Electromagnetic fields and boundary conditions at the interface of generalized transformation media

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 80, Issue 6, 2009, Pages

  Bergamin, Luzi ^a  

^a EUROPEAN SPACE AGENCY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

BASIC APPLICATION; GENERALIZED TRANSFORMATION; ISOTROPIC MEDIUM; TRANSFORMATION OPTICS;

BOUNDARY CONDITIONS; ELECTROMAGNETIC FIELD MEASUREMENT; ELECTROMAGNETIC FIELDS; VACUUM;

MAXWELL EQUATIONS;

EID: 72149088737     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.80.063835     Document Type: Article

References (17)

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4. Often the terms constitutive equation or constitutive relation refer to media, only. Since in the present context the constitutive relations of transformation media are closely related to the vacuum relation, we follow and call the following equation constitutive relation of vacuum.
5. Throughout, this paper all equations are written in natural units ε0 = μ0 =c=1. Furthermore, Einstein's summation convention is used, which implies a summation over all repeated indices. Further details of our notation are explained in Appendix x0.
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11. As is seen from Fig.f1, the explicit coordinates of laboratory space with medium are distinguished from the ones of empty laboratory space. This is necessary, since a particular point in spacetime may be represented by different values of the coordinates in empty laboratory space and in laboratory space with the medium.
12. This dispersion relation is the result of the mathematical manipulations of transformation optics and it is not claimed that it corresponds to any real medium.
13. These independent transformations do not establish a symmetry, since the constitutive relation is not invariant. Nevertheless, they are invariant transformations of the equations of motion.
14. From now on, the notation of generalized transformation optics will be used, unless explicitly mentioned differently. The case of standard transformation optics always follows by a simple identification x= μ = x̄ μ.
15. Since the two sets of equations in Eqs. d9 d10, depend on two different sets of fields, a rescaling of the fields of one set also represents an invariant transformation of the equations of motion. However, these rescalings change the constitutive relation as they change permittivity and permeability by a (not essentially positive) constant. This implies that the interpretation of a negative refractive index in transformation optics actually is the effect of an ambiguity. This is most easily seen in the relativistically covariant formulation: according to a negative refractive index is found if εijk in Eqs. d9 d10 changes sign under the transformation, as this sign has to be absorbed by a rescaling of E and H with a negative constant. However, in the relativistically covariant formulation the (four-dimensional) Levi-Civita symbol only appears as an overall factor in the constraint ε μνρσ ∂ μ Fρσ =0, and thus the change of sign is without any consequences. Thus, starting from the relativistically covariant formulation, the negative refractive index appears rather as an ambiguity.
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