Time Line in the Study of Optical Properties of the MFE

1854: The Maxwell Fisheye is described in, Cambridge and Dublin Mathematical Jour- nal, in one number of set puzzles, in 1853. The challenge is to find the refractive index as a function of radius, given that a ray describes a circular path, and further to prove the focusing properties of the lens. The solution is given in the 1854 edition of the same journal. The problems and solutions were originally published anonymously, but the solution to this puzzle (and one other) was included in Niven’s The Scientific Papers of James Clerk Maxwell, which was released eleven years after Maxwell’s death6. Maxwell concludes:

“ It would require a more accurate investigation into the law of the refractive index of the different coats of the lens to test its agreement with the supposed medium. On the other hand, we find that the law of the index of refraction which would give a minimum of the aberration for a sphere of this kind placed in water, gives results not discordant with facts, so far as they can be readily ascertained.”

1944: In his bookMathematical Theory of Optics R. K. Luneberg gives a complete anal- yse of the lens including the forms of the wavefront, trajectories, optical line elements, etc. The new geometrical interpretation is introduced, and relation with a complex plane is described.

1958: Chen. T. Tai derives the Maxwell equations for MFE. In his paper on electrody- namics of Fisheye medium, Tai mentioned the lack of sufficient numerical studies, which would result in detailed analyse on wave behaviour in focusing point in the Fisheye medium [50,51]

1971: Yu. N. Demkov et al. Consider the resemblance of the Maxwell Fisheye Medium with a potential problem as ”...the hydrogen atom of optics...” They conclude that MFE is a perfect imaging device in the full wave regime.

1990: David Shafer suggested a modification on the original MFE, to make it practically realisable.

2009: Ulf Leonhard brought it back again to the community’s attention by re-deriving the full Green’s function solution and suggested to place a drain in the image position to completing the time-reversal symmetry.

From 2009 to 2015 vast amount of work dedicated to verifying whether Fisheye is capable of making a perfect imagine beyond the geometrical optics or not.

2015: We published a model theory which describes a crucial effect which never consid- ered before: the internal interaction between sources and detectors. These interactions have a significant influence on the resolution of the image in the MFE.

The imaging properties of MFE in the geometrical optics are well studied and understood [7,47, 52], however, introducing the Fisheye as a diffraction-free imaging device in full

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Maxwell Fisheye Medium in Geometrical Optics Regime 52

electromagnetic regime [16] needs a careful deliberation while there is a subtle difference between geometrical optics and wave regime when in comes to image quality. In this research, we assume to see whether MFE, which makes the perfect imaging in geometrical optics, holds its significance in the full electromagnetic regime or not.

Chapter 4

Electrodynamics of The Maxwell

Fisheye Medium

4.1

Introduction

In Chapter 3 we studied the geometrical optics of the MFE. The MFE is a perfect imaging device in that regime. In this chapter, we study the MFE in wave regime to determine if its perfect imaging feature remains invariant under the transition to the wave regime. The standard Helmholtz equation, as the wave equation, is associated with the homogeneous medium, and therefore is not the full descriptive equation of wave propagation through an inhomogeneous medium like the MFE. There are two standard methods for deriving the proper form of wave equations in the MFE. First, we can consider the Fisheye as an inhomogeneous stratified dielectric in RN, through which waves travel with varying speed. Second, we can use the transformation optics interpretation for gradient refractive index profile interpreted as the curved vacuum. Under specific conditions [35,53], the corresponding metric of the curved vacuum can be derived from the refractive index profile. Then the wave propagation in the induced geometry can be analysed. The equivalency of the two models is discussed in various studies [34,53,54]. Nevertheless, recognising the equivalent geometry in a general case is far from trivial. Fortunately, for a few highly symmetrical gradient lenses including MFE, the geometry is known from geometrical optics. n-dimensional MFE medium performs a geometry of n-dimensional hypersphere [47]. The connecting map between manifolds is a stereographic projection which projects the Rn to Sna, the equivalency

being a valid base on electrodynamics conformal invariance.

To solve any electrodynamic problem, we need to to decouple the field equations and boundary conditions into independent scalar functions. The methods used for any par- ticular problem depends on physical the symmetries of the medium and the geometrical symmetries of the boundary conditions. In the following, we study both approaches for

Electrodynamics of The Maxwell Fisheye Medium 54

deriving the decoupled scalar functions in the Maxwell Fisheye problem. Our first ap- proach is based on Debye’s potentials while for the second approach we need to develop modified Debye potentials. In a medium such as the MFE, wave propagation has its most compact and comprehensive expression in the forms of dyadic Green’s functions. Dyadic Green’s functions are bitensorial functions rather than conventional vector func- tions. The additional information that they carry is the source’s spatial orientation. We will see that the polarisation of the source has effect on the formation of the image. In this chapter we deriving the Green’s function of the Maxwell Fisheye medium.

Remember that the Fisheye is an inhomogeneous medium with a radially symmetric refractive index profile as:

n(r) = 2n0a 2

a2_+r2 (4.1)

whereais the characteristic length associated with the stereographic projection andn0 is a constant denotes the refractive index in the centre of the MFE: as r increases the refractive index decreases.