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 M² 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 M², 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.

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:

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

Where $w_0=\underset{z}{\min }w\left(z\right)$ and λ is the laser wavelength.

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

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 M² 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.

2.3. Real Beams and M²Parameter

The beam quality M² parameter adapts the Gaussian model to a real beam. It quantifies the quality in relation to the similarities between a real beam and a Gaussian beam. For real lasers two parameters are necessary: M²_x and M²_y . M²_x is defined as:

where W_(0,x) and θ_(r,x) are the beam waist and the divergence angle of the real beam with respect to the x axis, after focalization; BPP_real and BPP_Gaussian are the Beam Product Parameters of the examined laser and the ideal Gaussian beam. Similar expressions can be derived for the y axis. An extended discussion is available in Paschotta (2011) and Alda (2013).

The lower case w is used for the Gaussian beam model; the upper case W is used for parameters obtained from operation of the apparatus and from simulations.

It can be noticed that M² represents the ratio between the beam waist radius of the real beam and the beam waist radius of a Gaussian beam with the same divergence angle. M² defines the shape of the real beam along the z-axis, in that it defines the specific curve of the family in (2) In addition, it includes the information on ( and on the product W₀ θ_r, which are known and constant for a given laser.

Most real lasers are not Gaussian and require a different formulation for the beam radius W_(z) by means of second moments, as detailed below.

2.4. Second moment definition of the beam radius

The parameters W₀ and M₂ completely define the beam propagation. When the beam is not axial-symmetric, as for most real lasers, the beam radius can be defined by the second moment of I(x,y), where the ( axis is substituted by the principal axes x and y as in Figure 1(d). The radius propagation function in (1) and (2) become:

Where:

which are used by the image analysis software developed for the apparatus. Similar expressions can be derived for the y axis.

2.5. Characterization methods

The beam parameters M²_x , M²_y , W_0,x and W_0,y in (4) are calculated from the measures of the beam radius W(z) at several positions (e.g., 10 positions, with current setup) along the beam path. To estimate the beam waist W₀ and to calculate M² two methods are available from ISO (2005) and Siegman (1997). The algorithms based on these methods have both been implemented for redundancy and are outlined for comparison in Table 2.

The first algorithm requires the separate estimation of the beam waist W_0,x and of the divergence angle θ_r,x. The second algorithm permits to evaluate the parameters by fitting from images distributed in a shorter part of the beam path.

3. System design

The measurement of the M² 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:

• the hardware layout reshapes the beam to take measures both in the far and near field;

• the software interface controls the image acquisition and analysis to characterize the laser beam.

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.

3.1. Design of the optical system

Among the main hardware modules, in addition to the rail guide and the digital camera, is the focusing lens (Figure 5). Commercial lasers are usually equipped with an internal optical system to minimize divergence and technical specifications include a nominal exit beam radius W_op in addition to the wavelength (.

As shown in Figure 4, to design the apparatus, the focal length f and rail guide length L are selected based on these latter parameters and simulations of the Gaussian beam model, described below.

The main purpose of the lens is to reshape the beam as in Figure 2.The focal length should satisfy the following trade-off:

• minimizing the Rayleigh range, influencing the rail guide length L and

• keeping the beam waist above a minimum measurable size (approximately 10 pixels) which depends on the camera sensor resolution.

The camera resolution is critical for this aim. A simple camera model derived from those discussed in Lanzetta & Culpepper (2010) has been implemented to simulate the effect on the estimation of the beam radius caused by the spatial discretization and the intensity level quantization. These simulations provide a lower limit for the beam waist, which is used for the lens selection

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.

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 ${\delta }_n^{255}$ 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 $p_{ij}^{255}$ 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.

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:

• smaller sensor, higher focalization and more compact apparatus;

• smaller and lower cost lenses;

• lower optical aberration and better accuracy.

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. M²estimation

Figure 8 shows the results for a simulated beam with M² = 1, obtained with the second method in Table 2. After fitting the data from 11 simulations, the error on M² is about 1%, which is considered acceptable.

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:

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:

• camera setting (exposure time, gain, …) and image acquisition (Figure 13).

• image analysis to output parameters $W_{x,0}, W_{y,0}, z_{R_x}, z_{R_y}, M_x^2, M_y^2$.

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 M². Multiple images reduce the background noise and increase accuracy but take about 200 ms per image.

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).

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).

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.

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).

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 (M² 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 M²_x values for the diode laser are consistent with the ones available in the literature.

9. Conclusions

The design and the realization of a beam propagation analyzer (hardware and software) have been presented. The M² 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.

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Notes

Author notes

* Corresponding author. E-mail address: lanzetta@unipi.it (M. Lanzetta).