Abstract

A difference equation obtained for the reassociated Green dyadic formalism is exploited to obtain solutions for the case of a multilayered dielectric medium. Computation of limits as layers become progressively thinner leads to a parallel development for the case of a dielectric varying continuously in a single direction. A demonstration example then shows how discrete and continuous techniques can be combined into a hybrid formulation. Finally, numerical computations are presented for the simple case of a dipole, illustrating convergence of the difference equation solutions to the differential equation solution.

© 2000 Optical Society of America

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References

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  1. A. Sommerfeld, Partial Differential Equations (Academic, New York, 1949).
  2. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
  3. C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1994).
  4. R. W. P. King, T. T. Wu, M. Owens, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992).
  5. J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).
  6. K. Warnick, D. Arnold, “Electromagnetic Green functions using differential forms,” J. Electromagn. Waves Appl. 10, 427–438 (1996).
    [CrossRef]
  7. K. Warnick, D. Arnold, “Green forms for anisotropic, inhomogeneous media,” J. Electromagn. Waves Appl. 11, 1145–1164 (1997).
    [CrossRef]
  8. G. P. S. Cavalcante, D. A. Rogers, A. J. Giardola, “Analysis of electromagnetic wave propagation in multi-layered media using dyadic Green’s functions,” Radio Sci. 17, 503–508 (1982).
    [CrossRef]
  9. L. W. Li, J. A. Bennett, P. L. Dyson, “Some methods of solving for the coefficients of dyadic Green’s functions in isotropic stratified media,” Int. J. Electron. 70, 803–814 (1991).
    [CrossRef]
  10. R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
    [CrossRef]
  11. R. L. Hartman, P. T. Leung, S. M. Cohen, “Molecular fluorescence in the vicinity of a gradient-index medium,” J. Opt. Soc. Am. A 17, 933–936 (2000).
    [CrossRef]
  12. E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965), pp. 74–75.

2000

1999

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

1997

K. Warnick, D. Arnold, “Green forms for anisotropic, inhomogeneous media,” J. Electromagn. Waves Appl. 11, 1145–1164 (1997).
[CrossRef]

1996

K. Warnick, D. Arnold, “Electromagnetic Green functions using differential forms,” J. Electromagn. Waves Appl. 10, 427–438 (1996).
[CrossRef]

1991

L. W. Li, J. A. Bennett, P. L. Dyson, “Some methods of solving for the coefficients of dyadic Green’s functions in isotropic stratified media,” Int. J. Electron. 70, 803–814 (1991).
[CrossRef]

1982

G. P. S. Cavalcante, D. A. Rogers, A. J. Giardola, “Analysis of electromagnetic wave propagation in multi-layered media using dyadic Green’s functions,” Radio Sci. 17, 503–508 (1982).
[CrossRef]

1978

R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).

Arnold, D.

K. Warnick, D. Arnold, “Green forms for anisotropic, inhomogeneous media,” J. Electromagn. Waves Appl. 11, 1145–1164 (1997).
[CrossRef]

K. Warnick, D. Arnold, “Electromagnetic Green functions using differential forms,” J. Electromagn. Waves Appl. 10, 427–438 (1996).
[CrossRef]

Bennett, J. A.

L. W. Li, J. A. Bennett, P. L. Dyson, “Some methods of solving for the coefficients of dyadic Green’s functions in isotropic stratified media,” Int. J. Electron. 70, 803–814 (1991).
[CrossRef]

Cavalcante, G. P. S.

G. P. S. Cavalcante, D. A. Rogers, A. J. Giardola, “Analysis of electromagnetic wave propagation in multi-layered media using dyadic Green’s functions,” Radio Sci. 17, 503–508 (1982).
[CrossRef]

Chance, R.

R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).

Coddington, E. A.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965), pp. 74–75.

Cohen, S. M.

R. L. Hartman, P. T. Leung, S. M. Cohen, “Molecular fluorescence in the vicinity of a gradient-index medium,” J. Opt. Soc. Am. A 17, 933–936 (2000).
[CrossRef]

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

Dyson, P. L.

L. W. Li, J. A. Bennett, P. L. Dyson, “Some methods of solving for the coefficients of dyadic Green’s functions in isotropic stratified media,” Int. J. Electron. 70, 803–814 (1991).
[CrossRef]

Giardola, A. J.

G. P. S. Cavalcante, D. A. Rogers, A. J. Giardola, “Analysis of electromagnetic wave propagation in multi-layered media using dyadic Green’s functions,” Radio Sci. 17, 503–508 (1982).
[CrossRef]

Hartman, R. L.

R. L. Hartman, P. T. Leung, S. M. Cohen, “Molecular fluorescence in the vicinity of a gradient-index medium,” J. Opt. Soc. Am. A 17, 933–936 (2000).
[CrossRef]

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

King, R. W. P.

R. W. P. King, T. T. Wu, M. Owens, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992).

Leung, P. T.

R. L. Hartman, P. T. Leung, S. M. Cohen, “Molecular fluorescence in the vicinity of a gradient-index medium,” J. Opt. Soc. Am. A 17, 933–936 (2000).
[CrossRef]

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

Levinson, N.

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965), pp. 74–75.

Li, L. W.

L. W. Li, J. A. Bennett, P. L. Dyson, “Some methods of solving for the coefficients of dyadic Green’s functions in isotropic stratified media,” Int. J. Electron. 70, 803–814 (1991).
[CrossRef]

Owens, M.

R. W. P. King, T. T. Wu, M. Owens, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992).

Prock, A.

R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).

Rogers, D. A.

G. P. S. Cavalcante, D. A. Rogers, A. J. Giardola, “Analysis of electromagnetic wave propagation in multi-layered media using dyadic Green’s functions,” Radio Sci. 17, 503–508 (1982).
[CrossRef]

Silbey, R.

R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations (Academic, New York, 1949).

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1994).

Wait, J. R.

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).

Warnick, K.

K. Warnick, D. Arnold, “Green forms for anisotropic, inhomogeneous media,” J. Electromagn. Waves Appl. 11, 1145–1164 (1997).
[CrossRef]

K. Warnick, D. Arnold, “Electromagnetic Green functions using differential forms,” J. Electromagn. Waves Appl. 10, 427–438 (1996).
[CrossRef]

Wu, T. T.

R. W. P. King, T. T. Wu, M. Owens, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992).

Adv. Chem. Phys.

R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).

Int. J. Electron.

L. W. Li, J. A. Bennett, P. L. Dyson, “Some methods of solving for the coefficients of dyadic Green’s functions in isotropic stratified media,” Int. J. Electron. 70, 803–814 (1991).
[CrossRef]

J. Chem. Phys.

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

J. Electromagn. Waves Appl.

K. Warnick, D. Arnold, “Electromagnetic Green functions using differential forms,” J. Electromagn. Waves Appl. 10, 427–438 (1996).
[CrossRef]

K. Warnick, D. Arnold, “Green forms for anisotropic, inhomogeneous media,” J. Electromagn. Waves Appl. 11, 1145–1164 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Radio Sci.

G. P. S. Cavalcante, D. A. Rogers, A. J. Giardola, “Analysis of electromagnetic wave propagation in multi-layered media using dyadic Green’s functions,” Radio Sci. 17, 503–508 (1982).
[CrossRef]

Other

A. Sommerfeld, Partial Differential Equations (Academic, New York, 1949).

E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1965), pp. 74–75.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. (Institute of Electrical and Electronics Engineers, New York, 1994).

R. W. P. King, T. T. Wu, M. Owens, Lateral Electromagnetic Waves (Springer-Verlag, New York, 1992).

J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1962).

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Figures (6)

Fig. 1
Fig. 1

Geometry and slab indexing.

Fig. 2
Fig. 2

Epsilon variation in the continuous case.

Fig. 3
Fig. 3

Continuous epsilon with jumps.

Fig. 4
Fig. 4

Convergence of slicing method results to differential result.

Fig. 5
Fig. 5

Epsilon near a typical singularity.

Fig. 6
Fig. 6

Integration contours in complex plane.

Equations (111)

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E(R)=iωμG(R, R)J(R)dV(R),
G(R, R)=δ(j, s)GU(R, R)+Gj(R, R),
GU(R, R)=-1ks2 zˆz^tδ(R-R)+i4π 0+dλn=0+ 2-δ(n, 0)λhs(λ) l=01Ml,n,λ(+hs)Ml,n,λt(-hs)+Nl,n,λ(+hs)Nl,n,λt(-hs)Ml,n,λ(-hs)Ml,n,λt(+hs)+Nl,n,λ(-hs)Nl,n,λt(+hs)zzzz,
Gj(R, R)J(R)=i4π λ=0+dλn=0+ 2-δ(n,0)λhs(λ) l=01[Ml,n,λ(+hj)Ml,n,λ(-hj)Nl,n,λ(+hj)Nl,n,λ(-hj)]Cl,n,λ,jFl,n,λ,j,
Cl,n,λ,jcl,n,λ,jcl,n,λ,j,Fl,n,λ,jfl,n,λ,jfl,n,λ,j
cl,n,λ,0=cl,n,λ,N=fl,n,λ,0=fl,n,λ,N=0.
×M=kN,×N=kM.
P(j, j)Cj+δ(j, s)0M(+hs)tJ
=P(j-1, j)Cj-1+δ(j-1, s)M(-hs)tJ0,
Q(j, j)Fj+δ(j, s)0N(+hs)tJ
=Q(j-1, j)Fj-1+δ(j-1, s)(N(-hs)tJ)0,
P(m, n)(hm exp(ihmzn))(-hm exp(-ihmzn))(exp(ihmzn))(exp(-ihmzn)),
Q(m, n)(km exp(ihmzn))(km exp(-ihmzn))hmkm exp(ihmzn)-hmkm exp(-ihmzn).
Πc(m, n)
Iifm=nP(n, n)-1P(n-1, n)Πc(m, n-1)ifm<nΠc(n, m)-1ifm>n.
Πc(n-1, n)=P(n, n)-1P(n-1, n),
Πc(m, n)=Πc(n-1, n) . . . Πc(m+1, m+2)×Πc(m, m+1).
Cj=Πc(j-1, j)Cj-1+δ(j, s+1)M(-hs)tJ0-δ(j, s)0M(hs)tJ.
Cn=Πc(m, n)Cm.
CN=Πc(0, N)C0+Πc(s, N)(1-δ(s, N))M(-hs)t(δ(s, 0)-1)M(+hs)tJ,
Πc(N, s)CN-Πc(0, s)C0=(1-δ(s, N))M(-hs)t(δ(s, 0)-1)M(+hs)tJ.
[(Πc(N, s))1(-Πc(0, s))2]cNc0
=(1-δ(s, N))M(-hs)t(δ(s, 0)-1)M(+hs)tJ,
Πf(m, n)
Iifm=nQ(n, n)-1Q(n-1, n)Πf(m, n-1)ifm<nΠf(n, m)-1ifm>n.
[(Πf(N, s))1(-Πf(0, s))2]fNf0
=(1-δ(s, N))N(-hs)t(δ(s, 0)-1)N(+hs)tJ;
Cj=P(j, j)-1P(j-1, j)Cj-1=hj+hj-12hj exp(-i(hj-hj-1)zj)hj-hj-12hj exp(-i(hj+hj-1)zj)hj-hj-12hj exp(i(hj+hj-1)zj)hj+hj-12hj exp(i(hj-hj-1)zj)Cj-1.
Cj=11+R j,j-1 [exp(-i(hj-hj-1)zj)R j,j-1exp(-i(hj+hj-1)zj)R j,j-1exp(i(hj+hj-1)zj)exp(i(hj-hj-1)zj)]Cj-1.
dCdz=dhdz -iz-12hexp(-2ihz)2hexp(2ihz)2hiz-12hCordCdz=TcC.
dCdz=TcC+δ(z-zs)M(-h(zs))t-M(+h(zs))tJ,
C(zb)=0c(zb),C(zt)=c(zt)0.
Fj=Q(j, j)-1Q(j-1, j)Fj-1=-12hj -hj-1kjkj-1+hjkj-1kjexp(-i(hj-hj-1)zj)hj-1kjkj-1-hjkj-1kjexp(-i(hj+hj-1)zj)hj-1kjkj-1-hjkj-1kjexp(i(hj+hj-1)zj)-hj-1kjkj-1+hjkj-1kjexp(i(hj-hj-1)zj)Fj-1.
Fj=kj-1kj(1-Rj, j-1)×exp(-i(hj-hj-1)zj)-Rj,j-1 exp(-i(hj+hj-1)zj)-Rj,j-1 exp(i(hj+hj-1)zj)exp(i(hj-hj-1)zj)Fj-1,
Ri,jihj-jhiihj+jhi=ki2hj-kj2hiki2hj+kj2hi.
dFdz=-iz+12h dhdz12h dhdz-1k dkdzexp(-2ihz)12h dhdz-1k dkdzexp(2ihz)iz-12h dhdzFordFdz=TfF.
Tf=Tc-1k dkdz 0exp(-2ihz)exp(2ihz)0.
dFdz=TfF+δ(z-zs)N(-h(zs))t-N(+h(zs))tJ,
F(zb)=0f(zb),F(zt)=f(zt)0.
dΦc(z0, z)dz=Tc(z)Φc(z0, z),Φc(z0, z0)=I.
C(z)=Φc(z0, z)C(z0),
C(z)=Φc(zs, z)Φc(zs, zb)-1C(zb)+z=zbzΦc(zs, z)-1δ(z-zs)×M(-h(zs))t-M(+h(zs))tJdz.
C(z)=Φc(zb, z)C(zb)+Φc(zs, z)H(z-zs)×M(-h(zs))t-M(h(zs))tJ,
H(z)=1ifz00otherwise.
C(z)=Φc(zt, z)C(zt)-Φc(zs, z)H(zs-z)×M(-h(zs))t-M(h(zs))tJ.
C(zt)=Φc(zb, zt)C(zb)+Φc(zs, zt)M(-h(zs))t-M(+h(zs))tJ.
Φc(zt, zs)C(zt)-Φc(zb, zs)C(zb)=M(-h(zs))t-M(+h(zs))tJ.
[(Φc(zt, zs))1(-Φc(zb, zs))2]c(zt)c(zb)
=M(-h(zs))t-M(+h(zs))tJ.
dΦf(z0, z)dz=Tf(z)Φf(z0, z),Φf(z0, z0)=I,
F(z)=Φf(zb, z)F(zb)+Φf(zs, z)H(z-zs)×N(-h(zs))t-N(+h(zs))tJ,
[(Φf(zt, zs))1(-Φf(zb, zs))2]f(zt)f(zb)
=N(-h(zs))t-N(+h(zs))tJ,
C(zb+)=Πc(zb-, zb+)0c(-)+M(-hs)tJ0.
C(zt-)=Φc(zb, zt)C(zb+).
C(+)=Πc(zt-, zt+)C(zt-).
Πc(zb+, zb-)Φc(zt, zb)Πc(zt+, zt-)10-1
×c(+)c(-)=M(-hs)tJ0.
C(z)=Φc(zb, z)Φc(zb, z)Πc(zb-, zb+)M(-hs)tJc(-),
C(z)=Φc(zt, z)Πc(zt+, zt-)c(+)0.
  (Πf(zb+, zb-)Φf(zt, zb)Πf(zt+, zt-))10-1
×f(+)f(-)=N(-hs)tJ0.
  (Πc(zb+, zb-)Φc(zt, zb)Πc(zt+, zt-))10-1
×c(+)c(-)=M(-hs)t-M(hs)tJ.
Φ(a, z)=Φ(a, z)ifz<bΦ(b, z)Π(b-, b+)Φ(a, b)ifz>b.
c(+)c(-)=Φc(+, -)10-1-1M(-hs)tJ-M(hs)tJ,
f(+)f(-)=Φf(+, -)10-1-1N(-hs)tJ-N(hs)tJ.
Re(n(z))1+2H(z),
Im(n(z))18 (z/d)2if0z4d0otherwise,
J(R)=-iω exp(-iωt)pzˆδ(R-zs).
C(-)=C(+)=F(-)=F(+)=0.
C(z)=F(z)=0.
N(±hs)tJ=-iω exp(-iωt)pδ(R-zs)×(λ2/ks)exp(±ihszs).
f(+)f(-)=(Φf)1,1-10(Φf)2,1/(Φf)1,1-1 exp(-ihszs)-exp(ihszs)×-iωp exp(-iωt) λ2ks δ(R-zs).
F0,0,λ(z)=Φf(+, z)f(+)0=exp(-ih(zs)zs)Φf(+, -)1,1 -iωp exp(-iωt)×λ2ks δ(R-zs)Φf(+, z)1.
z^tE(zzˆ)=iμω2(p exp(-iωt))4πk(zs)k(z) λ=0+ λ3 exp(-izsh(zs))hs(λ)Φf(+, -)1,1
×[exp(izh(z))exp(-izh(z))]Φf(+, z)1dλ.
Φf(+, -)=Πf(z1+, z1-)  Πf(zN-1+, zN-1-)×Πf(zN+, zN-),
Φf(+, z)=Πf(zj+1+, zj+1-)  Πf(zN-1+, zN-1-)×Πf(zN+, zN-),
Cj=Tc(Δ, zj)Cj-1,
Tc(Δ, zj)=1+i2 exp(-iωμΔ(1-i)zj)1-i2 exp(-iωμΔ(1+i)zj)1-i2 exp(iωμΔ(1+i)zj)1+i2 exp(iωμΔ(1-i)zj).
limΔ0Tc(Δ, zj)=12 1+i1-i1-i1+iT.
limΔ0Tf(Δ, zj)=12 1+i1-i1-i1+iT,
Fj=Tj(Δ, zj)Fj-1.
ΓG(λ)dλ=0,lima+Γ2G(λ)dλ=0,
λ=0+G(λ)dλ=lima+Γ1G(λ)dλ=-lima+Γ3+Γ4G(λ)dλ =iμ=0-bG(μi)dμ+μ=0+G(μ-bi)dμ.
Cc1c2andFf1f2.
(g(z))×M(h)=g(z)zˆ×M(h)=g(z){Mrφˆ-Mφrˆ}foranyfunctiong(z),
×M(h)=kN(h)+izh{Mrφˆ-Mφrˆ},
×N(h)=k-i kh-hk+izhhkk2M(h),
×(g(z)N(h))=-ihk g(z)+g(z)k-i kh-hk+izhhkk2M(h)foranyg(z),
exp(-2izh)M(h)=M(-h).
×([M(h)M(-h)]C)=k[N(h)N(-h)]C.
×(c1M(h))=(c1 +izhc1){Mr(h)φˆ-Mφ(h)rˆ}+c1kN(h).
×(c1M(h))=h2h (-c1+exp(-2ihz)c2) {Mr(h)φˆ-Mφ(h)rˆ}+c1kN(h).
×(c2M(-h))=h2h (exp(2ihz)c1-c2) {Mr(-h)φˆ-Mφ(-h)rˆ}+c2kN(-h).
×(k[N(h)N(-h)]C)=k2[M(h)M(-h)]C.
×(kc1N(h))=-ihk kc1+k-izh-h2hc1+h2h exp(-2ihz)c2+k-i kh-hk+izhhkk2×kc1M(h).
×(kc2N(-h))=ihk kc2+kizh-h2hc2+h2h exp(2ihz)c1+k-i hk-kh+izhhkk2×kc2M(-h).
×(×([M(h)M(-h)]C))=k2[M(h)M(-h)]C.
×([N(h)N(-h)]F)=k[M(h)M(-h)]F.
×(f1(z)N(h))=-ihk f1(z)M(h)+k-i kh-hk+izhhkk2×f1M(h).
×(f1(z)N(h))=k-ih2k+ihkk2×f1(z)M(h)-ihk h2h-kk×f2M(-h).
×(f2(z)N(-h))=ihk f2(z)M(-h)+k-i -kh+hk+izhhkk2×f2M(-h).
×(f2(z)N(-h))=ihk h2h-kkf1M(h)+k+ih2k-ihkk2f2(z)M(-h).
×(k[M(h)M(-h)]F)=k2[N(h)N(-h)]F.
×(kf1M(h))=(kf1+kf1+kf1izh)×{Mr(h)φˆ-Mφ(h)rˆ}+k2f1N(h).
×(kf1M(h))=k-hk2h(f1-exp(-2ihz)f2)×{Mr(h)φˆ-Mφ(h)rˆ}+k2f1N(h).
×(kf2M(-h))=(kf2+kf2-kf2izh)×{Mr(-h)φˆ-Mφ(-h)rˆ}+k2f2N(-h).
×(kf2M(-h))=k-hk2h(f2-exp(2ihz)f1)×{Mr(-h)φˆ-Mφ(-h)rˆ}+k2f2N(-h).
×(×([N(h)N(-h)]F))=k2[N(h)N(-h)]F.

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