1. Introduction

Since the seminal paper by Pendry et al.[1] there has been a lot of interest in using metamaterilas for control of electromagnetic fields in order to make objects invisible. By mathematically investigating the freedom of design they outlined ways of how to cloak an object by guiding the light around it and so making it invisible for the observer. This was later confirmed by full-wave simulations [2] and then experimentally [3] at microwave frequencies.

In this letter a different approach is proposed for cloaking objects from electromagnetic fields in the near infrared, based on scattering optical elements (SOE). Similar to metamaterials, SOEs have shown a great ability to control the propagation of electromagnetic waves [4, 5]. By inverse design (ID) one can calculate the positions and shapes of an array of scattering objects so the multiple scattering of an incident wave is tailored, through the principles of superposition, into almost any asked-for topography after passing the SOE structure. Here, two different devices are proposed based on optimal control of scattering of waves provided by the SOE technology. Although these first SOE designs show a very limited functionality range including cloaking of only specific incident angles of a single frequency and polarized plane wave, they demonstrate the feasibility of cloaking using developed fabrication methods applied for the visible and near infrared wavelengths.

The devices presented here are computed under the limitations of a specific micromanipulation fabrication method[15], which consists of slicing the full structure into several 2D photonic plates prepared by a semiconductor nanofabrication technique. Rods with square shaped cross section are etched out from the plates, supported by a frame. Subsequently, these plates are assembled into the 3D structure by micromanipulation. An illustration of such a devise is shown in Fig. 1.

2. Inverse design tool

The ID tool [6] used to design the SOEs is an integration of a two dimensional (2D) multiple scattering theory (MST) electromagnetic solver with a global optimization method, the genetic algorithm (GA) [9].

The MST method is a semi analytical field solver. The method is based on the T-matrix and expanding the scattered steady-state field from each individual scatterer in series of natural Bessel functions. The number of functions used in the simulation process was truncated at seven since no change was observed when increasing the size of the expansion. The MST is in general a faster method than numerical methods if dealing with discrete scattering systems, such as the one presented here. In addition, the method solves the scattering problem with very high accuracy since it is based on analytical solution to Maxwell’s equation as well as analytical solutions to the boundary condition of the scatterers. These qualities, i.e. the speed and accuracy, makes the MST a very good candidate for SOE design. However, when dealing with non-circular scatterer the method is limited by Graf’s formula. To ensure the validity of Grafs formula in the MST the scatterers are forced to be separated a minimum distance; the boundary between the two materials can not be placed inside the outer radius surrounding the rod scatterer. In accordance, two rods in neighboring plates are never placed exactly atop of one another. For a general review of the method the reader is referred to Ref. [8] and for a complete description on how to calculate the T-matrix for freely formed scatterers to Ref. [7].

 

Fig. 1. A schematic view of a 5-layers SOE cloaking device. The arrows illustrates the path of the incident light passing the device, set by the electric field E_inc. The volume marked by the blue box at the center of the SOE structure outlines the cloaked area, where E = 0 is obtained. The white plane, placed at x_f, is the plane of observations where the transmitted field has regained it initial shape, i.e. E_inc.

Download Full Size | PDF

Using the MST field solver in a 2D environment, the electromagnetic scattering problem can be solved within a very short CPU time. This low cost of computational power permits the use of a more powerful optimization algorithms such as the GA. The GA has been proved to be effective in complex problem solving in photonics [10, 11, 12], though basically any search algorithm could be used in the inverse design process. To facilitate the optimization process the position of each scattering rod is determined by a fixed array of lattice sites (LS). The full SOE device is then coded by a string of binary bits where each bit codes the presence or absence of a rod at each LS. This restricted freedom of design has been shown sufficient for engineering of very complex devices, e.g. providing full control of spontaneous emission [5].

Finally, if the fabrication method would allow more free positioning of the bars an optimization based on evolutionary strategies or gradient-based searches would be a good alternative [13]. These methods adapt better to real parameter optimization problems. However, it has to be mentioned that to prove the supremacy of one optimization method over another is not a trivial problem. The No-Free Lunch theorem by Wolpert and Macready [14] states that for any algorithm, any elevated performance over one class of problems is exactly paid for in bad performance over another class. In other words, it does not exist no better nor worse optimization algorithm with respect to its average performance on all possible classes of problems. In addition, for many multi parameter optimization problems, as the one treated here, there is no way of checking whether the final solution is the global optimum. The only way one can assure this is to calculate all possible configurations.

3. Design set up

The dielectric material used in the fabrication process of the plates is set to gallium arsenide (GaAs) that at the wavelength of 1550nm has the dielectric constant ε= 11.4. Even though the fabrication method provides a high variety for the size of the rods, the rod cross section is set fixed to 400nm × 400nm and with a minimum separation of 900nm between neighboring rods in the same plate plate.

Here, the design process will include a TM-polarized plane wave propagating in the positive x-direction of the wavelength λ = 1550nm Since the MST field solver gives the steady state solution to the electromagnetic scattering problem the single frequency polarized light is the most simple design set-up and was limited by the effective calculation time. However, it is a straight forward operation to include multiple discretized frequencies as well as both polarization. Optimization over a frequency range has previously been done using this same design tool for transparent acoustic lenses [16] and optical cavities [5]. The time needed to simulate each design is proportional to the number of frequencies times the number of polarizations used to characterize the functionality. For this kind of designs a parallel computation is a necessity to complete the design within reasonable time frame.

A schematic view of the SOE cloaking device is illustrated in Fig. 1. The figure shows how the incident light is carefully guided around the central area and to later recover its original shape. However, due to the scattering of the frames of the dielectric plates used in the fabrication technique to support the bars in the SOE structure it is not possible to make a general devices independent of the incident angle of the light. The devices presented here will address a plane incident wave propagating in the normal direction to the plates as indicated in the figure.

A perfect cloaking device can be identified by two specific quantities; First the electromagnetic field is annulled within the cloaked area and second, the wave after passing the device has regained its original shape. A perfect plane wave propagationg in the positive x-direction can be uniquely identified by the Poynting vector parallel to the x-axis, i.e. S→_inc(x,y) = x̂ in all space. If the average amplitude of the electric field within the cloaked area is set by α and the x and y-component of the Poynting vector of the transmitted wave, by β and γ, respectively, the quality number of one SOE cloaking device can be calculated as,

where s→ is the binary string of parameters that codes the rod distribution of the SOE device and a, b and c are weight constants. Now, the higher the value of f(s→) the better the cloaking functionality of the device, i.e. (1) low α or low field amplitude in the cloaking area, (2) low β or small y-component, and (3) unity γ or conserved x-component of the Poynting vector of the transmitted wave. By using the GA to maximizing Eq. (1) it is possible to computer generate the asked for SOE cloaking device. However, since the device is categorized by three independent magnitudes it is important to properly scale each of these values so an asked-for overall quality of the design is obtained. In accordance the three weight constants, a, b, and c are included in Eq. (1). For example, if an optimal device is obtained after the maximization process with a high y-dependency of the Poyninting vector one needs to increase the value of b in order to magnify the pressure on β in Eq. (1).

4. SOE devices

The first design setup includes 13 dielectric plates. By implementing the symmetry of the system with respect to the x-axis the computational time can be decreased considerably. This first device is coded by 69 symmetric LS resulting in a total of 5.9 × 10²⁰different possible designs. The total size of this device measures 5.2μm × 9.4μm. The cloaking area is set to cover a square area at the origin of the system of the size 1.0² μm. The α and β magnitudes are calculated as the average value of the Poynting vector at x_f = 8.0μm over the line segment -4. 1μm to 4.1μm.

 

Fig. 2. 13 layers single symmetric cloaking device. The figure shows the electromagnetic field distribution for, (a) the optimized SOE structure, (b) the optimized structure cloaking a rod placed at the origin. The black squares show a cross section of the SOE device. The black line at the origin marks the cloaked area and the black dashed line the plane of observation

Download Full Size | PDF

The quality function f(s→) was implemented with a = 4, b = 3 and c = 8. After approximately 12h CPU time on a 2.6GHz Pentium IV processor the following three optimized values were obtained, α ₁ = 0.05, β ₁ = 0.13 and (|1 -γ ₁|) = 0.25. This device with a total f value of 5.7 is pictured in Fig. 2(a) with the plane wave scattering response. The cloaking area is outlined with a black line and the black dashed line marks the plane of observation, i.e. the coordinates for the calculation of the Poynting vector magnitudes, β and γ. Since the observation window is finite the shape of the passed wave outside this area is not included in the quality estimation of the device. As a result of this it is observed in the picture that the field outside the window, i.e. at y = +/ − 5μm, show high dissimilarity with the incident wave.

To verify the cloaking functionality of the device an extra scattering rod, of the size 400nm × 400nm, was placed inside the cloaking area. The simulated experiment is illustrated in Fig. 2(b). The figure shows an identical scattering response as for the initial design confirming the cloaking ability.

Since this design lack symmetry with respect to the y-axis it can only cloak objects in one direction with the observer at x_f = 8μm. It should be of interest to cloak an object from observers at both sides of the SOE device (±x_f). To achieve this a double symmetry condition was imposed for the design setup, including symmetry with respect to both axis, x = 0 and y = 0. In order to account for the lower freedom of design an additional 8 plates were included, resulting in a 21 plates device, or 53 double symmetric LS. The weight constants, as well as the cloaking area, were set identical to the earlier design. The optimization converged to a device identified by α ₂ = 0.01, β ₂ = 0.08, and (∣1 − γ ₂∣) = 0.4, resulting in a total f value of 4.22. Please notice that even if β ₂ < β ₁ the phase of the wave over the sensor is shifted π over the segment y = −1 to y = +1. This is a possible scenario since β is only measured by the conservation of the Pointing vector and does not include the actual phase of the wave. If the designer want to ensure a constant phase a fourth parameter has to be included in quality function. Moreover, this quality value indicates a worse performing device than the first design. However, one might conclude when looking at the final result, pictured in Fig. 3(a), that the quality is increased, with the argument that the plane wave is almost completely reconstructed on the right hand side of the structure (β ₂ < β ₁), however with somewhat lower amplitude (|1 − γ ₂| > |1 − γ ₁|).

If this is the opinion of the designer the weight constants should be tuned in accordance by increasing the value of b and decrease the value of c. Figure 3(b) shows the response of the device when a scattering object is cloaked. Once again the scattered field is unchanged with respect to the initial design in agreement with the cloaking ability.

 

Fig. 3. 21 layers double symmetric cloaking device. The figure shows the electromagnetic field distribution for, (a) the optimized SOE structure, (b) the optimized structure cloaking a rod placed at the origin. The black squares show a cross section of the SOE device. The black line at the origin marks the cloaked area and the black dashed line the plane of observation

Download Full Size | PDF

The symmetry restriction in the design process can of course be further increased in this same manner including cloaking for multiple incident angles. Though, the higher fold symmetry the smaller the freedom of design. A four fold symmetry would including cloaking for four angles, 0°, 90°, 180° and 270°, and the design space would again be halved to a fourth of the initial design space, now including four symmetry axis, x = 0, y = 0, x = y and x = −y. When decreasing the freedom of design in this manner the simple approach using fixed LS would probably not be sufficient to achieve a good solution. To increase the freedom of design when dealing with high symmetry device design, the position of the scatterers should be chosen freely as well as the diameter. However, please notice that this impose using a more complete fabrication method such as direct laser writing [17].

5. Conclusions

In conclusion, initial results on shadowing specific areas from electromagnetic wave in the near infrared have been presented using the generic technology provided by scattering optical elements. Since these results only include low symmetry they do not compete with metamaterial for microwaves, however they make an initial step toward more complex and general devices for shorter wavelengths. Finally, the university of the inverse design approach makes it straight forward to address visible light and even acoustical wave for cloaking and local isolation of sound.

This study was performed through Special Coordination Funds for Promoting Science and Technology from the MEXT, Japan. The author would like to thank Dr. Hideki T. Miyazaki for discussions and Dr. D. Hill for suggesting the problem.