Abstract

In this paper, we use Clifford (geometric) algebra Cl3,0 to verify if electromagnetic energy–momentum density is still conserved for oblique superposition of two elliptically polarized plane waves with the same frequency. We show that energy–momentum conservation is valid at any time only for the superposition of two counter-propagating elliptically polarized plane waves. We show that the time-average energy–momentum of the superposition of two circularly polarized waves with opposite handedness is conserved regardless of the propagation directions of the waves. And, we show that the resulting momentum density of the superposed waves generally has a vector component perpendicular to the momentum densities of the individual waves.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Gauthier, “What happens to energy and momentum when two oppositely moving wave pulses overlap?,” Am. J. Phys. 71, 787–790 (2003).
    [CrossRef]
  2. E. A. Notte-Cuello and W. A. Rodrigues Jr., “Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields,” Rep. Math. Phys. 62, 91–101 (2008).
    [CrossRef]
  3. B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, 1988).
  4. C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, 2003).
  5. D. Hestenes, “Oersted medal lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
    [CrossRef]
  6. W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhäuser, 1999).
  7. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), p. 165.
  8. B. Thidé, Electromagnetic Field Theory, 2nd ed. (Dover, 2010), p. 22; ebook available from www.plasma.uu.se/CED.
  9. I. Bialynicki-Birula and Z. Bialynicki-Birula, “Beams of electromagnetic radiation carrying angular momentum: The Riemann-Silberstein vector and the classical-quantum correspondence,” Opt. Commun. 264, 342–351 (2006).
    [CrossRef]
  10. G. Kaiser, “Helicity, polarization, and Riemann-Silberstein vortices,” arXiv:math-ph/0309010v2.
  11. T. Vold, “An introduction to geometric calculus and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
    [CrossRef]
  12. Q. M. Sugon Jr. and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkäuser, 2002), pp. 297–306.
    [CrossRef]
  13. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 299–300.
  14. W. E. Baylis, R. Cabrera and J. D. Kaselica, “Quantum/classical interface: Classical geometric origin of fermion spin,” Adv. Appl. Clifford Algebras (published online) (2010); http://www.springerlink.com/content/d20h16078tw728rl/
    [CrossRef]
  15. P. Cornille, Advanced Electromagnetism and Vacuum Physics (World Scientific, 2003), pp. 367–387.
  16. S. V. Kukhlevsky, “Non-classical energy conservation in multi-wave systems: ’Extra energy,’ ’negative energy’ and ’annihilation of energy’”; arXiv:physics/0606055v2.
  17. S. V. Kukhlevsky, “Breaking of energy conservation law: Creating and destroying of energy by subwavelength nanosystems”; arXiv:physics/0610008v2.
  18. P. Lounesto, Clifford Algebra and Spinors, 2nd ed. (Cambridge University Press, 2002), pp. 29, 86.
  19. Q. M. Sugon Jr. and D. J. McNamara, “A geometric algebra reformulation of geometric optics,” Am. J. Phys. 72, 92–97 (2004).
    [CrossRef]

2010

B. Thidé, Electromagnetic Field Theory, 2nd ed. (Dover, 2010), p. 22; ebook available from www.plasma.uu.se/CED.

W. E. Baylis, R. Cabrera and J. D. Kaselica, “Quantum/classical interface: Classical geometric origin of fermion spin,” Adv. Appl. Clifford Algebras (published online) (2010); http://www.springerlink.com/content/d20h16078tw728rl/
[CrossRef]

2008

E. A. Notte-Cuello and W. A. Rodrigues Jr., “Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields,” Rep. Math. Phys. 62, 91–101 (2008).
[CrossRef]

2006

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Beams of electromagnetic radiation carrying angular momentum: The Riemann-Silberstein vector and the classical-quantum correspondence,” Opt. Commun. 264, 342–351 (2006).
[CrossRef]

2004

Q. M. Sugon Jr. and D. J. McNamara, “A geometric algebra reformulation of geometric optics,” Am. J. Phys. 72, 92–97 (2004).
[CrossRef]

2003

N. Gauthier, “What happens to energy and momentum when two oppositely moving wave pulses overlap?,” Am. J. Phys. 71, 787–790 (2003).
[CrossRef]

P. Cornille, Advanced Electromagnetism and Vacuum Physics (World Scientific, 2003), pp. 367–387.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, 2003).

D. Hestenes, “Oersted medal lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
[CrossRef]

2002

P. Lounesto, Clifford Algebra and Spinors, 2nd ed. (Cambridge University Press, 2002), pp. 29, 86.

Q. M. Sugon Jr. and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkäuser, 2002), pp. 297–306.
[CrossRef]

1999

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 299–300.

W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhäuser, 1999).

1994

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), p. 165.

1993

T. Vold, “An introduction to geometric calculus and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
[CrossRef]

1988

B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, 1988).

Baylis, W. E.

W. E. Baylis, R. Cabrera and J. D. Kaselica, “Quantum/classical interface: Classical geometric origin of fermion spin,” Adv. Appl. Clifford Algebras (published online) (2010); http://www.springerlink.com/content/d20h16078tw728rl/
[CrossRef]

W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhäuser, 1999).

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Beams of electromagnetic radiation carrying angular momentum: The Riemann-Silberstein vector and the classical-quantum correspondence,” Opt. Commun. 264, 342–351 (2006).
[CrossRef]

Bialynicki-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Beams of electromagnetic radiation carrying angular momentum: The Riemann-Silberstein vector and the classical-quantum correspondence,” Opt. Commun. 264, 342–351 (2006).
[CrossRef]

Cabrera, R.

W. E. Baylis, R. Cabrera and J. D. Kaselica, “Quantum/classical interface: Classical geometric origin of fermion spin,” Adv. Appl. Clifford Algebras (published online) (2010); http://www.springerlink.com/content/d20h16078tw728rl/
[CrossRef]

Cornille, P.

P. Cornille, Advanced Electromagnetism and Vacuum Physics (World Scientific, 2003), pp. 367–387.

Doran, C.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, 2003).

Gauthier, N.

N. Gauthier, “What happens to energy and momentum when two oppositely moving wave pulses overlap?,” Am. J. Phys. 71, 787–790 (2003).
[CrossRef]

Hestenes, D.

D. Hestenes, “Oersted medal lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 299–300.

Jancewicz, B.

B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, 1988).

Kaiser, G.

G. Kaiser, “Helicity, polarization, and Riemann-Silberstein vortices,” arXiv:math-ph/0309010v2.

Kaselica, J. D.

W. E. Baylis, R. Cabrera and J. D. Kaselica, “Quantum/classical interface: Classical geometric origin of fermion spin,” Adv. Appl. Clifford Algebras (published online) (2010); http://www.springerlink.com/content/d20h16078tw728rl/
[CrossRef]

Kukhlevsky, S. V.

S. V. Kukhlevsky, “Non-classical energy conservation in multi-wave systems: ’Extra energy,’ ’negative energy’ and ’annihilation of energy’”; arXiv:physics/0606055v2.

S. V. Kukhlevsky, “Breaking of energy conservation law: Creating and destroying of energy by subwavelength nanosystems”; arXiv:physics/0610008v2.

Lasenby, A.

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, 2003).

Lounesto, P.

P. Lounesto, Clifford Algebra and Spinors, 2nd ed. (Cambridge University Press, 2002), pp. 29, 86.

McNamara, D. J.

Q. M. Sugon Jr. and D. J. McNamara, “A geometric algebra reformulation of geometric optics,” Am. J. Phys. 72, 92–97 (2004).
[CrossRef]

Q. M. Sugon Jr. and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkäuser, 2002), pp. 297–306.
[CrossRef]

Notte-Cuello, E. A.

E. A. Notte-Cuello and W. A. Rodrigues Jr., “Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields,” Rep. Math. Phys. 62, 91–101 (2008).
[CrossRef]

Rodrigues, W. A.

E. A. Notte-Cuello and W. A. Rodrigues Jr., “Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields,” Rep. Math. Phys. 62, 91–101 (2008).
[CrossRef]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), p. 165.

Sugon, Q. M.

Q. M. Sugon Jr. and D. J. McNamara, “A geometric algebra reformulation of geometric optics,” Am. J. Phys. 72, 92–97 (2004).
[CrossRef]

Q. M. Sugon Jr. and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkäuser, 2002), pp. 297–306.
[CrossRef]

Thidé, B.

B. Thidé, Electromagnetic Field Theory, 2nd ed. (Dover, 2010), p. 22; ebook available from www.plasma.uu.se/CED.

Vold, T.

T. Vold, “An introduction to geometric calculus and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
[CrossRef]

Adv. Appl. Clifford Algebras

W. E. Baylis, R. Cabrera and J. D. Kaselica, “Quantum/classical interface: Classical geometric origin of fermion spin,” Adv. Appl. Clifford Algebras (published online) (2010); http://www.springerlink.com/content/d20h16078tw728rl/
[CrossRef]

Am. J. Phys.

T. Vold, “An introduction to geometric calculus and its application to electrodynamics,” Am. J. Phys. 61, 505–513 (1993).
[CrossRef]

N. Gauthier, “What happens to energy and momentum when two oppositely moving wave pulses overlap?,” Am. J. Phys. 71, 787–790 (2003).
[CrossRef]

D. Hestenes, “Oersted medal lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
[CrossRef]

Q. M. Sugon Jr. and D. J. McNamara, “A geometric algebra reformulation of geometric optics,” Am. J. Phys. 72, 92–97 (2004).
[CrossRef]

Opt. Commun.

I. Bialynicki-Birula and Z. Bialynicki-Birula, “Beams of electromagnetic radiation carrying angular momentum: The Riemann-Silberstein vector and the classical-quantum correspondence,” Opt. Commun. 264, 342–351 (2006).
[CrossRef]

Rep. Math. Phys.

E. A. Notte-Cuello and W. A. Rodrigues Jr., “Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields,” Rep. Math. Phys. 62, 91–101 (2008).
[CrossRef]

Other

B. Jancewicz, Multivectors and Clifford Algebra in Electrodynamics (World Scientific, 1988).

C. Doran and A. Lasenby, Geometric Algebra for Physicists (Cambridge University Press, 2003).

W. E. Baylis, Electrodynamics: A Modern Geometric Approach (Birkhäuser, 1999).

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1994), p. 165.

B. Thidé, Electromagnetic Field Theory, 2nd ed. (Dover, 2010), p. 22; ebook available from www.plasma.uu.se/CED.

G. Kaiser, “Helicity, polarization, and Riemann-Silberstein vortices,” arXiv:math-ph/0309010v2.

Q. M. Sugon Jr. and D. J. McNamara, “A Hestenes spacetime algebra approach to light polarization,” in Applications of Geometric Algebra in Computer Science and Engineering, L.Dorst, C.Doran, and J.Lasenby, eds. (Birkäuser, 2002), pp. 297–306.
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), pp. 299–300.

P. Cornille, Advanced Electromagnetism and Vacuum Physics (World Scientific, 2003), pp. 367–387.

S. V. Kukhlevsky, “Non-classical energy conservation in multi-wave systems: ’Extra energy,’ ’negative energy’ and ’annihilation of energy’”; arXiv:physics/0606055v2.

S. V. Kukhlevsky, “Breaking of energy conservation law: Creating and destroying of energy by subwavelength nanosystems”; arXiv:physics/0610008v2.

P. Lounesto, Clifford Algebra and Spinors, 2nd ed. (Cambridge University Press, 2002), pp. 29, 86.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Time-average energy density (projected at horizontal x and vertical z plane) of the superposition of two identical linearly (circularly) polarized waves propagating in the xz plane. One wave moves in the e 3 direction and the other in e 1 sin θ + e 3 cos θ , where θ is (a) 0, (b) π 4 , (c) π 2 , (d) 3 π 4 , and (e) π. Time-average energy density of each wave is a constant.

Fig. 2
Fig. 2

Time-average momentum density (projected at horizontal x and vertical z plane) of the superposition of two identical linearly polarized waves propagating in the xz plane. One wave moves in the e 3 direction and the other in e 1 sin θ + e 3 cos θ , where θ is (a) 0, (b) π 4 , (c) π 2 , (d) 3 π 4 , and (e) π. The time-average momentum density of the superposition has no e 2 component.

Fig. 3
Fig. 3

Time-average momentum density (in three-dimensional projection) of the superposition of two identical linearly polarized waves propagating in the xz plane. One wave moves in the e 3 direction and the other in e 1 sin θ + e 3 cos θ , where θ is (a) 0, (b) π 4 , (c) π 2 , (d) 3 π 4 , and (e) π. The time-average momentum density of the superposition has an e 2 component.

Equations (121)

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

a b = a b + i ( a × b ) ,
U + S c = 1 2 ε 0 ( E + i ζ H ) ( E + i ζ H ) ,
( 1 c t + ) ( E + i ζ H ) = 0 .
E a + i ζ H a = ( e 1 + i e 2 ) ( a ̂ + e i ( ω t k r ) + a ̂ e i ( ω t k r ) ) ,
U a + b U a + U b ,
S a + b S a + S b ,
e i e j + e j e i = 2 δ i j ,
a b = a b + i ( a × b ) ,
A ̂ = A 0 + A 1 + i A 2 + i A 3 .
A ̂ = A 0 A 1 + i A 2 i A 3 .
( A ̂ + B ̂ ) = A ̂ + B ̂ ,
( A ̂ B ̂ ) = A ̂ B ̂ .
e ± i e 2 φ = cos φ ± i e 2 sin φ .
e 1 e ± i e 2 φ = e i e 2 φ e 1 ,
e 2 e ± i e 2 φ = e ± i e 2 φ e 2 ,
e 3 e ± i e 2 φ = e i e 2 φ e 3 .
e 1 = e i e 2 θ 2 e 1 e i e 2 θ 2 = e 1 cos θ e 3 sin θ ,
e 2 = e i e 2 θ 2 e 2 e i e 2 θ 2 = e 2 ,
e 3 = e i e 2 θ 2 e 3 e i e 2 θ 2 = e 3 cos θ + e 1 sin θ .
e ̂ ± = e 1 ± i e 2 .
e ̂ + e ̂ + = e ̂ e ̂ = 0 ,
e ̂ ± e ̂ = 2 ( 1 ± e 3 ) .
e ̂ + e ̂ + e ̂ e ̂ + = 4 ,
e ̂ + e ̂ e ̂ e ̂ + = 4 e 3 .
e ̂ ± = e 1 ± i e 2 = e 1 cos θ e 3 sin θ ± i e 2 ,
e ̂ + e ̂ = ( 1 + e 1 e 1 ) + ( e 3 e 2 × e 1 ) + i e 1 × e 1 ,
e ̂ + e ̂ = ( 1 + e 1 e 1 ) + ( e 3 e 2 × e 1 ) i e 1 × e 1 ,
e ̂ + e ̂ = ( 1 + cos θ ) + ( e 3 + e 3 cos θ + e 1 sin θ ) + i e 2 sin θ ,
e ̂ + e ̂ = ( 1 + cos θ ) + ( e 3 + e 3 cos θ + e 1 sin θ ) i e 2 sin θ .
E ̂ r ̂ = 0 ,
r ̂ = 1 c t + ,
E ̂ = E + i ζ H ,
2 E ̂ r ̂ r ̂ = ( 1 c 2 2 t 2 2 ) ( E + i ζ H ) = 0 .
E ̂ E ̂ = | E | 2 ζ 2 | H | 2 2 ζ E × H .
S ̂ c = 1 2 ε 0 E ̂ E ̂ = U + S c ,
U = 1 2 ( ε 0 | E | 2 + μ 0 | H | 2 ) ,
S = E × H
E ̂ a + b = E ̂ a + E ̂ b ,
E ̂ a + b E ̂ a + b = E ̂ a E ̂ a + E ̂ b E ̂ b + E ̂ b E ̂ a + E ̂ a E ̂ b .
S ̂ a + b c = S ̂ a c + S ̂ b c + S ̂ a b c ,
S ̂ a b c = U a b + S a b c = 1 2 ε 0 ( E ̂ a E ̂ b + E ̂ b E ̂ a ) .
S ̂ a b c = ε 0 2 ( E b E a + ζ 2 H b H a + E a E b + ζ 2 H a H b ) + i 2 ε 0 ζ ( E b H a H b E a + E a H b H a E b ) .
U a b = ε 0 ( E a E b + ζ 2 H a H b ) ,
S a b c = ε 0 ( E a × H b + E b × H a ) .
E a E b = ζ 2 H a H b ,
E a × H b = E b × H a .
E ̂ a ± = E a ± + i ζ H a ± = e ̂ + a ̂ ± ψ ̂ a ± 1 ,
a ̂ ± = | a ̂ ± | e i α ± ,
ψ ̂ a = e i ( ω t k r ) ,
k = k e 3
( 1 c 2 2 t 2 2 ) E ̂ a ± = ( ω 2 c 2 + | k | 2 ) E ̂ a ± = 0 .
ω 2 c 2 | k | 2 = 0 ,
γ ± = ω t k r ± α ± ,
E a ± = e 1 | a ± | cos ( γ ± ) e 2 | a ± | sin ( γ ± ) ,
i ζ H a ± = i [ ± e 1 | a ± | sin ( γ ± ) + e 2 | a ± | cos ( γ ± ) ] .
E ̂ a ± E ̂ a ± = e ̂ + a ̂ ± ψ ̂ a ± 1 ( e ̂ a ̂ ± ψ ̂ a 1 ) = 2 ( 1 + e 3 ) | a ̂ ± | 2 ,
S ̂ a ± c = U a ± + S a ± c = ε 0 ( 1 + e 3 ) | a ̂ ± | 2 .
U a ± = ε 0 | a ̂ ± | 2 ,
S a ± c = e 3 ε 0 | a ̂ ± | 2 .
E ̂ a = E ̂ a + + E ̂ a = E a + i ζ H a = e ̂ + ( a ̂ ψ ̂ a 1 + a ̂ + ψ ̂ a ) .
E ̂ a = e ̂ ( a ̂ ψ ̂ a + a ̂ + ψ ̂ a 1 ) ,
S ̂ a c = ε 0 ( 1 + e 3 ) ( | a ̂ | 2 + | a ̂ + | 2 + a ̂ + a ̂ ψ ̂ a 2 + a ̂ a ̂ + ψ ̂ a 2 ) .
S ̂ a c = ε 0 ( 1 + e 3 ) [ | a ̂ | 2 + | a ̂ + | 2 + 2 | a + | | a | cos ( γ + + γ ) ] ,
S ̂ a c = 1 τ 0 τ S ̂ a c d t = ε 0 [ ( 1 + e 3 ) ( | a ̂ | 2 + | a ̂ + | 2 ) ] .
U a = ε 0 ( | a ̂ | 2 + | a ̂ + | 2 ) ,
S a c 2 = ε 0 c ( | a ̂ | 2 + | a ̂ + | 2 ) e 3 ,
E ̂ b = E b + i ζ H b = e ̂ + ( b ̂ ψ ̂ b 1 + b ̂ + ψ ̂ b ) ,
b ̂ ± = | b ̂ ± | e i β ± ,
ψ ̂ b = e i ( ω t k r ) ,
k = k e 3 .
E ̂ b = e ̂ ( b ̂ ψ ̂ b + b ̂ + ψ ̂ b 1 ) ,
S ̂ b c = ε 0 ( 1 + e 3 ) ( | b ̂ | 2 + | b ̂ + | 2 + b ̂ + b ̂ ψ ̂ b 2 + b ̂ b ̂ + ψ ̂ b 2 ) ,
S ̂ b c = 1 τ 0 τ S ̂ b c d t = ε 0 [ ( 1 + e 3 ) ( | b ̂ | 2 + | b ̂ + | 2 ) ] .
U b = ε 0 ( | b ̂ | 2 + | b ̂ + | 2 ) ,
S b c 2 = ε 0 c ( | b ̂ | 2 + | b ̂ + | 2 ) e 3 .
S ̂ a b c = 1 2 ε 0 ( E ̂ a E ̂ b + E ̂ b E ̂ a ) .
1 2 ε 0 E ̂ a E ̂ b = ε 0 2 e ̂ + e ̂ ( a ̂ b ̂ ψ ̂ a 1 ψ ̂ b + a ̂ b ̂ + ψ ̂ a 1 ψ ̂ b 1 + a ̂ + b ̂ ψ ̂ a ψ ̂ b + a ̂ + b ̂ + ψ ̂ a ψ ̂ b 1 ) ,
1 2 ε 0 E ̂ b E ̂ a = ε 0 2 e ̂ + e ̂ ( b ̂ a ̂ ψ ̂ b 1 ψ ̂ a + b ̂ a ̂ + ψ ̂ b 1 ψ ̂ a 1 + b ̂ + a ̂ ψ ̂ b ψ ̂ a + b ̂ + a ̂ + ψ ̂ b ψ ̂ a 1 ) ,
ψ ̂ a 1 ψ ̂ b = ψ ̂ b ψ ̂ a 1 = e i ( k k ) r ,
ψ ̂ a 1 ψ ̂ b 1 = ψ ̂ b 1 ψ ̂ a 1 = e i [ 2 ω t ( k + k ) r ] ,
ψ ̂ a ψ ̂ b = ψ ̂ b ψ ̂ a = e i [ 2 ω t ( k + k ) r ] ,
ψ ̂ a ψ ̂ b 1 = ψ ̂ b 1 ψ ̂ a = e i ( k k ) r .
S ̂ a b c = ε 0 [ ( 1 + e 1 e 1 + e 3 + e 1 × e 2 ) G + ( e 1 × e 1 ) H ] ,
G = | a ̂ | | b ̂ | cos [ α β + ( k k ) r ] + | a ̂ + | | b ̂ + | cos [ α + β + ( k k ) r ] + | a ̂ | | b ̂ + | cos [ α β + 2 ω t + ( k + k ) r ] + | a ̂ + | | b ̂ | cos [ α + β + 2 ω t ( k + k ) r ] ,
H = | a ̂ | | b ̂ | sin [ α β + ( k k ) r ] + | a ̂ + | | b ̂ + | sin [ α + β + ( k k ) r ] + | a ̂ | | b ̂ + | sin [ α β + 2 ω t + ( k + k ) r ] + | a ̂ + | | b ̂ | sin [ α + β + 2 ω t ( k + k ) r ] .
e ̂ + e ̂ = e ̂ + e ̂ = 0
S ̂ a + b τ c = S ̂ a τ c + S ̂ b τ c + S ̂ a b τ c ,
S ̂ a b τ c = ε 0 2 [ e ̂ + e ̂ ( a ̂ b ̂ e i ( k k ) r + a ̂ + b ̂ + e i ( k k ) r ) + e ̂ + e ̂ ( b ̂ a ̂ e i ( k k ) r + b ̂ + a ̂ + e i ( k k ) r ) ] .
S ̂ a b τ c = ε 0 [ ( 1 + e 1 e 1 + e 3 + e 1 × e 2 ) G + ( e 1 × e 1 ) H ] ,
G = | a ̂ | | b ̂ | cos [ α β + ( k k ) r ] + | a ̂ + | | b ̂ + | cos [ α + β + ( k k ) r ] ,
H = | a ̂ | | b ̂ | sin [ α β + ( k k ) r ] + | a ̂ + | | b ̂ + | sin [ α + β + ( k k ) r ] .
( k k ) r = k [ x sin θ + z ( 1 cos θ ) ] .
e 1 e 1 = cos θ ,
e 1 × e 2 = e 1 sin θ + e 3 cos θ ,
e 1 × e 1 = e 2 sin θ .
S ̂ a b τ c = ε 0 { ( 1 + cos θ ) G + [ e 3 ( 1 + cos θ ) + e 1 sin θ ] G e 2 sin θ H } ,
G = | a ̂ | | b ̂ | cos { α β + k [ x sin θ + z ( 1 cos θ ) ] } + | a ̂ + | | b ̂ + | cos { α + β + k [ x sin θ + z ( 1 cos θ ) ] } ,
H = | a ̂ | | b ̂ | sin { α β + k [ x sin θ + z ( 1 cos θ ) ] } + | a ̂ + | | b ̂ + | sin { α + β + k [ x sin θ + z ( 1 cos θ ) ] } .
U a b = ε 0 ( 1 + cos θ ) G ,
S a b c 2 = ε 0 { [ e 3 ( 1 + cos θ ) + e 1 sin θ ] G e 2 sin θ H } .
U a + b = U a + U b + U a b ,
S a + b c 2 = S a c 2 + S b c 2 + S a b c 2 .
U a + b = ε 0 { | a ̂ | 2 + | a ̂ + | 2 + | b ̂ | 2 + | b ̂ + | 2 + ( 1 + cos θ ) [ | a ̂ | | b ̂ | cos [ α β + | k | ( x sin θ + z ( 1 cos θ ) ) ] + | a ̂ + | | b ̂ + | cos [ α + β + | k | ( x sin θ + z ( 1 cos θ ) ) ] ] } ,
S a + b c 2 = ε 0 c { e 1 sin θ [ | a ̂ | | b ̂ | cos [ α β + | k | ( x sin θ + z ( 1 cos θ ) ) ] + | a ̂ + | | b ̂ + | cos [ α + β + | k | ( x sin θ + z ( 1 cos θ ) ) ] + | b ̂ | 2 + | b ̂ + | 2 ] e 2 sin θ [ | a ̂ | | b ̂ | sin [ α β + | k | ( x sin θ + z ( 1 cos θ ) ) ] + | a ̂ + | | b ̂ + | sin [ α + β + | k | ( x sin θ + z ( 1 cos θ ) ) ] ] + e 3 [ ( 1 + cos θ ) ( | a ̂ | | b ̂ | cos [ α β + | k | ( x sin θ + z ( 1 cos θ ) ) ] + | a ̂ + | | b ̂ + | cos [ α + β + | k | ( x sin θ + z ( 1 cos θ ) ) ] ) + | a ̂ | 2 + | a ̂ + | 2 + ( | b ̂ | 2 + | b ̂ + | 2 ) cos θ ] } .
U a = U b = 2 ε 0 ,
U a + b = ε 0 { 4 + 2 ( 1 + cos θ ) cos [ x sin θ z ( 1 cos θ ) ] } ,
S a c 2 = 2 ε 0 c e 3 ,
S b c 2 = 2 ε 0 c ( sin θ e 1 + cos θ e 3 ) ,
S a + b c 2 = 2 ε 0 c { e 1 sin θ [ 1 + cos ( x sin θ z ( 1 cos θ ) ) ] + e 3 ( 1 + cos θ ) [ 1 + cos ( x sin θ z ( 1 cos θ ) ) ] } .
U a + b U a + U b ,
S a + b c 2 S a c 2 + S b c 2 .
U a = U b = ε 0 ,
U a + b = ε 0 { 2 + ( 1 + cos θ ) cos [ x sin θ z ( 1 cos θ ) ] } ,
S a c 2 = ε 0 c e 3 ,
S b c 2 = ε 0 c ( sin θ e 1 + cos θ e 3 ) ,
S a + b c 2 = ε 0 c { e 1 sin θ [ 1 + cos ( x sin θ z ( 1 cos θ ) ) ] e 2 sin θ [ sin ( x sin θ z ( 1 cos θ ) ) ] + e 3 ( 1 + cos θ ) [ 1 + cos ( x sin θ z ( 1 cos θ ) ) ] } .
U a = U b = ε 0 ,
U a + b = 2 ε 0 ,
S a c 2 = ε 0 c e 3 ,
S b c 2 = ε 0 c ( sin θ e 1 + cos θ e 3 ) ,
S a + b c 2 = ε 0 c [ e 1 sin θ + e 3 ( 1 + cos θ ) ] .

Metrics