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Cloaking of irregularly shaped bodies using coordinate transformation
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Cloaking of irregularly shaped bodies using coordinate transformation
T
⁎
H.H Sidhwaa, , R.P.R.C. Aiyarb, Z. Kavehvashc,d a
Department of Mathematics, Sharif University of Technology, Tehran, Iran Department of Electronics Telecommunications Engineering, Fr. C. Rodrigues Institute of Technology, Mumbai, India Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran d Sharif University of Technology, Tehran, Iran b c
A R T IC LE I N F O
ABS TRA CT
MSC: 00-01 99-00
The inception of transformation optics has opened avenues for designing of a plethora of applications related to the propagation of electromagnetic waves in anisotropic media. In this paper, an algorithm is proposed using a coordinate transformation for the purpose of designing a cloak for a body having an arbitrary convex geometry. For the purpose of verification of the algorithm, a ray tracing process is carried out for an ellipsoid as well as an irregularly shaped body, both having axial symmetry. The ray tracing is carried out in a two dimensional plane since the transmitted ray vector would lie in the plane of incidence containing the incident ray vector and the normal vector to the cloak at the point of incidence. The approach not only reduces the numerical and analytical complexity compared to the conventional approach, but also reduces the order of singularity of the equation of Hamiltonian, improving the ability of the algorithm to cloak bodies with complex geometries which cannot be done using the conventional method.
Keywords: Transformation optics Cloaking Arbitrary geometry Geometrical optics Coordinate transformation
1. Introduction The idea of cloaking using transformation optics [1], wherein the path of a ray is bent around the body of interest so as to give an impression that the ray is travelling in a straight line uninterrupted by any obstacle [2] has been discussed and implemented in various ways in literature [3–12]. A generalised method for designing two dimensional arbitrarily shaped cloaks using the approach of coordinate transformation has been discussed by Li and Li in [13,14]. In another effort towards designing approximate cloaks of arbitrary shapes in two dimensions has been developed [15]. Also, a technique for designing spacetime cloaks is discussed in [16], [17], [18]. A novel designing approach for concentrators using folded transformation optics has been discussed in [19–21].[22,23]. A cloaking method for arbitrarily shaped bodies using NURBS has been discussed in [24]. The conventional method requires the calculation of the material characteristics which are used for the derivation of the Hamiltonian. Usage of this method for cloaking an irregularly shaped body leads to partial cloaking or complete failure of cloaking depending on the angle of incidence of the ray. The reason is that the Hamiltonian formed contains a term |Λ|, the determinant of the Jacobian in its denominator, which tends to zero in certain cases of geometry and position, causing the algorithm to become unstable. The algorithm explained here obviates this shortcoming by formulating the Hamiltonian such that the determinant of the Jacobian is eliminated from its denominator reducing the order of singularity of the equation. The improvement in the quality of cloaking is evident from the discussion in Section 5 where the cloaking of an irregular shaped body is carried out using the modified method. For
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Corresponding author at: Sharif University of Technology Tehran, Iran E-mail address: [email protected] (H.H. Sidhwa).
https://doi.org/10.1016/j.ijleo.2019.163201 Received 26 April 2019; Received in revised form 16 July 2019; Accepted 7 August 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
Fig. 1. Coordinate transformation for an arbitrary geometry.
the purpose of verification of the algorithm, two cases are considered. First, it is assumed that the arbitrary body is an ellipsoid having axial symmetry. Next, an irregularly shaped body having axial symmetry is considered for the purpose of cloaking. The cloaking is considered in the geometric optics limit. The physical size of the cloak and body under consideration is much smaller than the wavelength of the incident rays, hence the process of ray tracing has been used to verify the cloaking efficiency instead of carrying out a full wave simulation. The only requirement for geometry is that it should be a single valued function of the parameters θ, ϕ and should also be continuous. In order to cloak bodies with sharp corners, such as a square, the incident ray should not be incident on the corners since the concept of a normal is not defined at the discontinuity. Normals would then have to be defined on the sides close to the sharp corner and averaged to find a resultant normal for the point of incidence. It is conspicuous from the plots that the cloaking technique using the modified method is efficient enough to cloak irregular shaped bodies while the conventional method is not able to cloak such a bodies, effectively. Also, similar to the approach followed in [8], the need to calculate the material characteristics explicitly prior to the formulation of the Hamilton has been avoided. The organisation of the paper is as follows: Section 2 provides review of transformation optics and Hamiltonian Mechanics. The proposed reformulation of Hamiltonian for Arbitrary shaped bodies is described in Section 3. The application of the proposed algorithm for ellipsoid and arbitrary shaped cloaks are presented in Sections 4 and 5, respectively. The paper is concluded in Section 6. 2. A brief discussion on transformation optics and Hamiltonian mechanics Consider an arbitrarily shaped body to be cloaked which is enclosed by another body with the same topology. The annular region between the inner and outer bodies is the cloak. A coordinate transformation is carried out in accordance with the procedure explained in [8] to map any point lying within the region from the centre to the boundary of the outer body to the region lying between the two bodies. Consider O to be the centre of the body to be cloaked as shown in Fig. 1.The body to be cloaked and the cloak form concentric bodies. Let A be any point on the surface of the inner body and A′ be the corresponding point having the same value of (θ, ϕ) on the surface of the outer body in the radial direction. In order to map any point lying in the region OA to a point lying in the region AA′, the following transformation could be carried out:
r ′ = (1 − τ ) r + τR 0
(1a)
θ′ = θ
(1b)
ϕ′ = ϕ,
where τ = OA/OA′ is the thickness of the cloak, r is the position vector of a point on the surface of the body to be cloaked for a given (θ, ϕ), r′ is the position vector of the point on the surface of the cloak for the same (θ, ϕ), and R0 is a mathematical function that defines the nature of contour of the surfaces of the cloaked and outer bodies which is dependent on (θ, ϕ). Since the cloak parameters depend on (θ, ϕ), for the unique definition of the parameters, the function R0 must be a single valued function of (θ, ϕ). In accordance with formulations discussed in [7], but for an arbitrarily shaped cloak, a Jacobian has been formulated.
Λil′ =
∂x i′ r′ τR x ∂R 0 ˆ ∂R 0 ϕˆl ⎤ θl + = δil − 30 x i xl + τ 2i ⎡ . ⎢ r r r ⎣ ∂θ ∂xl ∂ϕ sin θ ⎥ ⎦
(2)
where θˆl and ϕˆl are the components of the unit vectors θˆ and ϕˆ . The Hamiltonian mechanics which is required in the ray-tracing approach for derivation and verification of the proposed cloaking method of arbitrary shaped bodies, is discussed next. A Hamiltonian is an operator corresponding to the total energy of the system under the condition that the coordinate system is not time-dependent and the potential is not velocity-dependent [27]. Since there is no loss of energy while the ray propagates through the 2
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
medium, the Hamiltonian can be found in order to trace the path of the ray in the cloaked medium [7]. Let k be the propagation vector in the medium. Also, let n, ϵ and μ be tensors. The Hamiltonian based on the conventional approach can be expressed as:
H = 〈k|μ′|k〉 − |μ′|
= < k|
(3a)
ΛΛT 1 |k > − , |Λ| |Λ|
(3b)
where Λ is the Jacobian matrix. As could be noticed from this relation, the determinant of the Jacobian Λ is in the denominator of the expression. Therefore when |Λ| → 0 in certain cases of geometry or position, the expression becomes unstable and the path of the ray diverges in some random direction. This can be seen from the plot in Fig. (8) where the conventional cloaking technique [7] is used for cloaking an irregularly shaped body. The ray in red has completely digressed from the path it should have followed for efficient cloaking. Since |Λ| is a function which is not dependent on the Hamiltonian, |Λ| can be eliminated. The Hamiltonian would now become:
H = 〈k|ΛΛT |k〉 − 1 = 0.
(4)
This Hamiltonian which circumvents the requirement of the calculation of the determinant of the Jacobian |Λ| will be used here for further analysis. The motion of a material particle is determined by the Hamilton-Jacobi equation [30]. The function of a ray vector in geometric optics is the same as that of the momentum of a particle in mechanics, while the frequency is akin to the Hamiltonian, i.e. the energy of the particle. Usage of the conventional approach for cloaking irregularly shaped bodies leads to poor quality of cloaking which can be evident from the plot in Fig. (8). The presence of the determinant of the Jacobian |Λ| in the denominator of the equation of the Hamilton causes it to become unstable and causes the path of the ray to diverge in a random direction, which is evident from path of the rays in Fig. (8).Thus, in the following section, we propose an appropriate modifications to the Hamiltonian method to make it applicable for arbitrary shape bodies.
3. Proposed reformulation of hamiltonian for arbitrarily shaped bodies The media properties in the transformed medium (μ′, ϵ′) can be expressed as:
μ′ = ϵ ′ =
ΛΛT ΛΛT ϵ= |Λ| |Λ|
(5)
where ϵ = μ = I, since it is in free-space. Since at each point in the transformed medium, μ′ = ϵ′, the problem of impedance mismatch would not arise. Further, the same design approach can be implemented for dielectrics instead of free space, by representing ϵ as a matrix. Rest of the approach would remain same. The determinant, when the original space is free space can be deduced as:
|μ′| = |ϵ ′| =
|Λ||ΛT | 1 = . (|Λ|)3 |Λ|
(6)
In this relation, the denominator has degree of 3, since the Jacobian Λ is a 3 times 3 matrix. If the conventional method [7] is used for calculation of Hamiltonian which requires the calculation of the material characteristics, the determinant of the Jacobian equation, Eq. (2), would have to be calculated. The complexity of this equation indicates the numerical and analytical difficulty which one would encounter to calculate the determinant. Even for a two dimensional cloak, there is immense difficulty for calculation of determinant as discussed in [2]. It is for this reason that a modified Hamiltonian is proposed and used for further analysis, Eq. (4), which circumvents the need for calculation of the determinant of the Jacobian. Even then, if the material properties are needed to be found for the design of metamaterial, it can be done using Eq. (5) which would provide the profile for the variation of material characteristics inside the cloaked medium. Therefore, in order to reduce computational complexity and overcome anomalies in ray tracing approach, especially in case of irregularly shaped bodies, it would be sagacious to propose a method that obviates the requirement of calculation of material properties in order to verify the proposed cloaking method. The proposed reformulation of Hamiltonian with the aim to eliminate the need to calculate material properties is explained as follows: The transformation equation, Eq. (2) could be expressed in terms of the primed coordinates-
Λil′ =
x ′ x ′ τR 0 x ′ ∂R r′ ⎡ ∂R 0 ϕˆl ⎞ 1 ⎤ δil − i l + τ i ⎛⎜ 0 θˆl + . ⎟ r ⎢ r′ r′ r′ r ′ ⎝ ∂θ ∂ϕ sin θ ⎠ r ′ ⎥ ⎣ ⎦
(7)
Further analysis will be carried out in terms of r′, but for lucidity, the primes will be dropped. For the sake of simplicity and without ∂R loss of generality, we consider an axisymmetric cloak i.e. ∂ϕ0 = 0 . The Hamiltonian is expressed in terms of the transformed coordinate system. Substituting the Jacobian expressed in Eq. (7) in the equation for the Hamiltonian, Eq. (4) and performing mathematical operations one would reach to the following relation for the Hamiltonian: 3
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
[(τR 0 )2 − 2rτR 0 ] (r − τR 0 )2 τ 2 ∂R 2 + 4 ⎛ 0 ⎞ (k·x)2 (k·x)2 − r4 (1 − τ )2r 2 r ⎝ ∂θ ⎠ 2τ ∂R τ 2R ∂R + 2 0 (k·x)(k·θˆ) − 5 0 0 (k·x)2 (x·θˆ). r ∂θ r ∂θ
H = k·k +
(8)
Having obtained the Hamiltonian relation, the normal vector, nin, for the incident ray on the outer surface of the cloak can be obtained as follows:
1 ∂R 0 ˆ nin = ⎡ θ − rˆ ⎤. ⎥ ⎢ ⎦ ⎣ R 0 ∂θ The transmitted ray vector kt must lie in the plane of incidence [28].
k t = kin + qnin where kin is the incident ray vector, nin is the normal vector pointing inwards to the plane of incidence, and q is a scalar quantity which is obtained by solving for H = 0 in Eq. (8). For the purpose of ray tracing, the path can be parameterised [29] using the Hamiltonian relation, Eq. (8), as:
dx ∂H = dς ∂k
(9a)
dk ∂H =− dς ∂x
(9b)
where ς is the parameterising variable and x is the position vector. Ray tracing is carried out by solving (x, k) as a function of ς starting from ς = 0 which corresponds to the point at which the incident ray touches the cloak. In a spherical coordinate system, the position vector x points in the direction of the radial vector rˆ , which means x = r = r rˆ . The Hamiltonian can be solved as per [29] as:
∂H [(τR 0 )2 − 2rτR 0 ] τ 2 ∂R 2 = 2k + 2(k·x) x + 2 4 ⎛ 0 ⎞ (k·x) x 4 ∂k r r ⎝ ∂θ ⎠
+
2τ ∂R 0 [(k·x) θˆ + (k·θˆ) x]. r 2 ∂θ
(10)
6τrR 0 − 4(τR 0 )2 ⎞ (r − τR 0 ) τR 0 ∂H (k·x)2 − 2 = ⎡⎛ ⎢ (1 − τ )2r 3 ∂x r5 ⎠ ⎣⎝ τ 2 ∂R 2 τ ∂R − 4 5 ⎛ 0 ⎞ (k·x)2 − 4 3 ⎛ 0 ⎞ (k·x)(k·θˆ) ⎤ rˆ ⎥ r ⎝ ∂θ ⎠ r ⎝ ∂θ ⎠ ⎦ ⎜
⎟
2τ ∂R (r − τ ) τR 0 ⎛ ∂R 0 ⎞ 2τ 2 ⎛ ∂R 0 ⎞ ⎛ ∂2R 0 ⎞ (k·x)2 + ⎡ 5 ⎛ 0 ⎞ (R 0 − r )(k·x)2 + 2 + ⎢ (1 − τ )2r 3 ⎝ ∂θ ⎠ r 5 ⎝ ∂θ ⎠ ⎝ ∂θ 2 ⎠ ⎣ r ⎝ ∂θ ⎠ (k·r)2 2τ ⎛ ∂R 0 ⎞ 2τ ⎛ ∂2R 0 ⎞ (k·r)(k·θˆ) ⎤ θˆ − + ⎥ r 2 r 2 ⎝ ∂θ ⎠ r 3 ⎝ ∂θ 2 ⎠ ⎦ ⎜
⎜
+⎡ ⎢ ⎣
⎟
⎟
((τR 0 )2 − 2rτR 0 ) 2τ 2 ∂R 2 2τ ∂R 2(k·r) + 4 ⎛ 0 ⎞ (k·r)+ 2 ⎛ 0 ⎞ (k·θˆ) ⎤ k. ⎥ r4 r ⎝ ∂θ ⎠ r ⎝ ∂θ ⎠ ⎦
(11)
Eqs. (10) and (11) explain the differentiation of the Hamilton with respect to the propagation vector k, and the position vector x respectively. These equations facilitate the parameterisation of the path necessary for ray tracing inside the cloaked medium. Furthermore, it can be observed from these equations for ray tracing, that the necessity of calculating the material properties for the purpose of cloaking has been eliminated. In order to calculate the Jacobian, a spherical coordinate system (θ, ϕ) has been used. However, for the case of ray tracing wherein the calculation of distance is involved, the vectors rˆ, θˆ, k are expressed in the corresponding Cartesian coordinate system as described by Pendry et al. [7]. In order to validate the proposed cloaking method, the obtained formulations are applied to two cases, ellipsoidal cloak in Section 4 and irregularly shaped cloak in Section 5. 4. Ellipsoidal cloak For the verification of the above algorithm, the outer contour is considered to be an ellipsoid with axial symmetry. The ellipsoid would look identical for any section containing the axis of symmetry. Given that we are assuming a spherical polar axis of symmetry which is the Z axis, we can carry out the ray tracing procedure in two dimensions for an ellipse with the major axis as ae and the minor axis as be. The outer contour of the ellipsoid can be expressed as:
R0 =
ae be be2cos2 θ + ae2sin2 θ
(12) 4
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
Fig. 2. Two-dimensional plot of an ellipsoidal cloak in the X-Z plane with variation in the position of point of incidence and the rays being incident from below the cloak.
sin θ cos θ (be2 − ae2 ) ae be ∂R 0 = ∂θ (be2cos2 θ + ae2sin2 θ)3/2
(13)
(be2 − ae2)sin2 2θ ∂ 2R 0 cos 2θ 3 ⎤. = (be2 − ae2 ) ae be ⎡ 2 2 + 2 2 2 2 3/2 ⎢ (be cos θ + ae sin θ) 4 (be cos2 θ + ae2sin2 θ)5/2 ⎥ ∂θe ⎣ ⎦
(14)
Shown in Fig. (2) is a two-dimensional plot of the ellipsoidal cloak in the X-Z plane with variation in the point of incidence starting with a point of incidence on right, at a large distance from the inner cloak and moving towards the inner body. It can be seen that rays are incident in the xz plane, but the point of incidence is altered in small steps so that the ray incident from below first hits the cloak at a large distance from the inner body and the point of incidence is altered so that it is incident just below the body under consideration, and finally it moves towards the other extreme. In Fig. (3), the rays are incident diagonally to the cloak, starting at the lower left side of it and propagating towards the right top side. Fig. (4) shows the variation in the angle of incidence at the same point of entry in the cloak and the paths traced by the three rays. 5. Irregularly shaped cloak In order to test the robustness of the algorithm in case of irregularly shaped bodies, the outer contour is considered to be an irregular shaped body having axial symmetry. Since we are assuming a spherical polar axis of symmetry which is the Z axis, we can carry out the ray tracing procedure in two dimensions for this body in the XZ plane:
R1 (θ) = α +
ar ar2 + br2
cos(5θ) +
br ar2 + br2
sin θ (15)
Fig. 3. Two-dimensional plot of an ellipsoidal cloak in the X-Z plane with the rays being incident on the lower left side of the cloak and travelling towards the top right side. 5
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
Fig. 4. Two-dimensional plot of an ellipsoidal cloak in the X-Z plane with variation in the angle of incidence on the same point at the lower left side of the cloak and travelling towards the top right side.
where α, ar and br are constants. The first and second order derivatives of R1(θ) can be calculated as below:
∂R1 ar =− 5 sin(5θ) + ∂θ ar2 + br2 ∂2R1 ar =− 25 cos(5θ) − ∂θ 2 ar2 + br2
br ar2 + br2
cos θ
br ar2 + br2
(16)
sin θ (17)
Fig. (5) depicts the path of the rays in a cloak with outer contour defined by Eq. (15). The rays are incident from the bottom and propagate in the cloak and travel from the top into free space. Shown in Fig. (6) is a plot in two dimensions of an irregular shaped axisymmetric cloak in the x-z plane with oblique incidence and variation in the point of incidence. Also, Fig. (7) shows the path traced in the irregular shaped cloak by three different rays obtained by varying the angle of incidence at the same point of entry in the cloak. The rays are incident at the lower left corner and propagate towards the right through the cloak into free space. The plots in the three Figs. (5 –7 ) clearly illustrates the ability to cloak random shaped bodies using the modified Hamiltonian equation, i.e. Eq. (4), where the term |Λ|, determinant of the Jacobian of the Hamiltonian Eq. (3b), has been eliminated from the denominator. This elimination has served two purposes. First, in certain cases of geometry and position, the determinant of the Jacobian which tends towards zero has been eliminated. Thus this approach has let to a reduction in the order of singularity of the equation. The second purpose has been the dispensation from the necessity to calculate the determinant of the Jacobian which can be very cumbersome as the geometry under consideration becomes more complex. As against this, the random shaped body under consideration is tried to be cloaked using the conventional method [2], [7]. The Hamiltonian Eq. (3b) is used which contains the term |Λ| in its denominator. Fig. (8) shows the resultant plots where ray depicted in red is able to trace the path correctly to a certain extent while the blue one is not able to travel in the right direction at all, indicating the inefficiency of the conventional method to correctly carry out ray tracing
Fig. 5. Two-dimensional plot of an irregular shaped axisymmetric cloak in the X-Z plane with variation in the point of incidence and rays travelling from bottom to top. 6
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
Fig. 6. Two-dimensional plot of an irregular shaped axisymmetric cloak in the X-Z plane with oblique incidence and variation in the point of incidence.
Fig. 7. Two-dimensional plot of an irregular shaped axisymmetric cloak in the X-Z plane with variation in the angle of incidence and rays travelling from bottom to top.
Fig. 8. Plot of the an arbitrarily shaped cloak in the X-Y plane for two different angles of incidence at different points of entry. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
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Optik - International Journal for Light and Electron Optics 197 (2019) 163201
H.H. Sidhwa, et al.
for all cases. This result confirms the above discussed point that the presence of the determinant causes the equation to become unstable since it tends to zero in certain cases of geometry and position. Thus the conventional approach of cloaking is not suitable for completely arbitrarily shaped bodies. Thus the method which requires the calculation of material characteristics prior to the formulation of Hamiltonian suffers from the problem of singularity and instability as well as the numerical and analytical complexity of calculating the determinant of Jacobian. The modified method for calculating the Hamiltonian discussed in this paper rectifies both these problems. 6. Conclusion A generalised coordinate transformation technique using spherical coordinates for the design of an arbitrarily shaped cloak in three dimensions has been demonstrated. A procedure for calculating the path of a ray through the cloak is described. This algorithm obviates the requirement of calculation of metamaterial characteristics (μ and ϵ tensors) prior to the formulation of Hamiltonian, as has been the procedure reported in literature.In order to demonstrate the validity of the algorithm, first it has been applied to an axisymmetric ellipsoid cloak. Next, to test its ruggedness, it is applied to an irregular shaped body which is efficiently cloaked. In comparison to this modified method, when the conventional approach for which the calculation of material characteristics is mandatory was used for cloaking the irregular shaped body under consideration, the ray completely digresses from the correct path leading to a failure in the cloaking process. The presence of the determinant of the Jacobian, |Λ|, in the denominator of the Hamiltonian causes the algorithm to become unstable since it tends to zero in certain cases of geometry and position. The algorithm explained here obviates these requirements. Also, the usage of generalised coordinates for the purpose of plotting compared to the usage of Cartesian coordinates used by Pendry et al. [7] gives the algorithm the flexibility to plot irregular shaped bodies and also the ability to plot rays which are close to the inner boundary of the cloak. Thus there are three distinct contributions of this research work. First, the ability to cloak irregular shaped objects, which cannot be achieved using the conventional approach, next the freedom from the necessity of calculating material properties explicitly, third, the utilisation of generalised coordinates. The other minor contribution is the verification of the algorithm using ray tracing in contrast to other algorithms proposed for cloaking arbitrary shaped bodies which have been verified using commercial soft-wares.This technique can be applied to any arbitrary surface for which the function R0(θ, ϕ) is single valued on it and is continuous. 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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Cloaking of irregularly shaped bodies using coordinate transformation
T
⁎
H.H Sidhwaa, , R.P.R.C. Aiyarb, Z. Kavehvashc,d a
Department of Mathematics, Sharif University of Technology, Tehran, Iran Department of Electronics Telecommunications Engineering, Fr. C. Rodrigues Institute of Technology, Mumbai, India Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran d Sharif University of Technology, Tehran, Iran b c
A R T IC LE I N F O
ABS TRA CT
MSC: 00-01 99-00
The inception of transformation optics has opened avenues for designing of a plethora of applications related to the propagation of electromagnetic waves in anisotropic media. In this paper, an algorithm is proposed using a coordinate transformation for the purpose of designing a cloak for a body having an arbitrary convex geometry. For the purpose of verification of the algorithm, a ray tracing process is carried out for an ellipsoid as well as an irregularly shaped body, both having axial symmetry. The ray tracing is carried out in a two dimensional plane since the transmitted ray vector would lie in the plane of incidence containing the incident ray vector and the normal vector to the cloak at the point of incidence. The approach not only reduces the numerical and analytical complexity compared to the conventional approach, but also reduces the order of singularity of the equation of Hamiltonian, improving the ability of the algorithm to cloak bodies with complex geometries which cannot be done using the conventional method.
Keywords: Transformation optics Cloaking Arbitrary geometry Geometrical optics Coordinate transformation
1. Introduction The idea of cloaking using transformation optics [1], wherein the path of a ray is bent around the body of interest so as to give an impression that the ray is travelling in a straight line uninterrupted by any obstacle [2] has been discussed and implemented in various ways in literature [3–12]. A generalised method for designing two dimensional arbitrarily shaped cloaks using the approach of coordinate transformation has been discussed by Li and Li in [13,14]. In another effort towards designing approximate cloaks of arbitrary shapes in two dimensions has been developed [15]. Also, a technique for designing spacetime cloaks is discussed in [16], [17], [18]. A novel designing approach for concentrators using folded transformation optics has been discussed in [19–21].[22,23]. A cloaking method for arbitrarily shaped bodies using NURBS has been discussed in [24]. The conventional method requires the calculation of the material characteristics which are used for the derivation of the Hamiltonian. Usage of this method for cloaking an irregularly shaped body leads to partial cloaking or complete failure of cloaking depending on the angle of incidence of the ray. The reason is that the Hamiltonian formed contains a term |Λ|, the determinant of the Jacobian in its denominator, which tends to zero in certain cases of geometry and position, causing the algorithm to become unstable. The algorithm explained here obviates this shortcoming by formulating the Hamiltonian such that the determinant of the Jacobian is eliminated from its denominator reducing the order of singularity of the equation. The improvement in the quality of cloaking is evident from the discussion in Section 5 where the cloaking of an irregular shaped body is carried out using the modified method. For
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Corresponding author at: Sharif University of Technology Tehran, Iran E-mail address: [email protected] (H.H. Sidhwa).
https://doi.org/10.1016/j.ijleo.2019.163201 Received 26 April 2019; Received in revised form 16 July 2019; Accepted 7 August 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
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Fig. 1. Coordinate transformation for an arbitrary geometry.
the purpose of verification of the algorithm, two cases are considered. First, it is assumed that the arbitrary body is an ellipsoid having axial symmetry. Next, an irregularly shaped body having axial symmetry is considered for the purpose of cloaking. The cloaking is considered in the geometric optics limit. The physical size of the cloak and body under consideration is much smaller than the wavelength of the incident rays, hence the process of ray tracing has been used to verify the cloaking efficiency instead of carrying out a full wave simulation. The only requirement for geometry is that it should be a single valued function of the parameters θ, ϕ and should also be continuous. In order to cloak bodies with sharp corners, such as a square, the incident ray should not be incident on the corners since the concept of a normal is not defined at the discontinuity. Normals would then have to be defined on the sides close to the sharp corner and averaged to find a resultant normal for the point of incidence. It is conspicuous from the plots that the cloaking technique using the modified method is efficient enough to cloak irregular shaped bodies while the conventional method is not able to cloak such a bodies, effectively. Also, similar to the approach followed in [8], the need to calculate the material characteristics explicitly prior to the formulation of the Hamilton has been avoided. The organisation of the paper is as follows: Section 2 provides review of transformation optics and Hamiltonian Mechanics. The proposed reformulation of Hamiltonian for Arbitrary shaped bodies is described in Section 3. The application of the proposed algorithm for ellipsoid and arbitrary shaped cloaks are presented in Sections 4 and 5, respectively. The paper is concluded in Section 6. 2. A brief discussion on transformation optics and Hamiltonian mechanics Consider an arbitrarily shaped body to be cloaked which is enclosed by another body with the same topology. The annular region between the inner and outer bodies is the cloak. A coordinate transformation is carried out in accordance with the procedure explained in [8] to map any point lying within the region from the centre to the boundary of the outer body to the region lying between the two bodies. Consider O to be the centre of the body to be cloaked as shown in Fig. 1.The body to be cloaked and the cloak form concentric bodies. Let A be any point on the surface of the inner body and A′ be the corresponding point having the same value of (θ, ϕ) on the surface of the outer body in the radial direction. In order to map any point lying in the region OA to a point lying in the region AA′, the following transformation could be carried out:
r ′ = (1 − τ ) r + τR 0
(1a)
θ′ = θ
(1b)
ϕ′ = ϕ,
where τ = OA/OA′ is the thickness of the cloak, r is the position vector of a point on the surface of the body to be cloaked for a given (θ, ϕ), r′ is the position vector of the point on the surface of the cloak for the same (θ, ϕ), and R0 is a mathematical function that defines the nature of contour of the surfaces of the cloaked and outer bodies which is dependent on (θ, ϕ). Since the cloak parameters depend on (θ, ϕ), for the unique definition of the parameters, the function R0 must be a single valued function of (θ, ϕ). In accordance with formulations discussed in [7], but for an arbitrarily shaped cloak, a Jacobian has been formulated.
Λil′ =
∂x i′ r′ τR x ∂R 0 ˆ ∂R 0 ϕˆl ⎤ θl + = δil − 30 x i xl + τ 2i ⎡ . ⎢ r r r ⎣ ∂θ ∂xl ∂ϕ sin θ ⎥ ⎦
(2)
where θˆl and ϕˆl are the components of the unit vectors θˆ and ϕˆ . The Hamiltonian mechanics which is required in the ray-tracing approach for derivation and verification of the proposed cloaking method of arbitrary shaped bodies, is discussed next. A Hamiltonian is an operator corresponding to the total energy of the system under the condition that the coordinate system is not time-dependent and the potential is not velocity-dependent [27]. Since there is no loss of energy while the ray propagates through the 2
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medium, the Hamiltonian can be found in order to trace the path of the ray in the cloaked medium [7]. Let k be the propagation vector in the medium. Also, let n, ϵ and μ be tensors. The Hamiltonian based on the conventional approach can be expressed as:
H = 〈k|μ′|k〉 − |μ′|
= < k|
(3a)
ΛΛT 1 |k > − , |Λ| |Λ|
(3b)
where Λ is the Jacobian matrix. As could be noticed from this relation, the determinant of the Jacobian Λ is in the denominator of the expression. Therefore when |Λ| → 0 in certain cases of geometry or position, the expression becomes unstable and the path of the ray diverges in some random direction. This can be seen from the plot in Fig. (8) where the conventional cloaking technique [7] is used for cloaking an irregularly shaped body. The ray in red has completely digressed from the path it should have followed for efficient cloaking. Since |Λ| is a function which is not dependent on the Hamiltonian, |Λ| can be eliminated. The Hamiltonian would now become:
H = 〈k|ΛΛT |k〉 − 1 = 0.
(4)
This Hamiltonian which circumvents the requirement of the calculation of the determinant of the Jacobian |Λ| will be used here for further analysis. The motion of a material particle is determined by the Hamilton-Jacobi equation [30]. The function of a ray vector in geometric optics is the same as that of the momentum of a particle in mechanics, while the frequency is akin to the Hamiltonian, i.e. the energy of the particle. Usage of the conventional approach for cloaking irregularly shaped bodies leads to poor quality of cloaking which can be evident from the plot in Fig. (8). The presence of the determinant of the Jacobian |Λ| in the denominator of the equation of the Hamilton causes it to become unstable and causes the path of the ray to diverge in a random direction, which is evident from path of the rays in Fig. (8).Thus, in the following section, we propose an appropriate modifications to the Hamiltonian method to make it applicable for arbitrary shape bodies.
3. Proposed reformulation of hamiltonian for arbitrarily shaped bodies The media properties in the transformed medium (μ′, ϵ′) can be expressed as:
μ′ = ϵ ′ =
ΛΛT ΛΛT ϵ= |Λ| |Λ|
(5)
where ϵ = μ = I, since it is in free-space. Since at each point in the transformed medium, μ′ = ϵ′, the problem of impedance mismatch would not arise. Further, the same design approach can be implemented for dielectrics instead of free space, by representing ϵ as a matrix. Rest of the approach would remain same. The determinant, when the original space is free space can be deduced as:
|μ′| = |ϵ ′| =
|Λ||ΛT | 1 = . (|Λ|)3 |Λ|
(6)
In this relation, the denominator has degree of 3, since the Jacobian Λ is a 3 times 3 matrix. If the conventional method [7] is used for calculation of Hamiltonian which requires the calculation of the material characteristics, the determinant of the Jacobian equation, Eq. (2), would have to be calculated. The complexity of this equation indicates the numerical and analytical difficulty which one would encounter to calculate the determinant. Even for a two dimensional cloak, there is immense difficulty for calculation of determinant as discussed in [2]. It is for this reason that a modified Hamiltonian is proposed and used for further analysis, Eq. (4), which circumvents the need for calculation of the determinant of the Jacobian. Even then, if the material properties are needed to be found for the design of metamaterial, it can be done using Eq. (5) which would provide the profile for the variation of material characteristics inside the cloaked medium. Therefore, in order to reduce computational complexity and overcome anomalies in ray tracing approach, especially in case of irregularly shaped bodies, it would be sagacious to propose a method that obviates the requirement of calculation of material properties in order to verify the proposed cloaking method. The proposed reformulation of Hamiltonian with the aim to eliminate the need to calculate material properties is explained as follows: The transformation equation, Eq. (2) could be expressed in terms of the primed coordinates-
Λil′ =
x ′ x ′ τR 0 x ′ ∂R r′ ⎡ ∂R 0 ϕˆl ⎞ 1 ⎤ δil − i l + τ i ⎛⎜ 0 θˆl + . ⎟ r ⎢ r′ r′ r′ r ′ ⎝ ∂θ ∂ϕ sin θ ⎠ r ′ ⎥ ⎣ ⎦
(7)
Further analysis will be carried out in terms of r′, but for lucidity, the primes will be dropped. For the sake of simplicity and without ∂R loss of generality, we consider an axisymmetric cloak i.e. ∂ϕ0 = 0 . The Hamiltonian is expressed in terms of the transformed coordinate system. Substituting the Jacobian expressed in Eq. (7) in the equation for the Hamiltonian, Eq. (4) and performing mathematical operations one would reach to the following relation for the Hamiltonian: 3
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[(τR 0 )2 − 2rτR 0 ] (r − τR 0 )2 τ 2 ∂R 2 + 4 ⎛ 0 ⎞ (k·x)2 (k·x)2 − r4 (1 − τ )2r 2 r ⎝ ∂θ ⎠ 2τ ∂R τ 2R ∂R + 2 0 (k·x)(k·θˆ) − 5 0 0 (k·x)2 (x·θˆ). r ∂θ r ∂θ
H = k·k +
(8)
Having obtained the Hamiltonian relation, the normal vector, nin, for the incident ray on the outer surface of the cloak can be obtained as follows:
1 ∂R 0 ˆ nin = ⎡ θ − rˆ ⎤. ⎥ ⎢ ⎦ ⎣ R 0 ∂θ The transmitted ray vector kt must lie in the plane of incidence [28].
k t = kin + qnin where kin is the incident ray vector, nin is the normal vector pointing inwards to the plane of incidence, and q is a scalar quantity which is obtained by solving for H = 0 in Eq. (8). For the purpose of ray tracing, the path can be parameterised [29] using the Hamiltonian relation, Eq. (8), as:
dx ∂H = dς ∂k
(9a)
dk ∂H =− dς ∂x
(9b)
where ς is the parameterising variable and x is the position vector. Ray tracing is carried out by solving (x, k) as a function of ς starting from ς = 0 which corresponds to the point at which the incident ray touches the cloak. In a spherical coordinate system, the position vector x points in the direction of the radial vector rˆ , which means x = r = r rˆ . The Hamiltonian can be solved as per [29] as:
∂H [(τR 0 )2 − 2rτR 0 ] τ 2 ∂R 2 = 2k + 2(k·x) x + 2 4 ⎛ 0 ⎞ (k·x) x 4 ∂k r r ⎝ ∂θ ⎠
+
2τ ∂R 0 [(k·x) θˆ + (k·θˆ) x]. r 2 ∂θ
(10)
6τrR 0 − 4(τR 0 )2 ⎞ (r − τR 0 ) τR 0 ∂H (k·x)2 − 2 = ⎡⎛ ⎢ (1 − τ )2r 3 ∂x r5 ⎠ ⎣⎝ τ 2 ∂R 2 τ ∂R − 4 5 ⎛ 0 ⎞ (k·x)2 − 4 3 ⎛ 0 ⎞ (k·x)(k·θˆ) ⎤ rˆ ⎥ r ⎝ ∂θ ⎠ r ⎝ ∂θ ⎠ ⎦ ⎜
⎟
2τ ∂R (r − τ ) τR 0 ⎛ ∂R 0 ⎞ 2τ 2 ⎛ ∂R 0 ⎞ ⎛ ∂2R 0 ⎞ (k·x)2 + ⎡ 5 ⎛ 0 ⎞ (R 0 − r )(k·x)2 + 2 + ⎢ (1 − τ )2r 3 ⎝ ∂θ ⎠ r 5 ⎝ ∂θ ⎠ ⎝ ∂θ 2 ⎠ ⎣ r ⎝ ∂θ ⎠ (k·r)2 2τ ⎛ ∂R 0 ⎞ 2τ ⎛ ∂2R 0 ⎞ (k·r)(k·θˆ) ⎤ θˆ − + ⎥ r 2 r 2 ⎝ ∂θ ⎠ r 3 ⎝ ∂θ 2 ⎠ ⎦ ⎜
⎜
+⎡ ⎢ ⎣
⎟
⎟
((τR 0 )2 − 2rτR 0 ) 2τ 2 ∂R 2 2τ ∂R 2(k·r) + 4 ⎛ 0 ⎞ (k·r)+ 2 ⎛ 0 ⎞ (k·θˆ) ⎤ k. ⎥ r4 r ⎝ ∂θ ⎠ r ⎝ ∂θ ⎠ ⎦
(11)
Eqs. (10) and (11) explain the differentiation of the Hamilton with respect to the propagation vector k, and the position vector x respectively. These equations facilitate the parameterisation of the path necessary for ray tracing inside the cloaked medium. Furthermore, it can be observed from these equations for ray tracing, that the necessity of calculating the material properties for the purpose of cloaking has been eliminated. In order to calculate the Jacobian, a spherical coordinate system (θ, ϕ) has been used. However, for the case of ray tracing wherein the calculation of distance is involved, the vectors rˆ, θˆ, k are expressed in the corresponding Cartesian coordinate system as described by Pendry et al. [7]. In order to validate the proposed cloaking method, the obtained formulations are applied to two cases, ellipsoidal cloak in Section 4 and irregularly shaped cloak in Section 5. 4. Ellipsoidal cloak For the verification of the above algorithm, the outer contour is considered to be an ellipsoid with axial symmetry. The ellipsoid would look identical for any section containing the axis of symmetry. Given that we are assuming a spherical polar axis of symmetry which is the Z axis, we can carry out the ray tracing procedure in two dimensions for an ellipse with the major axis as ae and the minor axis as be. The outer contour of the ellipsoid can be expressed as:
R0 =
ae be be2cos2 θ + ae2sin2 θ
(12) 4
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Fig. 2. Two-dimensional plot of an ellipsoidal cloak in the X-Z plane with variation in the position of point of incidence and the rays being incident from below the cloak.
sin θ cos θ (be2 − ae2 ) ae be ∂R 0 = ∂θ (be2cos2 θ + ae2sin2 θ)3/2
(13)
(be2 − ae2)sin2 2θ ∂ 2R 0 cos 2θ 3 ⎤. = (be2 − ae2 ) ae be ⎡ 2 2 + 2 2 2 2 3/2 ⎢ (be cos θ + ae sin θ) 4 (be cos2 θ + ae2sin2 θ)5/2 ⎥ ∂θe ⎣ ⎦
(14)
Shown in Fig. (2) is a two-dimensional plot of the ellipsoidal cloak in the X-Z plane with variation in the point of incidence starting with a point of incidence on right, at a large distance from the inner cloak and moving towards the inner body. It can be seen that rays are incident in the xz plane, but the point of incidence is altered in small steps so that the ray incident from below first hits the cloak at a large distance from the inner body and the point of incidence is altered so that it is incident just below the body under consideration, and finally it moves towards the other extreme. In Fig. (3), the rays are incident diagonally to the cloak, starting at the lower left side of it and propagating towards the right top side. Fig. (4) shows the variation in the angle of incidence at the same point of entry in the cloak and the paths traced by the three rays. 5. Irregularly shaped cloak In order to test the robustness of the algorithm in case of irregularly shaped bodies, the outer contour is considered to be an irregular shaped body having axial symmetry. Since we are assuming a spherical polar axis of symmetry which is the Z axis, we can carry out the ray tracing procedure in two dimensions for this body in the XZ plane:
R1 (θ) = α +
ar ar2 + br2
cos(5θ) +
br ar2 + br2
sin θ (15)
Fig. 3. Two-dimensional plot of an ellipsoidal cloak in the X-Z plane with the rays being incident on the lower left side of the cloak and travelling towards the top right side. 5
Optik - International Journal for Light and Electron Optics 197 (2019) 163201
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Fig. 4. Two-dimensional plot of an ellipsoidal cloak in the X-Z plane with variation in the angle of incidence on the same point at the lower left side of the cloak and travelling towards the top right side.
where α, ar and br are constants. The first and second order derivatives of R1(θ) can be calculated as below:
∂R1 ar =− 5 sin(5θ) + ∂θ ar2 + br2 ∂2R1 ar =− 25 cos(5θ) − ∂θ 2 ar2 + br2
br ar2 + br2
cos θ
br ar2 + br2
(16)
sin θ (17)
Fig. (5) depicts the path of the rays in a cloak with outer contour defined by Eq. (15). The rays are incident from the bottom and propagate in the cloak and travel from the top into free space. Shown in Fig. (6) is a plot in two dimensions of an irregular shaped axisymmetric cloak in the x-z plane with oblique incidence and variation in the point of incidence. Also, Fig. (7) shows the path traced in the irregular shaped cloak by three different rays obtained by varying the angle of incidence at the same point of entry in the cloak. The rays are incident at the lower left corner and propagate towards the right through the cloak into free space. The plots in the three Figs. (5 –7 ) clearly illustrates the ability to cloak random shaped bodies using the modified Hamiltonian equation, i.e. Eq. (4), where the term |Λ|, determinant of the Jacobian of the Hamiltonian Eq. (3b), has been eliminated from the denominator. This elimination has served two purposes. First, in certain cases of geometry and position, the determinant of the Jacobian which tends towards zero has been eliminated. Thus this approach has let to a reduction in the order of singularity of the equation. The second purpose has been the dispensation from the necessity to calculate the determinant of the Jacobian which can be very cumbersome as the geometry under consideration becomes more complex. As against this, the random shaped body under consideration is tried to be cloaked using the conventional method [2], [7]. The Hamiltonian Eq. (3b) is used which contains the term |Λ| in its denominator. Fig. (8) shows the resultant plots where ray depicted in red is able to trace the path correctly to a certain extent while the blue one is not able to travel in the right direction at all, indicating the inefficiency of the conventional method to correctly carry out ray tracing
Fig. 5. Two-dimensional plot of an irregular shaped axisymmetric cloak in the X-Z plane with variation in the point of incidence and rays travelling from bottom to top. 6
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Fig. 6. Two-dimensional plot of an irregular shaped axisymmetric cloak in the X-Z plane with oblique incidence and variation in the point of incidence.
Fig. 7. Two-dimensional plot of an irregular shaped axisymmetric cloak in the X-Z plane with variation in the angle of incidence and rays travelling from bottom to top.
Fig. 8. Plot of the an arbitrarily shaped cloak in the X-Y plane for two different angles of incidence at different points of entry. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)
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for all cases. This result confirms the above discussed point that the presence of the determinant causes the equation to become unstable since it tends to zero in certain cases of geometry and position. Thus the conventional approach of cloaking is not suitable for completely arbitrarily shaped bodies. Thus the method which requires the calculation of material characteristics prior to the formulation of Hamiltonian suffers from the problem of singularity and instability as well as the numerical and analytical complexity of calculating the determinant of Jacobian. The modified method for calculating the Hamiltonian discussed in this paper rectifies both these problems. 6. Conclusion A generalised coordinate transformation technique using spherical coordinates for the design of an arbitrarily shaped cloak in three dimensions has been demonstrated. A procedure for calculating the path of a ray through the cloak is described. This algorithm obviates the requirement of calculation of metamaterial characteristics (μ and ϵ tensors) prior to the formulation of Hamiltonian, as has been the procedure reported in literature.In order to demonstrate the validity of the algorithm, first it has been applied to an axisymmetric ellipsoid cloak. Next, to test its ruggedness, it is applied to an irregular shaped body which is efficiently cloaked. In comparison to this modified method, when the conventional approach for which the calculation of material characteristics is mandatory was used for cloaking the irregular shaped body under consideration, the ray completely digresses from the correct path leading to a failure in the cloaking process. The presence of the determinant of the Jacobian, |Λ|, in the denominator of the Hamiltonian causes the algorithm to become unstable since it tends to zero in certain cases of geometry and position. The algorithm explained here obviates these requirements. Also, the usage of generalised coordinates for the purpose of plotting compared to the usage of Cartesian coordinates used by Pendry et al. [7] gives the algorithm the flexibility to plot irregular shaped bodies and also the ability to plot rays which are close to the inner boundary of the cloak. Thus there are three distinct contributions of this research work. First, the ability to cloak irregular shaped objects, which cannot be achieved using the conventional approach, next the freedom from the necessity of calculating material properties explicitly, third, the utilisation of generalised coordinates. The other minor contribution is the verification of the algorithm using ray tracing in contrast to other algorithms proposed for cloaking arbitrary shaped bodies which have been verified using commercial soft-wares.This technique can be applied to any arbitrary surface for which the function R0(θ, ϕ) is single valued on it and is continuous. 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