Abstract

We present an exact calculation for the scattering of light from a single sphere made of a Faraday-active material into first order of the external magnetic field. When the size of the sphere is small compared with the wavelength, the known T matrix for a magneto-active Rayleigh scatterer is found. We address the issue of whether there is a so-called photonic Hall effect—a magneto-transverse anisotropy in light scattering—for one Mie scatterer. In the limit of geometrical optics, we compare our results with the Faraday effect in a Fabry–Perot etalon.

© 1998 Optical Society of America

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References

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  1. F. A. Erbacher, R. Lenke, G. Maret, “Multiple light scattering in magneto-optically active media,” Europhys. Lett. 21, 551–556 (1993).
    [CrossRef]
  2. G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature (London) 381, 54–55 (1996).
    [CrossRef]
  3. A. Sparenberg, G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of photonic magneto-resistance,” Phys. Rev. Lett. 79, 757–760 (1997).
    [CrossRef]
  4. F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
    [CrossRef]
  5. B. A. van Tiggelen, R. Maynard, T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
    [CrossRef]
  6. R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1982).
  7. L. Landau, E. Lifchitz, Quantum Mechanics (Mir, Moscow, 1967).
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1980).
  9. A. Bott, W. Zdunkowski, “Electromagnetic energy within dielectric spheres,” J. Opt. Soc. Am. A 4, 1361–1365 (1987).
    [CrossRef]
  10. H. Y. Ling, “Theoretical investigation of transmission through a Faraday-active Fabry–Perot étalon,” J. Opt. Soc. Am. A 11, 754–758 (1994).
    [CrossRef]
  11. R. Rosenberg, C. B. Rubinstein, D. R. Herriott, “Resonant optical Faraday rotator,” Appl. Opt. 3, 1079–1083 (1964).
    [CrossRef]
  12. V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
    [CrossRef] [PubMed]

1997 (1)

A. Sparenberg, G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of photonic magneto-resistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[CrossRef]

1996 (2)

G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature (London) 381, 54–55 (1996).
[CrossRef]

B. A. van Tiggelen, R. Maynard, T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[CrossRef]

1995 (1)

V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

F. A. Erbacher, R. Lenke, G. Maret, “Multiple light scattering in magneto-optically active media,” Europhys. Lett. 21, 551–556 (1993).
[CrossRef]

1988 (1)

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[CrossRef]

1987 (1)

1964 (1)

Bott, A.

Cuevas, E.

V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
[CrossRef] [PubMed]

Erbacher, F. A.

F. A. Erbacher, R. Lenke, G. Maret, “Multiple light scattering in magneto-optically active media,” Europhys. Lett. 21, 551–556 (1993).
[CrossRef]

Gasparian, V.

V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
[CrossRef] [PubMed]

Herriott, D. R.

John, S.

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[CrossRef]

Landau, L.

L. Landau, E. Lifchitz, Quantum Mechanics (Mir, Moscow, 1967).

Lenke, R.

F. A. Erbacher, R. Lenke, G. Maret, “Multiple light scattering in magneto-optically active media,” Europhys. Lett. 21, 551–556 (1993).
[CrossRef]

Lifchitz, E.

L. Landau, E. Lifchitz, Quantum Mechanics (Mir, Moscow, 1967).

Ling, H. Y.

MacKintosh, F. C.

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[CrossRef]

Maret, G.

F. A. Erbacher, R. Lenke, G. Maret, “Multiple light scattering in magneto-optically active media,” Europhys. Lett. 21, 551–556 (1993).
[CrossRef]

Maynard, R.

B. A. van Tiggelen, R. Maynard, T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[CrossRef]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1982).

Nieuwenhuizen, T. M.

B. A. van Tiggelen, R. Maynard, T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[CrossRef]

Ortuño, M.

V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
[CrossRef] [PubMed]

Rikken, G. L. J. A.

A. Sparenberg, G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of photonic magneto-resistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[CrossRef]

G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature (London) 381, 54–55 (1996).
[CrossRef]

Rosenberg, R.

Rubinstein, C. B.

Ruiz, J.

V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
[CrossRef] [PubMed]

Sparenberg, A.

A. Sparenberg, G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of photonic magneto-resistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1980).

van Tiggelen, B. A.

A. Sparenberg, G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of photonic magneto-resistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[CrossRef]

G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature (London) 381, 54–55 (1996).
[CrossRef]

B. A. van Tiggelen, R. Maynard, T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[CrossRef]

Zdunkowski, W.

Appl. Opt. (1)

Europhys. Lett. (1)

F. A. Erbacher, R. Lenke, G. Maret, “Multiple light scattering in magneto-optically active media,” Europhys. Lett. 21, 551–556 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nature (London) (1)

G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of magnetically induced transverse diffusion of light,” Nature (London) 381, 54–55 (1996).
[CrossRef]

Phys. Rev. B (1)

F. C. MacKintosh, S. John, “Coherent backscattering of light in the presence of time-reversal-noninvariant and parity-nonconserving media,” Phys. Rev. B 37, 1884–1897 (1988).
[CrossRef]

Phys. Rev. E (1)

B. A. van Tiggelen, R. Maynard, T. M. Nieuwenhuizen, “Theory for multiple light scattering from Rayleigh scatterers in magnetic fields,” Phys. Rev. E 53, 2881–2908 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

A. Sparenberg, G. L. J. A. Rikken, B. A. van Tiggelen, “Observation of photonic magneto-resistance,” Phys. Rev. Lett. 79, 757–760 (1997).
[CrossRef]

V. Gasparian, M. Ortuño, J. Ruiz, E. Cuevas, “Faraday rotation and complex-valued traversal time for classical light waves,” Phys. Rev. Lett. 75, 2312 (1995).
[CrossRef] [PubMed]

Other (3)

R. G. Newton, Scattering Theory of Waves and Particles (Springer-Verlag, New York, 1982).

L. Landau, E. Lifchitz, Quantum Mechanics (Mir, Moscow, 1967).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1980).

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Figures (7)

Fig. 1
Fig. 1

Schematic view of the magneto-scattering geometry. Generally, θ denotes the angle between incident and outgoing wave vectors; ϕ is the azymuthal angle in the plane of the magnetic field and the y axis. The latter is by construction the magneto-transverse direction defined as the direction perpendicular to both the magnetic field and the incident wave vector. Angle α coincides with angle θ in the special but relevant case that the incident vector is normal to the magnetic field.

Fig. 2
Fig. 2

Real part (solid curve) and imaginary part (dashed curve) of the magneto-forward scattering matrix Tkk1 in the circular basis of polarization, plotted versus size parameter x for index of refraction m=1.33 in units of W=V0Bλ.

Fig. 3
Fig. 3

Magneto-transverse scattering cross section F(θ) for a Rayleigh scatterer with index of refraction m=1.1 and size parameter x=0.1. Solid curve, positive correction; points, negative correction. The curve has been normalized by the parameter W. No net magneto-transverse scattering is expected in this case because the projections onto the y axis of these corrections cancel one another. Axis numbers in the format 4e-06 are equivalent to 4×10-6.

Fig. 4
Fig. 4

Magneto-transverse scattering cross section F(θ) for a Mie scatterer of size parameter x=5 and of index of refraction m=1.1. The curve has been normalized by the parameter W. Solid curve, positive correction; points, negative correction. In this case a net magneto-transverse scattering is expected because the projections onto the y axis of these corrections do not cancel one another.

Fig. 5
Fig. 5

Normalized magneto-transverse light current η as a function of size parameter x for an index of refraction m =1.0946. The curve is displayed in units of W.

Fig. 6
Fig. 6

Magneto-cross section for two Rayleigh scatterers each of size parameter ka=0.1 and separated by a distance corresponding to size parameter kr12=5. In this case the enhanced forward scattering leads also to a net magneto-transverse current along the vertical axis. Axis numbers in the format 4e-06 are equivalent to 4×10-6.

Fig. 7
Fig. 7

Magnetically induced change of phase δϕ—similar to Fabry–Perot modes of a cavity—as a function of size parameter x for the partial wave of J=1, the one with central impact. The curve is for m=10 and has been normalized by the value 2aV0B.

Equations (49)

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ε(B, r)-I=[(ε0-1)I+εFΦ]ϴ(|r|-a).
T(B, r, ω)=V(r, ω)+V(r, ω)·G0·V(r, ω)+.
T1=(1-ε0)εFϴT0·Φ·(I+G0·T0).
|Ψσ,k±=(I+G0±·T0)|σ,k.
Ψσ,-k-*(r)=(-1)1+σΨ-σ,k+(r).
Tkσ,kσ1=εFω2Ψσ,k-|ϴΦ|Ψσ,k+.
T-k-σ,-k-σ(B)=Tkσ,kσ(B),
T-kσ,-kσ(-B)=Tkσ,kσ(B).
Ψσ,k+(r)=2πρiJ+1YJMλ(rˆ)fλλJ(r)YJMλ*(kˆ)·χσ.
fJ(ρ)=m-iuJ(ρ)cJ00-iuJ(ρ)cJ[J(J+1)]1/2/ρ0000uJ(ρ)dJ.
YJ1M1λ1*(rˆ)·Φ·YJ2M2λ2(rˆ)dΩr
=δJ1J2δM1M2Qλ1λ2(J1, M1),
Q(J, M)=-MB1J(J+1)1J(J+1)01J(J+1)00001J(J+1)
Tk,k1=16πωWJ,M(-M)[CJYJ,Me(kˆ)YJ,Me*(kˆ)+DJYJ,Mm(kˆ)YJ,Mm*(kˆ)],
W=V0Bλ
CJ=-2cJ2*|uJ|2y3J(J+1)(y2-y*2)AJ*y*-AJy,
DJ=-2dJ2*|uJ|2y3J(J+1)(y2-y*2)AJ*y-AJy*,
CJ=-cJ2*uJ2yJ(J+1)AJy-J(J+1)y2+1+AJ2,
DJ=-dJ2*uJ2yJ(J+1)-AJy-J(J+1)y2+1+AJ2.
Tσσ0=2πiωJ1 2J+1J(J+1)(aJ*+σσbJ*)×[πJ,1(cos θ)+σστJ,1(cos θ)].
πJ,M(cos θ)=Msin θPJM(cos θ),
τJ,M(cos θ)=ddθPJM(cos θ).
Tkk1=(Bˆ·kˆ)(kˆ·kˆ)-Bˆ·kˆ(kˆ·kˆ)2-1TBˆ=kˆ1+(Bˆ·kˆ)(kˆ·kˆ)-Bˆ·kˆ(kˆ·kˆ)2-1TBˆ=kˆ1+(Bˆ·gˆ)TBˆ=gˆ1,
Tσσ1(Bˆ=kˆ)=2WωJ1 2J+1J(J+1)(-σ)(CJ+σσDJ)×[πJ,1(cos θ)+σστJ,1(cos θ)],
Tσσ1(Bˆ=kˆ)=2WωJ1 2J+1J(J+1)(-σ)(CJ+σσDJ)×[πJ,1(cos θ)+σστJ,1(cos θ)],
Tσσ1(Bˆ=gˆ)=4iWωJ1JM>0 2J+1J(J+1)M sin(Mθ) ×(J-M)!(J+M)!(σσCJ+DJ)×[πJ,M2(0)+σστJ,M2(0)].
Tkσ,kσ1=δσσ(Bˆ·kˆ)(-σ) 2WωJ1(2J+1)(CJ+DJ).
-Im(Tkσ,kσ)ω=σdΩk |Tkσ,kσ|2(4π)2.
18π2σdΩk Re(Tkσ,kσ0Tkσ,kσ1*).
d(cos θ)[πJ(cos θ)τK(cos θ)+τJ(cos θ)πK(cos θ)]
=0,
d(cos θ)[πJ(cos θ)πK(cos θ)+τJ(cos θ)τK(cos θ)]
=2J2(J+1)22J+1δJK.
Re[aJ*cJ2(2/i)]=Im(cJ2),
Re[bJ*dJ2(2/i)]=Im(dJ2).
2σσ Re(Tσσ0, Tσσ1*)02πdϕ0πd cos θσσ|Tσσ0|2=-sin ϕF(θ).
ηIup(B)-Idown(B)Iup(B=0)+Idown(B=0)=20πdϕ0πd cos θ sin θ sin ϕσσ Re(Tσσ0Tσσ1*)0πdϕ0πd cos θ sin θ sin ϕσσ|Tσσ0|2.
Tk,k=t0kˆ·kˆ+it1Bˆ·(kˆ×kˆ)it1Bˆ·kˆ-it1Bˆ·kˆt0,
FRayleigh(θ)=3mx34π2(m2+2)2cos θ sin θ.
ηV0Bkx3sin(kr12)kr122,ifkr121.
c1=2 exp[i(x-y)](m+1)[1+r exp(-2iy)],
d1=2 exp[i(x-y)](m+1)[1-r exp(-2iy)].
T=Tscatt0+T1Tscatt0 exp(T1/Tscatt0),
ImT1Tscatt0σσ=δϕ(-σ)Bˆ·kˆδσσ,
δϕ=2aV0B 1+R1-R1[1+M sin(2y)2],
A(y)=11+M sin(2y)2,
δϕ=V0B(τc0/m),
τdwellmax=(1+m2)a/c0
τdwellmin=4m2/(1+m2)a/c0.

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