1. Introduction

Illusionary effects like mirages have fascinated people for centuries and have inspired myths, novels, and films. With the development of metamaterials [1–4] and transformation optics (TO) [5–26], more fascinating illusionary effects have been proposed, including invisibility cloaking [12], transforming the images of one object into another [13, 14], opening virtual holes in a wall [13, 15], superscattering and gateway [16–22], super absorbers [24], illusion of shifted sources [25], etc. The creation of such intriguing illusionary effects requires materials with unusual properties such as negative index or anisotropy in both permittivity and permeability, which usually can only be realized by metamaterials [1–4] with the units in the deep subwavelength scale. Due to the difficulty in realizing metamaterials with the ideal TO parameters, “reduced parameters” that sacrifice the property of omnidirectional impedance matching have been generally adopted [7, 19–21].

Very recently, by using photonic crystals (PhCs) with the units at the wavelength scale, ultratransparency media with omnidirectional impedance matching and the ability of forming aberration-free virtual images have been realized, which provides another candidate materials for TO devices [27]. Interestingly, the equal frequency contours (EFCs) of such ultratransparent PhCs are not centered the Brillouin zone center, but shifted towards the $X$ point. In other words, such ultratransparent media are nonlocal or spatially dispersive, beyond the description of local TO media. Recently, we find that such ultratransparency effect can be achieved for both polarizations in a broad range of frequencies by using one-dimensional (1D) dielectric PhCs [27, 28] for a relatively narrow range of incident angles.

In this work, we propose and demonstrate the application of such 1D ultratransparent PhCs with wide-angle impedance matching and shifted elliptical EFCs in TO, and more specifically, in the creation of optical illusions, which is also known as illusion optics. We demonstrate two examples: one creates the illusion of a different-sized scatterer, and the other creates the illusion of a shifted source with an opposite phase. We show that the shift of the EFCs of the ultratransparent PhCs does not affect the refractive behaviors, but provides a new degree of freedom for additional phase modulation. With such 1D ultratransparent PhCs, TO and illusion optics become more feasible in the optical frequency regime.

2. TO media and ultratransparent PhCs

To begin with, we consider a TO medium obtained by stretching the coordinate along the $x$ direction in the background medium of air. For transverse electric (TE) polarization with electric fields polarized in the $z$ direction, the parameters of the TO medium are written as [23],

Thus, the dispersion of the TO medium, i.e. $\frac{k_x^2}{{\mu }_y}+\frac{k_y^2}{{\mu }_x}={\epsilon }_z{k}_0^2$, can be rewritten as $s^2{k}_x^2+k_y^2=k_0^2$, indicating that the EFC is an ellipse with the same height as the EFC of air in the $k_y$ direction, as shown in Fig. 1(a). Here, $k_x$ and $k_y$ are the $x$ and $y$ components of wave vectors in the TO medium, respectively. $k_0$ is the wave number in air. As an example, the stretching ratio is set to be larger than unity, i.e., $s>1$. Thus, the EFC of the TO medium (blue lines) is slimmer than the EFC of air (gray lines) in Fig. 1(a). Thus, the coordinate mesh is looser in the $x$ direction in the TO medium region, as illustrated in Fig. 1(a).

 

Fig. 1 (a) Upper inset: the EFCs of air (gray solid lines) and TO medium obtained by stretching air in the $x$ direction (blue solid lines). Lower inset: the scheme of forming the TO medium slab (with loose mesh) in the background medium of air (with dense mesh). (b) Upper: the EFC of the 1D ultratransparent PhCs (red solid lines), which has the same shape of the EFC of the TO media (blue dashed lines), but shifted in the $k_x$ direction. Lower inset: the 1D ultratransparent PhCs are used to replace the TO medium slab in (a).

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If a “shifted” EFC of the same shape as that of the TO medium can be obtained in a 1D ultratransparent PhC [see Fig. 1(b)], then the refraction of a slab of such 1D ultratransparent PhC would be the same as the refraction of a slab of TO media, because the shape of the EFC determines the refractive behaviors. The wide-angle impedance matching of ultratransparent PhCs ensures almost zero reflection and total transmission. Therefore, the functionality of such a slab of 1D ultratransparent PhC can be applied to replace a slab of the TO media. The only difference is that there may exist additional phases in the ultratransparent PhCs due to the shift of the EFC.

Through a trial-and-error iterative optimal algorithm, wide-angle impedance-matched 1D ultratransparent PhCs with the same shaped EFCs as that of the TO media can be obtained [28]. The design of such 1D ultratransparent PhC turns out to be surprisingly easy, due to the large degrees of freedoms in engineering the nonlocal effective parameters in PhCs [27]. In Fig. 2, we show a specific example of 1D ultratransparent PhCs. The designed PhC is composed of symmetric unit cells, which consist of dielectric components I and II with a lattice constant of $a$ [see Fig. 2(a)]. The relative permittivities $\epsilon$ and the thicknesses $d$ of the two components are presented in Fig. 2(a). In Fig. 2(b), by using software COMSOL Multiphysics based on finite-element methods, we plot the EFC of the PhC (red solid lines) at the normalized frequency $fa/c=0.402$ for TE polarization. $f$ and $c$ are the eigen-frequency and light velocity in vacuum, respectively. Evidently, the EFC can be approximately regarded as a part of an ellipse located at the X point. Compared with the EFC of the TO medium with $\left\{{\epsilon }_z,{\mu }_x,{\mu }_y\right\}=\left\{0.5,2,0.5\right\}$ at the same frequency [blue dashed lines in Fig. 2(b)], we see that the EFC of the PhC has the same height in the $k_y$ direction, which satisfies the requirement proposed above. In addition, in Fig. 2(c), we plot the transmittance through the 1D PhC slab composed of $n$ unit cells as the function of the incident angle at $fa/c=0.402$. We observe almost perfect transmission under the incident angle of 70° irrespective of the number of unit cells, indicating a wide-angle impedance matching effect. Thus, such a 1D ultratransparent PhC can approximately work as the TO medium with $\left\{{\epsilon }_z,{\mu }_x,{\mu }_y\right\}=\left\{0.5,2,0.5\right\}$, whose incident angle-dependent transmittance is denoted by the purple lines in Fig. 2(c).

 

Fig. 2 (a) Schematic graph and related parameters of the unit cell of the designed 1D ultratransparent PhC. (b) The EFCs of the PhC in the second band for TE polarization. The red solid line and blue dashed line denote the EFCs of the PhC and the TO medium at the normalized frequency $fa/c=0.402$, respectively. (c) Incident angle-dependent transmittance for waves propagating through the PhC slab composed of $n$ ( = 4, 5, 6, 15) unit cells at $fa/c=0.402$. [(d), (e)] The snapshot of electric fields for wave propagating through a TO medium slab (upper inset) and a PhC slab (lower inset) under an incident angle of 45°. The thickness of the TO medium slab is (d) $5a$, (e) $6a$. And the number of unit cells of the PhC slab is (d) 5, (e) 6.

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To further verify the above results and show the effect of the shift of the EFC, with the software COMSOL Multiphysics, we simulate the wave propagation through the TO medium slab with a thickness of $5a$ (upper inset) and a PhC slab with 5 unit cells (lower inset) in Fig. 2(d). The waves are incident from the left port boundary, and the upper and lower boundaries are set to be periodic boundary condition in the simulation setup. Simulation results show perfect transmission under the incident angle of 45°. Interestingly, we notice a $\pi$ phase difference in the transmission waves, which actually is caused by the shift of the PhC’s EFC. As the center of the PhC’s EFC is located at the $X$ point, an addition $\pi$ phase difference will be imposed on waves when passing through each unit cell. As a consequence, when the total number of the unit cells in the PhC slab is odd, a $\pi$ phase difference will be seen when compared with the TO medium. On the other hand, if the total number is even, there will be no phase difference, which is confirmed in the simulation results in Fig. 2(e). In Fig. 2(e), the thickness of the TO medium slab is $6a$ (upper inset), and the number of unit cells of the PhC slab is 6 (lower inset). Comparing the transmitted waves under the incident angle of 45°, we can see identical phases, in contrary to the case with an odd number of unit cells shown in Fig. 2(d).

3. Illusion devices via ultratransparent PhCs

Here, with the 1D ultratransparent PhCs, we can realize applications in illusion optics. As an example, we consider a cylindrical model in Fig. 3. Here, we start with an air cylinder with a radius of $R_1$, and assume a virtual core with a radius of $R_2$, as illustrated in Fig. 3(a). Then, we change the radius of the core to be $R_3$ through transforming the coordinate in the radial direction based on the method of TO, as illustrated in Fig. 3(b). Thus, the parameters of the shell will be gradient functions of the radius [8, 17], that is, ${\mu }_r=1/{\mu }_{\theta }=1+\frac{R_1}{r}\frac{R_2-R_3}{R_1-R_2}$ and ${\epsilon }_z=\frac{R_1-R_2}{r}\frac{r\left(R_1-R_2\right)+R_1\left(R_2-R_3\right)}{{\left(R_1-R_3\right)}^2}$ with $R_1\ge r\ge R_3$. And the parameters of the core are ${\mu }_r={\mu }_{\theta }=1$ and ${\epsilon }_z={\left(R_2/R_3\right)}^2$. Next, we discretize the gradient shell into four uniform layers, as denoted by A, B, C and D in Fig. 3(c). The parameters of the four layers are determined roughly by the arithmetic mean of the graded parameters, as presented in Fig. 3(e). Finally, we design four corresponding 1D ultratransparent PhCs to replace the four uniform layers, as illustrated in Fig. 3(d). The relative permittivities and thickness of the dielectric components I and II of the four 1D PhCs are presented in Fig. 3(e). All the PhCs have the same lattice constant of $a$. The numbers of unit cells in layers A, B, C and D are 4, 2, 2 and 1, respectively. The working frequency is chosen as $fa/c=0.3$.

 

Fig. 3 Schematic graph of coordinate transformation from (a) an air cylinder to [(b)-(d)] core-shell structured cylinders composed of (b) gradient TO media, (c) discretized TO media and (d) 1D ultratransparent PhCs. (e) The parameters of the discretized layers A, B, C and D in (c), and parameters of the unit cells of the PhCs in (d).

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In the first example, we set the core to be a perfect electric conductor (PEC) to achieve illusionary scattering effect. For comparison, in Fig. 4(a), we first simulate the wave scattering by a PEC cylinder with a radius of $R=4a$ under the illumination of a TE polarized Gaussian beam in the air background. Then, in Fig. 4(b), we coat the PEC cylinder with the ideally gradient TO medium shell, as that in Fig. 3(b). The simulation results show that the scattering pattern is changed into that of a relative larger cylinder, as shown in Fig. 4(d). In Fig. 4(c), the 1D ultratransparent PhCs are used to replace the TO media, showing almost the same scattering pattern. As a result, any observer will see an illusion of a larger cylinder from scattered waves.

 

Fig. 4 Snapshots of electric fields for a TE polarized Gaussian beam scattered by (a) a PEC cylinder with $R=4a$, (b) a PEC cylinder with $R=4a$ coated by the TO medium shell, (c) a PEC cylinder with $R=4a$ coated by the 1D PhC shell, and (d) a PEC cylinder with $R=6.5a$.

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Such an illusionary effect can be understood from the theory of TO. Since the TO medium shell is obtained by compressing an air shell in the radial direction by varying the inner radius from $6.5a$ to $4a$, the model consisting of a PEC core with $R=4a$ and the TO medium shell is equivalent to the model consisting of a PEC core with $R=6.5a$ and an air shell. As the background medium is air, the wave scattering in Figs. 4(b) and 4(c) will be the same as the wave scattering by a PEC cylinder with $R=6.5a$, as confirmed by the simulation results in Fig. 4(d).

In the second example, we place a TE polarized monopolar point source inside the core to show the source illusion effect. The relative permittivity of the dielectric core is 2.64. In Fig. 5(a), the point source has an offset distance $d^{\text{'}}$ ($=1.6a$) from the center of the core, and the shell is composed of the 1D ultratransparent PhCs proposed above. From the field distribution, we see relatively standard cylindrical radiated waves in air due to the wide-angle impedance matching effect of the PhC shell. Moreover, one interesting phenomenon is that by back-tracking the radiation waves, the point source should have an offset distance of $d^{\text{'}\text{'}}=2.6a$, as displayed in Fig. 5(b). This means that the PhC shell enables a real source to resemble an illusion source off its original position. By using the TO theory, the relationship of the offset distance of the real and illusion source can be derived as,

 

Fig. 5 Snapshots of electric fields radiated by a TE polarized monopolar point source placed (a) in the core of a PhC shell, (b) in the background medium of air. The offset distance of the source from the center is (a) $d^{\text{'}}=1.6a$, (b) $d^{\text{'}\text{'}}=2.6a$. The sources in (a) and (b) have opposite phases.

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Equation (2) reveals that the offset distance of the illusion source actually can be further controlled by varying the inner radius of the PhC shell. Another interesting observation is that the phase of the illusion source is opposite to that of the real source. This is induced by a total odd number of the unit cells of the ultratransparent PhCs. We note that such a source illusion effect has been demonstrated in [25] by using anisotropic negative-index materials. Interestingly, here only dielectric materials are required, which largely simplifies the design and fabrication.

Finally, it is worth noting the performances of the illusion devices consisting of ultratransparent PhCs can be further improved by making more discretized layers to achieve quasi-continuous parameters. The number of discretized layers is dependent on the size of the device. In fact, the above results reveal that four discretized layers are sufficient to obtain relatively good performances for most illusions devices.

4. Conclusions

In summary, we have demonstrated that 1D ultratransparent PhCs with shifted elliptical EFC and wide-angle impedance matching can approximately work as the TO media to realize optical illusionary effects. The shift of the EFC does not affect the refractive behaviors, but provide a new degree of freedom in phase modulation. Ultratransparent PhCs may provide a feasible platform for realization of more illusionary effects, especially in optical frequencies. We believe that there are diverse experimental methods to realize such illusionary effects. In the microwave frequency regime, a single material index with various filling ratios of holes is probably the simplest approach. In the optical frequency regime, the simplest method may be the thin-film fabrication, where multiple materials can be deposited alternatively to form the multiple film structure.