controlling images parameters in the reconstruction

8/3/2019 Controlling Images Parameters in the Reconstruction

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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 4, JULY/AUGUST 2004 829

Controlling Images Parameters in the ReconstructionProcess of Digital Holograms

Pietro Ferraro, Member, IEEE, Giuseppe Coppola, Domenico Alfieri, Sergio De Nicola, Andrea Finizio, andGiovanni Pierattini

AbstractDigital holograms recorded with a charge-coupleddevice array are numerically reconstructed in amplitude andphase through calculation of the FresnelKirchhoff integral.The flexibility offered by the reconstruction process in digitalholography allows exploitation of new possibilities of applica-tion in microscopy. Through the reconstruction process we willshow that it is possible to control image parameters as focusdistance, image size, and image resolution. Those explored po-tentialities open further the novel prospective of application ofdigital holography in single- and multiwavelengths operationeither for display or metrological applications. We demonstrate

the concept of controlling parameters in image reconstruction ofdigital holograms in some real situations for inspecting siliconmicroelectronicmechanical systems structures.

Index TermsHolographic interferometry, holography,metrology, microelectromechanical devices, microscopy.

I. INTRODUCTION

RECENT developments in solid-state image sensors and

digital computers have made it possible to directly record

holograms by charge coupled device (CCD) cameras and nu-

merical reconstruction of the object wave front by computer.

The idea of using a computer for reconstructing a hologram was

proposed for the first time by Goodman and Laurence [1] and

by Kronrod et al. [2].

This technique as well as the limitations imposed by the low

spatial resolution of the CCD camera array compared to that

of photographic material has been widely discussed and var-

ious configurations of digital holography (DH) have been pro-

posed and applied in various fields of applications [3][6]. DH

has been applied for deformation analysis, object contouring

[7][10], measurement of particle position [11][15], investi-

gation of large objects for contouring and comparative analysis,

for three-dimensional image formation, and for measuring dis-

tribution of complex amplitude for application to microscopy

[16][21].Two methods are usually adopted to reconstruct digital

holograms called Fresnel transformation method (FTM) and

the Convolution method (CM) [5], [6] even if new methods

have been developed [22].

In FTM, the reconstruction pixel increases with the recon-

struction distance so that the size of image, in terms of number

of pixels, is reduced for longer distances, limiting the resolution

Manuscript received January 6, 2004; revised June 8, 2004.The authors are with the National Research Council (CNR), Naples 80131,

Italy (e-mail: [email protected]).Digital Object Identifier 10.1109/JSTQE.2004.833876

of amplitude and phase reconstruction. In CM, by contrast, the

reconstruction pixel does not change and remains equal to the

pixel size of the recording array. However, the CM is more ap-

propriate for reconstruction at small distances, whereas the FTM

is useful for longer distances according to the paraxial approx-

imation necessary to apply it. This last reconstruction method

is particularly indicated for microscopic metrological applica-

tions such as, for example, in microelectromechanical systems

(MEMS) inspection and/or characterization [23]. In fact, DH

has been demonstrated to be a useful tool for inspecting micro-components and microstructures [24][31].

Recently, for metrological applications multiwavelength

DH has been employed [32], [33], even if a substantial lim-

itation occurs when FTM reconstruction is adopted. In fact,

the holograms recorded with different wavelengths produce

reconstructed images of different size and resolution hindering

a direct superimposition [32][35].

Here, we propose different approaches to adopt in the recon-

struction process of digital holograms. The proposed methods

add much more flexibility for DH. Through the proposed

methods different parameters involved in the reconstruction

images, such as correct focusing, image size, and image res-olution, can be simultaneously controlled. The possibility to

control such parameters is useful for some typical cases of

application of DH in microscopy and multiwavelength opera-

tions overcoming the above outlined limitations. In Section II,

principles of operation of DH will be described. In Section III,

by carefully managing the reconstruction process it is pos-

sible to track in real-time focus change due to the movements

of the sample during the inspection. In Section IV, we will

demonstrate how it is possible to control image size in the

reconstruction image plane independently from the recording

distance and wavelength. In that way, it is possible to apply

holographic interferometry between two holograms of the same

object even if they are recorded at different distances. In thesame way, the multiwavelength DH is applicable with perfect

superimposing for color application of DH or deformation

analysis. Finally, in Section V, we will demonstrate a way to

control image resolution by changing the reconstruction pixel

size for recovering undersampled wrapped phase obtaining the

correct profile of highly curved MEMS structure.

II. PRINCIPLE OF OPERATION OF DH

In holography, an object is illuminated by a collimated,

monochromatic, coherent light with a wavelength . The object

1077-260X/04$20.00 2004 IEEE


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830 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 10, NO. 4, JULY/AUGUST 2004

scatters the incoming light forming a complex wavefield (the

object beam)

(1)

where is the amplitude and the phase and and de-

note the Cartesian coordinates in the plane where the wave field

is recorded (hologram plane). The phase incorporates

information about the topographic profile of the MEMS under

investigation because it is related to the optical path difference

(OPD)

OPD (2)

where a reflection configuration has been considered. The

purpose of holography is to capture the complete wavefront,

in particular the phase , and reconstructs this wavefront in

order to obtain a quantitative information about the topo-

graphic profile of the object. Since all light-sensitive sensors

respond to intensity only, the phase is encoded in the inten-sity fringe pattern adding another coherent background wave

, called the reference beam. Both

waves interfere at the surface of the recording device. The

intensity of this interference pattern is calculated by

(3)

where denotes the conjugate complex. The hologram is pro-

portional to this intensity, .

In the reconstruction process, the hologram can be seenlike an amplitude transmittance that diffracts the reference

wave. In other words, the wavefront scattered by the object

under investigation is obtained through the propagation of

the product from the holographic plane to the

image plane evaluate by means of the Fresnel approximation of

the RayleighSommerfelds diffraction integral [4].

Generally, in DH the recording device is a CCD camera array;

it is a two-dimensional rectangular raster of pixels, with

pixel pitches and in the two directions. Thus, hologram

patters recorded by a CCD are nothing less than digitized ver-

sion of the wavefields that impinge on the CCD surface. There-

fore, and are the sampling intervals in the observationplane, and they define the resolution of the reconstructed im-

ages. DH is spatially sampled and stored as a numerical array in

a computer and, in the reconstruction process, the recorded in-

tensity distribution of the hologram is numerically multiplied by

the reference wave field in the hologram plane. Then, in order

to evaluate the intensity and the phase distribution of the re-

constructed real image, the diffracted field in the image plane

is determined by a discretization of the integral of diffraction.

Thus, the numerical reconstruction of the complex wave field

allows not only intensity but also the phase of the reconstructed

wave front. The possibility to manage the phase in DH is very

attractive because aberrations can be removed. In fact, it has

been proved that a wave front curvature introduced by micro-scope objective and lenses can be successfully removed and/or

Fig. 1. Experimental setup for recording digital holograms; BS is beam

splitter, M is mirror, MO is microscope objective, and BE is beam expander.

compensated as well as other types of aberrations like spher-

ical type, astigmatism, and anamorphism, or even longitudinal

image shifting introduced by the cube beam splitter [36][42].

In particular, by means of the FTM the reconstructed image

is obtained by

FFT (4)

where FFT denotes the fast Fourier transform algorithm.Thus, the reconstructed image isan matrixwith elements

( , ) and steps

(5)

along the two transversal directions.

Fig. 1 shows the experimental recording holographic setup.

The laser source wavelength is nm. The reference and

object beams are plane wave fronts obtained by a beam expander

(BE). The first beam splitter is a cube polarizing beam splitter

(BS) and a wave plate is in the reference beam to obtain

equal polarization directions for the two beams. In order to avoidthe problem of twin image and eliminate the zeroth diffraction

order, an off-axis configuration has been employed [43]. In the

off-axis geometry, the mirror which reflects the reference wave

is oriented such that the reflected wave reaches the CCD camera

array plane with a small incidence angle with respect to the

object wave, which propagates perpendicular to the hologram

plane .

The microscope objective is an aspheric lens. Different as-

pheric lenses are available with focal lengths and numerical

apertures, respectively, of mm and ;

mm and ; mm, .

The lenses are microscope objective equivalents, respectively, to

35X, 20X, and 10X. The CCD detector is a 1280 1024 pixelarray with pixel size m.


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FERRARO et al.: CONTROLLING IMAGES PARAMETERS IN RECONSTRUCTION PROCESS OF DIGITAL HOLOGRAMS 831

Fig. 2. Intensity of the signal recorded on a group of 4 2 4 pixels and related to the phase-shift of the fringes.

III. FOCUS TRACKING

The possibility to have an experimental setup with different

microscope objectives allows us to obtain magnified images of

the sample object. However, since in DH an imaging short focal

system is required, if the sample experiences even very small

displacements along the optical axis, a large change occurs in

the distance to the imaging plane and consequently the focus can

be lost. In fact, if isthe distance from the lens to the object, is

the distance of the image plane from the lens, and is the focal

length, then any axial displacement of the sample results in

a shift of the phase detected at time given by

(6)

and in a translation of the imaging plane in front of the CCD

given by

(7)

where is magnification of the imaging system.

In this case, the distance used in the numerical reconstruction

process must be changed to get an in-focus amplitude and/or

phase-contrast image. Any small axial displacement, even in

the micrometric range, displaces the image plane and the focus

can be lost. The effect of defocus influences the phase-contrast

reconstruction, thereby affecting the quantitative information.

Thus, the distance used in the numerical reconstruction process

must be changed to get an in-focus amplitude or phase-contrast

image. However, the tedious and cumbersome search for newfocal planes becomes intolerable, especially if there is a need to

visualize the phenomena in anything approximating real time.

Displacement of the object may occur for different reasons. One

unavoidable situation is encountered in case of thermal char-

acterization of objects. Naturally, temperature changes cause

thermal expansion of the object and/or its mechanical support

that may not be predictable.

In order to overcome this problem, it is possible to detect the

axial displacement of the object by measuring the phase shift of

the hologram fringes [30]. In fact, with reference to the config-

uration reported in Fig. 1, any axial displacement of the object

caused a shift in the fringe pattern of the hologram. By recording

the phase shift in a small flat portion of the object, it ispossible todetermine the displacement. In this way, the incremental change

of the reconstruction distance can be obtained and in-focus am-

plitude and phase-contrast images, for each recorded hologram,

can be reconstructed.

The method effectiveness has been demonstrated in a

quasi-real-time inspection of silicon MEMS structures while

their temperature is changed. Two different types of siliconMEMS structures having out-of-plane deformation due to the

residual stress induced by the microfabrication process have

been investigated [29]: a cantilever 50 50 m and bridges

10 m wide. The silicon wafer was mounted on a metal plate

and was held by a vacuum chuck system. The metal plate was

mounted (see Fig. 1) on a translation stage in proximity of an

MO with focal length mm and . The

sample was heated from 23 to 120 C, by a remote-con-

trolled heating element. An axial displacement due to the

overall thermal expansion of the metallic plate and the trans-

lation stage was observed. So, a first hologram was recorded

before raising the temperature. The numerical reconstructionfor a well-focused image was found at an initial distance of

100 mm with a magnification . While heating the

sample, the phase shift of the fringes in quasi-real time was

detected by measuring the average intensity change in a group

of 4 4 pixels. Fig. 2 shows the recorded intensity signal.

The signal had 3149 points sampled at a rate of 12.5 points/s.

At each ten points, a hologram was recorded, obtaining a se-

quence of 314 holograms. The signal of Fig. 2 was analyzed by

applying an FFT algorithm; so, for each added point recorded,

the wrapped phase was detected and the unwrapped phase was

calculated. The displacement was calculated considering

both (6) and (7). Fig. 3 shows the calculated displacement as

function of the sampled point. In this way, the numerical recon-

struction distance for each hologram was continuously updated

and, as shown in Fig. 3, the final hologram reconstruction dis-

tance differs from the initial one by about 40 mm.

Fig. 4(a)(c) shows the amplitude and phase-contrast recon-

structions for the cantilever beam obtained reconstructing three

different holograms of the recorded sequence of 314 holograms;

i.e., the holograms no. 1, no. 196, and no. 314, relative to three

different temperatures.

The reconstructions were performed automatically by ap-

plying the focus tracking procedure. Fig. 4(a) shows the

reconstruction of the hologram no. 1 at mm; Fig. 4(b)

shows that of the hologram no. 196 at mm, whileFig. 4(c) shows that of the hologram no. 314 at mm.


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Fig. 3. Displacement of the sample measured in quasi-real time by analyzing the phase shift of hologram fringes.

Fig. 4. In-focus amplitude and phase map for the cantilever beamreconstructed from three holograms of the record sequence and obtainedapplying the focus-tracking procedure.

The presence of fringes in the phase-map image of the

wrapped phase observed on the cantilever indicates it had

an intrinsic out-of-plane deformation. As expected, the re-

constructions in Fig. 4 are all in focus, even though the size

of the reconstructed object decreases from Fig. 4(a) to (c)

because of the increasing of the reconstruction distance. InFig. 5 is reported a different MEMS structure with a bridge

Fig. 5. Wrapped and unwrapped phase-contrast reconstruction for a bridgestructure: (a) and (b) applying the focus-tracking procedure and (c) and (d)without focus tracking.

shape and with smaller dimensions. Fig. 5(a) and (b) shows

the phase, wrapped and unwrapped, respectively, applying the

focus-tracking procedure; whereas in Fig. 5(c) and (d) the same

quantities obtained without focus tracking are reported.Fig. 5 shows that the effect of defocus influences the phase-

contrast reconstruction, thereby affecting the quantitative in-

formation. In particular, it is clearly visible that the edges of

the bridge are blurred when the focus tracking method is not

applied.

Thus, applying the proposed method, the corrected recon-

struction distance can be evaluated for each acquired hologram

and well-focused amplitude and phase-contrast images can be

obtained.

IV. CONTROLLING IMAGE SIZE

In Section III, it can be noted that in Fig. 4(a)(c) the size ofthe reconstructed object decreases because of the increasing of


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the larger reconstruction distance. Therefore, although well-fo-

cused images were obtained, it was not possible to compare two

of them directly since they had different sizes due to the different

width of the reconstruction pixel (RP). From (5), we note that

RP increases with the reconstruction distance so that the size of

the image, in terms of number of pixels, is reduced for longer

distances. On the other hand, direct subtraction of unwrappedphase maps from two holograms at two different distances is

important [15] in order to obtain quantitative information, e.g.,

on the deformations caused by the thermal load.

Similar difficulties arise in multiwavelength DH (MWDH)

used for color display and for applications in metrology. In

MWDH, for each wavelength the width of the RP increases

with the reconstruction wavelength for a fixed reconstruction

distance. Consequently, holograms recorded with different

wavelengths produce images with different sizes, when numer-

ically reconstructed by means of the FTM. A color DH display

requires simultaneous reconstruction of images recorded with

different wavelengths (colors) and the resulting reconstructed

images must be perfectly superimposed to get a correct color

display [34], [35]. This is prevented by the differing image

sizes, and this also prevents phase comparison required for

holographic interferometry [32], [33].

To avoid the above problems, we have developed a simple

method for controlling the image size of the reconstructed im-

ages generated by the FTM so that two images of the same ob-

ject recorded at two different distancesand/or wavelength can be

directly compared. The method is intrinsically embedded in the

holographic reconstruction process without the need of image

scaling at the end of the process. The size is controlled through

fictitious enlargement of the number of the pixels of the recorded

digital holograms. From (5), it is clear that the reconstructed

image is enlarged or contracted according to the reconstruction

distance and/or the reconstruction wavelength and that the

size of the RP depends from the lateral number of the pixels

and . Our method for controlling image size is based on

the RP changing by using a larger number of pixels in the re-

construction process. In fact, and can be augmented by

padding the matrix of the hologram with zeros in both the hori-

zontal and vertical directions such that

(8)

giving

(9)

where and are the number of pixels of the hologram

recorded at the nearer distance . In a similar way, in MWDH,if one hologram has been recorded with wavelength and a

Fig. 6. In-focus phase map for the cantilever beam relative to three differentreconstruction distances.

Fig. 7. Wrapped image phases reconstructed at different distances with

application of padding operation.

second with , where , at the same distance, then the

number of pixels of that hologram may be changed such that

(10)

in order to obtain the same width for the RP

(11)

where, now, and are the number of pixels of the holo-

gram recorded at the smaller wavelength .

The controlling size procedure was applied on the cantilever

reconstructions reposted in Section II. In particular, in Fig. 6, the

three reconstructed phases of Fig. 4 are reported in a different

way.

In this way, it is very clear that the size of the cantilever, in

terms of pixels, is reduced due to the three holograms being

recorded and reconstructed at different distances. The same re-

constructions, after the padding procedure has been applied tothe holograms, are shown in Fig. 7. In particular, the holograms


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Fig. 8. Unwrapped phase of a cantilever at distance (a)d = 1 1 5 : 5

mm, (b)d = 1 3 9 : 5

mm, and (c)d = 1 3 9 : 5

mm with the padding operation. (d) Phasemap subtraction between (a) and (c).

reconstructed at distance mm had 512 512 pixels;

the holograms at mm and mm have been

padded, according the (8), with zeros up to 614 614 pixels and

718 718 pixels, respectively. From Fig. 7, it can be noted that

the reconstructed phase images maintain the same size indepen-

dent of distance.

Thus, on these reconstructed images, direct phase subtraction

can be performed and possible differences can be emphasized.

An example of direct phase subtraction is reported in Fig. 8. In

Fig. 8(a) is reported the unwrapped phase image of the hologram

recorded at distance mm; Fig. 8(b) shows the un-

wrapped phase image of the MEMS at mm without

zero padding; whereas, Fig. 8(c) shows the unwrapped phase

of the MEMS at obtained by the padding operation. Thus,

phase subtraction between the phase maps of Fig. 8(a) and (c)

can be performed. In Fig. 8(d), we show the difference between

the unwrapped phase maps with equal size indicating the small

deformation caused by thermal load.

An example of amplitude reconstruction size control is re-

ported in Fig. 9. Fig. 9(a) shows portion of the reconstruction

along the longitudinal axis ( axis) for a Ronchi grating [44]

without the padding operation; as expected, the period of the

grating decreases.

Fig. 9(b) shows the results of the same reconstruction when

the padding operation applied. It is clear that the size of the

grating is kept constant. The Talbot effect is noticeable along the

axis. The initial distance ofreconstruction was mm,

while the final mm with steps of 10 mm and a total

number of 40 reconstructed holograms. The initial number of

pixels for the first reconstruction was ;

whereas, the final reconstruction was performed with padding

of zeros up to pixels. In Fig. 9(a) and (b),

a common number of pixels was extracted from reconstructed

images in order to prove the usefulness of the controlling sizeprocedure.

Fig. 9. Ronchi grating reconstructed at different distances: (a) pitch of thegrating decreases for longer distances and (b) size is kept unchanged withpadding operation.

In order to demonstrate that size can be controlled in MWDH

applications, we recorded holograms with two different wave-

lengths at nm and nm, respectively.

The test object was a double Ronchi grating with periods of

5.0 and 3.5 lines/mm, respectively. All holograms were initially

recorded with pixels. Fig. 10(a) and (b)

shows the reconstructed amplitude of the grating at , respec-

tively, without and with padding operation applied to the holo-

gram. Fig. 10(c) shows the amplitude image reconstruction at

(green). The red hologram (that at ) was reconstructed[Fig. 10(b)], according to (10), adds a number of zeros around

the hologram such that to obtain an image

having equal size in respect to that of Fig. 10(c).

RGB combination of the red and green images gives perfect

superimposition and new color (yellow) in Fig. 10(d).

From these examples, it is clear that by means of a simple

padding of the recorded digital holograms with zeros, it is pos-

sible to control the size of the reconstructed images independent

of distance and wavelength.

V. CONTROLLING RESOLUTION

The possibility to manage the RP by varying the numberof pixels of the hologram appears important not only for


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Fig. 10. MWDH reconstruction of the red hologram (a) without padding and (b) with padding. (c) Reconstruction of the green hologram and (d) superimpositionof red and green images (b) and (c).

controlling the size of the reconstructed images, but for en-

hancing the resolution of this image too. In fact, in FTM, the

spatial frequencies displayable in reconstructed images are

band limited by the size of the reconstruction pixel, which

represents the sampling gauge in the image plane so that spatialfrequencies higher then the Nyquist limit are undersampled and

reconstructed incorrectly. Depending on the objects, undersam-

pling can affect the correctness of reconstructed phase map.

In order to recover the necessary resolution in reconstructed

images that has been lost intrinsically by the application of the

FTM, we have developed an approach analogous to that of the

size controlling. From (5), it is clear that reconstruction pixel

and, consequently, resolution depend on the wavelength, the

distance, number of the pixels , of the sensor array and

their physical size, and the reconstructed image has a reduced

spatial lateral resolution for higher reconstruction distances.

Thus, the resolution can be easily improved through artificial

enlargement of the number of the pixels in recorded digital

holograms.

To demonstrate the method effectiveness, the silicon-based

membrane reported in Fig. 11 has been investigated. In silicon

MEMS inspection, undersampling can occur if the shape of

the MEMS structure grows too rapidly for a fixed distance,

wavelength, number and size of pixels of the CCD, and as con-

sequence the resulting profile can be incorrectly reconstructed.

Fig. 12(a) shows the phase map reconstructed obtained at a dis-

tance mm and nm from the recorded hologram

consisting of pixels; thus, the corre-

sponding reconstruction pixel size is m.

The plot of the unwrapped phase along the diagonal direc-tion of the membrane is shown superimposed in Fig. 12(a).

Fig. 11. SEM picture of a MEMS structure deformed by residual stress.

Although this plot is consistent with the expected deforma-

tion qualitatively shown in Fig. 11, some small undersampling

occurs at the extremity of the corner. The unwrapped phase

profile can be modeled by a parabolic function given by

(12)

where is the cantilever initial slope and depends on internal

stresses of the structure, and is its radius of curvature.

The parabolic increasing of the phase change implies that

the small phase increment corresponding to the reconstruc-

tion pixel size is

(13)

Proper sampling of the phase distribution on the recon-

struction image plane is obtained when the minimum phaseincrement that can be resolved is less than , i.e.,


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Fig. 12. Wrapped phase maps of the MEMS: (a) from the original hologramwith 1024 2 1024pixels,(b) from a selected central portionof 512 2 512 pixels,and (c) from the previous hologram with 512

2

512 pixel but padded with zerosup to 1024 2 1024 pixels.

(Nyquist criterion). According to (13), this condition leadsstraightforwardly to the estimate of the maximum distance

, along which proper sampling of the phase can be ob-

tained, namely

(14)

Considering the (5) for the RP, the maximum distance

along the direction of the deflected MEMS can be written interms of the CCD pixel size by

(15)

showing that, for a fixed reconstruction distance and recording

pixel size , increasing the sampling number through the

padding operation increases the maximum range . Thus,

in Fig. 12(a) the profile is correctly retrieved up to a certain

value, according to (15). We can assume safely that the profile

up to this value is the actual true profile of the membrane along

that line. By fitting the unwrapped phase shown in Fig. 12(a),

with a parabolic law along the direction, the fitted values of

the radius of curvature cm and of the cantilever slope

rad can be estimated. From (15), a value of

m for the maximum range can readily obtained.

Thus, the maximum value corresponding to correctly sampled

phase is radians. It can

be verified from Fig. 12(a) that this value is in agreement with

the maximum range that can be estimated from the unwrapped

phase data shown.

Then, we extracted a subhologram of 512 512 pixels

from the center of the original hologram recorded with 1024

1024 pixels. From this subhologram the phase map reported in

Fig. 12(b) is reconstructed employing the same reconstructiondistance utilized for the reconstruction of Fig. 12(a). From the

phase map it can be seen that the Nyquist limit was strongly ex-

ceeded and undersampling occurred in the wrapped phase map

as evidenced by the appearance of the circular fringe on the tip

of the membrane corner. The plot along the diagonal line of the

unwrapped phase demonstrates that the undersampling gives

an incorrect profile of the MEMS meaning that the phase has

been incorrectly retrieved. By using the previously determined

values of and , (15) predicts in this case ( ,

i.e., m) that the maximum linear range along

which a phase unwrapping procedure could work properly is

m and the corresponding maximum value of thephase is rad. This value is in good agreement

with the data shown in Fig. 12(b). In order to recover, the lost

resolution of the same hologram used in Fig. 12(b) has been

padded with zeros up to 1024 1024 pixels. The reconstructed

phase map is shown in Fig. 12(c). It can be seen that the

resolution has been recovered giving the correct profile of the

structure. In fact, in this case, the phase profile of the membrane

has been retrieved correctly up to the same value of the original

1024 1024 pixel hologram.

Finally, to obtain the complete correct profile of the mem-

brane, the 1024 1024 pixels hologram has been padded with

zeros up to 2048 2048 pixels, and the resulting phase map is

shown in Fig. 13(a). In Fig. 13(b), we show the same 512 512hologram but padded to 2048 2048 pixels. From the plot of the


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Fig. 13. Close-up of the wrapped phase map of: (a) hologram of Fig. 11(a)but padded to 2048

2

2048 pixels and (b) hologram of Fig. 11(c) but padded to2048

2

2048 pixels.

unwrapped phase along the central line of the MEMS in Fig. 13,

it is clear that the padding operation allows the recovery of the

correct phase map in both cases. In fact, in this case, according to

(15), no undersampling should occur up to the maximum phase

value of about 410 rad that is higher that the maximum height

of the membrane.

It is important to note that real content of information in

Fig. 13(b) is exactly the same as that producing the phase map of

Fig. 12(b) with only a padding operation making the difference.

Nevertheless, the profile of the MEMS is almost completely

correctly recovered in case of Fig. 13(a). That means the re-

quired information about phase map can be extracted even from

a reduced hologram. This can have an impact on compression

of digital holograms without loss of information in analogy to

which is proposed in [45].

Thus, with the proposed method, it is possible to recover the

necessary resolution for correct reconstruction of phase and/or

amplitude images by FTM in DH. Finally, it is important to

point out that even if the resolution can be improved by zeropadding on the digital hologram the corresponding field of view

remains unaffected as it depends on the real aperture of the op-

tical system, while the padding operation allows only a fictitious

enlargement of the aperture of the digital hologram.

VI. CONCLUSION

In this paper, we have shown how different parameters can be

simultaneously controlled in reconstructing digital holograms.

The flexibility offered by the proposed approaches in the recon-struction process of DH allows exploitation of new possibili-

ties of application in microscopy. Through the reconstruction

process we show that it is possible to control image parame-

ters as focus distance, image size, and image resolution. These

newly explored potentialities open further a novel prospective

of application of DH in single- and multiwavelengths operation

either for display or metrological applications. Specifically, wedemonstrated a method for tracking focus during the recording

of a sequence of holograms. In this way, a corrected recon-

struction distance, for each acquired hologram and well-focused

amplitude and phase-contrast image, can be obtained control-ling the correct focus. The method can even be applied as a

quasi-real-time procedure and it could constitute a significantstep forward in the implementation of a DH microscope for

quas-ireal-time observation in many fields of application.Furthermore, we have shown that by means of a simple

padding of the recorded digital holograms with zeros, it is pos-

sible to control the size of the reconstructed images independent

of distance and wavelength. By this controlling operation the

problem of the superimposition in MWDH applications has

been solved without the need for subsequent resizing of im-

ages. In metrological applications it is possible by the proposed

method to subtract directly reconstructed phase maps of the

same object to detect small deformations in different hologramsrecorded and reconstructed at different distances.

Finally, we have demonstrated that by controlling the recon-

struction pixel, in a real situation it is possible to recover the

necessary resolution for correct reconstruction of phase and/or

amplitude images by FTM in DH. The method allows correct

profile reconstruction of highly curved MEMS in which thewrapped phase map is beyond the Nyquist limit.

ACKNOWLEDGMENT

The authors would like to thank Dr. C. Magro of STMicro-

electronics, Catania, Italy, for useful discussions about silicon

MEMS structures.

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Pietro Ferraro (M97) received the B.S. degree inphysics from Naples University Federico II, Italy.

From 1988 to 1993, he was a Researcher withAlenia Aeronautics in the field of optical methodsfor nondestructive testing (NDT) of aerospacecomposite structures including holographic inter-

ferometry and fiber-optic sensors based on fiberBragg gratings (FBGs). He was a Physics and Optics

Teacher in 1993while being an Associate Researcherwith the Institute of Cybernetics E. Caianielloof the National Council of Research (CNR), Italy.

He is currently a Researcher with the National Institute of Optics, Naples,Italy. His current research interests include developing optical methods (digitalholography, interferometry) for characterization of materials, components, andFBG sensors for strain measurements.

Giuseppe Coppola received the M.Sc degree inelectronic engineering and the Ph. D. degree inelectronics and computer science in 1997 and 2001,respectively, both from the University of Napoli

Federico II, Italy.In 2001, he spent six months as Visiting Scien-

tist at Delft Institute of Microelectronics and Sub-micron Technologies, Technical University of Delft,The Netherlands, working on the design and realiza-tion of an optoelectronic modulator. Since 2002, he

has been a Researcher at the Institute for Microelec-tronics and Microsystems (IMM) of the National Research Council (CNR), in

Naples. His research interests include the design and characterization of sil-icon-based optoelectronics devices and characterization of MEMS structuresby digital holography.


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Domenico Alfieri received the M.Sc. degree intelecommunication engineering from the Universityof Naples, Federico II, Naples, Italy, in 2002.

He is currently a Researcher with the National In-stitute of Optics, Naples. His research interests in-clude developing optical methods for monitoring anddiagnostic noninvasive fMEMS and for characteriza-tion of materials by means of digital holography.

SergioDe Nicola received the B.S. degree in physics

from Naples University Federico II, Naples, Italy,in 1982.

From 1983 to 1987, he was a Systems Analystwith Elettronica SpA, Rome, Italy, and at AleniaSpA, Naples. Since 1988, he has been on the staffof the Institute of Cybernetics E. Caianiello ofthe National Council of Research, Italy, where heis currently a Senior Researcher. He has contributedto about 130 papers in international journals andconferences. His research interests include the de-

velopment and applications of interferometric and holographic techniques for

nondestructive materials evaluation, wavefront analysis, numerical modelingof laser beam propogation in heterogeneous media, nonlinear optics, and

quantum-like models for beam propogation analysis.

Andrea Finizio has been with the Optical Depart-ment, the Institute of Cybernetics E. Caianielloof the National Council of Research (CNR), Italy,managing the experimental activity of the opticallaboratories since 1968. He has senior knowledgeof laboratory techniques and the design systems forclassical and digital holography and interferometryapplied to nondestructive testing (NDT) and optical

testing. He is a coauthor of about 65 papers ininternational journals and conferences.

Giovanni Pierattini received the degree in physics from Naples UniversityFederico II, Naples, Italy, in 1968.

Since 1969, he has been a Researcher with the Institute of Cybernetics E.Caianiello of the National Council of Research, Naples, where he leads theOptical Department. Since 1970, he has worked for the noninvasive diagnosticsof biological samples of artworks through holography. His activities have ba-sically concerned the field of modern optics and he has been involved in manyresearch projects. His current interests are in nonlinear optics, interferometry,and holography. He is the author or coauthor of about 120 papers in interna-

tional journals. He has filed two patent applications.Mr. Pierattini has been a member of the Scientific Council of the National

Group of Quantum Electronics and Plasma (GNEQP), since 1997.

controlling images parameters in the reconstruction

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