Abstract

Lorenz–Mie multiple-scattering theory is used to perform semi-analytical calculations of the lossy dispersion relations of propagating modes in infinite chains of metallic spheres. Lossy modes are described by allowing the projection of the wavevector along the chain axis to be a complex number rather than the more common complex frequency description. We show that even when the constituent particles are much smaller than the wavelength, one generally needs to go well beyond the coupled dipole approximation to achieve stable predictions.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” Opt. Lett. 23, 1331–1333 (1998).
    [CrossRef]
  2. M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
    [CrossRef]
  3. D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
    [CrossRef]
  4. W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
    [CrossRef]
  5. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
    [CrossRef]
  6. M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
    [CrossRef]
  7. M. Conforti and M. Guasoni, “Dispersive properties of linear chains of lossy metal nanoparticles,” J. Opt. Soc. Am. B 27, 1576–1582 (2010).
    [CrossRef]
  8. S. M. Raeis, Z. Bajestani, M. Shahabadi, and N. Talebi, “Analysis of plasmon propagation along a chain of metal nanospheres using the generalized multipole technique,” J. Opt. Soc. Am. B 28, 937–943 (2011).
    [CrossRef]
  9. B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nanoantennas design: applications to Yagi-Uda configurations,” J. Opt. Soc. Am. B 28, 1213–1223 (2011).
    [CrossRef]
  10. I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex omega approach versus complex-k approach,” Phys. Rev. B 83, 115451 (2011).
    [CrossRef]
  11. A. Andrea and E. Nader, “Guided propagation along quadrupolar chains of plasmonic nanoparticles,” Phys. Rev. B 79, 235412 (2009).
    [CrossRef]
  12. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  13. W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
    [CrossRef]
  14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  15. I. Abramowitz and M. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).
  16. C. M. Linton and I. Thompson, “One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation,” J. Comput. Phys. 228, 1815–1829 (2009).
    [CrossRef]
  17. L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley1985).
  18. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1990).
  19. B. Stout, J. Auger, and A. Devilez, “Recursive T matrix algorithm for resonant multiple scattering: applications to localized plasmon excitations,” J. Opt. Soc. Am. A 25, 2549–2557(2008).
    [CrossRef]
  20. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  21. B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
    [CrossRef]
  22. S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
    [CrossRef]
  23. K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C 112, 15721–15728 (2008).
    [CrossRef]

2011 (3)

2010 (1)

2009 (2)

C. M. Linton and I. Thompson, “One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation,” J. Comput. Phys. 228, 1815–1829 (2009).
[CrossRef]

A. Andrea and E. Nader, “Guided propagation along quadrupolar chains of plasmonic nanoparticles,” Phys. Rev. B 79, 235412 (2009).
[CrossRef]

2008 (2)

B. Stout, J. Auger, and A. Devilez, “Recursive T matrix algorithm for resonant multiple scattering: applications to localized plasmon excitations,” J. Opt. Soc. Am. A 25, 2549–2557(2008).
[CrossRef]

K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C 112, 15721–15728 (2008).
[CrossRef]

2006 (2)

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
[CrossRef]

2004 (2)

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
[CrossRef]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

2002 (1)

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

2001 (1)

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

2000 (1)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

1998 (1)

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Abramowitz, I.

I. Abramowitz and M. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Albaladejo, S.

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
[CrossRef]

Andrea, A.

A. Andrea and E. Nader, “Guided propagation along quadrupolar chains of plasmonic nanoparticles,” Phys. Rev. B 79, 235412 (2009).
[CrossRef]

Atwater, H. A.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Auger, J.

Auger, J.-C.

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

Aussenegg, F. R.

Bajestani, Z.

Bohren, C. F.

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

Bonod, N.

Botten, L. C.

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

Brongersma, M. L.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1990).

Citrin, D. S.

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
[CrossRef]

Conforti, M.

Devilez, A.

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

Enoch, S.

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Gómez-Medina, R.

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
[CrossRef]

Guasoni, M.

Hartman, J. W.

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Koenderink, A. F.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

Kong, J. A.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley1985).

Krenn, J. R.

Lafait, J.

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

Laroche, M.

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
[CrossRef]

Leitner, A.

Linton, C. M.

C. M. Linton and I. Thompson, “One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation,” J. Comput. Phys. 228, 1815–1829 (2009).
[CrossRef]

McPhedran, R. C.

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Nader, E.

A. Andrea and E. Nader, “Guided propagation along quadrupolar chains of plasmonic nanoparticles,” Phys. Rev. B 79, 235412 (2009).
[CrossRef]

Nicorovici, N.

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

Nixon, J. N.

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

Polman, A.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

Premaratne, M.

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex omega approach versus complex-k approach,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

Quinten, M.

Raeis, S. M.

Rolly, B.

Rukhlenko, I. D.

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex omega approach versus complex-k approach,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

Senz, J. J.

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
[CrossRef]

Shahabadi, M.

Shin, R. T.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley1985).

Stegun, M.

I. Abramowitz and M. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Stout, B.

Talebi, N.

Tanabe, K.

K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C 112, 15721–15728 (2008).
[CrossRef]

Thompson, I.

C. M. Linton and I. Thompson, “One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation,” J. Comput. Phys. 228, 1815–1829 (2009).
[CrossRef]

Tsang, L.

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley1985).

Udagedara, I. B.

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex omega approach versus complex-k approach,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

J. Comput. Phys. (1)

C. M. Linton and I. Thompson, “One- and two-dimensional lattice sums for the three-dimensional Helmholtz equation,” J. Comput. Phys. 228, 1815–1829 (2009).
[CrossRef]

J. Math. Phys. (1)

S. Enoch, R. C. McPhedran, N. Nicorovici, L. C. Botten, and J. N. Nixon, “Sums of spherical waves for lattices, layers and lines,” J. Math. Phys. 42, 5859–5870 (2001).
[CrossRef]

J. Mod. Opt (1)

B. Stout, J.-C. Auger, and J. Lafait, “A transfer matrix approach to local field calculations in multiple scattering problems,” J. Mod. Opt. 49, 2129–2152 (2002).
[CrossRef]

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

J. Opt. Soc. Am. B (3)

J. Phys. Chem. C (1)

K. Tanabe, “Field enhancement around metal nanoparticles and nanoshells: a systematic investigation,” J. Phys. Chem. C 112, 15721–15728 (2008).
[CrossRef]

Nano Lett. (1)

D. S. Citrin, “Coherent excitation transport in metal-nanoparticle chains,” Nano Lett. 4, 1561–1565 (2004).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (7)

M. L. Brongersma, J. W. Hartman, and H. A. Atwater, “Electromagnetic energy transfer and switching in nanoparticle chain arrays below the diffraction limit,” Phys. Rev. B 62, R16356 (2000).
[CrossRef]

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852–9858 (1989).
[CrossRef]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70, 125429 (2004).
[CrossRef]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B 74, 033402 (2006).
[CrossRef]

M. Laroche, S. Albaladejo, R. Gómez-Medina, and J. J. Senz, “Tuning the optical response of nanocylinder arrays: an analytical study” Phys. Rev. B 74, 245422 (2006).
[CrossRef]

I. B. Udagedara, I. D. Rukhlenko, and M. Premaratne, “Complex omega approach versus complex-k approach,” Phys. Rev. B 83, 115451 (2011).
[CrossRef]

A. Andrea and E. Nader, “Guided propagation along quadrupolar chains of plasmonic nanoparticles,” Phys. Rev. B 79, 235412 (2009).
[CrossRef]

Other (5)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

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

I. Abramowitz and M. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing (Wiley1985).

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE, 1990).

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.


Figures (4)

Fig. 1.
Fig. 1.

Real part of the dispersion relations in the dipole approximation (dashed curves), and fully converged multipole calculations with nmax=10 (full lines). The longitudinal mode with positive imaginary part is in cyan (gray), and the “T1” mode with positive imaginary part is in blue (black line). The “T2” transverse mode with negative imaginary part is in orange (gray).

Fig. 2.
Fig. 2.

Imaginary part (b) of the dispersion relations with the dipole approximation (dashed curves), and with fully converged multipole calculations nmax=10 (full lines). The longitudinal mode with positive imaginary part is in cyan (gray), while the “T1” mode with positive imaginary part is in blue (black line). The “T2” transverse mode with negative imaginary part is in orange (gray).

Fig. 3.
Fig. 3.

Normalized extinction is a solid blue line and the scattering cross section is given by a dashed green line of a silver monomer in terms of frequency (a=25nm).

Fig. 4.
Fig. 4.

Positive and imaginary parts of the dispersion relations in the Re[β]>0 part of the Brillouin zone. Transverse modes with Im[β]>0 modes are solid blue (black) lines, while that with Im[β]<0 is given by a dashed blue (black) line. Longitudinal modes with Im[β]>0 are solid cyan (gray) lines, while that with Im[β]<0 is a dashed cyan (gray) line.

Equations (43)

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

Es(r,ω)=eikr4πεbϵ0r3{(1ikr)[3r^(r^·p)p]+k2r2((r^×p)×r^)},
p(ω)=ϵ0εbα(ω)Ee(ω).
α=6πik3t1(e)=i6πk3a1,
α(ω)4πa3α˜(ω).
Ee,=Ei,(ad)3α˜j*ei(|j|kd+jβd)|j|3(1i|j|kd(jkd)2)Ee,,Ee,=Ei,+(ad)3α˜j*2ei(|j|kd+jβd)|j|3(1i|j|kd)Ee,,
Ee=Ei+(ka)3α˜j*η(j,kd,βd)Ee,
η(j,kd,βd)1(kd)3(1|j|3ikd1j2(kd)21|j|)ei(|j|kd+jβd),η(j,kd,βd)2(kd)3(1|j|3ikd1j2)ei(|j|kd+jβd).
Σ(ω,β)(ka)3j*η(j,kd,βd),
Ee=Ei+Σ(ω,β)α˜Ee.
α˜Ee=Eiα˜1Σ(ω,β),
α˜1Σ(ω,β)=0,
Lin(z)=j=1zjjn,
Σ(ω,β,d)=(ad)3{[Li3(ei(kβ)d)+Li3(ei(k+β)d)]ikd[Li2(ei(kβ)d)+Li2(ei(k+β)d)](kd)2[Li1(ei(kβ)d)+Li1(ei(k+β)d)]}Σ(ω,β,d)=2(ad)3{[Li3(ei(kβ)d)+Li3(ei(k+β)d)]ikd[Li2(ei(kβ)d)+Li2(ei(k+β)d)]}.
ΔHn,m+k2Hn,m=0,
Hn,m(r)hn(kr)Yn,m(r^)n=0,1,;m=n,,n,
η(j,kd,βd)=13i[h2(kd|j|)2h0(kd|j|)]eijβd,η(j,kd,βd)=23i[h2(kd|j|)+h0(kd|j|)]eijβd.
ln(ω,β)j*Hn,0(jkdz^)eijβd=2n+14πj=1hn(jkd)[eijβd+(1)neijβd],
Σ(ω,β,d)=i(ka)323π[2l0(β)15l2(β)],Σ(ω,β,d)=i(ka)323π[2l0(β)+25l2(β)].
××ΨH,q,pk2ΨH,q,p=0.
ΨH,1,p(kr)×[rHp(kr)]n(n+1),ΨH,2,p(kr)×[ΨH,1,p(kr)]k,
Ei(r)=E0q=12p=1ΨJ,q,p(kr)aq,pE0ΨJt(kr)a,
f(j)=te(j),
f(j)Ta(j),
T=t+tΩ(ω,β)T,
Ω(ω,β)j=1[eijβdH(jkdz^)+eijβdH(jkdz^)],
eijβd[H(jkdz^)]q,p;qp+eijβd[H(jkdz^)]q,p;q,p=lCl(p,q;p,q)2l+14πhl(jkd)[eijβd+(1)neijβd],
[Ω]q,p;qp=lCl(p,q;p,q)ll(ω,β).
T=[t1Ω]1,
[t1Ω]ν=0.
α˜123i(ka)3[t1(e)]2,1,m;2,1,m1,Σ(ω,β)23i(ka)3[Ω(ω,β)]2,1,m;2,1,m.
ω¯ωd2πc=dλv,β¯βd2π,
hn(jkd)=p=0n{(n(pInt(p2)))!Int(p2)!(np)!(2n12Int(p2))!!(2Int(p+12)1)!!(i)p+1(kd)n+1p[(eikd)jjn+1p]},
h0(jkd)=ijkdeijkd,h1(jkd)=eijkd(1jkdi(jkd)2),h2(jkd)=eijkd(ijkd3(jkd)23i(jkd)3).
ln(β)=2n+14πp=0n[((i)p+1(n(pInt(p2)))!Int(p2)!(np)!(2n12Int(p2))!!(2Int(p+12)1)!!)(Lin+1pexp[i(k+β)d]+()nLin+1pexp[i(kβ)d])(kd)n+1p].
[ψ]q,q,p,p=δq,qδp,pψn(kR),[ξ]q,q,p,p=δq,qδp,pξn(kR),
a¯[ψ]a,f¯[ξ]f
t¯[ξ]t[ψ]1,T¯[ξ]T[ψ]1,Ω¯[ψ]Ω[ξ]1.
f¯(j)T¯a¯(j),
T¯a¯=t¯a¯+t¯Ω¯T¯a¯.
T¯=[t¯1Ω¯]1.
{ε(AG)}={{ε1(AG)}200nmλ<310nm,{ε2(AG)}310λ2000nm,{ε(AG)}={{ε1(AG)}200λ<330nm,{ε2(AG)}330λ2000nm,
{ε1(AG)}=1.308415×1011λ6+1.764343×108λ59.761668×106λ4+0.002832725λ30.4538023λ2+37.94213λ1288.348{ε1(AG)}=3.636188×1011λ65.443344×108λ5+3.365273×105λ41.100094×102λ3+2.005786λ2193.4021λ+7.706263×103
{ε2(AG)}=2.037181×1017λ6+1.183540×1013λ52.537882×1010λ4+2.430043×107λ31.420089×104λ2+8.990214×104λ+8.526028{ε2(AG)}=2.327098×1017λ6+1.471828×1013λ53.635520×1010λ4+4.530857×107λ32.946733×104λ2+9.56229×102λ11.49465.

Metrics