FINITE-EMBEDDED COORDINATE DESIGNED TRANSFORMATION-OPTICAL DEVICES
The design method for complex electromagnetic materials is expanded from form-invariant coordinate transformations of Maxwell's equations to finite embedded coordinate transformations. Embedded transformations allow the transfer of electromagnetic field manipulations from the transformation-optical medium to another medium, thereby allowing the design of structures that are not exclusively invisible. A topological criterion for the reflectionless design of complex media is also disclosed and is illustrated in conjunction with the topological criterion to design a parallel beam shifter and a beam splitter with unconventional electromagnetic behavior.
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This application is a Continuation application of U.S. application Ser. No. 12/268,295 filed Nov. 10, 2008, now U.S. Pat. No. 8,837,031, issued Sep. 16, 2014, which claims the benefit of priority from provisional application No. 60/987,014 filed Nov. 9, 2007, and from provisional application No. 60/987,127 filed Nov. 12, 2007, the contents of which are incorporated herein by reference.
FIELDThe technology herein relates to optical phenomena, and more particularly to complex electromagnetic materials that allow the transfer of electromagnetic field manipulations from the transformation-optical medium to another medium. The technology herein also relates to parallel beam shifters and beam splitters with unconventional electromagnetic behavior.
BACKGROUND AND SUMMARYMetamaterials offer an enormous degree of freedom for manipulating electromagnetic fields, as independent and nearly arbitrary gradients can be introduced in the components of the effective permittivity and permeability tensors. In order to exploit such a high degree of freedom, a viable method for the well-aimed design of complex materials would be desirable.
Pendry et al., Science 312, 1780 (2006) reported a methodology based on continuous form-invariant coordinate transformations of Maxwell's equations which allows for the manipulation of electromagnetic fields in a previously unknown and unconventional fashion. This method was successfully applied for the design and the experimental realization of an invisibility cloak and generated widespread interest specifically in the prospects of electromagnetic cloaking—a topic that has dominated much of the subsequent discussion.
The methodology presented in Pendry et al makes use of form-invariant continuous coordinate transformations of Maxwell's equations. The use of continuous transformations provides a complex transformation-optical material which is invisible to an external observer. In other words, the field modifications precipitated in the transformation-optical device generally may not be transferred to another medium and the original electromagnetic properties of waves impinging on the medium are restored as soon as the waves exit the optical component. Transformation-optical designs reported in the literature so far generally have in common that the electromagnetic properties of the incident waves are exclusively changed within the restricted region of the transformation-optical device. However, for the sake of the continuity of the transformation, the field manipulation cannot be transferred to another medium or free space and thus remains an, in many cases, undesired local phenomenon.
It would be desirable to have also a tool for the design of electromagnetic/optical components that takes advantage of the high degree of design freedom provided by the transformation-optical approach, but allows the transfer of field manipulations outside the transformation-optical material. Such a method would allow the creation of optical devices with unconventional electromagnetic/optical behavior and functionality that exceeds the abilities of conventional components like lenses, beam steerers, beam shifters, beam splitters and similar.
The technology herein expands the design method for complex electromagnetic materials from form-invariant coordinate transformations of Maxwell's equations to finite embedded coordinate transformations. In contrast to continuous transformations, embedded transformations allow the transfer of electromagnetic field manipulations from the transformation-optical medium to another medium, thereby allowing the design of structures that are not exclusively invisible. The illustrative exemplary non-limiting implementations provide methods to design such novel devices by a modified transformation-optical approach. The conceived electromagnetic/optical devices can be reflectionless under certain circumstances.
The exemplary illustrative non-limiting technology herein further delivers a topological criterion for the reflectionless design of complex media. This exemplary illustrative non-limiting expanded method can be illustrated in conjunction with the topological criterion to provide an example illustrative non-limiting parallel beam shifter and beam splitter with unconventional electromagnetic behavior.
The concept of embedded coordinate transformations significantly expands the idea of the transformation-optical design of metamaterials which itself was restricted to continuous coordinate transformations so far. The expansion to embedded transformations allows for non-reversibly change to the properties of electromagnetic waves in transformation media and for transmission of the changed electromagnetic properties to free space or to a different medium in general. In order to design the medium as reflectionless, a new topological criterion for the embedded transformations can be used to impose constraints to the metric of the spaces at the interface between the transformation-optical medium and the surrounding space. This metric criterion can be applied in the conception of a parallel beam shifter and a beam splitter and confirmed in 2D full wave simulations. Such exemplary illustrative non-limiting devices can provide an extraordinary electromagnetic behavior which is not achievable by conventional materials. Such examples clearly state the significance of embedded coordinate transformations for the design of new electromagnetic elements with tunable, unconventional optical properties.
These and other features and advantages will be better and more completely understood by referring to the following detailed description of exemplary non-limiting illustrative implementations in conjunction with the drawings of which:
The exemplary illustrative non-limiting technology herein provides a generalized approach to the method of form-invariant coordinate transformations of Maxwell's equations based on finite embedded coordinate transformations. The use of embedded transformations adds a significant amount of flexibility to the transformation design of complex materials. For example, with finite-embedded transformations, it is possible to transfer field manipulations from the transformation-optical medium to a second medium, eliminating the requirement that the transformation optical structure be invisible to an external observer. The finite-embedded transformation thus significantly broadens the range of materials that can be designed to include device-type structures capable of focusing or steering electromagnetic waves. Like transformation optical devices, the finite-embedded transform structures can be reflectionless under conditions that we describe below. The general methodology is graphically illustrated below for a parallel beam shifter.
x′(x,y,z)=x
y′(x,y,z)=y+a(x+b)
z′(x,y,z)=z (1-3)
As can be seen, the transformation only affects the y-coordinates while the x- and z-coordinates remain unchanged. The y-coordinates in
The material properties in regions II and III can be arbitrarily chosen and do not have to be considered in the following discussion. As pointed out in D. Schurig et al, Opt. Express 14, 9794 (2006) and E. J. Post, Formal Structure of Electromagnetics, Dover Publications (1997), the material properties of a transformation-optical medium can be calculated as
εi′j′=det(Aii′)−1Aii′Ajj′εij
μi′j′=det(Aii′)−1Aii′Ajj′μij (4-5)
for a given coordinate transformation xa′(xa)=Aaa′xa, where Aaa′ is the Jacobi matrix and det(Aaa′) its determinant.
The coordinate systems at the interface at x=d between region I and free space appear to be discontinuous. As a discontinuous coordinate transformation can be considered as the limit of a continuous coordinate transformation, the discontinuous transition at the boundary at x=d may be taken into account in order to rigorously apply the same formalism as described in J. B. Pendry, et al., Science 312, 1780 (2006); and D. Schurig et al, Opt. Express 14, 9794 (2006), for the continuous coordinate transformation. In this case, the material properties at the interface will carry the character of the discontinuity at the boundary. Mathematically, the discontinuity at the interface between region I and free space can be described by
x′(x,y,z)=x
y′(x,y,z)=θ(d−x)[y+a(x+d)]+θ(x−d)y
z′(x,y,z)=z (6-8)
where θ(ξ)=1 for ξ>0 and 0 for ξ<0. Calculating the permittivity and permeability tensors by use of equations (4-5), one obtains
Refer to the Appendix for a more detailed mathematical analysis.
The material properties in equations (9-11) above can be interpreted in two distinct ways. In the first case, the transformation is considered to be discontinuous at the boundary, so that the interface has to be taken into account and the material properties at the boundary contain all terms in equations (10) and (11) above. The delta distribution, being the derivative of the Heaviside function, carries the signature of the discontinuity at the boundary between region I and free space in
The exemplary illustrative non-limiting implementation herein provides an alternate interpretation of the calculated material parameters. In this context, the discontinuity at the boundary is not considered in the calculation of the material properties. This means that the delta distribution, which is responsible for the backshift of the beam to its old path at x=d, is not included in equation (10). In other words, the material properties are calculated only within the transformation-optical material without taking the interface to free space into account and then embedded into free space. The linear coordinate transformation (I=1) for the design of a parallel beam shifter illustrated in
In more detail, it is useful to explain the difference between “embedded transformations” and “discontinuous transformations”. Interpreting the transformation as discontinuous, the boundary must be taken into account and the transformation of the y-coordinate at the transition from region I and free space must read
y1(x,y,z)=θ(d−x)[y+ak1(x,y)]+θ(x−d)y (18)
so that a21|a=d∞δ(d−x). As in the ray approximation the y-coordinate lines in
This method is similar to the “embedded Green function” approach in the calculation of electron transport through interfaces (see, J;. E. Inglesfield, J. Phys. C: Solid State Phys. 14, 3795 (1981), so that we refer to it as an “embedded coordinate transformation”. For this case, the beam in
At this point, the question arises as to which conditions must hold for the embedded transformation in order to design a reflectionless optical device. We found as a necessary—and in our investigated cases also sufficient—topological condition for our exemplary illustrative non-limiting implementation is that the metric in and normal to the interface between the transformation-optical medium and the non-transformed medium (in this case free space) must be continuous to the surrounding space. In the case of an exemplary illustrative non-limiting beam shifter, this means that the distances as measured along the x-, y- and z-axis in the transformation optical medium and free space must be equal along the boundary (x=d). As can be clearly seen from
A second, more sophisticated example a beam splitter is shown in
2D full-wave simulations can be carried out to adequately predict the electromagnetic behavior of waves impinging on a beam shifter and a beam divider, respectively. The calculation domain can be bounded by perfectly matched layers. The polarization of the plane waves can be chosen to be perpendicular to the x-y plane. Such 2D full wave simulations can be used to confirm the propagation properties of waves impinging on a transformation-optical parallel beam shifter.
A similar behavior can be observed for waves with wave fronts of arbitrary curvature, as for example for convergent waves (
As a second example for the strength of the new design tool, a transformation-optical beam splitter was conceived.
In more detail,
In
A further example beam focusing and expanding unit is shown in
computing a local form-invariant coordinate transformation of Maxwell's Equations for a transformation-optical medium of an optical device (block 302);
computing form-invariant coordinate transformations for a region surrounding the transformation-optical medium of the device (block 304);
embedding the local form-invariant computed coordinate transformation into free space region or a second surrounding optical medium defining the device (block 306); and
providing the computed coordinate transformations to a metamaterial fabrication process equipment (block 308).
recording initial configurations of E-M fields in a medium on a Cartesian coordinate mesh (block 402);
distorting/changing the Cartesian coordinate mesh to a predetermined configuration represented by a distorted coordinate mesh (block 404);
recording the distortions as a coordinate transformation between the Cartesian mesh and the distorted coordinate mesh (block 406);
computing scaled E-M field values permittivity c and permeability p values in accordance with the coordinate transformations (block 408);
embedding the computed E-M field values into the coordinate space defining a transformation-optical device (block 410), whereby transformations for the finite region of space defining a transformation-optic device are defined;
using the scaled μ and ε values to set/define the two-dimensional and/or three-dimensional spacial permittivity and permeability properties of a reconfigurable or non-reconfigurable transformation-optical structure/device composed of metamaterials;
and
providing the computed transformed values defining the transformation-optical device to a metamaterial fabrication processing equipment that fabricates the metamaterial for the optical device (block 412).
As should be clear from the above, all of the devices illustrated above can be realized in a relatively straightforward manner using artificially structured metamaterials. Techniques for designing these basic elements for metamaterials are now well-known to the community. One of the advantages of using metamaterials is that tunability can be implemented, allowing the materials to change dynamically between several transformation optical states, or continuously over a range of transformation-optical states. In this way, a set of dynamically reconfigurable devices to control electromagnetic waves can be designed into a structure that can also be used for load-bearing applications.
All publications cited above are hereby incorporated herein by reference.
While the technology herein has been described in connection with exemplary illustrative non-limiting implementations, the invention is not to be limited by the disclosure. The invention is intended to be defined by the claims and to cover all corresponding and equivalent arrangements whether or not specifically disclosed herein.
APPENDIXThe mathematical formalism used for the calculation of the complex material properties is similar to the one reported in M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry (2007), arXiv:0706.2452v1 and D. Schurig, J. B. Pendry, and D. R. Smith, Opt. Expr. 14, 9794 (2006). For a given coordinate transformation xa′(xa)=Aaa′xa=(Aaa′: Jacobi matrix, a=1 . . . 3), the electric permittivity εi′j′ and the magnetic permeability μi′j′ of the resulting material can be calculated by
εi,j=[det(Aii′)]−1Aii′A (1)
μi,j=[det(Aii′)]−1Aii′A (2)
where det((Aii′) denotes the determinant of the Jacobi matrix. For all the transformations carried out in this letter, the mathematical starting point is 3-dimensional euclidian space expressed in cartesian coordinates with isotropic permittivities and permeabilities εi,j=ε0δi,j and μi,j=μ0δi,j Kronecker delta).
A possible coordinate transformation for the design of a parallel beam shifter and a beam splitter consisting of a slab with thickness 2d and height 2h can be expressed by
x′(x,y,z)=x (3)
x′(x,y,z)=θ(h1−|y|)[y+ak1(x,y)]+θ(|y|−h1)[y+γ(y)k1(x,y)(y−s2(y)h)] (4)
x′(x,y,z)=z (5)
with
where 2h1 is the maximum allowed width of the incoming beam, a determines the shift amount and l=1 . . . n is the order of the nonlinearity of the transformation.
The transformation equations are defined for (|x|≦d), (|y|≦h) and |z|<infinity). For the case p=1, equations (3)-(5) describe a parallel beam shifter whereas for p=2 the equations refer to a beam splitter. The Jacobi matrix of the transformation and its determinant are
with
a21=θ(h2−|f1′|)[lak′i−1]+(f2−|h1′|)[lγk′t-1 (f1′−s2(y′)h)] (12)
a22=θ(h2−|f1′|)+θ(h2−|f1′|)[+γkt′] (13)
where
By equations (1)-(2) immediately above, it is straightforward to calculate the tensors of the transformed relative electric permittivity εi,j=ε/ε0 and the relative magnetic permeability μp=μ/μ0, which in the material representation are obtained as
is the metric tensor of the coordinate transformation. At this point it should be mentioned that only the domain with θ(h1−|f′1|)=≡1 has to be considered in the material implementation which simplifies the mathematical expressions.
Claims
1. A process of fabricating a transformation-optical/transformation-electromagnetic device comprising:
- computing a local form-invariant finite coordinate transformation for a transformation-optical device;
- computing at least one form-invariant coordinate transformation for a region surrounding the device;
- embedding the local form-invariant finite computed coordinate transformation into the region surrounding the device; and
- fabricating the device based at least in part on the computed coordinate transformations.
2. The process of claim 1 wherein said transformation-optical/transformation-electromagnetic device comprises a metamaterial.
3. The process of claim 1 further comprising providing non-reversibly change to the properties of electromagnetic waves in said device.
4. The process of claim 2 further including transmitting the changed electromagnetic properties to free space.
5. The process of claim 2 further including transmitting the changed electromagnetic properties to a different medium.
6. The process of claim 1 further including enabling transference of electromagnetic field manipulations from said device to a further medium.
7. A computer implemented method of designing transformation-optical/transformation-electromagnetic devices using embedded coordinate transformations of electromagnetic permittivity and permeability tensors, comprising:
- computing a local form-invariant coordinate transformation of Maxwell's Equations for a transformation-optical medium of an optical device;
- computing form-invariant coordinate transformations for a region surrounding the transformation-optical medium of the device;
- embedding the local form-invariant computed coordinate transformation into free space region or a second surrounding optical medium defining the device; and
- providing the computed coordinate transformations to a metamaterial fabrication process equipment.
8. A method of implementing effective permittivity and permeability transformation design for complex electromagnetic materials used in transformation-optical/transformation-electromagnetic devices, comprising:
- computing a local coordinate transformation for the transformation optical medium of a device;
- computing a coordinate transformation for a region surrounding the transformation-optical medium of a device;
- embedding the computed local coordinate transformation into computed coordinate transformation for the region surrounding the transformation-optical medium device;
- wherein electromagnetic filed manipulations from a first transformation-optical medium can be transferred to a second medium or free space, and wherein continuity of the transformation design across a medium-to-medium interface is retained.
9. A method of implementing effective permittivity and permeability transformation design for complex electromagnetic materials used in transformation-optical/transformation-electromagnetic devices, comprising:
- computing a local coordinate transformation for the transformation optical medium of a device;
- computing a coordinate transformation for a region surrounding the transformation-optical medium of a device; and
- embedding the computed local coordinate transformation into computed coordinate transformation for the region surrounding the transformation-optical medium device,
- thereby enabling a transference of electromagnetic field manipulations from a first transformation-optical medium to a second medium.
10. An electromagnetic/photonic beam-splitter, comprising:
- a metamaterial fabricated by computing a local coordinate transformation, computing a coordinate transformation for a region surrounding the metamaterial, embedding the computed local coordinate transformation into computed coordinate transformation for the region surrounding the metamaterial, and incorporating the metamaterial into the beam splitter; and
- means for directing electromagnetic radiation to the metamaterial.
11. A parallel beam shifter, comprising:
- a metamaterial fabricated by computing a local coordinate transformation, computing a coordinate transformation for a region surrounding the metamaterial, embedding the computed local coordinate transformation into computed coordinate transformation for the region surrounding the metamaterial, and incorporating the metamaterial into the beam shifter; and
- means for directing electromagnetic radiation to the metamaterial.
12. A process of fabricating a transformation-optical/transformation-electromagnetic device comprising:
- recording initial configurations of E-M fields in a medium on a Cartesian coordinate mesh;
- distorting/changing the coordinate mesh to a predetermined configuration represented by a distorted coordinate mesh;
- recording the distortions as a coordinate transformation between the Cartesian mesh and the distorted coordinate mesh;
- computing scaled E-M field values in accordance with the coordinate transformations;
- embedding the computed values into the coordinate space defining a transformation-optical device; and
- fabricating the device based at least in part on said embedded computed values.
13. The process of claim 12 wherein computed E-M fields are represented by one or more of electric displacement field D, magnetic field intensity B, or Poynting vector S.
Type: Application
Filed: Aug 19, 2014
Publication Date: Mar 19, 2015
Applicant: Duke University (Durham, NC)
Inventors: Marco RAHM (Kaiserslautern-Erfenbach), David R. Smith (Durham, NC), David A. Schurig (Durham, NC)
Application Number: 14/463,449
International Classification: G02B 1/00 (20060101); G06F 17/50 (20060101); G05B 19/4097 (20060101); G02B 27/10 (20060101);