Abstract

Optical activity in the terahertz spectral domain has recently seen a growing interest, but fine understanding of these phenomena is not yet developed. In this article, we study analytically the response of a metallic helix in the terahertz regime and present a full nonlocal calculation of its chiroptical response. Because we do not use multipolar expansion, this calculation is very general and applies to the case where the helix size is comparable to the wavelength of the light. We calculate the circular birefringence and dichroism in three configurations: propagation along or perpendicular to the helix axis and response of an isotropic distribution of such helices. We obtain analytical expressions and can examine the consequence of the breakdown of the multipolar expansion and the wavelength-dependence of the chiroptical response, as well as give orders of magnitude that compare favorably with experimental data. This calculation is also comforted by a finite element calculation.

© 2012 Optical Society of America

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References

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  1. D. Mittleman, Sensing with Terahertz Radiation, Optical Sciences (Springer, 2003).
  2. K. J. Chau, M. C. Quong, and A. Y. Elezzabi, “Terahertz time-domain investigation of axial optical activity from a sub-wavelength helix,” Opt. Express 15, 3557–3567 (2007).
    [CrossRef]
  3. A. Y. Elezzabi and S. Sedergberg, “Optical activity in an artificial chiral media: a terahertz time-domain investigation of Karl F. Lindman’s 1920 pioneering experiment,” Opt. Express 17, 6600–6612 (2009).
    [CrossRef]
  4. K. J. Chau, “Investigation of the chiral origins of electromagnetic activity,” Opt. Lett. 35, 1187–1189 (2010).
    [CrossRef]
  5. L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge University, 2004).
  6. I. Tinoco, “Circular dichroism of large molecules,” Int. J. Quantum Chem. 16, 111–117 (1979).
    [CrossRef]
  7. L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).
  8. D. M. Wood and N. W. Ashcroft, “Quantum size effects in the optical properties of small metallic particles,” Phys. Rev. B 25, 6255–6274 (1982).
    [CrossRef]
  9. D. Moore and I. Tinoco, “The circular dichroism of large helices. A free particle on a helix,” J. Chem. Phys. 72, 3396–3400 (1980).
    [CrossRef]
  10. M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
    [CrossRef]
  11. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).
  12. D. J. Griffiths, Introduction to Electrodynamics (Prentice Hall, 1999).
  13. Handbook of Chemistry and Physics, 76th ed. (CRC Press, 1996).
  14. F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-medited resonance and quantum-size effect,” J. Opt. Soc. Am. B 3, 1647–1655 (1986).
    [CrossRef]
  15. B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B 24, 554–561 (1981).
    [CrossRef]
  16. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  17. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24, 4493–4499 (1985).
    [CrossRef]
  18. E. J. Zeman and G. C. Schatz, “An accurate electromagnetic study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cdv,” J. Phys. Chem. 91, 634–643 (1987).
    [CrossRef]
  19. Note that the sign of Δnc is not made explicit in [4] but it can be inferred from its Fig. 2.
  20. Comsol multiphysics 3.4 (Comsol, Sweden).

2010 (1)

2009 (1)

2007 (2)

K. J. Chau, M. C. Quong, and A. Y. Elezzabi, “Terahertz time-domain investigation of axial optical activity from a sub-wavelength helix,” Opt. Express 15, 3557–3567 (2007).
[CrossRef]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

1987 (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cdv,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

1986 (1)

1985 (1)

1982 (1)

D. M. Wood and N. W. Ashcroft, “Quantum size effects in the optical properties of small metallic particles,” Phys. Rev. B 25, 6255–6274 (1982).
[CrossRef]

1981 (1)

B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B 24, 554–561 (1981).
[CrossRef]

1980 (1)

D. Moore and I. Tinoco, “The circular dichroism of large helices. A free particle on a helix,” J. Chem. Phys. 72, 3396–3400 (1980).
[CrossRef]

1979 (1)

I. Tinoco, “Circular dichroism of large molecules,” Int. J. Quantum Chem. 16, 111–117 (1979).
[CrossRef]

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Alexander, R. W.

Ashcroft, N. W.

D. M. Wood and N. W. Ashcroft, “Quantum size effects in the optical properties of small metallic particles,” Phys. Rev. B 25, 6255–6274 (1982).
[CrossRef]

Barron, L.

L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge University, 2004).

Bell, R. J.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

Chau, K. J.

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Dasgupta, B. B.

B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B 24, 554–561 (1981).
[CrossRef]

Elezzabi, A. Y.

Flytzanis, C.

Fuchs, R.

B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B 24, 554–561 (1981).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics (Prentice Hall, 1999).

Hache, F.

Landau, L. D.

L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Lifchitz, E. M.

L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

Long, L. L.

Mittleman, D.

D. Mittleman, Sensing with Terahertz Radiation, Optical Sciences (Springer, 2003).

Moore, D.

D. Moore and I. Tinoco, “The circular dichroism of large helices. A free particle on a helix,” J. Chem. Phys. 72, 3396–3400 (1980).
[CrossRef]

Ordal, M. A.

Querry, M. R.

Quong, M. C.

Ricard, D.

Schatz, G. C.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cdv,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Sedergberg, S.

Silveirinha, M. G.

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

Tinoco, I.

D. Moore and I. Tinoco, “The circular dichroism of large helices. A free particle on a helix,” J. Chem. Phys. 72, 3396–3400 (1980).
[CrossRef]

I. Tinoco, “Circular dichroism of large molecules,” Int. J. Quantum Chem. 16, 111–117 (1979).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

Wood, D. M.

D. M. Wood and N. W. Ashcroft, “Quantum size effects in the optical properties of small metallic particles,” Phys. Rev. B 25, 6255–6274 (1982).
[CrossRef]

Zeman, E. J.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cdv,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Appl. Opt. (1)

Int. J. Quantum Chem. (1)

I. Tinoco, “Circular dichroism of large molecules,” Int. J. Quantum Chem. 16, 111–117 (1979).
[CrossRef]

J. Chem. Phys. (1)

D. Moore and I. Tinoco, “The circular dichroism of large helices. A free particle on a helix,” J. Chem. Phys. 72, 3396–3400 (1980).
[CrossRef]

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

J. Phys. Chem. (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic study of surface enhancement factors for Ag, Au, Cu, Li, Na, Al, Ga, In, Zn, and Cdv,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (3)

D. M. Wood and N. W. Ashcroft, “Quantum size effects in the optical properties of small metallic particles,” Phys. Rev. B 25, 6255–6274 (1982).
[CrossRef]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

B. B. Dasgupta and R. Fuchs, “Polarizability of a small sphere including nonlocal effects,” Phys. Rev. B 24, 554–561 (1981).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other (8)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, 1997).

D. J. Griffiths, Introduction to Electrodynamics (Prentice Hall, 1999).

Handbook of Chemistry and Physics, 76th ed. (CRC Press, 1996).

Note that the sign of Δnc is not made explicit in [4] but it can be inferred from its Fig. 2.

Comsol multiphysics 3.4 (Comsol, Sweden).

D. Mittleman, Sensing with Terahertz Radiation, Optical Sciences (Springer, 2003).

L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).

L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge University, 2004).

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Figures (5)

Fig. 1.
Fig. 1.

Sketch of the metallic helix. The framework x,y,z is attached to the helix. The inset represents a top view of the helix. The four outlined points (1), (1), (2), (2) are useful for the discussion.

Fig. 2.
Fig. 2.

Functions F (dashed line) and F=2F (solid line) as a function of the parameter ka.

Fig. 3.
Fig. 3.

Circular birefringence in arbitrary units as a function of the frequency for helices whose axis is perpendicular (dotted line) and parallel (dashed line) to the beam propagation and for an isotropic distribution of helices (solid line). Note that the isotropic signal goes to 0 when the frequency becomes very small, corresponding to the validity regime of the usual multipolar expansion.

Fig. 4.
Fig. 4.

Circular dichroism in arbitrary units as a function of the frequency for helices whose axis is perpendicular (dotted line) and parallel (dashed line) to the beam propagation and for an isotropic distribution of helices (solid line). Note that the isotropic signal goes to 0 when the frequency becomes very small, corresponding to the validity regime of the usual multipolar expansion.

Fig. 5.
Fig. 5.

FEM simulations of the optical activity in a metallic helix in the terahertz domain. Color map provides the value of the electric field along the x axis, for right (A) and left (B) incident circular polarizations. The color scale is identical for both polarizations. The red arrows show the electric field direction at the center axis of the helix.

Equations (70)

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H^=H^0+H^I,
H^I=e2mec(p^·A+A·p^).
j^=e2me[n^(r)p^c+p^cn^(r)]
j=Tr[ρ^j^].
j0=e2mecTr[ρ^(0)n^]A.
j1=e2meTr[ρ^(1)(n^p^+p^n^)].
j0(r,t)=e2mecm,nn|ρ^(0)|mm|n^A|n=e2mecA0eiωtmfmm|eik·r^n^(r)|m.
fm=1exp(EmEFkBT)+1.
g(k)=1Ωf(r)eikrdr.
j0(k,ω)=e2ΩmecωA0mfmm|ei(kk)·r^|m.
j0(k,ω)=Ne2ΩmecA0,
σ(ω)=iNe2Ωme(ω+iΓ),
ε(ω)=1ωp2ω(ω+iΓ)withωp2=Ne2Ωmeε0,
d=1πfμσ,
iħρ^t=[H^,ρ^].
iħρ^(1)t=[H^I,ρ^(0)]+[H^0,ρ^(1)].
n|ρ^(1)|m=fnfmħ(ωnmωiΓ)n|H^I|m,
n|H^I|m=e2mecA0·[n|eik·r^p^|m+n|p^eik·r^|m]eiωt,
j1(r,t)=e2mem,nn|ρ^(1)|mm|n^(r)p^+p^n^(r)|n.
j1(r,t)=e22me2cm,nfnfmħ(ωnmωiΓ)A0·[n|eik·r^p^|m+n|p^eik·r^|m]×[m|n^(r)p^|n+m|p^n^(r)|n]eiωt.
σ1̲̲(k,ω)=ie22Ωme2ωm,nfnfmħ(ωnmωiΓ)[m|eik·r^p^|n+m|p^eik·r^|n][n|eik·r^p^|m+n|p^eik·r^|m]·
ε1̲̲(k,ω)=iε0ωσ1̲̲(k,ω)·
(εYYεYZεZYεZZ).
ε±=12[(εYY+εZZ)±i(εYZεZY)]εiso±Δε.
Δnc=(Δε)n0,
Δα=2ω(Δε)n0c,
x=acosθ,y=asinθ,z=bθ.
ψl=1Kπsinlθ
El=l2E0withE0=ħ22meL2.
px=iħaL2[sinθθ+12cosθ],py=iħaL2[cosθθ12sinθ],pz=iħbL2θ.
Anm=n|eikyp^x|m,Bnm=n|eikyp^z|m,Cnm=[Anm+(A˜mn)*][Bmn+(B˜nm)*],
εzx=e24ε0Ωme2ω2m,nfnfmħ(ωnmωiΓ)Cnm.
εzx=e24ε0ħΩme2ω2m,n(fmfn)Cnmωnm(ω+iΓ)2ωnm2.
εzx=e24ε0ħΩme2ω21(ω+iΓ)2m,n(fmfn)Cnmωnm.
εzx=iωp2EFn02mec1ω(ω+iΓ)2a2bL4F(ka).
F(x)=4xloddl2Jl(x)Jl(x).
εzx=iωp2EFn02mec1ω(ω+iΓ)2a2bL4.
Dnm=n|eikzp^x|m,Enm=n|eikzp^y|m,Fnm=[Dnm+(D˜mn)*][Emn+(E˜nm)*],
εyx=e24ε0ħΩme2ω21(ω+iΓ)2m,n(fmfn)Fnmωnm.
εyx=iωp2EFn02mec1ω(ω+iΓ)2a2bL4.
Δε=i2(εYZεZY)=i2εzx,
Δε=i2(εXYεYX)=iεyx.
Δε,=ωp2EFn02mec1ω(ω+iΓ)2a2bL4F,(ka),
Δnc,=ωp2EF2meca2bL4F,(ka)ω2Γ2ω(ω2+Γ2)2,
Δα,=ωp2EF2mec2a2bL4F,(ka)ωΓ(ω2+Γ2)2.
Δε=23Δε+13Δε,
Δε=ωp2EFn06mec1ω(ω+iΓ)2a2bL4[1F].
Anm=n|eikyp^x|m·
Anm=iħaL21Kπ02Kπsinnθeikasinθ[sinθθ+12cosθ]sinmθdθ=iħa8L2[(2m1)(Fn+m1Fnm+1)+(2m+1)(Fnm1Fn+m+1)],
FM=1Kπ02KπeikasinθcosMθ.
eizsinθ=n=+einθJn(z),
Anm=iħa4L2[2m[Jnm1(ka)Jnm+1(ka)]+[Jnm1(ka)+Jnm+1(ka)]]=iħaL2[mJnm(ka)+nm2kaJnm(ka)],
Anm+(A˜mn)*=iħaL2(n+m)Jnm(ka).
Bnm=n|eikyp^z|m=iħbL21Kπ02Kπsinnθeikasinθθsinmθdθ=iħbL2m2(Gn+m+Gnm),
GM=1Kπ02KπeikasinθsinMθ·
Bnm=ħbL2mJnm(ka)
Bnm+(B˜mn)*=ħbL2(n+m)Jnm(ka).
Cnm=iħ2abL4(n+m)2Jnm(ka)Jnm(ka)
IC=m,n(fmfn)Cnmωnm·
ωnm=(n2m2)E02mFl,
IC=iħ2abL4E0×8mF3m(fmfm+l)loddlJl(ka)Jl(ka).
m(fmfm+l)=lmfm=4Kl,
IC=i2Nħ2kabL4EFF(ka)
Dnm=n|eikzp^x|m=iħaL21Kπ02Kπsinnθikbθ[sinθθ+12cosθ]sinmθdθ=iħa8L2[(2m1)(Hn+m1Hnm+1)+(2m+1)(Hnm1Hn+m+1)],
HM=1Kπ02KπeikbθcosMθdθ=12Kπ02Kπ[ei(kb+M)θ+ei(kbM)θ]dθ.
Enm=n|eikzp^y|m=iħaL21Kπ02Kπsinnθikbθ[cosθθ12sinθ]sinmθdθ=iħa8L2[(2m1)(In+m1+Inm+1)+(2m+1)(Inm1+In+m+1)],
IM=1Kπ02KπeikbθsinMθdθ=12iKπ02Kπ[ei(kb+M)θei(kbM)θ]dθ.
Fnm=i(ħaL2)2116(n+m)2[δ(l+kb+1)δ(lkb+1)δ(l+kb1)+δ(lkb1)].
IF=iħ2a216L4E0×8mF3m(fmfm+l)l[δ(l+kb+1)δ(lkb+1)δ(l+kb1)+δ(lkb1)].
IF=2iNħ2ka2bL4EF.

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