Abstract

The problem of a nonchiral object in an isotropic medium is analyzed through wave-field decomposition and treatment of the chiral medium as a combination of two nonchiral media. Integral equations for the internal wave fields of the object are formed, suitable for numerical computation. For an object with small contrast, solutions are found through the Born approximation. As a special case, a spherical object is considered and expressions for the scattered far fields are found in explicit analytical form. As numerical examples, certain measurable polarization quantities are presented together with their interpretation. The theory is applicable, for example, to the analysis of data obtained from the measurement of the glucose content of blood in terms of light scattered from blood cells.

© 1997 Optical Society of America

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References

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  1. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, London, 1982), pp. 14–18.
  2. M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
    [CrossRef]
  3. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [CrossRef]
  4. S. Bassiri, C. H. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric–chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
    [CrossRef]
  5. M. P. Silverman and J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).
    [CrossRef]
  6. M. P. Silverman and J. Badoz, “Interferometric enhancement of chiral asymmetrics: ellipsometry with an optically active Fabry–Perot interferometer,” J. Opt. Soc. Am. A 11, 1894–1917 (1994).
    [CrossRef]
  7. M. P. Silverman, J. Badoz, and B. Briat, “Chiral reflection from a naturally optically active medium,” Opt. Lett. 17, 886–889 (1992).
    [CrossRef] [PubMed]
  8. J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. 31, 6535–6546 (1992).
    [CrossRef] [PubMed]
  9. M. P. Silverman, W. Strange, J. Badoz, and A. Vitkin, “Rotation in turbid chiral fluids,” Bull. Am. Phys. Soc. 41, 705 (1996).
  10. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  11. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994).
  12. I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
    [CrossRef]
  13. J. Van Bladel, “Some remarks on Green's dyadic for infinite space,” IRE Trans. Antennas Propag. 9, 563–566 (1961).
  14. A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  15. I. V. Lindell, Methods in Electromagnetic Field Analysis (Clarendon, Oxford, 1992).
  16. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 544–555.
  17. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), pp. 325–326.
  18. For details of elliptical polarization see M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), pp. 25–28. Note, however, that the ellipsometric angles of Figure 1.6 (page 13) are labeled differently from those of our paper.
  19. M. P. Silverman, N. Ritchie, G. M. Cushman, and B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. A 5, 1852–1862 (1988).
    [CrossRef]
  20. J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of light beam using birefringence modulator,” J. Opt. 8, 373–384 (1977).
  21. M. P. Silverman, More Than One Mystery: Explorations in Quantum Interference (Springer, New York, 1995).

1996 (1)

M. P. Silverman, W. Strange, J. Badoz, and A. Vitkin, “Rotation in turbid chiral fluids,” Bull. Am. Phys. Soc. 41, 705 (1996).

1995 (1)

I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
[CrossRef]

1994 (1)

1992 (2)

1988 (2)

1986 (1)

1985 (1)

M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

1980 (1)

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1974 (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Badoz, J.

Bassiri, S.

Bohren, C. F.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Bonner, R. F.

Briat, B.

Cushman, G. M.

Engheta, N.

Fisher, B.

Gandjbakhche, A. H.

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
[CrossRef]

Papas, C. H.

Puska, P. P.

I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
[CrossRef]

Ritchie, N.

Ruotanen, L. H.

I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
[CrossRef]

Schmitt, J. M.

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
[CrossRef]

Silverman, M. P.

Strange, W.

M. P. Silverman, W. Strange, J. Badoz, and A. Vitkin, “Rotation in turbid chiral fluids,” Bull. Am. Phys. Soc. 41, 705 (1996).

Vitkin, A.

M. P. Silverman, W. Strange, J. Badoz, and A. Vitkin, “Rotation in turbid chiral fluids,” Bull. Am. Phys. Soc. 41, 705 (1996).

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Appl. Opt. (1)

Bull. Am. Phys. Soc. (1)

M. P. Silverman, W. Strange, J. Badoz, and A. Vitkin, “Rotation in turbid chiral fluids,” Bull. Am. Phys. Soc. 41, 705 (1996).

Chem. Phys. Lett. (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

J. Opt. Soc. Am. A (4)

Lett. Nuovo Cimento (1)

M. P. Silverman, “Specular light scattering from a chiral medium,” Lett. Nuovo Cimento 43, 378–382 (1985).
[CrossRef]

Microwave Opt. Tech. Lett. (1)

I. V. Lindell, A. H. Sihvola, P. P. Puska, and L. H. Ruotanen, “Conditions for the parameter dyadics of lossless bi-anisotropic media,” Microwave Opt. Tech. Lett. 8, 268–272 (1995).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green's functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Other (10)

I. V. Lindell, Methods in Electromagnetic Field Analysis (Clarendon, Oxford, 1992).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 544–555.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), pp. 325–326.

For details of elliptical polarization see M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, New York, 1970), pp. 25–28. Note, however, that the ellipsometric angles of Figure 1.6 (page 13) are labeled differently from those of our paper.

J. Van Bladel, “Some remarks on Green's dyadic for infinite space,” IRE Trans. Antennas Propag. 9, 563–566 (1961).

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge U. Press, London, 1982), pp. 14–18.

M. P. Silverman and J. Badoz, “Multiple reflection from isotropic chiral media and the enhancement of chiral asymmetry,” J. Electromagn. Waves Appl. 6, 587–601 (1992).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994).

J. Badoz, M. Billardon, J. C. Canit, and M. F. Russel, “Sensitive devices to determine the state and degree of polarization of light beam using birefringence modulator,” J. Opt. 8, 373–384 (1977).

M. P. Silverman, More Than One Mystery: Explorations in Quantum Interference (Springer, New York, 1995).

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Figures (9)

Fig. 1
Fig. 1

Geometry of the problem. The incident plane wave is decomposed into two wave fields with right-hand (E+i) and left-hand (E-i) circular polarizations with respect to the direction of propagation. Each of the wave fields sees the chiral medium as an effective nonchiral medium of its own. The scatterer can be replaced by equivalent sources that scatter the wave-field components E+s and E-s.

Fig. 2
Fig. 2

Definition of the basis vectors in the two scattering planes: (a) yz plane, (b) xz plane. For forward scattering (θ=0°) the basis is reduced to (u x, u y, u z).

Fig. 3
Fig. 3

Degree of circular polarization, τL(45) for circularly polarized incident field, varying scattering angles θ=0° to 180°, different size parameters ka=1, 3, and 5, and parameter values s=1.43, μs=1, and κr=0.1.

Fig. 4
Fig. 4

Same as Fig. 3, but for degree of linear polarization τL.

Fig. 5
Fig. 5

Same as Fig. 3, but for circular intensity difference δC.

Fig. 6
Fig. 6

Degree of circular polarization τL(45) for linearly polarized incident field, varying scattering angles θ=0° to 180°, different size parameters ka=1, 3, and 5, and parameter values s= 1.43, μs=1, and κr=0.1.

Fig. 7
Fig. 7

Same as Fig. 6, but for degree of linear polarization τL.

Fig. 8
Fig. 8

Effect of small chirality parameter κr on the degree of linear polarization τL(+) for a medium with ka=10.

Fig. 9
Fig. 9

Effect of large chirality parameter κr on degrees of linear and circular polarization τL(+) and τC(+), respectively, for a medium with ka=1. The values of κr are a: 0.1; b: 0.3; c: 0.7.

Equations (105)

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D=E-j kωκrH,
B=μH+j kωκrE.
E+=12(E-jηH),E-=12(E+jηH),
H+=12H-1jηE,H-=12H+1jηE,
η=μ.
H+=jηE+,H-=-jηE-.
+=(1+κr),-=(1-κr),
μ+=μ(1+κr),μ-=μ(1-κr).
p(E)=u·E×E*jE·E*,
p(E+)=+1,p(E-)=-1.
k+=ωμ++=k(1+κr),
k-=ωμ--=k(1-κr),
×E=-jωμsμH=-jωμH+kκrE-Jm,
×H=jωsE=jωE+kκrH+Je,
Jm=[jωμ(μs-1)H+kκrE]PV(r),
Je=[jω(s-1)E-kκrH]PV(r).
Je+=12Je+1jηJm=jω2s-+E+1jημsμ-μ+HPVr,
Je-=12Je-1jηJm=jω2s--E-1jημsμ-μ-HPVr,
Jm+=12(Jm+jηJe)=jω2[(μsμ-μ+)H+jη(s-+)E]PV(r),
Jm-=12(Jm-jηJe)=jω2[(μsμ-μ-)H-jη(s--)E]PV(r).
Je+Je-
=jω2s+μs-21+κrs-μss-μss+μs-21-κr
×E+E-PVr,
Jm+Jm-
=jωμ2μs+s-21+κrμs-s μs-sμs+s-21-κr
×H+H-PVr.
Jm±=±jηJe±,
G±er=I+1k±2G±|r|,
G±m(r)=-G±(|r|)×I,
G±(|r|)=exp(-jk±|r|)4π|r|.
G±e(r-r)=PVG±e(r-r)-1k±2Lδ(r-r),
E±(r)=E±i(r)+E±s(r).
G±(r)=G±e(r)±1k±G±m(r)=(I±1k±×I+1k±2)G±(|r|),
E±s(r)=-jωμ±PV V G±(r-r)·Je±(r)dV-13jω±Je±(r).
Es(r)=E+s(r)+E-s(r)
E+sE-sr=-jηV k+G+r-r00k-G-r-r·Je+rJe-rdV,
E±s(r)
=kk±2V {[s+μs-2(1±κr)]G±(r-r)·E±(r)
+(s-μs)G±(r-r)·E(r)}dV.
G±|r-r|-jk±usG±|r-r|,
G±|r-r|-k±2ususG±|r-r|,
G±(|r-r|)exp(jk±us·r)G±(r),
G±(|r-r|)T± exp(jk±us·r)G±(r),
T±=I±jus×I-usus.
c±·c±=0,us·c±=0.
E±s(usr)
kk±2G±(r)T±·{[s+μs-2(1±κr)]P±(us)
+(s-μs)Q±(us)},
P±(us)=V exp(jk±us·r)E±(r)dV,
Q±(us)=V exp(jk±us·r)E±(r)dV.
Ei(r)=E+i exp(-jk+z)+E-i exp(-jk-z),
Hi(r)=H+i exp(-jk+z)+H-i exp(-jk-z),
P±(us)=V exp(jk±us·r)E±i(r)dV=E±i  exp[j(k±us-k±uz)·r]dV,
Q±(us)=V exp(jk±us·r)Ei(r)dV=Ei exp[j(k±us-kuz)·r]dV.
V exp(jk·r)dV=F(ka)V,k=|k|,V=4πa33,
F(x)=3x3(sin x-x cos x).
P±(us)=F(p±)VE±i,p±=2k±a sin(θ/2),
Q±(us)=F(q)VEi,
q=2kasin2(θ/2)+κr2 cos2(θ/2),
E+sr, θE-sr, θ=k2V2G+r00G-r×s+μs-21+κr1+κrFp+T+s-μs1-κrFqT-s-μs1+κrFqT+s+μs-21-κr1-κrFp-T-·E=iE-i.
E±i=uxjuy2E±i=u±iE±i,
E±s=12(uxsjuys)E±s=u±sE±s,
E+sE-s=S++S-+S+-S--E+iE-i.
Sξζ=k2VGξ(r)αξζ=k2V exp(-jkξr)4πrαξζ,
uzs=uy sin θ+uz cos θ,uys=uy cos θ-uz sin θ,
uy=uzs sin θ+uys cos θ,uz=uzs cos θ-uys sin θ.
T±·E±i=u±s2E±i cos2(θ/2),
T±·Ei=u±s2Ei sin2(θ/2),
u±s=12(uxsjuys).
u±s·u±s=0,u±s·u±s*=1,u+s×u-s=juzs.
uzs=ux sin θ+uz cos θ,uxs=ux cos θ-uz sin θ,
ux=uzs sin θ+uxs cos θ,uz=uzs cos θ-uxs sin θ.
T±·E±i=u±s2E±i cos2(θ/2),
T±·Ei=-u±s2Ei sin2(θ/2),
α++yz=α++xz=[s+μs-2(1+κr)]×(1+κr)F(p+)cos2(θ/2),
α+-yz=-α+-xz=(s-μs)(1+κr)F(q)sin2(θ/2),
α-+yz=-α-+xz=(s-μs)(1-κr)F(q)sin2(θ/2),
α--yz=α--xz=[s+μs-2(1-κr)]×(1-κr)F(p-)cos2(θ/2).
AR=tan ψ.
τC=|E+s|2-|E-s|2|E+s|2+|E-s|2,
Es=u+sE+s+u-sE-s.
τC=uzs·Es×Es*jEs·Es*,
τC=VI.
τC=sin(2ψ).
τL(0)=|Exs|2-|Eys|2|Exs|2+|Eys|2,
Es=uxsExs+uysEys.
τL=[τL(0)]2+[τL(45)]2
τL=|Es·Es|Es·Es*.
τL=U2+Q2I.
τL=a2-b2a2+b2=cos(2ψ),
tan(2ϕ)=τL(45)τL(0)=UQ
cos ϕ=us·ui.
δC=I+-I-I++I-,
Es=u+sE+s+u-sE-s
E+s=S++E+i+S+-E-i,
E-s=S-+E+i+S--E-i,
τC=|S++E+s+S+-E-s|2-|S-+E+s+S--E-s|2|S++E+s+S+-E-s|2+|S-+E+s+S--E-s|2,
τL=2|(S++E+s+S+-E-s)(S-+E+s+S--E-s)||S++E+s+S+-E-s|2+|S-+E+s+S--E-s|2.
τC(+)=α++2-α-+2α++2+α-+2,
τL(+)=2α++α-+α++2+α-+2,
τC(x)=(α+++α+-)2-(α-++α--)2(α+++α+-)2+(α-++α--)2,
τL(x)=2(α+++α+-)(α-++α--)(α+++α+-)2+(α-++α--)2.
δC=α++2-α+-2+α-+2-α--2α++2+α+-2+α-+2+α--2.
ϕ=12(k+-k-)r=π(n+-n-)rλ=κrkr
δC=(1-κr)2-(1+κr)2(1-κr)2+(1+κr)2-2κr,|κr|1.

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