Abstract

A system of coupled quantum harmonic oscillators whose Hamiltonian conserves photon number begets a one-photon correspondence principle (OPCoP), which allows solutions to the classical linear Maxwell equations for propagation in matter to be reinterpreted as precise descriptions of one-photon states. With the help of the OPCoP, we derive the linear classical Maxwell equations from the Schrödinger equation for one-polariton state evolution. The role of the matter’s initial quantum state in setting the macroscopic medium parameters is made explicit. It is shown that most of the kinds of linear Maxwell equations possible follow from this model, thus showing that the vast extant body of linear, sourceless optical waveguide theory [Optical Waveguide Theory (Chapman and Hall, 1983)] can be applied to the exact analysis of one-photon propagation in optical fibers.

© 2007 Optical Society of America

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  1. R. W. C. Vance and F. Ladouceur, "One-photon electrodynamics in optical fiber with fluorophore systems. I. One-photon correspondence principle for electromagnetic field propagation in matter," J. Opt. Soc. Am. B 24, 928-941 (2007).
    [CrossRef]
  2. I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).
  3. I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.
  4. I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
    [CrossRef]
  5. R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).
  6. D. Marcuse, Engineering Quantum Electrodynamics (Harcourt Brace, 1970).
  7. M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).
  8. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press,2005).
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    [CrossRef] [PubMed]
  10. J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
    [CrossRef]
  11. T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  15. M. Hawton, "Photon wave functions in a localized coordinate space basis," Phys. Rev. A 59, 3223-3227 (1999).
    [CrossRef]
  16. M. Hawton, "Photon position operator with commuting components," Phys. Rev. A 59, 954-959 (1999).
    [CrossRef]
  17. M. Hawton and W. E. Baylis, "Photon position operators and localized bases," Phys. Rev. A 64, 012101 (2001).
    [CrossRef]
  18. I. Bialynicki-Birula, "Exponential localization of photons," Phys. Rev. Lett. 80, 5247-5250 (1998).
    [CrossRef]
  19. T. M. Monro, School of Chemistry & Physics. University of Adelaide, Adelaide, South Australia 5005, Australia (personal communication, 2005).
  20. C. H. Henry and Y. Shani, "Analysis of mode propagation in optical waveguide devices by fourier expansion," IEEE J. Quantum Electron. 27, 523-530 (1991).
    [CrossRef]
  21. S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
    [CrossRef]
  22. L. Poladian, N. A. Issa, and T. M. Munro, "Fourier decomposition algorithm for leaky modes of fibers with arbitrary geometry," Opt. Express 10, 449-454 (2002).
    [PubMed]
  23. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989).
    [PubMed]
  24. I. M. Bassett, Optical Fibre Technology Centre, University of Sydney, 206 National Innovation Centre, Australian Technology Park Eveleigh, New South Wales 1430, Australia (personal communication. 2006).

2007 (1)

2004 (1)

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

2002 (2)

J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
[CrossRef]

L. Poladian, N. A. Issa, and T. M. Munro, "Fourier decomposition algorithm for leaky modes of fibers with arbitrary geometry," Opt. Express 10, 449-454 (2002).
[PubMed]

2001 (1)

M. Hawton and W. E. Baylis, "Photon position operators and localized bases," Phys. Rev. A 64, 012101 (2001).
[CrossRef]

1999 (2)

M. Hawton, "Photon wave functions in a localized coordinate space basis," Phys. Rev. A 59, 3223-3227 (1999).
[CrossRef]

M. Hawton, "Photon position operator with commuting components," Phys. Rev. A 59, 954-959 (1999).
[CrossRef]

1998 (1)

I. Bialynicki-Birula, "Exponential localization of photons," Phys. Rev. Lett. 80, 5247-5250 (1998).
[CrossRef]

1996 (1)

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

1995 (1)

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

1994 (2)

1992 (1)

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

1991 (1)

C. H. Henry and Y. Shani, "Analysis of mode propagation in optical waveguide devices by fourier expansion," IEEE J. Quantum Electron. 27, 523-530 (1991).
[CrossRef]

Amann, U.

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Barnett, S. M.

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

Bassett, I. M.

I. M. Bassett, Optical Fibre Technology Centre, University of Sydney, 206 National Innovation Centre, Australian Technology Park Eveleigh, New South Wales 1430, Australia (personal communication. 2006).

Baylis, W. E.

M. Hawton and W. E. Baylis, "Photon position operators and localized bases," Phys. Rev. A 64, 012101 (2001).
[CrossRef]

Bennett, C. H.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (Bangalore, India, 1984). pp. 175-179.

Bialynicki-Birula, I.

I. Bialynicki-Birula, "Exponential localization of photons," Phys. Rev. Lett. 80, 5247-5250 (1998).
[CrossRef]

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).

I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989).
[PubMed]

Brassard, G.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (Bangalore, India, 1984). pp. 175-179.

Delaney, P. M.

Fiebig, M.

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Gerry, C.

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press,2005).

Harris, M. R.

Hawton, M.

M. Hawton and W. E. Baylis, "Photon position operators and localized bases," Phys. Rev. A 64, 012101 (2001).
[CrossRef]

M. Hawton, "Photon wave functions in a localized coordinate space basis," Phys. Rev. A 59, 3223-3227 (1999).
[CrossRef]

M. Hawton, "Photon position operator with commuting components," Phys. Rev. A 59, 954-959 (1999).
[CrossRef]

Henry, C. H.

C. H. Henry and Y. Shani, "Analysis of mode propagation in optical waveguide devices by fourier expansion," IEEE J. Quantum Electron. 27, 523-530 (1991).
[CrossRef]

Hewlett, S. J.

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

Hohlwein, D.

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Huttner, B.

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

Ihringer, J.

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Issa, N. A.

Kim, H.

J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
[CrossRef]

King, R. G.

Knight, P. L.

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press,2005).

Ladouceur, F.

R. W. C. Vance and F. Ladouceur, "One-photon electrodynamics in optical fiber with fluorophore systems. I. One-photon correspondence principle for electromagnetic field propagation in matter," J. Opt. Soc. Am. B 24, 928-941 (2007).
[CrossRef]

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

Lonkai, T.

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Lottermoser, T.

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall,1983).

Marcuse, D.

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt Brace, 1970).

Monro, T. M.

T. M. Monro, School of Chemistry & Physics. University of Adelaide, Adelaide, South Australia 5005, Australia (personal communication, 2005).

Munro, T. M.

Poladian, L.

Priya, S.

J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
[CrossRef]

Ryu, J.

J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
[CrossRef]

Scully, M. O.

M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).

Shani, Y.

C. H. Henry and Y. Shani, "Analysis of mode propagation in optical waveguide devices by fourier expansion," IEEE J. Quantum Electron. 27, 523-530 (1991).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall,1983).

Suhail Zubiary, M.

M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).

Uchino, K.

J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
[CrossRef]

Vance, R. W. C.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989).
[PubMed]

Acta Phys. Pol. A (1)

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

C. H. Henry and Y. Shani, "Analysis of mode propagation in optical waveguide devices by fourier expansion," IEEE J. Quantum Electron. 27, 523-530 (1991).
[CrossRef]

J. Electroceram. (1)

J. Ryu, S. Priya, K. Uchino, and H. Kim, "Magnetoelectric effect in composites of magnetostrictive and piezoelectric materials," J. Electroceram. 8, 107-119 (2002).
[CrossRef]

J. Lightwave Technol. (1)

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

T. Lottermoser, T. Lonkai, U. Amann, D. Hohlwein, J. Ihringer, and M. Fiebig, "Magnetic phase control by an electric field," Nature 430, 541-544 (2004).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Rev. A (4)

M. Hawton, "Photon wave functions in a localized coordinate space basis," Phys. Rev. A 59, 3223-3227 (1999).
[CrossRef]

M. Hawton, "Photon position operator with commuting components," Phys. Rev. A 59, 954-959 (1999).
[CrossRef]

M. Hawton and W. E. Baylis, "Photon position operators and localized bases," Phys. Rev. A 64, 012101 (2001).
[CrossRef]

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

I. Bialynicki-Birula, "Exponential localization of photons," Phys. Rev. Lett. 80, 5247-5250 (1998).
[CrossRef]

Prog. Opt. (1)

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

Other (10)

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt Brace, 1970).

M. O. Scully and M. Suhail Zubiary, Quantum Optics (Cambridge U. Press, 1997).

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge U. Press,2005).

I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.

T. M. Monro, School of Chemistry & Physics. University of Adelaide, Adelaide, South Australia 5005, Australia (personal communication, 2005).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall,1983).

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (Bangalore, India, 1984). pp. 175-179.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1989).
[PubMed]

I. M. Bassett, Optical Fibre Technology Centre, University of Sydney, 206 National Innovation Centre, Australian Technology Park Eveleigh, New South Wales 1430, Australia (personal communication. 2006).

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Figures (1)

Fig. 1
Fig. 1

Definition of basis vectors for free-space modes.

Equations (161)

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E ( k , r , ς , δ ) = E 0 cos ( k r ω t + δ ) e ( k , ς ) ,
H ( k , r , ς , δ ) = E 0 μ 0 c cos ( k r ω t + δ ) h ( k , ς ) ,
e ( k , ς ) = { i ( k ) ς = TE j ( k ) ς = TM ,
h ( k , ς ) = k k × e ( k , ς ) = { j ( k ) ς = TE i ( k ) ς = TM ,
i ( k ) = { x if k = k z z × k z × k otherwise , j ( k ) = k × i ( k ) .
( E ( k ) H ( k ) ) = [ 1 ε 0 ϕ T E ( k ) 1 ε 0 ϕ T M ( k ) 1 μ 0 ϕ T M ( k ) 1 μ 0 ϕ T E ( k ) ] ( i j ) ,
ϕ T E ( k ) = ε 0 E 0 ( k , 1 ) cos ( k r ω t + δ 1 ) ,
ϕ T M ( k ) = μ 0 E 0 ( k , 2 ) cos ( k r ω t + δ 2 ) .
F ± ( k , r , t ) = ε 0 2 E ( k , r , t ) ± i μ 0 2 H ( k , r , t ) = ϕ ± ( k , r , t ) f ± ( k ) ,
ϕ ± ( k , r , t ) = 1 2 ( ϕ T E ( k , r , t ) i ϕ T M ( k , r , t ) ) ,
f ± ( k ) = 1 2 ( i ( k ) ± i j ( k ) ) ,
R 0 ( z ) = I R 0 ( z ) = diag ( 1 1 1 ) ,
z = k k ,
x = R 0 ( k ) x ,
y = R 0 ( k ) y ;
z = + z ,
x = + x ,
y ̂ = + y ;
z = z ,
x = x ,
y = + y .
R k ( α ) R 0 ( k ) = [ r 11 r 12 cos ( ϕ ) sin ( θ ) r 21 r 22 sin ( ϕ ) sin ( θ ) cos ( ϕ + α ) sin ( θ ) sin ( ϕ + α ) sin ( θ ) cos ( θ ) ] ,
r 11 = cos ( ϕ ) ( cos ( θ ) 1 ) cos ( ϕ + α ) + cos ( α ) ,
r 12 = cos ( ϕ ) ( cos ( θ ) 1 ) sin ( ϕ + α ) + sin ( α ) ,
r 21 = cos ( ϕ ) ( cos ( θ ) 1 ) sin ( ϕ + α ) cos ( θ ) sin ( α ) ,
r 22 = cos ( ϕ ) ( cos ( θ ) 1 ) cos ( ϕ + α ) + cos ( θ ) cos ( α ) ,
k = k [ cos ( ϕ ) sin ( θ ) sin ( ϕ ) sin ( θ ) cos ( θ ) ] .
{ i ( k ) = R k ( α ( k k ) ) R 0 ( k ) x , j ( k ) = R k ( α ( k k ) ) R 0 ( k ) y , k } ,
α = π 2 ϕ R ( k ) = [ sin ( ϕ ) cos ( θ ) cos ( ϕ ) sin ( θ ) cos ( ϕ ) cos ( ϕ ) cos ( θ ) sin ( ϕ ) sin ( θ ) sin ( ϕ ) 0 sin ( θ ) cos ( θ ) ] ,
i ( k ) = R ( k ) x , j ( k ) = R ( k ) y ,
f ± ( k ) = i ( k ) ± i j ( k ) 2 = R ( k ) x ± i y 2 .
i ( k ) × j ( k ) = k k , k × i ( k ) = j ( k ) k , j ( k ) × k = i ( k ) k ,
f ± ( k ) × f ( k ) = i k k , k × f ± ( k ) = i k f ± ( k ) .
U = ε 0 E 0 ( k , ς ) 2 L x L y L z ,
E ̂ 0 = m 2 ε 0 L x L y L z ω x ̂ .
x ̂ = 2 m ω ( a exp ( i ξ ) + a exp ( i ξ ) ) ,
p ̂ = i m ω 2 ( a exp ( i ξ ) a exp ( i ξ ) ) .
E ̂ ( k , r , ς ) ( a ( k , ς ) exp ( i ξ ( r ) ) + a ( k , ς ) exp ( i ξ ( r ) ) ) e ( k , ς ) ,
H ̂ ( k , r , ς ) ( a ( k , ς ) exp ( i ξ ( r ) ) + a ( k , ς ) exp ( i ξ ( r ) ) ) h ( k , ς ) .
α cos ( ω t ξ ( r ) arg ( α ) ) ,
ξ ( r ) = k r with ω = k c .
E ̂ ( k , r , ς ) = 1 2 ω ε 0 L x L y L z ( a ( k , ς ) exp ( i k r ) + a ( k , ς ) exp ( i k r ) ) e ( k , ς ) ,
H ̂ ( k , r , ς ) = 1 2 ω μ 0 L x L y L z ( a ( k , ς ) exp ( i k r ) + a ( k , ς ) exp ( i k r ) ) h ( k , ς ) ,
E ̂ ( k , ς ) ( a ( k , ς ) + a ( k , ς ) ) e ( k , ς ) ,
H ̂ ( k , ς ) ( a ( k , ς ) + a ( k , ς ) ) h ( k , ς ) ,
exp ( α a ( k , ς ) e i ( ω t k r ) ) 0 ( 0 , 0 ) = e i ω t 2 ( 0 ( 0 , 0 ) + α e i ( ω t k r ) 1 ( 0 , 0 ) + α 2 2 ! e 2 i ( ω t k r ) 2 ( 0 , 0 ) + ) .
n ( r , t ) = exp ( n i ( ω t k r ) ) n ( 0 , 0 ) .
n T E , 1 n T E , 2 n T E , 3 , , , n T M , 1 n T M , 2 n T M , 3 ( r , t ) = exp ( i j ( ( n T E , j + n T M , j ) ( ω j t k j r ) ) ) n T E , 1 n T E , 2 n T E , 3 , , , n T M , 1 n T M , 2 n T M , 3 ( 0 , 0 ) ,
n T E , 1 n T E , 2 n T E , 3 , , , n T M , 1 n T M , 2 n T M , 3 ( 0 , 0 ) = ( n T E , 1 ! n T M , 1 ! n T E , 2 ! n T M , 2 ! ) 1 2 a ( k 1 , TE ) n T E , 1 a ( k 1 , TM ) n T M , 1 a ( k 2 , TE ) n T E , 2 a ( k 2 , TM ) n T M , 2 0 ( 0 , 0 ) .
H ̂ = K 0 I + K j k a j a k ,
K j k = δ j k ω j = c δ j k k j ,
{ ψ 1 , T E 1 0 0 , , , 0 0 0 ( 0 , 0 ) + ψ 1 , T M 0 0 0 , , , 1 0 0 ( 0 , 0 ) + ψ 2 , T E 0 1 0 , , , 0 0 0 ( 0 , 0 ) + ψ 2 , T M 0 0 0 , , , 0 1 0 ( 0 , 0 ) + ψ 3 , T E 0 0 1 , , , 0 0 0 ( 0 , 0 ) : ψ j C ψ j , T E 2 + ψ j , T M 2 = 1 }
i t ( ψ 1 , T E ( r , t ) ψ 1 , T M ( r , t ) ψ 2 , T E ( r , t ) ) = ( K 11 K 12 K 13 K 12 * K 22 K 23 K 13 * K 23 * K 33 ) ( ψ 1 , T E ( r , t ) ψ 1 , T M ( r , t ) ψ 2 , T E ( r , t ) ) ,
ψ j , ς ( r , t ) = ψ j , ς ( 0 , t ) exp ( i k j r ) ψ j , ς ( r , t ) = i k j ψ j , ς ( r , t ) .
ε 0 E ̂ ( k ) = ε 0 ( E ̂ ( k , TE ) + E ̂ ( k , TM ) ) ,
= 1 2 ω L x L y L z ( x ̂ ( k , TE ) i ( k ) + x ̂ ( k , TM ) j ( k ) ) ,
μ 0 H ̂ ( k ) = μ 0 ( H ̂ ( k , TE ) + H ̂ ( k , TM ) ) ,
= 1 2 ω L x L y L z ( x ̂ ( k , TM ) i ( k ) + x ̂ ( k , TE ) j ( k ) ) ,
x ̂ ( k , ς ) = a ( k , ς ) + a ( k , ς ) .
F ± ( k ) = 1 2 ( ε 0 E ̂ ( k ) ± i μ 0 H ̂ ( k ) ) = 1 2 ω L x L y L z ( x ̂ ( k , TE ) i x ̂ ( k , TM ) 2 ) i ( k ) ± i j ( k ) 2 , = 1 2 ω L x L y L z ( a ± ( k ) + a ( k ) ) f ± ( k ) ,
a ± ( k ) = a ( k , TE ) ± i a ( k , TM ) 2 .
n + , 1 n + , 2 , , n , 1 n , 2 ( 0 , 0 ) = a + ( k 1 ) n + , 1 a ( k 1 ) n , 1 a + ( k 2 ) n + , 2 a ( k 2 ) n + , 2 n + , 1 ! n , 1 ! n + , 2 ! n , 2 ! 0 ( 0 , 0 ) ,
exp ( α a + ( k j ) e i ( k r ω t ) ) 0 ( 0 , 0 ) = exp ( α a T E ( k j ) i a T M ( k j ) 2 e i ( k r ω t ) ) 0 ( 0 , 0 ) , = exp ( α 2 a T E ( k j ) e i ( k r ω t ) ) exp ( i α 2 a T M ( k j ) e i ( k r ω t ) ) 0 ( 0 , 0 ) ,
E ( k , r , t ) cos ( k r ω t ) i ( k ) + sin ( k r ω t ) j ( k ) ,
H ( k , r , t ) sin ( k r ω t ) i ( k ) + cos ( k r ω t ) j ( k ) .
ψ j , ± ( r , t ) = 1 2 ( ψ j , T E ( r , t ) i ψ j , T M ( r , t ) ) .
{ ψ 1 , + ( r , t ) , ψ 1 , ( r , t ) , ψ 2 , + ( r , t ) , } ( F + ( r , t ) = j ω j ψ j , + ( 0 , t ) f + ( k j ) exp ( i k j r ) F ( r , t ) = j ω j ψ j , ( 0 , t ) f ( k j ) exp ( i k j r ) ) ,
i F ± ( r , t ) t = j ω j ω j ψ j , ± ( 0 , t ) f ± ( k j ) exp ( i k j r ) = c j ω j ψ j , ± ( 0 , t ) k j f ± ( k j ) exp ( i k j r ) = ± c j ω j ψ j , ± ( 0 , t ) i k j × f ± ( k j ) exp ( i k j r ) = ± c × F ± ( r , t ) .
i F ± ( r , t ) t = ± c × F ± ( r , t ) , F ± ( r , t ) = 0 .
F ± ( r , t ) = 1 8 π 3 all k e i k r F ± ( r , t ) d 3 k ,
F ̃ ± ( k , t ) = 1 8 π 3 all r e i k r F ± ( r , t ) d 3 r .
F ̃ ± ( k , t ) t = ± c k × F ̃ ± ( k , t ) , k F ̃ ± ( k , t ) = 0 .
{ Ψ A , T E T M , Ψ B , T E T M } , { Ψ A , ± , Ψ B , ± } ,
{ F ̃ A , ± ( k , t ) , F ̃ B , ± ( k , t ) } , { F A , ± ( r , t ) , F B , ± ( r , t ) } ,
Ψ A , Ψ B = j 1 ( ψ A , T E , j * ψ B , T E , j + ψ A , T M , j * ψ B , T M , j ) = j 1 ( ψ A , + , j * ψ B , + , j + ψ A , , j * ψ B , , j ) , = 1 c all k k 1 ( F ̃ A , + ( k , t ) F ̃ B , + ( k , t ) + F ̃ A , ( k , t ) F ̃ B , ( k , t ) ) d 3 k , = all r ( F A , + ( r , t ) L B B F B , + ( r , t ) + F A , ( r , t ) L B B F B , ( r , t ) ) d 3 r ,
L B B X ( r ) = def 1 2 π 3 c all k exp ( i r k ) k all u exp ( i u k ) X ( u ) d 3 u d 3 k ,
ε 0 E ̆ ( k , r , t ) = 1 2 ω L x L y L z ( ψ T E ( k , r , t ) i ( k ) + ψ T M ( k , r , t ) j ( k ) ) ,
μ 0 H ̆ ( k , r , t ) = 1 2 ω L x L y L z ( ψ T M ( k , r , t ) i ( k ) + ψ T M ( k , r , t ) j ( k ) )
E ̆ ( r , t ) = 1 2 ε 0 ( F + ( r , t ) + F ( r , t ) ) , E ( r , t ) = Re ( E ̆ ( r , t ) ) ,
H ̆ ( t , r ) = 1 i 2 μ 0 ( F + ( r , t ) F ( r , t ) ) , H ( r , t ) = Re ( H ̆ ( r , t ) ) .
F ± = 1 2 ( ε 0 E ( r , t ) ± i μ 0 H ( r , t ) ) ,
i ψ T E ( k , t ) t = ω f ( k ) ψ T E ( k , t ) + exp ( i k r 0 ) C T E * ( k ) ψ m ( t ) , ω f ( k ) = c k ,
i ψ T M ( k , t ) t = ω f ( k ) ψ T M ( k , t ) + exp ( i k r 0 ) C T M * ( k ) ψ m ( t ) ,
i ψ m ( t ) t = ω m ψ m ( t ) + al k exp ( i k r 0 ) [ C T E ( k ) ψ T E ( k , t ) + C T M ( k ) ψ T M ( k , t ) ] ,
i ψ σ ( k , t ) t = ω f ( k ) ψ σ ( k , t ) + all atoms m ( C σ , m * ( k ) exp ( i k r m ) ψ m ( r m , t ) ) ,
i ψ m ( t ) t = ω m ψ m ( t ) + all k exp ( i k r m ) [ C T E ( k ) ψ T E ( k , t ) + C T M ( k ) ψ T M ( k , t ) ] ,
σ { TE , TM }
i F ̃ ± ( k , t ) t = ± c i k × F ̃ ± ( k , t ) + all atoms m ( exp ( i k r m ) C ± , m * ( k ) f ± ( k ) ψ m ( t ) ) ,
i ψ m ( t ) t = ω m ψ m ( t ) + all k exp ( i k r m ) [ C + ( k ) c k f + ( k ) F ̃ + ( k , t ) + C ( k ) c k f ( k ) F ̃ ( k , t ) ] ,
C ± , m ( k ) = C T E , m ( k ) ± i C T M , m ( k ) 2 c k ,
i F ± ( r , t ) t = ± c × F ± ( r , t ) + all atoms m ( C ± , m ( r m r ) ψ m ( t ) ) ,
i ψ m ( t ) t = ω m ψ m ( t )
+ 1 8 π 3 all k exp ( i k r m ) c k [ C + ( k ) f + ( k ) all r exp ( i k r ) F + ( r , t ) d 3 r
+ [ C ( k ) f ( k ) all r exp ( i k r ) F ( r , t ) d 3 r ] ,
C ± , m ( r ) = 1 8 π 3 all k e i k r C ± , m ( k ) f ± ( k ) d 3 k .
i F ± ( r , t ) t = ± c × F ± ( r , t ) + all atoms m ( C ± , m ( r m r ) ψ m ( t ) ) ,
i ψ m ( t ) t = ω m ψ m ( t ) + all r C + , m ( r m r ) F + ( r , t ) d 3 r + all r C , m ( r m r ) F ( r , t ) d 3 r ,
= ω m ψ m ( t ) + ( ( L B B C + , m ( r ) ) r F + ( r , t ) + ( L B B C , m ( r ) ) r F ( r , t ) ) r = r m ,
A ( r ) r B ( r ) = def all u A ( r u ) F ( u , t ) d 3 u ,
ψ m ( t ) = i exp ( i ω m t ) t ( ( L B B C + , m ( r ) ) r F + ( r , t ) + ( L B B C , m ( r ) ) r F ( r , t ) ) r = r m + ψ m ( 0 ) exp ( i ω m t ) , = i t ( i exp ( i ω m t ) ω m t ( ( L B B C + , m ( r ) ) r F + ( r , t ) + ( L B B C , m ( r ) ) r F ( r , t ) ) r = r m i exp ( i ω m t ) ω m ψ m ,
i F ± ( r , t ) t + i G ± ( r , t ) t = ± c × F ± ( r , t ) ,
G ± ( r , t ) = Φ ± ( r , t )
+ all atoms m ( C ± , m ( r m r ) ( i exp ( i ω m t ) ω m t ( ( L B B C + , m ( u ) ) r F + ( u , t ) + ( L B B C , m ( u ) ) r F ( u , t ) ) u = r m ) ) ,
Φ ± ( r , t ) = all atoms m ( exp ( i ω m t ) ω m C ± , m ( r m r ) ψ m ( 0 ) ) .
G ± ( r , t ) G ± ( r , t ) ϕ ± ( r , t ) , F ± ( r , t ) F ± ( r , t ) + ϕ ± ( r , t ) ,
ϕ ± ( r , t ) = all k ϕ ̃ ± ( k , t ) exp ( i k r ) d 3 k ,
× H ̆ ( r , t ) = t ( ε 0 ( E ̆ 0 ( r , t ) + E ̆ ( r , t ) ) + P ̆ ( r , t ) ) ,
P ̆ ( r , t ) = all atoms m ( exp ( i ω m t ) ω m t [ i ε 0 L 11 , m E ̆ ( r , t ) L 12 , m H ̆ ( r , t ) c ] ) ,
× E ̆ ( r , t ) = t ( μ 0 ( H ̆ ( r , t ) + H ̆ ( r , t ) ) + M ̆ ( r , t ) ) ,
M ̆ ( r , t ) = all atoms m ( exp ( i ω m t ) ω m t [ L 21 , m E ̆ ( r , t ) c + i μ 0 L 22 , m H ̆ ( r , t ) ] ) ,
E ̆ 0 ( r , t ) = 1 2 ε 0 ( Φ + ( r , t ) + Φ ( r , t ) ) ,
H ̆ 0 ( r , t ) = 1 i 2 μ 0 ( Φ + ( r , t ) Φ ( r , t ) ) ,
L 11 , m E ̆ ( r , t ) = 1 2 all atoms m ( ( C + , m ( r m r ) + C , m ( r m r ) )
( ( L B B ( C + , m ( u ) + C , m ( u ) ) ) r E ̆ ( u , t ) ) u = r m ) ,
L 12 , m H ̆ ( r , t ) = 1 2 all atoms m ( ( C + , m ( r m r ) + C , m ( r m r ) )
( ( L B B ( C + , m ( u ) C , m ( u ) ) ) r H ̆ ( u , t ) ) u = r m ) ,
L 21 , m E ̆ ( r , t ) = 1 2 all atoms m ( ( C + , m ( r m r ) C , m ( r m r ) )
( ( L B B ( C + , m ( u ) + C , m ( u ) ) ) r E ̆ ( u , t ) ) u = r m ) ,
L 22 , m H ̆ ( r , t ) = 1 2 all atoms m ( ( C + , m ( r m r ) C , m ( r m r ) )
( ( L B B ( C + , m ( u ) C , m ( u ) ) ) r H ̆ ( u , t ) ) u = r m ) .
× H ( r , t ) = ε 0 E 0 ( r , t ) t + ε 0 E ( r , t ) t + P ( r , t ) t ,
× E ( r , t ) = μ 0 H 0 ( r , t ) t + μ 0 H ( r , t ) t + M ( r , t ) t ,
P ( r , t ) = all atoms m [ ( P E + P ¯ E ) E ( r , t ) + ( P M + P ¯ M ) H ( r , t ) ] ,
P E = ε 0 ω m ( sin ( ω m t ) t Re ( L 11 , m ) cos ( ω m t ) t Im ( L 11 , m ) ) ,
P M = 1 c ω m ( cos ( ω m t ) t Re ( L 12 , m ) + sin ( ω m t ) t Im ( L 12 , m ) ) ,
P ¯ E = ε 0 ω m ( cos ( ω m t ) t Re ( L 11 , m ) + sin ( ω m t ) t Im ( L 11 , m ) ) H t ,
P ¯ M = 1 c ω m ( sin ( ω m t ) t Re ( L 12 , m ) + sin ( ω m t ) t Im ( L 12 , m ) ) H t ,
M ( r , t ) = all atoms m [ ( M M + M ¯ M ) H ( r , t ) + ( M E + M ¯ E ) E ( r , t ) ] ,
M M = μ 0 ω m ( sin ( ω m t ) t Re ( L 22 , m ) cos ( ω m t ) t Im ( L 22 , m ) ) ,
M E = 1 c ω m ( cos ( ω m t ) t Re ( L 21 , m ) + sin ( ω m t ) t Im ( L 21 , m ) ) ,
M ¯ M = μ 0 ω m ( cos ( ω m t ) t Re ( L 22 , m ) + sin ( ω m t ) t Im ( L 22 , m ) ) H t ,
M ¯ E = 1 c ω m ( sin ( ω m t ) t Re ( L 21 , m ) + sin ( ω m t ) t Im ( L 21 , m ) ) H t ,
H t X = H t ( X ̆ + X ̑ ) = def i ( X ̆ X ̑ ) ,
ε = ε 0 ( 1 + all atoms m ( sin ( ω m t ) t Re ( L 11 , m ) cos ( ω m t ) t Im ( L 11 , m ) ω m ) + all atoms m ( cos ( ω m t ) t Re ( L 11 , m ) + sin ( ω m t ) t Im ( L 11 , m ) ω m ) H t ) ,
μ = μ 0 ( 1 + all atoms m ( sin ( ω m t ) t Re ( L 22 , m ) cos ( ω m t ) t Im ( L 22 , m ) ω m ) + all atoms m ( cos ( ω m t ) t Re ( L 22 , m ) + sin ( ω m t ) t Im ( L 22 , m ) ω m ) H t ) ,
Γ 12 = all atoms m ( cos ( ω m t ) t Re ( L 12 , m ) + sin ( ω m t ) t Im ( L 12 , m ) c ω m )
all atoms m ( cos ( ω m t ) t Re ( L 12 , m ) + sin ( ω m t ) * t Im ( L 12 , m ) c ω m ) H t .
K ( r ) r X ( r ) all u K ( u ) d 3 u X ( r ) ,
all r K ( r ) d 3 r = 2 π 0 0 r k sin ( k r ) 0 π sin ( θ ) 0 2 π K ̃ ( k ( θ , ϕ ) ) d ϕ d θ d k d r ,
C ± , m ( r ) r X ( r ) ± i 2 π 3 0 0 u k sin ( k u ) C ± , m ( k ) d k d u ( 0 0 π 2 ) X ( r ) .
L 11 , m E ̆ ( r , t ) 1 2 ρ m ( r ) 2 π 3 0 0 uk sin ( k u ) ( C + , m ( k ) C , m ( k ) ) d k d u ( 0 0 π 2 )
× 1 2 π 3 0 0 u k sin ( k u ) ( C + , m ( k ) C , m ( k ) c k ) d k d u ( 0 0 π 2 ) E ̆ ( r ) ,
π ρ m ( r ) k 0 8 c ( 0 0 u sin ( k u ) ( C + , m ( k ) C , m ( k ) ) d k d u ) 2 diag ( 0 0 1 ) E ̆ ( r , t ) ,
L 22 , m H ̆ ( r , t ) π ρ m ( r ) k 0 8 c ( 0 0 u sin ( k u ) ( C + , m ( k ) C , m ( k ) ) d k d u ) 2 diag ( 0 0 1 ) H ̆ ( r , t ) .
exp ( i β ) ( i ± i j ) = ( i ± i j ) , ( i j ) = ( cos ( β ) sin ( β ) sin ( β ) cos ( β ) ) ( i j ) ;
F ̃ + ( k , t ) exp ( i ( β + δ ) ) F ̃ + ( k , t ) ,
F ̃ ( k , t ) exp ( i ( β δ ) ) F ̃ ( k , t ) ,
( ε 0 E ̆ ( k , t ) μ 0 H ̆ ( k , t ) ) exp ( i β ) 2 ( 1 1 i i ) ( e i δ 0 0 e i δ ) 1 2 ( 1 i 1 i ) ( ε 0 E ̆ ( k , t ) μ 0 H ̆ ( k , t ) ) ,
E ̆ ( k , t ) cos ( δ ) E ̆ ( k , t ) e i β μ 0 ε 0 sin ( δ ) H ̆ ( k , t ) e i β ,
H ̆ ( k , t ) cos ( δ ) H ̆ ( k , t ) e i β + ε 0 μ 0 sin ( δ ) E ̆ ( k , t ) e i β .
L 11 ; m E ̆ ( r , t ) π ρ m ( r ) k 0 8 c ( 0 0 u sin ( k u ) ( C + , m ( k ) C , m ( k ) ) d k d u ) 2 diag ( 1 3 1 3 1 3 ) E ̆ ( r , t ) ε ( ω ) = ( 1 + π ρ m ( r ) k 0 24 c ( ω 2 ω m 2 ) ( 1 + ω ω m ) ( 0 0 u sin ( k u ) ( C + , m ( k ) C , m ( k ) ) d k d u ) 2 ) ε 0 .
sin ( ω m t ) ω m t cos ( ω t + δ ) = 1 ω m 2 ω 2 cos ( ω t + δ ) + ω sin ( ω m t ) sin ( δ ) ω m cos ( ω m t ) cos ( δ ) ω m ( ω m 2 ω 2 ) ,
cos ( ω m t ) ω m t cos ( ω t + δ ) = ω ω m ( ω m 2 ω 2 ) sin ( ω t + δ ) + ω cos ( ω m t ) sin ( δ ) + ω m sin ( ω m t ) cos ( δ ) ω m ( ω m 2 ω 2 ) .
ε ( ω ) = ( 1 + all matter kinds m ( π ρ m ( r ) k 0 24 c ( ω 2 ω m 2 ) ( 0 0 u k sin ( k u ) ( C + , m ( k ) C , m ( k ) ) d k d u ) 2 ) ) ε 0 ,
μ ( ω ) = ( 1 + all matter kinds m ( π ρ m ( r ) k 0 24 c ( ω 2 ω m 2 ) ( 0 0 u k sin ( k u ) ( C + , m ( k ) + C , m ( k ) ) d k d u ) 2 ) ) μ 0 .
n ( λ ) 1 + all matter kinds m ε m ω ( λ ) 2 ω m ( λ ) 2 = 1 + all matter kinds m B m λ 2 λ 2 λ m 2 ,
ψ m ( t ) = i exp ( i ω m t ) t ( ε 0 2 ( L B B ( C + , m ( r ) + C , m ( r ) ) ) r E ̆ ( r , t ) + i μ 0 2 ( L B B ( C + , m ( r ) C , m ( r ) ) ) r H ̆ ( r , t ) ) r = r m ,
i exp ( i ω m t ) t ( ε 0 2 ( L B B ( C + , m ( r ) + C , m ( r ) ) ) r E ( r , t ) + i μ 0 2 ( L B B ( C + , m ( r ) C , m ( r ) ) ) r H ( r , t ) ) r = r m .
p ( r , t ) = 1 2 ( ε ( r ) ε 0 ) E ( r , t ) E ( r , t ) + 1 2 ( μ ( r ) μ 0 ) H ( r , t ) H ( r , t ) ,
1 2 all r ( ε ( r ) E ( r , t ) 2 + μ ( r ) H ( r , t ) 2 ) d 3 r = 1 .
( ε 0 E ( r , t ) 2 + μ 0 H ( r , t ) 2 ) d 3 r ( ε ( r ) E ( r , t ) 2 + μ ( r ) H ( r , t ) 2 ) d 3 r .

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