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Abstract
This paper presents analytical modeling of the speckle decorrelation noise in digital Fresnel holographic interferometry. The theoretical analysis is carried out by considering the complex coherence factor between two speckled images from two digitally reconstructed holograms at two different instants. The expression giving the modulus of the coherence factor is established and depends on the local surface deformation and parameters from the holographic setup. The analysis is supported by realistic simulations and experiments. Both simulations and experimental results exhibit a very good agreement with the theoretical prediction.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Introduction
Holography is a very powerful method for imaging and metrology [1–4]. When becoming digital [5], holographic recordings are able to directly yield the complex-valued wave front of any encoded object volume or surface. Holographic phase imaging measures the optical path length related to the scene of interest [6]. The relevant data is a wrapped in modulo $2\pi$ phase that can be advantageously used for several purposes: automatic refocusing [7–9], tracking refractive index changes [10], microscopy [11,12], tomography [13], roughness measurements [14,15], surface shape profiling [16–18], remote metrology [19], or also surface deformation measurements [20–22]. The comparison of the optical phases extracted from two digital holograms at two different instants refers to digital holographic interferometry [4,6]. The method of holographic interferometry has the advantage of being non-intrusive by the use of laser illumination but also to provide full-field measurements. In addition, with the advent of very high-speed sensors, both high spatial and temporal resolutions can be obtained [23].
From the practical point of view, the phase change is of interest and is calculated by the modulo $2\pi$ subtraction of two phases at two instants. This phase is also known as the Doppler phase and we refer to this term in this paper. Thus, the process of holographic interferometry is perfectly adapted to the measurement of deformations of any object when submitted to mechanical load [24], or to periodic or transient excitation such as vibrations [25–30]. However, the speckle pattern produced from the object is modified and changes from its initial state. This induces decorrelation noise in the Doppler phase, requiring advanced filtering in order to get noise-free phase data [31] or amplitude images [32–34]. The probability density of the phase noise induced by the speckle decorrelation is governed by the modulus of the complex correlation coefficient $|\mu |$ between the two speckle fields [31,35,36]. Note that there are other uncertainty sources inducing speckle noise decorrelation in the measured Doppler phase from digital Fresnel holography. For example, speckle decorrelation may be due to laser wavelength change between exposures [37], to defocusing of the reconstructed image [38] (the reconstruction distance is "not good"), to saturation of the recorded holograms [39], or also due to quantization with low number of bits [40].
From the theoretical point of view, description of speckle decorrelation has to consider the complex coherence factor, $\boldsymbol {\mu }$ [36], between two speckle fields when experimental parameters do change. The speckle decorrelation was discussed in several papers, for example in [41–45], for the case of wavelength changes. The decorrelation in speckle interferometers was discussed from the point of view of the fringe visibility in the correlation fringes [46–48]. This paper aims at investigating the phase noise by considering the theoretical model for the complex correlation coefficient in the case of digital Fresnel holographic interferometry. The case of objects submitted to deformations between the two digitally recorded holograms is examined.
The paper is organized as follows: section 1 presents the basic fundamentals of digital Fresnel holography and section 2 discusses on the theoretical modeling for the complex coherence factor. In section 3, simulations are carried out in order to compare results obtained with simulations and the analytical model, and finally section 4 provides experiments and comparisons with the predicted theory. Section 5 draws the conclusions of the study.
1. Theorical background
1.1 Digital Fresnel holography
Digital Fresnel holography is based on the coherent mixing from a reference optical path and from diffraction at surface/volume of any object illuminated by a laser beam. The specificity of Fresnel holography is that recording uses lens-less configuration as depicted in Fig. 1(a). The mixing between the reference wave $\mathcal {R}$ and the object wave $\mathcal {O}$ results in the digital hologram expressed as:
Fig. 1. (a) Basic scheme for digital Fresnel holography; the wave diffracted from the object propagates in the free space to the sensor area, and the reference wave impacts directly the matrix of pixels, (b) scheme of light propagation in digital holographic imaging with physical propagation from object to sensor plane, and numerical propagation from sensor plane to image plane, (c) numerical scheme for simulation of noisy Doppler phases by considering the point spread function of digital Fresnel holography ($\mathcal {FT}$ means Fourier Transform).
The illuminated object surface is at distance $d_0$ from the recording sensor. The object wave diffracted to the sensor plane can be expressed with the Fresnel approximations by Eq. (2) [49] ($i = \sqrt {-1}$):
The object wave front at the object plane is $A(x, y)=a(x, y) \exp \left [i \psi (x, y)\right ]$, $\lambda$ is the wavelength of light, $a$ is related to the object reflectance and $\psi$ is the optical phase related to the object surface shape and roughness. From the recorded holograms, the reconstruction of the object field at any distance $d_r$ from the recording plane is given by the discrete Fresnel transform in Eq. (3) [5,50]:
with $\textrm {FFT}$ the two-dimensional Fast Fourier Transform and $h_F$ the Fresnel kernel defined by Eq. (4),With Eq. (3), the complex-valued optical field can be obtained. When considering two consecutive instants, the phase variation (Doppler phase) is obtained by subtracting the two phases extracted from the two digitally reconstructed optical fields at the two instants.
1.2 PSF of digital Fresnel holography
When the reconstruction distance is set to $d_r = - d_0$ in Eq. (3), the initial object plane is recovered from the computation. In this case, the close relation between the initial object plane and the digitally reconstructed one can be described by the Point Spread Function (PSF) of digital Fresnel holography. Let $A(x,y)$ be the complex amplitude from the initial object plane and $A_r(X,Y)$ that of the complex-valued in-focus image calculated at the output of the reconstruction algorithm. Then, the two optical fields are linked by the convolution relation [49], with $\textrm {PSF}(x,y)$ being the Point Spread Function:
Basically, the full propagation scheme of digital Fresnel holography can be summarized in Fig. 1(b). The configuration is considered as lens-less, one half of the propagation physically exists (from the object to the sensor array), whereas the second half is pure numerical reconstruction from the sensor area to the image plane. By considering correct digital refocus, the general formulation of the $\textrm {PSF}(x,y)$ can be written as ($\otimes$ means convolution product) [50,51]:
In Eq. (8), $N,M$ are the number of pixels of the matrix sensor and $p_x,p_y$ are the pixel pitches respectively in $x$ and $y$ directions, $\lambda$ being the wavelength of light. Note that $p_x,p_y$ and $\Delta _{x},\Delta _{y}$ are linked to the fill factor of the sensor by $\xi =\Delta _{x}\Delta _{y}/p_x p_y$. In this paper, $\textrm {sinc}(x)=\sin (x)/x$.
1.3 Decorrelation noise in phase measurements
As pointed out before, the Doppler phase suffers from noise due to the speckle decorrelation between the two instants, when the object surface is submitted to any change. The decorrelation noise has specific properties compared to other noise sources in imaging systems: first the noise does not follow Gaussian statistics, then it exhibits amplitude-dependent statistics, last, the noise is not stationary in the field of view. That makes the decorrelation speckle noise a very particular source of random fluctuations in coherent phase imaging. The probability density of this noise is given by [35]:
Fig. 2. Probability density function of the speckle decorrelation noise as depending on $|\boldsymbol {\mu }|$.
As $|\boldsymbol {\mu }|$ decreases, broadening of the probability density is observed. The higher the value of $|\boldsymbol {\mu }|$ is, the less the signal is noisy and the more the curve tends to Gaussian shape. On the contrary, if $|\boldsymbol {\mu }|$ decreases, the phase data will be more and more noisy and the noise will be uniformly distributed between $-\pi$ and $\pi$. The non-stationary property of the speckle phase noise can be appreciated with Fig. 3 which considers the case of the Doppler phase obtained between two different instants when a structure is vibrating. Figure 3(a) shows the modulo $2\pi$ noisy phase fringe pattern and Fig. 3(b) provides the estimated noise using the two-dimensional windowed Fourier transform [52]. The standard deviation and the value of $|\boldsymbol {\mu }|$ both depend on the fringe density in Fig. 3(a). In order to get more quantitative noise appraisal, Fig. 3(c) provides the map of the standard deviation of noise in the phase map when locally calculated over the set of $13\times 13$ patches (each sized $18\times 18$ pixels). In Fig. 3(d) is shown the map of the local estimations of $|\boldsymbol {\mu }|$ over the same set of patches.
Fig. 3. Non-stationary property of the speckle phase noise, (a) mod $2\pi$ noisy digital fringes, (b) noise map from (a), (c) standard deviation of noise for each patch, (d) modulus of the coherence factor for the same patches; the grids in (a),(b),(c),(d) indicate the patches in the phase data.
One can observe that the speckle noise is not uniformly distributed over the field of view, but that it depends on the local fringe density, which is related to the elevation of the phase map. The elevation of the phase map is closely related to the displacement of the structure between the two exposures. The noise is stronger in the high displacement areas and less where there is no significant displacement. As can be observed, the areas in which the standard deviation is high in Fig. 3(c) is related to areas with high fringe density in Fig. 3(a), indicating a close link between the phase noise and the local slope of the surface deformation.
The standard deviation $\sigma _{\epsilon }$ of the decorrelation noise depends on the modulus of the complex coherence factor between the two speckle fields [36]. The relation is non trivial, but an approximate formula for $\sigma _{\epsilon }$ valid for $|\boldsymbol {\mu }| \in [0.7; 1]$ was provided [38]:
It follows that linking the fundamental parameters influencing $|\boldsymbol {\mu }|$ is of major interest in order to be able to predict the noise standard deviation in the Doppler phase maps.
Next section discusses on the theoretical expression of the modulus of the complex coherence factor by taking into account of the $\textrm {PSF}$ of digital Fresnel holography.
2. Theorical modelling
2.1 Complex coherence factor of the digital images
The complex correlation coefficient is defined by the mutual intensity between the two image fields $A_r(\mathbf {X_1})$ and $A_r(\mathbf {X_1})$ obtained for two instants $t_1$ and $t_2 = t_1 + \Delta t$ [36,53]:
In this way, the statistical average is applied only to the non-deterministic factor $\langle A(\mathbf {x_1})A^*(\mathbf {x_2}) \rangle$. According to Ruffin et al. [41,54], the phase of the object under illumination and observation can be written as:
The illumination of the object surface is considered as constant (illumination with a uniform extended light spot) to yield the amplitude of the object wave $a(\mathbf {x})\simeq a$. The surface height at any instant is $h_z(\mathbf {x})$, while $\rho (\mathbf {x})$ corresponds to the roughness of the object surface.
Figure 4 depicts the notations for illumination, observation, surface height and roughness. The surface height may change between instants $t_1$ and $t_2$ because of the surface deformation in case where the object is submitted to loads (pneumatic, acoustic, mechanic, thermal,…).
The roughness $\rho (\mathbf {x})$ is random and statistical approach has to be considered, whereas $h_z(\mathbf {x})$ is deterministic because it only depends on the surface height and deformation. It follows:
The joint-characteristic function of the roughness can be expressed as [43,54]:
In Eq. (18), $R_{hh}$ is the normalized surface roughness auto-correlation function. If the roughness standard deviation $S_q$ is larger than the wavelength $\lambda$ [55] and the width of the roughness auto-correlation is narrower than the width of the $\textrm {PSF}$ of the imaging system, Eq. (18) can be expressed as a Dirac distribution $\boldsymbol {\delta }(\mathbf {x})$. One obtains after a limited expansion of the exponential function [41,43,54,56]:
It comes:
It follows that the cross correlation of the two fields $A_r(\mathbf {X_1})$ and $A_r(\mathbf {X_2})$ does not depend on the surface roughness of the object, but only on the difference in elevation between the two instants.
2.2 Local surface deformation slopes
The deformation of the object can be seen as a local linear variation of the surface shape around its original position. In order to evaluate Eq. (20) and to obtain the expression of $\boldsymbol {\mu }$ as a function of the linear variation between the two consecutive instants, one considers the case of the out-of-plane surface deformation according to $h_z(x,y,t) = h_0 + \alpha _x(t) x + \alpha _y(t) y$. For the in-plane deformation, the coordinate is changing according to $x_0 + \beta _x(t) x$. So, $(\alpha _x(t),\alpha _y(t))$ are the local slopes due to out-of-plane deformation and $\beta _x(t)$ is that due to in-plane deformation. It then comes (for fluidity of calculation $h_z(\mathbf {x},t)= h_0 + \alpha (t) \mathbf {x}$):
Since the second integration is also a convolution product, one gets:
The integral in Eq. (24) is the Fourier transform of the product of the two $\textrm {PSF}$ evaluated at the spatial frequency $\mathbf {u} = \frac {\Omega _z}{2 \pi } \left (\alpha (t_1) - \alpha (t_2)\right ) + \frac {\Omega _x}{2 \pi } \left (\beta _x(t_1) - \beta _x(t_2)\right )$ depending on the deformation state at time $t_1$ and $t_2$. Noting $\Delta \alpha _{12} = \alpha (t_1) - \alpha (t_2)$ and $\Delta \beta _{12} = \beta _x(t_1) - \beta _x(t_2)$ respectively the local out-of-plane and in-plane slopes due to the surface deformation between instants $t_1$ and $t_2$, we have ($\otimes$ stands for convolution product):
2.3 Final expression of $|\boldsymbol {\mu }|$
For the sake of clarity, one notes $\mathbf {u}_{\alpha \beta } = \frac {\Omega _z}{2 \pi } \Delta \alpha _{12} + \frac {\Omega _x}{2 \pi } \Delta \beta _{12}$. Using Eq. (11) and Eq. (25) one obtains the expression of the complex correlation coefficient of the two fields which is expressed as a convolution product:
This expression is not easily manipulated as it is, except when performing numerical calculation of the convolution for special cases. However, an approximate expression for the modulus of the complex correlation coefficient can be provided by considering the sensor matrix as real-valued function (which it is a priori). Thus, one gets:
The modulus $|\boldsymbol {\mu }|$ depends on the slope variation between the two instants $\Delta \alpha _{12}$ and $\Delta \beta _{12}$. In addition, $|\boldsymbol {\mu }|$ is directly proportional to the auto-correlation of the analog-digital pupil-type function of digital Fresnel holography.
The PSF of the system being related to convolution of two functions (Eq. (6)), its Fourier transform is the multiplication of the Fourier transform of both functions. For the pixel function [51], one has (FT means Fourier Transform):
Similarly with $\mathbf {U_x} = \frac {\mathbf {N} p_{x}}{\lambda d_{0}}$, one obtains for the filtering function of the Fresnel transform:
Since the two shifted Dirac functions will compensate for after the convolution in Eq. (27), finally it comes for $|\boldsymbol {\mu }|$:
Note that the key parameter $\mathbf {U_x} = {\mathbf {N} \mathbf {p_{x}} }/{\lambda d_{0}}$ included in Eq. (31) is the cut-off frequency of digital Fresnel holography and refers to the inverse as what is usually considered to be the spatial resolution of the digital holographic reconstruction, $\mathbf {\rho _x} = 1/ \mathbf {U_x} = \lambda d_{0} / {\mathbf {N} \mathbf {p_{x}} }$ [4–6]. It only depends on the ideal in-focus reconstruction distance, the pixel pitch and the number of pixels in the recorded hologram. When $\mathbf {U_x}$ increases (extended sensor), $|\boldsymbol {\mu }|$ tends to be high for the same deformation state, meaning the standard deviation of noise is low. But when $\mathbf {U_x}$ decreases (narrow sensor), $|\boldsymbol {\mu }|$ also decreases and the standard deviation of noise increases. That means that speckle phase decorrelation in digital Fresnel holography is closely related to the dimensions of the sensor that is used for recording holograms. The larger the sensor, the weaker the decorrelation is. Inversely, the narrower the sensor is, the higher the noise in the phase data.
2.4 Summary
In this section, we aim at summarizing the hypothesis and results of the theoretical analysis. The modeling was conducted by the evaluation of the modulus of the complex coherence factor. For that, the local surface slopes of the in-plane and out-of-plane surface deformation are the relevant parameters. These local slopes must be considered in the sense of the spatial frequencies they produce for the holographic system according to Eq. (32):
2.5 Case of non-extended pixels
The impact of the pixel width (represented by the sinc function in Eq. (31)) may be negligible compared to the $\textrm {rect}$ function which depends on the sensor dimensions. Mathematically, that means that the $\textrm {sinc}$ function of the active surface of pixels can be reduce to 1. In this case, the expression of $|\boldsymbol {\mu }|$ is simply related to the two-dimensional convolution of the rectangular function, and is given by:
Here, $u$ and $v$ are respectively the spatial frequency in the $x$ and $y$ direction with $u = \frac {\Omega _z \Delta \alpha _x}{2\pi } + \frac {\Omega _x \Delta \beta _x}{2\pi }$ and $v = \frac {\Omega _z \Delta \alpha _y}{2\pi }$. Note that the curve representing Eq. (33) in the $x$ and $y$ directions is simply a straight line with negative slope. It follows that $|\boldsymbol {\mu }|$ exhibits anisotropy according to the sensor dimensions and fringe orientation. That means $|\boldsymbol {\mu }|$ depends on the inclination of the deformation slope with respect to the length of the sensor matrix. For the one-dimensional case, the hypothesis of the non extended pixels can be assumed by $1/\Delta _x >> {N p_{x}}/{\lambda d_{0}}$, and can also be expressed by $\sqrt {\xi } N p_x^2 / \lambda d_0 <<1$. The ratio depends on experimental parameters. If $d_0$ increases, that is the distance between sensor and object increases, the ratio decreases and the extended surface of pixels has a reduced influence. This corresponds to the case where the spatial resolution of the reconstructed image is much larger than the width of the pixels. So the farther the object, the less the influence of pixel width on phase decorrelation is. This results was also pointed out in [50]. Obviously, these rules go the other way when the object moves close to the sensor. That means that the influence of the pixel width on phase decorrelation is more significant in this case.
In order to appraise the impact of the sinc function on $|\boldsymbol {\mu }|$, a simulation of the sinc and the rect functions is considered with the following parameters: $d_0=300$ mm, $\lambda =0.532$ $\mu$m, $p_x=20$ $\mu$m, and $\xi =0.58$. A comparison of the two functions with different values $N=(128,256,512,1024)$ is shown in Fig. 5(a). The width of the rectangle function compared to the sinc is relevant for $N=512$ and $N=1024$. That means that for these cases, the shape of the curve of $|\boldsymbol {\mu }|$ is modified and is no longer a straight line with negative slope. This point is illustrated in Fig. 5(b) where $|\boldsymbol {\mu }|$ is represented for the four cases $N=(128,256,512,1024)$. In the last two curves for $N=512$ and $N=1024$, the effect of the pixel surface extension can be clearly observed since the plots become curved.
Fig. 5. (a) Comparison of the rect function from the sensor width and the sinc from the extended pixels for $N=(128,256,512,1024)$, (b) variation of $|\boldsymbol {\mu }|$ for the four cases $N=(128,256,512,1024)$. The color code for $N$ in (a) is conserved in (b).
Figure 5 shows that extended pixels only modify the shape of the curve of $|\boldsymbol {\mu }|$, but that they do not significantly change the sensitivity to the asymmetry of the sensor. This means that the anisotropy of noise is not modified by the extended surface of pixels. Fig. 6 shows the modulus of the complex coherence factor for different cut-off frequencies of digital Fresnel holography.
Next section presents a comparison between realistic numerical simulations and the theoretical expression in Eq. (31).
Fig. 6. Comparison of $|\boldsymbol {\mu }|$ for different pixels size sensor for $N=(128,256,512,1024)$ and $d_0=300$ mm, $\lambda =0.532$ $\mu$m, $p_x=20$ $\mu$m.
3. Confrontation to simulations
3.1 Object to image simulation
Simulations of phase images are carried out in order to simulate phase maps from digital holographic Fresnel interferometry with corruption from speckle decorrelation noise. The basic scheme for the simulation is depicted in Fig. 1(c) which corresponds to the numerical implementation of the convolution equation of digital Fresnel holography (Eq. (5)). The computation is carried out using fast Fourier transforms. The initial object plane is Fourier transformed and multiplied by the Fourier transform of the PSF (Eq. (30)), and then the inverse Fourier transform yields the image plane. For the object plane, a flat surface with Gaussian roughness (standard deviation at $10 \lambda$) is numerically simulated. The object submitted to surface deformation is simulated by adding the deformation modeled as a plane with a certain slope. The slope has amplitude $\Delta \alpha$ and an orientation described by angle $\phi$, so that the deformation is given by $S(x,y) = \Delta \alpha \cos (\phi ) x + \Delta \alpha \sin (\phi ) y$. For the sake of simplicity, only out-of-plane deformations are considered.
In order to mimic realistic practical situations, the parameters are chosen as follows: $d_0=2760$ mm, $\lambda =0.532$ $\mu$m, $p_x=4.4$ $\mu$m, $M=1200$ and $N=1600$ pixels. The sensor is voluntary chosen having rectangular dimensions in order to highlight the anisotropy of noise, which depends on the fringe orientation. For the simulation, the pixels are considered as non-extended (fill factor close to zero). With the parameters, one has $U_{max}\simeq 4.79$ mm$^{-1}$ and $V_{max}\simeq 3.59$ mm$^{-1}$. By considering illumination at normal incidence ($\theta _o = \theta _e =0$ in Fig. 4), so that $\Omega _z=4 \pi /\lambda$ and $\Omega _x =0$, the spatial frequency of the slope has amplitude equal to $2 \Delta \alpha / \lambda$. The slopes are chosen so that the corresponding spatial frequencies are in the maximum range $[0,\sqrt {U^2_{max}+V^2_{max}}]$. This leads to $\Delta \alpha \in [0,1.5 \times 10^{-3}]$. The orientation of the fringe pattern is adjusted by varying angle $\phi$ from 0 to 360$^{\circ }$. The noisy Doppler phase is finally obtained by computing the difference between two phase maps from computation described in Fig. 1(c).
3.2 Estimation of noise
The noise maps are extracted by calculating the difference between the original noise-free and noisy phases. The standard deviation and the probability density of noise are estimated. With Eq. (9), the data are fitted to the theoretical equation in order to estimated the value of $|\boldsymbol {\mu }|$.
3.3 Comparison between simulations and theory
Figure 7 presents the surface deformations as slopes in their noise-free and noisy versions for 5 inclination angles $\phi =(0,30,45,60,90)^\circ$.
Fig. 7. Extractions from the simulator for surface deformations as slopes. The modulo $2\pi$ phase maps are given for 5 inclination angles (0,30,45,60,90) and progressive spatial frequencies from $0.01mm^{-1}$ to $0.08mm^{-1}$. Noise-free and noisy surface deformations are displayed for each case.
Figure 8 shows the comparisons between the values of $|\boldsymbol {\mu }|$ estimated from the simulated noisy phase maps and that obtained with theory from Eq. (33). As can be observed, the simulation is in very good agreement with the theoretical expression. This confirms that the decorrelation noise is sensitive to the orientation of the fringes when the sensor exhibits asymmetry such as rectangular formats.
Fig. 8. Comparison between the analytical expression from Eq. (33) and the simulation results for different values of the slope amplitude and orientation.
4. Experiments
4.1 Experimental setup
In this section, comparison between analytical and experimental results is provided. The experimental set-up is presented in Fig. 9(a). The object surface and the camera sensor are illuminated by a continuous green laser (wavelength $\lambda =532$ nm, 6 W maximum power). The light emitted by the laser is split by a polarizing beam splitter (PBS) to produce the reference and illumination beams. The object wave is spatially expanded to illuminate the structure by using a dedicated DOE (Diffractive Optical Element) [57]. The DOE was designed with 8 subareas, each of them producing a particular laser beam shape [58]. Figure 9(b) illustrates the diversity of shapes that can be produced with the DOE: square area, elliptical areas, narrow and large rectangular beams (vertical and horizontal). Such beam shaping increases the photometry efficiency of the set-up by avoiding wasting light with classical lenses and mirror assembly. For further details on the realization of the DOE, the reader is invited to consider [59]. The sensor is from SONY ICX274AL and is inserted in a Imaging Source DMK 51buc02 camera. The pixel size is at $p_x = p_y = 4.4$ $\mu$m and maximum resolution is at 1200 $\times$ 1600 pixels. The exposure time is set at 100 $\mu$s and the configuration is off-axis digital Fresnel holography. The illumination beam impacts the object surface with angle $\theta _e = 15$ degrees and observation is at normal incidence $(\theta _o=0)$, so that $\Omega _z \simeq 1.96 \pi /\lambda$.
Fig. 9. (a) Experimental set-up for full-field holographic vibrometry (PBS: polarizing beam splitter, DOE: Diffractive Optical Element, $\lambda /2$ half-wave plate), (b) set of beam shape structures that can be produced by the DOE to illuminate the object surface. The z axis is oriented along the propagation of the light beam from the structure plane and the x-y axes are parallel to the structure plane.
For the comparison with theory, the tested object is a rectangular aluminum beam sized 26.4 cm high, 2.03 cm wide and 0.49 cm thick. The distance between sensor and object is set at $d_0\simeq 2760$ mm. This yields $U_{max}=4.79~mm^{-1}$ and $V_{max}=5.59~mm^{-1}$. The mechanical beam is submitted to a load force that produces bending and then a displacement. This is done using a screw with micrometric accuracy at one of its extremity, whereas the other one is clamped to the optical table. The precision screw is embedded in a non-deformable heavy beam, so that the screw will push the structure and apply a controlled static force to the top of the beam. Note that submitting the beam to load force at its extremity induces out-of-plane deformations. The advantage of such mechanical configuration is that the bending of the beam produces straight and parallel fringes. It follows that, in this set-up, the comparison is carried out by considering out-of-plane experimental deformations, such that there is no contribution from any in-plane movement. The mechanical beam is oriented in three positions, at 0 degree (horizontal beam), at 45 degrees and finally at 90 degrees (vertical beam). These three configurations are well suited for exhibiting the noise anisotropy of the holographic arrangement. Figures 10(a)–10(c) show schemes of the aluminum beam for the three orientations.
Fig. 10. Schemes for the three positions of the aluminum beam oriented at, (a) 90 degree for the vertical beam, (b) 45 degrees, (c) 0 degrees for the horizontal beam.
4.2 Data processing
Sets of digital holograms are recorded for the three beam inclinations when progressively applying the load force to the beam, starting from no load to a maximum load enabling fringe patterns to be processed. Then, the digital holograms are numerically reconstructed and the phase are extracted to compute the Doppler phases between the current load state and the initial static state. From that, a set of noisy phase fringe patterns is obtained. In order to estimate the experimental slope due to deformation, and thus the corresponding spatial frequency, robust and error-reduced de-noising using the 2-D windowed Fourier transform is performed [31,52,60]. The de-noised phase maps are then unwrapped and converted into physical data (multiplication by $\lambda /3.92\pi$) in order to estimate the slope of the surface deformation. The surface slope is estimated using least square minimization when fitting data with $\Delta \alpha _x x + \Delta \alpha _y y$. From $(\Delta \alpha _x , \Delta \alpha _y)$ the spatial frequencies of the slope are estimated. With the de-noised phase data, the decorrelation noise can be estimated by subtraction of the raw phase. Then, the probability density function and the modulus of the complex coherence factor are estimated. Note that the slope estimation is performed on local patches mapping the beam in order to get local slope estimations and local estimates of $|\boldsymbol {\mu }|$. Similarly, for each patch over which the slope deformation is measured, the value of $|\boldsymbol {\mu }|$ is obtained.
4.3 Experimental results
The recording of the holograms is carried out with the native resolution of 1200 $\times$ 1600 pixels. Figure 11 shows the image amplitudes of the reconstructed holograms in the three different inclinations. The off-axis images can be appreciated. For each beam inclination, Fig. 11 provides raw phase maps in which the progressive fringe density increases from the foot (clamped beam) to the top of the beam (load force).
Fig. 11. Examples of reconstructed amplitudes and phase maps for the three beam inclinations, (a) vertical beam, (b) beam oriented at 45 degrees, (c) horizontal beam. Four phase images for four different bending are provided.
These raw data are then processed to get estimations of the noise, the slope and finally of $|\boldsymbol {\mu }|$.
The theoretical profile of $|\boldsymbol {\mu }|$ is recalled in Fig. 12(a) and the three orientations of the beam are highlighted, respectively with the dashed blue line for the horizontal beam, the dashed red line for the vertical orientation and the dashed green line for the inclined one. Figures 12(b)–12(d) show the comparison between the theoretical $|\boldsymbol {\mu }|$ and the experimental results respectively for the vertical beam, the horizontal beam and the inclined beam.
Fig. 12. Comparison between theory and experiments, (a) theoretical values of $|\boldsymbol {\mu }|$, with the dashed blue line for the horizontal beam, the dashed red line for the vertical orientation and the dashed green line for the inclined one, (b) comparison between experimental estimations of $|\boldsymbol {\mu }|$ and theoretical values for the case of the vertical beam, (c) comparison between experimental estimations of $|\boldsymbol {\mu }|$ and theoretical values for the case of the horizontal beam, (d) comparison between experimental estimations of $|\boldsymbol {\mu }|$ and theoretical values for the case of the inclined beam.
Figure 12 shows the very good agreement between the theoretical expression in Eq. (31) and the experimental results. Note that from the experimental point of view, values of $|\boldsymbol {\mu }|$ lower than 0.3 are difficult to reach. This is why plots in Fig. 12(b)-(c) do not exhibit values of $|\boldsymbol {\mu }|$ down to 0. When $|\boldsymbol {\mu }|$ decreases, the fringe density increases and requires that phase jumps are sampled with at least 4 pixels per phase jump. In addition, the de-noising algorithm also requires a minimum of sampling data points to operate or this is not possible to process data further. Note that, in a near future, more powerful de-noising algorithms could eventually be considered for very low values of $\mu$, such as new approaches provided by deep learning [61]. The results in Fig. 12 confirm that the modulus of the complex correlation factor is closely related to the slope of the surface deformation. The theoretical value of $|\boldsymbol {\mu }|$ depends on the parameters of the experimental set-up and especially of the width of the sensor in the Fresnel configuration. These results also demonstrate the importance of carrying out holographic measurements at high spatial sensor resolution because the decorrelation noise will be lower compared to reduced sensor spatial extension.
5. Conclusion
This paper provides the theoretical expression of the modulus of the coherence factor controlling the speckle noise decorrelation in measurements from digital holographic interferometry. As main result, the correlation coefficient depends on the geometrical characteristics of the set-up such as the number of pixels of the recording matrix, the size and pitch of the pixels, the illumination wavelength, the distance between object and sensor, and the local slope of the surface deformation between the two considered instants. The theoretical analysis shows that the decorrelation is anisotropic and depends on both the local phase fringe pattern orientation and the widths of the sensor. The theory is confronted to realistic simulations confirming the relevance of the proposed analysis. Experiments are carried out for the case of a mechanical beam submitted to bending with a load force. From the digitally reconstructed holograms, the phase fringe patterns for three different orientations of the mechanical beam are processed and the local slopes of the induced surface deformations are evaluated. From those data, the modulus of the complex coherence factor is estimated. Experimental results confirm the very good agreement with the theoretical modeling and demonstrate the anisotropic characteristic of the decorrelation phase noise in digital Fresnel holography.
The interpretation of the theory can be approached from the point of view of spatial frequencies. In the Fresnel configuration, it can be considered that the sensor, due to its limited spatial extension, behaves as a low-pass filter and attenuates the high spatial frequencies corresponding to the strong slopes of the surface deformation. It follows that the speckle decorrelation noise is related to the modulation transfer function of digital Fresnel holography and that noise increases if the local slope of surface deformation increases. Holographic imaging, as a linear filtering between the physical object and the digitally reconstructed image, disturbs the propagation of the spatial frequencies related to the deformation slope and this has for consequence the increase of the speckle decorrelation in the Doppler phase. Consequently, the attenuation results in phase noise in the measured phase fringe pattern between the two considered instants.
The results presented in this paper open the way to new advanced approaches of de-noising in digital holographic metrology by considering prior knowledge on the surface deformation in order to predict the local noise and to adapt the noise processing. This would be very useful for high-speed holographic imaging in which the number of pixels of the sensor is reduced and the decorrelation noise higher than in conventional experiments.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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