Keywords: 
• geometric optics /
• phase retrieval /
• transport of intensity equation /
• image interpolation /
• optical processing

Abstract: Phase, as an important property of light field, is difficult to be obtained by the existing detection equipment, which can only detect the intensity information of the light field, however, losing the phase information, hence the phase problem can be summarized as the demand of retrieving a sample's complex-field from measurements of intensity. Transport of intensity equation (TIE) based method is one of the typical phase retrieval approaches. When the intensity distribution of the test plane and the axial intensity derivative are known, the phase distribution of the test plane can be calculated directly by solving the equation. Conventionally, intensity derivative is approximated by a finite difference between one in-focus image and one defocused image or two defocused images recorded symmetrically about the focal plane, therefore the proper selection of the distance parameter between defocused image and in-focus image becomes particularly important. A novel approach combining the image interpolation and lens-based TIE was proposed. Firstly, the relationship among two defocused images and one focused image captured was described in geometrical optics model. Secondly, new defocused images with different blur parameters were calculated by image interpolation. Lastly, these new images interpolated and the focused images captured were applied to calculate the phase information. The method could obtain the desired intensity distribution at any positions rapidly with only three captured intensity images, without the mechanical movement of CCD or sample, providing an available way for some special occasions that the intensity acquisitions at appropriate location existed a certain restriction. A practical image acquisition platform was also constructed, the interpolated intensity image was compared with the intensity image obtained by CCD to verify the correctness of the interpolation result, the phase retrieval results under two different computational intensity derivative conditions were given relatively. The experimental results presented verified the feasibility and effectiveness of the method.