1104 CHINESE OPTICS LETTERS / Vol. 7, No. 12 / Decemb er 10, 2009
Holography and speckle in phase-shifting
digital holography
Invited Pap er
Ichirou Yamaguchi
Toyo Seiki Seisakusyo, 1-2-6 Funado, Itabashi-ku, Tokyo 174-0041, Japan
E-mail: ichiyama1115@yaho o.co.jp
Received August 18, 2009
The correlation prop erties of the optical field diffusely reflected from a rough surface under coherent
illumination are analyzed numerically. The cross-correlations of the complex amplitudes and the intensities
b efore and after translation and/or tilt of the surface are calculated at an arbitrary observation plane in
three-dimensional (3D) space. The results provide us with the 3D distributions of phase changes and
sp eckle displacement that lead to the distributions and signal-to-noise ratio of the displacement to be
measured by holographic interferometry and speckle correlation technique. Comparisons with analytical
relationships and physical interpretation of them are also discussed.
OCIS co des: 030.0030, 090.0090, 120.0120.
doi: 10.3788/COL20090712.1104.
High coherence of laser light opened the new field of
interferometry that can be applied to diffusely reflect-
ing surfaces. Holography enables three-dimensional (3D)
imaging by recording amplitude and phase of the ob-
ject wave, while the sp eckle patterns appearing in the
laser scattered by a diffuse surface show high contrast
everywhere in 3D space. Holographic interferometry
and speckle metrology detect changes of phase and in-
tensity of the light diffusely reflected from diffuse sur-
faces and are closely related with each other because the
changes of intensity and phase are dep endent on each
other. Localization and interpretation of the fringe pat-
terns observed in holographic interferometry were widely
discussed
[1−4]
. Relationships between holographic in-
terferometry and speckle methods were later clarified
analytically
[5,6]
. These results were utilized in electronic
recording and processing of speckle patterns employed
for speckle metrology. Digital holography that comprises
both digital recording of holograms and digital recon-
struction of images brings these techniques even closer
to each other because of flexibility of digital recording
and processing of intensities obtained from comparatively
simple optical setups, realizing automatic and quantita-
tive measurements of surface shape and deformation of
3D diffuse objects. However, the effects of digital record-
ing such as those of number and pitch of charge-coupled
device (CCD) pixels and the bit-depth of video signals
cannot be investigated in a simple analytic way. Even
the optical influence of surface roughness requires compli-
cated assumptions on its interaction with incident light.
In this letter, we propose a comparatively simple model
based on the random phase scatterers model for an ob-
ject and the angular spectrum for propagation
[7]
. It has
been employed for computer simulation and is applicable
to general problems appearing in the measurement of
surface shape and deformation using coherent light.
In the former theory on fringe formation in holographic
and speckle interferometry, we expressed the complex
amplitude at an observation plane by a linear superpo-
sition of the complex amplitude at the object plane
[5]
.
The weighting function for the superposition is either
the parabolic wave or the diffraction at a lens aperture,
depending on the mutual position of an object and an
observation field. The observation field is either the
diffraction field or the image field. In the present analy-
sis, the both observation fields can be investigated from
the unified standpoint that is based on the angular spec-
trum expansion of optical fields
[7]
. We also assume a
process of digital holography where the complex ampli-
tude in the diffraction field is recorded and reconstructed
to produce an image of the object. In this case, hologram
size plays a role of a lens aperture, while the resolution of
the CCD limits a size of the recordable object. Thus we
can treat the general imaging systems including digital
holography and ordinary lens setups.
In digital holography, a laser beam is divided into two
paths, a reference and an object, and recombined to
generate a hologram as an interference pattern, which is
recorded by a CCD and stored in a computer. We start
from a digital holographic setup represented by the two-
dimensional (2D) coordinate system shown in Fig. 1.
A diffuse object is illuminated by a point mono chro-
matic source S with the wave number k = 2π/λ.
Fig. 1. Coordinate system.
1671-7694/2009/121104-05
c
° 2009 Chinese Optics Letters
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