Abstract

The dual wave–particle nature of light and the degree of polarization are fundamental concepts in quantum physics and optical science, but their exact relation has not been explored within a full vector-light quantum framework that accounts for interferometric polarization modulation. Here, we consider vector-light quantum complementarity in double-pinhole photon interference and derive a general link between the degree of polarization and wave–particle duality of light. The relation leads to an interpretation for the degree of polarization as a measure describing the complementarity strength between photon path predictability and so-called Stokes visibility, the latter taking into account both intensity and polarization variations in the observation plane. It also unifies results advanced in classical studies by showing that the degree of polarization can be viewed as the ability of a light beam to exhibit intensity and polarization-state fringes. The framework we establish thus provides novel aspects and deeper insights into the role of the degree of polarization in quantum-light complementarity and photon interference.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

2018 (3)

2017 (4)

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

A. Z. Goldberg and D. F. V. James, “Perfect polarization for arbitrary light beams,” Phys. Rev. A 96, 053859 (2017).
[Crossref]

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
[Crossref]

J. H. Eberly, X.-F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
[Crossref]

2016 (1)

2014 (4)

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, “Fair sampling perspective on an apparent violation of duality,” Proc. Natl. Acad. Sci. USA 111, 12337–12341 (2014).
[Crossref]

F. De Zela, “Relationship between the degree of polarization, indistinguishability, and entanglement,” Phys. Rev. A 89, 013845 (2014).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

2011 (1)

M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
[Crossref]

2010 (3)

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. Am. A 27, 1764–1769 (2010).
[Crossref]

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

2009 (1)

2008 (1)

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[Crossref]

2007 (1)

M. Jakob and J. A. Bergou, “Complementarity and entanglement in bipartite qudit systems,” Phys. Rev. A 76, 052107 (2007).
[Crossref]

2006 (2)

2005 (1)

2004 (1)

2000 (1)

B.-G. Englert and J. A. Bergou, “Quantitative quantum erasure,” Opt. Commun. 179, 337–355 (2000).
[Crossref]

1999 (1)

A. Zeilinger, “Experiment and the foundations of quantum physics,” Rev. Mod. Phys. 71, S288–S297 (1999).
[Crossref]

1996 (1)

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

1995 (1)

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

1991 (2)

L. Mandel, “Coherence and indistinguishability,” Opt. Lett. 16, 1882–1883 (1991).
[Crossref]

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[Crossref]

1988 (1)

D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
[Crossref]

Abouraddy, A. F.

Agarwal, G. S.

G. S. Agarwal, Quantum Optics (Cambridge University, 2013).

Bergou, J. A.

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

M. Jakob and J. A. Bergou, “Complementarity and entanglement in bipartite qudit systems,” Phys. Rev. A 76, 052107 (2007).
[Crossref]

B.-G. Englert and J. A. Bergou, “Quantitative quantum erasure,” Opt. Commun. 179, 337–355 (2000).
[Crossref]

Björk, G.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Blomstedt, K.

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

Bolduc, E.

E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, “Fair sampling perspective on an apparent violation of duality,” Proc. Natl. Acad. Sci. USA 111, 12337–12341 (2014).
[Crossref]

Boyd, R. W.

E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, “Fair sampling perspective on an apparent violation of duality,” Proc. Natl. Acad. Sci. USA 111, 12337–12341 (2014).
[Crossref]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

De Zela, F.

F. De Zela, “Hidden coherences and two-state systems,” Optica 5, 243–250 (2018).
[Crossref]

F. De Zela, “Relationship between the degree of polarization, indistinguishability, and entanglement,” Phys. Rev. A 89, 013845 (2014).
[Crossref]

Dogariu, A.

Eberly, J. H.

Ellis, J.

Englert, B.-G.

B.-G. Englert and J. A. Bergou, “Quantitative quantum erasure,” Opt. Commun. 179, 337–355 (2000).
[Crossref]

B.-G. Englert, “Fringe visibility and which-way information: an inequality,” Phys. Rev. Lett. 77, 2154–2157 (1996).
[Crossref]

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[Crossref]

Friberg, A. T.

J. J. Gil, A. Norrman, A. T. Friberg, and T. Setälä, “Polarimetric purity and the concept of degree of polarization,” Phys. Rev. A 97, 023838 (2018).
[Crossref]

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence [Invited],” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

Ghiu, I.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Gil, J. J.

J. J. Gil, A. Norrman, A. T. Friberg, and T. Setälä, “Polarimetric purity and the concept of degree of polarization,” Phys. Rev. A 97, 023838 (2018).
[Crossref]

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

Goldberg, A. Z.

A. Z. Goldberg and D. F. V. James, “Perfect polarization for arbitrary light beams,” Phys. Rev. A 96, 053859 (2017).
[Crossref]

Greenberger, D. M.

D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
[Crossref]

Hochrainer, A.

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
[Crossref]

Jaeger, G.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Jakob, M.

M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
[Crossref]

M. Jakob and J. A. Bergou, “Complementarity and entanglement in bipartite qudit systems,” Phys. Rev. A 76, 052107 (2007).
[Crossref]

James, D. F. V.

A. Z. Goldberg and D. F. V. James, “Perfect polarization for arbitrary light beams,” Phys. Rev. A 96, 053859 (2017).
[Crossref]

Klimov, A. B.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Korotkova, O.

Lahiri, M.

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
[Crossref]

M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
[Crossref]

Laing, A.

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

Lapkiewicz, R.

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
[Crossref]

Leach, J.

E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, “Fair sampling perspective on an apparent violation of duality,” Proc. Natl. Acad. Sci. USA 111, 12337–12341 (2014).
[Crossref]

Lemos, G. B.

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
[Crossref]

Leppänen, L.-P.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Leuchs, G.

E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, “Fair sampling perspective on an apparent violation of duality,” Proc. Natl. Acad. Sci. USA 111, 12337–12341 (2014).
[Crossref]

Luis, A.

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. Am. A 27, 1764–1769 (2010).
[Crossref]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[Crossref]

A. Luis, Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), Vol. 61, Chap. 5.

Mandel, L.

Marian, P.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Marian, T. A.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Mathews, J. C. F.

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

Miatto, F. M.

E. Bolduc, J. Leach, F. M. Miatto, G. Leuchs, and R. W. Boyd, “Fair sampling perspective on an apparent violation of duality,” Proc. Natl. Acad. Sci. USA 111, 12337–12341 (2014).
[Crossref]

Norrman, A.

J. J. Gil, A. Norrman, A. T. Friberg, and T. Setälä, “Polarimetric purity and the concept of degree of polarization,” Phys. Rev. A 97, 023838 (2018).
[Crossref]

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

O’Brien, J. L.

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

Ossikovski, R.

J. J. Gil and R. Ossikovski, Polarized Light and the Mueller Matrix Approach (CRC Press, 2016).

Qian, X.-F.

Réfrégier, P.

Roueff, A.

Saastamoinen, K.

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Saleh, B. E. A.

Sánchez-Soto, L. L.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Scully, M. O.

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[Crossref]

Setälä, T.

J. J. Gil, A. Norrman, A. T. Friberg, and T. Setälä, “Polarimetric purity and the concept of degree of polarization,” Phys. Rev. A 97, 023838 (2018).
[Crossref]

A. Norrman, K. Blomstedt, T. Setälä, and A. T. Friberg, “Complementarity and polarization modulation in photon interference,” Phys. Rev. Lett. 119, 040401 (2017).
[Crossref]

A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence [Invited],” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
[Crossref]

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Stokes parameters and polarization contrasts in Young’s interference experiment,” Opt. Lett. 31, 2208–2210 (2006).
[Crossref]

T. Setälä, J. Tervo, and A. T. Friberg, “Contrasts of Stokes parameters in Young’s interference experiment and electromagnetic degree of coherence,” Opt. Lett. 31, 2669–2671 (2006).
[Crossref]

Shadbolt, P.

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

Shimony, A.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Söderholm, J.

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

Tervo, J.

Vaidman, L.

G. Jaeger, A. Shimony, and L. Vaidman, “Two interferometric complementarities,” Phys. Rev. A 51, 54–67 (1995).
[Crossref]

Vamivakas, A. N.

Walther, H.

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[Crossref]

Wolf, E.

Yasin, A.

D. M. Greenberger and A. Yasin, “Simultaneous wave and particle knowledge in a neutron interferometer,” Phys. Lett. A 128, 391–394 (1988).
[Crossref]

Zeilinger, A.

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. B. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
[Crossref]

A. Zeilinger, “Experiment and the foundations of quantum physics,” Rev. Mod. Phys. 71, S288–S297 (1999).
[Crossref]

J. Opt. Soc. Am. A (2)

Nat. Phys. (1)

P. Shadbolt, J. C. F. Mathews, A. Laing, and J. L. O’Brien, “Testing foundations of quantum mechanics with photons,” Nat. Phys. 10, 278–286 (2014).
[Crossref]

Nature (London) (1)

M. O. Scully, B.-G. Englert, and H. Walther, “Quantum optical tests of complementarity,” Nature (London) 351, 111–116 (1991).
[Crossref]

New J. Phys. (1)

L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
[Crossref]

Opt. Commun. (3)

G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[Crossref]

B.-G. Englert and J. A. Bergou, “Quantitative quantum erasure,” Opt. Commun. 179, 337–355 (2000).
[Crossref]

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Figures (1)

Fig. 1.
Fig. 1. Double-pinhole vector-light interference. A generally partially polarized quantum-light beam in state $ \hat \rho $ is divided into two orthogonal modes (the associated vacuum modes are also shown) that undergo unitary operations $ {\hat u_1} $ and $ {\hat u_2} $ at the openings in screen $ {\cal A} $. The superposed quantum vector-light field is observed in plane $ {\cal B} $ where intensity fringes and/or polarization modulation may appear.

Equations (28)

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S ^ j ( x 1 , x 2 ) = E ^ ( ) ( x 1 ) σ j E ^ ( + ) ( x 2 ) , j { 0 , , 3 } .
S j ( x 1 , x 2 ) = t r [ ρ ^ S ^ j ( x 1 , x 2 ) ] , j { 0 , , 3 } ,
g 2 ( x 1 , x 2 ) = 1 2 j = 0 3 | s j ( x 1 , x 2 ) | 2 ,
s j ( x 1 , x 2 ) = S j ( x 1 , x 2 ) [ S 0 ( x 1 ) S 0 ( x 2 ) ] 1 / 2 ,
P 2 ( x ) = j = 1 3 s j 2 ( x )
D 0 ( r 1 , r 2 ) = | S 0 ( r 1 ) S 0 ( r 2 ) | S 0 ( r 1 ) + S 0 ( r 2 ) .
S j ( r ) = S j ( r ) + S j ( r ) + 2 [ S 0 ( r ) S 0 ( r ) ] 1 / 2 × | s j ( r 1 , r 2 ) | cos [ θ j ( r 1 , r 2 ) k ( r 1 r 2 ) ] ,
V j ( r ) = max [ S j ( r ) ] min [ S j ( r ) ] max [ S 0 ( r ) ] + min [ S 0 ( r ) ] , j { 0 , , 3 } ,
V j ( r ) = C ( r ) | s j ( r 1 , r 2 ) | , j { 0 , , 3 } ,
C ( r ) = 2 [ S 0 ( r ) S 0 ( r ) ] 1 / 2 S 0 ( r ) + S 0 ( r ) 2 [ S 0 ( r 1 ) S 0 ( r 2 ) ] 1 / 2 S 0 ( r 1 ) + S 0 ( r 2 ) .
V ( r ) = 1 2 [ V 0 2 ( r ) + V 1 2 ( r ) + V 2 2 ( r ) + V 3 2 ( r ) ] 1 / 2 ,
V ( r ) = C ( r ) g ( r 1 , r 2 ) .
D 0 2 ( r 1 , r 2 ) + V 2 ( r ) 1 ,
ρ ^ = γ ρ ^ + ( 1 γ ) ρ ^ ,
ρ ^ = | c x | 2 | ψ x ψ x | + | c y | 2 | ψ y ψ y | + c x c y | ψ x ψ y | + c y c x | ψ y ψ x |
ρ ^ = | c x | 2 | ψ x ψ x | + | c y | 2 | ψ y ψ y | ,
p ( r 0 ) = [ 1 4 p x p y ( 1 γ 2 ) ] 1 / 2 ,
d 0 ( r 1 , r 2 ) = | p x p y | , v ( r ) = 2 ( p x p y ) 1 / 2 γ ,
d 0 2 ( r 1 , r 2 ) + v 2 ( r ) = p 2 ( r 0 ) .
D 0 2 ( r 1 , r 2 ) + V 2 ( r ) = P 2 ( r 0 ) ,
P ( r 0 ) = [ 1 4 n ¯ x n ¯ y ( 1 | g xy | 2 ) ( n ¯ x + n ¯ y ) 2 ] 1 / 2 ,
D 0 ( r 1 , r 2 ) = | n ¯ x n ¯ y | n ¯ x + n ¯ y , V ( r ) = 2 ( n ¯ x n ¯ y ) 1 / 2 n ¯ x + n ¯ y | g xy | .
E ^ ( + ) ( r m ) = K ( a ^ mx a ^ my ) e i k r m , m { 0 , 1 , 2 } ,
S ^ j ( r m ) = | K | 2 ( a ^ mx a ^ my ) σ j ( a ^ mx a ^ my ) ,
S ^ j ( r 1 , r 2 ) = | K | 2 ( a ^ 1 x a ^ 1 y ) σ j ( a ^ 2 x a ^ 2 y ) e i k ( r 2 r 1 ) .
( a ^ 1 μ a ^ 2 μ ) = ( r 0 μ t v μ t 0 μ r v μ ) ( a ^ 0 μ a ^ v μ ) , μ { x , y } .
a ^ 1 x = e i φ 0 x a ^ 0 x , a ^ 2 x = e i φ v x a ^ v x ,
a ^ 1 y = e i φ v y a ^ v y , a ^ 2 y = e i φ 0 y a ^ 0 y ,