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High-negative effective refractive index of silver nanoparticles system in nanocomposite films
Optics Communications 285 (2012) 816–820
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
High-negative effective refractive index of silver nanoparticles system in nanocomposite films Konstantin K. Altunin a,⁎, Oleg N. Gadomsky b a b
I.N. Ulyanov Ulyanovsk State Pedagogical University, 432700 Ulyanovsk, Russia Ulyanovsk State University, 432700 Ulyanovsk, Russia
a r t i c l e
i n f o
Article history: Received 27 July 2011 Received in revised form 8 October 2011 Accepted 11 November 2011 Available online 24 November 2011 Keywords: Spherical silver nanoparticle Effective polarizability of electrons PMMA Nanocomposite film Quasi-zero refractive index Ideal optical antireflection
a b s t r a c t We have proved on the basis of the experimental optical reflection and transmission spectra of the nanocomposite film of poly(methyl methacrylate) with silver nanoparticles that (PMMA + Ag) nanocomposite films have quasi-zero refractive indices in the optical wavelength range. We show that to achieve quasi-zero values of the complex index of refraction of composite materials is necessary to achieve high-negative effective refractive index in the system of spherical silver nanoparticles. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction In our work [1-3] reported that the synthesis of new composite materials with quasi-zero values of effective refractive indices over a wide optical wavelength range. The conclusion that these materials have a quasi-zero refractive index is made based on an analysis of the experimental spectra of the optical reflection and transmission, as well as on the basis of the measurements of photovoltaic silicon solar cells, covered with a film of these materials. We show that in structures containing a film of this material, conditions are realized close to the ideal optical antireflection, when the reflection of light reaches the minimum value in the wide optical wavelength range, and the amplitude of the transmitted wave into the underlying medium is close to the amplitude of the incident wave [1]. On the basis of the analysis of photovoltaic measurement efficiency silicon solar cells the giant photovoltaic effect is detected [3], showing that the efficiency of solar cells coated with films of our materials enhanced by 1.6 times. Observed experimentally as a weak angular dependence of the photocurrent short circuit in silicon solar cells with antireflective coatings of our materials indicates that an additional reserve for increasing the efficiency of solar cells. Experimental investigations show that we are dealing with new composite materials, the refractive index which is close to zero and
⁎ Corresponding author. E-mail address: [email protected] (K.K. Altunin).
the values of the refractive index are realized in the wide optical wavelength range from 450 to 1100 nm. Synthesized by our technology composite materials are spherical silver nanoparticles in the polymer matrix PMMA. In Fig. 1 we have shown a typical distribution of nanoparticles in size in the films of PMMA + Ag, synthesized by our technology. In Fig. 2 we have shown the experimental spectrum of light reflection from the silicon surface and the silicon surface with PMMA + Ag film thickness of 50 μm. The reflection spectrum of structure (PMMA + Ag)/Si contains the interference maxima at the location where you can evaluate the refractive index of the film. In this work, we show that the quasi-zero values of the refractive index of the composite film due to the fact that the system of spherical nanoparticles forming a high-negative effective index of refraction, caused by negative variance effective polarizability of the valence electrons in spherical nanoparticles. As seen from Fig. 2, PMMA + Ag film, synthesized by our technology, is a high-efficiency antireflective coating that allows approximately seven times reducing light reflection from the silicon surface. 2. Quasi-zero refractive indices of the composite material The refractive index of the composite material (PMMA + Ag) is defined by the formula [4]:
m m n2 ¼ q1 np þ 1−q1 nm ;
0030-4018/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.033
ð1Þ
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
Fig. 1. The particle size distribution curve of nanoparticles of 100x100-nm (poly(methyl methacrylate)+ 1 wt.% Ag) nanocomposite film obtained using a JEM 100B (JEOL) transmitting electron microscope at 75 kV is presented on this figure. The weight maintenance of silver in the polymer film is nearby 1%.
where nm is the index of refraction of the polymer matrix (PMMA), np is the refractive index of the nanoparticles, q1m is the weight content of silver in the composite material, m
q1 ¼
ρA q1 ; ρA q1 þ ρm ð1−q1 Þ
ð2Þ
where ρA, ρm is density of solid silver and polymer (PMMA), respectively, ′
q1 ¼ N0
4π 3 a ; 3
ð3Þ
N0′ = 1/(RxRyRz) is the concentration of nanoparticles in the composite material, a is the radius of the nanoparticles of silver, Rγ = 2a + Δγ, Δγ is the mean distance between the surfaces of adjacent nanoparticles, γ = x, y, z. For a uniform distribution of spherical nanoparticles have ′
N0 ¼
1 ; ð2a þ ΔÞ3
ð5Þ
where κp is the absorption index, N is the concentration of valence p electrons in nanoparticles of silver (N = 5.8 ⋅ 10 22 cm − 3), αeff is the effective polarizability of the valence electrons of the form p
α eff ¼
α ; 1−aT Nα
Fig. 2. The reflection spectra of an amorphous silicon (curve 1) and the silicon surface with the antireflection of PMMA + Ag nanocomposite film (curve 2).
where α is quantum polarizability α¼
2jd0 j2 1 ℏ ω0 −ω−ı
1 τ′2
;
ð7Þ
corresponding to an isolated resonance at the frequency ω0. In this formula, d0 is the dipole moment of the quantum transition, ω is frequency of the external field, 1/τ2′ is resonance width. The parameters d0, ω0, 1/τ2′ for electric dipole quantum transitions in the quantum spherical nanoparticles of silver is defined by the experimental scattering spectra of the isolated region of silver [5]. In [2] we obtained the interpolation formulae for these transitions at different radii of silver nanoparticles. In the formula (6) aT is external geometric factor. For small particle radii aT ≈ − (4π/3)(1 + ıωa/c) [1]. Select in the formula (5) the real and imaginary parts. Then we obtain the following system of equations to calculate the np and κp: n2p −κ 2p ¼ ε1 ; 2np κ p ¼ ε2 ;
ð8Þ
where ð4Þ
where Δ is average distance between the surfaces of adjacent spherical nanoparticles in the composite. In our experiments, the weight content of silver was about 3%. This means that for a radius a = 2.5 nm, the distance between adjacent nanoparticles is equal to Δ ≈ 21.6 nm, i.e. with a high degree of accuracy can be considered as spherical nanoparticles in the composite insulated and neglect the interaction between them. The refractive index of the nanoparticles np in the formula (1) is defined as follows: 8π p 2 1 þ Nα eff 3 ; np þ ıκ p ¼ 4π p 1− Nα eff 3
817
ð6Þ
3 3 −χ 1 þ 2χ 1 −2χ 22 4π 4π ; ε1 ¼ 2 3 −χ 1 þ χ 22 4π 3 3χ 2 ε2 ¼ ; 2 4π 3 −χ 1 þ χ 22 4π p
χ 1 ¼ NReα eff ;
ð9Þ
p
χ 2 ¼ NImα eff :
The refractive index of the matrix of poly(methyl methacrylate) nm = 1.492, so, according to the formula (1), to achieve quasi-zero values of n2 it is necessary that the refractive index np was negative. It is possible, if ε2 b 0, κp > 0, i.e., necessary to achieve the negative dispersion when p
Reα eff > 0;
p
Imα eff b 0:
ð10Þ
In Fig. 3 we have shown the dispersion dependence of the effective polarizability of the valence electrons in a spherical silver nanoparticles with radius a = 2.5 nm. The condition (10) is well satisfied in the wide optical wavelength range.
818
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
where Δn2 is the uncertainty of the refractive index of nanocomposite films, Δd2 is the corresponding uncertainty in film thickness. The validity of Eq. (12) confirmed by measurements of reflection and transmission spectra in various matrices (PMMA, PHEMA, and others), when added to those of the matrix of silver nanoparticles. It was noted that the location of interference maxima do not vary with film thickness. The uncertainty of the refractive index of Δn2 in the ratio of Eq. (12) means that near zero values of the refractive index there is a region of admissible quasi-zero values of the refractive index of the nanocomposite films outside of accurately measuring the refractive index. The physical meaning of the uncertainty of the film thickness Δd2 is the following. A gradual change in the thickness of the nanocomposite film changes the range of permissible values ofΔn2, so that the product of these variables is equal to a constant value. Outside of quasi-zero values of the refractive index we find that the refractive index determined, that is taking the exact value and, consequently, the thickness of the film is a deterministic quantity. So when the film thickness Δd2=1 μm, according to the relation (12), we obtain the value of the refractive index of the nanocomposite films n2 = 1.6. Given that the refractive index of the matrix is equal to 1.492, it means that the addition of silver nanoparticles increases the refractive index of the nanocomposite films compared with the refractive index of the matrix. That is experimentally observed. Thus, according to the uncertainty relation (12) quasi-zero values of the refractive index can be achieved at the thickness about 50 μm of the nanocomposite films. Using relation (12) as shown below, can be properly interpreted experimental reflection spectra of various structures containing composite film of the new materials.
(a)
8 6 4
5
2
4 3
0 400
500
600
700
2 800
900 1000
1
(b)
3
0 -25
-2
Im
e ff
-4 -6
5 4
-8
3
400
500
600
700
2
800
900 1000
1
Fig. 3. Effective polarizability αeff of valence electrons in the spherical nanoparticles of silver as a function of wavelength and radius of the nanoparticles. (a) - the real part, and (b) - the imaginary part of the effective polarizability.
From the system (8) we obtain the following solution for np: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ε þ ε21 þ ε22 : np ¼ − 2 1 2
ð11Þ
The high-negative values of effective refractive index of nanoparticles np are obtained, as seen from (9), for the high values of ε1, which 4π in turn depend on the difference between 1− χ 1 . When 3 4π m χ 1 ¼ 0:999, q1 = 0.03, a = 2.5 nm we find that np = − 48.3 and the 3 refractive index of the composite vanishes. A slight deviation 4π NReα peff on 0.999, due to, for example, the distribution of the radii 3 of the nanoparticles in the composite, can reach values n2 = 0.039. 3. The uncertainty relation for the refractive indices of new composite materials and film thickness of the material 4π NReα peff in the 3 range of 0.99 and 0.999 leads to a change in the refractive index of (PMMA + Ag) composite material from zero to n2 = 0.039. Change of this magnitude due to random scatter of the radii of spherical silver nanoparticles in accordance with Fig. 1. This means that the refractive index of the new composite material is a random variable that varies in the range of Δn2, which includes the value of n2 = 0. Analysis of the experimental reflection and transmission spectra of optical radiation shows that the following uncertainty relation As shown above, a small change in the value
1 Δn Δd ≈1:6; λ 2 2
ð12Þ
4. The amplitudes of optical fields inside and outside of the composite film with quasi-zero refractive indices The possibility of a perfect optical antireflection when the reflected light is completely suppressed, incorporated in the classical formulae for single-layer coating on the surface of a semi-infinite optical medium [6]. In [7] it was shown that a monolayer of spherical nanoparticles capable to suppress the reflection of light from the surface of a semiinfinite medium in the visible wavelength range. There were formulae for the amplitudes of the fields inside and outside of the nanoparticle monolayer. On the basis of these formulae we have shown that the effect of an ideal optical antireflection of the surface is possible when the amplitude of the reflected light wave is zero, and the amplitude of the transmitted wave into the underlying medium is equal to the amplitude of the external waves at different incident angles. The formulae for the amplitudes of the light fields inside and outside of the monolayer of spherical metal nanoparticles are expressed through the effective polarizability of the valence electrons in the nanoparticles. The valence electrons are involved in the electric dipole quantum transitions. The ideal optical antireflection effect was predicted theoretically in [7] for example, a monolayer of spherical nanoparticles on the surface of a semi-infinite medium. In this work we have derived formulae for the amplitudes of optical fields inside and outside of a monolayer of particles with their discrete distribution. Based on these formulae, it was shown in particular that the amplitude of the transmitted wave inside the underlying medium is the amplitude of the incident wave, regardless of the angle of incidence of external radiation. The ideal optical antireflection effect explore in this work are based on classical formulae for the amplitudes of the reflected and transmitted waves inside the underlying medium. In the classical formulae of a layer optical properties are characterized by the complex
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
index of refraction of nanocomposite film and the refractive indices of surrounding media. The amplitude of the s-polarized waves reflected from the nanocomposite film, located on the surface of a semi-infinite optical medium with complex refractive index n3 + ıκ3 we define the classical formula [6]: s r 123
s s þ r expð2ıβ2 Þ s r : ¼ E0 12 s 23 1 þ r 12 r s23 expð2ıβ2 Þ
The amplitude of the s-polarized waves transmitted through the composite film at the medium 3, defined as s
s
t 123 ¼ E0
t s12 t s23 expðıβ2 Þ ; 1 þ r s12 r s23 expð2ıβ2 Þ
ð14Þ
where E0s is the amplitude of the s-polarized external wave, the propagation constant of the layer β2 ¼ k0 d2 ðn2 þ ıκ 2 Þcosθ2 ;
ð15Þ
k0 = 2π/λ, λ is the wavelength of incident light, d2 is nanocomposite film thickness, θ1 is angle of incidence of the external optical wave, θ2 is angle of refraction in medium 2, θ3 is angle of refraction in medis s um 3, r12 and r23 are the Fresnel amplitude coefficients of reflection at s s the boundaries 1–2 and 2–3, respectively, t12 and t23 are Fresnel amplitude coefficients of transmission of optical radiation on the boundaries of 1–2 and 2–3, respectively, s
r ik ¼ s
t ik ¼
ðni þ ıκ i Þcosθi −ðnk þ ıκ k Þcosθk ; ðni þ ıκ i Þcosθi þ ðnk þ ıκ k Þcosθk
ð16Þ
2ðni þ ıκ i Þcosθi ; ðni þ ıκ i Þcosθi þ ðnk þ ıκ k Þcosθk
ð17Þ
i, k are indices labeling medium. Relationship between the angles θ1 and θ2 determined from the equality ðn2 þ ıκ 2 Þsinθ2 ¼ sinθ1 ;
ð18Þ
that can be implemented in different physical situations. Upon reaching the n2 and κ2 quasi-zero values justice selection rules (18) is evident only in the case of normal incidence of light, when θ1 → 0. Below you will find comments, allows us to generalize our analysis and for the cases of oblique incidence of light with arbitrary angle of incidence θ1. Field in the film is composed of the refracted wave at the boundary between media 1 and 2 (amplitude of the E1s ) and the wave reflected from the boundary of 2–3 (the amplitude of the E2s ). The boundary conditions define the field at the 1–2 s
s
E1 ¼ E0
1þ
s t 12 ; s s r 12 r 23 expð2ıβ2 Þ
ð19Þ
as well as the electric field strength inside the composite film near the boundary of 2–3, s
s
E2 ¼ E0
r s23 t s12 expð2ıβ2 Þ : 1 þ r s12 r s23 expð2ıβ2 Þ
ð20Þ
We assume that the following equality s
s
r 23 expð2ıβ2 Þ ¼ −r12 :
ð21Þ
This equality can be achieved if the refractive index of the film is zero, that is n2 = κ2 = 0. Then from (14) if n2 = κ2 = 0, we obtain s
s
t 123 ¼ E0 expðıβ2 Þ;
which holds for all values of the angle of incidence of external radiation on the nanocomposite film, lying on the substrate. Substituting Eq. (21) in formula (13), we obtain the uncertainty of the type (0/0), which, however, can be easily removed. As a result, n2 = κ2 = 0 we get that s
s
s
r 123 ¼ 0; t 123 ¼ E0 ; ð13Þ
ð22Þ
819
ð23Þ
that is the ideal optical antireflection condition. The condition (23) is valid not only for different values of the angle of incidence of external radiation, but for different underlying optical media. This means that the transmittance of the layer is equal to T¼
n3 cosθ3 s 2 t ; n1 cosθ1 123
ð24Þ
and if n3 > 1, then the transmittance of the layer to more than one. If the underlying medium is nonabsorbing, then under the ideal optical antireflection conditions (23) film is invisible to the observer who is in medium 1. If the underlying medium 3 is absorbing, then the observer will perceive the medium 3 as a blackbody, regardless of film thickness and angle of incidence of external radiation. We now consider the physical situation in which the refractive index and absorption coefficient are different from zero, but take values close to zero. This case is quasi-zero values of n2 and κ2. Note that the substitution of near-zero values of n2 and κ2 without any restriction to the classical formula (13) leads to values of the reflection amplitude close to unity. At that time, as in n2 = κ2 = 0, as was shown above, the reflection amplitude vanishes. Moreover, the rate of change of R s, i.e. ∂ R s/∂ n2 at n2 = κ2 = 0 goes to infinity. To resolve these contradictions, we assume that the experimental spectra of the optical reflection and transmission of metal-polymer nanostructured composite films with quasi-zero values of the refractive index and absorption coefficient are formed by adding the amplitudes of the coherent reflection and transmission with complex ~ 2 ¼ n2 þ ıκ 2 . refractive indices in the region of small values of n Then, after the integration of the amplitudes (13) and (14) we obtain the following formula for the amplitude reflectance and transmittance of the film: s s 2 s 1− rs12 2 1 þ r 12 r 23 Φ2 r 123c s ~2 þ ı ; ¼ r 12 Δn ln Es0 2k0 d2 r s12 1 þ r s12 r s23 s s t s123c ı s s ¼ 1− t t F r r ; k0 d2 12 23 c 12 23 Es0
ð25Þ
ð26Þ
1 Φ 1 F c ðxÞ ¼ pffiffiffi arctan pffiffi2ffi −arctan pffiffiffi ; x x x where Φ2 = exp(ıβ2), the Fresnel amplitude reflection and transmission coefficients, and refraction angle θ2, defined by Eqs. (16)–(17), choosing a value of the complex refractive index of the interval ~ 2 Þ. The first term in Eq. (26) corresponds to the zero value of ð0; Δn the refractive index of the film from the region of integration ~ 2 Þ. In this case, the dependence of refraction angles θ2 vanishes ð0; Δn as the phase in different observation points of the film vanishes. This means that placing any restrictions on the angles of incidence and refraction does not make sense. At nonzero values of n2 and κ2 films were melted selection performed Eq. (18), therefore the values of cosθ2, close to zero cannot be implemented in quasi-zero values of n2 and κ2. Thus, formula (26) for the amplitudes of the transmitted wave inside the medium 3 can be applied with a high degree of accuracy at different values of angle of incidence of external radiation, given that the second and third terms in this formula is much less than unity. Similarly, in quasi-zero values of n2 and κ2 formula (25) for the amplitude of the reflected waves can be used with a high
820
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
degree of accuracy at different values of angle of incidence, given the weak dependence of the Fresnel coefficients of the angles θ1 and θ2. The derivation of formulae for the amplitudes p-polarized waves can be carried out in this way. In this case the form of the formula (25) remains valid if the Fresnel amplitude coefficients to replace s s the rik and tik the corresponding Fresnel amplitude coefficients of pp polarized waves. For the amplitude of the transmitted wave is t123 p-polarized waves should use any appropriate formula for the amplitude of the transmitted wave [8] and we integrate this expression ~ 2 Þ. over all values of the refractive index in the range ð0; Δn Similarly, we derive the formula for the amplitudes of the fields inside the layer with quasi-zero refractive indices, using the formula ~ 2 Þ. (19) and (20) with subsequent integration in the interval ð0; Δn 5. Theoretical description of the experimental spectra
2
s 2 t T2 ¼ 123c 2 T 1 ts 123c ′
2
jt j 31′ 2 t 13
ð28Þ
s by replacing the different from unity. Here t s123c ′ differs from t123c indices 1 → 3. Thus, the correct explanation of the experimental spectrum shown in Fig. 4, it is possible only when the (PMMA + Ag) nanocomposite layer has a quasi-zero refractive index and amplitude passing through this layer waves obey the formula (26). 6. Discussion
We apply the formula (25) and (26), as well as the uncertainty relation (12), for a theoretical description of the experimental reflection and transmission spectra of structures containing the nanocomposite layers with quasi-zero refractive indices. The experimental reflection spectrum (Fig. 2) can be represented s s as R = |r123c | 2/|E0s | 2, where r123c is given by (25). In this case d2 replaced by Δd2 in accordance with the uncertainty relation (12). A satisfactory theoretical description of the experimental reflection ~ 2 ¼ 0:039. The fact that the value spectrum in Fig. 2, is reached at n of uncertainty Δd2 smaller than the thickness of the nanocomposite films d2 = 50 μm means that the nanostructured layer behaves in the field of optical radiation as the layer with an effective thickness, which has uncertainty Δd2. In Fig. 4 we have shown the experimental spectrum of the ratio T2/T1 structure (PMMA+ Ag)/glass, where T2 is the optical transmittance of structure, when light falls from the side of the film, T1 is the optical transmittance, when the light falls from the substrate and the surrounding medium (media 1 and 4) for the structure is air. We consider the ratio T2/T1 in the traditional approach. Assume that the film of poly(methyl methacrylate) with silver nanoparticles can be described by the formula (14). In this case, the ratio 2
T 2 jt 12 t 23 j jt 31 j ¼ ¼ 1; T 1 jt 32 t 21 j2 jt 13 j2
Now we calculate the ratio of the direct optical transmission from the side of the film and the inverse optical transmission from the side of the substrate, using the formula (26). In this case, the ratio
ð27Þ
i.e. optical transmission, when direct optical radiation falls from the side of the nanocomposite film and from the side of the medium 4. This is consistent with the known result [9].
So, in this work we have proved that our technology for synthesizing (PMMA + Ag) nanocomposite films have the complex refractive ~ 2 Þ. An illustraindices in the quasi-zero refractive index range ð0; Δn tion of the unique optical properties of such films is, for example, optical reflection and transmission spectra in Fig. 2, Fig. 4. Further evidence of the unique properties of these films is measuring photovoltaic silicon solar cells, coated (PMMA + Ag) films with quasi-zero refractive indices. As shown in [3] the giant photovoltaic effect is detected in such solar cells, allowing a 1.6-fold their efficiency enhancement. According to the formula (1), the achievement of quasi-zero values of the effective refractive index of nanocomposite films, including zero, perhaps in the case where the system of silver nanoparticles formed by the high-negative effective refractive index. This is possible because of the negative dispersion of effective polarizability of the valence electrons in the silver nanoparticles. Metamaterials with negative refractive index in a large number of theoretical and experimental work [8,10-12]. A distinctive feature of our metamaterials is that the refractive index of the composite is non-negative value (n2 ≥ 0) and the refractive index of the nanoparticles, which is part of the nanocomposite material reaches the highnegative values in the wide optical wavelength range. Here we consider the spherical nanoparticles, which are implemented within the electric dipole quantum transitions of the valence electrons in the silver nanoparticles. In this article we discussed the nanocomposite metamaterials with a random refractive index range of permissible values which obeys the uncertainty relation (12). The physical meaning of the uncertainty of the film thickness Δd2 we explain the fact that in the film with the quasi-zero refractive index an area of the spatial coherence of the dipole oscillators is formed. References
Fig. 4. Relative optical transmittance of the coating nanocomposite film PMMA + Ag with weight silver maintenance of 1% with thickness of d2=50 μm on the glass with thickness of 2 mm on the wavelength of the external optical radiation, T2 is the optical transmittance from the film, T1 is the optical transmittance from the side of the underlying medium.
[1] O.N. Gadomsky, K.K. Altunin, N.M. Ushakov, JETP Letters 90 (2009) 251. [2] O.N. Gadomsky, K.K. Altunin, N.M. Ushakov, I.D. Kosobudskii, V.Y. Podvigalkin, D.M. Kulbatskii, Technical Physics 55 (2010) 996. [3] O.N. Gadomsky, K.K. Altunin, N.M. Ushakov, D.M. Kulbatskii, JETP Letters 93 (2011) 320. [4] N.V. Voshchinnikov, G. Videen, T. Henning, Applied Optics 46 (2007) 4065. [5] H. Tamaru, H. Kuwata, H.T. Miyazaki, K. Miyano, Applied Physics Letters 80 (2002) 1826. [6] L.D. Landau, E.M. Lifshitz, 2nd edition, Electrodynamics of continuous media, 8, Pergamon Press, Oxford, New York, Beijing, Frankfurt, 1984. 7 O.N. Gadomsky, A.S. Shalin, JETP 105 (2007) 761. 8 A.P. Vinogradov, A.V. Dorofeenko, S. Zouhdi, Physics Usp. 51 (2008) 485. 9 S.A. Furman, A.V. Tikhonravov, Basics of optics of multilayer systems, Gif-sur-Yvette Cedex1st edition, 1992. 10 M. Silveirinha, N. Engheta, Physical Review B 75 (2007) 075119. 11 M. Silveirinha, N. Engheta, Physical Review Letters 97 (2006) 157403. 12 J.-Q. Xi, J.K. Kim, E.F. Schubert, Nano Letters 5 (2005) 1385.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
High-negative effective refractive index of silver nanoparticles system in nanocomposite films Konstantin K. Altunin a,⁎, Oleg N. Gadomsky b a b
I.N. Ulyanov Ulyanovsk State Pedagogical University, 432700 Ulyanovsk, Russia Ulyanovsk State University, 432700 Ulyanovsk, Russia
a r t i c l e
i n f o
Article history: Received 27 July 2011 Received in revised form 8 October 2011 Accepted 11 November 2011 Available online 24 November 2011 Keywords: Spherical silver nanoparticle Effective polarizability of electrons PMMA Nanocomposite film Quasi-zero refractive index Ideal optical antireflection
a b s t r a c t We have proved on the basis of the experimental optical reflection and transmission spectra of the nanocomposite film of poly(methyl methacrylate) with silver nanoparticles that (PMMA + Ag) nanocomposite films have quasi-zero refractive indices in the optical wavelength range. We show that to achieve quasi-zero values of the complex index of refraction of composite materials is necessary to achieve high-negative effective refractive index in the system of spherical silver nanoparticles. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved.
1. Introduction In our work [1-3] reported that the synthesis of new composite materials with quasi-zero values of effective refractive indices over a wide optical wavelength range. The conclusion that these materials have a quasi-zero refractive index is made based on an analysis of the experimental spectra of the optical reflection and transmission, as well as on the basis of the measurements of photovoltaic silicon solar cells, covered with a film of these materials. We show that in structures containing a film of this material, conditions are realized close to the ideal optical antireflection, when the reflection of light reaches the minimum value in the wide optical wavelength range, and the amplitude of the transmitted wave into the underlying medium is close to the amplitude of the incident wave [1]. On the basis of the analysis of photovoltaic measurement efficiency silicon solar cells the giant photovoltaic effect is detected [3], showing that the efficiency of solar cells coated with films of our materials enhanced by 1.6 times. Observed experimentally as a weak angular dependence of the photocurrent short circuit in silicon solar cells with antireflective coatings of our materials indicates that an additional reserve for increasing the efficiency of solar cells. Experimental investigations show that we are dealing with new composite materials, the refractive index which is close to zero and
⁎ Corresponding author. E-mail address: [email protected] (K.K. Altunin).
the values of the refractive index are realized in the wide optical wavelength range from 450 to 1100 nm. Synthesized by our technology composite materials are spherical silver nanoparticles in the polymer matrix PMMA. In Fig. 1 we have shown a typical distribution of nanoparticles in size in the films of PMMA + Ag, synthesized by our technology. In Fig. 2 we have shown the experimental spectrum of light reflection from the silicon surface and the silicon surface with PMMA + Ag film thickness of 50 μm. The reflection spectrum of structure (PMMA + Ag)/Si contains the interference maxima at the location where you can evaluate the refractive index of the film. In this work, we show that the quasi-zero values of the refractive index of the composite film due to the fact that the system of spherical nanoparticles forming a high-negative effective index of refraction, caused by negative variance effective polarizability of the valence electrons in spherical nanoparticles. As seen from Fig. 2, PMMA + Ag film, synthesized by our technology, is a high-efficiency antireflective coating that allows approximately seven times reducing light reflection from the silicon surface. 2. Quasi-zero refractive indices of the composite material The refractive index of the composite material (PMMA + Ag) is defined by the formula [4]:
m m n2 ¼ q1 np þ 1−q1 nm ;
0030-4018/$ – see front matter. Crown Copyright © 2011 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.11.033
ð1Þ
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
Fig. 1. The particle size distribution curve of nanoparticles of 100x100-nm (poly(methyl methacrylate)+ 1 wt.% Ag) nanocomposite film obtained using a JEM 100B (JEOL) transmitting electron microscope at 75 kV is presented on this figure. The weight maintenance of silver in the polymer film is nearby 1%.
where nm is the index of refraction of the polymer matrix (PMMA), np is the refractive index of the nanoparticles, q1m is the weight content of silver in the composite material, m
q1 ¼
ρA q1 ; ρA q1 þ ρm ð1−q1 Þ
ð2Þ
where ρA, ρm is density of solid silver and polymer (PMMA), respectively, ′
q1 ¼ N0
4π 3 a ; 3
ð3Þ
N0′ = 1/(RxRyRz) is the concentration of nanoparticles in the composite material, a is the radius of the nanoparticles of silver, Rγ = 2a + Δγ, Δγ is the mean distance between the surfaces of adjacent nanoparticles, γ = x, y, z. For a uniform distribution of spherical nanoparticles have ′
N0 ¼
1 ; ð2a þ ΔÞ3
ð5Þ
where κp is the absorption index, N is the concentration of valence p electrons in nanoparticles of silver (N = 5.8 ⋅ 10 22 cm − 3), αeff is the effective polarizability of the valence electrons of the form p
α eff ¼
α ; 1−aT Nα
Fig. 2. The reflection spectra of an amorphous silicon (curve 1) and the silicon surface with the antireflection of PMMA + Ag nanocomposite film (curve 2).
where α is quantum polarizability α¼
2jd0 j2 1 ℏ ω0 −ω−ı
1 τ′2
;
ð7Þ
corresponding to an isolated resonance at the frequency ω0. In this formula, d0 is the dipole moment of the quantum transition, ω is frequency of the external field, 1/τ2′ is resonance width. The parameters d0, ω0, 1/τ2′ for electric dipole quantum transitions in the quantum spherical nanoparticles of silver is defined by the experimental scattering spectra of the isolated region of silver [5]. In [2] we obtained the interpolation formulae for these transitions at different radii of silver nanoparticles. In the formula (6) aT is external geometric factor. For small particle radii aT ≈ − (4π/3)(1 + ıωa/c) [1]. Select in the formula (5) the real and imaginary parts. Then we obtain the following system of equations to calculate the np and κp: n2p −κ 2p ¼ ε1 ; 2np κ p ¼ ε2 ;
ð8Þ
where ð4Þ
where Δ is average distance between the surfaces of adjacent spherical nanoparticles in the composite. In our experiments, the weight content of silver was about 3%. This means that for a radius a = 2.5 nm, the distance between adjacent nanoparticles is equal to Δ ≈ 21.6 nm, i.e. with a high degree of accuracy can be considered as spherical nanoparticles in the composite insulated and neglect the interaction between them. The refractive index of the nanoparticles np in the formula (1) is defined as follows: 8π p 2 1 þ Nα eff 3 ; np þ ıκ p ¼ 4π p 1− Nα eff 3
817
ð6Þ
3 3 −χ 1 þ 2χ 1 −2χ 22 4π 4π ; ε1 ¼ 2 3 −χ 1 þ χ 22 4π 3 3χ 2 ε2 ¼ ; 2 4π 3 −χ 1 þ χ 22 4π p
χ 1 ¼ NReα eff ;
ð9Þ
p
χ 2 ¼ NImα eff :
The refractive index of the matrix of poly(methyl methacrylate) nm = 1.492, so, according to the formula (1), to achieve quasi-zero values of n2 it is necessary that the refractive index np was negative. It is possible, if ε2 b 0, κp > 0, i.e., necessary to achieve the negative dispersion when p
Reα eff > 0;
p
Imα eff b 0:
ð10Þ
In Fig. 3 we have shown the dispersion dependence of the effective polarizability of the valence electrons in a spherical silver nanoparticles with radius a = 2.5 nm. The condition (10) is well satisfied in the wide optical wavelength range.
818
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
where Δn2 is the uncertainty of the refractive index of nanocomposite films, Δd2 is the corresponding uncertainty in film thickness. The validity of Eq. (12) confirmed by measurements of reflection and transmission spectra in various matrices (PMMA, PHEMA, and others), when added to those of the matrix of silver nanoparticles. It was noted that the location of interference maxima do not vary with film thickness. The uncertainty of the refractive index of Δn2 in the ratio of Eq. (12) means that near zero values of the refractive index there is a region of admissible quasi-zero values of the refractive index of the nanocomposite films outside of accurately measuring the refractive index. The physical meaning of the uncertainty of the film thickness Δd2 is the following. A gradual change in the thickness of the nanocomposite film changes the range of permissible values ofΔn2, so that the product of these variables is equal to a constant value. Outside of quasi-zero values of the refractive index we find that the refractive index determined, that is taking the exact value and, consequently, the thickness of the film is a deterministic quantity. So when the film thickness Δd2=1 μm, according to the relation (12), we obtain the value of the refractive index of the nanocomposite films n2 = 1.6. Given that the refractive index of the matrix is equal to 1.492, it means that the addition of silver nanoparticles increases the refractive index of the nanocomposite films compared with the refractive index of the matrix. That is experimentally observed. Thus, according to the uncertainty relation (12) quasi-zero values of the refractive index can be achieved at the thickness about 50 μm of the nanocomposite films. Using relation (12) as shown below, can be properly interpreted experimental reflection spectra of various structures containing composite film of the new materials.
(a)
8 6 4
5
2
4 3
0 400
500
600
700
2 800
900 1000
1
(b)
3
0 -25
-2
Im
e ff
-4 -6
5 4
-8
3
400
500
600
700
2
800
900 1000
1
Fig. 3. Effective polarizability αeff of valence electrons in the spherical nanoparticles of silver as a function of wavelength and radius of the nanoparticles. (a) - the real part, and (b) - the imaginary part of the effective polarizability.
From the system (8) we obtain the following solution for np: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ε þ ε21 þ ε22 : np ¼ − 2 1 2
ð11Þ
The high-negative values of effective refractive index of nanoparticles np are obtained, as seen from (9), for the high values of ε1, which 4π in turn depend on the difference between 1− χ 1 . When 3 4π m χ 1 ¼ 0:999, q1 = 0.03, a = 2.5 nm we find that np = − 48.3 and the 3 refractive index of the composite vanishes. A slight deviation 4π NReα peff on 0.999, due to, for example, the distribution of the radii 3 of the nanoparticles in the composite, can reach values n2 = 0.039. 3. The uncertainty relation for the refractive indices of new composite materials and film thickness of the material 4π NReα peff in the 3 range of 0.99 and 0.999 leads to a change in the refractive index of (PMMA + Ag) composite material from zero to n2 = 0.039. Change of this magnitude due to random scatter of the radii of spherical silver nanoparticles in accordance with Fig. 1. This means that the refractive index of the new composite material is a random variable that varies in the range of Δn2, which includes the value of n2 = 0. Analysis of the experimental reflection and transmission spectra of optical radiation shows that the following uncertainty relation As shown above, a small change in the value
1 Δn Δd ≈1:6; λ 2 2
ð12Þ
4. The amplitudes of optical fields inside and outside of the composite film with quasi-zero refractive indices The possibility of a perfect optical antireflection when the reflected light is completely suppressed, incorporated in the classical formulae for single-layer coating on the surface of a semi-infinite optical medium [6]. In [7] it was shown that a monolayer of spherical nanoparticles capable to suppress the reflection of light from the surface of a semiinfinite medium in the visible wavelength range. There were formulae for the amplitudes of the fields inside and outside of the nanoparticle monolayer. On the basis of these formulae we have shown that the effect of an ideal optical antireflection of the surface is possible when the amplitude of the reflected light wave is zero, and the amplitude of the transmitted wave into the underlying medium is equal to the amplitude of the external waves at different incident angles. The formulae for the amplitudes of the light fields inside and outside of the monolayer of spherical metal nanoparticles are expressed through the effective polarizability of the valence electrons in the nanoparticles. The valence electrons are involved in the electric dipole quantum transitions. The ideal optical antireflection effect was predicted theoretically in [7] for example, a monolayer of spherical nanoparticles on the surface of a semi-infinite medium. In this work we have derived formulae for the amplitudes of optical fields inside and outside of a monolayer of particles with their discrete distribution. Based on these formulae, it was shown in particular that the amplitude of the transmitted wave inside the underlying medium is the amplitude of the incident wave, regardless of the angle of incidence of external radiation. The ideal optical antireflection effect explore in this work are based on classical formulae for the amplitudes of the reflected and transmitted waves inside the underlying medium. In the classical formulae of a layer optical properties are characterized by the complex
K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
index of refraction of nanocomposite film and the refractive indices of surrounding media. The amplitude of the s-polarized waves reflected from the nanocomposite film, located on the surface of a semi-infinite optical medium with complex refractive index n3 + ıκ3 we define the classical formula [6]: s r 123
s s þ r expð2ıβ2 Þ s r : ¼ E0 12 s 23 1 þ r 12 r s23 expð2ıβ2 Þ
The amplitude of the s-polarized waves transmitted through the composite film at the medium 3, defined as s
s
t 123 ¼ E0
t s12 t s23 expðıβ2 Þ ; 1 þ r s12 r s23 expð2ıβ2 Þ
ð14Þ
where E0s is the amplitude of the s-polarized external wave, the propagation constant of the layer β2 ¼ k0 d2 ðn2 þ ıκ 2 Þcosθ2 ;
ð15Þ
k0 = 2π/λ, λ is the wavelength of incident light, d2 is nanocomposite film thickness, θ1 is angle of incidence of the external optical wave, θ2 is angle of refraction in medium 2, θ3 is angle of refraction in medis s um 3, r12 and r23 are the Fresnel amplitude coefficients of reflection at s s the boundaries 1–2 and 2–3, respectively, t12 and t23 are Fresnel amplitude coefficients of transmission of optical radiation on the boundaries of 1–2 and 2–3, respectively, s
r ik ¼ s
t ik ¼
ðni þ ıκ i Þcosθi −ðnk þ ıκ k Þcosθk ; ðni þ ıκ i Þcosθi þ ðnk þ ıκ k Þcosθk
ð16Þ
2ðni þ ıκ i Þcosθi ; ðni þ ıκ i Þcosθi þ ðnk þ ıκ k Þcosθk
ð17Þ
i, k are indices labeling medium. Relationship between the angles θ1 and θ2 determined from the equality ðn2 þ ıκ 2 Þsinθ2 ¼ sinθ1 ;
ð18Þ
that can be implemented in different physical situations. Upon reaching the n2 and κ2 quasi-zero values justice selection rules (18) is evident only in the case of normal incidence of light, when θ1 → 0. Below you will find comments, allows us to generalize our analysis and for the cases of oblique incidence of light with arbitrary angle of incidence θ1. Field in the film is composed of the refracted wave at the boundary between media 1 and 2 (amplitude of the E1s ) and the wave reflected from the boundary of 2–3 (the amplitude of the E2s ). The boundary conditions define the field at the 1–2 s
s
E1 ¼ E0
1þ
s t 12 ; s s r 12 r 23 expð2ıβ2 Þ
ð19Þ
as well as the electric field strength inside the composite film near the boundary of 2–3, s
s
E2 ¼ E0
r s23 t s12 expð2ıβ2 Þ : 1 þ r s12 r s23 expð2ıβ2 Þ
ð20Þ
We assume that the following equality s
s
r 23 expð2ıβ2 Þ ¼ −r12 :
ð21Þ
This equality can be achieved if the refractive index of the film is zero, that is n2 = κ2 = 0. Then from (14) if n2 = κ2 = 0, we obtain s
s
t 123 ¼ E0 expðıβ2 Þ;
which holds for all values of the angle of incidence of external radiation on the nanocomposite film, lying on the substrate. Substituting Eq. (21) in formula (13), we obtain the uncertainty of the type (0/0), which, however, can be easily removed. As a result, n2 = κ2 = 0 we get that s
s
s
r 123 ¼ 0; t 123 ¼ E0 ; ð13Þ
ð22Þ
819
ð23Þ
that is the ideal optical antireflection condition. The condition (23) is valid not only for different values of the angle of incidence of external radiation, but for different underlying optical media. This means that the transmittance of the layer is equal to T¼
n3 cosθ3 s 2 t ; n1 cosθ1 123
ð24Þ
and if n3 > 1, then the transmittance of the layer to more than one. If the underlying medium is nonabsorbing, then under the ideal optical antireflection conditions (23) film is invisible to the observer who is in medium 1. If the underlying medium 3 is absorbing, then the observer will perceive the medium 3 as a blackbody, regardless of film thickness and angle of incidence of external radiation. We now consider the physical situation in which the refractive index and absorption coefficient are different from zero, but take values close to zero. This case is quasi-zero values of n2 and κ2. Note that the substitution of near-zero values of n2 and κ2 without any restriction to the classical formula (13) leads to values of the reflection amplitude close to unity. At that time, as in n2 = κ2 = 0, as was shown above, the reflection amplitude vanishes. Moreover, the rate of change of R s, i.e. ∂ R s/∂ n2 at n2 = κ2 = 0 goes to infinity. To resolve these contradictions, we assume that the experimental spectra of the optical reflection and transmission of metal-polymer nanostructured composite films with quasi-zero values of the refractive index and absorption coefficient are formed by adding the amplitudes of the coherent reflection and transmission with complex ~ 2 ¼ n2 þ ıκ 2 . refractive indices in the region of small values of n Then, after the integration of the amplitudes (13) and (14) we obtain the following formula for the amplitude reflectance and transmittance of the film: s s 2 s 1− rs12 2 1 þ r 12 r 23 Φ2 r 123c s ~2 þ ı ; ¼ r 12 Δn ln Es0 2k0 d2 r s12 1 þ r s12 r s23 s s t s123c ı s s ¼ 1− t t F r r ; k0 d2 12 23 c 12 23 Es0
ð25Þ
ð26Þ
1 Φ 1 F c ðxÞ ¼ pffiffiffi arctan pffiffi2ffi −arctan pffiffiffi ; x x x where Φ2 = exp(ıβ2), the Fresnel amplitude reflection and transmission coefficients, and refraction angle θ2, defined by Eqs. (16)–(17), choosing a value of the complex refractive index of the interval ~ 2 Þ. The first term in Eq. (26) corresponds to the zero value of ð0; Δn the refractive index of the film from the region of integration ~ 2 Þ. In this case, the dependence of refraction angles θ2 vanishes ð0; Δn as the phase in different observation points of the film vanishes. This means that placing any restrictions on the angles of incidence and refraction does not make sense. At nonzero values of n2 and κ2 films were melted selection performed Eq. (18), therefore the values of cosθ2, close to zero cannot be implemented in quasi-zero values of n2 and κ2. Thus, formula (26) for the amplitudes of the transmitted wave inside the medium 3 can be applied with a high degree of accuracy at different values of angle of incidence of external radiation, given that the second and third terms in this formula is much less than unity. Similarly, in quasi-zero values of n2 and κ2 formula (25) for the amplitude of the reflected waves can be used with a high
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K.K. Altunin, O.N. Gadomsky / Optics Communications 285 (2012) 816–820
degree of accuracy at different values of angle of incidence, given the weak dependence of the Fresnel coefficients of the angles θ1 and θ2. The derivation of formulae for the amplitudes p-polarized waves can be carried out in this way. In this case the form of the formula (25) remains valid if the Fresnel amplitude coefficients to replace s s the rik and tik the corresponding Fresnel amplitude coefficients of pp polarized waves. For the amplitude of the transmitted wave is t123 p-polarized waves should use any appropriate formula for the amplitude of the transmitted wave [8] and we integrate this expression ~ 2 Þ. over all values of the refractive index in the range ð0; Δn Similarly, we derive the formula for the amplitudes of the fields inside the layer with quasi-zero refractive indices, using the formula ~ 2 Þ. (19) and (20) with subsequent integration in the interval ð0; Δn 5. Theoretical description of the experimental spectra
2
s 2 t T2 ¼ 123c 2 T 1 ts 123c ′
2
jt j 31′ 2 t 13
ð28Þ
s by replacing the different from unity. Here t s123c ′ differs from t123c indices 1 → 3. Thus, the correct explanation of the experimental spectrum shown in Fig. 4, it is possible only when the (PMMA + Ag) nanocomposite layer has a quasi-zero refractive index and amplitude passing through this layer waves obey the formula (26). 6. Discussion
We apply the formula (25) and (26), as well as the uncertainty relation (12), for a theoretical description of the experimental reflection and transmission spectra of structures containing the nanocomposite layers with quasi-zero refractive indices. The experimental reflection spectrum (Fig. 2) can be represented s s as R = |r123c | 2/|E0s | 2, where r123c is given by (25). In this case d2 replaced by Δd2 in accordance with the uncertainty relation (12). A satisfactory theoretical description of the experimental reflection ~ 2 ¼ 0:039. The fact that the value spectrum in Fig. 2, is reached at n of uncertainty Δd2 smaller than the thickness of the nanocomposite films d2 = 50 μm means that the nanostructured layer behaves in the field of optical radiation as the layer with an effective thickness, which has uncertainty Δd2. In Fig. 4 we have shown the experimental spectrum of the ratio T2/T1 structure (PMMA+ Ag)/glass, where T2 is the optical transmittance of structure, when light falls from the side of the film, T1 is the optical transmittance, when the light falls from the substrate and the surrounding medium (media 1 and 4) for the structure is air. We consider the ratio T2/T1 in the traditional approach. Assume that the film of poly(methyl methacrylate) with silver nanoparticles can be described by the formula (14). In this case, the ratio 2
T 2 jt 12 t 23 j jt 31 j ¼ ¼ 1; T 1 jt 32 t 21 j2 jt 13 j2
Now we calculate the ratio of the direct optical transmission from the side of the film and the inverse optical transmission from the side of the substrate, using the formula (26). In this case, the ratio
ð27Þ
i.e. optical transmission, when direct optical radiation falls from the side of the nanocomposite film and from the side of the medium 4. This is consistent with the known result [9].
So, in this work we have proved that our technology for synthesizing (PMMA + Ag) nanocomposite films have the complex refractive ~ 2 Þ. An illustraindices in the quasi-zero refractive index range ð0; Δn tion of the unique optical properties of such films is, for example, optical reflection and transmission spectra in Fig. 2, Fig. 4. Further evidence of the unique properties of these films is measuring photovoltaic silicon solar cells, coated (PMMA + Ag) films with quasi-zero refractive indices. As shown in [3] the giant photovoltaic effect is detected in such solar cells, allowing a 1.6-fold their efficiency enhancement. According to the formula (1), the achievement of quasi-zero values of the effective refractive index of nanocomposite films, including zero, perhaps in the case where the system of silver nanoparticles formed by the high-negative effective refractive index. This is possible because of the negative dispersion of effective polarizability of the valence electrons in the silver nanoparticles. Metamaterials with negative refractive index in a large number of theoretical and experimental work [8,10-12]. A distinctive feature of our metamaterials is that the refractive index of the composite is non-negative value (n2 ≥ 0) and the refractive index of the nanoparticles, which is part of the nanocomposite material reaches the highnegative values in the wide optical wavelength range. Here we consider the spherical nanoparticles, which are implemented within the electric dipole quantum transitions of the valence electrons in the silver nanoparticles. In this article we discussed the nanocomposite metamaterials with a random refractive index range of permissible values which obeys the uncertainty relation (12). The physical meaning of the uncertainty of the film thickness Δd2 we explain the fact that in the film with the quasi-zero refractive index an area of the spatial coherence of the dipole oscillators is formed. References
Fig. 4. Relative optical transmittance of the coating nanocomposite film PMMA + Ag with weight silver maintenance of 1% with thickness of d2=50 μm on the glass with thickness of 2 mm on the wavelength of the external optical radiation, T2 is the optical transmittance from the film, T1 is the optical transmittance from the side of the underlying medium.
[1] O.N. Gadomsky, K.K. Altunin, N.M. Ushakov, JETP Letters 90 (2009) 251. [2] O.N. Gadomsky, K.K. Altunin, N.M. Ushakov, I.D. Kosobudskii, V.Y. Podvigalkin, D.M. Kulbatskii, Technical Physics 55 (2010) 996. [3] O.N. Gadomsky, K.K. Altunin, N.M. Ushakov, D.M. Kulbatskii, JETP Letters 93 (2011) 320. [4] N.V. Voshchinnikov, G. Videen, T. Henning, Applied Optics 46 (2007) 4065. [5] H. Tamaru, H. Kuwata, H.T. Miyazaki, K. Miyano, Applied Physics Letters 80 (2002) 1826. [6] L.D. Landau, E.M. Lifshitz, 2nd edition, Electrodynamics of continuous media, 8, Pergamon Press, Oxford, New York, Beijing, Frankfurt, 1984. 7 O.N. Gadomsky, A.S. Shalin, JETP 105 (2007) 761. 8 A.P. Vinogradov, A.V. Dorofeenko, S. Zouhdi, Physics Usp. 51 (2008) 485. 9 S.A. Furman, A.V. Tikhonravov, Basics of optics of multilayer systems, Gif-sur-Yvette Cedex1st edition, 1992. 10 M. Silveirinha, N. Engheta, Physical Review B 75 (2007) 075119. 11 M. Silveirinha, N. Engheta, Physical Review Letters 97 (2006) 157403. 12 J.-Q. Xi, J.K. Kim, E.F. Schubert, Nano Letters 5 (2005) 1385.