# Purely numerical compensation for microscope objective phase curvature in digital holographic microscopy: influence of digital phase mask position.

**ABSTRACT** Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction. This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is introduced in the reconstruction process to perform this curvature correction. Different corrections are discussed in terms of their influence on the reconstructed image size and location in space. The results are presented according to two different expressions of the Fresnel transform, the single Fourier transform and convolution approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.

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**ABSTRACT:**The tendency of particles to aggregate depends on particle-particle and particle-fluid interactions. These interactions can be characterized but it requires accurate 3D measurements of particle distributions. We introduce the application of an off-axis digital holographic microscopy for measuring distributions of dense micrometer (2 μm) particles in a liquid solution. We demonstrate that digital holographic microscopy is capable of recording the instantaneous 3D position of particles in a flow volume. A new reconstruction method that aids identification of particle images was used in this work. About 62% of the expected number of particles within the interrogated flow volume was detected. Based on the 3D position of individual particles, the tendency of particle to aggregate is investigated. Results show that relatively few particles (around 5–10 of a cohort of 1500) were aggregates. This number did not change significantly with time.Optics and Lasers in Engineering 05/2015; 68. · 1.70 Impact Factor -
##### Conference Paper: Accurate quantitative phase imaging through telecentric digital holographic microscopy

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**ABSTRACT:**The use of non-telecentric imaging systems in quantitative phase digital holographic microscopy introduces strong inaccuracies. We show that even negligible errors in the radius and center of curvature of the remaining quadratic phase factor introduce big errors in the numerical phase measurements. The errors depend on the position of the object in the field-of-view. However, when a telecentric imaging system is utilized for the recording of the holograms, the hybrid imaging method shows shift-invariant behavior, and therefore accurate quantitative phase imaging can be performed.SPIE Sensing Technology + Applications; 06/2014 - SourceAvailable from: Valentin Raditoiu[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we present our results obtained using an interferometric technique to investigate a constant thickness sample, which presents inside regions with different values for the refractive index. The sample is prepared on a polycarbonate (PC) substrate. On its surface, some microstructures are imprinted using controlled values for temperature and forces, after we determined experimentally the value of the temperature for glass transition. These microstructures were then filled with an epoxy resin. Using the images acquired experimentally, we computed the phase shift map introduced by the sample in the optical path. It is proportional with the geometrical height of the microstructures and with the difference between the refractive indices of the polycarbonate and the epoxy resin. In this way, we find the phase shift profile inside of a transparent thin film of a sample with plane parallel surfaces.UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 01/2014; 76(1):177-186. · 0.28 Impact Factor

Page 1

Purely numerical compensation for microscope

objective phase curvature in digital

holographic microscopy: influence of digital phase

mask position

Frédéric Montfort, Florian Charrière, and Tristan Colomb

Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut d’Optique Appliquée, CH-1015 Lausanne, Switzerland

Etienne Cuche

Lyncée Tec SA, PSE-A, CH-1005 Lausanne, Switzerland

Pierre Marquet

Centre de Neurosciences Psychiatriques, Département de Psychiatrie DP-CHUV, Site de Cery,

1008 Prilly-Lausanne, Switzerland

Christian Depeursinge

Ecole Polytechnique Fédérale de Lausanne (EPFL), Institut d’Optique Appliquée, CH-1015 Lausanne, Switzerland

Received March 17, 2006; revised June 1, 2006; accepted June 3, 2006; posted June 8, 2006 (Doc. ID 69123)

Introducing a microscope objective in an interferometric setup induces a phase curvature on the resulting

wavefront. In digital holography, the compensation of this curvature is often done by introducing an identical

curvature in the reference arm and the hologram is then processed using a plane wave in the reconstruction.

This physical compensation can be avoided, and several numerical methods exist to retrieve phase contrast

images in which the microscope curvature is compensated. Usually, a digital array of complex numbers is in-

troduced in the reconstruction process to perform this curvature correction. Different corrections are discussed

in terms of their influence on the reconstructed image size and location in space. The results are presented

according to two different expressions of the Fresnel transform, the single Fourier transform and convolution

approaches, used to propagate the reconstructed wavefront from the hologram plane to the final image plane.

© 2006 Optical Society of America

OCIS codes: 090.1000, 090.1760, 100.3010, 110.0180.

1. INTRODUCTION

Because of the limited sampling capacity of the electronic

camera compared with the one of photosensitive materi-

als such as photographic plates, the spatial resolution of

reconstructed images in digital holography was formerly

limited compared with classical holography. Different ap-

proaches exist to achieve microscopic imaging with digital

holography. One can, for example, use spherical diverging

waves for the hologram recording, which allows a numeri-

cal enlargement of the object during the reconstruction

process, without any image-forming lens, as described in

Chap. 5 of Ref. 1. By the introduction of a microscope ob-

jective (MO), Cuche et al.2have demonstrated that digital

holographic microscopy (DHM) allows one to reconstruct,

with a lateral resolution below micrometers, the optical

topography of specimens with a nanometric accuracy.

Nevertheless, the introduction of a MO increases the com-

plexity of the reconstruction process. Indeed, the MO in-

troduces a phase curvature to the object wave that should

be compensated perfectly to perform accurate measure-

ment and imaging of the phase delay induced by the

specimen. There are two different main possibilities to

compensate for this phase curvature, either physically by

introducing the same curvature in the reference wave or

digitally as presented in Refs. 2–6.

The physical compensation, in standard interference

microscopy like the Linnik configuration (see, for ex-

ample, Chap. 20 in Ref. 7), is done experimentally by in-

serting the same MO in the reference arm, at equal dis-

tance from the exit of the interferometer. The curvature of

the object wave is then compensated by the reference

wavefront during interference. Nevertheless, this method

requires a precise alignment of all the optical elements.

Moreover, each modification in the object arm needs to be

precisely reproduced in the reference arm.

In the present paper, we call digital phase mask (DPM)

a complex numbers array, by which the reconstructed

wavefront is multiplied during the hologram processing.

Digitally, the definition and the position of the DPMs used

to compensate the phase curvature can be different. Fer-

raro et al. make the compensation in the image plane by

subtracting the reconstructed phase of a hologram ac-

2944J. Opt. Soc. Am. A/Vol. 23, No. 11/November 2006 Montfort et al.

1084-7529/06/112944-10/$15.00© 2006 Optical Society of America

Page 2

quired without a specimen.3Cuche et al. define a numeri-

cal quadratic curvature model2that could be automati-

cally computed in the image plane.4Finally, two recent

papers show that the compensation for the MO curvature

(and for phase aberrations) can be done in the hologram

plane by using a reference hologram5or polynomial DPM

models (standard or Zernike) computed automatically.6In

spite of the numerous phase retrieval techniques pro-

posed in the literature, no systematic study of the behav-

ior of the phase images obtained through these recon-

struction methods has, to our knowledge, been performed

yet. In this paper, we present analytically the influence of

the DPMs’ position (hologram or image plane) in the re-

construction process, in particular in terms of position

and size of the reconstructed specimen region of interest

(ROI).

2. BASES OF DIGITAL HOLOGRAPHY

A. Principle and Reconstruction

Digital holography allows one to retrieve the original

complex wavefront from an amplitude image, called a ho-

logram, recorded on an electronic camera such as a CCD

or complementary metal-oxide semiconductor camera.

This hologram is created by the interference, in off-axis

geometry, between two coherent waves: on one side the

wave of interest, called object wave O, coming from the

object, and on the other a reference wave R. In the holo-

gram plane, the two-dimensional recorded intensity dis-

tribution IH?x,y? can be written8as

IH?x,y? = ?O + R?2= ?O?2+ ?R?2+ RO*+ R*O,

?1?

where R*O and RO*are the interference terms with R*

and O*denoting the complex conjugate of the two waves.

After hologram apodization9and spatial filtering,10the

virtual interference term R*O (the same procedure can be

done with the real interference term RO*) is multiplied in

the hologram plane by a DPM ?H,5which should be ide-

ally equal to R, to reproduce the original wavefront ?H

=RR*O=?HR*O in the hologram plane. Once the wave-

front ?Hhas been retrieved, it has to be propagated to the

image plane to have a focused image. This propagation of

a monochromatic reconstructed wavefront ?Hat wave-

length ?=2?/k from the hologram plane to the image

plane overadistance

d

approximation,1–6,9–11which allows one to implement nu-

merically the propagation by simple fast Fourier trans-

forms (FFTs), as will be pointed out further:

is done inthe Fresnel

?I?x,y? =

exp?ikd?

i?d???H??,??

?exp?i

?

?d??? − x?2+ ?? − y?2??d?d?,

?2?

where ?Iis the corresponding wavefront propagated to

the image plane. Let us define the two-dimensional

Fresnel transform (FT) of parameter ?=??d of a given

function f?x,y? as

F??f?x,y?? =

1

?2??f??,??

?exp?i

?

?d??? − x?2+ ?? − y?2??d?d?.

?3?

Using this definition, Eq. (2) can be written as

?I?x,y? = − i exp?ikd?F??d??H?x,y??.

This analytical expression of propagation can be digi-

tized by using two different formulations: the single FT

and the convolution formulations.

?4?

1. Single Fourier-Transform Formulation

The propagation in the Fresnel approximation [Eq. (4)]

can be written using a single FT:

exp?i

?FT??H??,??exp?i

In its discrete formulation, the low time-consuming FFT

algorithm can be employed.2In the following text, this

formulation will be referred to as FT formulation.

In this case the sampling step of the propagated image

is not the same as the initial one. If the initial image is

given by Npts?Nptspoints with a sampling step T??T?,

the image propagated over a distance d is sampled with

the same number of points but with a sampling step given

by

?I?x,y? =

exp?ikd?

i?d

?

?d?x2+ y2??

?d??2+ ?2???.

?

?5?

Tx=

?d

NptsT?

,

Ty=

?d

NptsT?

.

?6?

2. Convolution Formulation

The Fresnel propagation given by Eq. (4) can be written

using a convolution formulation:

??H?x,y?? ? exp?i

?I?x,y? =

exp?ikd?

i?d

?

?d?x2+ y2??. ?7?

Its discrete form is a little more time-consuming than the

FT formulation when computed.1,12The convolution ex-

pression of the propagation in the Fresnel approximation

has the same sampling step before and after the propaga-

tion. Thus if the image in the hologram plane is sampled

in Npts?Nptspoints with a sampling step T??T?, the

propagated image is sampled with the same number of

points and a sampling step ?Tx?Ty?=?T??T??.

B. Microscope Objective Introduction

The introduction of a MO in the object arm opens the pos-

sibility of imaging at the submicrometer scale. As shown

in Fig. 1, the optical arrangement in the object arm is that

of an ordinary single-lens system producing a magnified

image of the specimen in an image plane. In comparison

with classical microscopy, the difference is that the CCD

camera is not in the image plane but is in the hologram

plane that is located between the MO and the image

plane, at a distance d from the image. This situation can

Montfort et al.

Vol. 23, No. 11/November 2006/J. Opt. Soc. Am. A2945

Page 3

be considered to be equivalent to a holographic configura-

tion without a MO with an object wave emerging directly

from the image and not from the object itself.

The MO produces a curvature of the wavefront in the

object arm. This deformation affects only the phase of the

object wave and does not disturb amplitude contrast im-

aging. However, to perform an accurate measurement of

the phase delay induced by the specimen only, the phase

curvature induced by the MO must be perfectly compen-

sated. This compensation can be done in the hologram

plane by a DPM ?Hand/or in the image plane by a DPM

?I. Therefore we can write the corrected wavefront gener-

ally as

?I?x,y? = − ?Ii exp?ikd?F??d??H?H?x,y??.

?8?

3. HOLOGRAM RECONSTRUCTION: THE

IDEAL CASE

Let us first express analytically the reconstruction pro-

cess that exactly reproduces the image resulting from the

object through the MO, as it would be performed on an op-

tical bench, without any scaling or lateral or axial shift-

ing. This ideal case formulation will serve as a gauge im-

age for comparison with the images obtained by the

different digital reconstruction methods.

The hologram is multiplied by an ideal DPM ?id

sponding to a replica of the reference wave:

Hcorre-

?id

H= ?id

HR*O = RR*O = O,

?9?

where the reference wave amplitude has been assumed to

be equal to one. The propagation over a distance didex-

pressed using Eq. (4) is given by

?id

I= − i exp?ikdid?F??did??id

H?,

?10?

=− i exp?ikdid?F??did?O?,

?11?

where ?id

front.

In a general approach, we can consider an off-axis mi-

croscopy setup (angle ? between the propagation direction

of the reference and object waves), in which the curva-

tures of the reference and object waves at the hologram

plane are different (Fig. 2). Let us define the centers of

the spherical reference and object waves as

I

corresponds to the exact initial object wave-

SR= ?SRx,SRy,?hr

2− SRx

2− SRy

2?1/2?,

?12?

SO= ?0,0,ho?,

?13?

where hrand hoare, respectively, the distances between

the source points of the reference and object waves and

the recombining location of the two beams. Note that the

source point of the spherical object wave is located at the

back focal plane of the MO. The reference wavefront in

the hologram plane is thus given by8

R?x,y? = exp?i

?

?hr

??x − SRx?2+ ?y − SRy?2??.

?14?

Let us now define a blank object wave O0(without a

specimen in the transmission configuration and with a

flat surface in the reflection configuration).4Because we

assumed that only phase curvature is induced by the MO,

the wavefront of the blank object wave at the hologram

plane is

O0?x,y? = exp?i

?

?ho

?x2+ y2??.

?15?

To recover the phase delay induced by the object only,

the phase curvature induced by both the MO and the ref-

erence beam curvatures can be compensated by multiply-

ing ?id

by a second DPM ?id

plane. The latter is determined by the complex conjugate

of the blank wave O0propagated to the image plane:

II

introduced in the image

?id

I= ?F??did??id,0

H??*,

?16?

=?F??did?O0??*.

?17?

The corrected wavefront ?id

Ibecomes

?id

I= ?id

I?id

I,

?18?

=?F??did?O0??*F??did?O?.

?19?

Inserting Eqs. (14) and (15), Eqs. (17) and (18) can be

written as

Fig. 1.

CCD defining the hologram plane is placed in front of the image

obtained though the microscope objective (MO). d is the recon-

struction distance.

Standard configuration in holographic microscopy: The

Fig. 2. Schema of the used notations.

2946J. Opt. Soc. Am. A/Vol. 23, No. 11/November 2006 Montfort et al.

Page 4

?id

I?x,y? = exp?− i

?

??ho+ did??x2+ y2??,

?20?

?id

I?x,y? = − i exp?ikdid?

?exp?− i

?

??ho+ did??x2+ y2??F??did??id

H?. ?21?

?id

tered in SOat a distance ho+didfrom the image plane.

This general development expresses the retrieval of the

object wavefront, in which the MO curvature is corrected.

Potentially this approach may also be used for correction

of optical aberrations of the holographic setup,4but the

development will be restricted to the case without aberra-

tions, focusing on the effect of the phase curvature com-

pensation.

Ilogically corresponds to the spherical wavefront cen-

4. PHASE CURVATURE COMPENSATION

DURING NUMERICAL RECONSTRUCTION

To illustrate the different reconstruction approaches and

propagation formulations, a hologram of a quartz micro-

lens recorded in a transmission DHM setup with a 20?

MO (numerical aperature of 0.5) is used. This microlens

has a diameter of 240 ?m and a height of 21.15 ?m. Fig-

ures 3(a)–3(c) present the raw phase reconstruction, the

two-dimensional unwrapped phase image, and its three-

dimensional representation, respectively. The different

notations for the approaches are done with a subscript

letter: no letter, general; i, image plane; h, hologram

plane; and m, mixed.

A. General Approach

Let us develop the general approach in which a DPM is

introduced both in the hologram plane and in the image

plane. The application of this general approach is devel-

oped in detail and illustrated with examples in Ref. 6. We

first apply a DPM ?Hto the interference term R*O in the

hologram plane,

?H= ?HR*O.

?22?

Propagating the resulting wave over a distance d to the

image plane yields ?I;

?I= − i exp?ikd?F??d??H?,

?23?

=− i exp?ikd?F??d??HR*O?.

?24?

Then a second DPM ?Iis applied in the image plane to

compensate for the curvature of the propagated wave-

front. The corrected wavefront ?Iis thus given by

?I= ?I?I,

?25?

=− i exp?ikd??IF??d??HR*O?.

To determine the effects of ?Hon the propagation, we will

compare ?Iwith ?id

(19) and (21)].

The DPM applied in the image plane for the phase cur-

vature compensation will of course depend on the DPM

introduced in the hologram plane. We defined the digital

reference wave using the same notation as in the ideal

case in Eqs. (14) and (12). In this way we can define a gen-

eral DPM in the hologram plane. As it is supposed to be a

curvature correction term, it is defined as the conjugate of

a spherical wave centered in SD:

?H= exp?− i

?26?

Iobtained in the ideal case [see Eqs.

?

?hd

??x − SDx?2+ ?y − SDy?2??,

?27?

SD= ?SDx,SDy,?hd

2− SDx

2− SDy

2?1/2?.

?28?

The DPM ?Iin the image plane that compensates the re-

sulting propagated wavefront is then given by

?I?x,y? = exp?− i

??hd− hr?

? exp?i

M2?y − h?

hr

? exp?− i

M2?y − d?

hr

???SDx− SRx?2+ ?SDy− SRy?2?

?

?

??h + d/M??

SRy

1

M2?x − h?

hd??

M2?x − d?

hd??

SRx

hr

−

SDx

hd??

2

+

1

−

SDy

2??

?

??h0+ d/M??

SRy

1

SRx

hr

−

SDx

hd??

2

+

1

−

SDy

2??,

?29?

where M and h are defined as

h =

hdhr

hd− hr

,

M =

h − d

h

.

?30?

Finally, the phase curvature-corrected wavefront in the

image plane ?Iis given by

Fig. 3.

transmission DHM setup (diameter 240 ?m, height 21.15 ?m),

(b) two-dimensional unwrap of (a), (c) perspective representation

of (b).

(a) Phase reconstruction of the microlens recorded in a

Montfort et al.

Vol. 23, No. 11/November 2006/J. Opt. Soc. Am. A2947

Page 5

?I?x,y? = ?I?x,y??I?x,y?

= − i exp?ikd?

?exp?− i

M2?y − h?

= exp?ik?d −

1

MF??d/M?x?,y??

??h + d/M??

SRy

hr

M??

?

1

M2?x − h?

hd??

SRx

hr

−

SDx

hd??

2

+

1

−

SDy

2??

d

1

M?id

H?x?,y??,

?31?

where

=?y−d?SR4?hr−SD4?hd??M. The propagation direction

is no longer parallel to the optical axis, but is given by the

angle ?:

x?=?x−d?SRx?hr−SDx?hd??M

and

y?

sin ? =

SRx

2+ SRy

hr

2

−

SDx

2+ SDy

hd

2

.

?32?

By analyzing the image plane DPM given by Eq. (29),

we can find that the first term is a phase constant of no

particular interest and can be suppressed. The second

term is compensating for the phase deformation induced

by the reference wave and the DPM in the hologram

plane. Finally the third term is the correction term of the

object wavefront curvature. The final image ?Iis a replica

of the ideal case image scaled by a factor M and laterally

shifted.

We note that one can retrieve the results of the ideal

approach by setting ?H=R:

?H= R ⇒ SD= SR,

hd= hr⇒ lim

hd→?

h = hr,

M = 1,

which gives the well-known results of Eqs. (10), (20), and

(21):

lim

?H→R

?I?x,y? = ?id

I?x,y?,

?33?

lim

?H→R

?I?x,y? = exp?− i

??x2+ y2?

??ho+ d??,

?34?

lim

?H→R

?I?x,y? = − i exp?ikd?

?exp?− i

??x2+ y2?

??ho+ d???id

I?x,y?.

?35?

B. Image Plane Approach

In the case of the image plane approach, no DPM is ap-

plied in the hologram plane and the propagating term is

R*O, the illumination wave being considered of unit in-

tensity. This can be seen as if the hologram would be re-

constructed with a plane wave propagating along the op-

tical axis (Fig. 4).

The phase curvature compensation process is therefore

applied to the propagated interference term R*O. The

DPM can be computed from known flat areas on the speci-

men with the procedure described in Ref. 4 or from the

propagation of a blank hologram as described in Ref. 3.

This DPM is given by

?i

I= ?F??di?R*O0??*.

?36?

The flattened wavefront ?i

Ican be written as

?i

I= − i exp?ikdi??F??di?R*O0??*F??di?R*O?.

?37?

The condition ?H=1 imposes the following:

SD= 0,lim

?H→1

hd= ?,

SD

hd

= 0,

Fig. 4.

phase curvature is compensated in the image plane. The reconstructed image is not the image of the object through the MO (shown by

a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations,

respectively.

(a) Reconstruction in the image plane approach: The illumination beam is a plane wave propagating along the optical axis. The

2948J. Opt. Soc. Am. A/Vol. 23, No. 11/November 2006 Montfort et al.

Page 6

⇒ lim

hd→?

h = hr,lim

hd→?

M =

hr− d

hr

= Mi.

Introducing these results in Eqs. (29) and (31), we can ex-

press the DPM ?i

rection leading to the expression of the corrected image

wavefront ?i

?I?x,y? = exp?i

1

Ithat expresses the phase curvature cor-

I:

?i

I?x,y? = lim

?H→1

?

??hr+ di/Mi??

??h0+ di/Mi??

2?y −

hr

1

Mi

2?x − SRx?2

+

Mi

2?y − SRy?2??exp?− i

??x −

hr

?I?x,y? = − i exp?ik?di−

I?

Mi

?

1

Mi

2

di

SRx?

2

+

1

Mi

di

SRy?

M??

2??,

?38?

?i

I?x,y? = lim

?H→1

di

?

1

Mi

?id

x −

di

hr

SRx

,

y −

di

hr

SRx

Mi?.

?39?

Equation (39) shows that the correction in the image

plane approach also introduces a resizing of the image in

comparison with the ideal case. The scale factor is a func-

tion of the reference beam curvature hr. This scaling is

due to the fact that, compared with the ideal solution, the

correction of the reference curvature is not performed in

the hologram plane as it is when the hologram is pro-

cessed with exactly the same reference wave used during

acquisition.

In the image plane approach, the image is also laterally

shifted in space, as mentioned in the general approach

and shown in Fig. 4. The shift is due to the fact that the

propagation direction is modified by an angle ? from the

optical axis of the object beam. ? is given by

sin ? =

SRx

2+ SRy

hr

2

.

?40?

This inclination of the propagation direction arises from

the fact that the illumination wave propagates along the

optical axis, which is precisely inclined of an angle ? from

the correct reference wave. This induced error corre-

sponds to a tilt of the wavefront that is not corrected in

the hologram plane and induced this propagation devia-

tion. The lateral shift is thus given by

Lshift=

di

hr

?SRx

2+ SRy

2?1/2= disin ?.

?41?

This shift is not convenient for the numerical propaga-

tion. Indeed, the image is no longer centered in the recon-

struction window. In a convolution formulation of the

propagation (see subsection 5.B), this results in a tailed

image [Fig. 4(c)]. In the case of the FT formulation, it may

not be a problem if the sampling step is small enough so

that the field of view of the window is large enough to

cover the off-axis propagating wavefront [Fig. 4(d)]. The

mixed approach will give a solution in which the recon-

structed image has the same size and sampling step as

the reconstructed image in the image plane approach, but

without lateral shift (Fig. 5).

C. Hologram Plane Approach

In this second digital approach, a single DPM is applied

in the hologram plane.5Thus the considered wavefront is

directly R*O. Let us suppose a recording of a reference

hologram, where no object is present in the object beam.

The recorded term is then given by R*O0. Its conjugate

defines perfectly the DPM to be applied in the hologram

plane:

?h

H= RO0

*.

?42?

By multiplying the interference term by the DPM and ex-

pressing the result as a function of the ideal retrieved

wavefront, we obtain [Fig. 6(b)]:

?h

H= ?h

HR*O = O0

*?id

H= ?h

H.

?43?

Thus the propagation of this resulting wavefront over a

distance dhto the image plane can be expressed as

?h

I= − i exp?ikdh?F??dh??h

H? = − i exp?ikdh?F??dh?O0

*O?.

?44?

The interference term has been at the same time multi-

plied by the illumination wave R and by the correction

term that compensates for the object wavefront curva-

ture. The result is a plane wave modulated by the object-

related phase variations. Its propagation will therefore be

a plane wave and no curvature compensation will be

needed at any reconstruction distance, in particular in

the focused image plane (Fig. 6).

Nevertheless, the multiplication, in the hologram plane

already, of the interference term by the curvature com-

pensation term has an influence on the image. Indeed, the

propagated wavefront is O0

case, which influences the focus distance, image size, etc.

In the case of a microscope without aberrations, the

term O0

sponds to the transfer function of a lens. The corrected

wavefront in the image plane ?h

*O instead of O in the ideal

*compensating the curvature of the MO corre-

Iis given by

?h

I?x,y? = − i exp?ikdh?

?F??dh?exp?− i

?

?ho

?x2+ y2???id

H?x,y??

?45?

=exp?ik?dh−

dh

Mh??

1

Mh

?id

I?

x

Mh

,

y

Mh?.

?46?

The algorithm compensating for the phase curvature is

thus equivalent to the insertion of a numerical lens in the

hologram plane. We note that the focal length is deter-

mined only by the object wave shape and is totally inde-

pendent of the reference wavefront, which has been com-

pensated by the DPM.

Montfort et al.

Vol. 23, No. 11/November 2006/J. Opt. Soc. Am. A2949

Page 7

The resulting image ?h

and magnified by a factor Mhgiven by the thin-lens rela-

tion:

His thus focused at a distance dh

1

ho

= −

1

did

+

1

dh

,

Mh=

dh

did

,

?47?

where hois the focal length of the introduced numerical

lens, and didis the focus distance of the reconstructed im-

age in the ideal case (equal to the distance between the

image of the object through the MO and the hologram

plane). We note that hocorresponds to the distance be-

tween the back focal plane of the MO and the hologram

plane.

D. Mixed Approach

The mixed approach is a method combining both the ho-

logram and the image plane approaches. It consists in de-

fining the DPM in the hologram plane keeping account of

only some selected polynomial orders for a partial holo-

gram plane correction. After propagation, an image plane

DPM is defined and the remaining polynomial orders are

corrected. Several combinations are possible depending

on which orders are corrected in the hologram plane. Nev-

Fig. 5.

first-order DPM is applied in the hologram plane and a second of higher orders in the image plane. The reconstructed image is not the

image of the object through the MO (shown by a dashed line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the

image plane in convolution and FT formulations, respectively. The white lines define the diameter of the microlens diameter to be com-

pared with Fig. 6

(a) Reconstruction in the mixed approach: The illumination beam is a plane wave propagating along the reference wave axis. The

Fig. 6.

vature is compensated in the hologram plane. The reconstructed image is not the image of the object through the MO (shown by a dashed

line). (b) Phase image in the hologram plane. (c) and (d) Phase images in the image plane in convolution and FT formulations, respec-

tively. The white lines define the diameter of the microlens reconstructed with the mixed approach (Fig. 5).

(a) Reconstruction in the hologram plane approach: The illumination beam is a replica of the reference beam. The phase cur-

2950J. Opt. Soc. Am. A/Vol. 23, No. 11/November 2006Montfort et al.

Page 8

ertheless, only the case of the first-order phase correction,

i.e., planar phase correction, in the hologram plane will be

discussed. This corresponds to illuminating the hologram

with a plane wave having the same propagation direction

as the reference wave. It is thus similar to the image

plane approach, except that the illumination wave has

the same propagation direction as the reference wave in-

stead of the object wave (Fig. 5).

In the hologram plane, the interference term R*O is

multiplied by a first-order DPM corresponding to a plane

wave ?PWm

H= exp?− i

H? [Fig. 5(b)]:

?m

H= PWm

2?

??

SRx

hr

x +

SRy

hr

y??.

?48?

The determination of ?m

Iresults in

?m

I?x,y? =?PWm

H?x,y?

?F??dm?R*O?x +

dm

hr

SRx,y +

dm

hr

SRy???

*

,

?49?

and the corrected wavefront ?m

given by

I

in the image plane is

?m

I?x,y? = − i exp?ikdm??i

?F??dm?R*O?x +

I?x +

dm

hr

SRx,y +

dm

hr

SRy?

SRy??.

dm

hr

SRx,y +

dm

hr

?50?

The comparison of ?m

that the reconstructed images are exactly the same in

both cases, but spatially located at different positions. In-

deed, the propagation in the mixed approach deviates by

an angle −? from the image plane approach, where ? is

given by Eq. (40), which means that the reconstructed

wavefront is again propagating along the optical axis. In

the case without aberrations, the mixed approach is a

particular case of the general approach in which ?H

=PWm

Iwith ?i

Igiven by Eq. (37) indicates

H:

?H= PWm

H⇒ SD→ ?,

hd→ ?,

SD

hd

=

SR

hr

,

⇒ lim

hd→?

h = hr, lim

hd→?

M =

hr− d

hr

= Mi.

Using Eqs. (29) and (31), we can express the DPM ?m

phase correction:

Ifor

?m

I?x,y? = ??I?x,y???H=PWm

= exp?i

? exp?− i

H

?

??hr+ dm/Mi??

??h0+ dm/Mi??

x2

Mi

2+

y2

Mi

2??

?

x2

Mi

2+

y2

Mi

2??, ?51?

which

compensated image wavefront ?m

leads totheexpressionofthecurvature-

I:

?m

I?x,y? = ??I?x,y???H=PWm

H

?52?

=exp?ik?dm−

dm

Mi??

1

Mi

?id

I?

x

Mi

,

y

Mi?.

?53?

These results show that the reconstructed image is the

same as the one issued from the image plane approach,

except that it is centered on the optical axis. The scaling

factor and propagation distance are the same. The effect

of the first-order correction in the hologram plane is to

center the image on the optical axis. This mixed approach

is thus interesting in the sense that it can be applied to

the convolution approach of the Fresnel propagation.

5. DISCUSSION

A. Analytical Formulation

It has been shown that in the image plane approach, the

reference wave may induce some differences compared

with the ideal case, as the phase curvature correction is

not performed in the hologram plane, but only in the im-

age plane. In the hologram plane approach, the difference

with the ideal case is due to the correction of the object

wave curvature, already performed in the hologram

plane, the consequences therefore depending on the shape

of the object wavefront. Finally, the mixed approach re-

duces the difference between the image approach and the

ideal case by correcting the propagation direction. Table 1

summarizes quantitatively the consequences in terms of

magnification and shift for each approach.

Each of these approaches has its own particularities.

The hologram plane approach has the advantage of

propagating, along the optical axis, a wave containing

only the phase deformations due to the object. The phase

modulations induced by the object are most often weak

and the wave propagates quite like a plane wave, mean-

ing that the phase curvature is compensated for any

propagation distance. As the DPM is applied in the fixed

hologram plane, it does not depend on the reconstruction

distance like in the image plane approach. Thus the DPM

can be determined once for a given setup, which is of

great interest in automated reconstruction processes. The

drawback of this solution is that the image is not focused

in the hologram plane. Thus, the areas used for the fit of

the DPM6are disturbed by the diffraction pattern of the

object. The DPM determined in the hologram approach

may thus be approximative in some cases, and may need

a minor adjustment in the image plane, creating a par-

ticular mixed approach.

The image plane approach has the opposite arguments.

It has the advantage of a focused image, and therefore

clear constant phase areas are available around the object

to perform the phase compensation procedure. This ad-

vantage is balanced with the fact that the phase curva-

ture is not compensated for any reconstruction distance

Montfort et al.

Vol. 23, No. 11/November 2006/J. Opt. Soc. Am. A2951

Page 9

and that the image is not centered in the reconstruction

window. Only the presented mixed approach corrects this

last disadvantage.

B. Discrete Formulation

All the considerations on the reconstruction approach are

done considering continuous functions. Nevertheless, the

sampling of the image and the propagation method have

an influence on the size of the reconstructed images.

The continuous expression of the resulting image ?Iin

the different propagation methods can be summarized by

the expression

?I?x,y? = exp?ik?d −

d

M??

1

M?id

H?

x − a

M

,

y − b

M?. ?54?

The ideal case corresponds to d=did, M=1, a=b=0. As de-

fined in the different reconstruction approaches, the fo-

cusing distance d and the magnification ratio M are given

by

M =

d

did

.

?55?

Considering these expressions, let us define the image

sizes in the discrete formulation using both the convolu-

tion and FFT expression of the propagation.

1. Fourier Transform Formulation

Using the FT formulation of the propagation [Eq. (5)], ?I

is expressed as

?I?n,m? = exp?ik?d −

M

H??n − a??

NptsT?

d

M??

1

??id

?d

1

M, ?m − b??

?d

NptsT?

1

M?.

?56?

In the digital reconstruction, the focusing distance is

given from Eq. (56):

d = Mdid.

?57?

Inserting Eq. (58) into Eq. (57), we obtain

?I?n,m? = exp?ik?d −

H??n − a??

d

M??

1

M

??id

?did

NptsT?

, ?m − b??

?did

NptsT??

= ?id

I?n − a?,m − b??.

?58?

This last result shows that the multiplication of the wave-

front by a quadratic DPM in the hologram plane has no

influence on the resulting image size. The FT formulation

of the propagation is thus not sensitive to scaling induced

by the different reconstruction methods [see Figs. 4(d),

5(d), and 6(d)].

As presented extensively in Ref. 1, one seems at first

sight to lose (or gain) resolution by applying the FFT ver-

sion. Oncloserexamination

?d/NptsT?corresponds to the resolution limit given by the

diffraction theory of optical systems: The hologram is the

aperture of the optical system with side length Npts?T?.

According to the theory of diffraction, at a distance d be-

hind the hologram a diffraction pattern develops. Tx

=?d/NptsT?is therefore the diameter of the Airy disk (or

speckle diameter) in the plane of the reconstructed image,

which limits the resolution. This can be regarded as the

automatic scaling algorithm, setting the resolution of the

image reconstructed in the Fresnel approximation by a

FFT always to the physical limit. The numerical lens in-

serted by the reconstruction algorithm thus has no effect.

onerecognizesthat

2. Convolution Formulation

It has already been shown that the convolution expres-

sion [Eq. (7)] of the propagation in the Fresnel approxi-

mation has the same sampling step before and after the

propagation. This means that the image is sampled in the

same manner in the hologram plane than in the image

plane. ?Ican be written as

Table 1. Summary of the Different Reconstructed Image Properties

Approach

?

MagnificationShift

Ideal

?id?x,y?

1(0, 0)

Hologram plane

exp?ik?d−did??

1

Mh

?id

I?

x

Mh

,

y

Mh?

ho−d

ho

(0, 0)

Image plane

exp?ik?d−did??

1

Mh

?id

I?

x−

d

hr

Mi

SRx

,

y−

d

hr

SRy

Mi?

hr−d

hr

?

d

hr

SRx,

d

hr

SRy?

Mixed

exp?ik?d−did??

1

Mi

?id

I?

x

Mi

,

y

Mi?

hr−d

hr

(0, 0)

2952J. Opt. Soc. Am. A/Vol. 23, No. 11/November 2006 Montfort et al.

Page 10

?I?n,m? = exp?ik?d −

d

M??

1

M?id

H?

?n − a??T?

M

,

?m − b??T?

M ?,

?59?

which means that the size of the images is magnified the

same way as in the continuous domain. Figure 6(c) is re-

constructed using the hologram plane approach. The im-

age has the same size as the hologram. Figure 5(c) is re-

constructed using the mixed approach. The size of the

images of Figs. 5(c) and 5(d) is not the same because the

magnification factor with respect to the ideal case is dif-

ferent. Reconstruction of holograms by the convolution

approach results indeed in images with more or fewer pix-

els per unit length than those reconstructed by the FFT.

However, because of the physical limits, the image reso-

lution does not change.

6. CONCLUSION

In digital holography, the access to numerical data allows

an easy compensation of the nondesired phase curvatures

in the reconstructed images. Nevertheless, procedures are

required to determine the correction to be applied. The re-

corded data, corresponding to the interference term R*O,

usually do not allow the retrieval of the original image,

but only a scaled replica displaced both laterally and axi-

ally. The reconstructed image can therefore be considered

to be an image of the object through a system, composed

of a MO and an additional numerical lens, that can be

analytically characterized to get the exact properties of

the complete holographic microscope. In this paper, we

have presented for the first time to our knowledge the in-

fluence of the phase masks’ position involved in the recon-

struction process, in particular in terms of position and

size of the reconstructed specimen ROI.

We hope that this study on the reconstruction methods,

in conjunction with the remarks on the reconstructed im-

age sampling regarding the FT or convolution formula-

tion, will clarify the relations subtended between the

available hologram processing techniques, facilitating the

user’s choice for each specific application of DHM.

ACKNOWLEDGMENT

This work was funded through research grants 205320-

103885/1 from the Swiss National Science Foundation.

Thee-mailaddress

frederic.montfort@a3.epfl.ch.

for F.Montfortis

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Vol. 23, No. 11/November 2006/J. Opt. Soc. Am. A2953