Longitudinal eld-induced polarized light transmittance of magnetic uids

Shengli Pu , Min Dai, Guoqing SunCollege of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

a b s t r a c ta r t i c l e i n f o

Article history:Received 30 November 2009Received in revised form 5 June 2010Accepted 7 June 2010

Keywords:Magnetic uidFaraday effectGeometric shadowing effectMagneto-optical devices

The complete optical transmittance for a polarized light passing through the magnetic uids is investigatedtheoretically and experimentally, when the externally magnetic eld is applied along the propagationdirection of the incident light. Hybrid effects due to the geometric shadowing and Faraday rotation areconsidered simultaneously. The Langevin-like functions are employed to describe the magnetic-eld-dependent volume concentration of the particle-aggregation () and the approximate number of magneticnanoparticles in the particle-aggregation (N0). Based on the experiments on the geometric shadowing effectof our magnetic uid sample, the analytical expression for the total transmitted power with externallymagnetic eld after an analyzer is derived. Theoretical simulations disclose the inuence of certain criticalparameters of the magnetic uids on the eld-dependent optical transmittance. For the entire polarized lighttransmittance, qualitative agreement between the calculations and the experiments is achieved. Applicationsof magnetic uids to several polarized devices operating in longitudinal eld arrangement are proposed anddiscussed. The results presented in this work may be useful for designing the corresponding magnetic-uid-based optical devices.

2010 Elsevier B.V. All rights reserved.

1. Introduction

Magnetic uids are two-phase stable systems of magneticallyultrane particles dispersed in suitable liquid base carriers, whichpossess both the uidity of liquids and the magnetism of solidmagnetic materials [1]. These peculiar properties imply that magneticuids are newfashioned functional materials with nanosized struc-ture. The applications of magnetic uids to machinery have beenexploited since the invention of magnetic uids and thecorresponding products have been commercialized nowadays. Theconventional optical properties of magnetic uids, e.g. linearbirefringence, linear dichroism and Faraday rotation, have beeninvestigated for several decades [211]. The recently boomingdevelopment of nanostructured materials and the optical informationtechnology evoke the renewed study of the late-model opticalproperties of magnetic uids such as magnetically tunable opticalscattering [1215], eld dependent refractive index [16] and opticaltransmission [1723], clustering of nanoparticles [24,25], tunablemagnetic photonic crystals [2632], etc. Several potential magnetic-uid-based optical devices have been proposed and demonstrated byresearchers [3338], which signify that the nanostructured magneticuids are promising optical functional materials.

The magnetic-eld-induced optical transmittance of magneticuids can be classied into two cases: the transverse eld opticaltransmittance and the longitudinal optical transmittance. The formercase means that the magnetic eld is perpendicular to the propaga-tion direction of the incident light while the latter one implies that themagnetic eld parallels the propagation direction of the incident light.For the transverse eld arrangement, the variations of light transmit-tance with magnetic eld are mainly due to the magnetic nanopar-ticles aggregation and the associated microstructure-induced opticalanisotropy of the magnetic uids. These aspects have been investi-gated extensively and intensively. For the longitudinal eld cong-uration, the magnetic-eld-dependent optical transmittance is chieyassigned to the column/chain formation of magnetic particles alongthe magnetic eld direction and the related geometric shadowingeffect [2022,39]. If the polarized light is applied for the longitudinaleld conguration, not only the Faraday rotation but also thegeometric shadowing effect will happen. Therefore, the hybrid effectsof Faraday rotation and geometric shadowing should be consideredsimultaneously when the magnetic uids are used for the systems oflongitudinal eld polarized light. To the best of our knowledge, all ofthe present studies have neglected either Faraday rotation orgeometric shadowing effect of the magnetic uids, which will resultin considerable errors for some situations. In this work, the theoryand experiments about the optical transmittance contributed tothe hybrid effects have been conducted and the applications tocorrelated optical devices have been discussed. The results in thiswork may be helpful for designing some polarized optical devices

Optics Communications 283 (2010) 40124016

Corresponding author. Tel.: +86 21 65666454; fax: +86 21 65667144.E-mail address: shlpu@usst.edu.cn (S. Pu).

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with high precision based on the longitudinal eld effects of magneticuids.

2. Theory

The longitudinal eld optical transmittance of magnetic uids dueto geometric shadowing effect will increase with the strength of theexternally applied magnetic eld. The involved physical mechanismshave been elaborated by Li et al. [22,4042]. Lately, Helseth hasestablished an equilibrium theory to dene this kind of opticaltransmittance quantitatively [39]. According to Helseth's theory, themagnetic-eld-dependent transmitted power of the magnetic uidsdue to geometric shadowing effect can be expressed as

P = PmaxP exp CH2

; 1

where P=PmaxPmin is the difference between the maximumtransmitted power (saturation power Pmax) when the magnetic eldis sufciently high and the one Pmin at zero eld, H is the externallymagnetic eld strength and C is a variable that depends on thetemperature of the magnetic uids, the initial particle-congurationand density. For the longitudinal eld system used for polarized light,the accompanying Faraday rotation should be considered and thenthe transmitted power after the analyzer is given by

P = PmaxP exp CH2

h i t cos2 ; 2

where t is the transmittivity of the analyzer, which is usually less thanunity because of the Fresnel reection and absorption of the analyzer. is the included angle between the polarization direction of the incidentlight and that of the analyzer (see Fig. 1). is the Faraday rotation angleof linearly polarized light passing through the magnetic uids and isfound to be [5]

=2 d = xyL

1 + Q 2 liquidp1 + P 1 + QL

=

q ; 3

where d is the thickness of the magnetic uids, is the wavelengthof the incident light in vacuum, xy is the off-diagonal term of thedielectric tensor of the magnetic uids, is the volume concentrationof the particle-aggregation, Q=solid/liquid and P = solidliquid

=

solid + liquid

. solid and liquid are the dielectric constants of theparticle-aggregation and the remanent liquid phase within themagnetic uids, respectively. L

= coth

1= and =

Heff/kT. k is the Boltzmann constant and T is the absolute temperatureof the magnetic uids. Heff=H+Hw is the effective eld within the

magnetic uids and Hw is the Weiss internal eld representingthe interparticle interactions. =0VMd is the dipole moment of theparticle-aggregation. 0 is the permeability in vacuum. Md is thesaturation magnetization of the magnetic nanoparticles. V=N0V0 isthe volume of the particle-aggregation, where is a constant of theorder of unity b1 [2], N0 is the number of magnetic nanoparticles inthe particle-aggregation, V0 = 4r

3= 3

and r are the volume and

radius of the single magnetic nanoparticle, respectively.It is well-known that the magnetic nanoparticles within the

magnetic uids will agglomerate into aggregation when the exter-nally magnetic eld is applied. Moreover, the stronger the externallymagnetic eld, the more the particles take part in the agglomeration.Then, both the volume concentration of the particle-aggregation ( )and the number of magnetic nanoparticles in the particle-aggregation(N0) become larger. So the eld-dependent and N0 should beincorporated into Eq. (3). The properties of magnetic uids aremostly related with the magnetic nanoparticles agglomeration andcan usually be described by the Langevin function, for example eld-dependent magnetization and refractive index [1,43]. Therefore, wesuppose that the variations in and N0 (approximately equals N0)withmagnetic eld H should be similar to Langevin functions, and canbe expressed as = sL and N0 = N0 sL . Here, the s andN0 s are the saturated values of and N0, respectively. =dH /kTand d=0V0Md is the dipole moment of the single magneticnanoparticle.

3. Experimental details

Themagnetic uid we utilize for experimental investigation in thispaper is oil-based ferrite magnetic uid with saturation magnetiza-tion of 100 Oe and viscosity of 9 mPa s, which is provided by FerrotecCorporation. The average diameter of the magnetic nanoparticles isabout 10 nm. The sample cell is composed of two plane glass platesand thin spacer. The magnetic uid is injected into the sample cellwith optical pathlength of around 200 m. Fig. 1 shows the diagram ofthe experimental setup for investigating the longitudinal eld-induced polarized light transmittance of magnetic uids, whichincludes the hybrid effects of geometric shadowing and Faradayrotation. The highly stable HeNe laser with wavelength 632.8 nm isused as the light source. Themagnetic uid sample is placed in the gapbetween the two poles of the electromagnet. The electromagnetgenerates a uniformmagnetic eld in the sample region. The strengthof the magnetic eld can be adjusted by tuning the magnitude of thesupply current and is monitored by a gaussmeter. The smalleststrength of the magnetic eld in our experiments is nonzero (seeFigs. 2 and 6). This is owed to the remanence of the electromagnet.Both poles of the electromagnet are drilled to obtain two collimatedsmall holes (about 5 mm in diameter) for the incident light to pass

Fig. 1. The schematics of experimental setup for studying the longitudinal eld-induced polarized light transmittance of magnetic uids. The solid and dashed lines represent thelight rays and the electric wires, respectively.

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through. The propagation direction of the incident light parallels themagnetic eld. The transmitted light after the analyzer are recordedby a digital power meter.

To obtain the maximum sensitivity of the transmitted power withrespect to the Faraday rotation angle (viz. the externally magneticeld strength H), especially under the small magnetic eld, the initialpolarization direction of the analyzer is set with =45 according toEq. (2). In our experimental arrangement, the negative sign in Eq. (2)is satised.

4. Results and discussion

For studying the optical transmittance only due to the geometricshadowing effect, the analyzer is removed from the experimentalsetup. And the transmitted power as a function of externally magneticeld is experimentally measured as shown in Fig. 2. From Fig. 2, wecan see that the transmitted power increases with the magnetic eldmonotonically. At low eld regime, the transmitted power increasesrapidly with the externally magnetic eld. However, saturation of thetransmitted power takes place for the high eld region. As it has beenpointed out that the increase of the transmitted power with magneticeld is assigned to the geometric shadowing effect, i.e. the chain-likeaggregation of the magnetic nanoparticles along the externallymagnetic eld. This aggregation results in decreased number ofmagnetic nanoparticles in the transverse cross section (parallels themagnetic uid sample surface), so the incident light will be lessblocked and the transmitted powerwill increase.When the externallymagnetic eld is applied beyond certain critical value (usually around10 Oe), the magnetic nanoparticle within the magnetic uids will bemagnetized and particles agglomeration happens. Under low eldregion, the higher the magnetic eld, the more the particles willparticipate in the agglomeration and the less the effective particlescan block the incident light. This leads to the sharp increase of thetransmitted power with the magnetic eld strength. Nevertheless,when the externally is sufciently high, the agglomeration processtends to stable state and seldom particles will further participate inthe agglomeration. This will bring about the saturation of thetransmitted power with the magnetic eld. This aggregation-inducedgeometric shadowing effect can fairly account for the experimentalresults as shown in Fig. 2.

Fitting the experimental data to Eq. (1), we can get the solid linein Fig. 2 and the following tting parameters are acquired: Pmax =76.58245, P=17.00193 and C=1.04473105. From Fig. 2, we cansee that the experimental results are in good agreement withHelseth's equilibrium theory, namely Eq. (1). So, the explicit function

for the geometric shadowing effect-induced optical transmittance ofour experimental magnetic uid can be written as

P = 76:5824517:00193exp 1:04473 105H2

: 4

From Eqs. (2) to (4) and the above analyses, the analytical ex-pression for the total transmitted power after the analyzer (for ourexperimental sample) is given by

P = 76:5824517:00193exp 1:04473 105H2 h i

t

cos2 45-2 d= xysL L

1 + Q 2 liquidp1 + sL P 1 + QL

=

q8>:

9>=>;

;

5

where = 0V0MdH = kT and = 0N0sV0MdL H + Hw =kT .

It should be pointed out that Eq. (1) was obtained for magneticcolloids with paramagnetic polystyrene beads of radius of 1.4 and0.5 m. They consist of nanoscale iron oxide grain in a polymer matrix[39]. Our magnetic uid consists of magnetic beads with diameter ofabout 10 nm. But Eq. (1) is fairly applicable to our sample as shown inFig. 2. So we believe that the geometric shadowing effect-inducedoptical transmittance is mainly assigned to the nanosized magneticparticles and the polystyrene has insignicant inuence. This canreadily explained by the transparency of the polystyrene and therelatively low index difference between the polystyrene and theliquid carrier compared to that between the magnetic materials andthe liquid carrier.

To further understand the total optical transmittance withexternally magnetic eld, numerical simulations based on Eq. (5)are done. For the calculations, the following reasonable values of theconcerned parameters are taken unless otherwise specially stated:d=200 m; =0.6328 m; xy=0.1 [44]; s=, where =Ms /Mdis the volume fraction of the magnetic nanoparticles within the mag-netic uid and Ms is the saturation magnetization of the magneticuid; r=5 nm, Ms=100 Oe and Md=4.46105 A/m [1]; solid=4.5and liquid=2.56 [3]; N0 sR2d = 4r3 = 3

=3108 (close to

3108 particles per aggregation), where R=0.5 m is the radius ofthe aggregated column [45]. T=295 K (room temperature) and

Fig. 2. The transmitted power P as a function of externally magnetic eld H when onlyconsidering the geometric shadowing effect. The inset shows the relationship betweenthe magnetic eld strength H and the applied current I.

Fig. 3. The total transmitted power as a function of externally magnetic eld H forsamples with different reduced thickness d /.

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Hw=200 Oe [3,46]. Because we care more about the magnetic-eld-and parameter-dependent optical transmittance rather than theabsolute transmitted power, the calculated optical transmittancehave been normalized and then the accurate value of the analyzer'stransmittivity (t) is unnecessary.

The magnetic-eld-dependent optical transmittance for sampleswith different reduced thickness (d /) is simulated and shown inFig. 3. d /=316 approximates to our experimental conditions (fourother values of d / around 316 are also selected for comparison inFig. 3), and the Faraday rotation angle is smaller than 45 for allcases. Fig. 3 reveals that the transmitted power increases fast withmagnetic eld until 500 Oe and then tends to saturate when theexternally magnetic eld is beyond 500 Oe. Moreover, the thicker thesample, the higher the transmitted power. The sensitivity of thetransmitted power with respect to the externally magnetic eld ishigh for thick sample, especially in the low eld region.

Fig. 4 shows the inuence of off-diagonal term of the dielectrictensor of magnetic particles (xy) on the eld-dependent opticaltransmittance, which is analogous to Fig. 3. Under a constantexternally magnetic eld, the increase of transmitted power with xyis due to the change of Faraday rotation angle . When increasing thesample thickness (d /) under a constant externallymagneticeld, theFaraday rotation angle and absorption of the sample can increasesimultaneously. So the increase of transmitted power with d /indicates that the change of Faraday rotation effect dominates overthat of the absorption effect. If the transmitted power decreases withd /, it means that the change of the absorption effect dominates overthat of the Faraday rotation effect.

The magnetic-eld-dependent optical transmittance for sampleswith different liquid phase dielectric constants (liquid) is simulatedand shown in Fig. 5. Fig. 5 reveals that the transmitted powerincreases fast with magnetic eld until 500 Oe for all samples and theincrease rate is almost independent of liquid. When the externallymagnetic eld is beyond 500 Oe, the samples with large liquid phasedielectric constants have slightly high transmitted power.

Apparently, some similar behaviors between Fig. 5 and Figs. 3and 4 are found. But the transmitted power is nearly independent ofliquid when Hb500 Oe and slightly dependent on liquid whenHN500 Oe. Moreover, the property of liquid is different from d /and xy, that is, liquid is magnetic-eld-dependent while d / and xyare independent of magnetic eld. It is well-known that moreparticles will agglomerate into aggregation when the externally

magnetic eld is increased, which results in the decreased value ofliquid with magnetic eld. So the eld-dependent optical transmit-tance curve will change gradually from the curve for relatively highvalue of liquid to that for relatively small value of liquid as denoted bythe arrow in Fig. 5, when the externally magnetic eld augments.Therefore, the eld-dependent variation of liquid phase refractiveindex feebly inuences the optical transmittance curves.

Theoretical calculations are also carried out with different Weissinternal elds in a large range. Results indicate the Weiss internaleld, at least for our magnetic uid parameters, does not affect theeld-dependent optical transmittance.

The experimentally optical transmittance after the analyzer as afunction of externally magnetic eld is displayed in Fig. 6. From Figs. 3to 6, qualitative agreement between the experimental data and thetheoretical calculations can be obtained if considering the possibleexperimental errors. In case the accurate values of the magnetic uidparameters and the optical components are obtained, the quantitativecomparison between the experiments and calculations can beimplemented.

The aforementioned results indicate thatmagnetic uids operatingin longitudinal eld conguration can be employed to designmagnetically controllable optical attenuator, optical switch [47], and

Fig. 4. The total transmitted power as a function of externally magnetic eld H forsamples with different off-diagonal terms of the dielectric tensor xy.

Fig. 5. The total transmitted power as a function of externally magnetic eld H fordifferent liquid phase dielectric constants liquid.

Fig. 6. The experimentally optical transmittance after the analyzer as a function ofexternally magnetic eld H.

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light modulator if the proper operating point is selected. The almostlinear relationships between the transmitted power and the exter-nally magnetic eld at low eld (b500 Oe) or high eld (N500 Oe)region are favorable for fabricating the relevant optical devices. Inaddition, the transmitted power is more sensitive to the externallymagnetic eld at low eld than high eld. So the quality of the opticaldevices can be improved when operating in low eld region. Theattenuation range and sensitivity of the attenuator can be enlarged byincreasing d / or xy that can be seen in Figs. 3 and 4.

5. Conclusions

In summary, the explicit relationship between the transmittedpower and the externally longitudinal (parallels incident light)magnetic eld is given when considering both the geometricshadowing and Faraday rotation effects. The experimental data arein qualitative agreement with the theoretical calculations. Boththeoretic and experimental results reveal that the transmittedpower increases almost linearly with the externally magnetic eld Hwhen Hb500 Oe (low eld) or HN500 Oe (high eld). Under the loweld regime, the transmitted power rises fast with H while saturatedtrend of transmitted power with regard to H occurs at high eld.Moreover, the magnetic-eld-dependent optical transmittance curveis comparatively sensitive to reduced thickness d / and off-diagonalterm of the dielectric tensor xy of the magnetic particles and slightlydepends on the liquid phase dielectric constant liquid. Theselongitudinal-induced polarized light properties of magnetic uidscan be exploited to fabricate optical attenuator, optical switch, lightmodulator, etc. Relatively better quality of the involved opticaldevices (e.g. high sensitivity, large tuning range and low signaldistortion) can be realized by choosing the low eld working region orincreasing the values of d / and xy.

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (No. 10704048). We are grateful to thereviewer(s) whose constructive comments have helped to improvethe quality of the paper considerably.

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