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Imaging properties of uniaxially anisotropic negative refractive index materials
Physics Letters A 313 (2003) 316–324 www.elsevier.com/locate/pla
Imaging properties of uniaxially anisotropic negative refractive index materials Liangbin Hu a,∗ , Zhifang Lin b a Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China b Department of Physics, Fudan University, Shanghai 200433, PR China
Received 17 March 2003; received in revised form 29 April 2003; accepted 1 May 2003 Communicated by V.M. Agranovich
Abstract The imaging properties of uniaxially anisotropic negative refractive index materials have been investigated theoretically. Based on an accurate theoretical formulation, we have discussed in detail whether and under what conditions the slabs of uniaxially anisotropic negative refractive index materials will have the power to focus perfectly all Fourier components of an image. It was shown that if the effects of absorption and dispersion can be neglected, under some particular conditions the slabs of uniaxially anisotropic negative refractive index materials will have the power to focus perfectly all E-mode or all H -mode or both E- and H -mode Fourier components of an image. It was also shown that perfect focus cannot occur in the presence of absorption and dispersion, but if the effects of absorption and dispersion are rather small, excellent focus can still occur. 2003 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 78.20.Ci; 42.30.Wb; 78.66.Bz
1. Introduction Recently a new type of metamaterials that have negative refractive index have attracted much experimental and theoretical interest [1]. The concept of negative refractive index was first proposed by Veselago over thirty years ago, who predicted that electromagnetic wave propagations in a material having simultaneously negative permittivity and permeability µ should give rise to several peculiar characteristics, including anomalous refraction (that is, the refractive index is negative), reversal of both the Doppler shift and the Cherenkov radiation, etc., and he termed such type of materials as negative refractive index (NRI) materials or left-handed materials (LHMs) [2]. Due to the absence of naturally occurring NRI materials, Veselago’s prediction did not attract much attention until very recently, when some researchers reported that a new type of metamaterials have been prepared successfully following the suggestion of Pendry et al. [3] and were demonstrated to be NRI materials [4]. This success has aroused great interest in the unusual electrodynamic properties of NRI materials [1]. One of the most * Corresponding author.
E-mail address: [email protected] (L. Hu). 0375-9601/03/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00761-8
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interesting peculiar properties of NRI materials is their unusual imaging properties [2,5]. It was first pointed out by Veselago that the simple planar slabs made of NRI materials will converge rather than diverge the radiation of a source, hence the slabs of NRI materials could be employed as an unconventional alternative to traditional lenses [2]. Recently Pendry further concluded that in some particular case the slabs of ideal lossless and dispersiveless NRI materials can amplify the evanescent waves, and due to this fact, the slabs of such NRI materials can act like perfect lens, which means that such a slab can focus perfectly all Fourier components of an image and hence the image will have maximum resolution [5]. Though in real NRI materials the effects of absorption and dispersion will decrease inevitably the resolution of an image, Pendry’s conclusion is still a rather interesting challenge to the traditional limitation on lens performance. Motivated by Pendry’s interesting conclusion, recently a number of theoretical works have been devoted to the investigation of the unusual imaging properties of NRI materials [6–11]. Basically in all previously published theoretical works, only isotropic NRI materials have been studied and the effects of anisotropy have been neglected completely. However, the recently realized NRI materials are actually anisotropic in nature, and up to now, no isotropic NRI materials have been prepared successfully in experiments [1,4]. In fact, from both the experimental and theoretical viewpoints, the task of preparing an isotropic NRI material is much more difficult than that of preparing an anisotropic NRI material [1]. As was well known before, the electrodynamic properties of anisotropic materials are usually very different from that of isotropic materials [12–14]. Due to the above reasons, in order to get a more clear understanding of the unusual imaging properties of NRI materials and to clarify whether Pendry’s proposal is applicable or not, the effects of anisotropy should be taken into account. The simplest and most common form of anisotropy is uniaxial anisotropy, and from the analysis of the symmetry of the recently realized NRI materials [4], they should also be uniaxially anisotropic. In this Letter we present a theoretical investigation on the unusual imaging properties of uniaxially anisotropic NRI materials. Based on an accurate theoretical formulation, we will study in detail whether the slabs of uniaxially anisotropic NRI materials can have the power to focus perfectly all Fourier components of an image. We will find out accurately the conditions under which the slabs of uniaxially anisotropic NRI materials will have the power to focus perfectly all E-mode or all H -mode or both E-mode and H -mode Fourier components of an image. We will also show that, in principle perfect focus cannot occur in the presence of absorption or dispersion, but excellent focus can still occur if the absorption and dispersion of a NRI material can be controlled to be rather small. The Letter is organized as follows: in Section 2, we will establish an accurate theoretical formulation for describing the imaging properties of the slabs of uniaxially anisotropic NRI materials. In Section 3, based on the accurate formulas established in Section 2, we will discuss whether and under what conditions the slabs of uniaxially anisotropic NRI materials will have the power to focus perfectly all Fourier components of an image and whether they can amplify the evanescent components if perfect focus occurs.
2. Theoretical formulation Consider a simple planar slab made of NRI materials embedded in an isotropic regular medium. The frequencydependent permittivity and permeability of the surrounding isotropic regular medium will be denoted as r (ω) and µr (ω). The thickness of the slab will be denoted as d. We choose the z-axis to be along the normal of the interfaces of the slab with the surrounding medium, and the two interfaces are located at z = 0 and z = d. The permittivity and permeability of the slab are assumed to be uniaxially anisotropic and its optical axis is along the z-axis. The permittivity and permeability of the slab will both be second-rank tensors and have the following forms [13]: ˆ (ω) = t (ω)lt + z (ω)lz ,
µ(ω) ˆ = µt (ω)lt + µz (ω)lz ,
(1)
where lt = ex ex + ey ey , lz = ez ez , and ex,y,z are unit vectors along the x, y and z axes. z (ω), µz (ω), and t (ω), µt (ω) are the frequency-dependent components of ˆ and µˆ in the directions parallel and perpendicular to the optical axis of the slab, respectively. (For the simplicity of notation, in the formulas given below the frequency dependence
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of the permittivity and permeability of the slab and the surrounding regular medium will not be denoted explicitly, remembering that they will be frequency-dependent in the presence of dispersion.) We assume that on the left-hand side of the slab there is a point source located at r0 = (x0 , y0 , z0 ), (z0 < 0). Physically the point source can be represented by electric and magnetic point dipoles with the following time-harmonic electric and magnetic source current density Jω (r , t) = J0 e−iωt δ(r − r0 ),
ω (r , t) = M 0 e−iωt δ(r − r0 ). M
(2)
The electromagnetic waves radiated by the point source can be obtained by solving the Maxwell’s equation, but it is unwieldy to solve directly the Maxwell’s equations for the geometry considered here. For a point dipole source described by Eq. (2), a more convenient approach is available. As was shown in Ref. [14], the electromagnetic fields radiated by a point dipole source can be derived exactly from two scalar Green’s functions, namely the E-mode Green’s function Gω and the H -mode Green’s function Gω . The E-mode Green’s function Gω describes the propagation of the E-mode components of the radiated waves of the point source, and the H -mode Green’s function Gω describes the propagation of the H -mode components. This method has been discussed in detail in Ref. [14]. According to Ref. [14], in the slab of the uniaxially anisotropic NRI material (i.e., in the region of 0 < z < d), Gω and Gω satisfy the following equations, respectively, µz ∂ 2 2 2 (3) + ∇ + k r , r0 ) = −δ(r − r0 ), 1 Gω ( t µt ∂z2 z ∂ 2 2 2 (4) + ∇t + k2 Gω (r , r0 ) = −δ(r − r0 ), t ∂z2 where k1 2 = ω2 t µz , k2 2 = ω2 µt z , ∇t is the invariant component of the gradient ∇ transverse to ez . In the surrounding isotropic regular medium (i.e., in the region of z < 0 and z > d), both Gω and Gω will satisfy the following equation 2 ∇ + kr2 G, (5) r , r0 ) = −δ(r − r0 ), kr2 = ω2 µr r . ω ( At the interfaces of the slab with the surrounding medium, the following boundary conditions must be satisfied: 1 ∂Gω are continuous at z = 0 and z = d, µt (z) ∂z 1 ∂Gω Gω and are continuous at z = 0 and z = d, t (z) ∂z Gω and
(6) (7)
where µt (z) = µr and t (z) = r in the region of z < 0 and z > d, and µt (z) = µt and t (z) = t in the region of 0 < z < d. If the scalar Green’s function Gω and Gω can be found, the electric and magnetic fields radiated by the point dipole sources can be obtained directly from Gω and Gω . For example, if the point dipoles are oriented 0 are parallel to the z-axis), one can show that the electric and magnetic fields can along the z-axis (i.e., J0 and M be obtained exactly from the scalar Green’s functions Gω and Gω by the following relations [14] J0 Q Gω (r , r0 ) + M0 ez × ∇t Gω (r , r0 ), iωt (z) M0 Hω (r ) = J0 ez × ∇t Gω (r , r0 ) − Qµ Gω (r , r0 ), iωµt (z)
E ω (r ) = −
(8) (9)
and Q µ are defined by where the operator Q = ∇t ∂ − ez t (z) ∇t2 , Q ∂z z (z)
µ = ∇t ∂ − ez µt (z) ∇t2 . Q ∂z µz (z)
(10)
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Since the electric and magnetic fields radiated by the point source can always be derived directly from the scalar Green’s functions Gω and Gω , in what follows we will only need to study the imaging properties of the scalar fields described by Gω and Gω . To find out the E-mode Green’s function Gω and the H -mode Green’s function Gω , we express them by plane wave spectrum, which is a well established technique in the electromagnetic radiation theory [14,15]: 1 G, (11) ( r , r ) = d 2 kt ei kt ·(r −r0 )t gω, (z, z0 ; kt ), 0 ω (2π)2 where the subscript ‘t’ denotes the transverse component of a vector, gω and gω are the one-dimensional E- and H -mode Green’s functions, respectively. From Eqs. (3)–(5), one can see that inside the slab of the NRI material, gω and gω will satisfy the following equations, respectively, µz d 2 2 2 + k − k (12) 1 t gω (z, z0 ; kt ) = −δ(z − z0 ), µt dz2 z d 2 2 2 (13) + k − k 1 t gω (z, z0 ; kt ) = −δ(z − z0 ), t dz2 and in the surrounding isotropic regular medium, gω and gω will both satisfy the following equation 2 d 2 2 , + k − k r t gω (z, z0 ; kt ) = −δ(z − z0 ). dz2 The solution for gω can be expressed by i ikz |z−z0 | + R(kt )e−ikz z , 2kz e gω = A(kt )eikz z + B(kt )e−ikz z , T (kt )eikz z ,
(14)
z < 0, (15)
0 < z < d, z > d,
where kz and kz are the z components of the wave vectors of the E-mode Fourier components, which are given by
kz = ω2 r µr − kt2 and kz ≡ ω2 t µt − µt kt2 /µz . In Eq. (15), the coefficients R, A, B, and T are functions of kt . These coefficients can be determined from the boundary condition shown in Eq. (6), and we get that
R(kt ) =
i(µ2t kz2 − µ2r kz 2 )(eikz |z0 |+ikz d − eikz |z0 |−ikz d )
2kz [(µr kz − µt kz )2 eikz d − (µr kz + µt kz )2 e−ikz d ]
,
(16)
A(kt ) = −
iµt (µr kz + µt kz )eikz |z0 |−ikz d
(µr kz − µt kz )2 eikz d − (µr kz + µt kz )2 e−ikz d
,
(17)
B(kt ) = T (kt ) =
iµt (µt kz − µr kz )eikz |z0 |+ikz d
,
(18)
.
(19)
(µr kz − µt kz )2 eikz d − (µr kz + µt kz )2 e−ikz d 2iµr µt kz eikz |z0 |−ikz d
(µr kz + µt kz )2 e−ikz d − (µr kz − µt kz )2 eikz d
Similarly, the solution for gω can be expressed by i ik |z−z | z 0 + R (k )e −ikz z , z < 0, t 2kz e z z ik −ik gω = A (kt )e z + B (kt )e z , 0 < z < d, z > d, T (kt )eikz z ,
(20)
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where kz is the z components of the wave vectors of the H -mode Fourier components and is given by kz =
ω2 t µt − t kt2 /z . The coefficients R , A , B , and T can be determined from the boundary condition shown in Eq. (7), and the results are
R (kt ) =
i(t2 kz2 − r2 kz 2 )(eikz |z0 |+ikz d − eikz |z0 |−ikz d )
2kz [(r kz − t kz )2 eikz d − (r kz + t kz )2 e−ikz d ]
,
(21)
A (kt ) =
−it (r kz + t kz )eikz |z0 |−ikz d
,
(22)
,
(23)
.
(24)
(r kz − t kz )2 eikz d − (r kz + t kz )2 e−ikz d
B (kt ) = T (kt ) =
it (t kz − r kz )eikz |z0 |+ikz d
(r kz − t kz )2 eikz d − (r kz + t kz )2 e−ikz d 2ir t kz eikz |z0 |−ikz d
(r kz + t kz )2 e−ikz d − (r kz − t kz )2 eikz d
The E- and H -mode Green’s functions Gω and Gω can be got by substituting Eqs. (15)–(19) and (20)–(24) into Eq. (11), respectively, and then the electromagnetic fields (both inside and outside the slab) can be obtained from Eqs. (8), (9).
3. Results and discussions The formulas established in Section 2 are exact and no approximation is involved. Based on these exact formulas, we can find out accurately whether and under what conditions the slabs of uniaxially anisotropic NRI materials can act like perfect lens. If a slab of a NRI material can act like perfect lens, then for a point source located on the left– right side of the slab, there should exist a focal point inside or on the right-hand side of the slab at which all Fourier components will have the same phase, i.e., inside or on the right-hand side of the slab all Fourier components of the electromagnetic waves should be perfectly focused at a certain foci. From Eqs. (15)–(24) and noticing that kz , kz , kz and the coefficients given in Eqs. (16)–(19) and (21)–(24) are all complex functions of kt (the transverse components of the wave vectors), one can see that in general cases no perfect focus can occur both inside and on the right-hand side of the slab. But from Eqs. (15)–(24), one can find that if some particular conditions are satisfied, a slab of a uniaxially anisotropic NRI material will have the power to focus perfectly all E-mode or all H -mode or both E- and H -mode Fourier components of the electromagnetic waves inside or on the right-hand side of the slab. By detailed analysis, the following results can be obtained from Eqs. (15)–(24): 1. If all components of the permittivity tensor ˆ and the permeability tensor µˆ of a uniaxially anisotropic NRI material are real and satisfy the following condition, a slab of such uniaxially anisotropic NRI material will have the power to focus perfectly all E-mode Fourier components of frequency ω at a certain foci inside or on the right-hand of the slab: µt (ω) < 0,
µz (ω) < 0,
t (ω)µz (ω) = r (ω)µr (ω),
t (ω) < 0 and µt (ω)µz (ω) = µr (ω)2 .
(25)
From Eqs. (16)–(19), one can see that if the above condition is satisfied, the coefficients given in Eqs. (16)–(19) will become R = 0,
A = 0,
B=
i ikz |z0 | e , 2kz
T=
i ikz |z0 |−ikz (1+|µt (ω)/µr (ω)|)d e , 2kz
(26)
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and from Eqs. (11) and (15), the E-mode waves of frequency ω (both inside and on the right-hand side of the slab) will be given by i d 2 kt k1z ei kt ·(r −r0 )t −ikz (z−z1,f (ω)) , 0 < z < d, 2 Gω (r , r0 ) = 8πi (27) d 2 kt 1 ei kt ·(r −r0 )t +ikz (z−z2,f (ω)) , z > d, 2 8π
kz
where z1,f (ω) ≡ |z0 µr (ω)/µt (ω)| for 0 < z < d (inside the slab) and z2,f (ω) ≡ d + d|µt (ω)/µr (ω)| − |z0 | for z > d (on the right-hand side of the slab). Eqs. (26), (27) show that: (1) All E-mode Fourier components of frequency ω will be perfectly transmitted through the slab, i.e., no reflection will occur at both surfaces of the slab for all E-mode Fourier components of frequency ω; (2) If z1,f (ω) < d, then inside the slab, all E-mode Fourier components of frequency ω will have the same phase at r1,f (ω) = (x0 , y0 , z1,f (ω)), i.e., they will be perfectly focused at the foci r1,f (ω) = (x0 , y0 , z1,f (ω)); (3) If z2,f (ω) > d, then on the right-hand side of the slab, all E-mode Fourier components of frequency ω will be perfectly focused at the foci r2,f (ω) = (x0 , y0 , z2,f (ω)); (4) If the condition (25) is satisfied but the condition z2,f (ω) > d or the condition z1,f (ω) < d is not satisfied, the foci on the right-hand side of the slab or the foci inside the slab will disappear. 2. If all components of the permittivity tensor ˆ and the permeability tensor µˆ of a uniaxially anisotropic NRI material are real and satisfy the following condition, a slab of such uniaxially anisotropic NRI material will have the power to focus perfectly all H -mode Fourier components of frequency ω at a certain foci inside or on the right-hand of the slab: t (ω) < 0,
z (ω) < 0,
z (ω)µt (ω) = r (ω)µr (ω),
µt (ω) < 0 and z (ω)t (ω) = r (ω)2 .
(28)
From Eqs. (21)–(24), one can see that if this condition is satisfied, the coefficients given in Eqs. (21)–(24) will become R = 0,
A = 1,
B =
i ikz |z0 | e , 2kz
T=
i ikz |z0 |−ikz (1+|t (ω)/r (ω)|)d e , 2kz
(29)
and from Eqs. (11) and (20), the H -mode waves of frequency ω (both inside and on the right-hand side of the slab) will be given by i d 2 kt k1z ei kt ·(r −r0 )t −ikz (z−z1,f (ω)) , 0 < z < d, 2 Gω (r , r0 ) = 8πi (30) d 2 kt 1 ei kt ·(r −r0 )t +ikz (z−z2,f (ω)) , z > d, 2 8π
kz
where z1,f (ω) ≡ |z0 r (ω)/t (ω)| for 0 < z < d and z2,f (ω) ≡ d + d|t (ω)/r (ω)| − |z0 | for z > d. Eqs. (29), (30) show that: (1) All H -mode Fourier components of frequency ω will be perfectly transmitted through the slab; (2) If z1,f (ω) < d, then inside the slab, all H -mode Fourier components of frequency ω will be perfectly focused at the foci r1,f (ω) = (x0 , y0 , z1,f (ω)); (2) If z2,f (ω) > d, then on the right-hand side of the slab, all H -mode Fourier components of frequency ω will be perfectly focused at the foci r2,f (ω) = (x0 , y0 , z2,f (ω)); (4) If the condition (28) is satisfied but the condition z2,f (ω) > d or the condition z1,f (ω) < d is not satisfied, the foci on the right-hand side of the slab or the foci inside the slab will disappear. 3. From Eqs. (25) and (28), one can see that unlike isotropic NRI materials, in the presence of uniaxial anisotropy, the conditions for the perfect focus of E- and H -mode Fourier components are usually very different. A slab of a uniaxially anisotropic NRI material can focus perfectly both E- and H -mode Fourier components of frequency ω at a certain foci inside or on the right-hand side of the slab only if the condition (25) and (28) are satisfied simultaneously, or equivalently, the following condition is satisfied: t (ω) < 0,
z (ω) < 0,
r (ω)/t (ω) = µr (ω)/µt (ω),
µt (ω) < 0,
µz (ω) < 0 and
t (ω)z (ω) = r (ω)2 .
(31)
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If this condition is satisfied, one can see that inside the slab both E- and H -mode Fourier components of frequency ω will be perfectly focused at the same foci z1,f (ω) = |z0 r (ω)/t (ω)| (providing that z1,f (ω) < d), and on the right-hand side of the slab both E- and H -mode Fourier components of frequency ω will be perfectly focused at the same foci z2,f (ω) = d + d|t (ω)/r (ω)| − |z0| (providing that z2,f (ω) > d). It should be noted that a slab of a NRI material can act like perfect lens only if the slab can focus perfectly all Fourier components of different frequencies at the same foci. From the results present above, one can see clearly that in the presence of dispersion (i.e., the components of the permittivity tensor ˆ and the permeability tensor µˆ of a NRI material are frequency dependent), a slab of a NRI material cannot act like perfect lens. The reason is that if the permittivity and permeability of a NRI material is frequency dependent, z1,f (ω) and z2,f (ω) will also be frequency dependent and hence the Fourier components of different frequencies will be focused at different places, so, in principle, a slab of a NRI material can act like perfect lens only if the NRI material is dispersiveless medium. But apparently, excellent focus can still occur if the dispersion of a NRI material is rather small. The reason is that if the dispersion of a NRI material is rather small, the permittivity and permeability of the NRI material will be nearly frequency independent and hence the frequency dependence of z1,f (ω) and z2,f (ω) can be neglected. In such cases, the Fourier components of different frequencies will be focused at approximately the same place. 4. Similarly, one can show that a slab of a NRI material can act like perfect lens only if the NRI material is lossless medium, i.e., no perfect focus can occur in the presence of absorption. To illustrate the influences of absorption, we assume that the permittivity and permeability of the slab have a small imaginary part: ˆ = ˆ0 (1 +iδ), µˆ = µˆ 0 (1 + iδ), where ˆ0 and µˆ 0 are the real parts of ˆ and µˆ and the imaginary parts are assumed to be much smaller than the real parts (i.e., δ 1). The real parts ˆ0 and µˆ 0 are assumed to satisfy the condition (31). Substituting ˆ = ˆ0 (1 + iδ) and µˆ = µˆ 0 (1 + iδ) into Eqs. (11)–(24), which are still valid in the presence of absorption, the fields both inside and outside the slab can be found. Let us taking the fields inside the slab, for example. In the case that δ 1 and the real parts ˆ0 and µˆ 0 satisfy the condition (31), the fields inside the slab will have the following form: , i ikt2 , 2 1 G (r , r0 ) = d kt 1 + 2 δ ei kt ·(r −r0 )t −ikz (z−z1,f ) 8π 2 kz 2kz
, k2 δ + (32) d 2 kt t3 ei kt ·(r −r0 )t +ikz (z+z1,f −2d| Re(µr )/ Re(µt )|) + O δ 2 , 2 16π kz in which z1,f = |z0 Re(µr )/ Re(µt )|. Compared with Eqs. (27) and (30), one can see that due to the presence of absorption, the perfect focus inside the slab is smeared, i.e., no foci exist inside the slab at which all Fourier components will have the same phase, so, in principle, a slab of a NRI material can act like perfect lens only if the NRI material is lossless medium. It should be noted that in actual NRI materials, for example, in the recently realized NRI materials reported in Ref. [4], both absorption and dispersion will present inevitably and also necessarily due to the requirement of the causality, so strictly speaking, no actual NRI materials can truly focus perfectly all Fourier components of an image [16]. But the above results also show that if the absorption and dispersion of a NRI material are rather small, excellent focus can still occur. In fact, if δ 1, the second term in Eq. (32) can be neglected compared with the first term, and a sharp image can still be focused at z = z1,f if δ is sufficiently small. Though the recently realized NRI materials are highly dispersive and absorptive [4], recent numerical simulations suggest that in principle it may be possible to control the absorptions and dispersions of NRI materials to be rather small [17]. Finally, we discuss whether the slabs of uniaxially anisotropic NRI materials will amplify evanescent components if perfect focus occur. Evanescent components are such Fourier components that kz (the z components of the wave vectors) are imaginary, i.e., kz = ±i|kz |, and the variation of the amplitude of evanescent components with distance is determined by the sign of kz . Previous theoretical investigations have shown that the slabs of isotropic NRI materials will amplify evanescent components if perfect focus occur [5]. Physically, to determine the sign of kz for evanescent components, one must take the analytical properties and the convergence of the wave
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functions into account. As has been discussed in Refs. [14,15], in order to keep the wave functions inside and on the right-hand side of the slab be convergent, for evanescent components a constraint need to be imposed on the choice of the sign of kz so that eikz z → 0 as kt → ∞. Considering this constraint plus the boundary conditions, one can get the following results: (1) For evanescent components inside the slab, if the condition (31) is satisfied and z1,f (ω) ≡ |r (ω)z0 /t (ω)| > d, the amplitudes of evanescent components will increase with distance in the slab, and no focus can occur inside the slab. If the condition (31) is satisfied and z1,f (ω) < d, then inside the slab perfect focus will occur in the image plane z = z1,f (ω), and the amplitudes of evanescent components will increase with distance before they are focused in the image plane z = z1,f (ω) and will decrease with distance in the region of z1,f (ω) < z < d; (2) For evanescent components on the right-hand side of the slab, if the condition (31) is satisfied and z2,f (ω) ≡ d + d|t (ω)/r (ω)| − |z0 | < d, the amplitudes of evanescent components will decrease with distance, and no focus can occur on the right-hand side of the slab. If the condition (31) is satisfied and z2,f (ω) > d, then on the right-hand side of the slab perfect focus will occur in the image plane z = z2,f (ω), and the amplitudes of evanescent components will increase with distance before they are focused in the image plane z = z2,f (ω) and will decrease with distance in the region from z = z2,f (ω) to infinity. In conclusion, we have presented a theoretical investigation on the imaging properties of uniaxially anisotropic negative refractive index materials. Based on an accurate theoretical formulation, we have found out accurately the conditions under which the slabs of uniaxially anisotropic negative refractive index materials will have the power to focus perfectly all E-mode or all H -mode or both E- and H -mode Fourier components of an image. We have show that in principle the slabs of NRI materials cannot focus perfectly all Fourier components of an image in the presence of absorption and dispersion, but excellent focus can still occur if the absorptions and dispersions of the NRI materials are rather small.
Acknowledgements L.B. Hu was supported by the Natural Science Foundation of Guangdong Province of China under Contract No. 011151.
References [1] See, for example, M.C.K. Wiltshire, Science 292 (2001) 77; J.B. Pendry, Phys. World 14 (2001) 47; R. Fitzgerald, Phys. Today 53 (2000) 17; P.F. Schewe, B. Stein, AIP Phys. Bull. Phys. News 476 (2000); A. Jones, NSF News, 21 March 2000. [2] V.G. Veselago, Sov. Phys. Usp. 10 (1968) 509. [3] J.B. Pendry, A.J. Holden, W.J. Stewart, I. Youngs, Phys. Rev. Lett. 76 (1996) 4773. [4] R.A. Shelby, D.R. Smith, S. Schultz, Science 292 (2001) 77; D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemet-Nasser, S. Schultz, Phys. Rev. Lett. 84 (2000) 4184; R.A. Shelby, D.R. Smith, S.C. Nemat-Nasser, S. Schultz, Appl. Phys. Lett. 78 (2001) 489; D.R. Smith, N. Kroll, Phys. Rev. Lett. 85 (2000) 2933. [5] J.B. Pendry, Phys. Rev. Lett. 85 (2000) 3966. [6] J.B. Pendry, S.A. Ramakrishna, J. Phys. C 14 (2002) 8463. [7] Z.M. Zhang, C.J. Fu, Appl. Phys. Lett. 80 (2002) 1097. [8] C. Luo, S.G. Johnson, J.D. Joannopoulos, J.B. Pendry, Phys. Rev. B 65 (2002) R201104. [9] J.T. Shen, P.M. Platzman, Appl. Phys. Lett. 80 (2002) 3286. [10] N. Fang, X. Zhang, Appl. Phys. Lett. 82 (2003) 161. [11] R.J. Blaikie, S.J. McNab, Microelectron. Eng. 61–62 (2002) 97. [12] R. Marques, F. Medina, R. Raffi-El-Idrissi, Phys. Rev. B 65 (2002) 144440. [13] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon, New York, 1960.
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L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall International, Englewood Cliffs, NJ, 1973. T.B. Hansen, A.D. Yaghjian, Plane-Wave Theory of Time-Domain Fields, IEEE Press, 1999. N. Garcia, M. Nieto-Vesperinas, Phys. Rev. Lett. 88 (2002) 207403. P. Markos, C.M. Soukoulis, Phys. Rev. E 65 (2002) 036622; P. Markos, I. Rousochatzakis, C.M. Soukoulis, Phys. Rev. E 66 (2002) 045601.
Imaging properties of uniaxially anisotropic negative refractive index materials Liangbin Hu a,∗ , Zhifang Lin b a Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China b Department of Physics, Fudan University, Shanghai 200433, PR China
Received 17 March 2003; received in revised form 29 April 2003; accepted 1 May 2003 Communicated by V.M. Agranovich
Abstract The imaging properties of uniaxially anisotropic negative refractive index materials have been investigated theoretically. Based on an accurate theoretical formulation, we have discussed in detail whether and under what conditions the slabs of uniaxially anisotropic negative refractive index materials will have the power to focus perfectly all Fourier components of an image. It was shown that if the effects of absorption and dispersion can be neglected, under some particular conditions the slabs of uniaxially anisotropic negative refractive index materials will have the power to focus perfectly all E-mode or all H -mode or both E- and H -mode Fourier components of an image. It was also shown that perfect focus cannot occur in the presence of absorption and dispersion, but if the effects of absorption and dispersion are rather small, excellent focus can still occur. 2003 Elsevier B.V. All rights reserved. PACS: 73.20.Mf; 78.20.Ci; 42.30.Wb; 78.66.Bz
1. Introduction Recently a new type of metamaterials that have negative refractive index have attracted much experimental and theoretical interest [1]. The concept of negative refractive index was first proposed by Veselago over thirty years ago, who predicted that electromagnetic wave propagations in a material having simultaneously negative permittivity and permeability µ should give rise to several peculiar characteristics, including anomalous refraction (that is, the refractive index is negative), reversal of both the Doppler shift and the Cherenkov radiation, etc., and he termed such type of materials as negative refractive index (NRI) materials or left-handed materials (LHMs) [2]. Due to the absence of naturally occurring NRI materials, Veselago’s prediction did not attract much attention until very recently, when some researchers reported that a new type of metamaterials have been prepared successfully following the suggestion of Pendry et al. [3] and were demonstrated to be NRI materials [4]. This success has aroused great interest in the unusual electrodynamic properties of NRI materials [1]. One of the most * Corresponding author.
E-mail address: [email protected] (L. Hu). 0375-9601/03/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)00761-8
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interesting peculiar properties of NRI materials is their unusual imaging properties [2,5]. It was first pointed out by Veselago that the simple planar slabs made of NRI materials will converge rather than diverge the radiation of a source, hence the slabs of NRI materials could be employed as an unconventional alternative to traditional lenses [2]. Recently Pendry further concluded that in some particular case the slabs of ideal lossless and dispersiveless NRI materials can amplify the evanescent waves, and due to this fact, the slabs of such NRI materials can act like perfect lens, which means that such a slab can focus perfectly all Fourier components of an image and hence the image will have maximum resolution [5]. Though in real NRI materials the effects of absorption and dispersion will decrease inevitably the resolution of an image, Pendry’s conclusion is still a rather interesting challenge to the traditional limitation on lens performance. Motivated by Pendry’s interesting conclusion, recently a number of theoretical works have been devoted to the investigation of the unusual imaging properties of NRI materials [6–11]. Basically in all previously published theoretical works, only isotropic NRI materials have been studied and the effects of anisotropy have been neglected completely. However, the recently realized NRI materials are actually anisotropic in nature, and up to now, no isotropic NRI materials have been prepared successfully in experiments [1,4]. In fact, from both the experimental and theoretical viewpoints, the task of preparing an isotropic NRI material is much more difficult than that of preparing an anisotropic NRI material [1]. As was well known before, the electrodynamic properties of anisotropic materials are usually very different from that of isotropic materials [12–14]. Due to the above reasons, in order to get a more clear understanding of the unusual imaging properties of NRI materials and to clarify whether Pendry’s proposal is applicable or not, the effects of anisotropy should be taken into account. The simplest and most common form of anisotropy is uniaxial anisotropy, and from the analysis of the symmetry of the recently realized NRI materials [4], they should also be uniaxially anisotropic. In this Letter we present a theoretical investigation on the unusual imaging properties of uniaxially anisotropic NRI materials. Based on an accurate theoretical formulation, we will study in detail whether the slabs of uniaxially anisotropic NRI materials can have the power to focus perfectly all Fourier components of an image. We will find out accurately the conditions under which the slabs of uniaxially anisotropic NRI materials will have the power to focus perfectly all E-mode or all H -mode or both E-mode and H -mode Fourier components of an image. We will also show that, in principle perfect focus cannot occur in the presence of absorption or dispersion, but excellent focus can still occur if the absorption and dispersion of a NRI material can be controlled to be rather small. The Letter is organized as follows: in Section 2, we will establish an accurate theoretical formulation for describing the imaging properties of the slabs of uniaxially anisotropic NRI materials. In Section 3, based on the accurate formulas established in Section 2, we will discuss whether and under what conditions the slabs of uniaxially anisotropic NRI materials will have the power to focus perfectly all Fourier components of an image and whether they can amplify the evanescent components if perfect focus occurs.
2. Theoretical formulation Consider a simple planar slab made of NRI materials embedded in an isotropic regular medium. The frequencydependent permittivity and permeability of the surrounding isotropic regular medium will be denoted as r (ω) and µr (ω). The thickness of the slab will be denoted as d. We choose the z-axis to be along the normal of the interfaces of the slab with the surrounding medium, and the two interfaces are located at z = 0 and z = d. The permittivity and permeability of the slab are assumed to be uniaxially anisotropic and its optical axis is along the z-axis. The permittivity and permeability of the slab will both be second-rank tensors and have the following forms [13]: ˆ (ω) = t (ω)lt + z (ω)lz ,
µ(ω) ˆ = µt (ω)lt + µz (ω)lz ,
(1)
where lt = ex ex + ey ey , lz = ez ez , and ex,y,z are unit vectors along the x, y and z axes. z (ω), µz (ω), and t (ω), µt (ω) are the frequency-dependent components of ˆ and µˆ in the directions parallel and perpendicular to the optical axis of the slab, respectively. (For the simplicity of notation, in the formulas given below the frequency dependence
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of the permittivity and permeability of the slab and the surrounding regular medium will not be denoted explicitly, remembering that they will be frequency-dependent in the presence of dispersion.) We assume that on the left-hand side of the slab there is a point source located at r0 = (x0 , y0 , z0 ), (z0 < 0). Physically the point source can be represented by electric and magnetic point dipoles with the following time-harmonic electric and magnetic source current density Jω (r , t) = J0 e−iωt δ(r − r0 ),
ω (r , t) = M 0 e−iωt δ(r − r0 ). M
(2)
The electromagnetic waves radiated by the point source can be obtained by solving the Maxwell’s equation, but it is unwieldy to solve directly the Maxwell’s equations for the geometry considered here. For a point dipole source described by Eq. (2), a more convenient approach is available. As was shown in Ref. [14], the electromagnetic fields radiated by a point dipole source can be derived exactly from two scalar Green’s functions, namely the E-mode Green’s function Gω and the H -mode Green’s function Gω . The E-mode Green’s function Gω describes the propagation of the E-mode components of the radiated waves of the point source, and the H -mode Green’s function Gω describes the propagation of the H -mode components. This method has been discussed in detail in Ref. [14]. According to Ref. [14], in the slab of the uniaxially anisotropic NRI material (i.e., in the region of 0 < z < d), Gω and Gω satisfy the following equations, respectively, µz ∂ 2 2 2 (3) + ∇ + k r , r0 ) = −δ(r − r0 ), 1 Gω ( t µt ∂z2 z ∂ 2 2 2 (4) + ∇t + k2 Gω (r , r0 ) = −δ(r − r0 ), t ∂z2 where k1 2 = ω2 t µz , k2 2 = ω2 µt z , ∇t is the invariant component of the gradient ∇ transverse to ez . In the surrounding isotropic regular medium (i.e., in the region of z < 0 and z > d), both Gω and Gω will satisfy the following equation 2 ∇ + kr2 G, (5) r , r0 ) = −δ(r − r0 ), kr2 = ω2 µr r . ω ( At the interfaces of the slab with the surrounding medium, the following boundary conditions must be satisfied: 1 ∂Gω are continuous at z = 0 and z = d, µt (z) ∂z 1 ∂Gω Gω and are continuous at z = 0 and z = d, t (z) ∂z Gω and
(6) (7)
where µt (z) = µr and t (z) = r in the region of z < 0 and z > d, and µt (z) = µt and t (z) = t in the region of 0 < z < d. If the scalar Green’s function Gω and Gω can be found, the electric and magnetic fields radiated by the point dipole sources can be obtained directly from Gω and Gω . For example, if the point dipoles are oriented 0 are parallel to the z-axis), one can show that the electric and magnetic fields can along the z-axis (i.e., J0 and M be obtained exactly from the scalar Green’s functions Gω and Gω by the following relations [14] J0 Q Gω (r , r0 ) + M0 ez × ∇t Gω (r , r0 ), iωt (z) M0 Hω (r ) = J0 ez × ∇t Gω (r , r0 ) − Qµ Gω (r , r0 ), iωµt (z)
E ω (r ) = −
(8) (9)
and Q µ are defined by where the operator Q = ∇t ∂ − ez t (z) ∇t2 , Q ∂z z (z)
µ = ∇t ∂ − ez µt (z) ∇t2 . Q ∂z µz (z)
(10)
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Since the electric and magnetic fields radiated by the point source can always be derived directly from the scalar Green’s functions Gω and Gω , in what follows we will only need to study the imaging properties of the scalar fields described by Gω and Gω . To find out the E-mode Green’s function Gω and the H -mode Green’s function Gω , we express them by plane wave spectrum, which is a well established technique in the electromagnetic radiation theory [14,15]: 1 G, (11) ( r , r ) = d 2 kt ei kt ·(r −r0 )t gω, (z, z0 ; kt ), 0 ω (2π)2 where the subscript ‘t’ denotes the transverse component of a vector, gω and gω are the one-dimensional E- and H -mode Green’s functions, respectively. From Eqs. (3)–(5), one can see that inside the slab of the NRI material, gω and gω will satisfy the following equations, respectively, µz d 2 2 2 + k − k (12) 1 t gω (z, z0 ; kt ) = −δ(z − z0 ), µt dz2 z d 2 2 2 (13) + k − k 1 t gω (z, z0 ; kt ) = −δ(z − z0 ), t dz2 and in the surrounding isotropic regular medium, gω and gω will both satisfy the following equation 2 d 2 2 , + k − k r t gω (z, z0 ; kt ) = −δ(z − z0 ). dz2 The solution for gω can be expressed by i ikz |z−z0 | + R(kt )e−ikz z , 2kz e gω = A(kt )eikz z + B(kt )e−ikz z , T (kt )eikz z ,
(14)
z < 0, (15)
0 < z < d, z > d,
where kz and kz are the z components of the wave vectors of the E-mode Fourier components, which are given by
kz = ω2 r µr − kt2 and kz ≡ ω2 t µt − µt kt2 /µz . In Eq. (15), the coefficients R, A, B, and T are functions of kt . These coefficients can be determined from the boundary condition shown in Eq. (6), and we get that
R(kt ) =
i(µ2t kz2 − µ2r kz 2 )(eikz |z0 |+ikz d − eikz |z0 |−ikz d )
2kz [(µr kz − µt kz )2 eikz d − (µr kz + µt kz )2 e−ikz d ]
,
(16)
A(kt ) = −
iµt (µr kz + µt kz )eikz |z0 |−ikz d
(µr kz − µt kz )2 eikz d − (µr kz + µt kz )2 e−ikz d
,
(17)
B(kt ) = T (kt ) =
iµt (µt kz − µr kz )eikz |z0 |+ikz d
,
(18)
.
(19)
(µr kz − µt kz )2 eikz d − (µr kz + µt kz )2 e−ikz d 2iµr µt kz eikz |z0 |−ikz d
(µr kz + µt kz )2 e−ikz d − (µr kz − µt kz )2 eikz d
Similarly, the solution for gω can be expressed by i ik |z−z | z 0 + R (k )e −ikz z , z < 0, t 2kz e z z ik −ik gω = A (kt )e z + B (kt )e z , 0 < z < d, z > d, T (kt )eikz z ,
(20)
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where kz is the z components of the wave vectors of the H -mode Fourier components and is given by kz =
ω2 t µt − t kt2 /z . The coefficients R , A , B , and T can be determined from the boundary condition shown in Eq. (7), and the results are
R (kt ) =
i(t2 kz2 − r2 kz 2 )(eikz |z0 |+ikz d − eikz |z0 |−ikz d )
2kz [(r kz − t kz )2 eikz d − (r kz + t kz )2 e−ikz d ]
,
(21)
A (kt ) =
−it (r kz + t kz )eikz |z0 |−ikz d
,
(22)
,
(23)
.
(24)
(r kz − t kz )2 eikz d − (r kz + t kz )2 e−ikz d
B (kt ) = T (kt ) =
it (t kz − r kz )eikz |z0 |+ikz d
(r kz − t kz )2 eikz d − (r kz + t kz )2 e−ikz d 2ir t kz eikz |z0 |−ikz d
(r kz + t kz )2 e−ikz d − (r kz − t kz )2 eikz d
The E- and H -mode Green’s functions Gω and Gω can be got by substituting Eqs. (15)–(19) and (20)–(24) into Eq. (11), respectively, and then the electromagnetic fields (both inside and outside the slab) can be obtained from Eqs. (8), (9).
3. Results and discussions The formulas established in Section 2 are exact and no approximation is involved. Based on these exact formulas, we can find out accurately whether and under what conditions the slabs of uniaxially anisotropic NRI materials can act like perfect lens. If a slab of a NRI material can act like perfect lens, then for a point source located on the left– right side of the slab, there should exist a focal point inside or on the right-hand side of the slab at which all Fourier components will have the same phase, i.e., inside or on the right-hand side of the slab all Fourier components of the electromagnetic waves should be perfectly focused at a certain foci. From Eqs. (15)–(24) and noticing that kz , kz , kz and the coefficients given in Eqs. (16)–(19) and (21)–(24) are all complex functions of kt (the transverse components of the wave vectors), one can see that in general cases no perfect focus can occur both inside and on the right-hand side of the slab. But from Eqs. (15)–(24), one can find that if some particular conditions are satisfied, a slab of a uniaxially anisotropic NRI material will have the power to focus perfectly all E-mode or all H -mode or both E- and H -mode Fourier components of the electromagnetic waves inside or on the right-hand side of the slab. By detailed analysis, the following results can be obtained from Eqs. (15)–(24): 1. If all components of the permittivity tensor ˆ and the permeability tensor µˆ of a uniaxially anisotropic NRI material are real and satisfy the following condition, a slab of such uniaxially anisotropic NRI material will have the power to focus perfectly all E-mode Fourier components of frequency ω at a certain foci inside or on the right-hand of the slab: µt (ω) < 0,
µz (ω) < 0,
t (ω)µz (ω) = r (ω)µr (ω),
t (ω) < 0 and µt (ω)µz (ω) = µr (ω)2 .
(25)
From Eqs. (16)–(19), one can see that if the above condition is satisfied, the coefficients given in Eqs. (16)–(19) will become R = 0,
A = 0,
B=
i ikz |z0 | e , 2kz
T=
i ikz |z0 |−ikz (1+|µt (ω)/µr (ω)|)d e , 2kz
(26)
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and from Eqs. (11) and (15), the E-mode waves of frequency ω (both inside and on the right-hand side of the slab) will be given by i d 2 kt k1z ei kt ·(r −r0 )t −ikz (z−z1,f (ω)) , 0 < z < d, 2 Gω (r , r0 ) = 8πi (27) d 2 kt 1 ei kt ·(r −r0 )t +ikz (z−z2,f (ω)) , z > d, 2 8π
kz
where z1,f (ω) ≡ |z0 µr (ω)/µt (ω)| for 0 < z < d (inside the slab) and z2,f (ω) ≡ d + d|µt (ω)/µr (ω)| − |z0 | for z > d (on the right-hand side of the slab). Eqs. (26), (27) show that: (1) All E-mode Fourier components of frequency ω will be perfectly transmitted through the slab, i.e., no reflection will occur at both surfaces of the slab for all E-mode Fourier components of frequency ω; (2) If z1,f (ω) < d, then inside the slab, all E-mode Fourier components of frequency ω will have the same phase at r1,f (ω) = (x0 , y0 , z1,f (ω)), i.e., they will be perfectly focused at the foci r1,f (ω) = (x0 , y0 , z1,f (ω)); (3) If z2,f (ω) > d, then on the right-hand side of the slab, all E-mode Fourier components of frequency ω will be perfectly focused at the foci r2,f (ω) = (x0 , y0 , z2,f (ω)); (4) If the condition (25) is satisfied but the condition z2,f (ω) > d or the condition z1,f (ω) < d is not satisfied, the foci on the right-hand side of the slab or the foci inside the slab will disappear. 2. If all components of the permittivity tensor ˆ and the permeability tensor µˆ of a uniaxially anisotropic NRI material are real and satisfy the following condition, a slab of such uniaxially anisotropic NRI material will have the power to focus perfectly all H -mode Fourier components of frequency ω at a certain foci inside or on the right-hand of the slab: t (ω) < 0,
z (ω) < 0,
z (ω)µt (ω) = r (ω)µr (ω),
µt (ω) < 0 and z (ω)t (ω) = r (ω)2 .
(28)
From Eqs. (21)–(24), one can see that if this condition is satisfied, the coefficients given in Eqs. (21)–(24) will become R = 0,
A = 1,
B =
i ikz |z0 | e , 2kz
T=
i ikz |z0 |−ikz (1+|t (ω)/r (ω)|)d e , 2kz
(29)
and from Eqs. (11) and (20), the H -mode waves of frequency ω (both inside and on the right-hand side of the slab) will be given by i d 2 kt k1z ei kt ·(r −r0 )t −ikz (z−z1,f (ω)) , 0 < z < d, 2 Gω (r , r0 ) = 8πi (30) d 2 kt 1 ei kt ·(r −r0 )t +ikz (z−z2,f (ω)) , z > d, 2 8π
kz
where z1,f (ω) ≡ |z0 r (ω)/t (ω)| for 0 < z < d and z2,f (ω) ≡ d + d|t (ω)/r (ω)| − |z0 | for z > d. Eqs. (29), (30) show that: (1) All H -mode Fourier components of frequency ω will be perfectly transmitted through the slab; (2) If z1,f (ω) < d, then inside the slab, all H -mode Fourier components of frequency ω will be perfectly focused at the foci r1,f (ω) = (x0 , y0 , z1,f (ω)); (2) If z2,f (ω) > d, then on the right-hand side of the slab, all H -mode Fourier components of frequency ω will be perfectly focused at the foci r2,f (ω) = (x0 , y0 , z2,f (ω)); (4) If the condition (28) is satisfied but the condition z2,f (ω) > d or the condition z1,f (ω) < d is not satisfied, the foci on the right-hand side of the slab or the foci inside the slab will disappear. 3. From Eqs. (25) and (28), one can see that unlike isotropic NRI materials, in the presence of uniaxial anisotropy, the conditions for the perfect focus of E- and H -mode Fourier components are usually very different. A slab of a uniaxially anisotropic NRI material can focus perfectly both E- and H -mode Fourier components of frequency ω at a certain foci inside or on the right-hand side of the slab only if the condition (25) and (28) are satisfied simultaneously, or equivalently, the following condition is satisfied: t (ω) < 0,
z (ω) < 0,
r (ω)/t (ω) = µr (ω)/µt (ω),
µt (ω) < 0,
µz (ω) < 0 and
t (ω)z (ω) = r (ω)2 .
(31)
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If this condition is satisfied, one can see that inside the slab both E- and H -mode Fourier components of frequency ω will be perfectly focused at the same foci z1,f (ω) = |z0 r (ω)/t (ω)| (providing that z1,f (ω) < d), and on the right-hand side of the slab both E- and H -mode Fourier components of frequency ω will be perfectly focused at the same foci z2,f (ω) = d + d|t (ω)/r (ω)| − |z0| (providing that z2,f (ω) > d). It should be noted that a slab of a NRI material can act like perfect lens only if the slab can focus perfectly all Fourier components of different frequencies at the same foci. From the results present above, one can see clearly that in the presence of dispersion (i.e., the components of the permittivity tensor ˆ and the permeability tensor µˆ of a NRI material are frequency dependent), a slab of a NRI material cannot act like perfect lens. The reason is that if the permittivity and permeability of a NRI material is frequency dependent, z1,f (ω) and z2,f (ω) will also be frequency dependent and hence the Fourier components of different frequencies will be focused at different places, so, in principle, a slab of a NRI material can act like perfect lens only if the NRI material is dispersiveless medium. But apparently, excellent focus can still occur if the dispersion of a NRI material is rather small. The reason is that if the dispersion of a NRI material is rather small, the permittivity and permeability of the NRI material will be nearly frequency independent and hence the frequency dependence of z1,f (ω) and z2,f (ω) can be neglected. In such cases, the Fourier components of different frequencies will be focused at approximately the same place. 4. Similarly, one can show that a slab of a NRI material can act like perfect lens only if the NRI material is lossless medium, i.e., no perfect focus can occur in the presence of absorption. To illustrate the influences of absorption, we assume that the permittivity and permeability of the slab have a small imaginary part: ˆ = ˆ0 (1 +iδ), µˆ = µˆ 0 (1 + iδ), where ˆ0 and µˆ 0 are the real parts of ˆ and µˆ and the imaginary parts are assumed to be much smaller than the real parts (i.e., δ 1). The real parts ˆ0 and µˆ 0 are assumed to satisfy the condition (31). Substituting ˆ = ˆ0 (1 + iδ) and µˆ = µˆ 0 (1 + iδ) into Eqs. (11)–(24), which are still valid in the presence of absorption, the fields both inside and outside the slab can be found. Let us taking the fields inside the slab, for example. In the case that δ 1 and the real parts ˆ0 and µˆ 0 satisfy the condition (31), the fields inside the slab will have the following form: , i ikt2 , 2 1 G (r , r0 ) = d kt 1 + 2 δ ei kt ·(r −r0 )t −ikz (z−z1,f ) 8π 2 kz 2kz
, k2 δ + (32) d 2 kt t3 ei kt ·(r −r0 )t +ikz (z+z1,f −2d| Re(µr )/ Re(µt )|) + O δ 2 , 2 16π kz in which z1,f = |z0 Re(µr )/ Re(µt )|. Compared with Eqs. (27) and (30), one can see that due to the presence of absorption, the perfect focus inside the slab is smeared, i.e., no foci exist inside the slab at which all Fourier components will have the same phase, so, in principle, a slab of a NRI material can act like perfect lens only if the NRI material is lossless medium. It should be noted that in actual NRI materials, for example, in the recently realized NRI materials reported in Ref. [4], both absorption and dispersion will present inevitably and also necessarily due to the requirement of the causality, so strictly speaking, no actual NRI materials can truly focus perfectly all Fourier components of an image [16]. But the above results also show that if the absorption and dispersion of a NRI material are rather small, excellent focus can still occur. In fact, if δ 1, the second term in Eq. (32) can be neglected compared with the first term, and a sharp image can still be focused at z = z1,f if δ is sufficiently small. Though the recently realized NRI materials are highly dispersive and absorptive [4], recent numerical simulations suggest that in principle it may be possible to control the absorptions and dispersions of NRI materials to be rather small [17]. Finally, we discuss whether the slabs of uniaxially anisotropic NRI materials will amplify evanescent components if perfect focus occur. Evanescent components are such Fourier components that kz (the z components of the wave vectors) are imaginary, i.e., kz = ±i|kz |, and the variation of the amplitude of evanescent components with distance is determined by the sign of kz . Previous theoretical investigations have shown that the slabs of isotropic NRI materials will amplify evanescent components if perfect focus occur [5]. Physically, to determine the sign of kz for evanescent components, one must take the analytical properties and the convergence of the wave
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functions into account. As has been discussed in Refs. [14,15], in order to keep the wave functions inside and on the right-hand side of the slab be convergent, for evanescent components a constraint need to be imposed on the choice of the sign of kz so that eikz z → 0 as kt → ∞. Considering this constraint plus the boundary conditions, one can get the following results: (1) For evanescent components inside the slab, if the condition (31) is satisfied and z1,f (ω) ≡ |r (ω)z0 /t (ω)| > d, the amplitudes of evanescent components will increase with distance in the slab, and no focus can occur inside the slab. If the condition (31) is satisfied and z1,f (ω) < d, then inside the slab perfect focus will occur in the image plane z = z1,f (ω), and the amplitudes of evanescent components will increase with distance before they are focused in the image plane z = z1,f (ω) and will decrease with distance in the region of z1,f (ω) < z < d; (2) For evanescent components on the right-hand side of the slab, if the condition (31) is satisfied and z2,f (ω) ≡ d + d|t (ω)/r (ω)| − |z0 | < d, the amplitudes of evanescent components will decrease with distance, and no focus can occur on the right-hand side of the slab. If the condition (31) is satisfied and z2,f (ω) > d, then on the right-hand side of the slab perfect focus will occur in the image plane z = z2,f (ω), and the amplitudes of evanescent components will increase with distance before they are focused in the image plane z = z2,f (ω) and will decrease with distance in the region from z = z2,f (ω) to infinity. In conclusion, we have presented a theoretical investigation on the imaging properties of uniaxially anisotropic negative refractive index materials. Based on an accurate theoretical formulation, we have found out accurately the conditions under which the slabs of uniaxially anisotropic negative refractive index materials will have the power to focus perfectly all E-mode or all H -mode or both E- and H -mode Fourier components of an image. We have show that in principle the slabs of NRI materials cannot focus perfectly all Fourier components of an image in the presence of absorption and dispersion, but excellent focus can still occur if the absorptions and dispersions of the NRI materials are rather small.
Acknowledgements L.B. Hu was supported by the Natural Science Foundation of Guangdong Province of China under Contract No. 011151.
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