## Abstract

this paper discusses on the influence of decorrelation noise induced by quantization and shot-noise when recording digital holograms at very high frame rate. A criterion based on the coherence factor of the hologram phase difference is proposed. The main parameters of interest are the ratio between the reference and the object waves and the sensor dynamics, depending on the photo-electron capacity of pixels. The study is based on a full numerical simulation of the holographic process, which provides useful rules. This leads to define the optimal conditions for recording at very-high frame rate with minimization of the decorrelation noise. Experimental results obtained with frame rate at 50kHz confirm the proposed approach.

© 2015 Optical Society of America

## 1. Introduction

Digital holography is a very powerful method for imaging and metrology [1,2]. Holographic phase imaging measures the optical path length map associated with transparent specimens (transmission illumination) or opaque surfaces (reflection illumination) and translates this data into relevant information. The measured field of interest is related to a wrapped modulo 2π phase map (also called phase fringe pattern). From a theoretical point of view, optimization rules have been established by taking into account many factors such as the pixel pitch of the sensor, the object-to-sensor distance, the numerical reconstruction [3–7], the properties of the reconstruction algorithm [8,9] or the sampling requirements [10–13]. In 2005, Mills and Yamaguchi evaluated the quality of the reconstructed images for different quantization levels [14]. They concluded that the object phase distribution does not substantially influence image appearance above the threshold of 4-bit quantization, and that, at bit depths lower than 4 bits, random phase modulation introduces a speckle noise effect. More recently, in 2011, Pandley and Hennelly studied the influence of quantization error in the recorded holograms on the fidelity of both the intensity and phase of the reconstructed image [15]. They have shown that quantization error is introduced as uniformly distributed additive noise in the recording plane and this manifests itself as a complex noise in the reconstruction plane with Gaussian distributed real and imaginary parts, Rayleigh distributed amplitude and uniformly distributed phase. These previous works were focused mainly on the quality of the image amplitude reconstructed from the digitally recorded hologram. In addition, the number of photo-electrons generated in each pixel strongly influences the speckle decorrelation in the phase. The quantum nature of light intervenes with the photon noise. Gross's group studied the influence of the noise sources in heterodyne holography [16–19]. They proposed a hybrid digital processing combining the phase shift method and the spatial Fourier transform in order to reduce artifacts due to parasitic orders and phase shift non linearity. Furthermore, they demonstrated that the ultimate noise in amplitude reconstruction was dominated by the photon noise.

From a practical point of view photon noise and quantization noise are mixed during the recording process and both have to be taken into account. In this paper, we are interested in the combined effect of quantization and shot-noise on the phase measurement. This is motivated by several reasons that are related to recording of holograms at a very high frame rate (up to 100kHz). As a general rule, in “normal conditions”, the exposure time, the ratio between the reference and the object waves, and the digitization of the holograms, are perfectly well controlled. This means, recordings are carried out with maximum occupation of the bit depth and the quantization effects are often quite irrelevant. The shot-noise is then related to the number of photo electrons per pixels and the signal-to-noise ratio can be very high if each pixel includes sufficient electron capacity [17,19]. However, ideal conditions are not always obtained when dealing with dynamic objects for which high-speed recording is required. Especially, the bit depth and the dynamics of the sensor may be not fully occupied, and noise may be included in the reconstructed data, both in the phase and the amplitude.

In this paper, we aim at considering the influence of quantization and shot-noise on quantitative phase measurement when the recording conditions are severe and performed at very high frame rate. Quantization and shot-noise induce a decorrelation phase noise when evaluating the phase difference between two recorded holograms. In order to quantify it, a criterion based on the coherence factor of the hologram phase is proposed. This leads to define the optimal conditions for recording at very-high frame rate with minimization of decorrelation noise. The study is based on a full numerical simulation of the holographic process. Experimental results obtained with a frame rate at 50kHz confirm these rules.

This paper is organized as follows; in section 2, we describe the theoretical basics and the phase decorrelation noise in the phase measurement; section 3 discusses on the particularity of high-speed recording of digital holograms. In section 4, the numerical simulation is used to establish criterion so as to evaluate the influence of combined quantization-shot-noise-induced decorrelation noise in the phase data. Section 5 discusses on experimental set-up and results. Finally, section 6 draws conclusions and perspectives to the study.

## 2. Theoretical basics

Let us consider the recording of digital holograms in a smooth-plane reference wave scheme. In the recording plane, at which an object wave interferes with a reference wave, the hologram is expressed as [1,2]:

where *R*(*x*,*y*) = *a _{R}*exp[2

*i*π(

*u*

_{0}

*x*+

*v*

_{0}

*y*)] is the reference wave with spatial frequencies {

*u*

_{0},

*v*

_{0}}, and

*O*is the wave diffracted in the recording plane by the object located at distance

*d*

_{0}from this plane. In the Fresnel approximations, we have [20] (

*i*= √−1):

where *A*(*X*,*Y*) = *A*_{0}(*X*,*Y*)exp[*iψ*_{0}(*X*,*Y*)] is the object wave front at the object plane and λ is the wavelength. Considering the discrete recording, the reconstruction of the object field at distance −*d*_{0} from the recording plane is given by the discrete Fresnel transform [1–3,5,8]:

with (*lp _{x}*,

*kp*) the pixel coordinate at which the digital hologram is recorded (

_{y}*l*,

*k*: integers;

*p*,

_{x}*p*: pixel pitches). From the numerical computation, the amplitude and phase of the diffracted field can be evaluated. In the case where the object under interest is “smooth” (e.g. biological samples, transparent fluids), the phase is not speckled, whereas for the case of a rough object surface, the phase is speckled. Quantitative measurement is carried out by a subtraction between the object phase and a “reference” phase. The phase difference is then obtained modulo 2π and requires phase unwrapping. In the case of digital holographic microscopy the reference phase may be the phase extracted without any object in the measurement beam. For the case of an object having a rough surface, the reference is that of the object at the “reference” state. When dealing with dynamic measurements, the subtraction is performed between phases calculated at each instant of the hologram sequence [21,22], the phase produced at each instant by the object being time-varying. In this paper we note

_{y}*ψ*

_{1}and

*ψ*

_{2}the two speckled phases used for the subtraction and extracted from the reconstruction (Eq. (3)) at two different instants

*t*

_{1}and

*t*

_{2}. The discussion in this paper is mainly related to speckle-type object wave fronts.

A limitation of this approach is related to the speckle decorrelation occurring when the object is time-varying (vibrating object for example). This decorrelation adds a high spatial frequency noise to the useful signal. The raw phase difference is then not directly suitable for visualization or comparison with some theoretical/simulation results. Smoothing methods based on sin-cos filtering [23], wavelets [24] of nonlocal means filtering [25] may be also used, resulting in an increase of the signal-to-noise ratio of the amplitude and phase difference map. Speckle correlation has been theoretically studied by several authors [26–28]. For studying speckle phase decorrelation, the second-order statistical description is of interest. So, the description of the correlation property is related to the second-order probability density function of the phase [29–31]. Note *ε* the noise induced by the speckle decorrelation between two object fields reconstructed for two different instants and Δ*φ* the phase change due to the time-varying dynamic object. Then *ψ*_{2} = *ψ*_{1} + *ε* + Δ*φ*, Δ*φ* being considered as a deterministic variable. The probability density function of phase noise *ε* depends on the modulus of the complex coherence factor |*μ*| between the two speckle fields. With *β* = |*μ*|cos(*ε*), the second-order probability density of the phase noise *ε* is given by [29–31]:

This probability density is centered (mean value at 0) and its width depends on |*μ*|. Practically the width has to be small enough in order to be able to unwrap the phase difference. Curves of Eq. (4) versus |*μ*| can be found in [31] on page 45, Fig. 2.18 (note that Fig. 3(c), Fig. 4(c), Fig. 4(f), Fig. 5(c) and Fig. 5(f) show some of such theoretical curves with red color). Equation (4) can be used as a pertinent indicator so as to compare the decorrelation sensitivity of digital holography, the correlation factor |*μ*| being a quality marker extracted from experimental data. When the correlation between the phases is high, the correlation factor tends toward 1, and the decorrelation noise probability density is very narrow, whereas when the correlation between the phases is weak, the correlation factor decreases below 0.7, and the decorrelation noise probability density is spread out.

The decorrelation noise that can be measured in the calculated phase difference may have several origins: modifications at the object surface (due to loading such as mechanical, vibrations, heating, pneumatic,…), laser wavelength change between exposures (e.g. surface shape measurement), reduced number of recording pixels (low resolution of the sensor), defocusing of the reconstructed image (the reconstruction distance is “not good”), saturation of the recorded hologram, quantization effect due to a low number of useful bits or shot-noise if the number of photo-electrons is too low. Although these origins are from different physical phenomena, their consequence is at the same level: decorrelation noise is added to the phase difference and the correlation factor |*μ*| can be used as a quality marker to qualify the decorrelation noise. In this paper, we are interested in the noise induced by recordings at very high-speed; the next section discusses the requirements of such recordings.

## 3. Specificities of very high-speed recordings

There are specificities of very high-speed recording that must be pointed out. Indeed, when considering a specimen with a rough surface (typically a non cooperative target), the photometric efficiency is very low since a few number of photons are really contributing to the object wave in the recording plane. The illumination produced by the object wave is inversely proportional to the square of *d*_{0} in the Fresnel configuration. So, bigger the object, longer the distance and weaker is the light flux from the object surface. Note that adding an imaging lens is not useful in off-axis digital holography [32]. Thanks to the coherent mixing by heterodyning with the reference wave, the object wave *O* is amplified by the reference wave *R*, because of the term *R*O* included in the recorded hologram (see the third term of Eq. (1)). So, a weak object wave may be balanced by a strong reference wave, if |*R*|^{2}>>|*O*|^{2}. In addition, the reference wave is directly impacting the sensor and this makes it easier to get large amount of photons. However, when recording at high frame rate, the histogram of the digital hologram tends to be narrower than if |*R*|^{2} = |*O*|^{2}, and the effective number of quantization bits is then low. The consequence is that this adds speckle decorrelation in the phase difference measurement. Note that the increase of the exposure time to capture more photons from the object wave constitutes a limited alternative since dynamic phenomena require high-speed recording. Thus, the exposure time can be equal, at maximum, to the inverse of the frame rate. In addition, the exposure time has to be smaller as possible to reduce the effect of time-averaging, which leads to phase distortion [22]. Another possibility is to dramatically increase the laser power but this has two counterparties: when using pulsed lasers the pulse repetition is not compatible with high pulse energy and when using a continuous wave (cw) laser the power would not be increased indefinitely on user demand.

So, in the case of high-speed recording, the experimental conditions strongly influence the decorrelation noise in the phase map. In order to investigate this influence, the parameters that have to be taken into account are: the ratio between |*R*|^{2} and |*O*|^{2}, and the used sensor dynamics. The choice of these two parameters can be also understood from the experimental point of view. In any experiments, the ratio between |*R*|^{2} and |*O*|^{2} can be easily controlled by simply measuring the light power. In addition, the used sensor dynamics is simply evaluated by considering the histogram peak of each wave. In this paper we note *Rc* the ratio between |*R*|^{2} and |*O*|^{2}, defined with Eq. (5):

and we note *α* (given in %) the used sensor dynamics. This parameter is closely related to the number of photo-electrons per pixel. Especially, we refer *α* to the photo-electrons provided by the reference wave according to Eq. (6):

where *N _{pe}* is the maximum number of photo-electrons of the pixels of the sensor (

*N*= 16000 for the sensor in this paper).

_{pe}When increasing parameter *Rc*, the histogram of the hologram tends to be narrow and thus the quantization dynamics is low, but this is compatible with very short exposure time. At contrary, if *Rc* is decreased, the reference wave tends to be equal to the object wave, the histogram is spread out, the quantization dynamics is high, thus diminishing the induced decorrelation noise. However, this situation is not compatible with short exposure time, since the object may be not a cooperative target. In order to evaluate how these parameters influence the quality of the phase measurement, section 4 discusses on a full numerical simulation of the digital holographic process.

## 4. Numerical approach

In order to evaluate the influence of the above mentioned critical parameters, a numerical simulation was developed by taking into account the full acquisition and reconstruction process, including the dynamic object excitation, optical wave propagation, interference recording and processing, and the final quantitative phase evaluation. The simulation starts with a known random surface in the object plane (which is the same at *t*_{1} and *t*_{2}), and with the mathematical modeling for the vibration field at *t*_{1} and *t*_{2} [22]. We set the object is illuminated with a uniform laser beam. The phase of the complex object field at *t*_{1}, *t*_{2} includes the random phase due to the random surface and the optical phase due to the vibration. The complex field at *t*_{1}, *t*_{2} is then propagated to the sensor plane using the discrete Fresnel transform. The diffracted fields are mixed with the reference wave to produce the holograms, and they are quantized with 8 bits. Then, the reconstruction process starts by computing the complex amplitude fields at the object plane using the discrete Fresnel transform anew. From the complex fields at the calculated object plane, the optical phases are estimated and the phase difference between phases at *t*_{1} and *t*_{2} is calculated. After recovering the phase differences between *t*_{1} and *t*_{2}, the error between the initial input phase difference and the one obtained with the simulation can be evaluated. This error is the key to highlight the critical parameters of the experimental set-up.

The dynamic object was chosen to be a simply supported rectangular aluminum plate (sized *L _{x}* ×

*L*, = 0.3m × 0.8m,

_{y}*h*= 5mm) submitted to a local impact. This choice of structure and boundary conditions was motivated by the fact that simply supported rectangular plates are structures having vibration modes with analytic solutions [33–35]. The impact generates a transient flexural vibration of the plate. This transient vibration includes a lot of modes, giving a temporal bandwidth to the dynamic phase encoded in the digital hologram. For the simulation, eigen frequencies of the plate modes are in the range [0Hz,20kHz], allowing precise synthesis of the transient vibration. The physical parameters required for simulating the dynamic optical phase are the applied force (Dirac impulsion of unitary amplitude

*F*

_{0}= 1N), the mass density

*ρ*= 2.6989g·cm

^{−3}, the Young modulus

*E*= 70Gpa, the Poisson ratio

*ν*= 0.33. Modal damping coefficients are supposed to be equal to

*η*= 0.5 × 10

^{−3}.

The frame rate is adjusted at 50kHz and the hologam includes *M* × *N* = 512 × 512 pixels, with 8 bits digitization, a full pixel capacity at *N _{pe}* = 16000 photo-electrons (data for a comercial high-speed sensor), the power ratio

*Rc*varies from 0.1 to 10000, and the percentage of the used dynamics varies from

*α*= 0% (no photons on the reference beam) to

*α*= 100% (the reference beam provides a number of photo-electrons close to the sensor saturation). In the simulation, the number of photo-electrons of the reference beam is adjusted, and then the amplitude of the object beam is calculated using the ratio

*Rc*. Note that adjusting

*α*in the simulation is also equivalent to adjust the exposure time in the experimental set-up. The advantage of adjusting

*α*instead of the exposure time is that problems related to time-averaging are by-passed [22]. The analysis can then be focused on the mixed quantization-shot-noise effects which are not mixed with time-averaging effects.

The photon noise is taken into account through a random Gaussian noise added to the hologram before quantization and whose standard deviation is √*H*. Generally the photon noise is described by Poisson statistics; but for numbers of photo-electrons greater than 10, the law can be approximated by a Gaussian distribution. The interferogram seen by the sensor takes the form of Eq. (7):

where *H _{n}* defines the hologram impacted by shot-noise and

*randn*means “normal random Gaussian noise” (i.e. centered with standard deviation equal to 1). Then, this hologram is quantized with 8 bits.

Two holograms were calculated, the first corresponds to instant *t*_{1} = 80ms and the second one to *t*_{2} = 80.01ms after the shock by the Dirac impulsion. In this way, we get two optical phases for two different states of the vibrating plate, including a low number of resolved 2*π* phase jumps. The deformation of the plate between these two instants induces a natural decorrelation in the computed phase difference. However, we are interested in evaluating the noise induced by recording conditions. So, adjusting parameters *Rc* and α yields different amounts of decorrelation noise, which is qualified by the coherence factor |*μ*|.

From the simulated digital holograms, the discrete Fresnel transform can be applied to recover both the amplitude and phase at the object plane. For each computed instant, the simulated measured phase difference can be calculated and is then subtracted from the exact initial phase difference to get the phase error. This phase error includes the decorrelation noise, the amount of which being closely related to the recording conditions. From the phase error, the histogram of the data is calculated and then the probability density is estimated. This probability density can be fitted with Eq. (4), and the coherence factor |*μ*| can be determined.

Figure 1(a) shows the results obtained from the simulation for the evaluation of the coherence factor. The coherence factor (from 0 to 1) is shown according to the ratio *Rc* and the dynamics *α*. The higher the value of |*μ*|, the lower is the noise due to quantization. Figure 1(a) exhibits the area that gives the less sensitivity to noise of the recording, according to the recording conditions given by the couple of values (*α*, *Rc*). From an experimental point of view, the phase map can be processed if the decorrelation noise is not too high. We consider that a minimum of |*μ*| = 0.90 is acceptable to minimize the combined effect of quantization-shot-noise decorrelation in the modulo 2π phase data. In Fig. 1(a), an optimal zone with coherence factor higher than 0.90 is clearly distinguishable and given by the dark red area. By applying a threshold on |*μ*| in the data, the optimal area can be identified and is shown in Fig. 1(b) as a binary picture (0 or 1). The white zone is related to |*μ*|>0.90 whereas the dark one corresponds to |*μ*|<0.90. So, the results of the simulation can be used to establish rules to optimize the experimental recording: a phase map of “correct quality” may be obtained from the high-speed recording if the experimental parameters (*Rc*,*α*) are adjusted in the white area in Fig. 1(b). In this area, the combined effect of quantization noise and shot-noise is reduced. In Fig. 1(a) and Fig. 1(b) are also plotted lines for |*O*|^{2} = {1,10,100,1000,10000} photo electrons (dashed yellow lines). As can be seen, when the object wave is two strong (i.e. >10000 photo-electrons) or two weak (∼few photo-electrons), then the coherence factor is decreased so that the phase may be not suitable.

In order to illustrate the quality of the phase map according to the decorrelation noise level, the probability density of noise and the value of the coherence factor, Fig. 2 shows data extracted from the simulation. Figure 2(a) exhibits modulo 2π simulated phase due to the vibrating plate between the two instants (reference phase). Figure 2(b) shows the reconstructed phase difference from the full data processing for the case *Rc* = 1000 and *α* = 5%, which corresponds to point n°2 in Fig. 1(b). Figure 2(c) exhibits the phase map for *Rc* = 300 and *α* = 60%, for point n°5 in Fig. 1(b). As can be observed, the phase map in Fig. 2(b) is quite noisy, whereas that in Fig. 2(c) is highly contrasted with low noise. Figure 3(a) shows the noise map extracted from the difference between the calculated phase map and the reference one for point n°2. The coherence factor is estimated to |*μ*| = 0.501. Figure 3(b) shows the phase noise map for point n°5, leading to |*μ*| = 0.993. The probability densities of the decorrelation noise for both phase data are plotted in Fig. 3(c). The very good agreement between estimated probability density and fitting with Eq. (4) can be seen.

Figures 4 and Fig. 5 show couples of results from the simulation. The 4 corresponding points are plotted in Fig. 1(b) as red spots and numbered 1,3 and 4,6 (complementary to points n°1 and n°5). Figures 4(a), Fig. 4(b) and Fig. 4(c) show the case *Rc* = 1 and *α* = 10%, corresponding to point n°1 in Fig. 1(b), with respectively, the probability density of the quantization gray levels of the digital hologram, the phase difference map between the two instants, and the probability density of the phase noise (extracted from simulation data and fitted with Eq. (4)). Figures 4(d), Fig. 4(e) and Fig. 4(f) show the case *Rc* = 8500 and *α* = 40% (point n°3). Figures 5(a), Fig. 5(b) and Fig. 5(c) show the case *Rc* = 60 and *α* = 20% (point n°4), and Fig. 5(d), Fig. 5(e) and Fig. 5(f) show the case *Rc* = 950 and *α* = 80% (point n°6). Data points n°4 and 6 are localized in the optimal zone in Fig. 1(b). Figure 5 demonstrates that the quality of the phase extracted from the holograms is high since the coherence factors are respectively |*μ*| = (0.989,0.988). Data points n°1 and 3 are outside the optimal area. Thus, the decorrelation noise is higher, the coherence factor is lower (|*μ*| = (0.823,0.824) respectively) and the phase map noisier. Note that the probability density obtained from the simulation is in very good agreement with the theoretical probability density of Eq. (4). This demonstrates that Eq. (4) is quite adapted to the description of quantization-shot-noise-induced decorrelation noise in phase measurement based on digital holography.

The use of the numerical simulation of the full digital holographic process permits to evaluate the influence of the induced decorrelation noise and to give criterion for the recording parameters. The next part is devoted to the experimental study with high-speed digital holographic recordings.

## 4. Experimental analysis

So as to study the decorrelation noise induced by the recording conditions, an experimental analysis was carried out by considering the parameters taken into account in Section 3. A set-up was arranged in a Fresnel configuration and the hologram processing is similar to that described in [22]. The experimental set-up is described in Fig. 6. A 6W cw laser at 532nm was used to illuminate the object. The half-wave plate at the output of the laser is used to adjust the power in the object and reference paths. The laser is separated into a reference wave and an object wave by use of a polarizing beam splitter (PBS). The polarization of the object wave is then rotated 90° to be parallel with that of the reference wave, so that interferences can occur. The object and reference waves are combined by the 50% beam splitter cube placed in front of the high-speed sensor. The reference wave is expanded, spatially filtered using a spatial filter (microscope objective and pinhole), and collimated to produce a smooth plane reference wave.

The object wave is spatially expanded to illuminate the structure, using a lens assembly and mirrors (not detailed in Fig. 6). A set of negative lenses are inserted in the object path, in front of the cube, between the object and the sensor and is used to increase the studied area [22,36,37]. This negative lens induces a change in the object-to-sensor distance, which becomes smaller than the initial distance *d*_{0}. The sensor is a high-speed camera (Photron), with pixel pitch at 20μm and a maximum spatial resolution including *M* × *N* = 1024 × 1024 pixels. The full pixel capacity is at 16000 photo-electrons. At the full spatial resolution, the maximum frame rate is 13500Hz. When increasing the frame rate, the spatial resolution is decreased. The dynamic object is an aluminum beam 89.5cm × 2cm × 0.2cm, suspended at one of its end, and excited by a mechanical shaker clamped at its center. In order to produce a reproductible movement at the surface of the object, a 0.4s sweep excitation with bandwidth [20Hz-10kHz] is applied. The frame rate of the sensor was set to 50kHz, the spatial resolution was 384 × 568 and the exposition time was adjusted from 0.3μs to 15μs.

The input laser power is adjusted from 0.5W to 5W. By increasing both the laser power and the exposure time, the power of the reference wave can be shifted to the higher bits, and saturation may be obtained. Using the half wave plate placed before the polarizing beam splitter (PBS), the ratio *Rc* can be optically adjusted. The power meter at the output of the 50% beam splitter cube is used to measure the power along the reference beam |*R*|^{2} (Fig. 6(b)), and the object beam |*O*|^{2} (Fig. 6(c)), so that the ratio *Rc* = |*R*|^{2}/|*O*|^{2} can be measured. For each recording, the sensor is systematically triggered at *t* = 5ms so as to record always holograms at the same vibration state of the object. So, the experimental decorrelation noise is measured for the same amount of initial deformation-induced decorrelation noise. Thus, variations in the decorrelation noise should be due to recording conditions. Note that the noise evaluation provides a global value and it is not possible to dissociate the different noise origins.

Parameter *Rc* was measured from 4.2 to 2570. One hundred holograms were recorded. Figure 7(a) shows the set of 100 measurements plotted in a 3D space with the exposition time, the ratio *Rc* and the saturation level (*α*,%). Each set is represented by a red dot and the number of the measurement. The set of measurements covers a large part of the (*Rc*,*α*) domain. Note that having such experimental configuration does not permit to record holograms with low *Rc* and high *α*., because of physical limits provided by the maximum laser power and exposure time. However, increasing *Rc* and the exposure time leads to record holograms with saturated gray levels (see for ex. set n°60). Figure 7(b) shows the set of recorded holograms plotted in the (*Rc*,*α*) reference axis and the optimal zone. Dots were colored according to the measured coherence factor in the experimental phase differences. Especially, we chose to add an intermediary threshold at 0.80. Red dots correspond to measurements for which the coherence factor is lower than 0.80, green dots those for which the coherence factor is greater than 0.9 and blues dots correspond to a coherence factor between 0.80 and 0.9. This is motivated by few practical considerations: from our experience, we consider that a minimum of |*μ*| = 0.80 is acceptable to process the modulo 2π phase data [32]. Since the experimental data is subjected to other decorrelation-induced noises, the constraint on |*μ*|, for this data, has to be slightly relaxed. To get the phase noise in the data, the processing of the phase difference map is the following: first, apply a low-pass filtering to the phase difference, then calculate the difference between the low-pass filtered phase and the raw phase, finally, this result is an estimation of the noise included in the data. This approach was discussed in ref [32,38]. From the noise map, the histogram of the data is calculated and then the noise probability density is estimated. This probability density can be fitted with Eq. (4), and the coherence factor |*μ*| can be obtained.

Figures 8 and 9 show a set of experimental results obtained with the recorded holograms. Since the excited structure is not exactly the same as for the numerical simulation, the phase patterns that were obtained are different. Modulo 2*π* phase map show the propagation of the mechanical wave in the beam 5ms after the shock provided by the shaker. Since recordings are triggered at always the same instant, these phase maps should be quite identical if there were no influence of quantization and shot-noise on the phase measurement. Figure 8(a) shows the case *Rc* = 7.81 and *α* = 2.7% for point n°1, with coherence factor |*μ*| = 0.900, with respectively from the left to the right, the probability density of the quantization gray levels of the recorded experimental hologram, the phase difference map between the two instants (mod 2π), and the probability density of the phase noise (extracted from experimental data and fitted with Eq. (4)). Figure 8(b) and Fig. 8(c) are respectively, point n°2, with *Rc* = 125, *α* = 17%, and |*μ*| = 0.913, point n°3, with *Rc* = 125, *α* = 48%, and |*μ*| = 0.909. Figure 9(a,b,c) shows the results for respectively point n°4, *Rc* = 4.2, *α* = 0.2%, and |*μ*| = 0.564, point n°5, *Rc* = 825, *α* = 13.8%, and |*μ*| = 0.772, and point n°6, *Rc* = 2185, *α* = 23.7%, and |*μ*| = 0.769. For points n°1,2,3 the decorrelation noise is low and the phase map is suitable for accurate metrology; however, this is not the case for points n°4,5,6 for which the phase map has low quality.

Note that in Fig. 7(b) a few red dots are localized inside the theoretical optimal zone, where they should not. The main parts of the other colored dots are in the “good place”. We attribute these differences to noise sources discussed in Section 2. On one hand, the experimental set-up includes more noise sources such as the read-out noise of the sensor (not known), or other possible technical noise like the noise on reference that may be not totally filter off because of the on-axis configuration of the experiment [22]. On the other hand, the parameters influencing the measurement results were considered separately from each other. Experimentally, it is impossible to bypass the effects of exposure time, spatial resolution of the hologram, fill factor, read-out noise, other technical noise, and so on… This may explain why the coherence factor |*μ*| is smaller in experimental results than expected from the numerical simulation: most of the blue points in Fig. 7(b) should be green instead of blue. Nevertheless, as a general rule, Fig. 8 shows the good agreement between the optimal area defined with the numerical simulation and the experimental results. These experimental results confirm that the optimization of the recording conditions have to be carefully checked when working with high-speed digital phase imaging. In these experiments, the optimal values for *R _{c}* and

*α*are

*R*≈100 and

_{c}*α*≈40-50%, ensuring a reduced influence of the combined quantization-shot-noise-induced decorrelation noise in high-speed digital holographic metrology.

## 6. Conclusion

This paper proposes a quality assessment of quantization and shot-noise induced decorrelation noise in high-speed digital holographic phase imaging. A criterion based on the coherence factor of the hologram phase difference is proposed and this leads to define the optimal conditions for recording at very-high frame rate. A full simulation of the holographic process establishes the best choice for two important parameters: the used sensor dynamics (*α*, depending on the maximum available photo-electrons per pixel) and the ratio between the reference wave and the object wave (*Rc*). By considering a threshold of the coherence factor such that |*μ*|>0.90, an optimal zone for the recording parameters (*Rc*,*α*) is obtained. In this zone the decorrelation noise due to quantization and shot-noise is minimized. Experimental holograms recorded at 50kHz frame rate were processed and the coherence factor measured. It was shown that noise-reduced measurements are possible with about 40-50% of the full sensor dynamics and ratio at about 100, thus yielding suitable phase maps for metrology purposes. In addition the experimental results are in good agreement with the numerical simulation. Using the rules defined with the proposed methodology, future works will concern application to the study of vibration fields having high dynamics as it is the case in acoustic black holes [39] and the propagation of acoustic waves in inhomogeneous materials, for which the frame rate has to be increased up to 100kHz.

## Acknowledgments

This study is part of the Chair program VIBROLEG (Vibroacoustics of Lightweight structures) supported by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic and Hybrid Structures). The authors wish to associate the industrial and academic partners of this project; respectively Airbus, Alstom Power, Bureau Veritas, CETIM, Daher, DCNS Research, STX and Université du Maine in France.

The authors are grateful to the reviewers for providing them very helpful remarks and comments.

## References and links

**1. **T. C. Poon, *Digital Holography and Three-Dimensional Display: Principles and Applications* (Springer-Verlag, 2010).

**2. **P. Picart, *New Techniques in Digital Holography* (ISTE-Wiley, 2015).

**3. **I. Yamaguchi, J. Kato, S. Ohta, and J. Mizuno, “Image formation in phase-shifting digital holography and applications to microscopy,” Appl. Opt. **40**(34), 6177–6186 (2001). [CrossRef] [PubMed]

**4. **D. P. Kelly, B. M. Hennelly, C. McElhinney, and T. J. Naughton, “A practical guide to digital holography and generalized sampling,” Proc. SPIE **7072**, 707215 (2008). [CrossRef]

**5. **P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A **25**(7), 1744–1761 (2008). [CrossRef] [PubMed]

**6. **D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng. **48**(9), 095801 (2009). [CrossRef]

**7. **D. P. Kelly, J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Quantifying the 2.5D imaging performance of digital holographic systems,” JEOS **6**, 11034 (2011). [CrossRef]

**8. **T. Kreis, “Frequency analysis of digital holography,” Opt. Eng. **41**(4), 771–778 (2002). [CrossRef]

**9. **T. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. **41**(8), 1829–1839 (2002). [CrossRef]

**10. **A. Stern and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. **43**(1), 239–250 (2004). [CrossRef]

**11. **L. Xu, X. Peng, Z. Guo, J. Miao, and A. Asundi, “Imaging analysis of digital holography,” Opt. Express **13**(7), 2444–2452 (2005). [CrossRef] [PubMed]

**12. **N. Demoli, H. Halaq, K. Sariri, M. Torzynski, and D. Vukicevic, “Undersampled digital holography,” Opt. Express **17**(18), 15842–15852 (2009). [CrossRef] [PubMed]

**13. **M. Leclercq and P. Picart, “Digital Fresnel holography beyond the Shannon limits,” Opt. Express **20**(16), 18303–18312 (2012). [CrossRef] [PubMed]

**14. **G. A. Mills and I. Yamaguchi, “Effects of quantization in phase-shifting digital holography,” Appl. Opt. **44**(7), 1216–1225 (2005). [CrossRef] [PubMed]

**15. **N. Pandey and B. Hennelly, “Quantization noise and its reduction in lensless Fourier digital holography,” Appl. Opt. **50**(7), B58–B70 (2011). [CrossRef] [PubMed]

**16. **M. Lesaffre, N. Verrier, and M. Gross, “Noise and signal scaling factors in digital holography in weak illumination: relationship with shot noise,” Appl. Opt. **52**(1), A81–A91 (2013). [CrossRef] [PubMed]

**17. **M. Gross, M. Atlan, and E. Absil, “Noise and aliases in off-axis and phase-shifting holography,” Appl. Opt. **47**(11), 1757–1766 (2008). [CrossRef] [PubMed]

**18. **M. Gross and M. Atlan, “Digital holography with ultimate sensitivity,” Opt. Lett. **32**(8), 909–911 (2007). [CrossRef] [PubMed]

**19. **F. Verpillat, F. Joud, M. Atlan, and M. Gross, “Digital holography at shot noise level,” J. Disp. Technol. **6**(10), 455–464 (2010). [CrossRef]

**20. **J. W. Goodman, *Introduction to Fourier Optics*, (McGraw-Hill Editions, 1996).

**21. **Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analyses of a digital hologram sequence,” Appl. Opt. **46**(23), 5719–5727 (2007). [CrossRef] [PubMed]

**22. **J. Poittevin, P. Picart, C. Faure, F. Gautier, and C. Pézerat, “Multi-point vibrometer based on high-speed digital in-line holography,” Appl. Opt. **54**(11), 3185–3196 (2015). [CrossRef] [PubMed]

**23. **H. A. Aebischer and S. Waldner, “A simple and effective method for filtering speckle-interferometric phase fringe patterns,” Opt. Commun. **162**(4-6), 205–210 (1999). [CrossRef]

**24. **A. Federico and G. H. Kaufmann, “Denoising in digital speckle pattern interferometry using wave atoms,” Opt. Lett. **32**(10), 1232–1234 (2007). [CrossRef] [PubMed]

**25. **A. Uzan, Y. Rivenson, and A. Stern, “Speckle denoising in digital holography by nonlocal means filtering,” Appl. Opt. **52**(1), A195–A200 (2013). [CrossRef] [PubMed]

**26. **M. Lehmann, “Decorrelation-induced phase errors in phase-shifting speckle interferometry,” Appl. Opt. **36**(16), 3657–3667 (1997). [CrossRef] [PubMed]

**27. **M. Lehmann, “Phase-shifting speckle interferometry with unresolved speckles: a theoretical investigation,” Opt. Commun. **128**(4-6), 325–340 (1996). [CrossRef]

**28. **M. Lehmann, “Optimization of wave-field intensities in phase-shifting speckle interferometry,” Opt. Commun. **118**(3-4), 199–206 (1995). [CrossRef]

**29. **D. Middleton, *Introduction to Statistical Communication Theory* (McGraw Hill, 1960).

**30. **W. B. Davenport and W. L. Root, *Random Signals and Noise* (Mc Graw Hill, 1958).

**31. **J. C. Dainty, *Laser Speckle and Related Phenomena* (Springer Verlag, 1984).

**32. **M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. **52**(9), 1275–1286 (2012). [CrossRef]

**33. **K. F. Graff, *Wave Motion in Elastic Solids* (Dover Publications, 1975).

**34. **J.-L. Guyader, *Vibration in Continuous Media*, (ISTE-Wiley, 2006).

**35. **A.W. Leissa, *Vibration of Plates*, (N70–18461, NASA, 1969).

**36. **U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. **35**(4), 977–982 (1996). [CrossRef]

**37. **J. Mundt and T. Kreis, “Digital holographic recording and reconstruction of large scale objects for metrology and display,” Opt. Eng. **49**(12), 125801 (2010). [CrossRef]

**38. **P. Picart, R. Mercier, M. Lamare, and J.-M. Breteau, “A simple method for measuring the random variation of an interferometer,” Meas. Sci. Technol. **12**(8), 1311–1317 (2001). [CrossRef]

**39. **V. Denis, A. Pelat, F. Gautier, and B. Elie, “Modal overlap factor of a beam with an Acoustic Black Hole termination,” J. Sound Vibrat. **333**(12), 2475–2488 (2014). [CrossRef]