**4. Digital holography for analysing unsteady wake flows**

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

**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**

The comparison is made by taking into account the difference of optical thickness.

**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 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

complex wave diffracted in the sensor plane.

306 Holographic Materials and Optical Systems

The unsteady wake flows generated in wind tunnel present a large scale of variations in refractive index from subsonic to supersonic domain. The feasibility of three‐wavelength digital holographic interferometry has been shown on two‐dimensional unsteady flows and the time evolution of the gas density field has been determined on the subsonic near wake flow downstream a circular cylinder [8]. But, when the flow regime reaches the transonic or supersonic domain, problems appear because refractive index gradients become very strong and a shadow effect superimposes to the micro‐fringes of interferences. Moreover, the displacement of vortices is very high compared to the exposure time (300 ns given by the acousto‐optical cell, **Figure 3**) what leads to blurred zones in interferograms and limits the interferograms analysis (**Figure 20**).

**Figure 20.** Highlighting of blurred areas and shadow effect—Mach 0.73.

#### **4.1. Michelson holographic interferometry**

At first, an ORCA Flash 2.8 camera from Hamamatsu with a matrix of 1920 × 1440 pixels, 3.65 μm2 , has been bought to increase the spatial resolution and, for the temporal resolution, the continuous laser light source of the interferometer has been replaced by a Quanta‐Ray pulsed laser, Model Lab 170‐10 Hz from Spectra‐Physics. This laser is injected through a 1064 nm laser diode and outputs a wavelength at 1064 nm having 3 m in coherence length (TEM00 mode). Here, the first harmonic is used (532 nm) and delivers about 400 mJ in 8 ns. The beam diameter is about 8–9 mm. **Figure 21** shows how the laser was installed in Michelson interferometer presented in **Figure 3**. The output beam is equipped with two sets '*λ*/2‐polarizing beam splitter cube' to significantly reduce the laser energy sent to the camera. It is seen in **Figure 21** that the beam splitter cube forms the reference wave which is reflected by the concave mirror on the camera and the measurement wave which passes through the test section. The second achromatic lens, 70 mm in focal length of 70 mm yields the magnification of the image on the CCD.

**Figure 21.** Digital Michelson holographic interferometer using a pulsed laser.

If *L*1 is the distance between the beam splitter cube and the concave mirror, and *L*2 the distance between the same beam splitter cube and the flat mirror located behind the test section, the laser coherence length must be greater than twice the difference (*L*2 − *L*1) for the interference fringes may be formed on the CCD. This difference is here of the order of 2.5 m.

**Figure 22** shows an interferogram of unsteady wake flow around a circular cylinder at Mach 0.73 with Michelson interferometer, the 2D FFT spectrum with the +1 order used to reconstruct the map of the modulo 2*π* phase difference. The interferogram exhibits a good quality indicating that vortex structures and small shock waves are well frozen. But, in the modulo 2*π* phase difference map shown on the right of **Figure 22**, phase jumps are still present. They are surrounded by black ellipses on the figure and they will cause phase shifts during the unwrapping of the modulo 2*π* phase map. To decrease the sensitivity of the measurement by a factor of 2, the optical bench has been modified to create a Mach‐Zehnder type bench.

**Figure 22.** Interferogram analysis at Mach 0.73—residual phase shifts.

#### **4.2. Mach‐Zehnder holographic interferometry**

**4.1. Michelson holographic interferometry**

308 Holographic Materials and Optical Systems

**Figure 21.** Digital Michelson holographic interferometer using a pulsed laser.

If *L*1 is the distance between the beam splitter cube and the concave mirror, and *L*2 the distance between the same beam splitter cube and the flat mirror located behind the test section, the laser coherence length must be greater than twice the difference (*L*2 − *L*1) for the interference

**Figure 22** shows an interferogram of unsteady wake flow around a circular cylinder at Mach 0.73 with Michelson interferometer, the 2D FFT spectrum with the +1 order used to reconstruct the map of the modulo 2*π* phase difference. The interferogram exhibits a good quality indicating that vortex structures and small shock waves are well frozen. But, in the modulo 2*π* phase difference map shown on the right of **Figure 22**, phase jumps are still present. They are surrounded by black ellipses on the figure and they will cause phase shifts during the unwrapping of the modulo 2*π* phase map. To decrease the sensitivity of the measurement by a factor of 2, the optical bench has been modified to create a Mach‐Zehnder type bench.

fringes may be formed on the CCD. This difference is here of the order of 2.5 m.

μm2

the CCD.

At first, an ORCA Flash 2.8 camera from Hamamatsu with a matrix of 1920 × 1440 pixels, 3.65

, has been bought to increase the spatial resolution and, for the temporal resolution, the continuous laser light source of the interferometer has been replaced by a Quanta‐Ray pulsed laser, Model Lab 170‐10 Hz from Spectra‐Physics. This laser is injected through a 1064 nm laser diode and outputs a wavelength at 1064 nm having 3 m in coherence length (TEM00 mode). Here, the first harmonic is used (532 nm) and delivers about 400 mJ in 8 ns. The beam diameter is about 8–9 mm. **Figure 21** shows how the laser was installed in Michelson interferometer presented in **Figure 3**. The output beam is equipped with two sets '*λ*/2‐polarizing beam splitter cube' to significantly reduce the laser energy sent to the camera. It is seen in **Figure 21** that the beam splitter cube forms the reference wave which is reflected by the concave mirror on the camera and the measurement wave which passes through the test section. The second achromatic lens, 70 mm in focal length of 70 mm yields the magnification of the image on

> In Mach‐Zehnder interferometer, shown in **Figure 23**, the measuring beam crosses only once the test section and the reference beam passes outside the test section so that the sensitivity is decreased by a factor 2.

**Figure 23.** Digital Mach‐Zehnder holographic interferometer using a pulsed laser.

In this optical set‐up, the reference beam is reflected successively by several little flat mirrors. That produces a polarization rotation of the reference wave which must be corrected by inserting a *λ*/2 plate in front of the spatial filter of the reference wave. The contrast of the interference fringes can thus be optimized on the interferogram. The cylinder is equipped with an unsteady pressure transducer at a 90° azimuth to the flow axis in order to correlate the laser pulse with the signal of unsteady pressure. In this manner, one period of the phenomenon can be sampled by 20° step with several different tests. First, in the enlarged part of reference and measurement interferograms of **Figure 24**, one can see the straight interference micro‐fringes distorted by the shear layer incoming from the upper of the cylinder. 2D FFT spectra show that the spatial carrier frequency (vertical fringes) is localized on horizontal axis (order +1 of hologram). After applying a spatial filter around the first order and subtracting the reference

to the measurement phase map, one obtains the modulo 2*π* phase difference map where no phase shifts appear.

**Figure 24.** Interferograms analysis at Mach 0.73.

Then, an unwrapping has to be applied to obtain the phase difference map *Δθ* and the gas density field *ρ*/*ρ*0 presented in **Figure 25** is deduced from the Gladstone‐Dale relationship and Eq. (8) is obtained:

$$\frac{\rho}{\rho\_0} = 1 - \left(\frac{\rho\_s}{\rho\_0} \frac{\lambda \Delta \phi}{2\pi eK}\right) \tag{8}$$

**Figure 25.** Instantaneous and averaged gas density fields (*ρ/ρo*)—Mach 0.73.

where is the standard gas density computed at 1 atm and 0°C, 0 the stagnation gas density, *λ* the wavelength of the interferometer, *e* the width of the test section and K the Gladstone‐ Dale constant: 296×10‐6.

The instantaneous interferogram of **Figure 25** shows that shock waves emitted by the vortices of the vortex shading are very well analysed (no phase shift) and the averaged gas density field exhibits a strong decreasing of the gas density just behind the cylinder up to 90% of 0.

For information, the shadow effect can be easily reduced. If the beam of the reference arm is blocked (see **Figure 23**), the Mach‐Zehnder interferometer looks like to shadowgraph optical set‐up. In these conditions, the sensor can be adjusted along the optical axis to focus and image the middle of the test section on the sensor. As this condition is reached, the shadow effect is minimized.
