## Abstract

A digital three-color holographic interferometer was designed to analyze the variations in refractive index induced by a candle flame. Color holograms are generated and recorded with a three layer photodiode stack sensor allowing a simultaneous recording with a high spatial resolution. Phase maps are calculated using Fourier transform and spectral filtering is applied to eliminate parasitic diffraction orders. Then, the contribution along each color is obtained with the simultaneous three wavelength measurement. Results in the case of the candle flame are presented. Zero order fringe, meaning zero optical path difference, can be easily extracted from the experimental data, either by considering a modeled colored fringe pattern or the wrapped phases along the three wavelengths.

© 2008 Optical Society of America

## 1. Introduction

Technological improvements performed in the area of matrix sensors offer new perspectives for holography. Hologram recording can be performed using CCD matrix with increased resolution and more important digital gain. Although the CCD resolution and size are not as good as that of holographic plates, digital approach is more accessible and versatile since the time for the hologram processing is greatly reduced and treatment is purely numerically.

Classically, digital holography was developed with a single wavelength. This was the case in papers relating time averaging and vibrometry [1,2], simultaneous two-dimensional full field vibration analysis [3,4], very high speed digital holography [5,6] or high amplitude vibration analysis [7]. Optical metrology performed with color holography was described in references [8,9] in which color image holographic interferometry with panchromatic plates was presented. Recently, Yamaguchi *et al* proposed phase shifting digital color holography [10]. They recorded three color digital holograms with a 1636×1238 pixel matrix with pixel pitches of 3.9×3.9µm^{2}. The sensor used a Bayer filter which consists in a spatial spectral filter. The authors showed the possibility for reconstructing a color amplitude image. However, the resolution was degraded since the effective pixel number along each wavelength was 818×619. In 2003, Demoli *et al* developed a quasi-Fourier off-axis experimental setup using a monochrome CCD sensor with 1008×1018 square pixels, 9×9µm^{2} in size [11]. Strategy consisted in a sequential recording of the reference and measurement holograms along the three red-green-blue channels. Then, a movie was built using the three color results by applying the subtraction method to the digital holograms [12]. The movie shows convective flows induced by the thermal dissipation in a tank filled with oil [10].

For several years ONERA has been developing optical metrology based on real-time color holographic interferometry using reflection panchromatic holographic plates. These plates are used as reference hologram and real time analysis is performed. The chemical treatment is a disadvantage because it produces variations in the gelatin thickness. Nevertheless, very nice results have been obtained for the visualization a candle flame [8]. The unsteady wake flow around a circular cylinder was also studied at Mach 0.4 [13]. A possibility to avoid chemical processing is related to digital three-color holographic interferometry. Furthermore, it seems to be a promising way for analyzing transparent phase objects. This paper presents, to the best of our knowledge, the first digital three color holographic interferometer for flow visualization, allowing a simultaneous high resolution recording along each color. In this technique, an optical setup has been designed to generate micro fringes in the observed field and the recording support (holographic plate) is replaced by a specific CMOS sensor constituted with three stacked photodiode layers. Then, interference micro fringes produced by the superimposition of three reference waves and three probe waves can be simultaneously recorded on the three spectral bands (red, green, blue). Phase and amplitude images are computed using Fourier transform in delayed time. Spectral filtering is applied on each Fourier plane in order to eliminate the parasitic diffraction orders. Then, phase differences are obtained by subtracting the reference phase to the probe phase. Several optical setups were tested and the best configuration allows the visualization of field about 70mm and increases the sensitivity since the probe wave crosses twice the test section. Interferences induced by a flame candle have been recorded and intensities have been computed from the phase differences. Fringes obtained with this process are those found with real-time color holographic interferometry using classical holographic plates. The very good comparison between classical and digital holography allows the validation of the full processing, that is: recording, filtering and phase reconstructions along the three colors.

A comparison can be made between digital and image color holographic interferometry as regards to previous obtained results [14–16]. In image color holographic interferometry, a panchromatic holographic plate (7000 to 10,000l/mm in spatial resolution) has to be illuminated with a total energy of 600 µJoule and the coherence length of the three lasers has to be more 2 meters. About 220 successive frames of the phenomenon are recorded at high framing rate (35 000 images per second with an exposure time of 750 ns for each). Each image has to be digitalized and processed. It is important to obtain a reference hologram of about 50% diffraction efficiency for the three lines and the implementation of the optical setup is not very easy. In digital color holographic interferometry, energy of 1µJoule is sufficient to illuminate the Foveon sensor (200l/mm in resolution). The laser length coherence can be reduced to several centimeters. The framing rate is limited to 8 frames per second, full size and some problems can be encountered with the RGB filters overlapping. The implementation is easy enough and the phase difference is entirely estimated with a computer.

## 2. Theoretical basics

Hologram analysis performed by direct and inverse two-dimensional FFT algorithms is very well adapted for the reconstruction of transparent phase objects. For any wavelength *λ*, the image plane hologram recorded can be expressed as:

where *O*
_{0}(*x*,*y*) and *R*(*x*,*y*) are the zero order diffraction and the reference wave respectively and *O*=*b*
_{λ}exp(*iφ _{λ}*) is the object wave. For convenience,

*R*(

*x*,

*y*) can be represented with unitary amplitude and zero phase. Subscript

*λ*refers to one of the three color, that is

*λ*=

*R*,

*G*or

*B*. In the case of in-line holography, computation by Fourier transform gives a broad spectrum centered at the zero spatial frequency. So, no relevant information can be extracted from such a spectrum. Consider now off-axis holography in which a spatial carrier is introduced along

*x*or

*y*or along the two directions. In a general case, 2

*π*(

*u*+

_{λ}x*v*) is the spatial carrier modulation along

_{λ}y*xy*, the hologram becomes:

By developing complex exponentials one obtains:

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}+{b}_{\lambda}(x,y)\mathrm{exp}[-i{\phi}_{\lambda}(x,y)]\mathrm{exp}\left[-2i\pi \left({u}_{\lambda}x+{\nu}_{\lambda}y\right)\right]$$

Fourier transform of Eq. (3) gives:

where *C _{λ}*(

*u*,

*v*) and

*A*(

_{λ}*u*,

*v*) are respectively the Fourier transform of

*b*(

_{λ}*x*,

*y*)exp[

*iφ*(

_{λ}*x*,

*y*)] and

*O*

_{0}(

*x*,

*y*). If the spatial frequencies

*u*and

_{λ}*v*are well chosen, the three orders are well separated in the Fourier plane. Applying a binary mask around the frequency +

_{λ}*u*, +

_{λ}*v*, respectively of width Δ

_{λ}*u*and Δ

*ν*, allows the extraction of the object optical phase

*φ*

_{λ}(

*x*,

*y*). Inverse Fourier transform applied to the selected region gives an estimation of the object complex amplitude:

where * means convolution and *h*(*x*,*y*) is the impulse response corresponding to the filtering applied in the Fourier domain. It is such that

After the filtering process, a biased optical phase can be estimated from relation (5):

and the amplitude of the fundamental harmonic at wavelength *λ* by:

In equations (7,8), ℑ_{m}[…]and ℜ_{e}[…] mean imaginary and real part of complex value.

To get equations (6,7), one have to discriminate the useful lobe in the Fourier plane. This is quite easy to do if the hologram is not too noisy or if the optical phase *φ _{λ}* has a narrow spectral expansion. Practically, the spectral lobe of the first order diffraction can be manually selected in order to make it more reliable. The spatial resolution of the measurement is affected by the choice of the width of the filtering function. If we consider the width at the first zeros of the impulse response then the spatial resolution can be estimated to be

*R*=1/Δ

_{x}*u*in the

*x*direction and

*R*=1/Δ

_{y}*ν*in the

*y*direction (see Eq. (6)). The compromise must be found between increasing the resolution of the method, which means extending the spatial bandwidth of the filtering, and filtering the unwanted parasitic orders, which means narrowing the spatial bandwidth.

Note that the suppression of the spatial carrier can be obtained by multiplying Eq. (5) by exp[-2*iπ*(*u _{λ}x*+

*ν*)]. When the test object is modified, for example by heating or with a flow circulation, this induces a modification in the refractive index along the probe beam and thus this modifies the optical path and then the optical phase. At any wavelength

_{λ}y*λ*, recording a new hologram

*H*(

_{λ}*x*,

*y*) leads to the computation of phase

*ξ*(

^{P}_{λ}*x*,

*y*)=

*φ*(

_{λ}*x*,

*y*)+Δ

*φ*(

_{λ}*x*,

*y*)+2

*πu*+2

_{λ}x*πν*. The phase change between the two states of the object is simply obtained by computing Δ

_{λ}y*φ*(

_{λ}*x*,

*y*)=

*ξ*

^{P}

_{λ}(

*x*,

*y*)-

*ξ*

^{R}

_{λ}(

*x*,

*y*). In a setup such that the probe beam crosses twice the tested region, the optical path difference due to the phenomena under interest is given by:

## 3. Optical setup designed for recording micro fringes

Figure 1 presents the optical setup. The three wavelengths are issued from two different lasers. A cw argon-krypton ionized laser delivers the red and green lines at *λ _{R}*=647nm and

*λ*=514nm and a DPSS laser delivers the last one at

_{G}*λ*=457nm. The coherence length of the red line is about 10m, it is several centimeters for the green line and about 5m for the blue line. A Fabry-Perot etalon is inserted in the argon-krypton laser in order to increase the coherence length of the green line. The three lines are combined by dichroic mirror (2) and the acousto-optical cell (3) diffracts the other unwanted lines. The three reference waves and the three probe waves are divided by the beam splitter cube (6). An optical right angle prism formed by the two small mirrors (1) is used to adjust the length of the reference and probe beams. The spatial filter (5), constituted by a microscope objective (×60) and a pinhole (25µm), expands and filters the beams. It is located just at the focal distance of the two achromatic lenses (7) in order to generate the reference and probe plane waves. The probe beam crosses the test section twice because the light is reflected by the flat mirror (11) located just behind the test section (8). Then, the beam splitter returns the probe beam into the sensor. The achromatic lens (9) forms an afocal beam expander system with lens (7). A polarizer (not shown) allows adjusting the beams in order to adjust the optical power in the recording plane. The displacement of one of the flat mirrors (1) is used to choose the fringe density and orientation in the detector plane (adjustment of

_{B}*u*and

_{λ}*ν*). As the image size and fringe resolution depend on the achromatic lens (9), several different lenses were tested (focal length: 30mm, 50mm and 70mm) and the best results were obtained with the last one. The second beam splitter (6) allows interferences to be formed on the sensor.

_{λ}The sensor used is a one-piece digital camera capable of acquiring accurate, high resolution color images using a Foveon X3-5M direct color image sensor with 1420×1064 pixels with pitches 5×5µm^{2} (http://www.hanvision.com). The used technology is different from that of the classical color mosaic sensor where one half of the pixels detect green and only one-quarter detect red or blue. In Bayer mosaic, offsets among color detection positions and artifacts are produced when the missing data is estimated. In the three layers of stacked photodiodes every pixel location detects simultaneously the three colors. So, the three color recording is performed simultaneously with the full spatial resolution of the sensor. However, the spectral sensitivity of the sensor is not very narrow along each wavelength since the color segmentation is based on penetration depth of the photons in silicon. Figure 2 shows the spectral sensitivity compared to the three laser lines used in the experiments. The sensitivity curves are given by the manufacturer. The curves indicate that there must be a diffusion of each three laser lines into each other three RGB channels. For example, it is possible to find red light in the blue channel or in the green channel.

## 4. Experimental results

The method of digital three-color holographic interferometry has been tested to determine the refractive index produced by a flame candle just placed in the front of the visualization windows. First, the power of each laser line is adjusted in order to not saturate histograms of recordings. Then, two micro fringe images are recorded with and without the flame in order to constitute the reference phase and the probe phase. Figure 3 shows such a three color hologram recorded with the three layer photodiode sensor.

Each red, green, blue hologram is extracted from the full RGB data. The Fourier transforms along each wavelength are presented in Fig. 4 for the reference and the probe data. Spectra are in dB units. As expected, it can be seen that the spectral selectivity of the sensor is not very excellent because a diffusion of the other colors into each spectral bandwidth can be observed. Particularly, there is a strong diffusion of green and red lines in the blue plane. In the red spectrum, one can see clearly the parasitic hologram due to the green line and in the green spectrum, the parasitic hologram due to the red line. In the blue spectra, there are the two parasitic holograms of the red and green lines. These observations are correlated to curves of Fig. 2. However, this diffusion of the laser lines does not prevent spectral filtering since the lobes are quite separated in the Fourier plane. This is due to the wavelength dependence on the spatial frequency. In the setup, the incidence angle of the laser rays in the sensor area is adjusted to the same value for the red, green and blue light. If *θ _{x}* and

*θ*are the incidence angles respectively in the

_{y}*x*and

*y*direction, then the spatial frequencies are respectively

*u*=sin(

_{λ}*θ*)/

_{x}*λ*and

*ν*=sin(

_{λ}*θ*)/

_{y}*λ*. The spectral distance between the three lobes in the Fourier plane are given for the green-blue distance by

and for the green-red distance by

The spectral distance is related to the increase in the incidence angles. The filtering will be possible if the bandwidth occupied by each color is smaller than the distance between two colors (*R-G* and *G-B*) in order that there is no overlapping. So, the lower spatial resolution of the method is *R _{y}*=1/2Δ

*uν*whereas the maximum spatial resolution can reach that of the sensor. With an incidence angle of about (|

_{GB}*θ*|,|

_{x}*θ*|)≈(0.36°,1.45°) in the setup, this leads to Δ

_{y}*uν*≈6.32mm

_{GB}^{-1}and Δ

*uν*≈10.42mm

_{GR}^{-1}giving the lower value for the resolution in the image plane

*R*≈79µm.

_{y}The power spectrum of the probe hologram includes the variations in the refractive index produced by the flame; thus this exhibits a horizontal spectral stretching in the three colors as shown in Fig. 4. Nevertheless, it is still possible to apply the spatial filtering and to calculate the three inverse Fourier transforms since there is no overlapping. In Fig. 4, the vertical bandwidth is limited by the color diffusion. The bandwidth of the filtering is indicated by colored line. Vertical resolution is about approximately equal to *R _{y}* whereas in the horizontal direction, the filtering can occupy the full available bandwidth that is 200mm

^{-1}. This means than the spatial resolution is that of the sensor in the horizontal direction, so

*R*=

_{x}*p*=5µm.

_{x}In the numerical processing, the spatial carriers were not removed. Phase difference was obtained by subtracting the reference to the probe phase for each color. Note that this operation leads to the automatic removal of the spatial carrier (see Section 2). Then, phase differences are unwrapped to eliminate the 2*π* phase jumps and the optical path difference is deduced from the phase difference. An example issued from Fig. 4 is shown in Fig. 5 for the blue light. Phase maps recorded without and with the flame and optical path difference are presented.

Note that the optical path difference is negative and that the peak-to-valley is 3.920µm. This is quite coherent with the observed phenomenon since the candle flame induces a diminution in refractive index.

Considering the characteristic of the afocal imaging of the test zone, the spatial resolution in the *x* direction after processing is about 57µm in the object plane (equivalent to 17.5mm^{-1}) and about 856µm in the *y* direction (equivalent to 1.16mm^{-1}). For a similar test object size, when using panchromatic holographic plates, the spatial resolution is about 7000mm^{-1} but it drops down to about 66.6mm^{-1} after recording the image with a slide 400ASA film. With the setup presented in the paper and for the *x* direction, the spatial resolution of digital holographic interferometry is about 4 times lower than that of classical analogical holography.

## 5. Reconstruction of fringes intensity – Comparison with modeling

From the phase differences, it is possible to calculate the intensity of interferences induced by the flame in the R-G-B colors. Note that the R-G-B optical path differences are almost identical due to the small dependence of *δ* with the wavelength *λ*. From the experimental results, it is computed the fringe intensity along each color according to the following relation

Then the color interferences can be synthesized by computing Eq. (13):

A color image of interference fringes can be next displayed on the monitor. Note that amplitude used in Eq. (12) can also be arbitrary amplitude instead of the measured ones. The choice for the amplitude depends on the relative ratios between the three colors. If the three channels have the same amplitude then the zero order fringe will be white, corresponding to the maximum of the RGB fringes of Eq. (13). In a different case, the zero order will have a color influenced by the ratios, but the position of the maximum of the RGB fringes remains invariant with these ratios. The image synthesized according to Eq. (13) can be compared to that calculated with the “MIDI” software developed by ONERA [14]. Taking into account the spectral characteristics of the light source and the three RGB camera filters, it is possible to analytically calculate the intensity of the interference fringes as a function of the optical path difference in order to reproduce the experimentally visualized colors on the monitor. Here, the comparison is made between the color fringes deduced from Eq. (12,13) and those given by the software. Figure 6 presents the color interferences given by Eq. (12,13) when the amplitude of R, G, B are chosen to be equal.

Colors obtained in the flame are identical to those given by software “MIDI”. Notes that MIDI calculates the color fringes versus the optical path difference and the experimental reconstructed color fringes include the double sensitivity due to the double pass in the test region. As in the modeling, it can be observed the white fringe and identical tints on each side. In the flame, the color stretching depends on the phase object while the stretching given by the software is linear. By comparing the reference color table with experimental tints, a variation of 3.55µm in the optical path difference was measured between the center of the flame and the right side of the field of view. Figure 7 presents the profiles of the synthesized fringes in the cases where the amplitude of the R, G, B components is chosen to be the measured one and when it is chosen to be equal.

Note that the maximum amplitudes of the two profiles correspond to the white fringe. The abscissa of the zero order fringe is found to be 6.9mm. The amplitude in the region of interest in Fig. 6 is estimated to be equal to 3.55µm for the three colors and this is in good agreement with the value estimated from the reference color table. Note that this value is very close to that obtained by real-time color holographic interferometry using panchromatic holographic plates [15].

The absolute zero order is found by observing the synthesized color fringes whereas it cannot be found by considering the unwrapped phase data. Indeed, each unwrapped phase map is obtained with an unknown value multiple of 2*π*. However, it can be found by considering the wrapped phase: the zero order fringe is that for which all the three wrapped phase values are identically null. Figure 8 illustrates this property since it can be seen that the three wrapped R, G, B phases have simultaneously a 0 value at the position of the white fringe (*x*=6.9mm).

The use of three colors is of a great interest if one looks at precisely the background tint observed outside the flame. In normal experimental conditions, the synthesized background color should be represented in white color (Δ*φ _{λ}*=0 and

*δ*=0) if there has been no disturbances between the reference and the probe recordings. In Fig. 6, it can be seen that the white fringe is shifted and located inside the flame. This means that the background pattern has slightly moved between the reference and probe recording. The value of the reconstructed background color quantifies this displacement. It is found to be about

*δ*=+1.872µm. This particularity could not be observed with experiments based on single wavelength interferences and shows the interest to use the color in digital holographic interferometry.

## 6. Conclusion

An optical setup based on digital three-color holographic interferometry has been designed for analyzing flows. Digital holograms are simultaneously recorded with a three layer photodiode sensor which allows a very good spatial resolution along each wavelength.

Data processing is performed by Fourier transform and spatial filtering. Due to the width spectral sensitivity of the sensor, adjustment of the incidence angles of the reference beam into the recording plane must be done in order to allow the spatial filtering in the Fourier plane. To the best of our knowledge, the results presented in the paper show the first metrological application with three wavelength digital holography. Interest of using color has been demonstrated since the zero order fringe can be easily determined and the variation in the background color due to disturbances can be quantified.

Limits of the methods are found in the wide spectral sensitivity of the sensor which produces a light diffusion in each monochromatic hologram. Works are currently in progress for removing the color diffusion using a segmentation approach. Success in the strategy will allow a large spatial filtering, increasing the spatial resolution in the reconstructed object.

Future works concern the extension of the proposed technique for analyzing unsteady wake flows around a circular cylinder.

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