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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

° 2009 Chinese Optics Letters

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