## Abstract

A color digital holographic interferometry movie was produced by applying the subtraction digital holography method in a quasi-Fourier off-axis experimental setup. The movie was numerically recorded and replayed from three sets of digital holograms obtained with three different laser lines (476 nm, 532 nm, and 647 nm). The movie shows convective flows induced by thermal dissipation in a tank filled with oil.

©2003 Optical Society of America

## 1. Introduction

Recent applications of digital holography (DH) in microscopy [1,2], correction of aberrated wavefronts [3,4], particle measurement [5], or synthetic aperture imaging [6], have demonstrated the usefulness of this method. DH also offers new perspectives in holographic interferometry: in endoscopic inspection [7], life science research [8], or remote shape control [9]. Furthermore, by utilizing the phase-shifting principle, DH has been applied in vibration measurements [10] and in color reconstruction [11].

In Ref. [12] an effort was reported towards realizing a system for digital holographic recording of fast events using a frequency-doubled Q-switched Nd:Yag laser, where three pulsed holograms were recorded within a 12 ns frame interval. However, to record a dynamic event, a practical limitation imposed by the DH method must be overcome. Digital holograms are recorded optically using a CCD camera and reconstructed numerically using computer software. Advantages of DH are that no wet processing is necessary and retrieving hologram information is possible practically in real-time. Drawbacks are related to the recording conditions of low numerical aperture which leads to large speckle noise and overlapping of the zero- and first-order terms. The known techniques for suppressing the zero-order disturbance such as high-pass filtering [13] or phase-shifting [14] are not well suited for recording and reconstructing dynamic processes [15]. The use of a new method called subtraction digital holography [15] enables suppressing of the zero-order disturbance in the off-axis DH by subtracting two stochastically changed speckled primary fringe patterns. With this method, only one hologram capture is required for each instant of time thus allowing the wavefront phase change to be captured at a rate limited only by the image acquisition system.

The use of multiple wavelengths in classical holographic interferometry brings several advantages compared to single wavelength measurements. They can be used for improving the resolution by speckle averaging and smoothing out the image [16], desensitizing the method [17], and providing absolute data throughout the entire field of observation [18, 19]. Moreover, we have previously also shown [20] that in quantitative holographic interferometry, color recording may solve the sign phase ambiguities that unavoidably appear in monochrome recording.

In this paper, we combine our new subtraction DH method in a quasi-Fourier off-axis experimental setup with three-color holographic recording to produce, to the best of our knowledge, the first color digital holographic interferometry movie.

## 2. Theory

Consider a quasi-Fourier setup with a reference point source and a three-dimensional phase object both located at the input plane and a CCD camera located at the hologram plane. A digital hologram is obtained by recording the primary fringes pattern onto the CCD array. A standard object reconstruction is obtained by numerically Fourier transforming the hologram data. To apply the subtraction DH method [15], two stochastically changed holograms first must be recorded, then subtracted, and finally Fourier transformed. If between two recordings the object has been changed not only stochastically, then the secondary fringes will dominate in the object reconstruction area.

Input can be described by

where *R*
_{0} and *S*
_{0} are the amplitudes of the reference and object signals, respectively, *n*(*x*
_{1},*y*
_{1},*z*) is the refractive index of the object, and λ is the laser wavelength. The reference signal is located at the position (*x*
_{0}, *y*
_{0}) in the input plane.

For a collimated input beam the amplitude distribution at the hologram plane distanced *d* from the input plane can be described by the Fresnel diffraction

where
$\psi ({x}_{2},{y}_{2})=\frac{\pi}{\lambda d}\left[{\left({x}_{2}-{x}_{0}\right)}^{2}+{\left({y}_{2}-{y}_{0}\right)}^{2}\right]$
and *C*
_{0}=(1/*i*λ*d*) exp (*i*2π*d*/λ) denote respectively the quadratic phase (or chirped phase) term and the complex constant, and *s*(*x*
_{2}, *y*
_{2})=|*s*(*x*
_{2}, *y*
_{2})| exp [*i*φ(*x*
_{2}, *y*
_{2})] denotes the complex wavefront emerging from the phase object. We assume that the wavefront variations are caused only by changes in refractive index along the integration path through the object.

The function |*s*(*x*
_{2}, *y*
_{2})| describes speckle noise with the property

where 〈〉 denotes an ensemble average.

Intensity at the hologram plane is given by squaring the modulus of the amplitude described by Eq. (2), i. e.

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\frac{{R}_{0}}{\mathit{\lambda d}}s({x}_{2},{y}_{2})\mathrm{exp}\left[-i\psi ({x}_{2},{y}_{2})\right]+\frac{{R}_{0}}{\mathit{\lambda d}}{s}^{*}({x}_{2},{y}_{2})\mathrm{exp}\left[i\psi ({x}_{2},{y}_{2})\right],$$

where * denotes a complex conjugate operation.

To describe a dynamic process with *N* subsequent images, we first include a time variable in Eq. (4), i.e.

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\frac{{R}_{0}}{\mathit{\lambda d}}s({x}_{2},{y}_{2},{t}_{n})\mathrm{exp}\left[-i\psi ({x}_{2},{y}_{2})\right]+\frac{{R}_{0}}{\mathit{\lambda d}}{s}^{*}({x}_{2},{y}_{2},{t}_{n})\mathrm{exp}\left[i\psi ({x}_{2},{y}_{2})\right]$$

where time instants are described by: *t _{n}*,

*n*=0,1,...,

*N*. Next, we assume that the nonstochastic phase changes of the object wavefront are dominant compared to the stochastic changes of the speckled amplitude and phase in Eq. (5). It follows that 〈|

*s*(

*x*

_{2},

*y*

_{2},

*t*)|〉=〈|

_{n}*s*(

*x*

_{2},

*y*

_{2},

*t*

_{0})|〉 and |

*s*(

*x*

_{2},

*y*

_{2},

*t*)| ≈ |

_{n}*s*(

*x*

_{2},

*y*

_{2},

*t*

_{0})|.

By numerical subtracting the *n*-*th* from the 0-*th* recorded fringe patterns we obtain

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\propto \left[s({x}_{2},{y}_{2},{t}_{0})-s({x}_{2},{y}_{2},{t}_{n})\right]\mathrm{exp}\left[-i\psi ({x}_{2},{y}_{2})\right]+CC$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\propto \mid s({x}_{2},{y}_{2},{t}_{n})\mid \mathrm{exp}\left[i\frac{\phi ({x}_{2},{y}_{2},{t}_{0})+\phi ({x}_{2},{y}_{2},{t}_{n})}{2}\right]\times \mathrm{sin}\left[\frac{\Delta \phi ({x}_{2},{y}_{2},{t}_{n})}{2}\right]$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}\left[-i\frac{\pi}{\mathit{\lambda d}}\left({x}_{2}^{2}+{y}_{2}^{2}\right)\right]\times \mathrm{exp}\left[-i\frac{2\pi}{\mathit{\lambda d}}\left({x}_{2}{x}_{0}+{y}_{2}{y}_{0}\right)\right]+CC,$$

where *CC* means the complex conjugate of the previous term.

The secondary fringes showing the phase difference φ(*x*
_{2}, *y*
_{2}, *t*
_{0})-φ(*x*
_{2}, *y*
_{2}, *t _{n}*)=Δφ(

*x*

_{2},

*y*

_{2},

*t*) with

_{n}where

are easily obtained by calculating the Fourier transform of Eq. (6). Since Δ*I*(*x*
_{2}, *y*
_{2}, *t _{n}*) is a real-valued function, two symmetrically reconstructed secondary fringe distributions appear, each of them containing the difference between two wavefronts recorded at two different instants.

Once the secondary fringe distribution is obtained, the physical parameters such as the refractive index or temperature distribution across the object area can be determined. The numerical extraction of the phase distributions from the fringe patterns as well as the further processing of the interference phase to determine the physical parameters is discussed in detail [21, 22].

From Eq. (6) it is evident that the zeros of the subtracted image at the time instant *t _{n}*,

depend only on the wavelength λ Since *m*
_{λ} (*x*
_{2}, *y*
_{2}, *t _{n}*) is the relative displacement of the fringes, from Eq. (9) follows that different λ will introduce a scaling in the primary fringe pattern. Thus, in forming digital interferograms with multiple wavelengths, scaling of each particular interferogram must be appropriately compensated.

## 3. Experiment

The experimental setup for recording digital holograms is schematically shown in Fig. 1. Three beams from an argon ion laser (476 nm), a cw frequency-doubled Nd:YAG laser (532 nm), and a krypton ion laser (647 nm) are combined by dichroic mirrors to form a three color beam. This beam is then split into an object and a reference beam.

The object beam is expanded and collimated onto a transparent tank filled with oil (Fig. 2). A 400 Ohms immerged resistor is used to dissipate electrical power and to create convective thermal flows which lead to refractive index variations in the oil. The object to be recorded (i.e., the oil and its index variations) being of a pure phase type, a ground glass plate is used to diffusely scatter the object beam.

The reference beam is expanded then focused off-axis onto the plane of the ground glass to form the reference point source. The angle between the two beams is 2° and the intensity ratio between reference and object beam is adjusted to 3:1 using the variable beamsplitter and a set of neutral density filters.

The digital holograms are recorded at a distance of 2 m from the object using a Kodak Megaplus ES 1.0 digital camera. This camera is equipped with a 8-bit monochrome CCD sensor of 1008×1018 square pixels (9×9 µm each). The digital output is fed to a Matrox Pulsar frame grabber. Acquisition, which is controlled using the Matrox MIL library and numerical processing, which is performed by proprietary programs, are done on a Pentium III/500 MHz PC.

## 4. Protocol

The experimental setup described above was used to record a three color digital holographic interferometry movie. This movie shows convective flows induced by thermal dissipation in the tank.

For each wavelength, a first hologram was recorded without heating as a “reference hologram” (in the interferometric sense) which records the “undisturbed” phase distribution. Heating of the oil through the resistor was then switched on and successive holograms were recorded during the heating process. The electrical power was 1 mW; at such a low power, the convective process was slow: a capture rate of 1 “color hologram” (composed of three single R, G, and B holograms recorded sequentially) every 10 seconds appears adequate. With a recording time of about 8 minutes, we obtain three datasets of 49 digital holograms, where each set is representative of the dynamic of the phase distribution under a given wavelength.

## 5. Results

To synthesize the digital three color interferometry movie from the experimental datasets, the following procedure was applied:

- To compensate for the “λ-scaling” of the primary fringes pattern, green and blue sets of holograms were normalized against red set and scaled for factors 1.2 and 1.47, respectively.
- In each set, the e
*n*-*th*hologram was subtracted from the first, reference, hologram. - In each set, the subtracted holograms were Fourier-transformed.
- Each component of the RGB images was weighted using the weights required to get a white flat field in the reference hologram (Fig. 3).
- The color movie was composed from the sequence of RGB images. At a playback frame rate of 5 images/s, the 8 minutes (real-time) recorded heating process is compressed into a 9.8 seconds (playback-time) movie.

Selected color images are shown in Figs. 3 and 4. Figure 3 illustrates the object reconstruction before the heating process, while Fig. 4 shows the first four interferograms at the start of the heating process. For comparison, Fig. 5 shows the red channel of the color interferograms from Fig. 4.

Figure 6 shows the last frame of the color digital holographic interferometry movie composed of 49 still interferograms of a total duration of 9.8 seconds.

## 6. Conclusion

We have produced a color movie composed from a sequence of still digital holographic interferograms. The interferograms are obtained by applying the subtraction digital holography method in a quasi-Fourier off-axis setup with three different laser wavelengths.

The comparison of multi- to single-color images (Figs. 4 and 5, respectively) confirm that the reconstructed polychromatic interferometric fringes contain less speckles and are of smoother appearance. It appears also that they are more informative than the monochromatic ones: from the sequence of the first four interferograms (Fig. 4), the wavelength dependence of the observed convective heat dissipation is clearly visible. At first a blue fringe appears (λ=476 nm), then a green one (λ=532 nm), and finally a red one (λ=647 nm), thus showing clearly a three step fringe evolution process instead of the single one that would be obtained for monochromatic recording (Fig. 5). The appearance of other colors than those used in recording indicate that the observed phase changes are represented as color-coded information.

In quantitative measurements, resolution would thus be increased not only thanks to the better image quality, but also by making use of the chromatic information contained in the interferogram.

## Acknowledgments

This work has been supported by the Ministry of Science and Technology, Croatia (project No. 0035005), and by public contract “Plan État-Région Alsace”, France (project “Imagerie Diffractive”). Dr. N. Demoli has been supported during his stay at LSP by an industrial contract (FOSSAV) administrated by Égide-France.

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