The problem of subwavelength imaging by Veselago lens

Solid State Communications 146 (2008) 198–201 www.elsevier.com/locate/ssc

The problem of subwavelength imaging by Veselago lens A.L. Efros ∗ University of Utah, Department of Physics, Salt Lake City, UT 84112, USA Received 31 March 2007; accepted 16 April 2007 by the Guest Editors Available online 24 August 2007

Abstract A brief discussion of the imaging by the Veselago lens is given with a special attention to the absorption inside the lens. The near field regime is considered as the best candidate for the subwavelength imaging. A near field lens with a virtual focus is discussed in the original part of the paper. It is shown that it does not have some disadvantages of a lens with a real focus. A small absorption is not crucial for such a lens. The theory can be created without taking into account the imaginary part of the dielectric constant . The field does not diverge near the focus as Im  ⇒ 0 as it does in the lens with the real focus. Thus, the total absorption in this case may be significantly less. The lens or the sequence of the lenses can be used for three-dimensional image transmission. c 2007 Elsevier Ltd. All rights reserved.

PACS: 42.25.Gg; 42.68.Sq; 74.78.Na Keywords: A. Nanostructures; D. Optical properties

1. Introduction Recently there has been a growing interest in the creation of lenses with unusually sharp foci (see the recent review [1]). The near field or “electrostatic” lenses are the most promising candidates for this case (see experimental works [2,3]). Theoretical research on these lenses goes back to the paper by Nicorovici et al. [4] of 1994 who discussed a two-dimensional cylindrical geometry. The active element of this lens is a cylindrical metallic coating that has dielectric constant  = −1 + iIm  with negative real part and small imaginary part. The absolute value of the real part coincides with the dielectric constant of either the internal or external region. In this geometry the problem of focusing and cloaking of a twodimensional dipole source was studied. The wavelength of the source was assumed to be much larger than the radius of the cylinder. Thus, this is the near field regime. Then one can neglect retardation. In this approximation the magnetic field is small and the electrostatic potential obeys the Laplace equation ∇((r)∇Φ(r)) = 0. ∗ Tel.: +1 801 585 5018; fax: +1 801 581 4803.

E-mail address: [email protected]. c 2007 Elsevier Ltd. All rights reserved. 0038-1098/$ - see front matter doi:10.1016/j.ssc.2007.04.048

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On the other hand, if the frequency ω obeys the condition ωτ  1, where τ is the relaxation time of electrons, and the size of the slab is not smaller than the Fermi length of the metal [5], the real part of dielectric constant of a metal can be written in the form  ω 2 p (2) (ω) = 1 − ω −1 while the imaginary part is small √ as (ωτ ) . Here ω p is the plasma frequency. At ω = ω p / 2 one gets  = −1. All these conditions may be fulfilled only if the system is small enough (20–60 nm). So the new physics of this approach is closely connected to the recent development of nanotechnology and rarely can be found in classical textbooks. It is important that the wavelength is the largest length in this problem and it does not enter into the equations. So, by definition, any focusing system that is described by an electrostatic equation provides subwavelength resolution. Nicorovici et al. [4] obtained the following important results:

A. The solution of the cylindrical problem does not exist at Im  = 0. B. There are “resonant regions”, where the field tends to infinity as Im  ⇒ 0.

A.L. Efros / Solid State Communications 146 (2008) 198–201

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Fig. 1. Electrostatic or Veselago slab lens with two real foci outside and inside the slab. S is the source, a is the distance from the source to the slab, d is the width of the slab.

Fig. 2. Same as in Fig. 1 with resonant regions shown by dark rectangles in the background.The meaning of d and a are the same as in Fig. 1. Due to Milton et al. [6] the field diverges in these regions when the absorption tends to zero.

The cylindrical problem may be easily transformed into a problem of a metallic slab lens (Fig. 1) [6]. A similar slab lens was proposed in 1967 by Veselago [7]. The slab of the Veselago lens is filled by the material with simultaneously negative  and µ (Left-Handed Material). The Veselago lens provides focusing in both near field and far field regimes. In the near field regime, depending on polarization, the problem of the Veselago lens is equivalent either to the problem of the electrostatic lens or to the problem of the magnetostatic lens. In this paper we consider the TM polarization and electrostatic lens, where the slab is made of a metal. In the wave regime the Veselago lens should also have the specific magnetic properties. The creation of such lenses is a much more difficult problem. Very often they are the periodic structures [1] and their magnetic permeability µ may have a spatial dispersion (permeability µ is a function of the wave vector k). This dispersion creates problems with the boundary conditions because they are formulated in the coordinate rather than in the k-space. Moreover, in this case the magnetic permeability is the property of an electromagnetic mode rather than the property of a material. Thus, µ might be different for the propagating and for the evanescent modes [8]. All these problems are outside this paper. I consider here only “the basic model” which is a homogeneous and isotropic “hypothetical” slab that has (do not ask why!) Re  = −1 and Re µ = −1 while the imaginary parts are small. In the far field regime, if the wavelength is small enough, the width of the focus is given by the diffraction theory (see [9–11]). The criterion that shows how small the wavelength should be includes not only the width of the slab but also the absorption (Im  and Im µ). At a smaller absorption the diffraction theory is applicable at smaller wavelength [12]. If Im  = Im µ = 0, the problem of the far field Veselago lens also does not have an integrable solution and the same resonant regions, as in the electrostatic lens exist in the far field case as well. This observation has been made by Garcia and NietoVesperinas [13], Pokrovsky and Efros [9], and by Haldane [14] in connection with the paper by Pendry [15], who coined

a term “the perfect lens” for both near field and far field cases. Apparently the crucial role of the absorption was not understood in Pendry’s paper but the terminological finding made a powerful impetus to the whole field. The main point of our understanding of the far field case is that two limits, namely, absorption ⇒ 0 and wavelength ⇒ 0 do not commute. If one takes the first limit keeping wavelength constant, one gets infinite fields in the resonant regions and no integrable solution. Taking the second limit at a constant absorption one gets diffraction regime with finite fields that are independent of absorption if the absorption is not large. The calculations show [12] that the diffraction theory is applicable in the far field regime unless the losses are extremely small. Fig. 2 shows the resonant regions by Nicorovici et al. for the two-dimensional slab lens. There are two regions with the width d − a, where d is the width of the lens and a is the distance from the source to the slab. One of them is located between two foci of the lens while the middle of the other region is located at the front edge of the slab. In both regions electric field diverges as Im  H⇒ 0. This means the absence of a quadratically integrable solution at Im  = 0. Since the electric field E in the resonant regions increases with decreasing Im  the total absorption does not depend linearly on Im . Milton et al. [6] showed that Im  E 2 ∼ Im  (2a/d−1) | ln Im |.

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This equation is valid in the near field regime and also in the far field regime if absorption is very small. If a < d/2, the resonant regions overlap and the source S is inside the resonant region. In this case the total absorption tends to infinity as Im  tends to zero. Milton et al. [6,16] showed that under this condition the electrostatic lens cloaks two-dimensional dipoles rather than imaging them “perfectly”. 2. Electrostatic lenses with quasi-virtual and virtual foci If d > a > d/2, it follows from Eq. (3) that the absorption in the electrostatic lens tends to zero as Im  tends to zero, but due to increase of the fields in the resonant regions it is proportional

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Fig. 3. Slab lens with a virtual focus outside the slab (a > 2d).

to Im  (2a/d−1) , where 0 < 2a/d − 1 < 1. Thus, at small Im  the absorption is much larger than in the system without resonant regions. It is important that the electrostatic lens cannot have any real focus at all because due to the well-known Earnshaw’s theorem the potential has neither a maximum nor a minimum at a point where the charge is absent. That is why the boundaries of the resonant region go through both foci (see Fig. 2). There is no maximum in the z-direction at the focal point. An observer located to the right of point z = 2d may see an image smeared in the lateral direction by a distance [17,18,11,19–21] ∆ ≈ 2π d/|ln(Im )|.

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The lateral width of the image tends to zero as Im  tends to zero. However, an observer located to the left of this point may see only a large and strongly oscillating potential. Thus, the “focus” of electrostatic lens may be called quasi-virtual. This picture makes a substantial difference with the picture of the far field focus of the Veselago lens if it is in the regime of diffraction. In this case the real three-dimensional focus exists that can be observed from all directions. But the focus is smeared in the framework of the diffraction theory. General theorems of Electrodynamics forbid “perfect” focusing in a sense that the image cannot exactly reproduce the object. Indeed, to reproduce the field near the source one needs the source. Outside the regime of diffraction we do not see much difference between the far field and near field foci. In this paper I would like to pay attention to the near field lenses with completely virtual foci. Virtual foci appear under condition a > d. If a > 2d, the virtual focus (VF) is outside the slab (Fig. 3), but if 2d > a > d, VF is inside the slab (Fig. 4). First I show that in both cases the electrostatic problem has an exact integrable solution that exists at Im  = 0. Suppose that electrostatic potential at point S (z = 0) in Figs. 3 or 4 has some arbitrary y-dependence Z ∞ 1 Φ(y, 0) = V (k) exp(iky)dk. (5) 2π −∞ The time exponent is omitted here. For z > 0, the potential has a form Z ∞ 1 Φ(y, z) = V (k) exp(iky)Fk (z)dk, (6) 2π −∞ where the function Fk (z) should be found in three different regions using the equation

Fig. 4. Slab lens with a virtual focus (a > d) at a distance η = a − d from the rear face of the slab. If a − d < d, the virtual focus is inside the slab.



 d2 2 − k Fk (z) = 0 dz 2

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and proper boundary conditions at the faces of the slab. Assuming that  = 1 outside the slab and  = −1 inside, one gets that F(z) is continuous while dF/dz should change sign at the boundaries of the regions. Then we get Fk (z) = exp −kz; 0 d + a.

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To find the potential behind the slab (z > d + a) one should substitute the Eq. (10) into Eq. (5). This potential has a form Z ∞ dk Φ(y, z) = V (k) exp[iky − |k|(z − 2d)]. (11) −∞ 2π One can see a single VF at z = 2d. An observer at any point to the right of the slab sees potential Φ(y, z) as if it were created by the object, given by Eq. (5) but this object is shifted without any distortion to the point z = 2d. In this sense the VF is “perfect”. The absence of any distortion is due to the amplification of potential with increasing z inside the slab (see Eq. (9)). Note that the existence of the perfect VF does not contradict any general theorem because at the point of the VF the potential does not have a maximum or a minimum. All integrals over k are finite at a > d because z − 2a < 0 inside the slab and z − 2d > 0 behind the slab. Thus, in this case the solution of Maxwell’s equations can be obtained without any regularization. It is interesting that the VF may be located infinitesimally close to the rear face of the slab (the point z = d + a). This happens if the object is at a distance a from the lens and a is only infinitesimally larger than the thickness of the slab d. Let a = d + η. Now the virtual image is inside the slab at a distance η from its rear face. At negative η the focus becomes real and

A.L. Efros / Solid State Communications 146 (2008) 198–201

the theory is ruined by divergent integrals. But at any positive η this does not happen. This means that at Im  = 0 the image can be created without any distortions infinitesimally close to the rear surface of the slab but still inside the slab. Now I take into account the imaginary part of . It is easy to show [11] that in this case the potential at z > d + a has a form Z ∞ dk V (k) exp[iky − |k|(z − 2d)] . (12) Φ(y, z) = 2 −∞ 2π 1 + (Im /2) exp[2|k|d]

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Consider the potential at the rear face of the slab at z = a+d. One can show that if the distance η between the VF and the rear face is such that

some applications. Using this argument Podolskiy et al. [21] proposed the lens with a focus at the rear face (η = 0) as the optimal. We think that the virtual lens (η > 0) might be better because it minimizes absorption. Note that in the experiment by Fang et al. [2] the focus is virtual with η = 5 nm as it follows from Fig. 1 of their paper. c. An observer behind the lens sees an object that is displaced by a distance 2d toward the observer without any distortion. The image of a three-dimensional object can be displaced in the same way. Due to the small absorption one can construct a system consisting of a sequence of the slabs to transmit the image over a larger distance.

(Im /2)2 exp[2d/η]  1,

Acknowledgments

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one can put Im  = 0 and return to Eq. (11). The size of the light spot at the rear end of the slab is of the order of η. The estimate from Eq. (13) implies that absorption can be dropped under the condition η > d/|ln Im |. 3. Conclusions 1. To achieve the subwavelength imaging with the Veselago lens one should work in the near field regime. In this case electrostatic (or magnetostatic) approximation is applicable. In both cases wavelength is irrelevant and it should be considered as the largest (infinite) length in the problem. In the far field regime the subwavelength imaging is possible only in the case of extremely small losses. 2. I considered an electrostatic lens with a virtual focus as shown in Figs. 3 and 4. I do not think that the resolution of such a lens may be substantially better than the resolution of a lens with a real focus, but a lens with a virtual focus has the following properties that might be useful: a. If Im  is small, the solution can be obtained without taking absorption into account and this solution is integrable in all space. Since the field does not diverge as Im  ⇒ 0 the total absorption in the slab is much less than in the geometry where the focus is real. b. The geometry of the virtual focus lens has a larger distance between the object and the lens. This might be useful for

We are grateful to Graeme Milton and Emmanuel Rashba for the helpful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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