## Dispersion and stability analysis for a finite difference beam propagation method

Optics Express, Vol. 16, Issue 12, pp. 8755-8768 (2008)

http://dx.doi.org/10.1364/OE.16.008755

Acrobat PDF (807 KB)

### Abstract

Applying continuous and discrete transformation techniques, new analytical expressions to calculate dispersion and stability of a Runge-Kutta Finite Difference Beam Propagation Method (RK-FDBPM) are obtained. These expressions give immediate insight about the discretization errors introduced by the numerical method in the plane-wave spectrum domain. From these expressions a novel strategy to adequately set the mesh steps sizes of the RK-FDBPM is presented. Assessment of the method is performed by studying the propagation in several linear and nonlinear photonic devices for different spatial discretizations.

© 2008 Optical Society of America

## 1. Introduction

1. A. Koster, E. Cassan, S. Laval, L. Vivien, and D. Pascal, “Ultracompact splitter for submicrometer silicon-on-insulator rib waveguides,” J. Opt. Soc. Am. A **21**, 2180–2185 (2004). [CrossRef]

2. Z. Chen, Z. Li, and B. Li, “A 2-to-4 decoder switch in SiGe/Si multimode interference,” Opt. Express **14**, 2671–2678 (2006). [CrossRef] [PubMed]

3. J. Yamauchi, K. Sumida, and H. Nakano, “Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method,” IEEE Photon. Technol. Lett. **18**, 1858–1860 (2006). [CrossRef]

4. D. Dai, J. He, and S. He, “Compact silicon-on-insulator-based multimode interference coupler with bilevel taper structure,” Appl. Opt. **44**, 5036–5041 (2005). [CrossRef] [PubMed]

1. A. Koster, E. Cassan, S. Laval, L. Vivien, and D. Pascal, “Ultracompact splitter for submicrometer silicon-on-insulator rib waveguides,” J. Opt. Soc. Am. A **21**, 2180–2185 (2004). [CrossRef]

2. Z. Chen, Z. Li, and B. Li, “A 2-to-4 decoder switch in SiGe/Si multimode interference,” Opt. Express **14**, 2671–2678 (2006). [CrossRef] [PubMed]

6. M. Takenaka and Y. Nakano, “Multimode interference bistable laser diode,” IEEE Photon. Technol. Lett. **15**, 1035–1037, (2003). [CrossRef]

7. M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, “Integrable multimode interference distributed Bragg reflector laser all-optical flip flop,” IEEE Photon. Technol. Lett. **18**, 1421–1423 (2006). [CrossRef]

3. J. Yamauchi, K. Sumida, and H. Nakano, “Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method,” IEEE Photon. Technol. Lett. **18**, 1858–1860 (2006). [CrossRef]

6. M. Takenaka and Y. Nakano, “Multimode interference bistable laser diode,” IEEE Photon. Technol. Lett. **15**, 1035–1037, (2003). [CrossRef]

8. N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, “Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods,” IEEE J. Lightwave Technol. **24**1834–1842 (2006). [CrossRef]

6. M. Takenaka and Y. Nakano, “Multimode interference bistable laser diode,” IEEE Photon. Technol. Lett. **15**, 1035–1037, (2003). [CrossRef]

7. M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, “Integrable multimode interference distributed Bragg reflector laser all-optical flip flop,” IEEE Photon. Technol. Lett. **18**, 1421–1423 (2006). [CrossRef]

10. C. Ma and E. Van Keuren, “A three-dimensional wide-angle BPM for optical waveguide structures,” Opt. Express **15**, 402–407 (2007). [CrossRef] [PubMed]

## 2. RK-FDBPM method

15. W. P. Huang and C. L. Xu
, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

*ē*=

*e*+

_{x}xô*e*is the slowly varying complex envelope of the electric field

_{y}ŷ*ε*, which can be written as

*k*is the vacuum propagation constant and

_{0}*n*is the BPM reference refractive index.

_{ref}15. W. P. Huang and C. L. Xu
, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

*M*is the semivectorial transversal differential operator

_{uu}*u*and

*v*can indistinctly be the

*x*and

*y*variables and

*n*is the device refractive index, which can be linear or non-linear [13

13. J. de-Oliva-Rubio and I. Molina-Fernández, “Fast semivectorial non-linear finite-difference beampropagation method,” Microwave Opt. Technol. Lett. **40**, 73–77 (2004). [CrossRef]

13. J. de-Oliva-Rubio and I. Molina-Fernández, “Fast semivectorial non-linear finite-difference beampropagation method,” Microwave Opt. Technol. Lett. **40**, 73–77 (2004). [CrossRef]

15. W. P. Huang and C. L. Xu
, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. **29**, 2639–2649 (1993). [CrossRef]

*z*) is a column vector containing either the e

_{x}or e

_{y}field components discretized in the transversal mesh points and

*e*(

^{m,n}_{u}*z*)=

*e*(

_{u}*m*Δ

*x*,

*n*Δ

*y*,

*z*) is the value of the field in the mesh coordinates (

*m*,

*n*), then the

*k*-th element of

_{k}(

*z*)=

*e*(

^{m,n}_{u}*z*), where

*k*=(

*m*-1)

*N*+

_{x}*n*and

*N*is the number of mesh points in the computational window on the

_{x}*x*axis. In other words, vector

13. J. de-Oliva-Rubio and I. Molina-Fernández, “Fast semivectorial non-linear finite-difference beampropagation method,” Microwave Opt. Technol. Lett. **40**, 73–77 (2004). [CrossRef]

16. M. Matsuhara, “A novel beam propagation method based on the Galerkin method,” Electron. Commun. Jpn. Pt. II **73**, 41–47 (1990). [CrossRef]

_{p}contains the discretized field at each numerical propagation step

*z*=

_{p}*p*Δ

*z*i.e.

_{p}=

*p*Δ

*z*) and

*z*

_{p+1}=(

*p*+1)Δ

*z*as a fourth order Taylor series expansion around

*z*=

_{p}*p*Δ

*z*.

## 3. RK-FDBPM dispersion analysis

### 3.1 Dispersion relations for the Fresnel equation in linear and homogenous media

*n*is now a constant. The spatial spectral analysis of any wave front can be carried out using the Fourier transform [17]. In the three-dimensional case the Fourier transform is

18. I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, “Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena,” IEEE Trans. Microwave Theory Tech. **42**, 1079–1085 (1994). [CrossRef]

*D*=0, so

_{u}*x*, the dispersion equation is

### 3.2 Dispersion relations for the RK-FDBPM in linear and homogenous media

*e*, which is sampled in the three spatial dimensions, as a discrete sequence, its three-dimensional Z transform can be defined as

^{m,n,p}_{u}_{x}, Ω

_{y}y Ω

_{z}are the corresponding angular frequencies expressed in rad/sample and A

_{z}is the attenuation function in Np/sample. Contrary to its continuous counterpart, in this case the Z variable has been allowed to become complex to take into account possible numerical losses of the discretization algorithm. Once again, the relation between the frequencies of the continuous and discrete domains is established by means of the sampling periods

*RK*differentiates the numerical method propagation constant from that of the continuous equation. Using Eq. (13) to Z transform the difference equation of the Runge-Kutta algorithm in Eq.(6) and substituting Eq. (14) and Eq. (15) into the transformed equation, it is possible to calculate the dispersion relations of the RK-FDBPM as [14]

*N*and

_{x}*N*are the number of samples of the computational window in the

_{y}*x*and

*y*direction, respectively. The size of

*N*×

_{x}N_{y}*N*, for linear homogeneous media it is symmetric and constant, meaning that all the elements in each subdiagonal are identical, so

_{x}N_{y}*k*,

*k*+

*p*)=

*k*,

*k*-

*p*), ∀

*k*∈[1,

*N*]. Each coefficient

_{x}N_{y}*M̿*)

_{T}^{n}, where

*M̿*

^{n}_{T,0}represents the diagonal elements.

*x*-invariant case, the obtained dispersion equation is,

## 4. Selecting RK-FDBPM discretization parameters

*L*can be calculated as

*3.1*can be readily applied to the wide angle continuous partial derivatives equation to calculate the continuous dispersion equation. Also, provided wide angle FDBPM algorithms have the form [12

12. S. Sujecki “Wide-angle, finite-difference beam propagation in oblique coordinate system,” J. Opt. Soc. Am. A **25**, 138–145 (2008). [CrossRef]

*3.2*. In fact, Eq. (20) has the same form as the ones obtained from paraxial Crank-Nicolson or ADI FDBPM, and the proposed dispersion characterization technique has already been applied in both cases [14].

## 5. Results

### 5.1. 2D linear devices

_{f}=1.571 in the film, n

_{s}=1.57 in the substrate and n

_{c}=1.569 in the cover. The film’s width is 10 µm, the slab propagation axis is tilted θ=10° from

*z*axis and the excitation wavelength is λ

_{0}=633 nm.

_{eff}) at λ

_{0}. Mesh steps are Δy=0.5 µm, Δz=0.1 λ

_{0}in the case of (a)-mesh, Δy=0.1 µm, Δz=0.1 λ

_{0}for (b)-mesh, and Δy=0.025 µm, Δz=0.0125 λ

_{0}in the case of (c)-mesh.

*z*=

*0*is the tilted slab fundamental mode, so the excitation spectrum is centered at β

_{y}=-k

_{0}

*n*sinθ=-2.7 rad/µm, and 90% of the spatial spectrum energy is located in the range from -3 rad/µm to -2.47 rad/µm, so it will be considered a spatial bandwidth of -3 rad/µm. For this βy value, one can see in Fig. 1 how the phase error is not much smaller than 2π when the (a)-mesh is used. In the case of the (b)-mesh, the phase error is approximately 0.085×2π, and for the (c)-mesh it is roughly 0.01×2π. In the latter two cases, the numerical dispersion is considered to be sufficiently small.

_{eff}_{0}excitation is shown. Fig. 2(a) shows the solution obtained using the (a)-mesh. It can be seen how the high numerical dispersion makes the BPM method unable to properly characterize the mode guidance along the structure. Fig. 2(b) shows both the (b)-mesh and the (c)-mesh solutions, which are indistinguishable. It can be seen how in these cases the numerical dispersion remains limited and the achieved solution is sufficiently accurate.

_{21}calculation, the cross-section overlap integral of the propagating field and the fundamental mode has been solved at each propagation step, as defined in [19].

*z*evolution of S

_{21}, where the curve’s ripple is caused by the lateral misalignment between the exact theoretical position of the slab longitudinal axis and the position of the transverse mesh points at each propagation step. In Fig. 3(a) it can be seen how the S

_{21}modulus remains near unity for the (b)-mesh and (c)-mesh. However, for the (a)-mesh, not far from the excitation the field profile begins to differ greatly from the fundamental mode, so the S

_{21}modulus dramatically decreases. In Fig. 3(b) one can see how the S

_{21}phase error agrees well with the values calculated in the dispersion analysis.

### 5.2. 3D linear devices

_{1}=1.4549 embedded in a medium with index n

_{2}=1.4440. At the design wavelength (λ

_{0}=1550nm) the multi mode section supports 10 guided modes. The device is designed to work as a 120° 2×3 coupler with S parameters as in [20

20. I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, “Planar lightwave circuit six-port technique for optical measurements and characterizations,” IEEE J. Lightwave Technol. **23**, 2148–2157 (2005). [CrossRef]

*β*,

_{x}*β*∈[-1,1](rad/µm). Therefore, this is the band in which the RK-FDBPM phase error is intended to be limited, considering a total propagation length L=3570 µm.

_{y}20. I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, “Planar lightwave circuit six-port technique for optical measurements and characterizations,” IEEE J. Lightwave Technol. **23**, 2148–2157 (2005). [CrossRef]

### 5.3. Non-linear devices

_{0}and TM

_{0}modes in a non-linear straight slab. The slab’s refractive index profile is the same as that of the linear slab analyzed in section 5.

*1*, but in this case the substrate index has an additional non-linear component [13

**40**, 73–77 (2004). [CrossRef]

21. K. Hayata, A. Misawa, and M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems,” J Opt Soc Am B **7**, 1268–1280 (1990). [CrossRef]

*n*

^{2}

_{NL}=

*n*

^{2}

_{s}+

*α*(|

_{s}*e*|

_{x}^{2}+|

*e*|

_{y}^{2}+|

*e*|

_{z}^{2}), and the non-linear coefficien0t is

*α*=10

_{s}^{-10}

*V*

^{2}/

*m*

^{2}. The guiding structure is excited simultaneously with the linear slab TE

_{0}and TM

_{0}modes at 633nm wavelength. The excitation power is 2.92 mW/mm, equally divided among both modes. The propagation length is

*L*=2400

*λ*

_{0}.

*y*=0.5

*µm*and Δ

*z*=8

*λ*

_{0}. At the upper band end (0.3 rad/µm), the normalized phase error is 1.5×10

^{-3}, so from previous considerations, negligible numerical phase dispersion is expected. However, in the device simulation results shown in Fig. 8 for this specific mesh size, a couple of solitons generated at z=400λ

_{0}and z=1200λ

_{0}are observed. These propagate in the non-linear substrate with a propagation angle smaller than 1,5° from the longitudinal axis. It is obvious that this behavior will make the plane-wave spectrum to be shifted in frequency, making it now be centered at 0.35 rad/µm, and thus its bandwidth is shifted up to 0.65 rad/µm.

*y*=0.25µm and Δ

*z*=4

*λ*

_{0}. However, the corresponding numerical attenuation curve (calculated as the real part of Eq. (17) and shown in blue line in Fig. 7(b)) is positive in the higher frequencies, meaning that the numerical technique is unstable for this mesh setup. In order to encourage the finite difference technique’s stability, the propagation step is reduced to Δ

*z*=2.5

*λ*

_{0}, resulting in the Fig. 7(b) red curve, which still yields instability problems. Finally, with Δ

*z*=2

*λ*

_{0}the green curve in Fig. 7(b) is obtained which, being negative for all frequencies, ensures the RK-FDBPM’s stability.

^{-3}, so numerical dispersion will not affect simulation results. To prove this assumption, a third finer mesh was used. For this case, mesh sizes were set to Δ

*y*=0.125

*µm*and Δ

*z*=

*λ*

_{0}/2, which also ensure stability. The green curve in Fig. 7(a) is the phase error for this third mesh setup, showing a normalized value of 1×10

^{-3}at 0.6 rad/µm.

_{0}for each of the proposed discretization meshes are plotted in Fig. 9. It can be seen that the simulation results obtained with the two last mesh setups are formally identical.

*y*=0 µm corresponds to the field guided by the linear slab; the pulse located at

*y*=- 35 µm comes from the first soliton emitted at approximately

*z*=400 λ

_{0}(as seen in Fig. 8); while the pulse located around

*y*=- 20 µm corresponds to the second soliton formed at

*z*=1200 λ

_{0}. The first pulse is not affected by numerical dispersion, as its plane-wave spectrum is centered at 0 where, as shown by Fig.7(a), numerical dispersion is negligible. This is the reason why all the results in Fig. 9 are almost the same for this pulse. On the other hand, the plane-wave spectrum of the two solitons is passband-type as a consequence of their propagation angle, but as their propagation angle is similar, they will suffer from the same amount of dispersion phase error per unit length. However, as the first soliton formed at

*z*=400 λ

_{0}suffers an accumulated phase error which is much higher than the second one generated at

*z*=1200 λ

_{0}, the total error for the first pulse is much more sensitive to mesh size settings. This explains why the error in Fig. 9 simulations with the coarser discretization mesh is higher for the first emitted soliton than for the second one.

^{-4}dB, so it can be stated that losses do not affect the simulation results.

## 4. Conclusions

## Acknowledgments

## References and Links

1. | A. Koster, E. Cassan, S. Laval, L. Vivien, and D. Pascal, “Ultracompact splitter for submicrometer silicon-on-insulator rib waveguides,” J. Opt. Soc. Am. A |

2. | Z. Chen, Z. Li, and B. Li, “A 2-to-4 decoder switch in SiGe/Si multimode interference,” Opt. Express |

3. | J. Yamauchi, K. Sumida, and H. Nakano, “Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method,” IEEE Photon. Technol. Lett. |

4. | D. Dai, J. He, and S. He, “Compact silicon-on-insulator-based multimode interference coupler with bilevel taper structure,” Appl. Opt. |

5. | S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed “Optically switched arrayed waveguide gratings using phase modulation,” IEEE J. Lightwave Technol. |

6. | M. Takenaka and Y. Nakano, “Multimode interference bistable laser diode,” IEEE Photon. Technol. Lett. |

7. | M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, “Integrable multimode interference distributed Bragg reflector laser all-optical flip flop,” IEEE Photon. Technol. Lett. |

8. | N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, “Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods,” IEEE J. Lightwave Technol. |

9. | M. Takenaka and Y. Nakano, “Simulation of all-optical flip-flops based on bistable laser diodes with nonlinear couplers,” in Proceedings of the 4 |

10. | C. Ma and E. Van Keuren, “A three-dimensional wide-angle BPM for optical waveguide structures,” Opt. Express |

11. | 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,” IEEE Photon. Technol. Lett. |

12. | S. Sujecki “Wide-angle, finite-difference beam propagation in oblique coordinate system,” J. Opt. Soc. Am. A |

13. | J. de-Oliva-Rubio and I. Molina-Fernández, “Fast semivectorial non-linear finite-difference beampropagation method,” Microwave Opt. Technol. Lett. |

14. | J. de-Oliva-Rubio, |

15. | W. P. Huang and C. L. Xu
, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. |

16. | M. Matsuhara, “A novel beam propagation method based on the Galerkin method,” Electron. Commun. Jpn. Pt. II |

17. | B. E. A. Saleh and M. C. Teich, |

18. | I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, “Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena,” IEEE Trans. Microwave Theory Tech. |

19. | C. Vassallo, |

20. | I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, “Planar lightwave circuit six-port technique for optical measurements and characterizations,” IEEE J. Lightwave Technol. |

21. | K. Hayata, A. Misawa, and M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems,” J Opt Soc Am B |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(130.3120) Integrated optics : Integrated optics devices

(130.4310) Integrated optics : Nonlinear

(220.2560) Optical design and fabrication : Propagating methods

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: April 2, 2008

Revised Manuscript: April 17, 2008

Manuscript Accepted: April 18, 2008

Published: May 30, 2008

**Citation**

J. de-Oliva-Rubio, I. Molina-Fernández, and R. Godoy-Rubio, "Dispersion and stability analysis for a finite difference beam propagation method," Opt. Express **16**, 8755-8768 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8755

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### References

- A. Koster, E. Cassan, S. Laval, L. Vivien, and D. Pascal, "Ultracompact splitter for submicrometer silicon-on-insulator rib waveguides," J. Opt. Soc. Am. A 21, 2180-2185 (2004). [CrossRef]
- Z. Chen, Z. Li, and B. Li, "A 2-to-4 decoder switch in SiGe/Si multimode interference," Opt. Express 14, 2671-2678 (2006). [CrossRef] [PubMed]
- J. Yamauchi, K. Sumida and H. Nakano, "Analysis of a polarization splitter with a multilayer filter using a padé-operator-based power-conserving fourth-order accurate beam-propagation method," IEEE Photon. Technol. Lett. 18, 1858-1860 (2006). [CrossRef]
- D. Dai, J. He, and S. He, "Compact silicon-on-insulator-based multimode interference coupler with bilevel taper structure," Appl. Opt. 44, 5036-5041 (2005). [CrossRef] [PubMed]
- S. T. Lee, C. E. Png, F. Y. Gardes, and G. T. Reed "Optically switched arrayed waveguide gratings using phase modulation," IEEE J. Lightwave Technol. 18, 1858-1860 (2006).
- M. Takenaka and Y. Nakano, "Multimode interference bistable laser diode," IEEE Photon. Technol. Lett. 15, 1035-1037, (2003). [CrossRef]
- M. Raburn, M. Takenaka, K. Takeda, X. Song, J. S. Barton, and Y. Nakano, "Integrable multimode interference distributed Bragg reflector laser all-optical flip flop," IEEE Photon. Technol. Lett. 18, 1421-1423 (2006). [CrossRef]
- N. N. Elkin, A. P. Napartovich, V. N. Troschieva, D. V. Vysotsky, T. Lee, S. C. Hagness, N. Kim, L. Bao, and J.L Mawst, "Antiresonant reflecting optical waveguide-type vertical-cavity surface emitting lasers: Comparison of full-vectorial finite difference time domain and 3D bidirectional beam propagation methods," IEEE J. Lightwave Technol. 24, 1834-1842 (2006). [CrossRef]
- M. Takenaka and Y. Nakano, "Simulation of all-optical flip-flops based on bistable laser diodes with nonlinear couplers," in Proceedings of the 4th Int. Conf. on Numerical Simulation of Optoelectronic Devices (NUSOD ???04), 15-18 (2004).
- C. Ma and E. Van Keuren, "A three-dimensional wide-angle BPM for optical waveguide structures," Opt. Express 15, 402-407 (2007). [CrossRef] [PubMed]
- 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," IEEE Photon. Technol. Lett. 18, 2535-2537 (2006). [CrossRef]
- S. Sujecki "Wide-angle, finite-difference beam propagation in oblique coordinate system," J. Opt. Soc. Am. A 25, 138-145 (2008). [CrossRef]
- J. de-Oliva-Rubio and I. Molina-Fernández, "Fast semivectorial non-linear finite-difference beam-propagation method," Microwave Opt. Technol. Lett. 40,73-77 (2004). [CrossRef]
- J. de-Oliva-Rubio, Desarrollo y validación de técnicas de diferencias finitas para el análisis de dispositivos ópticos lineales y no-lineales, Ph. D. Thesis (in Spanish), ISBN: 84-690-3321-2, (Universidad de Málaga, 2006).
- W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron. 29,2639-2649 (1993). [CrossRef]
- M. Matsuhara, "A novel beam propagation method based on the Galerkin method," Electron. Commun. Jpn. Pt. II 73, 41-47 (1990). [CrossRef]
- B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, 2nd Ed. (John Wiley & Sons, 2007), Chap. 4, pp. 105-112.
- I. Molina Fernández, C. Camacho Peñalosa, and J. I. Ramos, "Application of the two-dimensional Fourier transform to nonlinear wave propagation phenomena," IEEE Trans. Microwave Theory Tech. 42,1079-1085 (1994). [CrossRef]
- C. Vassallo, Optical waveguides concepts, ser. Optical waveguides sciences and technology (Elsevier, Amsterdam, 1991), Chap. 1, pp. 18-26.
- I. Molina-Fernández, J. G. Wangüemert-Pérez, A. Ortega-Moñux, R. G. Bosisio, and KeWu, "Planar lightwave circuit six-port technique for optical measurements and characterizations," IEEE J. Lightwave Technol. 23, 2148-2157 (2005) [CrossRef]
- K. Hayata, A. Misawa, and M. Koshiba, "Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems," J Opt Soc Am B 7, 1268-1280 (1990). [CrossRef]

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