1. INTRODUCTION

A. Motivation

With the invention of the laser in 1960, the laser speckle phenomenon was recognized and soon understood as a consequence of the surface roughness of the scattering object [1]. Since speckles occur due to surface roughness, speckles do contain information about surface roughness, and, thus, laser speckles can be used for optical surface roughness measurements. In contrast to other roughness measurement principles based on surface topography measurements, speckle-based principles cannot resolve the surface topography or surface height distribution $h(x,y)$, but only certain statistical parameters of surface roughness. For such a measurement task, the evaluation of laser speckles from the surface scattering of coherent light enables the statistical assessment of surface roughness structures on a scale below Abbe’s imaging diffraction limit, i.e., from large distances and with small apertures. The present study is focused on measuring the root mean square height, i.e., roughness parameter

B. State of the Art

Speckle-based roughness measurement principles can be subdivided according to the properties of illumination light. While some groups have investigated speckled illumination [2–5], the majority of the studied measurement setups have considered non-speckled illumination. For this reason, the present study concentrates on the case of non-speckled illumination.

Furthermore, as the illumination light, either monochromatic or polychromatic light is chosen. In contrast to monochromatic illumination, polychromatic illumination also enables the assessment of optically non-smooth surfaces, i.e., surfaces with a roughness parameter ${S_{\rm q}}$ larger than half of the average light wavelength [6]. For fundamental works regarding speckle-based roughness measurements with polychromatic illumination, refer to [5,7–10]. However, a particular strength of roughness measurements with laser speckles is the assessment of the roughness of optically smooth surfaces without the need to fulfill Abbe’s diffraction limit—in neither lateral nor axial direction. For a roughness parameter ${S_{\rm q}}$ below approximately one quarter of the average light wavelength, the highest sensitivity is achieved with monochromatic illumination and a respective evaluation of monochromatic speckles [6]. For this reason, the present study is focused on using monochromatic laser light for the illumination.

Speckle-based roughness measurement principles can be further divided according to the speckle evaluation technique. The speckle contrast evaluation [11] has been introduced together with a theoretical model that describes the dependency of speckle contrast on the roughness parameter ${S_{\rm q}}$ [12–15]. A standard textbook that summarizes the theoretical background is [16]. Theoretical studies and experiments on surfaces with different topographies demonstrated that the contrast evaluation technique also enables the distinction between different surface topographies [17]. In other words, speckle contrast is sensitive with respect to the roughness surface parameter ${S_{\rm q}}$ and surface topography, while the latter is here an undesired sensitivity, i.e., a cross-sensitivity. This phenomenon is well known, but its actual effect on the roughness measurement error has not been quantified yet. In addition, other fundamental sources of uncertainty such as photon shot noise, speckle noise, and the naturally present surface height fluctuation are known, but an investigation of the resulting measurement limits is missing. For instance, a total measurement uncertainty of the roughness parameter of about 1 nm is reported in [18]. This shows that precise roughness measurements of smooth surfaces are possible with contrast-based speckle evaluations, but the individual contributions from, e.g., photon shot noise, speckle noise, and a varying topography to the measurement uncertainty are currently not known. As a result, the potentially achievable measurement uncertainty for surface roughness measurements with the contrast-based evaluation of monochromatic speckles needs to be clarified.

The contrast evaluation technique was extended to a self-correlation evaluation by means of an auto-correlation of the speckle image [19,20]. Again, a cross-sensitivity with respect to the surface topography exists. For this reason, the speckle-based measurement approach was used in [21] to solely detect a deviation from a set value of surface roughness, which is important for in-line or in-process applications in manufacturing. The estimated and experimentally determined measurement uncertainty of the roughness parameter was below 1 nm. This supports the hypothesis that precise roughness measurements on smooth surfaces are possible with the evaluation of monochromatic speckles, but it is not clear whether the auto-correlation evaluation or the contrast evaluation technique performs better or whether both perform similarly. Furthermore, a theoretical error estimation is given in [20] that includes fluctuations of the scattered light intensity from shot noise and temporal laser intensity fluctuations, but the effect of speckle noise and a varying topography on the measurement result has not been studied to date.

Another evaluation technique is the cross-correlation evaluation of two speckle images with different illumination or observation directions (angular correlation) [22,23], or with different wavelengths (spectral correlation) [24,25]. An overview and theoretical framework for both cross-correlation techniques is presented in [26]. From a physical perspective, both techniques are identical, because the essential part is cross-correlating two speckle images with different wave vectors, which is realized by using either two different wavelengths or two different illumination/observation directions in the optical setup. The speckle cross-correlation technique enables roughness measurements with an extended measurement range and is particularly useful for optically non-smooth surfaces with a roughness value ${S_{\rm q}}$ of about 1 µm up to 50 µm [6]. Since such high roughness values are out of the scope of the present paper, the cross-correlation evaluation is not further considered here. For the sake of completeness, it is mentioned that not only the speckle cross-correlation evaluation technique but also the auto-correlation evaluation [5,9] and speckle contrast evaluation [7,8] are applicable for both monochromatic as well as polychromatic speckle images. While the use of polychromatic speckles enables an increase in measurement range, it lowers the measurement sensitivity in general.

In summary, the evaluation of monochromatic laser speckles is appropriate to measure the surface roughness parameter ${S_{\rm q}}$ on optically smooth surfaces with the highest sensitivity. To identify the potential of this measurement approach, the minimal achievable measurement uncertainty regarding fundamental sources of uncertainty, such as the natural surface height variation, photon shot noise, speckle noise, and a varying surface topography, has to be studied. In addition, which of the two available speckle image evaluation techniques, contrast based or auto-correlation based, is superior for achieving minimal measurement uncertainty has to be investigated.

C. Aim and Outline of the Paper

The aim of the paper is to derive a theoretical basis for the estimation of fundamental limits of roughness measurement uncertainty when using monochromatic speckles. To understand the influence of inevitable sources of uncertainty such as the natural randomness of roughness itself, quantum shot noise, speckle noise, as well as the typically unknown surface height distribution and mean surface shape, analytical error analyses and simulations of the measurement chain are performed. Numerical experiments are the only feasible way to identify the different contributions to measurement uncertainty, because the reference value of surface roughness is exactly known, and it can be varied without changing the stochastic surface topography. Furthermore, the two evaluation techniques on the basis of the contrast and the auto-correlation of the speckle image are compared with respect to the minimal achievable measurement uncertainty.

The measurement chain for monochromatic speckle roughness measurements is explained in Section 2, including the different speckle image evaluation techniques and their relation to each other. In Section 3 follows an analytical derivation of the fundamental limits of the achievable measurement uncertainty regarding the natural randomness of the surface height, photon shot noise, and speckle noise. In addition, the efficiency of each evaluation technique is studied. The respective findings are verified with numerical experiments in Section 4, while errors due to unpredictable variations of surface topography (type of surface height distribution, lateral correlation length, axial surface position, surface tilt/gradient, surface curvature) are investigated in Section 5. The paper closes with a conclusion and outlook in Section 6.

2. MEASUREMENT CHAIN

The considered measurement arrangement is depicted in Fig. 1. A monochromatic laser beam illuminates the surface of the measurement object. The complex-valued light field that occurs in the object plane immediately after the scattering takes place is

 

Fig. 1. 4-f measurement arrangement consisting of a beam splitter, two lenses with focal lengths $f$ and a pupil. It is based on the imaging system proposed in [16, page 99], but here with $(x,y)$ as object/surface plane coordinates, $(\alpha ,\beta)$ as pupil plane coordinates, and $(u,v)$ as speckle image plane coordinates. For the theoretical and numerical error analysis, a circular pupil with diameter $D$ is assumed here.

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As one known speckle image evaluation method to assess surface roughness, the contrast of the speckle image is calculated:

The contrast–roughness relation depends on the parameters of the measurement system (imaging system including illumination) and also on the topography of the studied rough surface. Here, throughout the paper, a plane-wave illumination with a constant intensity as well as the 4-f imaging system as depicted in Fig. 1 with a pupil with circular symmetry are studied. In addition, as a starting point for subsequent theoretical studies, an isotropic surface with Gaussian height fluctuations and a circularly symmetric Gaussian-shaped height correlation function is considered initially. Under these assumptions, an analytical expression of the relation between the speckle contrast and surface roughness can be found in [16], page 101:

The number $N \ge 1$ is the ratio of the equivalent area of an imaging resolution spot on the surface and the correlation area of the wave scattered at the surface, which follows from the equation

 

Fig. 2. Speckle contrast $C$ over the wavelength-normalized surface roughness ${S_{\rm q}}/\lambda$ for different values of parameter ${N_0}$, calculated with Eqs. (8) and (9).

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The speckle contrast is a scalar, i.e., a 0D, quantity. The alternative evaluation of the self-similarity of the speckle image by means of an auto-correlation function provides an extension to a 2D quantity. Since isotropic rough surfaces are considered, the mean auto-correlation function is circularly symmetric, and, thus, it can be represented by a 1D correlation function over the total distance $\tau$ between the non-shifted ($I$) and shifted (${I_\tau}$) speckle image. This correlation function of the speckle intensity image, where the image is normalized by its expectation value, reads

For the sake of completeness, it is emphasized that, in general, an auto-correlation-based speckle evaluation can provide more surface information beyond a single roughness parameter. It can also provide parameters of the height correlation such as the correlation length for isotropic surfaces [19], and it has a directional sensitivity that is useful to assess anisotropic surfaces [28]. However, both are beyond the scope of this paper, because the speckle-based surface roughness measurement is studied here for isotropic surfaces and regarding the sole measurement the roughness parameter ${S_{\rm q}}$.

3. THEORETICAL ERROR ANALYSIS OF FUNDAMENTAL MEASUREMENT LIMITS

In each of the following subsections, a fundamental unavoidable source of uncertainty is studied regarding its effect on the speckle-based measurement of the roughness parameter ${S_{\rm q}}$:

The expected essential parameters that characterize these uncertainties are:

The aim of the theoretical studies is to present analytical expressions of the corresponding measurement limits, so that concrete limits can be determined straightforwardly for each measurement configuration. For this purpose, the measurement arrangement with the plane-wave illumination presented in Section 2 as well as the introduced surface assumptions are considered here further, i.e., a plane surface with isotropic Gaussian height variations and a constant circular symmetric Gaussian-shaped height correlation function.

The derivation of each measurement uncertainty limit is based on the uncertainty propagation calculus (analytically and numerically with Monte Carlo simulations) and the direct calculation of moments for given probability density functions or probability mass functions. The alternative, information-theoretical approach of calculating the Fisher information and Cramér–Rao lower bound [29,30] was discarded here, because the respective handling with the model of the image signal as a function of the unknown quantities is considered as more difficult. While the calculation of the Cramér–Rao bound is a known appropriate approach to determine the measurement uncertainty limit due to shot noise and camera noise with a constant variance (see recent examples concerning speckle-based displacement measurements [31,32] and optical flow velocity measurements [33,34]), less respective experience is available regarding speckle noise. Since speckle noise can be expected to play a crucial role in the achievable roughness measurement uncertainty, not information theory but pure stochastics is applied.

In the following, random variables and their associated mean values need to be clearly distinguished. Therefore, as an example, the random variable that describes the detected speckle contrast in the interrogation window is subsequently denoted as $C$, and its mean value is $\bar C$. The random variable that describes the surface roughness in the respective interrogation area is denoted as ${S_{\rm q}}$, and its mean value is ${\bar S_{\rm q}}$.

A. Natural Measurement Limit for Surface Roughness

Since the interrogation area $A$ is finite when assessing the roughness parameter ${S_{\rm q}}$ as defined in Eq. (1), the randomness of the surface topography causes an unavoidable, natural measurement uncertainty of ${S_{\rm q}}$, even if the surface topography could be recorded with no height uncertainty as well as negligibly small lateral sampling steps and a negligible spatial averaging (imaging resolution). In this ideal case, the studied surface in the interrogation area $A$ is composed of $K$ surface height sampling points ${h_k}$, $k = 1,\ldots ,K$, and Eq. (1) yields

Note that ${h_k}$ denotes the height deviation from the mean height ${h_0}$.

With an uncertainty propagation calculation of the expression ${S_{\rm q}} = \sqrt {S_{\rm q}^2}$, we obtain at first

To obtain an analytic expression for ${\rm Var}({S_{\rm q}^2})$, the simplified case of ${N_{{\rm surface},{\rm id}}}$ independent height values (with zero mean), each with an identical standard deviation ${\bar S_{\rm q}}$, is studied first. Assuming further a Gaussian distribution of the height values, the normalized expression $S_{\rm q}^2 \cdot {N_{{\rm surface,id}}}/\bar S_{\rm q}^2$ [compare with the squared Eq. (11)] obeys a chi-squared distribution with ${N_{{\rm surface,id}}}$ degrees of freedom having the variance $2{N_{{\rm surface,id}}}$. As a result, it is

Since the height values are typically not uncorrelated, the concept of an equivalent topography consisting of ${N_{{\rm surface,id}}}$ independent height values is applied. Dividing the interrogation area $A$ by the equivalent area ${A_{\rm c}}$ of the height correlation function of the surface gives the approximate average number ${N_{{\rm surface}}}$ of uncorrelated height values. For the considered example of a radially symmetric 2D Gaussian correlation function with ${r_c}$ as the $\sigma$ parameter, the equivalent area ${A_{\rm c}}$ of the height correlation function amounts to $2\pi r_c^2$. For a respective 2D Gaussian height correlation function, it is found with numerical studies that ${N_{{\rm surface,id}}} = 2{N_{{\rm surface}}}$. Combining this finding with Eqs. (12) and (13) finally yields the natural limit of measurement uncertainty for ${S_q}$, which results from the stochastic nature of the surface height in a finite field of view:

Note that this limit means a lower bound of the relative uncertainty for any roughness measurement principle, including those based on laser speckles.

B. Speckle Noise

The natural randomness of surface roughness leads to a random speckle pattern and, thus, to speckle noise. To clarify the effect of speckle noise on the measurement uncertainty of the roughness value ${S_{\rm q}}$ for no photon shot noise and an ideal noise-free camera, we begin with the uncertainty propagation

While the derivative needs to be calculated numerically by evaluating the relation between mean roughness ${\bar S_{\rm q}}$ and mean contrast $\bar C$ given by Eq. (8) (see also Fig. 2), an analytic expression for the variance of $C$ is derived in the following.

According to Eq. (5), the contrast $C = s/m$ is calculated with the arithmetic mean value $m$ and the sample standard deviation $s$ of the speckle image intensity values in the imaged interrogation area. For $\bar C \ll 1$, i.e., $\bar s \ll \bar m$, the variance of $m$ and the covariance of $s$ and $m$ can be neglected, and an uncertainty propagation leads to

Note further that

Combining Eqs. (16)–(18) then yields for $\bar C \ll 1$

To get rid of the assumption $\bar C \ll 1$, the variance of the contrast is studied with a Monte Carlo simulation for contrast values in the entire range $0 \le \bar C \le 1$. Since the covariance of $s$ and $m$ cannot be neglected anymore, a Monte Carlo simulation is the most suitable approach. For this purpose, ${N_{{\rm speckle,id}}}$ uncorrelated intensity values are randomly generated for given values of $\bar C$ and $\bar m$. To simulate the intensity values, a gamma distribution with the shape parameter $1/{\bar C^2}$ and scale parameter ${\bar C^2}\bar m$ is applied, so that the expectation value amounts to $\bar m$ (shape parameter multiplied by the scale parameter), and the variance amounts to ${\bar C^2}{\bar m^2} = {\bar s^2}$ (shape parameter multiplied by the squared scale parameter). This gamma distribution models the expected conversion of Gaussian distribution for a negligible speckle contrast $\bar C \to 0$ to the exponential distribution for fully developed speckles at $\bar C = 1$. The resulting variance ${\rm Var}(C)$ normalized by $\frac{{{{\bar C}^2}}}{{2{N_{{\rm speckle,id}}}}}$ is shown over $\bar C$ in Fig. 3. The results (red dots) verify Eq. (19) for $\bar C \ll 1$ (dashed line) as well as the accordingly enhanced solution (solid line)

 

Fig. 3. Normalized variance of the speckle contrast $C$ for different mean speckle contrasts $\bar C$ as a result of a numerical Monte Carlo experiment (red dots) in comparison with Eq. (19) as approximation for $\bar C \ll 1$ (dashed line) and with Eq. (20) as enhanced analytic solution (solid line).

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To estimate the equivalent number of independent speckle intensity values, we adopt the finding from the previous section while approximating the correlation function of the speckle intensity with a 2D Gaussian function so that ${N_{{\rm speckle,id}}} = 2{N_{{\rm speckle}}}$. The symbol ${N_{{\rm speckle}}}$ denotes the average number of speckles, which is obtained by dividing the interrogation area $A$ by the equivalent area ${A_{\rm k}}$ of the correlation function of the speckle intensity. ${A_{\rm k}}$ is also the equivalent area of the point-spread function, and reads ${\lambda ^2}{f^2}/(\pi {(D/2)^2})$ for the optical system shown in Fig. 1, with $D$ as the diameter of the pupil; see [16], page 100. After inserting Eq. (20) into Eq. (15), we finally obtain the relative uncertainty of ${S_{\rm q}}$ in the form

As expected, the relative uncertainty is inversely proportional to the square root of the average number of speckles. The prefactor, which is shown in Fig. 4 over different true values ${\bar S_{\rm q}}$ of the roughness parameter (normalized by the light wavelength), is 0.5 for ${\bar S_{\rm q}}/\lambda \to 0$ and larger than or close to 0.5 otherwise. Hence, the lower limit due to speckle noise is a relative measurement uncertainty of approximately $\frac{1}{{4\sqrt {{N_{{\rm speckle}}}}}}$. The considered imaging regime is ${N_0} \gg 1$, with ${N_0} = \frac{{{N_{{\rm surface}}}}}{{{N_{{\rm speckle}}}}}$, i.e., the number of imaged equivalent surface correlation areas in the interrogation area is larger than the average number of speckles. As a consequence, the speckle noise limit is about a factor of $\sqrt {{N_0}} /2$ larger than the natural limit due to the surface roughness itself.

 

Fig. 4. Prefactor from Eq. (21) over ${\bar S_{\rm q}}/\lambda$.

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C. Photon Shot Noise

Using monochromatic coherent light and assuming an ideal noise-free camera means that the number of detected photons of each pixel obeys Poisson distribution. This photon shot noise leads to a measurement uncertainty of ${S_{\rm q}}$ even if the speckle pattern does not change, which is the subsequently studied case. To study photon shot noise, an ideal camera with a quantum efficiency of one is assumed, and the pixel intensity values are assumed to be expressed in unit number of photons, so that the variance and the expectation value of a pixel intensity are equal. Note that the final result can also be adopted for the case of quantum efficiency below one by substituting the intensity values in unit number of photons that fall onto the camera by the intensity values in unit number of detected photons.

To determine the resulting standard deviation of the random error of ${S_{\rm q}}$, Eq. (15) is used again and studied for the case of a small nominal speckle contrast $\tilde C \ll 1$. According to Eqs. (16) and (17), the variance of the contrast then reads

Here, it is important to distinguish between the nominal mean value ${\tilde s^2}$ of ${s^2}$ in the theoretical absence of shot noise and the actual mean value ${\bar s^2} = {\tilde s^2} + \tilde m$ in the case of shot noise, while $\tilde m = \bar m$. Thus, the actual mean contrast is $\bar C = \bar s/\bar m$, which has to be taken into account when evaluating the derivative of the ${\bar S_{\rm q}}$-$\bar C$ relation in Eq. (15). Before that, the variance of ${s^2}$ needs to be calculated by evaluating Eq. (7); see Appendix A.1:

Combining this finding with Eq. (22) leads to the result

To determine a valid expression also for nonnegligible contrast values, ${\rm Var}(C)$ is determined by means of a Monte Carlo simulation within the entire contrast range $0 \le \tilde C \le 1$, and for $\tilde m = 20,\!000$ as an example. The normalized results (red dots) are shown in Fig. 5 together with the approximation for $\tilde C \ll 1$ according to Eq. (24) (dashed line). The Monte Carlo results verify the approximation for $\tilde C \ll 1$, which correctly describes the $\tilde m$-dependent increase at $\tilde C \le 1/\sqrt {\tilde m}$. The larger the average number of photons $\tilde m$, the faster happens the transition from 0.5 to one. The Monte Carlo results further enable the identification and verification of the enhanced solution (solid line)

 

Fig. 5. Normalized variance of the speckle contrast $C$ for different nominal speckle contrasts $\tilde C$ as a result of a numerical Monte Carlo experiment (red dots) in comparison with Eq. (24) as approximation for $\tilde C \ll 1$ (dashed line) and with Eq. (25) as enhanced analytic solution (solid line).

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Inserting Eq. (25) into Eq. (15), the uncertainty of ${S_{\rm q}}$ normalized by $\lambda$ finally reads

The absolute uncertainty is inversely proportional to the square root of the average total number of photons, which is the average number of photons per pixel multiplied by the number of pixels in the interrogation window. To evaluate the prefactor, the aforementioned bias of $\bar C$ is taken into account by using $\bar C = \sqrt {{{\tilde C}^2} + \frac{1}{{\tilde m}}}$, although the effect is marginal for the contrast range shown in Fig. 6. According to the loss of sensitivity, the uncertainty increases for ${\bar S_{\rm q}} \to 0$ and ${\bar S_{\rm q}} \to \lambda /4$. In between, a surface roughness range occurs with a prefactor of (for instance) $\approx 1/4$ depending on ${N_0}$. Thus, an absolute roughness uncertainty of $\approx \frac{\lambda}{{4\sqrt {{N_{{\rm pixel}}}\tilde m}}}$ is identified for this measurement range as an approximation of the lower limit due to shot noise. As a consequence, shot noise leads to the larger uncertainty component in comparison to the relative uncertainty limit caused by speckle noise only for roughness values ${S_{\rm q}}/\lambda \lt \sqrt {\frac{{{N_{{\rm speckle}}}}}{{{N_{{\rm pixel}}}\tilde m}}}$. For larger roughness values, the shot noise limit is smaller than the speckle noise limit and becomes negligible.

 

Fig. 6. Prefactor from Eq. (26) over ${\bar S_{\rm q}}/\lambda$.

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D. Camera Noise with a Constant Variance

Each camera pixel intensity ${I_n}$ in Eqs. (6) and (7) is now considered to be superposed with additive white Gaussian noise with the constant variance ${\sigma _{{\rm noise}}} \ll \bar m$, while shot noise is neglected. To complete Eq. (15) under that condition, the variance of the image contrast $C$ in the interrogation window needs to be determined.

First, the case of a small nominal speckle contrast $\tilde C \ll 1$ is studied, so that Eqs. (16) and (17) can be applied:

Similar to the case of shot noise, it is again important to distinguish between the nominal mean value ${\tilde s^2}$ of ${s^2}$ in the absence of camera noise and the actual mean value ${\bar s^2} = {\tilde s^2} + \sigma _{{\rm noise}}^2$ for non-zero camera noise. The variance of ${s^2}$ is calculated on the basis of Eq. (7); see Appendix A.2:

Inserting the result into Eq. (27) yields

To determine a valid expression also for nonnegligible contrast values, first, ${\rm Var}(C)$ is determined by means of a Monte Carlo simulation within the entire contrast range $0 \le \tilde C \le 1$, and for $\eta = 100$ as an example. The normalized results (red dots) and the approximation for $\tilde C \ll 1$ according to Eq. (29) (dashed line) are shown in Fig. 7. As a result, the approximation for $\tilde C \ll 1$, which correctly describes the $\eta$-dependent increase at $\tilde C \le 1/\eta$, is verified. The larger the signal-to-noise ratio $\eta$, the faster is the transition from 0.5 to one. The Monte Carlo results also enable the identification and verification of the enhanced solution (solid line)

 

Fig. 7. Normalized variance of the speckle contrast $C$ for different nominal speckle contrasts $\tilde C$ as a result of a numerical Monte Carlo experiment (red dots) in comparison with Eq. (29) as an approximation for $\tilde C \ll 1$ (dashed line) and with Eq. (30) as the enhanced analytic solution (solid line).

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Inserting Eq. (30) into Eq. (15), the uncertainty of ${S_{\rm q}}$ normalized by $\lambda$ finally reads

 

Fig. 8. Prefactor from Eq. (31) over ${\bar S_{\rm q}}/\lambda$.

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The absolute uncertainty is inversely proportional to the total signal-to-noise ratio, i.e., the average signal-to-noise ratio of one pixel multiplied by the square root of the number of pixels in the interrogation window. To evaluate the prefactor, the bias of $\bar C$ is taken into account by using $\bar C = \sqrt {{{\tilde C}^2} + \frac{1}{{{\eta ^2}}}}$, although the effect is marginal for the shown cutout in Fig. 8. Similar to the effect of shot noise, the resulting measurement uncertainty increases for ${\bar S_{\rm q}} \to 0$ and ${\bar S_{\rm q}} \to \lambda /4$, and a measurement range occurs with a prefactor of (for instance) $\approx 1/4$ depending on ${N_0}$. Thus, an absolute roughness uncertainty of $\approx \frac{\lambda}{{4\sqrt {{N_{{\rm pixel}}}{\eta ^2}}}}$ is identified within this measurement range, which serves as a rough approximation of the lower limit due to camera noise. In comparison with the lower limit due to speckle noise, the camera noise limit is larger for smooth surfaces with ${\bar S_{\rm q}}/\lambda \lt \sqrt {\frac{{{N_{{\rm speckle}}}}}{{{\eta ^2}{N_{{\rm pixel}}}}}}$. However, the camera noise limit is relevant only in comparison to the shot noise limit, if the number of photons is too low, i.e., if $\tilde m \lt \sigma _{{\rm noise}}^2$ in unit number of photons. In the opposite case of a sufficiently high number of detected photons $\tilde m \gg \sigma _{{\rm noise}}^2$, the dominating source of uncertainty is shot noise (in addition to speckle noise), and the camera noise is negligible.

4. VERIFICATION

To verify the key results from natural noise [Eq. (14)] due to the random surface, speckle noise [Eq. (21)], photon shot noise [Eq. (26)], and camera noise [Eq. (31)], numerical experiments are performed. While surfaces with a different roughness parameter $0 \lt {S_{\rm q}} \lt \lambda /4$ are studied, other surface parameters and the experimental parameters for the optical measurement are kept constant.

A. Experimental Parameters

The interrogation area $A$ of the surface is $250 \times {10^{- 9}}\;{{\rm m}^2}$. This corresponds to a square with an edge length of 0.5 mm and is a small measurement area in comparison with what is possible with speckle-based surface measurements in general. However, the small interrogation area is chosen to limit the computation effort.

For the surface correlation function, a radially symmetric 2D Gaussian function is assumed. Using ${r_c} = 1.995\;{\unicode{x00B5}{\rm m}}$ as the $\sigma$ parameter, the equivalent area of the height correlation function amounts to ${A_{\rm c}} = 2\pi r_c^2 = 25 \times {10^{- 12}}\;{{\rm m}^2}$. Hence, the number of uncorrelated height values of an equivalent topography in the interrogation area is

The equivalent area ${A_{\rm k}}$ of the point-spread function of the optical system, which is shown in Fig. 1, is ${\lambda ^2}{f^2}/(\pi {(D/2)^2})$, with $D$ as the diameter of the pupil ([16], page 100). Assuming a laser light wavelength $\lambda = 532\;{\rm nm}$ and an effective numerical aperture $(D/2)/f = 0.019$ (with $f = 0.1\;{\rm m}$, $D = 0.0038\;{\rm m}$), it is ${A_{\rm k}} = 250 \times {10^{- 12}}\;{{\rm m}^2}$. As a result, the average number of speckles in the interrogation area is

The parameter ${N_0} = \frac{{{A_{\rm k}}}}{{{A_{\rm c}}}}$ thus reads

B. Numerical Implementation of the Measurement Chain

The sampling period to discretely model the surface is 0.25 µm, i.e., the interrogation area $A$ comprises $2000 \times 2000 = 4,\!000,\!000$ sampling points, and the equivalent area ${A_{\rm c}}$ of the height correlation function comprises 400 sampling points. While a lower sampling period and, thus, more sampling points would be desirable to accurately model the correlation function of the surface, the chosen value is a compromise to reach acceptable computation times.

The discrete surface is applied to the speckle image calculation described in Section 2, which is realized with MATLAB and the built-in 2D fast Fourier transform. A description of the algorithm is also provided in [16], so that the speckle image simulation is considered to be a valid emulation of the optical measurement.

A camera pixel size of 2.5 µm is assumed for the speckle image recording, i.e., all $10 \times 10$ sampling points are combined to represent one pixel. This means 40 pixels per speckle (w.r.t. the average speckle size), so that image discretization should not significantly disturb the calculation of speckle contrast. The resulting number ${N_{{\rm pixel}}}$ of pixels in the interrogation area is $200 \times 200 = 40,\!000$.

To investigate the measurement limits due to natural surface variations and speckle noise, the sample size of the Monte Carlo study amounts to 100, i.e., the standard deviation of 100 measurements on different surfaces is evaluated. To investigate the measurement limits due to photon shot noise and camera noise, each speckle image is superposed with noise 20 times, so that the total sample size is 2,000. For the photon shot noise analysis, an average number of photons per pixel of $\tilde m = 5,\!000$ is considered. The camera noise analysis is conducted for ${\sigma _{{\rm noise}}} = 20$, which means an average signal-to-noise ratio $\eta = \tilde m/{\sigma _{{\rm noise}}}$ of 250. Note that these numerical values are important for reproducing the calculations, as well as for applying the calculation results to other parameter values by rescaling in accordance with the analytical results.

C. Results

The mean speckle contrast in the simulated images is shown in Fig. 9 over the surface roughness (blue line), and it is compared with the theory according to Eq. (8) (red line). While the simulation result slightly differs from theory, the agreement is considered to be sufficient for subsequent investigations.

 

Fig. 9. Speckle contrast as a function of surface roughness from theory (red line) and simulation (blue line).

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In Fig. 10(a), the relative standard deviation of the natural noise of surface roughness is shown over ${S_{\rm q}}$. The simulation result agrees with theory [Eq. (14)], which means a constant relative measurement uncertainty. In contrast to this natural uncertainty limit, the speckle noise limit of the speckle-based roughness measurement is not constant. Simulation and theory [Eq. (21)] agree for surface roughness values ${S_{\rm q}} \gt \lambda /16$. For lower surface roughness values, however, the simulation shows an ongoing uncertainty increase, while the theory predicts a convergence to a relative uncertainty minimum. The reason for this discrepancy is the equivalent number of independent speckle intensity values ${N_{{\rm speckle,id}}}$. In contrast to what was assumed in the theoretical derivation, an increasing correlation of speckle patterns was identified in the simulation when ${S_{\rm q}}$ decreases. As was heuristically derived from the numerical investigations, Eq. (21) needs to be enhanced by the factor ${({1 + {{({\frac{\lambda}{{16{S_{\rm q}}}}})}^4}})^{\frac{1}{4}}}$ to correctly reproduce the simulation result; see the added dotted curve in Fig. 10. This means that a lower bound of the absolute uncertainty of $\sigma ({S_{\rm q}}) = \frac{\lambda}{{64\sqrt {{N_{{\rm speckle}}}}}}$ is asymptotically reached for ${S_{\rm q}} \to 0$. As a result, the relative measurement uncertainty of the speckle-based roughness measurement increases for ${S_{\rm q}} \to 0$ as well as for ${S_{\rm q}} \to \lambda /4$. In the measurement range in between, the uncertainty limit due to speckle noise is closest to the natural variation of ${S_{\rm q}}$. The lower bound of the relative uncertainty due to speckle noise is approximately $\frac{1}{{4\sqrt {{N_{{\rm speckle}}}}}}$, as derived in Section 3.B. As expected, however, the speckle noise limit of speckle-based roughness measurements is larger than the natural limit for all values of ${S_{\rm q}}$.

 

Fig. 10. (a) Natural variation of ${S_{\rm q}}$ and the effect of speckle noise on the relative uncertainty of speckle-based roughness measurements. (b) Relative measurement uncertainty due to photon shot noise and camera noise.

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The separately calculated relative uncertainty due to shot noise and camera noise is shown in Fig. 10(b). In each case, the simulation agrees with the theoretical finding, Eqs. (26) and (31), respectively. Similar to the effect of speckle noise, photon shot noise and camera noise cause a relative measurement uncertainty that increases for ${S_{\rm q}} \to 0$ and ${S_{\rm q}} \to \lambda /4$. For the given signal-to-noise ratio, the camera noise limit is lower than the photon shot noise limit. However, both are negligibly small in comparison to the speckle noise limit. As expected for shot-noise-limited photodetection and a sufficiently large amount of detected light energy, the dominating effect to deal with is the statistical behavior of laser speckles, which ultimately limits the minimal achievable measurement uncertainty.

5. CROSS-SENSITIVITY REGARDING SURFACE TOPOGRAPHY

Sections 3 and 4 dealt with the limits of speckle-based roughness measurements due to random effects. In addition, a cross-sensitivity with respect to surface topography exists, which is a systematic effect. If not corrected, the systematic error results in further measurement limitations that are studied in the following for surface roughness and shape features. For the numerical experiments, the setup is the same as described in Section 4. To not overload the discussion, a single true roughness value ${S_{\rm q}} = 0.1 \lambda$ is studied here, which is in the center of the measurement range (see Fig. 9) and is measurable with a minimal random error (see Fig. 10).

A. Influence of Surface Roughness

First, remember that the influence of the standard deviation ${S_{\rm q}}$ of the surface height distribution on speckle-based measurement is the desired sensitivity and no cross-sensitivity.

In addition, the lateral correlation length parameter ${r_c}$ of the surface height correlation function is well known to influence the measurement result; see Section 2. Keeping all other parameters constant, i.e., studying isotropic surfaces with a Gaussian height distribution with the same roughness value ${S_{\rm q}} = 0.1 \lambda$, the resulting relative measurement error over the correlation length extension factor is shown in Fig. 11(a). A reduced correlation length leads to a negative error, and an increased correlation length leads to a positive measurement error. However, for the studied range of correlation lengths, the error is about one order of magnitude lower than the error due to speckle noise and, thus, negligible.

 

Fig. 11. Influence of (a) lateral correlation length and (b) height distribution of the surface on the measurement of ${S_{\rm q}}$. The studied true value of ${S_{\rm q}}$ is $0.1 \lambda$. The error bars indicate the standard uncertainty of the shown error mean values.

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Note that for non-isotropic surfaces, which are not studied here, an error also results if only one of the two correlation length parameters varies.

Furthermore, the surface height distribution influences the measurement result. To vary the current Gaussian distribution from sub- to super-Gaussian, an extension factor is included in the exponent of the Gaussian probability function. The resulting relative measurement error over the height distribution extension factor is shown in Fig. 11(b). For sub-Gaussian distributions, the error is positive. For super-Gaussian distributions, a negative error results. Nevertheless, the error in the studied range of distributions is also negligibly small in comparison with the error due to speckle noise.

B. Influence of Surface Shape and Position

The influences of the axial surface position, a non-zero surface tilt angle, a cylindrical (1D), and a spherical (2D) surface curvature with a finite curvature radius are separately analyzed. The results of the numerical experiments are shown in Fig. 12. Note that the simulations contain only the different distances of the surface element from the sensor, i.e., the surface height value is varied in accordance with the axial position of the surface element. Respective changes of the illumination condition as well as a respective oblique perspective on surface roughness structures are neglected here, which is possible due to the interest in small incipient deviations from the initial surface position.

 

Fig. 12. Relative measurement error due to (a) axial position, (b) tilt, and (c) cylindrical and spherical curvature of the surface. The studied true value of ${S_{\rm q}}$ is $0.1 \lambda$. The error bars indicate the standard uncertainty of the shown error mean values.

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As a result of the simplified numerical experiments, no error occurs for the different axial surface positions. Note that the studied axial surface position values are smaller than the axial diffraction limit (depth resolution) of the optical system. According to [35], the axial diffraction limit for the numerical aperture $(D/2)/f = 0.019$ reads $2\lambda /{({\frac{{D/2}}{f}})^2} = 5540 \lambda$. In reality, the wavefront of the illumination will change for an increasing deviation of the axial surface position, which is currently not taken into account in the simulation. The resulting effect is similar to the effect of a surface curvature, which will cause a non-zero error. The influence of a surface curvature will be discussed below after the surface tilt influence.

A non-zero tilt angle leads to a positive relative measurement error, which (approximately linearly) increases with the absolute value of the tilt angle. For the considered range of tilt angles, the measurement error is negligibly small. Note that the studied tilt angle values are smaller than the acceptance half-angle of the sensor’s receiving optics, which is ${\rm arctan} ({\frac{{D/2}}{f}}){= 1.09^ \circ}$. This means that the specular reflection is not outside the sensor aperture and is always received here. For tilt angles larger than the acceptance half-angle, the main fraction of the scattered light signal is not detected.

The curvature-induced measurement error is positive, and it is inverse to the curvature radius, while the error of a spherical curvature is slightly larger than the error of a cylindrical curvature with the same curvature radius. As expected, the errors tend to zero for large curvature radii, while a sufficiently planar surface is obtained for curvature radii larger than the double focal length $f = 0.1\;{\rm m}$ of the first lens. For smaller curvature radii, the measurement error increases, but down to a curvature radius of $f/3$, it is one order of magnitude smaller in comparison with the error due to speckle noise.

6. CONCLUSIONS

By illuminating a surface with coherent light and evaluating laser speckles in the scattered light image, the measurement of the surface roughness parameter ${S_{\rm q}}$ is possible. The aim of the paper was to understand the ultimate limits of the achievable measurement uncertainty by studying the case of monochromatic speckles, a planar surface, and an isotropic surface roughness with a Gaussian height distribution and Gaussian correlation function.

First, the image parameter with the highest information content regarding the measurement quantity ${S_{\rm q}}$ is sought after, because two image evaluation approaches are commonly used, namely, by means of the speckle contrast and auto-correlation of the speckle image. Since both image parameters are directly related to each other, the same minimal achievable measurement uncertainty is obtained with either approach. Therefore, only one evaluation approach was studied further in the paper, which is the contrast evaluation approach.

Second, the fundamental measurement uncertainty limits due to (a) the natural randomness of the surface roughness itself, (b) speckle noise, (c) quantum shot noise, and (d) camera noise were derived by means of theory and are verified by well-established optical simulations. The identified analytic expressions show the expected behavior for the relative measurement uncertainty of ${S_{\rm q}}$, which is an inverse proportionality with the square root of the number of (a) independent surface height areas, (b) independent speckles, (c) photons, and (d) signal-to-noise ratio squared. In the entire measurement range, the error due to speckle noise is larger than the error due to natural surface height variations. For the numerically studied optical and camera configuration, speckle noise limits the minimal achievable measurement uncertainty. Due to speckle noise, an absolute uncertainty $\sigma ({S_{\rm q}}) = \frac{\lambda}{{64\sqrt {{N_{{\rm speckle}}}}}}$ is asymptotically reached for ${S_{\rm q}} \to 0$ in the region ${S_{\rm q}} \lt \lambda /16$. For ${S_{\rm q}} \gt \lambda /16$, the relative measurement uncertainty is bounded by the limit $\frac{1}{{4\sqrt {{N_{{\rm speckle}}}}}}$. For the studied setup, a minimal relative uncertainty of ${S_{\rm q}}$ between 1% and 2% resulted in the range ${S_{\rm q}} = (0.03 - 0.15) \lambda$, and the relative uncertainty strongly increases outside this measurement range.

All analytic expressions for measurement uncertainty were successfully verified by numerical experiments. For the speckle noise limit, the theoretical derivation should be enhanced in the future regarding the determination of the number of equivalent independent speckles. Note further that the investigations were focused on the resulting random errors, while non-zero systematic errors can also occur and need to be corrected with a calibration.

Third, the systematic influence of surface topography on the measurement error was quantified, studying a deviation of the roughness correlation length and height distribution of the roughness, as well as a deviation of the axial surface position, surface tilt, and surface curvature. As a result, all measurement errors were found to be negligibly small in comparison with the dominating error due to speckle noise. However, only small deviations of the influencing quantities were studied, and a nonnegligible systematic error can occur for larger values of the influencing quantities. Regarding the surface shape influence, a fundamental measurement limit occurs when surface gradients are so large that light is reflected in a direction outside the aperture of the receiving optics.

In conclusion, a theoretical basis for the estimation of the fundamental measurement limits is provided for the measurement quantity ${S_{\rm q}}$ when using monochromatic speckles. Further studies should focus on the influence of illumination with non-planar wavefronts, as well as on extending the findings towards the use of polychromatic speckles. Furthermore, the presented methodology should be applied next to study the measurement limits of laser speckle metrology with respect to the multi-parametric surface characterization. For instance, the additional determination of the correlation length parameter(s) is of interest for isotropic surfaces [18] as well as for anisotropic surfaces [19]. To finally reveal the full potential of speckle-based parametric characterization, surfaces with superposed structures with smaller and larger than the optical diffraction limit also need attention in future investigations [28].