Abstract

We study electromagnetic field propagation through a planar three-waveguide coupler with linear gain or loss in a configuration that is the optical analog of a quantum 𝒫𝒯-symmetric system. This model is experimentally feasible on at least four proven architectures: lossy waveguide couplers, pumped waveguides couplers, non-Hermitian electronics and coupled pumped whispering gallery mode resonators. We show that our device provides all-optical amplitude (phase) to phase (amplitude) conversion in the 𝒫𝒯-symmetric regime at given propagation lenghts. The device has a π amplitude to phase conversion range if an extra binary phase is allowed in the reference signal, and a phase to amplitude conversion range that depends linearly on the gain-to-coupling ratio of the system. Our scheme may prove valuable in implementing phase shift keying formats, which have longer unrepeated transmission distance than intensity modulation schemes.

© 2016 Optical Society of America

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References

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    [Crossref]
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2015 (3)

X. Yu, E. Arbabi, L. L. Goddard, X. Li, and X. Chen, “Monolithically integrated self-rolled-up microtube-based vertical coupler for threedimensional photonic integration,” Appl. Phys. Lett. 107, 031102 (2015).
[Crossref]

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

B. M. Rodríguez-Lara and J. Guerrero, “Optical finite representation of the Lorentz group,” Opt. Lett. 40, 5682–5685 (2015).
[Crossref] [PubMed]

2014 (3)

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

S. Longhi, “Bound states in the continuum in 𝒫𝒯-symmetric optical lattices,” Opt. Lett. 39, 1697–1700 (2014).
[Crossref] [PubMed]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

2013 (2)

J. A. G. Coelho, M. B. Costa, A. C. Ferreira, M. G. da Silva, M. L. Lyra, and A. S. B. Sombra, “Realization of all-optical logic gates in a triangular triple-core photonic crystal fiber,” J. Lightwave Technol. 31, 731–739 (2013).
[Crossref]

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

2012 (2)

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in 𝒫𝒯-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys A: Math. Theor. 45, 444029 (2012).
[Crossref]

2011 (1)

Q. Tao, F. Luo, L. Cao, J. Hu, D. Cai, Z. Wan, J. Zhang, and Z. Yu, “Thermal stability analysis of nonlinear optical switch based on assymetric three-core nonlinear directional coupler with variable Gaussian coupling coefficient,” Opt. Eng. 50, 071104 (2011).
[Crossref]

2010 (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

2009 (2)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009).
[Crossref]

2008 (1)

2007 (3)

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical 𝒫𝒯-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

J. W. M. Menezes, W. B. de Fraga, A. C. Ferreira, K. D. A. Saboia, A. F. G. F. Filho, G. F. Guimaraes, J. R. R. Sousa, H. H. B. Rocha, and A. S. B. Sombra, “Logic gates based in two- and three-modes nonlinear optical fiber couplers,” Opt. Quant. Electron. 39, 1191–1206 (2007).
[Crossref]

J. W. M. Menezes, W. B. de Fraga, G. F. Guimaraes, A. C. Ferreira, H. H. B. Rocha, M. G. da Silva, and A. S. B. Sombra, “Optical switches and all-fiber logical devices based on triangular and planar three-core nonlinear optical fiber couplers,” Opt. Commun. 276, 107–115 (2007).
[Crossref]

2005 (1)

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and S. M. Viana, “Por qué y cómo encontramos funciones de matrices: entropía en mecánica cuántica,” Rev. Mex. Fis. 51, 87–98 (2005).

2003 (1)

G. J. Liu, B. M. Lang, Q. Li, and G. L. Jin, “Three-core nonlinear directional coupler with variable coupling coefficient,” Opt. Eng. 42, 2930–2935 (2003).
[Crossref]

1997 (1)

Y. Chen, “Fiber and integrated optics,” Fiber and Integrated Optics 16, 287–301 (1997).
[Crossref]

1996 (1)

D. Artigas, J. Olivas, F. Dios, and F. Canal, “Supermode analysis of the three-waveguide nonlinear directional coupler: the critical power,” Opt. Commun. 131, 53–60 (1996).
[Crossref]

1990 (2)

N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990).
[Crossref]

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quant. Electron. 22, 95–122 (1990).
[Crossref]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Elect. 18, 1580–1583 (1982).
[Crossref]

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in 𝒫𝒯-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012).
[Crossref]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Arbabi, E.

X. Yu, E. Arbabi, L. L. Goddard, X. Li, and X. Chen, “Monolithically integrated self-rolled-up microtube-based vertical coupler for threedimensional photonic integration,” Appl. Phys. Lett. 107, 031102 (2015).
[Crossref]

Artigas, D.

D. Artigas, J. Olivas, F. Dios, and F. Canal, “Supermode analysis of the three-waveguide nonlinear directional coupler: the critical power,” Opt. Commun. 131, 53–60 (1996).
[Crossref]

Bagheri, H.

A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

Bender, C. M.

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Benner, A. F.

A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

Budd, R. A.

A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

Cai, D.

Q. Tao, F. Luo, L. Cao, J. Hu, D. Cai, Z. Wan, J. Zhang, and Z. Yu, “Thermal stability analysis of nonlinear optical switch based on assymetric three-core nonlinear directional coupler with variable Gaussian coupling coefficient,” Opt. Eng. 50, 071104 (2011).
[Crossref]

Canal, F.

D. Artigas, J. Olivas, F. Dios, and F. Canal, “Supermode analysis of the three-waveguide nonlinear directional coupler: the critical power,” Opt. Commun. 131, 53–60 (1996).
[Crossref]

Cao, L.

Q. Tao, F. Luo, L. Cao, J. Hu, D. Cai, Z. Wan, J. Zhang, and Z. Yu, “Thermal stability analysis of nonlinear optical switch based on assymetric three-core nonlinear directional coupler with variable Gaussian coupling coefficient,” Opt. Eng. 50, 071104 (2011).
[Crossref]

Chen, X.

X. Yu, E. Arbabi, L. L. Goddard, X. Li, and X. Chen, “Monolithically integrated self-rolled-up microtube-based vertical coupler for threedimensional photonic integration,” Appl. Phys. Lett. 107, 031102 (2015).
[Crossref]

Chen, Y.

Y. Chen, “Fiber and integrated optics,” Fiber and Integrated Optics 16, 287–301 (1997).
[Crossref]

Childers, D.

A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

Childers, E.

A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

Christodoulides, D. N.

B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015).
[Crossref]

B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014).
[Crossref]

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

K. R. Khan, T. X. Wu, D. N. Christodoulides, and G. I. Stegeman, “Soliton switching and multi-frequency generation in a nonlinear photonic crystal fiber coupler,” Opt. Express 16, 9417–9428 (2008).
[Crossref] [PubMed]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical 𝒫𝒯-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Coelho, J. A. G.

Costa, M. B.

da Silva, M. G.

J. A. G. Coelho, M. B. Costa, A. C. Ferreira, M. G. da Silva, M. L. Lyra, and A. S. B. Sombra, “Realization of all-optical logic gates in a triangular triple-core photonic crystal fiber,” J. Lightwave Technol. 31, 731–739 (2013).
[Crossref]

J. W. M. Menezes, W. B. de Fraga, G. F. Guimaraes, A. C. Ferreira, H. H. B. Rocha, M. G. da Silva, and A. S. B. Sombra, “Optical switches and all-fiber logical devices based on triangular and planar three-core nonlinear optical fiber couplers,” Opt. Commun. 276, 107–115 (2007).
[Crossref]

de Fraga, W. B.

J. W. M. Menezes, W. B. de Fraga, A. C. Ferreira, K. D. A. Saboia, A. F. G. F. Filho, G. F. Guimaraes, J. R. R. Sousa, H. H. B. Rocha, and A. S. B. Sombra, “Logic gates based in two- and three-modes nonlinear optical fiber couplers,” Opt. Quant. Electron. 39, 1191–1206 (2007).
[Crossref]

J. W. M. Menezes, W. B. de Fraga, G. F. Guimaraes, A. C. Ferreira, H. H. B. Rocha, M. G. da Silva, and A. S. B. Sombra, “Optical switches and all-fiber logical devices based on triangular and planar three-core nonlinear optical fiber couplers,” Opt. Commun. 276, 107–115 (2007).
[Crossref]

Dios, F.

D. Artigas, J. Olivas, F. Dios, and F. Canal, “Supermode analysis of the three-waveguide nonlinear directional coupler: the critical power,” Opt. Commun. 131, 53–60 (1996).
[Crossref]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009).
[Crossref]

Eisfeld, A.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

El-Ganainy, R.

R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013).
[Crossref]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010).
[Crossref]

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical 𝒫𝒯-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007).
[Crossref] [PubMed]

Ellis, F. M.

J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys A: Math. Theor. 45, 444029 (2012).
[Crossref]

Fan, S. H.

B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014).
[Crossref]

Fasano, B. V.

A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

Ferreira, A. C.

J. A. G. Coelho, M. B. Costa, A. C. Ferreira, M. G. da Silva, M. L. Lyra, and A. S. B. Sombra, “Realization of all-optical logic gates in a triangular triple-core photonic crystal fiber,” J. Lightwave Technol. 31, 731–739 (2013).
[Crossref]

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Q. Tao, F. Luo, L. Cao, J. Hu, D. Cai, Z. Wan, J. Zhang, and Z. Yu, “Thermal stability analysis of nonlinear optical switch based on assymetric three-core nonlinear directional coupler with variable Gaussian coupling coefficient,” Opt. Eng. 50, 071104 (2011).
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Figures (5)

Fig. 1
Fig. 1 Schematics for a PT-symmetric three waveguide coupler realized in four different architectures: (a) lossy waveguides, n0 = ω0 and nj = ωj + j with j=1,2, (b) pumped waveguides, (c) non-Hermitian electronics and (d) coupled whispering gallery microcavities, nj = ωjj with j=0,1, and n2 = ω2.
Fig. 2
Fig. 2 Squared response to impulse function, | E n ( m ) | 2 , at the zeroth (dotted blue), n = 0, first (solid black), n = 1, and second (dashed red), n = 2, waveguides for an initial field impinging at the (a) zeroth, m = 0, (b) first, m = 1, and (c) second, m = 2, waveguide of a coupler with gain-to-coupling ratio ξ = 0.5 that keeps ����-symmetry.
Fig. 3
Fig. 3 Same as Fig. 2 but for a gain-to-coupling ratio ξ = 1.25 2 that breaks ����-symmetry.
Fig. 4
Fig. 4 Conversion from the initial reference field amplitude, ��1, to the output field phase, argE2(ζf), for an initial reference field (a) E1(0) = ��1 and (b) E1(0) = −��1 in a ����-symmetric amplitude to phase converter with gain-to-coupling parameter ξ = 0.5.
Fig. 5
Fig. 5 Conversion from the initial phase difference, δ, between signal, E 0 ( 0 ) = e i δ / 2 , and reference, E 1 ( 0 ) = 1 / 2 , to the output field amplitude, |E2(ζf)|, in a ����-symmetric phase to amplitude converter with gain-to-coupling parameter ξ = 0.5. The squared output field amplitude, |E2(ζf)|2, is shown.

Equations (41)

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i d d z 0 ( z ) = n 0 0 ( z ) + g 1 ( z ) ,
i d d z 1 ( z ) = n 1 1 ( z ) + g [ 0 ( z ) + 2 ( z ) ] ,
i d d z 2 ( z ) = n 2 2 ( z ) + g 1 ( z ) ,
j ( z ) = e i ( ω 1 + i γ 1 ) z E j ( z ) ,
i d d z E 0 ( z ) = ( ω ω 1 + i γ ) E 0 ( z ) + g E 1 ( z ) ,
i d d z E 1 ( z ) = g [ E 0 ( z ) + E 2 ( z ) ] ,
i d d z E 2 = ( ω ω 1 i γ ) E 2 ( z ) + g E 1 ( z ) .
i d d ζ E ( ζ ) = E ( ζ ) ,
= ( i ξ 1 0 1 0 1 0 1 i ζ ) , E ( ζ ) = ( E 0 ( ζ ) , E 1 ( ζ ) , E 2 ( ζ ) ) ,
Ω = 2 ξ 2 .
E ( ζ ) = 𝕌 ( ζ ) E ( 0 ) , 𝕌 ( ζ ) = e i ζ ,
𝕌 ( ζ ) = j = 0 ( i ζ ) j j ! j ,
= { 𝟙 + i Ω sin Ω ζ + 1 2 ( cos Ω ζ 1 ) 2 , ξ 2 , 𝟙 + i ξ 1 2 ζ 2 2 , ξ = 2 ,
3 = Ω 2 .
E n ( m ) ( ζ ) = 𝕌 n , m ( ζ ) , n , m = 0 , 1 , 2 ,
= α m + n α 0 ( 2 m ) ( 2 m ) F 1 2 [ m , n , 2 , α 0 α 2 ] ,
α 0 = { Ω 2 / β 2 , ξ 2 2 / ( 2 ζ ) 2 , ξ = 2 ,
α = { i 2 / β , ξ 2 i ζ / ( 2 ζ ) , ξ = 2 ,
β = Ω cos 1 2 Ω ζ ξ sin 1 2 Ω ζ .
ζ f = arccos ( ξ 2 1 ) + 2 n π Ω , n = 0 , 1 , 2 , ,
E 0 ( ζ f ) = E 2 ( 0 ) ,
E 1 ( ζ f ) = E 1 ( 0 ) + i 2 ξ E 2 ( 0 ) ,
E 2 ( ζ f ) = E 0 ( 0 ) + 2 ξ [ i E 1 ( 0 ) + ξ E 2 ( 0 ) ] .
E 0 ( 0 ) = 𝒜 0 e i ϕ 1 , E 1 ( 0 ) = 𝒜 1 e i ϕ 1 , E 2 ( 0 = 0 ) ,
E 0 ( ζ f ) = 0 ,
E 1 ( ζ f ) = 𝒜 1 e i ( ϕ 1 + π ) ,
E 2 ( ζ f ) = 𝒜 0 e i ( ϕ 0 + π ) + 2 ξ 𝒜 1 e i ( ϕ 1 + π 2 ) .
| E 2 ( ζ f ) | = 𝒜 0 2 + 4 ξ 2 𝒜 1 2 4 ξ 𝒜 0 𝒜 1 sin δ ,
arg [ E 2 ( ζ f ) ] = arctan 𝒜 0 sin ϕ 0 2 ξ 𝒜 1 cos ϕ 1 𝒜 0 cos ϕ 0 2 ξ 𝒜 1 sin ϕ 1 ,
E 0 ( 0 ) = 1 + 𝒜 1 2 , E 1 ( 0 ) = ± 𝒜 1 , E 2 ( 0 ) = 0 .
| E 2 ( ζ f ) | = 1 ( 4 ξ 2 1 ) 𝒜 1 2 ,
arg [ E 2 ( ζ f ) ] = arctan 2 ξ η ,
η = 𝒜 1 1 𝒜 1 2 .
𝒱 ϕ = [ arg E 2 ( ζ f ) ] max [ arg E 2 ( ζ f ) ] min ,
= { π π / 2 E 1 ( 0 ) = 𝒜 1 , 3 π / 2 π , E 1 ( 0 ) = 𝒜 1 ,
= π 2 ,
E 0 ( 0 ) = e i δ 𝒜 1 E 1 ( 0 ) = 𝒜 1 , E 2 ( 0 ) = 0 .
| E 2 ( ζ f ) | = 1 + 4 ξ ( ξ 1 ) sin δ 𝒜 ,
arg [ E 2 ( ζ f ) ] = arctan 2 ξ sin δ cos δ .
𝒱 a = | E 2 ( ζ f ) | max 2 | E 2 ( ζ f ) | min 2 | E 2 ( ζ f ) | max 2 + | E 2 ( ζ f ) | min 2 ,
= 8 ξ 𝒜 0 𝒜 1 𝒜 0 2 + 4 ξ 2 𝒜 1 2 ,

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