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Coherence effects in digital in-line holographic
microscopy
Unnikrishnan Gopinathan,*Giancarlo Pedrini, and Wolfgang Osten
Institut für Technische Optik, Universitäat Stuttgart, Pfaffenwaldering 9, 70569 Stuttgart, Germany
*
Corresponding author: unni.gopinathan@gmail.com
Received June 12, 2008; accepted July 21, 2008;
posted August 12, 2008 (Doc. ID 97336); published September 17, 2008
We analyze the effects of partial coherence in the image formation of a digital in-line holographic microscope
(DIHM). The impulse response is described as a function of cross-spectral density of the light used in the space-
frequency domain. Numerical simulation based on the applied model shows that a reduction in coherence of
light leads to broadening of the impulse response. This is also validated by results from experiments wherein
a DIHM is used to image latex beads using light with different spatial and temporal coherence. © 2008 Op-
tical Society of America
OCIS codes: 030.1640, 030.1670, 030.4070, 090.1995, 170.0180.
1. INTRODUCTION
In-line holography with spherical waves using a lensless
configuration was originally proposed by Gabor [1] while
attempting to address the problem of lens aberrations in
electron microscopy. To address the practical issues asso-
ciated with the hologram reconstruction in the original
method, one of the approaches was to record the hologram
digitally and use numerical techniques for reconstruction.
These methods have been generically referred to as digi-
tal in-line holography (DIH) [2]. Many novel recording
and reconstruction methods have been proposed to ex-
tract information regarding three-dimensional complex
amplitude distribution [3–5] uncorrupted by twin image
and three-dimensional complex spatial coherence distri-
bution [6].
Over the years, DIH has been successfully applied to
imaging microscopic objects with electrons [7,8] and pho-
tons [9–14]. Most of these works use configurations with-
out a lens. In addition to eliminating aberrations, the
lensless configuration is well suited for a wide range of
sources including x rays and deep ultraviolet for improved
resolution [15–17].
In many applications using DIH, it is common to treat
the source as completely coherent, both spatially and tem-
porally. This assumption, though valid in many cases, will
not hold for all. For instance, x-ray sources are partially
coherent. Partially coherent light is also used for many
applications in optical microscopy [18–20]. These applica-
tions would benefit from a generalized treatment that
takes into account the state of coherence of light. Effects
of spatial coherence in lensless Fresnel holography
[21,22] and in-line Gabor holography [23] have been stud-
ied earlier. However these works have mainly dealt with
the spatial coherence effects for the case of quasi-
monochromatic light.
In this paper we provide a theoretical framework for
analyzing the partial coherence effects in the image for-
mation of a digital in-line holographic microscope
(DIHM). The analysis presented is valid for primary or
secondary sources of any state of coherence. The state of
coherence of the source is characterized by the cross-
spectral density (CSD) [24–26] in the space-frequency do-
main. The analysis in the space-frequency domain allows
us to account for any dispersive effect of the object. We
provide an expression for the impulse response in terms
of CSD of the source. The impulse response for the case of
a Gaussian Schell source is simulated. It is seen that a re-
duction in coherence of the light leads to broadening of
the impulse response. This is also validated by results
from experiments wherein a DIHM is used to image latex
beads using light with different spatial and temporal co-
herence.
2. IMPULSE RESPONSE FOR PARTIALLY
COHERENT LIGHT
A. Hologram Formation
The geometry of the system discussed is shown in Fig. 1.
Let
denote the illumination pupil of a planar secondary
source illuminated by a primary source directly or indi-
rectly (through an optical system) with a coherence state
characterized by the CSD function W共p1,p2,
兲of the
fluctuating field U共p1,t兲and U共p2,t兲at two points P1and
P2in the secondary source illumination pupil
;p1and p2
are the position vectors of two points P1and P2.Ifthe
fluctuations U共p1,t兲and U共p2,t兲are members of the sta-
tistical ensemble 兵U共p,t兲=U共p,
兲e−j
t其that is stationary
at least in the wide sense, Wolf [26] has shown that
W共p1,p2,
兲may be expressed as a correlation function in
the space-frequency domain as
W共p1,p2,
兲=具U*共p1,
兲U共p2,
兲典
.共1兲
To find out the impulse response of the system, we con-
sider the interference between the light emanating from
the source and the light scattered from a point object lo-
cated at the point Qwith a position vector ri. The scat-
Gopinathan et al. Vol. 25, No. 10/ October 2008/J. Opt. Soc. Am. A 2459
1084-7529/08/102459-8/$15.00 © 2008 Optical Society of America
tered wave from the object may be considered to be a
spherical wave emanating from the point Qwith an am-
plitude ⍜共ri,
兲. The scattered and unscattered compo-
nent of the light from the point P1reaches the point Pfol-
lowing two paths P1QP and P1Pgiven by
V1共r,
兲=⌰共ri,
兲U共p1,
兲ejk兩ri−p1兩
兩ri−p1兩
ejk兩r−ri兩
兩r−ri兩+U共p1,
兲ejk兩r−p1兩
兩r−p1兩.
共2兲
If the distance of the scatterer rifrom the source is quite
large compared to the source dimension, the term 兩ri
−p1兩may be approximated as ri−si·p1in the numerator
and as riin the denominator; siis the unit vector along
the vector ri. In the numerator, 兩ri−p1兩appears in the ex-
ponential term multiplied by the factor k. Hence, the ap-
proximation to 兩ri−p1兩used is different in the numerator
and the denominator. Similarly, if the point Plies suffi-
ciently far away from the source, the terms 兩r−ri兩and 兩r
−p1兩may be approximated as r−s·riand r−s·p1in the
numerator and as rin the denominator, respectively; sis
the unit vector along the vector r. Using the far field ap-
proximations given above we may write Eq. (2) as
V1共rs,
兲=⌰共ri,
兲U共p1,
兲ejkri
ri
ejkr
re−jk共si·p1+s·ri兲
+U共p1,
兲ejkr
re−jks·p1.共3兲
A similar expression may be written for the light from the
point P2reaching the point Pfollowing two paths P2QP
and P2Pgiven by
V2共rs,
兲=⌰共ri,
兲U共p2,
兲ejkri
ri
ejkr
re−jk共si·p2+s·ri兲
+U共p2,
兲ejkr
re−jks·p2.共4兲
The spectral density at the point Pis given by
S共rs,
兲=
冕
冕
具V1
*共rs,
兲V2共rs,
兲典d2p1d2p2.共5兲
The integration in Eq. (5) is over the source domain
.All
integrals in the following discussion unless otherwise
stated run from −⬁to +⬁. Substituting Eqs. (3)–(5) we ob-
tain
S共s,
兲=
冕
冕
W共p1,p2,
兲
冋
兩⌰共ri,
兲兩2
ri
2e−jksi·共p2−p1兲
+e−jks·共p2−p1兲+ejkri
ri
⌰共ri,
兲e−jk共si·p2+s·ri−s·p1兲
+e−jkri
ri
⌰*共ri,
兲ejk共si·p1+s·ri−s·p2兲
册
d2p1d2p2.共6兲
If we interchange p1and p2in the fourth term and use
the relation W共p2,p1,
兲=W*共p1,p2,
兲(valid for any CSD
function [24,25]), it may be noted that the fourth term in
Eq. (6) is the complex conjugate of the third term. It may
be seen that S共.兲is a function of only sand not rsince Pis
a point in the far field. The first term in Eq. (6) represents
the light reaching Pafter scattering at Q, whereas the
second term represents light reaching the point Pdirectly
from the source without any scattering. The last two
terms represent the interference of the scattered light
with the unscattered light.
The CSD of a statistically stationary source of any state
of coherence may be expanded as a linear combination of
orthonormal functions
n共p,
兲[26]:
W共p1,p2,
兲=兺
n
␣
n共
兲
n
*共p1,
兲
n共p2,
兲,共7兲
where the functions
n共p,
兲are the eigenfunctions and
␣
n共
兲are the eigenvalues of the homogeneous Fredholm
integral equation
冕
W共p1,p2,
兲
n共p1,
兲d2p1=
␣
n共
兲
n共p2,
兲.共8兲
In other words, Eq. (7) states that CSD may be expressed
as the sum of contributions from spatially completely co-
herent elementary sources if it is a continuous function
and bounded throughout the source volume [26]. Substi-
tuting Eq. (7) into Eq. (6) we obtain an expression for the
spectral density at a point Pin the far field,
S共s,
兲=兩⌰共ri,
兲兩2
ri
2兺
n
␣
n共
兲兩
˜
n共ksi⬜,
兲兩2
+兺
n
␣
n共
兲兩
˜
n共ks⬜,
兲兩2
+ejkri
ri
⌰共ri,
兲e−jks·ri兺
n
␣
n共
兲
˜
n共ksi⬜,
兲
˜
n
*共ks⬜,
兲
+e−jkri
ri
⌰*共ri,
兲ejks·ri兺
n
␣
n
*共
兲
˜
n
*共ksi⬜,
兲
˜
n共ks⬜,
兲,
共9兲
where s⬜and si⬜are the projections of the vectors sand
Fig. 1. Schematic illustrating the notations used in the text. SF
denotes spatial filter (see Fig. 4).
2460 J. Opt. Soc. Am. A/ Vol. 25, No. 10/ October 2008 Gopinathan et al.
sionto the z=0 plane and
˜
n共.兲is the spatial Fourier
transform of
n共.兲defined as
˜
n共.,
兲=
冕
n共p,
兲e−jp·共.兲d2p.共10兲
Taking a temporal Fourier transform of the spectral
density gives us the intensity at the point Precorded as a
hologram denoted by the function H共.兲and given by
H共s,.兲=
冕
S共s,
兲e−j
.d
.共11兲
The first term in the expression for the hologram, ob-
tained by substituting Eq. (9) into Eq. (11), is proportional
to the intensity of the scattered light from the particle,
and the second term is proportional to the reference beam
without contribution from the object. Often the reference
beam is recorded separately and subtracted from the ho-
logram [7,9] to minimize the effects of the zero-order
term. The third and the fourth terms result in the object
image and a twin image of the object during reconstruc-
tion.
B. Numerical Reconstruction
The image of the object is formed by reconstructing the
hologram numerically using the Helmholtz–Kirchhoff in-
tegral [7,9]. The reconstructed intensity at the point rcin
the reconstructed volume is given by
U共rc,
兲=
冕
⌳
H共s,
兲ejks·rcd2s⬜.共12兲
The integral in the above equation runs over the area ⌳of
the detector plane. Hereafter, we focus our attention on
the contribution of the third term in the expression for
the hologram given by Eq. (11) [in conjunction with Eq.
(9)]. This term results in the formation of an image of a
point object located at ridenoted as Uiand obtained by
substituting the expression for the hologram in Eq. (12):
Ui共rc,
兲=
冕
⌰共ri,
兲ejkri
ri兺
n
␣
n共
兲
˜
n共ksi⬜,
兲
⫻
冕
⌳
˜
n*共ks⬜,
兲ejks·共rc−ri兲d2s⬜e−j
d
.共13兲
Equation (13) describes the image formation in a DIHM
with light of any state of coherence. It may be seen that
the image formation for partially coherent light may be
described in terms of the sum of contributions from com-
pletely spatially coherent modes,
Ui共rc,
兲=兺
n
Ui
n共rc,
兲,共14兲
where
Ui
n共rc,
兲=
冕
⌰共ri,
兲ejkri
ri
␣
n共
兲
˜
n共ksi⬜,
兲
⫻
冕
⌳
˜
n*共ks⬜,
兲ejks·共rc−ri兲d2s⬜e−j
d
共15兲
is the contribution of the nth spatially coherent mode to
the image formed. If the object is considered to be consti-
tuted of a collection of point objects in a volume ⍀the im-
age of such an object may be obtained by integrating Eq.
(13) over all the object points in the volume ⍀:
U共rc,
兲=
冕
⍀
冕
⌰共ri,
兲ejkri
ri兺
n
␣
n共
兲
˜
n共ksi⬜,
兲
⫻
冕
⌳
˜
n*共ks⬜,
兲ejks·共rc−ri兲d2s⬜e−j
d
d3ri.
共16兲
We may write Eq. (16) as
U共rc,
兲=
冕
⍀
冕
⌰共ri,
兲ejkri
ri
hi共rc−ri,
兲e−j
d
d3ri,
共17兲
where hi共.,
兲is the impulse response of the system given
by
hi共.,
兲=兺
n
␣
n共
兲
˜
n共ksi⬜,
兲
冕
⌳
˜
n*共ks⬜,
兲ejks·共.兲d2s⬜.
共18兲
It may be seen that the impulse response is a function of
the position of the object through the term
n
˜
共ksi⬜,
兲,
which makes the system shift variant for partially coher-
ent light. We now proceed to discuss a few special cases.
C. Spatially Coherent Light
When the light is completely spatially coherent then the
CSD may be represented by the lowest order mode 共n
=0兲. For this case, the impulse response is given by
hi
coh共.,
兲=
␣
0共
兲
˜
0共ksi⬜,
兲
冕
⌳
˜
0*共ks⬜,
兲ejks·共.兲d2s⬜.
共19兲
D. Narrowband Light
For narrowband light the bandwidth ⌬
is very small
compared to center frequency
0. If the assumption that
the modulus and the phase of the CSD function W共.兲is
constant over ⌬
is valid, then the CSD function may be
approximated as T共
兲W共p1,p2,
0兲,T共
兲being the spec-
tral density of light. For this case the impulse response
may be written as
hi
nb共.,
兲=T共
兲兺
n
␣
n共
0兲
˜
n共k0si⬜,
0兲
⫻
冕
⌳
˜
n*共k0s⬜,
0兲ejk0s·共.兲d2s⬜.共20兲
Gopinathan et al. Vol. 25, No. 10/ October 2008/J. Opt. Soc. Am. A 2461
E. Gaussian Schell Sources
A Schell model source is characterized by CSD of the
form [25]
W共p1,p2,
兲=冑S共p1,
兲冑S共p2,
兲
共p1−p2,
兲,共21兲
where S共p,
兲is the spectral intensity at pand
共p1
−p2,
兲is the complex degree of spatial coherence. For the
Gaussian Schell model sources the spectral intensity dis-
tribution and degree of coherence are Gaussian functions
given by
S共p,
兲=A共
兲e−兩p兩2/2
s
2共
兲,共22兲
共p1−p2,
兲=e−兩p1−p2兩2/2
2共
兲.共23兲
For the Gaussian Schell model sources, the eigenfunc-
tions and eigenvalues,
n共p,
兲and n共
兲, that solve the
Fredholm integral equation [Eq. (8)] are given by [27]
n共x兲=
冉
2c
冊
1/4 1
冑2nn!Hn共x冑2c兲e−cx2,共24兲
␣
n=A
冉
a+b+c
冊冉
b
a+b+c
冊
n
,共25兲
where Hn共.兲is a hermite polynomial of order n,c
=冑a2+2ab,a=1/4
s
2共
兲, and b=1/2
2共
兲. In the above
equations the dependence of a,b, and con
is dropped
for convenience. The parameter

[27], defined as the ra-
tio of
to
s, is the measure of degree of global coherence
[27] of the source. When

Ⰷ1, the source may be said to
be globally coherent and is well approximated by the low-
est order mode. When

Ⰶ1 the source may be considered
to be globally incoherent, and a larger number of modes of
the order of 1/

are necessary to represent the source
properly. The Fourier transform of
n共x兲in Eq. (24) is
given by
˜
n共
兲=
冉
2c
冊
1/4 1
冑2nn!H
˜
n
冉
冑2c
冊
e−
2/4c.共26兲
Substituting Eq. (26) into Eq. (20) we get the impulse re-
sponse when a Gaussian Schell source with narrow tem-
poral bandwidth centered at frequency
0is used,
hi
Schell共.,
兲=T共
兲兺
n
冉
2c
冊
1/2
␣
n共
0兲
2nn!H
˜
n*
冉
k0
冑2csi⬜
冊
⫻
冕
⌳
H
˜
n
冉
k0
冑2cs⬜
冊
e−k0
2共兩s⬜兩2+兩s0⬜兩2兲/4cejk0s·共.兲d2s⬜.
共27兲
One could express the impulse response in the time
Fig. 2. Impulse response evaluated at rx=ry=0 for rzranging from −0.1 to 0.1 mm for a Gaussian Schell source with (a)

=5,

=0.2 and
temporal FWHM bandwidth equal to 7 nm, (b)

=5,

=0.2 and temporal FWHM bandwidth equal to 14 nm, (c) temporal FWHM band-
width equal to 7 and 14 nm with

=5, and (d) temporal FWHM bandwidth equal to 7 and 14 nm with

=0.2. The Xaxis in all four plots
indicates distance ⌬rz in millimeters. The Yaxis shows normalized amplitude.
2462 J. Opt. Soc. Am. A/ Vol. 25, No. 10/ October 2008 Gopinathan et al.
domain by taking the temporal Fourier transform of
Eq. (27).
hi
Schell共.,
兲=
共
兲兺
n
冉
2c
冊
1/2
␣
n共
0兲
2nn!H
˜
n*
冉
k0
冑2csi⬜
冊
⫻
冕
⌳
H
˜
n
冉
k0
冑2cs⬜
冊
e−k0
2共兩s⬜兩2+兩s0⬜兩2兲/4cejk0s·共.兲d2s⬜,
共28兲
where
共
兲=
冕
T共
兲ej
d
.共29兲
We simulate the impulse response of a DIHM as given in
Eq. (28) (for si⬜=0) when a narrowband Gaussian Schell
model source is used. The distance Dbetween the source
and CCD was assumed to be 15 mm. The CCD was as-
sumed to have a square pixel of width 6.7
m. The tem-
poral frequency distribution of the light T共
兲was as-
sumed to have a Gaussian distribution with a center
wavelength of 670 nm. The temporal coherence of the
light is varied by varying the temporal bandwidth mea-
sured as full bandwidth at half wavelength (FBHW) of
the Gaussian distribution. Figure 2shows the plots of the
absolute value of the impulse response h共r兲with the x
and ythe components of r,rx, and ryequal to zero and rz
ranging from −0.1 to 0.1 mm in steps of 5
m. In the
plots shown in Figs. 2(a) and 2(b) the value of

is
changed keeping the temporal coherence constant. As the
value of

decreases from 5 to 0.2 (decreasing spatial co-
herence), the impulse response broadens in the Zdirec-
tion. For

=5, the lowest order mode, n=0, was used to
calculate the impulse response. For the other values of

,
the first 21 modes (n=0 to 20) were used. In the plots
shown in Figs. 2(c) and 2(d) the value of

is kept con-
stant, and the temporal bandwidth of the light is
changed. It may be observed that an increase in temporal
bandwidth of the light (decreasing temporal coherence)
results in the broadening of the impulse response. Fig-
ures 3(a)–3(d) show the corresponding plots of the abso-
lute value of impulse response h共r兲when rz=0. Again one
observes a broadening of the impulse response in the X–Y
plane as the spatial and temporal coherence of light de-
creases.
3. EXPERIMENTS
A. System Description
The schematic of the setup used to study the effects of
temporal and spatial coherence in a lensless digital in-
line holographic is shown in Fig. 4. Light from a primary
source illuminates a spatial filter (SF) that serves as a
secondary source. The objects used in this paper are
spherical latex beads of 6
m in diameter mounted onto a
Fig. 3. (Color online) Impulse response evaluated in the plane rz= 0 for a Gaussian Schell source with (a)

=5,

=0.2 and temporal
FWHM bandwidth equal to 7 nm, (b)

=5,

=0.2 and temporal FWHM bandwidth equal to 14 nm, (c) temporal FWHM bandwidth equal
to 7 and 14 nm with

=5, and (d) temporal FWHM bandwidth equal to 7 and 14 nm with

=0.2. The Xand Yaxes in all four plots
indicate rxand ryin pixels. Each pixel translates to a physical distance 3
m. The Zaxis shows normalized amplitude.
Gopinathan et al. Vol. 25, No. 10/ October 2008/J. Opt. Soc. Am. A 2463
microscopic slide. The light emanating from the pinhole is
scattered by the object. The object beam interferes with
the unscattered reference beam, and the hologram is re-
corded by a CCD (1000 pixels ⫻1000 pixels, pixel size
6.7
m⫻6.7
m) placed at a distance 15 mm from the
source. From each object hologram the zero-order term is
removed by subtracting the reference wave intensity (re-
corded separately). The holograms are then used for the
numerical reconstruction performed using Eq. (12) over
volume ⌬x⫻⌬y⫻⌬zcentered on the position of the object.
The reconstruction space coordinates are given by ⌬r
=rc−r0and has an origin at rc=r0. The resolution in the
reconstructed volume is D/Nx
␦
dx in the Xdirection and
D/Ny
␦
dy in the Ydirection, where Dis the distance be-
tween the source and the CCD,
␦
dx and
␦
dy are the pixel
pitches of the CCD in the Xand the Ydirections, and Nx
and Nyare the number of hologram pixels in the Xand
the Ydirections.
B. Results and Discussion
Two light sources with different temporal coherence were
used in the experiment. (1) LD1, a laser diode with center
wavelength 670 nm and temporal full width at half-
maximum (FWHM) bandwidth of 2 nm. (2) LD2, a laser
diode with center wavelength 635 nm and FWHM band-
width of 12 nm. The spatial coherence was varied by us-
ing spatial filters of two different diameters, 1 and 5
m.
As the diameter of the pinhole increases, the spatial co-
herence of light decreases. The light emanating from the
pinhole can be considered to be spatially coherent if the
absolute value of degree of coherence of light stays close
to unity within the pinhole. If the degree of coherence of
light drops considerably from the unity value within the
pinhole, then the light is spatially incoherent. In light of
the discussion in Section 2, partially spatially coherent
Fig. 4. Schematic of a lensless DIH setup. An SF, which acts as
a secondary source, is illuminated via an imaging system by a
primary source (S). The light scattered by the micro-object and
the unscattered light forms an in-line hologram at the CCD
plane.
Fig. 5. Experimental result showing the reconstructed amplitude at ⌬rx =⌬ry = 0 for ⌬rz ranging from −0.1 to 0.1 mm for (a) source LD1
共FWHM= 2 nm兲with spatial filters of 1 and 5
m in diameter, (b) source LD2 共FWHM= 12 nm兲with spatial filters of 1 and 5
min
diameter, (c) source LD1 and LD2 with spatial filter of 1
m in diameter, and (d) source LD1 and LD2 with spatial filter of 5
min
diameter. The Xaxis in all the four plots indicates distance ⌬rz in millimeters. The Yaxis shows normalized amplitude.
2464 J. Opt. Soc. Am. A/ Vol. 25, No. 10/ October 2008 Gopinathan et al.
light can be described as the summation of a large num-
ber of spatially coherent modes. Shown in Figs. 5(a)–5(d)
are plots of the reconstructed amplitude at ⌬rx =⌬ry =0 for
⌬rz varying from −0.1 to 0.1 mm at intervals of 10
min
the reconstruction space. Figure 5(a) shows plots when
two different spatial filters of 1 and 5
m in diameter are
used with light source LD1. Figure 5(b) shows the corre-
sponding plots for light source LD2. Figure 5(c) shows the
plots when two light sources LD1 and LD2 are used with
the spatial filter of 1
m in diameter. Figure 5(d) shows
the corresponding plots for the spatial filter of 5
m in di-
ameter. For a given temporal coherence of the light, as the
spatial coherence decreases, the image of the bead is
broader along the Zdirection. Shown in Figs. 6(a)–6(d)
are three-dimensional plots of the reconstructed ampli-
tude in the plane ⌬rz =0 of the reconstruction space. Given
in Figs. 6(a) and 6(b) are the plots when the light source
LD1 is used with spatial filters of 1 and 5
m in diameter,
respectively. Figures 6(c) and 6(d) are the corresponding
plots for the light source LD2. It may be observed that a
decrease in the spatial and temporal coherence of light
leads to a broadening of the reconstructed image of beads.
4. CONCLUSION
We have theoretically analyzed the effects of partial co-
herence in a lensless DIH system. Our analysis is valid
for primary or secondary sources of any state of coher-
ence, though in this paper we have considered a planar
secondary source. It was found that the impulse response
of the system for light of any state of coherence is a func-
tion of the cross-spectral density of the light. The impulse
response was simulated for the case of a Gaussian Schell
model source. We have provided results from experiments
used to image a spherical latex bead using a lensless DIH
microscope using light from two different sources with
varying temporal bandwidth in conjunction with spatial
filters of two different sizes. The experimental results
were found to be in general agreement with the predic-
tions of theory and simulation.
ACKNOWLEDGMENTS
U. Gopinathan gratefully acknowledges the financial sup-
port of the Alexander Von Humboldt Foundation. This
work was also supported by the German Science Founda-
tion (DFG) grant OS. 111/19-2.
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