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Electromagnetic

Graduate Course (Communications) 1st Semester, 1388-1389 Sharif University of Technology Contents of lecture 9

� High frequency approximations: • Physical

� Maxwell equations at high frequencies

� Eikonal equation and rays

of rays from conductive surfaces

Geometrical optics 2 Introduction

� We have, so far, approached the problem of scattering by

• Exactly solution (cylinders, spheres, wedges,…)

equation method (surface and volume formulations)

� If objects cannot be treated by exact methods, we have to resort to numerical solution of integral equations k � This is OK if the object is not much i larger than the , otherwise the number of variables will be huge!

Geometrical optics 3 Introduction

� For such large objects, the of integral equations is impossible, given the limitations of today’s computers

� The usual way to handle these cases is to use approximations (asymptotic solutions)

Geometrical optics 4

� Consider the simple problem of reflection by a flat perfect conductor. As in case of scattering by layered objects, one can distinguish a TEz and a TMz case.

� Rotate the coordinate system such that z is along the surface ˆ n ˆ and y is along nˆ � k i

nˆ kˆ ˆ ˆ ˆ ˆ ˆ kˆ r kr� k i �2� k i � n� n i

Geometrical optics 5 Physical optics

� In this system:

TEz Incident wave Ei ( r )� yˆE0 exp � � j ki � r �

Reflected wave Er ( r )� � yˆE0 exp � � j kr � r �

z i�k�k i,, x, 0, k i z �

�k, 0, �k k nˆ r� i,, x i z � kˆ ˆ r k i x

Geometrical optics 6 Physical optics

1 ˆ H( r )� k � yˆE exp � � jk � r � Incident wave i� i 0 i 1 H( r )� � kˆ � yˆE exp � � jk� r � Reflected wave r� r 0 r z

H nˆ i kˆ ˆ r k i x H r

Geometrical optics 7 Physical optics

� On the surface of the conductor:

nˆ� H � n ˆ � H � H � i r � Hi �2nˆ � H i H r

� The same can be shown for the TM case

Hi ( r )� yˆH0 exp � � j ki � r � nˆ� H � n ˆ �� Hi � H r �

Hr ( r )� yˆH0 exp � � j kr � r � �2nˆ � H i

Geometrical optics 8 Physical optics

� In both cases the surface current density is given by

Js � nˆ � H �2 n ˆ � H i 1 H( r )� kˆ � yˆE exp � � jk � r � i� i 0 i 1 �kˆ � nˆ � k ˆ Eexp � � j k � r � z � i� i � 0 i nˆ kˆ ˆ r k i x

Geometrical optics 9 Physical optics

� Now consider a perfect conductor with an arbitrary shape

� The incident wave induced surface currents on such objects, which generate the scattering field

ˆ F rˆ�expjk r ˆ � r�J ( r � ) dS � Js � n � H � � �� � � s S exp �� jkr � Ef () r �jk� rˆ � � r ˆ� F� r ˆ � � 4� r � �

jkexp �� jkr � Hf () r � �rˆ � F� r ˆ � 4� r

Geometrical optics 10 Physical optics

optics approximation: if the curvature of the surface of the perfect conductor is much larger than wavelength then surface locally approximated by a flat tangential surface

� Current on the surface approximated by

J s � 0

�2nˆ � Hi in the illuminated region Js � � � 0 in the shadow region

nˆ Js�2 nˆ � H i F rˆ�expjk r ˆ � r�J ( r � ) dS � � � � � � s Sill

Geometrical optics 11 Physical optics

1 H� Hexp� �j k � r � , H �kˆ � E i i,0 i i,0� i i ,0

� � F rˆ �2 exp �j k �k � r� � nˆ () r � dS � � H � �� � � � s i � � � i,0 � Sill �

J s � 0 � Physics optics can also be generalized to include effects. For this

we need to compute additional surface Sill

currents at the edges of rapidly Js�2 nˆ � H i varying surfaces

Geometrical optics 12 ��������������������������������������������������������������������������� ��������������������������������������������������������������������������������� �����������������������������������������������������