[ieee propagation in wireless communications (iceaa) - torino, italy (2011.09.12-2011.09.16)] 2011...

Beam Tracing Theory in Minkowski Space

E. Hasanoglu (Hasanov)

Department of Mathematics, Ik University, Merutiyet Ky, niversite Sokak, ile/stanbul, Turkey34980, tel. +90 216 528 71 71, fax: +90 216 712 14 74 e-mail: [email protected]

Abstract This paper provides a novel approach to beam theory in homogeneous lossless mediun. The main idea is to interpret the classic eikonal equation in three dimensional Minkowski space.

1 INTRODUCTION

In this paper we extend the method described in [3] to the case when the initial value function for the eikonal equation is a complex valued function. The main idea is to interpret the classical ray theory in the three dimensional Minkowski space. It is a remarkable fact that the eikonal equation due its special form, may have complex solutions even when the initial function is real-valued. Complex solutions, called complex rays, describe the behavior of the field in the shaded region where the real rays cannot not penetrate. Complex rays provide also the correct mathematical tool for interpretation of Gaussian beams.

2 EIKONAL EQUATION WITH COMPLEX VALUED INITIAL CONDITION

Consider the standard Helmholth equation 0 1 with boundary condition , , 0 , , 2 on a domain in the plane 0. Physically this problem corresponds to the diffraction problem on the plane when sources are located in the space 0. For large Eqs. (1)-(2) are asymptotically equivalent to the equations S 1 (eikonal equation) (3) And 2 0 (transport equation) (4) We are mainly interested mainly in solutions of Eqs. (3)-(4) subject to complex valued initial conditions , , 0 , , ,

with , 0 and , , 0 which means that sources, located in the space 0 exciting beam like initial field , , , in the plane 0. Since , satisfies Eq. (3), then we have , , 1 and 0, and the quantity 1 1 2 2 Is just real, so 1 Is real or imaginary. We say that the beam is space like if 1 0 The vector , , 1 ) is normal to the surface , , , 0 in the Minkowski space with the indefinite metric , , . For real valued , ,0 the quantity

1 is the area elemet of , measured in Minkowski metric. Applying formally the characteristic method to the equation (3), the solution subject to the initial condition , , 0 0,

978-1-61284-978-2/11/$26.00 2011 IEEE1440

Can be written as

1

1 , , or in vector form , , , , , , , 1 , , 1 , , , That is the normal (i.e. the direction of the propagation) consists of two components. The imaginary part of the normal is responsible for decreasing part of the field. One can use these considerations for interpretation of Gaussian beams. If we set , and then 1 that is , is a time-like surface if

and space-like if

The characteristic equations gives rise to the mapping 1 , 1 and

Solving these equations for and we get ,

and substituting in the third equation we get the phase function , , in the form , , which is, so called, the complex distance ([2]). 3 CONCLUSON The eikonal equation is a particular case of the Hamiltton-Jacobi equation. It is known that this equation can be solved in the framework of the Lagrangian manifolds. Lagrangian manifold is an dimensonal isotropic submanifold of 2 dimensional simplectic manifolds. Unfortunately, exact complex solutions of the Hamilton- Jacobi equation cannot be fond by using Lagrange manifolds. V.P. Maslov in [4] offered a method for approximately solution of Hamilton-Jacobi equation using complex Lagrangian manifold jerm.

References

[1] Keller, J.B. & Strefier, W. Complex rays with an application to Gaussian beams. J. Opt.. Soc. Amer., 61, 40-43, 1971.

[2] Deschamps, G.A. Gaussian beam as a bundle

of complex rays. Electron. Lett., 7, 684-685, 1971. [3] Hasanov, E. A new theory of complex rays

IMA Journal of Applied Mathematics, 69, 521-537, 2004. [4] Maslov V.P. Complex WKB method in non linear equations, (in Russian), Moscow, Nauka, 1977. [5] Maslov V.P. Operational methods, Moscow, Mir Publishers, 1974.

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[ieee propagation in wireless communications (iceaa) - torino, italy (2011.09.12-2011.09.16)] 2011...

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