Abstract

Planar photonic integrated circuits based on four-port couplers offer enhanced sophistication and functionality. Each four-port coupler is characterized by sixteen signal coupling coefficients governed by ten energy constraints. The ability to generate the constrained sixteen coupling coefficients is needed in the analysis of the four-port coupler. However, the energy constraint equations are nonlinear and cumbersome to solve directly. We introduce two techniques to reduce these signal coupling coefficients to a set of six free parameters. Hence we can characterize all possible couplers in terms of their sixteen constrained coupling coefficients, or either of two sets of six free parameters. This reduction in parameters has significant ramifications for the design, specification, and empirical characterization of these useful building blocks.

© 2006 Optical Society of America

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  1. B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
    [Crossref]
  2. F. J. Fraile-Peláez, J. Capmany, and M. A. Muriel, "Transmission bistability in a double-coupler fiber ring resonator," Opt. Lett. 16, 907-909 (1991).
    [Crossref] [PubMed]
  3. K. Sasayama, M. Okuno, and K. Habara, "Coherent optical transversal filter using silica-based waveguides for high-speed signal processing," J. Lightwave Technol. 9, 1225-1230 (1991).
    [Crossref]
  4. D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994).
    [Crossref]
  5. E. M. Dowling and D. L. MacFarlane, "Lightwave lattice filters for optically multiplexed communication systems," J. Lightwave Technol. 12, 471-486 (1994).
    [Crossref]
  6. Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
    [Crossref]
  7. D. L. MacFarlane, E. M. Dowling, and V. Narayan, "Ring resonators with N×M couplers," Fiber Integr. Opt. 14, 195-210 (1995).
    [Crossref]
  8. C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
  9. L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).
  10. I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
    [Crossref]
  11. D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
    [Crossref] [PubMed]
  12. G. Griffel, "Synthesis of optical filters using ring resonator arrays," IEEE Photon. Technol. Lett. 12, 810-812 (2000).
    [Crossref]
  13. D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
    [Crossref]
  14. D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
    [Crossref]
  15. H. J. Carlin, "The scattering matrix in network theory," IRE Trans. Circuit Theory CT-3, 88-97 (1956).
  16. D. M. Pozar, Microwave Engineering, 2nd ed. (Wiley, 1998).
  17. J. Reed and G. J. Wheeler, "A method of analysis of symmetrical four port networks," IRE Trans. Microwave Theory Tech. MTT-4, 246-252 (1956).
    [Crossref]
  18. C. R. Boyd, Jr., "On a class of multiple line directional couplers," IRE Trans. Microwave Theory Tech. MTT-10, 287-294 (1962).
    [Crossref]
  19. K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans. Microwave Theory Tech. MTT-13, 194-202 (1965).
    [Crossref]
  20. S. Hagelin, "A flow graph analysis of 3- and 4-port junction circulators," IEEE Trans. Microwave Theory Tech. MTT-14, 243-249 (1966).
    [Crossref]
  21. J. J. Campbell, "Application of the solutions of certain boundary value problems to the symmetrical four-port junction and specially truncated bends in parallel-plate waveguides and balanced strip-transmission lines," IEEE Trans. Microwave Theory Tech. MTT-16, 165-176 (1968).
    [Crossref]
  22. R. Levy, "Analysis and synthesis of waveguide multi-aperture directional couplers," IEEE Trans. Microwave Theory Tech. MTT-16, 995-1006 (1968).
    [Crossref]
  23. G. P. Riblet, "A coupling theorem for matched symmetrical two-branch four-port networks," IEEE Trans. Circuits Syst. CAS-25, 145-148 (1978).
    [Crossref]
  24. O. Schwelb and R. Antepyan, "Conservation laws for distributed four-ports," IEEE Trans. Microwave Theory Tech. MTT-33, 157-160 (1985).
    [Crossref]
  25. J. Esteban and J. M. Rebollar, "Generalized scattering matrix of generalized two-port discontinuities: application to four-port and nonsymmetric six-port couplers," IEEE Trans. Microwave Theory Tech. 39, 1725-1734 (1991).
    [Crossref]
  26. K. Araki and Y. Naito, "On the properties of lossless reciprocal 4-port circuits with reflection symmetry," IEEE Trans. Circuits Syst. 39, 155-161 (1992).
    [Crossref]
  27. H.-C. Lu and T.-H. Chu, "Multiport scattering matrix measurement using a reduced-port network analyzer," IEEE Trans. Microwave Theory Tech. 51, 1525-1533 (2003).
    [Crossref]
  28. J. Martens, D. V. Judge, and J. A. Bigelow, "Uncertainties associated with many-port (>4) S-parameter measurements using a four-port vector network analyzer," IEEE Trans. Microwave Theory Tech. 52, 1361-1368 (2004).
    [Crossref]
  29. C. B. Fafadia, "Thick linear optical lattice filters," Master's thesis (University of Texas at Dallas, 2003).
  30. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991).
    [Crossref]
  31. F. R. Gantmacher, The Theory of Matrices (Chelsea, 1977), Vol. 1.
  32. F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, 1962).
  33. R. P. Stanley, Enumerative Combinatorics (Cambridge U. Press, 1999), Vol. 2.
    [Crossref]
  34. P. Lounesto, Clifford Algebras and Spinors (Cambridge U. Press, 2001).
    [Crossref]
  35. Ulf Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, 1997).

2005 (2)

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

2004 (2)

D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
[Crossref] [PubMed]

J. Martens, D. V. Judge, and J. A. Bigelow, "Uncertainties associated with many-port (>4) S-parameter measurements using a four-port vector network analyzer," IEEE Trans. Microwave Theory Tech. 52, 1361-1368 (2004).
[Crossref]

2003 (1)

H.-C. Lu and T.-H. Chu, "Multiport scattering matrix measurement using a reduced-port network analyzer," IEEE Trans. Microwave Theory Tech. 51, 1525-1533 (2003).
[Crossref]

2000 (1)

G. Griffel, "Synthesis of optical filters using ring resonator arrays," IEEE Photon. Technol. Lett. 12, 810-812 (2000).
[Crossref]

1995 (2)

Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

D. L. MacFarlane, E. M. Dowling, and V. Narayan, "Ring resonators with N×M couplers," Fiber Integr. Opt. 14, 195-210 (1995).
[Crossref]

1994 (2)

E. M. Dowling and D. L. MacFarlane, "Lightwave lattice filters for optically multiplexed communication systems," J. Lightwave Technol. 12, 471-486 (1994).
[Crossref]

D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994).
[Crossref]

1992 (1)

K. Araki and Y. Naito, "On the properties of lossless reciprocal 4-port circuits with reflection symmetry," IEEE Trans. Circuits Syst. 39, 155-161 (1992).
[Crossref]

1991 (3)

J. Esteban and J. M. Rebollar, "Generalized scattering matrix of generalized two-port discontinuities: application to four-port and nonsymmetric six-port couplers," IEEE Trans. Microwave Theory Tech. 39, 1725-1734 (1991).
[Crossref]

F. J. Fraile-Peláez, J. Capmany, and M. A. Muriel, "Transmission bistability in a double-coupler fiber ring resonator," Opt. Lett. 16, 907-909 (1991).
[Crossref] [PubMed]

K. Sasayama, M. Okuno, and K. Habara, "Coherent optical transversal filter using silica-based waveguides for high-speed signal processing," J. Lightwave Technol. 9, 1225-1230 (1991).
[Crossref]

1989 (1)

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

1985 (1)

O. Schwelb and R. Antepyan, "Conservation laws for distributed four-ports," IEEE Trans. Microwave Theory Tech. MTT-33, 157-160 (1985).
[Crossref]

1984 (1)

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
[Crossref]

1978 (1)

G. P. Riblet, "A coupling theorem for matched symmetrical two-branch four-port networks," IEEE Trans. Circuits Syst. CAS-25, 145-148 (1978).
[Crossref]

1968 (2)

J. J. Campbell, "Application of the solutions of certain boundary value problems to the symmetrical four-port junction and specially truncated bends in parallel-plate waveguides and balanced strip-transmission lines," IEEE Trans. Microwave Theory Tech. MTT-16, 165-176 (1968).
[Crossref]

R. Levy, "Analysis and synthesis of waveguide multi-aperture directional couplers," IEEE Trans. Microwave Theory Tech. MTT-16, 995-1006 (1968).
[Crossref]

1966 (1)

S. Hagelin, "A flow graph analysis of 3- and 4-port junction circulators," IEEE Trans. Microwave Theory Tech. MTT-14, 243-249 (1966).
[Crossref]

1965 (1)

K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans. Microwave Theory Tech. MTT-13, 194-202 (1965).
[Crossref]

1962 (1)

C. R. Boyd, Jr., "On a class of multiple line directional couplers," IRE Trans. Microwave Theory Tech. MTT-10, 287-294 (1962).
[Crossref]

1956 (2)

H. J. Carlin, "The scattering matrix in network theory," IRE Trans. Circuit Theory CT-3, 88-97 (1956).

J. Reed and G. J. Wheeler, "A method of analysis of symmetrical four port networks," IRE Trans. Microwave Theory Tech. MTT-4, 246-252 (1956).
[Crossref]

Amarasinghe, N. V.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

Antepyan, R.

O. Schwelb and R. Antepyan, "Conservation laws for distributed four-ports," IEEE Trans. Microwave Theory Tech. MTT-33, 157-160 (1985).
[Crossref]

Araki, K.

K. Araki and Y. Naito, "On the properties of lossless reciprocal 4-port circuits with reflection symmetry," IEEE Trans. Circuits Syst. 39, 155-161 (1992).
[Crossref]

Bigelow, J. A.

J. Martens, D. V. Judge, and J. A. Bigelow, "Uncertainties associated with many-port (>4) S-parameter measurements using a four-port vector network analyzer," IEEE Trans. Microwave Theory Tech. 52, 1361-1368 (2004).
[Crossref]

Boyd, C. R.

C. R. Boyd, Jr., "On a class of multiple line directional couplers," IRE Trans. Microwave Theory Tech. MTT-10, 287-294 (1962).
[Crossref]

Butler, J.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

Campbell, J. J.

J. J. Campbell, "Application of the solutions of certain boundary value problems to the symmetrical four-port junction and specially truncated bends in parallel-plate waveguides and balanced strip-transmission lines," IEEE Trans. Microwave Theory Tech. MTT-16, 165-176 (1968).
[Crossref]

Capmany, J.

Carlin, H. J.

H. J. Carlin, "The scattering matrix in network theory," IRE Trans. Circuit Theory CT-3, 88-97 (1956).

Chu, T.-H.

H.-C. Lu and T.-H. Chu, "Multiport scattering matrix measurement using a reduced-port network analyzer," IEEE Trans. Microwave Theory Tech. 51, 1525-1533 (2003).
[Crossref]

Dietrich, E.

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

Dowling, E. M.

D. L. MacFarlane, E. M. Dowling, and V. Narayan, "Ring resonators with N×M couplers," Fiber Integr. Opt. 14, 195-210 (1995).
[Crossref]

E. M. Dowling and D. L. MacFarlane, "Lightwave lattice filters for optically multiplexed communication systems," J. Lightwave Technol. 12, 471-486 (1994).
[Crossref]

D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994).
[Crossref]

Esteban, J.

J. Esteban and J. M. Rebollar, "Generalized scattering matrix of generalized two-port discontinuities: application to four-port and nonsymmetric six-port couplers," IEEE Trans. Microwave Theory Tech. 39, 1725-1734 (1991).
[Crossref]

Evans, G.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

Evans, G. A.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

Fafadia, C.

Fafadia, C. B.

C. B. Fafadia, "Thick linear optical lattice filters," Master's thesis (University of Texas at Dallas, 2003).

Fraile-Peláez, F. J.

Gantmacher, F. R.

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1977), Vol. 1.

Goodman, J. W.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
[Crossref]

Govindan, V.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
[Crossref] [PubMed]

Griffel, G.

G. Griffel, "Synthesis of optical filters using ring resonator arrays," IEEE Photon. Technol. Lett. 12, 810-812 (2000).
[Crossref]

Habara, K.

K. Sasayama, M. Okuno, and K. Habara, "Coherent optical transversal filter using silica-based waveguides for high-speed signal processing," J. Lightwave Technol. 9, 1225-1230 (1991).
[Crossref]

Hagelin, S.

S. Hagelin, "A flow graph analysis of 3- and 4-port junction circulators," IEEE Trans. Microwave Theory Tech. MTT-14, 243-249 (1966).
[Crossref]

Heidrich, H.

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

Henry, C.

Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

Hoffmann, D.

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

Horn, R. A.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991).
[Crossref]

Hunt, L. R.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
[Crossref] [PubMed]

I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
[Crossref]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991).
[Crossref]

Judge, D. V.

J. Martens, D. V. Judge, and J. A. Bigelow, "Uncertainties associated with many-port (>4) S-parameter measurements using a four-port vector network analyzer," IEEE Trans. Microwave Theory Tech. 52, 1361-1368 (2004).
[Crossref]

Kannan, G.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
[Crossref]

Kurokawa, K.

K. Kurokawa, "Power waves and the scattering matrix," IEEE Trans. Microwave Theory Tech. MTT-13, 194-202 (1965).
[Crossref]

Langenhorst, R.

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

Laskowski, E.

Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

Leonhardt, Ulf

Ulf Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, 1997).

Levy, R.

R. Levy, "Analysis and synthesis of waveguide multi-aperture directional couplers," IEEE Trans. Microwave Theory Tech. MTT-16, 995-1006 (1968).
[Crossref]

Li, Y.

Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

Lounesto, P.

P. Lounesto, Clifford Algebras and Spinors (Cambridge U. Press, 2001).
[Crossref]

Lu, H.-C.

H.-C. Lu and T.-H. Chu, "Multiport scattering matrix measurement using a reduced-port network analyzer," IEEE Trans. Microwave Theory Tech. 51, 1525-1533 (2003).
[Crossref]

MacFarlane, D. L.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
[Crossref] [PubMed]

D. L. MacFarlane, E. M. Dowling, and V. Narayan, "Ring resonators with N×M couplers," Fiber Integr. Opt. 14, 195-210 (1995).
[Crossref]

E. M. Dowling and D. L. MacFarlane, "Lightwave lattice filters for optically multiplexed communication systems," J. Lightwave Technol. 12, 471-486 (1994).
[Crossref]

D. L. MacFarlane and E. M. Dowling, "Z-domain techniques in the analysis of Fabry-Perot etalons and multilayer structures," J. Opt. Soc. Am. A 11, 236-245 (1994).
[Crossref]

I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
[Crossref]

Madsen, C.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).

Mak, C.

Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

Martens, J.

J. Martens, D. V. Judge, and J. A. Bigelow, "Uncertainties associated with many-port (>4) S-parameter measurements using a four-port vector network analyzer," IEEE Trans. Microwave Theory Tech. 52, 1361-1368 (2004).
[Crossref]

Masood, T.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

McWilliams, S.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

Moslehi, B.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
[Crossref]

Muriel, M. A.

Murnaghan, F. D.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, 1962).

Naito, Y.

K. Araki and Y. Naito, "On the properties of lossless reciprocal 4-port circuits with reflection symmetry," IEEE Trans. Circuits Syst. 39, 155-161 (1992).
[Crossref]

Narayan, V.

D. L. MacFarlane, E. M. Dowling, and V. Narayan, "Ring resonators with N×M couplers," Fiber Integr. Opt. 14, 195-210 (1995).
[Crossref]

Okuno, M.

K. Sasayama, M. Okuno, and K. Habara, "Coherent optical transversal filter using silica-based waveguides for high-speed signal processing," J. Lightwave Technol. 9, 1225-1230 (1991).
[Crossref]

Panahi, I.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
[Crossref] [PubMed]

Panahi, I. M. S.

I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
[Crossref]

Patterson, S.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

Pozar, D. M.

D. M. Pozar, Microwave Engineering, 2nd ed. (Wiley, 1998).

Rebollar, J. M.

J. Esteban and J. M. Rebollar, "Generalized scattering matrix of generalized two-port discontinuities: application to four-port and nonsymmetric six-port couplers," IEEE Trans. Microwave Theory Tech. 39, 1725-1734 (1991).
[Crossref]

Reed, J.

J. Reed and G. J. Wheeler, "A method of analysis of symmetrical four port networks," IRE Trans. Microwave Theory Tech. MTT-4, 246-252 (1956).
[Crossref]

Riblet, G. P.

G. P. Riblet, "A coupling theorem for matched symmetrical two-branch four-port networks," IEEE Trans. Circuits Syst. CAS-25, 145-148 (1978).
[Crossref]

Roh, D.

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
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Sasayama, K.

K. Sasayama, M. Okuno, and K. Habara, "Coherent optical transversal filter using silica-based waveguides for high-speed signal processing," J. Lightwave Technol. 9, 1225-1230 (1991).
[Crossref]

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B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
[Crossref]

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R. P. Stanley, Enumerative Combinatorics (Cambridge U. Press, 1999), Vol. 2.
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Tong, J.

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

D. L. MacFarlane, J. Tong, C. Fafadia, V. Govindan, L. R. Hunt, and I. Panahi, "Extended lattice filters enabled by four directoinal couplers," Appl. Opt. 43, 6124-6133 (2004).
[Crossref] [PubMed]

I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
[Crossref]

Tur, M.

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
[Crossref]

Wenke, G.

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

Wheeler, G. J.

J. Reed and G. J. Wheeler, "A method of analysis of symmetrical four port networks," IRE Trans. Microwave Theory Tech. MTT-4, 246-252 (1956).
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Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

Zhao, J.

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).

Appl. Opt. (1)

Electron. Lett. (1)

Y. Li, C. Henry, E. Laskowski, C. Mak, and H. Yaffe, "Waveguide EDFA gain equalization filter," Electron. Lett. 31, 2005-2006 (1995).
[Crossref]

EURASIP J. Appl. Signal Process. (1)

L. R. Hunt, V. Govindan, I. Panahi, J. Tong, G. Kannan, D. L. MacFarlane, and G. Evans, "Active optical lattice filters," EURASIP J. Appl. Signal Process. 10, 1-11 (2005).

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[Crossref]

IEEE Photon. Technol. Lett. (2)

G. Griffel, "Synthesis of optical filters using ring resonator arrays," IEEE Photon. Technol. Lett. 12, 810-812 (2000).
[Crossref]

D. Roh, T. Masood, S. Patterson, N. V. Amarasinghe, S. McWilliams, G. A. Evans, and J. Butler, "Dual-wavelength AllnGaAS-InP grating-outcoupled surface-emitting laser with an integrated two dimensional photonic lattice outcoupler," IEEE Photon. Technol. Lett. 17, 270-273 (2005).
[Crossref]

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K. Araki and Y. Naito, "On the properties of lossless reciprocal 4-port circuits with reflection symmetry," IEEE Trans. Circuits Syst. 39, 155-161 (1992).
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G. P. Riblet, "A coupling theorem for matched symmetrical two-branch four-port networks," IEEE Trans. Circuits Syst. CAS-25, 145-148 (1978).
[Crossref]

IEEE Trans. Microwave Theory Tech. (8)

O. Schwelb and R. Antepyan, "Conservation laws for distributed four-ports," IEEE Trans. Microwave Theory Tech. MTT-33, 157-160 (1985).
[Crossref]

J. Esteban and J. M. Rebollar, "Generalized scattering matrix of generalized two-port discontinuities: application to four-port and nonsymmetric six-port couplers," IEEE Trans. Microwave Theory Tech. 39, 1725-1734 (1991).
[Crossref]

H.-C. Lu and T.-H. Chu, "Multiport scattering matrix measurement using a reduced-port network analyzer," IEEE Trans. Microwave Theory Tech. 51, 1525-1533 (2003).
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J. Martens, D. V. Judge, and J. A. Bigelow, "Uncertainties associated with many-port (>4) S-parameter measurements using a four-port vector network analyzer," IEEE Trans. Microwave Theory Tech. 52, 1361-1368 (2004).
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IRE Trans. Microwave Theory Tech. (2)

J. Reed and G. J. Wheeler, "A method of analysis of symmetrical four port networks," IRE Trans. Microwave Theory Tech. MTT-4, 246-252 (1956).
[Crossref]

C. R. Boyd, Jr., "On a class of multiple line directional couplers," IRE Trans. Microwave Theory Tech. MTT-10, 287-294 (1962).
[Crossref]

J. Lightwave Technol. (3)

E. M. Dowling and D. L. MacFarlane, "Lightwave lattice filters for optically multiplexed communication systems," J. Lightwave Technol. 12, 471-486 (1994).
[Crossref]

D. Hoffmann, H. Heidrich, G. Wenke, R. Langenhorst, and E. Dietrich, "Integrated optics eight-port 90° hybrid on LiNbO3," J. Lightwave Technol. 7, 794-798 (1989).
[Crossref]

K. Sasayama, M. Okuno, and K. Habara, "Coherent optical transversal filter using silica-based waveguides for high-speed signal processing," J. Lightwave Technol. 9, 1225-1230 (1991).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Proc. IEEE (1)

B. Moslehi, J. W. Goodman, M. Tur, and H. J. Shaw, "Fiber optic lattice signal processing," Proc. IEEE 72, 909-930 (1984).
[Crossref]

Other (10)

C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).

I. M. S. Panahi, G. Kannan, L. R. Hunt, D. L. MacFarlane, and J. Tong, "Lattice filter with adjustable gains and its application in optical signal processing," in 2005 IEEE/SP 13th Workshop on Statistical Signal Processing (IEEE Press, 2005), pp. 321-326.
[Crossref]

D. M. Pozar, Microwave Engineering, 2nd ed. (Wiley, 1998).

C. B. Fafadia, "Thick linear optical lattice filters," Master's thesis (University of Texas at Dallas, 2003).

R. A. Horn and C. R. Johnson, Topics in Matrix Analysis (Cambridge U. Press, 1991).
[Crossref]

F. R. Gantmacher, The Theory of Matrices (Chelsea, 1977), Vol. 1.

F. D. Murnaghan, The Unitary and Rotation Groups (Spartan, 1962).

R. P. Stanley, Enumerative Combinatorics (Cambridge U. Press, 1999), Vol. 2.
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P. Lounesto, Clifford Algebras and Spinors (Cambridge U. Press, 2001).
[Crossref]

Ulf Leonhardt, Measuring the Quantum State of Light (Cambridge U. Press, 1997).

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Figures (5)

Fig. 1
Fig. 1

Two deep trenches perpendicular to each other and oriented at 45° to a 3 μ m wide ridge waveguide. The four-port coupler is milled with a focused ion beam.

Fig. 2
Fig. 2

Signal flow diagram of a four-direction coupler that reflects, transmits, routes left, and routes right. In our notation, the four ports are labeled W, N, E, and S. At each port, there is a reflected component ρ, a transmitted component τ, a right-directed component α, and a left-directed component β. Shown explicitly in this diagram are ρ w , τ w , α w , and β w , the coupling coefficients at the W port.

Fig. 3
Fig. 3

Transmission line interpretation of the matrix multiplication of a 2 × 2 decomposition into simple unitary matrices.

Fig. 4
Fig. 4

Transmission line interpretation of the matrix multiplication of a 4 × 4 decomposition into simple unitary matrices.

Fig. 5
Fig. 5

General transmission line interpretation of the matrix multiplication of an n × n decomposition into simple unitary matrices.

Equations (111)

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F = [ 0 f 1 f 2 f 3 f 1 0 f 4 f 5 f 2 f 4 0 f 6 f 3 f 5 f 6 0 ] .
s R 3 = 1 2 [ ( f 1 + f 6 ) , ( f 5 f 2 ) , ( f 3 + f 4 ) ] ,
t R 3 = 1 2 [ ( f 1 f 6 ) , ( f 5 + f 2 ) , ( f 3 f 4 ) ] .
λ 2 = ( s 2 + t 2 ) , μ = ( s 2 t 2 ) ,
S = 1 1 + λ 2 [ ( 1 λ 2 ) I + 2 F ] .
S = I + 2 1 + 4 s 2 ( F + F 2 ) .
S = 1 1 + 2 λ 2 + μ 2 [ ( 1 + 2 λ 2 μ 2 ) I + ( 2 + 4 λ 2 ) F + 2 F 2 + 2 F 3 ] .
S = 2 1 + λ 2 [ 1 λ 2 f 1 f 2 f 3 f 1 1 λ 2 f 4 f 5 f 2 f 4 1 λ 2 f 6 f 3 f 5 f 6 1 λ 2 ] .
S = Γ [ A B C D ] ,
A = [ ( 1 Γ f 1 2 f 2 2 f 3 2 ) ( f 1 f 2 f 4 f 3 f 5 ) ( f 2 + f 1 f 4 f 3 f 6 ) ( f 3 + f 2 f 6 + f 1 f 5 ) ] ,
B = [ ( f 1 f 2 f 4 f 3 f 5 ) ( 1 Γ f 1 2 f 4 2 f 5 2 ) ( f 4 f 1 f 2 f 5 f 6 ) ( f 5 f 1 f 3 + f 4 f 6 ) ] ,
C = [ ( f 2 + f 1 f 4 f 3 f 6 ) ( f 4 f 1 f 2 f 5 f 6 ) ( 1 Γ f 2 2 f 4 2 f 6 2 ) ( f 6 f 2 f 3 f 4 f 5 ) ] ,
D = [ ( f 3 + f 2 f 6 f 1 f 5 ) ( f 5 f 1 f 3 + f 4 f 6 ) ( f 6 f 2 f 3 f 4 f 5 ) ( 1 Γ f 3 2 f 5 2 f 6 2 ) ] ,
Γ = 2 1 + 4 s 2 .
S = E [ K L M N ] ,
K = [ 1 E + 1 + f 4 2 + f 5 2 + f 6 2 ( f 1 f 6 2 + f 1 + f 2 f 4 + f 3 f 5 + f 3 f 4 f 6 f 2 f 5 f 6 ) ( f 2 f 5 2 + f 2 f 1 f 4 + f 3 f 6 f 3 f 4 f 5 f 1 f 5 f 6 ) ( f 3 f 4 2 + f 3 f 2 f 6 f 1 f 5 f 2 f 4 f 5 + f 1 f 4 f 6 ) ] ,
L = [ ( f 1 f 6 2 + f 1 f 2 f 4 f 3 f 5 + f 3 f 4 f 6 f 2 f 5 f 6 ) 1 E + 1 + f 2 2 + f 3 2 + f 6 2 ( f 4 f 3 2 + f 4 + f 1 f 2 + f 5 f 6 + f 1 f 3 f 6 f 2 f 3 f 5 ) ( f 5 f 2 2 + f 5 + f 1 f 3 f 4 f 6 f 2 f 3 f 4 f 1 f 2 f 6 ) ] ,
M = [ ( f 2 f 5 2 + f 2 + f 1 f 4 f 3 f 6 f 3 f 4 f 5 f 1 f 5 f 6 ) ( f 4 f 3 2 f 4 + f 1 f 2 + f 5 f 6 f 1 f 3 f 6 + f 2 f 3 f 5 ) 1 E + 1 + f 1 2 + f 3 2 + f 5 2 ( f 6 f 1 2 + f 6 + f 2 f 3 + f 4 f 5 + f 1 f 3 f 4 f 1 f 2 f 5 ) ] ,
N = [ ( f 3 f 4 2 + f 3 + f 2 f 6 + f 1 f 5 + f 1 f 4 f 6 + f 1 f 4 f 6 f 2 f 4 f 5 ) ( f 5 f 2 2 f 5 + f 1 f 3 f 4 f 6 + f 2 f 3 f 4 + f 1 f 2 f 6 ) ( f 6 f 1 2 f 6 + f 2 f 3 + f 4 f 5 f 1 f 3 f 4 + f 1 f 2 f 5 ) 1 E + 1 + f 1 2 + f 2 2 + f 4 2 ] ,
E = 2 1 + 2 λ 2 + μ 2 .
f 1 = f 2 = f 3 = f 4 = f 5 = f 6 = 2 3 ,
G = [ 0.2077 0.0383 0.1917 0.9585 0.9585 0.2077 0.0383 0.1917 0.1917 0.9585 0.2077 0.0383 0.0383 .1917 0.9585 0.2077 ] ,
f 1 = f 2 = f 3 = 0 ,
f 4 = f 5 = f 6 = 1 2 ,
G = [ 1 0 0 0 0 0.4286 0.2857 0.8571 0 0.8571 0.4286 0.2857 0 0.2857 0.8571 0.4286 ] .
Tr ( S ) = 4 p 0 q 0 ,
κ = 6 p 0 2 q 0 2 + 2 p 0 2 q ̃ q ̃ 2 ( q ̃ q ̃ ) ( p ̃ p ̃ ) .
H = [ a 2 + b 2 c 2 d 2 2 b c 2 a d 2 a c + 2 b d 2 b c + 2 a d a 2 b 2 + c 2 d 2 2 c d 2 a b 2 b d 2 a c 2 a b + 2 c d a 2 b 2 c 2 + d 2 ] ,
a 2 + b 2 + c 2 + d 2 = 1 ,
a 2 + b 2 c 2 d 2 = h 11 ,
a 2 b 2 + c 2 d 2 = h 22 ,
a 2 b 2 c 2 + d 2 = h 33 .
S = [ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ]
p = 1 + 0 i + 0 j + 0 k ,
q = 1 2 + 1 2 i + 1 2 j + 1 2 k .
B = 1 1 + p 0 q 0 = 1 1 + 1 2 = 2 3 ,
A = B 1 = 1 3 .
F = [ 0 1 3 1 3 1 3 1 3 0 1 3 1 3 1 3 1 3 0 1 3 1 3 1 3 1 3 0 ] .
P = [ p 11 p 12 p 21 p 22 ] U ( 2 ) .
J ( γ 11 ) * [ 1 0 0 γ ¯ 21 ] P = [ γ ¯ 11 d 11 d 11 γ 11 ] [ 1 0 0 γ ¯ 21 ] [ γ 11 s 12 γ 21 d 11 s 22 ] = [ γ ¯ 11 d 11 d 11 γ 11 ] [ γ 11 s 12 d 11 γ ¯ 21 s 22 ] [ 1 A 0 B ] .
P = [ 1 0 0 γ 21 ] J ( γ 11 ) [ 1 0 0 γ 12 ] = [ γ 11 d 11 γ 12 γ 21 d 11 γ 21 γ ¯ 11 γ 12 ] .
S = [ s 11 s 12 s 13 s 14 s 21 s 22 s 23 s 24 s 31 s 32 s 33 s 34 s 41 s 42 s 43 s 44 ] U ( 4 ) .
d i j = ( 1 γ i j 2 ) 1 2 .
γ 11 = s 11 ,
γ 21 = s 21 d 11 .
γ 31 = s 31 d 21 d 11 .
γ 41 = s 41 d 31 d 21 d 11 .
S 41 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 γ ¯ 41 ] ,
S = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 γ ¯ 41 ] S = [ γ 11 s 12 41 s 13 41 s 14 41 γ 21 d 11 s 22 41 s 23 41 s 24 41 γ 31 d 21 d 11 s 32 41 s 33 41 s 34 41 d 31 d 21 d 11 s 42 41 s 43 41 s 44 41 ] .
S 31 * = [ 1 0 0 0 0 1 0 0 0 0 γ 31 d 31 0 0 d 31 γ ¯ 31 ] * ,
[ 1 0 0 0 0 1 0 0 0 0 γ ¯ 31 d 31 0 0 d 31 γ 31 ] [ γ 11 s 12 31 s 13 31 s 14 31 γ 21 d 11 s 22 31 s 23 31 s 24 31 γ 31 d 21 d 11 s 32 31 s 33 31 s 34 31 d 31 d 21 d 11 s 42 31 s 43 31 s 44 31 ] = [ γ 11 s 12 31 s 13 31 s 14 31 γ 21 d 11 s 22 31 s 23 31 s 24 31 ( γ 31 2 + d 31 2 ) d 21 d 11 s 32 31 s 33 31 s 34 31 ( d 31 γ 31 γ 31 d 31 ) d 21 d 11 s 42 31 s 43 31 s 44 31 ] = [ γ 11 s 12 31 s 13 31 s 14 31 γ 21 d 11 s 22 31 s 23 31 s 24 31 d 21 d 11 s 32 31 s 33 31 s 34 31 0 s 42 31 s 43 31 s 44 31 ] .
S 11 * S 21 * S 31 * S 41 * S = S 1 = [ 1 s 12 1 s 13 1 s 14 1 0 s 22 1 s 23 1 s 24 1 0 s 32 1 s 33 1 s 34 1 0 s 42 1 s 43 1 s 44 1 ] .
G 1 = [ 1 0 0 0 0 s 22 1 s 23 1 s 24 1 0 s 32 1 s 33 1 s 34 1 0 s 42 1 s 43 1 s 44 1 ] .
γ 22 = s 22 1 ,
γ 32 = s 32 1 d 22 .
γ 42 = s 42 1 d 32 d 22 .
S 22 * S 32 * S 42 * S 1 = S 2 = [ 1 0 0 0 0 1 s 23 2 s 24 2 0 0 s 33 2 s 34 2 0 0 s 43 2 s 44 2 ] ,
S 2 = [ 1 0 0 0 0 1 0 0 0 0 s 33 2 s 34 2 0 0 s 43 2 s 44 2 ] .
γ 33 = s 33 2 ,
γ 43 = s 43 2 d 33 ,
S 33 * S 43 * S 2 = S 3 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 s 44 3 ] ,
s 44 3 = 1 and γ 44 = s 44 3 .
S = S 41 S 31 S 21 S 11 S 42 S 32 S 22 S 43 S 33 S 44 ,
G S = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 γ 41 ] [ 1 0 0 0 0 1 0 0 0 0 γ 31 d 31 0 0 d 31 γ ¯ 31 ] [ 1 0 0 0 0 γ 21 d 21 0 0 d 21 γ ¯ 21 0 0 0 0 1 ] [ γ 11 d 11 0 0 d 11 γ ¯ 11 0 0 0 0 1 0 0 0 0 1 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 γ 42 ] [ 1 0 0 0 0 1 0 0 0 0 γ 32 d 32 0 0 d 32 γ ¯ 32 ] [ 1 0 0 0 0 γ 22 d 22 0 0 d 22 γ ¯ 22 0 0 0 0 1 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 γ 43 ] [ 1 0 0 0 0 1 0 0 0 0 γ 33 d 33 0 0 d 33 γ ¯ 33 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 γ 44 ] .
γ 41 = γ 42 = γ 43 = γ 44 = 1 ,
γ 11 = γ 21 = γ 31 = γ 22 = γ 32 = γ 33 = 0.05 ,
S = [ 0.05 0.0499 0.0499 0.9963 0.0499 0.0498 0.996 0.0549 0.0499 0.996 0.055 0.0497 0.9963 0.0549 0.0497 0.0448 ] ,
γ 41 = γ 42 = γ 43 = γ 44 = 1 ,
γ 11 = γ 21 = γ 31 = γ 22 = γ 32 = γ 33 = 1 2 ,
S = [ 0.5 0.433 0.375 0.6495 0.433 0.25 0.433 0.75 0.375 0.433 0.8125 0.1083 0.6495 0.75 0.1083 0.0625 ] .
S = [ 0.5 0.4330 0.375 0.6495 0.4330 0.25 0.4330 0.75 0.3750 0.4330 0.8125 0.1083 0.6495 0.75 0.1083 0.0625 ] ,
γ 11 = s 11 = 0.5 , d 11 = 0.866 ,
γ 21 = s 21 d 11 = 0.5 , d 21 = 0.866 ,
γ 31 = s 31 d 21 d 11 = 0.5 , d 31 = 0.866 ,
γ 41 = s 41 d 31 d 21 d 11 = 1 .
S 11 = [ 0.5 0.866 0 0 0.866 0.5 0 0 0 0 1 0 0 0 0 1 ] , S 21 = [ 1 0 0 0 0 0.5 0.866 0 0 0.866 0.5 0 0 0 0 1 ] ,
S 31 = [ 1 0 0 0 0 1 0 0 0 0 0.5 0.866 0 0 0.866 0.5 ] , S 41 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
S 1 = S 11 * S 21 * S 31 * S 41 * S = [ 1 0 0 0 0 0.5 0.433 0.75 0 0.433 0.625 0.6495 0 0.75 0.6495 0.125 ] .
γ 22 = s 11 = 0.5 , d 22 = 0.866 ,
γ 32 = s 32 d 22 = 0.5 , d 32 = 0.866 ,
γ 42 = s 42 d 32 d 22 = 1 .
S 22 = [ 1 0 0 0 0 0.5 0.866 0 0 0.866 0.5 0 0 0 0 1 ] , S 32 = [ 1 0 0 0 0 1 0 0 0 0 0.5 0.866 0 0 0.866 0.5 ] , S 42 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
S 2 = S 22 * S 32 * S 42 * S 1 = [ 1 0 0 0 0 1 0 0 0 0 0.5 0.866 0 0 0.866 0.5 ] .
γ 33 = s 33 2 = 0.5 , d 33 = 0.866 ,
γ 43 = s 43 2 d 33 = 1
S 33 = [ 1 0 0 0 0 1 0 0 0 0 0.5 0.866 0 0 0.866 0.5 ] , S 43 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
S 3 = S 33 * S 43 * S 2 = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ]
A = [ γ 11 γ 12 γ 1 , n 1 γ 1 n γ 21 γ 22 γ 2 , n 1 γ n 1 , 1 γ n 1 , 2 γ n 1 ]
Φ ( A ) ( = U ( A ) ) = [ 1 1 γ n 1 ] [ 1 1 J ( γ n 1 , 1 ) ] [ J ( γ 11 ) 1 1 ] [ 1 1 γ n 1 , 2 ] [ 1 1 J ( γ n 2 , 2 ) ] [ 1 J ( γ 1 , 2 ) 1 ] [ 1 1 γ 2 , n 1 ] [ 1 1 J ( γ 1 , n 1 ) ] [ 1 1 γ 1 n ] .
P = [ 1 0 0 1 ]
S = [ 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ] ,
γ 11 = s 11 = 0.5 , d 11 = 0.866 ,
γ 21 = s 21 d 11 = 0.5774 , d 21 = 0.8165 ,
γ 31 = s 31 d 21 d 11 = 0.7071 , d 31 = 0.7071 ,
γ 41 = s 41 d 31 d 21 d 11 = 1 .
S 1 = S 11 * S 21 * S 31 * S 41 * S = [ 1 0 0 0 0 0.5774 0.5774 0.5774 0 0.8165 0.4083 0.4083 0 0 0.7071 0.7071 ] .
γ 22 = s 22 1 = 0.5774 , d 22 = 0.8165 ,
γ 32 = s 32 1 d 22 = 1 , d 32 = 0 .
γ 33 d 22 d 11 = 1 2 ,
γ 44 d 33 d 22 d 11 = 1 2 ,
γ 41 d 31 γ 21 γ 32 γ 22 d 33 γ 44 γ 41 d 31 d 21 γ 11 d 22 d 33 γ 44 γ 41 γ 31 γ 42 γ 32 γ 43 γ 33 γ 44 = 1 2 ,
γ 42 γ 43 2 = 1 2 .
S = [ γ 11 d 11 γ 12 γ 21 d 11 γ 21 γ ¯ 11 γ 12 ] .
s 11 = a 11 ( p ) = p S 11 2 ( i , j ) B 2 a i j ( p ) ,
s 12 = d 11 γ 12 = a 11 ( Q ) a 12 ( Q ) = p S 12 2 ( i , j ) B 2 a i j ( p ) ,
s 21 = γ 21 d 11 = a 21 ( γ ) a 11 ( γ ) = p S 21 2 ( i , j ) B 2 a i j ( p ) ,
s 22 = γ 21 γ ¯ 11 γ 12 = a 21 ( s ) a 11 ( s ) a 12 ( s ) p S 22 2 ( i , j ) B 2 a i j ( p ) .
s i j ( S ) = p S i j n ( i j ) B n a i j ( p ) ,
# ( S i j n ) = c j + i 1 , j i + 1 , 1 i j n ,
# ( S i j n ) = c j + i 2 , n i , 1 i n ,
# ( S i j n ) = # ( S j i n ) ,

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