Abstract

We propose a synthesis method of optical Hadamard transformers using multimode interference (MMI) couplers. By using the signal transfer matrix of 2 × 2, 4 × 4, and 8 × 8 MMI couplers, we show that sum and difference units of input signals can be synthesized. An interchange unit of two signals can also be synthesized. One synthesis method of Hadamard transformers is a combination of only 2 × 2 units, and the other is a combination of N × N(N ≥ 4) units as well as 2 × 2 units. The design examples of operation units are shown, and the size and the output power of Hadamard transformers are estimated.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
    [CrossRef]
  2. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  3. R. Ulrich, T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
    [CrossRef]
  4. M. Bachmann, P. A. Besse, H. Melchior, “General self-imaging properties in N × M multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994).
    [CrossRef] [PubMed]
  5. M. Bachmann, P. A. Besse, H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
    [CrossRef] [PubMed]
  6. P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12, 1004–1009 (1994).
    [CrossRef]
  7. E. R. Thoen, L. A. Molter, J. P. Donnelly, “Analysis of N × M waveguide splitters and couplers with multimode guiding sections,” in Guided-Wave Optoelectronics: Device Characterization, Analysis, and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 143–153.
  8. M. Rajarajan, B. M. A. Rahman, K. T. V. Grattan, “Accurate numerical analysis of multimode-interference-based 3-dB couplers,” Appl. Opt. 37, 5672–5678 (1998).
    [CrossRef]
  9. M. Rajarajan, B. M. A. Rahman, K. T. V. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. 17, 243–248 (1999).
    [CrossRef]
  10. J. M. Heaton, R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
    [CrossRef]
  11. 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]
  12. 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, 1843–1850 (1992).
    [CrossRef]
  13. T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
    [CrossRef]
  14. J. E. Zucker, K. L. Jones, T. H. Chiu, K. Brown-Goebeler, “Strained quantum wells for polarization-independent electrooptic waveguide switches,” J. Lightwave Technol. 10, 1926–1930 (1992).
    [CrossRef]
  15. R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
    [CrossRef]
  16. M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
    [CrossRef]
  17. 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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
    [CrossRef]
  18. R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
    [CrossRef]
  19. T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
    [CrossRef]
  20. T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
    [CrossRef]
  21. F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).
  22. K. G. Beauchamp, Applications of Walsh and Related Functions with an Introduction to Sequency Theory (Academic, London, 1984), Chap. 2.
  23. E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Upper Saddle River, N.J., 1988), Chap. 8.
  24. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Sec. 5.2.
  25. B. E. Krivenkov, P. E. Tverdokhleb, Yu. V. Chugui, “Analysis of images by Hadamard optical transform,” Appl. Opt. 14, 1829–1834 (1975).
    [CrossRef] [PubMed]
  26. R. G. Hunsperger, Integrated Optics: Theory and Technology, 3rd ed. (Springer-Verlag, New York, 1991), Sec. 17.1.1.
    [CrossRef]
  27. M. N. Armenise, V. M. N. Passaro, “Optical signal processors and applications,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 303–312.
    [CrossRef]
  28. D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
    [CrossRef]
  29. D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
    [CrossRef]
  30. K. Tsutsumi, T. Sueta, “A proposed integrated-optical device for Hadamard transformation,” Trans. IECE Jpn. J60-C, 500–502 (1977).
  31. K. Tsutsumi, T. Sueta, “A synthesis of optical guided-wave signal transformers,” Trans. IECE Jpn. J62-C, 381–388 (1979).
  32. K. Tsutsumi, “Studies on synthesis of guided-wave optical signal transformers,” Ph.D. dissertation (Graduate School of Engineering Science, Osaka University, Osaka, Japan, 1980).
  33. M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12, 63–65 (1987).
    [CrossRef] [PubMed]

2001 (2)

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

1999 (4)

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

M. Rajarajan, B. M. A. Rahman, K. T. V. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. 17, 243–248 (1999).
[CrossRef]

J. M. Heaton, R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

1998 (1)

1995 (2)

M. Bachmann, P. A. Besse, H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
[CrossRef] [PubMed]

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

1994 (4)

M. Bachmann, P. A. Besse, H. Melchior, “General self-imaging properties in N × M multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994).
[CrossRef] [PubMed]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12, 1004–1009 (1994).
[CrossRef]

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[CrossRef]

1992 (3)

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (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, 1843–1850 (1992).
[CrossRef]

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

1991 (1)

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]

1987 (1)

1979 (1)

K. Tsutsumi, T. Sueta, “A synthesis of optical guided-wave signal transformers,” Trans. IECE Jpn. J62-C, 381–388 (1979).

1978 (1)

1977 (3)

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
[CrossRef]

K. Tsutsumi, T. Sueta, “A proposed integrated-optical device for Hadamard transformation,” Trans. IECE Jpn. J60-C, 500–502 (1977).

1975 (1)

1973 (1)

Anderson, D. B.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
[CrossRef]

Andreadakis, N. 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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (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]

Armenise, M. N.

M. N. Armenise, V. M. N. Passaro, “Optical signal processors and applications,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 303–312.
[CrossRef]

August, R. R.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
[CrossRef]

Bachmann, M.

Beauchamp, K. G.

K. G. Beauchamp, Applications of Walsh and Related Functions with an Introduction to Sequency Theory (Academic, London, 1984), Chap. 2.

Besse, P. A.

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]

Birbeck, J. C. H.

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Boyd, J. T.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
[CrossRef]

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Upper Saddle River, N.J., 1988), Chap. 8.

Brown-Goebeler, K.

J. E. Zucker, K. L. Jones, T. H. Chiu, 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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

Cappuzzo, M.

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

Chiu, T. H.

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

Chugui, Yu. V.

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

Davis, R. L.

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
[CrossRef]

Deri, R. J.

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (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]

Donnelly, J. P.

E. R. Thoen, L. A. Molter, J. P. Donnelly, “Analysis of N × M waveguide splitters and couplers with multimode guiding sections,” in Guided-Wave Optoelectronics: Device Characterization, Analysis, and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 143–153.

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, 1843–1850 (1992).
[CrossRef]

Earnshaw, M. P.

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

Goh, T.

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

Gomez, L.

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Sec. 5.2.

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

Grattan, K. T. V.

Hamilton, M. C.

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
[CrossRef]

Hawkins, R. J.

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (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]

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]

Heaton, J. M.

J. M. Heaton, R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Hilton, K. P.

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Himeno, A.

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology, 3rd ed. (Springer-Verlag, New York, 1991), Sec. 17.1.1.
[CrossRef]

Jenkins, R. M.

J. M. Heaton, R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Jones, K. L.

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

Jutamulia, S.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

Kamiya, T.

Krivenkov, B. E.

Laskowski, E.

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

Marhic, M. E.

Melchior, H.

Molter, L. A.

E. R. Thoen, L. A. Molter, J. P. Donnelly, “Analysis of N × M waveguide splitters and couplers with multimode guiding sections,” in Guided-Wave Optoelectronics: Device Characterization, Analysis, and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 143–153.

Okamoto, K.

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

Okuno, M.

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

Orihara, Y.

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

Parker, J. T.

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Passaro, V. M. N.

M. N. Armenise, V. M. N. Passaro, “Optical signal processors and applications,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 303–312.
[CrossRef]

Paunescu, A.

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

Pennings, E. C. M.

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (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, 1843–1850 (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]

Rahman, B. M. A.

Rajarajan, M.

Saida, T.

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

Scherer, A.

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (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]

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

Shibata, T.

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

Smit, M. K.

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12, 1004–1009 (1994).
[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, 1843–1850 (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]

Smith, G. W.

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Soldano, L. B.

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12, 1004–1009 (1994).
[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, 1843–1850 (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]

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

Soole, J. B. D.

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

Sueta, T.

K. Tsutsumi, T. Sueta, “A synthesis of optical guided-wave signal transformers,” Trans. IECE Jpn. J62-C, 381–388 (1979).

K. Tsutsumi, T. Sueta, “A proposed integrated-optical device for Hadamard transformation,” Trans. IECE Jpn. J60-C, 500–502 (1977).

Sugita, A.

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

Takiguchi, K.

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

Thoen, E. R.

E. R. Thoen, L. A. Molter, J. P. Donnelly, “Analysis of N × M waveguide splitters and couplers with multimode guiding sections,” in Guided-Wave Optoelectronics: Device Characterization, Analysis, and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 143–153.

Tsutsumi, K.

K. Tsutsumi, T. Sueta, “A synthesis of optical guided-wave signal transformers,” Trans. IECE Jpn. J62-C, 381–388 (1979).

K. Tsutsumi, T. Sueta, “A proposed integrated-optical device for Hadamard transformation,” Trans. IECE Jpn. J60-C, 500–502 (1977).

K. Tsutsumi, “Studies on synthesis of guided-wave optical signal transformers,” Ph.D. dissertation (Graduate School of Engineering Science, Osaka University, Osaka, Japan, 1980).

Tverdokhleb, P. E.

Uchiyama, K.

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

Ulrich, R.

van der Heijden, J. M. M.

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[CrossRef]

van Dongen, T.

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[CrossRef]

van Roijen, R.

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[CrossRef]

van Stralen, M. J. N.

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[CrossRef]

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, 1843–1850 (1992).
[CrossRef]

Verbeek, B. H.

R. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[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, 1843–1850 (1992).
[CrossRef]

Wight, D. R.

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[CrossRef]

Yamada, H.

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

Zucker, J. E.

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

Appl. Opt. (4)

Appl. Phys. Lett. (3)

R. M. Jenkins, J. M. Heaton, D. R. Wight, J. T. Parker, J. C. H. Birbeck, G. W. Smith, K. P. Hilton, “Novel 1 × N and N × N integrated optical switches using self-imaging multimode GaAs/AlGaAs waveguides,” Appl. Phys. Lett. 64, 684–686 (1994).
[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. van Roijen, E. C. M. Pennings, M. J. N. van Stralen, T. van Dongen, B. H. Verbeek, J. M. M. van der Heijden, “Compact InP-based ring lasers employing multimode interference couplers and combiners,” Appl. Phys. Lett. 64, 1753–1755 (1994).
[CrossRef]

Electron. Lett. (4)

T. Saida, Y. Orihara, H. Yamada, K. Takiguchi, T. Goh, K. Okamoto, “Integrated optical polarisation analyser on planar lightwave circuit,” Electron. Lett. 35, 1948–1949 (1999).
[CrossRef]

T. Saida, K. Okamoto, K. Uchiyama, K. Takiguchi, T. Shibata, A. Sugita, “Integrated optical digital-to-analogue converter and its application to pulse pattern recognition,” Electron. Lett. 37, 1237–1238 (2001).
[CrossRef]

T. Saida, A. Himeno, M. Okuno, A. Sugita, K. Okamoto, “Silica-based 2 × 2 multimode interference coupler with arbitrary power splitting ratio,” Electron. Lett. 35, 2031–2033 (1999).
[CrossRef]

M. P. Earnshaw, J. B. D. Soole, M. Cappuzzo, L. Gomez, E. Laskowski, A. Paunescu, “Compact, low-loss 4 × 4 optical switch matrix using multimode interferometers,” Electron. Lett. 37, 115–116 (2001).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. B. Anderson, J. T. Boyd, M. C. Hamilton, R. R. August, “An integrated-optical approach to the Fourier transform,” IEEE J. Quantum Electron. QE-13, 268–275 (1977).
[CrossRef]

D. B. Anderson, R. L. Davis, J. T. Boyd, R. R. August, “Comparison of optical-waveguide lens technologies,” IEEE J. Quantum Electron. QE-13, 275–282 (1977).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

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, “Ultracompact monolithic integration of balanced, polarization diversity photodetectors for coherent lightwave receivers,” IEEE Photon. Technol. Lett. 4, 1238–1240 (1992).
[CrossRef]

J. M. Heaton, R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999).
[CrossRef]

J. Lightwave Technol. (5)

J. E. Zucker, K. L. Jones, T. H. Chiu, 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, 1843–1850 (1992).
[CrossRef]

M. Rajarajan, B. M. A. Rahman, K. T. V. Grattan, “A rigorous comparison of the performance of directional couplers with multimode interference devices,” J. Lightwave Technol. 17, 243–248 (1999).
[CrossRef]

L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
[CrossRef]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12, 1004–1009 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Lett. (1)

Trans. IECE Jpn. (2)

K. Tsutsumi, T. Sueta, “A proposed integrated-optical device for Hadamard transformation,” Trans. IECE Jpn. J60-C, 500–502 (1977).

K. Tsutsumi, T. Sueta, “A synthesis of optical guided-wave signal transformers,” Trans. IECE Jpn. J62-C, 381–388 (1979).

Other (8)

K. Tsutsumi, “Studies on synthesis of guided-wave optical signal transformers,” Ph.D. dissertation (Graduate School of Engineering Science, Osaka University, Osaka, Japan, 1980).

R. G. Hunsperger, Integrated Optics: Theory and Technology, 3rd ed. (Springer-Verlag, New York, 1991), Sec. 17.1.1.
[CrossRef]

M. N. Armenise, V. M. N. Passaro, “Optical signal processors and applications,” in Advances in Integrated Optics, S. Martellucci, A. N. Chester, M. Bertolotti, eds. (Plenum, New York, 1994), pp. 303–312.
[CrossRef]

E. R. Thoen, L. A. Molter, J. P. Donnelly, “Analysis of N × M waveguide splitters and couplers with multimode guiding sections,” in Guided-Wave Optoelectronics: Device Characterization, Analysis, and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 143–153.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing, and Neural Networks (Wiley, New York, 1992).

K. G. Beauchamp, Applications of Walsh and Related Functions with an Introduction to Sequency Theory (Academic, London, 1984), Chap. 2.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, Upper Saddle River, N.J., 1988), Chap. 8.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Sec. 5.2.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

N × N MMI coupler. The width and length of the MMI waveguide are W M and L N , respectively. Access waveguides with width W IO and length l are located at pitch p.

Fig. 2
Fig. 2

N × N MMI coupler with phase shifters in access waveguides. Phase-shifting sections are shown by the shaded sections. Phase shift ϕ Ii is at the ith (i = 1, 2, … , N) input access waveguide and ϕ Oi is at the output access waveguide.

Fig. 3
Fig. 3

Synthesis of a Hadamard transformer (N = 8) by use of only 2 × 2 units. Sum and difference units are denoted by S & D. Interchange units are shown by arrows.

Fig. 4
Fig. 4

Synthesis of a Hadamard transformer (N = 8) by use of 4 × 4 and 8 × 8 units as well as 2 × 2 units. Sum and difference units are denoted by S & D.

Fig. 5
Fig. 5

Evolution of fields in a 2 × 2 sum and difference unit. Width W M and length L N of the MMI waveguide are 6 and 178 μm, respectively. The input and output ends are at the left- and right-hand sides, respectively. Absolute values of electric fields E y are shown.

Fig. 6
Fig. 6

Evolution of fields in an interchange unit. Width W M and length L N of the MMI waveguide are 6 and 356 μm, respectively. The input and output ends are at the left- and right-hand sides, respectively. Absolute values of electric fields E y are shown.

Fig. 7
Fig. 7

Evolution of fields in a 4 × 4 sum and difference unit. Width W M and length L N of the MMI waveguide are 12 and 655.6 μm, respectively. The input and output ends are at the left- and right-hand sides, respectively. Absolute values of electric fields E y are shown.

Fig. 8
Fig. 8

Evolution of fields in an 8 × 8 sum and difference unit. Width W M and length L N of the MMI waveguide are 24 and 2533.5 μm, respectively. The input and output ends are at the left- and right-hand sides, respectively. Absolute values of electric fields E y are shown.

Fig. 9
Fig. 9

Synthesis of a Hadamard transformer (N = 8) with 1 × 1 units as well as 2 × 2 units. Sum and difference units are denoted by S & D.

Fig. 10
Fig. 10

Evolution of fields in a 1 × 1 unit. Width W M and length L N of the MMI waveguide are 3 and 26.2 μm, respectively. The input and output ends are at the left- and right-hand sides, respectively. Absolute values of electric fields E y are shown.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

LN=3Lπ/N,
βνk0n2- ν+12πλ04n2We2,
We=WM+λ0πn12n22σ1n22-n121/2,
Lπ 4n2We23λ0.
AO1AO2AO3AON=PNAI1AI2AI3AIN,
PN=VNBNVNT.
gνx=sinν+1πWM x, ν=0, 1, 2,
Viν+1=2Nsin2i-1ν+1π2N for ν+1N, Viν+1= 1Nsin2i-1ν+1π2N for ν+1=N,
i=1N Viν+12=1.
Bν+1ν+1=expjν+12πN.
T=expjϕO1000expjϕO2000expjϕON ×PNexpjϕI1000expjϕI2000expjϕIN,
P2= 12111-1expjπ/200expj2π111-1 =12expjπ/4expj3π/4expj3π/4expjπ/4,
T= 12100exp-jπ/2×P2exp-jπ/400exp-j3π/4 =12111-1.
P22=P2P2=0-1-10.
T=0110.
P4= 12×expjπ/4-11expjπ/4-1expjπ/4expjπ/411expjπ/4expjπ/4-1expjπ/41-1expjπ/4.
P42=P4P4= 12expjπ/400expj3π/40expjπ/4expj3π/400expj3π/4expjπ/40expj3π/400expjπ/4.
T= 121001011001-10100-1.
P84= 12×expjπ/4000000expj3π/40expjπ/40000expj3π/4000expjπ/400expj3π/400000expjπ/4expj3π/4000000expj3π/4expjπ/400000expj3π/400expjπ/4000expj3π/40000expjπ/40expj3π/4000000expjπ/4.
T= 12100000010100001000100100000110000001-100000100-100010000-101000000-1
H8=111111111111-1-1-1-111-1-1-1-11111-1-111-1-11-1-111-1-111-1-11-111-11-11-1-11-111-11-11-11-1
H8=TSD1000000000001000010000000000010000100000000000100001000000000001 ×TSD1000000000100000010000000001000000001000000000100000010000000001TSD, TSD=110000001-100000000110000001-100000000110000001-100000000110000001-1.
H8=111111111-1-111-1-1111-1-1-1-1111-11-1-11-111111-1-1-1-11-1-11-111-111-1-111-1-11-11-11-11-1.
H8=110000001-100000000110000001-100000000110000001-100000000110000001-1100100000110000001-100000100-100000000100100000110000001-100000100-1100000010100001000100100000110000001-100000100-100010000-101000000-1.
δLNLN2 δWeWe- δλ0λ0 δn2n2

Metrics