1. Introduction

In nature, due to the refraction and reflection of light in the ambient there appears optical illusion phenomenon. Mirage effect is a typical paradigm, but it is complicate and not easy to be observed. Artificially, optical cloak and illusion can be created with the use of electromagnetic (EM) metamaterial which is a synthetic composite material with delicate structures and capable of molding the flow of light in a flexible manner. Several typical phenomena are realized theoretically and experimentally such as negative refraction [1–6], cloak [7–12], and subwavelength propagation [13–15] of an EM wave. In the perspective of transformation optics [7, 8, 16–18], the variation of EM constitutive parameters forms a curved space for an EM wave and thus guides it in a desired manner. As a result, the EM wave propagating within the meatamaterial manifests exotic behaviors, which can not be observed in the conventional materials.

Cloaking implies that an object is invisible to an observer outside a specified region, indicating that the EM wave undergoes nearly no scattering or absorption. To achieve this purpose, three schemes are proposed, one of which is to design a metamaterial shell so that the EM wave can detour round it, resulting in the cloaking of an object placed inside the shell [9–12]. Another scheme proposed is to cancel the scattering field from an object to be cloaked by different mechanisms of cancelation. Along this line Alu and Engheta proposed the scattering-cancelation-based plasmonic cloaking [19,20]. The complementary-medium-based cloaking is also proposed, where an “anti-object” is used to cancel the scattering effect of the object to be hidden [21]. Besides the cloaking device made of passive materials, active-source-based cloaking is suggested as well where active sources are introduced to cancel the incident EM wave in a predesigned region [22–26]. In this way, any passive object placed in this area is invisible.

Provided that an “anti-object” (active sources) can cancel the scattering effect from an object (incident EM wave) exactly, then ideal cloaking can be achieved. Interestingly, if the “anti-object” (active sources) can simultaneously generate the effect as if another object exists, then optical illusion can also be created. Generally speaking, cloaking is just a special case of the optical illusion. With metamaterials, Lai et al. [27] implemented the optical illusion where an “anti-object” is used to cancel the effect of an object, and meanwhile a “restoring object” is used to create the effect of an illusion object. However, as pointed out by Vasquez and coworkers [24, 25], for the cloaking devices made of passive material an intrinsic drawback is the limitation of its working frequency bandwidth. In addition, material loss is also a big challenge to be conquered in the design of metamaterials. Active cloaking can somehow avoid the above problems. The present work is devoted to designing an optical illusion device based on the lower order active sources. Furthermore, to make the design feasible in the experiment we even reduce the active sources to dipole ones. Our simulation results suggest that with a cluster of dipole active line sources both the cloaking and the illusion effects can be realized. In particular, it is shown that this active illusion device can even be operated as a beam rotator.

2. Theoretical framework

To demonstrate the cloaking and the illusion effects, we have employed the multiple scattering theory [28–30] to perform all the simulations. For simplicity, we expatiate the approach in two dimensional case, the idea can also be generalized to three dimensional case in a similar manner. The active sources radiating EM wave of transverse magnetic (TM) mode with its electric field polarized along z direction is considered and used to construct cloaking (illusion) devices. The external source that emanates EM wave outside the cloaking (illusion) device can be a plane wave, a Gaussian beam, or a line source, all of which can be considered as an incident wave. They can be expanded in the form

The scenarios to realize the cloaking and illusion with a cluster of active sources are as follows. Suppose the size of the cloaking (illusion) device is R_d. Then, to cloak an object the scattering field outside R_d must vanish or nearly vanish. A cluster of N active sources can achieve this purpose by creating a “quiet region” (the total EM field in this area equals to 0) with the radius R_c. In this “quiet region”, the EM field from the active sources is just the minus of the external field, so that they can cancel each other exactly. Accordingly, any object placed in the area is subject to no scattering. Simultaneously, if the EM field from all the active sources is zero outside R_d, then cloaking is realized. To achieve an illusion effect, besides the “quiet region” that conceal the object A, the active sources should simultaneously generate the EM field identical to that scattered from a disguised object B. In this manner, the illusion device can transform object A into another object B. Concretely, the active source based device can cloak a real object A by creating a “quiet region”. Meanwhile, it generates the scattering field that mimics the illusory object B, which determines what an observer can “see”. Accordingly, the key point in designing the active cloaking (illusion) device is to obtain the coefficient q_n,m of each active source in Eq. (2), which can be determined by the conditions on the circle r = R_c and r = R_d.

3. Result and discussion

It has already been shown that the active source based cloak can be realized with the high order active sources [24, 25]. But from Eq. (2) we can find that for each active source we must determine 2m′_c + 1 coefficients q_n,m. Therefore, for the high order active source it is a really tough job to decide all the coefficients simultaneously in the experiment. If we can reduce the order of the active source to a much lower degree, then the theoretical design could be a feasible one for the experimentalist. For this purpose, we have calculated the necessary order m′_c and the corresponding number N of the active sources to realize cloak for a plane wave incident upon the device. In Fig. 1, we present the result where the active source order m′_c is plotted as the function of the number N of the active sources. The size of the cloak device is R_d = 20λ and the effective cloaking region (quite region) is R_c = 2λ. The active sources are arranged uniformly around a circle with the radius r = 10λ. The scheme used to determine the number of the active sources and the corresponding order is as follows. First, we specify the order of the active sources; Second, we fix the initial number of the active sources as less as possible and then determine the number of the sampling points, which are used to obtain the overdetermined linear equations according to Eq. (7); Third, solve this linear equations and judge (i) whether the result is convergent and (ii) whether the cloak effect is realized. If the conditions are satisfied, the number of the active sources is what we want. If the conditions are not satisfied, we have to increase the number of the active sources and then repeat step 2–3 until the minimal number of the active sources satisfying the conditions are obtained. It can be found that when the order is 58, the source number should be 3 in order to realize the cloaking effect. This is the case considered in the literature [24, 25]. It can be seen from Fig. 1 that with the increase of the source number, the order of the active source can be reduced rapidly at first to a value less than 10, and then with further increase of the source number to 94, we can even reduce the source order to m′_c = 1, suggesting that a cluster of dipole line sources can be operated as a cloaking device. It should be noted that the result given in Fig. 1 is also workable for the other external sources such as a line source or a Gaussian beam.

 

Fig. 1 The order m_c and the corresponding number N of the active sources to realize cloak for an incident plane wave. The size of the device is R_d = 20λ, the size of the “quite region” is R_c = 2λ, they are marked in Fig. 2(a) and Fig. 2(c) by red solid line and white solid line, respectively. The active sources are arranged uniformly around a circle r = 10λ.

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Fig. 2 Cloaking an object with 3 higher order active sources of m′_c = 58 (a), (b) and 94 lower order active sources of m′_c = 1 (c), (d). Panels (a) and (c) correspond to the total electric field patterns and panels (b) and (d) correspond to the electric patterns of E_tot − E_ext. The incident TM Gaussian beam has the working wavelength λ, the waist radius 2λ, and the beam center located at (0,0). The cloaking objects are the perfect electric conductance (PEC) film with the length l = 5λ for (a), (b) and the length l′ = 10λ for (c), (d), respectively. The red solid circles and the white solid circles in panels (a) and (c) denote, respectively, the size of the cloak devices R_d = 20λ and the size of the “quiet region” R_c = 2λ.

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To exemplify the result shown in Fig. 1, we have compared the cloaking effects for two different cloaking devices. The first one consists of 3 higher order active sources with m′_c = 58 and the other one consists of 94 lower order active sources with m′_c = 1. The results are shown, respectively, in Figs. 2(a), 2(b) and Figs. 2(c), 2(d) where the red solid circles denote the boundary of the devices with the size R_d = 20λ and the white solid circles denote boundary of the “quiet region” with the size R_c = 2λ. It can be seen that the external incident TM Gaussian beam experiences no scattering after passing through the cloaking devices as indicated by the total field patterns in panels (a) and (c), resulting in a good cloaking effects. By observing the electric field patterns E_tot − E_ext shown in panels (b) and (d), we can explain the cloaking effect clearly. In the center region of the cloaking devices, the active sources create the field patterns of a Gaussian beam, but with a π phase difference compared to the external incident one so that they can cancel each other exactly, leading to the appearance of the “quiet region”. Compared to the high order active sources, the low order ones are easy to be implemented and convenient to operate in the experiment. Another negative aspect of the high order active device is the strong field around it, which can’t be ignored since it possibly cause the damage of the materials which make up of the active sources. While for the lower order case, the strong field only exists in the region close to the active sources. It should be pointed out that although the “quiet region” is only a circular area with the radius R_c = 2λ, outside the region the field is also very weak. Accordingly, the object with a larger size can be cloaked as can be seen in Fig. 2(a) and Fig. 2(b) where the object has a length l = 5λ. For the cloaking device made of lower order active sources, an even larger object can be cloaked as can be observed in Fig. 2(c) and Fig. 2(d) where the object has a length l′ = 10λ. As a result, it can be found out that the lower order active sources are more appropriate in designing the cloaking device. In what follows, all the simulations are to be performed with this dipole-source-based device.

Now let’s turn to examine the illusion effect created by the dipole-source-based device, which consists of 94 dipole line sources arranged uniformly on a circle with the radius r = 10λ. The object concerned is still a PEC film consisting of the nearly touched cylinders with the length 10λ, which is arranged in vertical and horizontal configuration as shown in Fig. 3. When the illusion device is switched off, the only EM wave incident on the object is the external source. In our case, the external source is a line source located at r₀ = (−14λ, 0), which radiates the TM wave with E_ext (k, r) = H₀[k|r − r₀|]. It is certain that when the PEC film is placed vertically the scattering cross section is larger so that a wider shadow region can be formed as shown in Fig. 3(a). While for the case when the PEC film is placed horizontally, only a small shadow region appears as can be observed in Fig. 3(b). The corresponding scattering fields are also presented, respectively, in Fig. 3(c) and Fig. 3(d) where we can find a remarkable difference. Then, we switch on the illusion device. It can be observed from Fig. 3(e) that for the vertical case the shadow region is turned into the smaller one as shown in Fig. 3(b). While for the horizontal case a large shadow region nearly identical to Fig. 3(a) appears as can be observed in Fig. 3(f). Therefore, with the illusion device we can turn one object into another one. To understand the illusion effect, we have presented the electric field patterns of E_tot − E_ext corresponding to Fig. 3(e) and Fig. 3(f). The results are shown in Fig. 3(g) and Fig. 3(h), respectively. It can be observed that in the area around the object the electric field patterns are nearly the same as that of the line source but with a π phase difference, which leads to the form of the “quiet region” inside the illusion device as shown in Fig. 3(e) and Fig. 3(f). Therefore, the illusion device shields the scattering effect from the object placed inside the “quiet region”. At the same time, outside the device (r > R_d) the active sources can create the same electric field pattern as that scattered from another object. This can be corroborated by comparing the electric field patterns in panels (g), (h) with those in panels (d), (c), respectively. Since the illusion device can be tuned dynamically by adjusting the coefficients q_nm, it is capable of transforming one object into arbitrary other ones as we want, which makes it very useful in the practical situations.

 

Fig. 3 The total electric field patterns for a PEC object of the length 10λ arranged vertically (a) and horizontally (b) under the radiation of a TM line source. Panels (c) and (d) are the corresponding scattering field patterns for the cases in (a) and (b), respectively. Panels (e) and (f) are the electric field patterns when the illusion devices are switched on. Panel (g) and (h) are the corresponding electric field patterns of E_tot − E_ext for the cases shown in panels (e) and (f), respectively.

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With a Gaussian beam as an external incident wave, we can demonstrate the illusion reflection and refraction in a clearer manner. Both the illusion device and the object to be considered are the same as those used in Fig. 3. Only the coefficients of each active dipole line source need to be adjusted accordingly. As we know, when a Gaussian beam is incident on a PEC film it will be totally reflected as can be observed from the simulation results shown in Fig. 4(a) and Fig. 4(b) where the white (yellow) solid arrows denote the incident (outgoing) Gaussian beam. If the illusion device is switched on, we can find that the total reflection to the left upward is turned into a total transmission to the right downward as can be seen in Fig. 4(c), which is actually the result shown in Fig. 4(b). This suggests that the PEC film with the vertical configuration behaves as the horizonal one. The illusion device used in Fig. 4(c) and Fig. 4(d) has the same function as that of an array of dielectric particles in a recent work [33]. Actually, with the active illusion device we can mimic the situation when the PEC film is placed in any orientation. In this way, the outgoing beam at any desired direction can be realized. Accordingly, the illusion device with this function is a beam rotator. If the illusion device in Fig. 4(b) is switched on, a similar transition from total reflection to total transmission occurs. By comparing the results shown in panels (a) and (b) with those in panels (d) and (c), we can find that the illusion device just transforms the reflection behavior of one object into that of another one. To understand our results, we present the corresponding electric field patterns of E_tot − E_ext in panels (e) and (f), respectively, for the cases shown in panels (c) and (d). It can be seen that inside the illusion device the active sources create a Gaussian beam like field patterns for both cases but with a π phase difference compared to the incident Gaussian beam. Therefore, a “quiet region” is formed, resulting in the cloak of the object. At the same time, the active sources can generate the outgoing Gaussian beams as those in Fig. 4(a) and Fig. 4(b). By adjusting the coefficients q_nm, the outgoing Gaussian beam at any other direction can be achieved.

 

Fig. 4 The electric field patterns for a TM Gaussian beam incident on a PEC film of the length 10λ placed vertically (a) and horizontally (b). Panels (c) and (d) correspond to the electric field patterns when the illusion device is switched on. Panels (e) and (f) are the electric field patterns of E_tot − E_ext that correspond to the cases in (c) and (d), respectively. The white solid arrows in panels (a)–(d) denote the direction of the incident Gaussian beam, while the yellow solid arrows in panels (a)–(d) denote the direction of the outgoing Gaussian beam. The positions of the active dipole line sources and the size of the illusion device are the same as those in Fig. 3.

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With the paradigms given above, we have demonstrated that the active-source-based cloaking (illusion) device is very flexible in controlling the cloaking (illusion) phenomena. It can avoid the loss issue occurring in the metamaterials based cloaking (illusion) device. Besides, the active-source-based cloaking (illusion) device can even overcome the drawback of narrow working frequency band inherent in the cloaking (illusion) device made of metamaterials, complementary media, or plasmonic media.

4. Conclusion

In summary, we have shown theoretically in present work that the optical cloaking (illusion) can be realized by appropriately arranging a cluster of active dipole line sources, which possibly makes the design realizable in the experiment. Our simulation suggests that with this active-dipole-line-source-based illusion device we can transform one object into another one. Besides, a beam rotator that capable of controlling the outgoing beam at any direction can also be implemented with the active illusion device. The flexibility and feasibility of our design makes the illusion device potentially applicable.