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All of the examples of holography thus far have involved the reconstruction of holograms that were generated and recorded using a reference wave. The recording of the hologram has been assumed to be at a certain plane, i.e., the detector or the CCD camera. The use of the reference allows for the recording of the phase (or depth information) of the object. However, as this chapter will demonstrate, the amplitude and phase of the optical field from an object are interrelated during propagation and obey the eikonal equations or the transport-of-intensity (TI) equations. It is therefore possible, in principle, to deduce the phase distribution of the original object if the amplitude (or intensity) is recorded at different distances during the propagation of the diffracting optical field. In fact, the amplitude and phase objects can be recorded and reconstructed using TI principles, as described in this section. Conventionally, phase-contrast microscopy is used to image weak phase objects. In cell biology, interferometric systems (along with phase-unwrapping algorithms) are usually employed to quantify the amplitude and phase of the field observed at a detector plane. If the complete 3D structure of the phase object is desired, then the data is acquired tomographically, and inversion algorithms are used to reconstruct the 3D complex index distribution of the object. The interferometric systems are usually bulky, sensitive to perturbations and noise, and usually require phase unwrapping, which may introduce artifacts. Intensity-based phase retrieval techniques, in which both the amplitude and phase of a field are retrieved from defocused intensity measurements, offer an experimentally simple solution to determine the phase quantitatively without phase-unwrapping algorithms. TI principles can be applied to record and reconstruct an asymmetric bulk phase object in 3D. Figures 10.1(a) and 10.1(b) show typical TI setups to tomographically record a 360-deg view for transmissive and reflective objects, respectively.
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