ISSN 00271349, Moscow University Physics Bulletin, 2011, Vol. 66, No. 1, pp. 5–11. © Allerton Press, Inc., 2011. Original Russian Text © E.V. Golovacheva, A.M. Lerer, N.G. Parkhomenko, 2011, published in Vestnik Moskovskogo Universiteta. Fizika, 2011, No. 1, pp. 6–11.
Diffraction of Electromagnetic Waves of Optical Range on a Metallic Nanovibrator E. V. Golovachevaa, A. M. Lerera, and N. G. Parkhomenkob a
Southern Federal University, ul. Sorge 5, RostovonDon, 344090 Russia b FGUP GKB Svyaz’, pr. Sokolova 96, RostovonDon, 344010 Russia email:
[email protected],
[email protected] Received July 22, 2010
Abstract—A solution of the twodimensional integro–differential equation describing diffraction of electro magnetic waves on metallic nanovibrators and on nanocrystals coated by a metallic film was obtained. Copper and gold nanoantenna characteristics were studied theoretically in the optical range. It was noted that the dependence of the scattered field on frequency is of a resonant character and the resonant wavelengths of nanovibrators are larger than those of an ideally conducting vibrator of the same size. DOI: 10.3103/S0027134911010103
tions for electromagnetic field components under given boundary conditions. These are the method of finite differences in the frequency–spatial and spa tial–temporal representations (FDTD) and the finite element method (FEM). In OA modeling these meth ods were used, in particular, in [6] (FDTD) and [8] (FEM). In the second group of the methods, the boundary problem is reduced to the solution of inte gral, integro–differential, paired integral, and paired summator equations. One undeniable advantage of the methods of the first group is their versatility. Their drawbacks are high demands on computers, long cal culation time, the necessity to discretize not only the scatterer but also the environmental space, and diffi culties in calculations of objects that contain small scale elements. In addition, problems in the passage to an open space with the satisfaction of the radiation condition occur. These problems do not occur when solving integral equations (IEs). The choice of the IE type is determined first of all by the object’s structure. This is why the methods that are based during the solu tion of IEs are not as versatile as the methods of the first group. However, special computer programs that are developed on their basis usually operate faster by several orders of magnitude. Several types of IE describing diffraction on dielectric bodies exist [11]. Most of these IEs can be divided into two groups, sur face (SIEs) and volume (VIEs) integral equations. In SIEs, the unknown variable is the field at the interface of dielectrics; in VIEs, it is the field in all the internal points of the body. VIEs have a set of advantages: they are simpler, nonhomogeneity and nonlinearity of the dielectric do not complicate their solution signifi cantly, and the electric field in a dielectric is obtained directly as a result of the solution. Applications of IEs for plasmon structures were described, for example, in [9] (SIEs) and [12] (VIEs).
INTRODUCTION Optical antennas (OAs) are used to increase the efficiency of energy transfer from an external field to a local one and back. In problems of microscopy, optical antennas make it possible to concentrate radiation in a size less than the diffraction limit [1] and replace tra ditional focusing lenses and object glasses. OAs create an extreme increase of the local electric field. This OA property can be used to increase the efficiency of pho tophysical processes in lightradiating devices and solar batteries, to determine DNA structure, and to detect individual molecules [2, 3]. Carbon nanotubes [4], as well as metal and metal dielectric vibrators and spheres are used as OAs [5–9]. One of most promising constructions is a nanocrystal coated with a metal film. OA properties are similar to radiorange antennas with some significant differ ences in physical characteristics and the impossibility of using the scalability principle. Most of the differ ences are connected with the fact that a metal in the optical range is not an ideal conductor but has proper ties of solid body plasma. This is caused by the pres ence of free electron gas. So, in solving diffraction problems for optical waves of the optical range on a metal object, it is necessary to take into account the field inside the sample. A wide spectrum of methods for simulating interactions between electromagnetic radiation and matter exists. For a restricted number of boundary problems that permit one to apply the sepa ration of variables, an analytical solution exists. For diffraction on a sphere, this is obtained via Mie theory. Mie theory was applied for modeling OAs in [5]. Most of the numerical methods for calculating the electro magnetic field in a resonance region of frequencies can be divided into two large groups. The first group is methods based on the direct solution of wave equa 5
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The aim of this work is to develop an efficient numericalanalytical method for solving VIEs and to apply it in studying diffraction of electromagnetic waves on metaldielectric nanostructures. An efficient method for calculating submillimeter carbon nanotube antennas is given in [13] and [14]. The method is based on the numericalanalytical solu tion of paired integral and integro–differential equa tions for an impedance vibrator. The integral repre sentation of the Green function is used as the basis of this method. The singularity of the IE kernel manifests itself in slow convergence of integrals in the matrix ele ments of the obtained systems of linear algebraic equa tions. Improving the convergence of integrals is easier than regularizing IEs in the spatial representation. In this paper, this approach is applied for solving VIEs. The complex dielectric permeability of metals and the refractive index of ZnO in the optical range of wave lengths were presented in [15]. These experimental results for metals are well approximated by the formula for dielectric permeability of plasma 2
ε 's = 1 – ( λ/λ p ) , ε ''s = – λ
3
2 G/ ( 2πcλ p ),
j ( r, z) E = 0 ( r, z ) τ(r)
∫∫
j ( r', z' )G ( r, r', z, z' )r ( dr' ) dz',
–l 0
where j(r, z) = τ(r)E(r, z), and the Green function has the form 2π
1 exp ( – ikR ) G ( r, r', z, z' ) = dϕ, 4π R
∫ 0
2
2
2
where R = r + r' – 2rr' cos ϕ + ( z – z' ) and E0(r, z) = 1 2π E ext (x, y, z)dϕ. z 2π 0 For an external field, we consider a plane wave. Its projection to the vibrator axis has the form E0(r, z) = E0 sinθexp[i(kxx + kyy + kzz)]; θ is the incidence angle, which is calculated from the z axis. The kernel of IDE (2) has a logarithmic singularity. Let us represent it, as in [13], in the form of the Fourier integral
∫
∞
1 K ( δκ )e –iγ ( z – z' ) dγ, g ( R ) = 0 2 4π –∞
∫
ε s = ε 's – iε ''s ,
where λp is the plasma wavelength and G is the fre quency of electron collisions. For copper, λp = 151.9 nm and G = –0.25 × 1015 Hz; for silver, λp = 147 nm and G = –0.135 × 1015 Hz [16]. 1. SOLUTION OF THE TWODIMENSIONAL INTEGRO–DIFFERENTIAL EQUATION The threedimensional integro–differential equa tion (IDE) for a dielectric body has the form [11] 2
E ( x, y, z ) = [ graddiv + k ]
∫
(2)
l a
2
2 d + 2 + k dz
ext
× τE ( x', y', z' )g ( R ) dv' + E ( x, y, z ),
2
2
2
∞
1 ˜g ( r, r', γ )e iγ ( z – z' ) dγ, G ( r, r', z, z' ) = 4π
∫
⎧ I 0 ( rδ )K 0 ( r'δ ), r ≤ r' . where ˜g (r, r', γ) = ⎨ ⎩ I 0 ( r'δ )K 0 ( rδ ), r ≥ r', We solve IDE (2) by the Galerkin method. Let us expand the unknown function j(r, z) in the weighted Chebyshev polynomials of the second kind: ∞
j ( r, z ) =
∑Z
⎛ z⎞ l
m ( r )U m ⎝ ⎠ ,
m=0
V
2
2
2
(GF), R = ( x – x' ) + ( y – y' ) + ( z – z' ) , k is the wave number, τ = ε – 1, ε is the dielectric permeability of the body, and Eext(x, y, z) is the external field. Let us consider a dielectric cylinder with the radius a and length 2l along the z axis with its center in the origin. In the case a Ⰶ l one can assume that the elec tric field strength has a single component parallel to the z axis and depends only on the coordinates r, z. In this case, Eq. (1) is reduced to the twodimensional IDE
(3)
–∞
(1)
1 exp ( – ikR) is the Green function where g(R) = 4π R
2
where κ = r + r' – 2rr' cos ϕ , δ = γ – k , K0 is the Macdonald function. Then
(4)
m 1 2 2 1/2 z 1 z U m ⎛ ⎞ = i ( l – z ) U m ⎛ ⎞ . ⎝ l⎠ ⎝ πl m + 1 l⎠
Substituting (4) into (2), we project it onto U n ⎛ z⎞ . As ⎝ l⎠ a result, we obtain a system of integral equations (IE) ∞
∑
m=0
Z m ( r )D nm 1 = B n ( r ) + 2π τ(r)
∞
∞
∑ ∫ (γ
2
2
–k )
m = 0 –∞
∞
J m + 1 ( γl ) J n + 1 ( γl ) × dγ r'Z m ( r' )g˜ ( r, r', γ ) dr', γ γ
∫ 0
n = 0, 1, 2…,
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DIFFRACTION OF ELECTROMAGNETIC WAVES OF OPTICAL RANGE
where
2
2π
where
l
1 z B n ( r ) = dϕ U n ⎛ ⎞ E 0 ( r, z ) dz ⎝ l⎠ 2π
∫ ∫ 0
2
2
=
2πk a r'τ (r')J0(r’sinϑ)dr' l 0
∫
∫
l –l
j (r',
ik cos ϑz'
z') e , ϑ is the observation angle accounted from the vibrator and F(ϑ) is the dimensionless scattering diagram. The obtained solution can be easily generalized for the case of diffraction on several dielectric cylinders.
–l
Jn + 1 ( kz l ) = E 0 sin θJ 0 ( k ⊥ r q ) , kz
F(ϑ)
7
2
k⊥ = kx + ky ,
l
z z D nm = U n ⎛ ⎞ U m ⎛ ⎞ ( dz ) ⎝ l⎠ ⎝ l⎠
∫
2. SOLUTION OF A ONEDIMENSIONAL INTEGRO–DIFFERENTIAL EQUATION Let us consider an approximate solution of IDE (2) for a homogeneous dielectric cylinder. We seek the solution in the form j ( r, z ) = I ( z )U ( r ), where I is the unknown function, U(r) =
–l
⎧ 0, m and n are of different evenness ⎪ ⎪1 1 π 1 1 1 – , ⎪ 2 cos ⎛⎝ q ⎞⎠ 2 2 2 m n = ⎨π p –1 q –1 ⎪ ⎪ m and n of different evenness ⎪ q = m – n, p = m + n. ⎩
2
We solve integral equation (5) by the collocation method. We use the quadrature a r'f (r')dr' =
∑
p Ap p=1
∫
0
f (rp) and require that (5) is satisfied in the
quadrature nodes. Let us denote A p Zm(rp) = Zmp (unknown coefficients), Bnq = Bn(rq), and take into account the first M equations in (5). As a result, we obtain a system of linear algebraic equations (SLAE) M
∑X
m=0
D nm mq Aq τ ( rq )
M
= B nq +
n = 0, 1, …, M,
P
∑ ∑X
qp mp A nm ,
m = 0p = 1
(6)
q = 1, …, P,
where
qp
A nm
⎧1∞ 2 J m ( γl ) J n ( γl ) ˜ ⎪ ( γ – k 2 ) g ( r p, r q, γ ) dγ, γ γ ⎪π = ⎨ 0 ⎪ m and n of the same evenness ⎪ ⎩ 0, m and n are of different evenness.
∫
(7)
The integral representation of the singular kernel (3) permits one to circumvent the difficulties connected with calculations of integrals of bisingular functions. In this case, the singularity of the IDE kernel mani fests itself in slow convergence of integrals in the spec tral space (7). To improve convergence of the integrals is easier (see [13]) than to regularize an IDE with a logarithmic kernel. Solving the obtained SLAE (6), we find the unknown function j(r, z). It is easy to obtain the expression for calculation of the field in the far zone: – ikr
e E ϑ = H ϕ Z 0 ≈ F ( ϑ ), 4πr/l MOSCOW UNIVERSITY PHYSICS BULLETIN
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J0(χr')/J0(χa), χ = k ε – k z , and J0 is the Bessel function. The function U(r) is the first term in the expansion of the field inside a dielectric cylinder infi nite along z. The integrals over r' in (2) can be taken analytically. Let us require the fulfillment of (2) for r = a. In this case, it follows from (2) that l
I ( z) d + k 2⎞ = E 0 ( a, z ) + ⎛ I ( z' )G 2 ( z, z' ) dz', (8) ⎝ dz 2 ⎠ τ
∫
–l
where ∞
χJ 1 ( χa )I 0 ( aδ ) + δJ 0 ( χa )I 1 ( aδ ) a G 2 ( z, z' ) = 2 2 2π χ +δ
∫
–∞
(9)
K 0 ( αδ ) –iγ ( z – z' ) dγ. × e J 0 ( χa ) The solution of (8) is sought by the abovemen tioned Galerkin method. In the method of collocations for metal nanovibra tors, the quadrature of the highest accuracy [17] was used. For nanocrystals coated by a film, one used the same quadrature in the internal domain and the rect angular formula in the external domain. The method for calculating integrals (7) and (9) is described in [13]. The obtained numerical–analytical solution has rapid internal convergence. For P = 4 and M = 5–10 in the SLAE (6), the error of the solution with respect to internal convergence is less than 1%. The calculation time for one point of the frequency characteristic is about 0.1 s on a computer with a Intel(R) Core(TM)2 CPU 4400 @ 2.00 GHz processor. 3. VERIFICATION OF THE MATHEMATICAL MODEL AND CALCULATION RESULTS The results are presented for the angles of inci dence θ = 90° and observation ϑ = 90°, which are nor mal to the vibrator (excluding Fig. 7). In Fig. 1, the solutions of the twodimensional (2) and onedimen No. 1
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GOLOVACHEVA et al. F(90°) 14
F(90°) 16 1
12
14 10
2
7
65 4
1
8
12
6
8 3
4 3
4
2
2 0
5
10
15
20
25
30 f, THz
Fig. 1. Comparison of the solutions of IDE (2) (solid curves) and (8) (dashed curves) for a dielectric cylinder with ε = 300, length L = 2l = 10 μm. Radius a (μm) of the vibrator is 0.2, 0.1, and 0.05 (curves 1–3, respectively).
F(90°) 12 10
432 1 S
8
2l
1.5
2.5
2.0
3.0
3.5 λ, μm
Fig. 2. Comparison of the characteristics of metal and dielectric nanovibrators. L = 0.7 μm, a = 0.01 μm. Metals (solid curves): (1) ideal; (2) gold; (3) copper. Dielectrics (dashed curves): (4) ε = –1000; (5) ε = –2000; (6) ε = –3000; (7) ε = –5000.
F(90°) 6
Z 2l
0 1.0
1 2
3
4
5
5
6
X 4
2a
6
3
5
4
2 2 1 0 1.00
1.25
1.50
1.75
2.00
2.25 λ, μm
0 0.5
1.0
1.5
2.0
2.5
3.0
3.5 λ, μm
Fig. 3. The characteristics of a connected copper NV. The length L = 0.35 μm, a = 0.01 μm, the clearance s (in μm) is 0.01 (curve 1), 0.02 (2), 0.03 (3), 0.1 (4). Curve 5 is a sin gle nanovibrator.
Fig. 4. Comparison of the characteristics of copper NVs of different lengths L. a = 0.01 μm. L varies from 0.3 μm (curve 1) to 0.8 μm (6) with a step of 0.1 μm.
sional (8) IDEs for a homogeneous dielectric cylinder are compared. The calculated curves coincide with a sufficiently high accuracy. The error of the solution of (8) certainly increases with an increase of the cylin der radius. For a cylinder with a radius of 0.1 μm, the resonance frequency calculated using the CST Micro wave Studio program developed by Computer Simula tion Technology is 25.4 THz, which coincides with our results. For nanovibrators, the error of the solution of IDE (8) reaches several percent. So all the results pre sented below are obtained from the solution of (2).
The passage from dielectric nanovibrators (DNVs) with negative dielectric permeability ε to metal nanovibrators (MNVs) has been studied (Fig. 2). As seen from the figure, the curves for DNVs approach those for an ideally conducting MNV with an increase of ε . It should be emphasized that the ideally con ducting MNV was calculated by the method described in [13] and based on the solution of integral equations that principally differ from the IDEs of this paper. In [9], the resonance wavelengths of golden nanovibrators are calculated by a method based on the
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DIFFRACTION OF ELECTROMAGNETIC WAVES OF OPTICAL RANGE F(90°) 16
9
F(90°) 15 4
14 1
12
12
3 2
10
2
9
6
8 6
6
3
1
4
5
3
2 0 0.5
1.0
1.5
2.0
2.5
3.0
0 0.5
3.5 λ, μm
Fig. 5. Comparison of the characteristics of copper NVs with different radii a. L = 0.7 μm. Curves 1–3 correspond to the following a (μm): 0.02; 0.015; 0.01.
solution of surface IEs for threedimensional dielec tric bodies. The first version of the NV is a single NV with a length of 100 nm. Its cross section is a square with an edge of 40 nm. The second version is two sim ilar NVs placed on the same axis with a clearance of 30 nm (the inset in Fig. 3).The calculated resonance wavelengths are λr = 580 nm and λr = 662 nm. The order of the SLAE exceeds 3000. For similar cylindri cal NVs (with the same cross section of 1600 nm2), we obtained λr = 579 nm and λr = 646 nm. The order of the SLAE does not exceed 50. The difference of 2.5% can be explained by both the difference in the shape of the cross section and by the small difference in the dielectric permeability of gold; we took the data from another source.
1.0
1.5
2.0
2.5
3.0
3.5 λ, μm
Fig. 6. Characteristic of ZnO NV (with a radius of 0.01 μm, L = 0.7 μm) with variation of the thickness of the copper coating. The coating thickness is 5, 10, 15, and 20 nm (curves 1–4, respectively). Curves 5 and 6 corre spond to copper NVs with radii of 15 and 30 nm.
with a change of the metal and its thickness. The intensity of the field scattered by the MNV increases with an increase of its radius (Figs. 5, 6) but it is less than for the ideally conducting vibrator (Fig. 2). One should note the form of the dependence of λr on the radius and thickness of the metal coating. For the dielectric (Fig. 1) and ideal metal vibrator, λr increases with an increase of the radius a; for metal ones, the dependence is inverse, λr decreases (Figs. 5, 6). When diffraction takes place on a ZnO nanocrystal coated by a metal film (Fig. 6), the resonance is observed even if the film thickness t is several nanom eters. The resonance wavelength is strongly affected by the coating thickness. For comparison, the character
NUMERICAL RESULTS Figures 2–5 and 7 present the results for homoge neous nanovibrators and Fig. 6 presents the results for twolayer ones; this is a ZnO nanocrystal coated with a copper film. It follows from the figures that the dependence of the scattered field on the wavelength is of resonance character. One can observe two resonances with the chosen sizes and excitation. The NV is overlapped by a single halfwave during the first resonance and by three halfwaves during the second one. The amplitude of the first resonance is vastly larger than that of the sec ond one. The resonance wavelength of nanovibrators is larger than those of ideally conducting vibrators of the same size (Fig. 2). In a connected MNV, λr increases with a decrease of the clearance (Fig. 3). Certainly, the resonance wavelength increases with an increase of the vibrator length (Fig. 4). For a given nanovibrator length, one can vary λr in a wide range MOSCOW UNIVERSITY PHYSICS BULLETIN
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1.2 1.0 0.8 150 0.6 0.4 0.2 0 180 0.2 0.4 0.6 210 0.8 1.0 1.2
90 120
60 2 4
1
30
0 3 33 240
300 270
Fig. 7. Scattering diagram of a copper nanovibrator at nor mal incidence, L = 0.7 μm, a = 0.01 μm. Curves 1–4 cor respond to the following λ (μm): 0.5; 1.0; 1.5; 2.0. No. 1
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istics of fully metal vibrators are presented in the same figure. One can see that, for t = 20 nm (curve 4), the nanocrystal and the copper nanovibrator (curve 6) have similar characteristics. For lesser t, the field pen etrates into the nanocrystal and increases its influence on the scattering diagram. The scattering diagram of metal nanovibrators (Fig. 7) is similar to that of ideally conducting longer nanovibrators. The abovementioned dependences of λr on the radius and thickness of the metal coating can be explained by a simple formula for estimating the reso (m) nance wavelength of the mth resonance λ r : (m)
λr
c 2L = e , v m
where c is the velocity of light in a vacuum, v is the rate of a wave propagating along the vibrator, and Le is the electric (efficient) length of the vibrator; Le > L and depends on the relation of L/a and frequency. For the (m) ideal vibrator, v = c, the dependence λ r (a) is explained by the dependence Le(a). For a dielectric and nonideal metal nanovibrator, this is explained first of all by the fact that the decelerating coefficient n = c is a function of the radius. For a wave propagating v along a dielectric cylinder, the field is localized inside the dielectric and decreases exponentially outside it. With an increase of the radius, the deceleration coeffi (m) cient increases and, consequently, λ r also increases. At the dielectric–plasma interface (in our case, the vacuum–electron plasma of the metal) near the sur face wave (plasmon), the field exponentially decreases when it travels away from the interface both into the dielectric and into the metal. With a decrease of metal thickness, the interaction of plasmons propagating at the opposite boundaries of the metal increases. In this case, the deceleration coefficient n and, consequently, (m) λ r increase with an increase of the radius (Figs. 4, 5). This dependence n(a) can be easily obtained from the analysis of the dispersion equation for the Ewave propagating in the plasma layer with a thickness of 2a: χ χ – 1 = 2 coth χ 2 a, ε1 ε2 2
(10)
where χj = k n – ε j , ε2 is the dielectric permeability of the layer and ε1 is that of the dielectric surrounding the layer. Equation (10) has a solution for Reχ2 > 0 (the plasmon solution) when Reε2 < 0. Let Reχ2 Ⰷ 1 (the plasmons propagate along the boundaries of the layer without interaction). In this case, cothχ2a ≈ 1, Eq. (10) has a solution that does not depend on a:
ε1 ε2 . ε1 – ε2 Let Reχ2a Ⰶ 1 (strong interaction of the plas ε1 ⎞ 2 . mons), then from (10) we obtain n2 = ε1 + ⎛ ⎝ kaε 2⎠ When Reε2 < 0, the real part of the deceleration coef ficient increases with a decrease of a. n =
CONCLUSIONS The properties of metal nanovibrators and nanoc rystalsvibrators coated by a metal film were studied theoretically. The solution of the boundary problem on the diffraction of electromagnetic waves of the optical range on a metaldielectric nanovibrator is reduced to the solution of integro–differential equa tions for a nonhomogeneous dielectric cylinder whose kernels are represented as Fourier integrals. The IDEs were solved by combining the Galerkin and colloca tion methods. The matrix elements in the obtained SLAEs are expressed as Fourier integrals. Such a rep resentation of the kernels and matrix elements easily permits one to circumvent the difficulties connected with the singularity of IDE kernels. The internal con vergence of the solution is shown to be rapid. The characteristics of copper and gold nanoantennas were studied in the optical range. It was noted that the dependence of the scattered field on frequency is of a resonance character and that the resonance wave lengths of nanovibrators are larger than those of an ideally conducting vibrator of the same size. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research, project no. 090213530 ofi_ts. REFERENCES 1. P. Bharadwaj, B. Deutsch, and L. Novotny, Adv. Opt. Photon. 1, 438 (2009). 2. V. V. Klimov, Nanoplasmonics (Fizmatlit, Moscow, 2009; Pan Stanford, 2011). 3. V. V. Klimov, Usp. Fiz. Nauk 178, 875 (2008) [Phys. Usp. 51, 839 (2008)]. 4. K. Kempa, J. Rybczynski, et al., Adv. Mater. 19, 421 (2007). 5. J. Li, A. Salandrino, and N. Engheta, Phys. Rev. B 76, 245403 (2007). 6. J.S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, Nano Lett. 9, 1897 (2009). 7. V. S. Zuev and C. Ya. Zueva, J. Russ. Laser Res. 28, 272 (2007). 8. J. Li and N. Engheta, IEEE Trans. Antennas Propagat. 55, 3018 (2007).
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DIFFRACTION OF ELECTROMAGNETIC WAVES OF OPTICAL RANGE 9. M. Kern and J. F. Martin, J. Opt. Soc. Am. 26, 732 (2009). 10. M. Born and E. Wolf, The Principles of Optics (Nauka, Moscow, 1970; 6th ed., Cambridge Univ., Cambridge, 1999). 11. N. G. Khizhnyak, Integral Equations of Macroscopic Electrodynamics (Naukova Dumka, Kiev, 1986) [in Russian]. 12. A. M. Lerer, V. V. Makhno, P. V. Makhno, and A. A. Yachmenov, Radiotekh. Elektron. 52, 424 (2007) [J. Commun. Technol. Electron. 52, 399 (2007)].
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13. A. M. Lerer, Vestn. Mosk. Univ., Fiz. Astron., No. 5, 43 (2010). 14. A. M. Lerer, Vestn. Mosk. Univ., Fiz. Astron., No. 6, 48 (2010). 15. http://www.luxpop.com 16. P. V. Makhno, Candidate’s Dissertation in Mathemat ics and Physics (Yuzhn. Fed. Univ., RostovonDon, 2008). 17. V. I. Krylov and L. T. Shul’gina, A Handbook of Numer ical Integration (Nauka, Moscow, 1966) [in Russian].
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