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the discrete dipole approximation ( dda ) provides a flexible and general method to calculate scattering and absorption of light by objects of arbitrary geometry @xcite . the approximation consists of replacing the continuum target of interest by an array of polarizable points , which acquire oscillating electric dipole moments in response to the electric field due to the incident wave plus all of the other dipoles . broadly speaking , there are two criteria determining the accuracy of the approximation :    * the interdipole separation @xmath1 should be small compared to the wavelength of the radiation in the material ( @xmath2 , where @xmath0 is the complex refractive index , and @xmath3 is the wavenumber _ in vacuo _ ) ; * the interdipole separation @xmath1 should be small enough to resolve structural dimensions in the target .    with modern workstations , it is now feasible to carry out dda calculations on targets containing up to @xmath4 dipoles @xcite . in addition to calculation of scattering and absorption cross sections , the dda has recently been applied to computation of forces and torques on illuminated particles @xcite    if the dipoles are located on a cubic lattice , then in the limit where the interdipole separation @xmath5 , the familiar clausius - mossotti relation ( see , e.g , @xcite ) can be used to determine the choice of dipole polarizabilities required so that the dipole array will approximate a continuum target with dielectric constant @xmath6 . @xcite showed how this estimate for the dipole polarizabilities should be modified to include radiative reaction corrections , and @xcite derived the @xmath7 $ ] corrections required so that an infinite cubic lattice would have the same dispersion relation as a continuum of given dielectric constant . nearly all dda calculations to date have assumed the dipoles to be located on a cubic lattice . if , instead , a rectangular lattice is used , it will still be possible to apply fft techniques to the discrete dipole approximation , in essentially the same way as has been done for a cubic lattice @xcite . however , the ability to use different lattice constants in different directions might be useful in representing certain target geometries . the objective of the present report is to obtain the dispersion relation for propagation of electromagnetic waves on a rectangular lattice of polarizable points . this dispersion relation will be expanded in powers of @xmath8 , where @xmath1 is the characteristic interdipole separation . for a lattice of specified dipole polarizabilities , this will allow us to determine the dispersion relation for electromagnetic waves propagating on the lattice when @xmath9 . alternatively , if we require the lattice to propagate waves with a particular dispersion relation , inversion of the lattice dispersion relation will provide a prescription for assigning dipole polarizabilities when seeking to approximate a continuum material with a rectangular lattice of polarizable points . the problem of wave propagation on an infinite polarizable cubic lattice of point dipoles has been solved previously by ( * ? ? ? * herafter dg93 ) . here we describe the generalization of this problem to rectangular lattices . consider an infinite rectangular lattice with lattice sites at _ = ( n_1 d_1,n_2 d_2,n_3 d_3 ) , where the @xmath10 are integers and the lattice constants @xmath11 are , in general , all different . the density @xmath12 of dipoles is just n = 1d_1d_2d_3  , and it will be convenient to define the characteristic lattice length scale d ( d_1d_2d_3)^1/3  . since the lattice is anisotropic , the polarizability @xmath13 of the dipoles located at lattice sites is a tensor , i.e. , the polarization is ( ) =( ) . thus the polarization vector is not , in general , parallel to the electric field or to the vector potential . from charge conservation we have = 0 , or , since = t , [ eq:4 ] we obtain the transversality condition = 0 . [ eq:5 ] we assume that the dipole moment at the lattice site @xmath14 is _ ( t)= _ 0(0 ) e^i_-it . [ eq:6 ] with the lorentz gauge condition + 1ct = 0 and equation ( [ eq:4 ] ) , the vector potential @xmath15 satisfies the wave equation @xmath16      using these dimensionless quantities , the mode equation ( [ eq:21 ] ) may be written _ l=1 ^ 3 m_il _ lj e_j = e_i , [ eq : mode_equation ] where m_ij _ - d^3^  , [ eq:31 ] and equation ( 25 ) becomes _ _ j=1 ^ 3 _ i _ ij e_j = 0 . [ eq:32 ] it should be noted that the mode equation for a cubic lattice has the same form as equation ( [ eq : mode_equation ] ) . the matrix elements @xmath30 in equation ( [ eq:31 ] ) also have the same form as the matrix elements for a cubic lattice but with the vector @xmath31 replaced by @xmath29 for a rectangular lattice . since the matrix elements for a cubic lattice have been calculated previously , it is convenient to rewrite ( [ eq:31 ] ) as m_ij = m_ij^c + t_ij , [ eq:33 ] where @xmath32 is the matrix element for a cubic lattice , and @xmath33 is t_ij _ - _  . [ eq:34 ] for a given polarizability tensor @xmath34 , the mode equation ( [ eq : mode_equation ] ) allows one to determine the dispersion relation @xmath35 . in the present case , the dispersion relation is known : = m , where @xmath36 is the ( complex ) refractive index and @xmath37 is a unit vector in the direction of propagation , which is fixed . from equation ( [ eq:32 ] ) for a rectangular lattice , the polarization rather than the vector potential is perpendicular to the direction of propagation , i.e. , _ j=1 ^ 3 a_i _ ij e_j = 0 . [ eq:36 ] therefore , the mode equation ( [ eq : mode_equation ] ) can be used to determine dipole polarizabilities @xmath38 such that equation ( [ eq:36 ] ) is satisfied . we seek the dipole polarizabilities in the limit @xmath39 . in this limit , one can write _ ij^(0 ) + _ ij^(1 ) + o(^4 )  , m_ij^c = m_ij^c(0 ) + m_ij^c(1 ) + o(^4 )  , t_ij = t_ij^(0 ) + t_ij^(1 ) + o(^4 )  , = ^(0 ) + ^(1 ) + o(^4 )  , where superscripts 0 and 1 designate the zero frequency limit and the leading order corrections , respectively . the low - frequency expansion of the matrix element @xmath30 for a cubic lattice was given by dg93 and is : m_ij^c(0 ) = m^2 + 23(m^2 - 1)_ij - m^2m^2 - 1a_i a_j  , [ eq:41 ]    m_ij^c(1 ) = ^2_ij + m^2 ^ 2(1-_ij)c_3a_i a_j- 4 ^ 2i3_ij^3  , [ eq:42 ] where @xmath40 from equation ( [ eq:34 ] ) , one finds t_ij^(0 ) = _ ij r_0 ( i )  , [ eq:43 ] where r_0(i)= _ - _ [ eq : r_0 ]  . note that @xmath41 , so that there are only two independent sums which require evaluation . numerical evaluation is discussed in the appendix . @xmath42   \nonumber\\               & - & \nu^2 \delta_{ij }     \left [ r_1 + ( m^2 -1 ) r_2(i ) + 8m^2 a_i^2 r_3 ( i , i )     - 4m^2 \sum_{l=1}^3 a_l^2 r_3(i , l ) \right ] \ , ,   \label{eq:44}\end{aligned}\ ] ] where r_1 = _ - _ , r_2(i ) = _ - _ , r_3(i , j ) = _ - _ . [ eq : r_3 ] in the limit of cubic lattice , @xmath43 , and all the sums @xmath44 in equations ( [ eq : r_0 ] ) - ( [ eq : r_3 ] ) vanish . for a general rectangular lattice , these sums must be evaluated for each set of new lattice constants . care must be used in evaluation of these sums as described in the next section . we note that @xmath45 and @xmath46 may be obtained from @xmath47 : r_1 = _ i=1 ^ 3_j=1 ^ 3 r_3(i , j ) [ eq : r_1fromr_3 ]  ; r_2(i ) = _ j=1 ^ 3 r_3(i , j ) [ eq : r_2fromr_3 ]  . since @xmath48 , six independent quantities suffice to determine @xmath45 , @xmath46 , and @xmath47 .      in the static limit the polarizability tensor @xmath49 is diagonal , and the mode equation and transversality condition become : _ j=1 ^ 3 _ jj^(0 ) e_j^(0 ) = e_i^(0 ) , [ eq:48 ] _ i=1 ^ 3 a_i _ ii^(0 ) e_i^(0 ) = 0 . [ eq:49 ] substituting ( [ eq:41 ] ) and ( [ eq:43 ] ) into ( [ eq:48 ] ) and taking into account condition ( [ eq:49 ] ) one gets _ ii^(0 ) e_i^(0 ) = e_i^(0 ) . [ eq:50 ] for @xmath50 , the static dipole polarizability for a rectangular lattice is _ ii^(0 ) = _ cm 1 + _ cm r_0(i ) , [ eq:51 ] where the clausius - mossotti polarizability is _ cm . it is evident from equation ( [ eq:51 ] ) that the static polarizability for a cubic lattice reduces to the clausius - mossotti result . if @xmath51 in one of the directions , then the polarizability in this direction is undetermined by equation ( [ eq:50 ] ) . however , since the result ( [ eq:51 ] ) is valid for values of @xmath52 arbitrarily close to zero , we take ( [ eq:51 ] ) to be the solution for all directions of the electric field subject to equation ( [ eq:49 ] ) . we now proceed to obtain the leading corrections for small @xmath53 . the first order mode equation and transversality condition are : _ l=1 ^ 3 m_il^(1 ) _ lj^(0 ) e_j^(0 ) + _ j=1 ^ 3 _ l=1 ^ 3 m_il^(0 ) = e_i^(1 ) , [ eq:53 ] _ j=1 ^ 3 a_i = 0 . [ eq : transversality ] from equations ( [ eq:41 ] ) , ( [ eq:43 ] ) , ( [ eq:50 ] ) , and ( [ eq : transversality ] ) one gets @xmath54    & = & \sum_{j=1}^3 \left [ { m^2 + 2 \over 3(m^2 -1 ) } + r_0(i ) \right]\ ,    \left [ \gamma_{ij}^{(1)}\ , e_j^{(0 ) }    +   \gamma_{ij}^{(0)}\ , e_j^{(1 ) } \right ] \nonumber \\ & = & \sum_{j=1}^3 { \gamma_{ij}^{(1 ) } \over \gamma_{ii}^{(0 ) } } e_j^{(0 ) } + e_i^{(1 ) } \ , .   \ ] ] therefore , equation ( [ eq:53 ] ) becomes [ eq:60 ] _ j=1 ^ 3 e_j^(0)=0   . [ eq : the_equation ] it is convenient to write @xmath55 in the form m_ij^(1 ) = ^2 _ ij l_ii + m^2 ^ 2 a_i a_j k_ij  , where , from ( [ eq:42 ] ) and ( [ eq:44 ] ) @xmath56 and k_ij = c_3 + r_1 -2 r_2(i ) -2 r_2(j ) + 8 r_3(i , j ) . we now observe that one can have the square bracket in equation ( [ eq : the_equation ] ) vanish for each @xmath57 if the first - order correction to the polarizability is @xmath58 \ , . \label{eq : gamma_ij^(1)}\end{aligned}\ ] ] equation ( [ eq : gamma_ij^(1 ) ] ) makes it explicit that the frequency - dependent term in the dipole polarizability depends on the direction of propagation of the wave . we observe , however , that this solution is not unique . in fact , the transversality condition ( [ eq : transversality ] ) permits the more general solution    _ ij^(1 ) = -^2 _ ii^(0 ) _ jj^(0 ) where @xmath59 could , in principle , be a function of @xmath60 , the direction of propagation , or the polarization . we note , however , that if @xmath61 then eq . ( [ eq : the_equation ] ) is no longer satisfied term by term , but only after summation . since we appear to have freedom in the choice of @xmath59 , numerical experiments using different choices for @xmath59 could be used to see what value of @xmath59 appears to be optimal for scattering calculations using the discrete dipole approximation ..      propagation of electromagnetic waves on a cubic lattice has been discussed by dg93 . in their analysis it was assumed that if the dielectric material is isotropic , the polarizability @xmath62 can be taken to be isotropic : @xmath63 and @xmath64 . with this assumption , contraction of eq . ( [ eq:60 ] ) with @xmath52 leads to [ eq : alpha_dg93 ] ^(1)_ij = -^2 ^ 2 _ ij  , s_j=1 ^ 3 ( a_je_j^(0))^2  . as in eq.(4.14 ) of @xcite . however , we have seen above that for a cubic lattice ( for which @xmath65 ) , @xmath66 can be made diagonal but not isotropic : if we choose @xmath67 so that @xmath66 is diagonal , we obtain [ eq : alpha_cubic ] ^(1)_ij = -^2 ^ 2 _ ij  . which differs from the result of @xcite since @xmath68 , except for the special case of propagation in the ( 1,1,1 ) direction , for which @xmath69 . we now compare discrete dipole approximation ( dda ) calculations of scattering and absorption using ( 1 ) the polarizability prescription ( [ eq : alpha_dg93 ] ) from dg93 and ( 2 ) the present result ( [ eq : alpha_cubic ] ) . we consider spheres where we can use mie theory to easily calculate the exact result for comparison . let @xmath70 and @xmath71 be the cross sections for scattering and absorption , and let @xmath72 and @xmath73 be the dimensionless scattering and absorption efficiency factors for a sphere of radius @xmath74 . figure [ fig : m=1.33 + 0.01i ] shows @xmath75 and @xmath76 calculated using the dda with a target consisting of @xmath77 dipoles , with radius @xmath78 . the dda calculations used a cubic lattice , and were carried out using the public domain code ddscat.6.1 @xcite . calculations were done for values of the scattering parameters @xmath79 satisfying @xmath80 , as this appears to be a good validity criterion for dda calculations . the upper panel shows scattering and absorption efficiencies calculated with mie theory and using the dda with the present prescription for polarizabilities @xmath13 . the middle and lower panels show fractional errors @xmath81 $ ] of the dda scattering and absorption cross sections . because the dda target ( dipoles on a cubic lattice ) is not rotationally symmetric , the scattering problem was solved for 12 different incident directions , in each case for two orthogonal polarizations . the cross sections shown in fig . [ fig : m=1.33 + 0.01i ] are averages over the 12 incident directions and two incident polarizations .    in the limit @xmath82 , identical results are obtained for both polarizability prescriptions ; this is expected since they differ only in terms of @xmath83 . note that even at @xmath84 ( i.e. , @xmath85 )  for which the polarizabilities are exactly given by the clausius - mossotti polarizabilities  the dda calculation has a nonzero error even as @xmath86 and @xmath87 . this is because ( 1 ) the dipole array only approximates a spherical geometry and ( 2 ) when the `` lattice dispersion relation '' polarizabilities are used , the dda does not accurately model the `` screening '' of the target material close to the surface by the polarization at the target surface . for modest dielectric functions , and sufficient numbers of dipoles , the errors are small  for the case shown in figure [ fig : m=1.33 + 0.01i ] the fractional errors in the @xmath86 limit are @xmath88 .    as @xmath89 increases , the two prescriptions result in differences in computed values of @xmath75 and @xmath76 , although the differences are small , and comparable in magnitude to the errors in the @xmath86 limit . in figure [ fig : m=1.33 + 0.01i ] our new prescription appears to result in small improvements in accuracy ( relative to dg93 ) for @xmath75 for @xmath90 , and likewise for @xmath76 for @xmath91 . for @xmath92 , however , the dg93 prescription appears to give more accurate results for @xmath76 . note , however , that for both polarizability prescriptions the fractional errors in @xmath75 and @xmath76 for @xmath93 are of the same order as the fractional errors for @xmath87 , so it is not clear that the errors should be attributed to the lattice dispersion relation  the errors in @xmath75 and @xmath76 may instead be associated with errors in the computed polarization field near the surfaces , arising from the fact that in the dda the dipoles in the surface layer are imperfectly `` screened '' from the external field by their neighbors .    for @xmath80 , the errors in @xmath75 using the new polarizabilities are of order @xmath94 for @xmath95 ( figure [ fig : m=1.33 + 0.01i ] ) , @xmath96 ( figure [ fig : m=1.70 + 0.10i ] ) and @xmath97 ( figure [ fig : m=2.00 + 1.00i ] ) , and of order @xmath98 for @xmath99 ( figure [ fig : m=3.00 + 4.00i ] ) . in all four cases , for wavelengths satisfying @xmath90 , the magnitude of the error in @xmath75 is of the same order as the error at zero frequency , suggesting that it is mainly due to the surface . the errors in @xmath76 tend to be somewhat larger than for @xmath75 , especially for large values of @xmath100 . for @xmath99 ( figure [ fig : m=3.00 + 4.00i ] ) , the new polarizabilities result in a fractional error of @xmath101 for @xmath102 and @xmath77 dipoles . figure [ fig : m=1.70_0.10ib ] shows results calculated for @xmath96 spheres represented by @xmath103 dipoles  8 times as many as in fig . [ fig : m=1.70_0.10i ] . note that the fractional error for @xmath82 is now only 0.5%  half as large as for @xmath77 . this can be ascribed to the fact that the fraction of the dipoles in the surface monolayer was reduced by a factor of 2 . note also that the errors for finite @xmath89 are also about a factor of 2 smaller than for @xmath77 . these results appear to confirm the suspicion that the errors arise primarily from the surface layer . @xcite recently proposed a method for assigning dipole polarizabilities that takes into account the effects of target geometry , resulting in dipole polarizabilties that are location - dependent even when the target being approximated has homogeneous composition . @xcite have extended this treatment by adding finite - wavelength corrections to the `` surface - corrected '' polarizability prescription , and used dda calculations for spheres and ellipsoids to show that accurate results for @xmath75 and @xmath76 can be obtained for modest numbers of dipoles even when the target material has a large refractive index @xmath104 . the current calculations does _ not _ include such `` surface corrections '' ,  the prescription ( [ eq : alpha_cubic ] ) depends only on local properties  so it comes as no surprise that there are systematic errors even in the limit @xmath86 . @xcite show that the error can be greatly reduced if the polarizabililites are assigned in a way that allows for both target geometry ( i.e. , surface corrections ) and the finite wavelength corrections obtained from the lattice dispersion relation analysis . using these `` surface - corrected lattice dispersion relation polarizabilities '' , @xcite show that for the case @xmath99 , the error in @xmath76 can be reduced to of order @xmath105 for @xmath106 using as few as @xmath107 dipoles , whereas for @xmath106 , the error is seen from figure [ fig : m=3.00 + 4.00i ] to be @xmath108 even with @xmath77 dipoles . such surface corrections will obviously be of value for dda calculations , but at the moment have been applied only to sperical , spheroidal , or ellipsoidal targets @xcite . the principal results of this paper are as follows :    1 . we have derived the exact mode equation ( [ eq : mode_equation ] ) for propagation of electromagnetic waves through a rectangular lattice of polarizable points . if the electric polarizability tensor @xmath109 for the lattice points is specified , then we can solve the mode equation to obtain the dispersion relation for waves propagating on the lattice . alternatively , if we wish the lattice to have the same dispersion relation as a continuum medium of complex refractive index @xmath0 , then we can solve the mode equation for the appropriate polarizability tensor . 3 .   in the case of a cubic lattice , our derived `` lattice dispersion relation '' polarizability tensor ( [ eq : alpha_cubic ] ) is diagonal but anisotropic , and differs from the isotropic polarizabilities obtained by dg93 . the new polarizability prescription has been tested in dda scattering calculations for spheres . the resulting cross sections have accuracies comparable to those obtained with the dg93 prescription . we thank robert lupton for making available the sm software package . this work was supported in part by nsf grants ast-9988126 and ast-0216105 . the coefficients @xmath110 are obtained numerically from the sums ( [ eq : r_0 ] ) . note that eq . ( [ eq : r_0 ] ) for @xmath113 involves the difference between two sums which separately diverge at infinity . to handle this divergence , one notes that in nature one is always interested in finite wavelengths and nonzero frequencies ; when dealing with finite wavelengths , the lattice sums converge because of the oscillations introduced at large distances by retardation effects . to obtain the proper behavior when going to the zero frequency limit , we modify the summations to suppress the contribution from large distances . the manner in which this is achieved is not critical . we compute r_0(i)=_0 [ eq : evalr_0 ] the exponential cutoff allows one to limit the summations to @xmath114 and @xmath115 . the summation ( [ eq : evalr_0 ] ) is carried out for various small values of @xmath109 and it is found , as anticipated , that for @xmath116 the numerical result for @xmath113 is insensitive to the choice of @xmath109 . each of the sums in ( [ eq : evalr_0 ] ) is over a lattice ; @xmath117 defines the distance from the origin for the cubic lattice , and @xmath118 is the distance from the origin for the reciprocal lattice . each sum diverges at infinity , where we can replace the discrete summation by an integration ; the reason why the divergences cancel is that the two lattices have the same density of lattice points , so that at large distances , both approach the same integration . collinge , m.j . , & draine , b.t . 2004 , `` discrete - dipole approximation with polarizabilities that account for both finite wavelength and target geometry '' , j. opt . am . , a , submitted ( http://xxx.arxiv.org/abs/astro-ph/0311304v1 )
we discuss the propagation of electromagnetic waves on a rectangular lattice of polarizable point dipoles . for wavelengths long compared to the lattice spacing 
 , we obtain the dispersion relation in terms of the lattice spacing and the dipole polarizabilities . 
 we also obtain the dipole polarizabilities required for the lattice to have the same dispersion relation as a continuum medium of given refractive index @xmath0 ; our result differs from previous work by draine & goodman ( 1993 ) . 
 our new prescription can be used to assign dipole polarizabilities when the discrete dipole approximation is used to study scattering by finite targets . 
 results are shown for selected cases .