Nano-Holographic Interferometry for In-Vivo Observations - Biomedical Optical Phase Microscopy and Nanoscopy - page 370

imposed by diffraction-limited optical instruments. This possibility was analyzed in many
publications [28 31] where examples are included. The full discussion of this problem is

beyond the scope of this chapter. This chapter also included FTHI holographic
interferograms of phase objects that were formed and analyzed along the lines utilized in
holographic interferometry.

In this chapter, procedures that are connected to numerical super-resolution methods
are utilized [32] . These procedures include basic theorems on analytic functions and the
numerical generation of additional pixels. The latter, called re-pixelation of the images, is
a standard procedure also utilized in numerical super-resolution. Furthermore, the fact that
the process of image formation includes different replications of the image due to the
different diffraction orders is an additional factor that results in an increase in the
information contained in an image. The effect of the successive images depends on the
structure of the image plane array of the CCD, i.e., on the so-called “factor of fullness”
of the detector array. However, it is equivalent to microscanning, which is to obtain
successive frames of displaced images. Finer sampling procedures produce higher
accuracies and better modulation transfer functions together with higher Nyquist
frequencies [33] .

The degree of accuracy in measurements was ascertained on the basis of nanocrystals of
sodium chloride. The actual sizes of crystals utilized in the theoretical computations
presented in Ref. [22] are of the same order of magnitude of the crystals observed in this
chapter. The geometric proportions of the crystals can be obtained from theoretical
considerations. The mean error of the measured aspects ratios is 4.6% with a standard
deviation of 6 6.6%. From the actual sizes of the measured sides, the mean absolute error in
the length measurements is on the order of 3 nm and the standard deviation is 6 3.7 nm. The
measured lengths of the crystals agree very well with the lengths computed from the
sodium chloride elementary cell size ( d 5 0.573 nm at room temperature). The overall
measurements performed show an overall standard deviation that is of the order of 6 5
elementary unit cells.

In this chapter, the problem of super-resolution was approached based on the theory of
information and it is possible to further appreciate the power of the Gabor's original idea
when he invented holography. The fact that with the holographic method one can measure
the amplitude and the phase of the signal leads to results showing that the obtainable
resolution depends basically on the atomic structure of the observed objects. In the case of
simple objects as the prismatic isomeric nanocrystals of sodium chloride, the average
standard deviation ( 6 5) of all of the performed measurements is within the abovementioned
quantity.

In a final summary, the possibility of observing near-field events in the far field was
experimentally verified for the first time ever. Mathematical models were developed and