Abstract

We introduce a method to estimate the coupling coefficients of the guided field amplitude and the corresponding angular bandwidth in a multimode slab waveguide. This scalar diffraction theory approach is simpler than the more rigorous electromagnetic treatment and is directly applicable to communications systems that use large (dimensions or numerical aperture) waveguides, as in substrate-mode interconnects. Moreover, this method provides conceptual insight as to a parameter’s effect on the field-amplitude mode distribution and angular bandwidth.

© 1998 Optical Society of America

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References

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  1. K.-H. Brenner, F. Sauer, “Diffractive–reflective optical interconnects,” Appl. Opt. 27, 4252–4254 (1988).
    [CrossRef]
  2. R. K. Kostuk, Y.-T. Huang, D. Hetherington, M. Kato, “Reduced alignment and chromatic sensitivity of holographic optical interconnects with substrate mode holograms,” Appl. Opt. 28, 4939–4944 (1990).
    [CrossRef]
  3. R. T. Chen, C. Zhao, T.-H. Oh, “Performance-optimized optical bidirectional backplane bus for multiprocessor systems,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 299–317.
  4. Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.
  5. B. L. Booth, “Recent developments in polymer waveguide technology and applications for data link and optical interconnect systems,” in Optoelectronic Packaging, M. R. Feldman, Y.-C. Lee, eds., Proc. SPIE2691, 2–8 (1996).
    [CrossRef]
  6. R. E. Epworth, “The phenomenon of modal noise in analogue and digital optical fibre systems,” in Fourth European Conference on Optical Communication, B. Catania, ed. (Istituto Internazionale delle Comunicazioni Via Pertinace, Villa Piaggio, Genova, Italy, 1978), pp. 492–501.
  7. J. W. Goodman, E. G. Rawson, “Statistics of modal noise in fibers: a case of constrained speckle,” Opt. Lett. 6, 324–326 (1981).
    [CrossRef] [PubMed]
  8. Y. Tremblay, B. S. Kawasaki, K. O. Hill, “Modal noise in optical fibers: open and closed speckle pattern regimes,” Appl. Opt. 20, 1652–1655 (1981).
    [CrossRef] [PubMed]
  9. T. Kanada, “Evaluation of modal noise in multimode fiber-optic systems,” J. Lightwave Technol. LT-2, 11–18 (1984).
    [CrossRef]
  10. J. P. G. Bristow, Y. Liu, J. A. Lehman, K. Johnson, “Multimode optical interconnects with low modal noise for spaceborne databases and commercial computers,” in Photonics for Space Environments III, E. W. Taylor, ed., Proc. SPIE2482, 189–204 (1995).
    [CrossRef]
  11. J. E. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979), Chap. 4.
  12. C. R. Pollock, Fundamentals of Optoelectronics (Irwin, Chicago, 1995), Chap. 11.

1990 (1)

1988 (1)

K.-H. Brenner, F. Sauer, “Diffractive–reflective optical interconnects,” Appl. Opt. 27, 4252–4254 (1988).
[CrossRef]

1984 (1)

T. Kanada, “Evaluation of modal noise in multimode fiber-optic systems,” J. Lightwave Technol. LT-2, 11–18 (1984).
[CrossRef]

1981 (2)

Booth, B. L.

B. L. Booth, “Recent developments in polymer waveguide technology and applications for data link and optical interconnect systems,” in Optoelectronic Packaging, M. R. Feldman, Y.-C. Lee, eds., Proc. SPIE2691, 2–8 (1996).
[CrossRef]

Brenner, K.-H.

K.-H. Brenner, F. Sauer, “Diffractive–reflective optical interconnects,” Appl. Opt. 27, 4252–4254 (1988).
[CrossRef]

Bristow, J. P.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Bristow, J. P. G.

J. P. G. Bristow, Y. Liu, J. A. Lehman, K. Johnson, “Multimode optical interconnects with low modal noise for spaceborne databases and commercial computers,” in Photonics for Space Environments III, E. W. Taylor, ed., Proc. SPIE2482, 189–204 (1995).
[CrossRef]

Chen, R. T.

R. T. Chen, C. Zhao, T.-H. Oh, “Performance-optimized optical bidirectional backplane bus for multiprocessor systems,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 299–317.

Eldada, L.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Epworth, R. E.

R. E. Epworth, “The phenomenon of modal noise in analogue and digital optical fibre systems,” in Fourth European Conference on Optical Communication, B. Catania, ed. (Istituto Internazionale delle Comunicazioni Via Pertinace, Villa Piaggio, Genova, Italy, 1978), pp. 492–501.

Goodman, J. W.

Hennessy, W. A.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Hetherington, D.

Hill, K. O.

Huang, Y.-T.

Johnson, K.

J. P. G. Bristow, Y. Liu, J. A. Lehman, K. Johnson, “Multimode optical interconnects with low modal noise for spaceborne databases and commercial computers,” in Photonics for Space Environments III, E. W. Taylor, ed., Proc. SPIE2482, 189–204 (1995).
[CrossRef]

Kanada, T.

T. Kanada, “Evaluation of modal noise in multimode fiber-optic systems,” J. Lightwave Technol. LT-2, 11–18 (1984).
[CrossRef]

Kato, M.

Kawasaki, B. S.

Kostuk, R. K.

Lee, S. H.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Lehman, J. A.

J. P. G. Bristow, Y. Liu, J. A. Lehman, K. Johnson, “Multimode optical interconnects with low modal noise for spaceborne databases and commercial computers,” in Photonics for Space Environments III, E. W. Taylor, ed., Proc. SPIE2482, 189–204 (1995).
[CrossRef]

Liu, Y.

J. P. G. Bristow, Y. Liu, J. A. Lehman, K. Johnson, “Multimode optical interconnects with low modal noise for spaceborne databases and commercial computers,” in Photonics for Space Environments III, E. W. Taylor, ed., Proc. SPIE2482, 189–204 (1995).
[CrossRef]

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Liu, Y. S.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Midwinter, J. E.

J. E. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979), Chap. 4.

Oh, T.-H.

R. T. Chen, C. Zhao, T.-H. Oh, “Performance-optimized optical bidirectional backplane bus for multiprocessor systems,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 299–317.

Osgood, R. M.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Peczalski, A.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Plotts, A.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Pollock, C. R.

C. R. Pollock, Fundamentals of Optoelectronics (Irwin, Chicago, 1995), Chap. 11.

Rawson, E. G.

Rowlette, J.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Sauer, F.

K.-H. Brenner, F. Sauer, “Diffractive–reflective optical interconnects,” Appl. Opt. 27, 4252–4254 (1988).
[CrossRef]

Scarmozzino, R.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Stack, J.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Tremblay, Y.

Uzguz, V.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Woinarowski, R. J.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Yardley, J.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

Zhao, C.

R. T. Chen, C. Zhao, T.-H. Oh, “Performance-optimized optical bidirectional backplane bus for multiprocessor systems,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 299–317.

Appl. Opt. (3)

J. Lightwave Technol. (1)

T. Kanada, “Evaluation of modal noise in multimode fiber-optic systems,” J. Lightwave Technol. LT-2, 11–18 (1984).
[CrossRef]

Opt. Lett. (1)

Other (7)

J. P. G. Bristow, Y. Liu, J. A. Lehman, K. Johnson, “Multimode optical interconnects with low modal noise for spaceborne databases and commercial computers,” in Photonics for Space Environments III, E. W. Taylor, ed., Proc. SPIE2482, 189–204 (1995).
[CrossRef]

J. E. Midwinter, Optical Fibers for Transmission (Wiley, New York, 1979), Chap. 4.

C. R. Pollock, Fundamentals of Optoelectronics (Irwin, Chicago, 1995), Chap. 11.

R. T. Chen, C. Zhao, T.-H. Oh, “Performance-optimized optical bidirectional backplane bus for multiprocessor systems,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 299–317.

Y. S. Liu, R. J. Woinarowski, W. A. Hennessy, J. P. Bristow, Y. Liu, A. Peczalski, J. Rowlette, A. Plotts, J. Stack, J. Yardley, L. Eldada, R. M. Osgood, R. Scarmozzino, S. H. Lee, V. Uzguz, “Polymer optical interconnect technology (POINT)—optoelectronic packaging and interconnect for board and backplane applications,” in Optoelectronic Interconnects and Packaging, R. T. Chen, P. S. Guifoyle, eds., Vol. CR62 of SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1996), pp. 405–414.

B. L. Booth, “Recent developments in polymer waveguide technology and applications for data link and optical interconnect systems,” in Optoelectronic Packaging, M. R. Feldman, Y.-C. Lee, eds., Proc. SPIE2691, 2–8 (1996).
[CrossRef]

R. E. Epworth, “The phenomenon of modal noise in analogue and digital optical fibre systems,” in Fourth European Conference on Optical Communication, B. Catania, ed. (Istituto Internazionale delle Comunicazioni Via Pertinace, Villa Piaggio, Genova, Italy, 1978), pp. 492–501.

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Figures (5)

Fig. 1
Fig. 1

Slab waveguide geometry and propagation-vector notation.

Fig. 2
Fig. 2

Coupling-coefficient distribution 10% angular bandwidth (BW) as a function of λ0/(n 1 d) from the exact electromagnetic treatment, C m BW, and the scalar diffraction theory approach, C mdiffBW.

Fig. 3
Fig. 3

Field-amplitude coupling coefficients with λ0 = 0.85 μm, n 1 = 1.5, and a plane-wave angle of incidence of Θ = 70° for (a) a thickness of d = 5 μm and (b) a thickness of d = 20 μm.

Fig. 4
Fig. 4

Field-amplitude coupling coefficients with λ0 = 0.85 μm, n 1 = 1.5, and a thickness of d = 10 μm for a plane-wave angle of incidence of (a) Θ = 90° and (b) Θ = 50°.

Fig. 5
Fig. 5

Field-amplitude coupling coefficients with λ0 = 0.85 μm, n 1 = 1.5, and a thickness of d = 10 μm for (a) a normally incident Gaussian field with a 2-μm waist and (b) a normally incident, shifted Gaussian field in which each lobe has a 0.1-μm waist and a 2-μm offset from the waveguide center.

Equations (17)

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tan n 1 k 0 d   cos   Θ - m π 2 = n 1 2 sin 2   Θ - n 2 2 n 1 2 cos 2   Θ 1 / 2 ,
E ym = B m exp - γ m x for   x > d 2 A m cos h m x - m π 2 for   | x | d , D m exp γ m x for   x < - d
A m = 2 cos 2 h m d - m π 2 + cos 2 - h m d - m π 2 γ m + 4 h m d + 1 2 sin 2 h m d cos m π h m 1 / 2 , B m = 2 A m exp γ m d cos h m d - m π 2 , D m = 2 A m exp γ m d cos - h m d - m π 2 .
E inc x = m = 0 | x | d N   C m E ym + m = 0 | x | > d N   C m E ym = E inc x rect x 2 d + E inc x 1 - rect x 2 d ,
E inc x rect x 2 d = m = 0 | x | d N   C m E ym = m = 0 N   C m A m exp - i   m π 2 exp ih m x + exp i   m π 2 exp - ih m x .
E inc x rect x 2 d - h crit h crit   G ˜ h exp ihx d h + - h crit h crit   G ˜ * h exp - ihx d h = -   G ˜ h exp ihx rect h 2 h crit d h + -   G ˜ * h exp - ihx rect h 2 h crit d h .
E inc x rect x 2 d -   G ˜ 2 π λ 0   n 1 cos   Θ × exp i   2 π λ 0   n 1 cos   Θ x × d 2 π λ 0   n 1 cos   Θ + -   G ˜ * 2 π λ 0   n 1 cos   Θ × exp - i   2 π λ 0   n 1 cos   Θ x × d 2 π λ 0   n 1 cos   Θ = 2 π - 1 G ˜ 2 π ξ + 2 π G ˜ * 2 π ξ ,
1 2 π   E inc x rect x 2 d ξ = n 1 cos   Θ λ 0 = ℑℑ - 1 G ˜ 2 π ξ + ℑℑ G ˜ * 2 π ξ = G ˜ 2 π ξ + G ˜ * - 2 π ξ = G ˜ 2 π ξ + G ˜ * 2 π ξ = 2 G ˜ 2 π ξ .
G ˜ 2 π ξ δ Θ Θ m C m A m - i m ,
δ Θ Θ m = 1 for   Θ = Θ m 0 for   Θ Θ m .
C m 1 4 π A m - i m   E inc x rect x 2 d ξ = n 1 cos   Θ λ 0   δ Θ Θ m C m diff ,     for   m = 0 ,   1 , ,   N .
1 = m = 0 N   C m diff 2 .
Δ Θ max < 2 diff ,
m π 2 n 1 k 0 d   cos   Θ m + 1 π 2 ,
Δ Θ max Θ m - Θ m + 1 = cos - 1 m λ 0 4 n 1 d - cos - 1 m + 1 λ 0 4 n 1 d .
2 2 d = ξ max - ξ min = n 1 λ 0 cos Θ 0 - diff - cos Θ 0 + diff = n 1 λ 0 2   sin   Θ 0 sin   diff .
2 diff = 2   sin - 1 λ 0 2 n 1 d   sin   Θ 0 .

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