Quantitative phase amplitude microscopy IV: imaging thick specimens

The ability to image phase distributions with high spatial resolution is a key capability of microscopy systems. Consequently, the development and use of phase microscopy has been an important aspect of microscopy research and development. Most phase microscopy is based on a form of interference. Some phase imaging techniques, such as differential interference microscopy or phase microscopy, have a low coherence requirement, which enables high-resolution imaging but in effect prevents the acquisition of quantitative phase information. These techniques are therefore used mainly for phase visualization. On the other hand, interference microscopy and holography are able to yield quantitative phase measurements but cannot offer the highest resolution. A new approach to phase microscopy, quantitative phase-amplitude microscopy (QPAM) has recently been proposed that relies on observing the manner in which intensity images change with small defocuses and using these intensity changes to recover the phase. The method is easily understood when an object is thin, meaning its thickness is much less than the depth of field of the imaging system. However, in practice, objects will not often be thin, leading to the question of what precisely is being measured when QPAM is applied to a thick object. The optical transfer function formalism previously developed uses three-dimensional (3D) optical transfer functions under the Born approximation. In this paper we use the 3D optical transfer function approach of Streibl not for the analysis of 3D imaging methods, such as tomography, but rather for the problem of analysing 2D phase images of thick objects. We go on to test the theoretical predictions experimentally. The two are found to be in excellent agreement and we show that the 3D imaging properties of QPAM can be reliably predicted using the optical transfer function formalism.