**3. Digital holography for analysing supersonic jet**

In this part, the supersonic flow of a small vertical jet has been analysed using three different techniques based on digital holography. The first one is based on Michelson digital holographic interferometer using three wavelengths as a luminous source [8], the second one uses the same source (three wavelengths) and Wollaston prisms to separate the reference waves and the measurement waves [11] and the last one is a little bit particular because a specific diffraction grating is manufactured to obtain several different diffractions of measurement waves and to avoid having the reference wave [12].

#### **3.1. Michelson holographic interferometry**

The optical set‐up presented in **Figure 3** is very simple and looks like a conventional Michelson interferometer in which a beam splitter cube (7) is inserted between the spatial filter (6) and the aerodynamic phenomenon under analysis (11). The light source consists of three diode‐ pumped solid‐state lasers, one red (R), one green (G) and one blue (B), emitting respectively at 660, 532 and 457 nm. A half wave plate (1) is used to rotate by 90° the polarization of the blue line (S to P) and a flat mirror (2) and two dichroic plates (3) allow the superimposing of the three wavelengths. An acousto‐optical cell (4) deflects the parasitic wavelengths in a mask (5) and diffracts the three wavelengths RGB using three characteristic frequencies injected into the crystal. The spatial filter (6), composed of microscope objective (60×) and a small hole of 25 μm, is placed at the focal length of the achromatic lens (9) in order to illuminate the phenomenon with a parallel beam. On‐going, 50% of the light is returned towards the concave mirror (8) to form the three reference beams and 50% of the light passes through the test section (between (9) and (12) to form the measuring waves. The flat mirror (12) placed behind the test section (11) returns the beams in the beam splitter cube (7). 25% of the light focused on the diaphragm which is placed in front of the achromatic lens (13). It is the same for the 25% of the reference beam which is focused on the same diaphragm by the concave mirror (8).

**Figure 3.** Michelson digital holographic interferometer.

Michelson digital holographic interferometer has been implemented around the ONERA wind tunnel and two optical tables isolate the optical set‐up from external vibrations. **Figure 4** shows the generation of micro‐fringes used as spatial carrier frequencies.

**Figure 4.** Micro‐fringes formation by the transparent object.

When the focal points of the reference and object waves are superimposed in the diaphragm which is placed in front of the lens (13), see **Figure 3**, a uniform background colour is observed on the screen for each colour. The combination of three background colour (R, G and B) produces a white colour on 3CCD camera. If the focusing point of the three reference waves is moved in the plane of the diaphragm, straight interference fringes are introduced into the field of visualization. This is achieved very simply by rotating the concave mirror (8). Without flow, these micro‐fringes are recorded on the 3CCD to calculate the three reference phase maps. Then, the wind tunnel is running and the three object waves are distorted by the aerodynamic phenomenon. Micro‐fringe interferences are again recorded 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. This optical technique was tested for analysing the supersonic flow of a small vertical jet, 5.56 mm in inner diameter at different pressures of injection. The location of the vertical jet in the middle of the test section is shown in **Figure 3**. The exposure time (10 ms) is given by the acousto‐optical cell noted (4) in **Figure 3**. The fringe space introduced in the field is much narrowed, about four or five pixels between two successive fringes, in order to generate three high spatial carrier frequencies. With this configuration, the sensitivity is increased. Each interferogram is processed with 2D fast Fourier transform and **Figure 5** shows the spectra computed for the reference and measurement for each colour plane. One can see that the generated spatial frequencies are respectively equal to 40.5, 30.9 and 28.4 lines per millimetre for the blue, green and red lines. Then, a filtering window is selected to cover the useful signal of the +1 order localized in the spectrum and an inverse 2D FFT is applied to reconstruct the amplitude and the phase of the signal.

pumped solid‐state lasers, one red (R), one green (G) and one blue (B), emitting respectively at 660, 532 and 457 nm. A half wave plate (1) is used to rotate by 90° the polarization of the blue line (S to P) and a flat mirror (2) and two dichroic plates (3) allow the superimposing of the three wavelengths. An acousto‐optical cell (4) deflects the parasitic wavelengths in a mask (5) and diffracts the three wavelengths RGB using three characteristic frequencies injected into the crystal. The spatial filter (6), composed of microscope objective (60×) and a small hole of 25 μm, is placed at the focal length of the achromatic lens (9) in order to illuminate the phenomenon with a parallel beam. On‐going, 50% of the light is returned towards the concave mirror (8) to form the three reference beams and 50% of the light passes through the test section (between (9) and (12) to form the measuring waves. The flat mirror (12) placed behind the test section (11) returns the beams in the beam splitter cube (7). 25% of the light focused on the diaphragm which is placed in front of the achromatic lens (13). It is the same for the 25% of the reference beam which is focused on the same diaphragm by the concave mirror (8).

Michelson digital holographic interferometer has been implemented around the ONERA wind tunnel and two optical tables isolate the optical set‐up from external vibrations. **Figure 4** shows

**Figure 3.** Michelson digital holographic interferometer.

294 Holographic Materials and Optical Systems

**Figure 4.** Micro‐fringes formation by the transparent object.

the generation of micro‐fringes used as spatial carrier frequencies.

**Figure 5.** 2D spectra computed from the reference and the measurement interferograms.

First, the phase maps are calculated from the three reference and three measurement spectra so that the modulo 2*π* phase difference maps shown in **Figure 6**. One can see that the structures of shocks and decompression appear in the jet. As the difference phase maps are computed modulo 2*π*, a phase unwrapping has to be conducted and the results given the unwrapped phase maps are also presented in **Figure 6**. At a pressure of 3 bar, we can note a phase variation of 12 rad.

**Figure 6.** Maps of RGB phase difference (modulo 2*π* and unwrapped), *P* = 3 bar.

Finally, the maps of light intensity and optical thickness difference are calculated from the phase difference maps. They are presented in **Figure 7** for pressures ranging from 2 to 5 bar.

**Figure 7.** Evolution of the luminous intensity and the optical thickness with the pressure.

Concerning the maps of the luminous intensity, they are corresponding to figures which will be obtained if a technique of image holographic interferometry using panchromatic plates has been used. Knowing the wavelength and the phase, the maps of optical thickness can be deduced. They are also presented in **Figure 7** from 2 to 5 bar. At 2 bar and in the middle of shock structures, the optical thickness varies up to 0.2 μm and at 5 bar, it varies up to 1μm.

#### **3.2. Three‐wavelength holographic interferometry using Wollaston prisms**

This part proposes an optical set‐up based on digital holographic interferometry using two widely shifting Wollaston prisms and a single crossing of the phenomenon. Each Wollaston prism is located at the focal point of 'Z' astigmatic optical set‐up. The second Wollaston is located in front of the camera and between the two sagittal and transverse focal lines so that a rotation around the optical axis generates interference micro‐fringes which are used as spatial carrier frequency.

#### *3.2.1. Definition of Wollaston prism characteristics*

modulo 2*π*, a phase unwrapping has to be conducted and the results given the unwrapped phase maps are also presented in **Figure 6**. At a pressure of 3 bar, we can note a phase variation

Finally, the maps of light intensity and optical thickness difference are calculated from the phase difference maps. They are presented in **Figure 7** for pressures ranging from 2 to 5 bar.

**Figure 6.** Maps of RGB phase difference (modulo 2*π* and unwrapped), *P* = 3 bar.

**Figure 7.** Evolution of the luminous intensity and the optical thickness with the pressure.

of 12 rad.

296 Holographic Materials and Optical Systems

Differential interferometry using Wollaston prism visualizes the light deviation of the refrac‐ tive index in a direction perpendicular to the direction of the interference fringes. Indeed, in the case of quartz prism having a very weak pasting angle, the gradient of the refractive index is measured because the birefringence angle is very weak and the distance between the two interfering beams is of the order of a few tenths of a millimetre or a few millimetres in the test section. Data integration is necessary to obtain the absolute refractive index. To avoid this integration, it was decided to manufacture two Wollaston prisms having a very high birefrin‐ gence angle so that the distance between the two interfering beams is greater than the dimen‐ sion of the measuring field (jet size). The interference measurement will be made between a beam which does not pass through the phenomenon (reference beam) and one which crosses the phenomenon under analysis. If ( − ) is the crystal birefringence and , the pasting angle of prisms, the birefringence angle can be expressed using the following equation:

$$
\varepsilon = \varepsilon \left( \lambda \right) = \mathfrak{Z} \left( \mathfrak{n}\_{\varepsilon} - \mathfrak{n}\_{o} \right) \mathfrak{t} \boxminus \mathfrak{n} \left( \alpha \right) \tag{1}
$$

If a very high birefringence angle is sought, the pasting angle and the crystal birefringence have to be as high as possible. To remember, the Δ birefringence values for quartz and calcite are, respectively, equal to +0.009 and −0.172. It can be seen that calcite birefringence is basically twenty times greater than quartz birefringence.

If is the radius curvature of the spherical mirror used in the optical set‐up, the shift between the two interfering beams can be written as:

$$
\varepsilon \, d\boldsymbol{x} = \varepsilon \boldsymbol{\mathcal{R}} = \mathcal{Z}\boldsymbol{\mathcal{R}}(n\_{\boldsymbol{\theta}} - n\_{\boldsymbol{\alpha}})\tan(\boldsymbol{\alpha})\tag{2}
$$

Thus, for a spherical mirror of 400 mm in diameter and 4 m in the radius of curvature, has to be near to 200 mm. By choosing calcite crystal, the pasting angle can be found from the following relationship:

$$\alpha = \arctan\left(\frac{d\mathbf{x}}{2R\Delta n}\right) = \arctan\left(\frac{0.2}{8\times 0.172}\right) = 8.27^{\circ}\tag{3}$$

Calcite Wollaston prisms with 8° pasting angle have been manufactured.

#### *3.2.2. Optical set‐up with single crossing of the test section*

**Figure 8** shows the principle of Z optical set‐up using Wollaston prisms. Here also, three different DPSS lasers (red, green and blue) constitute the luminous light source and the optical set‐up uses two spherical mirrors, 250 mm in diameter and 2.5 m in radius of curvature.

**Figure 8.** Digital holographic interferometer using very large Wollaston prisms in 'Z' set‐up.

**Figure 9.** Astigmatism represented by sectional views and Wollaston prism in the front of the camera.

As all optical pieces are not exactly on the optical axis of spherical mirrors, we can observe astigmatism in the optical arrangement. The first prism located at the focal length of the first spherical mirror produces two optical rays which are returned by parallel light beams onto the second spherical mirror. This one refocuses the light beam into the second Wollaston prism which is mounted 'tumble' with the first one. An analyser located behind the second prism allows visualizing the interference fringes in colour. The image of the object under analysis is formed by a field lens placed in front of the 3CCD sensor. Here, the advantage of astigmatic set‐up is used because the focusing point in the front of the camera is not unique. **Figure 9** shows this particularity: the optical beams are focused on the two focal images successively separated by a few millimetres. The first one gives the tangential image encountered when the beam focuses in the horizontal plane, and the second one, called the sagittal image, is obtained when the beam focuses in the vertical plane.

Thus, for a spherical mirror of 400 mm in diameter and 4 m in the radius of curvature, has to be near to 200 mm. By choosing calcite crystal, the pasting angle can be found from the

> 0.2 arctan arctan 8.27 2 8 0.172 æö æ ö = = = ° ç÷ ç ÷ èø è ø

**Figure 8** shows the principle of Z optical set‐up using Wollaston prisms. Here also, three different DPSS lasers (red, green and blue) constitute the luminous light source and the optical set‐up uses two spherical mirrors, 250 mm in diameter and 2.5 m in radius of curvature.

*Rn x* (3)

Δ

Calcite Wollaston prisms with 8° pasting angle have been manufactured.

**Figure 8.** Digital holographic interferometer using very large Wollaston prisms in 'Z' set‐up.

**Figure 9.** Astigmatism represented by sectional views and Wollaston prism in the front of the camera.

*dx*

following relationship:

298 Holographic Materials and Optical Systems

*α*

*3.2.2. Optical set‐up with single crossing of the test section*

**Figure 10** shows, on the reception side, the different figures of interference observed when the second Wollaston prism is moved along the optical axis from the tangential image (TI) towards the sagittal image (SI). The interference fringes which were horizontal and much narrowed, spread. When the interference fringes spread again, we can observe a rotation of 90° by them to give a quasi‐uniform vertical background colour, at half distance between the tangential and sagittal images. Then, they continue to rotate by 90° up to the sagittal image and they narrow to become horizontal. Interference fringes stay horizontal above the sagittal image and narrow more and more. Knowing this property, we can adjust the spatial carrier frequency by the axial displacement of the prism for its amplitude and by rotating the prism for its orientation. In our tests, the Wollaston prism is located at half distance between the tangential and sagittal images, so that the interference fringes are generated in the same direction as the direction of the two interfering beams (vertical shift and vertical fringes). Gontier et al. [13] has widely described this feature. If the number of fringes in the visualized field has to be increased, the Wollaston prism has to be turned on itself in the plane perpendicular to the optical axis. **Figure 10** shows two positions of rotation of the Wollaston prism (20 and 45°) with a maximum number of fringes obtained for the rotation of 45°.

**Figure 10.** Evolution of interference fringes when the second Wollaston is moved from the sagittal image to the tangen‐ tial image.

#### *3.2.3. Results obtained*

First, **Figure 11** shows the interferograms for the reference and the measurement with an enlarged view near the injection. For a pressure of 4 bar, for instance, one can see the horizontal interference fringes disturbed by the flow. The interferograms of **Figure 10** also show that the field is reduced on the right and left sides: this is the result of the rotation of the Wollaston prism at return which has a limited size (15 mm2 ). The polarization fields which were com‐ pletely separated on the way interfere with each other as the prism placed in front of the camera is rotated. It is also noteworthy that the polarizer is rotated exactly to the same amount as the Wollaston prism. The tightening of the fringes is maximal when the prism is rotated by 45°.

**Figure 11.** Interference micro‐fringes recorded for the reference and the measurement, *P* = 4 bar.

Then, 2D fast Fourier transform is applied to filter the zero and ‐1 orders on the three channels for the reference and the measurement interferograms. In **Figure 12**, one sees that the different window filtering size can be taken on the three channels and that the reduced frequencies are equal to 0.12, 0.10 and 0.9 mm‐1 for the blue, green and red channels that correspond to resolution of 18.6 lines/mm, 15.5 lines/mm and 13.9 lines/mm. The spatial resolution used is lower than that used in the technique of Michelson interferometry.

**Figure 12.** 2D spectra computed on the three channels for the reference interferogram, *P* = 4 bar.

**Figure 13** shows the spectrum of the measurement for P = 3 bar, the modulo 2*π* phase map, the superimposition of the three red, green and blue luminous intensities deduced from the phase difference maps and also the optical thickness map computed from the phase difference map. These maps are concerning the red channel. Moreover, a deconvolution of the optical thickness maps based on the assumption that the jet is axisymmetric has been applied. Thus, it is possible to obtain the radial distribution of the refractive index and the density in the jet according to the relation proposed by Gladstone‐Dale. This method is widely described by Rodriguez et al. [14]. In the treatment process, the optical thickness of maps calculated for each jet pressure is split into two parts, on either side of the axis of symmetry of the jet. If the results found by both sides of the jet are identical to the symmetry axis, the assumption of the axial jet symmetry is verified and the results can be considered correct. In **Figure 12**, the radial gas density is presented at 3 bar, and the density values found on the axis are very close, the analysis being done on the right or on the left.

**Figure 13.** Analysis of the case for red channel *P* = 3 bar (from spectrum to gas density).

### **3.3. Digital holography without reference wave**

*3.2.3. Results obtained*

300 Holographic Materials and Optical Systems

prism at return which has a limited size (15 mm2

First, **Figure 11** shows the interferograms for the reference and the measurement with an enlarged view near the injection. For a pressure of 4 bar, for instance, one can see the horizontal interference fringes disturbed by the flow. The interferograms of **Figure 10** also show that the field is reduced on the right and left sides: this is the result of the rotation of the Wollaston

pletely separated on the way interfere with each other as the prism placed in front of the camera is rotated. It is also noteworthy that the polarizer is rotated exactly to the same amount as the Wollaston prism. The tightening of the fringes is maximal when the prism is rotated by 45°.

**Figure 11.** Interference micro‐fringes recorded for the reference and the measurement, *P* = 4 bar.

lower than that used in the technique of Michelson interferometry.

**Figure 12.** 2D spectra computed on the three channels for the reference interferogram, *P* = 4 bar.

Then, 2D fast Fourier transform is applied to filter the zero and ‐1 orders on the three channels for the reference and the measurement interferograms. In **Figure 12**, one sees that the different window filtering size can be taken on the three channels and that the reduced frequencies are equal to 0.12, 0.10 and 0.9 mm‐1 for the blue, green and red channels that correspond to resolution of 18.6 lines/mm, 15.5 lines/mm and 13.9 lines/mm. The spatial resolution used is

). The polarization fields which were com‐

Digital holography without reference wave allows quantitative phase imaging by using a high‐ resolution holographic grating for generating a four‐wave shearing interferogram. The high‐ resolution holographic grating is designed in a 'kite' configuration so as to avoid parasitic mixing of diffraction orders. The selection of six diffraction orders in the Fourier spectrum of the interferogram allows reconstructing phase gradients along specific directions. The spectral analysis yields the useful parameters of the reconstruction process. The derivative axes are exactly determined whatever the experimental configurations of the holographic grating. The integration of the derivative yields the phase and the optical thickness [12].

#### *3.3.1. Base of digital holography without reference*

**Figure 14** shows the principle of the hologram recording of pure phase modulation where an incident plane crosses the phenomenon under analysis. This wave, disturbed by the phenom‐ enon, is simultaneously diffracted in several directions by a diffraction grating operating in reflection. The different images diffracted by the grating interfere with each other at a distance of the diffraction plane.

**Figure 14.** Principle of self‐referenced digital holography by reflection.

The sensor therefore records a digital hologram produced by the coherent superimposition of all the diffraction orders. Let (′) the complex object wave front in the object plane, and <sup>=</sup> (r)exp( <sup>=</sup> −1 the wave front diffracted from the object plane to the record‐ ing plane (**r** is the vector of the Cartesian coordinates {*x, y*} in the plane perpendicular to the *z* direction). The reflection of the incident wave on the holographic grating produces a set of replicated waves, whose propagation direction is given by their wave vector = 2/ ( is the unit vector of the propagation direction). The amplitude of the diffracted wave front at the recording plane is expressed as:

$$O(r, k\_n) = A\_o(r - r\_n) \exp(ik\_n r + i\varphi\_o(r - r\_n))\tag{4}$$

In Eq. (4), is the spatial shift produced by propagation along distance from the holographic plane and in the direction of unit vector e*n*. Due to the holographic grating, *P* = 4 wave fronts (*n* varying from 1 to *P* = 4) are diffracted in different directions. The interferogram recorded in the sensor plane is written as:

$$H(r) = \sum\_{n=1}^{n=P} A\_O^2 \left( r - r\_n \right) + 2\Re \left\{ \sum\_{n=1}^{P-1} \sum\_{m=n+1}^P \mathcal{O}(r, k\_n) \mathcal{O}^\* \left( r, k\_m \right) \right\} \tag{5}$$

In Eq. (5), the first term is related to the zero order, and the last one is related to coherent cross‐ mixing between the P diffracted orders. The last term includes the useful data related to the phase at the object plane. Noting Δ the phase of cross‐mixing (, )\*(, ), we get:

enon, is simultaneously diffracted in several directions by a diffraction grating operating in reflection. The different images diffracted by the grating interfere with each other at a

The sensor therefore records a digital hologram produced by the coherent superimposition of all the diffraction orders. Let (′) the complex object wave front in the object plane, and <sup>=</sup> (r)exp( <sup>=</sup> −1 the wave front diffracted from the object plane to the record‐ ing plane (**r** is the vector of the Cartesian coordinates {*x, y*} in the plane perpendicular to the *z* direction). The reflection of the incident wave on the holographic grating produces a set of replicated waves, whose propagation direction is given by their wave vector = 2/ ( is the unit vector of the propagation direction). The amplitude of the diffracted wave front at

In Eq. (4), is the spatial shift produced by propagation along distance from the holographic plane and in the direction of unit vector e*n*. Due to the holographic grating, *P* = 4 wave fronts (*n* varying from 1 to *P* = 4) are diffracted in different directions. The interferogram recorded in

( ) ( ) ( ) <sup>1</sup>

ì ü ï ï = -+ í ý ï ï î þ å åå <sup>R</sup>

*O n n m*

2 (, ) ,

*\**

*Hr A r r Ork O rk* (5)

1 1 1

= = =+

*n n mn*

= -

*n P P P*

2

(4)

distance of the diffraction plane.

302 Holographic Materials and Optical Systems

**Figure 14.** Principle of self‐referenced digital holography by reflection.

the recording plane is expressed as:

the sensor plane is written as:

$$
\Delta \phi\_{nm}(r) = (k\_n - k\_m)r + \varphi\_\mathbb{Q} - r - r\_n) - \varphi\_\mathbb{Q}(r - r\_m) \tag{6}
$$

Eq. (6) can be simplified by considering spatial derivatives of the object phase according to:

$$
\Delta\varphi p\_{nm}(r) \cong (k\_n - k\_m)r + |\mathbf{s}\_{nm}|\frac{\partial\varphi\_t(r)}{\partial r\,\epsilon\_{nm}}\tag{7}
$$

In Eq. (7), is the unit vector of vector <sup>=</sup> − and − is the spatial carrier phase modulation. In the Fourier plane of the interferogram, the diffraction orders are sep‐ arated from the zero‐order diffraction and localized by the spatial frequency vector ( − )/2. Since they are localized at different spatial frequencies in the Fourier domain, they can be filtered in the same way as for off‐axis interferometry. Then, the spatial carrier frequency is removed. Note that spatial derivatives of the object phase are provided along an axis given by the scalar product . The scaling of each spatial derivative is related to . Thus, it appears that the method provides spatial derivatives with different sensitivi‐ ties, which depend on the geometric configuration of the diffraction orders. After extracting each useful order, the term ∂0/ ∂ is extracted from the argument of the inverse Fourier transform of the filtered interferogram spectrum. We note ∂0/ ∂ <sup>=</sup> ∂0/ ∂, <sup>=</sup> and = (1/) ∂0/ ∂, with *<sup>q</sup>* varying from <sup>1</sup> to *<sup>Q</sup>*, *<sup>Q</sup>* being the num‐ ber of really independent axis amongst the set of the useful orders included in the spectrum. So, the spatial variable is simply the direction of the axis along which the de‐ rivative operator operates. In the spatial frequency domain, this axis has a corresponding axis which will be referred as . Note that, from <sup>a</sup> computational point of view, and are 2D vectors. In case that exceeds <sup>2</sup>*π*, phase jumps occur and phase unwrapping is required. The scaling coefficient depends on the distance and on the couple of in‐ volved wave vectors (, ). Then, the spatial integration of terms has to be carried out to get the quantity . The wave‐front reconstruction problem has already been discussed by many authors and the methods are based on least‐square estimations or modal estima‐ tions. Note that in these works the wave front differences are defined at each point accord‐ ing to the sensor sampling geometry. In the modal approach, the wave front and its differences are expanded in a set of functions and the optimal expansion coefficients are determined (for example, using Zernike and Legendre polynomials). Here, the numerical method is based on the weighted least square criterion and according to Refs. [15, 16], quantity can be recovered.

#### *3.3.2. Design of diffraction grating*

First, a holographic grating is recorded with the optical set‐up shown in **Figure 15**. The holographic plates are single‐layer silver‐halide holographic plates from Gentet (http:// www.ultimate‐holography.com/). The spatial resolution reaches 7000 lines per mm and the holographic plate has been preferred to the photopolymer which has lower spatial resolution (1000–2000 lines per mm).

**Figure 15.** Optical set‐up defined for recording of reflection holographic grating.

A first beam splitter cube (80/20) forms a reference beam (blue beam) with 20% of the incident light and 80% of the light is used to form the four‐object beams. Plane waves are obtained with two lenses and two spatial filters. Object waves are generated by three‐beam splitter cubes (50/50) so that the luminous intensities of each beam (reference and object) are all equal to 20% of the initial laser power. After several reflections on flat mirrors (MP), four small mirrors located around a square (configuration no. 1) and around a kite (configuration no. 2) returns each object beams towards the holographic plate. As the reference wave and the four object waves are incoming on each side of the hologram, the hologram is recorded by reflection and the angle *θ* formed by the reference and the object waves are equal to 27 mrad for configuration no. 1. After four sequential exposures, one for each object wave, the holographic plate is developed and bleached. Then, the grating is inserted in the optical set‐up which is used for analysing the small supersonic jet from a nozzle. **Figure 16** shows the optical set‐up with a single crossing of the phenomenon at the distance δz between the sensor and the image of the high‐resolution holographic grating (HRHG).

Digital Holographic Interferometry for Analysing High‐Density Gradients in Fluid Mechanics http://dx.doi.org/10.5772/66111 305

**Figure 16.** Digital holographic interferometer without reference.

*3.3.2. Design of diffraction grating*

304 Holographic Materials and Optical Systems

(1000–2000 lines per mm).

First, a holographic grating is recorded with the optical set‐up shown in **Figure 15**. The holographic plates are single‐layer silver‐halide holographic plates from Gentet (http:// www.ultimate‐holography.com/). The spatial resolution reaches 7000 lines per mm and the holographic plate has been preferred to the photopolymer which has lower spatial resolution

**Figure 15.** Optical set‐up defined for recording of reflection holographic grating.

high‐resolution holographic grating (HRHG).

A first beam splitter cube (80/20) forms a reference beam (blue beam) with 20% of the incident light and 80% of the light is used to form the four‐object beams. Plane waves are obtained with two lenses and two spatial filters. Object waves are generated by three‐beam splitter cubes (50/50) so that the luminous intensities of each beam (reference and object) are all equal to 20% of the initial laser power. After several reflections on flat mirrors (MP), four small mirrors located around a square (configuration no. 1) and around a kite (configuration no. 2) returns each object beams towards the holographic plate. As the reference wave and the four object waves are incoming on each side of the hologram, the hologram is recorded by reflection and the angle *θ* formed by the reference and the object waves are equal to 27 mrad for configuration no. 1. After four sequential exposures, one for each object wave, the holographic plate is developed and bleached. Then, the grating is inserted in the optical set‐up which is used for analysing the small supersonic jet from a nozzle. **Figure 16** shows the optical set‐up with a single crossing of the phenomenon at the distance δz between the sensor and the image of the

An interferogram without flow and another with flow are directly recorded on the sensor (2000 × 1500 pixels, 365 mm2 ), then analysed in delayed time by 2D fast Fourier transform in order to localize the different interference orders. For configuration no. 1, **Figure 17** shows the location of four mirrors used at the recording (square). Order 1 results of interaction of the beams incoming from M1 and M2 mirrors and the order 1′ between M3 and M4. Similarly, order 2 is generated by the interference between the waves incoming from M1 and M3 mirrors and order 2′ those issuing from M2 and M4. Order 3 is only produced by the interference between M1 and M4 and order 4 between M2 and M3. For configuration no. 1, order 1 or 2 has been enlarged in order to show that order 1 and 1′ or 2 and 2′ are not quite superimposed.

**Figure 17.** Position of four mirrors (square and kite) at the recording, localization of different diffraction orders in 2D FFT plane and zoom of +1 order.

In fact, one obtains two spectral signatures slightly shifted. It is not possible to separate them by filtering and to reconstruct the phase derivative map induced by only order 1. For this reason, the four mirrors have been set at the four tops of a kite configuration (no. 2). The problem encountered with configuration no. 1 does not exist and 2D FFT shows that it is very easy to localize all the different diffraction orders (on right in **Figure 17**). There is no spectral overlap and all orders useful for the reconstruction are well separated. Each order of interfer‐ ence is then selected successively and separately with a circular mask (0.05 mm−1 radius). Then, the phase gradient of reference image is calculated for each order of interference (**Figure 18**). Subtracting the reference image to the measurement image gives a modulo 2*π* map of phase gradient difference caused by the flow. Then, difference phase maps have to be unwrapped and results are presented in **Figure 18** for the six diffracted orders and for a value of the generating pressure equal to 5 bar. Finally, knowing the phase gradient difference in the six directions, a reconstruction of the absolute phase map is possible. For this processing of integration calculation, one can use one of integration methods proposed in the literature, for instance that proposed by Frankot and Chellappa [15]. The modulus of complex amplitude and the optical phase of the diffracted field by the object can be combined for obtaining the complex wave diffracted in the sensor plane.

**Figure 18.** Recorded interferogram and gradient phase maps obtained for the six interference orders.

#### **3.4. Comparisons with digital holographic interferometry using a reference wave**

**Figure 19** shows results obtained with digital holographic interferometry without reference and two other results obtained with digital holographic interferometry using a reference wave. The comparison is made by taking into account the difference of optical thickness.

The scale level is basically the same for the three results (from 0 up to 1.2 μm), and **Figure 19** shows at 5 bar that they are in good agreement because spatial locations of the structures of compression and expansion waves are similarly positioned in the three measurements. From a point of view of easiness and accuracy of results, the optical set‐up without reference is complicated to implement and must achieve a kite‐type reference hologram. It is also difficult to obtain a hologram with high diffraction efficiency. In addition, the data obtained must be integrated, which cause a certain imprecision in the measurement. For the optical set‐up using Wollaston prisms, it is very bulky and costly because the Wollaston prisms of 'large field' type are expensive and difficult to manufacture. On the other hand, the measured values are absolute values as those obtained with Michelson interferometer that seems the least restrictive optical arrangement of the three set‐ups tested.

**Figure 19.** Comparison of experimental results obtained for three different interferometric techniques for a pressure at P=5 bar. (a) Without reference set‐up, (b) Michelson set‐up, (c) Wollaston set‐up.
