Abstract

Self-imaging properties of generalized N × N multimode interference couplers are derived. Positions, amplitudes, and phases of the self-images are directly related to the lengths and widths of the coupler by solving the eigenmode superposition equation analytically for any arbitrary length. Devices of length (M/N) 3Lc, where M is the multiple occurrence of the N self-images, are analyzed in detail. The general formalism is applied to practical N × N couplers used in integrated optics, and simple phase relations are obtained.

© 1994 Optical Society of America

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    [CrossRef]
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  3. Th. Niemeier, R. Ulrich, “Quadrature outputs from fiber interferometer with 4 × 4 coupler,” Opt. Lett. 11, 677–679 (1986).
    [CrossRef] [PubMed]
  4. L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
    [CrossRef]
  5. E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
    [CrossRef]
  6. L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.
  7. A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.
  8. E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.
  9. J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
    [CrossRef]
  10. L. B. Soldano, M. Bouda, M. K. Smit, B. H. Verbeek, “New small-size single-mode optical power splitter based on multimode interference,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 465–468.
  11. J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
    [CrossRef]
  12. M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.
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  14. D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
    [CrossRef]
  15. P. A. Besse, M. Bachmann, H. Melchior, “Phase relations in multimode interference couplers and their application to generalized integrated Mach–Zehnder optical switches,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.
  16. R. Ulrich, T. Kamiya, “Resolution of self-image in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
    [CrossRef]
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    [CrossRef]

1992

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

1991

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

1986

1981

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

1978

1975

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

1973

Andreadakis, N. C.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.

Ankele, G.

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

Bachmann, M.

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

P. A. Besse, M. Bachmann, H. Melchior, “Phase relations in multimode interference couplers and their application to generalized integrated Mach–Zehnder optical switches,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

Besse, P. A.

L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

P. A. Besse, M. Bachmann, H. Melchior, “Phase relations in multimode interference couplers and their application to generalized integrated Mach–Zehnder optical switches,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

Bhat, R.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.

Birbeck, J. C. H.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Bouda, M.

L. B. Soldano, M. Bouda, M. K. Smit, B. H. Verbeek, “New small-size single-mode optical power splitter based on multimode interference,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 465–468.

Brown-Goebeler, K.

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

Bryngdahl, O.

Caneau, C.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Chandrasekharan, K.

K. Chandrasekharan, Introduction to Analytic Number Theory (Springer-Verlag, Berlin, 1968), pp. 34–44.
[CrossRef]

Chang, D. C.

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

Chiu, T. H.

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

Curtis, L.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

de Miguel, J. L.

A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.

Deri, R. J.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.

Dubost, A. H.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

Ferraras, A.

A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.

Gini, E.

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

Gómez-Salas, E.

A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.

Gozdz, A. S.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Hawkins, R. J.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Hayes, T. R.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.

Heaton, J. M.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Hernández-Gil, F.

A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.

Hilton, K. P.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Jenkins, R. M.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Jones, K. L.

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

Kamiya, T.

Kuester, E. F.

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

Melchior, H.

L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

P. A. Besse, M. Bachmann, H. Melchior, “Phase relations in multimode interference couplers and their application to generalized integrated Mach–Zehnder optical switches,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

Niemeier, Th.

Parker, J. T.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Pennings, E. C. M.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.

Rodríguez, F.

A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.

Scherer, A.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Shah, V.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Smit, M. K.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

L. B. Soldano, M. Bouda, M. K. Smit, B. H. Verbeek, “New small-size single-mode optical power splitter based on multimode interference,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 465–468.

Soldano, L. B.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

L. B. Soldano, M. Bouda, M. K. Smit, B. H. Verbeek, “New small-size single-mode optical power splitter based on multimode interference,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 465–468.

Song, J.-I.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Soole, J. B. D.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

Tell, B.

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

Ulrich, R.

Veerman, F. B.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

Verbeek, B. H.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

L. B. Soldano, M. Bouda, M. K. Smit, B. H. Verbeek, “New small-size single-mode optical power splitter based on multimode interference,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 465–468.

Wight, D. R.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

Zucker, J. E.

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

Appl. Phys. Lett.

E. C. M. Pennings, R. J. Deri, A. Scherer, R. Bhat, T. R. Hayes, N. C. Andreadakis, M. K. Smit, L. B. Soldano, R. J. Hawkins, “Ultracompact, low-loss directional couplers on InP based on self-imaging by multimode interference,” Appl. Phys. Lett. 59, 1926–1928 (1991).
[CrossRef]

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, K. P. Hilton, “Novel 1-to-N way integrated optical beam splitters using symmetric mode mixing in GaAs/AlGaAs multimode waveguides,” Appl. Phys. Lett. 61, 1754–1756 (1992).
[CrossRef]

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

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L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1849 (1992).
[CrossRef]

J. E. Zucker, K. L. Jones, T. H. Chiu, B. Tell, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Other

P. A. Besse, M. Bachmann, H. Melchior, “Phase relations in multimode interference couplers and their application to generalized integrated Mach–Zehnder optical switches,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

K. Chandrasekharan, Introduction to Analytic Number Theory (Springer-Verlag, Berlin, 1968), pp. 34–44.
[CrossRef]

M. Bachmann, M. K. Smit, P. A. Besse, E. Gini, H. Melchior, L. B. Soldano, “Polarization-insensitive low-voltage optical waveguide switch using InGaAsP/InP four-port Mach–Zehnder interferometer,” in Optical Fiber Communication Conference and International Conference on Integrated Optics and Optical Fiber Communication, Vol. 4 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 32–33.

R. J. Deri, E. C. M. Pennings, A. Scherer, A. S. Gozdz, C. Caneau, N. C. Andreadakis, V. Shah, L. Curtis, R. J. Hawkins, J. B. D. Soole, J.-I. Song, “Ultra-compact, monolithic integration of balanced, polarization-diversity photodetectors,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), pp. 457–460.

L. B. Soldano, M. Bouda, M. K. Smit, B. H. Verbeek, “New small-size single-mode optical power splitter based on multimode interference,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 465–468.

L. B. Soldano, M. Bachmann, P. A. Besse, M. K. Smit, H. Melchior, “Large optical bandwidth of InGaAsP/InP multimode interference 3-dB couplers,” presented at the Sixth European Conference on Integrated Optics, Neuchatel, Switzerland, 18–22 April 1993.

A. Ferraras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, F. Hernández-Gil, “Design and fabrication of a InP/InGaAsP multimode power splitter,” in Integrated Photonics Research, Vol. 10 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 151–154.

E. C. M. Pennings, R. J. Deri, R. Bhat, T. R. Hayes, N. C. Andreadakis, “Ultra-compact integrated all-passive optical 90° hybrid using self-imaging,” in Proceedings of the 18th European Conference on Optical Communication (VDE Verlag, Berlin, 1992), Vol. 1, pp. 461–464.

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Figures (6)

Fig. 1
Fig. 1

Schematic view of a practical 2 × 2 multimode interference (MMI) coupler. Light from the input waveguide is launched into the MMI section, propagated, and imaged into the output waveeuides.

Fig. 2
Fig. 2

General multimode interference coupler studied in this paper.

Fig. 3
Fig. 3

MMI section with input light distribution f(x) and decomposition into guided eigenmodes E i (x) of the structure. For theoretical purposes the functions E i (x) and f(x) are antisymmetrically extended to a virtual MMI section and periodically repeated in the ±x directions.

Fig. 4
Fig. 4

MMI section with length L N M produces N images of the extended input function fin(x). Here the case of M = 1 is illustrated. The real MMI section of width W contains N images of the real input f(x) at the output. Images are partly inverted and partly upright. They may even be cut into two parts.

Fig. 5
Fig. 5

MMI coupler with access waveguides illustrating the case of N even. Inputs are numbered bottom-up with index i and outputs are numbered top-down with j.

Fig. 6
Fig. 6

MMI coupler with access waveguides illustrating the case of N odd.

Equations (23)

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E i ( x ) = sin [ π ( i + 1 ) x W ] with i = 0 , 1 , 2 , .
β 0 n k 0 Δ β 01 3 , Δ β 01 = β 0 β 1 3 π 2 2 n k 0 W 2 , β i β 0 i ( i + 2 ) Δ β 01 3 ,
f in ( x ) = i = 0 a i E i ( x ) , a i = 2 W 0 W f ( x ) E i * ( x ) d x .
L N M = M N 3 L c = M N 3 π Δ β 01 ,
f out ( x ) = i = 0 a i E i ( x ) exp ( j β i L N M ) .
f out ( x ) = i = 0 a i A i E i ( x ) , A i = exp [ j β 0 L N M + j π M N i ( i + 2 ) ] .
A 0 = exp ( j β 0 L N M ) , A i = A i 1 exp [ j π M N ( 2 i + 1 ) ] .
f out ( x ) = 1 C q = 0 N 1 f in ( x x q ) exp ( j φ q ) .
x q = ( 2 q N ) M N W ,
φ q = q ( N q ) M N π .
C = exp ( j β 0 L N M ) q = 0 N 1 exp [ j π x q W + j φ q ] , C = exp ( j β 0 L N M + j π M N ) q = 0 N 1 exp [ j π M N q ( N q ) ] .
C = N exp [ j β 0 L N M = 1 + j π N + j π 4 ( N 1 ) ] .
B i = 1 C q = 0 N 1 exp ( j Φ i , q ) with Φ i , q = π ( i + 1 ) x q W + φ q .
Φ i , q 1 = Φ i 1 , q + π M N ( 2 i + 1 ) .
i + j even : φ i j = π + φ N ( j i ) / 2 = φ 0 + π + π 4 N × ( j i ) ( 2 N j + i ) , i + j odd : φ i j = φ N ( j + i 1 ) / 2 = φ 0 + π 4 N ( j + i 1 ) × ( 2 N j i + 1 ) ,
φ o = β 0 L N M = 1 π N π 4 ( N 1 ) .
j odd or i + j even : q = N ( j i ) / 2 , j even or i + j odd : q = N ( j + i 1 ) / 2 .
B i = B i 1 exp [ j π M N ( 2 i + 1 ) ] .
A i = 1 C q = 0 N 1 exp [ j π ( i + 1 ) x q W + j φ q ] .
A i = 1 C q = 0 N 1 exp [ j π ( i + 1 ) x q W + j φ q ] .
A i E i ( x ) = 1 2 j { A i exp [ j π ( i + 1 ) x W ] A j exp [ j π ( i + 1 ) x W ] } .
A i E i ( x ) = 1 C q = 0 N 1 1 2 j { exp [ j π ( i + 1 ) x x q W ] exp [ j π ( i + 1 ) x x q W ] } exp ( j φ q ) , A i E i ( x ) = 1 C q = 0 N 1 E i ( x x q ) exp ( j φ q ) .
f out ( x ) = 1 C q = 0 N 1 [ i = 0 a i E i ( x x q ) ] exp ( j φ q ) , f out ( x ) = 1 C q = 0 N 1 f in ( x x q ) exp ( j φ q ) .

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