Nano-Holographic Interferometry for In-Vivo Observations - Biomedical Optical Phase Microscopy and Nanoscopy - page 351

The microscopic slide acts as a Fabry P´rot interferometer and generates the multiple

diffraction orders that are shown in Figure 17.5 . Hence, the residual stresses and the
microscopic slide produce the effect equivalent of the grating shown in Figure 17.5 .

17.5 Determination of the Pitch of the Gratings

This section describes the fringe pitch measurement process formed by the wavefronts
corresponding to the ordinary and extraordinary beams. Whilst in the preceding analysis
only the real orders were considered, in this section, the orders corresponding to the
imaginary solutions of the diffraction equation derived by Toraldo di Francia [9] are
included. This information is retrieved from the FT of the image captured by the CCD
camera (see the inset of Figure 17.4 ). Sine values of the emerging wavefronts resulting
from the evanescent waves are plotted versus the fringe orders extracted from the FT shown
in Figure 17.4 .

As mentioned before, orders were measured in the FT starting from the zero order which is
taken as the origin of coordinates. Sine values are computed utilizing the equation
developed by Toraldo di Francia ( [9] , chapter III, section 47):

n g
n s

λ

sin φ 5 sin θ c 1 (17.18)

n s ð p o = N o Þ

where φ is the complex angle corresponding to the evanescent orders: the angle of
diffraction is evaluated with respect to the direction of the incoming laser beam, p o is the
pitch of the fringes, N o is the fringe order obtained from the FT, n s is the index of refraction
of the saline solution, and n g is the index of refraction of the microscope slide.

All the emerging orders except the first one are in the range of the complex sine function.
Since the sine values are greater than 1, the corresponding angles are complex numbers
with a real part and an imaginary part. Utilizing the plane-wave complex solutions of the
Maxwell equations for both ordinary and extraordinary wavefronts, one arrives to a system
of fringes whose variable intensity, which is finally recorded by the sensor, can be
expressed as follows:

x

2 p o N o n s

I ð x Þ 5 I o 1 I 1 cos (17.19)

n g

Figure 17.7 shows the fringe pitch variation for the different orders extracted from the FT.
The spatial frequency (i.e., pitch) was determined by dividing the field of view of the
recorded image by the fringe order. By fitting the experimental data, one obtains
hyperbolic-type trend functions that define the fundamental frequency of the fringe pattern.
A detailed frequency analysis of the observed images revealed that there are two spatial