Abstract

The dispersive properties of dielectric materials in both the time and the frequency domains are discussed. Special emphasis is placed on the treatment of heterogeneous materials, particularly two-phase mixtures. A time-domain Maxwell–Garnett rule is derived that differs from the corresponding frequency-domain formula in that it is expressed in terms of convolutions and inverse operators of the susceptibility kernels of the materials. Much of the analysis deals with the question of how the temporal dispersion of the dielectric responses of various physical materials is affected by the mixing process. Debye, Lorentz, Drude, and modified Debye susceptibility models are treated in detail.

© 1998 Optical Society of America

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  1. C. J. F. Böttcher, Theory of Electric Polarization, 2nd ed. (Elsevier, Amsterdam, 1973).
  2. A. Lakhtakia, “Size-dependent Maxwell–Garnett formula from an integral equation formalism,” Optik (Stuttgart) 91, 134–137 (1992).
  3. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  4. R. Landauer, “Electrical conductivity in inhomogeneous media,” in Electrical Transport and Optical Properties in Inhomogeneous Media, Vol. 40 of AIP Conference Proceedings, J. C. Garland, D. B. Tanner, eds. (American Institute of Physics, New York, 1978), pp. 2–45.
  5. A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
    [CrossRef]
  6. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  7. W. S. Weiglhofer, A. Lakhtakia, “On causality requirements for material media,” Arch. Elektrotech.Übertragungstech. (Int. J. Electron. Commun.) 50, 389–391 (1996).
  8. J. C. M. Garnett, “Colours in metal glasses and in metal films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).
    [CrossRef]
  9. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  10. U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Materials Science (Springer-Verlag, Berlin, 1995).
  11. B. K. P. Scaife, Principles of Dielectrics (Clarendon, Oxford, 1989).
  12. A. T. C. Chang, T. T. Wilheit, “Remote sensing of atmospheric water vapor, liquid water, and wind speed at the ocean surface by passive microwave techniques from the Nimbus 5 satellite,” Radio Sci. 14, 793–802 (1979).
    [CrossRef]
  13. U. Kaatze, “Microwave dielectric properties of water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 2, pp. 37–53.
  14. O. Barajas, H. A. Buckmaster, “Calculation of the temperature dependence of the Debye and relaxation activation parameters from complex permittivity data for light water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 3, pp. 55–66.
  15. G. C. Gerace, E. K. Smith, “A comparison of cloud models,” IEEE Antennas Propag. Mag. 32, 32–38 (1990).
    [CrossRef]
  16. H. J. Liebe, T. Manabe, G. A. Hufford, “Millimeter-wave attenuation and delay rates due to fog/cloud conditions,” IEEE Trans. Antennas Propag. 37, 1617–1623 (1989).
    [CrossRef]
  17. L. D. Landau, E. M. Lifshitz, L. P. Pitaevskiı̆, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).
  18. M. T. Hallikainen, F. T. Ulaby, M. Abdelrazik, “Dielectric properties of snow in the 3 to 37 GHz range,” IEEE Trans. Antennas Propag. 34, 1329–1340 (1986).
    [CrossRef]
  19. R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

1996 (1)

W. S. Weiglhofer, A. Lakhtakia, “On causality requirements for material media,” Arch. Elektrotech.Übertragungstech. (Int. J. Electron. Commun.) 50, 389–391 (1996).

1992 (2)

A. Lakhtakia, “Size-dependent Maxwell–Garnett formula from an integral equation formalism,” Optik (Stuttgart) 91, 134–137 (1992).

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

1990 (1)

G. C. Gerace, E. K. Smith, “A comparison of cloud models,” IEEE Antennas Propag. Mag. 32, 32–38 (1990).
[CrossRef]

1989 (1)

H. J. Liebe, T. Manabe, G. A. Hufford, “Millimeter-wave attenuation and delay rates due to fog/cloud conditions,” IEEE Trans. Antennas Propag. 37, 1617–1623 (1989).
[CrossRef]

1986 (1)

M. T. Hallikainen, F. T. Ulaby, M. Abdelrazik, “Dielectric properties of snow in the 3 to 37 GHz range,” IEEE Trans. Antennas Propag. 34, 1329–1340 (1986).
[CrossRef]

1980 (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1979 (1)

A. T. C. Chang, T. T. Wilheit, “Remote sensing of atmospheric water vapor, liquid water, and wind speed at the ocean surface by passive microwave techniques from the Nimbus 5 satellite,” Radio Sci. 14, 793–802 (1979).
[CrossRef]

1904 (1)

J. C. M. Garnett, “Colours in metal glasses and in metal films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).
[CrossRef]

Abdelrazik, M.

M. T. Hallikainen, F. T. Ulaby, M. Abdelrazik, “Dielectric properties of snow in the 3 to 37 GHz range,” IEEE Trans. Antennas Propag. 34, 1329–1340 (1986).
[CrossRef]

Barajas, O.

O. Barajas, H. A. Buckmaster, “Calculation of the temperature dependence of the Debye and relaxation activation parameters from complex permittivity data for light water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 3, pp. 55–66.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Böttcher, C. J. F.

C. J. F. Böttcher, Theory of Electric Polarization, 2nd ed. (Elsevier, Amsterdam, 1973).

Buckmaster, H. A.

O. Barajas, H. A. Buckmaster, “Calculation of the temperature dependence of the Debye and relaxation activation parameters from complex permittivity data for light water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 3, pp. 55–66.

Chang, A. T. C.

A. T. C. Chang, T. T. Wilheit, “Remote sensing of atmospheric water vapor, liquid water, and wind speed at the ocean surface by passive microwave techniques from the Nimbus 5 satellite,” Radio Sci. 14, 793–802 (1979).
[CrossRef]

Garnett, J. C. M.

J. C. M. Garnett, “Colours in metal glasses and in metal films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).
[CrossRef]

Gerace, G. C.

G. C. Gerace, E. K. Smith, “A comparison of cloud models,” IEEE Antennas Propag. Mag. 32, 32–38 (1990).
[CrossRef]

Hallikainen, M. T.

M. T. Hallikainen, F. T. Ulaby, M. Abdelrazik, “Dielectric properties of snow in the 3 to 37 GHz range,” IEEE Trans. Antennas Propag. 34, 1329–1340 (1986).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Hufford, G. A.

H. J. Liebe, T. Manabe, G. A. Hufford, “Millimeter-wave attenuation and delay rates due to fog/cloud conditions,” IEEE Trans. Antennas Propag. 37, 1617–1623 (1989).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

Kaatze, U.

U. Kaatze, “Microwave dielectric properties of water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 2, pp. 37–53.

Karlsson, A.

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

Kreibig, U.

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Materials Science (Springer-Verlag, Berlin, 1995).

Kress, R.

R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

Kristensson, G.

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

Lakhtakia, A.

W. S. Weiglhofer, A. Lakhtakia, “On causality requirements for material media,” Arch. Elektrotech.Übertragungstech. (Int. J. Electron. Commun.) 50, 389–391 (1996).

A. Lakhtakia, “Size-dependent Maxwell–Garnett formula from an integral equation formalism,” Optik (Stuttgart) 91, 134–137 (1992).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskiı̆, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

Landauer, R.

R. Landauer, “Electrical conductivity in inhomogeneous media,” in Electrical Transport and Optical Properties in Inhomogeneous Media, Vol. 40 of AIP Conference Proceedings, J. C. Garland, D. B. Tanner, eds. (American Institute of Physics, New York, 1978), pp. 2–45.

Liebe, H. J.

H. J. Liebe, T. Manabe, G. A. Hufford, “Millimeter-wave attenuation and delay rates due to fog/cloud conditions,” IEEE Trans. Antennas Propag. 37, 1617–1623 (1989).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskiı̆, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

Manabe, T.

H. J. Liebe, T. Manabe, G. A. Hufford, “Millimeter-wave attenuation and delay rates due to fog/cloud conditions,” IEEE Trans. Antennas Propag. 37, 1617–1623 (1989).
[CrossRef]

Pitaevskii?, L. P.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskiı̆, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

Scaife, B. K. P.

B. K. P. Scaife, Principles of Dielectrics (Clarendon, Oxford, 1989).

Smith, E. K.

G. C. Gerace, E. K. Smith, “A comparison of cloud models,” IEEE Antennas Propag. Mag. 32, 32–38 (1990).
[CrossRef]

Ulaby, F. T.

M. T. Hallikainen, F. T. Ulaby, M. Abdelrazik, “Dielectric properties of snow in the 3 to 37 GHz range,” IEEE Trans. Antennas Propag. 34, 1329–1340 (1986).
[CrossRef]

Vollmer, M.

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Materials Science (Springer-Verlag, Berlin, 1995).

Weiglhofer, W. S.

W. S. Weiglhofer, A. Lakhtakia, “On causality requirements for material media,” Arch. Elektrotech.Übertragungstech. (Int. J. Electron. Commun.) 50, 389–391 (1996).

Wilheit, T. T.

A. T. C. Chang, T. T. Wilheit, “Remote sensing of atmospheric water vapor, liquid water, and wind speed at the ocean surface by passive microwave techniques from the Nimbus 5 satellite,” Radio Sci. 14, 793–802 (1979).
[CrossRef]

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Arch. Elektrotech.Übertragungstech. (Int. J. Electron. Commun.) (1)

W. S. Weiglhofer, A. Lakhtakia, “On causality requirements for material media,” Arch. Elektrotech.Übertragungstech. (Int. J. Electron. Commun.) 50, 389–391 (1996).

IEEE Antennas Propag. Mag. (1)

G. C. Gerace, E. K. Smith, “A comparison of cloud models,” IEEE Antennas Propag. Mag. 32, 32–38 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

H. J. Liebe, T. Manabe, G. A. Hufford, “Millimeter-wave attenuation and delay rates due to fog/cloud conditions,” IEEE Trans. Antennas Propag. 37, 1617–1623 (1989).
[CrossRef]

M. T. Hallikainen, F. T. Ulaby, M. Abdelrazik, “Dielectric properties of snow in the 3 to 37 GHz range,” IEEE Trans. Antennas Propag. 34, 1329–1340 (1986).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. Karlsson, G. Kristensson, “Constitutive relations, dissipation and reciprocity for the Maxwell equations in the time domain,” J. Electromagn. Waves Appl. 6, 537–551 (1992).
[CrossRef]

Optik (Stuttgart) (1)

A. Lakhtakia, “Size-dependent Maxwell–Garnett formula from an integral equation formalism,” Optik (Stuttgart) 91, 134–137 (1992).

Philos. Trans. R. Soc. London Ser. A (1)

J. C. M. Garnett, “Colours in metal glasses and in metal films,” Philos. Trans. R. Soc. London Ser. A 203, 385–420 (1904).
[CrossRef]

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Radio Sci. (1)

A. T. C. Chang, T. T. Wilheit, “Remote sensing of atmospheric water vapor, liquid water, and wind speed at the ocean surface by passive microwave techniques from the Nimbus 5 satellite,” Radio Sci. 14, 793–802 (1979).
[CrossRef]

Other (10)

U. Kaatze, “Microwave dielectric properties of water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 2, pp. 37–53.

O. Barajas, H. A. Buckmaster, “Calculation of the temperature dependence of the Debye and relaxation activation parameters from complex permittivity data for light water,” in Microwave Aquametry, A. Kraszewski, ed. (IEEE, Piscataway, N.J., 1996), Chap. 3, pp. 55–66.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskiı̆, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

C. J. F. Böttcher, Theory of Electric Polarization, 2nd ed. (Elsevier, Amsterdam, 1973).

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Vol. 25 of Materials Science (Springer-Verlag, Berlin, 1995).

B. K. P. Scaife, Principles of Dielectrics (Clarendon, Oxford, 1989).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

R. Landauer, “Electrical conductivity in inhomogeneous media,” in Electrical Transport and Optical Properties in Inhomogeneous Media, Vol. 40 of AIP Conference Proceedings, J. C. Garland, D. B. Tanner, eds. (American Institute of Physics, New York, 1978), pp. 2–45.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

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Figures (7)

Fig. 1
Fig. 1

Geometry of the mixing problem: spherical isotropic dielectric inclusions in the isotropic dielectric environment.

Fig. 2
Fig. 2

Relaxation frequency of a water–air mixture as a function of the volume fraction of the water phase. The curves correspond to three different temperatures T.

Fig. 3
Fig. 3

Plasma frequency of a mixture (relative to the plasma frequency of inclusions) with Lorentz spheres in air as a function of the volume fraction of the inclusions. The high-frequency response is assumed to be ε=10.

Fig. 4
Fig. 4

Resonance frequency of a mixture (relative to the resonance frequency of inclusions) with Lorentz spheres in air as a function of the volume fraction of the inclusions. The high-frequency response is assumed to be ε=10, and two plasma-to-resonance frequency ratios of the inclusion phase are shown.

Fig. 5
Fig. 5

Cole–Cole plot of the mixture of spherical ethanol droplets in water (20% ethanol). Also shown are the Cole–Cole diagrams of pure water and pure ethanol. Two frequency points are marked on each curve: 1, the relaxation frequency of ethanol (1.33 GHz); and 2, the relaxation frequency of water (15.8 GHz). Note that the negative part of the imaginary part is given, because of the time convention exp(jωt).

Fig. 6
Fig. 6

Same as Fig. 5, but for a mixture of water drops in ethanol (80% ethanol).

Fig. 7
Fig. 7

Susceptibility kernels for water, ethanol, an ethanol-in-water mixture (20% ethanol), and a water-in-ethanol mixture (80% ethanol).

Tables (2)

Tables Icon

Table 1 Susceptibility Kernels χ(t) and Corresponding Frequency-Dependent Permittivity Functions for the Models Analyzed in Section 5a

Tables Icon

Table 2 Susceptibility Kernels χ(t) and Corresponding Resolvent Kernels χres(t) for the Models Analyzed in Section 5a

Equations (120)

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1ε0D(r, t)=ε(r)E(r, t)+(χ*E)(r, t),
B(r, t)=μ0H(r, t),
(χ*E)(t)=-tχ(t-t)E(t)dt.
1ε0D(ω)=ε(ω)E(ω),
D(ω)=-D(t)exp(-jωt)dt,
E(ω)=-E(t)exp(-jωt)dt,
ε(ω)-ε=0χ(t)exp(-jωt)dt.
1ε0D(t)=E(t)+-t[χf(t-t)+χs(t-t)]E(t)dt=E(t)+E(t)-tχf(t-t)dt+-tχs(t-t)E(t)dt=εE(t)+(χs*E)(t),
ε(ω=0)=ε+0χs(t)dt=1+0χ(t)dt,
εeff=εb+3fεb εi-εbεi+2εb-f(εi-εb).
εeff=εb+3fεb εi-εbεi+2εb.
[E](t)=[(ε+χ *)E](t)=[(εδ+χ)*E](t),
[(i+2b)Ei](t)=3[bEe](t).
Ei(t)=[(i+2b)-13bEe](t).
α=ε0V(i-b)(i+2b)-13b,
EL(t)=Ee(t)+13ε0[b-1P](t),
EL(t)=3b-nαε0-13bEe(t),
eff=b+nαε03b-nαε0-13b.
eff=b+nαε0=b+f(i-b)(i+2b)-13b,
[(a+A*)(b+B*)]-1=(a+A*)-1(b+B*)-1,
eff=b+f(i-b)[i+2b-f(i-b)]-13b.
limω εi(ω)εb(ω)=-2.
εi(ω)=αmiωmβniωn,
εb(ω)=αmbωmβnbωn.
εeff=εb+3fεb εi-εbγiεi+γbεb,
εeff(ω)=αmeωmβneωn,
αme=k,lαkb[αliβm-k-lb(γi+3f)+αlbβm-k-li(γb-3f)]α0iγi+α0bγb,
m=0, 1, 2, 3, ,
β0e=1,
βne=k,lβkb(αliβn-k-lbγi+αlbβn-k-liγb)α0iγi+α0bγb,
n=1, 2, 3, .
αme=εb(γi+3f)αmi+εb2(γb-3f)βmiγiα0i+εbγb,
m=0, 1, 2, 3, ,
β0e=1,
βne=γiαni+εbγbβniγiα0i+εbγb,n=1, 2, 3, .
αme=εb(1+3f)αmi+εb2(2-3f)βmiα0i+2εb,
m=0, 1, 2, 3, ,
β0e=1,
βne=αni+2εbβniα0i+2εb,n=1, 2, 3, ,
αme=εb(1+2 f)αmi+2εb2(1-f)βmi(1-f)α0i+εb(2+f),
m=0, 1, 2, 3, ,
β0e=1,
βne=(1-f)αni+εb(2+f)βni(1-f)α0i+εb(2+f),n=1, 2, 3, .
ε(ω)=ε+εs-ε1+jωτ,
εeff(ω)=ε,eff+εs,eff-ε,eff1+jωτeff.
ε,eff=εb+3fεb ε-εbε+2εb-f(ε-εb),
εs,eff=εb+3fεb εs-εbεs+2εb-f(εs-εb),
τeff=τ (1-f)ε+(2+f)εb(1-f)εs+(2+f)εb.
εs=190.0-0.375T,ε=4.90,
τ=(1.99/T)exp(2140/T)×10-12s,
ε(ω)=ε+ωp2ω02-ω2+jων,
ε,eff=εb+3fεb ε-εbε+2εb-f(ε-εb),
ωp,eff=f 3εb(1-f)ε+(2+f)εbωp,
ω0,eff2=ω02+1-f(1-f)ε+(2+f)εbωp2,
νeff=ν.
ε(ω)=ε-ωp2ω2-jων.
ε(ω)-jσωε0
ω0,eff=1-f(1-f)ε+(2+f)εb1/2ωp.
ω0,eff=ωp1+2εb.
α=ε0V(εi-εb) 3εbεi+2εb,
ε(ω)=ε+ωp2(ω0+jω)2
εeff(ω)=ε,eff+ωp,eff2ω0,eff2-ω2+jωνeff,
ε,eff=εb+3fεb ε-εbε+2εb-f(ε-εb),
ωp,eff=f 3εb(1-f)ε+(2+f)εbωp,
ω0,eff2=ω02+1-f(1-f)ε+(2+f)εbωp2,
νeff=2ω0.
εi(ω)=ε,i+Ωi(ω),
εb(ω)=ε,b+Ωb(ω),
εeff(ω)=ε,eff+Ωeff(ω),
Ωeff=Ωb+3f3ε,b2(Ωi-Ωb)+(1-f)(ε,i-ε,b)2Ωb+A(Ωi-Ωb)ΩbA[A+(1-f)Ωi+(2+f)Ωb],
ε,eff=ε,b+3fε,b ε,i-ε,b(1-f)ε,i+(2+f)ε,b.
εs=80.1,ε=4.9,τ=1.01×10-11s
εs=25.1,ε=4.4,τ=1.2×10-10s.
eff=εb+3εbf(ε-εb+χ*)×[ε+2εb-f(ε-εb)+(1-f)χ*]-1.
χ(t)=H(t)β exp(-t/τ),
(χ1*χ2)(t)=H(t)β1β2 exp(-t/τ1)-exp(-t/τ2)1/τ2-1/τ1.
(χ1*χ2)(t)=H(t)β1β2t exp(-t/τ).
χres(t)=-H(t)β exp[-(1+βτ)t/τ].
eff=ε,eff+χeff(t)*,
ε,eff=εb+3εbf ε-εbε+2εb-f(ε-εb),
χeff(t)=9fεb2βH(t)[ε+2εb-f(ε-εb)]2×exp-1τ+(1-f)βε+2εb-f(ε-εb)t.
χeff(t)=χb(t)+A(t)+(ε,eff-ε,b)Bres(t)+(A*Bres)(t).
A(t)=(ε,eff-ε,b)χb(t)ε,b+χi(t)-χb(t)ε,i-ε,b+χbε,b*χi-χbε,i-ε,b(t),
B(t)=fbχb(t)+fiχi(t),
fb=(2+f)(1-f)ε,i+(2+f)ε,b,
fi=(1-f)(1-f)ε,i+(2+f)ε,b.
χb(t)=εs,b-ε,bτbexp-tτbH(t)αb exp(-tβb)H(t),
χi(t)=εs,i-ε,iτiexp-tτiH(t)αi exp(-tβi)H(t).
A(t)=a1χb(t)+a2χi(t)+a3χb(t)(t/τb).
a1=ε,eff-ε,bε,b-ε,eff-ε,bε,i-ε,b×1+εs,i-ε,iε,bτbτi-τb,
a2=ε,eff-ε,bε,i-ε,b1+εs,b-ε,bε,bτiτi-τb,
a3=-(ε,eff-ε,b)(εs,b-ε,b)ε,b(ε,i-ε,b),
Bres(t)=-χ+(t)-χ-(t)-α+ exp(-β+t)H(t)-α- exp(-β-t)H(t),
2β±=fiαi+βi+fbαb+βb±[(fiαi-βi+fbαb+βb)2+4fiαi(βi-βb)]1/2,
α±=βi-βb(β-βb)(β±-βb)-(β-βi)(β±-βi).
χeff(t)=c1χb(t)+c2χi(t)+c3χ+(t)+c4χ-(t)+c5χb(t)(t/τb),
c1=1+a11-α+β+-βb-α-β--βb+a3α+βb(β+-βb)2+α-βb(β--βb)2,
c2=a21-α+β+-βi-α-β--βi,
c3=-ε,eff+ε,b-a1 αbβb-β+-a2 αiβi-β+-a3 αbβb(β+-βb)2,
c4=-ε,eff+ε,b-a1 αbβb-β--a2 αiβi-β--a3 αbβb(β--βb)2,
c5=a31-α+β+-βb-α-β--βb
min(βi, βb)<β-<max(βi, βb)<β+,α±>0.
-1=(1+χ*)-11+χres*=(δ+χres)*.
χres(t)+χ(t)+(χres*χ)(t)=0.
χres(t)=k=1(-1)k[(χ*)k-1χ](t).
ab+(aB+bA+A*B)*=1=δ*,
aB(t)+bA(t)+(A*B)(t)=0.
β exp(-t/τ)
βτ1+jωτ
ωp2ν0sin(ν0t)exp(-νt/2)
ωp2ω02-ω2+jων
ν02=ω02-(ν/2)2
ωp2ν[1-exp(-νt)]
ωp2-ω2+jων
ωp2t exp(-νt/2)
ωp2(ν/2+jω)2
ωp2ν0sin(ν0t)exp(-νt/2)
-ωp2ωrsin(ωrt)exp(-νt/2)
ωp2ν[1-exp(-νt)]
-ωp2ωrsin(ωrt)exp(-νt/2)

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