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350lect15 - 15 Plane-wave form of Maxwells equations...

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15 Plane-wave form of Maxwell’s equations, prop- agation in arbitrary direction Having seen how EM waves are generated by radiation sources and how spher- ical TEM waves develop a “planar” character over increasingly large regions as they propagate away from their sources, it is time to shift our attention to propagation and guidance phenomena using a plane-wave formalism. x y z D x 0 M HPBW = λ D x 2 D x 2 D 2 x λ Fresnel region r o x Perhaps the most “practical” rationalization of this switch from spherical to plane-wave emphasis is that waves produced by compact sources invariably “look” planar at the scales of practical receiving systems (that will study near the end of this course) situated afar. We wish to study wave solutions of Maxwell’s equations exhibiting the planar phasor form ˜ E = E o e - j k · r = ˆ eE o e - j k · r and time-domain variations Re { ˜ E e j ω t } = Re { E o e j ( ω t - k · r ) } = ˆ e | E o | cos( ω t - k · r + E o ) where wave vector k is to be found in compliance with ω and Maxwell’s equations according to some specific “dispersion relation” including the details of the propagation medium. 1
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For simplicity, the above phasor has been declared to be linearly polarized. Circular or elliptic polarized wave fields can be con- structed later on via superposition methods. k z x y z k r = ( x, y, z ) k x k y k · r = const. Linearly polarized wave field phasor above can be expanded as ˜ E = E o e - j k · r = E o e - j ( k x x + k y y + k z z ) assuming a wave vector k = ( k x , k y , k z ) = ˆ xk x + ˆ yk y + ˆ zk z expressed in terms of its projections ( k x , k y , k z ) along the Cartesian coordinate axes ( x, y, z ) . A special case we are familiar with is k x = k y = 0 , k z > 0 , when k = k z ˆ z = k ˆ z and e - j k · r = e - jkz as in plane TEM waves travelling in + z direction having a wavelength λ = 2 π k
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