1. Introduction

Early in 1968, Veselago predicted that a planar slab of negative-index materials(NIM), which possess both negative permittivity ε and negative permeability μ, could refocus the electro-magnetic (EM) waves [1]. Recently this research was further pushed by the works of Pendry and other scientists [2–21], they showed that the lens with such NIM could be a superlens which can break through or overcome the diffraction limit of conventional imaging system. After that, several other beyond-limit properties of NIM systems are found, such as the sub-wavelength cavities [22] and the waveguides [23], and some of the theoretical results are confirmed by experiments [6–12]. All of these beyond-limit properties give us new physical pictures and opportunities to design devices. In addition, new numerical [13, 14] and theoretical (such as Green’s function [15]) methods are also used to understand the phenomena in such systems. So far, almost all studies are done with the strictly single-frequency sources, so the coherent properties of EMwaves in the NIMsystems have not been studied to the best of our knowledge. However,the coherent properties are very important in the wave interference, the imaging, the signal processing and the telecommunication [24–26]

In this letter, the temporal coherence of the NIM slab image exicted by a random quasi-monochromatic source is investigated by the finite-difference time-domain (FDTD) algorithm. We find that the temporal-coherence gain is prominent even when the frequency-filtering effects are very weak. Based on the new physical picture of the signal propagation in NIM, we construct a phenomenological theory which we denominate signal superposition theory(SST) by which we can easily obtain the image field and derive the equation of the temporal-coherence relation between the source and its image. The results of SST agree well with that obtained by FDTD algorithm and strict Green’s function method. The key idea of the new coherence mechanism is that the signals propagating along the different paths have the different “group” retarded time.

2. Our model and the results of FDTD

Figure 1 shows the setup of our two-dimensional model. The thickness of the NIM slab is d.

 

Fig. 1. The schematic diagram of our model with ray paths(left); and the typical snapshot of electric field in our FDTD simulation (right).

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In this paper, only the TM modes are investigated(the TM modes have the electric field perpendicular to the two-dimensional plane of our model), the exciting source(its details will be introduced later) is located at d/2 in front of the slab and set as a quasi-monochromatic field which can be expressed as E_s (t)=U_s (t)exp(-iω ₀ t), where U_s (t) is a slow-varying random function, ω ₀=π/(20δ_t ) is the central frequency of our random source and δ_t =1.18×10^-15 s is the smallest time-step in FDTD simulation. To realize the negative ε and negative μ, the electric polarization density P⃗ and the magnetic moment density M⃗ are phenomenologically introduced in FDTD simulation [27]. The relative permittivity and the relative permeability of the NIM are ε_r (ω)=μ_r (ω)=1+${\omega }_P^2$/(${\omega }_a^2$-ω ²-iγ). In our model, ω_a =1.884×10¹³/s, γ=ω_a /100, ω_P =10×ω_a . At ω ₀, we have ε_r =μ_r =-1.00-i0.0029. Here, we emphasize that the distance (d/2=λ ₀, λ ₀ is the wavelength in vaccum corresponding to the central frequency ω ₀) between the source and the slab is too large to excite strong evanescent modes of NIM in FDTD simulation [13, 14, 16]. Actually the evanescent field in our simulation can be neglected comparing with the propagating field, and what we are studying is the property dominated by the propagating field.

For a random generated sine-wave pulse with the random starting phase φ_i , the starting time t_i and the random pulse length t_pi , we can represent it with sin(ω ₀(t-t_i )×φ_i ), t_i <t <t_i +t_pi , then our random source is composed of many such sine-wave pulses, i.e. Σ _i sin(ω ₀(t-t_i )+φ_i ), which possesses the average pulse length tp. The sources generated by this way are a quasimonochromatic field with the central frequency ω ₀. The larger t_p , the narrower its spectrum width. In the simulation, the fields of the source and the image are recorded for a duration of 4×10⁵ δ_t to obtain the data for analysis.

For the convenience, we can define <|E(ω)|> as the field spectrum (FS), where $E\left(\omega \right)={lim}_{T\to \infty }\frac{1}{T}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}E\left(t\right)\mathit{exp}\left(-i\omega t\right)dt$ and <> represents ensemble average(Obviously, FS is not a random function). It is noted that the width of a spectrum in this paper means the full width of its half maximum. In the beginning, the FS width of the random source is a little too large (Δω_s ≃ω ₀/20). The temporal-coherence gain of the image is observed, in the meantime the FS width of the image is also found narrower than that of the source (Δω_i <Δω_s ). It is obvious that there are the frequency-filtering effects caused by the NIM dispersion, such as the frequency-dependent interface reflection and focal length. Then we reduce the FS width of the source to Δω_s ≃ω ₀/80 by increasing the pulse-length t_p of the source, at this moment, the reflection and the focal-length difference are very small [28]. With such source, the FS widths of the source and its image are almost the same(Δω_i ≃Δω_s ), the difference between them is less than 5%, which is our criterion to judge whether the frequency-filtering effects can be neglected or not. Figure 2(a) shows the E(ω) of a certain random source field E_s (t) and its corresponding image field E_i (t) obtained by FDTD simulation, it is noted that we do not perform the ensemble average on Fig. 2(a) in order to make the phenomena of no frequency-filtering clear at a glance. Even so the prominent gain of temporal coherence is still observed. In Fig. 2(b), the source field (up) and the image field (down) obtained by FDTD simulation are compared. The profiles of them are genically similar, but the image profile is much smoother. The normalized temporalcoherence function g ⁽¹⁾(τ)=<E*(t)E(t+τ)>/<E*(t)E(t)> of the source (black) and the image (red) obtained by FDTD simulation are shown in Fig. 3. The temporal coherence of the image field is obviously better than that of the source. From g ⁽¹⁾, the image coherent time is obtained, $T_i^{\mathit{\text{co}}}$ =∫|$g_i^{\left(1\right)}$ (τ)|² dτ=1268δ_t , which is about 50% longer than the source coherent time $T_s^{\mathit{\text{co}}}$ =860δ_t .

 

Fig. 2. (a)The E(ω) of the source(up) and the image obtained by FDTD simulation(down) in unit of ω ₀. (b)The electric field of the source(up) and its image obtained by FDTD simulation(down) vs time. (c) The image field based on Eq.(1) (up), and the image field from the Green’s function method (down) vs time.

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Fig. 3. The normalized temporal-coherence function g ⁽¹⁾ of the source field (black) and the image fields obtained respectively from the FDTD simulation (red), Eq.(2) (blue) and the Green’s function method(green).

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We know that the temporal-coherence gain can be achieved by high-Q cavities[25, 26], which also means strong frequency-filtering effect. In our case, however, we observe the prominent gain of temporal coherence even when the frequency-filtering effects are very weak. To reveal the new mechanism of the temporal-coherence gain in NIMsystems, the numerical experiments in which only the ray near a certain incident angle (such as only paraxial ray)can pass through the NIM slab are performed. The results show that there is no gain of coherence anymore and the image field profile also looks like that of the source field very much. Therefore, we think that the gain of temporal coherence of the NIM slab image is not from one ray corresponding to a certain incident angle, but probably from the interference between the rays corresponding to the different incident angles.

3. The phenomenological theory

According to Fermat’s principle (or Snell’s law), all the rays originaing from source and reaching imaging point have the same optical path length(OPL) ∫nds which determines the wave phase and the refracted “paths” of rays, i.e. ∫nds=const. In this sense, the NIM slab and the traditional lens have the same focusing mechanism(just ∫ _pathsnds=0 for NIM slab). Howerver, there is something special for NIM slab. Because the temporal-coherence information is in the signals —fluctuation of random field, the signal propagating picture should be essential in our study. We know that the optical signals propagate in the group velocity v_g which is always positive. Obviously, if the path is longer (corresponding to larger incident angle), the signal will spend longer propagating time, which is called group retarded time (GRT) in this Letter. So we expect that the rays corresponding to the different incident angles have different retarded time. And it is really confirmed by our FDTD simulation. The larger the incident angle, the longer the retarded time. Inside the NIM, the GRT of a path is $\frac{d}{\cos \left(\theta \right)v_g}$ (this is also confirmed by our numerical experiments), where θ is the incident angle and v_g =c/3.04 is the group velocity of NIM around ω ₀ [29]. The total GRT from source to image is τ_r =τ ₀/cos(θ), where τ ₀=d/c+d/v_g is the GRT of the paraxial ray. Now, the new picture of a signal propagating through NIM slab is that a signal, generated at t_s from the source, will propagate along all the focusing paths and reach imaging point at different moment t_s +τ ₀/cos(θ) corresponding to different paths(this is schematically shown in Fig. 1). This picture is totally different from traditional lenses, whose images don’t have obvious temporal-coherence gain because their focusing rays have the same OPL and similar GRT.

Based on preceding analysis, we suppose that the image field of the random quasimonochromatic source in the NIM slab is the sum of all signals from different paths with different GRT. This is the key point of SST, and then the image field E_i (t) can be obtained:

where U ₀ is the normalization factor. The image field based on Eq.(1) is shown in Fig. 2(c) (up), we can see it excellently agrees with the FDTD result in Fig. 2(b)(down). To show the interference effect of different paths, we assume there are only two paths (such as A and B in Fig. 1). Based on Eq.(1), the image field $E_i=e^{-i{\omega }_{0}t}\left(U_s\left(t-{\tau }_r^A\right)+U_s\left(t-{\tau }_r^B\right)\right)$ , then the temporal coherence of the image $G\left(\tau \right)=<E_i^{*}\left(t\right)E_i\left(t+\tau \right)>=e^{-i{\omega }_0\tau }<U_s^{*}\left(t-{\tau }_r^A\right)U_s\left(t-{\tau }_r^A+\tau \right)+U_s^{*}\left(t-{\tau }_r^B\right)U_s\left(t-{\tau }_r^B+\tau \right)+U_s^{*}\left(t-{\tau }_r^A\right)U_s\left(t-{\tau }_r^B+\tau \right)+U_s^{*}\left(t-{\tau }_r^B\right)U_s\left(t-{\tau }_r^A+\tau \right)>$ . The first two terms are the same as the coherence function of the source field, so they do not contribute to the coherence gain. The last two terms are from the interference between two paths, they can be very large at the condition τ≃±(${\tau }_r^B$ -${\tau }_r^A$ ). 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.

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 from Eq. (1):

where $h_i\left(t\right)=\frac{{\left(\frac{{\tau }_0}{t}\right)}^2}{\sqrt{1-{\left(\frac{{\tau }_0}{t}\right)}^2}}$ 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. According to SST, we calculate the image coherence function g ⁽¹⁾ vs time (Fig. 3 blue) which agrees with our FDTD result (Fig. 3 red) pretty well (we will discuss the deviation later). Eq.(2) can also explain the temporal-coherence gain of the image. Even if the source field is totally temporal incoherent G_s (t ₂-t ₁)∝δ(t ₂-t ₁), based on Eq.(2) 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.

To further confirm SST and FDTD results, the strict Green’s function method [15] is used to check our results. We only include the propagating field (no evanescent wave) in Green’s function. The strict image field vs time is shown in Fig. 2(c)(down), and the image temporalcoherence function g ⁽¹⁾(τ) vs time is shown in Fig. 3 (green). In Fig. 3, we can see that the FDTD result (red) is almost the same as the strict Green’s function method (green). But the result of SST (blue) deviates from that of the Green’s function at very large τ(>3000δ_t ) which is corresponding to very long path(or very large incident angle). This is understandable since we neglect the dispersion of NIM totally and only use v_g (ω ₀) in SST. For the very-large-angle rays a small index difference (from the dispersion of NIM) can lead to a large focal-length difference. So the deviation is from the focus-filtering effects. When we reduce the FS width of the source further, the deviation of SST is smaller.

Although SST is only a good approximation generally, SST can help us to study more complex systems qualitatively and quantitatively owing to the simplicity of physical picture. The finite-long two-dimensional NIM slab is a good example which is hard to deal by Green’s function method. In Fig. 4, we plot the coherent time $T_i^{\mathit{\text{co}}}$ vs the length L of NIM slab obtained from the FDTD simulation (blue) and SST (red) respectively. They coincide with each other pretty well (the deviation reason has been discussed). The increase of $T_i^{\mathit{\text{co}}}$ with L can be explained simply by SST. Since the image field $E_i\left(t\right)=\frac{1}{U_0}{e}^{-i{\omega }_{0}t}{\int }_{{\theta }_{\mathit{min}}}^{{\theta }_{\mathit{max}}}{U}_s\left(t-\frac{{\tau }_0}{\cos \theta }\right)d\theta$ , the large-angle paths (θ>θ_max )and their contributions to the temporal-coherence gain are missed in the short NIM slab.

Owing to the fact that what we find is from the propagating field, so the temporal-coherence gain is not the near-field property. In addition, the new mechanism of the temporal-coherence gain is not limited to the n≃-1 NIM slab, also applicable to other NIM slabs, such as the photonic crystal slab with negative refractive index [9, 11, 21].

 

Fig. 4. The coherent time $T_i^{\mathit{\text{co}}}$ versus the NIM slab length L of the FDTD (blue) and our theory (red).

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4. Conclusion

In summary, we have numerically and theoretically studied the temporal coherence of the NIM slab image excited by the quasi-monochromatic source. we find that the temporal coherence of the image can be improved prominently even when the frequency-filtering effects are very weak. Based on the new physical picture,a phenomenological theory is constructed 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 corresponding to different paths. The results should have important impacts on the study of coherence about NIM systems, and we hope the no-frequency-filtering coherence gain of the NIM slab can be applied in the imaging, the coherent optical communication, and the signal processing.

Obviously, the temporal-coherence gain of NIMslab image is another evidence that the NIM phenomena are consistent with the causality [11]. In addition, SST can also be easily extended to three-dimensional systems.