COL 12(2), 020901(2014) CHINESE OPTICS LETTERS February 10, 2014
Phase retrieval from double axially displaced holograms for
dual-wavelength in-line holography
Yan Li (
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1∗
, Wen Xiao (
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)
1
, Feng Pan (
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)
1
, and Lu Rong (
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2
1
Key Laboratory of Precision Opto-mechatronics Technology, Ministry of Education,
Beihang University, Beijing 100191, China
2
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
∗
Corresponding author: lyx0514@aspe.buaa.edu.cn
Received September 25, 2013; accepted December 12, 2013; posted online January 27, 2014
A phase retrieval method for dual-wavelength in-line digital holography is presented with double axially
displaced holograms. A synthetic wavelength is used during iterative propagations to retrieve wrap-free
phase distributions with a much extended measurement range. The simulation and experimental results
demonstrate a better elimination of the twin image, a faster rate of convergence of the iterative routine and
less number of wavelengths are compared with previously reported multiple-wavelength in-line holography.
OCIS codes: 090.1995, 100.5070, 100.3010.
doi: 10.3788/COL201412.020901.
In-line holography
[1]
, owing to no optical ele ments be-
fore the detector, is a suitable imaging tool in vis-
ible spectrum
[2]
, short-wavelength spectrum
[3,4]
, and
particle-field analysis
[5]
. Howerver, the hologram is
formed by the interference between the free propagated
light as the reference beam and the diffracted object
wave, both o f which propagate along the same direction
downstream the light source. Therefore, the defocused
image and zero-order term are superimposed onto the
real image in the r e construction process
[2−5]
, which dete-
riorates the image quality and degrades the measurement
accuracy.
Various methods have been proposed to eliminate or
minimize this inherent limitation in in-line holography.
Early metho ds , including superposition of holograms
from multiple wavelengths
[6,7]
and holo gram subtraction
at different recording planes
[8,9]
, are only suitable for
electron digital holographic microscopy or X-ray digi-
tal holographic microscopy. Digital techniques, such as
linear filtering
[10]
, only suppress rather than eliminate
the twin image. Phase retrieval algorithms provide an
effective solution for twin ima ge elimination. By involv-
ing the iterative diffraction propagations back and forth
between the object and the r ecording domains, real image
is well separated from the unwanted conjugate image. For
some iterative algor ithms use square module constraint
on the recording plane without imposing any constraint
on the o bject plane
[11,12]
, the convergence of w hich is very
likely to be trapped into local minimum
[13]
. In other a l-
gorithms, the amplitude of single hologram is imposed on
the recording plane as the constraint, of which the c on-
vergence is guaranteed via other constraints on the ob-
ject plane
[14−16]
. We developed a phase retrieval method
using two axially displaced holograms combined with a
finite transmission constraint on the object plane to elim-
inate the conjugate image , and it has b e e n demonstrated
that this approach has much faster speed o f convergence
and better elimination effect
[17]
. The methods above
only allow for the determination of the phase modulo
2π, and in some cases this is not sufficient. For objects,
whose optical thickness variations are greater than the
wavelength, the phase images ar e wra pped in the range
of (−π, π] radians after acta ngent calculation
[18]
. Bao et
al. presented phas e retrieval technique based on multiple
wavelengths to extend the range of measurement
[19,20]
.
Since there is no object cons traint, it requires a large
number of illumination wavelengths and the convergence
is not guaranteed.
In this letter, a phase retrieval method for dual-
wavelength in-line digital hologra phy has been prop osed
for dispersion-fr e e objects, in which iteration calculation
is p e rformed using double axially displaced holo grams
recorded at dual- wavelengths respectively. A synthetic
wavelength is used to ca lculate the wrap-free phase, as
in two-wavelength interferometry
[18]
. The appropriate
constraints are imposed both on the hologram plane and
object plane, thus faster speed of convergence and better
elimination effect are r ealized with less number of wave-
lengths compared with those of previous reported ap-
proaches in Refs. [19,20]. Meanwhile, more accurate un-
ambiguous unwrapped phase distributions are achieved
because of synthetic wavelength. The feasibility of this
method is valida ted by the numerical simulations and re-
lated experiments.
The schematic diagram of the setup is illustrated in
Fig. 1. The b e am derived from a tunable laser was
expanded and collima ted by a beam expander (com-
posed of a spatial filter and a collimating lens), then
passed though the sample. A complementary metal-
oxide-semiconductor (CMOS) located consequtively at
the distance z
1
and z
2
downstream of the sample was uti-
lized to record four holograms at both dual wavelengths
of 643.0 and 658.0 nm, which determined that the cor-
responding synthetic wavelength Λ used in iteration was
28.2 µm. The background images were taken in the same
condition in the absence of the object. The corre spond-
ing normalized holo grams were carried out by dividing
the holograms by background images in every hologram
planes
[15,16]
.
The main iterative procedures are as follows.
1671-7694/2014/020901(4) 020901-1
c
2014 Chinese Optics Letters