**5. Digital holography for visualizing inside strongly refracting transparent objects**

High‐density gradients can also exist inside strongly refracting objects and the visualization and the measurement of these phenomena remain an open problem. For example, objects as a glass ball, a light bulb, a glass container, a glass flask, etc. are not opaque but they are strongly refracting light and measuring inside is not straightforward. It follows that observing phe‐ nomena, such as refractive index variations, convection currents, or thermal gradients, occurring inside the object requires specific methods. Different experimental methods are usually used to investigate fluids and to visualize/measure dynamic flows [7, 8, 17]. Never‐ theless, these approaches are appropriated when the envelope including the flow is relatively smooth and transparent (i.e. not strongly refracting). A suitable experimental method should be able to exhibit the phase changes inside the object without suffering from any image distortion. The experimental approach described here is based on stochastic digital holography to investigate flows inside a strongly refracting envelope [18]. It leads to the measurement of the phase change inside the object, so as to get a quantitative measurement. Experimental results are provided in the case of the visualization of refractive index variations inside a light bulb and a comparison with image transmission and reflection holography is also provided.

#### **5.1. Proposed method**

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

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

Δ

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‐

*ρ ρ πeK* (8)

2 æ ö <sup>f</sup> = - ç ÷ è ø

0 0 1

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

*<sup>s</sup> ρ ρ λ*

phase shifts appear.

310 Holographic Materials and Optical Systems

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

Eq. (8) is obtained:

where

Dale constant: 296×10‐6.

The approach adapted to visualize inside a strongly refracting object is described in **Figure 26**. The sensor includes *N* × *M*=1920 × 1440 pixels with pitches *px* = *py* = 3.65 μm. The main feature is that a diffuser is used to illuminate the object to provide a back illumination. The set‐up exhibits some similarity compared to a classical transmission microscope, although, no mi‐ croscope objective is used and the illuminating wave is quite a speckled wave. A negative lens is put in front of the cube to virtually reduce the object imaged by this lens. This leads to a more compact system compared to the case where the lens is not used. For example, for an object size of 10–15 cm, the distance *d*0 in **Figure 26** has to be greater than 2 m. The use of the negative lens produces a smaller image of the image, whose position is close to the sensor [19, 20]. Thus, the distance that has to be used in the algorithm is *d*'0 (see **Figure 25**). The optimization of the off‐axis the set‐up has to follow the basic rules about the Shannon conditions [21].

**Figure 26.** Stochastic digital holographic set‐up.

In particular, the focal length of the lens has to be judiciously chosen. Especially, the criterion is the observation angle max from the sensor, which has to fulfil this condition:

$$\Theta\_{\text{max}} = \frac{\lambda}{\left(4 - 2\alpha\right) \pi \alpha \times \left(p\_{x'} p\_y\right)}\tag{9}$$

where α is the accepted tolerance in the superposition of the useful +1 order and the 0 order. Here, the diffuser (considered here as a 'stochastic screen') is sized 10 cm × 20 cm and a superposition tolerance of *α* = 20% is accepted. The evaluation of the focal length and distance leads to *d*0 = 800 mm, *d*'0 = 100 mm and *f*'=−150 mm. Holograms can be reconstructed by the adjustable magnification method described in [22] or by the discrete Fresnel transform [19– 21]. After reconstruction of the complex amplitude in the virtual object plane, an amplitude image and a phase image can be calculated. The amplitude image is related to the image of the object given by the lens, whereas the phase is useful to investigate refractive index variations, convection currents, or thermal gradients, occurring inside the object. For this, one has to evaluate the temporal phase difference at different instants. A quantitative measurement can be obtained after unwrapping the phase differences. Since the refractive index variations are encoded in the unwrapped phase, the use of the Gladstone‐Dale relation allows determining density variations.

#### **5.2. Proof of principle**

The proposed method has been applied to the visualization and analysis of light bulb during its lighting. This bulb was submitted to a current to produce light and holograms were recorded at different instants after its lighting. **Figure 27** shows the recorded hologram when the bulb is off (a) and when the bulb is lighting (b). The speckle nature of the hologram is clearly observed. **Figure 27c** shows the amplitude image obtained with the discrete Fresnel transform. The stochastic screen and the ampoule can be clearly seen so that the strand of the bulb. **Figure 27d** and **e** shows respectively the modulo 2*π* digital fringes and unwrapped phase differences obtained between two instants (light off and light on). One can note a very large amplitude variation since the phase values are in the range 10–50 rad (see the colour bar in **Figure 27e**). This measurement includes the contribution due to the refractive index change in the bulb and also a contribution due to the dilatation of the envelope and its refractive index variation due to the temperate increase inside the lamp (≈ 500°C). The 'numerical fringes' observed in **Figure 27d** exhibits the refractive index variations integrated in the glass container.

**Figure 27.** Quantitative measurement inside the bulb, (a) virtual phase extracted from numerical reconstruction (bulb off), (b) bulb lighting, (c) image amplitude of the strand, (d) modulo 2*π* phase computed from (a) and (b), (e) unwrap‐ ping of (d).

#### **5.3. Comparison with silver‐halide plate holographic interferometry**

**Figure 26.** Stochastic digital holographic set‐up.

312 Holographic Materials and Optical Systems

density variations.

**5.2. Proof of principle**

In particular, the focal length of the lens has to be judiciously chosen. Especially, the criterion

where α is the accepted tolerance in the superposition of the useful +1 order and the 0 order. Here, the diffuser (considered here as a 'stochastic screen') is sized 10 cm × 20 cm and a superposition tolerance of *α* = 20% is accepted. The evaluation of the focal length and distance leads to *d*0 = 800 mm, *d*'0 = 100 mm and *f*'=−150 mm. Holograms can be reconstructed by the adjustable magnification method described in [22] or by the discrete Fresnel transform [19– 21]. After reconstruction of the complex amplitude in the virtual object plane, an amplitude image and a phase image can be calculated. The amplitude image is related to the image of the object given by the lens, whereas the phase is useful to investigate refractive index variations, convection currents, or thermal gradients, occurring inside the object. For this, one has to evaluate the temporal phase difference at different instants. A quantitative measurement can be obtained after unwrapping the phase differences. Since the refractive index variations are encoded in the unwrapped phase, the use of the Gladstone‐Dale relation allows determining

The proposed method has been applied to the visualization and analysis of light bulb during its lighting. This bulb was submitted to a current to produce light and holograms were recorded at different instants after its lighting. **Figure 27** shows the recorded hologram when the bulb is off (a) and when the bulb is lighting (b). The speckle nature of the hologram is clearly observed. **Figure 27c** shows the amplitude image obtained with the discrete Fresnel transform. The stochastic screen and the ampoule can be clearly seen so that the strand of the bulb. **Figure 27d** and **e** shows respectively the modulo 2*π* digital fringes and unwrapped phase differences obtained between two instants (light off and light on). One can note a very large

*α pp* (9)

is the observation angle max from the sensor, which has to fulfil this condition:

*<sup>λ</sup> <sup>θ</sup>*

( ) ( ) max 42 , max <sup>=</sup> - *x y*

> In order to check for the quality of the results obtained with the proposed method, the results obtained were compared with analogue image‐holography [23]. The two possible set‐ups are described in **Figure 28** and can be either transmission or reflection holographic interferometry. **Figure 28a** shows the transmission holography mode and **Figure 28b** that for reflection holography. Note that the set‐ups require the use of photographic plates and that the diffuser is also used to get a stochastic screen to illuminate the object. The process is as follows: record a transmission or reflection hologram, apply the chemical treatment to the plate to develop and bleach, dry the plate, put the holographic plate in the set‐up anew (exactly at the same location), at this step the holographic image of the ampoule is observable, adjust the camera lens to produce a focused image, then record real‐time interferences between the initial bulb and that currently submitted to the current. Note that only the luminous intensity of interfer‐ ence fringes can be obtained, and not the phase image as it is the case for the digital holographic approach.

**Figure 28.** Image transmission holography (a) and image reflection holography (b).

**Figure 29** shows a comparison between results obtained with digital holography and those obtained with image holography. **Figure 29a** shows the image obtained with the amplitude and phase change measured by digital holography, after calculating the intensity = (1 + cos()), where is the phase change and *A* the amplitude image. **Figure 29b** shows the interference fringes obtained with the set‐up of **Figures 28a** and **29c** shows those obtained with the set‐up of **Figure 28b**. A very good agreement can be observed. Furthermore, the image quality given by each method can be appreciated. Image holography provides the best spatial resolution: the strand of the lamp can be clearly seen in **Figure 29b** and **c**. However, digital holography is more flexible since no chemical processing is required and a phase image can be obtained.

**Figure 29.** Comparison between intensity of fringes (a) fringes calculated with digital holography, (b) fringes obtained with transmission holography and (c) fringes obtained with reflection holography.
