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We present a new scheme for the directional diffraction management of light with incompatible transformation optics. By introducing the concept of disclinations into transformation optics, we demonstrate that the rotational incompatible mapping violates the integrability condition of the coordinate transformation and gives rise to the non-vanishing Frank vector. It is revealed that such special coordinate transformations can produce rotational topological defects in physical space, which magnifies or compresses the diffraction of light beams propagating in arbitrary directions. We verify our theoretical analysis by numerical simulations of the light scattering from a cylinder of a generic disclination medium.
© 2015 Optical Society of America
1. Introduction
Transformation optics [1, 2] has attracted considerable interest due to its unprecedented ability of controlling the electromagnetic waves. The underlying physics of transformation optics relies on the formal invariance of Maxwell’s equations under spatial coordinate transformations [3,4]. Once the coordinate transformation is specified, the constitutive tensors, i.e., the permittivity ε, the permeability μ and the magnetoelectric coupling terms κ ± iγ [5] are normalized by the metric of the curved space. Physically, the metric tensor determines the light cone structure of the space-time and measures the local distance between two neighboring points. As a result, the propagation of the light ray follows the geodesic and the separation of nearby geodesics is determined by the local spatial curvature [4, 6]. This medium-geometry analogy, facilitated by the advance of modern nano-fabrication technology, has successfully enabled many exciting applications such as invisibility cloaks [7–13], light beam control [14–22] and a vast range of novel electromagnetic effects [23–27].
The diffraction control of a beam can be realized by the special coordinate transformations which compress or expand the space in the transversal direction. By introducing transversal compression or expansion that is linearly or nonlinearly dependent on the propagation distance, there are many theoretical and experimental works on the shaping of the waves, such as beam bending, expanding, collimating in open space [14–19] or waveguide geometry [20–22]. These results are based on compatible coordinate transformations which are continuously invertible one-to-one mappings, and hence preserve the same topology between the virtual and physical spaces [1, 2]. Furthermore, these transformations are symmetric with respect to a preassigned propagating direction and therefore the wave shaping for the outgoing beam is highly directional. However, the omnidirectional control of diffraction is crucial for applications of microwave antennas, energy harvesting and future on-chip integrated photonic circuits. Unfortunately, the paradigm based on compatible coordinate transformation cannot be used to deal with the omnidirectional diffraction because they cannot describe the intrinsic singularity at the device center.
In this work, we propose a novel incompatible coordinate transformation and demonstrate the omnidirectional diffraction control of the beams with metamaterials designed by transformation optics. The proposed incompatible coordinate transformation breaks the rotational symmetry of the virtual space, and the transformation Jacobian matrix is not the integrable function of the physical coordinates [28]. The central idea is to introduce a topological defect to describe the omnidirectional diffraction. Point topological defects were proposed to eliminate material singularities [29], and Y. G. Ma et. al. demonstrated an omnidirectional retroreflector by transmuting a dielectric singularity in virtual space into a mere topological defect in a real metamaterial [30]. In the continuum theory of defects, the topological defect such as screw or edge dislocations created by breaking the translational symmetry [31, 32], carries singular torsion on the defect core. We show that such angular coordinate transformations would lead to rotational topological defects, known as disclinations, and they can act as device engineering the diffraction of the electromagnetic beams in arbitrary directions. It is interesting to note that this device is local flat except the core. More importantly, due to the topological nature of the design, our scheme is protected against weak perturbations, and thus allows for tunability without loss of functionality.
This paper is organized as follows. In Section 2, we first review the beam broadening in free space due to diffraction with emphasis on its geometric nature. Section 3 introduces the concept of rotational topological defects in transformation optics, and construct the underlying geometric structure. In Section 4, we propose a generic model of wedge disclinations and study the physical effects on the diffraction management of light. A conclusion is finally drawn in Section 5.
2. Beam diffraction broadening and diffraction control by compatible transformation optic
Diffraction broadening is an intrinsic phenomenon for any classical wave, and is a big obstacle limiting the performance of many linear and nonlinear devices. For instance, the fundamental mode of a coherent Gaussian beam in free space spreads with a far field diffraction angle of ϕf = λ0/πω0 [33], where λ0 is the vacuum wavelength and ω0 is the beam waist. Generally speaking, diffraction is a geometrical effect and depends weakly on the refractive index of the host medium. It originates from the different phase accumulated by wave components of different spatial frequencies (wave vectors) during propagation. For traditional photonic devices, the diffraction broaden can be suppressed by utilizing photonic crystals [34, 35], waveguide arrays [36], nonlinear Kerr effect [37] or metasurface [38].
For simplicity, we consider the propagation of a finite light beam in two-dimensional (2D) free space with one propagating direction x and one transverse direction y. Using Fourier analysis, the spatial profile of the beam can be decomposed with the transversal wave vector ky. In homogeneous medium, the longitudinal component of the wave vector is determined by the trivially diffraction relation , where kx, ky are normalized by k = ω/c. For a paraxial beam with ky ≪ 1, the diffraction relation approximates to [36]
When the beam propagates over a distance L, the phase accumulation of each transverse component ky is φ(ky) = kx × kL. Similar to the treatment of group velocity in temporal dispersion, the diffraction relation leads to a diffraction angle , which describes that the transversal profile of the beam spreads as the beam propagates due to the different phases accumulations in different directions.In Fig. 1(a), we plot the diffraction relations in free space for the general beam (solid red) and the paraxial beam (dashed blue), respectively. It is obvious that the constant frequency surface is concave, meaning that there are only negative diffraction [36], i. e., the effective diffraction coefficient , in a homogeneous medium. The simulated field pattern for a fundamental Gaussian beam with λ0 = 0.4 and ω0 = 0.6 is shown in Fig. 1(b). The simulations were performed with a commercial finite element method solver Comsol Multiphysics. It is obvious that the simulated result for the broaden of the beam agrees well with the calculated diffraction angle (purple dashed).
Fig. 1 (a) Diffraction relations kx = kx(ky) of a real beam (solid red) and a paraxial beam in theory (dashed blue) propagating in air. It should be noted that the paraxial approximation fails for kx < 0.5. (b) Beam spreading of a 2D fundamental Gaussian beam in air due to diffraction, where ϕf ≈ 13.3° is the diffraction angle. (c) Diffraction suppression with a slab filled with transformed medium, the corresponding coordinate transformation is an angular mapping from the symmetric region bounded by the purple dashed lines to the region bounded by the green dashed lines.
As is known, it requires angular compression or expansion along the propagating axis to modify the diffraction of a monochromatic beam. For example, in Fig. 1(c) we present the propagation of a Gaussian beam in a sandwich like structure where the centered slab is composed of the transformed medium created by the following compatible coordinate transformation
here 2h is the height of the slab, k1 is the slop of the diffraction angle, and x0 is the x coordinate of the starting point. The mapping (2) describes the transformation from a trapezoid with one hypotenuse (dashed purple) to a rectangle (with one dashed blue side). It is shown that most of the energy is bounded in the transformed medium with nearly flat wave front. It should be noted that because the slab is not confined by perfect electric conductors, there are radiated waves leaky to the free space. This is a very important point to explain the asymmetric effects of the different parts with opposite Df in Sec. 4.3. The incompatible transformation optics and rotational defect structures
To realize omnidirectional diffraction management, it requires a symmetrically angular coordinate transformation around a symmetric (central) point. From the symmetry argument, this means that the central point is a topological defect in the physical space. The presence of topological defect indicates a topology transformation between the virtual and the physical space which cannot be described by conventional compatible transformation optics. To formulate this angular coordinate transformation, we begin by considering a general coordinate transformation between a flat virtual space denoted by coordinate r and a curved physical space denoted by r′
here u(r) = r′ − r is the displacement vector. According to the transformation optics, the constitutive parameters in the transformed space are given by where J is the transformation Jacobian matrix and |J| denotes the determinant of J.In general, the material electromagnetic parameters can be determined once the coordinate transformation or the displacement vector is given according to the specific aim of the device. However, it is important to note that the displacement vector is a multivalued function when there are topological defects in the physical space. To characterize the global nature of the coordinate transformation, we introduce the topological number Q as the increment of u along a closed circuit B in the physical space via [31, 32, 39]
with the Cartesian components (∇u)ij = ∂iuj. It can be readily shown that Q is invariant under a continuous coordinate transformation u → u + Λ(r). This means Q is a topological invariant, which is independent on the particular choice of the integral circuit.The deformed state of the original undeformed volume element is completely described by the displacement vector. As is well known that a general transformation in three dimensional space can be viewed as composite operations of pure translation together with pure rotation. In the linear region, ∇u is an asymmetric tensor which can be decomposed into the sum of a symmetric part (∂iuj + ∂jui)/2 and an antisymmetric part (∂iuj − ∂jui)/2. The symmetric part describes the infinitesimal strain of the lattice and corresponds to a change of the shape, while the antisymmetric part describes the infinitesimal rigid rotation of a local volume. Because of the decomposition of ∇u, Q can be expressed as the sum
where r0 is the reference point, and the nonvanishing Burgers vector b and Frank vector Ω correspond to the translational and rotational defect, respectively [32]. It should be noted that the topological number Q vanishes identically for the compatible mapping where the displacement vector is a single valued differentiable function. This implies that the image of any closed circuit in virtual space still closed in the physical space under compatible mapping. For a general topological defect, there exists both nonvanishing Burgers and Frank vector. For the special case of Ω = 0, it is a dislocation which breaks the translational symmetry. If b = 0, the topological defect is known as disclination breaking the rotational symmetry. In our previous work, we have shown the dislocation carrying torsion tensor can be as a general linear medium with bi-anisotropic constitutive law [28]. In the present work, we only consider the effect of disclination with nonzero Frank vector.The Frank vector measures the rotational closed failure of a Burgers circuit. Because any two order antisymmetric tensor is dual to a vector, we can define a dual vector ω of a from the curl of u as
ω is related to a by ωi = εijkajk, where εikl = 0, ±1 is the permutation symbol. Obviously, ω is a pseudo-vector field which measures the infinitesimal rotation of the displacement vector according to du = ω × dr. Consequently, the Frank vector Ω takes the following form with θ = ∇ × ∇ω is the disclination density. Here, we have used the Stokes theorem and n is the local unit normal vector on the surface. From Eq. (9), θ can be regarded as the density of topological current corresponding to the density of the disclinations on a closed surface. For the compatible transformation, the circuit integral of ω along any loop vanishes identically like the topological number Q. This statement is equivalent to the rotational compatible condition θ = 0 in the differential form [32]. If the physical space contains some rotational topological defects such as disclinations, the compatible condition is no longer satisfied and there would be non-vanishing angular closed failure. As pointed out in [28], the presence of dislocations violates the integrability condition of the transformation Jacobian matrix. The presence of disclination also violates the integrability condition (∂i∂j − ∂j∂i)ωk = 0.Wedge disclination is the simplest model of rotational topological defects, which can be created via a standard Volterra process: (a) first remove or insert a wedge of a constant deficit angle Ω from a domain in the virtual space; (b) then cement the remaining part by sticking the cutting boundaries. In Fig. 2, we show the physical spaces corresponding to disclinations of constant Frank angle Ω = π/2 and −π, respectively. The polar angles of the loops in Fig. 2(b) and Fig. 2(c) (dashed red) range from 0 to 2π in physical space. However, the ranges of their reverse maps in the virtual space cover from 0 to 3π/2 and 3π, respectively. The opposite case is also true for the map of the red loop in the virtual space, which is similar to the different choices of the Burgers circuit in crystalline solids. As a result, the coordinate transformations to create configurations with nonzero Frank vectors break the topology of the virtual and the physical space. This allows one to interpret Ω as a vectorial topological charge which characterize the degree of rotational symmetry breaking [40–42]. From the Velterra construction of disclinations, it is easily found that the displacement vector is a multivalued function whence it passes through the cutting surface. We note similar multivalued transformations are also found in conformal transformation optics when the exponential, logarithmic or power functions of a complex variable are involved [43]. Here, it should be noted that the Jacobian matrix, and hence the material parameters is single-valued on the cutting surface although the displacement vector is multivalued. We can understand this by observing that the cone resulting from the Volterra construction has a locally flat metric tensor except at the origin. Due to the naive geometry/material corresponding of transformation optics, the material parameters of the transformation medium must be continuous functions despite at the defect line. A heuristic argument for the single-valuedness of the metric based on elasticity can be also found in [44].
Fig. 2 (a) The virtual flat 2D space represented by a perfect lattice. The corresponding physical spaces produced by wedge disclinations of Frank angle (b) Ω = π/2 and (c) Ω = −π.
In Fig. 2, it is shown the curved space created by the rotational topological defects with respect to the defect core at origin,and the region along any radii (not diameter) is expanded or compressed depending on the sign of the topological charge. Thus we expect the rotational defects can act as omnidirectional devices for the manipulation of light beam. For simplicity, we only consider 2D geometry in the present work, where the central point (in 3D) corresponds to the symmetric axis of a cylinder. As an application of the theory in this part, we will investigate the effect of a generalized model of disclination on the radial and non-radial propagating light beams.
4. Demonstration of omnidirectional diffraction
Without loss of generality, we consider the following 2D coordinate transformation
where , ϕ = tan−1y/x. For a 2D system, the central point corresponds to the symmetric axis of a cylinder. This transformation describes a general wedge disclination whose Frank angle is a function of the radius r. For simplicity, we consider a linearly dependent function of the form where α is a constant coefficient, and R0 = 10 is a characteristic length which represents the maximum radius. So that the modulus of Ω lies in the range 0 < |Ω| < |α|. If the Frank vector is a constant (Ω(r) ≡ α), the resulted physical space describes the conical space shown in Fig. 2, whose curvature is a singular function R1212 = 2παδ(2)(r), where δ(2)(r) is the Dirac delta-function in the plane. For the coordinate transformation given by Eq. (10), the constitutive parameters of transformed medium can be calculated from the following Jacobian matrix where F = R0r/α.According to Eq. (4) and Eq. (13), the calculated constitutive parameters of permittivity ε are presented in Fig. 3. It can be seen that there are conical distribution for the material parameters which gives rise to the novel response for the incident beam.
Fig. 3 Distribution of the constitutive parameters of permittivity ε(=μ) in a cylinder with R=5. (a) εxx, (b) εxy(= εyx) and (c) εyy.
To get a deeper insight into the optical response of the general disclination, we examine the coordinate transformation (10) before performing numerical simulations. The sign difference before the displacement vectors comes from the SO(2) Lie algebra structure, which describes the rotation in 2D plane. When the polar angle of the position vector r makes a loop in the virtual space, the displacement vector due to the coordinate transformation is Δu = 2πΩ(r)(1, −1)T. It is this non-vanishing displacement vector that breaks the rotational symmetry of the virtual space. Moreover, if the topological charge Ω is a function of the radius from the defect core, we can obtain a generalized conical space with varying conical parameter. We will show that this position dependent conical parameter is crucial for the manipulating of the diffraction of the propagating waves.
Figure 4 shows the propagation of a light beam incident from left on a cylindrical transformation medium with different values of α. In Fig. 4(a), α = −0.5 where the negative α corresponds to the insertion of wedge of non-constant Frank angle. It is shown that the incident beam first broadens, then narrows rapidly in the cylinder. At the exit side, the beam becomes a nearly collimating beam with a narrow width. We note that the effect of the cylindrical medium can be divided into two semi-cylinder where the effective spatial diffraction Df in the first semi-cylinder medium is negative and smaller than air, so that the expansion of the beam in the transverse direction is stronger than that in air. On the contrary, the effective diffraction Df in the right semi-cylinder is positive which leads to the narrowing of the beam. This process is similar to the dispersion engineering in the optical glass fibers where the dispersion can be cancelled by alternate use of fibers of negative and positive dispersion. We can also engineer the diffraction with positive conical parameter. In Fig. 4(c), we present the result for α = 0.85. It is found that the effects of the left and right semi-cylinder cannot cancel each other. So that the exit beam is broaden in y-direction while keeping a near flat wave front. Simulation results for other values of α (not shown) show that the asymmetric influences of the left and right parts of the cylinder is general due to the wave nature of light. Hence, by carefully alternate use of negative and positive diffraction, we can cancel or change the sign of the spatial diffraction with a cylindrical device. Figures 4(b) and 4(d) show the field pattern for α = −0.9 and 1.2, respectively. It can be found that the diffraction is amplified for α = −0.9. While the beam is focused for α = 1.2 where the field pattern is similar to the case of convex lens.
Fig. 4 The radial propagation of a light beam through a cylindrical transformation medium with negative conical parameter α: (a) −0.5, (b) −0.9 and positive conical parameter α: (c) 0.85, (d) 1.2, respectively. Here, the beam incident from the left side, and the cylinder filled with transformation medium with R=5.
The conical cylinder has another interesting feature. It has been shown that the simplest cylindrical medium with constant Ω can bend the non-radical light rays due to the topological charge it carries: attract the ray when the charge is positive and repel the ray when the charge is negative [42]. For comparison, we calculated the scattering of the non-radical beam whose transverse center is not along the x-axis. In Figs. 5(a)–5(c), the scattering of the beam with different height h by the cylinders corresponding to negative α = −0.5 clearly show that the cylinder attracts the beam for negative α and the bending angles increase with h. However, it is seemed that the beam is also attracted for positive α = 1.2 shown in Figs. 5(d)–5(f). To solve this contradiction, we should noted that the observed beam bending from a cylinder region can be devided the effect of the transformed medium together with the effect of the cylindrical boundary between the transformed medium and air. It is found that the beam inside the cylinder is attracted and repelled by the core indeed, while the curved boundary give rise to opposite effects to the beam depending on the sign of α. It is the effect of these two factor giving rise to the observed pattern. A detailed analysis of the effect of the boundary between the transformation medium and air, and the ray dynamics in the conical space will be presented in another work. It should also be noted that the diffraction Df is not symmetric for the upper and lower parts of the beam due to the non-radical motion. Consequently, the profile of the beam changes under propagation. Especially, the profile changes strongly when it approaches the outer boundary of the cylinder, while changes slightly when it approaches the center due to the radius dependent Ω(r).
Fig. 5 Bending of the non-radial beam by the cylindrical transformation medium. The beam heights are h = 0.8 for (a, d), h = 1.6 for (b, e), and h = 2.4 for (c, f), respectively. The negative α = −0.5 for (a, b, c) and positive α = 1.2 for (d, e, f).
5. Summary
In conclusion, we introduced the concept of the disclinations into transformation optics and demonstrated the ability of diffraction engineering with such judicious designed topological defects. We theoretically investigated the underlying geometry due to the breaking of rotational symmetry in the coordinate transformation due to the disclinations. It’s found that the presence of disclination violates the integrability conditions and leads to non-vanishing Frank vector in the physical space. We then numerically demonstrated the omnidirectional diffraction management and the bending of light beams with a generic model of wedge disclination. We anticipate that the method presented in this work will open a new route for the control of diffraction of electromagnetic waves by use of more complex and multi-scaled rotational defects. Thus, it will allow us to explore fundamental physics of classical and quantum waves in topologically nontrivial manifolds.
Acknowledgments
This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 61205185, 61275170 and 61107032) and Beijing Training Project For The Leading Talents in S & T (Grant No. Z151100000315008). The work was also supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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