**2. Theory and principle of operation of digital holography**

The principle of the optical recording and reconstruction in classical holography is illustrated in Fig.1. The reference beam *R* interferes at the plane of the holographic plate at off-axis angle with respect to the object beam *O*. In this set-up the reconstructed image is spatially separated from the zero-order diffraction and the second image, the so-called 'twin image'. These three diffraction orders propagate along different directions and can be observed separately, leading to a significant improvement compared to the original in-line configuration originally developed by Gabor (Gabor, 1948; Goodman, 1996) where the zeroorder and the two conjugate images overlap.

The intensity distribution *I x* , *y* across the *x-y* holographic recording plane can be written as the modulus squared of the complex superposition *O x* , , *y R x y* , namely

$$\mathcal{A}\left(\mathbf{x},\mathbf{y}\right) = \left|\mathcal{O}\left(\mathbf{x},\mathbf{y}\right) + \mathcal{R}\left(\mathbf{x},\mathbf{y}\right)\right|^{2} = \left|\mathcal{R}\left(\mathbf{x},\mathbf{y}\right)\right|^{2} + \left|\mathcal{O}\left(\mathbf{x},\mathbf{y}\right)\right|^{2} + \mathcal{R}^{\*}\left(\mathbf{x},\mathbf{y}\right)\mathcal{O}\left(\mathbf{x},\mathbf{y}\right) + \mathcal{R}\left(\mathbf{x},\mathbf{y}\right)\mathcal{O}^{\*}\left(\mathbf{x},\mathbf{y}\right) \tag{1}$$

where the symbol \* denotes complex conjugation, *O x* , , exp , *y O x y i x y o* is the complex amplitude of the object wave with real amplitude *O x* , *y* and phase *<sup>o</sup> x*, *y* and

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

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

digitized interferogram.

of the beam from the digitized hologram.

order and the two conjugate images overlap.

off-axis angle

**2. Theory and principle of operation of digital holography** 

The principle of the optical recording and reconstruction in classical holography is illustrated in Fig.1. The reference beam *R* interferes at the plane of the holographic plate at

spatially separated from the zero-order diffraction and the second image, the so-called 'twin image'. These three diffraction orders propagate along different directions and can be observed separately, leading to a significant improvement compared to the original in-line configuration originally developed by Gabor (Gabor, 1948; Goodman, 1996) where the zero-

The intensity distribution *I x* , *y* across the *x-y* holographic recording plane can be written

<sup>222</sup> \* \* *I xy Oxy Rxy Rxy Oxy R xyOxy RxyO xy* , , , , , ,, , , (1)

as the modulus squared of the complex superposition *O x* , , *y R x y* , namely

where the symbol \* denotes complex conjugation, *O x* , , exp , *y O x y i x y*

complex amplitude of the object wave with real amplitude *O x* , *y* and phase

with respect to the object beam *O*. In this set-up the reconstructed image is

*o* is the

*<sup>o</sup> x*, *y* and

*R x* , , exp , *y R x y i x <sup>R</sup> y* is the complex amplitude of the reference wave with real amplitude *R x* , *y* and phase *<sup>R</sup> x*,*y* . Eq. (1) can be modified to the form

$$I(\mathbf{x}, y) = \left( \left| \mathbf{R}(\mathbf{x}, y) \right|^2 + \left| \mathbf{O}(\mathbf{x}, y) \right|^2 \right) + 2 \left| \mathbf{R}(\mathbf{x}, y) \right| \left| \mathbf{O}(\mathbf{x}, y) \right| \cos \left( \varphi\_o(\mathbf{x}, y) - \varphi\_\mathbf{R}(\mathbf{x}, y) \right) \tag{1a}$$

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

$$R(\mathbf{x}, y)I(\mathbf{x}, y) = R(\mathbf{x}, y) \left| \mathbf{R}(\mathbf{x}, y) \right|^2 + R(\mathbf{x}, y) \left| O(\mathbf{x}, y) \right|^2 + \left| \mathbf{R}(\mathbf{x}, y) \right|^2 O(\mathbf{x}, y) + R^2(\mathbf{x}, y) O^\*(\mathbf{x}, y) \tag{2}$$

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

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 161

, ; arctan Re , *Q x <sup>y</sup> xyd Q x <sup>y</sup>*

procedure can be employed to convert the phase modulo-2*π* into a continuous phase

In the following we will describe the main methods for numerical reconstruction of holograms (Kreis & Jueptner, 1997; Kronrod et al., 1972; Schnars & Jueptner, 1994, 2002).

The convolution integral given by Eq.(3) can be manipulated to obtain the reconstructed

 

Eq.(5) shows that the reconstruction field is determined essentially by the two-dimensional Fourier transformation of the multiplication of the hologram *I x* , *y* by the reference wave

> 2 2 *wxy i x y* ( , ) exp *<sup>d</sup>*

 1 2 22 1 *<sup>Q</sup>* , exp exp *id i F RxyI xywxy* ,, , , *i d <sup>d</sup>* 

where the direct (+1) or inverse (-1) continuous two dimensional Fourier transformations of

<sup>1</sup> *F f xy*, , *f x y i x y dxdy* , exp 2

reconstruction wavelength and to the reconstruction distance *d* by the following relations

 *d* 

*d*

   

  2 2

*R x y I x y i x y i x y dxdy d d*

 

*i d d*

, , exp exp

1 2 , exp exp

*Q id i*

Eq. (5) can be written in terms of the Fourier integral

 and 

the function *f x*,*y* are defined, respectively, by

> 

Eq. (4b) provides phase values wrapped in the interval

distribution in order to obtain a smooth phase image.

**2.1.1 Fresnel transformation method (FTM)** 

 ,

**2.1 Reconstruction methods** 

function. Indeed it results that

 

 

*R x* ,*y* and the chirp phase function

diffracted field *Q*

 

> and

spatial variables

In Eq. (7)

<sup>2</sup> *I x*,; , *<sup>y</sup> d Qx <sup>y</sup>* (4a)

(4b)

. A well-known unwrapping

in terms of the so-called Fresnel transformation of the hologram

2

 

in the reconstruction plane and they are related to the

(7)

are the spatial frequencies (Kreis, 2002a, 2002b) corresponding to the

 

 

(8)

 

 

(5a)

  (6)

 

(5)

2 2

 ,

Im ,

The first term on the right side of this equation is proportional to the reference wave field, the second one is a spatially varying "cloud" surrounding the first term. These two terms constitute the zero-order of diffraction or DC term. The third term represents, apart for a constant factor, an exact replica of the original wavefront *O x* , , exp , *y O x y i x <sup>o</sup> y* and for this reason it is called virtual image, or simply image of the object. The last term is another copy, the so called twin image of the original object wave or real image.

In holography the hologram can be regarded as an amplitude transmittance that diffracts the reference wave. In DH the object wave field is determined through the numerical calculation of the optical field propagation of *R x* , , *yh x y* from the holographic plane back to the object plane , . The numerical reconstruction of a digitally recorded hologram follows the scalar diffraction theory in the Fresnel approximation of the Rayleigh-Sommerfield diffraction integral. The reconstructed diffracted field *Q* , in the reconstruction plane , at distance *d* from the hologram plane can be written in the paraxial approximation in the following form

$$Q(\xi,\eta) = \frac{1}{i\lambda d} \exp\left(i\frac{2\pi}{\lambda}d\right) \int\_{-\alpha}^{\alpha} \int\_{-\alpha}^{\alpha} R(x,y)I(x,y) \exp\left[i\frac{\pi}{\lambda d}\left(\left(x-\xi\right)^2 + \left(y-\xi\right)^2\right)\right] dx dy \tag{3}$$

Eq.(3) is the starting point for reconstructing numerically the digitized hologram in the paraxial approximation, where the *x* and *y* values and the corresponding and values in the reconstruction plane are small compared to the distance *d* (see Fig.2).

Fig. 2. Optical set-up in off-axis digital holography

Once the complex field *Q* , has been calculated at distance *d*, the intensity *I x* , ; *y d* and phase distribution *x*, ; *y d* of the reconstructed image can be determined by the following relations

$$I(\mathbf{x}\_{\prime}y;d) = \left| Q(\mathbf{x}\_{\prime}y) \right|^2 \tag{4a}$$

$$\log(x, y; d) = \arctan \frac{\text{Im}\left[Q(x, y)\right]}{\text{Re}\left[Q(x, y)\right]} \tag{4b}$$

Eq. (4b) provides phase values wrapped in the interval , . A well-known unwrapping procedure can be employed to convert the phase modulo-2*π* into a continuous phase distribution in order to obtain a smooth phase image.

### **2.1 Reconstruction methods**

160 Advanced Holography – Metrology and Imaging

The first term on the right side of this equation is proportional to the reference wave field, the second one is a spatially varying "cloud" surrounding the first term. These two terms constitute the zero-order of diffraction or DC term. The third term represents, apart for a constant factor, an exact replica of the original wavefront *O x* , , exp , *y O x y i x*

and for this reason it is called virtual image, or simply image of the object. The last term is

In holography the hologram can be regarded as an amplitude transmittance that diffracts the reference wave. In DH the object wave field is determined through the numerical calculation of the optical field propagation of *R x* , , *yh x y* from the holographic plane

follows the scalar diffraction theory in the Fresnel approximation of the Rayleigh-

1 2 <sup>2</sup> <sup>2</sup> *<sup>Q</sup>* , exp *i d Rx*, , exp *<sup>y</sup> I x <sup>y</sup> i x <sup>y</sup> dxdy i d <sup>d</sup>*

Eq.(3) is the starting point for reconstructing numerically the digitized hologram in the

. The numerical reconstruction of a digitally recorded hologram

at distance *d* from the hologram plane can be written in the

 

(3)

has been calculated at distance *d*, the intensity *I x* , ; *y d* and

*x*, ; *y d* of the reconstructed image can be determined by the following

another copy, the so called twin image of the original object wave or real image.

Sommerfield diffraction integral. The reconstructed diffracted field *Q*

 

paraxial approximation, where the *x* and *y* values and the corresponding

the reconstruction plane are small compared to the distance *d* (see Fig.2).

back to the object plane

reconstruction plane

   ,

 ,

paraxial approximation in the following form

 

Fig. 2. Optical set-up in off-axis digital holography

 ,

Once the complex field *Q*

phase distribution

relations

*<sup>o</sup> y*

 ,

 

 and 

in the

values in

In the following we will describe the main methods for numerical reconstruction of holograms (Kreis & Jueptner, 1997; Kronrod et al., 1972; Schnars & Jueptner, 1994, 2002).

### **2.1.1 Fresnel transformation method (FTM)**

The convolution integral given by Eq.(3) can be manipulated to obtain the reconstructed diffracted field *Q* , in terms of the so-called Fresnel transformation of the hologram function. Indeed it results that

$$\begin{split} Q(\boldsymbol{\xi},\boldsymbol{\eta}) &= \frac{1}{i\lambda d} \exp\Big(i\frac{2\pi}{\lambda}d\Big) \exp\Big[i\frac{\pi}{\lambda d}\Big(\boldsymbol{\xi}^2 + \boldsymbol{\eta}^2\Big)\Big] \\ &\int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} R(\boldsymbol{x},\boldsymbol{y}) I(\boldsymbol{x},\boldsymbol{y}) \exp\Big[i\frac{\pi}{\lambda d}\Big(\boldsymbol{x}^2 + \boldsymbol{y}^2\Big)\Big] \exp\Big[-i\frac{2\pi}{\lambda d}(\boldsymbol{\xi}\boldsymbol{x} + \boldsymbol{y}\boldsymbol{\eta})\Big] d\boldsymbol{x} d\boldsymbol{y} \end{split} \tag{5}$$

Eq.(5) shows that the reconstruction field is determined essentially by the two-dimensional Fourier transformation of the multiplication of the hologram *I x* , *y* by the reference wave *R x* ,*y* and the chirp phase function

$$w(\mathbf{x}, y) = \exp\left[i\frac{\pi}{\lambda d}(\mathbf{x}^2 + y^2)\right] \tag{5a}$$

Eq. (5) can be written in terms of the Fourier integral

$$\mathcal{Q}(\boldsymbol{\xi},\boldsymbol{\eta}) = \frac{1}{i\lambda d} \exp\left(i\frac{2\pi}{\lambda}d\right) \exp\left[i\frac{\pi}{\lambda d}\left(\nu\_{\boldsymbol{\xi}}^{2} + \nu\_{\boldsymbol{\eta}}^{2}\right)\right] \left(\boldsymbol{F}^{+1} \left[\boldsymbol{R}\left(\mathbf{x},\boldsymbol{y}\right)\boldsymbol{I}\left(\mathbf{x},\boldsymbol{y}\right)\boldsymbol{w}\left(\mathbf{x},\boldsymbol{y}\right)\right] \left(\nu\_{\boldsymbol{\xi}},\nu\_{\boldsymbol{\eta}}\right)\right) \tag{6}$$

where the direct (+1) or inverse (-1) continuous two dimensional Fourier transformations of the function *f x*,*y* are defined, respectively, by

$$F^{\frac{\pm\pm1}{2}}\left[f\left(\mathbf{x},\boldsymbol{y}\right)\right]\left(\nu\_{\boldsymbol{\xi}},\nu\_{\eta}\right) = \int\_{-\infty}^{\infty}\int\_{-\infty}^{\infty}f\left(\mathbf{x},\boldsymbol{y}\right)\exp\left[\mp i2\pi\left(\nu\_{\boldsymbol{\xi}}\mathbf{x}+\nu\_{\eta}\boldsymbol{y}\right)\right]d\mathbf{x}d\boldsymbol{y}\tag{7}$$

In Eq. (7) and are the spatial frequencies (Kreis, 2002a, 2002b) corresponding to the spatial variables and in the reconstruction plane and they are related to the reconstruction wavelength and to the reconstruction distance *d* by the following relations

$$\nu\_{\xi} = \frac{\xi}{\lambda d} \qquad \qquad \nu\_{\eta} = \frac{\eta}{\lambda d} \tag{8}$$

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 163

Eq. (11) can allows to compute a matrix of *N M* complex numbers of the reconstructed field via the discrete two-dimensional fast Fourier transform algorithm. According to the theory of discrete Fourier transform, the sampling frequency intervals are given by

*<sup>d</sup>*

According to Eq. (12) the pixel width in the reconstructed plane is different from those of the digitized hologram and it scales inversely to the aperture of the optical system, i.e., to the side length *S Nx* of the hologram (limiting the analysis to the *x-*direction for the sake of simplicity). This result is in agreement with the theory of diffraction which predicts that at a distance *d* from the hologram plane the developed diffraction pattern is characterized by the

reconstructed image ( amplitude or phase image) is limited by the diffraction limit of the imaging system through the automatic scaling imposed by the Fresnel transform. Assuming that the hologram *I x* , *y* has spatial frequencies smaller than those in the quadratic phase factor *w x* ,*y* , the main problem when calculating Eq. (11) comes from an adequate sampling of the exponential function *w x* ,*y* inside the integral and of the global phase

sampling of *w x* ,*y* in the Nyquist limit, it is easy to obtain the approximate condition (limiting our analysis to one dimension only) that determines the range of distances *d* where

the discrete Fresnel reconstruction algorithm (cfr. Eq (11)) gives good results, namely

*c N x d d*

The same argument can be applied to the global phase factor 22 22 exp(*i dr s* )

which has too rapid variation with increasing spatial distance *d*, giving, this time, the condition *<sup>c</sup> d d* . A good reconstruction of both the amplitude and phase-contrast image is accomplished only if the equality *<sup>c</sup> d d* is assumed, while, if only the amplitude-contrast reconstruction is considered, the less restrictive condition *<sup>c</sup> d d* must hold. In fact, in this case the global phase factor is unessential for evaluating intensity distribution and the intensity profiles are of lower spatial frequency variation than the corresponding phase.

has to be greater than 24.5mm. From Eq. (12) it can be easily deduced that the width

*I dN <sup>S</sup> S* 

of the numerical reconstruction increases linearly with reconstruction distance *d*

of the reconstruction pixel, namely

*d N x* 

which, together with relations (8) allow us to determine the

*M y* 

*d S* of its Airy disk (or speckle diameter). Therefore the resolution of the

multiplying the expression in Eq.(11). Assuming the

2

of the reconstruction pixel at distance *<sup>c</sup> d d* in the Nyquist limit

(13)

 

(14)

*d Nx x* . As an example,

, the Fresnel method is valid for

 

 ,

 

, the distance

(12)

1 *N x* 

diameter

Note that the size *<sup>c</sup>*

according to the scaling law

*S N <sup>I</sup>* 

factor 22 22 exp *i dr s* 

 

coincides with pixel size of the sampled hologram, i.e., *c c*

distance greater than 98mm; for *N=1024* pixel, =532nm and *x y* 6.7 m

for *N=512* pixel, =632nm and pixel size *x y* 11 m

 

 

dimensions

 and 1 *M y* 

With the off-axis geometry the object wave and the reference wave arrive in the hologram plane with separate directions and, according to the above equations, the different terms of the numerically reconstructed wavefront propagate along different directions, owing to their different spatial frequencies. In fact, if Eq.(2) is substituted into Eq. (5), it is clear that the reconstruction of the DC term, the virtual and the real image are essentially governed by the frequency content of the respective spectra at the reconstruction distance *d*, which ultimately impose restrictions on the spatial bandwidth of the object and reference beam. If the reference field is given by *Rxy I ikx ky* , exp *R xy* where <sup>2</sup> , *RI Rx <sup>y</sup>* is the intensity of the reference field and **k** *kkk <sup>x</sup>* , , *<sup>y</sup> <sup>z</sup>* is the corresponding wave vector, the three terms are separated in the Fourier domain corresponding to the reconstruction plane , at distance *d*. The zero-order is located around the origin while the image and the twin image are symmetrically centred on *k k x y* 2, 2 and *k k x y* 2, 2 , respectively. To achieve good quality reconstruction in DH, the sampling theorem (Nyquist criterion) has to be fulfilled across the whole CCD array area. The criterion requires at least two pixels per fringe period and this implies that the maximum interference angle max between the spherical wavelet from each point of the object and the reference wave field is determined by the pixel size *x* according to the relation

$$a\_{\text{max}} = \frac{\lambda}{2\Delta x} \tag{9}$$

Relation (9) expresses the fact that for recording an hologram by a CCD array with pixel spacing *x* at least two pixels per fringe are needed. For example in case of a camera with pixel size *x* = 9 m, the maximum interference angle is max 1.7 for =532 nm. Mathematically, the two-dimensional spatial sampling *I n xm* , *y* of the hologram *I x* , *y* on a rectangular raster of *N* × *M* points can be described by the following relation

$$I\left(n\Delta\mathbf{x}, m\Delta y\right) = I\left(\mathbf{x}, y\right) \operatorname{rect}\left(\frac{\mathbf{x}}{N\Delta\mathbf{x}}, \frac{y}{M\Delta y}\right) \sum\_{n=1}^{N} \sum\_{m=1}^{M} \delta\left(\mathbf{x} \cdot n\Delta\mathbf{x}, y \cdot m\Delta y\right) \tag{10}$$

where *δ(x, y)* is the two-dimensional Dirac-delta function, *n* and *m* are integer numbers, *NxMy* is the area of the digitized hologram and *rect x* ,*y* is equal to one if the point of coordinated *x*, *y* is inside the area of the digitized hologram, and is zero elsewhere. *x* and y in Eq.(10) are the distances between the neighboring pixels on the CCD array in the horizontal and vertical directions, respectively. If the whole CCD array has a finite width given by *NxMy* , where *N* and *M* are the pixel numbers in each directions, the discrete representation of the Fresnel reconstruction integral given by Eq. (5) can be written as

$$\begin{split} Q\left(r\Delta\nu\_{\xi},s\Delta\nu\_{\eta}\right) &= \frac{1}{i\lambda d} \exp\left(i\frac{2\pi}{\lambda}d\right) \exp\left[i\pi d d\left(r^{2}\Delta\nu\_{\xi}^{2} + s^{2}\Delta\nu\_{\eta}^{2}\right)\right] \\ &\times \Delta\mathbf{x}\Delta\mathbf{y} \sum\_{n=-N/2}^{N/2-1} \sum\_{m=-M/2}^{M/2-1} I\left(n\Delta\mathbf{x},m\Delta\mathbf{y}\right) \mathbf{R}\left(n\Delta\mathbf{x},m\Delta\mathbf{y}\right) w(n\Delta\mathbf{x},m\Delta\mathbf{y}) \exp\left[i\pi \mathbf{2}\left(\frac{m}{N} + \frac{sm}{M}\right)\right]^{(11)} \end{split} (11)$$

With the off-axis geometry the object wave and the reference wave arrive in the hologram plane with separate directions and, according to the above equations, the different terms of the numerically reconstructed wavefront propagate along different directions, owing to their different spatial frequencies. In fact, if Eq.(2) is substituted into Eq. (5), it is clear that the reconstruction of the DC term, the virtual and the real image are essentially governed by the frequency content of the respective spectra at the reconstruction distance *d*, which ultimately impose restrictions on the spatial bandwidth of the object and reference beam. If the reference field is given by *Rxy I ikx ky* , exp *R xy* where <sup>2</sup> , *RI Rx <sup>y</sup>* is the intensity of the reference field and **k** *kkk <sup>x</sup>* , , *<sup>y</sup> <sup>z</sup>* is the corresponding wave vector, the three terms are separated in the Fourier domain corresponding to the reconstruction plane

at distance *d*. The zero-order is located around the origin while the image and the twin

achieve good quality reconstruction in DH, the sampling theorem (Nyquist criterion) has to be fulfilled across the whole CCD array area. The criterion requires at least two pixels per

spherical wavelet from each point of the object and the reference wave field is determined

max 2 *a*

Relation (9) expresses the fact that for recording an hologram by a CCD array with pixel spacing *x* at least two pixels per fringe are needed. For example in case of a camera with

Mathematically, the two-dimensional spatial sampling *I n xm* , *y* of the hologram *I x* , *y*


where *δ(x, y)* is the two-dimensional Dirac-delta function, *n* and *m* are integer numbers, *NxMy* is the area of the digitized hologram and *rect x* ,*y* is equal to one if the point of coordinated *x*, *y* is inside the area of the digitized hologram, and is zero elsewhere. *x* and y in Eq.(10) are the distances between the neighboring pixels on the CCD array in the horizontal and vertical directions, respectively. If the whole CCD array has a finite width given by *NxMy* , where *N* and *M* are the pixel numbers in each directions, the discrete

,, , ,

representation of the Fresnel reconstruction integral given by Eq. (5) can be written as

*<sup>x</sup> <sup>y</sup> I n x m y I x y rect x nxy my NxMy*

on a rectangular raster of *N* × *M* points can be described by the following relation

*x* 

 and *k k x y* 2, 2 

(10)

1 1

*N M*

*n m*

 

 

*rn sm x y I n xm y Rn xm y wn xm y i N M*

, , , exp 2

22 22

  , respectively. To

max between the

1.7 for =532 nm.

(9)

(11)

fringe period and this implies that the maximum interference angle

pixel size *x* = 9 m, the maximum interference angle is max

1 2 , exp exp

*i d*

 

 

21 21

*N M*

 

*Qr s i d i dr s*

*n Nm M*

2 2

 ,

image are symmetrically centred on *k k x y* 2, 2

by the pixel size *x* according to the relation

Eq. (11) can allows to compute a matrix of *N M* complex numbers of the reconstructed field via the discrete two-dimensional fast Fourier transform algorithm. According to the theory of discrete Fourier transform, the sampling frequency intervals are given by 1 *N x* and 1 *M y* which, together with relations (8) allow us to determine the dimensions of the reconstruction pixel, namely

$$
\Delta \xi = \frac{\lambda d}{N \Delta x} \qquad \qquad \Delta \eta = \frac{\lambda d}{M \Delta y} \tag{12}
$$

According to Eq. (12) the pixel width in the reconstructed plane is different from those of the digitized hologram and it scales inversely to the aperture of the optical system, i.e., to the side length *S Nx* of the hologram (limiting the analysis to the *x-*direction for the sake of simplicity). This result is in agreement with the theory of diffraction which predicts that at a distance *d* from the hologram plane the developed diffraction pattern is characterized by the diameter *d S* of its Airy disk (or speckle diameter). Therefore the resolution of the reconstructed image ( amplitude or phase image) is limited by the diffraction limit of the imaging system through the automatic scaling imposed by the Fresnel transform. Assuming that the hologram *I x* , *y* has spatial frequencies smaller than those in the quadratic phase factor *w x* ,*y* , the main problem when calculating Eq. (11) comes from an adequate sampling of the exponential function *w x* ,*y* inside the integral and of the global phase factor 22 22 exp *i dr s* multiplying the expression in Eq.(11). Assuming the sampling of *w x* ,*y* in the Nyquist limit, it is easy to obtain the approximate condition (limiting our analysis to one dimension only) that determines the range of distances *d* where the discrete Fresnel reconstruction algorithm (cfr. Eq (11)) gives good results, namely

$$d \ge d\_c = \frac{N \Delta x^2}{\lambda} \tag{13}$$

The same argument can be applied to the global phase factor 22 22 exp(*i dr s* ) , which has too rapid variation with increasing spatial distance *d*, giving, this time, the condition *<sup>c</sup> d d* . A good reconstruction of both the amplitude and phase-contrast image is accomplished only if the equality *<sup>c</sup> d d* is assumed, while, if only the amplitude-contrast reconstruction is considered, the less restrictive condition *<sup>c</sup> d d* must hold. In fact, in this case the global phase factor is unessential for evaluating intensity distribution and the intensity profiles are of lower spatial frequency variation than the corresponding phase. Note that the size *<sup>c</sup>* of the reconstruction pixel at distance *<sup>c</sup> d d* in the Nyquist limit coincides with pixel size of the sampled hologram, i.e., *c c d Nx x* . As an example, for *N=512* pixel, =632nm and pixel size *x y* 11 m , the Fresnel method is valid for distance greater than 98mm; for *N=1024* pixel, =532nm and *x y* 6.7 m , the distance has to be greater than 24.5mm. From Eq. (12) it can be easily deduced that the width *S N <sup>I</sup>* of the numerical reconstruction increases linearly with reconstruction distance *d* according to the scaling law

$$S\_I = \frac{\lambda dN}{S} \tag{14}$$

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 165

The fringe pattern that results from the interference of the reference beam and an object beam carries phase information on the object under test and any change in its state gives rise

holograms recorded at different states 1*s* and 2*s* of the object, the corresponding phase

 , ;*s s Ar* 2 1 *g Q s Ar* 

, , arctan Re , , *Q s Arg Q s Q s*

 

 

, ; Re , ; Re , ; Im , ; Im , ;

For instance, in the case of deformation measurement, *s1* and *s2* are states of deformation of the object under investigation and the calculated interference phase provides information about the displacement field onto the surface in case of opaque objects or the full optical path variation occurring in transparent objects (Rastogi, 1994). Digital Holographic interferometry has also been applied to measure deformations of either large and very small objects, to investigate the refractive index changes (Dubois et al., 1999; Zito et al., 2009) and for comparing accurately the shapes of objects and for removing spherical aberrations introduced by high numerical aperture objective lenses employed in digital holographic microscopy applications where the paraxial approximations implicit in the Fresnel treatment often fails (Jueptner et al, 1987; Kato et al., 2003; Kim, 2000; Pedrini et al., 2001).

In this section we present some applications of digital holography and of the previously described wavefront reconstruction technique in the infrared region as a high accuracy technique for testing reflective objects. The method makes use of a two-beam interferometric set-up (De Nicola et al., 2008) for recording digital off-axis interferograms of reflective

A conventional gas flowing CO2 laser emitting on the P(20) line at =10.6 m is employed as an infrared source. The laser cavity, 82cm long, is defined by a partially reflective flat mirror (R=95%) and an out-coupling mirror (R=90%) of 3m radius of curvature mounted upon a piezoelectric translator. The output laser beam is horizontally polarized by means of an intracavity ZnSe Brewster window. The system, pumped by an electric discharge of 10mA, when the laser threshold is approximately 9mA provides an output optical power in the range of 500-800mW. The spatial profile of the laser is set in the fundamental TEM00 Gaussian mode by means of an intra-cavity iris diaphragm. In this configuration, the laser

objects at wavelength 10.6m. A scheme of the set-up is shown in Fig. 3.

, ; <sup>2</sup> *g Q s*

 Im , ,

 

 

 21 21

Re , ; Im , ; Re , ; Im , ;

*Qs Qs Qs Qs*

*Qs Qs Qs Qs*

21 2 1

, ; <sup>2</sup> are the numerical reconstructions of two

(18a)

 

(19)

 

 , ; <sup>1</sup> (18)

**2.2 Digital holographic interferometry** 

If the complex fields *Q s*

change 2 1 

by taking into account that

2 1

**3. Infrared digital holography** 

*s s*

 

to a corresponding modification in the phase information.

, ; <sup>1</sup> and *Q s*

 

 

 

we can express the phase change in the following form

, ;*s s* is given by

 

where *N* and the width *S Nx* of the hologram are input parameters in the reconstruction process. Nevertheless this result is only compatible with condition given by Eq.(13), derived from the appropriate sampling of reconstructed amplitude image. This means that maintaining the number *N* as a constant may lead to a badly sampled reconstructed image if the reconstruction distance does not satisfy the above requirements. By padding the recorded hologram with zeros in the border, after Fresnel diffraction the external part of the reconstructed hologram can be prevented from leaving the support matrix and entering the opposite side of the matrix because of aliasing. This is of particular importance in the numerical reconstruction of off-axis recorded holograms where the use of the pixel area of the recording array is less efficient.

### **2.1.2 Convolution transformation method**

An alternative way of numerical reconstruction of holograms is through the calculation of the propagated angular spectrum, the so called "convolution approach" to digital holography. The reconstructed field, in the paraxial approximation, can be written in this case in the following form

$$Q\left(\left(\nu\_{\bar{\xi}},\nu\_{\eta}\right)\right) = \exp\left(i\frac{2\pi}{\lambda}d\right) \mathrm{F}^{\mathrm{-1}}\left\{\left(\exp\left[-i\pi\lambda d\left(\nu\_{\bar{\xi}}^{2} + \nu\_{\eta}^{2}\right)\right] \mathrm{F}^{\mathrm{+1}}\left[R\left(\mathbf{x},y\right)h\left(\mathbf{x},y\right)\right]\left(\nu\_{\bar{\xi}},\nu\_{\eta}\right)\right\}\right.\tag{15}$$

where the Fourier transform of the chirp function *wxy* (,) given by Eq. (4) has been used, namely

$$F^{+\mathbf{1}}\left[w(\mathbf{x},y)\right](\nu\_{\tilde{\boldsymbol{\varphi}}'},\nu\_{\eta}) = \operatorname{id}\lambda \exp\left[\mathbf{I}\cdot\mathbf{i}\pi\lambda d\left(\left(\nu\_{\tilde{\boldsymbol{\varphi}}}^2 + \nu\_{\eta}^2\right)\right)\right] \tag{16}$$

It can be shown that when the angular spectrum is used, the use of two Fourier transforms for computing Eq.(15), once for taking the Fourier transform of the hologram (multiplied by the reference wave) and another time for taking the inverse Fourier transform, leads to a cancellation of the scale factor between the input and output field to obtain that the pixel size of the reconstructed images equal to that of the sampled hologram, i.e., *x* and *y* and the actual sizes of h the input hologram and reconstructed image are identical *S S <sup>I</sup>* . We point out that, although Eq.(6) and Eq.(15) are formally equivalent, the different use of the DFT algorithm to perform the calculation of the same diffraction integral, makes the convolution-based algorithm valid for near distances *<sup>c</sup> d d* . Both methods overlap at distance *<sup>c</sup> d d* . Clearly this method is computationally more expensive than the direct evaluation of the Fresnel integral, since it requires two Fourier transforms (one direct and one inverse) but it is advantageous for keeping constant the length scales of the reconstructed images for all distances satisfying the near-field approximation (Zhang et al., 2004). From the discrete complex values of the reconstructed field, the intensity *I rsd* , ; and phase distribution *rsd* , ; of the reconstructed image can be determined according to the following two equations

$$I\left(r,s;d\right) = \left|Q\left(r\Delta\xi, s\Delta\eta\right)\right|^2\tag{17a}$$

$$\varphi(r,s;d) = \arctan\frac{\text{Im}\left[Q(r\Lambda\xi, s\Lambda\eta)\right]}{\text{Re}\left[Q(r\Lambda\xi, s\Lambda\eta)\right]}\tag{17b}$$

### **2.2 Digital holographic interferometry**

The fringe pattern that results from the interference of the reference beam and an object beam carries phase information on the object under test and any change in its state gives rise to a corresponding modification in the phase information.

If the complex fields *Q s* , ; <sup>1</sup> and *Q s* , ; <sup>2</sup> are the numerical reconstructions of two holograms recorded at different states 1*s* and 2*s* of the object, the corresponding phase change 2 1 , ;*s s* is given by

$$\Delta\phi\big(\xi,\eta;s\_2-s\_1\big) = \operatorname{Arg}\Big[\bigotimes\big(\xi,\eta;s\_2\big)\big] - \operatorname{Arg}\Big[\bigotimes\big(\xi,\eta;s\_1\big)\big)\Big]\tag{18}$$

by taking into account that

164 Advanced Holography – Metrology and Imaging

where *N* and the width *S Nx* of the hologram are input parameters in the reconstruction process. Nevertheless this result is only compatible with condition given by Eq.(13), derived from the appropriate sampling of reconstructed amplitude image. This means that maintaining the number *N* as a constant may lead to a badly sampled reconstructed image if the reconstruction distance does not satisfy the above requirements. By padding the recorded hologram with zeros in the border, after Fresnel diffraction the external part of the reconstructed hologram can be prevented from leaving the support matrix and entering the opposite side of the matrix because of aliasing. This is of particular importance in the numerical reconstruction of off-axis recorded holograms where the use of the pixel area of

An alternative way of numerical reconstruction of holograms is through the calculation of the propagated angular spectrum, the so called "convolution approach" to digital holography. The reconstructed field, in the paraxial approximation, can be written in this


where the Fourier transform of the chirp function *wxy* (,) given by Eq. (4) has been used, namely


 

It can be shown that when the angular spectrum is used, the use of two Fourier transforms for computing Eq.(15), once for taking the Fourier transform of the hologram (multiplied by the reference wave) and another time for taking the inverse Fourier transform, leads to a cancellation of the scale factor between the input and output field to obtain that the pixel size of the reconstructed images equal to that of the sampled hologram, i.e.,

 *y* and the actual sizes of h the input hologram and reconstructed image are identical *S S <sup>I</sup>* . We point out that, although Eq.(6) and Eq.(15) are formally equivalent, the different use of the DFT algorithm to perform the calculation of the same diffraction integral, makes the convolution-based algorithm valid for near distances *<sup>c</sup> d d* . Both methods overlap at distance *<sup>c</sup> d d* . Clearly this method is computationally more expensive than the direct evaluation of the Fresnel integral, since it requires two Fourier transforms (one direct and one inverse) but it is advantageous for keeping constant the length scales of the reconstructed images for all distances satisfying the near-field approximation (Zhang et al., 2004). From the discrete complex values of the reconstructed field, the intensity *I rsd* , ;

> <sup>2</sup> *I rsd Qr s* ,; ,

, ; arctan Re , *Qr s rsd*

 

 

 

 

> 

 

*rsd* , ; of the reconstructed image can be determined according to

Im ,

 *Qr s* 

 

(16)

(17a)

(17b)

*x* and

(15)

 

the recording array is less efficient.

case in the following form

 

 

and phase distribution

the following two equations

 

**2.1.2 Convolution transformation method** 

$$\operatorname{Arg}\left[Q\left(\xi,\eta,s\right)\right] = \arctan\left[\frac{\operatorname{Im}\left(Q\left(\xi,\eta,s\right)\right)}{\operatorname{Re}\left(Q\left(\xi,\eta,s\right)\right)}\right] \tag{18a}$$

we can express the phase change in the following form

$$\Delta\phi(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{2}-\boldsymbol{s}\_{1}) = \frac{\mathrm{Re}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{2})\Big]\mathrm{Im}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{1})\Big] - \mathrm{Re}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{2})\Big]\mathrm{Im}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{1})\Big]}{\mathrm{Re}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{2})\Big]\mathrm{Re}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{1})\Big] + \mathrm{Im}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{2})\Big]\mathrm{Im}\Big[\boldsymbol{Q}(\boldsymbol{\xi},\boldsymbol{\eta};\boldsymbol{s}\_{1})\Big]}}\tag{19}$$

For instance, in the case of deformation measurement, *s1* and *s2* are states of deformation of the object under investigation and the calculated interference phase provides information about the displacement field onto the surface in case of opaque objects or the full optical path variation occurring in transparent objects (Rastogi, 1994). Digital Holographic interferometry has also been applied to measure deformations of either large and very small objects, to investigate the refractive index changes (Dubois et al., 1999; Zito et al., 2009) and for comparing accurately the shapes of objects and for removing spherical aberrations introduced by high numerical aperture objective lenses employed in digital holographic microscopy applications where the paraxial approximations implicit in the Fresnel treatment often fails (Jueptner et al, 1987; Kato et al., 2003; Kim, 2000; Pedrini et al., 2001).

### **3. Infrared digital holography**

In this section we present some applications of digital holography and of the previously described wavefront reconstruction technique in the infrared region as a high accuracy technique for testing reflective objects. The method makes use of a two-beam interferometric set-up (De Nicola et al., 2008) for recording digital off-axis interferograms of reflective objects at wavelength 10.6m. A scheme of the set-up is shown in Fig. 3.

A conventional gas flowing CO2 laser emitting on the P(20) line at =10.6 m is employed as an infrared source. The laser cavity, 82cm long, is defined by a partially reflective flat mirror (R=95%) and an out-coupling mirror (R=90%) of 3m radius of curvature mounted upon a piezoelectric translator. The output laser beam is horizontally polarized by means of an intracavity ZnSe Brewster window. The system, pumped by an electric discharge of 10mA, when the laser threshold is approximately 9mA provides an output optical power in the range of 500-800mW. The spatial profile of the laser is set in the fundamental TEM00 Gaussian mode by means of an intra-cavity iris diaphragm. In this configuration, the laser

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 167

compensate for the loss of resolution with increasing reconstruction distance *d,* the digitized hologram was padded with value of zero to achieve a larger array of *N N* \* \* 256 256 points and a reconstruction pixel of size 103m ×103m at a distance *d=* 250mm. This process increases the number of 2D-FFT points, changing artificially the period, while holding the spatial sampling rate of the digitized hologram fixed. In this manner, spatial spectral components of the reconstructed images, originally hidden from view can be shifted to points where they can be observed. However aliasing created by the use of a 2DFFT may result from spreading or leakage of the spectral components away from the correct frequency leading to undesirable modification of the reconstructed image. Aliasing occurs during numerical implementations if the reconstruction distance is less than the

of the window function tends to increase min *d* . The condition min *d d* sets the maximum number of samples with value of zero at the end of the digitized holograms for a given hologram recording distance. In our case, for the infrared wavelength =10.6 m, the condition is satisfied since we have min *d* =241 mm for \* *N* 256. The reconstruction was

components *<sup>x</sup> k* and *<sup>y</sup> k* of the wave-vectors were adjusted to centre the image in the reconstruction plane. The determined phase values are wrapped (Yamaguchi & Zhang,

can be employed to convert the phase modulo-2 into a continuous phase distribution in order to obtain a smooth 3-D phase profile (Demoli et al., 2003; Gass et al., 2003; Schnars, 1994). The height distribution *h x* ,*y* of the object is the information to be retrieved. It is

letters inscribed in the first aluminium block. Amplitude images were reconstructed from the digitized infrared holograms according to method discussed in the previous section. In Fig.4(a), 4(c) the size of the reconstruction pixel is 213m×213m; Fig. 4(b), 4(d) shows the

a)

 , min *d N*

 , exp 

> 

. Fig.(4) shows the results of the reconstruction of the image of the

 

> 

. Increasing the size \* *N*

*ik k x y* where the two

. A well-known unwrapping procedure

, ;*d* by the simple relationship

minimum object-to-hologram recordable distance \* 2

obtained with a reference beam of the form *R*

1997; Yamaguchi et al., 2002) in the interval -

related to the reconstructed phase distribution

corresponding reconstructions with padding operations.

 

*hxy xyd* , ,; 4 

Fig. 3. Mach-Zehnder interferometer employed to record IR digital holograms of reflective samples: M1, M2 and M3, mirrors; BE beam expander; BS1 and BS2, beam splitters.

beam is characterized by a spot size of 6mm on the flat mirror and a divergence less than 2mrad. Figure 3 shows the DH optical set-up based on a Mach–Zehnder interferometer. The infrared beam is directed to a beam expander with magnification 2.5 × and 15 mm diameter of output beam. Two mirrors M1 and M2 and two beam splitters, BS1 (70T/30R at 450 angle of incidence ) and BS2 (50T/50R at 450 angle of incidence), are used to form the interferometer. The beam splitters are ZnSe coated windows with a diameter 50mm each. The interferometer allows to record the interference patterns between the two beams, the reference beam and that reflected by the test object on the detection plane of a pyroelectric videocamera \_Spiricon Pyrocam III, Model PY-III-C-A, which has a matrix of *N N* 124 × 124 pyroelectric sensor elements of LiTaO3, with square pixel size 85 85 *m m* and centerto-center spacing of the pixels, the pixel pitch, 100 100 *m m* . The reference beam interferes with the object at a small angle 2 3 , as required by the sampling theorem. The pyroelectric camera allows detection of CW infrared laser radiation by means of an internal chopper. It is connected to a personal computer to record digitized fringe patterns. The reflective objects used are two opaque aluminium blocks. The first one is a rectangle of size 20mm × 35mm that has letters inscribed. The letters are " UOR" and "XUO" ( about 3mm × 4 mm each). The second one is a disc of radius 25.4mm which has inscribed a set of concentric circular tracks. The aluminium blocks are located at distance *d*=250mm from the array. The discrete finite form of Eq. (11) is obtained through the pixel pitch 100 100 *m m* of the pyrocam array and the reconstructed object field *x*, ; *y d* is obtained by applying the 2D fast Fourier transform (2D-FFT) algorithm to the discrete samples of the quantity 2 2 *IR i d* , , exp . By the 2D-FFT algorithm, the size of the reconstruction pixel at distance *d* is *x ydN dN* . For a reconstruction distance *d=* 250 mm , the resolution of the reconstructed field distribution is limited by the size *xy m m* 213 213 of the reconstruction pixel. To

Fig. 3. Mach-Zehnder interferometer employed to record IR digital holograms of reflective samples: M1, M2 and M3, mirrors; BE beam expander; BS1 and BS2, beam splitters.

beam is characterized by a spot size of 6mm on the flat mirror and a divergence less than 2mrad. Figure 3 shows the DH optical set-up based on a Mach–Zehnder interferometer. The infrared beam is directed to a beam expander with magnification 2.5 × and 15 mm diameter of output beam. Two mirrors M1 and M2 and two beam splitters, BS1 (70T/30R at 450 angle of incidence ) and BS2 (50T/50R at 450 angle of incidence), are used to form the interferometer. The beam splitters are ZnSe coated windows with a diameter 50mm each. The interferometer allows to record the interference patterns between the two beams, the reference beam and that reflected by the test object on the detection plane of a pyroelectric videocamera \_Spiricon Pyrocam III, Model PY-III-C-A, which has a matrix of *N N* 124 ×

theorem. The pyroelectric camera allows detection of CW infrared laser radiation by means of an internal chopper. It is connected to a personal computer to record digitized fringe patterns. The reflective objects used are two opaque aluminium blocks. The first one is a rectangle of size 20mm × 35mm that has letters inscribed. The letters are " UOR" and "XUO" ( about 3mm × 4 mm each). The second one is a disc of radius 25.4mm which has inscribed a set of concentric circular tracks. The aluminium blocks are located at distance *d*=250mm from the array. The discrete finite form of Eq. (11) is obtained through the pixel

*x*, ; *y d* is obtained by applying the 2D fast Fourier transform (2D-FFT) algorithm to the

For a reconstruction distance *d=* 250 mm , the resolution of the reconstructed field

, , exp

 2 3 , as required by the sampling

 

100 100 *m m* of the pyrocam array and the reconstructed object field

 . By the 2D-FFT

*m m* 

*m m* 

of the reconstruction pixel. To

 .

. The reference beam

and center-

124 pyroelectric sensor elements of LiTaO3, with square pixel size 85 85

to-center spacing of the pixels, the pixel pitch, 100 100

 

distribution is limited by the size *xy m m* 213 213

discrete samples of the quantity 2 2 *IR i d*

 

algorithm, the size of the reconstruction pixel at distance *d* is *x ydN dN*

interferes with the object at a small angle

pitch 

compensate for the loss of resolution with increasing reconstruction distance *d,* the digitized hologram was padded with value of zero to achieve a larger array of *N N* \* \* 256 256 points and a reconstruction pixel of size 103m ×103m at a distance *d=* 250mm. This process increases the number of 2D-FFT points, changing artificially the period, while holding the spatial sampling rate of the digitized hologram fixed. In this manner, spatial spectral components of the reconstructed images, originally hidden from view can be shifted to points where they can be observed. However aliasing created by the use of a 2DFFT may result from spreading or leakage of the spectral components away from the correct frequency leading to undesirable modification of the reconstructed image. Aliasing occurs during numerical implementations if the reconstruction distance is less than the minimum object-to-hologram recordable distance \* 2 min *d N* . Increasing the size \* *N* of the window function tends to increase min *d* . The condition min *d d* sets the maximum number of samples with value of zero at the end of the digitized holograms for a given hologram recording distance. In our case, for the infrared wavelength =10.6 m, the condition is satisfied since we have min *d* =241 mm for \* *N* 256. The reconstruction was obtained with a reference beam of the form *R* , exp *ik k x y* where the two components *<sup>x</sup> k* and *<sup>y</sup> k* of the wave-vectors were adjusted to centre the image in the reconstruction plane. The determined phase values are wrapped (Yamaguchi & Zhang, 1997; Yamaguchi et al., 2002) in the interval - , . A well-known unwrapping procedure can be employed to convert the phase modulo-2 into a continuous phase distribution in order to obtain a smooth 3-D phase profile (Demoli et al., 2003; Gass et al., 2003; Schnars, 1994). The height distribution *h x* ,*y* of the object is the information to be retrieved. It is related to the reconstructed phase distribution , ;*d* by the simple relationship *hxy xyd* , ,; 4 . Fig.(4) shows the results of the reconstruction of the image of the letters inscribed in the first aluminium block. Amplitude images were reconstructed from the digitized infrared holograms according to method discussed in the previous section. In Fig.4(a), 4(c) the size of the reconstruction pixel is 213m×213m; Fig. 4(b), 4(d) shows the corresponding reconstructions with padding operations.

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 169

The size is 103m×103m, and the resolution is clearly improved, as it can be seen by comparing the reconstructed images shown Fig. 4(b), 4(d) with those of Fig. 4(a), 4(c). The hologram of the aluminium disk shaped object is shown in Fig. 5. The concentric

Fig. 5. Infrared hologram of an aluminium disk shaped object which has inscribed a set of

Figure (6) shows the phase images reconstructed form the disk shaped object. Fig. 6(a), 6(c) and 6(e) display the phase distribution of the object reconstructed without padding operation and Fig. 6(b), 6(d) and 6(f) show the corresponding reconstructions with zero padding operation. It is noticeable in the 3D perspectives plots of the reconstructed phase shown in Fig. 6(d) and 6(f) that the circular-shaped tracks inscribed in the steel disk are

concentric circular tracks.

better resolved with padding operation.

circular tracks superimposed on the interference fringes are clearly visible.

Fig. 4. Amplitude reconstruction of the letters "ROU" before padding operation (a), after zero padding (b); "XUO" before padding operation (c), after padding (d).

b)

c)

d) Fig. 4. Amplitude reconstruction of the letters "ROU" before padding operation (a), after

zero padding (b); "XUO" before padding operation (c), after padding (d).

The size is 103m×103m, and the resolution is clearly improved, as it can be seen by comparing the reconstructed images shown Fig. 4(b), 4(d) with those of Fig. 4(a), 4(c). The hologram of the aluminium disk shaped object is shown in Fig. 5. The concentric circular tracks superimposed on the interference fringes are clearly visible.

Fig. 5. Infrared hologram of an aluminium disk shaped object which has inscribed a set of concentric circular tracks.

Figure (6) shows the phase images reconstructed form the disk shaped object. Fig. 6(a), 6(c) and 6(e) display the phase distribution of the object reconstructed without padding operation and Fig. 6(b), 6(d) and 6(f) show the corresponding reconstructions with zero padding operation. It is noticeable in the 3D perspectives plots of the reconstructed phase shown in Fig. 6(d) and 6(f) that the circular-shaped tracks inscribed in the steel disk are better resolved with padding operation.

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 171

d)

e)

f) Fig. 6. Phase images of aluminium disc:(a) two-dimensional reconstruction without padding

operation; (b) 2D reconstruction with padding; (c), (e) two different 3D maps of the

reconstructed surface profile without padding; (d), (f) with zero padding.

c)

a)

b)

c)

Fig. 6. Phase images of aluminium disc:(a) two-dimensional reconstruction without padding operation; (b) 2D reconstruction with padding; (c), (e) two different 3D maps of the reconstructed surface profile without padding; (d), (f) with zero padding.

Infrared Holography for Wavefront Reconstruction and Interferometric Metrology 173

(a) (b) (c) Fig. 8. (a) Photo of the Perseus statue; (b) acquired interferogram; (c) numerical reconstruction.

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

azimuthal mode index *ℓ* (orbital helicity) is physically related to OAM, per photon, of the

*xyz p! w z w z w z*

<sup>1</sup> <sup>2</sup> , , exp <sup>π</sup>

2 2

*<sup>r</sup> ikr L i w z R z*

exp 2 1 arctan ,

*<sup>z</sup> i p*

 

2

2

*p*

*2p ! r r*

*r*

*z*

exp exp <sup>2</sup>

in the transverse distribution of the optical field. The

2 2

(20)

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

**Gaussian modes** 

angular dependence exp*i*

*p*

LG modes. The LG modes of azimuthal index *ℓ* and radial index *p* are
