Abstract

A triple-output Mach-Zehnder interferometer (MZI) operating with long-range surface plasmon-polariton waves, consisting of a MZI in cascade with a triple coupler, is demonstrated at a wavelength of ~1370 nm, using the thermo-optic effect to produce phase shifting. A theoretical model for three-waveguide coupling is also proposed and was applied to compute the transfer characteristic of various designs. Dimensions for the device were selected to optimize performance, experiments were performed, and the results were compared to theory. The outputs were sinusoidally related to the thermally-induced phase shift and separated by ~2π/3 rad, as expected theoretically. Four detection schemes that take advantage of the 3 times larger dynamic range and suppress time-varying common perturbations are proposed and analyzed in order to improve the detection limit of the device. A minimum detectable phase shift ~2/3 that of a single output was obtained from a power difference scheme and a normalization scheme. The smallest minimum detectable phase shift was 7.3 mrad. The device is promising for sensing applications, including (bio)chemical sensing.

© 2014 Optical Society of America

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2013 (4)

2012 (2)

H. Fan, R. Buckley, P. Berini, “Passive long-range surface plasmon-polariton devices in Cytop,” Appl. Opt. 51(10), 1459–1467 (2012).
[CrossRef] [PubMed]

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

2010 (2)

C. Chiu, E. Lisicka-Shrzek, R. N. Tait, P. Berini, “Fabrication of surface plasmon waveguides and devices in Cytop with integrated microfluidic channels,” J. Vac. Sci. Technol. B 28(4), 729–735 (2010).
[CrossRef]

J. Gosciniak, S. I. Bozhevolnyi, T. B. Andersen, V. S. Volkov, J. Kjelstrup-Hansen, L. Markey, A. Dereux, “Thermo-optic control of dielectric-loaded plasmonic waveguide components,” Opt. Express 18(2), 1207–1216 (2010).
[CrossRef] [PubMed]

2009 (2)

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1(3), 484–588 (2009).
[CrossRef]

K. C. Vernon, D. E. Gómez, T. J. Davis, “A compact interferometric sensor design using three waveguide coupling,” J. Appl. Phys. 106(10), 104306 (2009).
[CrossRef]

2006 (4)

2005 (1)

P. Berini, R. Charbonneau, N. Lahoud, G. Mattiussi, “Characterization of long-range surface plasmon-polariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

2004 (2)

S. Y. Wu, H. P. Ho, W. C. Law, C. Lin, S. K. Kong, “Highly sensitive differential phase-sensitive surface plasmon resonance biosensor based on the Mach-Zehnder configuration,” Opt. Lett. 29(20), 2378–2380 (2004).
[CrossRef] [PubMed]

T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, “Surface plasmon-polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
[CrossRef]

2002 (1)

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
[CrossRef]

2000 (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[CrossRef]

1999 (1)

R. G. Heideman, P. V. Lambeck, “Remote opto-chemical sensing with extreme sensitivity: Design, fabrication and performance of a pigtailed integrated optical phase-modulated Mach–Zehnder interferometer system,” Sens. Actuators B Chem. 61(1), 100–127 (1999).
[CrossRef]

1998 (1)

1987 (2)

S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23(5), 499–509 (1987).
[CrossRef]

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5(1), 174–183 (1987).
[CrossRef]

1982 (1)

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3x3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616 (1982).
[CrossRef]

1981 (1)

S. K. Sheem, “Optical fiber interferometers with [3x3] directional couplers: Analysis,” J. Appl. Phys. 52(6), 3865 (1981).
[CrossRef]

Andersen, T. B.

Berini, P.

H. Fan, P. Berini, “Thermo-optic characterization of long-range surface-plasmon devices in Cytop,” Appl. Opt. 52(2), 162–170 (2013).
[CrossRef] [PubMed]

A. Khan, O. Krupin, E. Lisicka-Skrzek, P. Berini, “Mach-Zehnder refractometric sensor using long-range surface plasmon waveguides,” Appl. Phys. Lett. 103(11), 111108 (2013).
[CrossRef]

H. Fan, P. Berini, “Noise cancellation in long-range surface plasmon dual-output Mach-Zehnder interferometers,” J. Lightwave Technol. 31(15), 2606–2612 (2013).
[CrossRef]

H. Fan, R. Buckley, P. Berini, “Passive long-range surface plasmon-polariton devices in Cytop,” Appl. Opt. 51(10), 1459–1467 (2012).
[CrossRef] [PubMed]

C. Chiu, E. Lisicka-Shrzek, R. N. Tait, P. Berini, “Fabrication of surface plasmon waveguides and devices in Cytop with integrated microfluidic channels,” J. Vac. Sci. Technol. B 28(4), 729–735 (2010).
[CrossRef]

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photon. 1(3), 484–588 (2009).
[CrossRef]

G. Gagnon, N. Lahoud, G. A. Mattiussi, P. Berini, “Thermally activated variable attenuation of long-range surface plasmon-polariton waves,” J. Lightwave Technol. 24(11), 4391–4402 (2006).
[CrossRef]

R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J. Lightwave Technol. 24(1), 477–494 (2006).
[CrossRef]

I. Breukelaar, R. Charbonneau, P. Berini, “Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides,” J. Appl. Phys. 100(4), 043104 (2006).
[CrossRef]

P. Berini, R. Charbonneau, N. Lahoud, G. Mattiussi, “Characterization of long-range surface plasmon-polariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[CrossRef]

Bernardin, T.

Boltasseva, A.

Bozhevolnyi, S. I.

Breukelaar, I.

I. Breukelaar, R. Charbonneau, P. Berini, “Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides,” J. Appl. Phys. 100(4), 043104 (2006).
[CrossRef]

R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J. Lightwave Technol. 24(1), 477–494 (2006).
[CrossRef]

Buckley, R.

Charbonneau, R.

I. Breukelaar, R. Charbonneau, P. Berini, “Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides,” J. Appl. Phys. 100(4), 043104 (2006).
[CrossRef]

R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J. Lightwave Technol. 24(1), 477–494 (2006).
[CrossRef]

P. Berini, R. Charbonneau, N. Lahoud, G. Mattiussi, “Characterization of long-range surface plasmon-polariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

Chiu, C.

C. Chiu, E. Lisicka-Shrzek, R. N. Tait, P. Berini, “Fabrication of surface plasmon waveguides and devices in Cytop with integrated microfluidic channels,” J. Vac. Sci. Technol. B 28(4), 729–735 (2010).
[CrossRef]

Chuang, S. L.

S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23(5), 499–509 (1987).
[CrossRef]

S. L. Chuang, “A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation,” J. Lightwave Technol. 5(1), 174–183 (1987).
[CrossRef]

Cluzel, B.

Dandridge, A.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3x3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616 (1982).
[CrossRef]

Davis, T. J.

K. C. Vernon, D. E. Gómez, T. J. Davis, “A compact interferometric sensor design using three waveguide coupling,” J. Appl. Phys. 106(10), 104306 (2009).
[CrossRef]

Dereux, A.

Fabricius, N.

Fafard, S.

Fainman, Y.

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

Fan, H.

Fatome, J.

Finot, C.

Gagnon, G.

Gómez, D. E.

K. C. Vernon, D. E. Gómez, T. J. Davis, “A compact interferometric sensor design using three waveguide coupling,” J. Appl. Phys. 106(10), 104306 (2009).
[CrossRef]

Gosciniak, J.

Hassan, K.

Heideman, R. G.

R. G. Heideman, P. V. Lambeck, “Remote opto-chemical sensing with extreme sensitivity: Design, fabrication and performance of a pigtailed integrated optical phase-modulated Mach–Zehnder interferometer system,” Sens. Actuators B Chem. 61(1), 100–127 (1999).
[CrossRef]

Ho, H. P.

Hollenbach, U.

Hua, P.

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
[CrossRef]

Ingenhoff, J.

Jonathan Luff, B.

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
[CrossRef]

Kawaguchi, K.

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
[CrossRef]

Kaya, S.

Khan, A.

A. Khan, O. Krupin, E. Lisicka-Skrzek, P. Berini, “Mach-Zehnder refractometric sensor using long-range surface plasmon waveguides,” Appl. Phys. Lett. 103(11), 111108 (2013).
[CrossRef]

Kjelstrup-Hansen, J.

Kong, S. K.

Koo, K. P.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3x3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616 (1982).
[CrossRef]

Krupin, O.

A. Khan, O. Krupin, E. Lisicka-Skrzek, P. Berini, “Mach-Zehnder refractometric sensor using long-range surface plasmon waveguides,” Appl. Phys. Lett. 103(11), 111108 (2013).
[CrossRef]

Lahoud, N.

Lambeck, P. V.

R. G. Heideman, P. V. Lambeck, “Remote opto-chemical sensing with extreme sensitivity: Design, fabrication and performance of a pigtailed integrated optical phase-modulated Mach–Zehnder interferometer system,” Sens. Actuators B Chem. 61(1), 100–127 (1999).
[CrossRef]

Law, W. C.

Leosson, K.

K. Leosson, T. Nikolajsen, A. Boltasseva, S. I. Bozhevolnyi, “Long-range surface plasmon polariton nanowire waveguides for device applications,” Opt. Express 14(1), 314–319 (2006).
[CrossRef] [PubMed]

T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, “Surface plasmon-polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
[CrossRef]

Lin, C.

Lisicka-Shrzek, E.

C. Chiu, E. Lisicka-Shrzek, R. N. Tait, P. Berini, “Fabrication of surface plasmon waveguides and devices in Cytop with integrated microfluidic channels,” J. Vac. Sci. Technol. B 28(4), 729–735 (2010).
[CrossRef]

Lisicka-Skrzek, E.

A. Khan, O. Krupin, E. Lisicka-Skrzek, P. Berini, “Mach-Zehnder refractometric sensor using long-range surface plasmon waveguides,” Appl. Phys. Lett. 103(11), 111108 (2013).
[CrossRef]

Luff, B. J.

Markey, L.

Mattiussi, G.

R. Charbonneau, C. Scales, I. Breukelaar, S. Fafard, N. Lahoud, G. Mattiussi, P. Berini, “Passive integrated optics elements based on long-range surface plasmon-polaritons,” J. Lightwave Technol. 24(1), 477–494 (2006).
[CrossRef]

P. Berini, R. Charbonneau, N. Lahoud, G. Mattiussi, “Characterization of long-range surface plasmon-polariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

Mattiussi, G. A.

Nikolajsen, T.

K. Leosson, T. Nikolajsen, A. Boltasseva, S. I. Bozhevolnyi, “Long-range surface plasmon polariton nanowire waveguides for device applications,” Opt. Express 14(1), 314–319 (2006).
[CrossRef] [PubMed]

T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, “Surface plasmon-polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
[CrossRef]

Pang, L.

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

Piehler, J.

Ptasinski, J.

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

Quigley, G. R.

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
[CrossRef]

Scales, C.

Sheem, S. K.

S. K. Sheem, “Optical fiber interferometers with [3x3] directional couplers: Analysis,” J. Appl. Phys. 52(6), 3865 (1981).
[CrossRef]

Slutsky, B.

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

Sun, P. C.

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

Tait, R. N.

C. Chiu, E. Lisicka-Shrzek, R. N. Tait, P. Berini, “Fabrication of surface plasmon waveguides and devices in Cytop with integrated microfluidic channels,” J. Vac. Sci. Technol. B 28(4), 729–735 (2010).
[CrossRef]

Tveten, A. B.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3x3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616 (1982).
[CrossRef]

Vernon, K. C.

K. C. Vernon, D. E. Gómez, T. J. Davis, “A compact interferometric sensor design using three waveguide coupling,” J. Appl. Phys. 106(10), 104306 (2009).
[CrossRef]

Volkov, V. S.

Weeber, J.-C.

Wilkinson, J. S.

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
[CrossRef]

B. J. Luff, J. S. Wilkinson, J. Piehler, U. Hollenbach, J. Ingenhoff, N. Fabricius, “Integrated optical Mach-Zehnder biosensor,” J. Lightwave Technol. 16(4), 583–592 (1998).
[CrossRef]

Wu, S. Y.

Zacharatos, F.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

Appl. Phys. Lett. (3)

T. Nikolajsen, K. Leosson, S. I. Bozhevolnyi, “Surface plasmon-polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004).
[CrossRef]

A. Khan, O. Krupin, E. Lisicka-Skrzek, P. Berini, “Mach-Zehnder refractometric sensor using long-range surface plasmon waveguides,” Appl. Phys. Lett. 103(11), 111108 (2013).
[CrossRef]

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3x3) fiber directional couplers,” Appl. Phys. Lett. 41(7), 616 (1982).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. L. Chuang, “Application of the strongly coupled-mode theory to integrated optical devices,” IEEE J. Quantum Electron. 23(5), 499–509 (1987).
[CrossRef]

IEEE Sens. J. (1)

J. Ptasinski, L. Pang, P. C. Sun, B. Slutsky, Y. Fainman, “Differential detection for nanoplasmonic resonance sensors,” IEEE Sens. J. 12(2), 384–388 (2012).
[CrossRef]

J. Appl. Phys. (4)

K. C. Vernon, D. E. Gómez, T. J. Davis, “A compact interferometric sensor design using three waveguide coupling,” J. Appl. Phys. 106(10), 104306 (2009).
[CrossRef]

I. Breukelaar, R. Charbonneau, P. Berini, “Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides,” J. Appl. Phys. 100(4), 043104 (2006).
[CrossRef]

P. Berini, R. Charbonneau, N. Lahoud, G. Mattiussi, “Characterization of long-range surface plasmon-polariton waveguides,” J. Appl. Phys. 98(4), 043109 (2005).
[CrossRef]

S. K. Sheem, “Optical fiber interferometers with [3x3] directional couplers: Analysis,” J. Appl. Phys. 52(6), 3865 (1981).
[CrossRef]

J. Lightwave Technol. (5)

J. Vac. Sci. Technol. B (1)

C. Chiu, E. Lisicka-Shrzek, R. N. Tait, P. Berini, “Fabrication of surface plasmon waveguides and devices in Cytop with integrated microfluidic channels,” J. Vac. Sci. Technol. B 28(4), 729–735 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[CrossRef]

Sens. Actuators B Chem. (2)

R. G. Heideman, P. V. Lambeck, “Remote opto-chemical sensing with extreme sensitivity: Design, fabrication and performance of a pigtailed integrated optical phase-modulated Mach–Zehnder interferometer system,” Sens. Actuators B Chem. 61(1), 100–127 (1999).
[CrossRef]

P. Hua, B. Jonathan Luff, G. R. Quigley, J. S. Wilkinson, K. Kawaguchi, “Integrated optical dual Mach-Zehnder interferometer sensor,” Sens. Actuators B Chem. 87(2), 250–257 (2002).
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H. Asiri, “Fabrication of surface plasmon biosensors in CYTOP,” Master’s thesis, Dep. Chem. Biol. Eng., Univ. Ottawa, Ottawa, Canada, 2012.

A. B. Buckman, Guided-Wave Photonics (Harcourt Brace Jovanovich, 1992).

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H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

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Figures (10)

Fig. 1
Fig. 1

Sketch of the structure of interest: (a) triple-output MZI, and (b) triple coupler structure with notation identifying the modes travelling thereon.

Fig. 2
Fig. 2

Calculated normalized Ey field distribution of (a) the long-range ssb0 mode supported by a single waveguide, (b) the long-range supermode (1 1 1), (c) the long-range supermode (−1 0 1), and (d) the long-range supermode (1 −1 1), supported by a triple coupler.

Fig. 3
Fig. 3

Calculated output powers compared to results of [17] for a dielectric slab triple coupler of core index 1.646 and cladding index 1.6, operating at 786 nm. The core width was 1 μm, the coupler separation was s = 1 μm and coupler length L = 0.75Lc.

Fig. 4
Fig. 4

Calculated β/β0 and MPA versus stripe separation s for the triple coupler.

Fig. 5
Fig. 5

Calculated normalized optical power output versus phase shift ϕ for triple-output MZIs for triple couplers designed with (a) s = 1 μm, L = 457 μm, (b) s = 2 μm, L = 828.57 μm, and (c) s = 3 μm, L = 1530 μm.

Fig. 6
Fig. 6

Microscope images of (a) the whole triple-output MZI device and (b) the triple coupler portion of the device under higher magnification.

Fig. 7
Fig. 7

Block diagram of the thermo-optic experimental setup.

Fig. 8
Fig. 8

Transfer characteristic measurements for a triple-output MZI: (a) output optical power versus dissipated electrical power, and (b) normalized optical power versus thermally induced phase shift. The theoretical results are superimposed on the measurements in Part (b). The inset is a mosaic of normalized mode images showing the power switching among the three outputs.

Fig. 9
Fig. 9

Output optical power difference versus dissipated electrical power obtained from the measurements of Fig. 8(a).

Fig. 10
Fig. 10

Experiment illustrating time-varying perturbation suppression by normalization: (a) unnormalized output optical power and power difference and (b) normalized optical power versus dissipated electrical power.

Tables (1)

Tables Icon

Table 1 Standard deviations σ and minimum detectable phase shifts Δϕmin for Pi, Di, Pi/(P1 + P2 + P3), and Di/(P1 + P2 + P3) of three time tracing measurements taken on the same device.

Equations (23)

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n( T )=n( T 0 )+( T T 0 ) dn dT
E y,in ( x,y,0 )= E y,s ( x+s,y,0 )+ E y,s ( xs,y,0 ) e j ϕ
C i = A E y,in ( x,y,0 ) E y,i ( x,y,0 )dA ( A E y,in ( x,y,0 ) E y,in ( x,y,0 )dA )( A E y,i ( x,y,0 ) E y,i ( x,y,0 )dA )
E y,out ( x,y,L )= i C i E y,i ( x,y,0 ) e ( α i +j β i )L
C 1 = A E y,out ( x,y,L ) E y,s ( x+s,y,L )dA ( A E y,out ( x,y,L ) E y,out ( x,y,L )dA )( A E y,s ( x+s,y,L ) E y,s ( x+s,y,L )dA )
C 2 = A E y,out ( x,y,L ) E y,s ( x,y,L )dA ( A E y,out ( x,y,L ) E y,out ( x,y,L )dA )( A E y,s ( x,y,L ) E y,s ( x,y,L )dA )
C 3 = A E y,out ( x,y,L ) E y,s ( xs,y,L )dA ( A E y,out ( x,y,L ) E y,out ( x,y,L )dA )( A E y,s ( xs,y,L ) E y,s ( xs,y,L )dA )
MPA=α 20 1000 log 10 e
P 1 =[ acos( ϕ 2π 3 )+ b 1 ]( P in + p i )+ p o
P 2 =( acosϕ+ b 2 )( P in + p i )+ p o
P 3 =[ acos( ϕ+ 2π 3 )+ b 3 ]( P in + p i )+ p o
D 2 =( 3acosϕ+2 b 2 b 1 b 3 )( P in + p i )
P 2 P 1 + P 2 + P 3 = (acosϕ+ b 2 )( P in + p i )+ p o ( b 1 + b 2 + b 3 )( P in + p i )+3 p o
P 2 P 1 + P 2 + P 3 = a b 1 + b 2 + b 3 cosϕ+ b 2 b 1 + b 2 + b 3
D 2 P 1 + P 2 + P 3 = (3acosϕ+2 b 2 b 1 b 3 )( P in + p i ) ( b 1 + b 2 + b 3 )( P in + p i )+3 p o
D 2 P 1 + P 2 + P 3 = 3a b 1 + b 2 + b 3 cosϕ+ 2 b 2 b 1 b 3 b 1 + b 2 + b 3
Δ ϕ min = Δ P min | P ϕ | = kσ | P ϕ |
P 2 =( acosϕ+ b 2 ) P in
P 2 ϕ =( asinϕ ) P in
Δ ϕ min | P 2 = k ( asinϕ ) P in σ| P 2
Δ ϕ min | D 2 = k ( 3asinϕ ) P in σ| D 2
Δ ϕ min | P 2 P 1 + P 2 + P 3 = k a b 1 + b 2 + b 3 sinϕ σ| P 2 P 1 + P 2 + P 3
Δ ϕ min | D 2 P 1 + P 2 + P 3 = k 3a b 1 + b 2 + b 3 sinϕ σ| D 2 P 1 + P 2 + P 3

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