**6. Digital colour holographic interferometry**

The fast development of technology, such as high resolution sensors, various DPSS lasers with large coherence, data post processing and computation power provide now opportunities to conceive new optical methods capable of simultaneous full field measurements with high spatial and temporal resolutions and giving absolute data. Digital holography with matrix sensors appeared in the last decade with cheap high resolution CCD cameras and the increasing power of computers. Image sensors have now size and spatial resolutions compatible with the needs for digital holographic recording. For example, matrices with 16361238 pixels of size 3.93.9m² are now available (Yamaguchi & Zhang, 1997). In the literature, only few papers concern works in digital colour holographic interferometry. In 2002, Yamaguchi et al. & Kato et al. demonstrated phase shifting digital colour holography using a multi-wavelength HeCd continuous wave laser (636nm, 537.8nm, 441.6nm) and a colour CCD equipped with a Bayer mosaic4. The authors demonstrated the

<sup>4</sup> Matrix structure in front of a sensor to create a colour information from a panchromatic monochrome

Real-Time Colour Holographic Interferometry (from Holographic Plate to Digital Hologram) 21

\*

 

\* ,, , ,

 

*(u,v)* are respectively the Fourier transform of *b*

Inverse Fourier transform applied to the selected region gives an estimation of the object complex amplitude and of the impulse response corresponding to the filtering applied in the

narrow spectrum. Practically, the spectral lobe of the first order diffraction can be manually selected in order to make it more reliable. When the test object is modified, for instance by a flow, this induces a modification in the refractive index along the probe beam and thus this

states of the object is simply obtained by computing. In a setup where the test section is

4

 

The assembly shown in Fig. 13a is very simple. It is like a conventional Michelson interferometer in which a beamsplitter cube is inserted between the spatial filter and the test section. The spatial filter is placed at the focal point of the achromatic lens so that the test section is illuminated with a parallel light beam as in previous optical setups. 50% of the light is reflected from the concave spherical mirror to form the three reference beams and 50% of light passes through the test section to form the three measurement waves. The flat mirror, placed just behind the test section, returns the beams towards the beam splitter. 25% of light is focused on the diaphragm which is placed in front of the camera lens. So, 25% of the reference beam intensities are focused in the same diaphragm by the concave mirror. The generation of micro-fringes used as spatial carrier frequencies is shown in Fig.14. When the focal points of the reference and object waves are superimposed in the diaphragm which is placed in front of the camera, a uniform background colour is observed on the screen. If the focusing point of reference waves is moved in the plane of the diaphragm, straight

*H xy O xy Oxy i ux vy O xy i ux vy* (6)

 

<sup>0</sup> , , , exp 2 , exp 2

<sup>0</sup> , , , exp , exp 2 , exp , exp 2

 

separated in the Fourier plane. Applying a binary mask around the frequency *+u*

This is quite easy to do if the hologram is not too noisy or if the optical phase

 and *v*

modifies the optical path and then the optical phase. At any wavelength

crossed twice, the optical path difference due to the phenomena is given by:

*H xy O xy b xy i xy i ux vy*

*x+v*

 

*b xy i xy i ux vy* (7)

 

 

*(x,y)* leads to the computation of phase. The phase change between the two

*H uv A uv C u u v v C u u v v* (8)

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

*y)* is the spatial carrier

*(x,y)]* and

, *+v*,

*(x,y)*.

has a

, recording a new

 

 

(9)

*(x,y)exp[i*

are well chosen, the three orders are well

 

 

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

modulation along *xy*, the hologram becomes:

Fourier transform of Eq. (7) gives:

*(u,v)* and *A*

*u* and 

**6.2 Digital Michelson holographic interferometer** 

*O0(x,y)*. If the spatial frequencies *u*

Fourier domain (Desse et al., 2008).

By developing complex exponentials one obtains:

where *C*

hologram *H*

respectively of width

possibility for the reconstruction of colour images but experimental results have a relatively low spatial resolution since the effective pixel number at each wavelength was 818×619, leading to an effective pixel pitch of 7.8m. Demoli et al., 2003, presented the first study on fluids using digital colour Fourier holography. They used a monochrome CCD sensor and three wavelengths from three continuous wave lasers (647nm, 532nm, 476nm). Their results show the evolution of thermal dissipation in an oil tanker with an excellent resolution, by using the amplitude image of the reconstructed holograms. Note that historically the use of several wavelengths for holographic interferometry was first described by Jeong et al., 1997. Furthermore Ecole Popytechnique Fédérale de Lausanne's researchers (Kuhn et al., 2007) and INOA (Ferraro et al., 2007) proposed the use of several wavelengths in digital holographic microscopy. For quantitative phase microscopy, Ferraro et al., 2004, showed that severe chromatic aberration can be eliminated. Since the main effect of the chromatic aberration is to shift the correct focal image plane differently at each wavelength, this can be readily compensated by adjusting the corresponding reconstruction distance for each wavelength. In these works the recording at each wavelength is sequential. Such approach was applied in three dimensional image fusion using sequential colour recording at several distances (Javidi et al., 2005). Such strategy was also considered by J. Zhao et al., 2008, by using three laser wavelengths as an imaging approach. All these methods use sequential recording at each wavelength and generally off axis reference waves are incident on the recording area at a constant angle. At the same time, two wavelength profilometry was proposed in a digital holographic microscope using two laser diodes (680nm, 760nm) and a monochrome sensor, giving a synthetic wavelength of 6.428m (Kuhn et al., 2007). The method is based on a spatial multiplexing by incoherent addition of single-wavelength interferograms, in the same way as was developed by Picart et al., 2003, 2005, with a single wavelength, each having different propagation directions for the reference waves, and recording with a monochrome CCD.

As regards these works, ONERA and LAUM5 decided to join theirs respective competences acquired in the past in order to develop adaptable and new optical imaging methods, firstly having properties such as full field imaging with high spatial and temporal resolutions, secondly giving absolute data after post processing and finally giving dynamic three dimensional measurements. These non-invasive optical methods are based on digital colour holography.

### **6.1 Theoretical basics**

In the case of Fluids Mechanics, hologram analysis performed by direct and inverse twodimensional FFT algorithms is very well adapted for the reconstruction of transparent phase objects. For any wavelength , the recorded image plane hologram can be expressed as:

$$H\_{\lambda}(\mathbf{x}, \mathbf{y}) = O\_0(\mathbf{x}, \mathbf{y}) + R(\mathbf{x}, \mathbf{y})O^\*(\mathbf{x}, \mathbf{y}) + R^\*(\mathbf{x}, \mathbf{y})O(\mathbf{x}, \mathbf{y}) \tag{5}$$

where *O0(x,y)* and *R(x,y)* are the zero order and the reference wave respectively and *O = bexp(i)* is the object wave. For convenience, *R(x,y)* can be represented with unit amplitude and zero phase. Subscript refers to one of the three colours, that is  *= R, G or B*. In the case of in-line holography, computation of the Fourier transform gives a broad spectrum centred 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* 

<sup>5</sup> **L**aboratoire d'**A**coustique de l'**U**niversité du **M**aine, Prof. P. Picart, Le Mans, France

or *y* or along the two directions. In a general case, *2π(ux+vy)* is the spatial carrier modulation along *xy*, the hologram becomes:

$$H\_{\boldsymbol{\lambda}}\left(\mathbf{x},\boldsymbol{y}\right) = O\_{0}\left(\mathbf{x},\boldsymbol{y}\right) + O\left(\mathbf{x},\boldsymbol{y}\right) \exp\left[2i\pi\left(\boldsymbol{\mu}\_{\boldsymbol{\lambda}}\mathbf{x} + \boldsymbol{\nu}\_{\boldsymbol{\lambda}}\mathbf{y}\right)\right] + O^{\prime}\left(\mathbf{x},\boldsymbol{y}\right) \exp\left[-2i\pi\left(\boldsymbol{\mu}\_{\boldsymbol{\lambda}}\mathbf{x} + \boldsymbol{\nu}\_{\boldsymbol{\lambda}}\mathbf{y}\right)\right] \tag{6}$$

By developing complex exponentials one obtains:

$$\begin{aligned} H\_{\boldsymbol{\lambda}}(\boldsymbol{\x},\boldsymbol{y}) &= O\_{\boldsymbol{\phi}}(\boldsymbol{\x},\boldsymbol{y}) + b\_{\boldsymbol{\lambda}}(\boldsymbol{\x},\boldsymbol{y}) \exp[-i\boldsymbol{\rho}\_{\boldsymbol{\lambda}}(\boldsymbol{x},\boldsymbol{y})] \exp[2i\pi(\boldsymbol{\mu}\_{\boldsymbol{\lambda}}\boldsymbol{x} + \boldsymbol{v}\_{\boldsymbol{\lambda}}\boldsymbol{y})] \\ &+ b\_{\boldsymbol{\lambda}}(\boldsymbol{x},\boldsymbol{y}) \exp[-i\boldsymbol{\rho}\_{\boldsymbol{\lambda}}(\boldsymbol{x},\boldsymbol{y})] \exp[-2i\pi(\boldsymbol{\mu}\_{\boldsymbol{\lambda}}\boldsymbol{x} + \boldsymbol{v}\_{\boldsymbol{\lambda}}\boldsymbol{y})] \end{aligned} \tag{7}$$

Fourier transform of Eq. (7) gives:

20 Advanced Holography – Metrology and Imaging

possibility for the reconstruction of colour images but experimental results have a relatively low spatial resolution since the effective pixel number at each wavelength was 818×619, leading to an effective pixel pitch of 7.8m. Demoli et al., 2003, presented the first study on fluids using digital colour Fourier holography. They used a monochrome CCD sensor and three wavelengths from three continuous wave lasers (647nm, 532nm, 476nm). Their results show the evolution of thermal dissipation in an oil tanker with an excellent resolution, by using the amplitude image of the reconstructed holograms. Note that historically the use of several wavelengths for holographic interferometry was first described by Jeong et al., 1997. Furthermore Ecole Popytechnique Fédérale de Lausanne's researchers (Kuhn et al., 2007) and INOA (Ferraro et al., 2007) proposed the use of several wavelengths in digital holographic microscopy. For quantitative phase microscopy, Ferraro et al., 2004, showed that severe chromatic aberration can be eliminated. Since the main effect of the chromatic aberration is to shift the correct focal image plane differently at each wavelength, this can be readily compensated by adjusting the corresponding reconstruction distance for each wavelength. In these works the recording at each wavelength is sequential. Such approach was applied in three dimensional image fusion using sequential colour recording at several distances (Javidi et al., 2005). Such strategy was also considered by J. Zhao et al., 2008, by using three laser wavelengths as an imaging approach. All these methods use sequential recording at each wavelength and generally off axis reference waves are incident on the recording area at a constant angle. At the same time, two wavelength profilometry was proposed in a digital holographic microscope using two laser diodes (680nm, 760nm) and a monochrome sensor, giving a synthetic wavelength of 6.428m (Kuhn et al., 2007). The method is based on a spatial multiplexing by incoherent addition of single-wavelength interferograms, in the same way as was developed by Picart et al., 2003, 2005, with a single wavelength, each having different propagation directions for the reference waves, and

As regards these works, ONERA and LAUM5 decided to join theirs respective competences acquired in the past in order to develop adaptable and new optical imaging methods, firstly having properties such as full field imaging with high spatial and temporal resolutions, secondly giving absolute data after post processing and finally giving dynamic three dimensional measurements. These non-invasive optical methods are based on digital colour

In the case of Fluids Mechanics, hologram analysis performed by direct and inverse twodimensional FFT algorithms is very well adapted for the reconstruction of transparent phase

,,,,,, *yxOyxRyxOyxRyxOyxH* \* \*

where *O0(x,y)* and *R(x,y)* are the zero order and the reference wave respectively and *O =* 

of in-line holography, computation of the Fourier transform gives a broad spectrum centred 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* 

<sup>5</sup> **L**aboratoire d'**A**coustique de l'**U**niversité du **M**aine, Prof. P. Picart, Le Mans, France

refers to one of the three colours, that is

*)* is the object wave. For convenience, *R(x,y)* can be represented with unit amplitude

, the recorded image plane hologram can be expressed as:

<sup>0</sup> (5)

 *= R, G or B*. In the case

recording with a monochrome CCD.

holography.

*bexp(i*

**6.1 Theoretical basics** 

objects. For any wavelength

and zero phase. Subscript

$$\tilde{H}\_{\lambda}(\boldsymbol{\mu}, \boldsymbol{\upsilon}) = \boldsymbol{A}\_{\lambda}(\boldsymbol{\mu}, \boldsymbol{\upsilon}) + \mathbf{C}\_{\lambda} \left(\boldsymbol{\mu} - \boldsymbol{\mu}\_{\lambda'} \boldsymbol{\upsilon} - \boldsymbol{\upsilon}\_{\lambda}\right) + \mathbf{C}\_{\lambda}^{\ast} \left(\boldsymbol{\mu} + \boldsymbol{\mu}\_{\lambda'} \boldsymbol{\upsilon} + \boldsymbol{\upsilon}\_{\lambda}\right) \tag{8}$$

where *C(u,v)* and *A(u,v)* are respectively the Fourier transform of *b(x,y)exp[i(x,y)]* and *O0(x,y)*. If the spatial frequencies *u* and *v* are well chosen, the three orders are well separated in the Fourier plane. Applying a binary mask around the frequency *+u*, *+v*, respectively of width *u* and *v*, allows the extraction of the object optical phase *(x,y)*. Inverse Fourier transform applied to the selected region gives an estimation of the object complex amplitude and of the impulse response corresponding to the filtering applied in the Fourier domain (Desse et al., 2008).

This is quite easy to do if the hologram is not too noisy or if the optical phase has a narrow spectrum. Practically, the spectral lobe of the first order diffraction can be manually selected in order to make it more reliable. When the test object is modified, for instance by a flow, 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 , recording a new hologram *H(x,y)* leads to the computation of phase. The phase change between the two states of the object is simply obtained by computing. In a setup where the test section is crossed twice, the optical path difference due to the phenomena is given by:

$$
\delta \mathcal{S} = \frac{\vec{\mathcal{A}}}{4\pi} \Delta \varphi\_{\vec{\lambda}} \tag{9}
$$

### **6.2 Digital Michelson holographic interferometer**

The assembly shown in Fig. 13a is very simple. It is like a conventional Michelson interferometer in which a beamsplitter cube is inserted between the spatial filter and the test section. The spatial filter is placed at the focal point of the achromatic lens so that the test section is illuminated with a parallel light beam as in previous optical setups. 50% of the light is reflected from the concave spherical mirror to form the three reference beams and 50% of light passes through the test section to form the three measurement waves. The flat mirror, placed just behind the test section, returns the beams towards the beam splitter. 25% of light is focused on the diaphragm which is placed in front of the camera lens. So, 25% of the reference beam intensities are focused in the same diaphragm by the concave mirror.

The generation of micro-fringes used as spatial carrier frequencies is shown in Fig.14. When the focal points of the reference and object waves are superimposed in the diaphragm which is placed in front of the camera, a uniform background colour is observed on the screen. If the focusing point of reference waves is moved in the plane of the diaphragm, straight

Real-Time Colour Holographic Interferometry (from Holographic Plate to Digital Hologram) 23

interferogram recording with the signal of the unsteady pressure measurement. The cycle of the vortex street was decomposed in eight different instants shifted by 76 s and at each instant, five interferograms were recorded from several cycles to average the unsteady maps. First, Fig. 15 shows two micro fringes images recorded with and without the flow in order to constitute reference and object interferograms. It can be seen in the zoomed image

The three Fourier transforms are calculated from each image in order to reconstruct the phase maps with the +1 order (the zero order and the -1 order are filtered). An example of reference and measurement spectra is given in Fig. 16 for the green line. One can see that the spectrum only exhibits a spot corresponding to the green spatial carrier frequency. No parasitic frequencies due to the blue and red lines are found. By subtracting the reference phase maps from the measurement phase maps, one obtains the modulo2 phase difference maps. After unwrapping, it possible to compute the refractive index maps and the gas

Fig. 16. FFT reference and object spectra and difference phase map for the green line

that micro-fringes are deformed by the shear layer of the upper side.

Fig. 15. Micro-fringes recording for the reference and object images

density field assuming the Gladstone-Dale relation.

Fig. 13. Digital colour holographic interferometer – Formation of spatial carrier frequencies

interference fringes are introduced into the field of visualization. These micro fringes are recorded on the CCD in order to calculate the three reference phase maps. Then the wind tunnel is started and the three object waves are distorted by the aerodynamic phenomenon. Micro-fringes interference is again recorded by the 3CCD sensor to enable calculation of the phase maps related to the object. For maps of phase difference, the reference phase is subtracted from the phase object.

Fig. 14. Generation and micro-fringes formation by the phenomenon studied
