Incoherent imaging of a dielectric object may be obtained by averaging out the multi-frequency amplitudes of the fields scattered from this object. The coherent scattered fields of the different optical paths or different phases produce the interference fringes. Since the periods of these interference fringes depend on the frequencies of the signal source, the multifrequency averaging of the intensity of the scattered fields may be shown to produce the average intensity without the interference fringes in the lit region, if the bandwidth in use is wide enough.
The measure of the visibility of the fringes may be given by the mutual coherence function or complex degree of coherence in the time domain. One may define the mutual coherence function as an ensemble average of the intensity of the time harmonic scattered fields in the frequency domain. One may obtain a closed form mutual coherence function for the scattered intensity from a conducting half plane. It is shown that the interference fringes generated by the addition of the incident and the edge diffracted fields of the conducting half plane are vanished completely if one takes the frequency averaging integral over the infinite frequency band. It is also shown that the average intensities in the lit and the shadow regions are that of the incident field and the zero shadow field, respectively, which clearly defines the location of the edge of the conducting half plane. Similar effects are confirmed from the fields scattered by the conducting and the dielectric circular cylinders.
Numerical calculations of the average intensities scattered from the conducting half plane, the conducting circular cylinder, and the dielectric circular cylinder are shown to converge to the intensity of the incident field in the lit region and zero shadow field without interference fringes as its bandwidth increases. Interference fringes generated from the two contributions such as the incident and the reflected waves of two different optical paths are shown to be reduced sufficiently provided that its coherence length defined by 2π multiplied by the inverse of the wavenumber bandwidth of the source signal is much smaller than the optical path difference. This definition of the coherence length is identical with that defined in the time domain coherence function.
Images of the dielectric cylinders of its circular, rectangular, and trapezoidal cross sections are obtained via the back projections of the frequency-averaged intensities of the scattered fields in either the circular rotational measurement or the cross-borehole measurement configuration. The closer the source or the field points to the circular dielectric cylinder and the wider the bandwidth, the closer reproduction of original cross section occurs in the circular rotational mode.
For the cross-borehole measurement of an air cylinder in a lossy opaque medium such as the underground medium, the lower frequency waves are preferred since the loss of the medium affects the higher frequency signals. It is shown that the more effective way utilizing the lower frequencies and the narrow bandwidth is to find the resonance frequencies that give the nulls or the deep dips in the shadow region of the air cylinder. The spherical cavity in the lossless background medium gives similar nulls and dips in its shadow region when the wavelength is comparable with the size of the spherical cavity. Various parameter dependences producing nulls and dips are calculated and it shows that the use of null frequencies with rather narrow bandwidth is very effective in the incoherent imaging of the cylindrical and spherical cavities in the lossy background medium. The dimension and the contrast of the objects obtained in the incoherent imaging exceeds far beyond the criterion of the validity of the Born inversion known as the diffraction tomography or the geophysical tomography which is the only available method to obtain the imaging of the dielectric objects to date. Incoherent imaging of such cavities in the cross-borehole configuration shows that their images are expanded considerably in the direction perpendicular to the vertical borehole. Similar expansion occurs in the diffraction tomography since the scattered fields are measured only in the boreholes and can not be measured along this direction.
The boundary element method is used for the calculation of the fields scattered from the dielectric cylinders of the rectangular, the square, and the trapezoidal cross sections. Its incoherent imaging shows the rounded corners due to the insufficient bandwidth used, which comes from the limitation of the numerical computation of the forward scattered fields.
