1. Introduction

The dream of light-behavior remote control has haunted human being for thousands of years and there are legends with such imagination. In reality, although different methods of light-behavior controlling have been successfully used, the technique of light-behavior remote control still does not come into being yet. One basic method to control light behavior in practice is using different cavities which are widely applied in electromagnetic/optical devices, such as resonators, modulators, switches and lasers [1]. Recently, a totally new direction of modern photonics/optics, i.e., the “transformation optics”, is setup by the research of invisibility cloak [2, 3], which is designed from geometry transformation [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. After that, the devices like super scatter [19] and anti-cloak [20] etc [21, 22, 23, 24], are designed based on transformation optics to change light propagation. Further more, a new concept of “complementary medium” [25, 26, 27] is generated in transformation optics, i.e. a designed medium which can cancel the scattering effect of another existing medium in distance so that the scattering effect or the reflecting effect of “the complementary medium and the existing medium” is zero. However, the light-behavior remote control by transformation optics has not yet been put forward and got further studied. In this paper, combining the cavity tuning methods and the theory of transformation optics, we propose a general light-behavior remote control method which is using a transformation optical device to remotely modify the cavity. The essential idea is that the transformation optical device is designed as the complementary medium of a part of optical cavity, e.g. the cavity mirror, then its reflection can be remotely tuned (even to zero-reflection), hence the optical property of a cavity could be totally changed as expected and the light-behavior remote control can be realized. In this study, the remote cloak [21] is widely used to tune the cavity, but we note that, besides the remote cloak, other kinds of devices designed from more general folded geometry transformation [28] could be utilized for light-behavior remote control as well.

The method of light-behavior remote control in this work, has some advantages. First, since the cavity is not modulated directly, there is no physical change of the cavity and no damage caused. Therefore the cavity could be conveniently used repeatedly. Second, the modulating range could be larger, for example, it can modulate the quality factor Q of a cavity in several orders (even to almost zero) or tune the resonant frequency in a large range. Third, since the cavity is modulated at a distance and the quality factor Q could be tuned to a very low value, even to almost zero so that there seems to be no cavity at all, both the controller and the cavity could be well hidden.

To demonstrate that our method works, three basic schemes by tuning the cavity at a distance with a remote cloak are showed. The numerical simulations are in finite-difference time-domain (FDTD) method and finite element method (FEM). The novel phenomena, such as “remotely enhancing the brightness of a cavity”, “hiding a laser which can be switched on remotely at anytime”, and “changing resonant-modes of a cavity in a large frequency range”, are shown to be theoretically possible.

2. The physical design and basic schemes

In this section, we will explain the basic design and three schemes of light-behavior remote control. Generally, for a cavity, the essential parameters are the quality factor Q and the resonant frequency ω ₀. We will show that both Q and ω ₀ could be remotely tuned. Our discussions are based on the Fabry-Perot (FP) cavity, since it is the simplest and most typical cavity which is widely used in optical/photonic devices. As the FP cavity shown in Fig. 1(a), there are two mirrors S ₁ and S ₂ at both sides of the cavity whose reflection coefficients are r ₁ and r ₂, respectively, and between two mirrors there is a kind filling medium in which the wave vector is k. So the cavity quality factor Q is determined by the value of r ₁ r ₂ and the resonance frequency ω ₀ is determined by kd = nπ where n is an integer number.

If we can tune Q and ω ₀ of the FP cavity at a distance by changing the key parameters r_i (i=1,2) and kd with the remote cloak, the light-behavior remote control could be realized. Although this design is a general one, our model is a two-dimensional (2D) one and our study is focusing on the E-polarization modes in which only the μ_r, μ_θ, and ε_z are considered, as shown in Fig. 1. For simplicity, r ₁ is kept unchanged in our study, only r ₂ and kd are tuned. For first two schemes in (Fig. 1(a) and (b)), the S ₃ in the remote cloaks are designed as the complementary medium of the right mirror S ₂, then the mirror can be totally canceled for optical effects. In other words, the reflection coefficient r ₂ can be tuned to zero, so the quality factor Q can be tuned considerably. Actually, with different remote cloak designs, Q can be tuned from the original value to zero, since S ₃ can be designed to cancel only part of mirror S ₂, then “physically effective thickness” of the mirror can be changed from the original thickness to zero. The difference between Fig. 1(a) and (b) is that Fig. 1(a) is with absorptive medium in the FP cavity, while Fig. 1(b) is with gain medium. In scheme 3 (Fig. 1(c)), the FP cavity which is made of a dielectric slab and S ₃ is designed to cancel part of the slab, then the effective optical thickness kd of cavity can be tuned and the resonant frequency ω ₀ is changed. The numerical simulation and detailed study will be showed in next section.

The remote cloaks in the three schemes are similar to the structure in Ref [21]. A remote cloak is composed of two parts which are showed in Fig. 1(a), the part I is the core area and the part II is the cloaking shell, where a and b are the inner and the outer radii of the cloaking shell, respectively, and c is the effective radius of the remote cloak. In scheme 1 and scheme 2, S ₃ inside the cloaking shell is designed as the complementary medium of mirror S ₂, so that r ₂ of S ₂ could be changed. In scheme 3, S ₃ is designed to counteract the part D _Δ of the dielectric layer D_d, then the resonant frequency of D_d will be shifted. For these effects, we choose the permeability and the permittivity parameters as: μ_r,I = 1, μ_θ,I = 1, and ${\epsilon }_{z,I}={\left(\frac{c}{a}\right)}^2$ in the core area, ${\mu }_{r,\mathrm{II}}=\frac{r-a}{r}+\frac{c}{r}\frac{b-a}{b-c},{\mu }_{\theta ,\mathrm{II}}=\frac{1}{{\mu }_{r,\mathrm{II}}},\mathrm{and}{\epsilon }_{z,\mathrm{II}}={\left(\frac{b-c}{b-a}\right)}^2{\mu }_{r,\mathrm{II}}$ in the cloaking shell region, and μ _r,S3 = μ_r,II μ_r,i, μ_θ,IIμ_θ,i, and ε _z,S3 = ε_z,IIε_z,i for S ₃, where μ_r,i, μ_θ,i and ε_z,i are the permeability and the permittivity of the part of FP cavity to be counteracted.

 

Fig. 1. Schemes of light-behavior remote control. (a) Scheme 1: the remote modification of the output energy current from an absorptive cavity. (b) Scheme 2: the remote control of lasing behavior. (c) Scheme 3: the remote tuning of the resonant frequency.

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3. Numerical simulations and discussions

To demonstrate the effect of our designs of light-behavior remote control, we have done the numerical simulations of the three schemes.

 

Fig. 2. The remote modification of the output energy current from an absorptive cavity. (a)Electric field of an absorptive cavity. (b)Electric field of an absorptive cavity and a remote cloak. (c)lnJ .vs. ε″ curves. J is the output energy current. Case 1: only an absorptive FP cavity. Case 2: an absorptive FP cavity and a remote cloak.

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In scheme 1, we illustrate that how to remotely tune the output energy current (the brightness) from an absorptive FP cavity. For the FP cavity in Fig. 2(a), S ₁ is a highly-reflected metal mirror (r ₁ ~ 1), while S ₂ is a dielectric mirror (ε _z,s2 > 0) whose effect can be modified by a remote cloak (with the same dispersive model in Ref [10]). Between two mirrors, there is the absorptive medium, whose permittivity is ε = ε′ + iε″, where ε′ = 1.01 and ε″ = 0.1. A linear light source is set at the middle position of cavity and the light emits from S ₂. And we detect the output energy current on the boundary B ₁. Without the remote cloak, the field by FDTD simulation is also shown in Fig. 2(a), which demonstrates that the light inside FP cavity is reflected by two mirrors and forms a resonant mode. Only a small part of light energy can emit from the mirror S ₂ since the absorption in the cavity. Next, we set a remote cloak on the right side to the FP cavity to modify the reflection coefficient r ₂ of mirror S ₂, as shown in Fig. 2(b). Since S ₃ is the complementary medium of the mirror S ₂, the effective thickness of S ₂ can be reduced. When S ₃ is design to exactly cancel the mirror S ₂ (the effective thickness of S ₂ is zero), the light from the source inside the cavity propagates rightward like in a free space, as depicted by the field shown in Fig. 2(b). Since the light is not reflected between two mirrors anymore, much less energy is absorbed, so that the output energy current could be larger than that of the original FP cavity.

We have also done the quantitative study of this effect. In Fig. 2(c), the output energy currents .vs. imaginary part of dielectric constant ε″ are compared for two cases, case 1 with only a FP cavity, case 2 with a FP cavity and a remote cloak. Since the output energy current J is proportional to the decay factor exp(-α) where α ∝ Qε″, the brightness of a cavity will exponentially depends on the quality factor Q and the material dissipation ε″. The numerical results of lnJ .vs. ε″ in Fig. 2 clearly show such dependence, and the lnJ of the case 2 with smaller Q since mirror S ₂ is cloaked is obviously larger (lnJ still decreases since the electromagnetic wave travels through the absorptive medium).

 

Fig. 3. The remote control of lasing behavior. (a) Electric field of a laser. (b) Electric field of a laser with a remote cloak on the right side. (c) lgJ .vs. t curves. J is the output energy current, and T is the period of the light.

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In scheme 2, we demonstrate how to remotely control the lasing behaviors by the remote cloak to modify the mirror S ₂. The setup of FP cavity in Fig. 3 is the same as scheme 1, but with gain material inside, which is simulated by four-level atomic model [29]. To simulate the spontaneous emission, some quasi-monochromatic point sources are set randomly inside the cavity. According the laser theory, when the gain is less than the threshold, there is only amplified spontaneous emission, but if the gain is over the threshold, the lasing will appear, which is shown in Fig. 3(a). The lasing threshold is proportional to the inverse of quality factor 1/Q. In Fig. 3(b), the remote cloak which can exactly cancel the mirror S ₂ is added to the right side of the laser cavity. Since the light are not reflected by S ₂ anymore and the Q is very small, the threshold becomes so high that the lasing mode is not excited, and only the amplified spontaneous emitting field is observed. To show the difference, we also show the output energy current versus time for the two cases in Fig. 3(c). The blue dash line depicts the lasing process in which J increases evidently (stable at about t/T = 2500), while the red solid line shows a constant low output energy for the amplified spontaneous emission. Our scheme 2 is not only a way to remotely switch off or on a laser, but also a good way to hide a prepared laser. If we cancel both mirrors S ₁ and S ₂ of our model by two complementary mirrors in cloaks, then observers at far away can not find any difference and such concealing could be a plot for future science fictions.

Besides the remote tuning of a mirror of FP cavity in scheme 1 and scheme 2, we also can tune the optical path kd to shift the resonant frequency, which is shown in scheme 3. As depicted in Fig. 1(c), the FP cavity in scheme 3 is a little different, only a thick dielectric layer, for which two mirrors are the two interfaces between the dielectric and vacuum. If a remote cloak is added on the right side of a dielectric, then the “effective optical path” of the cavity becomes kd"′, where d′ = d - Δ. Hence the resonant condition becomes kd′ = nπ. To demonstrate this effect, we calculate transmission spectrum in two cases by FEM, as shown in Fig. 4(a). In case 1, the spectrum of the original dielectric layer D_d is shown by the blue line marked with stars. We can see that there are three original resonant modes. In case 2, the spectrum of the FP cavity with a remote cloak on the right side is shown by the red dash line marked with circles. From the spectrum, we can observe that the resonant modes shift. Numerically, we also have calculated the spectrum of a dielectric layer with thickness d - Δ, shown by black line in Fig. 4(a). We can see that these resonant modes of a shorter cavity are exactly the same as case 2, just as we expected. In Fig. 4(a), the largest shift range is about 15%. In fact, by this method the resonant frequency can be tuned from a small shift to a very large one with different cloak design.

 

Fig. 4. (a) The remote tuning of the resonant frequency. Case 1: spectrum (J .vs. ω curves, J is the output energy current.) of cavity D_d. Case 2: spectrum of cavity D_d and a remote cloak. The thickness of the shorter cavity is d - Δ. (b) Light-behavior remote control of PhCs. The scatters (the yellow region) change the optical path (contribution from the region marked with “x”) in PhCs (the blue array).

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What’s more, we can extend light-behavior remote control method to remotely tune some complex cavities. For example, we can tune 1D photonic crystals (PhCs) by a transformation-optics device which is similar in Ref. [30]. The scheme is depicted in Fig. 4(b). By changing the effective optical path of every layer by transformation optical device, the photonic bandgap structure can be tuned in a wide range without any physical change PhCs. Similarly, the defect mode inside PhCs can also be tuned or generated by the transformation optical devices.

4. Conclusion

In conclusion, the light-behavior remote control method based on the transformation optics is proposed in this work. The tune of the quality factor Q and the resonant frequency ω ₀ of the cavities in wide ranges are demonstrated. By modifying these important characters of cavities, the light behavior can be remotely controlled without any physical change or damage to the cavities. In this work, we present three schemes, i.e. the output energy current of a absorptive cavity, the lasing behavior of a cavity with gain, and the cavity resonant frequency or the photonic band-gap of PhCs could be controlled by transformation optical devices. This work has proposed a new way for the application of transformation optical devices. With fabrication breakthrough of meta-material and transformation optical devices, we believe that the light-behavior remote control will be widely used, not only in optical/photonic devices, but also in electromagnetic devices, such as radar or antenna etc.