Optics Communications 248 (2005) 41–50

www.elsevier.com/locate/optcom

3D image reconstruction of transparent microscopicobjects using digital holography

Francisco Palacios a,*, Jorge Ricardo a, Daniel Palacios b, Edison Goncalves c,Jose L. Valin d, Rodrigo De Souza c

a Physics Department, Faculty of Natural Science, University of Orient, Patricio Lumumba SN, Santiago de Cuba, Cubab Nuclear Physics Department, University Simon Bolivar, Caracas, Venezuela

c Engineering Mecatronic and Mechanic System Department, Polytechnical School, Sao Paulo University (USP), Brazild Mechanics Department, High Polytechnical Institute ‘‘Jose Antonio Echeverrıa’’, CP 1930, AP 6028, Havana, Cuba

Received 20 July 2004; received in revised form 26 November 2004; accepted 26 November 2004

Abstract

In this paper, we present the potentialities of the digital holography microscopy for 3D image reconstruction of

transparent microscopic objects. A method for object volume reconstruction based on the capture of only one off-axis

hologram is discussed. We show that this technique can be efficiently used for obtaining quantitative information from

the intensity and the phase distributions of the reconstructed field at different locations along the propagation direction.

The potential of the method has been exploited by applying the 3D image reconstruction procedure to etched nuclear

tracks induced in the widely used polymer detector CR-39. Nowadays, confocal microscopy and atomic force micros-

copy are applied to study nuclear track morphology in CR-39 detectors. The micro-holographic method developed in

this work, and applied to that particular study, constitutes a new alternative procedure that overcomes the previously

described at least in technological simplicity.

� 2004 Elsevier B.V. All rights reserved.

PACS: 07.05.Pj; 42.30.RxKeywords: 3D phase reconstruction; Phase unwrapping; Digital holography; Holographic contouring

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.11.095

* Corresponding author. Tel.: +53 22 633011/2263; fax: 53 22

632689/2042.

E-mail address: palacios@cnt.uo.edu.cu (F. Palacios).

1. Introduction

Digital holography has several features that

make it an interesting alternative to conventional

microscopy. These features include an improved

focal depth, possibility to generate 3D images,

and phase contrast images [1–3].

ed.

42 F. Palacios et al. / Optics Communications 248 (2005) 41–50

One popular version of the experimental set-up

in digital holography is the digital Fresnel hologra-

phy, which uses quasi-parallel beans [4] to record

with a CCD camera the intensity distribution of

the hologram. The digital hologram is multipliedby the reference wavefield in the hologram plane

and the diffracted field in the image plane is deter-

mined by the usual Fresnel–Kirchoff integral [5] to

calculate the intensity and the phase distribution

of the reconstructed real image. From the numer-

ical standpoint, the Fresnel–Kirchoff integral can

be represented in terms of an inverse discrete Fou-

rier transformation.Although Fourier transform holographic meth-

od needs only one hologram for obtaining the

phase value corresponding to all points of the

reconstructed phase field, must contouring method

uses two holograms to obtain the phase interfero-

gram and later, after unwrapping procedure, the

object shape. This is the case of the two-wave-

length contouring [6], two-source contouringmethod [7], etc. In general, the determination of

phase values by Fourier method, as is necessary

to use the arctan function for extracting the phase

[8], are wrapped onto the range �p to p rad, and

the process of phase unwrapping, i.e., restoration

of the unknown multiple of 2p to all pixels, must

be carry out before the parameter can be deduced

from the phase distribution.A wide variety of partial solution to the

unwrapping procedure is discussed in the litera-

ture [9,10]. However, the temporal phase

unwrapping is recognized as the best by its sim-

plicity and the fact that it is guarantee to pro-

duce a correct unwrapping of phase maps

containing physical boundaries of the object. In

temporal phase unwrapping [11] the 3D phasefield is generated making holograms capture dur-

ing the time and the phase unwrapping consider

at t direction for phase accumulation. In this

case the 2D unwrapping is not performed, only

along t-axis the unwrapping works, but many

holograms must be captured. This method when

applied for shape contouring or dynamic defor-

mation is fast compared with any other methodthat uses 2D unwrapping and the results are not

exposed to corrupting unwrapping mistake. In

this case the unwrapping procedure takes into

consideration the phase difference for obtaining

the phase image.

Interesting applications of digital holography

rely on the possibility of carrying out whole

reconstruction of the recorded wave front andconsequently, the determination of the phase

distribution at any arbitrary plane located be-

tween the object and the recording plane, and

along the object reconstructed image. Grilli

et al. [12] use the digital holography possibility

of reconstructing the image field at different

planes from the object to find the object beam

phase form and to show how the experimentalset-up affects this, but they do not found the

3D object shape.

In this work, we employ a variant of unwrap-

ping algorithm that shares the same basic princi-

ple with the temporal phase unwrapping

technique but uses as t-axis the reconstructed

depth in the image field. This is the principal

idea applied in this work for to obtain the 3Dphase image corresponding to a nuclear track in-

duced in the CR-39 detector as microscopic

object.

The organization of the paper is as follows. In

Section 2, the general principle of digital hologra-

phy is presented. The experimental set-up is

described in Section 3. The 3D image reconstruc-

tion algorithm is discussed in Section 4. The resultsand discussion are discussed in Section 5.

2. General principles of digital holography

Holography is a method that allows reconstruc-

tion of whole optical wave fields. The principle of

recording and reconstruction of holograms isshown in Fig. 1. The light coming from the object

under test and a reference wave interfere in the

hologram plane.

In classical holography the reconstruction is

carry out by illumination of the hologram intensity

with the reference wave. A virtual and a real image

of the object are reconstructed.

In digital holography this reconstruction pro-cess is performed by multiplication of the stored

digital hologram with the numerical description

of the reference wave and by the convolution of

Fig. 1. Recording of the hologram (a) and image reconstruction (b).

F. Palacios et al. / Optics Communications 248 (2005) 41–50 43

the result with the impulse response function. Thediffracted field in the image plane is given by the

Rayleigh–Sommerfeld diffraction formula

wðx0,y0Þ ¼ 1

ik

Z Zhðn,gÞrðn,gÞf ðx0,y0,n,gÞ

� cosH dx dy ð1Þ

with

f ðx0,y0,n,gÞ ¼ expðikqÞq

ð2Þ

and

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid 02 þ ðn� x0Þ2 þ ðg� y 0Þ2

q, ð3Þ

where d 0 is the reconstruction distance, i.e., the dis-

tance backward measured from the hologram

plane n–g to the image plane, h(n,g) is the recordedhologram, r(n,g) represents the reference wave

field, k denotes the wave number and k is the wave-length of the laser source. Due to the small angles

between the hologram normal and the rays from

the hologram to the image points, the obliquity

factor cosH can normally be set cosH = 1.

Different approaches for solution of Eq. (1)

have been proposed [13]. The Fresnel approxima-

tion replaces the factor q by the distance d 0 in

the denominator of Eq. (2) and the square rootin the argument of the exponential function is re-

placed by the first terms of a binomial expansion.

This approximation is valid since the dimension

of the CCD chip is small in comparison to the dis-tance d 0 [13]. With these approximations Eq. (1)

takes the form:

wðx0,y0; d 0Þ ¼ exp½ipd 0kðm2 þ l2Þ�

�Z Z

hðn,gÞrðn,gÞgðn,gÞ

� exp½�2ipðnmþ glÞ� dn dg, ð4Þ

where the quadratic phase function g(n,g) is the

impulse response,

gðn,gÞ ¼ expði2pd 0=kÞikd 0 exp

ipd 0k

ðn2 þ g2Þ� �

ð5Þ

and the spatial frequencies are m ¼ x0

d 0k, l ¼ y0

d 0k.

The discrete finite form of Eq. (4) is obtainedthrough the pixel size (Dn,Dg) of the CCD array

[14], which is different from that (Dx 0,Dy 0) in the

image plane x 0–y 0 and related as follows:

Dx0 ¼ d 0kMDn

, Dy 0 ¼ d 0kNDg

,

where M and N are the pixels number of the CCDarray in x 0 and y 0 directions, respectively.

According to Eq. (4), the wavefield w(x 0,y 0,d 0) is

determined essentially by the 2D Fourier trans-

form of the quantity h(n,g)r(n,g)g(n,g). For rapid

numerical calculations a discrete formulation of

Eq. (4) involving a 2D fast Fourier transform algo-

rithm (FFT) as showed in:

Fig. 2. Experimental set-up.

44 F. Palacios et al. / Optics Communications 248 (2005) 41–50

wðm,n,d 0Þ ¼ expði2pd 0=kÞikd 0

� expipkd 0 ðm2Dx02þn2Dy02Þ

� �FFT

� IHðj,lÞrðj,lÞexpipkd 0 ðj2Dn2þ l2Dg2Þ

� �� �m,n

,

ð6Þ

where j, l, m, n are integers (�M/2 6 j, l 6M/2),

(�N/2 6 m, n 6 N/2) and IH is the digitalhologram.

In the formulation based on Eq. (6) the recon-

structed image is enlarged or contracted according

to the depth d 0. An alternative approach is useful

for keeping the size of the reconstructed image

constant [5]. In this formulation, the wavefield

w(x 0,y 0,d 0) can be calculated by,

wðx0,y 0,d 0Þ ¼ I�1fI½hðn,gÞrðn,gÞ�I½gðn,gÞ�g, ð7Þ

where I½gðn,gÞ� is the Fourier transform of the im-

pulse response, namely

I½gðn,gÞ� ¼Z 1

�1

Z 1

�1gðn,gÞ exp½�i2pðmnþ lgÞ� dn dg:

ð7aÞ

Taking into account the form of the impulse re-

sponse in Eq. (5) we have that its Fourier trans-

form is given by,

I½gðn,gÞ� ¼ expði2pd 0=kÞ exp½�ipkd 0ðm2 þ l2Þ�:ð8Þ

With this method the size of the reconstructed im-

age does not change, i.e., Dx 0 = Dn, Dy 0 = Dg and

is necessary one Fourier transform and one inverseFourier transform each to obtain one 2D recon-

structed image at a distance d 0. Although the com-

putational procedure is heavier in this case

compared to the Fresnel approximation approach

of Eq. (6), this method allows for easy comparison

of the reconstructed images at different distances d 0

since the size does not change with modifying the

reconstruction distance. Furthermore, in this casewe get an exact solution to the diffraction integral

as far as the sampling Nyquist theorem is not

violated.

3. Experimental set-up

Fig. 2 shows the experimental set-up used in

this work, corresponding to a digital holographic

microscope designed for transmission imagingwith transparent sample. The basic architecture is

that of a Mach-Zender interferometer. A linearly

polarized He–Ne laser (15 mW) is used as light

source. The expanded beam from the laser is di-

vided by the beam splitter (BS1) into reference

and object arms.

With the combinations of the dual polarizers P1

and P2 is adjusted the intensities in the referencearm and the object arm of the interferometer and

the same polarization state is also guaranteed for

both arms improving their interference.

The specimen M is illuminated by a plane wave

and a microscope objective (MO) that produces a

wave front called object wave O collects the trans-

mitted light. A condenser, not shown, is used to

intensify the light intensity of the objectillumination.

At the exit of the interferometer, the interfer-

ence between the object wave O and the reference

wave R creates the hologram intensity Ih(n,g). Theoff-axis geometry is considered, for this reason the

mirror E2, which reflects the reference wave is ori-

ented such that the reference wave reaches the

CCD camera with a small incidence angle with re-spect to the propagation direction of the object

wave.

A digital hologram is recorded by a standard

black and white CCD camera (PULNIX model

TM-9701) and transmitted to a computer by

means of a Frame Grabber card. The digital holo-

gram Ih(j,l) is an array of M · N = 640 · 480 8-bit-

F. Palacios et al. / Optics Communications 248 (2005) 41–50 45

encoded numbers that results from the 2D sam-

pling of Ih(j,l) by the CCD camera,

IHðj,lÞ ¼ Ihðn,gÞrectnLx

,gLy

� �

�XM=2

k¼�M=2

XN=2

l¼�N=2

dðn� jDn,g� lDgÞ, ð9Þ

where j,l are integers, Lx · Ly = 8.91 mm · 6.58

mm is the area of the sensitive chip, and

Dn = 11.6 lm, Dg = 13.6 lm defines the sampling

intervals in the hologram plane.

Holographic microscopy has been proposed in

various configurations [15], in this work we are

working with a geometry including a MO that

was proposed for the first time in 1966 by VanLig-ten and Osterberg [16]. In digital holography a

similar approach was used by Cuche et al. [17].

As shown if Fig. 3, the optical arrangement in

the object arm is an ordinary single-lens imaging

system.

The MO produce a magnified image of the

object, and the hologram plane 0n (the CCD

plane) is located between the MO and the imageplane (0x 0), at a distance d 0 from the image. This

situation can be considered to be equivalent to a

holographic configuration without magnification

optics with an object wave emerging directly

from the magnified image and not from the ob-

ject itself. For this reason the term image holog-

raphy is sometime used to designate this

procedure. Classical microscopy can be achievedwith this arrangement by translation of the ob-

ject or the hologram plane such that the image

is focused on the CCD.

Fig. 3. Configuration for holographic microscopy.

4. 3D image reconstruction

In this work, nuclear tracks induced in CR-39

polymeric detector are used as transparent micro-

scopic object. CR-39 detector is one of the mosthomogeneous, amorphous polymer with which

one can achieve high resolution and sensitivity in

nuclear track recording. The crossroad of nuclear

particles and heavy ions through most of the solid

semiconductors create fine patterns of intense

damage at atomic scale. This is the origin of nucle-

ar tracks induced in solids. These tracks have to be

etched chemically to make them visible with anoptical microscopy. Etching procedure consists in

chemical dissolution of material along the trace

of nuclear particle and a dissolution or general at-

tack of detector surface. In general, etched tracks

have conical forms whose dimensions depend on

the energy of particles and conditions of chemical

etching.

The variation of refractive index between thenot revealed part of CR-39 detector (nc = 1.5)

and the etched nuclear track (nair = 1) leads to a

change of the optical path length and thereby a

change of phase between the light waves passing

through track contour and reference wave. With

digital holography it is possible to reconstruct

the wave field of the object for different reconstruc-

tion distances, so that different optic sections orreconstructed 2D images can be established.

4.1. 2D image reconstruction

The 2D image reconstruction is digitally and

numerically carried out from the observed holo-

gram pattern, expressed by Ih(j,l), using the above

Fresnel diffraction equation (7). The result of thecalculations is an array of complex numbers called

wave front w(x 0,y 0,d 0), which represent the com-

plex amplitude of optical field in the observation

plane 0x 0y 0. The distance between the hologram

plane 0n,g and the observation plane is defined

by the reconstruction distance d 0. When the holo-

gram is recorded without a MO, d 0 must be equal

to the distance (or more precisely, to the opticalpath length) between the object and the CCD to

obtain in-focus reconstructed images. In holo-

graphic microscopy, image focussing occurs when

46 F. Palacios et al. / Optics Communications 248 (2005) 41–50

the reconstruction distance is equal to the distance

between the CCD and the image during the holo-

gram recording (d 0 in Fig. 3).

Since w(x 0,y 0,d 0) is an array of complex number,

we can obtain a 2D amplitude-contrast image bycalculating the intensity:

Iðx0,y0,d 0Þ ¼ Re½wðx0,y0,d 0Þ�2 þ Im½wðx0,y0,d 0Þ�2

ð11Þand a 2D phase-contrast image by calculating the

argument,

/ðx0,y 0,d 0Þ ¼ arctanIm½wðx0,y 0,d 0Þ�Re½wðx0,y 0,d 0Þ� : ð12Þ

For phase-contrast imaging the digital hologram

has to be multiplied by a digital reference wave

Rd [17], which must be a replica of experimental

reference wave R. In classical holography, thesame operation is performed optically when the

hologram is illuminated with the reference wave.

If we assume that a perfect plane wave is used as

reference for hologram recording, Rd is calculated

as follows:

Rdðj,lÞ ¼ AR exp i2pkðkxjDnþ kylDgÞ

� �, ð13Þ

where j and l are integers (�M/2 6 j 6 M/2), (�N/

2 6 l 6 N/2), AR is the amplitude, Dn and Dg are

the sampling intervals in the hologram plane,and kx, ky are the two components of the wave vec-

tor that must be adjusted such that the propaga-

tion direction of Rd matches as closely as

possible that of the experimental reference wave.

In practice a method to approximate Rd to R

consists in varying kx, ky until exist coincidence be-

tween the form and distribution of fringes of the

real part of Rd with the interference fringes ofthe hologram. A fine adjustment of kx and kycan be performed by removal all fringes of the

reconstructed phase distribution in some area of

the image where a constant phase is presumed.

4.2. 3D unwrapping algorithm

The proposed method of 3D reconstruction isbased on a similar procedure to that of temporal

unwrapping, where the t-axis is substituted by the

z-axis or rather in direction where varies the focus

of the reconstructed image. Starting from the recon-

struction of the2D phase-contrast image, the phase

at each pixel is measured as a function of the field

depth varying the reconstruction distance d 0.The phase values calculated by Eq. (12) lie auto-

matically in the range �p to +p; this means that

calculating the sequence of the complex map of

field w(x 0,y 0,d 0) for various reconstruction dis-

tances d 0 separated by a distance Dd 0 sufficiently

small, so that discontinuities do not exist between

two consecutive phase planes, the continuous

phase distribution can be obtained by computingthe finite sum of Eq. (12),

/ðx0,y0,d 0sÞ ¼

Xd 0sd 0¼d 0f

/ðx0,y 0,d 0Þ, ð14Þ

where d 0f and d 0

s are the distances from the CCD

to the bottom of nuclear tracks and to the upper

top of detector surface, respectively. These two

distances are measured regarding the magnified

image of the object and can be determined experi-

mentally using the intensity image reconstruction

and auto focus software techniques.

The first reconstructed phase field is carried outfor d 0 ¼ d 0

f , taking this like reference for the sum

of the successive phase maps. This way, the phase

maps calculated for values of d 0 increased they rep-

resent phase variations regarding that of reference,

and then the sum would give the integration of the

phase along direction where varies the focus.

Contrary to the traditional temporal unwrap-

ping, in this work the accumulation of phaseplanes / is used instead of the phase difference

D/ among adjacent planes because, in this case, di-

rectly with the phase values / the fringes of con-

tour of phase of the object are obtained for a

certain distance of reconstruction.

The simplicity of this algorithm resides in the

correct election of Dd 0 that guarantee the phase

difference between two consecutive planes shouldbe less that p/2 providing that discontinuities do

not exist between two consecutive phase planes.

Fig. 4(a) illustrates light crossroad through a

nuclear track induced in detector.

If light rays propagate along straight lines in a

transparent medium along the z-axis the change

Fig. 5. Scheme of the unwrapping path based on sum of 2D

phase planes algorithm in 3D space.

Fig. 4. (a) Light crossroad through a nuclear track induced in

CR-39 detector. (b) Object wave emerging directly from the

magnified image and not from the object itself.

F. Palacios et al. / Optics Communications 248 (2005) 41–50 47

of phase between two adjacent rays at MO,

which is maintained in the magnified image, is

given by:

D/i,iþ1 ¼ /i � /iþ1 ¼ kðdi � diþ1Þ, ð15Þ

where k denotes the wave vector (k = 2p/k), dI,dI + 1, the optical path lengths of the rays i andi + 1, respectively.

The ith phase changes can be expressed as

/i ¼2pkðdicnc þ nadiaÞ ð16Þ

and for (i + 1)th phase change,

/iþi ¼2pkðdðiþ1Þcnc þ dðiþ1ÞanaÞ: ð17Þ

Subtracting Eq. (17) from Eq. (16) and reorganiz-

ing the terms,

D/iþ1,i ¼2pkDdcðnc � 1Þ: ð18Þ

Considering D/ < p/2, then Ddc < k/2.To carry out previous analysis to the plane of

the magnified image of object, Fig. 4(b), it is nec-

essary to multiply Ddc by the axial magnify b of

the MO.

Dd 0 ¼ Ddcb,

where b = m2 [18]. In this work m = 90, so that fi-

nally Dd 0 < 2 mm.

As the reconstruction is carried out to the mag-

nified image of the object, corresponding to thefree-space of propagation, the reconstruction for-

mulas are valid, as well as the obtained phase

values.

Summarizing, Eq. 14 indicates a 3D phase

unwrapping algorithm when Dd 0 satisfy the previ-

ous condition. Fig. 5 illustrates graphically the

algorithm.

The values /i � 1 and /i,i � 1 represent the un-wrapped phase by calculating phase sum from

i = 1 to i � 1 and wrapped phase between i and

i � 1, respectively.

The distinctive feature of this method is that

only one hologram is needed to unwrap the whole

3D phase field, and the unwrapping procedure is

very simple in this condition.

5. Results and discussion

Fig. 6 shows a digital hologram of a selected

area of CR-39 detector recorded with the

Fig. 6. Digital hologram recorded by CCD camera.

Fig. 7. Sequence of a video clip presentation of the intensity

distribution, of a digitally reconstructed object wavefield,

obtained at different distances d 0 from the hologram plane

along z-axis direction.

48 F. Palacios et al. / Optics Communications 248 (2005) 41–50

experimental set-up presented in Fig. 2, the MO

magnification is m = 90 and numerical aperture

NA = 1.25.

The hologram appears as a pattern of interfer-

ence fringes. In this work, experimental conditions

were adjusted so that fringes of interference wereright and not curve, avoiding the necessity of

digital correction of the spherical aberration intro-

duced by MO. Another parameter that character-

izes the interference fringes constitutes its spacing,

which keeps close inverse relationship between ob-

ject and reference wavefield with incidence angle

on the CCD. There is an optimum spacing be-

tween interference fringes, for which maximumseparation among the terms that characterize the

reconstructed wavefield (zero order and unsharp

virtual image) is guaranteed, without requirements

of sampling imposed by the CCD resolution are

violated.

From the hologram, the reconstructed ampli-

tude-contrast images was determined for d 0 rang-

ing from 180 to 225 mm and with Dd 0 = 1 mm.The sequence of intensity distribution was com-

bined to obtain a clip video presented in Fig. 7.

The reconstructed wavefield contains only the real

image of the nuclear tracks without influence of

the other reconstructed terms because of the off-

axis geometry of the set-up (see Fig. 2).

The video clip shows that a point shaped inten-

sity pattern is obtained at the reconstruction dis-tances corresponding to the distance from the

hologramplane to focusing distance of each bottomtracks. This example shows that numerical recon-

struction of holograms provides an efficient method

for visualizing virtual focusing of microscopic sam-

ple, which constitutes the experimental counterpart

of traditional focusing of an optical microscope.

Quantitative analysis of geometrical shape and

its main parameters of sample rely on the ability

of digital holographic microscopy to provideinformation not only about intensity but also on

the phase distribution of the optical wavefield at

different planes from the recorded hologram.

F. Palacios et al. / Optics Communications 248 (2005) 41–50 49

Applying the 3D reconstruction method described

in Section 4.2 the continuous phase distribution

was obtained (Fig. 8).

Height distribution h(x,y) on the sample surface

is simply proportional to the reconstructed phasedistribution /(x,y) [19]:

hðx,yÞ ¼ k4p

/ðx,yÞ: ð19Þ

In [19] an experiment performed with a surface of

homogeneous optical properties demonstrated

that phase changes corresponding to optical path

less than 10 nm can be measured on the basis of

the reconstructed phase-contrast image.

As mentioned above, the reconstructed phase

distribution can be used for quantitative measure-

ments, as it is illustrated in Fig. 9. Dimensional

Fig. 8. Reconstructed continuous phase distribution corresponding t

Fig. 9. (a) 3D plot of the reconstructed surface of a nuclear track

volume parameters and the profile forms of the nu-

clear track can be estimated.

Etched nuclear tracks were obtained by irradia-

tion of CR-39 samples with 5.48 MeV a-particlesfrom 241Am. The detector was etched 4 h in 6 NNaOH solution at 70 �C.

By means of Eq. (19) the calculated phase distri-

bution values were converted to height measure-

ments to calculate the zi values in the graph.

Corresponding xi and yj values were calculated

as follows:

xi ¼iDnm

, i ¼ 0,1, . . . ,M ,

yj ¼iDgm

, j ¼ 0,1, . . . ,N ,

o the hologram presented in Fig. 6. (a) 2D, (b) 3D perspective.

. (b) Sagittal cut at the middle of the reconstructed surface.

50 F. Palacios et al. / Optics Communications 248 (2005) 41–50

where Dn and Dg are the sampling intervals in the

hologram plane, m is the MO magnification and

M,N are the pixels number of the CCD array in

x and y directions, respectively.

6. Conclusions

In this paper, we have proved the potentialities of

the digital holography microscopy for 3D image

reconstruction of transparent microscopic objects.

A method for object volume reconstruction based

on the capture of only one off-axis hologram wasdiscussed, showing that this technique can be effi-

ciently used for obtaining quantitative information

from the intensity and the phase distributions of the

reconstructed field at different locations along the

propagation direction. The proposed method was

applied to the 3D image reconstruction of etched

nuclear tracks induced in the widely used polymer

detector CR-39. The micro-holographic methoddeveloped in this work, applied to the study of nu-

clear tracks, constitutes a new alternative procedure

that overcomes the current techniques, such as con-

focal microscopy and atomic force microscopy, at

least in technological simplicity.

Acknowledgments

The present work was supported by the Brazil-

ian research agencies CAPES and FINEP.

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