**1. Introduction**

156 Advanced Holography – Metrology and Imaging

Zhang, S. H., Gan, Sh., Cao, D. Zh., Xiong, J., Zhang X., & Wang, K., (2009b). Phase-reversal

Zhang, S. H., Gan, Sh., Xiong, J., Zhang, X., & Wang, K., (2010). Illusion optics in chaotic light, *Phys. Rev. A* Vol.82, No.2, (August 2010) 021804(R), ISSN 1050-2947.

031895(R), ISSN 1050-2947.

diffraction in incoherent light, *Phys. Rev. A* Vol.80, No.3, (September 2009)

Long wavelength interferometry has been widely applied in different fields, such as infrared optics, infrared transmitting materials, high-reflective multilayer dielectric coatings for highpower laser systems. In optical metrology, long-wave interferometers are also employed for shape measurement of reflective rough surfaces and for testing optical systems that requires deep aspherics. An advantage of using longer wavelength is that the aspheric departure from the best fit reference-sphere, in unit of the probing wavelength, is reduced at longer wavelength, thus allowing one to obtain an interferogram of the deep aspheric under test. This leads to extension of the unambiguous distance measurement range, a well-known problem in interferometric metrology, where the essential difficulty relates to the interferometric fringe order, which cannot be determined unambiguously from a single measurement of phase interference. This problem is also of particular significance in digital holographic and interferomety based applications where digital processing of the recorded interferogram makes it possible to extract quantitatively amplitude and phase of the numerically reconstructed wavefront (Cuche el al., 1999a, 1999b; Vest, 1979; Yaroslavsky & Eden, 1996). Digital holography based imaging techniques provide real time capabilities to record also three-dimensional objects using interference between an object wave and a reference wave captured by an image sensor such a CCD sensor. Three-dimensional 3D information of the object can be obtained from the numerical reconstruction of a single digitally recorded hologram, since the information about the optically interfering waves is stored in the form of matrices. The numerical reconstruction process offers many more possibilities than conventional optical processing (Goodman & Lawrence, 1967; Stetson & Powell, 1966; Stetson & Brohinsky, 1985). For example, it is possible to numerically focus on any section of the three dimensional volume object without mechanical focusing adjustment (Grilli et al., 2001; Lai et al., 2000), correct optical components defects such as lens aberrations or compensate the limited depth of field of an high magnification microscope objective. Full digital processing of holograms requires high spatial resolution sensor arrays with demanding capabilities for imaging applications and non destructive testing. In this regard, several new recording materials and optoelectronic sensors have been devised for

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 159

2 2 , , , 2 , , cos , , *o R I xy Rxy Oxy Rxy Oxy xy xy*

which shows explicitly that the recorded hologram contains a term with an amplitude and phase modulated spatial carrier. For the numerical reconstruction of the recorded hologram, the interference pattern *I x* , *y* is illuminated by the reference wave *R x* ,*y* , i.e , we have

2 22 2 \* *R x*,, , , , , , , , , *<sup>y</sup> I x <sup>y</sup> R x <sup>y</sup> R x <sup>y</sup> R x <sup>y</sup> O x <sup>y</sup> R x <sup>y</sup> O x <sup>y</sup> R x <sup>y</sup> O x <sup>y</sup>*

a)

b)

Fig. 1. Optical configuration for recording (a) and for reconstruction (b) of off-axis

*<sup>R</sup> y* is the complex amplitude of the reference wave with real

 

(1a)

(2)

*<sup>R</sup> x*,*y* . Eq. (1) can be modified to the form

*R x* , , exp , *y R x y i x*

holograms.

amplitude *R x* , *y* and phase

recording holograms beyond the visible wavelengths, in the infrared. Coherent imaging based on digital processing of infrared holograms presents the great advantage of providing quantitative data from the recorded interference pattern (Lei et al., 2001a, 2001b; Leith & Upatnieks, 1965; Nilsson & Carlsson, 2000; Seebacker et al., 2001) but, the lower spatial resolution of the infrared sensor arrays compared to those working in the visible region, appears to be a major limiting factor to accurate numerical wavefront reconstruction for the digitized interferogram.

In this chapter we will describe potential applications of digital processing methods for whole optical wavefront reconstruction of infrared recorded holograms. The basic principle of numerical reconstruction of digitized holograms will be reviewed and we will discuss numerical methods to compensate the loss of spatial resolution at longer wavelength and the limited spatial resolution of the recording array We will present several examples to show that infrared digital holography can be exploited as a high accuracy technique for testing both reflective and transmissive objects. In order to demonstrate the feasibility of digital holography in the infrared region we will describe a two beam interferometric technique for recording holograms, both in reflection and transmission type geometry which employs a pyroelectric sensor array. The availability of pyroelectric sensor arrays for recording digital infrared holograms offers the advantage of a compact and efficient interferometric set-up for recording infrared hologram. We will also present application of digital holographic processing methods for numerical reconstruction of wavefronts of optical beams, such as Laguerre-Gauss beams possessing wavefront singularities in the infrared region. In fact, this technique 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 is a problem of crucial importance. We will present examples of characterization of the vortex signature of infrared Laguerre-Gauss beam through the numerical reconstruction of the singular phase of the beam from the digitized hologram.
