## Abstract

Invisibility carpet cloaks are usually used to hide an object beneath carpet. In this paper we propose and demonstrate a carpet filter to hide objects and create illusions above the filter by using a Fourier optics method. Instead of using transformation optics, we get electromagnetic parameters of the filter by optical transfer functions, which play the role of modulating the propagation of the scattering angular spectrum directly from an object above the filter. By further adding a functional layer onto the filter, we can even camouflage the object so that it appears to be a different object. The analytical results are confirmed by numerical simulations. Our method is completely different from the current coordinate transfer method and may provide another point of view to more clearly understand the mechanism of invisibility cloaks.

© 2010 OSA

## 1. Introduction

Recently, transformation optics-based invisibility cloaks [1–3] have attracted increasing interest. The currently proposed invisibility cloaks can be sorted into shell cloaks [4–16] and carpet cloaks [17–20]. The first cloaks can either bend electromagnetic waves around the cloak shells to avoid the waves touching the inside objects [4–13] or penetrate through the cloak shells to impinge upon the objects [14,15]. To overcome the limitation that objects to be hidden are generally enveloped by cloak shells, Lai *et al.* have proposed and demonstrated a way to hide an object outside of cloak shells and furthermore to create an optical illusion of changing one object into another object [21,22] by using negative index complementary media [23,24]. Carpet cloaks are used to hide objects beneath the carpet (exactly, under a curved mirror). The reflected light from the curved mirror is altered by the carpet so that it looks as if it is from a flat reflection-conducting sheet [17]. As a result, the objects under the curved mirror cannot be impinged upon by external light and hence are invisible. Invisibility carpet cloaks have been experimentally demonstrated at microwave frequencies [18] and infrared wavelengths [19,20], respectively. So far all of the objects to be concealed, however, are covered by carpet cloaks, and furthermore, electromagnetic parameters of all current invisibility cloaks are obtained by coordinate transformation. Here, we present a lossless carpet cloak to hide an object that is placed above the carpet. The carpet can be on a flat mirror or an absorber. The scattered electromagnetic waves by the object and, further, by the artificially constructed carpet filter under the object can be cancelled completely. Therefore, for a mirror substrate you see a reflection of yourself. For an absorber, you just see no reflection––just complete darkness. Both can serve to hide an object. Furthermore, the filter can also create illusions that camouflage an object above it so that it looks like a completely different object. Analytical results are confirmed by finite element method simulations [13].

## 2. Theoretical analysis

We begin our analysis from the angular spectrum (AS) theory of Fourier optics [25–27], which has been successfully employed for analyzing electromagnetic wave propagation in both near-field and far-field regions within linear systems [28]. As shown in Fig. 1(a)
, the AS of an object at a $y=0$ plane in air illuminated by a light with wavelength *λ* can be read as $A({f}_{x},{f}_{z},0)=F[U(x,z,0)]$, where $U(x,z,0)$ is the complex amplitude of the object, *F* represents the Fourier transform (FT) operator, and ${f}_{x}$ and ${f}_{z}$ are the spatial frequencies of the object in the *x* and *z* directions, respectively. After propagating to a $y=l$ plane in air, the AS becomes $A({f}_{x},{f}_{z},l)=A({f}_{x},{f}_{z},0)\mathrm{exp}[j(2\pi /\lambda )l\sqrt{1-{\lambda}^{2}({f}_{x}{}^{2}+{f}_{z}{}^{2})}]$, where $l>0$ means the transmission ${A}_{t}({f}_{x},{f}_{z},l)$, while $l<0$ means the reflection ${A}_{s}({f}_{x},{f}_{z},l)$. For mathematical convenience, we abbreviate the above formula as $A(l)=A(0)H(l)$, where $A(0)$ and $A(l)$ are the ASs of the object at the $y=0$ and $y=l$ planes, respectively, and $H(l)=\mathrm{exp}[j(2\pi /\lambda )l\sqrt{1-{\lambda}^{2}({f}_{x}{}^{2}+{f}_{z}{}^{2})}]$ is the transfer function (TF) of the $y=l$ thick air layer along the *y* direction. For the observing plane $y=d\text{'}$($<0$), which is generally above the object, the AS directly scattered by the object can be read as ${A}_{s}(d\text{'})=A(0)H(d\text{'})$. So, the complex amplitude $U(d\text{'})$ of the object at the observing plane (image) can be obtained by the inverse FT (denoted by ${F}^{-1}$) to the total AS ${A}_{s}(d\text{'})$ as $U(d\text{'})={F}^{-1}[{A}_{s}(d\text{'})]$.

From the above analysis we see that the AS of the image is determined exactly by the TF of an optical system. If the scattering AS ${A}_{s}(d\text{'})$ of an object can be completely compensated, then the object is invisible. To prevent the object $O({\epsilon}_{obj},{\mu}_{obj})$ with AS $A(0)$ above the carpet [Fig. 1(b), denoted as a star] from being visible and where ${\epsilon}_{obj}$ and ${\mu}_{obj}$ are the electromagnetic parameters of the object, we can construct a carpet filter that plays two roles consisting of two parts [Fig. 1(a), gray region divided by a dashed line)] between the object and a mirror [Fig. 1(a), hatched region]. One role is to achieve impedance matching with the object, which makes the scattering AS directly from the object destructively interfere with that from the carpet so as to cancel the reflection completely [29]. The second role is to synthesize the TF $H={A}_{t}^{*}(0)$ to compensate for the transmission AS ${A}_{t}(0)$ of the object to make ${A}_{t}({f}_{x},{f}_{z})H({f}_{x},{f}_{z})=1$ before reaching the total reflection mirror, where superscript ***** means the conjugation operation. As a result, at the observing plane $y=d\text{'}$, one can only see the surface of the reflector under the carpet, implying that the object above the carpet is invisible [Fig. 1(b)].

Next, instead of using the coordinate transformation method, we present a Fourier optics approach to get the electromagnetic parameters of such filters. Suppose that the refractive index of an *l* thick medium is *n*; then, its TF can be read as $H(l)=\mathrm{exp}[j(2\pi /\lambda )(nl)\sqrt{1-{(\lambda /n)}^{2}({f}_{x}{}^{2}+{f}_{z}{}^{2})}]$, where *n* can be a scalar or a tensor. By symmetrically putting a medium layer with thickness *l* and refractive index $-n$ in succession, we can get its TF to be $H\text{'}(l)=\mathrm{exp}[j(2\pi /\lambda )(-nl)\sqrt{1-{(\lambda /n)}^{2}({f}_{x}{}^{2}+{f}_{z}{}^{2})}]={H}^{*}(l)$, which can compensate for the former TF. This process is similar to the following optical phase conjugation. Suppose that the object $O({\epsilon}_{obj},{\mu}_{obj})$ is placed in the air; we can place another object $O\text{'}(-{\epsilon}_{obj},-{\mu}_{obj})$ in the negative index medium ($\epsilon =-1$, $\mu =-1$) in succession to make the information of $O({\epsilon}_{obj},{\mu}_{obj})$undetectable [30,31]. However, such conjugation matching cannot prevent an object from being visible, because the AS of the object misses the optical path $2l$. To compensate for the loss of the optical path, we can additionally put a *d* thick compressed medium with refractive index ${n}^{\u2033}=diag({n}_{x},{n}_{y},{n}_{z})$ in succession. The TF of this additional medium can be easily obtained as ${H}^{\u2033}(d)=\mathrm{exp}[j(2\pi /\lambda )(-{n}_{y}d)\sqrt{1-{\lambda}^{2}({f}_{x}{}^{2}/{n}_{x}^{2}+{f}_{z}{}^{2}/{n}_{z}^{2})}]$. For the purpose of completely hiding the object, the total TF of the above system should satisfy

Substituting **H**, $H\text{'}$, and ${H}^{\u2033}$into Eq. (1), we get

Further, taking account of the conditions of impedance matching ${\epsilon}_{x}/{\mu}_{x}={\epsilon}_{y}/{\mu}_{y}={\epsilon}_{z}/{\mu}_{z}=1$ and ${n}_{x}^{2}={\epsilon}_{z}{\mu}_{y}$,${n}_{y}^{2}={\epsilon}_{z}{\mu}_{x}$,${n}_{z}^{2}={\epsilon}_{x}{\mu}_{y}$, we get the permittivity and permeability of the compressed medium as

Now we consider that the object $O({\epsilon}_{obj},{\mu}_{obj})$ to be hidden is a square (with ${\epsilon}_{obj}=2$ and ${\mu}_{obj}=1$ and side length 0.5 unit) centered at point (0, 0) in the $x-y$ plane in air [Fig. 1(a), star logo]. From the above discussion we see that to construct a carpet filter to cancel this object on it, we can first construct a square structure centered at (0, −1) with a side length of 0.5 and ${\epsilon}_{ob{j}^{*}}=-2$, ${\mu}_{ob{j}^{*}}=-1$ embedded in a negative refractive-index medium with $\epsilon =-1$, $\mu =-1$ [$-1.5<y<-0.5$, Fig. 1(a), gray region above the dashed line], and then below the structure underlay a compressed inhomogeneous medium with anisotropic parameters ${\epsilon}_{z}$, ${\mu}_{x}$, ${\mu}_{y}$ [$-2.5<y<-1.5$, Fig. 1(a), gray region below the dashed line] on a mirror or an absorber [$\text{y}<-\text{2}.\text{5}$, Fig. 1(a), hatched region] to compensate for the optical path loss. Taking $d=l$ in Eq. (3), we can get the electromagnetic parameters of the compressed inhomogeneous medium as${\epsilon}_{z}=3$, ${\mu}_{x}=3$, and${\mu}_{y}=1/3$, respectively.

## 3. Numerical simulations

To demonstrate the invisibility effect of the above filter, we use the finite element method [13] to simulate the TE wave propagation (electric field **E** is parallel to the *z* direction, and the wavelength is a $\lambda =\text{0 .5}$unit). A perfectly matched layer is employed as the absorbing boundary condition. In the calculation, an illumination-point light source placed at (0, 1.5) is used, and the calculation grids are normalized to the wavelength. Figure 2(a)
shows the scattering field distribution of only the object $O({\epsilon}_{obj},{\mu}_{obj})$in front of a mirror in air (without the carpet). We can see that the incident light field is obviously destroyed by the object. Figure 2(b) presents the light field distribution as the carpet is inserted between the object and the mirror. The figure shows that the reflected light looks as if it is directly from the mirror, and only the light source and the mirror are observed. This means that the destroyed light field by the object $O({\epsilon}_{obj},{\mu}_{obj})$ is compensated for completely by the scattering light field of the carpet, indicating that the object is invisible, and there seems to be nothing on the carpet. For comparison, we calculate the light field distribution as both the object and the carpet are removed, and the light emitted from the point source directly illuminates the reflector [Fig. 2(c)]. We see that the total scattering-light field distribution above the object in Figs. 2(b) and 2(c) is exactly the same, further verifying the conclusion that the object $O({\epsilon}_{obj},{\mu}_{obj})$is completely hidden by the carpet under the object [schematically shown in Fig. 1(b)]. Figures 2(d)–2(f) show the light field distributions as the illumination light changes to an oblique plane wave (63 degrees with the *x* direction) in the $x-y$ plane, while the system is the same as Figs. 2(a)–2(c), respectively. The figures further confirm that the object above the filter is invisible.

Figure 3 shows the simulated scattering-light field distributions as: (a)–(c) a point light source and (d)–(f) an oblique plane wave illuminated on (a),(d) a square in front of an absorber; (b),(e) a square upon an invisibility filter; and (c),(f) nothing upon the filter underlaid by the absorber, respectively. The structures are corresponding to Figs. 2(a)–2(f), respectively, except that the mirror is replaced by an absorber. By comparing Figs. 3(b) and 3(c) and Figs. 3(e) and 3(f), we can see that in the two illumination conditions, no reflection but complete darkness is seen, and the object is hidden.

Furthermore, we can also construct another carpet filter on the mirror or absorber substrate as shown in Fig. 1(c) to create optical illusions; that is, to change the scattering AS of the object $O({\epsilon}_{obj},{\mu}_{obj})$ above the carpet to that of another completely different object, $O\text{'}({\epsilon}_{obj\text{'}},{\mu}_{obj\text{'}})$. In this case the carpet filter also consists of two parts [Fig. 1(c), dark gray region divided by a dashed line] but plays three roles. One role is to cancel the AS${A}_{s}(0)$ directly scattered from the object. The second role is to synthesize the TF ${H}_{1}={A}_{t}^{*}(0)$ to compensate for the transmission AS ${A}_{t}(0)$of the object to make ${A}_{t}(0){H}_{1}=1$ before reaching the substrate. The third role is to synthesize another TF ${H}_{2}={A}_{o\text{'}}(0)$ [Fig. 1(c), gray region below the dashed line] to make the AS change to ${A}_{o}(0){H}_{1}{H}_{2}={A}_{o\text{'}}(0)$, meaning that the AS of $O({\epsilon}_{obj},{\mu}_{obj})$is modulated to that of $O\text{'}({\epsilon}_{obj\text{'}},{\mu}_{obj\text{'}})$. By carrying out the inverse FT to ${A}_{o\text{'}}(0)$, we can “see” another object, $O\text{'}({\epsilon}_{obj\text{'}},{\mu}_{obj\text{'}})$ [Fig. 1(d), drip logo], instead of $O({\epsilon}_{obj},{\mu}_{obj})$above the carpet [Fig. 1(c), star logo].

Suppose that the object $O({\epsilon}_{obj},{\mu}_{obj})$ to be camouflaged is the same square as discussed in Fig. 2, while another object $O\text{'}({\epsilon}_{obj\text{'}},{\mu}_{obj\text{'}})$to be changed is a circle [center: (0, 0), radius: 1/4 unit]; then, the first part of the camouflaging carpet [Fig. 1(c), dark gray region upon the dashed line] is exactly the same as that of Fig. 1(a). The second part [Fig. 1(c), dark gray region below the dashed line], which is a compressed medium along the *y* direction for 3 times [refer to Eq. (1)], plays the roles of compensating for the optical path, matching the impedance, and restoring the optical field of the object $O\text{'}({\epsilon}_{obj\text{'}},{\mu}_{obj\text{'}})$ (${\epsilon}_{obj\text{'}}=2$,${\mu}_{obj\text{'}}=1$) simultaneously. Because the information of the circle to be observed above the carpet in air is restored in the compressed medium, the circle will become an elliptical structure in the compressed layer. The semi-minor axis (along the *y* direction) is one-third of the radius of the circle (1/12 unit). However, the semi-major axis (*x* direction) of the ellipse is the same as the radius of the circle, because in the *x* direction no compression occurs (1/4 unit). To make the observed circle above the carpet centered at (0, 0), the distance of the center of the circle to the substrate should be compressed to one-third of the original. Thus the embedding ellipse is centered at (0, −5/3). The electromagnetic parameters of the ellipse can be obtained from ${\epsilon}_{obj\text{\'}}\epsilon "$ and ${\mu}_{obj\text{\'}}\mu "$ as ${\epsilon}_{obj\text{'}z}=6$, ${\mu}_{obj\text{'}x}=3$, and ${\mu}_{obj\text{'}y}=1/3$, respectively, where $\epsilon "$ and $\mu "$ are obtained from Eq. (3). Figure 4(a)
shows the simulated optical field distribution of light scattered by the square object and the camouflaging filter. Compared with Fig. 2(a), we see that the observed reflected light by the system is different from that by the square object. To confirm whether the square is changed to a circle, we calculate the scattering field distribution of only the circular object $O\text{'}({\epsilon}_{obj\text{'}},{\mu}_{obj\text{'}})$ in front of a mirror in air [Fig. 4 (b), without the carpet]. By comparing Figs. 4 (a) and 4(b), we see that the total scattering-light field distribution above the object plane is exactly the same, indicating that the square object $O({\epsilon}_{obj},{\mu}_{obj})$is indeed changed into a circle. Figures 4(c) and 4(d) show the light field distributions as the illumination light is an oblique plane wave with a 63 degree angle to the *x* direction, while the system is the same as Figs. 4(a) and 4(b), respectively. The figures further confirm that the object upon the carpet is camouflaged to another object.

Figure 5 shows the simulated scattering-light field distributions as (a),(b) a point light source and (c),(d) an oblique plane wave illuminates on (a),(c) a square upon an absorber supported camouflaging carpet filter, and (b),(d) a circle in front of an absorber, respectively. The structures correspond to Figs. 4 (a)–4(d), respectively, except that the mirror is replaced by an absorber. From the figures we can see that in the two illumination conditions no reflection is seen, but that the square structure above the filter looks like a circle object occurs.

## 4. Conclusions and discussions

In conclusion, we have demonstrated both theoretically and numerically how to hide objects and create illusions above a carpet filter by using a Fourier optics approach instead of transformation optics. In terms of AS theory, we have designed a special carpet filter by introducing transfer functions to cancel the scattered AS of an object above the filter so as to make the object invisible. Furthermore, we have also demonstrated that by constructing a little more complicated carpet filter we can even change the scattering AS of an object to that of another completely different object. Consequently, one object is camouflaged as another object.

It should be pointed out that, although our theoretical analysis and numerical simulations are focused on the carpet filters underlain by a mirror or an absorber, which are two typical cases with total reflection and zero reflection, respectively, our results are also valid if the carpet filters are placed on other dielectric backgrounds [32]. What is different is that the illuminating light may be partly reflected or totally transmitted. The invisibility and illusion phenomena are true. Regarding loss of the cloaks, due to absorption the cloaks can be visible and, hence, complete invisibility may be impossible [13]. Finally, all the electromagnetic parameters gotten by our Fourier optics approach can also be mathematically gotten by coordinate transformation, meaning that our Fourier optics approach is equivalent to transformation optics for getting the geometric parameters and electromagnetic constants of the cloaks. However, what we hope to address is that our method may provide an intuitive point of view to more clearly understand the physical mechanism of invisibility cloaks.

## Acknowledgments

We are grateful to Prof. C. T. Chan and Dr. Y. Lai for helpful discussions and invaluable suggestions. This work is supported by the 973 Program (Grant 2007CB935300), the National Science Foundation of China (NSFC) (Grant Nos. 60925020, 60736041, and 10774116), and the Science and Technology Bureau of Wuhan City, Hubei, China (Grant No. 200951830552). K. D. Wu is also supported by the National Science Fund for Talent Training in Basic Science (Grant No. J0830310) and the Ph.D. Candidates’ Self-research Program of Wuhan University (Grant No. 20082020101000013).

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