VOLUME 84, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 14 FEBRUARY 2000

New Effects in Light Scattering in Disordered Media and Coherent Backscattering Cone:Systems of Magnetic Particles

F. A. Pinheiro,1,* A. S. Martinez,2 and L. C. Sampaio1

1Centro Brasileiro de Pesquisas Físicas�CNPq, Rua Dr. Xavier Sigaud, 150, Urca, CEP 22290-180, Rio de Janeiro, Rio deJaneiro, Brazil

2Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Avenida Bandeirantes 3900, CEP14040-901, Ribeirão Preto, São Paulo, Brazil

(Received 8 October 1999)

Single and multiple scattering of light by magnetic particles and their implications to the coherentbackscattering effect are reported. Single scattering of light by small magnetic particles presents unusualfeatures such as forward-backward asymmetry and resonance effects. In multiple scattering, this leadsto a global decrease in the localization parameter k��, which exhibits an oscillatory dependence on thescatterer magnetic permeability. By considering magnetic scatterers following a Curie-Weiss suscepti-bility law, we suggest that k�� can be tuned by varying the temperature.

PACS numbers: 42.25.Dd

In recent years, there has been a growing interest inthe field of multiple scattering of light in disordered me-dia. This fact was initially motivated by the realizationthat many electronic effects have their analog in multiplelight scattering systems [1–6]. In particular, the coherentbackscattering effect [3], the optical counterpart of elec-tronic weak localization, was the first evidence of the im-portant role played by interference effects in multiple lightscattering systems. Coherent backscattering manifests it-self by an enhancement, in the backward direction, of theintensity of light scattered by multiple scattering medium.This enhancement is essentially due to the constructive in-terference between the direct and reversed light paths [7].However, in all multiple light scattering phenomena re-ported so far, including the destruction of the coherentbackscattering effect by magneto-optical active materials[8,9], it is assumed that the scatterers are nonmagnetic.

When a disordered medium of identical spherical mag-netic scatterers of radius a and magnetic permeability ms

is illuminated by light with wavelength l, the localiza-tion parameter k�� depends on ms, where k � 2p�l and�� � ���1 2 �cosu�� is the transport mean free path. Theanisotropy factor �cosu� and the photon mean free path� � 1��Fs� (where F is the number density of scatterersand s is the total scattering cross section) are functions ofthe Mie coefficients, which depend on ms [10]. The widthof the backscattering cone is proportional to �k���21 andconsequently also depends on ms.

In the present Letter, we report on investigations of thelocalization parameter of light in the presence of magneticscatterers. In the small-particle limit (ka ø 1), we showboth analytically and numerically that (i) in certain rangesof ms, each single scattering event is predominantly back-wards, which leads to a reduction of the transport mean freepath, thus the localization parameter; and (ii) resonancesmay appear, which leads to an oscillatory behavior of k��

as a function of ms. The intermediate-particle limit ka � 1and large-particle limit ka ¿ 1 are treated numerically.

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Exploring the fact that the magnetic permeability of mag-netic scatterers follows a typical Curie-Weiss law near thecritical temperature, we suggest that the localization pa-rameter can be controlled by simply varying the tempera-ture of the disordered medium.

To investigate the influence of magnetic scatterers onthe localization parameter, it is essential to consider thesingle scattering of light by magnetic particles. In a singleMie scattering problem, a plane wave with wavelength l

is scattered by a homogeneous magnetic sphere of radiusa, complex magnetic permeability ms, and complex elec-tric permittivity es, leading to a complex refraction indexns �

pesms. The medium has a real electric permittivity

em (it does not absorb nor emit light) and vacuum mag-netic permeability m0, which leads to a real refractive in-dex nm �

pemm0. The relative refractive index is m �

ns�nm �p

me, where e � e0 1 ıe00 and m � m0 1 ım00

are the complex relative electric permittivity and the com-plex relative magnetic permeability, respectively. Boththe scatterers and the medium are considered to be non-magneto-optical active.

Although in standard texts on light scattering [11–13]the relative magnetic permeability m is usually assumed tobe unitary, this is no longer valid for magnetic particles.In ferromagnetic materials m can assume a rather largespectrum of values. For instance, for a nickel bulk at roomtemperature, m is of the order of 106 [14].

The quantities of interest in single scattering are the totalcross section s and anisotropy factor �cosu�, which canbe expressed in terms of the Mie coefficients an and bn

[11–13]. For magnetic scatterers, an and bn were derivedby Kerker [10]. In the small-particle limit, we have, in thelowest order in ka,

a1 �2ı�ka�3

3kac

01�mka� 2 2m̃c1�mka�

kac01�mka� 1 m̃c1�mka�

, (1a)

b1 �2ı�ka�3

3m̃kac

01�mka� 2 2c1�mka�

m̃kac01�mka� 1 c1�mka�

, (1b)

© 2000 The American Physical Society 1435

VOLUME 84, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 14 FEBRUARY 2000

where m̃ � m�m �p

e�m and cn�z� and c 0n�z� are the

Ricatti-Bessel function and its derivative with respect tothe argument, respectively. It is important to stress thefact that the small-particle limit does not mean pointlikescatterers.

To obtain the Rayleigh limit, one has also to ensure thatthe size of the scatterer is small compared to the wave-length inside the particle (jmjka ø 1), which leads toa1 � g��1 2 e���2 1 e�� and b1 � g��1 2 m���2 1 m��,where g � 22ı�ka�3�3 [10]. Notice that all the depen-dence on m is in the b1 term, which vanishes in the non-magnetic case. In this case, the leading term in the bn

expansion is of the order of �ka�5. The anisotropy factor toterms of lowest order in ka is �cosu� � Re�a1b�

1���ja1j2 1

jb1j2�. It is important to notice that in magnetic scattering

�cosu� does not vanish, in contrast with the nonmagne-tic case.

However, when jmj ¿ 1, which is exactly the case offerromagnetic materials, the small-particle limit must betreated in a different form. In this case, the limit jmjka ¿ 1must be considered using cn�z� cos�z 2 �n 1 1��2p�in Eqs. (1), leading to

a1 �2ı�ka�3

3p�m, ka� 1 2m̃p�m, ka� 2 m̃

, (2a)

b1 �2ı�ka�3

3m̃p�m, ka� 1 2m̃p�m, ka� 2 1

, (2b)

where p�m, ka� � ka tan�mka�. Different from the Ray-leigh limit, the a1 term also has a magnetic contribution forlarge values of jmj. The periodic function tan�mka�, whichdepends on m, is responsible for resonances in �cosu�.Furthermore, these oscillations are due to the real part ofm, since tan�ıx� � ı tanh�x� which is nonperiodic.

To analyze in more detail the problem of single magneticscattering, we numerically calculate the value of �cosu� asa function of the size parameter ka for dielectric (e00 � 0)scatterers for three different values of the real part of therelative magnetic permeability m0, as is exhibited in Fig. 1.The numerical code has been adapted from Ref. [15] toinclude the magnetic contribution. In the small-particlelimit, single scattering is isotropic (�cosu� � 0) for non-magnetic scatterers (m0 � 1). The appearance of usualMie resonances in �cosu� is observed as ka increases. Thissituation changes drastically when magnetic scatterers areconsidered. Magnetic single scattering is characterized bythe nonvanishing value of �cosu� even in the small-particlelimit, as it can be seen by m0 � 2 and m0 � 100 curves(Fig. 1). Furthermore, we observe the presence of reso-nance phenomena for large values of m in the scatteringanisotropy for the small-particle limit, which is absent inthe nonmagnetic case. This is in agreement with our ana-lytical result [see Eqs. (2) and comments that follow].

The consequences of the introduction of magnetic scat-terers in the single scattering problem in the small-particlelimit can be analyzed in detail in Fig. 2, where �cosu� is

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FIG. 1. The scattering anisotropy factor �cosu�, plotted as afunction of the size parameter ka for three different values ofthe real part of the relative magnetic permeability: nonmagneticcase m0 � 1 (dashed curve), m0 � 2 (dotted curve), and m0 �100 (solid curve). The other parameters used are e0 � 1.4161,e00 � 0, and m00 � 0.

plotted as a function of both the real (m0) and imaginary(m00) parts of m for ka 0.63. The scattering anisotropyhas an oscillatory dependence on m0, confirming the pres-ence of resonance phenomena. As m0 increases, the realpart of the refractive index m also increases. A large valueof m leads to the building up of standing waves inside thescatterer, which are responsible for the observed resonancephenomena. On the other hand, for a pure imaginarymagnetic permeability, �cosu� is a very slowly varyingfunction of m00 (Fig. 2). We also numerically obtain thelimiting value �cosu� � 20.3 when m00 tends towards in-finity. This means that a high level of magnetic dissipationof energy contributes only with a constant negative valueto �cosu� in the small-particle limit.

The fact that �cosu� can assume negative values means apredominant backward scattering. This is quite unusual in

FIG. 2. �cosu� plotted as a function of both the real m0 (solidcurve) and the imaginary m00 (dashed curve) parts of m in thesmall-particle limit ka 0.63 and in the large-particle limitka 63 (inset). The relative dielectric constant is e � e0 �1.4161.

VOLUME 84, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 14 FEBRUARY 2000

single light scattering problems by nonmagnetic scatterers.The reason for the appearance of a preferential backscat-tering configuration in the small-particle limit is the sig-nificant contribution given by magnetic dipole radiation inmagnetic scattering [10]. We also observe negative valuesof �cosu� for some values of m0 (see Fig. 2). However,in this case, a more striking feature is observed: not only�cosu� can be negative, but it can also oscillate betweenpositive and negative values. This means that, by varyingm0, we can change a preferential forward scattering con-figuration to a preferential backward one. In the light ofthe coherent backscattering effect, this leads to interestingfeatures in the angular profile of the backscattering cone.In the inset of Fig. 2 �cosu� is plotted as a function of m0

and m00 for a size parameter ka � 63. We observe the typi-cal Mie resonances for the real magnetic permeability case,as expected, but the scattering is always preferentially inthe forward direction, i.e., �cosu� . 0.

In the following, using the above results concerningsingle magnetic scattering, we numerically calculate1��k���, which is proportional to the width of the back-scattering cone, in a magnetic medium with typicalconcentration F � 0.01 and illuminated with light ofwavelength l � 633 nm. We consider only the scalaraspect of light propagation. In Fig. 3 we show the depen-dence of 1��k��� on both m0 and m00 in the small-particlelimit for dielectric scatterers. Notice the global decreaseof the localization parameter induced by the presenceof magnetic �m fi 1� scatterers. Furthermore, while m00

gives an almost constant contribution to the cone width,m0 contributes to a fast oscillation. It is also importantto point out the wide spectrum of values involved in thisoscillation: the cone width can vary over 3 orders ofmagnitude when jmj varies from 1 to 100. This interestingpattern of the angular profile of the backscattering coneis essentially due to the fact that the scattering anisotropy

FIG. 3. The inverse of the localization parameter 1��k���, pro-portional to the backscattering cone width, plotted as a func-tion of both the real m0 (solid curve) and the imaginary m00

(dashed curve) parts of m in the small-particle limit ka 0.63for e � e0 � 1.4161. The light wavelength is l � 633 nm andthe concentration of scatterers F � 0.01.

can oscillate between positive and negative values as m

varies.The oscillatory behavior for real values of m is also

present in the large-particle limit (see Fig. 4). Here, we ob-serve not only a global decrease in the localization parame-ter, when compared with the nonmagnetic case, but also thefact that k�� decreases as both m0 and m00 increases until itachieves a saturation minimum value. In particular, for m0

and typical parameters cited above (F � 0.01 and l �633 nm), this value is approximately k�� 3.1, whichsuggests further investigations on the onset of “strong” lo-calization of light in disordered magnetic media.

Let us now briefly discuss possible situations where thepredictions of our model could be experimentally verified.The basic idea is to exploit the temperature dependenceof the relative magnetic permeability. The magnetic sus-ceptibility (related to the relative permeability by m �1 1 x) of ferromagnetic materials above the critical tem-perature follows a typical Curie-Weiss temperature depen-dence x ~ �T 2 Tc�21 [14]. Thus, a precise tuning of therelative magnetic permeability m can be performed in thistemperature range. This alters the scattering anisotropyfactor and consequently the backscattering cone width. Asa result, the utilization of magnetic scatterers followingthe Curie-Weiss susceptibility law offers the possibility totune the value of the localization parameter k�� by vary-ing the sample temperature. An interesting experimentalaspect in the large-particle limit is that, as the temperatureapproaches the Curie-Weiss critical temperature, one canset k�� to a minimum value.

Another interesting application of our model is to con-sider the case of recurrent multiple scattering. In this case,as argued by Wiersma et al. [16], the enhancement factorof the backscattering cone for strong scattering systems isnot exactly 2, but depends on the value of k��. The factthat one can, by considering the Curie-Weiss susceptibil-ity of magnetic scatterers, tune the value of k��, makes our

FIG. 4. The inverse of the localization parameter 1��k���, plot-ted as a function of both the real m0 (solid curve) and theimaginary m00 (dashed curve) parts of m in the large particlelimit ka 63 for e � e0 � 1.4161. The light wavelength isl � 633 nm and the concentration of scatterers F � 0.01.

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model extremely promising in the sense of controlling theenhancement factor as well.

Finally, it is important to point out that the diffusion con-stant D � �1�3�yE�� will be strongly affected in the pres-ence of magnetic scatterers, not only through �� but alsothrough the “transport velocity” yE . The fact that singlescattering by ferromagnetic particles exhibits a characteris-tic resonant behavior, even in the small-particle limit, willintroduce an extra time delay in light propagation, causinga decrease of yE if compared with the nonmagnetic case[17]. This point, as well as polarization effects, is underinvestigation and will be reported soon.

In conclusion, we have studied the influence of mag-netic scatterers on single and multiple scattering of light,and their implications in the coherent backscattering effectand localization parameter. We have shown that, in thesmall-particle limit, single scattering exhibits not only aforward-backward spatial asymmetry, but also an unusualresonance effect. We have observed an oscillatory depen-dence of the backscattering cone width on the relative mag-netic permeability m, as well as a global decrease of thelocalization parameter induced by the presence of magneticscatterers. Furthermore, considering magnetic scatterersfollowing the Curie-Weiss susceptibility law, we have sug-gested a manner to tune the localization parameter k�� byvarying the temperature.

We thank I. S. Oliveira and L. G. Guimarães for fruitfuldiscussions. This work was supported by the Brazilianagencies CAPES and CNPq.

*Author to whom correspondence should be addressed.Electronic address: felipe@cbpf.br

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