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Grzegorczyk et a VoL23,No9/September2006/J.Opt.Soc.Am.A2325 axis of the cylinders, but the theory is directly generaliz N,(k, p)=ikN(kp)ein, able to oblique incidences and spherical particles as well The interactions between the particles are computed us- where the prime denotes a derivative with respect to the ing the Foldy-Lax exact multiple-scattering equations, argument and where the Hankel functions are replaced yielding the expansion coefficients of the total field in the by the Bessel function for Rg Mn and RgNn. Note that ylindrical coordinate system(P, c, 2). The advantage of Eqs. (2)the coefficients an are unknown and are solved for the method is the ability to model an arbitrary number of upon applying the boundary condition on the total tan tion other than the mode truncation in the cylindrical cylinder. Before this is done, the fields inside the cyln o particles of arbitrary sizes, without additional approxima- gential electric and magnetic fields at the boundary of ti wave expansion. The disadvantages are that the particles also need to be expressed as in Eqs. (2)with another set of are required to be of a canonical shape(cylinders in our unknown coefficients. However, since we shall use in this case)and of identical permittivity. Since the number of paper the approach based on the Maxwell stress tensor articles as well as the number of plane-wave incidences the computation of the force on the cylinder due to the in can be arbitrary, both trapping and binding phenomena cident field requires the knowledge of the field outside the can be predicted, as we shall illustrate hereafter. We cylinder only, hence we do not write the field inside for should note that the limitations of in-plane incidence and brevity. When multiple particles are present, Eqs. (2) identical radius(which we shall use later)are easily need to be generalized so that the scattered field of a lifted. In addition, the case of an electric field perpendicu- given particle includes the effects from the incident field lar to the axis of the cylinders is not discussed here but and from the scattered field of all the other particles, ir could be studied in a very similar manner cluding all interactions. The self-consistent formulation proposed by Foldy2 and Lax26 is used here and can be written as 2. CYLINDRICAL WAVE EXPANSION AND FOLDY-LAX APPROACH TO MULTIPLE E(p)=En(p)+∑ECn(p) SCATTERING The problem we are considering is expressed as follows: a total of L infinite cylinders with their axes parallel to Equation(4)states that the final exciting field of particle are located at positions(x(,yo(e=l, .. L). For the sake g is equal to the incident wave plus the scattered waves to of simplicity, all the cylinders are assumed to be identical, particle q from all the other particles e except q itself. A of radius a, real relative permittivity ee, wavenumber ke, a generalization of Eqs. (2), Escat(p) is written as and nonmagnetic. A plane wave with electric field Einc(p)=iEoeik'p is incident onto the system with k=kk Egat(p)=> asaN,(k, p-Pa xk+yky, where a bold symbol denotes a vector and a at denotes a vector of norm 1. It is already well known where the cylindrical functions Nn have to be evaluated at that such incidence, along with the associated magnetic translated origins for each particle q located at pg field,can be decomposed onto a cylindrical coordinate sys- =(xa, y). The coefficients ana) are the generalized version of an and reduce to an in the case of a single particle. Their expression is obtained from Eine(p)=iEoek'p=> aRgN,(k, p), (1a) where Tn is the known T coefficient of a cylinder for the zxKE given polarization, Hine(p) ×k ke n(kca) Jn(ka) k Hocan hl ohere an=Eoine-indi/k. The scattered field in the regg g tside the cylinder is decomposed in a similar fashion B,=-ckH(ka) ,(ka)-ky(ka)()]- n=一x (7b) and where wn are solved from Hap)=:∑aM(,p), an+ T here n is the impedance of the background medium. Ir Eqs.(1)and(2) with ra being the angle between the x axis and the vec M,(k, p)=p-h(kp)eind-akHn(kp)eine, tor joining the centers or particles (e) and(g). equation (8)has been obtained upon applying the translationalaxis of the cylinders, but the theory is directly generalizable to oblique incidences and spherical particles as well. The interactions between the particles are computed using the Foldy–Lax exact multiple-scattering equations, yielding the expansion coefficients of the total field in the cylindrical coordinate system ˆ ,ˆ ,zˆ. The advantage of the method is the ability to model an arbitrary number of particles of arbitrary sizes, without additional approximation other than the mode truncation in the cylindrical wave expansion. The disadvantages are that the particles are required to be of a canonical shape (cylinders in our case) and of identical permittivity. Since the number of particles as well as the number of plane-wave incidences can be arbitrary, both trapping and binding phenomena can be predicted, as we shall illustrate hereafter. We should note that the limitations of in-plane incidence and identical radius (which we shall use later) are easily lifted. In addition, the case of an electric field perpendicular to the axis of the cylinders is not discussed here but could be studied in a very similar manner. 2. CYLINDRICAL WAVE EXPANSION AND FOLDY–LAX APPROACH TO MULTIPLE SCATTERING The problem we are considering is expressed as follows: a total of L infinite cylinders with their axes parallel to zˆ are located at positions x ,y =1,...,L. For the sake of simplicity, all the cylinders are assumed to be identical, of radius a, real relative permittivity c, wavenumber kc, and nonmagnetic. A plane wave with electric field Einc=zˆE0 eik· is incident onto the system with k=kˆ k =xˆ kx+yˆ ky, where a bold symbol denotes a vector and a hat denotes a vector of norm 1. It is already well known that such incidence, along with the associated magnetic field, can be decomposed onto a cylindrical coordinate system as24 Einc = zˆE0 eik· = n=− + anRgNnk,, 1a Hinc = − zˆ kˆ zˆ kˆ E0 0 eik· , 1b where an=E0i n e−ini /k. The scattered field in the region outside the cylinder is decomposed in a similar fashion as Escat = n=− + an s Nnk,, 2a Hscat = 1 i n=− + an s Mnk,, 2b where is the impedance of the background medium. In Eqs. (1) and (2), Mnk, = ˆ in Hn 1 kein − ˆ kHn 1 kein , 3a Nnk, = zˆkNn 1 kein , 3b where the prime denotes a derivative with respect to the argument and where the Hankel functions are replaced by the Bessel function for RgMn and RgNn. Note that in Eqs. (2) the coefficients an s are unknown and are solved for upon applying the boundary condition on the total tangential electric and magnetic fields at the boundary of the cylinder. Before this is done, the fields inside the cylinder also need to be expressed as in Eqs. (2) with another set of unknown coefficients. However, since we shall use in this paper the approach based on the Maxwell stress tensor, the computation of the force on the cylinder due to the incident field requires the knowledge of the field outside the cylinder only; hence we do not write the field inside for brevity. When multiple particles are present, Eqs. (2) need to be generalized so that the scattered field of a given particle includes the effects from the incident field and from the scattered field of all the other particles, including all interactions. The self-consistent formulation proposed by Foldy25 and Lax26 is used here and can be written as24 Eexq = Einc +=1 lq L Escat . 4 Equation (4) states that the final exciting field of particle q is equal to the incident wave plus the scattered waves to particle q from all the other particles except q itself. As a generalization of Eqs. (2), Escat q is written as Escat q = n=− + an sq Nnk, − q, 5 where the cylindrical functions Nn have to be evaluated at translated origins for each particle q located at q =xq ,yq. The coefficients an sq are the generalized version of an s and reduce to an s in the case of a single particle. Their expression is obtained from an sq = Tnwn q , 6 where Tn is the known T coefficient of a cylinder for the given polarization, Tn = kc k Jnkca Hn 1 ka Bn − Jnka Hn 1 ka , 7a Bn = 2ik akc kHn 1 kaJnkca − kcJn kcaHn 1 ka−1, 7b and where wn q are solved from wn q = eik·qan + n=− + =1 lq L Hn−n 1 kl − qe−in−nlqTn wn , 8 with q being the angle between the xˆ axis and the vector joining the centers or particles and q. Equation (8) has been obtained upon applying the translational Grzegorczyk et al. Vol. 23, No. 9/September 2006/J. Opt. Soc. Am. A 2325