All-optical switch consisting of two-stage interferometers controlled by using saturable absorption of monolayer graphene

Masayuki Oya; Hiroki Kishikawa; Nobuo Goto; Shin-ichiro Yanagiya

doi:10.1364/OE.20.027322

All-optical switch consisting of two-stage interferometers controlled by using saturable absorption of monolayer graphene

Masayuki Oya, Hiroki Kishikawa, Nobuo Goto, and Shin-ichiro Yanagiya

Optics Express
Vol. 20,
Issue 24,
pp. 27322-27330
(2012)
•https://doi.org/10.1364/OE.20.027322

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Masayuki Oya, Hiroki Kishikawa, Nobuo Goto, and Shin-ichiro Yanagiya, "All-optical switch consisting of two-stage interferometers controlled by using saturable absorption of monolayer graphene," Opt. Express 20, 27322-27330 (2012)

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Abstract

At routing nodes in future photonic networks, pico-second switching will be a key function. We propose an all-optical switch consisting of two-stage Mach-Zehnder interferometers, whose arms contain graphene saturable absorption films. Optical amplitudes along the interferometers are controlled to perform switching between two output ports instead of phase control used in conventional switches. Since only absorption is used for realizing complete switching, insertion loss of 10.2 dB is accompanied in switching. Picosecond response can be expected because of the fast response of saturable absorption of graphene. The switching characteristics are theoretically analyzed and numerically simulated by the finite-difference beam propagation method (FD-BPM).

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X. Yang, A. K. Mishra, D. Lenstra, F. M. Huijskens, H. de Waardt, G. D. Khoe, and H. J. S. Dorren, “Sub-picosecond all-optical switch using a multi-quantum-well semiconductor optical amplifier,” Optics Commun. 236, 329–334 (2004).
[Crossref]
M. Nagase, Y. Shoji, S. Suda, K. Kintaka, H. Kawashima, R. Akimoto, H. Kuwatsuka, T. Hasama, and H. Ishikawa, “Ultrafast all-optical gating operation using michelson interferometer for hybrid integration of intersubband transition switch on Si platform,” IEEE Photon. Tech. Lett. 23(24), 1884–1886 (2011).
[Crossref]
H. Kishikawa and N. Goto, “Proposal of all-optical wavelength-selective switching using waveguide-type Raman amplifiers and 3dB couplers,” IEEE/OSA J. Lightwave Technol. 23(4), 1631–1636 (2005).
[Crossref]
H. Kishikawa, K. Kimiya, N. Goto, and S. Yanagiya, “All-optical wavelength-selective switch consisting of asymmetric X-junction couplers and Raman amplifiers for wide wavelength range,” IEEE/OSA J. Lightwave Technol. 28(1), 172–180 (2010).
[Crossref]
K. Suto, T. Saito, T. Kimura, J. Nishizawa, and T. Tanabe, “Semiconductor Raman amplifier for terahertz bandwidth optical communication,” IEEE/OSA J. Lightwave Technol. 20(4), 705–711 (2002).
[Crossref]
H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[Crossref]
A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6(3), 183–191 (2007).
[Crossref]
F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]
S. Yamashita,“A tutorial on nonlinear photonics applications of carbon nanotube and graphene” IEEE/OSA J. Lightwave Technol. 30(4), 427–447 (2012).
[Crossref]
T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]
Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Yang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]
Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q. H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4(3), 297–307 (2011).
[Crossref]
J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92(4), 042116-1–3 (2008).
[Crossref]
M. Izutsu, A. Enokihara, and T. Sueta,“Optical-waveguide hybrid coupler,” Opt. Lett. 7(11), 549–551 (1982).
[Crossref] [PubMed]
P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]
H. Hiura, N. Goto, and S. Yanagiya, “Wavelength-insensitive integrated-optic circuit consisting of asymmetric X-junction couplers for recognition of BPSK labels,” IEEE/OSA J. Lightwave Technol. 27(24), 5543–5551 (2009).
[Crossref]
W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11(1), 32–39 (1975).
[Crossref]
Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano, 4(2), 803–810 (2010).
[Crossref] [PubMed]
Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, and H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Tech. Lett. 6(5), 626–628 (1994).
[Crossref]
T. Hashimoto, T. Kurosaki, M. Yanagisawa, Y. Suzuki, Y. Akahori, Y. Inoue, Y. Tohmori, K. Kato, Y. Yamada, N. Ishihara, and K. Kato, “A 1.3/1.55-μm wavelength-division multiplexing optical module using a planar light-wave full duplex operation,” IEEE/OSA J. Lightwave Technol. 18(11), 1541–1547 (2000).
[Crossref]

2012 (1)

S. Yamashita,“A tutorial on nonlinear photonics applications of carbon nanotube and graphene” IEEE/OSA J. Lightwave Technol. 30(4), 427–447 (2012).
[Crossref]

2011 (2)

Q. Bao, H. Zhang, Z. Ni, Y. Wang, L. Polavarapu, Z. Shen, Q. H. Xu, D. Tang, and K. P. Loh, “Monolayer graphene as a saturable absorber in a mode-locked laser,” Nano Res. 4(3), 297–307 (2011).
[Crossref]

M. Nagase, Y. Shoji, S. Suda, K. Kintaka, H. Kawashima, R. Akimoto, H. Kuwatsuka, T. Hasama, and H. Ishikawa, “Ultrafast all-optical gating operation using michelson interferometer for hybrid integration of intersubband transition switch on Si platform,” IEEE Photon. Tech. Lett. 23(24), 1884–1886 (2011).
[Crossref]

2010 (3)

H. Kishikawa, K. Kimiya, N. Goto, and S. Yanagiya, “All-optical wavelength-selective switch consisting of asymmetric X-junction couplers and Raman amplifiers for wide wavelength range,” IEEE/OSA J. Lightwave Technol. 28(1), 172–180 (2010).
[Crossref]

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]

Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Ferrari, “Graphene mode-locked ultrafast laser,” ACS Nano, 4(2), 803–810 (2010).
[Crossref] [PubMed]

2009 (3)

H. Hiura, N. Goto, and S. Yanagiya, “Wavelength-insensitive integrated-optic circuit consisting of asymmetric X-junction couplers for recognition of BPSK labels,” IEEE/OSA J. Lightwave Technol. 27(24), 5543–5551 (2009).
[Crossref]

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Yang, “Atomic-layer graphene as a saturable absorber for ultrafast pulsed lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

2008 (1)

J. M. Dawlaty, S. Shivaraman, M. Chandrashekhar, F. Rana, and M. G. Spencer, “Measurement of ultrafast carrier dynamics in epitaxial graphene,” Appl. Phys. Lett. 92(4), 042116-1–3 (2008).
[Crossref]

2007 (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6(3), 183–191 (2007).
[Crossref]

2005 (1)

H. Kishikawa and N. Goto, “Proposal of all-optical wavelength-selective switching using waveguide-type Raman amplifiers and 3dB couplers,” IEEE/OSA J. Lightwave Technol. 23(4), 1631–1636 (2005).
[Crossref]

2004 (2)

X. Yang, A. K. Mishra, D. Lenstra, F. M. Huijskens, H. de Waardt, G. D. Khoe, and H. J. S. Dorren, “Sub-picosecond all-optical switch using a multi-quantum-well semiconductor optical amplifier,” Optics Commun. 236, 329–334 (2004).
[Crossref]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, “Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,” Appl. Phys. Lett. 85(12), 2196–2198 (2004).
[Crossref]

2002 (1)

K. Suto, T. Saito, T. Kimura, J. Nishizawa, and T. Tanabe, “Semiconductor Raman amplifier for terahertz bandwidth optical communication,” IEEE/OSA J. Lightwave Technol. 20(4), 705–711 (2002).
[Crossref]

2000 (1)

T. Hashimoto, T. Kurosaki, M. Yanagisawa, Y. Suzuki, Y. Akahori, Y. Inoue, Y. Tohmori, K. Kato, Y. Yamada, N. Ishihara, and K. Kato, “A 1.3/1.55-μm wavelength-division multiplexing optical module using a planar light-wave full duplex operation,” IEEE/OSA J. Lightwave Technol. 18(11), 1541–1547 (2000).
[Crossref]

1997 (1)

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]

1994 (1)

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, and H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Tech. Lett. 6(5), 626–628 (1994).
[Crossref]

1982 (1)

M. Izutsu, A. Enokihara, and T. Sueta,“Optical-waveguide hybrid coupler,” Opt. Lett. 7(11), 549–551 (1982).
[Crossref] [PubMed]

1975 (1)

W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11(1), 32–39 (1975).
[Crossref]

Akahori, Y.

Akimoto, R.

Anada, T.

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]

Ando, S.

Bao, Q.

Basko, D. M.

Benson, T. M.

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Burns, W. K.

W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11(1), 32–39 (1975).
[Crossref]

Chandrashekhar, M.

Cohen, O.

Dawlaty, J. M.

de Waardt, H.

Dorren, H. J. S.

Enokihara, A.

M. Izutsu, A. Enokihara, and T. Sueta,“Optical-waveguide hybrid coupler,” Opt. Lett. 7(11), 549–551 (1982).
[Crossref] [PubMed]

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6(3), 183–191 (2007).
[Crossref]

Goto, N.

Hak, D.

Hasama, T.

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Hashimoto, T.

Hiura, H.

Huijskens, F. M.

Inoue, Y.

Ishihara, N.

Ishikawa, H.

Izutsu, M.

M. Izutsu, A. Enokihara, and T. Sueta,“Optical-waveguide hybrid coupler,” Opt. Lett. 7(11), 549–551 (1982).
[Crossref] [PubMed]

Kato, K.

Kawachi, M.

Kawashima, H.

Kendall, P. C.

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]

Khoe, G. D.

Kimiya, K.

Kimura, T.

Kintaka, K.

Kishikawa, H.

Kurosaki, T.

Kuwatsuka, H.

Lenstra, D.

Liu, A.

Loh, K. P.

Milton, A. F.

W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11(1), 32–39 (1975).
[Crossref]

Mishra, A. K.

Nagase, M.

Ni, Z.

Nicolaescu, R.

Nishizawa, J.

Novoselov, K. S.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6(3), 183–191 (2007).
[Crossref]

Ohmori, Y.

Paniccia, M.

Polavarapu, L.

Popa, D.

Privitera, G.

Rana, F.

Rong, H.

Rozhin, A. G.

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Saito, T.

Sawada, T.

Sewell, P.

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]

Shen, Z.

Shen, Z. X.

Shivaraman, S.

Shoji, Y.

Spencer, M. G.

Suda, S.

Sueta, T.

M. Izutsu, A. Enokihara, and T. Sueta,“Optical-waveguide hybrid coupler,” Opt. Lett. 7(11), 549–551 (1982).
[Crossref] [PubMed]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Suto, K.

Suzuki, Y.

Takahashi, H.

Tan, P. H.

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Tanabe, T.

Tang, D.

Tohmori, Y.

Torrisi, F.

Wang, F.

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Wang, Y.

Xu, Q. H.

Yamada, Y.

Yamashita, S.

S. Yamashita,“A tutorial on nonlinear photonics applications of carbon nanotube and graphene” IEEE/OSA J. Lightwave Technol. 30(4), 427–447 (2012).
[Crossref]

Yan, Y.

Yanagisawa, M.

Yanagiya, S.

Yang, D. Y.

Yang, X.

Zhang, H.

ACS Nano (1)

Adv. Funct. Mater. (1)

Adv. Mater. (1)

T. Hasan, Z. Sun, F. Wang, F. Bonaccorso, P. H. Tan, A. G. Rozhin, and A. C. Ferrari, “Nanotube - polymer composites for ultrafast photonics,” Adv. Mater. 21(38–39), (2009).
[Crossref]

Appl. Phys. Lett. (2)

IEEE J. Quantum Electron. (1)

W. K. Burns and A. F. Milton, “Mode conversion in planar-dielectric separating waveguides,” IEEE J. Quantum Electron. QE-11(1), 32–39 (1975).
[Crossref]

IEEE Photon. Tech. Lett (1)

IEEE Photon. Tech. Lett. (1)

IEEE/OSA J. Lightwave Technol (4)

S. Yamashita,“A tutorial on nonlinear photonics applications of carbon nanotube and graphene” IEEE/OSA J. Lightwave Technol. 30(4), 427–447 (2012).
[Crossref]

IEEE/OSA J. Lightwave Technol. (3)

P. Sewell, T. M. Benson, T. Anada, and P. C. Kendall,“Bi-oblique propagation analysis of symmetric and asymmetric Y-junctions,” IEEE/OSA J. Lightwave Technol. 15(4), 688–696 (1997).
[Crossref]

Nano Res. (1)

Nature Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6(3), 183–191 (2007).
[Crossref]

Nature Photon. (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Photon. 4(9), 611–622 (2010).
[Crossref]

Opt. Lett. (1)

M. Izutsu, A. Enokihara, and T. Sueta,“Optical-waveguide hybrid coupler,” Opt. Lett. 7(11), 549–551 (1982).
[Crossref] [PubMed]

Optics Commun (1)

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Fig. 1

Optical switch operated by controlling the gain of Raman amplifiers.

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Fig. 2

Modeling of a sheet of monolayer graphene; (a) signal and control light perpendicularly incident to graphene, and (b) model of cascade connection of attenuator and amplifier.

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Fig. 3

Optical transmittance as a function of optical incident intensity for saturable absorption in monolayer graphene.

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Fig. 4

Modeling of two sheets of monolayer graphene; (a) signal and control light perpendicularly incident to graphene, and (b) model of cascade connection of attenuators and amplifiers.

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Fig. 5

Switch architecture consisting of two-stage interferometers having two-stage monolayer graphene in both arms in the first interferometer.

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Fig. 6

Optical intensities along the switch obtained by FD-BPM simulation: (a) switched to output port B_out by inserting control light to lower arm (α_A = 1, $α_{B} = 1 + \sqrt{2}$ ) and (b) switched to output port A_out by inserting control light to upper arm ( $α_{A} = 1 + \sqrt{2}$ , α_B = 1).

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Tables (1)

Table 1 Comparison of optical output intensities between theoretical analysis and smula-tion.

View Table

Equations (15)

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

(\begin{matrix} E_{out}^{lower} \\ E_{out}^{upper} \end{matrix}) = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) (\begin{matrix} E_{in}^{lower} \\ E_{in}^{upper} \end{matrix}),

\begin{matrix} (\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = {(\frac{1}{\sqrt{2}})}^{3} (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ 0 & β_{fix} \end{matrix}) (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) \\ \times (\begin{matrix} α_{B} & 0 \\ 0 & α_{A} \end{matrix}) (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) (\begin{matrix} A_{in} \\ B_{in} \end{matrix}) \\ = \frac{1}{2 \sqrt{2}} (\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}) (\begin{matrix} A_{in} \\ B_{in} \end{matrix}), \end{matrix}

{\begin{matrix} a_{11} & = α_{A} - α_{B} - β_{fix} (α_{A} + α_{B}) \\ a_{12} & = α_{A} + α_{B} + β_{fix} (- α_{A} + α_{B}) \\ a_{21} & = - (α_{A} - α_{B}) - β_{fix} (α_{A} + α_{B}) \\ a_{22} & = - (α_{A} + α_{B}) + β_{fix} (- α_{A} + α_{B}) \end{matrix} .

(\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = \frac{E_{in}}{2 \sqrt{2}} (\begin{matrix} a_{11} \\ a_{21} \end{matrix}) .

(\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = - E_{in} (\begin{matrix} 1 \\ 0 \end{matrix}) .

(\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = - E_{in} (\begin{matrix} 0 \\ 1 \end{matrix}) .

\begin{array}{l} (\begin{matrix} A_{m} \\ B_{m} \end{matrix}) & = \frac{1}{2} (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) (\begin{matrix} α_{B} & 0 \\ 0 & α_{A} \end{matrix}) (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix}) (\begin{matrix} A_{in} \\ B_{in} \end{matrix}) \\ = \frac{1}{2} (\begin{matrix} α_{B} - α_{A} & α_{B} + α_{A} \\ - α_{B} - α_{A} & - α_{B} + α_{A} \end{matrix}) (\begin{matrix} A_{in} \\ B_{in} \end{matrix}) . \end{array}

(\begin{matrix} A_{m} \\ B_{m} \end{matrix}) = \frac{E_{in}}{2} (\begin{matrix} α_{B} - α_{A} \\ - α_{B} - α_{A} \end{matrix}) .

(\begin{matrix} A_{m} \\ B_{m} \end{matrix}) = - \frac{E_{in}}{2} (\begin{matrix} \sqrt{2} \\ 2 + \sqrt{2} \end{matrix}) .

(\begin{matrix} A_{m} \\ B_{m} \end{matrix}) = - \frac{E_{in}}{2} (\begin{matrix} - \sqrt{2} \\ 2 + \sqrt{2} \end{matrix}) .

(\begin{matrix} A_{m} \\ β_{fix} B_{m} \end{matrix}) = - \frac{E_{in}}{2} (\begin{matrix} 1 \\ 1 \end{matrix}) .

(\begin{matrix} A_{m} \\ β_{fix} B_{m} \end{matrix}) = - \frac{E_{in}}{2} (\begin{matrix} - 1 \\ 1 \end{matrix}) .

(\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = \frac{β_{fix 0} E_{in}}{2 \sqrt{2}} (\begin{matrix} a_{11} \\ a_{21} \end{matrix}) .

(\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = - β_{fix 0} E_{in} (\begin{matrix} 1 \\ 0 \end{matrix}) .

(\begin{matrix} A_{out} \\ B_{out} \end{matrix}) = - β_{fix 0} E_{in} (\begin{matrix} 0 \\ 1 \end{matrix}) .

Control arm	Theory		Simulation
Control arm	\|A_out\|²	\|B_out\|²	\|A_out\|²	\|B_out\|²
$α_{B} = 1 + \sqrt{2}$	0	9.55× 10⁻²	3.36 × 10⁻⁴	9.83 × 10⁻²
$α_{A} = 1 + \sqrt{2}$	9.55 × 10⁻²	0	8.74 × 10⁻²	3.05 × 10⁻⁴