1. INTRODUCTION

Due to the technological advancement of the past decades, life without lasers is unimaginable. Many distinct laser source types exist, e.g., edge-emitting lasers (EELs) or vertical-cavity surface-emitting lasers (VCSELs), each having its own characteristics making it suitable for specific applications [1]. Additionally, beam shaping techniques allow to tailor the laser’s optical radiation to a desired irradiance profile [2]. To ensure the laser beam behaves according to its design, beam profiling is crucial. Laser beam profiling is the process to measure and quantify the (transverse) irradiance profile of a laser source. In general, laser beam profiling techniques can be subdivided into camera based approaches and scanning based approaches [3]. In the former technique, a CCD or CMOS sensor is used to obtain a 2D image of the irradiance profile, where the gray value of each pixel represents the local irradiance of the laser beam incident on the sensor. For the latter technique, optical power is measured by a power meter while a knife-edge, slit, or pinhole mechanically scans through the laser beam and partially blocks the light.

The most critical property examined through beam profiling is the laser’s beam width. Rather inconveniently, different applications use different beam width definitions. For example, during a photolithograpic etching step, beam width is defined as the spatial extent of the beam’s irradiance profile hitting the material and needs to be kept under control [4]. In the context of laser eye safety classification, the beam width of the irradiance profile is evaluated based on its thermal impact on the human eye [5].

A beam profile that is analytically well described in literature is the Gaussian beam profile, which corresponds to the fundamental or lowest order transverse mode of a laser resonator cavity [6]. Many methods exist to measure the beam width of a Gaussian beam [7–10]. However, for a highly irregular laser beam profile, those beam width methods do not necessarily give accurate results, and the definition of its beam width becomes ambiguous [5]. In contrast, a top-hat beam profile has by definition a sharp edge. Its beam width can intuitively be defined as the radial extent from the center of the beam profile to its sharp edge.

Thus far, there exists no single universal definition of laser beam width. Multiple beam width measurement methods co-exist, i.e., they yield different width values, and universal conversion factors among them do not exist [11]. The most common methods are the ${\rm{D}}4\sigma$ or second order moment method corresponding to the ISO 11146 standard [12], the D86 method [13], and the knife-edge method [14]. For all beam width methods listed above, the outcome is sensitive to noise or background offset levels on the measurement data [3]. In an industrial setting, the imaging conditions for a camera based beam profiling technique are typically less controlled, indicating the need for a laser beam width method that works well for various irradiance profiles under poor imaging conditions. Moreover, depending on the application, it might be of interest to have a method that can estimate the beam widths of multiple laser beams from a single image while the laser beams have a spatial extent of only a few pixels on the sensor.

For these purposes, we investigate the usefulness of the so-called maximal thermal hazard (MTH) method introduced by Schulmeister et al. (2006) in [5]. In this method, one searches for the most hazardous combination of power within an area of the irradiance profile and the diameter of that area. Laser beam width is defined as the diameter where the ratio of the power within an area to the area’s diameter is maximal. The MTH method lends itself effortlessly for a camera based implementation, as each pixel represents the local irradiance incident on the sensor, yielding a spatial distribution of the profile that can be analyzed through image processing. So far, the MTH method has been proposed as a method to determine the thermal damage diameter of irradiance profiles since the standardized ${\rm{D}}4\sigma$ method is not considered to be appropriate [5].

In this paper, we will explore the feasibility of the MTH method for camera based beam profiling, in particular, for beam width estimations in a single transverse plane of the laser beam. First, the performance of the MTH method to characterize the laser beam width will be examined on computer generated irradiance profiles and on experimental data from a VCSEL array. Realistic image noise and background offset levels will be added to the computer generated images of the laser beams to study their impact on the MTH method performance. We will compare the performance of the MTH method with other beam width measurement methods prominent in literature. Second, we will expand upon the MTH method to accurately determine beam widths that extend only a few pixels using a Monte Carlo technique.

2. MTH METHOD AS A BEAM WIDTH ESTIMATION METHOD

In this section, we will first introduce the computer generated and experimentally measured irradiance profiles that are used in this paper. Next, we will discuss the noise model applied to the computer generated irradiance profile images to simulate real life imaging conditions. Then, we will describe three beam width methods common in literature—${\rm{D}}4\sigma$, D86, and knife-edge—as well as the MTH method. Last, we will assess the performance of the MTH method by comparing its estimated beam widths to those obtained using the other beam width methods. All implementations and analyses have been done in MATLAB R2020b.

A. Irradiance Profiles

In this paper, we will use computer generated images of top-hat and Gaussian irradiance profiles. A top-hat laser beam is defined by an irradiance profile $I(x,y)$, which has a constant value across its transverse spatial extent. Mathematically, a circular top-hat profile is formulated as

Alongside the computer generated irradiance profiles, we will also use experimental data of the emission profile of a multi-mode VCSEL. The near field of such a VCSEL has generally a highly irregular irradiance profile due to the emission of multiple high order transverse modes and potential current crowding [15]. The images were taken by a picosecond high speed intensified camera (ICCD) (4Picos, Stanford Computer Optics). An image of the recorded profiles can be found in Section 2.E, Fig. 6.

B. Noise Model ICCD Camera

The CCD or CMOS sensors used in camera based beam profilers typically exhibit noise that disrupts the image information and is therefore undesired for image processing [16] Beam profiling techniques on noisy images are known to be unreliable, and appropriate measures, such as correcting offset levels due to background illumination or isolating the beam profile to limit the calculation area, need to be taken [3,17]. The image noise using a CCD or CMOS sensor can be modeled by a Poisson–Gaussian noise model given by

To apply this noise model on our computer generated beam profile images, we need to determine realistic values of noise components ${\eta _P}$ and ${\eta _N}$. We base ourselves on the ICCD camera used to obtain the experimental data of VCSELs in this paper. An ICCD camera consists of an image intensifier placed in front of a CCD sensor and has the advantage of very fast shutter times and detection of low light levels as compared to a generic CCD sensor. Similar to a CCD, the noise sources for an ICCD are signal-dependent photon shot noise and signal-independent dark current noise and readout noise [19]. Based on the procedures described in [20–22] and knowing that the ICCD camera captures 10-bit images, we have experimentally determined the sensor’s gain factor $\alpha$ to be equal to 0.63 LSB/electron. Using the gain factor $\alpha$ to convert the pixel’s digital signal values from LSB into electrons, the rate parameter of the Poisson distribution representing signal-dependent noise component ${\eta _P}$ is obtained. Additionally, signal-independent noise component ${\eta _N}$ is described by a zero-mean normal distribution with an experimentally determined standard deviation of 4.31 LSB, corresponding to 6.84 electrons.

The computer generated irradiance profile images are created using definitions of the top-hat and Gaussian irradiance profiles, given by Eqs. (1) and (2), respectively, with an irradiance amplitude ranging from zero to ${S_{{\max}}}= 2^{10} - 1 = 1023\; {\rm{LSB}}$. The value of each pixel $(i,j)$ in the computer generated image is proportional to the optical power incident on the pixel. To apply Poisson–Gaussian noise to these images, the pixels’ digital signal values are converted to electrons using the sensor’s gain factor $\alpha$. In this paper, the computer generated irradiance profile images are always normalized by dividing with ${S_{{\max}}}$ before estimating the beam widths using the methods described in Section 2.C.

Apart from the noise described by the Poisson–Gaussian noise model given above, images taken by CCD or CMOS sensors can have a positive offset level as the result of dark current in the sensor or background illumination. To have accurate beam width estimations, it is critical to correct for the offset level through accurate background correction. For example, in the case of the ${\rm{D}}4\sigma$ method, pixels with a positive offset level surrounding the laser beam can cause a severe overestimation of beam width. Therefore, a background image is created by averaging a set of images recorded without the sensor being illuminated by the laser source. This background image is then subtracted from the images under consideration. Due to the readout noise of the sensor, it is possible to have negative pixel values in the non-illuminated regions of the image [3]. But since it takes effort to perform precise background correction, it is of interest to have a beam width estimation method that is insensitive to these offset levels.

C. Beam Width Measurement Methods

1. D4$\sigma$ Method

The D4$\sigma$, or second order moment, beam width is defined as four times the standard deviation of the irradiance profile along the $x$ and $y$ axes. The ${\rm{D}}4\sigma$ beam width for elliptical beams, as set by the ISO 11146 standard [12], is given by

2. D86 Method

In the D86 method, one looks for the diameter of a circle centered around the beam’s center that encloses 86.5% of the beam power [13]. The enclosed power percentage may seem arbitrary, but stems from the fact that the D86 beam width of a Gaussian beam corresponds to the $1/{e^2}$ diameter.

3. Knife-Edge Method

Compared to the other beam width methods under consideration, the knife-edge method is not a camera based beam profiling technique. However, it can be extracted from camera images of the irradiance profile [3]. The optical power of a laser beam is measured by a detector while an opaque knife-edge slices through and partially blocks the beam. Considering an irradiance profile $I(x,y)$, the measured power in terms of the transverse position $x$ of the knife-edge is given by [8]

In the case of a Gaussian irradiance profile as defined in Eq. (2), Eq. (5) has a known solution:

If the laser beam has an irregular irradiance profile, Eq. (6) is no longer applicable. The beam width is then defined by a so-called clip width. Typically, this is chosen as the distance between the transverse positions $x$ of the knife-edge where 10% and 90% of the maximal power ${P_T}$ are measured [14]. To determine the beam width in both $x$ and $y$ directions, one has to repeat the measurement in both transverse directions. Moreover, if the beam profile is highly irregular, multiple scans in different directions are needed, increasing the measurement time [11]. In this paper, we will use the 10/90 clip width definition on the computer generated top-hat irradiance profiles, while the beam width for the computer generated Gaussian irradiance profiles is obtained through a fit of Eq. (6).

4. MTH Method

The MTH beam width of an irradiance profile at a single transverse plane is defined as the diameter of an evaluation window for which the ratio of the power within its area and the diameter of the window is maximal [5]. Since all irradiance profiles considered in this paper can be regarded as circularly symmetric, we focus on a circular evaluation window. Such a circular evaluation window yields a single diameter value for the irradiance profile. In general, the evaluation window can have any shape. For example, an elliptical evaluation window yields the transverse $x$ and $y$ beam widths along a rotation angle with respect to the sensor pixel axes.

 

Fig. 1. MTH ratio of (a) top-hat and (b) Gaussian irradiance profiles for a varying diameter of a circular evaluation window. The input beam diameter $2{r_o}$ is indicated by a gray line, while the beam diameter obtained through the MTH method is given by the black dashed line.

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A typical graph of the MTH method for a varying circular evaluation window applied to a computer generated top-hat profile and a Gaussian irradiance profile with a radius of ${r_o} = 50$ pixels is shown in Fig. 1. In this figure, we plot the ratio of the power within the evaluation window and the diameter of that evaluation window, henceforth referred to as the MTH ratio, as a function of the diameter of the evaluation window. For comparison purposes between top-hat and Gaussian profiles, we have normalized the MTH ratios. If the diameter of the evaluation window is smaller than the spatial extent of the irradiance profile, the MTH ratio increases with increasing size of the evaluation window. However, when the evaluation window becomes larger than the irradiance profile, the MTH ratio will decrease. In the case of the top-hat irradiance profile [Fig. 1(a)], this is easy to understand, as the power in the evaluation window will no longer increase for evaluation windows with a diameter larger than the beam diameter. In the case of the Gaussian irradiance profile [Fig. 1(b)], the maximum of the MTH ratio does not align with the input beam diameter $2{r_o}$. However, we will show in Section 2.D that the ratio between the input diameter and the MTH diameter is constant for Gaussian beams. Thus, the evaluation window diameter for which the MTH ratio is maximal can be considered a good estimate of the profile’s beam width. Moreover, this maximum can easily be determined for any irradiance profile, as illustrated in Fig. 1.

In the next section, we asses the quality of beam width estimation using the MTH method by comparing its performance with the other beam width methods. For varying sizes of the computer generated irradiance profiles, we study how the aforementioned methods perform when adding Poisson–Gaussian noise, when varying the amplitudes of the irradiance profiles, and when varying the background offset levels.

D. Beam Width Methods Tested on Computer Generated Irradiance Profiles

1. Performance without Added Noise or Background

Let us start to examine the performance of the MTH method on top-hat and Gaussian irradiance profiles for which no Poisson–Gaussian noise or background offset level is present in the computer generated images. The irradiance profiles have an amplitude ${I_o} = 1023\,{\rm{LSB}}$ and a beam radius ${r_o}$ varying from two to 100 pixels. The irradiance profiles are placed at the center of the computer generated image with a size of $640 \times 480$ pixels. For each beam radius ${r_o}$, we calculate the beam width using each method described in Section 2.C. All results are gathered in Fig. 2, where we plot the estimated beam diameter versus the input beam diameter, which corresponds to the double of the input radius ${r_o}$. In this figure, we want the estimated beam width to be equal to the input beam width.

 

Fig. 2. Estimated beam diameters for all considered beam width methods in function of the input beam diameter (${=} 2{r_o}$) for computer generated (a) top-hat and (b) Gaussian irradiance profiles. In (b), the beam diameter expected by the MTH method based on the numerically determined conversion factor is depicted by the gray line.

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For the top-hat irradiance profiles [Fig. 2(a)], both the standardized ${\rm{D}}4\sigma$ and MTH methods yield beam widths matching the input beam diameter. Contrarily, D86 and 10/90 knife-edge methods estimate the beam widths to be smaller than the input beam diameter with constant ratios of input versus estimated beam width of $0.912 \pm 0.037$ and $0.6869 \pm 0.0012$, respectively.

For the Gaussian irradiance profiles [Fig. 2(b)], ${\rm{D}}4\sigma$, D86, and knife-edge methods all measure beam widths equal to the $1/{e^2}$ diameter, which is expected by design. The MTH method, however, always yields a beam width smaller than the $1/{e^2}$ diameter, yet larger than the $1/e$ diameter [5]. The ratio between the diameter estimated by the MTH method and the $1/{e^2}$ diameter in Fig. 2(b) is fairly constant. Moreover, this ratio, or conversion factor, can be numerically determined by finding the local maxima of the MTH ratio, as illustrated in Fig. 1. Starting from the definition of the MTH diameter, the expression for a Gaussian irradiance profile given by Eq. (2), and switching to polar coordinates, the following expression is obtained:

As indicated by the gray line in Fig. 2(b), the estimated beam widths through the MTH method are in good agreement with the input beam diameter considering the conversion factor. Deviations between the input beam diameter and its estimate using this conversion factor are observed to be maximal at about three pixels large across the input beam diameter range considered in Fig. 2 and can be attributed to the sampling of the Gaussian irradiance profile through pixels with a limited resolution and the consequential loss of information.

2. Performance with Poisson–Gaussian Noise

To study the impact of image noise on the beam width methods, we now apply Poisson–Gaussian noise, as described in Section 2.B, to the computer generated irradiance profile images with an amplitude of ${I_o} = 1023\,{\rm{LSB}}$. No background offset level is added. We again examine the performance of the beam width methods by comparing their estimated beam diameters to the input beam diameters of the irradiance profiles. As noise will have an impact on the results, we generate the noisy images 100 times, repeat the beam width estimations for a distinct variety of input beam radii ${r_o}$ ranging from two to 100 pixels, and calculate the mean value and standard deviation. The results are summarized in Fig. 3.

 

Fig. 3. Estimated beam diameters for all considered beam width methods in function of the input beam diameter (${=} 2{r_o}$) for computer generated (a) top-hat and (b) Gaussian irradiance profiles with Poisson–Gaussian noise applied. In (b), the beam diameter expected by the MTH method based on the numerically determined conversion factor is depicted by the gray line.

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For both the top-hat [Fig. 3(a)] and Gaussian [Fig. 3(b)] irradiance profiles, all beam width methods yield precise and accurate beam diameters for input beam diameters above 150 pixels, with standard deviations smaller than a pixel’s width. For smaller input beam diameters, the ${\rm{D}}4\sigma$ method becomes progressively more imprecise and inaccurate as the mean value of and the spread on the estimated beam widths increases. Moreover, for input beam diameters of 10 pixels or less, even imaginary beam width values can be obtained. D86 and knife-edge methods are observed to be rather insensitive to image noise as long as the input beam diameter is 10 pixels or larger. Also, the MTH method yields precise and accurate results across the entire input beam diameter range for both top-hat and Gaussian irradiance profiles, taking into account the conversion factor discussed in Fig. 2(b).

While these results are obtained using the noise level of a specific ICCD camera, similar noise levels occur in all CCD and CMOS cameras. Hence, the excellent performance of the MTH method in noisy conditions will also apply to other cameras.

3. Performance for Varying Signal-to-Noise Ratios

Next, we want to examine the performance of the beam width methods for varying amplitudes of the irradiance profiles while Poisson–Gaussian noise is present in the image. This will reflect the methods’ performance as a function of the signal-to-noise ratio (SNR). If the signal level of an irradiance profile drops, the corresponding pixel values can be dominated by image noise. This can make beam width estimations inaccurate even though the beam shape itself is not altered. Therefore, having a beam width method that works well with a low SNR is very desirable.

The SNR is defined as the ratio of the signal level and noise level of the image. The signal level is taken as the maximal amplitude of the irradiance profile, expressed in LSB, converted to electrons using the gain factor $\alpha$ of the sensor. The noise level of the noisy image, generated through Eq. (3), is defined as the square root of the sum of the variances of image noise contributions. The SNR is then mathematically defined as

The performance of the studied beam width methods is assessed by comparing the estimated beam widths to the input beam diameter for top-hat and Gaussian irradiance profiles with amplitude levels ${I_o}$ ranging from 10 to 1023 LSB. By varying ${I_o}$ and keeping ${\sigma _N}$ fixed, we effectively change the SNR according to Eq. (8). Poisson–Gaussian noise is again applied on the images while no background offset level is added, and the beam width estimations are repeated 100 times. A fixed input beam diameter of 100 pixels is chosen, as each beam width method under consideration yields precise and accurate results when the irradiance profile has an amplitude ${I_o} = 1023$ LSB and when image noise is present (see Fig. 3). In Fig. 4, the mean values and standard deviations of the estimated beam diameters are plotted in function of the SNR.

 

Fig. 4. Estimated beam diameters for all considered beam width methods in function of the SNR of computer generated (a) top-hat and (b) Gaussian irradiance profiles with a beam diameter of 100 pixels when Poisson–Gaussian noise is applied.

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The ${\rm{D}}4\sigma$ method gets increasingly imprecise for decreasing SNR for both the top-hat and Gaussian irradiance profiles [Figs. 4(a) and 4(b), respectively]. For SNR below 15, the mean value of the ${\rm{D}}4\sigma$ method deviates strongly from the input beam diameter since the method can yield imaginary beam width values. On the other hand, D86, knife-edge, and MTH methods are much more robust against low SNR. In the case of top-hat irradiance profiles, these methods struggle with an SNR of only five or less, i.e., when the beam profile is submerged in image noise. This also leads to a slight deviation from the expected beam width for the MTH method seen in Fig. 4(a). The Gaussian irradiance profiles are inherently more difficult for beam width estimations with low SNRs, as their tails get subdued by image noise sooner.

Thus, even though a sufficient SNR is desired when using D86, knife-edge, and MTH methods on non-top-hat irradiance profiles, they are significantly less sensitive to low SNRs than the ${\rm{D}}4\sigma$ method.

4. Performance for Varying Background Offset Levels

Finally, we also study how the MTH method performs on computer generated images with positive non-zero offset levels. The strength of the added background level is expressed as a percentage. For example, if we apply a background offset level of 1% on an image of an irradiance profile with a maximal amplitude of 1023 LSB, the bias level of the image becomes non-zero with a value of 10 LSB while the computer generated irradiance profile still has a maximum equal to 1023 LSB. Depending on imaging conditions, the background level in real world images can vary across the image. For simplicity, we consider a uniform background level across the entire image ranging from 0% to 1%. The irradiance profiles have an amplitude ${I_o} = 1023 \,{\rm{LSB}}$ and a fixed input beam diameter of 100 pixels, as each beam width method under test yields precise and accurate results when image noise is present in the image (Figs. 3 and 4). Once the background offset level is added to the image, Poisson–Gaussian noise is subsequently applied, and beam width estimations are repeated 100 times. The performance of the MTH method is assessed by comparing estimated beam widths to those obtained through the other beam width methods. The results are gathered in Fig. 5 where we plot the mean values and standard deviations of the estimated beam diameters in function of the applied background offset level.

 

Fig. 5. Estimated beam diameters for all considered beam width methods in function of the added background offset level for computer generated (a) top-hat and (b) Gaussian irradiance profiles with a beam diameter of 100 pixels and with Poisson–Gaussian noise applied.

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For top-hat irradiance profiles [Fig. 5(a)], the common beam width methods are susceptible to the addition of background offset levels. Moreover, for any background offset level, ${\rm{D}}4\sigma$, D86, and knife-edge methods never yield the same results as when no background is added. As can be seen in the figure, both D86 and knife-edge methods have a tipping point, i.e., a background offset level from which their estimated beam widths suddenly become extremely inaccurate. For the D86 method, this tipping point is observed at a background offset level of about 0.35%, and for the knife-edge method, at about 0.75%. In the case of Gaussian irradiance profiles [Fig. 5(b)], these beam width methods are even more impacted by background offset levels. A tipping point is no longer observed for the D86 or knife-edge method.

In contrast, the MTH method yields consistently accurate beam diameters for each studied background offset level added to both types of irradiance profiles. Of course, the MTH method will not remain robust for any amount of added background offset level. In the case of an input beam diameter of 100 pixels and an irradiance amplitude equal to 1023 LSB, the MTH beam width for a top-hat irradiance profile remains stable up until a 30% background offset level, while for a Gaussian irradiance profile, it starts to become inaccurate from a 5% background offset level onward (not shown here).

The results shown in this subsection showcase the importance of proper background corrections for camera based beam profilers based on the ${\rm{D}}4\sigma$, D86, or knife-edge method. Furthermore, lowering the irradiance profile’s amplitude would amplify the sensitivity of these beam width methods to image noise and a background offset level, making them even more unreliable. The robustness of the MTH method against background offset levels is a significant improvement over the other beam width methods. The MTH method alleviates the need for very accurate background corrections using camera based beam profilers.

E. Beam Width Methods Tested on Experimental Data

Last, let us apply the beam width methods on experimental data of an array of identical multi-mode VCSEL emitters. Multiple VCSEL emitters are projected onto the 4Picos ICCD camera with an appropriate SNR and with a sufficient magnification to avoid being limited by the pixel resolution. An example of a background corrected image is shown in Fig. 6. Because of the clear spatial extent of each emitter, the nominal beam width of the captured VCSEL emitters is manually estimated to be about 130 pixels large. All studied beam width methods were applied on both background corrected and non-background corrected images. For each beam width method, the mean values and standard deviations of the beam widths of five VCSEL emitters are calculated. For neighboring emitters to not influence beam width calculations, the emitters’ beam profiles were cropped out of the image during processing by a 210 pixel wide square, as indicated by the green squares in Fig. 6. The results are summarized in Table 1 and illustrated in Fig. 6 where the beam width results are plotted on top of the captured image.

 

Fig. 6. Example of an image of a VCSEL array projected onto the 4Picos sensor, background corrected without removing the residual background offset level of 4%. The beam width estimations of the considered methods are superposed on the image. The green boxes represent the cropped image to single out the VCSEL emitter for analysis.

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Table 1. Computed Beam Widths in $\#$ Pixels on a Non-background Corrected Image and Background Corrected Image without Removing the Residual 4% Background Offset Level of Multi-Mode VCSEL Emitters Captured by the 4Picos ICCD Camera

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Both ${\rm{D}}4\sigma$ and D86 methods overestimate the nominal VCSEL beam width substantially, whether or not we perform background correction. As a top-hat irradiance profile is considered a reasonable approximation of VCSEL irradiance profiles, the background corrected results appear to not align with results discussed in Fig. 3(a). These results indicate that background correction, as described in Section 2.B, appears to be inadequate for images recorded using the ICCD camera. Indeed, by determining the average pixel signal value of all non-illuminated pixels in the cropped images, we obtain that the background offset level for the background corrected image is still 4%. This high residual background offset level can be attributed to possible scattering in the lens projecting the multi-mode VCSEL array onto the ICCD camera, or scattering in the lens coupling system and image intensifier of the ICCD camera itself. We remark that this residual background offset level is not accounted for through the initial background correction since the laser source is turned off there.

The knife-edge method yields beam width estimations closer to the nominal value of 130 pixels, but its results are affected by background correction as anticipated by the results shown in Fig. 5(a). However, the knife-edge method does not overestimate the VCSELs’ beam widths as expected from Fig. 5(a) for a background offset level of 4%. This is because the size of the cropped image to single out the VCSEL emitters has an impact on beam width estimation. We observed (not shown here) that the common beam width methods estimated larger beam widths if the image crop size is increased. This explains why the overestimations of ${\rm{D}}4\sigma$, D86, and knife-edge methods are rather limited compared to the results shown in Fig. 5(a). Beam width estimations using these common methods can be improved upon with an additional offset correction of the residual background offset level before determining the beam widths [17]. However, if the residual background offset is not homogeneous across the image, it is complex to obtain a good residual background offset level estimation. Moreover, properly estimating the residual background offset on an image of a dense laser beam pattern with a limited amount of non-illuminated pixels is difficult. Since the latter is a use case for which we are exploring the feasibility of the MTH method, we did not perform an additional residual background correction on the experimental data shown in Fig. 6.

In contrast, the beam widths estimated by the MTH method correspond well to the nominal beam width, independent of background correction. Furthermore, the MTH method was found to be unaffected by a varying crop size.

3. BEAM WIDTH ESTIMATIONS WITH SUB-PIXEL RESOLUTION

In the previous section, we demonstrated that the MTH method is robust to image noise and background offset levels and that it is an appropriate alternative to the common beam width methods described in literature for camera based beam profilers. A remaining drawback for all of the camera based techniques is the limited resolution when the spatial extent of an irradiance profile on the sensor is small compared to the pixel size of the sensor. As a result, it is generally recommended to fill the sensor active area as much as possible with a single laser beam [3]. In this section, we will demonstrate that the MTH method can open the path to determine beam widths of irradiance profiles that extend only a few pixels with a sub-pixel resolution. This is highly relevant when characterizing large arrays of laser beams with a single image.

A. Sub-Pixel Resolution through a Monte Carlo Technique

The idea proposed in this paper to increase the resolution of the beam width estimation is to assign weights to the pixels of the irradiance profile corresponding to the amount of overlap between the evaluation window and the pixels. If a pixel is entirely enclosed by the evaluation window, the pixel’s signal value is fully added to the enclosed power. We can state that the pixel has a weight of one. Contrarily, the pixels located at the border of the evaluation window are typically only partially enclosed. In the previous section, these pixels were either completely added or disregarded depending on the amount of overlap when calculating the enclosed power. Now, these edge pixels are given a weight between zero and one, depending on the overlap of the evaluation window with that pixel. To calculate the enclosed power of an irradiance profile, the signal values of all evaluated pixels are weighted before summing them. Since the total pixel signal value is weighted, we assume the irradiance across the pixel is uniformly distributed. This is true for pixel sampling a top-hat irradiance profile, except for the edge pixels where the irradiance profile only partially overlaps with the pixel. This could possibly lead to a deviation of the estimated beam width with respect to the actual beam diameter. Note that one could also apply this pixel weighting technique on the D86 method for sub-pixel resolution. However, it will not alleviate its sensitivity to improper background correction.

 

Fig. 7. (a) Illustration of the Monte Carlo technique used to assign weights to the pixels of a top-hat irradiance profile with an evaluation window with a diameter of 3.6 pixels. (b) Non-pixel weighted and pixel weighted MTH ratio of a top-hat irradiance profile with a diameter of $2{r_o} = 3.6$ pixels.

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The overlap between the evaluation window and a given pixel can be determined through numerical integration using a Monte Carlo technique, illustrated in Fig. 7(a). Here, a large number of points are randomly selected within the spatial extent of the pixel with a uniform spatial distribution. In the illustration, only 100 points are shown, while pixel weights are determined using 200,000 points. Next, the number of points located inside the evaluation window, depicted by the yellow solid line, is counted. The ratio of the number of points inside the evaluation window, denoted in red, and the total number of points in the pixel, denoted in blue and red, yields an estimate of the overlap of the evaluation window with the pixel. This ratio is chosen as the weight assigned to the pixel when calculating the power enclosed by the evaluation window.

To investigate this pixel weighted MTH method on small irradiance profiles, top-hat irradiance profiles are also generated using the same Monte Carlo integration technique. An example of the MTH ratio with a sub-pixel resolution of 0.04 pixels on a computer generated top-hat irradiance profile with a diameter of $2{r_o} = 3.6$ pixels is given in Fig. 7(b). A good agreement between the input beam diameter and the estimated sub-pixel beam width is observed, illustrated by the gray solid line and black dashed line, respectively. If one would have used the non-pixel weighted MTH method with the resolution of only a pixel, the beam width would have been firmly underestimated as depicted by the red dashed line. Once the evaluation window gets much larger than the irradiance profile itself, the MTH ratios obtained through pixel weighting and without pixel weighting align since the evaluation windows completely enclose the irradiance profile.

B. Performance of Pixel Weighted MTH Method

1. Performance on Computer Generated Irradiance Profiles

The goal is to accurately estimate the beam width of small top-hat and Gaussian irradiance profiles, defined by Eq. (1) and Eq. (2), respectively, where ${r_o}$ can now have any arbitrary positive real value. We study the performance of the MTH method on noise-free images without a background offset level as well as on images with a background offset level of 0.1% and with Poisson–Gaussian noise applied. We repeat the beam width estimations 100 times and calculate the mean value and standard deviation. We asses the quality of the beam width estimation of the pixel weighted MTH method by comparing the results with the non-pixel weighted MTH method. The irradiance profile amplitude ${I_o}$ is set to 1023 LSB, and the input beam diameter $2{r_o}$ is varied from two to 11 pixels in steps of 0.2 pixels. The sub-pixel resolution of the pixel weighted MTH method is set at 0.05 pixels. The results are shown in Fig. 8.

 

Fig. 8. Estimated beam widths for the pixel weighted and non-pixel weighted MTH method with sub-pixel resolution in function of the input beam diameter (${=} 2{r_o}$) for computer generated (a) top-hat and (b) Gaussian irradiance profiles. In (b), the expected beam width diameter estimated by the MTH method based on the numerically determined conversion factor is depicted by the gray line.

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Considering the top-hat irradiance profiles [Fig. 8(a)], the non-pixel weighted MTH method nicely returns the input beam diameter for integer input radii ${r_o}$, as expected from the results in Fig. 2. For non-integer input radii ${r_o}$, however, the method also yields integer beam radius values. While an error of the beam width estimation of one pixel can be acceptable for very large beam diameters, it yields poor beam width estimations for beam profiles extending only a few pixels. Represented in red in Fig. 8(a), the pixel weighted MTH method increases the accuracy of beam width estimation and removes the step-like behavior. For input beam diameters larger than three pixels, the maximal deviation between the beam width and its estimation is improved from 1.6 to 0.6 pixels, illustrated by the dashed lines in Fig. 8(a). When the background offset level and Poisson–Gaussian noise are present in the computer generated images, the pixel weighted MTH method maintains improved accuracy with high precision.

The accuracy of beam width estimations on Gaussian irradiance profiles [Fig. 8(b)] is also improved using the pixel weighted MTH method. The maximal deviation from the input beam diameter is decreased from about 2.5 to 0.5 pixels for input beam diameters above three pixels, indicated by the dashed lines in Fig. 8(b). Similar to the results on top-hat irradiance profiles, the improved accuracy with the high precision of the pixel weighted MTH method is maintained on noisy images with a background offset level.

Thus, the pixel weighting technique betters the accuracy of the MTH method with respect to the non-pixel weighted MTH method. Furthermore, the pixel weighted MTH method is a robust method to determine the beam widths of very small beam diameters in poor imaging conditions.

2. Performance on Experimental Data

Last, we investigate how the pixel weighted MTH method performs on experimental data for which the laser beams are not magnified adequately onto the sensor. In Section 2.E, we projected multi-mode VCSEL emitters onto the 4Picos ICCD sensor, and their nominal spatial extent was manually estimated to be 130 pixels large. Through binning of the pixels of the image shown in Fig. 6, we can emulate as if the sensor has recorded the VCSEL emitters with fewer pixels. For example, if we bin the image with a bin size equal to four, we combine $4 \times 4$ pixels into a single pixel and effectively make the image, and thus the beam profile, four times smaller.

To study the precision and accuracy of ${\rm{D}}4\sigma$ and MTH methods on these binned images for different bin sizes, we use the same background corrected image as shown in Fig. 6. We bin the image with bin sizes ranging from one to 32, and estimate beam widths on the binned image. The resulting beam width estimation multiplied with the bin size should be equal to the non-binned beam width estimation. The sub-pixel resolution through which we vary the evaluation window size is adjusted to the bin size accordingly, starting from a pixel resolution when the image is not binned. The 210 pixel crop size to single out individual irradiance profiles from the image will be appropriately adapted to the bin size as well. The results are summarized in Fig. 9 where we plot the estimated beam width converted back to the original image size using the known bin size, in function of the bin size.

 

Fig. 9. Estimated beam diameters for the ${\rm{D}}4\sigma$ and the pixel weighted MTH method for different bin sizes on the background corrected image without removing the residual background offset level of 4% of the experimental data of the VCSEL emitters shown in Fig. 6.

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The results are analogous to those obtained in Section 2.E. The ${\rm{D}}4\sigma$ method overestimates the VCSELs’ beam widths, while the MTH method obtains values close to the nominal value with high precision for all bin sizes, even with the presence of a residual background offset level of 4%. The variations of mean values for both methods for different bin sizes can be attributed to pixel binning, as it redistributes the pixel signal levels of the irradiance profiles. Therefore, the MTH method, expanded for sub-pixel resolution beam width estimations, is still found to be the superior beam width method on experimental data of VCSEL emitters compared to the other methods discussed.

C. Future Perspectives

While we succeeded to enhance the resolution of the MTH method on small irradiance profiles by weighting the pixels corresponding to the overlap with the evaluation windows, more work can be done on improving the pixel weighted MTH method.

First, the residual error on sub-pixel resolution can probably be decreased by handling the weighting of the edge pixels in a more advanced manner. Expanding the method to take into account the non-uniform irradiance distributions across the “edge” pixels, the pixel’s signal value added to the enclosed power can be more representative. Moreover, this could also result in better beam width estimations of non-top-hat irradiance profiles. This is, however, outside the scope of the current paper and is subject for future research.

Second, the computational time for the pixel weighted MTH method, as it was implemented in MATLAB for this paper, is estimated to be one or two orders of magnitude longer than for the non-pixel weighted MTH method, depending on the desired sub-pixel resolution. Therefore, we want to mention that other ways of determining the pixel weights can be interesting. For example, an integration technique where each pixel is sampled using a regular or rotated grid of points could be sufficient for the pixel weighting purpose and less computationally heavy. We remark that pixel weighting should be determined only for the edge pixels of the evaluation window. Additionally, it can be interesting to find the maximum of the MTH ratio more efficiently through a gradient ascent technique. In this paper, we typically let the evaluation window grow from small to large and stop the MTH ratio calculations once the first maximum is reached.

4. CONCLUSION

In this paper, we investigated which beam width method is suitable for a camera based beam profiling technique to yield precise and accurate estimations of the transverse extent of the laser beam in poor imaging conditions and has the capability to handle multiple laser beam profiles in a single image that extend only a few pixels on the sensor. For this purpose, we examined the feasibility of the so-called MTH method, introduced by Schulmeister et al. (2006) in [5], on computer generated irradiance profiles as well as on experimental data of a multi-mode VCSEL array.

The MTH method is found to accurately estimate the beam width for computer generated top-hat and Gaussian irradiance profiles. Compared to beam width methods commonly described in literature, such as the standardized ${\rm{D}}4\sigma$, the D86, or the knife-edge method, the MTH method persists in precise and accurate beam width estimations in noisy imaging conditions, simulated by a Poisson–Gaussian model, for images with low to reasonable SNRs, and for images with inaccurate background corrections. Therefore, the MTH method can ease up the analysis software in camera based laser beam profilers.

Furthermore, we successfully expanded upon the MTH method to estimate beam widths of only a few pixels with a sub-pixel resolution by weighting the pixels of the irradiance profile according to their overlap with the evaluation window. The pixel weighting was obtained using a Monte Carlo technique. We were able to increase the accuracy of the beam width estimations of irradiance profiles with sub-pixel beam diameters to a maximal deviation of 0.5 pixels. These results were also obtained on noisy images with a positive background offset level where the other beam width methods failed.

The results shown in this paper illustrate that the MTH method is well suited for applications that require precise and accurate beam width estimates, even on images of very small irradiance profiles. These applications can range from determining the beam widths of tailored irradiance profiles obtained through laser beam shaping techniques to determining the overall beam width of a laser beam pattern generated in multi-beam LiDAR.