(x,y) are the phase deviations and α is the carrier frequency. below an example is given for explanation. the deviations Ø are deviations from an equally - spaced and parallel fizeau pattern.

α. in fig.5 fourier series approach for t(x,y) is given with the coefficients c_ν . these coefficients are split into a phase term and an amplitude term.

for the present we neglect the amplitude term and consider only the phase term which contains the phase deviations of the initial interferograir ν-times as sensitive. further the phase-functions for c_ν and c_-ν, are the conjugate complex to each other.

obtain the coefficients c_ν. now we let pass c_ν y and c_-ν through two windows in the fourier plane and obtain a total amplitude t_±ν as shown in fig.3. if we record the intensity (t_±ν)².

then we get the carrier frequency 2ν_α and the phase deviation 2νØ(x,y).

the disadvantage of this filtered interferogram is the high carrier frequency. this difficulty can be overcome by moire-transformations either by a normal grating as done by bryngdahl /2/ or by using a second interferogram of the same object. as an example we take a fizeau-interferometer for a planeness test (see fig.4). we suppose the plates as flat, apart from a small region on one plate. now we make two interferograms with inversed wedge positions and an equal carrier frequency α. the moire-picture formed by both interferograms has two times the sensitivity of a single interferogram.

in pig.6 the initial interferogram was a multiple-wavelength interferogram with a λ/50-fringe separation. such interferograms can be obtained by an apparatus /3/. the subdivision of the λ/2-step is indicated by the bright fringes within the λ/2-range. the follow-

ing pictures constitute 5 stagea of the ± first order filtering. the adjustment of the fringes was controlled by a moire-test and the result is the nearly parallel cut or interference "contrast". the sensitivities are λ/200, λ/400, λ/800.

now we return to the amplitude effects caused by a nonuniform illumination a(x,y). in this case the width of the oraque or transparent regions depends on a(x,y). in fig.7 c_ν are

given in the dependence on the mean intensity a(x,y); the switch-exposure e₀ and exposure time t. for the sake of simplicity the visibility is put one. below c_ν is plotted against ξ.

c_ν is determined by the chebychev polynomials of the second kind. the amplitude t_ν has zeroes and phase changes π for ν>1. two kinds of experiments can be performed. the intensity is proportional to the squared polynomials and therefore no phase changes are to be considered in this case.

the light illuminates the combination a(x,y), sinusoidal grating and hard limiter. an example of such a modulated transparency can be seen here. after the development the hard print is put into the filtering apparatus. a window in the position of c_ν allows the light of the ν-th order to pass through it causing an isophotic picture of a(x,y) as can be seen in fig.9, where the outside part of a carton arc had been subject in the filtering process. the number of the diffraction orders is increasing from ν=1 to ν=6. the number of zeroes is increased by one from order to order. the isophotes are calibrated one by another according to the chebychev polynomials /4/.

the fourier coefficients c_ν are partly negative or in other words the phase of the light amplitudes has the value 0 and π. these phase changes can be detected through an interference experiment by superimposing a reference wave.

have a basic frequency of nearly 4α. then the zero order of the modulated object is diffracted in the direction of the ± fourth order of the object. also the first object orders are free from phase changes. therefore they can be used as reference waves for the fifth and the minus third object order. below three examples are given with ν=5,4,-3. the interference pattern indicates the phase changes.