4 OPTICS LETTERS / Vol. 20, No. 1 / January 1, 1995

Measuring the transport mean free pathusing a reference random medium

A. A. Lisyansky, J. H. Li, and A. Z. Genack

Center for Advanced Technology for Ultrafast Photonic Materials and Applications, Department of Physics,Queens College of the City University of New York, Flushing, New York 11367

Received September 12, 1994

The transport mean free path of microwave radiation in a random sample is determined from measurementsof transmission through a composite sample of the medium with unknown scattering characteristics and arandom medium of variable thickness and known scattering parameters. The method can be applied at opticalfrequencies by use of a ceramic wedge as the medium of variable thickness.

Determining the transport mean free path and theinternal reflection coefficient for electromagneticwaves in random media is of fundamental and prac-tical interest.1 These properties are important instatistical studies of wave propagation and locali-zation. They can also be used to infer dynamicalinformation regarding complex systems, and they de-termine the opacity of important scattering mediasuch as paints, the atmosphere, and biological tis-sues. Until recently the internal reflection of wavesinside the sample at is boundaries has been neglectedin descriptions of photon propagation. However, be-cause of the large angles at which scattered wavesmay strike the internal surface of the sample, theaverage reflection coefficient of these waves may bequite large. This can result in substantial discrep-ancies between theory and experiment.2,3 Recentlywe demonstrated that accurate measurements of allbasic microscopic parameters can be achieved in boththe microwave4 and visible5 parts of spectrum if scat-tering at the boundary is properly taken into account.One can obtain these parameters by measuring thetotal transmission through and/or reflection from aslab of a random medium for different thicknessesand for different reflection conditions. However, itis often not possible to vary the thickness of thesample under investigation or to modify the reflec-tion coefficient at the boundary. In this Letter wepropose a simple nondestructive method for deter-mining the basic microscopic parameters in a ran-dom slab. We measure the relative transmission insamples composed of two media juxtaposed along thelongitudinal direction as a function of the thicknessof one or another of the media. Because the inter-faces do not change with thickness, the influence ofinternal reflection largely factors out.

Let us consider a sample composed of two adja-cent slabs of random media with total thickness L ­L1 1 L2 measured along the z direction. If condi-tions of weak scattering are satisfied, then the steady-state photon intensities I1,2 inside the slabs obey thediffusion equation

=2I1,2srd 2 a1,2I1,2srd ­1

D1,2Qsrd , (1)

0146-9592/95/010004-03$6.00/0

where ai is the absorption coefficient, Di ­ 1/3ni, isthe diffusion coefficient, ,1 is the mean free path,ni is the transport velocity of the ith medium, andQsrd is the source function. We have found that onecan accurately describe reflection and transmissionby replacing the incoming coherent flux by a source ofisotropic radiation at a depth z ­ zp, with a strengthequal to the incident flux.6 For the case of a planewave incident upon the slab, the source function canthen be written as Qsrd ­ qndsz 2 zpd, where q isthe source intensity. To solve Eq. (1) we need tospecify the boundary conditions. Because there is noincoming diffusive flux through the boundaries in ourmodel, the only flux toward the interior of the slab isthe reflected part of the outgoing flux. This givestwo boundary conditions in the form6

J s1d1 sx, y, z ­ 01d ­ 2R1J s1d

2 sx, y, z ­ 01d ,

J s2d2 sx, y, z ­ L2d ­ 2R2J s2d

1 sx, y, z ­ L2d , (2a)

where J sid1 and J sid

2 are diffusive fluxes in the ithmedium in the positive and negative directions, re-spectively. The condition at the interface betweenthe slabs can be written as

J s1d1 sx, y, z ­ L1d 2 J s1d

2 sx, y, z ­ L1d

­ J s2d1 sx, y, z ­ L1d 2 J s2d

2 sx, y, z ­ L1d ,

J s2d1 sx, y, z ­ L1d ­ s1 2 R12dJ s1d

1 sx, y, z ­ L1d

1 R12J s2d2 sx, y, z ­ L1d , (2b)

where R12 is the reflection coefficient between thetwo media.

The solutions of Eq. (1) with the boundaryconditions (2) are

I1szd ­ 23q

a1,1usz 2 zpdsinhfa1sz 2 zpdg

1qSa1

coshfa1sL1 2 zpdgcoshfa1sL1 1 z1dg

(tanhfa2sL2 1 z2dg

1a2,2

a1,1tanhfa1sL1 2 zpdg 1 z12

)sinhfa1sz 1 z1dg ,

(3a)

1995 Optical Society of America

January 1, 1995 / Vol. 20, No. 1 / OPTICS LETTERS 5

I2szd ­qSa1

n1

n2

sinhfa1sz1 1 zpdgcoshfa1sL1 1 z1dgcoshfa2sL2 1 z2dg

3 sinhfa2sL 1 z2 2 zdg . (3b)

Here we have introduced the designations

zi ­1

2ailn

1 1 aiz0i

1 2 aiz0i

, z0i ­2,1

31 1 Ri

1 2 Ri

,

R12 ­43

R12

1 2 R12

,

S ­

(tanhfa2sL2 1 z2dg

1a2,2

a1,1tanhfa1sL1 1 z1dg 1 ,2R12

)21

. (4)

In experiments one usually measures the totaltransmission T through or the total reflection R froma slab. Using Eqs. (2) and (3), we can write thesequantities as

T ­1 2 R2

qn1J s2d

1 sL2d

­3S

2a1,1

1 2 R2

1 1 R2

sinhsa2z2dsinhfa1sz1 1 zpdgcoshfa1sL1 1 z1dgcoshfa2sL2 1 z2dg

,

(5a)

R ­1 2 R1

qn1J s1d

2 s01d

­3S

2a1,1

1 2 R1

1 1 R1

sinhsa1z1dcoshfa1sL1 2 zpdgcoshfa1sL1 1 z1dg

3

(tanhfa2sL2 1 z2dg

1a2,2

a1,1tanhfa1sL1 2 zpdg 1 a2,2R12

). (5b)

We now consider measurements by using arandom scattering reference sample with known scat-tering parameters for one of the media. Unknownmicroscopic parameters of the other medium can thenbe determined from Eqs. (5) from measurements oftransmission or reflection at different thicknesses.Because the reflection coefficient between the adja-cent random media R12 is generally small, the term~R12 can be neglected. This reduces the numberof unknown parameters to three: the mean freepath ,2, the absorption coefficient a2, and the re-flection coefficient R2. If absorption in medium 1is not negligible, the range of thickness L1 mustbe chosen such that a1sL1 1 z1d , 1 for the small-est thickness in order that tanhfa1sL1 1 z1dg notbe saturated over the entire range of variation inL1. There are no restrictions on L2. In our exper-iments we measure relative transmission, keepingthe thickness of the second medium L2 constant.Moreover, because we had a2sL2 1 z2d . 2 in thisexperiment, tanhfa2sL2 1 z2dg . 1, so the expressionfor the relative transmission through the sample can

be simplified as

T sL1, L2dT sL1

0, L2d

­coshfa1sL1

0 1 z1dg 1a2,2

a1,1sinhfa1sL1

0 1 z1dg

coshfa1sL1 1 z1dg 1a2,2

a1,1sinhfa1sL1 1 z1dg

, (6)

and we end up with only one unknown parameter ,2.Note that the penetration depth zp has dropped outof this expression.

We applied the method described here to mea-sure the mean free path of microwave radiationin a sample of 0.95-cm-diameter alumina spheres

Fig. 1. Schematic of the experimental setup used tomeasure the total transmission of microwave radiationthrough the combined sample.

Fig. 2. Relative transmission through a combination of asample of randomly packed polystyrene spheres of lengthL1 and a mixture of alumina and hollow polypropylenespheres for filling fraction f ­ 0.30 at a thickness ofL2 ­ 15 cm as a function of the thickness of the poly-styrene sample at frequencies of 20.0 and 25.5 GHz.

6 OPTICS LETTERS / Vol. 20, No. 1 / January 1, 1995

Table 1. Input Parameters and Mean Free Pathsa

n (GHz) a1 (cm21) ,1 (cm) a2 (cm21) ,2 (cm)

20.0 0.0286 6.40 0.129 1.8325.5 0.0463 2.60 0.164 1.03

aAbsorption coefficient a1 and transport mean free path ,1 arefrom the reference medium; absorption coefficient a2 is from thetest medium; the transport mean free path ,2 is from the testmedium, which is determined from the fit to the data shown inFig. 2.

with a volume filling fraction f ­ 0.30 and hollowpolypropylene spheres of the same diameter withwall thicknesses of 0.2 cm. The frequency was nearthe second Mie resonance of the alumina spheres.The sample is contained in a 7.3-cm inner-diametercopper tube. The absorption coefficient of thissample was determined from measurements of theexponential attenuation coefficient of transmission,but the transport mean free path was unknown. Asa reference medium with known parameters we useda random collection of 1.27-cm polystyrene spheresat a f ­ 0.56.4 The experimental arrangement isshown in Fig. 1. K-band radiation is produced byan Alfred microwave oscillator emitted from a hornplaced 20 cm in front of the combined sample. Theincident radiation impinges upon the polystyrenereference medium, whose thickness is varied. Thethickness of the alumina/polypropylene test mediumwas fixed at L2 ­ 15 cm. The signal is detectedby using a Schottky diode detector. The microwaveamplitude is modulated at 2 kHz, and the signal ismeasured by using a lock-in detector. The variationof the transmission through the combined samplewith L1 for two different frequencies is shown inFig. 2. We fitted Eq. (6) to the data using only asingle fitting parameter, ,2. From a comparison ofsuch measurements of the transport mean free pathand independent frequency domain measurementsof the diffusion coefficient it is possible to deter-mine the transport velocity in the presence of sphere

resonances.7 The values of the input parametersand of the mean free paths determined from the fitto the data shown in Fig. 2 are given in Table 1.

In conclusion, we have derived an expression forthe relative transmission and reflection coefficientsof a structure composed of two adjoining randommedia. When the parameters for one of the mediaare known, it is possible to find the mean free pathof the other even though the internal reflection coef-ficients at the sample’s boundaries are not accuratelyknown. Once the mean free path is known the reflec-tion coefficient can be obtained from measurement oftransmission.5 The method is illustrated with mi-crowave measurements but could be readily appliedto optical measurements of the propagation proper-ties of a slab by use of a wedged reference sample withknown propagation parameters to produce a compos-ite sample of variable length when the slab and thewedge are juxtaposed.

This research was supported by the National Sci-ence Foundation under grant DMR-9311605.

References

1. P. Sheng, ed., Scattering and Localization of Classi-cal Waves (World Scientific, Singapore, 1990); C. M.Soukoulis, ed., Photonic Band Gaps and Localization(Plenum, New York, 1993).

2. A. Lagendijk, R. Vreeker, and P. de Vries, Phys. Lett.A 136, 81 (1989).

3. R. Berkovits, M. Kaveh, and S. Feng, Phys. Rev. B 40,737 (1989).

4. N. Garcia, A. Z. Genack, and A. A. Lisyansky, Phys.Rev. B 46, 14475 (1992).

5. J. H. Li, A. A. Lisyansky, D. Livdan, T. D. Cheung, andA. Z. Genack, Europhys. Lett. 22, 675 (1993).

6. N. Garcia, J. H. Li, W. Polkosnik, T. D. Cheung,P. H. Tsang, A. A. Lisyansky, and A. Z. Genack, Phys-ica B 175, 9 (1991).

7. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk,and A. Tip, Phys. Rev. Lett. 66, 3132 (1991).