Abstract

We consider the propagation of guided waves in a planar waveguide that has smooth walls except for a finite length segment that has random roughness. Maxwell’s equations are solved in the frequency domain for both TE and TM polarization in 2-D by using modal expansion methods. Obtaining numerical solutions is facilitated by using perfectly matched boundary layers and the R-matrix propagator.

Varying lengths of roughness segments are considered and numerical results are obtained for guided wave propagation losses due to roughness induced scattering. The roughness on each waveguide boundary is numerically generated from an assumed Gaussian power spectrum. The guided waves are excited by a Gaussian beam incident on the waveguide aperture. Considerable numerical effort is given to determine the stability of the algorithm.

© 2001 Optical Society of America

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References

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  1. J. M. Elson and P. Tran, “R-matrix propagator with perfectly matched layers for the study of integrated optical components,” J. Opt. Soc. Am. A 162983–2989 (1999).
    [Crossref]
  2. J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 541711–1715 (1996).
    [Crossref]
  3. J. M. Elson and P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” NATO ASI Series E: Applied Sciences on Photonic Band Gap MaterialsVol. 315 341–354 Crete, Greece June 15–29 (1995).
  4. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 112816–2828 (1993).
    [Crossref]
  5. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 131024–1035 (1996).
    [Crossref]
  6. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114185–200 (1995).
    [Crossref]
  7. J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 127363–379 (1996).
    [Crossref]
  8. D. Marcuse, “Mode conversion caused by surface imperfections in a dielectric slab waveguide,” Bell Sys. Tech. J. 48, 3187–3215 (1969).
  9. D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Sys. Tech. J. 48, 3233–3242 (1969).
  10. F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. and Quantum Elec. 26977–986 (1994).
    [Crossref]
  11. J. P. R. Lacey and F. P. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137282–288 (1990).
  12. K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
    [Crossref]
  13. F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141242–248 (1994).
    [Crossref]
  14. F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 281321–1322 (1992).
    [Crossref]
  15. J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
    [Crossref]
  16. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. 13779–784 (1996).
    [Crossref]
  17. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 131870–1876 (1996).
    [Crossref]
  18. A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
    [Crossref]

2000 (1)

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

1999 (2)

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

J. M. Elson and P. Tran, “R-matrix propagator with perfectly matched layers for the study of integrated optical components,” J. Opt. Soc. Am. A 162983–2989 (1999).
[Crossref]

1996 (5)

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 541711–1715 (1996).
[Crossref]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 131024–1035 (1996).
[Crossref]

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 127363–379 (1996).
[Crossref]

P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. 13779–784 (1996).
[Crossref]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 131870–1876 (1996).
[Crossref]

1995 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114185–200 (1995).
[Crossref]

1994 (2)

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. and Quantum Elec. 26977–986 (1994).
[Crossref]

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141242–248 (1994).
[Crossref]

1993 (1)

1992 (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 281321–1322 (1992).
[Crossref]

1990 (2)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
[Crossref]

J. P. R. Lacey and F. P. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137282–288 (1990).

1969 (2)

D. Marcuse, “Mode conversion caused by surface imperfections in a dielectric slab waveguide,” Bell Sys. Tech. J. 48, 3187–3215 (1969).

D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Sys. Tech. J. 48, 3233–3242 (1969).

Agarwal, A.

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Berenger, J. P.

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 127363–379 (1996).
[Crossref]

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114185–200 (1995).
[Crossref]

Carrasco, S.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Crespo, R. D.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Elson, J. M.

J. M. Elson and P. Tran, “R-matrix propagator with perfectly matched layers for the study of integrated optical components,” J. Opt. Soc. Am. A 162983–2989 (1999).
[Crossref]

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 541711–1715 (1996).
[Crossref]

J. M. Elson and P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” NATO ASI Series E: Applied Sciences on Photonic Band Gap MaterialsVol. 315 341–354 Crete, Greece June 15–29 (1995).

Fernández, S.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Foresi, J.

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Ibáñez, I.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Kimerling, L.

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Lacey, J. P. R.

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. and Quantum Elec. 26977–986 (1994).
[Crossref]

J. P. R. Lacey and F. P. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137282–288 (1990).

Ladouceur, F.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141242–248 (1994).
[Crossref]

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 281321–1322 (1992).
[Crossref]

Lalanne, P.

P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. 13779–784 (1996).
[Crossref]

Lee, K. K.

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Li, L.

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 131024–1035 (1996).
[Crossref]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 131870–1876 (1996).
[Crossref]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 112816–2828 (1993).
[Crossref]

Lim, D. R.

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Love, J. D.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141242–248 (1994).
[Crossref]

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 281321–1322 (1992).
[Crossref]

Luan, H.

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
[Crossref]

Marcuse, D.

D. Marcuse, “Mode conversion caused by surface imperfections in a dielectric slab waveguide,” Bell Sys. Tech. J. 48, 3187–3215 (1969).

D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Sys. Tech. J. 48, 3233–3242 (1969).

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
[Crossref]

Mendez, E.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
[Crossref]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
[Crossref]

Morris, G.

P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. 13779–784 (1996).
[Crossref]

Olivares, J.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Pandavenes, J.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Payne, F. P.

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. and Quantum Elec. 26977–986 (1994).
[Crossref]

J. P. R. Lacey and F. P. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137282–288 (1990).

Rodríguez, J.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Senden, T. J.

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141242–248 (1994).
[Crossref]

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 281321–1322 (1992).
[Crossref]

Tran, P.

J. M. Elson and P. Tran, “R-matrix propagator with perfectly matched layers for the study of integrated optical components,” J. Opt. Soc. Am. A 162983–2989 (1999).
[Crossref]

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 541711–1715 (1996).
[Crossref]

J. M. Elson and P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” NATO ASI Series E: Applied Sciences on Photonic Band Gap MaterialsVol. 315 341–354 Crete, Greece June 15–29 (1995).

Virgós, J.

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Annals of Physics (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. Mendez, “Enhanced backscattering of light from a random grating,” Annals of Physics 203225–307 (1990).
[Crossref]

Appl. Phys. Lett. (1)

K. K. Lee, D. R. Lim, H. Luan, A. Agarwal, J. Foresi, and L. Kimerling, “Effect of size and roughness on light transmission on a Si/SiO2 waveguide: Experiment and model,” Appl. Phys. Lett. 771617–1619 (2000).
[Crossref]

Bell Sys. Tech. J. (2)

D. Marcuse, “Mode conversion caused by surface imperfections in a dielectric slab waveguide,” Bell Sys. Tech. J. 48, 3187–3215 (1969).

D. Marcuse, “Radiation losses of dielectric waveguides in terms of the power spectrum of the wall distortion function,” Bell Sys. Tech. J. 48, 3233–3242 (1969).

Electron. Lett. (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Measurement of surface roughness in buried channel waveguides,” Electron. Lett. 281321–1322 (1992).
[Crossref]

IEE Proc. J. (1)

J. P. R. Lacey and F. P. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. J. 137282–288 (1990).

IEE Proc.-Optoelectron. (1)

F. Ladouceur, J. D. Love, and T. J. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141242–248 (1994).
[Crossref]

J. Comp. Phys. (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 114185–200 (1995).
[Crossref]

J. P. Berenger, “Three-dimensional perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. 127363–379 (1996).
[Crossref]

J. Opt. Soc. Am. (3)

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. 131024–1035 (1996).
[Crossref]

P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. 13779–784 (1996).
[Crossref]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. 131870–1876 (1996).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. and Quantum Elec. (1)

F. P. Payne and J. P. R. Lacey, “A theoretical analysis of scattering loss from planar optical waveguides,” Opt. and Quantum Elec. 26977–986 (1994).
[Crossref]

Opt. Engr. (1)

J. Rodríguez, R. D. Crespo, S. Fernández, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibáñez, and J. Virgós, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Engr. 381896–1906 (1999).
[Crossref]

Phys. Rev. B (1)

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on a truncated photonic crystal,” Phys. Rev. B 541711–1715 (1996).
[Crossref]

Other (1)

J. M. Elson and P. Tran, “Band structure and transmission of photonic media: a real-space finite-difference calculation with the R-matrix propagator,” NATO ASI Series E: Applied Sciences on Photonic Band Gap MaterialsVol. 315 341–354 Crete, Greece June 15–29 (1995).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of waveguide with wall roughness. The waveguide channel, having nominal width ω and permittivity 2, is bounded by media with permittivity 1. In the z-direction, the waveguide channel consists of a smooth segment (L rzL t), a roughness segment (0≤zL r), and a semi-infinite substrate region z≤0. The x-dimension of the computational region is bounded by xL x/2 and the shaded regions at the extreme left and right denote the PML absorption layers.

Fig. 2.
Fig. 2.

Schematic showing modeling of waveguide roughness region. The waveguide channel has nominal width ω. For this example, the roughness region is divided into N r=5 sublayers of thickness δ and the channel width within each sublayer is varied in accordance with the roughness. By doing this, each sublayer is z-invariant and solutions of the form given in Eqs. (34) and (35) apply throughout the waveguide. In the numerical results of this work, the number of sublayers ranged from 0 to 1600, δ=0.1λ, and 0λ≤L r≤160λ. The permittivity of the light-shaded and dark-shaded regions are ∊1=(2.25, 0.) and ∊2=(2.50, 0.), respectively.

Fig. 3.
Fig. 3.

Normalized transmitted power versus length of roughness calculated at z=-10000λ. There are four groups of curves where each group has four curves. In each group the four curves, labeled 1–4, correspond to a different roughness realization. The upper and lower two sets of curves are for TE and TM polarization, respectively. The upper two sets of data have been displaced upward 0.4 units for clarity. The horizontal curves are for the normalized reflected power. The downward sloping curves are the normalized transmitted power. The number of discretization points N=299 and the nominal waveguide channel width ω=1λ. All realizations are generated with rms roughness 0.05λ and correlation length 1λ.

Fig. 4.
Fig. 4.

Normalized transmitted power versus length of roughness calculated at z=-10000λ. These data are analogous to those in Fig. 3 except that all realizations are generated with rms roughness 0.10λ.

Fig. 5.
Fig. 5.

Normalized transmitted power versus length of roughness calculated at z=-10000λ. These data are analogous to those in Fig. 3 except that all realizations are generated with rms roughness 0.20λ.

Fig. 6.
Fig. 6.

Convergence of normalized transmitted power for various values of discretization points N. For the same roughness realization 1 as used in Figs. 35, the transmitted power is calculated at z=-10000λ versus length of roughness. The upper set of curves, which have for clarity been displaced upward by 0.2, are for TM polarization and the values of N vary from 199 to 449. The lower set of curves are for TE polarization and the values of N vary from 199 to 399. For the corresponding polarization, the curves are essentially converged except for the N=199 curves. The rms roughness is 0.2λ.

Fig. 7.
Fig. 7.

Convergence of normalized TM transmitted power for various values of discretization points N and length L x. The ratio of L x/Nx is adjusted so that Δx≈0.055λ is nearly constant. For the same roughness realization 1 that was used in Figs. 35, the transmitted power is calculated at z=-10000λ versus length of roughness and the rms roughness is 0.2λ. Note that as L x increases, there is an increase in transmitted power due to a decrease in the absorption rate for the plane wave solutions. This is not a convergence related issue.

Fig. 8.
Fig. 8.

Normalized TM transmitted power for various values of PML absorption layers N PML. For 1≤N PML≤24, the value A PML=8. For numerical reasons, the results shown by the curve labeled N PML=0 was actually calculated with N PML=1 and A PML=0.1 (see Eq. (42)). Thus, the N PML=0 curve only approximates the absence of PML layers. The incident beam is at normal incidence, L x=16.3λ, N=299, and rms roughness is 0.2λ. These data are for the same roughness realization 1 that was used in Figs. 35. The group of horizontal lines are the normalized reflected power. It is clear that if there is sufficient PML absorption, then the solutions converge to a common result.

Fig. 9.
Fig. 9.

TM E-field intensity profile at various depths from the waveguide opening. There is no roughness region in this example and the 8 indicated z values are relative to the waveguide input aperture plane. The width L x=18.3λ and the angle of incidence is θ i=30°. For clarity, all curves, except the z=-205λ curve, have been displaced by multiples of 0.01 units. The z=0λ curve is the electric field intensity across the waveguide input aperture plane and the remaining z-value curves show the progression of the intensity curves at indicated depths into the waveguide. It is clear that the refracted part of the incident Gaussian profile is ultimately absorbed by the right-hand set of PML layers without reflection. In addition, the final z=-205λ curve is beginning to take shape as the residual guided wave since much of the transmitted energy has disappeared. The two vertical lines at x=±0.5λ indicate the boundaries of the waveguide channel. The two vertical lines at x=±7.65λ indicate the start of the PML absorbing layers.

Fig. 10.
Fig. 10.

Analytical curves derived from the data shown in Figs. 35. The four curves in each of these figures are each compiled into one analytical curve as a power of 10. This yields an estimate of the dB loss as a function of the length of roughness.

Equations (45)

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× E ( x , y , z ) = i ( ω / c ) μ ( x , y , z ) H ( x , y , z )
× H ( x , y , z ) = i ( ω / c ) ( x , y , z ) E ( x , y , z )
i ( ω / c ) E x ( x , z ) = α z ( x , z ) H y ( x , z ) z
i ( ω / c ) ε x ( x , z ) E z ( x , z ) = H y ( x , z ) x
i ( ω / c ) H y ( x , z ) = β x ( x , z ) E z ( x , z ) x β z ( x , z ) E x ( x , z ) z
i ( ω / c ) H x ( x , z ) = β z ( x , z ) E y ( x , z ) z
i ( ω / c ) μ x ( x , z ) H z ( x , z ) = E y ( x , z ) x
i ( ω / c ) E y ( x , z ) = α x ( x , z ) H z ( x , z ) x α z ( x , z ) H x ( x , z ) z
μ j ( x , z ) = 1 + 4 π i σ j * ( x , z ) / ω ; ε j ( x , z ) = ( x , z ) μ j ( x , z )
β j ( x , z ) = 1 / μ j ( x , z ) ; α j ( x , z ) = 1 / ε j ( x , z )
H E ; E H
( μ x , μ z ) ( ε x , ε z ) ; ( ε x , ε z ) ( μ x , μ z )
f ( x , z ) = F ( x , k ) f ( k , z ) ; f ( k , z ) = F 1 ( x , k ) f ( x , z )
i ( ω / c ) E x ( k , z ) = α z ( k , k ) H y ( k , z ) z
( ω / c ) ε x ( k , k ) E z ( k , z ) = k H y ( k , z )
i ( ω / c ) H y ( k , z ) = i β x ( k , k ) k E z ( k , z ) + β z ( k , k ) E x ( k , z ) z
μ j ( k , k ) = F 1 ( x , k ) μ j ( x ) F ( x , k ) ; ε j ( k , k ) = F 1 ( x , k ) ε j ( x ) F ( x , k )
β j ( k , k ) = F 1 ( x , k ) β j ( x ) F ( x , k ) ; α j ( k , k ) = F 1 ( x , k ) α j ( x ) F ( x , k )
2 H y ( k , z ) z 2 = α z 1 β z 1 [ β x k ε x 1 k I ( ω / c ) 2 ] H y ( k , z ) = M H y ( k , z )
H y ( k , z ) = S ( e ξ z C + + e ξ z C )
E x ( k , z ) = i ( c / ω ) α z S ξ ( e ξ z C + e ξ z C )
( E x ( k , z ) E x ( k , z + δ ) ) = ( r 11 ( δ ) r 12 ( δ ) r 21 ( δ ) r 22 ( δ ) ) ( H y ( k , z ) H y ( k , z + δ ) )
r 11 ( δ ) = r 22 ( δ ) = ( i c / ω ) α z S ξ ( e ξ δ + e ξ δ ) ( e ξ δ e ξ δ ) 1 S 1
r 12 ( δ ) = r 21 ( δ ) = ( 2 i c / ω ) α z S ξ ( e ξ δ e ξ δ ) 1 S 1
( E x ( k , z ) E x ( k , z + z t ) ) = ( R 11 ( z t ) R 12 ( z t ) R 21 ( z t ) R 22 ( z t ) ) ( H y ( k , z ) H y ( k , z + z t ) )
R 11 ( z t ) = R 11 ( z t δ ) + R 12 ( z t δ ) [ r 11 ( δ ) R 22 ( z t δ ) ] 1 R 21 ( z t δ )
R 12 ( z t ) = −R 12 ( z t δ ) [ r 11 ( δ ) R 22 ( z t δ ) ] 1 r 12 ( δ )
R 21 ( z t ) = r 21 ( δ ) [ r 11 ( δ ) R 22 ( z t δ ) ] 1 R 21 ( z t δ )
R 22 ( z t ) = r 22 ( δ ) r 21 ( δ ) [ r 11 ( δ ) R 22 ( z t δ ) ] 1 r 12 ( δ )
E x ( k , 0 ) = E x t ( k , 0 ) ; H y ( k , 0 ) = H y t ( k , 0 )
E x ( k , L t ) = E x r ( k , L t ) + E x i ( k , L t ) ; H y ( k , L t ) = H y r ( k , L t ) + H y i ( k , L t )
H y t ( k , z ) = S e ξ z C + H y t ( k , z ) = S e ξ z S 1 H y t ( k , 0 )
E x t ( k , z ) = i ( c / ω ) α z S ξ e ξ z C + E x t ( k , z ) = i ( c / ω ) α z S ξ e ξ z S 1 H y t ( k , 0 )
E x t ( k , z ) = i ( c / ω ) α z S ξ S 1 H y t ( k , z ) = T H y t ( k , z )
( E x i ( x , z ) H y i ( x , z ) ) = ω σ 2 c π π / 2 π / 2 d θ ( cos θ 1 )
× exp [ ( ω σ 2 c ) 2 ( θ θ i ) 2 ] exp [ i ( ω / c ) ( x sin θ z cos θ ) ]
E x i ( k n , z ) = σ π exp [ ( ω σ 2 c ) 2 ( θ n θ i ) 2 ] exp [ i ( ω / c ) z cos θ n ]
H y i ( k n , z ) = E x i ( k n , z ) / cos θ n
( E x r ( x , z ) H y r ( x , z ) ) = 1 2 π d k ( E x r ( k ) H y r ( k ) ) exp [ i ( k x + q z ) ]
= 1 2 π d k ( E x r ( k , z ) H y r ( k , z ) ) exp ( i k x )
E x r ( k , z ) = Z ( k ) H y r ( k , z )
Z ( k n ) = ( ω / c ) 2 k n 2 ( ω / c )
( T R 11 R 12 R 21 Z R 22 ) ( H y t ( k , 0 ) H y r ( k , L t ) ) = ( R 12 H y i ( k , L t ) R 22 H y i ( k , L t ) E x i ( k , L t ) )
4 π σ x * ( η ) / ω = A PML ( η / δ PML ) 2
P i = n = N / 2 N / 2 E x i ( k n , L t ) [ H y i ( k n , L t ) ] * ; P t = n = N / 2 N / 2 E x t ( k n , 0 ) [ H y t ( k n , 0 ) ] *

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