Abstract

Coherent-mode representation provides physical insight and computational simplification into the analysis of random optical signals. In this work, we present the coherent-mode decomposition for pulsed electromagnetic beam fields. We show that the mode decomposition can be done for any valid space–frequency or space–time coherence matrix representing nonstationary pulsed electric field, and moreover, the spectral and temporal modes are connected via a Fourier transform relation. The analysis also yields the coherent modes of electromagnetic time-domain signals in temporal optics. We present the overall degree of coherence as a measure of the average temporal or spectral and spatial coherence of the beam. Several illustrative examples are discussed analytically and numerically.

© 2013 Optical Society of America

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References

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  1. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  2. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  3. J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
    [CrossRef]
  4. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
    [CrossRef]
  5. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  6. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
    [CrossRef]
  7. T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in a spherical volume,” Phys. Rev. E 71, 036118 (2005).
  8. G. P. Agraval, Nonlinear Fiber Optics (Elsevier, 2007).
  9. T. Voipio, T. Setälä, and A. T. Friberg, “Polarization changes in temporal imaging with pulses of random light,” Opt. Express 21, 8987–9004 (2013).
    [CrossRef]
  10. M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, “Coherent-mode representation of supercontinuum,” Opt. Lett. 37, 169–171 (2012).
    [CrossRef]
  11. T. Shirai, T. Setälä, and A. T. Friberg, “Temporal ghost imaging with classical non-stationary pulsed light,” J. Opt. Soc. Am. B 27, 2549–2555 (2010).
    [CrossRef]
  12. H. Kellock, T. Setälä, T. Shirai, and A. T. Friberg, “Image quality in double- and triple-intensity ghost imaging with classical partially polarized light,” J. Opt. Soc. Am. A 29, 2459–2468 (2012).
    [CrossRef]
  13. D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. 39, 1681–1699 (2000).
    [CrossRef]
  14. R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
    [CrossRef]
  15. P. Martinsson, H. Lajunen, and A. T. Friberg, “Communication modes with partially coherent fields,” J. Opt. Soc. Am. A 24, 3336–3342 (2007).
    [CrossRef]
  16. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non stationary fields,” Opt. Express 14, 5007–5012 (2006).
    [CrossRef]
  17. A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15, 5160–5165 (2007).
    [CrossRef]
  18. A. Burvall, A. Smith, and C. Dainty, “Elementary functions: propagation of partially coherent light,” J. Opt. Soc. Am. A 26, 1721–1729 (2009).
    [CrossRef]
  19. J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Shifted-elementary-mode representation for partially coherent vectorial fields,” J. Opt. Soc. Am. A 27, 2004–2014 (2010).
    [CrossRef]
  20. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  21. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
    [CrossRef]
  22. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [CrossRef]
  23. F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, 1978).
  24. T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
    [CrossRef]
  25. M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
    [CrossRef]
  26. P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
    [CrossRef]
  27. A. Luis, “Overall degree of coherence for vectorial electromagnetic fields and the Wigner function,” J. Opt. Soc. Am. A 24, 2070–2074 (2007).
    [CrossRef]
  28. E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1986).
    [CrossRef]
  29. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).
  30. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1993).
  31. D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University, 2008).
  32. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009).
    [CrossRef]
  33. V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011).
    [CrossRef]
  34. L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
    [CrossRef]
  35. E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
    [CrossRef]
  36. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894–1899 (2003).
    [CrossRef]

2013 (2)

2012 (2)

2011 (1)

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011).
[CrossRef]

2010 (2)

2009 (2)

2007 (3)

2006 (1)

2005 (1)

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in a spherical volume,” Phys. Rev. E 71, 036118 (2005).

2004 (4)

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[CrossRef]

2003 (3)

2000 (2)

1995 (1)

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

1992 (1)

1986 (1)

1982 (1)

1981 (2)

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

1963 (1)

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
[CrossRef]

Agraval, G. P.

G. P. Agraval, Nonlinear Fiber Optics (Elsevier, 2007).

Andrés, P.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

Blomstedt, K.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in a spherical volume,” Phys. Rev. E 71, 036118 (2005).

Borghi, R.

Burvall, A.

Dainty, C.

Dorrer, C.

Erkintalo, M.

Friberg, A. T.

T. Voipio, T. Setälä, and A. T. Friberg, “Polarization changes in temporal imaging with pulses of random light,” Opt. Express 21, 8987–9004 (2013).
[CrossRef]

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
[CrossRef]

H. Kellock, T. Setälä, T. Shirai, and A. T. Friberg, “Image quality in double- and triple-intensity ghost imaging with classical partially polarized light,” J. Opt. Soc. Am. A 29, 2459–2468 (2012).
[CrossRef]

M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, “Coherent-mode representation of supercontinuum,” Opt. Lett. 37, 169–171 (2012).
[CrossRef]

T. Shirai, T. Setälä, and A. T. Friberg, “Temporal ghost imaging with classical non-stationary pulsed light,” J. Opt. Soc. Am. B 27, 2549–2555 (2010).
[CrossRef]

P. Martinsson, H. Lajunen, and A. T. Friberg, “Communication modes with partially coherent fields,” J. Opt. Soc. Am. A 24, 3336–3342 (2007).
[CrossRef]

A. T. Friberg, H. Lajunen, and V. Torres-Company, “Spectral elementary-coherence-function representation for partially coherent light pulses,” Opt. Express 15, 5160–5165 (2007).
[CrossRef]

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in a spherical volume,” Phys. Rev. E 71, 036118 (2005).

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
[CrossRef]

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[CrossRef]

Genty, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).

Gori, F.

Guattari, G.

Huttunen, J.

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Kellock, H.

Lajunen, H.

Lancis, J.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011).
[CrossRef]

Lindberg, J.

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in a spherical volume,” Phys. Rev. E 71, 036118 (2005).

Luis, A.

Mandel, L.

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinsson, P.

Miller, D. A. B.

Piestun, R.

Piquero, G.

Riesz, F.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, 1978).

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1993).

Santarsiero, M.

Setälä, T.

Shirai, T.

Simon, R.

Smith, A.

Surakka, M.

Sz.-Nagy, B.

F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, 1978).

Tervo, J.

Tervonen, E.

Torres-Company, V.

Turunen, J.

Vahimaa, P.

Voipio, T.

Walmsley, I. A.

Wolf, E.

E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1986).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wyrowski, F.

Adv. Opt. Photon. (1)

Appl. Opt. (1)

J. Opt. A (1)

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (12)

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[CrossRef]

A. Burvall, A. Smith, and C. Dainty, “Elementary functions: propagation of partially coherent light,” J. Opt. Soc. Am. A 26, 1721–1729 (2009).
[CrossRef]

J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Shifted-elementary-mode representation for partially coherent vectorial fields,” J. Opt. Soc. Am. A 27, 2004–2014 (2010).
[CrossRef]

H. Kellock, T. Setälä, T. Shirai, and A. T. Friberg, “Image quality in double- and triple-intensity ghost imaging with classical partially polarized light,” J. Opt. Soc. Am. A 29, 2459–2468 (2012).
[CrossRef]

T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
[CrossRef]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[CrossRef]

R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
[CrossRef]

E. Wolf, “Coherent-mode propagation in spatially band-limited wave fields,” J. Opt. Soc. Am. A 3, 1920–1924 (1986).
[CrossRef]

E. Tervonen, A. T. Friberg, and J. Turunen, “Gaussian Schell-model beams generated with synthetic acousto-optic holograms,” J. Opt. Soc. Am. A 9, 796–803 (1992).
[CrossRef]

A. Luis, “Overall degree of coherence for vectorial electromagnetic fields and the Wigner function,” J. Opt. Soc. Am. A 24, 2070–2074 (2007).
[CrossRef]

P. Martinsson, H. Lajunen, and A. T. Friberg, “Communication modes with partially coherent fields,” J. Opt. Soc. Am. A 24, 3336–3342 (2007).
[CrossRef]

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

Opt. Acta (1)

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

Opt. Commun. (2)

T. Setälä, J. Tervo, and A. T. Friberg, “Theorems on complete electromagnetic coherence in the space-time domain,” Opt. Commun. 238, 229–236 (2004).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. E (2)

J. Huttunen, A. T. Friberg, and J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

T. Setälä, J. Lindberg, K. Blomstedt, J. Tervo, and A. T. Friberg, “Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in a spherical volume,” Phys. Rev. E 71, 036118 (2005).

Proc. Phys. Soc. London (1)

L. Mandel, “Intensity fluctuations of partially polarized light,” Proc. Phys. Soc. London 81, 1104–1114 (1963).
[CrossRef]

Prog. Opt. (1)

V. Torres-Company, J. Lancis, and P. Andrés, “Space-time analogies in optics,” Prog. Opt. 56, 1–80 (2011).
[CrossRef]

Other (6)

G. P. Agraval, Nonlinear Fiber Optics (Elsevier, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

F. Riesz and B. Sz.-Nagy, Functional Analysis (Ungar, 1978).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2004).

J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, 1993).

D. A. B. Miller, Quantum Mechanics for Scientists and Engineers (Cambridge University, 2008).

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Figures (4)

Fig. 1.
Fig. 1.

Ten largest (normalized) eigenvalues λn/λ0 of the coherent-mode decomposition of a pulsed beam with blackbody radiation spectrum (thermal frequency ωT) and Gaussian spectral correlations, with different spectral coherence widths Ωc. The curves correspond to the indicated values of the ratio rc=Ωc/ωT.

Fig. 2.
Fig. 2.

First fifteen eigenvalues of the coherent-mode decomposition of the spectral GSM pulse. The red crosses indicate eigenvalues which correspond to modes with same sign in the x and y components, and the blue dots indicate the orthogonal companion modes. First three modes of each type are shown in the insets, with solid and dotted curves indicating the spectral densities of the x and y components, respectively. We note that the eigenvalues decrease in magnitude but not very rapidly, with the ratio between successive eigenvalues between 0.6 and 1.0.

Fig. 3.
Fig. 3.

Modes associated with the eight largest eigenvalues. Solid and dashed curves denote values of Φn;x(ωω0) and Φn;y(ωω0), respectively, and the mode number n is indicated in each graph.

Fig. 4.
Fig. 4.

Reconstruction of the spectral coherence matrix elements Wxx(ω,ω) (top) and Wxy(ω,ω) (bottom) using 1, 2, 5, and 15 first modes, indicated with thin dotted, dash-dotted, dashed, and solid curves, respectively. The thick dashed curves show the exact values of the matrix elements.

Equations (70)

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Utot=D[|Ex(ρ,t)|2+|Ey(ρ,t)|2]d2ρdt<,
E(r,t)=0E˜(r,ω)exp(iωt)dω,
E˜(r,ω)=12πE(r,t)exp(iωt)dt.
W(r1,r2,ω1,ω2)=E˜*(r1,ω1)E˜T(r2,ω2),
Js(r,ω)=W(r,r,ω,ω).
Γ(r1,r2,t1,t2)=E*(r1,t1)ET(r2,t2),
Jt(r,t)=Γ(r,r,t,t).
Γ(r1,r2,t1,t2)=0W(r1,r2,ω1,ω2)×exp[i(ω1t1+ω2t2)]dω1dω2,
W(r1,r2,ω1,ω2)=14π2Γ(r1,r2,t1,t2)×exp[i(ω1t1+ω2t2)]dt1dt2,
W(r1,r2,ω1,ω2)=W(r2,r1,ω2,ω1),
Γ(r1,r2,t1,t2)=Γ(r2,r1,t2,t1),
g(ξ1)W(ξ1,ξ2)g(ξ2)d3ξ1d3ξ20,
f(η1)Γ(η1,η2)f(η2)d3η1d3η20,
Ps(r,ω)=14detJs(r,ω)tr2Js(r,ω),
μEM2(r1,r2,ω1,ω2)=W(r1,r2,ω1,ω2)F2trJs(r1,ω1)trJs(r2,ω2),
ρ1,ω1,α1|ρ2,ω2,α2=δ(2)(ρ2ρ1)δ(ω2ω1)δα2α1,
α0D|ρ,ω,αρ,ω,α|d2ρdω=1,
|ψ=α0Dcα(ρ,ω)|ρ,ω,αd2ρdω,
ρ1,ω1,α1|W^|ρ2,ω2,α2=Wα1α2(ρ1,ρ2,ω1,ω2),
W^=nλn|nn|,
W(ρ1,ρ2,ω1,ω2)=nλnΦn*(ρ1,ω1)ΦnT(ρ2,ω2),
Φn;α(ρ,ω)=n|ρ,ω,α.
m|n=0DΦmT(ρ,ω)Φn*(ρ,ω)d2ρdω={Φm(ρ,ω),Φn(ρ,ω)},
{Φm(ρ,ω),Φn(ρ,ω)}=δmn.
0DΦnT(ρ1,ω1)W(ρ1,ρ2,ω1,ω2)d2ρ1dω1=λnΦnT(ρ2,ω2).
(2xki2+ωk2c2)Wij(ρ1,ρ2,ω1,ω2)=(2xkj2+ωk2c2)Wij(ρ1,ρ2,ω1,ω2)=0,
xkiWij(ρ1,ρ2,ω1,ω2)=xkjWij(ρ1,ρ2,ω1,ω2)=0,
μ¯EM2=0DW(ρ1,ρ2,ω1,ω2)F2d2ρ1d2ρ2dω1dω2[0DtrJs(ρ,ω)d2ρdω]2=nλn2(nλn)2.
E˜(ρ,z,ω)=exp(ikz)iλz×E˜(0)(ρ,ω)exp[ik2z(ρρ)2]d2ρ.
W(ρ1,ρ2,ω1,ω2)=nλnζn*(ρ1,ω1)ζnT(ρ2,ω2),
Γ(ρ1,ρ2,t1,t2)=nλnΨn*(ρ1,t1)ΨnT(ρ2,t2),
{Ψm(ρ,t),Ψn(ρ,t)}=DΨmT(ρ,t)Ψn*(ρ,t)d2ρdt,
DΨmT(ρ1,t1)Γ(ρ1,ρ2,t1,t2)d2ρ1dt1=λmΨmT(ρ2,t2),
Γ(ρ1,ρ2,t1,t2)=nλnΘn*(ρ1,t1)ΘnT(ρ2,t2),
Θn(ρ,t)=0Φn(ρ,ω)exp(iωt)dω.
Ψn(ρ,t)=(2π)1/20Φn(ρ,ω)exp(iωt)dω,
λn=2πλn.
|t=(2π)1/20exp(iωt)|ωdω.
W(ρ1,ρ2,ω1,ω2)=J0W(ρ1,ρ2)M(ω1,ω2).
(Φn0)TJ0=λn(e)(Φn0)T,
Dχp(ρ1)W(ρ1,ρ2)d2ρ1=λp(ρ)χp(ρ2),
0ϕq(ω1)M(ω1,ω2)dω1=λq(s)ϕq(ω2),
Φpq(ρ,ω)=Φ0χp(ρ)ϕq(ω),
λpq=λp(ρ)λq(s),
W(ρ1,ρ2,ω1,ω2)=p,qλpqΦpq*(ρ1,ω1)ΦpqT(ρ2,ω2).
J0=I1e^1*e^1T+I2e^2*e^2T,
W(ρ1,ρ2,ω1,ω2)=n=12p,qλnpqΦnpq*(ρ1,ω1)ΦnpqT(ρ2,ω2),
Φnpq(ρ,ω)=e^nχp(ρ)ϕq(ω),
λnpq=Inλp(ρ)λq(s).
W(ρ1,ρ2)=exp(ρ12+ρ224w2)exp[(ρ2ρ1)22σ2],
P(u1,u2)=exp(u12+u224w2)exp[(u2u1)22σ2].
χp(u1)P(u1,u2)du1=λp(ρ)χp(u2),
χp(u)=(2cρ)1/4hp[(2cρ)1/2u],
λp(ρ)=(πaρ+bρ+cρ)1/2(bρaρ+bρ+cρ)p,
χp*(u)χp(u)du=δpp.
ξu=λp+1(ρ)/λp(ρ)=bρ/(aρ+bρ+cρ)=[rρ+1+(rρ2+2rρ)1/2]1,
W(ρ1,ρ2)=p,qλp(ρ)λq(ρ)[χp(x1)χq(y1)]*χp(x2)χq(y2).
M(ω1,ω2)=[S(ω1)S(ω2)]1/2μ(ω2ω1),
S(ω2)0ϕr(ω1)S(ω1)μ(ω2ω1)dω1=λr(s)ϕr(ω2),
S(ω)=Aω3[exp(ω/ωT)1]1,
μ(Δω)=exp(Δω2/Ωc2),
ω0ϕ˜r(ω˜1)exp(ω˜12+ω˜224Ω02)exp[(ω˜2ω˜1)22Ωc2]dω˜1=λr(s)ϕ˜r(ω˜2),
ϕr(ω)=(2cs)1/4hr[(2cs)1/2(ωω0)],
λr(s)=(πas+bs+cs)1/2(bsas+bs+cs)r,
W(x1,x2,y1,y2,ω1,ω2)=n=12p=0q=0r=0λnpqrΦnpqr*(x1,y1,ω1)ΦnpqrT(x2,y2,ω2),
Φnpqr(x,y,ω)=e^nχp(x)χq(y)ϕr(ω),
λnpqr=Inλp(x)λq(y)λr(e),
μ¯EM2=I12+I22I02p=0ξx2pq=0ξy2qr=0ξs2r(p=0ξxp)2(q=0ξyq)2(q=0ξsq)2=I12+I22I02(1ξx)(1ξy)(1ξs)(1+ξx)(1+ξy)(1+ξs),
Wij(ω1,ω2)=Bij[Si(ω1)Sj(ω2)]1/2exp[(ω2ω1)2/Ωcij2],
Ps2(ω)=14Sx(ω)Sy(ω)[Sx(ω)+Sy(ω)]2(1|Bxy|2),

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