1. Introduction

The use of two- or multiple- wavelengths in holography have been shown to be a relevant tool for desensitized testing of steep optical surfaces (aspheric mirrors and lenses) [1,2], large deformation of structures [3] or also surface shape profiling [4–7]. Such an approach can also be used for surface roughness measurements when the roughness is large compared to the wavelength [8,9]. With the advent of digital holography [10], a wide range of applications of multi-wavelength holography was demonstrated, such as endoscopic imaging [11,12], micro-component evaluation [13], microscopy [14], laser radar [15], strain fields [16], calibration of mechanical structures [17], erosion measurements [18], or also in-line industrial inspection [19–21]. Two-wavelength digital holography has many advantages over single wavelength holography. With a unique wavelength, the measured surface height of any natural object (non polished) is ambiguous if larger than the wavelength. Since the surface of the object also includes roughness, then the surface shape cannot be reconstructed because the phase extracted from the digital holograms results from a speckle pattern. The ambiguity and randomness of this phase can be mitigated with the use of another/several wavelengths leading to what is known as the synthetic wavelength. It follows that the unambiguous range becomes increased from microns to millimeter or larger. Considering two wavelengths $\lambda _1$ and $\lambda _2$, the synthetic wavelength is given by $\Lambda =\lambda _1 \lambda _2/|\lambda _1 -\lambda _2|$ [22]. However, there is an inherent issue of using several wavelengths that is related to the speckle decorrelation arising when the primary illumination wavelength is changed to extend to the next one. This induces a phase noise in the phase difference calculated from the two phases at the two wavelengths. The decorrelation noise is related to both the surface roughness and shape. Basically, the theoretical description of speckle decorrelation is supported by the analysis of the complex coherence factor $\boldsymbol{\mu}$ [23] between two speckle fields when experimental parameters do change, such as for example when the wavelength changes. Experimental evidence of the speckle decorrelation was highlighted in many papers, for example, such as those of Refs. [5,13,24]. From the theoretical point of view, the phenomenon of speckle decorrelation depending on the wavelength and the roughness was described by several authors [25–30], whereas others proposed approaches to describe the influence of the surface shape [24,31]. Note that the influence of the slope from surface deformation was discussed in [32] in the case of digital speckle image correlation. In the former, the influence of roughness is related to the standard deviation of the random height fluctuations, whereas in the latter was pointed out the role of the slope of the shape. Especially, the higher the surface slope, the larger the decorrelation noise is, and so the noise is related to the fringe density in the phase map. But in the literature, no behavioral analytical modelling was provided for the influence of surface slopes. It follows that further analysis has to be carried out in order to yield practical modelling to predict speckle decorrelation from both roughness and shape.

So, this paper proposes an analytical modelling for describing the speckle noise decorrelation in phase data from two- or multiple- wavelength digital holography. The theoretical analysis is supported by a realistic simulation including the surface roughness and surface shape. Comparison between theory and simulation demonstrates the relevance of the proposed analytical modelling. The paper is organized as follows: section 2 gives the theoretical basics and definitions for surface shape imaging by digital holography, section 3 discusses on the evaluation of the complex coherence factor by taking into account the roughness and the surface topography. In section 4, a realistic simulator is presented and section 5 provides the comparison between outputs from the simulator and the analytical modelling. Section 6 draws the conclusion of the paper.

2. Imaging surface topography and roughness with digital holography

At the scale of the visible radiation range, [0.4; 0.8] $\mu$m, any natural and unpolished object surface exhibits, due to its roughness and shape, a diversity of heights much higher than the wavelength. This has the direct consequence that the optical phase from a wave-front reflected/diffracted by such a surface also exhibits a diversity of values uniformly covering the interval $[-\pi ,+\pi ]$. The phase is then random and uniformly distributed over the interval and the field propagated from the rough surface is thus a speckle pattern [23]. The roughness of the surface, with zero-mean, corresponds to its irregularities. In this paper, the topography of the surface, i.e. the height of the object in the absence of roughness, is the average surface. In the following, we note $\rho (x,y)$ the roughness and $h_0(x,y)$ the topography. The random variable $\rho$ follows a Gaussian statistic with standard deviation $S_q$. The surface height of the object is represented by the sum of the topography and the roughness according to [27]:

Figure 1(a) illustrates the notations. The topography is deterministic and cannot, strictly speaking, be characterized by any statistics. However, it generates a diversity of local slopes along the surface shape. The roughness is characterized by parameter $S_q$ which is the root mean square of the values of $\rho$ and by $\rho _r$ which is the correlation length of $\rho$ (i.e. in plane $(x,y)$). Generally $\rho _r$ is estimated with the autocorrelation function of $\rho$.

 

Fig. 1. (a) Notations for surface topography and roughness, (b) notations for illumination and observation angles through an optical system, (c) imaging the surface through $2f_1-2f_2$ image-plane digital holography architecture.

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In any optical system, for any illumination wavelength $\lambda _n$, the optical phase corresponding to surface height $h(x,y)$ (topography+roughness) depends on the incidence and observation angles, respectively noted $\theta _e$ and $\theta _o$ as illustrated in Fig. 1(b). If one considers that the illumination and observation directions are coplanar to the figure, the optical phase is given by Eq. (2) [23]:

Note that Eq. (2) shows that the surface height irregularities seen by the optical beam decreases as the angle of incidence of the beam deviates from the normal to the surface, meaning that the apparent roughness decreases. For the optical system, the apparent surface shape is then $(\cos (\theta _e)+\cos (\theta _o))(h_0+\rho )$. This modelling was also used by J.W. Goodman [23] when ignoring shading effects and multiple reflections induced by surface roughness. It is therefore only valid for moderate angles $\theta _{e}, \theta _{o}$ and when the single reflection hypothesis remains valid [26,33]. The second term in Eq. (2) is related to illumination and observation angles and does not depend on the surface height properties. It follows that for studying decorrelation properties from the surface this term can be omitted. Note also that it vanishes when $\theta _e=\theta _o = 0$ (illumination and observation at normal incidence). The imaging system through which the surface can be observed may take different modalities: microscope with or without tube lens, single lens or telecentric imaging system. This last configuration is the most interesting for surface shape holographic imaging as depicted in Fig. 1(c) [30]. In the case of digital holography, a reference beam is added and directly impacts the sensor after reflecting from a beam splitter cube. In this paper, we will not discuss on the way to record and process the digital holograms to get the useful phase for surface shape measurement. Whatever the imaging modality, the optical system can be characterized by its point spread function. For the sake of compactness of the formulas, vector notation is adopted to designate the coordinates of a point located in the object plane $(x,y)$ and in the image plane $(X,Y)$, i.e. $\mathbf {x}$ and $\mathbf {X}$ respectively. Let $A(\mathbf {x})$ be the complex amplitude from the surface of the object and $O(\mathbf {X})$ the optical field emerging from the optical system at the image plane (Fig. 1(c)). Both two fields are linked by the following convolution relation [33] (Eq. (3)), with $PSF(\mathbf {x})$ being the point spread function of the optical system:

In the case of surface shape measurement with multi-wavelength digital holography, the surface is illuminated with different wavelengths and the complex-valued image field is obtained from numerical processing of digital holograms [22,34]. As mentioned above, since the surface is rough, the image field is a speckle pattern and its phase is uniformly distributed over the interval $[-\pi ,+\pi ]$. Especially, phase differences between different wavelengths are of interest. Those phase differences exhibit speckle decorrelation noise that is due to the natural decorrelation from the wavelength change and that from the surface shape. The characterization of the speckle decorrelation can be investigated by considering the complex coherence factor of the speckle fields [23] .

3. Correlation of speckle fields

3.1 Complex coherence factor

In the literature, several authors have studied the effect of the variation of the illumination wavelength of flat and rough objects on the correlation of the two fields of speckles obtained at the output of the observation system [25–30]. These works were based on the calculation of the complex coherence factor $\boldsymbol{\mu}$ which has for origin the evaluation of the mutual intensity between the two speckle fields. The complex coherence factor is a scaled version of the mutual intensity between two fields $O_1$ and $O_2$ obtained for two different wavelengths $\lambda _1$ and $\lambda _2$ at the output of the imaging system, such as [23] :

In Eq. (5), $PSF_1$ and $PSF_2$ are the point spread functions seen by wavelengths $\lambda _1$ and $\lambda _2$. By adopting the same formalism as B. Ruffing [35], the field $A(\mathbf {x})$ can be written in the form:

The phase term $\phi _n$ is random and can be estimated through a statistical analysis, whereas phase term $\Phi _n$ is deterministic because it only depends on the surface topography and does not a priori exhibit any random behavior as that can be for roughness. It follows that:

When the standard deviation of roughness $S_q$ is larger than the wavelength $\lambda _{1,2}$, that is $S_q \geq \lambda _{1,2}/(2(\cos (\theta _e)+\cos (\theta _o)))$ [36], and if the correlation length of the roughness is smaller than the width of the $PSF$ of the optical system, then several authors demonstrated that we have ($\boldsymbol {\delta }(\mathbf {x})$ is the Dirac distribution) [26–28,35]:

It follows that:

Equation (12) depends on the $PSF$ of the optical system seen by each of the wavelengths $\lambda _1$ and $\lambda _2$, on the roughness with the negative exponential including $S_q^2$ and on surface topography with the complex exponential with $h_0(\mathbf {x_1})$ and $h_0(\mathbf {x_2})$. The analytical evaluation of such an equation is not straigthforward and the result depends on the surface topography $h_0(\mathbf {x})$.

3.2 Influence of local slopes of the surface shape

A particular case of great interest is the calculation of the correlation factor by taking into account the local slope of the surface. Indeed, any surface topography can be seen, at first approximation, as a collection of local slopes (or facets) of variable amplitudes. The local slope of the surface produces parallel fringes in the phase fringe pattern and the speckle decorrelation is thus directly related to this slope. The phase noise is higher in areas where the fringes are tight, i.e. areas where the surface slope is steep. In order to evaluate Eq. (12) and to get an expression of $\boldsymbol{\mu}$ taking into account of the surface topography, let’s consider the case of the surface slope written as $h_0(x,y)=\alpha _x x + \alpha _y y$, so for the sake of simplicity $h_0(\mathbf {x})=\alpha \mathbf {x}$. This approach is useful in order to allow comparing the different local fringe densities versus noise for any local surface slopes.

It follows that:

The integral in Eq. (15) is the Fourier transform of the product of the two $PSF$ at both wavelengths, calculated at $\alpha \left ( \Omega _1 - \Omega _2\right )/2 \pi$. The result of the integral is thus the convolution product of two pupil functions $p_1$ and $p_2$ corresponding to the pupil of the optical system actually seen by each of the wavelengths $\lambda _1$ and $\lambda _2$. Thus, we have ($\otimes$ means convolution product) :

With Eqs. (4), (15), (16) and after suitable normalisation, one gets the expression of the complex coherence factor of the two speckles given in the form of a convolution equation:

In its present form, this expression is not easily manipulable, except if performing a numerical calculation of the convolution for few special cases. Nevertheless, one can provide an approximate expression for the modulus of the complex coherence factor.

3.3 Expression of $|\boldsymbol{\mu}|$

The modulus of the complex coherence factor $|\boldsymbol{\mu}|$ is of great interest because it governs the probability density function of the phase noise [37]. In addition, $|\boldsymbol{\mu}|$ is related to the standard deviation of the decorrelation noise by a non-trivial relation [23,38]. In order to get an estimation, Eq. (17) can be approached through its modulus, by considering the pupil functions as real and even functions (which they are a priori if the pupil is circular or rectangular and transparent). It follows that:

It is reasonable to assume that $\Delta \lambda =|\lambda _1-\lambda _2|$ and that $\lambda _1 \approx \lambda _2 = \lambda$, since the difference between the two wavelengths will not exceed a few 15 or 20 nanometers. The cut-off frequencies associated to the two pupils $p_1$ and $p_2$ are then equal to $R_u=r_{pup}/\lambda f_2$. The module $|\boldsymbol{\mu}|$ is thus directly proportional to the autocorrelation of the pupil function calculated at the value

Noting $\mu _0=\exp {\left [-\frac {S_q^2}{2} \Delta \Omega ^2\right ]}$, $K=\cos \theta _e + \cos \theta _o$, and given the expression of the autocorrelation function of the circular aperture [33] for a cut-off frequency at $R_u$, one finally gets Eq. (20), valid for $K\sqrt {\alpha _x^2+\alpha _y^2}/\Lambda \le 2 R_u$ (0 if not):

The condition for Eq. (20) yields the maximum surface slope that could be accepted by the optical system that is given by $\sqrt {\alpha _x^2+\alpha _y^2} \le 2 \Lambda R_u/ K$. This means that the maximum slope depends on the synthetic wavelength and on the number of pixels per speckle grains. Note that the autocorrelation function of the aperture, leading to Eq. (20), is related to the modulation transfer function of the optical system which represents the attenuation of spatial frequencies as they pass through the optical system. The coefficients $(\alpha _x,\alpha _y)$ represent the local slope of the surface as $h_0(x,y)=\alpha _x x + \alpha _y y$, and they correspond to spatial frequencies $(u_0,v_0)\cong (\alpha _x /\Lambda ,\alpha _y /\Lambda )$. We can therefore interpret Eq. (20) for $|\boldsymbol{\mu}|$ as the evidence that the correlation of the speckles between the two wavelengths is impacted by the modulation transfer function of the optical system in the sense that the correlation decreases if the local slope increases. Indeed, the filtering function perturbs the propagation of the spatial frequencies of this slope through the optical system. This filtering effect is consistent with the analysis of M. Sjödahl [24] in which is specified that, first, the solid angle due to the aperture of the optical system, in the image space, reduces the available spatial frequencies contributing to a speckle grain, and last that the gradient of the phase has to be considered, that is related to the surface slope.

Note that Eq. (20) can be approximated for $|\boldsymbol{\mu}| \ge 0.5$ with a linear approximation by:

By considering $|\boldsymbol{\mu}|$ as a quality marker for the phase data, authors considered that phase data is suitable for metrology purpose when $|\boldsymbol{\mu}| \ge 0.85$ [39].

Next section is devoted to a realistic simulator of speckle fields at two wavelengths and presents a confrontation between the analytical model of Eq. (20) and the simulator.

4. Realistic simulation

4.1 Basic fundamentals

The goal of the realistic simulation is to produce phase maps supposed to be obtained from two-wavelength digital holography with corruption by speckle decorrelation noise. The speckle noise must have the adequate probability density function [37] as well as correlation length of typically 4 to 8 pixels per speckle grain (usual values for off-axis digital holography). The more the height variations of the surface topography, the higher the number of phase fringes, and thus the higher the phase noise decorrelation between the two wavelengths should be. The simulator was designed in order to simulate the optical scheme in Fig. 1(c). The $PSF$ of the optical system for wavelength $\lambda$ is given by Eq. (22) [33,37]:

The surface shape with the given topography and roughness is supposed to be illuminated by wavelengths in the visible or near-infrared range of light for which we have $S_q > \lambda /4$, an illumination angle $\theta _e$, and an average observation angle $\theta _o$. The surface shape is at the focal plane of lens L1 in Fig. 1(c). The circular aperture of radius $r_{pup}$ is inserted in the image focal plane of the first lens L$_1$ and has the role of the aperture diaphragm. This diaphragm is thus located in the Fourier plane of the telecentric system. The value of $r_{pup}$ is adjusted to control the size of the speckle grain in the image plane and to simulate realistic phase images. The aperture is a binary spatial frequency filter with cutoff frequency $R_u=r_{pup}/\lambda f_2$. The average size of the speckle gain produced by the system is equal to $\rho _s=0.61 \lambda f_2/r_{pup}=0.61 /R_u$. With the control of the number of pixels per speckle grain, here noted $N_s$, one gets $R_u = 0.61/(N_s p_x)$. In order to generate speckle fields at two different wavelengths, optical fields are successively computed for $\lambda _1$ and $\lambda _2$ keeping constant the topography and surface roughness. According to Eq. (21), note that the maximum acceptable surface slope for high quality phase data is now given by:

The condition for Eq. (23) is that the roughness ensures $\mu _0 > 0.85$. Equation (23) shows that the maximum acceptable surface slope depends on the ratio between the synthetic wavelength and the number of pixels per speckle grain.

4.2 Simulation of surface roughness

The roughness is numerically simulated starting from Gaussian random variable with standard deviation $\sigma _r$, in agreement with the order of magnitude that one aims at simulating (typ. some multiples of $\lambda$). This primary roughness has a narrow Dirac autocorrelation function meaning that there is no correlation between the roughness heights of juxtaposed pixels. The spatial correlation length can be modified by applying a low-pass filtering with a finite impulse response Gaussian filter of standard deviation $\sigma _h=\sigma _{rg} p_x$, $\sigma _{rg}$ being a number of pixels and $p_x$ the pixel pitch, and whose impulse response is:

After filtering, on gets the surface roughness $\rho$ with standard deviation given by:

The Gaussian filter induces an extension of the spatial correlation length of the roughness which now has a value given by the autocorrelation function of the filter. Therefore, the correlation length of the roughness is $\rho _r=\sqrt {2} \sigma _h$. As illustrations, Figs. 2(a) and (c) show simulated surface roughness for $S_q=10\lambda _1$ and for respectively spatial correlation lengths $\rho _r=p_x$ and $\rho _r=5p_x$. Figures 2(b) and (d) show the autorrelation function of the roughness in Figs. 2(a) and (c). As can be seen, Fig. 2(b) has Dirac autocorrelation function (no spatial correlation between pixels) and Fig. 2(d) has an extended spatial correlation almost equal to five pixels.

 

Fig. 2. Simulation of surfaces, (a) surface roughness with $S_q=10\lambda _1$ and spatial correlation lengths $\rho _r=p_x$, (b) autocorrelation of roughness in (a), (c) surface roughness with $S_q=10\lambda _1$ and spatial correlation lengths $\rho _r=5p_x$, (d) autocorrelation of roughness in (c), (e) pseudo 3D view of example of the faceted surface with $8\times 8$ patchs and progressive surface slope.

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4.3 Simulation of the surface topography

The surface topography can be simulated using analytical functions such as, for example, a combination of Gaussian functions, polynomials, MATLAB peaks or membrane functions, or any combination of those. The maximum amplitude of the topography maybe adjustable from several hundredth microns (micro-object) to few millimeters, and its minimum is set to $0$. In this paper, simulations of topography are provided by the MATLAB membrane function and by a faceted surface composed of $8\times 8$ patches, which of each corresponding to a slope according to $h_0(x,y)=\alpha _x x + \alpha _y y$, with parameters $(\alpha _x,\alpha _y)$ varying between each patch. The slopes thus vary from facet to facet but generate phase fringes with spatial frequencies $(\alpha _x/\Lambda ,\alpha _y/\Lambda )$. As an illustration, let’s consider a surface topography defined with $(M,N)=(\textrm {1024},\textrm {1024})$ pixels of pitch $p_x=5$ $\mu$m. The size of each facets includes $128\times 128$ pixels, thus 64 patches in total constitute the surface. Figure 2(e) shows an example of the simulated faceted surface.

4.4 Simulation of the optical phases and complex-valued fields

For any wavelength $\lambda _n$, the optical phase corresponding to the surface is calculated taking into account the topography of the surface $h_0$, the roughness $\rho$, the angles of incidence $\theta _e$ and observation $\theta _o$ of the illumination wave, according to Eq. (2). Thus, the phase of the optical field has the characteristics of a random phase since the surface roughness is much larger than the wavelength $\lambda _n$ $(S_q > \lambda _n/4)$. The complex field, $A_n$, at the surface is calculated by considering a uniform amplitude for illumination, such as $A_n(x,y) = A_0\exp \left (i\psi _n(x,y)\right )$, according to Eq. (2), Eq. (7), and Eq. (8). The flow in the simulation algorithm is organized as follows [37,40,41]: 1- simulation of the surface with topography and roughness, 2- calculation of the optical phase resulting from the illumination and observation conditions, 3- calculation of the complex-valued field $A_n$, 4- propagation at the focal plane image $F_1$ using $FFT$, 5- multiplication by the aperture function (refer to Fig. 1(c)), 6- propagation to the image focal plane $F_2$ using $FFT$. In order to obtain the phase change due to illumination at two wavelengths, and including speckle decorrelation, one first computes the complex-valued field $O_1$ obtained at wavelength $\lambda _1$, then computes the complex-valued field $O_2$ for wavelength $\lambda _2$. From that, the optical phases of the two fields can be obtained with the argument of $O_2 O^*_1$. The phase difference is calculated modulo $2\pi$ and includes the phase fringe pattern due to the surface profile and the speckle decorrelation due to both topography and roughness.

5. Results

5.1 Surface slope versus speckle decorrelation

In this section, the simulator is used for comparison with the theoretical modelling of Eq. (20). The faceted surface composed of patches with varying slopes is used to evaluate the speckle decorrelation as depending on surface roughness and local slope of the topography. The initial wavelength is chosen at $\lambda _1=0.6328$ $\mu$m, with illumination and observation at normal incidence $(\theta _e = \theta _o= 0)$. We set $\Delta \lambda =|\lambda _2-\lambda _1|=2$ nm so that $\Lambda =200.9$ $\mu$m. The number of pixels per grain is set to $N_s=$ 4 and $N_s=$ 8. For topography, the slopes correspond to spatial frequencies ranging from $(u_0,v_0)=(0,0)$ mm$^{-1}$ to $(u_0, v_0)=(1/4p_x,1/4p_x)$ mm$^{-1}$, that is $(\alpha _x,\alpha _y)=(\Lambda u_0,\Lambda v_0)=(\Lambda /4p_x,\Lambda /4p_x)$. For the synthetic wavelength $\Lambda$ these slopes generate between 0 and 31.75 fringes in the facets, with variable inclinations, i.e. a maximum amplitude of topography of $\sim$ 6 mm. For roughness, the value $S_q = 10\lambda _1$ and two spatial correlation lengths $\rho _r=p_x$ and $\rho _r=5p_x$ are considered. So, results can be compared with constant roughness but with different spatial correlation lengths. Figure 3(a) shows the top-view of the faceted surface with slopes varying from patch to patch. Figure 3(b) shows the modulo $2\pi$ noise-free numerical fringes generated for the synthetic wavelength $\Lambda$. The slope gradient from one patch to another can be appreciated and the fringe density increases as the slope increases. The maximum slope corresponds to 4 pixels per digital fringe (lower right corner patch), while the upper left corner patch corresponds to a flat surface (zero slope).

 

Fig. 3. Faceted surface, (a) top-view of the faceted surface, (b) modulo $2\pi$ noise-free fringes for synthetic wavelength $\Lambda$.

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For parameters $N_s=4$, $S_q = 10\lambda _1$ and $\rho _r=p_x$, Fig. 4(a) exhibits the modulo $2 \pi$ noisy phase fringes, Fig. 4(b) shows the decorrelation phase noise for each patch, Fig. 4(c) yields the standard deviation of noise over the patches and Fig. 4(d) displays the modulus of the coherence factor for each patch. The coherence factor decreases as the slope increases inversely to the standard deviation of the noise which increases with the slope. The maximum correlation factor is $\sim 0.89$ and corresponds to the patch of the flat surface. The standard deviation of the maximum noise is $1.81$ rad ($\sim 2\pi /\sqrt {12}$) and corresponds to the patch with maximum slope. The noise reaches the maximum possible value for full decorrelation of speckles. For the flat surface patch, the speckle decorrelation is due to the surface roughness, whereas for the other patches the decorelation is combination of both decorrelation from roughness and from slope.

 

Fig. 4. Simulation results for number of pixels per speckle grain at $N_s=4$, $S_q = 10\lambda _1$ and $\rho _r=p_x$, (a) mod $2 \pi$ noisy digital fringes, (b) decorrelation noise for each patch, (c) standard deviation of noise for each patch, (d) modulus of the coherence factor for the patches.

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Similarly, Fig. 5 shows the results of the simulation for parameters $N_s=8$, $S_q = 10\lambda _1$ and $\rho _r=p_x$. The same trends are observed with the increase of the correlation factor for the decrease of the slope. The maximum correlation coefficient is $\sim 0.87$ and the maximum noise standard deviation is $1.81$ rad. The noise also reaches the maximum possible value for full decorrelation of the speckles but this limit is reached for more patches than in the previous case with $N_s=4$.

 

Fig. 5. Simulation results for number of pixels per speckle grain at $N_s=8$, $S_q = 10\lambda _1$ and $\rho _r=p_x$, (a) mod $2 \pi$ noisy digital fringes, (b) decorrelation noise for each patch, (c) standard deviation of noise for each patch, (d) modulus of the coherence factor for the patches.

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These results show, that at constant roughness, the local slope of the surface strongly influences the speckle decorrelation noise. From these results, the evolution of $|\boldsymbol{\mu}|$ as a function of the spatial frequencies $(u_0,v_0)$ of the patches can be plotted. The average spatial frequency in each patch, $\sqrt {u_0^2+v_0^2}$, is of interest, and varies over the interval [0; 70] mm$^{-1}$. Figure 6 shows the evolution of $|\boldsymbol{\mu}|$ for different cases and the theoretical curves provided by Eq. (20) are also displayed. Case with $\rho _r=5p_x$ is considered for $N_s=(4,8)$ and $S_q = 10\lambda _1$. One can observe the very strong dependence of the decorrelation with the size of the speckle grain. The larger the speckle grain size, the more the decorrelation increases. Therefore, the more the optical system is closed, the more the decorrelation increases. This result is related to the fact that the more the pupil is closed, the less the cone of light captured by the system is extended. Any inclination of the beam tends to strongly change the average phase collected by the aperture. In addition, Fig. 6 shows that the spatial correlation length of the surface roughness does not significantly influence the decorrelation. The most important roughness parameter is therefore the $S_q$ value which yields the baseline of decorrelation for no slope. The simulation curves show a decay that is perfectly superimposed on the decay predicted by Eq. (20). This result is important because it clearly demonstrates by the simulation, on one hand, that the proposed decorrelation model is valid, and on the other hand, that the speckle decorrelation is naturally controlled by the numerical aperture of the holographic device, by the local slope of topography and by the surface roughness.

 

Fig. 6. Modulus of the coherence factor vs roughness and slope represented as spatial frequencies for cases $N_s=(4,8)$, $S_q = 10\lambda _1$, $\rho _r=(1p_x,5p_x)$; dashed lines: Eq. (20).

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Figure 6 confirms the interpretation of Eq. (20) for $|\boldsymbol{\mu}|$ stating that for two-wavelength holography, speckle decorrelation is related to the modulation transfer function of the optical system and that it increases if the local slope increases. The optical system as a linear filtering disturbs the propagation of the spatial frequencies related to the slope and the consequence is an increase of the speckle decorrelation.

5.2 Estimation of speckle decorrelation from surface shape

From the previous section follows that the speckle noise decorrelation can be estimated by considering the surface shape topography and the roughness. For this, let’s consider the continuous membrane surface with total height almost at 2000 $\mu$m, two wavelengths with $\lambda _1=0.6328$ $\mu$m and $\Delta \lambda =2$ nm, giving $\Lambda =200.9$ $\mu$m (illumination and observation angles are set to $\theta _e = \theta _o= 0)$. Thus a large number of phase fringes can be generated in order to get a large variety of slopes. From the surface topography the average slope can be estimated over a set of patches and the corresponding spatial frequencies can be calculated. Thus, with Eq. (20), the value of $|\boldsymbol{\mu}|$ can be predicted by considering the surface slope and the parameter $S_q$ from roughness. Figure 7(a) shows the surface with its topography and roughness. From the continuous surface, the local slope is estimated for each of the $16\times 16$ patches. With the estimated local slopes, the theoretical prediction of the decorrelation noise for each patch can be obtained from Eq. (20). Figure 7(b) shows the pseudo-3D plot of the local slopes estimated over the $16\times 16$ patches. The modulo $2\pi$ fringes generated by the two-wavelength holographic contouring are displayed in Fig. 7(c) and the speckle decorrelation noise map is provided in Fig. 7(d). One clearly observes the close link between the surface slope, the fringe density and the noise amount. The standard deviation of noise and the modulus of the coherence factor can be estimated from the noise map in each patch corresponding to the surface slope evaluation. The modulus of the coherence factor can be estimated from Eq. (20). Figure 8(a) exhibits the standard deviation of noise over the set of patches and Fig. 8(b) gives the map of the corresponding values of $|\boldsymbol{\mu}|$. Finally, Fig. 8(c) compares the estimated values of $|\boldsymbol{\mu}|$ in the patch and that from the theory with Eq. (20). As can be seen, the prediction of decorrelation from the estimated local surface slopes matches very well the theoretical modelling with the local slopes (Eq. (20)).

 

Fig. 7. (a) Membrane surface topography and roughness for $N_s=4$, $S_q = 10\lambda _1$, and $\rho _r=1p_x$ with maximum height at 2000 $\mu$m, (b) local slopes of membrane surface calculated over $16 \times 16$ patches, (c) noisy modulo $2\pi$ phase fringe pattern obtained with $\Lambda =200.85\ \mu$m, (d) map of the speckle decorrrelation phase noise.

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Fig. 8. (a) Standard deviation of noise for each patch for $N_s=4$, $S_q = 10\lambda _1$ and $\rho _r=p_x$, (b) modulus of the coherence factor for the patches for $N_s=4$, $S_q = 10\lambda _1$ and $\rho _r=p_x$, (c) modulus of the coherence factor vs roughness and slope represented as spatial frequencies for cases $N_s=(4,8)$, $S_q = 10\lambda _1$, $\rho _r=(1p_x,5p_x)$; dots: coherence factor measured in patches, dashed lines: Eq. (20).

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At this point, the theory is demonstrated to be very well correlated with the realistic numerical simulations. One important question is related to the experimental verification of the theory. This is not a straightforward task since this requires mastering both the surface roughness and the shape of the structure, but also this requires a diversity of the surface slopes. The best way would be to design and realize a specific object exhibiting slope diversity, similarly as the faceted surface. The surface polishing would require being well mastered and controlled in order to get a constant roughness over the different sub-parts of the object. Accurate measurements of the roughness and surface slopes should have to be performed in order to feed the theoretical modeling of Eq. (20).

6. Conclusion

This paper establishes a new expression of the modulus of the coherence factor in the case of multi-wavelength digital holography in order to estimate the speckle decorrelation phase noise in surface shape measurements. The theoretical modelling includes the contribution of the roughness of the surface and of the local slopes of the shape and depends on the $PSF$ of the imaging system. As a result the maximum slope that could be measured by the system depends on the synthetic wavelength and on the number of pixels per speckle grains. The model also permits to evaluate the synthetic wavelength according to the slope distribution of the object of interest. A realistic simulation in the case of a telecentric optical configuration with circular aperture, and taking into account of the surface slope and roughness, has been carried out. The simulator can be used for any object shape. The results demonstrate the very good agreement between the modulus of the coherence factor estimated with the simulation and the one calculated with theory. Interpretation of the results can be considered if the local surface slope is approached from the point of view of spatial frequencies. Thus, one can consider that the pupil of the system due to its limited spatial extension behaves as a low-pass filter and will attenuate or suppress the high spatial frequency components corresponding to the strong slopes. Therefore, this attenuation results in phase noise in the measured phase fringe pattern. An important results is that when knowing the topography and the roughness of any object surface shape, the calculation of the modulus of the coherence factor in each sub-area of the object permits to predict the noise in the phase data. This opens the way for a predictive noise analysis in order to better adapt the de-noising strategies [42] in the data processing in multi-wavelength digital holography.