1. Introduction

High power lasers in manufacturing are used for cutting, welding, marking, drilling, surface treatments and powder processing on a variety of materials ranging from metals, to polymers and natural materials like leather or stone. Low power lasers are increasingly applied for optical inspection, by measuring position, distance, speed and surface properties like roughness and gloss.

The optimal selection for a given application of a laser source and its processing parameters, namely spot size and energy distribution, requires the characterization of the laser beam geometry and quality.

This paper introduces the most relevant laser beam parameters in manufacturing and describes the development of an apparatus to optically control the laser beam and measure these parameters by image analysis.

^{International Organization for Standardization (2005)} methods and standards for the laser beam characterization have been described in the literature (^{López & Villagómez, 2015}) and are commercially available. The beam quality can be measured by a Shack-Hartmann wavefront sensor. Among the most consolidated are ^{Yan et al. (2018)}, ^{Sheldakova et al. (2007)} and ^{Schäfer et al. (2006)}. More sophisticated (and complex) methods have no moving components (Schulze et al., 2015), are based on entropy (^{Porras et al., 2017}) or optical resonation (^{Kwee et al., 2007}). A general review of commercial instruments used in beam analysis is available in ^{Roundy (2000)}, however several technical aspects to achieve a fully functional setup are not available; in particular a camera sensor model is proposed to design the noise filter. Some works describe methods for the analysis of the beam profile with camera sensors. A CCD camera is used in ^{Schweinsberg & Van Woerkom (1999)} to characterize the M ² of a Ti-Sapphire laser; a movable lens permits the modification of the beam section scanned by the camera. Another lens provides the spot magnification, imposing a long distance between lens and camera. A method to estimate the deviation of the beam from an ideal envelope is presented in ^{Denisov & Karasik (2009)}. A general principle for layout design of a camera-based laser analyzer is described in ^{Alsultanny (2006)}. Educational experiments with low cost setup, including the characterization of a Gaussian beam, are described in ^{Andrews et al. (2007)}.

This work starts from recalling mathematically and physically the laser quality, namely beam waist, divergence, Rayleigh range and M ², which have greatest interest in industrial applications. This article focuses on the design and development of a simple and low-cost laboratory apparatus to optically control the laser beam and measure these parameters by image analysis. Possible measuring strategies are critically compared, sensor design is discussed and experimental validation is provided.

2. Laser beam parameters

2.1. Gaussian beam model and quality parameters

Laser beams are spatial and time high coherent electromagnetic waves presenting a preferred direction of propagation.

The fundamental model for a laser beam is the Gaussian beam with TEM₀₀ intensity distribution shown in Figure 1(a). This simplest case can be described by an axial-symmetric model with just two (polar) coordinates.

Figure 1 Intensity distribution of a Gaussian beam (a) and lower order modes (b, c). Intensity distribution of a real mixed mode and/or optically distorted beam (d). 

The propagation and the transverse axes z and ρ are represented in Figure 2. The intensity function I(ρ,z) (Figure 2, detail) is the time-averaged power density with Gaussian shape:

Iρ,z=I0w0wz2e-2ρ2wz2 (1)

Figure 2 Gaussian beam propagation functions. 

The beam radius w(z) is the conventional beam width on the ρ axis and is expressed as

wz=w01+λπw02z2 (2)

Where w0=minz⁡wz and λ is the laser wavelength.

This function is defined by three parameters that characterize the Gaussian beam summarized in Table 1.

Table 1 Laser parameters for a Gaussian beam. 

Parameter	Symbol	Description
Beam Waist	w0=minz⁡w(z)	W 0 represents the smallest radius reached by the beam along its path
Divergence (half-)Angle	tan⁡θ=λπw0	The region in which limz→∞⁡wz=θz. is called far field and can be approximated with a cone. A small beam waist implies a large divergence angle, and vice versa
Rayleigh Range	zR=πλw02	At a distance equal to zR from the beam waist, the beam radius is wzR=2w0 The interval centered in z0 and with semi-aperture of zR is called near field and can be approximated with a cylinder

These parameters affect the laser beam quality and can be estimated by the apparatus described in this paper and are synthesized by the standard Beam Quality parameter M² defined below.

2.2. Beam quality

The most common parameter to quantify the beam quality is the M ² parameter. It can be used to characterize also lower order laser beams, with the intensity distribution shown respectively in Figure 1(b, c and d). As described in ^{Siegman (1997)}, in a resumption of the previous work in ^{Siegman (1993)}, M² is related" and keep going with to the focusing capability of the laser on a narrow spot at a given divergence angle. The best possible beam quality (lower M²) is achieved for a diffraction-limited Gaussian beam, which is a particular solution of the paraxial approximation of the equation wave in (1) and (2) having M² = 1 (M² minimum). Many real lasers approach this value, especially low power single mode lasers. High power lasers can have a very large M² of more than 100 or even well above 1000.

Better beams (lower M²) have the following benefits: they allow locating the laser source farther from the working surface, preventing damages to the laser source due to heat and material ejection (Figure 3, top); in laser delivery systems (like optic fibers) higher quality beams propagate at narrower beam size; hence smaller and cheaper optics can be used (Figure 3, bottom); beams remain collimated for a longer distance, increasing the tolerance for longitudinal alignment.

Figure 3 Laser focalization (top) and collimation (bottom) by a convex lens. 

3. System design

The measurement of the M ² parameter requires the acquisition of images of the beam sections along the beam path. This task is accomplished reshaping the laser beam using an optical hardware and by collecting a series of images by means of a computer-controlled camera.

Figure 4 outlines the strategy for the development of the apparatus, which includes two modules described in detail below, the optical system (hardware) and the image analysis algorithm:

Figure 4 General scheme for the design of the apparatus. 

The laser beam intensity distribution is measured by a CMOS sensor. A CCD has not been selected for the higher cost.

Figure 5 shows a prism mounted on a rail guide. The linear displacement of the prism varies the laser section being acquired by the camera. The beam reflection by the prism permits to use a shorter rail guide, saving space on the platform, as opposed to facing the camera directly towards the laser source. Moreover, it reduces the apparatus size also in the transverse direction as opposed to a couple of mirrors. It also allows keeping the camera fixed, avoiding the movement of this cabled device.

Figure 5 Scheme of the hardware layout. The camera records the beam intensity. The movement of the prism permits the collection of images along the beam path. 

4. Simulated image analysis

The main input for the apparatus design is the smallest measurable beam radius and it has been estimated by computer simulations according to the following model.

To simulate the image acquisition, each pixel p _ij is treated like an integrator of the light intensity generated by the laser

for all the rows and columns of the simulated image p(i,j), 1≤ i≤ N, 1≤ j ≤ M. The intensity from (1) has been simulated for a set of 10 radii W _(x,sim)=(1,2,..,10) pixels (Figure 6). p _ij is proportional to the energy collected during the exposure time T, on the area A _ij of the incident pixel.

Figure 6 Simulated images with random noise δn255. The estimated beam radius W _x,est is indicated. 

The simulated intensity p _ij is then quantized to set the output range between 0 and 255, simulating the compensation in the exposure time to avoid image saturation. Finally, random noise δn255 between 0 and 5 levels is added, based on the actual background noise of the sensor used:

The conventional radius along the principal axis x estimated from simulated images in discrete form, derived from (5) and (6) is:

where i₀ is the x coordinate of the centroid of image p²⁵⁵ (i,j) and is calculated as x₀ from expression (7) A similar expression to (10) can be derived for the y axis.

To eliminate the noise from image regions far from the beam axis from the computation in expression (10), a cutoff function defined as:

removes all the pixels pij255 below the threshold cut.

The moment-based expression (10) is also used to measure the radius W _x and W _y from real images. It weighs each pixel at position (i,j) by its intensity p _ij.

The expression in (10) is independent on the absolute intensity; therefore, quantization in (8) is possible. In addition, it is not influenced by the sensor gain, optical filtering or laser intensity density change as the beam radius changes with the distance from the sensor. It is also independent on the intensity distribution, as opposed to fitting or gradient-based methods.

However, this expression is very sensitive (quadratically) on the intensity of pixels far from the centroid (noise). For this reason, windowing is necessary around the estimated radius size, in order to limit the number of unnecessary pixels.

Figure 7 shows the estimation error:

as a function of the simulated radius of the laser. A similar plot can be obtained for ε_y.

Figure 7 Cutoff effect with random noise. 

It can be noticed that the minimum cutoff level must be above the top noise level, which was set to 5, as shown by the curves at cut = 6 ÷ 8 with lower ε_x. The threshold level of the cut parameter is fixed and only depends on the sensor noise level, which is upper limited and can be defined with current method for each new apparatus. In the image analysis, cut is set to 5.

By this simulation it is possible to estimate the minimum laser radius (beam waist) providing sufficient accuracy.

A lower laser beam waist has several benefits:

From Figure 7, it can be observed that the minimum error ε_x is achieved for cut = 6, for a radius greater than 5 pixels.

By this simulation, the minimum measurable radius for a real laser has been conservatively set to 10 pixels. The selected CMOS sensor has a pixel size of 5.2 µm, corresponding to a minimum beam waist with square envelope of 52(52 µm ².

4.1. M ² estimation

Figure 8 shows the results for a simulated beam with M ² = 1, obtained with the second method in Table 2. After fitting the data from 11 simulations, the error on M ² is about 1%, which is considered acceptable.

Figure 8 M ² estimation from the simulated image set in Figure 6. The blue stars represent the simulated values Wx,est2, the green squares the estimated values Wx,est2 and the red line the polynomial fitting of the estimated radius along the z axis. The pixel dimension is 5.2 µm. 

By the M² estimation from the simulated images the cutoff level (of 5) and the number of acquired images (conservatively set greater than 10) are selected.

5. Lens and stage length selection

Once the minimum measurable beam waist is defined, the focal length of the lens can be selected. The lens is selected analyzing the behavior of the beam waist and the Rayleigh range against the variation of the wavelength and the nominal exit beam radius W _op . For a Gaussian beam, under certain conditions (^{Self, 1983}), it holds:

From image simulation the minimum beam waist of 10 pixels, corresponding to 52 µm is used. Expression (13) is plotted in Figure 11. Once f is selected for a given W _op and λ, the stage length L can be estimated from Figure 12 for the same λ and for an estimated W _0,x using the expression:

Figure 9 CAD model of the developed apparatus: (1) diode laser CPS 196, (2u-d) protected silver mirror PF10¬03-P01; (2) complete periscope assembly RS99/M; (3) kinematic mirror mount KM100; (4) mounted standard iris, 15 mm ID15/M; (5) D=25.4 F=500.0 N¬BK7 biconvex lens LB1869; (6) mounted standard iris, 15 mm ID15/M; (7) 60mm right angle prism PS913; (8) micrometric translation stage PT1; (8s) micrometric drive; (9) dovetail rail guide RLA600/M; (9s) rail guide ruler; (10) CMOS camera DCC1545M. 

Figure 10 Scheme of the hardware layout. The analyzed laser beam is routed by an optical track (2, 3, 5 and 7) to the camera 10. The aligning optics (2 and 3) must redirect the beam into the focusing lens (5), following a predetermined path. 

Figure 11 Focal length f required for W₀>52μm. 

Figure 12 Suggested stage length L with an optimal lens. 

with a rail length of 5 times the Rayleigh range. To reduce the rail guide length, a right angle prism has been used as retro-reflector (Figure 5): the prism translation doubles the beam path, allowing space saving for the apparatus.

6. The final system

The conceptual scheme for the use of the apparatus is presented in Figure 10. The final system was built using off-the-shelf components, with special care to the affordability of the apparatus.

In addition to those listed in Figure 9, based on the described simulation and design criteria, the following components have been chosen:

-monochrome CMOS sensor DCC1545 from µ-Eye with 1024x1280 square pixels with side 5.2 µm, operative λ = 400 ÷ 800 nm; -biconvex lens with diameter 25 mm and focal length f = 500 mm; -micrometer rail guide with 600 mm of travel length.

The developed setup is transportable (less than 5 kg).

In a real application it is very difficult, even impossible, to move the laser source to adjust the beam path to match a determinate trajectory. The external laser beam is received by aligning optics and redirected on a predetermined trajectory along an optical track in Figure 9.

The CAD model of the apparatus, realized with ^{Dassault Systèmes (2011)}, is shown in Figure 9 and commercial components are listed and described below.

The laser to be tested should be placed in the free space between units (1) and (2), facing the upper mirror (2u). Any external laser can also be tested, if it can be directed to (2u). High power lasers up to 1 kW can be characterized using a beam splitter.

The elements (2u), (2d), (3), (4), (6) compose the aligning optics. The beam travels through the air from (1) to (10), following an optical track formed by (2), (3), (4) and (7). The mirrors of the periscope (2) and the kinematic mirror (3) can be moved to align the beam to the correct path marked by the apertures (4) and (6). Before the beam strikes the prism it is focalized by a convex lens (4). The prism position is controlled by a micrometer rail guide (8s) with 50 µm resolution. The prism is supported by the translation stage (8) on the rail guide (9), allowing the movement of the prism along the propagation axis and the variation of the scanned section by the camera (10).

7. Beam characterization software

The measuring process is controlled via PC, with image analysis software implemented using the sensor software libraries ^{Thorlabs (2010)}.

Two software modules and interfaces are available:

Figure 13 Interface of the acquisition module. Real image (top left) and 3D view with projections (top right). Preview of the propagation functions W_x(z) and W_y(z) estimated from a single image at each position. 

The image analysis module (Figure 14) processes the images extracting the beam radii and reconstructing the propagation functions W _x (z) and W _y (z) estimated from multiple images at each position (between 5 and 10). The two methods in Table 2 are available to estimate the beam parameters and to calculate M ². Multiple images reduce the background noise and increase accuracy but take about 200 ms per image.

Figure 14 Interface of the image analysis module. 

The total procedure including manual positioning takes about 10 minutes.

Figure 15 and Figure 16 describe the procedures for the image acquisition performed with the acquisition software (Figure 13).

Figure 15 Image acquisition procedure. Multiple images are stored at each position. 

Figure 16 Autoexposure procedure. This procedure automatically sets the exposure time to avoid image saturation, (max p_ij < 255). procedure. Multiple images are stored at each position. 

For each position, several pictures (default is 5) are taken and processed to measure the beam radius. Each image is obtained by averaging 5 frames to reduce noise. The whole procedure is detailed in Figure 15.

The exposure time is dynamically modified to prevent image saturation according to the scheme in Figure 16 to compensate the intensity increase with the radius reduction.

Special care must be taken to cleaning the optical elements (Figure 17).

Figure 17 Detail of a real laser intensity image. Diagonal waves (1) caused by the diffraction of the neutral density (ND) filter. Round interference fringes (2) caused by dust grains on the CMOS sensor. 

To reduce risks of faulty images with fingerprints or dust, morphology operation with kernel radius of 15 pixels size provides an alarm.

8. Validation of the apparatus

A visualization of the laser beam propagation, associated with radii functions, is offered in Figure 19.

Figure 18 Measured beam radii on the two main axes from 50 images (cut=5). The mean and the standard deviation result [pixels]: Wx-=63.01 σWx=0.27 Wy-=63.40 σWy=0.36  

Figure 19 Laser beam sections along the beam path. The red line represents the squared radius variation. The images evidence that W_0,x< W_0,y. 

The apparatus has been tested on a HeNe red laser and on a diode laser (^{Thorlabs, 2015}). Tests on real images evidence a small relative standard deviation in Figure 18, coming from both the laser and the sensor in the radius estimation, if an adequate cutoff level is selected (Figure 21).

Figure 20 Relative standard deviation of the measured beam radius along the beam axis. The measurement error is less than 1.8%. 

Figure 21 Effect of the cutoff level on the relative standard deviation. The beam radius is estimated over 50 pictures of the same section to evaluate the effect of the cutoff level on the mean and the standard deviation. 

It can be noticed that for cut = 4 ÷ 9 the relative standard deviation is lower than 1% of the measured beam radius.

A statistical analysis to estimate the variance of the measured parameters (M ² and W) from different data presents a low variability (Figure 20).

The radius propagation curves obtained with the apparatus show the expected parabola behavior (Figure 22). The M ² _x values for the diode laser are consistent with the ones available in the literature.

Figure 22 Radius propagation values and polynomial fitting (blue) from different data sets with error bars (red) indicating the mean and the standard deviations of measures and corresponding M_x ² parameter values for the diode laser ^{Thorlabs (2012)}. 

9. Conclusions

The design and the realization of a beam propagation analyzer (hardware and software) have been presented. The M ² and other relevant parameters measured by the apparatus have been characterized.

A camera sensor model has been proposed to estimate the minimum measurable beam radius (beam waist) to design the noise filter. The moment-based radius assessment is independent on the absolute intensity and distribution of the laser.

Current uses of the developed setup are laser beam characterization and staff training. Experimenters and equipment designers can benefit of the described design. The equipment and the software can be easily integrated into an existing system. The proposed system can be an alternative, in both industry and laboratory, to the more expensive commercial measurement systems. The system evolution includes a motor driven axis for mass production measurement; attenuating filters could be fitted if necessary in case of lasers exceeding the sensor power.