## Abstract

We analyze statistical properties of dynamic speckles formed when an optically rough surface is illuminated by a fast-deflecting laser beam. The modified space-time correlation function of the light-intensity fluctuations has been introduced to estimate the correlation parameters of a dynamic speckle pattern. Dynamic speckles are considered in their application to range sensing using evaluation of the light-power-modulation frequency of a signal obtained from the integrating photodetector after spatial filtering of the scattered light. Multichannel configuration is suggested to improve the system accuracy. Conditions that should be fulfilled to get uncorrelated responses of photodiode are found. Proper averaging of the multichannel data allows designing a non-interferometric range sensor capable for measuring distance with accuracy of 1 εm during as short time as 1 ms.

©2008 Optical Society of America

## 1. Introduction

Statistical properties of dynamic speckles produced by light scattering from moving objects attract considerable attention of many research groups since sixties of the last century [1,2] when it was recognized that changes of the speckle pattern is somehow related to the object motion. The interest to deep understanding of dynamic-speckle behavior was stimulated by a number of promising applications of this technique such as non-contact velocity [3–5] or profile [6,7] measurements enhanced by digital signal processing. Stochastic nature of the signal produced by dynamic speckles suggests relatively large fluctuations in the estimated signal parameters, which leads to a large measurements error. It is evident that the accuracy of measurements can be increased by averaging of parameters estimated from independent uncorrelated signals. To minimize the time of measurements, an averaging is preferably to be performed in a system with a multichannel signal receiving and processing. In any case the important task for the system optimization is searching the conditions at which the received signals are uncorrelated.

Correlation properties of dynamic speckles and signals obtained after their spatial filtering have been analyzed [8,9] for the situation when the object surface is moving in one direction in respect to a laser beam, which in its turn is fixed in respect to an observer. However, in a recently proposed technique for *z*-distance measurements [10,11], the configuration of the experiment is different: dynamic speckles are created when a laser beam is moving in respect to both the object and the observer repeatedly scanning the object surface. Consequently, the statistical properties of dynamic speckles have some specific features. Particularly, two sequential scans of the same rough surface results in a formation of two dynamic speckle patterns, which are highly correlated in both time and space domains. Moreover, in the case of scanning beam, the relative movement of the object surface and laser beam can be two-dimensional. This requires the analysis of the correlation properties of speckle-patterns with respect to two coordinate systems. One coordinate system is related to the scanned surface while another is the observation plane where dynamic speckles are spatially filtered to be collected into a photo-detector. This paper is just devoted to the analysis of the correlation properties of dynamic speckles generated by a deflecting laser beam after their spatial filtering. We demonstrate that properly performed averaging of the data after multiple scanning an arbitrarily rough surface using multi-channel receiving with several photodiodes allows distance measurements with one-micrometer accuracy within one-millisecond-time window.

The paper is organized as following. First, we briefly describe the principle of distance measurements using dynamic speckles. Second, the space-time correlation function of the intensity fluctuations of dynamic speckles is adapted to the case of the deflecting laser beam. Then we present experiments devoted to the estimation of the correlation parameters important for optimal averaging of the data. Modifications of a dynamic-speckle distance-sensor aimed to improve the accuracy of measurements are discussed in the last section.

## 2. Dynamic speckles for distance measurements

When a divergent/convergent coherent beam is moving in respect to an optically rough surface, the speckles becomes dynamic often manifesting a translation motion in the space [12]. Velocity of this motion is a function of the radius of the illuminating wavefront, *R _{W}*, and of the distance,

*D*, between the object surface and observer. Both these parameters depend on the distance between the optical head and the object surface. Therefore, by measuring the velocity of dynamic speckles one can calculate the distance to the object and estimate the profile of the object surface. First, this technique was applied to profile measurements of moving objects [6] and then to measurements of the distance to a static surface by fast deflecting the laser beam [10]. When dynamic speckles are generated by a deflecting laser beam, their statistical parameters (such as mean velocity of speckles, coherence time, and others) depend on the geometry of the experiment in a different way as compared to the case of light scattering from a moving surface. Reason for this difference is following. During laser-beam scanning the object surface is immovable in respect to the observer while the laser spot is moving with the speed of

_{S}*V*. This is vice versa in the case of moving surface as it is schematically shown in Fig. 1, which contains videos for both right and left parts. It is easy to show that in the first case an average speckle velocity,

_{BS}*V*, is expressed as

_{SP}One can compare Eq. (1) with widely used expression for the velocity of dynamic speckles created by moving surface [6,8,13]:

In both cases the speed, *V _{BS}*, of the light spot relatively to the object surface is the same.

Let us consider a typical geometry of the dynamic-speckle sensor as shown in Fig. 2 (with incorporated video). It is convenient to arrange the beam deflector and the photodetector inside the same optical head. Therefore, the position of the illuminating beam waist in respect to the observer (distance between them is denoted by *D*
_{0}) is known a constant parameter defined by a particular geometry of the setup. The wavefront radius, *R _{W}*, is equal to the sought

*z*-distance between the beam waist and the object surface if

*z*is larger than the Rayleigh range of the Gaussian beam in TEM

_{00}mode (this condition is usually fulfilled in the experiment). Distance

*D*can be approximated as

_{S}*D*=

_{s}*D*

_{0}+

*z*assuming that the angle between the axes of illumination and observation is not very large (<30°). Anyway, for larger angles it is always possible to introduce a geometrical correction factor. It should be noted that the speed of scanning,

*V*, may also depend on

_{BS}*z*-distance. However, by proper design of the focusing optics this dependence can be minimized. Within these approximations we obtain a simple equation for

*z*-distance:

Here the parameters *D*
_{0} and *V _{BS}* are known while

*V*is to be measured. The fastest technique for speckle-velocity measurement is spatial filtering [13] in which the speckle pattern is filtered in the simplest case by a grating or Ronchi rulings. Radiation transmitted through the filter is modulated at a frequency,

_{SP}*f*, which is proportional to the velocity of dynamic speckles:

_{SP}*f*=

_{SP}*V*/Λ, where Λ is the period of the spatial filter. Finally the equation for calculating z-distance to the object surface in a dynamic-speckle sensor is

_{SP}Since the parameters of Eq. (4) are either preliminary known (*D*
_{0}, Λ, and *V _{BS}*) or online measured (

*f*), the dynamic-speckle sensor provides absolute measurements of the distance between rough surface and the beam waist position.

_{SP}Arrangement of a dynamic-speckle sensor is very simple and it does not include any sophisticated elements as one can see in Fig. 2. The laser beam is generated by a laser-diode and deflected either by a fast-rotating polygonal mirror or an acousto-optic deflector. After passing a focusing lens, the divergent laser beam rapidly and repeatedly scans the object surface. Light scattered from the object is filtered by a spatial filter (Ronchi rulings) and then collected into a photodiode by means of conventional lens. There are no optical elements between the surface and spatial filter: the plane of the filter is the observation plane and *D*
_{0} in Eq. (3), (4) is the distance between the filter and the focus position of the illuminated beam.

Experimental study of *z*-distance accuracy achieved in a dynamic-speckle sensor with a fast acousto-optic scanner was carried out in [14]. It was shown that the standard deviation from the mean *z*-distance is 110 μm when the averaging is completed over a single scan with the duration of 2.5 μs. According to the central limiting theorem, the deviation decreases as the square root of the ensemble length. Since a single scan can be performed very quickly, averaging after many scans allows accuracy improvement within still short time window. Averaging must be done over statistically independent traces. However, if the same area of the object surface is scanned several times the photocurrent traces are always exactly the same. Therefore, searching conditions which should be fulfilled to get uncorrelated responses of photodiode is the key problem for optimal averaging.

## 3. Correlation function of dynamic speckles

Statistical properties of dynamic speckles can be described by space-time correlation function [4,8]. When the moving surface is illuminated by a Gaussian beam and scattered light forms a speckle pattern in the observation plane after propagation in free space, the space-time correlation function of speckle-intensity fluctuations is [8]:

Here **r** is the radius-vector between two points of the observation and *τ* is the time difference between two moments of the observation. The parameters *τ _{C}* and

*τ*are the coherence time and the time delay required for a speckle to translate a distance

_{d}**r**, respectively. These parameters are defined as following in [8]:

It can be shown that in the case of light-beam scanning, the correlation function is expressed by the same Eq. (5) but the parameters *τ _{C}* and

*τ*are denoted in a slightly different form:

_{d}This modification is required due to a difference between the schemes with moving surface and scanning beam shown in Fig. 1. where both speckle displacement and its speed are calculated with different scale factors (cp. Eqs. (1) and (2)). In the Eqs. (5)–(8), the radius *r _{SP}* of an average speckle is defined by

Here *λ* is the laser wavelength and *r _{B}* is the half-width of the illuminating spot at the object surface.

*D*and

_{S}*R*were denoted before Eq. (1). Note that the definition of the speckle radius is the same in both cases (moving surface and laser-beam scanning) because it is the statistical parameter for static speckle pattern and signifies the correlation length at the spatial correlation function

_{W}*g*

^{(2)}(

**r**,0)-1 becomes 1/

*e*times the peak value.

For configuration of the system with arbitrary 2D scanning it is convenient to change the temporal variable *τ* in the correlation function for a new spatial variable **r**
_{0}=**V**
* _{BS}τ*, which represents the shift of the scanning beam in respect to the object surface. Considering Eqs. (7) and (8), with this variable the correlation function of Eq. (5) can be rewritten:

The equation for the correlation function given by Eq. (10) is more suitable for cross-correlation analysis of speckle patterns generated by reflection of the scanning beam from different parts of the object surface. First term in the product shows the spatial correlation properties of the speckle in the observation plane. The maximum of the correlation function at the coordinate **r**=**r**
_{0}
*D _{S}*/

*R*in the observation plane obviously corresponds to the shift of the speckle pattern at this distance when the scanning beam is shifted for a distance of

_{W}**r**

_{0}at the surface. At the same time, the correlation radius of the speckle pattern itself is always equal to

**r**

*. Second term of Eq. (10) shows that the speckle pattern significantly changes (in meaning of the correlation function decreasing) only when scanning the object surface is shifted by a half of the beam width at the object. The decay of the correlation function when the scanning beam is shifted for*

_{SP}*r*at the object surface with simultaneous shift of the speckle pattern defines a correlation parameter, the translation length in the observation plane, as

_{B}*L*=

_{T}*r*/

_{B}D_{S}*R*. This parameter indicates the distance in the observation plane at which a moving speckle pattern significantly changes its spatial structure [8].

_{W}In a dynamic-speckle distance-sensor the speckle pattern is spatially filtered and collected into a photodiode from which we get an informative signal as shown in the chart of Fig. 2. When designing a multichannel system with averaging of estimated signal frequencies, the following correlation properties are to be considered: correlation of a signal from a single scan, correlation between separate scans, and correlation of signals obtained from photodiodes at different locations in the observation plane. Statistical properties of the signal from the single scan are evidently determined by the dynamic speckle pattern itself and by the aperture of the spatial filter. It is clear that if the aperture size of the spatial filter is greater than the translation length *L _{T}*, the latter together with the mean speckle velocity will determine the correlation time,

*τ*, of the signal. This was shown by Veselov and Popov [9]. They estimated the spectral width of light-power modulation of dynamic speckle pattern generated by a moving scattering surface and transmitted through a spatial filter. With notations adopted in this paper and after corrections to the case of light beam scanning, the spectral width of light-power modulation frequency centered at

_{CP}*f*=

_{SP}*V*/Λ can be rewritten from [9] as:

_{SP}where *D _{F}* is the size of the spatial filter in the direction of speckles motion (orthogonal to the grooves of the Ronchi rulings). The correlation time,

*τ*, of a harmonic photodiode signal is inversely proportional to the spectral width:

_{CP}*τ*=1/

_{CP}*π*Δ

*f*. Consequently, when the scanning beam passes the distance

*L*=

_{CS}*V*the next part of the photocurrent trace becomes uncorrelated with the previous one. Using Eq. (7), the correlation length

_{BS}τ_{CP}*L*is expressed as

_{CS}In all our experiment, the second term of Eq. (12) is much smaller than the first one that simplifies expression for the correlation length: *L _{CS}* is simply equal to a half of the illuminated spot size,

*r*. As one can see this coincides with the conclusion obtained from the analysis of Eq. (12). The oscilloscope trace shown in the chart of Fig. 2 does also confirm this conclusion. It was recorded when the object surface was scanned by the laser beam (the spot width at the surface is 400 μm in the direction of scanning) at the speed of 60 m/s. As one can see from the chart of Fig. 2 the average length of harmonic oscillations is

_{B}*τ*=2.6 μs. The same value was also obtained by statistical analysis of a large number of similar traces. This corresponds to the correlation length

_{CP}*L*=156 μm, which is quite close to the half-width of the laser spot.

_{CS}As suggested above, for the optimization of the dynamic-speckle sensor it is important to find other correlation parameters related to the plane of both object and observer. Here we introduce two additional parameters. First, *L _{CD}*, defines the minimal distance on the object surface at which the adjacent scans should be displaced to result in uncorrelated responses of the photodiode. According to Eq. (10)

*L*is equal to

_{CD}*r*. Second,

_{B}*L*, is the minimal distance between two non-overlapped areas at the observation plane situated so that the light collected from them into different photodiodes generates uncorrelated responses. According to Eq. (10), the speckle patterns can be assumed statistically independent if two spatial filters (from which we collect the light into different photodiodes) are located at the distance of

_{CP}*r*along the axis perpendicular to the scanning direction, or at the distance of

_{SP}*L*along the scanning direction. In the next section we describe experiments devoted to the estimations of these parameters.

_{T}## 4. Experimental results

#### 4.1. Experimental setup

Basic schematic of the experimental setup is shown in Fig. 2. A light beam was generated by a laser diode RLDB808-350-3 at the wavelength of 808 nm with the output power of 350 mW. The coherence length of the laser diode was rather small (≈1 mm) but it was enough to obtain a speckle pattern with high visibility. In the experiments described below, an acousto-optic deflector (AOD) manufactured by Scientific Instruments, Ltd., Russia was used as a scanner. AOD provides periodical deflection of the laser beam with a period of 20 μs for our particular deflector. Conventional biconvex lens with a focal length of 50 mm was used to focus the light beam out of the object surface. Typical scanning speed achieved in our experiments is in the range from 50 to 150 m/s. Grounded plain metallic plate was used as an object. It was fixed in a computer-controlled translation stage, which is capable for precise positioning of the object in three dimensions with accuracy of 0.5 μm. The light scattered from the object was filtered by a spatial filter (Ronchi Rulings with Λ=50 μm) and then collected into the photodiode by using conventional lens. In some experiments cylindrical lenses were added into focusing or collecting optical systems. The signal from the photodiode is either stored in the digital oscilloscope (Agilent Infiniium 54831B) or processed in real time in a laboratory-made digital frequency-meter.

After scanning a certain area of the object, it was displaced for a small distance and scanned again. Thereafter, cross-correlation of the recorded photocurrent traces was calculated. We carried out experiments with both transversal and longitudinal displacement of the object.

#### 4.2. Transverse displacement of the scan

In this experiment we added a cylindrical lens to the focusing system so that the light beam at the surface had the elliptical shape sizing 600 μm in the direction of scanning and 60 μm in the orthogonal one. Each following scan was shifted for 2 μm in respect to the previous one. We have recorded 2300 oscilloscope traces by displacing sequentially the object surface orthogonal to the scanning beam. Figure 3 shows three typical responses from the recorded set of the data. The surface was shifted by 2 μm between the moments of recording of the traces 1 and 2, while the shift was 50 μm between the traces 1 and 3. It is seen that the shape of the trace 2 is almost the same as of trace 1 but the trace 3 has the shape completely different from both previous traces.

To estimate the correlation parameter *L _{CD}* we calculated the cross-correlation of a selected trace with all other traces. A typical result for one of the scan in the middle of the set is shown in Fig. 4. Here zero in the abscissa axis is referred to the scan position at the surface corresponding to the chosen trace. Repeating calculations of cross-correlations for differently chosen traces, we have found that the average width of the correlation peak is 30 μm. Remembering that the size of the illuminating beam in the direction of surface displacement is 60 μm, we obtain that the parameter

*L*is equal to

_{CD}*r*in accordance with predictions of Eq. (10). The experiment was repeated with different cylindrical lenses that provide changing the beam size in the direction orthogonal to scanning but keeping the other size the same (600 μm). With beam sizes of 400 and 1400 μm we measure the correlation parameter

_{B}*L*has been measured as 200 and 700 μm, respectively, i.e. equal to

_{CD}*r*.

_{B}#### 4.3. Longitudinal displacement of the scan

No cylindrical lens was used in this experiment, which results in the beam size of 600 μm in the direction of scanning and 400 μm in another direction due to astigmatism of our laser diode. Each subsequent scan was shifted from the previous by 3.5 μm in the direction of scanning. 400 oscilloscope traces of sequentially shifted scans were recorded. For better visualization of the correlation properties both instantaneous frequency and signal amplitude were calculated for every scan and presented as a color image shown in Fig. 5. The signal amplitude is coded as image intensity and the frequency is presented by pseudo-color according to the color map shown on the right. Figure 5 shows a collection of 400 sequential scans, which corresponds to the longitudinal displacement of 1.4 mm. Here the abscissa-axis indicates the scan number and the axis of ordinates is the current time of the photocurrent trace, which is readily recalculated into a position of the scanning beam at the surface as *x*=*V _{BS}t*. One can clearly see the tilted structure at approximately the same angle to the abscissa-axis over the whole image. This pattern appears due to repeated signatures of the same signal, which is shifted in time domain because of sequential displacement of scans in time in the direction of scanning. Since every next scan starts at a different point and spacing between these start points is less than the length of the scan, consequent scan probes the part of the surface which was already scanned by the previous trace. The moving speckle formed by scattered light from the same surface area generates almost the same response of the photodiode after spatial filtering. However, the signal signature is shifted with respect to the previous one due to shift of the scan window. The average number of adjacent scans correlating each other was found to be about 80. Since the longitudinal shift between the adjacent scans is 3.5 μm, we get the width of the correlation peak being equal to 280 μm. Therefore, the correlation parameter

*L*in the case of longitudinal displacement of the surface is again about a half of the full size of the illuminated spot at the surface,

_{CD}*r*.

_{B}## 5. Multi-channel sensor

In previous sections we discussed correlation properties of filtered dynamic speckles in the case when measurements are performed by a single photodiode. However, many kinds of objects provide a light scattering in a wide solid angle, while only a small part of the scattered light is collected into a photodiode. For example, with our metallic plate about 0.15% of the reflected light power was collected. The rest of the scattered light was not used. Installation of the additional photodiodes behind larger spatial filter (or several spatial filters with respective photodiodes) with averaging of instant frequency data would allow the improvement of a distance-measurement accuracy if responses of these photodiodes are not correlated. In this section we report an experimental study of correlation properties of dynamic speckles when scattered and filtered light is collected into at least two photodiodes.

#### 5.1. Experimental arrangement

To find the minimal distance between the adjacent photodiodes at which their responses are uncorrelated (correlation parameter *L _{CP}*) we modified our experimental setup. Schematic layout of the modified setup is shown in Fig. 6. The laser beam is scanning the object surface in the direction perpendicular to the plane of the drawing. Correspondingly, the grooves of the spatial filter are parallel to the plane of the drawing. In contrast with the arrangement shown in Fig. 2, the scattered light here is collected by cylindrical lens, whose axis lies in the plane of the drawing. Insert in Fig. 6 shows 3D-view of the collecting optics and the spatial filter. Scattered and filtered light power is measured by a photodiode fixed in a translation stage, which provides its movement along the line focused by a cylindrical lens. Displacement of the photodiode is orthogonal to the direction of scanning. The photodiode aperture is 0.25×1.9 mm2 where the first size is measured along the direction of its displacement.

Two cylindrical lenses were added in the illuminating part so that their axes were perpendicular to the plane of the drawing in Fig. 6. By varying their position we were able to change the size of the scanning beam in the direction orthogonal to the scan while the other size was kept the same since it is defined by the biconvex lens. Maintaining the second size constant is very important because it allows achieving a high signal-to-noise ratio in all the experiments. As known [11], SNR of the photocurrent in dynamic-speckle sensor depends on the beam size in the direction of the scan because it defines the average size of the speckles in the plane of the spatial filter.

#### 5.2. Correlation parameter L_{CP} versus the beam size

For each position of the cylindrical lenses in the illuminating arm we recorded oscilloscope traces of the photocurrent at different positions of the photodiode. Cross correlations of the photodiode responses were calculated in a similar way at it is described in Sect. 4.2. Thereafter, the correlation parameter *L _{CP}* was estimated as a correlation-peak width. Repeating the measurements for different positions of the cylindrical lenses we obtain the dependence of the parameter

*L*on the beam size 2

_{CP}*r*, which is shown in Fig. 7.

_{B}As one can see, in contrast with the previous case, the correlation parameter is inversely proportional to the beam size. It is worth noting that the correlation displacement is equal to the mean speckle size (also measured in the direction orthogonal to the scan) at the plane of the spatial filter: *D _{SP}*=2

*D*/

_{S}λ*πr*. The latter dependence is shown in Fig. 7 by the solid line. Therefore, the half-width of the correlation peak in the observation plane is equal to the radius of the speckles,

_{B}*r*, as it is predicted by the correlation function of Eq. (10).

_{SP}## 6. Discussion

The parameter *L _{CP}* defines the minimum distance between the centers of two separated parts of the spatial filter after which the filtered light power displays completely different excursions in time. It is worth noting that the distance between the respective photodiodes can be either larger or smaller depending on the design of the collecting optical system. In this sense it is more convenient to use the parameter of the angular separation of the scattered light

*α*≅

_{COR}*L*/

_{CP}*D*, which depends only on the beam size at the surface:

_{S}*α*=2

_{COR}*λ*/

*πr*. From one hand,

_{B}*α*defines an optimal amount of the scattered light to be collected into the photodiode. When the light is collected from a larger angle, summation of uncorrelated signals leads to decrease of the photocurrent’s modulated part with respect to its dc component thus diminishing the signal-to-noise ratio. From the other hand, optimal performance of a multi-channel dynamic-speckle sensor requires that two adjacent photodiodes receive the light scattered at different angles with a difference larger than

_{COR}*α*.

_{COR}In the general case the multi-channel sensor may consist of a number of spatial filters (optically connected to respective photodiodes) having different spatial periods, Λ* _{i}*, and situated at different distances,
${D}_{{F}_{i}}$
, from the beam waist of the scanning beam. The main requirement for the achievement of the improved accuracy is that the angular separation (as it is viewed from the object side) between centers of any pair neighboring filters should be larger than

*α*. When the object surface is scanned by a laser beam, the signal in each channel will be modulated at a frequency of ${f}_{{\mathrm{PD}}_{i}}$ (here

_{COR}*i*indicates the channel number) near the mean frequency

*f*=

_{SP}*V*/Λ. Since all modulation frequencies are measured simultaneously, they will correspond to the same distance

_{SP}*z*between the object surface and the beam waist. Assuming that multi-channel sensor consists of

*N*photodiodes, the mean distance

*z̄*is calculated with help of Eq. (4):

Here calibration coefficients *C _{i}* take into account geometrical positions of the spatial filters. In the case of a small angle between the axis of the illumination and the line connecting the illuminated area of the object with the center of

*i*-filter,

*C*is close to unity.

_{i}In Sect. 5 we analyzed the configuration in which multiple channels are positioned orthogonal to the direction of the laser-beam scanning. However, these channels can be also situated along the scan direction. In the latter case the minimal distance, which leads to uncorrelated signals, is defined by another correlation parameter, the translation length of dynamic speckles in the plane of the spatial filter, *L _{T}*=

*r*

_{B}*D*/

_{S}*R*[8]. Therefore, one can use two-dimensional array of photodiodes for multi-channel receiving of spatially filtered scattered light.

_{W}Note that *z*-distance in a multi-channel sensor is measured during a single scan of the object. Since photodiodes responses of different channels are statistically independent, accuracy of *z*-distance measurements will be √*N* -folds better than for single-photodiode dynamic-speckles sensor. Short duration of a single scan implies fast parallel data processing for the calculation of mean modulation frequency. Different mathematical algorithms such as zero-crossing, fast Fourier transfer (FFT), instantaneous frequency, etc. can be applied for frequency calculation. We have processed a large number of recorded photodiode responses (such as shown in Fig. 3.) using three abovementioned algorithms and found that FFT and instantaneous frequency algorithms give smaller error of *z*-estimation than zero-crossing. However, the typical frequency estimation errors, for both zero-crossing algorithm and FFT, as it follows from our experiments, are of the same order of magnitude. Since zero-crossing algorithm is the simplest and the fastest, it is the most suitable for real-time data processing in a dynamic-speckle sensor.

Further increase of *z*-distance accuracy can be achieved by multiple scanning providing that each subsequent scan is shifted from the previous one by the distance of *L _{CD}*=

*r*. Suppose that during the scan numbered by “

_{B}*j*” the mean frequency of the photocurrent in

*i*-channel is measured to be ${f}_{{\mathrm{PD}}_{\mathrm{ij}}}$ . Then after

*M*scans over the object surface,

*z*-distance will be

Multiple scanning of the object surface can be readily implemented with continuously moving production lines. By scanning the surface in the direction orthogonal to the line movement one can adjust the beam size and scanning-repetition rate with actual speed of the line so that optimal covering of the surface with beam scans will be achieved. It is also possible to install additional deflector in the optical head to provide necessary shift of the scan.

Basing on the experimental study of a single-photodiode dynamic-speckle sensor [14], which reports *z*-distance accuracy Δ*z*=100 μm for scan duration *t _{SC}*=2.5 μs, we may estimate accuracy that can be achieved in a sensor with 25 channels. 400 statistically independent measurements will be performed in each channel during a time-window of 1 ms because the average length of the harmonic oscillation is about 2.5 μs as seen in the chart of Fig. 2. Multiplying by 25 independent channels, we obtain 10000 independent measurements. Since accuracy increases as the square root of the number of independent measurements, we may expect 100-folds better accuracy, i.e. a multichannel sensor is capable for measuring the distance with accuracy of 1 μm within one-millisecond response time.

## 7. Conclusion

Statistical properties of dynamic speckles formed during illumination of an optically rough surface by a deflecting laser beam have been analyzed. The technique based on these speckles is very promising for fast measurements of the profile of any optically rough surface. Scanning an object surface by a coherent laser beam is more advantageous compare to the previously proposed and widely analyzed configuration in which an object is moving in respect to the laser beam and the photo-detector. It has higher flexibility and it allows fast acquiring a large amount of useful data that leads to improvement of the measurements accuracy.

The space-time correlation function of the light-intensity fluctuations has been modified to make more convenient an analysis of the correlation parameters of the light scattered from an object surface during its scanning by a coherent beam. Experimentally estimated correlation parameters of the photo-detector responses on the spatially filtered dynamic speckles coincide with theoretical predictions obtained from the modified space-time correlation function. These parameters allow us to find an optimal geometry of a dynamic-speckle sensor possessing both high accuracy of measurements and fast response time. Proposed sensor perfectly works with the most of optically rough surfaces (metals, papers, plastics, etc.) and allows achieving micron accuracy within a millisecond response time. The arrangement of the sensor is quite simple and it can be useful for various industrial applications because the rough surfaces are typical objects there.

## Acknowledgments

The Authors from Finland acknowledge the financial support of the Academy of Finland under the project 107554.

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