Abstract

We present an absorbing boundary condition for electromagnetic frequency domain simulations of photonic crystals and metamaterials. This boundary condition can simultaneously absorb multiple Bloch–Floquet eigenmodes of a periodic crystal, including both propagating and evanescent modes. The photonic crystal or metamaterial in question can include lossy, active, anisotropic, and even bi-anisotropic inclusions. The absorbing boundary condition is dependent on an orthogonality condition for Bloch–Floquet eigenmodes, a generalized version of which is presented here. We test this absorbing boundary condition numerically and present the results.

© 2013 Optical Society of America

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References

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  1. A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microw. Guided Wave Lett. 9, 502–504 (1999).
    [CrossRef]
  2. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  3. W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
    [CrossRef]
  4. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), Chap. 9.6.
  5. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
    [CrossRef]
  6. Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20, 463–468 (2002).
    [CrossRef]
  7. M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. 11, 152–154 (2001).
    [CrossRef]
  8. E. P. Kosmidou, T. I. Kosmanis, and T. D. Tsiboukis, “A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures,” IEEE Trans. Magn. 39, 1191–1194 (2003).
    [CrossRef]
  9. A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
    [CrossRef]
  10. A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, “The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers,” Opt. Express 16, 11376–11392 (2008).
    [CrossRef]
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    [CrossRef]
  12. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), Chap. 11.1.1.
  13. J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).
  14. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljaičić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
    [CrossRef]
  15. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannoppoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
    [CrossRef]
  16. M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide: fundamental connection between polarization-mode dispersion and group-velocity dispersion,” J. Opt. Soc. Am. B 19, 2867–2875 (2002).
    [CrossRef]
  17. D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
    [CrossRef]
  18. W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B 82, 235306 (2010).
    [CrossRef]
  19. M. Davanco, Y. Urzhumov, and G. Shvets, “The complex bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction,” Opt. Express 15, 9681–9691 (2007).
    [CrossRef]
  20. C. Fietz, Y. Urzhumov, and G. Shvets, “Complex k band diagrams of 3D metamaterial/photonic crystals,” Opt. Express 19, 19027–19041 (2011).
    [CrossRef]

2011

2010

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B 82, 235306 (2010).
[CrossRef]

2008

2007

2004

A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
[CrossRef]

2003

E. P. Kosmidou, T. I. Kosmanis, and T. D. Tsiboukis, “A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures,” IEEE Trans. Magn. 39, 1191–1194 (2003).
[CrossRef]

D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
[CrossRef]

2002

2001

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. 11, 152–154 (2001).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljaičić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
[CrossRef]

1999

A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microw. Guided Wave Lett. 9, 502–504 (1999).
[CrossRef]

1996

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

1994

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Adibi, A.

Askari, M.

Avniel, Y.

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bienstman, P.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannoppoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Braeuer, A.

D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Davanco, M.

Engeness, T. D.

Esselle, K. P.

A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
[CrossRef]

Fan, S.

A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microw. Guided Wave Lett. 9, 502–504 (1999).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Fietz, C.

Fink, Y.

Horbath, L.

A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
[CrossRef]

Ibanescu, M.

Integlia, R. A.

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B 82, 235306 (2010).
[CrossRef]

Jacobs, S. A.

Jiang, W.

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B 82, 235306 (2010).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), Chap. 11.1.1.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), Chap. 9.6.

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljaičić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
[CrossRef]

A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microw. Guided Wave Lett. 9, 502–504 (1999).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Joannoppoulos, J. D.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannoppoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Johnson, S. G.

Koshiba, M.

Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20, 463–468 (2002).
[CrossRef]

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. 11, 152–154 (2001).
[CrossRef]

Kosmanis, T. I.

E. P. Kosmidou, T. I. Kosmanis, and T. D. Tsiboukis, “A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures,” IEEE Trans. Magn. 39, 1191–1194 (2003).
[CrossRef]

Kosmidou, E. P.

E. P. Kosmidou, T. I. Kosmanis, and T. D. Tsiboukis, “A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures,” IEEE Trans. Magn. 39, 1191–1194 (2003).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Lidorikis, E.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannoppoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Mekis, A.

A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microw. Guided Wave Lett. 9, 502–504 (1999).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Michaelis, D.

D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
[CrossRef]

Momeni, B.

Oskooi, A. F.

Peschel, U.

D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
[CrossRef]

Reinke, C. M.

Sanders, B. C.

A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
[CrossRef]

Sasaki, S.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. 11, 152–154 (2001).
[CrossRef]

Shvets, G.

Skorobogatiy, M.

Skorobogatiy, M. A.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannoppoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Soljacic, M.

Soljaicic, M.

Song, W.

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B 82, 235306 (2010).
[CrossRef]

Tsiboukis, T. D.

E. P. Kosmidou, T. I. Kosmanis, and T. D. Tsiboukis, “A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures,” IEEE Trans. Magn. 39, 1191–1194 (2003).
[CrossRef]

Tsuji, Y.

Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20, 463–468 (2002).
[CrossRef]

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. 11, 152–154 (2001).
[CrossRef]

Urzhumov, Y.

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Waechter, C.

D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Weily, A.

A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
[CrossRef]

Weisberg, O.

Zhang, L.

Appl. Opt.

IEEE Microw. Guided Wave Lett.

A. Mekis, S. Fan, and J. D. Joannopoulos, “Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides,” IEEE Microw. Guided Wave Lett. 9, 502–504 (1999).
[CrossRef]

IEEE Microw. Wirel. Compon.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wirel. Compon. 11, 152–154 (2001).
[CrossRef]

IEEE Trans. Magn.

E. P. Kosmidou, T. I. Kosmanis, and T. D. Tsiboukis, “A comparative FDTD study of various PML configurations for the termination of nonlinear photonic bandgap waveguide structures,” IEEE Trans. Magn. 39, 1191–1194 (2003).
[CrossRef]

J. Comput. Phys.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Microw. Opt. Technol. Lett.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microw. Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

A. Weily, L. Horbath, K. P. Esselle, and B. C. Sanders, “Performance of PML absorbing boundary conditions in 3D photonic crystal waveguides,” Microw. Opt. Technol. Lett. 40, 1–3 (2004).
[CrossRef]

Opt. Express

Phys. Rev. B

W. Song, R. A. Integlia, and W. Jiang, “Slow light loss due to roughness in photonic crystal waveguides: an analytic approach,” Phys. Rev. B 82, 235306 (2010).
[CrossRef]

Phys. Rev. E

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannoppoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Phys. Rev. Lett.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787–3790 (1996).
[CrossRef]

Proc. SPIE

D. Michaelis, U. Peschel, C. Waechter, and A. Braeuer, “Coupling coefficients of photonic crystal waveguides,” Proc. SPIE 4987, 114–125 (2003).
[CrossRef]

Other

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), Chap. 11.1.1.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002).

J. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), Chap. 9.6.

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Figures (4)

Fig. 1.
Fig. 1.

(a) |Hz| of a ωa/c=0.5, ky=ω/csin(π/6) eigenmode propagating through five unit cells of the PC. The cylinder in the center of the unit cell is vacuum with permittivity ϵ=1. The surrounding area is a dielectric with permittivity ϵ=5i·106. The eigenmode is excited at the left boundary and absorbed at the right boundary. The negligible reflection can be seen visually from the absence of interference in the field profile. (b) Power reflection from the right boundary normalized to the incident power for three different values of ky. Both incident and reflected powers were calculated using Eq. (8).

Fig. 2.
Fig. 2.

(a) Diagram of the scattering simulation. Plane waves are incident upon the vacuum–PC interface from the vacuum, and Bloch eigenmodes are incident upon the interface from the PC, with ky=π/6 for all incident waves. The amplitudes of the outgoing waves are measured using Eq. (8) and normalized to the incident amplitudes to define the scattering amplitudes of the interface. (b) Diagram of the simulation of reflection and transmission of plane waves through a five layered PC slab. (c) Real and (d) imaginary parts of kx(ω) calculated from a complex wavenumber eigenvalue simulation [19]. (e) Absolute value and (f) argument (phase) of the transmission and reflection amplitudes returned by the five layer PC slab simulation diagramed in (b) (solid lines) and calculated using the interface scattering amplitudes (dotted lines).

Fig. 3.
Fig. 3.

(a) Diagram of the 2D PC waveguide simulation, calculating the reflection from a 90° turn in the PC waveguide. (b) Band diagram of a E=Eze^ eigenmode of the PC. The bandgap is indicated by the shaded area. (c) Complex wavenumber dispersion curve [19] of the PC waveguide eigenmode. The bandgap of the PC is again indicated by the shaded area. Note that inside the bandgap there is a cutoff frequency for the PC waveguide mode. Also, at the high frequency end of the bandgap the PC waveguide mode remains confined to the waveguide even as the frequency exits the bandgap. At frequencies above the bandgap, there is an additional nonevanescent leaky mode in the PC waveguide that is not shown here.

Fig. 4.
Fig. 4.

(a) |Ez| for the PC waveguide bend simulation at frequency ωa/(2πc)=0.41. The waveguide eigenmode is excited at the left boundary of the domain and absorbed at both the left and top boundaries. Note the interference in the horizontal branch of the waveguide due to the reflection at the 90° bend and the lack of interference in the vertical branch of the PC waveguide due to negligible reflection at the Bloch-ABC boundary. (b) Power reflection from the top Bloch-ABC boundary normalized to power incident upon that boundary. (c) Power reflection from the 90° bend in the PC waveguide normalized to power incident upon the 90° bend. The shaded area indicates the PC bandgap shown in Fig. 3(b).

Equations (21)

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

ϵω2c2E×(1μ×E)=0.
FE(v,E)=ϵω2c2v·E(×v)1μ(×E).
Ωd3xFE(v,E)=Ωd3xv·[ϵω2c2E×(1μ×E)]+Ωdav·[n^×(1μ×E)],
Bdn·[E˜i*×Hj+Ej×H˜i*]=Siδij.
n^×(1μ×E)=iωcn^×[n+AnincHn+nAnreflHn],
Anrefl=Bdn·[E˜n*×H+E×H˜n*]Bdn·[E˜n*×Hn+En×H˜n*].
B(v,E)=v·iωc[n+AnincHn+nHnBdn·[E˜n*×H+E×H˜n*]Bdn·[E˜n*×Hn+En×H˜n*]].
An=12Bdn·[E˜n*×H+E×H˜n*](12Bdn·[E˜n*×Hn+En×H˜n*])1/2.
E(t,x)=e(x)ei(ωtk·x),H(t,x)=h(x)ei(ωtk·x),
e(x+N·a)=e(x),h(x+N·a)=h(x),
(A^iB^n)·ψi=βiB^ψi,(A^iB^n)·ψ˜i=βi*B^ψ˜i,A^=(ϵ^ωcξ^ωc+it×+k0×ζ^ωcit×k0×μ^ωc),B^=(0n^×n^×0),ψi=(eihi),ψ˜i=(e˜ih˜i).
ψ˜i|O^|ψjΩdn(eihi)·O^·(ejhj),
ψ˜i|(A^iB^n)|ψj=ψj|(A^iB^n)|ψ˜i*inψ˜i|B^|ψj,
nψ˜i|B^|ψj=i(βiβj)ψ˜i|B^|ψj.
ψ˜i|B^|ψj=Ωdn·[e˜i*×hj+ej×h˜i*]=siδij,
Ωdn·[E˜i*×Hj+Ej×H˜i*]=Siδij,
A^·ui=λiB^·ui.
vi·A^=λivi·B^,
vi·(B^·uj)=1λj(vi·A^)·uj=λiλjvi·B^·uj.
vi·B^·uj=biδij,
vi·A^·uj=aiδij,

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