The missing ingredient in effective-medium theories: Standard deviations USA. University Park, PA 16802, USA

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1 The missing ingredient in effective-medium theories: Stndrd devitions Crig F. Bohren 1,*, Xuerong Xio 2, nd Akhlesh Lkhtki 2 1 Deprtment of Meteorology, Pennsylvni Stte University, University Prk, PA 16802, USA 2 Deprtment of Engineering Science nd Mechnics, Pennsylvni Stte University, University Prk, PA 16802, USA Abstrct: Effective-medium theories for electromgnetic constitutive prmeters of prticulte composite mterils re theories of verges. Stndrd devitions re bsent becuse of the lck of rigorous theories. But ensemble verges nd stndrd devitions cn be clculted from rigorous theory of reflection by plnr multilyers. Averge reflectivities t ll ngles of incidence nd two orthogonl polriztion sttes for multilyer composed of two kinds of electriclly thin lyers gree well with reflectivities for single lyer with the sme overll thickness nd volume-weighted verge of the reltive permittivities of these two components. But the reltive stndrd devition cn be pprecible depending on the ngle of incidence nd the polriztion stte of the incident illumintion, nd increses with incresing difference between the constitutive prmeters of the two lyers. This suggests tht verge constitutive prmeters obtined from effective-medium theories do not hve uniform vlidity for ll clcultions in which they might be used. Keywords: Effective-medium theories; composite mterils; reflection by multilyers

2 2 Electromgnetic constitutive prmeters such s permittivity nd permebility re verge response functions. For pure moleculr or tomic mteril such s wter or glss or gold, the verging volume is of order the cube of the wvelength of n exciting electromgnetic wve. At wvelengths well into the ultrviolet, the number of molecules per cubic wvelength is so lrge, even for gses, tht these verges re usully sufficient for describing reflection nd refrction becuse of opticlly smooth interfces for ny ngle of incidence nd polriztion stte, reflection nd trnsmission by thin films nd multilyers, nd scttering nd bsorption by prticles of ny size, shpe, nd composition. Becuse there re so mny molecules in the verging volume, the stndrd devition of the response function reltive to its verge is often negligibly smll. To n electromgnetic wve, prticle much smller thn the wvelength in both the mteril of the prticle nd the surrounding mteril (electriclly smll) is no different from gint molecule with very lrge polrizbility. Thus composite mteril consisting of, sy, mny smll prticles suspended in continuous mtrix (e.g., colloidl gold in queous suspension) should be chrcterized to good pproximtion by n verge or effective permittivity (we tke the permebility to be tht of free spce). But becuse the number density of prticles in composite mteril is much less thn moleculr number densities, we expect stndrd devitions for such mteril to be pprecibly greter thn those for moleculr mteril. By prticle we men bound ggregtion of sufficiently mny toms or molecules tht it cn be ssigned mcroscopic properties such s temperture, pressure, density, nd permittivity. Effective-medium theories for composite mterils hve long history, with contributions from Poisson, Mossotti, Clusius, Mxwell, Ryleigh, Mxwell Grnett,

3 3 Bruggemn, nd others. For compendium of clssic ppers on effective-medium theories s well s more modern ppers, see Ref. [1]. It is often implicitly ssumed tht effective-medium theories pply to rndomly inhomogeneous mterils (s opposed to, sy, n rry of identicl spheres t sites on regulr lttice). But rndom mteril is not single mteril, rther the nme of n ensemble of mny systems with the sme volume frction of prticles of given composition suspended in given mteril, distributed rndomly in spce nd possibly in size, shpe, nd (if non-sphericl) orienttion. Becuse n effective-medium theory yields only verges, two such theories for the sme rndomly inhomogeneous mteril cnnot be legitimtely compred, or prticulr theory compred with mesurements, without knowing stndrd devitions. And there s the rub. To our knowledge, stndrd devitions for composite rndom mterils hve not been clculted, nd for good reson: lck of rigorous theory of such mterils. Fced with this lck, to obtin some insights we turn to composite system for which rigorous theory does exist: multilyer. Reflection nd trnsmission by ny number of lyers cn be clculted using the mtrix method [2]. For exmple, for normlly incident electromgnetic wve ( E, H ), the electromgnetic wve reflected ( E, H ) nd trnsmitted ( E, H ) by N lyers is t t i i r r : Ei + Er Et = MM M N, (1) Hi Hr Ht where the chrcteristic mtrix of lyer of thickness d, with wvenumber k nd intrinsic impednce Z is cos kd iz sin kd M =. (2) isin k d/ Z cos kd

4 4 (Similr expressions hold for two orthogonl incident wves t rbitrry incidence.) If 2 π n d / λ= 1, M is pproximtely 1 i2 πdz0 / λ, (3) i2 πε ( / ε0) d / λz0 1 where λ is the free-spce wvelength, d is the thickness of the th lyer with permittivityε nd refrctive index n = ε / ε, ε 0 0 is the permittivity of free spce, nd Z 0 is the impednce of free spce. Hrmonic time-dependence exp( iωt) with circulr frequency ω is ssumed, nd the permebility of ll lyers is tht of free spce µ 0. If qudrtic nd higher powers of 2 π n d / λre neglected, the mtrix of multilyer with totl thickness h= d is independent of the order of the lyers nd pproximtely, 1 iz02 πh / λ, (4) i2 πε ( v / ε0) h/ λz0 1 where ε = ε f (5) v nd f = d / h. ε is weighted verge depending on only the volume frctions nd v permittivities of the lyers. Of the geometricl properties of the multilyer, only the totl thickness (reltive to h ) of ll lyers with the sme composition, not their individul thicknesses, determinesε v. For two-component multilyer ε = fε + (1 f) ε, (6) v b

5 5 where f is the volume frction of the component with permittivity ε. This eqution, subect to ssumptions underlying its derivtion, is vlid for ll ngles of incidence nd both liner polriztion sttes of the incident illumintion provided tht the permebilities of the lyers re equl nd the rtioε / ε b is neither too lrge nor too smll. Eqution (6) is correct in the limits f 0nd f 1, which suggests tht it is likely to be most ccurte for f = 1 nd f 1, lest ccurte for intermedite vlues, sy, 0.3 < f < 0.7. The verge Eq. (6) is n nlyticl expression. Although no such expression exists for the stndrd devition, we cn compute it s follows: The reflection coefficient r%nd trnsmission coefficient t%for plne wve incident t ngle θ on two-component N-lyer in free spce cn be clculted using M11 cos θ M12 / Z0 cosθ r% p M11 cos θ + M12 / Z0 = M21 cos θ M22 / Z0 1/Z t 0 % p M21 cos θ + M22 / Z0 (7) for p-polriztion nd M11 + M12 cos θ / Z0 1 r% s M11 + M12 cos θ / Z0 = M + M cos θ / Z cos θ/z t% M + M cos θ / Z s (8) for s-polriztion. The mtrix elements M re obtined from i M M M = where M exp( ia d )exp( ia d )...exp( ia d ), (9) N N N 1 N ωµ 0{1 ( ε0/ ε )sin θ} A = (10) ωε 0 for p-polriztion, nd

6 6 0 ωµ 0 A = 2. (11) ωε0 sin θ ωε 0 for s-polriztion. Either ε = ε orε εb =. The totl number of lyers is N = N + N, b where N is the number with permittivityε nd thickness d, nd permittivityε b nd thickness d b.the volume frction f of the -component is N is the number with b f = Nd Nd + Nd b b. (12) If d nd db re fixed, then for fixed N nd N, the totl thickness of the multilyer is fixed. But the order of the lyers is vrible, nd two (or more) lyers of the sme component mteril cn be dcent to ech other. This corresponds to clumping of prticles in prticulte composite mteril, which often is difficult to eliminte completely. To ensure tht ech lyer is electriclly thin we tke d = 0.1 λ/ 2π n ( =, b). (13) We generte n N-element rry of permittivities, chosen rndomly to be ε or ε b subect to the constrint tht N is fixed. For ech such rry, nd fixed ngle of incidenceθ, the reflectivity 2 R= r% is clculted for the two polriztion sttes from Eqs. (7)-(11). The verge reflectivity R nd its stndrd devition re clculted for mny such rrys nd compred with R( ε v), where ε is the weighted verge Eq. (6). v Figure 1 shows clcultions for 250 rrys with N = 50, n = 1.95, n b = 1.4, nd λ = 550 nm. These refrctive indices re for hypotheticl mterils with permittivity rtio of bout 2. N = 30 is chosen to give volume frction f = For both incident

7 7 polriztion sttes R is pproximtely equl to R( ε ) for llθ. But the reltive stndrd devition is pprecible, s high s 20%. Perhps more importnt, the reltive stndrd devition is not uniform, vrying both with the ngle of incidence nd the polriztion stte. Clcultions for higher nd lower f re similr. Clcultions lso were done t λ = 650 nm for two lyers composed of rel mterils, silicon dioxide (SiO 2 ) nd cuprous oxide (Cu 2 O). The refrctive index of SiO 2 is [3], nd tht of Cu 2 O is i0.1 [4]; hence, the rtio of their permittivities is bout 4. Figure 2 shows clcultions for multilyers with cuprous oxide volume frction The mximum reltive stndrd devition, 40%, is even higher thn in Fig. 1. Indirect evidence tht these stndrd devitions re relistic is mesurements of 90 o scttering by single evporting glycerol droplets (dimeter d 6µmnd 18µm ) contining bout 1% by volume of polystyrene ltex spheres (d = nm) [5]. A consequence of this inhomogeneity is fluctutions bout the men (up to 30%) of scttering s function of droplet dimeter, which increse with incresing ltex sphere size. Wht these simple clcultions suggest, but do not prove, is tht verge permittivities of composite prticulte mterils my be ccompnied by pprecible stndrd devitions. Also, such permittivities do not necessrily hve the sme unrestricted vlidity s those of moleculr mterils (which lso re verges). Permittivities of moleculr mterils re often used without hesittion for clculting mny different quntities. But our results suggest tht the reltive error in clcultions using verge permittivities of composite prticulte mterils cn depend on wht they re used for. Reflectivity clcultions using Eq (6) need not gree exctly or nerly so with clcultions of the men. If the former clcultions lie within stndrd devition of the v

8 8 men they cn be sid to gree with reflectivities for rndom medium. It is esy to lose sight of this becuse we re ccustomed to looking t two curves nd sying tht they gree or disgree. But for sttisticl clcultions exct greement doesn t exist. All clcultions lying within n intervl re eqully correct. And the sme is true of mesurements mde on only one member of lrge ensemble of similr smples. We cnnot know how close n individul mesurement is to the men of lrge number of similr mesurements unless they re mde, which they often re not.. *bohren@meteo.psu.edu

9 9 References [1] Lkhtki, A. (Ed.), Selected Ppers on Liner Opticl Composite Mterils, (SPIE Press, Bellinghm, WA 1996). [2] Hecht, E. Optics, 4 th ed., Addison-Wesley: Boston, 2002; [3] Philipp, H. R. Silicon Dioxide (SiO 2 ) Glss. In Hndbook of Opticl Constnts of Solids Vol. I; Plik, E., Ed.; Acdemic: Sn Diego, 1998, [4] Ribbing, C. G.; A. Roos. Copper Oxides (Cu 2 O, CuO) In Hndbook of Opticl Constnts of Solids Vol. II; Plik, E., Ed.; Acdemic: Sn Diego, 1998, [5] Ngo, D.; R. G. Pinnick. Suppression of Scttering Resonnces in Inhomogeneous Prticles. J. Opt. Soc. Am. A ,

10 10 Figure Cptions 1. Reflectivity s function of ngle of incidence for two-component 50-lyer multilyer verged over 250 rndom rrys. () Incident p-polriztion. (b) Incident s- polriztion. Dshed lines show the verge R plus or minus one stndrd devition. The volume frction of the component with refrctive index n = 1.95 is 0.52; n b = 1.4 nd the free-spce wvelength is 550 nm. 2. Reflectivity t 650 nm s function of ngle incidence for two-component 50-lyer multilyer composed of cuprous oxide (volume frction 0.43) nd silicon dioxide. () Incident p-polriztion. (b) Incident s-polriztion. Dshed lines show the verge R plus or minus one stndrd devition.

11 Figure 1 11

12 Figure 2 12

2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm

2.57/2.570 Midterm Exam No. 1 March 31, :00 am -12:30 pm 2.57/2.570 Midterm Exm No. 1 Mrch 31, 2010 11:00 m -12:30 pm Instructions: (1) 2.57 students: try ll problems (2) 2.570 students: Problem 1 plus one of two long problems. You cn lso do both long problems,