Nano-Holographic Interferometry for In-Vivo Observations - Biomedical Optical Phase Microscopy and Nanoscopy - page 355

The orders 0 and 1 overlap in an area that depends on the process of formation of the image
(see Figure 17.8 ). The order 0 produces an image on the sensor, which is centered at a
value x of the horizontal coordinate; let us call this image S ( x ). The order 1 1 will create a
shifted image S ( x 2 Δ x ). The shift implies a change in the optical path between

corresponding points of the surface. In the present case, the trajectories of the beams inside
the prismatic crystals are straight lines and the resulting phase changes are proportional to
the observed image shifts.

The formation of the diffraction pattern was described for the X direction: a similar
argument can be made for the Y direction. The diffraction pattern of a prismatic object was
obtained. The goal is to extract metrological information from this pattern.

17.7 Procedures to Extract Metrological Information from
the Recorded Images

A digital form of the observed objects FT was obtained and it is necessary to develop
digital procedures to obtain metrological information on the object. There are a number of
ways to process the recorded images. Two direct procedures to obtain metrological
information are outlined in this section.

17.7.1 Image Shift

The zero-order image S ( x ) was defined in Section 17.6 and it was remarked that the
order 1 1 creates a shifted image of the particle, S ( x 2 Δ x ). The shift implies a change in

optical path between corresponding points of the surface. Since beams inside the prismatic
nano-sized object are straight lines, the resulting phase change Δφ is proportional to the

observed image shift:

Δφð x ; Δ x Þ 5 K p ½ S ð x Þ 2 S ð x 2 Δ x Þ (17.20)

where K p is a coefficient of proportionality. Equation (17.20) corresponds to a shift of
the image of the amount x . If the FT of the image is computed numerically, one can

apply the shift theorem of the FT. For a function f ( x ) shifted by the amount

Δ

Δ x , the

Fourier spectrum remains the same but the linear term ω Δ x is added to the phase— ω

sp
is the angular frequency of the FT. It is necessary to evaluate this phase change. The shift
can be measured on the image by determining the number of pixels representing the
displacement between corresponding points of the image (see Figure 17.8 , Part 4). Through
this analysis and by using FT, it is possible to compute the thickness t of the prismatic
nano-object.

sp