Geometrical Optics

Electromagnetic scattering Graduate Course Electrical Engineering (Communications) 1st Semester, 1388-1389 Sharif University of Technology Contents of lecture 9 � High frequency approximations: • Physical optics • Geometrical optics � Maxwell equations at high frequencies � Eikonal equation and rays � Reflection of rays from conductive surfaces Geometrical optics 2 Introduction � We have, so far, approached the problem of scattering by • Exactly solution (cylinders, spheres, wedges,…) • Integral 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 wavelength, 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 Physical optics � Consider the simple problem of wave 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 normal 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 )� yÊ0 exp � � j ki � r � Reflected wave Er ( r )� � yÊ0 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 � Magnetic field 1 ˆ H( r )� k � yÊ exp � � jk � r � Incident wave i� i 0 i 1 H( r )� � kˆ � yÊ 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 � yÊ 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 � Physics 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 diffraction 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 �� .

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