Methods of imaging phase objects are considered. First the square-root filter is inferred from a definition of fractional-order derivatives given in terms of the integration of a fractional order called the Riemann-Liouville integral. Then we present a comparison of the performance of three frequency-domain real filters: square root, Foucault, and Hoffman. The phase-object imaging method is useful as a phase-shift measurement technique under the condition that the output image intensity is a known function of object phase. For the square-root filter it is the first derivative of the object phase function. The Foucault filter, in spite of its position, gives output image intensities expressed by Hilbert transforms. The output image intensity obtained with the Hoffman filter is not expressed by an analytical formula. The performance of the filters in a 4f imaging system with coherent illumination is simulated by use of VirtualLab 1.0 software.

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We propose three amplitude filters for visualization of phase objects. They interact with the spectra of pure-phase objects in the frequency plane and are based on tangent and error functions as well as antisymmetric combination of square roots. The error function is a normalized form of the Gaussian function. The antisymmetric square-root filter is composed of two square-root filters to widen its spatial frequency spectral range. Their advantage over other known amplitude frequency-domain filters, such as linear or square-root graded ones, is that they allow high-contrast visualization of objects with large variations of phase gradients.

There is a need for a frequency-domain real filter that visualizes pure-phase objects with thickness either considerably smaller or much bigger than 2π rad and gives output image irradiance proportional to the first derivative of object phase function for a wide range of phase gradients. We propose to construct a nonlinearly graded filter as a combination of Foucault and the square-root filters. The square root filter in frequency plane corresponds to the semiderivative in object space. Between the two half-planes with binary values of amplitude transmittance a segment with nonlinearly varying transmittance is located. Within this intermediate sector the amplitude transmittance is given with a biased antisymmetrical function whose positive and negative frequency branches are proportional to the square-root of spatial frequencies contained therein. Our simulations show that the modified square root filter visualizes both thin and thick pure phase objects with phase gradients from 0.6π up to more than 60π rad/mm.

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