Inspired by the branch cut that can link two Riemann sheets in complex function theory, we utilize the branch cut to mimic an electromagnetic ‘wormhole’ linking two 2D ‘parallel spaces’ in a reference space. With the help of optical conformal mapping, we design a time-varying inhomogeneous medium that can effectively perform like an electromagnetic ‘wormhole’ in the real space. Based on this method, we can simulate the evolutionary process of an electromagnetic ‘wormhole’ and the wave propagation from one space to another in a laboratory environment. The proposed device may also be applied in light capture, light modulators, and absorption with directional dependence.
© 2017 Optical Society of America
A wormhole or Einstein-Rosen Bridge is a tunnel that links two parallel space-times . Space transference or time travel can be achieved by passing through the wormhole. However, negative energy is required to maintain the opening of the wormhole, and there has been no observation of a real wormhole (study on wormholes is mainly limited to theoretical works).
The wormhole is caused by the warp of space and time. The materials can be transferred between two parallel space-times when the wormhole is open, which makes it possible for instant transportation or time travel. The two parallel space-times are isolated again when the wormhole is closed. The wormhole, a tunnel linking two parallel space-times, is often vividly described as a virtual ‘tube’. In actuality the essence of the wormhole is highly warped space-time, which can be treated as the boundary between two parallel space-times. The influence of a curved space-time on electromagnetic waves can be equivalently mimicked by the influence of the medium (e.g. the permittivity and the permeability or the refractive index) on the electromagnetic wave, which is the essential idea of Transformation Optics (TO): the propagation of the light in a curved space-time is the same as the propagation of the light in the transformation medium that is calculated by the metric tensor of this curved space-time [2–4]. In fact, TO has been widely applied to mimic astronomical phenomena predicted by general relativity, e.g. black holes [5–7], time travelling , wormholes [9–11], space-time cloaking [12,13], etc., by which many complex astronomical phenomena can be mimicked with laboratory environment through the propagation of light in the effective transformation medium calculated by TO. Greenleaf et al. first theoretically studied how to mimic an electromagnetic wormhole by TO . Based on these studies, wormholes for plasmonic  and DC magnetic fields  are proposed. In these studies, the wormhole was described as “invisible tunnels”, and the regions they enclose are not detectable to lateral electromagnetic observations. Actually this description cannot comprehensively reflect all aspects of the wormhole: a wormhole is a time-related phenomenon, which means that it only opens within a limited period of time. All previous studies on wormholes do not involve time (i.e. the materials for modeling wormholes are time invariant), and hence cannot vividly describe the process of a wormhole’s opening and closing. In this study, we first introduce the time-varying factor in optical conformal mapping to mimic the phenomenon of light travelling between two ‘parallel spaces’, which can be an analogy to the process of an electromagnetic wormhole’s building up and vanishing. With the help of TO, we can use some time-varying inhomogeneous media to effectively mimic an electromagnetic wormhole’s opening and closing processes in a laboratory environment.
Our method can vividly mimic the full process of an electromagnetic ‘wormhole’. Light waves are spread separately into two ‘parallel spaces’ before the electromagnetic ‘wormhole’ opens (i.e. two ‘parallel spaces’ are isolated and do not influence each other). The electromagnetic ‘wormhole’ opens at a certain time, and its radius gradually increases over time (i.e. the common region that links the two different ‘parallel spaces’ gradually increases). In this case, the light in one space will be coupled into the other parallel space and propagate into that space once it touches the ‘wormhole’ (i.e. the common boundary between the two spaces). Later the ‘wormhole’ will gradually close, and the light that is coupled into another space will stay in that space once the ‘wormhole’ is fully closed. Different from previous methods to design a wormhole by transformation optics [9–11], we use optical conformal mapping with a time-varying factor to mimic the propagation of the light between two Riemann sheets, which can be a new way to effectively mimic the propagation of light between two ‘parallel spaces’ linked by an optical ‘wormhole’.
We note that here we only give a vivid analogy between branch cut and optical ‘wormhole’, and our model is not identical to the evolution of wormhole space in Ref. 1 (the theory of relativity is not included in this study). Note that real wormholes describe exotic curved spacetimes, and the manifold given in this study is flat. The paper is organized as follow: Firstly we briefly review the function of the Zhukovski transformation which is an essential component to design the electromagnetic ‘wormhole’. We then explain how to introduce a time-varying factor during the conformal mapping to achieve a time-related phenomenon that can mimic an electromagnetic ‘wormhole’. The required permittivity to mimic an electromagnetic ‘wormhole’ is theoretically deduced in this section. Advantages, other applications and further works are given in the discussion part.
2.1 Optical conformal mapping
Some concepts in complex analysis  can help us to describe an electromagnetic ‘wormhole’. For some multi-value functions, one variable may correspond to more than one function value. To describe such one-to-many relation, Riemann sheets are introduced to describe analytical multiple-valued function: each Riemann sheet corresponds to one value domain. Branch cut are curves (often line segments) that connects different Riemann sheets. More examples and details on Riemann sheets and branch cut can be found in textbook for Complex Analysis (e.g. Ref. 14). An optical wormhole is an invisible tunnel that links two or more parallel spaces. Light keeps propagating in one space when it does not touch the wormhole. However, if the light touches the wormhole, it will enter into another space and propagate in that space.
Riemann sheets can be modeled as many ‘parallel spaces’, and the branch cut that links different Riemann sheets can mimic the ‘wormhole’ that connects these different ‘parallel spaces’. In complex analysis, Riemann sheets are connected by an abstract concept, namely, a branch cut. If the size of the branch cut is changing with time, we can use this concept to mimic a wormhole’s opening and closing between two or more ‘parallel spaces’. For simplicity, we only consider the ‘wormhole’ to link two ‘parallel spaces’ (‘wormholes’ linking many ‘parallel spaces’ can be analyzed by choosing some other special analytic transformations). Another thing we should mention is that we do not introduce any time-coordinate transformation in this paper (i.e. we only involve the space-coordinate transformation and introduce a time-varying factor), and hence the light just travels between two ‘parallel spaces’ without time flowing backwards or forwards.
Our theoretical basis is the optical conformal mapping [4, 15–18], which can be treated as a special case of general TO [3, 19] and has been applied to design many novel optical devices [15–18]. There are two spaces in TO: the real space and the virtual space (often referred to as the reference space). The light and medium in the two spaces establish a corresponding relationship with the help of the coordinate transformation. We can study the light’s propagation in the reference space (e.g. a curved free space) to deduce the light’s path in the real space (e.g. a flat space filled with some special medium). We only consider the 2D space in this study and use complex coordinates z = x + iy and w = u + iv as the coordinates in the real space and the reference space, respectively. We should note that the reference space and the real space here are not the ‘parallel spaces’ connected by the ‘wormhole’. In our method, we have two parallel 2D spaces (i.e. two Riemann sheets) connected by the branch cut in the reference space (see Fig. 1(a)). Correspondingly, we also have two parallel 2D spaces connected by the ‘wormhole’ in the real space (see Fig. 1(b)). In the real space, the gray region outside the yellow circle (corresponding to the upper Riemann sheet in the reference space) is one space, and the blue region inside the yellow circle (corresponding to the lower Riemann sheet in the reference space) is the other parallel space. Two regions in the real space stand for two ‘parallel spaces’, which are connected by the yellow boundary (i.e. the ‘wormhole’). We begin with the Zhukovski transformation, which can be written as [4,16]:
The basic function of the Zhukovski transformation is shown in Fig. 1. Note that the parameter a in Eq. (1) determines the length of the branch cut in the reference space (i.e. the line segment colored yellow with length 4a) and in the real space (i.e. the circle colored yellow with radius a). In the reference space there are two Riemann sheets (i.e. the upper sheet and the lower sheet, colored black and blue, respectively) connected by the branch cut (i.e. an abstract tunnel colored yellow). The two Riemann sheets can be treated as two ‘parallel spaces’. The branch cut performs as a tunnel connecting the two ‘parallel spaces’ and is invisible to the outside viewer, and can be treated as an electromagnetic ‘wormhole’ connecting two ‘parallel spaces’. Both Riemann sheets are free space, and hence the light propagates in a straight path in each sheet. The green light never touches the branch cut and keeps propagating on the upper Riemann sheet. The red light initially propagates on the upper Riemann sheet, and then it falls into the lower Riemann sheet when it touches the branch cut. This model can vividly reflect the process of light passing through a ‘wormhole’. In the real space (see Fig. 1(b)), the outer and the inner parts of a circle with radius a correspond to the upper and the lower Riemann sheets in the reference space, respectively, and can be treated as two ‘parallel spaces’ The yellow circle is the branch cut and can be treated as a ‘wormhole’. If the ‘wormhole’ is closed (i.e., the size of the yellow circle reduces to a point), only the gray space is kept, which does not couple with the other blue parallel space (which is a whole ‘parallel space’ below the grey space as Eq. (1) becomes the identity transformation when a = 0).
To establish the corresponding relationship between the reference space and the real space, the refractive index distribution in the real space can be calculated by [16, 17]:Eq. (2). The relation between a coordinate in the real space (z = x + iy) and the reference space (w = u + iv) has been given by Eq. (1). Combining Eqs. (1) and (2), we can obtain the required medium distribution of the ‘wormhole’:
From Eq. (3) we also see that the refractive index in the whole space is non-negative and thus no need for any negative refractive index material. The refractive index material is zero only at two discrete points z = a and z = -a. As we can see from Eq. (3), the refractive index at the yellow circle (i.e. the ‘wormhole’) is n = 2|sinθ| (θ∈[0,2π) is defined by θ = arctan(y/x)), which means the refractive index changes continuously on the ‘wormhole’.
2.2 Time-varying medium
The size of the branch cut changes with time (e.g. a is not a constant but gradually changes with time a = a(t) in Eq. (3)) to reflect the fact that the size of the ‘wormhole’ changes with the time. At the beginning (e.g. t = 0), the ‘wormhole’ is closed (i.e. a = 0). In the reference space, the light propagates independently on the upper and the lower Riemann sheets (i.e. there is no coupling between the two ‘parallel spaces’). Equation (3) reduces to the identity transformation, and hence the situation in the real space is the same (i.e. the ‘parallel spaces’ are isolated). At some time t = t0, the size of the branch cut gradually increases (i.e. a gradually increases with time), which means that the electromagnetic ‘wormhole’ gradually opens up. In the reference space, the length of the branch cut (the yellow line segment in Fig. 1(a)) increases with time, which leads to the light coupling between the two Riemann sheets. In the real space, the size of the ‘wormhole’ (the yellow circle in Fig. 1(b)) also increases with time. The region inside the yellow circle just corresponds to another parallel space different from the space outside the yellow circle. The light propagation between two ‘parallel spaces’ can be achieved by the ‘wormhole’ with a non-zero radius. At a later time (e.g. t = t0 + Δt/2), the size of the branch cut begins to decrease (i.e. the ‘wormhole’ gradually closes). The common region connecting two ‘parallel spaces’ gradually decreases both in the reference space and the real space. After a certain time (e.g. t = t0 + Δt), the size of the branch cut is zero again (i.e. the ‘wormhole’ is fully closed and the tunnel between the two ‘parallel spaces’ disappears). The two parallel Riemann sheets are isolated again in the reference space. In the real space, the circular ‘wormhole’ reduces to a point. Now the light can only propagate in the free space outside this point and cannot be coupled into the region inside the point (e.g. another parallel space).
In the real space, the ‘wormhole’ is equivalently realized by the inhomogeneous refractive index given by Eq. (3), which equivalently reflects the warped space-time effect. At the beginning (e.g. t = 0), the light propagates in the free space without any scattering. After the ‘wormhole’ is opened up, the light propagating around the ‘wormhole’ (the yellow circle in Fig. 1(b)) will be attracted into the ‘wormhole’ and propagates into another parallel space (i.e. the region inside the yellow circle colored blue in Fig. 1(b)). The yellow circle with radius a(t) is the electromagnetic ‘wormhole’, which is the boundary of the two ‘parallel spaces’ (the regions inside and outside the circle with different colors are two different spaces). We should note that we do not introduce any time-coordinate transformation in Eq. (1), and hence the time axes in two ‘parallel spaces’ are the same (i.e. without time delay between two spaces).
2.3 Wormhole in time domain (The monochromatic wave case)
For a monochromatic light, we do not need to consider any dispersion. We can introducing the time factor in Eq. (3) to build the ‘wormhole’, e.g. time-varying factor a can be assumed by:Equations (3) and (4) show the required refraction index in the time domain to mimic the ‘wormhole’ for a monochromatic light.
2.4 The polychromatic light case
For a pulse in the time domain, we need to choose a suitable material dispersion relation that satisfies the following two conditions: first the refractive index at a specific frequency of interest (e.g. ω = ω0) should be exactly the same as the one given in Eq. (3), and secondly the causality should be satisfied (i.e. the Kramer-Kronig relation should be satisfied ). We can assume the dispersion relation of the permittivity fulfills the Drude model (so that the Kramer-Kronig relation is naturally satisfied ):Eq. (3)) at the target frequency ω = ω0:
For simplicity, we ignore the loss (i.e. γ = 0) and obtain the expression of factor A from Eq. (6):
The electric polarizability can be obtained from Eqs. (5) and (7) with γ = 0:
The electric polarizability in the time domain can be obtained by applying the inverse Fourier transformation of Eq. (8):
Here H(t) is the Heaviside step function, which reflects the fact that the causality is naturally satisfied here. The constitutive relation in the time domain can be obtained from Eq. (9):Eq. (4). As we can see from Eq. (10), there are always some spatial positions inside the ‘wormhole’ where ε is negative. We can rewrite Eq. (10) by using z = ρ exp(iθ) with ρ∈[0,∞], θ∈[0,2π]:
As we can see from the above formula that factor (2cos2θ-a2/ρ2) is less than zero at some spatial positions (e.g. θ∈[π/4,3π/4] or [5π/4,7π/4]). It means that as time increases the negative factor ω02a2(2cos2θ-a2/ρ2)tH(t)/ρ2 also increases at these spatial positions, which leads the value of [1 + ω02 a2(2cos2θ-a2/ρ2)tH(t)/ρ2] (i.e. the relative permittivity) to be more negative at these spatial positions.
3. Numerical simulations
3.1 Frequency domain simulations (The monochromatic wave case)
We used the finite element method (FEM) with COMSOL Multiphysics to simulate the wave propagation between two ‘parallel spaces’ when the size of the ‘wormhole’ slowly changes with time. The ray tracing model is chosen to mimic the ray’s propagation when the size of the ‘wormhole’ changes in Fig. 2 (a is set as a varying parameter in this case). The refractive index distribution is also plotted when a changes (see Fig. 3). As shown in Fig. 2, more light is coupled into the parallel space as the size of the ‘wormhole’ a increases. The rays that are attracted into the center (i.e. another parallel space) will never come back to the original space as the refractive index increases rapidly to infinity in the center region. Note that the wave propagation for the wormhole’s closing is the inverse process of the wormhole’s opening (we do not show it in Fig. 2).
Another thing we should mention is how slow it should be if we want to use the time-harmonic model in the frequency domain to mimic this time-varying process. We can analyze this in the reference space as we do not introduce any time coordinate transformation between the two spaces. In the reference space, both Riemann sheets are free space. Only the size of the branch cut changes with time as the function described by Eq. (4). The light will immediately enter to the other Riemann sheet when it touches the branch cut. Provided that the changing speed of the ‘wormhole’ size (i.e. da(t)/dt = 2aM/Δt) is much less than the speed of light in the free space, we can use the frequency domain simulations by setting a as a scanning parameter to mimic the light propagation since a is a time-varying factor in the time domain. Note that the requirement of the changing speed of ‘wormhole’ is only for the frequency domain simulation. Next we will use the time-domain simulation to mimic the performance of our ‘wormhole’. In this case, we do not have any requirement on the changing speed of ‘wormhole’.
3.2 Time domain simulations (The monochromatic wave case)
To mimic the light propagation in the ‘wormhole’ in the time domain, we need make some simplifications on the required medium in Eq. (3) to avoid the singularity in the center. As the infinity in refraction index corresponds to the infinity on the lower Riemann sheet in the reference space, this reduction will influence the performance of the ‘wormhole’ in some degree. We can rewrite Eq. (3) as:Fig. 4. For the opening process of the ‘wormhole’, the numerical simulation results in the time domain are consistent with those in the frequency domain. In the time domain simulation, some part of the light changes its direction even when the ‘wormhole’ is closed. This is caused by the singularity reduction (see Eq. (12)) on the wormhole. As the singularity in the center is removed, the ‘wormhole’ cannot keep the light in another parallel space when it is closing. Some lights will return to the original space (whose propagation directions are also changed). When the light propagates from the upper Riemann sheet to the lower Riemann sheet, it should keep propagating in the lower Riemann sheet (i.e. without coming back to the upper Riemann sheet) even if the branch cut is close. However, this requires the refractive index in the center of the inhomogeneous medium to increase very fast to infinity. It cannot be realized in numerical simulations (i.e. we have to make some reductions in our numerical model and see Eqs. (12) and (13)), which are the reasons why there are several deviating rays (i.e. comes back from the other Riemann sheet even if the ‘wormhole’ is close) in Figs. 4 and 5.
Note that there are some regions where the refraction index is smaller than 1 in our wormhole, which makes it difficult to realize. We make an additional simulation when any medium with the refraction index smaller than 1 is replaced by air (see Fig. 5). Compared with Fig. 4, we can see that rays are also attracted into the center when the wormhole is opening (e.g. from t = 50 ns to 70 ns), even if the refraction index smaller than 1 are replaced by air in Fig. 5. All rays that have been attracted to the center keep propagating forward when the wormhole begins to close in Fig. 4 (i.e. no rays are reflected). However, some rays that have been attracted to the center are reflected when the ‘wormhole’ begins to close for the case that refraction index smaller than 1 are replaced by air in Fig. 5.
In this study, the time system in two ‘parallel spaces’ are the same (i.e. we use the same time system in both the upper and lower Riemann sheets in the reference space), and no time-coordinate transformation is introduced (we only involve the space-coordinate transformation and introduce a time-varying factor). In other words, there is no clock back and clock forward effect between the two ‘parallel spaces’ connected by the ‘wormhole’ (only the spatial connection is set up). If we want to achieve a clock back or clock forward effect, the medium that we use to mimic the electromagnetic wormhole may be more complicated, and more general TO theory is required.
We have linked two ‘parallel spaces’ by the Zhukovski transformation in Eq. (1). If we want to connect more ‘parallel spaces’ together, some other kinds of analytic transformations can be considered (e.g. an exponential transformation). To demonstrate the ‘wormhole’ effect of the proposed inhomogeneous medium in the time domain experimentally, we need some time-varying medium (i.e. to achieve the required refractive index at different times), which may be realized by tunable metamaterials [21, 22].
Another thing we should mention is that our ‘wormhole’ is a directional device (i.e. it performs like a ‘wormhole’ when the incident light comes from the y direction as shown in Fig. 1). The same device can perform as a directional cloaking when the light comes from the x direction, which has been explored in previous studies [16, 23]. This can be understood from Fig. 1: the green light from the x direction cannot touch the branch cut, and hence will never be coupled into another parallel space. It only propagates around the ‘wormhole’, which is also verified by the ray tracing method and full wave simulation (see Fig. 6). Previous experimental result also shows that the directional cloaking effect does not only work for the geometrical ray, but also for the electromagnetic wave in a broad frequency range . Such a directional device may also be applied in capturing and shielding the light beam with directional selectivity.
If we add some loss in the center region (i.e. |z|<a), the proposed device can also perform as a directional absorber for waves incident from the y direction. For example, we can introduce some gradual loss in the region inside the circle with radius a in the reference space (i.e. the lower Riemann sheet in the reference space), and rewrite the refractive index in the real space as:Eq. (14) is shown in Fig. 7. Such a device whose function is directionally dependent (i.e. cloaking effect from the x direction and wormhole or absorber effect from the y direction) may have some applications in directional light energy capturing (e.g. the light is shielded or absorbed when it comes from the x or y direction, respectively). a can be a fixed number in this application (e.g. no need to be a time-dependent parameter).
We should note that even the Zhukovski transformation has been utilized to design many other optical devices (e.g. invisibility cloaking [23–27]) in previous studies, in the present paper we propose how to use this transformation (combined with a time-varying factor) to design an electromagnetic ‘wormhole’ for the first time. Compared with the wormholes designed by previous methods [9–11], the electromagnetic ‘wormhole’ designed by our method has many advantages: firstly the electromagnetic ‘wormhole’ designed in this paper can mimic light traveling between two ‘parallel spaces’, instead of two different spatial regions in the same space. Secondly the permittivity to mimic the electromagnetic ‘wormhole’ is a time-varying medium (see Eq. (11)) in our model but not time-invariant medium like electromagnetic ‘wormhole’ designed by previous methods [9–11]. Thirdly, our method can be extended to an electromagnetic ‘wormhole’ that links many ‘parallel spaces’ (i.e. more than two ‘parallel spaces’) at the same time based on the multiple Riemann sheets in the complex analysis. The method proposed in this paper will open up a new way to mimic an electromagnetic ‘wormhole’, which is closer to the real wormhole. The most likely way to realize the proposed ‘wormhole’ is utilizing some tunable meta-materials whose response to the incident wave (i.e. effective permittivity and permeability) can be tuned with time [21, 22, 28, 29].
The refractive index’s distribution influences the propagation of the light, which performs effectively as the curved space’s influence on the light. Highly warped space-time can produce a wormhole effect (i.e. it can connect two parallel spaces and couple the light from one space to the other). In this study, we have established an effective refractive index distribution that can achieve an electromagnetic ‘wormhole’ effect similar to the wormhole produced by the wrapped space-time. The concept of an electromagnetic ‘wormhole’ has been equivalently mimicked by the inhomogeneous refractive index. Numerical simulations have been given to mimic the propagation of the light from the wormhole’s opening to its closing. Our study will yield a new way to design and study the electromagnetic ‘wormhole’ by TO and optical conformal mapping. Our model in this paper may have some other potential applications in directivity-dependent tunable light absorbers, directivity-dependent tunable cloaking, and we will make a further step on mimicking a real optical wormhole.
National High Technology Research and Development Program (863 Program) of China (No. 2012AA030402); National Natural Science Foundation of China for Young Scholars (No. 11604292); National Natural Science Foundation of China (Nos. 61178062 and 60990322); Fundamental research funds for the central universities (No. 2017FZA5001); Program of Zhejiang Leading Team of Science and Technology Innovation; Postdoctoral Science Foundation of China (No. 2013M541774); Preferred Postdoctoral Research Project Funded by Zhejiang Province (No. BSH1301016); Swedish VR grant (# 621-2011-4620) and AOARD.
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