Tomographic Phase Microscopy (TPM) - Biomedical Optical Phase Microscopy and Nanoscopy - page 227

Numerical simulations have demonstrated that the Born approximation is valid when the
total phase delay of the E-field induced by the specimen is less than π /2 [21] . The thickness

of a single biological cell is typically about 10 m, with index difference with respect to the

medium about 0.03. Thus, the phase delay induced by typical cells is approximately

μ

π at a

source wavelength of λ 5 633 nm. Therefore, the Born approximation is not expected to be

valid for imaging biological cells. In this respect, the Rytov approximation is more relevant
than the Born approximation. It is not sensitive to the size of the sample or the total phase
delay, but rather to the gradient of the refractive index. Specifically, the Rytov
approximation is valid when the following condition is satisfied:

!

2

Uð !

n δ c rφ ð S Þ 2

Þ

φ ð S Þ 5 ln ; with (12.7)

!

π

U ð I Þ ð Þ

and n δ is the index variation in the sample over the length scale of wavelength. This
condition basically asserts that the Rytov approximation is independent of the specimen size
and only limited by the phase gradient rφ

(S)

. For a weakly scattering sample such as a

(S)

biological cell, the phase change rφ is linearly proportional to n δ to a first

approximation, such that the relation is valid when n δ { 1. The index variation n δ is in the

range of 0.03 0.04 for biological cells. Therefore, we can expect that the Rytov

approximation is legitimate in imaging biological cells, while the Born approximation is
subject to significant distortions in the reconstructed image.

As suggested by Devaney [23] , the implementation of the Rytov approximation in the
Fourier diffraction theorem requires a slightly different approach. Following Devaney's
method, we introduce the complex phase, ! ! Þ 5 e φð ! Þ ; and substitute this

into the wave equation ( Eq. (12.1) ). After applying the approximation given by Eq. (12.7) ,
we again obtain the Fourier diffraction theorem ( Eq. (12.6) ), but with U (S) replaced by
U ð S Þ

φð Þ; defined by U ð

Rytov defined as:

!

Uð ! Þ

U ð I Þ ð !

! U ð S Þ 5 U ð I Þ ð Þ ln (12.8)

Rytov

Þ

The rest of the reconstruction is the same as described in the text following Eq. (12.6) .

12.3 Experimental Implementation

12.3.1 Experimental Setup

ATPM instrument is shown in Figure 12.1 . It is designed to record complex E-field images
at various angles of illumination for the sample stationary at the sample stage of the
microscope [13] . The heterodyne Mach Zehnder interferometer [13] is used in the

instrument to record amplitude images and phase images. An He Ne laser beam