Abstract

We show that it is possible to obtain 2×2 waveguide couplers with arbitrary power splitting ratios by interconnecting a pair of unequal-width waveguides as the phase-tuning section into the middle of two short MMI sections. These couplers have simple geometry and low loss. They offer valuable new possibilities for designing waveguide-based photonic integrated circuits.

© 2008 Optical Society of America

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References

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  1. L. B. Soldano and 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. D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
    [CrossRef]
  3. S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995).
    [CrossRef]
  4. S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
    [CrossRef]
  5. V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
    [CrossRef]
  6. P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
    [CrossRef]
  7. M. Bachmann, P. A. Besse, and 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]
  8. J. Leuthold and C. H. Joyner, "Multimode interference couplers with tunable power splitting ratios," J. Lightwave Technol. 19, 700-707 (2001).
    [CrossRef]
  9. N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999).
    [CrossRef]
  10. H. Ohe, H. Shimizu, and Y. Nakano, "InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion," IEEE Photon. Technol. Lett. 19, 1816-1818 (2007).
    [CrossRef]
  11. Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
    [CrossRef]
  12. T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
    [CrossRef]
  13. S. Y. Tseng, C. Fuentes-Hernandez, D. Owens, and B. Kippelen, "Variable splitting ratio 2�?2 MMI couplers using multimode waveguide holograms," Opt. Express 15, 9015-9021 (2007).
    [CrossRef] [PubMed]
  14. D. J. Y. Feng, T. S. Lay, and T. Y. Chang, "Waveguide couplers with new power splitting ratios made possible by cascading of short multimode interference sections," Opt. Express 15, 1588-1593 (2007).
    [CrossRef] [PubMed]
  15. BeamPROP, version 5.1, RSoft Inc., NY (2005).
  16. FimmProp, version 4.3, Photon Design, Oxford, UK (2004).
  17. D. J. Y. Feng, T. S. Lay, and T. Y. Chang, "Compact 2x2 Couplers for Unequal Splitting of Power Obtained by Cascading of Short MMI Sections," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JThA25, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-JThA25

2007 (3)

2003 (2)

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

2002 (1)

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
[CrossRef]

2001 (1)

1999 (2)

T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
[CrossRef]

N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999).
[CrossRef]

1996 (2)

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
[CrossRef]

1995 (3)

S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995).
[CrossRef]

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

M. Bachmann, P. A. Besse, and 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]

Bachmann, M.

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
[CrossRef]

M. Bachmann, P. A. Besse, and 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]

Besse, P. A.

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
[CrossRef]

M. Bachmann, P. A. Besse, and 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]

Chang, T. Y.

Feng, D. J. Y.

Forrest, S. R.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

Fuentes-Hernandez, C.

Gini, E.

P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
[CrossRef]

Glesk, I.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

Hamacher, M.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
[CrossRef]

Heidrich, H.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
[CrossRef]

Hibino, Y.

S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995).
[CrossRef]

Himeno, A.

T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
[CrossRef]

Hunziker, W.

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

Joyner, C. H.

Kippelen, B.

Lagali, N. S.

N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999).
[CrossRef]

Lai, Q.

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

Lay, T. S.

Leuthold, J.

Li, C.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

MacDonald, R. I.

N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999).
[CrossRef]

Matsuo, S.

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

Melchior, H.

P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
[CrossRef]

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

M. Bachmann, P. A. Besse, and 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]

Menon, V. M.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

Nakano, Y.

H. Ohe, H. Shimizu, and Y. Nakano, "InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion," IEEE Photon. Technol. Lett. 19, 1816-1818 (2007).
[CrossRef]

Oda, K.

S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995).
[CrossRef]

Ohe, H.

H. Ohe, H. Shimizu, and Y. Nakano, "InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion," IEEE Photon. Technol. Lett. 19, 1816-1818 (2007).
[CrossRef]

Ohiso, Y.

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

Okamoto, H.

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

Okamoto, K.

T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 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, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
[CrossRef]

Owens, D.

Paiam, M. R.

N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999).
[CrossRef]

Pennings, E. C. M.

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

Prucnal, P. R.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

Rabus, D. G.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
[CrossRef]

Saida, T.

T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
[CrossRef]

Segawa, T.

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

Shimizu, H.

H. Ohe, H. Shimizu, and Y. Nakano, "InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion," IEEE Photon. Technol. Lett. 19, 1816-1818 (2007).
[CrossRef]

Soldano, L. B.

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

Sugita, A.

T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
[CrossRef]

Suzuki, S.

S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995).
[CrossRef]

Tong, W.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

Troppenz, U.

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
[CrossRef]

Tseng, S. Y.

Xia, F.

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

Yoshikuni, Y.

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (2)

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, "Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers," Electron. Lett. 32, 1576-1577 (1996).
[CrossRef]

T. Saida, A. Himeno, M. Okuno, A. Sugita, and K. Okamoto, "Silica-based 2x2 multimode interference coupler with arbitrary power splitting ratio," Electron. Lett.  35, 2031-2033 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

D. G. Rabus, M. Hamacher, U. Troppenz, and H. Heidrich, "Optical filters based on ring resonators with integrated semiconductor optical amplifiers in GaInAsP-InP," IEEE J. Sel. Top. Quantum Electron. 8, 1405-1411 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

V. M. Menon, W. Tong, C. Li, F. Xia, I. Glesk, P. R. Prucnal, and S. R. Forrest, "All-optical wavelength conversion using a regrowth-free monolithically integrated Sagnac interferometer," IEEE Photon. Technol. Lett. 15, 254-256 (2003).
[CrossRef]

N. S. Lagali, M. R. Paiam, and R. I. MacDonald, "Theory of variable-ratio power splitters using multimode interference couplers," IEEE Photon. Technol. Lett. 11, 665-667 (1999).
[CrossRef]

H. Ohe, H. Shimizu, and Y. Nakano, "InGaAlAs multiple-quantum-well optical phase modulators based on carrier depletion," IEEE Photon. Technol. Lett. 19, 1816-1818 (2007).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

S. Matsuo, Y. Yoshikuni, T. Segawa, Y. Ohiso, and H. Okamoto, "A widely tunable optical filter using ladder-type structure," IEEE Photonics Technol. Lett. 15, 1114-1116 (2003).
[CrossRef]

J. Lightwave Technol. (4)

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

S. Suzuki, K. Oda, and Y. Hibino, "Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz," J. Lightwave Technol. 13, 1766-1770 (1995).
[CrossRef]

P. A. Besse, E. Gini, M. Bachmann and H. Melchior, "New 2x2 and 1x3 Multimode Interference Couplers with Free Selection of Power Splitting Ratios," J. Lightwave Technol. 14, 2286-2293 (1996).
[CrossRef]

J. Leuthold and C. H. Joyner, "Multimode interference couplers with tunable power splitting ratios," J. Lightwave Technol. 19, 700-707 (2001).
[CrossRef]

Opt. Express (2)

Other (3)

BeamPROP, version 5.1, RSoft Inc., NY (2005).

FimmProp, version 4.3, Photon Design, Oxford, UK (2004).

D. J. Y. Feng, T. S. Lay, and T. Y. Chang, "Compact 2x2 Couplers for Unequal Splitting of Power Obtained by Cascading of Short MMI Sections," in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JThA25, http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2007-JThA25

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagrams of a 2×2 coupler based on two short MMI sections interconnected by a pair of phase-shifter waveguides in their middle: (a) In-plan view of the 2×2 coupler; and (b) Cross-section view of the basic layer structure in the waveguide ridge.

Fig. 2.
Fig. 2.

K vs. length for all possible shortest 2×2 MMI couplers made up by simple rectangular geometry. Number by Italic: the device length in unit of As2 . (A+D): MMI-A+MMI-D, (2A): MMI-A+MMI-A, hD: half of MMI-D.

Fig. 3.
Fig. 3.

The simulated 2-D field maps for seven different cascaded MMI couplers with (a) K=1 (A+A), (b) K=0 (A+B), (c) K=0.15 (A+C), (d) K=0.07 (A+D), (e) K=0.93 (B+D), (f) K=0.64 (C+D), and (g) K=0.80 (D+D). (0.96 Total): 96% transmittance in total. The values for total transmittance and K are obtained by 3-D BPM.

Fig. 4.
Fig. 4.

The MPA analyzed and 2-D EME simulated results for K vs. total length (LU+LPS+LV) for the 2×2 couplers with phase shifter of (a) βa >βb (Δ : δab=0.2/0 µm; o : δab=0.1/0 µm), and (b) βa <βb (Δ : δab=0/0.2 µm; o : δab=0/0.1 µm).

Tables (1)

Tables Icon

Table I Transfer functions of MMI-A, MMI-B, MMI-C, and MMI-D

Equations (4)

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

L π = 4 n s W e 2 3 λ 0 = 4 n s r 2 s 2 3 λ 0 A r 2 s 2 ,
H a ( b ) = exp ( j β a ( b ) L PS ) for waveguide a and b
[ HB U ( PS ) V HX ± U ( PS ) V HX U ( PS ) V HB ± U ( PS ) V ] = [ HB U HX ± U HX U HB U ] [ H b 0 0 H a ] [ HB V HX ± V HX V HB V ]
K U ( PS ) V = HX ± U ( PS ) V 2 HX ± U ( PS ) V 2 + HB ± U ( PS ) V 2

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