Nano-Holographic Interferometry for In-Vivo Observations - Biomedical Optical Phase Microscopy and Nanoscopy - page 361

to the brightest region of the image in Figure 17.10A which is the zero order; (ii) the bright
areas shifted in the 6 X direction corresponding to orders 6 1; (iii) the bright areas shifted
in the 6 Y direction corresponding to orders 6 1; and (iv) a sinusoidal fringe. The actual
image ( Figure 17.10A ), the sum of the filtered orders ( Figure 17.10F ), and the numerical
simulation ( Figure 17.11B ) show a close resemblance. The model of image formation
outlined in Section 17.6 ( Figure 17.8 ) is fully supported by the previously presented
developments. The details of the procedure to relate observed shifts to the thickness of the
crystal are provided in Ref. [23] .
17.9.2 Determination of Thickness of Nanocrystals Through Evaluation
of the Optical Path Change
The procedure to obtain depth information from the recorded images is now described. For
each analyzed crystal, a particular frequency is selected. This frequency must be present in
the FT of the image and should be such that the necessary operations for frequency
separation are feasible. This means that frequencies that depend on the thickness of
nanocrystal must not be near the selected frequency. The frequency of interest is
individualized in the background of the observed object. Then, the selected frequency is
located in the FT of the object. A proper filter size is then selected to pass a number of
harmonics around the chosen frequency. Those additional frequencies carry the information
on the change in phase produced by the change in the optical path. In the next step, the
phases of the modulated and unmodulated carriers are computed. The change of phase is
introduced in Eq. (17.24) and the value of t is determined. The phase difference is not
constant throughout the prismatic crystal face since it is unlikely that the crystal face is
parallel to the image plane of the camera. Therefore, an average thickness is computed.
This process was repeated for all prismatic nanocrystals analyzed in the study.
Figure 17.12A shows the recorded image for a prismatic nanocrystal, while Figure 17.12B
shows the corresponding FT of the image. The crystal does not have central symmetry.
The “theoretical” phase pattern corresponding to a carrier composed of straight fringes is
obtained by taking the filter 1 3 1 about the selected frequency. The phase pattern is
modulated because of the presence of the nanocrystal. The phase pattern thus obtained must
be masked in order to match with the edges of the nanocrystal detected in the image
recorded by the microscope. The value of thickness can be obtained by averaging the
thickness distributions over the whole surface. By filtering the FT pattern of nanocrystal
shown in Figure 17.12B , one can obtain fringe patterns corresponding to different spatial
frequencies, e.g., 8.3 nm ( Figure 17.13A ), 5.6 nm ( Figure 17.13B ), and 5.3 nm
( Figure 17.13C ), respectively. These pitches are fractions of the pixels of the original
figure and hence are fractional orders resulting from the bicubic spline interpolation of the
recorded image. To determine the thickness of the nanocrystal, harmonics were selected