**2. Experimental set-up**

184 Advanced Holography – Metrology and Imaging

spherical aberration (Stadelmaier & Massig, 2000), astigmatism (Grilli et al., 2001) and

Compensation of aberrations is fundamental when quantitative phase determination is used in microscopic metrological applications. Several approaches have been proposed to remove the aberrations. A method proposed by Cuche (Cuche et al., 1999) used a single hologram, which involved the computation of a digital replica of the reference wave depending on two reconstruction parameters. A double-exposure technique (Ferraro et al., 2003) can compensate completely the inherent wave front curvature in quantitative phase contrast imaging, although it needs two hologram recordings with and without the sample and a subtraction procedure between the two holograms. In the paper (Colomb et al., 2006) the authors developed a method to compensate the tilt aberration by recording a hologram corresponding to a blank image to compute the first-order parameters directly from the hologram. However, the method is limited to selecting profiles or areas known to be flat in the hologram plane. A method was proposed by Miccio (Miccio et al., 2007) who performed a two-dimensional fitting with the Zernike polynomials of the reconstructed unwrapped phase, although the application of the method is limited to be

For image reconstruction different algorithms have been developed. Among them, the Single Fourier Transform Formulation (SFTF) (Schnars, 1994), the convolution-based algorithm (CV) (Demetrakopoulos & Mittra, 1974) and the angular spectrum-based algorithm (ASA) (Yu & Kim, 2005) are most commonly used. The SFTF algorithm is fast and can be used with objects larger than a CCD. However, variation of the size of the reconstructed image as result of a change in the reconstruction depth poses problems in applications such as reconstruction of color holograms (Yamaguchi et al., 2002) and particle sizing (Pan & Meng, 2003). In contrast, the CV algorithm keeps the size of the reconstructed image the same as that of the CCD. However, it is applicable only to objects that are smaller than the CCD. In addition, when the CV algorithm is used for objects much smaller than a CCD, it degrades the image quality, since the image is represented by only a small number of pixels. The CV algorithm was extended to large objects by zero padding the holograms before reconstruction (Kreis et al., 1997). This approach, however, led to an increase in

To avoid a number of limitations of the previous algorithms, the double Fresnel-transform algorithm (DBFT) that allows the reconstruction of digital holograms with adjustable magnification was developed (Zhang et al., 2004)**.** This algorithm involves two reconstruction steps implemented by a conventional single Fourier-transform algorithm. Through the adjustment of the distance parameter in the first stages, it is possible to control the size of the reconstructed images, independent of distance and wavelength, even for

The method of image reconstruction proposed here is similar to the DBFT algorithm, since we also use two steps for image reconstruction, but they differ in the objectives and meaning of the two steps. Our formulation is specific for digital holographic microscopy; this way the objective of the first stage is the calculation of the objects' Fourier transform plane, where the complex wavefield contains all the information about the phase and intensity of object wavefield. Then, the second step is to reconstruct the complex amplitudes of the image wavefield starting from the objects' Fourier transform plane. We will describe and show

objects larger than a CCD without any computational penalty.

anamorphism (De Nicola et al., 2005).

the special case of thin objects.

computational load.

advantages of the proposed method.

Figure 1 shows the experimental set-up used in this work. It is a Digital Holographic Microscope designed for transmission imaging with transparent sample. The basic architecture is that of a Mach-Zehnder interferometer. A linearly polarized He-Ne laser (15 mW) is used as light source. The expanded beam from the laser is divided by the beam splitter BS1 into reference and object beams. The microscope produces a magnified image of the object and the hologram plane is located between the microscope objective MO and the image plane *(x'-y')* which is at a distance *d'* from the recording hologram plane *(-)*. In digital holographic microscopy we can consider the object wave emerging from the magnified image and not from the object itself (VanLigten & Osterberg, 1966).

Fig. 1. Experimental set-up: BE, bean expander; BS, bean splitter (the splitting ratio of BS1 and BS2 are 90/10 and 50/50 respectively); M, mirror; MO, microscope objective; S, sample; P, dual polarizer; CCD, charge coupled device.

With the combinations of the dual polarizer *P1* and *P2* the intensities are adjusted in the reference arm and the object arm of the interferometer and the same polarization state is also guaranteed for both arms improving their interference. The specimen S is illuminated by a plane wave and a microscope objective, that produces a wave front called object wave *O*, collects the transmitted light**.** A condenser, not shown, is used to concentrate the light or focus the light in order that the entire beam passes into the MO. At the exit of the interferometer the two beams are combined by beam splitter BS2 being formed at the CCD plane the interference pattern between the object wave *O* and the reference wave *R*, which is recorded as the hologram of intensity *IH(,)*,

$$I\_H\left(\mathbb{\mathcal{G}}\,\eta\right) = \left|\mathrm{O}\right|^2 + \left|R\right|^2 + R^\*O + RO^\* \tag{1}$$

where *R\** and *O\** are the complex conjugates of the reference and object waves, respectively. The two first terms form the zero-order, the third and fourth terms are respectively the virtual (or conjugate image) and real image, which correspond to the interference terms. The off-axis geometry is considered; for this reason the mirror M2, which reflects the reference wave, is oriented so that the reference wave reaches the CCD camera with a small incidence angle with respect to the propagation direction of the object wave. A digital hologram is recorded by the CCD camera HDCE-10 with 1024x768 square pixels of size 4.65 µm, and

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 187

Applying the (ASA) algorithm the distance *D* is calculated, as shown in section 3.1.1**.** After the distance *D* has been calculated, the first propagation is carried out by means of the Fresnel approximation method, specifically the S*ingle* F*ourier* T*ransform* F*ormulation* (SFTF),

2 2 2 2

*A=exp(i2πz/λ)/(iπλ)* and considering a plane reference wave with unit strength perpendicular

At z *= D*, the reconstructed wavefield, consists of a zero order and the twin images (real and

2 2 2 2 , ; exp , exp *f f*

to the complex field distribution *õ(u,v)* on the back focal plane of the objective lens. From Abbe's theory of image formation (Lipson S. & Lipson H., 1981), the field on the back focal

> ,; , , , , *<sup>f</sup> SFTF o o uvz D o uv S uv ox y f*

<sup>2</sup> *o o* , , *o o* , exp *o o oo o x y f o x <sup>y</sup> i xu <sup>y</sup> v dx dy <sup>f</sup>*

2 2 , exp exp 1

where *So* is the distance from the object to the lens, *f* the focal distance of the lens,

From the wave theory of image formation, after the objective lens producing the diffraction pattern of the object in its back focal plane, a second Fourier transformation performed on the diffraction pattern it is associated with the image of the object (Goodman, 1968). Consequently, all the information about image wavefield at the hologram plane is contained in the complex wavefront *o(u,v)* on the back focal plane, therefore the reconstruction of the optical wavefield *o(x',y';d')* can be carried out from the *(u-v)* plane, instead of traditionally

*<sup>i</sup> i k <sup>S</sup> S uv ik S f u v*

*f*)} is the Fourier transform of the object wave distribution at the plane *(u-*

(6)

2 *<sup>o</sup> <sup>o</sup>*

*f f f*

*D D*

(4)

*SFTF* 

*z z*

(3)

 

   

is an operator that denotes the Fourier transform,

*uvz D* can be expressed by Eq. (4) by

*)* with a filtered hologram \* , *<sup>f</sup>*

 

 

*SFTF* 

> 

(5)

 

*uvz D* is equivalent

*HI RO* 

(7)

the

*SFTF* , ; exp *<sup>H</sup>* , exp *i i uvz A u v I*

*,*

*i i uvz D A u v I*

containing only spatial components of the real image (Cuche et al., 2000),

As has been proven (Palacios et al., 2008), the complex field , ; *<sup>f</sup>*

*SFTF H*

conjugate). The filtered complex wavefield , ; *<sup>f</sup>*

to the recording plane.

where {*o*(*xo*, *yo,* 

and *S*

where z is the reconstruction distance,

replacing the specimen hologram *IH(*

plane can be represented by the expression,

*(u,v)* is a quadratic phase factor,

wavelength of the incident plane wave.

the hologram plane *(ξ-η)*, figure 4.

*v),* with *o(xo,yo)*, the amplitude transmittance of the object,

transmitted to the computer by means of the IEEE 1394 interface. The digital hologram *IH(j,l)* is an array of *M x N = 1024 x 768* 8-bit-encoded numbers that results from the twodimensional sampling of *IH(,)* by the CCD camera,

$$I\_H\left(j,l\right) = I\_H\left(\xi,\eta\right) \text{rect}\left[\frac{\xi}{L\_\chi},\frac{\eta}{L\_y}\right] \sum\_{j=-M/2}^{M/2} \sum\_{l=-N/2}^{N/2} \delta\left(\xi - j\Delta\xi,\eta - l\Delta\eta\right) \tag{2}$$

where *j, l* are integers defining the positions of the hologram pixels and = = *4.65 m*  defines the sampling intervals in the hologram plane.

### **3. Basic principles of the alternative reconstruction method**

In Digital Holographic Microscopy the field produced by the objective lens can be reconstructed in any plane along the field propagation direction. Traditionally the optical field  *(x',y')* on the image plane *(x'-y')* is calculated by propagation of the wavefront *(,)*, a distance *z = d'* from the hologram plane *(-)*, figure 2.

Fig. 2. Schematic diagram of traditional image reconstruction methods in DHM.

In our approach the reconstruction of the complex wave distribution *o(x',y') (x',y';z=d')* consists basically of two stages that involve two wavefield propagations. In the first stage, figure 3, we reconstruct the wave distribution *õ(u,v)* on the *(u-v)* plane at reconstruction distance z *= D* (first propagation)*.* 

Fig. 3. Reconstruction of the wave distribution *õ(u,v)* on the *(u-v)* plane at reconstruction distance z *= D* (first propagation).

Applying the (ASA) algorithm the distance *D* is calculated, as shown in section 3.1.1**.** After the distance *D* has been calculated, the first propagation is carried out by means of the Fresnel approximation method, specifically the S*ingle* F*ourier* T*ransform* F*ormulation* (SFTF),

$$\mathcal{W}\_{\rm SFT}\left(\mu,\upsilon;z\right) = A \exp\left[\frac{i\pi}{\lambda z}\left(\mu^2 + \upsilon^2\right)\right] \Im\left\{I\_H\left(\xi,\eta\right)\exp\left[\frac{i\pi}{\lambda z}\left(\xi^2 + \eta^2\right)\right]\right\} \tag{3}$$

where z is the reconstruction distance, is an operator that denotes the Fourier transform, *A=exp(i2πz/λ)/(iπλ)* and considering a plane reference wave with unit strength perpendicular to the recording plane.

At z *= D*, the reconstructed wavefield, consists of a zero order and the twin images (real and conjugate). The filtered complex wavefield , ; *<sup>f</sup> SFTF uvz D* can be expressed by Eq. (4) by replacing the specimen hologram *IH(,)* with a filtered hologram \* , *<sup>f</sup> HI RO* containing only spatial components of the real image (Cuche et al., 2000),

$$\mathcal{W}\_{\rm SFT}\left(\boldsymbol{u},\boldsymbol{v};\boldsymbol{z}=\boldsymbol{D}\right) = A \exp\left[\frac{i\pi}{\lambda D}\left(\boldsymbol{u}^{2}+\boldsymbol{v}^{2}\right)\right] \Im\left\{I\_{H}^{\boldsymbol{f}}\left(\boldsymbol{\xi},\boldsymbol{\eta}\right)\exp\left[\frac{i\pi}{\lambda D}\left(\boldsymbol{\xi}^{2}+\boldsymbol{\eta}^{2}\right)\right]\right\}\tag{4}$$

As has been proven (Palacios et al., 2008), the complex field , ; *<sup>f</sup> SFTF uvz D* is equivalent to the complex field distribution *õ(u,v)* on the back focal plane of the objective lens. From Abbe's theory of image formation (Lipson S. & Lipson H., 1981), the field on the back focal plane can be represented by the expression,

$$\left\|\boldsymbol{\psi}\_{\mathrm{SFT}}^{\boldsymbol{f}}\left(\boldsymbol{\mu},\boldsymbol{\upsilon};\boldsymbol{z}=\boldsymbol{D}\right)\right\| \cong \left\|\left(\boldsymbol{\mu},\boldsymbol{\upsilon}\right)=\mathrm{S}\_{\phi}\left(\boldsymbol{\mu},\boldsymbol{\upsilon}\right)\mathfrak{S}\left[o\left(\mathbf{x}\_{o},\boldsymbol{y}\_{o},\boldsymbol{\lambda}\boldsymbol{f}\right)\right] \tag{5}$$

where {*o*(*xo*, *yo, f*)} is the Fourier transform of the object wave distribution at the plane *(uv),* with *o(xo,yo)*, the amplitude transmittance of the object,

$$\mathfrak{S}\left[o\left(\mathbf{x}\_{o'}\boldsymbol{y}\_{o'}\,\mathcal{A}f\right)\right] = \iint o(\mathbf{x}\_{o'}\boldsymbol{y}\_{o})\exp\left[-i\frac{2\pi}{\lambda f}(\mathbf{x}\_{o}\boldsymbol{\mu}+\boldsymbol{y}\_{o}\boldsymbol{\nu})\right]d\mathbf{x}\_{o}d\boldsymbol{y}\_{o}\tag{6}$$

and *S(u,v)* is a quadratic phase factor,

186 Advanced Holography – Metrology and Imaging

transmitted to the computer by means of the IEEE 1394 interface. The digital hologram *IH(j,l)* is an array of *M x N = 1024 x 768* 8-bit-encoded numbers that results from the two-

rect *M/2 N/2*

In Digital Holographic Microscopy the field produced by the objective lens can be reconstructed in any plane along the field propagation direction. Traditionally the optical

 *(x',y')* on the image plane *(x'-y')* is calculated by propagation of the wavefront

*)*, figure 2.

*-*

Fig. 2. Schematic diagram of traditional image reconstruction methods in DHM. In our approach the reconstruction of the complex wave distribution *o(x',y')*

consists basically of two stages that involve two wavefield propagations. In the first stage, figure 3, we reconstruct the wave distribution *õ(u,v)* on the *(u-v)* plane at reconstruction

Fig. 3. Reconstruction of the wave distribution *õ(u,v)* on the *(u-v)* plane at reconstruction

*<sup>ξ</sup> <sup>η</sup> I j,l I <sup>ξ</sup>,<sup>η</sup> , <sup>δ</sup> - j , -l L L*

*j M/2l N /2 x y*

 

  

> =

(2)

 = *4.65 m* 

> *(,)*,

 *(x',y';z=d')*

*)* by the CCD camera,

where *j, l* are integers defining the positions of the hologram pixels and

**3. Basic principles of the alternative reconstruction method** 

dimensional sampling of *IH(*

field  *,*

defines the sampling intervals in the hologram plane.

*H H*

a distance *z = d'* from the hologram plane *(*

distance z *= D* (first propagation)*.* 

distance z *= D* (first propagation).

$$S\_{\phi}(u,v) = \frac{i}{\lambda f} \exp[-ik\left(S\_o + f\right)] \exp\left|\frac{ik}{2f}\left(u^2 + v^2\right)\left(1 - \frac{S\_o}{f}\right)\right|\tag{7}$$

where *So* is the distance from the object to the lens, *f* the focal distance of the lens, the wavelength of the incident plane wave.

From the wave theory of image formation, after the objective lens producing the diffraction pattern of the object in its back focal plane, a second Fourier transformation performed on the diffraction pattern it is associated with the image of the object (Goodman, 1968). Consequently, all the information about image wavefield at the hologram plane is contained in the complex wavefront *o(u,v)* on the back focal plane, therefore the reconstruction of the optical wavefield *o(x',y';d')* can be carried out from the *(u-v)* plane, instead of traditionally the hologram plane *(ξ-η)*, figure 4.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 189

As we will demonstrate in next section, the formulation based on Eq. (9) guarantees that the reconstructed image maintains its size independently of depth *d'* and the phase curvature

Figure 5(a) shows the hologram of a USAF resolution target recorded by the experimental set-up. In figure 5(b) are represented the components of the reconstructed wavefields at *z = D*. The undiffracted reconstruction wave forms a zero-order image which is located in the center (already filtered), and corresponds to the first two terms on the right-hand side

Due to off-axis geometry the real image (delimited by the circle) and the conjugate image (delimited by the rectangle) are positioned at different locations on the reconstruction plane and they correspond to the interference terms on the right-hand side of Eq. (1). The separation of the interference terms depends on the angle between the reference and object

dynamic range of pixel values, which improves the visualization of the intensity distribution

*SFTF(x'y',d'= D)|2]* was used to compress the

compensation can be done easily by techniques of image background subtraction**.**

**3.1 Experimental validation of the method: advantages and limitations** 

(a) (b)

reconstructed, figure 6, rather than the image of the objects.

Fig. 5. (a) Hologram of a USAF resolution target, (b) Components of the reconstructed wavefield at *z = D*. The circle delimits the real image and the rectangle the conjugate image. At *z = D* the reconstruction of real image is different of the genuine appearance of object image because at this distance the Fraunhofer diffraction pattern of the objects is

of Eq. (1).

wave. In figure 5b the expression *log[1+|(*

of the reconstructed wavefields (Lim, 1990).

Fig. 4. Reconstruction of the optical wavefield *o(x',y';d')* in the image plane from the *(u-v)* plane (second propagation).

In the second stage of the method, the complex wavefield *(x',y';d')* at an arbitrary distance *d'* can be obtained by propagation of the wavefield *õ(u,v)* through a distance *d'* and the result is inverse Fourier transformed,

$$\Psi\left(\mathbf{x'},\mathbf{y'};d'\right) = \mathfrak{T}^{-1}\left[\nu\_{\mathrm{SFT}}^{\zeta}\left(\mathbf{u},\upsilon;z=D\right)\exp\left(i d'\sqrt{k^2 + k\_{\mathrm{u}}^2 + k\_{\nu}^2}\right)\right] \tag{8}$$

where *-1* symbolizes the inverse Fourier Transform, *k=2/λ*, *ku* and *kv* are corresponding spatial frequencies of *u* and *v* respectively. The numerical implementation of Eq. (8), that we call the D*ouble-*P*ropagation* algorithm (DPA), is given by,

$$\begin{split} \left| \boldsymbol{\nu}\_{\text{DPA}} \left( \boldsymbol{m}, \boldsymbol{n}; \boldsymbol{d}' \right) = \text{FFT}^{-1} \left\{ \boldsymbol{\nu}\_{\text{SFTF}}^{\text{f}} \left( \boldsymbol{l}, \boldsymbol{j}; \boldsymbol{z} = \boldsymbol{D} \right) \right. \\ \left. \times \exp \left[ 2 \pi \, \boldsymbol{i} \, \text{d}" \sqrt{\left( 1 \, \text{/} \boldsymbol{\lambda} \right)^{2} + \left( \boldsymbol{l} \, \text{/} \boldsymbol{N} \, \text{\boldsymbol{\Delta}}\_{\text{F}}^{\text{f}} \right)^{2} + \left( \boldsymbol{j} \, \text{/} \boldsymbol{M} \, \text{\boldsymbol{\Delta}} \boldsymbol{\eta} \right)^{2}} \right] \right\} \end{split} \tag{9}$$

where *j, l, m, n* are integers *(-M/2 < j, l < M/2), (-N/2 < m, n < N/2*) and , ; *<sup>f</sup> SFTF l j z D* is the discrete formulation of Eq. (4). From Eq. (9) we can obtain the intensity image *IDPA(x',y';d')* by calculating *|DPA(m,n;d')|2* and the phase image *DPA(x',y';d')* by calculating arg[*DPA(m,n;d')*].

In short, the following steps describe the general procedure of the alternative reconstruction method:


Fig. 4. Reconstruction of the optical wavefield *o(x',y';d')* in the image plane from the *(u-v)*

*d'* can be obtained by propagation of the wavefield *õ(u,v)* through a distance *d'* and the

 <sup>1</sup> <sup>222</sup> , ; , ; exp *<sup>f</sup> SFTF <sup>u</sup> x y d u v z D id k k k*

spatial frequencies of *u* and *v* respectively. The numerical implementation of Eq. (8), that we

exp 2 1 / / /

discrete formulation of Eq. (4). From Eq. (9) we can obtain the intensity image *IDPA(x',y';d')*

In short, the following steps describe the general procedure of the alternative reconstruction

5. Determining the intensity image or the phase image by calculating the square modulus

2 2 2

 

*i d lN jM*

*SFTF uvz D* , ; using Eq. (3), i.e., the complex amplitudes of the wavefield

*uvz D* by filtering the spatial components that correspond to the

*(x',y';d')* at an arbitrary distance

*/λ*, *ku* and *kv* are corresponding

(8)

(9)

*l j z D* is the

*DPA(x',y';d')* by calculating

 

> *SFTF*

plane (second propagation).

where

by calculating *|*

*DPA(m,n;d')*].

arg[

method:

2. Calculate

4. Calculate

at *z =D*.

result is inverse Fourier transformed,

*DPA*

1. Determining the distance *D*.

3. Obtaining , ; *<sup>f</sup> SFTF* 

or the argument of Eq. (9).

In the second stage of the method, the complex wavefield

call the D*ouble-*P*ropagation* algorithm (DPA), is given by,

 

*-1* symbolizes the inverse Fourier Transform, *k=2*

 

*DPA(m,n;d')|2* and the phase image

*DPA(m,n;d')*, using Eq. (9), at an arbitrary distance *d'*.

where *j, l, m, n* are integers *(-M/2 < j, l < M/2), (-N/2 < m, n < N/2*) and , ; *<sup>f</sup>*

*<sup>f</sup> m n d FFT l j z D SFTF*

1

, ; , ;

complex amplitudes of the Fourier transform of the objects.

As we will demonstrate in next section, the formulation based on Eq. (9) guarantees that the reconstructed image maintains its size independently of depth *d'* and the phase curvature compensation can be done easily by techniques of image background subtraction**.**

### **3.1 Experimental validation of the method: advantages and limitations**

Figure 5(a) shows the hologram of a USAF resolution target recorded by the experimental set-up. In figure 5(b) are represented the components of the reconstructed wavefields at *z = D*. The undiffracted reconstruction wave forms a zero-order image which is located in the center (already filtered), and corresponds to the first two terms on the right-hand side of Eq. (1).

Due to off-axis geometry the real image (delimited by the circle) and the conjugate image (delimited by the rectangle) are positioned at different locations on the reconstruction plane and they correspond to the interference terms on the right-hand side of Eq. (1). The separation of the interference terms depends on the angle between the reference and object wave. In figure 5b the expression *log[1+|(SFTF(x'y',d'= D)|2]* was used to compress the dynamic range of pixel values, which improves the visualization of the intensity distribution of the reconstructed wavefields (Lim, 1990).

Fig. 5. (a) Hologram of a USAF resolution target, (b) Components of the reconstructed wavefield at *z = D*. The circle delimits the real image and the rectangle the conjugate image.

At *z = D* the reconstruction of real image is different of the genuine appearance of object image because at this distance the Fraunhofer diffraction pattern of the objects is reconstructed, figure 6, rather than the image of the objects.

Alternative Reconstruction Method and Object Analysis in Digital Holographic Microscopy 191

Because the reference wave is plane, the distance D is the physical distance from the

Using the hologram of figure 5a, the behavior of the wavefield near the back focal plane is visualized. With *z* ranging from *141 mm* to *173 mm* and incremental step of *8 mm* a sequence of image reconstructions is presented in figure 8. The sequence shows the change of phase

z 141 mm 149 mm 157 mm 165 mm 173 mm

focal plane and it is increased as the wavefield propagates away from this plane.

From figure 8 it is corroborated that as the reconstruction plane approaches the focal plane, the phase jumps between the reference and object waves gradually disappear. These phase jumps totally disappear for z *= D = 173 mm*, where the focal plane is reconstructed. This behaviour allows us to conclude that the curvature of the wavefront has a minimum on the

*SFTF(l,j;z)])* and

Fig. 7. Distance D determination using *ASA*.

*SFTF(l,j;z)|2]* respectively.

*log[1+|(*

A

B

hologram to back focal point of the objective lens.

**3.1.2 Behavior of the wavefield near to the back focal plane** 

(A) and intensity (B) calculated from the expressions: arg(*FFT-1[*

Fig. 8. Behavior of the wavefield in the region of the back focal plane.

Fig. 6. Intensity of field distribution on the focal plane.

Larger or lower spatial frequencies of object decomposition will be represented by intensity in the focal plane that is farther or closer from optical axis or equivalently farther or closer from the center of the pattern where is contained the undiffracted object wavefield.
