Abstract

Metal–insulator–metal (MIM) waveguide mesh structures utilize X-junctions as power distribution elements to create interference and feedback effects, thereby providing rich device functionality. We present a generalized analytical model for MIM mesh structures by incorporating a modified characteristic impedance model for MIM junctions into the scattering matrix formalism. The modified impedance model accounts for metal absorption and provides accurate prediction of plasmonic field distribution at X-junctions in terms of both magnitude and phase. Closed-form expressions for 2×1 and 2×2 MIM mesh architectures as well as MIM stub structures are then obtained and are dependent only on waveguide geometry and junction configuration. The model does not require numerically extracted parameters, and results agree, within a few percent, with those obtained from finite-difference time-domain method for both two-dimensional and three-dimensional waveguide geometries. The capability of the model for efficient design and optimization of junction-based MIM devices is demonstrated through the development of various filter and resonant devices.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2012 (2)

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012).
[CrossRef]

H. Nejati and A. Beirami, “Theoretical analysis of the characteristic impedance in metal-insulator-metal plasmonic transmission lines,” Opt. Lett. 37, 1050–1052 (2012).
[CrossRef]

2011 (2)

2010 (4)

2009 (4)

J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
[CrossRef]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009).
[CrossRef]

D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for nano-scale light confinement,” Opt. Express 17, 16646–16653 (2009).
[CrossRef]

2008 (3)

2007 (1)

2006 (6)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006).
[CrossRef]

Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006).
[CrossRef]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14, 1957–1964 (2006).
[CrossRef]

L. Chen, J. Shakya, and M. Lipson, “Subwavelength confinement in an integrated metal slot waveguide on silicon,” Opt. Lett. 31, 2133–2135 (2006).
[CrossRef]

2005 (2)

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

F. Hu and Z. Zhou, “Wavelength filtering and demultiplexing structure based on aperture-coupled plasmonic slot cavities,” J. Opt. Soc. Am. B 28, 2518–2523 (2005).
[CrossRef]

2003 (1)

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef]

Agrawal, G. P.

Atwater, H.

E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Beirami, A.

Berini, P.

P. Berini, “Bulk and surface sensitivity of surface plasmon waveguides,” New J. Phys. 10, 105010 (2008).
[CrossRef]

Brongersma, M. L.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010).
[CrossRef]

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006).
[CrossRef]

Cai, W.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010).
[CrossRef]

Chandran, A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006).
[CrossRef]

Chen, L.

Dai, D.

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Fan, S.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010).
[CrossRef]

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

Feigenbaum, E.

E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef]

E. Feigenbaum and M. Orenstein, “Perfect 4-way splitting in nano plasmonic X-junctions,” Opt. Express 15, 17948–17953 (2007).
[CrossRef]

Forsberg, E.

Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006).
[CrossRef]

Guo, Y.

Guo, Z.

Han, Z.

Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006).
[CrossRef]

Hattori, H. T.

He, S.

Helmy, A. S.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012).
[CrossRef]

B. Lau, M. A. Swillam, and A. S. Helmy, “Hybrid orthogonal junctions: wideband plasmonic slot-silicon waveguide couplers,” Opt. Express 18, 27048–27059 (2010).
[CrossRef]

M. A. Swillam and A. S. Helmy, “Filter response of feedback plasmonic junctions,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper ITuD4.

Homola, J.

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef]

Hosseini, A.

Hu, F.

Huang, X.

Kocabas, S. E.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Lau, B.

Li, H.

Lin, X.

Lipson, M.

Liu, L.

Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006).
[CrossRef]

Luo, B.

Luo, X.

Maier, S. A.

Massoud, Y.

Miller, D. A. B.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Min, C.

Nejati, H.

Orenstein, M.

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

Pan, W.

Pannipitiya, A.

Premaratne, M.

Qiu, M.

J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
[CrossRef]

Rukhlenko, I. D.

Schuller, J. A.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006).
[CrossRef]

Shakya, J.

Shin, W.

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010).
[CrossRef]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Swillam, M. A.

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012).
[CrossRef]

B. Lau, M. A. Swillam, and A. S. Helmy, “Hybrid orthogonal junctions: wideband plasmonic slot-silicon waveguide couplers,” Opt. Express 18, 27048–27059 (2010).
[CrossRef]

M. A. Swillam and A. S. Helmy, “Filter response of feedback plasmonic junctions,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper ITuD4.

Tian, J.

J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
[CrossRef]

Veronis, G.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

C. Min and G. Veronis, “Absorption switches in metal-dielectric-metal plasmonic waveguides,” Opt. Express 17, 10757–10766 (2009).
[CrossRef]

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

Wen, K.

Yan, L.

Yi, H.

Yu, S.

J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
[CrossRef]

Zhou, Z.

Zia, R.

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006).
[CrossRef]

Adv. Mater. (1)

W. Cai, W. Shin, S. Fan, and M. L. Brongersma, “Elements for plasmonic nanocircuits with three-dimensional slot waveguides,” Adv. Mater. 22, 5120–5124 (2010).
[CrossRef]

Anal. Bioanal. Chem. (1)

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef]

Appl. Phys. Lett. (2)

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

J. Tian, S. Yu, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
[CrossRef]

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

Mater. Today (1)

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9, 20–27 (2006).
[CrossRef]

New J. Phys. (1)

P. Berini, “Bulk and surface sensitivity of surface plasmon waveguides,” New J. Phys. 10, 105010 (2008).
[CrossRef]

Opt. Commun. (1)

Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259, 690–695 (2006).
[CrossRef]

Opt. Express (8)

Opt. Lett. (4)

Photon. Technol. Lett. (1)

M. A. Swillam and A. S. Helmy, “Feedback effects in plasmonic slot waveguides examined using a closed-form model,” Photon. Technol. Lett. 24, 497–499 (2012).
[CrossRef]

Phys. Rev. B (2)

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmonic slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

E. Feigenbaum and H. Atwater, “Resonant guided wave networks,” Phys. Rev. Lett. 104, 147402 (2010).
[CrossRef]

Science (1)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

Other (2)

M. A. Swillam and A. S. Helmy, “Filter response of feedback plasmonic junctions,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper ITuD4.

Lumerical, “Multicoefficient material modelling in FDTD,” http://www.lumerical.com/solutions/whitepapers/fdtd_multicoefficient_material_modeling.html .

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Figures (17)

Fig. 1.
Fig. 1.

(a) Top view and (b) cross-sectional view of the total electric field intensity of the propagating MIM fundamental mode at λ=1550nm. The waveguide is 340 nm tall with a 50 nm core.

Fig. 2.
Fig. 2.

Schematic and parameters of an MIM X-junction.

Fig. 3.
Fig. 3.

Comparison between FDTD and S-matrix modeling results of a symmetric X-junction with 100 nm cores and 400 nm arms. The junction is excited from the left horizontal arm, and both the normalized transmission and phase of the plasmonic waves in each junction arm are plotted.

Fig. 4.
Fig. 4.

Comparison between FDTD and S-matrix modeling results of an asymmetric X-junction with 50 nm horizontal cores, 100 nm vertical cores, and 400 nm arms. The junction is excited from the left horizontal arm, and both the transmission and phase of the plasmonic waves in each junction arm are plotted.

Fig. 5.
Fig. 5.

Model limitation demonstrated by the percent deviation from equal power splitting (25%) in a symmetric X-junction at (a) λ=1100nm, (b) λ=1550nm, and (c) λ=2000nm.

Fig. 6.
Fig. 6.

Schematic and S-matrix setup 2×1 MIM mesh structure.

Fig. 7.
Fig. 7.

Comparison between the S-matrix model and FDTD simulation results of a 2×1 MIM resonator. The structure consists of MIM waveguides with 50 nm cores and a 750 nm long interconnect.

Fig. 8.
Fig. 8.

Schematic and S-matrix setup of a 2×2 MIM mesh structure.

Fig. 9.
Fig. 9.

Comparison between the S-matrix model and FDTD simulation results of a 2×2 MIM resonator network. The structure consists of MIM waveguides with 50 nm cores and a 750 nm long interconnect.

Fig. 10.
Fig. 10.

Schematic and S-matrix setup of (a) a MIM X-junction with closed vertical ports and (b) a MIM mesh with two double-side stubs.

Fig. 11.
Fig. 11.

(a) Intensity and (b) phase comparison between FDTD and S-matrix modeling results for a homogeneous double-stubbed MIM structure where L1=L2=325nm, L3=L4=100nm, and Larm=200nm. The structure is optimized for a band-reject response centered at λ=1550nm.

Fig. 12.
Fig. 12.

Schematic of a 2D MIM mesh structure with widened internal network waveguides. The structure can be modeled analytically using the S-matrix setup illustrated in Fig. 8.

Fig. 13.
Fig. 13.

Normalized transmission at different output ports of a 2D MIM mesh with widened internal waveguide interconnects. The network output arms have 100 nm cores, and the internal interconnects are 800 nm long with 150 nm cores.

Fig. 14.
Fig. 14.

Normalized transmission of output port 7 of a 2D MIM mesh with (a) 150 nm wide network interconnects with varying lengths and (b) 800 nm long network interconnects with varying widths. The widths of the output arms are kept constant at 100 nm.

Fig. 15.
Fig. 15.

Schematic of a 2×2 MIM mesh with (a) width discontinuity in the interconnects, (b) a modified X-junction that includes width discontinuity, and (c) an interconnect discontinuity.

Fig. 16.
Fig. 16.

Comparison of transmission spectra between FDTD simulation and the S-matrix model for an enhanced 2×2 mesh with width discontinuity along the interconnect (Design 2) for output ports (a) 1, (b) 2, (c) 5, and (d) 6. The FDTD spectrum for a regular 2×2 mesh without discontinuity is also plotted as a reference (Design 1).

Fig. 17.
Fig. 17.

Model limitation demonstrated by the percent deviation from equal power splitting (25%) in a symmetric 50 nm X-junction at (a) λ=1100nm, (b) λ=1550nm, and (c) λ=2000nm. The data are extracted from 3D FDTD simulations.

Equations (18)

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

Zi(ω,di)βMIM(ω,di)din2ωεo,
r^=|ZLZOZL+ZO|andti^=|2ZoZLZo+ZLPi|=|2ZoZLZo+ZLZiZL|,
r=r^exp((α4+jβ4)(2L4)),ti=t^iexp([(αi+jβi)(Li)+(α4+jβ4)(L4)]),
[Y1Y2]=[S11S12S21S22][X1X2],
S11=rJ1+(tJ1,2rJ2tJ1,2exp(ϕ))1rJ1rJ2exp(2ϕ),S12=S21=tJ1,2tJ2,2exp(ϕ)1rJ1rJ2exp(2ϕ),andS22=rJ2+tJ2,2rJ1tJ2,2exp(ϕ)1rJ1rJ2exp(2ϕ).
Eout=[exp(ϕ)+rJ1rJ2exp(3ϕ)+rJ12rJ22exp(5ϕ)+rJ14rJ24exp(7ϕ)+]Ein=exp(ϕ)1rJ1rJ2exp(2ϕ)Ein.
T=Y2X1=t22exp(ϕ)1r2exp(2ϕ).
[Y1Y2Y3Y4]=[S11S12S13S14S21S22S23S24S31S32S33S34S41S42S43S44][X1X2X3X4],
S11=rJ1+rJ4tJ1,22exp(2ϕh)1rJ1rJ4exp(2ϕh),S12=jtJ1,3+jrJ4tJ1,1tJ1,2exp(2ϕh)1rJ1rJ4exp(2ϕh),S13=jtJ1,2tJ4,3exp(ϕh)1rJ1rJ4exp(2ϕh),S14=tJ1,2tJ4,2exp(ϕh)1rJ1rJ4exp(2ϕh),S21=jtJ1,1+jrJ4tJ1,2tJ1,3exp(2ϕh)1rJ1rJ4exp(2ϕh),S22=r1+rJ4tJ1,1tJ1,3exp(2ϕh)1rJ1rJ4exp(2ϕh),S23=tJ1,3tJ4,3exp(ϕh)1rJ1rJ4exp(2ϕh),S24=jtJ1,3tJ4,2exp(ϕh)1rJ1rJ4exp(2ϕh),S31=jtJ1,2tJ4,1exp(ϕh)1rJ1rJ4exp(2ϕh),S32=tJ1,1tJ4,1exp(ϕh)1rJ1rJ4exp(2ϕh),S33=rJ4+rJ1tJ4,1tJ4,3exp(2ϕh)1rJ1rJ4exp(2ϕh),S34=jtJ4,3+jrJ1tJ4,1tJ4,2exp(2ϕh)1rJ1rJ4exp(2ϕh),S41=tJ1,2tJ4,2exp(ϕh)1rJ1rJ4exp(2ϕh),S42=jtJ1,1tJ4,2exp(ϕh)1rJ1rJ4exp(2ϕh),S43=jtJ4,1+jrJ1tJ4,2tJ4,3exp(2ϕh)1rJ1rJ4exp(2ϕh),S44=r4+rJ1tJ4,22exp(2ϕh)1rJ1rJ4exp(2ϕh).
X2=x1Y2+y1Y3,X3=x2Y3+y2Y2,
x1=rJ2exp(2ϕv)+tJ2,3rJ3tJ2,1exp(2ϕv2ϕh)1rJ2rJ3exp(2ϕh),x2=rJ3exp(2ϕv)+tJ3,1rJ2tJ3,3exp(2ϕv2ϕh)1rJ2rJ3exp(2ϕh),y1=tJ3,1tJ2,1exp(2ϕvϕh)1rJ2rJ3exp(2ϕh),andy2=tJ2,3tJ3,3exp(2ϕvϕh)1rJ2rJ3exp(2ϕh).
T=Y4X1=S41+S21(S42x1+S43y2)1S22x1S23y2+[S42y1+S43x2+(S42x1+S43y2)(S22y1+S23x2)1S22x1S23y2][S31+S21(S32x1+S33y2)1S22x1S23y21S32y1S33x2(S32x1+S33y2)(S22y1+S23x2)1S22x1S23y2].
[Y1Y2Y3Y4]=[S11=rS12=t3S13=t1S14=0S21=t1S22=rS23=t2S24=0S31=t3S32=t2S33=rS34=0S41=t2S42=t1S43=t3S44=0][X1X2X3X4],
X2=rtopY2,X3=rbottomY3,
T=S41+S42S21rtop1S22rtop+rbottom(S42S23rtop1S22rtop+S43)(S31+S32S21rtop1S22rtop1rbottomS33+S32S23rtop1S22rtop).
R=R1+R3T1T2exp(2ϕB)1R2R3exp(2ϕB)andT=1R2,
R1=|ZBZAZB+ZA|,R2=|ZAZBZB+ZA|,R3=|ZCZBZB+ZC|,T1=1R12,T2=1R22.
r=r+t22Rexp(2ϕA)1Rr,t1=t1+t2Rt3exp(2ϕA)1Rr,t2=t2T+t2RrTexp(3ϕA)1Rr,t3=t3+t2Rt1exp(2ϕA)1Rr,

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