Y. Chiou and H. Chang, "Analysis of optical waveguide discontinuities using Padè approximants," IEEE Photon. Technol. Lett. 9, 964-966 (1997). [CrossRef]
H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999). [CrossRef]
Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002). [CrossRef]
N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003). [CrossRef]
M. Couture, "On the numerical solution of fields in cavities using the magnetic Hertz vector," IEEE Trans. Microwave Theory and Technol. 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), Chap. 2. [CrossRef]
C. G. Someda, Onde elettromagnetiche, (UTET Ed., Torino 1996), Chap.1.
R.-C. Tyan, A. A. Salvekar, H.-Pu Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, "Design, fabrication, and characterization of form-birefringent multilayer polarizing beam splitter," J. Opt. Soc. Am. A 14, 1627-1636 (1997). [CrossRef]
K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998). [CrossRef]
T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985). [CrossRef]
N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003). [CrossRef]
N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004). [CrossRef]
N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).
P. Zorabedian, "Axial-mode instability in tunable external-cavity semiconductor lasers," IEEE J. Quantum Electron. 40, 778-783 (2004).
A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998). [CrossRef]
N. Marcuvitz and J. Schwinger, "On the representation of the electric and magnetic field produced by currents and discontinuities in waveguides," J. Appl. Phys. 22, 806-820 (1951). [CrossRef]
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]
A. Yariv, Quantum Electron., 3rd ed. (John Wiley & Sons, Canada, 1989), Chap. 22.
A. Taflove, and S. C. Hagness, Computational Electrodynamic: the Finite-difference time-domain method, 2nd. ed. (Arthec House Publishers, London 2000), Chaps. 2, 4, and 7.
G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981). [CrossRef]
M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 55-62.

Bossert, D. J.
Chang, H.
Chiou, Y.
Couture, M.
De Castro, B.
Egan, A.
Fang, D.-G.
Feng, N.-N.
Frateschi, N. C.
Fujii, M.
Hoefer, W J. R.
Hoefer, W. J. R.
Huang, W.-P.
Indik, R. A.
Lu, Y. Y.
Marcuvitz, N.
McInerney, J. G.
Mikami, O.
Moloney, J. V.
Mukai, T.
Mur, G.
Muro, K.
Nikoskinen, K. I.
Ning, C. Z.
Osgood, R. M.
Rao, H.
Rubens, A.
Saitoh, T.
Salvekar, A. A.
Scarmozzino, R.
Schwinger, J.
Shiraishy, K.
Tyan, R.-C.
Wei, S. H.
Wright, M. W.
Xu, C.
Yee, K. S.
Zhou, G.-R.
Zorabedian, P.

IEEE J. Lightwave Technol.

N.-N. Feng, C. Xu, W.-P. Huang, and D.-G. Fang, "A new preconditioner based paraxial approximation for stable and efficient reflective beam propagation method," IEEE J. Lightwave Technol. 21, 1996-2001 (2003). [CrossRef]
K. Muro and K. Shiraishy, "Poly-Si/SiO2 laminated walk-off polarizer having a beam-splitting angle of more than 20°," IEEE J. Lightwave Technol. 16, 127-133 (1998). [CrossRef]
T. Saitoh, T. Mukai, and O. Mikami "Theoretical analysis and fabrication of antireflection coatings on laser-diode facets," IEEE J. Lightwave Technol. LT-3, 288-293 (1985). [CrossRef]
N.-N. Feng, G.-R. Zhou, and W.-P. Huang, "Space mapping technique for design optimization of antireflection coatings in photonic devices," IEEE J. Lightwave Technol. 21, 281-285 (2003). [CrossRef]

IEEE J. of Quantum Electron.

N.-N. Feng and W.-P. Huang, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

IEEE J. Quantum Electron.

P. Zorabedian, "Axial-mode instability in tunable external-cavity semiconductor lasers," IEEE J. Quantum Electron. 40, 778-783 (2004).
A. Egan, C. Z. Ning, J. V. Moloney, R. A. Indik, M. W. Wright, D. J. Bossert, and J. G. McInerney, "Dynamic instabilities in master oscillator power amplifiers semiconductors," IEEE J. Quantum Electron. 34, 166-170 (1998). [CrossRef]
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]

IEEE Photon. Technol. Lett.

N.-N. Feng and W.-P. Huang, "An efficient computation scheme for time-domain reflection at optical waveguide discontinuities," IEEE Photon. Technol. Lett. 16, 461-463 (2004). [CrossRef]
Y. Chiou and H. Chang, "Analysis of optical waveguide discontinuities using Padè approximants," IEEE Photon. Technol. Lett. 9, 964-966 (1997). [CrossRef]
H. Rao, R. Scarmozzino, and R. M. Osgood, "A bi-directional beam propagation method for multiple dielectric interfaces," IEEE Photon. Technol. Lett. 11, 830-832 (1999). [CrossRef]
Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002). [CrossRef]

IEEE Trans. Antennas Propag.

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

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

IEEE Trans. Microwave Theory and Technol.

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

IEEE Trans. Microwave Theory Technol.

W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 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 Technol. 46, 2463-2475 (1998). [CrossRef]

J. Appl. Phys.

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

J. Opt. Soc. Am. A

R.-C. Tyan, A. A. Salvekar, H.-Pu Chou, C.-C. Cheng, A. Scherer, P.-C. Sun, F. Xu, and Y. Fainman, "Design, fabrication, and characterization of form-birefringent multilayer polarizing beam splitter," J. Opt. Soc. Am. A 14, 1627-1636 (1997). [CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), pp. 55-62.
A. Yariv, Quantum Electron., 3rd ed. (John Wiley & Sons, Canada, 1989), Chap. 22.
A. Taflove, and S. C. Hagness, Computational Electrodynamic: the Finite-difference time-domain method, 2nd. ed. (Arthec House Publishers, London 2000), Chaps. 2, 4, and 7.
T. Rozzi and M. Farina, Advanced electromagnetic analysis of passive and active planar structures, (IEE Electromagnetic wave series 46, London. 1999), Chap. 2. [CrossRef]
C. G. Someda, Onde elettromagnetiche, (UTET Ed., Torino 1996), Chap.1.

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