The Group Velocity Picture of Metamaterial Systems

and plays the key role in the metamaterial abnormal properties. For these strange beams, such as negative refraction beams, with dispersion we can obtain the group velocity (energy velocity) which determine the beams propagating direction. So the group velocity should be the basic picture for us to understand these strange beams and help us to design related devices. More In-depth analysis of the theory, properties and description of the most potential technological applications of metamaterials for the realization of novel devices such as subwavelength lenses, invisibility cloaks, dipole and reflector antennas, high frequency telecommunications, new designs of bandpass filters, absorbers and concentrators of EM waves etc. In order to create a new devices it is necessary to know the main electrodynamical characteristics of metamaterial structures on the basis of which the device is supposed to be created. The electromagnetic wave scattering surfaces built with metamaterials are primarily based on the ability of metamaterials to control the surrounded electromagnetic fields by varying their permeability and permittivity characteristics. The book covers some solutions for microwave wavelength scales as well as exploitation of nanoscale EM wavelength such as visible specter using recent advances of nanotechnology, for instance in the field of nanowires, nanopolymers, carbon nanotubes and graphene. Metamaterial is suitable for scholars from extremely large scientific domain and therefore given to engineers, scientists, graduates and other interested professionals from photonics to nanoscience and from material science to antenna engineering as a comprehensive reference on this artificial materials of tomorrow. Group Picture of

seriously, the dispersion is related with some very basic limitations of this world, e.g.t h e causality limitation that the group velocity in metamaterial should be less than the vacuum light speed. If our design of devices is based on the metamaterials which violate these basic limitations, the design surely can not work since such metamaterial could not exist in this world. From this view, the group velocity picture is not only needed in understanding and explanation, but also required in research of some topics and design of devices. For instance, in the research of the limitation of the cloak [7,8] and abnormal phenomena on the interface of the hyperbolic metamaterial [9], if we neglect the dispersion of material, from group velocity picture we will immediately find that we have fallen into a superluminal trap, since the energy velocity in such artificial systems is divergent. So, the group velocity picture can help us avoid such traps.
As a basic value for revealing the complex propagating process, abnormal group velocity has been studied for decades. In the early 1960's the group velocity in material has been studied by Brillouin [10] and the group velocity in strongly scattering media is investigated by J. H. Page, Ping Sheng et al. in 1996[11], which indicate that the physical origin of the remarkably low velocities of propagation lies in the renormalization of the effective medium by strong resonant scattering. So far, metamaterials generally are composite of "photonic atoms" which can scatter light coherently. And all abnormal properties of metamaterials, e.g. these strange beams, are from these complex coherent scattering. Another byproduct of these scattering is the (abnormal) group velocity. In other words, the strange beam and the abnormal group velocity are two sides of a same coin. Further more, with some abnormal group velocity, such as the extremely slow light, we can design new signal-processing devices or new detecting devices. Hence, exploring the group velocity in metamaterial is very vital for revealing mechanism and the design of the real optical devices.
The numerical simulation takes an important role in research for modern photonics. For metamaterial, since the difficulties of experimental realization, the numerical tools become very essential for researchers. But, in some frequency domain simulation softwares, the dispersion is neglected totally. As we discussed above, we think such softwares can misleading researchers to some imaginary metamaterial which can not exist in this world. Such as for cloaking study, these softwares could present perfect invisibility very easily, but from our study [7,8], that is misleading one. We strongly recommend the time-domain softwares, such as finite-difference time-domain (FDTD) method or finite-element time-domain(FETD) method.
Their simulating results are much more convincible since they are generally with physical dispersion in the simulation and fit for metamaterial studies. This paper is organized as following. The first section is the introduction in which we generally introduce the group velocity picture of the metamaterial study. As we have discussed, the group velocity is the key for understanding these abnormal properties of metamaterials and also can help us to avoid some traps of basic physical limit. We have also commended the softwares fit for metamaterial studies.
In the second section, the optical properties of the interface between hyperbolic meta-material (with anisotropic hyperbolic dispersion) and common dielectric is investigated. With material dispersion, a comprehensive theory is constructed, and the hyperlens effect that the evanescent wave can be converted into the radiative wave is confirmed. At the inverse process of hyperlens, we find a novel mechanism to compress and stop (slow) light at wide frequency range, which can be used as a removable memory or a light trap. All theoretical results are demonstrated by finite-difference time-domain simulation.
In the third section we propose general evanescent-mode-sensing methods to probe the quantum electrodynamics (QED) vacuum polarization. The methods are based on the phase change and the energy time delay of evanescent wave caused by small dissipation. From our methods, high sensitivity can be achieved even though the external field, realizable in contemporary experiments, is much smaller than the Schwinger critical field.
In the forth section the image field of the negative-index superlens with the quasi-monochromatic random source is discussed, and dramatic temporal-coherence gain of the image in the numerical simulation is observed, even if there is almost no reflection and no frequency filtering effects. From the new physical picture, a theory is constructed to obtain the image field and demonstrate that the temporal coherence gain is from different "group" retarded time of different optical paths. Our theory agrees excellently with the numerical simulation and strict Green's function method. These study should have important consequences in the coherence studies in the related systems and the design of novel devices.
In the fifth section, the dynamical processes of dispersive cloak by finite-difference time-domain numerical simulation are carried out. It is found that there is a strong scattering process before achieving the stable state and its time length can be tuned by the dispersive strength. Poynting-vector directions show that the stable cloaking state is constructed locally while an intensity front sweeps through the cloak. Deeper studies demonstrate that the group velocity tangent component V gθ is the dominant factor in the process. This study is helpful not only for clear physical pictures but also for designing better cloaks to defend passive radars.
In the sixth section, the limitation of the electromagnetic cloak with dispersive material is investigated based on causality. The results show that perfect invisibility can not be achieved because of the dilemma that either the group velocity V g diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method is performed to validate the analysis.
In the last section, we give a summary of our works. asymptote of HM dispersion, abnormal omnidirectional transmission occurs [17]. Although some theoretical and experimental works [16] have demonstrated that the EW really can be converted into RW by HI of the layered cylindrical HM, a full theory involving the "material dispersion"(will be explained later) has not been given so far. For meta-material systems, if without physical dispersion, some abnormal optical properties can not be clearly explained and the dynamical study of wave propagation can not be carried out [20]. Even more seriously, the causality violation because of the superluminal group velocity (v g > c)inHMispointed out [18], which makes the observed hyper-lens effect doubtful. To solve these problems and predict new phenomena, more robust theory with stronger base is needed.
On the other hand, to compress and to stop (slow) light pulses are very essential for modern optical/photonics research and signal processing. Hence, a new mechanism, which can compress and stop (slow) light pulses and is frequency and direction insensitive, would induce wide interest in related directions.
In this Letter, we theoretically and numerically investigate the novel optical properties of flat HI [22], in which, unlike the cylindrical HI, the translational symmetry guarantees the simple physical picture for intuitive understanding, the quantitative study of the conversion between EW and RW etc.
A general theory of HI is constructed with physical dispersion of HM. On the HI, not only the conversion from EW to RW (CER) of hyperlens is confirmed, when RW is incident from HM to dielectric(the inverse process of hyper-lens), but also the almost total conversion from RW to EW (CRE) can occur, i.e. there is "no-transmission and no-refelection" (NTNR). More important we find that this is a new mechanism to compress and stop (slow) light pulses in wide frequency and direction range with many potential applications. Theoretically and numerically we demonstrate that the superluminal group velocity in hyperlens is artificial since the HM material dispersion is neglected in previous study [18]. At last, the feasibility to realize these functions on real structures is discussed. All theoretical results are demonstrated by finite-difference-time-domain (FDTD) simulations.
Our model is as follows. Assuming two plane waves are incident to HI from HM and isotropic dielectric, and scattered from HI, as shown in the upper-right insert of Fig.(1). The HI is in the x-z plane, while the incident surface and both HM optical axes lie in the x-y plane. The incident waves are chosen as TM wave with field components (E x , E y , H z ) and same "parallel wave-vector" k x . The HM is with the permittivity tensor aŝ in its principle axes coordinate, where ǫ 1 < 0andǫ 2 > 0 are assumed. And the permittivity of isotropic dielectric material is ǫ. The essential point of our model is that the negative diagonal component is dispersive ǫ 1 = ǫ 1 (ω), which is called material dispersion in our study. It is well known that dispersion is physically required for real meta-materials with abnormal effective constitutive coefficients, such as negative permittivity. We will see that the material dispersion will help us to obtain self-consistent explanation of abnormal optical properties of HI and to avoid causality violation.
For simplicity, the HM is nonmagnetic and Gaussian unit is employed throughout the paper. We define the angle between the HI (or x axis) and the positive-ǫ principle axis of HM is θ, In the HM region and the isotropic dielectric region the fields can be expressed uniformly as where the coefficients are defined as c σ νξ = E σ νξ /H σ νz with ν = i, s for the incident fields to HI or the scattered fields from HI; ξ = x, y; σ = h, d for the HM region or dielectric region. Since the translation symmetry of flat HI, the wave-vector parallel component k x is continuous at both regions. In following discussion, the k vectors are normalized byk σ νξ = k σ νξ /k 0 where k 0 = ω/c.
To explore the transmission and reflection properties of HI, we define a scattering matrix associated with the incident fields and out-going fields at HI.
where the superscript T means the matrix transpose, r ef a c t o r sα, γ are defined as α =( ǫ 1 sin 2 θ − ǫ 2 cos 2 θ) 1 2 ; γ =( |ǫ 1 | + ǫ 2 ) sin 2θ/2α, and the values ofk d i(s)y = ± ǫ −k 2 x andk h i(s)y andk h i(s)y =(±(|ǫ 1 |ǫ 2 (k 2 x + α 2 )) 1/2 +k x αγ)/α 2 are uniquely determined by Eq.(2), respectively. From Eq.(3) we can easily get the reflection and transmission coefficients across the HI from upper to down, or inverse. For the case of hyperlens that the wave is incident from the isotropic medium to the HM, t dh = S 22 ; r dh = S 12 .W h e nk 2 x > ǫ, the incident and the reflected waves in the isotropic dielectric are EWs with y-component wave-vectors We note that, although single EW can not carry net energy current(time averaged), two EWs, i.e. the incident and reflected EWs, can carry net energy current S iy in isotropic dielectric medium, since the reflected EW gains an extra-phase from complex reflecting coefficient r dh . The energy current S iy carried by two EWs can be converted by HI into the RW energy current S ty in the HM: From Eq(4), the hyper-lens effect and the image-improving by CER could be quantitatively studied.
After confirming CER on HI, it is natural to wonder if CRE can occur too, or if there are other novel phenomena on HI. Next we will study the inverse process of hyper-lens, i.e. the RW is incident from HM and the transmitted field is in the dielectric. For such inverse processes, there is a critical condition θ = θ c ≡ arctan ǫ 2 /|ǫ 1 |,whichmeansHI(x axis) perpendicular to one of hyperbola-dispersion asymptotes, or in other words, the asymptote is parallel with y axis now, as shown by the the solid lines in Fig.(1). At this critical condition, especially when the transmitted wave is EW, we will find CRE with NTNR, compressing and stopping light pulses, etc.
Before we get into detailed derivation, for the critical case (θ = θ c )w efi r s tp r e s e n tt w o seemingly conflicting conclusions of reflected wave from two different arguments, which will clearly show the most tricky point of HI.
The first argument is from the "intuitive way" which is based on Fig.(1). Since there is no reflection wave-vector on the dispersion curve to satisfy the k x continuity, we intuitively expect that there should be no reflected wave with omnidirectional incidence. If the incident angle is large enoughk 2 x > ǫ so that the transmitted field is EW, and since a single EW can not carry energy current, NTNR is the only possible choice and we expect that CRE will occur on HI. But from the second argument based on Eq.(4), we will obtain a different result. Since θ = θ c is a critical case, we should be more careful and discuss in a more subtle and strict way.
We first suppose the θ = θ c as shown by dashed lines Fig.(1), so the finitek ry of reflected field for a fixedk x can be found. Next we let the angle θ to approach θ c continuously (which can be realized physically by choosing different direction of HI), then we find thatk ry → ∞ when θ → θ c for a fixedk x .
To explain the conflicting results, we need to calculate the group velocity inside HM with material dispersion, which will also show that the superluminal group velocity is artificial. The most general expression of the group velocity of the reflected wave (which is also the group velocity of transmitted field in HM of hyperlens case) can be obtained from Eq.(2) as: wherek p x =( k x cos θ −k h y sin θ) andk p y =( k h y cos θ +k x sin θ) are the "k components" in the principal-axes coordinate of HM and β =( ǫ 2 sin 2 θ − ǫ 1 cos 2 θ) 1 2 . From Eq.(5), we find that, if the material dispersion of HM is neglected ∂ǫ 1 /∂ω = ∂ǫ 2 /∂ω = 0, then we will obtain the superluminal group velocity as shown in Fig.2(a). When θ → θ c ,thev g even diverges.
But with material dispersion, the x and y components of v g is recalculated, and we find that there is no v g > c at all cases, as shown in Fig.2(b) in which two components of v g versus θ − θ c based on Eq.(5), with the parameters ǫ ′ 2 = 0, [21] and ω = ω p / √ 2, θ c = π/4. When approaching the critical angle θ → θ c , two components can be approximated as: .S i n c ek h ry → ∞ at the critical angle θ c , the group velocity of the reflected wave should be zero v g = 0 at the critical angle, as shown in Fig.2 What does the zero-group velocity of reflected wave mean? The analysis will give us clear answer. As we have pointed out,since the reflected energy current S r is not zero and S r = v g W where W is the energy density of reflected wave, hence the energy density W must be infinite large at the critical angle. From Eq.(2), we can obtain that the electric field of reflected wave |E h r | is really divergent at the critical angle. The divergent field strength means that it need infinite long time to accumulate energy at HI for the reflected field. In other words, there is no reflected wave physically, as our intuition has told us. When the incident angle is large enough   x > ǫ, since the energy of incident RW can not be transmitted, also can not be reflected, the only answer is that the energy is stored at the HI or CRE occurs. Thus, we can have a self-consistent explanation for our seemingly conflicting results.
To confirm our theoretical discussion at θ c , the FDTD simulation [79] with strict physical HM dispersion (Drude mode) which satisfies Kramar-Kronig relation, is done. The parameter of HM and dielectric are ǫ 1 = −3, ǫ 2 = 3,andǫ l = 1.1. For the case(k x > ǫ), as shown Fig.3(a), a light beam is incident from HM to HI in 45 0 angle,as we predicted, there is no reflection and no transmission, and the field energy is accumulated at HI and stopped there. More detailed observation shows that at the boundary the field energy is mainly at the dielectric side, as shown Fig.3

(b).
We also has checked the group velocities of hyperlens cases and and find no violation of the causality. Actually, in FDTD simulation, if there is superluminal group velocity the program will be numerically unstable.
The dynamical study, such as with the pulse incidence, can reveal more interesting phenomena of HI. Since the group velocity along HI is zero at NTNR case as discussed, we expect that the pulse energy will accumulate on HI and stay at the incident position until it is dissipated because of absorption of HM.
The numerical experiments with incident pulses by FDTD are also done. As shown in Fig.3(c) and (d), two pulses arrive at the HI at different time, then they stop at the incident positions on HI. The pulse vertical length is compressed to almost zero, but their width keeps same so that they are still well separated in Fig.3(d). We emphasize at here that this is a novel mechanism to compress and stop (slow) light pulses with special advantages. The first advantage is that this mechanism works at very wide frequency and wide incident-angle range, which is confirmed by FDTD simulation in Fig.3 with incident of pretty short pulses. The frequency and incident-angle insensitivity is because the mechanism is from a simple geometry property, i.e. the HI (x axis) perpendicular to one of hyperbola-dispersion asymptotes. The second is that the decay (because of dissipation) of trapped field on HI is much slower than in common metallic material since the trapped field energy is mainly in the dielectric side as shown in Fig.3(b). The third is that the trapped signals are easy to take out (read) since they are on the interface. Because of these advantages, HI could be used as a removable recorder (dynamical memory) in optical/photonic signal processing, or as a wide-frequency wide-angle light trapper in photovoltaic devices.
However, we should point out that the above theoretical and numerical studies are with the assumption of the ideal hyper-dispersion, which is still void when k ry → ∞. In reality, such HM does't exist, so that we need to study the limit of hyper-dispersion of realizable HM. HM can be realized by many structures, i.e. one-dimensional (1D) periodic metal-dielectric binary layers [24,25] or two-dimensional (2D) periodic metallic lines [26], as shown in Fig.4(a). For these structures, the dispersion relation can be calculated exactly. In Fig.4, the calculated frequency contour of a 1D metal-dielectric binary layers is shown, from which we can see that the effective HM medium is not available anymore when |k| approaches π/a. Based on this limit, we can roughly estimate the slow limit of group velocity by v gx ∼ 1 w h e r eγ s = k ry /k 0 is the slowing coefficient. For the 2D metallic-line structure, from the modern technical limit we assume the smallest lattice constant as a = 10nm. If the incident is the micro-wave ω = 5.8GHz (γ s ∼ 10 7 )a n dǫ ′ 1 (ω)=6.9 × 10 −10 s as in Ref [27], we obtain v gx ∼ 4.6m/s which means considerably slow light although not totally stopped, and v gy ∼ 7.07 × 10 −8 m/s which means that the strongly-compressed light pulses can be easily achieved.
In conclusion, we have theoretically and numerically investigate the optical properties of HI. The theory with dispersion of meta-material is constructed and the hyperlens effect of CER is confirmed. At the inverse process of hyperlens, the abnormal phenomena of CRE with NTNR and a novel mechanism to compress and stop light in wide frequency range are revealed. Based the calculated group velocity, we demonstrate that the previously-pointed-out superluminal group velocity in HM is artificial since the material dispersion is neglected. FDTD simulations confirm that the HI has potential to be a removable optical/photonic The Group Velocity Picture of Metamaterial Systems www.intechopen.com recorder, or a wide-frequency wide-angle light trapper. At last the realizability of these phenomena on the real metallic structures is discussed. Obviously, the new mechanism works not only for electromagnetic waves, but also for acoustic or matter waves if hyperbolic dispersion is available, so that more interesting phenomena and applications are waiting for further theoretical and experimental research.

The methods to detect vacuum polarization by evanescent modes
Vacuum is one of the most fundamental concepts in all quantum fields [30][31][32] since all excitations are from the vacuum and determined by vacuum in some way. Modern vacuum concept is started from quantum electrodynamics (QED), which describes the interaction between light and matter (including vacuum), and has been continually studied both experimentally and theoretically [33][34][35][36][37][38]. According to QED, the vacuum becomes weakly anisotropic, dispersive, dissipative and even nonlinear optical medium, when external electric field is approaching the Schwinger critical value E c ≃ 10 18 V/m.I n o t h e r w o r d s , the real and imaginary parts of vacuum refractive index could deviate from unit and zero [34,35], respectively. Physically, the deviation of the imaginary part is mainly from the electron-positron pair generation. However, the electron-positron pair generation, also generally called as vacuum polarization (VP) processes [34], which is schematically shown in Fig.1, has not been directly observed for over half century since very high E c is beyond the contemporary technical limit. Therefore, it is natural to wonder if we can find an approach to probe VP with external field E ext much smaller than E c .
In this work, we propose evanescent-mode-sensing methods based on new mechanism to detect the QED VP, which is based on the measuring the phase change and the energy time delay of evanescent wave (EW). We find that the required external field could be one order weaker than E c , which may be realizable by contemporary experiments. The idea is from the "dual roles" of real and imaginary parts of refractive index n. Supposing a medium with complex index n = n ′ + in ′′ , our goal is to detect the very tiny change of n ′ or n ′′ . For radiative waves, since n ′ determines the real part of wavevector k ≃ n ′ ω/c and it is easy to measure the phase change or group delay, so, it is natural to choose the radiative wave to probe small change of n ′ . On the other hand, for radiative waves, tiny change of n ′′ causes an extremely small decay change which is very hard to measure in the limited lab space. However, for the evanescent waves, the roles of n ′ and n ′′ are totally exchanged, i.e., n ′ dominates the decay rate, while the n ′′ introduces a phase change which is much easier to detect. Further more, we will demonstrate that n ′′ can also introduce the energy propagation for EWs whose energy velocity v e ∝ n ′′ c a nb ee x t r e m e l ys l o w . S u c has l o ww a v ec a nb e detected by measuring the delay time τ at a short distance.
Actually, the tenneling mechanism of EW has been widely studied [39][40][41]. We would like to emphasize the mechanism difference between ours and that in the previous works. In Ref [41], they are based on "two interfaces" structure (a slab). Such "two-interfaces" structure will generate both evanescent modes exp(±κx) and such two evanescent modes can carry energy current [9], which called as "tunneling mechanism". So, even if the material dissipation is neglected [41] , the energy propagation is still available. However, in our model, since there is only a single interface (Fig.2), obviously if without dissipation there will be no the energy current at all [42], then, no phase change and no the energy delay time. So, our mechanism is based on the dynamical picture and the dissipation is critical.
Here we note that, because the probing light is much weaker than the external field in our model, the nonlinear effect is negligible. For a linear system, all dynamical processes can be solved numerically by sum of multi-frequency componentsčňwhich can be obtained by Green's function methods [42].
Our model is schematically shown in Fig.6, based on the total internal reflection (TIR) at the interface between a dielectric media n 1 (region I) and vacuum (region II). When the incident angle θ i > θ c = arcsin(1/n 1 ), the TIR will occur and the transmitted wave in the vacuum is the EW. We choose θ i is a little larger than θ c to make sure that almost all frequency components  are totally reflected when the incidence is the slowly-varying quasi-monochromatic wave. An interferometer or a photon detector is set at distance L from the interface so that the phase and intensity change can be detected.
The time-dependent Maxwell equations are given by ∇×E = −µ(z)µ 0 ∂H/∂t and ∇×H = ǫ(z)ǫ 0 ∂E/∂t,w h e r eǫ(z) and µ(z) are the relative permittivity and the relative permeability, respectively, and c = 1/ √ ǫ 0 µ 0 . To obtain the concrete results, the system parameters are chosen as following, the incident angle θ i = 0.1667π, the refractive index of region In 1 = √ ǫ 1 = 2, and the vacuum refractive index of region II n 2 = √ ǫ 2 µ 2 = 1 + δ + in ′′ ,w h e r e δ << 1a n dn ′′ << 1 are the real and imaginary index deviations of vacuum, because of VP processes caused by strong external field. If the incident probing light is a plane wave, the transmitted wave in the vacuum region can be generally written in the form E(x, z, t)= Eexp(ik z z + ik r − iωt),w h e r ek = n 1 sin θ i ω/c and k z = (n 2 ω/c) 2 − k 2 are the wave vectors parallel and perpendicular to the interface. For the EW, k z is described as: The physical meaning of k z is very clear that the imaginary part Im(k z )=κ z corresponds to the exponential decay of the field, and the real part Re(k z ) causes a phase change because of VP. The phase change at distance z = L is which could be measured by interferometers [43].
Besides the phase change Δφ, with the same model as shown in Fig.6, there is another way to detect the tiny n ′′ by measuring the time delay of irradiance fluctuation [44] of the evanescent wave. It is a dynamic process as following. First, we suppose that the incident light is not a plane wave anymore, but with a slow intensity fluctuation, as shown in Fig.7(a). Then, the question is "What will happen for the EW in region II ?" Numerically, from the strict Green's function method with physical dissipation and dispersion, it is found that the fluctuation will propagate on the EW from the interface to far away, as shown in Fig.7(b). So, we can measure the time delay τ of the fluctuation propagation on the EW to detect the VP effect. The propagation speed of irradiance fluctuation can be obtained by the energy velocity v e which is defined as is the local energy density of the electromagnetic wave. In our model, the energy velocity is obtained as: 2 , when the dissipation and dispersion are very weak. The physical meaning of v e can be understood as the "propagation" speed of irradiance fluctuation of the EW, which can be measured [44].
Hence, experimentally the time delay τ of the irradiance fluctuation at distance L can be measured: Since it is near field phenomenon, the detecting should be very near the interface. For the VP effect, since n ′′ is extremely small, the "propagation" speed of the irradiance fluctuation is so slow that τ gets to peco-second level when the distance is one tenth of the wavelength L = 60nm.
Therefore, either the phase change Δφ o rt h et i m ed e l a yτ are very sensitive for n ′′ ,a n dt h e EW is a good candidate to probe the VP effect. Here, we note that the famous Kramers-Kronig relations still fit for QED vacuum [34]. Hence, the observation of imaginary part of vacuum index also confirms the dispersion of QED vacuum.
Next, we will quantitatively study the VP detect by our methods. Supposing that an external homogeneous constant electric field E ext , which is perpendicular to the xz plane and smaller than the Schwinger critical electric field E c , is applied to the vacuum (region II)only,asshown in Fig.6, then, the optical properties of the vacuum can be described by the Euler-Heisenberg Lagrangian L eff [34,37]. Physically, the imaginary part of Euler-Heisenberg Lagrangian L eff is related to the imaginary part of VP operator, and therefore corresponds to the electron-positron pair generation.
Consequently, the vacuum refractive index can be deduced from the Lagrangian L eff [34,37,38].
In our model, since the external magnetic field is supposed to be zero, thus the vacuum refractive index is determined only by the external homogeneous constant electric field E ext . We use n and n ⊥ to refer the effective refractive index of vacuum when the electric field of probing light are parallel and perpendicular to the field E ext , respectively. n and n ⊥ can be obtained from the reference [38]: π exp(−nπ/y),w h e r ey = |E ext |/E c , and α ≃ 1/137 is the fine-structure constant. Therefore we have δ = Re n (⊥) − 1, n ′′ = Im n (⊥) for n and n ⊥ when we solve the equations such as Eq. (7) in this letter.  The parameters of our model in Fig.6 are chosen as following. The wavelength of probing light is λ 0 = 600nm, the dielectric constant in the region I is ε = 4, and the incident angle is θ inc = 0.1667π > θ c , so that the field in vacuum is evanescent. The distance L for the phase detecting is L p = 6µm = 10 × λ 0 , while for τ detecting is L τ = 60nm = 0.1 × λ 0 , respectively. The QED theoretical results of real and imaginary part of n and n ⊥ are shown in Fig.8(a) and Fig.8(b), respectively. Bring these results into Eq.(8) and Eq.(10), the phase change Δφ and the delayed time τ canbeobtained,whichareshowninFig8 (c)andFig.8 (d), respectively. Numerically, the phase change with plane wave incidence and the time delay of local amplitude maximum are calculated by Green's function method, which are also shown in Fig.8(c) and Fig.8 (d). Comparing the analytical results from Eq.(8) and Eq.(9) and numerical results, we can find that they agree with each other very well.
Next, we will analyze the possibility to observe the VP effect in experimental conditions. The recent experimental advances [45] have raised hopes that lasers may achieve fields just one or two orders of magnitude below the Schwinger critical field strength. In this case E ext ∼ 0.1E c , from our numerical and analytical results in Fig.8, we can see the Δφ can get to ∼ 10 −1 mrad order, which are in measuring limit of contemporary interferometer [43]. Very recently, it is supposed that the electric field E could be effectively amplified 4 times larger by coherent constructive interference of laser beams [36]. If E ext can get to 0.5E c by this method, not only Δφ can be one order larger, but also the delay time τ can get to sub peco-second level and may be measured by contemporary photon detectors.

The temporal coherence gain of the negative-index superlens image
Veselago predicted that the negative-index material (NIM) has some unusual properties, such as a flat slab of the NIM could function as a lens for electromagnetic (EM) waves [1]. This research direction was further pushed by works of Pendry and others [4,5,[46][47][48][49][50][51][52][53][54][55][56][57][58] who showed the lens with such NIM (i.e. ǫ = µ = −1 + δ)c o u l db easuperlens whose image resolution can go beyond the usual diffraction limit. After that, several beyond-limit properties of NIM systems are found, such as, the sub-wavelength cavity [59] and the waveguide [60]. Some of the theoretical results are confirmed by experiments [4,46,49]. And these beyond-limit properties give us new physical pictures and opportunities to design devices. Recently, new numerical [50,51] and theoretical Green's function [52] methods are used to understand the phenomena in such systems. But so far almost all studies are done with the strictly single-frequency sources, so that the coherent properties of EM waves (or photons) in the NIM systems have not been studied to the best of our knowledge. Even more seriously, there is no theory for the propagation of coherent functions in NIM systems. The importance of coherence research can not be over-estimated since the coherence is essential in the wave interference , the imaging , the signal processing and the telecommunication [61,62]. Can we find new frontier to go beyond at the coherent properties in NIM systems? If so, can we develop a simple theoretical method to deal with the image coherence of superlens? Fig. 9. The schematic diagram of our model with ray paths(left); and the typical snapshot of electric field in our FDTD simulation (right).
In this section, the finite difference time domain (FDTD) method is used in the two-dimensional (2D) numerical experiments to study the temporal coherence of the superlens image with random quasi-monochromatic sources. We observe the dramatic temporal-coherence gain of the superlens image even if the reflection and frequency-filtering effects are very weak. Based on the new physical picture of the signal (the fluctuation of random source) propagation in NIM, we construct a theory to obtain the image field and derive the equation of the temporal-coherence relation between the source and its image. The new mechanism of the temporal-coherent gain can be explained by the key idea that the signals on different paths have different "group" retarded time. Our theory excellently agrees with numerical results and the strict Green's function results.
The setup of the 2D system is shown in Fig.9. The thickness of the infinite-long NIM slab is d.
To realize the negative ǫ and negative µ, the electric polarization density P and the magnetic
The random source is composed of the randomly generated plane-wave pulses, with the average pulse length t p and the random starting phase and starting time. In the simulation, we record the field of the source and the image for a duration of 4 × 10 5 δ t to obtain the data for analysis. For the convenience, we define E(ω)=lim T→∞ T −T E(t)exp(−iωt) as the field spectrum (FS).
Unusual phenomena.−At first, the FS width of the random source is a little too large (Δω s ≃ ω 0 /20). When we observe the image temporal-coherence gain, we also find that the FS width of the image is sharper than the source (Δω i < Δω s ). It is obvious that there are the frequency-filtering effects because of the NIM dispersion, such as the frequency-dependent interface reflection and focal length. After increasing the pulse-length t p of the source, we reduce the source FS width to Δω s ≃ ω 0 /100, then the reflection and focal-length difference are very small [64]. With such source, the FS widths of source and image are almost same Δω i ≃ Δω s , as shown in Fig.10a. The difference between two widths is < 5%, which is our criterion of the quasi-monochromatic source. Even so the dramatic gain of temporal coherence is still observed. In Fig.10b, the source field (up) and the image field (down) vs time of FDTD simulation are compared. The profiles of them are genically similar, but the image profile is much smoother.
The normalized temporal-coherence function g (1) (τ)= < E * (t)E(t + τ) > /< E * (t)E(t) > (<> means the ensemble average) of the source (black) and the image (red) from FDTD simulation are shown in Fig. 11. The temporal coherence of the image field is obvious better than the source. From g (1) , the image coherent time is obtained T co i = g (1) i (τ)dτ = 1268δ t , which is about 50% longer than the source coherent time T co s = 860δ t . Although the gain of the spatial coherence only by propagation is well-known [62], the dramatic gain of temporal coherence is generally from the high-Q cavities, contrary to our case, which have strong filtering effects. To reveal the new mechanism of the temporal coherence gain in NIM systems, we also have done more numerical experiments in which only the ray near a certain incident angle (shown in Fig.9), such as only paraxial rays (θ ≃ 0), can pass through the superlens. Then the image field profile vs time looks very like the source field and has no gain of coherence anymore. Therefore, the gain of temporal coherence of the superlens image is not from one ray with certain incident angle, but probably from the interference between the rays with different incident angles. Then, what is different between the rays with different incident angles? After carefully checking the field profiles of different-incident-angle cases, we find that the profiles have different retarded time. The larger incident angle the longer retarded time.
Physical pictures.−To deeper understand the new mechanism of coherence gain and construct our theory, we need make two physical pictures clear. The first one is about the optical path length (OPL) nds which determines the wave phase and the refracted "paths" of rays in Fig.9 according to Fermat's principle (or Snell's law). Based on ray optics, the superlens and traditional lenses have same focusing mechanism, that all focusing rays have same OPL paths nds = const ( paths nds = 0 for superlens) from source to image [1]. But this picture is so well-known that it suppresses the other important picture. Because the temporal-coherence information is in the fluctuation signals of random field, the signal propagating picture should be essential for our study. The optical signals propagate in the group velocity v g which is always positive. Obviously, if the path (in Fig.9) is longer (larger incident angle), the signal need a longer propagating time, which is called group retarded time (GRT) in this section Inside the NIM, the GRT of a path should be The total GRT from source to image is τ r = τ 0 /cos(θ) where the τ 0 = d/c + d/v g is the GRT of the paraxial ray. Now, the new propagating picture for a signal through superlens is that a signal, generated at at t s from the source, will propagate on all focusing paths and arrive at  image position at very different time t s + τ 0 /cos(θ) from different paths(this is schematically shown in Fig.9). This picture is totally different from traditional lenses, whose images don't have obvious temporal-coherence gain because their focusing rays have same OPL and similar GRT.
Our theory.−Based on these analysis, we suppose that the superlens image field of the random quasi-monochromatic source is the sum of all signals from different paths with different GRT. This is the key point of our theory, and then the image field can be obtained: )dθ (11) where U s (t) is the slowly-varying profile function of the source and U 0 is the normalization factor. In Fig.10c (up), we show the result of the image field based on Eq.(11), we can see it is in excellent agreement with the FDTD result in 2b(down). To show the interference effect of different paths, we assume there are only two paths (such as A and B in Fig.9). Based on Eq.(11) the image field is The first two terms are same as the source field (just a time-shift) so they don't contribute to the coherence gain. The last two terms are from interference between two paths. The third (or the forth) term could be very large at the condition . This condition can always be satisfied between any two paths since τ is a continuous variable. So the interfering terms between the paths are responsible for the image temporal-coherence gain.
From Eq. (11), after the variable transformation t s = t − τ 0 /cosθ and some algebra, the relation of the temporal coherence between the image and the source can be obtained:  Fig. 11. The normalized temporal-coherence function g (1) vs time of the source field (black), and of the image field which obtained from the FDTD simulation (red), from Eq.(2) (blue) and from the Green's function method(green). where h i (t)=( τ 0 /t) 2 / 1 − (τ 0 /t) 2 is the response function of different incident angles, and G s (t 2 − t 1 )= < E * s (t 1 )E s (t 2 ) > is the temporal-coherence function of the source. Eq.(12) can explain the temporal-coherence gain of the image too. Even if the source field is totally temporal incoherent G s (t 2 − t 1 ) ∝ δ(t 2 − t 1 ), based on Eq.(12) we can find that G i (τ) is not a δ-function anymore, so the image is partial temporal coherent. The product of h * i (t 1 )h i (t 2 + τ) includes the interference between paths. According to our theory, we calculate the image coherence function g (1) vs time (Fig.11 blue) which agree with our FDTD result (Fig.11 red) pretty well (we will discuss the deviation later).
To further confirm our theory and FDTD results, the strict Green's function method [52] is engaged to check our results. We only include the radiating field (no evanescent wave) in Green's function. The strict image field vs time is shown in Fig.10c(down), and the image temporal-coherence function g (1) vs time is shown in Fig.11 (blue). In Fig.11, we can see that the FDTD result (red) is almost exactly same as the strict Green's function method (green). But our theory (blue) deviates from the strict result at very large τ > 3000δ t which corresponding to very long path(or very large incident angle). This is understandable since in our theory we neglect the dispersion of NIM totally and only use v g (ω 0 ). For the very-large-angle rays a small index difference (from the dispersion of NIM) can cause large focal-length difference. Hence the deviation is from the focus-filtering effect. When we reduce the FS width of source to an even smaller value (i.e. Δω s = ω 0 /500), the deviation of our theory is smaller.
Although our theory is only a good approximation generally, owing to the picture simplicity and clarity the theory can help us to study more complex systems qualitatively and quantitatively. The finitely-long 2D superlens is a good example which is hard to deal by Green's function method. In Fig.(11), we plot the coherent time T co i vs superlens length L of the FDTD simulation (black) and of our theory (red) , respectively. They coincide with each other pretty well (the deviation reason has been discussed). The increase of the T co i with the increase of L can be explained simply according our theory. Since the image field is E i (t)= 1 U 0 e −iω 0 t θ max θ min U s (t − τ 0 cosθ )dθ, the large-angle paths (θ > θ max and θ < θ min )and their contribution to the temporal-coherence gain are missed in the short superlens.
Obviously, Eq.(11) is suitable not only for random quasi-monochromatic source, but also for all quasi-monochromatic fields, such as the slowly-varying Gaussian pulses and slowly switching-on process mentioned by [52]. Our theory can be easily extended to 3D systems too. And owing to the fact that what we find is from the radiating field, so the temporal-coherence gain is not the near-field property. Actually, the new mechanism of the temporal-coherence gain is not limited for the n ≃− 1 superlens, also applicable to other superlenses, such as the photonic crystal superlens in [46,49,58]. But the specialities of n ≃− 1 superlens, such as almost no frequency-filtering (no frequency loss) and no reflection (no energy loss), can be used to design novel optical/photonic coherence-gain devices.
In summary, for the first time we have numerically and theoretically studied the temporal coherence of the superlens image with the quasi-monochromatic source. Numerically, we observe that the temporal coherence of the image can be improved considerably even almost without reflection and filtering effects. Based on new physical picture, we construct a theory to calculate the image field and temporal-coherence function, which excellently agree with the FDTD results and strict Green's function results. The mechanism of the temporal coherence gain is theoretically explained by the different GRT of different paths. Although the evanescent wave is very weak in this study, the coherence of evanescent wave in NIM systems is a very interesting topic which will be discussed elsewhere [66]. Other related topics, such as the spatial coherence which is very essential for the image quality of the superlens, can also be studied through the similar methods. Although our study is within the confinement of classic optics, similar investigation can be extended to the quantum optics [62], and interesting results can be expected. Obviously, the temporal-coherence gain of superlens is another evidence that the NIM phenomena are consistent with the causality [49]. We suppose that the temporal-coherence gain phenomena could be observed in micro-wave experiments [4,46]. Therefore, this study should have important consequences in the future studies of coherence in NIM systems. The no-reflection and no-frequency-filtering coherence gain of the superlens has some potential applications in the imaging, the coherent optical communication, and the signal processing.

The physical picture and the essential elements of the dynamical process for dispersive cloaking structures
Recently, the theory [67,68] has been developed based on the geometry transformation to realize a cloaking structure (CS), in which objects become invisible from outside. Then a two-dimensional (2D) cylindrical CS [69] and a nonmagnetic optical CS [70,71] are designed.
More surprisingly, the experiment [72] demonstrates that such a 2D CS really works with a "reduced" design made of split-ring resonators. These pioneers' works are really attractive and open a new window to realize the invisibility of human dream. However, so far almost all theoretical [67][68][69][70][71][73][74][75] studies of the CS are done in the frequency domain and the geometry transformation idea is supposed to work only for a single frequency, so that the effects of the dispersion have not been intensively studied. As pointed out in Ref. [68] and the quantitatively study in our recent work [7], the dispersion is required for the cloaking material to avoid the divergent group velocity. For the dispersive CS, new topics, such as the dynamical process, can be introduced in. Dynamical study is essential for the cloaking study since without it we can not answer the questions, such as how can the field gets to its stable state?, is there any strong scattering or oscillation in the process?, and how long is the process?, etc. More important, because the real radars generally are pulsive ones, the dynamical process is critical for the cloaking effect around the goal frequency. So the dynamical study not only gives us whole physical picture of the cloaking, but also helps us to design more effective cloaks.
In this section, the dynamical process of the electromagnetic (EM) CS is investigated by finite-difference time-domain (FDTD) numerical experiments. In our simulation, the Lorentzian dispersion relations are introduced into the permittivity and the permeability models, then the real dynamical process can be simulated [76][77][78]. Based on numerical simulation, we can follow the details of the dynamical process, such as the time-dependent scattered field, the building-up process of the cloaking effect, and the final stable cloaking state. By tuning the dispersion parameters and observing their effects on the dynamical process and the scattered field, we can find the essential elements which dominate the process. Theoretical analysis of these essential elements can help us to have a deeper physical picture beyond the phenomena and to design more effective cloaks.
The setup of the system is shown in Fig. 13(a), similar as the one in Ref. [69]. R 1 and R 2 = 2R 1 are the inner and the outer cylindrical radii of the CS, respectively. A perfect electric conductor (PEC) shell is pressed against the inner surface of the CS. The CS is surrounded by the free space with ε 0 = µ 0 = 1. From the left side, an incident plane wave with working frequency ω 0 is scattered by the CS, the total field and the scattered field can be recorded inside and outside B1 respectively by the numerical technique [79]. So the scattering cross-section σ can be calculated easily. Our study is focused on the E-polarized modes, for which only the permittivity and the permeability components ε z , µ r ,andµ θ are needed to be considered (For H-polarized modes, considering the corresponding components µ z , ε r ,andε θ , we can obtain the same numerical results in the dynamical process.). All of them are supposed to have the form 1 + F j (r) × f j (ω),wheresubscriptj could be z, r,andθ for ε z , µ r ,andµ θ , respectively. The filling factors F j (r) are only r-dependent, ω p is the plasma frequency which set to be a constant ω p = 10ω 0 ,and f j (ω)=ω 2 p /(ω 2 aj − ω 2 − iωγ) are the Lorentzian dispersive functions, where   γ is the "resonance width" or called as "dissipation factor," ω aj are the resonant frequency of "atoms" (resonant units) in metamaterials.
For the study of the dispersive CS, we suppose that the real parts of the ε z , µ r ,andµ θ always satisfy the geometry transformation of Ref. [69] at ω 0 : Then the filling factors F j (r) at different r can be obtained: To investigate the dispersive effect on the dynamical process, we tune the dispersion parameters ω aj in our numerical experiments. We use the working frequency ω 0 as the frequency unit since it is the same for all cases in this section so the ratio A j = ω aj /ω 0 represents ω aj . Obviously, for the Lorentzian dispersive relation, the dispersion is stronger when ω 0 and ω aj are closer to each other (the working frequency is near the resonant frequency), or in other words, when A j approaches one. Since there are singular values of real part of ε and µ, in our numerical simulation we have done some approximations, [80] such as we set the maximum and the minimum for ε and µ. Although such approximations will affect the cloaking effect of stable state, [74] we find that the influence of these approximations on the dynamical process is very small and can be neglected.
First, we show an example of evolving electronic field during the dynamical process in Fig.  13 with concrete parameters of A r , A θ , A z ,andγ. In Fig. 13(a), the plane wave arrives at the left side of the CS, and is ready to enter the CS. From Fig. 13(b)-13(e), the cloaking effect is built up step by step, at last, the field gets to the stable state shown in Fig. 13(f). Because of the dispersion, there is an obvious time delay in the cloaking effect and the strong scattered field is observed.
We introduce a time-dependent scattering cross-section σ(t) to quantitatively study the dynamical process, which is defined as where t = n × T, n = 0, 1, 2, ..., T is the period of the incident wave,J scat (t) is the one-period-average energy flow of scattered field, andS inct is the averaged energy flow density of incident field. To observe the dispersive effect on σ(t) during the dynamical process, at the first step, we keep A z and γ constant and change A r and A θ , the results are shown in Fig. 14(a). From the σ versus t curves, we can find the general properties of the dynamical process. First, there is strong scattering in the dynamical process. At the beginning, σ increases rapidly when the wave gets to the CS, then reaches its maximum (at about ninth period). After that, σ starts to decay until it gets to the stable value (of the stable cloaking state). Second, unlike other systems, there is no oscillation in the process. This property will be discussed later. Third, the time length of dynamical process, called as "relaxation time" generally, can be tuned by the dispersion. From Fig. 14(a), we can see that the main dispersive effect is on the decaying process. From case 1 to case 5, A r and A θ become closer to one,sothat the dispersion is stronger. We find that the stronger the dispersion, the longer the relaxation time. For comparing with the cloaking cases, we also show the σ(t) of the naked PEC shell in the case 6. From the definition of σ, we know that the area covered by these curve in Fig.  14(a) is proportional to the total scattered energy in the dynamical process. So the CS with the weaker dispersion will scatter less field (better cloaking effect) in the dynamical process. But, such a general conclusion is still not enough for us to get a clear physical picture to understand the cloaking dynamical process.
Next, we check whether the absorption of the CS is important in the process. The absorption is determined by the imaginary part of ε and µ. To study this effect, we hold A r , A θ ,a n dA z constant but modify the dispassion factor γ. We modify the filling factors F j simultaneously, so that the real parts of ε and µ are kept unchanged at ω 0 . In such way, we can keep the dispersion strength almost unchanged, but with the imaginary parts of ε and µ changed. Results in Fig.  14(b) show that the stronger absorption only leads to larger stable value of σ,l e a v i n gt h e relaxation time nearly unchanged. Thus, we can exclude the absorption from the relevant parameter list, since it only influences the σ(t) of stable state considerably.
To obtain deeper insight of the dynamical process, we need to study the dynamical process more carefully. From Figs. 13(b)-13(e), we can see that the "field intensity" (shown by different color in the figures) propagates slower inside the CS than that in the outside vacuum. And when the inside field intensity "catches up" the outside one [in Fig. 13 Fig. 15, we see that there is the "intensity front"(shown by red dashed curve) which separates two regions of the CS. At the right side of the front, the field intensity in the CS is much weaker than the outside and the Poynting vector directions are not regular(especially near the front). But at the left-side region which is swept by the intensity front, the Poynting vectors are very regular and nearly along the "cloaking rays" which was predicted at the coordinate transformation [68]. Since the cloaking effect can be interpreted by the mimic picture that the light runs around the cloaking area through these curved cloaking rays, it is not surprising to find that the stable cloaking state is achieved when the intensity front sweeps through the whole CS and these optical rays are well constructed. The surprising thing is that the stable cloaking state seems to be constructed locally. We believe this property is related the original cloaking recipe, [68] which makes the cloaking material is almost impedance matched layer by layer. This also explains why there is no oscillation in the cloaking dynamical process generally. This picture also can interpret the strong scattered field in the dynamical process, since these "irregular rays" at the right-side region of the intensity front must be scattered strongly. Further, we can use this picture to analysis the dynamical process of other incident waves, such as the Gaussian beams, which are composed of different plane-wave components.
With these understanding, now we are ready to find the correlation between the relaxation time and the CS dispersion. It is well known that the field intensity (or energy) propagates at the group velocity V g , which is controlled by the material dispersion. So the intensity front, which determines the dynamical process, should move in V g . Thus, we can explain the results in Fig. 14, since our modification of the dispersive parameters can cause the V g changed. But, because the cloaking material is the strong anisotropic material, the V g at different directions could be very different. Can we predict more precisely which component dominates the relaxation time? The answer is "yes." In Fig. 15(d), we can see that the stable energy flow in the CS is nearly along the θ direction at most regions of the CS. Then it is reasonable for us to argue that it is the component along the θ direction V gθ , not the component along the r direction V gr , that dominates the relaxation time and the total scattered energy in the dynamical process.
For the anisotropic cloaking material, the V gθ and V gr can be expressed as: ,wherec is the velocity of light in vacuum.
In order to illustrate our prediction, the σ(t) under different V gθ and V gr are investigated, respectively. First, we keep the V gr unvaried by holding A θ , A z and γ constant [keep dε z /dω and dµ θ /dω unchanged], only modify A r to change the V gθ . The results are shown in Fig.14(c), when A r is closer to one,theV gθ becomes smaller (with larger dµ r /dω), the relaxation time is longer and more energy scattered in the dynamical process. So the larger V gθ means the better cloaking effect in the dynamical process. On the other hand, when we keep the V gθ unvaried and change V gr by holding A r , A z ,a n dγ constant and modifying A θ , the results are shown in Fig. 14(d). We find that the relaxation time is almost unchanged with the change of V gr . Obviously, V gθ is the dominant element in the dynamical process. This conclusion can help us to design a better CS to defend the pulsive radars. In the expression of V gθ ,i ti sa l s os h o w n how to tune V gθ by modifying dispersion parameters.
It seems that the larger V gθ , the better cloaking effect in the dynamical process. However, since the V g (and its components) cannot exceed c generally, there is a minimum limit for the relaxation time of the cloaking dynamical process. We can estimate it through dividing the mean length of the propagation rays by V gθ .Inourmodel,themeanlengthisπ(R 2 + R 1 )/2, about three wavelengths. So the relaxation time can not be shorter than three periods. Figure  14 shows that our estimation is coincident with our simulation results. Actually, here we are facing a very basic conflict to make a "better" CS, which is more discussed in our other works. [7,81] The conflict is from the fact that the pretty strong dispersion is required to realize a good stable cloaking effect at a certain frequency, [7,68] but at this research we show that the weaker dispersion can realize a better cloaking effect in the dynamical process. At real design of the CS, there should be an optimized trade-off.
Based on causality, the limitation of the electromagnetic cloak with dispersive material is investigated in this section The results show that perfect invisibility can not be achieved because of the dilemma that either the group velocity V g diverges or a strong absorption is imposed on the cloaking material. It is an intrinsic conflict which originates from the demand of causality. However, the total cross section can really be reduced through the approach of coordinate transformation. A simulation of finite-difference time-domain method is performed to validate our analysis.

Limitation of the electromagnetic cloak with dispersive material
Through the ages, people have dreamed to have a magic cloak whose owner can not be seen by others. For this fantastic dream, plenty of work has been done by scientists all over the world. For example, the researchers diminished the scattering or the reflection from objects by absorbing screens [82] and small, non-absorbing, compound ellipsoids [83]. More recently, based on the coordinate transformation, J. B. Pendry, et al theoretically proposed a general recipe for designing an electromagnetic cloak to hide an object from the electromagnetic(EM) wave [68]. An arbitrary object may be hidden because it remains untouched by external radiation. Meanwhile, Ulf Leonhardt described a similar method where the Helmholtz equation is transformed to produce similar effects in the geometric limit [67,75] [72]. In addition, Wenshan Cai , et al proposed an electromagnetic cloak using high-order transformation to create smooth moduli at the outer interface and presented a design of a non-magnetic cloak operating at optical frequencies [70,71]. According to the general recipe, the electromagnetic cloak is supposed to be perfect or "fully functioned" at certain frequency as long as we can get very close to the ideal design although there is a singularity in the distribution, which has been elucidated further in several literatures [84,85]. However, in all these pioneering works, the interests are mainly focused on single-frequency EM waves, so that the effects of the dispersion, which is related with very basic physical laws, are not well studied. If the dispersion is introduced into the study, can we have a deeper insight into the cloaking physics?
In this section we will show the ideal cloaking can not be achieved because of another more basic physical limitation-the causality limitation (based on the same limitation, Chen et al obtained a constraint of the band width that limit the design of an invisibility cloak [86]). Starting from dispersion relation and combining with the demand of causality, we will demonstrate that the ideal cloaking will lead to the dilemma that either the group velocity V g diverges or a strong absorption is imposed on the cloaking material. Our derivation and numerical experiments based on the finite-difference-time-domain(FDTD) methods will show that the absorption cross section will be pretty large and dominate the total cross section for a dispersive cloak, even with very small imaginary parts of permittivity and permeability.
Let's consider a more general coordinate transformation on an initial homogeneous medium with ε i = µ i in r space: r ′ = f (r), θ ′ = θ, ϕ ′ = ϕ, following the approach in Ref. [87] and [89], we get the following radius-dependent, anisotropic relative permittivity and permeability: dr . We emphasize that since the transformation is directly acted on the Maxwell equations, the above equations are also suited for the imaginary parts of constitutive parameters, and all physical properties of wave propagation in r space should be inherited in r ′ space, such as the absorption. This is very important for us to have consistent physical pictures in both spaces. At working frequency ω 0 , for a propagating mode with k-vector as {k r ′ ,k θ ′ ,k ϕ ′ } inside the cloak, we have the dispersion relation of the anisotropic material [87] as: T h e nw ec a nd e fi n ek r ′ = ω c n r ′ cosα and k t ′ = ω c n t ′ sinα, the group velocity can be obtained as: Where m r ′ = n r ′ + ω dn r ′ dω and m t ′ = n t ′ + ω dn t ′ dω . If the transformation has the following characteristic: f (r = 0)=R 1 , f (r = R 2 )=R 2 ,t h en when r ′ → R 1 (or r → 0), n t ′ will tend to zero, and the group velocity is approximated as: We will discuss Eq.15 in two cases. The first case is with the finite dn t ′ dω . Obviously, V g will diverge when sinα → 0 for any finite dn t ′ dω . Such divergence is shown in figure 16 for a concrete example, in which the transformation is r ′ = f (r)=( R 2 − R 1 )r/R 2 + R 1 as Ref. [68], R 2 = 2R 1 ,thusn r ′ = 2andn t ′ = 2 − 4/(r/R 1 + 2). The dispersion parameters are set as m r ′ = 2.5, ω dn t ′ dω = 4 at working frequency. In figure 16, the curves of V g vs α are plotted for different R 1 /r values. We can see that, for large R 1 /r(r → 0), the group velocity (more precisely, the tangential component of V g ) will diverge at both peaks around α = 0.
Because of the causality limitation, it is well-known that the group velocity can not exceed c except in the "strong dispersion" frequency range (or called "resonant range"). But, if the working frequency is in the "strong-dispersion" range of the cloaking material, the absorption must be very strong and it will destroy the ideal cloaking obviously. So perfect invisibility can not be achieved for the finite dn t ′ dω because it will lead to superluminal velocity or strong absorption.
In addition, the curves with the criterion condition V g = c on the plane [R 1 /r, α] are plotted for different ω dn t ′ dω in figure 17. The region to the left of curves is corresponding to V g < c and the region to the right is corresponding to V g > c. There exists a maximum max{R 1 /r} for each curve in order that V g ≤ c can be hold for all the α. Especially, for the no-dispersion case ω dn t ′ dω = 0, we can see that V g > c at all R 1 /r for large α values,whichmeansthewhole cloak is not physical if there is no dispersion. This "dispersion-is-required" conclusion can be generally derived from Eq.14, and it is consistent with the analysis in Ref. [68]. From figure 17 dω (generally true) at all α values except α = 0(or π), so that the group velocity difficulty seems to be overcome. But, since the causality limitation, the non-zero dε r ′ dω means non-zero imaginary part of permittivity (non-zero dissipation). The non-zero dissipation and the almost-zero group velocity will result in very strong absorption. This means that the energy of rays near the inner cloaking radius R 1 is almost totally absorbed by the cloaking material. As we pointed out at the beginning that the absorption in r ′ space should also appear in r space, because of the consistence between two spaces. The strong absorption in r space can be interpreted in the following way. From the transformation (which is also suited for imaginary part), we can find that when r → 0, the finite imaginary part in r ′ space corresponds to the infinite imaginary part in r space, which also means very strong absorption in the initial homogeneous medium. So the perfect cloaking is still impossible because of the strong absorption which is enforced by the causality limitation.
For a two-dimensional coordinate transformation: r ′ = f (r), θ ′ = θ, z ′ = z,t h es a m e conclusions of the causality limitation can be obtained through the similar analysis, although the coordinate transformation and the singularities are different from the three-dimensional case.
Next, we will discuss the physical meaning of the dilemma that either the group velocity V g diverges or a strong absorption is imposed on the cloaking material. First, it is an intrinsic conflict which can not be solved by the methods, for example, "the system is imbedded in a medium" [68]. We believe that the ideal cloaking is impossible because of the causality limitation and this conclusion is consistent with the statement of previous studies [89] that the perfect invisibility is unachievable because of the wave nature of light. Second, we have to face the question: " Why the causality is violated for ideal cloaking which is based on the simple coordinate transformation?". Our answer is that the causality is only guaranteed by the Lorentz co-variant transformation, but the coordinate transformation for ideal cloaking is not Lorentz co-variant. Such violation is obvious if we suppose the initial medium in the r space is not dispersive, such as the vacuum, but as we have pointed out (also mentioned in Ref. [68], the cloaking material (in r ′ space) must be dispersive to avoid the group velocity over c. Such Lorentz co-variant violation is generally true forąřtransformation opticsąśsince material parameters are non-relativistic, so the causality limitation should be checked widely. Third, from Eq.14, we can find that not only the inner layers of the cloak(r ′ → R 1 )b u ta l s o the other layers(r ′ > R 1 ) must be dispersive. For every layer, a certain dispersive strength is needed to avoid V g > c.
In the following, we will validate that the total cross section can be reduced drastically, and that the perfect cloaking cannot be achieved because of strong absorption by FDTD numerical experiments. Compared with other frequency-domain simulation methods, such as the finite element methods or the transfer-matrix methods, the FDTD simulation can better reflect the real physical process of cloaking. For example, we note that the FDTD calculation will be numerically unstable when the dispersion is not included in the cloak's material. For simplicity, the simulation is limited to two-dimensional cloak [69]. Without lost of generality, only TE modes are investigated in this study (TE modes have the electric field perpendicular to the two-dimensional plane of our model). Thus the constitutive parameters involved here are ǫ z ′ , µ r ′ , µ θ ′ . The dispersion is introduced into our FDTD by standard Lorentz model: Where ω pz ′ , ω pr ′ , ω pθ ′ are plasma frequencies, ω az ′ , ω ar ′ , ω aθ ′ are atom resonated frequencies, γ z ′ , γ r ′ , γ θ ′ are damping factors and F pz ′ , F pr ′ , F pθ ′ are filling factors. In our simulation, an E-polarized time-harmonic uniform plane wave whose wavelength λ 0 in vacuum is 3.75cm is incident from left to right. The real parts of the constitutive parameters at ω 0 = 2πc/λ 0 satisfy the cloaking coordinate transformation [69,91], and they are µ r ′ = r ′ −R 1 r ′ , µ θ ′ = 1 µ r ′ , ε z ′ =( R 2 R 2 −R 1 ) 2 r ′ −R 1 r ′ ,whereR 1 is 0.665λ 0 , R 2 is 1.33λ 0 . And the dispersive parameters are set as follows: if ε z ′ > 1thenω az ′ = 1.4ω 0 ,elseω az ′ = 0.6ω 0 , ω ar ′ = 0.6ω 0 , ω aθ ′ = 1.4ω 0 , γ z ′ = γ r ′ = γ θ ′ = ω 0 /100. ω pz ′ = ω pr ′ = ω pθ ′ = 4ω 0 , F z ′ (r)=( ε z ′ − 1) . In fact, these parameters have many possible choices. The different groups of parameters correspond to different dynamic processes which we will discuss in another section [92]. Figure 18 shows the snapshots of the electric-field distribution in two cases: the cloak with the perfect electric conductor (PEC) at radius R 1 (left), and the naked PEC with radius R 1 (right). Obviously, the cloak is very effective. Quantitatively, we calculate the absorption cross section and the scattering cross section of the cloak at the stable state, and they are 0.67λ 0 and 0.24λ 0 respectively, while the scattering cross section of the naked PEC is 3.14λ 0 .S o ,w i t h dispersive cloak, the total cross section is three times smaller, and the absorption cross section dominates as we predicted. To emphasize the huge absorption of the cloak, we use a common homogeneous isotropic media, with ε = µ = 1.1 but all other parameters are the same as the cloak, to replace the cloaking material. Then we find the absorption cross section is only 0.089λ 0 which is about one order smaller. The reason of strong absorption has been discussed before.
Now we can have a full view of cloaking recipe based on the coordinate transformation. First, the cloaking material must be dispersive, and the strong absorption can not be avoided because of the causality limitation. Thus it is not perfectly invisible. Second, the scattering cross section of the dispersive cloak could be small, so that the scattered field is weak. Although the ideal invisibility is impossible, the cloaking recipe still has a main advantage. The "strong-absorption and weak-scattering" property means that the cloak almost can not be observed except from the forward direction. so that such cloak can well defend thę ařpassive radarsąś which detect the perturbation of the original field. It is well-known that at the Rayleigh scattering case, where the radius of the scatterer is much smaller than the wavelength, the absorption cross section could be larger than the scattering cross section because of the diffraction. The cloaking can be thought as giant Rayleigh scattering case, where the light rays are forced to "diffract" around the cloaked area.
In conclusion, the properties of the dispersive cloak are investigated, and the limitation of causality is revealed. Our study shows that the superluminal velocity or a strong absorption can not be overcome since the intrinsic conflict between the coordinate transformation to obtain the cloaking and the causality limitation. In addition, we validate the results using a numerical simulation which is performed in FDTD algorithm with physical parameters. The numerical experiments show that the absorption cross section is dominant and the scattering cross section can be reduced significantly. The study gives us a full view of the cloaking recipe based on the coordinate transformation, and will have further profound influence on the related topics.

Summary
In summary, we have investigate the metamaterial systems from group velocity(energy velocity) picture. From these topics, we demonstrate the importance of group velocity in metamaterial studies. From group velocity, we can find the physical origin of abnormal optical phenomena of metamaterials, such as the "no-transmission no-reflection" on the hyper-medium surface which is from a zero-group-velocity reflecting mode, and the coherence gain of superlens image which is from the different group delay on different paths. From group velocity, we can avoid some traps of violating basic physical limitation, such as the violation of causality limitation in cloaking study. These traps are very serious since the metamaterial from our imagination could exist in this world if violating basic limitations. From group velocity, we can find the key parameter of cloaking dynamical process and help us to optimizing the design of cloak design. From group velocity of evanescent wave, new detecting methods could be found, for example the detecting of QED vacuum polarization by phase change or delay time of evanescent wave. We believe that only with the well-constructed group velocity picture, the deeper understanding of the abnormal optical/photonic properties of metamaterials is possible. All these research works also show that the group velocity study of metamaterials can lead us to many new interesting topics, which are still waiting for further research. In-depth analysis of the theory, properties and description of the most potential technological applications of metamaterials for the realization of novel devices such as subwavelength lenses, invisibility cloaks, dipole and reflector antennas, high frequency telecommunications, new designs of bandpass filters, absorbers and concentrators of EM waves etc. In order to create a new devices it is necessary to know the main electrodynamical characteristics of metamaterial structures on the basis of which the device is supposed to be created. The electromagnetic wave scattering surfaces built with metamaterials are primarily based on the ability of metamaterials to control the surrounded electromagnetic fields by varying their permeability and permittivity characteristics. The book covers some solutions for microwave wavelength scales as well as exploitation of nanoscale EM wavelength such as visible specter using recent advances of nanotechnology, for instance in the field of nanowires, nanopolymers, carbon nanotubes and graphene. Metamaterial is suitable for scholars from extremely large scientific domain and therefore given to engineers, scientists, graduates and other interested professionals from photonics to nanoscience and from material science to antenna engineering as a comprehensive reference on this artificial materials of tomorrow.

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