Abstract

Dispersion of electromagnetic waves is usually described in terms of an integrodifferential equation. We show that whenever a differential operator can be found that annihilates the susceptibility kernel of the medium, dispersion can be modeled by a partial differential equation without nonlocal operators.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974).
  2. R. S. Beezley, R. J. Krueger, “An electromagnetic inverse problem for dispersive media,” J. Math. Phys. (New York) 26, 317–325 (1985).
    [CrossRef]
  3. 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]
  4. T. M. Roberts, P. G. Petropoulos, “Asymptotics and energy estimates for electromagnetic pulses in dispersive media,” J. Opt. Soc. Am. A 13, 1204–1217 (1996).
    [CrossRef]
  5. J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1979).
  6. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  7. J. Chazarain, A. Piriou, Introduction to the Theory of Linear Partial Differential Equations (North-Holland, Amsterdam, 1982).
  8. P. Henrici, Applied and Computational Complex Analysis (Wiley-Interscience, New York, 1974), Vol. 1.
  9. S. Rikte, “Propagation of transient electromagnetic waves in stratified bi-isotropic media and related inverse scattering problems,” Ph.D. dissertation (Lund University, Lund, Sweden, 1994).

1996 (1)

1992 (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]

1985 (1)

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

Beezley, R. S.

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

Chazarain, J.

J. Chazarain, A. Piriou, Introduction to the Theory of Linear Partial Differential Equations (North-Holland, Amsterdam, 1982).

Christy, R. W.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1979).

Henrici, P.

P. Henrici, Applied and Computational Complex Analysis (Wiley-Interscience, New York, 1974), Vol. 1.

Jackson, J. D.

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

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]

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]

Krueger, R. J.

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

Milford, F. J.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1979).

Petropoulos, P. G.

Piriou, A.

J. Chazarain, A. Piriou, Introduction to the Theory of Linear Partial Differential Equations (North-Holland, Amsterdam, 1982).

Reitz, J. R.

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1979).

Rikte, S.

S. Rikte, “Propagation of transient electromagnetic waves in stratified bi-isotropic media and related inverse scattering problems,” Ph.D. dissertation (Lund University, Lund, Sweden, 1994).

Roberts, T. M.

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974).

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]

J. Math. Phys. (New York) (1)

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

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

Other (6)

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974).

J. R. Reitz, F. J. Milford, R. W. Christy, Foundations of Electromagnetic Theory (Addison-Wesley, Reading, Mass., 1979).

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

J. Chazarain, A. Piriou, Introduction to the Theory of Linear Partial Differential Equations (North-Holland, Amsterdam, 1982).

P. Henrici, Applied and Computational Complex Analysis (Wiley-Interscience, New York, 1974), Vol. 1.

S. Rikte, “Propagation of transient electromagnetic waves in stratified bi-isotropic media and related inverse scattering problems,” Ph.D. dissertation (Lund University, Lund, Sweden, 1994).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (151)

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

t+c1 x  t+cn xϕ+t+a1 x  t+am xϕ=0,
t+c xt-c xϕ+0tχ(t-t)ϕ(x, t)dt=0,
D(r, t)=0E(r, t)+0tE(r, t-t)χ(t)dt,
H(r, t)=1μ0B(r, t).
χ(t)=α exp(-t/τ).
{L1(χ)}(t)τχt(t)+χ(t)=0,t>0,
χ(0)=α.
χ(t)=ωp2 exp-νt2 sin ν0tν0,ν02=ω02-ν24.
{L2(χ)}(t)χtt(t)+νχt(t)+ω02χ(t)=0,
t>0,χ(0)=0,χt(0)=ωp2.
{M(E)}(z, t)Ezz(z, t)-1c2Ett(z, t)+0tχ(t-t)Ett(z, t)dt
=0,z>0,t>0,
E(z, 0)=0,Et(z, 0)=0,z>0,
E(0, t)=f(t),t>0.
{L1(M(E))}(z, t)=τ tEzz(z, t)-1c2Ett(z, t)+Ezz(z, t)-1a2Ett(z, t),
a2c2ατ+1<c2.
{L1(M(E))}(z, t)=0,z>0,t>0,
{M(E)}(z, 0)=0,z>0
{M(E)}(z, t)=0,z>0,t>0.
{M(E)}(z, 0)=Ezz(z, 0)-1c2Ett(z, 0)=0,z>0.
E(z, 0)=0,z>0,
Ett(z, 0)=0,z>0.
τ tEzz(z, t)-1c2Ett(z, t)
+Ezz(z, t)-1a2Ett(z, t)=0,z>0,t>0,
E(z, 0)=0,Et(z, 0)=0,Ett(z, 0)=0,z>0,
E(0, t)=f(t),t>0.
{L2(M(E))}(z, t)=2t2Ezz(z, t)-1c2Ett(z, t)+ν tEzz(z, t)-1c2Ett(z, t)+ω02Ezz(z, t)-1a2Ett(z, t),
a2c2ω02ωp2+ω02<c2.
2t2Ezz(z, t)-1c2Ett(z, t)
+ν tEzz(z, t)-1c2Ett(z, t)
+ω02Ezz(z, t)-1a2Ett(z, t)=0,z>0,t>0,
E(z, 0)=0,Et(z, 0)=0,Ett(z, 0)=0,
Ettt(z, 0)=0,z>0,E(0, t)=f(t),t>0.
{K1(E)}(z, t){L1(M(E))}(z, t)=τ tEzz(z, t)-1c2Ett(z, t)+Ezz(z, t)-1a2Ett(z, t).
P(x, y)iτy-x2+y2c2+-x2+y2a2.
P3(x, y)iτy-x2+y2c2.
ny=0,nx=±nyc,
z=const.,t=±zc+const.
P(x, y+λ)=iτc2(y+λ)3+1a2(y+λ)2-iτx2(y+λ)-x2=0.
p(z)τc2(z-γ)3+1a2(z-γ)2+τx2(z-γ)+x2=0.
α3=τc2,α2=-3 τc2γ+1α2,
α1=3 τc2γ2-2 γa2+τx2,
α0=-τc2γ3+γ2a2-τx2γ+x2.
α3>0,α2>0,α0>0.
α1α2-α0α3=-8 τ2c4γ3+8 τc2a2γ2
-2a4+2τ2x2c2γ+ατ2c2x2>0
Im λ=Im(y+λ)=-Re z+γ>γ
{K2(E)}(z, t){L2(M(E))}(z, t)=2t2Ezz(z, t)-1c2Ett(z, t)+ν tEzz(z, t)-1c2Ett(z, t)+ω02Ezz(z, t)-1a2Ett(z, t)
P(x, y)=-y2(-x2+y2/c2)+iνy(-x2+y2/c2)+ω02(-x2+y2/a2),
P4(x, y)=-y2(-x2+y2/c2).
z=const.,t=±zc+const.
p(z)(z-γ)4+ν(z-γ)3+c2x2+ω02a2(z-γ)2+νc2x2(z-γ)+ω02c2x2=0.
ηt+c z tt-c z E(z, t)+t+a zt-a z E(z, t)=0,
η=a2c2τ=τατ+1.
E(z, t)=12πiBE˜(z, p)exp (pt)dp,t>0,
E˜(z, p)=F(p)exp[zP1(p)]+G(p)exp[zP2(p)],
P1(p)=-pc1+ηpa2/c2+ηp1/2,
P2(p)=pc1+ηpa2/c2+ηp1/2.
P1(p)=-pc-α2c+O1p,
P2(p)=pc+α2c+O1p.
E(z, t)=12πiBF(p)exp[pt+P1(p)z]dp.
f(t)=E(0, t)=12πiBF(p)exp(pt)dp,
F(p)=0f(t)exp(-pt)dt.
E(z, t)ft-zcexp-αz2c
E(z, t)=n=0ηnEn(ξ, σ),
ξ=η-2(z-ct),σ=η-1t,
t+c tE0(z, t)+1ηc2-a22c2E0(z, t)=0,z>0,t>0,
E0(z, 0)=0,z>0,E0(0, t)=f(t),t>0.
E0(z, t)=ft-zcexp-αz2c.
0=ddp[pt+P1(p)z]=t+zP1(pˆ).
E(z, t)exp[tpˆ+zP1(pˆ)] 12πiCF(p)×exp12zP1(pˆ)(p-pˆ)2dp.
pˆ=0orpˆ=-1η.
P1(p)=-pa+η2c2-a2a3p2+O(p3),
E(z, t)12πiCF(p)exppt-za+p2η(c2-a2)2a3zdp.
E(z, t)ft-za.
t+a zϕ(z, t)=η(c2-a2)2a22t2ϕ(z, t),
z>0,t>0,
ϕ(z, 0)=0,ϕt(z, 0)=0,z>0,
ϕ(0, t)=f(t),t>0.
t+a zϕ(z, t)=η2(c2-a2) 2z2ϕ(z, t),
z>0,t>0,
ϕ(z, 0)=0,z>0,ϕ(0, t)=f(t),t>0.
ϕ(z, t)=z4πd0t f(τ)(t-τ)3×exp-14d[z-a(t-τ)]2t-τdτft-za,
d=η2(c2-a2).
ft-z0cexp-αz02c
E=n=0ηn/2En(ξ, t),
ξ=η-1/2(z-at),
E0(z, t)=z4πd0t f(τ)(t-τ)3
×exp-14d[z-a(t-τ)]2t-τdτ.
η2t2+ν tt+c zt-c zE(z, t)+t+a zt-a zE(z, t)=0,
η=a2c2ω02=1ωp2+ω02.
E˜(z, p)=F(p)exp[zP1(p)]+G(p)exp[zP2(p)],
P1(p)=-pc1+ηνp+ηp2a2/c2+ηνp+ηp21/2,
P2(p)=pc1+ηνp+ηp2a2/c2+ηνp+ηp21/2.
P1(p)=-pc-c2-a22ηc3 1p+O1p2,
P2(p)=pc+c2-a22ηc3 1p+O1p2.
E(z, t)=12πiB F(p)exp[pt+P1(p)z]dp,
E(z, t)12πiB F(p)exppt-zc-c2-a22ηc3 zpdp=ft-zc-c2-a22ηc3z×0t ft-s-zcsJ12c2-a22ηc3zsds.
f(t)=(sin ωt)H(t)ωtH(t)for0<t1/ω,
E(z, t)ωt-z/cγz J1  2γz(t-z/c)  ,
γ=c2-a22ηc3=ωp22c.
E(z, t)=n=0ηnEn(ξ, t),
ξ=η-1(z-ct),
zt+c zE0(z, t)-c2-a22ηc3E0(z, t)=0,
z>0,t>0,
E0(z, 0)=0,z>0,E0(0, t)=f(t),t>0.
E0(z, t)12πiB F(p)×exppt-zc-c2-a22ηc3 zpdp.
E(z, t)exp[tpˆ+zP1(pˆ)]×12πiC F(p)exp12zP1(pˆ)(p-pˆ)2dp,
pˆ=0orpˆ=-ν2±i1η-ν241/2.
P1(p)=-pa+ην2c2-a2a3p2+O(p3),
E(z, t)12πiC F(p)exppt-za+p2ην(c2-a2)2a3zdp.
t+a zϕ(z, t)=ην2(c2-a2) 2z2ϕ(z, t),
z>0,t>0,
ϕ(z, 0)=0,t>0,ϕ(0, t)=f(t),t>0.
ϕ(z, t)=z4πd0t f(τ)(t-τ)3×exp-14d[z-a(t-τ)]2t-τdτ
ft-za.
E(z, t)f(t-z/a)
E(z, t)=n=0ηnEn(z, t)
E0(z, t)=f(t-z/a).
D(r, t)=0E(r, t)+0tE(r, t-t)G(t)dt+c0tB(r, t-t)K(t)dt,
H(r, t)=0c0tE(r, t-t)K(t)dt+c2B(r, t),c2=10μ0,
G(t)=ωp2 exp-νt2 sin ν0tν0,ν02=ω02-ν24,
K(t)=ωp2fν0exp-νt2ν0 cos ν0t-ν2sin ν0t.
{L2(G)}(t) Gtt(t)+νGt(t)+ω02G(t)=0,t>0,
G(0)=0,Gt(0)=ωp2,
{L2(K)}(t)=0,t>0,
K(0)=ωp2f,Kt(0)=-ωp2νf.
E(z, t)=xˆ1E1(z, t)+xˆ2E2(z, t)=E1(z, t)E2(z, t).
G(t)=G(t)00G(t),K(t)=0-K(t)K(t)0.
{T(E)}(z, t)Ett(z, t)-1c2Ett(z, t)-1c20tG(t-t)Ett(z, t)dt+2c0tK(t-t)Ezt(z, t)dt=0,
z>0,t>0,
E(z, 0)=0,z>0,
Et(z, 0)=0,z>0,
E(0, t)=f(t),t>0.
{L2(T(E))}(z, t)=2t2Ezz(z, t)-1c2Ett(z, t)+ν tEzz(z, t)-1c2Ett(z, t)+ω02E zz(z, t)-1a2E tt(z, t)-2cωp2fJEztt(z, t),
a2=c2ω02ωp2+ω02,J=0-110.
{L2(T(E))}(z, t)=0,z>0,t>0,
{T(E)}(z, 0)=0,z>0,
t{T(E)}(z, 0)=0,z>0
{T(E)}(z, t)=0,z>0,t>0.
{T(E)}(z, 0)=Ezz(z, 0)-1c2Ett(z, 0)=0,z>0,
t{T(E)}(z, 0)=Ezzt(z, 0)-1c2Ettt(z, 0)+2cK(0)Ezt(z, 0)=0,z>0.
E(z, 0)=0,Et(z, 0),z>0,
Ett(z, 0)=0,Ettt(z, 0)=0,z>0.
2t2Ezz(z, t)-1c2Ett(z, t)
+ν tEzz(z, t)-1c2Ett(z, t)
+ω02Ezz(z, t)-1a2Ett(z, t)
+2cωp2fJEztt(z, t)=0,z>0,t>0,
E(z, 0)=0,Et(z, 0)=0,Ett(z, 0)=0,
Ettt(z, 0)=0,z>0,
E(0, t)=f(t),t>0.

Metrics