Th e op tic a l c r oss-s e c tion th e ore m w ith in c i d e n t e ld s

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1 journal of modern optc, 1999, vol. 46, no. 5, 891± 899 Th e op tc a l c r o- e c ton th e ore m th n c d e n t e ld c on ta n n g e v a n e c e n t c o m p on e n t P. SCOTT CARNEY Department of Phyc and Atronomy and the Rocheter Theory Center for Optcal Scence and Engneerng Unverty of Rocheter, Rocheter, Ne York 14627, USA (Receved 8 Aprl 1998; reved 4 October 1998) Abtrac t. The optcal cro-ecton theorem extended to cae n hch the ncdent eld contan evanecent ave. Phycal nterpretaton are dcued. Some explct example are gven and poble applcaton are propoed. 1. In tro d u c to n The optcal theorem generally formulated for the cae here the ncdent eld a ngle homogeneou plane ave [1, 2] and for par of plane ave ncdent on non-aborbng meda [3, p. 47]. More recently the theorem ha been generalzed for arbtrary ncdent-free eld [4], that eld contanng homogeneou plane ave mode only. In th paper a generalzaton of the theorem derved hch apple hen evanecent ave are preent n the ncdent eld. The generalzaton of the optcal theorem obtaned n [4] ha the form here P e 4pk I, f, d2 d 2, 1, a a 2 and P e the poer extnguhed from the ncdent beam by catterng and aborpton. the angular correlaton functon of the ncdent eld hch decrbed tattcally by an enemble, each member of hch a uperpoton of plane ave th randomly dtrbuted ampltude a and I denote the magnary part. Further, f, the catterng ampltude n the drecton pec ed by the unt vector hen the ncdent ave a plane ave hch propagate n the drecton pec ed by. More explctly, the total eld gven by r r r, 3 here Journal of Modern Optc ISSN 0950± 0340 prnt/issn 1362± 3044 onlne Ñ 1999 Taylor & Franc Ltd

2 R 892 P. S. Carney r a exp kr d 2, 4 r a f, exp kr r d 2, 5 r r r r, 6 p^x q^y 1 p 2 q 2 1/ 2^z, 7 p^x q^y, 8 d 2 dp dq, 9 and ^x, ^y and ^z are the Cartean unt vector n the x, y and z drecton repectvely. The aborbed poer gven by the expreon P a 1 k I R Ñ dr, 10 herer a urface hch completely encloed the catterer. The poer carred by the cattered eld gven by the expreon Notng that nce P 1 k I R Ñ dr. 11 at e the homogeneou Helmholtz equaton I Ñ dr 0, 12 e nd that for any determntc eld the extnguhed poer gven by the expreon P e 1 k I R Ñ Ñ dr. 13 The theorem gven by equaton (1) ha been derved by ntegratng the rghthand de of equaton (13) over the urface R of a large phere of radu R, centred at ome pont n the catterer. The extnguhed poer ndependent of the ze of the phere over hch the ux ntegrated o long a the phere encloe the catterer. For kr u cently large the ntegral may be evaluted by the method of tatonary phae [5]. Such a dervaton vald only th free ncdent eld, that th ncdent eld of the form gven by equaton (4) for hch the angular pectrum ampltude a may be non-zero only for real unt vector o that 2 p 2 q 2 < It hould be noted that, although equaton (1) a generalzaton of hat uually referred to a the optcal cro-ecton theorem, e have abandoned the concept of cro-ecton and ntead deal th the unnormalzed extnguhed poer P e. The cro-ecton de ned a the extnguhed poer dvded by the ncdent poer per unt area. The ncdent poer per unt area for anythng other than a ngle plane ave an ambguou quantty hch depend not only on the orentaton of the plane of projecton but alo on t abolute poton.

3 2. De r v a t on o f th e th e or e m fo r e v an e c e n t n c d e n t e ld We hall generalze the theorem expreed by equaton (1) to nclude any ncdent eld hch propagate nto the half-pace z > 0. To do th e hall rt extend the optcal theorem to tuaton here the ncdent eld evanecent, that eld for hch the ncdent plane ave ha the form exp kr 0, 15 here the z component of the unt vector 0 ( 0 0 1) trctly magnary. Such a ave decay exponentally n ampltude th ncreang value of z. The extnguhed poer n th cae tll taken to be the um of the aborbed poer and the poer carred by the cattered eld alone and equaton (13) tll hold. Hoever, one mut no note that, n order to calculate the poer aborbed by the catterer a done n equaton (10), the ource of the ncdent eld mut le entrely outde the urface R of ntegraton. Snce any ncdent eld contanng evanecent component mut be generated at nte dtance from the catterer, the urface R cannot mply be taken a large a e lke a done n calculaton nvolvng aymptotc method for ncdent eld hch are homogeneou. The ource of homogeneou eld may be regarded a beng located n ntely far from the catterer and thu alay outde R. Thu R mut le at a nte dtance from the catterer and e abandon the aymptotc method employed to derve the theorem for homogeneou eld. We can determne the poer extnguhed from an ncdent evanecent ave by dentfyng the varable n hch the extnguhed poer mut be analytc and then proceed to contnue analytcally, n everal varable, the reult pertanng to ncdent homogeneou ave. It ha been hon[6] that, f the unt vector j expreed n polar coordnate (µ j, u j), ^x j nµ j co u j ^y n µ j n u j ^z co µ j, 16 here ^x, ^y and ^z are the unt vector along the x, y and z drecton repectvely, then the catterng ampltude f 1, 2 an analytc functon of µ 1. Furthermore, n ve of the recprocty relaton hch hold for both real and complex drecton of propagaton [7], f 1, 2 f 2, 1, 17 the catterng ampltude f 1, 2 mut alo be an analytc functon of µ 2. The complex unt vector correpondng to the evanecent component of the eld may be expreed n term of complex µ and real u. The analytc contnuaton of f 1, 2 to complex drecton of propagaton unque and unambguou and gve u a complete decrpton of the cattered eld ncludng the near zone.² Let u conder the poer extnguhed from a coherent beam contng of to plane ave Optcal cro-ecton theorem 893 a1 exp kr 1 a2 exp kr ² We refer the reader to[6], pec cally equaton (2.10). The ampltude n that reference related to the catterng ampltude n our notaton by A a, b f ^z, here gven by equaton (2.9) of [6]. For applcaton of analytc contnuaton to the calculaton of the cattered eld and the extncton coe cent aocated th the reacton of evanecent ave, ee reference[8].

4 894 P. S. Carney We denote the extnguhed poer n th cae a P e a 1, a 2. Makng ue of equaton (13)) th quantty may be expreed n the form P e a1, a2 r 4p 1 a1a 1 f, 1 exp kr 1 2 a 2a 2 f, 2 exp kr 2 2 a 1 a 2 f, 1 exp kr 2 1 a 2 a 1 f, 2 exp kr 1 dx, 19 here dx n µdµdu, µ and u beng the polar coordnate of the unt vector. Becaue the real part of the ntegral taken, th quantty not analytc n the angular varable µ 1 or µ 2. In order to explot the analytc properte of the catterng ampltude, t ll prove ueful to de ne a quantty P e 1, P e 1, P e 1, 1 P e 1, Ung equaton (19) hch apple for real a ell a complex 1 and 2, e nd that 2r 4p 1 f, 2 exp kr 1 2 f, 1 exp kr 2 dx, 21 for all, generally complex, unt vector 1 and 2. In ve of the analytc properte of f 1, 2 t follo from equaton (21) and Theorem of [9] that an analytc functon of the to complex varable µ 1 and µ 2 eparately. By Hartog theorem [10, theorem 2, p. 32] t follo that analytc n the pace of the to complex varable µ 1 and µ 2. The angular correlaton functon of the ncdent eld contng of to monochromatc plane ave de ned by equaton (18) gven by the expreon, a1d a 1 d 2 1 a2d 2 1 a 2 d , 22 here d 2 the to-dmenonal Drac deltafuncton. It follo from equaton (1) that, for real 1 and 2, P e 4p a1, a2 k I a1 2 f 1, 1 a 1 a2 f 1, 2 a 2 a1f 2, 1 a2 2 f 2, 2. Ung equaton (23) and the de nton (20) e nd that 8p k f 1, 2 f 2, 1, 24 for real 1 and 2. Snce, gven by equaton (21), an analytc functon of µ 1 and µ 2 the expreon on the rght-hand de of equaton (24) the boundary value on the real µ 1 and µ 2 axe of an analytc functon of to complex varable. We eek the contnuaton of equaton (24) n the pace of complex µ 1 and µ 2. We recall that f g z an analytc functon of z, then o g z and that g z an analytc functon of z. We ee that there precely one functon, analytc n 23

5 Optcal cro-ecton theorem 895 µ 1 and µ 2, hoe boundary value on the real µ 1 and µ 2 axe equaton (24), namely the functon 8p k f 1, 2 f 2, The unquene of th contnuaton follo from a bac theorem of analytc functon theory. If e take 1 2 0, e deduce from the de nton (20) that 4P e 1, The poer extnguhed from a ngle ncdent plane ave, be t evanecent or homogeneou, therefore gven by the expreon P e 4pk If 0, Makng ue of the lnearty of the problem, a a done n [4] to obtan equaton (1), e nd that the generalzed optcal theorem th arbtrary ncdent eld ha the form P e 4pk I, f, d2 d 2, 28 hch reduce, a t hould, to equaton (1) hen the ncdent eld doe not contan evanecent ave. 3. Ph yc al n te r p r e tat on When the ncdent eld homogeneou, equaton (27) ha a ell-knon phycal nterpretaton. Conervaton of energy requre that the extnguhed poer P e mut be removed from the ncdent eld, evdently by an nterference mechanm. A one move progrevely farther from the catterer, there only one component of the cattered eld hch propagate along th the ncdent plane ave, namely the forard-cattered eld. We may therefore conclude thout any calculaton that P e mut be a functon of the forard± catterng ampltude f 0, 0. The exact functonal relatonhp ha to be orked out and gven by equaton (27) th 0 beng real for homogeneou ave [1± 4]. If the ncdent ave evanecent, the phycal pcture become more complcated. Evanecent ave cannot ext n a ource-free unbounded three-dmenonal pace. We conder then, a a partcular model for the generaton of evanecent ave, to half-pace, the left half-pace beng unformly lled th a delectrc th a real ndex of refracton greater than unty at the frequency hch e conder. The rght half-pace taken to be vacuum, except for the preence of a catterer of nte extent. In the abence of a catterer, a ngle evanecent ave may be produced n the rght half-pace by the total nternal re ecton of a homogeneou plane ave ncdent from the left. The nteracton of the evanecent ave and the catterer produce a cattered eld that carre a nte (non-zero) amount of poer to the far zone.² Furthermore, the catterer telf may aborb ome of the ncdent poer. ² The catterng ampltude f 1, 2 n th cae mut take nto account the preence of the delectrc half-pace. Wthn the accuracy of the rt Born approxmaton (ngle catterng) the catterng ampltude unchanged from that of the catterer n free pace.

6 896 P. S. Carney Fgure 1. An evanecent plane ave n the rght half-pace generated by total nternal re ecton of a ave hch propagate n the drecton pec ed by the unt vector p^x q^y m^z n the left half-pace. The reultng evanecent ave correpond to a complex drecton of propagaton pec ed by the unt vector np^x nq^y m ^z th m n 2 p 2 n 2 q 2 1 1/ 2, aumng that n 2 < p 2 q 2. The re ected ave propagate n the drecton pec ed by the unt vector ~ p^x q^y m^z. A and B are contant hch depend on the value of the ndex of refracton n the left half-pace. The proce of catterng generate another evanecent ave hch correpond to a complex drecton of propagaton and couple back nto the delectrc half-pace a a homogeneou plane ave hch propagate n the drecton of the re ected ave, ~. We mut account for the poer cattered and aborbed by the catterer n the rght half-pace by depleton of the re ected beam n the left half-pace. There precely one component of the cattered eld hch may couple back nto the left halfpace and propagate concurrently th the re ected beam and that the evanecent component of the cattered eld correpondng to the complex drecton of propagaton pec ed by 0. The poer mut therefore be a functon of the catterng ampltude n the drecton pec ed by 0 for an ncdent evanecent plane ave th complex drecton pec ed by 0 ( gure 1). Equaton (27) expree th relatonhp. There are other ay that one mght envage the generaton of evanecent plane ave. One mght conder the evanecent ave to be a component of a eld produced by a ource near the catterer hch doe not telf apprecably nteract th the cattered eld. It ha been hon that an evanecent ave may alo be repreented a a ngular lmt of a uperpoton of homogeneou plane ave, a partcular example beng the tuaton here evanecent ave are generated n the at plane of a narro Gauan beam [11]. 4. Exa m p le of ap p lc a to n To llutrate the man reult derved n ecton 2, e conder the catterng of an evanecent ave by a delectrc medum characterzed by a delectrc uceptblty h r o that 2 r k 2 r 4p k 2 h r r. 29

7 Aumng that the uceptblty mall compared th unty and from pont to pont change moothly throughout the catterer, e may calculate the poer perturbatvely (.e. by mean of the Born ere). We nd that, to rt order n h, the poer extnguhed from a ngle plane ave gven by the expreon P e 4p ki h r exp kr 0 0 d 3 r. 30 When the ncdent eld a homogeneou plane ave, the extnguhed poer yeld nformaton only about the volume ntegral of the magnary part of the uceptblty. We can ee that, hen the ncdent eld evanecent, the extnguhng poer related to a component of the analytc contnuaton of the Fourer tranform of the magnary part of the ueptblty. If t knon a pror that the medum may be decrbed by a functon hch eparable n the patal varable n the form Ih r a z b q, 31 here z the dtance along the drecton of decay of the evanecent ave and q a vector orthogonal to the z ax, then, by performng experment th evanecent ave hch decay at varou rate, e may contruct a Laplace tranform of the z dependence of the magnary part of the uceptblty, that the z dependence of the aborptve part a z of the medum. A a pec c example of th tuaton, conder a lab of materal hoe uceptblty vare moothly from zero to ome maxmum value and then back to zero over the thckne of the lab (th n keepng th the condton requred for the valdty of the rt Born approxmaton), for example a z a 0 n p z 32 t for 0 < z < t here t the thckne of the lab. Wrtng e nd that the poer gven by the expreon b q d 2 q Ab, 33 P e 1 exp k 0 t 4ktAb a 0 1 k 0z t 2, 34 hch proportonal to the Laplace tranform of a z th the varable of tranformaton beng k 0z ( gure 2). We mght alo apply th reult to catterng on a hghly dordered nonaborbable catterer. Explctly, e conder the tuaton here and Optcal cro-ecton theorem 897 I h r 0 35 C r, r C h r d r r, 36 h h h here C r, r r r the to-pont correlaton functon of the uceptblty. It ha been hon [4] that for uch a medum the catterng ampltude to the econd order of perturbaton gven by the expreon

8 898 P. S. Carney Fgure 2. The extnguhed poer a a functon of the decay contant of the evanecent ave, 0z, for the catterer decrbed by equaton (32) th the avelength of the ncdent eld beng uch that kt 5. f, 0 k 4 V V C r, r exp k r r r r Ung equaton (37) and (27) e ee that n th cae exp k r 0 r d 3 r d 2 r. 37 P e 4p k 4 C h r exp k 0 0 r d 3 r, 38 V hch gve the famlar reult that, hen the ncdent eld homogeneou, the extnguhed poer proportonal to the zero patal frequency component of the uctuaton of the medum. When the ncdent eld evanecent, the extnguhed poer related to a component of the analytc contnuaton of the Fourer tranform of the patal uctuaton of the medum. In both example gven here, the nformaton obtaned n meaurng the extnguhed poer may be ueful n calculatng a uper-reolved mage of the catterng object. Tradtonal method of uper-reoluton rely, ether explctly or mplctly, on analytc contnuaton of the Fourer component of the catterer outde the Eald lmtng phere. Thee method are computatonally untable becaue of the exponental groth of noe n the proce. The ablty to meaure the Fourer tranform of the object functon at pont along the magnary axe of the complex Fourer varable may allo for a check agant run-aay exponental error. 5. Co n c lu o n We have found a generalzaton of the optcal theorem to tuaton n hch the ncdent ave evanecent. In mot practcal tuaton the evanecent component of the ncdent eld may be neglected becaue they fall o exponentally th ncreang propagaton dtance. Hoever, there are ome tuaton, notably n near- eld optc, n hch the catterer n cloe proxmty to a ource of nhomogeneou eld. In thee tuaton the evanecent component of the

9 Optcal cro-ecton theorem 899 ncdent eld may nteract th the catterer and contrbute gn cantly to the poer radated to the far eld. A c kn o l e d g m e n t The author ould lke to thank Dr Eml Wolf for careful readng of everal draft of th paper and for many helpful dcuon and comment. He alo oblged to Dr Bor Zel dovch for ome helpful comment and dcuon. Th reearch a upported by the US Department of Energy under grant DE-FG02-90ER and by the Ar Force O ce of Scent c Reearch under grant F R e fe r e n c e [1] Feenberg, E., 1932, Phy. Rev., 40, 40. [2] van der Hult, H. C., 1949, Phyca, 15, 740. [3] Neton, R. G., 1982, Scatterng Theory of Wave and Partcle (Ne York: Sprnger). [4] Carney, P. S., Wolf, E., and Agaal, G. S., 1997, J. opt. Soc. Am. A, 14, [5] Bleten, N., and Handelman, R. A., 1975, Aymptotc Expanon of Integral (Ne York: Holt, Rnehart and Wnton). [6] Wolf, E., and Neto-Veperna, M., 1985, J. opt. Soc. Am. A, 2, 886. [7] Carmnat, R., Neto-Veperna, M., and Grffet, J., 1998, J. opt. Soc. Am. A, 15, 706. [8] Qunten, M., Pack, A., and Wannemacher,R., 1999, Appl. Phy. B, 68, 87; Zvyagn, A. V. and Goto, K., 1998, J. opt. Soc. Am. A, 15, 3003; Lu, C., Kaer, J., Lange, S. and Van Labeke, D., 1993, J. mod. Opt., 40, 1239; Che, H., Wang, D.-S. and Kerker, M., 1979, Appl. Opt., 18, [9] Dettman, J. W., 1965, Appled Complex V arable (Ne York: Dover Publcaton). [10] Bochner, S., and Martn, W. T., 1948, Several Complex V arable (Prnceton Unvrty Pre). [11] Berry, M., 1994, J. Phy. A, 27, L391.