Quasi-separable t-scattering operator approach to local field direct calculations in multiple scattering problems

Recurrent relation (98) for determinant of Eqs.(96) system matrix in Jacobi's approximation enables one to get solution to this system analytically, provided the incident electric field excites the current inside the fist single vibrator-dipole only

(see some details to solution in Appendix C)

, (100)

The second Eq.(100) is dispersion equation for the complex variable . In the middle part of a long linear array of vibrator-dipoles, when and vibrator position number is fixed, Eq.(100) gives asymptotically

, (101)

where and is a wave number. Having known the wave number, one can evaluate as usually [41] corresponding phase velocity and group velocity putting

, , (102)

The frequency dependence of wave number is obtained by solution of dispersion Eq.(100). Being complicate in general, this dispersion equation is simple resolved analytically in physically interesting case of and when wave number takes a form

, (103)

Apply the obtained form (103) for dispersion equation to the limit of small distances between linear array vibrator-dipoles tuned to half wavelength when according Eq.(88) the coupling factor approximates to minus one and wave phase velocity in Eq.(102) due to Eq.(103) is given by

(104)

where denotes the light speed in free space. To evaluate the group velocity one needs considering the case of weakly detuned vibrator-dipoles, which length is lightly different from half wavelength and vibrator-dipole input impedance is defined according to [48] via relations

(105)

As one can verify at resonance frequency the input impedance becomes pure real and at tuned frequency the input impedance value is consistent with Eq.(88). Substituting now the weakly detuned vibrator-dipole impedance (105) into expression for in Eq.(88) and the last into Eq.(102) with accounting Eq.(103) gives for group velocity at resonance frequency a value

(106)

The obtained value shows that group velocity of currents' exciting propagation between close packed resonance vibrator-dipoles is negative in sign and small in magnitude compared with light speed in vacuum.

Return to analytic solution (100) to Eq.(96) system. Bearing in mind definition (101) for wave number, this solution describes rather standing wave of currents' exciting along linear array of vibrator-dipoles than propagating wave, though asymptotics (101) for the middle part of a long linear array of vibrator-dipoles when and is fixed describes propagating wave. Consider now a limit for the right end part of the long linear array corresponding to solution in Eq.(97). Asymptotics of analytic solution (100) for the right end describes a standing wave, with exponentially decreasing amplitude as in accordance to Fig. 6(c). Especially interesting from physical point of view to note a special case , which realized for dispersion equation (103) and when above amplitude exponentially decreasing effect is canceled. In this case the asymptotics for expansion coefficient (97) of current excited inside the vibrator-dipole with number takes a form

(107)

This asymptotics substantially oscillates in dependence on as was noticed on the Fig. 6.

Conclusion

In this work, we have presented obtained for the first time analytic solution to fundamental in electromagnetic wave multiple scattering theory Lippmann-Schwinger (LS) integral equation for T-scattering operator of electric wave field by dielectric and conducting nonmagnetic particle of arbitrary size and shape. The solution is derived with the aid of a chosen vector expansion functions' basis and written as sum of separable scattering operators weighted by inverse of a generating matrix and named quasi-separable (QS) T-scattering operator. The QS solution to LS equation is generalized for ensemble of coupled particles in free space as well as for coupled particles inside unit cell of electromagnetic crystal. An equations' system for self consistent currents excited inside coupled particles was derived also in QS form.

In the case of a single spherical particle we have verified that vector expansion functions' basis chosen as vector spherical wave functions the QS T-scattering operator gives the Mie solution for incident transverse plane wave scattered from particle and transmitted inside particle. The elements of diagonal generating matrix are presented for this case in terms of Mie scattering coefficients and special bilinear functionals of spherical vector wave functions on spherical particle volume. We considered also a principally another choosing the vector expansion functions' defined on finite elements of particle volume and named conditionally pre-Haar basis. It was shown that such “basis”, even including finite number of expansions' functions, leads to QS simplified scattering potential operator and automatically to a corresponding exact solution to LS equation, which tends to solution of LS equation with actual, not simplified scattering potential operator when number expansions' function of pre-Haar basis becomes infinite.

Mentioned above equations' system for self consistent currents excited inside coupled particles can be resolved in general case with the aid of recursive procedure, which is appeared at a particle attachment in spirit of invariant imbedding method. But for some interesting special cases this system is resolved via simple methods analytically. On this way were considered such phenomena as artificial double diamagnetic-paramagnetic narrow peak in metamaterial with unit cell of coupled plasmonic particles; creation for eigenmodes with overtones in periodic arrays of particles with coupling matrix of stochastic property; extinction rate for transfer of currents' exciting in linear array of particles with coupling matrix of Jacobi's property and standing and propagation wave phenomenon for such transfer.

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Apendix A

Diagonal generating matrix (55) in terms of Mie scattering coefficients and bilinear -functional of spherical vector wave functions

Let us denote a bilinear functional of spherical vector wave functions on spherical particle volume defined by

(A1)

and similarly

(A2)

Here an auxiliary matrix is used

(A3)

and is Riccati-Bessel function (see [43], page 129). The basic result of this Appendix A consists in the following formulas for diagonal generating matrix (55) elements

(A4)

There are also simple relations between Mie scattering and transmission coefficients

(A5)

that can help to understand the derivation of Eqs.(58) and (59) with the aid of QS T-scattering operator (56).

In electric and magnetic dipole approximation the QS T-scattering operator (56) becomes equal to a sum

(A6)

where the RHS first “electric” and second “magnetic” terms are created by Eq.(56) second and first terms, respectively, taken in approximation = 1, = 0, 1. For the case of small spherical particle , we find

(A7)

with and being the spherical particle electric and magnetic susceptibilities, respectively.

In the first formula (A7) for QS T-scattering electric dipole approximation a Dirac delta-function is appeared, actually, as expression

(A8)

and similarly when argument is replaced to , in accordance with Dirac delta-function analog definition in Eqs.(63) by considering functions defined on particle finite elements. Formulas (A7) were obtained in ref.[17] via the Hertz's vector of electric dipole and magnetic dipole scattering study. One can rewrite the electric and magnetic dipole approximations (A7) for T-scattering operator in QS form (13) as

(A9)

where unit vectors are defined in subsection 7.2. Note that electric dipole vector expansion function in Eq.(A9) with analog Dirac delta-function (A8) satisfies automatically the solenoidal restriction in the last Eq.(6) for points laying strictly inside spherical particle domain. With accounting points on spherical particle surface or in the case of true Dirac delta-function the selenoidal restriction on electric dipole vector expansion function is verified with the aid of generalized function theory (see next Appendix B). Magnetic dipole vector expansion function in Eq.(A9) satisfies the solenoidal restriction automatically.

Apendix B

Solenoidal restriction on finite element vector expansions' functions (65)

Let us verify that vector expansions' functions (65) defined on finite elements of particle volume satisfy with corresponding accuracy to solenoidal restriction in the last Eq. (6) in weak sense of theory of generalized functions [51]. For points lying strictly inside particle volume domain the solenoidal restriction is satisfied exactly.

Take divergence of a vector function (65) and integrate this differential operation with product by a test smooth function of compact support. We obtain

(B1)

We suppose for simplicity that all subdomains have form of cubes with their edges being parallel to the axes of the Cartesian coordinate system. Denoting the test function averaged over cube 2D section perpendicular to axis one can transform

(B2)

where is subdomain centre coordinate and denotes the cube edge length, with . In the limit of Eq.(B2) RHS tends to zero as

Apendix С

Analytic solution to Eqs.(96) system in Jacobis' approximation

Turn to recurrent relation (98) for Eqs.(96) system matrix determinant. One can directly verify the following solution to this recurrent relation

(C1)

Solution in Eq.(100) to Eqs.(96) system is written out first as

(C2)

that is verified directly again. Transformation of solution (C2) to the form in Eq.(100) is performed using Eq.(C1).

quantum wave electric field

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