Abstract

The time-domain theory of forerunners (precursors) in temporally dispersive, nonmagnetic, isotropic materials is developed with the propagator technique. Specifically, the impulse response at a (comparatively) large propagation depth is expanded in two different ways: (a) with respect to the wave front and (b) with respect to slowly varying field components. A few numerical examples illustrating the theory are given.

© 1998 Optical Society of America

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References

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  1. L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).
  2. K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).
  3. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).
  4. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [CrossRef]
  5. L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [CrossRef]
  6. S. Shen, K. E. Oughstun, “Dispersive pulse propagation in a double-resonance Lorentz medium,” J. Opt. Soc. Am. B 6, 948–963 (1989).
    [CrossRef]
  7. P. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
    [CrossRef]
  8. T. Roberts, P. Petropoulos, “Asymptotics and energy estimates for electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. A 13, 1204–1217 (1996).
    [CrossRef]
  9. G. Kristensson, “Direct and inverse scattering problems in dispersive media—Green’s functions and invariant imbedding techniques,” in R. Kleinman, R. Kress, E. Martensen, eds. Direct and Inverse Boundary Value Problems, Vol. 37 of Methoden und Verfahren der Mathematischen Physik (Peter Lang, Frankfurt am Main, 1991), pp. 105–119.
  10. M. Kelbert, I. Sazonov, Pulses and Other Wave Processes in Fluids (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  11. R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys. 26, 317–325 (1985).
    [CrossRef]
  12. A. Karlsson, “Wave propagators for transient waves in one-dimensional media,” Wave Motion 24, 85–99 (1996).
    [CrossRef]
  13. S. Rikte, “The theory of the propagation of TEM-pulses in dispersive bisotropic slabs,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1995).
  14. T. M. Roberts, “Causality theorems,” in J. P. Corones, G. Kristensson, P. Nelson, D. L. Seth, ed., Invariant Imbedding and Inverse Problems (Society for Industrial and Applied Mathematics, Boulder, Colo.1992), pp. 114–128.
  15. A. Karlsson, R. Stewart, “Wave propagators for transient waves in periodic media,” J. Opt. Soc. Am. A 12, 1513–1521 (1995).
    [CrossRef]
  16. S. Rikte, “Sommerfeld’s forerunner in stratified isotropic and bi-isotropic media.” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1994).
  17. A. Karlsson, “Inverse scattering for viscoelastic media using transmission data,” Inverse Probl. 3, 691–709 (1987).
    [CrossRef]
  18. P. Fuks, A. Karlsson, G. Larson, “Direct and inverse scattering for dispersive media,” Inverse Probl. 10, 555–571 (1994).
    [CrossRef]
  19. K. E. Oughstun, G. C. Sherman, “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
    [CrossRef]
  20. L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren der matematischen Wissenshaften 256. (Springer-Verlag, Berlin, 1983).
  21. S. Rikte, “One-dimensional pulse propagation in temporally dispersive media—exact solutions versus numerical results,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1996).
  22. C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

1996 (2)

1995 (2)

A. Karlsson, R. Stewart, “Wave propagators for transient waves in periodic media,” J. Opt. Soc. Am. A 12, 1513–1521 (1995).
[CrossRef]

P. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
[CrossRef]

1994 (1)

P. Fuks, A. Karlsson, G. Larson, “Direct and inverse scattering for dispersive media,” Inverse Probl. 10, 555–571 (1994).
[CrossRef]

1989 (1)

1988 (1)

1987 (1)

A. Karlsson, “Inverse scattering for viscoelastic media using transmission data,” Inverse Probl. 3, 691–709 (1987).
[CrossRef]

1985 (1)

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys. 26, 317–325 (1985).
[CrossRef]

1914 (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Beezley, R. S.

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys. 26, 317–325 (1985).
[CrossRef]

Bender, C. M.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

Fuks, P.

P. Fuks, A. Karlsson, G. Larson, “Direct and inverse scattering for dispersive media,” Inverse Probl. 10, 555–571 (1994).
[CrossRef]

Hörmander, L.

L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren der matematischen Wissenshaften 256. (Springer-Verlag, Berlin, 1983).

Jackson, J. D.

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

Karlsson, A.

A. Karlsson, “Wave propagators for transient waves in one-dimensional media,” Wave Motion 24, 85–99 (1996).
[CrossRef]

A. Karlsson, R. Stewart, “Wave propagators for transient waves in periodic media,” J. Opt. Soc. Am. A 12, 1513–1521 (1995).
[CrossRef]

P. Fuks, A. Karlsson, G. Larson, “Direct and inverse scattering for dispersive media,” Inverse Probl. 10, 555–571 (1994).
[CrossRef]

A. Karlsson, “Inverse scattering for viscoelastic media using transmission data,” Inverse Probl. 3, 691–709 (1987).
[CrossRef]

Kelbert, M.

M. Kelbert, I. Sazonov, Pulses and Other Wave Processes in Fluids (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Kristensson, G.

G. Kristensson, “Direct and inverse scattering problems in dispersive media—Green’s functions and invariant imbedding techniques,” in R. Kleinman, R. Kress, E. Martensen, eds. Direct and Inverse Boundary Value Problems, Vol. 37 of Methoden und Verfahren der Mathematischen Physik (Peter Lang, Frankfurt am Main, 1991), pp. 105–119.

Krueger, R. J.

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys. 26, 317–325 (1985).
[CrossRef]

Larson, G.

P. Fuks, A. Karlsson, G. Larson, “Direct and inverse scattering for dispersive media,” Inverse Probl. 10, 555–571 (1994).
[CrossRef]

Orszag, S. A.

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Oughstun, K. E.

Petropoulos, P.

Rikte, S.

S. Rikte, “One-dimensional pulse propagation in temporally dispersive media—exact solutions versus numerical results,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1996).

S. Rikte, “The theory of the propagation of TEM-pulses in dispersive bisotropic slabs,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1995).

S. Rikte, “Sommerfeld’s forerunner in stratified isotropic and bi-isotropic media.” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1994).

Roberts, T.

Roberts, T. M.

T. M. Roberts, “Causality theorems,” in J. P. Corones, G. Kristensson, P. Nelson, D. L. Seth, ed., Invariant Imbedding and Inverse Problems (Society for Industrial and Applied Mathematics, Boulder, Colo.1992), pp. 114–128.

Sazonov, I.

M. Kelbert, I. Sazonov, Pulses and Other Wave Processes in Fluids (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Shen, S.

Sherman, G. C.

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

Stewart, R.

Ann. Phys. (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[CrossRef]

L. Brillouin, “Über die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[CrossRef]

Inverse Probl. (2)

A. Karlsson, “Inverse scattering for viscoelastic media using transmission data,” Inverse Probl. 3, 691–709 (1987).
[CrossRef]

P. Fuks, A. Karlsson, G. Larson, “Direct and inverse scattering for dispersive media,” Inverse Probl. 10, 555–571 (1994).
[CrossRef]

J. Math. Phys. (1)

R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys. 26, 317–325 (1985).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Wave Motion (2)

P. Petropoulos, “The wave hierarchy for propagation in relaxing dielectrics,” Wave Motion 21, 253–262 (1995).
[CrossRef]

A. Karlsson, “Wave propagators for transient waves in one-dimensional media,” Wave Motion 24, 85–99 (1996).
[CrossRef]

Other (11)

S. Rikte, “The theory of the propagation of TEM-pulses in dispersive bisotropic slabs,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1995).

T. M. Roberts, “Causality theorems,” in J. P. Corones, G. Kristensson, P. Nelson, D. L. Seth, ed., Invariant Imbedding and Inverse Problems (Society for Industrial and Applied Mathematics, Boulder, Colo.1992), pp. 114–128.

S. Rikte, “Sommerfeld’s forerunner in stratified isotropic and bi-isotropic media.” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1994).

G. Kristensson, “Direct and inverse scattering problems in dispersive media—Green’s functions and invariant imbedding techniques,” in R. Kleinman, R. Kress, E. Martensen, eds. Direct and Inverse Boundary Value Problems, Vol. 37 of Methoden und Verfahren der Mathematischen Physik (Peter Lang, Frankfurt am Main, 1991), pp. 105–119.

M. Kelbert, I. Sazonov, Pulses and Other Wave Processes in Fluids (Kluwer Academic, Dordrecht, The Netherlands, 1996).

L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Grundlehren der matematischen Wissenshaften 256. (Springer-Verlag, Berlin, 1983).

S. Rikte, “One-dimensional pulse propagation in temporally dispersive media—exact solutions versus numerical results,” (Lund Institute of Technology, Department of Electromagnetic Theory, P.O. Box 118, S-211 00 Lund, Sweden, 1996).

C. M. Bender, S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

L. Brillouin, Wave Propagation and Group Velocity (Academic, New York, 1960).

K. E. Oughstun, G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, Berlin, 1994).

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

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Figures (10)

Fig. 1
Fig. 1

Scattering geometry with the incident, scattered, and internal electric fields indicated.

Fig. 2
Fig. 2

Sign of the coefficients nk for a single-resonance Lorentz medium with ωp=20×1016 rad/s, ω0=4×1016 rad/s and ν=0.56×1016 s-1. The signature +--+ is violated at k=23. This means that the expansion (6.8) has to be truncated at k<23. The coefficients χk have the same signs as nk for the k in this figure. The corresponding propagator kernel is depicted in Fig. 3.

Fig. 3
Fig. 3

Propagator kernel P(z; t) for a single-resonance Lorentz medium characterized by Brillouin’s parameters (z=10-6 m, ωp=20×100/3×c/z, ω0=400/3×c/z, ν=56/3×c/z). 32,768 data points were used at the equidistant discretization of the time interval 0<t<2×z/c. The propagator rule was used once in the computation. Approximations to the second precursor obtained by the time-domain method are shown also. Brillouin’s forerunner is an appropriate approximation to the field in the neighborhood of the quasilatent time t1=1/2×z/c only. Higher-order approximations must be used to obtain a good approximation to the “tail” of the second precursor. These approximations lie almost on top of the numerical result.

Fig. 4
Fig. 4

Refractive index N(t) for the single-resonance Lorentz medium characterized by Brillouin’s parameters; see Fig. 3.

Fig. 5
Fig. 5

Sign of the coefficients nk for a double-resonance Lorentz medium with ωp1=5×1016 rad/s, ω01=1016×rad/s, ν1=0.2×1016 s-1, and ωp2=20×1016 rad/s, ω02=10×1016 rad/s, ν2=0.56×1016 s-1. The signature +--+ is violated at k=17. This means that the expansion (6.8) has to be truncated at k<17. The coefficients χk have the same signs as nk for the k in this figure. The corresponding propagator is depicted in Fig. 6.

Fig. 6
Fig. 6

Propagator kernel P(z; t) for a double-resonance Lorentz medium characterized by the Shen–Oughstun parameters z=48π×10-8 m, ωp1=5×16π×c/z, ω01=16π×c/z, ν1=0.2×16π×c/z, ωp2=20×16π×c/z, ω02=10×16π×c/z, ν2=0.56×16π×c/z. 65,536 data points were used at the equidistant discretization of the time interval 0<t<2×z/c. The propagator rule was used twice. Approximations to the second forerunner are shown as well. As in the single-resonance case, Brillouin’s forerunner is an appropriate approximation to the field in the vicinity of the quasi-latent time t1=3/2×z/c only. To obtain a good approximation to the “tail,” higher-order approximations must be used. The higher-order approximations lie almost on top of the numerical result.

Fig. 7
Fig. 7

Refractive index N(t) for the double-resonance Lorentz medium characterized by the Shen–Oughstun parameters; see Fig. 6.

Fig. 8
Fig. 8

Propagator kernel Q(z)P(z; t) for a Debye half-space characterized by α=100×c/z and β=40×c/z. 4096 data points were used at the equidistant discretization of the time interval 0<t<2×z/c.

Fig. 9
Fig. 9

Hyper-Airy functions B2(x), B4(x), and B6(x).

Fig. 10
Fig. 10

Hyper-Airy functions B3(x), B5(x), and B7(x).

Equations (194)

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cηD(r, t)=E(r, t)+(χ*E)(r, t),
cB(r, t)=ηH(r, t),
(χ*E)(r, t)=-tχ(t-t)E(r, t)dt.
limω 0 exp(-iωt)χ(t)dt=0
(theRiemannLebesguelemma).
Ei(t)=exEi(t),Hi(t)=eyHi(t),
Hi(t)=Ei(t)/η.
×E(r, t)=-tB(r, t),×H(r, t)=tD(r, t),
E(r, t)=exEx(z, t),H(r, t)=eyHy(z, t).
c zEx-ηHy=t011+χ*0Ex-ηHy,
0<z<d.
Er(t)=exEr(t),Hr(t)=eyHr(t),
Hr(t)=-Er(t)/η.
Et(t)=exEt(t),Ht(t)=eyHt(t),
Ht(t)=Et(t)/η.
Ei(t)+Er(t)=Ex(0, t),Et(t)=Ex(d, t),
Hi(t)+Hr(t)=Hy(0, t),Ht(t)=Hy(d, t).
E+E-=WEx-ηHy,W=121-Z1Z.
Ex-ηHy=W-1E+E-,W-1=11-NN.
Z1+Z*,N1+N*
ZN=1(N=Z-1),
N2=Er,whereEr1+χ*
2N(t)+(N*N)(t)=χ(t),
Z(t)+N(t)+(Z*N)(t)=0.
N(t)=k=112k[(χ*)k-1χ](t).
Z(t)=k=1(-1)k[(N*)k-1N](t).
Ex=E++E-,-ηHy=-E++E-.
R=(ηZ+η1)-1(ηZ-η1)=(1+N)-1(1-N),
RR*,
2R(t)-Z(t)+(Z*R)(t)=0,
2R(t)+N(t)+(N*R)(t)=0,
4R(t)+2(χ*R)(t)+χ(t)+[χ*(R*R)](t)=0.
T=1+R,
(cz±t)E±=tN*E±,0<z<d.
E±(z)E±(z, t),EiEi(t),
ErEr(t),EtEt(t),
EiEr=SSRSRSE+(0)E-(0),
Et0=SSRSRSE+(d)E-(d),
S1+S*,ST=1(S=T-1)
E±(z2, t±(z2-z1)/c)={P[±(z2-z1)]E±(z1,·)}(t)
zP(z)=-KP(z),P(0)=1,
K=1ctN*=1cN(0)+K(·)*,K(t)=1cN(t).
P(z)=exp(-zK).
P0(z)=exp(-zc-1t)=δ(·-z/c)*
p(z)=exp[-ik(ω)z],
P(z1+z2)=P(z1)P(z2),
P(0)=1,
P-1(z)=P(-z),
P(z)=Q(z)[1+P(z;·)*],
zQ(z)=-N(0)Q(z)/c,Q(0)=1.
Q(z)=exp-zcN(0).
zP(z; t)=-K(t)-[P(z;·)*K(·)](t),
P(0; t)=0,
P(z; t)+P(-z; t)+[P(z;·)*P(-z;·)](t)=0,
P(z; t)=k=1 (-z)kk![(K*)k-1K](t).
1+P(z;·)*=exp(-zK*).
E+(z)=P(z)E+(0),E-(z)=P(d-z)E-(d).
Et=M(1-R2)P(d)(δd/c*Ei),
Er=REi-M(1-R2)RP(2d)(δ2(d/c)*Ei),
M=(1-R2P(2d)δ2(d/c)*)-1
(δa*Ei)(t):=Ei(t-a)
E+(z)=MTP(z)(δz/c*Ei),
E-(z)=-MTRP(2d-z)(δ2d-z/c*Ei).
E+(z)=TP(z)(δz/c*Ei)=(1+R)P(z)(δz/c*Ei),E-(z)=0.
Ex(z)=(1+R)P(z)(δz/c*Ei),
ηHy(z)=(1-R)P(z)(δz/c*Ei),
[P(z)δ](t)=Q(z){δ(t)+P(z; t)},
K(t)=H(t)j=0k-1 tjj!djKdtj(+0)+H(t)0t (t-t)k-1(k-1)!dkKdtk(t)dt,
k=1, 2, 3, ,
2N(k)(+0)=χ(k)(+0)-j=0k-1N(j)(+0)N(k-1-j)(+0),
N(+0)=12χ(+0),
K(+0)=12c[χ(+0)-χ2(+0)/4)].
P(z)=Q(z)exp(-zK*)=Q(z)j=0k-1Qj(z)Q˜k(z),
Qj(z)=exp-zK(j)(+0)tjH(t)j!*,
j=0, 1, 2, , k-1
Q˜k(z)=exp-z(k-1)!H(t)0t(t-t)k-1×K(k)(t)dt*.
Qj(z)=1+Qj(z;·)*,j=0, 1, 2, , k-1,
Qj(z; t)=H(t)i=1[-zK(j)(+0)]i ti(1+j)-1[i(1+j)-1]!i!.
[PS(z)δ](t)=Q(z){δ(t)+PS(z, t)},
PS(z)=Q(z)Q0(z)=Q(z)[1+PS(z;·)*].
PS(z; t)=-zK(+0){I0[2-zK(+0)t]-I2[2-zK(+0)t]}H(t)=-zK(+0){J0[2zK(+0)t]+J2[2zK(+0)t]}H(t).
Ei(t)=j=0k-1 (t-t)jj!djdtjEi(t)+tt (t-t)k-1(k-1)!dkdtkEi(t)dt,
k=1, 2, 3, .
(χ*Ei)(t)=j=1kχj d(j-1)dt(j-1)Ei(t)+Xk*dkdtkEi(t),
χj=(-1)j-1(j-1)!0tj-1χ(t)dt
Xk(t)=(-1)k(k-1)!t(τ-t)k-1χ(τ)dτH(t).
N*=j=1knj d(j-1)dt(j-1)+Nk*dkdtk.
χ*=k=0χk+1 dkdtkδ*,N*=k=0nk+1 dkdtkδ*.
χk+1=2nk+1+i=0knk-i+1ni+1.
n1=1+χ1-1,
n2=χ221+χ1,
nk+1=χk+1-i=1k-1nk-i+1ni+12(1+n1),k>1.
n4k+10,n4k+20,
n4k+30,n4k+40.
P(z)=exp-zcddtj=1knj d(j-1)dt(j-1)Pk+1(z)=j=1kPj(z)Pk+1(z),
Pj(z)=exp-zcnj djdtj,j=1, 2, 3, , k
Pk+1(z)=exp-zcddtNk*dkdtk.
Pj(z)=Pj(z;·)*,j=1, 2, 3, , k.
P1(z; t)=δ(t-t1),t1=n1z/c,
Pj(z; t)=1tjBj(t/tj),j=2, 3, 4, , k,
tj=j|nj|zc1/j,j=1, 2, 3, , k.
P=P1*P2*P3*P4*Pk*Pk+1,
-Pk(z; t)dt=1,
limz0 Pk(z; t)=δ(t),
Pk(z1; t)*Pk(z2; t)=Pk(z1+z2; t).
PB=P1*P2*P3,
-czPB(z; t)=n1tPB(z; t)+n2t2PB(z; t)+n3t3PB(z; t),PB(+0; t)=δ(t).
PB(z; t)=expn2327n32zc-n23n3[t-t1(z)]×Aisign(n3) [t-t1(z)]t3(z)t3(z),
t1(z)=n1-n223n3 zc,t3(z)=3|n3|zc1/3
χ(t)=ωp2ν0sin(ν0t)exp-ν2tH(t),
χ(k)(+0)=(-1)k ωp22iν0(bk-b¯k)=(-1)k+1 ωp2ω0kν0sink arcsinν0ω0,
χ(k+2)(+0)=-[νχ(k+1)(+0)+ω02χ(k)(+0)].
Xk(t)=(-1)k ωp2ω0kν0sinν0t+k arcsinν0ω0×exp-ν2tH(t),
χk=(-1)k+1 ωp2ω0kν0sink arcsinν0ω0.
χk+2=-1ω02(χk+νχk+1).
χ1=ωp2ω02,χ2=-νωp2ω04,
χ3=-ωp2(ω02-ν2)ω06,χ4=ωp2ν(2ω02-ν2)ω08.
n1=1+ωp2ω021/2-1>0,
n2=-11+ωp2ω021/2νωp22ω04<0,
n3=-121+ωp2ω021/2ωp2(ω02-ν2)ω06+ν2ωp44ω0811+ωp2ω02<0.
χ(t)=m=1M ωpm2ν0msin(ν0mt)exp-νm2tH(t).
χk=(-1)k+1m=1M ωpm2ω0mkν0msink arcsinν0mω0m.
χ(t)=α exp(-βt)H(t).
χk=(-1)k-1αβ-k
n1=1+αβ1/2-1,n2=-α2β2(1+α/β)1/2,
n3=4βα+3α38β4(1+α/β)3/2.
χ(t)=ωp2ν[1-exp(-νt)]H(t).
P(z; t)=-1t[F(·)*P(·)](t)-zK(t),
F(t)=ztK(t).
P(z1+z2; t)=P(z1; t)+P(z2; t)+[P(z1;·)*P(z2;·)](t)
z=48π×10-8 m,ωp1=5×16π×c/z,
ω01=16π×c/z,ν1=0.2×16π×c/z,
ωp2=20×16π×c/z,ω02=10×16π×c/z,
ν2=0.56×16π×c/z.
J(r, t)=exJx(z, t),
×E=-tB,×H=J+tD,
zE±=±c-1tNE±±ηZJx/2.
±z+c-1tN.
(±z+c-1tN)E±=δ(z)δ(t).
E±(z, t)=-12E±(z-z; t-t)×(ηZJx)(z, t)dtdz.
E±(z; t)=H(±z)E(|z|; t),
E(|z|; t)=Q(|z|)[δ(t-|z|/c)+P(|z|; t-|z|/c)]
[±z+c-1t(1+N*)]E±(z, t)=E0(t)δ(z),
E±(z, t)=E±(z, t-t)E0(t)dt,
E±(z, t+|z|/c)=H(±z)[P(|z|)E0](t),
Er=(1-R)E-(0),Et=(1-R)E+(d),
E+(0)=-RE-(0),E-(d)=-RE+(d)
E+(z, t)=E+(z, t-t)E+(0, t)dt-120dE+(z-z; t-t)(ηZJx)×(z; t)dtdz,0<z<d,
E-(z, t)=E-(z-d, t-t)E-(d, t)dt-120dE-(z-z; t-t)(ηZJx)×(z, t)dtdz,0<z<d,
E+(d)=-120dMP(d-z)ηZ[δ(d-z)c*Jx(z)]dz+120dMRP(d+z)×ηZ[δ(d+z)c*Jx(z)]dz,
E-(0)=120dMRP(2d-z)×ηZ[δ(2d-z)c*Jx(z)]dz-120dMP(z)ηZ[δz/c*Jx(z)]dz,
supx|xβϕ(α)(x)|<
A2k(x)=12π exp(-ξ2k/(2k)+ixξ)dξ,
-<x<+.
A2k+1(x, η)=12π exp(iζ2k+1/(2k+1)+ixζ)dζ,
-<x<+,
exp[iζ2k+1/(2k+1)]exp[iξ2k+1/(2k+1)]
inSasη+0,
Ak(x)dx=1.
A2(x)=12πexp-x22,
A3(x)=Ai(x),-<x<+.
A2k(2k-1)(x)=(-1)kxA2k(x),-<x<+,
A2k+1(2k)(x)=(-1)k+1xA2k+1(x),-<x<+
Ai(x)=xAi(x),-<x<+,
A2k(m)(0)=12π(iξ)m exp-ξ2k2kdξ=im2π[1+(-1)m)]×0tm exp-t2k2kdt
Γ(x)=0tx-1 exp(-t)dt:
A2k(2m)(0)=(-1)mπ(2k)2m+12k-1Γ2m+12k,
A2k(2m-1)(0)=0.
A2k+1(m)(0)=12π(iζ)m exp[iζ2k+1/(2k+1)]dζ
ζ(t)=t expiπ(2k+1)2,t>0
im2πexpi(m+1)π(2k+1)2+(-1)m exp-i(m+1)π(2k+1)2×0tm exp-t2k+12k+1dt
A2k+1(2m)(0)=(-1)mπcos(2m+1)π(2k+1)2×(2k+1)2m+12k+1-1Γ2m+12k+1,
A2k+1(2m-1)(0)=(-1)mπsinmπ2k+1×(2k+1)2m2k+1-1Γ2m2k+1.
A2k(x)=(-1)k(2k-2)!0x(x-ξ)2k-2×ξA2k(ξ)dξ+m=02k-2 xmm!A2k(m)(0),
A2k+1(x)=(-1)k+1(2k-1)!0x(x-ξ)2k-1×ξA2k+1(ξ)dξ+m=02k-1 xmm!A2k+1(m)(0),
ωAk(ωx)=0.
A3(x)x-1/42πexp-23x3/2,x+.
A3(-x)x-1/4πcos23x3/2-π4,x+.
A2k(x)=I2k(ix)+I2k(-ix),
Ik(z)=12π0 exp-tkk+tzdt
Ik(x)x-k-22(k-1)[2π(k-1)]1/2expk-1kxkk-1,x+;
A2k(x)2π(2k-1)1/2|x|-k-12k-1×exp-2k-12kcosk-12k-1π|x|2k2k-1×cos2k-12ksink-12k-1π×|x|2k2k-1-k-12k-1π2,x±.
A2k+1(x)=exp-iπ(2k+1)2×I2k+1-ix×exp-iπ(2k+1)2+expiπ(2k+1)2×I2k+1ix expiπ(2k+1)2.
A2k+1(x)x-2k-14kπkexp-2k2k+1×cosk-1kπ2x2k+12k×cos2k2k+1sink-1kπ2x2k+12k-k-12kπ2x+
A2k+1(-x)x-2k-14kπkcos2k2k+1x2k+12k-π4,
x+
Ik(x)x-k-22(k-1)[2π(k-1)]1/2expk-1kxkk-1×1+(2k-1)(k-2)24(k-1)x-kk-1,x+;
(4k-1)(k-1)12(2k-1)2π(2k-1)1/2|x|-3k-12k-1
×exp-2k-12kcosk-12k-1π|x|2k2k-1
×cos2k-12ksink-12k-1π|x|2k2k-1-3k-12k-1π2.
(4k+1)(2k-1)48kπkx-6k+14k
×exp-2k2k+1cosk-1kπ2x2k+12k
×cos2k2k+1sink-1kπ2x2k+12k-3k+12kπ2.
(4k+1)(2k-1)48kπkx-6k+14k cos2k2k+1x2k+12k-3π4.

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