**5. Conclusions**

176 Advanced Holography – Metrology and Imaging

lenses (L1 and L2) of focal length *f*=3*cm* and a pinhole (P) of 250*µm* placed in between the two mirrors M1 and M2, as shown in Fig. 9. The two beam splitters (BS1 and BS2) are ZnSe and *p=0*, the lowest order vortex, with a beam waist w*o=*2.4*mm* at the out-coupling mirror position. The LG mode *ψ*10 represents the object beam. The reference beam consists of a plane wave obtained with the spatial filter/expansion system L1PL2 that comprises two coated windows with a diameter of 50*mm*. The vortex output beam is split into reference and object beams at the BS1 position. The interference pattern between the plane wave reference and the object beam is recorded on the detection plane of an internally chopped pyroelectric video camera (Spiricon Pyrocam III Model PY-III-C-A), having an infrared sensor of 124*×*124 pixel elements of LiTaO3 with square size of 85*µm* and a pixel pitch Δ*x*=100*µm*. The reference beam interferes with the object beam at a small angle (α≤λ/(2Δ*x*)3°) as required by the sampling theorem. In Fig. 10(a) we can see the intensity distribution of the LG vortex beam *ψ*10 (*ℓ*=1 and *p*=0) with a dark central core, recorded by the pyroelectric array at distance *d=*52*cm* from the out-coupling mirror of the cavity. The digitized carrier modulated infrared interference pattern *I10(x,y)* in the (*x*, *y*)-plane of the

Because of the off-axis recording geometry, the two-dimensional Fourier transform of the fringe pattern is characterized by three dominant diffraction orders, as shown in Fig. 10(c) where the amplitude of the Fourier transform has been represented. The full complex two-dimensional Fourier transform of the fringe pattern can be written in the

<sup>10</sup> <sup>10</sup> <sup>10</sup> exp 2 dd, *x y x y I , I x, y I x, y i x y x y*

background illumination (Cuche et al., 2000; Liu et al., 2002; Onural, 2000), corresponding to

<sup>10</sup> *x xc - , - y yc* and

the Fourier spectrum of the vortex field and its complex conjugate, respectively, shifted by

two terms contribute to the two outer peaks of Fig. 10(c), and correspond to the positive and negative components of the vortex field spectrum. By taking the shifted inverse Fourier

The retrieved phase distribution in Fig. 11 shows a spiral profile with a discontinuity line typical of single topological charge vortex (White et al., 1991). We point out that the described method can be usefully employed for characterizing the vorticity of infrared beams of potential use in optical telecommunication applications, where the preservation of the purity of the mode along the propagation direction (Indebetouw, 1993) is a problem of

*I, I, - , -*

*x y dc x y x xc y yc*

 

10 10

*xc* and

 

*x xc y yc*

 <sup>10</sup> 

 

> 

*yc* along the *x*- and *y*-direction, respectively. These

(21)

 *x xc + , +y yc* represent

<sup>10</sup> ,

 

(22)

is the contribution to the spectrum given by the low-frequency

<sup>10</sup> *x xc - , - y yc* , the amplitude of the vortex field can be determined, as

*+ , +*

camera is presented in Fig. 10(b).

 

 

a peak at low spatial frequencies;

the carrier spatial frequencies

shown in Fig 10(d), respectively.

In eq. (21) *dc x <sup>y</sup> I ,*

form

whereas

transform of

crucial importance.

In this chapter we have shown that the numerical reconstruction of a whole optical wavefield through digital holography can be successfully performed in the mid-infrared regime using pyroelectric and microbolometric sensors. Amplitude and phase reconstructions were obtained by back-Fresnel propagation from the hologram recording plane to the object plane. Digital holography is closely related to digital image processing and to the mathematical models of imaging. We have described methods for improving the accuracy of the reconstruction which allows us to compensate for the loss of resolution at longer wavelength and the low spatial resolution of the pyroelectric camera array. It is worth pointing out that the improved spatial resolution of digital holography in the mid infrared regime is a significant improvement in a number of biologically relevant measurements related to biological cell and tissue analysis, where electric potential or light induced phase changes are expected to play a significant role in the characterization of complex biological structures. Infrared digital holography has also been applied for large object investigation.
