Tomographic Phase Microscopy (TPM) - Biomedical Optical Phase Microscopy and Nanoscopy - page 225

12.2 Theory—Optical Diffraction Tomography

12.2.1 Inverse Radon Transform

For the illumination of plane waves on a thin sample with small index contrast, the phase of
the transmitted wave is to a good approximation equal to the line integral of the refractive
index along the path of beam propagation. Therefore, the phase image can simply be
interpreted as the projection of refractive index, analogous to the projection of absorption in
X-ray tomography. Then the Fourier slice theorem, also known as inverse Radon transform,
can be used to reconstruct the 3D map. Since the theorem is relatively well documented and
the application of the theorem to the experimental data is rather straightforward, we guide
the readers to consult Ref. [21] .

12.2.2 Optical Diffraction Tomography

It is relatively straightforward to implement a deconvolution algorithm for creating a 3D
fluorescent image from a stack of full-field fluorescent images taken while scanning an
objective lens. Each fluorescent particle acts as a point source, and there is negligible
interference among the particles. The point spread function can be defined only by the
imaging system. On the other hand, it is more complicated to implement 3D deconvolution
for absorption and refractive index. Unlike fluorescent imaging, both amplitude and phase
images of the transmitted field must be recorded since samples affect both amplitude and
phase of the field. Moreover, interference among scatterers complicates the deconvolution
process. To fully describe the effect of interference, the wave equation must be solved. This
requires extensive computation time, and it is even more difficult to extract the structure of
objects from the transmitted E-field images.

Approximations such as Born and Rytov have been employed in the past to make this
problem relatively easy to solve [2,21] . According to these approximations, the relationship
between the 3D scattering potential and the 2D measured field can be simplified by
assuming that the scattered field is weak compared to the incident field. Using the Born
approximation, Wolf derived a formulation that enables reconstruction of a 3D object from
2D measured E-fields [1] . For each illumination angle, the Fourier transform of the 2D
measured E-field is mapped onto a spherical surface in the frequency domain of the 3D
scattering potential. This spherical surface is called the Ewald sphere. In this section, we
briefly introduce Wolf's original theory and Devaney's modification to adopt the first
Rytov approximation [23] .
With scalar field assumption, the propagation of light field, Uð ! Þ; through the medium can

be described by the wave equation as follows:

! ! !

2 U

2 U

Þ 1 k 0 n r ð ð Þ ð Þ 5 0 (12.1)