The discrete dipole approximation: an overview and recent developments

Transcription

1 The dcrete dpole approxmaton: an overvew and recent development M.A. Yurkn a,b, and A.G. Hoektra a a Secton Computatonal Scence, Faculty of Scence, Unverty of Amterdam, Krulaan 40, 1098 SJ, Amterdam, The Netherland b Inttute of Chemcal Knetc and Combuton, Sberan Branch of the Ruan Academy of Scence, Inttutkaya Str., 60090, Novobrk, Rua Abtract We preent a revew of the dcrete dpole approxmaton (DDA), whch a general method to mulate lght catterng by arbtrarly haped partcle. We put the method n htorcal context and dcu recent development, takng the vewpont of a general framework baed on the ntegral equaton for the electrc feld. We revew both the theory of the DDA and t numercal apect, the latter beng of crtcal mportance for any practcal applcaton of the method. Fnally, the poton of the DDA among other method of lght catterng mulaton hown and poble future development are dcued. Keyword: dcrete dpole approxmaton, revew, lght catterng mulaton Correpondng author: Tel.: ; fax: E-mal addre: myurkn@cence.uva.nl

2 Content 1 Introducton... General framework... Varou DDA model Theoretcal bae of the DDA Accuracy of DDA mulaton The DDA for cluter of phere Modfcaton and extenon of the DDA Numercal conderaton Drect v. teratve method Scatterng order formulaton Block-Toepltz FFT Fat multpole method Orentaton averagng and repeated calculaton Comparon of the DDA to other method Concludng remark... 8 Acknowledgement... 8 Appendx. Decrpton of ued acronym and ymbol... 8 Reference Introducton The dcrete dpole approxmaton (DDA) a general method to compute catterng and aborpton of electromagnetc wave by partcle of arbtrary geometry and compoton. Intally the DDA wa propoed by Purcell and Pennypacker (PP) [1], who replaced the catterer by a et of pont dpole. Thee dpole nteract wth each other and the ncdent feld, gvng re to a ytem of lnear equaton, whch olved to obtan dpole polarzaton. All the meaured catterng quantte can be obtaned from thee polarzaton. The DDA wa further developed by Drane and coworker [-5], who popularzed the method by developng a publcly avalable computer code DDSCAT [6]. Later t wa hown that the DDA alo can be derved from the ntegral equaton for the electrc feld, whch dcretzed by dvdng the catterer nto mall cubcal ubvolume. Th dervaton wa apparently frt performed by Goedecke and O'Bren [7] and further developed by other (ee, for ntance, [8-11]). It mportant to note that the fnal equaton, produced by both lne of dervaton of the DDA are eentally the ame. The only dfference that dervaton baed on the ntegral equaton gve more mathematcal nght nto the approxmaton, thu pontng at way to mprove the method, whle the model baed on pont dpole phycally clearer. The DDA called the coupled dpole method or approxmaton by ome reearcher [1,1]. There are alo other method, uch a the volume ntegral equaton formulaton [14] and the dgtzed Green functon (DGF) [7], whch were developed completely ndependently from PP. However, later they were hown to be equvalent to DDA [8,15]. In th revew we wll ue the term DDA to refer to all uch method, nce we decrbe them n term of one general framework. However, t dffcult to eparate unambguouly the DDA from other mlar method, baed on the volume ntegral equaton for the electromagnetc feld, uch a a broad range of method of moment (MoM) wth dfferent bae and tetng functon [16-19]. In our opnon, one fundamental apect of the DDA that the oluton for the phycally meanngful nternal feld or ther drect dervatve, e.g. polarzaton, play an ntegral role n the proce. In other word, any DDA formulaton can be nterpreted a replacng a catterer by a et of nteractng dpole; th further dcued n Secton. An

3 example of method that not condered DDA the MoM wth hgher-order herarchcal Legendre ba functon [17]. The DDA a popular method n the lght-catterng communty and t ha been revewed by everal author. An extenve revew by Drane and Flatau [4] cover almot all DDA development up to A more recent revew by Drane [5] manly concern applcaton and numercal conderaton. DDA theory wa dcued together wth other method for lght catterng mulaton n revew by Wredt [0], Chappetta and Torrean [1], and Kahnert [15] and n book by Mhchenko et al. [] and Tang et al. []. Jone [4] placed the DDA n context of dfferent method wth repect to partcle characterzaton. However, many mportant DDA development nce 1994 are not mentoned n any of thee manucrpt. Thoe that are mentoned are uually condered a de-tep, and are not placed nto a general framework. Moreover, to the bet of our knowledge numercal apect of the DDA have never been revewed extenvely each paper dcue only a few partcular apect. In th revew we try to fll th gap. A general framework developed n Secton to eae the further dcuon of dfferent DDA model. Th framework baed on the ntegral equaton becaue t allow a unform decrpton of all the DDA development. However, connecton to a phycally clearer model of pont dpole dcued throughout the ecton. The ource of error n the DDA formulaton are alo dcued there. In Secton the phycal prncple of the DDA are revewed and reult of dfferent model are compared. In Subecton.1 dfferent mprovement of polarzablte and nteracton term are revewed from a theoretcal pont of vew. Dfferent expreon for C ab alo are dcued. Comparon of mulaton reult ung dfferent formulaton gven n Subecton.. Subecton. cover the pecal cae of a cluter of phere that allow partcular mprovement and mplfcaton. In Secton.4 dfferent gnfcant modfcaton are revewed, whch do not fall completely nto the general framework decrbed n Secton. Enhancement of the DDA for ome pecal purpoe alo are dcued. Dfferent numercal apect of the DDA are revewed n Secton 4. Thee are concerned prmarly wth olvng very large ytem of lnear equaton (Subecton 4.1). Subecton 4. decrbe the mplet teratve procedure to olve DDA lnear ytem, whch ha a clear phycal meanng. The pecal tructure of the DDA nteracton matrx for a rectangular grd and t applcaton to decreae computatonal cot are decrbed n ubecton 4. and 4.4 repectvely. General method to accelerate calculaton, whch do not requre a rectangular grd, are dcued n Subecton 4.5. Subecton 4.6 cover pecal technque to ncreae the effcency of repeated calculaton (e.g. n orentaton averagng). A numercal comparon of the DDA wth other method revewed n Secton 5; t trong and weak pont are dcued. Secton 6 conclude the revew and dcue future development of the DDA. General framework The exp( ω t) tme dependence of all feld aumed throughout th revew. The catterer aumed delectrc but not magnetc (magnetc permttvty μ = 1). The electrc permttvty aumed otropc to mplfy the dervaton; however, extenon to arbtrary delectrc tenor traghtforward. 1 The general form of the ntegral equaton governng the electrc feld nde the delectrc catterer the followng [8,15]: nc E( r) = E ( r) + d r G( r, r ) χ( r ) E( r ) + M( V0, r) L( V0, r) χ( r) E( r), (1) V \ V 0 1 In mot formulae calar value can be replaced drectly by tenor, but there are excepton. Extenon of DDA to optcally anotropc catterer are dcued n Secton.4.

4 where E nc (r) and E(r) are the ncdent and total electrc feld at locaton r; χ( r ) = ( ε ( r) 1) 4π the uceptblty of the medum at pont r (ε(r) relatve permttvty). V the volume of the partcle,.e., the volume that contan all pont where the uceptblty not zero. V 0 a maller volume uch that V 0 V, r V 0 \ V 0. G ( r, r ) the free pace dyadc Green functon, defned a ˆ ˆ ˆ ˆ = ( + ˆ ˆ exp(kr) exp(kr) ) = RR 1 kr RR G( r, r ) k I k I I, () R R R R R where I the dentty dyadc, k = ω c the free pace wave vector, R = r r, R = R, and R ˆ R ˆ a dyadc defned a R ˆR ˆ = R R (μ and ν are Cartean component of the vector μν μ ν or tenor). M the followng ntegral aocated wth the fntene of the excluon volume V0 M( V = ( 0, r) d r G( r, r ) χ ( r ) E( r ) G ( r, r ) χ( r) E( r) ), () V 0 where G ( r, r ) the tatc lmt ( k 0 ) of G ( r, r ) : ˆ ˆ = ˆ ˆ 1 1 RR G ( r, r ) = I R R. (4) R L the o-called elf-term dyadc: nˆ Rˆ L ( V = 0, r) d r, (5) R V0 where n ˆ an external normal to the urface V0 at pont r'. L alway a real, ymmetrc dyadc wth trace equal to 4π [5]. It mportant to note that L doe not depend on the ze of the volume V 0, but only on t hape (and locaton of the pont r nde t). On the contrary, M doe depend on the ze of the volume, moreover t approache zero when the ze of the volume decreae [8] (f both χ(r) and E(r) are contnuou nde V 0 ). When dervng Eq. (1) the ngularty of the Green functon ha been treated explctly, therefore t preferable to the commonly ued formulaton [8,15]: nc E( r) = E ( r) + d r G( r, r ) χ ( r ) E( r ). (6) V Moreover, Yanghjan noted [5] that there ext everal method for treatng the ngularty n Eq. (6) leadng to dfferent reult. He alo proved that the dervaton of Eq. (6) fale n the vcnty of the ngularty of G ( r, r ). Hence t can be condered correct only f the ngularty then treated n a way mlar to that of Lakhtaka [8], reultng n the correct Eq. (1). U N V =1 Dcretzaton of Eq. (1) done n the followng way [15]. Let V =, V I = /0 V j for j ; N denote number of ubvolume. Although the formulaton applcable to any et of ubvolume V, n mot applcaton tandard (equal) cell are ued. Then the hape of the catterer cannot alway be decrbed exactly by uch tandard cell. Hence, the dcretzaton may be only approxmately correct. Aumng r V and choong V 0 = V, Eq. (1) can be rewrtten a nc E( r) = E ( r) + d r G( r, r ) χ( r ) E( r ) + M( V, r) L( V, r) χ( r) E( r). (7) j V j In the framework of the DDA we uually call a ubvolume a dpole. 4

5 The et of Eq. (7) (for all ) exact. Further, one fxed pont r nde each V (t center) choen and r = r et. In many cae the followng aumpton can be made: d r G( r, r ) χ ( r ) E( r ) = V jgjχ( rj ) E( rj ), (8) V j M( V, r ) = M χ( r ) E( r ), (9) whch tate that ntegral n Eq. (7) lnearly depend upon the value of χ and E at pont r. Eq. (7) can then be rewrtten a nc E = E + G jv j χ je j + ( M L ) χ E, (10) nc where E = E r ), E = E nc ( ), χ = χ r ), L = L V, r ). ( r j ( ( The uual approxmaton [15] to conder E and χ contant nde each ubvolume: E( r) = E, χ ( r) = χ for r V, (11) whch automatcally mple Eq. (8), (9) and (0) M = ( d r G( r, r ) G ( r, r )), (1) V 1 (0) G = j d r G( r, r ). V (1) j V j Supercrpt (0) denote approxmate value of the dyadc. A further approxmaton, whch ued n almot all formulaton of the DDA, ncludng e.g. [8], (0) G j = G( r, r j ). (14) Th aumpton made mplctly by all formulaton that tart by replacng the catterer wth a et of pont dpole. It mportant to note that Eq. (10) and dervaton reultng from t requre weaker aumpton (Eq. (8), (9)) than mpoed by Eq. (11) and, moreover, Eq. (14). It poble to formulate the DDA baed on Eq. (10), e.g. the Peltonem formulaton [6] that decrbed n Secton.1. We potulate Eq. (10) a a dtnctve feature of the DDA,.e. a method called the DDA f and only f t man equaton equvalent to Eq. (10) wth any V, χ, M, L, and G j. Kahnert [15] dtnguhed the DDA from the MoM by the fact that the MoM olve drectly Eq. (10) for unknown E, whle the DDA eek not the total, but the exctng electrc feld exc elf E = ( I + ( L M ) χ ) E = E E, (15) elf E = ( M L ) χ E, (16) elf where E the feld nduced by the ubvolume on telf. Eq. (10) then equvalent to nc exc exc E = E G jα je j, (17) where j α the polarzablty tenor defned a ( I + ( L M ) ) 1 α = V χ χ. (18) However, an alternatve formulaton of the DDA ext [4] eekng a oluton for unknown polarzaton P : exc P = αe = V χ E, (19) nc 1 E = α P G jp j. (0) It mportant to note that P, defned by Eq. (19), only an approxmaton to the polarzaton of the ubvolume V. Th approxmaton exact only under the aumpton of Eq. (11), whle the formulaton telf doe not requre t. The formulaton, ung Eq. (0), can be thought a an ntermedary between the DDA and the MoM a clafed by Kahnert [15], therefore j 5

6 revealng complete equvalence of thee two formulaton. The pecal tructure of the matrx G make Eq. (0) preferable over Eq. (10), (17) to fnd a numercal oluton. Th j dcued n Secton 4. Lakhtaka [8] clafed trong and weak form of the DDA a thoe accountng for or neglectng M repectvely. The weak form approache the trong form when the ze of the cell decreae, becaue M approache zero. For a cubcal cell V and wth r located at the center of the cell, L can be calculated analytcally yeldng [5] 4π L = I. (1) Ung Eq. (18), th reult n the well-known Clauu-Moott (CM) polarzablty (ued orgnally by Purcell and Pennypacker [1]) for the weak form of the DDA: CM ε 1 α = Iα = Id, () 4π ε + where ε = ε ( r ), and d the ze of the cubcal cell. After the nternal electrc feld are determned, the cattered feld and cro ecton can be calculated. The cattered feld are obtaned by takng the lmt r of the ntegral n Eq. (1) (ee e.g. [7]): ca exp(kr) E ( r) = F( n), () kr where n = r r the unt vector n the catterng drecton, and F the catterng ampltude: d r exp( r n) ( r ) E( r F( n) = k ( I nn ˆ ˆ) χ ). V k All other dfferental catterng properte, uch a ampltude and Mueller catterng matrce, and aymmetry parameter < co θ > can be derved from F(n), calculated for two ncdent polarzaton [7]. Radaton force alo can be calculated [8-0]. Conder an ncdent polarzed plane wave nc 0 E ( r) = e exp(k r), (5) where k = ka, a the ncdent drecton, and e 0 = 1. The catterng cro ecton C ca [7] = 1 Ω C ca d F( n). (6) k Aborpton and extncton cro ecton (C ab, C ext ) are derved [7,14] drectly from the nternal feld: C = k ( ) ab 4π d Im χ( r ) E( r ), (7) C V r nc 4 0 ( χ( r ) E( r ) [ E ( r )] ) = Re( F( a ) (4) 4π k d r Im e, (8) k π ext = ) V where * denote a complex conjugate. Conervaton of energy necetate that Cca = Cext Cab. (9) However, a wa noted by Drane [], ue of Eq. (9) for evaluaton of C ca can lead to larger error than Eq. (6), epecally when C ab > C ca. The eaet way to expre Eq. (4) and (7) n term of the nternal feld n the ubvolume center to aume Eq. (11), yeldng (0) F ( n) = k ( I nn ˆ ˆ) χ E d r exp( kr n), (0) V DDA can be ued for any ncdent wave, e.g. Gauan beam [1]; however, we do not dcu th here. 6

7 π k V Im( χ ) E = 4πk Im( PE. (1) (0) C = 4 ) ab Further approxmaton of Eq. (0), leavng only the lowet order expanon of the exponent around r, lead to (0) F ( n) = k ( I nn ˆ ˆ) P exp( kr n), () whch together wth Eq. (8), lead to (0) nc Cext = 4π k Im( P E ). () Eq. () and () are dentcal to thoe ued by Purcell and Pennypacker [1] and then by Drane [], whle expreon for C ab (compared to Eq. (1)) are lghtly dfferent. Thee dfference are dcued n Subecton.1. Unfortunately, many reearcher do not pecfy explctly how the catterng quantte are obtaned from the computed nternal feld or polarzaton. Thoe who do uually ue Drane precrpton (Eq. (6), (), (), and (5)). Error of the formulaton can be clafed a aocated wth the fnte cell ze d (dcretzaton error), and wth approxmatng the partcle hape wth a et of tandard cell, e.g. cubcal (hape error). Dcretzaton error reult from conderng E contant nde each cell and the approxmate evaluaton of M and G. Shape error alo can be condered a reultng from the aumpton of contant χ and E nde borderng cell, whch fale nce the edge of the partcle croe thee cell. On the other hand, hape error can be vewed a a dfference of the reult for the exact partcle hape and for that compred of the et of tandard cell. Both error approach zero when N, whle the geometry of the catterer and parameter of the ncdent feld are fxed. However, the ame doe not apply f kd 0 whle N fxed,.e. the DDA not exact n the long-wavelength lmt. Moreover, both error are entve to the ze of the catterer n the reonance regon (ee dcuon n Subecton.). The behavor of thee error wa tuded by Yurkn et al. []. Varou DDA model.1 Theoretcal bae of the DDA Snce the orgnal manucrpt by Purcell and Pennypacker [1], many attempt have been made to mprove the DDA. The frt tage ( ) of thee mprovement wa revewed by Drane and Flatau [4]. It ha been noted [] that Eq. () doe not atfy energy conervaton, and reult obtaned ung th formulaton do not atfy the optcal theorem. Baed on the well-known [] radatve reacton (RR) electrc feld, a correcton to the polarzablty for a fnte dpole wa added []: CM RR α α =. CM (4) 1 ( )k α Drane [] alo propoed the followng expreon for the aborpton cro ecton: (0) exc* * Cab = 4π k [ Im( P E ) ( ) k P P ], (5) derved from Eq. (9) appled to a ngle pont dpole. The PP formulaton ue Eq. (5) wthout the econd part. It can be verfed that Eq. (5) reult n zero aborpton for any catterer f the polarzablty of the followng form: 1 H α = A ( )k I, A = A, (6) where H denote the conjugate tranpoe of a tenor. For real refractve ndex m, RR and all other expreon pecfed below reult n α atfyng Eq. (6), whch make Eq. (5) clearly favorable over e.g. the PP formulaton. It mut be noted however that the orgnal PP formulaton, where CM polarzablty wa ued, alo reult n zero aborpton for real m. j 7

8 The correcton n Eq. (4) O ( kd ) ). Several other correcton of ( ) ) O kd have been propoed. The frt one wa propoed by Goedecke and O Bren [7] and ndependently n two other manucrpt [4,5]. They tarted from Eq. (10)-(1) and ued the followng mplfyng fact for a cubcal cell (alo vald for phercal cell), reultng from ymmetry: RR ˆ ˆ 1 d Rf ( R) = d Rf ( R) I, (7) R cube cube where the orgn n the center of the cube. Eq. (7) vald for any f(r) that ha a ngularty of le than thrd order for R 0,.e. the ntegral on both de are defned. They obtaned ( 0) exp(kr) M = I k d R. (8) R cube By expandng exp(kr) n Taylor ere one can obtan (0) d R 4 M = ( ) I k + kd + O k d. (9) R cube The remanng ntegral wa evaluated by approxmatng the cube by a volume-equvalent phere, reultng n (0) DGF M = I b ( kd) + ( )( kd) O ( kd) 4, (40) ( ( ) 1 + DGF 1 1 = (4π ) b (41) An exact evaluaton, obtaned wthout expandng the exponent, of Eq. (8) for the equvolume 1 phere wth radu a = d( 4π ) wa performed by Lvenay and Chen [6] and mplemented nto the DGF formulaton of the DDA by Hage and Greenberg [14,5] and later Lakhtaka [7]: (0) M = (8π ) I[ (1 ka)exp(ka) 1]. (4) In term of the frt two order of expanon, th yeld an dentcal reult a Eq. (40). Fnally the polarzablty obtaned a CM DGF α α =. (4) CM DGF 1 ( α d )( b1 ( kd) + ( )( kd) ) We denote the method baed on Eq. (4) a LAK. Dfference between LAK and DGF hould be notceable only for large value of kd. Dungey and Bohren [8], ung reult by Doyle [9], propoed the followng treatment of the polarzablty. Frt, each cubc cell replaced by the ncrbed phere that called a dpolar ubunt wth a hgher relatve electrc permttvty ε a determned by the Maxwell- Garnett effectve medum theory [7]: ε 1 ε 1 f =, (44) ε + ε + where f = π 6 the volume fllng factor. Other effectve medum theore alo may be ued [40]. Next, the dpole moment of the equvalent phere determned ung the Me theory, and the polarzablty defned a [9] M α = α 1, (45) k where α 1 the electrc dpole coeffcent from the Me theory (ee e.g. [41]): mψ 1( m x ) ψ 1( x ) ψ 1( x ) ψ 1( m x ) α 1 =, (46) m ψ ( m x ) ξ ( x ) ξ ( x ) ψ ( m x ) 1 1 where ψ, ξ are Rccat-Beel functon; x = kd and m = ε are the ze parameter and the relatve refractve ndex of the equvalent phere. We denote th formulaton for the polarzablty a the a 1 -term method (note that th termnology wa ntroduced later [4]). It 1 1 8

9 M CM ha the partcular property that α α cont 1 when m 1, contrary to all other polarzaton precrpton, for whch th rato approache 1. It hould be noted that the Me theory baed on the aumpton that the external electrc feld a plane wave. In mot applcaton of the DDA th true for the ncdent electrc feld, but not for the feld created by other ubvolume. Therefore the a1-term method expected to be correct only for very mall cell ze. Hence t not clear whether th method ha advantage even compared to CM. On the other hand, th method may be more jutfed for cluter of mall phere, where each phere can be condered a a dpole (ee Subecton.). Drane and Goodman [] ponted out that conderng electrc feld contant for evaluatng ntegral over a cell ntroduce error of order O( kd ) ). Th repreent a problem for many polarzablty correcton, baed on ntegral equaton. Drane and Goodman approached th problem from a dfferent angle. They determned the optmal polarzablty n the ene that an nfnte lattce of pont dpole wth uch polarzablty would lead to the ame propagaton of plane wave 4 a n a medum wth a gven refractve ndex. Th polarzablty wa called LDR (Lattce Dperon Relaton) and, a expected, CM plu hgh-order correcton. Thee correcton n turn depend on the drecton of propagaton a and the polarzaton of the ncdent feld e 0 : CM LDR α α =, (47) CM LDR LDR LDR 1 ( α d )( [ b1 + b m + b m S )( kd) + ( )( kd) ] LDR LDR LDR b , b , b , (48) 1 0 = ( a e ) μ μ μ S. (49) We ue a revere gn conventon n the denomnator of Eq. (47) and the LDR coeffcent a compared to the orgnal paper []. Recently t ha been hown [4] that the LDR dervaton not completely accurate, nce the reultng dpole moment doe not atfy the tranveralty condton, for whch a correcton wa propoed. Th corrected LDR (CLDR) dffer prncpally n the fact that the polarzablty tenor can not be made otropc but only dagonal [4], though not dependent on the ncdent polarzaton: CM CLDR α δ μν α μν =. (50) CM LDR LDR LDR 1 ( α d )( [ b1 + b m + b m aμ )( kd) + ( )( kd) ] Another flaw of LDR that t evdently not correct for dpole near the partcle urface. However, t not clear how to evaluate the effect of thee mtreated urface dpole on the overall reult, e.g. on the catterng cro ecton. Further mprovement of the DDA wa ntated by Peltonem [6] (PEL) who howed that the term M(V ) n Eq. (7) can be evaluated exactly up to the thrd order of kd by expandng the term χ ( r ) E( r ) under the ntegral n a Taylor ere over the pont r = r, yeldng M 1 μ V ( V ) = ν M (0), μν exp(kr) d R R χe ν + 1 V exp(kr) d R R ( k R + kr 1) ν ν 4 ( k R + kr ) Rμ Rν Rρ Rτ ρ τ χeν + O ( kd) χe), νρτ ν R χe where χ, E and ther dervatve are all condered at the pont r. Eq. (51) correct up to the thrd order of kd nce the thrd term n the Taylor ere vanhe becaue of ymmetry. For phercal V of radu a, the ntegral can be evaluated exactly [6] n a way mlar to μ (51) 4 wth certan drecton of propagaton and polarzaton tate. 9

10 obtanng Eq. (4), but only term of le than fourth order of kd are gnfcant, whch reult n 4 V π 1 M( ) = ( ka) ( ka) χ a χ ( χ ) + O ( ka) 4 χe + E E E ). (5) If χ contant nde the cell then the Maxwell equaton tate that E = m k E, E = 0. (5) Hence Eq. (9) vald up to the thrd order of ka and [( 1+ (1 10) m )( ka) ( )( ) ] M = ( 4π ) I + ka. (54) Pller and Martn [44] propoed ung amplng theory to evaluate the ntegral n Eq. (1). The electrc feld and the uceptblty ampled: r χ ( r ) E( r ) = h ( r r ) χ( r ) E( r ), (55) where h r (r) the mpule repone functon of an antalang flter defned a r n( qr) qr co( qr) h ( r ) =, (56) π r where q = π d. Eq. (1) then tranformed to Eq. (10) wth the o-called fltered Green functon, defned a 1 r G = d (, ) ( j r G r r h r rj ). (57) V j R / V0 Eq. (57) can be vewed a a generalzaton of Eq. (1). The latter obtaned f a pule functon condered ntead of h r. The ntegral n Eq. (57) evaluated analytcally [44], takng V 0 to be nfntemally mall. The fltered Green functon doe not have a ngularty when r = r j, therefore M = VG. It wa hown that the Fourer pectrum of E(r) le on a phere wth radu m(r)k, f m contant n the vcnty of r. Therefore at leat two amplng pont per wavelength n the catterer are requred. The uceptblty alo fltered, ether by a mean value flter or a more complcated one, e.g. a Hannng wndow. Th approach called FCD (fltered coupled dpole), and a computer code lbrary for evaluaton of fltered Green functon avalable [45]. Chaumet et al. [11] propoed drect ntegraton of the Green tenor (IT) n Eq. (1), (1). A Weyl expanon of the Green tenor performed, tranformng t to a form allowng effcent numercal computaton of the elf-term ( M L ). They alo propoed a correcton to the econd term n Drane expreon for C ab (Eq. (5)). Extenon of ther reult to a nonotropc elf-term (0) exc* * * C ab = 4π k [ Im( P E ) + Im( P ( M L ) P )/ V ], (58) elf The corrected econd term baed on radaton energy of a fnte dpole [11]: Im( E P ), n contrat to a pont dpole ued n the dervaton of Eq. (5). One can ee that Eq. (58) and (1) are equvalent. Moreover, both of them are equvalent to Eq. (5) f and only f M A )k H = + ( V I, A = A. (59) Th condton mlar, but not equvalent, to Eq. (6) and alway atfed for RR, DGF, and LAK. Other polarzablty precrpton atfy Eq. (59) for real m, then both Eq. (58) and (5) reult n zero aborpton. Rahman, Chaumet, and Bryant [46] propoed a new method (RCB) to determne polarzablty baed on the known oluton of the electrotatc problem for the ame catterer. In the tatc lmt the electrc feld at any pont lnearly related to the ncdent feld 1 0 E( r) = C ( r) E ( r). (60) Subttutng Eq. (60) nto Eq. (0) wth the tatc Green tenor, one can obtan the polarzablty, whch would gve an exact oluton n the tatc lmt, a 10

11 Λ RCB 1 α = V χλ, (61) 1 = C + G ( r, r j ) χ C j C, (6) j where C = C( r ). Th tatc polarzablty then replace the CM polarzablty, and the RR (Eq. (4)) appled to t [46] to obtan the fnal polarzablty for DDA mulaton. It wa later hown that RCB polarzablte dffer gnfcantly from CM only for dpole cloer than d to the nterface [47]. In ther next manucrpt [48] Rahman et al. tated that the prevou dervaton correct only f the tenor C contant nde the partcle (e.g. for ellpod), nce otherwe the polarzablty tenor obtaned from Eq. (61) generally not ymmetrc, whch phycally mpoble n the tatc cae. Th how that a partcle wth a non-contant C not equvalent to any et of phycal pont dpole even n the tatc regme. However, t equvalent to a et of non-phycal dpole wth an aymmetrc polarzablty. Therefore, the polarzaton defned by Eq. (61) formally can be ued, by telf or wth RR, even when C not contant. Collnge and Drane [47] emprcally combned the RCB precrpton wth CLDR to get the urface-corrected LDR (SCLDR): RCB ( ( I α ) ) 1 SCLDR RCB = α B α d, (6) where B the correcton matrx (analogou to Eq. (50)): LDR LDR LDR [( b + b m + b m a )( kd) ( )( kd ] B + μν = δ μν 1 μ ). (64) All method baed on the paper by Rahman et al. [46] are ntally lmted to very pecfc hape of the catterer (ellpod, nfnte lab and cylnder). Expanon of t applcablty to other hape debatable [48] and would anyway requre a prelmnary oluton of the electrotatc problem for the ame hape, whch generally not trval. All DDA formulaton are chematcally depcted n Fg. 1, whch alo how nterrelaton between them. Some formulaton can be compared unambguouly n term of theoretcal oundne: one an mprovement of the other,.e. t employ fewer approxmaton. Such formulaton are depcted n the ame column on Fg. 1, whle other cannot be compared drectly wth each other; they gve re to dfferent column. Comparon between formulaton from dfferent column can and ha been made almot excluvely emprcally by comparng the accuracy of the mulaton reult (ee Subecton.). All the above technque are amed at reducng dcretzaton error; only a few am at reducng hape error. Some of them employ adaptve dcretzaton (dfferent dpole ze) to better decrbe the hape of the catterer (ee Subecton.4). Another approach to average uceptblty n boundary ubvolume. The mplet averagng ung the Lorentz-Lorenz mxng rule wa propoed by Evan and Stephen [49] for the cae of the boundary between the catterer and t urroundng medum e χ χ = f, e (65) 4πχ 4πχ + + e where χ the effectve uceptblty, and f the volume fracton of the ubvolume actually occuped by catterer. A more advanced averagng, called the weghted dcretzaton (WD), wa propoed by Pller [1]. It modfe the uceptblty and elf-term of the boundary ubvolume. 5 The partcle urface, crong the ubvolume V, aumed lnear and dvde the ubvolume nto p two part: the prncpal that contan the center and a econdary wth uceptblte V V 5 any ubvolume that ha non-zero nterecton wth both the catterer and the outer medum. All uch ubvolume are accounted for. 11

12 Integral Eq. (1) dcretzaton (no aumpton) amplng wth antalang flter Eq. (7) Eq. (8), (9) FCD removng antalang flter comple General formulaton of DDA Eq. (0) Eq. (11) CLDR LDR SCLDR PEL Eq. (14) Eq. (14) IT RR a 1 term RCB mplfe to DGF, LAK M = 0 (weak form) CM mprovng polarzablty tartng from dpole formulaton Fg. 1. Scheme of nterrelaton between the dfferent DDA model dcued n Secton.1. Arrow down correpond to aumpton employed. Vertcal poton of the method qualtatvely correpond to t accuracy (hgher = better), however method n dfferent column cannot be compared drectly. p p χ, χ and electrc feld E E, E repectvely. Electrc feld are condered contant nde each part and related to each other va a boundary condton tenor T : E = T E. (66) Then the total polarzaton of the ubvolume can be evaluated a follow: p p e P = d r χ ( r ) E( r ) = V χ E + V χ E = V χ E, (67) V p p ( V χ I V χ T ) V e χ = +. (68) The uceptblty of the boundary ubvolume replaced by an effectve one. The effectve elf-term evaluated drectly tartng from Eq. (), conderng χ and E contant nde each part: p V p ( G( r, r ) G ( r, r )) χ + d r ( G( r, r ) G ( r, r )) χ T e e M χ = d r Pller [1] evaluated the ntegral n Eq. (69) numercally. The fnal equaton are the ame a Eq. (0), where polarzablte are obtaned from Eq. (18) ung effectve uceptblte and elf-term for boundary ubvolume. Hence, WD doe not modfy the general numercal cheme. Currently, there are no rgorou theoretcal reaon for preferrng one formulaton over other. However, theoretcal analye of DDA convergence when refnng dcretzaton recently conducted by Yurkn et al. [], howed that IT and WD gnfcantly mprove the convergence of hape and dcretzaton error, repectvely. Expermental verfcaton of thee theoretcal concluon tll to be performed. V. (69) 1

13 Table 1. Accuracy of dfferent DDA formulaton for a phere. a Value Method x a/d y m Error, % Ref. C ext a 1 -term 1 4 c [8] CSec, S 11 LAK 9 1 c , 7 [56] c 8 c , 5 4, 15 C ca, C ab DGF. c , 10 0 [] CSec LDR 8 c , 0.1 m-1 1 1, C ca C ab LDR 7 c , , 4 CSec LDR 16 c 10 c [51] CSec S 11 LDR any any 1 m [4] e C ca LDR 10 c 16 kd [148] S 11 LDR 10 c 4 kd RMS C ext, S 11 LDR c d , 8 [11] c d , c d 1.4 1, c d , c d , c 0.6 d.0, 86 Ψ FCD π, π.8, 5.6 c [44] Ψ WD-FCD 0.5. c 5 y Re =0.6 m <7 b 0.1 [10] c 6 m <.5 b c 6 m <4 b 1 CSec IT 5. c [11] C.1 c ab C 1.1 c ext CSec RCB-RR 8. c [48] 7.5 c c.5+.4 c c CSec SCLDR SCLDR RCB 7. c 1.5 c 1.5 c a RMS All error are relatve. CSec denote the maxmum error over all cro ecton, S 11 and S 11 correpond to maxmum and root mean quare error over the range of catterng angle, Ψ the normalzed mean error of the far-feld electrc feld [44]. In ome cae two error are hown n one cell eparated by a coma. They correpond to two value of one of the parameter n the ame row. b approxmate decrpton of the range. c th value determned by other value n the ame row. d th value lghtly dfferent for dfferent ze parameter. e th correpond to the rule of thumb for phere. 5 7 [47]. Accuracy of DDA mulaton Over the year many reult on the accuracy of DDA mulaton have been publhed. It, however, generally hard to ytematcally compare the relevant manucrpt becaue they all ue dfferent ndependent parameter, uch a the ze parameter x, refractve ndex m, or dcretzaton, a a functon of whch the error meaured. We wll decrbe dcretzaton by the parameter y = m kd or y Re = Re( m) kd. The former ued wherever poble; however, n ome cae a decrpton of reult more traghtforward n term of y Re. Accuracy reult 1

14 for catterng by a phere are ummarzed n Table 1. All manucrpt on th ubject can be dvded nto two clae: thoe that fx x and vary N (or equvalently, the number of dpole per phere radu a/d) wth y, and thoe that fx a/d and vary the ze parameter wth y. The former eaer to nterpret; the latter eaer to mulate. To facltate comparon between dfferent method we provde both x and a/d, however one of them dependent on the other. Some addtonal nformaton on thee reult follow below. Drane and Goodman [] compared RR, DGF, and LDR for cro ecton of a phere wth a / d = 16. DGF generally more accurate than RR. For m 1 1 LDR gve uperor or comparable reult to DGF, for m = + LDR and DGF are comparable, and for m = 4 + DGF preferable over LDR. In the revew of LDR DDA, Drane and Flatau [4] ummarzed that for m cro ecton can be evaluated to accurace of a few percent provded y 1. In that cae dfferental cro ecton have atfactory accuracy: relatve error up to 0-0%, but only where the abolute value of the dfferental cro ecton mall. For phere, uch reult are obtaned even for m. Comparon of CLDR to LDR [4] only reult n mnor dfference. Generally CLDR reult n lghtly better accuracy for C ca, but wore for C ab. Pller and Martn [44] compared FCD to LAK by tudyng the dependence of the mean relatve error of the far-feld electrc feld (Ψ) on y for phere wth x = π, π and m =1.5. It wa hown that FCD (wth a Hannng wndow flter for the electrc permttvty ε) roughly tme more accurate than LAK n the range 0.7 y.5 and gve mlar accurace for y 0.4 (for larger phere). Comparon of WD to tradtonal method [1] wa performed for phere wth x = π, π and m = 1., LAK wa ued to determne polarzablte. For m = 1. n the range 0.4 y 1. overall accuracy wa only lghtly mproved, but error peak for certan value of y were moothed out. For m = accuracy wa mproved 4-5 tme over the whole range y 1.. Pller alo howed [10] that a combnaton of WD and FCD gve even better reult. Generally FCD decreae the negatve effect of Re(ε) on accuracy and WD thoe of Im(ε). Rahman et al. [48] howed that RCB wa clearly uperor to CM n calculatng cro ecton for fxed a / d = 16 and m from to n the range y 1. Two correcton (LDR and RR) over the tatc cae were compared, and they gave mlar overall reult. Improvement of overall accuracy compared wth CM wa -5 tme n all cae tuded. For a thn lab, t wa hown [46,48] that the nternal feld calculated ung RCB dffer from thoe by CM motly near the nterface, where RCB yeld much maller error, almot the ame a far from nterface. Collnge and Drane [47] compared LDR, RCB, and SCLDR n calculaton of cro ecton of phere wth a / d = 1. It wa hown that for m = , LDR and SCLDR are uperor n the range y 0.8, whle for m = 5 + 4, SCLDR and RCB are uperor. Convergence of cro ecton for phere and ellpod for ncreang N wth fxed x and dfferent m (from to ) alo wa tuded. SCLDR howed the mot table reult for all cae, beng the mot or cloe to the mot accurate one; however, for ellpod wth large Im(m) RCB gave gnfcantly more accurate reult for C ca, epecally for larger y. Performance of the DDA for more complex hape alo wa tuded by dfferent author. Flatau et al. [50] compared DDA mulaton for a bphere wth an exact oluton from a multpole expanon. For m = , a / d = 16, and y 0. 8, LDR wa everal tme more accurate than DGF and reulted n error of le than 0.5% for both C ca and C ab. Xu and Gutafon [51] made a mlar but much more extended tudy of LDR. For m = , a / d = 5, and y 0.4, error n Cext, C ab, and co θ are wthn 10%. For y = 0.81, error n the angular dependence of S11 are up to 0% whle S 1 and S 1 were completely wrong. For m = , error n cro ecton exceed 10% for y

15 Error n the angular dependence of the Mueller matrx element are wthn 10-0% for y = 0. and ncreae rapdly wth ncreang y. For a fxed x = and m = , error n Cext, C ab, and < co θ > decreae from 10% to 1% whle y decreae from 1 to 0.. For y = 0., the angular dependence of S11 n good agreement wth the rgorou oluton, whle S 1 and S 1 dffer gnfcantly for certan orentaton of the bphere. Hage and Greenberg [14] compared LAK to expermental reult obtaned from mcrowave experment on porou cube. Ung m = , y = and N = 5504, they obtaned a dfference of le than 40% wth the expermental reult of angular catterng pattern, except for deep mnma. Lght catterng of cube, tle, and cylnder wth mlar parameter alo wa tuded and comparable dfference between experment and theory were obtaned. Theoretcal error were etmated to be le than 10%, except for deep mnma. Ikander et al. [4] conducted a lmted tet of LAK for mall elongated pherod, comparng the reult to thoe obtaned ung an teratve extended boundary condton method. Ung N = 64, calculaton were performed for apect rato up to 0 wth maxmum ze parameter of the long ax beng 10 and 0.5 for m = and repectvely. Error n catterng cro ecton were 1% and 11%, repectvely. Ku [5] compared LAK wth CM and the a1-term for dfferent hape, but h concluon are baed on a large parameter y (up to ), and are therefore upcou and not further dcued here. Anderen et al. [5] tuded the performance of the DDA for Raylegh-zed cluter of a few phere (mot DDA formulaton are then equvalent to CM). Several conttuent materal were teted, all wth hgh refractve ndce n the tuded regon. It wa hown that the DDA faled to converge ung the fxed computatonal reource for very hgh (up to 1.0) and very low (down to 0.1) Re(m); up to 0 dpole were ued per dameter of a ngle phere. It can be concluded that partcle wth more complex hape than phere are more dffcult to model wth the DDA, leadng to larger error for the ame m and y. Th effect can be explaned n general by the ncreae of urface to volume rato and hence larger fracton of boundary ubvolume []. Another poble reaon complex regon, e.g. contact between two partcle n a cluter, where rapd varaton of the electrc feld deterorate the overall accuracy. There, however, a notable excepton from th general tendency. Shape, whch can be modeled exactly by a et of cubcal dpole, e.g. a cube, can be mulated ung the DDA much more accurately than phere, epecally for mall y []. Drane and Flatau [4] have ntroduced a rule of thumb for dcretzaton: ue 10 dpole per wavelength n the medum (.e. ether y or y Re equal to 0.6, dependng on the nterpretaton). Though t wdely ued, the accuracy of the reult, when ung uch dcretzaton, hard to deduce a pror. Drane and Flatau themelve derved an etmate of the error baed on a et of tet mulaton. Th etmate decrbed above and mentoned n Table 1; t uually cted a a few percent accuracy n cro ecton. However, t may gnfcantly over- or under-etmate the error, epecally for large ze parameter. Moreover, t doe not completely account for the dependence on m, even n the tated range of t applcaton ( m ), nce DDA accuracy deterorate rapdly wth ncreang m (ee Table 1). Stll, the rule of thumb good frt gue for many applcaton. Mot tude of DDA accuracy are lmted to ntegral catterng quantte and, at mot, the angular dependence of S 11. In only a few manucrpt are other catterng quantte tuded. For ntance, Sngham [54] mulated the angular dependence of Mueller matrx element S 4 for phere and le compact partcle, ung CM polarzablty. It wa hown that an accurate mulaton of th element requre maller value of y than for S 11. For x =1. 55 and m =1. a calculaton of S11 wa accurate already for y = 0. 8, whle y 0. wa requred for S4. It wa alo reported that for le compact object lke dc and rod, the requred y wa larger, 0.4 and 0.55 repectvely, becaue of the maller nteracton between the dpole. However, Hoektra and Sloot argued [55] that th effect motly caued by the 15

16 pronounced S 4 entvty to urface roughne, whch gnfcant for maller ze f y fxed. They howed that for x =10.7 and m = 1. 05, very hgh accuracy acheved wth y = 0.66 becaue of the larger number of dpole ued. Internal feld are an ntermedate reult n the DDA. They cannot be drectly compared to the expermental reult; however, all meaured catterng quantte are derved from them. Therefore, a tudy of ther accuracy can reveal greater undertandng of the nature of DDA error. Hoektra et al. [56] performed uch a tudy for LAK polarzablty. Three phere were examned wth x = 9, 9, 5 and m =1. 05, , repectvely. Value of y were 0.44, 0.4, and 0.51 repectvely. The mot gnfcant error n the ampltude of the nternal feld were localzed at the boundary of the phere wth maxmum relatve error of.4%, 19%, and 10% repectvely. Error n S1, S, S 4 were gnfcant only for the thrd phere. It wa hown that for a gven y Re thee error rapdly ncreae wth m but only lghtly depend upon x n the range from 1 to 10. Moreover, the DDA capable of reproducng reonance of Me theory, although ther poton are lghtly hfted (le than 1% n m). Druger and Bronk [57] tuded the accuracy of the nternal feld for ngle and coated phere. They ued x =1. 5, m 1.8, and CM polarzablty. Error n the nternal feld were localzed at the nterface, wth average error larger than 0% for a ngle phere wth m = 1.8 and y = 0. 17, and le than 7% for a ngle and concentrc phere wth m = 1. and y = The core of the concentrc phere ha m =1. 1 and t dameter half the total dameter. The angular dependence of the abolute value of S 1 and S had gnfcant error n the de- and backcatterng. It can be concluded that hape error contrbute motly to the nternal feld near the boundary, and ncreae wth m. All the lterature dcung DDA accuracy how error a a functon of nput parameter and dcretzaton, whch the mot traghtforward way. The only excepton o far the rule of thumb, whch too general and approxmate to be appled n many partcular cae. A more ueful way to preent error to fx the dered accuracy for certan nput parameter and fnd the dcretzaton that reult n uch accuracy. Such an analy can be appled drectly to practcal calculaton and can be ued to derve rgorou etmate of DDA computatonal requrement [58]. In a number of manucrpt the orgn of error n the DDA wa examned to try to eparate and compare hape and dcretzaton error [49,59-6]; however, no defnte concluon were reached. The uncertanty wa due to the ndrect method ued that have nherent nterpretaton problem. Recently, Yurkn et al. [6] propoed a drect method to eparate hape and dcretzaton error, whch can be ued to tudy ther fundamental properte. Th method alo can be appled to tudy the performance of dfferent formulaton amed at decreang hape error, e.g. WD. For example, t ha been hown that the maxmum error of S 11 (θ) for a phere wth x = 5 and m = 1. 5, dcretzed ung 16 dpole per dameter ( y = 0.9 ), are motly due to hape error. However the ame not true for all meaured quantte. In another manucrpt [] t wa uggeted that the dcretzaton error hould decreae more rapdly wth decreang y than hape error. However, t tll hard to deduce a pror the mportance of hape error for a certan catterer and y; hence, further ytematc quanttatve tudy requred.. The DDA for cluter of phere There are two man pecularte when the DDA appled to cluter of phere. Frt, uch partcle are generally le compact, yeldng maller nteracton between dpole. Th lead to a maller condton number of the DDA nteracton matrx and hence fater convergence of the teratve olver (ee Secton 4.1). Second, when the conttuent phere are mall compared to the wavelength, each phere can be modeled a one phercal ubvolume, yeldng ome theoretcal mplfcaton. 16

17 A general theory ext [64] baed on the Me theory (generalzed multpartcle Me oluton (GMM) [65]) that allow for hghly accurate mulaton of cluter of phere. However, when many mall phere are ued one want to mnmze the number of unknown n the lnear ytem. Drect reducton of the GMM to the lowet order (ung only the frt order expanon coeffcent) lead to DDA + CM [64]. Improvng accuracy n the GMM done by accountng for hgher multpole moment, whle the DDA ntroduce hgher order correcton to the coeffcent of the lnear ytem. It not clear how the accuracy of thee two method compare wth each other; however, the former hould lead to a formulaton mlar to a coupled multpole method (Subecton.4) wth a larger number of unknown. DDA-baed method (tartng uually wth the ntegral equaton ntroduced n Secton ) hould be ucceful n makng the formulaton more accurate wthout ncreang the number of unknown, whch the goal for large cluter of mall phere. Moreover, the DDA may employ fat algorthm for olvng the lnear ytem. In th ettng, the fat multpole method (FMM) (ee Subecton 4.5) eem mot promng. It hould be noted, however, that a cluter havng a mall ze parameter (.e. n the electrotatc approxmaton) doe not mply that all expanon coeffcent, except the frt one, are neglgble. Th becaue the ze of the conttuent partcle alo very mall and the feld nde them are far from contant, epecally when the phere are located cloe to each other and have large refractve ndce [66]. Therefore, the DDA doe have ome prncpal dffculte of calculatng catterng by cluter of phere. Mackowk [67], for ntance, found that for ome ytem compoed of phere much maller than the wavelength, up to 10 expanon term were neceary to acheve convergence. In tude of oculatng phere, Ngo et al. [68] proved that the GMM could be chaotc and were able to calculate Lyapunov exponent, and that the low convergence for the touchng phere wa the reult of the ytem lyng n an attractor regon. A recent paper by Markel et al. [69] preented computatonally effcent modfcaton of the GMM n the tatc lmt and demontrated the nuffcency of the DDA to compute catterng properte of fractal aggregate accurately. However, Km et al. [70] howed that the DDA atfactory n calculatng the tatc polarzablty of delectrc nanocluter, epecally of cluter wth a large number of conttuent. The development of DDA-baed method for calculatng lght catterng by cluter of mall phere wa tarted by Jone [71,7], who developed a method mlar to CM. Ikander et al. [4] ued a method equvalent to LAK to calculate catterng of chaned aerool cluter. Th ubject wa further nvetgated by Koaza [7,74]. Lou and Charalampopoulo [75] (LC) further mproved the calculaton of the nteracton term and catterng quantte. Startng from an ntegral equaton for the nternal feld equvalent to Eq. (1), they aumed Eq. (11). After that the ntegral n Eq. (1) and (1) over phercal ubvolume can be evaluated analytcally. The reult for the nteracton term the followng: (0) G j = η( ka) G( r, r j ), (70) where a correcton functon η defned a n x x co x 4 η ( x ) = = 1 (1 10) x + O( x ). (71) x Eq. (0) alo evaluated analytcally, yeldng (0) F ( n) = k η ( ka)( I nn ˆ ˆ) P exp( kr n), (7) C (0) ext nc ( P ) = 4π kη( ka) Im E. (7) The followng expreon for C ab tated wthout dervaton: (0) C ab = 4π kη( ka) Im( PE ). (74) Markel et al. [76] appled the DDA to fractal cluter of phere, and tuded ther optcal properte. However, they have not fxed the polarzablty of a ngle dpole but rather 17

18 treated t a a varable, calculatng the dependence of a cluter optcal charactertc upon t. Putovt et al. [77] argued that the DDA naccurate for touchng phere. They developed a hybrd of the DDA and the GMM, whch conder only par nteracton between phere (a the DDA) but, when calculatng them, account for hgher multpole term. Th formulaton can be condered a the one provdng a more accurate evaluaton of the nteracton term (Eq. (1)), and hence mlar to LC. LC wa compared to DGF and LAK n a C ca computaton of a cluter of 10 partcle for m = and 0.05 ka Dfference between DGF and LAK are le than 1% (a expected), whle the dfference between LC and LAK ncreae quadratcally wth ka, reachng 10% for ka = 0.5. However, a no exact (e.g. GMM) oluton preented, the accuracy of each ndvdual method not clear. Okamoto [4] teted the a 1 -term method for cluter of up to touchng phere. No effectve medum needed n th cae, makng the method ounder. It wa hown that the a 1 - term clearly uperor to LDR n cro-ecton calculaton, when each phere treated a a ngle dpole. Error of the a 1 -term are le than 10% for y 1. when m = For three collnear touchng phere the error are 0% and 40% for y 1.9 and.8 when m = and + repectvely. However, error do not eem to dmnh gnfcantly for mall y (reult are preented only down to y = 0. ). Therefore, the a 1 -term eem utable for obtanng quck crude etmaton of cro ecton. In the equel of th ubecton we menton everal applcaton of the DDA to catterng from cluter of phere. It wa appled to decrbe the catterng by atrophycal dut aggregate [78,79] ung the a 1 -term method. Hull et al. [80] appled CM DDA to Deel oot partcle. LC wa appled [81] to the computaton of lght catterng by randomly branched chan aggregate. Lumme and Rahola [40] tuded catterng properte of cluter of large phere (each modeled by a et of dpole) wth the a 1 -term method conderng atrophycal applcaton. Hage and Greenberg [5] tuded catterng by porou partcle, whch were modeled a cluter of cubcal cell makng ther method equvalent to tandard LAK. Recently the DDA wth LDR wa ued [8] to model catterng by porou dut gran and compare them to approxmate theore, e.g. effectve medum theore. It alo wa ued to tudy lght catterng by fractal aggregate [8], epecally t dependence on the nternal tructure [84]..4 Modfcaton and extenon of the DDA Bourrely et al. [85] propoed to ue mall d to mnmze urface roughne, but larger dpole nde the partcle. Startng wth mall dpole wth CM polarzablty, one dpole combned wth 6 adjacent one (f they all have the ame polarzablty) producng a dpole, located at the ame pont but wth a 7 tme larger polarzablty. Th operaton repeated whle poble. Interacton term are condered n ther mplet form (Eq. (14)). Th method allow the decreae of the hape error wth only a mnor ncreae n the number of dpole. The author howed that th method more than two tme more accurate than CM for ome tet cae. Rouleau and Martn [86] propoed a generalzed em-analytcal method. A dynamc grd ued to evaluate the ntegral n Eq. (1). Frt, a tatc grd bult nde the partcle. Then each pont on the tatc grd ued a an orgn of a phercal coordnate ytem, and the partcle approxmated by an enemble of volume element n thee phercal coordnate. A uual, the polarzaton nde each ubvolume aumed contant, but Eq. (1) can be evaluated analytcally n phercal coordnate. Polarzaton nde a ubvolume obtaned by nterpolaton of t value at the pont of the tatc grd. In addton, adaptve grddng employed, where maller ubvolume are ued at the boundary of the partcle. Mulholland et al. [87] propoed a coupled electrc and magnetc dpole method (CEMD), where a magnetc dpole condered at each ubvolume together wth an electrc 18