Abstract

Amplified electromagnetic fields generated by a surface of finitely sized metal or dielectric particles are calculated. Regular arrays of particles produced by lithographic techniques and stochastic particle distributions that occur, e.g., in island films, are discussed. Retarded dipolar interactions between the particles are explicitly taken into account. Particles of finite size are considered for which dynamic depolarization and radiation damping effects are important. Limits of validity of the present approach are indicated. The total electromagnetic field from the surface is calculated by superposition: Scalar potentials characterizing a single particle are convoluted with a distribution function describing the particle positions. The surface Hertz vector is obtained from the single-particle Hertz vector by convolution with a two-dimensional Shah function representing the array or with the autocorrelation function of the stochastic surface. A plane-wave description of the dipolar fields is used, whereby the convolution is transformed into a simple multiplication in Fourier space. Cylindrical, general spheroidal, and spherical shapes are considered for the individual particle. Particle dipole moments are obtained by a self-consistent procedure. Dipolar interactions result in shifts and broadening of the particle plasmon resonances, which are responsible for the local intensity enhancement. A set of universal curves is given from which shift and broadening can be calculated for particles of all sizes and shapes. Extrema in the dipolar interactions arise when grating orders change from radiative to evanescent character. The strong variation of the Raman enhancement with angle and wavelength in the vicinity of these extrema is clearly predicted from the Hertz-vector calculation. The formalism described permits one to calculate electromagnetic properties of the surface and enhancement factors for any electromagnetic process occurring at or near the surface. As examples, the calculation of reflectivities for s-and p-polarized excitation and surface-enhanced Raman-scattering cross sections are discussed.

© 1985 Optical Society of America

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1983 (12)

D. J. Ehrlich and J. Y. Tsao, J. Vac. Sci. Technol. B 1, 969 (1983).
[CrossRef]

F. A. Houle, Chem. Phys. Lett. 95, 5 (1983).
[CrossRef]

D. V. Murphy and S. R. J. Brueck, Opt. Lett. 8, 494 (1983).
[CrossRef] [PubMed]

M. Weber and D. L. Mills, Phys. Rev. B 27, 2698 (1983).
[CrossRef]

A. Wokaun, H. P. Lutz, A. P. King, U. P. Wild, and R. R. Ernst, J. Chem. Phys. 79, 509 (1983).
[CrossRef]

D. A. Weitz, S. Garoff, J. I. Gersten, and A. Nitzan, J. Chem. Phys. 78, 5324 (1983).
[CrossRef]

B. N. J. Persson and A. Liebsch, Phys. Rev. B 28, 4247 (1983).
[CrossRef]

P. W. Barber, R. K. Chang, and H. Massoudi, Phys. Rev. Lett. 50, 997 (1983);Phys. Rev. B 27, 7251 (1983).
[CrossRef]

M. Meier and A. Wokaun, Opt. Lett. 8, 581 (1983).
[CrossRef] [PubMed]

C. K. Chen, T. F. Heinz, D. Ricard, and Y. R. Shen, Phys. Rev. B 27, 1965 (1983).
[CrossRef]

D. S. Chemla, J. P. Heritage, P. F. Liao, and E. D. Isaacs, Phys. Rev. B 27, 4553 (1983).
[CrossRef]

W. R. Holland and D. G. Hall, Phys. Rev. B 27, 7765 (1983);Phys. Rev. Lett. 52, 1041 (1984).
[CrossRef]

1982 (16)

P. F. Liao and A. Wokaun, J. Chem. Phys. 76, 751 (1982).
[CrossRef]

G. Rasigni, F. Varnier, M. Rasigni, J. P. Palmari, and A. Llebaria, Phys. Rev. B 25, 2315 (1982);Phys. Rev. B 27, 819 (1983);J. Opt. Soc. Am. 73, 1235 (1983).
[CrossRef]

R. M. Hart, J. G. Bergman, and A. Wokaun, Opt. Lett. 7, 105 (1982);P. F. Liao and M. B. Stern, Opt. Lett. 7, 483 (1982).
[CrossRef] [PubMed]

H. Nichols and R. M. Hexter, J. Chem. Phys. 76, 5595 (1982).
[CrossRef]

A. Wokaun, J. P. Gordon, and P. F. Liao, Phys. Rev. Lett. 48, 957 (1982).
[CrossRef]

U. Laor and G. C. Schatz, J. Chem. Phys. 76, 2888 (1982).
[CrossRef]

M. Kerker and C. G. Blatchford, Phys. Rev. B 26, 4052 (1982):M. Kerker, Acc. Chem. Res. 17, 271 (1984).
[CrossRef]

P. M. Fauchet and A. E. Siegman, Appl. Phys. Lett. 40, 824 (1982).
[CrossRef]

D. J. Ehrlich, S. R. J. Brueck, and J. Y. Tsao, Appl. Phys. Lett. 41, 630 (1982).
[CrossRef]

F. Keilmann and Y. H. Bai, Appl. Phys. A 29, 9 (1982);F. Keilmann, Phys. Rev. Lett. 51, 2097 (1983).
[CrossRef]

G. M. Goncher and C. B. Harris, J. Chem. Phys. 77, 3767 (1982);G. M. Goncher, C. A. Parsons, and C. B. Harris, J. Phys. Chem. 88, 4200 (1984).
[CrossRef]

R. J. von Gutfeld and R. T. Hodgson, Appl. Phys. Lett. 40, 352 (1982).
[CrossRef]

M. Neviere and R. Reinisch, Phys. Rev. B 26, 5403 (1982).
[CrossRef]

R. E. Kunz, J. G. Gordon, and M. R. Philpott, J. Chem. Phys. 77, 646 (1982).
[CrossRef]

T. J. Chuang, J. Chem. Phys. 76, 3828 (1982);T. J. Chuang and H. Seki, Phys. Rev. Lett. 49, 382 (1982).
[CrossRef]

A. M. Glass, P. F. Liao, D. H. Olson, and L. M. Humphrey, Opt. Lett. 7, 575 (1982).
[CrossRef] [PubMed]

1981 (17)

J. E. Sipe and J. Becher, J. Opt. Soc. Am. 71, 1286 (1981).
[CrossRef]

R. J. Nemanich, C. C. Tsai, and T. W. Sigmon, Phys. Rev. B 23, 6828 (1981);M. J. Thompson, N. M. Johnson, R. J. Nemanich, and C. C. Tsai, Appl. Phys. Lett. 39, 274 (1981);R. J. Nemanich, C. C. Tsai, M. J. Thompson, and T. W. Sigmon, J. Vac. Sci. Technol. 19, 685 (1981).
[CrossRef]

A. Nitzan and L. E. Brus, J. Chem. Phys. 74, 5321 (1981);J. Chem. Phys. 75, 2205 (1981).
[CrossRef]

A. Adams and P. K. Hansma, Phys. Rev. B 23, 3597 (1981).
[CrossRef]

J. Kirtley, T. N. Theis, and J. C. Tsang, Phys. Rev. B 24, 5650 (1981);T. N. Theis, J. R. Kirtley, D. J. DiMaria, and D. W. Wong, Phys. Rev. Lett. 50, 750 (1983).
[CrossRef]

T. H. Wood, D. A. Zwemer, C. V. Shank, and J. E. Rowe, Chem. Phys. Lett. 82, 5 (1981).
[CrossRef]

C. K. Chen, A. R. B. de Castro, and Y. R. Shen, Phys. Rev. Lett. 46, 145 (1981);C. K. Chen, T. F. Heinz, D. Ricard, and Y. R. Shen, Phys. Rev. Lett. 46, 1010 (1981);H. W. K. Tom, T. F. Heinz, and Y. R. Shen, Phys. Rev. Lett. 51, 1983 (1983);H. W. K. Tom, C. M. Mate, X. D. Zhu, J. E. Crowell, T. F. Heinz, G. A. Somorjai, and Y. R. Shen, Phys. Rev. Lett. 52, 348 (1984).
[CrossRef]

G. L. Eesley, Phys. Rev. B 24, 5477 (1981).
[CrossRef]

E. Borgarello, J. Kiwi, E. Pelizzetti, M. Visca, and H. Graetzel, J. Am. Chem. Soc. 103, 6324 (1981);R. Humphry-Baker, J. Lilie, and M. Graetzel, J. Am. Chem. Soc. 104, 422 (1982);D. Duonghong, J. Ramsden, and M. Graetzel, J. Am. Chem. Soc. 104, 2977 (1982).
[CrossRef]

J. F. Owen, P. W. Barber, B. J. Messinger, and R. K. Chang, Opt. Lett. 6, 272 (1981);J. F. Owen, P. W. Barber, P. B. Dorain, and R. K. Chang, Phys. Rev. Lett. 47, 1075 (1981);J. F. Owen, R. K. Chang, and P. W. Barber, Opt. Lett. 6, 540 (1981).
[CrossRef] [PubMed]

J. G. Bergman, D. S. Chemla, P. F. Liao, A. M. Glass, A. Pinczuk, R. M. Hart, and D. H. Olson, Opt. Lett. 6, 33 (1981).
[CrossRef] [PubMed]

S. Garoff, D. A. Weitz, T. J. Gramila, and C. D. Hanson, Opt. Lett. 6, 245 (1981).
[CrossRef] [PubMed]

D. S. Wang and M. Kerker, Phys. Rev. B 24, 1777 (1981).
[CrossRef]

P. F. Liao, J. G. Bergman, D. S. Chemla, A. Wokaun, J. Melngailis, A. M. Hawryluk, and N. P. Economou, Chem. Phys. Lett. 82, 355 (1981).
[CrossRef]

P. K. Aravind, A. Nitzan, and H. Metiu, Surf. Sci. 110, 189 (1981);R. Ruppin, Phys. Rev. B 26, 3440 (1982).
[CrossRef]

B. N. J. Persson and R. Ryberg, Phys. Rev. B 24, 6954 (1981).
[CrossRef]

W. Lukosz and M. Meier, Opt. Lett. 6, 251 (1981).
[CrossRef] [PubMed]

1980 (14)

J. I. Gersten, J. Chem. Phys. 72, 5779 (1980).
[CrossRef]

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F. R. Aussenegg, A. Leitner, and M. E. Lippitsch, eds., Surface Studies with Lasers (Springer-Verlag, Berlin, 1983).
[CrossRef]

A. Otto, in Light Scattering in Solids, M. Cardona, ed. (Springer-Verlag, Berlin, 1983), Vol. 4, p. 289

A. Wokaun, in Solid State Physics, H. Ehrenreich, D. Turnbull, and F. Seitz, eds. (Academic, New York, 1984), Vol. 38, p. 223.
[CrossRef]

J. Vac. Sci. Technol. B1(4) (1983) (Proceedings of the International Symposium on Electron, Ion, and Photon Beams).

T. J. Chuang, Surf. Sci. Rep.3, 1 (1983).
[CrossRef]

F. Claro, Phys. Rev. B (to be published);L. C. Chu and S. Y. Wang, Phys. Rev. (to be published).

H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer-Verlag, Berlin, 1980).

J. P. Gordon, AT&T Bell Laboratories, Crawfords Corner Road, Holmdel, New Jersey 07733;A. Wokaun and P. F. Liao, (personal communication).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

The plasmon resonance frequency ωres is obtained by maximizing the particle dipole moment pi [Eq. (43)]. If both the intrinsic losses of the material [∊2(ω)] and the radiation damping [Im(Aeff,iarray)] are small, pi is maximum where the real part of the denominator [Eq. (43)] vanishes, i.e., where 1+[∊1(ω)−1]Re(Aeff,iarray)−∊2(ω)Im(Aeff,iarray)=0.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

A. Wokaun and M. Meier, Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH Zentrum, CH 8092 Zurich, Switzerland;and P. F. Liao and M. B. Stern, Bell Communications Research, Crawfords Corner Road, Holmdel, New Jersey 07733 (personal communication;manuscript in preparation).

M. Meier and A. Wokaun, Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH Zentrum, CH 8092 Zurich, Switzerland (personal communication;manuscript in preparation).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).

E. J. Zeman and G. C. Schatz, in Proceedings of the Seventeenth Jerusalem Symposium in Quantum Chemistry and Biochemistry (Reidel, Dordrecht, The Netherlands. 1984).

For the special case of point dipoles of polarizability α, the local field Eloc consists of the incident field E0 and fields from all particle dipoles pi. These dipoles are in turn being induced by the local field pi = αEloc exp(iφi).

C. J. F. Boettcher, Theory of Electric Polarization, 2nd ed. (Elsevier, Amsterdam, 1973), Vol. 1.

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Figures (10)

Fig. 1
Fig. 1

Fourier-series representation of the Hertz vector ϕA(r) characterizing a square-particle array. The single-particle Hertz vector ϕ ( k x , k y ) is sampled [Eq. (15)] at discrete values of k, indicated by dots in the figure that correspond to grating orders. Orders within the light circle |k| = k are radiative, whereas all orders outside the light circle are evanescent. The inverse lattice of the array (lattice constant d) is shifted by the component ki of the incident wave vector in the plane of the surface. Incidence in the xz plane at an angle of 60° was chosen.

Fig. 2
Fig. 2

Finitely sized particle, confined by the surface A. For calculating the Hertz vector, the particle is sectioned into slices of thickness dz′ parallel to the xy plane. The unit step function S(r, z′) describes the contour of the slice at z = z′.

Fig. 3
Fig. 3

Depolarization constants for isolated large particles. (a) The difference between the static value Ai(ka = 0) and Re(Aeff,i) (i = x, y, z) is plotted against k(abc)1/3, where a, b, and c are the half-axes of the spheroid. (b) The imaginary parts Im(Aeff,i), normalized by the limiting value 2/9 k3abc that represents radiation damping for small particles, are plotted against k(abc)1/3. Solid lines, spheres; dashed lines, 3:1 ellipsoids, i = z; dashed–dotted lines, 3:1 ellipsoids, i = x.

Fig. 4
Fig. 4

Convergence of the k space summation. Calculated values for Re(Aeff,y) and Re(Aeff,z) are shown for d/λ = 0.7 as a function of the radius kmax of a circle in k space over which orders are summed. The sphere radius a is indicated as a parameter. As the Hertz vector decrease is determined by |k|a, the radius kmax required to achieve a given accuracy (1 × 10−4) is inversely proportional to the particle radius.

Fig. 5
Fig. 5

Pictorial representation of the calculation of the array Hertz vector as a Fourier sum [Eq. (15)]. A one-dimensional cross section along kx of the function ϕ ( k x , k y ) is shown. For a point dipole, ϕ is imaginary (dashed line) for kx < k and real (solid line) for kx > k. In the Fourier sum [Eq. (15)] ϕ is sampled at equidistant values of the argument (shown by arrows) and multiplied by the area of the lattice cell (2π/d)2. This corresponds to a coarse-grained integration of ϕ shown by the step function. The single-particle Hertz vector is obtained from continuous integration of ϕ ( k ). Differences between continuous and stepwise integration are small for k > kmax, where the integrand is approximately linear.

Fig. 6
Fig. 6

Real part of the dipolar-interaction constant Ci, i = x, y, z, defined by Eqs. (44), for spheres. Light is incident in the xz plane at an angle of 60° from the surface normal z. The rows of the array, with lattice constant d, are parallel to the plane of incidence.

Fig. 7
Fig. 7

Imaginary parts of the dipolar-interaction constants Ci, i = x, y, z for spheres and spheroids (geometry as in Fig. 6).

Fig. 8
Fig. 8

Condition for extrema in the dipolar interaction [Eq. (54)]. A cusp arises when a grating order crosses the light circle |k| = k. Light incidence in the xz plane at an angle of 60° was chosen as in Fig. 1. Crossing of the orders (a) (m = −1, n = 0) for d/λ = 0.536; (b) (m = −1, n = ±1) for d/λ = 1.008; and (c) (m = −2, n = 0) for d/λ = 1.072.

Fig. 9
Fig. 9

Real parts of Ci, i = x, y, z, for 3:1 ellipsoids. The orientation of the symmetry axis of the spheroid is denoted by x (dots), y (crosses), or z (solid lines). The geometry is the same as in Fig. 6.

Fig. 10
Fig. 10

Local intensity enhancement for prolate Ag spheroids on a 260-nm-period square lattice. The particles of volume 3 × 104 nm3 are oriented with their long axes parallel to the surface normal. Relative enhancement values (arbitrary units) are plotted as a function of excitation energy; normalization is equal for all curves. The variation of the static-depolarization constant A from 0.16 to 0.11 corresponds to aspect ratios a/b ranging from 2.1 to 3.

Tables (2)

Tables Icon

Table 1 Depolarization Constants for Large Spheres and Ellipsoidsa

Tables Icon

Table 2 Parameters Used in the Calculationsa

Equations (94)

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ϕ ( r ) = 1 r e i k r , r = | r r 0 | , k 2 = ( ω / c ) 2
Π ( E ) ( r ) = ϕ ( r ) p .
D ( r ) = × × Π ( E ) ( r ) ,
H ( r ) = i ω × Π ( E ) ( r )
( 2 + k 2 ) Π ( E ) ( r ) = 0 .
B ( r ) = × × Π ( H ) ( r ) ,
E ( r ) = + i ω × Π ( H ) ( r ) .
ϕ ( r ) = d k x d k y ϕ ( k x , k y ) × exp [ i k z ( z z 0 ) ] exp ( i k · r ) ,
k z = { + ( k 2 k 2 ) 1 / 2 sgn ( z z 0 ) for k 2 < k 2 + i ( k 2 k 2 ) 1 / 2 sgn ( z z 0 ) for k 2 > k 2 .
ϕ ( k ) = i 2 π 1 k z ,
ϕ A ( r ) = m n exp [ i ( k x i m d + k y i n d ) ] ϕ ( r r m n ) .
ϕ A ( r ) = d x d y m n δ ( x m d ) δ ( y n d ) × exp [ i ( k x i x + k y i y ) ] ϕ ( r r ) ,
S ( u ) S ( υ ) = Σ m Σ n δ ( u m ) δ ( υ n ) .
ϕ A ( r ) = [ 1 d 2 S ( x d ) exp ( i k x i x ) S ( y d ) exp ( i k y i y ) ] * * ϕ ( r ) .
ϕ A ( k ) = { S [ ( k x k x i ) d 2 π ] S [ ( k y k y i ) d 2 π ] } ϕ ( k ) .
ϕ A ( r ) = ( 2 π d ) 2 m n ϕ ( k x i + m 2 π d , k y i + n 2 π d ) × exp { i [ ( k x i + m 2 π d ) x + ( k y i + n 2 π d ) y + k z z ] } ,
A ( E ) ( r ) = ϕ A ( r ) p
E A ( r ) = D ( r ) p ,
p = V 4 π ( 1 ) E ins ,
4 π p = V ( 1 ) [ E 0 + D ( 0 ) p ] .
p = V ( 1 ) [ 4 π V ( 1 ) D ( 0 ) ] 1 E 0 .
ϕ V ( r ) p = V ϕ ( r r ) d p ( r ) = P V ϕ ( r r ) d V .
ϕ V ( r ) = 1 V d z S ( z ) ϕ ( r r ) d x d y .
ϕ V ( r ) = 1 V d z d r S ( r ; z ) × ϕ ( r r , z z ) .
ϕ V ( r ) = 1 V d z S ( r ; z ) * * ϕ ( r , z z ) .
ϕ V ( k , z ) = 1 V d z S ( k ; z ) · ϕ ( k ) exp [ i k z ( z z ) ] ,
S c ( r ; z ) = { 1 for | r | < a ( z ) 0 for | r | > a ( z ) ,
S c ( k ; z ) = 2 π a 2 J 1 ( k a ) k a , a = a ( z ) .
d ϕ z ( k , z ) = 1 V d z 2 π a 2 J 1 ( k a ) k a i 2 π k z exp [ i k z ( z z ) ] .
S e ( k ; z ) = 2 π a b J 1 [ ( a 2 k x 2 + b 2 k y 2 ) 1 / 2 ] ( a 2 k x 2 + b 2 k y 2 ) 1 / 2 ,
ϕ cyl ( k , z ) = 2 1 V a 2 k z 2 J 1 ( k a ) k a [ 1 exp ( i k z c ) cos ( k z z ) ] .
E z ( k , z ) = ( 2 z 2 + k 2 ) ϕ cyl ( k , z ) p z ,
E z ( k , z ) = 2 a 2 J 1 ( k a ) k a { exp ( i k z c ) cos ( k z z ) k 2 k z 2 [ 1 exp ( i k z c ) cos ( k z z ) ] } p z V .
E z ( k , z ) = 2 a 2 J 1 ( k a ) k a [ k 2 k z 2 ( 1 + k 2 k z 2 ) × exp ( k z c ) cosh ( k z z ) ] p z V .
E z ( k , 0 ) 2 k 2 a J 1 ( k a ) k 3 p z V .
d ϕ z ( k , z ) = 1 V d z 2 π r ( z ) 2 J 1 [ k r ( z ) ] k r ( z ) × i 2 π k z exp [ i k z ( z z ) ]
ϕ sphere ( k , z ) = a a d ϕ z ( k , z ) .
ϕ z ellipsoid ( k , z ) = c c d ϕ z ( k , z ) .
ϕ spheroid ( k , z ) = 1 V c c d z 2 π a ( z ) b ( z ) × J 1 [ ( k x 2 a ( z ) 2 + k y 2 b ( z ) 2 ) 1 / 2 ] ( k x 2 a ( z ) 2 + k y 2 b ( z ) 2 ) 1 / 2 × i 2 π k z exp [ i k z ( z z ) ] .
p i = V 4 π ( ω ) 1 1 + [ ( ω ) 1 ] A i E 0 , i ,
P = 1 4 π [ ( ω ) 1 ] [ E 0 + E dep ( 0 ) ] ,
A eff , i = 1 4 π E dep , i ( 0 ) P i , i = x , y , z .
d E 1 ( r 2 ) = 1 r 3 { ( 1 i k r ) ( n · d p 1 ) n + ( k r ) 2 [ ( n × d p 1 ) × n ] } e i k r ,
p i = V 4 π ( ω ) 1 1 + [ ( ω ) 1 ] A eff , i array E 0 , i , i = x , y , z .
Re ( C i ) = Re ( A eff , i array A eff , i isol . particle ) d 3 V , Im ( C i ) = Im ( A eff , i array ) d 3 V ,
E 1 ( r 2 ) = n · p 1 r 3 n ,
A eff , i array = V 4 π D ( 0 ) i i ,
E ins ( r ) = m n E ins ( K m n , r ) ,
k m n = ( k x i + m 2 π d , k y i + n 2 π d ) ,
E ins = m n E ins k m n ,
E ins ( k m n ) = 1 F S e ( r , z = 0 ) E ins ( k m n , r ) d r
E ins ( k m n ) = E ins ( k m n , r = 0 ) × 2 J 1 ( u ) / u ,
u = ( k m n , x 2 a 2 + k m n , y 2 b 2 ) 1 / 2
k max k max E ins ( k ) d k x d k y = Δ k 2 p k max / Δ k q E ins ( p Δ k , q Δ k ) ,
( k x i + m 2 π d ) 2 + ( k y i + n 2 π d ) 2 = k 2 .
( sin θ i + m λ d ) 2 + ( n λ d ) 2 = 1 .
= A Re ( S · n ) d a = ½ V Re ( J * · E ) d V ,
J = t P = i ω p V .
E ( 0 ) = D ( 0 ) p = m n D m n p .
m n = ½ Re ( i ω i j x , y , z p i * D m n , i j p j ) .
= 2 π ω j = x , y , z Im ( 1 V A eff , j array ) p j 2 .
Im ( A eff , j array ) = V 4 π ( 2 π d ) 2 radiating orders ( k k j 2 ) Im { ϕ V ( k m n ) } .
Im ( C j ) = d π radiating orders ( k 2 k j 2 ) Im { ϕ V ( k m n ) } .
= 2 π ω 1 d 3 j = x , y , z Im ( C j ) | p j | 2 .
d E i ( r ) = j ( 2 r i r j + δ i j k 2 ) ϕ V ( r r ) p j w ( r ) d r , i , j = x , y , z ,
E dep , i ( r ) = lim a a a d r j ( 2 r i r j + δ i j k 2 ) × ϕ V ( r r ) p j w ( r ) .
E dep , i ( k ; r ) = lim a a a d r j ( k i k j + δ i j k 2 ) × ϕ V ( k ) p j exp [ i k ( r r ) ] w ( r ) .
E dep , i = 1 N lim a 1 4 a 2 a a d r E dep , i ( r ) w ( r ) ,
N = lim a 1 4 a 2 a a d r w ( r ) .
E dep , i ( k ) = 1 N j ( k i k j + δ i j k 2 ) ϕ V ( k ) p j × lim a 1 4 a 2 a a d r exp ( i k · r ) × w ( r ) a a d r exp ( i k · r ) w ( r ) .
à ( k ) = lim a 1 4 a 2 | a a d r w ( r ) exp ( i k r ) | 2 .
E V , i ( k ) = j ( k i k j + δ i j k 2 ) ϕ V ( k ) p j .
E dep , i ( k ) = 1 N E V , i ( k ) · Ã ( k ) .
ϕ ( k ) = 1 N ϕ V ( k ) · Ã ( k ) .
A ( s ) = J 1 [ A ( k ) ] = lim a 1 4 a 2 a a d x w ( x s ) w ( x ) .
E dep , i ( s ) = 1 N E V , i ( s ) * * A ( s ) .
E dep , i ( s ) = 1 N d k j ( k i k j + δ i j k 2 ) ϕ V × ( k ) p j à ( k ) exp ( i k s ) exp ( i k z z ) .
S m n = m n N / cos θ m n = m n N k / k z ,
S m n = S m n ( k + ) + S m n ( k ) .
S m n ( k ± ) = c 8 π | E m n ( k ± ) 2 | ,
S m n ( k ± ) = F m n G m n ( k ± ) ,
F m n = 2 π 3 N 2 c k 2 | ϕ V ( k m n ) | 2 .
G m n ( k ± ) = i j x , y , z ( k i k j δ i j k 2 ) p i p j * .
I transmitted = c 8 π | E inc ( k 0 ) + E array ( k 0 ) | 2 .
r m n ( α β ) = E m n ( α ) / E 0 ( β ) , α , β = s , p .
ψ ( E ) ( r ; k ) = ( k x k z p x + k y k z p y k 2 + p z ) × ϕ V ( k ) exp ( i k r ) exp ( i k z z ) ,
ψ ( H ) ( r ; k ) = ω k x p y k y p x k 2 × ϕ V ( k ) exp ( i k r ) exp ( i k z z ) ,
E ( p ) ( r ; k ) = × × ψ ( E ) ( r ; k ) ,
E ( s ) ( r ; k ) = i ω × ψ ( H ) ( r ; k ) .
E dep , i ( s ) = 1 N lim a 1 4 a 2 a a d r E dep , i ( r + s ) w ( r ) = 1 N lim a 1 4 a 2 a a d r E dep , i ( r ) w ( r s ) ,
2 r i r j ϕ V ( r r ) = 2 y i y j ϕ V ( y ) .
E dep , i ( s ) = 1 N lim a a a d y E V , i ( y ) 1 4 a 2 × a a d x w ( x s ) w ( x y ) ,
E V , i ( y ) = j ( 2 y i y j + δ i j k 2 ) ϕ V ( y ) p j .
E dep , i ( s ) = 1 N d y E V , i ( y ) A ( s y ) = 1 N E V , i ( s ) * * A ( s ) ,

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