Transformation Fluctuation Electrodynamics: Application of Transformation Optics upon Thermal Radiation Illusion 

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Scattering cancellation using mantle meta-surfaces

Despite the success of metamaterials to realize TO based cloaks, the performance of such cloaks is low such that they are impractical for real life usages. The main draw backs is the media dispersion, the cloaking can be almost perfect but just at one fre-quency. Another draw back is that the TO based cloaks are bulky, highly anisotropic and in-homogeneous. Such drawback motivated the research on the usage of meta-surfaces to replace the TO based cloaks. A meta-surface is a sub-wavelength layer (ideally zero thickness) with its surface properties are controlled by the homogenized behaviour of surface allocated sub-wavelength unit cells. In what follows we summa-rize the state of art concerning the usage of meta-surface (also called mantle cloaks) in invisibility and illusion purposes. The technique of using mantle metasurfaces in order to achieve invisibility and electromagnetic illusion, is often called scattering cancellation technique. The reader is referred to the following review papers [32] and [22] for detailed literature surveys concerning scattering cancellation technique.
Invisibility cloaking using scattering-cancellation-technique is classi ed into such plasmonic and mantle cloaking. The main theme is to design a cloak that scatters the incident eld such that a destructive interference occurs between the scattering due to the cloaked object and the scattered eld due to the cloak itself. Plasmonic cloaks are mainly bulky cloaks with negative permittivity to coat positive permittivity dielectric object so that the total induced electric polarization, of the cloak and the cloaked object togather due to an incident eld, is cancelled so the scattered eld is minimized. On the other hand, mantle cloaks are optically thin conformal metasurface which is de ned such that the surface impedance of the cloak with the coated object, is the same as the surface impedance incident eld.

Thermal radiation from meta-surfaces

Fluctuation electrodynamics theory is the extension of Nyquist noise theorem [90] of thermal noise in electrical conductors which was later formally reformulated by Callen and Welton [13] as Fluctuation dissipation theorem (FDT). Actually lot of physical phenomena are explained by FDT, among others, thermal radiation [102], thermal noise in electrical circuits [115]. Actually both phenomena are electromag-netic uctuations with their physical origins are the same but occur at di erent wave length ranges. It is interesting to note that one can even use the same models to compute and control such behavior for example, electrical engineers design low noise electrical circuit components while physicists design photonic crystals with photonic band gaps to prohibits thermal radiation with the frequency range of the band-gap [75],[24] or just to tailor the emission properties [28] and [86]. Such similarity between the two phenomena propose that both can be analyzed by the same theoretical tools. Thermal noise in electrical systems is typically analyzed using electrical network the-ory, while thermal radiation is calculated by uctuation electrodynamics by solving Maxwell’s equation. Actually distributed and lumped circuit models are known to be used electromagnetic problems, for example, spectral transverse equivalent network is used to compute electromagnetic radiation from dipoles embedded in multilayer dielectric or modal transmission line model is used to compute for electromagnetic di raction from multilayer di raction gratings [96]. Other circuit models had been used to model mutual coupling between cavities, proximity antennas and even elec-tromagnetic propagation. Actually, it was S. Masloviski et al who proposed treating thermal radiation as thermal noise [78], [109] and using the former circuit models in both lumped forms to compute thermal radiation from arbitrary planar magneto-dielectric slabs [78] and for spherical shaped thermal emitters in [80]. A distributed circuit model for thermal noise from one dimensional transmission lines was proposed in [113]. In [79], without computations, Masloviski et al showed in analytically that the thermally radiated time averaged Poynting vector from arbitrary thermal emitter can computed in terms of the surface impedance in a similar manner the thermal noise power from the lumped N-port network was investigated in [115] and [40]. The main focus of the research work [80],[79], [109] and [78] is to maximize the thermal energy transfer between a thermal reservoir and energy collecting heat sink.
On the other hand, we are concerned with thermal radiation illusion. In chap-ter V, we propose using using lossy mantle metasurface in order to achieve thermal radiation illusion. Mantle metasurface were successful to achieve invisibility and scat-tering illusion using scattering cancellation technique. In a similar manner, scattering cancellation technique can be extended such that mantle meta-surfaces are used to achieve thermal radiation illusion. The electromagnetic properties of metasurface is typically quanti ed by its own surface impedance. Also, as it will be shown in chapter IV, meta-surfaces are the higher dimensional generalization of the lumped components of the one dimensional circuit network. Accordingly the former circuit models can be used to compute thermal radiation from a mantle meta-surface. Ac-tually in chapter V, we will show that the surface impedance is the sole factor to control thermal radiation signature. In other words, in order to duplicate a thermal radiation pattern or signature it is enough to design the surface impedance.

Near eld thermal radiation computation

A lot numerical schemes had been proposed to compute thermal radiation from thermal emitters with arbitrary geometries based on FE [92]. First, analytical com- putations of thermal radiations are restricted to simple geometries that posses sym- metries,i.e: planar [98], cylindrical and spherical [87] geometries. Thermal discrete dipole approximation (TDDA) is a discrete dipole approximation technique with the  dipoles are uctuating according to Eq.(1.21) while thermally radiated power are computed using the electric and magnetic green functions. Method of moments (also called boundary element method) had been reported, where thermal radiation is orig- inated from surface uctuating current [101]. Mode matching techniques had been reported, for instance, in [87] radiated heat transfer between two spheres is computed based on the coupling between the spherical bases of both spheres, and coupling (the inner product) between these di erent set of bases is done using addition theorem. Finally, nite time di erence scheme had been used where thermally radiated is time averaged over many simulation loops taking into consideration the statistical nature of the thermal sources.
In this work we make use of TDDA method with Maxwell’s equations are solved by COMSOL 3.5A Multiphysics solver which uses nite element method technique. TDDA method with its COMSOl numerical implementation is summarized in ap-pendix D. TDDA method is utilized for its relative simplest and easiness to implement and it is used to verify the di erent theoretical models proposed in this thesis.

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Duality Transformation

Duality transformation [70], [68] which also recently referred to as Field trans-formation [71], is transformation in the vector eld space. Duality transformation is based on the symmetry of electric and magnetic quantities in the Maxwell equations. Similar to transformation optics, it can be applied to obtain a solution for the dual problem through transforming the solution of the original problem [69]. The di er-ence between transformation optics and duality transformation is shown in Fig.1.13. Duality transformation is limited in the sense that a dielectric media can be only transformed into non reciprocal media and also, in the sense that the transforma-tion is space independent. Actually in [71], eld transformation was combined with transformation optics to design an invisibility cloak for a purely dielectric cylinder completely transparent in air for both TE and TM polarizations. This is why in chap-ter VI, eld transformation is investigated while investigating its e ect on uctuation electrodynamics theory.

Inverse Transformation at the Static regime

Since the constraints on the cloak are mainly imposed by the objects !and , so in the rst part of this section, the analytical formulation of the two-dimensional inverse problem in polar coordinates is outlined , then we study the analytically traceable cases where the two objects to cloak ( ) and to mimic (!) are prede ned in conductivity, together with the @ frontier, which forces the set of possible H transformations and thus the range of @! shapes.In what follows we focus on the case where ! is of isotropic thermal conductivity.
Analytical formulation of the two-dimensional problem in polar coordi-nates Here, we focus on searching for the H transformations that can satisfy Eq. (2.6), when the two conductivity matrices are forced. For this purpose, the MH matrix is rst analyzed in detail. An analytical calculation can be developed if we simplify the problem by considering a 2D geometry with polar coordinates (r; ). The transformation is written as: H(r; ) = (r0; 0) = [fH (r; ); gH (r; )] (2.12).
where r and r0 are the radial distances in the departure and arrival spaces, re-spectively, and and 0 are the polar angles in the departure and arrival spaces, respectively. The transformation H is expressed in terms of the r0 = fH (r; ) and 0 = gH (r; ) where fH and gH are the transformation functions to be speci ed. The MH matrix is the Jacobian matrix of the transformation and it relates the gradient operators of polar coordinate systems in the departure and arrival spaces as following: rr; T (r; ) = [M] rr0; 0 T (r0; 0).

Table of contents :

DEDICATION
LIST OF FIGURES
LIST OF APPENDICES
LIST OF ABBREVIATIONS
PREFACE
LIST OF PUBLICATIONS
CHAPTER I. Introduction
1.1 Motivation
1.2 Invisibility and Illusion Cloaking
1.2.1 Transformation optics cloaks using metamaterials .
1.2.2 Scattering cancellation using mantle meta-surfaces .
1.3 Thermal Energy Transfer
1.3.1 Heat conduction
1.3.2 Thermal radiation
1.3.3 Thermal radiation from meta-surfaces
1.3.4 Near eld thermal radiation computation
1.4 Duality Transformation
II. Controlling Temperature Signature using Transformation Optics
2.1 Introduction
2.2 Inverse Transformation
2.3 Inverse Transformation at the Static regime
2.4 Cloak Design Procedure
2.5 Numerical Simulation
2.6 Conclusion
III. Transformation Fluctuation Electrodynamics: Application of Transformation Optics upon Thermal Radiation Illusion 
3.1 Introduction
3.2 Transformation optics and Fluctuation Electrodynamics
3.3 Transformation Fluctuation Electrodynamics
3.4 Two Dimensional E/H-polarization Thermal Radiation
3.5 Two Dimensional E/H-Polarization Camou ages
3.6 Limitations and Conclusions
IV. Cloaking and Scattering Camou age using Transformation Optics based Mantle Meta-surfaces
4.1 Introduction
4.2 Meta-surface models: GSTC versus Impedance Matrix
4.2.1 Impedance matrix representation
4.2.2 GSTC
4.3 Equivalence between Scattering Cancellation and Discontinuous T.O
4.3.1 1D discontinuous space transformation
4.3.2 2D polar/cylindrical discontinuous space transformation
4.3.3 Compression of non cylindrical shell into an inhomogeneous and spatial dispersive MS:
4.4 Conclusion
V. Circuit Model for Thermal Radiation from Arbitrary Ther- mal Emitter at Constant Temperature
5.1 Introduction
5.2 TR Pattern for Arbitrary Thermal Emitter
5.3 Equivalent Surface Admittance
5.3.1 Surface impedance/admittance: Quick Review
5.3.2 Equivalent surface admittance for arbitrary shaped emitter
5.3.3 Admittance matrix representation
5.3.4 Reciprocal surfaces
5.4 GSTC description of an Equivalent Surface
5.5 TR from Circularly Symmetric Object
5.5.1 Equivalence between surface admittance and GSTC descriptions:
5.5.2 Numerical example
5.6 TR from arbitrary thermal emitter characterized by a Surface Impedance
5.6.1 Nyquist thermal sources
5.6.2 TR from Nyquist sources
5.7 Numerical Calculation
5.8 Conclusions and Limitations
VI. Fluctuation Electrodynamics in Reciprocal Chiral Media using Field Transformation and Discontinuous Field Transfor- mation based Mantle Cloaks
6.1 Introduction
6.2 Field Transformation
6.2.1 Decomposition into longitudinal and transverse components
6.2.2 Transforming between virtual and physical spaces . 210
6.3 Transformation Fluctuation Electrodynamics
6.3.1 Covariance of FE under FT transformation
6.3.2 FE in the physical space
6.3.3 Thermal illusion using FT based cloaks
6.4 Discontinuous Field Transformation
6.4.1 GSTC model for a discontinuous FT based mantle cloak
6.4.2 Circuit model for the discontinuous FT mantle cloak
6.5 Conclusion
VII. Conclusions and Future Work
APPENDICES
A.1 Introduction
A.2 Helmholtz Equation in Isotropic and Homogeneous Media
A.3 Thermal Radiation from Planar Slabs
A.3.1 Telegraphic equation
A.3.2 TR using Z-matrix
A.3.3 Masloviski’s model
A.4 Thermal Radiation from Cylindrical Shells
A.4.1 Radial transmission line
A.4.2 TR from cylindrical shell using circuit model
A.5 Thermal Radiation from Semi-Innite Planar or Cylindrical Emitters with Constant Temperature
B.1 Informal introduction
B.1.1 Covariant vs contravariant components of a vector .
B.1.2 Geometric algebra objects
B.2 Tensor calculus convention
B.3 Space Transformation
B.3.1 Tensors of rst order
B.3.2 Tensors of second order
B.4 Coordinate Transformation of Maxwell’s and Fourier’s law of Heat Conduction
B.4.1 Maxwell’s Equations
B.4.2 Space Transformation of Fourier’s law of heat conduction
C.1 Generalized Kircho’s laws for arbitrary Thermal Emitters .
C.2 Proof for the circuit model using Landauer Formalism
E.1 Fluctuation Dissipation Theorem
E.2 Fluctuation Electrodynamics correlations: Simplied derivation
BIBLIOGRAPHY

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