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.
Phase object visualization method is useful as a phase shift measurement technique when output image intensity signal is a known function of object phase or its derivative. This paper presents a comparison of performances of three real frequency domain filters: Foucault frequently called knife edge filter, Hoffman known in microscopy as a modulation contrast method and the semi-derivative filter. Its performance is simulated using Virtual Lab 1.O software in
4f imaging system with coherent illumination.
The space-time correlation functions of amplitude and intensity fluctuations are derived for a dynamic speckle generated by a flat diffuse object illuminated with a TEM<SUB>10</SUB> Hermite- Gaussian laser mode. The object is assumed to move in a plane with a constant velocity. The considerations are limited to the free propagation geometry. The speckle complex amplitude is assumed to obey circular Gaussian statistics. It is shown that intensity fluctuation is a space- time stationary process and its correlation function derived for both near and far diffraction fields depends on the angle which forms the vector of velocity and one of the axes of symmetry of the TEM<SUB>10</SUB> laser mode. This enables the simultaneous measurement of the direction and the magnitude of the object velocity. The velocimetry formulae are derived for a boiling far field speckle detected with a single point detector or with a circularly symmetric soft Gaussian aperture.
The main purpose of this review is to present the statistical properties of dynamic speckle produced by a diffuse object that moves with a constant velocity and the velocimetry applications that result from these properties. In this way the space-time correlation function of the speckle intensity variation is examined. Two typical speckle motions: boiling and translation are identified and discussed for various optical configurations in the diffraction and image fields in the case of the in plane object motion. The following measurement methods for the diffuse object velocity are introduced: the autocorrelation method, cross-correlation method, spatial filtering method, and zero-crossing method. There are several versions of the above mentioned methods. Special attention is paid to those which allow us to measure both the magnitude and direction of the velocity. In the vast majority of methods the Gaussian illuminating beam is used. In this review the TEM<SUB>10</SUB> illuminating beam is also taken into consideration. Then the autocorrelation methods of velocity measurement for the case of longitudinal motion are presented. It is shown that for this particular case the moving object need not to be a diffuse one, it can be a reflecting object as well. Possibility of the measurement of velocity which is arbitrarily directed with respect to the axis of the illuminating beam is also discussed. The speckle velocimetry methods in which the velocity is determined by means of a direct tracking and recording of the translating speckle generated by the moving object are mentioned. Finally some measurement systems in which laser speckles are used to evaluate subcutaneous flow of liquids are presented. Such systems were found useful in food conservation and medicine.
Two novel algorithms for the binarization of continuous rotationally symmetric real positive pupil filters are presented. Both algorithms are based on 1-D error diffusion concept. The original gray-tone apodizer is substituted by a set of transparent and opaque concentric annular zones. Depending on the algorithm the resulting binary mask consists of either equal width or equal area zones. The diffractive behavior of binary filters is evaluated. It is shown that the pupils with equal width zones give Fraunhofer diffraction pattern more similar to that of the original continuous-tone pupil than those with equal area zones, assuming in both cases the same resolution limit of printing device.
The secondorder statistical properties of the fully developed Gaussi an speck I e pattern generated by TEM1C mode have been consi dereci. The autocorrelation function of the complex speckle amplitude is calculated. The average visibility of interference fringes was measured and the experimental data are compared with the theoretical resul t.