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.

© 2006 Optical Society of America

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], transmission line matrix (TLM) [2], and wavelet- Galerkin [3] 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]–[7] 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]–[10] starting from the Helmholtz scalar equation. [10]–[11]. 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]–[10]. 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]–[13] ,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]–[17] (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]–[19]. In the frequency domain, the transmission and reflection properties of dielectric discontinuities may be derived by means of an equivalent circuit [20], [21], [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.

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Fig. 2. General ASR optical multilayer structure.

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

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2. Hertzian field produced by dielectric discontinuities

The Hertzian electric and magnetic vectors are [8]–[10] in rectangular coordinates represented by

Πe̅=aψe(x,y,z,t)
Πh̅=aψh(x,y,z,t)

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

E̅=Π̅eεμ2t2Πe̅μt(×Πh̅)
H̅=Πh̅εμ2t2Πh̅+εt(×Πe̅)

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

Ex=Ae+μtzψh
Ey=Be+εμt2ψe
Ez=Ceμtxψh
Hx=Ahεtzψe
Hy=Bhεμ2tψh
Hz=Ch+εtxψe

where

Ae,h=x2ψe,h+yxψe,h+zxψe,h
Be,h=xyψe,h+2yψe,h+zyψe,h
Ce,h=xzψe,h+yzψe,h+2zψe,h

Ψe, h(x, y, z, and t) represents the solution of the homogeneous wave equation for a non-dissipative medium [7], [8]

2ψe,h(x,y,z,t)με2ψe,h(x,y,z,t)t2=0

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] (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]

2ψe,h(x,y,z,t)με2ψe,h(x,y,z,t)t2μ2Ppert(x,y,z,t)t2=0

where

P(x,y,z,t)=Δε(x,y,z,t)ψe,h(x,y,z,t)

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

Δε=εi+1εii=1,mlayer.

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]

2ψe,h(x,y,z,t)με2ψe,h(x,y,z,t)t2=0.

Or the inhomogeneous scalar equation

2ψe,h(x,y,z,t)με2ψe,h(x,y,z,t)t2μ2Ppert(x,y,z,t)t2=0

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

ψn(i,j)x=ψn(i+12,j)ψn(i12,j)Δx+O(Δ2x)
ψn(i,j)y=ψn(i,j+12)ψn(i,j12)Δy+O(Δ2y)
ψn(i,j)t=ψn+12(i,j)ψn12(i,j)Δt+O(Δ2t)

and for the second order by

2ψn(i,j)x2ψn(i+1,j)2ψn(i,j)+ψn(i1,j)Δ2x
2ψn(i,j)y2ψn(i,j+1)2ψn(i,j)+ψn(i,j1)Δ2y
2ψn(i,j)t2ψn+1(i,j)2ψn(i,j)+ψn1(i,j)Δ2t

where i and j are the geometrical coordinates and n represents the time. We use for the implementation of Eq. (9) and Eq. (10) a matrix structure

position(ABCψ0(j=1)ψ0(j=0)ψ0(j=+1)ABCABCψ1(j=0)ABCABCABCABCABCABCABCABCψn(j=0)ABC)nxnumberofnodestimesteps

Where the first and the last column represents the ABCs Mur boundary condition [23], [24] that considers the wave equation:

2ψe,hxt1c2ψe,ht2+c22ψe,hy2=0x=0
2ψe,hxt1c2ψe,ht2+c22ψe,hy2=0x=h
2ψe,hyt1c2ψe,ht2+c22ψe,hx2=0y=0
2ψe,hyt1c2ψe,ht2+c22ψe,hx2=0y=h

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 Ψe, 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:

ψn+1(j)(με(Δt)2)=ψn(j+1)(1(Δz)2)+ψn(j)(2με(Δt)22(Δz)2)+
+ψn1(j)(με(Δt)2)+ψn(j1)(1(Δz)2)

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

ψn+1(j)(με(Δt)2+μΔε(Δt)2)=ψn(j+1)(1(Δz)2)+ψn(j)(2με(Δt)22(Δz)2+2μΔε(Δt)2)+
+ψn1(j)(με(Δt)2μΔε(Δt)2)+ψn(j1)(1(Δz)2)

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

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

with

a=με(Δt)2
a'=a+μΔε(Δt)2
b=1(Δz)2

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

 

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

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5. Results

The first simulated structure is the dielectric multilayer structure of Fig. 1. In particular Fig. 5 shows the Ey 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)=(m11+m12pl+1)p0(m21+m22pl+1)(m11+m12pl+1)p0+(m21+m22pl+1)

where l is the number of the layers, pj = njcos (θj) (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…Mj…Ml [25]. We then extend the solution of (10) to a bi-dimensional structure: the ASR simulated structure of Fig. 2 is characterized by ns=3.17, nc=3.524, dc=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].

 

Fig. 5. Time evolution of Ey field component.

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Fig. 6. Comparison between theoretical and simulated dielectric stack

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Fig. 7. Comparison between theoretical and simulated dielectric stack

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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.

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

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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., Torino1996), 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, London2000), 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.

References

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  1. 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).
  2. W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 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 Technol. 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 Technol. 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.

2004 (2)

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]

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

2003 (2)

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, 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]

2002 (1)

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

1999 (1)

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]

1998 (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 Technol. 46, 2463-2475 (1998).
[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]

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]

1997 (2)

1994 (1)

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

1991 (1)

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]

1987 (1)

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]

1986 (1)

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]

1985 (2)

W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 33, 882-893 (1985).
[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]

1981 (1)

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]

1966 (1)

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).

1951 (1)

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]

Bossert, D. J.

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]

Chang, H.

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

Chiou, Y.

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

Couture, M.

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]

De Castro, B.

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]

Egan, A.

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]

Fang, D.-G.

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]

Feng, N.-N.

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, 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]

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, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

Frateschi, N. C.

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]

Fujii, M.

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]

Hoefer, W J. R.

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]

Hoefer, W. J. R.

W. J. R. Hoefer, "The transmission-line matrix method-Theory and applications," IEEE Trans. Microwave Theory Technol. 33, 882-893 (1985).
[CrossRef]

Huang, W.-P.

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, 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]

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, "Time-domain reflective beam propagation method," IEEE J. of Quantum Electron. 30, 1542-1552 (1994).

Indik, R. A.

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]

Lu, Y. Y.

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Marcuvitz, N.

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]

McInerney, J. G.

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]

Mikami, O.

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]

Moloney, J. V.

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]

Mukai, T.

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]

Mur, G.

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]

Muro, K.

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]

Nikoskinen, K. I.

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]

Ning, C. Z.

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]

Osgood, R. M.

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]

Rao, H.

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]

Rubens, A.

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]

Saitoh, T.

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]

Salvekar, A. A.

Scarmozzino, R.

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]

Schwinger, J.

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]

Shiraishy, K.

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]

Tyan, R.-C.

Wei, S. H.

Y. Y. Lu and S. H. Wei, "A new iterative bi-directional beam propagation method," IEEE Photon. Technol. Lett. 14, 1533-1535 (2002).
[CrossRef]

Wright, M. W.

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]

Xu, C.

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]

Yee, K. S.

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).

Zhou, G.-R.

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]

Zorabedian, P.

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

IEEE J. Lightwave Technol. (4)

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. (1)

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. (3)

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. (4)

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. (2)

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. (1)

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. (1)

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. (2)

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. (1)

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 (1)

Other (5)

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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Figures (9)

Fig. 1.
Fig. 1.

General optical multilayer structure.

Fig. 2.
Fig. 2.

General ASR optical multilayer structure.

Fig. 3.
Fig. 3.

Equivalent circuit of the perturbed solution in the multilayered structure.

Fig. 4.
Fig. 4.

Finite difference algorithm of the simulated structure.

Fig. 5.
Fig. 5.

Time evolution of Ey field component.

Fig. 6.
Fig. 6.

Comparison between theoretical and simulated dielectric stack

Fig. 7.
Fig. 7.

Comparison between theoretical and simulated dielectric stack

Fig. 8.
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.
Fig. 9.

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

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

Π e ̅ = a ψ e ( x , y , z , t )
Π h ̅ = a ψ h ( x , y , z , t )
E ̅ = Π ̅ e εμ 2 t 2 Π e ̅ μ t ( × Π h ̅ )
H ̅ = Π h ̅ εμ 2 t 2 Π h ̅ + ε t ( × Π e ̅ )
E x = A e + μ t z ψ h
E y = B e + εμ t 2 ψ e
E z = C e μ t x ψ h
H x = A h ε t z ψ e
H y = B h εμ 2 t ψ h
H z = C h + ε t x ψ e
A e , h = x 2 ψ e , h + yx ψ e , h + zx ψ e , h
B e , h = xy ψ e , h + 2 y ψ e , h + zy ψ e , h
C e , h = xz ψ e , h + yz ψ e , h + 2 z ψ e , h
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 = 0
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 μ 2 P pert ( x , y , z , t ) t 2 = 0
P ( x , y , z , t ) = Δ ε ( x , y , z , t ) ψ e , h ( x , y , z , t )
Δε = ε i + 1 ε i i = 1 , m layer .
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 = 0 .
2 ψ e , h ( x , y , z , t ) με 2 ψ e , h ( x , y , z , t ) t 2 μ 2 P pert ( x , y , z , t ) t 2 = 0
ψ n ( i , j ) x = ψ n ( i + 1 2 , j ) ψ n ( i 1 2 , j ) Δ x + O ( Δ 2 x )
ψ n ( i , j ) y = ψ n ( i , j + 1 2 ) ψ n ( i , j 1 2 ) Δy + O ( Δ 2 y )
ψ n ( i , j ) t = ψ n + 1 2 ( i , j ) ψ n 1 2 ( i , j ) Δt + O ( Δ 2 t )
2 ψ n ( i , j ) x 2 ψ n ( i + 1 , j ) 2 ψ n ( i , j ) + ψ n ( i 1 , j ) Δ 2 x
2 ψ n ( i , j ) y 2 ψ n ( i , j + 1 ) 2 ψ n ( i , j ) + ψ n ( i , j 1 ) Δ 2 y
2 ψ n ( i , j ) t 2 ψ n + 1 ( i , j ) 2 ψ n ( i , j ) + ψ n 1 ( i , j ) Δ 2 t
position ( 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 ) n x number of nodes time s teps
2 ψ e , h x t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h y 2 = 0 x = 0
2 ψ e , h x t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h y 2 = 0 x = h
2 ψ e , h y t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h x 2 = 0 y = 0
2 ψ e , h y t 1 c 2 ψ e , h t 2 + c 2 2 ψ e , h x 2 = 0 y = h
ψ n + 1 ( j ) ( με ( Δt ) 2 ) = ψ n ( j + 1 ) ( 1 ( Δz ) 2 ) + ψ n ( j ) ( 2 με ( Δt ) 2 2 ( Δz ) 2 ) +
+ ψ n 1 ( j ) ( με ( Δt ) 2 ) + ψ n ( j 1 ) ( 1 ( Δz ) 2 )
ψ n + 1 ( j ) ( με ( Δt ) 2 + μΔε ( Δt ) 2 ) = ψ n ( j + 1 ) ( 1 ( Δz ) 2 ) + ψ n ( j ) ( 2 με ( Δt ) 2 2 ( Δz ) 2 + 2 μΔε ( Δt ) 2 ) +
+ ψ n 1 ( j ) ( με ( Δt ) 2 μΔε ( Δt ) 2 ) + ψ n ( j 1 ) ( 1 ( Δz ) 2 )
ψ n + 1 ( j ) = ψ n ( j + 1 ) ( b a ) + ψ n ( j ) ( 2 a 2 b a ) + ψ n 1 ( j ) ( 1 ) + ψ n ( j 1 ) ( b a )
ψ n + 1 ( j ) = ψ n ( j + 1 ) ( b a ' ) + ψ n ( j ) ( 2 a ' 2 b a ' ) + ψ n 1 ( j ) ( 1 ) + ψ n ( j 1 ) ( b a ' )
a = με ( Δt ) 2
a ' = a + μΔε ( Δt ) 2
b = 1 ( Δz ) 2
r ( s ) = ( 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 )

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