**4. Digital holography for wavefront reconstruction of infrared Laguerre-Gaussian modes**

In this section we present some applications of digital holography for analyzing the spatial distribution and signature vorticity of Laguerre-Gaussian (LG) modes. LG modes have been largely employed for interesting application such as optical trapping, rotational frequency shift and optical manipulation of micrometric systems (Allen et al., 1992, 1999; Arecchi et al., 1991). The concept of singularities in electromagnetic fields possessing an orbital angular momentum (Soskin & Vasnetov, 2001; Leach et al., 2002) has an extensive literature because of their interesting properties and potential applications. Laguerre-Gaussian beams are examples of optical fields, that possess wave-front singularities (optical vortices) of topological charge *ℓ*, where *ℓ* can take any integer value, that are related to the azimuthal angular dependence exp*i* in the transverse distribution of the optical field. The azimuthal mode index *ℓ* (orbital helicity) is physically related to OAM, per photon, of the LG modes. The LG modes of azimuthal index *ℓ* and radial index *p* are

$$\mathcal{W}\_{\ell p}(x,y,z) = \sqrt{\frac{2p}{\text{m}\left(\ell+p\right)}} \frac{1}{w\left(z\right)} \left(\frac{r\sqrt{2}}{w\left(z\right)}\right)^{\ell} \exp\left(-\frac{r^{2}}{w\left(z\right)^{2}}\right) \times$$

$$L\_{p}^{\ell}\left(\frac{2r^{2}}{w\left(z\right)^{2}}\right) \exp\left(-\frac{ikr^{2}}{2\mathcal{R}\left(z\right)}\right) \exp\left(-i\ell\phi\right) \times \tag{20}$$

$$\exp\left[-i\left(2p+\ell+1\right)\arctan\left(\frac{z}{z\_{r}}\right)\right],$$

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 175

of the fundamental mode or higher order cavity modes. The CO2 laser source operates at the wavelength λ=10.6*µm*. The laser cavity is defined by a partially reflective flat mirror (R=95%) and an out-coupling mirror (R=90%) mounted upon a piezoelectric translator. As aforementioned, the LG mode is selected with a diaphragm and is horizontally polarized by

The interferometric configuration comprises two mirrors, indicated as M1 and M2, and two beam splitters, BS1 (70T/30R at 45° angle of incidence) and BS2 (50T/50R at 45° angle of incidence) as depicted in figure 9. The output laser beam is a linear superposition of TEM10

Fig. 10. Infrared Laguerre-Gaussian beam analysis based on Fresnel reconstruction of offaxis digital hologram: (a) intensity distribution of the LG mode *ψ*10 recorded at the IR camera plane; (b) digitized interference intensity profile between LG object beam and plane wave

reference; (c) Fourier transform of the carrier modulated interference pattern; (d)

reconstructed vortex beam amplitude

01) corresponding to =1

means of an intracavity ZnSe Brewster window.

and TEM01 Hermite-Gaussian modes (the doughnut mode TEM\*

where *r* is the radial coordinate, arctan( / ) *y x* the azimuthal coordinate and *L x <sup>p</sup>* are the generalized Laguerre polynomials with azimuthal and radial mode numbers *ℓ* and *p*, respectively. The production and characterization of free space Laguerre-Gaussian beams at comparatively longer wavelengths gives the possibility of using optical vortices for observing the transfer of angular momentum to relatively larger objects, since, for a fixed power, the angular momentum in the beam is proportional to the wavelength. Quantitative characterizations of the structure and signature of the vorticity of infrared LG beams can be obtained by using an interferometer for recording the interference pattern of the LG mode (the object beam) and an external plane-wave reference beam (De Nicola et al., 2010). The experimental set-up is shown in Fig. 9.

Fig. 9. Outline of the Mach-Zehnder interferometric setup. A CO2 laser source operating at the wavelength λ=10.6*µm* produces an LG mode *ψ*10, divided at the BS1 (beam splitter) position into an object beam (LG mode) and a reference beam. The latter is a plane wave beam obtained after having spatially filtered and expanded the vortex field with the collimating lenses and pinhole system L1-P-L2. M1 and M2 are mirrors. The object beam interferes with the reference beam at the beam splitter BS2 position. The interference pattern is recorded with an IR pyroelectric camera.

A vortex optical field in the mid infrared range is produced by inserting a circular diaphragm of wide aperture (Fresnel number larger than one) inside the optical cavity of a CO2 laser. This operation easily leads to laser emission in the doughnut mode TEM\* 01 ,but the optical field sometimes presents impurities originating from the residual contributions

the generalized Laguerre polynomials with azimuthal and radial mode numbers *ℓ* and *p*, respectively. The production and characterization of free space Laguerre-Gaussian beams at comparatively longer wavelengths gives the possibility of using optical vortices for observing the transfer of angular momentum to relatively larger objects, since, for a fixed power, the angular momentum in the beam is proportional to the wavelength. Quantitative characterizations of the structure and signature of the vorticity of infrared LG beams can be obtained by using an interferometer for recording the interference pattern of the LG mode (the object beam) and an external plane-wave reference beam (De Nicola et al., 2010). The

Fig. 9. Outline of the Mach-Zehnder interferometric setup. A CO2 laser source operating at the wavelength λ=10.6*µm* produces an LG mode *ψ*10, divided at the BS1 (beam splitter) position into an object beam (LG mode) and a reference beam. The latter is a plane wave beam obtained after having spatially filtered and expanded the vortex field with the collimating lenses and pinhole system L1-P-L2. M1 and M2 are mirrors. The object beam interferes with the reference beam at the beam splitter BS2 position. The interference pattern

A vortex optical field in the mid infrared range is produced by inserting a circular diaphragm of wide aperture (Fresnel number larger than one) inside the optical cavity of a CO2 laser. This operation easily leads to laser emission in the doughnut mode TEM\*

the optical field sometimes presents impurities originating from the residual contributions

01 ,but

arctan( / ) *y x* the azimuthal coordinate and *L x <sup>p</sup>* are

where *r* is the radial coordinate,

experimental set-up is shown in Fig. 9.

is recorded with an IR pyroelectric camera.

of the fundamental mode or higher order cavity modes. The CO2 laser source operates at the wavelength λ=10.6*µm*. The laser cavity is defined by a partially reflective flat mirror (R=95%) and an out-coupling mirror (R=90%) mounted upon a piezoelectric translator. As aforementioned, the LG mode is selected with a diaphragm and is horizontally polarized by means of an intracavity ZnSe Brewster window.

The interferometric configuration comprises two mirrors, indicated as M1 and M2, and two beam splitters, BS1 (70T/30R at 45° angle of incidence) and BS2 (50T/50R at 45° angle of incidence) as depicted in figure 9. The output laser beam is a linear superposition of TEM10 and TEM01 Hermite-Gaussian modes (the doughnut mode TEM\* 01) corresponding to =1

Fig. 10. Infrared Laguerre-Gaussian beam analysis based on Fresnel reconstruction of offaxis digital hologram: (a) intensity distribution of the LG mode *ψ*10 recorded at the IR camera plane; (b) digitized interference intensity profile between LG object beam and plane wave reference; (c) Fourier transform of the carrier modulated interference pattern; (d) reconstructed vortex beam amplitude

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 177

Fig. 11. Infrared Laguerre-Gaussian beam analysis based on Fresnel reconstruction of off-

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

Allaria, E., Brugioni, S., De Nicola, S., Ferraro, P., Grilli, S. & Meucci, R. (2003). Digital Holography at 10.6 µm. *Optics Communications*, Vol. 215, pp. 257-262.

axis digital hologram: phase distribution of the lowest order vortex beam

**5. Conclusions** 

object investigation.

**6. References** 

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 camera is presented in Fig. 10(b).

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 form

$$\begin{split} \tilde{I}\_{10} \left( \nu\_{\mathbf{x'}} \nu\_{\mathbf{y}} \right) = \tilde{I}\_{dc} \left( \nu\_{\mathbf{x'}} \nu\_{\mathbf{y}} \right) + \tilde{\nu}\_{10} \left( \nu\_{\mathbf{x}} \cdot \nu\_{\mathbf{x}c'} \nu\_{\mathbf{y}} \cdot \nu\_{\mathbf{y}c} \right) + \\ + \overline{\tilde{\nu}}\_{10} \left( \nu\_{\mathbf{x}} + \nu\_{\mathbf{x}c'} \nu\_{\mathbf{y}} + \nu\_{\mathbf{y}c} \right) \end{split} \tag{21}$$

whereas

$$\tilde{I}\_{10}\left(\nu\_x,\nu\_y\right) = \mathcal{F}\left\{I\_{10}\left(\mathbf{x},\ y\right)\right\} = \iint I\_{10}\left(\mathbf{x},\ y\right) \exp\left[2\pi i \left(\nu\_x \mathbf{x} + \nu\_y y\right)\right] \text{dxd}y,\tag{22}$$

In eq. (21) *dc x <sup>y</sup> I ,* is the contribution to the spectrum given by the low-frequency background illumination (Cuche et al., 2000; Liu et al., 2002; Onural, 2000), corresponding to a peak at low spatial frequencies; <sup>10</sup> *x xc - , - y yc* and <sup>10</sup> *x xc + , +y yc* represent the Fourier spectrum of the vortex field and its complex conjugate, respectively, shifted by the carrier spatial frequencies *xc* and *yc* along the *x*- and *y*-direction, respectively. These 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 transform of <sup>10</sup> *x xc - , - y yc* , the amplitude of the vortex field can be determined, as shown in Fig 10(d), respectively.

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 crucial importance.

Fig. 11. Infrared Laguerre-Gaussian beam analysis based on Fresnel reconstruction of offaxis digital hologram: phase distribution of the lowest order vortex beam
