## 1. Introduction

Early in 1968, Veselago[1] predicted that a new type of artificial metamaterial, which possesses simultaneously negative permittivity and permeability, could function as a lens to focus electromagnetic waves. These research direction was promoted by Pendry’s work[2, 3] and other latter works [4, 5, 15, 16, 23, 22, 7, 8, 9, 10, 11, 12, 13, 14, 21, 20, 17, 18, 24, 19, 25, 6, 26]. They show that with such a metamaterial lens, not only the radiative waves but also the evanescent waves, can be collected at its image, so the lens could be a superlens which can break through or overcome the diffraction limit of the conventional imaging system. This beyond-limit property gives us a new window to design devices.

It is well-known that evanescent wave plays an important role in the beyond-limit property of the metamaterial superlens. Furthermore, evanescent waves become more and more important when the metamaterial devices enter sub-wavelength scales[28, 27]. Therefore, the quantitatively study of *pure* evanescent waves in the metamaterial superlens is very significant. However, the quantitatively effects of *pure* evanescent wave in the metamaterial superlens have not been intensively studied, since so far almost all studies were only interested in the image properties with global field[8, 9], which is the summation of radiative wave and evanescent wave. On the other hand, many theoretical works were performed to study the metamaterial superlens, employing either finite-difference-time domain (FDTD) simulations[16] or some approximate approaches[29, 30]. However, one cannot obtain the rigorous *pure* evanescent wave by these numerical methods, since the image field of the metamaterial superlens obtained by FDTD is global field, and other approximate approaches cannot be rigorous.

In reviewing these existing efforts, we feel desirable to develop a rigorous method that can be used to study quantitatively the transient phenomena of the evanescent wave in the image of the metamaterial superlens. In this paper, we will present a new method based on the Green’s function[7, 8] to serve this purpose. Our method can be successfully used to calculate the evanescent wave, as well as the radiative wave and the global field. The main idea of our method can be briefly illustrated as follows. Since the metamaterial superlens is a linear system, so all dynamical processes can be solved by sum of multi-frequency components. And each frequency component can be solved by sum of multi-wavevector components. So we can use Green’s function of multi-frequency components to obtain the strict numerical results. Therefore, our method based on the Green’s function is strict, and it is quite a universe method in any linear system, for example, it can be used to study the two dimensional (2D) and three dimensional (3D) metamaterial superlens.

The content of the chapter is organized as follows. We mainly focus on the details of our theory and method in Section 2. After that, in Section 3, we will calculate the field of the image of a NIR superlens by using our method, including radiative waves, evanescent waves, SEWs, and global field. The method will be confirmed by using FDTD simulations. In Section 4, we will present our study on the group delay of evanescent wave in the superlens by using our method. Finally, conclusions are presented in Section 5.

## 2. Theoretical method

In this section, we will focus on the theoretical details of our method. First, a time-dependent Green’s function will be introduced. Then, based on the Green’s function, the method to obtain evanescent waves as well as radiative waves will be presented.

### 2.1. A time-dependent Green’s function

First, we would like to introduce a very useful time-dependent Green’s function for the solution of inhomogeneous media. The time-dependent Green’s function can be applied to study the dynamical scattering processes[7, 22]. In the inhomogeneous media, the problem we study can be solved by Maxwell equations:

where

To solve Eq.(2), we introduce a dynamic Green’s function

where

Therefore, Eq.(3) becomes:

By Fourier transforming Eq.(5) from time domain to frequency domain, we obtain

where

By Fourier transforming the electric field

Considering the general properties:

Substituting Eq.(8), Eq.(9), Eq(10) and Eq.(11) into Eq.(6), we can obtain

Eq.(12) exhibits the role of

where

So the field

where

Now the remaining problem is to solve Eq(7) to get

As a typical example, the time-dependent Green’s function for the three-layer inhomogeneous media is presented, which is shown in Fig.1. The inhomogeneous media of the system can be described by:

where

where

For the 2D case, the wave vector

And for the 3D case, we have

After

### 2.2. The Green’s function for radiative waves and evanescent waves

Now, we will apply a time-dependent Green’s function for a radiative wave and an evanescent wave. This Green’s function can be directly developed from the Green’s function introduced in the Sec.2.1. The schematic model is shown in Fig.1. As we know, the plane solution wave for the electric field in vacuum is of the form

For simplicity, we first consider the 2D system, in which

and

where

From Eq.(22) to Eq.(24), we can directly calculate the radiative wave and evanescent wave. In the case of radiative wave (

for radiative waves.

In the case of evanescent wave (

for evanescent waves.

And for the global field, the global field Green’s function

Additionally, we can also focus our observation on the *subdivided evanescent wave* (SEW), with a certain integral range

As an typical example, a SEW with an integral range [

In this way, we can obtain the Green’s function for the SEW with any integral range. Obviously, evanescent wave could be regarded as the superposition of SEWs. Therefore, we have

From Eq.(28), Eq.(26), Eq.(25), and Eq.(27), one can obtain the SEW Green’s function

For the 3D system, obviously, the methods to get Green’s function for the SEW, the evanescent wave, the radiative wave, and the global field are respectively very similar with the above discussion, i.e., just replace

## 3. Electromagnetic waves in the image of the superlens

In this section, we will discuss the image’s field of a 2D metamaterial superlens, which is shown in Fig.2. The thickness of the metamaterial slab is

The inhomogeneous media of the metamaterial superlens system are described by:

The negative relative permittivity

where

In order to excite the evanescent wave strong enough in the image of the metamaterial superlens, the distance

For this metamaterial superlens system, it is very easy to obtain

respectively, where

The numerical results calculated by our method are shown in Fig.3. Fig.3(c)(up, the blue one), (d) and (e) show the global field, the evanescent wave, and two typical SEWs respectively. The integral

In order to convince our method, FDTD simulation is also applied to calculate the field of the image, which is shown in Fig.3(c) (down, the green one). Comparison with the results calculated by our method and FDTD shown in Fig.3(c), we can see they coincide with each other very well. In addition, we also calculate the frequency sepctrum of the image by our method, as shown in Fig.3(b) (down, the red one). Comparing the spectra of source and image, we can find they are very close to each other. This result also agrees with the Ref.[29]. Therefore, our method is convincible, which can be used to obtain the pure evanescent waves, the SEWs, and the global field effectively.

## 4. Group delay time of SEWs and its impacts on the temporal coherence

### 4.1. Group delay time of SEWs

From Figs.3(c)-(e), we can find that the profile of evanescent wave and that of SEWs look like that of radiative wave with a group delay time

where

Here

Similarly, we can also study the group delay time of the SEWs. We rewrite Eq.(34) as follows:

where

In our numerical experiment, in order to calculate the group delay time of the SEWs, we choose 70 SEWs with integral

Since the field profile of SEWs looks like that of a radiative wave, so we can write the field of the SEW with the integral range

where

The physical meaning of

### 4.2. Impacts of the group delay of SEWs on temporal coherence gain in the image of the superlens

One of the most interesting impacts of the group delay of SEWs is related to the first-order temporal coherence gain (CG). Here, we would like to discuss the CG caused by the SEWs in the image of the superlens. In our previous work [29, 30], we have investigated a prominent CG of the image by the radiative waves even when the frequency-filtering effects are very weak. Then, a natural question is what about the role of the evanescent waves play in the CG of a superlens? In this section, we will show that not only the radiative waves but also the evanescent waves, and the SEWs that can be responsible for the CG. Furthermore, we will show that the total CG in the image of a superlens is the weighted averaged of evanescent-wave coherence gain (ECG), radiative-wave coherence gain (RCG), radiative-wave and evanescent-wave coherence gain (RECG).

First of all, let’s consider the contributions of the evanescent waves on the CG. For this, we calculate the normalized first-order temporal coherence

where

or

here

To show how the ECG is produced from the interference of the SEWs with different group delay time, we assume there are only two SEWs, such as

From Eq.(38), we have the evanescent wave field:

then the temporal coherence of the evanescent wave in the image is given by

The first two terms are the same as the coherence function of the radiative wave, so they do not contribute to ECG. The last two terms are from the interference between SEWs, they can be very large at the condition

Therefore, ECG can always exist in the superlens when two conditions are satisfied: (1)* only the interference of those SEWs with close integral variable J→(r,t)=σ(r,t)E→(r,t) can produce the ECG.*

Here the integral variable “

Therefore, the coherence gain from the interference of SEWs is limited by

To convince it, three SEWs (SWE

## 5. Conclusion

In conclusion, based on the Green’s function, we have numerically and theoretically obtained the evanescent wave, as well as the SEWs, separating from the global field. This study could help us to investigate the effect of an evanescent wave on a metamaterial superlens directly and give us a new way to design new devices.