Abstract

A simple formalism to estimate modal power loss coefficients for an overmoded rectangular waveguide with rough surfaces is presented. The method is based on small index differences where the true radiation modes are approximated by free space modes. Loss coefficients are important in order to establish more accurate channel models for, e.g., optical backplane communication systems. The theory is validated by comparing the loss coefficients of a squeezed rectangular waveguide with the loss coefficients of a slab waveguide.

© 2004 Optical Society of America

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References

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  1. J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
    [Crossref]
  2. D. Marcuse, Light Transmission Optics, 2nd ed. (Krieger, 1982).
  3. F. Ladouceur, F. Love, and T. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141, 242–48 (1994).
    [Crossref]
  4. J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288 (1990).
  5. S. Lee, D. Mui, and L. Coldren, “Explicit Formulas of Normalized Radiation Modes in Multilayer Waveguides,” J. Lightwave Technol. 12, 2073–2079 (1194).
  6. Z. Wang, “Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides,” IEEE J. Quantum Electron. 34, 680–685 (1998).
    [Crossref]
  7. D. Marcuse, Theory of Dielectric Optical Waveguides, Quantum Electronics-Principles and Applications, 2nd ed. (Academic Press, Boston, 1997).
  8. D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
    [Crossref]
  9. A. S. Sudbø, “Why Are Accurate Computations of Mode Fields in Rectangular DielectricWaveguides Difficult?” J. Lightwave Technol. 10, 418–419 (1992).
    [Crossref]
  10. E. Marcatili, “Dielectric RectangularWaveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2103 (1969).
  11. D. Marcuse, “Radiation losses of step-tapered channel waveguides,” Appl. Opt. 19, 3676–3681 (1980).
    [Crossref] [PubMed]

2000 (1)

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

1998 (1)

Z. Wang, “Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides,” IEEE J. Quantum Electron. 34, 680–685 (1998).
[Crossref]

1994 (1)

F. Ladouceur, F. Love, and T. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141, 242–48 (1994).
[Crossref]

1992 (1)

A. S. Sudbø, “Why Are Accurate Computations of Mode Fields in Rectangular DielectricWaveguides Difficult?” J. Lightwave Technol. 10, 418–419 (1992).
[Crossref]

1990 (1)

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288 (1990).

1980 (1)

1969 (1)

E. Marcatili, “Dielectric RectangularWaveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2103 (1969).

1194 (1)

S. Lee, D. Mui, and L. Coldren, “Explicit Formulas of Normalized Radiation Modes in Multilayer Waveguides,” J. Lightwave Technol. 12, 2073–2079 (1194).

Bächtold, W.

D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
[Crossref]

Bogenberger, R.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Coldren, L.

S. Lee, D. Mui, and L. Coldren, “Explicit Formulas of Normalized Radiation Modes in Multilayer Waveguides,” J. Lightwave Technol. 12, 2073–2079 (1194).

Erni, D.

D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
[Crossref]

Guttmann, J.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Huber, H.-P.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Krumpholz, O.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Kuhn, K.-P.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Lacey, J.

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288 (1990).

Ladouceur, F.

F. Ladouceur, F. Love, and T. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141, 242–48 (1994).
[Crossref]

Lee, S.

S. Lee, D. Mui, and L. Coldren, “Explicit Formulas of Normalized Radiation Modes in Multilayer Waveguides,” J. Lightwave Technol. 12, 2073–2079 (1194).

Lenz, D.

D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
[Crossref]

Love, F.

F. Ladouceur, F. Love, and T. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141, 242–48 (1994).
[Crossref]

Marcatili, E.

E. Marcatili, “Dielectric RectangularWaveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2103 (1969).

Marcuse, D.

D. Marcuse, “Radiation losses of step-tapered channel waveguides,” Appl. Opt. 19, 3676–3681 (1980).
[Crossref] [PubMed]

D. Marcuse, Light Transmission Optics, 2nd ed. (Krieger, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides, Quantum Electronics-Principles and Applications, 2nd ed. (Academic Press, Boston, 1997).

Moisel, J.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Mui, D.

S. Lee, D. Mui, and L. Coldren, “Explicit Formulas of Normalized Radiation Modes in Multilayer Waveguides,” J. Lightwave Technol. 12, 2073–2079 (1194).

Payne, F.

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288 (1990).

Rankov, B.

D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
[Crossref]

Rode, M.

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Senden, T.

F. Ladouceur, F. Love, and T. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141, 242–48 (1994).
[Crossref]

Sudbø, A. S.

A. S. Sudbø, “Why Are Accurate Computations of Mode Fields in Rectangular DielectricWaveguides Difficult?” J. Lightwave Technol. 10, 418–419 (1992).
[Crossref]

Wang, Z.

Z. Wang, “Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides,” IEEE J. Quantum Electron. 34, 680–685 (1998).
[Crossref]

Wittneben, A.

D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. Marcatili, “Dielectric RectangularWaveguide and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2103 (1969).

IEE Proc. (1)

J. Lacey and F. Payne, “Radiation loss from planar waveguides with random wall imperfections,” IEE Proc. 137, 282–288 (1990).

IEE Proc.-Optoelectron. (1)

F. Ladouceur, F. Love, and T. Senden, “Effect of side wall roughness in buried channel waveguides,” IEE Proc.-Optoelectron. 141, 242–48 (1994).
[Crossref]

IEEE J. Quantum Electron. (1)

Z. Wang, “Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides,” IEEE J. Quantum Electron. 34, 680–685 (1998).
[Crossref]

J. Lightwave Technol. (2)

A. S. Sudbø, “Why Are Accurate Computations of Mode Fields in Rectangular DielectricWaveguides Difficult?” J. Lightwave Technol. 10, 418–419 (1992).
[Crossref]

S. Lee, D. Mui, and L. Coldren, “Explicit Formulas of Normalized Radiation Modes in Multilayer Waveguides,” J. Lightwave Technol. 12, 2073–2079 (1194).

Opt. Eng. (1)

J. Moisel, J. Guttmann, H.-P. Huber, O. Krumpholz, M. Rode, R. Bogenberger, and K.-P. Kuhn, “Optical backplanes with integrated polymer waveguides,” Opt. Eng. 39, 673–679 (2000).
[Crossref]

Other (3)

D. Marcuse, Light Transmission Optics, 2nd ed. (Krieger, 1982).

D. Marcuse, Theory of Dielectric Optical Waveguides, Quantum Electronics-Principles and Applications, 2nd ed. (Academic Press, Boston, 1997).

D. Lenz, B. Rankov, D. Erni, W. Bächtold, and A. Wittneben, “MIMO Channel for Modal Multiplexing in Highly Overmoded Optical Waveguides,” in International Zurich Seminar on Communications, pp. 196 (Zurich, 2004).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1.

Perturbed rectangular waveguide.

Fig. 2.
Fig. 2.

Simulation results in order to illustrate the applicability of the proposed method.

Fig. 3.
Fig. 3.

Radiation patterns for two different correlation lengths D with σ=4×10-9 using a Gaussian ACF (b=d=30 µm).

Equations (20)

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K μ , ρ = ω ε 0 4 i P ( n 2 n 0 2 ) ( E μ t * E ρ t + n 0 2 n 2 E μ z * E ρ z ) d x d y
K μ , ρ = ω ε 0 4 i P ( n 2 ( x , y , z ) n 0 2 ( x , y ) ) E μ x * E ρ x d x d y
α μ = { s = 1 4 K ̂ μ , ρ ( s ) ( κ , σ ) 2 F ( s ) ( β μ β ρ ) 2 } d κ d σ
F ( s ) ( β μ β ρ ) 2 = f ( s ) ( z ) f ( s ) ( z u ) exp [ i ( β μ β ρ ) u ] d u .
K ρ μ = ω ε 0 4 i P 0 b 0 f ( 1 ) ( z ) n f δ n ( n 1 n 2 ) 2 E ρ x * E μ x d x d y Integral I 1 + ω ε 0 4 i P f ( 2 ) ( z ) 0 0 d δ n E ρ x * E μ x d x d y Integral I 2
E μ x = a ( μ ) sin [ k x ( μ ) ( x + ξ ( μ ) ) ] cos [ k y ( μ ) ( y + η ( μ ) ) ] exp ( i β μ z )
P = 1 = 1 2 β μ ω μ 0 E μ x E μ x * d x d y .
E ρ x = u ( ρ ) exp [ i ( σ x + κ y + ρ z ) ] .
1 2 ε 0 ω β k 2 E ρ x ( σ , κ ) E ρ x * ( σ , κ ) d x d y = δ ( σ σ , κ κ )
I 1 = 0 b 0 f ( z ) δ n ( n 1 n 2 ) 2 E ρ x * E μ x d x d y
= δ n ( n 1 n 2 ) 2 u * ( ρ ) a ( μ )
× 0 f ( z ) exp ( i σ x ) sin [ k x ( μ ) ( x + ξ ( μ ) ) ] d x 0 b exp ( i κ y ) cos [ k y ( μ ) ( y + η ( μ ) ) ] d y χ ( k y , η , κ )
= δ n ( n 1 n 2 ) 2 u * ( ρ ) a ( μ ) sin ( k x ( μ ) ξ ( μ ) ) χ ( k y , η , κ ) f ( 1 ) ( z )
I 2 = δ n u * ( ρ ) a ( μ ) cos ( k y ( μ ) η ( μ ) ) ζ ( k x , σ , ξ ) ( f ( 2 ) ( z ) )
K ρ μ ( z ) = K ̂ ρ μ ( 1 ) f ( 1 ) ( z ) + K ̂ ρ μ ( 2 ) f ( 2 ) ( z )
K ̂ ρ μ ( 1 ) = ω ε 0 2 4 i P u * ( ρ ) a ( μ ) δ n ( n 1 n 2 ) 2 χ ( k y , η , κ ) sin ( k x ( μ ) ξ ( μ ) )
K ̂ ρ μ ( 2 ) = ω ε 0 2 4 i P u * ( ρ ) a ( μ ) δ n ζ ( k x , σ , ξ ) cos ( k y ( μ ) η ( μ ) )
χ ( k y , η , κ ) = 0 b exp ( i κ y ) cos [ k y ( μ ) ( y + η ( μ ) ) ] d y
ζ ( k x , σ , ξ ) = 0 d exp ( i σ x ) sin [ k x ( μ ) ( x + ξ ( μ ) ) ] d x
α tot = 1 10 z lg ( Σ P μ ( 0 ) Σ P μ ( 0 ) exp ( α μ z ) ) dB / cm

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