Diffraction, Fourier Optics and Imaging by Okan K. Ersoy

By Okan K. Ersoy

This publication offers present theories of diffraction, imaging, and similar subject matters according to Fourier research and synthesis thoughts, that are crucial for realizing, interpreting, and synthesizing sleek imaging, optical communications and networking, in addition to micro/nano platforms. purposes lined contain tomography; magnetic resonance imaging; artificial aperture radar (SAR) and interferometric SAR; optical communications and networking units; computer-generated holograms and analog holograms; and instant structures utilizing EM waves.

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Mechanical waves, such as longitudinal sound waves in a compressible fluid, are governed by the wave equation @ 2 uðx; tÞ & @ 2 uðx; tÞ ¼ ; @x2 Kc @t2 ð3:2-15Þ where & is the fluid density, and Kc is the compressibility. Hence, the phase velocity is given by sffiffiffiffiffi Kc v¼ & ð3:2-16Þ WAVES 29 B. Heat diffuses under steady-state conditions according to the wave equation Kh @ 2 uðx; tÞ @uðx; tÞ ; ¼ @x2 @t s ð3:2-17Þ where Kh is the thermal conductivity, s is the specific heat per unit volume, and uðx; tÞ is the local temperature.

7-1) in to r  E ¼ Àm @H @t gives the following: kEy^ex À kEx^ey ¼ Àmw½Hx^ex þ Hy^ey Š ð3:7-3Þ 1 Hx ¼ À E y  1 H y ¼ Ex ;  ð3:7-4Þ Hence, where  is called the characteristic impedance of the medium. It is given by pffiffiffiffiffiffiffi w ð3:7-5Þ  ¼ m ¼ vm ¼ m=e k Note that E  H ¼ ðEx Hx þ Ey Hy ÞejðkzþwtÞ ¼ 0 ð3:7-6Þ Thus, the electric and magnetic fields are orthogonal to each other. The Poynting vector S is defined by S ¼ E  H; ð3:7-7Þ which has units of W/m2, indicating power flow per unit area in the direction of propagation.

5 Derive Eq. 8-5). 8-2) becomes Uðr; fÞ ¼ 2ðp 1 ð dy 0 uðr; yÞeÀj2prrðcos y cos fþsin y sin fÞ Jdr ð2:8-9Þ 0 where J is the Jacobian given by   @x   J ¼  @r  @y  @r  @x    @y  ¼  cos y @y   sin y  @y  Àr sin y  ¼r r cos y  ð2:8-10Þ where j  j indicates determinant. Substituting J ¼ r in Eq. 8-9) gives the desired result. 6 (a) Find the Hankel transform of the cylinder function cylðrÞ defined by 8 1 > > >1 0 r < 2 > > < 1 1 cylðrÞ ¼ ð2:8-11Þ r¼ > 2 2 > > > > 1 :0 r> 2 (b) Find the Hankel transform of cylðr=DÞ.

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